namely_o the_o part_n great_a than_o the_o whole_a which_o be_v impossible_a wherefore_o the_o circumference_n shall_v not_o cut_v the_o line_n cd_o now_o i_o say_v that_o it_o shall_v not_o pass_v above_o the_o line_n cd_o and_o not_o touch_v it_o in_o the_o point_n d._n for_o if_o it_o be_v possible_a let_v it_o pass_v above_o it_o and_o extend_v the_o line_n cd_o to_o the_o circumference_n and_o let_v it_o cut_v it_o in_o the_o point_n â_o and_o draw_v the_o line_n fb_o and_o favorina_n and_o it_o shall_v follow_v as_o before_o that_o the_o line_n cd_o be_v great_a than_o the_o line_n cf_o which_o be_v impossible_a wherefore_o that_o be_v manifest_v which_o be_v require_v to_o be_v prove_v ¶_o second_v assumpt_n if_o there_o be_v a_o right_a angle_n unto_o which_o a_o base_a be_v subtend_v and_o if_o upon_o the_o same_o be_v describe_v a_o semicircle_n the_o circumference_n thereof_o shall_v pass_v by_o the_o point_n of_o the_o right_a angle_n the_o converse_n of_o this_o be_v add_v after_o the_o demonstration_n of_o the_o 31._o of_o the_o three_o out_o of_o pelitarius_n and_o these_o two_o assumpte_n of_o campane_n be_v necessary_a for_o the_o better_a understanding_n of_o the_o demonstration_n of_o the_o secoâd_a part_n of_o this_o 13._o proposition_n wherein_o be_v prove_v that_o the_o pyramid_n be_v contain_v in_o the_o sphere_n give_v ¶_o certain_a corollarye_n add_v by_o flussas_n first_o corollary_n the_o diameter_n of_o the_o sphere_n be_v in_o power_n quadruple_a sesquialtera_fw-la to_o the_o line_n which_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o base_a of_o the_o pyramid_n for_o forasmuch_o as_o it_o have_v be_v prove_v that_o the_o diameter_n kl_o be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o and_o it_o be_v prove_v also_o by_o the_o 12._o of_o this_o book_n that_o the_o side_n of_o be_v in_o power_n triple_a to_o the_o line_n eh_o which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n contain_v the_o triangle_n efg_o but_o the_o proportion_n of_o the_o extreme_n namely_o of_o the_o diameter_n to_o the_o line_n eh_o consist_v of_o the_o proportion_n of_o the_o mean_n namely_o of_o the_o proportion_n of_o the_o diameter_n to_o the_o line_n of_o and_o of_o the_o proportion_n of_o the_o line_n of_o to_o the_o line_n eh_o by_o the_o 5._o definition_n of_o the_o six_o which_o proportion_n namely_o triple_a and_o sesquialter_fw-la add_v together_o make_v quadruple_a sesquialter_fw-la as_o it_o be_v easy_a to_o prove_v by_o that_o which_o be_v teach_v in_o the_o declaration_n of_o the_o 5._o definition_n of_o the_o six_o book_n wherefore_o the_o corollary_n be_v manifest_a ¶_o second_v corollary_n only_o the_o line_n which_o be_v draw_v from_o the_o angle_n of_o the_o pyramid_n to_o the_o base_a opposite_a unto_o it_o campane_n &_o pass_v by_o the_o centre_n of_o the_o sphere_n be_v perpendicular_a to_o the_o base_a and_o fall_v upon_o the_o centre_n of_o the_o circle_n which_o contain_v the_o base_a for_o if_o any_o other_o line_n than_o the_o line_n kmh_o which_o be_v draw_v by_o the_o centre_n of_o the_o sphere_n to_o the_o centre_n of_o the_o circle_n shall_v fall_v perpendicular_o upon_o the_o plain_a of_o the_o base_a then_o from_o one_o and_o the_o self_n same_o point_n shall_v be_v draw_v to_o one_o and_o the_o self_n same_o plain_a two_o perpendicular_a line_n contrary_a to_o the_o 13._o of_o the_o eleven_o which_o be_v impossible_a far_o if_o from_o the_o top_n king_n shall_v be_v draw_v to_o the_o centre_n of_o the_o base_a namely_o to_o the_o point_n h_o any_o other_o right_a line_n not_o pass_v by_o the_o centre_n m_o two_o right_a life_n shall_v include_v a_o superficies_n contrary_n to_o the_o last_o common_a sentence_n which_o be_v absurd_a wherefore_o only_o the_o line_n which_o be_v draw_v by_o the_o centre_n of_o the_o sphere_n to_o the_o centre_n of_o the_o base_a be_v perpendicular_a to_o the_o say_v base_a and_o the_o line_n which_o be_v draw_v from_o the_o angle_n perpendicular_o to_o the_o base_a shall_v pass_v by_o the_o centre_n of_o the_o sphere_n three_o corollary_n the_o perpendicular_a line_n which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o base_a of_o the_o pyramid_n book_n be_v equal_a to_o the_o six_o part_n of_o the_o diameter_n of_o the_o sphere_n for_o it_o be_v before_o prove_v that_o the_o line_n mh_o which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o centre_n of_o the_o base_a be_v equal_a to_o the_o line_n nc_n which_o line_n nc_n be_v the_o six_o part_n of_o the_o diameter_n ab_fw-la and_o therefore_o the_o line_n mh_o be_v the_o six_o part_n of_o the_o diameter_n of_o the_o sphere_n for_o the_o diameter_n ab_fw-la be_v equal_a to_o the_o diameter_n of_o the_o sphere_n as_o have_v also_o before_o be_v prove_v ¶_o the_o 2._o problem_n the_o 14._o proposition_n to_o make_v a_o octohedron_n and_o to_o comprehend_v it_o in_o the_o sphere_n give_v namely_o that_o wherein_o the_o pyramid_n be_v comprehend_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n construction_n take_v the_o diameter_n of_o the_o former_a sphere_n give_v which_o let_v be_v the_o line_n ab_fw-la and_o divide_v it_o by_o the_o 10._o of_o the_o first_o into_o two_o equal_a part_n in_o the_o point_n c._n and_o describe_v upon_o the_o line_n ab_fw-la a_o semicircle_n adb_o and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n c_o raise_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n cd_o and_o draw_v a_o right_a line_n from_o the_o point_n d_o to_o the_o point_n b._n and_o describe_v a_o square_a efgh_o have_v every_o one_o of_o his_o side_n equal_a to_o the_o line_n bd._n and_o draw_v the_o diagonal_a line_n fh_a &_o eglantine_n cut_v the_o one_o the_o other_o in_o the_o point_n k._n and_o by_o the_o 12._o of_o the_o eleven_o from_o the_o point_n king_n namely_o the_o point_n where_o the_o line_n fh_o and_o eglantine_n cut_v the_o one_o the_o other_o raise_v up_o to_o the_o plain_a superficies_n wherein_o the_o square_a efgh_o be_v a_o perpendicular_a line_n kl_o and_o extend_v the_o line_n kl_o on_o the_o other_o side_n of_o the_o plain_a superficies_n to_o the_o point_n m._n and_o let_v each_o of_o the_o line_n kl_o and_o km_o be_v put_v equal_a to_o one_o of_o these_o line_n ke_o kf_o kh_o or_o kg_n and_o draw_v these_o right_a line_n le_fw-fr lf_n lg_n lh_o menander_n mf_n mg_o and_o mh_o demonstration_n now_o forasmuch_o as_o the_o line_n ke_o be_v by_o the_o corollary_n of_o the_o 34._o of_o the_o first_o equal_a to_o the_o line_n kh_o and_o the_o angle_n angle_n ekh_n be_v a_o right_a angle_n therefore_o the_o square_a of_o he_o be_v double_a to_o the_o square_n of_o eke_o by_o the_o 47._o of_o the_o first_o again_o forasmuch_o as_o the_o line_n lk_n be_v equal_a to_o the_o line_n ke_o by_o position_n and_o the_o angle_n lke_n be_v by_o the_o second_o definition_n of_o the_o eleven_o a_o right_a angle_n therefore_o the_o square_n of_o the_o line_n el_n be_v double_a to_o the_o square_n of_o the_o line_n eke_o and_o it_o be_v prove_v that_o the_o square_a of_o the_o line_n he_o be_v double_a to_o the_o square_n of_o the_o line_n eke_o wherefore_o the_o square_a of_o the_o line_n le_fw-fr be_v equal_a to_o the_o square_n of_o the_o line_n eh_o wherefore_o also_o the_o line_n le_fw-fr be_v equal_a to_o the_o line_n eh_o and_o by_o the_o same_o reason_n the_o line_n lh_o be_v also_o equal_a to_o the_o line_n he._n wherefore_o the_o triangle_n lhe_n be_v equilater_n in_o like_a sort_n may_v we_o prove_v that_o every_o one_o of_o the_o rest_n of_o the_o triangle_n who_o base_n be_v the_o side_n of_o the_o square_a efgh_o and_o top_n the_o point_n l_o and_o m_o be_v equilater_n the_o say_v eight_o triangle_n also_o be_v equal_a the_o one_o to_o the_o other_o for_o every_o side_n of_o each_o be_v equal_a to_o the_o side_n of_o the_o square_a efgh_o wherefore_o there_o be_v make_v a_o octohedron_n contain_v under_o eight_o triangle_n who_o side_n be_v equal_a now_o it_o be_v require_v to_o comprehend_v it_o in_o the_o sphere_n give_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n demonstration_n forasmuch_o as_o these_o three_o line_n lk_v km_o and_o ke_o be_v equal_a the_o one_o to_o the_o other_o therefore_o a_o semicircle_n describe_v upon_o the_o line_n lm_o shall_v pass_v also_o by_o the_o point_n e._n and_o by_o the_o same_o reason_n if_o the_o semicircle_n be_v turn_v round_o about_o until_o it_o return_v unto_o the_o self_n same_o
two_o line_n hif_n and_o tio_n cut_v the_o one_o the_o other_o be_v in_o one_o and_o the_o self_n same_o '_o plain_n by_o the_o 2._o of_o the_o eleven_o and_o therefore_o the_o point_n h_o t_o f_o oh_o be_v in_o one_o &_o the_o self_n same_o plain_a wherforeâ_n the_o rectangle_n figure_n hoff_z be_v quadrilater_n and_o equilater_n and_o in_o one_o and_o the_o self_n same_o plain_n be_v a_o square_a by_o the_o definition_n of_o a_o square_a and_o by_o the_o same_o reason_n may_v the_o rest_n of_o the_o base_n of_o the_o solid_a be_v prove_v to_o be_v square_n equal_a and_o plain_a or_o superficial_a now_o than_o the_o solid_a be_v comprehend_v of_o 6._o equal_a square_n which_o be_v contain_v of_o 12._o equal_a side_n which_o square_n make_v 8._o solid_a angle_n of_o which_o four_o be_v in_o the_o centre_n of_o the_o base_n oâ_n the_o pyramid_n and_o the_o other_o 4._o be_v in_o the_o middle_a section_n of_o the_o four_o perdendiculars_n wherefore_o the_o solid_a hoftpgrn_n be_v a_o cube_fw-la by_o the_o 21._o definition_n of_o the_o eleven_o and_o be_v inscribe_v in_o the_o pyramid_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o trilater_n equilater_n pyramid_n give_v be_v inscribe_v a_o cube_fw-la ¶_o a_o corollary_n the_o line_n which_o cut_v into_o two_o equal_a part_n the_o opposite_a side_n of_o the_o pyramid_n be_v triple_a to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n and_o pass_v by_o the_o centre_n of_o the_o cube_fw-la for_o the_o line_n sev_n who_o three_o part_n the_o line_n si_fw-it be_v cut_v the_o opposite_a side_n cd_o and_o ab_fw-la into_o two_o equll_a part_n but_o the_o line_n ei_o which_o be_v draw_v from_o the_o centre_n of_o the_o cube_fw-la to_o the_o base_a be_v prove_v to_o be_v a_o three_o part_n of_o the_o line_n es_fw-ge wherefore_o the_o side_n of_o the_o cube_fw-la which_o be_v double_a to_o the_o line_n ei_o shall_v be_v a_o three_o part_n of_o the_o whole_a line_n we_o which_o be_v as_o have_v be_v prove_v double_a to_o the_o line_n es._n the_o 19_o problem_n the_o 19_o proposition_n in_o a_o trilater_n equilater_n pyramid_n give_v to_o inscribe_v a_o icosahedron_n svppose_v that_o the_o pyramid_n be_v give_v ãâã_d abâdâ_n every_o one_o of_o who_o sâdes_n ãâã_d be_v divide_v into_o two_o equal_a part_n in_o the_o poyâââââ_n m_o king_n l_o p_o n._n construction_n and_o iâ_z every_o one_o of_o the_o base_n of_o that_o pyramid_n descry_v the_o trianglââ_n lââ_n pmn_n nkl_n and_o ãâ¦ã_o which_o triangle_n shall_v be_v equilater_n by_o the_o 4._o of_o the_o firât_o âor_a the_o side_n subtend_v equal_a angle_n of_o the_o pyramid_n contain_v under_o the_o half_n of_o the_o side_n of_o the_o same_o pyramisâ_n wherefore_o the_o side_n of_o the_o say_a triangle_n be_v equal_a let_v those_o side_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n by_o the_o 30._o of_o the_o six_o in_o the_o point_n c_o e_o q_o r_o s_o t_o h_o i_o oh_o five_o a'n_z x._o now_o then_o those_o side_n be_v cut_v into_o the_o self_n same_o proportion_n by_o the_o 2._o of_o the_o fourtenth_n and_o therefore_o they_o make_v the_o liâe_z section_n equal_a by_o the_o â_o part_n of_o the_o nine_o of_o the_o five_o now_o i_o say_v that_o the_o foresay_a point_n doâ_n receive_v the_o angle_n of_o the_o icosahedron_n inscribe_v in_o the_o pyramid_n abâd_n in_o the_o foresay_a triangle_n let_v there_o again_o be_v make_v other_o triangle_n by_o couple_v the_o section_n and_o let_v those_o triangle_n be_v tr_n joh_n ceq_n and_o vxy_n which_o shall_v be_v equilater_n for_o every_o one_o of_o their_o side_n do_v subâââd_v equal_a angle_n of_o equilater_n triangle_n and_o those_o say_v equal_a angle_n be_v contain_v under_o equal_a sideâ_n namely_o under_o the_o great_a segment_n and_o the_o less_o â_o and_o therefore_o the_o side_n which_o subtend_v those_o angle_n be_v equal_a by_o the_o 4._o of_o the_o first_o now_o let_v we_o prove_v that_o at_o each_o of_o the_o foresay_a point_n as_o for_o example_n at_o t_o be_v set_v the_o solid_a angle_n of_o a_o icosahâdronâ_n demonstration_n forasmuch_o as_o the_o triangle_n trs_n and_o tqo_n be_v equilater_n and_o equal_a the_o 4._o right_a line_n tr_z it_o be_v tq_n and_o to_o shall_v be_v equal_a and_o forasmuch_o as_o âpnk_v be_v a_o square_n cut_v the_o pyramid_n abâd_v into_o two_o equal_a paââââ_n by_o the_o corollay_n of_o the_o second_o of_o this_o bookeâ_n the_o line_n th._n shall_v be_v in_o power_n duple_n to_o the_o line_n tn_n or_o nh_v by_o the_o 47._o of_o the_o first_o for_o the_o line_n tn_n or_o nh_v be_v equal_a for_o that_o by_o construction_n they_o be_v each_o less_o segment_n and_o the_o line_n rt_n or_o it_o be_v be_v in_o power_n duple_n to_o the_o same_o line_n tn_n or_o nh_v by_o the_o corollary_n of_o the_o 16._o of_o this_o book_n for_o it_o subtend_v the_o angle_n of_o the_o triangle_n contain_v under_o the_o two_o segment_n wherefore_o the_o line_n th._n it_o be_v tr_z tq_n and_o to_o be_v equal_a and_o so_o also_o be_v the_o line_n his_o sir_n rq_n qo_n and_o oh_o which_o subtend_v the_o angle_n at_o the_o point_n t_o equal_a for_o the_o line_n qr_o contain_v in_o power_n the_o two_o line_n pq_n and_o pr._n the_o less_o segment_n which_o two_o line_n the_o line_n th._n also_o contain_v in_o power_n and_o the_o rest_n of_o the_o line_n do_v subtend_v angle_n of_o equilater_n triangle_n contain_v under_o the_o great_a segment_n and_o the_o less_o wherefore_o the_o five_o triangle_n trs_n tsh_o tho_o toq_fw-la tqr_n be_v equilater_n and_o equal_a make_v the_o solid_a angle_n of_o a_o icosahedron_n at_o the_o point_n t_o by_o the_o 16._o of_o the_o thirteen_o in_o the_o side_n pn_n of_o the_o triangle_n p_o nm_o and_o by_o the_o same_o reason_n in_o the_o other_o side_n of_o the_o 4._o triangle_n pnm_n nkl_n fmk_n &_o lfp_n which_o be_v inscribe_v in_o the_o base_n of_o the_o pyramid_n which_o side_n be_v 12â_n in_o number_n shall_v be_v set_v 12._o angle_n of_o the_o icosahedron_n contain_v under_o 20._o equal_a &_o equilater_n triangle_n of_o which_o fowere_n be_v set_v in_o the_o 4._o base_n of_o the_o pyramid_n namely_o these_o four_o triangle_n trs_n hoi_n ceq_n vxy_n 4._o triangle_n be_v under_o 4._o angle_n of_o the_o pyramid_n that_o be_v the_o four_o triangle_n cix_o ysh_a erv_n tqo_n and_o under_o every_o one_o of_o the_o six_o side_n of_o the_o pyramid_n be_v set_v two_o triangle_n namely_o under_o the_o side_n of_o the_o triangle_n this_o and_o thoâ_n under_o the_o side_n db_o the_o triangle_n rqe_n and_o rqt_n under_o the_o side_n da_z the_o triangle_n coq_fw-la and_o coi_fw-fr under_o the_o side_n ab_fw-la the_o triangle_n exc_a and_o exuâ_n under_o the_o side_n bg_o the_o triangle_n sur_n and_o svy_n and_o under_o the_o side_n agnostus_n the_o triangle_n jyh_n and_o jyx._n wherefore_o the_o solid_a be_v contain_v under_o 20._o equilater_n and_o equal_a triangle_n shall_v be_v a_o icosahedron_n by_o the_o 23._o definition_n of_o the_o eleven_o and_o shall_v be_v inscribe_v in_o the_o pyramid_n abâd_v by_o the_o first_o definition_n of_o this_o book_n for_o all_o his_o angle_n do_v at_o one_o time_n touch_v the_o base_n of_o the_o pyramid_n wherefore_o in_o a_o trilater_n equilater_n pyramid_n give_v we_o have_v inscribe_v a_o icosahedron_n ¶_o the_o 20._o proposition_n the_o 20._o problem_n in_o a_o trilater_n equilater_n pyramid_n give_v to_o inscribe_v a_o dodecahedron_n svppose_v that_o the_o pyramid_n give_v be_v abgd_v âche_fw-mi of_o who_o side_n let_v be_v cut_v into_o two_o equal_a part_n and_o draw_v the_o line_n which_o couple_v the_o section_n which_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o right_a line_n be_v draw_v by_o the_o section_n shall_v receive_v 20._o triangle_n make_v a_o icosahedron_n as_o in_o the_o former_a proposition_n it_o be_v manifest_a now_o than_o if_o we_o take_v the_o centre_n of_o those_o triangle_n we_o shall_v there_o find_v the_o 20._o angle_n of_o the_o dodecahedron_n inscribe_v in_o it_o by_o the_o 5._o of_o this_o book_n and_o forasmuch_o as_o 4._o base_n of_o the_o foresay_a icosahedron_n be_v concentricall_a with_o the_o base_n of_o the_o pyramid_n as_o it_o be_v prove_v in_o the_o 2._o corollary_n of_o the_o 6._o of_o this_o book_n there_o shall_v be_v place_v 4â_o angle_n of_o the_o dodecahedron_n namely_o the_o 4._o angle_n e_o f_o h_o d_o in_o the_o 4._o centre_n of_o the_o base_n and_o of_o the_o other_o 16._o angle_n under_o every_o one_o of_o the_o 6._o side_n of_o the_o pyramid_n be_v subtend_v two_o namely_o under_o the_o side_n ad_fw-la the_o angle_n ck_o under_o the_o side_n bd_o the_o angle_n li_n under_o the_o
proportion_n which_o be_v in_o power_n duple_n to_o of_o the_o side_n of_o the_o octohedron_n inscribe_v in_o the_o dodecahedron_n draw_v the_o diameter_n el_n and_o fk_o of_o the_o octohedron_n now_o they_o couple_v the_o middle_a section_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n ab_fw-la and_o gd_o by_o the_o 9_o of_o the_o fifteen_o &_o 3._o corollary_n of_o the_o 17._o of_o the_o thirteen_o &_o every_o one_o of_o those_o diameter_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n do_v make_v the_o less_o segment_n the_o side_n of_o the_o dodecahedron_n by_o the_o 4._o corollary_n of_o the_o same_o wherefore_o the_o side_n ab_fw-la be_v the_o less_o segment_n of_o the_o line_n fk_o but_o the_o line_n fk_o contain_v in_o power_n the_o two_o equal_a line_n of_o &_o eke_o by_o the_o 47._o of_o the_o first_o for_o the_o angle_n fek_n be_v a_o right_a angle_n of_o the_o square_a fekl_n of_o the_o octohedron_n wherefore_o the_o line_n fk_o be_v in_o power_n duple_n to_o the_o line_n ef._n wherefore_o the_o line_n ab_fw-la the_o side_n of_o the_o dodecahedron_n be_v the_o less_o segment_n of_o the_o line_n fk_o which_o be_v in_o power_n duple_n to_o of_o the_o sidâ_n of_o the_o octohedron_n the_o side_n therefore_o of_o a_o dodecahedron_n iâ_z the_o less_o segment_n of_o that_o right_a line_n which_o be_v in_o power_n duple_n to_o the_o side_n of_o the_o octohedron_n inscribe_v in_o the_o same_o dodâcahedron_n ¶_o the_o 22._o proposition_n the_o diameter_n of_o a_o icosahedron_n be_v in_o power_n sesquitertia_fw-la to_o the_o side_n of_o the_o same_o icosahedron_n and_o also_o be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o icosahedron_n for_o forasmuch_o as_o it_o have_v be_v prove_v by_o the_o 10._o of_o this_o book_n that_o if_o from_o the_o power_n of_o the_o diameter_n of_o the_o icosahedron_n be_v take_v away_o the_o triple_a of_o the_o power_n of_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o it_o there_o shall_v be_v leave_v a_o square_a sesquitertia_fw-la to_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n but_o the_o power_n of_o the_o side_n of_o the_o cube_fw-la triple_a be_v the_o diameter_n of_o the_o same_o cube_fw-la by_o the_o 15._o of_o the_o thirteen_o and_o the_o cube_fw-la &_o the_o pyramid_n inscribe_v in_o it_o be_v contain_v in_o one_o &_o the_o self_n same_o sphere_n by_o the_o first_o of_o this_o book_n and_o in_o one_o &_o the_o self_n same_o icosahedron_n by_o the_o corollary_n of_o the_o same_o wherefore_o one_o and_o the_o self_n same_o diameter_n of_o the_o cube_fw-la or_o of_o the_o sphere_n which_o contain_v the_o cube_fw-la and_o the_o pyramid_n be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o the_o pyramid_n by_o the_o 13._o of_o the_o thirteen_o wherefore_o it_o follow_v that_o if_o from_o the_o diameter_n of_o the_o icosahedron_n be_v take_v away_o the_o triple_a power_n of_o the_o side_n of_o the_o cube_fw-la or_o the_o sesquialter_fw-la power_n of_o the_o side_n of_o the_o pyramid_n which_o be_v the_o power_n of_o one_o and_o the_o self_n same_o diameter_n there_o shall_v be_v leave_v the_o sesquitertia_fw-la power_n of_o the_o side_n of_o the_o icosahedron_n the_o diameter_n therefore_o of_o a_o icosahedron_n be_v in_o power_n sesquitertia_fw-la to_o the_o side_n of_o the_o same_o icosahedron_n and_o also_o be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o icosahedron_n the_o 23._o proposition_n the_o side_n of_o a_o dodecahedron_n be_v to_o the_o side_n of_o a_o icosahedron_n inscribe_v in_o it_o as_o the_o less_o segment_n of_o the_o perpendicular_a of_o the_o pentagon_n be_v to_o that_o line_n which_o be_v draw_v from_o the_o centre_n to_o the_o side_n of_o the_o same_o pentagon_n constrution_n let_v there_o be_v take_v a_o dodecahedron_n abgdfso_n who_o side_n let_v be_v as_o or_o so_o and_o let_v the_o icosahedron_n inscribe_v in_o it_o be_v klnmne_a who_o side_n let_v be_v kl_o from_o the_o two_o angle_n of_o the_o pentagonâ_n bas_n and_o fas_fw-la of_o the_o dodecahedron_n namely_o from_o the_o angleââ_n and_o fletcher_n let_v there_o be_v draw_v to_o the_o common_a base_a as_o perpendicular_a line_n bc_o &_o fc_n which_o shall_v pass_v by_o the_o centre_n king_n &_o l_o of_o the_o say_v pentagon_n by_o the_o corollary_n of_o the_o 10._o of_o the_o thirteen_o draw_v the_o line_n bf_o and_o ro._n now_o forasmuch_o as_o the_o line_n ro_n subtend_v the_o angle_n ofr_n of_o thâ_z pentagon_n of_o the_o dodecahedron_n it_o shall_v cut_v the_o line_n fc_o by_o a_o extreme_a and_o mean_a proportion_n by_o the_o 3._o of_o this_o book_n let_v it_o cut_v it_o in_o the_o point_n i_o and_o forasmuch_o as_o the_o line_n kl_o be_v the_o side_n of_o the_o icosahedron_n inscribe_v in_o the_o dodecahedron_n it_o couple_v the_o centre_n of_o the_o base_n of_o the_o dodecahedron_n for_o the_o angle_n of_o the_o icosahedron_n be_v set_v in_o the_o centre_n of_o the_o base_n of_o the_o dodecahedron_n by_o the_o 7._o of_o the_o fifteen_o now_o i_o say_v that_o so_o the_o side_n of_o the_o dodecahedron_n be_v to_o kl_o the_o side_n of_o the_o icosahedron_n as_o the_o less_o segment_n if_o of_o the_o perpendicular_a line_n cf_o be_v to_o the_o line_n lc_n which_o be_v draw_v from_o the_o centre_n l_o to_o as_o the_o side_n of_o the_o pentagon_n for_o forasmuch_o as_o in_o the_o triangle_n bcf_n the_o two_o side_n cb_o and_o cf_o be_v in_o the_o centre_n l_o and_o king_n cut_v like_o proportional_o demonstration_n the_o line_n bf_o and_o kl_o shall_v be_v parellel_n by_o the_o 2._o of_o the_o six_o wherefore_o the_o triangle_n bcf_n and_o kcl_n shall_v be_v equiangle_n by_o the_o corollary_n of_o the_o same_o wherefore_o as_o the_o line_n cl_n be_v to_o the_o line_n klâ_n so_o be_v the_o line_n cf_o to_o the_o line_n bf_o by_o the_o 4._o of_o the_o six_o but_o cf_o make_v the_o less_o segment_n the_o line_n if_o by_o the_o 3._o of_o this_o book_n and_o the_o linâ_n bf_o make_v the_o less_o segment_n the_o line_n so_o namely_o the_o side_n of_o the_o dodecahedron_n by_o the_o 2._o corollary_n of_o the_o 13._o of_o the_o fifteen_o for_o the_o line_n bf_o which_o couple_v the_o angle_n b_o and_o f_o of_o the_o base_n of_o the_o dodecahedron_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la which_o contain_v the_o dodecahedron_n by_o the_o .13_o of_o the_o fifteen_o wherefore_o as_o the_o whole_a line_n câ_n be_v to_o the_o whole_a line_n bf_o so_o be_v the_o less_o segment_n if_o to_o the_o less_o segment_n so_o by_o the_o 2._o of_o the_o 14_o but_o as_o the_o line_n cf_o be_v to_o the_o line_n bf_o so_o be_v the_o line_n cl_n prove_v to_o be_v to_o the_o line_n kl_o wherefore_o as_o the_o line_n if_o be_v to_o the_o line_n so_o so_o be_v the_o line_n cl_n to_o the_o line_n kl_o wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n if_o the_o less_o segment_n of_o the_o perpendicular_a of_o the_o pentagon_n fas_fw-la be_v to_o the_o line_n lc_n which_o be_v draw_v from_o the_o centre_n of_o the_o pentagon_n to_o the_o base_a so_o be_v the_o line_n so_o the_o side_n of_o the_o dodecahedron_n to_o thâ_z line_n kl_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o the_o side_n therefore_o of_o a_o dodecahedron_n be_v to_o the_o side_n of_o a_o icosahedron_n inscribe_v in_o it_o as_o the_o less_o segment_n of_o the_o perpendicular_a of_o the_o pentagon_n be_v to_o that_o line_n which_o be_v draw_v from_o the_o cenâre_n to_o the_o side_n of_o the_o same_o pentagon_n ¶_o the_o 24._o proposition_n if_o half_a of_o the_o side_n of_o a_o icosahedron_n be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n and_o if_o the_o less_o segment_n thereof_o be_v take_v away_o from_o the_o whole_a side_n and_o again_o from_o the_o residue_n be_v take_v away_o the_o three_o part_n that_o which_o remain_v shall_v be_v equal_a to_o the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o icosahedron_n svppose_v that_o abgdf_n be_v a_o pentagon_n construction_n contain_v five_o side_n of_o the_o icosahedron_n by_o the_o 16._o of_o the_o thirteen_o and_o let_v it_o be_v inscribe_v in_o a_o circle_n who_o centre_n let_v be_v the_o point_n e._n and_o upon_o the_o side_n of_o the_o pentagon_n let_v there_o be_v rear_v up_o triangle_n make_v a_o solid_a angle_n of_o the_o icosahedron_n at_o the_o point_n i_o by_o the_o 16._o of_o the_o thirteen_o and_o in_o the_o circle_n abdella_n inscribe_v a_o equilater_n triangle_n ahk_n from_o the_o centre_n e_o draw_v to_o hk_v the_o side_n of_o the_o triangle_n and_o gd_o the_o side_n of_o the_o pentagon_n a_o perpendicular_a line_n which_o
same_o be_v treble_a it_o be_v manifest_a by_o the_o 12._o of_o the_o five_o that_o all_o the_o prism_n be_v to_o all_o the_o pyramid_n treble_a wherefore_o parallelipipedon_n be_v treble_a to_o pyramid_n set_v upon_o the_o self_n same_o base_a with_o they_o and_o under_o the_o same_o altitude_n they_o for_o that_o they_o contain_v two_o prism_n three_o corollary_n if_o two_o prism_n be_v under_o one_o and_o the_o self_n same_o altitude_n have_v to_o their_o base_n either_o both_o triangle_n or_o both_o parallelogram_n the_o prism_n be_v the_o one_o to_o the_o other_o as_o their_o base_n be_v for_o forasmuch_o as_o those_o prism_n be_v equemultiqlice_n unto_o the_o pyramid_n upon_o the_o self_n same_o base_n and_o under_o the_o same_o altitude_n which_o pyramid_n be_v in_o proportion_n as_o their_o base_n it_o be_v manifest_a by_o the_o 15._o of_o the_o five_o that_o the_o prism_n be_v in_o the_o proportion_n of_o the_o base_n for_o by_o the_o former_a corollary_n the_o prism_n be_v treble_a to_o the_o pyramid_n sât_v upon_o the_o triangular_a base_n four_o corollary_n prism_n be_v in_o sesquealtera_fw-la proportion_n to_o pyramid_n which_o have_v the_o self_n same_o quadrangled_a base_a that_o the_o prism_n have_v and_o be_v under_o the_o self_n same_o altitude_n for_o that_o pyramid_n contain_v two_o pyramid_n set_v upon_o a_o triangular_a base_a of_o the_o same_o prism_n for_o it_o be_v prove_v that_o that_o prism_n be_v treble_a to_o the_o pyramid_n which_o be_v set_v upon_o the_o half_a of_o his_o quadrangled_a base_a unto_o which_o the_o other_o which_o be_v set_v upon_o the_o whole_a base_a be_v double_a by_o the_o six_o of_o this_o book_n five_v corollary_n wherefore_o we_o may_v in_o like_a sort_n conclude_v that_o solid_n mention_v in_o the_o second_o corollary_n which_o solid_n campane_n call_v side_v column_n be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n which_o be_v poligonon_o figure_n for_o they_o be_v in_o the_o proportion_n of_o the_o pyramid_n or_o prism_n set_v upon_o the_o self_n same_o base_n and_o under_o the_o self_n same_o altitude_n that_o be_v they_o be_v in_o the_o proportion_n of_o the_o base_n of_o the_o say_a pyramid_n or_o prism_n for_o those_o solid_n may_v be_v divide_v into_o prism_n have_v the_o self_n same_o altitude_n when_o as_o their_o opposite_a base_n may_v be_v divide_v into_o triangle_n by_o the_o 20_o of_o the_o six_o upon_o which_o triangle_n prism_n be_v set_v be_v in_o proportion_n as_o their_o base_n by_o this_o 7._o proposition_n it_o plain_o appear_v that_o âuâlide_a as_o it_o be_v before_o note_v in_o the_o diffinitionââ_n under_o the_o definition_n of_o a_o prism_n comprehend_v also_o those_o kind_n of_o solid_n which_o campane_n call_v side_v column_n for_o in_o that_o he_o say_v every_o prism_n have_v a_o triangle_n to_o his_o base_a may_v be_v devidedâ_n etc_n etc_n he_o need_v not_o take_v a_o prism_n in_o that_o sense_n which_o campane_n and_o most_o man_n take_v it_o to_o have_v add_v that_o particle_n have_v to_o his_o base_a a_o triangle_n for_o by_o their_o sense_n there_o be_v no_o prism_n but_o it_o may_v have_v to_o his_o base_a a_o triangleâ_n and_o so_o it_o may_v seem_v that_o euclid_n ought_v without_o exception_n have_v say_v that_o every_o prism_n whatsoever_o may_v be_v divide_v into_o three_o pyramid_n equal_a the_o one_o to_o the_o other_o have_v also_o triangle_n to_o âheir_a base_n for_o so_o do_v campane_n and_o flussas_n put_v the_o proposition_n leave_v out_o the_o former_a particle_n have_v to_o his_o base_a a_o triangle_n which_o yet_o be_v red_a in_o the_o greek_a copy_n &_o not_o leât_v out_o by_o any_o other_o interpreter_n know_v abroad_o except_o by_o campane_n and_o flussas_n yea_o and_o the_o corollary_n follow_v of_o this_o proposition_n add_v by_o theon_n or_o euclid_n and_o amend_v by_o m._n dee_n seem_v to_o confirm_v this_o sense_n of_o this_o âs_a ãâã_d make_v manifest_a that_o every_o pyramid_n be_v the_o three_o part_n of_o the_o prism_n have_v the_o same_o base_a with_o it_o and_o equal_a altitude_n for_o and_o if_o the_o base_a of_o the_o prism_n have_v any_o other_o right_o line_v figure_n than_o a_o triangle_n and_o also_o the_o superficies_n opposite_a to_o the_o base_a the_o same_o figure_n that_o prism_n may_v be_v divide_v into_o prism_n have_v triangle_v base_n and_o the_o superficiece_n to_o those_o base_n opposite_a also_o triangle_v a_o ââike_a and_o equal_o for_o there_o as_o we_o see_v be_v put_v these_o word_n âor_a and_o if_o the_o base_a of_o the_o prism_n be_v any_o other_o right_o line_v figureâ_n etc_n etc_n whereof_o a_o man_n may_v well_o infer_v that_o the_o base_a may_v be_v any_o other_o rectiline_a figure_n whatsoever_o &_o not_o only_o a_o triangle_n or_o a_o parallelogram_n and_o it_o be_v true_a also_o in_o that_o sense_n as_o it_o be_v plain_a to_o see_v by_o the_o second_o corollary_n add_v out_o of_o flussas_n which_o corollary_n as_o also_o the_o first_o of_o his_o corollary_n be_v in_o a_o manner_n all_o one_o with_o the_o corollary_n add_v by_o theon_n or_o euclid_n far_o theon_n in_o the_o demonstration_n of_o the_o 10._o proposition_n of_o this_o book_n as_o we_o shall_v afterwards_o see_v most_o plain_o call_v not_o only_o side_v column_n prism_n but_o also_o parallelipipedon_n and_o although_o the_o 40._o proposition_n of_o the_o eleven_o book_n may_v seem_v hereunto_o to_o be_v a_o lât_n for_o that_o it_o can_v be_v understand_v of_o those_o prism_n only_o which_o have_v triangle_n to_o their_o like_a equal_a opposite_a and_o parallel_v side_n or_o but_o of_o some_o side_v column_n and_o not_o of_o all_o yet_o may_v that_o let_v be_v thus_o remove_v away_o to_o say_v that_o euclid_n in_o that_o proposition_n use_v genus_fw-la pro_fw-la specie_fw-la that_o be_v the_o general_a word_n for_o some_o special_a kind_n thereof_o which_o thing_n also_o be_v not_o rare_a not_o only_o with_o he_o but_o also_o with_o other_o learned_a philosopher_n thus_o much_o i_o think_v good_a by_o the_o way_n to_o note_v in_o far_a defence_n of_o euclid_n definition_n of_o a_o prism_n the_o 8._o theorem_a the_o 8._o proposition_n pyramid_n be_v like_o &_o have_a triangle_n to_o their_o base_n be_v in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o their_o side_n of_o like_a proportion_n be_v svppose_v that_o these_o pyramid_n who_o base_n be_v the_o triangle_n gbc_n and_o hef_n and_o top_n the_o point_n a_o and_o d_o be_v like_o and_o in_o like_a sort_n describe_v and_o let_v ab_fw-la and_o de_fw-fr be_v side_n of_o like_a proportion_n then_o i_o say_v that_o the_o pyramid_n abcg_o be_v to_o the_o pyramid_n defh_o in_o treble_a proportion_n of_o that_o in_o which_o the_o side_n ab_fw-la be_v to_o the_o side_n de._n make_v perfect_a the_o parallelipipedon_n namely_o the_o solid_n bckl_n &_o efxo_n and_o forasmuch_o as_o the_o pyramid_n abcg_o be_v like_a to_o the_o pyramid_n defh_o construction_n therefore_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n def_n demonstration_n &_o the_o angle_n gbc_n to_o the_o angle_n hef_fw-fr and_o moreover_o the_o angle_n abg_o to_o the_o angle_n deh_fw-it and_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n de_fw-fr so_o be_v the_o line_n bc_o to_o the_o line_n of_o and_o the_o line_n bg_o to_o the_o line_n eh_o and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n de_fw-fr so_o be_v the_o line_n bc_o to_o the_o line_n of_o and_o the_o side_n about_o the_o equal_a angle_n be_v proportional_a therefore_o the_o parallelogram_n bm_n be_v like_a to_o the_o parallelogram_n ep_n and_o by_o the_o same_o reason_n the_o parallelogram_n bn_o be_v like_a to_o the_o parallelogram_n er_fw-mi and_o the_o parellelogramme_n bk_o be_v like_a unto_o the_o parallelogram_n exit_fw-la wherefore_o the_o three_o parallelogram_n bm_n kb_o and_o bn_o be_v like_a to_o the_o three_o parallelogram_n ep_n exit_fw-la and_o er._n but_o the_o three_o parallelogram_n bm_n kb_o and_o bn_o be_v equal_a and_o like_a to_o the_o three_o opposite_a parallelogram_n and_o the_o three_o parallelogram_n ep_n exit_fw-la and_o ere_o be_v equal_a and_o like_a to_o the_o three_o opposite_a parallelogram_n wherefore_o the_o parallelipipedon_n bckl_n and_o efxo_n be_v comprehend_v under_o plain_a superficiece_n like_a and_o equal_a in_o multitude_n wherefore_o the_o solid_a bckl_n be_v like_a to_o the_o solid_a efxo_n but_o like_o parallelipipedon_n be_v by_o the_o 33._o of_o the_o eleven_o in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o side_n of_o like_a proportion_n be_v to_o side_n of_o like_a proportion_n wherefore_o the_o solid_a bckl_n be_v to_o the_o solid_a efxo_n in_o treble_a
fd._n wherefore_o cones_fw-la &_o cylinder_n consist_v upon_o equal_a base_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o altitude_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 15._o theorem_a the_o 15._o proposition_n in_o equal_a cones_n and_o cylinder_n the_o base_n be_v reciprocal_a to_o their_o altitude_n and_o cones_fw-la and_o cylinder_n who_o base_n be_v reciprocal_a to_o their_o altitude_n be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o these_o cones_fw-la acl_n egn_n or_o these_o cylinder_n axe_n eo_fw-la who_o base_n be_v the_o circle_n abcd_o efgh_o and_o axe_n kl_o and_o mn_v which_o axe_n be_v also_o the_o altitude_n of_o the_o cones_fw-la &_o cylinder_n be_v equal_a the_o one_o to_o the_o other_o cones_n then_o i_o say_v that_o the_o base_n of_o the_o cylinder_n xa_n &_o eo_fw-la be_v reciprokal_n to_o their_o altitude_n that_o be_v that_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o proposition_n so_o the_o altitude_n mn_v to_o the_o altitude_n kl_o for_o the_o altitude_n kl_o be_v either_o equal_a to_o the_o altitude_n mn_a or_o not_o first_o let_v it_o be_v equal_a case_n but_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eq_n but_o cones_fw-la and_o cylinder_n consist_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n be_v by_o the_o 11._o of_o the_o twelve_o wherefore_o the_o base_a abcd_o be_v equal_a to_o the_o base_a efgh_o wherefore_o also_o they_o be_v reciprokal_a as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o but_o now_o suppose_v that_o the_o altitude_n lk_n be_v not_o equal_a to_o the_o altitude_n m._n n_o construction_n but_o let_v the_o altitude_n mn_v be_v great_a and_o by_o the_o 3._o of_o the_o first_o from_o the_o altitude_n mn_v take_v away_o pm_v equal_a to_o the_o altitude_n kl_o so_o that_o let_v the_o line_n pm_n be_v put_v equal_a to_o the_o line_n kl_o and_o by_o the_o point_n p_o let_v there_o be_v extend_v a_o plain_a superficies_n tus_z which_o let_v cut_v the_o cylinder_n eo_fw-la and_o be_v a_o parallel_n to_o the_o two_o opposite_a plain_a superâicieces_n that_o be_v to_o the_o circle_n efgh_o and_o ro._n cylinder_n and_o make_v the_o base_a the_o circle_n efgh_o &_o the_o altitude_n mp_n imagine_v a_o cylinder_n es._n and_o for_o that_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eo_fw-la and_o there_o be_v a_o other_o cylinder_n es_fw-ge therefore_o by_o the_o 7._o of_o the_o five_o as_o the_o cylinder_n axe_n be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o cylinder_n eo_fw-la to_o the_o cylinder_n es._n but_o as_o the_o cylinder_n axe_n be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o base_a abcd_o to_o the_o base_a efgh_o for_o the_o cylinder_n axe_n and_o es_fw-ge be_v under_o one_o and_o the_o self_n same_o altitude_n and_o as_o the_o cylinder_n eo_fw-la be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o altitude_n mn_v to_o the_o altitude_n mp_n for_o cylinder_n consist_v upon_o equal_a base_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o altitude_n wherefore_o as_o the_o base_a abcd_o be_v âo_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n mp_n but_o the_o altitude_n pm_n be_v equal_a to_o the_o altitude_n kl_o wherefore_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o wherefore_o in_o the_o equal_a cylinder_n axe_n and_o eo_fw-la the_o base_n be_v reciprocal_a to_o their_o altitude_n but_o now_o suppose_v that_o the_o base_n of_o the_o cylinder_n axe_n and_o eo_fw-la be_v reciprokal_n to_o their_o altitude_n that_o be_v as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o demonstrate_v then_o i_o say_v that_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eo_fw-la for_o the_o self_n same_o order_n of_o construction_n remain_v for_o that_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o but_o the_o altitude_n kl_o be_v equal_a to_o the_o altitude_n pm_n wherefore_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n pm_n but_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o cylinder_n axe_n to_o the_o cylinder_n es_fw-ge for_o they_o be_v under_o equal_a altitude_n and_o as_o the_o altitude_n mn_o be_v to_o the_o altitude_n pm_n so_o be_v the_o cylinder_n eo_fw-la to_o the_o cylinder_n es_fw-ge by_o the_o 14._o of_o the_o twelve_o wherefore_o also_o as_o the_o cylinder_n axe_n be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o cylinder_n eo_fw-la to_o the_o cylinder_n es._n wherefore_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eo_fw-la by_o the_o 9_o of_o the_o five_o and_o so_o also_o be_v it_o in_o the_o cones_fw-la which_o haââ_n the_o self_n same_o base_n and_o altitude_n with_o the_o cylinder_n wherefore_o in_o equal_a cones_fw-la and_o cylinder_n the_o base_n be_v reciprocal_a to_o their_o altitude_n etc_n etc_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o campane_n and_o flussas_n hitherto_o have_v be_v show_v the_o passion_n and_o propriety_n of_o cones_fw-la and_o cylinder_n who_o altitude_n fall_v perpendicular_o upon_o the_o base_n now_o will_v we_o declare_v that_o cones_fw-la and_o cilinder_n who_o altitude_n fall_v oblique_o upon_o their_o base_n have_v also_o the_o self_n same_o passion_n and_o propriety_n which_o the_o foresay_a cones_fw-la and_o cilinder_n have_v forasmuch_o as_o in_o the_o ten_o of_o this_o book_n it_o be_v say_v that_o every_o cone_n be_v the_o three_o part_n of_o a_o cilind_a have_v one_o and_o the_o self_n same_o base_a &_o one_o &_o the_o self_n same_o altitude_n with_o it_o which_o thing_n be_v demonstrate_v by_o a_o cilind_a give_v who_o base_a be_v cut_v by_o a_o square_n inscribe_v in_o it_o and_o upon_o the_o side_n of_o the_o square_n be_v describe_v isosceles_a triangle_n make_v a_o polygonon_n figure_n and_o again_o upon_o the_o side_n of_o this_o polygonon_n figure_n be_v infinite_o after_o the_o same_o manner_n describe_v other_o isosceles_a triangle_n take_v away_o more_o they_o the_o half_a as_o have_v oftentimes_o be_v declare_v therefore_o it_o be_v manifest_a that_o the_o solid_n set_v upon_o these_o base_n be_v under_o the_o same_o altitude_n that_o the_o cilind_a incline_v be_v and_o be_v also_o include_v in_o the_o same_o cilind_a do_v take_v away_o more_o than_o the_o half_a of_o the_o cilind_a and_o also_o more_o they_o the_o half_a of_o the_o residue_n as_o it_o have_v be_v prove_v in_o erect_a cylinder_n for_o these_o incline_n solid_n be_v under_o equal_a altitude_n and_o upon_o equal_a base_n with_o the_o erect_a solid_n be_v equal_a to_o the_o erect_a solid_n by_o the_o corollary_n of_o the_o _o of_o the_o eleven_o wherefore_o they_o also_o in_o like_a sort_n as_o the_o erect_a take_v away_o more_o than_o the_o half_a if_o therefore_o we_o compare_v the_o incline_v cilind_a to_o a_o cone_n set_v upon_o the_o self_n same_o base_a and_o have_v his_o altitude_n erect_v and_o reason_n by_o a_o argument_n lead_v to_o a_o impossibility_n by_o the_o demonstration_n of_o the_o ten_o of_o this_o book_n we_o may_v prove_v that_o the_o side_v solid_a include_v in_o the_o incline_v cylinder_n be_v great_a than_o the_o triple_a of_o his_o pyramid_n and_o it_o be_v also_o equal_a to_o the_o same_o which_o be_v impossible_a and_o this_o be_v the_o first_o case_n wherein_o it_o be_v prove_v that_o the_o cilind_a not_o be_v equal_a to_o the_o triple_a of_o the_o cone_n be_v not_o great_a than_o the_o triple_a of_o the_o same_o and_o as_o touch_v the_o second_o case_n we_o may_v after_o the_o same_o manner_n conclude_v that_o that_o âided_v solid_a contain_v in_o the_o cylinder_n be_v great_a than_o the_o cylinder_n which_o be_v very_o absurd_a wherefore_o if_o the_o cylinder_n be_v neither_o great_a than_o the_o triple_a of_o the_o cone_n nor_o less_o it_o must_v needs_o be_v equal_a to_o the_o same_o the_o demonstration_n of_o these_o incline_v cylinder_n most_v plain_o follow_v the_o demonstration_n of_o the_o erect_a cylinder_n for_o it_o have_v already_o be_v prove_v that_o pyramid_n and_o side_v solid_n which_o be_v also_o call_v general_o prism_n be_v set_v upon_o equal_a base_n and_o under_o one_o and_o the_o self_n same_o altitude_n whether_o the_o altitude_n be_v erect_v or_o incline_v be_v equal_a the_o one_o to_o the_o other_o namely_o be_v in_o proportion_n as_o their_o base_n be_v by_o
place_n from_o whence_o first_o it_o begin_v to_o be_v move_v it_o shall_v pass_v by_o the_o point_n f_o g_o h_o and_o the_o octohedron_n shall_v be_v comprehend_v in_o a_o sphere_n i_o say_v also_o that_o it_o be_v comprehend_v in_o the_o sphere_n give_v for_o forasmuch_o as_o the_o line_n lk_n be_v equal_a to_o the_o line_n km_o by_o position_n and_o the_o line_n ke_o be_v common_a to_o they_o both_o and_o they_o contain_v right_a angle_n by_o the_o 3._o definition_n of_o the_o eleven_o therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a le_fw-fr be_v equal_a to_o the_o base_a em_n and_o forasmuch_o as_o the_o angle_n lem_n be_v a_o right_a angle_n by_o the_o 31._o of_o the_o three_o for_o it_o be_v in_o a_o semicircle_n as_o have_v be_v prove_v therefore_o the_o square_a of_o the_o line_n lm_o be_v double_a to_o the_o square_n of_o the_o line_n le_fw-fr by_o the_o 47._o of_o the_o first_o again_o forasmuch_o as_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n bc_o therefore_o the_o line_n ab_fw-la be_v double_a to_o the_o line_n bc_o by_o the_o definition_n of_o a_o circle_n but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n bd_o by_o the_o corollary_n of_o the_o 8._o and_o _o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v double_a to_o the_o square_n of_o the_o line_n bd._n and_o it_o be_v prove_v that_o the_o square_a of_o the_o line_n lm_o be_v double_a to_o the_o square_n of_o the_o line_n le._n wherefore_o the_o square_a of_o the_o line_n bd_o be_v equal_a to_o the_o square_n of_o the_o line_n le._n for_o the_o line_n eh_o which_o be_v equal_a to_o the_o line_n lf_n be_v put_v to_o be_v equal_a to_o the_o line_n db._n wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n lm_o wherefore_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n lm_o and_o the_o line_n ab_fw-la be_v the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o line_n lm_o be_v equal_a to_o the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o octoedron_n be_v contain_v in_o the_o sphere_n give_v le._n and_o it_o be_v also_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n wherefore_o there_o be_v make_v a_o octohedron_n and_o it_o be_v comprehend_v in_o the_o sphere_n give_v wherein_o be_v comprehend_v the_o pyramid_n and_o it_o be_v prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedrn_n which_o be_v require_v to_o be_v do_v and_o to_o be_v prove_v certain_a corollary_n add_v by_o flussas_n first_o corollary_n the_o side_n of_o a_o pyramid_n be_v in_o power_n sesquitertia_fw-la to_o the_o side_n of_o a_o octahedron_n inscribe_v in_o the_o same_o sphere_n for_o forasmch_v as_o the_o diameter_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n therefore_o of_o what_o part_n the_o diameter_n contain_v in_o power_n 6._o of_o the_o same_o the_o side_n of_o the_o octohedron_n contain_v in_o power_n 3._o but_o of_o what_o part_n the_o diameter_n contain_v 6._o of_o the_o same_o the_o side_n of_o the_o pyramid_n contain_v 4._o by_o the_o 13._o of_o this_o book_n wherefore_o of_o what_o part_n the_o side_n of_o the_o pyramid_n contain_v 4._o of_o the_o same_o the_o side_n of_o the_o octohedron_n contain_v 3._o second_o corollary_n a_o octohedron_n be_v divide_v into_o two_o equal_a and_o like_a pyramid_n the_o common_a base_n of_o these_o pyramid_n be_v set_v upon_o every_o square_n contain_v of_o the_o side_n of_o the_o octohedron_n upon_o which_o square_a be_v set_v the_o ââ_o triangle_n of_o the_o octohedron_n campane_n which_o pyramid_n be_v by_o the_o â_o definition_n of_o the_o eleven_o equal_a and_o like_a and_o the_o foresay_a square_n common_a to_o those_o pyramid_n be_v the_o half_a of_o the_o square_n of_o the_o diameter_n of_o the_o sphere_n for_o it_o be_v the_o square_a of_o the_o side_n of_o the_o octohedron_n three_o corollary_n the_o three_o diameter_n of_o the_o octohedron_n do_v cut_v the_o one_o the_o other_o perpendicular_o into_o two_o equal_a part_n in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o say_a octohedron_n as_o it_o be_v manifest_a by_o the_o three_o diameter_n eglantine_n fh_o and_o lm_o which_o cut_v the_o one_o the_o other_o in_o the_o centre_n king_n equal_o and_o perpendicular_o ¶_o the_o 3._o problem_n the_o 15._o proposition_n to_o make_v a_o solid_a call_v a_o cube_fw-la and_o to_o comprehend_v it_o in_o the_o sphere_n give_v namely_o that_o sphere_n wherein_o the_o former_a two_o solid_n be_v comprehendâdâ_n and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n treble_a to_o the_o side_n of_o the_o cube_fw-la take_v the_o diameter_n of_o the_o sphere_n give_v namely_o ab_fw-la and_o divide_v it_o in_o the_o point_n câ_n so_o that_o let_v the_o line_n ac_fw-la be_v double_a to_o the_o line_n bc_o by_o the_o 9_o of_o the_o six_o and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n adb_o and_o by_o the_o 11._o of_o the_o first_o from_o the_o pâynt_n c_o râyse_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n cd_o and_o draw_v a_o right_a linâ_n db._n and_o describe_v a_o square_a efgh_o have_v every_o one_o of_o his_o side_n equal_a to_o the_o line_n db_o and_o from_o the_o point_n e_o f_o g_o h_o raise_v up_o by_o the_o 12._o of_o the_o eleven_o unto_o the_o plain_a superficies_n of_o the_o square_a efgh_o perpendicular_a line_n eke_o fl_fw-mi gm_n and_o hn_o and_o let_v every_o one_o of_o the_o line_n eke_o fl_fw-mi gm_n and_o hn_o be_v put_v equal_a to_o one_o of_o the_o line_n of_o fg_o gh_o or_o he_o which_o be_v the_o side_n of_o the_o square_n and_o draw_v these_o right_a line_n kl_o lm_o mn_a and_o nk_v demonstration_n wherefore_o there_o be_v make_v a_o cube_fw-la namely_o fn_o which_o be_v contain_v under_o six_o equal_a square_n construction_n now_o it_o be_v require_v to_o comprehend_v the_o same_o cube_fw-la in_o the_o sphere_n give_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n ble_o to_o the_o side_n of_o the_o cube_fw-la demonstration_n draw_v these_o right_a line_n kg_v and_o eglantine_n and_o forasmuch_o as_o the_o angle_n keg_n be_v a_o right_a angle_n for_o that_o the_o line_n ke_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n eâ_n and_o therefore_o also_o to_o the_o right_a line_n eglantine_n by_o the_o 2._o definition_n of_o the_o eleven_o wherefore_o a_o semicircle_n describe_v upon_o the_o line_n kg_v shall_v book_n pass_v by_o the_o point_n e._n again_o forasmuch_o as_o the_o line_n fg_o be_v erect_v perpendicular_o to_o either_o of_o these_o line_n fl_fw-mi and_o fe_o by_o the_o definition_n of_o a_o square_a &_o by_o the_o 2._o definition_n of_o the_o eleven_o therefore_o the_o line_n fg_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n fk_o by_o the_o 4._o of_o the_o eleven_o wherefore_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n f_o to_o the_o point_n king_n the_o line_n gf_o shall_v be_v erect_v perpendicular_o to_o the_o line_n kf_o by_o the_o 2._o definition_n of_o the_o eleven_o and_o by_o the_o same_o reason_n again_o a_o semicircle_n describe_v upon_o the_o line_n gk_o shall_v pass_v also_o by_o the_o point_n f._n and_o likewise_o shall_v it_o pass_v by_o the_o rest_n of_o the_o point_n of_o the_o angle_n of_o that_o cube_fw-la if_o now_o the_o diameter_n kg_v abide_v fix_v the_o semicircle_n be_v turn_v round_o about_o until_o it_o return_v into_o the_o self_n same_o place_n from_o whence_o it_o begin_v first_o to_o be_v move_v the_o cube_fw-la shall_v be_v comprehend_v in_o a_o sphere_n i_o say_v also_o that_o it_o be_v comprehend_v in_o the_o sphere_n give_v demonstration_n for_o forasmuch_o as_o the_o line_n gf_o be_v equal_a to_o the_o linââe_n and_o the_o angle_n fletcher_n be_v a_o right_a angle_n therefore_o the_o square_a of_o the_o line_n eglantine_n be_v by_o the_o 47._o of_o the_o first_o double_a to_o the_o square_n of_o the_o line_n âf_n but_o the_o line_n of_o be_v equal_a to_o the_o line_n eke_o wherefore_o the_o square_a of_o the_o line_n eglantine_n be_v double_a to_o the_o square_n of_o the_o line_n eke_o wherefore_o the_o square_n of_o eglantine_n and_o eke_o that_o be_v the_o square_a of_o the_o line_n gk_o by_o the_o 47._o of_o the_o first_o be_v treble_a to_o the_o square_n of_o the_o line_n eke_o and_o forasmuch_o as_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n bc_o but_o
as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n bd_o by_o the_o corollary_n of_o the_o 8._o and_o 20._o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v treble_a to_o the_o square_n of_o the_o line_n bd._n and_o it_o be_v prove_v that_o the_o square_a of_o the_o line_n gk_o be_v treble_a to_o the_o square_n of_o the_o line_n ke_o and_o the_o line_n ke_o be_v put_v equal_a to_o the_o line_n bd._n wherefore_o the_o line_n kg_n be_v also_o equal_a to_o the_o line_n ab_fw-la and_o the_o line_n ab_fw-la be_v the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o line_n kg_n be_v equal_a to_o the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o cube_fw-la be_v comprehend_v in_o the_o sphere_n give_v and_o it_o be_v also_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n treble_a to_o the_o side_n of_o the_o cube_fw-la which_o be_v require_v tââe_o do_v and_o to_o be_v prove_v an_o other_o demonstration_n after_o flussas_n suppose_v that_o the_o diameter_n of_o the_o sphere_n give_v in_o the_o former_a proposition_n be_v the_o line_n aâ_z and_o let_v the_o centre_n be_v the_o point_n c_o upon_o which_o describe_v a_o semicircle_n adb_o and_o from_o the_o diameter_n ab_fw-la cut_v of_o a_o three_o part_n bg_o by_o the_o 9_o of_o the_o six_o and_o from_o the_o point_n g_o raise_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n dg_o by_o the_o 11._o of_o the_o first_o and_o draw_v these_o right_a line_n dam_n dc_o and_o db._n and_o unto_o the_o right_a line_n db_o put_v a_o equal_a right_a line_n zi_n and_o upon_o the_o line_n zi_n describe_v a_o square_a ezit_n and_o from_o the_o point_n e_o z_o i_o t_o erect_v unto_o the_o superficies_n ezit_n perpendicular_a line_n eke_o zh_n in_o tn_n by_o the_o 12._o of_o the_o eleven_o and_o put_v every_o one_o of_o those_o perpendicular_a line_n equal_a to_o the_o line_n zi_n and_o draw_v these_o right_a line_n kh_o hm_n mn_o and_o nk_v each_o of_o which_o shall_v be_v equal_a and_o parallel_n to_o the_o line_n zi_n and_o to_o the_o rest_n of_o the_o line_n of_o the_o square_n by_o the_o 33._o of_o the_o first_o and_o moreover_o they_o shall_v contain_v equal_a angle_n by_o the_o 10._o of_o the_o eleven_o and_o therefore_o the_o angle_n be_v right_a angle_n for_o that_o ezit_n be_v a_o square_n wherefore_o the_o rest_n of_o the_o base_n shall_v be_v square_n wherefore_o the_o solid_a ezitkhmn_n be_v contain_v under_o 6._o equal_a square_n be_v a_o cube_fw-la by_o the_o 21._o definition_n of_o the_o eleven_o extend_v by_o the_o opposite_a side_n ke_o and_o mi_n of_o the_o cube_fw-la a_o plain_a keim_n and_o again_o by_o the_o other_o opposite_a side_n nt_v and_o hz_n extend_v a_o other_o plain_a hztn_n now_o forasmuch_o as_o each_o of_o these_o plain_n divide_v the_o solid_a into_o two_o equal_a part_n namely_o into_o two_o prism_n equal_a and_o like_a by_o the_o 8._o definition_n of_o the_o eleven_o therefore_o those_o plain_n shall_v cut_v the_o cube_fw-la by_o the_o centre_n by_o the_o corollary_n of_o the_o 39_o of_o the_o eleven_o wherefore_o the_o common_a section_n of_o those_o plain_n shall_v pass_v by_o the_o centre_n let_v that_o common_a section_n be_v the_o line_n lf_n and_o forasmuch_o as_o the_o side_n hn_o and_o km_o of_o the_o superficiece_n keim_n and_o hztn_n do_v divide_v the_o one_o the_o other_o into_o two_o equal_a part_n by_o the_o corollary_n of_o the_o 34._o of_o the_o first_o and_o so_o likewise_o do_v the_o side_n zt_v and_o ei_o therefore_o the_o common_a section_n lf_o be_v draw_v by_o these_o section_n and_o divide_v the_o plain_n keim_n and_o hztn_n into_o two_o equal_a part_n by_o the_o first_o of_o the_o six_o for_o their_o base_n be_v equal_a and_o the_o altitude_n be_v one_o and_o the_o âame_n namely_o the_o altitude_n of_o the_o cube_fw-la wherefore_o the_o line_n lf_n shall_v divide_v into_o two_o equal_a part_n the_o diameter_n of_o his_o plain_n namely_o the_o right_a line_n ki_v em_n zn_n and_o nt_v which_o be_v the_o diameter_n of_o the_o cube_fw-la wherefore_o those_o diameter_n shall_v concur_v and_o cut_v one_o the_o other_o in_o one_o and_o the_o self_n same_o point_n let_v the_o same_o be_v o._n wherefore_o the_o right_a line_n ok_n oe_n oi_o om_o oh_o oz_n ot_fw-mi and_o on_o shall_v be_v equal_a the_o onâ_n to_o the_o other_o for_o that_o they_o be_v the_o half_n of_o the_o diameter_n of_o equal_a and_o like_a rectangle_n parallelogram_n wherefore_o make_v the_o centre_n the_o point_n oh_o and_o the_o space_n any_o of_o these_o line_n oe_z or_o ok_n etc_n etc_n a_o sphere_n describe_v shall_v pass_v by_o every_o one_o of_o the_o angle_n of_o the_o cube_fw-la namely_o which_o be_v at_o the_o point_n e_o z_o i_o t_o edward_n h_o m_o n_o by_o the_o 12._o definition_n of_o the_o eleven_o for_o that_o all_o the_o line_n draw_v from_o the_o point_n oh_o to_o the_o angle_n of_o the_o cube_fw-la be_v equal_a but_o the_o right_a line_n ei_o contain_v in_o power_n the_o two_o equal_a right_a line_n ez_o and_o zi_n by_o the_o 47._o of_o the_o first_o wherefore_o the_o square_a of_o the_o line_n ei_o be_v double_a to_o the_o square_n of_o the_o line_n zi_n and_o forasmuch_o as_o the_o right_a line_n ki_v subtend_v the_o right_a angle_n kei_n for_o that_o the_o right_a line_n keâ_n be_v erect_v perpendicular_o to_o the_o plaiâe_a superficies_n of_o the_o right_a line_n ez_o and_o zt_n by_o the_o 4._o of_o the_o eleven_o â_o therefore_o the_o square_a of_o the_o line_n ki_v be_v equal_a to_o the_o square_n of_o the_o line_n ei_o and_o eke_o but_o the_o square_n of_o the_o line_n ei_o be_v double_a to_o the_o square_n of_o the_o line_n eke_o for_o it_o be_v double_a to_o the_o square_n of_o the_o line_n zi_n as_o have_v be_v prove_v and_o the_o base_n of_o the_o cube_fw-la be_v equal_a square_n wherefore_o the_o square_a of_o the_o line_n ki_v be_v triple_a to_o the_o square_n of_o the_o line_n ke_o that_o be_v to_o the_o square_n of_o the_o line_n zi_n but_o the_o right_a line_n zi_n be_v equal_a to_o thâ_z right_a line_n db_o by_o position_n unto_o who_o square_a the_o square_n of_o the_o diameter_n ab_fw-la be_v triple_a by_o that_o which_o be_v demonstrate_v in_o the_o 13._o proposition_n of_o this_o book_n wherefore_o the_o diameter_n ki_v &_o db_o be_v equal_a wherefore_o there_o be_v describe_v a_o cube_fw-la king_n and_o it_o be_v comprehend_v in_o the_o sphere_n give_v wherein_o the_o other_o solid_n be_v contain_v the_o diameter_n of_o which_o sphere_n be_v the_o line_n ab_fw-la and_o the_o diameter_n ki_v or_o ab_fw-la of_o the_o same_o sphere_n be_v prove_v to_o be_v in_o power_n triple_a to_o the_o side_n of_o the_o cube_fw-la namely_o to_o the_o line_n db_o or_o zi_n ¶_o corollaryes_fw-mi add_v by_o flussas_n first_o corollary_n hereby_o it_o be_v manifest_a that_o the_o diameter_n of_o a_o sphere_n contain_v in_o power_n the_o side_n both_o of_o a_o pyramid_n and_o of_o a_o cube_fw-la inscribe_v in_o it_o for_o the_o power_n of_o the_o side_n of_o the_o pyramid_n be_v two_o three_o of_o the_o power_n of_o the_o diameter_n by_o the_o 13._o of_o this_o book_n and_o the_o power_n of_o the_o side_n of_o the_o cube_fw-la be_v by_o this_o proposition_n one_o three_o of_o the_o power_n of_o the_o say_a diameter_n wherefore_o the_o diameter_n of_o the_o sphere_n contain_v in_o power_n the_o side_n of_o the_o pyramid_n and_o of_o the_o cube_fw-la .._o ¶_o second_v corollary_n all_o the_o diameter_n of_o a_o cube_fw-la cut_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o cube_fw-la and_o moreover_o those_o diameter_n do_v in_o the_o self_n same_o point_n cut_v into_o two_o equal_a part_n the_o right_a line_n which_o join_v together_o the_o centre_n of_o the_o opposite_a base_n as_o it_o be_v manifest_a to_o see_v by_o the_o right_a line_n lof_o for_o the_o angle_n lko_n and_o fio_n be_v equal_a by_o the_o 29._o of_o the_o first_o and_o it_o be_v prove_v that_o they_o be_v contain_v under_o equal_a side_n wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_n lo_o and_o foe_n be_v equal_a in_o like_a sort_n may_v be_v prove_v that_o the_o rest_n of_o the_o right_a line_n which_o join_v together_o the_o centre_n of_o the_o opposite_a base_n do_v cut_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o the_o centre_n o._n ¶_o the_o 4._o problem_n the_o 16._o proposition_n to_o make_v a_o icosahedron_n and_o to_o comprehend_v it_o in_o the_o sphere_n
f._n and_o draw_v these_o right_a line_n favorina_n fb_o fc_o fd_o fe_o wherefore_o those_o line_n do_v divide_v the_o angle_n of_o the_o pentagon_n into_o two_o equal_a part_n in_o the_o point_n a_o b_o c_o d_o e_o by_o the_o 4._o of_o the_o first_o and_o forasmuch_o as_o the_o five_o angle_n that_o be_v at_o the_o point_n f_o aâe_fw-fr equal_a to_o four_o right_a angle_n by_o the_o corollary_n of_o the_o 15._o of_o the_o first_o and_o they_o be_v equal_v the_o one_o to_o the_o other_o by_o the_o 8._o of_o the_o first_o therefore_o one_o of_o those_o angle_n as_o âor_a example_n sake_n the_o angle_n afb_o be_v a_o fiâth_v part_n less_o than_o a_o right_a angle_n wherefore_o the_o angle_n remain_v namely_o fab._n &_o abf_n be_v one_o right_a angle_n and_o a_o five_o part_n over_o but_o the_o angle_n fab._n be_v equal_a to_o the_o angle_n fbc_n wherefore_o the_o whole_a angle_n abc_n be_v one_o of_o the_o angle_n of_o the_o pentagon_n be_v a_o right_a angle_n and_o a_o five_o part_v more_o than_o a_o right_a angle_n which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o flussas_n now_o let_v we_o teach_v how_o those_o five_o solid_n have_v each_o like_a inclination_n of_o their_o base_n âiâst_v let_v we_o take_v a_o pyramid_n and_o divide_v one_o of_o the_o side_n thereof_o into_o two_o equal_a part_n and_o from_o the_o two_o angle_n opposite_a unto_o that_o side_n dâaw_v perpendiculars_n which_o shall_v fall_v upon_o the_o section_n by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o rational_a and_o at_o the_o say_a point_n of_o division_n as_o may_v easy_o be_v prove_v wherefore_o they_o shall_v contain_v the_o angâe_n of_o the_o inclination_n of_o the_o plain_n by_o the_o 4._o definition_n of_o the_o eleven_o which_o angle_n be_v subtend_v of_o the_o opposite_a side_n of_o the_o pyramid_n now_o forasmuch_o as_o the_o rest_n of_o the_o angle_n of_o the_o inclination_n of_o the_o plain_n of_o the_o pyramid_n be_v contain_v under_o two_o perpendicular_a line_n of_o the_o triangle_n and_o be_v subtend_v of_o the_o side_n of_o the_o pyramid_n it_o follow_v by_o the_o 8._o of_o the_o firât_o that_o those_o angle_n be_v equal_a wherâfoâe_o by_o the_o 5._o definition_n of_o the_o eleven_o the_o superficiece_n be_v in_o like_a sort_n incline_v the_o one_o to_o the_o other_o ãâ¦ã_o one_o of_o the_o side_n of_o a_o cube_n be_v divide_v into_o two_o equal_a part_n if_o from_o the_o say_a section_n be_v draw_v in_o two_o of_o the_o base_n thereof_o two_o perpendicular_a line_n they_o shall_v be_v parallel_n and_o equal_a to_o the_o side_n of_o the_o square_n which_o contain_v a_o right_a angle_n and_o forasmuch_o as_o all_o the_o angle_n of_o the_o base_n of_o the_o cube_n be_v right_a angle_n therefore_o those_o perpendicular_n fall_v upon_o the_o section_n of_o the_o side_n common_a to_o the_o two_o base_n shall_v contâyne_v a_o right_a angle_n by_o the_o 10._o of_o the_o eleven_o which_o self_n angle_n be_v the_o angle_n of_o inclination_n by_o the_o 4._o definition_n of_o the_o eleven_o and_o be_v subtend_v of_o the_o diameter_n of_o the_o base_a of_o the_o cube_n and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o rest_n of_o the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o the_o cube_fw-la be_v right_a angle_n wherefore_o the_o inclination_n of_o the_o superficiece_n of_o the_o cube_fw-la the_o one_o to_o the_o other_o be_v equal_a by_o the_o 5._o definition_n of_o the_o eleven_o in_o a_o octohedron_n take_v the_o diameter_n which_o couple_v the_o two_o opposite_a angle_n incline_v and_o from_o those_o opposite_a angle_n draw_v to_o one_o and_o the_o selâe_a same_o side_n of_o the_o octohedron_n in_o two_o base_n thereof_o two_o perpendicular_a line_n which_o shall_v divide_v that_o side_n into_o two_o equal_a part_n and_o perpendicular_o by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o those_o perpendicular_n shall_v contain_v the_o angle_n of_o the_o inclination_n of_o the_o base_n by_o the_o 4._o definition_n of_o the_o eleven_o and_o the_o same_o angle_n be_v subtend_v of_o the_o diameter_n of_o the_o octohedron_n wherefore_o the_o rest_n of_o the_o angle_n after_o the_o same_o manner_n describe_v in_o the_o rest_n of_o the_o base_n be_v comprehend_v and_o subtend_v of_o equal_a side_n shall_v by_o the_o 8._o of_o the_o first_o be_v equal_a the_o one_o to_o the_o other_o and_o therefore_o the_o inclination_n of_o the_o plain_n in_o the_o octohedron_n shall_v by_o the_o 5._o definition_n of_o the_o eleven_o be_v equal_a in_o a_o icosahedron_n let_v there_o be_v draw_v from_o the_o angle_n of_o two_o of_o the_o base_n incline_v to_o one_o side_n common_a to_o both_o the_o say_v base_n perpendicular_n which_o shall_v contain_v the_o angle_n of_o the_o inclination_n of_o the_o base_n by_o the_o 4._o definition_n of_o the_o eleven_o which_o angle_n be_v subtend_v of_o the_o right_a line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n which_o contain_v five_o side_n of_o the_o icosahedron_n by_o the_o 16._o of_o this_o book_n for_o it_o couple_v the_o two_o opposite_a angle_n of_o the_o triangle_n which_o be_v join_v together_o wherefore_o the_o rest_n of_o the_o angle_n of_o the_o inclination_n of_o the_o base_n be_v after_o the_o same_o manner_n find_v out_o they_o shall_v be_v contain_v under_o equal_a side_n and_o subtend_v of_o equal_a base_n and_o therefore_o by_o the_o 8._o of_o the_o fiâst_n those_o angle_n shall_v be_v equal_a wherefore_o also_o all_o the_o inclination_n of_o the_o base_n of_o the_o icosahedron_n the_o one_o to_o the_o other_o shalbâ_n equal_a by_o the_o 5._o definition_n of_o the_o eleven_o in_o a_o dodecahedron_n incline_v from_o the_o two_o opposite_a angle_n of_o two_o next_o pentagon_n draw_v to_o their_o common_a side_n perpendicular_a line_n pass_v by_o the_o centre_n of_o the_o say_a pentagon_n which_o shall_v where_o they_o fall_v divide_v the_o side_n into_o two_o equal_a part_n by_o the_o 3._o of_o the_o three_o for_o the_o base_n of_o a_o dodecahedron_n be_v contain_v in_o a_o circle_n and_o the_o angle_n contaynâd_v under_o those_o perpendicular_a line_n be_v the_o inclination_n of_o those_o base_n by_o the_o 4._o definition_n of_o the_o eleven_o and_o the_o foresay_a opposite_a angle_n be_v couple_v by_o a_o right_a line_n equal_a to_o the_o right_a line_n which_o couple_v the_o opposite_a section_n into_o two_o equal_a part_n of_o the_o side_n of_o the_o dodecahedron_n by_o the_o 33._o of_o the_o first_o for_o they_o couple_v together_o the_o half_a side_n of_o the_o dodecahedron_n which_o half_n be_v parallel_n and_o equal_a by_o the_o 3._o corollary_n of_o the_o 17._o of_o this_o book_n which_o couple_v line_n also_o be_v equal_a by_o the_o same_o corollary_n wherefore_o the_o angle_n be_v contain_v of_o equal_a perpendicular_a line_n and_o subtend_v of_o equal_a couple_a line_n shall_v by_o the_o 8._o of_o the_o first_o be_v equal_a and_o they_o be_v the_o angle_n of_o the_o inclination_n wherefore_o the_o base_n of_o the_o dodecahedron_n be_v in_o like_a sort_n incline_v the_o one_o to_o the_o other_o by_o the_o 5._o definition_n of_o the_o eleven_o flussas_n after_o this_o teach_v how_o to_o know_v the_o rationality_n or_o irrationality_n of_o the_o side_n of_o the_o triangle_n which_o contain_v the_o angle_n of_o the_o inclination_n of_o the_o superficiece_n of_o the_o foresay_a body_n in_o a_o pyramid_n the_o angle_n of_o the_o inclination_n be_v contain_v under_o two_o perpâdicular_a line_n of_o the_o triangle_n rational_a and_o be_v subtend_v of_o the_o side_n of_o the_o pyramid_n now_o the_o side_n of_o the_o pyramid_n be_v in_o power_n sesquitertia_fw-la to_o the_o perpendicular_a line_n by_o the_o corollary_n of_o the_o 12._o of_o this_o book_n and_o therefore_o the_o triangle_n contain_v of_o those_o perpendicular_a line_n and_o the_o side_n of_o pyramid_n have_v his_o side_n rational_a &_o commensurable_a in_o power_n the_o one_o to_o the_o other_o forasmuch_o as_o the_o two_o side_n of_o a_o cube_n or_o right_a line_n equal_a to_o they_o subtend_v under_o the_o diameter_n of_o one_o of_o the_o base_n rational_a do_v make_v the_o angle_n of_o the_o inclination_n and_o the_o diameter_n of_o the_o cube_fw-la be_v in_o power_n sesquialter_fw-la to_o the_o diameter_n of_o the_o base_a which_o diameter_n of_o the_o base_a be_v in_o power_n double_a to_o the_o side_n by_o the_o 47._o of_o the_o first_o therefore_o those_o line_n be_v rational_a and_o commensurable_a in_o power_n in_o a_o octohedron_n rational_a who_o two_o perpendicular_n of_o the_o base_n contain_v the_o angle_n of_o the_o inclination_n of_o the_o octohedron_n which_o angle_n also_o be_v subtend_v of_o the_o diameter_n of_o the_o octohedron_n the_o diameter_n be_v in_o power_n
