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understand_v as_o be_v that_o before_o set_v as_o if_o the_o line_n a_o be_v equal_a to_o the_o line_n b_o and_o if_o the_o line_n c_o be_n also_o equal_a to_o the_o line_n b_o then_o of_o necessity_n the_o line_n a_o and_o c_o shall_v be_v equal_a the_o one_o to_o the_o other_o so_o be_v it_o in_o all_o superâ—iciesses_n angle_n &_o number_n &_o in_o all_o other_o thing_n of_o one_o kind_n that_o may_v be_v compare_v together_o 2_o and_o if_o you_o add_v equal_a thing_n to_o equal_a thing_n the_o whole_a shall_v be_v equal_v as_o if_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n cd_o &_o to_o the_o line_n ab_fw-la be_v add_v the_o line_n be_v &_o to_o the_o line_n cd_o be_v add_v also_o a_o other_o line_n df_o be_v equal_a to_o the_o line_n be_v so_o that_o two_o equal_a line_n namely_o be_v and_o df_o be_v add_v to_o two_o equal_a line_n ab_fw-la &_o cd_o then_o shall_v the_o whole_a line_n ae_n be_v equal_a to_o the_o whole_a line_n cf_o and_o so_o of_o all_o quantity_n general_o 3_o and_o if_o from_o equal_a thing_n you_o take_v away_o equal_a thing_n the_o thing_n remain_v shall_v be_v equal_a as_o if_o from_o the_o two_o line_n ab_fw-la and_o cd_o be_v equal_a you_o take_v away_o two_o equal_a line_n namely_o ebb_n and_o fd_o then_o may_v you_o conclude_v by_o this_o common_a sentence_n that_o the_o part_n remain_v namely_o ae_n and_o cf_o be_v equal_a the_o one_o to_o the_o other_o and_o so_o of_o all_o other_o quantity_n 4_o and_o if_o from_o unequal_a thing_n you_o take_v away_o equal_a thing_n the_o thing_n which_o remain_v shall_v be_v unequal_a as_o if_o the_o line_n ab_fw-la and_o cd_o be_v unequal_a
eglantine_n the_o side_n which_o include_v the_o equal_a angle_n be_v proportional_a wherefore_o the_o parallelogram_n abcd_o be_v by_o the_o first_o definition_n of_o the_o six_o like_v unto_o the_o parallelogram_n eglantine_n and_o by_o the_o same_o reason_n also_o the_o parallelogram_n abcd_o be_v like_a to_o the_o parallelogram_n kh_o abcd_o wherefore_o either_o of_o these_o parallelogram_n eglantine_n and_o kh_o be_v like_a unto_o the_o parallelogram_n abcd._n but_o rectiline_a figure_n which_o be_v like_a to_o one_o and_o the_o same_o rectiline_a figure_n be_v also_o by_o the_o 21._o of_o the_o six_o like_o the_o one_o to_o the_o other_o wherefore_o the_o parallelogram_n eglantine_n be_v like_a to_o the_o parallelogram_n hk_o other_o wherefore_o in_o every_o parallelogram_n the_o parallelogram_n about_o the_o dimecient_a be_v like_a unto_o the_o whole_a and_o also_o like_o the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o more_o brief_a demonstration_n after_o flussates_n suppose_v that_o there_o be_v a_o parallelogram_n abcd_o who_o dimetient_a let_v bâ—_z aâ—_z flussates_n about_o which_o let_v consist_v these_o parallelogram_n eke_o and_o ti_o have_v the_o angle_n at_o the_o point_n â—_o and_o 〈…〉_o with_o the_o whole_a parallelogram_n abcd._n then_o i_o say_v that_o those_o parallelogram_n eke_o and_o ti_fw-mi be_v like_a to_o the_o whole_a parallelogram_n db_o and_o also_o alâ—_z like_o the_o one_o to_o the_o other_o for_o forasmuch_o as_o bd_o eke_o and_o ti_fw-mi be_v parallelogram_n therefore_o the_o right_a line_n azg_v fall_v upon_o these_o parallel_a line_n aeb_fw-mi kzt_n and_o di_z g_z or_o upon_o these_o parallel_a line_n akd_v ezi_n and_o btg_n make_v these_o angle_n equal_a the_o one_o to_o the_o other_o namely_o the_o angle_n eaz_n to_o the_o angle_n kza_n &_o the_o angle_n eza_n to_o the_o angle_n kaz_n and_o the_o angle_n tzg_v to_o the_o angle_n zgi_n and_o the_o angle_n tgz_n to_o the_o angle_n izg_v and_o the_o angle_n bag_n to_o the_o angle_n agd_v and_o final_o the_o angle_n bga_n to_o the_o angle_n dag_n wherefore_o by_o the_o first_o corollary_n of_o the_o 32._o of_o the_o first_o and_o by_o the_o 34._o of_o the_o first_o the_o angle_n remain_v be_v equal_v the_o one_o to_o the_o other_o namely_o the_o angle_n b_o to_o the_o angle_n d_o and_o the_o angle_n e_o to_o the_o angle_n king_n and_o the_o angle_n t_o to_o the_o angle_n i_o wherefore_o these_o triangle_n be_v equiangle_n and_o therefore_o like_o the_o one_o to_o the_o other_o namely_o the_o triangle_n abg_o to_o the_o triangle_n gda_n and_o the_o triangle_n aez_n to_o the_o triangle_n zka_n &_o the_o triangle_n ztg_n to_o the_o triangle_n giz._n wherefore_o as_o the_o side_n ab_fw-la be_v to_o the_o side_n bg_o so_o be_v the_o side_n ae_n to_o the_o side_n ez_fw-fr and_o the_o side_n zt_n to_o the_o side_n tg_n wherefore_o the_o parallelogram_n contain_v under_o those_o right_a line_n namely_o the_o parallelogram_n abgd_v eke_o &_o ti_fw-mi be_v like_o the_o one_o to_o the_o other_o by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o every_o parallelogram_n the_o parallelogram_n etc_n etc_n as_o before_o which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o problem_n add_v by_o pelitarius_n two_o equiangle_n parallelogrammes_n be_v give_v so_o that_o they_o be_v not_o like_a to_o cut_v of_o from_o one_o of_o they_o a_o parallelogram_n like_a unto_o the_o other_o pelitarius_n suppose_v that_o the_o two_o equiangle_n parallelogram_n be_v abcd_o and_o cefg_n which_o let_v not_o be_v like_o the_o one_o to_o the_o other_o it_o be_v require_v from_o the_o parallelogram_n abcd_o to_o cut_v of_o a_o parallelogram_n like_a unto_o the_o parallelogram_n cefg_n let_v the_o angle_n c_o of_o the_o one_o be_v equal_a to_o the_o angle_n c_o of_o the_o other_o and_o let_v the_o two_o parallelogram_n be_v so_o 〈◊〉_d that_o the_o line_n bc_o &_o cg_o may_v make_v both_o one_o right_a line_n namely_o bg_o wherefore_o also_o the_o right_a line_n dc_o and_o ce_fw-fr shall_v both_o make_v one_o right_a line_n namely_o de._n and_o draw_v a_o line_n from_o the_o point_n f_o to_o the_o point_n c_o and_o produce_v the_o line_n fc_o till_o it_o concur_v with_o the_o line_n ad_fw-la in_o the_o point_n h._n and_o draw_v the_o line_n hk_o parallel_v to_o the_o line_n cd_o by_o the_o 31._o of_o the_o first_o then_o i_o say_v that_o from_o the_o parallelogram_n ac_fw-la be_v cut_v of_o the_o parallelogram_n cdhk_n like_v unto_o the_o parallelogram_n eglantine_n which_o thing_n be_v manifest_a by_o this_o 24._o proposition_n for_o that_o both_o the_o say_v parallelogram_n be_v describe_v about_o one_o &_o the_o self_n same_o dimetient_fw-la and_o to_o the_o end_n it_o may_v the_o more_o plain_o be_v see_v i_o have_v make_v complete_a the_o parallelogram_n abgl_n ¶_o a_o other_o problem_n add_v by_o pelitarius_n between_o two_o rectiline_a superficiece_n to_o find_v out_o a_o mean_a superficies_n proportional_a pelitarius_n suppose_v that_o the_o two_o superficiece_n be_v a_o and_o b_o between_o which_o it_o be_v require_v to_o place_v a_o mean_a superficies_n proportional_a reduce_v the_o say_v two_o rectiline_a figure_n a_o and_o b_o unto_o two_o like_a parallelogram_n by_o the_o 18._o of_o this_o book_n or_o if_o you_o think_v good_a reduce_v either_o of_o they_o to_o a_o square_a by_o the_o last_o of_o the_o second_o and_o let_v the_o say_v two_o parallelogram_n like_o the_o one_o to_o the_o other_o and_o equal_a to_o the_o superficiece_n a_o and_o b_o be_v cdef_n and_o fghk_n and_o let_v the_o angle_n fletcher_n in_o either_o of_o they_o be_v equal_a which_o two_o angle_n let_v be_v place_v in_o such_o sort_n that_o the_o two_o parallelogram_n ed_z and_o hg_o may_v be_v about_o one_o and_o the_o self_n same_o dimetient_fw-la ck_o which_o be_v do_v by_o put_v the_o right_a line_n of_o and_o fg_o in_o such_o sort_n that_o they_o both_o make_v one_o right_a line_n namely_o eglantine_n and_o make_v complete_a the_o parallelogram_n clk_n m._n then_o i_o say_v that_o either_o of_o the_o supplement_n fl_fw-mi &_o fm_o be_v a_o mean_a proportional_a between_o the_o superficiece_n cf_o &_o fk_o that_o be_v between_o the_o superficiece_n a_o and_o b_o namely_o as_o the_o superficies_n hg_o be_v to_o the_o superficies_n fl_fw-mi so_o be_v the_o same_o superficies_n fl_fw-mi to_o the_o superficies_n ed._n for_o by_o this_o 24._o proposition_n the_o line_n hf_o be_v to_o the_o line_n fd_o as_o the_o line_n gf_o be_v to_o the_o line_n fe_o but_o by_o the_o first_o of_o this_o book_n as_o the_o line_n hf_o be_v to_o the_o line_n fd_o so_o be_v the_o superficies_n hg_o to_o the_o superficies_n fl_fw-mi and_o as_o the_o line_n gf_o be_v to_o the_o line_n fe_o so_o also_o by_o the_o same_o be_v the_o superficies_n fl_fw-mi to_o the_o superficies_n ed._n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o superficies_n hg_o be_v to_o the_o superficies_n fl_fw-mi so_o be_v the_o same_o superficies_n fl_fw-mi to_o the_o superficies_n ed_z which_o be_v require_v to_o be_v do_v the_o 7._o problem_n the_o 25._o proposition_n unto_o a_o rectiline_a figure_n give_v to_o describe_v a_o other_o figure_n like_o which_o shall_v also_o be_v equal_a unto_o a_o other_o rectiline_a figure_n give_v svppose_v that_o the_o rectiline_a figure_n give_v whereunto_o be_v require_v a_o other_o to_o be_v make_v like_o be_v abc_n and_o let_v the_o other_o rectiline_a figure_n whereunto_o the_o same_o be_v require_v to_o be_v make_v equal_a be_v d._n now_o it_o be_v require_v to_o describe_v a_o rectiline_a figure_n like_o unto_o the_o figure_n abc_n and_o equal_a unto_o the_o figure_n d._n construction_n upon_o the_o line_n bc_o describe_v by_o the_o 44._o of_o the_o first_o a_o parallelogram_n be_v equal_a unto_o the_o triangle_n abc_n and_o by_o the_o same_o upon_o the_o line_n ce_fw-fr describe_v the_o parallelogram_n c_o m_o equal_a unto_o the_o rectiline_a figure_n d_o and_o in_o the_o say_a parallelogram_n let_v the_o angle_n fce_n be_v equal_a unto_o the_o angle_n cbl_n and_o forasmuch_o as_o the_o angle_n fce_n be_v by_o construction_n equal_a to_o the_o angle_n cbl_n demonstration_n add_v the_o angle_n be_v common_a to_o they_o both_o wherefore_o the_o angle_n lbc_n and_o be_v be_v equal_a unto_o the_o angle_n be_v and_o ecf_n but_o the_o angle_n lbc_n and_o be_v be_v equal_a to_o two_o right_a angle_n by_o the_o 29._o of_o the_o first_o wherefore_o also_o the_o angle_n be_v and_o ecf_n be_v equal_a to_o two_o right_a angle_n wherefore_o the_o line_n bc_o and_o cf_o by_o the_o 14._o of_o the_o first_o make_v both_o one_o right_a line_n namely_o bf_o and_o in_o like_a sort_n do_v the_o line_n le_fw-fr and_o em_n make_v both_o one_o right_a line_n namely_o lm_o then_o by_o the_o
great_a perfection_n than_o be_v a_o line_n but_o here_o in_o the_o definition_n of_o a_o solid_a or_o body_n euclid_n attribute_v unto_o it_o all_o the_o three_o dimension_n length_n breadth_n and_o thickness_n wherefore_o a_o solid_a be_v the_o most_o perfect_a quantity_n quantity_n which_o want_v no_o dimension_n at_o all_o pass_v a_o line_n by_o two_o dimension_n and_o pass_v a_o superficies_n by_o one_o this_o definition_n of_o a_o solid_a be_v without_o any_o designation_n of_o â—orme_n or_o figure_n easy_o understand_v only_o conceive_v in_o mind_n or_o behold_v with_o the_o eye_n a_o piece_n of_o timber_n or_o stone_n or_o what_o matter_n so_o ever_o else_o who_o dimension_n let_v be_v equal_a or_o unequal_a for_o example_n let_v the_o length_n thereof_o be_v 5._o inch_n the_o breadth_n 4._o and_o the_o thickness_n 2._o if_o the_o dimension_n be_v equal_a the_o reason_n be_v like_a and_o all_o one_o as_o it_o be_v in_o a_o sphere_n and_o in_o â—_o cube_fw-la for_o in_o that_o respect_n and_o consideration_n only_o that_o it_o be_v long_o broad_a and_o thick_a it_o bear_v the_o name_n of_o a_o solid_a or_o body_n â—nd_n have_v the_o nature_n and_o property_n thereof_o there_o be_v add_v to_o the_o endâ—_n of_o the_o definition_n of_o a_o solid_a that_o the_o term_n and_o limit_v of_o a_o solid_a â—s_n a_o superficies_n of_o thing_n infinity_n there_o iâ—_z no_o art_n or_o science_n all_o quantity_n therefore_o in_o this_o art_n entreat_v of_o be_v imagine_v to_o be_v finite_a infinite_a and_o to_o have_v their_o end_n and_o border_n as_o have_v be_v show_v in_o the_o first_o book_n that_o the_o limit_n and_o end_n of_o a_o line_n be_v point_n and_o the_o limit_n or_o border_n of_o a_o superficies_n be_v line_n so_o now_o he_o say_v thaâ—_z the_o end_n limit_n or_o border_n of_o a_o solideâ—_n be_v superficiece_n as_o the_o side_n of_o any_o square_a piece_n of_o timber_n or_o of_o a_o table_n or_o die_v or_o any_o other_o like_a be_v the_o term_n and_o limit_n of_o they_o 2_o a_o right_a line_n be_v then_o erect_v perpendicular_o to_o a_o pl_z 〈…〉_o erficies_n when_o the_o right_a line_n make_v right_a angle_n with_o all_o the_o line_n 〈…〉_o it_o definition_n and_o be_v draw_v upon_o the_o ground_n plain_a superficies_n suppose_v that_o upon_o the_o ground_n plain_a superficies_n cdef_n from_o the_o point_n b_o be_v erect_v a_o right_a line_n namely_o â—a_o so_o that_o let_v the_o point_v a_o be_v a_o loâ—e_n in_o the_o air_n draw_v also_o from_o the_o point_n â—_o in_o the_o plain_a superficies_n cdbf_n as_o many_o right_a line_n as_o you_o listen_v as_o the_o line_n bc_o bd_o â—â—_o bf_n bg_o hk_o bh_o and_o bl_n if_o the_o erect_a line_n basilius_n with_o all_o these_o line_n draw_v in_o the_o superficies_n cdef_n make_v a_o right_a angle_n so_o that_o all_o those_o â—ngles_v aâ—â—_n aâ—d_n aâ—e_n abfâ—_n aâ—g_v aâ—k_n abh_n abl_n and_o so_o of_o other_o be_v right_a angle_n then_o by_o this_o definition_n the_o line_n ab_fw-la iâ—_z a_o line_n â—â—â—cted_v upon_o the_o superficies_n cdef_n it_o be_v also_o call_v common_o a_o perpendicular_a line_n or_o a_o plumb_n line_n unto_o or_o upon_o a_o superficies_n definition_n 3_o a_o plain_a superficies_n be_v then_o upright_o or_o erect_v perpendicular_o to_o a_o plain_a superficies_n when_o all_o the_o right_a line_n draw_v in_o one_o of_o the_o plain_a superficiece_n unto_o the_o common_a section_n of_o those_o two_o plain_a superficiece_n make_v therewith_o right_a angle_n do_v also_o make_v right_a angle_n to_o the_o other_o plain_a superficies_n inclination_n or_o lean_v of_o a_o right_a line_n to_o a_o plain_a superficies_n be_v a_o acute_a angle_n contain_v under_o a_o right_a line_n fall_v from_o a_o point_n above_o to_o the_o plain_a superficies_n and_o under_o a_o other_o right_a line_n from_o the_o low_a end_n of_o the_o say_a line_n let_v down_o draw_v in_o the_o same_o plain_a superficies_n by_o a_o certain_a point_n assign_v where_o a_o right_a line_n from_o the_o first_o point_n above_o to_o the_o same_o plain_a superficies_n fall_v perpendicular_o touch_v in_o this_o three_o definition_n be_v include_v two_o definition_n the_o first_o be_v of_o a_o plain_a superficies_n erect_v perpendicular_o upon_o a_o plain_a superficies_n definition_n the_o second_o be_v of_o the_o inclination_n or_o lean_v of_o a_o right_a line_n unto_o a_o superficies_n of_o the_o first_o take_v this_o example_n suppose_v you_o have_v two_o superâ—icieces_n abcd_o and_o cdef_a of_o which_o let_v the_o superficies_n cdef_n be_v a_o ground_n plain_a superficies_n and_o let_v the_o superficies_n abcd_o be_v erect_v unto_o it_o and_o let_v the_o line_n cd_o be_v a_o common_a term_n or_o intersection_n to_o they_o both_o that_o be_v let_v it_o be_v the_o end_n or_o bind_v of_o either_o of_o they_o part_n &_o be_v draw_v in_o either_o of_o they_o in_o which_o line_n note_n at_o pleasure_n certain_a point_n as_o the_o point_n g_o h._n from_o which_o point_n unto_o the_o line_n cd_o draw_v perpendicular_a line_n in_o the_o superficies_n abcd_o which_o let_v be_v gl_n and_o hk_o which_o fall_v upon_o the_o superficies_n cdef_n if_o they_o cause_v right_a angle_n with_o it_o that_o be_v with_o line_n draw_v in_o it_o from_o the_o same_o point_n g_z and_o h_o as_o if_o the_o angle_n lgm_n or_o the_o angle_n lgn_v contain_v under_o the_o line_n â—g_z draw_v in_o the_o superficies_n erect_v and_o under_o the_o gm_n or_o gn_n draw_v in_o the_o ground_n superficies_n cdef_n lie_v flat_a be_v a_o right_a angle_n then_o by_o this_o definition_n the_o superficies_n abcd_o be_v upright_o or_o erect_v upon_o the_o superficies_n cdef_n it_o be_v also_o common_o call_v a_o superficies_n perpendicular_a upon_o or_o unto_o a_o superficies_n part_n for_o the_o second_o part_n of_o this_o definition_n which_o be_v of_o the_o inclination_n of_o a_o right_a line_n unto_o a_o plain_a superficies_n take_v this_o example_n let_v abcd_o be_v a_o ground_n plain_a superficies_n upon_o which_o from_o a_o point_n be_v a_o loft_n namely_o the_o point_n e_o suppose_v a_o right_a line_n to_o fall_v which_o let_v be_v the_o line_n eglantine_n touch_v the_o plain_a superficies_n abcd_o at_o the_o point_n g._n again_o from_o the_o point_n e_o be_v the_o top_n or_o high_a limit_n and_o end_n of_o the_o incline_a line_n eglantine_n let_v a_o perpendicular_a line_n fall_v unto_o the_o plain_a superficies_n abcd_o which_o let_v be_v the_o line_n of_o and_o let_v fletcher_n be_v the_o point_n where_o of_o touch_v the_o plain_a superficies_n abcd._n then_o from_o the_o point_n of_o the_o fall_n of_o the_o line_n incline_v upon_o the_o superficies_n unto_o the_o point_n of_o the_o fall_n of_o the_o perpendicular_a line_n upon_o the_o same_o superficies_n that_o be_v from_o the_o point_n g_o to_o the_o point_n fletcher_n draw_v a_o right_a line_n gf_o now_o by_o this_o definition_n the_o acute_a angle_n egf_n be_v the_o inclination_n of_o the_o line_n eglantine_n unto_o the_o superficies_n abcd._n because_o it_o be_v contain_v of_o the_o incline_a line_n and_o of_o the_o right_a line_n draw_v in_o the_o superficies_n from_o the_o point_n of_o the_o fall_n of_o the_o line_n incline_v to_o the_o point_n of_o the_o fall_n of_o the_o perpendicular_a line_n which_o angle_n must_v of_o necessity_n be_v a_o acute_a angle_n for_o the_o angle_n efg_o be_v by_o construction_n a_o right_a angle_n and_o three_o angle_n in_o a_o triangle_n be_v equâ—_fw-la 〈…〉_o â—ight_a angle_n wherefore_o the_o other_o two_o angle_n namely_o the_o angle_n egf_n and_o gef_n be_v equâ—_fw-la 〈…〉_o right_n angle_n wherefore_o either_o of_o they_o be_v less_o than_o a_o right_a angle_n wherefore_o the_o angle_n egf_n be_v a_o 〈…〉_o gle_a definition_n 4_o inclination_n of_o a_o plain_a superficies_n to_o a_o plain_a superficies_n be_v a_o acute_a angle_n contain_v under_o the_o right_a line_n which_o be_v draw_v in_o either_o of_o the_o plain_a superficiece_n to_o one_o &_o the_o self_n same_o point_n of_o the_o common_a section_n make_v with_o the_o section_n right_a angle_n suppose_v that_o there_o be_v two_o superficiece_n abcd_o &_o efgh_a and_o let_v the_o superficies_n abcd_o be_v suppose_v to_o be_v erect_v not_o perpendicular_o but_o somewhat_o lean_v and_o incline_v unto_o the_o plain_a superficies_n efgh_o as_o much_o or_o as_o little_a as_o you_o will_v the_o common_a term_n or_o section_n of_o which_o two_o superficiece_n let_v be_v the_o line_n cd_o from_o some_o one_o point_n aâ—_z from_o the_o point_n m_o assign_v in_o the_o common_a section_n of_o the_o two_o superficiece_n namely_o in_o the_o line_n cd_o draw_v a_o perpendicular_a line_n in_o either_o superficies_n in_o the_o ground_n superficies_n efgh_o draw_v the_o line_n mk_o and_o in_o the_o superficies_n abcd_o draw_v the_o line_n ml_o now_o
the_o word_n of_o psellus_n in_o his_o epitome_n of_o geometry_n where_o he_o entreat_v of_o the_o production_n and_o constitution_n of_o these_o body_n his_o word_n be_v these_o psellus_n all_o rectiline_v figure_n be_v erect_v upon_o their_o plain_n or_o base_n by_o right_a angle_n make_v prism_n who_o perceive_v not_o but_o that_o a_o pentagon_n erect_v upon_o his_o base_a of_o â—ive_a side_n make_v by_o his_o motion_n a_o side_v column_n of_o five_o side_n likewise_o a_o hexagon_n erect_v at_o right_a angle_n produce_v a_o column_n have_v six_o side_n and_o so_o of_o all_o other_o rectillne_v figure_n all_o which_o solid_n or_o body_n so_o produce_v whether_o they_o be_v side_v column_n or_o parallelipipedon_n be_v here_o in_o most_o plain_a word_n of_o this_o excellent_a and_o ancient_a greek_a author_n psellus_n call_v prism_n wherefore_o if_o the_o definition_n of_o a_o prism_n give_v of_o euclid_n shall_v extend_v itself_o so_o large_o as_o flussas_n imagine_v and_o shall_v include_v such_o figure_n or_o body_n as_o he_o note_v he_o ought_v not_o yet_o for_o all_o that_o so_o much_o to_o be_v offend_v and_o so_o narrow_o to_o have_v seek_v fault_n for_o euclid_n in_o so_o define_v may_v have_v that_o meaning_n &_o sense_n of_o a_o prism_n which_o psellus_n have_v so_o you_o see_v that_o euclid_n may_v be_v defend_v either_o of_o these_o two_o way_n either_o by_o that_o that_o the_o definition_n extend_v not_o to_o these_o figure_n and_o so_o not_o to_o be_v over_o general_n nor_o stretch_v far_o than_o it_o ought_v or_o else_o by_o that_o that_o if_o it_o shall_v stretch_v so_o far_o it_o be_v not_o so_o heinous_a for_o that_o as_o you_o see_v many_o have_v takâ—_n it_o in_o that_o sense_n in_o deed_n common_o a_o prism_n be_v take_v in_o that_o signification_n and_o meaning_n in_o which_o campaâ—â—â—_n flussas_n and_o other_o take_v it_o in_o which_o sense_n it_o seem_v also_o that_o in_o diverse_a proposition_n in_o these_o book_n follow_v it_o ought_v of_o necessity_n to_o be_v take_v 12_o a_o sphere_n be_v a_o figure_n which_o be_v make_v definition_n when_o the_o diameter_n of_o a_o semicircle_n abide_v fix_v the_o semicircle_n be_v turn_v round_o about_o until_o it_o return_v unto_o the_o self_n same_o place_n from_o whence_o it_o begin_v to_o be_v move_v to_o the_o end_n we_o may_v full_o and_o perfect_o understand_v this_o definition_n how_o a_o sphere_n be_v produce_v of_o the_o motion_n of_o a_o semicircle_n it_o shall_v be_v expedient_a to_o consider_v how_o quantity_n mathematical_o be_v by_o imagination_n conceive_v to_o be_v produce_v by_o flow_v and_o motion_n as_o be_v somewhat_o touch_v in_o the_o begin_n of_o the_o first_o book_n ever_o the_o less_o quantity_n by_o his_o motion_n bring_v forâ—h_v the_o quantity_n next_o above_o it_o as_o a_o point_n move_v flow_a or_o glide_v bring_v forth_o a_o line_n which_o be_v the_o first_o quantity_n and_o next_o to_o a_o point_n a_o line_n move_v produce_v a_o superficies_n which_o be_v the_o second_o quantity_n and_o next_o unto_o a_o line_n and_o last_o of_o all_o a_o superficies_n move_v bring_v forth_o a_o solid_a or_o body_n which_o be_v the_o three_o &_o last_o quantity_n these_o thing_n well_o mark_v it_o shall_v not_o be_v very_o hard_a to_o attain_v to_o the_o right_a understanding_n of_o this_o definition_n upon_o the_o line_n ab_fw-la be_v the_o diameter_n describe_v a_o semicircle_n acb_o who_o centre_n let_v be_v d_o the_o diameter_n ab_fw-la be_v six_v on_o his_o end_n orâ—_n point_n imagine_v the_o whole_a superficies_n of_o the_o semicircle_n to_o move_v round_o from_o some_o one_o point_n assign_v till_o it_o return_v to_o the_o same_o point_n again_o so_o shall_v it_o produce_v a_o perfect_a sphere_n or_o globe_n the_o form_n whereof_o you_o see_v in_o a_o ball_n or_o bowl_n and_o it_o be_v full_o round_a and_o solid_a for_o that_o it_o be_v describe_v of_o a_o semicircle_n which_o be_v perfect_o round_a as_o our_o country_n man_n johannes_n de_fw-fr sacro_fw-la busco_n in_o his_o book_n of_o the_o sphere_n busco_n of_o this_o definition_n which_o he_o take_v out_o of_o euclid_n do_v well_o collect_n but_o it_o be_v to_o be_v note_v and_o take_v heed_n of_o that_o none_o be_v deceive_v by_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 a_o sphere_n say_v he_o be_v the_o passage_n or_o move_v of_o the_o circumference_n of_o a_o semicircle_n till_o it_o return_v unto_o the_o place_n where_o it_o begin_v which_o agree_v not_o with_o euclid_n euclid_n plain_o say_v that_o a_o sphere_n be_v the_o passage_n or_o motion_n of_o a_o semicircle_n and_o not_o the_o passage_n or_o motion_n of_o the_o circumference_n of_o a_o semicircle_n neither_o can_v it_o be_v true_a that_o the_o circumference_n of_o a_o semicircle_n which_o be_v a_o line_n shall_v describe_v a_o body_n it_o be_v before_o note_v that_o every_o quantity_n move_v describe_v and_o produce_v the_o quantity_n next_o unto_o it_o wherefore_o a_o line_n move_v can_v not_o bring_v forth_o a_o body_n but_o a_o superficies_n only_o as_o if_o you_o imagine_v a_o right_a line_n fasten_v at_o one_o of_o his_o end_n to_o move_v about_o from_o some_o one_o point_n till_o it_o return_v to_o the_o same_o again_o it_o shall_v describe_v a_o plain_a superficies_n namely_o a_o circle_n so_o also_o if_o you_o likewise_o conceive_v of_o a_o crooked_a line_n such_o as_o be_v the_o circumference_n of_o a_o semicircle_n that_o his_o diameter_n fasten_v on_o both_o the_o end_n it_o shall_v move_v from_o a_o point_n assign_v till_o it_o return_v to_o the_o same_o again_o it_o shall_v describe_v &_o produce_v a_o â—ound_a superficies_n only_o which_o be_v the_o superficies_n and_o limit_n of_o the_o sphere_n and_o shall_v not_o produce_v the_o body_n and_o solidity_n of_o the_o sphere_n but_o the_o whole_a semicircle_n which_o be_v a_o superficies_n by_o his_o motion_n as_o be_v before_o say_v produce_v a_o body_n that_o be_v a_o perfect_a sphere_n so_o see_v you_o the_o error_n of_o this_o definition_n of_o the_o author_n of_o the_o sphere_n which_o whether_o it_o happen_v by_o the_o author_n himself_o which_o i_o think_v not_o or_o that_o that_o particle_n be_v thrust_v in_o by_o some_o one_o after_o he_o which_o be_v more_o likely_a it_o it_o not_o certain_a but_o it_o be_v certain_a that_o it_o be_v unapt_o put_v in_o and_o make_v a_o untrue_a definition_n which_o thing_n be_v not_o here_o speak_v any_o thing_n to_o derogate_v the_o author_n of_o the_o book_n which_o assure_o be_v a_o man_n of_o excellent_a knowledge_v neither_o to_o the_o hindrance_n or_o diminish_n of_o the_o worthiness_n of_o the_o book_n which_o undoubted_o be_v a_o very_a necessary_a book_n than_o which_o i_o know_v none_o more_o mere_a to_o be_v teach_v and_o read_v in_o school_n touch_v the_o ground_n and_o principle_n of_o astronomy_n and_o geography_n but_o only_o to_o admonish_v the_o young_a and_o unskilful_a reader_n of_o not_o fall_v into_o error_n sphere_n theodosius_n in_o his_o book_n de_fw-fr sphericis_n a_o book_n very_o necessary_a for_o all_o those_o which_o will_v see_v the_o ground_n and_o principle_n of_o geometry_n and_o astronomy_n which_o also_o i_o have_v translate_v into_o our_o vulgar_a tongue_n ready_a to_o the_o press_n define_v a_o sphere_n after_o this_o manner_n a_o sphere_n be_v a_o solid_a or_o body_n contain_v under_o one_o superficies_n in_o the_o middle_n whereof_o there_o be_v a_o point_n from_o which_o all_o line_n draw_v to_o the_o circumference_n be_v equal_a this_o definition_n of_o theodosius_n be_v more_o essential_a and_o natural_a then_o be_v the_o other_o give_v by_o euclid_n the_o other_o do_v not_o so_o much_o declare_v the_o inward_a nature_n and_o substance_n of_o a_o sphere_n as_o it_o show_v the_o industry_n and_o knowledge_n of_o the_o produce_v of_o a_o sphere_n and_o therefore_o be_v a_o causal_n definition_n give_v by_o the_o cause_n efficient_a or_o rather_o a_o description_n then_o a_o definition_n but_o this_o definition_n be_v very_o essential_a declare_v the_o nature_n and_o substance_n of_o a_o sphere_n as_o if_o a_o circle_n shall_v be_v thus_o define_v as_o it_o well_o may_v a_o circle_n be_v the_o passage_n or_o move_v of_o a_o line_n from_o a_o point_n till_o it_o return_v to_o the_o same_o point_n against_o it_o be_v a_o causal_n definition_n show_v the_o efficient_a cause_n whereof_o a_o circle_n be_v produce_v namely_o of_o the_o motion_n of_o a_o line_n and_o it_o be_v a_o very_a good_a description_n full_o show_v what_o a_o circle_n be_v such_o like_a description_n be_v the_o definition_n of_o a_o sphere_n give_v oâ—_n euclideâ—_n â—_z by_o the_o motion_n of_o a_o semicircle_n but_o when_o a_o circle_n be_v define_v to_o be_v a_o plain_a superficies_n
in_o the_o midst_n whereof_o be_v a_o point_n from_o which_o all_o line_n draw_v to_o the_o circumference_n thereof_o be_v equal_a this_o definition_n be_v essential_a and_o formal_a and_o declare_v the_o very_a nature_n of_o a_o circle_n and_o unto_o this_o definition_n of_o a_o circle_n be_v correspondent_a the_o definition_n of_o a_o sphere_n give_v by_o theodosius_n say_v that_o it_o be_v a_o solid_a oâ—_n body_n in_o the_o midst_n whereof_o there_o be_v a_o point_n from_o which_o all_o the_o line_n draw_v to_o the_o circumference_n be_v equal_a so_o see_v you_o the_o affinity_n between_o a_o circle_n and_o a_o sphere_n for_o what_o a_o circle_n be_v in_o a_o plain_a that_o be_v a_o sphere_n in_o a_o solid_a the_o fullness_n and_o content_n of_o a_o circle_n be_v describe_v by_o the_o motion_n of_o a_o line_n move_v about_o but_o the_o circumference_n thereof_o which_o be_v the_o limit_n and_o border_n thereof_o be_v describe_v of_o the_o end_n and_o point_n of_o the_o same_o line_n move_v about_o so_o the_o fullness_n content_n and_o body_n of_o a_o sphere_n or_o globe_n be_v describe_v of_o a_o semicircle_n move_v about_o but_o the_o spherical_a superficies_n which_o be_v the_o limit_n and_o border_n of_o a_o sphere_n sphere_n be_v describe_v of_o the_o circumference_n of_o the_o same_o semicircle_n move_v about_o and_o this_o be_v the_o superficies_n mean_v in_o the_o definition_n when_o it_o be_v say_v that_o it_o be_v contain_v under_o one_o superficies_n which_o superficies_n be_v call_v of_o johannes_n de_fw-fr â—acro_fw-la busco_n &_o other_o the_o circumference_n of_o the_o sphere_n sphâ—râ—_n galene_n in_o his_o book_n de_fw-fr diffinitionibus_fw-la mediciâ—â—_n â—_z give_v yet_o a_o other_o definition_n of_o a_o sphere_n by_o his_o property_n or_o common_a accidence_n of_o move_v which_o be_v thus_o a_o sphere_n be_v a_o figure_n most_o apt_a to_o all_o motion_n as_o have_v no_o base_a whereon_o the_o stay_n this_o be_v a_o very_a plain_a and_o witty_a definition_n declare_v the_o dignity_n thereof_o above_o all_o figure_n general_o sphere_n all_o other_o body_n or_o solid_n as_o cube_n pyramid_n and_o other_o have_v side_n base_n and_o angle_n all_o which_o be_v stay_v to_o rest_v upon_o or_o impediment_n and_o let_n to_o motion_n but_o the_o sphere_n have_v no_o side_n or_o base_a to_o stay_v one_o nor_o angle_n to_o let_v the_o course_n thereof_o but_o only_o in_o a_o point_n touch_v the_o plain_n wherein_o 〈◊〉_d stand_v move_v free_o and_o full_o with_o out_o let_v and_o for_o the_o dignity_n and_o worthiness_n thereof_o this_o circular_a and_o spherical_a motion_n be_v attribute_v to_o the_o heaven_n which_o be_v the_o most_o worthy_a body_n wherefore_o there_o be_v ascribe_v unto_o they_o this_o chief_a kind_n of_o motion_n this_o solid_a or_o bodily_a figure_n be_v also_o common_o call_v a_o globe_n globe_n 13_o the_o axe_n of_o a_o sphere_n be_v that_o right_a line_n which_o abide_v fix_v about_o which_o the_o 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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
semicircle_n to_o be_v the_o plain_a superficies_n which_o pass_v by_o it_o shall_v make_v in_o the_o superficies_n of_o the_o sphere_n a_o circle_n and_o it_o be_v manifest_a that_o be_v also_o a_o great_a circle_n for_o the_o diameter_n of_o the_o sphere_n which_o be_v also_o the_o diameter_n of_o the_o semicircle_n and_o therefore_o also_o of_o the_o circle_n be_v by_o the_o 15._o of_o the_o three_o great_a then_o all_o the_o right_a line_n draw_v in_o the_o circle_n or_o sphere_n which_o circle_n shall_v have_v both_o one_o centre_n be_v both_o also_o in_o that_o one_o plain_a superficies_n be_v by_o which_o the_o sphere_n be_v cut_v suppose_v that_o that_o section_n or_o circle_n in_o the_o great_a sphere_n be_v bcde_v and_o in_o the_o less_o be_v the_o circle_n fgh_n draw_v the_o diameter_n of_o those_o two_o circle_n in_o such_o sort_n that_o they_o make_v right_a angle_n and_o let_v those_o diameter_n be_v bd_o and_o ce._n construction_n and_o let_v the_o line_n agnostus_n be_v part_n of_o the_o line_n ab_fw-la be_v the_o semidiameter_n of_o the_o less_o sphere_n and_o circle_n as_z ab_fw-la be_v the_o semidiameter_n of_o the_o great_a sphere_n and_o great_a circle_n both_o the_o sphere_n and_o circle_n have_v one_o and_o the_o same_o centre_n now_o two_o circle_n that_o be_v bcde_v and_o fgh_a consist_v both_o about_o one_o and_o the_o self_n same_o centre_n be_v give_v let_v there_o be_v describe_v by_o the_o proposition_n next_o go_v before_o in_o the_o great_a circle_n bcde_v a_o polygonon_n figure_n consist_v of_o equal_a and_o even_a side_n not_o touch_v the_o less_o circle_n fgh_a and_o let_v the_o side_n of_o that_o figure_n in_o the_o four_o part_n of_o the_o circle_n namely_o in_o be_v be_v bk_o kl_o lm_o and_o me._n and_o draw_v a_o right_a line_n from_o the_o point_n king_n to_o the_o point_n a_o and_o extend_v it_o to_o the_o point_n n._n and_o by_o the_o 12._o of_o the_o eleven_o from_o the_o a_o raise_v up_o to_o the_o superficies_n of_o the_o circle_n bcde_v a_o perpendicular_a line_n axe_n and_o let_v it_o light_v upon_o the_o superficies_n of_o the_o great_a sphere_n in_o the_o point_n x._o noteâ—_n and_o by_o the_o line_n axe_n and_o by_o either_o of_o these_o line_n bd_o and_o kn_n extend_v plain_a superficiece_n now_o by_o that_o which_o be_v before_o speak_v those_o plain_a superficiece_n shall_v in_o the_o perceive_v superficies_n of_o the_o sphere_n make_v two_o great_a circle_n let_v their_o semicircle_n consist_v upon_o the_o diameter_n bd_o and_o kn_n be_v bxd_v and_o kxn_o and_o forasmuch_o satisfy_v as_o the_o line_n xa_n be_v erect_v perpendicular_o to_o the_o plain_a superficies_n of_o the_o circle_n bcde_v therefore_o all_o the_o plain_a superficiece_n which_o be_v draw_v by_o the_o line_n xa_n be_v erect_v perpendicular_o to_o the_o superficies_n of_o the_o circle_n bcde_v by_o the_o 18._o of_o the_o eleven_o wherefore_o the_o semicircle_n bxd_v and_o kxn_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n of_o the_o circle_n bcde_v and_o forasmuch_o as_o the_o semicircle_n bed_n bxd_v and_o kxn_o be_v equal_a for_o they_o consist_v upon_o equal_a diameter_n â—d_a and_o kn_n therefore_o also_o the_o four_o part_n or_o quarter_n of_o those_o circle_n namely_o be_v bx_n and_o kx_n be_v equal_a the_o one_o to_o the_o other_o wherefore_o how_o many_o side_n of_o a_o polygonon_n figure_n there_o be_v in_o the_o four_o part_n or_o quarter_n be_v so_o many_o also_o be_v there_o in_o the_o other_o four_o part_n or_o quarter_n bx_n and_o kx_n equal_a to_o the_o right_a line_n bk_o kl_o lm_o and_o me._n let_v those_o side_n be_v describe_v and_o let_v they_o be_v boy_n open_v pr._n rx_n k_n st_z tu_fw-la and_o vx_n follow_v and_o draw_v these_o right_a line_n so_o tâ—_n and_o vâ—_n and_o from_o the_o point_n oh_o and_o â—_o draw_v to_o the_o plain_a superficies_n of_o the_o circle_n bcde_v perpendicular_a line_n which_o perpendicular_a line_n will_v fall_v upon_o the_o common_a section_n of_o the_o plain_a superficiece_n namely_o upon_o the_o line_n bd_o &_o kn_n by_o the_o 38._o of_o the_o eleven_o for_o that_o the_o plain_a superficiece_n of_o the_o semicircle_n bxd_v kxn_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n of_o the_o circle_n bcde_v let_v those_o perpendicular_a line_n be_v gz_n demonstration_n and_o sweet_a and_o draw_v a_o right_a line_n from_o the_o point_n z_o to_o the_o point_n w._n and_o forasmuch_o as_o in_o the_o equal_a semicircle_n bxd_v and_o kxn_o the_o right_a line_n boy_n and_o k_n be_v equal_a from_o the_o end_n whereof_o be_v draw_v perpendicular_a line_n oz_n and_o sweet_a therefore_o by_o the_o corollary_n of_o the_o 35._o of_o the_o eleven_o the_o line_n oz_n be_v equal_a to_o the_o line_n sweet_a &_o the_o line_n bz_o be_v equal_a to_o the_o line_n kw_n flussas_n prove_v this_o a_o other_o way_n thus_o forasmuch_o as_o in_o the_o triangle_n swk_n and_o ozb_n the_o two_o angle_n swk_n and_o ozb_n be_v equal_a for_o that_o by_o construction_n they_o be_v right_a angle_n and_o by_o the_o 27._o of_o three_o the_o angle_n wk_n and_o zbo_n be_v equal_a for_o they_o subtend_v equal_a circumference_n sxn_v and_o oxd_v and_o the_o side_n sk_n be_v equal_a to_o the_o side_n ob_fw-la as_o it_o have_v before_o be_v prove_v wherefore_o by_o the_o 26._o of_o the_o first_o the_o other_o side_n &_o angle_n be_v equal_a namely_o the_o line_n oz_n to_o the_o line_n sweet_a and_o the_o line_n bz_o to_o the_o line_n kw_n but_o the_o whole_a line_n basilius_n be_v equal_a to_o the_o whole_a line_n ka_o by_o the_o definition_n of_o a_o circle_n wherefore_o the_o residue_n za_n be_v equal_a to_o the_o residue_n wa_n wherefore_o the_o line_n zw_n be_v a_o parallel_n to_o the_o line_n bk_o by_o the_o 2._o of_o the_o six_o and_o forasmuch_o as_o either_o of_o these_o line_n oz_n &_o sweet_a be_v erect_v perpendicular_o to_o the_o plain_a superficies_n of_o the_o circle_n bcde_v therefore_o the_o line_n oz_n be_v a_o parallel_n to_o the_o line_n sweet_a by_o the_o 6._o of_o the_o eleven_o and_o it_o be_v prove_v that_o it_o be_v also_o equal_a unto_o it_o wherefore_o the_o line_n wz_n and_o so_o be_v also_o equal_a and_o parallel_n by_o the_o 7_o of_o the_o eleven_o and_o the_o 33._o of_o the_o first_o and_o by_o the_o 3._o of_o the_o first_o and_o forasmuch_o as_o wz_n be_v a_o parallel_n to_o so_o but_o zw_n be_v a_o parallel_n be_v to_o kb_v wherefore_o so_o be_v also_o a_o parallel_n to_o kb_v by_o the_o 9_o of_o the_o eleven_o and_o the_o line_n boy_n and_o k_n do_v knit_v they_o together_o wherefore_o the_o four_o side_v figure_n book_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n for_o by_o the_o 7._o of_o the_o eleven_o if_o there_o be_v any_o two_o parallel_n right_a line_n and_o if_o in_o either_o of_o they_o be_v take_v a_o point_n at_o allaventure_n a_o right_a line_n draw_v by_o these_o point_n be_v in_o one_o &_o the_o self_n same_o plain_a superficies_n with_o the_o parallel_n and_o by_o the_o same_o reason_n also_o every_o one_o of_o the_o four_o side_v figure_n sopt_v and_o tpkv_n be_v in_o one_o and_o thâ—_z self_n same_o plain_a superficies_n and_o the_o triangle_n vrx_n be_v also_o in_o one_o and_o the_o self_n same_o plain_a superficies_n by_o the_o â—_o of_o the_o eleven_o now_o if_o we_o imagine_v right_a line_n draw_v from_o the_o point_n oh_o s_o p_o t_o r_o five_o to_o the_o point_n 〈◊〉_d there_o shall_v be_v describe_v a_o polyhedron_n or_o a_o solid_a figure_n of_o many_o side_n between_o the_o circumference_n bx_n and_o kx_n compose_v of_o pyramid_n who_o base_n be_v the_o four_o side_v figure_n bkos_n sopt_v tprv_n and_o the_o triangle_n vrx_n and_o top_n the_o point_n a._n and_o if_o in_o every_o one_o of_o the_o side_n kâ—l_o lm_o and_o i_o we_o use_v the_o self_n same_o construction_n that_o we_o do_v in_o bk_o and_o moreover_o in_o the_o other_o three_o quadrant_n or_o quarter_n and_o also_o in_o the_o other_o half_n of_o the_o sphere_n there_o shall_v then_o be_v make_v a_o polyhedron_n or_o solid_a figure_n consist_v of_o many_o side_n describe_v in_o the_o sphere_n which_o polyhedron_n be_v make_v of_o the_o pyramid_n who_o base_n be_v the_o foresay_a four_o side_v figure_n and_o the_o triangle_n vrk_n and_o other_o which_o be_v in_o the_o self_n same_o order_n with_o they_o and_o common_a top_n to_o they_o all_o in_o the_o point_n a._n now_o i_o say_v that_o the_o foresaid_a polihedron_n solid_a of_o many_o side_n touch_v not_o the_o superficies_n of_o the_o less_o sphere_n in_o which_o be_v the_o circle_n fgh_o construction_n draw_v by_o the_o 11._o of_o the_o eleven_o from_o the_o point_n a_o to_o the_o
the_o circumference_n on_o which_o the_o icosahedron_n be_v describe_v and_o that_o the_o diameter_n of_o the_o sphere_n be_v compose_v of_o the_o side_n of_o a_o hexagon_n and_o of_o two_o side_n of_o a_o decagon_n describe_v in_o one_o and_o the_o self_n same_o circle_n flussas_n prove_v the_o icosahedron_n describe_v to_o be_v contain_v in_o a_o sphere_n by_o draw_v right_a line_n from_o the_o point_n a_o to_o the_o point_n p_o and_o g_o after_o this_o manner_n forasmuch_o as_o the_o line_n zw_n wp_n be_v put_v equal_a to_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumferenceâ—_n and_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n be_v double_a to_o the_o line_n awe_n by_o construction_n therefore_o the_o line_n wp_n be_v also_o double_a to_o the_o same_o line_n awe_n wherefore_o the_o square_a of_o the_o line_n wp_n be_v quadruple_a to_o the_o square_n of_o the_o line_n awe_n by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o and_o those_o line_n pw_o and_o wa_n contain_v a_o right_a angle_n pwa_n as_o have_v before_o be_v prove_v be_v subtend_v of_o the_o right_a line_n ap_fw-mi wherefore_o by_o the_o 47._o of_o the_o first_o the_o line_n ap_fw-mi contain_v in_o power_n the_o line_n pw_o and_o wa._n wherefore_o the_o right_a line_n ap_fw-mi be_v in_o power_n quintuple_a to_o the_o line_n wa._n wherefore_o the_o right_a line_n ap_fw-mi and_o aq_n be_v quintuple_a to_o one_o and_o the_o same_o line_n wa_n be_v by_o the_o 9_o of_o the_o flu_v equal_a in_o like_a sort_n also_o may_v we_o prove_v that_o unto_o those_o line_n ap_fw-mi and_o aq_n be_v equal_a the_o rest_n of_o the_o line_n draw_v from_o the_o point_n a_o to_o the_o rest_n of_o the_o angle_n r_o s_o t_o v._o for_o they_o subtend_v right_a angle_n contain_v of_o the_o line_n wa_n and_o of_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n and_o forasmuch_o as_o unto_o the_o line_n wa_n be_v equal_a the_o line_n valerio_n which_o be_v likewise_o erect_v perpendicular_o unto_o the_o other_o plain_a superficies_n olmnx_n therefore_o line_n draw_v from_o the_o point_n a_o to_o the_o angle_n oh_o l_o m_o n_o x_o and_o subtend_v right_a angle_n at_o the_o point_n z_o contain_v under_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n and_o under_o the_o line_n az_n be_v equal_a not_o only_o the_o one_o to_o the_o other_o but_o also_o to_o the_o line_n draw_v from_o the_o say_a point_n a_o to_o the_o former_a angle_n at_o the_o point_n p_o r_o s_o t_o vâ—_n for_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n of_o each_o circle_n be_v equal_a &_o the_o line_n awe_n be_v equal_a to_o the_o line_n az._n but_o the_o line_n ap_fw-mi be_v prove_v equal_a to_o the_o line_n aq_n which_o be_v the_o half_a of_o the_o whole_a qy_n therefore_o the_o residue_n ay_o be_v equal_a to_o the_o foresay_a line_n ap_fw-mi aq_n etc_n etc_n wherefore_o make_v the_o centre_n the_o point_n a_o and_o the_o space_n one_o of_o those_o line_n aq_n ap_fw-mi etc_n etc_n extend_v the_o superficies_n of_o a_o sphere_n &_o it_o shall_v touch_v the_o 12._o angle_n of_o the_o icosahedron_n which_o be_v at_o the_o point_n oh_o l_o m._n n_o x_o pea_n king_n s_o t_o five_o q_o y_fw-fr which_o sphere_n be_v describe_v if_o upon_o the_o diameter_n qy_n be_v draw_v a_o semicircle_n and_o the_o say_a semicircle_n be_v move_v about_o till_o it_o return_v unto_o the_o same_o place_n from_o whence_o it_o begin_v first_o to_o be_v move_v ¶_o a_o corollary_n add_v by_o flussas_n the_o opposite_a side_n of_o a_o icosahedron_n be_v parallel_n for_o the_o diameter_n of_o the_o sphere_n do_v fall_v upon_o the_o opposite_a angle_n of_o the_o icosahedron_n as_o it_o be_v manifest_a by_o the_o right_a line_n qy_n if_o therefore_o there_o be_v imagine_v to_o be_v draw_v the_o two_o diameter_n pn_n and_o om_o they_o shall_v concur_v in_o the_o point_n f_o wherefore_o the_o right_a line_n which_o join_v they_o together_o pv_n and_o ln_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n by_o the_o 2._o of_o the_o eleven_o and_o forasmuch_o as_o the_o alternate_a angle_n at_o the_o end_n of_o the_o diameter_n be_v equal_a by_o the_o 8._o of_o the_o first_o for_o the_o triangle_n contain_v under_o equal_a semidiameteâ—s_n and_o the_o side_n of_o the_o icosahedron_n be_v equiangle_n therefore_o by_o the_o 28._o of_o the_o first_o the_o line_n pv_n and_o ln_o be_v paralle_n ¶_o the_o 5._o problem_n the_o 17._o proposition_n to_o make_v a_o dodecahedron_n and_o to_o comprehend_v it_o in_o the_o sphere_n give_v wherein_o be_v comprehend_v the_o foresay_a solid_n and_o to_o prove_v that_o the_o side_n of_o the_o dodecahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o residual_a line_n superficies_n now_o i_o say_v that_o it_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n forasmuch_o as_o the_o line_n zv_n be_v a_o parallel_n to_o the_o line_n sir_n as_o be_v before_o prove_v but_o unto_o the_o same_o line_n sir_n be_v the_o line_n cb_o a_o parallel_n by_o the_o 28._o of_o the_o first_o wherefore_o by_o the_o 9_o of_o the_o eleven_o the_o line_n we_o be_v a_o parallel_n to_o the_o line_n cb._n wherefore_o by_o the_o seven_o of_o the_o eleven_o the_o right_a line_n which_o join_v they_o together_o be_v in_o the_o self_n same_o plain_n wherein_o be_v the_o parallel_a line_n wherefore_o the_o trapesium_n buzc_n be_v in_o one_o plain_n and_o the_o triangle_n bwc_n be_v in_o one_o plain_n by_o the_o 2._o of_o the_o eleven_o now_o to_o prove_v that_o the_o trapesium_n buzc_n &_o the_o triangle_n bwc_n be_v in_o one_o and_o the_o self_n same_o plain_a we_o must_v prove_v that_o the_o right_a line_n yh_n and_o hw_o be_v make_v direct_o one_o right_a line_n which_o thing_n be_v thus_o prove_v forasmuch_o as_o the_o line_n hp_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n t_o and_o his_o great_a segment_n be_v the_o line_n pt_v therefore_o as_o the_o line_n hp_n be_v to_o the_o line_n pt_v so_o be_v the_o line_n pt_v to_o the_o line_n th._n but_o the_o line_n hp_n be_v equal_a to_o the_o line_n ho_o and_o the_o line_n pt_v to_o either_o of_o these_o line_n tw_n and_o oy_o wherefore_o as_o the_o line_n ho_o be_v to_o the_o line_n oy_fw-it so_o be_v the_o line_n with_o to_o the_o line_n th._n but_o the_o line_n ho_o and_o tw_n be_v side_n of_o like_a proportion_n be_v parallel_n by_o the_o 6._o of_o the_o eleven_o for_o either_o of_o they_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n bd_o â—_o and_o the_o line_n th._n and_o oy_o be_v parallel_n which_o be_v also_o side_n of_o like_a proportion_n by_o the_o same_o 6._o of_o the_o eleven_o for_o either_o of_o they_o be_v also_o erect_v perpendicular_o to_o the_o plain_a superficies_n bf_o but_o when_o there_o be_v two_o triangle_n have_v two_o side_n proportional_a to_o two_o side_n so_o set_v upon_o one_o angle_n that_o their_o side_n of_o like_a proportion_n be_v also_o parallel_n as_o the_o triangle_n yoh_n and_o htw_n be_v â—_o who_o two_o side_n oh_o &_o ht_o be_v in_o the_o two_o base_n of_o the_o cube_fw-la make_v a_o angle_n at_o the_o point_n h_o the_o side_n remain_v of_o those_o triangle_n shall_v by_o the_o 32._o of_o the_o six_o be_v in_o one_o right_a line_n wherefore_o the_o line_n yh_n &_o hw_o make_v both_o one_o right_a line_n but_o every_o right_a line_n be_v by_o the_o 3._o of_o the_o eleven_o in_o one_o &_o the_o self_n same_o plain_a superficies_n wherefore_o if_o you_o draw_v a_o right_a line_n from_o b_o to_o a'n_z there_o shall_v be_v make_v a_o triangle_n bye_o which_o shall_v be_v in_o one_o and_o the_o self_n same_o plain_a by_o the_o 2._o of_o the_o eleven_o and_o therefore_o the_o whole_a pentagon_n figure_n vbwcz_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n now_o also_o i_o say_v that_o it_o be_v equiangle_n equiangle_n for_o forasmuch_o as_o the_o right_a line_n no_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n r_o and_o his_o great_a segment_n be_v or_o therefore_o as_o both_o the_o line_n no_o and_o or_o add_v together_o be_v to_o the_o line_n on_o so_o by_o the_o 5._o of_o this_o book_n be_v the_o line_n on_o to_o the_o line_n or_o but_o the_o line_n or_o be_v equal_a to_o the_o line_n os_fw-mi wherefore_o as_o the_o line_n sn_a be_v to_o the_o line_n no_o so_o be_v the_o line_n no_o to_o the_o line_n os_fw-mi wherefore_o the_o line_n sn_a be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n oh_o
a_o right_a line_n couple_v their_o centre_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n the_o right_a line_n which_o couple_v the_o centre_n of_o the_o next_o base_n if_o by_o the_o centre_n of_o five_o base_n set_v upon_o one_o base_a be_v draw_v a_o plain_a superficies_n and_o by_o the_o centre_n of_o the_o base_n which_o be_v set_v upon_o the_o opposite_a base_a be_v draw_v also_o a_o plain_a superficies_n and_o then_o be_v draw_v a_o right_a line_n couple_v the_o centre_n of_o the_o opposite_a base_n that_o right_a line_n be_v so_o cut_v that_o each_o of_o his_o part_n set_v without_o the_o plain_a superficies_n be_v the_o great_a segment_n of_o that_o part_n which_o be_v contain_v between_o the_o plain_n the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o the_o dodecahedron_n to_o one_o of_o the_o base_n be_v in_o power_n quintuple_a to_o half_a the_o line_n which_o be_v between_o the_o plain_n and_o therefore_o the_o whole_a line_n which_o couple_v the_o centre_n of_o the_o opposite_a base_n be_v in_o power_n quintuple_a to_o the_o whole_a line_n which_o be_v between_o the_o say_a plain_n the_o line_n which_o subtâ—deth_v the_o angle_n of_o the_o base_a of_o the_o dodecahedron_n together_o with_o the_o side_n of_o the_o base_a be_v in_o power_n quintuple_a to_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n which_o contain_v the_o base_a to_o the_o circumference_n a_o section_n of_o a_o sphere_n contain_v three_o base_n of_o the_o dodecahedron_n take_v a_o three_o part_n of_o the_o diameter_n of_o the_o say_a sphere_n the_o side_n of_o the_o dodecahedron_n and_o the_o line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n be_v equal_a to_o the_o right_a line_n which_o couple_v the_o middle_a section_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n ¶_o the_o end_n of_o the_o element_n of_o geometry_n of_o the_o most_o ancient_a philosopher_n 〈◊〉_d of_o megara_n the_o intent_n of_o this_o preface_n number_n note_v the_o word_n unit_fw-la to_o express_v the_o greek_a monaâ—_n &_o not_o unity_n as_o we_o hauâ—_n all_o common_o till_o now_o use_v magnitude_n a_o point_n a_o line_n magnitude_n ano._n 1488._o ☞_o arithmetic_n note_n note_n anno._n 1550._o r._n b._n note_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d ☞_o this_o noble_a earl_n die_v anno._n 1554._o scarce_o of_o 24._o year_n of_o ageâ—_n have_v no_o issue_n by_o his_o wife_n daughter_n to_o the_o duke_n of_o somerset_n justice._n ☞_o ☞_o plato_n 7._o de_fw-fr rep._n ☞_o ☞_o note_n the_o difference_n between_o strataruhmetrie_n and_o tacticie_n i.d._n fâ—endâ—â—_n you_o will_v find_v it_o hard_o to_o perform_v my_o description_n of_o â—his_n fâ—ate_n but_o by_o châ—râ—graphieâ—_n you_o may_v help_v yourself_o some_o â—hat_n wherâ—_n thâ—_z figure_v know_v in_o sidâ—â—â—nd_a angle_n be_v not_o regular_a and_o â—_z resolution_n into_o triangle_n can_v sâ—â—uâ—_n etc_n etc_n and_o yet_o you_o will_v find_v it_o strange_a to_o deal_v thus_o general_o with_o arithmetical_a figure_n and_o that_o for_o battle_n â—ay_n their_o coâ—tentâ—â—â—_n differ_v so_o much_o from_o like_o geometrâ—call_a â—_z a_o marvelous_a glass_n ☞_o s.w.p._n ☞_o note_n 1._o 2._o 3._o 5._o 6._o 7._o 8._o i.d._n read_v in_o aristotle_n his_o 8._o book_n of_o politic_n the_o 5_o 6_o and_o 7._o chapter_n where_o you_o shall_v have_v some_o occasion_n far_o to_o think_v of_o music_n than_o common_o be_v think_v ☜_o ☜_o anno._n 1548_o and_o 1549._o in_o lovayn_n note_n i.d._n the_o cut_v of_o a_o sphere_n according_a to_o any_o proportion_n assign_v may_v by_o this_o proposition_n be_v do_v mechanical_o by_o temper_v liquour_n to_o a_o certain_a weight_n in_o respect_n of_o the_o weight_n of_o the_o sphere_n 〈◊〉_d swyâ—â—â—ng_n a_o common_a error_n note_v a_o paradox_n n._n t._n the_o wonderful_a use_n of_o these_o proposition_n the_o practice_n statical_a to_o know_v the_o proportion_n between_o the_o cube_n and_o the_o sphere_n i_o d._n d._n for_o so_o have_v you_o 256._o part_n of_o a_o grain_n grain_n the_o proportion_n of_o the_o square_n to_o the_o circle_n inscribe_v inscribe_v the_o square_n of_o the_o circle_v mechanical_o mechanical_o to_o any_o squirâ—_n gâ—uenâ—_n to_o 〈…〉_o note_v square_n of_o the_o circle_n without_o knowledge_n of_o the_o proportion_n between_o circumference_n and_o diameter_n to_o double_a the_o cube_n ready_o by_o art_n mechanical_a depend_v upon_o demonstration_n mathematical_a i_o d._n the_o 4._o side_n of_o this_o pyramiâ—_n must_v be_v 4._o isosceles_a triangle_n â—_o like_o and_o squall_v i_o d._n d._n in_o all_o workinger_n with_o this_o pyramid_n or_o cone_n let_v their_o situation_n be_v in_o all_o pointeâ—_n and_o condition_n a_o like_a oâ—_n all_o one_o while_o you_o be_v about_o â—ne_a work_n else_o you_o will_v 〈◊〉_d i_o d._n d._n consider_v well_o when_o you_o must_v put_v your_o wateâ—â—_n together_o and_o when_o you_o must_v empty_a youâ—_n first_o waterâ—_n out_o of_o your_o pyramiâ—_n or_o cone_n elâ—_n you_o will_v 〈◊〉_d 〈◊〉_d vitrwius_fw-la lib._n 9_o cap._n 3._o ☞_o god_n bâ—_z thank_v â—or_a this_o invention_n &_o the_o fruit_n ensue_v ensue_v note_n note_v as_o concern_v the_o spherical_a superficies_n of_o the_o water_n ☞_o ☞_o note_n note_v this_o abridgement_n of_o double_v 〈◊〉_d cube_n â—â—_o â—â—_o note_n *_o ☜_o to_o give_v cube_n one_o to_o the_o other_o in_o any_o proportion_n rational_a or_o irrational_a irrational_a empty_v the_o first_o the_o demonstration_n of_o this_o double_v of_o the_o cube_n and_o oâ—_n the_o rest_n i.d._n i.d._n here_o 〈…〉_o of_o the_o water_n water_n by_o the_o 33._o of_o the_o eleven_o book_n of_o euclid_n i.d._n i.d._n and_o your_o diligence_n in_o practice_n can_v â—o_o in_o weight_n of_o wateâ—â—_n perform_v it_o therefore_o now_o you_o arâ—_n able_a to_o â—eve_v good_a reason_n of_o your_o whole_a do_v do_v note_v this_o corollary_n corollary_n the_o great_a commodity_n follow_v of_o these_o new_a invention_n invention_n ☞_o such_o be_v the_o fruit_n of_o the_o mathematical_a science_n and_o artes._n man_n be_v the_o less_o world._n world._n ☜_o microcosmus_n microcosmus_n lib._n 3._o cap._n 1._o ☞_o saw_n milles._n milles._n atheneus_n lib._n 5._o cap._n 8._o proclus_n pag._n 18._o to_o go_v to_o the_o bottom_n of_o the_o sea_n without_o danger_n plutâ—â—â—bus_fw-la in_o marco_n mâ—rcello_n syâ—asius_n in_o epistolis_fw-la polybius_n plinius_n quintâ—lianus_n t._n livius_n livius_n athenaâ—s_n athenaâ—s_n galeâ—us_fw-la anthemius_n burn_v glass_n gun_n 4._o reg._n 20._o a_o perpetual_a motion_n a_o objection_n the_o answer_n ☜_o a_o mathematicien_n vitrunius_n who_o be_v a_o architect_n architect_n the_o immateriality_n of_o perfect_a architecture_n what_o lineament_n be_v note_n anno._n 1559._o 1559._o anno._n 1551_o de_fw-fr his_fw-la quae_fw-la mundo_fw-la mirabiliter_fw-la eveniunt_fw-la cap._n 8._o tusc._n â—_o â—_o ☞_o a_o digression_n apologetical_a apologetical_a a_o proverb_n fair_a fish_v and_o catch_v a_o frog_n ☞_o psal._n 140._o act._n 7._o c._n lib._n 30._o cap._n 1._o ☜_o r._n b._n ☞_o university_n ☜_o ☞_o the_o ground_n plat_z of_o this_o preface_n in_o a_o table_n the_o argumâ—â—â—_n of_o the_o first_o book_n definition_n of_o a_o point_n definition_n of_o a_o point_n after_o pythagoras_n definition_n of_o a_o liâ—â—_n an_o other_o definition_n of_o a_o line_n a_o other_o the_o end_n of_o a_o line_n difference_n of_o a_o point_n frâ—â—nity_n unity_n be_v a_o part_n of_o number_n a_o point_n be_v no_o part_n of_o quantity_n definition_n of_o a_o right_a line_n definition_n thereof_o after_o plato_n an_o other_o definition_n a_o other_o a_o other_o a_o other_o a_o other_o why_o euclid_n here_o define_v not_o a_o crooked_a line_n definition_n of_o a_o superficies_n a_o superficies_n 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of_o the_o sun_n say_v astronomer_n then_o happen_v when_o the_o sun_n the_o moon_n &_o our_o eye_n be_v in_o one_o right_a line_n for_o the_o moon_n then_o be_v in_o the_o midst_n between_o we_o and_o the_o sun_n cause_v it_o to_o be_v darken_v diverse_a other_o define_v a_o right_a line_n diverse_o definition_n as_o follow_v a_o right_a line_n be_v that_o which_o stand_v firm_a between_o his_o extreme_n other_o again_o a_o right_a line_n be_v that_o which_o with_o a_o other_o line_n of_o like_a form_n can_v make_v a_o figure_n again_o other_o a_o right_a line_n be_v that_o which_o have_v not_o one_o part_n in_o a_o plain_a superficies_n and_o a_o other_o erect_v on_o high_a again_o other_o a_o right_a line_n be_v that_o all_o who_o part_n agree_v together_o with_o all_o his_o other_o part_n again_o other_o a_o right_a line_n be_v that_o who_o extreme_n abide_v can_v be_v alter_v euclid_n do_v not_o here_o define_v a_o crooked_a line_n line_n for_o it_o need_v not_o it_o may_v easy_o be_v understand_v by_o the_o definition_n of_o a_o right_a line_n for_o every_o contrary_n be_v well_o manifest_v &_o set_v forth_o by_o his_o contrary_n one_o crooked_a line_n may_v be_v more_o crooked_a than_o a_o other_o and_o from_o one_o point_n to_o a_o other_o may_v be_v draw_v infinite_a crooked_a line_n but_o one_o right_a line_n can_v be_v right_a than_o a_o other_o and_o therefore_o from_o one_o point_n to_o a_o other_o there_o may_v be_v draw_v but_o one_o tight_a line_n as_o by_o figure_n above_o set_n you_o may_v see_v 5_o a_o superficies_n be_v that_o which_o have_v only_a length_n and_o breadth_n superficies_n a_o superficies_n be_v the_o second_o kind_n of_o quantity_n and_o to_o it_o be_v attribute_v two_o dimension_n namely_o length_n and_o breadth_n way_n as_o in_o the_o
