the_o 17._o proposition_n of_o the_o 14._o book_n construction_n â—rist_n part_n of_o the_o demonstration_n demonstration_n for_o the_o 4._o angle_n at_o the_o point_n king_n be_v equal_a to_o four_o right_a angle_n by_o the_o corollary_n of_o tâ—e_z 15._o of_o the_o first_o and_o those_o 4._o angle_n be_v equal_a the_o one_o to_o the_o other_o by_o the_o â—_o of_o the_o â—irst_n and_o therefore_o each_o be_v a_o right_a angle_n second_o part_n of_o the_o demonstration_n demonstration_n for_o the_o square_n of_o the_o line_n ab_fw-la which_o be_v prove_v equal_a to_o the_o square_n of_o the_o line_n lm_o be_v double_a to_o the_o square_n of_o the_o line_n bd_o which_o be_v also_o equal_a to_o the_o square_n of_o the_o line_n le._n this_o corollary_n be_v the_o 16._o proposition_n of_o the_o 14._o book_n after_o campane_n first_o part_n of_o the_o demonstration_n second_o part_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n demonstration_n by_o the_o 2._o assumpt_v of_o the_o 13._o of_o this_o book_n three_o part_n of_o the_o demonstration_n second_o part_n of_o the_o demonstration_n for_o the_o line_n qw_o be_v equal_a to_o the_o line_n iz_n &_o the_o line_n zw_n be_v common_a to_o they_o both_o this_o part_n be_v again_o afterward_o demonstrate_v by_o flussas_n the_o pentagon_n vbwcr_n prove_v to_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n the_o pentagon_n vbwcz_n it_o prove_v equiangle_n equiangle_n look_v for_o a_o far_a construction_n after_o flussas_n at_o the_o end_n of_o the_o demonstration_n that_o the_o side_n of_o the_o dodecahedron_n be_v a_o residual_a line_n draw_v in_o the_o former_a figure_n these_o line_n cta_n ctl_n câ—d_v the_o side_n of_o a_o pyramid_n the_o side_n of_o a_o cube_fw-la the_o side_n of_o a_o dodecahedron_n comparison_n of_o the_o five_o side_n of_o the_o foresay_a body_n an_o other_o demommonstration_n to_o prove_v that_o the_o side_n of_o the_o icosahedron_n be_v great_a than_o the_o side_n of_o the_o dodecahedron_n that_o 3._o square_n of_o the_o line_n fb_o be_v great_a they_o 6._o square_n of_o the_o line_n nb._n that_o there_o can_v be_v no_o other_o solid_a beside_o these_o five_o contain_v under_o equilater_n and_o equiangle_n base_n that_o the_o angle_n of_o a_o equilater_n and_o equiangle_n pentagon_n be_v one_o right_a angle_n and_o a_o 〈◊〉_d part_n 〈◊〉_d which_o thing_n be_v also_o before_o prove_v in_o the_o 〈◊〉_d of_o the_o 32._o of_o the_o â—irst_n the_o side_n of_o the_o angle_n of_o the_o inclâ—â—â—tion_n of_o the_o 〈◊〉_d of_o the_o 〈◊〉_d be_v 〈◊〉_d rational_a the_o side_n of_o the_o angle_n of_o the_o inclination_n of_o the_o 〈◊〉_d â—f_n tâ—e_z 〈…〉_o that_o the_o plain_n of_o a_o octohedron_n be_v in_o liâ—e_z sort_n incline_v that_o the_o plain_n of_o a_o icosahedron_n be_v in_o like_a sort_n incline_v that_o the_o plain_n of_o a_o dâ—â—â—â—â—hedron_n be_v 〈◊〉_d like_o sort_n incline_v the_o side_n of_o the_o angle_n of_o the_o inclination_n of_o the_o supeâ—ficieces_n of_o the_o tetrahedron_n be_v prove_v rational_a the_o side_n of_o the_o angle_n of_o the_o inclination_n of_o the_o superficiece_n of_o the_o cube_fw-la prove_v rational_a the_o side_n of_o thâ—_z angle_n etc_n etc_n of_o the_o octohedron_n prove_v rational_a the_o side_n of_o the_o angle_n etc_n etc_n of_o the_o icosahedron_n prove_v irrational_a how_o to_o know_v whether_o the_o angle_n of_o the_o inclination_n be_v a_o right_a angle_n a_o acute_a angle_n or_o a_o oblique_a angle_n the_o argument_n of_o the_o fourteen_o book_n first_o proposition_n after_o flussas_n construction_n demonstration_n demonstration_n this_o be_v manifest_a by_o the_o 12._o proposition_n of_o the_o thirtenh_o book_n as_o campane_n well_o gather_v in_o a_o corollary_n of_o the_o same_o the_o 4._o pâ—posâ—tion_n after_o flussas_n flussas_n this_o be_v afterward_o prove_v in_o the_o 4._o proposition_n this_o assumpt_n be_v the_o 3._o proposition_n after_o flussas_n construction_n of_o the_o assumpt_n demonstration_n of_o the_o assumpt_n construction_n of_o the_o proposition_n demonstration_n of_o the_o proposition_n proposition_n thâ—_z 5._o proposition_n aâ—tâ—r_a 〈◊〉_d construction_n demonstration_n the_o 5._o proposition_n aâ—ter_a fâ—ussas_n demonstration_n demonstration_n this_o be_v the_o reason_n of_o the_o corollary_n follow_v a_o corollary_n which_o also_o flussas_n put_v as_o a_o corollary_n after_o the_o 5._o proposition_n in_o his_o order_n the_o 6._o pâ—â—positionâ—â—ter_a flussas_n construction_n demonstration_n demonstration_n this_o be_v not_o hard_a to_o prove_v by_o the_o 15._o 16._o and_o 19_o of_o the_o â—â—â—eth_v â—â—â—eth_v in_o the_o corollary_n of_o the_o 17._o of_o the_o tâ—irtenth_o tâ—irtenth_o 〈◊〉_d again_o be_v require_v the_o assumpt_n which_o be_v afterward_o prove_v in_o this_o 4_o proposition_n proposition_n but_o first_o the_o assumpt_n follow_v the_o construction_n whereof_o here_o begin_v be_v to_o be_v prove_v the_o assumpt_n which_o also_o flussas_n put_v as_o a_o assumpt_n aâ—ter_v the_o 6._o proposition_n demonstration_n of_o the_o assumpt_n construction_n pertain_v to_o the_o second_o demonstration_n of_o the_o 4._o proposition_n second_o demonstration_n oâ—_n the_o 4._o proposition_n the_o 7â—_n proposition_n after_o flussas_n construction_n demonstration_n demonstration_n here_o again_o be_v require_v the_o assumpt_n afterward_o prove_v in_o this_o 4._o proposition_n proposition_n as_o may_v by_o the_o assumpt_n afterward_o in_o this_o proposition_n be_v plain_o prove_v the_o 8._o prodition_n aâ—ter_a flussas_n flussas_n by_o the_o corollary_n add_v by_o flussas_n after_o have_v assumpt_v put_v after_o the_o 17._o proposition_n of_o the_o 12._o book_n corollary_n of_o the_o 8._o after_o flussas_n this_o assumpt_n be_v the_o 3._o proposition_n aâ—ter_v â—lussas_n and_o be_v it_o which_o 〈◊〉_d time_n have_v be_v take_v aâ—_z grant_v in_o this_o book_n and_o oâ—ce_n also_o in_o the_o last_o proposition_n of_o the_o 13._o book_n as_o we_o have_v beâ—ore_o note_v demonstration_n demonstration_n in_o the_o 4._o section_n â—f_o this_o proposition_n proposition_n in_o the_o 1._o and_o 3_o section_n of_o the_o same_o proposition_n proposition_n in_o the_o 5._o section_n of_o the_o same_o proposition_n a_o corollary_n the_o first_o proposition_n after_o campane_n construction_n demonstration_n the_o 2._o proposition_n after_o campane_n demonstration_n lead_v to_o a_o impossibility_n the_o 4._o proposition_n after_o campane_n construction_n demonstration_n this_o corollary_n campane_n also_o â—utteth_v after_o the_o 4._o proposition_n in_o his_o order_n the_o 5._o proposition_n after_o campane_n construction_n demonstration_n this_o be_v the_o 6._o and_o 7._o proposition_n after_o campane_n construction_n demonstration_n this_o corollary_n campane_n also_o add_v after_o the_o 7._o proposition_n iâ—_z his_o order_n the_o 5._o proposition_n aâ—ter_v campane_n construction_n demonstration_n this_o assumpt_a campane_n also_o have_v after_o the_o 8._o proposition_n in_o his_o order_n construction_n demonstration_n the_o 9_o proposition_n after_o campane_n construction_n demonstration_n this_o campane_n put_v aâ—_z a_o corollary_n in_o the_o 9_o proposition_n after_o his_o order_n this_o corollary_n be_v the_o 9_o proposition_n after_o campane_n the_o 12._o proposition_n after_o campane_n construction_n demonstration_n the_o 13._o proposition_n after_o campane_n the_o 14._o proposition_n after_o campane_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n the_o 17._o proposition_n after_o campane_n firâ—t_o part_n of_o the_o construction_n first_o part_n of_o the_o demonstration_n second_o parâ—_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n the_o 18._o proposition_n after_o campane_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n the_o corollary_n of_o the_o 8._o proposition_n after_o campane_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n the_o argument_n of_o the_o 15._o book_n book_n in_o this_o proposition_n as_o also_o in_o all_o the_o other_o follow_v by_o the_o name_n of_o a_o pyramid_n understand_v a_o tetrahedron_n as_o i_o have_v before_o admonish_v construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n that_o which_o here_o follow_v concern_v the_o inclination_n of_o the_o plain_n of_o the_o five_o solid_n be_v before_o teach_v â—hough_o not_o altogether_o after_o the_o same_o manner_n out_o of_o flussas_n in_o the_o latter_a â—nde_n of_o the_o 13_o book_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n first_o part_n of_o the_o construction_n first_o part_n of_o the_o demonstration_n second_o part_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n three_o part_n of_o the_o construction_n three_o part_n of_o the_o demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n produce_v in_o the_o figure_n the_o line_n tf_n to_o the_o point_n b._n construction_n demonstration_n this_o proposition_n campane_n have_v &_o be_v the_o last_o also_o in_o order_n of_o the_o 15._o book_n with_o he_o the_o argument_n of_o the_o 16._o book_n construction_n demonstration_n demonstration_n by_o a_o pyramid_n understand_v a_o tetrahedron_n throughout_o all_o this_o book_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n demonstration_n that_o be_v aâ—_z 18._o to_o 1._o demonstration_n demonstration_n that_o iâ—_z as_o 9_o to_o 2._o 2._o that_o be_v as_o 18._o to_o 2._o or_o 9_o to_o 1._o draw_v in_o the_o figure_n a_o line_n from_o b_o to_o h._n h._n what_o the_o duple_n of_o a_o extreme_a and_o mean_a proportion_n be_v construction_n demonstration_n demonstration_n constrution_n demonstration_n construction_n demonstration_n demonstration_n demonstration_n that_o be_v at_o 13._o 1_o â—_o be_v toâ—_n â—_z demonstration_n demonstration_n construction_n demonstration_n construction_n demonstration_n extend_v in_o the_o figure_n a_o line_n â—rom_o the_o point_n e_o to_o the_o point_n b._n extend_v in_o the_o figure_n a_o line_n from_o the_o point_n e_o to_o the_o point_n b._n demonstration_n construction_n demonstration_n second_o part_n of_o the_o demonstration_n icosidodecahedron_n exoctohedron_n that_o the_o exoctohedron_n be_v contain_v in_o a_o sphere_n that_o the_o exoctohedron_n be_v contain_v in_o the_o sphere_n give_v that_o the_o diaâ—â—â—ter_n of_o the_o sphere_n be_v doâ—ble_a to_o the_o side_n â—f_o the_o exoctohedron_n that_o the_o icosidodecahedron_n be_v contain_v in_o the_o sphere_n give_v give_v that_o be_v as_o 8._o 103â—_n â—_z fault_n escape_v â—cl_n â—ag_n line_n faultesâ—_n co_n 〈◊〉_d 〈◊〉_d  _o  _o  _o errata_fw-la lib._n 1._o  _o 1_o 2_o 41_o point_n b._n at_o campane_n point_n c_o aâ—_z campane_n 3_o 1_o 22_o aâ—l_a line_n draw_v all_o righâ—_n 〈…〉_o 3_o 1_o 28_o line_n draw_v to_o the_o superficies_n right_a line_n draw_v to_o the_o circumference_n 9_o 1_o 42_o liâ—es_n ab_fw-la and_o ac_fw-la line_n ab_fw-la and_o bc_o 15_o 1_o 35_o be_v equal_a be_v prove_v equal_a 20_o 2_o 28_o by_o the_o first_o by_o the_o four_o 21_o 1_o 39_o tâ—e_z centre_n c._n the_o centre_n e_o 2d_o 2_o â—_o iâ—â—ower_a right_n if_o two_o right_a 25_o 2_o 3_o fâ—â—â—_n petition_n five_o petition_n 49_o 2_o 7_o 14._o â—â—_o 32.64_o etc_n etc_n 4.8.16.32.64_o &_o 53_o 1_o 39_o the_o triangle_n ng_z the_o triangle_n kâ—_n 54_o 2_o 25_o by_o the_o 44_o by_o the_o 42_o 57_o 2_o 23_o and_o câ—g_v in_o the_o and_o cgb_n be_v thâ—_z  _o  _o  _o in_o stead_n of_o â—lussates_n through_o out_o 〈◊〉_d whole_a book_n read_v â—lusâ—as_v  _o  _o  _o errata_fw-la lib._n 2._o  _o 60_o 2_o 29_o gnomon_n fgeh_a gnomon_n ahkd_v  _o  _o 30_o gnomon_n ehfg_n gnomon_n â—ckd_v 69_o 1_o 18_o the_o whole_a line_n the_o whole_a â—igure_n 76_o 2_o 9_o the_o diameter_n cd_o the_o diameter_n ahf_n  _o  _o  _o errata_fw-la lib._n 3._o  _o 82_o 2_o 36_o angle_n equal_a to_o the_o angle_n 92_o 1_o last_o the_o line_n ac_fw-la be_v the_o line_n of_o 〈◊〉_d  _o  _o  _o errata_fw-la lib._n 4._o  _o 110_o 2_o 10_o cd_o touch_v the_o ed_z touch_v the_o  _o  _o 12_o side_n of_o the_o other_o angle_n of_o the_o other_o 115_o 1_o 21_o and_o hb_o and_o he_o 117_o 2_o 44_o the_o angle_n acd_o the_o angle_n acb_o 118_o 1_o 2_o into_o ten_o equal_a into_o two_o equal_a 121_o 1_o 3â—_o cd_o and_o ea_fw-la cd_o de_fw-fr &_o ea_fw-la â—â—â—_o 1_o 29_o the_o first_o the_o three_o  _o  _o  _o errata_fw-la lib._n 5._o  _o 126_o 1_o 43_o it_o make_v 12._o more_o than_o 17._o by_o 5._o it_o make_v 24._o more_o than_o 17._o by_o 7._o 129_o 1_n  _o in_o stead_n of_o the_o figure_n of_o the_o 6._o definition_n draw_v in_o the_o magâ—â—_n a_o figure_n like_o unto_o thâ—s_n 134_o 2_o 4_o as_o ab_fw-la be_v to_o a_o so_o be_v cd_o to_o c_o as_z ab_fw-la be_v to_o b_o so_o be_v cd_o to_o d_o 141_o 2_o last_v but_o if_o king_n exceed_v m_o but_o if_o h_o exceed_v m_o lief_a be_v death_n and_o death_n be_v lief_a aetatis_fw-la svae_fw-la xxxx_n at_o london_n print_v by_o john_n day_n dwell_v over_o aldersgate_n beneath_o saint_n martins_n ¶_o these_o book_n be_v to_o be_v sell_v at_o his_o shop_n under_o the_o gate_n 1570._o
side_n the_o rectiline_a angle_n mab_n either_o of_o which_o be_v equal_a to_o the_o rectiline_a angle_n give_v cde_o which_o be_v require_v to_o be_v do_v an_o other_o construction_n also_o and_o demonstration_n after_o pelitarâ—us_n and_o if_o the_o perpendicular_a line_n chance_n to_o fall_v without_o the_o angle_n give_v namely_o if_o the_o angle_n give_v be_v a_o acute_a angle_n the_o self_n same_o manner_n of_o demonstration_n will_v serve_v but_o only_o that_o in_o stead_n of_o the_o second_o common_a sentence_n must_v be_v use_v the_o 3._o common_a sentence_n appollonius_n put_v another_o construction_n &_o demonstration_n of_o this_o proposition_n which_o though_o the_o demonstration_n thereof_o depend_v of_o proposition_n put_v in_o the_o three_o book_n yet_o for_o that_o the_o construction_n be_v very_o good_a for_o he_o that_o will_v ready_o and_o mechanical_o without_o demonstration_n describe_v upon_o a_o line_n give_v and_o to_o a_o point_n in_o it_o give_v a_o rectiline_a angle_n equal_a to_o a_o rectiline_a angle_n give_v i_o think_v not_o amiss_o here_o to_o place_v it_o and_o it_o be_v thus_o proposition_n oenopides_n be_v the_o first_o inventor_n of_o this_o proposition_n as_o witness_v eudemius_fw-la the_o 15._o theorem_a the_o 24._o proposition_n if_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_a to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n and_o if_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o one_o be_v great_a than_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o other_o the_o base_a also_o of_o the_o same_o shall_v be_v great_a than_o the_o base_a of_o the_o other_o svppose_v that_o there_o be_v two_o triangle_n abc_n and_o def_n have_v two_o side_n of_o the_o one_o that_o be_v ab_fw-la and_o ac_fw-la equal_a to_o two_o side_n of_o the_o other_o that_o be_v to_o de_fw-fr and_o df_o each_o to_o his_o correspondent_a side_n that_o be_v the_o side_n ab_fw-la to_o the_o side_n de_fw-fr and_o the_o side_n ac_fw-la to_o the_o side_n df_o and_o suppose_v that_o the_o angle_n bac_n be_v great_a than_o the_o angle_n edf_o then_o i_o say_v that_o the_o base_a bc_o be_v great_a than_o the_o base_a ef._n for_o forasmuch_o as_o the_o angle_n bac_n be_v great_a than_o the_o angle_n edf_o construction_n make_v by_o the_o 23._o proposition_n upon_o the_o right_a line_n de_fw-fr and_o to_o the_o point_n in_o it_o give_v d_o a_o angle_n edge_v equal_a to_o the_o angle_n give_v bac_n and_o to_o one_o of_o these_o line_n that_o be_v either_o to_o ac_fw-la or_o df_o put_v a_o equal_a line_n dg_o and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n g_o to_o the_o point_n e_o and_o a_o other_o from_o the_o point_n fletcher_n demonstration_n to_o the_o point_n g._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n de_fw-fr and_o the_o line_n ac_fw-la to_o the_o line_n dg_o the_o one_o to_o the_o outstretch_o and_o the_o angle_n bac_n be_v by_o construction_n equal_a to_o the_o angle_n edg_n therefore_o by_o the_o 4._o proposition_n the_o base_a bc_o be_v equal_a to_o the_o base_a eglantine_n again_o for_o as_o much_o as_o the_o line_n dg_o be_v equal_a to_o the_o line_n df_o therâ—_n by_o the_o 5._o proposition_n the_o angle_n dgf_n be_v equal_a to_o the_o angle_n dfg_n wherefore_o the_o angle_n dfg_n be_v great_a than_o the_o angle_n egf_n wherefore_o the_o angle_n efg_o be_v much_o great_a than_o the_o angle_n egf_n and_o forasmuch_o as_o efg_o be_v a_o triangle_n have_v the_o angle_n efg_o great_a than_o the_o angle_n egf_n and_o by_o the_o 18._o prâ—position_n under_o the_o great_a angle_n be_v subtend_v the_o great_a side_n therefore_o the_o side_n eglantine_n be_v great_a than_o the_o side_n ef._n but_o the_o side_n eglantine_n be_v equal_a to_o the_o side_n bc_o wherefore_o the_o side_n bc_o be_v great_a than_o the_o side_n ef._n if_o therefore_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_a to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n and_o if_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o one_o be_v great_a than_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o other_o the_o base_a also_o of_o the_o same_o shall_v be_v great_a than_o the_o base_a of_o the_o other_o which_o be_v require_v to_o be_v prove_v in_o this_o theorem_a may_v be_v three_o case_n prâ—position_n for_o the_o angle_n edg_n be_v put_v equal_a to_o the_o angle_n bac_n and_o the_o line_n dg_o be_v put_v equal_a to_o the_o line_n ac_fw-la and_o a_o line_n be_v draw_v from_o e_o to_o g_z the_o line_n eglantine_n shall_v either_o fall_v above_o the_o line_n gf_o or_o upon_o it_o or_o under_o it_o euclides_n demonstration_n serve_v 2._o when_o the_o line_n ge_z fall_v above_o the_o line_n gf_o as_o we_o have_v before_o manifest_o see_v but_o now_o let_v the_o