one_o and_o the_o self_n same_o sphere_n let_v the_o diameter_n of_o the_o sphere_n give_v by_o ab_fw-la and_o let_v the_o base_n of_o the_o icosahedron_n and_o dodecahedron_n describe_v in_o it_o construction_n be_v the_o triangle_n mnr_n and_o the_o pentagon_n fkh_n and_o about_o they_o let_v there_o be_v describe_v circle_n by_o the_o 5._o and_o 14._o of_o the_o four_o and_o let_v the_o line_n draw_v from_o the_o centre_n of_o those_o circle_n to_o the_o circumference_n be_v ln_o and_o ok_n then_o i_o say_v that_o the_o line_n ln_o and_o ok_n be_v equal_a and_o therefore_o one_o and_o the_o self_n same_o circle_n contain_v both_o those_o figure_n let_v the_o right_a line_n ab_fw-la be_v in_o power_n quintuple_a to_o some_o one_o right_a line_n as_o to_o the_o line_n cg_o by_o the_o corollary_n of_o the_o 6._o of_o the_o ten_o and_o make_v the_o centre_n the_o point_n c_o &_o the_o space_n cg_o describe_v a_o circle_n dzg_v and_o let_v the_o side_n of_o a_o pentagon_n inscribe_v in_o that_o circle_n by_o the_o 11._o of_o the_o four_o be_v the_o line_n zg_v and_o let_v eglantine_n subtend_v half_o of_o the_o ark_n zg_v be_v the_o side_n of_o a_o decagon_n inscribe_v in_o that_o circle_n and_o by_o the_o 30._o of_o the_o six_o divide_v the_o line_n cg_o by_o a_o extreme_a &_o mean_a proportion_n in_o the_o point_n i_o demonstration_n now_o forasmuch_o as_o in_o the_o 16._o of_o the_o thirteen_o it_o be_v prove_v that_o this_o line_n cg_o unto_o who_o the_o diameter_n ab_fw-la of_o the_o sphere_n be_v in_o power_n quintuple_a be_v the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n which_o contain_v five_o angle_n of_o the_o icosahedron_n and_o the_o side_n of_o the_o pentagon_n describe_v in_o that_o circle_n dzg_v namely_o the_o line_n zg_n be_v side_n of_o the_o icosahedron_n describe_v in_o the_o sphere_n who_o diameter_n be_v the_o line_n ab_fw-la therefore_o the_o right_a line_n zg_v be_v equal_a to_o the_o line_n mn_o which_o be_v put_v to_o be_v the_o side_n of_o the_o icosahedron_n or_o of_o his_o triangular_a base_a moreover_o by_o the_o 17._o of_o the_o thirteen_o it_o be_v manifest_a that_o the_o right_a line_n âh_a which_o subtend_v the_o angle_n of_o the_o pentagon_n of_o the_o dodecahedron_n inscribe_v in_o the_o foresay_a sphere_n be_v the_o side_n of_o the_o cube_n inscribe_v in_o the_o self_n same_o sphere_n for_o upon_o the_o angle_n of_o the_o cube_fw-la be_v make_v the_o angle_n of_o the_o dodecahedron_n wherefore_o the_o diameter_n ab_fw-la be_v in_o power_n triple_a to_o fh_v the_o side_n of_o the_o cube_n by_o the_o 15._o of_o the_o thirteen_o but_o the_o same_o line_n ab_fw-la be_v by_o supposition_n in_o power_n quintuple_a to_o the_o line_n cg_o wherefore_o five_o square_n of_o the_o line_n cg_o be_v equal_a to_o three_o square_n of_o the_o line_n fh_o for_o each_o be_v equal_a to_o one_o and_o the_o self_n same_o square_n of_o the_o line_n ab_fw-la and_o forasmuch_o as_o eglantine_n the_o side_n of_o the_o decagon_n cut_v the_o right_a line_n cg_o by_o a_o extreme_a and_o mean_a proportion_n by_o the_o corollary_n of_o the_o 9_o of_o the_o thirteen_o likewise_o the_o line_n hk_o cut_v the_o line_n fh_o the_o side_n of_o the_o cube_n by_o a_o extreme_a and_o mean_a proportion_n by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o therefore_o the_o line_n cg_o and_o fh_o be_v divide_v into_o the_o self_n same_o proportion_n by_o the_o second_o of_o this_o book_n and_o the_o right_a line_n ci_o and_o eglantine_n which_o be_v the_o great_a segment_n of_o one_o and_o the_o self_n same_o line_n cg_o be_v equal_a and_o forasmuch_o as_o five_o square_n of_o the_o line_n cg_o be_v equal_a to_o three_o square_n of_o the_o line_n fh_o therefore_o five_o square_n of_o the_o line_n ge_z be_v equal_a to_o three_o square_n of_o the_o line_n hk_o for_o the_o line_n ge_z and_o hk_o be_v the_o great_a segment_n of_o the_o line_n cg_o and_o fh_o wherefore_o five_o squâreâ_n of_o the_o lineâ_z cg_o &_o ge_z be_v equal_a to_o the_o square_n of_o the_o ãâã_d âh_a &_o hk_a by_o the_o 1â_o of_o the_o âift_n but_o unto_o the_o square_n of_o the_o line_n cg_o and_o geâ_n be_v âqual_a the_o squâre_n of_o thetine_a zg_n by_o the_o 10._o of_o the_o thirteen_o and_o unto_o the_o line_n zg_v the_o line_n mn_o be_v equal_a wherefore_o five_o square_n of_o the_o line_n mn_o be_v equal_a to_o three_o square_n of_o the_o line_n fh_o hk_o but_o the_o square_n of_o the_o line_n ââ_o and_o hk_o ãâã_d quintuple_a to_o the_o square_n of_o the_o line_n ok_n which_o be_v draw_v from_o the_o centre_n by_o the_o three_o of_o this_o book_n wherefore_o three_o square_n of_o the_o line_n fh_o and_o hk_o make_v 15._o square_n of_o the_o line_n ok_n and_o forasmuch_o as_o the_o square_n of_o the_o line_n mn_o be_v triple_a to_o the_o square_n of_o the_o line_n ln_o which_o be_v draw_v from_o the_o centre_n by_o the_o 12._o of_o the_o thirteen_o therefore_o five_o square_n of_o the_o line_n mn_o be_v equal_a to_o 15._o square_n of_o the_o line_n ln_o but_o five_o square_n of_o the_o line_n mn_o be_v equal_a unto_o three_o square_n of_o the_o line_n fh_o and_o hk_o wherefore_o one_o square_a of_o the_o line_n ln_o be_v equal_a to_o one_o square_a of_o the_o line_n ok_n be_v each_o the_o fifteen_o part_n of_o equal_a magnitude_n by_o the_o 15._o of_o the_o fifââ_n wherefore_o the_o line_n ln_o and_o ok_n which_o be_v draw_v from_o the_o centre_n be_v equal_a wherefore_o also_o the_o circle_n nrm_n and_o fkh_n which_o be_v describe_v of_o those_o line_n be_v equal_a and_o those_o circle_n contain_v by_o supposition_n the_o base_n of_o the_o dodecahedron_n and_o of_o the_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n wherefore_o one_o and_o the_o self_n same_o circle_n etc_n etc_n aâ_z in_o thâ_z proposition_n which_o be_v require_v to_o be_v prove_v the_o 5._o proposition_n if_o in_o a_o circle_n be_v inscribe_v the_o pentagon_n of_o a_o dodecahedron_n and_o the_o triangle_n of_o a_o icosahedron_n campane_n and_o from_o the_o centre_n to_o one_o of_o their_o side_n be_v draw_v a_o perpendicular_a line_n that_o which_o be_v contain_v 30._o time_n under_o the_o side_n &_o the_o perpendicular_a line_n fall_v upon_o it_o be_v equal_a to_o the_o superficies_n of_o that_o solid_a upon_o who_o side_n the_o perpendicular_a line_n fall_v svppose_v that_o in_o the_o circle_n age_n be_v describe_v the_o pentagon_n of_o a_o dodecahedron_n which_o let_v be_v abgde_v and_o the_o triangle_n of_o a_o icosahedron_n describe_v in_o the_o same_o sphere_n which_o let_v be_v afh_n construction_n and_o let_v the_o centre_n be_v the_o point_n c._n ââon_n which_o draw_v perpendicular_o the_o line_n ci_o to_o the_o side_n of_o the_o pentagon_n and_o the_o line_n cl_n to_o the_o side_n of_o the_o triangle_n then_o i_o say_v that_o the_o rectangle_n figure_n contain_v under_o the_o line_n ci_o and_o gd_o 30._o time_n be_v equal_a to_o the_o superficies_n of_o the_o dodecahedron_n and_o that_o that_o which_o be_v contain_v under_o the_o line_n cl_n &_o of_o 30._o time_n be_v equal_a to_o the_o superficies_n of_o the_o icosahedron_n describe_v in_o the_o same_o sphere_n draw_v these_o right_a line_n ca_n cf_o cg_o and_o cd_o demonstration_n now_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o base_a gd_o &_o the_o altitude_n i_fw-mi be_v double_a to_o the_o triangle_n gcd_n by_o the_o 41._o of_o the_o first_o and_o five_o triangle_n like_a and_o equal_a to_o the_o triangle_n gcd_n do_v make_v the_o pentagon_n abgde_v of_o the_o dodecahedron_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n gd_o and_o i_fw-mi five_o time_n be_v equal_a to_o two_o pentagon_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n gd_o and_o i_fw-mi _o time_n be_v equal_a to_o the_o 12._o pentagon_n which_o contain_v the_o superficies_n of_o the_o dodecahedron_n again_o that_o which_o be_v contain_v under_o the_o line_n cl_n and_o of_o be_v double_a to_o the_o triangle_n acf_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n cl_n and_o of_o three_o time_n be_v equal_a to_o two_o such_o triangle_n as_o afh_n be_v which_o be_v one_o of_o the_o base_n of_o the_o icosahedron_n for_o the_o triangle_n acf_n be_v the_o three_o part_n of_o the_o triangle_n afh_v as_o it_o be_v easy_a to_o prove_v by_o the_o 8._o &_o 4._o of_o the_o first_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n cl_n and_o af._n 30_o time_n time_n be_v equal_a to_o 10._o such_o triangle_n as_o afh_v iâ_z which_o contain_v the_o superficies_n of_o the_o icosahedron_n and_o forasmuch_o as_o one_o and_o the_o self_n same_o
and_o have_v the_o same_o altitude_n with_o it_o name_o the_o altitude_n of_o the_o parallel_a base_n as_o it_o be_v manifest_a by_o the_o former_a be_v equal_a to_o three_o of_o those_o pyramid_n of_o the_o octohedron_n by_o the_o first_o corollary_n of_o the_o seven_o of_o the_o twelve_o wherefore_o that_o prism_n shall_v have_v to_o the_o other_o prism_n under_o the_o same_o altitude_n compose_v of_o the_o 4._o pyramid_n of_o the_o whole_a octohedron_n the_o proportion_n of_o the_o triangular_a base_n by_o the_o 3._o corollary_n of_o the_o same_o and_o forasmuch_o as_o 4._o pyramid_n be_v unto_o 3._o pyramid_n in_o sesquitercia_fw-la proportion_n therefore_o the_o triangule_a base_a of_o the_o prism_n which_o contain_v 4._o pyramid_n be_v in_o sesquitertia_fw-la proportion_n to_o the_o base_a of_o the_o prism_n which_o contain_v three_o pyramid_n of_o the_o same_o octohedron_n and_o be_v set_v upon_o the_o base_a of_o the_o octohedron_n and_o under_o the_o altitude_n thereof_o that_o be_v in_o sesquitercia_fw-la proportion_n to_o the_o base_a of_o the_o octohedron_n but_o the_o base_a of_o the_o same_o octohedron_n be_v in_o sesquitertia_fw-la proportion_n to_o the_o base_a of_o the_o pyramid_n by_o the_o âenth_n of_o this_o book_n wherefore_o the_o triangular_a base_n namely_o of_o the_o prism_n which_o contain_v four_o pyramid_n of_o the_o octohedron_n and_o be_v under_o the_o altitude_n thereof_o be_v equal_a to_o the_o triangular_a base_n of_o the_o prism_n which_o contain_v three_o pyramid_n under_o the_o altitude_n of_o the_o pyramid_n efgh_o but_o the_o prism_n of_o the_o octohedron_n be_v equal_a to_o the_o octohedron_n and_o the_o prism_n of_o the_o pyramid_n efgh_o be_v prove_v triple_a to_o the_o same_o pyramid_n efgh_o now_o than_o the_o prism_n set_v upon_o equal_a base_n be_v the_o one_o to_o the_o other_o as_o their_o altitude_n be_v by_o the_o corollary_n of_o the_o 25._o of_o the_o eleven_o namely_o as_o be_v the_o parallelipidedon_n their_o double_n by_o the_o corollary_n of_o the_o 31._o of_o the_o eleven_o but_o the_o altitude_n of_o the_o octohedron_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la contain_v in_o the_o same_o sphere_n by_o the_o corollary_n of_o the_o 13._o of_o this_o book_n and_o the_o side_n of_o the_o cube_fw-la be_v in_o power_n to_o the_o altitude_n of_o the_o tetrahedon_n in_o that_o proportion_n that_o 12._o be_v to_o 16_o by_o the_o 18._o of_o the_o thirteen_o and_o the_o side_n of_o the_o octohedron_n be_v to_o the_o side_n of_o the_o pyramid_n in_o that_o proportion_n that_o 18._o be_v to_o 24._o by_o the_o same_o 18._o of_o the_o thirteen_o which_o proportion_n be_v one_o &_o the_o self_n same_o with_o the_o proportion_n of_o 12._o to_o 16._o wherefore_o that_o prism_n which_o be_v equal_a to_o the_o octohedron_n be_v to_o the_o prism_n which_o be_v triple_a to_o the_o tetrahedron_n in_o that_o proportion_n that_o the_o altitude_n or_o that_o the_o side_n be_v wherefore_o a_o octohedron_n be_v to_o the_o triple_a of_o a_o tetrahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n in_o that_o proportion_n that_o their_o side_n be_v which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n the_o side_n of_o a_o tetrahedron_n &_o of_o a_o octohedron_n be_v proportional_a with_o their_o altitude_n for_o the_o side_n &_o altitude_n be_v in_o power_n sesquitercia_fw-la moreover_o the_o diameter_n of_o the_o sphere_n be_v to_o the_o side_n of_o the_o tetrahedron_n as_o the_o side_n of_o the_o octohedron_n be_v to_o the_o ââde_n of_o the_o cubeâ_n namely_o the_o power_n of_o each_o be_v in_o sesquialter_fw-la proportion_n by_o the_o 18._o of_o the_o thirteen_o the_o 15._o proposition_n if_o a_o rational_a line_n contain_v in_o power_n two_o line_n make_v the_o whole_a and_o the_o great_a segment_n and_o again_o contain_v in_o power_n two_o line_n make_v the_o whole_a and_o the_o less_o segment_n the_o great_a segment_n shall_v be_v the_o side_n of_o the_o icosahedron_n and_o the_o less_o segment_n shall_v be_v the_o side_n of_o the_o dodecahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n svppose_v that_o agnostus_n be_v the_o diameter_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n abgc_n and_o let_v bg_o subtend_v the_o side_n of_o the_o pentagon_n describe_v of_o the_o side_n of_o the_o icosahedron_n by_o the_o 16._o of_o the_o thirteen_o moreover_o upon_o the_o same_o diameter_n agnostus_n or_o df_o equal_a unto_o it_o construction_n let_v there_o be_v describe_v a_o dodecahedron_n defh_o by_o the_o 1â_o of_o the_o thirteen_o who_o opposite_n side_n ed_z and_o fh_o let_v be_v cut_v into_o two_o equal_a part_n in_o the_o point_n i_o and_o king_n and_o draw_v a_o line_n from_o i_o to_o k._n and_o let_v the_o line_n of_o couple_n two_o of_o the_o opposite_a angle_n of_o the_o base_n which_o be_v join_v together_o then_o i_o say_v that_o ab_fw-la the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n which_o the_o diameter_n agnostus_n contain_v in_o power_n together_o with_o the_o whole_a line_n and_o line_n ed_z be_v the_o less_o segment_n which_o the_o same_o diameter_n agnostus_n or_o df_o contain_v in_o power_n together_o with_o the_o whole_a demonstration_n for_o forasmuch_o as_o the_o opposite_a side_n ab_fw-la and_o gc_o of_o the_o icosahedron_n be_v couple_v by_o the_o diameter_n agnostus_n and_o bc_o be_v equal_a &_o parallel_n by_o the_o 2._o corollary_n of_o the_o 16._o of_o the_o thirteen_o the_o right_a line_n bg_o &_o ac_fw-la which_o couple_v they_o together_o be_v equal_a &_o parallel_n by_o the_o 33._o of_o the_o first_o moreover_o the_o angle_n bac_n &_o abg_o be_v subtend_v of_o equal_a diamâters_n shall_v by_o the_o 8._o of_o the_o first_o be_v equal_a &_o by_o the_o 29_o of_o the_o ãâã_d they_o shall_v be_v right_a angle_n wherefore_o the_o right_a line_n agnostus_n ãâã_d in_o power_n the_o too_o line_n ab_fw-la and_o bg_o by_o the_o 47._o of_o ãâ¦ã_o and_o forasmuch_o as_o the_o line_n bg_o subtend_v the_o angle_n of_o the_o pentagon_n compose_v of_o the_o side_n of_o the_o icosahedron_n the_o great_a segment_n of_o the_o right_a line_n bg_o shall_v be_v the_o right_a line_n ab_fw-la by_o the_o â_o of_o the_o thirteen_o which_o line_n ab_fw-la together_o with_o the_o whole_a line_n bg_o the_o line_n agnostus_n contain_v in_o power_n and_o forasmuch_o aâ_z the_o line_n ik_fw-mi couple_v the_o opposite_a and_o parallel_a side_n ed_z and_o fh_o of_o the_o dodecahedron_n make_v at_o those_o point_n right_a angle_n by_o the_o 3._o corollary_n of_o the_o 17._o of_o tâe_z thirteen_o the_o right_a line_n of_o which_o couple_v together_o equal_a and_o parallel_a line_n ei_o &_o fk_v shall_v be_v equal_a to_o the_o same_o line_n ik_fw-mi by_o the_o 33._o of_o the_o first_o wherefore_o the_o angle_n def_n shall_v be_v â_o right_n angle_n by_o the_o 29._o of_o the_o first_o wherefore_o the_o diameter_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o ef._n but_o the_o less_o segment_n of_o the_o line_n ik_fw-mi be_v ed_z the_o side_n of_o the_o dodecahedron_n by_o the_o 4._o corollary_n of_o the_o 17â_n of_o the_o thirteen_o wherefore_o the_o same_o line_n ed_z be_v also_o the_o less_o segment_n of_o the_o line_n of_o which_o be_v equal_a unto_o the_o line_n ik_fw-mi wherefore_o the_o diameter_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o of_o by_o the_o 47._o of_o the_o first_o contain_v in_o powârâ_n ed_z the_o side_n of_o the_o dodecahedron_n the_o less_o segment_n together_o with_o the_o whole_a if_o therefore_o a_o rational_a line_n agnostus_n or_o df_o contain_v in_o power_n two_o line_n ab_fw-la and_o bg_o do_v make_v the_o whole_a line_n and_o the_o great_a segment_n and_o again_o contain_v in_o power_n two_o line_n of_o and_o ed_z do_v make_v the_o whole_a line_n and_o the_o less_o segment_n the_o great_a segment_n ab_fw-la shall_v be_v the_o side_n of_o the_o icosahedron_n and_o the_o less_o segment_n ed_z shall_v be_v the_o side_n of_o the_o dodecahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n the_o 16._o proposition_n if_o the_o power_n of_o the_o side_n of_o a_o octohedron_n be_v express_v by_o two_o right_a lineâ_z join_v together_o by_o a_o extreme_a and_o meâne_a proportion_n the_o side_n of_o the_o icosahedron_n contain_v in_o the_o same_o sphere_n shall_v be_v duple_v to_o the_o less_o segment_n let_v ab_fw-la the_o side_n of_o the_o octohedron_n abg_o contain_v in_o power_n the_o two_o line_n c_o and_o h_o which_o let_v have_v that_o proportion_n that_o the_o whole_a have_v to_o the_o great_a segment_n by_o the_o corollarye_a of_o the_o first_o proposition_n add_v by_o flussas_n after_o the_o last_o proposition_n of_o the_o six_o book_n construction_n and_o let_v the_o icosahedron_n contain_v in_o the_o same_o sphere_n be_v
be_v demonstrate_v fd_o be_v the_o diameter_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n which_o diameter_n be_v in_o power_n sesquialter_fw-la to_o ab_fw-la the_o side_n of_o the_o tetrahedron_n inscribe_v in_o theâ_z same_o sphere_n by_o the_o 13._o of_o the_o thirteen_o wherefore_o the_o line_n ed_z the_o side_n of_o the_o icosahedron_n be_v in_o power_n sesquialter_fw-la to_o gâ_n the_o great_a segment_n or_o less_o line_n if_o therefore_o the_o side_n of_o a_o tetrahedron_n contain_v in_o power_n two_o right_a line_n join_v together_o a_o extreme_a and_o mean_a proportion_n the_o side_n of_o a_o icosahedron_n describe_v in_o the_o self_n same_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o less_o right_a line_n ¶_o the_o 19_o proposition_n the_o superficies_n of_o a_o cube_n be_v to_o the_o superficies_n of_o a_o octohedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n in_o that_o proportion_n that_o the_o solid_n be_v construction_n svppose_v that_o abcde_v be_v a_o cube_n who_o four_o diameter_n let_v be_v the_o line_n ac_fw-la bc_o dc_o and_o ec_o produce_v on_o each_o side_n let_v also_o the_o octohedron_n inscribe_v in_o the_o self_n same_o sphere_n be_v fghk_v who_o three_o diameter_n let_v be_v fh_o gk_o and_o on_o then_o i_o say_v that_o the_o cube_fw-la abdella_n be_v to_o the_o octohedron_n fgh_n as_o the_o superficies_n of_o the_o cube_fw-la be_v to_o the_o superficies_n of_o the_o octohedron_n draw_v from_o the_o centre_n of_o the_o cube_fw-la to_o the_o base_a abed_o a_o perpendicular_a line_n cr._n and_o from_o the_o centre_n of_o the_o octohedron_n draw_v to_o the_o base_a gnh_o a_o perpendicular_a line_n âl_o demonstration_n and_o forasmuch_o as_o the_o three_o diameter_n of_o the_o cube_fw-la do_v pass_n by_o the_o ãâã_d c_o therefore_o by_o the_o 2._o corollary_n of_o the_o 15._o of_o the_o thirteen_o âhere_o shall_v be_v make_v of_o the_o cube_fw-la six_o pyramid_n as_o this_o pyramid_n abdec_n equal_a to_o the_o whole_a cube_fw-la for_o there_o be_v in_o the_o cube_fw-la âixe_v base_n upon_o which_o fall_v equal_a perpendicular_n from_o the_o cenâââ_n by_o the_o corollary_n of_o the_o assumpt_n of_o the_o 16._o of_o the_o twelve_o for_o the_o base_n be_v contain_v in_o equal_a circle_v of_o the_o sphere_n but_o in_o the_o octohedron_n the_o three_o diameter_n do_v make_v upon_o the_o 8._o base_n 8._o pyramid_n have_v their_o top_n in_o the_o centre_n by_o the_o 3._o corollary_n of_o the_o 14â_o of_o the_o thirteen_o now_o the_o base_n of_o the_o cube_fw-la and_o of_o the_o octohedron_n be_v contain_v in_o equal_a circle_n of_o the_o sphere_n by_o the_o 13._o of_o this_o book_n wherefore_o they_o shall_v be_v equal_o distant_a from_o the_o centre_n and_o the_o perpendicular_a line_n cr_n and_o â_o shall_v be_v equal_a by_o the_o corollary_n of_o the_o assumpt_n of_o the_o 16._o of_o the_o twelve_o wherefore_o the_o pyramid_n of_o the_o cube_fw-la shall_v be_v under_o one_o and_o the_o self_n same_o altitude_n with_o the_o pyramid_n of_o the_o octohedron_n namely_o under_o the_o perpendicular_a line_n draw_v from_o the_o centre_n to_o the_o base_n wherefore_o six_o pyramid_n of_o the_o cube_fw-la be_v to_o 8._o pyramid_n of_o the_o octohedron_n be_v under_o one_o and_o the_o same_o altitude_n in_o that_o proportion_n that_o their_o base_n be_v by_o the_o 6._o of_o the_o twelve_o that_o be_v one_o pyramid_n set_v upon_o six_o base_n of_o the_o cube_fw-la and_o have_v to_o his_o altitude_n the_o perpendicular_a line_n which_o pyramid_n be_v equal_a to_o the_o six_o pyramid_n by_o the_o same_o 6._o of_o the_o twelve_o be_v to_o one_o pyramid_n set_v upon_o the_o 8._o base_n of_o the_o octohedron_n be_v equal_a to_o the_o octohedron_n and_o also_o under_o onâ_n and_o the_o self_n same_o altitude_n in_o that_o proportion_n that_o six_o base_n of_o the_o cube_fw-la which_o contain_v the_o whole_a superficies_n of_o the_o cube_fw-la be_v to_o 8._o base_n of_o the_o octohedronâ_n which_o contain_v the_o whole_a superficies_n of_o the_o octohedron_n for_o the_o solid_n of_o those_o pyramid_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n be_v by_o the_o self_n same_o 6._o of_o the_o twelve_o wherefore_o âhe_z superficies_n of_o the_o cube_fw-la be_v to_o the_o superficies_n of_o the_o octohedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n in_o that_o proportion_n that_o the_o solid_n be_v which_o be_v require_v to_o be_v prove_v ¶_o the_o 20._o proposition_n if_o a_o cube_n and_o a_o octohedron_n be_v contain_v in_o one_o &_o the_o self_n same_o sphere_n they_o shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o side_n of_o the_o cube_n be_v to_o the_o semidiameter_n of_o the_o sphere_n svppose_v that_o the_o octohedron_n aecdb_n be_v inscribe_v in_o the_o sphere_n abcd_o and_o let_v the_o cube_fw-la inscribe_v in_o the_o same_o sphere_n be_v fghim_n who_o diameter_n let_v be_v high_a construction_n which_o be_v equal_a to_o the_o diameter_n ac_fw-la by_o the_o 15._o of_o the_o thirteen_o let_v the_o half_a of_o the_o diameter_n be_v ae_n then_o i_o say_v that_o the_o cube_fw-la fghim_n be_v to_o the_o octohedron_n aecdb_n as_o the_o side_n mg_o be_v to_o the_o semidiameter_n ae_n forasmuch_o as_o the_o diameter_n ac_fw-la be_v in_o power_n double_a to_o bk_o the_o side_n of_o the_o octohedron_n by_o the_o 14._o of_o the_o thirteen_o and_o be_v in_o power_n triple_a to_o mg_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o the_o same_o therefore_o the_o square_a bkdl_n shall_v be_v sesquialter_n to_o fm_o the_o square_a of_o the_o cube_fw-la from_o the_o line_n ae_n cut_v of_o a_o three_o part_n a_o and_o from_o the_o line_n mg_o cut_v of_o likewise_o a_o three_o part_n go_v by_o the_o 9_o of_o the_o six_o demonstration_n now_o than_o the_o line_n en_fw-fr shall_v be_v two_o three_o part_n of_o the_o line_n ae_n and_o so_o also_o shall_v the_o line_n more_n be_v of_o the_o line_n mg_o wherefore_o the_o parallelipipedon_n set_v upon_o the_o base_a bkdl_n and_o have_v his_o altitude_n the_o line_n ea_fw-la be_v triple_a to_o the_o parallelipipedon_n set_v upon_o the_o same_o base_a and_o have_v his_o altitude_n the_o line_n a_fw-la by_o the_o corollary_n of_o the_o 31._o of_o the_o eleven_o but_o it_o be_v also_o triple_a to_o the_o pyramid_n abkdl_n which_o be_v set_v upon_o the_o same_o base_a and_o be_v under_o the_o same_o altitude_n by_o the_o second_o corollary_n of_o the_o 7._o of_o the_o twelve_o wherefore_o the_o pyramid_n abkdl_n be_v equal_a to_o the_o parallelipipedon_n which_o be_v set_v upon_o the_o base_a bkdl_n and_o have_v to_o his_o altitude_n the_o line_n an._n but_o unto_o that_o parallelipipedom_n be_v double_a the_o parallelipipedon_n which_o be_v set_v upon_o the_o same_o base_a bkdl_n and_o have_v to_o his_o altitude_n a_o line_n double_a to_o the_o line_n en_fw-fr by_o the_o corollary_n of_o the_o 31._o of_o the_o first_o and_o unto_o the_o pyramid_n be_v double_a the_o octohedron_n abkldc_n by_o the_o 2._o corollary_n of_o the_o 14._o of_o the_o thirteen_o wherefore_o the_o octohedron_n abkdlc_n be_v equal_a to_o the_o parallelipipedon_n set_v upon_o the_o base_a bkld_a &_o have_v his_o altitude_n the_o line_n en_fw-fr by_o the_o 15._o of_o the_o five_o but_o the_o parallelipipedon_n set_v upon_o the_o base_a bkdl_n which_o be_v sesquialter_fw-la to_o the_o base_a fm_o and_o have_v to_o his_o altitude_n the_o line_n more_n which_o be_v two_o three_o part_n of_o the_o side_n of_o the_o cube_fw-la mg_o be_v equal_a to_o the_o cube_fw-la fg_o by_o the_o 2._o part_n of_o the_o 34._o of_o the_o eleven_o for_o it_o be_v before_o prove_v that_o the_o base_a bkdl_n be_v sesquialter_fw-la to_o the_o base_a fm_o now_o than_o these_o two_o parallelipipedon_n namely_o the_o parallelipipedon_n which_o be_v set_v upon_o the_o base_a bkdl_n which_o be_v sesquialter_fw-la to_o the_o base_a of_o the_o cube_fw-la and_o have_v to_o his_o altitude_n the_o line_n more_n which_o be_v two_o three_o part_n of_o mg_o the_o side_n of_o the_o cube_fw-la which_o parallelipipedon_n be_v prove_v equal_a to_o the_o cube_fw-la and_o the_o parallelipipedon_n set_v upon_o the_o same_o base_a bkdl_n and_o have_v his_o altitude_n the_o line_n en_fw-fr which_o parallelipipedon_n be_v prove_v equal_a to_o the_o octohedron_n these_o two_o parallelipipedon_n i_o say_v be_v the_o one_o to_o the_o other_o as_o the_o altitude_n more_n be_v to_o the_o altitude_n en_fw-fr by_o the_o corollary_n of_o the_o 31._o of_o the_o eleven_o wherefore_o also_o as_o the_o altitude_n more_n be_v to_o the_o altitude_n en_fw-fr so_o be_v the_o cube_fw-la fghim_n to_o the_o octohedron_n abkdlc_n by_o the_o 7._o of_o the_o five_o but_o as_o the_o line_n more_n be_v to_o the_o line_n en_fw-fr so_o be_v
overthrow_v and_o overwhelm_v the_o whole_a world_n he_o be_v utter_o rude_a and_o ignorant_a in_o the_o greek_a tongue_n so_o that_o certain_o he_o never_o read_v euclid_n in_o the_o greek_a nor_o of_o like_a translate_v out_o of_o the_o greek_a but_o have_v it_o translate_v out_o of_o the_o arabike_a tongue_n the_o arabian_n be_v man_n of_o great_a study_n and_o industry_n and_o common_o great_a philosopher_n notable_a physician_n and_o in_o mathematical_a art_n most_o expert_a so_o that_o all_o kind_n of_o good_a learning_n flourish_v and_o reign_v amongst_o they_o in_o a_o manner_n only_o these_o man_n turn_v whatsoever_o good_a author_n be_v in_o the_o greek_a tongue_n of_o what_o art_n and_o knowledge_n so_o ever_o it_o be_v into_o the_o arabike_a tongue_n and_o from_o thence_o be_v many_o of_o they_o turn_v into_o the_o latin_a and_o by_o that_o mean_v many_o greek_a author_n come_v to_o the_o hand_n of_o the_o latin_n and_o not_o from_o the_o first_o fountain_n the_o greek_a tongue_n wherein_o they_o be_v first_o write_v as_o appear_v by_o many_o word_n of_o the_o arabike_a tongue_n yet_o remain_v in_o such_o book_n as_o be_v zenith_n nadir_n helmuayn_fw-mi helmuariphe_n and_o infinite_a such_o other_o which_o arabian_n also_o in_o translate_n such_o greek_a work_n be_v accustom_v to_o add_v as_o they_o think_v good_a &_o for_o the_o full_a understanding_n of_o the_o author_n many_o thing_n as_o be_v to_o be_v see_v in_o diverse_a author_n as_o namely_o in_o theodosius_n de_fw-la sphera_fw-la where_o you_o see_v in_o the_o old_a translation_n which_o be_v undoubteldy_a out_o of_o the_o arabike_a many_o proposition_n almost_o every_o three_o or_o four_o leaf_n some_o such_o copy_n of_o euclid_n most_o likely_a do_v campanus_n follow_v wherein_o he_o find_v those_o proposition_n which_o he_o have_v more_o &_o above_o those_o which_o be_v find_v in_o the_o greek_a set_v out_o by_o hypsicles_n and_o that_o not_o only_o in_o this_o 15._o book_n but_o also_o in_o the_o 14._o book_n wherein_o also_o you_o find_v many_o proposition_n more_o they_o be_v find_v in_o the_o greek_a set_v out_o also_o by_o hypsicles_n likewise_o in_o the_o book_n before_o you_o shall_v find_v many_o proposition_n add_v and_o many_o invert_v and_o set_v out_o of_o order_n far_o otherwise_o than_o they_o be_v place_v in_o the_o greek_a examplar_n flussas_n also_o a_o diligent_a restorer_n of_o euclid_n a_o man_n also_o which_o have_v well_o deserve_v of_o the_o whole_a art_n of_o geometry_n have_v add_v moreover_o in_o this_o book_n as_o also_o in_o the_o former_a 14._o book_n he_o add_v 8._o proposition_n 9_o proposition_n of_o his_o own_o touch_v the_o inscription_n and_o circumscription_n ãâ¦ã_o body_n very_o singular_a ândoubtedly_o and_o witty_a all_o which_o for_o that_o nothing_o shall_v want_v to_o the_o desirous_a lover_n of_o knowledge_n i_o have_v faithful_o with_o no_o small_a pain_n turn_v and_o whereas_o flâssââ_n in_o the_o begin_n of_o the_o eleven_o book_n namely_o in_o the_o end_n of_o the_o definition_n there_o âeâ_n put_v two_o definition_n of_o the_o inscription_n and_o circumscription_n of_o solid_n or_o corporal_a figure_n within_o or_o about_o the_o one_o the_o other_o which_o certain_o be_v not_o to_o be_v reject_v yet_o for_o that_o until_o this_o present_a 15._o book_n there_o be_v no_o mention_n make_v of_o the_o inscription_n or_o circumscription_n of_o these_o body_n i_o think_v it_o not_o so_o convenient_a thârâ_n to_o place_v they_o but_o to_o refer_v they_o to_o the_o begin_n of_o this_o 15._o book_n where_o they_o be_v in_o manner_n of_o necessity_n require_v to_o the_o elucidation_n of_o the_o proposition_n and_o dâmonstrationâ_n of_o the_o same_o the_o definition_n be_v these_o definition_n 1._o a_o solid_a figure_n be_v then_o âaid_v to_o be_v inscribe_v in_o a_o solid_a figure_n when_o the_o angle_n of_o the_o figure_n inscribe_v touch_n together_o at_o one_o time_n either_o the_o angle_n of_o the_o figure_n circumscribe_v or_o the_o superficiece_n or_o the_o side_n definition_n 2._o a_o solid_a figure_n be_v then_o say_v to_o be_v circumscribe_v about_o a_o solid_a figure_n when_o together_o at_o one_o time_n either_o the_o angle_n or_o the_o superficiece_n or_o the_o side_n of_o the_o figure_n circumscribe_v âouch_v the_o angle_n of_o the_o figure_n inscribe_v in_o the_o fourââ_n book_n in_o the_o definition_n of_o the_o inscription_n or_o circumscription_n of_o plain_a rectiline_n figure_v one_o with_o in_o or_o about_o a_o other_o be_v require_v that_o all_o the_o angle_n of_o the_o figuââ_n inscribe_v shall_v at_o one_o time_n touch_v all_o the_o side_n of_o the_o figure_n circumscribe_v but_o in_o the_o five_o regular_a solid_n âo_o who_o chief_o these_o two_o definition_n pertain_v for_o that_o the_o number_n of_o their_o angle_n superficiece_n &_o side_n be_v not_o equal_a one_o compare_v to_o a_o other_o it_o be_v not_o of_o necessity_n that_o all_o the_o angle_n of_o the_o solid_a inscribe_v shall_v together_o at_o one_o time_n touch_v either_o all_o the_o angle_n or_o all_o the_o superficiece_n or_o all_o the_o side_n of_o the_o solid_a circumscribe_v but_o it_o be_v sufficient_a that_o those_o angle_n of_o the_o inscribe_v solid_a which_o touch_n do_v at_o one_o time_n together_o each_o touch_n some_o one_o angle_n of_o the_o figure_n circumscribe_v or_o some_o one_o base_a or_o some_o one_o side_n so_o that_o if_o the_o angle_n of_o the_o inscribe_v figure_n do_v at_o one_o time_n touch_v the_o angle_n of_o the_o figure_n circumscribe_v none_o of_o they_o may_v at_o the_o same_o time_n touch_v either_o the_o base_n or_o the_o side_n of_o the_o same_o circumscribe_v figure_n and_o so_o if_o they_o touch_v the_o base_n they_o may_v touch_v neither_o angle_n nor_o side_n and_o likewise_o if_o they_o touch_v the_o side_n they_o may_v touch_v neither_o angle_n nor_o base_n and_o although_o sometime_o all_o the_o angle_n of_o the_o figure_n inscribe_v can_v not_o touch_v either_o the_o angle_n or_o the_o base_n or_o the_o side_n of_o the_o figure_n circumscribe_v by_o reason_n the_o number_n of_o the_o angle_n base_n or_o side_n of_o the_o say_a figure_n circumscribe_v want_v of_o the_o number_n of_o the_o angle_n of_o the_o âigure_n inscribe_v yet_o shall_v those_o angle_n of_o the_o inscribe_v figure_n which_o touch_n so_o touch_v that_o the_o void_a place_n leave_v between_o the_o inscribe_v and_o circumscribe_v figure_n shall_v on_o every_o side_n be_v equal_a and_o like_a as_o you_o may_v afterward_o in_o this_o fifteen_o book_n most_o plain_o perceive_v ¶_o the_o 1._o proposition_n the_o 1._o problem_n in_o a_o cube_n give_v to_o describe_v admonish_v a_o trilater_n equilater_n pyramid_n svppose_v that_o the_o cube_fw-la give_v be_v abcdefgh_o in_o the_o same_o cube_fw-la it_o be_v require_v to_o inscribe_v a_o tetrahedron_n draw_v these_o right_a line_n ac_fw-la construction_n ce_fw-fr ae_n ah_o eh_o hc_n demonstration_n now_o it_o be_v manifest_a that_o the_o triangle_n aec_o ahe_o ahc_n and_o i_o be_v equilater_n for_o their_o side_n be_v the_o diameter_n of_o equal_a square_n wherefore_o aech_n be_v a_o trilater_n equilater_n pyramid_n or_o tetrahedron_n &_o it_o be_v inscribe_v in_o the_o cube_fw-la give_v by_o the_o first_o definition_n of_o this_o book_n which_o be_v require_v to_o be_v do_v ¶_o the_o 2._o proposition_n the_o 2._o problem_n in_o a_o trilater_n equilater_n pyramid_n give_v to_o describe_v a_o octohedron_n svppose_v that_o the_o trilater_n equilater_n pyramid_n give_v be_fw-mi abcd_o who_o side_n let_v be_v divide_v into_o two_o equal_a part_n in_o the_o point_n e_o z_o i_o king_n l_o t._n construction_n and_o draw_v these_o 12._o right_a line_n ez_o zi_n je_n kl_o lt_n tk_n eke_o kz_o zl_n li_n it_o and_o te_fw-la which_o 12._o right_a line_n be_v demonstration_n by_o the_o 4._o of_o the_o first_o equal_a for_o they_o subtend_v equal_a plain_a angle_n of_o the_o base_n of_o the_o pyramid_n and_o those_o equal_a angle_n be_v contain_v under_o equal_a side_n namely_o under_o the_o half_n of_o the_o side_n of_o the_o pyramid_n wherefore_o the_o triangle_n tkl_n tli_n tie_v tek_v zkl_n zli_n zib_n zek_n be_v equilater_n and_o they_o limitate_v and_o contain_v the_o solid_a tklezi_n wherefore_o the_o solid_a tklezi_n be_v a_o octohedron_n by_o the_o 23._o definition_n of_o the_o eleven_o and_o the_o angle_n of_o the_o same_o octohedron_n do_v touch_v the_o side_n of_o the_o pyramid_n abcd_o in_o the_o point_n e_o z_o i_o t_o edward_n l._n wherefore_o the_o octohedron_n be_v inscribe_v in_o the_o pyramid_n by_o the_o 1._o definition_n of_o this_o book_n wherefore_o in_o the_o trilater_n equilater_n pyramid_n give_v be_v inscribe_v a_o octohedron_n which_o be_v require_v to_o be_v do_v a_o corollary_n add_v by_o flussas_n hereby_o it_o be_v manifest_a that_o a_o pyramid_n be_v cut_v into_o two_o
to_o the_o base_n of_o the_o octohedron_n for_o forasmuch_o as_o the_o side_n of_o the_o base_n of_o the_o pyramid_n touch_v the_o one_o the_o other_o be_v parallel_n to_o the_o side_n of_o the_o octohedron_n which_o also_o touch_v the_o one_o the_o other_o as_o for_o example_n hl_o be_v prove_v to_o be_v a_o parallel_n to_o give_v and_o lc_a to_o diego_n therefore_o by_o the_o 15._o of_o the_o eleven_o the_o plain_a superficies_n which_o be_v draw_v by_o the_o line_n hl_o and_o lc_n be_v a_o parallel_n to_o the_o plain_a superficies_n draw_v by_o the_o line_n give_v and_o di._n and_o so_o likewise_o of_o the_o rest_n second_o corollary_n a_o right_a line_n join_v together_o the_o centre_n of_o the_o opposite_a base_n of_o the_o octohedron_n be_v sesquialter_fw-la to_o the_o perpendicular_a line_n draw_v from_o the_o angle_n of_o the_o inscribe_v pyramid_n to_o the_o base_a thereof_o for_o forasmuch_o as_o the_o pyramid_n and_o the_o cube_fw-la which_o contain_v it_o do_v in_o the_o self_n same_o point_n end_v their_o angle_n by_o the_o 1._o of_o this_o book_n therefore_o they_o shall_v both_o be_v enclose_v in_o one_o and_o the_o self_n same_o octohedron_n by_o the_o 4._o of_o this_o book_n but_o the_o diameter_n of_o the_o cube_fw-la join_v together_o the_o centre_n of_o the_o opposite_a base_n of_o the_o octohedron_n and_o therefore_o be_v the_o diameter_n of_o the_o sphere_n which_o contain_v the_o cube_fw-la and_o the_o pyramid_n inscribe_v in_o the_o cube_fw-la by_o the_o 13._o and_o 14._o of_o the_o thirteen_o which_o diameter_n be_v sesquialter_fw-la to_o the_o perpendicular_a which_o be_v draw_v from_o the_o angle_n of_o the_o pyramid_n to_o the_o base_a thereof_o for_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o base_a of_o the_o pyramid_n be_v the_o six_o part_n of_o the_o diameter_n by_o the_o 3._o corollary_n of_o the_o 13._o of_o the_o thirteen_o wherefore_o of_o what_o part_n the_o diameter_n contain_v six_o of_o the_o same_o part_n the_o perpendicular_a contain_v four_o ¶_o the_o 7._o proposition_n the_o 7._o problem_n in_o a_o dodecahedron_n give_v to_o inscribe_v a_o icosahedron_n svppose_v that_o the_o dodecahedron_n give_v be_v abcde_o and_o let_v the_o centre_n of_o the_o circle_n which_o contain_v six_o base_n of_o the_o same_o dodecahedron_n be_v the_o polnes_n l_o m_o n_o p_o q_o o._n construction_n and_o draw_v these_o right_a line_n ol_n om_o on_o open_v oq_fw-la and_o moreover_o these_o right_a line_n lm_o mn_a np_n pq_n ql_n and_o now_o forasmuch_o as_o equal_v and_o equilater_n pentagon_n be_v contain_v in_o equal_a circle_n therefore_o perpendicular_a line_n draw_v from_o their_o centre_n to_o the_o side_n shall_v be_v equal_a by_o the_o 14._o of_o the_o three_o and_o shall_v divide_v the_o side_n of_o the_o dodecahedron_n into_o two_o equal_a part_n by_o the_o 3._o of_o the_o same_o wherefore_o the_o foresay_a perpendicular_a line_n shall_v coâoutre_v in_o the_o point_n of_o the_o section_n demonstration_n wherein_o the_o side_n be_v divide_v into_o two_o equal_a part_n as_o lf_a and_o mf_o do_v and_o they_o also_o contain_v equal_a angle_n namely_o the_o inclination_n of_o the_o base_n of_o the_o dodecahedron_n by_o the_o 2._o corollary_n of_o the_o 18._o of_o the_o thirteen_o wherefore_o the_o right_a line_n lm_o mn_a np_n pq_n ql_n and_o the_o rest_n of_o the_o right_a line_n which_o join_v together_o two_o centre_n of_o the_o base_n and_o which_o subtende_v the_o equal_a angle_n contain_v under_o the_o say_a equal_a perpendicular_a line_n be_v equal_a the_o one_o to_o the_o other_o by_o the_o 4._o of_o the_o first_o wherefore_o the_o triangle_n olm_n omnes_n onp_n opq_n oql_n and_o the_o rest_n of_o the_o triangle_n which_o be_v set_v at_o the_o centre_n of_o the_o pentagon_n be_v equilater_n and_o equal_a now_o forasmuch_o as_o the_o 12._o pentagon_n of_o a_o dodecahedron_n contain_v 60._o plain_a superficial_a angle_n of_o which_o 60._o every_n âhre_o make_v one_o solid_a angle_n of_o the_o dodecahedron_n it_o follow_v that_o a_o dodecahedron_n have_v 20._o solid_a angle_n but_o each_o of_o those_o solid_a angle_n be_v subtend_v of_o each_o of_o the_o triangle_n of_o the_o icosahedron_n namely_o of_o each_o of_o those_o triangle_n which_o join_v together_o the_o centre_n of_o the_o pentagon_n which_o make_v the_o solid_a angle_n as_o we_o have_v before_o prove_v wherefore_o the_o 20_o equal_a and_o equilater_n triangle_n which_o subtende_v the_o 20._o solid_a angle_n of_o the_o dodecahedron_n and_o have_v their_o side_n which_o be_v draw_v from_o the_o centre_n of_o the_o pentagon_n common_a do_v make_v a_o icosahedron_n by_o the_o 25._o definition_n of_o the_o eleven_o and_o it_o be_v inscribe_v in_o the_o dodecahedron_n give_v by_o the_o first_o definition_n of_o this_o book_n for_o that_o the_o angle_n thereof_o do_v all_o at_o one_o time_n touch_v the_o base_n of_o the_o dodecahedron_n wherefore_o in_o a_o dodecahedron_n gevenâ_n iâ_z inscribe_v a_o icosahedron_n which_o be_v require_v to_o be_v do_v ¶_o the_o 8._o proposition_n the_o 8._o problem_n in_o a_o dodecahedron_n give_v to_o include_v a_o cube_fw-la describe_v by_o the_o 17._o of_o the_o thirteen_o a_o dodecahedron_n construction_n and_o by_o the_o same_o take_v the_o 12._o side_n of_o the_o cube_fw-la each_o of_o which_o subtend_v one_o angle_n of_o each_o of_o the_o 12._o base_n of_o the_o dodecahedron_n for_o the_o side_n of_o the_o cube_fw-la subtend_v the_o angle_n of_o the_o pentagon_n of_o the_o dodecahedron_n by_o the_o 2._o corollary_n of_o the_o 17._o of_o the_o thirteen_o if_o therefore_o in_o the_o dodecahedron_n describe_v by_o the_o self_n same_o 17._o proposition_n we_o draw_v the_o 12._o right_a line_n subtend_v under_o the_o foresay_a 12._o angle_n and_o end_v in_o 8._o angle_n of_o the_o dodecahedron_n and_o concur_v together_o in_o such_o sort_n that_o they_o be_v in_o like_a sort_n situate_a as_o it_o be_v plain_o prove_v in_o that_o proposition_n then_o shall_v it_o be_v manifest_a that_o the_o right_a line_n draw_v in_o this_o dodecahedron_n from_o the_o foresay_a 8._o angle_n thereof_o do_v make_v the_o foresay_a cube_fw-la which_o therefore_o be_v include_v in_o the_o dodecahedron_n for_o that_o the_o side_n of_o the_o cube_fw-la be_v draw_v in_o the_o side_n of_o the_o dodecahedron_n and_o the_o angle_n of_o the_o same_o cube_fw-la be_v set_v in_o the_o angle_n of_o the_o say_a dodecahedron_n as_o for_o example_n take_v 4._o pentagon_n of_o a_o dodecahedron_n demonstration_n namely_o agibo_n bhcno_n ckedn_n and_o dfaon_n and_o draw_v these_o right_a line_n ab_fw-la bc_o cd_o da._n which_o four_o right_a line_n make_v a_o square_n for_o that_o each_o of_o those_o right_a line_n do_v subtend_v equal_a angle_n of_o equal_a pentagon_n &_o the_o angle_n which_o those_o 4._o right_a line_n contain_v be_v right_a angle_n as_o we_o prove_v in_o the_o construction_n of_o the_o dodecahedron_n in_o the_o 17._o proposition_n before_o allege_a wherefore_o the_o six_o base_n be_v square_n do_v make_v a_o cube_fw-la by_o the_o 21._o definition_n of_o the_o eleven_o and_o for_o that_o the_o 8._o angle_n of_o the_o say_v cube_fw-la be_v set_v in_o 8._o angle_n of_o the_o dodecaheeron_n therefore_o be_v the_o say_v cube_fw-la inscribe_v in_o the_o dodecahedron_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o dodecahedron_n be_v inscribe_v a_o cube_fw-la which_o be_v require_v to_o be_v do_v ¶_o the_o 9_o proposition_n the_o 9_o problem_n in_o a_o dodecahedron_n give_v to_o include_v a_o octohedron_n svppose_v that_o the_o dodecahedron_n give_v be_v abgd_v construction_n now_o by_o the_o 3._o corollary_n of_o the_o 17._o of_o the_o thirteen_o take_v the_o 6._o side_n which_o be_v opposite_a the_o one_o to_o the_o other_o those_o 6._o side_n i_o say_v who_o section_n wherein_o they_o be_v divide_v into_o two_o equal_a part_n be_v couple_v by_o three_o right_a line_n which_o in_o the_o centre_n of_o the_o sphere_n wherein_o the_o dodecahedron_n be_v contain_v do_v cut_v the_o one_o the_o other_o perpendicular_o and_o let_v the_o point_n wherein_o the_o foresay_a side_n be_v cut_v into_o two_o equal_a part_n be_v a_o b_o g_o d_o c_o i._n and_o let_v the_o foresay_a three_o right_a line_n join_v together_o the_o say_a section_n be_v ab_fw-la gd_o and_o ci._n and_o let_v the_o centre_n of_o the_o sphere_n be_v e._n demonstration_n now_o forasmuch_o as_o by_o the_o foresay_a corollary_n those_o three_o right_a line_n be_v equal_a it_o follow_v by_o the_o 4._o of_o the_o first_o that_o the_o right_a line_n subtend_v the_o right_a angle_n which_o they_o make_v at_o the_o centre_n of_o the_o sphere_n which_o right_a angle_n be_v contain_v under_o the_o half_n of_o the_o say_v three_o right_a line_n be_v equal_a the_o one_o to_o the_o other_o
side_n gd_v the_o angle_n m_o n_o under_o the_o side_n ab_fw-la the_o angle_n t_o s_o under_o the_o side_n bg_o the_o angle_n p_o oh_o and_o under_o the_o side_n agnostus_n the_o angle_n r_o q_o so_o there_o rest_v 4._o angle_n who_o true_a place_n we_o will_v now_o appoint_v forasmuch_o as_o a_o cube_fw-la contain_v in_o one_o and_o the_o self_n same_o sphere_n with_o a_o dodecahedron_n be_v inscribe_v in_o the_o same_o dodecahedron_n as_o it_o be_v manifest_a by_o the_o 17._o of_o the_o thirteen_o and_o 8._o of_o this_o book_n it_o follow_v that_o a_o cube_fw-la and_o a_o dodecahedron_n circumscribe_v about_o it_o be_v contain_v in_o one_o and_o the_o self_n same_o body_n for_o that_o their_o angle_n concur_v in_o one_o and_o the_o self_n same_o point_n and_o it_o be_v prove_v in_o the_o 18._o of_o this_o book_n that_o 4._o angle_n of_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n be_v set_v in_o the_o middle_a section_n of_o the_o perpendicularâ_n which_o be_v draw_v from_o the_o solid_a angle_n of_o the_o pyramid_n to_o the_o opposite_a base_n wherefore_o the_o other_o 4._o angle_n of_o the_o dodecahedron_n be_v also_o as_o the_o angle_n of_o the_o cube_fw-la set_v in_o those_o middle_a section_n of_o the_o perpendicular_n namely_o the_o angle_n v_o be_v set_v in_o the_o midst_n of_o the_o perpendicular_a ahâ_n the_o angle_n y_fw-fr in_o the_o midst_n of_o the_o perpendicular_a bf_o the_o angle_n x_o in_o the_o midst_n of_o the_o perpendicular_a ge_z and_o last_o the_o angle_n d_o in_o the_o midst_n of_o the_o perpendicular_a d_o which_o be_v draw_v from_o the_o top_n of_o the_o pyramid_n to_o the_o opposite_a base_a wherefore_o those_o 4._o angle_n of_o the_o dodecahedron_n may_v be_v say_v to_o be_v direct_o under_o the_o solid_a angle_n of_o the_o pyramid_n or_o they_o may_v be_v say_v to_o be_v set_v at_o the_o perpendicular_n wherefore_o the_o dodecahedron_n after_o this_o manner_n set_v be_v inscribe_v in_o the_o pyramid_n give_v by_o the_o first_o definition_n of_o this_o book_n for_o that_o upon_o every_o one_o of_o the_o base_n of_o the_o pyramid_n be_v set_v a_o angle_n of_o the_o dodecahedron_n inscribe_v wherefore_o in_o a_o trilater_n equilater_n pyramid_n be_v inscribe_v a_o dodecahedron_n the_o 21._o problem_n the_o 21._o proposition_n in_o every_o one_o of_o the_o regular_a solid_n to_o inscribe_v a_o sphere_n in_o the_o 13._o of_o thâ_z thirteen_o and_o thâ_z other_o 4._o proposition_n follow_v he_o iâ_z be_v declare_v that_o âhe_n ââ_o regular_a solidesââre_o so_o contain_v in_o a_o sphere_n that_o âight_a linââ_n draw_v from_o the_o cenâââ_n oâ_n the_o ãâ¦ã_o of_o ãâã_d solid_a inscribe_v be_v equal_a which_o right_a line_n therefore_o make_v pyramid_n who_o âoppes_n be_v the_o centre_n of_o the_o sphere_n or_o of_o the_o solid_a and_o the_o basââââe_a cuâââ_n one_o of_o the_o base_n of_o those_o solid_n and_o ãâ¦ã_o solid_a squall_n and_o like_v the_o one_o to_o the_o other_o and_o describe_v in_o equal_a circle_n those_o circle_n shall_v cut_v the_o sphere_n for_o the_o angle_n which_o touch_v the_o circumference_n of_o the_o circle_n touch_v also_o the_o superficies_n of_o the_o sphere_n wherefore_o perpendiculars_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o base_n or_o to_o the_o plain_a superficiece_n of_o the_o equal_a circle_n be_v equal_a by_o the_o corollary_n of_o the_o assumpt_n of_o the_o 1â_o of_o the_o twelve_o wherefore_o make_v the_o centre_n the_o ãâã_d of_o the_o sphere_n which_o ãâã_d the_o solid_a and_o thâ_z space_n some_o one_o of_o the_o equal_a perpendicularâ_n dâscribâ_n a_o sphere_n and_o it_o shall_v touch_v every_o one_o of_o the_o base_n of_o ãâã_d solid_a ãâ¦ã_o perficies_fw-la of_o the_o sphere_n pass_v beyond_o those_o base_n when_o as_o those_o pââpeâdiculars_n ãâ¦ã_o be_v draw_v from_o the_o centre_n to_o the_o base_n by_o the_o 3._o corollary_n of_o the_o saâââââumpt_n wherâfore_o âe_v have_v iâ_z every_o one_o of_o the_o regular_a body_n inscribe_v a_o sphere_n which_o regular_a boâââ_n be_v in_o number_n one_o iâ_z ãâã_d by_o the_o corollary_n of_o the_o 1â_o of_o the_o ãâã_d a_o corollary_n the_o regular_a figure_n inscribe_v in_o sphere_n and_o also_o the_o sphere_n circumscribe_v about_o they_o or_o contain_v they_o have_v one_o and_o the_o self_n same_o centre_n namely_o their_o pyramid_n the_o ângles_n of_o who_o base_n touch_v the_o superââââââ_n of_o thâââhere_o do_v from_o those_o angle_n cause_v equal_a right_a line_n to_o be_v drawââ_n to_o one_o and_o âhe_n self_n ãâã_d poynâ_n make_v the_o topâââ_n of_o the_o pyramid_n in_o the_o same_o point_n and_o therefore_o they_o ãâ¦ã_o thâ_z cââtres_n of_o the_o sphere_n in_o the_o self_n same_o top_n when_o ãâã_d the_o right_a line_n draw_v from_o those_o angle_n to_o the_o croââed_a superficies_n wherein_o be_v ãâã_d the_o angle_n of_o the_o base_n of_o the_o pyramid_n be_v equal_o a_o advertisement_n of_o flussasâ_n â_z of_o these_o solid_n only_o the_o octohedron_n receive_v the_o other_o solid_n inscribe_v one_o with_o ãâ¦ã_o other_o for_o the_o octohedron_n contain_v the_o icosahedron_n inscribe_v in_o it_o and_o the_o same_o icosahedron_n contain_v the_o dodecahedron_n inscribe_v in_o the_o same_o icosahedron_n and_o the_o same_o dodecahedron_n contain_v the_o cube_fw-la inscribe_v in_o the_o same_o octohedron_n and_o ãâ¦ã_o âârââmscribeth_v the_o pyramid_n inscribe_v in_o the_o say_v octohedron_n but_o this_o happen_v not_o in_o the_o other_o solid_n the_o end_n of_o the_o fivetenth_fw-mi book_n of_o euclides_n elemenâââ_n after_o caâpaââ_n and_o ãâã_d ¶_o the_o sixteen_o book_n of_o the_o element_n of_o geometry_n add_v by_o flussas_n in_o the_o former_a fivetenth_fw-mi book_n have_v be_v teach_v how_o to_o inscribe_v the_o five_o regular_a solid_n one_o with_o in_o a_o other_o now_o seem_v to_o rest_n to_o compare_v those_o solid_a so_o inscribe_v one_o to_o a_o other_o and_o to_o set_v forth_o their_o passion_n and_o propriety_n which_o thing_n flussas_n consider_v in_o this_o sixteen_o book_n add_v by_o he_o book_n have_v excellent_o well_o and_o most_o cunning_o perform_v for_o which_o undoubted_o he_o have_v of_o all_o they_o which_o have_v a_o love_n to_o the_o mathematicals_n deserve_v much_o praise_n and_o commendation_n both_o for_o the_o great_a traâailes_n and_o paynâs_n which_o it_o be_v most_o likely_a he_o have_v taâân_v in_o invent_v such_o strange_a and_o wonderful_a proposition_n with_o their_o demonstration_n in_o this_o book_n contain_v as_o also_o for_o participate_v and_o communicate_v abroad_o the_o same_o to_o other_o which_o book_n also_o that_o the_o reader_n shall_v want_v nothing_o conduce_v to_o the_o perfection_n of_o euclides_n element_n i_o have_v with_o some_o travail_n translate_v &_o for_o the_o worthiness_n âhereof_o have_v add_v it_o aâ_z a_o sixteen_o book_n to_o the_o 15._o book_n of_o euclid_n vouchsafe_v therefore_o gentle_a reader_n diligent_o to_o read_v and_o poise_v it_o for_o in_o it_o shall_v you_o find_v noâ_n only_a matter_n strange_a and_o delectable_a but_o also_o occasion_n of_o invention_n of_o great_a thing_n pertain_v to_o the_o nature_n of_o the_o five_o regular_a solidâsâ_n ¶_o the_o 1._o proposition_n a_o dodecahedron_n and_o a_o cube_fw-la inscribe_v in_o it_o and_o a_o pyramid_n inscribe_v in_o the_o same_o cube_fw-la be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n for_o the_o angle_n of_o the_o pyramiâ_n be_v seâ_z in_o the_o angel_n of_o the_o cube_fw-la wherein_o it_o be_v inscribe_v by_o the_o first_o of_o the_o fivetenthâ_n and_o all_o the_o angle_n of_o the_o cube_fw-la be_v set_v in_o the_o angle_n of_o the_o dodecahedâââ_n circumscribe_v ãâ¦ã_o ãâã_d the_o 8._o of_o the_o fifteen_o and_o all_o the_o angle_n of_o the_o dodecahedron_n be_v set_v in_o the_o superficies_n of_o the_o sphere_n by_o the_o 17._o of_o the_o thirteen_o wherefore_o those_o three_o solid_n inscribe_v one_o within_o a_o other_o be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o first_o definition_n of_o the_o fifteen_o a_o dodecahedron_n therefore_o and_o a_o cube_fw-la inscribe_v in_o it_o and_o a_o pyramid_n inscribe_v in_o the_o same_o cube_fw-la be_v contain_v ãâ¦ã_o ââlfe_n same_o sphere_n ãâ¦ã_o these_o three_o solid_n livre_n ãâ¦ã_o elf_n same_o icosahedron_n or_o octohedron_n or_o pyramid_n ãâ¦ã_o i_o icosahedron_n by_o the_o 5.11_o &_o 12._o of_o the_o fifteen_o and_o they_o be_v ãâ¦ã_o ctohedron_n by_o the_o 4._o 6._o and_o 16._o of_o the_o same_o last_o they_o be_v inscribe_v in_o ãâ¦ã_o the_o first_o 18._o and_o 19_o of_o the_o same_o for_o the_o angle_n of_o all_o these_o solid_a ãâ¦ã_o the_o circumscribe_v icosahedron_n or_o octohedron_n or_o pyramid_n ¶_o the_o ãâ¦ã_o the_o proportion_n of_o a_o dodecahedron_n circumscribe_v about_o a_o cube_fw-la to_o a_o dodecahedron_n inscribe_v in_o the_o same_o cube_fw-la be_v
same_o the_o cube_fw-la contain_v 12_o namely_o be_v sesquialter_fw-la to_o the_o pyramid_n wherefore_o of_o what_o part_n the_o cube_fw-la contain_v 12_o of_o the_o same_o the_o whole_a octohedron_n which_o be_v double_a to_o the_o pyramid_n abdfc_n contain_v 54._o which_o 54._o have_v to_o 12._o quadruple_a sesquialter_fw-la proportion_n wherefore_o the_o whole_a octohedron_n be_v to_o the_o cube_fw-la inscribe_v in_o it_o in_o quadruple_a sesquialter_fw-la proportion_n wherefore_o we_o have_v prove_v that_o a_o octohedron_n give_v be_v quadruple_a sesquialter_fw-la to_o a_o cube_fw-la inscribe_v in_o it_o ¶_o a_o corollary_n a_o octohedron_n be_v to_o a_o cube_fw-la inscribe_v in_o it_o in_o that_o proportion_n that_o the_o square_n of_o their_o side_n be_v for_o by_o the_o 14._o of_o this_o book_n the_o side_n of_o the_o octohedron_n be_v in_o power_n quadruple_a sesquialter_fw-la to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o it_o ¶_o the_o 29._o proposition_n to_o prove_v that_o a_o octohedron_n give_v be_v â_z tredecuple_n sesquialter_fw-la to_o a_o trilater_n equilater_n pyramid_n inscribe_v in_o it_o let_v the_o octohedron_n geâen_o be_v ab_fw-la in_o which_o let_v there_o be_v inscribe_v a_o cube_fw-la fced_fw-la by_o the_o 4._o of_o the_o fifteen_o and_o in_o the_o cube_fw-la let_v there_o be_v inscribe_v a_o pyramid_n fegd_v by_o the_o â_o of_o the_o fifteen_o and_o forasmuch_o as_o the_o angle_n of_o the_o pyramid_n be_v by_o the_o same_o first_o of_o the_o fifteen_o set_v in_o the_o angle_n of_o the_o cube_fw-la and_o the_o angle_n of_o the_o cube_fw-la be_v set_v in_o the_o centre_n of_o the_o base_n of_o the_o octohedron_n namely_o in_o the_o point_n f_o e_o c_o d_o g_o by_o the_o 4._o of_o the_o fifteen_o wherefore_o the_o angle_n of_o the_o pyramid_n be_v set_v in_o the_o centre_n f_o c_o e_o d_o of_o the_o octohedron_n wherefore_o the_o pyramid_n fedg_n be_v inscribe_v in_o the_o octohedron_n by_o the_o 6._o of_o the_o fifteen_o and_o forasmuch_o as_o the_o octohedron_n ab_fw-la be_v to_o the_o cube_fw-la fced_fw-la inscribe_v in_o it_o quadruple_a sesquialter_fw-la by_o the_o former_a proposition_n and_o the_o cube_fw-la cdef_n be_v to_o the_o pyramid_n fedg_v inscribe_v in_o it_o triple_a by_o the_o 25._o of_o book_n wherefore_o three_o magnitude_n be_v give_v namely_o the_o octohedron_n the_o cube_fw-la and_o the_o pyramid_n the_o proportion_n of_o the_o extreme_n namely_o of_o the_o octohedron_n to_o the_o pyramid_n be_v make_v of_o the_o proportion_n of_o the_o mean_n namely_o of_o the_o octohedron_n to_o the_o cube_fw-la and_o of_o the_o cube_fw-la to_o the_o pyramid_n as_o it_o be_v easy_a to_o see_v by_o the_o declaration_n upon_o the_o 10._o definition_n of_o the_o five_o now_o then_o multiply_v the_o quantity_n or_o denomination_n of_o the_o proportion_n namely_o of_o the_o octohedron_n to_o the_o cube_fw-la which_o be_v 4_o 1_o â_o and_o of_o the_o cube_fw-la to_o the_o pyramid_n which_o be_v 3_o as_o be_v teach_v in_o the_o definition_n of_o the_o six_o there_o shall_v be_v produce_v 13_o 1_o â_o namely_o the_o proportion_n of_o the_o octohedron_n to_o the_o pyramid_n inscribe_v in_o it_o for_o 4_o ½_n multiply_v by_o 3._o produce_v 13_o ½_n wherefore_o the_o octohedron_n be_v to_o the_o pyramid_n inscribe_v in_o it_o in_o tredecuple_n sesquialter_fw-la proportion_n wherefore_o we_o have_v prove_v that_o a_o octohedron_n be_v to_o a_o trilater_n equilater_n pyramid_n inscribe_v in_o it_o in_o tredecuple_n sesquialter_fw-la proportion_n ¶_o the_o 30._o proposition_n to_o prove_v that_o a_o trilater_n equilater_n pyramid_n be_v noncuple_n to_o a_o cube_fw-la inscribe_v in_o it_o svppose_v that_o the_o pyramid_n give_v be_v abcd_o who_o two_o base_n let_v be_v abc_n and_o dbc_n and_o let_v their_o centre_n be_v the_o point_n g_o and_o i._n and_o from_o the_o angle_n a_o draw_v unto_o the_o base_a bc_o a_o perpendicular_a ae_n likewise_o from_o the_o angle_n d_o draw_v unto_o the_o same_o base_a bc_o a_o perpendicular_a de_fw-fr and_o they_o shall_v concur_v in_o the_o section_n e_o by_o the_o 3._o of_o the_o three_o and_o in_o they_o shall_v be_v the_o centre_n g_o an_o i_o by_o the_o corollary_n of_o the_o first_o of_o the_o three_o and_o forasmuch_o as_o the_o line_n ad_fw-la be_v the_o side_n of_o the_o pyramid_n the_o same_o ad_fw-la shall_v be_v the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la which_o contain_v the_o pyramid_n by_o the_o 1_o of_o the_o fivetenth_n demonstration_n draw_v the_o line_n give_v and_o forasmuch_o as_o the_o line_n give_v couple_v the_o centreâ_n of_o the_o base_n of_o the_o pyramid_n the_o say_a line_n give_v shall_v be_v the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n by_o the_o 18._o of_o the_o fifteen_o and_o forasmuch_o as_o the_o line_n agnostus_n be_v double_a to_o the_o line_n ge_z by_o the_o corollarye_a of_o the_o twelve_o of_o the_o thirteen_o the_o whole_a line_n ae_n shall_v be_v triple_a to_o the_o line_n ge_z and_o so_o be_v also_o the_o line_n de_fw-fr to_o the_o line_n je_n wherefore_o the_o line_n ad_fw-la and_o give_v be_v parallel_n by_o the_o 2._o of_o the_o six_o and_o therefore_o the_o triangle_n aed_n and_o gei_n be_v likeâ_n by_o the_o corollary_n of_o the_o same_o and_o forasmuch_o as_o the_o triangle_n aed_n and_o gei_n be_v like_a the_o line_n adâ_n shall_v be_v treble_v to_o the_o line_n give_v by_o the_o 4._o of_o the_o six_o but_o the_o line_n ad_fw-la be_v the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la circumscribe_v about_o the_o pyramid_n abcd_o and_o the_o line_n give_v be_v the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n abcd_o but_o the_o diameter_n of_o the_o base_n be_v equemultiplices_fw-la to_o the_o side_n namely_o be_v in_o power_n duple_n wherefore_o the_o side_n of_o the_o cube_fw-la circumscribe_v about_o the_o pyramid_n abcd_o be_v triple_a to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o same_o pyramid_n by_o the_o 15._o of_o the_o five_o but_o like_o cube_n be_v in_o triple_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o their_o side_n be_v by_o the_o 33._o of_o the_o eleven_o and_o the_o side_n be_v in_o triple_a proportion_n the_o one_o to_o the_o other_o wherefore_o triple_a take_v three_o time_n bring_v forth_o twenty_o sevencuple_n which_o be_v 27._o to_o 1_o for_o the_o 4._o term_n 27.9.3.1_o be_v set_v in_o triple_a proportion_n the_o proportion_n of_o the_o first_o to_o the_o four_o namely_o of_o 27._o to_o 1._o shall_v be_v treble_v to_o the_o proportion_n of_o the_o first_o to_o the_o second_o namely_o of_o 27._o to_o 9_o by_o the_o 10._o definition_n of_o the_o five_o which_o proportion_n of_o 27._o to_o 1._o be_v the_o proportion_n of_o the_o side_n triple_a which_o proportion_n also_o be_v find_v in_o like_o solides_fw-la wherefore_o of_o what_o part_n the_o cube_fw-la circumscribe_v contain_v 27._o of_o the_o same_o the_o cube_fw-la inscribe_v contain_v one_o but_o of_o what_o part_n the_o cube_fw-la circumscribe_v contain_v 27._o of_o the_o same_o the_o pyramid_n inscribe_v in_o it_o contain_v 9_o by_o the_o 25._o of_o this_o book_n wherefore_o of_o what_o part_n the_o pyramid_n ab_fw-la cd_o contain_v 9_o of_o the_o same_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n contain_v one_o wherefore_o we_o have_v prove_v that_o a_o trilater_n and_o equilater_n pyramid_n be_v nonâcuple_n to_o a_o cube_fw-la inscribe_v in_o it_o ¶_o the_o 31._o proposition_n a_o octohedron_n have_v to_o a_o icosohedron_n inscribe_v in_o it_o that_o proportion_n which_o two_o base_n of_o the_o octohedron_n have_v to_o five_o base_n of_o the_o icosahedron_n svppose_v that_o the_o octohedron_n give_v be_v abcd_o and_o let_v the_o icosahedron_n inscribe_v in_o it_o be_v fghmklio_n then_o i_o say_v that_o the_o octohedron_n be_v to_o the_o icosahedron_n as_o two_o base_n of_o the_o octohedron_n be_v to_o five_o base_n of_o the_o icosahedron_n for_o forasmuch_o as_o the_o solid_a of_o the_o octohedron_n consist_v of_o eight_o pyramid_n set_v upon_o the_o base_n of_o the_o octohedron_n demonstration_n and_o have_v to_o their_o altitude_n a_o perpendicular_a line_n draw_v from_o the_o centre_n to_o the_o base_a let_v that_o perpendicular_a be_v er_fw-mi or_o es_fw-ge be_v draw_v from_o the_o centre_n e_o which_o centre_n be_v common_a to_o either_o of_o the_o solid_n by_o the_o corollary_n of_o the_o 21._o of_o the_o fifteen_o to_o the_o centre_n of_o the_o base_n namely_o to_o the_o point_n r_o and_o s._n wherefore_o for_o that_o three_o pyramid_n be_v equal_a and_o like_a they_o shall_v be_v equal_a to_o a_o prism_n set_v upon_o the_o self_n same_o base_a and_o under_o the_o self_n same_o altitude_n by_o the_o corollary_n of_o the_o seven_o of_o the_o twelve_o but_o
in_o the_o point_n e._n and_o unto_o the_o line_n cg_o put_v the_o line_n cl_n equal_a now_o forasmuch_o as_o the_o line_n agnostus_n and_o gc_o be_v the_o great_a segâââtes_n of_o half_a the_o line_n ab_fw-la for_o âche_fw-mi of_o they_o be_v the_o half_a of_o the_o great_a segment_n of_o the_o whole_a line_n ab_fw-la the_o line_n ebb_v and_o ec_o shall_v be_v the_o less_o segment_n of_o half_a the_o line_n ab_fw-la wherefore_o the_o whole_a line_n câ_n be_v the_o great_a segment_n and_o the_o line_n ce_fw-fr be_v the_o less_o segment_n but_o as_o the_o line_n cl_n be_v to_o the_o line_n ce_fw-fr so_o be_v the_o line_n ce_fw-fr to_o the_o residue_n el._n wherefore_o the_o line_n el_n be_v the_o great_a segment_n of_o the_o line_n ce_fw-fr or_o of_o the_o line_n ebb_v which_o be_v equal_a unto_o it_o wherefore_o the_o residue_n lb_n be_v the_o less_o segment_n of_o the_o same_o ebb_n which_o be_v the_o lesâe_n segment_n of_o halfa_n the_o side_n of_o the_o cube_fw-la but_o the_o line_n agnostus_n gc_o and_o cl_n be_v three_o great_a segment_n of_o the_o half_a of_o the_o whole_a line_n ab_fw-la which_o three_o great_a segment_n make_v the_o altitude_n of_o the_o foresay_a solid_a wherefore_o the_o altitude_n of_o the_o say_v solid_a want_v of_o ab_fw-la the_o side_n of_o the_o cube_fw-la by_o the_o line_n lb_n which_o be_v the_o less_o segment_n of_o the_o line_n be._n which_o line_n be_v again_o be_v the_o less_o segment_n of_o half_a the_o side_n ab_fw-la of_o the_o cube_fw-la wherefore_o the_o foresay_a solid_a consist_v of_o the_o six_o solid_n whereby_o the_o dodecahedron_n exceed_v the_o cube_fw-la inscribe_v in_o it_o be_v set_v upon_o a_o base_a which_o want_v of_o the_o base_a of_o the_o cube_fw-la by_o a_o three_o part_n of_o the_o less_o segment_n and_o be_v under_o a_o altitude_n want_v of_o the_o side_n of_o the_o cube_fw-la by_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o side_n of_o the_o cube_fw-la the_o solid_a therefore_o of_o a_o dodecahedron_n exceed_v the_o solid_a of_o a_o cube_fw-la inscribe_v in_o it_o by_o a_o parallelipipedon_n who_o base_a want_v of_o the_o base_a of_o the_o cube_fw-la by_o a_o three_o part_n of_o the_o less_o segment_n and_o who_o altitude_n want_v of_o the_o altitude_n of_o the_o cube_fw-la by_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o side_n of_o the_o cube_fw-la ¶_o a_o corollary_n a_o dodecahedron_n be_v double_a to_o a_o cube_n inscribe_v in_o it_o take_v away_o the_o three_o part_n of_o the_o less_o segment_n of_o the_o cube_fw-la and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a of_o that_o excess_n for_o if_o there_o be_v give_v a_o cube_fw-la from_o which_o be_v cut_v of_o a_o solid_a set_v upon_o a_o three_o part_n of_o the_o less_o segment_n of_o the_o base_a and_o under_o one_o and_o the_o same_o altitude_n with_o the_o cube_fw-la that_o solid_a take_v away_o have_v to_o the_o whole_a solid_a the_o proportion_n of_o the_o section_n of_o the_o base_a to_o the_o base_a by_o the_o 32._o of_o the_o eleven_o wherefore_o from_o the_o cube_fw-la be_v take_v away_o a_o three_o âart_n of_o the_o less_o segment_n far_o forasmuch_o as_o the_o residue_n want_v of_o the_o altitude_n of_o the_o cube_fw-la by_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o altitude_n or_o side_n and_o that_o residue_n be_v a_o parallelipipedon_n if_o it_o be_v cut_v by_o a_o plain_a superficies_n parallel_v to_o the_o opposite_a plain_a superficiece_n cut_v the_o altitude_n of_o the_o cube_fw-la by_o a_o point_n it_o shall_v take_v away_o from_o that_o parallelipipedon_n a_o solid_a have_v to_o the_o whole_a the_o proportion_n of_o the_o section_n to_o the_o altitude_n by_o the_o 3._o corollary_n of_o the_o 25._o of_o the_o eleven_o wherefore_o the_o excess_n want_v of_o the_o same_o cube_fw-la by_o the_o thiâd_o part_n of_o the_o less_o segment_n and_o moreover_o by_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a of_o that_o excess_n ¶_o the_o 34._o proposition_n the_o proportion_n of_o the_o solid_a of_o a_o dodecahedron_n to_o the_o solid_a of_o a_o icosahedron_n inscribe_v in_o it_o consist_v of_o the_o proportion_n triple_a of_o the_o diameter_n to_o that_o line_n which_o couple_v the_o opposite_a base_n of_o the_o dodecahedron_n and_o of_o the_o proportion_n of_o the_o side_n of_o the_o cube_n to_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n svppose_v that_o ahbck_n be_v a_o dodecahedronâ_n who_o diameter_n let_v be_v ab_fw-la and_o let_v the_o line_n which_o couple_v the_o centre_n of_o the_o opposite_a base_n be_v khâ_n and_o let_v the_o icosahedron_n inscribe_v in_o the_o dodecahedron_n abc_n be_v do_v you_o who_o diameter_n let_v be_v de._n now_o forasmuch_o aâ_z oâe_n and_o the_o self_n same_o circle_n contain_v the_o pentagon_n of_o a_o dodecahedron_n &_o the_o triangle_n of_o a_o icosahedroâ_n describe_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o 14._o of_o the_o fourteen_o let_v that_o circle_n be_v igo._n wherefore_o io_o be_v the_o side_n of_o the_o cube_fw-la and_o ig_v the_o side_n of_o the_o icosahedron_n by_o the_o same_o then_o i_o say_v that_o the_o proportion_n of_o the_o dodecahedron_n ahbck_n to_o the_o icosahedron_n def_n inscribe_v in_o it_o consist_v of_o the_o proportion_n triple_a of_o the_o line_n ab_fw-la to_o the_o line_n kh_o and_o of_o the_o proportion_n of_o the_o line_n io_o to_o the_o line_n ig_n for_o forasmuch_o as_o the_o icosahedron_n def_n be_v inscribe_v in_o the_o dodecahedron_n abc_n demonstration_n by_o supposition_n the_o diameter_n de_fw-fr shall_v be_v equal_a to_o the_o line_n kh_o by_o the_o 7._o of_o the_o fifteen_o wherefore_o the_o dodecahedron_n set_v upon_o the_o diameter_n kh_o shall_v be_v inscribe_v in_o the_o same_o sphere_n wherein_o the_o icosahedron_n def_n be_v inscribe_v but_o the_o dodecahedron_n ahbck_n be_v to_o the_o dodecahedron_n upon_o the_o diameter_n kh_o in_o triple_a proportion_n of_o that_o in_o which_o the_o diameter_n ab_fw-la be_v to_o the_o diameter_n kh_o by_o the_o corollary_n of_o the_o 17._o of_o the_o twelve_o and_o the_o same_o dodecahedron_n which_o be_v set_v upon_o the_o diameter_n kh_o have_v to_o the_o icosahedron_n def_n which_o be_v set_v upon_o the_o same_o diameter_n or_o upon_o a_o diameter_n equal_a unto_o it_o namely_o de_fw-fr that_o proportion_n which_o io_o the_o side_n of_o the_o cube_fw-la have_v toâ_n ig_n the_o side_n of_o the_o icosahedron_n inscribe_v in_o one_o &_o the_o self_n same_o sphere_n by_o the_o 8_o of_o the_o fourteen_o wherefore_o the_o proportion_n of_o the_o dodecahedron_n ahbck_n to_o the_o icosahedron_n def_n inscribe_v in_o it_o consist_v of_o the_o proportion_n triple_a of_o the_o diameter_n ab_fw-la to_o the_o line_n kh_o which_o couple_v the_o centre_n of_o the_o opposite_a base_n of_o the_o dodecahedron_n which_o proportion_n be_v that_o which_o the_o dodecahedron_n ahbck_n have_v to_o the_o dodecahedron_n set_v upon_o the_o diameter_n kh_o and_o of_o the_o proportion_n of_o io_o the_o side_n of_o the_o cube_fw-la to_o ig_o the_o side_n of_o the_o icosahedron_n which_o be_v the_o proportion_n of_o the_o dodecahedron_n set_v upon_o the_o diameter_n kh_o to_o the_o icosahedron_n def_n describe_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o 5._o definition_n of_o the_o six_o the_o proportion_n therefore_o of_o the_o solid_a of_o a_o dodecahedron_n to_o the_o solid_a of_o a_o icosahedron_n inscribe_v in_o it_o consi_v of_o the_o proportion_n triple_a of_o the_o diameter_n to_o that_o line_n which_o couple_v the_o opposite_a base_n of_o the_o dodecahedron_n and_o of_o the_o proportion_n of_o the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n the_o 35._o proposition_n the_o solid_a of_o a_o dodecahedron_n contain_v of_o a_o pyramid_n circumscribe_v about_o it_o two_o nine_o part_n take_v away_o a_o three_o part_n of_o one_o nine_o part_n of_o the_o less_o segment_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o residue_n it_o have_v be_v prove_v that_o the_o dodecahedron_n together_o with_o the_o cube_fw-la inscribe_v in_o it_o be_v contain_v in_o one_o and_o the_o self_n same_o pyramid_n by_o the_o corollary_n of_o the_o first_o of_o this_o book_n and_o by_o the_o corollary_n of_o the_o 33._o of_o this_o book_n it_o be_v manifest_a that_o the_o dodecahedron_n be_v double_a to_o the_o same_o cube_fw-la take_v away_o the_o three_o part_n of_o the_o less_o segment_n and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a