namely_o ab_fw-la ac_fw-la and_o bc_o and_o therefore_o it_o be_v a_o equilater_n triangle_n by_o the_o definition_n of_o a_o equilater_n triangle_n and_o this_o be_v the_o first_o reason_n reason_n that_o the_o three_o line_n be_v equal_a be_v thus_o prove_v the_o line_n ac_fw-la and_o cb_o be_v equal_a to_o the_o line_n ab_fw-la wherefore_o they_o be_v equal_v the_o one_o to_o the_o other_o and_o this_o be_v the_o second_o reason_n reason_n that_o the_o line_n abâ—_n and_o bc_o be_v equal_a be_v thus_o prove_v the_o line_n ab_fw-la and_o ac_fw-la be_v draw_v from_o the_o centre_n of_o the_o circle_n ace_n to_o the_o circumference_n of_o the_o same_o wherefore_o they_o be_v equal_a by_o the_o definition_n of_o a_o circle_n and_o this_o be_v the_o three_o reason_n reason_n likewise_o that_o the_o line_n ac_fw-la and_o ab_fw-la resolution_n be_v equal_a be_v prove_v by_o the_o same_o reason_n for_o the_o line_n ac_fw-la and_o ab_fw-la be_v draw_v from_o the_o centre_n of_o the_o circle_n bcd_v wherefore_o they_o be_v equal_a by_o the_o same_o definition_n of_o a_o circle_n this_o be_v the_o four_o reason_n or_o syllogism_n and_o thus_o be_v end_v the_o whole_a resolution_n for_o that_o you_o be_v come_v to_o a_o principle_n which_o be_v indemonstrable_a &_o can_v not_o be_v resolve_v of_o a_o â—emonstration_n lead_v to_o a_o impossibility_n or_o to_o a_o absurdity_n you_o may_v have_v a_o example_n in_o the_o forth_o proposition_n of_o this_o book_n a_o addition_n of_o campanus_n but_o now_o if_o upon_o the_o same_o line_n give_v namely_o ab_fw-la you_o will_v describe_v the_o other_o two_o kind_n of_o triangle_n namely_o a_o isosceles_a or_o a_o triangle_n of_o two_o equal_a side_n &_o a_o scalenon_fw-mi or_o a_o triangle_n of_o three_o unequal_a side_n first_o for_o the_o describe_v of_o a_o isosceles_a triangle_n produce_v the_o line_n ab_fw-la on_o either_o side_n triangle_n until_o it_o concur_v with_o the_o circumference_n of_o both_o the_o circle_n in_o the_o point_n d_o and_o f_o and_o make_v the_o centre_n the_o point_n a_o describe_v a_o circle_n hfg_n according_a to_o the_o quantity_n of_o the_o line_n af._n likewise_o make_v the_o centre_n the_o point_n b_o describe_v a_o circle_n hdg_n according_a to_o the_o quantity_n of_o the_o line_n bd._n now_o they_o these_o circle_n shall_v cut_v the_o one_o the_o other_o in_o two_o point_n which_o let_v be_v h_o and_o g_z and_o lâ—t_o the_o â—ndâ—s_n of_o the_o line_n give_v be_v join_v with_o one_o of_o the_o say_a section_n by_o two_o right_a line_n which_o let_v be_v agnostus_n and_o bg_o and_o forasmuch_o as_o these_o two_o line_n ab_fw-la and_o ad_fw-la be_v draw_v from_o the_o centre_n of_o the_o circle_n cde_o unto_o the_o circumference_n thereof_o therefore_o be_v they_o equal_a likewise_o the_o line_n basilius_n and_o bf_o for_o that_o they_o be_v draw_v from_o the_o centre_n of_o the_o circle_n eacf_n to_o the_o circumference_n thereof_o aâ—e_v equal_a and_o forasmuch_o as_o either_o of_o the_o line_n ad_fw-la and_o bf_o be_v equal_a to_o the_o line_n ab_fw-la therefore_o they_o be_v equal_v the_o one_o to_o the_o other_o wherefore_o put_v the_o line_n ab_fw-la common_a to_o they_o both_o the_o whole_a line_n bd_o shall_v be_v equal_a to_o the_o whole_a line_n af._n but_o bd_o be_v equal_a to_o bg_o for_o they_o be_v both_o draw_v from_o the_o centre_n of_o the_o circle_n hdg_n to_o the_o circumference_n thereof_o and_o likewise_o by_o the_o same_o reason_n the_o line_n of_o be_v equal_a to_o the_o line_n ag._n wherefore_o by_o the_o common_a sentence_n the_o line_n agnostus_n and_o bg_o be_v equal_a the_o one_o to_o the_o other_o and_o either_o of_o they_o be_v great_a than_o the_o line_n ab_fw-la for_o that_o either_o of_o the_o two_o line_n bd_o and_o of_o be_v great_a than_o the_o line_n ab_fw-la wherefore_o upon_o the_o line_n give_v be_v describe_v a_o isosceles_a or_o triangle_n of_o two_o equal_a side_n scalenum_n you_o may_v also_o describe_v upon_o the_o self_n same_o line_n a_o scaleâ—on_n or_o triangle_n of_o three_o unequal_a side_n if_o by_o two_o right_a line_n you_o join_v both_o the_o end_n of_o the_o line_n gruen_n to_o some_o one_o point_n that_o be_v in_o the_o circumference_n of_o one_o of_o the_o two_o great_a circle_n so_o that_o that_o point_n be_v not_o in_o one_o of_o the_o two_o section_n and_o that_o the_o line_n df_o do_v not_o concur_v with_o it_o when_o it_o be_v on_o either_o side_n produce_v continual_o and_o direct_o for_o let_v the_o point_n king_n be_v take_v in_o the_o circumference_n of_o the_o circle_n hdg_n and_o let_v it_o not_o be_v in_o any_o of_o the_o section_n neither_o let_v the_o line_n df_o concur_v with_o it_o when_o it_o be_v produce_v continual_o and_o direct_o unto_o the_o circumference_n thereof_o and_o draw_v these_o line_n ak_o and_o bk_o and_o the_o line_n ak_o shall_v cut_v the_o circumference_n of_o the_o circle_n hfg_n let_v it_o cut_v it_o in_o the_o point_n l_o now_o then_o by_o the_o common_a sentence_n the_o line_n bk_o shall_v be_v equal_v to_o the_o line_n all_o for_o by_o the_o definition_n of_o a_o circle_n the_o line_n bk_o be_v equal_a to_o the_o line_n bg_o and_o the_o line_n all_o be_v equal_a to_o the_o line_n agnostus_n which_o be_v equal_a to_o the_o line_n bg_o wherefore_o the_o line_n ak_o be_v great_a than_o the_o line_n bk_o and_o by_o the_o same_o reason_n may_v it_o be_v prove_v that_o the_o line_n bk_o be_v great_a than_o the_o line_n ab_fw-la wherefore_o the_o triangle_n abk_n consist_v of_o three_o unequal_a side_n and_o so_o have_v you_o upon_o the_o line_n give_v describe_v all_o the_o kind_n of_o triangle_n this_o be_v to_o be_v note_v that_o if_o a_o man_n will_v mechanical_o and_o ready_o not_o regard_v demonstration_n upon_o a_o line_n give_v describe_v a_o triangle_n of_o three_o equal_a side_n mechanical_o he_o need_v not_o to_o describe_v the_o whole_a foresay_a circle_n but_o only_o a_o little_a part_n of_o each_o namely_o where_o they_o cut_v the_o one_o the_o other_o and_o so_o from_o the_o point_n of_o the_o section_n to_o draw_v the_o line_n to_o the_o end_n of_o the_o line_n gevenâ—_n as_o in_o this_o figure_n here_o put_v and_o likewise_o if_o upon_o the_o say_a line_n he_o will_v describe_v a_o triangle_n of_o two_o equal_a side_n ready_o let_v he_o extend_v the_o compass_n according_a to_o the_o quantity_n that_o he_o will_v have_v the_o side_n to_o be_v whether_o long_o than_o the_o line_n give_v or_o short_a and_o so_o draw_v only_o a_o little_a part_n of_o each_o circle_n where_o they_o cut_v the_o one_o the_o other_o &_o from_o the_o point_n of_o the_o section_n draw_v the_o line_n to_o the_o end_n of_o the_o line_n give_v as_o in_o the_o figure_n here_o put_v note_v that_o in_o this_o the_o two_o side_n must_v be_v such_o that_o be_v join_v together_o they_o be_v long_o than_o the_o line_n give_v and_o so_o also_o if_o upon_o the_o say_a right_a line_n he_o will_v describe_v a_o triangle_n of_o three_o unequal_a side_n ready_o let_v he_o extend_v the_o compass_n first_o according_a to_o the_o quantity_n that_o he_o will_v have_v one_o of_o the_o unequal_a side_n to_o be_v and_o so_o draw_v a_o little_a part_n of_o the_o circle_n &_o then_o extend_v it_o according_a to_o the_o quantity_n that_o he_o will_v have_v the_o other_o unequal_a side_n to_o be_v and_o draw_v likewise_o a_o little_a part_n of_o the_o circle_n and_o that_o do_v from_o the_o point_n of_o the_o section_n draâ—_n the_o line_n to_o the_o end_n of_o the_o line_n give_v as_o in_o the_o figure_n here_o put_v note_v that_o in_o this_o the_o two_o side_n must_v be_v such_o that_o the_o circle_n describe_v according_a to_o their_o quantity_n may_v cut_v the_o one_o the_o other_o the_o second_o problem_n the_o second_o proposition_n fron_n a_o point_n give_v to_o draw_v a_o right_a line_n equal_a to_o a_o rightline_v give_v svppose_v that_o the_o point_n give_v be_v a_o &_o let_v the_o right_a line_n give_v be_v bc._n it_o be_v require_v from_o the_o point_n a_o to_o draw_v a_o right_a line_n equal_a to_o the_o line_n bc._n draw_v by_o the_o first_o petition_n from_o the_o point_n a_o to_o the_o point_n b_o a_o right_a line_n ab_fw-la construâ—tiâ—â—_n and_o upon_o the_o line_n ab_fw-la describe_v by_o the_o first_o proposition_n a_o equilater_n triangle_n and_o let_v the_o same_o be_v dab_n and_o extend_v by_o the_o second_o petition_n the_o right_a line_n dam_n &_o db_o to_o the_o point_n e_o and_o f_o &_o by_o the_o three_o petition_n make_v the_o centre_n b_o and_o the_o space_n bc_o describe_v a_o circle_n cgh_n &_o again_o by_o the_o same_o make_v the_o centre_n d_o and_o the_o space_n dg_o describe_v a_o circle_n gkl_n
and_o forasmuch_o as_o the_o point_n b_o be_v the_o centre_n of_o the_o circle_n cgh_n demonstration_n therefore_o by_o the_o 15._o definition_n the_o line_n bc_o be_v equal_a to_o the_o line_n bg_o and_o forasmuch_o as_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n gkl_n therefore_o by_o the_o same_o the_o line_n dl_o be_v equal_a to_o the_o line_n dg_o of_o which_o the_o line_n dam_n be_v equal_a to_o a_o line_n db_o by_o the_o proposition_n go_v before_o wherefore_o the_o residue_n namely_o the_o line_n all_o be_v equal_a to_o the_o residue_n namely_o to_o the_o line_n bg_o by_o the_o three_o common_a sentence_n and_o it_o be_v prove_v that_o the_o line_n bc_o be_v equal_a to_o the_o line_n bg_o wherefore_o either_o of_o these_o line_n all_o &_o bc_o be_v equal_a to_o the_o line_n bg_o but_o thing_n which_o be_v equal_a to_o one_o and_o the_o same_o thing_n be_v also_o equal_a the_o one_o to_o the_o other_o by_o the_o first_o common_a sentence_n wherefore_o the_o line_n all_o be_v equal_a to_o the_o line_n bc._n wherefore_o from_o the_o point_n give_v namely_o a_o be_v draw_v a_o right_a line_n all_o equal_a to_o the_o right_a line_n give_v bc_o which_o be_v require_v to_o be_v do_v of_o problem_n and_o theoremesâ—_n as_o we_o have_v before_o note_v some_o have_v no_o case_n at_o all_o which_o be_v those_o which_o have_v only_o one_o position_n and_o construction_n and_o other_o some_o have_v many_o and_o diverse_a case_n which_o be_v such_o proposition_n which_o have_v diverse_a description_n &_o construction_n and_o change_v their_o position_n of_o which_o sort_n be_v this_o second_o proposition_n which_o be_v also_o a_o problem_n this_o proposition_n have_v two_o thing_n give_v proposition_n namely_o a_o point_n and_o a_o line_n the_o thing_n require_v be_v that_o from_o the_o point_n give_v wheresoever_o it_o be_v put_v be_v draw_v a_o line_n equal_a to_o the_o line_n give_v now_o this_o point_n give_v may_v have_v diverse_a position_n for_o it_o may_v be_v place_v either_o without_o the_o right_a line_n give_v or_o in_o some_o point_n in_o it_o if_o it_o be_v without_o it_o either_o it_o be_v on_o the_o side_n of_o it_o so_o that_o the_o right_a line_n draw_v from_o it_o to_o the_o end_n of_o the_o right_a line_n give_v make_v a_o angleâ—_n or_o else_o it_o be_v put_v direct_o unto_o itâ—_n so_o that_o the_o right_a line_n give_v be_v produce_v shall_v fall_v upon_o the_o point_n give_v which_o be_v without_o but_o if_o it_o be_v in_o the_o line_n give_v then_o either_o it_o be_v in_o one_o of_o the_o end_n or_o extreme_n thereof_o or_o in_o some_o place_n between_o the_o extreme_n so_o be_v there_o four_o diverse_a position_n of_o the_o point_n in_o respect_n of_o the_o line_n whereupon_o follow_v diverse_a delineation_n and_o construction_n and_o consequent_o variety_n of_o case_n case_n to_o the_o second_o case_n the_o figure_n here_o on_o the_o side_n set_v belong_v and_o as_o touch_v the_o order_n both_o oâ—_n construction_n and_o of_o demonstration_n it_o be_v â—ll_a one_o with_o the_o first_o the_o four_o case_n as_o touch_v construction_n herein_o differ_v from_o the_o two_o first_o case_n for_o thâ—t_v whereas_o in_o they_o you_o be_v will_v to_o draw_v a_o right_a line_n from_o the_o point_n give_v namely_o a_o to_o the_o point_n b_o which_o be_v one_o of_o the_o end_n of_o the_o line_n give_v here_o you_o shall_v not_o need_v to_o draw_v that_o line_n for_o that_o it_o be_v already_o draw_v as_o touch_v the_o rest_n both_o in_o construction_n and_o demonstration_n you_o may_v proceed_v as_o in_o the_o two_o firsteâ—_n as_o it_o be_v manifest_a to_o see_v in_o this_o figure_n here_o on_o the_o side_n put_v the_o 3._o problem_n the_o 3._o proposition_n two_o unequal_a right_a line_n be_v give_v to_o cut_v of_o from_o the_o great_a a_o right_a line_n equal_a to_o the_o less_o svppose_v that_o the_o two_o unequal_a right_a line_n give_v be_v ab_fw-la &_o c_o of_o which_o lâ—t_v the_o line_n ab_fw-la be_v the_o great_a it_o be_v require_v from_o the_o line_n ab_fw-la be_v the_o great_a to_o cut_v of_o a_o right_a line_n equal_a to_o the_o right_a line_n c_o which_o be_v the_o less_o line-drawâ—_a on_o by_o the_o second_o proposition_n from_o the_o point_n a_o a_o right_a line_n equal_a to_o the_o line_n c_o and_o let_v the_o same_o be_v ad_fw-la and_o make_v the_o centre_n a_o and_o the_o space_n ad_fw-la describe_v by_o the_o three_o petition_n a_o circle_n def_n and_o forasmuch_o as_o the_o point_n a_o be_v the_o centre_n of_o the_o circle_n def_n therefore_o ae_n be_v equal_a to_o ad_fw-la but_o the_o line_n c_o be_v equal_a to_o the_o line_n ad._n wherefore_o either_o of_o these_o line_n ae_n and_o c_o be_v equal_a to_o ad_fw-la wherefore_o the_o line_n ae_n be_v equal_a to_o the_o line_n c_o wherefore_o two_o unequal_a right_a line_n be_v give_v namely_o ab_fw-la and_o c_o the_fw-mi be_v cut_v of_o from_o ab_fw-la being_n the_o great_a a_o right_a line_n ae_n equal_a to_o the_o less_o line_n namely_o to_o c_o which_o be_v require_v to_o be_v do_v this_o proposition_n which_o be_v a_o problem_n have_v two_o thing_n give_v 〈◊〉_d namely_o two_o unequal_a right_a line_n the_o thing_n require_v be_v from_o the_o great_a to_o cut_v of_o a_o line_n equal_a to_o the_o less_o it_o have_v also_o diverse_a case_n for_o the_o line_n give_v either_o be_v distinct_a the_o one_o from_o the_o other_o or_o be_v join_v together_o at_o one_o of_o their_o end_n or_o they_o cut_v the_o one_o the_o other_o or_o the_o one_o cut_v the_o other_o in_o one_o of_o the_o extreme_n which_o may_v be_v two_o way_n for_o either_o the_o great_a cut_v the_o less_o or_o the_o less_o the_o great_a if_o they_o cut_v the_o one_o the_o other_o either_o â—ch_n cut_v the_o other_a into_o equal_a part_n or_o into_o unequal_a part_n or_o the_o one_o into_o equal_a part_n and_o the_o other_o into_o unequal_a part_n which_o may_v hap_v in_o two_o sort_n for_o the_o great_a may_v be_v cut_v into_o equal_a part_n and_o the_o less_o into_o unequal_a part_n or_o contrariwise_o case_n when_o the_o unequal_a line_n give_v be_v distinct_a the_o one_o from_o the_o other_o the_o figure_n before_o put_v serve_v but_o if_o the_o one_o cut_v the_o other_o in_o one_o of_o the_o extreme_n case_n as_o for_o example_n suppose_v that_o the_o unequal_a right_a line_n give_v be_v ab_fw-la and_o cd_o of_o which_o let_v the_o line_n cd_o be_v the_o great_a and_o let_v the_o line_n cd_o cut_v the_o line_n ab_fw-la in_o his_o extreme_a c._n then_o make_v the_o centre_n a_o and_o the_o space_n ab_fw-la describe_v a_o circle_n be._n and_o upon_o the_o line_n ac_fw-la describe_v a_o equilater_n triangle_n by_o the_o â—irst_n which_o let_v bâ—_z aec_o &_o produce_v the_o line_n ea_fw-la and_o ec_o and_o again_o make_v the_o centre_n e_o and_o the_o space_n of_o describe_v a_o ciâ—â—â—e_n gf_o likewise_o make_v the_o centre_n c_o and_o the_o space_n cg_o describe_v a_o circle_n gl_n now_o forasmuch_o as_o the_o line_n of_o be_v equal_a to_o the_o line_n eglantine_n â—or_a e_o be_v the_o centre_n of_o which_o the_o line_n ea_fw-la be_v equal_a to_o the_o line_n eâ—_n therefore_o the_o residue_n of_o be_v equal_v to_o the_o residue_n cg_o â—ut_fw-la the_o line_n of_o be_v equal_a to_o the_o line_n ab_fw-la for_o a_o be_v the_o centre_n whereâ—oâ—e_v also_o the_o line_n cg_o be_v equal_a to_o the_o line_n ab_fw-la but_o the_o line_n cg_o be_v also_o equal_a â—o_o the_o line_n cl_n for_o the_o point_n c_o be_v the_o centre_n wherefore_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n cl._n wherefore_o from_o the_o line_n cd_o be_v cut_v of_o the_o line_n cl_n which_o be_v equal_a to_o the_o line_n ab_fw-la case_n or_o it_o be_v less_o than_o the_o half_a and_o then_o make_v the_o centre_n c_o &_o the_o space_n cd_o describe_v a_o circle_n which_o shall_v cut_v of_o from_o the_o line_n ab_fw-la a_o line_n equal_a to_o the_o line_n cd_o case_n or_o it_o be_v great_a than_o the_o half_a and_o they_o unto_o the_o point_n a_o put_v the_o line_n of_o equal_a to_o the_o line_n cd_o by_o the_o second_o and_o make_v the_o centre_n a_o &_o the_o space_n of_o describe_v a_o circle_n which_o shall_v cut_v of_o from_o the_o line_n ab_fw-la a_o line_n equal_a to_o the_o line_n of_o that_o be_v unto_o the_o line_n cd_o but_o if_o it_o be_v great_a than_o the_o half_a case_n then_o again_o unto_o the_o point_n a_o put_v the_o line_n of_o equal_a to_o the_o line_n cd_o by_o the_o second_o proposition_n &_o make_v the_o centre_n a_o and_o the_o space_n of_o describe_v a_o circle_n which_o shall_v cut_v of_o from_o the_o line_n
suppose_v that_o the_o angle_n at_o the_o base_a be_v equal_a which_o in_o the_o former_a proposition_n be_v the_o conclusion_n and_o the_o conclusion_n be_v that_o the_o two_o side_n subtend_v the_o two_o angle_n be_v equal_a which_o in_o the_o former_a proposition_n be_v the_o supposition_n this_o be_v the_o chief_a kind_n of_o conversion_n uniform_a and_o certain_a there_o be_v a_o other_o kind_n of_o conversion_n first_o but_o not_o so_o full_a a_o conversion_n nor_o so_o perfect_a as_o the_o first_o be_v which_o happen_v in_o compose_a proposition_n that_o be_v in_o such_o which_o have_v more_o supposition_n than_o one_o and_o pass_v from_o these_o supposition_n to_o one_o conclusion_n in_o the_o connuerse_n of_o such_o proposition_n you_o pass_v from_o the_o conclusion_n of_o the_o first_o proposition_n with_o one_o or_o more_o of_o the_o supposition_n of_o the_o same_o &_o conclude_v some_o other_o supposition_n of_o the_o self_n first_o proposition_n of_o this_o kind_n there_o be_v many_o in_o euclid_n thereof_o you_o may_v have_v a_o example_n in_o the_o 8._o proposition_n be_v the_o converse_n of_o the_o fourâ—h_n this_o conversion_n be_v not_o so_o uniform_a as_o the_o other_o but_o more_o diverse_a and_o uncertain_a according_a to_o the_o multitude_n of_o the_o thing_n give_v or_o supposition_n in_o the_o proposition_n but_o because_o in_o the_o five_o proposition_n there_o be_v two_o conclusion_n proposition_n the_o first_o that_o the_o two_o angle_n at_o the_o baâ—e_n be_v equal_a the_o second_o that_o the_o angle_n under_o the_o baâ—e_n be_v equal_a this_o be_v to_o be_v note_v that_o this_o six_o proposition_n be_v the_o converse_n of_o the_o â—ame_n five_o as_o touch_v the_o first_o conclusion_n only_o conclusion_n you_o may_v in_o like_a manner_n make_v a_o converse_n of_o the_o same_o proposition_n touch_v the_o second_o conclusion_n thereof_o and_o that_o after_o this_o manner_n the_o two_o side_n of_o a_o triangle_n be_v produce_v if_o the_o angle_n under_o the_o base_a be_v equal_a the_o say_a triangle_n shall_v be_v a_o isosceles_a triangle_n in_o which_o proposition_n the_o supposition_n be_v that_o the_o angle_n under_o the_o base_a be_v equal_a which_o in_o the_o five_o proposition_n be_v the_o conclusionâ—_n &_o the_o conclusion_n in_o this_o proposition_n be_v that_o the_o two_o side_n of_o the_o triangle_n be_v equal_a which_o in_o the_o five_o proposition_n be_v the_o supposition_n but_o now_o for_o proof_n of_o the_o say_a proposition_n for_o suppose_v that_o ac_fw-la be_v equal_a to_o ad_fw-la and_o produce_v the_o line_n ca_n to_o the_o point_n e_o and_o put_v the_o line_n ae_n equal_a to_o the_o line_n db_o by_o the_o three_o proposition_n wherefore_o the_o whole_a line_n ce_fw-fr be_v equal_a to_o the_o whole_a line_n ab_fw-la by_o the_o second_o common_a sentence_n draw_v a_o line_n from_o the_o point_n e_o to_o the_o point_n b._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n ec_o and_o the_o line_n bc_o be_v common_a to_o they_o both_o and_o the_o angle_n acb_o be_v suppose_v to_o be_v equal_a to_o the_o angle_n abc_n wherefore_o by_o the_o four_o proposition_n the_o triangle_n ebc_n be_v equal_a to_o the_o triangle_n abc_n name_o the_o whole_a to_o the_o part_n which_o be_v impossible_a the_o 4._o theorem_a the_o 7._o proposition_n if_o from_o the_o end_n of_o one_o line_n be_v draw_v two_o right_a line_n to_o any_o point_n there_o can_v not_o from_o the_o self_n same_o end_n on_o the_o same_o side_n be_v draw_v two_o other_o line_n equal_a to_o the_o two_o first_o line_n the_o one_o to_o the_o other_o unto_o any_o other_o point_n for_o if_o it_o be_v possible_a then_o from_o the_o end_n of_o one_o &_o the_o self_n same_o right_a line_n namely_o ab_fw-la from_o the_o point_n i_o say_v a_o and_o b_o let_v there_o be_v draw_v two_o right_a line_n ac_fw-la and_o cb_o to_o the_o point_n c_o and_o from_o the_o same_o end_n of_o the_o line_n ab_fw-la let_v there_o be_v draw_v two_o other_o right_n right_a line_n ad_fw-la and_o db_o equal_a to_o the_o line_n ac_fw-la and_o cb_o the_o one_o to_o the_o other_o be_v each_o to_o his_o correspondent_a line_n and_o on_o one_o and_o the_o same_o side_n and_o to_o a_o other_o point_n namely_o to_o d_o so_o that_o let_v ca_n be_v equal_a to_o dam_n be_v both_o draw_v from_o one_o end_n that_o be_v a_o &_o let_v cb_o be_v equal_a to_o db_o be_v both_o also_o draw_v from_o one_o end_n that_o be_v b._n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n c_o to_o the_o point_n d._n absurdity_n now_o forasmuch_o as_o ac_fw-la be_v equal_a to_o ad_fw-la the_o angle_n acd_v also_o be_v by_o the_o 5._o proposition_n equal_a to_o the_o angle_n adc_o wherefore_o the_o angle_n acd_o be_v less_o they_o the_o angle_n bdc_o wherefore_o the_o angle_n bcd_o be_v much_o less_o than_o the_o angle_n bdc_o again_o forasmuch_o as_o bc_o be_v equal_a to_o bd_o and_o therefore_o also_o the_o angle_n bcd_o be_v equal_a to_o the_o angle_n bdc_o and_o it_o be_v prove_v that_o it_o be_v much_o less_o than_o it_o which_o be_v impossible_a if_o therefore_o from_o the_o end_n of_o one_o line_n be_v draw_v too_o right_a line_n to_o any_o point_n there_o can_v not_o from_o the_o self_n same_o end_n on_o the_o same_o side_n be_v draw_v two_o other_o line_n equal_a to_o the_o two_o first_o line_n the_o one_o to_o the_o other_o unto_o any_o other_o point_n which_o be_v require_v to_o be_v demonstrate_v in_o this_o proposition_n the_o conclusion_n be_v a_o negation_n which_o very_o rare_o happen_v in_o the_o mathematical_a art_n art_n for_o they_o ever_o for_o the_o most_o part_n use_v to_o conclude_v affirmative_o &_o not_o negative_o for_o a_o proposition_n universal_a affirmative_a be_v most_o agreeable_a to_o science_n as_o say_v aristotle_n and_o be_v of_o itself_o strong_a and_o need_v no_o negative_a to_o his_o proof_n but_o a_o universal_a proposition_n negative_a must_v of_o necessity_n have_v to_o his_o proof_n a_o affirmative_a for_o of_o only_o negative_a proposition_n there_o can_v be_v no_o demonstration_n and_o therefore_o science_n use_v demonstration_n conclude_v affirmative_o and_o very_o seldom_o use_v negative_a conclusion_n an_o other_o demonstration_n after_o campanus_n suppose_v that_o there_o be_v a_o line_n ab_fw-la from_o who_o end_n a_o and_o b_o let_v there_o be_v draw_v two_o line_n ac_fw-la and_o bc_o on_o one_o side_n which_o let_v concur_v in_o the_o point_n c._n then_o i_o say_v that_o on_o the_o same_o side_n there_o can_v be_v draw_v two_o other_o line_n from_o the_o end_n of_o the_o line_n ab_fw-la which_o shall_v concur_v at_o any_o other_o point_n so_o that_o that_o which_o be_v draw_v from_o the_o point_n a_o shall_v be_v equal_a to_o the_o line_n ac_fw-la and_o that_o which_o be_v draw_v from_o the_o point_n b_o shall_v be_v equal_a to_o the_o line_n bc._n for_o if_o it_o be_v possible_a let_v there_o be_v draw_v two_o other_o line_n on_o the_o self_n same_o side_n which_o let_v concur_v in_o the_o point_n d_o and_o let_v the_o line_n ad_fw-la be_v equal_a to_o the_o line_n ac_fw-la &_o the_o line_n bd_o equal_a to_o the_o line_n bc._n demonstration_n wherefore_o the_o point_n d_o shall_v fall_v either_o within_o the_o triangle_n abc_n or_o without_o for_o it_o can_v fall_v in_o one_o of_o the_o side_n for_o then_o a_o part_n shall_v be_v equal_a to_o his_o whole_a if_o therefore_o it_o fall_v without_o then_o either_o one_o of_o the_o line_n ad_fw-la and_o db_o shall_v cut_v one_o of_o the_o line_n ac_fw-la and_o cb_o or_o else_o neither_o shall_v cut_v neither_o case_n first_o let_v one_o cut_v the_o other_o and_o draw_v a_o right_a line_n from_o c_o to_o d._n now_o forasmuch_o as_o in_o the_o triangle_n acd_o the_o two_o side_n ac_fw-la and_o ad_fw-la be_v equal_a therefore_o the_o angle_n acd_o be_v equal_a to_o the_o angle_n adc_o by_o the_o five_o proposition_n likewise_o forasmuch_o as_o in_o the_o triangle_n bcd_o the_o two_o side_n bc_o and_o bd_o be_v equal_a therefore_o by_o the_o same_o the_o angle_n bcd_o &_o bdc_o be_v also_o equal_a and_o forasmuch_o as_o the_o angle_n bdc_o be_v great_a they_o the_o angle_n adc_o it_o follow_v that_o the_o angle_n bcd_o be_v great_a than_o the_o angle_n acd_o namely_o the_o part_n great_a than_o the_o whole_a which_o be_v impossible_a but_o if_o the_o point_n d_o fall_n without_o the_o triangle_n abc_n case_n so_o that_o the_o line_n cut_v not_o the_o one_o the_o other_o draw_v a_o line_n from_o d_z to_o c._n and_o produce_v the_o line_n bd_o &_o bc_o beyond_o the_o base_a cd_o unto_o the_o point_n e_o &_o f._n and_o forasmuch_o as_o the_o line_n ac_fw-la and_o ad_fw-la be_v equal_a the_o angle_n
acd_o and_o adc_o shall_v also_o be_v equal_a by_o the_o five_o proposition_n likewise_o for_o asmuch_o as_o the_o line_n bc_o and_o bd_o be_v equal_a the_o angle_n under_o the_o base_a namely_o the_o angle_n fdc_n and_o ecd_n be_v equal_a by_o the_o second_o part_n of_o the_o same_o proposition_n and_o for_o as_o much_o as_o the_o angle_n ecd_n be_v less_o than_o the_o angle_n acd_o it_o follow_v that_o the_o angle_n fdc_n be_v less_o they_o the_o angle_n adc_o which_o be_v impossible_a for_o that_o the_o angle_n ad_fw-la c_z be_v a_o part_n of_o the_o angle_n fd_o c._n and_o the_o same_o inconvenience_n will_v follow_v if_o the_o point_n d_o fall_n within_o the_o triangle_n abc_n the_o five_o theorem_a the_o 8._o proposition_n if_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_n to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n &_o have_v also_o the_o base_a of_o the_o one_o equal_a to_o the_o base_a of_o the_o other_o they_o shall_v have_v also_o the_o angle_n contain_v under_o the_o equal_a right_a line_n of_o the_o one_o equal_a to_o the_o angle_n contain_v under_o the_o equal_a right_a line_n of_o the_o other_o svppose_v that_o there_o be_v two_o triangle_n abc_n and_o def_n &_o let_v these_o two_o side_n of_o the_o one_o ab_fw-la and_o ac_fw-la be_v equal_a to_o these_o two_o side_n of_o the_o other_o de_fw-fr and_o df_o each_o to_o his_o correspondent_a side_n that_o be_v ab_fw-la to_o d_o e_o and_o ac_fw-la to_o df_o &_o let_v the_o base_a of_o the_o one_o namely_o bc_o be_v equal_a to_o the_o base_a of_o the_o other_o namely_o to_o ef._n then_o i_o say_v that_o the_o angle_n bac_n be_v equal_a to_o the_o angle_n edf_o for_o the_o triangle_n abc_n exact_o agree_v with_o the_o triangle_n de_fw-fr fletcher_n impossibility_n and_o the_o point_n b_o be_v put_v upon_o the_o point_n e_o and_o the_o right_a line_n bc_o upon_o the_o right_a line_n of_o the_o point_n c_o shall_v exact_o agree_v with_o the_o point_n fletcher_n for_o the_o line_n bc_o be_v equal_a to_o the_o line_n of_o and_o bc_o exact_o agree_v with_o of_o the_o line_n also_o basilius_n and_o ac_fw-la shall_v exact_o agree_v with_o the_o line_n ed_z &_o df._n for_o if_o the_o base_a bc_o do_v exact_o agree_v with_o the_o base_a fe_o but_o the_o side_n basilius_n &_o ac_fw-la do_v not_o exact_o agree_v with_o the_o side_n ed_z &_o df_o but_o differ_v as_o fg_o &_o gf_o do_v they_o from_o the_o end_n of_o one_o line_n shall_v be_v draw_v two_o right_a line_n to_o a_o point_n &_o from_o the_o self_n same_o end_n on_o the_o same_o side_n shall_v be_v draw_v then_o other_o line_n equal_a to_o the_o two_o first_o line_n the_o one_o to_o the_o other_o and_o unto_o a_o other_o point_n but_o that_o be_v impossible_a by_o the_o seven_o proposition_n wherefore_o the_o base_a bc_o exact_o agree_v with_o the_o base_a of_o the_o side_n also_o basilius_n and_o ac_fw-la do_v exact_o agree_v with_o the_o side_n ed_z and_o df._n wherefore_o also_o the_o angle_n bac_n shall_v exact_o agree_v with_o the_o angle_n edf_o and_o therefore_o shall_v also_o be_v equal_a to_o it_o 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 have_v also_o the_o base_a of_o the_o one_o equal_a to_o the_o base_a of_o the_o other_o they_o shall_v have_v also_o the_o angle_n contain_v under_o the_o equal_a right_a line_n of_o the_o one_o equal_a to_o the_o angle_n contain_v under_o the_o equal_a right_a line_n of_o the_o other_o which_o be_v require_v to_o be_v prove_v conversion_n this_o theorem_a be_v the_o converse_n of_o the_o four_o but_o it_o be_v not_o the_o chief_a and_o principal_a kind_a oâ—_n conversion_n for_o it_o turn_v not_o the_o whole_a supposition_n into_o the_o conclusion_n and_o the_o whole_a conclusion_n into_o the_o supposition_n for_o the_o four_o proposition_n who_o converse_n this_o be_v be_v a_o compound_n theorem_a have_v two_o thing_n give_v or_o suppose_v which_o be_v these_o the_o one_o that_o two_o side_n of_o the_o one_o triangle_n be_v equal_a to_o two_o side_n of_o the_o other_o triangle_n the_o other_a that_o the_o angle_n contain_v of_o the_o two_o side_n of_o the_o one_o be_v equal_a to_o the_o angle_n contain_v of_o the_o two_o side_n of_o the_o one_o but_o have_v amongst_o other_o one_o thing_n require_v which_o be_v that_o the_o base_a of_o the_o one_o be_v equal_a to_o the_o base_a of_o the_o other_o now_o in_o this_o 8._o proposition_n be_v the_o converse_n therofâ—_n that_o the_o base_a of_o the_o one_o be_v equal_a to_o the_o base_a of_o the_o other_a be_v the_o supposition_n or_o the_o thing_n give_v which_o in_o the_o former_a proposition_n be_v the_o conclusion_n and_o this_o that_o two_o side_n of_o the_o one_o be_v equal_a to_o two_o side_n of_o the_o other_o be_v in_o this_o proposition_n also_o a_o supposition_n like_a as_o it_o be_v in_o the_o former_a proposition_n so_o that_o it_o be_v a_o thing_n give_v in_o either_o proposition_n the_o conclusion_n of_o this_o proposition_n be_v that_o the_o angle_n enclose_v of_o the_o two_o equal_a side_n of_o the_o one_o triangle_n be_v equal_a to_o the_o angle_n enclose_v of_o the_o two_o equal_a side_n of_o the_o other_o triangle_n which_o in_o the_o former_a proposition_n be_v one_o of_o the_o thing_n give_v philo_n and_o his_o scholas_fw-la demonstrate_v this_o proposition_n without_o the_o help_n of_o the_o former_a proposition_n in_o this_o manner_n first_o let_v it_o fall_v direct_o case_n and_o forasmuch_o as_o the_o line_n de_fw-fr be_v equal_a to_o the_o line_n e_o g_o and_o dfg_n be_v one_o right_a line_n therefore_o deg_n be_v a_o isosceles_a triangle_n and_o so_o by_o the_o five_o proposition_n the_o angle_n at_o the_o point_n d_o be_v equal_a to_o the_o angle_n at_o the_o point_n g_o which_o be_v require_v to_o be_v prove_v the_o 4._o problem_n the_o 9_o proposition_n to_o divide_v a_o rectiline_a angle_n give_v into_o two_o equal_a part_n svppose_v that_o the_o rectiline_a angle_n give_v be_v bac_n it_o be_v require_v to_o divide_v the_o angle_n bac_n into_o two_o equal_a part_n construction_n in_o the_o line_n ab_fw-la take_v a_o point_n at_o all_o adventure_n &_o let_v the_o same_o be_v d._n and_o by_o the_o three_o proposition_n from_o the_o line_n ac_fw-la cut_v of_o the_o line_n ae_n equal_a to_o ad._n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n d_o to_o the_o point_n e._n and_o by_o the_o first_o proposition_n upon_o the_o line_n de_fw-fr describe_v a_o equilater_n triangle_n and_o let_v the_o same_o be_v dfe_n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n f._n then_o i_o say_v that_o the_o angle_n bac_n be_v by_o the_o line_n of_o divide_v into_o two_o equal_a part_n demonstration_n for_o forasmuch_o as_o ad_fw-la be_v equal_a to_o ae_n and_o of_o be_v common_a to_o they_o both_o therefore_o these_o two_o dam_n and_o of_o be_v equal_a to_o these_o two_o ea_fw-la and_o of_o the_o one_o to_o the_o other_o but_o by_o the_o first_o proposition_n the_o base_a df_o be_v equal_a to_o the_o base_a of_o wherefore_o by_o the_o 8._o proposition_n the_o angle_n daf_n be_v equal_a to_o the_o angle_n fae_z wherefore_o the_o rectiline_a angle_n give_v namely_o b_o ac_fw-la be_v divide_v into_o two_o equal_a part_n by_o the_o right_a line_n afâ—_n which_o be_v require_v to_o be_v do_v in_o this_o proposition_n be_v not_o teach_v to_o divide_v a_o right_n line_v angle_n into_o more_o part_n than_o two_o albeit_o to_o divide_v a_o angle_n so_o it_o be_v a_o right_a angle_n into_o three_o part_n it_o be_v not_o hard_a and_o it_o be_v teach_v of_o viâ—ellio_n in_o his_o first_o book_n of_o perspective_n nature_n the_o 28._o proposition_n â—or_a to_o divide_v a_o acute_a angle_n into_o three_o equal_a part_n be_v as_o say_v proclus_n impossible_a unless_o it_o be_v by_o the_o help_n of_o other_o line_n which_o be_v of_o a_o mix_a nature_n which_o thing_n nicomedes_n do_v by_o such_o line_n which_o be_v call_v concoideâ—_n linea_fw-la who_o first_o search_v out_o the_o invention_n nature_n &_o property_n of_o such_o line_n and_o other_o do_v it_o by_o other_o mean_v as_o by_o the_o help_n of_o quadrant_a line_n invent_v by_o hippias_n &_o nicomedes_n other_o by_o helices_n or_o spiral_n line_n invent_v of_o archimedes_n but_o these_o be_v thing_n of_o much_o difficulty_n and_o hardness_n and_o not_o here_o to_o be_v entreat_v of_o here_o against_o this_o proposition_n may_v of_o the_o adversary_n be_v bring_v a_o manifest_a instance_n for_o he_o may_v cavil_v that_o the_o
head_n of_o the_o equilater_n triangle_n shall_v not_o fall_v between_o the_o two_o right_a line_n but_o in_o one_o of_o they_o or_o without_o they_o both_o as_o for_o example_n there_o may_v also_o in_o this_o proposition_n be_v diverse_a casesâ—_n proposition_n â—f_n it_o so_o happen_v that_o there_o be_v no_o space_n under_o the_o base_a de_fw-fr to_o describe_v a_o equilater_n triangle_n but_o that_o of_o necessity_n you_o must_v describe_v it_o on_o the_o same_o side_n that_o the_o line_n ab_fw-la and_o ac_fw-la be_v for_o then_o the_o side_n of_o the_o equilater_n triangle_n either_o exact_o agree_v with_o the_o line_n ad_fw-la and_o ae_n if_o the_o say_a line_n ad_fw-la and_o ae_n be_v equal_a with_o the_o base_a de._n or_o they_o fall_v without_o they_o if_o the_o line_n ad_fw-la and_o ae_n be_v less_o than_o the_o base_a de._n or_o they_o fall_v within_o they_o if_o the_o say_a line_n be_v great_a theâ—_z the_o base_a de._n the_o 5._o problem_n the_o 10._o proposition_n to_o divide_v a_o right_a line_n give_v be_v finite_a into_o two_o equal_a part_n svppose_v that_o the_o right_a line_n give_v be_v ab_fw-la it_o be_v require_v to_o divide_v the_o line_n ab_fw-la into_o two_o equal_a part_n construction_n describe_v by_o the_o first_o proposition_n upon_o the_o line_n ab_fw-la a_o equilater_n triangle_n and_o let_v the_o same_o be_v abc_n and_o by_o the_o former_a proposition_n divide_v the_o angle_n acb_o into_o two_o equal_a part_n by_o the_o right_a line_n cd_o then_o i_o say_v that_o the_o right_a line_n ab_fw-la be_v divide_v into_o two_o equal_a part_n in_o the_o point_n d._n demonstration_n for_o forasmuch_o as_o by_o the_o first_o proposition_n ac_fw-la be_v equal_a to_o cb_o and_o cd_o be_v common_a to_o they_o both_o therefore_o these_o two_o line_n ac_fw-la &_o cd_o be_v equal_a to_o these_o two_o line_n bc_o &_o cd_o the_o one_o to_o the_o outstretch_o and_o the_o angle_n acd_o be_v equal_a to_o the_o angle_n bcd_o wherefore_o by_o the_o 4._o proposition_n the_o base_a ad_fw-la be_v equal_a to_o the_o base_a bd._n wherefore_o the_o right_a line_n give_v ab_fw-la be_v divide_v into_o two_o equal_a part_n in_o the_o point_n d_o which_o be_v require_v to_o be_v do_v apollonius_n reach_v to_o divide_v a_o right_a line_n be_v finite_a into_o two_o equal_a part_n after_o this_o manner_n apollonius_n the_o 6._o problem_n the_o 11._o proposition_n upon_o a_o right_a line_n give_v to_o raise_v up_o from_o a_o point_n give_v in_o the_o same_o line_n a_o perpendicular_a line_n svppose_v that_o the_o right_a line_n give_v be_v ab_fw-la &_o let_v the_o point_n in_o it_o give_v be_v c._n it_o be_v require_v from_o the_o point_n c_o to_o raise_z up_o unto_o the_o right_a line_n ab_fw-la a_o perpendicular_a line_n take_v in_o the_o line_n ac_fw-la a_o point_n at_o all_o adventure_n construction_n &_o let_v the_o same_o be_v d_o and_o by_o the_o 3._o proposition_n put_v unto_o dc_o a_o equal_a line_n ce._n and_o by_o the_o first_o proposition_n upon_o the_o line_n de_fw-fr describe_v a_o equilater_n triangle_n fde_v &_o draw_v a_o line_n from_o fletcher_n to_o c._n then_o i_o say_v that_o unto_o the_o right_a line_n give_v ab_fw-la and_o from_o the_o point_n in_o it_o give_v namely_o c_o be_v raise_v up_o a_o perpendicular_a line_n fc_o for_o forasmuch_o as_o dc_o be_v equal_a to_o ce_fw-fr demonstration_n &_o the_o line_n cf_o be_v common_a to_o they_o both_o therefore_o these_o two_o dc_o and_o cf_o be_v equal_a to_o these_o two_o ec_o &_o cf_o the_o one_o to_o the_o other_o and_o by_o the_o first_o proposition_n the_o base_a df_o be_v equal_a to_o the_o base_a of_o wherefore_o by_o the_o 8._o proposition_n the_o angle_n dcf_n be_v equal_a to_o the_o angle_n ecf_n and_o they_o be_v side_n angle_n but_o when_o a_o right_a line_n stand_v upon_o a_o right_a line_n do_v make_v the_o two_o side_n angle_n equal_a the_o one_o to_o the_o other_o either_o of_o those_o equal_a angle_n be_v by_o the._n 10._o definition_n a_o right_a angle_n &_o the_o line_n stand_v upon_o the_o right_a line_n be_v call_v a_o perpendicular_a line_n wherefore_o the_o angle_n dcf_n &_o thangle_v fce_n be_v right_a angle_n wherefore_o unto_o the_o right_a line_n give_v ab_fw-la &_o from_o the_o point_n in_o it_o c_o be_v raise_v up_o a_o perpendicular_a line_n cf_o which_o be_v require_v to_o be_v do_v although_o the_o point_n give_v shall_v be_v set_v in_o one_o of_o the_o end_n of_o the_o right_a line_n give_v it_o be_v easy_a so_o do_v it_o as_o it_o be_v before_o for_o produce_v the_o line_n in_o length_n from_o the_o point_n by_o the_o second_o petition_n you_o may_v work_v as_o you_o do_v before_o but_o if_o one_o require_v to_o erect_v a_o right_a line_n perpendicular_o from_o the_o point_n at_o the_o end_n of_o the_o line_n without_o produce_v the_o rightlyne_v that_o also_o may_v well_o be_v do_v after_o this_o manner_n appollonius_n teach_v to_o raise_v up_o unto_o a_o line_n give_v from_o a_o point_n in_o it_o give_v a_o perpendicular_a line_n after_o this_o manner_n construction_n the_o 7._o problem_n the_o 12._o proposition_n unto_o a_o right_a line_n give_v be_v infinite_a and_o from_o a_o point_n give_v not_o be_v in_o the_o same_o line_n to_o draw_v a_o perpendicular_a line_n let_v the_o right_a line_n give_v be_v infinite_a be_v ab_fw-la &_o let_v the_o point_n give_v not_o be_v in_o the_o say_a line_n ab_fw-la be_v c._n it_o be_v require_v from_o the_o point_n give_v namely_o c_o to_o draw_v unto_o the_o right_a line_n give_v ab_fw-la a_o perpendicular_a line_n construction_n take_v on_o the_o other_o side_n of_o the_o line_n ab_fw-la namely_o on_o that_o side_n wherein_o be_v not_o the_o point_n c_o a_o point_n at_o all_o adventure_n and_o let_v the_o same_o be_v d._n and_o make_v the_o centre_n c_o and_o the_o space_n cd_o describe_v by_o the_o three_o petition_n a_o circle_n and_o let_v the_o same_o be_v efg_o which_o let_v cut_v the_o line_n ab_fw-la in_o the_o point_n e_o and_o g._n and_o by_o the_o x._o proposition_n divide_v the_o line_n eglantine_n into_o two_o equal_a part_n in_o the_o point_n h._n and_o by_o the_o first_o petition_n draw_v these_o right_a line_n cg_o change_z and_o ce._n then_o i_o say_v that_o unto_o the_o right_a line_n give_v ab_fw-la &_o from_o the_o point_n give_v not_o be_v in_o it_o namely_o c_o be_v draw_v a_o perpendicular_a line_n ch._n demonstration_n for_o forasmuch_o as_o gh_o be_v equal_a to_o he_o and_o hc_n be_v common_a to_o they_o both_o therefore_o these_o two_o side_n gh_o and_o hc_n be_v equal_a to_o these_o two_o side_n eh_o &_o hc_n the_o one_o to_o the_o otherâ—_n and_o by_o the_o 15_o definition_n the_o base_a cg_o be_v equal_a to_o the_o base_a ce_fw-fr wherefore_o by_o the_o 8._o proposition_n the_o angle_n chg_n be_v equal_a to_o the_o angle_n i_o and_o they_o be_v side_n angle_n but_o when_o a_o right_a line_n stand_v upon_o a_o right_a line_n make_v the_o two_o side_n angle_n equal_a the_o one_o to_o the_o other_o either_o of_o those_o equal_a angle_n be_v by_o the_o 10._o definition_n a_o right_a angle_n and_o the_o line_n stand_v upon_o the_o say_a right_a line_n be_v call_v a_o perpendicular_a line_n wherefore_o unto_o the_o right_a line_n give_v ab_fw-la and_o from_o the_o point_n give_v c_o which_o be_v not_o in_o the_o line_n ab_fw-la be_v draw_v a_o perpendicular_a line_n change_z which_o be_v require_v to_o be_v do_v this_o problem_n do_v oenâ—pides_n first_o find_v out_o problem_n consider_v the_o necessary_a use_n thereof_o to_o the_o study_n of_o astronomy_n there_o arâ—_n two_o kind_n of_o perpendicular_a line_n solid_a whereof_o one_o be_v a_o plain_a perpendicular_a line_n the_o other_o be_v a_o solid_a a_o plain_a perpendicular_a line_n be_v when_o the_o point_n from_o whence_o the_o perpendiâ—uler_a line_n iâ—_z draw_v be_v in_o the_o same_o plain_a superficies_n with_o the_o line_n whereunto_o it_o be_v a_o perpendicular_a a_o solid_a perpendicular_a line_n be_v when_o the_o point_n from_o whence_o the_o perpendicular_a be_v draw_v be_v on_o high_a and_o without_o the_o plain_a superficies_n so_o that_o a_o plain_a perpendicular_a line_n be_v draw_v to_o a_o right_a line_n &_o a_o solid_a perpendicular_a line_n be_v draw_v to_o a_o superficies_n a_o plainâ—_n perpendicular_a line_n cause_v right_a angle_n with_o one_o only_a line_n namely_o with_o that_o upon_o who_o it_o fall_v but_o a_o solid_a perpendicular_a line_n cause_v right_a anâ—leâ—_n not_o only_o with_o one_o line_n but_o with_o as_o many_o line_n as_o may_v be_v draw_v in_o that_o superficies_n by_o the_o touch_n thereof_o line_n this_o proposition_n teach_v to_o draw_v a_o plain_a perpendicular_a line_n for_o it_o be_v draw_v to_o one_o line_n and_o suppose_v to_o
be_v require_v to_o be_v demonstrate_v this_o proposition_n may_v also_o be_v demonstrate_v without_o produce_v any_o of_o the_o side_n aâ—ter_v this_o manner_n the_o same_o may_v yet_o also_o be_v demonstrate_v a_o other_o way_n this_o proposition_n may_v yet_o moreover_o be_v demonstrate_v by_o a_o argument_n lead_v to_o a_o absurdity_n and_o that_o after_o this_o manner_n line_n a_o man_n may_v also_o more_o brief_o demonstrate_v this_o proposition_n by_o campanus_n definition_n of_o a_o right_a line_n which_o as_o we_o have_v before_o declare_v be_v thus_o a_o right_a line_n be_v the_o short_a extension_n or_o drawght_fw-mi that_fw-mi be_v or_o may_v be_v from_o one_o point_n to_o another_o wherefore_o any_o one_o side_n of_o a_o triangle_n for_o that_o it_o be_v a_o right_a line_n draw_v from_o some_o one_o point_n to_o some_o other_o one_o point_n be_v of_o necessity_n short_a than_o the_o other_o two_o side_n draw_v from_o and_o to_o the_o same_o point_n understanding_n epicurus_n and_o such_o as_o follow_v he_o deride_v this_o proposition_n not_o count_v it_o worthy_a to_o be_v add_v in_o the_o number_n of_o proposition_n of_o geometry_n for_o the_o easiness_n thereof_o for_o that_o it_o be_v manifest_a even_o to_o the_o sense_n but_o not_o all_o thing_n manifest_a to_o sense_n be_v straight_o way_n manifest_a to_o reason_n and_o understanding_n it_o pertain_v to_o one_o that_o be_v a_o teacher_n of_o science_n by_o proof_n and_o demonstration_n to_o render_v a_o certain_a and_o undoubted_a reason_n why_o it_o so_o appear_v to_o the_o senseâ—_n and_o in_o that_o only_o consist_v science_n the_o 14._o theorem_a the_o 21._o proposition_n if_o from_o the_o end_n of_o one_o of_o the_o side_n of_o a_o triangle_n be_v draw_v to_o any_o point_n within_o the_o say_a triangle_n two_o right_a line_n those_o right_a line_n so_o draw_v shall_v be_v less_o then_o the_o two_o other_o side_n of_o the_o triangle_n but_o shall_v contain_v the_o great_a angle_n svppose_v that_o abc_n be_v a_o triangle_n and_o from_o the_o end_n of_o the_o side_n bc_o namely_o from_o the_o point_n b_o and_o c_o let_v there_o be_v draw_v within_o the_o triangle_n two_o right_a line_n bd_o and_o cd_o to_o the_o point_n d._n then_o i_o say_v that_o the_o line_n bd_o and_o cd_o be_v less_o than_o the_o other_o side_n of_o the_o triangle_n namely_o than_o the_o side_n basilius_n and_o ac_fw-la and_o that_o the_o angle_n which_o they_o contain_v namely_o bdc_o be_v great_a than_o the_o angle_n bac_n extend_v by_o the_o second_o petition_n the_o line_n bd_o to_o the_o point_n e._n demonstration_n and_o forasmuch_o as_o by_o the_o 20._o proposition_n in_o every_o triangle_n the_o two_o side_n be_v great_a than_o the_o side_n remain_v therefore_o the_o two_o side_n of_o the_o triangle_n abe_n namely_o the_o side_n ab_fw-la and_o ae_n be_v great_a than_o the_o side_n ebb_n put_v the_o line_n ec_o common_a to_o they_o both_o wherefore_o the_o line_n basilius_n and_o ac_fw-la be_v great_a than_o the_o line_n be_v and_o ecâ—_n again_o forasmuch_o as_o by_o the_o same_o in_o the_o triangle_n ce_z the_o two_o side_n ce_fw-fr and_o ed_z be_v great_a than_o the_o side_n dc_o put_v the_o line_n db_o common_a to_o they_o bothâ—_n wherefore_o the_o line_n ce_fw-fr and_o ed_z be_v great_a than_o the_o line_n cd_o and_o db._n but_o it_o be_v prove_v that_o the_o line_n basilius_n and_o ac_fw-la be_v great_a than_o the_o line_n be_v and_o ec_o wherefore_o the_o line_n basilius_n and_o ac_fw-la be_v much_o great_a than_o the_o line_n bd_o and_o dc_o again_o forasmuch_o as_o by_o the_o 16._o proposition_n in_o every_o triangle_n the_o outward_a angle_n be_v great_a than_o the_o inward_a and_o opposite_a angle_n therefore_o the_o outward_a angle_n of_o the_o triangle_n cde_o namely_o bdc_o be_v great_a than_o the_o angle_n ce_z wherefore_o also_o by_o the_o same_o the_o outward_a angle_n of_o the_o triangle_n abe_n namely_o the_o angle_n ceb_n be_v great_a than_o the_o angle_n bac_n but_o it_o be_v prove_v that_o the_o angle_n bdc_o be_v great_a than_o the_o angle_n ceb_fw-mi wherefore_o the_o angle_n bdc_o be_v much_o great_a than_o the_o angle_n bac_n wherefore_o if_o from_o the_o end_n of_o one_o of_o the_o side_n of_o a_o triangle_n be_v draw_v to_o any_o point_n within_o the_o say_a triangle_n two_o right_a line_n those_o right_a line_n so_o draw_v shall_v be_v less_o than_o the_o two_o other_o side_n of_o the_o triangle_n but_o shall_v contain_v the_o great_a angle_n which_o be_v require_v to_o be_v demonstrate_v in_o this_o proposition_n be_v express_v that_o the_o two_o right_a line_n draw_v within_o the_o triangle_n have_v their_o beginning_n at_o the_o extreme_n of_o the_o side_n of_o the_o triangle_n for_o from_o the_o one_o extreme_a of_o the_o side_n of_o the_o triangle_n and_o from_o some_o one_o point_n of_o the_o same_o side_n may_v be_v draw_v two_o right_a line_n within_o the_o triangle_n which_o shall_v be_v long_o they_o the_o two_o outward_a line_n which_o be_v wonderful_a and_o seem_v strange_a that_o two_o right_a line_n draw_v upon_o a_o part_n of_o a_o line_n shall_v be_v great_a than_o two_o right_a line_n draw_v upon_o the_o whole_a line_n and_o again_o it_o be_v possible_a from_o the_o one_o extreme_a of_o the_o side_n of_o a_o triangle_n and_o from_o some_o one_o point_n of_o the_o same_o side_n to_o draw_v two_o right_a line_n within_o the_o triangle_n which_o shall_v contain_v a_o angle_n less_o than_o the_o angle_n contain_v under_o the_o two_o outward_a line_n as_o touch_v the_o first_o part_n the_o 8._o problem_n the_o 22._o proposition_n of_o three_o right_a line_n which_o be_v equal_a to_o three_o right_a line_n give_v to_o make_v a_o triangle_n but_o it_o behove_v two_o of_o those_o line_n which_o then_o soever_o be_v take_v to_o be_v great_a than_o the_o three_o for_o that_o in_o every_o triangle_n two_o side_n which_o two_o side_n soever_o be_v take_v be_v great_a than_o the_o side_n remain_v svppose_v that_o the_o three_o right_a line_n give_v be_v a_o b_o c_o of_o which_o let_v two_o of_o they_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 line_n a_o b_o be_v great_a than_o the_o line_n c_o and_o the_o line_n a_o c_o then_z the_o line_n b_o and_o the_o line_n b_o c_o then_z the_o line_n a._n it_o be_v require_v of_o three_o right_a line_n equal_a to_o the_o right_a line_n a_o b_o c_o to_o make_v a_o triangle_n construction_n take_v a_o right_a line_n have_v a_o appoint_a end_n on_o the_o side_n d_o and_o be_v infinite_a on_o the_o side_n e._n and_o by_o the_o 3._o proposition_n put_v unto_o the_o line_n a_o a_o equal_a line_n df_o and_o put_v unto_o the_o line_n b_o a_o equal_a line_n fg_o and_o unto_o the_o line_v c_o a_o equal_a line_n gh_o and_o make_v the_o centre_n fletcher_n and_o the_o space_n df_o describe_v by_o the_o 3._o petition_n a_o circle_n dkl_n again_o make_v the_o centre_n g_o and_o the_o space_n g_o h_o describe_v by_o the_o same_o a_o circle_n hkl_n and_o let_v the_o point_n of_o the_o intersection_n of_o the_o say_a circle_n be_v king_n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n king_n to_o the_o point_fw-fr fletcher_n &_o a_o other_o from_o the_o point_n king_n to_o the_o point_n g._n then_o i_o say_v that_o of_o three_o right_a line_n equal_a to_o the_o line_n a_o b_o c_o be_v make_v a_o triangle_n kfg_n demonstration_n for_o forasmuch_o as_o the_o point_n fletcher_n be_v the_o centre_n of_o the_o circle_n dkl_n therefore_o by_o the_o 15._o definition_n the_o line_n fd_o be_v equal_a to_o the_o line_n fk_o but_o the_o line_n a_o be_v equal_a to_o the_o line_n fd_o wherefore_o by_o the_o first_o common_a sentence_n the_o line_n fk_o be_v equal_a to_o the_o line_n a._n again_o forasmuch_o as_o the_o point_n g_o be_v the_o centre_n of_o the_o circle_n lkh_n therefore_o by_o the_o same_o definition_n the_o line_n gk_o be_v equal_a to_o the_o line_n ghâ—_n but_o the_o line_n c_o be_v equal_a to_o the_o line_n gh_o wherefore_o by_o the_o first_o common_a sentence_n the_o line_n kg_n be_v equal_a to_o the_o line_n c._n but_o the_o line_n fg_o be_v by_o supposition_n equal_a to_o the_o line_n b_o wherefore_o these_o three_o right_a line_n gf_o fk_o and_o kg_v be_v equal_a to_o these_o three_o right_a line_n a_o b_o c._n wherefore_o of_o three_o right_a line_n that_o be_v kf_n fg_o and_o gk_o which_o be_v equal_a to_o the_o three_o right_a line_n give_v that_o be_v to_o a_o b_o c_o be_v make_v a_o triangle_n kfg_n which_o be_v require_v to_o be_v do_v an_o other_o construction_n and_o demonstration_n after_o flussates_n suppose_v that_o the_o
into_o two_o equal_a part_n but_o also_o they_o divide_v the_o angle_n into_o two_o equal_a part_n but_o in_o parallogramme_n who_o side_n be_v not_o equal_a such_o as_o be_v a_o figure_n on_o the_o one_o side_n long_o and_o a_o rhomboide_n it_o be_v not_o so_o prâ—â—lus_n the_o converse_n of_o the_o first_o and_o second_o part_n of_o this_o proposition_n after_o proclus_n be_v 〈◊〉_d figure_n whatsoever_o have_v his_o opposite_a side_n and_o angle_n equal_a then_o be_v a_o parallellogram_n flussates_n a_o corollary_n take_v out_o of_o flussates_n a_o right_a line_n cut_v a_o parallelogram_n which_o way_n soever_o into_o two_o equal_a part_n shall_v also_o divide_v the_o diameter_n thereof_o into_o two_o equal_a part_n for_o if_o it_o be_v possible_a let_v the_o right_a line_n gc_o divide_v the_o parallelogram_n aebd_a into_o two_o equal_a part_n but_o let_v it_o divide_v the_o diameter_n de_fw-fr into_o two_o unequal_a part_n in_o the_o point_n i_o and_o lâ—t_v the_o part_n je_n be_v great_a than_o the_o part_n id_fw-la and_o unto_o the_o line_n id_fw-la put_v the_o line_n io_o equal_v by_o th3_n proposition_n absurdity_n and_o by_o the_o point_n oh_o draw_v unto_o the_o line_n ad_fw-la and_o be_v a_o parallel_a line_n of_o by_o the_o 31._o proposition_n wherefore_o in_o the_o triangle_n foi_fw-fr and_o cdi_o two_o angle_n of_o the_o one_o be_v equal_a to_o two_o angle_n of_o the_o other_o namely_o the_o angle_n jof_o and_o idc_n by_o the_o 29._o proposition_n &_o the_o angle_n fio_n &_o cid_n by_o the_o 15._o proposition_n &_o the_o side_n id_fw-la be_v equal_a to_o the_o side_n io._n wherefore_o by_o the_o 26._o proposition_n the_o triangle_n be_v equal_a wherefore_o the_o whole_a triangle_n eig_n be_v great_a than_o the_o triangle_n dic._n and_o forasmuch_o as_o the_o trapesium_n gbdc_n be_v suppose_v to_o be_v the_o half_a of_o the_o parrallelograme_n and_o the_o half_a of_o the_o same_o parallelogram_n be_v the_o triangle_n ebb_v by_o this_o proposition_n from_o the_o trapesium_n gbdc_n and_o the_o triangle_n ebb_v which_o be_v equal_a take_v away_o the_o trapesium_n gbdi_n which_o be_v common_a to_o they_o both_o and_o the_o residue_n namely_o the_o triangle_n dic_fw-la shall_v be_v equal_a to_o the_o residue_n namely_o to_o the_o triangle_n eig_n but_o it_o be_v also_o less_o as_o have_v before_o be_v prove_v which_o be_v impossible_a wherefore_o a_o right_a line_n devide_v a_o parallelogram_n into_o two_o equal_a part_n shall_v not_o divide_v the_o diameter_n thereof_o unequal_o wherefore_o it_o shall_v divide_v it_o equal_o which_o be_v require_v to_o be_v prove_v a_o addition_n of_o pelitarius_n between_o two_o right_a line_n be_v infinite_a and_o make_v a_o angle_n give_v pelitarius_n to_o place_v a_o line_n equal_a to_o a_o line_n give_v in_o such_o sort_n that_o it_o shall_v make_v with_o one_o of_o those_o line_n a_o angle_n equal_a to_o a_o other_o angle_n give_v now_o it_o behove_v that_o the_o two_o angle_n give_v be_v less_o than_o two_o right_a angle_n though_o this_o addition_n of_o pelitarius_n be_v not_o so_o much_o pertain_v to_o the_o proposition_n yet_o because_o it_o be_v witty_a and_o seem_v somewhat_o difficult_a i_o think_v it_o good_a here_o to_o anexe_v it_o the_o 25._o theorem_a the_o 35._o proposition_n parallelogram_n consist_v upon_o one_o and_o the_o same_o base_a and_o in_o the_o self_n same_o parallel_a line_n be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o these_o parallelogram_n abcd_o and_o ebb_n cf_o do_v consist_v upon_o one_o and_o the_o same_o base_a that_o be_v upon_o bc_o and_o in_o the_o self_n same_o parallel_a line_n that_o be_v of_o and_o bc._n then_o i_o say_v that_o the_o parallelogram_n abcd_o be_v equal_a to_o the_o parallelogram_n ebcf._n demonstration_n for_o forasmuch_o as_o abcd_o be_v a_o parallelogram_n therefore_o by_o the_o 34._o proposition_n the_o side_n ad_fw-la be_v equal_a to_o the_o side_n bc_o and_o by_o the_o same_o reason_n also_o the_o side_n of_o be_v equal_a to_o the_o side_n bc_o wherefore_o ad_fw-la be_v equal_a to_o of_o and_o de_fw-fr be_v common_a to_o they_o both_o wherefore_o the_o whole_a line_n ae_n be_v equal_a to_o the_o whole_a line_n df._n and_o the_o side_n ab_fw-la be_v equal_a to_o the_o side_n dc_o wherefore_o these_o two_o ea_fw-la and_o ab_fw-la be_v equal_a to_o these_o two_o fd_a and_o dc_o the_o one_o to_o the_o other_o and_o the_o angle_n fdc_n be_v equal_a to_o the_o angle_n eab_n namely_o the_o outward_a angle_n to_o the_o inward_a angle_n by_o the_o â—9_o proposition_n wherefore_o by_o the_o 4_o proposition_n the_o base_a ebb_n be_v equal_a to_o the_o base_a fc_n and_o the_o triangle_n eab_n be_v equal_a to_o the_o triangle_n fdc_n take_v away_o the_o triangle_n dge_n which_o be_v common_a to_o they_o both_o wherefore_o the_o residue_n namely_o the_o trapesium_n abgd_v be_v equal_a to_o the_o residue_n that_o be_v to_o the_o trapesium_n egcf_n put_v the_o triangle_n gbc_n common_a to_o they_o both_o wherefore_o the_o whole_a parallelogram_n abcd_o be_v equal_a to_o the_o whole_a parallelogram_n ebcf._n wherefore_o parallelogram_n consist_v upon_o one_o and_o the_o same_o base_a and_o in_o the_o self_n same_o parallel_a line_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v demonstrate_v parallelogram_n be_v say_v to_o be_v in_o the_o self_n same_o parallel_a line_n when_o their_o base_n and_o the_o opposite_a side_n unto_o they_o be_v one_o and_o the_o self_n same_o line_n with_o the_o parallel_n case_n in_o this_o proposition_n be_v three_o case_n for_o the_o line_n be_v may_v cut_v the_o line_n of_o either_o beyond_o the_o point_n d_o or_o in_o the_o point_n d_o or_o on_o this_o side_n the_o point_n d._n when_o it_o cut_v the_o line_n of_o beyond_o the_o point_n d_o the_o demonstration_n before_o put_v serve_v the_o 26._o theorem_a the_o 36._o proposition_n parallelogram_n consist_v upon_o equal_a base_n and_o in_o the_o self_n same_o parallel_a line_n be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o these_o parallelogram_n abcd_o and_o efgh_o do_v consist_v upon_o equal_a base_n that_o be_v upon_o bc_o and_o fg_o and_o in_o the_o self_n same_o parallel_a line_n that_o be_v ah_o and_o bg_o then_o i_o say_v that_o the_o parallelogram_n abcd_o be_v equal_a to_o the_o parallelogram_n efgh_o construction_n draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n e_o and_o a_o other_o from_o the_o point_n c_o to_o the_o point_n h._n demonstration_n and_o forasmuch_o as_o bc_o be_v equal_a to_o fg_o but_o fg_o be_v equal_a to_o eh_o therefore_o bc_o also_o be_v equal_a to_o eh_o and_o they_o be_v parallel_a line_n and_o the_o line_n be_v and_o change_z join_v they_o together_o but_o two_o right_a line_n join_v together_o two_o equal_a right_a line_n be_v parallel_n be_v themselves_o also_o by_o the_o 33_o proposition_n equal_v the_o one_o to_o the_o other_o and_o parallel_n wherefore_o ebch_n be_v a_o parallelogram_n and_o be_v equal_a to_o the_o parallelogram_n abcd_o for_o they_o have_v both_o one_o and_o the_o same_o base_a that_o be_v bc._n and_o be_v in_o the_o self_n same_o parallel_a line_n that_o be_v bc_o &_o eh_o and_o by_o the_o same_o reason_n also_o the_o parallelogram_n efgh_o be_v equal_a to_o the_o parallelogram_n ebch_o wherefore_o the_o parallelogram_n abcd_o be_v equal_a to_o the_o parallelogram_n efgh_o wherefore_o parallelogram_n consist_v upon_o equal_a base_n and_o in_o the_o self_n same_o parallel_v line_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v proposition_n in_o this_o proposition_n also_o be_v three_o case_n for_o the_o equal_a base_n may_v either_o be_v utter_o separate_v a_o sunder_o or_o they_o may_v touch_v aâ—_z one_o of_o the_o end_n or_o they_o may_v have_v one_o part_n common_a to_o they_o both_o case_n euclides_n demonstration_n serve_v when_o the_o base_n be_v utter_o separate_v a_o sunder_o which_o yet_o may_v happen_v seven_o diverse_a way_n way_n for_o the_o base_n be_v separate_v a_o sunder_o their_o opposite_a side_n also_o may_v be_v utter_o separate_v a_o sunder_o beyond_o the_o point_n d_o as_o the_o side_n ad_fw-la and_o eh_o in_o the_o first_o figure_n or_o they_o may_v touch_v together_o in_o one_o of_o the_o end_n and_o the_o whole_a side_n may_v be_v beyond_o the_o point_n d_o as_o the_o side_n ad_fw-la and_o eh_o do_v in_o the_o second_o figure_n or_o they_o may_v just_o agree_v the_o one_o with_o the_o other_o that_o be_v the_o point_n a_o and_o d_o may_v fall_v upon_o the_o point_n e_o and_o h_o as_o in_o the_o four_o figure_n or_o the_o side_n ad_fw-la be_v produce_v on_o this_o side_n the_o point_n a_o part_n of_o the_o opposite_a side_n unto_o the_o base_a fg_o may_v be_v on_o this_o