line_n eglantine_n case_n fall_v under_o the_o line_n e_o fletcher_n as_o in_o the_o figure_n here_o put_v and_o forasmuch_o as_o these_o two_o line_n ab_fw-la and_o ac_fw-la be_v equal_a to_o these_o two_o line_n de_fw-fr and_o dg_o the_o one_o to_o the_o other_o and_o they_o contain_v equal_a angle_n therefore_o by_o the_o 4._o proposition_n the_o base_a bc_o be_v equal_a to_o the_o base_a eglantine_n and_o forasmuch_o as_o within_o the_o triangle_n deg_n the_o two_o linne_n df_o and_o fe_o be_v set_v upon_o the_o side_n de_fw-fr therefore_o by_o the_o 21._o proposition_n the_o line_n df_o and_o fâ—_n be_v less_o than_o the_o outward_a line_n dg_o and_o ge_z but_o the_o line_n dg_o be_v equal_a to_o the_o line_n df._n wherefore_o the_o line_n ge_z be_v great_a than_o the_o line_n fe_o but_o ge_z be_v equal_a to_o bc._n wherefore_o the_o line_n bc_o be_v great_a the_o the_o line_n ef._n which_o be_v require_v to_o be_v prove_v it_o may_v peradventure_o semeâ—_n that_o euclid_n shall_v here_o in_o this_o proposition_n have_v prove_v that_o not_o only_o the_o base_n of_o the_o triangle_n be_v unequal_a but_o also_o that_o the_o area_n of_o the_o same_o be_v unequal_a for_o so_o in_o the_o four_o proposition_n after_o he_o have_v prove_v the_o base_a to_o be_v equal_a he_o prove_v also_o the_o area_n to_o be_v equal_a but_o hereto_o may_v be_v answer_v &c._a that_o in_o equal_a angle_n and_o base_n and_o unequal_a angle_n and_o base_n the_o consideration_n be_v not_o like_a for_o the_o angle_n and_o base_n be_v equal_a the_o triangle_n also_o shall_v of_o necessity_n be_v equal_a but_o the_o angle_n and_o base_n be_v unequal_a the_o area_n shall_v not_o of_o necessity_n be_v equal_a for_o the_o triangle_n may_v both_o be_v equal_a and_o unequal_a and_o that_o may_v be_v the_o great_a which_o have_v the_o great_a angle_n and_o the_o great_a base_n and_o it_o may_v also_o be_v the_o less_o and_o for_o that_o cause_n euclid_n make_v not_o mention_v of_o the_o comparison_n of_o the_o triangle_n whereof_o this_o also_o may_v be_v a_o cause_n for_o that_o to_o the_o demonstration_n thereof_o be_v require_v certain_a proposition_n concern_v parallel_a line_n which_o we_o be_v not_o as_o yet_o come_v unto_o etc_n howbeit_o after_o the_o 37â—_n proposition_n of_o his_o book_n you_o shall_v find_v the_o comparison_n of_o the_o area_n of_o triangle_n which_o have_v their_o side_n equal_a and_o their_o base_n and_o angle_n at_o the_o top_n unequal_a the_o 16._o theorem_a the_o 25._o proposition_n if_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_a to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n and_o if_o the_o base_a of_o the_o one_o be_v great_a than_o the_o base_a of_o the_o other_o the_o angle_n also_o of_o the_o same_o contain_v under_o the_o equal_a right_a linesâ—_n shall_v be_v great_a than_o the_o angle_n of_o the_o other_o svppose_v that_o there_o be_v two_o triangle_n a_o b_o c_o and_o def_n have_v two_o side_n of_o tb'one_n that_o be_v ab_fw-la and_o ac_fw-la equal_a to_o two_o side_n of_o the_o other_o that_o be_v to_o de_fw-fr and_o df_o each_o to_o his_o correspondent_a side_n namely_o the_o side_n ab_fw-la to_o the_o side_n df_o and_o the_o side_n ac_fw-la to_o the_o side_n df._n but_o let_v the_o base_a bc_o be_v great_a than_o the_o base_a ef._n then_o i_o say_v they_o the_o angle_n bac_n be_v great_a than_o the_o angle_n edf_o for_o if_o not_o absurdity_n then_o be_v it_o either_o equal_a unto_o it_o or_o less_o than_o it_o but_o the_o angle_n bac_n be_v not_o equal_a to_o the_o angle_n edf_o for_o if_o it_o be_v equal_a the_o base_a also_o bc_o shall_v by_o the_o 4._o proposition_n be_v equal_a to_o the_o base_a of_o but_o by_o supposition_n it_o be_v not_o wherefore_o
signify_v last_o of_o all_o a_o dodecahedron_n heaven_n for_o that_o it_o be_v make_v of_o pâ—ntagoâ—_n who_o angle_n be_v more_o ample_a and_o large_a than_o the_o angle_n of_o the_o other_o body_n and_o by_o that_o â—eaâ—â—â—_n draw_v more_o â—â—_o roundness_n 〈◊〉_d &_o to_o the_o form_n and_o nature_n of_o a_o sphere_n they_o assign_v to_o a_o sphere_n namely_o 〈…〉_o who_o so_o will_v 〈…〉_o in_o his_o tineus_n shall_v â—ead_a of_o these_o figure_n and_o of_o their_o mutual_a proportionâ—â—â—raunge_a maâ—terâ—_n which_o hâ—re_o be_v not_o to_o be_v entreat_v of_o this_o which_o be_v say_v shall_v be_v sufficient_a for_o the_o 〈◊〉_d of_o they_o and_o for_o thâ—_z declaration_n of_o their_o definition_n after_o all_o these_o definition_n here_o set_v of_o euclid_n flussas_n have_v add_v a_o other_o definition_n which_o 〈◊〉_d of_o a_o parallelipipedon_n which_o because_o it_o have_v not_o hitherto_o of_o euclid_n in_o any_o place_n be_v define_v and_o because_o it_o be_v very_o good_a and_o necessary_a to_o be_v have_v i_o think_v good_a not_o to_o omit_v it_o thus_o it_o be_v a_o parallelipipedon_n be_v a_o solid_a figure_n comprehend_v under_o four_o plain_a quadrangle_n figure_n parallelipipedon_n of_o which_o those_o which_o be_v opposite_a be_v parallel_n because_o these_o five_o regular_a body_n here_o define_v be_v not_o by_o these_o figure_n here_o set_v so_o full_o and_o lively_o express_v that_o the_o studious_a beholder_n can_v thorough_o according_a to_o their_o definition_n conceive_v they_o i_o have_v here_o give_v of_o they_o other_o description_n draw_v in_o a_o plain_n by_o which_o you_o may_v easy_o attain_v to_o the_o knowledge_n of_o they_o for_o if_o you_o draw_v the_o like_a form_n in_o matter_n that_o will_v bow_v and_o geve_v place_n as_z most_o apt_o you_o may_v do_v in_o fine_a past_v paper_n such_o as_o pastwive_n make_v woman_n paste_n of_o &_o they_o with_o a_o knife_n cut_v every_o line_n fine_o not_o through_o but_o half_a way_n only_o if_o 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 hadâ—â—tore_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 〈…〉_o the_o form_n of_o other_o body_n as_o prism_n and_o parallelipopedon_n 〈…〉_o set_v forth_o in_o these_o five_o book_n follow_v and_o see_v the_o very_a 〈◊〉_d of_o thâ—se_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 twâ—luâ—â—quilâ—â—â—â—_n and_o equiangle_n pâ—ntâ—â—â—â—â—_n upoâ—_n the_o foresay_a matter_n and_o fine_o cut_v as_o before_o be_v â—â—ught_v tâ—â—â—lâ—uâ—n_v line_n contain_v within_o thâ—_z figurâ—_n and_o bow_n and_o fold_n the_o penâ—â—gonâ—_n according_o and_o they_o will_v so_o close_o together_o thaâ—_z thâ—y_n will_v â—â—kâ—_n thâ—_z very_a form_n of_o a_o dodecahedron_n dâ—dâ—â—â—â—edron_n icosaâ—edron_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 ninâ—tâ—ne_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 â—ake_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 â—ase_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 〈◊〉_d point_n then_o one_o uâ—lesse_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 flâ—sâ—â—s_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 〈…〉_o the_o outward_a angâ—â—_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 coâ—â—â—â—e_v this_o figure_n which_o ulyss_n put_v for_o his_o demonstration_n â—_o wherein_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
proposition_n after_o prâ—â—lus_n a_o corollary_n take_v out_o of_o flussates_n demonstration_n lead_v to_o 〈◊〉_d absurdity_n a_o addition_n oâ—_n pelitarius_n demonstration_n three_o case_n in_o this_o proposition_n the_o first_o case_n construction_n demonstration_n three_o case_n in_o this_o proposition_n the_o first_o case_n every_o case_n may_v happen_v seven_o diverse_a way_n the_o like_a variety_n in_o each_o of_o the_o other_o two_o case_n euclides_n construction_n and_o demostration_n serve_v in_o all_o these_o case_n and_o in_o their_o varity_n also_o construction_n demonstration_n how_o triangle_n be_v say_v to_o be_v in_o the_o self_n same_o parallel_a line_n comparison_n of_o two_o triangle_n who_o side_n be_v equal_a their_o base_n and_o angle_n at_o the_o top_n be_v unequal_a when_o they_o be_v less_o than_o two_o right_a angle_n construction_n demonstration_n three_o case_n in_o this_o proposition_n each_o of_o these_o case_n also_o may_v be_v diverse_o note_n an_o other_o addition_n of_o pelitarius_n construction_n demonstration_n this_o theorem_a the_o converse_n of_o the_o 37._o proposition_n a_o addition_n of_o flâ—ssases_n a_o addition_n of_o campanus_n construction_n demonstration_n lead_v to_o a_o absurdity_n this_o proposition_n be_v the_o converse_n of_o the_o 38._o proposition_n demonstration_n two_o case_n in_o this_o proposition_n a_o corollary_n the_o self_n same_o demonstration_n will_v serve_v if_o the_o triangle_n &_o the_o parallelogram_n be_v upon_o equal_a base_n the_o converse_n of_o this_o proposition_n an_o other_o converse_n of_o the_o same_o proposition_n comparison_n of_o a_o triangle_n and_o a_o trapesium_n be_v upon_o one_o &_o the_o self_n same_o base_a and_o in_o the_o self_n same_o parallel_a line_n construction_n demonstration_n supplement_n &_o complement_n three_o case_n in_o this_o theorem_a the_o first_o case_n this_o proposition_n call_v gnomical_a and_o mystical_a the_o converse_n of_o this_o proposition_n construction_n demonstration_n application_n of_o space_n with_o excess_n or_o want_v a_o ancient_a invention_n of_o pythagoras_n how_o a_o figure_n be_v say_v to_o be_v apply_v to_o a_o line_n three_o thing_n give_v in_o this_o proposition_n the_o converse_n of_o this_o proposition_n construction_n demonstration_n a_o addition_n of_o pelitarius_n to_o describe_v a_o square_n mechanical_o a_o addition_n of_o procâ—â—â—_n the_o converse_n thereof_o construction_n demonstration_n pythagoras_n the_o first_o inventor_n of_o this_o proposition_n a_o addition_n of_o pâ—lâ—tariâ—â—_n an_o other_o addition_n of_o pelitarius_n an_o other_o addition_n of_o pelitarius_n an_o other_o addition_n of_o pelitarius_n a_o corollary_n this_o proposition_n be_v the_o converse_n of_o the_o former_a the_o argument_n of_o the_o second_o book_n what_o be_v the_o power_n of_o a_o line_n many_o compendious_a rule_n of_o reckon_v gather_v one_o of_o this_o book_n and_o also_o many_o rule_n of_o algebra_n two_o wonderful_a proposition_n in_o this_o book_n first_o definition_n what_o a_o parallelogram_n be_v four_o kind_n of_o parallelogram_n second_o definition_n a_o proposition_n add_v by_o campane_n after_o the_o last_o proposition_n of_o the_o first_o book_n construction_n demonstration_n barlaam_n barlaam_n construction_n demonstration_n barlaam_n construction_n demonstration_n barlaam_n construction_n demonstration_n a_o corollary_n barlaam_n construction_n demonstration_n constrâ—ction_n demonstration_n construction_n demonstration_n construction_n demonstration_n many_o and_o singular_a use_n of_o this_o proposition_n this_o proposition_n can_v not_o be_v reduce_v unto_o number_n demonstration_n demonstration_n a_o corollary_n this_o proposition_n true_a in_o all_o kind_n of_o triangle_n construction_n demonstration_n the_o argument_n of_o this_o book_n the_o first_o definition_n definition_n of_o unequal_a circle_n second_o definition_n a_o contigent_a line_n three_o definition_n the_o touch_n of_o circle_n be_v 〈◊〉_d in_o one_o poâ—â—â—_n only_o circle_n may_v touch_v together_o two_o maâ—â—â—_n of_o way_n four_o definition_n five_o definition_n six_o definition_n mix_v angle_n arke_n chord_n seven_o definition_n difference_n of_o a_o angle_n of_o a_o section_n and_o of_o a_o angle_n in_o a_o section_n eight_o definition_n nine_o definition_n ten_o definition_n two_o definition_n first_o second_o why_o euclid_n define_v not_o equal_a section_n constuction_n demonstration_n lead_v to_o a_o impossibility_n correlary_a demonstration_n lead_v to_o a_o impossibility_n the_o first_o para_fw-it of_o this_o proposition_n construction_n demonstration_n the_o second_o part_n converse_v of_o the_o first_o demonstration_n demonstration_n lead_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n construction_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibility_n two_o caseâ—_n in_o this_o proposition_n construction_n the_o first_o part_n of_o this_o proposition_n demonstration_n second_o part_n three_o part_n this_o demonstrate_v by_o a_o argument_n lead_v to_o a_o impossibilie_n an_o other_o demonstration_n of_o the_o latter_a part_n of_o the_o proposition_n lead_v also_o to_o a_o impossibility_n a_o corollary_n three_o part_n an_o other_o demonstration_n of_o the_o latter_a part_n lead_v also_o to_o a_o impossibility_n this_o proposion_n be_v common_o call_v caâ—dâ—_n panonis_fw-la a_o corollary_n construction_n demonstration_n an_o other_o demonstration_n of_o the_o same_o lead_v also_o to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n of_o the_o same_o lead_v also_o to_o a_o impossibility_n construction_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n of_o the_o same_o lead_v also_o to_o a_o impossibility_n the_o same_o again_o demonstrate_v by_o a_o argument_n lead_v to_o a_o absurdititie_n demonstratiâ—_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n after_o pelitarius_n lead_v also_o to_o a_o absurdity_n of_o circle_n which_o touch_v the_o one_o the_o other_o inward_o of_o circle_n which_o touch_v the_o one_o the_o other_o outward_o an_o other_o demonstration_n after_o pelitarius_n &_o flussates_n of_o circle_n which_o touch_n the_o one_o the_o other_o outward_o of_o circle_n which_o touch_n the_o one_o the_o other_o inward_o the_o first_o part_n of_o this_o theorem_a construction_n demonstration_n demonstration_n the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o an_o other_o demonstration_n of_o the_o first_o part_n after_o campane_n construction_n demonstration_n an_o other_o demonstration_n after_o campane_n the_o first_o part_n of_o this_o theorem_a demonstration_n lead_v to_o a_o absurdity_n second_o part_n three_o part_n construction_n demonstration_n a_o addition_n of_o pelitarius_n this_o problem_n commodious_a for_o the_o inscribe_v and_o circumscribe_v of_o figure_n in_o or_o abouâ—_n circle_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n after_o orontius_n demonstration_n lead_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n the_o one_o when_o the_o angle_n set_v at_o the_o circumference_n include_v the_o centre_n demonstration_n the_o other_o when_o the_o same_o angle_n set_v at_o the_o circumference_n include_v not_o the_o centre_n construction_n demonstration_n three_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n the_o three_o case_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n a_o addition_n of_o campane_n demonstrate_v by_o pelitariâ—s_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n construction_n three_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n the_o second_o case_n the_o three_o case_n a_o addition_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n the_o converse_n of_o the_o former_a proposition_n construction_n demonstration_n construction_n demonstration_n second_o part_n thirâ—_n part_n the_o five_o and_o last_o part_n an_o other_o demonstration_n to_o prove_v that_o the_o angâ—e_n in_o a_o semicircle_n be_v a_o right_a angle_n a_o corollary_n a_o addition_n of_o pâ—litarius_n demonstration_n leaâ—ing_v to_o a_o absurdity_n a_o addition_n of_o campane_n construction_n demonstration_n two_o case_n in_o this_o proposition_n three_o case_n in_o this_o proposition_n the_o first_o case_n construction_n demonstration_n the_o second_o case_n construction_n demonstration_n the_o three_o case_n construction_n demonstration_n construction_n demonstration_n two_o case_n in_o this_o proposition_n first_o case_n demonstration_n the_o second_o câ—se_n construction_n demonstration_n three_o case_n in_o this_o proposition_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n the_o second_o case_n construction_n demonstration_n first_o corollary_n second_o corollary_n three_o corollary_n this_o proposition_n be_v the_o converse_n of_o the_o former_a construction_n demonstration_n an_o other_o demonstration_n after_o pelitarius_n the_o argument_n of_o this_o book_n first_o definition_n second_o definition_n the_o inscriptition_n and_o circumscription_n of_o rectiline_a figure_n pertain_v only_o to_o regular_a figure_n the_o three_o definition_n the_o four_o definition_n the_o five_o definition_n the_o six_o devition_n seven_o definition_n 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 peliâ—arius_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 â—ther_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 fiâ—st_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 multiplicatiâ—a_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 constitâ—ens_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 tâ—ird_o definition_n rational_a proportion_n divide_v â—â—to_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 superquadâ—ipartiens_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 propâ—rtionalitie_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 â—n_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 twelfâ—h_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 fiâ—tâ—ne_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 differâ—câ—_n of_o the_o first_o part_n demonstratiâ—_n of_o tâ—e_z same_o first_o difference_n second_o difference_n three_o difference_n the_o second_o part_n â—f_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 constrâ—ction_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 demonstrationâ—eâ—aing_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 thâ—r_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 aâ—cdeâ—gh_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 be_v the_o converse_n of_o the_o first_o construction_n demonstration_n this_o be_v the_o converse_n of_o the_o former_a proposition_n construction_n demonstration_n construction_n the_o first_o part_n of_o this_o proposition_n demonstration_n lead_v to_o a_o impossibility_n the_o second_o part_n of_o this_o proposition_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o corollary_n out_o of_o flussates_n by_o this_o and_o the_o former_a proposition_n may_v a_o right_a line_n be_v divide_v into_o what_o part_n soever_o you_o will._n construction_n demonstration_n an_o other_o way_n after_o pelitarius_n a_o otâ—eâ—_n way_n after_o pelitarius_n construction_n demonstration_n an_o other_o way_n after_o campane_n construction_n demonstration_n a_o proposition_n add_v by_o pelitarius_n the_o
