signify_v last_o of_o all_o a_o dodecahedron_n heaven_n for_o that_o it_o be_v make_v of_o pântagoâ_n who_o angle_n be_v more_o ample_a and_o large_a than_o the_o angle_n of_o the_o other_o body_n and_o by_o that_o âeaâââ_n draw_v more_o ââ_o roundness_n ãâã_d &_o to_o the_o form_n and_o nature_n of_o a_o sphere_n they_o assign_v to_o a_o sphere_n namely_o ãâ¦ã_o who_o so_o will_v ãâ¦ã_o in_o his_o tineus_n shall_v âead_a of_o these_o figure_n and_o of_o their_o mutual_a proportionâââraunge_a maâterâ_n which_o hâre_o be_v not_o to_o be_v entreat_v of_o this_o which_o be_v say_v shall_v be_v sufficient_a for_o the_o ãâã_d of_o they_o and_o for_o thâ_z declaration_n of_o their_o definition_n after_o all_o these_o definition_n here_o set_v of_o euclid_n flussas_n have_v add_v a_o other_o definition_n which_o ãâã_d of_o a_o parallelipipedon_n which_o because_o it_o have_v not_o hitherto_o of_o euclid_n in_o any_o place_n be_v define_v and_o because_o it_o be_v very_o good_a and_o necessary_a to_o be_v have_v i_o think_v good_a not_o to_o omit_v it_o thus_o it_o be_v a_o parallelipipedon_n be_v a_o solid_a figure_n comprehend_v under_o four_o plain_a quadrangle_n figure_n parallelipipedon_n of_o which_o those_o which_o be_v opposite_a be_v parallel_n because_o these_o five_o regular_a body_n here_o define_v be_v not_o by_o these_o figure_n here_o set_v so_o full_o and_o lively_o express_v that_o the_o studious_a beholder_n can_v thorough_o according_a to_o their_o definition_n conceive_v they_o i_o have_v here_o give_v of_o they_o other_o description_n draw_v in_o a_o plain_n by_o which_o you_o may_v easy_o attain_v to_o the_o knowledge_n of_o they_o for_o if_o you_o draw_v the_o like_a form_n in_o matter_n that_o will_v bow_v and_o geve_v place_n as_z most_o apt_o you_o may_v do_v in_o fine_a past_v paper_n such_o as_o pastwive_n make_v woman_n paste_n of_o &_o they_o with_o a_o knife_n cut_v every_o line_n fine_o not_o through_o but_o half_a way_n only_o if_o they_o you_o bow_v and_o bend_v they_o according_o you_o shall_v most_o plain_o and_o manifest_o see_v the_o form_n and_o shape_n of_o these_o body_n even_o as_o their_o definition_n show_v and_o it_o shall_v be_v very_o necessary_a for_o you_o to_o hadââtore_v of_o that_o past_v paper_n by_o you_o for_o so_o shall_v yoâ_n upon_o it_o ãâ¦ã_o the_o form_n of_o other_o body_n as_o prism_n and_o parallelipopedon_n ãâ¦ã_o set_v forth_o in_o these_o five_o book_n follow_v and_o see_v the_o very_a ãâã_d of_o thâse_a body_n there_o mention_v which_o will_v make_v these_o book_n concern_v body_n as_o easy_a unto_o you_o as_o be_v the_o other_o book_n who_o figure_n you_o may_v plain_o see_v upon_o a_o plain_a superficies_n describe_v thiâ_z figurâ_n which_o consi_v of_o twâluââquilââââ_n and_o equiangle_n pântâââââ_n upoâ_n the_o foresay_a matter_n and_o fine_o cut_v as_o before_o be_v ââught_v tâââlâuân_v line_n contain_v within_o thâ_z figurâ_n and_o bow_n and_o fold_n the_o penââgonâ_n according_o and_o they_o will_v so_o close_o together_o thaâ_z thây_n will_v ââkâ_n thâ_z very_a form_n of_o a_o dodecahedron_n dâdââââedron_n icosaâedron_n if_o you_o describe_v this_o figure_n which_o consist_v of_o twenty_o equilater_n and_o equiangle_n triangle_n upon_o the_o foresay_a matter_n and_o fine_o cut_v as_o before_o be_v show_v the_o ninâtâne_a line_n which_o be_v contain_v within_o the_o figure_n and_o then_o bow_n and_o fold_n they_o according_o they_o will_v in_o such_o sort_n close_o together_o that_o therâ_n will_v be_v make_v a_o perfect_v form_n of_o a_o icosahedron_n because_o in_o these_o five_o book_n there_o be_v sometime_o require_v other_o body_n beside_o the_o foresay_a five_o regular_a body_n as_o pyramid_n of_o diverse_a form_n prism_n and_o other_o i_o have_v here_o set_v forth_o three_o figure_n of_o three_o sundry_a pyramid_n one_o have_v to_o his_o base_a a_o triangle_n a_o other_o a_o quadrangle_n figure_n the_o other_o â_o pentagonâ_n which_o if_o you_o describe_v upon_o the_o foresay_a matter_n &_o fine_o cut_v as_o it_o be_v before_o teach_v the_o line_n contain_v within_o each_o figure_n namely_o in_o the_o first_o three_o line_n in_o the_o second_o four_o line_n and_o in_o the_o three_o five_o line_n and_o so_o bend_v and_o fold_v they_o according_o they_o will_v so_o close_o together_o at_o the_o top_n that_o they_o will_v âake_v pyramid_n of_o that_o form_n that_o their_o base_n be_v of_o and_o if_o you_o conceive_v well_o the_o describe_v of_o these_o you_o may_v most_o easy_o describe_v the_o body_n of_o a_o pyramid_n of_o what_o form_n so_o ever_o you_o will._n because_o these_o five_o book_n follow_v be_v somewhat_o hard_o for_o young_a beginner_n by_o reason_n they_o must_v in_o the_o figure_n describe_v in_o a_o plain_a imagine_v line_n and_o superficiece_n to_o be_v elevate_v and_o erect_v the_o one_o to_o the_o other_o and_o also_o conceive_v solid_n or_o body_n which_o for_o that_o they_o have_v not_o hitherto_o be_v acquaint_v with_o will_v at_o the_o first_o sight_n be_v somewhat_o strange_a unto_o they_o i_o have_v for_o their_o more_o âase_a in_o this_o eleven_o book_n at_o the_o end_n of_o the_o demonstration_n of_o every_o proposition_n either_o set_v new_a figure_n if_o they_o concern_v the_o elevate_n or_o erect_v of_o line_n or_o superficiece_n or_o else_o if_o they_o concern_v body_n i_o have_v show_v how_o they_o shall_v describe_v body_n to_o be_v compare_v with_o the_o construction_n and_o demonstration_n of_o the_o proposition_n to_o they_o belong_v and_o if_o they_o diligent_o weigh_v the_o manner_n observe_v in_o this_o eleven_o book_n touch_v the_o description_n of_o new_a figure_n agree_v with_o the_o figure_n describe_v in_o the_o plain_a it_o shall_v not_o be_v hard_o for_o they_o of_o themselves_o to_o do_v the_o like_a in_o the_o other_o book_n follow_v when_o they_o come_v to_o a_o proposition_n which_o concern_v either_o the_o elevate_n or_o erect_v of_o line_n and_o superficiece_n or_o any_o kind_n of_o body_n to_o be_v imagine_v ¶_o the_o 1._o theorem_a the_o 1._o proposition_n that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n &_o part_v elevate_v upward_o be_v impossible_a for_o if_o it_o be_v possible_a let_v part_n of_o the_o right_a line_n abc_n namely_o the_o part_n ab_fw-la be_v in_o a_o ground_n plain_a superficies_n and_o the_o other_o part_n thereof_o namely_o bc_o be_v elevate_v upward_o and_o produce_v direct_o upon_o the_o ground_n plain_a superficies_n the_o right_a line_n ab_fw-la beyond_o the_o point_n b_o unto_o the_o point_n d._n wherefore_o unto_o two_o right_a line_n give_v abc_n and_o abdella_n the_o line_n ab_fw-la be_v a_o common_a section_n or_o part_n which_o be_v impossible_a impossibility_n for_o a_o right_a line_n can_v not_o touch_v a_o right_a line_n in_o ãâã_d point_n then_o one_o uâlesse_o those_o right_n be_v exact_o agree_v and_o lay_v the_o one_o upon_o the_o other_o wherefore_o that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n and_o part_v elevate_v upward_o be_v impossible_a which_o be_v require_v to_o be_v prove_v this_o figure_n more_o plain_o set_v forth_o the_o foresay_a demonstration_n if_o you_o elevate_v the_o superficies_n wherein_o the_o line_n bc._n an_o other_o demonstration_n after_o flâsââs_n if_o it_o be_v possible_a let_v there_o be_v a_o right_a line_n abg_o who_o part_n ab_fw-la let_v be_v in_o the_o ground_n plain_a superficies_n aed_n flussas_n and_o let_v the_o rest_n thereof_o bg_o be_v elevate_v on_o high_a that_o be_v without_o the_o plain_n aed_n then_o i_o say_v that_o abg_o be_v not_o one_o right_a line_n for_o forasmuch_o as_o aed_n be_v a_o plain_a superficies_n produce_v direct_o &_o equal_o upon_o the_o say_a plain_n aed_n the_o right_a line_n ab_fw-la towards_o d_z which_o by_o the_o 4._o definition_n of_o the_o first_o shall_v be_v a_o right_a line_n and_o from_o some_o one_o point_n of_o the_o right_a line_n abdella_n namely_o from_o c_o draâ_n unto_o the_o point_n g_o a_o right_a line_n cg_o wherefore_o in_o the_o triangle_n ãâ¦ã_o the_o outward_a angââ_n abâ_n be_v equal_a to_o the_o two_o inward_a and_o opposite_a angle_n by_o the_o 32._o of_o the_o first_o and_o therefore_o it_o be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o same_o wherefore_o the_o line_n abg_o forasmuch_o as_o it_o make_v a_o angle_n be_v not_o â_o right_n line_n wherefore_o that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n and_o part_v elevate_v upward_o be_v impossible_a if_o you_o mark_v well_o the_o figure_n before_o add_v for_o the_o plain_a declaration_n of_o euclides_n demonstration_n iâ_z will_v not_o be_v hard_o for_o you_o to_o coââââe_v this_o figure_n which_o ulyss_n put_v for_o his_o demonstration_n â_o wherein_o
triangle_n abc_n acd_o &_o cde_o and_o likewise_o the_o base_a fghkl_o into_o these_o triangle_n fgh_o fhl_n and_o hkl_n and_o imagine_v that_o upon_o every_o one_o of_o those_o triangle_n be_v set_v a_o pyramid_n of_o equal_a altitude_n with_o the_o two_o pyramid_n put_v at_o the_o beginning_n demonstration_n and_o for_o that_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n adc_o so_o be_v the_o pyramid_n abcm_n to_o the_o pyramid_n adcm_n by_o the_o 5._o of_o this_o book_n wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o the_o four_o side_v figure_n abcd_o be_v to_o the_o triangle_n acd_o so_o be_v the_o pyramid_n abcdm_n to_o the_o pyramid_n acdm_n but_o as_o the_o triangle_n acd_o be_v to_o the_o triangle_n cde_o so_o be_v the_o pyramid_n acdm_n to_o the_o pyramid_n cdem_fw-la wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o the_o base_a abcd_o be_v to_o thâ_z base_a cde_o so_o be_v the_o pyramid_n abcdm_n to_o the_o pyramid_n cdem_fw-la wherefore_o again_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o the_o base_a abcde_o be_v to_o the_o base_a cde_o so_o be_v the_o pyramid_n abcedm_n to_o the_o pyramid_n cdem_fw-la and_o by_o the_o same_o reason_n also_o as_o the_o base_a fghkl_o be_v to_o the_o base_a hkl_n so_o be_v the_o pyramid_n fghkln_v to_o the_o pyramid_n hkln_n and_o forasmuch_o as_o there_o be_v two_o pyramid_n cdem_n and_o hkln_n have_v triangle_n to_o their_o base_n and_o be_v under_o one_o and_o the_o self_n same_o altitude_n therefore_o by_o the_o 5._o of_o the_o twelve_o as_o the_o base_a cde_o be_v to_o the_o base_a hkl_n so_o be_v the_o pyramid_n cdem_n to_o the_o pyramid_n hkln_n now_o for_o that_o as_o the_o base_a abce_v be_v to_o the_o base_a cde_o so_o be_v the_o pyramid_n abcedm_n to_o the_o pyramid_n cdem_fw-la but_o as_o the_o base_a cde_o be_v to_o the_o base_a hkl_n so_o be_v the_o pyramid_n cdem_n to_o the_o pyramid_n hkln_n wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o the_o base_a abce_v be_v to_o the_o base_a hkl_n so_o be_v the_o pyramid_n abcedm_n to_o the_o pyramid_n hkln_n but_o also_o as_o the_o base_a hkl_n be_v the_o base_a fghkl_o so_o be_v the_o pyramid_n hkln_v to_o to_o the_o pyramid_n fghkln_n wherefore_o again_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o the_o base_a abce_v be_v to_o the_o base_a fghkl_o so_o be_v the_o pyramid_n abcedm_n to_o the_o pyramid_n fghkln_n wherefore_o pyramid_n consist_v under_o one_o and_o the_o self_n same_o altitude_n and_o have_v polygonon_n figure_n to_o their_o base_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v which_o be_v require_v to_o be_v prove_v the_o 7._o theorem_a the_o 7._o proposition_n every_o prism_n have_v a_o triangle_n to_o his_o base_a may_v be_v divide_v into_o three_o pyramid_n equal_a the_o one_o to_o the_o other_o have_v also_o triangle_n to_o their_o base_n svppose_v that_o abcdef_n be_v a_o prism_n have_v to_o his_o base_a the_o triangle_n abc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n def_n then_o i_o say_v that_o the_o prism_n abcdef_n may_v be_v divide_v into_o three_o pyramid_n equal_a the_o one_o to_o the_o other_o and_o have_v triangle_n to_o their_o base_n demonstration_n draw_v these_o right_a line_n bd_o ec_o and_o cd_o and_o forasmuch_o as_o abed_o be_v a_o parallelogram_n and_o his_o diameter_n be_v the_o line_n bd_o therefore_o the_o triangle_n abdella_n be_v equal_a to_o the_o triangle_n edb_n wherefore_o also_o the_o pyramid_n who_o base_a be_v the_o triangle_n abdella_n and_o top_n the_o point_n c_o be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n edb_n &_o top_n the_o point_n c_o by_o the_o 5._o of_o this_o book_n but_o the_o pyramid_n who_o base_a be_v the_o triangle_n edb_n and_o top_n the_o point_n c_o be_v one_o and_o the_o same_o which_o the_o pyramid_n who_o base_a be_v the_o triangle_n ebc_n and_o top_n the_o point_n d_o for_o they_o be_v comprehend_v of_o the_o self_n same_o plain_a superficiece_n namely_o of_o the_o triangle_n bdedec_n dbc_n and_o ebc_n wherefore_o also_o the_o pyramid_n who_o base_a be_v the_o triangle_n abdella_n and_o top_n the_o point_n c_o be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n ebc_n and_o top_n the_o point_n d._n again_o forasmuch_o as_o bcfe_n be_v a_o parallelogram_n and_o the_o diameter_n thereof_o be_v ec_o therefore_o the_o triangle_n ecf_n be_v equal_a to_o the_o triangle_n cbe_n wherefore_o also_o the_o pyramid_n who_o base_a be_v the_o triangle_n ebc_n and_o top_n the_o point_n d_o be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n ecf_n and_o top_n the_o point_n d_o by_o the_o 5._o of_o this_o book_n but_o the_o pyramid_n who_o base_a be_v the_o triangle_n bec_n and_o top_n the_o point_n d_o be_v prove_v to_o be_v equal_a to_o the_o pyramid_n who_o base_a iâ_z the_o triangle_n abdella_n and_o top_n the_o point_n c._n wherefore_o also_o the_o pyramid_n who_o base_a be_v the_o triangle_n cef_n and_o top_n the_o point_n d_o be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n abdella_n &_o top_n the_o point_n c._n wherefore_o the_o prism_n abdef_n be_v divide_v into_o three_o equal_a pyramid_n have_v triangle_n to_o their_o base_n and_o forasmuch_o as_o the_o pyramid_n who_o base_a be_v the_o triangle_n abdella_n and_o top_n the_o point_n c_o be_v one_o &_o the_o self_n same_o with_o the_o pyramid_n who_o base_a be_v the_o triangle_n cab_n &_o top_n the_o point_n d_o for_o they_o be_v contain_v under_o the_o self_n same_o plain_a superficiece_n but_o it_o have_v be_v prove_v that_o the_o pyramid_n who_o base_a be_v the_o triangle_n abdella_n and_o top_n the_o point_n c_o be_v the_o three_o pyramid_n of_o the_o prism_n who_o base_a be_v the_o triangle_n abc_n aâd_z the_o opposite_a side_n unto_o it_o the_o triangle_n def_n wherefore_o the_o pyramid_n who_o base_a be_v the_o triangle_n abc_n and_o top_n the_o point_n d_o be_v the_o three_o pyramid_n of_o the_o prism_n who_o base_a be_v the_o triangle_n abc_n and_o opposite_a side_n the_o triangle_n def_n wherefore_o every_o prism_n have_v a_o triangle_n to_o his_o base_a may_v be_v divide_v into_o three_o pyramid_n equal_a the_o one_o to_o the_o other_o have_v also_o triangle_n to_o their_o base_n 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 every_o pyramid_n be_v the_o three_o part_n of_o a_o prism_n have_v one_o and_o the_o same_o base_a with_o it_o and_o also_o be_v under_o the_o self_n same_o altitude_n with_o it_o for_o if_o the_o base_a of_o the_o prism_n be_v any_o other_o rectiline_a figure_n they_o a_o triangle_n that_o also_o may_v be_v divide_v into_o prism_n which_o shall_v have_v triangle_n to_o their_o base_n here_o campane_n and_o flussas_n add_v certain_a corollarye_n first_o corollary_n every_o prism_n be_v treble_a to_o the_o pyramid_n which_o have_v the_o self_n same_o triangle_n to_o his_o base_a that_o the_o prism_n have_v and_o the_o self_n same_o altitude_n as_o it_o be_v manifest_a by_o this_o proposition_n where_o the_o prism_n be_v divide_v into_o three_o equal_a pyramid_n of_o which_o two_o be_v upon_o one_o and_o the_o self_n same_o base_a and_o under_o one_o and_o the_o self_n same_o altitude_n but_o if_o the_o prism_n have_v to_o his_o âase_n a_o parallelogram_n and_o if_o the_o pyramid_n have_v to_o his_o base_a the_o half_a of_o the_o same_o parallelogram_n and_o their_o altitude_n be_v equal_a then_o again_o the_o pyramid_n shall_v be_v the_o three_o part_n of_o the_o prism_n for_o it_o be_v manifest_a by_o the_o 40._o of_o the_o cleveth_n that_o prism_n be_v under_o equal_a altitude_n and_o the_o one_o have_v to_o his_o base_a a_o triangle_n and_o the_o other_o a_o parallelogram_n double_a to_o the_o same_o triangle_n be_v equal_a the_o one_o to_o the_o other_o whereof_o follow_v the_o former_a conclusion_n second_o corollary_n if_o there_o be_v many_o prism_n under_o one_o and_o the_o same_o altitude_n and_o have_v triangle_n to_o their_o base_n they_o and_o if_o the_o triangular_a base_n be_v so_o join_v together_o upon_o one_o and_o the_o same_o plain_n that_o they_o compose_v a_o poligonon_n figure_n a_o pyramid_n set_v upon_o that_o base_a be_v a_o poligonon_n figure_n and_o under_o the_o same_o altitude_n be_v the_o three_o part_n of_o that_o solid_a which_o be_v compose_v of_o all_o the_o prism_n add_v together_o for_o forasmuch_o as_o euâry_v one_o of_o the_o prism_n which_o have_v to_o his_o base_a a_o triangle_n to_o every_o one_o of_o the_o pyramid_n set_v upon_o the_o same_o base_a the_o altitude_n be_v always_o one_o and_o the_o