word_n or_o two_o hereafter_o shall_v be_v say_v how_o immaterial_a and_o free_a from_o all_o matter_n number_n be_v who_o do_v not_o perceive_v yeâ_z who_o do_v not_o wonderful_o wonder_v at_o it_o for_o neither_o pure_a element_n nor_o aristoteles_n quinta_fw-la essentia_fw-la be_v able_a to_o serve_v for_o number_n as_o his_o proper_a matter_n nor_o yet_o the_o purity_n and_o simpleness_n of_o substance_n spiritual_a or_o angelical_a will_v be_v find_v proper_a enough_o thereto_o and_o therefore_o the_o great_a &_o godly_a philosopher_n anitius_n boetius_fw-la say_v omnia_fw-la quaecunque_fw-la a_o primava_fw-la rerum_fw-la naââra_fw-la constructa_fw-la sunt_fw-la numerorum_fw-la videntur_fw-la ratione_fw-la formata_fw-la hoc_fw-la enim_fw-la fuit_fw-la principale_z in_fw-la animo_fw-la conditoris_fw-la exemplar_n that_o be_v all_o thing_n which_o from_o the_o very_a first_o original_a be_v of_o thing_n have_v be_v frame_v and_o make_v do_v appear_v to_o be_v form_v by_o the_o reason_n of_o number_n for_o this_o be_v the_o principal_a example_n or_o pattern_n in_o the_o mind_n of_o the_o creator_n oh_o comfortable_a allurement_n oh_o ravish_a persuasion_n to_o deal_v with_o a_o science_n who_o subject_n iâ_z so_o ancient_a so_o pure_a so_o excellent_a so_o surmount_v all_o creature_n so_o use_v of_o the_o almighty_a and_o incomprehensible_a wisdom_n of_o the_o creator_n in_o the_o distinct_a creation_n of_o all_o creaturesâ_n in_o all_o their_o distinct_a part_n property_n nature_n and_o virtue_n by_o order_n and_o most_o absolute_a number_n bring_v from_o nothing_o to_o the_o formality_n of_o their_o be_v and_o state_n by_o number_n property_n therefore_o of_o we_o by_o all_o possible_a mean_n to_o the_o perfection_n of_o the_o science_n learned_a we_o may_v both_o wind_v and_o draw_v ourselves_o into_o the_o inward_a and_o deep_a search_n and_o vow_v of_o all_o creature_n distinct_a virtue_n nature_n property_n and_o former_a and_o also_o far_o arise_v 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idaâ_n â_z this_o mountain_n of_o contemplation_n and_o more_o than_o contemplation_n and_o also_o though_o number_n be_v a_o thing_n so_o immaterial_a so_o divine_a and_o eternal_a yet_o by_o degree_n by_o little_a and_o little_a stretch_v forth_o and_o apply_v some_o likeness_n of_o it_o as_o first_o to_o thing_n spiritual_a and_o then_o bring_v it_o low_o to_o thing_n sensible_o perceive_v as_o of_o a_o momentany_a sound_n iterate_v then_o to_o the_o least_o thing_n that_o may_v be_v see_v numerable_a and_o at_o length_n most_o gross_o to_o a_o multitude_n of_o any_o corporal_a thing_n see_v or_o feel_v and_o so_o of_o these_o gross_a and_o sensible_a thing_n we_o be_v train_v to_o learn_v a_o certain_a image_n or_o likeness_n of_o number_n and_o to_o use_v art_n in_o they_o to_o our_o pleasure_n and_o profit_n so_o gross_a be_v our_o conversation_n and_o dull_a be_v our_o apprehension_n while_o mortal_a sense_n in_o we_o rule_v the_o common_a wealth_n of_o our_o little_a world_n 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our_o own_o name_n glorious_o exemplify_v and_o register_v in_o the_o book_n of_o the_o trinity_n most_o bless_v and_o eternal_a but_o far_o understand_v that_o vulgar_a practiser_n have_v number_n otherwise_o in_o sundry_a consideration_n and_o extend_v their_o name_n far_o then_o to_o number_n who_o least_o part_n be_v a_o vnit._n for_o the_o common_a logist_n reckenmaster_n or_o arithmeticien_n in_o his_o use_n of_o number_n of_o a_o unit_fw-la imagine_v less_o parte_v and_o call_v they_o fraction_n as_o of_o a_o unit_fw-la he_o make_v a_o half_a and_o thus_o note_v it_o ½_n and_o so_o of_o other_o infinite_o diverse_a part_n of_o a_o unit_fw-la yea_o and_o far_o have_v fraction_n of_o fraction_n etc_n etc_n and_o forasmuch_o as_o addition_n substraction_n multiplication_n division_n and_o extraction_n of_o root_n be_v the_o chief_a and_o sufficient_a part_n of_o arithmetic_n arithmetic_n which_o be_v the_o science_n that_o demonstrate_v the_o property_n of_o number_n and_o all_o operation_n in_o number_n to_o be_v perform_v
diameter_n be_v double_a to_o that_o square_n who_o diameter_n it_o be_v corollary_n the_o 34._o theorem_a the_o 48._o proposition_n if_o the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n of_o a_o triangle_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o two_o other_o side_n of_o the_o same_o triangle_n the_o angle_n comprehend_v under_o those_o two_o other_o side_n be_v a_o right_a angle_n svppose_v that_o abc_n be_v a_o triangle_n and_o let_v the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n there_o namely_o of_o the_o side_n bc_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o side_n basilius_n and_o ac_fw-la then_o i_o say_v that_o the_o angle_n bac_n be_v a_o right_a angle_n raise_v up_o by_o the_o 11._o proposition_n from_o the_o point_n a_o unto_o the_o right_a line_n ac_fw-la a_o perpendicular_a line_n ad._n and_o by_o the_o third_o proposition_n unto_o the_o line_n ab_fw-la put_v a_o equal_a line_n ad._n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n d_o to_o the_o poinâ_n c._n and_o forasmuch_o as_o the_o line_n dam_n be_v equal_a to_o the_o line_n ab_fw-la the_o square_n which_o be_v make_v of_o the_o line_n dam_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n ab_fw-la put_v the_o square_n of_o the_o line_n ac_fw-la common_a to_o they_o both_o wherefore_o the_o square_n of_o the_o line_n dam_n and_o ac_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n basilius_n and_o ac_fw-la but_o by_o the_o proposition_n go_v before_o the_o square_a of_o the_o line_n dc_o be_v equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o ac_fw-la for_o the_o angle_n dac_o be_v a_o right_a angle_n and_o the_o square_a of_o bc_o be_v by_o supposition_n equal_a to_o the_o square_n of_o ab_fw-la and_o ac_fw-la wherefore_o the_o square_a of_o dc_o be_v equal_a to_o the_o square_n of_o bc_o wherefore_o the_o side_n dc_o be_v equal_a to_o the_o 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parallelogram_n which_o have_v the_o same_o diameter_n which_o the_o whole_a parallelogram_n have_v and_o supplement_n be_v such_o which_o be_v without_o the_o diameter_n of_o the_o whole_a parallelogram_n as_o of_o the_o parallelogram_n abcd_o the_o partial_a or_o particular_a parallelogram_n gkcf_n and_o ebkh_n be_v parallelogram_n about_o the_o diameter_n for_o that_o each_o of_o they_o have_v for_o his_o diameter_n a_o part_n of_o the_o diameter_n of_o the_o whole_a parallelogram_n as_o ck_v and_o kb_v the_o particular_a diameter_n be_v part_n of_o the_o line_n cb_o which_o be_v the_o diameter_n of_o the_o whole_a parallelogram_n and_o the_o two_o parallelogram_n aegk_n and_o khfd_n be_v supplement_n because_o they_o be_v without_o the_o diameter_n of_o the_o whole_a parallelogram_n namely_o cb._n now_o any_o one_o of_o those_o partial_a parallelogram_n
number_n of_o the_o number_n ad_fw-la and_o db_o be_v double_a to_o the_o square_a number_n of_o ac_fw-la and_o cd_o for_o forasmuch_o as_o the_o number_n ad_fw-la be_v divide_v into_o the_o number_n ab_fw-la and_o bd_o therefore_o the_o square_a number_n of_o the_o number_n ad_fw-la and_o db_o be_v equal_a to_o the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n ad_fw-la and_o db_o the_o on_o into_o the_o other_o twice_o together_o with_o the_o square_n of_o the_o number_n ab_fw-la by_o the_o 7_o proposition_n but_o the_o square_n of_o the_o number_n ab_fw-la be_v equal_a to_o four_o square_n of_o either_o of_o the_o number_n ac_fw-la or_o cb_o for_o ac_fw-la be_v equal_a to_o the_o number_n cb_o wherefore_o also_o the_o square_n of_o the_o number_n ad_fw-la and_o db_o be_v equal_a to_o the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n ad_fw-la and_o db_o the_o one_o into_o the_o other_o twice_o and_o to_o four_o square_n of_o the_o number_n bc_o or_o ca._n and_o forasmuch_o as_o the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n ad_fw-la and_o db_o the_o one_o into_o the_o other_o together_o with_o the_o square_n of_o the_o number_n cb_o be_v equal_a to_o square_v of_o the_o number_n cd_o by_o the_o 6_o proposition_n therefore_o the_o number_n produce_v of_o the_o multiplication_n of_o the_o number_n ad_fw-la and_o db_o the_o one_o into_o the_o other_o twice_o together_o with_o two_o square_n of_o the_o number_n cb_o be_v equal_a to_o two_o square_n of_o the_o number_n cd_o wherefore_o the_o square_n of_o the_o number_n ad_fw-la and_o db_o be_v equal_a to_o two_o square_n of_o the_o number_n cd_o and_o to_o two_o square_n of_o the_o number_n ac_fw-la wherefore_o they_o be_v double_a to_o the_o square_n of_o the_o number_n ac_fw-la and_o cd_o and_o the_o square_a of_o the_o number_n ad_fw-la be_v the_o square_n of_o the_o whole_a and_o of_o the_o number_n add_v and_o the_o square_a of_o db_o be_v the_o square_a of_o the_o number_n add_v the_o square_a also_o of_o the_o number_n cd_o be_v the_o square_a of_o the_o number_n compose_v of_o the_o half_a and_o of_o the_o number_n add_v if_o therefore_o a_o even_a number_n be_v divide_v etc_n etc_n which_o be_v require_v to_o be_v prove_v the_o 1._o problem_n the_o 11._o proposition_n to_o divide_v a_o right_a line_n give_v in_o such_o sort_n that_o the_o rectangle_n figure_n comprehend_v under_o the_o whole_a and_o one_o of_o the_o part_n shall_v be_v equal_a unto_o the_o square_n make_v of_o the_o other_o part_n svppose_v that_o the_o right_a line_n give_v be_v ab_fw-la now_o it_o be_v require_v to_o divide_v the_o line_n ab_fw-la in_o such_o sort_n that_o the_o rectangle_n figure_n contain_v under_o the_o whole_a and_o one_o of_o the_o part_n shall_v be_v equal_a unto_o the_o square_n which_o be_v make_v of_o the_o other_o part_n describe_v by_o the_o 46._o of_o the_o first_o upon_o ab_fw-la a_o square_a abcd._n construction_n and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n ac_fw-la into_o two_o equal_a part_n in_o the_o point_n e_o and_o draw_v a_o line_n from_o b_o to_o e._n and_o by_o the_o second_o petition_n extend_v ca_n unto_o the_o point_n f._n and_o by_o the_o 3._o of_o the_o first_o put_v the_o line_n of_o equal_a unto_o the_o line_n be._n and_o by_o the_o 46._o of_o the_o first_o upon_o the_o line_n of_o describe_v a_o square_a fgah_n and_o by_o the_o 2._o petition_n extend_v gh_o unto_o the_o point_n k._n then_o i_o say_v that_o the_o line_n ab_fw-la be_v divide_v in_o the_o point_n h_o in_o such_o sort_n that_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o ab_fw-la and_o bh_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o ah_o demonstration_n for_o forasmuch_o as_o the_o right_a line_n ac_fw-la be_v divide_v into_o two_o equal_a part_n in_o the_o point_n e_o and_o unto_o it_o be_v add_v a_o other_o right_a line_n af._n therefore_o by_o the_o 6._o of_o the_o second_o the_o rectangle_n figure_n contain_v under_o cf_o and_o favorina_n together_o with_o the_o square_n which_o be_v make_v of_o ae_n be_v equal_a to_o the_o square_v which_o be_v make_v of_o ef._n but_o of_o be_v equal_a unto_o ebb_n wherefore_o the_o rectangle_n figure_n contain_v under_o cf_o and_o favorina_n together_o with_o the_o square_n which_o be_v make_v of_o ea_fw-la be_v equal_a to_o the_o square_n which_o be_v make_v of_o ebb_n but_o by_o 47._o of_o the_o first_o unto_o the_o square_n which_o be_v make_v of_o ebb_n be_v equal_a the_o square_n which_o be_v make_v of_o basilius_n and_o ae_n for_o the_o angle_n at_o the_o point_n a_o be_v a_o right_a angle_n wherefore_o that_o which_o be_v contain_v under_o cf_o and_o favorina_n together_o with_o the_o square_n which_o be_v make_v of_o ae_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o basilius_n and_o ae_n take_v away_o the_o square_n which_o be_v make_v of_o ae_n which_o be_v common_a to_o they_o both_o wherefore_o the_o rectangle_n figure_n remain_v contain_v under_o cf_o and_o favorina_n be_v equal_a unto_o the_o square_n which_o be_v make_v of_o ab_fw-la and_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o favorina_n be_v the_o figure_n fk_o for_o the_o line_n favorina_n be_v equal_a unto_o the_o line_n fg._n and_o the_o square_n which_o be_v make_v of_o ab_fw-la be_v the_o figure_n ad._n wherefore_o the_o figure_n fk_o be_v equal_a unto_o the_o figure_n ad._n take_v away_o the_o figure_n ak_o which_o be_v common_a to_o they_o both_o wherefore_o the_o residue_n namely_o the_o figure_n fh_o be_v equal_a unto_o the_o residue_n namely_o unto_o the_o figure_n hd_v but_o the_o figure_n hd_v be_v that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bh_o for_o ab_fw-la be_v equal_a unto_o bd._n and_o the_o figure_n fh_o be_v the_o square_n which_o be_v make_v of_o ah_o wherefore_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n ab_fw-la and_o bh_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n ha._n wherefore_o the_o right_a line_n give_v ab_fw-la be_v divide_v in_o the_o point_n h_o in_o such_o sort_n that_o the_o rectangle_n figure_n contain_v under_o ab_fw-la and_o bh_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o ah_o which_o be_v require_v to_o be_v do_v this_o proposition_n have_v many_o singular_a use_n upon_o it_o depend_v the_o demonstration_n of_o that_o worthy_a problem_n the_o 10._o proposition_n of_o the_o 4._o book_n proposition_n which_o teach_v to_o describe_v a_o isosceles_a triangle_n in_o which_o either_o of_o the_o angle_n at_o the_o base_a shall_v be_v double_a to_o the_o angle_n at_o the_o top_n many_o and_o diverse_a use_n of_o a_o line_n so_o divide_v shall_v you_o find_v in_o the_o 13._o book_n of_o euclid_n this_o be_v to_o be_v note_v that_o this_o proposition_n can_v not_o as_o the_o former_a proposition_n of_o this_o second_o book_n be_v reduce_v unto_o number_n number_n for_o the_o line_n ebb_n have_v unto_o the_o line_n ae_n no_o proportion_n that_o can_v be_v name_v and_o therefore_o it_o can_v not_o be_v express_v by_o number_n for_o forasmuch_o as_o the_o square_n of_o ebb_n be_v equal_a to_o the_o two_o square_n of_o ab_fw-la and_o ae_n by_o the_o 47._o of_o the_o first_o and_o ae_n be_v the_o half_a of_o ab_fw-la therefore_o the_o line_n be_v be_v irrational_a for_o even_o as_o two_o equal_a square_a number_n join_v together_o can_v not_o make_v a_o square_a number_n so_o also_o two_o square_a number_n of_o which_o the_o one_o be_v the_o square_a of_o the_o half_a root_n of_o the_o other_o can_v not_o make_v a_o square_a number_n as_o by_o a_o example_n take_v the_o square_n of_o 8._o which_o be_v 64._o which_o double_v that_o be_v 128._o make_v not_o a_o square_a number_n so_o take_v the_o half_a of_o 8._o which_o be_v 4._o and_o the_o square_n of_o 8._o and_o 4._o which_o be_v 64._o and_o 16._o add_v together_o likewise_o make_v not_o a_o square_a number_n for_o they_o make_v 80._o who_o have_v no_o root_n square_v which_o thing_n must_v of_o necessity_n be_v if_o this_o problem_n shall_v have_v place_n in_o number_n but_o in_o irrational_a number_n it_o be_v true_a and_o may_v by_o this_o example_n be_v declare_v let_v 8._o be_v so_o divide_v that_o that_o which_o be_v produce_v of_o the_o whole_a into_o one_o of_o his_o part_n shall_v be_v equal_a to_o the_o square_a number_n produce_v of_o the_o other_o part_n multiply_v 8._o into_o himself_o and_o there_o shall_v be_v produce_v 64._o that_o be_v the_o square_a abcd._n divide_v 8._o into_o two_o equal_a part_n that_o be_v into_o 4_o and_o 4._o as_o the_o line_n
the_o other_o in_o the_o point_n d_o and_o that_o only_a point_n be_v common_a to_o they_o both_o neither_o do_v the_o one_o enter_v into_o the_o other_o if_o any_o part_n of_o the_o one_o enter_v into_o any_o part_n of_o the_o other_o than_o the_o one_o cut_v and_o divide_v the_o other_o and_o touch_v the_o one_o the_o other_o not_o in_o one_o point_n only_o as_o in_o the_o other_o before_o but_o in_o two_o pointâs_n and_o have_v also_o a_o superficies_n common_a to_o they_o both_o as_o the_o circle_n ghk_v and_o hlk_v cut_v the_o one_o the_o other_o in_o two_o point_n h_o and_o edward_n and_o the_o one_o enter_v into_o the_o other_o also_o the_o superficies_n hk_o be_v common_a to_o they_o both_o for_o it_o be_v a_o part_n of_o the_o circle_n ghk_o and_o also_o it_o be_v a_o part_n of_o the_o circle_n hlk._n definition_n right_o line_n in_o a_o circle_n be_v say_v to_o be_v equal_o distant_a from_o the_o centre_n when_o perpendicular_a line_n draw_v from_o the_o centre_n unto_o 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line_n ab_fw-la and_o dc_o in_o the_o circle_n abcd_o be_v equidistant_v from_o the_o centre_n gâ_n â_z because_o the_o line_n og_n and_o gp_n perpendicular_o draw_v from_o the_o centre_n g_o upon_o the_o say_a line_n ab_fw-la and_o dc_o be_v equal_a and_o the_o line_n ab_fw-la have_v great_a distance_n from_o the_o centre_n g_o than_o have_v the_o the_o line_n of_o because_o the_o line_n og_n perpendicular_o drowen_v from_o the_o centre_n g_o to_o the_o line_n ab_fw-la be_v greâter_n than_o the_o line_n hg_o which_o be_v perpendicular_o draw_v from_o the_o cââtre_n g_o to_o the_o line_n ef._n definition_n a_o section_n or_o segment_n of_o a_o circle_n be_v a_o figure_n comprehend_v under_o a_o right_a line_n and_o a_o portion_n of_o the_o circumference_n of_o a_o circle_n as_o the_o figure_n abc_n be_v a_o section_n of_o a_o circle_n because_o it_o be_v comprehend_v under_o the_o right_a line_n ac_fw-la and_o the_o circumference_n of_o a_o circle_n abc_n likewise_o the_o figure_n def_n be_v a_o section_n of_o a_o circle_n for_o that_o it_o be_v comprehend_v under_o the_o right_a line_n df_o and_o the_o circumference_n def_n and_o the_o figure_n abc_n for_o that_o it_o contain_v within_o it_o the_o centre_n of_o the_o circle_n be_v call_v the_o great_a section_n of_o a_o circle_n and_o the_o figure_n def_n be_v the_o less_o section_n of_o a_o circle_n because_o it_o be_v whole_o without_o the_o centre_n of_o the_o circle_n as_o it_o be_v note_v in_o the_o 16._o definition_n of_o the_o first_o book_n a_o angle_n of_o a_o section_n or_o segment_n be_v that_o angle_n which_o be_v contain_v under_o a_o right_a line_n and_o the_o circumference_n of_o the_o circle_n definition_n as_o the_o angle_n abc_n in_o the_o section_n abc_n be_v a_o angle_n of_o a_o section_n because_o it_o be_v contain_v of_o the_o circumference_n bac_n and_o the_o right_a line_n bc._n likewise_o the_o angle_n cbd_v be_v a_o angle_n of_o the_o section_n bdc_o because_o it_o be_v contain_v 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section_n have_v in_o it_o the_o great_a angle_n but_o when_o the_o right_a line_n which_o comprehend_v the_o angle_n do_v receive_v any_o circumference_n of_o a_o circle_n definition_n than_o that_o angle_n be_v say_v to_o be_v correspondent_a and_o to_o pertain_v to_o that_o circumference_n as_o the_o right_a line_n basilius_n and_o bc_o which_o contain_v the_o angle_n ab_fw-la c_z and_o receive_v the_o circumference_n adc_o therefore_o the_o angle_n abc_n be_v say_v to_o subtend_v and_o to_o pertain_v to_o the_o circumference_n adc_o and_o if_o the_o right_a line_n which_o cause_n the_o angle_n concur_v in_o the_o centre_n of_o a_o circle_n then_o the_o angle_n be_v say_v to_o be_v in_o the_o centre_n of_o a_o circle_n as_o the_o angle_n efd_n be_v say_v to_o be_v in_o the_o centre_n of_o a_o circle_n for_o that_o it_o be_v comprehend_v of_o two_o right_a line_n fe_o and_o fd_v which_o concur_v and_o touch_v in_o the_o centre_n f._n and_o this_o angle_n likewise_o subtend_v the_o circumference_n egg_v which_o circumference_n also_o be_v the_o measure_n of_o the_o greatness_n of_o the_o angle_n efd_n a_o sector_n of_o a_o circle_n be_v a_o angle_n be_v set_v at_o the_o centre_n of_o a_o circle_n a_o figure_n contain_v under_o the_o right_a line_n which_o make_v that_o angle_n definition_n and_o the_o part_n of_o the_o circumference_n receive_v of_o they_o as_o the_o figure_n abc_n be_v a_o sector_n of_o a_o circle_n for_o that_o it_o have_v a_o angle_n at_o the_o centre_n namely_o the_o angle_n bac_n &_o be_v contain_v of_o the_o two_o right_a line_n ab_fw-la and_o ac_fw-la which_o contain_v that_o angle_n and_o the_o circumference_n receive_v by_o they_o definition_n like_a segment_n or_o section_n of_o a_o circle_n be_v those_o which_o have_v equal_a angle_n or_o in_o who_o be_v equal_a angle_n definition_n here_o be_v set_v two_o definition_n of_o like_a section_n of_o a_o circle_n the_o one_o pertain_v to_o the_o angle_n which_o be_v set_v in_o the_o centre_n of_o the_o circle_n and_o receive_v the_o circumference_n of_o the_o say_a section_n first_o the_o other_o pertain_v to_o the_o angle_n in_o the_o section_n which_o as_o before_o be_v say_v be_v ever_o in_o the_o circumference_n as_o if_o the_o angle_n bac_n be_v in_o the_o centre_n a_o and_o receive_v of_o the_o circumference_n blc_n be_v equal_a to_o the_o angle_n feg_n be_v also_o in_o the_o centre_n e_o and_o receive_v of_o the_o circumference_n fkg_v then_o be_v the_o two_o section_n bcl_n and_o fgk_n like_v by_o the_o first_o definition_n by_o the_o same_o definition_n also_o be_v the_o other_o two_o section_n like_a namely_o bcd_o and_o fgh_a for_o that_o the_o angle_n bac_n be_v equal_a to_o the_o
say_v that_o the_o line_n gfh_v which_o by_o the_o corollary_n of_o the_o 16._o of_o this_o book_n touch_v the_o circle_n be_v a_o parallel_n unto_o the_o line_n ab_fw-la circle_n for_o forasmuch_o as_o the_o right_a line_n cf_o fall_a upon_o either_o of_o these_o line_n ab_fw-la &_o gh_o make_v all_o the_o angle_n at_o the_o point_n â_o right_n angle_n by_o the_o 3._o of_o this_o book_n and_o the_o two_o angle_n at_o the_o point_n fare_v suppose_v to_o be_v right_a angle_n therefore_o by_o the_o 29._o of_o the_o first_o the_o line_n ab_fw-la and_o gh_o be_v parallel_n which_o be_v require_v to_o be_v do_v and_o this_o problem_n be_v very_o commodious_a for_o the_o inscribe_v or_o circumscribe_v of_o figure_n in_o or_o about_o circle_n the_o 16._o theorem_a the_o 18._o proposition_n if_o a_o right_a line_n touch_v a_o circle_n and_o from_o the_o centre_n to_o the_o touch_n be_v draw_v a_o right_a line_n that_o right_a line_n so_o draw_v shall_v be_v a_o perpendicular_a line_n to_o the_o touch_n line_n svppose_v that_o the_o right_a line_n de_fw-fr do_v touch_v the_o circle_n abc_n in_o the_o point_n c._n and_o take_v the_o centre_n of_o the_o circle_n abc_n and_o let_v the_o same_o be_v f._n impossibility_n and_o by_o the_o first_o petition_n from_o the_o point_n fletcher_n to_o the_o point_n c_o draw_v a_o right_a line_n fc_o then_o i_o say_v that_o cf_o be_v a_o perpendicular_a line_n to_o de._n for_o if_o not_o draw_v by_o the_o 12._o of_o the_o first_o from_o the_o point_n fletcher_n to_o the_o line_n de_fw-fr a_o perpendicular_a line_n fg._n and_o for_o asmuch_o as_o the_o angle_n fgc_n be_v a_o right_a angle_n therefore_o the_o angle_n gcf_n be_v a_o acute_a angle_n wherefore_o the_o angle_n fgc_n be_v great_a than_o the_o angle_n fcg_n but_o unto_o the_o great_a angle_n be_v subtend_v the_o great_a side_n by_o the_o 19_o of_o the_o first_o wherefore_o the_o line_n fc_o be_v great_a than_o the_o line_n fg._n but_o the_o line_n fc_o be_v equal_a to_o the_o line_n fb_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n wherefore_o the_o line_n fb_o also_o be_v great_a than_o the_o line_n fg_o namely_o the_o less_o than_o the_o great_a which_o be_v impossible_a wherefore_o the_o line_n fg_o be_v not_o a_o perpendicular_a line_n unto_o the_o line_n de._n and_o in_o like_a sort_n may_v we_o prove_v that_o no_o other_o line_n be_v a_o perpendicular_a line_n unto_o the_o line_n de_fw-fr beside_o the_o line_n fc_o wherefore_o the_o line_n fc_o be_v a_o perpendicular_a line_n to_o de._n if_o therefore_o a_o right_a line_n touch_v a_o circle_n &_o from_o you_o centre_n to_o the_o touch_n be_v draw_v a_o right_a line_n that_o right_a line_n so_o draw_v shall_v be_v a_o perpendicular_a line_n to_o the_o touch_n line_n which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n after_o orontius_n suppose_v that_o the_o circle_n give_v be_v abc_n which_o let_v the_o right_a line_n de_fw-fr touch_n in_o the_o point_n c._n orontius_n and_o let_v the_o centre_n of_o the_o circle_n be_v the_o point_n f._n and_o draw_v a_o right_a line_n from_o fletcher_n to_o c._n then_o i_o say_v that_o the_o line_n fc_o be_v perpendicular_a unto_o the_o line_n de._n for_o if_o the_o line_n fc_o be_v not_o a_o perpendicule_a unto_o the_o line_n de_fw-fr then_o by_o the_o converse_n of_o the_o x._o definition_n of_o the_o first_o book_n the_o angle_n dcf_n &_o fce_n shall_v be_v unequal_a &_o therefore_o the_o one_o be_v great_a than_o a_o right_a angle_n and_o the_o other_o be_v less_o than_o a_o right_a angle_n for_o the_o angle_n dcf_n and_o fce_n be_v by_o the_o 13._o of_o the_o first_o equal_a to_o two_o right_a angle_n let_v the_o angle_n fce_n if_o it_o be_v possible_a be_v great_a than_o a_o right_a angle_n that_o be_v let_v it_o be_v a_o obtuse_a angle_n wherefore_o the_o angle_n dcf_n âhal_v be_v a_o acute_a angle_n and_o forasmuch_o as_o by_o supposition_n the_o right_a line_n de_fw-fr touch_v the_o circle_n abc_n therefore_o it_o cut_v not_o the_o circle_n wherefore_o the_o circumference_n bc_o fall_v between_o the_o right_a line_n dc_o &_o cf_o &_o therefore_o the_o acute_a and_o rectiline_a angle_n dcf_n shall_v be_v great_a than_o the_o angle_n of_o the_o semicircle_n bcf_n which_o be_v contain_v under_o the_o circumference_n bc_o &_o the_o right_a line_n cf._n and_o so_o shall_v there_o be_v give_v a_o rectiline_n &_o acute_a angle_n great_a than_o the_o angle_n of_o a_o semicircle_n which_o be_v contrary_a to_o the_o 16._o proposition_n of_o this_o book_n wherefore_o the_o angle_n dcf_n be_v not_o less_o than_o a_o right_a angle_n in_o like_a sort_n also_o may_v we_o prove_v that_o it_o be_v not_o great_a than_o a_o right_a angle_n wherefore_o it_o be_v a_o right_a angle_n and_o therefore_o also_o the_o angle_n fce_n be_v a_o right_a angle_n wherefore_o the_o right_a line_n fc_o be_v a_o perpendicular_a unto_o the_o right_a line_n de_fw-fr by_o the_o 10._o definition_n of_o the_o firstâ_o which_o be_v require_v to_o be_v prove_v the_o 17._o theorem_a the_o 19_o proposition_n if_o a_o right_a line_n do_v touch_v a_o circle_n and_o from_o the_o point_n of_o the_o touch_n be_v raise_v up_o unto_o the_o touch_n line_n a_o perpendicular_a line_n in_o that_o line_n so_o raise_v up_o be_v the_o centre_n of_o the_o circle_n svppose_v that_o the_o right_a line_n de_fw-fr do_v touch_v the_o circle_n abc_n in_o the_o point_n c._n and_o from_o c_o raise_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o line_n de_fw-fr a_o perpendicular_a line_n ca._n then_o i_o say_v that_o in_o the_o line_n ca_n be_v the_o centre_n of_o the_o circle_n for_o if_o not_o then_o if_o it_o be_v possible_a let_v the_o centre_n be_v without_o the_o line_n ca_n as_o in_o the_o point_n f._n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o c_o to_o f._n impossibility_n and_o for_o asmuch_o as_o a_o certain_a right_a line_n de_fw-fr touch_v the_o circle_n abc_n and_o from_o the_o centre_n to_o the_o touch_n be_v draw_v a_o right_a line_n cf_o therefore_o by_o the_o 18._o of_o the_o three_o fc_o be_v a_o perpendicular_a line_n to_o de._n wherefore_o the_o angle_n fce_n be_v a_o right_a angle_n but_o the_o angle_n ace_n be_v also_o a_o right_a angle_n wherefore_o the_o angle_n fce_n be_v equal_a to_o the_o angle_n ace_n namely_o the_o less_o unto_o the_o great_a which_o be_v impossibleâ_n wherefore_o the_o point_n fletcher_n be_v not_o the_o centre_n of_o the_o circle_n abc_n and_o in_o like_a sort_n may_v we_o prove_v that_o it_o be_v no_o other_o where_o but_o in_o the_o line_n ac_fw-la if_o therefore_o a_o right_a line_n do_v touch_v a_o circle_n and_o from_o the_o point_n of_o the_o touch_n be_v raise_v up_o unto_o the_o touch_n line_v a_o perpendicular_a line_n in_o that_o line_n so_o raise_v up_o be_v the_o centre_n of_o the_o circle_n which_o be_v require_v to_o be_v prove_v the_o 18._o theorem_a the_o 20._o proposition_n in_o a_o circle_n a_o angle_n set_v at_o the_o centre_n be_v double_a to_o a_o angle_n set_v at_o the_o circumference_n so_o that_o both_o the_o angle_n have_v to_o their_o base_a one_o and_o the_o same_o circumference_n svppose_v that_o there_o be_v a_o circle_n abc_n and_o at_o the_o centre_n thereof_o centre_n namely_o the_o point_n e_o let_v the_o angle_n bec_n be_v set_v &_o at_o the_o circumference_n let_v there_o be_v set_v the_o angle_n bac_n and_o let_v they_o both_o have_v one_o and_o the_o same_o base_a namely_o the_o circumference_n bc._n then_o i_o say_v that_o the_o angle_n bec_n be_v double_a to_o the_o angle_n bac_n draw_v the_o right_a line_n ae_n and_o by_o the_o second_o petition_n extend_v it_o to_o the_o point_n f._n now_o for_o asmuch_o as_o the_o line_n ae_n be_v equal_a to_o the_o line_n ebb_n demonstration_n for_o they_o be_v draw_v from_o the_o centre_n unto_o the_o circumference_n the_o angle_n eab_n be_v equal_a to_o the_o angle_n eba_n by_o the_o 5._o of_o the_o first_o wherefore_o the_o angle_n eab_n and_o eba_n be_v double_a to_o the_o angle_n eab_n but_o by_o the_o 32._o of_o the_o same_o the_o angle_n bef_n be_v equal_a to_o the_o angle_n eab_n and_o eba_n wherefore_o the_o angle_n bef_n be_v double_a to_o the_o angle_n eab_n and_o by_o the_o same_o reason_n the_o angle_n fec_n be_v double_a to_o the_o angle_n eac_n wherefore_o the_o whole_a angle_n bec_n be_v double_a to_o the_o whole_a angle_n bac_n again_o suppose_v that_o there_o be_v set_v a_o other_o angle_n at_o the_o circumference_n centre_n and_o let_v the_o same_o be_v bdc_o and_o by_o the_o âirst_n petition_n draw_v a_o line_n from_o d_z to_o e._n and_o by_o the_o second_o petition_n extend_v
poligonon_n figure_n of_o 24._o side_n likewise_o of_o the_o hexagon_n ab_fw-la and_o of_o the_o pentagon_n ac_fw-la shall_v be_v make_v a_o poligonon_n figure_n of_o 30._o side_n one_o of_o who_o side_n shall_v subtend_v the_o ark_n bc._n for_o the_o denomination_n of_o ab_fw-la which_o be_v 6._o exceed_v the_o denomination_n of_o ac_fw-la which_o be_v 5._o only_a by_o unity_n so_o also_o forasmuch_o as_o the_o denomination_n of_o ab_fw-la which_o be_v 6._o exceed_v the_o denomination_n of_o ae_n which_o be_v 3._o by_o 3._o therefore_o the_o ark_n be_v shall_v contain_v 3._o side_n of_o a_o poligonon_n figure_n of_o .18_o side_n and_o observe_v this_o self_n same_o method_n and_o order_n a_o man_n may_v find_v out_o infinite_a side_n of_o a_o poligonon_n figure_n the_o end_n of_o the_o four_o book_n of_o euclides_n element_n ¶_o the_o five_o book_n of_o euclides_n element_n this_o five_o book_n of_o euclid_n be_v of_o very_o great_a commodity_n and_o use_n in_o all_o geometry_n book_n and_o much_o diligence_n ought_v to_o be_v bestow_v therein_o it_o ought_v of_o all_o other_o to_o be_v thorough_o and_o most_o perfect_o and_o ready_o know_v for_o nothing_o in_o the_o book_n follow_v can_v be_v understand_v without_o it_o the_o knowledge_n of_o they_o all_o depend_v of_o it_o and_o not_o only_o they_o and_o other_o write_n of_o geometry_n but_o all_o other_o science_n also_o and_o art_n as_o music_n astronomy_n perspective_n arithmetic_n the_o art_n of_o account_n and_o reckon_n with_o other_o such_o like_a this_o book_n therefore_o be_v as_o it_o be_v a_o chief_a treasure_n and_o a_o peculiar_a jewel_n much_o to_o be_v account_v of_o it_o entreat_v of_o proportion_n and_o analogy_n or_o proportionality_n which_o pertain_v not_o only_o unto_o line_n figure_n and_o body_n in_o geometry_n but_o also_o unto_o sound_n &_o voice_n of_o which_o music_n entreat_v as_o witness_v boetius_fw-la and_o other_o which_o write_v of_o music_n also_o the_o whole_a art_n of_o astronomy_n teach_v to_o measure_v proportion_n of_o time_n and_o move_n archimedes_n and_o jordan_n with_o other_o write_v of_o weight_n affirm_v that_o there_o be_v proportion_n between_o weight_n and_o weight_n and_o also_o between_o place_n &_o place_n you_o see_v therefore_o how_o large_a be_v the_o use_n of_o this_o five_o book_n wherefore_o the_o definition_n also_o thereof_o be_v common_a although_o hereof_o euclid_n they_o be_v accommodate_v and_o apply_v only_o to_o geometry_n the_o first_o author_n of_o this_o book_n be_v as_o it_o be_v affirm_v of_o many_o one_o eudoxus_n who_o be_v plato_n scholar_n eudoxus_n but_o it_o be_v afterward_o frame_v and_o put_v in_o order_n by_o euclid_n definition_n definition_n a_o part_n be_v a_o less_o magnitude_n in_o respect_n of_o a_o great_a magnitude_n when_o the_o less_o measure_v the_o great_a as_o in_o the_o other_o book_n before_o so_o in_o this_o the_o author_n first_o set_v orderly_a the_o definition_n and_o declaration_n of_o such_o term_n and_o word_n which_o be_v necessary_o require_v to_o the_o entreaty_n of_o the_o subject_n and_o matter_n thereof_o which_o be_v proportion_n and_o comparison_n of_o proportion_n or_o proportionality_n and_o first_o he_o show_v what_o a_o part_n be_v here_o be_v to_o be_v consider_v that_o all_o the_o definition_n of_o this_o five_o book_n be_v general_a to_o geometry_n and_o arithmetic_n and_o be_v true_a in_o both_o art_n even_o as_o proportion_n and_o proportionality_n be_v common_a to_o they_o both_o and_o chief_o appertain_v to_o number_v neither_o can_v they_o apt_o be_v apply_v to_o matter_n of_o geometry_n but_o in_o respect_n of_o number_n and_o by_o number_n yet_o in_o this_o book_n and_o in_o these_o definition_n here_o set_v euclid_n seem_v to_o speak_v of_o they_o only_o geometrical_o as_o they_o be_v apply_v to_o quantity_n continual_a as_o to_o line_n superficiece_n and_o body_n for_o that_o he_o yet_o continue_v in_o geometry_n i_o will_v notwithstanding_o for_o facility_n and_o far_a help_n of_o the_o reader_n declare_v they_o both_o by_o example_n in_o number_n and_o also_o in_o line_n for_o the_o clear_a understanding_n of_o a_o part_n it_o be_v to_o be_v note_v way_n that_o a_o part_n be_v take_v in_o the_o mathematical_a science_n two_o manner_n of_o way_n way_n one_o way_n a_o part_n be_v a_o less_o quantity_n in_o respect_n of_o a_o great_a whether_o it_o measure_v the_o great_a oâ_n no._n the_o second_o way_n way_n a_o part_n be_v only_o that_o less_o quantity_n in_o respect_n of_o the_o great_a which_o measure_v the_o great_a a_o less_o quantity_n be_v say_v to_o measure_n or_o number_v a_o great_a quantity_n great_a when_o it_o be_v oftentimes_o take_v make_v precise_o the_o great_a quantity_n without_o more_o or_o less_o or_o be_v as_o oftentimes_o take_v from_o the_o great_a as_o it_o may_v there_o remain_v nothing_o as_o suppose_v the_o line_n ab_fw-la to_o contain_v 3._o and_o the_o line_n cd_o to_o contain_v 9_o they_o do_v the_o line_n ab_fw-la measure_n the_o line_n cd_o for_o that_o if_o it_o be_v take_v certain_a time_n namely_o 3._o time_n it_o make_v precise_o the_o line_n cd_o that_o be_v 9_o without_o more_o or_o less_o again_o if_o the_o say_v less_o line_n ab_fw-la be_v take_v from_o the_o great_a cd_o as_o often_o as_o it_o may_v be_v namely_o 3._o time_n there_o shall_v remain_v nothing_o of_o the_o great_a so_o the_o number_n 3._o be_v say_v to_o measure_v 12._o for_o that_o be_v take_v certain_a time_n namely_o four_o time_n it_o make_v just_a 12._o the_o great_a quantity_n and_o also_o be_v take_v from_o 12._o as_o often_o as_o it_o may_v namely_o 4._o time_n there_o shall_v remain_v nothing_o and_o in_o this_o meaning_n and_o signification_n do_v euclid_n undoubted_o here_o in_o this_o define_v a_o part_n part_n say_v that_o it_o be_v a_o less_o magnitude_n in_o comparison_n of_o a_o great_a when_o the_o less_o measure_v the_o great_a as_o the_o line_n ab_fw-la before_o set_v contain_v 3_n be_v a_o less_o quantity_n in_o comparison_n of_o the_o line_n cd_o which_o contain_v 9_o and_o also_o measure_v it_o for_o it_o be_v certain_a time_n take_v namely_o 3._o time_n precise_o make_v it_o or_o take_v from_o it_o as_o often_o as_o it_o may_v there_o remain_v nothing_o wherefore_o by_o this_o definition_n the_o line_n ab_fw-la be_v a_o part_n of_o the_o line_n cd_o likewise_o in_o number_n the_o number_n 5._o be_v a_o part_n of_o the_o number_n 15._o for_o it_o be_v a_o less_o number_n or_o quantity_n compare_v to_o the_o great_a and_o also_o it_o measure_v the_o great_a for_o be_v take_v certain_a time_n namely_o 3._o time_n it_o make_v 15._o and_o this_o kind_n of_o part_n be_v call_v common_o pars_fw-la metiens_fw-la or_o mensurans_fw-la mensuranâ_n that_o be_v a_o measure_a part_n some_o call_v it_o pars_fw-la multiplicatina_fw-la multiplicatiâa_fw-la and_o of_o the_o barbarous_a it_o be_v call_v pars_fw-la aliquota_fw-la aliquota_fw-la that_o be_v a_o aliquote_v part_n and_o this_o kind_n of_o part_n be_v common_o use_v in_o arithmetic_n arithmetic_n the_o other_o kind_n of_o a_o part_n part_n be_v any_o less_o quantity_n in_o comparison_n of_o a_o great_a whether_o it_o be_v in_o number_n or_o magnitude_n and_o whether_o it_o measure_v or_o no._n as_o suppose_v the_o line_n ab_fw-la to_o be_v 17._o and_o let_v it_o be_v divide_v into_o two_o part_n in_o the_o point_n c_o namely_o into_o the_o line_n ac_fw-la &_o the_o line_n cb_o and_o let_v the_o line_n ac_fw-la the_o great_a part_n contain_v 12._o and_o let_v the_o line_n bc_o the_o less_o part_n contain_v 5._o now_o either_o of_o these_o line_n by_o this_o definition_n be_v a_o part_n of_o the_o whole_a line_n ab_fw-la for_o either_o of_o they_o be_v a_o less_o magnitude_n or_o quantity_n in_o comparison_n of_o the_o whole_a line_n ab_fw-la but_o neither_o of_o they_o measure_v the_o whole_a line_n ab_fw-la for_o the_o less_o line_n cb_o contain_v 5._o take_v as_o often_o as_o you_o listen_v will_v never_o make_v precise_o ab_fw-la which_o contain_v 17._o if_o you_o take_v it_o 3._o time_n it_o make_v only_o 15._o so_o lack_v it_o 2._o of_o 17._o which_o be_v to_o little_a if_o you_o take_v it_o 4._o time_n so_o make_v it_o 20._o them_z be_v there_o three_o to_o much_o so_o it_o never_o make_v precise_o 17._o but_o either_o to_o much_o or_o to_o little_a likewise_o the_o other_o part_n ac_fw-la measure_v not_o the_o whole_a line_n ab_fw-la for_o take_v once_o it_o make_v but_o 12._o which_o be_v less_o than_o 17._o and_o take_v twice_o it_o make_v 24._o which_o be_v more_o than_o 17._o by_o â_o so_o it_o never_o precise_o make_v by_o take_v thereof_o the_o whole_a ab_fw-la but_o either_o more_o or_o less_o and_o this_o kind_n of_o part_n they_o common_o call_v pars_fw-la
eglantine_n the_o side_n which_o include_v the_o equal_a angle_n be_v proportional_a wherefore_o the_o parallelogram_n abcd_o be_v by_o the_o first_o definition_n of_o the_o six_o like_v unto_o the_o parallelogram_n eglantine_n and_o by_o the_o same_o reason_n also_o the_o parallelogram_n abcd_o be_v like_a to_o the_o parallelogram_n kh_o abcd_o wherefore_o either_o of_o these_o parallelogram_n eglantine_n and_o kh_o be_v like_a unto_o the_o parallelogram_n abcd._n but_o rectiline_a figure_n which_o be_v like_a to_o one_o and_o the_o same_o rectiline_a figure_n be_v also_o by_o the_o 21._o of_o the_o six_o like_o the_o one_o to_o the_o other_o wherefore_o the_o parallelogram_n eglantine_n be_v like_a to_o the_o parallelogram_n hk_o other_o wherefore_o in_o every_o parallelogram_n the_o parallelogram_n about_o the_o dimecient_a be_v like_a unto_o the_o whole_a and_o also_o like_o the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o more_o brief_a demonstration_n after_o flussates_n suppose_v that_o there_o be_v a_o parallelogram_n abcd_o who_o dimetient_a let_v bâ_z aâ_z flussates_n about_o which_o let_v consist_v these_o parallelogram_n eke_o and_o ti_o have_v the_o angle_n at_o the_o point_n â_o and_o ãâ¦ã_o with_o the_o whole_a parallelogram_n abcd._n then_o i_o say_v that_o those_o parallelogram_n eke_o and_o ti_fw-mi be_v like_a to_o the_o whole_a parallelogram_n db_o and_o also_o alâ_z like_o the_o one_o to_o the_o other_o for_o forasmuch_o as_o bd_o eke_o and_o ti_fw-mi be_v parallelogram_n therefore_o the_o right_a line_n azg_v fall_v upon_o these_o parallel_a line_n aeb_fw-mi kzt_n and_o di_z g_z or_o upon_o these_o parallel_a line_n akd_v ezi_n and_o btg_n make_v these_o angle_n equal_a the_o one_o to_o the_o other_o namely_o the_o angle_n eaz_n to_o the_o angle_n kza_n &_o the_o angle_n eza_n to_o the_o angle_n kaz_n and_o the_o angle_n tzg_v to_o the_o angle_n zgi_n and_o the_o angle_n tgz_n to_o the_o angle_n izg_v and_o the_o angle_n bag_n to_o the_o angle_n agd_v and_o final_o the_o angle_n bga_n to_o the_o angle_n dag_n wherefore_o by_o the_o first_o corollary_n of_o the_o 32._o of_o the_o first_o and_o by_o the_o 34._o of_o the_o first_o the_o angle_n remain_v be_v equal_v the_o one_o to_o the_o other_o namely_o the_o angle_n b_o to_o the_o angle_n d_o and_o the_o angle_n e_o to_o the_o angle_n king_n and_o the_o angle_n t_o to_o the_o angle_n i_o wherefore_o these_o triangle_n be_v equiangle_n and_o therefore_o like_o the_o one_o to_o the_o other_o namely_o the_o triangle_n abg_o to_o the_o triangle_n gda_n and_o the_o triangle_n aez_n to_o the_o triangle_n zka_n &_o the_o triangle_n ztg_n to_o the_o triangle_n giz._n wherefore_o as_o the_o side_n ab_fw-la be_v to_o the_o side_n bg_o so_o be_v the_o side_n ae_n to_o the_o side_n ez_fw-fr and_o the_o side_n zt_n to_o the_o side_n tg_n wherefore_o the_o parallelogram_n contain_v under_o those_o right_a line_n namely_o the_o parallelogram_n abgd_v eke_o &_o ti_fw-mi be_v like_o the_o one_o to_o the_o other_o by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o every_o parallelogram_n the_o parallelogram_n etc_n etc_n as_o before_o which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o problem_n add_v by_o pelitarius_n two_o equiangle_n parallelogrammes_n be_v give_v so_o that_o they_o be_v not_o like_a to_o cut_v of_o from_o one_o of_o they_o a_o parallelogram_n like_a unto_o the_o other_o pelitarius_n suppose_v that_o the_o two_o equiangle_n parallelogram_n be_v abcd_o and_o cefg_n which_o let_v not_o be_v like_o the_o one_o to_o the_o other_o it_o be_v require_v from_o the_o parallelogram_n abcd_o to_o cut_v of_o a_o parallelogram_n like_a unto_o the_o parallelogram_n cefg_n let_v the_o angle_n c_o of_o the_o one_o be_v equal_a to_o the_o angle_n c_o of_o the_o other_o and_o let_v the_o two_o parallelogram_n be_v so_o ãâã_d that_o the_o line_n bc_o &_o cg_o may_v make_v both_o one_o right_a line_n namely_o bg_o wherefore_o also_o the_o right_a line_n dc_o and_o ce_fw-fr shall_v both_o make_v one_o right_a line_n namely_o de._n and_o draw_v a_o line_n from_o the_o point_n f_o to_o the_o point_n c_o and_o produce_v the_o line_n fc_o till_o it_o concur_v with_o the_o line_n ad_fw-la in_o the_o point_n h._n and_o draw_v the_o line_n hk_o parallel_v to_o the_o line_n cd_o by_o the_o 31._o of_o the_o first_o then_o i_o say_v that_o from_o the_o parallelogram_n ac_fw-la be_v cut_v of_o the_o parallelogram_n cdhk_n like_v unto_o the_o parallelogram_n eglantine_n which_o thing_n be_v manifest_a by_o this_o 24._o proposition_n for_o that_o both_o the_o say_v parallelogram_n be_v describe_v about_o one_o &_o the_o self_n same_o dimetient_fw-la and_o to_o the_o end_n it_o may_v the_o more_o plain_o be_v see_v i_o have_v make_v complete_a the_o parallelogram_n abgl_n ¶_o a_o other_o problem_n add_v by_o pelitarius_n between_o two_o rectiline_a superficiece_n to_o find_v out_o a_o mean_a superficies_n proportional_a pelitarius_n suppose_v that_o the_o two_o superficiece_n be_v a_o and_o b_o between_o which_o it_o be_v require_v to_o place_v a_o mean_a superficies_n proportional_a reduce_v the_o say_v two_o rectiline_a figure_n a_o and_o b_o unto_o two_o like_a parallelogram_n by_o the_o 18._o of_o this_o book_n or_o if_o you_o think_v good_a reduce_v either_o of_o they_o to_o a_o square_a by_o the_o last_o of_o the_o second_o and_o let_v the_o say_v two_o parallelogram_n like_o the_o one_o to_o the_o other_o and_o equal_a to_o the_o superficiece_n a_o and_o b_o be_v cdef_n and_o fghk_n and_o let_v the_o angle_n fletcher_n in_o either_o of_o they_o be_v equal_a which_o two_o angle_n let_v be_v place_v in_o such_o sort_n that_o the_o two_o parallelogram_n ed_z and_o hg_o may_v be_v about_o one_o and_o the_o self_n same_o dimetient_fw-la ck_o which_o be_v do_v by_o put_v the_o right_a line_n of_o and_o fg_o in_o such_o sort_n that_o they_o both_o make_v one_o right_a line_n namely_o eglantine_n and_o make_v complete_a the_o parallelogram_n clk_n m._n then_o i_o say_v that_o either_o of_o the_o supplement_n fl_fw-mi &_o fm_o be_v a_o mean_a proportional_a between_o the_o superficiece_n cf_o &_o fk_o that_o be_v between_o the_o superficiece_n a_o and_o b_o namely_o as_o the_o superficies_n hg_o be_v to_o the_o superficies_n fl_fw-mi so_o be_v the_o same_o superficies_n fl_fw-mi to_o the_o superficies_n ed._n for_o by_o this_o 24._o proposition_n the_o line_n hf_o be_v to_o the_o line_n fd_o as_o the_o line_n gf_o be_v to_o the_o line_n fe_o but_o by_o the_o first_o of_o this_o book_n as_o the_o line_n hf_o be_v to_o the_o line_n fd_o so_o be_v the_o superficies_n hg_o to_o the_o superficies_n fl_fw-mi and_o as_o the_o line_n gf_o be_v to_o the_o line_n fe_o so_o also_o by_o the_o same_o be_v the_o superficies_n fl_fw-mi to_o the_o superficies_n ed._n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o superficies_n hg_o be_v to_o the_o superficies_n fl_fw-mi so_o be_v the_o same_o superficies_n fl_fw-mi to_o the_o superficies_n ed_z which_o be_v require_v to_o be_v do_v the_o 7._o problem_n the_o 25._o proposition_n unto_o a_o rectiline_a figure_n give_v to_o describe_v a_o other_o figure_n like_o which_o shall_v also_o be_v equal_a unto_o a_o other_o rectiline_a figure_n give_v svppose_v that_o the_o rectiline_a figure_n give_v whereunto_o be_v require_v a_o other_o to_o be_v make_v like_o be_v abc_n and_o let_v the_o other_o rectiline_a figure_n whereunto_o the_o same_o be_v require_v to_o be_v make_v equal_a be_v d._n now_o it_o be_v require_v to_o describe_v a_o rectiline_a figure_n like_o unto_o the_o figure_n abc_n and_o equal_a unto_o the_o figure_n d._n construction_n upon_o the_o line_n bc_o describe_v by_o the_o 44._o of_o the_o first_o a_o parallelogram_n be_v equal_a unto_o the_o triangle_n abc_n and_o by_o the_o same_o upon_o the_o line_n ce_fw-fr describe_v the_o parallelogram_n c_o m_o equal_a unto_o the_o rectiline_a figure_n d_o and_o in_o the_o say_a parallelogram_n let_v the_o angle_n fce_n be_v equal_a unto_o the_o angle_n cbl_n and_o forasmuch_o as_o the_o angle_n fce_n be_v by_o construction_n equal_a to_o the_o angle_n cbl_n demonstration_n add_v the_o angle_n be_v common_a to_o they_o both_o wherefore_o the_o angle_n lbc_n and_o be_v be_v equal_a unto_o the_o angle_n be_v and_o ecf_n but_o the_o angle_n lbc_n and_o be_v be_v equal_a to_o two_o right_a angle_n by_o the_o 29._o of_o the_o first_o wherefore_o also_o the_o angle_n be_v and_o ecf_n be_v equal_a to_o two_o right_a angle_n wherefore_o the_o line_n bc_o and_o cf_o by_o the_o 14._o of_o the_o first_o make_v both_o one_o right_a line_n namely_o bf_o and_o in_o like_a sort_n do_v the_o line_n le_fw-fr and_o em_n make_v both_o one_o right_a line_n namely_o lm_o then_o by_o the_o
namely_o the_o circumference_n blakoc_n be_v equal_a unto_o the_o circumference_n remain_v of_o the_o self_n same_o circle_n abc_n namely_o to_o the_o circumference_n cpblak_n wherefore_o the_o angle_n bpc_n be_v equal_a unto_o the_o angle_n cok_n by_o the_o 27._o of_o the_o three_o wherefore_o by_o the_o 10._o definition_n of_o the_o three_o the_o segment_n bpc_n be_v like_a unto_o the_o segment_n cok_n and_o they_o be_v set_v upon_o equal_a right_a line_n bc_o and_o kc_o but_o like_o segment_n of_o circle_n which_o consist_v upon_o equal_a right_a line_n be_v also_o equal_a the_o one_o to_o the_o other_o by_o the_o 24._o of_o the_o three_o wherefore_o the_o segment_n bpc_n be_v equal_a unto_o the_o segment_n cok._n and_o the_o triangle_n gbc_n be_v equal_a unto_o the_o triangle_n gck._n wherefore_o the_o sector_n gbc_n be_v equal_a unto_o the_o sector_n gck._n and_o by_o the_o same_o reason_n also_o the_o sector_n gkl_n be_v equal_a unto_o either_o of_o the_o sector_n gbc_n and_o gck._n wherefore_o the_o three_o sector_n gbc_n and_o gck_n and_o gkl_n be_v equal_a the_o one_o to_o the_o outstretch_o and_o by_o the_o same_o reason_n also_o the_o sector_n hef_n and_o hfm_n and_o hmn_v be_v equal_a the_o one_o to_o the_o other_o wherefore_o how_o multiplex_n the_o circumference_n bl_o be_v to_o the_o circumference_n bc_o so_o multiplex_n be_v the_o sector_n glb_n to_o the_o sector_n gbc_n and_o by_o the_o same_o reason_n how_o multiplex_n the_o circumference_n ne_fw-fr be_v to_o the_o circumference_n of_o so_o multiplex_n be_v the_o sector_n hen_n to_o the_o sector_n hef_fw-fr if_o therefore_o the_o circumference_n bl_o be_v equal_a unto_o the_o circumference_n en_fw-fr the_o sector_n also_o bgl_n be_v equal_a unto_o the_o sector_n ehn._n and_o if_o the_o circumference_n bl_o exceed_v the_o circumference_n en_fw-fr the_o sector_n also_o bgl_n exceed_v the_o sector_n ehn._n and_o if_o the_o circumference_n be_v less_o the_o sector_n also_o be_v less_o now_o they_o there_o be_v four_o magnitude_n namely_o the_o two_o circumference_n bc_o and_o of_o and_o the_o two_o sector_n gbc_n &_o hef_n and_o to_o the_o circumference_n bc_o &_o to_o the_o sector_n gbc_n namely_o to_o the_o first_o and_o the_o three_o be_v take_v equemultiplices_fw-la that_o be_v the_o circumference_n bl_o and_o the_o sector_n gbl_n and_o likewise_o to_o the_o circumference_n of_o and_o to_o the_o sector_n hef_fw-fr namely_o to_o the_o second_o and_o four_o be_v take_v certain_a other_o equimultiplice_n namely_o the_o circumference_n en_fw-fr and_o the_o sector_n hen._n and_o it_o be_v prove_v that_o if_o the_o circumference_n bl_o exceed_v the_o circumference_n en_fw-fr the_o sector_n also_o bgl_n exceed_v the_o sector_n ehn._n and_o if_o the_o circumference_n be_v equal_a the_o segment_n also_o be_v equal_a and_o if_o the_o circumference_n be_v less_o the_o segment_n also_o be_v less_o wherefore_o by_o the_o conversion_n of_o the_o six_o definition_n of_o the_o five_o as_o the_o circumference_n bc_o be_v to_o the_o circumference_n of_o so_o be_v the_o sector_n gbc_n unto_o the_o sector_n hef_fw-fr which_o be_v all_o that_o be_v require_v to_o be_v prove_v corollary_n and_o hereby_o it_o be_v manifest_a that_o as_o the_o sector_n be_v to_o the_o sector_n so_o be_v angle_n to_o angle_v by_o the_o 11._o of_o the_o five_o flussates_n here_o add_v five_o proposition_n whereof_o one_o be_v a_o problem_n have_v three_o corollarye_n follow_v of_o it_o and_o the_o rest_n be_v theorem_n which_o for_o that_o they_o be_v both_o witty_a &_o also_o serve_v to_o great_a use_n as_o we_o shall_v afterward_o see_v i_o think_v not_o good_a to_o omit_v but_o have_v here_o place_v they_o but_o only_o that_o i_o have_v not_o put_v they_o to_o follow_v in_o order_n with_o the_o proposition_n of_o euclid_n as_o he_o have_v do_v ¶_o the_o first_o proposition_n add_v by_o flussates_n to_o describe_v two_o rectiline_a figure_n equal_a and_o like_a unto_o a_o rectiline_a figure_n give_v and_o in_o like_a sort_n situate_a which_o shall_v have_v also_o a_o proportion_n give_v suppose_v that_o the_o rectiline_a figure_n give_v be_v abh_n problem_n and_o let_v the_o proportion_n give_v be_v the_o proportion_n of_o the_o line_n gc_o and_o cd_o and_o by_o the_o 10._o of_o this_o book_n divide_v the_o line_n ab_fw-la like_a unto_o the_o line_n gd_o in_o the_o point_n e_o so_o that_o as_o the_o line_n gc_o be_v to_o the_o line_n cd_o so_o let_v the_o line_n ae_n be_v to_o the_o line_n ebb_n and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n afb_o and_o from_o the_o point_n e_o erect_v by_o the_o 11._o of_o the_o first_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n of_o cut_v the_o circumference_n in_o the_o point_n f._n and_o draw_v these_o line_n of_o and_o fb_o and_o upon_o either_o of_o these_o line_n describe_v rectiline_a figure_n like_o unto_o the_o rectiline_a figure_n ahb_n and_o in_o like_a sort_n situate_a by_o the_o 18._o of_o the_o six_o which_o let_v be_v akf_n &_o fib_n then_o i_o say_v that_o the_o rectiline_a figure_n akf_a and_o fib_n have_v the_o proportion_n give_v namely_o the_o proportion_n of_o the_o line_n gc_o to_o the_o line_n cd_o and_o be_v equal_a to_o the_o rectiline_a figure_n give_v abh_n unto_o which_o they_o be_v describe_v like_o and_o in_o like_a sort_n situate_a for_o forasmuch_o as_o afb_o be_v a_o semicircle_n therefore_o the_o angle_n afb_o be_v a_o right_a angle_n by_o the_o 31._o of_o the_o three_o and_o fe_o be_v a_o perpendicular_a line_n same_o wherefore_o by_o the_o 8._o of_o this_o book_n the_o triangle_n afe_a and_o fbe_n be_v like_a both_o to_o the_o whole_a triangle_n afb_o and_o also_o the_o one_o to_o the_o other_o wherefore_o by_o the_o 4._o of_o this_o book_n as_o the_o line_n of_o be_v to_o the_o line_n fb_o so_o be_v the_o line_n ae_n to_o the_o line_n of_o and_o the_o line_n of_o to_o the_o line_n ebb_n which_o be_v side_n contain_v equal_a angle_n wherefore_o by_o 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number_n even_o even_o be_v that_o which_o may_v be_v divide_v into_o two_o even_a part_n and_o that_o part_n again_o into_o two_o even_a part_n and_o so_o continual_o devide_v without_o stay_n ãâã_d come_v to_o unity_n as_o by_o exampleâ_n 64._o may_v be_v divide_v into_o 31_o and._n 32._o and_o either_o of_o these_o part_n may_v be_v divide_v into_o two_o even_a part_n for_o 32_o may_v be_v divide_v into_o 16_o and_o 16._o again_o 16_o may_v be_v divide_v into_o 8_o and_o 8_o which_o be_v even_o part_n and_o 8_o into_o 4_o and_o 4._o again_o 4_o into_o â_o and_o â_o and_o last_o of_o all_o may_v â_o be_v divide_v into_o one_o and_o one_o 9_o a_o number_n even_o odd_a call_v in_o latin_a pariter_fw-la impar_fw-la be_v that_o which_o a_o even_a number_n measure_v by_o a_o odd_a number_n definition_n as_o the_o number_n 6_o which_o 2_o a_o even_a number_n measure_v by_o 3_o a_o odd_a number_n three_o time_n 2_o be_v 6._o likewise_o 10._o which_o 2._o a_o even_a number_n measure_v by_o 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no_o part_n to_o measure_v they_o but_o only_a unity_n definition_n 13_o number_n prime_a the_o one_o to_o the_o other_o be_v they_o which_o only_a unity_n do_v measure_n be_v a_o common_a measure_n to_o they_o as_o 15._o and_o 22._o be_v number_n prime_a the_o one_o to_o the_o other_o .15_o of_o itself_o be_v no_o prime_a number_n for_o not_o only_a unity_n do_v measure_v it_o but_o also_o the_o numbeââ_n 5._o and_o 3_o for_o â_o time_n 5._o be_v lx_o likewise_o 22._o be_v of_o itself_o no_o prime_a number_n for_o it_o be_v measure_v by_o 2._o and_o 11_o beside_o unity_n for_o 11._o twice_o or_o 2._o eleven_o time_n make_v 22._o so_o that_o although_o neither_o of_o these_o two_o number_n 15._o and_o 22._o be_v a_o prime_n or_o incomposed_a number_n but_o either_o have_v part_n of_o his_o own_o whereby_o it_o may_v be_v measure_v beside_o unity_n yet_o compare_v together_o they_o be_v prime_a the_o one_o to_o the_o other_o for_o no_o one_o number_n do_v as_o a_o common_a measure_n 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be_v they_o which_o some_o one_o number_n be_v a_o common_a measure_n to_o they_o both_o do_v measure_n definition_n as_o 12._o and_o 8._o be_v two_o compose_a number_n the_o one_o to_o the_o other_o for_o the_o number_n 4_o be_v a_o common_a measure_n to_o they_o both_o 4._o take_v three_o time_n make_v 12_o and_o the_o same_o 4._o take_v two_o time_n make_v 8._o so_o be_v 9_o and_o 15_o 3._o measure_v they_o both_o also_o 10._o and_o 25_o for_o 5._o measure_v both_o of_o they_o and_o so_o infinite_o of_o other_o in_o this_o do_v number_n compose_v the_o one_o to_o the_o other_o or_o second_o number_n differre_fw-la from_o number_n prime_a the_o one_o to_o the_o other_o for_o that_o two_o number_n be_v compose_v the_o one_o to_o the_o other_o each_o of_o they_o several_o be_v of_o necessity_n a_o compose_a number_n as_o in_o the_o example_n before_o 8._o and_o 12._o be_v compose_v number_n likewise_o 9_o and_o 15_o also_o 10_o and_o 25_o but_o if_o they_o be_v two_o number_n prime_a the_o one_o to_o the_o other_o âit_n be_v not_o of_o necessity_n that_o each_o of_o they_o several_o be_v a_o prime_a number_n as_o 9_o and_o 22._o be_v two_o number_n prime_a the_o one_o to_o the_o other_o no_o one_o number_n measure_v both_o of_o they_o and_o yet_o neither_o of_o they_o in_o itself_o and_o in_o his_o own_o nature_n be_v a_o prime_a number_n but_o each_o of_o they_o be_v a_o compose_a number_n for_o 3._o measure_v 9_o and_o 11._o and_o 2._o measure_n 22._o 16_o a_o number_n be_v say_v to_o multiply_v a_o number_n when_o the_o number_n multiply_v definition_n be_v so_o oftentimes_o add_v to_o itself_o as_o there_o be_v in_o the_o number_n multiply_v unity_n and_o a_o other_o number_n be_v produce_v in_o multiplication_n be_v ever_o require_v two_o number_n the_o one_o be_v whereby_o you_o multiply_v common_o call_v the_o multiply_o or_o multiplicant_fw-la the_o other_o be_v that_o which_o be_v multiply_v multiplication_n the_o number_n by_o which_o a_o other_o be_v multiply_v namely_o the_o multiplyer_n be_v say_v to_o multiply_v as_o if_o you_o will_v multiply_v 4._o by_o 3_o then_o be_v three_o say_v to_o multiply_v 4_o therefore_o according_a to_o this_o definition_n because_o in_o 3._o there_o be_v three_o unity_n add_v 4,3_o time_n to_o itself_o say_v 3._o time_n 4_o so_o shall_v you_o bring_v forth_o a_o other_o number_n namely_o 12_o which_o be_v the_o sum_n produce_v of_o that_o multiplication_n and_o so_o of_o all_o other_o multiplication_n 17_o when_o two_o number_n multiply_v themselves_o the_o one_o the_o other_o produce_v a_o other_o definition_n the_o number_n produce_v be_v call_v a_o plain_a or_o superficial_a number_n and_o the_o number_n which_o muliply_v themselves_o the_o one_o by_o the_o other_o be_v the_o side_n of_o that_o number_n as_o let_v these_o two_o number_n 3._o and_o 6_o multiply_v the_o one_o the_o other_o say_v 3._o time_n 6_o or_o six_o time_n 3_o they_o shall_v produce_v 18._o this_o number_n 18._o thus_o produce_v be_v call_v a_o plain_a number_n or_o a_o superficial_a number_n and_o the_o two_o multiply_a number_n which_o produce_v 12_o namely_o 3._o and_o 6_o be_v the_o side_n of_o the_o same_o superficial_a or_o plain_a number_n that_o be_v the_o length_n and_o breadth_n thereof_o likewise_o if_o 9_o multiply_v 11_o or_o eleven_o nine_o there_o shall_v be_v produce_v 99_o a_o plain_a number_n who_o side_n be_v the_o two_o number_n 9_o and_o 11_o â_o as_o the_o length_n and_o breadth_n of_o the_o same_o they_o be_v call_v plain_a and_o superficial_a number_n number_n because_o be_v describe_v by_o their_o unity_n on_o a_o plain_a superficies_n they_o represent_v some_o superficial_a form_n or_o figure_n geometrical_a have_v length_n and_o breadth_n as_o you_o see_v of_o this_o example_n and_o so_o of_o other_o and_o all_o such_o plain_a or_o superficial_a number_n do_v ever_o represent_v right_o angle_v figure_n as_o appear_v in_o the_o example_n definition_n 18_o when_o three_o number_n multiply_v together_o the_o one_o into_o the_o other_o produce_v any_o number_n the_o number_n produce_v be_v call_v a_o solid_a number_n and_o the_o number_n multiply_v themselves_o the_o one_o into_o the_o outstretch_o be_v the_o side_n thereof_o as_o take_v these_o three_o number_n 3.4_o 5._o multiply_v the_o one_o into_o the_o other_o first_o 4._o into_o 5._o say_n four_o time_n 5._o be_v 20_o then_o multiply_v that_o number_n produce_v name_o 20._o into_o 3_o which_o be_v the_o three_o number_n so_o shall_v you_o produce_v 60._o which_o be_v a_o solid_a number_n and_o the_o three_o number_n which_o produce_v the_o number_n namely_o 3.4_o and_o 5._o be_v the_o side_n of_o the_o same_o and_o they_o be_v call_v solid_a number_n number_n because_o be_v describe_v by_o their_o unity_n they_o represent_v solid_a and_o bodylicke_a figure_n of_o geometry_n which_o have_v length_n breadth_n and_o thickness_n as_o you_o see_v this_o number_n 60._o express_v here_o by_o his_o unity_n who_o length_n be_v his_o side_n 5_o his_o breadth_n be_v 3_o and_o thickness_n 4._o and_o thus_o may_v you_o do_v of_o all_o other_o three_o number_n multiply_v the_o one_o the_o other_o definition_n 19_o a_o square_a number_n be_v that_o which_o be_v equal_o equal_a or_o that_o which_o be_v contain_v under_o two_o equal_a number_n as_o multiply_v two_o equal_a number_n the_o one_o into_o the_o other_o as_o 9_o by_o 9_o you_o shall_v produce_v 81_o which_o be_v a_o square_a number_n euclid_n call_v it_o a_o number_n equal_o equal_a because_o it_o be_v produce_v of_o the_o multiplication_n of_o two_o equal_a number_n the_o one_o into_o the_o other_o which_o number_n be_v also_o say_v in_o the_o second_o definition_n to_o contain_v a_o square_a number_n as_o in_o the_o definition_n of_o the_o second_o book_n two_o line_n be_v say_v to_o contain_v a_o square_a or_o a_o parallelogram_n figure_n it_o be_v call_v a_o square_a number_n number_n because_o be_v describe_v by_o his_o unity_n it_o represent_v the_o figure_n of_o a_o square_a in_o geometry_n as_o you_o here_o see_v do_v the_o number_n 81._o who_o side_n that_o be_v to_o say_v who_o length_n and_o breadth_n be_v 9_o and_o 9_o equal_a number_n which_o also_o be_v say_v to_o contain_v the_o square_a number_n 81_o and_o so_o of_o other_o definition_n 20_o a_o cube_fw-la number_n be_v that_o which_o be_v equal_o equal_a equal_o or_o that_o which_o be_v contain_v under_o three_o equal_a number_n as_o multiply_v three_o equal_a number_n the_o one_o into_o the_o other_o as_z 9_z 9_z and_o 9_z first_o 9_o by_o 9_o so_o shall_v you_o have_v 81_o which_o again_o multiply_v by_o 9_o so_o shall_v you_o produce_v 729._o which_o be_v a_o cube_fw-la number_n and_o euclid_n call_v
denomination_n of_o the_o number_n b._n for_o how_o often_o b_o measure_v a_o so_o many_o unity_n let_v there_o be_v in_o c._n and_o let_v d_o be_v unity_n and_o forasmuch_o as_o b_o measure_v a_o by_o those_o unity_n which_o be_v in_o c_o and_o unity_n d_z measure_v c_o by_o those_o unity_n which_o be_v in_o c_o therefore_o unity_n d_z so_o many_o time_n measure_v the_o number_n c_o as_z b_o do_v measure_n a._n demonstration_n wherefore_o alternate_o by_o the_o 15._o of_o the_o seven_o unity_n d_o so_o many_o time_n measure_v b_o as_z c_z do_v measure_n a._n wherefore_o what_o part_n unity_n d_z be_v of_o the_o number_n b_o the_o same_o part_n be_v c_o of_o a._n but_o unity_n d_o be_v a_o part_n of_o b_o have_v his_o denomination_n of_o b._n wherefore_o c_o also_o be_v a_o part_n of_o a_o have_v his_o denomination_n of_o b._n wherefore_o a_o have_v c_z as_o a_o part_n take_v his_o denomination_n of_o b_o which_o be_v require_v to_o be_v prove_v the_o meaning_n of_o this_o proposition_n be_v that_o if_o three_o measure_n any_o number_n that_o number_n have_v a_o three_o part_n and_o if_o four_o measure_n any_o number_n the_o say_a number_n have_v a_o four_o part_n and_o so_o forth_o ¶_o the_o 35._o theorem_a the_o 40._o proposition_n if_o a_o number_n have_v any_o part_n the_o number_n whereof_o the_o part_n take_v his_o denomination_n shall_v measure_v it_o svppose_v that_o the_o number_n a_o have_v a_o part_n namely_o b_o and_o let_v the_o part_n b_o have_v his_o denomination_n of_o the_o number_n c._n then_o i_o say_v that_z c_z measure_v a._n let_v d_o be_v unity_n and_o forasmuch_o as_o b_o be_v a_o part_n of_o a_o have_v his_o denomination_n of_o c_o proposition_n and_o d_z be_v unity_n be_v also_o a_o part_n of_o the_o number_n c_o have_v his_o denomination_n of_o c_o therefore_o what_o part_n unity_n d_z be_v of_o the_o number_n c_o the_o same_o part_n be_v also_o b_o of_o a_o wherefore_o unity_n d_z so_o many_o âââes_n measure_v the_o number_n c_o demonstration_n as_z b_o measurâââ_n a._n wherefore_o alternate_o by_o ââe_z 15._o âf_n the_o sââânth_n unity_n d_z so_o many_o ââmes_n mââââreth_v the_o nuââbeâ_n b_o ãâã_d c_o measââeth_fw-mi a._n wherefâââ_n c_o measure_v a_o which_o be_v requiââd_v to_o bâ_z prove_v this_o proposition_n be_v the_o converse_n of_o the_o former_a and_o the_o meaning_n thereof_o be_v that_o every_o number_n have_v a_o three_o part_n be_v measure_v of_o three_o and_o have_v a_o four_o part_n be_v measure_v of_o four_o and_o so_o forth_o ¶_o the_o 6._o problem_n the_o 41._o proposition_n to_o find_v out_o the_o least_o number_n that_o contain_v the_o part_n give_v svppose_v that_o the_o part_n give_v be_v a_o b_o c_o namely_o let_v a_o be_v a_o half_a part_n b_o a_o three_o part_n &_o c_o a_fw-fr four_o part_n construction_n now_o it_o be_v require_v to_o find_v out_o the_o least_o number_n which_o contain_v the_o part_n a_o b_o c._n let_v the_o say_a part_n a_o b_o c_o have_v their_o denomination_n of_o the_o number_n d_o e_o f._n and_o take_v by_o the_o 38._o of_o the_o seven_o âhe_z least_o number_n which_o the_o number_n d_o e_o f_o measure_n and_o let_v the_o same_o be_v g._n and_o forasmuch_o as_o the_o number_n d_o e_o f_o measure_v the_o number_n g_o therefore_o the_o number_n g_z have_v payte_n denominate_v of_o the_o number_n d_o e_o f_o by_o the_o 39_o of_o the_o seven_o but_o the_o part_n a_o b_o c_o have_v their_o denomination_n of_o the_o number_n d_o e_o f._n wherefore_o g_z have_v those_o part_n a_o b_o c._n i_o say_v also_o that_o it_o be_v the_o least_o number_n which_o have_v these_o part_n absurdity_n for_o if_o g_z be_v not_o the_o least_o number_n which_o contain_v those_o part_n a_o b_o c_o then_o let_v there_o be_v some_o number_n less_o than_o g_o which_o contain_v the_o say_a part_n a_o b_o c._n and_o suppose_v the_o same_o to_o be_v the_o number_n h._n and_o forasmuch_o as_o h_o have_v the_o say_a part_n a_o b_o c_o therefore_o the_o number_n that_o the_o part_n a_o b_o c_o take_v their_o denomination_n of_o shall_v measure_v h_o by_o the_o 40._o of_o the_o seven_o but_o the_o number_n wheroâ_n the_o part_n a_o b_o c_o take_v their_o denomination_n of_o be_v d_z e_o f._n wherefore_o the_o number_n d_o e_o f_o measure_n thâ_z number_n h_o which_o be_v less_o than_o g_o which_o be_v impossible_a for_o g_z be_v suppose_v to_o be_v tââ_n lââsâ_n number_n that_o the_o number_n d_o e_o f_o do_v measure_n wherefore_o there_o be_v no_o number_n less_o than_o g_o which_o contain_v these_o part_n a_o b_o c_o which_o be_v require_v to_o be_v do_v corrolary_n campane_n hereby_o it_o be_v manifest_a that_o if_o there_o be_v take_v the_o least_o number_n that_o number_n how_o many_o soever_o do_v measure_n the_o say_a number_n shall_v be_v the_o least_o which_o have_v the_o part_n denominate_v of_o the_o say_a number_n how_o many_o soever_o campane_n after_o he_o have_v teach_v to_o find_v out_o the_o first_o lest_o number_n that_o contain_v the_o part_n give_v teach_v also_o to_o find_v out_o the_o second_o lest_o number_n infinite_o that_o be_v which_o except_o the_o least_o of_o all_o be_v less_o than_o all_o other_o and_o also_o the_o three_o least_o and_o the_o four_o etc_n etc_n the_o second_o be_v find_v out_o by_o double_v the_o number_n g._n for_o the_o number_n which_o measure_v the_o number_n gâhall_v âhall_z also_o measure_v the_o double_a thereof_o by_o the_o 5._o commoâ_n sentence_n of_o the_o seven_o but_o there_o can_v be_v give_v a_o number_n great_a than_o the_o number_n g_o &_o less_o than_o the_o double_a thereof_o who_o the_o part_n give_v shall_v ââasureâ_n for_o forasmuch_o as_o the_o part_n give_v do_v âeasurâ_n the_o whole_a namely_o which_o be_v less_o than_o the_o double_a and_o they_o also_o measure_v the_o part_n take_v away_o namely_o the_o number_n g_o they_o shall_v also_o measure_v the_o residue_n namely_o a_o number_n less_o than_o g_o which_o be_v prove_v to_o be_v the_o least_o number_n that_o they_o do_v measure_n which_o be_v impossibleâ_n wherefore_o the_o second_o number_n which_o the_o say_v part_n give_v do_v measureâ_n must_v exceed_v g_z needs_o reach_v to_o the_o double_a of_o g_o and_o the_o three_o to_o the_o treble_a and_o the_o four_o to_o the_o quadruple_a and_o so_o infinite_o for_o those_o part_n can_v never_o measure_v any_o number_n less_o than_o the_o number_n g._n by_o this_o proposition_n also_o it_o be_v easy_a to_o find_v out_o the_o least_o number_n contain_v the_o part_n give_v of_o part_n as_o if_o we_o will_v find_v out_o the_o least_o number_n which_o contain_v one_o three_o ãâã_d ââ_o a_o halâe_a part_n part_n and_o one_o four_o part_n of_o a_o three_o part_n reduce_v the_o say_v dââers_n fraction_n into_o simple_a fraction_n by_o the_o common_a ãâã_d of_o reduce_v of_o frâââions_n namely_o the_o ãâã_d of_o a_o halâe_n into_o a_o ãâã_d part_n of_o a_o wholey_n and_o âhe_n four_o of_o thiâd_o into_o a_o twelve_o part_n of_o anâ_z whole_a and_o then_o by_o this_o problem_n search_v out_o the_o least_o number_n which_o contain_v a_o sixâ_o part_n and_o a_o twelve_o part_n and_o so_o have_v you_o do_v the_o end_n of_o the_o seven_o book_n of_o euclides_n element_n ¶_o the_o eighthe_a book_n of_o euclides_n element_n after_o that_o euclid_n have_v in_o the_o seven_o book_n entreat_v of_o the_o propriety_n of_o number_n in_o general_a and_o of_o certain_a kind_n thereof_o more_o special_o and_o of_o prime_n and_o compose_a number_n with_o other_o now_o in_o this_o eight_o book_n he_o prosecute_v far_a and_o find_v out_o and_o demonstrate_v the_o property_n and_o passion_n of_o certain_a other_o kind_n of_o number_n book_n as_o of_o the_o least_o number_n in_o proportion_n and_o how_o such_o may_v be_v find_v out_o infinite_o in_o whatsoever_o proportion_n which_o thing_n be_v both_o delectable_a and_o to_o great_a use_n also_o here_o be_v entreat_v of_o plain_a number_n and_o solid_a and_o of_o their_o side_n and_o proportion_n of_o they_o likewise_o of_o the_o passion_n of_o number_n square_a and_o cube_fw-la and_o of_o the_o nature_n and_o condition_n of_o their_o side_n and_o of_o the_o mean_a proportional_a number_n of_o plain_n solid_a square_a and_o cube_fw-la number_n with_o many_o other_o thing_n very_o requisite_a and_o necessary_a to_o be_v know_v ¶_o the_o first_o theorem_a the_o first_o proposition_n if_o there_o be_v number_n in_o continual_a proportion_n howmanysoever_o and_o if_o their_o extreme_n be_v prime_a the_o one_o to_o the_o other_o they_o be_v the_o least_o of_o all_o number_n that_o have_v one_o and_o the_o same_o proportion_n with_o they_o svppose_v that_o the_o number_n