first_o part_n of_o this_o proposition_n demonstration_n of_o the_o of_o the_o same_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o the_o first_o parâ—_n of_o this_o proposition_n demonstration_n of_o the_o same_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o demonstration_n of_o the_o first_o part_n the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o the_o first_o part_n of_o thâ—â—_n theorem_a the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o a_o corollary_n description_n of_o the_o rectiline_a figure_n require_v demonstration_n demonstration_n a_o corollary_n the_o first_o parâ—_n of_o this_o theorem_a the_o second_o part_n demonstrate_v the_o three_o part_n the_o first_o corollary_n the_o second_o corollary_n demonstration_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o first_o note_v that_o this_o be_v prove_v in_o the_o assumpt_n follow_v a_o assumpt_n an_o other_o demonstration_n of_o the_o second_o part_n after_o flussates_n an_o other_o demonstration_n after_o flussate_n demonstration_n of_o this_o proposition_n wherein_o be_v first_o prove_v that_o the_o parallegramme_n eglantine_n be_v like_a to_o the_o whole_a parallelogram_n abcd._n that_o the_o parallelogram_n kh_o be_v like_a to_o the_o whole_a parallelogram_n abcd_o that_o the_o parallelogram_n eglantine_n and_o kh_o be_v like_o the_o one_o to_o the_o other_o an_o other_o demonstration_n after_o flussates_n a_o addition_n of_o pelitarius_n another_o addition_n of_o pelitarius_n construction_n demonstration_n demonstration_n demonstration_n by_o the_o dimetient_fw-la be_v understand_v here_o the_o dimetient_fw-la which_o be_v draw_v from_o the_o angle_n which_o be_v common_a to_o they_o both_o to_o the_o opposite_a angle_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o way_n after_o flussates_n in_o this_o proposition_n be_v two_o case_n in_o the_o first_o the_o parallelogram_n compare_v to_o the_o parallelogram_n describe_v of_o the_o half_a line_n be_v describe_v upon_o a_o line_n great_a they_o the_o half_a line_n in_o the_o second_o upon_o a_o line_n less_o the_o first_o case_n where_o the_o parellelogramme_n compare_v namely_o of_o be_v describe_v upon_o the_o line_n ak_o which_o be_v great_a than_o the_o half_a line_n ac_fw-la demonstration_n of_o this_o case_n the_o second_o case_n where_o the_o parallelogram_n compare_v namely_o ae_n be_v describe_v upon_o the_o line_n ad_fw-la which_o be_v less_o than_o the_o line_n ac_fw-la demonstration_n of_o the_o second_o case_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n a_o corollary_n add_v by_o flussates_n and_o be_v put_v of_o theon_n as_o a_o assumpt_n beâ—ore_o the_o 17._o proposition_n of_o the_o ten_o book_n which_o â—or_a that_o it_o follow_v of_o this_o proposition_n i_o think_v it_o not_o amiss_o here_o to_o place_n construction_n demonstration_n construction_n demonstration_n an_o other_o way_n construction_n demonstration_n the_o converse_n of_o the_o former_a proposition_n demonstration_n that_o the_o angle_n at_o thâ—_z centre_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o circumference_n whereon_o they_o be_v that_o the_o angle_n at_o the_o circumference_n be_v so_o also_o that_o the_o sector_n be_v so_o also_o construction_n of_o the_o problem_n demonstartion_n of_o the_o same_o the_o first_o corollary_n the_o second_o corollary_n the_o three_o corollary_n demonstration_n of_o this_o proposition_n demonstration_n of_o this_o proposition_n demonstration_n of_o this_o proposition_n demonstration_n of_o the_o first_o part_n of_o this_o proposition_n demonstration_n of_o the_o second_o part_n why_o euclid_n in_o the_o midst_n of_o his_o work_n be_v compel_v to_o add_v these_o three_o book_n of_o number_n arithmetic_n of_o more_o excellency_n than_o geometry_n thing_n intellectual_a of_o more_o worthiness_n theâ—_z thing_n sensible_a arithmetic_n minister_v prinâ—ciples_n and_o ground_n in_o a_o manner_n to_o all_o science_n boetius_fw-la cap._n 2._o lib._n prim_fw-la arithmeti_n timaus_n the_o argument_n of_o the_o seven_o book_n the_o first_o definition_n without_o unity_n shall_v be_v confusion_n of_o thing_n â—oetius_fw-la in_o his_o book_n dâ—_z unitate_fw-la &_o uno_fw-la an_o other_o desinition_n of_o unity_n the_o second_o definition_n difference_n between_o a_o point_n and_o unity_n boetius_fw-la an_o other_o desinition_n of_o number_n jordane_n an_o other_o definition_n of_o number_n unity_n have_v in_o it_o the_o virtue_n and_o power_n of_o all_o number_n number_n consider_v three_o manner_n of_o wayâ—_n the_o three_o definition_n the_o four_o definition_n the_o five_o definition_n the_o six_o definition_n boetius_fw-la an_o other_o definition_n of_o a_o even_a number_n note_n pithagoraâ—_n an_o other_o definition_n an_o other_o definition_n an_o other_o definition_n the_o seven_o definition_n an_o other_o definition_n of_o a_o odd_a number_n an_o other_o definition_n the_o eight_o definition_n campane_n an_o other_o definition_n of_o a_o even_o even_o number_n flussates_n an_o other_o definition_n boetius_fw-la an_o other_o definition_n the_o nine_o definition_n campane_n an_o other_o definition_n flussates_n an_o other_o definition_n an_o other_o definition_n the_o ten_o definition_n this_o definition_n not_o find_v in_o the_o greek_n an_o other_o definition_n boeâ—ius_n definition_n of_o a_o number_n even_o even_o and_o even_o â—d_a the_o eleven_o definition_n flusâ—ates_n an_o other_o definition_n the_o twelve_o definition_n prime_n number_n call_v incomposed_a number_n the_o thirteen_o definition_n the_o fourteen_o definition_n the_o fifteen_o definition_n the_o sixteen_o definition_n two_o number_n require_v in_o multiplication_n the_o seventeen_o definition_n why_o they_o be_v call_v superficial_a number_n the_o eighteen_o definition_n why_o they_o be_v call_v solid_a number_n the_o nineteen_o definition_n why_o it_o be_v call_v a_o square_a number_n the_o twenty_o definition_n why_o it_o be_v call_v a_o cube_fw-la number_n the_o twenty_o one_o definition_n why_o the_o definition_n of_o proportional_a magnitude_n be_v unlike_a to_o the_o definitio_fw-la of_o proportional_a number_n the_o twenty_o two_o definition_n the_o twenty_o three_o definition_n perfect_a number_n rare_a &_o of_o great_a use_n in_o magic_a &_o in_o secret_a philosophy_n in_o what_o respect_n a_o number_n be_v perfect_a two_o kind_n of_o imperfect_a number_n a_o â—â—mber_n wanâ—â—ngâ—_n common_a sentence_n â—irst_n common_a sentence_n â—â—cond_v â—ommon_a sentence_n three_o common_a sentence_n forth_o common_a sentence_n â—iâ—th_o common_a sentence_n six_o common_a sentence_n seven_o comâ—mon_a sentence_n constrâ—ctioâ—_n demonstratiâ—â—_n lead_v to_o a_o absurdity_n the_o converse_n of_o â—his_n proposition_n how_o to_o â—now_v whether_o two_o number_n give_v be_v prime_a the_o one_o to_o the_o other_o two_o case_n in_o this_o problem_n the_o first_o case_n the_o second_o case_n demonstration_n of_o the_o second_o case_n that_o cf_o be_v a_o common_a measure_n to_o the_o number_n ab_fw-la and_o cd_o that_o cf_o be_v the_o great_a common_a measure_n to_o ab_fw-la and_o cd_o the_o second_o case_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n this_o proposition_n and_o the_o 6._o proposition_n in_o discrete_a quantity_n answer_v to_o the_o first_o of_o the_o five_o in_o continual_a quantity_n demonstration_n construction_n demonstration_n thiâ—_n proposition_n and_o the_o next_o follow_v in_o discreet_a quantity_n answer_v to_o the_o five_o proposition_n of_o the_o five_o book_n in_o continual_a quantity_n construction_n demonstration_n constuâ—ction_n demonstration_n an_o other_o demonstration_n after_o flussates_n construction_n demonstration_n construction_n demonstration_n this_o proposition_n iâ—_z discreet_a quantity_n answer_v to_o the_o nine_o proposition_n of_o the_o five_o book_n in_o continual_a quantity_n demonstration_n this_o in_o discreet_a quantity_n answer_v to_o the_o twelve_o proposition_n of_o the_o five_o in_o continual_a quantity_n demonstration_n this_o in_o discrete_a quantity_n answer_v to_o the_o sixteen_o proposition_n of_o the_o five_o book_n in_o continual_a quantity_n note_n this_o in_o discrete_a quantity_n answer_v to_o tâ—â—_n twenty_o one_o proposition_n oâ—_n the_o five_o book_n in_o continual_a quantity_n demonstration_n certain_a addition_n of_o campane_n the_o second_o case_n proportionality_n divide_v proportionality_o compose_v euerse_a proportionality_n the_o conversâ—_n of_o the_o same_o prâ—position_n demonstration_n a_o corollary_n follow_v thâ—se_a proposition_n adâ—ed_v by_o campane_n coâ—strâ—ctioâ—_n demonstration_n demonstration_n â—emonstraâ—ion_n a_o corollary_n add_v by_o flussâ—tes_n demonstration_n this_o proposition_n and_o the_o former_a may_v be_v extend_v to_o number_n how_o many_o soever_o the_o second_o part_n of_o this_o proposition_n which_o be_v the_o converse_n of_o the_o first_o demonstration_n a_o assumpt_v add_v by_o campane_n this_o proposition_n in_o number_n demonstrate_v that_o which_o the_o 17._o of_o the_o six_o demonstrate_v in_o line_n demonstration_n the_o second_o part_n which_o be_v the_o converse_n of_o the_o
first_o demonstration_n demonstration_n lead_v to_o a_o impossibility_n this_o proposition_n in_o discreet_a quantity_n answer_v to_o the_o 23._o proposition_n of_o the_o five_o book_n in_o continual_a quantity_n this_o and_o the_o eleven_o proposition_n follow_v declare_v the_o passion_n and_o property_n ofâ—_n prime_a number_n demonstration_n lead_v to_o a_o impossibility_n this_o be_v the_o converse_n of_o the_o former_a proposition_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n demonstration_n demonstration_n demonstration_n of_o the_o first_o part_n lead_v to_o a_o absurdity_n demonstration_n of_o the_o second_o part_n which_o be_v the_o conâ—câ—se_n of_o the_o first_o lean_v also_o to_o a_o absurdity_n demonstrasion_n lead_v to_o a_o absurdity_n demonstrasion_n a_o corollary_n â—â—ded_v by_o campave_n demonstration_n lâ—ading_v to_o a_o impossibility_n an_o other_o demonstration_n demonstration_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n add_v by_o campane_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o absurdity_n the_o second_o caseâ—_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n leaâ—iâ—g_v â—o_o a_o absurdity_n the_o second_o case_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n demonstration_n the_o converse_n of_o the_o former_a proposition_n demonstration_n construction_n demonstration_n leâ—ding_v to_o a_o absurdity_n a_o corollary_n adâ—ed_v by_o campane_n how_o to_o â—inde_v out_o the_o second_o least_o number_n and_o the_o three_o and_o so_o â—orth_o infinite_o how_o to_o siâ—â—_n out_o the_o least_o â—â—mâ—_n a_o conâ—ayâ—â—g_n â—â—e_z paâ—â—s_n of_o part_n the_o arguâ—â—â—â—_n of_o the_o eight_o book_n demonstration_n lead_v to_o a_o absurdity_n construction_n demonstration_n this_o proposition_n be_v the_o â—â—uerse_a of_o the_o first_o demonstrationâ—_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o absurdity_n the_o second_o case_n demonstration_n this_o proposition_n in_o number_n answer_v to_o the_o of_o the_o six_o touch_v parellelogramme_n construction_n demonstration_n an_o other_o demonstration_n after_o campane_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n a_o corollary_n add_v by_o flussates_n construction_n demonstration_n this_o proposition_n be_v the_o converse_n of_o the_o former_a construction_n demonstration_n the_o first_o part_n of_o this_o proposition_n demonstrate_v the_o second_o part_n demonstrate_v construction_n the_o first_o part_n of_o this_o prâ—position_n demonstrate_v the_o second_o part_n demonstrate_v construction_n demonstration_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o a_o negative_a proportion_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o a_o negative_a proposition_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o demonstration_n of_o the_o fiâ—st_a part_n of_o this_o proposition_n demonstration_n of_o the_o second_o part_n demonstration_n of_o the_o first_o part_n of_o this_o proposition_n the_o second_o part_n this_o proposition_n be_v the_o converse_n of_o the_o 18._o proposition_n construction_n demonstration_n this_o proposition_n be_v the_o converse_n of_o the_o 19_o proposition_n construction_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n a_o corollary_n add_v by_o flussates_n construction_n construction_n demonstration_n a_o corollary_n add_v by_o flussates_n another_o corollary_n add_v by_o flussates_n the_o argument_n of_o the_o niâ—th_o book_n demonstration_n this_o proposition_n be_v the_o converse_n oâ—_n tâ—e_z formâ—â—_n demonstration_n a_o corollary_n aâ—ded_v by_o campane_n demonstration_n demonstration_n demonstration_n a_o corollary_n add_v by_o campane_n demonstration_n demonstration_n demonstration_n of_o the_o first_o part_n the_o second_o part_n demonstrate_v demostration_n of_o the_o three_o part_n demostration_n of_o the_o first_o part_n of_o this_o proposition_n the_o second_o pâ—rt_n demonstrate_v demonstration_n of_o the_o first_o part_n leave_v to_o a_o absurdity_n demonstration_n of_o the_o â—â—cond_a pâ—â—â—_n lead_v alâ—o_o to_o a_o absurdity_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o demonstration_n aâ—ter_n flussates_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o demonstration_n after_o campane_n demonstration_n lead_v to_o a_o absurdity_n a_o proposition_n add_v by_o campane_n construction_n demonstration_n demonstration_n to_o prove_v that_o the_o number_n a_o and_o c_o be_v prime_a to_o b._n demonstratiou_n this_o proposition_n be_v the_o converse_n of_o the_o former_a demonstration_n this_o answer_v to_o the_o 2._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 3._o of_o the_o three_o demonstration_n this_o answer_v to_o thâ—_z 4._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 5._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 6._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 7._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 8._o of_o the_o second_o demonstratition_n this_o answer_v to_o thâ—_z 9_o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 10._o oâ—_n the_o second_o demonstration_n a_o negative_a proposition_n demonstration_n leaâ—ing_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o abjurditie_n three_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n the_o three_o case_n divert_v case_n â—n_v this_o proposition_n the_o first_o case_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n a_o proposition_n add_v by_o campane_n a_o other_o add_v by_o he_o demonstration_n lead_v to_o a_o absurdity_n demonstration_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o demonstration_n demonstration_n demonstration_n this_o proposition_n teachâ—th_v how_o to_o find_v out_o a_o perfect_a number_n construction_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n the_o argument_n of_o the_o ten_o book_n difference_n between_o number_n and_o magnitude_n a_o line_n be_v not_o make_v of_o point_n as_o number_n be_v make_v of_o unity_n this_o book_n the_o hard_a to_o understand_v of_o all_o the_o book_n of_o euclid_n in_o this_o book_n be_v entreat_v of_o a_o strange_a manner_n of_o matter_n then_o in_o the_o former_a many_o even_o of_o the_o well_o learn_v have_v think_v that_o this_o book_n can_v not_o well_o be_v understand_v without_o algebra_n the_o nine_o former_a book_n &_o the_o principle_n of_o this_o â—ooke_n well_o understand_v this_o book_n will_v not_o be_v hard_a to_o understand_v the_o fâ—rst_a definition_n the_o second_o definition_n contraries_n make_v manifest_a by_o the_o compare_v of_o the_o one_o to_o the_o other_o the_o third_o definition_n what_o the_o power_n of_o a_o line_n be_v the_o four_o definition_n unto_o the_o suppose_a line_n first_o set_v may_v be_v compare_v infinite_a line_n why_o some_o mislike_n that_o the_o line_n first_o set_v shall_v be_v call_v a_o rational_a line_n flussates_n call_v this_o line_n a_o line_n certain_a this_o rational_a line_n the_o ground_n in_o a_o manner_n of_o all_o the_o proposition_n in_o this_o ten_o book_n note_n the_o line_n rational_a of_o purpose_n the_o six_o definition_n campâ—nus_n â—ath_z cause_v much_o obscurity_n in_o this_o ten_o book_n the_o seven_o definition_n flussates_n in_o steed_n of_o this_o word_n irrational_a use_v this_o word_n uncertain_a why_o they_o be_v call_v irrational_a line_n the_o cause_n of_o the_o obscurity_n and_o confusedness_n in_o this_o book_n the_o eight_o definition_n the_o nine_o definition_n the_o ten_o definition_n the_o eleven_o definition_n construction_n demonstration_n a_o corollary_n construction_n demonstration_n this_o proposition_n teach_v that_o incontinuall_a quantity_n which_o the_o first_o of_o the_o seven_o teach_v in_o discrete_a quantity_n construction_n demonstration_n lead_v to_o a_o absurdity_n two_o case_n in_o this_o proposition_n the_o first_o case_n this_o proposition_n teach_v that_o in_o continual_a quantity_n which_o the_o 2._o of_o the_o sâ—â—ith_v teach_v in_o number_n the_o second_o case_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n this_o problem_n reduce_v to_o a_o theorem_a this_o proposition_n teach_v that_o in_o continual_a quantity_n which_o the_o 3._o of_o the_o second_o teach_v in_o number_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o absurdity_n the_o second_o case_n a_o leâ—ma_n necessary_a