side_n the_o rectiline_a angle_n mab_n either_o of_o which_o be_v equal_a to_o the_o rectiline_a angle_n give_v cde_o which_o be_v require_v to_o be_v do_v an_o other_o construction_n also_o and_o demonstration_n after_o pelitarâus_n and_o if_o the_o perpendicular_a line_n chance_n to_o fall_v without_o the_o angle_n give_v namely_o if_o the_o angle_n give_v be_v a_o acute_a angle_n the_o self_n same_o manner_n of_o demonstration_n will_v serve_v but_o only_o that_o in_o stead_n of_o the_o second_o common_a sentence_n must_v be_v use_v the_o 3._o common_a sentence_n appollonius_n put_v another_o construction_n &_o demonstration_n of_o this_o proposition_n which_o though_o the_o demonstration_n thereof_o depend_v of_o proposition_n put_v in_o the_o three_o book_n yet_o for_o that_o the_o construction_n be_v very_o good_a for_o he_o that_o will_v ready_o and_o mechanical_o without_o demonstration_n describe_v upon_o a_o line_n give_v and_o to_o a_o point_n in_o it_o give_v a_o rectiline_a angle_n equal_a to_o a_o rectiline_a angle_n give_v i_o think_v not_o amiss_o here_o to_o place_v it_o and_o it_o be_v thus_o proposition_n oenopides_n be_v the_o first_o inventor_n of_o this_o proposition_n as_o witness_v eudemius_fw-la the_o 15._o theorem_a the_o 24._o proposition_n if_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_a to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n and_o if_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o one_o be_v great_a than_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o other_o the_o base_a also_o of_o the_o same_o shall_v be_v great_a than_o the_o base_a of_o the_o other_o svppose_v that_o there_o be_v two_o triangle_n abc_n and_o def_n have_v two_o side_n of_o the_o one_o that_o be_v ab_fw-la and_o ac_fw-la equal_a to_o two_o side_n of_o the_o other_o that_o be_v to_o de_fw-fr and_o df_o each_o to_o his_o correspondent_a side_n that_o be_v the_o side_n ab_fw-la to_o the_o side_n de_fw-fr and_o the_o side_n ac_fw-la to_o the_o side_n df_o and_o suppose_v that_o the_o angle_n bac_n be_v great_a than_o the_o angle_n edf_o then_o i_o say_v that_o the_o base_a bc_o be_v great_a than_o the_o base_a ef._n for_o forasmuch_o as_o the_o angle_n bac_n be_v great_a than_o the_o angle_n edf_o construction_n make_v by_o the_o 23._o proposition_n upon_o the_o right_a line_n de_fw-fr and_o to_o the_o point_n in_o it_o give_v d_o a_o angle_n edge_v equal_a to_o the_o angle_n give_v bac_n and_o to_o one_o of_o these_o line_n that_o be_v either_o to_o ac_fw-la or_o df_o put_v a_o equal_a line_n dg_o and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n g_o to_o the_o point_n e_o and_o a_o other_o from_o the_o point_n fletcher_n demonstration_n to_o the_o point_n g._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n de_fw-fr and_o the_o line_n ac_fw-la to_o the_o line_n dg_o the_o one_o to_o the_o outstretch_o and_o the_o angle_n bac_n be_v by_o construction_n equal_a to_o the_o angle_n edg_n therefore_o by_o the_o 4._o proposition_n the_o base_a bc_o be_v equal_a to_o the_o base_a eglantine_n again_o for_o as_o much_o as_o the_o line_n dg_o be_v equal_a to_o the_o line_n df_o therâ_n by_o the_o 5._o proposition_n the_o angle_n dgf_n be_v equal_a to_o the_o angle_n dfg_n wherefore_o the_o angle_n dfg_n be_v great_a than_o the_o angle_n egf_n wherefore_o the_o angle_n efg_o be_v much_o great_a than_o the_o angle_n egf_n and_o forasmuch_o as_o efg_o be_v a_o triangle_n have_v the_o angle_n efg_o great_a than_o the_o angle_n egf_n and_o by_o the_o 18._o prâposition_n under_o the_o great_a angle_n be_v subtend_v the_o great_a side_n therefore_o the_o side_n eglantine_n be_v great_a than_o the_o side_n ef._n but_o the_o side_n eglantine_n be_v equal_a to_o the_o side_n bc_o wherefore_o the_o side_n bc_o be_v great_a than_o the_o side_n ef._n if_o therefore_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_a to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n and_o if_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o one_o be_v great_a than_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o other_o the_o base_a also_o of_o the_o same_o shall_v be_v great_a than_o the_o base_a of_o the_o other_o which_o be_v require_v to_o be_v prove_v in_o this_o theorem_a may_v be_v three_o case_n prâposition_n for_o the_o angle_n edg_n be_v put_v equal_a to_o the_o angle_n bac_n and_o the_o line_n dg_o be_v put_v equal_a to_o the_o line_n ac_fw-la and_o a_o line_n be_v draw_v from_o e_o to_o g_z the_o line_n eglantine_n shall_v either_o fall_v above_o the_o line_n gf_o or_o upon_o it_o or_o under_o it_o euclides_n demonstration_n serve_v 2._o when_o the_o line_n ge_z fall_v above_o the_o line_n gf_o as_o we_o have_v before_o manifest_o see_v but_o now_o let_v the_o line_n eglantine_n case_n fall_v under_o the_o line_n e_o fletcher_n as_o in_o the_o figure_n here_o put_v and_o forasmuch_o as_o these_o two_o line_n ab_fw-la and_o ac_fw-la be_v equal_a to_o these_o two_o line_n de_fw-fr and_o dg_o the_o one_o to_o the_o other_o and_o they_o contain_v equal_a angle_n therefore_o by_o the_o 4._o proposition_n the_o base_a bc_o be_v equal_a to_o the_o base_a eglantine_n and_o forasmuch_o as_o within_o the_o triangle_n deg_n the_o two_o linne_n df_o and_o fe_o be_v set_v upon_o the_o side_n de_fw-fr therefore_o by_o the_o 21._o proposition_n the_o line_n df_o and_o fâ_n be_v less_o than_o the_o outward_a line_n dg_o and_o ge_z but_o the_o line_n dg_o be_v equal_a to_o the_o line_n df._n wherefore_o the_o line_n ge_z be_v great_a than_o the_o line_n fe_o but_o ge_z be_v equal_a to_o bc._n wherefore_o the_o line_n bc_o be_v great_a the_o the_o line_n ef._n which_o be_v require_v to_o be_v prove_v it_o may_v peradventure_o semeâ_n that_o euclid_n shall_v here_o in_o this_o proposition_n have_v prove_v that_o not_o only_o the_o base_n of_o the_o triangle_n be_v unequal_a but_o also_o that_o the_o area_n of_o the_o same_o be_v unequal_a for_o so_o in_o the_o four_o proposition_n after_o he_o have_v prove_v the_o base_a to_o be_v equal_a he_o prove_v also_o the_o area_n to_o be_v equal_a but_o hereto_o may_v be_v answer_v &c._a that_o in_o equal_a angle_n and_o base_n and_o unequal_a angle_n and_o base_n the_o consideration_n be_v not_o like_a for_o the_o angle_n and_o base_n be_v equal_a the_o triangle_n also_o shall_v of_o necessity_n be_v equal_a but_o the_o angle_n and_o base_n be_v unequal_a the_o area_n shall_v not_o of_o necessity_n be_v equal_a for_o the_o triangle_n may_v both_o be_v equal_a and_o unequal_a and_o that_o may_v be_v the_o great_a which_o have_v the_o great_a angle_n and_o the_o great_a base_n and_o it_o may_v also_o be_v the_o less_o and_o for_o that_o cause_n euclid_n make_v not_o mention_v of_o the_o comparison_n of_o the_o triangle_n whereof_o this_o also_o may_v be_v a_o cause_n for_o that_o to_o the_o demonstration_n thereof_o be_v require_v certain_a proposition_n concern_v parallel_a line_n which_o we_o be_v not_o as_o yet_o come_v unto_o etc_n howbeit_o after_o the_o 37â_n proposition_n of_o his_o book_n you_o shall_v find_v the_o comparison_n of_o the_o area_n of_o triangle_n which_o have_v their_o side_n equal_a and_o their_o base_n and_o angle_n at_o the_o top_n unequal_a the_o 16._o theorem_a the_o 25._o proposition_n if_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_a to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n and_o if_o the_o base_a of_o the_o one_o be_v great_a than_o the_o base_a of_o the_o other_o the_o angle_n also_o of_o the_o same_o contain_v under_o the_o equal_a right_a linesâ_n shall_v be_v great_a than_o the_o angle_n of_o the_o other_o svppose_v that_o there_o be_v two_o triangle_n a_o b_o c_o and_o def_n have_v two_o side_n of_o tb'one_n that_o be_v ab_fw-la and_o ac_fw-la equal_a to_o two_o side_n of_o the_o other_o that_o be_v to_o de_fw-fr and_o df_o each_o to_o his_o correspondent_a side_n namely_o the_o side_n ab_fw-la to_o the_o side_n df_o and_o the_o side_n ac_fw-la to_o the_o side_n df._n but_o let_v the_o base_a bc_o be_v great_a than_o the_o base_a ef._n then_o i_o say_v they_o the_o angle_n bac_n be_v great_a than_o the_o angle_n edf_o for_o if_o not_o absurdity_n then_o be_v it_o either_o equal_a unto_o it_o or_o less_o than_o it_o but_o the_o angle_n bac_n be_v not_o equal_a to_o the_o angle_n edf_o for_o if_o it_o be_v equal_a the_o base_a also_o bc_o shall_v by_o the_o 4._o proposition_n be_v equal_a to_o the_o base_a of_o but_o by_o supposition_n it_o be_v not_o wherefore_o
number_n of_o the_o number_n ad_fw-la and_o db_o be_v double_a to_o the_o square_a number_n of_o ac_fw-la and_o cd_o for_o forasmuch_o as_o the_o number_n ad_fw-la be_v divide_v into_o the_o number_n ab_fw-la and_o bd_o therefore_o the_o square_a number_n of_o the_o number_n ad_fw-la and_o db_o be_v equal_a to_o the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n ad_fw-la and_o db_o the_o on_o into_o the_o other_o twice_o together_o with_o the_o square_n of_o the_o number_n ab_fw-la by_o the_o 7_o proposition_n but_o the_o square_n of_o the_o number_n ab_fw-la be_v equal_a to_o four_o square_n of_o either_o of_o the_o number_n ac_fw-la or_o cb_o for_o ac_fw-la be_v equal_a to_o the_o number_n cb_o wherefore_o also_o the_o square_n of_o the_o number_n ad_fw-la and_o db_o be_v equal_a to_o the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n ad_fw-la and_o db_o the_o one_o into_o the_o other_o twice_o and_o to_o four_o square_n of_o the_o number_n bc_o or_o ca._n and_o forasmuch_o as_o the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n ad_fw-la and_o db_o the_o one_o into_o the_o other_o together_o with_o the_o square_n of_o the_o number_n cb_o be_v equal_a to_o square_v of_o the_o number_n cd_o by_o the_o 6_o proposition_n therefore_o the_o number_n produce_v of_o the_o multiplication_n of_o the_o number_n ad_fw-la and_o db_o the_o one_o into_o the_o other_o twice_o together_o with_o two_o square_n of_o the_o number_n cb_o be_v equal_a to_o two_o square_n of_o the_o number_n cd_o wherefore_o the_o square_n of_o the_o number_n ad_fw-la and_o db_o be_v equal_a to_o two_o square_n of_o the_o number_n cd_o and_o to_o two_o square_n of_o the_o number_n ac_fw-la wherefore_o they_o be_v double_a to_o the_o square_n of_o the_o number_n ac_fw-la and_o cd_o and_o the_o square_a of_o the_o number_n ad_fw-la be_v the_o square_n of_o the_o whole_a and_o of_o the_o number_n add_v and_o the_o square_a of_o db_o be_v the_o square_a of_o the_o number_n add_v the_o square_a also_o of_o the_o number_n cd_o be_v the_o square_a of_o the_o number_n compose_v of_o the_o half_a and_o of_o the_o number_n add_v if_o therefore_o a_o even_a number_n be_v divide_v etc_n etc_n which_o be_v require_v to_o be_v prove_v the_o 1._o problem_n the_o 11._o proposition_n to_o divide_v a_o right_a line_n give_v in_o such_o sort_n that_o the_o rectangle_n figure_n comprehend_v under_o the_o whole_a and_o one_o of_o the_o part_n shall_v be_v equal_a unto_o the_o square_n make_v of_o the_o other_o part_n svppose_v that_o the_o right_a line_n give_v be_v ab_fw-la now_o it_o be_v require_v to_o divide_v the_o line_n ab_fw-la in_o such_o sort_n that_o the_o rectangle_n figure_n contain_v under_o the_o whole_a and_o one_o of_o the_o part_n shall_v be_v equal_a unto_o the_o square_n which_o be_v make_v of_o the_o other_o part_n describe_v by_o the_o 46._o of_o the_o first_o upon_o ab_fw-la a_o square_a abcd._n construction_n and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n ac_fw-la into_o two_o equal_a part_n in_o the_o point_n e_o and_o draw_v a_o line_n from_o b_o to_o e._n and_o by_o the_o second_o petition_n extend_v ca_n unto_o the_o point_n f._n and_o by_o the_o 3._o of_o the_o first_o put_v the_o line_n of_o equal_a unto_o the_o line_n be._n and_o by_o the_o 46._o of_o the_o first_o upon_o the_o line_n of_o describe_v a_o square_a fgah_n and_o by_o the_o 2._o petition_n extend_v gh_o unto_o the_o point_n k._n then_o i_o say_v that_o the_o line_n ab_fw-la be_v divide_v in_o the_o point_n h_o in_o such_o sort_n that_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o ab_fw-la and_o bh_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o ah_o demonstration_n for_o forasmuch_o as_o the_o right_a line_n ac_fw-la be_v divide_v into_o two_o equal_a part_n in_o the_o point_n e_o and_o unto_o it_o be_v add_v a_o other_o right_a line_n af._n therefore_o by_o the_o 6._o of_o the_o second_o the_o rectangle_n figure_n contain_v under_o cf_o and_o favorina_n together_o with_o the_o square_n which_o be_v make_v of_o ae_n be_v equal_a to_o the_o square_v which_o be_v make_v of_o ef._n but_o of_o be_v equal_a unto_o ebb_n wherefore_o the_o rectangle_n figure_n contain_v under_o cf_o and_o favorina_n together_o with_o the_o square_n which_o be_v make_v of_o ea_fw-la be_v equal_a to_o the_o square_n which_o be_v make_v of_o ebb_n but_o by_o 47._o of_o the_o first_o unto_o the_o square_n which_o be_v make_v of_o ebb_n be_v equal_a the_o square_n which_o be_v make_v of_o basilius_n and_o ae_n for_o the_o angle_n at_o the_o point_n a_o be_v a_o right_a angle_n wherefore_o that_o which_o be_v contain_v under_o cf_o and_o favorina_n together_o with_o the_o square_n which_o be_v make_v of_o ae_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o basilius_n and_o ae_n take_v away_o the_o square_n which_o be_v make_v of_o ae_n which_o be_v common_a to_o they_o both_o wherefore_o the_o rectangle_n figure_n remain_v contain_v under_o cf_o and_o favorina_n be_v equal_a unto_o the_o square_n which_o be_v make_v of_o ab_fw-la and_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o favorina_n be_v the_o figure_n fk_o for_o the_o line_n favorina_n be_v equal_a unto_o the_o line_n fg._n and_o the_o square_n which_o be_v make_v of_o ab_fw-la be_v the_o figure_n ad._n wherefore_o the_o figure_n fk_o be_v equal_a unto_o the_o figure_n ad._n take_v away_o the_o figure_n ak_o which_o be_v common_a to_o they_o both_o wherefore_o the_o residue_n namely_o the_o figure_n fh_o be_v equal_a unto_o the_o residue_n namely_o unto_o the_o figure_n hd_v but_o the_o figure_n hd_v be_v that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bh_o for_o ab_fw-la be_v equal_a unto_o bd._n and_o the_o figure_n fh_o be_v the_o square_n which_o be_v make_v of_o ah_o wherefore_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n ab_fw-la and_o bh_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n ha._n wherefore_o the_o right_a line_n give_v ab_fw-la be_v divide_v in_o the_o point_n h_o in_o such_o sort_n that_o the_o rectangle_n figure_n contain_v under_o ab_fw-la and_o bh_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o ah_o which_o be_v require_v to_o be_v do_v this_o proposition_n have_v many_o singular_a use_n upon_o it_o depend_v the_o demonstration_n of_o that_o worthy_a problem_n the_o 10._o proposition_n of_o the_o 4._o book_n proposition_n which_o teach_v to_o describe_v a_o isosceles_a triangle_n in_o which_o either_o of_o the_o angle_n at_o the_o base_a shall_v be_v double_a to_o the_o angle_n at_o the_o top_n many_o and_o diverse_a use_n of_o a_o line_n so_o divide_v shall_v you_o find_v in_o the_o 13._o book_n of_o euclid_n this_o be_v to_o be_v note_v that_o this_o proposition_n can_v not_o as_o the_o former_a proposition_n of_o this_o second_o book_n be_v reduce_v unto_o number_n number_n for_o the_o line_n ebb_n have_v unto_o the_o line_n ae_n no_o proportion_n that_o can_v be_v name_v and_o therefore_o it_o can_v not_o be_v express_v by_o number_n for_o forasmuch_o as_o the_o square_n of_o ebb_n be_v equal_a to_o the_o two_o square_n of_o ab_fw-la and_o ae_n by_o the_o 47._o of_o the_o first_o and_o ae_n be_v the_o half_a of_o ab_fw-la therefore_o the_o line_n be_v be_v irrational_a for_o even_o as_o two_o equal_a square_a number_n join_v together_o can_v not_o make_v a_o square_a number_n so_o also_o two_o square_a number_n of_o which_o the_o one_o be_v the_o square_a of_o the_o half_a root_n of_o the_o other_o can_v not_o make_v a_o square_a number_n as_o by_o a_o example_n take_v the_o square_n of_o 8._o which_o be_v 64._o which_o double_v that_o be_v 128._o make_v not_o a_o square_a number_n so_o take_v the_o half_a of_o 8._o which_o be_v 4._o and_o the_o square_n of_o 8._o and_o 4._o which_o be_v 64._o and_o 16._o add_v together_o likewise_o make_v not_o a_o square_a number_n for_o they_o make_v 80._o who_o have_v no_o root_n square_v which_o thing_n must_v of_o necessity_n be_v if_o this_o problem_n shall_v have_v place_n in_o number_n but_o in_o irrational_a number_n it_o be_v true_a and_o may_v by_o this_o example_n be_v declare_v let_v 8._o be_v so_o divide_v that_o that_o which_o be_v produce_v of_o the_o whole_a into_o one_o of_o his_o part_n shall_v be_v equal_a to_o the_o square_a number_n produce_v of_o the_o other_o part_n multiply_v 8._o into_o himself_o and_o there_o shall_v be_v produce_v 64._o that_o be_v the_o square_a abcd._n divide_v 8._o into_o two_o equal_a part_n that_o be_v into_o 4_o and_o 4._o as_o the_o line_n