side_n the_o point_n a_o and_o a_o other_o part_n may_v be_v common_a with_o the_o line_n ad_fw-la as_o in_o the_o five_o figure_n or_o one_o end_n of_o the_o side_n eh_o may_v light_v upon_o the_o point_n a_o and_o the_o whole_a side_n on_o this_o side_n of_o it_o as_o in_o the_o six_o figure_n or_o the_o say_a side_n eh_o may_v utter_o be_v separate_v a_o sunder_o on_o this_o side_n the_o point_n a_o as_o in_o the_o seven_o figure_n and_o the_o two_o other_o case_n also_o may_v in_o like_a manner_n have_v seven_o variety's_n case_n as_o in_o the_o figure_n here_o underneath_o and_o on_o the_o other_o side_n of_o this_o leaf_n set_v it_o be_v manifest_a and_o here_o be_v to_o be_v note_v that_o in_o these_o three_o case_n and_o in_o all_o their_o variety_n also_o also_o the_o construction_n &_o demonstration_n put_v by_o euclid_n namely_o the_o draw_v of_o line_n from_o the_o point_n b_o to_o the_o point_n e_o &_o from_o the_o point_n c_o to_o the_o point_n h_o and_o so_o prove_v it_o by_o the_o former_a proposition_n will_v serve_v only_o in_o the_o four_o variety_n of_o each_o case_n there_o need_v no_o far_a construction_n for_o that_o the_o conclusion_n straight_o way_n follow_v by_o the_o former_a proposition_n the_o 27._o theorem_a the_o 37._o proposition_n triangle_n consist_v upon_o one_o and_o the_o self_n same_o base_a and_o in_o the_o self_n same_o paralle_n be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o these_o triangle_n abc_n and_o dbc_n do_v consist_v upon_o one_o and_o the_o same_o base_a namely_o bc_o and_o in_o the_o self_n same_o parallel_a line_n that_o be_v ad_fw-la and_o bc._n then_o i_o say_v that_o the_o triangle_n abc_n be_v equal_a to_o the_o triangle_n bdc_o construction_n produce_v by_o the_o 2._o petition_n the_o line_n ad_fw-la on_o each_o side_n to_o the_o point_n e_o and_o f._n and_o by_o the_o 31._o proposition_n by_o the_o point_n b_o draw_v unto_o the_o line_n ca_n a_o parallel_a line_n be_v and_o by_o the_o same_o by_o the_o point_n c_o draw_v unto_o the_o line_n bd_o a_o parallel_a line_n cf._n demonstration_n wherefore_o ebca_n and_o dbcf_n be_v parallelogram_n and_o the_o parallelogram_n ebca_n be_v by_o the_o 35._o proposition_n equal_a to_o the_o parallelogram_n dbcf_n for_o they_o consist_v upon_o one_o and_o the_o self_n same_o base_a namely_o bc_o and_o be_v in_o the_o self_n same_o parallel_a line_n that_o be_v bc_o and_o ef._n but_o the_o triangle_n abc_n be_v by_o the_o 34._o proposition_n the_o half_a of_o the_o parallelogram_n ebca_n for_o the_o diameter_n ab_fw-la divide_v it_o into_o two_o equal_a part_n &_o by_o the_o same_o the_o triangle_n dbc_n be_v the_o half_a of_o the_o parallelogram_n dbcf_n for_o the_o diameter_n dc_o divide_v it_o into_o two_o equal_a part_n but_o the_o half_n of_o thing_n equal_a be_v also_o equal_a the_o one_o to_o the_o other_o by_o the_o 7._o common_a sentence_n wherefore_o ehe_fw-ge triangle_n abc_n be_v equal_a to_o the_o triangle_n dbc_n wherefore_o triangle_n consist_v upon_o one_o and_o the_o self_n same_o base_a and_o in_o the_o self_n same_o parallel_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v demonstrate_v those_o triangle_n be_v say_v to_o be_v contain_v within_o the_o self_n same_o parallel_a line_n line_n which_o have_v their_o base_n in_o one_o of_o the_o parallel_a line_n have_v their_o â—oppes_n in_o the_o other_o here_o as_o i_o promise_v will_v i_o show_v out_o of_o proclus_n the_o comparison_n of_o two_o triangle_n which_o have_v their_o side_n equal_a have_v the_o base_n and_o angle_n at_o the_o top_n unequal_a unequal_a and_o first_o i_o say_v that_o the_o unequal_a angle_n at_o the_o top_n be_v equal_a to_o two_o right_a angle_n the_o triangle_n shall_v be_v equal_a as_o for_o example_n angle_n but_o now_o let_v the_o angle_n bac_n and_o edf_o be_v less_o than_o two_o right_a angle_n and_o again_o let_v the_o angle_n at_o the_o point_n a_o be_v great_a than_o the_o angle_n at_o the_o point_n d._n then_o i_o say_v that_o the_o triangle_n abc_n be_v great_a than_o the_o triangle_n def_n let_v the_o same_o construction_n be_v also_o here_o that_o be_v in_o the_o two_o former_a and_o forasmuch_o as_o the_o angle_n bac_n and_o edf_o that_o be_v the_o angle_n edge_n &_o edf_o be_v less_o than_o two_o right_a angle_n but_o the_o angle_n edge_n and_o gdk_n be_v equal_a to_o two_o right_a angle_n take_v away_o the_o angle_n fdg_fw-mi which_o be_v common_a to_o they_o both_o wherefore_o the_o angle_n remain_v namely_o edf_o be_v less_o than_o the_o angle_n remain_v namely_o then_o gdk_a that_o be_v the_o angle_n hdk_n which_o by_o the_o 1â—_o proposition_n be_v equal_a to_o the_o angle_n ed_z f_o be_v less_o than_o the_o angle_n gdk_n wherefore_o the_o whole_a angle_n gd_o h_o be_v less_o than_o double_a to_o the_o angle_n gd_o k._n but_o it_o be_v double_a to_o the_o angle_n dgf_n as_o before_o it_o be_v prove_v wherefore_o the_o angle_n gd_v king_n be_v great_a than_o the_o angle_n dgf_n put_v the_o angle_n dgl_n equal_a to_o the_o angle_n gd_v king_n by_o the_o 23._o proposition_n and_o produce_v the_o line_n gl_n till_o it_o concur_v with_o the_o line_n of_o also_o producedâ—_n &_o let_v the_o concourse_n be_v in_o the_o point_n l._n and_o draw_v a_o line_n from_o d_z to_o l._n and_o for_o as_o much_o as_o the_o angle_n d_o gl_n be_v equal_a to_o the_o angle_n gd_v king_n and_o they_o be_v alternate_a angle_n therefore_o the_o line_n gl_n be_v a_o parallel_n to_o d_z e_o by_o the_o 27._o proposition_n wherefore_o by_o this_o proposition_n the_o triangle_n gd_v e_o and_o ld_n e_o be_v equal_a but_o the_o triangle_n i_o will_v be_v great_a than_o the_o triangle_n fd_o e_o and_o the_o triangle_n gd_o e_o be_v equal_a to_o the_o triangle_n abc_n wherefore_o the_o triangle_n abc_n be_v great_a than_o the_o triangle_n d_o of_o which_o be_v require_v to_o be_v prove_v the_o 28._o theorem_a the_o 38._o proposition_n triangle_n which_o consist_v upon_o equal_a base_n and_o in_o the_o self_n same_o parallel_a line_n be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o these_o triangle_n abc_n and_o def_n do_v consist_v upon_o equal_a base_n that_o be_v upon_o bc_o and_o of_o and_o in_o the_o self_n same_o parallel_a line_n that_o be_v bf_o and_o ad._n then_o i_o say_v that_o the_o triangle_n abc_n be_v equal_a to_o the_o triangle_n abc_n be_v equal_a to_o the_o triangle_n def_n construction_n produce_v by_o the_o second_o petition_n the_o line_n ad_fw-la on_o each_o side_n to_o the_o point_n g_o and_o h._n and_o by_o the_o 31._o proposition_n by_o the_o point_n b_o draw_v unto_o ca_n a_o parallel_a line_n bg_o and_o by_o the_o same_o by_o the_o point_n fletcher_n draw_v unto_o de_fw-fr a_o parallel_a line_n fh_o demonstration_n wherefore_o gbca_n and_o defh_n be_v parallelogram_n but_o the_o parallelogram_n gbca_n be_v by_o the_o 36_o proposition_n equal_a to_o the_o parallelogram_n defh_o for_o they_o consist_v upon_o equal_a base_n that_o be_v bc_o and_o of_o and_o be_v in_o the_o self_n same_o parallel_a line_n that_o be_v bf_o and_o gh_o but_o by_o the_o 34._o proposition_n the_o triangle_n abc_n be_v the_o half_a of_o the_o parallelogram_n gbca_n for_o the_o diameter_n ab_fw-la divide_v it_o into_o two_o equal_a part_n and_o the_o triangle_n def_n be_v by_o the_o same_o the_o half_a of_o the_o parallelogram_n defh_o for_o the_o diameter_n fd_o divide_v it_o into_o two_o equal_a part_n but_o the_o half_n of_o thing_n equal_a be_v by_o the_o 7._o common_a sentence_n equal_a â—he_n one_o to_o the_o other_o wherefore_o the_o triangle_n abc_n be_v equal_a to_o the_o triangle_n def_n wherefore_o triangle_n which_o consist_v upon_o equal_a base_n and_o in_o the_o self_n same_o parallel_a line_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v in_o this_o proposition_n be_v three_o case_n proposition_n for_o the_o base_n of_o the_o triangle_n either_o have_v one_o part_n common_a to_o they_o both_o or_o the_o base_a of_o the_o one_o touch_v the_o base_a of_o the_o other_o only_o in_o a_o point_n or_o their_o base_n be_v utter_o sever_v a_o sunder_o and_o each_o of_o these_o case_n may_v also_o be_v diverse_o diverse_o as_o we_o before_o have_v see_v in_o parallelogram_n consist_v on_o equal_a base_n and_o be_v in_o the_o self_n same_o parallel_a line_n so_o that_o he_o which_o diligent_o note_v the_o variety_n that_o be_v there_o put_v touch_v they_o may_v also_o easy_o frame_v the_o same_o variety_n to_o each_o case_n in_o this_o proposition_n wherefore_o i_o think_v it_o needle_n here_o to_o repeat_v the_o same_o again_o for_o how_o soever_o the_o base_n be_v put_v
the_o other_o in_o the_o point_n d_o and_o that_o only_a point_n be_v common_a to_o they_o both_o neither_o do_v the_o one_o enter_v into_o the_o other_o if_o any_o part_n of_o the_o one_o enter_v into_o any_o part_n of_o the_o other_o than_o the_o one_o cut_v and_o divide_v the_o other_o and_o touch_v the_o one_o the_o other_o not_o in_o one_o point_n only_o as_o in_o the_o other_o before_o but_o in_o two_o pointâ—s_n and_o have_v also_o a_o superficies_n common_a to_o they_o both_o as_o the_o circle_n ghk_v and_o hlk_v cut_v the_o one_o the_o other_o in_o two_o point_n h_o and_o edward_n and_o the_o one_o enter_v into_o the_o other_o also_o the_o superficies_n hk_o be_v common_a to_o they_o both_o for_o it_o be_v a_o part_n of_o the_o circle_n ghk_o and_o also_o it_o be_v a_o part_n of_o the_o circle_n hlk._n definition_n right_o line_n in_o a_o circle_n be_v say_v to_o be_v equal_o distant_a from_o the_o centre_n when_o perpendicular_a line_n draw_v from_o the_o centre_n unto_o those_o line_n be_v equal_a and_o that_o line_n be_v say_v to_o be_v more_o distant_a upon_o who_o fall_v the_o great_a perpendicular_a line_n as_o in_o the_o circle_n abcd_o who_o centre_n be_v e_o the_o two_o line_n ab_fw-la and_o cd_o have_v equal_a distance_n from_o the_o centre_n e_o because_o that_o the_o line_n of_o draw_v from_o the_o centre_n e_o perpendicular_o upon_o the_o line_n ab_fw-la and_o the_o line_n eglantine_n draw_v likewise_o perpendilar_o from_o the_o centre_n e_o upon_o the_o line_n cd_o be_v equal_a the_o one_o to_o the_o other_o but_o in_o the_o circle_n hklm_n who_o centre_n be_v no_o the_o line_n hk_o have_v great_a distance_n from_o the_o centre_n n_o than_o have_v the_o line_n lm_o for_o that_o the_o line_n on_o draw_v from_o the_o centre_n n_o perpendicular_o upon_o the_o line_n hk_o be_v great_a than_o the_o line_n np_v which_o be_v draw_v from_o the_o centre_n n_o perpendicular_o upon_o the_o line_n lm_o so_o likewise_o in_o the_o other_o figure_n the_o line_n ab_fw-la and_o dc_o in_o the_o circle_n abcd_o be_v equidistant_v from_o the_o centre_n gâ—_n â—_z because_o the_o line_n og_n and_o gp_n perpendicular_o draw_v from_o the_o centre_n g_o upon_o the_o say_a line_n ab_fw-la and_o dc_o be_v equal_a and_o the_o line_n ab_fw-la have_v great_a distance_n from_o the_o centre_n g_o than_o have_v the_o the_o line_n of_o because_o the_o line_n og_n perpendicular_o drowen_v from_o the_o centre_n g_o to_o the_o line_n ab_fw-la be_v greâ—ter_n than_o the_o line_n hg_o which_o be_v perpendicular_o draw_v from_o the_o câ—â—tre_n g_o to_o the_o line_n ef._n definition_n a_o section_n or_o segment_n of_o a_o circle_n be_v a_o figure_n comprehend_v under_o a_o right_a line_n and_o a_o portion_n of_o the_o circumference_n of_o a_o circle_n as_o the_o figure_n abc_n be_v a_o section_n of_o a_o circle_n because_o it_o be_v comprehend_v under_o the_o right_a line_n ac_fw-la and_o the_o circumference_n of_o a_o circle_n abc_n likewise_o the_o figure_n def_n be_v a_o section_n of_o a_o circle_n for_o that_o it_o be_v comprehend_v under_o the_o right_a line_n df_o and_o the_o circumference_n def_n and_o the_o figure_n abc_n for_o that_o it_o contain_v within_o it_o the_o centre_n of_o the_o circle_n be_v call_v the_o great_a section_n of_o a_o circle_n and_o the_o figure_n def_n be_v the_o less_o section_n of_o a_o circle_n because_o it_o be_v whole_o without_o the_o centre_n of_o the_o circle_n as_o it_o be_v note_v in_o the_o 16._o definition_n of_o the_o first_o book_n a_o angle_n of_o a_o section_n or_o segment_n be_v that_o angle_n which_o be_v contain_v under_o a_o right_a line_n and_o the_o circumference_n of_o the_o circle_n definition_n as_o the_o angle_n abc_n in_o the_o section_n abc_n be_v a_o angle_n of_o a_o section_n because_o it_o be_v contain_v of_o the_o circumference_n bac_n and_o the_o right_a line_n bc._n likewise_o the_o angle_n cbd_v be_v a_o angle_n of_o the_o section_n bdc_o because_o it_o be_v contain_v under_o the_o circumference_n bdc_o and_o the_o right_a line_n bc._n and_o these_o angle_n be_v common_o call_v mix_v angle_n angle_n because_o they_o be_v contain_v under_o a_o right_a line_n and_o a_o crooked_a and_o these_o portion_n of_o circumference_n be_v common_o call_v arke_n arke_n and_o the_o right_a line_n be_v call_v chord_n chord_n or_o right_a line_n subtend_v and_o the_o great_a section_n have_v ever_o the_o great_a angle_n and_o the_o less_o section_n the_o less_o angle_n a_o angle_n be_v say_v to_o be_v in_o a_o section_n definition_n when_o in_o the_o circumference_n be_v take_v any_o point_n and_o from_o that_o point_n be_v draw_v right_a line_n to_o the_o end_n of_o the_o right_a line_n which_o be_v the_o base_a of_o the_o segment_n the_o angle_n which_o be_v contain_v under_o the_o right_a line_n draw_v from_o the_o point_n be_v i_o say_v say_v to_o be_v a_o angle_n in_o a_o section_n as_o the_o angle_n abc_n be_v a_o angle_n be_v the_o section_n abc_n because_o from_o the_o point_n b_o be_v a_o point_n in_o the_o circumference_n abc_n be_v draw_v two_o right_a line_n bc_o and_o basilius_n to_o the_o end_n of_o the_o line_n ac_fw-la which_o be_v the_o base_a of_o the_o section_n abc_n likewise_o the_o angle_n adc_o be_v a_o angle_n in_o the_o section_n adc_o because_o from_o the_o point_n d_o be_v in_o the_o circumference_n adc_o be_v draw_v two_o right_a line_n namely_o dc_o &_o da_z to_o the_o end_n of_o the_o right_a line_n ac_fw-la which_o be_v also_o the_o base_a to_o the_o say_a section_n adc_o so_o you_o see_v it_o be_v not_o all_o one_o to_o say_v section_n a_o angle_n of_o a_o section_n and_o a_o angle_n in_o a_o section_n a_o angle_n of_o a_o section_n consist_v of_o the_o touch_n of_o a_o right_a line_n and_o a_o crooked_a and_o a_o angle_n in_o a_o section_n be_v place_v on_o the_o circumference_n and_o be_v contain_v of_o two_o right_a line_n also_o the_o great_a section_n have_v in_o it_o the_o less_o angle_n and_o the_o less_o section_n have_v in_o it_o the_o great_a angle_n but_o when_o the_o right_a line_n which_o comprehend_v the_o angle_n do_v receive_v any_o circumference_n of_o a_o circle_n definition_n than_o that_o angle_n be_v say_v to_o be_v correspondent_a and_o to_o pertain_v to_o that_o circumference_n as_o the_o right_a line_n basilius_n and_o bc_o which_o contain_v the_o angle_n ab_fw-la c_z and_o receive_v the_o circumference_n adc_o therefore_o the_o angle_n abc_n be_v say_v to_o subtend_v and_o to_o pertain_v to_o the_o circumference_n adc_o and_o if_o the_o right_a line_n which_o cause_n the_o angle_n concur_v in_o the_o centre_n of_o a_o circle_n then_o the_o angle_n be_v say_v to_o be_v in_o the_o centre_n of_o a_o circle_n as_o the_o angle_n efd_n be_v say_v to_o be_v in_o the_o centre_n of_o a_o circle_n for_o that_o it_o be_v comprehend_v of_o two_o right_a line_n fe_o and_o fd_v which_o concur_v and_o touch_v in_o the_o centre_n f._n and_o this_o angle_n likewise_o subtend_v the_o circumference_n egg_v which_o circumference_n also_o be_v the_o measure_n of_o the_o greatness_n of_o the_o angle_n efd_n a_o sector_n of_o a_o circle_n be_v a_o angle_n be_v set_v at_o the_o centre_n of_o a_o circle_n a_o figure_n contain_v under_o the_o right_a line_n which_o make_v that_o angle_n definition_n and_o the_o part_n of_o the_o circumference_n receive_v of_o they_o as_o the_o figure_n abc_n be_v a_o sector_n of_o a_o circle_n for_o that_o it_o have_v a_o angle_n at_o the_o centre_n namely_o the_o angle_n bac_n &_o be_v contain_v of_o the_o two_o right_a line_n ab_fw-la and_o ac_fw-la which_o contain_v that_o angle_n and_o the_o circumference_n receive_v by_o they_o definition_n like_a segment_n or_o section_n of_o a_o circle_n be_v those_o which_o have_v equal_a angle_n or_o in_o who_o be_v equal_a angle_n definition_n here_o be_v set_v two_o definition_n of_o like_a section_n of_o a_o circle_n the_o one_o pertain_v to_o the_o angle_n which_o be_v set_v in_o the_o centre_n of_o the_o circle_n and_o receive_v the_o circumference_n of_o the_o say_a section_n first_o the_o other_o pertain_v to_o the_o angle_n in_o the_o section_n which_o as_o before_o be_v say_v be_v ever_o in_o the_o circumference_n as_o if_o the_o angle_n bac_n be_v in_o the_o centre_n a_o and_o receive_v of_o the_o circumference_n blc_n be_v equal_a to_o the_o angle_n feg_n be_v also_o in_o the_o centre_n e_o and_o receive_v of_o the_o circumference_n fkg_v then_o be_v the_o two_o section_n bcl_n and_o fgk_n like_v by_o the_o first_o definition_n by_o the_o same_o definition_n also_o be_v the_o other_o two_o section_n like_a namely_o bcd_o and_o fgh_a for_o that_o the_o angle_n bac_n be_v equal_a to_o the_o
two_o side_n eh_o and_o eke_o of_o the_o triangle_n ehk_n and_o the_o angle_n feg_n be_v great_a than_o the_o angle_n hek_n therefore_o by_o the_o 24._o of_o the_o first_o the_o base_a fg_o be_v great_a than_o the_o base_a hk_n and_o by_o the_o same_o reason_n may_v it_o be_v prove_v that_o the_o line_n ac_fw-la be_v great_a than_o the_o line_n ab_fw-la and_o so_o be_v manifest_v the_o whole_a proposition_n the_o 15._o theorem_a the_o 16._o proposition_n if_o from_o the_o end_n of_o the_o diameter_n of_o a_o circle_n be_v draw_v a_o right_a line_n make_v right_a angle_n it_o shall_v fall_v without_o the_o circle_n and_o between_o that_o right_a line_n and_o the_o circumference_n can_v not_o be_v draw_v a_o other_o right_a line_n and_o the_o angle_n of_o the_o semicircle_n be_v great_a than_o any_o acute_a angle_n make_v of_o right_a line_n but_o the_o other_o angle_n be_v less_o than_o any_o acute_a angle_n make_v of_o right_a line_n svppose_v that_o there_o be_v a_o circle_n abc_n who_o centre_n let_v be_v the_o point_n d_o and_o let_v the_o diameter_n thereof_o be_v ab_fw-la then_o i_o say_v that_o a_o right_a line_n draw_v from_o the_o point_n a_o make_v with_o the_o diameter_n ab_fw-la right_a angle_n shall_v fall_v without_o the_o circle_n theorem_a for_o if_o it_o do_v not_o then_o if_o it_o be_v possible_a let_v it_o fall_v within_o the_o circle_n as_o the_o line_n ac_fw-la do_v and_o draw_v a_o line_n from_o the_o point_n d_o to_o the_o point_n c._n absurdity_n and_o for_o asmuch_o as_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_n dam_n be_v equal_a to_o the_o line_n dc_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n therefore_o the_o angle_n dac_o be_v equal_a to_o the_o angle_n acd_o but_o the_o angle_n dac_o be_v by_o supposition_n a_o right_a angle_n wherefore_o also_o the_o angle_n acd_o be_v a_o right_a angle_n wherefore_o the_o angle_n dac_o and_o acd_o be_v equal_a to_o two_o right_a angle_n which_o by_o the_o 17._o of_o the_o first_o be_v impossible_a wherefore_o a_o right_a line_n draw_v from_o the_o point_n a_o make_v with_o the_o diameter_n ab_fw-la right_a angle_n shall_v not_o fall_v within_o the_o circle_z in_o like_a sort_n also_o may_v we_o prove_v that_o it_o fall_v not_o in_o the_o circumference_n wherefore_o it_o fall_v without_o as_o the_o line_n ae_n do_v i_o say_v also_o part_n that_o between_o the_o right_a line_n ae_n and_o the_o circumference_n acb_o can_v not_o be_v draw_v a_o other_o right_a line_n for_o if_o it_o be_v possible_a let_v the_o line_n of_o so_o be_v draw_v and_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n d_o draw_v unto_o the_o line_n favorina_n a_o perpendicular_a line_n dg_o and_o for_o asmuch_o as_o agd_n be_v a_o right_a angle_n but_o dag_n be_v less_o than_o a_o right_a angle_n therefore_o by_o the_o 19_o of_o the_o first_o the_o side_n ad_fw-la be_v great_a than_o the_o side_n dg_o but_o the_o line_n dam_n be_v equal_a to_o the_o line_n dh_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n wherefore_o the_o line_n dh_o be_v great_a than_o the_o line_n dg_o namely_o the_o less_o great_a than_o the_o great_a which_o be_v impossible_a wherefore_o between_o the_o right_a line_n ae_n and_o the_o circumference_n acb_o can_v not_o be_v draw_v a_o other_o right_a line_n i_o say_v moreover_o part_n that_o the_o angle_n of_o the_o semicircle_n contain_v under_o the_o right_a line_n ab_fw-la and_o the_o circumference_n cha_fw-mi be_v great_a than_o any_o acute_a angle_n make_v of_o right_a line_n and_o the_o angle_n remain_v contain_v under_o the_o circumference_n cha_fw-mi and_o the_o right_a line_n ae_n be_v less_o than_o any_o acute_a angle_n make_v of_o right_a line_n for_o if_o there_o be_v any_o angle_n make_v of_o right_a line_n great_a than_o that_o angle_n which_o be_v contain_v under_o the_o right_a line_n basilius_n and_o the_o circumference_n cha_fw-mi or_o less_o than_o that_o which_o be_v contain_v under_o the_o circumference_n cha_fw-mi and_o the_o right_a line_n ae_n then_o between_o the_o circumference_n cha_fw-mi and_o the_o right_a line_n ae_n there_o shall_v fall_v a_o right_a line_n which_o make_v the_o angle_n contain_v under_o the_o right_a line_n great_a than_o that_o angle_n which_o be_v contain_v under_o the_o right_a line_n basilius_n and_o the_o circumference_n cha_fw-mi and_o less_o than_o the_o angle_n which_o be_v contain_v under_o the_o circumference_n cha_fw-mi and_o the_o right_a line_n ae_n but_o there_o can_v fall_v no_o such_o line_n as_o it_o have_v before_o be_v prove_v wherefore_o no_o acute_a angle_n contain_v under_o right_a line_n be_v great_a than_o the_o angle_n contain_v under_o the_o right_a line_n basilius_n and_o the_o circumference_n cha_fw-mi nor_o also_o less_o than_o the_o angle_n contain_v under_o the_o circumference_n cha_fw-mi and_o the_o line_n ae_n correlary_a hereby_o it_o be_v manifest_a that_o a_o perpendicular_a line_n draw_v from_o the_o end_n of_o the_o diameter_n of_o a_o circle_n touch_v the_o circle_n and_o that_o a_o right_a line_n touch_v a_o circle_n in_o one_o point_n only_o for_o it_o be_v prove_v by_o the_o 2._o of_o the_o three_o that_o a_o right_a line_n draw_v from_o two_o point_n take_v in_o the_o circumference_n of_o a_o circle_n shall_v fall_v within_o the_o circle_n which_o be_v require_v to_o be_v demonstrate_v the_o 2._o problem_n the_o 17._o proposition_n from_o a_o point_n give_v to_o draw_v a_o right_a line_n which_o shall_v touch_v a_o circle_n give_v construction_n svppose_v that_o the_o point_n give_v be_v a_o and_o let_v the_o circle_n give_v be_v bcd_o it_o be_v require_v from_o the_o point_n a_o to_o draw_v a_o right_a line_n which_o shall_v touch_v the_o circle_n bcd_o take_v by_o the_o first_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 by_o the_o first_o petition_n draw_v the_o right_a line_n ade_n and_o make_v the_o centre_n e_o and_o the_o space_n ae_n describe_v by_o the_o three_o petition_n a_o circle_n afg_n and_o from_o the_o point_n d_o raise_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o line_n ea_fw-la a_o perpendicular_a line_n df._n and_o by_o the_o first_o petition_n draw_v these_o line_n ebf_a and_o ab_fw-la then_o i_o say_v that_o from_o the_o point_n a_o be_v draw_v to_o the_o circle_n bcd_v a_o touch_n line_n ab_fw-la demonstration_n for_o for_o asmuch_o as_o the_o point_n e_o be_v the_o centre_n of_o the_o circle_n bcd_o and_o also_o of_o the_o circle_n afg_n therefore_o the_o line_n ea_fw-la be_v equal_a to_o the_o line_n of_o and_o the_o line_n ed_z to_o the_o line_n ebb_n for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n wherefore_o to_o these_o two_o line_n ae_n and_o ebb_n be_v equal_a these_o two_o line_n of_o &_o ed_z and_o the_o angle_n at_o the_o point_n e_o be_v common_a to_o they_o both_o wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a df_o be_v equal_a to_o the_o base_a ab_fw-la and_o the_o triangle_n def_n be_v equal_a to_o the_o triangle_n eba_n and_o the_o rest_n of_o the_o angle_n remain_v to_o the_o rest_n of_o the_o angle_n remain_v wherefore_o the_o angle_n edf_o be_v equal_a to_o the_o angle_n eba_n but_o the_o angle_n edf_o be_v a_o right_a angle_n wherefore_o also_o the_o angle_n eba_n be_v a_o right_a angle_n and_o the_o line_n ebb_n be_v draw_v from_o the_o centre_n but_o a_o perpendicular_a line_n draw_v from_o the_o end_n of_o the_o diameter_n of_o a_o circle_n touch_v the_o circle_n by_o the_o corellary_n of_o the_o 16._o of_o the_o three_o wherefore_o the_o line_n ab_fw-la touch_v the_o circle_n bcd_o wherefore_o from_o the_o point_n give_v namely_o a_o be_v draw_v unto_o the_o circle_n give_v bcd_o a_o touch_n line_n ab_fw-la which_o be_v require_v to_o be_v do_v ¶_o a_o addition_n of_o pelitarius_n pelitarius_n unto_o a_o right_a line_n which_o cut_v a_o circle_n to_o draw_v a_o parallel_n line_n which_o shall_v touch_v the_o circle_n suppose_v that_o the_o right_a line_n ab_fw-la do_v out_o the_o circle_n abc_n in_o the_o point_n a_o and_o b._n it_o be_v require_v to_o draw_v unto_o the_o line_n ab_fw-la a_o parallel_n line_n which_o shall_v touch_v the_o circle_n let_v the_o centre_n of_o the_o circle_n be_v the_o point_n d._n and_o divide_v the_o line_n ab_fw-la into_o two_o equal_a part_n in_o the_o point_n e._n and_o by_o the_o point_n e_o and_o by_o the_o centre_n d_o draw_v the_o diameter_n cdef_n and_o from_o the_o point_n fletcher_n which_o be_v the_o end_n of_o the_o diameter_n raise_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o diameter_n cf_o a_o perpendicular_a line_n gfh_v then_o i_o
the_o touch_n and_o cut_v the_o circle_n may_v either_o pass_v by_o the_o centre_n or_o not_o if_o it_o pass_v by_o the_o centre_n then_o be_v it_o manifest_a by_o the_o 18._o of_o this_o book_n that_o it_o fall_v perpendicular_o upon_o the_o touch_n line_n and_o divide_v the_o circle_n into_o two_o equal_a part_n so_o that_o all_o the_o angle_n in_o each_o semicircle_n be_v by_o the_o former_a proposition_n right_a angle_n and_o therefore_o equal_a to_o the_o alternate_a angle_n make_v by_o the_o say_v perpendicular_a line_n and_o the_o touch_n line_n if_o it_o pass_v not_o by_o the_o centre_n then_o follow_v the_o construction_n and_o demonstration_n before_o put_v the_o 5._o problem_n the_o 33._o proposition_n upon_o a_o right_a line_n give_v to_o describe_v a_o segment_n of_o a_o circle_n which_o shall_v contain_v a_o angle_n equal_a to_o a_o rectiline_a angle_n give_v svppose_v that_o the_o right_a line_n give_v be_v ab_fw-la and_o let_v the_o rectiline_a angle_n geâ—â—n_v be_v c._n it_o be_v require_v upon_o the_o right_a line_n give_v ab_fw-la to_o describe_v a_o segment_n of_o a_o circle_n which_o shall_v contain_v a_o angle_n equal_a to_o the_o angle_n c._n now_o the_o angle_n c_o be_v either_o a_o acute_a angle_n or_o a_o right_a angle_n or_o a_o obtuse_a angle_n case_n first_o let_v it_o be_v a_o acute_a angle_n as_o in_o the_o first_o description_n and_o by_o the_o 23_o of_o the_o first_o upon_o the_o right_a line_n ab_fw-la and_o to_o the_o point_n in_o it_o a_o describe_v a_o angle_n equal_a to_o the_o angle_n c_o construction_n and_o let_v the_o same_o be_v dab_n wherefore_o the_o angle_n dab_n be_v a_o acute_a angle_n from_o the_o point_n a_o raise_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o line_n ad_fw-la a_o perpendicular_a line_n af._n and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n ab_fw-la into_o two_o equal_a part_n in_o the_o point_n f._n and_o by_o the_o 11._o of_o the_o same_o from_o the_o point_n fletcher_n raise_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n fg_o and_o draw_v a_o line_n from_o g_z to_o b._n demonstration_n and_o forasmuch_o as_o the_o line_n of_o be_v equal_a to_o the_o line_n fb_o and_o the_o line_n fg_o be_v common_a to_o they_o both_o therefore_o these_o two_o line_n of_o and_o fg_o be_v equal_a to_o these_o two_o line_n fb_o and_o fg_o and_o the_o angle_n afg_n be_v by_o the_o 4._o petition_n equal_a to_o the_o angle_n gfb_n wherefore_o by_o the_o 4._o of_o the_o same_o the_o base_a agnostus_n be_v equal_a to_o the_o base_a gb_o wherefore_o make_v the_o centre_n g_o and_o the_o space_n ga_o describe_v by_o the_o 3._o petition_n a_o circle_n and_o it_o shall_v pass_v by_o the_o point_n b_o describe_v such_o a_o circle_n &_o let_v the_o same_o be_v abe_n and_o draw_v a_o line_n from_o e_o to_o b._n now_o forasmuch_o as_o from_o the_o end_n of_o the_o diameter_n ae_n namely_o from_o the_o point_n a_o be_v 〈◊〉_d a_o right_a line_n ad_fw-la make_v together_o with_o the_o right_a line_n ae_n a_o right_a angle_n therefore_o by_o the_o corollary_n of_o the_o 16._o of_o the_o three_o the_o line_n ad_fw-la touch_v the_o circle_n abe_n and_o forasmuch_o as_o a_o certain_a right_a line_n ad_fw-la touch_v the_o circle_n abe_n &_o from_o the_o point_n a_o where_o the_o touch_n be_v be_v draw_v into_o the_o circle_v a_o certain_a right_a line_n ab_fw-la therefore_o by_o the_o 32._o of_o the_o three_o the_o angle_n dab_n be_v equal_a to_o the_o angle_n aeb_fw-mi which_o be_v in_o the_o alternate_a segment_n of_o the_o circle_n but_o the_o angle_n dab_n be_v equal_a to_o the_o angle_n c_o wherefore_o the_o angle_n c_o be_v equal_a to_o the_o angle_n aeb_fw-mi wherefore_o upon_o the_o right_a line_n give_v ab_fw-la be_v describe_v a_o segment_n of_o a_o circle_n which_o contain_v the_o angle_n aeb_fw-mi which_o be_v equal_a to_o the_o angle_n give_v namely_o to_o c._n case_n but_o now_o suppose_v that_o the_o angle_n c_o be_v a_o right_a angle_n it_o be_v again_o require_v upon_o the_o right_a line_n ab_fw-la to_o describe_v a_o segment_n of_o a_o circle_n which_o shall_v contain_v a_o angle_n equal_a to_o the_o right_a angle_n c._n construction_n describe_v again_o upon_o the_o right_a line_n ab_fw-la and_o to_o the_o point_n in_o it_o a_o a_o angle_n bid_v equal_a to_o the_o rectiline_a angle_n give_v c_o by_o the_o 23._o of_o the_o first_o as_o it_o be_v set_v forth_o in_o the_o second_o description_n and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n ab_fw-la into_o two_o equal_a part_n in_o the_o point_n f._n and_o make_v the_o centre_n the_o point_n f_o and_o the_o space_n favorina_n or_o fb_o describe_v by_o the_o 3._o petition_n the_o circle_z aeb_fw-mi demonstration_n wherefore_o the_o right_a line_n ad_fw-la touch_v the_o circle_n aeb_fw-mi for_o that_o the_o angle_n bad_a be_v a_o right_a angle_n wherefore_o the_o angle_v bad_a be_v equal_a to_o the_o angle_n which_o be_v in_o the_o segment_n aeb_fw-mi for_o the_o angle_n which_o be_v in_o a_o semicircle_n be_v a_o right_a angle_n by_o the_o 31._o of_o the_o three_o but_o the_o angle_n bad_a be_v equal_a to_o the_o angle_n c._n wherefore_o there_o be_v again_o describe_v upon_o the_o line_n ab_fw-la a_o segment_n of_o a_o circle_n namely_o aeb_n which_o contain_v a_o angle_n equal_a to_o the_o angle_n give_v namely_o to_o c._n but_o now_o suppose_v that_o the_o angle_n c_o be_v a_o obtuse_a angle_n case_n upon_o the_o right_a line_n ab_fw-la and_o to_o the_o point_n in_o it_o a_o describe_v by_o the_o 23._o of_o the_o first_o a_o angle_n bid_v equal_a to_o the_o angle_n c_o as_o it_o be_v in_o the_o three_o description_n construction_n and_o from_o the_o point_n a_o raise_v up_o unto_o the_o line_n ad_fw-la a_o perpendicular_a line_n ae_n by_o the_o 11._o of_o the_o first_o and_o again_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n ab_fw-la into_o two_o equal_a part_n in_o the_o point_n f._n and_o from_o the_o point_n f._n raâ—se_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n fg_o by_o the_o 11._o of_o the_o same_o &_o draw_v a_o line_n from_o g_z to_o b._n demonstration_n and_o now_o forasmuch_o as_o the_o line_n of_o be_v equal_a to_o the_o line_n fb_o and_o the_o line_n fg_o be_v common_a to_o they_o both_o therefore_o these_o two_o line_n of_o and_o fg_o be_v equal_a to_o these_o two_o line_n bf_o and_o fg_o and_o the_o angle_n afg_n be_v by_o the_o 4._o petition_n equal_a to_o the_o angle_n bfg_n wherefore_o by_o the_o 4._o of_o the_o same_o the_o base_a agnostus_n be_v equal_a to_o the_o base_a gb_o wherefore_o make_v the_o centre_n g_o and_o the_o space_n ga_o describe_v by_o the_o 3._o petition_n a_o circle_n and_o it_o shall_v pass_v by_o the_o point_n b_o let_v it_o be_v describe_v as_o the_o circle_n aeb_fw-mi be_v and_o forasmuch_o as_o from_o the_o end_n of_o the_o diameter_n ae_n be_v draw_v a_o perpendicular_a line_n ad_fw-la therefore_o by_o the_o corollary_n of_o the_o 16._o of_o the_o three_o the_o line_n ad_fw-la touch_v the_o circle_n aeb_fw-mi &_o from_o the_o point_n of_o the_o touch_n namely_o a_o be_v extend_v the_o line_n ab_fw-la wherefore_o by_o the_o 32._o of_o the_o three_o the_o angle_n bad_a be_v equal_a to_o the_o angle_n ahb_n which_o be_v in_o the_o alternate_a segment_n of_o the_o circle_n but_o the_o angle_n bad_a be_v equal_a to_o the_o angle_n c._n wherefore_o the_o angle_n which_o be_v in_o the_o segment_n ahb_n be_v equal_a to_o the_o angle_n c._n wherefore_o upon_o the_o right_a line_n give_v ab_fw-la be_v describe_v a_o segment_n of_o a_o circle_n ahb_n which_o contain_v a_o angle_n equal_a to_o the_o angle_n give_v namely_o c_o which_o be_v require_v to_o be_v do_v the_o 6._o problem_n the_o 34._o proposition_n from_o a_o circle_n give_v to_o cut_v away_o a_o section_n which_o shall_v contain_v a_o angle_n equal_a to_o a_o rectiline_a angle_n give_v svppose_v that_o the_o circle_n give_v be_v ac_fw-la and_o let_v the_o rectiline_a angle_n give_v be_v d._n it_o be_v require_v from_o the_o circle_n abc_n to_o cut_v away_o a_o segment_n which_o shall_v contain_v a_o angle_n equal_a to_o the_o angle_n d._n construction_n draw_v by_o the_o 17._o of_o the_o three_o a_o line_n touch_v the_o circle_n and_o let_v the_o same_o be_v of_o and_o let_v it_o touch_v in_o the_o point_n b._n and_o by_o the_o 23._o of_o the_o first_o upon_o the_o right_a line_n of_o and_o to_o the_o point_n in_o it_o b_o describe_v the_o angle_n fbc_n equal_a to_o the_o angle_n d._n demonstration_n now_o forasmuch_o as_o a_o certain_a right_a line_n of_o touch_v the_o circle_n abc_n in_o the_o point_n b_o and_o frora_fw-la
construction_n draw_v by_o the_o 17._o of_o the_o three_o a_o right_a line_n touch_v the_o circle_n abc_n and_o let_v the_o same_o be_v gah_n and_o let_v it_o touch_v in_o the_o point_n a._n and_o by_o the_o 23._o of_o the_o first_o unto_o the_o right_a line_n ah_o and_o unto_o the_o point_n in_o it_o a_o describe_v a_o angle_n hac_fw-la equal_a unto_o the_o angle_n def_n and_o by_o the_o self_n same_o unto_o the_o right_a line_n agnostus_n and_o unto_o the_o point_n in_o it_o a_o make_v a_o angle_n gab_n equal_a unto_o the_o angle_n dfe_n and_o draw_v a_o right_a line_n from_o b_o to_o c._n and_o for_o asmuch_o as_o a_o certain_a right_a line_n gah_n touch_v the_o circle_n abc_n demonstration_n and_o from_o the_o point_n where_o it_o touch_v namely_o a_o be_v draw_v into_o the_o circle_n a_o right_a line_n ac_fw-la therefore_o by_o the_o 31._o of_o the_o three_o the_o angle_n hac_fw-la be_v equal_a unto_o the_o angle_n abc_n which_o be_v in_o the_o alternate_a segment_n of_o the_o circle_n but_o the_o angle_n hac_fw-la be_v equal_a to_o the_o angle_n def_n wherefore_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n def_n and_o by_o the_o same_o reason_n the_o angle_n acb_o be_v equal_a to_o the_o angle_n dfe_n wherefore_o the_o angle_n remain_v bac_n be_v equal_a unto_o the_o angle_n remain_v edf_o wherefore_o the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n def_n and_o it_o be_v describe_v in_o the_o circle_n give_v abc_n wherefore_o in_o a_o circle_n give_v be_v describe_v a_o triangle_n equiangle_n unto_o a_o triangle_n give_v which_o be_v require_v to_o be_v do_v the_o 3._o problem_n the_o 3._o proposition_n about_o a_o circle_n give_v to_o describe_v a_o triangle_n equiangle_n unto_o a_o triangle_n give_v svppose_v that_o the_o circle_n give_v be_v abc_n and_o let_v the_o triangle_n give_v be_v def_n it_o be_v require_v about_o the_o circle_n abc_n to_o describe_v a_o triangle_n equiangle_n unto_o the_o triangle_n def_n construction_n extend_v the_o line_n of_o on_o each_o side_n to_o the_o point_n g_o and_o h._n and_o by_o the_o first_o of_o the_o three_o take_v the_o centre_n of_o the_o circle_n abc_n and_o let_v the_o same_o be_v the_o point_n k._n and_o then_o draw_v a_o right_a line_n kb_o and_o by_o the_o 23._o of_o the_o first_o unto_o the_o right_a line_n kb_o and_o unto_o the_o point_n in_o it_o king_n make_v a_o angle_n bka_fw-mi equal_a unto_o the_o angle_n deg_n and_o likewise_o make_v the_o angle_n bkc_n equal_a unto_o the_o angle_n dfh._n and_o by_o the_o 17._o of_o the_o three_o draw_v right_a line_n touch_v the_o circle_n a_o b_o c_o in_o the_o point_n a_o b_o c._n and_o let_v the_o same_o be_v lam_n mbn_n and_o ncl_n demonstration_n and_o for_o asmuch_o as_o the_o right_a line_n lm_o mn_a &_o nl_o do_v touch_v the_o circle_n abc_n in_o the_o point_n a_o b_o c_o and_o from_o the_o centre_n king_n unto_o the_o point_n a_o b_o c_o be_v draw_v right_a line_n ka_o kb_o and_o kc_o therefore_o the_o angle_n which_o be_v at_o the_o point_n a_o b_o c_o be_v right_a angle_n by_o the_o 18._o of_o the_o three_o and_o for_o asmuch_o as_o the_o four_o angle_n of_o the_o four_o side_v figure_n ambk_n be_v equal_a unto_o four_o right_a angle_n who_o angle_n kam_fw-mi &_o kbm_n be_v two_o right_a angle_n therefore_o the_o angle_n remain_v akb_n and_o ambiguity_n be_v equal_a to_o two_o right_a angle_n and_o the_o angle_n deg_n &_o def_n be_v by_o the_o 13._o of_o the_o first_o equal_a to_o two_o right_a angle_n wherefore_o the_o angle_n akb_n and_o ambiguity_n be_v equal_a unto_o the_o angle_n deg_n and_o def_n of_o which_o two_o angle_n the_o angle_n akb_n be_v equal_a unto_o the_o angle_n deg_n wherefore_o the_o angle_n remain_v amb_n be_v equal_a unto_o the_o angle_n remain_v def_n in_o like_a sort_n may_v it_o be_v prove_v that_o the_o angle_n lnm_n be_v equal_a to_o the_o angle_n dfe_n wherefore_o the_o angle_n remain_v mln_v be_v equal_a unto_o the_o angle_n remain_v edf_o wherefore_o the_o triangle_n lmn_o be_v equiangle_n unto_o the_o triangle_n def_n and_o it_o be_v describe_v about_o the_o circle_n abc_n wherefore_o about_o a_o circle_n give_v be_v describe_v a_o triangle_n equiangle_n unto_o a_o triangle_n give_v which_o be_v require_v to_o be_v do_v ¶_o a_o other_o way_n after_o pelitarius_n peliâ—arius_n in_o the_o circle_n abc_n inscribe_v a_o triangle_n ghk_v equiangle_n to_o the_o triangle_n edf_o by_o the_o former_a proposition_n so_o that_o let_v the_o angle_n at_o the_o point_n g_o be_v equal_a to_o the_o angle_n d_o and_o let_v the_o angle_n at_o the_o point_n h_o be_v equal_a to_o the_o angle_n e_o and_o let_v also_o the_o angle_n at_o the_o point_n king_n be_v equal_a to_o the_o angle_n f._n then_o draw_v the_o line_n lm_o parallel_n to_o the_o line_n gh_o which_o let_v touch_v the_o circle_n in_o the_o point_n a_o which_o may_v be_v do_v by_o the_o proposition_n add_v of_o the_o say_v pelitarius_n after_o the_o 17._o proposition_n construction_n draw_v likewise_o the_o line_n mn_a parallel_n unto_o the_o line_n hk_o and_o touch_v the_o circle_n in_o the_o point_n b_o and_o also_o draw_v the_o line_n ln_o parallel_n unto_o the_o line_n gk_o and_o touch_v the_o circle_n in_o the_o point_n c._n and_o these_o three_o line_n shall_v undoubted_o concur_v as_o in_o the_o point_n l_o m_o and_o n_o demonstration_n which_o may_v easy_o be_v prove_v if_o you_o produce_v on_o either_o side_n the_o line_n gh_o gk_o and_o hk_o until_o they_o cut_v the_o line_n lm_o ln_o and_o mn_v in_o the_o point_n oh_o p_o q_o r_o s_o t._n now_o i_o say_v that_o the_o triangle_n lmn_o circumscribe_v about_o the_o circle_n abc_n be_v equiangle_n to_o the_o triangle_n def_n for_o it_o be_v manifest_a that_o it_o be_v equiangle_n unto_o the_o triangle_n ghk_o by_o the_o propriety_n of_o parallel_a line_n for_o the_o angle_n mtq_n be_v equal_a to_o the_o angle_n at_o the_o point_n g_o of_o the_o triangle_n ghk_o by_o the_o 29._o of_o the_o first_o and_o therefore_o also_o the_o angle_n at_o the_o point_n l_o be_v equal_a to_o the_o self_n same_o angle_n at_o the_o point_n g_z for_o the_o angle_n at_o the_o point_n l_o be_v by_o the_o same_o 29._o proposition_n equal_a to_o the_o angle_n mtq_n and_o by_o the_o same_o reason_n the_o angle_n at_o the_o point_n m_o be_v equal_a to_o the_o angle_n at_o the_o point_n h_o of_o the_o self_n same_o triangle_n and_o the_o angle_n at_o the_o point_n n_o to_o the_o angle_n at_o the_o point_n k._n wherefore_o the_o whole_a triangle_n lmn_o be_v equiangle_n to_o the_o whole_a triangle_n ghk_o wherefore_o also_o it_o be_v equiangle_n to_o the_o triangle_n def_n which_o be_v require_v to_o be_v do_v the_o 4._o problem_n the_o 4._o proposition_n in_o a_o triangle_n give_v to_o describe_v a_o circle_n svppose_v that_o the_o triangle_n give_v be_v abc_n it_o be_v require_v to_o describe_v a_o circle_n in_o the_o triangle_n abc_n construction_n divide_v by_o the_o 9_o of_o the_o first_o the_o angle_n abc_n and_o acb_o into_o two_o equal_a parâ—es_n by_o two_o right_a line_n bd_o and_o cd_o and_o let_v these_o right_a line_n meet_v together_o in_o the_o point_n d._n and_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n d_o draw_v unto_o the_o right_a line_n ab_fw-la bc_o and_o ca_n perpendicular_a line_n namely_o de_fw-fr df_o and_o dg_o demonstration_n and_o forasmuch_o as_o the_o angle_n abdella_n be_v equal_a to_o though_o angle_v cbd_v and_o the_o right_a angle_n bed_n be_v equal_a unto_o the_o right_a angle_n bfd_n now_o then_o there_o be_v two_o triangle_n ebb_v and_o fbd_v have_v two_o angle_n equal_a to_o two_o angle_n and_o one_o side_n equal_a to_o one_o side_n namely_o bd_o which_o be_v common_a to_o they_o both_o and_o subtend_v one_o of_o the_o equal_a angle_n wherefore_o by_o the_o 26._o of_o the_o first_o the_o rest_n of_o the_o side_n be_v equal_a unto_o the_o rest_n of_o the_o side_n wherefore_o the_o line_n de_fw-fr be_v equal_a unto_o the_o line_n df_o and_o by_o the_o same_o reason_n also_o the_o â—ne_a dg_o be_v equal_a unto_o the_o line_n df._n wherefore_o these_o three_o right_a line_n de_fw-fr df_o &_o dg_o be_v equal_a the_o one_o to_o the_o other_o by_o the_o first_o common_a sentence_n wherefore_o make_v the_o centre_n the_o point_n d_o and_o the_o space_n de_fw-fr or_o df_o or_o dg_o describe_v a_o circle_n and_o it_o will_v pass_v through_o the_o point_n e_o f_o g_o and_o will_v touch_v the_o right_a line_n ab_fw-la bc_o and_o ca._n for_o the_o angle_n make_v at_o the_o point_n e_o f_o g_o be_v right_a angle_n for_o if_o the_o circle_z cut_v those_o right_a line_n then_o from_o the_o end_n of_o
of_o the_o first_o either_o of_o these_o line_n ab_fw-la and_o ad_fw-la into_o two_o equal_a part_n in_o the_o point_n e_o and_o f._n and_o by_o the_o point_n e_o by_o the_o 31._o of_o the_o first_o draw_v a_o line_n eh_o parallel_v unto_o either_o of_o these_o line_n ab_fw-la and_o dc_o demonstration_n and_o by_o the_o same_o by_o the_o point_n fletcher_n draw_v a_o line_n fk_a parallel_n unto_o either_o of_o these_o line_n ad_fw-la and_o bc._n wherefore_o every_o one_o of_o these_o figure_n ak_o kb_o ah_o hd_n agnostus_n gc_o bg_o and_o gd_o be_v a_o parallelogram_n and_o the_o side_n which_o be_v opposite_a the_o one_o to_o the_o other_o be_v by_o the_o 34._o of_o the_o first_o equal_a the_o one_o to_o the_o other_o and_o forasmuch_o as_o the_o line_n ad_fw-la be_v equal_a unto_o the_o line_n ab_fw-la and_o the_o half_a of_o the_o line_n ad_fw-la be_v the_o line_n ae_n and_o the_o half_a of_o the_o line_n ab_fw-la be_v the_o line_n of_o therefore_o the_o line_n ae_n be_v equal_a unto_o the_o line_n of_o wherefore_o by_o the_o same_o the_o side_n which_o be_v opposite_a be_v equal_a wherefore_o the_o line_n fg_o be_v equal_a unto_o the_o line_n eglantine_n in_o like_a sort_n may_v we_o prove_v that_o either_o of_o these_o line_n gh_o and_o gk_o be_v equal_a to_o either_o of_o these_o line_n fg_o and_o ge._n wherefore_o by_o the_o first_o common_a sentence_n these_o four_o line_n ge_z gf_o gh_o and_o gk_o be_v equal_a the_o one_o to_o the_o other_o wherefore_o make_v the_o centre_n g_o and_o the_o space_n either_o ge_z or_o gf_o gh_o or_o gk_o describe_v a_o circle_n and_o it_o will_v pass_v by_o the_o point_n e_o f_o h_o edward_n and_o will_v touch_v the_o right_a line_n ab_fw-la bc_o cd_o and_o da._n for_o the_o angle_n at_o the_o point_n e_o f_o h_o edward_n be_v right_a angle_n for_o if_o the_o circle_n do_v cut_v the_o right_a line_n ab_fw-la bc_o cd_o and_o dam_n than_o the_o line_n which_o be_v draw_v by_o the_o end_n of_o the_o diameter_n of_o the_o circle_n make_v right_a angle_n shall_v fall_v within_o the_o circle_n which_o be_v impossible_a by_o the_o 16._o of_o the_o three_o wherefore_o the_o centre_n be_v the_o point_n g_o and_o the_o space_n be_v ge_z or_o gf_o or_o gh_o or_o gk_o if_o a_o circle_n be_v describe_v it_o shall_v not_o cut_v the_o rigâ—t_a line_n ab_fw-la bc_o cd_o and_o da._n wherefore_o it_o shall_v touch_v they_o and_o it_o be_v describe_v in_o the_o square_n abcd_o wherefore_o in_o a_o square_n give_v be_v describe_v a_o circle_n which_o be_v require_v to_o be_v do_v the_o 9_o problem_n the_o 9_o proposition_n about_o a_o square_n give_v to_o describe_v a_o circle_n svppose_v that_o the_o square_n give_v be_v ab_fw-la cd_o it_o be_v require_v about_o the_o square_n abcd_o to_o describe_v a_o circle_n draw_v right_a line_n from_o a_o to_o c_o and_o from_o d_z to_o b_o &_o let_v they_o cut_v the_o one_o the_o other_o in_o the_o point_n e._n construction_n and_o forasmuch_o as_o the_o line_n dam_n be_v equal_a unto_o the_o line_n ab_fw-la demonstration_n and_o the_o line_n ac_fw-la be_v common_a unto_o they_o both_o therefore_o these_o two_o line_n dam_n and_o ac_fw-la be_v equal_a unto_o these_o two_o line_n basilius_n and_o ac_fw-la the_o one_o to_o the_o other_o and_o the_o base_a dc_o be_v equal_a unto_o the_o base_a bc._n wherefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n dac_o be_v equal_a unto_o the_o angle_n bac_n wherefore_o the_o angle_n dab_n be_v divide_v into_o two_o equal_a part_n by_o the_o line_n ac_fw-la and_o in_o like_a sorâ—_n may_v we_o prove_v that_o every_o one_o of_o these_o angle_n abc_n bcd_o and_o cda_n be_v divide_v into_o two_o equal_a part_n by_o the_o right_a line_n ac_fw-la and_o db._n and_o forasmuch_o as_o the_o angle_n dab_n be_v equal_a unto_o the_o angle_n abc_n and_o of_o the_o angle_n dab_n the_o angle_n eab_n be_v the_o half_a and_o of_o the_o angle_n abc_n the_o angle_n eba_n be_v the_o half_a therefore_o the_o angle_n eab_n be_v equal_a unto_o the_o angle_n eba_n wherefore_o by_o the_o 6._o of_o the_o first_o the_o side_n ea_fw-la be_v equal_a unto_o the_o side_n ebb_n in_o like_a sort_n may_v we_o prove_v that_o either_o of_o these_o right_a line_n ea_fw-la and_o ebb_n be_v equal_a unto_o either_o of_o these_o line_n ec_o and_o ed._n wherefore_o these_o four_o line_n ea_fw-la ebb_n ec_o and_o ed_z be_v equal_a the_o one_o to_o the_o other_o wherefore_o make_v the_o centre_n e_o and_o the_o space_n any_o of_o these_o line_n ea_fw-la ebb_n ec_o or_o ed._n describe_v a_o circle_n and_o it_o will_v pass_v by_o the_o point_n a_o b_o c_o d_o and_o shall_v be_v describe_v about_o the_o square_n abcd_o as_o it_o be_v evident_a in_o the_o figure_n abcd._n wherefore_o about_o a_o square_a give_v be_v describe_v a_o circle_n which_o be_v require_v to_o be_v do_v ¶_o a_o proposition_n add_v by_o pelitarius_n a_o square_n circumscribe_v about_o a_o circle_n be_v double_a to_o the_o square_n inscribe_v in_o the_o same_o circle_n this_o may_v also_o be_v demonstrate_v by_o the_o equality_n of_o the_o triangle_n and_o square_n contain_v in_o the_o great_a square_n the_o 10._o problem_n the_o 10._o proposition_n to_o make_v a_o triangle_n of_o two_o equal_a side_n call_v isosceles_a which_o shall_v have_v either_o of_o the_o angle_n at_o the_o base_a double_a to_o the_o other_o angle_n take_v a_o right_a line_n at_o all_o adventure_n which_o let_v be_v ab_fw-la &_o by_o the_o 11._o of_o the_o second_o let_v it_o be_v so_o divide_v in_o the_o point_n c_z construction_n that_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a unto_o the_o square_v which_o be_v make_v of_o the_o line_n ac_fw-la and_o make_v the_o centre_n the_o point_n a_o &_o the_o space_n ab_fw-la describe_v by_o the_o 3._o petition_n a_o circle_n bde_v and_o by_o the_o 1._o of_o the_o four_o into_o the_o circle_n bde_v apply_v a_o right_a line_n bd_o equal_a to_o the_o right_a line_n ac_fw-la which_o be_v not_o great_a than_o the_o diameter_n of_o the_o circle_n bde_v and_o draw_v line_n from_o a_o to_o d_z and_o from_o d_z to_o c._n and_o by_o the_o 5._o of_o the_o fourth_z about_o the_o triangle_n acd_v describe_v a_o circle_n acdf_n demonstration_n and_o forasmuch_o as_o the_o rectangle_n figure_n contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n ac_fw-la for_o that_o be_v by_o supposition_n but_o the_o line_n ac_fw-la be_v equal_a unto_o the_o line_n bd._n wherefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n bd._n and_o forasmuch_o as_o without_o the_o circle_n acdf_n be_v take_v a_o point_n b_o and_o from_o b_o unto_o the_o circle_n acdf_n be_v draw_v two_o right_a line_n bca_o and_o bd_o in_o such_o sort_n that_o the_o one_o of_o they_o cut_v the_o circle_n and_o the_o other_o end_v at_o the_o circumference_n and_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n bd_o therefore_o by_o the_o 17._o of_o the_o three_o the_o line_n bd_o touch_v the_o circle_n acdf_n and_o forasmuch_o as_o the_o line_n bd_o touch_v in_o the_o point_n d_o and_o from_o d_o where_o the_o touch_n be_v be_v draw_v a_o right_a line_n dc_o therefore_o by_o the_o 32._o of_o the_o same_o the_o angle_n bdc_o be_v equal_a unto_o the_o angle_n dac_o which_o be_v in_o the_o alternate_a segment_n of_o the_o circle_n and_o forasmuch_o as_o the_o angle_n bdc_o be_v equal_a unto_o the_o angle_v dac_o put_v the_o angle_n cda_n common_a unto_o they_o both_o wherefore_o the_o whole_a angle_n bda_n be_v equal_a to_o these_o two_o angle_n cda_n &_o dac_o but_o unto_o the_o angle_n cda_n &_o dac_o be_v equal_a the_o outward_a angle_n bcd_o by_o the_o 32._o of_o the_o 1._o wherefore_o the_o angle_n bda_n be_v equal_a unto_o the_o angle_v bcd_o but_o the_o angle_v bda_n be_v by_o the_o 5._o of_o the_o first_o equal_a unto_o the_o angle_n cbd_v for_o by_o the_o 15._o definition_n of_o y_fw-es first_o the_o side_n ad_fw-la be_v equal_a unto_o the_o side_n ab_fw-la wherefore_o by_o the_o 1._o common_a sentence_n the_o angle_n dba_n be_v equal_a unto_o the_o angle_n bcd_o wherefore_o these_o three_o angle_n bda_n dba_n and_o bcd_o be_v equal_v the_o one_o to_o the_o other_o and_o forasmuch_o as_o the_o angle_n dbc_n be_v equal_a unto_o the_o angle_n bcd_o the_o side_n therefore_o bd_o be_v equal_a unto_o the_o side_n dc_o but_o the_o line_n bd_o be_v by_o supposition_n
touch_v the_o circle_n for_o unto_o the_o point_n where_o the_o line_n lm_o touch_v the_o circle_n which_o let_v be_v n_o draw_v the_o line_n fn_o and_o it_o be_v manifest_a by_o the_o 18._o of_o the_o three_o that_o either_o of_o the_o angleâ—_n at_o the_o point_n n_o be_v a_o right_a angle_n wherâ—fore_o for_o asmuch_o as_o the_o angle_n l_o of_o the_o triangle_n fln_n be_v equal_a to_o the_o angle_n m_o of_o the_o triangle_n fmn_n &_o the_o angle_n n_o of_o the_o one_o be_v equal_a to_o the_o anglâ—_n n_o of_o the_o other_o and_o the_o line_n fn_o be_v common_a to_o they_o both_o the_o line_n nl_o shall_v by_o the_o 26._o of_o the_o first_o be_v equal_a to_o the_o line_n nm_o and_o so_o be_v the_o line_n ml_o divide_v equal_o in_o the_o point_n n._n and_o forasmuch_o as_o the_o three_o side_n of_o the_o triangle_n fgp_fw-mi be_v equal_a to_o the_o three_o side_n of_o the_o triangle_n fmp_n the_o angle_n p_o of_o the_o one_o shall_v be_v equal_a to_o the_o angle_n p_o of_o the_o other_o by_o the_o 8._o of_o the_o first_o wherefore_o either_o of_o they_o be_v a_o right_a angle_n by_o the_o 13._o of_o the_o same_o and_o forasmuch_o as_o the_o two_o angle_n fmp_n and_o fpm_n of_o the_o triangle_n fmp_n be_v equal_a to_o the_o two_o angle_n fmn_a &_o fnm_n of_o the_o triangle_n fmn_n and_o the_o side_n fm_o be_v common_a to_o they_o both_o therefore_o the_o line_n fp_n be_v equal_a to_o the_o line_n fn_o but_o the_o line_n fn_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n wherefore_o also_o the_o line_n fp_n be_v draw_v from_o the_o centre_n to_o the_o circumference_n and_o forasmuch_o as_o the_o line_n mg_o be_v perpendicular_a to_o the_o line_n fp_n therefore_o by_o the_o corollary_n of_o the_o 16._o of_o the_o three_o it_o touch_v the_o circle_n wherefore_o the_o pentagon_n ghklm_v circumscribe_v about_o the_o circle_n be_v equilater_n it_o be_v also_o equiangle_n aâ—_z it_o be_v easy_a to_o prove_v by_o the_o equality_n of_o the_o half_n which_o be_v require_v to_o be_v do_v the_o 13._o problem_n the_o 13._o proposition_n a_o equilater_n and_o equiangle_n pentagon_n figure_n be_v give_v to_o describe_v in_o it_o a_o circle_n svppose_v that_o the_o equilater_n &_o equiangle_n pentagon_n figure_n give_v be_v abcde_o it_o be_v require_v in_o the_o say_v five_o angle_a figure_n abcde_o to_o describe_v a_o circle_n divide_v by_o the_o the_o 9_o of_o the_o first_o either_o of_o these_o angle_n bcd_o and_o cde_o into_o two_o equal_a part_n by_o these_o right_a line_n cf_o and_o fd_o demonstration_n and_o from_o the_o point_n f_o where_o thâ—se_a right_a line_n cf_o and_o df_o meet_v draw_v these_o right_a line_n fb_o favorina_n &_o fâ—_n and_o forasmuch_o as_o bc_o be_v equal_a unto_o cd_o â—nd_v cf_o be_v common_a to_o they_o both_o wherefore_o these_o two_o line_n bc_o and_o cf_o be_v equal_a to_o these_o two_o line_n dc_o and_o cf_o and_o the_o angle_n bcf_n be_v equal_a to_o the_o angle_n dcf_n wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a bf_o be_v equal_a to_o the_o base_a df_o and_o the_o triangle_n bcf_n be_v equal_a to_o the_o triangle_n dcf_n and_o the_o angle_n remain_v be_v equal_a unto_o the_o angle_n remain_v the_o one_o to_o the_o other_o under_o which_o be_v subtend_v equal_a side_n wherefore_o the_o angle_n cbf_n be_v equal_a to_o the_o angle_n cdf_n and_o forasmuch_o as_o the_o angle_n cde_o be_v double_a to_o the_o angle_n cdf_n but_o the_o angle_n cde_o be_v equal_a to_o the_o angle_n abc_n and_o the_o angle_n cdf_n be_v equal_a to_o the_o angle_n cbf_n wherefore_o the_o angle_n cba_o be_v double_a to_o the_o angle_n cbf_n wherefore_o the_o angle_n abf_n be_v equal_a to_o the_o angle_n fbc_n wherefore_o the_o angle_n abc_n be_v divide_v into_o two_o equal_a part_n by_o the_o right_a line_n bf_o in_o like_o sort_n also_o may_v it_o be_v provedâ—_n that_o either_o of_o these_o angle_n bae_o and_o aed_n be_v divide_v into_o two_o equal_a part_n by_o either_o of_o these_o line_n favorina_n and_o ef._n draw_v by_o the_o 12._o of_o the_o first_o from_o the_o point_n e_o to_o the_o right_a line_n ab_fw-la bc_o cd_o and_o ea_fw-la perpendicular_a lineâ—_z fg_o fh_o fk_o fl_fw-mi and_o fm_o and_o forasmuch_o as_o the_o angle_n hcf_n be_v equal_a 〈◊〉_d the_o anglâ—_n kcf_n and_o the_o right_a angle_n fhc_n be_v equal_a to_o the_o right_a angle_n fkc_n now_o then_o there_o be_v two_o triangle_n fhc_n and_o fkcâ—_n have_v two_o angle_n equal_a to_o two_o angle_n the_o one_o to_o the_o other_o and_o one_o side_n equal_a to_o one_o side_n for_o fc_n be_v commoâ—_n unto_o they_o both_o and_o be_v subtend_v under_o one_o of_o the_o equal_a angle_n wherefore_o by_o the_o 2d_o of_o the_o first_o the_o side_n remayâ—â—ng_v be_v equal_a unto_o the_o side_n remain_v wherefore_o the_o perpendicular_a fh_o be_v equal_a unto_o the_o perpendicular_a fk_n and_o in_o like_a sort_n also_o may_v it_o be_v prouâ—d_v that_o every_o one_o of_o these_o line_n fl_fw-mi fm_o and_o fg_o be_v equal_a to_o every_o one_o of_o thâ—â—â—_n line_n fh_o and_o fk_o wherefore_o these_o five_o right_a line_n fg_o fh_o fk_o fl_fw-mi and_o fm_o be_v equal_a the_o one_o to_o the_o other_o wherefore_o make_v the_o centre_n fletcher_n and_o the_o space_n fg_o or_o fh_o or_o fk_o or_o fl_fw-mi or_o fm_o describe_v a_o circle_n and_o it_o will_v pass_v by_o the_o point_n g_o h_o edward_n l_o m._n and_o shall_v touch_v the_o right_a line_n ab_fw-la bc_o cd_o de_fw-fr and_o ea_fw-la by_o the_o corollary_n of_o the_o 16._o of_o the_o three_o for_o the_o angle_n which_o be_v at_o the_o point_n g_o h_o edward_n l_o m_o be_v right_a angle_n absurdity_n for_o if_o it_o do_v not_o touch_v they_o but_o cut_v then_o from_o the_o end_n of_o the_o diameter_n of_o the_o circle_n shall_v be_v draw_v a_o right_a line_n make_v two_o right_a angle_n and_o fall_v with_o in_o the_o circle_n which_o be_v by_o the_o 16._o of_o the_o three_o prove_v to_o be_v impossible_a wherefore_o make_v fletcher_n the_o centre_n and_o the_o space_n one_o of_o these_o line_n fg_o fh_o fk_o fl_fw-mi fm_o describe_v a_o circle_n and_o it_o shall_v not_o cut_v the_o right_a line_n ab_fw-la bc_o cd_o de_fw-fr and_o ea_fw-la wherefore_o by_o the_o corollary_n of_o the_o 16._o of_o the_o three_o it_o shall_v touch_v they_o as_o it_o be_v manifest_a in_o the_o circle_n ghklm_n wherefore_o in_o the_o equilater_n and_o equiangle_n pentagon_n figure_n give_v be_v describe_v a_o circle_n which_o be_v require_v to_o be_v doneâ—_n by_o this_o proposition_n and_o the_o former_a it_o be_v manifest_v that_o perpendicular_a line_n draw_v from_o the_o middle_a point_n of_o the_o side_n of_o a_o equilater_n and_o equiangle_n pentagon_n figure_n corollary_n and_o produce_v shall_v pass_v by_o the_o centre_n of_o the_o circle_n in_o which_o the_o say_a pentagon_n figure_n be_v inscribe_v and_o shall_v also_o divide_v the_o opposite_a angle_n equal_o as_o here_o ak_n be_v one_o right_a line_n and_o divide_v the_o side_n cd_o and_o the_o angle_n a_o equal_o and_o so_o of_o the_o rest_n which_o be_v thus_o prove_v the_o space_n about_o the_o centre_n fletcher_n be_v equal_a to_o four_o right_a angle_n which_o be_v divide_v into_o ten_o equal_a angle_n by_o ten_o right_a line_n meet_v together_o in_o the_o point_n f._n wherefore_o the_o five_o angle_n afm_n mfe_n efl_fw-mi lfd_n and_o dfk_n be_v equal_a to_o two_o right_a angle_n wherefore_o by_o the_o 14._o of_o the_o first_o the_o line_n of_o and_o fk_o make_v one_o right_a line_n the_o like_a proof_n also_o will_v serve_v touch_v the_o reâ—t_n of_o the_o line_n and_o this_o be_v always_o true_a in_o all_o equilater_n figure_n which_o consist_v of_o uneven_a side_n the_o 14._o problem_n the_o 14._o proposition_n about_o a_o pentagon_n or_o figure_n of_o five_o angle_n give_v be_v equilater_n and_o equiangle_n to_o describe_v a_o circle_n svppose_v that_o the_o pentagon_n or_o figure_n of_o five_o angle_n give_v be_v of_o equal_a side_n and_o of_o equal_a angle_n be_v abcde_o it_o be_v require_v about_o the_o say_a pentagon_n abcde_o to_o describe_v a_o circle_n divide_v by_o the_o 9_o of_o the_o first_o either_o of_o these_o angle_n bcd_o construction_n &_o cde_o into_o two_o equal_a part_n by_o either_o of_o these_o right_a line_n cf_o and_o df._n and_o from_o the_o point_n f_o where_o those_o right_a line_n meet_v draw_v unto_o the_o point_n b_o a_o e_o these_o right_a line_n fb_o favorina_n fe_o demonstration_n and_o in_o like_a sort_n by_o the_o proposition_n go_v before_o may_v it_o be_v prove_v that_o every_o one_o of_o these_o angle_n cba_o bae_o and_o aed_n be_v divide_v into_o two_o equal_a part_n by_o these_o right_a