to_o be_v prâ—â—â—d_v beâ—oâ—e_n 〈◊〉_d â—all_n to_o the_o demonstration_n construction_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n this_o problem_n reduce_v to_o a_o theorem_a construction_n demonstration_n how_o magnitude_n be_v say_v to_o be_v in_o proportion_n the_o onâ—_n to_o the_o other_o as_o number_n be_v to_o number_v this_o proposition_n be_v the_o converse_n of_o forme_a construction_n demonstration_n a_o corollary_n construction_n demonstration_n construction_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n this_o be_v the_o 〈…〉_o demonstration_n the_o first_o part_n demonstrate_v an_o other_o demonstration_n of_o the_o first_o part_n an_o other_o demonstration_n oâ—_n the_o same_o first_o part_n after_o montaureus_n demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o former_a an_o other_o demonstration_n of_o the_o second_o part_n this_o assumpt_n follow_v as_o a_o corollary_n of_o the_o 25_o but_o so_o as_o it_o may_v also_o be_v here_o in_o method_n place_v you_o shall_v â—inde_v it_o after_o the_o 53._o of_o this_o book_n absolute_o demonstrate_v for_o there_o it_o serve_v to_o the_o 54._o his_o demonstration_n demonstration_n of_o the_o three_o part_n demonstration_n of_o the_o four_o part_n which_o be_v the_o converse_n of_o the_o â—_o conclusion_n of_o the_o whole_a proposition_n a_o corollary_n proâ—e_v of_o the_o first_o part_n of_o the_o corollary_n proof_n of_o the_o second_o part_n proof_n of_o the_o three_o pâ—rt_n proâ—e_a oâ—_n the_o four_o part_n certain_a annotation_n â—ut_z of_o montauâ—â—us_fw-la rule_v to_o know_v whether_o two_o superficial_a number_n be_v like_a or_o no._n this_o assumpt_n be_v the_o converse_n of_o the_o 26._o of_o the_o eight_o demonstration_n oâ—_n the_o first_o part_n demonstration_n of_o the_o second_o partâ—_n a_o corollary_n to_o find_v out_o the_o first_o line_n incommensurable_a in_o length_n only_o to_o the_o line_n give_v to_o find_v out_o the_o second_o line_n incommensurable_a both_o in_o length_n and_o in_o power_n to_o the_o line_n give_v construction_n demonstration_n tâ—is_n be_v wiâ—h_n zambert_n a_o aâ—â—â—mpt_n but_o uâ—â—eâ—ly_o improper_o â—lâ—ssateâ—_n mane_v iâ—_z a_o corollary_n but_o the_o greeâ—e_n and_o montaureus_n maâ—e_n it_o a_o proposition_n but_o every_o way_n a_o â—nfallible_a truth_n 〈…〉_o demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n a_o corollary_n demonstration_n an_o other_o way_n to_o prove_v that_o the_o line_n a_o e_o c_o f_o be_v proportional_a demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o paâ—t_n which_o be_v the_o converse_n of_o the_o first_o a_o corollary_n demonstration_n of_o the_o first_o part_n by_o a_o argument_n leadindg_fw-mi to_o a_o absurdity_n demonstration_n of_o the_o second_o paâ—t_n lead_v also_o to_o a_o impossibility_n and_o this_o second_o part_n be_v the_o converse_n of_o the_o first_o demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o how_o to_o divide_v the_o line_n bc_o ready_o in_o such_o sort_n as_o iâ—_z require_v in_o the_o proposition_n demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o tâ—e_z former_a a_o other_o demonstrationâ—y_v a_o argument_n lead_v to_o a_o absurdity_n a_o assumpt_n a_o corollary_n add_v by_o montaureuâ—_n cause_n cause_o of_o increase_v the_o difficulty_n of_o this_o book_n note_n construction_n demonstration_n diverse_a caâ—es_n in_o this_o proposition_n the_o second_o case_n the_o first_o kind_n of_o rational_a line_n commensurable_a in_o length_n this_o particle_n in_o the_o proposition_n according_a to_o any_o of_o the_o foresay_a way_n be_v not_o in_o vain_a put_v the_o second_o kind_n of_o rational_a line_n commensurable_a in_o length_n the_o three_o case_n the_o three_o kind_n of_o rational_a line_n commensurable_a in_o length_n the_o four_o case_n this_o proposition_n be_v the_o converse_n of_o the_o former_a proposition_n construction_n demonstration_n a_o assumpt_n construction_n demonstration_n definition_n of_o a_o medial_a line_n a_o corollary_n this_o assumpt_n be_v nothing_o else_o but_o a_o part_n of_o the_o first_o proposition_n of_o the_o six_o book_n 〈◊〉_d how_o a_o square_n be_v say_v to_o be_v apply_v upon_o a_o line_n construction_n demonstration_n construction_n demonstration_n note_n a_o corollary_n construction_n demonstration_n lead_v to_o a_o absurdity_n construction_n demonstration_n construction_n demonstration_n demonstration_n a_o corollary_n to_o find_v out_o two_o square_a number_n exceed_v the_o one_o the_o other_o by_o a_o square_a â—umber_n a_o assumpt_n construction_n demonstration_n montaureus_n make_v this_o a_o assumpt_n as_o the_o grecke_n text_n seem_v to_o do_v likewise_o but_o without_o a_o cause_n construction_n demonstration_n this_o assumpt_n set_v foâ—th_o nothing_o â—_z but_o that_o which_o the_o first_o oâ—_n the_o sâ—â—t_n jet_v â—orth_o and_o therefore_o in_o sâ—me_a examplar_n it_o be_v not_o find_v construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o corollary_n i_o dee_n dee_n the_o second_o corollary_n corollary_n therefore_o if_o you_o divide_v the_o square_n of_o the_o side_n ac_fw-la by_o the_o side_n bc_o the_o portion_n dc_o will_v be_v the_o product_n etc_n etc_n as_o in_o the_o former_a corollary_n i_o dâ—e_n dâ—e_n the_o third_o corollary_n corollary_n therefore_o if_o the_o parallelogram_n of_o basilius_n and_o ac_fw-la be_v divide_v by_o bc_o the_o product_n will_v geve_v the_o perpendicular_a d_o a._n these_o three_o corollarye_n in_o practice_n logistical_a and_o geometrical_a be_v profitable_a an_o other_o demonstration_n of_o this_o four_o part_n of_o the_o determination_n a_o assumpt_n construction_n demonstration_n the_o first_o part_n of_o the_o determination_n conclude_v the_o second_o part_n conclude_v the_o total_a conclusion_n construction_n demonstration_n the_o first_o part_n of_o the_o determination_n conclude_v the_o second_o part_n conclude_v the_o total_a conclusion_n construction_n demonstration_n the_o first_o part_n conclude_v the_o second_o part_n conclude_v the_o three_o part_n conclude_v the_o total_a conclusion_n the_o first_o senary_a by_o composition_n definition_n of_o a_o binomial_a line_n six_o kind_n of_o binomial_a line_n demonstration_n definition_n of_o a_o first_o bimedial_a line_n construction_n demonstration_n definition_n of_o a_o second_o bimedial_a line_n demonstration_n definition_n of_o a_o great_a line_n definition_n of_o a_o line_n who_o power_n be_v rational_a and_o medial_a definition_n of_o a_o liâ—e_z contain_v in_o power_n two_o medial_n a_o assumpt_n the_o second_o senary_a by_o composition_n demonstration_n lead_v to_o a_o impossibility_n a_o corollary_n demonstration_n lead_v to_o a_o impossibilâ—â—e_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibiâ—â—â—e_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n lead_v to_o a_o absurdity_n six_o kind_n of_o binomial_a line_n a_o binomial_a line_n consi_v of_o two_o paâ—tâ—s_n firsâ—_o definition_n seconâ—_o definition_n three_o â—â—â—â—â—â—ition_n four_o definition_n five_o definition_n six_o definition_n the_o three_o senary_a by_o composition_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o corollary_n add_v by_o flussates_n m._n do_v you_o his_o book_n call_v tyâ—â—câ—niâ—m_fw-la mathematicum_fw-la this_o assumpt_n as_o be_v before_o note_v follow_v most_o â—riâ—fly_o without_o far_a demonstration_n of_o the_o 25._o of_o this_o book_n demonstration_n a_o assumpt_n the_o four_o senary_a by_o composition_n construction_n demonstration_n the_o first_o part_n of_o this_o demonstration_n conclude_v the_o second_o part_n of_o the_o demonstration_n conclude_v the_o three_o part_n conclude_v the_o total_a conclusion_n demonstration_n the_o first_o part_n of_o this_o demonstration_n conclude_v the_o three_o part_n conclude_v the_o four_o part_n concludeâ—_n the_o five_o part_n conclude_v the_o total_a conclusion_n demonstration_n construction_n demonstration_n demonstration_n demonstration_n demonstration_n look_v after_o the_o assumpt_n conclude_v at_o this_o mark_n for_o plain_a open_n of_o this_o place_n the_o use_n of_o this_o assumpt_n be_v in_o the_o next_o proposition_n &_o other_o follow_v the_o five_o senary_a by_o composition_n construction_n demonstration_n conclude_v that_o dg_o be_v a_o binomial_a line_n construction_n demonstration_n conclude_v that_o dg_o be_v a_o binomial_a line_n construction_n demonstration_n †_o *_o ‡_o dg_o conclude_v a_o binomial_a line_n a_o corollary_n add_v by_o m._n dee_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n the_o six_o senary_n construction_n demonstration_n construction_n demonstration_n a_o corollary_n add_v by_o flussetes_n note_n construction_n demonstration_n an_o other_o demonstration_n after_o p._n montaureus_n an_o other_o demonstration_n after_o campane_n construction_n demonstration_n an_o other_o demonstration_n afâ—â—r_v campane_n construction_n demonstration_n a_o assumpt_n an_o other_o demonstration_n after_o campanâ—_n note_n
second_o part_n of_o the_o first_o case_n the_o second_o case_n first_o part_n of_o the_o second_o case_n second_o part_n of_o the_o second_o case_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o first_o part_n of_o the_o first_o case_n 〈◊〉_d second_o 〈◊〉_d of_o the_o 〈◊〉_d case_n the_o second_o case_n a_o corollary_n the_o first_o senary_a by_o substraction_n demonstration_n an_o other_o demonstration_n after_o campane_n definition_n of_o the_o eight_o irrational_a line_n definition_n of_o 〈◊〉_d â—inth_o irrational_a line_n an_o other_o demonstration_n after_o campane_n construction_n demonstration_n definition_n of_o the_o ten_o irrational_a line_n definition_n of_o the_o eleven_o irrational_a line_n â—â—â—â—iâ—ition_n of_o the_o twelve_o irraâ—ionall_a line_n definition_n of_o the_o thirteen_o and_o last_o irrational_a line_n a_o assumpt_n of_o campane_n i_o dee_n though_o campane_n lemma_n be_v true_a yeâ—_z the_o manner_n of_o demonstrate_v it_o narrow_o consider_v be_v not_o artificial_a second_o senary_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o absurdity_n construction_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n 〈◊〉_d a_o abjurdâ—tâ—â—â—_n six_o kind_n of_o reâ—iduall_a line_n first_o definition_n second_o definition_n three_o definition_n four_o definition_n five_o definition_n six_o definition_n three_o senary_n construction_n demonstration_n construction_n demoâ—stratiâ—â—_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o other_o more_o ready_a way_n to_o find_v out_o the_o six_o residual_a line_n four_o senary_n the_o â—irst_n parâ—_n of_o the_o construction_n the_o first_o part_n of_o the_o demonstration_n note_n a_z and_o fk_o conclude_v rational_a parallelogram_n note_n dh_o and_o fk_o parallelogram_n medial_a second_o part_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n ln_o be_v the_o only_a liâ—e_z â—hat_n we_o seek_v &_o consider_v first_o part_n of_o the_o construction_n the_o first_o part_n of_o thâ—_z demonstration_n a_z and_o fk_o conclude_v parallelogram_n medial_a dh_o &_o eke_o rational_a the_o second_o part_n of_o the_o construction_n the_o second_o part_n of_o the_o demonstration_n demonstration_n analytical_o the_o proâ—e_n hereof_o follow_v amongâ—_n other_o thing_n the_o line_n ln_o find_v which_o be_v the_o principal_a drift_n of_o all_o the_o former_a discourse_n the_o first_o part_n of_o the_o construction_n the_o fiâ—st_a part_n of_o the_o demonstration_n note_n a_z and_o fk_o medial_a note_n dh_o and_o eke_o medial_a note_n ai_fw-fr incommensurable_a to_o eke_o second_o part_n of_o the_o construction_n the_o principal_a line_n ln_o find_v find_v because_o the_o line_n of_o and_o â—g_z be_v prove_v commensurable_a in_o length_n length_n by_o the_o first_o oâ—_n the_o six_o and_o ten_o of_o the_o ten_o the_o first_o part_n of_o the_o construction_n the_o first_o part_n of_o the_o demonstration_n note_n ak_o rational_a note_n dk_o medial_a a_z and_o fk_o incommensurable_a the_o second_o part_n of_o the_o construction_n the_o second_o part_n of_o the_o demonstration_n ln_o the_o chief_a line_n of_o this_o theorem_a find_v demonstration_n the_o line_n ln_o demonstration_n the_o five_o senary_a these_o six_o proposition_n follow_v be_v the_o conuâ—rses_n of_o the_o six_o former_a proposition_n construction_n demonstration_n demonstration_n by_o the_o 20._o of_o the_o ten_o ten_o by_o the_o 21._o of_o the_o ten_o ten_o by_o the_o 22._o of_o the_o ten_o â—f_o conclude_v a_o residual_a line_n construction_n demonstration_n cf_o conclude_v a_o residual_a line_n construction_n demonstration_n cf_o conclude_v a_o residual_a line_n construction_n demonstration_n cf_o prove_v a_o residual_a line_n cf_o prove_v a_o residual_a line_n construction_n demonstration_n cf_o â—roved_v â—_o residual_a the_o six_o senary_n construction_n demonstration_n cd_o conclude_v a_o residual_a line_n note_n construction_n demonstration_n cd_o prove_v a_o medial_a construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n seven_o senary_n constraction_n demonstration_n construction_n demonstration_n demonstration_n construction_n demonstration_n on_o lead_v to_o a_o impossibility_n a_o corollary_n the_o determination_n have_v sundry_a part_n orderly_a to_o be_v prove_v construction_n demonstration_n this_o be_v a_o assumpt_n problematical_a artificial_o use_v and_o demonstrate_v demonstrate_v therefore_o those_o three_o line_n be_v in_o continual_a proportion_n fe_o conclude_v a_o residual_a liâ—â—_n which_o be_v sumwhat_o prepâ—â—icroâ—sly_o in_o respect_n oâ—_n the_o â—â—der_n propound_v both_o in_o the_o proposition_n and_o also_o in_o the_o determination_n construction_n demonstration_n construction_n demonstration_n here_o be_v the_o â—ower_a part_n of_o the_o propositiâ—_n more_o orderly_a huddle_v theâ—_z in_o the_o former_a demöstration_n construction_n demonstration_n a_o assumpt_n an_o other_o demonstration_n after_o flussas_n construction_n demonstration_n this_o be_v in_o a_o manner_n the_o converse_n of_o both_o the_o former_a proposition_n joint_o construction_n demonstration_n construction_n demonstration_n demonstration_n an_o other_o demonstration_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n lead_v to_o a_o impossibility_n the_o argument_n of_o the_o eleven_o book_n a_o point_n the_o beginning_n of_o all_o quantity_n continual_a the_o method_n use_v by_o euclid_n in_o the_o ten_o â—â—â—mer_a boot_n â—irst_n boâ—â—e_n second_o â—â—oâ—e_n three_o booâ—e_n forth_o bâ—oâ—e_n â—iveth_v boâ—â—e_n six_o booâ—e_n seven_o bookâ—_n â—ight_a booâ—â—_n nine_o book_n ten_o booâ—e_n what_o be_v entreaâ—ea_n of_o in_o the_o fiâ—e_a boot_n follow_v 〈◊〉_d â—â—â—ular_a bodiesâ—_n the_o â—â—all_a end_n 〈…〉_o oâ—_n i_o uâ—â—â—â—es_v â—eomeâ—â—â—all_a element_n comparison_n â—â—_o the_o 〈◊〉_d â—â—oâ—e_n and_o 〈◊〉_d book_n 〈◊〉_d first_o definition_n a_o solid_a the_o most_o perfect_a quantity_n no_o science_n of_o thing_n infinite_a second_o definition_n three_o definition_n two_o difâ—initions_n include_v in_o this_o definition_n declaration_n of_o the_o first_o part_n declaration_n of_o the_o second_o part_n four_o definition_n five_o definition_n six_o definition_n seven_o defâ—inition_n eight_o definition_n nine_o dissilition_n ten_o definition_n eleven_o definition_n an_o other_o definition_n of_o a_o prism_n which_o be_v a_o special_a definition_n of_o a_o prism_n as_o it_o be_v common_o call_v and_o use_v this_o body_n call_v figura_fw-la serratilis_fw-la psellus_n twelve_o definition_n what_o be_v to_o be_v taâ—â—n_v heed_n of_o in_o the_o definition_n of_o a_o sphere_n give_v by_o johannes_n de_fw-fr sacro_fw-la busco_n theodosiuâ—_n definition_n of_o a_o sphere_n the_o circumference_n of_o a_o sphere_n galens_n definition_n 〈◊〉_d a_o sphâ—râ—_n the_o dignity_n of_o a_o sphere_n a_o sphere_n call_v a_o globe_n thirteen_o definition_n theodosius_n definition_n of_o the_o axe_n of_o a_o sphere_n fourtenth_n definition_n theodosius_n definition_n of_o the_o centre_n of_o a_o sphere_n flussas_n definition_n of_o the_o centre_n of_o a_o sphere_n fivetenth_fw-mi definition_n difference_n between_o the_o diameter_n &_o axe_n of_o a_o sphere_n sevententh_n definition_n first_o kind_n of_o cones_n a_o cone_n call_v of_o campane_n a_o roâ—â—de_a pyramid_n sevententh_n definition_n a_o conical_a superficies_n eightenth_n definition_n ninetenth_fw-mi definition_n a_o cillindricall_a superficies_n corollary_n a_o roundâ—_n column_n or_o sphere_n a_o corollary_n add_v by_o campane_n twenty_o definition_n twenty_o one_o definition_n twenty_o two_o definition_n a_o tetrahedron_n one_o of_o the_o five_o regular_a body_n difference_n between_o a_o tetrahedron_n and_o a_o pyramid_n psellus_n call_v a_o tetrahedron_n a_o pyramid_n twenty_o three_o definition_n twenty_o â—oâ—er_v definition_n twenty_o five_o definition_n five_o regular_a body_n the_o dignity_n of_o these_o body_n a_o tetrahedron_n ascribe_v unto_o the_o fire_n a_o octohedron_n ascribe_v unto_o the_o air_n a_o ikosahedron_n assign_v unto_o the_o water_n a_o cube_fw-la assign_v unto_o the_o earth_n a_o dodecahedron_n assign_v to_o heaven_n definition_n of_o a_o parallelipipedon_n a_o dâ—dâ—â—â—â—edron_n a_o icosaâ—edron_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n after_o flussas_n construction_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n construction_n construction_n a_o assumpt_n as_o m._n dee_n prove_v it_o demonstration_n demonstration_n lead_v to_o a_o impossibility_n this_o proposition_n be_v as_o it_o be_v the_o converse_n of_o the_o six_o construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n john_n dee_n dee_n this_o require_v the_o imagination_n of_o a_o plain_a superficies_n pass_v by_o the_o point_n a_o and_o the_o straight_a line_n bc._n and_o so_o help_v yourself_o in_o the_o like_a case_n either_o mathematical_o imagine_v or_o mechanical_o practise_v second_o