proportion_n than_o have_v bc_o to_o ef._n wherefore_o a_o have_v to_o d_o a_o great_a proportion_n than_o have_v bc_o to_o ef._n wherefore_o alternate_o a_o have_v to_o bc_o a_o great_a proportion_n than_o have_v d_z to_o of_o wherefore_o by_o composition_n abc_n have_v to_o bc_o a_o great_a proportion_n than_o have_v def_n to_o ef._n wherefore_o again_o alternate_o abc_n have_v to_o def_n a_o great_a proportion_n than_o have_v bc_o to_o ef._n wherefore_o by_o the_o former_a proposition_n the_o proportion_n of_o a_o to_o d_z be_v great_a than_o the_o proportion_n of_o abc_n to_o def_n which_o be_v require_v to_o be_v prove_v the_o end_n of_o the_o five_o book_n of_o euclides_n element_n ¶_o the_o six_o book_n of_o euclides_n element_n this_o spaniard_o book_n be_v for_o use_v and_o practice_v book_n a_o most_o special_a book_n in_o it_o be_v teach_v the_o proportion_n of_o one_o figure_n to_o a_o other_o figure_n &_o of_o their_o side_n the_o one_o to_o the_o other_o and_o of_o the_o side_n of_o one_o to_o the_o side_n of_o a_o other_o likewise_o of_o the_o angle_n of_o the_o one_o to_o the_o angle_n of_o the_o other_o moreover_o it_o teach_v the_o description_n of_o figure_n like_o to_o â_z give_v and_o marvelous_a application_n of_o figure_n to_o line_n even_o or_o with_o decrease_n or_o excess_n with_o many_o other_o theorem_n not_o only_o of_o the_o propoâtions_n of_o right_n line_v figure_n but_o also_o of_o sector_n of_o circle_n with_o their_o angle_n on_o the_o theorem_n and_o problem_n of_o this_o book_n depend_v for_o the_o most_o part_n the_o composition_n of_o all_o instrument_n of_o measure_v length_n breadth_n or_o deepen_n and_o also_o the_o reason_n of_o the_o use_n of_o the_o same_o instrument_n as_o of_o the_o geometrical_a âquarâ_n geometry_n the_o scale_n of_o the_o astrolabe_n the_o quadrant_a the_o staff_n and_o such_o other_o the_o use_n of_o which_o instrument_n beside_o all_o other_o mechanical_a instrument_n of_o raise_v up_o of_o move_v and_o draw_v huge_a thing_n incredible_a to_o the_o ignorant_a and_o infinite_a other_o begin_v which_o likewise_o have_v their_o ground_n out_o of_o this_o book_n be_v of_o wonderful_a and_o unspeakable_a profit_n beside_o the_o inestimable_a pleasure_n which_o be_v in_o they_o definition_n 1._o like_o rectiline_a figure_n be_v such_o definition_n who_o angle_n be_v equal_a the_o one_o to_o the_o other_o and_o who_o side_n about_o the_o equal_a angle_n be_v proportional_a as_o if_o you_o take_v any_o two_o rectiline_a figure_n as_o for_o example_n two_o triangle_n abc_n and_o def_n ãâ¦ã_o of_o the_o one_o triangle_n be_v equal_a to_o the_o angle_n of_o the_o other_o namely_o if_o the_o angle_n a_o be_v equal_a to_o the_o angle_n d_o and_o the_o angle_n b_o equal_v to_o the_o angle_n e_o &_o also_o the_o angle_n c_o equal_v to_o the_o angle_n f._n and_o moreover_o iâ_z the_o side_n which_o contain_v the_o equal_a angle_n be_v proportional_a as_o if_o the_o side_n ab_fw-la have_v that_o proportion_n to_o the_o side_n bc_o whâch_v the_o side_n de_fw-fr have_v to_o the_o side_n of_o and_o also_o if_o the_o side_n bc_o be_v unto_o the_o side_n ca_n as_o âhe_n side_n of_o be_v to_o the_o side_n fd_o and_o moreover_o if_o the_o side_n ca_n be_v to_o the_o side_n ab_fw-la as_o the_o side_n fd_o be_v to_o the_o side_n de_fw-fr then_o be_v these_o two_o triangle_n say_v to_o be_v like_o and_o so_o judge_v you_o of_o any_o other_o kind_n of_o figure_n as_o if_o in_o the_o parallelogram_n abcd_o and_o efgh_a the_o angle_n a_o be_v equal_a to_o the_o angle_n e_o and_o the_o angle_n b_o equal_v to_o the_o angle_n fletcher_n and_o the_o angle_n c_o equal_v to_o the_o angle_n g_o and_o the_o angle_n d_o equal_v to_o the_o angle_n h._n and_o farthermore_o if_o the_o side_n ac_fw-la have_v that_o proportion_n to_o the_o side_n cd_o which_o the_o side_n eglantine_n have_v to_o the_o side_n gh_o and_o if_o also_o the_o side_n cd_o be_v to_o the_o side_n db_o as_o the_o side_n gh_o be_v to_o the_o side_n hf_o and_o moreover_o if_o the_o side_n db_o be_v to_o the_o side_n basilius_n as_o the_o side_n hf_o be_v to_o the_o side_n fe_o and_o final_o if_o the_o side_n basilius_n be_v to_o the_o side_n ac_fw-la as_o the_o side_n fe_o be_v to_o the_o side_n eglantine_n then_o be_v these_o parallelogram_n like_a definition_n 2._o reciprocal_a figure_n be_v those_o when_o the_o termeâ_n of_o proportion_n be_v both_o antecedentes_fw-la and_o consequentes_fw-la in_o either_o figure_n as_o if_o you_o have_v two_o parallelogram_n abcd_o and_o efgh_o if_o the_o side_n ab_fw-la to_o the_o side_n of_o a_o antecedent_n of_o the_o first_o figure_n to_o a_o consequent_a of_o the_o second_o figure_n have_v mutual_o the_o same_o proportion_n which_o the_o side_n eglantine_n have_v to_o the_o side_n ac_fw-la a_o antecedent_n of_o the_o second_o figure_n to_o a_o consequent_a of_o the_o first_o figure_n then_o be_v these_o two_o figure_n reciprocal_a they_o be_v call_v of_o some_o figure_n of_o mutual_a side_n and_o that_o undoubted_o not_o amiss_o nor_o unapt_o figure_n and_o to_o make_v this_o definition_n more_o plain_a campane_n and_o pestitarius_n and_o othersâ_n thus_o put_v it_o reciprocal_a figure_n be_v when_o the_o side_n of_o other_o ãâã_d mutual_o proportional_a as_o in_o the_o example_n and_o declaration_n before_o give_v among_o the_o barbarous_a they_o be_v call_v mutekesia_n reserve_v still_o the_o arabike_a word_n definition_n 3._o a_o right_a line_n be_v say_v to_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n when_o the_o whole_a be_v to_o the_o great_a part_n as_o the_o great_a part_n be_v to_o the_o less_o as_o if_o the_o line_n ab_fw-la be_v so_o divide_v in_o the_o point_n c_o that_o the_o whole_a line_n ab_fw-la have_v the_o same_o proportion_n to_o the_o great_a part_n thereof_o namely_o to_o ac_fw-la which_o the_o same_o great_a part_n ac_fw-la have_v to_o the_o less_o part_n thereof_o namely_o to_o cb_o then_o be_v the_o line_n ab_fw-la divide_v by_o a_o extreme_a and_o mean_a proportion_n common_o it_o be_v call_v a_o line_n divide_v by_o proportion_n haâing_v a_o mean_a and_o two_o extreme_n how_o to_o divide_v a_o line_n in_o such_o sort_n be_v teach_v in_o the_o 11._o proposition_n of_o the_o second_o book_n but_o not_o under_o this_o form_n of_o proportion_n 4._o the_o alitude_n of_o a_o figure_n be_v a_o perpendicular_a line_n draw_v from_o the_o top_n to_o the_o base_a definition_n as_o the_o altitude_n or_o height_n of_o the_o triangle_n abc_n be_v the_o line_n ad_fw-la be_v draw_v perpendicular_o from_o the_o point_n a_o be_v the_o top_n or_o high_a part_n of_o the_o triangle_n to_o the_o base_a thereof_o bc._n so_o likewise_o in_o other_o figure_n as_o you_o see_v in_o the_o example_n here_o set_v that_o which_o here_o âee_n call_v the_o altitude_n or_o height_n of_o a_o figure_n in_o the_o first_o book_n in_o the_o 35._o proposition_n and_o certain_a other_o follow_v he_o teach_v to_o be_v contain_v within_o two_o equidistant_a line_n so_o that_o figure_v to_o have_v one_o altitude_n and_o to_o be_v contain_v within_o two_o equidistant_a line_n be_v all_o one_o so_o in_o all_o these_o example_n if_o from_o the_o high_a point_n of_o the_o figure_n you_o draw_v a_o equidistant_a line_n to_o the_o base_a thereof_o and_o then_o from_o that_o point_n draw_v a_o perpendicular_a to_o the_o same_o base_a that_o perpendicular_a be_v the_o altitude_n of_o the_o figure_n 5._o a_o proportion_n be_v say_v to_o be_v make_v of_o two_o proportion_n or_o more_o when_o the_o quantity_n of_o the_o proportion_n multiply_v the_o one_o into_o the_o other_o produce_v a_o other_o quantity_n definition_n an_o other_o example_n where_o the_o great_a inequality_n and_o the_o less_o inequality_n be_v mix_v together_o 6._o 4._o 2._o 3._o the_o denomination_n of_o the_o proportion_n of_o 6._o to_o 4_o be_v 1_o â_o â_o example_n of_o 4._o to_o 2_o be_v â_o â_o and_o of_o 2._o to_o 3_o be_v â_o â_o now_o if_o you_o multiply_v as_o you_o ought_v all_o these_o denomination_n together_o you_o shall_v produce_v 12._o to_o 6_o namely_o dupla_fw-la proportion_n forasmuch_o as_o so_o much_o have_v hitherto_o be_v speak_v of_o addition_n of_o proportion_n it_o shall_v not_o be_v unnecessary_a somewhat_o also_o to_o say_v of_o substraction_n of_o they_o proportion_n where_o it_o be_v to_o be_v note_v that_o as_o addition_n of_o they_o be_v make_v by_o multiplication_n of_o their_o denomination_n the_o one_o into_o the_o other_o so_o be_v the_o substraction_n of_o the_o one_o from_o the_o other_o do_v by_o division_n of_o the_o denomination_n of_o the_o one_o by_o the_o denomination_n of_o the_o other_o as_o if_o you_o will_v from_o sextupla_fw-la proportion_n subtrahe_fw-la dupla_fw-la proportion_n take_v
and_o the_o self_n same_o top_n namely_o e_o and_o therefore_o be_v under_o one_o and_o the_o same_o altitude_n and_o by_o the_o same_o reason_n as_o the_o triangle_n cde_o be_v to_o the_o triangle_n ade_n so_o be_v the_o line_n ce_fw-fr to_o the_o line_n ea_fw-la wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o line_n bd_o be_v to_o the_o line_n dam_fw-ge so_o be_v the_o line_n ce_fw-fr to_o the_o line_n ea_fw-la but_o now_o suppose_v that_o in_o the_o triangle_n abc_n the_o side_n ab_fw-la &_o ac_fw-la be_v cut_v proportional_o so_o that_o as_o bd_o be_v to_o da_z so_o let_v ce_fw-fr be_v to_o ea_fw-la &_o draw_v a_o line_n from_o d_z to_o e._n then_o second_o i_o say_v that_o the_o line_n de_fw-fr be_v a_o parallel_n to_o the_o line_n bc._n part_n for_o the_o same_o order_n of_o construction_n be_v keep_v for_o that_o as_z bd_o be_v to_o da_z so_o be_v ce_fw-fr to_o ea_fw-la but_o as_o bd_o be_v to_o da_z so_o be_v the_o triangle_n bde_v to_o the_o triangle_n ade_n by_o the_o 1._o of_o the_o six_o &_o as_o ce_fw-fr be_v to_o ea_fw-la so_o by_o the_o same_o be_v the_o triangle_n cde_o to_o the_o triangle_n ade_n therefore_o by_o the_o 11._o of_o the_o five_o as_o the_o triangle_n bde_v be_v to_o the_o triangle_n ade_n so_o be_v the_o triangle_n cde_o to_o the_o triangle_n ade_n wherefore_o either_o of_o these_o triangle_n bde_v and_o cde_o have_v to_o the_o triangle_n ade_n one_o and_o the_o same_o proportion_n wherefore_o by_o the_o 9_o of_o the_o five_o the_o triangle_n bde_v be_v equal_a unto_o the_o triangle_n cde_o and_o they_o be_v upon_o one_o and_o the_o self_n base_a namely_o de._n but_o triangle_n equal_a and_o set_v upon_o one_o base_a be_v also_o contain_v within_o the_o same_o parallel_a line_n by_o the_o 39_o of_o the_o first_o wherefore_o the_o line_n de_fw-fr be_v unto_o the_o line_n bc_o a_o parallel_n if_o therefore_o to_o any_o one_o of_o the_o side_n of_o a_o triangle_n be_v draw_v a_o parallel_a line_n it_o cut_v the_o other_o side_n of_o the_o same_o triangle_n proportional_o and_o if_o the_o side_n of_o a_o triangle_n be_v cut_v proportional_o a_o right_a line_n draw_v from_o section_n to_o section_n be_v parallel_n to_o the_o other_o side_n of_o the_o triangle_n which_o thing_n be_v require_v to_o be_v demonstrate_v ¶_o here_o also_o flussates_n add_v a_o corollary_n if_o a_o line_n parallel_n to_o one_o of_o the_o side_n of_o a_o triangle_n do_v cut_v the_o triangle_n it_o shall_v cut_v of_o from_o the_o whole_a triangle_n a_o triangle_n like_a to_o the_o whole_a triangle_n flussates_n for_o as_o it_o have_v be_v prove_v it_o divide_v the_o side_n proportional_o so_o that_o as_o ec_o be_v to_o ea_fw-la so_o be_v bd_o to_o dam_fw-ge wherefore_o by_o the_o 18._o of_o the_o five_o as_z ac_fw-la be_v to_o ae_n so_o be_v ab_fw-la to_o ad._n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o ac_fw-la be_v to_o ab_fw-la so_o be_v ae_n to_o ad_fw-la wherefore_o in_o the_o two_o triangle_n ead_n and_o cab_n the_o side_n about_o the_o common_a angle_n a_o be_v proportional_a the_o say_v triangle_n also_o be_v equiangle_n for_o forasmuch_o as_o the_o right_a line_n aec_o and_o adb_o do_v fall_v upon_o the_o parallel_a line_n ed_z and_o cb_o therefore_o by_o the_o 29._o of_o the_o firsâ_o they_o make_v the_o angle_n aed_n and_o ade_n in_o the_o triangle_n ade_n equal_a to_o the_o angle_n acb_o and_o abc_n in_o the_o triangle_n acb_o wherefore_o by_o the_o first_o definition_n of_o this_o book_n the_o whole_a triangle_n abc_n be_v like_a unto_o the_o triangle_n cut_v of_o ade_n the_o 3._o theorem_a the_o 3._o proposition_n if_o a_o angle_n of_o a_o triangle_n be_v divide_v into_o two_o equal_a part_n and_o if_o the_o right_a line_n which_o divide_v the_o angle_n divide_v also_o the_o base_a the_o segment_n of_o the_o base_a shall_v be_v in_o the_o same_o proportion_n the_o one_o to_o the_o other_o that_o the_o other_o side_n of_o the_o triangle_n be_v and_o if_o the_o segmente_n of_o the_o base_a be_v in_o the_o same_o proportion_n that_o the_o other_o side_n of_o the_o say_a triangle_n be_v a_o right_n draw_v from_o the_o top_n of_o the_o triangle_n unto_o the_o section_n shall_v divide_v the_o angle_n of_o the_o triangle_n into_o two_o equal_a part_n svppose_v that_o there_o be_v a_o triangle_n abc_n and_o by_o the_o 9_o of_o the_o first_o let_v the_o angle_n bac_n be_v divide_v into_o two_o equal_a part_n by_o the_o right_a line_n ad_fw-la which_o let_v cut_v also_o the_o base_a bc_o in_o the_o point_n d._n then_o i_o say_v that_o as_o the_o segment_n bd_o be_v to_o the_o segment_n dc_o so_o be_v the_o side_n basilius_n to_o the_o side_n ac_fw-la construction_n for_o by_o the_o point_n c_o by_o the_o 31._o of_o the_o first_o draw_v unto_o the_o line_n da_z a_o parallel_a line_n ce_fw-fr and_o extend_v the_o line_n basilius_n till_o it_o concur_v with_o the_o line_n ce_fw-fr in_o the_o point_n e_o and_o do_v make_v the_o triangle_n bec_n