now_o forasmuch_o as_o d_z measure_v ab_fw-la and_o bc_o it_o also_o measure_v the_o whole_a magnitude_n ac_fw-la and_o it_o measure_v ab_fw-la wherefore_o d_z measure_v these_o magnitude_n ca_n and_o ab_fw-la wherefore_o ca_n &_o ab_fw-la be_v commensurable_a and_o they_o be_v suppose_v to_o be_v incommensurableâ_n which_o be_v impossible_a wherefore_o no_o magnitude_n measure_v ab_fw-la and_o bc._n wherefore_o the_o magnitude_n ab_fw-la and_o bc_o be_v incommensurable_a and_o in_o like_a sort_n may_v they_o be_v prove_v to_o be_v incommensurable_a if_o the_o magnitude_n ac_fw-la be_v suppose_v to_o be_v incommensurable_a unto_o bc._n if_o therefore_o there_o be_v two_o magnitude_n incommensurable_a compose_v the_o whole_a also_o shall_v be_v incommensurable_a unto_o either_o of_o the_o two_o part_n component_a and_o if_o the_o whole_a be_v incommensurable_a to_o one_o of_o the_o part_n component_n those_o first_o magnitude_n shall_v be_v incommensurable_a which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o montaureus_n if_o a_o whole_a magnitude_n be_v incommensurable_a to_o one_o of_o the_o two_o magnitude_n which_o make_v the_o whole_a magnitude_n it_o shall_v also_o be_v incommensurable_a to_o the_o other_o of_o the_o two_o magnitude_n for_o if_o the_o whole_a magnitude_n ac_fw-la be_v incommensurable_a unto_o the_o magnitude_n bc_o then_o by_o the_o 2_o part_n of_o this_o 16._o theorem_o the_o magnitude_n ab_fw-la and_o bc_o shall_v be_v incommensurable_a wherefore_o by_o the_o first_o part_n of_o the_o same_o theorem_a the_o magnitude_n ac_fw-la shall_v be_v incommensurable_a to_o either_o of_o these_o magnitude_n ab_fw-la and_o bc._n this_o corollary_n ãâã_d use_v in_o the_o demonstration_n of_o the_o â3_o theorem_a &_o also_o of_o other_o proposition_n ¶_o a_o assumpt_n if_o upon_o a_o right_a line_n be_v apply_v a_o parallelogram_n want_v in_o figure_n by_o a_o square_n the_o parallelogram_n so_o apply_v be_v equal_a to_o that_o parallelogram_n which_o be_v contain_v under_o the_o segment_n of_o the_o right_a line_n which_o segment_n be_v make_v by_o reason_n of_o that_o application_n this_o assumpt_n i_o before_o add_v as_o a_o corollary_n out_o of_o flussates_n after_o the_o 28._o proposition_n of_o the_o six_o book_n ¶_o the_o 14._o theorem_a the_o 17._o proposition_n if_o there_o be_v two_o right_a line_n unequal_a and_o if_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o less_o line_n and_o want_v in_o figure_n by_o a_o square_a if_o also_o the_o parallelogram_n thus_o apply_v divide_v the_o line_n where_o upon_o it_o be_v apply_v into_o part_n commensurable_a in_o length_n then_o shall_v the_o great_a line_n be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o great_a and_o if_o the_o great_a be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o right_a line_n commensurable_a in_o length_n unto_o the_o great_a and_o if_o also_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o less_o line_n and_o want_v in_o figure_n by_o a_o square_n then_o shall_v it_o divide_v the_o great_a line_n into_o part_n commensurable_a but_o now_o suppose_v that_o the_o line_n bc_o be_v in_o power_n more_o than_o the_o line_n a_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n bc._n first_o and_o upon_o the_o line_n bc_o let_v there_o be_v apply_v a_o rectangle_n parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o line_n a_o and_o want_v in_o figure_n by_o a_o square_a and_o let_v the_o say_a parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n bd_o and_o dc_o then_o must_v we_o prove_v that_o the_o line_n bd_o be_v unto_o the_o line_n dc_o commensurable_a in_o length_n the_o same_o construction_n and_o supposition_n that_o be_v before_o remain_v we_o may_v in_o like_a sort_n prove_v that_o the_o line_n bc_o be_v in_o power_n more_o than_o the_o line_n a_o by_o the_o square_n of_o the_o line_n fd._n but_o by_o supposition_n the_o line_n bc_o be_v in_o power_n more_o they_o the_o line_n a_o by_o the_o square_n of_o a_o line_n commensurable_a unto_o it_o in_o length_n wherefore_o the_o line_n bc_o be_v unto_o the_o line_n fd_v commensurable_a in_o length_n wherefore_o the_o line_n compose_v of_o the_o two_o line_n bf_o and_o dc_o be_v commensurable_a in_o length_n unto_o the_o line_n fd_o by_o the_o second_o part_n of_o the_o 15._o of_o the_o ten_o wherefore_o by_o the_o 12._o of_o the_o ten_o or_o by_o the_o first_o part_n of_o the_o 15._o of_o the_o ten_o the_o line_n bc_o be_v commensurable_a in_o length_n to_o the_o line_n compose_v of_o bf_o and_o dc_o but_o the_o whole_a line_n conpose_v bf_o and_o dc_o be_v commensurable_a in_o length_n unto_o dc_o for_o bf_o as_o before_o have_v be_v prove_v be_v equal_a to_o dc_o wherefore_o the_o line_n bc_o be_v commensurable_a in_o length_n unto_o the_o line_n dc_o by_o the_o 12._o of_o the_o ten_o whâââfore_o also_o the_o line_n bd_o be_v commensurable_a in_o length_n unto_o the_o line_n dc_o by_o the_o second_o part_n of_o thâ_z 15._o of_o the_o teâth_n if_o therefore_o there_o be_v two_o right_a line_n unequal_a and_o if_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o less_o and_o want_v in_o figure_n by_o a_o square_a if_o also_o the_o parallelogram_n thus_o apply_v divide_v the_o line_n whereupon_o it_o be_v apply_v into_o part_n commensurable_a in_o length_n then_o shall_v the_o great_a line_n be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o great_a and_o if_o the_o great_a be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o great_a and_o if_o also_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n make_v of_o the_o less_o and_o want_v in_o figure_n by_o a_o square_n then_o shall_v it_o divide_v the_o great_a line_n into_o part_n commensurable_a in_o length_n which_o be_v require_v to_o be_v prove_v campanâ_n after_o this_o proposition_n reach_v how_o we_o may_v ready_o apply_v upon_o the_o line_n bc_o a_o parallelogram_n equal_a to_o the_o four_o part_n of_o the_o square_n of_o half_n of_o the_o line_n a_o and_o want_v in_o figure_n by_o a_o square_a after_o this_o manner_n divide_v the_o line_n bc_o into_o two_o line_n in_o such_o sort_n that_o half_a of_o the_o line_n a_o shall_v be_v the_o mean_a proportional_a between_o those_o two_o line_n which_o be_v possible_a when_o as_o the_o line_n bc_o be_v suppose_v to_o be_v great_a than_o the_o line_n a_o and_o may_v thus_o be_v do_v proposition_n divide_v the_o line_n bc_o into_o two_o equal_a part_n in_o the_o point_n e_o and_o describe_v upon_o the_o line_n bc_o a_o semicircle_n bhc_n and_o unto_o the_o line_n bc_o and_o from_o the_o point_n c_o erect_v a_o perpâdicular_a line_n ck_o and_o put_v the_o line_n ck_o equal_a to_o half_o of_o the_o line_n aâ_z and_o by_o the_o point_n king_n draw_v unto_o the_o line_n ec_o a_o parallel_n line_n kh_o cut_v the_o semicircle_n in_o the_o point_n h_o which_o it_o must_v needs_o cut_v forasmuch_o as_o the_o line_n bc_o be_v great_a than_o the_o line_n a_o and_o from_o the_o point_n h_o draw_v unto_o the_o line_n bc_o a_o perpendicular_a liâe_z hd_v which_o line_n hdâ_n forasmuch_o as_o by_o the_o 34_o of_o the_o first_o it_o be_v equal_a unto_o the_o line_n kc_o shall_v also_o be_v equal_a to_o half_o of_o the_o line_n a_o draw_v the_o line_n bh_o and_o hc_n now_o then_o by_o the_o ââ_o of_o the_o three_o the_o angle_n bhc_n be_v a_o right_a aâgle_n wherefore_o by_o the_o corollary_n of_o the_o eight_o of_o the_o six_o book_n the_o line_n hd_v be_v the_o mean_a proportional_a between_o the_o line_n bd_o and_o dc_o wherefore_o the_o half_a of_o the_o line_n a_o which_o be_v equal_a unto_o the_o line_n hd_v be_v the_o mean_a proportional_a between_o the_o line_n bd_o and_o dc_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n bd_o and_o dc_o be_v equal_a to_o the_o four_o part_n of_o the_o square_n of_o the_o line_n a._n and_o so_o if_o upon_o the_o line_n bd_o be_v describe_v a_o rectangle_n parallelogram_n have_v his_o other_o side_n equal_a to_o the_o line_n dc_o there_o shall_v be_v apply_v upon_o the_o line_n bc_o a_o rectangle_n parallelogram_n equal_a unto_o the_o square_n of_o half_n of_o the_o line_n a_o and_o want_v in_o figure_n by_o
if_o the_o line_n cb_o be_v irrational_a they_o shall_v be_v incommensurable_a but_o as_o the_o line_n bd_o which_o be_v equal_a to_o the_o line_n basilius_n be_v to_o the_o line_n bc_o so_o be_v the_o square_a ad_fw-la to_o the_o parallelogram_n ac_fw-la wherefore_o the_o parallelogram_n ac_fw-la shall_v be_v incommensurable_a to_o the_o square_a ad_fw-la which_o be_v rational_a for_o that_o the_o line_n ab_fw-la whereupon_o it_o be_v describe_v be_v suppose_v to_o be_v rational_a wherefore_o the_o parallelogram_n ac_fw-la which_o be_v contain_v under_o the_o rational_a line_n ab_fw-la and_o the_o irrational_a line_n bc_o be_v irrational_a ¶_o a_o assumpt_n if_o there_o be_v two_o right_a line_n as_o the_o first_o be_v to_o the_o second_o so_o be_v the_o square_n which_o be_v describe_v upon_o the_o first_o to_o the_o parallelogram_n which_o be_v contain_v under_o the_o two_o right_a line_n suppose_v that_o there_o be_v two_o right_a line_n ab_fw-la and_o bc._n book_n then_o i_o say_v that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la ââ_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc._n describe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n ab_fw-la a_o square_a ad._n and_o make_v perfect_a the_o parallelogram_n ac_fw-la now_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o for_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n bd_o so_o be_v the_o square_a ad_fw-la to_o the_o parallelogram_n ca_n by_o the_o first_o of_o the_o sixâ_o and_o ad_fw-la be_v the_o square_n which_o be_v make_v of_o the_o line_n ab_fw-la and_o ac_fw-la be_v that_o which_o be_v contain_v under_o the_o line_n bd_o and_o bc_o that_o be_v under_o the_o line_n ab_fw-la &_o bc_o therefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_n describe_v upon_o the_o the_o line_n ab_fw-la to_o the_o rectangle_n figure_n contain_v under_o the_o line_n ab_fw-la &_o bc._n and_o converse_o as_o the_o parallelogram_n which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v to_o the_o square_n of_o the_o line_n ab_fw-la so_o be_v the_o line_n cb_o to_o the_o line_n basilius_n ¶_o the_o 19_o theorem_a the_o 22._o proposition_n if_o upon_o a_o rational_a line_n be_v apply_v the_o square_a of_o a_o medial_a line_n the_o other_o side_n that_o make_v the_o breadth_n thereof_o shall_v be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n whereupon_o the_o parallelogram_n be_v apply_v svppose_v that_o a_o be_v a_o medial_a line_n and_o let_v bc_o be_v a_o line_n rational_a and_o upon_o the_o line_n bc_o describe_v a_o rectangle_n parallelogram_n equal_a unto_o the_o square_n of_o the_o line_n a_o and_o let_v the_o same_o be_v bd_o make_v in_o breadth_n the_o line_n cd_o then_o i_o say_v that_o the_o line_n cd_o be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n cb._n ãâã_d for_o forasmuch_o as_o a_o be_v a_o medial_a line_n it_o contain_v in_o power_n by_o the_o 21._o of_o the_o ten_o a_o rectangle_n parallelogram_n comprehend_v under_o rational_a right_a line_n commensurable_a in_o power_n only_o suppose_v that_o be_v contain_v in_o power_n the_o parallelogram_n gf_o and_o by_o supposition_n it_o also_o contain_v in_o power_n the_o parallelogram_n bd._n wherefore_o the_o parallelogram_n bd_o be_v equal_a unto_o the_o parallelogram_n gf_o and_o it_o be_v also_o equiangle_n unto_o it_o for_o that_o they_o be_v each_o rectangle_n but_o in_o parallelogram_n equal_a and_o equiangle_n the_o side_n which_o contain_v the_o equal_a angle_n be_v reciprocal_a by_o the_o 14._o of_o the_o six_o wherefore_o what_o proportion_n the_o line_n bc_o have_v to_o the_o line_n eglantine_n the_o same_o have_v the_o line_n of_o to_o the_o line_n cd_o therefore_o by_o the_o 22._o of_o the_o six_o as_o the_o square_n of_o the_o line_n bc_o be_v to_o the_o square_n of_o the_o line_n eglantine_n so_o be_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n cd_o but_o the_o square_n of_o the_o line_n bc_o be_v commensurable_a unto_o the_o square_n of_o the_o line_n eglantine_n by_o supposition_n for_o either_o of_o they_o be_v rational_a wherefore_o by_o the_o the_o 10._o of_o the_o ten_o the_o square_a of_o the_o line_n of_o be_v commensurable_a unto_o the_o square_n of_o the_o linâ_n cd_o but_o the_o square_n of_o the_o line_n of_o be_v rational_a wherefore_o the_o square_a of_o the_o line_n cd_o be_v likewise_o rational_a wherefore_o the_o line_n cd_o be_v rational_a and_o forasmuch_o as_o the_o line_n of_o be_v incommensurable_a in_o length_n unto_o the_o line_n eglantine_n for_o they_o be_v suppose_v to_o be_v commensurable_a in_o power_n only_o but_o as_o the_o line_n of_o be_v to_o the_o line_n eglantine_n so_o by_o the_o assumpt_n go_v before_o be_v the_o square_a of_o the_o line_n of_o to_o the_o parallelogram_n which_o be_v contain_v under_o the_o line_n of_o and_o eglantine_n wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a of_o the_o line_n of_o be_v incommensurable_a unto_o the_o parallelogram_n which_o be_v contain_v under_o the_o line_n fe_o and_o eglantine_n but_o unto_o the_o square_n of_o the_o line_n of_o the_o square_n of_o the_o line_n cd_o be_v commensurable_a for_o it_o be_v prove_v that_o âither_o of_o they_o be_v a_o rational_a linâ_n and_o that_o which_o be_v contain_v under_o the_o line_n dc_o and_o cb_o be_v commensurable_a unto_o that_o which_o be_v contain_v under_o the_o line_n fe_o and_o eglantine_n for_o they_o be_v both_o equal_a to_o the_o square_n of_o the_o line_n a._n wherefore_o by_o the_o 13._o of_o the_o ten_o the_o square_a of_o the_o line_n cd_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n dc_o and_o cb._n but_o as_o the_o square_n of_o the_o line_n cd_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n dc_o and_o cb_o so_o by_o the_o assumpt_n go_v before_o be_v the_o line_n dc_o to_o the_o line_n cb._n wherefore_o the_o line_n dc_o be_v incommensurable_a in_o length_n unto_o the_o line_n cb._n wherefore_o the_o line_n cd_o be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n cb._n if_o therefore_o upon_o a_o rational_a line_n be_v apply_v the_o square_a of_o a_o medial_a line_n the_o other_o side_n that_o make_v the_o breadth_n thereof_o shall_v be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n whereupon_o the_o parallelogram_n be_v apply_v which_o be_v require_v to_o be_v prove_v a_o square_n be_v say_v to_o be_v apply_v upon_o a_o line_n when_o it_o or_o a_o parallelogram_n equal_a unto_o it_o be_v apply_v upon_o the_o say_a line_n line_n if_o upon_o a_o rational_a line_n give_v we_o will_v apply_v a_o rectangle_n parallelogram_n equal_a to_o the_o square_n of_o a_o medial_a line_n give_v and_o so_o of_o any_o line_n give_v we_o must_v by_o the_o 11._o of_o the_o six_o find_v out_o the_o three_o line_n proportional_a with_o the_o rational_a line_n and_o the_o medial_a line_n give_v so_o yet_o that_o the_o rational_a line_n be_v the_o first_o and_o the_o medial_a line_n give_v which_o contain_v in_o power_n the_o square_a to_o be_v apply_v be_v the_o second_o for_o then_o the_o superficies_n contain_v under_o the_o first_o and_o the_o three_o shall_v be_v equal_a to_o the_o square_n of_o the_o middle_a line_n by_o the_o 17._o of_o the_o six_o ¶_o the_o 20._o theorem_a the_o 23._o proposition_n a_o right_a line_n commensurable_a to_o a_o medial_a line_n be_v also_o a_o medial_a line_n svppose_v that_o a_o be_v a_o medial_a line_n and_o unto_o the_o line_n a_o let_v the_o line_n b_o be_v commensurable_a either_o in_o length_n &_o in_o power_n or_o in_o power_n only_o construction_n then_o i_o say_v that_o b_o also_o be_v a_o medial_a line_n let_v there_o be_v put_v a_o rational_a line_n cd_o and_o upon_o the_o line_n cd_o apply_v a_o rectangle_n parallelogram_n ce_fw-fr equal_a unto_o the_o square_n of_o the_o line_n a_o and_o make_v in_o breadth_n the_o line_n ed._n wherefore_o by_o the_o proposition_n go_v before_o the_o line_n ed_z be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n cd_o demonstration_n and_o again_o upon_o the_o line_n cd_o apply_v a_o rectangle_n parallelogram_n cf_o equal_a unto_o the_o square_n of_o the_o line_n b_o and_o make_v in_o breadth_n the_o line_n df._n and_o forasmuch_o as_o the_o line_n a_o be_v commensurable_a unto_o the_o line_n b_o therefore_o the_o square_a of_o the_o line_n a_o be_v commensurable_a to_o the_o square_n of_o the_o line_n b._n but_o the_o parallelogram_n ec_o be_v equal_a to_o the_o square_n of_o the_o linâ_n a_o and_o the_o parallelogram_n cf_o be_v equal_a to_o
parallelogram_n shall_v afterward_o be_v teach_v in_o the_o 27._o and_o 28._o proposition_n of_o this_o book_n ¶_o a_o corollary_n hereby_o it_o be_v manifest_a that_o a_o rectangle_n parallelogram_n contain_v under_o two_o right_a line_n be_v the_o mean_a proportional_a between_o the_o square_n of_o the_o say_v line_n corollary_n as_o it_o be_v manifest_a by_o the_o first_o of_o the_o six_o that_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v the_o mean_a proportional_a between_o the_o square_n ad_fw-la and_o cx_o this_o corollary_n be_v put_v after_o the_o 53._o proposition_n of_o this_o book_n as_o a_o assumpt_n and_o there_o demonstrate_v which_o there_o in_o his_o place_n you_o shall_v find_v but_o because_o it_o follow_v of_o this_o proposition_n so_o evident_o and_o brief_o without_o far_a demonstration_n i_o think_v it_o not_o amiss_o here_o by_o the_o way_n to_o note_v it_o ¶_o the_o 23._o theorem_a the_o 26._o proposition_n a_o medial_a superficies_n exceed_v not_o a_o medial_a superficies_n by_o a_o rational_a superficies_n for_o if_o it_o be_v possible_a let_v ab_fw-la be_v a_o medial_a superficies_n exceed_v ac_fw-la be_v also_o a_o medial_a superficies_n by_o db_o be_v a_o rational_a superficies_n and_o let_v there_o be_v put_v a_o rational_a right_a line_n ef._n construction_n and_o upon_o the_o line_n of_o apply_v a_o rectangle_n parallelogram_n fh_o equal_a unto_o the_o medial_a superficies_n ab_fw-la who_o other_o side_n let_v be_v eh_o and_o from_o the_o parallelogram_n fh_o take_v away_o the_o parallelogram_n fg_o equal_a unto_o the_o medial_a superficies_n ac_fw-la absurdity_n wherefore_o by_o the_o three_o common_a sentence_n the_o residue_n bd_o be_v equal_a to_o the_o residue_n kh_o but_o by_o supposition_n the_o superficies_n db_o be_v rational_a wherefore_o the_o superficies_n kh_o be_v also_o rational_a and_o forasmuch_o as_o either_o of_o these_o superficiece_n ab_fw-la and_o ac_fw-la be_v medial_a and_o ab_fw-la be_v equal_a unto_o fh_o &_o ac_fw-la unto_o fg_o therefore_o either_o of_o these_o superficiece_n fh_a and_o fg_o be_v medial_a and_o they_o be_v apply_v upon_o the_o rational_a line_n ef._n wherefore_o by_o the_o 22._o of_o the_o ten_o either_o of_o these_o line_n he_o and_o eglantine_n be_v rational_a &_o incommensurable_a in_o length_n unto_o the_o line_n ef._n and_o forasmuch_o as_o the_o superficies_n db_o be_v rational_a and_o the_o superficies_n kh_o be_v equal_a unto_o it_o therefore_o kh_o be_v also_o rational_a and_o it_o be_v apply_v upon_o the_o rational_a line_n of_o for_o it_o be_v apply_v upon_o the_o line_n gk_o which_o be_v equal_a to_o the_o line_n of_o wherefore_o by_o the_o 20._o of_o the_o ten_o the_o line_n gh_o be_v rational_a and_o commensurable_a in_o length_n unto_o the_o line_n gk_o but_o the_o line_n gk_o be_v equal_a to_o the_o line_n ef._n wherefore_o the_o line_n gh_o be_v rational_a and_o commensurable_a in_o length_n unto_o the_o line_n ef._n but_o the_o line_n eglantine_n be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ef._n wherefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n eglantine_n be_v incommensurable_a in_o length_n unto_o the_o line_n gh_o and_o as_o the_o line_n eglantine_n be_v to_o the_o line_n gh_o so_o be_v the_o square_a of_o the_o line_n eglantine_n to_o the_o parallelogram_n contain_v under_o the_o line_n eglantine_n and_o gh_o by_o the_o assumpt_n put_v before_o the_o 21._o of_o the_o ten_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a of_o the_o line_n eglantine_n be_v incommensurable_a unto_o the_o parallelogram_n contain_v under_o the_o line_n eglantine_n and_o gh_o but_o unto_o the_o square_n of_o the_o line_n eglantine_n be_v commensurable_a the_o square_n of_o the_o line_n eglantine_n and_o gh_o for_o either_o of_o they_o be_v rational_a as_o have_v before_o be_v prove_v wherefore_o the_o square_n of_o the_o line_n eglantine_n and_o gh_o be_v incommensurable_a unto_o the_o parallelogram_n contain_v under_o the_o line_n eglantine_n and_o gh_o but_o unto_o the_o parallelogram_n contain_v under_o the_o line_n eglantine_n and_o gh_o be_v commensurable_a that_o which_o be_v contain_v under_o the_o line_n fg_o and_o gh_o twice_o for_o they_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n namely_o as_o unity_n be_v to_o the_o number_n 2_o or_o as_o 2._o be_v to_o 4_o and_o therefore_o by_o the_o 6._o of_o this_o book_n they_o be_v commensurable_a wherefore_o by_o the_o 13._o of_o the_o ten_o the_o square_n of_o the_o line_n eglantine_n and_o gh_o be_v incommensurable_a unto_o that_o which_o be_v contain_v under_o the_o line_n eglantine_n and_o gh_o twice_o this_o be_v more_o brieâly_o conclude_v by_o the_o corollary_n of_o the_o 13._o of_o the_o ten_o but_o the_o square_n of_o the_o line_n eglantine_n and_o gh_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n eglantine_n and_o gh_o twice_o be_v equal_a to_o the_o square_n of_o the_o line_n eh_o by_o the_o 4._o of_o the_o second_o wherefore_o the_o square_a of_o the_o line_n eh_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n eglantine_n and_o gh_o by_o the_o 16._o of_o the_o ten_o but_o the_o square_n of_o the_o line_n fg_o &_o gh_o be_v rational_a wherefore_o the_o square_a of_o the_o line_n eh_o be_v irrational_a wherefore_o the_o line_n also_o eh_o be_v irrational_a but_o it_o have_v before_o be_v prove_v to_o be_v rational_a which_o be_v impossible_a wherefore_o a_o medial_a superficies_n exceed_v not_o a_o medial_a superficies_n by_o a_o rational_a superficies_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 4._o problem_n the_o 27._o proposition_n to_o find_v out_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o rational_a parallelogram_n let_v there_o be_v put_v two_o rational_a line_n commensurable_a in_o power_n only_o namely_o a_o and_o b._n construction_n and_o by_o the_o 13._o of_o the_o six_o take_v the_o mean_a proportional_a between_o the_o line_n a_o and_o b_o and_o let_v the_o same_o line_n be_v c._n and_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o by_o the_o 12._o of_o the_o six_o let_v the_o line_n c_o be_v to_o the_o line_n d._n demonstration_n and_o forasmuch_o as_o a_o and_o b_o be_v rational_a line_n commensurable_a in_o power_n only_o therefore_o by_o the_o 21._o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o that_o be_v the_o square_a of_o the_o line_n c._n for_o the_o square_n of_o the_o line_n c_o be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n a_o anâ_z b_o by_o the_o 17._o of_o the_o six_o be_v medial_a âherfore_o c_z also_o be_v a_o medial_a line_n and_o for_o that_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o line_n c_o to_o the_o line_n d_o therefore_o as_o the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o so_o be_v the_o square_a of_o the_o line_n c_o to_o the_o square_n of_o the_o line_n d_o by_o the_o 22._o of_o the_o six_o but_o the_o square_n of_o the_o line_n a_o and_o b_o be_v commensurable_a for_o the_o liââs_v a_o and_o b_o aâe_fw-fr suppose_a to_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o also_o the_o square_n of_o the_o line_n c_o and_o d_o be_v commensurable_a by_o the_o 10._o of_o the_o ten_o wherefore_o the_o line_n c_o and_o d_o be_v commensurable_a in_o power_n only_o and_o c_o be_v a_o medial_a line_n wherefore_o by_o the_o 23._o of_o the_o ten_o d_z also_o be_v a_o medial_a line_n wherefore_o c_z and_o d_o be_v medial_a line_n commensurable_a in_o power_n only_o now_o also_o i_o say_v that_o they_o contain_v a_o rational_a parallelogram_n for_o for_o that_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o line_n c_o to_o the_o line_n d_o therefore_o alternate_o also_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n a_o be_v to_o the_o line_n c_o so_o be_v the_o line_n b_o to_o the_o line_n d._n but_o as_o the_o line_n a_o be_v to_o the_o line_n c_o so_o be_v the_o line_n c_o to_o the_o line_n b_o wherefore_o as_o the_o line_n c_o be_v to_o the_o line_n b_o so_o be_v the_o line_n b_o to_o the_o line_n d._n wherefore_o the_o parallelogram_n contain_v under_o the_o line_n c_o and_o d_o be_v equal_a to_o the_o square_n of_o the_o line_n b._n but_o the_o square_n of_o the_o line_n b_o be_v rational_a wherefore_o the_o parallelogram_n which_o be_v contain_v under_o the_o line_n c_o and_o d_o be_v also_o rational_a wherefore_o there_o be_v find_v out_o medial_a line_n commensurablâ_n in_o powâr_n only_o contain_v a_o rational_a parallelogrammeâ_n which_o ãâã_d require_v to_o be_v do_v the_o 5._o problem_n the_o
28._o proposition_n to_o find_v out_o medial_a right_a line_n commensurable_a in_o power_n only_o contain_v a_o medial_a parallelogram_n let_v there_o be_v put_v three_o rational_a right_a line_n commensurable_a in_o power_n only_o namely_o a_o b_o and_o c_o construction_n and_o by_o the_o 13._o of_o the_o six_o take_v the_o mean_a proportional_a between_o the_o line_n a_o and_o b_o &_o let_v thâ_z same_o be_v d._n and_o as_o the_o line_n b_o be_v to_o the_o line_n c_o so_o by_o the_o 12._o of_o the_o six_o let_v the_o line_n d_o be_v to_o the_o line_n e._n and_o forasmuch_o as_o the_o line_n a_o and_o b_o be_v rational_a commensurable_a in_o power_n only_o demonstration_n therefore_o by_o the_o 21._o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o that_o be_v the_o square_a of_o the_o line_n d_o be_v medial_a wherefore_o d_z be_v a_o medial_a line_n and_o forasmuch_o as_o the_o line_n b_o and_o c_o be_v commensurable_a in_o power_n only_o and_o as_o the_o line_n b_o be_v to_o the_o line_n c_o so_o be_v the_o line_n d_o to_o the_o line_n e_o wherefore_o the_o line_n d_z and_o e_o be_v commensurable_a in_o power_n only_o by_o the_o corollary_n of_o the_o ten_o of_o this_o book_n but_o d_o be_v a_o medial_a line_n wherefore_o e_o also_o be_v a_o medial_a line_n by_o the_o 23._o of_o this_o book_n wherefore_o d_z &_o e_o be_v medial_a line_n commensurable_a in_o power_n only_o i_o say_v also_o that_o they_o contain_v a_o medial_a parallelogram_n for_o for_o that_o as_o the_o line_n b_o be_v to_o the_o line_n c_o so_o be_v the_o line_n d_o to_o the_o line_n e_o therefore_o alternate_o by_o the_o 16_o of_o the_o five_o as_o the_o line_n b_o be_v to_o the_o line_n d_o so_o be_v the_o line_n c_o to_o the_o line_n e._n but_o as_o the_o line_n b_o be_v to_o the_o line_n d_o so_o be_v the_o line_n d_o to_o the_o line_n aâ_z by_o converse_n proportion_n which_o be_v prove_v by_o the_o corollary_n of_o the_o four_o of_o the_o five_o wherefore_o as_o the_o line_n d_o be_v to_o the_o line_n a_o so_o be_v the_o line_n c_o to_o the_o line_n e._n wherefore_o that_o which_o be_v contain_v under_o the_o line_n a_o &_o c_o be_v by_o the_o 16._o of_o the_o sixâ_o equal_a to_o that_o which_o be_v contain_v under_o the_o line_n d_o &_o e._n but_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o c_o be_v medial_a by_o the_o 21._o of_o the_o ten_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o be_v medial_a wherefore_o there_o be_v find_v out_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o medial_a superficies_n which_o be_v require_v to_o be_v do_v a_o assumpt_n to_o find_v out_o two_o square_a number_n which_o add_v together_o make_v a_o square_a number_n let_v there_o be_v put_v two_o like_o superficial_a number_n ab_fw-la and_o bc_o which_o how_o to_o find_v out_o have_v be_v teach_v after_o the_o 9_o proposition_n of_o this_o book_n and_o let_v they_o both_o be_v either_o even_a number_n or_o odd_a and_o let_v the_o great_a number_n be_v ab_fw-la and_o forasmuch_o as_o if_o from_o any_o even_a number_n be_v take_v away_o a_o even_a number_n or_o from_o a_o odd_a number_n be_v take_v away_o a_o odd_a number_n the_o residue_n shall_v be_v even_o by_o the_o 24._o and_o 26_o of_o the_o nine_o if_o therefore_o from_o ab_fw-la be_v a_o even_a number_n be_v take_v away_o bc_o a_o even_a number_n or_o from_o ab_fw-la be_v a_o odd_a number_n be_v take_v away_o bc_o be_v also_o odd_a the_o residue_n ac_fw-la shall_v be_v even_o divide_v the_o number_n ac_fw-la into_o two_o equal_a part_n in_o d_o wherefore_o the_o number_n which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o cd_o be_v by_o the_o six_o of_o the_o second_o as_o barlaam_n demonstrate_v it_o in_o number_n equal_a to_o the_o square_a number_n of_o bd._n but_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v a_o square_a number_n for_o it_o be_v prove_v by_o the_o first_o of_o the_o nine_o that_o if_o two_o like_o plain_a number_n multiply_v the_o one_o the_o other_o produce_v any_o number_n the_o number_n produce_v shall_v be_v a_o square_a number_n wherefore_o there_o be_v find_v out_o two_o square_a number_n the_o one_o be_v the_o square_a number_n which_o be_v produce_v of_o ab_fw-la into_o bc_o and_o the_o other_o the_o square_a number_n produce_v of_o cd_o which_o add_v together_o make_v a_o square_a number_n namely_o the_o square_a number_n produce_v of_o bd_o multiply_v into_o himself_o forasmuch_o as_o they_o be_v demonstrate_v equal_a to_o it_o corollary_n a_o corollary_n âumber_n and_o hereby_o it_o be_v manifest_a that_o there_o be_v find_v out_o two_o square_a number_n namely_o the_o ãâã_d the_o square_a number_n of_o bd_o and_o the_o other_o the_o square_a number_n of_o cd_o so_o that_o that_o number_n wherein_o the_o one_o exceed_v the_o other_o the_o number_n i_o say_v which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v also_o a_o square_a number_n namely_o when_o aâ_z &_o bc_o be_v like_o plain_a number_n but_o when_o they_o be_v not_o like_o plain_a number_n then_o be_v there_o find_v out_o two_o square_a number_n the_o square_a number_n of_o bd_o and_o the_o square_a number_n of_o dc_o who_o excess_n that_o be_v the_o number_n whereby_o the_o great_a exceed_v the_o less_o namely_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v not_o a_o square_a number_n ¶_o a_o assumpt_n to_o find_v out_o two_o square_a number_n which_o add_v together_o make_v not_o a_o square_a number_n assumpt_n let_v ab_fw-la and_o bc_o be_v like_o plain_a number_n so_o that_o by_o the_o first_o of_o the_o nine_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v a_o square_a number_n and_o let_v ac_fw-la be_v a_o even_a number_n and_o divide_v câ_n into_o two_o equal_a parâes_n in_o d._n now_o by_o that_o which_o have_v before_o be_v say_v in_o the_o former_a assumpt_n it_o be_v manifest_a that_o the_o square_a number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o cd_o be_v equal_a to_o the_o square_a number_n of_o bd._n take_v away_o from_o cd_o unity_n de._n wherefore_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_n of_o ce_fw-fr be_v less_o than_o the_o square_a number_n of_o bd._n now_o then_o i_o say_v that_o the_o square_a numâer_n produce_v of_o ab_fw-la into_o bc_o add_v to_o the_o square_a number_n of_o ce_fw-fr make_v not_o a_o square_a number_n for_o if_o they_o do_v make_v a_o square_a number_n than_o that_o square_a number_n which_o they_o make_v be_v either_o great_a they_o the_o square_a number_n of_o be_v or_o equal_a unto_o it_o or_o less_o than_o it_o first_o great_a it_n can_v be_v for_o it_o be_v already_o prove_v that_o the_o square_a number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v less_o than_o the_o square_a number_n of_o bd._n but_o between_o the_o square_a number_n of_o bd_o and_o the_o square_a number_n of_o be_v there_o be_v no_o mean_a square_a number_n for_o the_o number_n bd_o exceed_v the_o number_n be_v only_o by_o unity_n which_o unity_n can_v by_o no_o mean_n be_v divide_v into_o number_n or_o if_o the_o number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_n of_o the_o number_n ce_fw-fr shall_v be_v great_a than_o the_o square_a of_o the_o number_n be_v then_o shall_v the_o self_n same_o number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_n of_o the_o number_n ce_fw-fr be_v equal_a to_o the_o square_n of_o the_o number_n bd_o the_o contrary_a whereof_o be_v already_o prove_v wherefore_o if_o it_o be_v possible_a let_v that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o the_o number_n ce_fw-fr be_v equal_a to_o the_o square_a number_n of_o be._n and_o let_v ga_o be_v double_a to_o unity_n de_fw-fr that_o be_v let_v it_o be_v the_o number_n two_o now_o forasmuch_o as_o the_o whole_a number_n ac_fw-la be_v by_o supposition_n double_a to_o the_o whole_a number_n cd_o of_o which_o the_o number_n agnostus_n be_v double_a to_o unity_n de_fw-fr therefore_o by_o the_o 7._o of_o the_o seven_o the_o residue_n namely_o the_o number_n gc_o be_v double_a to_o the_o residue_n namely_o to_o the_o number_n ec_o wherefore_o the_o number_n gc_o be_v divide_v into_o two_o equal_a part_n in_o e._n wherefore_o that_o which_o be_v produce_v of_o gb_o into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v equal_a to_o the_o square_a number_n
to_o the_o number_n ce_fw-fr so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n of_o therefore_o by_o conversion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o as_o the_o number_n cd_o be_v to_o the_o number_n de_fw-fr so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n to_o the_o line_n fb_o but_o the_o number_n c_o d_o have_v not_o to_o the_o number_n de_fw-fr that_o proportion_n that_o a_o square_a nâmbeâ_n hâth_v to_o a_o square_a number_n wherefore_o neither_o also_o the_o square_a of_o the_o line_n ab_fw-la have_v to_o the_o square_n of_o the_o line_n bf_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n ab_fw-la be_v by_o the_o 9_o of_o the_o ten_o incommensurable_a in_o length_n to_o the_o line_n bf_o and_o the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n of_o by_o the_o square_n of_o the_o right_a line_n bf_o which_o be_v incommensurable_a in_o length_n unto_o the_o line_n ab_fw-la wherefore_o the_o line_n ab_fw-la and_o of_o be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n of_o by_o the_o square_n of_o the_o line_n fb_o which_o be_v commensurable_a in_o length_n unto_o the_o line_n abâ_n â_z which_o be_v require_v to_o be_v do_v ¶_o a_o assumpt_n if_o there_o be_v two_o right_a line_n have_v between_o themselves_o any_o proportion_n as_o the_o one_o right_a line_n be_v to_o the_o other_o so_o be_v the_o parallelogram_n contain_v under_o both_o the_o right_a line_n to_o the_o square_n of_o the_o less_o of_o those_o two_o line_n suppose_v that_o these_o two_o right_a ab_fw-la and_o bc_o be_v in_o some_o certain_a proportion_n find_v then_o i_o say_v that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o parallelogram_n contain_v under_o ab_fw-la and_o bc_o to_o the_o square_n of_o bc._n describe_v the_o square_a of_o the_o line_n bc_o and_o let_v the_o same_o be_v cd_o and_o make_v perfect_a the_o parallelogram_n ad_fw-la now_o it_o be_v manifest_a that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o parallelogram_n ad_fw-la to_o the_o parallelogram_n or_o square_n be_v by_o the_o first_o of_o the_o six_o but_o the_o parallelogram_n ad_fw-la be_v that_o which_o be_v bontain_v under_o the_o line_n ab_fw-la and_o bc_o for_o the_o line_n bc_o be_v equal_a to_o the_o line_n bd_o and_o the_o parallelogram_n be_v be_v the_o square_n of_o the_o line_n bc._n wherefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o parallelogram_n coutain_v under_o the_o line_n ab_fw-la and_o bc_o to_o the_o square_n of_o the_o line_n bc_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 8._o problem_n the_o 31._o proposition_n to_o find_v out_o two_o medial_a line_n commensurable_a in_o power_n only_o comprehend_v a_o rational_a superficies_n so_o that_o the_o great_a shall_v be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o great_a let_v there_o be_v take_v by_o the_o 29._o of_o the_o ten_o two_o rational_a line_n commensurable_a in_o power_n only_o a_o and_o b_o construction_n so_o that_o let_v the_o line_n a_o be_v the_o great_a be_v in_o power_n more_o than_o the_o line_n b_o be_v the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n aâ_n â_z and_o let_v the_o square_n of_o the_o line_n c_o be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n a_o and_o b_o which_o be_v do_v by_o find_v out_o the_o mean_a proportional_a line_n namely_o the_o line_n c_o between_o the_o line_n a_o and_o b_o by_o the_o 13._o of_o the_o six_o now_o the_o parallelogram_n contain_v under_o the_o line_n a_o and_o b_o be_v medial_a by_o the_o 21._o of_o this_o book_n wherefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o the_o square_a also_o of_o the_o line_n c_o be_v medial_a wherefore_o the_o line_n c_o also_o be_v medial_a unto_o the_o square_n of_o the_o line_n b_o let_v the_o parallelogram_n contain_v under_o the_o line_n c_o and_o d_o be_v equal_a by_o find_v out_o a_o three_o line_n proportional_a namely_o the_o line_n d_o to_o the_o two_o line_n c_o and_o b_o by_o the_o 11._o of_o the_o six_o but_o the_o square_n of_o the_o line_n b_o be_v rational_a demonstration_n wherefore_o the_o parallelogram_n contain_v under_o the_o line_n c_o and_o d_o be_v rational_a and_o for_o that_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n a_o and_o b_o to_o the_o square_n of_o the_o line_n b_o by_o the_o assumpt_n go_v before_o but_o unto_o the_o parallelogram_n contain_v under_o the_o line_n a_o and_o b_o be_v equal_v the_o square_n of_o the_o line_n c_o and_o unto_o the_o square_n of_o the_o line_n b_o be_v equal_a the_o parallelogram_n contain_v under_o the_o line_n c_o and_o d_o as_o it_o have_v now_o be_v prove_v therefore_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o square_a of_o the_o line_n c_o to_o the_o parallelogram_n contain_v under_o the_o line_n c_o d._n but_o as_o the_o square_n of_o the_o line_n c_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n c_o and_o d_o so_o be_v the_o line_n c_o to_o the_o line_n d._n wherefore_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o line_n c_o to_o the_o line_n d._n but_o by_o supposition_n the_o line_n a_o be_v commensurable_a unto_o the_o line_n b_o in_o power_n only_o wherefore_o by_o the_o 11._o of_o the_o ten_o the_o line_n c_o also_o be_v unto_o the_o line_n d_o commensurable_a in_o power_n only_o but_o the_o line_n c_o be_v medial_a wherefore_o by_o the_o 23â_n of_o the_o ten_o the_o line_n d_o also_o be_v medial_a and_o for_o that_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o line_n c_o to_o the_o line_n d_o but_o the_o line_n a_o be_v in_o power_n more_o than_o the_o line_n b_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n a_o by_o supposition_n wherefore_o the_o line_n c_o also_o be_v in_o power_n more_o than_o the_o line_n d_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n c._n wherefore_o there_o be_v find_v out_o two_o medial_a line_n c_o and_o d_o commensurable_a in_o power_n only_o comprehend_v a_o rational_a superficies_n and_o the_o line_n c_o be_v in_o power_n more_o than_o the_o line_n d_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n c._n and_o in_o like_a sort_n may_v be_v find_v out_o two_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o rational_a superficies_n so_o that_o the_o great_a shall_v be_v in_o power_n more_o they_o the_o less_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o great_a namely_o when_o the_o line_n a_o be_v in_o power_n more_o they_o the_o line_n b_o by_o the_o square_n of_o a_o line_n incommensurable_n in_o length_n unto_o the_o line_n a_o which_o to_o do_v be_v teach_v by_o the_o 30._o of_o this_o book_n the_o self_n same_o construction_n remain_v that_o part_n of_o this_o proposition_n from_o these_o word_n and_o for_o that_o as_o the_o line_n a_o be_v to_o the_o line_n b_o to_o these_o word_n but_o by_o supposition_n the_o line_n a_o be_v commensurable_a unto_o the_o line_n b_o may_v more_o easy_o be_v demonstrate_v after_o this_o manner_n the_o line_n c_o b_o d_o be_v in_o continual_a proportion_n by_o the_o second_o part_n of_o the_o 17._o of_o the_o six_o but_o the_o line_n a_o c_o d_o be_v also_o in_o continual_a proportion_n by_o the_o same_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o line_n a_o be_v to_o the_o line_n c_o so_o be_v the_o line_n b_o to_o the_o line_n d._n wherefore_o alternate_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_fw-mi be_v the_o line_n c_o to_o the_o line_n d._n etc_n etc_n which_o be_v require_v to_o be_v do_v ¶_o a_o assumpt_n if_o there_o be_v three_o right_a line_n have_v between_o themselves_o any_o proportion_n as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o parallelogram_n contain_v under_o the_o first_o and_o the_o second_o to_o the_o parallelogram_n contain_v under_o the_o second_o and_o the_o three_o suppose_v that_o these_o three_o line_n ab_fw-la b_o c_o and_o cd_o be_v in_o some_o certain_a proportion_n then_o i_o say_v that_o as_o the_o line_n ab_fw-la be_v to_o
contain_v under_o the_o line_n bd_o and_o dc_o be_v equal_a to_o the_o square_n of_o the_o line_n da._n as_o touch_v the_o four_o that_o the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o ad_fw-la be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n basilius_n and_o ac_fw-la be_v thus_o prove_v for_o forasmuch_o as_o as_o we_o have_v already_o declare_v the_o triangle_n abc_n be_v like_a and_o therefore_o equiangle_n to_o the_o triangle_n abdella_n therefore_o as_o the_o line_n bc_o be_v to_o the_o line_n ac_fw-la dâe_n so_o be_v the_o line_n basilius_n to_o the_o line_n ad_fw-la by_o the_o 4._o of_o the_o six_o corollary_n but_o if_o there_o be_v four_o right_a line_n proportional_a that_o which_o be_v contain_v under_o the_o first_o and_o the_o last_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o two_o mean_n by_o the_o 16._o of_o the_o six_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n bc_o and_o ad_fw-la be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la profitable_a i_o say_v moreover_o that_o if_o there_o be_v make_v a_o parallelogram_n complete_a determination_n contain_v under_o the_o line_n bc_o and_o ad_fw-la which_o let_v be_v ec_o and_o if_o likewise_o be_v make_v complete_a the_o parallelogram_n contain_v under_o the_o line_n basilius_n and_o ac_fw-la which_o let_v be_v of_o it_o may_v by_o a_o other_o way_n be_v prove_v that_o the_o parallelogram_n ec_o be_v equal_a to_o the_o parallelogram_n af._n for_o forasmuch_o as_o either_o of_o they_o be_v double_a to_o the_o triangle_n acb_o by_o the_o 41._o of_o the_o first_o and_o thing_n which_o be_v double_a to_o one_o and_o the_o self_n same_o thing_n be_v equal_a the_o one_o to_o the_o other_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n bc_o and_o ad_fw-la be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la 2._o ¶_o a_o assumpt_n if_o a_o right_a line_n be_v divide_v into_o two_o unequal_a part_n as_o the_o great_a part_n be_v to_o the_o less_o assumpt_n so_o be_v the_o parallelogram_n contain_v under_o the_o whole_a line_n and_o the_o great_a part_n to_o the_o parallelogram_n contain_v under_o the_o whole_a line_n and_o the_o less_o part_n this_o assumpt_n differ_v little_a from_o the_o first_o proposition_n of_o the_o six_o book_n 3._o ¶_o a_o assumpt_n if_o there_o be_v two_o unequal_a right_a line_n and_o if_o the_o less_o be_v divide_v into_o two_o equal_a part_n the_o parallelogram_n contain_v under_o the_o two_o unequal_a line_n be_v double_a to_o the_o parallelogram_n contain_v under_o the_o great_a line_n &_o half_a of_o the_o less_o line_n suppose_v that_o there_o be_v two_o unequal_a right_a line_n ab_fw-la and_o bc_o of_o which_o leâ_z ab_fw-la be_v the_o great_a and_o divide_v the_o line_n bc_o into_o two_o equal_a part_n in_o the_o point_n d._n thân_o i_o say_v that_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la &_o bc_o be_v double_a to_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bd._n from_o the_o point_n b_o raise_v up_o upon_o the_o right_a line_n bc_o a_o perpendicular_a line_n be_v and_o let_v be_v be_v equal_a to_o the_o line_n basilius_n and_o draw_v from_o the_o point_n c_o and_o d_o the_o line_n cg_o and_o df_o parallel_v and_o equal_a to_o be_v and_o then_o draw_v the_o right_a line_n gfe_n the_o figure_n be_v complete_a nââ_n for_o that_o aââhe_a line_n db_o be_v to_o the_o line_n dc_o so_o be_v the_o parallelogram_n bf_o to_o the_o parallelogram_n dg_o by_o the_o 1._o of_o the_o six_o therefore_o by_o composition_n of_o proportion_n as_o the_o whole_a line_n bc_o be_v to_o the_o line_n dc_o so_o be_v the_o parallelogram_n bg_o to_o the_o parallelogram_n dg_o by_o the_o 18._o of_o the_o five_o but_o the_o line_n bc_o be_v double_a to_o the_o line_n dc_o wherefore_o the_o parallelogram_n bg_o be_v double_a to_o the_o parallelogram_n dg_o but_o the_o parallâlogramme_n bg_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o for_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n be_v and_o the_o parallelogram_n dg_o be_v contain_v under_o the_o line_n ab_fw-la and_o bd_o for_o the_o line_n bd_o be_v equal_a to_o the_o line_n dc_o and_o the_o line_n ab_fw-la to_o the_o line_n df_o which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 10._o problem_n the_o 33._o proposition_n to_o âinde_v out_o two_o right_a line_n incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o rational_a superficies_n and_o the_o parallelogram_n contain_v under_o they_o make_v a_o medial_a superficies_n take_v by_o the_o 30._o of_o the_o ten_o two_o rational_a right_a line_n commensurable_a in_o power_n only_o namely_o ab_fw-la and_o bc_o so_o that_o let_v the_o line_n ab_fw-la be_v the_o great_a be_v in_o power_n more_o than_o the_o line_n bc_o be_v the_o less_o construction_n by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n ab_fw-la and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n bc_o into_o two_o equal_a part_n in_o the_o point_n d._n and_o upon_o the_o line_n ab_fw-la apply_v a_o parallelogram_n equal_a to_o the_o square_a either_o of_o the_o line_n bd_o or_o of_o the_o line_n dc_o and_o want_v in_o figure_n by_o a_o square_a by_o the_o 28._o of_o the_o six_o and_o let_v that_o parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n afb_o and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n e_o raise_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n of_o cut_v the_o circumference_n in_o the_o point_n f._n and_o draw_v line_n from_o a_o to_o fletcher_n demonstration_n and_o from_o fletcher_n to_o b._n and_o forasmuch_o as_o there_o be_v two_o unequal_a right_a line_n ab_fw-la and_o bc_o and_o the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n bc_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o ab_fw-la and_o upon_o the_o line_n ab_fw-la be_v apply_v a_o parallelogram_n equal_a to_o the_o four_o part_n oâ_n the_o square_a of_o the_o line_n bc_o that_o be_v to_o the_o square_n of_o the_o half_a of_o the_o line_n bc_o and_o want_v in_o âigure_n by_o a_o square_a and_o the_o say_a parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n wherefore_o by_o the_o 2._o part_n of_o the_o 18._o of_o the_o ten_o the_o line_n ae_n be_v incommensurable_a in_o length_n unto_o the_o line_n ebb_n but_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o parallelogram_n contain_v under_o the_o line_n basilius_n and_o ae_n to_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o be_v by_o the_o second_o assumpt_n before_o put_v and_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ae_n be_v equal_a to_o the_o square_n of_o the_o line_n of_o by_o the_o second_o part_n of_o the_o first_o assumpt_v before_o put_v and_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o be_v be_v by_o the_o first_o part_n of_o the_o same_o assumpt_n equal_a to_o the_o square_n of_o the_o line_n bf_o wherefore_o the_o square_a of_o the_o line_n of_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n bf_o wherefore_o the_o line_n of_o and_o bf_o be_v incommensurable_a in_o power_n and_o forasmuch_o as_o ab_fw-la be_v a_o rational_a line_n by_o supposition_n therefore_o by_o the_o 7_o definition_n of_o the_o ten_o the_o square_a of_o the_o line_n ab_fw-la be_v rational_a conclude_v wherefore_o also_o the_o square_n of_o the_o line_n of_o and_o fb_o add_v together_o make_v a_o rational_a superficies_n for_o by_o the_o 47._o of_o the_o first_o they_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la again_o forasmuch_o as_o by_o the_o three_o part_n of_o the_o first_o assumpt_n go_v before_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v equal_a to_o the_o square_n of_o the_o line_n ef._n but_o by_o supposition_n that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v equal_a to_o the_o square_n of_o the_o line_n bd._n wherefore_o the_o line_n fe_o be_v equal_a to_o the_o line_n bd._n wherefore_o the_o linâ_n bg_o be_v double_a to_o the_o line_n â_o e._n wherefore_o by_o the_o three_o assumpt_v go_v before_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v double_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o ef._n but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v by_o supposition_n medial_a
wherefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o of_o be_v also_o medial_a but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o of_o conclude_v be_v by_o the_o last_o part_n of_o the_o first_o assumpt_n go_v before_o equal_a to_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o fb_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n of_o &_o fb_o be_v a_o medial_a superficies_n and_o it_o be_v prove_v that_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n of_o and_o fb_o add_v together_o be_v rational_a conclusion_n wherefore_o there_o be_v find_v out_o two_o right_a line_n of_o and_o fb_o incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o rational_a superficies_n and_o the_o parallelogram_n contain_v under_o they_o be_v a_o medial_a superficiâsâ_n which_o be_v require_v to_o be_v do_v ¶_o the_o 11._o problem_n the_o 34._o proposition_n to_o find_v out_o two_o right_a line_n incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o medial_a superficies_n and_o the_o parallelogram_n contain_v under_o they_o make_v a_o rational_a superficies_n take_v by_o the_o 31._o of_o the_o ten_o two_o medial_a line_n ab_fw-la and_o bc_o commensurable_a in_o power_n only_o comprehend_v a_o rational_a superficies_n construction_n so_o that_o let_v the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n bc_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n ab_fw-la and_o describe_v upon_o the_o line_n ab_fw-la a_o semicircle_n adb_o and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n bc_o unto_o two_o equal_a part_n in_o the_o point_n e._n and_o by_o the_o 28._o of_o the_o six_o upon_o the_o line_n ab_fw-la apply_v a_o parallelogram_n equal_a to_o the_o square_n of_o the_o line_n be_v and_o want_v in_o figure_n by_o a_o square_a and_o let_v that_o parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n of_o and_o fb_o wherefore_o the_o line_n of_o be_v incommensurable_a in_o length_n unto_o the_o line_n fb_o by_o the_o 2._o part_n of_o the_o 18._o of_o the_o ten_o and_o from_o the_o point_n f_o unto_o the_o right_a line_n ab_fw-la raise_v up_o by_o the_o 11._o of_o the_o first_o a_o perpendicular_a line_n fd_o and_o draw_v line_n from_o a_o to_o d_o and_o from_o d_z to_o b._n demonstration_n and_o forasmuch_o as_o the_o line_n of_o be_v incommensurable_a unto_o the_o line_n fb_o but_o by_o the_o second_o assumpt_v go_v before_o the_o 33._o of_o the_o ten_o as_o the_o line_n of_o be_v to_o the_o line_n fb_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n basilius_n and_o of_o to_o the_o parallelogram_n contain_v under_o the_o line_n basilius_n and_o bf_o wherefore_o by_o the_o ten_o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o of_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bf_o but_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o of_o be_v equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bf_o be_v also_o equal_a to_o the_o square_n of_o the_o line_n db_o by_o the_o second_o part_n of_o the_o first_o assumpt_n go_v before_o the_o 33._o of_o the_o ten_o wherefore_o the_o square_a of_o the_o line_n ad_fw-la be_v incommensurable_a to_o the_o square_n of_o the_o line_n db._n wherefore_o the_o line_n ad_fw-la and_o db_o be_v incommensurable_a in_o power_n and_o forasmuch_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v medial_a therefore_o also_o the_o superficies_n make_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o be_v medial_a conclude_v for_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o be_v by_o the_o 47._o of_o the_o first_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o forasmuch_o as_o the_o line_n bc_o be_v double_a to_o the_o line_n fd_o as_o it_o be_v prove_v in_o the_o proposition_n go_v before_o therefore_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o and_o bc_o be_v double_a to_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o fd_v by_o the_o three_o assumpt_v go_v before_o the_o 33._o proposition_n wherefore_o it_o be_v also_o commensurable_a unto_o it_o by_o the_o six_o of_o the_o ten_o but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v suppose_v to_o be_v rational_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd_o be_v also_o rational_a but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd_v be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o by_o the_o last_o part_n of_o the_o first_o assumpt_n go_v before_o the_o 33._o of_o the_o ten_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o be_v also_o rational_a conclusion_n wherefore_o there_o be_v âound_v out_o two_o right_a line_n ad_fw-la and_o db_o incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o medial_a superficies_n and_o the_o parallelogram_n contain_v under_o they_o make_v a_o rational_a superficies_n which_o be_v require_v to_o be_v do_v ¶_o the_o 12._o problem_n the_o 35._o proposition_n to_o find_v out_o two_o right_a line_n incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o medial_a superficies_n and_o the_o parallelogram_n contain_v under_o they_o make_v also_o a_o medial_a superficies_n which_o parallelogram_n moreover_o shall_v be_v incommensurable_a to_o the_o superficies_n make_v of_o the_o square_n of_o those_o line_n add_v together_o take_v by_o the_o 32._o of_o the_o ten_o two_o medial_a line_n ab_fw-la and_o bc_o commensurable_a in_o power_n only_o construction_n comprehend_v a_o medial_a superficies_n so_o that_o let_v the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n bc_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n ab_fw-la and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n adb_o and_o let_v the_o rest_n of_o the_o construction_n be_v as_o it_o be_v in_o the_o two_o former_a proposition_n and_o forasmuch_o as_o by_o the_o 2_o part_n of_o the_o 18._o of_o the_o ten_o the_o line_n of_o be_v incommensurable_a in_o length_n unto_o the_o line_n fb_o demonstration_n therefore_o the_o line_n ad_fw-la be_v incommensurable_a in_o power_n unto_o the_o line_n db_o by_o that_o which_o be_v demonstrate_v in_o the_o proposition_n go_v before_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v medial_a therefore_o that_o also_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o which_o square_n be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la by_o the_o 47._o of_o the_o first_o be_v medial_a conclude_v and_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o fb_o be_v equal_a to_o either_o of_o the_o square_n of_o the_o line_n ebb_v and_o fd_o for_o by_o supposition_n the_o parallelogram_n contain_v under_o the_o line_n of_o and_o fb_o be_v equal_a to_o the_o square_n of_o the_o line_n ebb_n and_o the_o same_o parallelogram_n be_v equal_a to_o the_o square_n of_o the_o line_n df_o by_o the_o three_o part_n of_o the_o first_o assumpt_n go_v before_o the_o 33._o of_o the_o ten_o wherefore_o the_o line_n be_v be_v equal_a to_o the_o line_n df._n wherefore_o the_o line_n bc_o be_v double_a to_o the_o line_n fd._n wherefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v double_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd._n wherefore_o they_o be_v commensurable_a by_o the_o six_o of_o this_o book_n but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v medial_a by_o supposition_n wherefore_o also_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd_o be_v medial_a by_o the_o corollary_n of_o the_o 23_o of_o the_o ten_o but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd_v be_v by_o the_o four_o part_n of_o the_o first_o assumpt_n go_v before_o the_o 33._o of_o the_o ten_o equal_v to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o conclude_v wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o be_v also_o medial_a and_o forasmuch_o as_o the_o line_n ab_fw-la be_v incommensurable_a in_o length_n unto_o the_o line_n bc._n but_o the_o line_n bc_o be_v commensurable_a in_o length_n unto_o the_o line_n be._n wherefore_o by_o the_o 13â_n of_o
the_o ten_o the_o line_n ab_fw-la be_v incommensurable_a in_o length_n unto_o the_o line_n be._n wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o be_v by_o the_o first_o of_o the_o six_o and_o 10._o of_o this_o book_n but_o unto_o the_o square_n of_o the_o line_n ab_fw-la be_v equal_a the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o by_o the_o 47._o of_o the_o first_o and_o unto_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o be_v be_v equal_a that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd_v that_o be_v which_o be_v contain_v under_o ad_fw-la and_o db._n for_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o fd_o be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n ad_fw-la and_o db_o by_o the_o last_o part_n of_o the_o first_o assumpt_n go_v before_o the_o 33._o of_o this_o ten_o book_n conclude_v wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o db._n wherefore_o there_o be_v find_v out_o two_o right_a line_n ad_fw-la and_o db_o incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o medial_a superficies_n conclusion_n and_o the_o parallelogram_n contain_v under_o they_o make_v also_o a_o medial_a superficies_n which_o parallelogram_n moreover_o be_v incommensurable_a to_o the_o superficies_n compose_v of_o the_o square_n of_o those_o line_n add_v together_o which_o be_v require_v to_o be_v do_v the_o beginning_n of_o the_o senaries_n by_o composition_n ¶_o the_o 2d_o theorem_a the_o 36._o proposition_n if_o two_o rational_a line_n commensurable_a in_o power_n only_o be_v add_v together_o composition_n the_o whole_a line_n be_v irrational_a and_o be_v call_v a_o binomium_n or_o a_o binomial_a line_n ãâ¦ã_o b_o and_o bc_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n bc._n but_o unto_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v commensurable_a the_o parallelogram_n contain_v under_o ab_fw-la and_o bc_o twice_o by_o the_o 6._o of_o the_o ten_o wherefore_o that_o which_o be_v contain_v under_o ab_fw-la and_o bc_o twice_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n bc_o by_o the_o 13_o of_o the_o ten_o but_o unto_o the_o square_n of_o the_o line_n bc_o be_v commensurable_a that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o by_o the_o 15._o of_o the_o ten_o for_o by_o supposition_n the_o line_n ab_fw-la and_o bc_o be_v commensurable_a in_o power_n only_o wherefore_o by_o the_o 13._o of_o the_o ten_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o wherefore_o by_o the_o 16._o of_o the_o ten_o that_o which_o be_v contain_v under_o ab_fw-la and_o bc_o twice_o together_o with_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o which_o by_o the_o 4._o of_o the_o second_o be_v equal_a to_o the_o square_n of_o the_o whole_a line_n ac_fw-la be_v incommensurable_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o ab_fw-la and_o bc_o add_v together_o but_o that_o which_o be_v compose_v of_o the_o square_n of_o ab_fw-la and_o bc_o add_v together_o be_v rational_a for_o it_o be_v commensurable_a to_o either_o of_o the_o square_n of_o the_o line_n ab_fw-la and_o bg_o of_o which_o either_o of_o they_o be_v rational_a by_o supposition_n wherefore_o the_o square_a of_o the_o line_n ac_fw-la be_v by_o the_o 10._o definition_n of_o the_o ten_o irrational_a wherefore_o the_o line_n ac_fw-la also_o be_v irrational_a and_o be_v call_v a_o binomial_a line_n line_n this_o proposition_n show_v the_o generation_n and_o production_n of_o the_o second_o kind_n of_o irrational_a line_n which_o be_v call_v a_o binomium_n or_o a_o binomial_a line_n the_o definition_n whereof_o be_v full_o gather_v out_o of_o this_o proposition_n and_o that_o thus_o a_o binomium_n or_o a_o binomial_a line_n be_v a_o irrational_a line_n compose_v of_o two_o rational_a line_n commensurable_a the_o one_o to_o the_o other_o in_o power_n only_o and_o it_o be_v call_v a_o binomium_n that_o be_v have_v two_o name_n because_o it_o be_v make_v of_o two_o such_o line_n as_o of_o his_o part_n which_o be_v only_o commensurable_a in_o power_n and_o not_o in_o length_n and_o therefore_o each_o part_n or_o line_n or_o at_o the_o least_o the_o one_o of_o they_o as_o touch_v length_n be_v uncertain_a and_o unknown_a wherefore_o be_v join_v together_o their_o quantity_n can_v be_v express_v by_o any_o one_o number_n or_o name_n but_o each_o part_n remain_v to_o be_v several_o name_v in_o such_o sort_n as_o it_o may_v and_o of_o these_o binomial_a line_n there_o be_v six_o several_a kind_n line_n the_o first_o binomiall_n the_o second_o the_o three_o the_o four_o the_o five_o and_o the_o six_o of_o what_o nature_n and_o condition_n each_o of_o these_o be_v shall_v be_v know_v by_o their_o definitious_a which_o be_v afterward_o set_v in_o their_o due_a place_n ¶_o the_o 25._o theorem_a the_o 37._o proposition_n if_o two_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o rational_a superficies_n be_v add_v together_o the_o whole_a line_n be_v irrational_a and_o be_v call_v a_o first_o bimedial_a line_n let_v these_o two_o medial_a line_n ab_fw-la and_o bc_o be_v commensurable_a in_o power_n only_o and_o contain_v a_o rational_a superficies_n the_o 27._o of_o the_o ten_o teach_v to_o find_v out_o two_o such_o line_n be_v compose_v then_o i_o say_v that_o the_o whole_a line_n ac_fw-la be_v irrational_a for_o as_o ãâã_d say_v in_o the_o proposition_n next_o go_v before_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o ãâã_d ab_fw-la and_o bc_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twisâ_n wherefore_o by_o the_o 16._o of_o the_o ten_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o demonstration_n that_o be_v the_o square_a of_o the_o line_n ac_fw-la be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o iâ_z commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o once_o by_o the_o 6._o of_o the_o ten_o wherefore_o the_o square_a of_o the_o whole_a line_n ac_fw-la be_v by_o the_o 13â_n of_o the_o ten_o incommensurable_a âo_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o once_o but_o by_o supposition_n the_o line_n ab_fw-la and_o bc_o comprehend_v a_o rational_a superficies_n wherefore_o the_o square_a of_o the_o whole_a line_n ac_fw-la be_v irrational_a wherefore_o also_o the_o line_n ac_fw-la be_v irrational_a and_o it_o be_v call_v a_o first_o bimedial_a line_n the_o three_o irrational_a line_n which_o be_v call_v a_o first_o bimedial_a line_n be_v show_v by_o this_o proposition_n and_o the_o definition_n thereof_o be_v by_o it_o make_v manifest_a line_n which_o be_v this_o a_o first_o bimedial_a line_n be_v a_o irrational_a line_n which_o be_v compose_v of_o two_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o rational_a parallelogram_n it_o be_v call_v a_o first_o bimedial_a line_n by_o cause_n the_o two_o medial_a line_n or_o part_n whereof_o it_o be_v compose_v contain_v a_o rational_a superficies_n which_o be_v prefer_v before_o a_o irrational_a ¶_o the_o 26._o theorem_a the_o 38._o proposition_n if_o two_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o medial_a superficies_n be_v add_v together_o the_o whole_a line_n be_v irrational_a and_o be_v call_v a_o second_o bimedial_a line_n let_v these_o two_o medial_a line_n ab_fw-la and_o bc_o be_v commensurable_a in_o power_n only_o and_o contain_v a_o medial_a superficies_n the_o 28._o of_o the_o ten_o teach_v to_o findâ_n out_o two_o such_o line_n be_v add_v together_o construction_n then_o i_o say_v that_o the_o whole_a line_n ac_fw-la be_v irrational_a take_v a_o rational_a line_n de._n and_o by_o the_o 44._o of_o the_o first_o upon_o the_o line_n de_fw-fr apply_v the_o parallelogram_n df_o equal_a to_o the_o square_n of_o the_o line_n ac_fw-la who_o be_v other_o side_n let_v be_v the_o line_n dg_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la be_v by_o the_o 4._o of_o the_o second_o equal_a to_o that_o which_o be_v compose_v of_o