casâ—_n demonstration_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n note_v this_o manner_n of_o imagination_n mathematical_a demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n in_o tâ—is_fw-la â—ronoâ—â—â—oâ—â—â—_n must_v understand_v the_o proportional_a â—artes_n or_o sâ—â—â—ions_n to_o be_v thâ—se_a which_o be_v contain_v 〈◊〉_d the_o parallel_a superficies_n construction_n demonstration_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n construction_n demonstration_n two_o case_n in_o this_o proposition_n the_o first_o case_n second_o case_n construction_n demonstration_n an_o other_o demonstration_n construction_n demonstration_n construction_n three_o case_n in_o this_o proposition_n the_o first_o case_n a_o necessary_a thing_n to_o be_v prove_v before_o he_o procee_v any_o â—arther_a in_o the_o construction_n of_o the_o problem_n problem_n which_o how_o to_o find_v out_o be_v teach_v at_o the_o end_n of_o this_o demonstration_n and_o also_o be_v teach_v in_o the_o assumpt_n put_v before_o the_o 14._o proposition_n of_o the_o ten_o book_n demonstration_n of_o the_o first_o case_n second_o case_n three_o case_n an_o other_o demonstration_n to_o prove_v that_o thâ—_z line_n ab_fw-la be_v not_o less_o they_o the_o line_n lx._n this_o be_v before_o taâ—ght_n in_o the_o ten_o book_n in_o the_o assumpt_n put_v before_o the_o 14._o proposition_n proposition_n m._n dee_n to_o avoid_v cavillation_n add_v to_o euclides_n proposition_n this_o word_n six_o who_o i_o have_v follow_v according_o and_o not_o zamberts_n in_o this_o this_o kind_n of_o body_n mention_v in_o the_o proposition_n be_v call_v a_o parallelipipedom_n according_a to_o the_o definition_n before_o give_v thereof_o demonstration_n that_o the_o opposite_a side_n be_v parallelogram_n demonstration_n that_o the_o opposite_a superficies_n be_v equal_a equal_a ab_fw-la be_v equal_a to_o dc_o because_o the_o superficies_n ac_fw-la be_v prove_v a_o parallelogram_n and_o by_o the_o same_o reason_n be_v bh_o equal_a to_o cf_o because_o the_o superficies_n fb_o be_v prove_v a_o parallelogram_n therefore_o the_o 34._o of_o the_o first_o be_v our_o proof_n fâ—â—â—â—_n corollary_n second_o corolry_n these_o solid_n which_o he_o speak_v of_o in_o this_o corollary_n be_v of_o some_o call_v side_v column_n three_o corollary_n constrâ—ction_n demonstration_n demonstration_n look_v at_o the_o end_n of_o the_o demonstration_n what_o be_v understand_v by_o stand_a line_n john_n do_v you_o his_o figure_n by_o this_o figure_n it_o appear_v why_o â—uch_v prism_n be_v call_v pledge_n of_o 〈◊〉_d uâ—ry_a shape_n of_o a_o wedge_n as_o be_v the_o solid_a defgac_n etc_n etc_n stand_v line_n construction_n demonstration_n i_o dees_n figure_n two_o case_n in_o this_o proposition_n thâ—_z first_o case_n construction_n we_o be_v behold_v to_o m._n do_v you_o for_o invent_v this_o figure_n with_o other_o which_o till_o his_o reform_n be_v as_o much_o misshapen_a as_o this_o be_v and_o so_o both_o in_o the_o greek_a and_o latin_a copy_n remain_v demonstration_n demonstration_n note_v now_o how_o the_o base_a respective_o be_v takend_v for_o so_o may_v alteration_n of_o respect_n alter_v the_o name_n of_o the_o bound_n either_o of_o solid_n or_o plain_n second_o case_n case_n there_o you_o perceive_v how_o the_o base_a be_v diverse_o consider_v &_o choose_v as_o before_o we_o advertise_v you_o construction_n demonstration_n construction_n demonstration_n *_o *_o note_v this_o famous_a lemma_n the_o double_a oâ—â—he_n cube_n cube_n note_n note_n 〈…〉_o lemma_n note_v what_o iâ—_z yet_o lack_v requisite_a to_o the_o double_n of_o the_o cube_n the_o converse_n of_o both_o the_o part_n of_o the_o first_o case_n the_o converse_n of_o the_o second_o case_n the_o general_a conclusion_n construction_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n to_o find_v two_o middle_a proportional_n between_o two_o number_n give_v note_v the_o practice_n of_o approach_v to_o precieâ—es_n in_o cubik_fw-mi root_n root_n this_o be_v the_o way_n to_o apply_v any_o square_n give_v to_o a_o line_n also_o give_v sufficienty_fw-mi extend_v a_o problem_n worth_a the_o search_n for_o construction_n demonstration_n construction_n demonstration_n demonstration_n it_o be_v evident_a that_o those_o perpendicular_n be_v all_o one_o with_o the_o stand_a line_n of_o the_o solid_n if_o their_o solid_a angle_n be_v make_v of_o superâ—icâ—â—il_n right_a angle_n only_o double_v of_o the_o 〈◊〉_d etc_n etc_n etc_n dâ—mâ—â—â—â—ratioâ—â—f_a possibility_n in_o the_o problem_n an_o other_o argument_n to_o comâ—oâ—t_v the_o studious_a demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o part_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n construction_n demonstration_n demonstration_n which_o of_o some_o be_v call_v side_v column_n side_v column_n construction_n demonstration_n the_o square_n of_o the_o circle_n demonstration_n â—eading_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n the_o first_o case_n that_o a_o square_a within_o any_o circle_n describe_v be_v big_a than_o half_a the_o circle_n that_o the_o isosceles_a triangle_n without_o the_o square_n be_v great_a than_o half_a the_o segment_n wherein_o they_o be_v second_o case_n case_n this_o as_o 〈…〉_o afterwards_o at_o the_o end_n of_o the_o demâ—straâ—ion_n prove_v construction_n demonstration_n construction_n demonstration_n diâ—â—reence_n between_o the_o first_o problem_n and_o the_o second_o construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n note_v this_o well_o for_o it_o iâ—_z of_o great_a use_n an_o other_o way_n of_o demonstration_n of_o the_o fâ—â—st_a problem_n of_o thâ—â—_n addition_n note_v this_o properâ—ie_n of_o a_o triangle_n rectangle_n construction_n demonstration_n demonstration_n though_o i_o say_v without_o the_o square_n yet_o you_o must_v think_v that_o it_o may_v be_v also_o within_o the_o square_n &_o that_o diverse_o wherefore_o this_o problem_n may_v have_v diverse_a case_n so_o but_o brief_o to_o aside_o all_o may_v thus_o be_v say_v cut_v any_o side_n of_o that_o square_n into_o 3_n parte_v in_o the_o proportion_n of_o x_o to_o y._n note_v the_o manner_n of_o the_o drift_n in_o this_o demonstration_n and_o construction_n mix_o and_o with_o no_o determination_n to_o the_o construction_n as_o common_o iâ—_z in_o problemeâ—â—_n which_o be_v here_o of_o i_o so_o usedâ—_n for_o a_o example_n to_o young_a student_n of_o variety_n in_o art_n construction_n demonstration_n demonstration_n note_v and_o remember_v one_o â—eâ—th_n in_o these_o solid_n the_o conclusion_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n namely_o that_o it_o be_v divide_v moreover_o into_o two_o equal_a prism_n conclusion_n of_o the_o second_o part_n demonstration_n of_o the_o last_o part_n that_o the_o two_o prism_n be_v great_a than_o the_o half_a of_o the_o whole_a pyramid_n conclusion_n of_o the_o last_o part_n conclusion_n of_o the_o whole_a proposition_n proposition_n a_o assumpt_n a_o assumpt_n conclusion_n of_o the_o whole_a demonstration_n lead_v to_o a_o impossibility_n impossibility_n in_o the_o assuâ—pâ—â—â—llowinâ—_n the_o second_o proposition_n of_o this_o bâ—â—ke_n construction_n demonstration_n demonstration_n note_n side_v column_n sometime_o call_v prism_n be_v triple_a to_o pyramid_n have_v one_o base_a and_o equal_a heâ—th_n with_o they_o note_v â—arallelipipedons_n treble_v to_o pyramid_n of_o one_o base_a and_o heith_n with_o they_o construction_n demonstration_n a_o addition_n by_o campane_n and_o flussas_n demonstration_n of_o the_o first_o part_n demonstration_n â—f_n the_o second_o part_n which_o iâ—_z the_o converse_n of_o the_o first_o constr_n 〈◊〉_d 〈◊〉_d parallelipipedon_n call_v prism_n prism_n by_o this_o it_o be_v manifest_a that_o euclid_n comprehend_v side_v column_n also_o under_o the_o name_n of_o a_o prism_n prism_n a_o prism_n have_v for_o his_o base_a a_o polygonon_n figure_n as_o we_o have_v often_o before_o note_v unto_o you_o note_v m._n do_v you_o his_o chief_a purpose_n in_o his_o addition_n demonstration_n touch_v cylinder_n second_o case_n second_o parâ—_n which_o concern_v cillinder_n construction_n demonstration_n construction_n demonstration_n touch_v cylinder_n demonstration_n touch_v cones_n first_o part_n of_o the_o proposition_n demonstrate_v touch_v cones_n two_o case_n in_o this_o proposition_n the_o first_o case_n second_o case_n construction_n demonstration_n touch_v cylinder_n second_o part_n demonstrate_v construction_n construction_n note_v this_o lm_o because_o of_o kz_o in_o the_o next_o proposition_n and_o here_o the_o point_n m_o for_o the_o point_n z_o in_o the_o next_o demonstration_n i_o dee_n dee_n for_o that_o the_o section_n be_v make_v by_o the_o number_n two_o that_o be_v by_o take_v half_n and_o of_o the_o residue_n the_o halâ—eâ—_n and_o so_o to_o ld_n be_v a_o half_a and_o a_o residue_n which_o shall_v be_v a_o common_a measure_n back_o again_o to_o make_v side_n of_o the_o poligonon_n figure_n construction_n
demonstration_n demonstration_n the_o circle_n so_o make_v or_o so_o consider_v in_o the_o sphere_n be_v call_v the_o great_a circle_n all_o other_o not_o have_v the_o centre_n of_o the_o sphere_n to_o be_v their_o centre_n alsoâ—_n be_v call_v less_o circle_n note_v these_o description_n description_n an_o other_o corollary_n corollary_n an_o other_o corollary_n construction_n construction_n this_o be_v also_o prove_v in_o the_o assumpt_n before_o add_v out_o oâ—_n flussas_n note_v what_o a_o great_a or_o great_a circle_n in_o a_o spear_n be_v first_o part_n of_o the_o construction_n noteâ—_n noteâ—_n you_o know_v full_a well_o that_o in_o the_o superficies_n of_o the_o sphere_n â—â—â—ly_o the_o circumference_n of_o the_o circle_n be_v but_o by_o thâ—se_a circumference_n the_o limitation_n and_o assign_v of_o circle_n be_v use_v and_o so_o the_o circumference_n of_o a_o circle_n usual_o call_v a_o circle_n which_o in_o this_o place_n can_v not_o offend_v this_o figure_n be_v restore_v by_o m._n dee_n his_o diligence_n for_o in_o the_o greek_a and_o latin_a euclides_n the_o line_n gl_n the_o line_n agnostus_n and_o the_o line_n kz_o in_o which_o three_o line_n the_o chief_a pinch_n of_o both_o the_o demonstration_n do_v stand_v be_v untrue_o draw_v as_o by_o compare_v the_o studious_a may_v perceive_v note_n you_o must_v imagine_v 〈◊〉_d right_a line_n axe_n to_o be_v perpendicular_a upon_o the_o diameter_n bd_o and_o ce_fw-fr though_o here_o ac_fw-la the_o semidiater_n seem_v to_o be_v part_n of_o ax._n and_o so_o in_o other_o point_n in_o this_o figure_n and_o many_o other_o strengthen_v your_o imagination_n according_a to_o the_o tenor_n of_o construction_n though_o in_o the_o delineation_n in_o plain_a sense_n be_v not_o satisfy_v note_n boy_n equal_a to_o bk_o in_o respect_n of_o m._n dee_n his_o demonstration_n follow_v follow_v note_v â—his_fw-la point_n z_o that_o you_o may_v the_o better_a understand_v m._n dee_n his_o demonstration_n second_o part_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n demonstration_n which_o of_o necessity_n shall_v fall_v upon_o z_o as_o m._n dee_n prove_v it_o and_o his_o proof_n be_v set_v after_o at_o this_o mark_n ✚_o follow_v i_o dee_n dee_n but_o az_o be_v great_a them_z agnostus_n as_o in_o the_o former_a proposition_n km_o be_v evident_a to_o be_v great_a than_o kg_v so_o may_v it_o also_o be_v make_v manifest_a that_o kz_o do_v neither_o touch_n nor_o cut_v the_o circle_n fgâ—h_n a_o other_o prove_v that_o the_o line_n ay_o be_v great_a they_o the_o line_n ag._n ag._n this_o as_o a_o assumpt_n be_v present_o prove_v two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o impossibility_n second_o case_n case_n as_o it_o be_v â—asiâ—_n to_o gather_v by_o the_o assumpt_n put_v after_o the_o secoâ—â—_n of_o this_o booâ—â—_n note_v a_o general_a rule_n the_o second_o part_n of_o the_o problem_n two_o way_n execute_v a_o upright_a cone_n the_o second_o part_n of_o the_o problem_n the_o second_o â—aâ—â—_n oâ—_n the_o problem_n ☜_o ☜_o this_o may_v easy_o be_v demonstrate_v as_o in_o thâ—_z 17._o proposition_n the_o section_n of_o a_o sphere_n be_v prove_v to_o be_v a_o circle_n circle_n for_o take_v away_o all_o doubt_n this_o aâ—_z a_o lemma_n afterward_o be_v demonstrate_v a_o lemma_n as_o it_o be_v present_o demonstrate_v construction_n demonstration_n the_o second_o part_n of_o the_o problem_n *_o construction_n demonstration_n an_o other_o way_n of_o execute_v this_o problem_n the_o converse_n of_o the_o assumpt_n a_o great_a error_n common_o maintain_v between_o straight_o and_o crooked_a all_o manner_n of_o proportion_n may_v be_v give_v construction_n demonstration_n the_o definition_n of_o a_o circled_a â—â—apâ—â—d_n in_o a_o spâ—erâ—_n construction_n demonstration_n this_o be_v manifest_a if_o you_o consider_v the_o two_o triangle_n rectangle_v hom_o and_o hon_n and_o then_o with_o all_o use_v the_o 47._o of_o the_o first_o of_o euclid_n construction_n demonstration_n construction_n demonstration_n this_o in_o manner_n of_o a_o lemmâ—_n be_v present_o prove_v note_v here_o of_o axe_n base_a &_o solidity_n more_o than_o i_o need_n to_o bring_v any_o far_a proof_n for_o note_n note_n i_o say_v half_a a_o circular_a revolution_n for_o that_o suffice_v in_o the_o whole_a diameter_n n-ab_n to_o describe_v a_o circle_n by_o iâ—_z it_o be_v move_v â—â—out_v his_o centre_n q_o etc_n etc_n lib_fw-la 2_o prop_n 2._o de_fw-fr spheâ—a_n &_o cylindrâ—_n note_n note_n a_o rectangle_n parallelipipedon_n give_v equal_a to_o a_o sphere_n give_v to_o a_o sphere_n or_o to_o any_o part_n of_o a_o sphere_n assign_v as_o a_o three_o four_o five_o etc_o etc_o to_o geve_v a_o parallelipipedon_n equal_a side_v colume_n pyramid_n and_o prism_n to_o be_v give_v equal_a to_o a_o sphere_n or_o to_o any_o certain_a part_n thereof_o to_o a_o sphere_n or_o any_o segment_n or_o sector_n of_o the_o same_o to_o geve_v a_o cone_n or_o cylinder_n equal_a or_o in_o any_o proportion_n assign_v far_o use_v of_o spherical_a geometry_n the_o argument_n of_o the_o thirteen_o book_n construction_n demonstration_n demonstration_n the_o assumpt_v prove_v prove_v because_o ac_fw-la be_v suppose_v great_a than_o ad_fw-la therefore_o his_o residue_n be_v less_o than_o the_o residue_n of_o ad_fw-la by_o the_o common_a sentence_n wherefore_o by_o the_o supposition_n db_o be_v great_a than_o â—c_z the_o chieâ—e_a line_n in_o all_o euclides_n geometry_n what_o be_v mean_v here_o by_o a_o section_n in_o one_o only_a poiâ—t_n construction_n demonstration_n demonstration_n note_v how_o ce_fw-fr and_o the_o gnonom_n xop_n be_v prove_v equal_a for_o it_o serve_v in_o the_o converse_n demonstrate_v by_o m._n dee_n here_o next_o after_o this_o proposition_n â—the_v converse_v of_o the_o former_a former_a as_o we_o haâ—e_z note_v the_o place_n of_o the_o peculiar_a proâ—e_n there_o â—in_a the_o demonstration_n of_o the_o 3._o 3._o therefore_o by_o my_o second_o theorem_a add_v upon_o the_o second_o proposition_n dc_o be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n a._n and_o because_o ac_fw-la be_v big_a than_o cb_o therefore_o dam_n be_v great_a than_o ac_fw-la wherefore_o if_o a_o right_a line_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v demonstrate_v therefore_o by_o my_o second_o theorem_a add_v upon_o the_o second_o proposition_n dc_o be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n a._n and_o because_o ac_fw-la be_v big_a than_o cb_o therefore_o dam_n be_v great_a than_o ac_fw-la wherefore_o if_o a_o right_a line_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v construction_n construction_n though_o i_o say_v perpendicular_a yes_o you_o may_v perceive_v how_o infinite_a other_o position_n will_v serve_v so_o that_o diego_n and_o ad_fw-la make_v a_o angle_n for_o a_o triangle_n to_o have_v his_o side_n proportional_o cut_v etc_n etc_n demonstration_n demonstration_n i_o dee_n this_o be_v most_o evident_a of_o my_o second_o theorem_a add_v to_o the_o three_o proposition_n for_o to_o add_v to_o a_o whole_a line_n a_o line_n equal_a to_o the_o great_a segment_n &_o to_o add_v to_o the_o great_a segment_n a_o line_n equal_a to_o the_o whole_a line_n be_v all_o one_o thing_n in_o the_o line_n produce_v by_o the_o whole_a line_n i_o mean_v the_o line_n divide_v by_o extreme_a and_o mean_a proportion_n this_o be_v before_o demonstrate_v most_o evident_o and_o brief_o by_o m._n dee_n after_o the_o 3._o proposition_n note_n note_v 4._o proportional_a line_n note_v two_o middle_a proportional_n note_v 4._o way_n of_o progression_n in_o the_o proportion_n of_o a_o line_n divide_v by_o extreme_a and_o middle_a proportion_n what_o resolution_n and_o composition_n be_v have_v before_o be_v teach_v in_o the_o begin_n of_o the_o first_o book_n book_n proclus_n in_o the_o greek_a in_o the_o 58._o page_n construction_n demonstration_n two_o case_n in_o this_o proposition_n construction_n thâ—_z first_o case_n demonstration_n the_o second_o case_n construction_n demonstration_n construction_n demonstration_n this_o corollary_n be_v the_o 3._o proposition_n of_o theâ—4_n â—4_o book_n after_o campane_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n construction_n demonstration_n construstion_n demonstration_n constrâ—yction_n demonstration_n this_o corollary_n be_v the_o 11._o proposition_n of_o the_o 14._o book_n after_o campane_n this_o corollary_n be_v the_o 3._o corollary_n after_o the_o 17._o proposition_n of_o the_o 14_o book_n after_o campane_n campane_n by_o the_o name_n oâ—_n a_o pyramid_n both_o here_o &_o iâ—_z this_o book_n follow_v understand_v a_o tetrahedron_n an_o other_o construction_n and_o demonstration_n of_o the_o second_o part_n after_o fâ—ussas_n three_o part_n of_o the_o demonstration_n this_o corollary_n be_v the_o 15._o proposition_n of_o the_o 14._o book_n after_o campane_n this_o corollary_n campane_n put_v as_o a_o corollary_n after_o