part_n but_o the_o line_n basilius_n shall_v concur_v with_o the_o line_n ce_fw-fr by_o the_o 5._o petition_n for_o that_o the_o angle_n ebc_n and_o be_v be_v less_o than_o two_o right_a angle_n for_o the_o angle_n ecb_n be_v equal_a to_o the_o outward_a and_o opposite_a angle_n adb_o by_o the_o 29._o of_o the_o first_o and_o the_o two_o angle_n adb_o and_o dba_n of_o the_o triangle_n bad_a be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o first_o now_o forasmuch_o as_o upon_o the_o parallel_n ad_fw-la and_o ec_o fall_v the_o right_a line_n ac_fw-la therefore_o by_o the_o 29._o of_o the_o first_o the_o angle_n ace_n be_v equal_a unto_o the_o angle_n god_n but_o unto_o the_o angle_n god_n be_v the_o angle_n bid_v suppose_v to_o be_v equal_a wherefore_o the_o angle_n bad_a be_v also_o equal_a unto_o the_o angle_n ace_n again_o because_o upon_o the_o parallel_n ad_fw-la and_o ec_o fall_v the_o right_a line_n bae_o the_o outward_a angle_n bad_a by_o the_o 28._o of_o the_o first_o be_v equal_a unto_o the_o inward_a angle_n aec_o but_o before_o it_o be_v provell_v that_o the_o angle_n ace_n be_v equal_a unto_o the_o angle_v bad_a wherefore_o the_o angle_n ace_n be_v equal_a unto_o the_o angle_v aec_o wherefore_o by_o the_o 6._o of_o the_o first_o the_o side_z ae_n be_v equal_a unto_o the_o side_n ac_fw-la and_o because_o to_o one_o of_o the_o side_n of_o the_o triangle_n be_v namely_o to_o ec_o be_v draw_v a_o parallel_a line_n ad_fw-la therefore_o by_o the_o 2._o of_o the_o six_o as_o bd_o be_v to_o dc_o so_o be_v basilius_n to_o ae_n but_o ae_n be_v equal_a unto_o ac_fw-la therefore_o as_o bd_o be_v to_o dc_o so_o be_v basilius_n to_o ac_fw-la but_o now_o suppose_v that_o as_o the_o segment_n bd_o be_v to_o the_o segment_n dc_o first_o so_o be_v the_o side_n basilius_n to_o the_o side_n ac_fw-la &_o draw_v a_o line_n from_o a_o to_o d._n then_o i_o say_v that_o the_o angle_n bac_n be_v by_o the_o right_a line_n ad_fw-la divide_v into_o two_o equal_a part_n for_o the_o same_o order_n of_o construction_n remain_v for_o that_o as_o bd_o be_v to_o dc_o so_o be_v basilius_n to_o ac_fw-la but_o as_o bd_o be_v to_o dc_o so_o be_v basilius_n to_o ae_n by_o the_o 2._o of_o the_o six_o for_o unto_o one_o of_o the_o side_n of_o the_o triangle_n be_v namely_o unto_o the_o side_n ec_o be_v draw_v a_o parallel_a line_n ad._n wherefore_o also_o as_o basilius_n be_v to_o ac_fw-la so_o be_v basilius_n to_o ae_n by_o the_o 11._o of_o the_o five_o wherefore_o by_o the_o 9_o of_o the_o five_o ac_fw-la be_v equal_a unto_o ae_n wherefore_o also_o by_o the_o 5._o of_o the_o first_o the_o angle_n aec_o be_v equal_a unto_o the_o angle_n ace_n but_o the_o angle_n aec_o by_o the_o 29._o of_o the_o first_o be_v equal_a unto_o the_o outward_a angle_n bad_a and_o the_o angle_n ace_n be_v equal_a unto_o the_o angle_n god_n which_o be_v alternate_a unto_o he_o wherefore_o the_o angle_n bad_a be_v equal_a unto_o the_o angle_n god_n wherefore_o the_o angle_n bac_n be_v by_o the_o right_a line_n ad_fw-la divide_v into_o two_o equal_a part_n wherefore_o if_o a_o angle_n of_o a_o triangle_n be_v divide_v into_o two_o equal_a part_n and_o if_o the_o right_a line_n which_o divide_v the_o angle_n cut_v also_o the_o base_a the_o segment_n of_o the_o base_a shall_v be_v in_o the_o same_o proportion_n the_o one_o to_o the_o other_o that_o the_o other_o side_n of_o the_o say_a triangle_n be_v and_o if_o the_o segment_n of_o the_o base_a be_v in_o the_o same_o proportion_n that_o the_o other_o side_n of_o the_o say_a triangle_n be_v a_o right_a line_n draw_v from_o the_o top_n of_o the_o
same_o be_v treble_a it_o be_v manifest_a by_o the_o 12._o of_o the_o five_o that_o all_o the_o prism_n be_v to_o all_o the_o pyramid_n treble_a wherefore_o parallelipipedon_n be_v treble_a to_o pyramid_n set_v upon_o the_o self_n same_o base_a with_o they_o and_o under_o the_o same_o altitude_n they_o for_o that_o they_o contain_v two_o prism_n three_o corollary_n if_o two_o prism_n be_v under_o one_o and_o the_o self_n same_o altitude_n have_v to_o their_o base_n either_o both_o triangle_n or_o both_o parallelogram_n the_o prism_n be_v the_o one_o to_o the_o other_o as_o their_o base_n be_v for_o forasmuch_o as_o those_o prism_n be_v equemultiqlice_n unto_o the_o pyramid_n upon_o the_o self_n same_o base_n and_o under_o the_o same_o altitude_n which_o pyramid_n be_v in_o proportion_n as_o their_o base_n it_o be_v manifest_a by_o the_o 15._o of_o the_o five_o that_o the_o prism_n be_v in_o the_o proportion_n of_o the_o base_n for_o by_o the_o former_a corollary_n the_o prism_n be_v treble_a to_o the_o pyramid_n sât_v upon_o the_o triangular_a base_n four_o corollary_n prism_n be_v in_o sesquealtera_fw-la proportion_n to_o pyramid_n which_o have_v the_o self_n same_o quadrangled_a base_a that_o the_o prism_n have_v and_o be_v under_o the_o self_n same_o altitude_n for_o that_o pyramid_n contain_v two_o pyramid_n set_v upon_o a_o triangular_a base_a of_o the_o same_o prism_n for_o it_o be_v prove_v that_o that_o prism_n be_v treble_a to_o the_o pyramid_n which_o be_v set_v upon_o the_o half_a of_o his_o quadrangled_a base_a unto_o which_o the_o other_o which_o be_v set_v upon_o the_o whole_a base_a be_v double_a by_o the_o six_o of_o this_o book_n five_v corollary_n wherefore_o we_o may_v in_o like_a sort_n conclude_v that_o solid_n mention_v in_o the_o second_o corollary_n which_o solid_n campane_n call_v side_v column_n be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n which_o be_v poligonon_o figure_n for_o they_o be_v in_o the_o proportion_n of_o the_o pyramid_n or_o prism_n set_v upon_o the_o self_n same_o base_n and_o under_o the_o self_n same_o altitude_n that_o be_v they_o be_v in_o the_o proportion_n of_o the_o base_n of_o the_o say_a pyramid_n or_o prism_n for_o those_o solid_n may_v be_v divide_v into_o prism_n have_v the_o self_n same_o altitude_n when_o as_o their_o opposite_a base_n may_v be_v divide_v into_o triangle_n by_o the_o 20_o of_o the_o six_o upon_o which_o triangle_n prism_n be_v set_v be_v in_o proportion_n as_o their_o base_n by_o this_o 7._o proposition_n it_o plain_o appear_v that_o âuâlide_a as_o it_o be_v before_o note_v in_o the_o diffinitionââ_n under_o the_o definition_n of_o a_o prism_n comprehend_v also_o those_o kind_n of_o solid_n which_o campane_n call_v side_v column_n for_o in_o that_o he_o say_v every_o prism_n have_v a_o triangle_n to_o his_o base_a may_v be_v devidedâ_n etc_n etc_n he_o need_v not_o take_v a_o prism_n in_o that_o sense_n which_o campane_n and_o most_o man_n take_v it_o to_o have_v add_v that_o particle_n have_v to_o his_o base_a a_o triangle_n for_o by_o their_o sense_n there_o be_v no_o prism_n but_o it_o may_v have_v to_o his_o base_a a_o triangleâ_n and_o so_o it_o may_v seem_v that_o euclid_n ought_v without_o exception_n have_v say_v that_o every_o prism_n whatsoever_o may_v be_v divide_v into_o three_o pyramid_n equal_a the_o one_o to_o the_o other_o have_v also_o triangle_n to_o âheir_a base_n for_o so_o do_v campane_n and_o flussas_n put_v the_o proposition_n leave_v out_o the_o former_a particle_n have_v to_o his_o base_a a_o triangle_n which_o yet_o be_v red_a in_o the_o greek_a copy_n &_o not_o leât_v out_o by_o any_o other_o interpreter_n know_v abroad_o except_o by_o campane_n and_o flussas_n yea_o and_o the_o corollary_n follow_v of_o this_o proposition_n add_v by_o theon_n or_o euclid_n and_o amend_v by_o m._n dee_n seem_v to_o confirm_v this_o sense_n of_o this_o âs_a ãâã_d make_v manifest_a that_o every_o pyramid_n be_v the_o three_o part_n of_o the_o prism_n have_v the_o same_o base_a with_o it_o and_o equal_a altitude_n for_o and_o if_o the_o base_a of_o the_o prism_n have_v any_o other_o right_o line_v figure_n than_o a_o triangle_n and_o also_o the_o superficies_n opposite_a to_o the_o base_a the_o same_o figure_n that_o prism_n may_v be_v divide_v into_o prism_n have_v triangle_v base_n and_o the_o superficiece_n to_o those_o base_n opposite_a also_o triangle_v a_o ââike_a and_o equal_o for_o there_o as_o we_o see_v be_v put_v these_o word_n âor_a and_o if_o the_o base_a of_o the_o prism_n be_v any_o other_o right_o line_v figureâ_n etc_n etc_n whereof_o a_o man_n may_v well_o infer_v that_o the_o base_a may_v be_v any_o other_o rectiline_a figure_n whatsoever_o &_o not_o only_o a_o triangle_n or_o a_o parallelogram_n and_o it_o be_v true_a also_o in_o that_o sense_n as_o it_o be_v plain_a to_o see_v by_o the_o second_o corollary_n add_v out_o of_o flussas_n which_o corollary_n as_o also_o the_o first_o of_o his_o corollary_n be_v in_o a_o manner_n all_o one_o with_o the_o corollary_n add_v by_o theon_n or_o euclid_n far_o theon_n in_o the_o demonstration_n of_o the_o 10._o proposition_n of_o this_o book_n as_o we_o shall_v afterwards_o see_v most_o plain_o call_v not_o only_o side_v column_n prism_n but_o also_o parallelipipedon_n and_o although_o the_o 40._o proposition_n of_o the_o eleven_o book_n may_v seem_v hereunto_o to_o be_v a_o lât_n for_o that_o it_o can_v be_v understand_v of_o those_o prism_n only_o which_o have_v triangle_n to_o their_o like_a equal_a opposite_a and_o parallel_v side_n or_o but_o of_o some_o side_v column_n and_o not_o of_o all_o yet_o may_v that_o let_v be_v thus_o remove_v away_o to_o say_v that_o euclid_n in_o that_o proposition_n use_v genus_fw-la pro_fw-la specie_fw-la that_o be_v the_o general_a word_n for_o some_o special_a kind_n thereof_o which_o thing_n also_o be_v not_o rare_a not_o only_o with_o he_o but_o also_o with_o other_o learned_a philosopher_n thus_o much_o i_o think_v good_a by_o the_o way_n to_o note_v in_o far_a defence_n of_o euclid_n definition_n of_o a_o prism_n the_o 8._o theorem_a the_o 8._o proposition_n pyramid_n be_v like_o &_o have_a triangle_n to_o their_o base_n be_v in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o their_o side_n of_o like_a proportion_n be_v svppose_v that_o these_o pyramid_n who_o base_n be_v the_o triangle_n gbc_n and_o hef_n and_o top_n the_o point_n a_o and_o d_o be_v like_o and_o in_o like_a sort_n describe_v and_o let_v ab_fw-la and_o de_fw-fr be_v side_n of_o like_a proportion_n then_o i_o say_v that_o the_o pyramid_n abcg_o be_v to_o the_o pyramid_n defh_o in_o treble_a proportion_n of_o that_o in_o which_o the_o side_n ab_fw-la be_v to_o the_o side_n de._n make_v perfect_a the_o parallelipipedon_n namely_o the_o solid_n bckl_n &_o efxo_n and_o forasmuch_o as_o the_o pyramid_n abcg_o be_v like_a to_o the_o pyramid_n defh_o construction_n therefore_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n def_n demonstration_n &_o the_o angle_n gbc_n to_o the_o angle_n hef_fw-fr and_o moreover_o the_o angle_n abg_o to_o the_o angle_n deh_fw-it and_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n de_fw-fr so_o be_v the_o line_n bc_o to_o the_o line_n of_o and_o the_o line_n bg_o to_o the_o line_n eh_o and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n de_fw-fr so_o be_v the_o line_n bc_o to_o the_o line_n of_o and_o the_o side_n about_o the_o equal_a angle_n be_v proportional_a therefore_o the_o parallelogram_n bm_n be_v like_a to_o the_o parallelogram_n ep_n and_o by_o the_o same_o reason_n the_o parallelogram_n bn_o be_v like_a to_o the_o parallelogram_n er_fw-mi and_o the_o parellelogramme_n bk_o be_v like_a unto_o the_o parallelogram_n exit_fw-la wherefore_o the_o three_o parallelogram_n bm_n kb_o and_o bn_o be_v like_a to_o the_o three_o parallelogram_n ep_n exit_fw-la and_o er._n but_o the_o three_o parallelogram_n bm_n kb_o and_o bn_o be_v equal_a and_o like_a to_o the_o three_o opposite_a parallelogram_n and_o the_o three_o parallelogram_n ep_n exit_fw-la and_o ere_o be_v equal_a and_o like_a to_o the_o three_o opposite_a parallelogram_n wherefore_o the_o parallelipipedon_n bckl_n and_o efxo_n be_v comprehend_v under_o plain_a superficiece_n like_a and_o equal_a in_o multitude_n wherefore_o the_o solid_a bckl_n be_v like_a to_o the_o solid_a efxo_n but_o like_o parallelipipedon_n be_v by_o the_o 33._o of_o the_o eleven_o in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o side_n of_o like_a proportion_n be_v to_o side_n of_o like_a proportion_n wherefore_o the_o solid_a bckl_n be_v to_o the_o solid_a efxo_n in_o treble_a
proportion_n of_o that_o in_o which_o the_o side_n of_o like_a proportion_n ab_fw-la be_v to_o the_o side_n of_o like_a proportion_n de._n but_o as_o the_o solid_a bckl_n be_v to_o the_o solid_a efxo_n so_o be_v the_o pyramid_n abcg_o to_o the_o pyramid_n defh_o by_o the_o 15._o of_o the_o five_o for_o that_o the_o pyramid_n be_v the_o six_o part_n of_o this_o solid_a for_o the_o prism_n be_v the_o half_a of_o the_o parallelipipedon_n be_v treble_a to_o the_o pyramid_n wherefore_o the_o pyramid_n abcg_o be_v to_o the_o pyramid_n defh_o in_o treble_a proportion_n of_o that_o in_o which_o the_o side_n ab_fw-la be_v to_o the_o side_n de._n which_o be_v require_v to_o be_v prove_v corollary_n hereby_o it_o be_v manifest_v that_o like_a pyramid_n have_v to_o their_o base_n poligonon_o figure_n be_v in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o side_n of_o like_a proportion_n be_v to_o side_n of_o like_a proportion_n for_o if_o they_o be_v divide_v into_o pyramid_n have_v triangle_n to_o their_o base_n for_o like_o poligonon_fw-fr figure_n be_v divide_v into_o like_a triangle_n and_o equal_a in_o multitude_n and_o the_o side_n be_v of_o like_a proportion_n as_o one_o of_o the_o pyramid_n of_o the_o one_o have_v a_o triangle_n to_o his_o base_a be_v to_o one_o of_o the_o pyramid_n of_o the_o other_o have_v also_o a_o triangle_n to_o his_o base_a so_o also_o be_v all_o the_o pyramid_n of_o the_o one_o pyramid_n have_v triangle_n to_o their_o base_n to_o all_o the_o pyramid_n of_o the_o other_o pyramid_n have_v also_o triangle_n to_o their_o base_n that_o be_v the_o pyramid_n have_v to_o his_o base_a a_o poligononâigure_n to_o the_o pyramid_n have_v also_o to_o his_o base_a a_o poligononâigure_n but_o a_o pyramid_n have_v a_o triangle_n to_o his_o base_a be_v to_o a_o pyramid_n have_v also_o a_o triangle_n to_o his_o base_a &_o be_v like_a unto_o it_o in_o treble_a proportion_n of_o that_o in_o which_o side_n of_o like_a proportion_n be_v to_o side_n of_o like_a proportion_n wherefore_o a_o pyramid_n have_v to_o his_o base_a a_o poligonon_n figure_n be_v to_o a_o pyramid_n have_v also_o a_o polygonon_n figure_n to_o his_o base_a the_o say_a pyramid_n be_v like_o the_o one_o to_o the_o other_o in_o treble_a proportion_n of_o that_o in_o which_o side_n of_o like_a proportion_n be_v to_o side_n of_o like_a proportion_n likewise_o prism_n and_o side_v column_n be_v set_v upon_o the_o base_n of_o those_o pyramid_n flussas_n and_o under_o the_o same_o altitude_n forasmuch_o as_o they_o be_v equemultiplices_fw-la unto_o the_o pyramid_n namely_o triple_n by_o the_o corollary_n of_o the_o 7._o of_o this_o book_n shall_v have_v the_o âormer_a porportion_n that_o the_o pyramid_n have_v by_o the_o 15_o of_o the_o five_o and_o therefore_o they_o shall_v be_v in_o treble_a proportion_n of_o that_o in_o which_o the_o side_n of_o like_a proportion_n be_v ¶_o the_o 9_o theorem_a the_o 9_o proposition_n in_o equal_a pyramid_n have_v triangle_n to_o their_o base_n the_o base_n be_v reciprocal_a to_o their_o altitude_n and_o pyramid_n have_v triangle_n to_o their_o base_n who_o base_n be_v reciprocal_a to_o their_o altitude_n be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o bcga_n and_o efhd_n be_v equal_a pyramid_n have_v to_o their_o base_n the_o triangle_n bcg_n and_o efh_n and_o the_o top_n the_o point_n a_o and_o d._n then_o i_o say_v that_o the_o base_n of_o the_o two_o pyramid_n bcga_n and_o efhd_n be_v reciprocal_a to_o their_o altitude_n that_o be_v as_o the_o base_a bcg_n be_v to_o the_o base_a efh_n so_o be_v the_o altitude_n of_o the_o pyramid_n efhd_v to_o the_o altitude_n of_o the_o pyramid_n bcga_n make_v perfect_a the_o parallelipipedon_n namely_o bgml_n and_o ehpo_n and_o forasmuch_o as_o