13._o of_o the_o six_o take_v the_o mean_a proportional_a between_o the_o line_n bc_o and_o cf_o which_o let_v be_v gh_o and_o by_o the_o 18._o of_o the_o six_o upon_o the_o line_n gh_o let_v there_o be_v describe_v a_o rectiline_a figure_n khg_n like_a unto_o the_o rectiline_a figure_n abc_n and_o in_o like_a sort_n situate_a and_o for_o that_o as_o the_o line_n bc_o be_v to_o the_o line_n gh_o so_o be_v the_o line_n gh_o to_o the_o line_n cf_o demonstration_n but_o if_o there_o be_v three_o right_a line_n proportional_a as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o figure_n which_o be_v describe_v of_o the_o first_o unto_o the_o figure_n which_o be_v describe_v of_o the_o second_o the_o say_a figure_n be_v like_o and_o in_o like_a sort_n situate_a by_o the_o second_o corollary_n of_o the_o 20._o of_o the_o six_o wherefore_o as_o the_o line_n bc_o be_v to_o the_o line_n cf_o so_o be_v the_o triangle_n abc_n to_o the_o triangle_n kgh_n but_o as_o the_o line_n bc_o be_v to_o the_o line_n cf_o so_o be_v the_o parallelogram_n be_v to_o the_o parallelogram_n of_o by_o the_o 1._o of_o the_o six_o wherefore_o as_o the_o triangle_n a_o bc_o be_v to_o the_o triangle_n kgh_o so_o be_v the_o parallelogram_n be_v to_o the_o parallelogram_n ef._n wherefore_o alternate_o also_o by_o the_o 16._o of_o the_o five_o as_o the_o triangle_n abc_n be_v to_o the_o parallelogram_n be_v so_o be_v the_o triangle_n kgh_o to_o the_o parallelogram_n of_o but_o the_o triangle_n abc_n be_v equal_a unto_o the_o parallelogram_n be_v wherefore_o also_o the_o triangle_n kgh_n be_v equal_a unto_o the_o parallelogram_n of_o but_o the_o parallelogram_n fe_o be_v equal_a unto_o the_o rectiline_a figure_n d._n wherefore_o also_o the_o rectiline_a figure_n kgh_o be_v equal_a unto_o the_o rectiline_a figure_n d_o and_o the_o rectiline_a figure_n kgh_o be_v by_o supposition_n like_o unto_o the_o rectiline_a figure_n abc_n wherefore_o there_o be_v describe_v a_o rectiline_a figure_n kgh_o like_o unto_o the_o rectiline_a figure_n give_v abc_n and_o equal_a unto_o the_o other_o rectiline_a figure_n give_v d_z which_o be_v require_v to_o be_v do_v the_o 19_o theorem_a the_o 26._o proposition_n if_o from_o a_o parallelogram_n be_v take_v away_o a_o parallelogram_n like_a unto_o the_o whole_a and_o in_o like_a sort_n set_v have_v also_o a_o angle_n common_a with_o it_o then_o be_v the_o parallelogram_n about_o one_o and_o the_o self_n same_o dimecient_a with_o the_o whole_a svppose_v that_o there_o be_v a_o parallelogram_n abcd_o and_o from_o the_o parallelogram_n abcd_o take_v away_o a_o parallelogram_n of_o like_a unto_o the_o parallelogram_n abcd_o and_o in_o like_a sort_n situate_a have_v also_o the_o angle_n dab_n common_a with_o it_o then_o i_o say_v that_o the_o parallelogram_n abcd_o and_o of_o be_v both_o about_o one_o and_o the_o self_n same_o absurdity_n dimecient_a afc_n that_o be_v that_o the_o dimecient_a afc_n of_o the_o whole_a parallelogram_n abcd_o pass_v by_o the_o angle_n f_o of_o the_o parallelogram_n of_o and_o be_v common_a to_o either_o of_o the_o parallelogram_n for_o if_o ac_fw-la do_v not_o pass_v by_o the_o point_n fletcher_n then_o if_o it_o be_v possible_a let_v it_o pass_v by_o some_o other_o point_n as_o ahc_n do_v now_o than_o the_o dimetient_fw-la ahc_n shall_v cut_v either_o the_o side_n gf_o or_o the_o side_n of_o of_o the_o parallelogram_n af._n let_v it_o cut_v the_o side_n gf_o in_o the_o point_n h._n and_o by_o the_o 31._o of_o the_o first_o by_o the_o point_n h_o let_v there_o be_v draw_v to_o either_o of_o these_o line_n ad_fw-la and_o bc_o a_o parallel_a line_n hk_o wherefore_o gk_o be_v a_o parallelogram_n and_o be_v about_o one_o and_o the_o self_n same_o dimetient_fw-la with_o the_o parallelogram_n abcd._n and_o forasmuch_o as_o the_o parallelogram_n abcd_o and_o gk_o be_v about_o one_o and_o the_o self_n same_o dimecient_a therefore_o by_o the_o 24._o of_o the_o six_o the_o parallelogram_n abcd_o be_v like_a unto_o the_o parallelogram_n gk_o wherefore_o as_o the_o line_n dam_n be_v to_o the_o line_n ab_fw-la so_o be_v the_o line_n ga_o to_o the_o line_n ak_o by_o the_o conversion_n of_o the_o first_o definition_n of_o the_o six_o and_o for_o that_o the_o parallelogram_n abcd_o and_o eglantine_n be_v by_o supposition_n like_v therefore_o as_o the_o line_n dam_n be_v to_o the_o line_n ab_fw-la so_o be_v the_o line_n ga_o to_o the_o line_n ae_n wherefore_o the_o line_n ga_o have_v one_o and_o the_o self_n proportion_n to_o either_o of_o these_o line_n ak_o and_o ae_n wherefore_o by_o the_o 9_o of_o the_o five_o the_o line_n ak_o be_v equal_a unto_o the_o line_n ae_n namely_o the_o less_o to_o the_o great_a which_o be_v impossible_a the_o self_n same_o inconvenience_n also_o will_v follow_v if_o you_o put_v the_o dimetient_fw-la ac_fw-la to_o cut_v the_o side_n fe_o wherefore_o ac_fw-la the_o dimetient_n of_o the_o whole_a parallelogram_n abcd_o pass_v by_o the_o angle_n and_o point_n f._n and_o therefore_o the_o parallelogram_n aefg_n be_v about_o one_o and_o the_o self_n same_o dimetient_fw-la with_o the_o whole_a parallelogram_n abcd._n wherefore_o if_o from_o a_o parallelogram_n be_v take_v away_o a_o parallelogram_n like_o unto_o the_o whole_a and_o in_o like_a sort_n situate_a have_v also_o a_o angle_n common_a with_o it_o then_o be_v that_o parallelogram_n about_o one_o and_o the_o self_n same_o dimetient_fw-la with_o the_o whole_a which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n after_o flussates_n which_o prove_v this_o proposition_n affirmative_o from_o the_o parallelogram_n abgd_v let_v there_o be_v take_v away_o the_o parallelogram_n aezk_v like_a and_o in_o like_a sort_n situate_a with_o the_o whole_a parallelogram_n abgd_v and_o have_v also_o the_o angle_n a_o common_a with_o the_o whole_a parallelogram_n flussates_n then_o i_o say_v that_o both_o their_o diameter_n namely_o az_o and_o azg_n do_v make_v one_o and_o the_o self_n same_o right_a line_n divide_v the_o side_n ab_fw-la and_o bâ—_n into_o two_o equal_a part_n in_o the_o point_n c_o and_o f_o by_o the_o 10._o of_o the_o first_o and_o draw_v a_o line_n from_o c_o to_o f._n wherefore_o the_o line_n cf_o be_v a_o parallel_n to_o the_o right_a line_n agnostus_n by_o the_o corollary_n add_v by_o campane_n after_o the_o 29._o of_o the_o first_o wherefore_o the_o angle_n bag_n and_o bcf_n be_v equal_a by_o the_o 29._o of_o the_o first_o but_o the_o angle_n eaz_n be_v equal_a unto_o the_o angle_n bag_n by_o reason_n the_o parallelogram_n be_v suppose_v to_o be_v like_o wherefore_o the_o same_o angle_n eaz_n be_v equal_a to_o the_o angle_n bcf_n namely_o the_o outward_a angle_n to_o the_o inward_a and_o opposite_a angle_n wherefore_o by_o the_o 28._o of_o the_o first_o the_o line_n az_o and_o cf_o be_v parallel_a line_n now_o than_o the_o line_n az_o and_o agnostus_n be_v parallel_n to_o one_o and_o the_o self_n same_o line_n namely_o to_o cf_o do_v concur_v in_o the_o point_n a._n wherefore_o they_o be_v set_v direct_o the_o one_o to_o the_o other_o so_o that_o they_o both_o make_v one_o right_a line_n by_o that_o which_o be_v add_v in_o the_o end_n of_o the_o 30._o proposition_n of_o the_o first_o wherefore_o the_o parallelogram_n abgd_v and_o aezk_n be_v about_o one_o and_o the_o self_n fame_n dimetient_fw-la which_o be_v require_v to_o be_v prove_v the_o 20._o theorem_a the_o 27._o proposition_n of_o all_o parallelogram_n apply_v to_o a_o right_a line_n want_v in_o figure_n by_o parallelogram_n like_a and_o in_o like_a sort_n situate_a to_o that_o parallelogram_n which_o be_v describe_v of_o the_o half_a line_n the_o great_a parallelogram_n be_v that_o which_o be_v describe_v of_o the_o half_a line_n be_v like_a unto_o the_o want_n ac_fw-la let_v there_o be_v a_o right_a line_n ab_fw-la and_o by_o the_o 10._o of_o the_o first_o divide_v it_o in_o two_o equal_a part_n in_o the_o point_n c._n and_o unto_o the_o right_a line_n ab_fw-la apply_v a_o parallelogram_n ad_fw-la want_v in_o figure_n by_o the_o parallelogram_n db_o which_o let_v be_v like_a and_o in_o like_a sort_n describe_v unto_o the_o parallelogram_n describe_v of_o half_a the_o line_n ab_fw-la which_o be_v bc._n then_o i_o say_v that_o of_o all_o the_o parallelogram_n which_o may_v be_v apply_v unto_o the_o line_n ab_fw-la and_o which_o want_n in_o figure_n by_o parallelogram_n like_a and_o in_o like_a sort_n situate_a unto_o the_o parallelogram_n db_o the_o great_a be_v the_o parallelogram_n ad._n for_o unto_o the_o right_a line_n ab_fw-la let_v there_o be_v apply_v a_o parallelogram_n of_o want_v in_o figure_n by_o the_o parallelogram_n fb_o which_o let_v be_v like_a and_o in_o like_a sort_n situate_a unto_o the_o parallelogram_n db._n then_o i_o say_v that_o the_o parallelogram_n ad_fw-la be_v
proportion_n that_o the_o first_o of_o the_o three_o line_n put_v be_v to_o the_o 〈◊〉_d â—or_a tâ—e_z 〈◊〉_d line_n to_o the_o three_o namely_o the_o line_n ae_n to_o the_o line_n ebb_n be_v in_o double_a proportion_n that_o it_o be_v to_o the_o second_o by_o the_o 10._o definition_n of_o the_o fiâ—t_z ¶_o the_o second_o corollary_n hereby_o may_v we_o learn_v how_o from_o a_o rectiline_a â—igure_n give_v to_o take_v away_o a_o part_n appoint_v leaâ—ing_v the_o rest_n of_o the_o rectiline_a â—igure_n like_o unto_o the_o whole_a corollary_n for_o if_o from_o the_o right_a line_n ab_fw-la be_v cut_v of_o a_o part_n appoint_v namely_o ebb_n by_o the_o 9_o of_o this_o book_n as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o rectiline_a â—igure_n describe_v of_o the_o line_n of_o to_o the_o rectiline_a figure_n describe_v of_o the_o line_n fb_o the_o say_v figure_n being_n suppose_v to_o be_v like_o both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o rectiline_a â—igure_n describe_v of_o the_o line_n ab_fw-la and_o be_v also_o in_o like_a sort_n situate_a wherefore_o take_v away_o â—rom_o the_o rectiline_a â—igure_n describe_v of_o the_o line_n ab_fw-la the_o rectiline_a figure_n describe_v of_o the_o line_n fb_o the_o residue_n namely_o the_o rectiline_a figure_n describe_v of_o the_o line_n of_o shall_v be_v both_o like_a unto_o the_o whole_a rectiline_a â—igure_n give_v describe_v of_o the_o line_n ab_fw-la and_o in_o like_a sort_n situate_a ¶_o the_o three_o corollary_n to_o compose_v two_o like_o rectiline_a â—_z into_o one_o rectiline_a figure_n like_o and_o equal_a to_o the_o same_o figure_n let_v their_o side_n of_o like_a proportion_n be_v set_v so_o that_o they_o make_v a_o right_a angle_n corollary_n as_o the_o line_n of_o and_o fb_o be_v and_o upon_o the_o line_n subtend_v the_o say_a angle_n namely_o the_o line_n ab_fw-la describe_v a_o rectiline_a â—igure_n like_o unto_o the_o rectiline_a figure_n give_v and_o in_o like_a sort_n situate_a by_o the_o 18._o of_o this_o book_n and_o the_o same_o shall_v be_v equal_a to_o the_o two_o rectiline_a figure_n give_v by_o the_o 31._o of_o this_o book_n ¶_o the_o second_o proposition_n if_o two_o right_a line_n cut_v the_o one_o the_o other_o obtuseangle_v wise_a and_o from_o the_o end_n of_o the_o line_n which_o â—ut_z the_o one_o the_o other_o be_v draw_v perpendicular_a line_n to_o either_o line_n the_o line_n which_o be_v between_o the_o end_n and_o the_o perpendicular_a line_n be_v cut_v reciprocal_a suppose_v that_o there_o be_v two_o right_a line_n ab_fw-la and_o gd_o cut_v the_o one_o the_o other_o in_o the_o point_n e_o and_o make_v a_o obtuse_a angle_n in_o the_o section_n e._n and_o from_o the_o end_n of_o the_o line_n namely_o a_o and_o g_o let_v there_o be_v draw_v to_o either_o line_n perpendicular_a line_n namely_o from_o the_o point_n a_o to_o the_o line_n gd_o which_o let_v be_v ad_fw-la and_o from_o the_o point_n g_o to_o the_o right_a line_n ab_fw-la which_o let_v be_v gb_o then_o i_o say_v that_o the_o right_a line_n ab_fw-la and_o gd_o do_v between_o the_o end_n a_o and_o the_o perpendicular_a b_o and_o the_o end_n g_o and_o the_o perpendicular_a d_o cut_v the_o one_o the_o other_o reciprocal_a in_o the_o point_n e_o so_o that_o as_o the_o line_n ae_n be_v to_o the_o line_n ed_z so_o be_v the_o line_n ge_z to_o the_o line_n ebb_n proposition_n for_o forasmuch_o as_o the_o angle_n ade_n and_o gbe_n be_v right_a angle_n therefore_o they_o be_v equal_a but_o the_o angle_n aed_n and_o geb_fw-mi be_v also_o equal_a by_o the_o 15._o of_o the_o first_o wherefore_o the_o angle_n remain_v namely_o ead_n &_o egb_n be_v equal_a by_o the_o corollary_n of_o the_o 32._o of_o the_o first_o wherefore_o the_o triangle_n aed_n and_o geh_a be_v equiangle_n wherefore_o the_o side_n about_o the_o equal_a angle_n shall_v be_v proportional_a by_o the_o 4._o of_o the_o six_o wherefore_o as_o the_o line_n ae_n be_v to_o the_o line_n ed_z so_o be_v the_o line_n ge_z to_o the_o line_n ebb_n if_o therefore_o two_o right_a line_n cut_v the_o one_o the_o other_o obtuseangled_a wife_n etc_o etc_o which_o be_v require_v to_o be_v prove_v ¶_o the_o three_o proposition_n if_o two_o right_a line_n make_v a_o acute_a angle_n and_o from_o their_o end_n be_v draw_v to_o each_o line_n perpendicular_a line_n cut_v they_o the_o two_o right_a line_n give_v shall_v be_v reciprocal_a proportional_a as_o the_o segment_n which_o be_v about_o the_o angle_n suppose_v that_o there_o be_v two_o right_a line_n ab_fw-la and_o gb_o make_v a_o acute_a angle_n abg_o and_o from_o the_o point_n a_o and_o g_o let_v there_o be_v draw_v unto_o the_o line_n ab_fw-la and_o gb_o perpendicular_a line_n ac_fw-la and_o ge_z cut_v the_o line_n ab_fw-la and_o gb_o in_o the_o point_n e_o and_o â—_o then_o i_o say_v that_o the_o line_n namely_o ab_fw-la to_o gb_o be_v reciprocal_a proportional_a as_o the_o segment_n namely_o cb_o to_o ebb_v which_o be_v about_o the_o acute_a angle_n b._n proposition_n for_o forasmuch_o as_o thâ—_z right_o angle_n acb_o and_o gerard_n be_v equal_a and_o the_o angleâ—_n abg_o be_v common_a to_o the_o triangle_n abc_n and_o gbeâ—_n â—_z therefore_o the_o angle_n remain_v bac_n and_o egb_n be_v equal_a by_o the_o corollary_n of_o the_o 32._o of_o the_o first_o wherefore_o the_o triangle_n abc_n and_o gbe_n be_v equiangle_n wherefore_o the_o sideâ—_n about_o the_o equal_a angle_n be_v proportional_a by_o the_o 4._o of_o the_o six_o â—_o so_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n fc_o so_o be_v the_o line_n gb_o to_o the_o line_n be._n wherefore_o alternate_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n gb_o so_o be_v the_o line_n cb_o to_o the_o line_n be._n if_o therefore_o two_o right_a line_n makâ—_n aâ—â—câ—te_a angleâ—_n &_o câ—_n which_o be_v require_v to_o be_v prove_v â—_o the_o four_o proposition_n if_o in_o a_o circle_n be_v draw_v two_o right_a line_n cut_v the_o one_o the_o other_o the_o section_n of_o the_o one_o to_o the_o section_n of_o the_o other_o shall_v be_v reciprocal_a proportional_a in_o the_o circle_n agb_n let_v these_o two_o right_a line_n 〈…〉_o one_o the_o other_o in_o the_o point_n e._n proposition_n thâ—â—_n i_o say_v that_o reciprocal_a 〈◊〉_d â—hâ—_n line_n ae_n be_v to_o the_o line_n ed_z so_o be_v the_o line_n ge_z to_o the_o line_n ebb_n for_o forasmuch_o as_o by_o the_o 35._o of_o the_o three_o the_o rectangle_n figure_n contain_v under_o the_o line_n ae_n and_o ebb_n be_v equal_a to_o the_o rectangle_n figure_n contain_v under_o the_o line_n ge_z and_o ed_z but_o in_o equal_a rectangle_n parallelogram_n the_o side_n about_o the_o equal_a angle_n be_v reciprocal_a by_o the_o 14._o of_o the_o six_o therefore_o the_o line_n ae_n be_v to_o the_o line_n ed_z reciprocal_a as_o the_o line_n ge_z be_v to_o the_o line_n ebb_n by_o the_o second_o definition_n of_o the_o six_o if_o therefore_o in_o a_o circle_n be_v draw_v two_o right_a line_n etc_o etc_o which_o be_v require_v to_o be_v prove_v ¶_o the_o five_o proposition_n if_o from_o a_o point_n give_v be_v draw_v in_o a_o plain_a superficies_n two_o right_a line_n to_o the_o concave_a circumference_n of_o a_o circle_n they_o shall_v be_v reciprocal_a proportional_a with_o their_o part_n take_v without_o the_o circle_n and_o moreover_o a_o right_a line_n draw_v from_o the_o say_a point_n &_o touch_v the_o circle_n shall_v be_v a_o mean_a proportional_a between_o the_o whole_a line_n and_o the_o utter_a segment_n suppose_v that_o there_o be_v a_o circle_n abdella_n and_o without_o it_o take_v a_o certain_a point_n namely_o g._n and_o from_o the_o point_n g_o draw_v unto_o the_o concave_a circumference_n two_o right_a line_n gb_o and_o gd_o cut_v the_o circle_n in_o the_o point_n c_o and_o e._n and_o let_v the_o line_n ga_o touch_v the_o circle_n in_o the_o point_n a._n then_o i_o say_v that_o the_o line_n namely_o gb_o to_o gd_o be_v reciprocal_a as_o their_o part_n take_v without_o the_o circle_n namely_o as_o gc_o to_o ge._n proposition_n for_o forasmuch_o as_o by_o the_o corollary_n of_o the_o 36._o of_o the_o three_o the_o rectangle_n figure_n contain_v under_o the_o line_n gb_o and_o ge_z be_v equal_a to_o the_o rectangle_n figure_n contain_v under_o the_o line_n gd_o and_o gc_o therefore_o by_o the_o 14._o of_o the_o six_o reciprocal_a as_o the_o line_n gb_o be_v to_o the_o line_n gd_o so_o be_v the_o line_n gc_o to_o the_o line_n ge_z for_o they_o be_v side_n contain_v equal_a angle_n i_o say_v moreover_o that_o between_o the_o line_n gb_o and_o ge_z or_o between_o the_o line_n gd_o and_o gc_o the_o touch_n line_n ga_o be_v a_o mean_a proportional_a part_n for_o forasmuch_o as_o the_o rectangle_n
figure_n comprehend_v under_o the_o line_n gb_o and_o ge_z be_v equal_a to_o the_o square_n make_v of_o the_o line_n agnostus_n by_o the_o 36._o of_o the_o three_o it_o follow_v that_o the_o touch_n line_n ga_o be_v a_o mean_a proportional_a between_o the_o extreme_n gb_o and_o ge_z by_o the_o second_o part_n of_o the_o 17._o of_o the_o six_o for_o that_o by_o that_o proposition_n the_o line_n gb_o ga_o and_o ge_z be_v proportional_a and_o by_o the_o same_o reason_n may_v it_o be_v prove_v that_o the_o line_n ga_o be_v a_o mean_a proportional_a between_o the_o line_n gd_o and_o gc_o and_o so_o of_o all_o other_o if_o therefore_o from_o a_o point_n gevenâ—_n &_o câ—_n which_o be_v require_v to_o be_v demonstrate_v the_o end_n of_o the_o six_o book_n of_o euclides_n element_n ¶_o the_o seven_o book_n of_o euclides_n element_n hitherto_o in_o the_o six_o book_n before_o have_v euclid_n pass_v through_o and_o entreat_v of_o the_o element_n of_o geometry_n without_o the_o aid_n and_o succour_v of_o number_n but_o the_o matter_n which_o remain_v to_o be_v teach_v and_o to_o be_v speak_v of_o in_o these_o his_o geometrical_a book_n which_o follow_v as_o in_o the_o ten_o eleven_o and_o so_o forth_o he_o can_v by_o no_o mean_n full_o and_o clear_o make_v plain_a &_o demonstrate_v without_o the_o help_n and_o aid_n of_o number_n in_o the_o ten_o be_v entreat_v of_o line_n irrational_a and_o uncertain_a and_o that_o of_o many_o &_o sundry_a kind_n and_o in_o the_o eleven_o &_o the_o other_o follow_v he_o teach_v the_o nature_n of_o body_n and_o compare_v their_o side_n and_o line_n together_o all_o which_o for_o the_o most_o part_n be_v also_o irrational_a and_o as_o rational_a quantity_n and_o the_o comparison_n and_o proportion_n of_o they_o can_v be_v know_v nor_o exact_o try_v but_o by_o the_o mean_a of_o number_n in_o which_o they_o be_v first_o see_v and_o perceive_v even_o so_o likewise_o can_v irrational_a quantity_n be_v know_v and_o find_v out_o without_o number_n as_o straightness_n be_v the_o trial_n of_o crookedness_n and_o inequality_n be_v try_v by_o equality_n so_o be_v quantity_n irrational_a perceive_v and_o know_v by_o quantity_n rational_a which_o be_v â—irst_n and_o chief_o find_v among_o number_n wherefore_o in_o these_o three_o book_n follow_v be_v as_o it_o be_v in_o the_o midst_n of_o his_o element_n he_o be_v compel_v of_o necessity_n to_o entreat_v of_o number_n number_n although_o not_o so_o full_o as_o the_o nature_n of_o number_n require_v yet_o so_o much_o as_o shall_v seem_v to_o be_v fit_a and_o sufficient_o to_o serve_v for_o his_o purpose_n whereby_o be_v see_v the_o necessity_n that_o the_o one_o art_n namely_o geometry_n have_v of_o the_o other_o namely_o of_o arithmetic_n and_o also_o of_o what_o excellency_n and_o worthiness_n arithmetic_n be_v above_o geometry_n geometry_n in_o that_o geometry_n borrow_v of_o it_o principle_n aid_n and_o succour_n and_o be_v as_o it_o be_v maim_v with_o out_o it_o whereas_o arithmetic_n be_v of_o itself_o sufficient_a and_o nead_v not_o at_o all_o any_o aid_n of_o geometry_n but_o be_v absolute_a and_o perfect_a in_o itself_o and_o may_v well_o be_v teach_v and_o attain_a unto_o without_o it_o again_o the_o matter_n or_o subject_n where_o about_o geometry_n be_v occupy_v which_o be_v line_n figure_n and_o body_n be_v such_o as_o offer_v themselves_o to_o the_o sense_n as_o triangle_n square_n circle_n cube_n and_o other_o be_v see_v &_o judge_v to_o be_v such_o as_o they_o be_v by_o the_o sight_n but_o number_n which_o be_v the_o subject_n and_o matter_n of_o arithmetic_n fall_v under_o no_o sense_n nor_o be_v represent_v by_o any_o shape_n form_n or_o figure_n and_o therefore_o can_v be_v judge_v by_o any_o sense_n but_o only_o by_o consideration_n of_o the_o mind_n and_o understanding_n now_o thing_n sensible_a be_v far_o under_o in_o degree_n then_o be_v thing_n intellectual_a sensible_a and_o be_v of_o nature_n much_o more_o gross_a than_o they_o wherefore_o number_n as_o be_v only_o intellectual_a be_v more_o pure_a more_o immaterial_a and_o more_o subtle_a far_o then_o be_v magnitude_n and_o extend_v itself_o far_a for_o arithmetic_n not_o only_o aid_v geometry_n but_o minister_v principle_n and_o ground_n to_o many_o other_o nay_o rather_o to_o all_o other_o science_n and_o art_n science_n as_o to_o music_n astronomy_n natural_a philosophy_n perspective_n with_o other_o what_o other_o thing_n be_v in_o music_n entreat_v of_o than_o number_n contract_v to_o sound_n and_o voice_n in_o astronomy_n who_o without_o the_o knowledge_n of_o number_n can_v do_v any_o thing_n either_o in_o search_v out_o of_o the_o motion_n of_o the_o heaven_n and_o their_o course_n either_o in_o judge_v and_o foreshow_v the_o effect_n of_o they_o in_o natural_a philosophy_n it_o be_v of_o no_o small_a force_n the_o wise_a and_o best_a learned_a philosopher_n that_o have_v be_v as_o pythagoras_n timeus_n plato_n and_o their_o follower_n find_v out_o &_o teach_v most_o pithy_o and_o pure_o the_o secret_a and_o hide_a knowledge_n of_o the_o nature_n and_o condition_n of_o all_o thing_n by_o number_n and_o by_o the_o propriety_n and_o passion_n of_o they_o of_o what_o force_n number_n be_v in_o perspective_n let_v he_o declare_v and_o judge_v who_o have_v any_o thing_n travel_v therein_o yea_o to_o be_v short_a what_o can_v be_v worthy_o and_o with_o praise_n practise_v in_o common_a life_n of_o any_o man_n of_o any_o condition_n without_o the_o knowledge_n of_o number_n yea_o it_o have_v be_v teach_v of_o the_o chief_a amongst_o philosopher_n that_o all_o natural_a thing_n be_v frame_v and_o have_v their_o constitution_n of_o number_n boetius_fw-la say_v hoc_fw-la fuit_fw-la principal_n in_o animâ—_n câ—â—ditoris_fw-la exemplar_n arithmeti_n number_n say_v he_o be_v the_o principal_a example_n and_o patron_n in_o the_o mind_n of_o the_o creator_n of_o the_o world_n do_v not_o that_o great_a philosop_n timaus_n in_o his_o book_n timaus_n &_o also_o plato_n in_o his_o timâ—â—_n follow_v he_o show_v how_o the_o soul_n be_v compose_v of_o harmonical_a number_n and_o consonantes_fw-la of_o music_n number_n compass_v all_o thing_n and_o be_v after_o these_o man_n the_o be_v and_o very_a essence_n of_o all_o thing_n and_o minister_v aid_n and_o help_v as_o to_o all_o other_o knowledge_n so_o also_o no_o small_a to_o geometry_n which_o thing_n cause_v euclidâ—_n in_o the_o midst_n of_o his_o book_n of_o geometry_n to_o insert_v and_o place_n these_o three_o book_n of_o arithmetic_n as_o without_o the_o aid_n of_o which_o he_o can_v not_o well_o pass_v any_o father_n in_o this_o seven_o book_n he_o â—irst_n place_v the_o general_a principle_n and_o first_o ground_n of_o arithmetic_n and_o set_v the_o definition_n of_o number_n or_o multitude_n and_o the_o kind_n thereof_o as_o in_o the_o first_o book_n he_o do_v of_o magnitude_n and_o the_o kind_n and_o part_n thereof_o book_n after_o that_o he_o entreat_v of_o number_n general_o and_o of_o their_o part_n and_o search_v and_o demonstrate_v in_o general_a the_o most_o common_a passion_n and_o propriety_n of_o the_o same_o and_o chief_o of_o number_n prime_a or_o incommensurable_a and_o of_o number_n compose_v or_o commensurable_a and_o of_o their_o propriety_n and_o partly_o also_o of_o the_o comparison_n oâ—_n proportion_n of_o one_o number_n to_o a_o other_o ¶_o definition_n definition_n 1_o unity_n be_v that_o whereby_o every_o thing_n that_o be_v be_v say_v to_o be_v on_o as_o a_o point_n in_o magnitude_n be_v the_o least_o thing_n in_o magnitude_n and_o no_o magnitude_n at_o all_o &_o yet_o the_o ground_n and_o beginning_n of_o all_o magnitude_n even_o so_o be_v unity_n in_o multitude_n or_o number_n the_o least_o thing_n in_o number_n and_o no_o number_n at_o all_o and_o yet_o the_o ground_n and_o beginning_n of_o all_o number_n and_o therefore_o it_o be_v here_o in_o this_o place_n of_o euclid_n first_o define_v as_o in_o the_o first_o book_n for_o the_o like_a reason_n and_o cause_n be_v a_o point_n first_o define_v unity_n say_v euclid_n be_v that_o whereby_o every_o thing_n be_v say_v to_o be_v one_o that_o be_v unity_n be_v that_o whereby_o every_o thing_n be_v divide_v and_o separate_v from_o a_o other_o and_o remain_v on_o in_o itself_o pure_a and_o distinct_a from_o all_o other_o otherwise_o be_v not_o this_o unity_n thing_n whereby_o all_o thing_n be_v sejoin_v the_o on_o from_o the_o other_o all_o thing_n shall_v suffer_v mixtion_n and_o be_v in_o confusion_n and_o where_o confusion_n be_v there_o be_v no_o order_n nor_o any_o thing_n can_v be_v exact_o know_v either_o what_o it_o be_v or_o what_o be_v the_o nature_n and_o what_o be_v the_o property_n thereof_o unity_n therefore_o be_v that_o which_o make_v every_o thing_n to_o be_v that_o which_o it_o be_v boetius_fw-la say_v very_o apt_o
which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o by_o a_o rational_a superficies_n for_o either_o of_o they_o be_v rational_a wherefore_o also_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o exceed_v the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o by_o a_o rational_a superficies_n when_o yet_o each_o of_o they_o be_v a_o medial_a superficies_n which_o be_v impossible_a wherefore_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a be_v in_o one_o point_n only_o divide_v into_o his_o name_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 35._o theorem_a the_o 47._o proposition_n a_o line_n contain_v in_o power_n two_o medial_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n svppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n be_v divide_v into_o his_o name_n in_o the_o point_n c_o so_fw-mi that_o let_v the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la &_o cb_o medial_a and_o that_o also_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o medial_a and_o moreover_o incommensurable_a â—o_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb._n then_o i_o say_v construction_n that_o the_o line_n ab_fw-la can_v in_o no_o other_o point_n be_v divide_v into_o his_o name_n but_o only_o in_o the_o point_n c._n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n d_o so_fw-mi that_o let_v not_o the_o line_n ac_fw-la be_v one_o and_o the_o same_o that_o be_v equal_a with_o the_o line_n db_o but_o by_o supposition_n let_v the_o line_n ac_fw-la be_v the_o great_a and_o take_v a_o rational_a line_n ef._n and_o by_o the_o 43._o of_o the_o first_o upon_o the_o line_n of_o apply_v a_o rectangle_n parallelogram_n eglantine_n equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o and_o likewise_o upon_o the_o line_n hc_n which_o be_v equal_a to_o the_o line_n of_o apply_v the_o parallelogram_n hk_o equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o wherefore_o the_o whole_a parallelogram_n eke_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la absurdity_n again_o upon_o the_o same_o line_n of_o describe_v the_o parallelogram_n el_n equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o db._n wherefore_o the_o residue_n namely_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o be_v equal_a to_o the_o parallelogram_n remain_v namely_o to_o mk_o and_o forasmuch_o as_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v by_o supposition_n medial_a therefore_o the_o parallelogram_n eglantine_n which_o be_v equal_a unto_o it_o be_v also_o medial_a and_o it_o be_v apply_v upon_o the_o rational_a line_n ef._n wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n he_o be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n ef._n and_o by_o the_o same_o reason_n also_o the_o line_n hn_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o same_o line_n ef._n and_o forasmuch_o as_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o for_o it_o be_v suppose_v to_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o once_o therefore_o the_o parallelogram_n eglantine_n be_v incommensurable_a to_o the_o parallelogram_n h_o â—_o wherefore_o the_o line_n eh_o also_o be_v incommensurable_a in_o length_n to_o the_o line_n hn_o and_o they_o be_v rational_a line_n wherefore_o the_o line_n eh_o and_o hn_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n en_fw-fr be_v a_o binomial_a line_n and_o be_v divide_v into_o his_o name_n in_o the_o point_n h._n and_o in_o like_a sort_n may_v we_o prove_v that_o the_o same_o binomial_a line_n en_fw-fr be_v divide_v into_o his_o name_n in_o the_o point_n m_o and_o that_o the_o line_n eh_o be_v not_o one_o and_o the_o same_o that_o be_v equal_a with_o the_o line_n mn_o as_o it_o be_v prove_v in_o the_o end_n of_o the_o demonstration_n of_o the_o 44._o of_o this_o book_n wherefore_o a_o binomial_a line_n be_v divide_v into_o his_o name_n in_o two_o sundry_a point_n which_o be_v impossible_a by_o the_o 42._o of_o the_o ten_o wherefore_o a_o line_n contain_v in_o power_n two_o medial_n be_v not_o in_o sundry_a point_n divide_v into_o his_o name_n wherefore_o it_o be_v divide_v in_o one_o point_n only_o which_o be_v require_v to_o be_v demonstrate_v ¶_o second_a definition_n it_o be_v show_v before_o that_o of_o binomial_a line_n there_o be_v six_o kind_n line_n the_o definition_n of_o all_o which_o be_v here_o now_o set_v and_o be_v call_v second_o definition_n all_o binomial_a line_n as_o all_o other_o kind_n of_o irrational_a line_n be_v conceive_v consider_v and_o perfect_o understand_v only_o in_o respect_n of_o a_o rational_a line_n who_o part_n as_o before_o be_v teach_v be_v certain_a and_o know_v and_o may_v be_v distinct_o express_v by_o number_n unto_o which_o line_n they_o be_v compare_v this_o rationalâ—_fw-la line_n must_v you_o ever_o have_v before_o your_o eye_n in_o all_o these_o definition_n so_o shall_v they_o all_o be_v â—asie_a enough_o paâ—tâ—s_n a_o binomial_a line_n you_o know_v be_v make_v of_o two_o part_n or_o name_n whereof_o the_o one_o be_v great_a than_o the_o other_o wherefore_o the_o power_n or_o square_v also_o of_o the_o one_o be_v great_a than_o the_o power_n or_o square_a oâ—_n the_o other_o the_o three_o first_o kind_n of_o binomial_a line_n namely_o the_o first_o the_o seconâ—_o &_o the_o three_o be_v produce_v when_o the_o square_a of_o the_o great_a name_n or_o part_n of_o a_o binomâ—all_n eâ—cedeth_v the_o square_a of_o the_o less_o name_n or_o part_n by_o the_o square_n of_o a_o line_n which_o be_v commensurable_a in_o length_n to_o it_o namely_o to_o the_o great_a the_o three_o last_o kind_n namely_o the_o four_o the_o â—iâ—t_n and_o the_o six_o be_v produce_v when_o the_o square_a of_o the_o great_a name_n or_o part_n â—â—â—â—edeth_v the_o square_a of_o the_o less_o name_n or_o part_n by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o it_o that_o be_v to_o the_o great_a part_n definition_n a_o first_o binomial_a line_n be_v who_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o tâ—e_z less_o part_n â—y_a the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o great_a part_n and_o the_o great_a part_n be_v also_o commensurable_a in_o length_n to_o tâ—e_z rational_a line_n first_o set_v as_o lâ—t_v the_o rational_a line_n first_o set_v be_v ab_fw-la who_o part_n be_v distinct_o know_v suppose_v also_o that_o the_o line_n ce_fw-fr be_v a_o binomial_a line_n who_o name_n or_o part_n let_v be_v cd_o and_o de._n and_o let_v the_o square_n of_o the_o line_n cd_o the_o great_a part_n exceed_v the_o square_n of_o the_o line_n de_fw-fr the_o less_o part_n by_o the_o square_n of_o the_o line_n fg_o which_o line_n fg_o let_v bâ—e_o commensurable_a in_o length_n to_o the_o line_n cd_o which_o be_v the_o great_a part_n of_o the_o binomial_a line_n and_o moreover_o let_v the_o line_n cd_o the_o great_a paâ—t_n be_v commensurble_n in_o length_n to_o the_o rational_a line_n first_o set_v namely_o to_o ab_fw-la so_o by_o this_o dâ—â—inition_n the_o binomial_a line_n ce_fw-fr be_v a_o first_o binomial_a line_n definition_n a_o second_o binomial_a line_n be_v when_o the_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o the_o less_o part_n by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o it_o and_o the_o less_o part_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n first_o set_v as_o suppose_v ever_o the_o rational_a line_n let_v ce_fw-fr be_v a_o binomial_a line_n divide_v in_o the_o point_n d._n the_o square_n of_o who_o great_a part_n cd_o let_v exceed_v the_o square_n of_o the_o less_o part_n de_fw-fr by_o the_o square_n of_o the_o line_n fg_o which_o line_n â—g_z let_v be_v commensurable_a in_o length_n unto_o the_o line_n cd_o tâ—e_z great_a pâ—â—â—_n oâ—_n the_o binomial_a line_n and_o let_v also_o the_o line_n de_fw-fr the_o less_o part_n of_o the_o binomial_a line_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n first_o set_v ab_fw-la so_o by_o this_o definition_n the_o binomial_a line_n ce_fw-fr be_v a_o second_o binomial_a line_n â—â—â—â—â—â—ition_n a_o three_o binomial_a
draw_v by_o the_o 11._o of_o the_o firsâ—_o a_o perpendicular_a line_n ab_fw-la and_o let_v ab_fw-la be_v a_o rational_a line_n and_o make_v perfectâ—_n the_o parallelogram_n bc._n wherefore_o bg_o be_v irrational_a by_o that_o which_o be_v declare_v and_o prove_v in_o manner_n of_o a_o assumpt_n in_o the_o end_n of_o the_o demonstration_n of_o the_o 38_o and_o the_o line_n that_o contain_v it_o iâ—_z power_n be_v also_o irrational_a let_v the_o line_n cd_o contain_v in_o power_n the_o superficies_n bc._n wherefore_o cd_o be_v irrational_a &_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 the_o line_n cd_o apply_v to_o a_o rational_a line_n namely_o ab_fw-la make_v the_o breadth_n a_o medial_a line_n namely_o ac_fw-la but_o the_o square_n of_o none_o of_o the_o foresay_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n a_o medial_a line_n again_o make_v perfect_v the_o parallelogram_n ed._n wherefore_o the_o parallelogram_n ed_z be_v also_o irrational_a by_o the_o say_v assumpt_v in_o the_o end_n of_o the_o 98._o his_o demonstration_n brieâ—ly_o prove_v and_o the_o line_n which_o contain_v it_o in_o power_n be_v irrational_o let_v the_o line_n which_o contain_v it_o in_o power_n be_v df._n wherefore_o df_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 the_o foresay_a irrational_a line_n for_o the_o square_n of_o none_o of_o the_o foresay_a irrational_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n the_o line_n cd_o wherefore_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 demonstrate_v ¶_o the_o 92._o theorem_a the_o 116._o proposition_n now_o let_v we_o prove_v that_o in_o square_a figure_n the_o diameter_n be_v incommensurable_a in_o length_n to_o the_o side_n svppose_v that_o abcd_o be_v a_o square_a and_o let_v the_o diameter_n thereof_o be_v ac_fw-la then_o i_o say_v that_o the_o diameter_n ac_fw-la be_v incommensurable_a in_o length_n to_o the_o side_n ab_fw-la for_o if_o it_o be_v possible_a impossibility_n let_v it_o be_v commensurable_a in_o length_n i_o say_v that_o they_o this_o will_v follow_v that_o one_o and_o the_o self_n same_o number_n shall_v be_v both_o a_o even_a number_n &_o a_o odd_a number_n it_o be_v manifest_a by_o the_o 47._o of_o the_o first_o that_o the_o square_a of_o the_o line_n ac_fw-la be_v double_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o for_o that_o the_o line_n ac_fw-la be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la by_o supposition_n therefore_o the_o line_n ac_fw-la have_v unto_o the_o line_n ab_fw-la that_o proportion_n that_o a_o number_n have_v to_o a_o number_n by_o the_o 5._o of_o the_o ten_o let_v the_o line_n ac_fw-la have_v unto_o the_o line_n ab_fw-la that_o proportion_n that_o the_o number_n of_o have_v to_o the_o number_n g._n and_o let_v of_o and_o g_z be_v the_o least_o number_n that_o have_v one_o and_o the_o same_o proportion_n with_o they_o wherefore_o of_o be_v not_o unity_n for_o if_o of_o be_v unity_n and_o it_o have_v to_o the_o number_n g_o that_o proportion_n that_o the_o line_n ac_fw-la have_v to_o the_o line_n ab_fw-la and_o the_o line_n ac_fw-la be_v great_a than_o the_o line_n ab_fw-la wherefore_o unity_n of_o be_v great_a than_o the_o number_n g_o which_o be_v impossible_a wherefore_o fe_o be_v not_o unity_n wherefore_o it_o be_v a_o number_n and_o for_o that_o as_o the_o square_n of_o the_o line_n ac_fw-la be_v to_o the_o square_n of_o the_o line_n ab_fw-la so_o be_v the_o square_a number_n of_o the_o number_n of_o to_o the_o square_a number_n of_o the_o number_n g_o for_o in_o each_o be_v the_o proportion_n of_o their_o side_n double_v by_o the_o corollary_n of_o the_o 20_o of_o the_o six_o and_o 11._o of_o the_o eight_o and_o the_o proportion_n of_o the_o line_n ac_fw-la to_o the_o line_n ab_fw-la double_v be_v equal_a to_o the_o proportion_n of_o the_o number_n of_o to_o the_o number_n g_o double_a for_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n ab_fw-la so_o be_v the_o number_n of_o to_o the_o number_n g._n but_o the_o square_n of_o the_o line_n ac_fw-la be_v double_a to_o the_o square_n of_o the_o line_n ab_fw-la wherefore_o the_o square_a number_n produce_v of_o the_o number_n of_o be_v double_a to_o the_o square_a number_n produce_v of_o the_o number_n g._n wherefore_o the_o square_a number_n produce_v of_o of_o be_v a_o even_a number_n wherefore_o of_o be_v also_o a_o even_a number_n for_o if_o of_o be_v a_o odd_a number_n the_o square_a number_n also_o produce_v of_o it_o shall_v by_o the_o 23._o and_o 29._o of_o the_o nine_o be_v a_o odd_a number_n for_o if_o odd_a number_n how_o many_o soever_o be_v add_v together_o and_o if_o the_o multitude_n of_o they_o be_v odd_a the_o whole_a also_o shall_v be_v odd_a wherefore_o of_o be_v a_o even_a number_n divide_v the_o number_n of_o into_o two_o equal_a part_n in_o h._n and_o forasmuch_o as_o the_o number_n of_o and_o g_z be_v the_o least_a number_n in_o that_o proportion_n therefore_o by_o the_o 24._o of_o the_o seven_o they_o be_v prime_a number_n the_o one_o to_o the_o other_o and_o of_o be_v a_o even_a number_n wherefore_o g_z be_v a_o odd_a number_n for_o if_o g_z be_v a_o even_a number_n the_o number_n two_z shall_v measure_v both_o the_o number_n of_o and_o the_o number_n g_z for_o every_o even_a number_n have_v a_o half_a part_n by_o the_o definition_n but_o these_o number_n of_o &_o g_o be_v prime_a the_o one_o to_o the_o other_o wherefore_o it_o be_v impossible_a that_o they_o shall_v be_v measure_v by_o two_o or_o by_o any_o other_o number_n beside_o unity_n wherefore_o g_z be_v a_o odd_a number_n and_o forasmuch_o as_o the_o number_n of_o be_v double_a to_o the_o number_n eh_o therefore_o the_o square_a number_n produce_v of_o of_o be_v quadruple_a to_o the_o square_a number_n produce_v of_o eh_o and_o the_o square_a number_n produce_v of_o of_o be_v double_a to_o the_o square_a number_n produce_v of_o g._n wherefore_o the_o square_a number_n produce_v of_o g_o be_v double_a to_o the_o square_a number_n produce_v of_o eh_o wherefore_o the_o square_a number_n produce_v of_o g_o be_v a_o even_a number_n wherefore_o also_o by_o those_o thing_n which_o have_v be_v before_o speak_v the_o number_n g_z be_v a_o even_a number_n but_o it_o be_v prove_v that_o it_o be_v a_o odd_a number_n which_o be_v impossible_a wherefore_o the_o line_n ac_fw-la be_v not_o commensurable_a in_o length_n to_o the_o line_n ab_fw-la wherefore_o it_o be_v incommensurable_a an_o other_o demonstration_n we_o may_v by_o a_o other_o demonstration_n prove_v that_o the_o diameter_n of_o a_o square_n be_v incommensurable_a to_o the_o side_n thereof_o impossibility_n suppose_v that_o there_o be_v a_o square_a who_o diameter_n let_v be_v a_o and_o let_v the_o side_n thereof_o be_v b._n then_o i_o say_v that_o the_o line_n a_o be_v incommensurable_a in_o length_n to_o the_o line_n b._n for_o if_o it_o be_v possible_a let_v it_o be_v commensurable_a in_o length_n and_o again_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o let_v the_o number_n of_o be_v to_o the_o number_n g_o and_o let_v they_o be_v the_o least_o that_o have_v one_o and_o the_o same_o proportion_n with_o they_o wherefore_o the_o number_n of_o and_o g_z be_v prime_a the_o one_o to_o the_o other_o first_o i_o say_v that_o g_z be_v not_o unity_n for_o if_o it_o be_v possible_a let_v it_o be_v unity_n and_o for_o that_o the_o square_a of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o as_o the_o square_a number_n produce_v of_o of_o be_v to_o the_o square_a number_n produce_v of_o g_z as_o it_o be_v prove_v in_o the_o â—ormer_a demonstration_n but_o the_o square_n of_o the_o line_n a_o be_v double_a to_o the_o square_n of_o the_o line_n b._n wherefore_o the_o square_a number_n produce_v of_o of_o be_v double_a to_o the_o square_a number_n produce_v of_o g._n and_o by_o your_o supposition_n g_z be_v unity_n wherefore_o the_o square_a number_n produce_v of_o of_o be_v the_o number_n two_o which_o be_v impossible_a wherefore_o g_z be_v not_o unity_n wherefore_o it_o be_v a_o number_n and_o for_o that_o as_o the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o so_o be_v the_o square_a number_n produce_v of_o of_o to_o the_o square_a number_n produce_v of_o g._n wherefore_o the_o square_a number_n produce_v of_o of_o be_v double_a to_o the_o square_a number_n produce_v of_o g._n wherefore_o the_o square_a number_n produce_v of_o g._n
at_o all_o adventure_n namely_o d_o five_o g_o s_o and_o a_o right_a line_n be_v draw_v from_o the_o point_n d_o to_o the_o point_n g_o and_o a_o other_o from_o the_o point_n five_o to_o the_o point_n s._n wherefore_o by_o the_o 7._o of_o the_o eleven_o the_o line_n dg_o and_o we_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n and_o forasmuch_o as_o the_o line_n de_fw-fr be_v a_o parallel_n to_o the_o line_n bg_o therefore_o by_o the_o 24._o of_o the_o first_o the_o angle_n edt_n be_v equal_a to_o the_o angle_n bgt_fw-mi for_o they_o be_v alternate_a angle_n and_o likewise_o the_o angle_n dtv_n be_v equal_a to_o the_o angle_n gts_fw-fr now_o then_o there_o be_v two_o triangle_n that_o be_v dtv_n and_o gts_fw-fr have_v two_o angle_n of_o the_o one_o equal_a to_o two_o angle_n of_o the_o other_o and_o one_o side_n of_o the_o one_o equal_a to_o one_o side_n of_o the_o other_o namely_o the_o side_n which_o subtend_v the_o equal_a angle_n that_o be_v the_o side_n dv_o to_o the_o side_n g_n for_o they_o be_v the_o half_n of_o the_o line_n de_fw-fr and_o bg_o wherefore_o the_o side_n remain_v be_v equal_a to_o the_o side_n remain_v wherefore_o the_o line_n dt_n be_v equal_a to_o the_o line_n tg_n and_o the_o line_n vt_fw-la to_o the_o line_n it_o be_v if_o therefore_o the_o opposite_a side_n of_o a_o parallelipipedon_n be_v divide_v into_o two_o equal_a part_n and_o by_o their_o section_n be_v extend_v plain_a superficiece_n the_o common_a section_n of_o those_o plain_a superficiece_n and_o the_o diameter_n of_o the_o parallelipipedon_n do_v divide_v the_o one_o the_o other_o into_o two_o equal_a part_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o flussas_n every_o plain_a superficies_n extend_v by_o the_o centre_n of_o a_o parallelipipedon_n divide_v that_o solid_a into_o two_o equal_a part_n and_o so_o do_v not_o any_o other_o plain_a superficies_n not_o extend_v by_o the_o centre_n for_o every_o plain_n extend_v by_o the_o centre_n cut_v the_o diameter_n of_o the_o parallelipipedon_n in_o the_o centre_n into_o two_o equal_a part_n for_o it_o be_v prove_v that_o plain_a superficiece_n which_o cut_v the_o solid_a into_o two_o equal_a part_n do_v cut_v the_o dimetient_fw-la into_o two_o equal_a part_n in_o the_o centre_n wherefore_o all_o the_o line_n draw_v by_o the_o centre_n in_o that_o plain_a superficies_n shall_v make_v angle_n with_o the_o dimetient_fw-la and_o forasmuch_o as_o the_o diameter_n fall_v upon_o the_o parallel_n right_a line_n of_o the_o solid_a which_o describe_v the_o opposite_a side_n of_o the_o say_v solid_a or_o upon_o the_o parallel_n plain_a superficiece_n of_o the_o solid_a which_o make_v angel_n at_o the_o end_n of_o the_o diameter_n the_o triangle_n contain_v under_o the_o diameter_n and_o the_o right_a line_n extend_v in_o that_o plain_n by_o the_o centre_n and_o the_o right_a line_n which_o be_v draw_v in_o the_o opposite_a superficiece_n of_o the_o solid_a join_v together_o the_o end_n of_o the_o foresay_a right_a line_n namely_o the_o end_n of_o the_o diameter_n and_o the_o end_n of_o the_o line_n draw_v by_o the_o centre_n in_o the_o superficies_n extend_v by_o the_o centre_n shall_v always_o be_v equal_a and_o equiangle_n by_o the_o 26._o of_o the_o first_o for_o the_o opposite_a right_a line_n draw_v by_o the_o opposite_a plain_a superficiece_n of_o the_o solid_a do_v make_v equal_a angle_n with_o the_o diameter_n forasmuch_o as_o they_o be_v parallel_a line_n by_o the_o 16._o of_o this_o book_n but_o the_o angle_n at_o the_o centre_n be_v equal_a by_o the_o 15._o of_o the_o first_o for_o they_o be_v head_n angle_n &_o one_o side_n be_v equal_a to_o one_o side_n namely_o half_o the_o dimetient_fw-la wherefore_o the_o triangle_n contain_v under_o every_o right_a line_n draw_v by_o the_o centre_n of_o the_o parallelipipedon_n in_o the_o superficies_n which_o be_v extend_v also_o by_o the_o say_a centre_n and_o the_o diameter_n thereof_o who_o end_n be_v the_o angle_n of_o the_o solid_a be_v equal_a equilater_n &_o equiangle_n by_o the_o 26._o of_o the_o first_o wherefore_o it_o follow_v that_o the_o plain_a superficies_n which_o cut_v the_o parallelipipedon_n do_v make_v the_o part_n of_o the_o base_n on_o the_o opposite_a side_n equal_a and_o equiangle_n and_o therefore_o like_a and_o equal_a both_o in_o multitude_n and_o in_o magnitude_n wherefore_o the_o two_o solid_a section_n of_o that_o solid_a shall_v be_v equal_a and_o like_a by_o the_o 8._o definition_n of_o this_o book_n and_o now_o that_o no_o other_o plain_a superficies_n beside_o that_o which_o be_v extend_v by_o the_o centre_n divide_v the_o parallelipipedon_n into_o two_o equal_a part_n it_o be_v manifest_a if_o unto_o the_o plain_a superficies_n which_o be_v not_o extend_v by_o the_o centre_n we_o extend_v by_o the_o centre_n a_o parallel_n plain_a superficies_n by_o the_o corollary_n of_o the_o 15._o of_o this_o book_n for_o forasmuch_o as_o that_o superficies_n which_o be_v extend_v by_o the_o centre_n do_v divide_v the_o parallelipipedom_n into_o two_o equal_a parâ—â—_n it_o be_v manifest_a that_o the_o other_o plain_a superficies_n which_o be_v parallel_n to_o the_o superficies_n which_o divide_v the_o solid_a into_o two_o equal_a part_n be_v in_o one_o of_o the_o equal_a part_n of_o the_o solid_a wherefore_o see_v that_o the_o whole_a be_v ever_o great_a than_o his_o part_n it_o must_v needs_o be_v that_o one_o of_o these_o section_n be_v less_o than_o the_o half_a of_o the_o solid_a and_o therefore_o the_o other_o be_v great_a for_o the_o better_a understanding_n of_o this_o former_a proposition_n &_o also_o of_o this_o corollary_n add_v by_o flussas_n it_o shall_v be_v very_o needful_a for_o you_o to_o describe_v of_o past_v paper_n or_o such_o like_a matter_n a_o parallelipipedom_n or_o a_o cube_n and_o to_o divide_v all_o the_o parallelogram_n thereof_o into_o two_o equal_a part_n by_o draw_v by_o the_o centre_n of_o the_o say_a parallelogram_n which_o centre_n be_v the_o point_n make_v by_o the_o cut_n of_o diagonal_a line_n draw_v from_o thâ—_z opposite_a angle_n of_o the_o say_a parallelogram_n line_n parallel_n to_o the_o side_n of_o the_o parallelogram_n as_o in_o the_o former_a figure_n describe_v in_o a_o plain_a you_o may_v see_v be_v the_o six_o parallelogram_n de_fw-fr eh_o ha_o ad_fw-la dh_o and_o cg_o who_o these_o parallel_a line_n draw_v by_o the_o centre_n of_o the_o say_a parallelogram_n namely_o xo_o or_o pr._n and_o px_n do_v divide_v into_o two_o equal_a part_n by_o which_o four_o line_n you_o must_v imagine_v a_o plain_a superficies_n to_o be_v extend_v also_o these_o parallel_a line_n kl_o ln_o nm_o and_o mâ—_n by_o which_o four_o line_n likewise_o yâ—_n must_v imagine_v a_o plain_a superficies_n to_o be_v extend_v you_o may_v if_o you_o will_v put_v within_o your_o body_n make_v thus_o of_o past_v paper_n two_o superficiece_n make_v also_o of_o the_o say_a paper_n have_v to_o their_o limit_n line_n equal_a to_o the_o foresay_a parallel_n line_n which_o superficiece_n must_v also_o be_v divide_v into_o two_o equal_a part_n by_o parallel_a line_n draw_v by_o their_o centre_n and_o must_v cut_v the_o one_o the_o other_o by_o these_o parallel_a line_n and_o for_o the_o diameter_n of_o this_o body_n extend_v a_o thread_n from_o one_o angle_n in_o the_o base_a of_o the_o solid_a to_o his_o opposite_a angle_n which_o shall_v pass_v by_o the_o centre_n of_o the_o parallelipipedon_n as_o do_v the_o line_n dg_o in_o the_o figure_n before_o describe_v in_o the_o plain_n and_o draw_v in_o the_o base_a and_o the_o opposite_a superficies_n unto_o it_o diagonal_a line_n from_o the_o angle_n from_o which_o be_v extend_v the_o diameter_n of_o the_o solid_a as_o in_o the_o former_a description_n be_v the_o line_n bg_o and_o de._n and_o when_o you_o have_v thus_o describe_v this_o body_n compare_v it_o with_o the_o former_a demonstration_n and_o it_o will_v make_v it_o very_o plain_a unto_o you_o so_o your_o letter_n agree_v with_o the_o letter_n of_o the_o figure_n describe_v in_o the_o book_n and_o this_o description_n will_v plain_o set_v forth_o unto_o you_o the_o corollary_n follow_v that_o proposition_n for_o where_o as_o to_o the_o understanding_n of_o the_o demonstration_n of_o the_o proposition_n the_o superficiece_n put_v within_o the_o body_n be_v extend_v by_o parallel_a line_n draw_v by_o the_o centre_n of_o the_o base_n of_o the_o parallelipipedon_n to_o the_o understanding_n of_o the_o say_a corollary_n you_o may_v extend_v a_o superficies_n by_o any_o other_o line_n draw_v in_o the_o say_a base_n so_o that_o yet_o it_o pass_v through_o the_o midst_n of_o the_o thread_n which_o be_v suppose_v to_o be_v the_o centre_n of_o the_o parallelipipedon_n ¶_o the_o 35._o theorem_a the_o 40._o proposition_n if_o there_o be_v