triangle_n unto_o the_o section_n divide_v the_o angle_n of_o the_o triangle_n into_o two_o equal_a part_n this_o construction_n be_v the_o half_a part_n of_o that_o gnomical_a figure_n describe_v in_o the_o 43._o proposition_n of_o the_o first_o book_n which_o gnomical_a figure_n be_v of_o great_a use_n in_o a_o manner_n in_o all_o geometrical_a demonstration_n the_o 4._o theorem_a the_o 4._o proposition_n in_o equiangle_n triangle_n the_o side_n which_o contain_v the_o equal_a angle_n be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n svppose_v that_o there_o be_v two_o equiangle_n triangle_n abc_n and_o dce_n and_o let_v the_o angle_n abc_n of_o the_o one_o triangle_n be_v equal_a unto_o the_o angle_v dce_n of_o the_o other_o triangle_n and_o the_o angle_n bac_n equal_a unto_o the_o angle_v cde_o and_o moreover_o the_o angle_n acb_o equal_a unto_o the_o angle_n dec_n then_o i_o say_v that_o those_o side_n of_o the_o triangle_n abc_n &_o dce_n which_o include_v the_o equal_a angle_n be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n construction_n for_o let_v two_o side_n of_o the_o say_a triangle_n namely_o two_o of_o those_o side_n which_o be_v subtend_v under_o equal_a angle_n as_o for_o example_n the_o side_n bc_o and_o ce_fw-fr be_v so_o set_v that_o they_o both_o make_v one_o right_a line_n and_o because_o the_o angle_n abc_n &_o acb_o be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o first_o but_o the_o angle_n acb_o be_v equal_a unto_o the_o angle_n dec_n therefore_o the_o angle_n abc_n &_o dec_n be_v less_o they_o two_o right_a angle_n wherefore_o the_o line_n basilius_n &_o ed_z be_v produce_v will_v at_o the_o length_n meet_v together_o let_v they_o meet_v and_o join_v together_o in_o the_o point_n f._n demonstration_n and_o because_o by_o supposition_n the_o angle_n dce_n be_v equal_a unto_o the_o angle_n abc_n therefore_o the_o line_n bf_o be_v by_o the_o 28._o of_o the_o first_o a_o parallel_n unto_o tâ—e_z line_n cd_o and_o forasmuch_o as_o by_o supposition_n the_o angle_n acb_o be_v equal_a unto_o the_o angle_n dec_n therefore_o again_o by_o the_o 28._o of_o the_o first_o the_o line_n ac_fw-la be_v a_o parallel_n unto_o the_o line_n fe_o wherefore_o fadc_n be_v a_o parallelogram_n wherefore_o the_o side_n favorina_n be_v equal_a unto_o the_o side_n dc_o and_o the_o side_n ac_fw-la unto_o the_o side_n fd_o by_o the_o 34._o of_o the_o first_o and_o because_o unto_o one_o of_o the_o side_n of_o the_o triangle_n bfe_n namely_o to_o fe_o be_v draw_v a_o parallel_a line_n ac_fw-la therefore_o as_o basilius_n be_v to_o of_o so_o be_v bc_o to_o ce_fw-fr by_o the_o 2._o of_o the_o six_o but_o of_o be_v equal_a unto_o cd_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o basilius_n be_v to_o cd_o so_o be_v bc_o to_o ce_fw-fr which_o be_v side_n subtend_v under_o equal_a angle_n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o ab_fw-la be_v to_o bc_o so_o be_v dc_o to_o ce._n again_o forasmuch_o as_o cd_o be_v a_o parallel_n unto_o bf_o therefore_o again_o by_o the_o 2._o of_o the_o six_o as_o bc_o be_v to_o ce_fw-fr so_o be_v fd_v to_o de._n but_o fd_o be_v equal_a unto_o ac_fw-la wherefore_o as_o bc_o be_v to_o ce_fw-fr so_o be_v ac_fw-la to_o de_fw-fr which_o be_v also_o side_n subtend_v under_o equal_a angle_n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o â—s_v bc_o be_v to_o ca_n so_o be_v ce_fw-fr to_o edâ—_n wherefore_o forasmuch_o as_o it_o have_v be_v demonstrate_v that_o as_o ab_fw-la be_v unto_o be_v so_o be_v dc_o unto_o ceâ—_n but_o as_o dc_o be_v unto_o ca_n so_o be_v ce_fw-fr unto_o edâ—_n it_o follow_v of_o equality_n by_o the_o 22._o of_o the_o five_o that_o â—s_a basilius_n be_v unto_o ac_fw-la so_o be_v cd_o unto_o deâ—_n wherefore_o in_o equiangle_n triangleâ—_n the_o side_n which_o include_v the_o equal_a angle_n be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n â—hich_z be_v require_v to_o be_v demonstrate_v the_o 5._o theorem_a the_o 5._o proposition_n if_o two_o triangle_n have_v their_o side_n proportional_a the_o triangâ—â—s_n be_v equiangle_n and_o those_o angle_n in_o they_o be_v equal_a under_o which_o be_v subtend_v side_n of_o like_a proportion_n svppose_v that_o there_o be_v two_o triangle_n abc_n &_o def_n have_v their_o side_n proportional_a as_o ab_fw-la be_v to_o bc_o so_o let_v de_fw-fr be_v to_o of_o proposition_n &_o as_o bc_o be_v to_o ac_fw-la so_o let_v of_o be_v to_o df_o and_o moreover_o as_o basilius_n be_v to_o ac_fw-la so_o let_v ed_z be_v to_o df._n then_o i_o say_v that_o the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n def_n and_o those_o angle_n in_o they_o be_v equal_a under_o which_o be_v subtend_v side_n of_o like_a proportion_n that_o be_v the_o angle_n abc_n be_v equal_a unto_o the_o angle_n def_n and_o the_o angle_n bca_o unto_o the_o angle_n efd_n and_o moreover_o the_o angle_n bac_n to_o the_o angle_n edf_o upon_o the_o right_a line_n of_o construction_n and_o unto_o the_o point_n in_o it_o e_o &_o f_o describe_v by_o the_o 23._o of_o the_o first_o angle_n equal_a unto_o the_o angle_n abc_n &_o acb_o which_o let_v be_v feg_fw-mi and_o efg_o namely_o let_v the_o angle_n feg_fw-mi be_v equal_a unto_o the_o angle_n abc_n and_o let_v the_o angle_n efg_o be_v equal_a to_o the_o angle_n acb_o demonstration_n and_o forasmuch_o as_o the_o angle_n abc_n and_o acb_o be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o first_o therefore_o also_o the_o angle_n feg_v and_o efg_o be_v less_o than_o two_o right_a angle_n wherefore_o by_o the_o 5._o petition_n of_o the_o first_o the_o right_a line_n eglantine_n &_o fg_o shall_v at_o the_o length_n concur_v let_v they_o concur_v in_o the_o point_n g._n wherefore_o efg_o be_v a_o triangle_n wherefore_o the_o angle_n remain_v bac_n be_v equal_a unto_o the_o angle_n remain_v egf_n by_o the_o first_o corollary_n of_o the_o 32._o of_o the_o first_o wherefore_o the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n gef_n wherefore_o in_o the_o triangle_n abc_n and_o egf_n the_o side_n which_o include_v the_o equal_a angle_n by_o the_o 4._o of_o the_o six_o be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n wherefore_o as_o ab_fw-la be_v to_o bc_o so_o be_v ge_z to_o ef._n but_o as_o ab_fw-la be_v to_o bc_o so_o by_o supposition_n be_v de_fw-fr to_o ef._n wherefore_o as_o de_fw-fr be_v to_o of_o so_o be_v ge_z to_o of_o by_o the_o 11._o of_o the_o five_o wherefore_o either_o of_o these_o de_fw-fr and_o eglantine_n have_v to_o of_o one_o and_o the_o same_o proportion_n wherefore_o by_o the_o 9_o of_o the_o five_o de_fw-fr be_v equal_a unto_o eglantine_n and_o by_o the_o same_o reason_n also_o df_o be_v equal_a unto_o fg._n now_o forasmuch_o as_o de_fw-fr be_v equal_a to_o eglantine_n and_o of_o be_v common_a unto_o they_o both_o therefore_o these_o two_o side_n de_fw-fr &_o of_o be_v equal_a unto_o these_o two_o side_n ge_z and_o of_o and_o the_o base_a df_o be_v equal_a unto_o the_o base_a fg._n wherefore_o the_o angle_n def_n by_o the_o 8._o of_o the_o first_o be_v equal_a unto_o the_o angle_n gef_n and_o the_o triangle_n def_n by_o the_o 4._o of_o the_o first_o be_v equal_a unto_o the_o triangle_n gef_n and_o the_o rest_n of_o the_o angle_n of_o the_o one_o triangle_n be_v equal_a unto_o the_o rest_n of_o the_o angle_n of_o the_o other_o triangle_n the_o one_o to_o the_o outstretch_o under_o which_o be_v subtend_v equal_a side_n wherefore_o the_o angle_n dfe_n be_v equal_a unto_o the_o angle_n gfe_n and_o the_o angle_n edf_o unto_o the_o angle_n egf_n and_o becauseâ—_n the_o angle_n feed_v be_v equal_a unto_o the_o angle_n gef_n but_o the_o angle_n gef_n be_v equal_a unto_o the_o angle_n abc_n therefore_o the_o angle_n abc_n be_v also_o equal_a unto_o the_o angle_n feed_v and_o by_o the_o same_o reason_n the_o angle_n acb_o be_v equal_a unto_o the_o angle_v dfeâ—_n and_o moreover_o the_o angle_n bac_n unto_o the_o angle_n edf_o wherefore_o the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n def_n if_o two_o triangle_n therefore_o have_v their_o side_n proportional_a the_o triangle_n shall_v be_v equiangle_n &_o those_o angle_n in_o they_o shall_v be_v equal_a under_o which_o be_v subtend_v side_n of_o like_a proportion_n which_o be_v require_v to_o be_v demonstrate_v the_o 6._o theorem_a the_o 6._o proposition_n if_o there_o be_v two_o triangle_n whereof_o the_o one_o have_v one_o angle_n equal_a to_o one_o angle_n of_o
the_o square_a of_o the_o line_n a_o have_v not_o unto_o the_o square_n of_o the_o line_n b_o the_o same_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n then_o i_o say_v that_o the_o line_n a_o and_o b_o be_v incommensurable_a in_o length_n for_o if_o the_o line_n a_o and_o b_o be_v commensurable_a in_o length_n than_o the_o square_a of_o the_o line_n a_o shall_v have_v unto_o the_o square_n of_o the_o line_n b_o the_o same_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n by_o the_o first_o part_n of_o this_o proposition_n but_o by_o supposition_n it_o have_v not_o wherefore_o the_o line_n a_o and_o b_o be_v not_o commensurable_a in_o length_n proposition_n wherefore_o they_o be_v incomensurable_a in_o length_n wherefore_o square_n make_v of_o right_a line_n commensura_fw-la in_o length_n have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o square_n which_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o squaâ—e_a number_n shall_v also_o have_v the_o side_n commensurable_a in_o length_n but_o square_n describe_v of_o right_a line_n incommensurable_a in_o length_n have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o square_n which_o have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n have_v not_o also_o their_o side_n commensurable_a in_o length_n which_o be_v all_o that_o be_v require_v to_o be_v prove_v ¶_o corollary_n hereby_o it_o be_v manifest_a that_o right_a line_n commensurable_a in_o length_n be_v also_o ever_o commensurable_a in_o power_n but_o right_a line_n commensurable_a in_o power_n be_v not_o always_o commensurable_a in_o length_n corollary_n and_o right_a line_n incommensurable_a in_o length_n be_v not_o always_o incommensurable_a in_o power_n but_o right_a line_n incommensurable_a in_o power_n be_v ever_o also_o incommensurable_a in_o length_n for_o forasmuch_o as_o square_n make_v of_o right_a line_n commensurable_a in_o length_n have_v that_o proportion_n the_o one_o to_o the_o other_o corollary_n that_o a_o square_a number_n have_v to_o a_o square_a number_n by_o the_o first_o part_n of_o this_o proposition_n but_o magnitude_n which_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o number_n simple_o have_v to_o number_n be_v by_o the_o six_o of_o the_o ten_o commensurable_a wherefore_o right_a line_n commensurable_a in_o length_n be_v commensurable_a not_o only_o in_o length_n but_o also_o in_o power_n part_n again_o forasmuch_o as_o there_o be_v certain_a square_n which_o have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n but_o yet_o have_v that_o proportion_n the_o one_o to_o the_o other_o which_o number_n simple_o have_v to_o number_v their_o side_n in_o deed_n be_v in_o power_n commensurable_a for_o that_o they_o describe_v square_n which_o have_v that_o proportion_n which_o number_n simple_o have_v to_o number_n which_o square_n be_v therefore_o commensurable_a by_o the_o 6._o of_o this_o book_n but_o the_o say_v side_n be_v incommensurable_a in_o length_n by_o the_o latter_a part_n of_o this_o proposition_n wherâ—fore_o it_o be_v true_a that_o line_n commensurable_a in_o power_n be_v not_o straight_o way_n commensurable_a in_o length_n also_o pâ—rt_n and_o by_o the_o selâ—e_a same_o reason_n be_v prove_v also_o that_o three_o part_n of_o the_o corollary_n that_o line_n incommensurable_a in_o length_n be_v not_o always_o incommensurable_a in_o power_n for_o they_o may_v be_v incommensurable_a in_o length_n but_o yet_o commensurable_a in_o power_n as_o in_o those_o square_n which_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 but_o not_o as_o a_o square_a number_n be_v to_o a_o square_a number_n part_n but_o right_a line_n incommensurable_a in_o power_n be_v always_o also_o incommensurable_a in_o length_n for_o iâ—_z they_o be_v commensurable_a in_o length_n they_o shall_v also_o be_v commensurable_a in_o power_n by_o the_o first_o part_n of_o this_o corollary_n but_o they_o be_v suppose_v to_o be_v incommensurable_a in_o length_n which_o be_v absurd_a wherefore_o right_a line_n incommensurable_a in_o power_n be_v ever_o incommensurable_a in_o lengthy_a for_o the_o better_a understanding_n of_o this_o proposition_n and_o the_o other_o follow_v i_o have_v here_o add_v certain_a annotation_n take_v out_o of_o montaureus_n montauâ—â—us_fw-la and_o first_o as_o touch_v the_o signification_n oâ—_n word_n and_o term_n herein_o use_v whâ—ch_a arâ—_n such_o that_o unless_o they_o be_v well_o mark_v and_o poise_v the_o matter_n will_v be_v obscure_a and_o hard_a and_o in_o a_o manner_n inexplicable_a first_o this_o you_o must_v note_v that_o line_n to_o be_v commensurable_a in_o length_n and_o line_n to_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 be_v all_o one_o so_o that_o whatsoever_o line_n be_v commensurable_a in_o length_n be_v also_o in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_v and_o converse_o what_o so_o ever_o line_n 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 be_v also_o commensurable_a in_o length_n as_o it_o be_v manifest_a by_o the_o 5_o and_o 6_o of_o this_o book_n likewise_o line_n to_o be_v incommensurable_a in_o length_n and_o not_o to_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 be_v all_o one_o as_o it_o be_v manifest_a by_o the_o 7._o and_o 8._o of_o this_o book_n wherefore_o that_o which_o be_v say_v in_o this_o theorem_a aught_o to_o be_v understand_v of_o line_n commensurable_a in_o length_n and_o incommensurable_a in_o length_n this_o moreover_o be_v to_o be_v note_v that_o it_o be_v not_o all_o one_o numbers_z to_o be_v square_a number_n and_o to_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n for_o although_o square_a number_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n yet_o be_v not_o all_o those_o number_n which_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n square_a number_n for_o they_o may_v be_v like_o superficial_a number_n and_o yet_o not_o square_a number_n which_o yet_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n although_o all_o square_a number_n be_v like_o superficial_a number_n for_o between_o two_o square_a number_n there_o fall_v one_o mean_a proportional_a number_n by_o the_o 11._o of_o the_o eight_o but_o if_o between_o two_o number_n there_o fall_v one_o mean_v proportional_a number_n those_o two_o number_n be_v like_o superficial_a number_n by_o the_o 20._o of_o the_o eight_o so_o also_o if_o two_o number_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n they_o shall_v be_v like_o superficial_a number_n by_o the_o first_o corollary_n add_v after_o the_o last_o proposition_n of_o the_o eight_o book_n and_o now_o to_o know_v whether_o two_o superficial_a number_n give_v be_v like_o superficial_a number_n or_o no_o no._n it_o be_v thus_o find_v out_o first_o if_o between_o the_o two_o number_n give_v there_o fall_v no_o mean_a proportional_a then_o be_v not_o these_o two_o number_n like_o superficial_a number_n by_o the_o 18._o of_o the_o eight_o but_o if_o there_o do_v fall_v between_o they_o a_o mean_v proportional_a then_o be_v they_o like_v superâ—iciall_a number_n by_o the_o 20._o of_o the_o eight_o moreover_o two_o like_o superficial_a number_n multiply_v the_o one_o into_o the_o other_o do_v produce_v a_o square_a number_n by_o the_o firsâ—_o of_o the_o nine_o wherefore_o if_o they_o do_v not_o produce_v a_o square_a number_n then_o be_v they_o not_o like_o superficial_a number_n and_o if_o the_o one_o be_v multiply_v into_o the_o other_o they_o produce_v a_o square_a number_n then_o be_v they_o like_v superficial_a by_o the_o 2._o of_o the_o nine_o moreover_o if_o the_o say_v two_o superficial_a number_n be_v in_o superperticular_a or_o superbipartient_fw-fr proportion_n then_o be_v they_o not_o like_o superficial_a number_n for_o if_o they_o shall_v be_v like_a then_o shall_v there_o be_v a_o mean_a proportional_a between_o they_o by_o the_o 20._o of_o the_o eight_o but_o that_o be_v contrary_a to_o the_o corollary_n of_o the_o 20._o of_o the_o eight_o and_o the_o easy_o to_o conceive_v the_o demonstration_n follow_v take_v this_o example_n of_o that_o which_o we_o have_v say_v ¶_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 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