the_o pyramid_n bcga_n be_v equal_a to_o the_o pyramid_n efhd_v part_n &_o the_o solid_a bgml_n be_v sextuple_a to_o the_o pyramid_n bcga_n for_o the_o parallelipipedon_n be_v duple_n to_o the_o prism_n set_v upon_o the_o base_a of_o the_o pyramid_n &_o the_o prism_n be_v triple_a to_o the_o pyramid_n and_o likewise_o the_o solid_a ehpo_n be_v sextuple_a to_o the_o pyramid_n efhd_v wherefore_o the_o solid_a bgml_n be_v equal_a to_o the_o solid_a ehpo_n but_o in_o equal_a parallelipipedon_n the_o base_n be_v by_o the_o 34._o of_o the_o eleven_o reciprocal_a to_o their_o altitude_n wherefore_o as_o the_o base_a bn_o be_v to_o the_o base_a eq_n so_o be_v the_o altitude_n of_o the_o solid_a ehpâ_n to_o the_o altitude_n of_o the_o solid_a bgml_n but_o as_o the_o base_a bn_o be_v to_o the_o base_a eq_n so_o be_v the_o triangle_n gbc_n to_o the_o triangle_n hef_n by_o the_o 15._o of_o the_o âifth_o for_o the_o triangle_n gbc_n &_o hef_n be_v the_o half_n of_o the_o parallelogram_n bn_o and_o eq_n â_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o triangle_n gbc_n be_v to_o the_o triangle_n hef_fw-fr so_o be_v the_o altitude_n of_o the_o solid_a ehpo_n to_o the_o altitude_n of_o the_o solid_a bgml_n but_o the_o altitude_n of_o the_o solid_a ehpo_n be_v one_o and_o the_o same_o with_o the_o altitude_n of_o the_o pyramid_n efhd_v and_o the_o altitude_n of_o the_o solid_a bgml_n be_v one_o and_o the_o same_o with_o the_o altitude_n of_o the_o pyramid_n bcga_n wherefore_o as_o the_o base_a gbc_n be_v to_o the_o base_a hef_fw-fr so_o be_v the_o altitude_n of_o the_o pyramid_n efhd_v to_o the_o altitude_n of_o the_o pyramid_n bcga_n wherefore_o the_o base_n of_o the_o two_o pyramid_n bcga_n and_o efhd_n be_v reciprocal_a to_o their_o altitude_n but_o now_o suppose_v that_o the_o base_n of_o the_o pyramid_n bcga_n and_o efhd_n be_v reciprocal_a to_o their_o altitude_n first_o that_o be_v as_o the_o base_a gbc_n be_v to_o the_o base_a hef_fw-fr so_o let_v the_o altitude_n of_o the_o pyramid_n efhd_v be_v to_o the_o altitude_n of_o the_o pyramid_n bcga_n then_o i_o say_v that_o the_o pyramid_n bcga_n be_v equal_a to_o the_o pyramid_n efhd_v for_o the_o self_n same_o order_n of_o construction_n remain_v for_o that_o as_o the_o base_a gbc_n be_v to_o the_o base_a âef_n so_o be_v the_o altitude_n of_o the_o pyramid_n efhd_v to_o the_o altitude_n of_o the_o pyramid_n bcga_n but_o as_o the_o base_a gbc_n be_v to_o the_o base_a hef_fw-fr so_o be_v the_o parallelogram_n gc_o to_o the_o parallelogram_n hf._n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o parallelogram_n gc_o be_v to_o the_o parallegoramme_n hf_o so_o be_v the_o altitude_n of_o the_o pyramid_n efhd_v to_o the_o altitude_n of_o the_o pyramid_n bcga_n but_o the_o altitude_n of_o the_o pyramid_n efnd_v and_o of_o the_o solid_a ehpo_n be_v one_o and_o the_o self_n same_o and_o the_o altitude_n of_o the_o pyramid_n bcga_n and_o of_o the_o solid_a bgml_n be_v also_o one_o and_o the_o same_o wherefore_o as_o the_o base_a gc_o be_v to_o the_o base_a hf_o so_o be_v the_o altitude_n of_o the_o solid_a ehpo_n to_o the_o altitude_n of_o the_o solid_a bgml_n but_o parallelipipedon_n who_o base_n be_v reciprocal_a to_o their_o altitude_n be_v by_o the_o 34._o of_o the_o eleven_o equal_v the_o one_o to_o the_o other_o wherefore_o the_o parallelipipedon_n bgml_n be_v equal_a to_o the_o parallelipipedon_n ehpo_n but_o the_o pyramid_n bcga_n be_v the_o six_o part_n of_o the_o solid_a bgml_n and_o likewise_o the_o pyramid_n efhd_v be_v the_o six_o part_n of_o the_o solid_a ehpo_n wherefore_o the_o pyramid_n bcga_n be_v equal_a to_o the_o pyramid_n efhd_v wherefore_o in_o equal_a pyramid_n have_v triangle_n to_o their_o base_n the_o base_n be_v reciprocal_a to_o their_o altitude_n and_o pyramid_n have_v triangle_n to_o their_o base_n who_o base_n be_v reciprocal_a to_o their_o altitude_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o campane_n and_o flussas_n hereby_o it_o be_v manifest_v that_o equal_a pyramid_n have_v to_o their_o base_n poligonon_n figure_v have_v their_o base_n reciprocal_a with_o their_o altitude_n and_o pyramid_n who_o base_n be_v poligonon_o figure_n be_v reciprocal_a with_o their_o altitude_n be_v equal_a the_o one_o to_o the_o other_o suppose_v that_o upon_o the_o polygonon_n figure_v a_o and_o b_o be_v set_v equal_a pyramid_n then_o i_o say_v that_o their_o base_n a_o and_o b_o be_v reciprocal_a with_o their_o altitude_n describe_v by_o the_o 25._o of_o the_o six_o triangle_n equal_a to_o the_o base_n a_o and_o b._n which_o let_v be_v c_o and_o d._n upon_o which_o let_v there_o be_v set_v pyramid_n equal_a in_o altitude_n with_o the_o pyramid_n a_o and_o b._n wherefore_o the_o pyramid_n c_o and_o d_o be_v set_v upon_o base_n equal_a with_o the_o base_n of_o the_o pyramid_n a_o and_o b_o and_o have_v also_o their_o altitude_n equal_a
namely_o the_o part_n great_a than_o the_o whole_a which_o be_v impossible_a wherefore_o the_o circumference_n shall_v not_o cut_v the_o line_n cd_o now_o i_o say_v that_o it_o shall_v not_o pass_v above_o the_o line_n cd_o and_o not_o touch_v it_o in_o the_o point_n d._n for_o if_o it_o be_v possible_a let_v it_o pass_v above_o it_o and_o extend_v the_o line_n cd_o to_o the_o circumference_n and_o let_v it_o cut_v it_o in_o the_o point_n â_o and_o draw_v the_o line_n fb_o and_o favorina_n and_o it_o shall_v follow_v as_o before_o that_o the_o line_n cd_o be_v great_a than_o the_o line_n cf_o which_o be_v impossible_a wherefore_o that_o be_v manifest_v which_o be_v require_v to_o be_v prove_v ¶_o second_v assumpt_n if_o there_o be_v a_o right_a angle_n unto_o which_o a_o base_a be_v subtend_v and_o if_o upon_o the_o same_o be_v describe_v a_o semicircle_n the_o circumference_n thereof_o shall_v pass_v by_o the_o point_n of_o the_o right_a angle_n the_o converse_n of_o this_o be_v add_v after_o the_o demonstration_n of_o the_o 31._o of_o the_o three_o out_o of_o pelitarius_n and_o these_o two_o assumpte_n of_o campane_n be_v necessary_a for_o the_o better_a understanding_n of_o the_o demonstration_n of_o the_o secoâd_a part_n of_o this_o 13._o proposition_n wherein_o be_v prove_v that_o the_o pyramid_n be_v contain_v in_o the_o sphere_n give_v ¶_o certain_a corollarye_n add_v by_o flussas_n first_o corollary_n the_o diameter_n of_o the_o sphere_n be_v in_o power_n quadruple_a sesquialtera_fw-la to_o the_o line_n which_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o base_a of_o the_o pyramid_n for_o forasmuch_o as_o it_o have_v be_v prove_v that_o the_o diameter_n kl_o be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o and_o it_o be_v prove_v also_o by_o the_o 12._o of_o this_o book_n that_o the_o side_n of_o be_v in_o power_n triple_a to_o the_o line_n eh_o which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n contain_v the_o triangle_n efg_o but_o the_o proportion_n of_o the_o extreme_n namely_o of_o the_o diameter_n to_o the_o line_n eh_o consist_v of_o the_o proportion_n of_o the_o mean_n namely_o of_o the_o proportion_n of_o the_o diameter_n to_o the_o line_n of_o and_o of_o the_o proportion_n of_o the_o line_n of_o to_o the_o line_n eh_o by_o the_o 5._o definition_n of_o the_o six_o which_o proportion_n namely_o triple_a and_o sesquialter_fw-la add_v together_o make_v quadruple_a sesquialter_fw-la as_o it_o be_v easy_a to_o prove_v by_o that_o which_o be_v teach_v in_o the_o declaration_n of_o the_o 5._o definition_n of_o the_o six_o book_n wherefore_o the_o corollary_n be_v manifest_a ¶_o second_v corollary_n only_o the_o line_n which_o be_v draw_v from_o the_o angle_n of_o the_o pyramid_n to_o the_o base_a opposite_a unto_o it_o campane_n &_o pass_v by_o the_o centre_n of_o the_o sphere_n be_v perpendicular_a to_o the_o base_a and_o fall_v upon_o the_o centre_n of_o the_o circle_n which_o contain_v the_o base_a for_o if_o any_o other_o line_n than_o the_o line_n kmh_o which_o be_v draw_v by_o the_o centre_n of_o the_o sphere_n to_o the_o centre_n of_o the_o circle_n shall_v fall_v perpendicular_o upon_o the_o plain_a of_o the_o base_a then_o from_o one_o and_o the_o self_n same_o point_n shall_v be_v draw_v to_o one_o and_o the_o self_n same_o plain_a two_o perpendicular_a line_n contrary_a to_o the_o 13._o of_o the_o eleven_o which_o be_v impossible_a far_o if_o from_o the_o top_n king_n shall_v be_v draw_v to_o the_o centre_n of_o the_o base_a namely_o to_o the_o point_n h_o any_o other_o right_a line_n not_o pass_v by_o the_o centre_n m_o two_o right_a life_n shall_v include_v a_o superficies_n contrary_n to_o the_o last_o common_a sentence_n which_o be_v absurd_a wherefore_o only_o the_o line_n which_o be_v draw_v by_o the_o centre_n of_o the_o sphere_n to_o the_o centre_n of_o the_o base_a be_v perpendicular_a to_o the_o say_v base_a and_o the_o line_n which_o be_v draw_v from_o the_o angle_n perpendicular_o to_o the_o base_a shall_v pass_v by_o the_o centre_n of_o the_o sphere_n three_o corollary_n the_o perpendicular_a line_n which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o base_a of_o the_o pyramid_n book_n be_v equal_a to_o the_o six_o part_n of_o the_o diameter_n of_o the_o sphere_n for_o it_o be_v before_o prove_v that_o the_o line_n mh_o which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o centre_n of_o the_o base_a be_v equal_a to_o the_o line_n nc_n which_o line_n nc_n be_v the_o six_o part_n of_o the_o diameter_n ab_fw-la and_o therefore_o the_o line_n mh_o be_v the_o six_o part_n of_o the_o diameter_n of_o the_o sphere_n for_o the_o diameter_n ab_fw-la be_v equal_a to_o the_o diameter_n of_o the_o sphere_n as_o have_v also_o before_o be_v prove_v ¶_o the_o 2._o problem_n the_o 14._o proposition_n to_o make_v a_o octohedron_n and_o to_o comprehend_v it_o in_o the_o sphere_n give_v namely_o that_o wherein_o the_o pyramid_n be_v comprehend_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n construction_n take_v the_o diameter_n of_o the_o former_a sphere_n give_v which_o let_v be_v the_o line_n ab_fw-la and_o divide_v it_o by_o the_o 10._o of_o the_o first_o into_o two_o equal_a part_n in_o the_o point_n c._n and_o describe_v upon_o the_o line_n ab_fw-la a_o semicircle_n adb_o and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n c_o raise_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n cd_o and_o draw_v a_o right_a line_n from_o the_o point_n d_o to_o the_o point_n b._n and_o describe_v a_o square_a efgh_o have_v every_o one_o of_o his_o side_n equal_a to_o the_o line_n bd._n and_o draw_v the_o diagonal_a line_n fh_a &_o eglantine_n cut_v the_o one_o the_o other_o in_o the_o point_n k._n and_o by_o the_o 12._o of_o the_o eleven_o from_o the_o point_n king_n namely_o the_o point_n where_o the_o line_n fh_o and_o eglantine_n cut_v the_o one_o the_o other_o raise_v up_o to_o the_o plain_a superficies_n wherein_o the_o square_a efgh_o be_v a_o perpendicular_a line_n kl_o and_o extend_v the_o line_n kl_o on_o the_o other_o side_n of_o the_o plain_a superficies_n to_o the_o point_n m._n and_o let_v each_o of_o the_o line_n kl_o and_o km_o be_v put_v equal_a to_o one_o of_o these_o line_n ke_o kf_o kh_o or_o kg_n and_o draw_v these_o right_a line_n le_fw-fr lf_n lg_n lh_o menander_n mf_n mg_o and_o mh_o demonstration_n now_o forasmuch_o as_o the_o line_n ke_o be_v by_o the_o corollary_n of_o the_o 34._o of_o the_o first_o equal_a to_o the_o line_n kh_o and_o the_o angle_n angle_n ekh_n be_v a_o right_a angle_n therefore_o the_o square_a of_o he_o be_v double_a to_o the_o square_n of_o eke_o by_o the_o 47._o of_o the_o first_o again_o forasmuch_o as_o the_o line_n lk_n be_v equal_a to_o the_o line_n ke_o by_o position_n and_o the_o angle_n lke_n be_v by_o the_o second_o definition_n of_o the_o eleven_o a_o right_a angle_n therefore_o the_o square_n of_o the_o line_n el_n be_v double_a to_o the_o square_n of_o the_o line_n eke_o and_o it_o be_v prove_v that_o the_o square_a of_o the_o line_n he_o be_v double_a to_o the_o square_n of_o the_o line_n eke_o wherefore_o the_o square_a of_o the_o line_n le_fw-fr be_v equal_a to_o the_o square_n of_o the_o line_n eh_o wherefore_o also_o the_o line_n le_fw-fr be_v equal_a to_o the_o line_n eh_o and_o by_o the_o same_o reason_n the_o line_n lh_o be_v also_o equal_a to_o the_o line_n he._n wherefore_o the_o triangle_n lhe_n be_v equilater_n in_o like_a sort_n may_v we_o prove_v that_o every_o one_o of_o the_o rest_n of_o the_o triangle_n who_o base_n be_v the_o side_n of_o the_o square_a efgh_o and_o top_n the_o point_n l_o and_o m_o be_v equilater_n the_o say_v eight_o triangle_n also_o be_v equal_a the_o one_o to_o the_o other_o for_o every_o side_n of_o each_o be_v equal_a to_o the_o side_n of_o the_o square_a efgh_o wherefore_o there_o be_v make_v a_o octohedron_n contain_v under_o eight_o triangle_n who_o side_n be_v equal_a now_o it_o be_v require_v to_o comprehend_v it_o in_o the_o sphere_n give_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n demonstration_n forasmuch_o as_o these_o three_o line_n lk_v km_o and_o ke_o be_v equal_a the_o one_o to_o the_o other_o therefore_o a_o semicircle_n describe_v upon_o the_o line_n lm_o shall_v pass_v also_o by_o the_o point_n e._n and_o by_o the_o same_o reason_n if_o the_o semicircle_n be_v turn_v round_o about_o until_o it_o return_v unto_o the_o self_n same_o
f._n and_o draw_v these_o right_a line_n favorina_n fb_o fc_o fd_o fe_o wherefore_o those_o line_n do_v divide_v the_o angle_n of_o the_o pentagon_n into_o two_o equal_a part_n in_o the_o point_n a_o b_o c_o d_o e_o by_o the_o 4._o of_o the_o first_o and_o forasmuch_o as_o the_o five_o angle_n that_o be_v at_o the_o point_n f_o aâe_fw-fr equal_a to_o four_o right_a angle_n by_o the_o corollary_n of_o the_o 15._o of_o the_o first_o and_o they_o be_v equal_v the_o one_o to_o the_o other_o by_o the_o 8._o of_o the_o first_o therefore_o one_o of_o those_o angle_n as_o âor_a example_n sake_n the_o angle_n afb_o be_v a_o fiâth_v part_n less_o than_o a_o right_a angle_n wherefore_o the_o angle_n remain_v namely_o fab._n &_o abf_n be_v one_o right_a angle_n and_o a_o five_o part_n over_o but_o the_o angle_n fab._n be_v equal_a to_o the_o angle_n fbc_n wherefore_o the_o whole_a angle_n abc_n be_v one_o of_o the_o angle_n of_o the_o pentagon_n be_v a_o right_a angle_n and_o a_o five_o part_v more_o than_o a_o right_a angle_n which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o flussas_n now_o let_v we_o teach_v how_o those_o five_o solid_n have_v each_o like_a inclination_n of_o their_o base_n âiâst_v let_v we_o take_v a_o pyramid_n and_o divide_v one_o of_o the_o side_n thereof_o into_o two_o equal_a part_n and_o from_o the_o two_o angle_n opposite_a unto_o that_o side_n dâaw_v perpendiculars_n which_o shall_v fall_v upon_o the_o section_n by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o rational_a and_o at_o the_o say_a point_n of_o division_n as_o may_v easy_o be_v prove_v wherefore_o they_o shall_v contain_v the_o angâe_n of_o the_o inclination_n of_o the_o plain_n by_o the_o 4._o definition_n of_o the_o eleven_o which_o angle_n be_v subtend_v of_o the_o opposite_a side_n of_o the_o pyramid_n now_o forasmuch_o as_o the_o rest_n of_o the_o angle_n of_o the_o inclination_n of_o the_o plain_n of_o the_o pyramid_n be_v contain_v under_o two_o perpendicular_a line_n of_o the_o triangle_n and_o be_v subtend_v of_o the_o side_n of_o the_o pyramid_n it_o follow_v by_o the_o 8._o of_o the_o firât_o that_o those_o angle_n be_v equal_a wherâfoâe_o by_o the_o 5._o definition_n of_o the_o eleven_o the_o superficiece_n be_v in_o like_a sort_n incline_v the_o one_o to_o the_o other_o ãâ¦ã_o one_o of_o the_o side_n of_o a_o cube_n be_v divide_v into_o two_o equal_a part_n if_o from_o the_o say_a section_n be_v draw_v in_o two_o of_o the_o base_n thereof_o two_o perpendicular_a line_n they_o shall_v be_v parallel_n and_o equal_a to_o the_o side_n of_o the_o square_n which_o contain_v a_o right_a angle_n and_o forasmuch_o as_o all_o the_o angle_n of_o the_o