superficies_n or_o solidity_n in_o the_o hole_n or_o in_o partâ—_n such_o certain_a knowledge_n demonstrative_a may_v arise_v and_o such_o mechanical_a exercise_n thereby_o be_v devise_v that_o sure_a i_o be_o to_o the_o sincere_a &_o true_a student_n great_a light_n aid_n and_o comfortable_a courage_n far_a to_o wade_v will_v enter_v into_o his_o heart_n and_o to_o the_o mechanical_a witty_a and_o industrous_a deviser_n new_a manner_n of_o invention_n &_o execution_n in_o his_o work_n will_v with_o small_a travail_n for_o foot_n application_n come_v to_o his_o perceiveraunce_n and_o understanding_n therefore_o even_o a_o manifold_a speculation_n &_o practice_n may_v be_v have_v with_o the_o circle_n his_o quantity_n be_v not_o know_v in_o any_o kind_n of_o small_a certain_a measure_n so_o likewise_o of_o the_o sphere_n many_o problem_n may_v be_v execute_v and_o his_o precise_a quantity_n in_o certain_a measure_n not_o determine_v or_o know_v yet_o because_o both_o one_o of_o the_o first_o humane_a occasion_n of_o invent_v and_o stablish_v this_o art_n be_v measure_v of_o the_o earth_n and_o therefore_o call_v geometria_n that_o be_v earthmeasure_v and_o also_o the_o chief_a and_o general_a end_n in_o deed_n be_v measure_n and_o measure_n require_v a_o determination_n of_o quantity_n in_o a_o certain_a measure_n by_o number_n express_v it_o be_v needful_a for_o mechanical_a earthmeasure_n not_o to_o be_v ignorant_a of_o the_o measure_n and_o content_n of_o the_o circle_n neither_o of_o the_o sphere_n his_o measure_n and_o quantity_n as_o near_o as_o sense_n can_v imagine_v or_o wish_v and_o in_o very_a deed_n the_o quantity_n and_o measure_n of_o the_o circle_n be_v know_v make_v not_o only_o the_o cone_n and_o cylinder_n but_o also_o the_o sphere_n his_o quantity_n to_o be_v as_o precise_o know_v and_o certain_a therefore_o see_v in_o respect_n of_o the_o circle_n quantity_n by_o archimedes_n specify_v this_o theorem_a be_v note_v unto_o you_o i_o will_v by_o order_n upon_o that_o as_o a_o supposition_n infer_v the_o conclusion_n of_o this_o our_o theorem_n note_n 1._o wherefore_o if_o you_o divide_v the_o one_o side_n as_o tq_n of_o the_o cube_fw-la tx_n into_o 21._o equal_a part_n and_o where_o 11._o part_n do_v end_n reckon_v from_o t_o suppose_v the_o point_n p_o and_o by_o that_o point_n p_o imagine_v a_o plain_a pass_v parallel_n to_o the_o opposite_a base_n to_o cut_v the_o cube_fw-la tx_n and_o thereby_o the_o cube_fw-la tx_n to_o be_v divide_v into_o two_o rectangle_n parallelipipedon_n namely_o tn_n and_o px_n it_o be_v manifest_a give_v tn_n to_o be_v equal_a to_o the_o sphere_n a_o by_o construction_n and_o the_o 7._o of_o the_o five_o note_n 2._o second_o the_o whole_a quantity_n of_o the_o sphere_n a_o assign_v be_v contain_v in_o the_o rectangle_n parallelipipedon_n tn_n you_o may_v easy_o transform_v the_o same_o quantity_n into_o other_o parallelipipedon_n rectangle_v of_o what_o height_n and_o of_o what_o parallelogram_n base_a you_o listen_v by_o my_o first_o and_o second_o problem_n upon_o the_o 34._o of_o this_o book_n and_o the_o like_a may_v you_o do_v to_o any_o assign_a part_n of_o the_o sphere_n a_o by_o the_o like_a mean_n devide_v the_o parallelipipedon_n tn_n as_o the_o part_n assign_v do_v require_v as_o if_o a_o three_o four_o five_o or_o six_o part_n of_o the_o sphere_n a_o be_v to_o be_v have_v in_o a_o parallelipipedon_n of_o any_o parallelograâ—â—e_n base_a assign_v or_o of_o any_o heith_n assign_v then_o devide_v tp_n into_o so_o many_o part_n as_o into_o 4._o if_o a_o four_o part_n be_v to_o be_v transform_v or_o into_o five_o if_o a_o five_o part_n be_v to_o be_v transform_v etc_n etc_n and_o then_o proceed_v â—s_v you_o do_v with_o cut_v of_o tn_n from_o tx_n and_o that_o i_o say_v of_o parallelipipedon_n may_v in_o like_a sort_n by_o my_o â—â—yd_n two_o problem_n add_v to_o the_o 34._o of_o this_o book_n be_v do_v in_o any_o side_v column_n pyramid_n and_o prismeâ—_n so_o thâ—â—_n in_o pyramid_n and_o some_o prism_n you_o use_v the_o caution_n necessary_a in_o respect_n of_o their_o quan_fw-mi 〈…〉_o odyes_n have_v parallel_n equal_a and_o opposite_a base_n who_o part_n 〈…〉_o re_fw-mi in_o their_o proposition_n be_v by_o euclid_n demonstrate_v and_o final_o 〈…〉_o addition_n you_o have_v the_o way_n and_o order_n how_o to_o geve_v to_o a_o sphere_n or_o any_o segment_n oâ—_n the_o same_o cones_n or_o cylinder_n equal_a or_o in_o any_o proportion_n between_o two_o right_a line_n give_v with_o many_o other_o most_o necessary_a speculation_n and_o practice_n about_o the_o sphere_n i_o trust_v that_o i_o have_v sufficient_o â—raughted_v your_o imagination_n for_o your_o honest_a and_o profitable_a study_n herein_o and_o also_o give_v you_o reaâ—â—_n â—â—tter_n wheâ—â—_n with_o to_o sâ—â—p_v the_o mouth_n of_o the_o malicious_a ignorant_a and_o arrogant_a despiser_n of_o the_o most_o excellent_a discourse_n travail_n and_o invention_n mathematical_a sting_v aswell_o the_o heavenly_a sphere_n &_o star_n their_o spherical_a solidity_n geometry_n with_o their_o convene_n spherical_a superficies_n to_o the_o earth_n at_o all_o time_n respect_v and_o their_o distance_n from_o the_o earth_n as_o also_o the_o whole_a earthly_a sphere_n and_o globe_n itself_o and_o infinite_a other_o case_n concern_v sphere_n or_o globe_n may_v hereby_o with_o as_o much_o ease_n and_o certainty_n be_v determine_v of_o as_o of_o the_o quantity_n of_o any_o bowl_n ball_n or_o bullet_n which_o we_o may_v gripe_v in_o our_o hand_n reason_n and_o experience_n be_v our_o witness_n and_o without_o these_o aid_n such_o thing_n of_o importance_n never_o able_a of_o we_o certain_o to_o be_v know_v or_o attain_a unto_o here_o end_n m._n john_n do_v you_o his_o addition_n upon_o the_o last_o proposition_n of_o the_o twelve_o book_n a_o proposition_n add_v by_o flussas_n if_o a_o sphere_n touch_v a_o plain_a superficiesâ—_n a_o right_a line_n draw_v from_o the_o centre_n to_o the_o touch_n shall_v be_v erect_v perpendicular_o to_o the_o plain_a superficies_n suppose_v that_o there_o be_v a_o sphere_n bcdl_n who_o centre_n let_v be_v the_o point_n a._n and_o let_v the_o plain_a superficies_n gci_n touch_v the_o spear_n in_o the_o point_n c_o and_o extend_v a_o right_a line_n from_o the_o centre_n a_o to_o the_o point_n c._n then_o i_o say_v that_o the_o line_n ac_fw-la be_v erect_v perpendicular_o to_o tâ—e_z plain_a gic._n let_v the_o sphere_n be_v cut_v by_o plain_a superficiece_n pass_v by_o the_o right_a line_n lac_n which_o plain_n let_v be_v abcdl_n and_o acel_n which_o let_v cut_v the_o plain_n gci_n by_o the_o right_a line_n gch_o and_o kci_fw-la now_o it_o be_v manifest_a by_o the_o assumpt_n put_v before_o the_o 17._o of_o this_o book_n that_o the_o two_o section_n of_o the_o sphere_n shall_v be_v circle_n have_v to_o their_o diameter_n the_o line_n lac_n which_o be_v also_o the_o diameter_n of_o the_o sphere_n wherefore_o the_o right_a line_n gch_o and_o kci_fw-la which_o be_v draw_v in_o the_o plain_n gci_n do_v at_o the_o point_n c_o fall_n without_o the_o circle_n bcdl_n and_o ecl._n wherefore_o they_o touch_v the_o circle_n in_o the_o point_n c_o by_o the_o second_o definition_n of_o the_o three_o wherefore_o the_o right_a line_n lac_n make_v right_a angle_n with_o the_o line_n gch_v and_o kci_fw-la by_o the_o 16._o of_o the_o three_o wherefore_o by_o the_o 4._o of_o the_o eleven_o the_o right_a line_n ac_fw-la be_v erect_v perpendicular_o to_o to_o the_o plain_a superficies_n gci_n wherein_o be_v draw_v the_o line_n gch_v and_o kci_fw-la if_o therefore_o a_o sphere_n touch_v a_o plain_a superficies_n a_o right_a line_n draw_v from_o the_o centre_n to_o the_o touch_n shall_v be_v erect_v perpendicular_o to_o the_o plain_a superficies_n which_o be_v require_v to_o be_v prove_v the_o end_n of_o the_o twelve_o book_n of_o euclides_n element_n ¶_o the_o thirteen_o book_n of_o euclides_n element_n in_o this_o thirteen_o book_n be_v set_v forth_o certain_a most_o wonderful_a and_o excellent_a passion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n book_n a_o matter_n undoubted_o of_o great_a and_o infinite_a use_n in_o geometry_n as_o you_o shall_v both_o in_o this_o book_n and_o in_o the_o other_o book_n follow_v most_o evident_o perceive_v it_o teach_v moreover_o the_o composition_n of_o the_o five_o regular_a solid_n and_o how_o to_o inscribe_v they_o in_o a_o sphere_n give_v and_o also_o set_v forth_o certain_a comparison_n of_o the_o say_a body_n both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o sphere_n wherein_o they_o be_v describe_v the_o 1._o theorem_a the_o 1._o proposition_n if_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o to_o the_o great_a segment_n be_v add_v the_o half_a of_o the_o whole_a line_n the_o square_n make_v of_o those_o two_o
and_o mean_a proportion_n but_o in_o one_o only_a point_n which_o be_v requisite_a to_o be_v demonstrate_v a_o theorem_a 2._o what_o right_a line_n so_o ever_o be_v divide_v into_o two_o part_n have_v those_o his_o two_o part_n proportional_a to_o the_o two_o segment_n of_o a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n be_v also_o itself_o divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o those_o his_o two_o part_n be_v his_o two_o segment_n of_o the_o say_a proportion_n suppose_v ab_fw-la to_o be_v a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c_o and_o ac_fw-la to_o be_v the_o great_a segment_n suppose_v also_o the_o right_a line_n de_fw-fr to_o be_v divide_v into_o two_o part_n in_o the_o point_n f_o and_o that_o the_o part_n df_o be_v to_o fe_o as_o the_o segment_n ac_fw-la be_v to_o cb_o or_o df_o to_o be_v to_o ac_fw-la as_o fe_o be_v to_o cb._n for_o so_o these_o payte_n be_v proportional_a to_o the_o say_a segment_n i_o say_v now_o that_o de_fw-fr be_v also_o divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n f._n and_o that_o df_o fe_o be_v his_o segment_n of_o the_o say_a proportion_n for_o see_v as_o ac_fw-la be_v to_o cb_o so_o be_v df_o to_o fe_o by_o supposition_n therefore_o as_o ac_fw-la and_o cb_o which_o be_v ab_fw-la be_v to_o cb_o so_o be_v df_o and_o fe_o which_o be_v de_fw-fr to_o fe_o by_o the_o 18._o of_o the_o five_o wherefore_o alternate_o as_o ab_fw-la be_v to_o de_fw-fr so_o be_v cb_o to_o fe_o and_o therefore_o the_o residue_n ac_fw-la be_v to_o the_o residue_n df_o as_o ab_fw-la be_v to_o de_fw-fr by_o the_o five_o of_o the_o five_o and_o then_o alternate_o ac_fw-la be_v to_o ab_fw-la as_o de_fw-fr be_v to_o df._n now_o therefore_o backward_o ab_fw-la be_v to_o ac_fw-la as_o de_fw-fr be_v to_o df._n but_o as_o ab_fw-la be_v to_o ac_fw-la so_o be_v ac_fw-la to_o cb_o by_o the_o three_o definition_n of_o the_o six_o book_n wherefore_o de_fw-fr be_v to_o df_o as_o ac_fw-la be_v to_o cb_o by_o the_o 11._o of_o the_o five_o and_o by_o supposition_n as_o ac_fw-la be_v to_o cb_o so_o be_v df_o to_o fe_o wherefore_o by_o the_o 11._o of_o the_o five_o as_z de_fw-fr be_v to_o df_o so_o be_v df_o to_o fe_o wherefore_o by_o the_o 3._o definition_n of_o the_o six_o de_fw-fr be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n f._n wherefore_o df_o and_o fe_o be_v the_o segment_n of_o the_o say_a proportion_n therefore_o what_o right_a line_n so_o ever_o be_v divide_v into_o two_o part_n have_v those_o his_o two_o part_n proportional_a to_o the_o two_o segment_n of_o a_o line_n divide_v by_o extreme_a and_o mean_a proportioned_a be_v also_o itself_o divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o those_o his_o two_o part_n be_v his_o two_o segment_n of_o the_o say_v proportion_v which_o be_v requisite_a to_o be_v demonstrate_v note_n many_o way_n these_o two_o theorem_n may_v be_v demonstrate_v which_o i_o leave_v to_o the_o exercise_n of_o young_a student_n but_o utter_o to_o want_v these_o two_o theorem_n and_o their_o demonstration_n in_o so_o principal_a a_o line_n or_o rather_o the_o chief_a pillar_n of_o euclides_n geometrical_a palace_n geometry_n be_v hitherto_o and_o so_o will_v remain_v a_o great_a disgrace_n also_o i_o think_v it_o good_a to_o note_v unto_o you_o what_o we_o mean_v by_o one_o only_a point_n we_o mâ—â—â—â—_n that_o the_o quantity_n of_o the_o two_o segment_n can_v not_o be_v alter_v the_o whole_a line_n be_v once_o give_v and_o though_o from_o either_o end_n of_o the_o whole_a line_n the_o great_a segment_n may_v begin_v poiâ—t_n and_o so_o as_o it_o be_v the_o point_n of_o section_n may_v seem_v to_o be_v alter_v yet_o with_o we_o that_o be_v no_o alteration_n forasmuch_o as_o the_o quantity_n of_o the_o segment_n remain_v all_o one_o i_o mean_v the_o quantity_n of_o the_o great_a segment_n be_v all_o one_o at_o which_o end_n so_o ever_o it_o be_v take_v and_o therefore_o likewise_o the_o quantity_n of_o the_o less_o segment_n be_v all_o one_o etc_n etc_n the_o like_a consideration_n may_v be_v have_v in_o euclides_n ten_o book_n in_o the_o binomial_a line_n etc_n etc_n joâ—n_n dee_n 1569._o decemb._n 18._o the_o 3._o theorem_a the_o 3._o proposition_n if_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o to_o the_o less_o segment_n be_v add_v the_o half_a of_o the_o gerater_n segment_n the_o square_n make_v of_o those_o two_o line_n add_v together_o be_v quintuple_a to_o the_o square_n make_v of_o the_o half_a line_n of_o the_o great_a segment_n svppose_v that_o the_o right_a line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c._n and_o let_v the_o great_a segment_n thereof_o be_v ac_fw-la and_o divide_v ac_fw-la into_o two_o equal_a part_n in_o the_o point_n d._n then_o i_o say_v that_o the_o square_a of_o the_o line_n bd_o it_o quintuple_a to_o the_o square_n of_o the_o line_n dc_o construction_n describe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n ab_fw-la a_o square_a ae_n and_o describe_v and_o make_v perfect_a the_o figure_n that_o be_v divide_v the_o line_n at_o like_a unto_o the_o division_n of_o the_o line_n ab_fw-la by_o the_o 10._o of_o the_o six_o in_o the_o point_n r_o h_o by_o which_o point_n draw_v by_o the_o 31._o of_o the_o first_o unto_o the_o line_n ab_fw-la parallel_a line_n rm_n and_o hn._n so_o likewise_o draw_v by_o the_o point_n d_o c_o unto_o the_o line_n be_v these_o parallel_a line_n dl_o and_o c_n &_o draw_v the_o diameter_n bt_n demonstration_n and_o forasmuch_o as_o the_o line_n ac_fw-la be_v double_a to_o the_o line_n dc_o therefore_o the_o square_a of_o ac_fw-la be_v quadruple_a to_o the_o square_n of_o dc_o by_o the_o 20._o of_o the_o six_o that_o be_v the_o square_a r_n to_o the_o square_a fg_o and_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o parallelogram_n ce_fw-fr &_o the_o square_a of_o the_o line_n ac_fw-la be_v the_o square_a r_n wherefore_o the_o parallelogram_n ce_fw-fr be_v equal_a to_o the_o square_a rs._n but_o the_o square_a r_n be_v quadruple_a to_o the_o square_a fg_o wherefore_o the_o parallelogram_n ce_fw-fr also_o be_v quadruple_a to_o the_o square_a fg._n again_o forasmuch_o as_o the_o line_n ad_fw-la be_v equal_a to_o the_o line_n dc_o therefore_o the_o line_n hk_o be_v equal_a the_o line_n kf_o wherefore_o also_o the_o square_a gf_o be_v equal_a to_o the_o square_a hl_o wherefore_o the_o line_n gk_o be_v equal_a to_o the_o line_n kl_o that_o be_v the_o line_n mn_v to_o the_o line_n ne_fw-fr wherefore_o the_o parallelogram_n mf_n be_v equal_a to_o the_o parallelogram_n fe_o but_o the_o parallelogram_n mf_n be_v equal_a to_o the_o parallelogram_n cg_o wherefore_o the_o parallelogram_n cg_o be_v also_o equal_a to_o the_o parallelogram_n fe_o put_v the_o parallelogram_n cn_fw-la common_a to_o they_o both_o after_o wherefore_o the_o gnomon_n xop_n be_v equal_a to_o the_o parallelogram_n ck_o but_o the_o parallelogram_n ce_fw-fr be_v prove_v to_o be_v quadruple_a to_o gâ—_n the_o square_a wherefore_o the_o gnomon_n xop_n be_v quadruple_a to_o the_o square_a gf_o wherefore_o the_o square_a dn_o be_v quintuple_a to_o the_o square_a fg_o and_o dn_o be_v the_o square_a of_o the_o line_n db_o and_o gf_o the_o square_a of_o the_o line_n dc_o wherefore_o the_o square_a of_o the_o line_n db_o be_v quintuple_a to_o the_o square_n of_o the_o line_n dc_o if_o therefore_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o to_o the_o less_o segment_n be_v add_v the_o half_a of_o the_o great_a segment_n the_o square_n make_v of_o those_o two_o line_n add_v together_o be_v quintuple_a to_o the_o square_n make_v of_o the_o half_a line_n of_o the_o great_a segment_n which_o be_v require_v to_o be_v demonstrate_v you_o shall_v find_v this_o proposition_n a_o other_o way_n demonstrate_v after_o the_o five_o proposition_n of_o this_o book_n here_o follow_v m._n dee_n his_o addition_n ¶_o a_o theorem_a 1._o if_o a_o right_a line_n give_v be_v quintuple_a in_o power_n to_o the_o power_n of_o a_o segment_n of_o himself_o the_o double_a of_o that_o segment_n former_a and_o the_o other_o part_n remain_v of_o the_o first_o give_v line_n make_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n and_o that_o double_a of_o the_o segment_n be_v the_o great_a part_n thereof_o forasmuch_o as_o this_o be_v the_o converse_n of_o euclides_n
line_n thereby_o adjoin_v let_v be_v bq_n i_o say_v that_o cq_n be_v a_o line_n also_o divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n b_o and_o that_o bq_n the_o line_n adjoin_v be_v the_o less_o segment_n for_o by_o the_o third_o it_o be_v prove_v that_o half_a ac_fw-la which_o let_v be_v cd_o with_o cb_o as_o one_o line_n compose_v have_v his_o power_n or_o square_v quintuple_a to_o the_o power_n of_o the_o segment_n cd_o wherefore_o by_o the_o second_o of_o this_o book_n the_o double_a of_o c_o d_o be_v divide_v by_o extreme_a and_o middle_a proportioned_a and_o the_o great_a segment_n thereof_o shall_v be_v cb._n but_o by_o construction_n cq_n be_v the_o double_a of_o cd_o for_o it_o be_v equal_a to_o ac_fw-la wherefore_o cq_n be_v divide_v by_o extreme_a and_o middle_a proportion_n in_o the_o point_n b_o and_o the_o great_a segment_n thereof_o shall_v be_v cb._n wherefore_o bq_n be_v the_o less_o segment_n which_o be_v the_o line_n adjoin_v therefore_o a_o line_n be_v divide_v by_o extreme_a and_o middle_a proportion_n if_o the_o less_o segment_n be_v produce_v equal_o to_o the_o length_n of_o the_o great_a segment_n the_o line_n thereby_o adjoin_v together_o with_o the_o say_v less_o segment_n make_v a_o new_a line_n divide_v by_o extreme_a &_o mean_a proportion_n whoâ—e_v less_o segment_n be_v the_o line_n adjoin_v which_o be_v to_o be_v demonstrate_v ¶_o a_o corollary_n 2._o ifâ—_n from_o the_o great_a segment_n of_o a_o line_n divide_v by_o extreme_a and_o middle_a proportion_n a_o line_n equal_a to_o the_o less_o segment_n be_v cut_v of_o the_o great_a segment_n thereby_o be_v also_o divide_v by_o extreme_a and_o mean_a proportion_n who_o great_a segmentâ—_n shall_v be_v 〈◊〉_d that_o part_n of_o it_o which_o be_v cut_v of_o for_o take_v from_o ac_fw-la a_o line_n equal_a to_o cb_o let_v be_v remain_v i_o say_v that_o ac_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n r_o and_o that_o cr_n the_o line_n cut_v of_o be_v the_o great_a segment_n for_o it_o be_v prove_v in_o the_o former_a corollary_n that_o cq_n be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n b._n but_o ac_fw-la be_v equal_a to_o cq_n by_o construction_n and_o cr_n be_v equal_a to_o cb_o by_o construction_n wherefore_o the_o residue_n ar_n be_v equal_a to_o bq_n the_o residue_n see_v therefore_o the_o whole_a ac_fw-la be_v equal_a to_o the_o whole_a cq_n and_o the_o great_a part_n of_o ac_fw-la which_o be_v cr_n be_v equal_a to_o cb_o the_o great_a part_n of_o cq_n and_o the_o less_o segment_n also_o equal_a to_o the_o less_o and_o withal_o see_v cq_n be_v prove_v to_o be_v divide_v by_o extreme_a &_o mean_a proportion_n in_o the_o point_n b_o it_o follow_v of_o necessity_n that_o ac_fw-la be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n r._n and_o see_v cb_o be_v the_o great_a segment_n of_o cq_n cr_n shall_v be_v the_o great_a segment_n of_o ac_fw-la which_o be_v to_o be_v demonstrate_v a_o corollary_n 3._o it_o be_v evident_a thereby_o a_o line_n be_v divide_v by_o extreme_a and_o mean_a proportion_n that_o the_o line_n whereby_o the_o great_a segment_n exceed_v the_o less_o together_o with_o the_o less_o segment_n do_v make_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n who_o less_o segment_n be_v the_o say_a line_n of_o exceesse_n or_o difference_n between_o the_o segment_n john_n dee_n ¶_o two_o new_a way_n to_o divide_v any_o right_a line_n give_v by_o a_o extreme_a and_o mean_a proportion_n demonstrate_v and_o add_v by_o m._n dee_n a_o problem_n to_o divide_v by_o a_o extreme_a and_o mean_a proportion_n any_o right_a line_n give_v in_o length_n and_o position_n suppose_v a_o line_n give_v in_o length_n and_o position_n to_o be_v ab_fw-la i_o say_v that_o ab_fw-la be_v to_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n construction_n divide_v ab_fw-la into_o two_o equal_a part_n as_o in_o the_o point_n c._n produce_v ab_fw-la direct_o from_o the_o point_n b_o to_o the_o point_n d_o make_v bd_o equal_a to_o bc._n to_o the_o line_n ad_fw-la and_o at_o the_o point_n d_o let_v a_o line_n be_v draw_v etc_n perpendicular_a by_o the_o 11._o of_o the_o first_o which_o let_v be_v df_o of_o what_o length_n you_o will_v from_o df_o and_o at_o the_o point_n d_o cut_v of_o the_o six_o part_n of_o df_o by_o the_o 9_o of_o the_o six_o and_o let_v that_o six_o part_n be_v the_o line_n dg_o upon_o df_o as_o a_o diameter_n describe_v a_o semicircle_n which_o let_v be_v dhf_n from_o the_o point_n g_o rear_v a_o line_n perpendicular_a to_o df_o which_o suppose_v to_o be_v gh_o and_o let_v it_o come_v to_o the_o circumference_n of_o dhf_n in_o the_o point_n h._n draw_v right_a line_n hd_v and_o hf._n produce_v dh_o from_o the_o point_n h_o so_o long_o till_o a_o line_n adjoin_v with_o dh_o be_v equal_a to_o hf_o which_o let_v be_v di_fw-it equal_a to_o hf._n from_o the_o point_n h_o to_o the_o point_n b_o the_o one_o end_n of_o our_o line_n give_v let_v a_o right_a line_n be_v draw_v as_o hb_o from_o the_o point_n i_o let_v a_o line_n be_v draw_v to_o the_o line_n ab_fw-la so_o that_o it_o be_v also_o parallel_a to_o the_o line_n hb_o which_o parallel_n line_n suppose_v to_o be_v ik_fw-mi cut_v the_o line_n ab_fw-la at_o the_o point_n k._n i_o say_v that_o ab_fw-la be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n in_o the_o point_n k._n for_o the_o triangle_n dki_n have_v hb_o parallel_n to_o ik_fw-mi have_v his_o side_n dk_o and_o diego_n demonstration_n cut_v proportional_o by_o the_o 2._o of_o the_o six_o wherefore_o as_o ih_v be_v to_o hd_v so_o be_v kb_o to_o bd._n and_o therefore_o compound_o by_o the_o 18._o of_o the_o five_o as_o diego_n be_v to_o dh_o so_o be_v dk_o to_o db._n but_o by_o construction_n diego_n be_v equal_a to_o hf_o wherefore_o by_o the_o 7._o of_o the_o five_o di_z be_v to_o dh_o as_o hf_o be_v to_o dh_o wherefore_o by_o the_o 11._o of_o the_o five_o dk_o be_v to_o db_o as_o hf_o be_v to_o dh_o wherefore_o the_o square_a of_o dk_o be_v to_o the_o square_n of_o db_o as_o the_o square_n of_o hf_o be_v to_o the_o square_n of_o dh_o by_o the_o 22._o of_o the_o six_o but_o the_o square_n of_o hf_o be_v to_o the_o square_n of_o dh_o as_o the_o line_n gf_o be_v to_o the_o line_n gdâ—_n by_o my_o corollary_n upon_o the_o 5_o problem_n of_o my_o addition_n to_o the_o second_o proposition_n of_o the_o twelve_o wherefore_o by_o the_o 11._o of_o the_o five_o the_o square_a of_o dk_o be_v to_o the_o square_n of_o db_o as_o the_o line_n gf_o be_v to_o the_o line_n gd_o but_o by_o construction_n gf_o be_v quintuple_a to_o gd_o wherefore_o the_o square_a of_o dk_o be_v quintuple_a to_o the_o square_n of_o db_o and_o therefore_o the_o double_a of_o db_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o bk_o be_v the_o great_a segment_n thereof_o by_o the_o 2._o of_o this_o thirteen_o wherefore_o see_v ab_fw-la be_v the_o double_a of_o db_o by_o construction_n the_o line_n ab_fw-la be_v divide_v 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 bk_o wherefore_o ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n k._n we_o have_v therefore_o divide_v by_o extreme_a and_o mean_a proportion_n any_o line_n give_v in_o length_n and_o position_n which_o be_v requisite_a to_o be_v do_v the_o second_o way_n to_o execute_v this_o problem_n suppose_v the_o line_n give_v to_o be_v ab_fw-la divide_v aâ—_z into_o two_o equal_a part_n as_o suppose_v it_o to_o be_v do_v in_o the_o point_n c._n produce_v ab_fw-la from_o the_o point_n b_o adjoin_v a_o line_n equal_a to_o bc_o which_o let_v be_v bd._n to_o the_o right_a line_n ad_fw-la and_o at_o the_o point_n d_o erect_v a_o perpendicular_a line_n equal_a to_o bd_o let_v that_o be_v de._n produce_v ed_z from_o the_o point_n d_o to_o the_o point_n f_o make_v df_o to_o contain_v five_o such_o equal_a part_n as_o de_fw-fr be_v one_o now_o upon_o of_o as_o a_o diameter_n describe_v a_o semicircle_n which_o let_v 〈◊〉_d ekf_n and_o let_v the_o point_n where_o the_o circumference_n of_o ekf_n do_v cut_v the_o line_n ab_fw-la be_v the_o point_n k._n i_o say_v that_o ab_fw-la be_v divide_v in_o the_o point_n king_n by_o a_o extreme_a and_o mean_a proportion_n for_o by_o the_o 13._o of_o the_o six_o ed_z dk_o &_o df_o be_v three_o line_n in_o continual_a proportion_n dk_o be_v the_o middle_n proportional_a â—_o wherefore_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o as_z ed_z be_v to_o df_o so_o be_v
line_n kg_v into_o two_o equal_a part_n in_o the_o point_n m._n fâ—ussas_n and_o draw_v a_o line_n from_o m_o to_o g._n and_o forasmuch_o as_o by_o construction_n the_o line_n kh_o be_v equal_a to_o the_o line_n ac_fw-la and_o the_o line_n hl_o to_o the_o line_n cb_o therefore_o the_o whole_a line_n ab_fw-la be_v equal_a to_o the_o whole_a line_n kl_o wherefore_o also_o the_o half_a of_o the_o line_n kl_o namely_o the_o line_n lm_o be_v equal_a to_o the_o semidiameter_n bn_o wherefore_o take_v away_o from_o those_o equal_a line_n equal_a part_n bc_o and_o lh_o the_o residue_v nc_n and_o mh_o shall_v be_v equal_a wherefore_o in_o the_o two_o triangle_n mhg_n and_o ncd_a the_o two_o side_n about_o the_o equal_a right_n angle_n dcn_n and_o ghm_n namely_o the_o side_n dc_o cn_fw-la and_o gh_o he_o be_v equal_a wherefore_o the_o base_n mg_o and_o nd_o be_v equal_a by_o the_o 4._o of_o the_o first_o and_o by_o the_o same_o reason_n may_v it_o be_v prove_v that_o right_a line_n draw_v from_o the_o point_n m_o to_o the_o point_n e_o and_o f_o be_v equal_a to_o the_o line_n nd_o but_o the_o right_a line_n nd_o be_v equal_a to_o the_o line_n a_fw-la which_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n wherefore_o the_o line_n mg_o be_v equal_a to_o the_o line_n mk_o &_o also_o to_o the_o line_n i_o mf_n and_o ml_o wherefore_o make_v the_o centre_n the_o point_n m_o and_o the_o space_n mk_o or_o mg_o describe_v a_o semicircle_n kgl_n and_o the_o diameter_n kl_o abide_v fix_v let_v the_o say_a semicircle_n kgl_n be_v move_v round_o about_o until_o it_o return_v to_o the_o same_o place_n from_o whence_o it_o begin_v to_o be_v move_v and_o there_o shall_v be_v describe_v a_o sphere_n about_o the_o centre_n m_o by_o the_o 12._o definition_n of_o the_o eleven_o touch_v every_o one_o of_o the_o angle_n of_o the_o pyramid_n which_o be_v at_o the_o point_n king_n e_o f_o g_o for_o those_o angle_n be_v equal_o distant_a from_o the_o centre_n of_o the_o sphere_n namely_o by_o the_o semidiameter_n of_o the_o say_a sphere_n as_o have_v before_o be_v prove_v wherefore_o in_o the_o sphere_n give_v who_o diameter_n be_v the_o line_n kl_o or_o the_o line_n ab_fw-la be_v inscribe_v a_o tetrahedron_n efgk_n now_o i_o say_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n sesquialtera_fw-la to_o the_o side_n of_o the_o pyramid_n demonstration_n for_o forasmuch_o as_o the_o line_n ac_fw-la be_v double_a to_o the_o line_n cb_o by_o construction_n therefore_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n bc._n wherefore_o by_o conversion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o the_o line_n ab_fw-la be_v sesquialtera_fw-la to_o the_o line_n ac_fw-la but_o as_o the_o line_n basilius_n be_v to_o the_o line_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n basilius_n to_o the_o square_n of_o the_o line_n ad._n for_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n d_o as_o the_o line_n bd_o be_v to_o the_o line_n ad_fw-la so_o be_v the_o same_o ad_fw-la to_o the_o line_n ac_fw-la by_o reason_n of_o the_o likeness_n of_o the_o ttriangle_n dab_n &_o dac_o by_o the_o 8._o of_o the_o six_o &_o by_o reason_n also_o that_o as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o square_a of_o the_o first_o to_o the_o square_n of_o the_o second_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n basilius_n be_v sesquialter_fw-la to_o the_o square_n of_o the_o line_n ad._n but_o the_o line_n basilius_n be_v equal_a to_o the_o diameter_n of_o the_o sphere_n give_v namely_o to_o the_o line_n kl_o as_o have_v be_v prove_v &_o the_o line_n ad_fw-la be_v equal_a to_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o sphere_n wherefore_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o the_o pyramid_n wherefore_o there_o be_v make_v a_o pyramid_n comprehend_v in_o a_o sphere_n give_v and_o the_o diameter_n of_o the_o sphere_n be_v sesquialtera_fw-la to_o the_o side_n of_o the_o pyramid_n which_o be_v require_v to_o be_v do_v and_o prove_v ¶_o a_o other_o demonstration_n to_o prove_v that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ad_fw-la to_o the_o square_n of_o the_o line_n dc_o let_v the_o description_n of_o the_o semicircle_n adb_o be_v as_o in_o the_o first_o description_n and_o upon_o the_o line_n ac_fw-la describe_v by_o the_o 46._o of_o the_o first_o a_o square_a ec_o and_o make_v perfect_v the_o parallelogram_n fb_o now_o forasmuch_o as_o the_o triangle_n dab_n be_v equiangle_n to_o the_o triangle_n dac_o by_o the_o 32._o of_o the_o six_o therefore_o as_o the_o line_n basilius_n be_v to_o the_o line_n ad_fw-la so_o be_v the_o line_n dam_fw-ge to_o the_o line_n ac_fw-la by_o the_o 4._o of_o the_o six_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n ad_fw-la by_o the_o 17._o of_o the_o six_o and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o parallelogram_n ebb_v to_o the_o parallelogram_n fb_o by_o the_o 1._o of_o the_o six_o and_o the_o parallelogram_n ebb_n be_v that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la for_o the_o line_n ea_fw-la be_v equal_a to_o the_o line_n ac_fw-la and_o the_o parallelogram_n bf_o be_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o bc._n wherefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb._n but_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n ad_fw-la by_o the_o corollary_n of_o the_o 8._o of_o the_o six_o and_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o be_v equal_a to_o the_o square_n of_o the_o line_n cd_o for_o the_o perpendicular_a line_n dc_o be_v the_o mean_a proportional_a between_o the_o segment_n of_o the_o base_a namely_o ac_fw-la and_o cb_o by_o the_o former_a corollary_n of_o the_o 8._o of_o the_o six_o for_o that_o the_o angle_n adb_o be_v a_o right_a angle_n wherefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ad_fw-la to_o the_o square_n of_o the_o line_n dc_o by_o the_o 11._o of_o the_o five_o which_o be_v require_v to_o be_v prove_v ¶_o two_o assumpte_n add_v by_o campane_n first_o assumpt_n suppose_v that_o upon_o the_o line_n ab_fw-la be_v erect_v perpendicular_o the_o line_n dc_o which_o line_n dc_o let_v be_v the_o mean_a proportional_a between_o the_o part_n of_o the_o line_n ab_fw-la namely_o ac_fw-la &_o cb_o so_o that_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cd_o so_o let_v the_o line_n cd_o be_v to_o the_o line_n cb._n and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n then_o i_o say_v that_o the_o circumference_n of_o that_o semicircle_n shall_v pass_v by_o the_o point_n d_o which_o be_v the_o end_n of_o the_o perpendicular_a line_n but_o if_o not_o than_o it_o shall_v either_o cut_v the_o line_n cd_o or_o it_o shall_v pass_v above_o it_o and_o include_v it_o not_o touch_v it_o first_o let_v it_o cut_v it_o in_o the_o point_n e._n and_o draw_v these_o right_a line_n ebb_v and_o ea_fw-la wherefore_o by_o the_o 31._o of_o the_o three_o the_o whole_a angle_n aeb_n be_v a_o right_a angle_n wherefore_o by_o the_o first_o part_n of_o the_o corollary_n of_o the_o 8._o of_o the_o six_o the_o line_n ac_fw-la be_v to_o the_o line_n â—c_z as_z the_o line_n ec_o be_v to_o the_o line_n cb._n but_o by_o the_o 8._o of_o the_o five_o the_o proportion_n of_o the_o line_n ac_fw-la to_o the_o line_n ec_o be_v great_a than_o the_o proportion_n of_o the_o same_o line_n ac_fw-la to_o the_o line_n cd_o for_o the_o line_n ce_fw-fr be_v less_o than_o the_o line_n cd_o now_o for_o that_o the_o line_n ce_fw-fr be_v to_o thâ—_z line_n cb_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n ce_fw-fr and_o the_o line_n cd_o be_v to_o the_o line_n cb_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cd_o therefore_o by_o the_o 13._o of_o the_o five_o the_o proportion_n of_o the_o line_n ec_o to_o the_o line_n cb_o be_v great_a than_o the_o proportion_n of_o the_o line_n cd_o to_o the_o line_n cb._n wherefore_o by_o the_o 10._o of_o the_o five_o the_o line_n ec_o be_v great_a than_o the_o line_n dc_o
for_o their_o use_n devise_v the_o rule_n of_o alligation_n in_o how_o sundry_a case_n do_v it_o conclude_v for_o they_o such_o precise_a verity_n as_o neither_o by_o natural_a wit_n nor_o other_o experience_n they_o be_v able_a else_o to_o know_v and_o with_o the_o merchant_n then_o to_o make_v a_o end_n how_o ample_a &_o wonderful_a be_v the_o rule_n of_o false_a position_n especial_o as_o it_o be_v now_o by_o two_o excellent_a mathematician_n of_o my_o familer_n acquaintance_n in_o their_o life_n time_n enlarge_v i_o mean_v gemma_fw-la frisius_n and_o simon_n jacob_n who_o can_v either_o in_o brief_a conclude_v the_o general_a and_o capital_a rule_n or_o who_o can_v imagine_v the_o myriad_o of_o sundry_a case_n and_o particular_a example_n in_o act_n and_o earnest_n continual_o wrought_v try_v and_o conclude_v by_o the_o forename_a rule_n only_o how_o sundry_a other_o arithmetical_a practice_n be_v common_o in_o merchant_n hand_n and_o knowledge_n they_o themselves_o can_v at_o large_a testify_v the_o mintmaster_n and_o goldsmith_n in_o their_o mixture_n of_o metal_n either_o of_o diverse_a kind_n or_o diverse_a value_n how_o be_v they_o or_o may_v they_o exact_o be_v direct_v and_o marvelous_o pleasure_v if_o arithmetic_n be_v their_o guide_n and_o the_o honourable_a physician_n will_v glad_o confess_v themselves_o much_o behold_v to_o the_o science_n of_o arithmetic_n and_o that_o sundry_a way_n but_o chief_o in_o their_o art_n of_o graduation_n and_o compound_v medicine_n and_o though_o galenus_n auerâ—ois_n arnoldus_fw-la lullus_fw-la and_o other_o have_v publish_v their_o position_n aswell_o in_o the_o quantity_n of_o the_o degree_n above_o temperament_n as_o in_o the_o rule_n conclude_v the_o new_a form_n result_v yet_o a_o more_o precise_a commodious_a and_o easy_a method_n be_v extant_a b._n by_o a_o countryman_n of_o we_o above_o 200_o year_n ago_o invent_v and_o forasmuch_o as_o i_o be_o uncertain_a who_o have_v the_o same_o or_o when_o that_o little_a latin_a treatise_n as_o the_o author_n write_v it_o shall_v come_v to_o be_v print_v both_o to_o declare_v the_o desire_n i_o have_v to_o pleasure_v my_o country_n wherein_o i_o may_v and_o also_o for_o very_o good_a proof_n of_o number_n use_v in_o this_o most_o subtle_a and_o fruitful_a philosophical_a conclusion_n i_o intend_v in_o the_o mean_a while_n most_o brief_o and_o with_o my_o far_a help_n to_o communicate_v the_o pith_n thereof_o unto_o you_o first_o describe_v a_o circle_n who_o diameter_n let_v be_v a_o inch_n divide_v the_o circumference_n into_o four_o equal_a part_n fron_n the_o centre_n by_o those_o 4._o section_n extend_v 4._o right_a line_n each_o of_o 4._o inch_n and_o a_o half_a long_o or_o of_o as_o many_o as_o you_o list_v above_o 4._o without_o the_o circumference_n of_o the_o circle_n so_o that_o they_o shall_v be_v of_o 4._o inch_n long_a at_o the_o leaâ—t_n without_o the_o circleâ—_n make_v good_a evident_a mark_n at_o every_o inch_n end_n if_o you_o listen_v you_o may_v subdivide_v the_o inch_n again_o into_o 10._o or_o 12._o small_a part_n equal_a at_o the_o end_n of_o the_o line_n write_v the_o name_n of_o the_o 4._o principal_a elemental_a quality_n hight_z and_o coldâ—_n one_o against_o the_o other_o and_o likewise_o moist_a and_o dry_a one_o against_o the_o other_o and_o in_o the_o circle_n write_v temperate_a which_o temperature_n have_v a_o good_a latitude_n as_o appear_v by_o the_o complexion_n of_o man_n and_o therefore_o we_o have_v allow_v unto_o it_o the_o foresay_a circle_n and_o not_o a_o point_n mathematical_a or_o physical_a and_o though_o i_o here_o speak_v only_o of_o two_o thing_n miscible_a and_o most_o common_o more_o than_o three_o four_o five_o or_o six_o etc_n etc_n be_v to_o be_v mix_v and_o in_o one_o compound_v to_o be_v reducedâ—_n &_o the_o form_n result_v of_o the_o same_o to_o serve_v the_o turn_n yet_o these_o rule_n be_v sufficient_a due_o repeat_v and_o iterate_v note_n in_o proceed_v first_o with_o any_o two_o and_o then_o with_o the_o fonne_z result_v and_o a_o other_o &_o so_o forth_o for_o the_o last_o work_n conclude_v the_o form_n result_v of_o they_o all_o i_o need_v nothing_o to_o speak_v of_o the_o mixture_n here_o suppose_v what_o it_o be_v common_a philosophy_n have_v define_v it_o say_v mixtio_fw-la est_fw-la miscibilium_fw-la alteratorum_fw-la per_fw-la minima_fw-la coniunctorum_fw-la vnio_n every_o word_n in_o the_o definition_n be_v of_o great_a importance_n i_o need_n not_o also_o spend_v any_o time_n to_o show_v how_o the_o other_o manner_n of_o distribute_v of_o degree_n do_v agree_v to_o these_o rule_n neither_o need_n i_o of_o the_o far_a use_n belong_v to_o the_o cross_n of_o graduation_n before_o describe_v in_o this_o place_n declare_v unto_o such_o as_o be_v capable_a of_o that_o which_o i_o have_v all_o ready_a say_v neither_o yet_o with_o example_n specify_v the_o manifold_a variety's_n by_o the_o foresay_a two_z general_a rule_n to_o be_v order_v the_o witty_a and_o studious_a here_o have_v sufficient_a and_o they_o which_o be_v not_o able_a to_o attain_v to_o this_o without_o lively_a teach_n and_o more_o in_o particular_a will_v have_v large_a discourse_n then_o be_v mete_v in_o this_o place_n to_o be_v deal_v withal_o and_o other_o perchance_o with_o a_o proud_a snuff_n will_v disdain_v this_o litlâ—â—_n and_o will_v be_v unthankful_a for_o much_o more_o ay_o therefore_o conclude_v and_o wish_v such_o as_o have_v modest_a and_o earnest_a philosophical_a mind_n to_o laud_n god_n high_o for_o this_o and_o to_o marvel_v that_o the_o profound_a and_o subtle_a point_n concern_v mixture_n of_o form_n and_o quality_n natural_a be_v so_o matchâ—_n and_o marry_v with_o the_o most_o simple_a easy_a and_o short_a way_n of_o the_o noble_a rule_n of_o algiebar_n who_o can_v remain_v therefore_o unpersuade_v to_o love_v halloo_o and_o honour_v the_o excellent_a science_n of_o arithmetic_n for_o here_o you_o may_v perceive_v that_o the_o little_a finger_n of_o arithmetic_n be_v of_o more_o might_n and_o contrive_n then_o a_o hundred_o thousand_o man_n wit_n of_o the_o middle_a sort_n be_v able_a to_o perform_v or_o true_o to_o conclude_v with_o out_o help_v thereof_o now_o will_v we_o fâ—rder_v by_o the_o wise_a and_o valiant_a captain_n be_v certify_v what_o help_v he_o have_v by_o the_o rule_n of_o arithmetic_n in_o one_o of_o the_o art_n to_o he_o appertain_v and_o of_o the_o greek_n name_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d that_o be_v the_o skill_n of_o order_v soldier_n in_o battle_n ray_n after_o the_o best_a manner_n to_o all_o purpose_n this_o art_n so_o much_o depend_v upon_o number_n use_n and_o the_o mathematicals_n that_o aelianus_n the_o best_a writer_n thereof_o in_o his_o work_n tö_o the_o emperor_n hadrianus_n by_o his_o perfection_n in_o the_o mathematicals_n be_v great_a than_o other_o before_o he_o have_v think_v his_o book_n to_o pass_v all_o other_o the_o excellent_a work_n write_v of_o that_o art_n unto_o his_o day_n for_o of_o it_o have_v write_v aeneas_n gyneas_n of_o thessaly_n pyrrhus_n epirota_n and_o alexander_n his_o son_n clearchus_n pausanias_n euangelus_n polybiâ—s_n familiar_a friend_n to_o scipio_n eupolemus_n iphicrates_n possidonius_fw-la and_o very_o many_o other_o worthy_a captain_n philosopher_n and_o prince_n of_o immortal_a fame_n and_o memory_n who_o fair_a flower_n of_o their_o garland_n in_o this_o feat_n be_v arithmetic_n and_o a_o little_a perceiverance_n in_o geometrical_a figure_n but_o in_o many_o other_o case_n do_v arithmetic_n stand_v the_o captain_n in_o great_a stead_n as_o in_o proportion_v of_o victual_n for_o the_o army_n either_o remain_v at_o a_o stay_n or_o sudden_o to_o be_v increase_v with_o a_o certain_a number_n of_o soldier_n and_o for_o a_o certain_a time_n or_o by_o good_a art_n to_o diminish_v his_o company_n to_o make_v the_o victual_n long_o to_o serve_v the_o remanent_fw-la &_o for_o a_o certain_a determine_a time_n 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proposition_n which_o require_v some_o action_n or_o do_v as_o the_o make_v of_o some_o â—igure_n or_o to_o divide_v a_o figure_n or_o line_n to_o apply_v figure_n to_o â—igure_n to_o add_v figure_n together_o or_o to_o subtrah_n one_o from_o a_o other_o to_o describe_v to_o inscribe_v to_o circumscribe_v one_o figure_n within_o or_o without_o another_o and_o such_o like_a as_o of_o the_o first_o proposition_n of_o the_o first_o book_n be_v a_o problem_n which_o be_v thuss_v upon_o a_o right_a line_n give_v not_o be_v infinite_a to_o describe_v a_o equilater_n triangle_n or_o a_o triangle_n of_o three_o equal_a side_n for_o in_o it_o beside_o the_o demonstration_n and_o contemplation_n of_o the_o mind_n be_v require_v somewhat_o to_o be_v do_v name_o to_o make_v a_o equilater_n triangle_n upon_o a_o line_n give_v and_o in_o the_o end_n of_o every_o problem_n after_o the_o demonstration_n be_v conclude_v aâ—ter_v this_o manâ—er_n which_o be_v the_o thing_n which_o be_v require_v to_o be_v do_v be_v a_o theorem_a be_v a_o proposition_n which_o require_v the_o search_a ouâ—_n and_o demonstration_n of_o some_o property_n or_o passion_n of_o some_o figure_n wherein_o be_v only_a speculation_n and_o contemplation_n of_o mind_n without_o do_v or_o work_v of_o any_o thing_n as_o the_o five_o proposition_n of_o the_o first_o book_n which_o be_v thus_o a_o isosceles_a or_o triangle_n of_o two_o equal_a side_n have_v his_o angle_n at_o thâ—_z base_a equal_a the_o one_o to_o the_o other_o etc_n etc_n be_v a_o theorem_a for_o in_o it_o be_v require_v only_o to_o be_v prove_v and_o make_v plain_a by_o reason_n and_o demonstration_n that_o these_o two_o angle_n be_v equal_a without_o further_a work_n or_o do_v and_o in_o the_o endâ—_n of_o â—uâ—ry_a theorem_a after_o the_o demonstration_n be_v conclude_v after_o this_o manner_n which_o thing_n be_v require_v to_o be_v demonstrate_v or_o prove_v the_o first_o problem_n the_o first_o proposition_n upon_o a_o right_a line_n give_v not_o be_v infinite_a to_o describe_v a_o equilater_n triangle_n or_o a_o triangle_n of_o three_o equal_a side_n svppose_v that_o the_o right_a line_n give_v be_v ab_fw-la it_o be_v require_v upon_o the_o line_n ab_fw-la to_o describe_v a_o equilater_n triangle_n namely_o a_o triangle_n of_o three_o equal_a side_n construction_n now_o therefore_o make_v the_o centre_n the_o point_n a_o and_o the_o space_n ab_fw-la describe_v by_o the_o three_o petition_n a_o circle_n bcd_o and_o again_o by_o the_o same_o make_v the_o centre_n the_o point_n b_o and_o the_o space_n bâ—_n describe_v a_o other_o circle_n ace_n and_o by_o the_o first_o petition_n from_o the_o point_n c_o wherein_o the_o circle_n cut_v the_o one_o the_o other_o draw_v one_o right_a line_n to_o the_o point_n a_o and_o a_o other_o right_a line_n to_o the_o point_n b._n and_o forasmuch_o as_o the_o point_n a_o be_v the_o centre_n of_o the_o circle_n cbd_v demonstration_n therefore_o by_o the_o 15._o definition_n the_o line_n ac_fw-la be_v equal_a to_o the_o line_n ab_fw-la again_o forasmuch_o as_o the_o point_n b_o be_v the_o centre_n of_o the_o circle_n caesar_n therefore_o by_o the_o same_o definition_n the_o line_n bc_o be_v equal_a to_o the_o line_n basilius_n and_o it_o be_v prove_v that_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n ab_fw-la wherefore_o either_o of_o these_o line_n ca_n and_o cb_o be_v equal_a to_o the_o line_n ab_fw-la but_o thing_n which_o be_v equal_a to_o one_o and_o the_o same_o thing_n be_v also_o equal_a the_o one_o to_o the_o other_o by_o the_o first_o common_a sentence_n wherefore_o the_o line_n ca_n also_o be_v equal_a to_o the_o line_n cb._n wherefore_o these_o three_o right_a line_n ca_n ab_fw-la and_o bc_o be_v equal_a the_o one_o to_o the_o other_o wherefore_o the_o triangle_n abc_n be_v equilater_n wherefore_o upon_o the_o line_n ab_fw-la be_v describe_v a_o equilater_n triangle_n abc_n wherefore_o upon_o a_o line_n give_v not_o be_v infinite_a there_o be_v describe_v a_o equilater_n triangle_n which_o be_v the_o thing_n which_o be_v require_v to_o be_v do_v a_o triangle_n or_o any_o other_o rectiline_v figure_n be_v then_o say_v to_o be_v set_v or_o describe_v upon_o a_o line_n when_o the_o line_n be_v one_o of_o the_o side_n of_o the_o figure_n this_o first_o proposition_n be_v a_o problem_n because_o it_o require_v act_n or_o do_v namely_o to_o describe_v a_o triangle_n and_o this_o be_v to_o be_v note_v that_o every_o proposition_n whether_o it_o be_v a_o problem_n or_o a_o theorem_a common_o contain_v in_o it_o a_o thing_n give_v and_o a_o thing_n require_v to_o be_v search_v out_o although_o it_o be_v not_o always_o so_o and_o the_o thing_n give_v give_v be_v ever_o
ab_fw-la a_o line_n equal_a to_o the_o line_n of_o that_o be_v to_o the_o line_n cd_o but_o if_o they_o cut_v the_o one_o the_o other_o as_o the_o line_n cd_o &_o ab_fw-la do_v case_n then_o make_v the_o centre_n b_o &_o the_o space_n basilius_n describe_v a_o circle_n of_o &_o draw_v a_o line_n from_o the_o point_n b_o to_o the_o point_n c_o &_o produce_v it_o to_o the_o point_n f._n and_o forasmuch_o as_o the_o two_o right_a line_n bf_o and_o cd_o be_v unequal_a and_o the_o line_n cd_o cut_v the_o line_n bf_o by_o one_o of_o his_o extreme_n therefore_o it_o be_v possible_a to_o cut_v of_o from_o cd_o a_o line_n equal_a to_o the_o line_n bf_o for_o how_o to_o do_v it_o we_o have_v before_o declare_v wherefore_o it_o be_v possible_a from_o the_o line_n cd_o to_o cut_v of_o a_o line_n equal_a to_o the_o line_n ab_fw-la or_o ab_fw-la and_o bf_o be_v equal_a the_o one_o to_o the_o other_o this_o be_v to_o be_v note_v that_o in_o all_o these_o case_n serve_v a_o man_n may_v both_o as_o touch_v construction_n and_o demonstration_n proceed_v as_o in_o the_o first_o case_n for_o it_o be_v possible_a in_o any_o position_n to_o put_v to_o the_o end_n of_o the_o great_a line_n a_o line_n equal_a to_o the_o less_o line_n and_o so_o make_v the_o centre_n the_o say_a end_n and_o the_o space_n the_o less_o line_n to_o describe_v a_o circle_n which_o shall_v cut_v of_o from_o the_o great_a line_n a_o line_n equal_a to_o the_o line_n put_v namely_o to_o the_o less_o line_n give_v as_o it_o be_v manifest_a to_o see_v in_o the_o figure_n partly_o here_o under_o set_n and_o partly_o in_o the_o begin_n of_o the_o other_o side_n put_v the_o first_o theorem_a the_o 4._o proposition_n if_o there_o be_v two_o triangle_n of_o which_o two_o side_n of_o the_o one_o be_v equal_v to_o two_o side_n of_o the_o other_o each_o side_n to_o his_o correspondent_a side_n and_o have_v also_o on_o angle_n of_o the_o one_o equal_a to_o one_o angle_n of_o the_o other_o namely_o that_o angle_n which_o be_v contain_v under_o the_o equal_a right_a line_n the_o base_a also_o of_o the_o one_o shall_v be_v equal_a to_o the_o base_a of_o the_o other_o and_o the_o one_o triangle_n shall_v be_v equal_a to_o the_o other_o triangle_n and_o the_o other_o angle_n remain_v shall_v be_v equal_a to_o the_o other_o angle_n remain_v the_o one_o to_o the_o other_o under_o which_o be_v subtend_v equal_a side_n svppose_v that_o there_o be_v two_o triangle_n abc_n &_o def_n have_v two_o side_n of_o the_o one_o namely_o ab_fw-la and_o ac_fw-la equal_a to_o two_o side_n of_o the_o other_o namely_o to_o de_fw-fr and_o df_o the_o one_o to_o the_o other_o that_o be_v ab_fw-la to_o de_fw-fr and_o ac_fw-la to_o df_o have_v also_o the_o angle_n bac_n equal_a to_o the_o angle_n edf_o then_o i_o say_v that_o the_o base_a also_o bc_o be_v equal_a to_o the_o base_a of_o &_o that_o the_o triangle_n abc_n be_v equal_a to_o the_o triangle_n def_n and_o that_o the_o other_o angle_n remain_v be_v equal_a to_o the_o other_o angle_n remain_v the_o one_o to_o the_o other_o under_o which_o be_v subtend_v equal_a side_n that_o be_v that_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n def_n and_o that_o the_o angle_n acb_o be_v equal_a to_o to_o the_o angle_n dfe_n absurdity_n for_o the_o triangle_n abc_n exact_o agree_v with_o the_o triangle_n def_n and_o the_o point_n a_o be_v put_v upon_o the_o point_n d_o &_o the_o right_a line_n ab_fw-la upon_o the_o right_a line_n de_fw-fr the_o point_n b_o also_o shall_v exact_o agree_v with_o the_o point_n e_o for_o that_o by_o supposition_n the_o line_n ab_fw-la be_v equal_a to_o the_o line_n de._n and_o the_o line_n ab_fw-la exact_o agree_a with_o the_o line_n de_fw-fr the_o right_a line_n also_o ac_fw-la exact_o agree_v with_o the_o right_a line_n df_o for_o that_o by_o supposition_n the_o angle_n bac_n be_v equal_a to_o the_o angle_n edf_o and_o forasmuch_o as_o the_o right_a line_n ac_fw-la be_v also_o by_o supposition_n equal_a to_o the_o right_a line_n df_o therefore_o the_o point_n c_o exact_o agree_v with_o the_o point_n f._n again_o forasmuch_o as_o the_o point_n c_o exact_o agree_v with_o the_o point_n fletcher_n and_o the_o point_n b_o exact_o agree_v with_o the_o point_n e_o therefore_o the_o base_a bc_o shall_v exact_o agree_v with_o the_o base_a ef._n for_o if_o the_o point_n b_o do_v exact_o agree_v with_o the_o point_n e_o and_o the_o point_n c_o with_o the_o point_n fletcher_n and_o the_o base_a bc_o do_v not_o exact_o agree_v with_o the_o base_a of_o than_o two_o right_a line_n do_v include_v a_o superficies_n which_o by_o the_o 10._o common_a sentence_n be_v impossible_a wherefore_o the_o base_a bc_o exact_o agree_v with_o the_o base_a of_o and_o therefore_o be_v equal_a unto_o it_o wherefore_o the_o whole_a triangle_n abc_n exact_o agree_v with_o the_o whole_a triangle_n def_n &_o therefore_o by_o the_o 8._o common_a sentence_n be_v equal_a unto_o it_o and_o by_o the_o same_o the_o other_o angle_n remain_v exact_o agree_v with_o the_o other_o angle_n remain_v and_o be_v equal_a the_o one_o to_o the_o other_o that_o be_v the_o angle_n abc_n to_o the_o angle_n def_n and_o the_o angle_n acb_o to_o the_o angle_n dfe_n if_o therefore_o there_o be_v two_o triangle_n of_o which_o two_o side_n of_o the_o one_o be_v 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 have_v also_o one_o angle_n of_o the_o one_o equal_a to_o one_o angle_n of_o the_o other_o namely_o that_o angle_n which_o be_v contain_v under_o the_o equal_a right_a line_n the_o base_a also_o of_o the_o one_o shall_v be_v equal_a to_o the_o base_a of_o the_o other_o and_o the_o one_o triangle_n shall_v be_v equal_a to_o the_o other_o triangle_n and_o the_o other_o angle_n remain_v shall_v be_v equal_a to_o the_o other_o angle_n remain_v the_o one_o to_o the_o other_o under_o which_o be_v subtend_v eqaull_a side_n which_o thing_n be_v require_v to_o be_v demonstrate_v this_o proposition_n which_o be_v a_o theorem_a 〈…〉_o have_v two_o thing_n give_v namely_o the_o equality_n of_o two_o side_n of_o the_o one_o triangle_n to_o two_o side_n of_o the_o other_o triangle_n and_o the_o equality_n of_o two_o angle_n contain_v under_o the_o equal_a side_n in_o it_o also_o be_v three_o thing_n require_v it_o the_o equality_n of_o base_a to_o base_a the_o equality_n of_o field_n to_o field_n and_o the_o equality_n of_o the_o other_o angle_n of_o the_o one_o triangle_n to_o the_o other_o angle_n of_o the_o other_o triangle_n under_o which_o be_v subtend_v equal_a side_n other_o one_o side_n of_o a_o plain_a figure_n be_v equal_a to_o a_o other_o and_o so_o general_o one_o right_a line_n be_v equal_a to_o a_o other_o when_o the_o one_o be_v apply_v to_o the_o other_o their_o extreme_n agree_v together_o for_o otherwise_o every_o right_a line_n apply_v to_o any_o right_a line_n agree_v therewith_o but_o equal_a right_a line_n only_o agree_v in_o the_o extreme_n other_o one_o rectiline_v angle_n be_v equal_a to_o a_o other_o rectiline_v angle_n when_o one_o of_o the_o side_n which_o comprehend_v the_o one_o angle_n be_v set_v upon_o one_o of_o the_o side_n which_o comprehend_v the_o other_o angle_n the_o other_o side_n of_o the_o one_o agree_v with_o the_o other_o side_n of_o the_o other_o and_o that_o angle_n be_v the_o great_a who_o side_n fall_v without_o and_o that_o the_o less_o who_o side_n fall_v within_o put_v where_o as_o in_o this_o proposition_n be_v put_v this_o particle_n each_o to_o his_o correspondent_a side_n instead_n whereof_o often_o time_n afterward_o be_v use_v this_o phrase_n the_o one_o to_o the_o other_o it_o be_v of_o necessity_n so_o put_v for_o otherwise_o two_o side_n of_o one_o triangle_n add_v together_o may_v be_v equal_a to_o two_o side_n of_o a_o other_o triangle_n add_v together_o and_o the_o angle_n also_o contain_v under_o the_o equal_a side_n may_v be_v equal_a and_o yet_o the_o two_o triangle_n may_v notwithstanding_o be_v unequal_a other_o where_o note_n that_o a_o triangle_n be_v say_v to_o be_v equal_a to_o a_o other_o triangle_n when_o the_o field_n or_o area_n of_o the_o one_o be_v equal_a to_o the_o area_n of_o the_o other_o and_o the_o area_n of_o a_o triangle_n 〈◊〉_d be_v that_o space_n which_o be_v contain_v within_o the_o side_n of_o a_o triangle_n and_o the_o circuit_n or_o compass_n of_o a_o triangle_n be_v a_o line_n compose_v of_o all_o the_o side_n of_o a_o triangle_n and_o so_o may_v you_o think_v of_o all_o oâ—her_n rectiline_v figure_n figure_n and_o now_o to_o prove_v that_o there_o may_v be_v two_o triangle_n two_o side_n of_o one_o of_o which_o be_v add_v together_o
square_n which_o be_v make_v of_o the_o line_n ab_fw-la and_o ac_fw-la construction_n describe_v by_o the_o 46._o proposition_n upon_o the_o line_n bc_o a_o square_a bdce_n and_o by_o the_o same_o upon_o the_o line_n basilius_n and_o ac_fw-la describe_v the_o square_n abfg_n and_o ackh_n and_o by_o the_o point_n a_o draw_v by_o the_o 31._o proposition_n to_o either_o of_o these_o line_n bd_o and_o ce_fw-fr a_o parallel_a line_n al._n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n d_o and_o a_o other_o from_o the_o point_n c_o to_o the_o point_n e._n demonstration_n and_o forasmuch_o as_o the_o angle_n bac_n and_o bag_n be_v right_a angle_n therefore_o unto_o a_o right_a line_n basilius_n and_o to_o a_o point_n in_o it_o give_v a_o be_v draw_v two_o right_a line_n ac_fw-la and_o ag_z not_o both_o on_o one_o and_o the_o same_o side_n make_v the_o two_o side_n angle_n equal_a to_o two_o right_a angle_n wherefore_o by_o the_o 14._o proposition_n the_o line_n ac_fw-la and_o ag_z make_v direct_o one_o right_a line_n and_o by_o the_o same_o reason_n the_o line_n basilius_n and_o ah_o make_v also_o direct_o one_o right_a line_n and_o forasmuch_o as_o the_o angle_n dbc_n be_v equal_a to_o the_o angle_n fba_n for_o either_o of_o they_o be_v a_o right_a angle_n put_v the_o angle_n abc_n common_a to_o they_o both_o wherefore_o the_o whole_a angle_n dba_n be_v equal_a to_o the_o whole_a angle_n fbc_n and_o forasmuch_o as_o these_o two_o line_n ab_fw-la and_o bd_o be_v equal_a to_o these_o two_o line_n bf_o and_o bc_o the_o one_o to_o the_o other_o and_o the_o angle_n dba_n be_v equal_a to_o the_o angle_n fbc_n therefore_o by_o the_o 4._o proposition_n the_o base_a ad_fw-la be_v equal_a to_o the_o base_a fc_n and_o the_o triangle_n abdella_n be_v equal_a to_o the_o triangle_n fbc_n but_o by_o the_o 31._o proposition_n the_o parallelogram_n bl_o be_v double_a to_o the_o triangle_n abdella_n for_o they_o have_v both_o one_o and_o the_o same_o base_a namely_o bd_o and_o be_v in_o the_o self_n same_o parallel_a line_n that_o be_v bd_o and_o all_o and_o by_o the_o same_o the_o square_a gb_o be_v double_a to_o the_o triangle_n fbc_n for_o they_o have_v both_o one_o and_o the_o self_n same_o base_a that_o be_v bf_o and_o be_v in_o the_o self_n same_o parallel_a line_n that_o be_v fb_o and_o gc_o but_o the_o double_n of_o thing_n equal_a be_v by_o the_o six_o common_a sentence_n equal_v the_o one_o to_o the_o other_o wherefore_o the_o parallelogram_n bl_o be_v equal_a to_o the_o square_a gb_o and_o in_o like_a sort_n if_o by_o the_o first_o petition_n there_o be_v draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n e_o and_o a_o other_o from_o the_o point_n b_o to_o the_o point_n king_n we_o may_v prove_v that_o the_o parallelogram_n cl_n be_v equal_a to_o the_o square_a hc_n wherefore_o the_o whole_a square_a bdec_n be_v equal_a to_o the_o two_o square_n gb_o and_o hc_n but_o the_o square_a bdec_n be_v describe_v upon_o the_o line_n bc_o and_o the_o square_n gb_o and_o hc_n be_v describe_v upon_o the_o line_n basilius_n &_o ac_fw-la wherefore_o the_o square_a of_o the_o side_n bc_o be_v equal_a to_o the_o square_n of_o the_o side_n basilius_n and_o ac_fw-la wherefore_o in_o rectangle_n triangle_n the_o square_n which_o be_v make_v of_o the_o side_n that_o subtend_v the_o right_a angle_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o side_n contain_v the_o right_a angle_n which_o be_v require_v to_o be_v demonstrate_v this_o most_o excellent_a and_o notable_a theorem_a be_v first_o invent_v of_o the_o great_a philosopher_n pythagoras_n proposition_n who_o for_o the_o exceed_a joy_n conceive_v of_o the_o invention_n thereof_o offer_v in_o sacrifice_n a_o ox_n as_o record_n hiâ—rone_n proclus_n lycius_n &_o vitruutus_fw-la and_o it_o have_v be_v common_o call_v of_o barbarous_a writer_n of_o the_o latter_a time_n dulcarnon_n pâ—lâ—tariâ—â—_n a_o addition_n of_o pelitarius_n to_o reduce_v two_o unequal_a square_n to_o two_o equal_a square_n suppose_v that_o the_o square_n of_o the_o line_n ab_fw-la and_o ac_fw-la be_v unequal_a it_o be_v require_v to_o reduce_v they_o to_o two_o equal_a square_n join_v the_o two_o line_n ab_fw-la and_o ac_fw-la at_o their_o end_n in_o such_o sort_n that_o they_o make_v a_o right_a angle_n bac_n and_o draw_v a_o line_n from_o b_o to_o c._n then_o upon_o the_o two_o end_n b_o and_o c_o make_v two_o angle_n each_o of_o which_o may_v be_v equal_a to_o half_a a_o right_a angle_n this_o be_v do_v by_o erect_v upon_o the_o line_n bc_o perpendicule_a line_n from_o the_o point_n b_o and_o c_o and_o so_o by_o the_o 9_o proposition_n devide_v â—che_fw-mi of_o the_o right_a angle_n into_o two_o equal_a part_n and_o let_v the_o angle_n bcd_o and_o cbd_v be_v either_o of_o they_o half_o of_o a_o right_a angle_n and_o let_v the_o line_n bd_o and_o cd_o concur_v in_o the_o point_n d._n then_o i_o say_v that_o the_o two_o square_n of_o the_o side_n bd_o and_o cd_o be_v equal_a to_o the_o two_o square_n of_o the_o side_n ab_fw-la and_o ac_fw-la for_o by_o the_o 6_o proposition_n the_o two_o side_n db_o and_o dc_o be_v equal_a and_o the_o angle_n at_o the_o point_n d_o be_v by_o the_o 32._o proposition_n a_o right_a angle_n wherefore_o the_o square_a of_o the_o side_n bc_o be_v equal_a to_o the_o square_n of_o the_o two_o side_n db_o and_o dc_o by_o the_o 47._o proposition_n but_o it_o be_v also_o equal_a to_o the_o square_n of_o the_o two_o side_n ab_fw-la and_o ac_fw-la by_o the_o self_n same_o proposition_n wherefore_o by_o the_o common_a sentence_n the_o square_n of_o the_o two_o side_n bd_o and_o dc_o be_v equal_a to_o the_o square_n of_o the_o two_o side_n ab_fw-la and_o ac_fw-la which_o be_v require_v to_o be_v do_v an_o other_o addition_n of_o pelitarius_n pelitarius_n if_o two_o right_a angle_a triangle_n have_v equal_a base_n the_o square_n of_o the_o two_o side_n of_o the_o one_o be_v equal_a to_o the_o square_n of_o the_o two_o side_n of_o the_o other_o this_o be_v manifest_a by_o the_o former_a construction_n and_o demonstration_n an_o other_o addition_n of_o pelitarius_n pelitarius_n two_o unequal_a line_n be_v give_v to_o know_v how_o much_o the_o square_n of_o the_o one_o be_v great_a than_o the_o square_a of_o the_o other_o suppose_v that_o there_o be_v two_o unequal_a line_n ab_fw-la and_o bc_o of_o which_o let_v ab_fw-la be_v the_o great_a it_o be_v require_v to_o search_v out_o how_o much_o the_o square_n of_o ab_fw-la exceed_v the_o square_a of_o bc._n that_o be_v i_o will_v find_v out_o the_o square_n which_o with_o the_o square_n of_o the_o line_n bc_o shall_v be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la put_v the_o line_n a_o b_o and_o bc_o direct_o that_o they_o make_v both_o one_o right_a line_n and_o make_v the_o centre_n the_o point_n b_o and_o the_o space_n basilius_n describe_v a_o circle_n ade_n and_o produce_v the_o line_n ac_fw-la to_o the_o circumference_n and_o let_v it_o concur_v with_o it_o in_o the_o point_n e._n and_o upon_o the_o line_n ae_n and_o from_o the_o point_n c_o erect_v by_o the_o 11._o proposition_n a_o perpendicular_a line_n cd_o which_o produce_v till_o it_o concur_v with_o the_o circumference_n in_o the_o point_n d_o &_o draw_v a_o line_n from_o b_o to_o d._n then_o i_o say_v that_o the_o square_n of_o the_o line_n cd_o be_v the_o excess_n of_o the_o square_n of_o the_o line_n ab_fw-la above_o the_o square_n of_o the_o line_n bc._n for_o forasmuch_o as_o in_o the_o triangle_n bcd_o the_o angle_n at_o the_o point_n c_o be_v a_o right_a angle_n the_o square_a of_o the_o base_a bd_o be_v equal_a to_o the_o square_n of_o the_o two_o side_n bc_o and_o cd_o by_o this_o 47._o proposition_n wherefore_o also_o the_o square_a of_o the_o line_n ab_fw-la be_v equal_a to_o the_o self_n same_o square_n of_o the_o line_n bc_o and_o cd_o wherefore_o the_o square_a of_o the_o line_n bc_o be_v so_o much_o less_o than_o the_o square_a of_o the_o line_n ab_fw-la as_o be_v the_o square_a of_o the_o line_n cd_o which_o be_v require_v to_o search_v out_o an_o other_o addition_n of_o pelitarius_n pelitarius_n the_o diameter_n of_o a_o square_n be_v give_v to_o geve_v the_o square_a thereof_o this_o be_v easy_a to_o be_v do_v for_o if_o upon_o the_o two_o end_n of_o the_o line_n be_v draw_v two_o half_a right_a angle_n and_o so_o be_v make_v perfect_a the_o triangle_n then_o shall_v be_v describe_v half_a of_o the_o square_n the_o other_o half_o whereof_o also_o be_v after_o the_o same_o manner_n easy_a to_o be_v describe_v hereby_o it_o be_v manifest_a that_o the_o square_n of_o the_o