the_o square_n of_o the_o line_n ab_fw-la and_o bc_o demonstration_n together_o with_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o but_o the_o square_n of_o the_o line_n ac_fw-la be_v equal_a to_o the_o parallelogram_n df._n wherefore_o the_o parallelogram_n df_o be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o now_o then_o again_o by_o the_o 44_o of_o the_o first_o upon_o the_o line_n de_fw-fr apply_v the_o parallelogram_n eh_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o bc._n wherefore_o the_o parallelogram_n remain_v namely_o hf_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o and_o forasmuch_o as_o either_o of_o these_o line_n ab_fw-la and_o bc_o be_v medial_a therefore_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v also_o medial_a and_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v by_o the_o corollary_n of_o the_o â4_o of_o the_o ten_o medial_a for_o by_o the_o 6._o of_o this_o book_n it_o be_v commensurable_a ââ_o that_o ãâã_d be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o once_o which_o be_v by_o supposition_n medial_a ãâ¦ã_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v equal_a the_o parallelogram_n eh_o and_o unto_o that_o ãâã_d contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v equal_a the_o parallelogram_n he_o ãâ¦ã_o either_o of_o these_o parallelogram_n he_o and_o of_o be_v medial_a and_o they_o be_v apply_v upon_o the_o rational_a line_n ed._n wherefore_o by_o the_o 22._o of_o the_o ten_o either_o of_o these_o line_n dh_o and_o hg_o be_v a_o rational_a line_n and_o incommensurable_a in_o length_n unto_o the_o line_n de._n and_o forasmuch_o as_o by_o supposition_n the_o line_n ab_fw-la be_v incommensurable_a in_o length_n unto_o the_o line_n bc._n but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n be_v â_z so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o parallelogram_n which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10_o of_o this_o bookâ_n the_o square_a of_o the_o line_n ab_fw-la be_v incommensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bc._n but_o to_o the_o square_n of_o the_o line_n ab_fw-la be_v commensâable_a that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o by_o the_o 15._o of_o the_o ten_o for_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v commensurable_a when_o as_o the_o line_n ab_fw-la and_o bc_o be_v put_v to_o be_v commensurable_a in_o power_n only_o and_o to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v commensurable_a that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o by_o the_o 6_o of_o the_o ten_o wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o but_o to_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v equal_a the_o parallelogram_n eh_o and_o to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v equal_a the_o parallelogram_n fh_o wherefore_o the_o parallelogram_n fh_o be_v incommensurable_a to_o the_o parallelogram_n he._n wherefore_o the_o line_n dh_o be_v incommensurable_a in_o length_n to_o the_o line_n hg_o by_o the_o 1_o of_o the_o six_o and_o 10_o of_o this_o book_n and_o it_o be_v prove_v that_o they_o be_v rational_a line_n wherefore_o the_o line_n dh_o &_o hg_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o by_o the_o 36._o of_o the_o ten_o the_o whole_a line_n dg_o be_v irrational_o and_o the_o line_n de_fw-fr be_v rational_a but_o a_o rectangle_n superficies_n comprehend_v under_o a_o rational_a line_n and_o a_o irrational_a line_n be_v by_o the_o corollary_n add_v after_o the_o 21_o of_o the_o ten_o irrational_a wherefore_o the_o superficies_n df_o be_v irrational_a and_o the_o line_n also_o which_o contain_v it_o in_o power_n be_v irrational_a but_o the_o line_n ac_fw-la contain_v in_o power_n the_o superficies_n df._n wherefore_o the_o line_n ac_fw-la be_v irrational_a and_o it_o be_v call_v a_o second_o bimedial_a line_n this_o proposition_n show_v the_o generation_n of_o the_o four_o irrational_a line_n call_v a_o second_o bimedial_a line_n line_n the_o definition_n whereof_o be_v evident_a by_o this_o proposition_n which_o be_v thus_o a_o second_o bimedial_a line_n be_v a_o irrational_a line_n which_o be_v make_v of_o two_o medial_a line_n commensurable_a in_o power_n only_o join_v together_o which_o comprehend_v a_o medial_a superficies_n and_o it_o be_v call_v a_o second_o bimediall_n because_o the_o two_o medial_a line_n of_o which_o it_o be_v compose_v contain_v a_o medial_a superficies_n and_o not_o a_o rational_a now_o a_o medial_a be_v by_o nature_n &_o in_o knowledge_n after_o a_o rational_a ¶_o the_o 27._o theorem_a the_o 39_o proposition_n if_o two_o right_a line_n incommensurable_a in_o power_n be_v add_v together_o have_v that_o which_o be_v compose_v of_o the_o square_n of_o they_o rational_a and_o the_o parallelogram_n contain_v under_o they_o medial_a the_o whole_a right_a line_n be_v irrational_a and_o be_v call_v a_o great_a line_n let_v tâese_n two_o right_a line_n ab_fw-la and_o bc_o be_v incommensurable_a in_o power_n only_o and_o make_v that_o which_o be_v require_v in_o the_o proposition_n the_o 33._o of_o the_o ten_o teach_v to_o find_v out_o two_o such_o line_n be_v add_v together_o then_o i_o say_v that_o the_o whole_a line_n ac_fw-la be_v irrational_a demonstration_n for_o forasmuch_o as_o by_o supposition_n the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v medial_a therefore_o the_o parallelogram_n contain_v twice_o under_o the_o line_n ab_fw-la and_o bc_o be_v medial_a for_o that_o which_o be_v contain_v under_o ab_fw-la and_o bc_o twice_o be_v commensurable_a to_o that_o which_o be_v contain_v under_o ab_fw-la and_o bc_o once_o by_o the_o 6._o of_o the_o ten_o wherefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o that_o which_o be_v contain_v under_o ab_fw-la &_o bc_o twice_o be_v medial_a but_o by_o supposition_n that_o which_o iâ_z compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v rational_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v incommensurable_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc._n wherefore_o by_o the_o 16._o of_o the_o ten_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la &_o bc_o twice_o which_o be_v by_o the_o 4._o of_o the_o second_o equal_a to_o the_o square_n of_o the_o line_n ac_fw-la be_v incommensurable_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc._n but_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v rational_a wherefore_o the_o square_a of_o the_o whole_a line_n ac_fw-la be_v irrational_a wherefore_o the_o line_n ac_fw-la also_o be_v irrational_a and_o be_v call_v a_o great_a line_n and_o it_o be_v call_v a_o great_a line_n for_o that_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la &_o bc_o which_o be_v rational_a be_v great_a than_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o which_o be_v medial_a now_o it_o be_v meet_v that_o the_o name_n shall_v be_v give_v according_a to_o the_o property_n of_o the_o rational_a a_o assumpt_n this_o proposition_n teach_v the_o production_n of_o the_o five_o irrational_a line_n which_o be_v call_v a_o great_a line_n which_o be_v by_o the_o sense_n of_o this_o proposition_n thus_o define_v a_o great_a line_n be_v a_o irrational_a line_n which_o be_v compose_v of_o two_o right_a line_n which_o be_v incommensurable_a in_o power_n line_n the_o square_n of_o which_o add_v together_o make_v a_o rational_a superficies_n and_o the_o parallelogram_n which_o they_o contain_v be_v medial_a it_o be_v therefore_o call_v a_o great_a line_n as_o theon_n say_v because_o the_o square_n of_o the_o two_o line_n of_o which_o it_o be_v compose_v add_v together_o be_v rational_a be_v great_a than_o the_o medial_a superficies_n contain_v under_o they_o twice_o and_o it_o be_v convenient_a that_o the_o denomination_n be_v take_v of_o the_o propriety_n of_o the_o
let_v the_o same_o be_v e._n and_o as_o the_o number_n d_o be_v to_o the_o number_n ab_fw-la so_o let_v the_o square_n of_o the_o line_n e_o be_v to_o the_o square_n of_o fg._n wherefore_o by_o the_o 6._o of_o the_o ten_o the_o line_n e_o be_v commensurable_a in_o power_n to_o the_o line_n fg_o &_o the_o line_n e_o be_v rational_a wherefore_o also_o the_o line_n fg_o be_v rational_a and_o for_o that_o the_o number_n d_o have_v not_o âo_o the_o number_n ab_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o neither_o also_o shall_v the_o square_a of_o the_o line_n e_o have_v to_o the_o square_n of_o the_o line_n fg_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n e._n again_o as_o the_o number_n basilius_n be_v to_o the_o number_n ac_fw-la so_o let_v the_o square_n of_o the_o line_n fg_o be_v to_o the_o square_n of_o the_o line_n gh_o demonstration_n wherefore_o by_o the_o 6._o of_o the_o ten_o the_o square_a of_o the_o line_n fg_o be_v commensurable_a to_o the_o square_n of_o the_o line_n gh_o and_o the_o square_a of_o the_o line_n fg_o be_v rational_a wherefore_o the_o square_a of_o the_o line_n gh_o be_v also_o rational_a wherefore_o also_o the_o line_n gh_o be_v rational_a and_o for_o that_o the_o number_n ab_fw-la have_v not_o to_o the_o number_n ac_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o neither_o also_o have_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n gh_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n gh_o wherefore_o the_o line_n fg_o and_o gh_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n fh_o be_v a_o binomial_a line_n i_o say_v moreover_o that_o it_o be_v a_o six_o binomial_a line_n for_o for_o that_o as_o the_o number_n d_o be_v to_o the_o number_n ab_fw-la so_o be_v the_o square_a of_o the_o line_n e_o to_o the_o square_n of_o the_o line_n fg._n and_o as_o the_o number_n basilius_n be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n gh_o wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o the_o number_n d_z be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n e_o to_o the_o square_n of_o the_o line_n gh_o but_o the_o number_n d_o have_v not_o to_o the_o number_n ac_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o neither_o also_o have_v the_o square_a of_o the_o line_n e_o to_o the_o square_n of_o the_o line_n gh_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n e_o be_v incommensurable_a in_o length_n to_o the_o line_n gh_o and_o it_o be_v already_o prove_v that_o the_o line_n fg_o be_v also_o incommensurable_a in_o length_n to_o the_o line_n e._n wherefore_o either_o of_o these_o line_n fg_o and_o gh_o be_v incommensurable_a in_o length_n to_o the_o line_n e._n and_o for_o that_o as_o the_o number_n âa_o be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n gh_o therefore_o the_o square_n of_o the_o line_n fg_o be_v great_a than_o the_o square_a of_o the_o line_n gh_o unto_o the_o square_n of_o the_o line_n fg_o let_v the_o square_n of_o the_o line_n gh_o and_o king_n be_v equal_a wherefore_o by_o eversion_n of_o proportion_n as_o the_o number_n ab_fw-la be_v to_o the_o number_n bc_o so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n k._n but_o the_o number_n ab_fw-la have_v not_o to_o the_o number_n bc_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o neither_o also_o have_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n king_n that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n fg_o be_v incommensurable_a in_o length_n unto_o the_o line_n k._n wherefore_o the_o line_n fg_o be_v in_o power_n more_o than_o the_o line_n gh_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o it_o and_o the_o line_n fg_o and_o gh_o be_v rational_a commensurable_a in_o power_n only_o and_o neither_o of_o the_o line_n fg_o &_o gh_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v namely_o to_o e._n wherefore_o the_o line_n fh_o be_v a_o six_o binomial_a line_n which_o be_v require_v to_o be_v find_v out_o ¶_o a_o corollary_n add_v out_o of_o flussates_n by_o the_o 6._o forme_a proposition_n it_o iâ_z manifest_a hoâ_n ãâã_d divide_v any_o right_a line_n give_v into_o the_o name_n of_o every_o one_o of_o the_o sixâ_o foresay_a binomiall_n line_n flussates_n for_o if_o it_o be_v require_v to_o divide_v a_o right_a line_n give_v into_o a_o first_o binomial_a line_n then_o by_o the_o 48â_n of_o this_o book_n find_v out_o a_o first_o binomial_a line_n and_o this_o right_a line_n be_v so_o find_v out_o divide_v into_o his_o name_n you_o may_v by_o the_o 10._o of_o the_o six_o divide_v the_o right_a line_n give_v in_o like_a sort_n and_o so_o in_o the_o other_o five_o follow_v although_o i_o here_o note_n unto_o you_o this_o corollary_n out_o of_o ãâ¦ã_o in_o very_a conscience_n and_o of_o grateful_a âindeâ_n i_o be_o enforce_v to_o certify_v you_o that_o iâ_z any_o year_v before_o the_o travail_n of_o flussas_n upon_o euâliâââ_n geometrical_a element_n be_v publish_v the_o order_n how_o to_o divide_v not_o only_o the_o 6._o binomial_a line_n into_o their_o name_n but_o also_o to_o add_v to_o the_o 6._o residââls_v their_o due_a part_n ând_v fârthermore_o to_o divide_v all_o the_o other_o irrational_a line_n of_o this_o ten_o book_n into_o the_o part_n distinct_a of_o which_o they_o be_v compose_v with_o many_o other_o strange_a conclusion_n mathematical_a to_o the_o better_a understanding_n of_o this_o ten_o book_n and_o other_o mathematical_a book_n most_o necessary_a be_v by_o m._n john_n dee_fw-mi invent_v and_o demonstrate_v mathematicum_fw-la as_o in_o his_o book_n who_o title_n be_v tyrocinium_fw-la mathematicum_fw-la dedicate_v to_o petruâ_n nonnius_n an._n 1559._o may_v at_o large_a appear_v where_o also_o be_v one_o new_a art_n with_o sundry_a particular_a point_n whereby_o the_o mathematical_a science_n great_o may_v be_v enrich_v which_o his_o book_n i_o hope_v god_n will_v one_o day_n allow_v he_o opportunity_n to_o publish_v with_o diverse_a other_o his_o mathematical_a and_o metaphysical_a labour_n and_o invention_n ¶_o a_o assumpt_n be_v a_o right_a line_n be_v divide_v into_o two_o part_n how_o soever_o the_o rectangle_n parallelogram_n contain_v under_o both_o the_o part_n be_v the_o mean_a proportional_a between_o the_o square_n of_o the_o same_o part_n and_o the_o rectangle_n parallelogram_n contain_v under_o the_o whole_a line_n and_o one_o of_o the_o part_n be_v the_o mean_a proportional_a between_o the_o square_n of_o the_o whole_a line_n and_o the_o square_a of_o the_o say_a part_n book_n suppose_v that_o there_o be_v two_o square_n ab_fw-la and_o bc_o and_o let_v the_o line_n db_o and_o be_v so_o be_v put_v that_o they_o both_o make_v one_o right_a line_n wherefore_o by_o the_o 14._o of_o the_o first_o the_o line_n fb_o and_o bg_o make_v also_o both_o one_o right_a line_n and_o make_v perfect_a the_o parallelogram_n ac_fw-la then_o i_o say_v that_o the_o rectangle_n parallelogram_n dg_o be_v the_o mean_a proportional_a between_o the_o square_n ab_fw-la and_o bc_o and_o moreover_o that_o the_o parallelogram_n dc_o be_v the_o mean_a proportional_a between_o the_o square_n ac_fw-la and_o cb._n first_o the_o parallelogram_n agnostus_n be_v a_o square_a for_o forasmuch_o as_o the_o line_n db_o be_v equal_a to_o the_o line_n bf_o and_o the_o line_n be_v unto_o the_o line_n bg_o therefore_o the_o whole_a line_n de_fw-fr be_v equal_a to_o the_o whole_a line_n fg._n but_o the_o line_n de_fw-fr be_v equal_a to_o either_o of_o these_o line_n ah_o &_o kc_o and_o the_o line_n fg_o be_v equal_a to_o either_o of_o these_o line_n ak_o and_o hc_n by_o the_o 34._o of_o the_o first_o demonstration_n wherefore_o the_o parallelogram_n ac_fw-la be_v equilater_n it_o be_v also_o rectangle_n by_o the_o 29._o of_o the_o first_o wherefore_o by_o the_o 46._o of_o the_o first_o the_o parallelogram_n ac_fw-la be_v a_o square_a now_o for_o that_o as_o the_o line_n fb_o be_v to_o the_o line_n bg_o so_o be_v the_o line_n db_o
the_o definition_n of_o a_o first_o binomial_a line_n seâ_z before_o the_o 48._o proposition_n of_o this_o book_n the_o line_n dg_o be_v a_o first_o binomial_a line_n which_o be_v require_v to_o be_v prove_v this_o proposition_n and_o the_o five_o follow_v be_v the_o converse_n of_o the_o six_o former_a proposition_n ¶_o the_o 43._o theorem_a the_o 61._o proposition_n the_o square_a of_o a_o first_o bimedial_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o second_o binomial_a line_n svppose_v that_o the_o line_n ab_fw-la be_v a_o first_o bimedial_a line_n and_o let_v it_o be_v suppose_v to_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o of_o which_o let_v ac_fw-la be_v the_o great_a part_n take_v also_o a_o rational_a line_n de_fw-fr and_o by_o the_o 44._o of_o the_o first_o apply_v to_o the_o line_n de_fw-fr the_o parallelogram_n df_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la &_o make_v in_o breadth_n the_o line_n dg_o construction_n then_o i_o say_v that_o the_o line_n dg_o be_v a_o second_o binomial_a line_n let_v the_o same_o construction_n be_v in_o this_o that_o be_v in_o the_o proposition_n go_v before_o and_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o first_o bimedial_a line_n demonstration_n and_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o therefore_o by_o the_o 37._o of_o the_o ten_o the_o line_n ac_fw-la and_o cb_o be_v medial_a commensurable_a in_o power_n only_o comprehend_v a_o rational_a superficies_n wherefore_o also_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v medial_a wherefore_o the_o parallelogram_n dl_o be_v medial_a by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o and_o it_o be_v apply_v upon_o the_o rational_a line_n de._n wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n md_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n de._n again_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v rational_a therefore_o also_o the_o parallelogram_n mf_n be_v rational_a and_o it_o be_v apply_v unto_o the_o rational_a line_n ml_o wherefore_o the_o line_n mg_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n ml_o that_o be_v to_o the_o line_n de_fw-fr by_o the_o 20._o of_o the_o ten_o wherefore_o the_o line_n dm_o be_v incommensurable_a in_o length_n to_o the_o line_n mg_o and_o they_o be_v both_o rational_a wherefore_o the_o line_n dm_o and_o mg_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n dg_o be_v a_o binomial_a line_n line_n now_o rest_v to_o prove_v that_o it_o be_v a_o second_o binomial_a line_n forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v great_a than_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o the_o assumpt_n before_o the_o 60._o of_o this_o book_n therefore_o the_o parallelogram_n dl_o be_v great_a than_o the_o parallelogrrmme_a mf_n wherefore_o also_o by_o the_o first_o of_o the_o six_o the_o line_n dm_o be_v great_a than_o the_o line_n mg_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la be_v commensurable_a to_o the_o square_n of_o the_o line_n cb_o therefore_o the_o parallelogram_n dh_o be_v commensurable_a to_o the_o parallelogram_n kl_o wherefore_o also_o the_o line_n dk_o be_v commensurable_a in_o length_n to_o the_o line_n km_o and_o that_o which_o be_v contain_v under_o the_o line_n dk_o and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n mn_o that_o be_v to_o the_o four_o part_n of_o the_o square_n of_o the_o line_n mg_o wherefore_o by_o the_o 17._o of_o the_o ten_o the_o line_n dm_o be_v in_o power_n more_o than_o the_o line_n mg_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n dm_o and_o the_o line_n mg_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v namely_o to_o de._n wherefore_o the_o line_n dg_o be_v a_o second_o binomial_a line_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 44._o theorem_a the_o 62._o proposition_n the_o square_a of_o a_o second_o bimedial_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n thereof_o a_o three_o binomial_a line_n svppose_v that_o ab_fw-la be_v a_o second_o bimedial_a line_n and_o let_v ab_fw-la be_v suppose_v to_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o so_fw-mi that_o let_v ac_fw-la be_v the_o great_a part_n and_o take_v a_o rational_a line_n de._n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n de_fw-fr apply_v the_o parallelogram_n df_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o make_v in_o breadth_n the_o line_n dg_o then_o i_o say_v that_o the_o line_n dg_o be_v a_o three_o binomial_a line_n let_v the_o self_n same_o construction_n be_v in_o this_o that_o be_v in_o the_o proposition_n next_o go_v before_o and_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o second_o bimedial_a line_n construction_n and_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o demonstration_n therefore_o by_o the_o 38._o of_o the_o ten_o the_o line_n ac_fw-la and_o cb_o be_v medial_n commensurable_a in_o power_n only_o comprehend_v a_o medial_a superficies_n wherefore_o â _o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o be_v medial_a and_o it_o be_v equal_a to_o the_o parallelogram_n dl_o by_o construction_n wherefore_o the_o parallelogram_n dl_o be_v medial_a and_o be_v apply_v unto_o the_o rational_a line_n de_fw-fr wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n md_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n de._n and_o by_o the_o like_a reason_n also_o *_o the_o line_n mg_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ml_o that_o be_v to_o the_o line_n de._n wherefore_o either_o of_o these_o line_n dm_o and_o mg_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n de._n and_o forasmuch_o as_o the_o line_n ac_fw-la be_v incommensurable_a in_o length_n to_o the_o line_n cb_o but_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cb_o so_o by_o the_o assumpt_n go_v before_o the_o 22._o of_o the_o ten_o be_v the_o square_a of_o the_o line_n ac_fw-la to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb._n wherefore_o the_o square_a of_o the_o line_n ac_fw-la be_v incâmmmensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb._n wherefore_o that_o â¡_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o that_o be_v the_o parallelogram_n dl_o to_o the_o parallelogram_n mf_n wherefore_o by_o the_o first_o of_o the_o six_o and_o 10._o of_o the_o ten_o the_o line_n dm_o be_v incommensurable_a in_o length_n to_o the_o line_n mg_o and_o they_o be_v prove_v both_o rational_a line_n wherefore_o the_o whole_a line_n dg_o be_v a_o binomial_a line_n by_o the_o definition_n in_o the_o 36._o of_o the_o ten_o now_o rest_v to_o prove_v that_o it_o be_v a_o three_o binomial_a line_n as_o in_o the_o former_a proposition_n so_o also_o in_o this_o may_v we_o conclude_v that_o the_o line_n dm_o be_v great_a than_o the_o line_n mg_o and_o that_o the_o line_n dk_o be_v commensurable_a in_o length_n to_o the_o line_n km_o and_o that_o that_o which_o be_v contain_v under_o the_o line_n dk_o and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n mn_v wherefore_o the_o line_n dm_o be_v in_o power_n more_o than_o the_o line_n mg_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n dm_o and_o neither_o of_o the_o line_n dm_o nor_o mg_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n de._n wherefore_o by_o the_o definition_n of_o a_o three_o binomiâll_v line_v the_o line_n dg_o be_v a_o three_o binomial_a line_n which_o be_v require_v to_o be_v prove_v ¶_o here_o follow_v certain_a annotation_n by_o m._n dee_n make_v upon_o three_o place_n in_o the_o demonstration_n which_o be_v not_o very_o evident_a to_o young_a beginner_n â _o the_o square_n of_o the_o line_n ac_fw-la and_o câ_n be_v medial_n ãâã_d iâ_z teach_v after_o the_o 21â_n of_o this_o ten_o and_o therefore_o forasmuch_o as_o they_o be_v by_o supposition_n commensurable_a the_o one_o to_o the_o other_o by_o the_o 15._o of_o the_o ten_o the_o compound_n of_o they_o both_o be_v commensurable_a to_o each_o part_n but_o the_o part_n be_v medial_n therefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o the_o compound_n shall_v be_v
five_o binomial_a line_n but_o if_o neither_o of_o the_o line_n ae_n nor_o ebb_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v neither_o also_o of_o the_o line_n cf_o nor_o fd_o shall_v be_v commensurable_a in_o length_n to_o the_o same_o and_o so_o either_o of_o the_o line_n ab_fw-la and_o cd_o shall_v be_v a_o six_o binomial_a line_n a_o line_n therefore_o commensurable_a in_o length_n to_o a_o binomial_a line_n be_v also_o a_o binomial_a line_n of_o the_o self_n same_o order_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 49._o theorem_a the_o 67._o proposition_n a_o line_n commensurable_a in_o length_n to_o a_o bimedial_a line_n be_v also_o a_o bimedial_a line_n and_o of_o the_o self_n same_o order_n svppose_v that_o the_o line_n ab_fw-la be_v a_o bimedial_a line_n and_o unto_o the_o line_n ab_fw-la let_v the_o line_n cd_o be_v commensurable_a in_o length_n then_o i_o say_v that_o the_o line_n cd_o be_v a_o bimedial_a line_n and_o of_o the_o self_n order_n that_o the_o line_n ab_fw-la be_v divide_v the_o line_n ab_fw-la into_o his_o part_n in_o the_o point_n e._n construction_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o bimedial_a line_n and_o be_v divide_v into_o his_o part_n in_o the_o point_n e_o therefore_o by_o the_o 37._o and_o 38._o of_o the_o ten_o the_o line_n ae_n and_o ebb_n be_v medial_n commensurable_a in_o power_n only_o and_o by_o the_o 12._o of_o the_o six_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o let_v the_o line_n ae_n be_v to_o the_o line_n cf._n wherefore_o by_o the_o 19_o of_o the_o five_o the_o residue_n demonstration_n namely_o the_o line_n ebb_n be_v to_o the_o residue_n namely_o to_o the_o line_n fd_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o but_o the_o line_n ab_fw-la be_v commensurable_a in_o length_n to_o the_o line_n cd_o wherefore_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o line_n cf_o and_o the_o line_n ebb_n to_o the_o line_n fd._n now_o the_o line_n ae_n and_o ebb_n be_v medial_a wherefore_o by_o the_o 23._o of_o the_o ten_o the_o line_n cf_o and_o fd_o be_v also_o medial_a and_o for_o that_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd._n but_o the_o line_n ae_n and_o ebb_n be_v commensurable_a in_o power_n only_o wherefore_o the_o line_n cf_o and_o fd_o be_v also_o commensurable_a in_o power_n only_o and_o it_o be_v prove_v that_o they_o be_v medial_a wherefore_o the_o line_n cd_o be_v a_o bimedial_a line_n i_o say_v also_o that_o it_o be_v of_o the_o self_n same_o order_n that_o the_o line_n ab_fw-la be_v for_o for_o that_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd_o but_o as_o the_o line_n cf_o be_v to_o fd_o so_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd_o by_o the_o first_o of_o the_o six_o therefore_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o by_o the_o 11._o of_o the_o five_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd_o but_o as_o ae_n be_v to_o ebb_v so_o by_o the_o 1._o of_o the_o six_o be_v the_o square_a of_o the_o line_n ae_n to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n therefore_o by_o the_o 11._o of_o the_o five_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n so_o be_v the_o square_a of_o the_o line_n cf_o to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o square_n of_o the_o line_n cf_o so_o be_v that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n to_o that_o which_o be_v contain_v under_o the_o line_n cf_o &_o fd._n but_o the_o square_n of_o the_o line_n ae_n be_v commensurable_a to_o the_o square_n of_o the_o line_n cf_o because_o ae_n and_o cf_o be_v commensurable_a in_o length_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n in_o commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd._n if_o therefore_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v rational_a that_o be_v if_o the_o line_n ab_fw-la be_v a_o first_o bimedial_a line_n that_o also_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o be_v rational_a wherefore_o also_o the_o line_n cd_o be_v a_o first_o bimedial_a line_n but_o if_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v medial_a that_o be_v if_o the_o line_n ab_fw-la be_v a_o second_o bimedial_a line_n that_o also_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o be_v medial_a wherefore_o also_o the_o line_n cd_o be_v a_o second_o bimedial_a line_n wherefore_o the_o line_n ab_fw-la and_o cd_o be_v both_o of_o one_o and_o the_o self_n same_o order_n which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o flussates_n but_o first_o note_v by_o p._n montaâreus_n a_o line_n commensurable_a in_o power_n only_o to_o a_o bimedial_a line_n be_v also_o a_o bimedial_a line_n and_o of_o the_o self_n same_o order_n suppose_v that_o ab_fw-la be_v a_o bimedial_a line_n either_o a_o first_o or_o a_o second_o whereunto_o let_v the_o line_n gd_v be_v commensurable_a in_o power_n only_o flussetes_n take_v also_o a_o rational_a line_n ez_o upon_o which_o by_o the_o 45._o of_o the_o first_o apply_v a_o rectangle_n parallelogram_n equal_a to_o the_o square_n of_o the_o line_n ab_fw-la which_o let_v be_v ezfc_a and_o let_v the_o rectangle_n parallelogram_n cfih_n be_v equal_a to_o the_o square_n of_o the_o line_n gd_o and_o forasmuch_o as_o upon_o the_o rational_a line_n ez_o be_v apply_v a_o rectangle_n parallelogram_n of_o equal_a to_o the_o square_n of_o a_o first_o bimedial_a line_n therefore_o the_o other_o side_n thereof_o namely_o ec_o be_v a_o second_o binomial_a line_n by_o the_o 61._o of_o this_o book_n and_o forasmuch_o as_o by_o supposition_n the_o square_n of_o the_o line_n ab_fw-la &_o gd_o be_v commensurable_a therefore_o the_o parallelogram_n of_o and_o ci_o which_o be_v equal_a unto_o they_o be_v also_o commensurable_a and_o therefore_o by_o the_o 1._o of_o the_o six_o the_o line_n ec_o and_o change_z be_v commensurable_a in_o length_n but_o the_o line_n ec_o be_v a_o second_o binomial_a line_n wherefore_o the_o line_n change_z be_v also_o a_o second_o binomial_a line_n by_o the_o 66._o of_o this_o book_n and_o forasmuch_o as_o the_o superficies_n ci_o be_v contain_v under_o a_o rational_a line_n ez_o or_o cf_o and_o a_o second_o binomial_a line_n change_z therefore_o the_o line_n which_o contain_v it_o in_o power_n namely_o the_o line_n gd_o be_v a_o first_o bimedial_a line_n by_o the_o 55._o of_o this_o book_n and_o so_o be_v the_o line_n gd_o in_o the_o self_n same_o order_n of_o bimedial_a line_n that_o the_o line_n ab_fw-la be_v the_o like_a demonstration_n also_o will_v serve_v if_o the_o line_n ab_fw-la be_v suppose_v to_o bâ_z a_o second_o bimedial_a line_n for_o so_o shall_v it_o make_v the_o breadth_n ec_o a_o three_o binomial_a line_n whereunto_o the_o line_n change_z shall_v be_v commensurable_a in_o length_n and_o therefore_o ch_n also_o shall_v be_v a_o three_o binomial_a line_n by_o mean_n whereof_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ci_o namely_o the_o line_n gd_o shall_v also_o be_v a_o second_o bimedial_a line_n wherefore_o a_o line_n commensurable_a either_o in_o length_n or_o in_o power_n only_o to_o a_o bimedial_a line_n be_v also_o a_o bimedial_a line_n of_o the_o self_n same_o order_n but_o so_o be_v it_o not_o of_o necessity_n in_o binomial_a line_n for_o if_o their_o power_n only_o be_v commensurable_a it_o follow_v not_o of_o necessity_n that_o they_o be_v binomiall_n of_o one_o and_o the_o self_n same_o order_n note_n but_o they_o be_v each_o binomialls_n either_o of_o the_o three_o first_o kind_n or_o of_o the_o three_o last_o as_o for_o example_n suppose_v that_o ab_fw-la be_v a_o first_o binomial_a line_n who_o great_a name_n let_v be_v agnostus_n and_o unto_o ab_fw-la let_v the_o dz_o be_v commensurable_a in_o power_n only_o then_o i_o say_v that_o the_o line_n dz_o be_v not_o of_o the_o self_n same_o order_n that_o the_o line_n ab_fw-la be_v for_o if_o it_o be_v possible_a let_v the_o line_n dz_o be_v of_o the_o self_n same_o order_n that_o the_o line_n ab_fw-la be_v wherefore_o the_o line_n dz_o may_v
in_o like_a sort_n be_v divide_v as_o the_o line_n ab_fw-la be_v by_o that_o which_o have_v be_v demonstrate_v in_o the_o 66._o proposition_n of_o this_o bookeâ_n let_v it_o be_v so_o divide_v in_o the_o point_n e._n wherefore_o it_o can_v not_o be_v so_o divide_v in_o any_o other_o point_n by_o the_o 42â_n of_o this_o book_n and_o for_o that_o the_o line_n ab_fw-la ââ_o to_o the_o line_n dz_o as_o the_o line_n agnostus_n be_v to_o the_o line_n de_fw-fr but_o the_o line_n agnostus_n &_o de_fw-fr namely_o the_o great_a name_n be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o by_o the_o 10._o of_o this_o book_n for_o that_o they_o be_v commensurable_a in_o length_n to_o ãâã_d and_o the_o self_n same_o rational_a line_n by_o the_o first_o definition_n of_o binomial_a line_n wherefore_o the_o line_n ab_fw-la and_o dz_o be_v commensurable_a in_o length_n by_o the_o 13._o of_o this_o book_n but_o by_o supposition_n they_o be_v commensurable_a in_o power_n only_o which_o be_v impossible_a the_o self_n same_o demonstration_n also_o will_v serve_v if_o we_o suppose_v the_o line_n ab_fw-la to_o be_v a_o second_o binomial_a line_n for_o the_o less_o name_n gb_o and_o ez_o be_v commensurable_a in_o length_n to_o one_o and_o the_o self_n same_o rational_a line_n shall_v also_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o and_o therefore_o the_o line_n ab_fw-la and_o dz_o which_o be_v in_o the_o self_n same_o proportion_n with_o they_o shall_v also_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o which_o be_v contrary_a to_o the_o supposition_n far_o if_o the_o square_n of_o the_o line_n ab_fw-la and_o dz_o be_v apply_v unto_o the_o rational_a line_n cf_o namely_o the_o parallelogram_n et_fw-la and_o hl_o they_o shall_v make_v the_o breadthes_fw-mi change_z and_o hk_o first_o binomial_a line_n of_o what_o order_n soever_o the_o line_n ab_fw-la &_o dz_o who_o square_n be_v apply_v unto_o the_o rational_a line_n be_v by_o the_o 60._o of_o this_o book_n wherefore_o it_o be_v manifest_a that_o under_o a_o rational_a line_n and_o a_o first_o binomial_a line_n be_v confuse_o contain_v all_o the_o power_n of_o binomial_a line_n by_o the_o 54._o of_o this_o book_n wherefore_o the_o only_a commensuration_n of_o the_o power_n do_v not_o of_o necessity_n bring_v forth_o one_o and_o the_o self_n same_o order_n of_o binomial_a line_n the_o self_n same_o thing_n also_o may_v be_v prove_v if_o the_o line_n ab_fw-la and_o dz_o be_v suppose_v to_o be_v a_o four_o or_o five_o binomial_a line_n who_o power_n only_o be_v conmmensurable_a namely_o that_o they_o shall_v as_o the_o first_o bring_v forth_o binomial_a line_n of_o diverse_a order_n now_o forasmuch_o as_o the_o power_n of_o the_o line_n agnostus_n and_o gb_o and_o de_fw-fr and_o ez_o be_v commensurable_a &_o proportional_a it_o be_v manifest_a that_o if_o the_o line_n agnostus_n be_v in_o power_n more_o than_o the_o line_n gb_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o agnostus_n the_o line_n de_fw-fr also_o shall_v be_v in_o power_n more_o than_o the_o line_n ez_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n de_fw-fr by_o the_o 16._o of_o this_o book_n and_o so_o shall_v the_o two_o line_n ab_fw-la and_o dz_o be_v each_o of_o the_o three_o first_o binomial_a line_n but_o if_o the_o line_n agnostus_n be_v in_o power_n more_o than_o the_o line_n gb_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n agnostus_n the_o line_n de_fw-fr shall_v also_o be_v in_o powâr_a ãâã_d than_o the_o line_n ez_o by_o the_o square_n of_o a_o line_n incomensurable_a in_o length_n unto_o the_o line_n de_fw-fr by_o the_o self_n same_o proposition_n and_o so_o shall_v eke_v of_o the_o line_n ab_fw-la and_o dz_o be_v of_o the_o three_o last_o binomial_a line_n but_o why_o it_o be_v not_o so_o in_o the_o three_o and_o six_o binomial_a line_n the_o reason_n be_v for_o that_o in_o they_o neither_o of_o the_o nameâ_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v fc_n ¶_o the_o 50._o theorem_a the_o 68_o proposition_n a_o line_n commensurable_a to_o a_o great_a line_n be_v also_o a_o great_a line_n svppose_v that_o the_o line_n ab_fw-la be_v a_o great_a line_n and_o unto_o the_o line_n ab_fw-la let_v the_o line_n cd_o be_v commensurable_a construction_n then_o i_o say_v that_o the_o line_n cd_o also_o be_v a_o great_a line_n divide_v the_o line_n ab_fw-la into_o his_o part_n in_o the_o point_n e._n wherefore_o by_o the_o 39_o of_o the_o ten_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a and_o let_v the_o rest_n of_o the_o construction_n be_v in_o this_o as_o it_o be_v in_o the_o former_a and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o demonstration_n so_o be_v the_o line_n ae_n to_o the_o line_n cf_o &_o thâ_z line_n ebb_n to_o the_o line_n fd_o but_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n cd_o by_o supposition_n wherefore_o the_o line_n ae_n be_v commensurable_a to_o the_o line_n cf_o and_o the_o line_n ebb_n to_o the_o line_n fd._n and_o for_o that_o as_o the_o line_n ae_n be_v to_o the_o line_n cf_o so_o be_v the_o line_n ebb_n to_o the_o line_n fd._n therefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd._n wherefore_o by_o composition_n also_o by_o the_o 18._o of_o the_o five_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cd_o to_o the_o line_n fd._n wherefore_o by_o the_o 22._o of_o the_o six_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v to_o the_o square_n of_o the_o line_n ebb_n so_o be_v the_o square_a of_o the_o line_n cd_o to_o the_o square_n of_o the_o line_n fd._n and_o in_o like_a sort_n may_v we_o prove_v that_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v to_o the_o square_n of_o the_o line_n ae_n so_o be_v the_o square_a of_o the_o line_n cd_o to_o the_o square_n of_o the_o line_n cf._n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v to_o the_o square_n of_o the_o line_n ae_n and_o ebb_n so_o be_v the_o square_a of_o the_o line_n cd_o to_o the_o square_n of_o the_o line_n cf_o and_o fd._n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v to_o the_o square_n of_o the_o line_n cd_o so_o be_v the_o square_n of_o the_o line_n ae_n and_o ebb_n to_o the_o square_n of_o the_o line_n cf_o and_o fd._n but_o the_o square_n of_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o square_n of_o the_o line_n cd_o for_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n cd_o by_o supposition_n wherefore_o also_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v commensurable_a to_o the_o square_n of_o the_o line_n cf_o and_o fd._n but_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a and_o be_v add_v together_o be_v rational_a wherefore_o the_o square_n of_o the_o line_n cf_o and_o fd_o be_v incommensurable_a &_o be_v add_v together_o be_v also_o rational_a and_o in_o like_a sort_n may_v we_o prove_v that_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n twice_o be_v commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o twice_o but_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n twice_o be_v medial_a wherefore_o also_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o twice_o be_v medial_a wherefore_o the_o line_n cf_o and_o fd_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a wherefore_o by_o the_o 39_o of_o the_o ten_o the_o whole_a line_n cd_o be_v irrational_a &_o be_v call_v a_o great_a line_n a_o line_n therefore_o commensurable_a to_o a_o great_a line_n be_v also_o a_o great_a line_n an_o other_o demonstration_n of_o peter_n montaureus_n to_o prove_v the_o same_o suppose_v that_o the_o line_n ab_fw-la be_v a_o great_a line_n and_o unto_o it_o let_v the_o line_n cd_o be_v commensurable_a any_o way_n that_o be_v either_o both_o in_o length_n and_o in_o power_n or_o else_o in_o power_n only_o montaureus_n then_o i_o say_v that_o the_o line_n cd_o also_o be_v a_o great_a
power_n a_o rational_a and_o a_o medial_a which_o be_v require_v to_o be_v demonstrate_v an_o other_o demonstration_n of_o the_o same_o after_o campane_n suppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a campane_n whereunto_o let_v the_o line_n gd_v be_v commensurable_a either_o in_o length_n and_o power_n or_o in_o power_n only_o then_o i_o say_v that_o the_o line_n gd_o be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a take_v a_o rational_a line_n ez_o upon_o which_o by_o the_o 45._o of_o the_o first_o apply_v a_o rectangle_n parallelogram_n ezfc_n equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o upon_o the_o line_n cf_o which_o be_v equal_a to_o the_o line_n ez_o apply_v the_o parallelogram_n fchi_n equal_a to_o the_o square_n of_o the_o line_n gdâ_n and_o let_v the_o breadth_n of_o the_o say_a parallelogram_n be_v the_o line_n eglantine_n and_o ch._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n gd_o at_o the_o least_o in_o power_n only_o therefore_o the_o parallelogram_n of_o and_o fh_o which_o be_v equal_a to_o their_o square_n shall_v be_v commensurable_a wherefore_o by_o the_o 1._o of_o the_o six_o the_o right_a line_n ec_o and_o change_z be_v commensurable_a in_o length_n and_o forasmuch_o as_o the_o parallelogram_n of_o which_o be_v equal_a to_o the_o square_n of_o the_o line_n aâ_z which_o contain_v in_o power_n â_o rational_a and_o a_o medial_a be_v apply_v upon_o the_o rational_a ez_o make_v in_o breadth_n the_o line_n ec_o therefore_o the_o line_n ec_o be_v a_o five_o binomial_a line_n by_o the_o 64._o of_o this_o book_n unto_o which_o line_n ec_o the_o line_n change_z be_v commensurable_a in_o length_n wherefore_o by_o the_o 66._o of_o this_o book_n the_o line_n change_z be_v also_o a_o five_o binomial_a line_n and_o forasmuch_o as_o the_o superficies_n ci_o be_v contain_v under_o the_o rational_a line_n ez_o that_o be_v cf_o and_o a_o five_o binomall_a line_n change_z therefore_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ci_o which_o by_o supposition_n be_v the_o line_n gd_o be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a by_o the_o 58._o of_o this_o book_n a_o line_n therefore_o commensurable_a to_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a etc_n etc_n ¶_o the_o 52._o theorem_a the_o 70._o proposition_n a_o line_n commensurable_a to_o a_o line_n contain_v in_o power_n two_o medial_n be_v also_o a_o line_n contain_v in_o power_n two_o medial_n svppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n and_o unto_o the_o line_n ab_fw-la let_v the_o line_n cd_o be_v commensurable_a whether_o in_o length_n &_o power_n or_o in_o power_n only_o then_o i_o say_v that_o the_o line_n cd_o be_v a_o line_n contain_v in_o power_n two_o medial_n forasmuch_o as_o the_o line_n ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n construction_n let_v it_o be_v divide_v into_o his_o part_n in_o the_o point_n e._n wherefore_o by_o the_o 41._o of_o the_o ten_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o that_o also_o which_o be_v contain_v under_o they_o medial_a and_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n &_o ebb_n be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n let_v the_o self_n same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a demonstration_n and_o in_o like_a sort_n may_v we_o prove_v that_o the_o line_n cf_o &_o fd_o be_v incommensurable_a in_o power_n and_o that_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_v add_v together_o be_v commensurable_a to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o and_o that_o that_o also_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o be_v medial_a by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o and_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o be_v medial_a by_o the_o same_o corollary_n â_o and_o moreover_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o &_o fd_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o the_o line_n cd_o be_v a_o line_n contain_v in_o power_n two_o medial_n which_o be_v require_v to_o be_v prove_v ¶_o a_o assumpt_n add_v by_o montaureus_n assumpt_n that_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o be_v thus_o prove_v for_o because_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_v add_v together_o be_v to_o the_o square_n of_o the_o line_n ae_n so_o be_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o to_o the_o square_n of_o the_o line_n cf_o as_o it_o be_v prove_v in_o the_o proposition_n go_v before_o therefore_o alternate_o as_o that_o which_o be_v make_v of_o the_o square_n of_o ae_n and_o ebb_v add_v together_o be_v to_o that_o which_o be_v make_v of_o the_o square_n of_o cf_o and_o fd_o add_v together_o so_o be_v the_o square_a of_o the_o line_n ae_n to_o the_o square_n of_o the_o line_n cf._n but_o before_o namely_o in_o the_o 68_o proposition_n it_o be_v prove_v that_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o square_n of_o the_o line_n cf_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o alternate_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n so_o be_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd._n but_o by_o supposition_n that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n &_o ebb_n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o be_v incommensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd_o which_o be_v require_v to_o be_v prove_v an_o other_o demonstration_n after_o campane_n suppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n whereunto_o let_v the_o line_n gd_v be_v commensurable_a either_o in_o length_n and_o in_o power_n or_o in_o power_n only_o campanâ_n then_o i_o say_v that_o the_o line_n gd_o be_v a_o line_n contain_v in_o power_n two_o medial_n let_v the_o same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a and_o forasmuch_o as_o the_o parallelogram_n of_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o be_v apply_v upon_o a_o rational_a line_n ez_o it_o make_v the_o breadth_n ec_o a_o six_o binomial_a line_n by_o the_o 65._o of_o this_o book_n and_o forasmuch_o as_o the_o parallelogram_n of_o &_o ci_o which_o be_v equal_a unto_o the_o square_n of_o the_o line_n ab_fw-la and_o gd_o which_o be_v suppose_v to_o be_v commensurable_a be_v commensurable_a therefore_o the_o line_n ec_o and_o change_z be_v commensurable_a in_o length_n by_o the_o first_o of_o the_o six_o but_o ec_o be_v a_o six_o binomial_a line_n wherefore_o change_z also_o be_v a_o six_o binomial_a line_n by_o the_o 66._o of_o this_o book_n and_o forasmuch_o as_o the_o superficies_n ci_o be_v contain_v under_o the_o rational_a line_n cf_o and_o a_o six_o binomial_a line_n change_z therefore_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ci_o namely_o the_o line_n gd_o be_v a_o line_n contain_v in_o power_n two_o medial_n by_o the_o 59_o of_o
two_o other_o proposition_n go_v next_o before_o it_o so_o far_o misplace_v that_o where_o they_o be_v word_n for_o word_n before_o duâly_o place_v be_v the_o 105._o and_o 106._o yet_o here_o after_o the_o book_n end_v they_o be_v repeat_v with_o the_o number_n of_o 116._o and_o 117._o proposition_n zambert_n therein_o be_v more_o faithful_a to_o follow_v as_o he_o find_v in_o his_o greek_n example_n than_o he_o be_v skilful_a or_o careful_a to_o do_v what_o be_v necessary_a yea_o and_o some_o greek_n write_v ancient_a copy_n have_v they_o not_o so_o though_o in_o deed_n they_o be_v well_o demonstrate_v yet_o truth_n disord_v be_v half_a disgracedâ_n especial_o where_o the_o pattern_n of_o good_a order_n by_o profession_n be_v avouch_v to_o be_v but_o through_o ignorance_n arrogancy_n and_o âemerltie_n of_o unskilful_a method_n master_n many_o thing_n remain_v yet_o in_o these_o geometrical_a element_n undue_o tumble_v in_o though_o true_a yet_o with_o disgrace_n which_o by_o help_n of_o so_o many_o wit_n and_o hability_n of_o such_o as_o now_o may_v have_v good_a cause_n to_o be_v skilful_a herein_o will_v i_o hope_v ere_o long_o be_v take_v away_o and_o thing_n of_o importance_n want_v supply_v the_o end_n of_o the_o ten_o book_n of_o euclides_n element_n ¶_o the_o eleven_o book_n of_o euclides_n element_n book_n hitherto_o have_v âvclidâ_n in_o thâsâ_n former_a book_n with_o a_o wonderful_a method_n and_o order_n entreat_v of_o such_o kind_n of_o figure_n superficial_a which_o be_v or_o may_v be_v describe_v in_o a_o superficies_n or_o plain_a and_o have_v teach_v and_o set_v forth_o their_o property_n nature_n generation_n and_o production_n even_o from_o the_o first_o root_n ground_n and_o beginning_n of_o they_o namely_o from_o a_o point_n which_o although_o it_o be_v indivisible_a yet_o be_v it_o the_o begin_n of_o all_o quantity_n continual_a and_o of_o it_o and_o of_o the_o motion_n and_o slow_v thereof_o be_v produce_v a_o line_n and_o consequent_o all_o quantity_n continual_a as_o all_o figure_n plain_a and_o solid_a what_o so_o ever_o euclid_n therefore_o in_o his_o first_o book_n begin_v with_o it_o boot_n and_o from_o thence_o go_v he_o to_o a_o line_n as_o to_o a_o thing_n most_o simple_a next_o unto_o a_o point_n then_o to_o a_o superficies_n and_o to_o angle_n and_o so_o through_o the_o whole_a first_o book_n boââe_n he_o entreat_v of_o these_o most_o simple_a and_o plain_a ground_n in_o the_o second_o book_n he_o entreat_v further_o ââoâe_n and_o go_v unto_o more_o hard_a matter_n and_o teach_v of_o division_n of_o line_n and_o of_o the_o multiplication_n of_o line_n and_o of_o their_o part_n and_o of_o their_o passion_n and_o property_n and_o for_o that_o rightline_v â_z be_v far_o distant_a in_o nature_n and_o property_n from_o round_a and_o circular_a figure_n in_o the_o three_o book_n he_o instruct_v the_o reader_n of_o the_o nature_n and_o condition_n of_o circle_n booâe_n in_o the_o four_o book_n he_o compare_v figure_n of_o right_a line_n and_o circle_n together_o bâoâe_n and_o teach_v how_o to_o describe_v a_o figure_n of_o right_a line_n with_o in_o or_o about_o a_o circle_n and_o contrariwise_o a_o circle_n with_o in_o or_o about_o a_o rectiline_a figure_n in_o the_o five_o book_n he_o search_v out_o the_o nature_n of_o proportion_n a_o matter_n of_o wonderful_a use_n and_o deep_a consideration_n boââe_n for_o that_o otherwise_o he_o can_v not_o compare_v âigure_n with_o figure_n or_o the_o side_n of_o figure_n together_o for_o whatsoever_o be_v compare_v to_o any_o other_o thing_n be_v compare_v unto_o it_o undoubted_o under_o some_o kind_n of_o proportion_n wherefore_o in_o the_o six_o book_n he_o compare_v figure_n together_o booâe_n one_o to_o a_o other_o likewise_o their_o side_n and_o for_o that_o the_o nature_n of_o proportion_n can_v not_o be_v full_o and_o clear_o see_v without_o the_o knowledge_n of_o number_n wherein_o it_o be_v first_o and_o chief_o find_v in_o the_o seven_o bookâ_n eight_o booââ_n and_o nine_o book_n book_n he_o entreatâth_v of_o number_n &_o of_o the_o kind_n and_o property_n thereof_o and_o because_o that_o the_o side_n of_o solid_a body_n for_o the_o most_o part_n be_v of_o such_o sort_n that_o compare_v together_o they_o have_v such_o proportion_n the_o one_o to_o the_o other_o booâe_n which_o can_v not_o be_v express_v by_o any_o number_n certain_a and_o therefore_o be_v call_v irrational_a line_n he_o in_o the_o ten_o book_n have_v write_v &_o teach_v which_o lineâ_z be_v commensurable_a or_o incommensurable_a the_o one_o to_o the_o other_o and_o of_o the_o diversity_n of_o kind_n of_o irrational_a line_n with_o all_o the_o condition_n &_o propriety_n of_o they_o and_o thus_o have_v euclid_n in_o these_o ten_o foresay_a book_n full_o &_o most_o plenteous_o in_o a_o marvelous_a order_n teach_v whatsoever_o seem_v necessary_a and_o requisite_a to_o the_o knowledge_n of_o all_o superficial_a figure_n of_o what_o sort_n &_o form_n so_o ever_o 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they_o you_o bow_v and_o bend_v they_o according_o you_o shall_v most_o plain_o and_o manifest_o see_v the_o form_n and_o shape_n of_o these_o body_n even_o as_o their_o definition_n show_v and_o it_o shall_v be_v very_o necessary_a for_o you_o to_o hadââtore_v of_o that_o past_v paper_n by_o you_o for_o so_o shall_v yoâ_n upon_o it_o ãâ¦ã_o the_o form_n of_o other_o body_n as_o prism_n and_o parallelipopedon_n ãâ¦ã_o set_v forth_o in_o these_o five_o book_n follow_v and_o see_v the_o very_a ãâã_d of_o thâse_a body_n there_o mention_v which_o will_v make_v these_o book_n concern_v body_n as_o easy_a unto_o you_o as_o be_v the_o other_o book_n who_o figure_n you_o may_v plain_o see_v upon_o a_o plain_a superficies_n describe_v thiâ_z figurâ_n which_o consi_v of_o twâluââquilââââ_n and_o equiangle_n pântâââââ_n upoâ_n the_o foresay_a matter_n and_o fine_o cut_v as_o before_o be_v ââught_v tâââlâuân_v line_n contain_v within_o thâ_z figurâ_n and_o bow_n and_o fold_n the_o penââgonâ_n according_o and_o they_o will_v so_o close_o together_o thaâ_z thây_n will_v ââkâ_n thâ_z very_a form_n of_o a_o dodecahedron_n dâdââââedron_n icosaâedron_n if_o you_o describe_v this_o figure_n which_o consist_v of_o twenty_o equilater_n and_o equiangle_n triangle_n upon_o the_o foresay_a matter_n and_o fine_o cut_v as_o before_o be_v show_v the_o ninâtâne_a line_n which_o be_v contain_v within_o the_o figure_n and_o then_o bow_n and_o fold_n they_o according_o they_o will_v in_o such_o sort_n close_o together_o that_o therâ_n will_v be_v make_v a_o perfect_v form_n of_o a_o icosahedron_n because_o in_o these_o five_o book_n there_o be_v sometime_o require_v other_o body_n beside_o the_o foresay_a five_o regular_a body_n as_o pyramid_n of_o diverse_a form_n prism_n and_o other_o i_o have_v here_o set_v forth_o three_o figure_n of_o three_o sundry_a pyramid_n one_o have_v to_o his_o base_a a_o triangle_n a_o other_o a_o quadrangle_n figure_n the_o other_o â_o pentagonâ_n which_o if_o you_o describe_v upon_o the_o foresay_a matter_n &_o fine_o cut_v as_o it_o be_v before_o teach_v the_o line_n contain_v within_o each_o figure_n namely_o in_o the_o first_o three_o line_n in_o the_o second_o four_o line_n and_o in_o the_o three_o five_o line_n and_o so_o bend_v and_o fold_v they_o according_o they_o will_v so_o close_o together_o at_o the_o top_n that_o they_o will_v âake_v pyramid_n of_o that_o form_n that_o their_o base_n be_v of_o and_o if_o you_o conceive_v well_o the_o describe_v of_o these_o you_o may_v most_o easy_o describe_v the_o body_n of_o a_o pyramid_n of_o what_o form_n so_o ever_o you_o will._n because_o these_o five_o book_n follow_v be_v somewhat_o hard_o for_o young_a beginner_n by_o reason_n they_o must_v in_o the_o figure_n describe_v in_o a_o plain_a imagine_v line_n and_o superficiece_n to_o be_v elevate_v and_o erect_v the_o one_o to_o the_o other_o and_o also_o conceive_v solid_n or_o body_n which_o for_o that_o they_o have_v not_o hitherto_o be_v acquaint_v with_o will_v at_o the_o first_o sight_n be_v somewhat_o strange_a unto_o they_o i_o have_v for_o their_o more_o âase_a in_o this_o eleven_o book_n at_o the_o end_n of_o the_o demonstration_n of_o every_o proposition_n either_o set_v new_a figure_n if_o they_o concern_v the_o elevate_n or_o erect_v of_o line_n or_o superficiece_n or_o else_o if_o they_o concern_v body_n i_o have_v show_v how_o they_o shall_v describe_v body_n to_o be_v compare_v with_o the_o construction_n and_o demonstration_n of_o the_o proposition_n to_o they_o belong_v and_o if_o they_o diligent_o weigh_v the_o manner_n observe_v in_o this_o eleven_o book_n touch_v the_o description_n of_o new_a figure_n agree_v with_o the_o figure_n describe_v in_o the_o plain_a it_o shall_v not_o be_v hard_o for_o they_o of_o themselves_o to_o do_v the_o like_a in_o the_o other_o book_n follow_v when_o they_o come_v to_o a_o proposition_n which_o concern_v either_o the_o elevate_n or_o erect_v of_o line_n and_o superficiece_n or_o any_o kind_n of_o body_n to_o be_v imagine_v ¶_o the_o 1._o theorem_a the_o 1._o proposition_n that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n &_o part_v elevate_v upward_o be_v impossible_a for_o if_o it_o be_v possible_a let_v part_n of_o the_o right_a line_n abc_n namely_o the_o part_n ab_fw-la be_v in_o a_o ground_n plain_a superficies_n and_o the_o other_o part_n thereof_o namely_o bc_o be_v elevate_v upward_o and_o produce_v direct_o upon_o the_o ground_n plain_a superficies_n the_o right_a line_n ab_fw-la beyond_o the_o point_n b_o unto_o the_o point_n d._n wherefore_o unto_o two_o right_a line_n give_v abc_n and_o abdella_n the_o line_n ab_fw-la be_v a_o common_a section_n or_o part_n which_o be_v impossible_a impossibility_n for_o a_o right_a line_n can_v not_o touch_v a_o right_a line_n in_o ãâã_d point_n then_o one_o uâlesse_o those_o right_n be_v exact_o agree_v and_o lay_v the_o one_o upon_o the_o other_o wherefore_o that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n and_o part_v elevate_v upward_o be_v impossible_a which_o be_v require_v to_o be_v prove_v this_o figure_n more_o plain_o set_v forth_o the_o foresay_a demonstration_n if_o you_o elevate_v the_o superficies_n wherein_o the_o line_n bc._n an_o other_o demonstration_n after_o flâsââs_n if_o it_o be_v possible_a let_v there_o be_v a_o right_a line_n abg_o who_o part_n ab_fw-la let_v be_v in_o the_o ground_n plain_a superficies_n aed_n flussas_n and_o let_v the_o rest_n thereof_o bg_o be_v elevate_v on_o high_a that_o be_v without_o the_o plain_n aed_n then_o i_o say_v that_o abg_o be_v not_o one_o right_a line_n for_o forasmuch_o as_o aed_n be_v a_o plain_a superficies_n produce_v direct_o &_o equal_o upon_o the_o say_a plain_n aed_n the_o right_a line_n ab_fw-la towards_o d_z which_o by_o the_o 4._o definition_n of_o the_o first_o shall_v be_v a_o right_a line_n and_o from_o some_o one_o point_n of_o the_o right_a line_n abdella_n namely_o from_o c_o draâ_n unto_o the_o point_n g_o a_o right_a line_n cg_o wherefore_o in_o the_o triangle_n ãâ¦ã_o the_o outward_a angââ_n abâ_n be_v equal_a to_o the_o two_o inward_a and_o opposite_a angle_n by_o the_o 32._o of_o the_o first_o and_o therefore_o it_o be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o same_o wherefore_o the_o line_n abg_o forasmuch_o as_o it_o make_v a_o angle_n be_v not_o â_o right_n line_n wherefore_o that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n and_o part_v elevate_v upward_o be_v impossible_a if_o you_o mark_v well_o the_o figure_n before_o add_v for_o the_o plain_a declaration_n of_o euclides_n demonstration_n iâ_z will_v not_o be_v hard_o for_o you_o to_o coââââe_v this_o figure_n which_o ulyss_n put_v for_o his_o demonstration_n â_o wherein_o
two_o prism_n under_o equal_a altitude_n &_o the_o one_o have_v to_o his_o base_a a_o parallelogram_n and_o the_o other_o a_o triangle_n and_o if_o the_o parallelogram_n be_v double_a to_o the_o triangle_n those_o prism_n be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o these_o two_o prism_n abcdef_n ghkmon_n be_v under_o equal_a altitude_n and_o let_v the_o one_o have_v to_o his_o base_a the_o parallelogram_n ac_fw-la and_o the_o other_o the_o triangle_n ghk_o and_o let_v the_o parallelogram_n ac_fw-la be_v double_a to_o the_o triangle_n ghk_o then_o i_o say_v that_o the_o prism_n abcdef_n be_v equal_a to_o the_o prism_n ghkmon_n construction_n make_v perfect_v the_o parallelipipedon_n axe_n &_o go_v and_o forasmuch_o as_o the_o parallelogram_n ac_fw-la be_v double_a to_o the_o triangle_n ghk_o demonstration_n but_o the_o parallelogram_n gh_o be_v also_o by_o the_o 41._o of_o the_o first_o double_a to_o the_o triangle_n ghk_o wherefore_o the_o parallelogram_n ac_fw-la be_v equal_a to_o the_o parallelogram_n gh_o but_o parallelipipedon_n consist_v upon_o equal_a base_n and_o under_o one_o and_o the_o self_n same_o altitude_n be_v equal_a the_o one_o to_o the_o other_o by_o the_o 31._o of_o the_o eleven_o wherefore_o the_o solid_a axe_n be_v equal_a to_o the_o solid_a go_v but_o the_o half_a of_o the_o solid_a axe_n be_v the_o prism_n abcdef_n and_o the_o half_a of_o the_o solid_a go_v be_v the_o prism_n ghkmon_n wherefore_o the_o prism_n abcdef_n be_v equal_a to_o the_o prism_n ghkmon_n if_o therefore_o there_o be_v two_o prism_n under_o equal_a altitude_n and_o the_o one_o have_v to_o his_o base_a a_o parallelogram_n &_o the_o other_o a_o triangle_n and_o if_o the_o parallelogram_n be_v double_a to_o the_o triangle_n those_o prism_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v this_o proposition_n and_o the_o demonstration_n thereof_o be_v not_o hard_a to_o conceive_v by_o the_o former_a figure_n but_o you_o may_v for_o your_o full_a understanding_n of_o they_o take_v two_o equal_a parallelipipedon_n equilater_n and_o equiangle_n the_o one_o to_o the_o other_o describe_v of_o past_v paper_n or_o such_o like_a matter_n and_o in_o the_o base_a of_o the_o one_o parallelipipedon_n draw_v a_o diagonal_a line_n and_o draw_v a_o other_o diagonal_a line_n in_o the_o upper_a superficies_n opposite_a unto_o the_o say_a diagonal_a line_n draw_v in_o the_o base_a and_o in_o one_o of_o the_o parallelogram_n which_o be_v set_v upon_o the_o base_a of_o the_o other_o parallelipipedon_n draw_v a_o diagonal_a line_n and_o draw_v a_o other_o diagonal_a line_n in_o the_o parallelogram_n opposite_a to_o the_o same_o for_o so_o if_o you_o extend_v plain_a superficiece_n by_o those_o diagonal_a line_n there_o will_v be_v make_v two_o prism_n in_o each_o body_n you_o must_v take_v heed_n that_o you_o put_v for_o the_o base_n of_o each_o of_o these_o parallelipipedon_n equal_a parallelogram_n and_o then_o note_v they_o with_o letter_n according_a to_o the_o letter_n of_o the_o figure_n before_o describe_v in_o the_o plain_a and_o compare_v they_o with_o the_o demonstration_n and_o they_o will_v make_v both_o it_o and_o the_o proposition_n very_o clear_o unto_o you_o they_o will_v also_o geve_v great_a light_n to_o the_o corollary_n follow_v add_v by_o flussas_n a_o corollary_n add_v by_o flussas_n by_o this_o and_o the_o former_a proposition_n it_o be_v manifest_a that_o prism_n and_o solid_n column_n contain_v under_o two_o poligoâon_a figure_n equal_a like_a and_o parallel_n and_o the_o rest_n parallelogram_n may_v be_v compare_v the_o one_o to_o the_o other_o after_o the_o self_n same_o manner_n that_o parallelipipedon_n be_v for_o forasmuch_o as_o by_o this_o proposition_n and_o by_o the_o second_o corollary_n of_o the_o 2d_o of_o this_o book_n it_o be_v manifest_a that_o every_o parallelipipedon_n may_v be_v resolve_v into_o two_o like_a and_o equal_a prism_n of_o one_o and_o the_o same_o altitude_n who_o base_n shall_v be_v one_o and_o the_o self_n same_o with_o the_o base_a of_o the_o parallelipipedon_n or_o the_o half_a thereof_o which_o prisâes_v also_o shall_v be_v contâyned_v under_o the_o self_n same_o sideâ_n with_o the_o parallelipipedom_n the_o say_a sideâ_n be_v also_o side_n of_o like_a proportion_n i_o say_v that_o prismeâ_n may_v be_v compare_v together_o after_o the_o like_a manner_n that_o their_o parallelipipedonâ_n areâ_n for_o if_o we_o will_v divide_v a_o prism_n like_a unto_o his_o foliâe_n by_o the_o 25._o of_o this_o book_n you_o shall_v find_v in_o the_o corollaryes_fw-mi of_o the_o 25._o proposition_n that_o that_o which_o be_v set_v forth_o touch_v a_o parallelipipedon_n follow_v not_o only_o in_o a_o prism_n but_o also_o in_o any_o side_v column_n who_o opposite_a base_n be_v equal_a and_o like_a and_o his_o side_n parallelogram_n if_o it_o be_v require_v by_o the_o 27._o proposition_n upon_o a_o right_a line_n give_v to_o describe_v a_o prism_n like_a and_o in_o like_a sort_n situate_a to_o a_o prism_n give_v describe_v âââst_n the_o whole_a parallelipipedon_n whereof_o the_o prism_n give_v be_v the_o half_n which_o thing_n you_o see_v by_o this_o 40._o proposition_n may_v be_v do_v and_o unto_o that_o parallelipipedom_n describe_v upon_o the_o right_a line_n give_v by_o the_o say_v 27._o proposition_n a_o other_o parallelipipedon_n like_o and_o the_o half_a thereof_o shall_v be_v the_o prism_n which_o you_o seek_v for_o namely_o shall_v be_v a_o prism_n describe_v upon_o the_o right_a line_n give_v and_o like_v unto_o the_o prism_n give_v in_o deed_n prism_n can_v not_o be_v cut_v according_a to_o the_o 28._o proposition_n for_o that_o in_o their_o opposite_a side_n can_v be_v draw_v no_o diagonal_a line_n howbeit_o by_o that_o 28._o proposition_n those_o prism_n be_v manifest_o confirm_v to_o be_v equal_a and_o like_a which_o be_v the_o half_n of_o one_o and_o the_o self_n same_o parallelipipedon_n and_o as_o touch_v the_o 29._o proposition_n and_o the_o three_o follow_v it_o which_o prove_v that_o parallelipipedon_n under_o one_o and_o the_o self_n same_o altitude_n and_o upon_o equal_a base_n or_o the_o self_n same_o base_n be_v equal_a or_o if_o they_o be_v under_o one_o and_o the_o self_n same_o altitude_n they_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n areâ_n to_o apply_v these_o comparison_n unto_o ãâã_d it_o be_v to_o ãâã_d require_v that_o the_o base_n of_o the_o prism_n compare_v together_o be_v either_o all_o parallelogram_n or_o all_o triaâgles_n for_o so_o one_o and_o the_o self_n altitude_n remain_v the_o comparison_n of_o thing_n equal_a ãâã_d one_o and_o thââ_n self_n same_o and_o the_o half_n of_o the_o base_n be_v ever_o the_o one_o to_o the_o other_o in_o the_o same_o proportion_n that_o their_o whole_n be_v wherefore_o prism_n which_o be_v the_o half_n of_o the_o parallelipipedon_n and_o which_o have_v the_o same_o proportion_n the_o one_o to_o the_o other_o that_o the_o whole_a parallelipipedon_n have_v which_o be_v under_o one_o and_o thâ_z selfââame_v altitude_n must_v needs_o cause_v that_o their_o base_n be_v the_o half_n of_o the_o baseâ_n of_o the_o parallelipâpââââââe_n in_o the_o same_o proportion_n the_o one_o to_o the_o other_o that_o their_o whole_a parallelipipâdonâ_n be_v if_o therefore_o the_o wâole_a parallelipipedon_n be_v in_o the_o proportion_n of_o the_o whole_a base_n their_o hâlâââ_n also_o which_o be_v prism_n shall_v be_v in_o the_o proportion_n either_o of_o the_o whole_n if_o their_o base_n be_v parallelâgrâmmâââ_n or_o of_o the_o halââââf_n they_o be_v triangle_n which_o be_v ever_o all_o one_o by_o the_o 15._o of_o the_o five_o and_o forasmuch_o as_o by_o the_o 33._o proposition_n like_o parallelipipedon_n which_o be_v the_o double_n of_o their_o prism_n be_v in_o treble_a proportion_n the_o one_o to_o the_o other_o that_o their_o side_n of_o like_a proportion_n be_v it_o be_v manifest_a that_o prism_n be_v their_o half_n which_o have_v the_o one_o to_o the_o other_o the_o same_o proportion_n that_o their_o whole_n have_v by_o the_o 15_o of_o the_o five_o and_o have_v the_o self_n same_o side_n that_o theiâ_z parallelipipedon_n have_v be_v the_o one_o to_o the_o other_o in_o treble_a proportion_n of_o that_o which_o the_o side_n of_o like_a proportion_n be_v and_o for_o that_o prism_n be_v the_o one_o to_o the_o other_o in_o the_o same_o proportion_n that_o their_o parallelipipedon_n be_v and_o the_o base_n of_o the_o prism_n be_v all_o either_o triangle_n or_o parallelogram_n be_v the_o one_o to_o the_o other_o in_o the_o same_o proportion_n that_o the_o base_n of_o the_o parallelipipedon_n be_v who_o altitude_n also_o be_v always_o equal_a we_o may_v by_o the_o 34._o proposition_n conclude_v that_o the_o base_n of_o the_o prism_n and_o the_o base_n of_o the_o parallelipipedon_n their_o double_n be_v each_o the_o one_o to_o the_o