that_o superficies_n be_v rational_a svppose_v that_o a_o parallelogram_n be_v contain_v under_o a_o residual_a line_n ab_fw-la and_o a_o binomial_a line_n cd_o and_o let_v the_o great_a name_n of_o the_o binomial_a line_n be_v ce_fw-fr and_o the_o less_o name_n be_v ed_z and_o let_v the_o name_n of_o the_o binomial_a line_n namely_o ce_fw-fr and_o ed_z be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n namely_o to_o of_o and_o fâ—_n and_o in_o the_o self_n same_o proportion_n and_o let_v the_o line_n which_o contain_v in_o power_n that_o parallelogram_n be_v g._n then_o i_o say_v that_o the_o line_n g_o be_v rational_a take_v a_o rational_a line_n namely_o h._n and_o unto_o the_o line_n cd_o apply_v a_o parallelogram_n equal_a to_o the_o square_v of_o the_o line_n h_o construction_n and_o make_v in_o breadth_n the_o line_n kl_o wherefore_o by_o the_o 112._o of_o the_o ten_o kl_o be_v a_o residual_a line_n demonstration_n who_o name_n let_v be_v km_o and_o ml_o which_o be_v by_o the_o same_o commensurable_a to_o the_o name_n of_o the_o binomial_a line_n that_o be_v to_o ce_fw-fr and_o ed_z and_o be_v in_o the_o self_n same_o proportion_n but_o by_o position_n the_o line_n ce_fw-fr and_o ed_z be_v commensurable_a to_o the_o line_n of_o and_o fb_o and_o be_v in_o the_o self_n same_o proportion_n wherefore_o by_o the_o 12._o of_o the_o ten_o as_o the_o line_n of_o be_v to_o the_o line_n fbâ—_n so_o be_v the_o line_n km_o to_o the_o line_n ml_o wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n of_o be_v to_o the_o line_n km_o so_o be_v the_o line_n bf_o to_o the_o line_n lm_o wherefore_o the_o residue_n ab_fw-la be_v to_o the_o residue_n kl_o as_o the_o whole_a of_o be_v to_o the_o whole_a km_o but_o the_o line_n of_o be_v commensurable_a to_o the_o line_n km_o for_o either_o of_o the_o line_n of_o and_o km_o be_v commensurable_a to_o the_o line_n ce._n wherefore_o also_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n kl_o and_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n kl_o so_o by_o the_o first_o of_o the_o six_o be_v the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o ab_fw-la to_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o kl_o wherefore_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o ab_fw-la be_v commensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o kl_o but_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o kl_o be_v equal_a to_o the_o square_n of_o the_o line_n h._n wherefore_o the_o parallelogram_n contain_v under_o the_o line_n cd_o &_o ab_fw-la be_v commensurable_a to_o the_o square_n of_o the_o line_n h._n but_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n g._n wherefore_o the_o square_a of_o the_o line_n h_o be_v commensurable_a to_o the_o square_n of_o the_o line_n g._n but_o the_o square_n of_o the_o line_n h_o be_v rational_a wherefore_o the_o square_a of_o the_o line_n g_o be_v also_o rational_a wherefore_o also_o the_o line_n g_o be_v rational_a and_o it_o contain_v in_o power_n the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o cd_o if_o therefore_o a_o parallelogram_n be_v contain_v under_o a_o residual_a line_n and_o a_o binomial_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n and_o in_o the_o self_n same_o proportion_n the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v rational_a which_o be_v require_v to_o be_v prove_v ¶_o corollary_n hereby_o it_o be_v manifest_a that_o a_o rational_a parallelogram_n may_v be_v contain_v under_o irrational_a line_n ¶_o a_o otâ—â—r_n 〈…〉_o flussas_n 〈…〉_o line_n â—d_a whosâ—_n name_v aâ—_z and_o â—d_a let_v be_v commensurable_a in_o length_n unto_o the_o name_n of_o the_o residual_a line_n aâ—_z which_o let_v be_v of_o and_o fb_o and_o let_v the_o liâ—e_z aeâ—_n be_v to_o the_o line_n edâ—_n in_o the_o same_o proportion_n that_o the_o line_n of_o be_v to_o the_o line_n fâ—_n and_o let_v the_o right_a line_n â—_o contain_v in_o power_n the_o superficies_n dâ—_n then_o i_o say_v thaâ—_z the_o liâ—e_z â—_o be_v a_o rational_a linâ—_n 〈…〉_o lâ—ne_n construction_n which_o lâ—â—_n bâ—â—_n and_o upon_o the_o line_n â—â—_o describe_v by_o the_o 4â—_o of_o the_o first_o a_o parallelogram_n equal_a to_o the_o square_n of_o the_o line_n â—â—_o and_o make_n in_o breadth_n the_o line_n dc_o wherefore_o by_o the_o â—12_n of_o this_o book_n cd_o be_v a_o residuâ—ll_a lineâ—_z who_o name_n which_o let_v be_v â—â—_o and_o odd_a shall_v be_v coâ—mensurablâ—_n in_o length_n unto_o the_o name_n aâ—_z and_o â—d_a and_o the_o line_n c_o o_fw-fr shall_v be_v unto_o the_o line_n odd_a demonstration_n in_o the_o same_o proportion_n that_o the_o line_n ae_n be_v to_o the_o line_n edâ—_n but_o as_o the_o line_n aâ—_z be_v to_o the_o line_n â—d_a so_o by_o supposition_n be_v the_o line_n of_o to_o the_o line_n fe_o wherefore_o as_o the_o line_n county_n be_v to_o the_o line_n odd_a so_o be_v the_o line_n of_o to_o the_o line_n fâ—â—_v wherefore_o the_o line_n county_n and_o odd_a be_v commensurable_a with_o the_o line_n aâ—_z and_o â—â—_o by_o the_o â—â—_o of_o this_o book_n wherefore_o the_o residue_n namely_o the_o line_n cd_o be_v to_o the_o residue_n namely_o to_o the_o line_n aâ—_z as_o the_o line_n county_n be_v to_o the_o line_n of_o by_o the_o 19_o of_o the_o five_o but_o it_o be_v prove_v that_o the_o line_n county_n be_v commensurable_a unto_o the_o line_n af._n wherefore_o the_o line_n cd_o be_v commensurable_a unto_o the_o line_n ab_fw-la wherefore_o by_o the_o first_o of_o the_o six_o the_o parallelogram_n ca_n be_v commensurable_a to_o the_o parallelogram_n dâ—_n but_o the_o parallelogram_n â—â—_o iâ—_z by_o construction_n rational_a for_o it_o be_v equal_a to_o the_o square_n of_o the_o rational_a line_n â—_o wherefore_o the_o parallelogram_n â—d_a â—s_n also_o rational_o wherâ—fore_o the_o line_n â—_o which_o by_o supposition_n contain_v in_o power_n the_o superficies_n â—dâ—_n be_v also_o rational_a if_o therefore_o a_o parallelogram_n be_v contain_v etc_o etc_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 91._o theorem_a the_o 115._o proposition_n of_o a_o medial_a line_n be_v produce_v infinite_a irrational_a line_n of_o which_o none_o be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o svppose_v that_o a_o be_v a_o medial_a line_n then_o i_o say_v that_o of_o the_o line_n a_o may_v be_v produce_v infinite_a irrational_a line_n of_o which_o none_o shall_v be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o take_v a_o rational_a line_n b._n and_o unto_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o let_v the_o square_n of_o the_o line_n c_o be_v equal_a by_o the_o 14._o of_o the_o second_o â—_o demonstration_n wherefore_o the_o line_n c_o be_v irrational_a for_o a_o superficies_n contain_v under_o a_o rational_a line_n and_o a_o irrational_a line_n be_v by_o the_o assumpt_n follow_v the_o 38._o of_o the_o ten_o irrational_a and_o the_o line_n which_o contain_v in_o power_n a_o irrational_a superficies_n be_v by_o the_o assumpt_n go_v before_o the_o 21._o of_o the_o ten_o irrational_a and_o it_o be_v not_o one_o and_o the_o self_n same_o with_o any_o of_o those_o thirteen_o that_o be_v before_o for_o none_o of_o the_o line_n that_o be_v before_o apply_v to_o a_o rational_a line_n make_v the_o breadth_n medial_a again_o unto_o that_o which_o be_v contain_v under_o the_o line_n b_o and_o c_o let_v the_o square_n of_o d_o be_v equal_a wherefore_o the_o square_a of_o d_o be_v irrational_a wherefore_o also_o the_o line_n d_o be_v irrational_a and_o not_o of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o for_o the_o square_n of_o none_o of_o the_o line_n which_o be_v before_o apply_v to_o a_o rational_a line_n make_v the_o breadth_n the_o line_n c._n in_o like_a sort_n also_o shall_v it_o so_o follow_v if_o a_o man_n proceed_v infinite_o wherefore_o it_o be_v manifest_a that_o of_o a_o medial_a line_n be_v produce_v infinite_a irrational_a line_n of_o which_o none_o be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o which_o be_v require_v to_o be_v prove_v an_o other_o demonstration_n demonstration_n suppose_v that_o ac_fw-la be_v a_o medial_a line_n then_o i_o say_v that_o of_o the_o line_n ac_fw-la may_v be_v produce_v infinite_a irrational_a line_n of_o which_o none_o shall_v be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o irrational_a line_n before_o name_v unto_o the_o line_n ac_fw-la and_o from_o the_o point_n a_o
same_o superficies_n wherefore_o these_o right_a line_n ab_fw-la bd_o and_o dc_o be_v in_o one_o and_o the_o self_n same_o superficies_n and_o either_o of_o these_o angle_n abdella_n and_o bdc_o be_v a_o right_a angle_n by_o supposition_n wherefore_o by_o the_o 28._o of_o the_o first_o the_o line_n ab_fw-la be_v a_o parallel_n to_o the_o line_n cd_o if_o therefore_o two_o right_a line_n be_v erect_v perpendicular_o to_o one_o and_o the_o self_n same_o plain_a superficies_n those_o right_a line_n be_v parallel_n the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v here_o for_o the_o better_a understanding_n of_o this_o 6._o proposition_n i_o have_v describe_v a_o other_o figure_n as_o touch_v which_o if_o you_o erect_v the_o superficies_n abdella_n perpendicular_o to_o the_o superficies_n bde_v and_o imagine_v only_o a_o line_n to_o be_v draw_v from_o the_o point_n a_o to_o the_o point_n e_o if_o you_o will_v you_o may_v extend_v a_o thread_n from_o the_o say_a point_n a_o to_o the_o point_n e_o and_o so_o compare_v it_o with_o the_o demonstration_n it_o will_v make_v both_o the_o proposition_n and_o also_o the_o demonstration_n most_o clear_a unto_o you_o ¶_o a_o other_o demonstration_n of_o the_o six_o proposition_n by_o m._n dee_n suppose_v that_o the_o two_o right_a line_n ab_fw-la &_o cd_o be_v perpendicular_o erect_v to_o one_o &_o the_o same_o plain_a superficies_n namely_o the_o plain_a superficies_n open_v then_o i_o say_v that_o â—â—_o and_o cd_o be_v parallel_n let_v the_o end_n point_v of_o the_o right_a line_n ab_fw-la and_o cd_o which_o touch_v the_o plain_a superficies_n oâ—_n be_v the_o point_n â—_o and_o d_o fronâ—_n to_o d_o let_v a_o straight_a line_n be_v draw_v by_o the_o first_o petition_n and_o by_o the_o second_o petition_n let_v the_o straight_a line_n â—d_a be_v extend_v as_o to_o the_o point_n m_o &_o n._n now_o forasmuch_o as_o the_o right_a line_n ab_fw-la from_o the_o point_n â—_o produce_v do_v cut_v the_o line_n mn_v by_o construction_n therefore_o by_o the_o second_o proposition_n of_o this_o eleven_o book_n the_o right_a line_n ab_fw-la &_o mn_o be_v in_o one_o plainâ—_n superficies_n which_o let_v be_v qr_o cut_v the_o superficies_n open_v in_o the_o right_a line_n mn_o by_o the_o same_o mean_n may_v we_o conclude_v the_o right_a line_n cd_o to_o be_v in_o one_o plain_a superficies_n with_o the_o right_a line_n mn_o but_o the_o right_a line_n mn_o by_o supposition_n be_v in_o the_o plain_a superficies_n qr_o wherefore_o cd_o be_v in_o the_o plain_a superficies_n qr_o and_o aâ—_z the_o right_a line_n be_v prove_v to_o be_v in_o the_o same_o plain_a superficies_n qr_o therefore_o ab_fw-la and_o cd_o be_v in_o one_o plain_a superficies_n namely_o qr_o and_o forasmuch_o as_o the_o line_n aâ—_z and_o cd_o by_o supposition_n be_v perpendicular_a upon_o the_o plain_a superficies_n open_v therefore_o by_o the_o second_o definition_n of_o this_o book_n with_o all_o the_o right_a line_n draw_v in_o the_o superficies_n open_v and_o touch_v ab_fw-la and_o cd_o the_o same_o perpendiculars_n aâ—_z and_o cd_o do_v make_v right_a angle_n but_o by_o construction_n mn_v be_v draw_v in_o the_o plain_a superficies_n open_a touch_v the_o perpendicular_n ab_fw-la and_o cd_o at_o the_o point_n â—_o and_o d._n therefore_o the_o perpendicular_n aâ—_z and_o cd_o make_v with_o the_o right_a line_n mn_v two_o right_a angle_n namely_o abn_n and_o cdm_o and_o mn_v the_o right_a line_n be_v prove_v to_o be_v in_o the_o one_o and_o the_o same_o plain_a superficies_n with_o the_o right_a line_n ab_fw-la &_o cd_o namely_o in_o the_o plain_a superficies_n qr_o wherefore_o by_o the_o second_o part_n of_o the_o 28._o proposition_n of_o the_o first_o book_n the_o right_a lineâ—_z ab_fw-la and_o cd_o be_v parallel_n if_o therefore_o two_o right_a line_n be_v erect_v perpendicular_o to_o one_o and_o the_o self_n same_o plain_a superficies_n those_o right_a line_n be_v parallel_n the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o m._n dee_n hereby_o it_o be_v evident_a that_o any_o two_o right_a line_n perpendicular_o erect_v to_o one_o and_o the_o self_n same_o plain_a superficies_n be_v also_o themselves_o in_o one_o and_o the_o same_o plain_a superficies_n which_o be_v likewise_o perpendicular_o erect_v to_o the_o same_o plain_a superficies_n unto_o which_o the_o two_o right_a line_n be_v perpendicular_a the_o first_o part_n hereof_o be_v prove_v by_o the_o former_a construction_n and_o demonstration_n that_o the_o right_a line_n ab_fw-la and_o cd_o be_v in_o one_o and_o the_o same_o plain_a superficies_n qâ—_n the_o second_o part_n be_v also_o manifest_a that_o be_v that_o the_o plain_a superficies_n qr_o be_v perpendicular_o erect_v upon_o the_o plain_a superficies_n open_v for_o that_o aâ—_z and_o cd_o be_v in_o the_o plain_a superficies_n qr_o be_v by_o supposition_n perpendicular_a to_o the_o plain_a superficies_n open_v wherefore_o by_o the_o three_o definition_n of_o this_o book_n qr_o be_v perpendicular_o erect_v to_o or_o upon_o open_a which_o be_v require_v to_o be_v prove_v io._n do_v you_o his_o advice_n upon_o the_o assumpt_n of_o the_o 6._o as_o concern_v the_o make_n of_o the_o line_n de_fw-fr equal_a to_o the_o right_a line_n ab_fw-la very_o the_o second_o of_o the_o first_o without_o some_o far_a consideration_n be_v not_o proper_o enough_o allege_v and_o no_o wonder_n it_o be_v for_o that_o in_o the_o former_a bookeâ—_n whatsoeâ—â—â—â—aâ—h_v of_o line_n be_v speak_v the_o same_o have_v always_o be_v imagine_v to_o be_v in_o one_o only_a plain_a superficies_n consider_v or_o execute_v but_o here_o the_o perpendicular_a line_n ab_fw-la be_v not_o in_o the_o same_o plaint_n superficies_n that_o the_o right_a line_n db_o be_v therefore_o some_o other_o help_n must_v be_v put_v into_o the_o hand_n of_o young_a beginner_n how_o to_o bring_v this_o problem_n to_o execution_n which_o be_v this_o most_o plain_a and_o brief_a understand_v that_o bd_o the_o right_n line_n be_v the_o common_a section_n of_o the_o plain_a superficies_n wherein_o the_o perpendicular_n ab_fw-la and_o cd_o be_v &_o of_o the_o other_o plain_a superficies_n to_o which_o they_o be_v perpendicular_n the_o first_o of_o these_o in_o my_o former_a demonstration_n of_o the_o 6_o â—_o i_o note_v by_o the_o plain_a superficies_n qr_o and_o the_o other_o i_o note_v by_o the_o plain_a superficies_n open_v wherefore_o bd_o be_v a_o right_a line_n common_a to_o both_o the_o plain_a supâ—rficieces_n qr_o &_o op_n thereby_o the_o ponit_n b_o and_o d_o be_v common_a to_o the_o plain_n qr_o and_o op_n now_o from_o bd_o sufficient_o extend_v cut_v a_o right_a line_n equal_a to_o ab_fw-la which_o suppose_v to_o be_v bf_o by_o the_o three_o of_o the_o first_o and_o orderly_a to_o bf_o make_v de_fw-fr equal_a by_o the_o 3._o oâ—_n the_o first_o if_o de_fw-fr be_fw-mi great_a than_o bf_o which_o always_o you_o may_v cause_v so_o to_o be_v by_o produce_v of_o de_fw-fr sufficient_o now_o forasmuch_o as_o bf_o by_o construction_n be_v cut_v equal_a to_o ab_fw-la and_o de_fw-fr also_o by_o construction_n put_v equal_a to_o bf_o therefore_o by_o the_o 1._o common_a sentence_n de_fw-fr be_v put_v equal_a to_o ab_fw-la which_o be_v require_v to_o be_v do_v in_o like_a sort_n if_o de_fw-fr be_v a_o line_n give_v to_o who_o ab_fw-la be_v to_o be_v cut_v and_o make_v equal_a first_o out_o of_o the_o line_n db_o sufficient_o produce_v cut_v of_o dg_o equal_a to_o de_fw-fr by_o the_o three_o of_o the_o first_o and_o by_o the_o same_o 3._o cut_v from_o basilius_n sufficient_o produce_v basilius_n equal_a to_o dg_v then_o be_v it_o evident_a that_o to_o the_o right_a line_n de_fw-fr the_o perpendicular_a line_n ab_fw-la be_v put_v equal_a and_o though_o this_o be_v easy_a to_o conceive_v yet_o i_o have_v design_v the_o figure_n according_o whereby_o you_o may_v instruct_v your_o imagination_n many_o such_o help_n be_v in_o this_o book_n requisite_a as_o well_o to_o inform_v the_o young_a student_n therewith_o as_o also_o to_o master_n the_o froward_a gaynesayer_n of_o our_o conclusion_n or_o interrupter_n of_o our_o demonstration_n course_n ¶_o the_o 7._o theorem_a the_o 7._o proposition_n if_o there_o be_v two_o parallel_n right_a line_n and_o in_o either_o of_o they_o be_v take_v a_o point_n at_o all_o adventure_n a_o right_a line_n draw_v by_o the_o say_a point_n be_v in_o the_o self_n same_o superficies_n with_o the_o parallel_n right_a line_n svppose_v that_o these_o two_o right_a line_n ab_fw-la and_o cd_o be_v parallel_n and_o in_o either_o of_o they_o take_v a_o point_n at_o all_o adventure_n namely_o e_o and_o f._n then_o i_o say_v that_o a_o right_a line_n draw_v from_o the_o point_n e_o to_o the_o point_n fletcher_n be_v in_o the_o self_n same_o plain_a superficies_n that_o the_o