base_n of_o the_o cube_n be_v right_a angle_n therefore_o those_o perpendicular_n fall_v upon_o the_o section_n of_o the_o side_n common_a to_o the_o two_o base_n shall_v contâyne_v a_o right_a angle_n by_o the_o 10._o of_o the_o eleven_o which_o self_n angle_n be_v the_o angle_n of_o inclination_n by_o the_o 4._o definition_n of_o the_o eleven_o and_o be_v subtend_v of_o the_o diameter_n of_o the_o base_a of_o the_o cube_n and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o rest_n of_o the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o the_o cube_fw-la be_v right_a angle_n wherefore_o the_o inclination_n of_o the_o superficiece_n of_o the_o cube_fw-la the_o one_o to_o the_o other_o be_v equal_a by_o the_o 5._o definition_n of_o the_o eleven_o in_o a_o octohedron_n take_v the_o diameter_n which_o couple_v the_o two_o opposite_a angle_n incline_v and_o from_o those_o opposite_a angle_n draw_v to_o one_o and_o the_o selâe_a same_o side_n of_o the_o octohedron_n in_o two_o base_n thereof_o two_o perpendicular_a line_n which_o shall_v divide_v that_o side_n into_o two_o equal_a part_n and_o perpendicular_o by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o those_o perpendicular_n shall_v contain_v the_o angle_n of_o the_o inclination_n of_o the_o base_n by_o the_o 4._o definition_n of_o the_o eleven_o and_o the_o same_o angle_n be_v subtend_v of_o the_o diameter_n of_o the_o octohedron_n wherefore_o the_o rest_n of_o the_o angle_n after_o the_o same_o manner_n describe_v in_o the_o rest_n of_o the_o base_n be_v comprehend_v and_o subtend_v of_o equal_a side_n shall_v by_o the_o 8._o of_o the_o first_o be_v equal_a the_o one_o to_o the_o other_o and_o therefore_o the_o inclination_n of_o the_o plain_n in_o the_o octohedron_n shall_v by_o the_o 5._o definition_n of_o the_o eleven_o be_v equal_a in_o a_o icosahedron_n let_v there_o be_v draw_v from_o the_o angle_n of_o two_o of_o the_o base_n incline_v to_o one_o side_n common_a to_o both_o the_o say_v base_n perpendicular_n which_o shall_v contain_v the_o angle_n of_o the_o inclination_n of_o the_o base_n by_o the_o 4._o definition_n of_o the_o eleven_o which_o angle_n be_v subtend_v of_o the_o right_a line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n which_o contain_v five_o side_n of_o the_o icosahedron_n by_o the_o 16._o of_o this_o book_n for_o it_o couple_v the_o two_o opposite_a angle_n of_o the_o triangle_n which_o be_v join_v together_o wherefore_o the_o rest_n of_o the_o angle_n of_o the_o inclination_n of_o the_o base_n be_v after_o the_o same_o manner_n find_v out_o they_o shall_v be_v contain_v under_o equal_a side_n and_o subtend_v of_o equal_a base_n and_o therefore_o by_o the_o 8._o of_o the_o fiâst_n those_o angle_n shall_v be_v equal_a wherefore_o also_o all_o the_o inclination_n of_o the_o base_n of_o the_o icosahedron_n the_o one_o to_o the_o other_o shalbâ_n equal_a by_o the_o 5._o definition_n of_o the_o eleven_o in_o a_o dodecahedron_n incline_v from_o the_o two_o opposite_a angle_n of_o two_o next_o pentagon_n draw_v to_o their_o common_a side_n perpendicular_a line_n pass_v by_o the_o centre_n of_o the_o say_a pentagon_n which_o shall_v where_o they_o fall_v divide_v the_o side_n into_o two_o equal_a part_n by_o the_o 3._o of_o the_o three_o for_o the_o base_n of_o a_o dodecahedron_n be_v contain_v in_o a_o circle_n and_o the_o angle_n contaynâd_v under_o those_o perpendicular_a line_n be_v the_o inclination_n of_o those_o base_n by_o the_o 4._o definition_n of_o the_o eleven_o and_o the_o foresay_a opposite_a angle_n be_v couple_v by_o a_o right_a line_n equal_a to_o the_o right_a line_n which_o couple_v the_o opposite_a section_n into_o two_o equal_a part_n of_o the_o side_n of_o the_o dodecahedron_n by_o the_o 33._o of_o the_o first_o for_o they_o couple_v together_o the_o half_a side_n of_o the_o dodecahedron_n which_o half_n be_v parallel_n and_o equal_a by_o the_o 3._o corollary_n of_o the_o 17._o of_o this_o book_n which_o couple_v line_n also_o be_v equal_a by_o the_o same_o corollary_n wherefore_o the_o angle_n be_v contain_v of_o equal_a perpendicular_a line_n and_o subtend_v of_o equal_a couple_a line_n shall_v by_o the_o 8._o of_o the_o first_o be_v equal_a and_o they_o be_v the_o angle_n of_o the_o inclination_n wherefore_o the_o base_n of_o the_o dodecahedron_n be_v in_o like_a sort_n incline_v the_o one_o to_o the_o other_o by_o the_o 5._o definition_n of_o the_o eleven_o flussas_n after_o this_o teach_v how_o to_o know_v the_o rationality_n or_o irrationality_n of_o the_o side_n of_o the_o triangle_n which_o contain_v the_o angle_n of_o the_o inclination_n of_o the_o superficiece_n of_o the_o foresay_a body_n in_o a_o pyramid_n the_o angle_n of_o the_o inclination_n be_v contain_v under_o two_o perpâdicular_a line_n of_o the_o triangle_n rational_a and_o be_v subtend_v of_o the_o side_n of_o the_o pyramid_n now_o the_o side_n of_o the_o pyramid_n be_v in_o power_n sesquitertia_fw-la to_o the_o perpendicular_a line_n by_o the_o corollary_n of_o the_o 12._o of_o this_o book_n and_o therefore_o the_o triangle_n contain_v of_o those_o perpendicular_a line_n and_o the_o side_n of_o pyramid_n have_v his_o side_n rational_a &_o commensurable_a in_o power_n the_o one_o to_o the_o other_o forasmuch_o as_o the_o two_o side_n of_o a_o cube_n or_o right_a line_n equal_a to_o they_o subtend_v under_o the_o diameter_n of_o one_o of_o the_o base_n rational_a do_v make_v the_o angle_n of_o the_o inclination_n and_o the_o diameter_n of_o the_o cube_fw-la be_v in_o power_n sesquialter_fw-la to_o the_o diameter_n of_o the_o base_a which_o diameter_n of_o the_o base_a be_v in_o power_n double_a to_o the_o side_n by_o the_o 47._o of_o the_o first_o therefore_o those_o line_n be_v rational_a and_o commensurable_a in_o power_n in_o a_o octohedron_n rational_a who_o two_o perpendicular_n of_o the_o base_n contain_v the_o angle_n of_o the_o inclination_n of_o the_o octohedron_n which_o angle_n also_o be_v subtend_v of_o the_o diameter_n of_o the_o octohedron_n the_o diameter_n be_v in_o power_n
be_v demonstrate_v fd_o be_v the_o diameter_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n which_o diameter_n be_v in_o power_n sesquialter_fw-la to_o ab_fw-la the_o side_n of_o the_o tetrahedron_n inscribe_v in_o theâ_z same_o sphere_n by_o the_o 13._o of_o the_o thirteen_o wherefore_o the_o line_n ed_z the_o side_n of_o the_o icosahedron_n be_v in_o power_n sesquialter_fw-la to_o gâ_n the_o great_a segment_n or_o less_o line_n if_o therefore_o the_o side_n of_o a_o tetrahedron_n contain_v in_o power_n two_o right_a line_n join_v together_o a_o extreme_a and_o mean_a proportion_n the_o side_n of_o a_o icosahedron_n describe_v in_o the_o self_n same_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o less_o right_a line_n ¶_o the_o 19_o proposition_n the_o superficies_n of_o a_o cube_n be_v to_o the_o superficies_n of_o a_o octohedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n in_o that_o proportion_n that_o the_o solid_n be_v construction_n svppose_v that_o abcde_v be_v a_o cube_n who_o four_o diameter_n let_v be_v the_o line_n ac_fw-la bc_o dc_o and_o ec_o produce_v on_o each_o side_n let_v also_o the_o octohedron_n inscribe_v in_o the_o self_n same_o sphere_n be_v fghk_v who_o three_o diameter_n let_v be_v fh_o gk_o and_o on_o then_o i_o say_v that_o the_o cube_fw-la abdella_n be_v to_o the_o octohedron_n fgh_n as_o the_o superficies_n of_o the_o cube_fw-la be_v to_o the_o superficies_n of_o the_o octohedron_n draw_v from_o the_o centre_n of_o the_o cube_fw-la to_o the_o base_a abed_o a_o perpendicular_a line_n cr._n and_o from_o the_o centre_n of_o the_o octohedron_n draw_v to_o the_o base_a gnh_o a_o perpendicular_a line_n âl_o demonstration_n and_o forasmuch_o as_o the_o three_o diameter_n of_o the_o cube_fw-la do_v pass_n by_o the_o ãâã_d c_o therefore_o by_o the_o 2._o corollary_n of_o the_o 15._o of_o the_o thirteen_o âhere_o shall_v be_v make_v of_o the_o cube_fw-la six_o pyramid_n as_o this_o pyramid_n abdec_n equal_a to_o the_o whole_a cube_fw-la for_o there_o be_v in_o the_o cube_fw-la âixe_v base_n upon_o which_o fall_v equal_a perpendicular_n from_o the_o cenâââ_n by_o the_o corollary_n of_o the_o assumpt_n of_o the_o 16._o of_o the_o twelve_o for_o the_o base_n be_v contain_v in_o equal_a circle_v of_o the_o sphere_n but_o in_o the_o octohedron_n the_o three_o diameter_n do_v make_v upon_o the_o 8._o base_n 8._o pyramid_n have_v their_o top_n in_o the_o centre_n by_o the_o 3._o corollary_n of_o the_o 14â_o of_o the_o thirteen_o now_o the_o base_n of_o the_o cube_fw-la and_o of_o the_o octohedron_n be_v contain_v in_o equal_a circle_n of_o the_o sphere_n by_o the_o 13._o of_o this_o book_n wherefore_o they_o shall_v be_v equal_o distant_a from_o the_o centre_n and_o the_o perpendicular_a line_n cr_n and_o â_o shall_v be_v equal_a by_o the_o corollary_n of_o the_o assumpt_n of_o the_o 16._o of_o the_o twelve_o wherefore_o the_o pyramid_n of_o the_o cube_fw-la shall_v be_v under_o one_o and_o the_o self_n same_o altitude_n with_o the_o pyramid_n of_o the_o octohedron_n namely_o under_o the_o perpendicular_a line_n draw_v from_o the_o centre_n to_o the_o base_n wherefore_o six_o pyramid_n of_o the_o cube_fw-la be_v to_o 8._o pyramid_n of_o the_o octohedron_n be_v under_o one_o and_o the_o same_o altitude_n in_o that_o proportion_n that_o their_o base_n be_v by_o the_o 6._o of_o the_o twelve_o that_o be_v one_o pyramid_n set_v upon_o six_o base_n of_o the_o cube_fw-la and_o have_v to_o his_o altitude_n the_o perpendicular_a line_n which_o pyramid_n be_v equal_a to_o the_o six_o pyramid_n by_o the_o same_o 6._o of_o the_o twelve_o be_v to_o one_o pyramid_n set_v upon_o the_o 8._o base_n of_o the_o octohedron_n be_v equal_a to_o the_o octohedron_n and_o also_o under_o onâ_n and_o the_o self_n same_o altitude_n in_o that_o proportion_n that_o six_o base_n of_o the_o cube_fw-la which_o contain_v the_o whole_a superficies_n of_o the_o cube_fw-la be_v to_o 8._o base_n of_o the_o octohedronâ_n which_o contain_v the_o whole_a superficies_n of_o the_o octohedron_n for_o the_o solid_n of_o those_o pyramid_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n be_v by_o the_o self_n same_o 6._o of_o the_o twelve_o wherefore_o âhe_z superficies_n of_o the_o cube_fw-la be_v to_o the_o superficies_n of_o the_o octohedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n in_o that_o proportion_n that_o the_o solid_n be_v which_o be_v require_v to_o be_v prove_v ¶_o the_o 20._o proposition_n if_o a_o cube_n and_o a_o octohedron_n be_v contain_v in_o one_o &_o the_o self_n same_o sphere_n they_o shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o side_n of_o the_o cube_n be_v to_o the_o semidiameter_n of_o the_o sphere_n svppose_v that_o the_o octohedron_n aecdb_n be_v inscribe_v in_o the_o sphere_n abcd_o and_o let_v the_o cube_fw-la inscribe_v in_o the_o same_o sphere_n be_v fghim_n who_o diameter_n let_v be_v high_a construction_n which_o be_v equal_a to_o the_o diameter_n ac_fw-la by_o the_o 15._o of_o the_o thirteen_o let_v the_o half_a of_o the_o diameter_n be_v ae_n then_o i_o say_v that_o the_o cube_fw-la fghim_n be_v to_o the_o octohedron_n aecdb_n as_o the_o side_n mg_o be_v to_o the_o semidiameter_n ae_n forasmuch_o as_o the_o diameter_n ac_fw-la be_v in_o power_n double_a to_o bk_o the_o side_n of_o the_o octohedron_n by_o the_o 14._o of_o the_o thirteen_o and_o be_v in_o power_n triple_a to_o mg_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o the_o same_o therefore_o the_o square_a bkdl_n shall_v be_v sesquialter_n to_o fm_o the_o square_a of_o the_o cube_fw-la from_o the_o line_n ae_n cut_v of_o a_o three_o part_n a_o and_o from_o the_o line_n mg_o cut_v of_o likewise_o a_o three_o part_n go_v by_o the_o 9_o of_o the_o six_o demonstration_n now_o than_o the_o line_n en_fw-fr shall_v be_v two_o three_o part_n of_o the_o line_n ae_n and_o so_o also_o shall_v the_o line_n more_n be_v of_o the_o line_n mg_o wherefore_o the_o parallelipipedon_n set_v upon_o the_o base_a bkdl_n and_o have_v his_o altitude_n the_o line_n ea_fw-la be_v triple_a to_o the_o parallelipipedon_n set_v upon_o the_o same_o base_a and_o have_v his_o altitude_n the_o line_n a_fw-la by_o the_o corollary_n of_o the_o 31._o of_o the_o eleven_o but_o it_o be_v also_o triple_a to_o the_o pyramid_n abkdl_n which_o be_v set_v upon_o the_o same_o base_a and_o be_v under_o the_o same_o altitude_n by_o the_o second_o corollary_n of_o the_o 7._o of_o the_o twelve_o wherefore_o the_o pyramid_n abkdl_n be_v equal_a to_o the_o parallelipipedon_n which_o be_v set_v upon_o the_o base_a bkdl_n and_o have_v to_o his_o altitude_n the_o line_n an._n but_o unto_o that_o parallelipipedom_n be_v double_a the_o parallelipipedon_n which_o be_v set_v upon_o the_o same_o base_a bkdl_n and_o have_v to_o his_o altitude_n a_o line_n double_a to_o the_o line_n en_fw-fr by_o the_o corollary_n of_o the_o 31._o of_o the_o first_o and_o unto_o the_o pyramid_n be_v double_a the_o octohedron_n abkldc_n by_o the_o 2._o corollary_n of_o the_o 14._o of_o the_o thirteen_o wherefore_o the_o octohedron_n abkdlc_n be_v equal_a to_o the_o parallelipipedon_n set_v upon_o the_o base_a bkld_a &_o have_v his_o altitude_n the_o line_n en_fw-fr by_o the_o 15._o of_o the_o five_o but_o the_o parallelipipedon_n set_v upon_o the_o base_a bkdl_n which_o be_v sesquialter_fw-la to_o the_o base_a fm_o and_o have_v to_o his_o altitude_n the_o line_n more_n which_o be_v two_o three_o part_n of_o the_o side_n of_o the_o cube_fw-la mg_o be_v equal_a to_o the_o cube_fw-la fg_o by_o the_o 2._o part_n of_o the_o 34._o of_o the_o eleven_o for_o it_o be_v before_o prove_v that_o the_o base_a bkdl_n be_v sesquialter_fw-la to_o the_o base_a fm_o now_o than_o these_o two_o parallelipipedon_n namely_o the_o parallelipipedon_n which_o be_v set_v upon_o the_o base_a bkdl_n which_o be_v sesquialter_fw-la to_o the_o base_a of_o the_o cube_fw-la and_o have_v to_o his_o altitude_n the_o line_n more_n which_o be_v two_o three_o part_n of_o mg_o the_o side_n of_o the_o cube_fw-la which_o parallelipipedon_n be_v prove_v equal_a to_o the_o cube_fw-la and_o the_o parallelipipedon_n set_v upon_o the_o same_o base_a bkdl_n and_o have_v his_o altitude_n the_o line_n en_fw-fr which_o parallelipipedon_n be_v prove_v equal_a to_o the_o octohedron_n these_o two_o parallelipipedon_n i_o say_v be_v the_o one_o to_o the_o other_o as_o the_o altitude_n more_n be_v to_o the_o altitude_n en_fw-fr by_o the_o corollary_n of_o the_o 31._o of_o the_o eleven_o wherefore_o also_o as_o the_o altitude_n more_n be_v to_o the_o altitude_n en_fw-fr so_o be_v the_o cube_fw-la fghim_n to_o the_o octohedron_n abkdlc_n by_o the_o 7._o of_o the_o five_o but_o as_o the_o line_n more_n be_v to_o the_o line_n en_fw-fr so_o be_v