about_o the_o diameter_n together_o with_o the_o two_o supplement_n make_v a_o gnomon_n as_o the_o parallelogram_n ebkh_n with_o the_o two_o supplement_n aegk_n and_o khfd_n make_v the_o gnomon_n fgeh_a likewise_o the_o parallelogram_n gkcf_n with_o the_o same_o two_o supplement_n make_v the_o gnomon_n ehfg_n and_o this_o definition_n of_o a_o gnomon_n extend_v itself_o and_o be_v general_a to_o all_o kind_n of_o parallelogram_n whether_o they_o be_v square_n or_o figure_n of_o one_o side_n long_o or_o rhombus_fw-la or_o romboide_n to_o be_v short_a if_o you_o take_v away_o from_o the_o whole_a parallelogram_n one_o of_o the_o partial_a parallelogram_n which_o be_v about_o the_o diameter_n whether_o you_o will_v the_o rest_n of_o the_o figure_n be_v a_o gnomon_n campane_v after_o the_o last_o proposition_n of_o the_o first_o book_n add_v this_o proposition_n book_n two_o square_n be_v give_v to_o adjoin_v to_o one_o of_o they_o a_o gnomon_n equal_a to_o the_o other_o square_n which_o for_o that_o as_o than_o it_o be_v not_o teach_v what_o a_o gnomon_n be_v i_o there_o omit_v think_v that_o it_o may_v more_o apt_o be_v place_v here_o the_o do_v and_o demonstration_n whereof_o be_v thus_o suppose_v that_o there_o be_v two_o square_n ab_fw-la and_o cd_o unto_o one_o of_o which_o namely_o unto_o ab_fw-la it_o be_v require_v to_o add_v a_o gnomon_n equal_a to_o the_o other_o square_n namely_o to_o cd_o produce_v the_o side_n bf_o of_o the_o square_a ab_fw-la direct_o to_o the_o point_n e._n and_o put_v the_o line_n fe_o equal_a to_o the_o side_n of_o the_o square_a cd_o and_o draw_v a_o line_n from_o e_o to_o a._n now_o then_o forasmuch_o as_o efa_n be_v a_o rectangle_n triangle_n therefore_o by_o the_o 47._o of_o the_o first_o the_o square_a of_o the_o line_n ea_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n of_o &_o fa._n but_o the_o square_n of_o the_o line_n of_o be_v equal_a to_o the_o square_a cd_o &_o the_o square_a of_o the_o side_n favorina_n be_v the_o square_a ab_fw-la wherefore_o the_o square_a of_o the_o line_n ae_n be_v equal_a to_o the_o two_o square_n cd_o and_o ab_fw-la but_o the_o side_n of_o and_o favorina_n be_v by_o the_o 21._o of_o the_o first_o long_o than_o the_o side_n ae_n and_o the_o side_n favorina_n be_v equal_a to_o the_o side_n fb_o wherefore_o the_o side_n of_o and_o fb_o be_v long_o they_o the_o side_n ae_n wherefore_o the_o whole_a line_n be_v be_v long_o than_o the_o line_n ae_n from_o the_o line_n be_v cut_v of_o a_o line_n equal_a to_o the_o line_n ae_n which_o let_v be_v bc._n and_o by_o the_o 46._o proposition_n upon_o the_o line_n bc_o describe_v a_o square_a which_o let_v be_v bcgh_n which_o shall_v be_v equal_a to_o the_o square_n of_o the_o line_n ae_n but_o the_o square_n of_o the_o line_n ae_n be_v equal_a to_o the_o two_o square_n ab_fw-la and_o dc_o wherefore_o the_o square_a bcgh_n be_v equal_a to_o the_o same_o square_n wherefore_o forasmuch_o as_o the_o square_a bcgh_n be_v compose_v of_o the_o square_a ab_fw-la and_o of_o the_o gnomon_n fgah_n the_o say_a gnomon_n shall_v be_v equal_a unto_o the_o square_a cd_o which_o be_v require_v to_o be_v do_v a_o other_o more_o ready_a way_n after_o pelitarius_n suppose_v that_o there_o be_v two_o square_n who_o side_n let_v be_v ab_fw-la and_o bc._n it_o be_v require_v unto_o the_o square_n of_o the_o line_n ab_fw-la to_o add_v a_o gnomon_n equal_a to_o the_o square_n of_o the_o line_n bc._n set_v the_o line_n ab_fw-la and_o bc_o in_o such_o sort_n that_o they_o make_v a_o right_a angle_n abc_n and_o draw_v a_o line_n from_o a_o to_o c._n and_o upon_o the_o line_n ab_fw-la describe_v a_o square_n which_o let_v be_v abde_n and_o produce_v the_o line_n basilius_n to_o the_o point_n fletcher_n and_o put_v the_o line_n bf_o equal_a to_o the_o line_n ac_fw-la and_o upon_o the_o line_n bf_o describe_v a_o square_n which_o let_v be_v bfgh_o which_o shall_v be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la when_o as_o the_o line_n bf_o and_o ac_fw-la be_v equal_a and_o therefore_o it_o be_v equal_a to_o the_o square_n of_o the_o two_o line_n ab_fw-la and_o bc._n now_o forasmuch_o as_o the_o square_a bfgh_o be_v make_v complete_a by_o the_o square_a abde_n and_o by_o the_o gnomon_n fegd_v the_o gnomon_n fegd_v shall_v be_v equal_a to_o the_o square_n of_o the_o line_n bc_o which_o be_v require_v to_o be_v do_v the_o 1._o theorem_a the_o 1._o proposition_n if_o there_o be_v two_o right_a line_n and_o if_o the_o one_o of_o they_o be_v divide_v into_o part_n how_o many_o soever_o the_o rectangle_n figure_n comprehend_v under_o the_o two_o right_a line_n be_v equal_a to_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n undivided_a and_o under_o every_o one_o of_o the_o part_n of_o the_o other_o line_n svppose_v that_o there_o be_v two_o right_a line_n a_o and_o bc_o and_o let_v one_o of_o they_o namely_o bc_o be_v divide_v at_o all_o adventure_n in_o the_o point_n d_o and_o e._n then_o i_o say_v that_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n a_o and_o bc_o be_v equal_a unto_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n a_o and_o bd_o &_o unto_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n a_o and_o de_fw-fr and_o also_o unto_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n a_o and_o ec_o construction_n for_o from_o the_o point_n brayse_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o right_a line_n bc_o a_o perpendicular_a line_n bf_o &_o unto_o the_o line_n a_o by_o the_o three_o of_o the_o first_o put_v the_o line_n bg_o equal_a and_o by_o the_o point_n g_z by_o the_o 31._o of_o the_o first_o draw_v a_o parallel_n line_n unto_o the_o right_a line_n bc_o and_o let_v the_o same_o be_v gm_n and_o by_o the_o self_n same_o by_o the_o poor_n d_z e_o and_o c_o draw_v unto_o the_o line_n bg_o these_o parallel_a line_n dk_o demonstration_n el_n and_o ch._n now_o then_o the_o parallelogram_n bh_o be_v equal_a to_o these_o parallelogram_n bk_o dl_o and_o eh_o but_o the_o parallelogram_n bh_o be_v equal_a unto_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o bc._n for_o it_o be_v comprehend_v under_o the_o line_n gb_o &_o bc_o and_o the_o line_n gb_o be_v equal_a unto_o the_o line_n a_o and_o the_o parallelogram_n bk_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o bd_o for_o it_o be_v comprehend_v under_o the_o line_n gb_o and_o bd_o and_o bg_o be_v equal_a unto_o a_o and_o the_o parallelogram_n dl_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o de_fw-fr for_o the_o line_n dk_o that_o be_v bg_o be_v equal_a unto_o a_o and_o moreover_o likewise_o the_o parallelogram_n eh_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n a_o &_o ec_o wherefore_o that_o which_o be_v comprehend_v under_o the_o line_n a_o &_o bc_o be_v equal_a to_o that_o which_o be_v comprehend_v under_o the_o line_n a_o &_o bd_o &_o unto_o that_o which_o be_v comprehend_v under_o the_o line_n a_o and_o de_fw-fr and_o moreover_o unto_o that_o which_o be_v comprehend_v under_o the_o line_n a_o and_o ec_o if_o therefore_o there_o be_v two_o right_a line_n and_o if_o the_o one_o of_o they_o be_v divide_v into_o part_n how_o many_o soever_o the_o rectangle_n figure_n comprehend_v under_o the_o two_o right_a line_n be_v equal_a to_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n undivided_a and_o under_o every_o one_o of_o the_o part_n of_o the_o other_o line_n which_o be_v require_v to_o be_v demonstrate_v because_o that_o all_o the_o proposition_n of_o this_o second_o book_n for_o the_o most_o part_n be_v true_a both_o in_o line_n and_o in_o number_n and_o may_v be_v declare_v by_o both_o therefore_o have_v i_o have_v add_v to_o every_o proposition_n convenient_a number_n for_o the_o manifestation_n of_o the_o same_o and_o to_o the_o end_n the_o studious_a and_o diligent_a reader_n may_v the_o more_o full_o perceive_v and_o understand_v the_o agreement_n of_o this_o art_n of_o geometry_n with_o the_o science_n of_o arithmetic_n and_o how_o never_o &_o dear_a sister_n they_o be_v together_o so_o that_o the_o one_o can_v without_o great_a blemish_n be_v without_o the_o other_o i_o have_v here_o also_o join_v a_o little_a book_n of_o arithmetic_n write_v by_o one_o barlaam_n a_o greek_a author_n a_o man_n of_o great_a knowledge_n in_o which_o book_n be_v by_o the_o author_n demonstrate_v many_o of_o the_o self_n same_o propriety_n and_o passion_n in_o number_n which_o euclid_n in_o this_o his_o second_o book_n have_v demonstrate_v in_o magnitude_n
triangle_n it_o can_v not_o fall_v within_o nor_o in_o acuteangle_a triangle_n without_o for_o then_o the_o outward_a angle_n of_o a_o triangle_n shall_v be_v less_o than_o the_o inward_a and_o opposite_a angle_n which_o be_v contrary_a to_o the_o 16._o of_o the_o first_o triangle_n and_o this_o be_v to_o be_v note_v that_o although_o proper_o a_o acuteangle_n triangle_n by_o the_o definition_n thereof_o give_v in_fw-ge the_o first_o book_n be_v that_o triangle_n who_o angle_n be_v all_o acute_a yet_o forasmuch_o as_o there_o be_v no_o triangle_n but_o that_o it_o have_v a_o acute_a angle_n this_o proposition_n be_v to_o be_v understand_v &_o be_v true_a general_o in_o all_o kind_n of_o triangle_n whatsoever_o and_o may_v be_v declare_v by_o they_o as_o you_o may_v easy_o prove_v the_o 2._o problem_n the_o 14._o proposition_n unto_o a_o rectiline_a figure_n give_v to_o make_v a_o square_a equal_a svppose_v that_o the_o rectiline_a figure_n give_v be_v a._n it_o be_v require_v to_o make_v a_o square_a equal_a unto_o the_o rectiline_a figure_n a._n construction_n make_v by_o the_o 45._o of_o the_o first_o unto_o the_o rectiline_a figure_n a_o a_o equal_a rectangle_n parallelogram_n bcde_v now_o if_o the_o line_n be_v be_v equal_a unto_o the_o line_n ed_z then_z be_v the_o thing_n do_v which_o be_v require_v for_o unto_o the_o rectiline_a figure_n a_o be_v make_v a_o equal_a square_a bd._n but_o if_o not_o one_o of_o these_o line_n be_v &_o be_v ed_z the_o great_a let_v be_v be_v the_o great_a and_o let_v it_o be_v produce_v unto_o the_o point_fw-fr f._n and_o by_o the_o 3._o of_o the_o first_o put_v unto_o ed_z a_o equal_a line_n ef._n and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n bf_o into_o two_o equal_a part_n in_o the_o point_n g._n and_o make_v the_o centre_n the_o point_n g_o and_o the_o space_n gb_o or_o gf_o describe_v a_o semicircle_n bhf_n and_o by_o the_o 2._o petition_n extend_v the_o line_n de_fw-fr unto_o the_o point_n h._n demonstration_n and_o by_o the_o 1._o petition_n draw_v a_o line_n from_o g_z to_o h._n and_o forasmuch_o as_o the_o right_a line_n fb_o be_v divide_v into_o two_o equal_a part_n in_o the_o point_n g_o and_o into_o two_o unequal_a part_n in_o the_o point_n e_o therefore_o by_o the_o 5._o of_o the_o second_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n be_v and_o of_o together_o with_o the_o square_n which_o be_v make_v of_o the_o line_n eglantine_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n gf_o but_o the_o line_n gf_o be_v equal_a unto_o the_o line_n gh_o wherefore_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n be_v and_o of_o together_o with_o the_o square_n which_o be_v make_v of_o the_o line_n ge_z be_v equal_a to_o square_v which_o be_v make_v of_o the_o line_n gh_o but_o unto_o the_o square_n which_o be_v make_v of_o the_o line_n gh_o be_v equal_a the_o square_n which_o be_v make_v of_o the_o line_n he_o and_o ge_z by_o the_o 47._o of_o the_o first_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n be_v and_o of_o together_o with_o the_o square_v which_o be_v make_v of_o ge_z be_v equal_a to_o the_o square_n which_o be_v make_v of_o he_o and_o ge._n take_v away_o the_o square_a of_o the_o line_n eglantine_n common_a to_o they_o both_o wherefore_o the_o rectangle_n figure_n contain_v under_o the_o line_n be_v &_o of_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n eh_o but_o that_o which_o be_v contain_v under_o the_o line_n be_v and_o of_o be_v the_o parallelogram_n bd_o for_o the_o line_n of_o be_v equal_a unto_o the_o line_n ed._n wherefore_o the_o parallelogram_n bd_o be_v equal_a to_o the_o square_v which_o be_v make_v of_o the_o line_n he._n but_o the_o parallelogram_n bd_o be_v equal_a unto_o the_o rectiline_a figure_n a._n wherefore_o the_o rectiline_a figure_n a_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n he._n wherefore_o unto_o the_o rectiline_a figure_n give_v a_o be_v make_v a_o equal_a square_n describe_v of_o the_o line_n eh_o which_o be_v require_v to_o be_v do_v ¶_o the_o end_n of_o the_o second_o book_n of_o euclides_n element_n ¶_o the_o three_o book_n of_o euclides_n element_n book_n this_o three_o book_n of_o euclid_n entreat_v of_o the_o most_o perfect_a figure_n which_o be_v a_o circle_n wherefore_o it_o be_v much_o more_o to_o be_v esteem_v then_o the_o two_o book_n go_v before_o in_o which_o he_o do_v set_v forth_o the_o most_o simple_a propriety_n of_o rightline_v figure_n for_o science_n take_v their_o dignity_n of_o the_o worthiness_n of_o the_o matter_n that_o they_o entreat_v of_o but_o of_o all_o figure_n the_o circle_n be_v of_o most_o absolute_a perfection_n who_o propriety_n and_o passion_n be_v here_o set_v forth_o and_o most_o certain_o demonstrate_v here_o also_o be_v entreat_v of_o right_a line_n subtend_v to_o arke_n in_o circle_n also_o of_o angle_n set_v both_o at_o the_o circumference_n and_o at_o the_o centre_n of_o a_o circle_n and_o of_o the_o variety_n and_o difference_n of_o they_o wherefore_o the_o read_v of_o this_o book_n be_v very_o profitable_a to_o the_o attain_v to_o the_o knowledge_n of_o chord_n and_o arke_n it_o teach_v moreover_o which_o be_v circle_n contingent_a and_o which_o be_v cut_v the_o one_o the_o other_o and_o also_o that_o the_o angle_n of_o contingence_n be_v the_o least_o of_o all_o acute_a rightline_v angle_n and_o that_o the_o diameter_n in_o a_o circle_n be_v the_o long_a line_n that_o can_v be_v draw_v in_o a_o circle_n far_o in_o it_o may_v we_o learn_v how_o three_o point_n be_v give_v how_o soever_o so_o that_o they_o be_v not_o set_v in_o a_o right_a line_n may_v be_v draw_v a_o circle_n pass_v by_o they_o all_o three_o again_o how_o in_o a_o solid_a body_n as_o in_o a_o sphere_n cube_n or_o such_o like_a may_v be_v find_v the_o two_o opposite_a point_n which_o be_v a_o thing_n very_o necessary_a and_o commodious_a chief_o for_o those_o that_o shall_v make_v instrument_n serve_v to_o astronomy_n and_o other_o art_n definition_n definition_n equal_a circle_n be_v such_o who_o diameter_n be_v equal_a or_o who_o line_n draw_v from_o the_o centre_n be_v equal_a the_o circle_n a_o and_o b_o be_v equal_a if_o their_o diameter_n namely_o of_o and_o cd_o be_v equal_a or_o if_o their_o semidiameter_n which_o be_v line_n draw_v from_o the_o centre_n to_o the_o circumferenceâ—_n namely_o of_o and_o bd_o be_v equal_a by_o this_o also_o be_v know_v the_o definition_n of_o unequal_a circle_n circle_n circle_n who_o diameter_n or_o semidiameter_n be_v unequal_a be_v also_o unequal_a and_o that_o circled_a which_o have_v the_o great_a diameter_n or_o semidiameter_n be_v the_o great_a circle_n and_o that_o circle_v which_o have_v the_o less_o diameter_n or_o semidiameter_n be_v the_o less_o circle_n a_o right_a line_n be_v say_v to_o touch_v a_o circle_n definition_n which_o touch_v the_o circle_n and_o be_v produce_v cut_v it_o not_o as_o the_o right_a line_n of_o draw_v from_o the_o point_n e_o and_o pass_a by_o a_o point_n of_o the_o circle_n namely_o by_o the_o point_n g_o to_o the_o point_n fletcher_n only_o touch_v the_o circle_n gh_o and_o cut_v it_o not_o nor_o enter_v within_o it_o for_o a_o right_a line_n enter_v within_o a_o circle_n cut_v and_o divide_v the_o circle_n as_o the_o right_a line_n kl_o divide_v and_o cut_v the_o circle_n klm_n and_o enter_v within_o it_o and_o therefore_o touch_v it_o in_o two_o place_n but_o a_o right_a line_n touch_v a_o circle_n which_o be_v common_o call_v a_o contingent_a line_n line_n touch_v the_o circle_n only_o in_o one_o point_n circle_n be_v say_v to_o touch_v the_o one_o the_o other_o definition_n which_o touch_v the_o one_o the_o other_o cut_v not_o the_o one_o the_o other_o as_o the_o two_o circle_n ab_fw-la and_o bc_o touch_v the_o one_o the_o other_o for_o their_o circumference_n touch_v together_o in_o the_o point_n b._n but_o neither_o of_o they_o cut_v or_o divide_v the_o other_o neither_o do_v any_o part_n of_o the_o one_o enter_v within_o the_o other_o only_o and_o such_o a_o touch_n of_o circle_n be_v ever_o in_o one_o point_n only_o which_o point_n only_o be_v common_a to_o they_o both_o as_o the_o point_n b_o be_v in_o the_o conference_n of_o the_o circle_n ab_fw-la and_o also_o 〈…〉_o the_o â—â—â—â—â—ference_n of_o the_o circle_n bc._n way_n circle_n may_v touch_v together_o two_o manner_n of_o way_n either_o outward_o the_o one_o whole_o without_o the_o other_o or_o else_o the_o one_o be_v contain_v within_o the_o other_o as_o the_o circle_n de_fw-fr and_o df_o of_o which_o the_o one_o de_fw-fr contain_v the_o other_o namely_o df_o and_o touch_v the_o one_o
a_o circle_n be_v take_v any_o point_n which_o be_v not_o the_o centre_n of_o the_o circle_n and_o from_o that_o point_n be_v draw_v unto_o the_o circumference_n certain_a right_a line_n the_o great_a of_o those_o line_n shall_v be_v that_o line_n wherein_o be_v the_o centre_n and_o the_o lest_o shall_v be_v the_o residue_n of_o the_o same_o line_n and_o of_o all_o the_o other_o line_n that_o which_o be_v nigh_o to_o the_o line_n which_o pass_v by_o the_o centre_n be_v great_a than_o that_o which_o be_v more_o distant_a and_o from_o that_o point_n can_v fall_v within_o the_o circle_n on_o each_o side_n of_o the_o least_o line_n only_o two_o equal_a right_a line_n svppose_v that_o there_o be_v a_o circle_n abcd_o and_o let_v the_o diameter_n thereof_o be_v ad._n and_o take_v in_o it_o any_o point_n beside_o the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v f._n construction_n and_o let_v the_o centre_n of_o the_o circle_n by_o the_o 1._o of_o the_o three_o be_v the_o point_n e._n and_o from_o the_o point_n f_o let_v there_o be_v draw_v unto_o the_o circumference_n abcd_o these_o right_a line_n fd_v fc_o and_o fg._n then_o i_o say_v that_o the_o line_n favorina_n be_v the_o great_a and_o the_o line_n fd_o be_v the_o jest_n and_o of_o the_o other_o line_n the_o line_n fb_o be_v great_a than_o the_o line_n fc_o and_o the_o line_n fc_o be_v great_a than_o the_o line_n fg._n proposition_n draw_v by_o the_o first_o petition_n these_o right_a line_n be_v ce_fw-fr and_o ge._n and_o for_o asmuch_o as_o by_o the_o 20._o of_o the_o first_o in_o every_o triangle_n two_o side_n be_v great_a than_o the_o three_o demonstration_n therefore_o the_o line_n ebb_v and_o of_o be_v great_a than_o the_o residue_n namely_o then_o the_o line_n fb_o but_o the_o line_n ae_n be_v equal_a unto_o the_o line_n be_v by_o the_o 15._o definition_n of_o the_o first_o wherefore_o the_o line_n be_v and_o of_o be_v equal_a unto_o the_o line_n af._n wherefore_o the_o line_n of_o be_v great_a than_o then_o the_o line_n bf_o again_o for_o asmuch_o as_o the_o line_n be_v be_v equal_a unto_o ce_fw-fr by_o the_o 15._o definition_n of_o the_o first_o and_o the_o line_n fe_o be_v common_a unto_o they_o both_o therefore_o these_o two_o line_n be_v and_o of_o be_v equal_a unto_o these_o two_o ce_fw-fr and_o ef._n but_o the_o angle_n bef_n be_v great_a than_o the_o angle_n cef_n wherefore_o by_o the_o 24._o of_o the_o first_o the_o base_a bf_o be_v great_a than_o the_o base_a cf_o and_o by_o the_o same_o reason_n the_o line_n cf_o be_v great_a than_o the_o line_n fg._n again_o for_o asmuch_o as_o the_o line_n gf_o and_o fe_o be_v great_a than_o the_o line_n eglantine_n by_o the_o 20._o of_o the_o first_o part_n but_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_n eglantine_n be_v equal_a unto_o the_o line_n ed_z wherefore_o the_o line_n gf_o and_o fe_o be_v great_a than_o the_o line_n ed_z take_v away_o of_o which_o be_v common_a to_o they_o both_o wherefore_o the_o residue_n gf_o be_v great_a than_o the_o residue_n fd._n wherefore_o the_o line_n favorina_n be_v the_o great_a and_o the_o line_n fd_o be_v the_o jest_n and_o the_o line_n fb_o be_v great_a than_o the_o line_n fc_o and_o the_o line_n fc_o be_v great_a than_o the_o line_n fg._n part_n now_o also_o i_o say_v that_o from_o the_o point_n f_o there_o can_v be_v draw_v only_o two_o equal_a right_a line_n into_o the_o circle_n abcd_o on_o each_o side_n of_o the_o least_o line_n namely_o fd._n for_n by_o the_o 23._o of_o the_o first_o upon_o the_o right_a line_n give_v of_o and_o to_o the_o point_n in_o it_o namely_o e_o make_v unto_o the_o angle_n gef_n a_o equal_a angle_n feh_n and_o by_o the_o first_o petition_n draw_v a_o line_n from_o fletcher_n to_o h._n now_o for_o asmuch_o as_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_n eglantine_n be_v equal_a unto_o the_o line_n eh_o and_o the_o line_n of_o be_v common_a unto_o they_o both_o therefore_o these_o two_o line_n ge_z and_o of_o be_v equal_a unto_o these_o two_o line_n he_o and_o of_o and_o by_o construction_n the_o angle_n gef_n be_v equal_a unto_o the_o angle_n hef_fw-fr wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a fg_o be_v equal_a unto_o the_o base_a fh_o impossibilie_n i_o say_v moreover_o that_o from_o the_o point_n f_o can_v be_v draw_v into_o the_o circle_n no_o other_o right_a line_n equal_a unto_o the_o line_n fg._n for_o if_o it_o possible_a let_v the_o lineâ—_z fk_o be_v equal_a unto_o the_o line_n fg._n and_o for_o asmuch_o as_o fk_n be_v equal_a unto_o fg._n but_o the_o line_n fh_o be_v equal_a unto_o the_o line_n fg_o therefore_o the_o line_n fk_o be_v equal_a unto_o the_o line_n fh_o wherefore_o the_o line_n which_o be_v nigh_o to_o the_o line_n which_o pass_v by_o the_o centre_n be_v equal_a to_o that_o which_o be_v far_o of_o which_o we_o have_v before_o prove_v to_o be_v impossible_a impossibility_n or_o else_o it_o may_v thus_o be_v demonstrate_v draw_v by_o the_o first_o petition_n a_o line_n from_o e_o to_o king_n and_o for_o asmuch_o as_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_v ge_z be_v equal_a unto_o the_o line_n eke_o and_o the_o line_n fe_o be_v common_a to_o they_o both_o and_o the_o base_a gf_o be_v equal_a unto_o the_o base_a fk_n therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n gef_n be_v equal_a to_o the_o angle_n kef_n but_o the_o angle_n gef_n be_v equal_a to_o the_o angle_n hef_fw-fr wherefore_o by_o the_o first_o common_a sentence_n the_o angle_n hef_fw-fr be_v equal_a to_o the_o angle_n kef_n the_o less_o unto_o the_o great_a which_o be_v impossible_a wherefore_o from_o the_o point_n fletcher_n there_o can_v be_v draw_v into_o the_o circle_n no_o other_o right_a line_n equal_a unto_o the_o line_n gf_o wherefore_o but_o one_o only_a if_o therefore_o in_o the_o diameter_n of_o a_o circle_n be_v take_v any_o point_n which_o be_v not_o the_o centre_n of_o the_o circle_n and_o from_o that_o point_n be_v draw_v unto_o the_o circumference_n certain_a right_a line_n the_o great_a of_o those_o right_a line_n shall_v be_v that_o wherein_o be_v the_o centre_n and_o the_o least_o shall_v be_v the_o residue_n and_o of_o all_o the_o other_o line_n that_o which_o be_v nigh_o to_o the_o line_n which_o pass_v by_o the_o centre_n be_v great_a than_o that_o which_o be_v more_o distant_a and_o from_o that_o point_n can_v fall_v within_o the_o circle_n on_o each_o side_n of_o the_o least_o line_n only_o two_o equal_a right_a line_n which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n corollary_n hereby_o it_o be_v manifest_a that_o two_o right_a line_n be_v draw_v from_o any_o one_o point_n of_o the_o diameter_n the_o one_o of_o one_o side_n and_o the_o other_o of_o the_o other_o side_n if_o with_o the_o diameter_n they_o make_v equal_a angle_n the_o say_v two_o right_a line_n be_v equal_a as_o in_o this_o place_n be_v the_o two_o line_n fg_o and_o fh_o the_o 7._o theorem_a the_o 8._o proposition_n if_o without_o a_o circle_n be_v take_v any_o point_n and_o from_o that_o point_n be_v draw_v into_o the_o circle_n unto_o the_o circumference_n certain_a right_a line_n of_o which_o let_v one_o be_v draw_v by_o the_o centre_n and_o let_v the_o rest_n be_v draw_v at_o all_o adventure_n the_o great_a of_o those_o line_n which_o fall_v in_o the_o concavity_n or_o hollowness_n of_o the_o circumference_n of_o the_o circle_n be_v that_o which_o pass_v by_o the_o centre_n and_o of_o all_o the_o other_o line_n that_o line_n which_o be_v nigh_o to_o the_o line_n which_o pass_v by_o the_o centre_n be_v great_a than_o that_o which_o be_v more_o distant_a but_o of_o those_o right_a line_n which_o end_n in_o the_o convexe_n part_n of_o the_o circumference_n that_o be_v the_o least_o which_o be_v draw_v from_o the_o point_n to_o the_o diameter_n and_o of_o the_o other_o line_n that_o which_o be_v nigh_o to_o the_o least_o be_v always_o less_o than_o that_o which_o be_v more_o distant_a and_o from_o that_o point_n can_v be_v draw_v unto_o the_o circumference_n on_o each_o side_n of_o the_o least_o only_o two_o equal_a right_a line_n part_n now_o also_o i_o say_v that_o from_o the_o point_n d_o can_v be_v draw_v unto_o the_o circumference_n on_o each_o side_n of_o dg_o the_o least_o only_o two_o equal_a right_a line_n upon_o the_o right_a line_n md_o and_o unto_o the_o point_n in_o it_o m_o make_v by_o the_o 23._o of_o the_o first_o unto_o the_o angle_n kmd_v a_o equal_a angle_n dmb._n and_o by_o the_o first_o petition_n draw_v a_o line_n from_o d_z to_o b._n and_o for_o asmuch_o as_o by_o the_o 15._o
definition_n of_o the_o first_o the_o line_n mb_n be_v equal_a unto_o the_o line_n mk_o put_v the_o line_n md_o common_a to_o the_o both_o wherefore_o these_o two_o line_n mk_o and_o md_o be_v equal_a to_o these_o two_o line_n bm_n and_o md_o the_o one_o to_o the_o other_o and_o the_o angle_n kmd_v be_v by_o the_o 23._o of_o the_o first_o equal_a to_o the_o angle_n bmd_v wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a dk_o be_v equal_a to_o the_o base_a db._n or_o it_o may_v thus_o be_v demonstrate_v impossibility_n draw_v by_o the_o first_o petition_n a_o line_n from_o m_o to_o n._n and_o for_o asmuch_o as_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_n km_o be_v equal_a unto_o the_o line_n mn_o and_o the_o line_n md_o be_v common_a to_o they_o both_o and_o the_o base_a kd_v be_v equal_a to_o the_o base_a dn_o by_o supposition_n therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n kmd_v be_v equal_a to_o the_o angle_n dmn_n but_o the_o angle_n kmd_v be_v equal_a to_o the_o angle_n bmd_v wherefore_o the_o angle_n bmd_v be_v equal_a to_o the_o angle_n nmd_v the_o less_o unto_o the_o great_a which_o be_v impossible_a wherefore_o from_o the_o point_n d_o can_v not_o be_v draw_v unto_o the_o circumference_n abc_n on_o each_o side_n of_o dg_o the_o jest_n more_o than_o two_o equal_a right_a line_n if_o therefore_o without_o a_o circle_n be_v take_v any_o point_n and_o from_o that_o point_n be_v draw_v into_o the_o circle_n unto_o the_o circumference_n certain_a right_a line_n of_o which_o let_v one_o be_v draw_v by_o the_o centre_n and_o let_v the_o rest_n be_v draw_v at_o all_o adventure_n the_o great_a of_o those_o right_a line_n which_o fall_v in_o the_o concavity_n or_o hollowness_n of_o the_o circumference_n of_o the_o circle_n be_v that_o which_o pass_v by_o the_o centre_n and_o of_o all_o the_o other_o line_n that_o line_n which_o be_v nigh_o to_o the_o line_n which_o pass_v by_o the_o centre_n be_v great_a than_o that_o which_o be_v more_o distant_a but_o of_o those_o right_a line_n which_o end_n in_o the_o convexe_n part_n of_o the_o circumference_n that_o line_n be_v the_o lest_o which_o be_v draw_v from_o the_o point_n to_o the_o dimetient_fw-la and_o of_o the_o other_o line_n that_o which_o be_v nigh_o to_o the_o least_o be_v always_o less_o than_o that_o which_o be_v more_o distant_a and_o from_o that_o point_n can_v be_v draw_v unto_o the_o circumference_n on_o each_o side_n of_o the_o lest_o only_a two_o equal_a right_a line_n which_o be_v require_v to_o be_v prove_v this_o proposition_n be_v call_v common_o in_o old_a book_n amongst_o the_o barbarous_a caâ—dâ—_n panonis_n panonis_fw-la that_o be_v the_o peacock_n tail_n ¶_o a_o corollary_n hereby_o it_o be_v manifest_a corollary_n that_o the_o right_a line_n which_o be_v draw_v from_o the_o point_n give_v without_o the_o circle_n and_o fall_v within_o the_o circle_n be_v equal_o distant_a from_o the_o least_o or_o from_o the_o great_a which_o be_v draw_v by_o the_o centre_n be_v equal_a the_o one_o to_o the_o other_o but_o contrariwise_o if_o they_o be_v unequal_o distant_a whether_o they_o light_v upon_o the_o concave_n or_o convexe_a circumference_n of_o the_o circle_n they_o be_v unequal_a the_o 8._o theorem_a the_o 9_o proposition_n if_o within_o a_o circle_n be_v take_v a_o point_n and_o from_o that_o point_n be_v draw_v unto_o the_o circumference_n more_o than_o two_o equal_a right_a line_n the_o point_n take_v be_v the_o centre_n of_o the_o circle_n svppose_v that_o the_o circle_n be_v abc_n and_o within_o it_o let_v there_o be_v take_v the_o point_n d._n and_o from_o d_z let_v there_o be_v draw_v unto_o the_o circumference_n abc_n more_o than_o two_o equal_a right_a line_n construction_n that_o be_v da_z db_o and_o dc_o then_o i_o say_v that_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n abc_n draw_v by_o the_o first_o petition_n these_o right_a line_n ab_fw-la and_o bc_o demonstration_n and_o by_o the_o 10._o of_o the_o first_o divide_v they_o into_o two_o equal_a part_n in_o the_o point_n e_o and_o f_o namely_o the_o line_n ab_fw-la in_o the_o point_n e_o and_o the_o line_n bc_o in_o the_o point_n f._n and_o draw_v the_o line_n ed_z and_o fd_o and_o by_o the_o second_o petition_n extend_v the_o line_n ed_z and_o fd_v on_o each_o side_n to_o the_o point_n king_n g_o and_o h_o l._n and_o for_o asmuch_o as_o the_o line_n ae_n be_v equal_a unto_o the_o line_n ebb_n and_o the_o line_n ed_z be_v common_a to_o they_o both_o therefore_o these_o two_o side_n ae_n and_o ed_z be_v equal_a unto_o these_o two_o side_n be_v and_o ed_z and_o by_o supposition_n the_o base_a dam_n be_v equal_a to_o the_o base_a db._n wherefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n aed_n be_v equal_a to_o the_o angle_n bed_n wherefore_o either_o of_o these_o angle_n aed_n and_o bed_n be_v a_o right_a angle_n wherefore_o the_o line_n gk_o divide_v the_o line_n ab_fw-la into_o two_o equal_a part_n and_o make_v right_a angle_n and_o for_o asmuch_o as_z if_o in_o a_o circle_n a_o right_a line_n divide_v a_o other_o right_a line_n into_o two_o equal_a part_n in_o such_o sort_n that_o it_o make_v also_o right_a angle_n in_o the_o line_n that_o divide_v be_v the_o centre_n of_o the_o circle_n by_o the_o corollary_n of_o the_o first_o of_o the_o three_o therefore_o by_o the_o same_o corollary_n in_o the_o line_n gk_o be_v the_o centre_n of_o the_o circle_n abc_n and_o by_o the_o same_o reason_n may_v we_o prove_v that_o in_o the_o line_n hl_o be_v the_o centre_n of_o the_o circle_n abc_n and_o the_o right_a line_n gk_o and_o hl_o have_v no_o other_o point_n common_a to_o they_o both_o beside_o the_o point_n d._n wherefore_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n abc_n if_o therefore_o within_o a_o circle_n be_v take_v a_o point_n and_o from_o that_o point_n be_v draw_v unto_o the_o circumference_n more_o than_o two_o equal_a right_a line_n the_o point_n take_v be_v the_o centre_n of_o the_o circle_n which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n let_v there_o be_v take_v within_o the_o circle_n abc_n the_o point_n d._n impossibility_n and_o from_o the_o point_n d_o let_v there_o be_v draw_v unto_o the_o circumference_n more_o than_o two_o equal_a right_a line_n namely_o dam_n db_o and_o dc_o then_o i_o say_v that_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n for_o if_o not_o then_o if_o it_o be_v possible_a let_v the_o point_n e_o be_v the_o centre_n and_o draw_v a_o line_n from_o d_z to_o e_o and_o extend_v de_fw-fr to_o the_o point_n fletcher_n and_o g._n wherefore_o the_o line_n fg_o be_v the_o diameter_n of_o the_o circle_n abc_n and_o for_o asmuch_o as_o in_o fg_o the_o diameter_n of_o the_o circle_n abc_n be_v take_v a_o point_n namely_o d_z which_o be_v not_o the_o centre_n of_o that_o circle_n therefore_o by_o the_o 7._o of_o the_o three_o the_o line_n dg_o be_v the_o great_a and_o the_o line_n dc_o be_v great_a than_o the_o line_n db_o and_o the_o line_n db_o be_v greateâ—_n than_o the_o line_n da._n but_o the_o line_n dc_o db_o dam_n be_v also_o equal_a by_o supposition_n which_o be_v impossible_a wherefore_o the_o point_n e_o be_v not_o the_o centre_n of_o the_o circle_n abc_n and_o in_o like_a sort_n may_v we_o prove_v that_o no_o other_o point_n beside_o d._n wherefore_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n abc_n which_o be_v require_v to_o be_v prove_v the_o 9_o theorem_a the_o 10._o proposition_n a_o circle_n cut_v not_o a_o circle_n in_o more_o point_n than_o two_o for_o if_o it_o be_v possible_a let_v the_o circle_n abc_n cut_v the_o circle_n def_n in_o more_o point_n than_o two_o impossibility_n that_o be_v in_o b_o g_o h_o &_o f._n and_o draw_v line_n from_o b_o to_o g_z and_o from_o b_o to_o h._n and_o by_o the_o 10._o of_o the_o first_o divide_v either_o of_o the_o line_n bg_o &_o bh_o into_o two_o equal_a part_n in_o the_o point_n king_n and_o l._n and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n king_n raise_v up_o unto_o the_o line_n bh_o a_o perpendicular_a line_n kc_o and_o likewise_o from_o the_o point_n l_o raise_v up_o unto_o the_o line_n bg_o a_o perpendicular_a line_n lm_o and_o extend_v the_o line_n ck_o to_o the_o point_n a_o and_o lnm_n to_o the_o point_n x_o and_o e._n and_o for_o asmuch_o as_o in_o the_o circle_n abc_n the_o right_a line_n ac_fw-la divide_v the_o right_a line_n bh_o into_o two_o equal_a part_n and_o make_v right_a angle_n therefore_o by_o the_o 3._o of_o the_o three_o in_o the_o line_n ac_fw-la be_v the_o centre_n of_o
when_o perpendicular_a line_n draw_v from_o the_o centre_n to_o those_o line_n be_v equal_a by_o the_o 4._o definition_n of_o the_o three_o wherefore_o the_o line_n ab_fw-la and_o cd_o be_v equal_o distant_a from_o the_o centre_n but_o now_o suppose_v that_o the_o right_a line_n ab_fw-la and_o cd_o be_v equal_o distant_a from_o the_o centre_n first_o that_o be_v let_v the_o perpendicular_a line_n of_o be_v equal_a to_o the_o perpendicular_a line_n eglantine_n then_o i_o say_v that_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n cd_o for_o the_o same_o order_n of_o construction_n remain_v we_o may_v in_o like_a sort_n prove_v that_o the_o line_n ab_fw-la be_v double_a to_o the_o line_n of_o and_o that_o the_o line_n cd_o be_v double_a to_o the_o line_n cg_o and_o for_o asmuch_o as_o the_o line_n ae_n be_v equal_a to_o the_o line_n ce_fw-fr for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n therefore_o the_o square_a of_o the_o line_n ae_n be_v equal_a to_o the_o square_n of_o the_o line_n ce._n but_o by_o the_o 47._o of_o the_o first_o to_o the_o square_n of_o the_o line_n ae_n be_v equal_a the_o square_n of_o the_o line_n of_o and_o fa._n and_o by_o the_o self_n same_o to_o the_o square_n of_o the_o line_n ce_fw-fr be_v equal_v the_o square_n of_o the_o line_n eglantine_n and_o gc_o wherefore_o the_o square_n of_o the_o line_n of_o and_o favorina_n be_v equal_a to_o the_o square_n of_o the_o line_n eglantine_n and_o gc_o of_o which_o the_o square_a of_o the_o line_n eglantine_n be_v equal_a to_o the_o square_n of_o the_o line_n of_o for_o the_o line_n of_o be_v equal_a to_o the_o line_n eglantine_n wherefore_o by_o the_o three_o common_a sentence_n the_o square_a remain_v namely_o the_o square_a of_o the_o line_n of_o be_v equal_a to_o the_o square_n of_o the_o line_n cg_o wherefore_o the_o line_n ac_fw-la be_v equal_a unto_o the_o line_n cg_o but_o the_o line_n ab_fw-la be_v double_a to_o the_o line_n of_o and_o the_o line_n cd_o be_v double_a to_o the_o line_n cg_o wherefore_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n cd_o wherefore_o in_o a_o circle_n equal_a right_a line_n be_v equal_o distant_a from_o the_o centre_n and_o line_n equal_o distant_a from_o the_o centre_n be_v equal_a the_o one_o to_o the_o outstretch_o which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n for_o the_o first_o part_n after_o campane_n campane_n suppose_v that_o there_o be_v a_o circle_n abdc_n who_o centre_n let_v be_v the_o point_n e._n and_o draw_v in_o it_o two_o equal_a line_n ab_fw-la and_o cd_o then_o i_o say_v that_o they_o be_v equal_o distant_a from_o the_o centre_n draw_v from_o the_o centre_n unto_o the_o line_n ab_fw-la and_o cd_o these_o perpendicular_a line_n of_o and_o eglantine_n and_o by_o the_o 2._o part_n of_o the_o 3._o of_o this_o book_n the_o line_n ab_fw-la shall_v be_v equal_o divide_v in_o the_o point_n f._n and_o the_o line_n cd_o shall_v be_v equal_o divide_v in_o the_o point_n g._n and_o draw_v these_o right_a line_n ea_fw-la ebb_n ec_o and_o ed._n and_o for_o asmuch_o as_o in_o the_o triangle_n aeb_fw-mi the_o two_o side_n ab_fw-la and_o ae_n be_v equal_a to_o the_o two_o side_n cd_o and_o ce_fw-fr of_o the_o triangle_n ce_z &_o the_o base_a ebb_n be_v equal_a to_o the_o base_a ed._n therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n at_o the_o point_n a_o shall_v be_v equal_a to_o the_o angle_n at_o the_o point_n c._n and_o for_o asmuch_o as_o in_o the_o triangle_n aef_n the_o two_o side_n ae_n and_o of_o be_v equal_a to_o the_o two_o side_n ce_fw-fr and_o cg_o of_o the_o triangle_n ceg_n and_o the_o angle_n eaf_n be_v equal_a to_o the_o angle_n ceg_n therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a of_o iâ—_z equal_a to_o the_o base_a eglantine_n which_o for_o asmuch_o as_o they_o be_v perpendicular_a line_n therefore_o the_o line_n ab_fw-la &_o cd_o be_v equal_o distant_a from_o the_o centre_n by_o the_o 4._o definition_n of_o this_o book_n the_o 14._o theorem_a the_o 15._o proposition_n in_o a_o circle_n the_o great_a line_n be_v the_o diameter_n and_o of_o all_o other_o line_n that_o line_n which_o be_v nigh_o to_o the_o centre_n be_v always_o great_a than_o that_o line_n which_o be_v more_o distant_a svppose_v that_o there_o be_v a_o circle_n abcd_o and_o let_v the_o diameter_n thereof_o be_v the_o line_n ad_fw-la and_o let_v the_o centre_n thereof_o be_v the_o point_n e._n and_o unto_o the_o diameter_n ad_fw-la let_v the_o line_n bc_o be_v nigh_a than_o the_o line_n fg._n then_o i_o say_v that_o the_o line_n ad_fw-la be_v the_o great_a and_o the_o line_n bc_o be_v great_a than_o the_o line_n fg._n draw_v by_o the_o 12._o of_o the_o first_o from_o the_o centre_n e_o to_o the_o line_n bc_o and_o fg_o perpendicular_a line_n eh_o and_o eke_o construction_n and_o for_o asmuch_o as_o the_o line_n bc_o be_v nigh_a unto_o the_o centre_n than_o the_o line_n fg_o therefore_o by_o the_o 4._o definition_n of_o the_o three_o the_o line_n eke_o be_v great_a than_o the_o line_n eh_o and_o by_o the_o three_o of_o the_o first_o put_v unto_o the_o line_n eh_o a_o equal_a line_n el._n and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n l_o raise_v up_o unto_o the_o line_n eke_o a_o perpendicular_a line_n lm_o and_o extend_v the_o line_n lm_o to_o the_o point_n n._n and_o by_o the_o first_o petition_n draw_v these_o right_a line_n em_n en_fw-fr of_o and_o eglantine_n and_o for_o asmuch_o as_o the_o line_n eh_o be_v equal_a to_o the_o line_n el_fw-es therefore_o by_o the_o 14._o of_o the_o three_o demonstration_n and_o by_o the_o 4._o definition_n of_o the_o same_o the_o line_n bc_o be_v equal_a to_o the_o line_n mn_v again_o for_o asmuch_o as_o the_o line_n ae_n be_v equal_a to_o the_o line_n em_n and_o the_o line_n ed_z to_o the_o line_n en_fw-fr therefore_o the_o line_n ad_fw-la be_v equal_a to_o the_o line_n menander_n and_o en_fw-fr but_o the_o line_n i_o and_o en_fw-fr be_v by_o the_o 20._o of_o the_o first_o great_a then_o the_o line_n mn_v wherefore_o the_o line_n ad_fw-la be_v great_a than_o the_o line_n mn_v and_o for_o asmuch_o as_o these_o two_o line_n menander_n and_o en_fw-fr be_v equal_v to_o these_o two_o line_n fe_o and_o eglantine_n by_o the_o 15._o definition_n of_o the_o first_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n and_o the_o angle_n men_n be_v great_a than_o the_o angle_n feg_n therefore_o by_o the_o 24._o of_o the_o first_o the_o base_a mn_n be_v great_a than_o the_o base_a fg._n but_o it_o be_v prove_v that_o the_o line_n mn_o be_v equal_a to_o the_o line_n bc_o wherefore_o the_o line_n bc_o also_o be_v great_a than_o the_o line_n fg._n wherefore_o the_o diameter_n ad_fw-la be_v the_o great_a and_o the_o line_n bc_o be_v great_a than_o the_o line_n fg._n wherefore_o in_o a_o circle_n the_o great_a line_n be_v the_o diameter_n and_o of_o all_o the_o other_o line_n that_o line_n which_o be_v nigh_o to_o the_o 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of_o other_o rectiline_a figure_n consider_v figure_n by_o these_o two_o definition_n it_o be_v manifest_a that_o the_o inscription_n and_o circumscription_n of_o rectiline_a figure_n here_o speak_v of_o pertain_v to_o such_o rectiline_a figure_n only_o which_o have_v equal_a side_n and_o equal_a angle_n which_o be_v common_o call_v regular_a it_o be_v also_o to_o be_v note_v that_o rectiline_a figure_n only_o of_o one_o kind_n or_o form_n can_v be_v inscribe_v or_o circumscribe_v the_o one_o within_o or_o about_o the_o other_o as_o a_o triangle_n within_o or_o about_o a_o triangle_n a_o square_a within_o or_o about_o a_o square_n and_o so_o a_o pentagon_n within_o or_o about_o a_o pentagon_n &_o likewise_o of_o other_o of_o one_o form_n but_o a_o triangle_n can_v not_o be_v inscribe_v or_o circumscribe_v within_o or_o about_o a_o square_n nor_o a_o square_a within_o or_o about_o a_o pentagon_n and_o so_o of_o other_o of_o diverse_a kind_n for_o every_o plain_a rectiline_a figure_n 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it_o and_o therefore_o any_o regular_a rectiline_a figure_n may_v be_v inscribe_v within_o it_o and_o also_o be_v circumscribe_v about_o it_o and_o again_o a_o circle_n may_v be_v both_o inscribe_v within_o any_o regular_a rectiline_a figure_n and_o also_o be_v circumscribe_v about_o it_o as_o the_o triangle_n abc_n be_v inscribe_v in_o the_o circle_n abcâ—_n â—_z for_o that_o every_o angle_n touch_v some_o one_o point_n of_o the_o circumference_n of_o the_o circle_n as_o the_o angle_n cab_n of_o the_o triangle_n abc_n touch_v the_o point_n a_o of_o the_o circumference_n of_o the_o circle_n and_o the_o angle_n abc_n of_o the_o triangle_n touch_v the_o point_n b_o of_o the_o circumference_n of_o the_o circle_n and_o also_o the_o angle_n acb_o of_o the_o triangle_n 〈…〉_o the_o point_n â—_o of_o the_o circumference_n of_o the_o circle_n in_o like_a manner_n the_o square_a adef_n be_v inscribe_v in_o the_o same_o circle_n abc_n for_o that_o every_o angle_n of_o the_o square_n inscribe_v 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of_o the_o circle_n touch_v every_o one_o of_o the_o side_n of_o the_o figure_n within_o which_o it_o be_v inscribe_v as_o the_o circle_n abcd_o be_v inscribe_v within_o the_o triangle_n efg_o because_o the_o circumference_n of_o the_o circle_n touch_v every_o side_n of_o the_o triangle_n in_o which_o it_o be_v inscribedâ—_n namely_o the_o side_n of_o in_o the_o point_n b_o and_o the_o side_n gf_o in_o the_o point_n c_o and_o the_o side_n ge_z in_o the_o point_n d._n likewise_o by_o the_o same_o reason_n the_o same_o circle_n be_v inscribe_v within_o the_o square_a hikl_n and_o so_o may_v you_o judge_v of_o other_o rectiline_a figure_n a_o rectilined_a figure_n be_v say_v to_o be_v circumscribe_v about_o a_o circle_n devition_n when_o every_o one_o of_o the_o side_n of_o the_o figure_n circumscribe_v touch_v the_o circumference_n of_o the_o circle_n as_o in_o the_o former_a figure_n of_o the_o five_o definition_n the_o triangle_n efg_o be_v circumscribe_v about_o the_o circle_n abcd_o 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be_v say_v to_o be_v coapt_v or_o apply_v in_o a_o circle_n when_o the_o extreme_n or_o end_n thereof_o fall_v upon_o the_o circumference_n of_o the_o circle_n as_o the_o line_n bc_o be_v say_v to_o be_v coapt_v or_o to_o be_v apply_v to_o the_o circle_n abc_n for_o that_o both_o his_o extreme_n fall_v upon_o the_o circumference_n of_o the_o circle_n in_o the_o point_n b_o and_o c._n likewise_o the_o line_n de._n this_o definition_n be_v very_o necessary_a and_o be_v proper_o to_o be_v take_v of_o any_o line_n give_v to_o be_v coapt_v and_o apply_v into_o a_o circle_n so_o â—hat_n it_o exceed_v not_o the_o diameter_n of_o the_o circle_n give_v the_o 1._o problem_n the_o 1._o proposition_n in_o a_o circle_n give_v to_o apply_v a_o right_a line_n equal_a unto_o a_o right_a line_n give_v which_o exceed_v not_o the_o diameter_n of_o a_o circle_n svppose_v that_o the_o circle_n give_v be_v abc_n and_o let_v the_o right_a line_n give_v exceed_v not_o the_o diameter_n of_o the_o same_o circle_n be_v d._n now_o it_o be_v require_v in_o the_o circle_n give_v abc_n to_o apply_v a_o right_a line_n equal_a unto_o the_o right_a line_n d._n construction_n draw_v the_o diameter_n of_o the_o circle_n abc_n and_o let_v the_o same_o be_v bc._n now_o if_o the_o line_n bc_o be_v equal_a unto_o the_o line_n d_o then_o be_v that_o do_v which_o be_v require_v for_o in_o the_o circle_n give_v abc_n be_v apply_v a_o right_a line_n bc_o equal_a unto_o the_o right_a line_n d._n proposition_n but_o if_o not_o caseâ—_n then_o be_v the_o line_n bc_o great_a than_o the_o line_n d._n case_n and_o by_o the_o three_o of_o the_o first_o put_v unto_o the_o line_n d_o a_o equal_a line_n ce._n and_o make_v the_o centre_n c_o and_o the_o space_n ce_fw-fr describe_v by_o the_o three_o petition_n a_o circle_n egf_n cut_v the_o circle_n abc_n in_o the_o point_n fletcher_n &_o draw_v a_o line_n from_o c_o to_o f._n and_o for_o asmuch_o as_o the_o point_n c_o be_v the_o centre_n of_o the_o circle_n egf_n demonstration_n therefore_o by_o the_o 13._o definition_n of_o the_o first_o the_o line_n cf_o be_v equal_a unto_o the_o line_n ce._n but_o the_o line_n ce_fw-fr be_v equal_a unto_o the_o line_n d._n wherefore_o by_o the_o first_o common_a sentence_n the_o line_n cf_o also_o be_v equal_a unto_o the_o line_n d._n wherefore_o in_o the_o circle_n give_v abc_n be_v apply_v a_o right_a line_n ca_n equal_a unto_o the_o right_a line_n give_v d_z which_o be_v require_v to_o be_v do_v the_o 2._o problem_n the_o 2._o proposition_n in_o a_o circle_n give_v to_o describe_v a_o triangle_n equiangle_n unto_o a_o triangle_n give_v svppose_v that_o the_o circle_n give_v be_v abc_n and_o let_v the_o triangle_n give_v be_v def_n now_o it_o be_v require_v in_o the_o circle_n give_v abc_n to_o describe_v a_o triangle_n equiangle_n unto_o the_o triangle_n give_v def_n
rational_a part_n rather_o than_o of_o the_o medial_a part_n ¶_o the_o 28._o theorem_a the_o 40._o proposition_n if_o two_o right_a line_n incommensurable_a in_o power_n be_v add_v together_o have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o the_o parallelogram_n contain_v under_o they_o rational_a the_o whole_a right_a line_n be_v irrational_a and_o be_v call_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n in_o this_o proposition_n be_v teach_v the_o generation_n of_o the_o six_o irrational_a line_n which_o be_v call_v a_o line_n who_o power_n be_v rational_a and_o medial_a the_o definition_n of_o which_o be_v gather_v of_o this_o proposition_n after_o this_o manner_n a_o line_n who_o power_n be_v rational_a and_o medial_a be_v a_o irrational_a line_n which_o be_v make_v of_o two_o right_a line_n incommensurable_a in_o power_n add_v together_o medial_a who_o square_n add_v together_o make_v a_o medial_a superficies_n but_o that_o supersicy_n which_o they_o contain_v be_v rational_a the_o reason_n of_o the_o name_n be_v before_o set_v forth_o in_o the_o proposition_n ¶_o the_o 29._o theorem_a the_o 41._o proposition_n if_o two_o right_a line_n incommensurable_a in_o power_n be_v add_v together_o have_v that_o which_o be_v compose_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o the_o parallelogram_n contain_v under_o they_o medial_a and_o also_o incommensurable_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o they_o add_v togethers_o the_o whole_a right_a line_n be_v irrational_a and_o be_v call_v a_o line_n contain_v in_o power_n two_o medial_n in_o this_o proposition_n be_v teach_v the_o nature_n of_o the_o 7._o kind_n of_o irrational_a line_n which_o be_v call_v a_o line_n who_o power_n be_v two_o medial_n the_o definition_n whereof_o be_v take_v of_o this_o proposition_n after_o this_o manner_n a_o line_n who_o power_n be_v two_o medial_n be_v a_o irrational_a line_n which_o be_v compose_v of_o two_o right_a line_n incommensurable_a in_o power_n medial_n the_o square_n of_o which_o add_v together_o make_v a_o medial_a superficies_n and_o that_o which_o be_v contain_v under_o they_o be_v also_o medial_a and_o moreover_o it_o be_v incommensurable_a to_o that_o which_o be_v compose_v of_o the_o two_o square_n add_v together_o the_o reason_n why_o this_o line_n be_v call_v a_o line_n who_o power_n be_v two_o medial_n be_v before_o in_o the_o end_n of_o the_o demonstration_n declare_v and_o that_o the_o say_v irrational_a line_n be_v divide_v one_o way_n only_o that_o be_v in_o one_o point_n only_o into_o the_o right_a line_n of_o which_o they_o be_v compose_v and_o which_o make_v every_o one_o of_o the_o kind_n of_o those_o irrational_a line_n shall_v straight_o way_n be_v demonstrate_v but_o first_o will_v we_o demonstrate_v two_o assumpte_n here_o follow_v ¶_o a_o assumpt_n take_v a_o right_a line_n and_o let_v the_o same_o be_v ab_fw-la and_o divide_v it_o into_o two_o unequal_a part_n in_o the_o point_n c_o assumpt_n and_o again_o divide_v the_o same_o line_n ab_fw-la into_o two_o other_o unequal_a part_n in_o a_o other_o point_n namely_o in_o d_o and_o let_v the_o line_n ac_fw-la by_o supposition_n be_v great_a than_o the_o line_n db._n then_o i_o say_v that_o the_o square_n of_o the_o line_n ac_fw-la and_o bc_o add_v together_o be_v great_a than_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o divide_v the_o line_n ab_fw-la by_o the_o 10._o of_o the_o first_o into_o two_o equal_a part_n in_o the_o point_n e._n and_o forasmuch_o as_o the_o line_n ac_fw-la be_v great_a than_o the_o line_n db_o take_v away_o the_o line_n dc_o which_o be_v common_a to_o they_o both_o wherefore_o the_o residue_n ad_fw-la be_v great_a than_o the_o residue_n cb_o but_o the_o line_n ae_n be_v equal_a to_o the_o line_n ebb_n wherefore_o the_o line_n de_fw-fr be_v less_o than_o the_o line_n ec_o wherefore_o the_o point_n c_o and_o d_o be_v not_o equal_o distant_a from_o the_o point_n e_o which_o be_v the_o point_n of_o the_o section_n into_o two_o equal_a part_n and_o forasmuch_o as_o by_o the_o 5._o of_o the_o second_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o together_o with_o the_o square_n of_o the_o line_n ec_o be_v equal_a to_o the_o square_n of_o the_o line_n ebb_n and_o by_o the_o same_o reason_n that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o together_o with_o the_o square_n of_o the_o line_n de_fw-fr be_v also_o equal_a to_o the_o self_n same_o square_n of_o the_o line_n ebb_v wherefore_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o together_o with_o the_o square_n of_o the_o line_n ec_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o together_o with_o the_o square_n of_o the_o line_n de_fw-fr of_o which_o the_o square_a of_o the_o line_n de_fw-fr be_v less_o than_o the_o square_a of_o the_o line_n ec_o for_o it_o be_v prove_v that_o the_o line_n de_fw-fr be_v less_o than_o the_o line_n ec_o wherefore_o the_o parallelogram_n remain_v contain_v under_o the_o line_n ac_fw-la and_o cb_o be_v less_o they_o the_o parallelogram_n remain_v contain_v under_o the_o line_n ad_fw-la and_o db._n wherefore_o also_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v less_o than_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o but_o by_o the_o four_o of_o the_o second_o the_o square_a of_o the_o whole_a line_n ab_fw-la be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o and_o by_o the_o same_o reason_n the_o square_n of_o the_o whole_a line_n ab_fw-la be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o but_o it_o be_v already_o prove_v that_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v less_o than_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la &_o db_o twice_o wherefore_o the_o residue_n namely_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v great_a than_o the_o residue_n namely_o then_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o assumpt_n a_o rational_a superficies_n exceed_v a_o rational_a superficies_n by_o a_o rational_a superficies_n let_v ad_fw-la be_v a_o rational_a superficies_n and_o let_v it_o exceed_v of_o be_v also_o a_o rational_a superficies_n by_o the_o superficies_n ed._n then_o i_o say_v that_o the_o superficies_n ed_z be_v also_o rational_a for_o the_o parallelogram_n ad_fw-la be_v commensurable_a to_o the_o parallelogram_n of_o for_o that_o either_o of_o they_o be_v rational_a wherefore_o by_o the_o second_o part_n of_o the_o 15._o of_o the_o ten_o the_o parallelogram_n of_o be_v commensurable_a to_o the_o parallelogram_n ed._n but_o the_o the_o parallelogram_n of_o be_v rational_a wherefore_o also_o the_o parallelogram_n ed_z be_v rational_a ¶_o the_o 30._o theorem_a the_o 42._o proposition_n a_o binomial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n composition_n svppose_v that_o ab_fw-la be_v a_o binomial_a line_n and_o in_o the_o point_n g_o let_v it_o be_v divide_v into_o his_o name_n that_o be_v into_o the_o line_n whereof_o the_o whole_a line_n ab_fw-la be_v compose_v wherefore_o these_o line_n ac_fw-la and_o cb_o be_v rational_a commensurable_a in_o power_n only_o now_o i_o say_v that_o the_o line_n ab_fw-la can_v in_o any_o other_o point_n beside_o c_o be_v divide_v into_o two_o rational_a line_n commensurable_a in_o power_n only_o impossibility_n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v in_o the_o point_n d_o so_fw-mi that_o let_v the_o line_n ad_fw-la and_o de_fw-fr bâ—_z rational_a commensurable_a in_o power_n only_o first_o this_o manifest_a that_o neither_o of_o these_o point_n c_o and_o d_o divide_v the_o right_a line_n ab_fw-la into_o two_o equal_a part_n otherwise_o the_o line_n ac_fw-la and_o cb_o shall_v be_v rational_a commensurable_a in_o