angle_n bac_n god_n &_o dab_n be_v equal_a the_o one_o to_o the_o other_o then_o be_v it_o manifest_v that_o two_o of_o they_o which_o two_z so_o ever_o be_v take_v be_v great_a than_o the_o three_o but_o if_o not_o let_v the_o angle_n bac_n be_v the_o great_a of_o the_o three_o angle_n and_o unto_o the_o right_a line_n ab_fw-la and_o from_o the_o point_n a_o make_v in_o the_o plain_a superficies_n bac_n unto_o the_o angle_n dab_n a_o equal_a angle_n bae_o and_o by_o the_o 2._o of_o the_o first_o make_v the_o line_n ae_n equal_a to_o the_o line_n ad._n now_o a_o right_a line_n bec_n draw_v by_o the_o point_n e_o shall_v cut_v the_o right_a line_n ab_fw-la and_o ac_fw-la in_o the_o point_n b_o and_o c_o draw_v a_o right_a line_n from_o d_z to_o b_o and_o a_o outstretch_o from_o d_z to_o c._n and_o forasmuch_o as_o the_o line_n dam_n be_v equal_a to_o the_o line_n ae_n demonstration_n and_o the_o line_n ab_fw-la be_v common_a to_o they_o both_o therefore_o these_o two_o line_n dam_n and_o ab_fw-la be_v equal_a to_o these_o two_o line_n ab_fw-la and_o ae_n and_o the_o angle_n dab_n be_v equal_a to_o the_o angle_n bae_o wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a db_o be_v equal_a to_o the_o base_a be._n and_o forasmuch_o as_o these_o two_o line_n db_o and_o dc_o be_v great_a than_o the_o line_n bc_o of_o which_o the_o line_n db_o be_v prove_v to_o be_v equal_a to_o the_o line_n be._n wherefore_o the_o residue_n namely_o the_o line_n dc_o be_v great_a than_o the_o residue_n namely_o than_o the_o line_n ec_o and_o forasmuch_o as_o the_o line_n dam_n be_v equal_a to_o the_o line_n ae_n and_o the_o line_n ac_fw-la be_v common_a to_o they_o both_o and_o the_o base_a dc_o be_v great_a than_o the_o base_a ec_o therefore_o the_o angle_n dac_o be_v great_a than_o the_o angle_n eac_n and_o it_o be_v prove_v that_o the_o angle_n dab_n be_v equal_a to_o the_o angle_n bae_o wherefore_o the_o angle_n dab_n and_o dac_o be_v great_a than_o the_o angle_n bac_n if_o therefore_o a_o solid_a angle_n be_v contain_v under_o three_o plain_a superficial_a angle_n every_o two_o of_o those_o three_o angle_n which_o then_o so_o ever_o be_v take_v be_v great_a than_o the_o three_o which_o be_v require_v to_o be_v prove_v in_o this_o figure_n you_o may_v plain_o behold_v the_o former_a demonstration_n if_o you_o elevate_v the_o three_o triangle_n abdella_n aâ—c_n and_o acd_o in_o such_o â—orâ—that_o they_o may_v all_o meet_v together_o in_o the_o point_n a._n the_o 19_o theorem_a the_o 21._o proposition_n every_o solid_a angle_n be_v comprehend_v under_o plain_a angle_n less_o than_o four_o right_a angle_n svppose_v that_o a_o be_v a_o solid_a angle_n contain_v under_o these_o superficial_a angle_n bac_n dac_o and_o dab_n then_o i_o say_v that_o the_o angle_n bac_n dac_o and_o dab_n be_v less_o than_o four_o right_a angle_n construction_n take_v in_o every_o one_o of_o these_o right_a line_n acab_n and_o ad_fw-la a_o point_n at_o all_o adventure_n and_o let_v the_o same_o be_v b_o c_o d._n and_o draw_v these_o right_a line_n bc_o cd_o and_o db._n demonstration_n and_o forasmuch_o as_o the_o angle_n b_o be_v a_o solid_a angle_n for_o it_o be_v contain_v under_o three_o superficial_a angle_n that_o be_v under_o cba_o abdella_n and_o cbd_n therefore_o by_o the_o 20._o of_o the_o eleven_o two_o of_o they_o which_o two_z so_o ever_o be_v take_v be_v great_a than_o the_o three_o wherefore_o the_o angle_n cba_o and_o abdella_n be_v great_a than_o the_o angle_n cbd_v and_o by_o the_o same_o reason_n the_o angle_n bca_o and_o acd_o be_v great_a than_o the_o angle_n bcdâ—_n and_o moreover_o the_o angle_n cda_n and_o adb_o be_v great_a than_o the_o angle_n cdb_n wherefore_o these_o six_o angle_n cba_o abdella_n bca_o acd_o cda_n and_o adb_o be_v great_a they_o these_o three_o angle_n namely_o cbd_v bcd_o &_o cdb_n but_o the_o three_o angle_n cbd_v bdc_o and_o bcd_o be_v equal_a to_o two_o right_a angle_n wherefore_o the_o six_o angle_n cba_o abdella_n bca_o acd_o cda_n and_o adb_o be_v great_a they_o two_o right_a angle_n and_o forasmuch_o as_o in_o every_o one_o of_o these_o triangle_n abc_n and_o abdella_n and_o acd_o three_o angle_n be_v equal_a two_o right_a angle_n by_o the_o 32._o of_o the_o first_o wherefore_o the_o nine_o angle_n of_o the_o three_o triangle_n that_o be_v the_o angle_n cba_o acb_o bac_n acd_o dac_o cda_n adb_o dba_n and_o bad_a be_v equal_a to_o six_o right_a angle_n of_o which_o angle_n the_o six_o angle_n abc_n bca_o acd_o cda_n adb_o and_o dba_n be_v great_a than_o two_o right_a angle_n wherefore_o the_o angle_n remain_v namely_o the_o angle_n bac_n god_n and_o dab_n which_o contain_v the_o solid_a angle_n be_v less_o than_o sour_a right_a angle_n wherefore_o every_o solid_a angle_n be_v comprehend_v under_o plain_a angle_n less_o than_o four_o right_a angle_n which_o be_v require_v to_o be_v prove_v if_o you_o will_v more_o full_o see_v this_o demonstration_n compare_v it_o with_o the_o figure_n which_o i_o put_v for_o the_o better_a sight_n of_o the_o demonstration_n of_o the_o proposition_n next_o go_v before_o only_o here_o be_v not_o require_v the_o draught_n of_o the_o line_n ae_n although_o this_o demonstration_n of_o euclid_n be_v here_o put_v for_o solid_a angle_n contain_v under_o three_o superficial_a angle_n yet_o after_o the_o like_a manner_n may_v you_o proceed_v if_o the_o solid_a angle_n be_v contain_v under_o superficial_a angle_n how_o many_o so_o ever_o as_o for_o example_n if_o it_o be_v contain_v under_o four_o superficial_a angle_n if_o you_o follow_v the_o former_a construction_n the_o base_a will_v be_v a_o quadrangled_a figure_n who_o four_o angle_n be_v equal_a to_o four_o right_a angle_n but_o the_o 8._o angle_n at_o the_o base_n of_o the_o 4._o triangle_n set_v upon_o this_o quadrangled_a figure_n may_v by_o the_o 20._o proposition_n of_o this_o book_n be_v prove_v to_o be_v great_a than_o those_o 4._o angle_n of_o the_o quadrangled_a figure_n as_o we_o see_v by_o the_o discourse_n of_o the_o former_a demonstration_n wherefore_o those_o 8._o angle_n be_v great_a than_o four_o right_a angle_n but_o the_o 12._o angle_n of_o those_o four_o triangle_n be_v equal_a to_o 8._o right_a angle_n wherefore_o the_o four_o angle_n remain_v at_o the_o top_n which_o make_v the_o solid_a angle_n be_v less_o than_o four_o right_a angle_n and_o observe_v this_o course_n you_o may_v proceed_v infinite_o ¶_o the_o 20._o theorem_a the_o 22._o proposition_n if_o there_o be_v three_o superficial_a plain_a angle_n of_o which_o two_o how_o soever_o they_o be_v take_v be_v great_a than_o the_o three_o and_o if_o the_o right_a line_n also_o which_o contain_v those_o angle_n be_v equal_a then_o of_o the_o line_n couple_v those_o equal_a right_a line_n together_o it_o be_v possible_a to_o make_v a_o triangle_n svppose_v that_o there_o be_v three_o superficial_a angle_n abc_n def_n and_o ghk_v of_o which_o let_v two_o which_o two_o soever_o be_v take_v be_v great_a than_o the_o three_o that_o be_v let_v the_o angle_n abc_n and_o def_n be_v great_a than_o the_o angle_n ghk_o and_o let_v the_o angle_n def_n and_o ghk_o be_v great_a than_o the_o angle_n abc_n and_o moreover_o let_v the_o angle_n ghk_v and_o abc_n be_v great_a than_o the_o angle_n def_n and_o let_v the_o right_a line_n ab_fw-la bc_o de_fw-fr of_o gh_o and_o hk_o be_v equal_a the_o one_o to_o the_o other_o and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n c_o and_o a_o other_o from_o the_o point_n d_o to_o the_o point_n fletcher_n and_o moreover_o a_o other_o from_o the_o point_n g_o to_o the_o point_n k._n proposition_n then_o i_o say_v that_o it_o be_v possible_a of_o three_o right_a line_n equal_a to_o the_o line_n ac_fw-la df_o and_o gk_o to_o make_v a_o triangle_n that_o be_v that_o two_o of_o the_o right_a line_n ac_fw-la df_o and_o gk_o which_o then_o soever_o be_v take_v be_v great_a than_o the_o three_o now_o if_o the_o angle_n abc_n def_n case_n and_o ghk_v be_v equal_a the_o one_o to_o the_o other_o it_o be_v manifest_a that_o these_o right_a line_n ac_fw-la df_o and_o gk_o be_v also_o by_o the_o 4._o of_o the_o first_o equal_a the_o one_o to_o the_o other_o it_o be_v possible_a of_o three_o right_a line_n equal_a to_o the_o line_n ac_fw-la df_o and_o gk_o to_o make_v a_o triangle_n case_n but_o if_o they_o be_v not_o equal_a let_v they_o be_v unequal_a and_o by_o the_o 23._o of_o the_o first_o unto_o the_o right_a line_n hk_o construction_n and_o at_o the_o point_n in_o it_o h_o make_v unto_o the_o angle_n abc_n a_o equal_a angle_n khl._n and_o by_o the_o â—_o of_o the_o first_o to_o one_o of_o the_o line_n
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 â—fg_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 spâ—erâ—_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
triangle_n wherefore_o six_o such_o triangle_n as_o dbc_n be_v be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o bc_o thrice_o but_o six_o sâ—ch_a triangle_n as_o dbc_n be_v be_v equal_a to_o two_o such_o triangle_n as_o abc_n be_v wherefore_o that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o bc_o thrice_o be_v equal_a to_o two_o such_o triangle_n as_o abc_n be_v but_o two_o of_o those_o triangle_n take_v ten_o time_n contain_v the_o whole_a icosahedron_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n de_fw-fr &_o bc_o thirty_o time_n be_v equal_a to_o twenty_o such_o triangle_n as_o the_o triangle_n abc_n be_v that_o be_v to_o the_o whole_a superficies_n of_o the_o icosahedron_n follow_v wherefore_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 that_o which_o be_v contain_v under_o the_o line_n cd_o and_o fg_o to_o that_o which_o be_v contain_v under_o the_o line_n bc_o and_o de._n ¶_o corollary_n by_o this_o it_o be_v manifest_a 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 order_n so_o be_v that_o which_o be_v contain_v under_o the_o side_n of_o the_o pentagon_n and_o the_o perpendicular_a line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n describe_v about_o the_o pentagon_n to_o the_o same_o side_n to_o that_o which_o be_v contain_v under_o the_o side_n of_o the_o icosahedron_n and_o the_o perpendicular_a line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n describe_v about_o the_o triangle_n to_o the_o same_o side_n so_o that_o the_o icosahedron_n and_o dodecahedron_n be_v both_o describe_v in_o one_o and_o the_o self_n same_o sphere_n ¶_o the_o 4._o theorem_a the_o 4._o proposition_n flussas_n this_o be_v do_v now_o be_v to_o be_v prove_v 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 side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n construction_n take_v by_o the_o 2._o theorem_a of_o this_o book_n a_o circle_n contain_v both_o the_o pentagon_n of_o a_o dodecahedron_n and_o the_o triangle_n of_o a_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 let_v the_o same_o circle_n be_v dbc_n and_o in_o the_o circle_n dbc_n describe_v the_o side_n of_o a_o equilater_n triangle_n namely_o cd_o and_o the_o side_n of_o a_o equilater_n pentagon_n namely_o ac_fw-la and_o take_v by_o the_o 1._o of_o the_o three_o the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v e._n and_o from_o the_o point_n e_o draw_v unto_o the_o line_n dc_o and_o ac_fw-la perpendicular_a line_n of_o and_o eglantine_n and_o extend_v the_o line_n eglantine_n direct_o to_o the_o point_n b._n and_o draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n c._n and_o let_v the_o side_n of_o the_o cube_fw-la be_v the_o line_n h._n now_o i_o say_v 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 line_n h_o to_o the_o line_n cd_o demonstration_n forasmuch_o as_o the_o line_n make_v of_o the_o line_n ebb_v and_o bc_o add_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 a_o decagon_n be_v by_o the_o 9_o of_o the_o thirteen_o divide_a by_o a_o extreme_a and_o mean_a proportion_n and_o his_o great_a segment_n be_v the_o line_n be_v and_o the_o line_n eglantine_n be_v also_o by_o the_o 1._o of_o the_o foâ—retenth_n the_o half_a of_o the_o same_o line_n and_o the_o line_n of_o be_v the_o half_a of_o the_o line_n be_v by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o the_o line_n eglantine_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n â—â—â—eth_v his_o great_a segment_n shall_v be_v the_o line_n ef._n and_o the_o line_n h_o also_o be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n his_o great_a segment_n be_v the_o line_n ca_n as_o it_o be_v prove_v tâ—irtenth_o in_o the_o dodecahedron_n proposition_n wherefore_o as_o the_o line_n h_o be_v to_o the_o line_n ca_n so_o be_v the_o line_n eglantine_n to_o the_o line_n ef._n wherefore_o by_o the_o 16._o of_o the_o six_o that_o which_o be_v contain_v under_o the_o line_n h_o and_o of_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ca_n and_o eglantine_n and_o for_o that_o as_o the_o line_n h_o be_v to_o the_o line_n cd_o so_o be_v that_o which_o be_v contain_v under_o the_o line_n h_o and_o of_o to_o that_o which_o be_v contain_v under_o the_o line_n cd_o and_o of_o by_o the_o 1._o of_o the_o six_o but_o unto_o that_o which_o be_v contain_v under_o the_o line_n h_o and_o of_o be_v equal_a that_o which_o be_v contain_v under_o the_o line_n ca_n and_o eglantine_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o line_n h_o be_v to_o the_o line_n cd_o so_o be_v that_o which_o be_v contain_v under_o the_o line_n ca_n and_o eglantine_n to_o that_o which_o be_v contain_v under_o the_o line_n cd_o and_o of_o that_o be_v by_o the_o corollary_n next_o go_v before_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 line_n h_o to_o the_o line_n cd_o an_o other_o demonstration_n to_o prove_v 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 side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n prove_v let_v there_o be_v a_o circle_n abc_n and_o in_o it_o describe_v two_o side_n of_o a_o equilater_n pentagon_n by_o the_o 11._o of_o the_o five_o namely_o ab_fw-la and_o ac_fw-la and_o draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n c._n and_o by_o the_o 1._o of_o the_o three_o take_v the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v d._n and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n d_o and_o extend_v it_o direct_o to_o the_o point_n e_o and_o let_v it_o cut_v the_o line_n bc_o in_o the_o point_n g._n and_o let_v the_o line_n df_o be_v half_a to_o the_o line_n dam_fw-ge and_o let_v the_o line_n gc_o be_v treble_a to_o the_o line_n hc_n by_o the_o 9_o of_o the_o six_o proposition_n now_o i_o say_v that_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o bh_o be_v equal_a to_o the_o pentagon_n inscribe_v in_o the_o circle_n abc_n draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n d._n now_o forasmuch_o as_o the_o line_n ad_fw-la be_v double_a to_o the_o line_n df_o therefore_o the_o line_n of_o be_v sesquialter_fw-la to_o the_o line_n ad._n again_o assumpt_n forasmuch_o as_o the_o line_n gc_o be_v treble_a to_o the_o line_n change_z therefore_o the_o line_n gh_o be_v double_a to_o the_o line_n ch._n wherefore_o the_o line_n gc_o be_v sesquialter_fw-la to_o the_o line_n hg_o wherefore_o as_o the_o line_n favorina_n be_v to_o the_o line_n ad_fw-la so_o be_v the_o line_n gc_o to_o the_o line_n gh_o wherefore_o by_o the_o 16._o of_o the_o six_o that_o which_o be_v contain_v under_o the_o line_n of_o &_o hg_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n dam_n and_o gc_o but_o the_o line_n gc_o be_v equal_a to_o the_o line_n bg_o by_o the_o 3._o of_o the_o three_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o bg_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o gh_o but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o bg_o be_v equal_a to_o two_o such_o triangle_n as_o the_o triangle_n abdella_n be_v by_o the_o 41._o of_o the_o first_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o gh_o be_v equal_a to_o two_o such_o triangle_n as_o the_o triangle_n abdella_n be_v wherefore_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o gh_o â—ive_a time_n be_v equal_a to_o ten_o triangle_n but_o ten_o triangle_n be_v two_o pentagon_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o gh_o five_o time_n be_v equal_a to_o two_o pentagon_n and_o forasmuch_o as_o the_o line_n gh_o be_v double_a to_o the_o line_n hc_n therefore_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o gh_o be_v double_a to_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o hc_n by_o the_o 1._o of_o the_o six_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o change_z twice_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n
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 timâ—s_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 sâ—des_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 â—g_z iâ—_z to_o the_o liâ—â—_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 â—râ—_n cut_v proportional_o by_o the_o second_o of_o this_o bâ—oke_n whâ—râ—fore_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 wherâ—fore_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 contaynâ—d_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 linâ—â—_n cg_o and_o â—z_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 oâ—â—_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 liâ—â—_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 â—â—_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 aâ—cig_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 aâ—h_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 â—ases_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 aâ—câ—g_n for_o forasmuch_o as_o the_o line_n â—z_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 â—2â—_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 〈◊〉_d â—t_a wherefore_o that_o which_o be_v contain_v undeâ—_n the_o 〈◊〉_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 â—k_v â—roved_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 〈…〉_o equal_a â—o_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 〈…〉_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 â—â—tâ—_n therefore_o that_o which_o be_v contain_v under_o the_o line_n az_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 〈◊〉_d abâ—dâ—_n every_o one_o of_o who_o sâ—des_n 〈◊〉_d be_v divide_v into_o two_o equal_a part_n in_o the_o poyâ—â—â—â—â—_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 trianglâ—â—_n lâ—â—_n pmn_n nkl_n and_o 〈…〉_o which_o triangle_n shall_v be_v equilater_n by_o the_o 4._o of_o the_o firâ—t_o â—or_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 liâ—e_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 abâ—d_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 subâ—â—â—d_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 icosahâ—dronâ—_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 â—pnk_v be_v a_o square_n cut_v the_o pyramid_n abâ—d_v into_o two_o equal_a paâ—â—â—â—_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 abâ—d_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 â—che_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