that_o be_v the_o right_a line_n agnostus_n gb_o bd_o dam_fw-ge ca_n cg_o cb_o cd_o and_o ja_z ig_n ib_n id_fw-la be_v equal_a the_o one_o to_o the_o other_o wherefore_o also_o the_o 8._o triangle_n cag_n cgb_n cbd_v cda_n jag_n igb_n ibd_v &_o ida_n be_v equal_a and_o equilater_n and_o therefore_o agbdci_n be_v a_o octohedron_n by_o the_o 23._o definition_n of_o the_o eleven_o and_o the_o say_v octahedron_n be_v include_v in_o the_o dodecahedron_n by_o the_o first_o definition_n of_o this_o book_n for_o that_o all_o the_o angle_n thereof_o do_v at_o one_o time_n touch_v the_o side_n of_o the_o dodecahedron_n wherefore_o in_o the_o dodecahedron_n give_v be_v include_v a_o octohedron_n which_o be_v require_v to_o be_v do_v ¶_o the_o 10._o proposition_n the_o 10._o problem_n in_o a_o dodecahedron_n give_v to_o inscribe_v a_o equilater_n trilater_n pyramid_n svppose_v that_o the_o dodecahedron_n give_v be_v abcd_o of_o which_o dodecahedron_n take_v three_o base_n meet_v at_o the_o point_n s_o namely_o these_o three_o base_n alsik_n dnsle_n and_o sibrn_n construction_n and_o of_o those_o three_o base_n take_v the_o three_o angle_n at_o the_o point_n a_o b_o d_o and_o draw_v these_o right_a line_n ab_fw-la bd_o and_o dam_fw-ge and_o let_v the_o diameter_n of_o the_o sphere_n contain_v the_o dodecahedron_n be_v so_o and_o then_o draw_v these_o right_a line_n ao_o boy_n and_o do_v now_o forasmuch_o as_o by_o the_o 17._o of_o the_o thirteen_o the_o angle_n of_o the_o dodecahedron_n be_v set_v in_o the_o superficies_n of_o the_o sphere_n describe_v about_o the_o dodecahedronâ_n demonstration_n therefore_o if_o upon_o the_o diameter_n so_o and_o by_o the_o point_n a_o be_v describe_v a_o semicircle_n it_o shall_v make_v the_o angle_n sao_n a_o right_a angle_n by_o the_o 31._o of_o the_o three_o and_o likewise_o if_o the_o same_o semicircle_n be_v draw_v by_o the_o point_n d_o and_o b_o it_o shall_v also_o make_v the_o angle_n sbo_n and_o sdo_v right_a angle_n wherefore_o the_o diameter_n so_o contain_v in_o power_n both_o the_o line_n sa_o ao_o or_o the_o line_n sb_n boy_n or_o else_o sd_z do_v but_o the_o line_n sa_o sd_z sb_n be_v equal_a the_o one_o to_o the_o other_o for_o they_o each_o subtend_v one_o of_o the_o angle_n of_o equal_a pentagon_n wherefore_o the_o other_o line_n remain_v namely_o ao_o boy_n do_v be_v equal_a the_o one_o to_o the_o other_o and_o by_o the_o same_o reason_n may_v be_v prove_v that_o the_o diameter_n hd_v which_o subtend_v the_o two_o right_a line_n ha_o ad_fw-la contain_v in_o power_n both_o the_o say_v two_o right_a line_n and_o also_o contain_v in_o power_n both_o the_o right_a line_n hb_o and_o bd_o which_o two_o right_a line_n it_o also_o suhtend_v and_o moreover_o by_o the_o same_o reason_n the_o diameter_n ac_fw-la which_o subtend_v the_o right_a line_n cb_o and_o basilius_n contain_v in_o power_n both_o the_o say_v right_a line_n câ_n and_o basilius_n but_o the_o right_a line_n ha_o hb_o and_o cb_o be_v equal_a the_o one_o to_o the_o other_o for_o that_o each_o of_o they_o also_o subtend_v one_o of_o the_o angle_n of_o equal_a pentagonsâ_n wherefore_o the_o right_a line_n remain_v namely_o ad_fw-la bd_o and_o basilius_n be_v equal_a the_o one_o to_o the_o other_o and_o by_o the_o same_o reason_n may_v be_v prove_v that_o each_o of_o those_o right_a line_n ad_fw-la bd_o and_o basilius_n be_v equal_a to_o each_o of_o the_o right_a line_n ao_o boy_n and_o do_v wherefore_o the_o six_o right_a line_n ab_fw-la bd_o dam_fw-ge ao_o boy_n &_o do_v be_v equal_a the_o one_o to_o the_o other_o and_o therefore_o the_o triangle_n which_o be_v make_v of_o they_o namely_o the_o triangle_n abdella_n aob_n aod_n and_o bod_fw-mi be_v equal_a and_o equilater_n which_o triangle_n therefore_o do_v make_v a_o pyramid_n abdo_n who_o base_a be_v abdella_n and_o top_n the_o point_n o._n each_o of_o the_o angle_n of_o which_o pyramid_n namely_o the_o angle_n at_o the_o point_n a_o b_o d_o oh_o do_v in_o the_o self_n same_o point_n touch_v the_o angle_n of_o the_o dodecahedron_n wherefore_o the_o say_a pyramid_n be_v inscribe_v in_o the_o dodecahedron_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o dodecahedron_n give_v be_v inscribe_v a_o trilater_n equilater_n pyramid_n which_o be_v require_v to_o be_v do_v ¶_o the_o 11._o proposition_n the_o 11._o problem_n in_o a_o icosahedron_n give_v to_o inscribe_v a_o cube_fw-la it_o be_v manifest_a by_o the_o 7._o of_o this_o book_n that_o the_o angle_n of_o a_o dodecahedron_n be_v set_v in_o the_o centre_n of_o the_o base_n of_o the_o icosahedron_n and_o by_o the_o 8._o of_o this_o book_n it_o be_v prove_v that_o the_o angle_n of_o a_o cube_fw-la be_v set_v in_o the_o angle_n of_o a_o dodecahedron_n wherefore_o the_o self_n same_o angle_n of_o the_o cube_fw-la shall_v of_o necessity_n be_v set_v in_o the_o centre_n of_o the_o base_n of_o icosahedron_n wherefore_o the_o cube_fw-la shall_v be_v inscribe_v in_o the_o icosahedron_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o icosahedron_n give_v be_v include_v a_o cube_fw-la which_o be_v require_v to_o be_v do_v ¶_o the_o 12._o proposition_n the_o 12._o problem_n in_o a_o icosahedron_n give_v to_o inscribe_v a_o trilater_n equilater_n pyramid_n by_o the_o former_a proposition_n it_o be_v manifest_a that_o the_o angle_n of_o a_o cube_fw-la be_v set_v in_o the_o centre_n of_o the_o base_n of_o the_o icosâhedron_n and_o by_o the_o first_o of_o this_o book_n it_o be_v plain_a that_o the_o four_o angle_n of_o a_o pyramid_n be_v set_v in_o four_o angle_n of_o a_o cube_fw-la wherefore_o it_o be_v evident_a by_o the_o first_o definition_n of_o this_o book_n that_o a_o pyramid_n describe_v of_o right_a line_n join_v together_o these_o four_o centre_n of_o the_o base_n of_o the_o icosahedron_n shall_v be_v inscribe_v in_o the_o same_o icosahedron_n wherefore_o in_o a_o icosadron_n give_v be_v inscribe_v a_o equilater_n trilater_n pyramid_n which_o be_v require_v to_o be_v do_v ¶_o the_o 13._o problem_n the_o 13._o proposition_n in_o a_o cube_n give_v to_o inscribe_v a_o dodecahedron_n take_v a_o cube_n adfl_fw-mi construction_n and_o divide_v every_o one_o of_o the_o side_n thereof_o into_o two_o equal_a part_n in_o the_o point_n t_o h_o edward_n p_o g_o l_o m_o f_o and_o pkq_n and_o draw_v these_o right_a line_n tk_n gf_o pq_n hk_n ps_n and_o lm_o which_o line_n again_o divide_v into_o two_o equal_a part_n in_o the_o point_n n_o five_o a'n_z ay_o z_o x._o and_o draw_v these_o right_a line_n nigh_o vx_n and_o iz_n now_o the_o three_o line_n nigh_o vx_n and_o iz_n together_o with_o the_o diameter_n of_o the_o cube_fw-la shall_v cut_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o the_o centre_n of_o the_o cube_fw-la by_o the_o 3â_o of_o the_o eleven_o let_v that_o centre_n be_v the_o point_n o._n and_o not_o to_o stand_v long_o about_o the_o demonstration_n understand_v all_o these_o right_a line_n to_o be_v equal_a and_o parallel_n to_o the_o side_n of_o the_o cube_fw-la and_o to_o cut_v the_o one_o the_o other_o right_o angle_a wise_a by_o the_o 29._o of_o the_o first_o let_v their_o half_n namely_o fv_n gv_n high_a and_o king_n and_o the_o rest_n such_o like_a be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n by_o the_o 30._o of_o the_o six_o who_o great_a segment_n let_v be_v the_o line_n f_n gb_o hc_n and_o ke_o etc_n etc_n and_o draw_v these_o right_a line_n give_v ge_z bc_o and_o be._n now_o forasmuch_o as_o the_o line_n give_v be_v equal_a to_o the_o whole_a line_n gv_n which_o be_v the_o half_a of_o the_o side_n of_o the_o cube_fw-la demonstration_n and_o the_o line_n je_n be_v equal_a to_o the_o line_n bv_n that_o be_v to_o the_o less_o segment_n therefore_o the_o square_n of_o the_o line_n give_v and_o je_n be_v triple_a to_o the_o square_n of_o the_o line_n gb_o by_o the_o 4._o of_o the_o thirteen_o but_o unto_o the_o square_n of_o the_o line_n give_v and_o je_n the_o square_a of_o the_o line_n ge_z be_v equal_a by_o the_o 47._o of_o the_o firstâ_o for_o the_o angle_n gie_n be_v a_o right_a angle_n wherefore_o the_o square_a of_o the_o line_n ge_z be_v triple_a to_o the_o square_n of_o the_o line_n gb_o and_o forasmuch_o as_o the_o line_n fg_o be_v erect_v perpendicular_o to_o the_o plain_a agkl_n by_o the_o 4._o of_o the_o eleven_o for_o it_o be_v erect_v perpendicular_o to_o the_o two_o line_n agnostus_n and_o give_v therefore_o the_o angle_n bge_n be_v a_o right_a angle_n for_o the_o line_n ge_z be_v draw_v in_o the_o plain_a agkl_n wherefore_o the_o line_n be_v contain_v in_o power_n the_o two_o line_n bg_o and_o ge_z by_o the_o 47._o of_o the_o first_o be_v in_o power_n quadruple_a to_o the_o
the_o pyramisâ_n and_o they_o shall_v âall_n perpendicular_o upon_o the_o base_n and_o shall_v also_o fall_v upon_o the_o centre_n of_o the_o circle_n which_o contain_v the_o base_n by_o the_o corollary_n of_o the_o 13._o of_o the_o thirteen_o let_v the_o cenâre_n of_o the_o triangle_n abc_n be_v the_o point_n g_o and_o let_v the_o centre_n of_o the_o triangle_n adc_o be_v the_o point_n hâ_n and_o of_o the_o triangle_n adb_o let_v the_o point_n n_o be_v the_o centre_n and_o final_o let_v the_o point_n f_o be_v the_o centre_n of_o the_o other_o triangle_n dbc_n and_o let_v the_o right_a line_n fall_v upon_o those_o centre_n be_v deg_n beh_n cen_n &_o aef_n and_o by_o those_o centre_n g_o h_o n_o f_o let_v there_o be_v draw_v from_o the_o angle_n to_o the_o opposite_a side_n these_o right_a line_n agl_n dhk_n bnm_fw-la and_o dfl_fw-mi which_o shall_v fall_v perpendicular_o upon_o the_o side_n bc_o ca_n ad_fw-la and_o cb_o by_o the_o corollary_n of_o the_o 1â_o of_o the_o thirteen_o and_o therefore_o they_o shall_v cuâ_n they_o into_o two_o equal_a part_n in_o the_o point_n king_n l_o m_o by_o the_o 3._o of_o the_o three_o again_o let_v the_o line_n which_o be_v draw_v from_o the_o solid_a angle_n to_o the_o opposite_a base_n be_v divide_v into_o two_o equal_a part_n namely_o the_o line_n dg_o in_o the_o point_n t_o the_o line_n cn_fw-la in_o the_o point_n oh_o the_o line_n of_o in_o the_o point_n p_o and_o the_o line_n bh_o in_o âhe_n point_n r_o and_o draw_v the_o line_n ht_v ft_n ho_o and_o foyes_n now_o forasmuch_o as_o the_o line_n gk_o demonstration_n and_o gl_n which_o be_v draw_v from_o the_o centre_n of_o one_o and_o the_o self_n same_o triangle_n abc_n to_o the_o side_n be_v equal_a and_o the_o line_n dk_o and_o dl_o be_v equal_a for_o they_o be_v the_o perpendicular_n of_o equal_a &_o like_a triangle_n b._n and_o the_o line_n dg_o be_v common_a to_o they_o wherefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n kdg_n &_o ldg_n be_v equal_a and_o forasmuch_o as_o the_o line_n hd_v &_o df_o be_v draw_v from_o the_o centre_n of_o equal_a circle_n which_o contain_v the_o equal_a triangle_n adc_o &_o dbc_n therefore_o they_o be_v equal_a &_o the_o line_n dt_n be_v common_a to_o they_o both_o and_o they_o contain_v equal_a angle_n as_o before_o have_v be_v prove_v wherefore_o the_o base_n ht_v and_o ft_v be_v equal_a by_o the_o 4._o of_o the_o first_o and_o by_o the_o same_o reason_n if_o we_o draw_v the_o line_n cf_o and_o change_z may_v we_o prove_v that_o the_o other_o line_n ho_o and_o foe_n be_v equal_a to_o the_o same_o line_n ht_v and_o ft_v and_o also_o the_o one_o to_o the_o other_o wherefore_o also_o after_o the_o same_o manner_n may_v be_v prove_v that_o the_o rest_n of_o the_o line_n which_o couple_n the_o centre_n of_o the_o triangle_n and_o the_o section_n of_o the_o perpendicular_n into_o two_o equal_a part_n as_o the_o line_n np_v graccus_n gp_n rn_n nt_v ph_n go_v and_o rf_n be_v equal_a and_o forasmuch_o as_o from_o every_o one_o of_o the_o centre_n of_o the_o base_n be_v draw_v three_o right_a line_n to_o the_o section_n into_o two_o equal_a part_n of_o the_o perpendiculers_n and_o there_o be_v four_o centre_n it_o follow_v that_o these_o equal_a right_a line_n so_o draw_v be_v twelve_o in_o number_n of_o which_o every_o three_o and_o three_o make_v a_o solid_a angle_n in_o the_o four_o centre_n of_o the_o base_n and_o in_o the_o four_o section_n into_o two_o equal_a part_n of_o the_o perpendicular_n wherefore_o that_o solid_a have_v 8._o angle_n contain_v under_o 12._o equal_a side_n which_o make_v six_o quadrangled_a figure_n namely_o hoff_z pgrn_n phog_n gofr_n frnt_v and_o tnph_n now_o let_v we_o prove_v that_o those_o quadrangled_a figure_n be_v rectangle_n forasmuch_o as_o upon_o dc_o the_o common_a base_a of_o the_o triangle_n adc_o and_o bdc_o fall_v the_o perpendicular_n as_o and_o b_n which_o be_v draw_v by_o the_o centre_n h_o and_o fletcher_n either_o of_o these_o line_n his_o and_o sf_n shall_v be_v the_o three_o part_n of_o either_o of_o these_o line_n as_o and_o sb_n for_o the_o line_n ah_o be_v duple_n to_o the_o line_n his_o and_o divide_v the_o base_a dc_o into_o two_o equal_a part_n by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o in_o the_o triangle_n abs_fw-la the_o side_n as_o and_o b_n be_v cut_v proportional_o in_o the_o point_v h_o and_o fletcher_n and_o therefore_o the_o line_n hf_o be_v a_o parallel_n to_o the_o side_n ab_fw-la by_o the_o 2._o of_o the_o six_o wherefore_o the_o triangle_n asb_n and_o hsf_n be_v equiangle_n by_o the_o 6._o of_o the_o six_o wherefore_o the_o base_a hf_o shall_v be_v the_o three_o part_n of_o the_o base_a ab_fw-la by_o the_o 4._o of_o the_o six_o we_o may_v also_o prove_v that_o the_o line_n to_o be_v the_o three_o part_n of_o the_o line_n dc_o for_o the_o line_n ec_o and_o ed_z which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o pyramid_n be_v equal_a and_o the_o line_n en_fw-fr which_o be_v draw_v from_o the_o centre_n to_o the_o base_a be_v the_o three_o part_n of_o the_o line_n ec_o so_o also_o be_v the_o line_n ge_z the_o three_o part_n of_o the_o line_n ed_z by_o the_o corollary_n of_o the_o 13._o of_o the_o thirteen_o for_o it_o be_v the_o six_o part_n of_o the_o diameter_n of_o the_o sphere_n which_o contain_v the_o pyramid_n and_o the_o line_n on_o be_v the_o half_a of_o the_o whole_a line_n nc_n wherefore_o the_o residue_n eo_fw-la be_v the_o three_o part_n of_o the_o line_n ecâ_n and_o so_o also_o be_v the_o line_n et_fw-fr the_o three_o part_n of_o the_o line_n ed._n wherefore_o the_o line_n to_o in_o the_o triangle_n dec_n be_v a_o parallel_n to_o the_o line_n dc_o and_o be_v a_o three_o part_n of_o the_o same_o by_o the_o former_a 2._o and_o 4._o of_o the_o six_o as_o the_o line_n hf_o be_v prove_v the_o three_o part_n of_o the_o line_n ab_fw-la but_o ab_fw-la and_o dc_o be_v side_n of_o the_o pyramid_n be_v equal_a wherefore_o the_o line_n hf_o and_o to_o be_v the_o three_o part_n of_o equal_a line_n be_v equal_a by_o the_o 15._o of_o the_o five_o wherefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n htf_n and_o tfo_n be_v equal_a and_o by_o the_o same_o reason_n the_o angle_n opposite_a unto_o they_o namely_o the_o angle_n foh_o and_o oht_fw-mi be_v equal_v the_o one_o to_o the_o other_o and_o also_o be_v equal_a to_o the_o say_a angle_n htâ_n and_o tfo_n but_o these_o four_o angle_n be_v equal_a to_o 4._o right_a angle_n by_o the_o corollary_n of_o the_o 32._o of_o the_o first_o wherefore_o the_o angle_n of_o the_o quadrangle_n hoff_z be_v right_a angle_n and_o by_o the_o same_o reason_n may_v the_o angle_n of_o the_o other_o five_o quadrangled_a figure_n be_v prove_v right_a anglesâ_n now_o rest_v to_o prove_v that_o the_o foresay_a quadrangle_v be_v each_o in_o one_o and_o the_o self_n same_o plain_a take_v the_o quadrangle_n hoff_z and_o forasmuch_o as_o in_o the_o triangle_n asb_n the_o line_n hf_o be_v prove_v a_o parallel_n to_o the_o line_n ab_fw-la therefore_o it_o cut_v the_o line_n sv_n and_o sb_n proportional_o in_o the_o point_n i_o and_o f._n by_o the_o 2._o of_o the_o six_o now_o then_o forasmuch_o as_o sf_n be_v prove_v the_o three_o part_n of_o the_o line_n sb_n the_o line_n si_fw-it shall_v also_o be_v the_o third_o part_n of_o the_o line_n su._n moreover_o forasmuch_o as_o the_o line_n we_o which_o couple_v the_o section_n into_o equal_a part_n of_o the_o opposite_a âides_n of_o the_o pyramid_n namely_o of_o the_o side_n ab_fw-la and_o dc_o be_v by_o the_o centre_n e_o divide_v into_o two_o equal_a part_n by_o the_o corollary_n of_o the_o second_o of_o this_o book_n for_o it_o be_v the_o diameter_n of_o the_o octohedron_n inscribe_v in_o the_o pyramid_n therefore_o the_o line_n si_fw-it be_v two_z three_o part_n of_o the_o half_a line_n se._n and_o by_o the_o same_o reason_n forasmuch_o as_o in_o the_o triangle_n dec_n the_o line_n to_o be_v prove_v to_o be_v a_o parallel_n to_o the_o side_n dc_o it_o shall_v in_o the_o self_n same_o triangle_n cut_v the_o line_n ce_fw-fr and_o see_fw-la proportional_o in_o the_o point_n oh_o and_o i_o by_o the_o same_o 2._o of_o the_o six_o but_o the_o line_n eo_fw-la be_v prove_v to_o be_v a_o three_o part_n of_o the_o line_n ec_o wherefore_o the_o line_n ei_o be_v also_o a_o three_o part_n of_o the_o line_n es._n wherefore_o the_o residue_n be_v shall_v be_v two_o three_o part_n of_o the_o whole_a line_n es._n wherefore_o the_o point_n i_o cut_v of_o the_o line_n to_o and_o hf._n wherefore_o the_o