last_o proposition_n in_o the_o second_o book_n and_o also_o of_o the_o 31._o in_o the_o six_o book_n ¶_o a_o problem_n 5._o two_o unequal_a circle_n be_v give_v to_o find_v a_o circle_n equal_a to_o the_o excess_n of_o the_o great_a to_o the_o less_o suppose_v the_o two_o unequal_a circle_n give_v to_o be_v abc_n &_o def_n &_o let_v abc_n be_v the_o great_a construction_n who_o diameter_n suppose_v to_o be_v ac_fw-la &_o the_o diameter_n of_o def_n suppose_v to_o be_v df._n i_o say_v a_o circle_n must_v be_v find_v equal_a to_o that_o excess_n in_o magnitude_n by_o which_o abc_n be_v great_a thâ_z def_n by_o the_o first_o of_o the_o four_o in_o the_o circle_n abc_n apply_v a_o right_a line_n equal_a to_o df_o who_o one_o end_n let_v be_v at_o c_o and_o the_o other_o let_v be_v at_o b._n fron_n b_o to_o a_o draw_v a_o right_a line_n by_o the_o 30._o of_o the_o three_o it_o may_v appear_v demonstration_n that_o abc_n be_v a_o right_a angle_n and_o thereby_o abc_n the_o triangle_n be_v rectangle_v wherefore_o by_o the_o first_o of_o the_o two_o corollary_n here_o before_o the_o circle_z abc_n be_v equal_a to_o the_o circle_n def_n for_o bc_o by_o construction_n be_v equal_a to_o df_o and_o more_o over_o to_o the_o circle_n who_o diameter_n be_v ab_fw-la that_o circle_n therefore_o who_o diameter_n be_v ab_fw-la be_v the_o circle_n contain_v the_o magnitude_n by_o which_o abc_n be_v great_a than_o def_n wherefore_o two_o unequal_a circle_n be_v give_v we_o have_v find_v a_o circle_n equal_a to_o the_o excess_n of_o the_o great_a to_o the_o less_o which_o ought_v to_o be_v do_v a_o problem_n 6._o a_o circle_n be_v give_v to_o find_v two_o circle_n equal_a to_o the_o same_o which_o find_v circle_n shall_v have_v the_o one_o to_o the_o other_o any_o proportion_n give_v in_o two_o right_a line_n suppose_v abc_n a_o circle_n give_v and_o the_o proportion_n give_v let_v it_o be_v that_o which_o be_v between_o the_o two_o right_a line_n d_z and_o e._n i_o say_v that_o two_o circle_n be_v to_o be_v find_v equal_a to_o abc_n and_o with_o all_o one_o to_o the_o other_o construction_n in_o the_o proportion_n of_o d_o to_o e._n let_v the_o diameter_n of_o abc_n be_v ac_fw-la as_z d_z be_v to_o e_o so_o let_v ac_fw-la be_v divide_v by_o the_o 10._o of_o the_o six_o in_o the_o point_n f._n at_o fletcher_n to_o the_o line_n ac_fw-la let_v a_o perpendicular_a be_v draw_v fb_o and_o let_v it_o mete_v the_o circumference_n at_o the_o point_n b._n from_o the_o point_n b_o to_o the_o point_v a_o and_o c_o let_v right_a line_n be_v draw_v ba_o and_o bc._n i_o say_v that_o the_o circle_n who_o diameter_n be_v the_o line_n basilius_n and_o bc_o be_v equal_a to_o the_o circle_n abc_n and_o that_o those_o circle_n have_v to_o their_o diameter_n the_o line_n basilius_n and_o bc_o be_v one_o to_o the_o other_o in_o the_o proportion_n of_o the_o line_n d_o to_o the_o line_n e._n for_o first_o that_o they_o be_v equal_a it_o be_v evident_a by_o reason_n that_o abc_n be_v a_o triangle_n rectangle_n wherefore_o by_o the_o 47._o of_o the_o first_o the_o square_n of_o basilius_n and_o bc_o be_v equal_a to_o the_o square_n of_o ac_fw-la and_o so_o by_o this_o second_o it_o be_v manife_a the_o two_o circle_n to_o be_v equal_a to_o the_o circle_n abc_n second_o as_o d_o be_v to_o â_o so_o be_v of_o to_o fc_n by_o construction_n and_o as_o the_o line_n of_o be_v to_o the_o line_n fc_o so_o be_v the_o square_a of_o the_o line_n âa_o to_o the_o square_n of_o the_o line_n bc._n which_o thing_n we_o will_v brief_o prove_v thus_o the_o parallelogram_n contain_v under_o ac_fw-la and_o of_o use_n be_v equal_a to_o the_o square_n of_o basilius_n by_o the_o lemma_n after_o the_o 32._o of_o the_o ten_o book_n and_o by_o the_o same_o lemma_n or_o assumpt_n the_o parallelogram_n contain_v under_o ac_fw-la and_o âc_z be_v equal_a to_o the_o square_n of_o the_o line_n bc._n wherefore_o as_o the_o first_o parallelogram_n have_v itself_o to_o the_o secondâ_n so_o have_v the_o square_a of_o basilius_n equal_a to_o the_o first_o parallelogram_n itself_o to_o the_o square_n of_o bc_o equal_a to_o the_o second_o parallelogram_n but_o both_o the_o parallelogram_n have_v one_o height_n namely_o the_o line_n ac_fw-la and_o base_n the_o line_n of_o and_o fc_n wherefore_o as_o of_o be_v to_o fc_n so_o be_v the_o parallelogâamme_n contain_v under_o ac_fw-la of_o to_o the_o parallelogram_n contain_v under_o ac_fw-la fc_fw-la by_o the_o fiâst_n of_o the_o six_o and_o therefore_o as_o of_o be_v to_o fc_n so_o be_v the_o square_a of_o basilius_n to_o the_o square_n of_o bc._n and_o as_o the_o square_n of_o basilius_n be_v to_o the_o square_n of_o bc_o so_o be_v the_o circle_n who_o diameter_n be_v basilius_n to_o the_o circle_n who_o diameter_n be_v bc_o by_o this_o second_o of_o the_o twelve_o wherefore_o the_o circle_n who_o diameter_n be_v basilius_n be_v to_o the_o circle_n who_o diameter_n be_v bc_o as_o d_z be_v to_o e._n and_o before_o we_o prove_v they_o equal_a to_o the_o circle_n abc_n wherâfore_o a_o circle_n be_v give_v we_o have_v find_v two_o circle_n equal_a to_o the_o same_o which_o have_v the_o one_o to_o the_o other_o any_o proportion_n give_v in_o two_o right_a line_n which_o ought_v to_o be_v do_v note_n addition_n heâe_o may_v you_o perceive_v a_o other_o way_n how_o to_o execute_v my_o first_o problem_n for_o if_o you_o make_v a_o right_a angle_n contain_v of_o the_o diameter_n give_v as_o in_o this_o figure_n suppose_v they_o basilius_n and_o bc_o and_o then_o subtend_v the_o right_a angle_n with_o the_o line_n ac_fw-la and_o from_o the_o right_a angle_n let_v fall_v a_o line_n perpendicular_a to_o the_o base_a ac_fw-la that_o perpendicular_a at_o the_o point_n of_o his_o fall_n divide_v ac_fw-la into_o of_o and_o fc_n of_o the_o proportion_n require_v a_o corollary_n it_o follow_v of_o thing_n manifest_o prove_v in_o the_o demonstration_n of_o this_o problem_n that_o in_o a_o triangle_n rectangle_n if_o from_o the_o right_a angle_n to_o the_o base_a a_o perpendicular_a be_v let_v fall_v the_o same_o perpendicular_a cut_v the_o base_a into_o two_o part_n rectangle_n in_o that_o proportion_n one_o to_o the_o other_o that_o the_o square_n of_o the_o righâ_n line_n contain_v the_o right_a angle_n be_v in_o one_o to_o the_o other_o those_o on_o the_o one_o side_n the_o perpendicular_a be_v compare_v to_o those_o on_o the_o other_o both_o square_a and_o segment_n a_o problem_n 7._o between_o two_o circle_n give_v to_o find_v a_o circle_n middle_a proportional_a let_v the_o two_o circle_n give_v be_v acd_o and_o bef_a i_o say_v that_o a_o circle_n be_v to_o be_v find_v which_o between_o acd_a and_o bef_n be_v middle_a proportional_a construction_n let_v the_o diameter_n of_o acd_a be_v ad_fw-la and_o of_o bef_n let_v bâ_n be_v the_o diameter_n between_o ad_fw-la and_o bf_o find_v a_o line_n middle_a proportional_a by_o the_o 13._o of_o the_o six_o which_o let_v be_v hk_a i_o say_v that_o a_o circle_n who_o diameter_n be_v hk_n be_v middle_a proportional_a between_o acd_a and_o bef_a to_o ad_fw-la hk_o and_o bf_o three_o right_a line_n in_o continual_a proportion_n by_o construction_n let_v a_o four_o line_n be_v find_v to_o which_o bf_o shall_v have_v that_o proportion_n that_o ad_fw-la have_v to_o hk_n by_o the_o 12._o of_o the_o six_o &_o let_v that_o line_n be_v â_o demonstration_n it_o be_v manifest_a that_o the_o âower_a line_n ad_fw-la hk_o bf_o and_o l_o be_v in_o continual_a proportion_n for_o by_o construction_n as_o ad_fw-la be_v to_o hk_n so_o be_v bâ_n to_o l._n and_o by_o construction_n on_o as_z ad_fw-la be_v to_o hk_n so_o be_v hk_v to_o bf_o wherefore_o hk_n be_v to_o bf_o as_o bf_o be_v to_o l_o by_o the_o 11._o of_o the_o five_o wherefore_o the_o 4._o line_n be_v in_o continual_a proportion_n wherefore_o as_o the_o first_o be_v to_o the_o three_o that_o be_v ad_fw-la to_o bf_o so_o be_v the_o square_a of_o the_o first_o to_o the_o square_n of_o the_o second_o that_o be_v the_o square_a of_o ad_fw-la to_o the_o square_n of_o hk_n by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o and_o by_o the_o same_o corollary_n as_o hk_n be_v to_o l_o so_o be_v the_o square_a of_o hk_n to_o the_o square_n of_o bf_o but_o by_o alternate_a proportion_n the_o line_n ad_fw-la be_v to_o bf_o as_o hk_n be_v to_o l_o wherefore_o the_o square_a of_o ad_fw-la be_v to_o the_o square_n of_o hk_n as_o the_o square_n of_o hk_n be_v to_o the_o square_n of_o bf_o wherefore_o the_o square_a of_o hk_n be_v middle_a proportional_a between_o the_o square_n of_o ad_fw-la and_o the_o square_n of_o bf_o but_o as_o the_o square_n be_v
a_o triangle_n and_o if_o the_o parallelogram_n be_v double_a to_o the_o triangle_n those_o prism_n be_v by_o the_o 40._o of_o the_o eleven_o equal_v the_o one_o to_o the_o other_o therefore_o the_o prism_n contain_v under_o the_o two_o triangle_n bkf_a and_o ehg_n and_o under_o the_o three_o parallelogram_n ebfg_n ebkh_n and_o khfg_n part_n be_v equal_a to_o the_o prism_n contain_v under_o the_o two_o triangle_n gfc_a and_o hkl_n and_o under_o the_o three_o parallelogram_n kfcl_n lcgh_n and_o hkfg_n and_o it_o be_v manifest_a that_o both_o these_o prism_n of_o which_o the_o base_a of_o one_o be_v the_o parallelogram_n ebfg_n pyramid_n and_o the_o opposite_a unto_o it_o the_o line_n kh_o and_o the_o base_a of_o the_o other_o be_v the_o triangle_n gfc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n klh_n be_v great_a than_o both_o these_o pyramid_n who_o base_n be_v the_o triangle_n age_n and_o hkl_n and_o top_n the_o point_n h_o &_o d._n for_o if_o we_o draw_v these_o right_a line_n of_o and_o eke_o the_o prism_n who_o base_a be_v the_o parallelogram_n ebfg_n and_o the_o opposite_a unto_o it_o the_o right_a line_n hk_o be_v great_a than_o the_o pyramid_n who_o base_a be_v the_o triangle_n ebf_n &_o top_n the_o point_n k._n but_o the_o pyramid_n who_o base_a be_v the_o triangle_n ebf_n and_o top_n the_o point_n king_n be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h_o for_o they_o be_v contain_v under_o equal_a and_o like_a plain_a superficiece_n wherefore_o also_o the_o prism_n who_o base_a be_v the_o parallelogram_n ebfg_n and_o the_o opposite_a unto_o it_o the_o right_a line_n hk_o be_v great_a than_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h._n but_o the_o prism_n who_o base_a be_v the_o parallelogram_n ebfg_n and_o the_o opposite_a unto_o it_o the_o right_a line_n hk_o be_v equal_a to_o the_o prism_n who_o base_a be_v the_o triangle_n gfc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n hkl_n and_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h_o be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n hkl_n and_o top_n the_o point_n d._n part_n wherefore_o the_o foresay_a two_o prism_n be_v great_a than_o the_o foresay_a two_o pyramid_n who_o base_n be_v the_o triangle_n aeg_n hkl_n and_o top_n the_o point_n h_o and_o d._n wherefore_o the_o whole_a pyramid_n who_o base_a be_v the_o triangle_n abc_n proposition_n and_o top_n the_o point_n d_o be_v divide_v into_o two_o pyramid_n equal_a and_o like_v the_o one_o to_o the_o other_o and_o like_v also_o unto_o the_o whole_a pyramid_n have_v also_o triangle_n to_o their_o base_n and_o into_o two_o equal_a prism_n and_o the_o two_o prism_n be_v great_a than_o half_a of_o the_o whole_a pyramid_n which_o be_v require_v to_o be_v demonstrate_v if_o you_o will_v with_o diligence_n read_v these_o four_o book_n follow_v of_o euclid_n which_o concern_v body_n and_o clear_o see_v the_o demonstration_n in_o they_o contain_v it_o shall_v be_v requisite_a for_o you_o when_o you_o come_v to_o any_o proposition_n which_o concern_v a_o body_n or_o body_n whether_o they_o be_v regular_a or_o not_o first_o to_o describe_v of_o pâsâed_a paper_n according_a as_o i_o teach_v you_o in_o the_o end_n of_o the_o definition_n of_o the_o eleven_o book_n such_o a_o body_n or_o body_n as_o be_v there_o require_v and_o have_v your_o body_n or_o body_n thus_o describe_v when_o you_o have_v note_v it_o with_o letter_n according_a to_o the_o figure_n set_v forth_o upon_o a_o plain_a in_o the_o proposition_n follow_v the_o construction_n require_v in_o the_o proposition_n as_o for_o example_n in_o this_o three_o proposition_n it_o be_v say_v that_o every_o pyramid_n have_v a_o triangle_n to_o âis_n base_a may_v be_v divide_v into_o two_o pyramid_n etc_n etc_n here_o first_o describe_v a_o pyramid_n of_o past_v paper_n haâing_v his_o base_a triangle_v it_o skill_v not_o whether_o it_o be_v equilater_n or_o equiangle_v or_o not_o only_o in_o this_o proposition_n be_v require_v that_o the_o base_a be_v a_o triangle_n then_o for_o that_o the_o proposition_n suppose_v the_o base_a of_o the_o pyramid_n to_o be_v the_o triangle_n abc_n note_v the_o base_a of_o your_o pyramid_n which_o you_o have_v describe_v with_o the_o letter_n abc_n and_o the_o top_n of_o your_o pyramid_n with_o the_o letter_n d_o for_o so_o be_v require_v in_o the_o proposition_n and_o thus_o have_v you_o your_o body_n order_v ready_a to_o the_o construction_n now_o in_o the_o construction_n it_o be_v require_v that_o you_o divide_v the_o line_n ab_fw-la bc_o ca._n etc_o etc_o namely_o the_o six_o line_n which_o be_v the_o side_n of_o the_o four_o triangle_n contain_v the_o pyramid_n into_o two_o equal_a part_n in_o the_o poyntet_fw-la â_o f_o g_o etc_n etc_n that_o be_v you_o must_v divide_v the_o line_n ab_fw-la of_o your_o pyramid_n into_o two_o equal_a part_n and_o note_v the_o point_n of_o the_o division_n with_o the_o letter_n e_o and_o so_o the_o line_n bc_o be_v divide_v into_o two_o equal_a part_n note_v the_o point_n of_o the_o division_n with_o the_o letter_n f._n and_o so_o the_o rest_n and_o this_o order_n follow_v you_o as_o touch_v the_o rest_n of_o the_o construction_n there_o put_v and_o when_o you_o have_v finish_v the_o construction_n compare_v your_o body_n thus_o describe_v with_o the_o demonstration_n and_o it_o will_v make_v it_o very_o plain_a and_o easy_a to_o be_v understand_v whereas_o without_o such_o a_o body_n describe_v of_o matter_n it_o be_v hard_o for_o young_a beginner_n unless_o they_o have_v a_o very_a deep_a imagination_n full_o to_o conceive_v the_o demonstration_n by_o the_o sigâe_n as_o it_o be_v describe_v in_o a_o plain_a here_o for_o the_o better_a declaration_n of_o that_o which_o i_o have_v say_v have_v i_o set_v a_o figure_n who_o form_n if_o you_o describe_v upon_o past_v paper_n note_v with_o the_o like_a letter_n and_o cut_v the_o line_n âa_o dam_n dc_o and_o fold_v it_o according_o it_o will_v make_v a_o pyramid_n describe_v according_a to_o the_o construction_n require_v in_o the_o proposition_n and_o this_o order_n follow_v you_o as_o touch_v all_o other_o proposition_n which_o concern_v body_n ¶_o a_o other_o demonstration_n after_o campane_n of_o the_o 3._o proposition_n suppose_v that_o there_o be_v a_o pyramid_n abcd_o have_v to_o his_o base_a the_o triangle_n bcd_o and_o let_v his_o top_n be_v the_o solid_a angle_n a_o from_o which_o let_v there_o be_v draw_v three_o subtended_a line_n ab_fw-la ac_fw-la and_o ad_fw-la to_o the_o three_o angle_n of_o the_o base_a and_o divide_v all_o the_o side_n of_o the_o base_a into_o two_o equal_a part_n in_o the_o three_o point_n e_o f_o g_o divide_v also_o the_o three_o subtended_a line_n ab_fw-la ac_fw-la and_o ad_fw-la in_o two_o equal_a part_n in_o the_o three_o point_n h_o edward_n l._n and_o draw_v in_o the_o base_a these_o two_o line_n of_o and_o eglantine_n so_o shall_v the_o base_a of_o the_o pyramid_n be_v divide_v into_o three_o superficiece_n whereof_o two_o be_v the_o two_o triangle_n bef_n and_o egg_v which_o be_v like_o both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o whole_a base_a by_o the_o 2_o part_n of_o the_o second_o of_o the_o six_o &_o by_o the_o definition_n of_o like_a superâiciecâs_n &_o they_o be_v equal_v the_o one_o to_o the_o other_o by_o the_o 8._o of_o the_o first_o the_o three_o superficies_n be_v a_o quadrangled_a parallelogram_n namely_o efgc_n which_o be_v double_a to_o the_o triangle_n egg_v by_o the_o 40._o and_o 41._o of_o the_o first_o now_o then_o again_o from_o the_o point_n h_o draw_v unto_o the_o point_n e_o and_o f_o these_o two_o subtendent_fw-la line_n he_o and_o hf_o draw_v also_o a_o subtended_a line_n from_o the_o point_n king_n to_o the_o point_n g._n and_o draw_v these_o line_n hk_o kl_o and_o lh_o wherefore_o the_o whole_a pyramid_n abcd_o be_v divide_v into_o two_o pyramid_n which_o be_v hbef_n and_o ahkl_n and_o into_o two_o prism_n of_o which_o the_o one_o be_v ehfgkc_n and_o be_v set_v upon_o the_o quadrangled_a base_a cfge_n the_o other_o be_v egdhkl_n and_o have_v to_o his_o base_a the_o triangle_n egg_v now_o as_o touch_v the_o two_o pyramid_n hbef_a and_o ahkl_n that_o they_o be_v equal_v the_o one_o to_o the_o other_o and_o also_o that_o they_o be_v like_o both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o whole_a it_o be_v manifest_a by_o the_o definition_n of_o equal_a and_o like_a body_n and_o by_o the_o 10._o of_o the_o eleven_o and_o by_o 2._o part_n of_o the_o second_o of_o the_o six_o and_o that_o the_o two_o prism_n be_v equal_a it_o
the_o â_o of_o this_o book_n wherefore_o a_o cylinder_n incline_n shall_v be_v triple_a to_o every_o cone_n although_o also_o the_o cone_n be_v erect_v set_v upon_o one_o and_o the_o same_o base_a with_o it_o and_o be_v under_o the_o same_o altitude_n but_o the_o cilind_a erect_v be_v the_o triple_a of_o the_o same_o cone_fw-mi by_o the_o ten_o of_o this_o book_n wherefore_o the_o cilind_a incline_v be_v equal_a to_o the_o cilind_a erect_v being_n both_o set_v upon_o one_o and_o the_o self_n same_o base_a and_o have_v one_o and_o the_o self_n same_o altitude_n the_o same_o also_o come_v to_o pass_v in_o cones_fw-la which_o be_v the_o three_o part_n of_o equal_a cilinder_n &_o therefore_o be_v equal_a the_o one_o to_o the_o other_o wherefore_o according_a to_o the_o eleven_o of_o this_o book_n it_o follow_v that_o cylinder_n and_o cones_fw-la incline_v or_o erect_v be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n be_v for_o forasmuch_o as_o the_o erect_a be_v in_o proportion_n as_o their_o base_n be_v and_o to_o the_o erect_a cilinder_n the_o incline_n be_v equal_a therefore_o they_o also_o shall_v be_v in_o proportion_n as_o their_o base_n be_v and_o therefore_o by_o the_o 12._o of_o this_o book_n like_o cones_fw-la and_o cylinder_n be_v incline_v be_v in_o triple_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o the_o base_n be_v for_o forasmuch_o as_o they_o be_v equal_a to_o the_o erect_a which_o have_v the_o proportion_n by_o the_o 12._o of_o this_o book_n and_o their_o base_n also_o be_v equal_a with_o the_o base_n of_o the_o erect_v therefore_o they_o also_o shall_v have_v the_o same_o proportion_n wherefore_o it_o follow_v by_o the_o 13._o of_o this_o book_n thaâ_z cylinder_n incline_v be_v cut_v by_o a_o plain_a superficies_n parallel_v to_o the_o opposite_a plain_a superficiece_n thereof_o shall_v be_v cut_v according_a to_o the_o proportion_n of_o the_o axe_n for_o suppose_v that_o upon_o one_o and_o the_o self_n same_o base_a âe_n set_v a_o erect_a cylinder_n and_o a_o incline_v cylinder_n be_v both_o under_o one_o and_o the_o self_n same_o altitude_n which_o ãâ¦ã_o a_o plain_a superficies_n parallel_v to_o the_o opposite_a base_n now_o it_o be_v manifest_a that_o the_o section_n of_o the_o one_o cylinder_n be_v equal_a to_o the_o section_n of_o the_o other_o cylinder_n for_o they_o be_v set_v upon_o equal_a base_n and_o under_o one_o and_o the_o self_n same_o altitude_n namely_o between_o the_o parallel_n plain_a superficiece_n and_o their_o axe_n also_o be_v by_o those_o parallel_a plain_n superficiâ_fw-la ãâã_d proportional_o by_o the_o 1st_a of_o âhe_n âlevenâh_o wherefore_o the_o incline_v cylinder_n be_v equal_a to_o the_o erect_a cylinder_n shall_v have_v the_o proportion_n of_o theiâ_z axe_n aâ_z also_o have_v the_o erect_v for_o in_o echâ_n the_o proportion_n of_o the_o axe_n be_v one_o and_o the_o same_o wherefore_o incline_v cones_n and_o cylinder_n be_v set_v upon_o equal_a base_n shall_v by_o the_o 14._o of_o this_o book_n be_v in_o ãâã_d as_o their_o altitude_n ãâ¦ã_o forasmuch_o aâ_z the_o incline_v be_v equal_a to_o the_o erect_a which_o have_v the_o self_n same_o base_n and_o altitude_n and_o the_o erect_v be_v iâ_z proportion_n as_o their_o altitude_n therefore_o the_o incline_n shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_o âhe_z self_n same_o altitude_n which_o be_v common_a to_o each_o namely_o to_o the_o incline_v and_o to_o the_o erect_v and_o therefore_o in_o equal_a cones_fw-la and_o cylinder_n whether_o they_o be_v incline_v or_o erect_v the_o base_n shall_v be_v reciprocal_a proportional_a with_o the_o altitude_n and_o contrariwise_o by_o the_o 15._o of_o this_o book_n for_o forasmuch_o as_o we_o have_v oftentimes_o show_v that_o the_o incline_v cones_fw-la and_o cylinder_n be_v equal_a to_o the_o erect_a have_v the_o self_n same_o base_n and_o altitude_n with_o they_o and_o the_o erect_v unto_o who_o the_o incline_v be_v equal_a haâe_z their_o base_n reciprocal_a proportional_o with_o their_o altitude_n therefore_o it_o follow_v that_o the_o incline_v be_v equal_a to_o the_o erect_a have_v also_o their_o base_n and_o altitude_n which_o be_v common_a to_o each_o reciprocal_a proportional_a likewise_o if_o theiâ_z altitude_n &_o base_n be_v reciprocal_a proportional_a they_o themselves_o also_o shall_v be_v equal_a for_o that_o they_o be_v equal_a to_o the_o erect_a cylinder_n and_o cones_fw-la set_v upon_o the_o same_o base_n and_o be_v under_o the_o same_o altitude_n which_o erect_a cylinder_n be_v equal_a the_o one_o to_o the_o other_o by_o the_o same_o 15._o of_o this_o book_n wherefore_o we_o may_v conclude_v that_o those_o passion_n &_o propriety_n which_o in_o this_o twelve_o book_n have_v be_v prove_v to_o be_v in_o cones_fw-la and_o cylinder_n who_o altitude_n be_v erect_v perpendicular_o to_o the_o ãâ¦ã_o set_v oblique_o upoâ_n their_o base_n howbeit_o this_o be_v to_o be_v note_v that_o such_o incline_v cones_fw-la or_o cylinder_n be_v not_o perfect_a round_o as_o be_v the_o erect_v so_o that_o if_o they_o be_v cut_v by_o a_o plain_a superficies_n pass_v at_o right_a angle_n with_o their_o altitude_n this_o section_n be_v a_o conical_a section_n which_o be_v call_v ellipsis_n and_o shall_v not_o describe_v in_o their_o superficies_n a_o circle_n as_o it_o do_v in_o erect_a cylinder_n &_o cones_fw-la but_o a_o certain_a figure_n who_o less_o diameter_n be_v in_o cylinder_n equal_a to_o the_o dimetient_fw-la of_o the_o base_a that_o be_v be_v one_o and_o the_o same_o with_o it_o and_o the_o same_o thing_n happen_v also_o in_o cones_fw-la incline_v be_v cut_v after_o the_o same_o manner_n the_o 1._o problem_n the_o 16._o proposition_n two_o circle_n have_v both_o one_o and_o the_o self_n same_o centre_n be_v give_v to_o inscribe_v in_o the_o great_a circle_n a_o polygonon_n figure_n which_o shall_v consist_v of_o equal_a and_o even_a side_n and_o shall_v not_o touch_v the_o superficies_n of_o the_o less_o circle_n svppose_v that_o there_o be_v two_o circle_n abcd_o and_o efgh_a have_v one_o &_o the_o self_n same_o centre_n namely_o k._n it_o be_v require_v in_o the_o great_a circle_n which_o let_v be_v abcd_o to_o inscribe_v a_o polygonon_n figure_n which_o shall_v be_v of_o equal_a and_o even_a side_n and_o not_o touch_v the_o circle_n efgh_o draw_v by_o the_o centre_n king_n a_o right_a line_n bd._n construction_n and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n g_o raise_v up_o unto_o the_o right_a line_n bd_o a_o perpendicular_a line_n agnostus_n and_o extend_v it_o to_o the_o point_n c._n wherefore_o the_o line_n ac_fw-la touch_v the_o circle_n efgh_o by_o the_o 15._o of_o the_o three_o now_o therefore_o if_o by_o the_o 30._o of_o the_o three_o we_o divide_v the_o circumference_n bad_a into_o two_o equal_a part_n and_o again_o the_o half_a of_o that_o into_o two_o equal_a part_n and_o thus_o do_v continual_o we_o shall_v by_o the_o corollary_n of_o the_o 1._o of_o the_o ten_o at_o the_o length_n leave_v a_o certain_a circumference_n less_o than_o the_o circumference_n ad._n let_v the_o circumference_n leave_v be_v ld_n and_o from_o the_o point_n l._n draw_v by_o the_o 12._o of_o the_o first_o unto_o the_o line_n bd_o a_o perpendiculare_a line_n demonstration_n lm_o dee_n and_o extend_v it_o to_o the_o point_n n._n and_o draw_v these_o right_a line_n ld_n and_o dn_o and_o forasmuch_o as_o the_o angle_n dml_n and_o dmn_v be_v right_a angle_n therefore_o by_o the_o 3._o of_o the_o three_o the_o right_a line_n bd_o divide_v the_o right_a line_n ln_o into_o two_o equal_a part_n in_o the_o point_n m._n wherefore_o by_o the_o 4._o of_o the_o first_o the_o rest_n of_o the_o side_n of_o the_o triangle_n dml_n and_o dmn_v namely_o the_o line_n dl_o and_o dn_o shall_v be_v equal_v and_o forasmuch_o as_o the_o line_n ac_fw-la be_v a_o parallel_n to_o the_o ln_o by_o the_o 28._o of_o the_o first_o but_o ac_fw-la touch_v the_o circle_n efgh_o wherefore_o the_o line_n lm_o touch_v not_o the_o circle_n efgh_o and_o much_o less_o do_v the_o line_n ld_n and_o dn_o touch_v the_o circle_n efgh_o if_o therefore_o there_o be_v apply_v right_a line_n equal_a to_o the_o line_n ld_n continual_o into_o the_o circle_n abcd_o by_o the_o 1._o of_o the_o four_o there_o shall_v be_v describe_v in_o the_o circle_n abcd_o a_o polygonon_n figure_n which_o shall_v be_v of_o equal_a and_o figure_n even_o side_n and_o shall_v not_o touch_v the_o less_o circle_n namely_o efgh_o by_o the_o 14._o of_o the_o three_o or_o by_o the_o 29._o which_o be_v require_v to_o be_v do_v ¶_o corollary_n hereby_o it_o be_v manifest_a that_o a_o perpendicular_a line_n draw_v from_o the_o point_n l_o to_o the_o line_n bd_o touch_v not_o
line_n which_o subtend_v the_o angle_n zob_n to_o the_o three_o line_n which_o subtend_v the_o angle_n zkb_n but_o by_o construction_n boy_n be_v equal_a to_o bk_o therefore_o oz_n be_v equal_a to_o kz_o and_o the_o three_o alâo_o be_v equal_a to_o the_o three_o wherefore_o the_o point_n z_o in_o respect_n of_o the_o two_o triangle_n rectangle_v ozb_n and_o kzb_n determine_v one_o and_o the_o same_o magnitude_n iâ_z the_o line_n bz_o which_o can_v not_o be_v if_o any_o other_o point_n in_o the_o line_n bz_o be_v assign_v near_o or_o far_o of_o from_o the_o point_n b._n one_o only_a point_n therefore_o be_v that_o at_o which_o the_o two_o perpendicular_n kz_o and_o oz_n fall_v but_o by_o construction_n oz_n fall_v at_o z_o the_o point_n and_o therefore_o at_o the_o same_o z_o do_v the_o perpendicular_a draw_v from_o king_n fall_v likewise_o which_o be_v require_v to_o be_v demonstrate_v although_o a_o brief_a monition_n may_v herein_o have_v serve_v for_o the_o pregnant_a or_o the_o humble_a learner_n yet_o for_o they_o that_o be_v well_o please_v to_o have_v thing_n make_v plain_a with_o many_o word_n and_o for_o the_o stiffnecked_a busy_a body_n it_o be_v necessary_a with_o my_o controlment_n of_o other_o to_o annex_v the_o cause_n &_o reason_n thereof_o both_o invincible_a and_o also_o evident_a a_o corollary_n 1._o hereby_o it_o be_v manifest_a that_o two_o equal_a circle_n cut_v one_o the_o other_o by_o the_o whole_a diameter_n if_o from_o one_o and_o the_o same_o end_n of_o their_o common_a diameter_n equal_a portion_n of_o their_o circumference_n be_v take_v and_o from_o the_o point_n end_v those_o equal_a portion_n two_o perpendicular_n be_v let_v down_o to_o their_o common_a diameter_n those_o perpendicular_n shall_v fall_v upon_o one_o and_o the_o same_o point_n of_o their_o common_a diameter_n 2._o second_o it_o follow_v that_o those_o perpendicular_n be_v equal_a ¶_o note_n from_o circle_n in_o our_o first_o supposition_n each_o to_o other_o perpendicular_o erect_v we_o proceed_v and_o infer_v now_o these_o corollary_n whether_o they_o be_v perpendicular_o erect_v or_o no_o by_o reasou_v the_o demonstration_n have_v a_o like_a force_n upon_o our_o supposition_n here_o use_v ¶_o the_o 16._o theorem_a the_o 18._o proposition_n sphere_n be_v in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o their_o diameter_n be_v svppose_v that_o there_o be_v two_o sphere_n abc_n and_o def_n and_o let_v their_o diameter_n be_v bc_o and_o ef._n then_o i_o say_v that_o the_o sphere_n abc_n be_v to_o the_o sphere_n def_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n ef._n proposition_n for_o if_o not_o than_o the_o sphere_n abc_n be_v in_o treble_a proportion_n of_o that_o in_o which_o bc_o be_v to_o of_o either_o to_o some_o sphere_n less_o than_o the_o sphere_n def_n or_o to_o some_o sphere_n great_a first_o let_v it_o be_v unto_o a_o less_o namely_o to_o ghk_v case_n and_o imagine_v that_o the_o sphere_n def_n and_o ghk_o be_v both_o about_o one_o and_o the_o self_n same_o centre_n impossibility_n and_o by_o the_o proposition_n next_o go_v before_o describe_v in_o the_o great_a sphere_n def_n a_o polihedron_n or_o a_o solid_a of_o many_o side_n not_o touch_v the_o superficies_n of_o the_o less_o sphere_n ghk_v and_o suppose_v also_o that_o in_o the_o sphere_n abc_n be_v inscribe_v a_o polihedron_n like_a to_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n wherefore_o by_o the_o corollary_n of_o the_o same_o the_o polihedron_n which_o be_v in_o the_o sphere_n abc_n be_v to_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n ef._n but_o by_o supposition_n the_o sphere_n abc_n be_v to_o the_o sphere_n ghk_v in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n ef._n wherefore_o as_o the_o sphere_n abc_n be_v to_o the_o sphere_n ghk_o so_o be_v the_o polihedron_n which_o be_v describe_v in_o the_o sphere_n abc_n to_o the_o polihedron_n which_o be_v describe_v in_o the_o sphere_n def_n by_o the_o 11._o of_o the_o five_o wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o sphere_n abc_n be_v to_o the_o polihedron_n which_o be_v describe_v in_o it_o so_o be_v the_o sphere_n ghk_v to_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n but_o the_o sphere_n abc_n be_v great_a than_o the_o polihedron_n which_o be_v describe_v in_o it_o wherefore_o also_o the_o sphere_n ghk_o be_v great_a than_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n by_o the_o 14._o of_o the_o five_o but_o it_o be_v also_o less_o for_o it_o be_v contain_v in_o it_o which_o impossible_a wherefore_o the_o sphere_n abc_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n of_o to_o any_o sphere_n less_o than_o the_o sphere_n def_n in_o like_a sort_n also_o may_v we_o prove_v that_o the_o sphere_n def_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o be_v to_o the_o diameter_n bc_o to_o any_o sphere_n less_o than_o the_o sphere_n abc_n case_n now_o i_o say_v that_o the_o sphere_n abc_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n of_o to_o any_o sphere_n great_a they_o the_o sphere_n def_n for_o if_o it_o be_v possible_a let_v it_o be_v to_o a_o great_a namely_o to_o lmn_o wherefore_o by_o conversion_n the_o sphere_n lmn_o be_v to_o the_o sphere_n abc_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o be_v to_o the_o diameter_n bc._n but_o as_o the_o sphere_n lmn_o be_v to_o the_o sphere_n abc_n so_o be_v the_o sphere_n def_n to_o some_o sphere_n less_o they_o the_o sphere_n abc_n booââ_n as_o it_o have_v before_o be_v prove_v for_o the_o sphere_n lmn_o be_v great_a than_o the_o sphere_n def_n wherefore_o the_o sphere_n def_n be_v in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o be_v to_o the_o diameter_n bc_o to_o some_o sphere_n less_o they_o the_o sphere_n abc_n which_o be_v prove_v to_o be_v impossible_a wherefore_o the_o sphere_n abc_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o be_v be_v to_o of_o to_o any_o sphere_n great_a they_o the_o sphere_n def_n and_o it_o be_v also_o prove_v that_o it_o be_v not_o to_o any_o less_o wherefore_o the_o sphere_n abc_n be_v to_o the_o sphere_n def_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n of_o which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o flussas_n hereby_o it_o be_v manifest_a that_o sphere_n be_v the_o one_o to_o the_o other_o as_o like_o polihedrons_n and_o in_o like_a sort_n describe_v in_o they_o be_v namely_o each_o be_v in_o triple_a proportion_n of_o that_o in_o which_o the_o diameter_n a_o corollary_n add_v by_o mâ_n dee_n it_o be_v then_o evident_a how_o to_o geve_v two_o right_a line_n have_v that_o proportion_n between_o they_o which_o any_o two_o sphere_n give_v have_v the_o one_o to_o the_o other_o for_o if_o to_o their_o diameter_n as_o to_o the_o first_o and_o second_o line_n of_o four_o in_o continual_a proportion_n you_o adjoin_v a_o three_o and_o a_o four_o line_n in_o continual_a proportion_n as_o i_o have_v teach_v before_o the_o first_o and_o four_o line_n shall_v answer_v the_o problem_n rule_n how_o general_a this_o rule_n be_v in_o any_o two_o like_a solid_n with_o their_o correspondent_a or_o omologall_n line_n i_o need_v not_o with_o more_o word_n declare_v ¶_o certain_a theorem_n and_o problem_n who_o use_n be_v manifold_a in_o sphere_n cones_n cylinder_n and_o other_o solid_n add_v by_o joh._n dee_n a_o theorem_a 1._o the_o whole_a superficies_n of_o any_o sphere_n be_v quadrupla_fw-la to_o the_o great_a circle_n in_o the_o same_o sphere_n contain_v it_o be_v needless_a to_o bring_v archimedes_n demonstration_n hereof_o into_o this_o place_n see_v his_o book_n of_o the_o sphere_n and_o cylinder_n with_o other_o his_o woâkes_n be_v every_o where_o to_o be_v have_v and_o the_o demonstration_n thereof_o easy_a a_o theorem_a 2._o every_o sphere_n be_v quadruplâ_n to_o that_o cone_n who_o base_a be_v the_o great_a circle_n &_o height_n the_o semidiameter_n of_o the_o same_o sphere_n this_o be_v the_o 32._o proposition_n of_o archimedes_n fiâst_n book_n of_o the_o sphere_n and_o cylinder_n a_o problem_n 1._o a_o sphere_n be_v give_v to_o make_v a_o upright_a cone_n equal_a to_o the_o same_o or_o in_o any_o other_o proportion_n give_v between_o two_o right_a line_n and_o as_o concern_v the_o other_o part_n of_o
also_o divide_v into_o two_o equal_a part_n be_v cylinder_n which_o two_o equal_a cylinder_n let_v be_v ig_o and_o fk_v the_o axe_n of_o ig_n suppose_v to_o be_v hn_o and_o of_o fk_v the_o axe_n to_o be_v nm_o and_o for_o that_o fg_o be_v a_o upright_o cylinder_n and_o at_o the_o point_n n_o cut_v by_o a_o plain_a superficies_n parallel_v to_o his_o opposite_a base_n the_o common_a section_n of_o that_o plain_a superficies_n and_o the_o cylinder_n fg_o must_v be_v a_o circle_n circle_n equal_a to_o his_o base_a flb_n and_o have_v his_o centre_n the_o point_n n._n which_o circle_n let_v be_v jok_v and_o see_v that_o flb_n be_v by_o supposition_n equal_a to_o the_o great_a circle_n in_o a_o jok_n also_o shall_v be_v equal_a to_o the_o great_a circle_n in_o a_o contain_v also_o by_o reason_n mh_o be_v by_o supposition_n equal_a to_o the_o diameter_n of_o a_o and_o nh_o by_o construction_n half_a of_o mh_o it_o be_v manifest_a that_o nh_v be_v equal_a to_o the_o semidiameter_n of_o a._n if_o therefore_o you_o suppose_v a_o cone_n to_o have_v the_o circle_n jok_v to_o hiâ_z base_a and_o nh_v to_o his_o heith_n the_o sphere_n a_o shall_v be_v to_o that_o cone_n quadrupla_fw-la by_o the_o 2._o theorem_a let_v that_o cone_n be_v hiok_v wherefore_o a_o be_v quadrupla_fw-la to_o hiok_n and_o the_o cylinder_n ig_n have_v the_o same_o base_a with_o hiok_n the_o circle_n jok_v and_o the_o same_o heith_n the_o right_a line_n nh_v be_v triple_a to_o the_o cone_n hiok_v by_o the_o 10._o of_o this_o twelve_o book_n but_o to_o ig_v the_o whole_a cylinder_n fg_o be_v double_a as_o be_v prove_v wherefore_o fg_o be_v triple_a and_o triple_a to_o the_o cone_n hiok_v that_o be_v sextuple_a and_o a_o be_v prove_v quadrupla_fw-la to_o the_o same_o hiok_n wherefore_o fg_o be_v to_o hiok_v as_o 6._o to_o 1_o and_o a_o be_v to_o hiok_v as_o 4._o to_o 1_o demonstrate_v therefore_o fg_o be_v to_o a_o as_z 6_o to_o 4_o which_o in_o the_o least_o term_n be_v as_z 3_o to_o 2._o but_o 3_o to_o 2_o be_v the_o term_n of_o sesquialtera_fw-la proportion_n wherefore_o the_o cylinder_n fg_o be_v to_o a_o sesquialtera_fw-la in_o proportion_n second_o forasmuch_o as_o the_o superficies_n of_o a_o cylinder_n his_o two_o opposite_a base_n except_v be_v equal_a to_o that_o circle_n who_o semidiameter_n be_v middle_a proportional_a between_o the_o side_n of_o the_o cylinder_n and_o the_o diameter_n of_o his_o base_a as_o unto_o the_o 10._o of_o this_o book_n i_o have_v add_v but_o of_o fg_o the_o side_n bg_o be_v parallel_n and_o equal_a to_o the_o axe_n mh_o must_v also_o be_v equal_a to_o the_o diameter_n of_o a._n and_o the_o base_a flb_n be_v by_o supposition_n equal_a to_o the_o great_a circle_n in_o a_o contain_v must_v have_v his_o diameter_n fb_o equal_a to_o the_o say_a diameter_n of_o a._n the_o middle_a proportional_a therefore_o between_o bg_o and_o fb_o be_v equal_a each_o to_o other_o shalâ_n be_v a_o line_n equal_a to_o either_o of_o they_o as_o iâ_z ãâã_d set_n bg_o and_o fb_o together_o as_o one_o line_n and_o upon_o that_o line_n compose_v as_o a_o diameter_n make_v a_o semicircle_n and_o from_o the_o centre_n to_o the_o circumference_n draw_v a_o linâ_n perpendicular_a to_o the_o say_a diameter_n demonstrate_v by_o the_o ãâã_d of_o the_o six_o that_o perpendicular_a be_v middle_n proportional_a between_o fb_o and_o bg_o the_o semidiametersâ_n and_o he_o himself_o also_o a_o semidiameter_n and_o therefore_o by_o the_o definition_n of_o a_o circle_n equal_a to_o fb_o and_o likewise_o to_o bg_o and_o a_o circle_n have_v his_o semidiameter_n equal_a to_o the_o diameter_n fb_o be_v quadruple_a to_o the_o circle_n flb_n for_o the_o square_n of_o every_o whole_a line_n be_v quadruple_a to_o the_o squâre_n of_o his_o half_a line_n as_o may_v be_v prove_v by_o the_o 4._o of_o the_o second_o and_o by_o the_o second_o of_o this_o twelve_o circle_n be_v one_o to_o the_o other_o as_o the_o square_n of_o their_o diameter_n be_v wherefore_o the_o superficies_n cylindricall_fw-mi of_o fg_o alone_o be_v quadrupla_fw-la to_o his_o base_a flb_n but_o if_o a_o certain_a quantity_n be_v dupla_fw-la to_o one_o thing_n and_o a_o other_o quadrupla_fw-la to_o the_o same_o one_o thing_n those_o two_o quantity_n together_o be_v sextupla_fw-la to_o the_o same_o one_o thing_n therefore_o see_v the_o base_a opposite_a to_o flb_n be_v equal_a to_o to_o flb_n add_v to_o flb_n make_v that_o compound_n double_a to_o flb_n that_o double_a add_v to_o the_o cylindrical_a superficies_n of_o fg_o do_v make_v a_o superficies_n sextupla_fw-la to_o flb_n and_o the_o superficies_n of_o a_o be_v quadrupla_fw-la to_o the_o same_o flb_n by_o the_o first_o theorem_a therefore_o the_o cylindrical_a superficies_n of_o fg_o with_o the_o superficiece_n of_o his_o two_o base_n be_v to_o the_o superficies_n flb_n as_o 6_o to_o 1_o and_o the_o superficies_n of_o a_o to_o flb_n be_v as_o 4_o to_o 1._o wherefore_o the_o cylindrical_a superficies_n of_o fg_o &_o his_o two_o base_n together_o be_v to_o the_o superficies_n of_o a_o as_z 6_o to_o 4_o that_o be_v in_o the_o small_a term_n as_o 3_o to_o 2._o which_o be_v proper_a to_o sesâuialtera_fw-la proportion_n three_o it_o be_v already_o make_v evident_a that_o the_o superficies_n cylindrical_a of_o fg_o only_o by_o itself_o be_v quadrupla_fw-la to_o flb_n and_o also_o it_o be_v prove_v that_o the_o superficies_n of_o the_o sphere_n a_o be_v quadrupla_fw-la to_o the_o same_o flb_n wherefore_o by_o the_o 7._o of_o the_o five_o the_o cylindrical_a superficies_n of_o fg_o be_v equal_a to_o the_o superficies_n of_o a._n therefore_o every_o cylinder_n which_o have_v his_o base_a the_o great_a circle_n in_o a_o sphere_n and_o heith_n equal_a to_o the_o diameter_n of_o that_o sphere_n be_v sesquialtera_fw-la to_o that_o spear_n also_o the_o superficies_n of_o that_o cylinder_n with_o his_o two_o base_n be_v sesquialtera_fw-la to_o the_o superficies_n of_o the_o sphere_n and_o without_o his_o two_o base_n be_v equal_a to_o the_o superficies_n of_o the_o sphere_n which_o be_v to_o be_v demonstrate_v the_o lemma_n if_o a_o be_v to_o c_o as_z 6_o to_o 1_o and_o b_o to_o c_o as_z 4_o to_o 1_o a_o be_v to_o b_o as_z 6_o to_o 4._o for_o see_v b_o be_v to_o c_o as_z 4_o to_o 1_o by_o supposition_n therefore_o backward_o by_o the_o 4._o of_o the_o five_o c_z be_v to_o b_o as_z 1_o to_o 4._o imagine_v now_o two_o order_n of_o qnantity_n the_o first_o a_o c_o and_o b_o the_o second_o 6_o 1_o and_o 4._o forasmuch_o as_o a_o be_v to_o c_o as_z 6_o to_o 1_o by_o supposition_n and_o c_z be_v to_o b_o as_z 1_o to_o 4_o as_o we_o have_v prove_v wherefore_o a_o be_v to_o b_o as_z 6_o to_o 4_o by_o the_o 22_o of_o the_o five_o therefore_o if_o a_o be_v to_o c_o as_z 6_o to_o 1_o and_o b_o to_o c_z as_z 4_o to_o 1_o a_o be_v to_o b_o as_z 6_o to_o 4._o which_o be_v to_o be_v prove_v note_n slight_a thing_n some_o time_n lack_v evident_a proof_n breed_n doubt_n or_o ignorance_n and_o i_o need_n not_o warnâ_n you_o how_o genârall_a this_o demonstration_n be_v for_o if_o you_o put_v in_o the_o place_n of_o 6_o and_o 4_o any_o other_o number_n the_o like_a manner_n of_o conclusion_n will_v follow_v so_o likewise_o in_o place_n of_o 1._o any_o other_o one_o number_n may_v be_v as_z if_o a_o be_v to_o c_o as_o â_o to_o 5_o and_o b_o unto_o c_o be_v as_z 7_o to_o 5_o a_o shall_v be_v to_o b_o as_z 6_o to_o 7._o etc_n etc_n a_o problem_n 4._o to_o a_o sphere_n give_v to_o make_v a_o cylinder_n equal_a or_o in_o any_o proportion_n give_v between_o two_o right_a line_n suppose_v the_o give_v sphere_n to_o be_v a_o and_o the_o proportion_n give_v to_o be_v that_o between_o x_o and_o y._n i_o say_v that_o a_o cylinder_n be_v to_o be_v make_v equal_a to_o aâ_z or_o else_o in_o the_o same_o proportion_n to_o a_o that_o be_v between_o x_o to_o y._n let_v a_o cylinder_n be_v make_v such_o one_o as_o the_o theorem_a next_o before_o suppose_a that_o shall_v have_v his_o base_a equal_a to_o the_o great_a circle_n in_o a_o construction_n and_o height_n equal_a to_o the_o diameter_n of_o a_o let_v that_o cylinder_n bâ_z the_o upright_o cylinder_n bc._n leâ_n the_o one_o side_n of_o bc_o be_v the_o right_a line_n qc_n divide_v qc_n into_o three_o equal_a partâ_n of_o which_o let_v qe_n contain_v two_o and_o let_v the_o three_o part_n be_v ce._n by_o the_o point_n e_o suppose_v a_o plain_a parallel_n to_o the_o base_n of_o bc_o to_o pass_v through_o the_o cylinder_n bc_o cut_v the_o same_o by_o the_o circle_n de._n i_o say_v that_o the_o cylinder_n be_v be_v equal_a to_o the_o sphere_n a._n for_o see_v bc_o be_v a_o
contain_v in_o a_o the_o sphere_n be_v the_o circle_n bcd_o construction_n and_o by_o the_o problem_n of_o my_o addition_n upon_o the_o second_o proposition_n of_o this_o book_n as_o x_o be_v to_o you_o so_o let_v the_o circle_n bcd_v be_v to_o a_o other_o circle_n find_v let_v that_o other_o circle_n be_v efg_o and_o his_o diameter_n eglantine_n i_o say_v that_o the_o spherical_a superficies_n of_o the_o sphere_n a_o have_v to_o the_o spherical_a superficies_n of_o the_o sphere_n who_o great_a circle_n be_v efg_o or_o his_o equal_a that_o proportion_n which_o x_o have_v to_o y._n demonstration_n for_o by_o construction_n bcd_o be_v to_o efg_o as_o x_o be_v to_o you_o and_o by_o the_o theorem_a next_o beforeâ_n as_o bcd_o be_v to_o âfg_n so_o be_v the_o spherical_a superficies_n of_o a_o who_o great_a circle_n be_v bcd_o by_o supposition_n to_o the_o spherical_a superficies_n of_o the_o sphere_n who_o great_a circle_n be_v efg_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o x_o be_v to_o you_o so_o be_v the_o spherical_a superficies_n of_o a_o to_o the_o spherical_a superficies_n of_o the_o sphere_n who_o great_a circle_n be_v efg_o wherefore_o the_o sphere_n who_o diameter_n be_v eglantine_n the_o diameter_n also_o of_o efg_o be_v the_o sphere_n to_o who_o spherical_a superficies_n the_o spherical_a superficies_n of_o the_o sphere_n a_o have_v that_o proportion_n which_o x_o have_v to_o y._n a_o sphere_n be_v give_v therefore_o we_o have_v give_v a_o other_o sphere_n to_o who_o spherical_a superficies_n the_o superficies_n spherical_a of_o the_o sphere_n geum_z have_v any_o proportion_n give_v between_o two_o right_a line_n which_o ought_v to_o be_v do_v a_o problem_n 10._o a_o sphere_n be_v give_v and_o a_o circle_n less_o than_o the_o great_a circle_n in_o the_o same_o sphere_n contain_v to_o cope_v in_o the_o sphere_n give_v a_o circle_n equal_a to_o the_o circle_n give_v suppose_v a_o to_o be_v the_o sphere_n give_v and_o the_o circle_n give_v less_o than_o the_o great_a circle_n in_o a_o contain_v to_o be_v fkg_v spâerâ_n i_o say_v that_o in_o the_o sphere_n a_o a_o circle_n equal_a to_o the_o circle_n fkg_v be_v to_o be_v coapt_v first_o understand_v what_o we_o mean_v here_o by_o coapt_a of_o a_o circle_n in_o a_o sphere_n we_o say_v that_o circle_n to_o be_v coapt_v in_o a_o sphere_n who_o whole_a circumference_n be_v in_o the_o superficies_n of_o the_o same_o sphere_n let_v the_o great_a circle_n in_o the_o sphere_n a_o contain_v be_v the_o circle_n bcd_o who_o diameter_n suppose_v to_o be_v bd_o and_o of_o the_o circle_n fkg_v let_v fg_o be_v the_o diameter_n construction_n by_o the_o 1._o of_o the_o four_o let_v a_o line_n equal_a to_o fg_o be_v coapt_v in_o the_o circle_n bcd_o which_o line_n coapt_v let_v be_v be._n and_o by_o the_o line_n be_v suppose_v a_o plain_a to_o pass_v cut_v the_o sphere_n a_o and_o to_o be_v perpendicular_o erect_v to_o the_o superficies_n of_o bcd_o demonstration_n see_v that_o the_o portion_n of_o the_o plain_n remain_v in_o the_o sphere_n be_v call_v their_o common_a section_n the_o say_a section_n shall_v be_v a_o circle_n as_o before_o be_v prove_v and_o the_o common_a section_n of_o the_o say_a plain_n and_o the_o great_a circle_n bcd_o which_o be_v be_v by_o supposition_n shall_v be_v the_o diameter_n of_o the_o same_o circle_n as_o we_o will_v prove_v for_o let_v that_o circle_n be_v blem_n let_v the_o centre_n of_o the_o sphere_n a_o be_v the_o point_n h_o which_o h_o be_v also_o the_o centre_n of_o the_o circle_n bcd_o because_o bcd_o be_v the_o great_a circle_n in_o a_o contain_v from_o h_o the_o centre_n of_o the_o sphere_n a_o let_v a_o line_n perpendicular_o be_v let_v fall_n to_o the_o circle_n blem_n let_v that_o line_n be_v ho_o and_o it_o be_v evident_a that_o ho_o shall_v fall_v upon_o the_o common_a section_n be_v by_o the_o 38._o of_o the_o eleven_o and_o it_o divide_v be_v into_o two_o equal_a part_n by_o the_o second_o part_n of_o the_o three_o proposition_n of_o the_o three_o book_n by_o which_o point_n oh_o all_o other_o line_n draw_v in_o the_o circle_n blem_n be_v at_o the_o same_o point_n oh_o divide_v into_o two_o equal_a part_n as_o if_o from_o the_o point_n m_o by_o the_o point_n oh_o a_o right_a line_n be_v draw_v one_o the_o other_o side_n come_v to_o the_o circumference_n at_o the_o point_n n_o it_o be_v manifest_a that_o nom_fw-la be_v divide_v into_o two_o equal_a part_n at_o the_o point_n oh_o euclid_n by_o reason_n if_o from_o the_o centre_n h_o to_o the_o point_n n_o and_o m_o right_a line_n be_v draw_v hn_o and_o he_o the_o square_n of_o he_o and_o hn_o be_v equal_a for_o that_o all_o the_o semidiameter_n of_o the_o sphere_n be_v equal_a and_o therefore_o their_o square_n be_v equal_a one_o to_o the_o other_o and_o the_o square_a of_o the_o perpendicular_a ho_o be_v common_a wherefore_o the_o square_a of_o the_o three_o line_n more_n be_v equal_a to_o the_o square_n of_o the_o three_o line_n no_o and_o therefore_o the_o line_n more_n to_o the_o line_n no_o so_o therefore_o be_v nm_o equal_o divide_v at_o the_o point_n o._n and_o so_o may_v be_v prove_v of_o all_o other_o right_a line_n draw_v in_o the_o circle_n blem_n pass_v by_o the_o point_n oh_o to_o the_o circumference_n one_o both_o side_n wherefore_o o_n be_v the_o centre_n of_o the_o circle_n blem_n and_o therefore_o be_v pass_v by_o the_o point_n oh_o be_v the_o diameter_n of_o the_o circle_n blem_n which_o circle_n i_o say_v be_v equal_a to_o fkg_v for_o by_o construction_n be_v be_v equal_a to_o fg_o and_o be_v be_v prove_v the_o diameter_n of_o blem_n and_o fg_o be_v by_o supposition_n the_o diameter_n of_o the_o circle_n fkg_v wherefore_o blem_n be_v equal_a to_o fkg_v the_o circle_n give_v and_o blem_n be_v in_o a_o the_o sphere_n give_v wherefore_o we_o have_v in_o a_o sphere_n give_v coapt_v a_o circle_n equal_a to_o a_o circle_n give_v which_o be_v to_o be_v do_v a_o corollary_n beside_o our_o principal_a purpose_n in_o this_o problem_n evident_o demonstrate_v this_o be_v also_o make_v manifest_a that_o if_o the_o great_a circle_n in_o a_o sphere_n be_v cut_v by_o a_o other_o circle_n erect_v upon_o he_o at_o right_a angle_n that_o the_o other_o circle_n be_v cut_v by_o the_o centre_n and_o that_o their_o common_a section_n be_v the_o diameter_n of_o that_o other_o circle_n and_o therefore_o that_o other_o circle_n divide_v be_v into_o two_o equal_a part_n a_o problem_n 11._o a_o sphere_n be_v give_v and_o a_o circle_n less_o than_o double_a the_o great_a circle_n in_o the_o same_o sphere_n contain_v to_o cut_v of_o a_o segment_n of_o the_o same_o sphere_n who_o spherical_a superficies_n shall_v be_v equal_a to_o the_o circle_n give_v suppose_v king_n to_o be_v a_o sphere_n give_v who_o great_a circle_n let_v be_v abc_n and_o the_o circle_n give_v suppose_v to_o be_v def_n construction_n i_o say_v that_o a_o segment_n of_o the_o sphere_n king_n be_v to_o be_v cut_v of_o so_o great_a that_o his_o spherical_a superficies_n shall_v be_v equal_a to_o the_o circle_n def_n let_v the_o diameter_n of_o the_o circle_n abc_n be_v the_o line_n ab_fw-la at_o the_o point_n a_o in_o the_o circle_n abc_n cope_v a_o right_a line_n equal_a to_o the_o semidiameter_n of_o the_o circle_n def_n by_o the_o first_o of_o the_o four_o which_o line_n suppose_v to_o be_v ah_o from_o the_o point_n h_o to_o the_o diameter_n ab_fw-la let_v a_o perpendicular_a line_n be_v draw_v which_o suppose_v to_o be_v high_a produce_v high_a to_o the_o other_o side_n of_o the_o circumference_n and_o let_v it_o come_v to_o the_o circumference_n at_o the_o point_n l._n by_o the_o right_a line_n hil_n perpendicular_a to_o ab_fw-la suppose_v a_o plain_a superficies_n to_o pass_v perpendicular_o erect_v upon_o the_o circle_n abc_n and_o by_o this_o plain_a superficies_n the_o sphere_n to_o be_v cut_v into_o two_o segment_n one_o less_o than_o the_o half_a sphere_n namely_o hali_n and_o the_o other_o great_a than_o the_o half_a sphere_n namely_o hbli_n i_o say_v that_o the_o spherical_a superficies_n of_o the_o segment_n of_o the_o sphere_n king_n in_o which_o the_o segment_n of_o the_o great_a circle_n hali_n be_v contain_v who_o base_a be_v the_o circle_n pass_v by_o hil_n and_o top_n the_o point_n a_o be_v equal_a to_o the_o circle_n def_n demonstration_n for_o the_o circle_n who_o semidiameter_n be_v equal_a to_o the_o line_n ah_o be_v equal_a to_o the_o spherical_a superficies_n of_o the_o segment_n hal_n by_o the_o 4._o theorem_a here_o add_v and_o by_o construction_n ah_o be_v equal_a to_o the_o semidiameter_n of_o the_o circle_n def_n therefore_o the_o spherical_a superficies_n of_o the_o segment_n of_o the_o sphere_n king_n cut_v of_o by_o the_o
superficies_n or_o solidity_n in_o the_o hole_n or_o in_o partâ_n such_o certain_a knowledge_n demonstrative_a may_v arise_v and_o such_o mechanical_a exercise_n thereby_o be_v devise_v that_o sure_a i_o be_o to_o the_o sincere_a &_o true_a student_n great_a light_n aid_n and_o comfortable_a courage_n far_a to_o wade_v will_v enter_v into_o his_o heart_n and_o to_o the_o mechanical_a witty_a and_o industrous_a deviser_n new_a manner_n of_o invention_n &_o execution_n in_o his_o work_n will_v with_o small_a travail_n for_o foot_n application_n come_v to_o his_o perceiveraunce_n and_o understanding_n therefore_o even_o a_o manifold_a speculation_n &_o practice_n may_v be_v have_v with_o the_o circle_n his_o quantity_n be_v not_o know_v in_o any_o kind_n of_o small_a certain_a measure_n so_o likewise_o of_o the_o sphere_n many_o problem_n may_v be_v execute_v and_o his_o precise_a quantity_n in_o certain_a measure_n not_o determine_v or_o know_v yet_o because_o both_o one_o of_o the_o first_o humane_a occasion_n of_o invent_v and_o stablish_v this_o art_n be_v measure_v of_o the_o earth_n and_o therefore_o call_v geometria_n that_o be_v earthmeasure_v and_o also_o the_o chief_a and_o general_a end_n in_o deed_n be_v measure_n and_o measure_n require_v a_o determination_n of_o quantity_n in_o a_o certain_a measure_n by_o number_n express_v it_o be_v needful_a for_o mechanical_a earthmeasure_n not_o to_o be_v ignorant_a of_o the_o measure_n and_o content_n of_o the_o circle_n neither_o of_o the_o sphere_n his_o measure_n and_o quantity_n as_o near_o as_o sense_n can_v imagine_v or_o wish_v and_o in_o very_a deed_n the_o quantity_n and_o measure_n of_o the_o circle_n be_v know_v make_v not_o only_o the_o cone_n and_o cylinder_n but_o also_o the_o sphere_n his_o quantity_n to_o be_v as_o precise_o know_v and_o certain_a therefore_o see_v in_o respect_n of_o the_o circle_n quantity_n by_o archimedes_n specify_v this_o theorem_a be_v note_v unto_o you_o i_o will_v by_o order_n upon_o that_o as_o a_o supposition_n infer_v the_o conclusion_n of_o this_o our_o theorem_n note_n 1._o wherefore_o if_o you_o divide_v the_o one_o side_n as_o tq_n of_o the_o cube_fw-la tx_n into_o 21._o equal_a part_n and_o where_o 11._o part_n do_v end_n reckon_v from_o t_o suppose_v the_o point_n p_o and_o by_o that_o point_n p_o imagine_v a_o plain_a pass_v parallel_n to_o the_o opposite_a base_n to_o cut_v the_o cube_fw-la tx_n and_o thereby_o the_o cube_fw-la tx_n to_o be_v divide_v into_o two_o rectangle_n parallelipipedon_n namely_o tn_n and_o px_n it_o be_v manifest_a give_v tn_n to_o be_v equal_a to_o the_o sphere_n a_o by_o construction_n and_o the_o 7._o of_o the_o five_o note_n 2._o second_o the_o whole_a quantity_n of_o the_o sphere_n a_o assign_v be_v contain_v in_o the_o rectangle_n parallelipipedon_n tn_n you_o may_v easy_o transform_v the_o same_o quantity_n into_o other_o parallelipipedon_n rectangle_v of_o what_o height_n and_o of_o what_o parallelogram_n base_a you_o listen_v by_o my_o first_o and_o second_o problem_n upon_o the_o 34._o of_o this_o book_n and_o the_o like_a may_v you_o do_v to_o any_o assign_a part_n of_o the_o sphere_n a_o by_o the_o like_a mean_n devide_v the_o parallelipipedon_n tn_n as_o the_o part_n assign_v do_v require_v as_o if_o a_o three_o four_o five_o or_o six_o part_n of_o the_o sphere_n a_o be_v to_o be_v have_v in_o a_o parallelipipedon_n of_o any_o parallelograââe_n base_a assign_v or_o of_o any_o heith_n assign_v then_o devide_v tp_n into_o so_o many_o part_n as_o into_o 4._o if_o a_o four_o part_n be_v to_o be_v transform_v or_o into_o five_o if_o a_o five_o part_n be_v to_o be_v transform_v etc_n etc_n and_o then_o proceed_v âs_v you_o do_v with_o cut_v of_o tn_n from_o tx_n and_o that_o i_o say_v of_o parallelipipedon_n may_v in_o like_a sort_n by_o my_o ââyd_n two_o problem_n add_v to_o the_o 34._o of_o this_o book_n be_v do_v in_o any_o side_v column_n pyramid_n and_o prismeâ_n so_o thââ_n in_o pyramid_n and_o some_o prism_n you_o use_v the_o caution_n necessary_a in_o respect_n of_o their_o quan_fw-mi ãâ¦ã_o odyes_n have_v parallel_n equal_a and_o opposite_a base_n who_o part_n ãâ¦ã_o re_fw-mi in_o their_o proposition_n be_v by_o euclid_n demonstrate_v and_o final_o ãâ¦ã_o addition_n you_o have_v the_o way_n and_o order_n how_o to_o geve_v to_o a_o sphere_n or_o any_o segment_n oâ_n the_o same_o cones_n or_o cylinder_n equal_a or_o in_o any_o proportion_n between_o two_o right_a line_n give_v with_o many_o other_o most_o necessary_a speculation_n and_o practice_n about_o the_o sphere_n i_o trust_v that_o i_o have_v sufficient_o âraughted_v your_o imagination_n for_o your_o honest_a and_o profitable_a study_n herein_o and_o also_o give_v you_o reaââ_n ââtter_n wheââ_n with_o to_o sââp_v the_o mouth_n of_o the_o malicious_a ignorant_a and_o arrogant_a despiser_n of_o the_o most_o excellent_a discourse_n travail_n and_o invention_n mathematical_a sting_v aswell_o the_o heavenly_a sphere_n &_o star_n their_o spherical_a solidity_n geometry_n with_o their_o convene_n spherical_a superficies_n to_o the_o earth_n at_o all_o time_n respect_v and_o their_o distance_n from_o the_o earth_n as_o also_o the_o whole_a earthly_a sphere_n and_o globe_n itself_o and_o infinite_a other_o case_n concern_v sphere_n or_o globe_n may_v hereby_o with_o as_o much_o ease_n and_o certainty_n be_v determine_v of_o as_o of_o the_o quantity_n of_o any_o bowl_n ball_n or_o bullet_n which_o we_o may_v gripe_v in_o our_o hand_n reason_n and_o experience_n be_v our_o witness_n and_o without_o these_o aid_n such_o thing_n of_o importance_n never_o able_a of_o we_o certain_o to_o be_v know_v or_o attain_a unto_o here_o end_n m._n john_n do_v you_o his_o addition_n upon_o the_o last_o proposition_n of_o the_o twelve_o book_n a_o proposition_n add_v by_o flussas_n if_o a_o sphere_n touch_v a_o plain_a superficiesâ_n a_o right_a line_n draw_v from_o the_o centre_n to_o the_o touch_n shall_v be_v erect_v perpendicular_o to_o the_o plain_a superficies_n suppose_v that_o there_o be_v a_o sphere_n bcdl_n who_o centre_n let_v be_v the_o point_n a._n and_o let_v the_o plain_a superficies_n gci_n touch_v the_o spear_n in_o the_o point_n c_o and_o extend_v a_o right_a line_n from_o the_o centre_n a_o to_o the_o point_n c._n then_o i_o say_v that_o the_o line_n ac_fw-la be_v erect_v perpendicular_o to_o tâe_z plain_a gic._n let_v the_o sphere_n be_v cut_v by_o plain_a superficiece_n pass_v by_o the_o right_a line_n lac_n which_o plain_n let_v be_v abcdl_n and_o acel_n which_o let_v cut_v the_o plain_n gci_n by_o the_o right_a line_n gch_o and_o kci_fw-la now_o it_o be_v manifest_a by_o the_o assumpt_n put_v before_o the_o 17._o of_o this_o book_n that_o the_o two_o section_n of_o the_o sphere_n shall_v be_v circle_n have_v to_o their_o diameter_n the_o line_n lac_n which_o be_v also_o the_o diameter_n of_o the_o sphere_n wherefore_o the_o right_a line_n gch_o and_o kci_fw-la which_o be_v draw_v in_o the_o plain_n gci_n do_v at_o the_o point_n c_o fall_n without_o the_o circle_n bcdl_n and_o ecl._n wherefore_o they_o touch_v the_o circle_n in_o the_o point_n c_o by_o the_o second_o definition_n of_o the_o three_o wherefore_o the_o right_a line_n lac_n make_v right_a angle_n with_o the_o line_n gch_v and_o kci_fw-la by_o the_o 16._o of_o the_o three_o wherefore_o by_o the_o 4._o of_o the_o eleven_o the_o right_a line_n ac_fw-la be_v erect_v perpendicular_o to_o to_o the_o plain_a superficies_n gci_n wherein_o be_v draw_v the_o line_n gch_v and_o kci_fw-la if_o therefore_o a_o sphere_n touch_v a_o plain_a superficies_n a_o right_a line_n draw_v from_o the_o centre_n to_o the_o touch_n shall_v be_v erect_v perpendicular_o to_o the_o plain_a superficies_n which_o be_v require_v to_o be_v prove_v the_o end_n of_o the_o twelve_o book_n of_o euclides_n element_n ¶_o the_o thirteen_o book_n of_o euclides_n element_n in_o this_o thirteen_o book_n be_v set_v forth_o certain_a most_o wonderful_a and_o excellent_a passion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n book_n a_o matter_n undoubted_o of_o great_a and_o infinite_a use_n in_o geometry_n as_o you_o shall_v both_o in_o this_o book_n and_o in_o the_o other_o book_n follow_v most_o evident_o perceive_v it_o teach_v moreover_o the_o composition_n of_o the_o five_o regular_a solid_n and_o how_o to_o inscribe_v they_o in_o a_o sphere_n give_v and_o also_o set_v forth_o certain_a comparison_n of_o the_o say_a body_n both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o sphere_n wherein_o they_o be_v describe_v the_o 1._o theorem_a the_o 1._o proposition_n if_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o to_o the_o great_a segment_n be_v add_v the_o half_a of_o the_o whole_a line_n the_o square_n make_v of_o those_o two_o
line_n add_v together_o shall_v be_v quintuple_a to_o the_o square_n make_v of_o the_o half_a of_o the_o whole_a line_n svppose_v that_o the_o right_a line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c._n and_o let_v the_o great_a segment_n thereof_o be_v ac_fw-la and_o unto_o ac_fw-la add_v direct_o a_o right_a line_n ad_fw-la and_o let_v ad_fw-la be_v equal_a to_o the_o half_a of_o the_o line_n ab_fw-la construction_n then_o i_o say_v that_o the_o square_a of_o the_o line_n cd_o be_v quintuple_a to_o the_o square_n of_o the_o line_n da._n describe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n ab_fw-la and_o dc_o square_n namely_o ae_n &_o df._n and_o in_o the_o square_a df_o describe_v and_o make_v complete_a the_o figure_n and_o extend_v the_o line_n fc_o to_o the_o point_n g._n demonstration_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c_o therefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v the_o parallelogram_n ce_fw-fr and_o the_o square_a of_o the_o line_n ac_fw-la be_v the_o square_a hf._n wherefore_o the_o parallelogram_n ce_fw-fr be_v equal_a to_o the_o square_a hf._n and_o forasmuch_o as_o the_o line_n basilius_n be_v double_a to_o the_o line_n ad_fw-la by_o construction_n ãâã_d the_o line_n basilius_n be_v equal_a to_o the_o line_n ka_o and_o the_o line_n ad_fw-la to_o the_o line_n ah_o therefore_o also_o the_o line_n ka_o be_v double_a to_o the_o line_n ah_o but_o as_o the_o line_n ka_o be_v to_o the_o line_n ah_o so_o be_v the_o parallelogram_n ck_o to_o the_o parallelogram_n change_z wherefore_o the_o parallelogram_n ck_o be_v double_a to_o the_o parallelogram_n ch._n and_o the_o parallelogram_n lh_o and_o ch_n be_v double_a to_o the_o parallelogram_n change_z for_o supplement_n of_o parallelogram_n be_v bâ_z the_o 4â_o of_o the_o first_o equal_v the_o one_o to_o the_o other_o wherefore_o the_o parallelogram_n ck_o be_v equal_a to_o the_o parallelogram_n lh_o &_o ch._n and_o it_o be_v prove_v that_o the_o parallelogram_n ce_fw-fr be_v equal_a to_o the_o square_a fh_o wherefore_o the_o whole_a square_a ae_n be_v equal_a to_o the_o gnomon_n mxn_n and_o forasmuch_o as_o the_o line_n basilius_n iâ_z double_a to_o the_o line_n ad_fw-la therefore_o the_o square_a of_o the_o line_n basilius_n be_v by_o the_o 20._o of_o the_o six_o quadruple_a to_o the_o square_n of_o the_o line_n dam_fw-ge that_o be_v the_o square_a ae_n to_o the_o square_a dh_o but_o the_o square_a ae_n be_v equal_a to_o the_o gnomon_n mxn_n wherefore_o the_o gnomon_n mxn_n be_v also_o quadruple_a to_o the_o square_a dh_o wherefore_o the_o whole_a square_a df_o be_v quintuple_a to_o the_o square_a dh_o but_o the_o square_a df_o iâ_z the_o square_a of_o the_o line_n cd_o and_o the_o square_a dh_o be_v the_o square_a of_o the_o line_n da._n wherefore_o the_o square_a of_o the_o line_n cd_o be_v quintuple_a to_o the_o square_n of_o the_o line_n da._n if_o therefore_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o to_o the_o great_a segment_n be_v add_v the_o half_a of_o the_o whole_a line_n the_o square_n make_v of_o those_o two_o line_n add_v together_o shall_v be_v quintuple_a to_o the_o square_n make_v of_o the_o half_a of_o the_o whole_a line_n which_o be_v require_v to_o be_v demonstrate_v this_o proposition_n be_v a_o other_o way_n demonstrate_v after_o the_o five_o proposition_n of_o this_o book_n the_o 2._o theorem_a the_o â_o proposition_n if_o a_o right_a line_n be_v in_o power_n quintuple_a to_o a_o segment_n of_o the_o same_o line_n the_o double_a of_o the_o say_a segment_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o the_o great_a segment_n thereof_o be_v the_o other_o part_n of_o the_o line_n give_v at_o the_o beginning_n prove_v now_o that_o the_o double_a of_o the_o line_n ad_fw-la that_o be_v ab_fw-la be_v great_a than_o the_o line_n ac_fw-la may_v thus_o be_v prove_v for_o if_o not_o then_o if_o if_o it_o be_v possible_a let_v the_o line_n ac_fw-la be_v double_a to_o the_o line_n ad_fw-la wherefore_o the_o square_a of_o the_o line_n ac_fw-la be_v quadruple_a to_o the_o square_n of_o the_o line_n ad._n wherefore_o the_o square_n of_o the_o line_n ac_fw-la and_o ad_fw-la be_v quintuple_a to_o the_o square_n of_o the_o line_n ad._n and_o it_o be_v suppose_v that_o the_o square_a of_o the_o line_n dc_o be_v quintuple_a to_o the_o square_n of_o the_o line_n ad_fw-la wherefore_o the_o square_a of_o the_o line_n dc_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o ad_fw-la which_o be_v impossible_a by_o the_o 4._o of_o the_o second_o wherefore_o the_o line_n ac_fw-la be_v not_o double_a to_o the_o line_n ad._n in_o like_a sort_n also_o may_v we_o prove_v that_o the_o double_a of_o the_o line_n ad_fw-la be_v not_o less_o than_o the_o line_n ac_fw-la for_o this_o be_v much_o more_o absurd_a wherefore_o the_o double_a of_o the_o line_n ad_fw-la be_v great_a they_o the_o line_n acâ_n which_o be_v require_v to_o be_v prove_v this_o proposition_n also_o be_v a_o other_o way_n demonstrate_v after_o the_o five_o proposition_n of_o this_o book_n two_o theorem_n in_o euclides_n method_n necessary_a add_v by_o m._n dee_n a_o theorem_a 1._o a_o right_a line_n can_v be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n but_o in_o one_o only_a point_n suppose_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n to_o be_v ab_fw-la and_o let_v the_o great_a segment_n be_v ac_fw-la i_o say_v that_o ab_fw-la can_v not_o be_v divide_v by_o the_o say_a proportion_n in_o any_o other_o point_n then_o in_o the_o point_n c._n if_o a_o adversary_n will_v contend_v that_o it_o may_v in_o like_a sort_n be_v divide_v in_o a_o other_o point_n let_v his_o other_o point_n be_v suppose_v to_o be_v d_o make_v ad_fw-la the_o great_a segment_n of_o his_o imagine_a division_n which_o ad_fw-la also_o let_v be_v less_o than_o our_o ac_fw-la for_o the_o first_o discourse_n now_o forasmuch_o as_o by_o our_o adversary_n opinion_n ad_fw-la be_v the_o great_a segment_n of_o his_o divide_a lineâ_z the_o parallelogram_n contain_v under_o ab_fw-la and_o db_o be_v equal_a to_o the_o square_n of_o ad_fw-la by_o the_o three_o definition_n and_o 17._o proposition_n of_o the_o six_o book_n and_o by_o the_o same_o definition_n and_o proposition_n the_o parallelogram_n under_o ab_fw-la and_o cb_o contain_v be_v equal_a to_o the_o square_n of_o our_o great_a segment_n ac_fw-la wherefore_o as_o the_o parallelogram_n under_o ab_fw-la and_o dâ_n be_v to_o the_o square_n of_o ad_fw-la so_o iâ_z ãâã_d parallelogram_n under_o ab_fw-la and_o cb_o to_o the_o square_n of_o ac_fw-la for_o proportion_n of_o equality_n be_v conclude_v in_o they_o both_o but_o forasmuch_o as_o dââ_n iâ_z byâ_n âc_z supposition_n great_a them_z cb_o the_o parallelogram_n under_o ab_fw-la and_o db_o be_v great_a than_o the_o parallelogram_n under_o ac_fw-la and_o cb_o by_o the_o first_o of_o the_o six_o for_o ab_fw-la be_v their_o equal_a heith_n wherefore_o the_o square_a of_o ad_fw-la shall_v be_v great_a than_o the_o square_a of_o ac_fw-la by_o the_o 14._o of_o the_o five_o but_o the_o line_n ad_fw-la be_v less_o than_o the_o line_n ac_fw-la by_o supposition_n wherefore_o the_o square_a of_o ad_fw-la be_v less_o than_o the_o square_a of_o ac_fw-la and_o it_o be_v conclude_v also_o to_o be_v great_a than_o the_o square_a of_o ac_fw-la wherefore_o the_o square_a of_o ad_fw-la be_v both_o great_a than_o the_o square_a of_o acâ_n and_o also_o less_o which_o be_v a_o thing_n impossible_a the_o square_a therefore_o of_o ad_fw-la be_v not_o equal_a to_o the_o parallelogram_n under_o ab_fw-la and_o db._n and_o therefore_o by_o the_o three_o definition_n of_o the_o six_o ab_fw-la be_v not_o divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n d_o as_o our_o adversary_n imagine_v and_o second_o in_o like_a sort_n will_v the_o inconueniency_n fall_v out_o if_o we_o assign_v ad_fw-la our_o adversary_n great_a segment_n to_o be_v great_a than_o our_o ac_fw-la therefore_o see_v neither_o on_o the_o one_o side_n of_o our_o point_n c_o neither_o on_o the_o other_o side_n of_o the_o same_o point_n c_o any_o point_n can_v be_v have_v at_o which_o the_o line_n ab_fw-la can_v be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n it_o follow_v of_o necessity_n that_o ab_fw-la can_v be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c_o only_a therefore_o a_o right_a line_n can_v be_v divide_v by_o a_o extreme_a