length_n and_o so_o likewise_o shall_v the_o line_n ad_fw-la and_o db_o be_v for_o every_o line_n measure_v itself_o and_o any_o other_o line_n equal_a to_o itself_o moreover_o the_o line_n db_o be_v either_o one_o and_o the_o same_o with_o the_o line_n acâ—_n that_o be_v be_v equal_a to_o the_o line_n ac_fw-la oâ—_n else_o it_o be_v greater-then_a the_o line_n ac_fw-la either_o else_o it_o be_v less_o than_o it_o if_o db_o be_v equal_a to_o the_o line_n ac_fw-la then_o put_v the_o line_n db_o upon_o the_o line_n ac_fw-la each_o end_n of_o the_o one_o shall_v agree_v with_o each_o end_n of_o the_o other_o wherefore_o put_v the_o point_n b_o upon_o the_o point_n a_o the_o point_n d_o also_o shall_v fall_v upon_o the_o point_n c_o and_o the_o line_n ad_fw-la which_o be_v the_o rest_n of_o the_o line_n ac_fw-la shall_v also_o be_v equal_a to_o the_o line_n cb_o which_o be_v the_o rest_n of_o the_o line_n db._n wherefore_o the_o line_n ab_fw-la be_v divide_v into_o his_o name_n in_o the_o point_n c._n and_o so_o also_o shall_v the_o line_n ab_fw-la be_v divide_v in_o the_o point_n d_o be_v divide_v in_o the_o self_n â—ame_n point_n that_o the_o self_n same_o line_n ab_fw-la be_v before_o divide_v in_o the_o point_n c_o which_o be_v contrary_a to_o the_o supposition_n for_o by_o supposition_n it_o be_v divide_v in_o sundry_a point_n namely_o in_o c_o &_o d._n but_o if_o the_o line_n db_o be_v greaterâ—_n the_o the_o line_n ac_fw-la let_v the_o line_n ab_fw-la be_v deâ—ided_v into_o two_o equal_a part_n in_o the_o point_n e._n wherefore_o the_o point_n c_o &_o d_o shall_v not_o equal_o be_v distant_a from_o the_o point_n e_o now_o by_o the_o first_o assupt_v go_v before_o this_o proposition_n that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la &_o db_o be_v great_a they_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la &_o cbâ—_n but_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la &_o db_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la &_o db_o twice_o be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la &_o cb_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o for_o either_o of_o they_o be_v equal_a to_o the_o square_n of_o the_o whole_a line_n ab_fw-la by_o the_o 4._o of_o the_o second_o wherefore_o how_o much_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o be_v great_a than_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o so_o much_o be_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o great_a than_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o but_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o exceed_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o by_o a_o rational_a superficies_n by_o the_o 2._o assumpt_n go_v before_o this_o proposition_n for_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o be_v rational_a and_o so_o also_o be_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o for_o the_o line_n ad_fw-la and_o db_o be_v put_v to_o be_v rational_a commensurable_a in_o power_n only_o and_o so_o likewise_o be_v the_o line_n ac_fw-la and_o cb._n wherefore_o also_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la &_o db_o twice_o by_o a_o rational_a superficies_n when_o yet_o notwithstanding_o they_o be_v both_o medial_a superficiece_n by_o the_o 21._o of_o the_o ten_o which_o by_o the_o 26._o of_o the_o same_o be_v impossible_a and_o if_o the_o line_n db_o be_v less_o than_o the_o line_n ac_fw-la we_o may_v by_o the_o like_a demonstration_n prove_v the_o self_n same_o impossibility_n wherefore_o a_o binomial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n which_o be_v require_v to_o be_v demonstrate_v 〈…〉_o ollary_n add_v by_o flussates_n two_o ration_n 〈…〉_o surable_a in_o power_n only_o be_v add_v together_o can_v be_v equal_a to_o two_o other_o rational_a line_n 〈…〉_o in_o power_n only_o add_v together_o corollary_n for_o either_o of_o they_o shall_v make_v a_o binomia_fw-la 〈…〉_o so_o shall_v a_o binomial_a line_n be_v divide_v into_o his_o name_n in_o more_o point_n than_o onâ—â—â—ch_v by_o this_o proposition_n be_v prove_v to_o be_v impossible_a the_o like_a shall_v follow_v in_o the_o five_o 〈◊〉_d irrational_a line_n as_o touch_v their_o two_o name_n ¶_o the_o 31._o problem_n the_o 43._o proposition_n a_o first_o bimedial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n svppose_v that_o ab_fw-la be_v a_o first_o bimedial_a line_n and_o let_v it_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o so_fw-mi that_o let_v the_o line_n ac_fw-la and_o cb_o be_v medial_a commensurable_a in_o power_n only_o and_o contain_v a_o rational_a superficies_n then_o i_o say_v that_o the_o line_n ab_fw-la can_v not_o be_v divide_v into_o his_o name_n in_o any_o other_o point_n then_o in_o c._n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n d_o impossibilâ—â—e_n so_fw-mi that_o let_v ad_fw-la &_o db_o be_v medial_a line_n commensurable_a in_o power_n only_o comprehend_v a_o rational_a superficies_n now_o forasmuch_o as_o how_o much_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o diâ—ferreth_v from_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o so_o much_o differ_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o from_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o differ_v from_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o a_o rational_a superficies_n by_o the_o second_o assumpt_v go_v before_o the_o 41._o of_o the_o ten_o for_o either_o of_o those_o superficiece_n be_v rational_a wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o differ_v from_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o by_o a_o rational_a superficies_n when_o yet_o they_o be_v both_o medial_a superficiece_n which_o be_v impossible_a wherefore_o a_o first_o bimedial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 32._o theorem_a the_o 44._o proposition_n a_o second_o bimedial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n svppose_v that_o the_o line_n ab_fw-la be_v a_o second_o bimedial_a line_n be_v divide_v into_o his_o name_n in_o the_o point_n c_o so_o that_o let_v the_o line_n ac_fw-la and_o cb_o be_v medial_a line_n commensurable_a in_o power_n only_o comprehend_v a_o medial_a superficies_n it_o be_v manifest_a that_o the_o point_n c_o divide_v not_o the_o whole_a line_n ab_fw-la into_o two_o equal_a part_n impossibility_n for_o the_o line_n ac_fw-la and_o cb_o be_v not_o commensurable_a in_o length_n the_o one_o to_o the_o other_o now_o i_o say_v that_o the_o line_n ab_fw-la can_v be_v divide_v into_o his_o name_n in_o any_o other_o point_n but_o only_o in_o c._n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n d_o so_fw-mi that_o let_v not_o the_o line_n ac_fw-la be_v one_o and_o the_o same_o that_o be_v let_v it_o not_o be_v equal_a with_o the_o line_n db._n but_o let_v it_o be_v great_a than_o it_o now_o it_o be_v manifest_a by_o the_o first_o assumpt_n go_v before_o the_o 42._o proposition_n of_o this_o book_n that_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v great_a than_o the_o square_n of_o the_o line_n ad_fw-la and_o db._n and_o also_o that_o the_o line_n ad_fw-la and_o db_o be_v medial_a line_n commensurable_a in_o power_n only_o comprehend_v a_o medial_a supersicy_n take_v a_o rational_a line_n ef._n and_o by_o the_o 44._o of_o the_o first_o upon_o the_o line_n of_o apply_v a_o rectangle_n parallelogram_n eke_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la from_o which_o parallelogram_n take_v away_o the_o parallelogram_n eglantine_n equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o wherefore_o the_o residue_n namely_o the_o parallelogram_n hk_o
fg_o be_v commensurable_a to_o the_o square_n of_o the_o line_n gh_o but_o the_o square_n of_o the_o line_n fg_o be_v rational_a wherefore_o the_o square_a also_o of_o the_o line_n gh_o be_v rational_a wherefore_o the_o line_n gh_o be_v also_o rational_a and_o for_o that_o the_o number_n bâ—_n have_v not_o to_o the_o number_n cd_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n gh_o and_o they_o be_v both_o rational_a wherefore_o the_o line_n fg_o and_o gh_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o line_n fh_o be_v a_o residual_a line_n i_o say_v moreover_o that_o it_o be_v a_o six_o residual_a line_n for_o for_o that_o as_o the_o number_n e_o be_v to_o the_o number_n bc_o so_o be_v the_o square_a of_o the_o line_n a_o to_o the_o square_n of_o the_o line_n fg_o and_o as_o the_o number_n bc_o be_v to_o the_o number_n cd_o so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n gh_o therefore_o by_o equality_n of_o proportion_n as_o the_o number_n e_o be_v to_o the_o number_n cd_o so_o be_v the_o square_a of_o the_o line_n a_o to_o the_o square_n of_o the_o line_n gh_o but_o the_o number_n e_o have_v not_o to_o the_o number_n cd_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n a_o be_v incommensurable_a in_o length_n to_o the_o line_n gh_o and_o neither_o of_o these_o line_n fg_o nor_o gâ—_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n a._n and_o forasmuch_o as_o the_o square_n of_o the_o line_n fg_o be_v great_a than_o the_o square_a of_o the_o line_n gh_o unto_o the_o square_n of_o the_o line_n fg_o let_v the_o the_o square_n of_o the_o line_n gh_o and_o king_n be_v equal_a now_o therefore_o for_o that_o as_o the_o number_n bâ—_n be_v to_o the_o number_n cd_o so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n gh_o therefore_o by_o conversion_n of_o proportion_n as_o the_o number_n bc_o be_v to_o the_o number_n bd_o so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n k._n but_o the_o number_n bc_o have_v not_o to_o the_o number_n bd_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n k._n wherefore_o the_o line_n fg_o be_v in_o power_n more_o than_o the_o line_n gh_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n fg_o and_o neither_o of_o the_o line_n fg_o nor_o gh_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n a._n wherefore_o the_o line_n fh_o be_v a_o six_o residual_a line_n wherefore_o there_o be_v find_v out_o a_o six_o residual_a line_n which_o be_v require_v to_o be_v do_v line_n there_o be_v also_o a_o certain_a other_o ready_a way_n to_o find_v out_o every_o one_o of_o the_o foresaid_a six_o residual_a line_n which_o be_v after_o this_o manner_n suppose_v that_o it_o be_v require_v to_o find_v out_o a_o first_o residual_a line_n take_v a_o first_o binomial_a line_n ac_fw-la &_o let_v the_o great_a name_n thereof_o be_v ab_fw-la and_o unto_o the_o line_n bc_o let_v the_o line_n bd_o be_v equal_a wherefore_o the_o line_n ab_fw-la and_o bc_o that_o be_v the_o line_n ab_fw-la and_o bd_o be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n bc_o that_o be_v than_o the_o line_n bd_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n ab_fw-la and_o the_o line_n ab_fw-la be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v for_o the_o line_n ac_fw-la be_v put_v to_o be_v a_o first_o binomial_a line_n wherefore_o the_o line_n ad_fw-la be_v a_o first_o residual_a line_n and_o in_o like_a manner_n may_v you_o find_v out_o a_o second_o a_o three_o a_o four_o a_o five_o and_o a_o six_o residual_a line_n if_o you_o take_v for_o each_o a_o binomial_a line_n of_o the_o same_o order_n ¶_o the_o 67._o theorem_a the_o 91._o proposition_n if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n &_o a_o first_o residual_a line_n the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v a_o residual_a line_n svppose_v that_o there_o be_v a_o rectangle_n superficies_n ab_fw-la contain_v under_o a_o rational_a line_n ac_fw-la and_o a_o first_o residual_a line_n ad._n senary_n then_o i_o say_v that_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ab_fw-la be_v a_o residual_a line_n for_o forasmuch_o as_o ad_fw-la be_v a_o first_o residual_a line_n construction_n let_v the_o line_n join_v unto_o it_o be_v dg_o by_o the_o line_n join_v unto_o it_o understand_v such_o a_o line_n as_o be_v speak_v of_o in_o the_o end_n of_o the_o 79._o proposition_n wherefore_o the_o line_n agnostus_n and_o gd_o be_v rational_a commensurable_a in_o power_n only_o &_o the_o whole_a line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n ac_fw-la and_o the_o line_n agnostus_n be_v in_o power_n more_o than_o the_o line_n gd_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o agnostus_n by_o the_o definition_n of_o a_o first_o residual_a line_n divide_v the_o line_n gd_o into_o two_o equal_a part_n in_o the_o point_n e._n and_o upon_o the_o line_n agnostus_n apply_v a_o parallelogram_n equal_a to_o the_o square_n of_o the_o line_n eglantine_n and_o want_v in_o figure_n by_o a_o square_a and_o let_v the_o say_a parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n of_o and_o fg._n demonstration_n wherefore_o the_o line_n of_o be_v commensurable_a in_o length_n to_o the_o line_n fg_o by_o the_o 17._o of_o the_o ten_o â—_o and_o by_o the_o point_n e_o f_o and_o g_o draw_v unto_o the_o line_n ac_fw-la these_o parallel_a line_n eh_o fi_n and_o gk_o and_o make_v perfect_a the_o parallelogram_n ak_o and_o forâ—as_v much_o as_o the_o line_n of_o be_v commensurable_a in_o length_n to_o the_o line_n fg_o therefore_o also_o the_o whole_a line_n agnostus_n be_v commensurable_a iâ—_z length_n to_o either_o of_o the_o line_n of_o and_o fg_o by_o the_o 15._o of_o the_o ten_o but_o the_o line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o line_n ac_fw-la wherefore_o either_o of_o the_o line_n of_o and_o fg_o be_v commensurable_a in_o length_n to_o the_o line_n ac_fw-la but_o the_o line_n ac_fw-la be_v rational_a wherefore_o either_o of_o the_o line_n of_o and_o fg_o be_v also_o rational_a wherefore_o by_o the_o 19_o of_o the_o ten_o either_o of_o the_o parallelogram_n ai_fw-fr and_o fk_o be_v also_o rational_a parallelogram_n and_o forasmuch_o as_o the_o line_n de_fw-fr be_v commensurable_a in_o length_n to_o the_o line_n eglantine_n therefore_o also_o by_o the_o 15._o of_o the_o ten_o the_o line_n dg_o be_v commensurable_a in_o length_n to_o either_o of_o the_o line_n de_fw-fr and_o eglantine_n but_o the_o line_n dg_o be_v rational_a wherefore_o either_o of_o the_o line_n de_fw-fr and_o eglantine_n be_v rational_a and_o the_o self_n same_o line_n dg_o be_v incommensurable_a in_o length_n to_o the_o line_n ac_fw-la by_o the_o definition_n of_o a_o first_o residual_a line_n or_o by_o the_o 13._o of_o the_o tenthâ—_n for_o the_o line_n dg_o be_v incommensurable_a in_o length_n to_o the_o line_n agnostus_n which_o line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o line_n ac_fw-la medial_a wherefore_o either_o of_o the_o line_n de_fw-fr and_o eglantine_n be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n acâ—_n â—_z wherefore_o by_o the_o 21._o of_o the_o ten_o either_o of_o these_o parallelogram_n dh_o and_o eke_o be_v medial_a unto_o the_o parallelogram_n ai_fw-fr let_v the_o square_v lm_o be_v equal_a and_o unto_o the_o parallelogram_n fk_o let_v the_o square_a nx_n be_v equal_a be_v take_v away_o from_o the_o square_a lmâ—_n â—_z and_o haâ—ing_v the_o angle_n lom_o common_a to_o they_o both_o construction_n and_o to_o do_v this_o there_o must_v be_v find_v out_o the_o mean_a proportional_a between_o the_o line_n fi_n and_o fg._n for_o the_o square_n of_o the_o mean_a proportional_a be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n fi_n and_o fg._n and_o from_o the_o line_n lo_o cut_v of_o a_o line_n equal_a to_o the_o mean_a proportional_a so_o find_v out_o and_o descriâ—e_v the_o square_a thereof_o wherefore_o both_o the_o square_n lm_o and_o nx_o be_v about_o one_o and_o the_o self_n same_o diameter_n by_o the_o 20._o of_o the_o six_o let_v their_o diameter_n be_v or_o and_o describe_v the_o figure_n as_o
parallelogram_n fk_o wherefore_o also_o the_o square_a of_o the_o line_n lo_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n on_o wherefore_o the_o line_n lo_o and_o on_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o their_o square_n add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o twice_o rational_a wherefore_o the_o line_n ln_o be_v that_o irrational_a line_n which_o be_v call_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a and_o it_o contain_v in_o power_n the_o superficies_n ab_fw-la wherefore_o the_o line_n contain_v in_o power_n the_o superficies_n ab_fw-la be_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a if_o therefore_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n &_o a_o five_o residual_a line_n the_o line_n that_o contain_v in_o power_n the_o same_o superficies_n be_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a which_o be_v require_v to_o be_v prove_v ¶_o the_o 72._o theorem_a the_o 96._o proposition_n if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o six_o residual_a line_n the_o line_n which_o contain_v in_o power_n the_o same_o superficies_n be_v a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a svppose_v that_o ab_fw-la be_v a_o superficies_n contain_v under_o a_o rational_a line_n ac_fw-la &_o a_o six_o residual_a line_n ad._n then_o i_o say_v shall_v the_o line_n which_o contain_v in_o power_n the_o superficies_n ab_fw-la be_v a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a for_o unto_o the_o line_n ad_fw-la let_v the_o line_n dg_o be_v join_v demonstration_n and_o let_v the_o rest_n be_v as_o in_o the_o proposition_n go_v before_o and_o forasmuch_o as_o the_o line_n of_o be_v incommensurable_a in_o length_n to_o the_o line_n fg_o therefore_o the_o parallelogram_n ai_fw-fr be_v incommensurable_a to_o the_o parallelogram_n fkâ—_n and_o forasmuch_o as_o the_o line_n agnostus_n and_o ac_fw-la be_v rational_a commensurable_a in_o power_n only_o therefore_o the_o parallelogram_n ak_o be_v medial_a and_o in_o like_a manner_n the_o parallelogram_n dk_o be_v medial_a now_o forasmuch_o as_o the_o line_n agnostus_n and_o gd_o be_v commensurable_a in_o power_n only_o therefore_o they_o be_v incommensurable_a in_o length_n the_o one_o to_o the_o other_o but_o as_o the_o line_n agnostus_n be_v to_o the_o line_n gd_o so_o be_v the_o parallelogram_n ak_o to_o the_o parallelogram_n dk_o therefore_o the_o parallelogram_n ak_o be_v incommensurable_a to_o the_o parallelogram_n dk_o describe_v the_o like_a figure_n that_o be_v describe_v in_o the_o former_a proposition_n and_o we_o may_v in_o like_a sort_n prove_v that_o the_o line_n ln_o contain_v in_o power_n the_o superficies_n ab_fw-la i_o say_v moreover_o that_o it_o be_v a_o line_n make_v wâ—th_v a_o medial_a superficies_n the_o whole_a superficies_n medial_a for_o the_o parallelogram_n ak_o be_v medial_a wherefore_o that_o which_o be_v equal_a unâ—â—_n it_o namely_o that_o which_o be_v make_v of_o the_o sqâ—ares_n of_o the_o line_n lo_o and_o on_o add_v together_o be_v also_o medial_a and_o forasmuch_o as_o the_o parallelogram_n â—k_n be_v medial_a therefore_o that_o which_o be_v equal_a unto_o â—t_a namely_o that_o which_o be_v contain_v under_o the_o line_n lo_o and_o on_o twice_o be_v also_o medial_a and_o forasmuch_o as_o the_o parallogramme_n ak_o be_v incommensurable_a to_o the_o parallelogram_n dk_o therefore_o the_o square_n of_o the_o line_n lo_o and_o on_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n lo_o and_o on_o twice_o and_o forasmuch_o as_o the_o parallelogram_n ai_fw-fr be_v incommensurable_a to_o the_o parallelogram_n fk_o therefore_o also_o the_o square_a of_o the_o line_n lo_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n on_o wherefore_o the_o line_n lo_o and_o on_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n lo_o and_o on_o medial_a and_o that_o which_o be_v contain_v under_o they_o twice_o medial_a and_o moreover_o that_o which_o be_v make_v of_o the_o square_n of_o they_o be_v incommensurable_a to_o thate_v which_o be_v contain_v under_o they_o twice_o wherefore_o the_o line_n ln_o be_v that_o irrational_a line_n which_o be_v call_v a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a and_o it_o contain_v in_o power_n the_o superficies_n ab_fw-la wherefore_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ab_fw-la be_v a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a if_o therefore_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o six_o residual_a line_n the_o line_n which_o contain_v in_o power_n the_o same_o superficies_n be_v a_o line_n make_v which_o a_o medial_a superficies_n the_o whole_a superficies_n mediallâ—_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 73._o theorem_a the_o 97._o proposition_n the_o square_a of_o a_o residual_a line_n apply_v unto_o a_o rational_a line_n senary_a make_v the_o breadth_n or_o other_o side_n a_o first_o reâ—iduall_a line_n svppose_v that_o ab_fw-la be_v a_o residual_a line_n proposition_n and_o let_v cd_o be_v a_o rational_a line_n and_o unto_o the_o line_n cd_o apply_v the_o parallelogram_n ce_fw-fr equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o make_v the_o breadth_n the_o line_n cf._n then_o i_o say_v that_o the_o line_n cf_o be_v a_o first_o residual_a lineâ—_z construction_n for_o unto_o the_o line_n ab_fw-la let_v the_o line_n convenient_o join_v be_v suppose_v to_o be_v bâ—_n which_o self_n line_n be_v also_o call_v a_o line_n join_v as_o we_o declare_v in_o the_o end_n of_o the_o 79._o proposition_n wherefore_o the_o â—ines_v agnostus_n and_o gb_o be_v rational_a commensurable_a in_o power_n only_o and_o unto_o the_o line_n cd_o apply_v the_o parallelogram_n ch_n equal_a to_o the_o square_n of_o the_o line_n agnostus_n and_o unto_o the_o line_n kh_o which_o be_v equal_a to_o the_o line_n cd_o apply_v the_o parallelogram_n kl_o equal_a to_o the_o square_n of_o the_o line_n bg_o demonstration_n wherefore_o the_o whole_a parallelogram_n cl_n be_v equal_a to_o the_o square_n of_o the_o linâ—s_n aâ—_z and_o gbâ—_n and_o the_o parallelogrâ—mmâ—_n ce_fw-fr be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la wherefore_o the_o parallelogram_n remain_v namely_o the_o parallelogram_n fl_fw-mi be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o twice_o for_o by_o the_o 7._o of_o the_o second_o the_o square_n of_o the_o line_n agnostus_n and_o gb_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n &_o gb_o twice_o and_o to_o the_o square_n of_o the_o line_n ab_fw-la divide_v the_o line_n fm_o into_o two_o equal_a part_n in_o the_o point_n n._n and_o by_o the_o point_n n_o draw_v unto_o the_o line_n cd_o a_o parallel_a line_n nx_o wherefore_o either_o of_o the_o parallelogram_n fx_n and_o nl_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o once_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n agnostus_n and_o gb_o be_v rational_a unto_o which_o square_n the_o parallelogram_n cl_n be_v equal_a therefore_o the_o parallelogram_n cl_n also_o be_v rational_a wherefore_o the_o line_n gm_n be_v rational_a and_o ten_o commensurable_a in_o length_n to_o the_o line_n cd_o again_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o twice_o be_v ten_o medial_a therefore_o the_o parallelogram_n equal_a unto_o it_o namely_o the_o paralâ—elogramme_n fl_fw-mi be_v also_o medial_a wherefore_o the_o line_n fm_o be_v rational_a and_o ten_o incommensurable_a in_o length_n to_o the_o line_n cd_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n agnostus_n and_o gb_o be_v rational_a and_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o twice_o be_v medial_a therefore_o the_o square_n of_o the_o line_n agnostus_n and_o gb_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o twice_o but_o unto_o the_o square_n of_o the_o line_n agnostus_n and_o gb_o be_v equal_a the_o parallelogram_n cl_n and_o to_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o twice_o be_v equal_a the_o parallelogram_n fl_fw-mi wherefore_o the_o parallelogram_n cl_n be_v incommensurable_a to_o the_o parallelogram_n fl._n wherefore_o also_o the_o line_n cm_o be_v incommensurable_a in_o length_n to_o the_o line_n fm_o and_o they_o be_v both_o rational_a wherefore_o the_o line_n cm_o and_o fm_o be_v rational_a commensurable_a in_o power_n only_o and_o therefore_o the_o line_n cf_o be_v
of_o the_o ten_o the_o line_n de_fw-fr be_v a_o first_o binomial_a line_n divide_v it_o into_o his_o name_n in_o the_o point_n g._n and_o let_v dg_o be_v the_o great_a name_n wherefore_o the_o line_n dg_o and_o â—e_n be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n dg_o be_v in_o power_n more_o than_o the_o line_n ge_z by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n dg_o and_o the_o line_n dg_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v dc_o wherefore_o the_o line_n df_o be_v commensurable_a in_o length_n to_o the_o line_n dg_o wherefore_o by_o the_o 13._o of_o the_o ten_o the_o whole_a line_n df_o be_v commensurable_a in_o length_n to_o the_o line_n remain_v namely_o â—o_o the_o line_n gf_o and_o forasmuch_o as_o the_o line_n df_o be_v commensurable_a to_o the_o line_n fg_o but_o the_o line_n fd_o be_v rational_a wherefore_o the_o line_n fg_o be_v also_o rational_a and_o forasmuch_o as_o the_o line_n fd_o be_v commensurable_a in_o length_n to_o the_o line_n fg_o but_o the_o line_n df_o be_v incommensurable_a in_o length_n to_o the_o line_n fe_o wherefore_o the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n fe_o by_o the_o 13._o of_o the_o ten_o and_o they_o be_v both_o rational_a line_n wherefore_o the_o line_n gf_o and_o fe_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o by_o the_o 73._o of_o the_o ten_o the_o line_n eglantine_n be_v a_o residual_a line_n but_o it_o be_v also_o rational_a as_o before_o have_v be_v prove_v which_o be_v impossible_a namely_o that_o one_o &_o the_o same_o line_n shall_v be_v both_o rational_a and_o irrational_a wherefore_o a_o residual_a line_n be_v not_o one_o and_o the_o same_o with_o a_o binomial_a line_n that_o be_v be_v not_o a_o binomial_a line_n which_o be_v require_v to_o be_v demonstrate_v corollary_n ¶_o a_o corollary_n a_o residual_a line_n and_o the_o other_o five_o irrational_a line_n follow_v it_o be_v neither_o medial_a line_n nor_o one_o and_o the_o same_o between_o themseluesâ—_n that_o be_v one_o be_v utter_o of_o a_o diverse_a kind_n from_o a_o other_o for_o the_o square_n of_o a_o medial_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n rational_a and_o incommensurable_a in_o length_n to_o the_o rational_a line_n whereunto_o it_o be_v apply_v by_o the_o 22._o of_o the_o ten_o the_o square_a of_o a_o residual_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n a_o first_o residual_a line_n by_o the_o 97._o of_o the_o ten_o the_o square_a of_o a_o first_o medial_a residual_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n a_o second_o residual_a line_n by_o the_o 98._o of_o the_o ten_o the_o square_a of_o a_o second_o medial_a residual_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n a_o three_o residual_a line_n by_o the_o 99_o of_o the_o ten_o the_o square_a of_o a_o less_o line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n a_o four_o residual_a line_n by_o the_o 100_o of_o the_o ten_o the_o square_a of_o a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a apply_v to_o a_o rational_a line_n make_v the_o breadth_n a_o â—ift_n residual_a line_n by_o the_o 101._o of_o the_o ten_o and_o the_o square_a of_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a apply_v to_o a_o rational_a line_n make_v the_o breadth_n a_o six_o residual_a line_n by_o the_o 102._o of_o the_o ten_o now_o forasmuch_o as_o these_o foresay_a side_n which_o be_v the_o breadthes_fw-mi differ_v both_o from_o the_o first_o breadth_n sore_o that_o it_o be_v rational_a and_o differ_v also_o the_o one_o from_o the_o other_o for_o that_o they_o be_v residual_n of_o diverse_a order_n and_o kind_n it_o be_v manifest_a that_o those_o irrational_a line_n differ_v also_o the_o one_o from_o the_o other_o and_o forasmuch_o as_o it_o have_v be_v prove_v in_o the_o 111._o proposition_n that_o 〈◊〉_d residual_a 〈◊〉_d be_v not_o one_o and_o the_o same_o with_o a_o binomial_a line_n and_o it_o have_v also_o be_v prove_v that_o the_o 〈…〉_o of_o a_o residual_a line_n and_o of_o the_o five_o irrational_a line_n that_o follow_v it_o be_v apply_v to_o a_o rational_a line_n do_v make_v their_o breadthes_fw-mi one_o of_o the_o residual_n of_o that_o order_n of_o which_o they_o be_v who_o squareâ—_n be_v apply_v to_o the_o rational_a line_n likewise_o also_o the_o square_n of_o a_o binomial_a line_n and_o of_o the_o five_o irrational_a line_n which_o follow_v it_o be_v apply_v to_o a_o rational_a line_n do_v make_v the_o breadthes_fw-mi one_o of_o the_o binomials_n of_o that_o order_n of_o which_o they_o be_v who_o square_n be_v apply_v to_o the_o rational_a line_n wherefore_o the_o irrational_a line_n which_o follow_v the_o binomial_a line_n and_o the_o irrational_a line_n which_o follow_v the_o residual_a line_n differ_v the_o one_o from_o the_o other_o so_o that_o all_o the_o irrational_a line_n be_v 13._o in_o number_n namely_o these_o 1_o a_o medial_a line_n 2_o a_o binomial_a line_n 3_o a_o first_o bimedial_a line_n 4_o a_o second_o bimedial_a line_n 5_o a_o great_a line_n 6_o a_o line_n contain_v in_o power_n a_o rational_a superficies_n and_o a_o medial_a superficies_n 7_o a_o line_n contain_v in_o power_n two_o medial_a superficiece_n 8_o a_o residual_a line_n 9_o a_o first_o medial_a residual_a line_n 10_o a_o second_o medial_a residual_a line_n 11_o a_o less_o line_n 12_o a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a 13_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a ¶_o the_o 88_o theorem_a the_o 112._o proposition_n the_o square_a of_o a_o rational_a line_n apply_v unto_o a_o binomial_a line_n make_v the_o breadth_n or_o other_o side_n a_o residual_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o binomial_a line_n &_o in_o the_o self_n same_o proportion_n &_o moreover_o that_o residual_a line_n be_v in_o the_o self_n same_o order_n of_o residual_a line_n that_o the_o binomial_a line_n be_v of_o binomial_a line_n svppose_v that_o a_o be_v a_o rational_a line_n prove_v and_o bc_o a_o binomial_a line_n who_o great_a name_n let_v be_v cd_o and_o unto_o the_o square_n of_o the_o line_n a_o let_v the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o of_o so_o that_o of_o be_v the_o breadth_n be_v equal_a then_o i_o say_v that_o of_o be_v a_o residual_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o binomial_a line_n bc_o which_o name_n let_v be_v cd_o and_o db_o and_o be_v in_o the_o same_o proportion_n with_o they_o and_o moreover_o the_o line_n of_o be_v in_o the_o self_n same_o order_n of_o residual_a line_n that_o the_o line_n bc_o be_v of_o binomial_a line_n unto_o the_o square_n of_o the_o line_n a_o let_v the_o parallelogram_n contain_v under_o the_o line_n bd_o and_o g_o be_v equal_a construction_n now_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n bc_o &_o of_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n bd_o and_o g_o therefore_o reciprocal_o by_o the_o 14._o of_o the_o six_o as_o the_o line_n cb_o be_v to_o the_o bd_o so_o be_v the_o line_n g_o to_o the_o line_n ef._n but_o the_o line_n bc_o be_v great_a than_o the_o line_n bd_o wherefore_o the_o line_n g_o be_v great_a than_o the_o line_n ef._n demonstration_n unto_o the_o line_n g_o let_v the_o line_n eh_o be_v equal_a wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o line_n cb_o be_v to_o the_o line_n bd_o so_o be_v the_o line_n he_o to_o the_o line_n fe_o wherefore_o by_o division_n by_o the_o 17._o of_o the_o five_o as_o the_o line_n cd_o be_v to_o the_o line_n bd_o so_o be_v the_o line_n hf_o to_o the_o line_n fe_o demonstrate_v as_o the_o line_n hf_o be_v to_o the_o fe_o so_o let_v the_o line_n fk_o be_v to_o the_o line_n ke_o how_o this_o be_v to_o be_v do_v we_o will_v declare_v at_o the_o end_n of_o this_o demonstration_n wherefore_o by_o the_o 12._o of_o the_o five_o the_o whole_a line_n hk_o proportion_n be_v to_o the_o whole_a line_n kf_n as_o the_o line_n fk_o be_v to_o the_o line_n ke_o for_o as_o one_o of_o the_o antecedentes_fw-la be_v to_o one_o of_o the_o consequentes_fw-la so_o be_v all_o the_o antecedentes_fw-la to_o all_o the_o consequentes_fw-la but_o as_o the_o line_n fk_o be_v the_o line_n ke_o so_o be_v the_o line_n cd_o to_o the_o line_n db_o for_o fk_n be_v to_o eke_o as_o hf_o be_v to_o fe_o and_o hf_o be_v to_o fe_o as_o cd_o be_v db_o wherefore_o by_o the_o
of_o the_o one_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v neither_o name_n of_o the_o other_o also_o shall_v be_v commensurable_a in_o length_n to_o the_o same_o rational_a line_n put_v by_o the_o 13._o of_o the_o same_o wherefore_o the_o residual_a line_n gz_n shall_v be_v in_o the_o self_n same_o order_n of_o residual_a line_n that_o the_o binomial_a line_n gb_o be_v of_o binomial_a line_n by_o the_o definition_n of_o residual_a and_o binomial_a line_n the_o square_a therefore_o of_o a_o rational_a line_n apply_v to_o a_o binomial_a lineâ—_z &_o â—_o which_o be_v require_v 〈◊〉_d be_v prove_v ¶_o the_o 89._o theorem_a the_o 113._o proposition_n the_o square_a of_o a_o rational_a line_n apply_v unto_o a_o residual_a make_v the_o breadth_n or_o other_o side_n 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 and_o moreover_o that_o binomial_a line_n be_v in_o the_o self_n same_o order_n of_o binomial_a line_n that_o the_o residual_a line_n be_v of_o residual_a line_n svppose_v that_o a_o be_v a_o rational_a line_n and_o bd_o a_o residual_a line_n and_o unto_o the_o square_n of_o the_o line_n a_o let_v that_o which_o be_v contain_v under_o the_o line_n bd_o and_o kh_o be_v equal_a wherefore_o the_o square_a of_o the_o rational_a line_n a_o apply_v unto_o the_o residual_a line_n bd_o make_v the_o breadth_n or_o other_o side_n kh_o then_o i_o say_v that_o the_o line_n kh_o be_v 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 bd_o and_o in_o the_o self_n same_o proportion_n and_o that_o the_o line_n kh_o be_v in_o the_o self_n same_o order_n of_o binomial_a line_n that_o the_o line_n bd_o be_v of_o residual_a line_n unto_o the_o line_n bd_o let_v the_o line_n convenient_o join_v be_v dc_o construction_n wherefore_o the_o line_n bc_o and_o dc_o be_v rational_a commensurable_a in_o power_n only_o and_o unto_o the_o square_n of_o the_o line_n a_o let_v the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o g_o be_v equal_a but_o the_o square_n of_o the_o line_n a_o be_v rational_a demonstration_n wherefore_o the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o g_o be_v also_o rational_a wherefore_o also_o the_o line_n g_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n bc_o by_o the_o 20._o of_o the_o ten_o now_o forasmuch_o as_o the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o g_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n bd_o and_o kh_o therefore_o by_o the_o 16._o of_o the_o six_o as_o the_o line_n bc_o be_v to_o the_o line_n bd_o so_o iâ—_z the_o line_n kh_o to_o the_o line_n g._n but_o the_o line_n bc_o be_v great_a than_o the_o line_n bd._n wherefore_o also_o the_o line_n kh_o be_v great_a than_o the_o line_n g._n unto_o the_o line_n g_o lâ—t_fw-fr the_o line_n ke_o be_v equal_a wherefore_o the_o line_n ke_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n bc_o as_o also_o the_o line_n g_o be_v by_o the_o 12._o of_o the_o ten_o and_o for_o that_o as_o bc_o be_v to_o bd_o so_o be_v kh_o to_o ke_o wherefore_o by_o â—duersion_n of_o proportion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o as_o bc_o be_v to_o dc_o so_o be_v kh_o to_o eh_o assumpt_n â—kh_n into_o eh_o so_o let_v the_o line_n fh_o be_v to_o the_o line_n of_o how_o this_o be_v to_o be_v do_v we_o will_v decare_v at_o the_o end_n of_o this_o demonstration_n wherefore_o the_o residue_n kf_o be_v to_o the_o residue_n fh_o as_o the_o whole_a kh_o be_v to_o the_o whole_a he_o by_o the_o 19_o of_o the_o five_o that_o be_v as_o the_o line_n bc_o be_v to_o the_o line_n cd_o but_o the_o line_n bc_o and_o cd_o be_v commensurable_a in_o power_n only_o wherefore_o also_o the_o line_n kf_a and_o fh_o be_v commensurable_a in_o power_n only_o and_o for_o that_o as_o kh_o be_v to_o he_o so_o be_v kf_a to_o fh_v but_o as_o kh_o be_v to_o he_o so_o be_v also_o hf_o to_o fe_o therefore_o as_o kf_n be_v to_o fh_v so_o be_v fh_o to_o fe_o wherefore_o by_o the_o corollary_n of_o the_o 19_o of_o the_o six_o as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o square_a of_o the_o first_o to_o the_o square_n of_o the_o second_o wherefore_o as_o kf_n be_v to_o fe_o so_o be_v the_o square_a of_o the_o line_n kf_n to_o the_o square_n of_o the_o line_n fh_o but_o these_o square_n be_v commensurable_a for_o the_o line_n kf_a and_o fh_o be_v commensurable_a in_o power_n wherefore_o the_o line_n kf_a and_o fe_o be_v commensurable_a in_o length_n wherefore_o by_o the_o second_o part_n of_o the_o 15._o of_o the_o ten_o the_o line_n ke_o and_o of_o be_v commensurable_a in_o length_n wherefore_o by_o the_o same_o the_o line_n kf_a and_o fe_o be_v commensurable_a in_o length_n but_o the_o line_n ke_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n bc_o wherefore_o the_o line_n kf_o be_v also_o rational_a and_o commensurable_a in_o length_n to_o the_o line_n bc._n and_o for_o that_o as_o the_o line_n bc_o be_v to_o the_o line_n cd_o so_o it_o kf_n to_o eh_o therefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o bc_o be_v to_o kf_n so_o be_v cd_o to_o fh_o but_o the_o line_n bc_o be_v commensurable_a in_o length_n to_o the_o line_n kf_o wherefore_o the_o line_n cd_o be_v commensurable_a in_o length_n to_o the_o line_n fh_o but_o the_o line_n cd_o be_v rational_a wherefore_o also_o the_o line_n fh_o be_v rational_a and_o the_o line_n bc_o and_o cd_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o line_n kf_a and_o fh_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o line_n kh_o be_v 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 same_o proportion_n i_o say_v moreover_o that_o it_o be_v a_o binomial_a of_o the_o self_n same_o order_n of_o binomial_a line_n that_o the_o line_n bd_o be_v of_o residual_a line_n for_o if_o the_o line_n bc_o be_v in_o power_n more_o than_o the_o line_n cd_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n bc_o the_o line_n kf_o be_v also_o in_o power_n more_o than_o the_o line_n fh_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n kf_o by_o the_o 14._o of_o the_o ten_o and_o if_o the_o line_n bc_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v the_o line_n kf_o be_v also_o by_o the_o 12._o of_o the_o ten_o commensurable_a in_o length_n to_o the_o rational_a line_n and_o so_o the_o line_n bd_o be_v a_o first_o residual_a line_n and_o the_o line_n kh_o be_v in_o like_a sort_n a_o first_o binomial_a line_n if_o the_o line_n cd_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n the_o line_n fh_o be_v also_o commensurable_a in_o length_n to_o the_o same_o line_n and_o so_o the_o line_n bd_o be_v a_o second_o residual_a line_n and_o the_o line_n kh_o a_o second_o binomial_a line_n and_o if_o neither_o of_o the_o line_n bc_o nor_o cd_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n neither_o also_o of_o the_o line_n kf_n nor_o fh_o be_v commensurable_a in_o length_n to_o the_o same_o and_o so_o the_o line_n bd_o be_v a_o three_o residual_a line_n and_o the_o line_n kh_o a_o three_o binomial_a line_n but_o if_o the_o line_n bc_o be_v in_o power_n more_o than_o the_o line_n cd_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n bc_o the_o line_n kf_o be_v in_o power_n more_o they_o the_o line_n fh_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n kf_o by_o the_o 14._o of_o the_o ten_o and_o if_o the_o line_n bc_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v the_o line_n kf_o be_v also_o commensurable_a in_o length_n to_o the_o same_o line_n and_o so_o the_o line_n bd_o be_v a_o four_o residual_a line_n and_o the_o line_n kh_o a_o four_o binomial_a line_n and_o if_o the_o line_n cd_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n the_o line_n fh_o be_v also_o commensurable_a in_o length_n to_o the_o same_o &_o so_o the_o line_n bd_o be_v a_o five_o residual_a line_n &_o the_o line_n kh_o a_o five_o binomial_a line_n and_o if_o neither_o of_o the_o line_n bc_o nor_o cd_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n neither_o also_o of_o the_o line_n kf_n nor_o fh_o be_v commensurable_a in_o length_n
two_o other_o proposition_n go_v next_o before_o it_o so_o far_o misplace_v that_o where_o they_o be_v word_n for_o word_n before_o duâ—ly_o place_v be_v the_o 105._o and_o 106._o yet_o here_o after_o the_o book_n end_v they_o be_v repeat_v with_o the_o number_n of_o 116._o and_o 117._o proposition_n zambert_n therein_o be_v more_o faithful_a to_o follow_v as_o he_o find_v in_o his_o greek_n example_n than_o he_o be_v skilful_a or_o careful_a to_o do_v what_o be_v necessary_a yea_o and_o some_o greek_n write_v ancient_a copy_n have_v they_o not_o so_o though_o in_o deed_n they_o be_v well_o demonstrate_v yet_o truth_n disord_v be_v half_a disgracedâ—_n especial_o where_o the_o pattern_n of_o good_a order_n by_o profession_n be_v avouch_v to_o be_v but_o through_o ignorance_n arrogancy_n and_o â—emerltie_n of_o unskilful_a method_n master_n many_o thing_n remain_v yet_o in_o these_o geometrical_a element_n undue_o tumble_v in_o though_o true_a yet_o with_o disgrace_n which_o by_o help_n of_o so_o many_o wit_n and_o hability_n of_o such_o as_o now_o may_v have_v good_a cause_n to_o be_v skilful_a herein_o will_v i_o hope_v ere_o long_o be_v take_v away_o and_o thing_n of_o importance_n want_v supply_v the_o end_n of_o the_o ten_o book_n of_o euclides_n element_n ¶_o the_o eleven_o book_n of_o euclides_n element_n book_n hitherto_o have_v â—vclidâ—_n in_o thâ—sâ—_n former_a book_n with_o a_o wonderful_a method_n and_o order_n entreat_v of_o such_o kind_n of_o figure_n superficial_a which_o be_v or_o may_v be_v describe_v in_o a_o superficies_n or_o plain_a and_o have_v teach_v and_o set_v forth_o their_o property_n nature_n generation_n and_o production_n even_o from_o the_o first_o root_n ground_n and_o beginning_n of_o they_o namely_o from_o a_o point_n which_o although_o it_o be_v indivisible_a yet_o be_v it_o the_o begin_n of_o all_o quantity_n continual_a and_o of_o it_o and_o of_o the_o motion_n and_o slow_v thereof_o be_v produce_v a_o line_n and_o consequent_o all_o quantity_n continual_a as_o all_o figure_n plain_a and_o solid_a what_o so_o ever_o euclid_n therefore_o in_o his_o first_o book_n begin_v with_o it_o boot_n and_o from_o thence_o go_v he_o to_o a_o line_n as_o to_o a_o thing_n most_o simple_a next_o unto_o a_o point_n then_o to_o a_o superficies_n and_o to_o angle_n and_o so_o through_o the_o whole_a first_o book_n boâ—â—e_n he_o entreat_v of_o these_o most_o simple_a and_o plain_a ground_n in_o the_o second_o book_n he_o entreat_v further_o â—â—oâ—e_n and_o go_v unto_o more_o hard_a matter_n and_o teach_v of_o division_n of_o line_n and_o of_o the_o multiplication_n of_o line_n and_o of_o their_o part_n and_o of_o their_o passion_n and_o property_n and_o for_o that_o rightline_v â—_z be_v far_o distant_a in_o nature_n and_o property_n from_o round_a and_o circular_a figure_n in_o the_o three_o book_n he_o instruct_v the_o reader_n of_o the_o nature_n and_o condition_n of_o circle_n booâ—e_n in_o the_o four_o book_n he_o compare_v figure_n of_o right_a line_n and_o circle_n together_o bâ—oâ—e_n and_o teach_v how_o to_o describe_v a_o figure_n of_o right_a line_n with_o in_o or_o about_o a_o circle_n and_o contrariwise_o a_o circle_n with_o in_o or_o about_o a_o rectiline_a figure_n in_o the_o five_o book_n he_o search_v out_o the_o nature_n of_o proportion_n a_o matter_n of_o wonderful_a use_n and_o deep_a consideration_n boâ—â—e_n for_o that_o otherwise_o he_o can_v not_o compare_v â—igure_n with_o figure_n or_o the_o side_n of_o figure_n together_o for_o whatsoever_o be_v compare_v to_o any_o other_o thing_n be_v compare_v unto_o it_o undoubted_o under_o some_o kind_n of_o proportion_n wherefore_o in_o the_o six_o book_n he_o compare_v figure_n together_o booâ—e_n one_o to_o a_o other_o likewise_o their_o side_n and_o for_o that_o the_o nature_n of_o proportion_n can_v not_o be_v full_o and_o clear_o see_v without_o the_o knowledge_n of_o number_n wherein_o it_o be_v first_o and_o chief_o find_v in_o the_o seven_o bookâ—_n eight_o booâ—â—_n and_o nine_o book_n book_n he_o entreatâ—th_v of_o number_n &_o of_o the_o kind_n and_o property_n thereof_o and_o because_o that_o the_o side_n of_o solid_a body_n for_o the_o most_o part_n be_v of_o such_o sort_n that_o compare_v together_o they_o have_v such_o proportion_n the_o one_o to_o the_o other_o booâ—e_n which_o can_v not_o be_v express_v by_o any_o number_n certain_a and_o therefore_o be_v call_v irrational_a line_n he_o in_o the_o ten_o book_n have_v write_v &_o teach_v which_o lineâ—_z be_v commensurable_a or_o incommensurable_a the_o one_o to_o the_o other_o and_o of_o the_o diversity_n of_o kind_n of_o irrational_a line_n with_o all_o the_o condition_n &_o propriety_n of_o they_o and_o thus_o have_v euclid_n in_o these_o ten_o foresay_a book_n full_o &_o most_o plenteous_o in_o a_o marvelous_a order_n teach_v whatsoever_o seem_v necessary_a and_o requisite_a to_o the_o knowledge_n of_o all_o superficial_a figure_n of_o what_o sort_n &_o form_n so_o ever_o they_o be_v now_o in_o these_o book_n follow_v he_o entreat_v of_o figure_n of_o a_o other_o kind_n namely_o of_o bodily_a figure_n follow_v as_o of_o cube_n pyramid_n cones_n column_n cilinder_n parallelipipedon_n sphere_n and_o such_o othersâ—_n and_o show_v the_o diversity_n of_o they_o the_o generation_n and_o production_n of_o they_o and_o demonstrate_v with_o great_a and_o wonderful_a art_n their_o propriety_n and_o passion_n with_o all_o their_o nature_n and_o condition_n he_o also_o compare_v one_o oâ—_n they_o to_o a_o other_o whereby_o to_o know_v the_o reason_n and_o proportion_n of_o the_o one_o to_o the_o other_o chief_o of_o the_o five_o body_n which_o be_v call_v regular_a body_n element_n and_o these_o be_v the_o thing_n of_o all_o other_o entreat_v of_o in_o geometry_n most_o worthy_a and_o of_o great_a dignity_n and_o as_o it_o be_v the_o end_n and_o final_a intent_n of_o the_o whole_a be_v of_o geometry_n and_o for_o who_o cause_n have_v be_v write_v and_o speak_v whatsoever_o have_v hitherto_o in_o the_o former_a book_n be_v say_v or_o write_v as_o the_o first_o book_n be_v a_o ground_n and_o a_o necessary_a entry_n to_o all_o the_o râ—st_o â—ollowing_v so_o be_v this_o eleven_o book_n a_o necessary_a entry_n and_o ground_n to_o the_o rest_n which_o follow_v 〈◊〉_d and_o as_o that_o contain_v the_o declaration_n of_o word_n and_o definition_n of_o thinger_n requisite_a to_o the_o knowledge_n of_o superficial_a figure_n and_o entreat_v of_o line_n and_o of_o their_o division_n and_o section_n which_o be_v the_o term_n and_o limit_n of_o superficial_a figure_n so_o in_o this_o book_n be_v set_v forth_o the_o declaration_n of_o word_n and_o definition_n of_o thing_n pertain_v to_o solid_a and_o corporal_a figure_n and_o also_o of_o superficiece_n which_o be_v the_o term_n &_o limit_n of_o solid_n moreover_o of_o the_o division_n and_o intersection_n of_o they_o and_o diverse_a other_o thing_n without_o which_o the_o 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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
and_o one_o side_n of_o the_o one_o equal_a to_o one_o side_n of_o the_o other_o namely_o that_o side_n which_o subtend_v one_o of_o the_o equal_a angle_n that_o be_v the_o side_n ha_o be_v equal_a to_o the_o side_n dm_o by_o construction_n wherefore_o the_o side_n remain_v be_v by_o the_o 26._o of_o the_o first_o equal_a to_o the_o side_n remain_v wherefore_o the_o side_n ac_fw-la be_v equal_a to_o the_o side_n df._n in_o like_a sort_n may_v we_o prove_v that_o the_o side_n ab_fw-la be_v equal_a to_o the_o side_n de_fw-fr if_o you_o draw_v a_o right_a line_n from_o the_o point_n h_o to_o the_o point_n b_o and_o a_o other_o from_o the_o point_n m_o to_o the_o point_n e._n for_o forasmuch_o as_o the_o square_n of_o the_o line_n ah_o be_v by_o the_o 47._o of_o the_o first_o equal_v to_o the_o square_n of_o the_o line_n ak_o and_o kh_o and_o by_o the_o same_o unto_o the_o square_n of_o the_o line_n ak_o be_v equal_a the_o square_n of_o the_o line_n ab_fw-la and_o bk_o wherefore_o the_o square_n of_o the_o line_n ab_fw-la bk_o and_o kh_o be_v equal_a to_o the_o square_n of_o the_o line_n ah_o but_o unto_o the_o square_n of_o the_o line_n bk_o and_o kh_o be_v equal_a the_o square_n of_o the_o line_n bh_o by_o the_o 47._o of_o the_o first_o for_o the_o angle_n hkb_n be_v a_o right_a angle_n for_o that_o the_o line_n hk_o be_v erect_v perpendicular_o to_o the_o ground_n plain_a superficies_n wherefore_o the_o square_a of_o the_o line_n ah_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o bh_o wherefore_o by_o the_o 48._o of_o the_o first_o the_o angle_n abh_n be_v a_o right_a angle_n and_o by_o the_o same_o reason_n the_o angle_n dem_n be_v a_o right_a angle_n now_o the_o angle_n bah_o be_v equal_a to_o the_o angle_n edm_n for_o it_o be_v so_o suppose_a and_o the_o line_n ah_o be_v equal_a to_o the_o line_n dm_o wherefore_o by_o the_o 26._o of_o the_o first_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n de._n now_o forasmuch_o as_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n df_o and_o the_o line_n ab_fw-la to_o the_o line_n de_fw-fr therefore_o these_o two_o line_n ac_fw-la and_o ab_fw-la be_v equal_a to_o these_o two_o line_n fd_v and_o de._n but_o the_o angle_n also_o cab_n be_v by_o supposition_n equal_a to_o the_o angle_n fde_v wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a bc_o be_v equal_a to_o the_o base_a of_o and_o the_o triangle_n to_o the_o triangle_n and_o the_o rest_n of_o the_o angle_n to_o the_o rest_n of_o the_o angle_n wherefore_o the_o angle_n acb_o be_v equal_a to_o the_o angle_n dfe_n and_o the_o right_a angle_n ack_z be_v equal_a to_o the_o right_a angle_n dfn._n wherefore_o the_o angle_n remain_v namely_o bck_n be_v equal_a to_o the_o angle_n remain_v namely_o to_o efn_n and_o by_o the_o same_o reason_n also_o the_o angle_n cbk_n be_v equal_a to_o the_o angle_n fen_n wherefore_o there_o be_v two_o triangle_n bck_o &_o efn_n have_v two_o angle_n of_o the_o one_o equal_a to_o two_o angle_n of_o the_o other_o each_o to_o his_o correspondent_a angle_n and_o one_o side_n of_o the_o one_o equal_a to_o one_o side_n of_o the_o other_o namely_o that_o side_n that_o lie_v between_o the_o equal_a angle_n that_o be_v the_o side_n bc_o be_v equal_a to_o the_o side_n of_o wherefore_o by_o the_o 26._o of_o the_o first_o the_o side_n remain_v be_v equal_a to_o the_o side_n remain_v wherefore_o the_o side_n ck_o be_v equal_a to_o the_o side_n fn_o but_o the_o side_n ac_fw-la be_v equal_a to_o the_o side_n df._n wherefore_o these_o two_o side_n ac_fw-la and_o ck_o be_v equal_a to_o these_o two_o side_n df_o and_o fn_o and_o they_o contain_v equal_a angle_n wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a ak_n be_v equal_a to_o the_o base_a dn_o and_o forasmuch_o as_o the_o line_n ah_o be_v equal_a to_o the_o line_n dm_o therefore_o the_o square_a of_o the_o line_n ah_o be_v equal_a to_o the_o square_n of_o the_o line_n dm_o but_o unto_o the_o square_n of_o the_o line_n ah_o be_v equal_a the_o square_n of_o the_o line_n ak_o and_o kh_o by_o the_o 47._o of_o the_o first_o for_o the_o angle_n akh_n be_v a_o right_a angle_n and_o to_o the_o square_n of_o the_o line_n dm_o be_v equal_a the_o square_n of_o the_o line_n dn_o and_o nm_o for_o the_o angle_n dnm_n be_v a_o right_a angle_n wherefore_o the_o square_n of_o the_o line_n ak_o and_o kh_o be_v equal_a to_o the_o square_n of_o the_o line_n dn_o and_o nm_o of_o which_o two_o the_o square_a of_o the_o line_n ak_o be_v equal_a to_o the_o square_n of_o the_o line_n dn_o for_o the_o line_n ak_o be_v prove_v equal_a to_o the_o line_n a_fw-la wherefore_o the_o residue_n namely_o the_o square_a of_o the_o line_n kh_o be_v equal_a to_o the_o residue_n namely_o to_o the_o square_n of_o the_o line_n nm_o wherefore_o the_o line_n hk_o be_v equal_a to_o the_o line_n mn_v and_o forasmuch_o as_o these_o two_o line_n ha_o and_o ak_o be_v equal_a to_o these_o two_o line_n md_o and_o dn_o the_o one_o to_o the_o other_o and_o the_o base_a hk_n be_v equal_a to_o the_o base_a mn_n therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n hak_n be_v equal_a to_o the_o angle_n mdn._n if_o therefore_o there_o be_v two_o superficial_a angle_n equal_a and_o from_o the_o point_n of_o those_o angle_n be_v elevate_v on_o high_a right_a line_n comprehend_v together_o with_o those_o right_a line_n which_o be_v put_v at_o the_o beginning_n equal_a angle_n each_o to_o his_o corespondent_a angle_n and_o if_o in_o each_o of_o the_o erect_a line_n be_v take_v a_o point_n at_o all_o adventure_n and_o from_o those_o point_n be_v draw_v perpendicular_a line_n to_o the_o plain_a superficiece_n in_o which_o be_v the_o angle_n give_v at_o the_o beginning_n and_o frâ—â—_n the_o point_n which_o be_v by_o the_o perpendicular_a line_n make_v in_o the_o two_o plain_a superficiece_n be_v join_v right_a line_n to_o those_o angle_n which_o be_v put_v at_o the_o beginning_n those_o right_a line_n shall_v together_o with_o the_o line_n elevate_v on_o high_a make_v equal_a angle_n which_o be_v require_v to_o be_v prove_v because_o the_o figure_n of_o the_o former_a demonstration_n be_v somewhat_o hard_a to_o conceive_v as_o they_o be_v there_o draw_v in_o a_o plain_a by_o reason_n of_o the_o line_n that_o be_v imagine_v to_o be_v elevate_v on_o high_a i_o have_v here_o set_v other_o figure_n wherein_o you_o must_v erect_v perpendicular_o to_o the_o ground_n superficiece_v the_o two_o triangle_n bhk_v and_o emn_a and_o then_o elevate_v the_o triangle_n dfm_n &_o ache_n in_o such_o sort_n that_o the_o angle_n m_o and_o h_o of_o these_o triangle_n may_v concur_v with_o the_o angle_n m_o and_o h_o of_o the_o other_o erect_v triangle_n and_o then_o imagine_v only_o a_o line_n to_o be_v draw_v from_o the_o point_n g_o of_o the_o line_n agnostus_n to_o the_o point_n l_o in_o the_o ground_n superficies_n compare_v it_o with_o the_o former_a construction_n &_o demonstration_n and_o it_o will_v make_v it_o very_o easy_a to_o conceive_v ¶_o corollary_n by_o this_o it_o be_v manifest_a that_o if_o there_o be_v two_o rectiline_v superficial_a angle_n equal_a and_o upon_o those_o angle_n be_v elevate_v on_o high_a equal_a right_a line_n contain_v together_o with_o the_o right_a line_n put_v at_o the_o beginning_n equal_a angle_n perpendicular_a line_n draw_v from_o those_o elevate_v line_n to_o the_o ground_n plain_a superficiece_n wherein_o be_v the_o angle_n put_v at_o the_o beginning_n be_v equal_a the_o one_o to_o the_o other_o for_o it_o be_v manifest_a that_o the_o perpendicular_a line_n hk_o &_o mn_a which_o be_v draw_v from_o the_o end_n of_o the_o equal_a elevate_v line_n ah_o and_o dm_o to_o the_o ground_n superficiece_n be_v equal_a ¶_o the_o 31._o theorem_a the_o 36._o proposition_n if_o there_o be_v three_o right_a line_n proportional_a a_o parallelipipedon_n describe_v of_o those_o three_o right_a line_n be_v equal_a to_o the_o parallelipipedon_n describe_v of_o the_o middle_a line_n so_o that_o it_o consist_v of_o equal_a side_n and_o also_o be_v equiangle_n to_o the_o foresay_a parallelipipedon_n svppose_v that_o these_o three_o line_n a_o b_o c_o be_v proportional_a as_o a_o be_v to_o b_o so_o let_v b_o be_v to_o c._n then_o i_o say_v that_o the_o parallelipipedon_n make_v of_o the_o line_n a_o b_o c_o be_v equal_a to_o the_o parallelipipedon_n make_v of_o the_o line_n b_o so_fw-mi that_o the_o solid_a make_v of_o the_o line_n b_o consist_v of_o equal_a side_n and_o be_v also_o equiangle_n to_o the_o solid_a make_v of_o the_o line_n a_o b_o c._n describe_v by_o the_o 23._o of_o the_o