line_n ai._n wherefore_o the_o square_a of_o the_o line_n bw_o be_v equal_a to_o the_o square_n of_o the_o line_n ai_fw-fr wherefore_o also_o the_o line_n bt_n be_v equal_a to_o the_o line_n ai._n and_o by_o the_o same_o reason_n be_v the_o line_n id_fw-la and_o wc_n equal_a to_o the_o same_o line_n now_o forasmuch_o as_o the_o line_n ai_fw-fr and_o id_fw-la and_o the_o line_n all_o and_o ld_n be_v equal_a and_o the_o base_a il_fw-fr be_v common_a to_o they_o both_o the_o angle_n ali_n and_o dli_o shall_v be_v equal_a by_o the_o 8._o of_o the_o first_o and_o therefore_o they_o be_v right_a angle_n by_o the_o 10._o definition_n of_o the_o first_o and_o by_o the_o same_o reason_n be_v the_o angle_n whb_n and_o whc_n right_a angle_n and_o forasmuch_o as_o the_o two_o line_n ht_v and_o tw_n be_v equal_a to_o the_o two_o line_n lct_n and_o cti_n and_o they_o contain_v equal_a angle_n that_o be_v right_a angle_n by_o supposition_n therefore_o the_o angle_n wht_v and_o ilct_n be_v equal_a by_o the_o 4._o of_o the_o first_o wherefore_o the_o plain_a superficies_n aid_n be_v in_o like_a sort_n incline_v to_o the_o plain_a superficies_n abcd_o as_o the_o plain_a superficies_n bwc_n be_v incline_v to_o the_o same_o plain_n abcd_o by_o the_o 4._o definition_n of_o the_o eleven_o in_o like_a sort_n may_v we_o prove_v that_o the_o plain_n wcdi_n be_v in_o like_a sort_n incline_v to_o the_o plain_n abcd_o as_o the_o plain_a buzc_n be_v to_o the_o plain_n ebcf._n for_o that_o in_o the_o triangle_n yoh_n and_o ctpk_n which_o consist_v of_o equal_a side_n each_o to_o his_o correspondent_a side_n the_o angle_n yho_n and_o ctkp_n which_o be_v the_o angle_n of_o the_o inclination_n be_v equal_a and_o now_o if_o the_o right_a line_n ctk_n be_v extend_v to_o the_o point_n a_o and_o the_o pentagon_n cwida_n be_v make_v perfect_a we_o may_v by_o the_o same_o reason_n prove_v that_o that_o plain_n be_v equiangle_n and_o equilater_n that_o we_o prove_v the_o pentagon_n buzcw_o to_o be_v equaliter_fw-la and_o equiangle_n and_o likewise_o if_o the_o other_o plain_n bwia_fw-la and_o aid_v be_v make_v perfect_a they_o may_v be_v prove_v to_o be_v equal_a and_o like_a pentagon_n and_o in_o like_a sort_n situate_a and_o they_o be_v set_v upon_o these_o common_a right_a line_n bw_o wc_n wi_n ai_fw-fr and_o id_fw-la and_o observe_v this_o method_n there_o shall_v upon_o every_o one_o of_o the_o 12._o âidâs_n of_o the_o cube_fw-la be_v set_v every_o one_o of_o the_o 12._o pentagon_n which_o compose_v the_o dodecahedron_n ¶_o certain_a corollarye_n add_v by_o flussas_n first_o corollary_n the_o side_n of_o a_o cube_fw-la be_v equal_a to_o the_o right_a line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n of_o a_o dodecahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n with_o the_o cube_fw-la for_o the_o angle_n bwc_n and_o aid_n be_v subtend_v of_o the_o line_n bc_o and_o ad._n which_o be_v side_n of_o the_o cubeâ_n ¶_o second_v corollary_n in_o a_o dodecahedron_n there_o be_v six_o side_n every_o two_o of_o which_o be_v parallel_n and_o opposite_a who_o section_n into_o two_o equal_a part_n be_v couple_v by_o three_o right_a line_n which_o in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o dodecahedron_n divide_v into_o two_o equal_a part_n and_o perpendicular_o both_o themselves_o and_o also_o the_o side_n for_o upon_o the_o six_o base_n of_o the_o cube_fw-la be_v set_v six_o side_n of_o the_o dodecahedron_n as_o it_o have_v be_v prove_v by_o the_o line_n zv_o wi_n &c._a &c._a which_o be_v cut_v into_o two_o equal_a part_n by_o right_a line_n which_o join_v together_o the_o centre_n of_o the_o base_n of_o the_o cube_fw-la as_o the_o line_n the_fw-mi produce_v and_o the_o other_o like_a which_o line_n couple_v together_o the_o centre_n of_o the_o base_n be_v three_o in_o number_n cut_v the_o one_o the_o other_o perpendicular_o for_o they_o be_v parallel_n to_o the_o side_n of_o the_o cube_fw-la and_o they_o cut_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o cube_fw-la by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n and_o unto_o these_o equal_a line_n join_v together_o the_o centre_n of_o the_o base_n of_o the_o cube_fw-la be_v without_o the_o base_n add_v equal_a part_n oy_o p_o ct_v and_o the_o other_o like_a which_o by_o supposition_n be_v equal_a to_o half_n of_o the_o side_n of_o the_o dodecahedron_n wherefore_o the_o whole_a line_n which_o join_v together_o the_o sectoin_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n be_v equal_a and_o they_o cut_v those_o side_n into_o two_o equal_a part_n and_o perpendicular_o three_o corollary_n a_o right_a line_n join_v together_o the_o point_n of_o the_o section_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n into_o two_o equal_a part_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n the_o great_a segment_n thereof_o shall_v be_v the_o side_n of_o the_o cube_fw-la and_o the_o less_o segment_n the_o side_n of_o the_o dodecahedron_n contain_v in_o the_o self_n same_o sphere_n for_o it_o be_v prove_v that_o the_o right_a line_n yq_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n oh_o and_o that_o his_o great_a segment_n oq_fw-la be_v half_a the_o side_n of_o the_o cube_fw-la and_o his_o less_o segment_n oy_o be_v half_o of_o the_o side_n we_o which_o be_v the_o side_n of_o the_o dodecahedron_n wherefore_o it_o follow_v by_o the_o 15._o of_o the_o five_o that_o their_o double_n be_v in_o the_o same_o proportion_n wherefore_o the_o double_a of_o the_o line_n yq_n which_o join_v the_o point_n opposite_a unto_o the_o line_n y_fw-fr be_v the_o whole_a and_o the_o great_a segment_n be_v the_o double_a of_o the_o line_n oq_fw-la which_o be_v the_o side_n of_o the_o cube_fw-la &_o the_o less_o segment_n be_v the_o double_a of_o the_o line_n the_fw-mi which_o be_v equal_a to_o the_o side_n of_o the_o dodecahedron_n namely_o to_o the_o side_n we_o ¶_o the_o 6._o problem_n the_o 18._o proposition_n to_o find_v out_o the_o side_n of_o the_o foresay_a five_o body_n and_o to_o compare_v they_o together_o take_v the_o diameter_n of_o the_o sphere_n give_v and_o let_v the_o same_o be_v ab_fw-la and_o divide_v it_o in_o the_o point_n c_o so_fw-mi that_o let_v the_o line_n ac_fw-la be_v equal_a to_o cb_o by_o the_o 10._o of_o the_o first_o and_o in_o the_o point_n d_o so_fw-mi that_o let_v ad_fw-la be_v double_a to_o db_o by_o the_o 9_o of_o the_o six_o and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n aeb_fw-mi and_o from_o the_o point_n c_o and_o d_o raise_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o line_n ab_fw-la perpendicular_a line_n ce_fw-fr and_o df._n and_o draw_v these_o right_a line_n of_o fb_o and_o be._n now_o forasmuch_o as_o the_o line_n ad_fw-la be_v double_a to_o the_o line_n db_o therefore_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n db._n wherefore_o the_o line_n basilius_n be_v sesquialter_fw-la to_o the_o line_n ad_fw-la for_o it_o be_v as_o 3._o to_o 2._o but_o as_o the_o line_n basilius_n be_v to_o the_o line_n ad_fw-la so_o be_v the_o square_a of_o the_o line_n basilius_n to_o the_o square_n of_o the_o line_n of_o by_o the_o 6._o of_o the_o six_o or_o by_o the_o corollary_n of_o the_o same_o and_o by_o the_o corollary_n of_o the_o 20._o of_o the_o same_o for_o the_o triangle_n afb_o be_v equiangle_n to_o the_o triangle_n afd_v wherefore_o the_o square_a of_o the_o line_n basilius_n be_v sesquialter_fw-la to_o the_o square_n of_o the_o line_n af._n but_o the_o diameter_n of_o a_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o sidâ_n of_o the_o pyramid_n pyramid_n by_o the_o 13._o of_o this_o book_n and_o the_o line_n ab_fw-la be_v the_o diameter_n of_o the_o sphere_n wherefore_o the_o line_n of_o be_v equal_a to_o the_o side_n of_o the_o pyramid_n again_o forasmuch_o as_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n bd_o but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bd_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n fb_o by_o the_o corollary_n of_o the_o 8._o and_o 20._o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v treble_a to_o the_o square_n of_o the_o line_n fb_o but_o the_o diameter_n oâ_n a_o sphere_n be_v in_o power_n treble_a to_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o this_o book_n and_o the_o diameter_n of_o the_o sphere_n be_v the_o line_n ab_fw-la wherefore_o the_o line_n bf_o be_v the_o side_n of_o the_o cube_fw-la cube_fw-la and_o forasmuch_o as_o the_o line_n fb_o be_v the_o
double_a to_o the_o side_n of_o the_o octohedron_n &_o the_o side_n be_v in_o power_n sequitertia_fw-la to_o the_o perpendiclar_a line_n by_o the_o 12._o of_o this_o book_n wherefore_o the_o diameter_n thereof_o be_v in_o power_n duple_fw-fr superbipartiens_fw-la tertias_fw-la to_o the_o perpendicular_a line_n wherefore_o also_o the_o diameter_n and_o the_o perpendicular_a line_n be_v rational_a and_o commensurable_a by_o the_o 6._o of_o the_o ten_o as_o touch_v a_o icosahedron_n it_o be_v prove_v in_o the_o 16._o of_o this_o book_n that_o the_o side_n thereof_o be_v a_o less_o line_n when_o the_o diameter_n of_o the_o sphere_n be_v rational_a and_o forasmuch_o as_o the_o angle_n of_o the_o inclination_n of_o the_o base_n thereof_o be_v contain_v of_o the_o perpendicular_a line_n of_o the_o triangle_n and_o subtend_v of_o the_o right_a line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n which_o contain_v five_o side_n of_o the_o icosahedron_n and_o unto_o the_o perpendicular_a line_n the_o side_n be_v commensurable_a namely_o be_v in_o power_n sesquitertia_fw-la unto_o they_o by_o the_o corollary_n of_o the_o 12._o of_o this_o book_n therefore_o the_o perpendicular_a line_n which_o contain_v the_o angle_n be_v irrational_a line_n namely_o less_o line_n by_o the_o 105._o of_o the_o ten_o book_n and_o forasmuch_o as_o the_o diameter_n contain_v in_o power_n both_o the_o side_n of_o the_o icosahedron_n and_o the_o line_n which_o subtend_v the_o foresay_a angle_n if_o from_o the_o power_n of_o the_o diameter_n which_o be_v rational_a be_v take_v away_o the_o power_n of_o the_o side_n of_o the_o icosahedron_n which_o be_v irrational_a it_o be_v manifest_a that_o the_o residue_n which_o be_v the_o power_n of_o the_o subtend_a line_n shall_v be_v irrational_a for_o if_o it_o shall_v be_v rational_a the_o number_n which_o measure_v the_o whole_a power_n of_o the_o diameter_n and_o the_o part_n take_v away_o of_o the_o subtend_a line_n shall_v also_o by_o the_o 4._o common_a sentence_n of_o the_o seven_o measure_n the_o residue_n namely_o the_o power_n of_o the_o side_n irrational_a which_o be_v irrational_a for_o that_o it_o be_v a_o less_o line_n which_o be_v absurd_a wherefore_o it_o be_v manifest_a that_o the_o right_a line_n which_o compose_v the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o the_o icosahedron_n be_v irrational_a line_n for_o the_o subtend_a line_n have_v to_o the_o line_n contain_v a_o great_a proportion_n than_o the_o whole_a have_v to_o the_o great_a segment_n the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o a_o dodecahedron_n be_v contain_v under_o two_o perpendicular_n of_o the_o base_n of_o the_o dodecahedron_n and_o be_v subtend_v of_o that_o right_a line_n who_o great_a segment_n be_v the_o side_n of_o a_o cube_n inscribe_v in_o the_o dodecahedron_n which_o right_a line_n be_v equal_a to_o the_o line_n which_o couple_v the_o section_n into_o two_o equal_a part_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n and_o this_o couple_a line_n we_o say_v be_v a_o irrational_a line_n for_o that_o the_o diameter_n of_o the_o sphere_n contain_v in_o power_n both_o the_o couple_a line_n and_o the_o side_n of_o the_o dodecahedron_n but_o the_o side_n of_o the_o dodecahedron_n be_v a_o irrational_a line_n namely_o a_o residual_a line_n by_o the_o 17._o of_o this_o book_n wherefore_o the_o residue_n namely_o the_o couple_a line_n be_v a_o irrational_a line_n as_o it_o be_v âasy_a to_o prove_v by_o the_o 4._o common_a sentence_n of_o the_o seven_o and_o that_o the_o perpendicular_a line_n which_o contain_v the_o angle_n of_o the_o inclination_n be_v irrational_a be_v thus_o prove_v by_o the_o proportion_n of_o the_o subtend_a line_n of_o the_o foresay_a angle_n of_o inclination_n to_o the_o line_n which_o contain_v the_o angle_n be_v find_v out_o the_o obliquity_n of_o the_o angle_n angle_n for_o if_o the_o subtend_a line_n be_v in_o power_n double_a to_o the_o line_n which_o contain_v the_o angle_n then_o be_v the_o angle_n a_o right_a angle_n by_o the_o 48._o of_o the_o first_o but_o if_o it_o be_v in_o power_n less_o than_o the_o double_a it_o be_v a_o acute_a angle_n by_o the_o 23._o of_o the_o second_o but_o if_o it_o be_v in_o power_n more_o than_o the_o double_a or_o have_v a_o great_a proportion_n than_o the_o whole_a have_v to_o the_o great_a segmentâ_n the_o angle_n shall_v be_v a_o obtuse_a angle_n by_o the_o 12._o of_o the_o second_o and_o 4._o of_o the_o thirteen_o by_o which_o may_v be_v prove_v that_o the_o square_a of_o the_o whole_a be_v great_a than_o the_o double_a of_o the_o square_n of_o the_o great_a segment_n this_o be_v to_o be_v note_v that_o that_o which_o flussas_n have_v here_o teach_v touch_v the_o inclination_n of_o the_o base_n of_o the_o âive_a regular_a body_n hypsicles_n teach_v after_o the_o 5_o proposition_n of_o the_o 15._o book_n where_o he_o confess_v that_o he_o receive_v it_o of_o one_o isidorus_n and_o seek_v to_o make_v the_o matter_n more_o clear_a he_o endeavour_v himself_o to_o declare_v that_o the_o angle_n of_o the_o inclination_n of_o the_o solid_n be_v give_v and_o that_o they_o be_v either_o acute_a or_o obtuse_a according_a to_o the_o nature_n of_o the_o solid_a although_o euclid_v in_o all_o his_o 15._o book_n have_v not_o yet_o show_v what_o a_o thing_n give_v be_v wherefore_o flussas_n frame_v his_o demonstration_n upon_o a_o other_o ground_n proceed_v after_o a_o other_o manner_n which_o seem_v more_o plain_a and_o more_o apt_o hereto_o be_v place_v then_o there_o albeit_o the_o reader_n in_o that_o place_n shall_v not_o be_v frustrate_a of_o his_o also_o the_o end_n of_o the_o thirteen_o book_n of_o euclides_n element_n ¶_o the_o fourteen_o book_n of_o euclides_n element_n in_o this_o book_n which_o be_v common_o account_v the_o 14._o book_n of_o euclid_n be_v more_o at_o large_a entreat_v of_o our_o principal_a purpose_n book_n namely_o of_o the_o comparison_n and_o proportion_n of_o the_o five_o regular_a body_n customable_o call_v the_o 5._o figure_n or_o form_n of_o pythagoras_n the_o one_o to_o the_o other_o and_o also_o of_o their_o side_n together_o each_o to_o other_o which_o thing_n be_v of_o most_o secret_a use_n and_o inestimable_a pleasure_n and_o commodity_n to_o such_o as_o diligent_o search_v for_o they_o and_o attain_v unto_o they_o which_o thing_n also_o undoubted_o for_o the_o worthiness_n and_o hardness_n thereof_o for_o thing_n of_o most_o price_n be_v most_o hard_a be_v first_o search_v and_o find_v out_o of_o philosopher_n not_o of_o the_o inferior_a or_o mean_a sort_n but_o of_o the_o deep_a and_o most_o ground_a philosopher_n and_o best_a exercise_v in_o geometry_n and_o albeit_o this_o book_n with_o the_o book_n follow_v namely_o the_o 15._o book_n have_v be_v hitherto_o of_o all_o man_n for_o the_o most_o part_n and_o be_v also_o at_o this_o day_n number_v and_o account_v among_o euclides_n book_n and_o suppose_v to_o be_v two_o of_o he_o namely_o the_o 14._o and_o 15._o in_o order_n as_o all_o exemplar_n not_o only_o new_a and_o late_o set_v abroad_o but_o also_o old_a monument_n write_v by_o hand_n do_v manifest_o witness_v yet_o it_o be_v think_v by_o the_o best_a learned_a in_o these_o day_n that_o these_o two_o book_n be_v none_o of_o euclides_n but_o of_o some_o other_o author_n no_o less_o worthy_a nor_o of_o less_o estimation_n and_o authority_n notwithstanding_o than_o euclid_n apollonius_n a_o man_n of_o deep_a knowledge_n a_o great_a philosopher_n and_o in_o geometry_n marvellous_a who_o wonderful_a book_n write_v of_o the_o section_n of_o cones_fw-la which_o exercise_n &_o occupy_v thewitte_v of_o the_o wise_a and_o best_a learned_a be_v yet_o remain_v be_v think_v and_o that_o not_o without_o just_a cause_n to_o be_v the_o author_n of_o they_o or_o as_o some_o think_v hypsicles_n himself_o for_o what_o can_v be_v more_o plain_o then_o that_o which_o he_o himself_o witness_v in_o the_o preface_n of_o this_o book_n basilides_n of_o tire_n say_v hypsicles_n and_o my_o father_n together_o scan_v and_o peyse_v a_o writing_n or_o book_n of_o apollonius_n which_o be_v of_o the_o comparison_n of_o a_o dodecahedron_n to_o a_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n and_o what_o proportion_n these_o figure_n have_v the_o one_o to_o the_o other_o find_v that_o apollonius_n have_v fail_v in_o this_o matter_n but_o afterward_o say_v he_o i_o find_v a_o other_o copy_n or_o book_n of_o apollonius_n wherein_o the_o demonstration_n of_o that_o matter_n be_v full_a and_o perfect_a and_o show_v it_o unto_o they_o whereat_o they_o much_o rejoice_v by_o which_o word_n it_o seem_v to_o be_v manifest_a that_o apollonius_n be_v the_o first_o author_n of_o this_o book_n which_o be_v afterward_o set_v forth_o by_o hypsicles_n for_o so_o his_o own_o word_n after_o in_o
22._o of_o the_o six_o wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o both_o the_o line_n ab_fw-la &_o bc_o add_v the_o one_o to_o the_o other_o together_o with_o the_o line_n ac_fw-la that_o be_v as_z two_z such_o line_n as_o ab_fw-la be_v be_v to_o the_o line_n ac_fw-la so_o be_v both_o the_o line_n de_fw-fr and_o of_o add_v the_o one_o to_o the_o other_o together_o with_o the_o line_n df_o that_o be_v two_o such_o line_n as_o de_fw-fr be_v to_o the_o line_n df._n and_o in_o the_o same_o proportion_n be_v the_o half_n of_o the_o antecedent_n by_o the_o 15._o of_o the_o five_o wherefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n ac_fw-la so_o be_v the_o line_n de_fw-fr to_o the_o line_n df._n and_o therefore_o by_o the_o 19_o of_o the_o five_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o line_n df_o to_o the_o line_n fe_o wherefore_o also_o by_o division_n by_o the_o 17._o of_o the_o five_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cb_o so_o be_v the_o line_n df_o to_o the_o line_n de_fw-fr now_o that_o we_o have_v prove_v proposition_n that_o any_o right_a line_n whatsoever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n what_o proportion_n the_o line_n contain_v in_o power_n the_o square_n make_v of_o the_o whole_a line_n and_o of_o the_o great_a segment_n add_v together_o have_v to_o the_o line_n contain_v in_o power_n the_o square_n make_v of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o the_o same_o proportion_n have_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n now_o also_o that_o we_o have_v prove_v proposition_n that_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n so_o be_v the_o superficies_n of_o the_o dodecahedron_n to_o the_o superficies_n of_o the_o icosahedron_n be_v both_o describe_v in_o one_o and_o the_o self_n same_o sphere_n and_o moreover_o see_v that_o we_o have_v prove_v proposition_n that_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v the_o dodecahedron_n to_o the_o icosahedron_n for_o that_o both_o the_o pentagon_n of_o the_o dodecahedron_n and_o the_o triangle_n of_o the_o icosahedron_n be_v comprehend_v in_o one_o and_o the_o self_n same_o circle_n corollary_n all_o these_o thing_n i_o say_v be_v prove_v it_o be_v manifest_a that_o if_o in_o one_o and_o the_o self_n same_o sphere_n be_v describe_v a_o dodecahedron_n and_o a_o icosahedron_n they_o shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_z a_o right_a line_n whatsoever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o great_a segment_n add_v together_o be_v to_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o for_o for_o that_o as_o the_o dodecahedron_n be_v to_o the_o icosahedron_n so_o be_v the_o superficies_n of_o the_o dodecahedron_n to_o the_o superficies_n of_o the_o icosahedron_n that_o be_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n but_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n so_o any_o right_a line_n what_o so_o ever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n be_v the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o great_a segment_n add_v together_o to_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o wherefore_o as_o a_o dodecahedron_n be_v to_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n so_o any_o right_a line_n what_o so_o ever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n be_v the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n &_o of_o the_o great_a segment_n add_v together_o to_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o the_o end_n of_o the_o fourteen_o book_n of_o euclides_n element_n after_o hypsicles_n ¶_o the_o fourteen_o book_n of_o euclides_n element_n after_o flussas_n for_o that_o the_o fouretenth_fw-mi book_n as_o it_o be_v set_v forth_o by_o flussas_n contain_v in_o it_o more_o proposition_n than_o be_v find_v in_o hypsicles_n &_o also_o some_o of_o those_o proposition_n which_o hypsicles_n have_v be_v by_o he_o somewhat_o otherwise_o demonstrate_v i_o think_v my_o labour_n well_o bestow_v for_o the_o reader_n sake_n to_o turn_v it_o also_o all_o whole_a notwithstanding_o my_o travail_n before_o take_v in_o turn_v the_o same_o book_n after_o hypsicles_n where_o note_v you_o that_o here_o in_o this_o 14._o book_n after_o flussas_n and_o in_o the_o other_o book_n follow_v namely_o the_o 15._o and_o 16._o i_o have_v in_o allege_v of_o the_o proposition_n of_o the_o same_o 14._o book_n follow_v the_o order_n and_o number_n of_o the_o proposition_n as_o flussas_n have_v place_v they_o ¶_o the_o first_o proposition_n a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o a_o circle_n to_o the_o side_n of_o a_o pentagon_n inscribe_v in_o the_o same_o circle_n campane_n be_v the_o half_a of_o these_o two_o line_n take_v together_o namely_o of_o the_o side_n of_o the_o hexagon_n and_o of_o the_o side_n of_o the_o decagon_n inscribe_v in_o the_o same_o circle_n take_v a_o circle_n abc_n and_o inscribe_v in_o it_o the_o side_n of_o a_o pentagon_n which_o let_v be_v bc_o and_o take_v the_o centre_n of_o the_o circle_n which_o let_v be_v the_o point_n d_o construction_n and_o from_o it_o draw_v unto_o the_o side_n bc_o a_o perpendicular_a line_n de_fw-fr which_o produce_v to_o the_o point_n â_o and_o unto_o the_o line_n e_o fletcher_n put_v the_o line_n eglantine_n equal_a and_o draw_v these_o right_a line_n cg_o cd_o and_o cf._n then_o i_o say_v that_o the_o right_a line_n de_fw-fr which_o be_v draw_v from_o the_o centre_n to_o bc_o the_o side_n of_o the_o pentagon_n be_v the_o half_a of_o âhe_n sideâ_n of_o the_o decagon_n and_o hexagon_n take_v together_o demonstration_n forasmuch_o as_o the_o line_n de_fw-fr be_v a_o perpendicular_a ânto_n the_o line_n bc_o therefore_o the_o section_n be_v and_o ec_o shall_v be_v equal_a by_o the_o 3._o of_o the_o three_o and_o the_o line_n of_o be_v common_a unto_o they_o both_o and_o the_o angle_n fec_n and_o feb_n be_v right_a angle_n by_o supposition_n wherefore_o the_o base_n bf_o and_o fc_o be_v equal_a by_o the_o 4._o of_o the_o first_o but_o the_o line_n bc_o be_v the_o side_n of_o a_o pentagon_n by_o construction_n wherefore_o fc_o which_o subtend_v the_o half_a of_o the_o side_n of_o the_o pentagon_n be_v the_o side_n of_o the_o decagon_n inscribe_v in_o the_o circle_n abc_n but_o unto_o the_o line_n fc_o be_v by_o the_o 4._o of_o the_o first_o equal_a the_o line_n cg_o for_o they_o subtend_v right_a angle_n ceg_v and_o cef_n which_o be_v contain_v under_o equal_a side_n wherefore_o also_o the_o angle_n cge_n and_o cfe_n of_o the_o triangle_n cfg_n be_v equal_a by_o the_o 5._o of_o the_o first_o and_o forasmuch_o as_o the_o ark_n fc_o be_v subtend_v of_o the_o side_n of_o a_o decagon_n the_o ark_n ca_n shall_v be_v quadruple_a to_o the_o ark_n cf_o wherefore_o also_o the_o angle_n cda_n shall_v be_v quadruple_a to_o the_o angle_n cdf_n by_o the_o last_o of_o the_o sixâ_o and_o forasmuch_o as_o the_o same_o angle_n cda_n which_o be_v set_v at_o the_o centre_n be_v double_a to_o the_o angle_n cfa_n which_o be_v set_v at_o the_o circumference_n by_o the_o 20._o of_o the_o three_o therefore_o the_o angle_n cfa_n or_o cfd_n be_v double_a to_o the_o angle_n cdâ_n namely_o the_o half_a of_o quadruple_a but_o unto_o the_o angle_n cfd_n or_o cfg_n be_v prove_v equal_a the_o angle_n cgf_n wherefore_o the_o outward_a angle_n cgf_n be_v double_a to_o the_o angle_n cdf_n wherefore_o the_o angle_n cdg_n and_o dcg_n shall_v be_v equal_a for_o unto_o those_o two_o angle_n the_o angle_n cgf_n be_v equal_a by_o the_o 32._o of_o the_o first_o wherefore_o the_o side_n gc_o and_o gd_o be_v equal_a by_o the_o 6._o of_o the_o first_o wherefore_o also_o the_o line_n gd_o be_v equal_a to_o the_o line_n fc_o which_o be_v the_o side_n of_o the_o decagon_n but_o unto_o the_o right_a line_n fe_o be_v equal_a the_o line_n eglantine_n by_o construction_n wherefore_o the_o whole_a line_n de_fw-fr be_v equal_a to_o the_o two_o line_n câ_n and_o fe_o wherefore_o those_o line_n take_v together_o namely_o the_o line_n df_o and_o fc_o shall_v
spherâ_n contain_v the_o dodecahedron_n of_o this_o pentagon_n and_o the_o icosahedron_n of_o this_o triangle_n by_o the_o 4._o of_o this_o book_n â_o and_o the_o line_n cl_n fall_v perpendicular_o upon_o the_o side_n of_o the_o icosahedron_n and_o the_o line_n ci_o upon_o the_o side_n of_o the_o dodecahedron_n that_o which_o be_v 30._o time_n contain_v under_o the_o side_n and_o the_o perpendicular_a line_n fall_v upon_o it_o be_v equal_a to_o the_o superficies_n of_o that_o solid_a upon_o who_o side_n the_o perpendicular_a fall_v if_o therefore_o in_o a_o circle_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n order_n the_o superficiece_n of_o a_o dodecahedron_n and_o of_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n be_v the_o one_o to_o the_o other_o as_o that_o which_o be_v contain_v under_o the_o side_n of_o the_o one_o and_o the_o perpendicular_a line_n draw_v unto_o it_o from_o the_o centre_n of_o his_o base_a to_o that_o which_o be_v contain_v under_o the_o side_n of_o the_o other_o and_o the_o perpendicular_a line_n draw_v to_o it_o from_o the_o centre_n of_o his_o base_a for_o aâ_z thirtyâ_n timâs_o be_v to_o thirty_o time_n so_o be_v once_o to_o once_o by_o the_o 15._o of_o thâ_z five_o the_o 6._o proposition_n the_o superficies_n of_o a_o dodecahedron_n be_v to_o the_o superficies_n of_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n campane_n in_o that_o proportion_n that_o the_o side_n of_o the_o cube_n be_v to_o the_o side_n of_o the_o icosahedron_n contain_v in_o the_o self_n same_o sphere_n construction_n svppose_v that_o there_o be_v a_o circle_n abg_o &_o in_o it_o by_o the_o 4._o of_o this_o book_n let_v there_o be_v inscribe_v the_o sideâ_n of_o a_o dodecahedron_n and_o of_o a_o icosahedron_n contain_v in_o onâ_n and_o the_o self_n same_o sphere_n and_o let_v the_o side_n oâ_n the_o dodecahedron_n be_v agnostus_n and_o the_o side_n of_o the_o icosahedron_n be_v dg_o and_o let_v the_o centre_n be_v the_o point_n e_o from_o which_o draw_v unto_o those_o sâdes_n perpendicular_a line_n ei_o and_o ez_o and_o produce_v the_o line_n ei_o to_o the_o point_n b_o and_o draw_v the_o linâ_n bg_o and_o let_v the_o side_n of_o the_o cube_fw-la contain_v in_o the_o self_n same_o sphere_n be_v gc_o then_o i_o say_v that_o the_o superficies_n of_o the_o dodecahedron_n iâ_z to_o the_o superficies_n of_o the_o icosahedron_n as_o the_o line_n âg_z iâ_z to_o the_o liââ_n gd_o for_o forasmuch_o as_o the_o line_n ei_o beinâ_n divide_v by_o a_o extreme_a and_o mean_a proportion_n the_o great_a segment_n thereof_o shall_v be_v the_o linâ_n ez_o by_o the_o corollary_n of_o the_o first_o of_o this_o book_n demonstration_n and_o the_o line_n cg_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n his_o great_a segment_n be_v the_o line_n agnostus_n by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o wherefore_o the_o right_a line_n ei_o and_o cg_o ârâ_n cut_v proportional_o by_o the_o second_o of_o this_o bâoke_n whârâfore_o as_o the_o line_n cg_o be_v to_o the_o line_n agnostus_n so_o be_v the_o line_n ei_o to_o the_o line_n ez_fw-fr wherâfore_o that_o which_o it_o contain_v under_o the_o extreme_n cg_o and_o ez_o be_v squall_n to_o that_o which_o iâ_z contaynâd_v under_o the_o mean_n agnostus_n and_o ei._n by_o the_o 16._o of_o the_o six_o but_o as_o that_o which_o iâ_z contain_v under_o the_o linââ_n cg_o and_o âz_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n dg_o and_o ez_o so_o by_o the_o first_o of_o the_o six_o iâ_z the_o linâ_n cg_o to_o the_o line_n dg_o for_o both_o those_o parallelogram_n have_v oââ_n and_o the_o self_n same_o altitude_n namely_o the_o line_n ez_fw-fr wherefore_o as_o that_o which_o be_v contain_v under_o the_o line_n ei_o and_o agnostus_n which_o iâ_z prove_v equal_a to_o that_o which_o be_v contain_v under_o the_o lineâ_z cg_o and_o ez_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n dg_o and_o ez_o so_o be_v the_o line_n cg_o to_o the_o liââ_n dg_o but_o as_o that_o which_o be_v contain_v under_o the_o line_n ei_o and_o agnostus_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n dg_o and_o ez_o so_o by_o the_o corollary_n of_o the_o former_a proposition_n be_v the_o superficies_n of_o the_o dodecahedron_n to_o the_o superficies_n of_o the_o icosahedron_n wherefore_o as_o the_o superficies_n ââ_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v cg_o the_o side_n of_o the_o cube_fw-la to_o gd_v the_o side_n of_o the_o icosahedron_n the_o superficies_n therefore_o of_o a_o dodecahedron_n be_v to_o the_o superficiesâ_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v require_v to_o be_v prove_v a_o assumpt_n the_o pentagon_n of_o a_o dodecahedron_n order_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o perpendicular_a line_n which_o fall_v upon_o the_o base_a of_o the_o triangle_n of_o the_o icosahedron_n and_o five_o six_o part_n of_o the_o side_n of_o the_o cube_fw-la the_o say_v three_o solid_n be_v describe_v in_o one_o and_o the_o self_n same_o sphere_n suppose_v that_o in_o the_o circle_n abeg_n the_o pentagon_n of_o a_o dodecahedron_n be_v aâcig_a and_o let_v two_o side_n thereof_o ab_fw-la and_o agnostus_n be_v subtend_v of_o the_o right_a line_n bg_o and_o let_v the_o triangle_n of_o the_o icosahedron_n inscribe_v in_o the_o self_n same_o sphere_n construction_n by_o the_o 4._o of_o this_o book_n be_v afh_n and_o let_v the_o centre_n of_o the_o circle_n be_v the_o point_n d_o and_o let_v the_o diameter_n be_v ade_n cut_v fh_o the_o side_n of_o the_o triangle_n in_o the_o point_n z_o and_o cut_v the_o line_n bg_o in_o the_o point_n k._n and_o draw_v the_o right_a line_n bd._n and_o from_o the_o right_a line_n kg_v cut_v of_o a_o three_o part_n tg_n by_o the_o 9_o of_o the_o six_o now_o than_o the_o line_n bg_o subtend_v two_o side_n of_o the_o dodecahedron_n shall_v be_v the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o same_o sphere_n demonstration_n by_o the_o 17._o of_o the_o thirteen_o and_o the_o triangle_n of_o the_o icosahedron_n of_o the_o same_o sphere_n shall_v be_v aâh_a by_o the_o 4._o of_o this_o book_n and_o the_o line_n az_o which_o pass_v by_o the_o centre_n d_o shall_v fall_v perpendicular_o upon_o the_o side_n of_o the_o triangle_n for_o forasmuch_o as_o the_o angle_n gay_a &_o bae_o be_v equal_a by_o the_o 27._o of_o the_o thirdâ_o for_o they_o be_v see_v upon_o equal_a circumference_n therefore_o the_o âases_v bk_o and_o kg_n be_v by_o the_o â_o of_o the_o first_o equal_a wherefore_o the_o line_n bt_n contain_v 5._o six_o part_n of_o the_o line_n bg_o then_o i_o say_v that_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o bt_n be_v equal_a to_o the_o pentagon_n aâcâg_n for_o forasmuch_o as_o the_o line_n âz_n be_v sesquialter_n to_o the_o line_n ad_fw-la for_o the_o line_n dâ_n be_v divide_v into_o two_o equal_a part_n in_o the_o point_n z_o by_o the_o corollary_n of_o the_o â2â_n of_o the_o thirteen_o likewise_o by_o construction_n the_o line_n kg_n be_v sesquialter_fw-la to_o the_o line_n kt_n therefore_o as_o the_o line_n az_o be_v to_o the_o line_n ad_fw-la so_o be_v the_o line_n kg_v to_o the_o ãâã_d ât_a wherefore_o that_o which_o be_v contain_v undeâ_n the_o ãâã_d az_o and_o kt_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o mean_n ad_fw-la and_o kg_n by_o the_o 16._o of_o the_o six_o but_o unto_o the_o line_n kg_n be_v the_o line_n âk_v âroved_v equal_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o bk_o but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o bk_o be_v by_o the_o 41._o of_o the_o first_o double_a to_o the_o triangle_n abdella_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n be_v double_a to_o the_o same_o triangle_n abdella_n and_o forasmuch_o as_o the_o pentagon_n abcig_n contaynethâ_n ãâ¦ã_o equal_a âo_o the_o triangle_n abdella_n and_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n contain_v two_o such_o triangle_n therefore_o the_o pentagon_n abcig_n be_v duple_fw-fr sesquialter_fw-la to_o the_o rectangle_n parallelogram_n contain_v under_o the_o line_n az_o and_o kt_n and_o ãâ¦ã_o 1._o of_o the_o six_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o bt_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n as_o the_o base_a bt_n be_v to_o the_o base_a ââtâ_n therefore_o that_o which_o be_v contain_v under_o the_o line_n az_o
together_o and_o a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o any_o base_a of_o the_o cube_fw-la be_v equal_a to_o half_a the_o side_n of_o the_o cube_fw-la which_o be_v require_v to_o be_v prouâd_v ¶_o a_o corollary_n if_o two_o three_o of_o the_o power_n of_o the_o diameter_n of_o the_o sphere_n be_v multiply_v into_o the_o perpendicular_a line_n equal_a to_o half_a the_o side_n of_o the_o cube_fw-la campane_n there_o shall_v be_v produce_v a_o solid_a equal_a to_o the_o solid_a of_o the_o cube_fw-la for_o it_o be_v before_o manifest_v that_o two_o three_o part_n of_o the_o power_n of_o the_o diameter_n of_o the_o sphere_n be_v equal_a to_o two_o base_n of_o the_o cube_fw-la if_o therefore_o unto_o each_o of_o those_o two_o three_o be_v apply_v half_o the_o altitude_n of_o the_o cube_fw-la they_o shall_v make_v each_o of_o those_o solid_n equal_a to_o half_o of_o the_o cube_fw-la by_o the_o 31._o of_o the_o eleven_o for_o they_o have_v equal_a base_n wherefore_o two_o of_o those_o solid_n be_v equal_a to_o the_o whole_a cube_fw-la you_o shall_v understand_v gentle_a reader_n that_o campane_n in_o his_o 14._o book_n of_o euclides_n element_n have_v 18._o proposition_n with_o diverse_a corollary_n follow_v of_o they_o some_o of_o which_o proposition_n and_o corollary_n i_o have_v before_o in_o the_o twelve_o and_o thirteen_o book_n add_v out_o of_o flussas_n as_o corollary_n which_o thing_n also_o i_o have_v note_v on_o the_o side_n of_o those_o corollary_n namely_o with_o what_o proposition_n or_o corollary_n of_o campane_n 14._o book_n they_o do_v agree_v the_o rest_n of_o his_o 18._o proposition_n and_o corollary_n be_v contain_v in_o the_o twelve_o former_a proposition_n and_o corollary_n of_o this_o 14._o book_n after_o flussas_n where_o you_o may_v see_v on_o the_o side_n of_o each_o proposition_n and_o corollary_n with_o what_o proposition_n and_o corollary_n of_o campane_n they_o agree_v but_o the_o eight_o proposition_n follow_v together_o with_o their_o corollary_n flussas_n have_v add_v of_o himself_o as_o he_o himself_o affirm_v the_o 13._o proposition_n one_o and_o the_o self_n same_o circle_n contain_v both_o the_o square_n of_o a_o cube_fw-la and_o the_o triangle_n of_o a_o octohedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n svppose_v that_o there_o be_v a_o cube_fw-la abg_o and_o a_o octohedron_n def_n describe_v in_o one_o and_o the_o self_n same_o sphere_n who_o diameter_n let_v be_v ab_fw-la or_o dh_o and_o let_v the_o line_n draw_v from_o the_o centre_n that_o be_v the_o semidiameter_n of_o the_o circle_n which_o ctonaine_v the_o base_n of_o those_o solid_n â_o be_v ca_n and_o id_fw-la then_o i_o say_v that_o the_o line_n ca_n and_o id_fw-la be_v equal_a forasmuch_o as_o ab_fw-la the_o diameter_n of_o the_o sphere_n which_o contain_v the_o cube_fw-la demonstration_n be_v in_o power_n triple_a to_o bg_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o the_o thirteen_o unto_o which_o side_n ag_z the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la be_v in_o power_n double_a by_o the_o 47._o of_o the_o first_o which_o line_n agnostus_n be_v also_o the_o diameter_n of_o the_o circle_n which_o contain_v the_o base_a by_o the_o 9_o of_o the_o four_o therefore_o ab_fw-la the_o diameter_n of_o the_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o line_n agnostus_n namely_o of_o what_o part_n the_o line_n ab_fw-la contain_v in_o power_n 12._o of_o the_o same_o the_o line_n agnostus_n shall_v contain_v in_o power_n 8._o and_o therefore_o the_o right_a line_n ac_fw-la which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n to_o the_o circumference_n contain_v in_o power_n of_o the_o same_o part_n 2._o wherefore_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n sextuple_a to_o the_o line_n which_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o square_a of_o the_o cube_fw-la but_o the_o diameter_n of_o the_o self_n same_o sphere_n which_o contain_v the_o octohedron_n be_v one_o and_o the_o self_n same_o with_o the_o diameter_n of_o the_o cube_fw-la namely_o dh_o be_v equal_a to_o ab_fw-la and_o the_o same_o diameter_n be_v also_o the_o diameter_n of_o the_o square_n which_o be_v make_v of_o the_o side_n of_o the_o octohedron_n wherefore_o the_o say_a diameter_n be_v in_o power_n double_a to_o the_o side_n of_o the_o same_o octohedron_n by_o the_o 14._o of_o the_o thirteen_o but_o the_o side_n df_o be_v in_o power_n triple_a to_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o triangle_n of_o the_o octohedron_n namely_o to_o the_o line_n id_fw-la by_o the_o 12._o of_o the_o thirteen_o wherefore_o the_o self_n same_o diameter_n ab_fw-la or_o dh_o which_o be_v in_o power_n sextuple_a to_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o square_a of_o the_o cube_fw-la be_v also_o sextuple_a to_o the_o line_n id_fw-la draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o triangle_n of_o the_o octohedron_n wherefore_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n to_o the_o circumference_n which_o contain_v the_o base_n of_o the_o cube_fw-la and_o of_o the_o octohedron_n be_v equal_a and_o therefore_o the_o circle_n be_v equal_a by_o the_o first_o definition_n of_o the_o three_o wherefore_o one_o and_o the_o self_n same_o circle_n contain_v &c._a &c._a as_o in_o the_o proposition_n which_o be_v require_v to_o be_v prove_v a_o corollary_n hereby_o it_o be_v manifest_a that_o perpendicular_n couple_v together_o in_o a_o sphere_n the_o centre_n of_o the_o circle_n which_o contain_v the_o opposite_a base_n of_o the_o cube_fw-la and_o of_o the_o octohedron_n be_v equal_a for_o the_o circle_n be_v equal_a by_o the_o second_o corollary_n of_o the_o assumpt_n of_o the_o 16._o of_o the_o twelve_o and_o the_o line_n which_o pass_v by_o the_o centre_n of_o the_o sphere_n couple_v together_o the_o centre_n of_o the_o base_n be_v also_o equal_a by_o the_o first_o corollary_n of_o the_o same_o wherefore_o the_o perpendicular_a which_o couple_v together_o the_o opposite_a base_n of_o the_o octohedron_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la for_o either_o of_o they_o be_v the_o altitude_n erect_v the_o 14._o proposition_n a_o octohedron_n be_v to_o the_o triple_a of_o a_o tetrahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n in_o that_o proportion_n that_o their_o side_n be_v svppose_v that_o there_o be_v a_o octohedron_n abcd_o and_o a_o tetrahedron_n efgh_o upon_o who_o base_a fgh_n erect_v a_o prism_n construction_n which_o be_v do_v by_o erect_v from_o the_o angle_n of_o the_o base_a perpendicular_a line_n equal_a to_o the_o altitude_n of_o the_o tetrahedron_n which_o prism_n shall_v be_v triple_a to_o the_o tetrahedron_n efgh_o by_o the_o first_o corollary_n of_o the_o 7._o of_o the_o twelve_o then_o i_o say_v that_o the_o octohedron_n abcd_o be_v to_o the_o prism_n which_o be_v triple_a to_o the_o tetrahedron_n efgh_o as_o the_o side_n bc_o be_v to_o the_o side_n fg._n for_o forasmuch_o as_o the_o side_n of_o the_o opposite_a base_n of_o the_o octohedron_n demonstration_n be_v right_a line_n touch_v the_o one_o the_o other_o and_o be_v parellel_n to_o other_o right_a line_n touch_v the_o one_o the_o other_o for_o the_o side_n of_o the_o square_n which_o be_v compose_v of_o the_o side_n of_o the_o octohedron_n be_v opposite_a wherefore_o the_o opposite_a plain_a triangle_n namely_o abc_n &_o kid_n shall_v be_v parallel_n and_o so_o the_o rest_n by_o the_o 15._o of_o the_o eleven_o let_v the_o diameter_n of_o the_o octohedron_n be_v the_o line_n ad._n now_o then_o the_o whole_a octohedron_n be_v cut_v into_o four_o equal_a and_o like_a pyramid_n set_v upon_o the_o base_n of_o the_o octohedron_n and_o have_v the_o same_o altitude_n with_o it_o &_o be_v about_o the_o diameter_n ad_fw-la namely_o the_o pyramid_n set_v upon_o the_o base_a bid_v and_o have_v his_o top_n the_o point_n a_o and_o also_o the_o pyramid_n set_v upon_o the_o base_a bcd_o have_v his_o top_n the_o same_o point_n a._n likewise_o the_o pyramid_n set_v upon_o the_o base_a ikd_v &_o have_v his_o top_n the_o same_o point_n a_o and_o moreover_o the_o pyramid_n set_v upon_o the_o base_a ckd_v and_o have_v his_o top_n the_o former_a point_n a_o which_o pyramid_n shall_v be_v equal_a by_o the_o 8._o definition_n of_o the_o eleven_o for_o they_o each_o consist_v of_o two_o base_n of_o the_o octohedron_n and_o of_o two_o triangle_n contain_v under_o the_o diameter_n ad_fw-la and_o two_o side_n of_o the_o octohedron_n wherefore_o the_o prism_n which_o be_v set_v upon_o the_o base_a of_o the_o octohedron_n
def_n who_o side_n let_v be_v de_fw-fr and_o let_v the_o right_a line_n subtend_v the_o angle_n of_o the_o pentagon_n make_v of_o the_o side_n of_o the_o icosahedron_n be_v the_o line_n ef._n then_o i_o say_v that_o the_o side_n ed_z be_v in_o power_n double_a to_o the_o line_n h_o the_o less_o of_o those_o segment_n forasmuch_o as_o by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n demonstration_n it_o be_v manifest_a that_o ed_z the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n of_o the_o line_n efâ_n and_o that_o the_o diameter_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o of_o namely_o the_o whole_a and_o the_o great_a segment_n but_o by_o supposition_n the_o side_n ab_fw-la contain_v in_o power_n the_o two_o line_n c_o &_o h_o join_v together_o in_o the_o self_n same_o proportion_n wherefore_o the_o line_n of_o be_v to_o the_o line_n ed_z as_z the_o line_n c_o be_v to_o the_o line_n h_o by_o the_o â_o oâ_n this_o bokeâ_n and_o alternally_n by_o the_o 16._o of_o the_o five_o the_o line_n of_o be_v to_o the_o line_n c_o as_o the_o line_n ed_z be_v to_o the_o line_n h._n and_o forasmuch_o as_o the_o line_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o of_o and_o the_o line_n ab_fw-la contain_v in_o power_n the_o two_o line_n c_o and_o h_o therefore_o the_o square_n of_o the_o line_n of_o and_o ed_z be_v to_o the_o square_n of_o the_o line_n df_o as_o the_o square_n of_o the_o line_n c_o and_o h_o to_o the_o square_a ab_fw-la and_o alternate_o the_o square_n of_o the_o line_n of_o and_o âd_a be_v to_o the_o square_n of_o the_o line_n c_o and_o h_o as_o the_o square_n of_o the_o line_n df_o be_v to_o the_o square_n of_o the_o line_n abâ_n but_o df_o the_o diameter_n be_v by_o the_o 14._o of_o the_o thirtenâh_o iâ_z power_n double_a to_o ab_fw-la the_o side_n of_o the_o octohedron_n inscribe_v by_o supposition_n in_o the_o same_o sphere_n wherefore_o the_o square_n of_o the_o line_n of_o and_o ed_z be_v double_a to_o the_o square_n of_o the_o line_n c_o and_o h._n and_o therefore_o one_o square_a of_o the_o line_n ed_z be_v double_a to_o one_o square_a of_o the_o line_n h_o by_o the_o 12._o of_o the_o five_o wherefore_o ed_z the_o side_n of_o the_o icosahedron_n be_v in_o power_n duple_n to_o the_o line_n h_o which_o be_v the_o less_o segment_n if_o therefore_o the_o poweâ_n of_o the_o side_n of_o a_o octohedron_n be_v express_v by_o two_o right_a line_n join_v together_o by_o a_o extreme_a and_o mean_a proportion_n the_o side_n of_o the_o icosahedron_n contain_v in_o the_o same_o sphere_n shall_v be_v duple_v to_o the_o less_o segment_n the_o 17._o proposition_n if_o the_o side_n of_o a_o dodecahedron_n and_o the_o right_a line_n of_o who_o the_o say_a side_n be_v the_o less_o segment_n be_v so_o set_v that_o they_o make_v a_o right_a angle_n the_o right_a line_n which_o contain_v in_o power_n half_o the_o line_n subtend_v the_o angle_n be_v the_o side_n of_o a_o octohedron_n contain_v in_o the_o self_n same_o sphere_n svppose_v that_o ab_fw-la be_v the_o side_n of_o a_o dodecahedron_n and_o let_v the_o right_a line_n of_o which_o that_o side_n be_v the_o less_o segment_n be_v agnostus_n namely_o which_o couple_v the_o opposite_a side_n of_o the_o dodecahedron_n by_o the_o 4._o corollary_n of_o the_o 17._o of_o the_o thirteen_o construction_n and_o let_v those_o line_n be_v so_o set_v that_o they_o make_v a_o right_a angle_n at_o the_o point_n a._n and_o draw_v the_o right_a line_n bg_o and_o let_v the_o line_n d_o contain_n in_o power_n half_o the_o line_n bg_o by_o the_o first_o proposition_n add_v by_o flussas_n after_o the_o last_o of_o the_o six_o then_o i_o say_v that_o the_o line_n d_o be_v the_o side_n of_o a_o octohedron_n contain_v in_o the_o same_o sphere_n forasmuch_o as_o the_o line_n agnostus_n make_v the_o great_a segment_n gc_o the_o side_n of_o the_o cube_fw-la contain_v in_o the_o same_o sphere_n by_o the_o same_o 4._o corollary_n of_o the_o 17._o of_o the_o thirteen_o demonstration_n and_o the_o square_n of_o the_o whole_a line_n ag._n and_o of_o the_o less_o segment_n ab_fw-la be_v triple_a to_o the_o square_n of_o the_o great_a segment_n gc_o by_o the_o 4._o of_o the_o thirteen_o moreover_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n triple_a to_o the_o same_o line_n gc_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o the_o thirteen_o wherefore_o the_o line_n bg_o be_v equal_a to_o the_o ãâã_d for_o it_o contain_v in_o power_n the_o two_o line_n ab_fw-la and_o ag_z by_o the_o 47._o of_o the_o first_o and_o therefore_o it_o contain_v in_o power_n the_o triple_a of_o the_o line_n gc_o but_o the_o side_n of_o the_o octohedron_n contain_v in_o the_o same_o sphere_n be_v in_o power_n triple_a to_o half_a the_o diameter_n of_o the_o sphere_n by_o the_o 14._o of_o the_o thirteen_o and_o by_o supposition_n the_o line_n d_o contaiââââ_n in_o powââ_n the_o half_a of_o the_o line_n bg_o wherefore_o the_o line_n d_o contain_v in_o power_n the_o half_a of_o the_o same_o diameter_n be_v the_o side_n of_o a_o octohedron_n if_o therefore_o the_o side_n of_o a_o dodecahedron_n and_o the_o right_a line_n of_o who_o the_o say_a side_n be_v the_o less_o segment_n be_v so_o set_v that_o they_o make_v a_o right_a angle_n the_o right_a line_n which_o contain_v in_o power_n half_o the_o line_n subtend_v the_o angle_n be_v the_o side_n of_o a_o ocâââedron_n contain_v in_o the_o self_n same_o sphere_n which_o be_v require_v to_o be_v prove_v a_o corollary_n unto_o what_o right_a line_n the_o side_n of_o the_o octoâedron_n be_v in_o power_n sesquialter_fw-la unto_o the_o same_o line_n the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o sphere_n be_v the_o great_a segment_n for_o the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o the_o segment_n cg_o unto_o which_o d_z the_o side_n of_o the_o octohedron_n be_v in_o power_n sesquialter_n that_o be_v be_v half_a of_o the_o power_n of_o the_o line_n bg_o which_o be_v triple_a unto_o the_o line_n cg_o ¶_o the_o 18._o proposition_n if_o the_o side_n of_o a_o tetrahedron_n contain_v in_o power_n two_o right_a line_n join_v together_o by_o a_o extreme_a and_o mean_a proportion_n the_o side_n of_o a_o icosahedron_n describe_v in_o the_o self_n same_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o less_o right_a line_n svppose_v that_o abc_n be_v a_o tetrahedron_n and_o let_v his_o side_n be_v ab_fw-la construction_n who_o power_n let_v be_v divide_v into_o the_o line_n agnostus_n and_o gb_o join_v together_o by_o a_o extreme_a and_o mean_a proportion_n namely_o let_v it_o be_v divide_v into_o agnostus_n the_o whole_a line_n and_o gb_o the_o great_a seâment_n by_o the_o corollary_n of_o the_o first_o proposition_n add_v by_o flussas_n after_o the_o last_o of_o the_o six_o and_o let_v ed_z be_v the_o side_n of_o the_o icosahedron_n edf_o contain_v in_o the_o self_n same_o sphere_n and_o let_v the_o line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n describe_v of_o the_o side_n of_o the_o icosahedron_n be_v ef._n then_o i_o say_v that_z ed_z the_o side_n of_o the_o icosahedron_n be_v in_o power_n sesquialter_fw-la to_o the_o less_o line_n gb_o demonstration_n forasmuch_o as_o by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n the_o side_n ed_z be_v the_o greâter_a segment_n of_o the_o line_n of_o which_o subtend_v the_o angle_n of_o the_o pentagon_n but_o as_o the_o whole_a line_n of_o be_v to_o the_o great_a segment_n ed_z so_o be_v the_o same_o grââter_a segment_n to_o the_o less_o by_o the_o 30._o of_o the_o six_o and_o by_o supposition_n agnostus_n be_v the_o whole_a line_n and_o gâ_n the_o great_a segment_n wherefore_o as_o of_o be_v to_o ed_z so_o be_v agnostus_n to_o gâ_n by_o the_o second_o of_o the_o fourteen_o and_o alternate_o the_o line_n of_o be_v to_o the_o line_n agnostus_n as_o the_o line_n ed_z be_v to_o the_o line_n gb_o and_o forasmuch_o as_o by_o supposition_n the_o line_n ab_fw-la contain_v in_o power_n the_o two_o line_n agnostus_n and_o gb_o therefore_o by_o the_o 4â_o of_o the_o first_o the_o angle_n agb_n be_v a_o right_a angle_n but_o the_o angle_n def_n be_v a_o right_a angle_n by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n wherefore_o the_o triangle_n agâ_n and_o feed_v be_v equiangle_n by_o the_o â_o of_o the_o six_o wherefore_o their_o side_n be_v proportional_a namely_o as_o the_o line_n ed_z be_v to_o the_o line_n gb_o so_o be_v the_o line_n fd_o to_o the_o line_n ab_fw-la by_o the_o 4._o of_o the_o six_o but_o by_o that_o which_o have_v before_o
the_o whole_a line_n mg_o to_o the_o whole_a line_n ea_fw-la by_o the_o 18._o of_o the_o five_o wherefore_o as_o mg_o the_o side_n of_o the_o cube_fw-la be_v to_o ea_fw-la the_o semidiameter_n so_o be_v the_o line_n fghim_n to_o the_o octohedron_n abkdlc_n inscribe_v in_o one_o &_o the_o self_n same_o sphere_n if_o therefore_o a_o cube_fw-la and_o a_o octohedron_n be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n they_o shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o semidiameter_n of_o the_o sphere_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n distinct_o to_o notify_v the_o power_n of_o the_o side_n of_o the_o five_o solid_n by_o the_o power_n of_o the_o diameter_n of_o the_o sphere_n the_o side_n of_o the_o tetrahedron_n and_o of_o the_o cube_fw-la do_v cut_v the_o power_n of_o the_o diameter_n of_o the_o sphere_n into_o two_o square_n which_o be_v in_o proportion_n double_a the_o one_o to_o the_o other_o the_o octohedron_n cut_v the_o power_n of_o the_o diameter_n into_o two_o equal_a square_n the_o icosahedron_n into_o two_o square_n who_o proportion_n be_v duple_n to_o the_o proportion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n who_o less_o segment_n be_v the_o side_n of_o the_o icosahedron_n and_o the_o dodecahedron_n into_o two_o square_n who_o proportion_n be_v quadruple_a to_o the_o proportion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n who_o less_o segment_n be_v the_o side_n of_o the_o dodecahedron_n for_o ad_fw-la the_o diameter_n of_o the_o sphere_n contain_v in_o power_n ab_fw-la the_o side_n of_o the_o tetrahedron_n and_o bd_o the_o side_n of_o the_o cube_fw-la which_o bd_o be_v in_o power_n half_a of_o the_o side_n ab_fw-la the_o diameter_n also_o of_o the_o sphere_n contain_v in_o power_n ac_fw-la and_o cd_o two_o equal_a side_n of_o the_o octohedron_n but_o the_o diameter_n contain_v in_o power_n the_o whole_a line_n ae_n and_o the_o great_a segment_n thereof_o ed_z which_o be_v the_o side_n of_o the_o icosahedron_n by_o the_o 15._o of_o this_o book_n wherefore_o their_o power_n be_v in_o duple_n proportion_n of_o that_o in_o which_o the_o side_n be_v by_o the_o first_o corollary_n of_o the_o 20._o of_o the_o six_o have_v their_o proportion_n duple_n to_o the_o proportion_n of_o a_o extreme_a &_o mean_a proportion_n far_o the_o diameter_n contain_v in_o power_n the_o whole_a line_n of_o and_o his_o less_o segment_n fd_o which_o be_v the_o side_n of_o the_o dodecahedron_n by_o the_o same_o 15._o of_o this_o book_n wherefore_o the_o whole_a have_v to_o the_o less_o â_o double_a proportion_n of_o that_o which_o the_o extreme_a have_v to_o the_o mean_a namely_o of_o the_o whole_a to_o the_o great_a segment_n by_o the_o 10._o definition_n of_o the_o five_o it_o follow_v that_o the_o proportion_n of_o the_o power_n be_v double_a to_o the_o double_a proportion_n of_o the_o side_n by_o the_o same_o first_o corollary_n of_o the_o 20._o of_o the_o six_o that_o be_v be_v quadruple_a to_o the_o proportion_n of_o the_o extreme_a and_o of_o the_o mean_a by_o the_o definition_n of_o the_o six_o a_o advertisement_n add_v by_o flussas_n by_o this_o mean_n therefore_o the_o diameter_n of_o a_o sphere_n be_v give_v there_o shall_v be_v give_v the_o side_n of_o every_o one_o of_o the_o body_n inscribe_v and_o forasmuch_o as_o three_o of_o those_o body_n have_v their_o side_n commensurable_a in_o power_n only_o and_o not_o in_o length_n unto_o the_o diameter_n give_v for_o their_o power_n be_v in_o the_o proportion_n of_o a_o square_a number_n to_o a_o number_n not_o square_a wherefore_o they_o have_v not_o the_o proportion_n of_o a_o square_a number_n to_o a_o square_a number_n by_o the_o corollary_n of_o the_o 25._o of_o the_o eight_o wherefore_o also_o their_o side_n be_v incommensurabe_n in_o length_n by_o the_o 9_o of_o the_o ten_o therefore_o it_o be_v sufficient_a to_o compare_v the_o power_n and_o not_o the_o length_n of_o those_o side_n the_o one_o to_o the_o otherâ_n which_o power_n be_v contain_v in_o the_o power_n of_o the_o diameter_n namely_o from_o the_o power_n of_o the_o diameter_n let_v there_o ble_v take_v away_o the_o power_n of_o the_o cube_fw-la and_o there_o shall_v remain_v the_o power_n of_o the_o tetrahedron_n and_o take_v away_o the_o power_n of_o the_o tetrahedron_n there_o remain_v the_o power_n of_o the_o cube_fw-la and_o take_v away_o from_o the_o power_n of_o the_o diameter_n half_o the_o power_n thereof_o there_o shall_v be_v leave_v the_o power_n of_o the_o side_n of_o the_o octohedron_n but_o forasmuch_o as_o the_o side_n of_o the_o dodecahedron_n and_o of_o the_o icosahedron_n be_v prove_v to_o be_v irrational_a for_o the_o side_n of_o the_o icosahedron_n be_v a_o less_o line_n by_o the_o 16._o of_o the_o thirteen_o and_o the_o side_n of_o the_o dedocahedron_n be_v a_o residual_a line_n by_o the_o 17._o of_o the_o same_o therefore_o those_o side_n be_v unto_o the_o diameter_n which_o be_v a_o rational_a line_n set_v incommensurable_a both_o in_o length_n and_o in_o power_n wherefore_o their_o comparison_n can_v not_o be_v define_v or_o describe_v by_o any_o proportion_n express_v by_o number_n by_o the_o 8._o of_o the_o ten_o neither_o can_v they_o be_v compare_v the_o one_o to_o the_o other_o for_o irrational_a line_n of_o diverse_a kind_n be_v incommensurable_a the_o one_o to_o the_o other_o for_o if_o they_o shall_v be_v commensurable_a they_o shall_v be_v of_o one_o and_o the_o self_n same_o kind_n by_o the_o 103._o and_o 105._o of_o the_o ten_o which_o be_v impossible_a wherefore_o we_o seek_v to_o compare_v they_o to_o the_o power_n of_o the_o diameter_n think_v they_o can_v not_o be_v more_o apt_o express_v then_o by_o such_o proportion_n which_o cut_v that_o rational_a power_n of_o the_o diameter_n according_a to_o their_o side_n namely_o divide_v the_o power_n of_o the_o diameter_n by_o line_n which_o have_v that_o proportion_n that_o the_o great_a segment_n have_v to_o the_o less_o to_o put_v the_o less_o segment_n to_o be_v the_o side_n of_o the_o icosahedron_n &_o devide_v the_o say_a power_n of_o the_o diameter_n by_o line_n have_v the_o proportion_n of_o the_o whole_a to_o the_o less_o segment_n to_o express_v the_o side_n of_o the_o dodecahedron_n by_o the_o less_o segment_n which_o thing_n may_v well_o be_v do_v between_o magnitude_n incommensurable_a the_o end_n of_o the_o fourteen_o book_n of_o euclides_n element_n after_o flussas_n ¶_o the_o fifteen_o book_n of_o euclides_n element_n this_o finetenth_n and_o last_o book_n of_o euclid_n or_o rather_o the_o second_o book_n of_o appollonius_n or_o hypsicles_n book_n teach_v the_o inscription_n and_o circumscription_n of_o the_o five_o regular_a body_n one_o within_o and_o about_o a_o other_o a_o thing_n undouted_o pleasant_a and_o delectable_a in_o mind_n to_o contemplate_v and_o also_o profitable_a and_o necessary_a in_o act_n to_o practice_v for_o without_o practice_n in_o act_n it_o be_v very_o hard_a to_o see_v and_o conceive_v the_o construction_n and_o demonstration_n of_o the_o proposition_n of_o this_o book_n unless_o a_o man_n have_v a_o very_a deep_a sharp_a &_o fine_a imagination_n wherefore_o i_o will_v wish_v the_o diligent_a student_n in_o this_o book_n to_o make_v the_o study_n thereof_o more_o pleasant_a unto_o he_o to_o have_v present_o before_o his_o eye_n the_o body_n form_v &_o frame_v of_o past_v paper_n as_o i_o teach_v after_o the_o definition_n of_o the_o eleven_o book_n and_o then_o to_o draw_v and_o describe_v the_o line_n and_o division_n and_o superficiece_n according_a to_o the_o construction_n of_o the_o proposition_n in_o which_o description_n if_o he_o be_v wary_a and_o diligent_a he_o shall_v find_v all_o thing_n in_o these_o solid_a matter_n as_o clear_v and_o as_o manifest_v unto_o the_o eye_n as_o be_v thing_n before_o teach_v only_o in_o plain_a or_o superficial_a figure_n and_o although_o i_o have_v before_o in_o the_o twelve_o book_n admonish_v the_o reader_n hereof_o yet_o because_o in_o this_o book_n chief_o that_o thing_n be_v require_v i_o think_v it_o shall_v not_o be_v irksome_a unto_o he_o again_o to_o be_v put_v in_o mind_n thereof_o far_o this_o be_v to_o be_v note_v that_o in_o the_o greek_a exemplar_n be_v find_v in_o this_o 15._o book_n only_o 5._o proposition_n which_o 5._o be_v also_o only_o touch_v and_o set_v forth_o by_o hypsicy_n unto_o which_o campane_n add_v 8._o and_o so_o make_v up_o the_o number_n of_o 13._o campane_n undoubted_o although_o he_o be_v very_o well_o learn_v and_o that_o general_o in_o all_o kind_n of_o learning_n yet_o assure_o be_v bring_v up_o in_o a_o time_n of_o rudeness_n when_o all_o good_a letter_n be_v darken_v &_o barberousnes_n have_v
equal_a part_n by_o every_o one_o of_o the_o three_o equal_a square_n which_o divide_v the_o octohedron_n into_o two_o equal_a part_n and_o perpendicular_o for_o the_o three_o diameter_n of_o those_o square_n do_v in_o the_o centre_n cut_v the_o one_o the_o other_o into_o two_o equal_a part_n and_o perpendicular_o by_o the_o three_o corollary_n of_o the_o 1â_o of_o the_o thirteen_o which_o square_n as_o for_o example_n the_o square_a ekli_n do_v divide_v in_o sunder_o the_o pyramid_n and_o the_o prism_n namely_o the_o pyramid_n kltd_v and_o the_o prism_n klteia_n from_o the_o pyramid_n ekzb_n and_o the_o prism_n ekzilg_n which_o pyramid_n be_v equal_a the_o one_o to_o the_o other_o and_o so_o also_o be_v the_o prism_n equal_v the_o one_o to_o the_o other_o by_o the_o 3._o of_o the_o twelve_o and_o in_o like_a sort_n do_v the_o rest_n of_o the_o square_n namely_o kzit_n and_o zlte_n which_o square_n by_o the_o second_o corollary_n of_o the_o 14._o of_o the_o thirteen_o do_v divide_v the_o octohedron_n into_o two_o equal_a part_n ¶_o the_o 3._o proposition_n the_o 3._o problem_n in_o a_o cube_fw-la give_v to_o describe_v a_o octohedron_n take_v a_o cube_n namely_o abcdefgh_n and_o divide_v every_o one_o of_o the_o side_n thereof_o into_o two_o equal_a part_n construction_n and_o draw_v right_a line_n couple_v together_o the_o section_n as_o for_o example_n these_o right_a line_n pq_n and_o r_n which_o shall_v be_v equal_a unto_o the_o side_n of_o the_o cube_fw-la by_o the_o 33._o of_o the_o first_o and_o shall_v divide_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o the_o midst_n of_o the_o diameter_n agnostus_n in_o the_o point_n i_o by_o the_o corollary_n of_o the_o 34._o of_o the_o first_o wherefore_o the_o point_n i_o be_v the_o centre_n of_o the_o base_a of_o the_o cube_fw-la and_o by_o the_o same_o reason_n may_v be_v find_v out_o the_o centre_n of_o the_o rest_n of_o the_o base_n which_o let_v be_v the_o point_n king_n l_o oh_o n_o m._n and_o draw_v these_o right_a line_n li_n in_o more_n ol_n king_n kl_o km_o ko_o ni_fw-fr nl_o nm_o &_o no_o demonstration_n and_o now_o forasmuch_o as_o the_o angle_n ipl_n be_v a_o right_a angle_n by_o the_o 10._o of_o the_o eleven_o for_o the_o line_n ip_n and_o pl_z be_v parallel_n to_o the_o line_n ramires_n and_o ab_fw-la and_o the_o right_a line_n il_fw-fr subtend_v the_o right_a angle_n ipl_n namely_o it_o subtend_v the_o half_a side_n of_o the_o cube_fw-la which_o contain_v the_o right_a angle_n ipl_n and_o likewise_o the_o right_a line_n in_o subtend_v the_o angle_n iqm_n which_o be_v equal_a to_o the_o same_o angle_n ipl_n and_o be_v contain_v under_o right_a line_n equal_a to_o the_o right_a line_n which_o contain_v the_o angle_n ipl_n wherefore_o the_o right_a line_n in_o be_v equal_a to_o the_o right_a line_n ill_n by_o the_o 4._o of_o the_o first_o and_o by_o the_o same_o reason_n may_v we_o prove_v that_o every_o one_o of_o the_o right_a line_n more_o ol_n king_n kl_o km_o ko_o ni_fw-fr nl_o nm_o and_o no_o which_o subtend_v angle_n equal_a to_o the_o self_n same_o angle_n ipl_n and_o be_v contain_v under_o side_n equal_a to_o the_o side_n which_o contain_v the_o angle_n ipl_n be_v equal_a to_o the_o right_a line_n il._n wherefore_o the_o triangle_n kli_n klo_n kmi_n kmo_fw-la and_o nli_n nlo_n nmi_n nmo_fw-la be_v equilater_n and_o equal_a and_o they_o contain_v the_o solid_a iklonm_n wherefore_o iklonm_n be_v a_o octohedron_n by_o the_o 23._o definition_n of_o the_o eleven_o and_o forasmuch_o as_o the_o angle_n thereof_o do_v altogether_o in_o the_o point_n i_o king_n l_o oh_o n_o m_o touch_v the_o base_n of_o the_o cube_fw-la which_o contain_v it_o it_o follow_v that_o the_o octohedron_n be_v inscribe_v in_o the_o cube_fw-la by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o the_o cube_fw-la give_v be_v describe_v a_o octohedron_n which_o be_v require_v to_o be_v do_v ¶_o a_o corollary_n aâded_v by_o âlussââ_n hereby_o it_o be_v manifest_a that_o right_a line_n join_v together_o the_o centre_n of_o the_o opposite_a base_n of_o the_o cube_fw-la do_v cut_v the_o one_o the_o other_o into_o two_o equal_a part_n and_o perpendicular_o in_o the_o centre_n of_o the_o cube_fw-la or_o in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o cube_fw-la for_o forasmuch_o as_o the_o right_a line_n lm_o and_o io_o which_o knââ_n together_o the_o centre_n of_o the_o opposite_a base_n of_o the_o cube_fw-la do_v also_o knit_v together_o the_o opposite_a angel_n of_o the_o octahedron_n inscribe_v in_o the_o cube_fw-la it_o follow_v by_o the_o 3._o corollary_n of_o the_o 14._o of_o the_o thirteen_o that_o those_o line_n lm_o and_o io_o do_v cut_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o a_o point_n but_o the_o diameter_n of_o the_o cube_fw-la do_v also_o cut_v the_o one_o the_o other_o into_o two_o equal_a part_n by_o the_o 39_o of_o the_o eleven_o wherefore_o that_o point_n shall_v be_v the_o centre_n of_o the_o sphere_n which_o contain_v the_o câââ_n for_o make_v that_o point_n the_o centre_n and_o the_o space_n some_o one_o of_o the_o semidiameter_n describe_v a_o sphere_n and_o it_o shall_v pass_v by_o the_o angle_n of_o the_o cube_fw-la and_o likewise_o make_v the_o same_o point_n the_o centre_n and_o the_o space_n half_o of_o the_o line_n lm_o describe_v a_o sphere_n and_o it_o shall_v also_o pass_v by_o the_o angle_n of_o the_o octohedron_n ¶_o the_o 4._o proposition_n the_o 4._o problem_n in_o a_o octohedron_n give_v to_o describe_v a_o cube_n svppose_v that_o the_o octohedron_n give_v be_v abgdez_n and_o let_v the_o two_o pyramid_n thereof_o be_v abgde_v and_o zbgde_v construction_n and_o take_v the_o centre_n of_o the_o triangle_n of_o the_o pyramid_n abgde_v that_o be_v take_v the_o centre_n of_o the_o circle_n which_o contain_v those_o triangle_n and_o let_v those_o centre_n be_v the_o pointâs_n t_o i_o king_n l._n and_o by_o these_o centre_n let_v there_o be_v draw_v parallel_n line_n âo_o the_o side_n of_o the_o square_n bgde_v which_o parallel_a âigââ_n linââ_n let_v be_v mtn_n nlx_n xko_n &_o oim._n demonstration_n and_o forasmuch_o as_o thâse_a parallel_n right_a line_n do_v by_o the_o 2._o of_o the_o six_o cut_v the_o equal_a right_a line_n ab_fw-la agnostus_n ad_fw-la and_o ae_n proportional_o therefore_o they_o concur_v in_o the_o point_n m_o n_o x_o o._n wherefore_o the_o right_a line_n mn_v nx_o xo_o and_o om_fw-la which_o subtend_v equal_a angle_n set_v at_o the_o point_n a_o &_o contain_v under_o squall_n right_a line_n be_v equal_a by_o the_o 4._o of_o the_o first_o and_o moreover_o see_v that_o they_o be_v parallel_n unto_o the_o line_n bg_o gd_o de_fw-fr eâ_n which_o make_v a_o square_a therefore_o mnxo_n be_v also_o a_o square_a by_o the_o 10._o of_o the_o eleven_o wherefore_o also_o by_o the_o 15._o of_o the_o âame_n the_o square_a mnxo_n be_v parallel_v to_o the_o square_n bgde_v for_o all_o tâe_z right_a line_n touch_v the_o one_o the_o other_o in_o the_o point_n of_o their_o section_n from_o the_o centre_n t_o i_o king_n l_o draw_v these_o right_a line_n ti_o ik_fw-mi kl_o ltâ_n and_o draw_v the_o right_a line_n aic_n and_o forasmuch_o as_o i_o be_v the_o centre_n of_o the_o equilater_n triangle_n abe_n therefore_o the_o right_a line_n ai_fw-fr be_v extend_v cut_v the_o right_a line_n be_v into_o two_o equal_a part_n by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o and_o forasmuch_o as_o more_n be_v a_o parallel_n to_o be_v therefore_o the_o triangle_n aio_fw-la be_v like_a to_o the_o whole_a triangle_n ace_n by_o the_o corollary_n of_o the_o 2._o of_o the_o six_o and_o the_o right_a line_n more_n be_v divide_v into_o two_o equal_a part_n in_o the_o point_n i_o by_o the_o 4._o of_o the_o six_o and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o right_a line_n mn_v nx_o xo_o be_v divide_v into_o two_o equal_a part_n in_o the_o point_n t_o l_o k._n wherefore_o also_o again_o the_o base_n ti_fw-mi ik_fw-mi kl_o lt_n which_o subtend_v the_o angle_n set_v at_o the_o point_n m_o oh_o x_o n_o which_o angle_n be_v right_a angle_n and_o be_v contain_v under_o equal_a side_n those_o base_n i_o say_v be_v equal_a and_o forasmuch_o as_o tim_n be_v a_o isosceles_a triangle_n therefore_o the_o angle_n set_v at_o the_o base_a namely_o the_o angle_n mti_n and_o mit_fw-fr be_v equal_a by_o the_o ââ_o of_o the_o first_o but_o the_o angle_n m_o be_v a_o right_a angle_n wherefore_o each_o of_o the_o angle_n mit_fw-fr and_o mti_n be_v the_o half_a of_o a_o right_a angle_n and_o by_o the_o same_o reason_n the_o angle_n oik_a &_o oki_n be_v equal_a wherefore_o the_o angle_n remain_v namely_o tik_fw-mi be_v a_o right_a angle_n
wherefore_o the_o line_n ce_fw-fr be_v to_o the_o line_n egg_n as_o the_o great_a segment_n to_o the_o less_o and_o therefore_o their_o proportion_n be_v as_o the_o whole_a line_n i_fw-mi be_v to_o the_o great_a segment_n ce_fw-fr and_o as_o the_o great_a segment_n ce_fw-fr be_v to_o the_o less_o segment_n egg_n wherefore_o the_o whole_a line_n ceg_n which_o make_v the_o great_a segment_n and_o the_o less_o be_v equal_a to_o the_o whole_a line_n i_fw-mi or_o je_n and_o forasmuch_o as_o two_o parallel_n plain_a superficiece_n namely_o that_o which_o be_v extend_v by_o job_n and_o that_o which_o be_v extend_v by_o the_o line_n ag_fw-mi be_v cut_v by_o the_o plain_n of_o the_o triangle_n be_v which_o pass_v by_o the_o line_n ag_fw-mi and_o ib_n their_o common_a section_n ag_v and_o ib_n shall_v be_v parallel_n by_o the_o 16._o of_o the_o eleven_o but_o the_o angle_n buy_v or_o bic_n be_v a_o right_a angle_n wherefore_o the_o angle_n agc_n be_v also_o a_o right_a angle_n by_o the_o 29._o of_o the_o first_o and_o those_o right_a angle_n be_v contain_v under_o equal_a side_n namely_o the_o line_n gc_n be_v equal_a to_o the_o line_n ci_o and_o the_o line_n ag_fw-mi to_o the_o line_n by_o by_o the_o 33._o of_o the_o first_o wherefore_o the_o base_n ca_o and_o cb_o be_v equal_a by_o the_o 4._o of_o the_o first_o but_o of_o the_o line_n cb_o the_o line_n ce_fw-fr be_v prove_v to_o be_v the_o great_a segment_n wherefore_o the_o same_o line_n ce_fw-fr be_v also_o the_o great_a segment_n of_o the_o line_n ca_o but_o cn_fw-la be_v also_o the_o great_a segment_n of_o the_o same_o line_n ca._n wherefore_o unto_o the_o line_n ce_fw-fr the_o line_n cn_fw-la which_o be_v the_o side_n of_o the_o dodecahedron_n and_o be_v set_v at_o the_o diameter_n be_v equal_a and_o by_o the_o same_o reason_n the_o rest_n of_o the_o side_n which_o be_v set_v at_o the_o diameter_n may_v be_v prove_v eguall_a to_o line_n equal_a to_o the_o line_n ce._n wherefore_o the_o pentagon_n inscribe_v in_o the_o circle_n where_o in_o be_v contain_v the_o triangle_n be_v be_v by_o the_o 11._o of_o the_o four_o equiangle_n and_o equilater_n and_o forasmch_v as_o two_o pentagon_n set_v upon_o every_o one_o of_o the_o base_n of_o the_o cube_fw-la do_v make_v a_o dodecahedron_n and_o six_o base_n of_o the_o cube_fw-la do_v receive_v twelve_o angle_n of_o the_o dodecahedron_n and_o the_o 8._o semidiameter_n do_v in_o the_o point_n where_o they_o be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n receive_v the_o rest_n therefore_o the_o 12._o pentagon_n base_n contain_v 20._o solid_a angle_n do_v inscribe_v the_o dodecahedron_n in_o the_o cube_fw-la by_o the_o 1._o definition_n of_o this_o book_n wherefore_o in_o a_o cube_fw-la give_v be_v inscribe_v a_o dodecahedron_n which_o be_v require_v to_o be_v do_v first_o corollary_n the_o diameter_n of_o the_o sphere_n which_o contain_v the_o dodecahedron_n contain_v in_o power_n these_o two_o side_n namely_o the_o side_n of_o the_o dodecahedron_n and_o the_o side_n of_o the_o cube_fw-la wherein_o the_o dodecahedron_n be_v inscribe_v for_o in_o the_o first_o figure_n a_o line_n draw_v from_o the_o centre_n oh_o to_o the_o point_n b_o the_o angle_n of_o the_o dodecahedron_n namely_o the_o line_n ob_fw-la contain_v in_o power_n these_o two_o line_n ov_v the_o half_a side_n of_o the_o cube_fw-la and_o ub_z the_o half_a side_n of_o the_o dodecahedron_n by_o the_o 47._o of_o the_o first_o wherefore_o by_o the_o 15._o of_o the_o five_o the_o double_a of_o the_o line_n ob_fw-la which_o be_v the_o diameter_n of_o the_o sphere_n contain_v the_o dodecahedron_n contain_v in_o power_n the_o double_a of_o the_o other_o line_n ov_n and_o ub_z which_o be_v the_o side_n of_o the_o cube_fw-la and_o of_o the_o dodecahedron_n ¶_o second_v corollary_n the_o side_n of_o a_o cube_fw-la divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o less_o segment_n the_o side_n of_o the_o dodecahedron_n inscribe_v in_o it_o and_o the_o great_a segment_n the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o same_o dodecahedron_n for_o it_o be_v before_o prove_v that_o the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o be_v the_o side_n of_o the_o triangle_n becâ_n but_o the_o side_n be_v which_o be_v equal_a to_o the_o lineâ_z gb_o and_o sf_n be_v the_o great_a segment_n of_o gf_o the_o side_n of_o the_o cube_fw-la which_o line_n âe_v subtend_v thâ_z angle_n of_o the_o pentagon_n be_v by_o the_o â_o of_o this_o book_n the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o dodecahedron_n three_o corollary_n the_o side_n of_o a_o cube_fw-la be_v equal_a to_o the_o side_n of_o a_o dodecahedron_n inscribe_v in_o it_o and_o circumscribe_v about_o it_o for_o it_o be_v manifest_a by_o this_o proposition_n that_o the_o side_n of_o a_o cube_fw-la make_v the_o less_o segment_n the_o side_n of_o a_o dodecahedron_n inscribe_v in_o it_o namely_o as_o in_o the_o first_o figure_n the_o line_n b_v the_o side_n of_o the_o dodecahedron_n inscribe_v be_v the_o less_o segment_n of_o the_o line_n gf_o the_o side_n of_o the_o cube_fw-la and_o it_o be_v prove_v in_o the_o 17._o of_o the_o thirteen_o that_o the_o same_o side_n of_o the_o cube_fw-la subtend_v the_o angle_n of_o the_o pentagon_n of_o the_o dodecahedron_n circumscribe_v and_o therefore_o it_o make_v the_o great_a segment_n the_o side_n of_o the_o dodecahedron_n or_o of_o the_o pentagon_n by_o the_o first_o corollary_n of_o the_o same_o wherefore_o it_o be_v equal_a to_o both_o those_o segment_n the_o 14._o problem_n the_o 14._o proposition_n in_o a_o cube_fw-la give_v to_o inscribe_v a_o icosahedron_n svppose_v that_o the_o cube_fw-la give_v by_o abc_n construction_n the_o centre_n of_o who_o base_n let_v be_v the_o point_n d_o e_o g_o h_o i_o king_n by_o which_o point_n draw_v in_o the_o base_n unto_o the_o other_o side_n parallel_v not_o touch_v the_o one_o the_o other_o and_o divide_v the_o line_n draw_v from_o the_o centre_n as_o the_o line_n dt_n etc_n etc_n by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n a_o f_o l_o m_o n_o b_o p_o q_o r_o s_o c_o oh_o by_o the_o 30._o of_o the_o six_o and_o let_v the_o great_a segment_n be_v about_o the_o centre_n and_o draw_v these_o right_a line_n all_o agnostus_n be_o and_o tg_n and_o forasmuch_o as_o the_o line_n cut_v be_v parallel_n to_o the_o side_n of_o the_o cube_fw-la demonstration_n they_o shall_v make_v right_a angle_n the_o one_o with_o the_o other_o by_o the_o 29._o of_o the_o first_o and_o forasmuch_o as_o they_o be_v equal_a their_o section_n shall_v be_v equal_a for_o that_o the_o section_n be_v like_a by_o the_o 2._o of_o the_o fourteen_o wherefore_o the_o line_n tg_n be_v equal_a to_o the_o line_n dt_n for_o they_o be_v each_o half_a side_n of_o the_o cube_fw-la wherefore_o the_o square_a of_o the_o whole_a line_n tg_n and_o of_o the_o less_o segment_n ta_n be_v triple_a to_o the_o square_n of_o the_o line_n ad_fw-la the_o great_a segment_n by_o the_o 4._o of_o the_o thirteen_o but_o the_o line_n agnostus_n contain_v in_o power_n the_o line_n at_o &_o tg_n for_o the_o angle_n atg_n be_v a_o right_a angle_n wherefore_o the_o square_a of_o the_o line_n agnostus_n be_v triple_a to_o the_o square_n of_o the_o line_n ad._n and_o forasmuch_o as_o the_o line_n mgl_n be_v erect_v perpendicular_o to_o the_o plain_a pass_v by_o the_o line_n at_o &_o which_o be_v parallel_n to_o the_o base_n of_o the_o cube_fw-la by_o the_o corollary_n of_o the_o 14._o of_o the_o eleven_o therefore_o the_o angle_n agl_n be_v a_o right_a angle_n but_o the_o line_n lg_n be_v equal_a to_o the_o line_n ad_fw-la for_o they_o be_v the_o great_a segment_n of_o equal_a line_n wherefore_o the_o line_n agnostus_n which_o be_v in_o power_n triple_a to_o the_o line_n ad_fw-la be_v in_o power_n triple_a to_o the_o line_n lg_n wherefore_o add_v unto_o the_o same_o square_n of_o the_o line_n agnostus_n the_o square_a of_o the_o line_n lg_n the_o square_a of_o the_o line_n all_o which_o by_o the_o 47._o of_o the_o first_o contain_v in_o power_n the_o two_o line_n agnostus_n and_o gl_n shall_v be_v quadruple_a to_o the_o line_n ad_fw-la or_o lg_n wherefore_o the_o line_n all_o be_v double_a to_o the_o line_n ad_fw-la by_o the_o 20._o of_o the_o six_o and_o therefore_o be_v equal_a to_o the_o line_n of_o or_o to_o the_o line_n lm_o and_o by_o the_o same_o reason_n may_v we_o prove_v that_o every_o one_o of_o the_o other_o line_n which_o couple_v the_o next_o section_n of_o the_o line_n cut_v as_o the_o line_n be_o pf_n pm_n mq_n and_o the_o rest_n be_v equal_a wherefore_o the_o triangle_n alm_n apf_n ample_n pmq_n and_o the_o rest_n such_o like_a be_v equal_a equiangle_n and_o equilater_n
by_o the_o 4._o and_o eight_o of_o the_o first_o and_o forasmuch_o as_o upon_o every_o one_o of_o the_o line_n cut_v of_o the_o cube_fw-la be_v set_v two_o triangle_n as_o the_o triangle_n alm_n and_o blmâ_n there_o shall_v be_v make_v 12._o triangle_n and_o forasmuch_o as_o under_o every_o one_o of_o the_o â_o angle_n of_o the_o cube_fw-la be_v subtend_v the_o other_o 8._o triangle_n as_o the_o triangle_n amp._n etc_n etc_n of_o 1â_o and_o 8._o triangle_n shall_v be_v produce_v 20._o triangle_n equal_a and_o equilater_n contain_v the_o solid_a of_o a_o icosahedron_n by_o the_o 25._o definition_n of_o the_o eleven_o which_o shall_v be_v inscribe_v in_o the_o cube_fw-la give_v abc_n by_o the_o first_o definition_n of_o this_o book_n the_o invention_n of_o the_o demonstration_n of_o this_o depend_v of_o the_o ground_n of_o the_o former_a wherefore_o in_o a_o cube_fw-la give_v we_o have_v describe_v a_o icosahedron_n which_o be_v require_v to_o be_v do_v first_o corollary_n the_o diameter_n of_o a_o sphere_n which_o contain_v a_o icosahedron_n contain_v two_o side_n namely_o the_o side_n of_o the_o icosahedron_n and_o the_o side_n of_o the_o cube_fw-la which_o contain_v the_o icosahedron_n for_o if_o we_o draw_v the_o line_n ab_fw-la it_o shall_v make_v the_o angle_n at_o the_o point_n a_o right_a angle_n for_o that_o it_o be_v a_o parallel_n to_o the_o side_n of_o the_o cube_fw-la wherefore_o the_o linâ_n which_o couple_v the_o opposite_a angleâ_n of_o the_o icosahedron_n at_o the_o point_v fletcher_n and_o b_o contain_v in_o power_n the_o line_n ab_fw-la the_o sidâ_n of_o thâ_z cube_fw-la and_o the_o line_n of_o the_o side_n of_o the_o icosahedron_n by_o the_o 47._o of_o the_o first_o which_o line_n fb_o be_v equal_a to_o the_o âiameter_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n by_o the_o demonstration_n of_o the_o ââ_o of_o the_o thirteen_o second_o corollary_n the_o six_o opposite_a side_n of_o the_o icosahedron_n divide_v into_o two_o equal_a part_n their_o section_n be_v couple_v by_o three_o equal_a right_a line_n cut_v the_o one_o the_o other_o into_o two_o equal_a part_n and_o perpendicular_o in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n for_o those_o three_o line_n be_v the_o three_o line_n which_o couple_v the_o centre_n of_o the_o base_n of_o the_o cube_fw-la which_o do_v in_o such_o sort_n in_o the_o centre_n of_o the_o cube_fw-la cut_v the_o one_o the_o other_o by_o the_o corollary_n of_o the_o three_o of_o this_o book_n and_o therefore_o be_v equal_a to_o the_o side_n of_o the_o cube_fw-la but_o right_a line_n draw_v from_o the_o centre_n of_o the_o cube_fw-la to_o the_o angle_n of_o the_o icosahedron_n every_o one_o of_o they_o shall_v subtend_v the_o half_a side_n of_o the_o cube_fw-la and_o the_o half_a side_n of_o the_o icosahedron_n which_o half_a side_n contain_v a_o right_a angle_n wherefore_o those_o line_n be_v equal_a whereby_o it_o be_v manifest_a that_o the_o foresay_a centre_n be_v the_o centre_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n three_o corollary_n the_o side_n of_o a_o cube_fw-la divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n the_o side_n of_o a_o icosahedron_n describe_v in_o it_o for_o the_o half_a side_n of_o the_o cube_fw-la make_v the_o half_a of_o the_o side_n of_o the_o icosahedron_n the_o great_a segment_n wherefore_o also_o the_o whole_a side_n of_o the_o cube_fw-la make_v the_o whole_a side_n of_o the_o icosahedron_n the_o great_a segment_n by_o the_o 15._o of_o the_o fifthe_o for_o the_o section_n be_v like_a by_o the_o â_o of_o the_o fourteen_o ¶_o four_o corollary_n the_o side_n and_o base_n of_o the_o icosahedron_n which_o be_v opposite_a the_o one_o to_o the_o other_o be_v parallel_n forasmuch_o as_o every_o one_o of_o the_o opposite_a side_n of_o the_o icosahedron_n may_v be_v in_o the_o parallel_a line_n of_o the_o cube_fw-la namely_o in_o those_o parallel_n which_o be_v opposite_a in_o the_o cube_fw-la and_o the_o triangle_n which_o be_v make_v of_o parallel_a line_n be_v parallel_n by_o the_o 15._o of_o the_o eleven_o therefore_o the_o opposite_a triângleâ_n of_o the_o icosahedron_n as_o also_o the_o side_n be_v parallel_n the_o one_o to_o the_o other_o ¶_o the_o 15._o problem_n the_o 15._o proposition_n in_o a_o icosahedron_n give_v to_o inscribe_v a_o octohedron_n svppose_v that_o the_o icosahedron_n give_v be_v acdf_n and_o by_o the_o former_a second_o corollary_n construction_n let_v there_o be_v take_v the_o three_o right_a line_n which_o cut_v the_o one_o the_o other_o into_o two_o equal_a part_n perpendicular_o and_o which_o couple_n the_o section_n into_o two_o equal_a part_n of_o the_o side_n of_o the_o icosahedron_n which_o let_v be_v be_v gh_o and_o kl_o cut_v the_o one_o the_o other_o in_o the_o point_n i_o and_o drawâ_n these_o riâht_a line_n âg_z ge_z eh_o and_o hb_o and_o forasmuch_o as_o the_o angle_n at_o the_o point_n i_o be_v by_o construction_n right_a angle_n demonstration_n ând_v be_v conââined_v under_o equal_a linesâ_n the_o ãâã_d gâ_n and_o ââ_o shall_v ãâ¦ã_o squâre_o by_o the_o â_o of_o the_o âirst_n likewise_o ânto_o thoâe_a ãâã_d shall_v be_v squall_v the_o line_n drâwân_v from_o ãâã_d point_n king_n and_o â_o to_o every_o one_o of_o the_o poinââs_n â_o g._n â_o h_o and_o therefore_o the_o triangle_n which_o ãâã_d the_o ââramis_n dgenk_n shall_v be_v equal_a ãâ¦ã_o and_o by_o ãâ¦ã_o ¶_o the_o 16._o problem_n the_o 16._o proposition_n in_o a_o octohedron_n give_v to_o inscribe_v a_o icosahedron_n let_v there_o be_v take_v a_o octohedron_n who_o 6._o angle_n let_v be_v a_o b_o c_o f_o p_o l._n and_o draw_v the_o line_n ac_fw-la bf_o pl_z construction_n cut_v the_o one_o the_o other_o perpendicular_o in_o the_o point_n r_o by_o the_o 2._o corollary_n of_o the_o 14._o of_o the_o thirteen_o and_o let_v every_o one_o of_o the_o 12._o side_n of_o the_o octohedron_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n h_o x_o m_o king_n d_o s_o n_o g_o five_o e_o q_o t._n and_o let_v the_o great_a segment_n be_v the_o line_n bh_o bx_n fm_o fk_o ad_fw-la aq_n cs_n et_fw-la pn_n pg_n lv_o le_fw-fr and_o draw_v these_o line_n hk_o xm_o ge_z nv_n d_n qt_n now_o forasmuch_o as_o in_o the_o triangle_n abf_n the_o side_n be_v cut_v proportional_o namely_o as_o the_o line_n bh_o be_v to_o the_o line_n ha_o so_o be_v the_o line_n fk_o to_o the_o line_n ka_o by_o the_o 2._o of_o the_o fourteen_o demonstration_n therefore_o the_o line_n hk_o shall_v be_v a_o parallel_n to_o the_o line_n bf_o by_o the_o 2._o of_o the_o six_o and_o forasmuch_o as_o the_o line_n ac_fw-la cut_v the_o line_n hk_o in_o the_o point_n z_o and_o the_o line_n zk_n be_v a_o parallel_n unto_o the_o line_n rf_n the_o line_n ramires_n shall_v be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n z_o by_o the_o 2._o of_o the_o six_o namely_o shall_v be_v cut_v like_o unto_o the_o line_n favorina_n and_o the_o great_a segment_n thereof_o shall_v be_v the_o line_n zr_n unto_o the_o line_n zk_v put_v the_o line_n ro_n equal_a by_o the_o 3._o of_o the_o first_o and_o draw_v the_o line_n ko_o now_o then_o the_o line_n ko_o shall_v be_v equal_a to_o the_o line_n zr_n by_o the_o 33._o of_o the_o âirst_n draw_v the_o line_n kg_v ke_o and_o ki_v and_o forasmuch_o as_o the_o triangle_n arf_n and_o azk_v be_v equiangle_n by_o the_o 6._o of_o the_o six_o the_o side_n az_o and_o zk_v shall_v be_v equal_a the_o one_o to_o the_o other_o by_o the_o 4._o of_o the_o six_o for_o the_o side_n be_v and_o rf_n be_v equal_a wherefore_o the_o line_n zk_n shall_v be_v the_o less_o segment_n of_o the_o line_n ra._n but_o if_o the_o great_a segment_n rz_n be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n the_o great_a segment_n thereof_o shall_v be_v the_o line_n zk_n which_o be_v the_o less_o segment_n of_o the_o whole_a line_n ramires_n by_o the_o 5._o of_o the_o thirâenth_o and_o forasmuch_o as_o the_o two_o line_n fe_o and_o fg_o be_v equal_a to_o the_o two_o line_n ah_o and_o ak_o namely_o each_o be_v less_o segment_n of_o equal_a side_n of_o the_o octohedron_n and_o the_o angle_n hak_n and_o efg_o be_v equal_a namely_o be_v right_a angle_n by_o the_o 14._o of_o the_o thirteen_o the_o base_n hk_o and_o gf_o shall_v be_v equal_a by_o the_o 4._o of_o the_o first_o and_o by_o the_o same_o reason_n unto_o they_o may_v be_v prove_v equal_a the_o line_n xm_o nv_n d_n and_o qt_o and_o forasmuch_o as_o the_o line_n ac_fw-la bf_o and_o pl_z do_v cut_v the_o one_o the_o other_o into_o two_o equal_a part_n and_o perpendicular_o by_o construction_n the_o line_n hk_o and_o ge_z which_o
subtend_v angle_n of_o triangle_n like_v unto_o the_o triangle_n who_o angle_n the_o line_n ac_fw-la bf_o and_o pl_z subtend_n be_v cut_v into_o two_o equal_a part_n in_o the_o point_n z_o an_o i_o by_o the_o 4._o of_o the_o six_o so_o also_o be_v the_o other_o line_n nv_n xm_o d_n qt_fw-la which_o be_v equal_a unto_o the_o line_n hk_o &_o ge_z cut_v in_o like_a sort_n and_o they_o shall_v cut_v the_o line_n ac_fw-la bf_o and_o pl_z like_z wherefore_o the_o line_n ko_o which_o be_v equal_a to_o rz_n shall_v make_v the_o great_a segment_n the_o line_n ro_n which_o be_v equal_a to_o the_o line_n zk_n for_o the_o great_a segment_n of_o the_o rz_n be_v the_o line_n zk_n and_o therefore_o the_o line_n oi_o shall_v be_v the_o less_o segment_n when_o as_o the_o whole_a line_n ri_n be_v equal_a to_o the_o whole_a line_n rz_n wherefore_o the_o square_n of_o the_o whole_a line_n ko_o and_o of_o the_o less_o segment_n oi_o be_v triple_a to_o the_o square_n of_o the_o great_a segment_n ro_n by_o the_o 4._o of_o the_o thirteen_o wherefore_o the_o line_n ki_v which_o contain_v in_o power_n the_o two_o line_n ko_o and_o oi_o be_v in_o power_n triple_a to_o the_o line_n ro_n by_o the_o 47._o of_o the_o first_o for_o the_o angle_n koi_n be_v a_o right_a angle_n and_o forasmuch_o as_o the_o line_n fe_o and_o fg_o which_o be_v the_o less_o segment_n of_o the_o side_n of_o the_o octohedron_n be_v equal_a and_o the_o line_n fk_o be_v common_a to_o they_o both_o and_o the_o angle_n kfg_n and_o kfe_n of_o the_o triangle_n of_o the_o octohedron_n be_v equal_a the_o base_n kg_v and_o ke_o shall_v by_o the_o 4_o of_o the_o first_o be_v equal_a and_o therefore_o the_o angle_n kie_fw-mi and_o kig_n which_o they_o subtend_v be_v equal_a by_o the_o 8._o of_o the_o first_o wherefore_o they_o be_v right_a angle_n by_o the_o 13._o of_o the_o first_o wherefore_o the_o right_a line_n ke_o which_o contain_v in_o power_n the_o two_o line_n ki_v and_o âe_n by_o the_o 47._o of_o the_o first_o be_v in_o power_n quadruple_a to_o the_o line_n ro_n or_o je_n for_o the_o line_n ri_n be_v prove_v to_o be_v in_o power_n triple_a to_o the_o same_o line_n ro_n but_o the_o line_n ge_z be_v double_a to_o the_o line_n je_n wherefore_o the_o line_n ge_z be_v also_o in_o power_n ãâ¦ã_o pf_n and_o by_o the_o same_o reason_n may_v be_v prove_v that_o the_o âest_n of_o the_o eleven_o solid_a angle_n of_o the_o ãâã_d be_v ãâ¦ã_o the_o section_n of_o every_o one_o of_o the_o side_n of_o the_o octohedron_n namely_o in_o the_o point_n e_o n_o five_o h_o â_o m_o â_o d_o s_o q_o t._n wherefore_o there_o be_v 12._o angle_n of_o the_o icosahedron_n moreover_o forasmuch_o as_o every_o one_o of_o the_o base_n of_o the_o octohedron_n do_v each_o contain_v triangle_n of_o the_o icosahedron_n ãâ¦ã_o pyramid_n abcâfp_n which_o be_v the_o half_a of_o the_o octohedron_n the_o triangle_n fcp_n receive_v in_o thâ_z section_n of_o his_o side_n the_o â_o triangle_n gm_n and_o the_o triangle_n cpb_n contain_v the_o triangle_n nxs_n and_o thâ_z triangle_n âap_n contain_v the_o triangle_n hnd_n and_o moreover_o the_o triangle_n apf_n contain_v the_o triangle_n edge_n and_o the_o same_o may_v be_v prove_v in_o the_o opposite_a pyramid_n abcfl_fw-mi wherefore_o there_o shall_v be_v eight_o triangleâ_n and_o forasmuch_o as_o beside_o these_o triangle_n to_o every_o one_o of_o the_o solid_a angle_n of_o the_o octohedron_n ãâã_d subtend_v two_o triangle_n as_o the_o triangle_n keg_n and_o meg_n to_o the_o angle_n f_o and_o the_o triangle_n hnv_n and_o xnv_n to_o the_o angle_n b_o also_o the_o triangle_n nd_n and_o âds_n to_o the_o angle_n p_o likewise_o the_o triangleâ_n dhk_n and_o qhk_v to_o the_o angle_n a_o moreover_o the_o triangle_n eqt_a and_o vqt_a to_o the_o angle_n l_o and_o final_o the_o triangle_n sxm_n and_o txm_n to_o the_o angle_n c_o these_o 12._o triangle_n be_v add_v to_o thââ_n for_o ãâã_d triangle_n shall_v produce_v _o triangle_n equal_a and_o equilater_v couple_v together_o which_o shall_v male_a a_o icosahedron_n by_o the_o 25._o definition_n of_o the_o eleven_o and_o it_o shall_v be_v inscribe_v in_o the_o octohedron_n give_v abcââl_o by_o the_o first_o definition_n of_o this_o book_n for_o the_o 1â_o angle_n thereof_o be_v set_v in_o 1â_o like_a section_n of_o the_o side_n of_o the_o octohedron_n wherefore_o in_o a_o octohedron_n give_v be_v inscribe_v a_o icosahedron_n ¶_o first_n corollary_n the_o side_n of_o a_o equilater_n triangle_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n a_o right_a line_n subtend_v within_o the_o triangle_n the_o angle_n which_o be_v contain_v under_o the_o great_a segment_n and_o the_o less_o be_v in_o power_n duple_n to_o the_o less_o segment_n of_o the_o same_o side_n for_o the_o line_n ke_o which_o subtend_v the_o angle_n kfe_n of_o the_o triangle_n afl_fw-mi which_o angle_n kfe_n be_v contain_v under_o the_o two_o segment_n kf_a &_o fe_o be_v prove_v equal_a ãâã_d the_o line_n hk_o which_o contain_v in_o power_n the_o two_o less_o segment_n ha_o and_o ak_o by_o the_o 47._o of_o the_o âârst_a foâ_n ãâã_d angle_n hak_n be_v ãâ¦ã_o second_o corollary_n the_o base_n of_o the_o icosahedron_n be_v concentrical_a that_o be_v have_v one_o and_o the_o self_n same_o centre_n with_o the_o base_n of_o the_o octohedron_n which_o contain_v it_o for_o suppose_v that_o ãâ¦ã_o octohedron_n ãâã_d ecd_v the_o base_a of_o a_o icosahedron_n and_o let_v the_o centre_n of_o the_o base_a abg_o be_v the_o point_n f._n and_o draw_v these_o right_a line_n favorina_n fb_o fc_o and_o fe_o now_o than_o the_o ãâ¦ã_o to_o the_o two_o line_n fb_o and_o bc_o for_o they_o be_v line_n draw_v from_o the_o centre_n and_o be_v also_o less_o segment_n and_o they_o contain_v the_o ãâ¦ã_o ¶_o the_o 17._o problem_n the_o 17._o proposition_n in_o a_o octohedron_n give_v to_o inscribe_v a_o dodecahedron_n construction_n svppose_v that_o the_o octohedron_n give_v be_v abgdec_n who_o 12._o âides_n let_v be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n as_o in_o the_o former_a proposition_n it_o be_v manifest_a that_o of_o the_o right_a line_n which_o couple_v thâse_a section_n be_v make_v 20._o triangle_n of_o which_o 8._o be_v concentrical_a with_o the_o base_n of_o the_o octohedron_n by_o the_o second_o corollary_n of_o the_o former_a proposition_n if_o therefore_o in_o every_o one_o of_o the_o centre_n of_o the_o 20._o triangle_n be_v inscribe_v by_o the_o 1._o of_o this_o book_n every_o one_o of_o the_o ââ_o ââgles_n of_o the_o dodecahedron_n demonstration_n we_o shall_v find_v that_o â_o angle_n of_o the_o dodecahedron_n be_v set_v in_o the_o 8._o centre_n of_o the_o base_n of_o the_o octohedron_n namely_o these_o angle_n i_o u._fw-mi ct_n oh_o m_o a_o p_o and_o x_o and_o of_o the_o other_o 12._o solid_a angle_n there_o be_v two_o in_o the_o centre_n of_o the_o two_o triangle_n which_o have_v one_o side_n common_a under_o every_o one_o of_o the_o solid_a angle_n of_o the_o octohedron_n namely_o under_o the_o solid_a angle_n a_o the_o two_o solid_a angle_n king_n z_o under_o the_o solid_a angle_n b_o the_o two_o solid_a angle_n h_o t_o under_o the_o solid_a angle_n g_o the_o two_o solid_a angle_n in_o v_z under_o the_o solid_a angle_n d_o the_o two_o solid_a angle_n f_o l_o under_o the_o solid_a angle_n e_o the_o two_o solid_a angle_n s_o n_o under_o the_o solid_a angle_n c_o the_o two_o solid_a angle_n queen_n r_o and_o forasmuch_o as_o in_o the_o octohedron_n be_v six_o solid_a angle_n under_o they_o shall_v be_v subtend_v 12._o solid_a angle_n of_o the_o dodâcahedron_n and_o so_o be_v mâde_v 20._o solid_a angle_n compose_v of_o 12._o equal_a and_o equilater_v superficial_a pentagon_n as_o it_o be_v ãâã_d by_o the_o 5._o of_o this_o book_n which_o therefore_o contain_v a_o dodecahedron_n by_o the_o 24._o definition_n of_o the_o eleven_o and_o it_o be_v inscribe_v in_o the_o octohedron_n by_o the_o 1._o definition_n of_o this_o book_n for_o that_o every_o one_o of_o the_o base_n of_o the_o octohedron_n do_v receive_v angle_n thereof_o wherefore_o in_o a_o octohedron_n give_v be_v inscribe_v a_o dodecahedron_n ¶_o the_o 18._o problem_n the_o 18._o proposition_n in_o a_o trilater_n and_o equilater_n pyramid_n to_o inscribe_v a_o cube_n construction_n svppose_v that_o there_o be_v a_o trilater_n equilater_n pyramid_n who_o base_a let_v be_v abc_n and_o âoppe_v the_o point_n d._n and_o let_v it_o be_v comprehend_v in_o a_o sphereâ_n by_o the_o 13._o of_o the_o ãâã_d and_o lââ_n the_o centre_n of_o that_o sphere_n be_v the_o point_n e._n and_o from_o the_o solid_a angle_n a_o b_o c_o d_o draw_v right_a line_n pass_v by_o the_o centre_n e_o unto_o the_o opposite_a base_n of_o
cube_n for_o forasmuch_o as_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o cube_fw-la subtend_v two_o side_n of_o the_o cube_fw-la which_o contain_v a_o right_a angle_n by_o the_o 1._o of_o the_o fifteen_o it_o be_v manifest_a by_o the_o 47._o of_o the_o first_o that_o the_o side_n of_o the_o pyramid_n subtend_v the_o say_a side_n be_v in_o power_n duple_n to_o the_o side_n of_o the_o cube_fw-la wherefore_o also_o the_o square_a of_o the_o side_n of_o the_o cube_fw-la be_v the_o half_a of_o the_o square_n of_o the_o side_n of_o the_o pyramid_n the_o side_n therefore_o of_o a_o cube_fw-la contain_v in_o power_n half_o the_o side_n of_o a_o equilater_n triangular_a pyramid_n inscribe_v in_o the_o say_v cube_fw-la ¶_o the_o 7._o proposition_n the_o side_n of_o a_o pyramid_n be_v duple_n to_o the_o side_n of_o a_o octohedron_n inscribe_v in_o it_o forasmuch_o as_o by_o the_o 2._o of_o the_o fivetenth_fw-mi it_o be_v prove_v that_o the_o side_n of_o the_o octohedron_n inscribe_v in_o a_o pyramid_n couple_v the_o middle_a section_n of_o the_o side_n of_o the_o pyramid_n wherefore_o the_o side_n of_o the_o pyramid_n and_o of_o the_o octohedron_n be_v parallel_n by_o the_o corollary_n of_o the_o 39_o of_o the_o first_o and_o therefore_o by_o the_o corollary_n of_o the_o 2._o of_o the_o six_o they_o subtend_v like_o triangle_n wherefore_o by_o the_o 4._o of_o the_o six_o the_o side_n of_o the_o pyramid_n be_v double_a to_o the_o side_n of_o the_o octohedron_n namely_o in_o the_o proportion_n of_o the_o side_n the_o side_n therefore_o of_o a_o pyramid_n be_v duple_n to_o the_o side_n of_o a_o octohedron_n inscribe_v in_o it_o ¶_o the_o 8._o proposition_n the_o side_n of_o a_o cube_n be_v in_o power_n duple_n to_o the_o side_n of_o a_o octohedron_n inscribe_v in_o it_o it_o be_v prove_v in_o the_o 3._o of_o the_o fifteen_o that_o the_o diameter_n of_o the_o octohedron_n inscribe_v in_o the_o cube_fw-la couple_v the_o centre_n of_o the_o opposite_a base_n of_o the_o cube_fw-la wherefore_o the_o say_a diameter_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la but_o the_o same_o be_v also_o the_o diameter_n of_o the_o square_n make_v of_o the_o side_n of_o the_o octohedron_n namely_o be_v the_o diameter_n of_o the_o sphere_n which_o contain_v it_o by_o the_o 14._o of_o the_o thirteen_o wherefore_o that_o diameter_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la be_v in_o power_n double_a to_o the_o side_n of_o that_o square_n or_o to_o the_o side_n of_o the_o octohedron_n inscribe_v in_o it_o by_o the_o 47._o of_o the_o first_o the_o side_n therefore_o of_o a_o cube_n be_v in_o power_n duple_n to_o the_o side_n of_o a_o octohedron_n inscribe_v in_o it_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 9_o proposition_n the_o side_n of_o a_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n which_o contain_v in_o power_n half_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o say_v dodecahedron_n svppose_v that_o of_o the_o dodecahedron_n abgd_v the_o side_n be_v ab_fw-la and_o let_v the_o base_a of_o the_o cube_fw-la inscribe_v in_o the_o dodecahedron_n be_v ecfh_n by_o the_o ââ_o of_o the_o fifteen_o and_o let_v the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o cube_fw-la be_fw-mi change_z by_o the_o 1._o of_o the_o fifteen_o construction_n wherefore_o the_o same_o pyramid_n be_v inscribe_v in_o the_o dodecahedron_n by_o the_o 10._o of_o the_o fifteen_o then_o i_o say_v that_o ab_fw-la the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n which_o contain_v in_o power_n half_o the_o line_n change_z which_o be_v the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o dodecahedron_n demonstration_n for_o forasmuch_o as_o ec_o the_o side_n of_o the_o cube_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n the_o line_n ab_fw-la the_o side_n of_o the_o dodecahedron_n by_o the_o âârst_a corollary_n of_o the_o 17._o of_o the_o thirteen_o for_o they_o be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o first_o of_o this_o book_n and_o the_o line_n ec_o the_o side_n of_o the_o cube_fw-la contain_v in_o power_n the_o half_a of_o the_o side_n change_z by_o the_o 6._o of_o this_o book_n wherefore_o ab_fw-la the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n ec_o which_o contain_v in_o power_n the_o half_a of_o the_o line_n change_z which_o be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o pyramid_n the_o side_n therefore_o of_o a_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n which_o contain_v in_o power_n half_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o say_v dodecahedron_n ¶_o the_o 10._o proposition_n the_o side_n of_o a_o icosahedron_n be_v the_o mean_a proportional_a between_o the_o side_n of_o the_o cube_n circumscribe_v about_o the_o icosahedron_n and_o the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o cube_n svppose_v that_o there_o be_v a_o cube_fw-la abfd_n in_o which_o let_v there_o be_v inscribe_v a_o icosahedron_n cligor_fw-la by_o the_o 14._o of_o the_o fifteen_o construction_n let_v also_o the_o dodecahedron_n inscribe_v in_o the_o same_o be_v edmnp_n by_o the_o 13._o of_o the_o same_o now_o forasmuch_o as_o cl_n the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n of_o ab_fw-la the_o side_n of_o the_o cube_fw-la circumscribe_v about_o it_o by_o the_o 3._o corollary_n of_o the_o 14._o of_o the_o fifteen_o demonstration_n and_o the_o side_n ed_z of_o the_o dodecahedron_n inscribe_v in_o thesame_o cube_fw-la be_v the_o less_o segment_n of_o the_o same_o side_n ab_fw-la of_o the_o cube_fw-la by_o the_o 2._o corollary_n of_o the_o 13._o of_o the_o fifteen_o it_o follow_v that_o ab_fw-la the_o side_n of_o the_o cube_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n cl_n the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o and_o the_o less_o segment_n ed_z the_o side_n of_o the_o dodecahedron_n likewise_o inscribe_v in_o it_o wherefore_o as_o the_o whole_a line_n ab_fw-la the_o side_n of_o the_o cube_fw-la be_v to_o the_o great_a segment_n cl_n the_o side_n of_o the_o icosahedron_n so_o be_v the_o great_a segment_n cl_n the_o side_n of_o the_o icosahedron_n to_o the_o less_o segment_n edâ_n the_o side_n of_o the_o dodecahedron_n by_o the_o three_o definition_n of_o the_o six_o wherefore_o the_o side_n of_o a_o icosahedron_n be_v the_o mean_a proportional_a between_o the_o side_n of_o the_o cube_fw-la circumscribe_v about_o the_o icosahedron_n and_o the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o cube_fw-la ¶_o the_o 11._o proposition_n the_o side_n of_o a_o pyramid_n be_v in_o power_n 1._o octodecuple_n to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o it_o for_o by_o that_o which_o be_v demonstrate_v in_o the_o 18._o of_o the_o fifteen_o the_o side_n of_o the_o pyramid_n be_v triple_a to_o the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la inscribe_v in_o it_o demonstration_n and_o therefore_o it_o be_v in_o power_n nonecuple_n to_o the_o same_o diameter_n by_o the_o 20._o of_o the_o six_o but_o the_o diamer_n be_v in_o power_n double_a to_o the_o side_n of_o the_o cube_fw-la by_o the_o 47._o of_o the_o first_o and_o the_o double_a of_o nonecuple_n make_v octodecuple_n wherefore_o the_o side_n of_o the_o pyramid_n be_v in_o power_n octodecuple_n to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o it_o ¶_o the_o 12._o proposition_n the_o side_n of_o a_o pyramid_n be_v in_o power_n octodecuple_n to_o that_o right_a line_n who_o great_a segment_n be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o pyramid_n forasmuch_o as_o the_o dodecahedron_n and_o the_o cube_fw-la inscribe_v in_o it_o be_v set_v in_o one_o and_o the_o sâlfâ_n same_o pyramid_n by_o the_o corollary_n of_o the_o first_o of_o this_o book_n and_o the_o side_n of_o the_o pyramid_n circumscribe_v about_o the_o cube_fw-la be_v in_o power_n octodecuple_n to_o the_o side_n of_o the_o cube_fw-la inscribe_v by_o the_o former_a proposition_n but_o the_o great_a segment_n of_o the_o self_n same_o side_n of_o the_o cube_fw-la be_v the_o side_n of_o the_o dodecahedron_n which_o contain_v the_o cube_fw-la by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o wherefore_o the_o side_n of_o the_o pyramid_n be_v in_o power_n octodecuple_n to_o that_o right_a line_n namely_o to_o the_o side_n of_o the_o cube_fw-la who_o great_a segment_n be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o pyramid_n ¶_o the_o 13._o proposition_n the_o side_n of_o a_o icosahedron_n inscribe_v in_o a_o octohedron_n be_v in_o power_n duple_n to_o the_o less_o segment_n of_o the_o side_n of_o the_o same_o
construction_n two_o case_n in_o this_o proposition_n first_o caseâ_n second_o case_n demonstration_n construction_n demonstration_n construction_n demonstration_n an_o other_o way_n after_o peliâarius_n construction_n demonstration_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n three_o case_n in_o this_o proposition_n the_o three_o case_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o proposition_n add_v by_o petarilius_n note_n construction_n demonstration_n an_o other_o way_n also_o after_o pelitarius_n construction_n demonstration_n an_o other_o way_n to_o do_v the_o samâ_n after_o pelitarius_n demonstration_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n construction_n demonstration_n demonstration_n a_o âther_n way_n to_o do_v the_o same_o after_o orontius_n an_o other_o way_n after_o pelitarius_n construction_n demonstration_n a_o addition_n of_o flussates_n flussates_n a_o poligonon_n figure_n be_v a_o figure_n consist_v of_o many_o side_n the_o argument_n of_o this_o five_o book_n the_o first_o author_n of_o this_o book_n eudoxus_n the_o first_o definition_n a_o part_n take_v two_o manner_n of_o way_n the_o fiâst_a way_n the_o second_o way_n how_o a_o less_o quantity_n be_v say_v to_o measure_v a_o great_a in_o what_o signification_n euclid_n here_o take_v a_o part_n parâ_n metienâ_n or_o mensuranâ_n pars_fw-la multiplicatiâa_fw-la pars_fw-la aliquota_fw-la this_o kind_n of_o part_n common_o use_v in_o arithmetic_n the_o other_o kind_n of_o part_n pars_fw-la constitâens_fw-la or_o componens_fw-la pars_fw-la aliquanta_fw-la the_o second_o definition_n number_n very_o necessary_a for_o the_o understanding_n of_o this_o book_n and_o the_o other_o book_n follow_v the_o tâird_o definition_n rational_a proportion_n divide_v ââto_o two_o kind_n proportion_n of_o equality_n proportion_n of_o inequality_n proportion_n of_o the_o great_a to_o the_o less_o multiplex_fw-la duple_n proportion_n triple_a quadruple_a quintuple_a superperticular_a sesquialtera_fw-la sesquitertia_fw-la sesquiquarta_fw-la superpartiens_fw-la superbipartiens_fw-la supertripartiens_fw-la superquadripartiens_fw-la superquintipartiens_fw-la multiplex_fw-la superperticular_a dupla_fw-la sesquialtera_fw-la dupla_fw-la sesquitertia_fw-la tripla_fw-la sesquialtera_fw-la multiplex_fw-la superpartiens_fw-la dupla_fw-la superbipartiens_fw-la dupla_fw-la supertripartiens_fw-la tripla_fw-la superbipartiens_fw-la tripla_fw-la superquadâipartiens_fw-la how_o to_o knoâ_n the_o denomination_n of_o any_o proportion_n proportion_n of_o the_o less_o in_o the_o great_a submultiplex_fw-la subsuperparticular_a subsuperpertient_fw-fr etc_n etc_n the_o four_o definition_n example_n of_o this_o definition_n in_o magnitude_n example_n thereof_o in_o number_n note_n the_o five_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n why_o euclid_n in_o define_v of_o proportion_n use_v multiplication_n the_o six_o definition_n a_o example_n of_o this_o definitiin_n in_o magnitude_n a_o example_n in_o number_n an_o other_o example_n in_o number_n an_o other_o example_n in_o number_n note_v this_o particle_n according_a to_o any_o multiplication_n a_o example_n where_o the_o equimultiplice_n of_o the_o first_o and_o three_o excee_v the_o equimultiplice_n of_o the_o second_o and_o four_o and_o yet_o the_o quantity_n give_v be_v not_o in_o one_o and_o the_o self_n same_o proportion_n a_o rule_n to_o produce_v equimultiplice_n of_o the_o first_o and_o three_o equal_a to_o the_o equimultiplice_n of_o the_o secondâ_n and_o forth_o example_n thereof_o the_o seven_o definition_n 9_z 12_o 3_o 4_o proportionality_n of_o two_o sort_n continual_a and_o discontinuall_a a_o example_n of_o continual_a proportionality_n in_o number_n 16.8.4.2.1_o in_o coutinnall_a proportionality_n the_o quantity_n can_v be_v of_o one_o kind_n discontinuall_a propârtionalitie_n example_n of_o discontinual_a proportionality_n in_o number_n in_o discontinual_a proportionality_n the_o proportion_n may_v be_v of_o diverse_a kind_n the_o eight_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n a_o example_n in_o number_n note_n the_o nine_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n example_n ân_a number_n the_o ten_o definition_n a_o rule_n to_o add_v proportion_n to_o proportion_n 8._o 4._o 2._o 1._o 2_o 2_o 2_o 1_o 1_o 1_o the_o eleven_o definition_n example_n of_o this_o definition_n in_o magnitude_n example_n in_o number_n the_o twelfâh_o definition_n example_n of_o this_o definition_n in_o magnitud_n example_n in_o number_n the_o thirteen_o definition_n example_n of_o this_o definition_n in_o magnitud_n example_n in_o number_n the_o fourteen_o definition_n example_n of_o this_o definition_n in_o magnitud_n example_n in_o number_n the_o fiâtâne_a definition_n this_o be_v the_o converse_n of_o the_o former_a definition_n example_n in_o magnitude_n example_n in_o number_n the_o sixteen_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n a_o example_n in_o number_n the_o sevententh_n definition_n a_o example_n of_o this_o definition_n in_o magnitude_n a_o example_n in_o number_n note_n the_o eighttenth_n definition_n a_o example_n of_o this_o definition_n in_o magnitude_n example_n in_o number_n the_o nineteen_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n example_n in_o number_n the_o 20._o definition_n the_o 2d_o definition_n these_o two_o last_o definition_n not_o find_v in_o the_o greek_a exampler_n construction_n demonstration_n demonstrationâ_n construction_n demonstration_n construction_n demonstration_n alemmae_fw-la or_o a_o assumpt_n a_o corollary_n converse_v proportion_n construction_n demonstration_n two_o case_n in_o this_o propotion_n the_o second_o the_o second_o part_n demonstrate_v the_o first_o part_n of_o this_o proposition_n demonstrate_v the_o second_o part_n of_o the_o proposition_n demonstrate_v first_o differâcâ_n of_o the_o first_o part_n demonstratiâ_n of_o tâe_z same_o first_o difference_n second_o difference_n three_o difference_n the_o second_o part_n âf_o this_o proposition_n the_o first_o parâ_n of_o this_o proposition_n demonstrate_v the_o second_o part_n prove_v the_o first_o part_n of_o this_o proposition_n prove_v the_o second_o part_n demonstrate_v construction_n demonstrationâ_n constrâction_n demonstrationâ_n construction_n demonstration_n a_o addition_n of_o campane_n demonstration_n construction_n demonstration_n demonstration_n of_o alternate_a proportion_n construction_n demonstration_n demonstration_n of_o proportion_n by_o division_n construction_n demonstration_n demonstration_n of_o proportion_n by_o composition_n this_o proposition_n be_v the_o converse_n of_o the_o former_a demonstrationâeâaing_v to_o a_o impossibility_n that_o which_o the_o five_o of_o this_o book_n prove_v only_o touch_v multiplices_fw-la this_o prove_v general_o of_o all_o magnitude_n alemma_n a_o corollary_n conversion_n of_o proportion_n this_o proposition_n pertain_v to_o proportion_n of_o equality_n inordinate_a proportionality_n the_o second_o difference_n the_o three_o difference_n thâr_a proposition_n pertain_v to_o proportion_n of_o equality_n in_o perturbate_n proportionality_n the_o three_o difference_n proportion_n of_o equality_n in_o ordinate_a proportionality_n construction_n demonstration_n when_o there_o be_v more_o than_o three_o magnitude_n in_o either_o order_n aâcdeâgh_n proportion_n of_o equality_n in_o perturbate_n proprotionalitie_n construction_n demonstration_n note_n that_o which_o the_o second_o proposition_n of_o this_o book_n prove_v only_o touch_v multiplices_fw-la be_v here_o prove_v general_o touch_v magnitude_n an_o other_o demonstration_n of_o the_o same_o affirmative_o an_o other_o demonstration_n of_o the_o same_o affirmative_o an_o other_o demonstration_n of_o the_o same_o demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n of_o the_o same_o affirmative_o demonstration_n demonstrationâ_n demonstration_n the_o argument_n of_o this_o six_o book_n this_o book_n necessary_a for_o the_o use_n of_o instrument_n of_o geometry_n the_o first_o definition_n the_o second_o definition_n reciprocal_a figure_n call_v mutual_a figure_n the_o three_o definition_n the_o four_o definition_n the_o five_o definition_n an_o other_o example_n of_o substraction_n of_o proportion_n the_o six_o definition_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n a_o corollary_n add_v by_o flussates_n the_o first_o part_n of_o this_o theorem_a demonstration_n of_o the_o second_o part_n a_o corollary_n add_v by_o flussates_n construction_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n which_o 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the_o square_n of_o the_o line_n b_o wherefore_o the_o parallelogram_n ec_o be_v commensurable_a unto_o the_o parallelogram_n cf._n but_o as_o the_o parallelogram_n ec_o be_v to_o the_o parallelogram_n cf_o so_o be_v the_o line_n ed_z to_o the_o line_n df_o by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o line_n ed_z be_v commensurable_a in_o length_n unto_o the_o line_n df._n but_o the_o line_n ed_z be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n dc_o wherefore_o the_o line_n df_o be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n dc_o by_o the_o 13._o of_o the_o ten_o wherefore_o the_o line_n cd_o and_o df_o be_v rational_a commensurable_a in_o power_n only_o but_o a_o rectangle_n figure_n comprehend_v under_o rational_a right_a line_n commensurable_a in_o power_n only_o be_v by_o the_o â1_n of_o the_o ten_o irrational_a and_o the_o line_n that_o contain_v it_o in_o power_n be_v irrational_a and_o 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commensurable_a in_o power_n only_o to_o the_o same_o and_o these_o line_n be_v also_o call_v medial_a for_o that_o they_o be_v commensurable_a in_o power_n to_o the_o medial_a line_n and_o in_o aâ_z muâh_o as_o they_o be_v medial_a line_n they_o be_v commensurable_a in_o power_n the_o one_o to_o the_o other_o but_o be_v compare_v the_o one_o to_o the_o other_o they_o may_v be_v commensurable_a either_o in_o length_n and_o therefore_o in_o power_n or_o else_o in_o power_n only_o and_o then_o if_o they_o be_v commensurable_a in_o length_n they_o be_v call_v also_o medial_a line_n commensurable_a in_o length_n and_o so_o consequent_o they_o be_v understand_v to_o be_v commensurable_a in_o power_n but_o iâ_z they_o be_v commensurable_a in_o power_n only_o yet_o notwithstanding_o they_o also_o be_v call_v medial_a line_n commensurable_a in_o power_n only_o flussates_n after_o this_o proposition_n teach_v how_o to_o come_v to_o the_o understanding_n of_o medial_a superficiece_n and_o line_n by_o surd_a number_n after_o this_o manner_n namely_o to_o express_v the_o medial_a superficiece_n by_o the_o root_n of_o number_n which_o be_v not_o square_a number_n and_o the_o line_n contain_v in_o power_n such_o medial_a superficiece_n by_o the_o root_n of_o root_n of_o number_n not_o square_a medial_a line_n also_o commensurable_a be_v express_v by_o the_o root_n of_o root_n of_o like_a superficial_a number_n but_o yet_o not_o square_a but_o such_o as_o have_v that_o proportion_n that_o the_o square_n of_o square_a number_n have_v for_o the_o root_n of_o those_o number_n and_o the_o root_n of_o root_n be_v in_o proportion_n as_o number_n be_v namely_o if_o the_o square_n be_v proportional_a the_o side_n also_o shall_v be_v proportional_a by_o the_o 22._o of_o the_o six_o but_o medial_a line_n incommensurable_a in_o power_n be_v the_o root_n of_o root_n of_o number_n which_o have_v not_o that_o proportion_n that_o square_a number_n have_v for_o their_o root_n be_v the_o power_n of_o medial_a line_n which_o be_v incommensurable_a by_o the_o 9_o of_o the_o ten_o but_o medial_a line_n commensurable_a in_o power_n only_o be_v the_o root_n of_o root_n of_o number_n which_o have_v that_o proportion_n that_o simple_a square_a number_n have_v and_o not_o which_o the_o square_n of_o square_n have_v for_o the_o root_n which_o be_v the_o power_n of_o the_o medial_a line_n be_v commensurable_a but_o the_o root_n of_o root_n which_o express_v the_o say_v medial_a line_n be_v incommensurable_a wherefore_o there_o may_v be_v find_v out_o infinite_a medial_a line_n incommensurable_a in_o powâr_n by_o compare_v infinite_a unlike_o plain_a number_n the_o one_o to_o the_o other_o for_o unlike_o plain_a number_n which_o have_v not_o the_o proportion_n of_o square_a number_n do_v make_v the_o root_n which_o express_v the_o superficiece_n of_o medial_a line_n incommensurable_a by_o the_o 9_o of_o the_o ten_o and_o therefore_o the_o medial_a line_n contain_v in_o power_n those_o superficiece_n be_v incommensurable_a in_o length_n for_o line_n incommensurable_a in_o power_n be_v always_o incommensurable_a in_o length_n by_o the_o corollary_n of_o the_o 9_o of_o the_o ten_o ¶_o the_o 21._o theorem_a the_o 24._o proposition_n a_o rectangle_n parallelogram_n comprehend_v under_o medial_a line_n commensurable_a in_o length_n be_v a_o medial_a rectangle_n parallelogram_n svppose_v that_o the_o rectangle_n parallelogram_n agnostus_n be_v comprehend_v under_o these_o medial_a right_a line_n ab_fw-la and_o bc_o which_o let_v be_v commensurable_a in_o length_n construction_n then_o i_o say_v that_o ac_fw-la be_v a_o medial_a rectangle_n parallelogram_n describe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n ab_fw-la a_o square_a ad._n demonstration_n wherefore_o the_o square_a ad_fw-la be_v a_o medial_a superficies_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v commensurablâ_n in_o length_n unto_o the_o line_n bc_o and_o the_o line_n ab_fw-la be_v equal_a unto_o the_o line_n bd_o therefore_o the_o line_n bd_o be_v commensurable_a in_o length_n unto_o the_o line_n bc._n but_o ãâã_d the_o line_n db_o be_v to_o the_o line_n bc_o so_o be_v the_o square_a dam_n to_o the_o parallelogram_n ac_fw-la by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a dam_n be_v commensurable_a unto_o the_o parallelogram_n ac_fw-la but_o the_o square_a dam_n be_v medial_a for_o that_o it_o be_v describe_v upon_o a_o medial_a line_n wherefore_o ac_fw-la also_o be_v a_o medial_a parallelogram_n by_o the_o former_a corollary_n a_o rectangle_v etc_o etc_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 22â_n theorem_a the_o 25._o proposition_n a_o rectangle_n parallelogram_n comprehend_v under_o medial_a right_a line_n commensurable_a in_o power_n only_o be_v either_o rational_a or_o medial_a and_o now_o if_o the_o line_n hk_o be_v commensurable_a in_o length_n unto_o the_o line_n he_o that_o be_v unto_o the_o line_n fg_o note_n which_o be_v equal_a to_o the_o line_n he_o than_o by_o the_o 19_o of_o the_o ten_o the_o parallelogram_n nh_o be_v rational_a but_o if_o it_o be_v incommensurable_a in_o length_n unto_o the_o line_n fg_o than_o the_o line_n hk_o and_o he_o be_v rational_a commensurable_a in_o power_n only_o and_o so_o shall_v the_o parallelogram_n hn_o be_v medial_a wherefore_o the_o parallelogram_n hn_o be_v either_o rational_a or_o medial_a but_o the_o parallelogram_n hn_o be_v equal_a to_o the_o parallelogram_n ag._n wherefore_o the_o parallelogram_n ac_fw-la be_v either_o rational_a or_o medial_a a_o rectangle_n parallelogram_n therefore_o comprehend_v under_o medial_a right_a line_n commensurable_a in_o power_n only_o be_v either_o rational_a or_o medial_a which_o be_v require_v to_o be_v demonstrate_v how_o to_o find_v medial_a line_n commensurable_a in_o power_n only_o contain_v a_o rational_a parallelogram_n and_o also_o other_o medial_a line_n commensurable_a in_o power_n contain_v a_o medial_a
this_o book_n wherefore_o a_o line_n commensurable_a to_o a_o line_n contain_v in_o power_n two_o medial_n etc_n etc_n a_o annotation_n if_o other_o to_o have_v be_v speak_v of_o six_o senary_n of_o which_o the_o first_o senary_n contain_v the_o production_n of_o irrational_a line_n by_o composition_n the_o second_o the_o division_n of_o they_o namely_o that_o those_o line_n be_v in_o one_o poinâ_n only_a devideâ_n the_o three_o the_o find_v out_o of_o binomial_a line_n of_o the_o first_o i_o say_v the_o second_o the_o three_o the_o four_o the_o five_o and_o the_o six_o after_o that_o begin_v the_o forth_o senary_a contain_v the_o difference_n of_o irrational_a line_n between_o themselves_o for_o by_o the_o nature_n of_o every_o one_o of_o the_o binomial_a line_n be_v demonstrate_v the_o difference_n of_o irrational_a line_n the_o five_o entreâteth_v of_o the_o application_n of_o the_o square_n of_o every_o irrational_a line_n namely_o what_o irrational_a line_n be_v the_o breadthes_fw-mi of_o every_o superficies_n so_o apply_v in_o the_o six_o senary_a be_v prove_v that_o any_o line_n commensurable_a to_o any_o irrational_a line_n be_v also_o a_o irrational_a line_n of_o the_o same_o nature_n and_o now_o shall_v be_v speak_v of_o the_o seven_o senary_a wherein_o again_o be_v plain_o set_v forth_o the_o rest_n of_o the_o difference_n of_o the_o say_a line_n between_o themselves_o and_o theâe_n be_v even_o in_o those_o irrational_a line_n a_o arithmetical_a proportionality_n note_n and_o that_o line_n which_o be_v the_o arithmetical_a mean_a proportional_a between_o the_o part_n of_o any_o irrational_a line_n be_v also_o a_o irrational_a line_n of_o the_o self_n same_o kind_n first_o it_o be_v certain_a that_o there_o be_v a_o arithmetical_a proportion_n between_o those_o part_n for_o suppose_v that_o the_o line_n ab_fw-la be_v any_o of_o the_o foresay_a irrational_a line_n as_o for_o example_n let_v it_o be_v a_o binomial_a line_n &_o let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n c._n and_o let_v ac_fw-la be_v the_o great_a name_n from_o which_o take_v away_o the_o line_n ad_fw-la equal_a to_o the_o less_o name_n namely_o to_o cb._n and_o divide_v the_o line_n cd_o into_o two_o equal_a part_n in_o the_o point_n e._n it_o be_v manifest_a that_o the_o line_n ae_n be_v equal_a to_o the_o line_n ebb_n let_v the_o line_n fg_o be_v equal_a to_o either_o of_o they_o it_o be_v plain_a that_o how_o much_o the_o line_n ac_fw-la differ_v from_o the_o line_n fg_o so_o much_o the_o same_o line_n fg_o differ_v from_o the_o line_n cb_o for_o in_o each_o be_v the_o difference_n of_o the_o line_n de_fw-fr or_o ec_o which_o be_v the_o property_n of_o arithmetical_a proportionality_n and_o it_o be_v manifest_a that_o the_o line_n fg_o be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la for_o it_o be_v the_o half_a thereof_o wherefore_o by_o the_o 66._o of_o the_o ten_o the_o line_n fg_o be_v a_o binomial_a line_n and_o after_o the_o self_n same_o manner_n may_v it_o be_v prove_v touch_v the_o rest_n of_o the_o irrational_a line_n ¶_o the_o 53._o theorem_a the_o 71._o proposition_n if_o two_o superficiece_n namely_o a_o rational_a and_o a_o medial_a superficies_n be_v compose_v together_o the_o line_n which_o contain_v in_o power_n the_o whole_a superficies_n be_v one_o of_o these_o four_o irrational_a line_n either_o a_o binomial_a line_n or_o a_o first_o bimedial_a line_n or_o a_o great_a line_n or_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n but_o now_o let_v the_o line_n eh_o be_v in_o power_n more_o than_o the_o line_n hk_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n eh_o case_n now_o the_o great_a name_n that_o be_v eh_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v ef._n wherefore_o the_o line_n eke_o be_v afourth_o binomial_a line_n and_o the_o line_n of_o be_v rational_a but_o if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o afourth_o binomial_a line_n the_o line_n that_o contain_v in_o power_n the_o same_o superficies_n be_v by_o the_o 57_o of_o the_o ten_o irrational_a and_o be_v a_o great_a line_n wherefore_o the_o line_n which_o contain_v in_o power_n the_o parallelogram_n ei_o be_v a_o great_a line_n wherefore_o also_o the_o line_n contain_v in_o power_n the_o superficies_n ad_fw-la be_v a_o great_a line_n case_n but_o now_o suppose_v that_o the_o superficies_n ab_fw-la which_o be_v rational_a be_v less_o than_o the_o superficies_n cd_o which_o be_v medial_a wherefore_o also_o the_o parallelogram_n eglantine_n be_v less_o than_o the_o parallelogram_n high_a wherefore_o also_o the_o line_n eh_o be_v less_o than_o the_o line_n hk_o now_o the_o line_n hk_o be_v in_o power_n more_o than_o the_o line_n eh_o either_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n hk_o or_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n hk_o first_o let_v it_o be_v in_o power_n more_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o hk_n case_n now_o the_o less_o name_n that_o be_v eh_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v of_o as_o it_o be_v before_o prove_v wherefore_o the_o whole_a line_n eke_o be_v a_o second_o binomial_a line_n and_o the_o line_n of_o be_v a_o rational_a line_n but_o if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o second_o binomial_a line_n the_o line_n that_o contain_v in_o power_n the_o same_o superficies_n be_v by_o the_o 55._o of_o the_o ten_o a_o first_o bimedial_a line_n wherefore_o the_o line_n which_o contain_v in_o power_n the_o parallelogram_n ei_o be_v a_o first_o bimedial_a line_n wherefore_o also_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ad_fw-la be_v a_o first_o bimedial_a line_n case_n but_o now_o let_v the_o line_n hk_o be_v in_o power_n more_o than_o the_o line_n eh_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n hk_o now_o the_o less_o name_n that_o be_v eh_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v ef._n wherefore_o the_o whole_a line_n eke_o be_v a_o five_o binomial_a line_n and_o the_o line_n of_o be_v rational_a but_o if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o five_o binomial_a line_n the_o line_n that_o contain_v in_o power_n the_o same_o superficies_n be_v by_o the_o 58._o of_o the_o ten_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a wherefore_o the_o line_n that_o contain_v in_o power_n the_o parallelogram_n ei_o be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a wherefore_o also_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ad_fw-la be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a if_o therefore_o a_o rational_a and_o a_o medial_a superficies_n be_v add_v together_o the_o line_n which_o contain_v in_o power_n the_o whole_a superficies_n be_v one_o of_o these_o four_o irrational_a line_n namely_o either_o a_o binomial_a line_n or_o a_o first_o bimedial_a line_n or_o a_o great_a line_n or_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 54._o theorem_a the_o 72._o proposition_n if_o two_o medial_a superficiece_n incommensurable_a the_o one_o to_o the_o other_o be_v compose_v together_o the_o line_n contain_v in_o power_n the_o whole_a superficies_n be_v one_o of_o the_o two_o irrational_a line_n remain_v namely_o either_o a_o second_o bimedial_a line_n or_o a_o line_n contain_v in_o power_n two_o medial_n let_v these_o two_o medial_a superficiece_n ab_fw-la and_o cd_o be_v incommensurable_a the_o one_o to_o the_o other_o be_v add_v together_o then_o i_o say_v that_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ad_fw-la be_v either_o a_o second_o bimedial_a line_n or_o a_o line_n contain_v in_o power_n two_o medial_n construction_n for_o the_o superficies_n ab_fw-la be_v either_o great_a or_o less_o than_o the_o superficies_n cd_o for_o they_o can_v by_o no_o mean_n be_v equal_a when_o as_o they_o be_v incommensurable_a case_n first_o let_v the_o superficies_n ab_fw-la be_v great_a than_o the_o superficies_n cd_o and_o take_v a_o rational_a line_n ef._n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n of_o apply_v the_o parallelogram_n eglantine_n equal_a to_o the_o superficies_n ab_fw-la and_o make_v in_o breadth_n the_o line_n eh_o and_o unto_o the_o same_o line_n of_o that_o be_v to_o the_o line_n hg_o apply_v the_o parallelogram_n high_a equal_a to_o the_o superficies_n cd_o &_o make_v in_o breadth_n the_o line_n hk_o and_o forasmuch_o as_o
the_o residue_n or_o of_o this_o excess_n but_o a_o pyramid_n be_v to_o the_o same_o cube_fw-la inscribe_v in_o it_o nonecuple_n by_o the_o 30._o of_o this_o book_n wherefore_o the_o dodecahedron_n inscribe_v in_o the_o pyramid_n and_o contain_v the_o same_o cube_fw-la twice_o take_v away_o the_o self_n same_o three_o of_o the_o less_o segment_n and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o residue_n shall_v contain_v two_o nine_o part_n of_o the_o solid_a of_o the_o pyramid_n of_o which_o nine_o part_v each_o be_v equal_a unto_o the_o cube_fw-la take_v away_o this_o self_n same_o excess_n the_o solid_a therefore_o of_o a_o dodecahedron_n contain_v of_o a_o pyramid_n circumscribe_v about_o it_o two_o nine_o part_n take_v away_o a_o three_o part_n of_o one_o nine_o part_n of_o the_o less_o segment_n of_o a_o line_n divide_v by_o a_o extmere_n and_o mean_a proportion_n and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o residue_n ¶_o the_o 36._o proposition_n a_o octohedron_n exceed_v a_o icosahedron_n inscribe_v in_o it_o by_o a_o parallelipipedon_n set_v upon_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n and_o have_v to_o his_o altitude_n the_o line_n which_o be_v the_o great_a segment_n of_o half_a the_o semidiameter_n of_o the_o octohedron_n svppose_v that_o there_o be_v a_o octohedron_n abcfpl_n construction_n in_o which_o let_v there_o be_v inscribe_v a_o icosahedron_n hkegmxnudsqtâ_n by_o the_o â6_o of_o the_o fifteen_o and_o draw_v the_o diameter_n azrcbroif_n and_o the_o perpendicular_a ko_o parallel_n to_o the_o line_n azr_n then_o i_o say_v that_o the_o octohedron_n abcfpl_n be_v great_a thân_n the_o icosahedron_n inscribe_v in_o it_o by_o a_o parallelipipedon_n set_v upon_o the_o square_n of_o the_o side_n hk_o or_o ge_z and_o have_v to_o his_o altitude_n the_o line_n ko_o or_o rz_n which_o be_v the_o great_a segment_n of_o the_o semidiameter_n ar._n forasmuch_o as_o in_o the_o same_o 16._o it_o have_v be_v prove_v that_o the_o triangle_n kdg_n and_o keq_n be_v describe_v in_o the_o base_n apf_n and_o alf_n of_o the_o octohedron_n demonstration_n therefore_o about_o the_o solid_a angle_n there_o remain_v upon_o the_o base_a feg_fw-mi three_o triangle_n keg_n kfe_n and_o kfg_n which_o contain_v a_o pyramid_n kefg_n unto_o which_o pyramid_n shall_v be_v equal_a and_o like_v the_o opposite_a pyramid_n mefg_n set_v upon_o the_o same_o base_a feg_n by_o the_o 8._o definition_n of_o the_o eleven_o and_o by_o the_o âame_n reason_n shall_v there_o at_o every_o solid_a angle_n of_o the_o octohedron_n remain_v two_o pyramid_n equal_a and_o like_a namely_o two_o upon_o the_o base_a ahk_n two_o upon_o the_o base_a bnv_n two_o upon_o the_o base_a dp_n and_o moreover_o two_o upon_o the_o base_a qlt._n now_o they_o there_o shall_v be_v make_v twelve_o pyramid_n set_v upon_o a_o base_a contain_v of_o the_o side_n of_o the_o icosahedron_n and_o under_o two_o leââe_a segment_n of_o the_o side_n of_o the_o octohedron_n contain_v a_o right_a angle_n as_o for_o example_n the_o base_a gef_n and_o forasmuch_o as_o the_o side_n ge_z subtend_v a_o right_a angle_n be_v by_o the_o 47._o of_o the_o âirst_n in_o power_n duple_n to_o either_o of_o the_o line_n of_o and_o fg_o and_o so_o the_o ââdeâ_n kh_o be_v in_o power_n duple_n to_o either_o of_o the_o side_n ah_o and_o ak_o and_o either_o of_o the_o line_n ah_o ak_o or_o of_o fg_o be_v in_o power_n duple_n to_o either_o of_o the_o line_n az_o or_o zk_v which_o contain_v a_o right_a angle_n make_v in_o the_o triangle_n or_o base_a ahk_n by_o the_o perpendicular_a az_o wherefore_o it_o follow_v that_o the_o side_n ge_z or_o hk_o be_v in_o power_n quadruple_a to_o the_o triangle_n efg_o or_o ahk_n but_o the_o pyramid_n kefg_n have_v his_o base_a efg_o in_o the_o plain_a flbp_n of_o the_o octohedron_n shall_v have_v to_o his_o altitude_n the_o perpendicular_a ko_o by_o the_o 4._o definition_n of_o the_o six_o which_o be_v the_o great_a segment_n of_o the_o semidiameter_n of_o the_o octohedron_n by_o the_o 16._o of_o the_o fifteen_o wherefore_o three_o pyramid_n set_v under_o the_o same_o altitude_n and_o upon_o equal_a base_n shall_v be_v equal_a to_o one_o prism_n set_v upon_o the_o same_o base_a and_o under_o the_o same_o altitude_n by_o the_o 1._o corollary_n of_o the_o 7._o of_o the_o twelve_o wherefore_o 4._o prism_n set_v upon_o the_o base_a gef_n quadruple_v which_o be_v equal_a to_o the_o square_n of_o the_o side_n ge_z and_o under_o the_o altitude_n ko_o or_o rz_n the_o great_a segment_n which_o be_v equal_a to_o ko_o shall_v contain_v a_o solid_a equal_a to_o the_o twelve_o pyramid_n which_o twelve_o pyramid_n make_v the_o excess_n of_o the_o octohedron_n above_o the_o icosahedron_n inscribe_v in_o it_o a_o octohedron_n therefore_o exceed_v a_o icosahedron_n inscribe_v in_o it_o by_o a_o parallelipipedon_n set_v upon_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n and_o have_v to_o his_o altitude_n the_o line_n which_o be_v the_o great_a segment_n of_o half_a the_o semidiameter_n of_o the_o octohedron_n ¶_o a_o corollary_n a_o pyramid_n exceed_v the_o double_a of_o a_o icosahedron_n inscribe_v in_o it_o by_o a_o solid_a set_v upon_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o and_o have_v to_o his_o altitude_n that_o whole_a line_n of_o which_o the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n for_o it_o be_v manifest_a by_o the_o 19_o of_o the_o fivetenth_n that_o a_o octohedron_n &_o a_o icosahedron_n inscribe_v in_o it_o be_v inscribe_v in_o one_o &_o the_o self_n same_o pyramid_n it_o have_v moreover_o be_v prove_v in_o the_o 26._o of_o this_o book_n that_o a_o pyramid_n be_v double_a to_o a_o octohedron_n inscribe_v in_o it_o wherefore_o the_o two_o excess_n of_o the_o two_o octohedron_n unto_o which_o the_o pyramid_n be_v equal_a above_o the_o two_o icosahedrons_n inscribe_v in_o the_o say_v two_o octohedron_n be_v bring_v into_o a_o solid_a the_o say_v solid_a shall_v be_v set_v upon_o the_o self_n same_o square_n of_o the_o side_n of_o the_o icosahedron_n and_o shall_v have_v to_o his_o altitude_n the_o perpendicular_a ko_o double_a who_o double_a couple_v the_o opposite_a side_n hk_o and_o xm_o make_v the_o great_a segment_n the_o same_o side_n of_o the_o icosahedron_n by_o the_o first_o and_o second_o corollary_n of_o the_o 14._o of_o the_o fiuââenâh_n the_o 37._o proposition_n if_o in_o a_o triangle_n have_v to_o his_o base_a a_o rational_a line_n set_v the_o side_n be_v commensurable_a in_o power_n to_o the_o base_a and_o from_o the_o top_n be_v draw_v to_o the_o base_a a_o perpendicular_a line_n cut_v the_o base_a the_o section_n of_o the_o base_a shall_v be_v commensurable_a in_o length_n to_o the_o whole_a base_a and_o the_o perpendicular_a shall_v be_v commensurable_a in_o power_n to_o the_o say_v whole_a base_a and_o now_o that_o the_o perpendicular_a ap_n be_v commensurable_a in_o power_n to_o the_o base_a bg_o demonstration_n iâ_z thus_o prove_v forasmuch_o as_o the_o square_n of_o ab_fw-la be_v by_o supposition_n commensurable_a to_o the_o square_n of_o bg_o and_o unto_o the_o rational_a square_n of_o ab_fw-la be_v commensurable_a the_o rational_a square_n of_o bp_o by_o the_o 12._o of_o the_o eleven_o wherefore_o the_o residue_n namely_o the_o square_a of_o pa_o be_v commensurable_a to_o the_o same_o square_n of_o bp_o by_o the_o 2._o part_n of_o the_o 15._o of_o the_o eleven_o wherefore_o by_o the_o 12._o of_o the_o ten_o the_o square_a of_o pa_o be_v commensurable_a to_o the_o whole_a square_n of_o bg_o wherefore_o the_o perpendicular_a ap_n be_v commensurable_a in_o power_n to_o the_o base_a bg_o by_o the_o 3._o definition_n of_o the_o ten_o which_o be_v require_v to_o be_v prove_v in_o demonstrate_v of_o this_o we_o make_v no_o mention_n at_o all_o of_o the_o length_n of_o the_o side_n ab_fw-la and_o ag_z but_o only_o of_o the_o length_n of_o the_o base_a bg_o for_o that_o the_o line_n bg_o be_v the_o rational_a line_n first_o set_v and_o the_o other_o line_n ab_fw-la and_o ag_z be_v suppose_v to_o be_v commensurable_a in_o power_n only_o to_o the_o line_n bg_o wherefore_o if_o that_o be_v plain_o demonstrate_v when_o the_o side_n be_v commensurable_a in_o power_n only_o to_o the_o base_a much_o more_o easy_o will_v it_o follow_v if_o the_o same_o side_n be_v suppose_v to_o be_v commensurable_a both_o in_o length_n and_o in_o power_n to_o the_o base_a that_o be_v if_o their_o lengthe_n be_v express_v by_o the_o root_n of_o square_a number_n ¶_o a_o corollary_n 1._o by_o the_o former_a thing_n demonstrate_v it_o be_v manifest_a that_o if_o from_o the_o power_n of_o the_o base_a and_o of_o one_o of_o the_o side_n be_v take_v away_o the_o