last_o proposition_n in_o the_o second_o book_n and_o also_o of_o the_o 31._o in_o the_o six_o book_n ¶_o a_o problem_n 5._o two_o unequal_a circle_n be_v give_v to_o find_v a_o circle_n equal_a to_o the_o excess_n of_o the_o great_a to_o the_o less_o suppose_v the_o two_o unequal_a circle_n give_v to_o be_v abc_n &_o def_n &_o let_v abc_n be_v the_o great_a construction_n who_o diameter_n suppose_v to_o be_v ac_fw-la &_o the_o diameter_n of_o def_n suppose_v to_o be_v df._n i_o say_v a_o circle_n must_v be_v find_v equal_a to_o that_o excess_n in_o magnitude_n by_o which_o abc_n be_v great_a thâ—_z def_n by_o the_o first_o of_o the_o four_o in_o the_o circle_n abc_n apply_v a_o right_a line_n equal_a to_o df_o who_o one_o end_n let_v be_v at_o c_o and_o the_o other_o let_v be_v at_o b._n fron_n b_o to_o a_o draw_v a_o right_a line_n by_o the_o 30._o of_o the_o three_o it_o may_v appear_v demonstration_n that_o abc_n be_v a_o right_a angle_n and_o thereby_o abc_n the_o triangle_n be_v rectangle_v wherefore_o by_o the_o first_o of_o the_o two_o corollary_n here_o before_o the_o circle_z abc_n be_v equal_a to_o the_o circle_n def_n for_o bc_o by_o construction_n be_v equal_a to_o df_o and_o more_o over_o to_o the_o circle_n who_o diameter_n be_v ab_fw-la that_o circle_n therefore_o who_o diameter_n be_v ab_fw-la be_v the_o circle_n contain_v the_o magnitude_n by_o which_o abc_n be_v great_a than_o def_n wherefore_o two_o unequal_a circle_n be_v give_v we_o have_v find_v a_o circle_n equal_a to_o the_o excess_n of_o the_o great_a to_o the_o less_o which_o ought_v to_o be_v do_v a_o problem_n 6._o a_o circle_n be_v give_v to_o find_v two_o circle_n equal_a to_o the_o same_o which_o find_v circle_n shall_v have_v the_o one_o to_o the_o other_o any_o proportion_n give_v in_o two_o right_a line_n suppose_v abc_n a_o circle_n give_v and_o the_o proportion_n give_v let_v it_o be_v that_o which_o be_v between_o the_o two_o right_a line_n d_z and_o e._n i_o say_v that_o two_o circle_n be_v to_o be_v find_v equal_a to_o abc_n and_o with_o all_o one_o to_o the_o other_o construction_n in_o the_o proportion_n of_o d_o to_o e._n let_v the_o diameter_n of_o abc_n be_v ac_fw-la as_z d_z be_v to_o e_o so_o let_v ac_fw-la be_v divide_v by_o the_o 10._o of_o the_o six_o in_o the_o point_n f._n at_o fletcher_n to_o the_o line_n ac_fw-la let_v a_o perpendicular_a be_v draw_v fb_o and_o let_v it_o mete_v the_o circumference_n at_o the_o point_n b._n from_o the_o point_n b_o to_o the_o point_v a_o and_o c_o let_v right_a line_n be_v draw_v ba_o and_o bc._n i_o say_v that_o the_o circle_n who_o diameter_n be_v the_o line_n basilius_n and_o bc_o be_v equal_a to_o the_o circle_n abc_n and_o that_o those_o circle_n have_v to_o their_o diameter_n the_o line_n basilius_n and_o bc_o be_v one_o to_o the_o other_o in_o the_o proportion_n of_o the_o line_n d_o to_o the_o line_n e._n for_o first_o that_o they_o be_v equal_a it_o be_v evident_a by_o reason_n that_o abc_n be_v a_o triangle_n rectangle_n wherefore_o by_o the_o 47._o of_o the_o first_o the_o square_n of_o basilius_n and_o bc_o be_v equal_a to_o the_o square_n of_o ac_fw-la and_o so_o by_o this_o second_o it_o be_v manife_a the_o two_o circle_n to_o be_v equal_a to_o the_o circle_n abc_n second_o as_o d_o be_v to_o â—_o so_o be_v of_o to_o fc_n by_o construction_n and_o as_o the_o line_n of_o be_v to_o the_o line_n fc_o so_o be_v the_o square_a of_o the_o line_n â—a_o to_o the_o square_n of_o the_o line_n bc._n which_o thing_n we_o will_v brief_o prove_v thus_o the_o parallelogram_n contain_v under_o ac_fw-la and_o of_o use_n be_v equal_a to_o the_o square_n of_o basilius_n by_o the_o lemma_n after_o the_o 32._o of_o the_o ten_o book_n and_o by_o the_o same_o lemma_n or_o assumpt_n the_o parallelogram_n contain_v under_o ac_fw-la and_o â—c_z be_v equal_a to_o the_o square_n of_o the_o line_n bc._n wherefore_o as_o the_o first_o parallelogram_n have_v itself_o to_o the_o secondâ—_n so_o have_v the_o square_a of_o basilius_n equal_a to_o the_o first_o parallelogram_n itself_o to_o the_o square_n of_o bc_o equal_a to_o the_o second_o parallelogram_n but_o both_o the_o parallelogram_n have_v one_o height_n namely_o the_o line_n ac_fw-la and_o base_n the_o line_n of_o and_o fc_n wherefore_o as_o of_o be_v to_o fc_n so_o be_v the_o parallelogâ—amme_n contain_v under_o ac_fw-la of_o to_o the_o parallelogram_n contain_v under_o ac_fw-la fc_fw-la by_o the_o fiâ—st_n of_o the_o six_o and_o therefore_o as_o of_o be_v to_o fc_n so_o be_v the_o square_a of_o basilius_n to_o the_o square_n of_o bc._n and_o as_o the_o square_n of_o basilius_n be_v to_o the_o square_n of_o bc_o so_o be_v the_o circle_n who_o diameter_n be_v basilius_n to_o the_o circle_n who_o diameter_n be_v bc_o by_o this_o second_o of_o the_o twelve_o wherefore_o the_o circle_n who_o diameter_n be_v basilius_n be_v to_o the_o circle_n who_o diameter_n be_v bc_o as_o d_z be_v to_o e._n and_o before_o we_o prove_v they_o equal_a to_o the_o circle_n abc_n wherâ—fore_o a_o circle_n be_v give_v we_o have_v find_v two_o circle_n equal_a to_o the_o same_o which_o have_v the_o one_o to_o the_o other_o any_o proportion_n give_v in_o two_o right_a line_n which_o ought_v to_o be_v do_v note_n addition_n heâ—e_o may_v you_o perceive_v a_o other_o way_n how_o to_o execute_v my_o first_o problem_n for_o if_o you_o make_v a_o right_a angle_n contain_v of_o the_o diameter_n give_v as_o in_o this_o figure_n suppose_v they_o basilius_n and_o bc_o and_o then_o subtend_v the_o right_a angle_n with_o the_o line_n ac_fw-la and_o from_o the_o right_a angle_n let_v fall_v a_o line_n perpendicular_a to_o the_o base_a ac_fw-la that_o perpendicular_a at_o the_o point_n of_o his_o fall_n divide_v ac_fw-la into_o of_o and_o fc_n of_o the_o proportion_n require_v a_o corollary_n it_o follow_v of_o thing_n manifest_o prove_v in_o the_o demonstration_n of_o this_o problem_n that_o in_o a_o triangle_n rectangle_n if_o from_o the_o right_a angle_n to_o the_o base_a a_o perpendicular_a be_v let_v fall_v the_o same_o perpendicular_a cut_v the_o base_a into_o two_o part_n rectangle_n in_o that_o proportion_n one_o to_o the_o other_o that_o the_o square_n of_o the_o righâ—_n line_n contain_v the_o right_a angle_n be_v in_o one_o to_o the_o other_o those_o on_o the_o one_o side_n the_o perpendicular_a be_v compare_v to_o those_o on_o the_o other_o both_o square_a and_o segment_n a_o problem_n 7._o between_o two_o circle_n give_v to_o find_v a_o circle_n middle_a proportional_a let_v the_o two_o circle_n give_v be_v acd_o and_o bef_a i_o say_v that_o a_o circle_n be_v to_o be_v find_v which_o between_o acd_a and_o bef_n be_v middle_a proportional_a construction_n let_v the_o diameter_n of_o acd_a be_v ad_fw-la and_o of_o bef_n let_v bâ—_n be_v the_o diameter_n between_o ad_fw-la and_o bf_o find_v a_o line_n middle_a proportional_a by_o the_o 13._o of_o the_o six_o which_o let_v be_v hk_a i_o say_v that_o a_o circle_n who_o diameter_n be_v hk_n be_v middle_a proportional_a between_o acd_a and_o bef_a to_o ad_fw-la hk_o and_o bf_o three_o right_a line_n in_o continual_a proportion_n by_o construction_n let_v a_o four_o line_n be_v find_v to_o which_o bf_o shall_v have_v that_o proportion_n that_o ad_fw-la have_v to_o hk_n by_o the_o 12._o of_o the_o six_o &_o let_v that_o line_n be_v â—_o demonstration_n it_o be_v manifest_a that_o the_o â—ower_a line_n ad_fw-la hk_o bf_o and_o l_o be_v in_o continual_a proportion_n for_o by_o construction_n as_o ad_fw-la be_v to_o hk_n so_o be_v bâ—_n to_o l._n and_o by_o construction_n on_o as_z ad_fw-la be_v to_o hk_n so_o be_v hk_v to_o bf_o wherefore_o hk_n be_v to_o bf_o as_o bf_o be_v to_o l_o by_o the_o 11._o of_o the_o five_o wherefore_o the_o 4._o line_n be_v in_o continual_a proportion_n wherefore_o as_o the_o first_o be_v to_o the_o three_o that_o be_v ad_fw-la to_o bf_o so_o be_v the_o square_a of_o the_o first_o to_o the_o square_n of_o the_o second_o that_o be_v the_o square_a of_o ad_fw-la to_o the_o square_n of_o hk_n by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o and_o by_o the_o same_o corollary_n as_o hk_n be_v to_o l_o so_o be_v the_o square_a of_o hk_n to_o the_o square_n of_o bf_o but_o by_o alternate_a proportion_n the_o line_n ad_fw-la be_v to_o bf_o as_o hk_n be_v to_o l_o wherefore_o the_o square_a of_o ad_fw-la be_v to_o the_o square_n of_o hk_n as_o the_o square_n of_o hk_n be_v to_o the_o square_n of_o bf_o wherefore_o the_o square_a of_o hk_n be_v middle_a proportional_a between_o the_o square_n of_o ad_fw-la and_o the_o square_n of_o bf_o but_o as_o the_o square_n be_v
one_o of_o the_o circle_n ¶_o a_o assumpt_n add_v by_o flussas_n if_o a_o sphere_n be_v cut_v of_o a_o plain_a superficies_n the_o common_a section_n of_o the_o superficiece_n shall_v be_v the_o circumference_n of_o a_o circle_n suppose_v that_o the_o sphere_n abc_n be_v cut_v by_o the_o plain_a superficies_n aeb_n and_o let_v the_o centre_n of_o the_o sphere_n be_v the_o point_n d._n construction_n and_o from_o the_o point_n d_o let_v there_o be_v draw_v unto_o the_o plain_a superficies_n aeb_n a_o perpendicular_a line_n by_o the_o 11._o of_o the_o eleven_o which_o let_v be_v the_o line_n de._n and_o from_o the_o point_n e_o draw_v in_o the_o plain_a superficies_n aeb_n unto_o the_o common_a section_n of_o the_o say_a superficies_n and_o the_o sphere_n line_n how_o many_o so_o ever_o namely_o ea_fw-la and_o ebb_n and_o draw_v these_o line_n dam_fw-ge and_o db._n now_o forasmuch_o as_o the_o right_a angle_n dea_n and_o deb_n be_v equal_a for_o the_o line_n de_fw-fr be_v erect_v perpendicular_o to_o the_o plain_a superficies_n demonstration_n and_o the_o right_a line_n dam_n &_o db_o which_o subtend_v those_o angle_n be_v by_o the_o 12._o definition_n of_o the_o eleven_o equal_a which_o right_a line_n moreover_o by_o the_o 47._o of_o the_o first_o do_v contain_v in_o power_n the_o square_n of_o the_o line_n de_fw-fr ea_fw-la and_o de_fw-fr ebb_n if_o therefore_o from_o the_o square_n of_o the_o line_n de_fw-fr ea_fw-la and_o de_fw-fr ebb_v you_o take_v away_o the_o square_a of_o the_o line_n de_fw-fr which_o be_v common_a unto_o they_o the_o residue_n namely_o the_o square_n of_o the_o line_n ea_fw-la &_o ebb_n shall_v be_v equal_a wherefore_o also_o the_o line_n ea_fw-la and_o ebb_n be_v equal_a and_o by_o the_o same_o reason_n may_v we_o prove_v that_o all_o the_o right_a line_n draw_v from_o the_o point_n e_o to_o the_o line_n which_o be_v the_o common_a section_n of_o the_o superficies_n of_o sphere_n and_o of_o the_o plain_a superficies_n be_v equal_a wherefore_o that_o line_n shall_v be_v the_o circumference_n of_o a_o circle_n by_o the_o 15._o definition_n of_o the_o first_o circle_n but_o if_o it_o happen_v the_o plain_a superficies_n which_o cut_v the_o sphere_n to_o pass_v by_o the_o centre_n of_o the_o sphere_n the_o right_a line_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o their_o common_a section_n shalâ—_n be_v equal_a by_o the_o 12._o definition_n of_o the_o eleven_o for_o that_o common_a section_n be_v in_o the_o superficies_n of_o the_o sphere_n wherefore_o of_o necessity_n the_o plain_a superficies_n comprehend_v under_o that_o line_n of_o the_o common_a section_n shall_v be_v a_o circle_n and_o his_o centre_n shall_v be_v one_o and_o the_o same_o with_o the_o centre_n of_o the_o sphere_n john_n dee_n euclid_n have_v among_o the_o definition_n of_o solid_n omit_v certain_a which_o be_v easy_a to_o conceive_v by_o a_o kind_n of_o analogy_n as_o a_o segment_n of_o a_o sphere_n a_o sector_n of_o a_o sphere_n the_o vertex_fw-la or_o top_n of_o the_o segment_n of_o a_o sphere_n with_o such_o like_a but_o that_o if_o need_n be_v some_o fartheâ—_n light_n may_v be_v give_v in_fw-ge this_o figure_n next_o before_o description_n understand_v a_o segment_n of_o the_o sphere_n abc_n to_o be_v that_o part_n of_o the_o sphere_n contain_v between_o the_o circle_n ab_fw-la who_o centre_n be_v e_o and_o the_o spherical_a superficies_n afb_o to_o which_o be_v a_o less_o segment_n add_v the_o cone_n adb_o who_o base_a be_v the_o former_a circle_n and_o top_n the_o centre_n of_o the_o sphere_n and_o you_o have_v dafb_n a_o sector_n of_o a_o sphere_n or_o solid_a sector_n as_o i_o call_v it_o de_fw-fr extend_a to_o fletcher_n show_v the_o top_n or_o vertex_fw-la of_o the_o segment_n to_o be_v the_o point_n f_o and_o of_o be_v the_o altitude_n of_o the_o segment_n spherical_a of_o segment_n some_o be_v great_a they_o the_o half_a sphere_n some_o be_v less_o as_o before_o abf_n be_v less_o the_o remanent_fw-la abc_n be_v a_o segment_n great_a than_o the_o half_a sphere_n ¶_o a_o corollary_n add_v by_o the_o same_o flussas_n by_o the_o foresay_a assumpt_n it_o be_v manifest_a that_o if_o from_o the_o centre_n of_o a_o sphere_n the_o line_n draw_v perpendicular_o unto_o the_o circle_n which_o cut_v the_o sphere_n be_v equal_a those_o circle_n be_v equal_a and_o the_o perpendicular_a line_n so_o draw_v fall_n upon_o the_o centre_n of_o the_o same_o circle_n for_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n contain_v in_o power_n the_o power_n of_o the_o perpendicular_a line_n and_o the_o power_n of_o the_o line_n which_o join_v together_o the_o end_n of_o those_o line_n wherefore_o from_o that_o square_a or_o power_n of_o the_o line_n from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n or_o common_a section_n draw_v which_o be_v the_o semidiameter_n of_o the_o sphere_n take_v away_o the_o power_n of_o the_o perpendicular_a which_o be_v common_a to_o they_o â—ound_v about_o it_o follow_v that_o the_o residue_v how_o many_o so_o ever_o they_o be_v be_v equal_a power_n and_o therefore_o the_o line_n be_v equal_a the_o one_o to_o the_o other_o wherefore_o they_o will_v describe_v equal_a circle_n by_o the_o first_o definition_n of_o the_o three_o and_o upon_o their_o centre_n fall_v the_o perpendicular_a line_n by_o the_o 9_o of_o the_o three_o corollary_n and_o those_o circle_n upon_o which_o fall_v the_o great_a perpendicular_a line_n be_v the_o less_o circle_n for_o the_o power_n of_o the_o line_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n be_v always_o one_o and_o equal_a to_o the_o power_n of_o the_o perpendicular_a line_n and_o also_o to_o the_o power_n of_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n to_o their_o circumference_n the_o great_a that_o the_o power_n of_o the_o perpendicular_a line_n take_v away_o from_o the_o power_n contain_v they_o both_o be_v the_o less_o be_v the_o power_n and_o therefore_o the_o line_n remain_v which_o be_v the_o semidiameter_n of_o the_o circle_n and_o therefore_o the_o less_o be_v the_o circle_n which_o they_o describe_v wherefore_o if_o the_o circle_n be_v equal_a the_o perpendicular_a line_n fall_v from_o the_o centre_n of_o the_o sphere_n upon_o they_o shall_v also_o be_v equal_a for_o if_o they_o shall_v be_v great_a or_o less_o the_o circle_n shall_v be_v unequal_a as_o it_o be_v before_o manifest_a but_o we_o suppose_v the_o perpendicular_n to_o be_v equal_a corollary_n also_o the_o perpendicular_a line_n fall_v upon_o those_o base_n be_v the_o least_o of_o all_o that_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n for_o the_o other_o draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n of_o the_o circle_n be_v in_o power_n equal_a both_o to_o the_o power_n of_o the_o perpendicular_n and_o to_o the_o power_n of_o the_o line_n join_v these_o perpendicular_n and_o these_o subtendent_fw-la line_n together_o make_v triangle_n rectangleround_v about_o as_o most_o easy_o you_o may_v conceive_v of_o the_o figure_n here_o annex_v a_o the_o centre_n of_o the_o sphere_n ab_fw-la the_o line_n from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n of_o the_o circle_n make_v by_o the_o section_n bcb_n the_o diameter_n of_o circle_n make_v by_o the_o section_n ac_fw-la the_o perpendicular_n from_o the_o centre_n of_o the_o sphere_n to_o the_o circlesâ—_n who_o diameter_n bcb_n be_v one_o both_o side_n or_o in_o any_o situation_n else_o cb_o the_o semidiameter_n of_o the_o circle_n make_v by_o the_o section_n ao_o a_o perpendicular_a long_o than_o ac_fw-la and_o therefore_o the_o semidiameter_n ob_fw-la be_v less_o acb_o &_o aob_n triangle_n rectangle_n ¶_o the_o 2._o problem_n the_o 17._o proposition_n two_o sphere_n consist_v both_o about_o one_o &_o the_o self_n same_o centre_n be_v give_v to_o inscribe_v in_o the_o great_a sphere_n a_o solid_a of_o many_o side_n which_o be_v call_v a_o polyhedron_n which_o shall_v not_o touch_v the_o superficies_n of_o the_o less_o sphere_n svppose_v that_o there_o be_v two_o sphere_n about_o one_o &_o the_o self_n same_o centre_n namely_o about_o a._n it_o be_v require_v in_o the_o great_a sphere_n to_o inscribe_v a_o polyhedron_n or_o a_o solid_a of_o many_o side_n which_o shall_v not_o with_o his_o superficies_n touch_v the_o superficies_n of_o the_o less_o sphere_n let_v the_o sphere_n be_v cut_v by_o some_o one_o plain_a superficies_n pass_v by_o the_o centre_n a._n construction_n then_o shall_v their_o section_n be_v circle_n flussas_n for_o by_o the_o 12._o definition_n of_o the_o eleven_o the_o diameter_n remain_v fix_v and_o the_o semicircle_n be_v turn_v round_o about_o make_v a_o sphere_n wherefore_o in_o what_o position_n so_o ever_o you_o imagine_v the_o
upright_o cylinder_n be_v cuâ—_n by_o a_o plain_a parallel_n to_o his_o base_n by_o construction_n therefore_o as_o the_o cylinder_n dc_o be_v to_o the_o cylinder_n be_v so_o be_v the_o axe_n of_o dc_o to_o the_o axe_n of_o be_v by_o the_o 13._o of_o this_o twelve_o demonstration_n wherefore_o as_o the_o axe_n be_v to_o the_o axe_n so_o be_v cylinder_n to_o cylinder_n but_o axe_n be_v to_o axe_n as_o side_n to_o side_n namely_o ce_fw-fr to_o qe_n because_o the_o axe_n be_v parallel_n to_o any_o side_n of_o a_o upright_o cylinder_n by_o the_o definition_n of_o a_o cylinder_n and_o the_o circle_n of_o the_o section_n be_v parallel_n to_o the_o base_n by_o construction_n wherefore_o in_o the_o parallelogram_n make_v of_o the_o axe_n and_o of_o two_o semidiameter_n on_o one_o side_n parallel_n one_o to_o the_o other_o be_v couple_v together_o by_o a_o line_n draw_v between_o their_o end_n in_o their_o circumference_n which_o line_n be_v the_o side_n qc_n it_o be_v evident_a that_o the_o axe_n of_o bc_o be_v cut_v in_o like_a proportion_n that_o the_o side_n qc_n be_v cut_v wherefore_o the_o cylinder_n dc_o be_v to_o the_o cylinder_n be_v as_o ec_o be_v to_o qe_n wherefore_o by_o composition_n the_o cylinder_n dc_o and_o be_v that_o be_v whole_a bc_o be_v to_o the_o cylinder_n be_v as_o ce_fw-fr and_o qe_n the_o whole_a right_a line_n qc_n be_v to_o qe_n but_o by_o construction_n qc_n be_v of_o 3._o such_o part_n as_o qe_n contain_v 2._o wherefore_o the_o cylinder_n bc_o be_v of_o 3._o such_o part_n as_o be_v contain_v 2._o wherefore_o bc_o the_o cylinder_n be_v to_o be_v as_o 3._o to_o 2_o which_o be_v sesquialtera_fw-la proportion_n but_o by_o the_o former_a theorem_a bc_o be_v sesquialtera_fw-la to_o the_o sphere_n a_o wherefore_o by_o the_o 7._o of_o the_o five_o be_v be_v equal_a to_o a._n therefore_o to_o a_o sphere_n give_v we_o have_v make_v a_o cylinder_n equal_a or_o thus_o more_o brief_o omit_v all_o cut_n of_o the_o cylinder_n forasmuch_o as_o bc_o be_v a_o upright_o cylinder_n his_o side_n be_v equal_a to_o his_o axe_n or_o heith_n therefore_o the_o two_o cylinder_n whereof_o one_o have_v the_o heith_n qc_n and_o the_o other_o the_o heith_n qe_n have_v both_o their_o base_n the_o great_a circle_n in_o the_o sphere_n a_o be_v one_o to_o the_o other_o as_o qc_n be_v to_o qe_n by_o the_o 14._o of_o this_o twelve_o but_o qc_n be_v to_o qe_n as_o 3._o to_o 2_o by_o construction_n and_o 3._o to_o 2._o be_v in_o sesquialtera_fw-la proportion_n therefore_o the_o cylinder_n bc_o have_v his_o heith_n qc_n &_o his_o base_a the_o great_a circle_n in_o a_o contain_v be_v sesquialtera_fw-la to_o the_o cylinder_n which_o have_v his_o base_a the_o great_a circle_n in_o a_o contain_v and_o heith_n the_o line_n qe_n but_o by_o the_o former_a theorem_a bc_o be_v also_o sesquialtera_fw-la to_o a_o wherefore_o the_o cylinder_n have_v the_o base_a bq_n which_o by_o supposition_n problem_n be_v equal_a to_o the_o great_a circle_n in_o a_o contain_v and_o heith_n qe_n be_v equal_a to_o the_o sphere_n a_o by_o the_o 7._o of_o the_o five_o and_o now_o it_o can_v not_o be_v hard_a to_o geve_v a_o cylinder_n to_o a_o in_o that_o proportion_n which_o be_v between_o x_o and_o y._n for_o let_v the_o side_n qe_n be_v to_o qp_v as_o you_o be_v to_o x_o by_o the_o 12._o of_o the_o six_o therefore_o backward_o qp_n be_v to_o qe_n as_o x_o to_o y._n wherefore_o the_o cylinder_n have_v the_o base_a the_o great_a circle_n in_o a_o and_o heith_n the_o line_n qp_n *_o be_v to_o the_o cylinder_n have_v the_o same_o base_a and_o heith_n the_o line_n qe_n as_o x_o be_v to_o you_o by_o the_o 14._o of_o this_o twelve_o but_o the_o cylinder_n have_v the_o heith_n qe_n &_o his_o base_a the_o great_a circle_n in_o a_o contain_v be_v prove_v equal_a to_o the_o sphere_n a._n wherefore_o by_o the_o 7._o of_o the_o five_o the_o cylinder_n who_o heith_n be_v qp_n and_o base_a the_o great_a circle_n in_o a_o contain_v be_v to_o the_o sphere_n a_o as_o x_o to_o y._n therefore_o to_o a_o sphere_n give_v we_o have_v make_v a_o cylinder_n in_o any_o proportion_n give_v between_o two_o right_a line_n and_o also_o before_o we_o have_v to_o a_o sphere_n give_v make_v a_o cylinder_n equal_a therefore_o to_o a_o sphere_n give_v we_o have_v make_v a_o cylinder_n equal_a or_o in_o any_o proportion_n give_v between_o two_o right_a line_n a_o problem_n 5._o a_o sphere_n be_v give_v and_o a_o circle_n upon_o that_o circle_n as_o a_o base_a to_o rear_v a_o cylinder_n equal_a to_o the_o sphere_n give_v or_o in_o any_o proportion_n give_v between_o two_o right_a line_n a_o problem_n 6._o a_o sphere_n be_v give_v and_o a_o right_a line_n to_o make_v a_o cylinder_n equal_a to_o the_o sphere_n give_v or_o in_o any_o other_o proportion_n between_o two_o right_a line_n give_v in_o this_o 5._o and_o 6._o problem_n first_o make_v a_o cylinder_n equal_a to_o the_o sphere_n give_v by_o the_o 4._o problem_n and_o then_o by_o the_o order_n of_o the_o 2._o and_o 3._o problem_n in_o cones_fw-la execute_v these_o accordingly_z in_o cylinder_n a_o problem_n 7._o two_o unlike_o cones_n or_o cylinder_n be_v give_v to_o find_v two_o right_a line_n which_o have_v the_o same_o proportion_n one_o to_o the_o other_o that_o the_o two_o give_v cones_fw-la or_o cylinder_n have_v one_o to_o the_o other_o upon_o one_o of_o their_o base_n rear_v a_o cone_n if_o cones_fw-la be_v compare_v or_o a_o cylinder_n if_o cylinder_n be_v compare_v equal_a to_o the_o other_o by_o the_o order_n of_o the_o second_o and_o three_o problem_n and_o the_o heith_n of_o the_o cone_n or_o cylinder_n on_o who_o base_a you_o rear_v a_o equal_a cone_fw-mi or_o cylinder_n with_o the_o new_a heith_n find_v have_v that_o proportion_n which_o the_o cones_fw-la or_o cylinder_n have_v one_o to_o the_o other_o by_o the_o 14._o of_o this_o twelve_o book_n a_o problem_n 8._o a_o upright_a cone_n and_o cylinder_n be_v give_v to_o find_v two_o right_a line_n have_v that_o proportion_n the_o one_o to_o the_o other_o which_o the_o cone_n and_o cylinder_n have_v one_o to_o the_o other_o suppose_v qek_v a_o upright_a cone_fw-mi and_o ab_fw-la a_o upright_o cylinder_n give_v i_o say_v two_o right_a line_n be_v to_o be_v give_v which_o shall_v have_v that_o proportion_n one_o to_o the_o other_o construction_n which_o qek_n and_o ab_fw-la have_v one_o to_o the_o other_o upon_o the_o base_a bh_o erect_v a_o cone_n equal_a to_o qek_v by_o the_o order_n of_o the_o second_o problem_n which_o let_v be_v obh_n and_o his_o heith_n let_v be_v oc_n demonstration_n and_o let_v the_o heith_n of_o ab_fw-la the_o cylinder_n be_v c_n produce_v c_n to_o p_o so_o that_o cp_n be_v tripla_fw-la to_o c_n &_o make_v perfect_a the_o cone_n pbh_n i_o say_v that_o fc_n and_o oc_n have_v that_o proportion_n which_o ab_fw-la have_v to_o qek_n for_o by_o constrâ—ction_n obh_n be_v equal_a to_o qek_v and_o pbh_n be_v equal_a to_o ab_fw-la as_o we_o will_v prove_v assumpt_v wise_a and_o pbh_n and_o obh_n be_v upon_o one_o base_a namely_o bh_o wherefore_o by_o the_o 14._o of_o this_o twelve_o â—bh_n and_o obh_n be_v one_o to_o the_o other_o as_o their_o heithe_n pc_n and_o oc_n be_v one_o to_o the_o other_o wherefore_o the_o cylinder_n and_o cone_n equal_a to_o pbh_n and_o obh_n be_v as_o pc_n be_v to_o oc_n by_o the_o 7._o of_o the_o five_o but_o ab_fw-la the_o cylinder_n qek_n the_o cone_n be_v equal_a to_o pbh_n and_o obh_n by_o construction_n wherefore_o ab_fw-la the_o cylinder_n be_v to_o qek_v the_o cone_n as_o pc_n be_v to_o oc_n wherefore_o we_o have_v find_v two_o right_a line_n have_v that_o proportion_n that_o a_o cone_fw-mi and_o a_o cylinder_n give_v have_v one_o to_o the_o other_o which_o thing_n we_o may_v execute_v upon_o the_o base_a of_o the_o cone_fw-it give_v as_o we_o do_v upon_o the_o base_a of_o the_o cylinder_n give_v on_o this_o manner_n upon_o the_o base_a of_o the_o cone_fw-it qek_n problem_n which_o base_a let_v be_v eke_o erect_v a_o cylinder_n equal_a to_o ab_fw-la by_o the_o order_n of_o my_o second_o problem_n which_o cylinder_n let_v be_v ed_z and_o gt_v his_o heith_n and_o let_v the_o heith_n of_o the_o cone_fw-it qek_n be_v qg_n take_v the_o line_n graccus_n the_o â—hird_n part_n oâ—_n qg_v by_o the_o 9_o of_o the_o six_o and_o with_o a_o plain_n pass_v by_o r_o parallel_n to_o eke_o cut_v of_o the_o cylinder_n eâ—_n which_o â—hall_v be_v equal_a to_o the_o cone_n qek_n by_o the_o assumpt_n follow_v i_o say_v now_o that_o ab_fw-la the_o cylinder_n be_v to_o qek_v the_o cone_n as_o gt_n be_v to_o gr._n for_o the_o cylinder_n ed_z be_v to_o the_o cylinder_n of_o as_o gt_n be_v
22._o of_o the_o six_o wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o both_o the_o line_n ab_fw-la &_o bc_o add_v the_o one_o to_o the_o other_o together_o with_o the_o line_n ac_fw-la that_o be_v as_z two_z such_o line_n as_o ab_fw-la be_v be_v to_o the_o line_n ac_fw-la so_o be_v both_o the_o line_n de_fw-fr and_o of_o add_v the_o one_o to_o the_o other_o together_o with_o the_o line_n df_o that_o be_v two_o such_o line_n as_o de_fw-fr be_v to_o the_o line_n df._n and_o in_o the_o same_o proportion_n be_v the_o half_n of_o the_o antecedent_n by_o the_o 15._o of_o the_o five_o wherefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n ac_fw-la so_o be_v the_o line_n de_fw-fr to_o the_o line_n df._n and_o therefore_o by_o the_o 19_o of_o the_o five_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o line_n df_o to_o the_o line_n fe_o wherefore_o also_o by_o division_n by_o the_o 17._o of_o the_o five_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cb_o so_o be_v the_o line_n df_o to_o the_o line_n de_fw-fr now_o that_o we_o have_v prove_v proposition_n that_o any_o right_a line_n whatsoever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n what_o proportion_n the_o line_n contain_v in_o power_n the_o square_n make_v of_o the_o whole_a line_n and_o of_o the_o great_a segment_n add_v together_o have_v to_o the_o line_n contain_v in_o power_n the_o square_n make_v of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o the_o same_o proportion_n have_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n now_o also_o that_o we_o have_v prove_v proposition_n that_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n so_o be_v the_o superficies_n of_o the_o dodecahedron_n to_o the_o superficies_n of_o the_o icosahedron_n be_v both_o describe_v in_o one_o and_o the_o self_n same_o sphere_n and_o moreover_o see_v that_o we_o have_v prove_v proposition_n that_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v the_o dodecahedron_n to_o the_o icosahedron_n for_o that_o both_o the_o pentagon_n of_o the_o dodecahedron_n and_o the_o triangle_n of_o the_o icosahedron_n be_v comprehend_v in_o one_o and_o the_o self_n same_o circle_n corollary_n all_o these_o thing_n i_o say_v be_v prove_v it_o be_v manifest_a that_o if_o in_o one_o and_o the_o self_n same_o sphere_n be_v describe_v a_o dodecahedron_n and_o a_o icosahedron_n they_o shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_z a_o right_a line_n whatsoever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o great_a segment_n add_v together_o be_v to_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o for_o for_o that_o as_o the_o dodecahedron_n be_v to_o the_o icosahedron_n so_o be_v the_o superficies_n of_o the_o dodecahedron_n to_o the_o superficies_n of_o the_o icosahedron_n that_o be_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n but_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n so_o any_o right_a line_n what_o so_o ever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n be_v the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o great_a segment_n add_v together_o to_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o wherefore_o as_o a_o dodecahedron_n be_v to_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n so_o any_o right_a line_n what_o so_o ever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n be_v the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n &_o of_o the_o great_a segment_n add_v together_o to_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o the_o end_n of_o the_o fourteen_o book_n of_o euclides_n element_n after_o hypsicles_n ¶_o the_o fourteen_o book_n of_o euclides_n element_n after_o flussas_n for_o that_o the_o fouretenth_fw-mi book_n as_o it_o be_v set_v forth_o by_o flussas_n contain_v in_o it_o more_o proposition_n than_o be_v find_v in_o hypsicles_n &_o also_o some_o of_o those_o proposition_n which_o hypsicles_n have_v be_v by_o he_o somewhat_o otherwise_o demonstrate_v i_o think_v my_o labour_n well_o bestow_v for_o the_o reader_n sake_n to_o turn_v it_o also_o all_o whole_a notwithstanding_o my_o travail_n before_o take_v in_o turn_v the_o same_o book_n after_o hypsicles_n where_o note_v you_o that_o here_o in_o this_o 14._o book_n after_o flussas_n and_o in_o the_o other_o book_n follow_v namely_o the_o 15._o and_o 16._o i_o have_v in_o allege_v of_o the_o proposition_n of_o the_o same_o 14._o book_n follow_v the_o order_n and_o number_n of_o the_o proposition_n as_o flussas_n have_v place_v they_o ¶_o the_o first_o proposition_n a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o a_o circle_n to_o the_o side_n of_o a_o pentagon_n inscribe_v in_o the_o same_o circle_n campane_n be_v the_o half_a of_o these_o two_o line_n take_v together_o namely_o of_o the_o side_n of_o the_o hexagon_n and_o of_o the_o side_n of_o the_o decagon_n inscribe_v in_o the_o same_o circle_n take_v a_o circle_n abc_n and_o inscribe_v in_o it_o the_o side_n of_o a_o pentagon_n which_o let_v be_v bc_o and_o take_v the_o centre_n of_o the_o circle_n which_o let_v be_v the_o point_n d_o construction_n and_o from_o it_o draw_v unto_o the_o side_n bc_o a_o perpendicular_a line_n de_fw-fr which_o produce_v to_o the_o point_n â—_o and_o unto_o the_o line_n e_o fletcher_n put_v the_o line_n eglantine_n equal_a and_o draw_v these_o right_a line_n cg_o cd_o and_o cf._n then_o i_o say_v that_o the_o right_a line_n de_fw-fr which_o be_v draw_v from_o the_o centre_n to_o bc_o the_o side_n of_o the_o pentagon_n be_v the_o half_a of_o â—he_n sideâ—_n of_o the_o decagon_n and_o hexagon_n take_v together_o demonstration_n forasmuch_o as_o the_o line_n de_fw-fr be_v a_o perpendicular_a â—nto_n the_o line_n bc_o therefore_o the_o section_n be_v and_o ec_o shall_v be_v equal_a by_o the_o 3._o of_o the_o three_o and_o the_o line_n of_o be_v common_a unto_o they_o both_o and_o the_o angle_n fec_n and_o feb_n be_v right_a angle_n by_o supposition_n wherefore_o the_o base_n bf_o and_o fc_o be_v equal_a by_o the_o 4._o of_o the_o first_o but_o the_o line_n bc_o be_v the_o side_n of_o a_o pentagon_n by_o construction_n wherefore_o fc_o which_o subtend_v the_o half_a of_o the_o side_n of_o the_o pentagon_n be_v the_o side_n of_o the_o decagon_n inscribe_v in_o the_o circle_n abc_n but_o unto_o the_o line_n fc_o be_v by_o the_o 4._o of_o the_o first_o equal_a the_o line_n cg_o for_o they_o subtend_v right_a angle_n ceg_v and_o cef_n which_o be_v contain_v under_o equal_a side_n wherefore_o also_o the_o angle_n cge_n and_o cfe_n of_o the_o triangle_n cfg_n be_v equal_a by_o the_o 5._o of_o the_o first_o and_o forasmuch_o as_o the_o ark_n fc_o be_v subtend_v of_o the_o side_n of_o a_o decagon_n the_o ark_n ca_n shall_v be_v quadruple_a to_o the_o ark_n cf_o wherefore_o also_o the_o angle_n cda_n shall_v be_v quadruple_a to_o the_o angle_n cdf_n by_o the_o last_o of_o the_o sixâ—_o and_o forasmuch_o as_o the_o same_o angle_n cda_n which_o be_v set_v at_o the_o centre_n be_v double_a to_o the_o angle_n cfa_n which_o be_v set_v at_o the_o circumference_n by_o the_o 20._o of_o the_o three_o therefore_o the_o angle_n cfa_n or_o cfd_n be_v double_a to_o the_o angle_n cdâ—_n namely_o the_o half_a of_o quadruple_a but_o unto_o the_o angle_n cfd_n or_o cfg_n be_v prove_v equal_a the_o angle_n cgf_n wherefore_o the_o outward_a angle_n cgf_n be_v double_a to_o the_o angle_n cdf_n wherefore_o the_o angle_n cdg_n and_o dcg_n shall_v be_v equal_a for_o unto_o those_o two_o angle_n the_o angle_n cgf_n be_v equal_a by_o the_o 32._o of_o the_o first_o wherefore_o the_o side_n gc_o and_o gd_o be_v equal_a by_o the_o 6._o of_o the_o first_o wherefore_o also_o the_o line_n gd_o be_v equal_a to_o the_o line_n fc_o which_o be_v the_o side_n of_o the_o decagon_n but_o unto_o the_o right_a line_n fe_o be_v equal_a the_o line_n eglantine_n by_o construction_n wherefore_o the_o whole_a line_n de_fw-fr be_v equal_a to_o the_o two_o line_n câ—_n and_o fe_o wherefore_o those_o line_n take_v together_o namely_o the_o line_n df_o and_o fc_o shall_v
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
right_a linâ—s_n now_o then_o multiply_v the_o 20._o triangle_n into_o the_o side_n of_o one_o of_o the_o triangle_n and_o so_o shall_v there_o be_v produce_v 6â—_o â—he_z half_a of_o which_o be_v 30._o and_o so_o many_o side_n have_v a_o icosahedron_n and_o in_o like_a sort_n in_o a_o dodecahedron_n forasmuch_o as_o 12._o pentagon_n make_v a_o dodecahedron_n and_o every_o pentagon_n contain_v â—_o right_a linesâ—_n multiply_v â—â—_o into_o 12._o and_o there_o shall_v be_v produce_v 60._o the_o half_a of_o which_o be_v 30._o and_o so_o many_o be_v the_o side_n of_o a_o dodecahedron_n and_o the_o reason_n why_o we_o take_v the_o half_a iâ—_z for_o that_o every_o side_n whether_o it_o be_v of_o a_o triangle_n or_o of_o a_o pentagon_n or_o of_o a_o square_a as_o in_o a_o cube_fw-la â—s_v take_v twice_o and_o by_o the_o same_o reason_n may_v you_o find_v out_o how_o many_o side_n be_v in_o a_o cube_fw-la and_o in_o a_o pyramid_n and_o in_o a_o octohedron_n but_o now_o again_o if_o you_o will_v find_v out_o the_o number_n of_o the_o angle_n of_o every_o one_o of_o the_o solid_a figure_n when_o you_o have_v do_v the_o same_o multiplication_n that_o you_o do_v before_o divide_v the_o same_o side_n by_o the_o number_n of_o the_o plain_a superficiece_n which_o comprehend_v one_o of_o the_o angle_n of_o the_o solid_n as_o for_o example_n forasmuch_o as_o 5._o triangle_n contain_v the_o solid_a angle_n of_o a_o icosahedron_n divide_v 60._o by_o 5._o and_o there_o will_v come_v forth_o 12._o and_o so_o many_o solid_a angle_n have_v a_o icosahedron_n in_o a_o dodecahedron_n forasmuch_o as_o three_o pentagon_n comprehend_v a_o angle_n divide_v 60._o by_o 3._o and_o there_o will_v come_v forth_o 20_o and_o so_o many_o be_v the_o angle_n of_o a_o dodecahedron_n and_o by_o the_o same_o reason_n may_v you_o find_v out_o how_o many_o angle_n be_v in_o each_o of_o the_o rest_n of_o the_o solid_a figure_n book_n if_o it_o be_v require_v to_o be_v know_v how_o one_o of_o the_o plain_n of_o any_o of_o the_o five_o solid_n be_v give_v there_o may_v be_v find_v out_o the_o inclination_n of_o the_o say_a plain_n the_o one_o to_o the_o other_o which_o contain_v each_o of_o the_o solid_n this_o as_o say_v isidorus_n our_o great_a master_n be_v foâ—â—d_v out_o after_o this_o manner_n it_o be_v manifest_a that_o in_o a_o cube_fw-la the_o plain_n which_o contain_v iâ—_z doâ—_n 〈◊〉_d the_o one_o the_o other_o by_o a_o right_a angle_n but_o in_o a_o tetrahedron_n one_o of_o the_o triangle_n be_v give_v let_v the_o end_n of_o one_o of_o the_o side_n of_o the_o say_a triangle_n be_v the_o centre_n and_o let_v the_o space_n be_v the_o perpendicular_a line_n draw_v from_o the_o top_n of_o the_o triangle_n to_o the_o base_a and_o describe_v circumferâ—nces_n of_o a_o circle_n which_o shall_v cut_v the_o one_o the_o other_o and_o from_o the_o intersection_n to_o the_o centre_n draw_v right_a line_n which_o shall_v contain_v the_o inclination_n of_o the_o plain_n contain_v the_o tetrahedron_n in_o a_o octoâ—edron_n take_v one_o of_o the_o side_n of_o the_o triangle_n thereof_o and_o upon_o it_o describe_v a_o square_a and_o draw_v the_o diagonal_a line_n and_o make_v the_o centre_n the_o end_n of_o the_o diagonal_a line_n and_o the_o space_n likewise_o the_o perpendicular_a line_n draw_v from_o the_o top_n of_o the_o triangle_n to_o the_o base_a describe_v circumference_n and_o again_o from_o the_o common_a section_n to_o the_o centre_n draw_v right_a line_n and_o they_o shall_v contain_v the_o inclination_n seek_v for_o in_o a_o icosahedron_n upon_o the_o side_n of_o one_o of_o the_o triangle_n thereof_o describe_v a_o pentagon_n and_o draw_v the_o line_n which_o subtend_v one_o of_o the_o angle_n of_o the_o say_a pentagon_n and_o make_v the_o centre_n the_o end_n of_o that_o line_n and_o the_o space_n the_o perpendicular_a line_n of_o the_o triangle_n describe_v circumference_n and_o draw_v from_o the_o common_a intersection_n of_o the_o circumference_n unto_o the_o centre_n right_a line_n and_o they_o shall_v contain_v likewise_o the_o inclination_n of_o the_o plain_n of_o the_o icosahedron_n in_o a_o dodecahedron_n take_v one_o of_o the_o pentagon_n and_o draw_v likewise_o the_o line_n which_o subtend_v one_o of_o the_o angle_n of_o the_o pentagon_n and_o make_v the_o centre_n the_o end_n of_o that_o line_n and_o the_o space_n the_o perpendicular_a line_n draw_v from_o the_o section_n into_o two_o equal_a part_n of_o that_o line_n to_o the_o side_n of_o the_o pentagon_n which_o be_v parallel_n unto_o it_o describe_v circumference_n and_o from_o the_o point_n of_o the_o intersection_n of_o the_o circumference_n draw_v unto_o the_o centre_n right_a line_n and_o they_o shall_v also_o contain_v the_o inclination_n of_o the_o plain_n of_o the_o dodecahedron_n thus_o do_v this_o most_o singular_a learned_a man_n reason_n think_v the_o demonstration_n in_o every_o one_o of_o they_o to_o be_v plain_a and_o clear_a but_o to_o make_v the_o demonstration_n of_o they_o manifest_a i_o think_v it_o good_a to_o declare_v and_o make_v open_a his_o wordesâ—_n and_o first_o in_o a_o tâ—trahedronâ—_n the_o end_n of_o the_o fivetenth_fw-mi book_n of_o euclides_n element_n after_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d ¶_o the_o 6._o proposition_n the_o 6._o problem_n in_o a_o octohedron_n give_v to_o inscribe_v a_o trilater_n equilater_n pyramid_n svppose_v thaâ—_z the_o octohedron_n whereâ—â—_n the_o tetrahedron_n be_v require_v to_o be_v insâ—riâ—â—â—_n be_v abgdei_n take_v 〈…〉_o base_n of_o the_o octoâ—â—dron_n that_o be_v construction_n 〈…〉_o close_o in_o the_o loweâ—â—_n triangle_n bgd_v namely_o aeâ—_n head_n igd_v and_o let_v the_o four_o be_v aib_n which_o be_v opposite_a to_o the_o low_a triangle_n before_o put_v namely_o to_o egg_v and_o take_v the_o centre_n of_o those_o four_o base_n which_o let_v be_v the_o point_n h_o c_o n_o â—_o and_o upon_o the_o triangle_n hcn_n erect_v a_o pyramid_n hcnl_n now_o forasmuch_o as_o these_o two_o base_n of_o the_o octohedron_n namely_o age_n and_o abi_n be_v set_v upon_o the_o right_a line_n eglantine_n and_o by_o which_o be_v opposite_a the_o one_o to_o the_o otherâ—_n in_o the_o square_a gebi_n of_o the_o octohedron_n from_o the_o poinâ—_n a_o draâ—e_v by_o the_o centre_n of_o the_o base_n namely_o by_o the_o centre_n h_o l_o perpendicular_a line_n ahf_n alk_v cut_v the_o line_n eglantine_n and_o by_o 〈◊〉_d two_o equal_a part_n in_o the_o point_n f_o edward_n by_o the_o corollary_n of_o the_o 1â—_o of_o the_o thirteen_o wherefore_o a_o right_a line_n draw_v from_o the_o point_n f_o to_o the_o point_n king_n demonstration_n shall_v be_v a_o parallel_n and_o equal_a to_o the_o side_n of_o the_o octohedron_n namely_o to_o â—â—_o and_o give_v by_o the_o 33._o of_o the_o first_o and_o the_o right_a line_n hl_o which_o cut_v the_o 〈…〉_o of_o ak_o proportional_o for_o ah_o and_o all_o be_v draw_v from_o the_o centre_n of_o equal_a circle_n to_o the_o circumference_n be_v a_o parallel_n to_o the_o right_a line_n fk_o by_o the_o 2._o of_o the_o six_o and_o also_o to_o the_o side_n of_o the_o octohedron_n namely_o to_o eâ—_n and_o ig_n by_o the_o 9_o of_o the_o eleven_o wherefore_o as_o the_o line_n of_o be_v to_o the_o line_n ah_o so_o be_v the_o line_n fk_o to_o the_o line_n hl_o by_o the_o 4._o of_o the_o six_o for_o the_o triangle_n afk_v and_o ahl_n be_v like_o by_o thâ—_z corollary_n of_o the_o 2._o of_o the_o six_o but_o the_o line_n of_o be_v in_o sesquialter_fw-la proportion_n to_o the_o line_n ah_o for_o the_o side_n eglantine_n make_v hf_o the_o half_a of_o the_o right_a line_n ah_o by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o fk_n or_o give_v the_o side_n of_o the_o octohedron_n be_v sesquialter_fw-la to_o the_o righâ—line_n hl._n and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o side_n of_o the_o octohedron_n be_v sesquialter_fw-la to_o the_o rest_n of_o the_o right_a line_n which_o make_v the_o pyramid_n hnci_n namely_o to_o the_o right_a lineâ—â—_n n_o nc_n ci_o ln_o and_o change_z wherefore_o those_o right_a line_n be_v equal_a and_o therefore_o the_o triangleâ—_n which_o be_v describe_v of_o they_o namely_o the_o triangle_n hcn_n hnl_n ncl_n and_o chl._n which_o make_v the_o pyramid_n hncl_n be_v equal_a and_o equilater_n and_o forasmuch_o as_o the_o angle_n of_o the_o same_o pyramid_n namely_o the_o angle_n h_o c_o n_o l_o do_v end_n in_o the_o centre_n of_o the_o base_n of_o the_o octohedron_n therefore_o it_o be_v inscribe_v â—o_o the_o same_o octohedron_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o octohedron_n â—even_v be_v inscribe_v a_o trilate_a equilaâ—â—â—â—â—â—amisâ—_n which_o be_v require_v to_o â—e_a donâ—_n a_o corollary_n the_o base_n of_o a_o pyramid_n inscribe_v in_o a_o octohedron_n be_v parallel_n
power_n of_o the_o other_o side_n and_o if_o the_o half_a of_o the_o power_n remain_v be_v apply_v upon_o the_o whole_a base_a it_o shall_v make_v the_o breadth_n that_o section_n of_o the_o base_a which_o be_v couple_v to_o the_o first_o side_n for_o from_o the_o power_n of_o the_o base_a bg_o and_o of_o one_o of_o the_o side_n agnostus_n that_o be_v from_o the_o square_n bd_o and_o gl_n the_o power_n of_o the_o other_o side_n ab_fw-la namely_o the_o square_a bk_o or_o the_o parallelogram_n em_n be_v take_v away_o and_o of_o the_o residue_n namely_o of_o the_o square_n bd_o and_o of_o the_o parallelogram_n odd_a or_o doctor_n which_o by_o supposition_n be_v equal_a unto_o odd_a the_o half_a name_o of_o the_o whole_a fr_z which_o be_v pd_v for_o the_o line_n graccus_n and_o pb_n be_v equal_a to_o the_o line_n go_v and_o po_n be_v apply_v to_o the_o whole_a line_n bg_o or_o gd_o and_o make_v the_o bredthe_a the_o line_n pg_n the_o section_n of_o the_o base_a bg_o which_o section_n be_v couple_v to_o the_o first_o side_n ag._n and_o by_o the_o same_o reason_n in_o the_o other_o side_n if_o from_o the_o square_n bd_o and_o bk_o be_v take_v away_o the_o square_a gl_n there_o shall_v remain_v the_o rectangle_n parallelogram_n foyes_n for_o the_o parallalelogramme_n em_n be_v equal_a to_o the_o square_a bk_o and_o the_o parallelogram_n gm_n to_o the_o parallelogram_n odd_a wherefore_o fp_n the_o half_a of_o the_o residue_n foyes_n make_v the_o breadth_n bp_o which_o be_v couple_v to_o the_o first_o side_n take_v ab_fw-la a_o corollary_n 2._o if_o a_o perpendicular_a draw_v from_o a_o angle_n of_o a_o triangle_n do_v cut_v the_o base_a the_o section_n be_v to_o the_o other_o side_n in_o power_n proportional_a by_o a_o arithmetical_a proportion_n for_o it_o be_v prove_v that_o the_o excess_n of_o the_o power_n of_o the_o line_n agnostus_n and_o ab_fw-la be_v one_o and_o the_o same_o with_o the_o excess_n of_o the_o power_n of_o the_o line_n pg_n and_o pb_n if_o therefore_o the_o power_n do_v equal_o exceed_v the_o one_o the_o other_o they_o shall_v by_o a_o arithmetical_a proportion_n be_v proportional_a the_o end_n of_o the_o sixteen_o book_n of_o the_o element_n of_o geometry_n add_v 〈…〉_o a_o brief_a treatise_n add_v by_o flussas_n of_o mix_a and_o compose_v regular_a solid_n regular_a solid_n be_v say_v to_o be_v compose_v and_o mix_v when_o each_o of_o they_o be_v transform_v into_o other_o solid_n keep_v still_o the_o form_n number_n and_o inclination_n of_o the_o base_n which_o they_o before_o have_v one_o to_o the_o other_o some_o of_o which_o yet_o be_v transform_v into_o mix_a solid_n and_o other_o some_o into_o simple_a into_o mix_v as_o a_o dodecahedron_n and_o a_o icosahedron_n which_o be_v transform_v or_o alter_v if_o you_o divide_v their_o side_n into_o two_o equal_a part_n and_o take_v away_o the_o solid_a angle_n subtend_v of_o plain_a superficial_a figure_n make_v by_o the_o line_n couple_v those_o middle_a section_n for_o the_o solid_a remain_v after_o the_o take_v away_o of_o those_o solid_a angle_n be_v call_v a_o icosidodecahedron_n icosidodecahedron_n if_o you_o divide_v the_o side_n of_o a_o cube_fw-la and_o of_o a_o octohedron_n into_o two_o equal_a part_n and_o couple_v the_o section_n the_o solid_a angle_n subtend_v of_o the_o plain_a superficiece_n make_v by_o the_o couple_a line_n be_v take_v away_o there_o shall_v be_v leave_v a_o solid_a which_o be_v call_v a_o exoctohedron_n exoctohedron_n so_o that_o both_o of_o a_o dodecahedron_n and_o also_o of_o a_o icosahedron_n the_o solid_a which_o be_v make_v shall_v be_v call_v a_o icosidodecahedron_n and_o likewise_o the_o solid_a make_v of_o a_o cube_n &_o also_o of_o a_o octohedron_n shall_v be_v call_v a_o exoctohedron_n but_o the_o other_o solid_a namely_o a_o pyramid_n or_o tetrahedron_n be_v transform_v into_o a_o simple_a solid_a for_o if_o you_o divide_v into_o two_o equal_a part_n every_o one_o of_o the_o side_n of_o the_o pyramid_n triangle_n describe_v of_o the_o line_n which_o couple_v the_o section_n and_o subtend_v and_o take_v away_o solid_a angle_n of_o the_o pyramid_n be_v equal_a and_o like_a unto_o the_o equilater_n triangle_n leave_v in_o every_o one_o of_o the_o base_n of_o all_o which_o triangle_n be_v produce_v a_o octohedron_n namely_o a_o simple_a and_o not_o a_o compose_a solid_a for_o the_o octohedron_n have_v four_o base_n like_v in_o number_n fonne_z and_o mutual_a inclination_n with_o the_o base_n of_o the_o pyramid_n and_o have_v the_o other_o four_o base_n with_o like_a situation_n opposite_a and_o parallel_a to_o the_o former_a wherefore_o the_o application_n of_o the_o pyramid_n take_v twice_o make_v a_o simple_a octohedron_n as_o the_o other_o solid_n make_v a_o mix_a compound_n solid_a ¶_o first_n definition_n a_o exoctohedron_n be_v a_o solid_a figure_n contain_v of_o six_o equal_a square_n and_o eight_o equilater_n and_o equal_a triangle_n ¶_o second_v definition_n a_o icosidodecahedron_n be_v a_o solid_a figure_n contain_v under_o twelve_o equilater_n equal_a and_o equiangle_n pentagons_n and_o twenty_o equal_a and_o equilater_n triangle_n for_o the_o better_a understanding_n of_o the_o two_o former_a definition_n and_o also_o of_o the_o two_o proposition_n follow_v i_o have_v here_o set_v two_o figure_n who_o form_n if_o you_o first_o describe_v upon_o past_v paper_n or_o such_o like_a matter_n and_o then_o cut_v they_o and_o fold_v they_o according_o they_o will_v represent_v unto_o you_o the_o perfect_a form_n of_o a_o exoctohedron_n and_o of_o a_o icosidodecahedron_n ¶_o the_o first_o problem_n to_o describe_v a_o equilater_n and_o equiangle_n exoctohedron_n and_o to_o contain_v it_o in_o a_o sphere_n give_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v double_a to_o the_o side_n of_o the_o say_a exoctohedron_n and_o now_o forasmuch_o as_o the_o opposite_a side_n and_o diameter_n of_o the_o base_n of_o the_o cube_fw-la be_v parallel_n sphere_n the_o plain_n extend_v by_o the_o right_a line_n qt_o ur_fw-la shall_v be_v a_o parallelogram_n and_o for_o that_o also_o in_o that_o plain_n lie_v qr_o the_o diameter_n of_o the_o cube_fw-la and_o in_o the_o same_o plain_n also_o be_v the_o line_n mh_o which_o divide_v the_o say_a plain_n into_o two_o equal_a part_n and_o also_o couple_v the_o opposite_a angle_n of_o the_o exoctohedron_n this_o line_n mh_o therefore_o divide_v the_o diameter_n into_o two_o equal_a part_n by_o the_o corollary_n of_o the_o 34._o of_o the_o first_o and_o also_o divide_v itself_o in_o the_o same_o point_n which_o let_v be_v s_o into_o two_o equal_a part_n by_o the_o 4_o of_o the_o first_o and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o rest_n of_o the_o line_n which_o couple_n the_o opposite_a angle_n of_o the_o exoctohedron_n do_v in_o saint_n the_o centre_n of_o the_o cube_fw-la divide_v thâ—_z one_o the_o other_o into_o two_o equal_a part_n for_o every_o one_o of_o the_o angle_n of_o the_o exoctohedron_n be_v set_v in_o every_o one_o of_o the_o base_n of_o the_o cube_fw-la wheâ—â—â—ore_o make_v the_o centre_n the_o point_n s_o and_o the_o space_n shall_v or_o sm_n describe_v a_o sphere_n and_o it_o shall_v touch_v every_o one_o of_o the_o angle_n eqâ—edistant_a from_o the_o point_n s_o and_o forasmuch_o as_o ab_fw-la the_o diameter_n of_o the_o sphere_n give_v be_v put_v equal_a to_o the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la give_v namely_o to_o the_o line_n rt_n and_o the_o same_o line_n rt_n be_v equal_a to_o the_o line_n mh_o by_o the_o 33._o of_o the_o first_o which_o line_n mh_o couple_a the_o opposite_a angle_n of_o the_o exoctohedron_n be_v draw_v by_o the_o centre_n wherefore_o it_o be_v the_o diameter_n of_o the_o sphere_n give_v which_o contain_v the_o exoctohedron_n final_o forasmuch_o as_o in_o the_o triangle_n rft_v the_o line_n po_n do_v cut_v the_o side_n into_o two_o equal_a part_n exoctohedron_n it_o shall_v cut_v they_o proportional_o with_o the_o base_n namely_o as_o fr_z be_v to_o fp_n so_o shall_v rt_n be_v to_o po_n by_o the_o 2._o of_o the_o six_o but_o fr_z be_v double_a to_o fp_n by_o supposition_n wherefore_o rt_n or_o the_o diameter_n he_o be_v also_o double_a to_o the_o line_n po_n the_o side_n of_o the_o exoctohedron_n wherefore_o we_o have_v describe_v a_o equilater_n &_o equiangle_n exoctohedron_n and_o comprehend_v it_o in_o a_o sphere_n give_v and_o have_v prove_v that_o the_o diameter_n of_o the_o sphere_n be_v double_a to_o the_o side_n of_o the_o exoctohedron_n ¶_o the_o 2._o problem_n to_o describe_v a_o equilater_n &_o equiangle_n icosidodecahedron_n &_o to_o comprehend_v it_o in_o a_o sphere_n give_v and_o to_o prove_v that_o the_o diameter_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n double_a to_o the_o side_n
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
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
the_o 7._o of_o the_o five_o our_o conclusion_n be_v infer_v the_o superficies_n spherical_a of_o the_o segment_n caesar_n to_o be_v to_o the_o superficies_n spherical_a of_o the_o segment_n fgh_o as_o ad_fw-la be_v to_o give_v a_o theorem_a 6._o to_o any_o solid_a sector_n of_o a_o sphere_n that_o upright_o câ—â—e_n be_v equal_a who_o base_a be_v equal_a to_o the_o connex_v spherical_a superficies_n of_o that_o sector_n and_o heith_n equal_a to_o the_o semidiameter_n of_o the_o same_o sphere_n hereof_o the_o demonstration_n in_o respect_n of_o the_o premise_n and_o the_o common_a argument_n of_o inscription_n and_o circumscription_n of_o figure_n be_v easy_a and_o nevertheless_o if_o your_o own_o write_n will_v not_o help_v you_o sufficient_o you_o may_v take_v help_n at_o archimedes_n hand_n in_o his_o first_o book_n &_o last_o proposition_n of_o the_o sphere_n and_o cylinder_n whether_o if_o you_o have_v recourse_n you_o shalâ—_n perceive_v how_o your_o theorem_a here_o amend_v the_o common_a translation_n there_o and_o also_o our_o delineation_n give_v more_o sâ—uâ—y_a show_n of_o the_o chief_a circumstance_n necessary_a to_o the_o construction_n than_o there_o you_o shall_v find_v of_o the_o sphere_n here_o imagine_v to_o be_v a_o we_o note_v a_o solid_a sector_n by_o the_o letterâ—_n pqro._n so_o that_o pqr_n do_v signify_v the_o spherical_a superficies_n to_o that_o solid_a sector_n belong_v note_n which_o be_v also_o common_a to_o the_o segment_n of_o the_o same_o sphere_n prq_n and_o therefore_o a_o line_n draw_v from_o the_o top_n of_o that_o segmentâ—_n which_o top_n suppose_v to_o be_v queen_n be_v the_o semidiameter_n of_o the_o circle_n which_o be_v equal_a to_o the_o spherical_a superficies_n of_o the_o say_v solid_a sector_n or_o segmentâ—_n as_o before_o be_v teach_v let_v that_o line_n be_v qp_n by_o q_n draw_v a_o line_n contingent_a which_o let_v be_v sqt._n at_o the_o point_n q_o from_o the_o line_n q_n cut_v a_o line_n equal_a to_o pq_n which_o let_v be_v sq_n and_o unto_o sq_n make_v qt_o equal_a then_o draw_v the_o right_a line_n osoâ—_n and_o oq_fw-la about_o which_o oq_fw-la as_o a_o axe_n faâ—â—ened_v if_o you_o imagine_v the_o triangle_n ost_n to_o make_v a_o etc_n half_a circular_a revolution_n you_o shall_v have_v the_o upright_a cone_fw-mi ost_n who_o heith_n be_v oq_fw-la the_o semidiameter_n of_o the_o sphere_n and_o base_a the_o circle_n who_o diameter_n be_v n-ab_n equal_a to_o the_o solid_a sector_n pqro._n a_o theorem_a 7._o to_o any_o segment_n or_o portion_n of_o a_o sphere_n that_o cone_fw-mi iâ—_z equal_a which_o have_v that_o circle_n to_o his_o base_a which_o be_v the_o bâ—se_a of_o the_o segment_n and_o heith_n a_o right_a line_n which_o unto_o the_o heith_n of_o the_o segment_n have_v that_o proportion_n which_o the_o semidiameter_n of_o the_o sphere_n together_o with_o the_o heith_n of_o the_o other_o segment_n remayâ—â—â—g_o have_v to_o the_o heith_n of_o the_o same_o other_o segment_n remaynâ—ng_v this_o be_v well_o demonstrate_v by_o archiâ—â—des_n &_o therefore_o need_v no_o invention_n of_o i_o cylindrâ—_n to_o confirm_v the_o same_o and_o for_o that_o the_o say_a demonstration_n be_v over_o long_o here_o to_o be_v add_v i_o will_v refer_v you_o thither_o for_o the_o demonstration_n and_o here_o supply_v that_o which_o to_o archimedes_n demonstration_n shall_v geve_v light_a and_o to_o your_o far_a speculation_n and_o practice_n shall_v be_v a_o great_a aid_n and_o direction_n suppose_v king_n to_o be_v a_o sphere_n &_o the_o great_a circle_n king_n in_o contain_v let_v be_v abce_n and_o his_o diameter_n be_v &_o centre_n d._n let_v the_o sphere_n king_n be_v cut_v by_o a_o plain_a superficies_n perpendicular_o erect_v upon_o the_o say_v great_a circle_n abce_n let_v the_o section_n be_v the_o circle_n about_o ac_fw-la and_o let_v the_o segment_n of_o the_o sphere_n be_v the_o one_o that_o wherein_o be_v abc_n who_o â—oppe_n be_v bâ—_n and_o the_o other_o let_v be_v that_o wherein_o be_v aec_o and_o his_o top_n let_v be_v e_o i_o say_v that_o a_o cone_fw-mi which_o have_v his_o base_a the_o circle_n about_o ac_fw-la &_o hold_v a_o line_n which_o to_o bf_o the_o heith_n of_o the_o segment_n who_o top_n be_v b_o have_v that_o proportion_n that_o a_o line_n compoâ—ed_v of_o de_fw-fr the_o semidiameter_n of_o the_o sphere_n and_o of_o the_o heith_n of_o the_o other_o remain_v segment_n who_o top_n be_v e_o have_v to_o of_o the_o heith_n of_o that_o other_o segment_n remain_v be_v equal_a to_o the_o segment_n of_o the_o sphere_n king_n who_o top_n be_v b._n to_o make_v this_o cone_fw-mi take_v my_o easy_a order_n thus_o frame_v your_o work_n for_o the_o find_n of_o the_o four_o proportional_a lineâ—_z by_o make_v of_o the_o first_o and_o a_o line_n compose_v of_o de_fw-fr and_o of_o the_o secondâ—_n and_o the_o three_o let_v be_v bf_o then_o by_o the_o 12._o of_o the_o sixâ—h_o let_v the_o four_o proportional_a line_n be_v find_v which_o let_v be_v fgâ—_n upon_o fletcher_n the_o centre_n of_o the_o base_a of_o the_o segment_n who_o top_n be_v b_o erect_v a_o line_n perpendicular_a equal_a to_o fg_o find_v and_o draw_v the_o line_n ga_o and_o gc_o and_o so_o make_v perfect_a the_o cone_n gac_n i_o say_v that_o the_o cone_fw-mi gac_n be_v equal_a to_o the_o segment_n of_o the_o sphere_n king_n who_o top_n be_v b._n in_o like_a manner_n for_o the_o other_o segment_n who_o top_n be_v e_o to_o find_v the_o heith_n due_a for_o a_o cone_fw-mi equal_a to_o it_o by_o the_o order_n of_o the_o theorem_a you_o must_v thus_o frame_v your_o line_n let_v the_o first_o be_v bf_o the_o second_o db_o and_o bf_o compose_v in_o one_o right_a line_n and_o the_o three_o must_v be_v of_o where_o by_o the_o 12._o of_o the_o six_o find_v the_o four_o it_o shall_v be_v the_o heith_n to_o rear_v upon_o the_o base_a the_o circle_n about_o ac_fw-la to_o make_v a_o upright_a cone_fw-mi equal_a to_o the_o segment_n who_o top_n be_v e._n ¶_o logistical_o ¶_o the_o logistical_a find_v hereof_o be_v most_o easy_a the_o diameter_n of_o the_o sphere_n be_v give_v and_o the_o portion_n of_o the_o diameter_n in_o the_o segment_n contain_v or_o axe_n of_o the_o segment_n be_v know_v then_o order_n your_o number_n in_o the_o rule_n of_o proportion_n as_o i_o here_o have_v make_v most_o plain_a in_o order_v of_o the_o line_n for_o the_o â—ought_v heith_n will_v be_v the_o producte_v a_o corollary_n 1._o hereby_o and_o other_o the_o premise_n it_o be_v evident_a that_o to_o any_o segment_n of_o a_o sphere_n note_n who_o whole_a diameter_n be_v know_v and_o the_o axe_n of_o the_o segment_n give_v a_o upright_a cone_n may_v be_v make_v equal_a or_o in_o any_o proportion_n between_o two_o right_a line_n assignedâ—_n and_o therefore_o also_o a_o cylinder_n may_v to_o the_o say_a segment_n of_o the_o sphere_n be_v make_v equal_a â—r_n in_o any_o proportion_n give_v between_o two_o right_a line_n a_o corollary_n 2._o manifest_o also_o of_o the_o former_a theorem_a it_o may_v be_v infer_v that_o a_o sphere_n and_o his_o diameter_n be_v divide_v by_o one_o and_o the_o same_o plain_a superficies_n to_o which_o the_o say_a diameter_n be_v perpendicularâ—_n the_o two_o segment_n of_o the_o sphere_n be_v one_o to_o the_o other_o in_o that_o proportion_n in_o which_o a_o rectangle_n parallelipipedon_n have_v for_o his_o base_a the_o square_n of_o the_o great_a part_n of_o the_o diameter_n and_o his_o heith_n a_o line_n compose_v of_o the_o less_o portion_n of_o the_o diameter_n and_o the_o semidiameter_n to_o the_o rectangle_n parallelipâ—pedon_n have_v for_o his_o base_a the_o square_n of_o the_o less_o portion_n of_o the_o diameter_n &_o his_o heith_n a_o line_n compose_v of_o the_o semidiameter_n &_o the_o great_a part_n of_o the_o diameter_n a_o theorem_a 8._o every_o sphere_n to_o the_o cube_fw-la make_v of_o his_o diameter_n be_v in_o manner_n as_o 11._o to_o 21._o as_o upon_o the_o first_o and_o second_o proposition_n of_o this_o book_n i_o begin_v my_o addition_n with_o the_o circle_n be_v the_o chief_a among_o plain_a figure_n and_o therein_o bring_v manifold_a consideration_n about_o circle_n as_o of_o the_o proportion_n between_o their_o circumference_n and_o their_o diameter_n of_o the_o content_a or_o area_n of_o circle_n of_o the_o proportion_n of_o circle_n to_o the_o square_n describe_v of_o their_o diameter_n &_o of_o circle_n to_o be_v give_v in_o all_o proâ—portions_n to_o other_o circle_n with_o diverse_a other_o most_o necessary_a problem_n who_o use_n be_v partly_o there_o specify_v so_o have_v i_o in_o the_o end_n of_o this_o book_n add_v some_o such_o problem_n &_o theoremesâ—_n about_o the_o sphere_n be_v among_o solid_n the_o chief_a as_o of_o the_o same_o either_o in_o itself_o consider_v or_o to_o cone_n and_o cylinder_n compare_v by_o reason_n of_o