two_o line_n hif_n and_o tio_n cut_v the_o one_o the_o other_o be_v in_o one_o and_o the_o self_n same_o '_o plain_n by_o the_o 2._o of_o the_o eleven_o and_o therefore_o the_o point_n h_o t_o f_o oh_o be_v in_o one_o &_o the_o self_n same_o plain_a wherforeâ_n the_o rectangle_n figure_n hoff_z be_v quadrilater_n and_o equilater_n and_o in_o one_o and_o the_o self_n same_o plain_n be_v a_o square_a by_o the_o definition_n of_o a_o square_a and_o by_o the_o same_o reason_n may_v the_o rest_n of_o the_o base_n of_o the_o solid_a be_v prove_v to_o be_v square_n equal_a and_o plain_a or_o superficial_a now_o than_o the_o solid_a be_v comprehend_v of_o 6._o equal_a square_n which_o be_v contain_v of_o 12._o equal_a side_n which_o square_n make_v 8._o solid_a angle_n of_o which_o four_o be_v in_o the_o centre_n of_o the_o base_n oâ_n the_o pyramid_n and_o the_o other_o 4._o be_v in_o the_o middle_a section_n of_o the_o four_o perdendiculars_n wherefore_o the_o solid_a hoftpgrn_n be_v a_o cube_fw-la by_o the_o 21._o definition_n of_o the_o eleven_o and_o be_v inscribe_v in_o the_o pyramid_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o trilater_n equilater_n pyramid_n give_v be_v inscribe_v a_o cube_fw-la ¶_o a_o corollary_n the_o line_n which_o cut_v into_o two_o equal_a part_n the_o opposite_a side_n of_o the_o pyramid_n be_v triple_a to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n and_o pass_v by_o the_o centre_n of_o the_o cube_fw-la for_o the_o line_n sev_n who_o three_o part_n the_o line_n si_fw-it be_v cut_v the_o opposite_a side_n cd_o and_o ab_fw-la into_o two_o equll_a part_n but_o the_o line_n ei_o which_o be_v draw_v from_o the_o centre_n of_o the_o cube_fw-la to_o the_o base_a be_v prove_v to_o be_v a_o three_o part_n of_o the_o line_n es_fw-ge wherefore_o the_o side_n of_o the_o cube_fw-la which_o be_v double_a to_o the_o line_n ei_o shall_v be_v a_o three_o part_n of_o the_o whole_a line_n we_o which_o be_v as_o have_v be_v prove_v double_a to_o the_o line_n es._n the_o 19_o problem_n the_o 19_o proposition_n in_o a_o trilater_n equilater_n pyramid_n give_v to_o inscribe_v a_o icosahedron_n svppose_v that_o the_o pyramid_n be_v give_v ãâã_d abâdâ_n every_o one_o of_o who_o sâdes_n ãâã_d be_v divide_v into_o two_o equal_a part_n in_o the_o poyâââââ_n m_o king_n l_o p_o n._n construction_n and_o iâ_z every_o one_o of_o the_o base_n of_o that_o pyramid_n descry_v the_o trianglââ_n lââ_n pmn_n nkl_n and_o ãâ¦ã_o which_o triangle_n shall_v be_v equilater_n by_o the_o 4._o of_o the_o firât_o âor_a the_o side_n subtend_v equal_a angle_n of_o the_o pyramid_n contain_v under_o the_o half_n of_o the_o side_n of_o the_o same_o pyramisâ_n wherefore_o the_o side_n of_o the_o say_a triangle_n be_v equal_a let_v those_o side_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n by_o the_o 30._o of_o the_o six_o in_o the_o point_n c_o e_o q_o r_o s_o t_o h_o i_o oh_o five_o a'n_z x._o now_o then_o those_o side_n be_v cut_v into_o the_o self_n same_o proportion_n by_o the_o 2._o of_o the_o fourtenth_n and_o therefore_o they_o make_v the_o liâe_z section_n equal_a by_o the_o â_o part_n of_o the_o nine_o of_o the_o five_o now_o i_o say_v that_o the_o foresay_a point_n doâ_n receive_v the_o angle_n of_o the_o icosahedron_n inscribe_v in_o the_o pyramid_n abâd_n in_o the_o foresay_a triangle_n let_v there_o again_o be_v make_v other_o triangle_n by_o couple_v the_o section_n and_o let_v those_o triangle_n be_v tr_n joh_n ceq_n and_o vxy_n which_o shall_v be_v equilater_n for_o every_o one_o of_o their_o side_n do_v subâââd_v equal_a angle_n of_o equilater_n triangle_n and_o those_o say_v equal_a angle_n be_v contain_v under_o equal_a sideâ_n namely_o under_o the_o great_a segment_n and_o the_o less_o â_o and_o therefore_o the_o side_n which_o subtend_v those_o angle_n be_v equal_a by_o the_o 4._o of_o the_o first_o now_o let_v we_o prove_v that_o at_o each_o of_o the_o foresay_a point_n as_o for_o example_n at_o t_o be_v set_v the_o solid_a angle_n of_o a_o icosahâdronâ_n demonstration_n forasmuch_o as_o the_o triangle_n trs_n and_o tqo_n be_v equilater_n and_o equal_a the_o 4._o right_a line_n tr_z it_o be_v tq_n and_o to_o shall_v be_v equal_a and_o forasmuch_o as_o âpnk_v be_v a_o square_n cut_v the_o pyramid_n abâd_v into_o two_o equal_a paââââ_n by_o the_o corollay_n of_o the_o second_o of_o this_o bookeâ_n the_o line_n th._n shall_v be_v in_o power_n duple_n to_o the_o line_n tn_n or_o nh_v by_o the_o 47._o of_o the_o first_o for_o the_o line_n tn_n or_o nh_v be_v equal_a for_o that_o by_o construction_n they_o be_v each_o less_o segment_n and_o the_o line_n rt_n or_o it_o be_v be_v in_o power_n duple_n to_o the_o same_o line_n tn_n or_o nh_v by_o the_o corollary_n of_o the_o 16._o of_o this_o book_n for_o it_o subtend_v the_o angle_n of_o the_o triangle_n contain_v under_o the_o two_o segment_n wherefore_o the_o line_n th._n it_o be_v tr_z tq_n and_o to_o be_v equal_a and_o so_o also_o be_v the_o line_n his_o sir_n rq_n qo_n and_o oh_o which_o subtend_v the_o angle_n at_o the_o point_n t_o equal_a for_o the_o line_n qr_o contain_v in_o power_n the_o two_o line_n pq_n and_o pr._n the_o less_o segment_n which_o two_o line_n the_o line_n th._n also_o contain_v in_o power_n and_o the_o rest_n of_o the_o line_n do_v subtend_v angle_n of_o equilater_n triangle_n contain_v under_o the_o great_a segment_n and_o the_o less_o wherefore_o the_o five_o triangle_n trs_n tsh_o tho_o toq_fw-la tqr_n be_v equilater_n and_o equal_a make_v the_o solid_a angle_n of_o a_o icosahedron_n at_o the_o point_n t_o by_o the_o 16._o of_o the_o thirteen_o in_o the_o side_n pn_n of_o the_o triangle_n p_o nm_o and_o by_o the_o same_o reason_n in_o the_o other_o side_n of_o the_o 4._o triangle_n pnm_n nkl_n fmk_n &_o lfp_n which_o be_v inscribe_v in_o the_o base_n of_o the_o pyramid_n which_o side_n be_v 12â_n in_o number_n shall_v be_v set_v 12._o angle_n of_o the_o icosahedron_n contain_v under_o 20._o equal_a &_o equilater_n triangle_n of_o which_o fowere_n be_v set_v in_o the_o 4._o base_n of_o the_o pyramid_n namely_o these_o four_o triangle_n trs_n hoi_n ceq_n vxy_n 4._o triangle_n be_v under_o 4._o angle_n of_o the_o pyramid_n that_o be_v the_o four_o triangle_n cix_o ysh_a erv_n tqo_n and_o under_o every_o one_o of_o the_o six_o side_n of_o the_o pyramid_n be_v set_v two_o triangle_n namely_o under_o the_o side_n of_o the_o triangle_n this_o and_o thoâ_n under_o the_o side_n db_o the_o triangle_n rqe_n and_o rqt_n under_o the_o side_n da_z the_o triangle_n coq_fw-la and_o coi_fw-fr under_o the_o side_n ab_fw-la the_o triangle_n exc_a and_o exuâ_n under_o the_o side_n bg_o the_o triangle_n sur_n and_o svy_n and_o under_o the_o side_n agnostus_n the_o triangle_n jyh_n and_o jyx._n wherefore_o the_o solid_a be_v contain_v under_o 20._o equilater_n and_o equal_a triangle_n shall_v be_v a_o icosahedron_n by_o the_o 23._o definition_n of_o the_o eleven_o and_o shall_v be_v inscribe_v in_o the_o pyramid_n abâd_v by_o the_o first_o definition_n of_o this_o book_n for_o all_o his_o angle_n do_v at_o one_o time_n touch_v the_o base_n of_o the_o pyramid_n wherefore_o in_o a_o trilater_n equilater_n pyramid_n give_v we_o have_v inscribe_v a_o icosahedron_n ¶_o the_o 20._o proposition_n the_o 20._o problem_n in_o a_o trilater_n equilater_n pyramid_n give_v to_o inscribe_v a_o dodecahedron_n svppose_v that_o the_o pyramid_n give_v be_v abgd_v âche_fw-mi of_o who_o side_n let_v be_v cut_v into_o two_o equal_a part_n and_o draw_v the_o line_n which_o couple_v the_o section_n which_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o right_a line_n be_v draw_v by_o the_o section_n shall_v receive_v 20._o triangle_n make_v a_o icosahedron_n as_o in_o the_o former_a proposition_n it_o be_v manifest_a now_o than_o if_o we_o take_v the_o centre_n of_o those_o triangle_n we_o shall_v there_o find_v the_o 20._o angle_n of_o the_o dodecahedron_n inscribe_v in_o it_o by_o the_o 5._o of_o this_o book_n and_o forasmuch_o as_o 4._o base_n of_o the_o foresay_a icosahedron_n be_v concentricall_a with_o the_o base_n of_o the_o pyramid_n as_o it_o be_v prove_v in_o the_o 2._o corollary_n of_o the_o 6._o of_o this_o book_n there_o shall_v be_v place_v 4â_o angle_n of_o the_o dodecahedron_n namely_o the_o 4._o angle_n e_o f_o h_o d_o in_o the_o 4._o centre_n of_o the_o base_n and_o of_o the_o other_o 16._o angle_n under_o every_o one_o of_o the_o 6._o side_n of_o the_o pyramid_n be_v subtend_v two_o namely_o under_o the_o side_n ad_fw-la the_o angle_n ck_o under_o the_o side_n bd_o the_o angle_n li_n under_o the_o
mark_v in_o your_o line_n what_o number_n the_o water_n cut_v take_v the_o weight_n of_o the_o same_o cube_n against_o in_o the_o same_o kind_n of_o water_n which_o you_o have_v before_o put_v that_o ãâã_d also_o into_o the_o pyramid_n or_o cone_n where_o you_o do_v put_v the_o first_o mark_v now_o again_o in_o what_o number_n or_o place_n of_o the_o line_n the_o water_n cut_v they_o two_o way_n you_o may_v conclude_v your_o purpose_n it_o be_v to_o weet_v either_o by_o number_n or_o line_n by_o number_n as_o if_o you_o divide_v the_o side_n of_o your_o fundamental_a cube_n into_o so_o many_o equal_a part_n as_o it_o be_v capable_a of_o convenient_o with_o your_o ease_n and_o preciseness_n of_o the_o division_n for_o as_o the_o number_n of_o your_o first_o and_o less_o line_n in_o your_o hollow_a pyramid_n or_o cone_n be_v to_o the_o second_o or_o great_a both_o be_v count_v from_o the_o vertex_fw-la so_o shall_v the_o number_n of_o the_o side_n of_o your_o fundamental_a cube_n be_v to_o the_o number_n belong_v to_o the_o radical_a side_n of_o the_o cube_n double_a to_o your_o fundamental_a cube_n which_o be_v multiply_v cubik_n wise_a will_v soon_o show_v itself_o whether_o it_o be_v double_a or_o no_o to_o the_o cubik_a number_n of_o your_o fundamental_a cube_n by_o line_n thus_o as_o your_o less_o and_o first_o line_n in_o your_o hollow_a pyramid_n or_o cone_n be_v to_o the_o second_o or_o great_a so_o let_v the_o radical_a side_n of_o your_o fundamental_a cube_n be_v to_o a_o four_o proportional_a line_n by_o the_o 12._o proposition_n of_o the_o six_o book_n of_o euclid_n which_o four_o line_n shall_v be_v the_o rote_n cubik_fw-mi or_o radical_a side_n of_o the_o cube_n double_a to_o your_o fundamental_a cube_n which_o be_v the_o thing_n we_o desire_v for_o this_o may_v i_o with_o joy_n say_v eyphka_n eyphka_n eyphka_n thank_v the_o holy_a and_o glorious_a trinity_n have_v great_a cause_n thereto_o then_o ensue_v archimedes_n have_v for_o find_v the_o fraud_n use_v in_o the_o king_n crown_n of_o gold_n as_o all_o man_n may_v easy_o judge_n by_o the_o diversity_n of_o the_o fruit_n follow_v of_o the_o one_o and_o the_o other_o where_o i_o speak_v before_o of_o a_o hollow_a cubik_fw-mi coffin_n the_o like_a use_n be_v of_o it_o and_o without_o weight_n thus._n fill_v it_o with_o water_n precise_o full_a and_o pour_v that_o water_n into_o your_o pyramid_n or_o cone_n and_o here_o note_v the_o line_n cut_v in_o your_o pyramid_n or_o cone_n again_o fill_v your_o coffin_n like_a as_o you_o do_v before_o put_v that_o water_n also_o to_o the_o firstâ_o mark_v the_o second_o cut_v of_o your_o line_n now_o as_o you_o proceed_v before_o so_o must_v you_o here_o proceed_v note_n and_o if_o the_o cube_n which_o you_o shall_v double_v be_v never_o so_o great_a you_o have_v thus_o the_o proportion_n in_o small_a between_o your_o two_o little_a cube_n and_o then_o the_o side_n of_o that_o great_a cube_n to_o be_v double_v be_v the_o three_o will_v have_v the_o four_o find_v to_o it_o proportional_a by_o the_o 12._o of_o the_o six_o of_o euâlide_n water_n note_v that_o all_o this_o while_n i_o forget_v not_o my_o first_o proposition_n statical_a here_o rehearse_v that_o the_o supersicy_n of_o the_o water_n be_v spherical_a wherein_o use_v your_o discretion_n to_o the_o first_o line_n add_v a_o small_a hear_v breadth_n more_o and_o to_o the_o second_o half_o a_o hear_v breadth_n more_o to_o his_o length_n for_o you_o will_v easy_o perceive_v that_o the_o difference_n can_v be_v no_o great_a in_o any_o pyramid_n or_o cone_n of_o you_o to_o be_v handle_v which_o you_o shall_v thus_o try_v for_o âinding_v the_o swell_n of_o the_o water_n above_o level_a square_n the_o semidiameter_n â_o from_o the_o centre_n of_o the_o earth_n to_o your_o first_o water_n superficies_n square_n then_o half_o the_o subtendent_fw-la of_o that_o watery_a superficies_n which_o subtendent_fw-la must_v have_v the_o equal_a part_n of_o his_o measure_n all_o one_o 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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 construâtiââ_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
diameter_n be_v double_a to_o that_o square_n who_o diameter_n it_o be_v corollary_n the_o 34._o theorem_a the_o 48._o proposition_n if_o the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n of_o a_o triangle_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o two_o other_o side_n of_o the_o same_o triangle_n the_o angle_n comprehend_v under_o those_o two_o other_o side_n be_v a_o right_a angle_n svppose_v that_o abc_n be_v a_o triangle_n and_o let_v the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n there_o namely_o of_o the_o side_n bc_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o side_n basilius_n and_o ac_fw-la then_o i_o say_v that_o the_o angle_n bac_n be_v a_o right_a angle_n raise_v up_o by_o the_o 11._o proposition_n from_o the_o point_n a_o unto_o the_o right_a line_n ac_fw-la a_o perpendicular_a line_n ad._n and_o by_o the_o third_o proposition_n unto_o the_o line_n ab_fw-la put_v a_o equal_a line_n ad._n and_o by_o 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side_n bc._n and_o forasmuch_o as_o ab_fw-la be_v equal_a to_o ad_fw-la ând_n ac_fw-la be_v common_a to_o they_o both_o therefore_o these_o two_o side_n dam_n and_o ac_fw-la be_v equal_a to_o these_o two_o side_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 to_o the_o base_a be_v wherefore_o by_o the_o 8._o proposition_n the_o angle_n dac_o be_v equal_a to_o the_o angle_n bac_n but_o the_o angle_n dac_o be_v a_o right_a angle_n wherefore_o also_o the_o angle_n bac_n be_v a_o right_a angle_n if_o therefore_o the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n of_o a_o triangle_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o two_o other_o side_n of_o the_o same_o triangle_n the_o angle_n comprehend_v under_o those_o two_o other_o side_n be_v a_o right_a angle_n which_o be_v require_v to_o be_v prove_v former_a this_o proposition_n be_v the_o converse_n of_o the_o former_a and_o be_v of_o pelitarius_n 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produce_v a_o plain_a and_o right_a angle_a superficial_a number_n as_o you_o see_v in_o the_o figure_n here_o set_v where_o the_o number_n of_o six_o or_o six_o unity_n be_v multiply_v by_o the_o number_n of_o five_o or_o by_o five_o unity_n of_o which_o multiplication_n be_v produce_v 30._o which_o number_n be_v set_v down_o and_o describe_v by_o his_o unity_n represent_v a_o plain_n and_o a_o right_a angle_a number_n wherefore_o even_o as_o equal_a number_n multiple_v by_o equal_a number_n produce_v number_n equal_v the_o one_o to_o the_o other_o so_o rectangle_n parallelogram_n which_o be_v comprehend_v under_o equal_a line_n be_v equal_a the_o one_o to_o the_o other_o definition_n 2._o in_o every_o parallelogram_n one_o of_o those_o parallelogram_n which_o soever_o it_o be_v which_o be_v about_o the_o diameter_n together_o with_o the_o two_o supplement_n be_v call_v a_o gnomon_n those_o particular_a parallelogram_n be_v say_v to_o be_v about_o the_o diameter_n of_o the_o parallelogram_n which_o have_v the_o same_o diameter_n which_o the_o whole_a parallelogram_n have_v and_o supplement_n be_v such_o which_o be_v without_o the_o diameter_n of_o the_o whole_a parallelogram_n as_o of_o the_o parallelogram_n abcd_o the_o partial_a or_o particular_a parallelogram_n gkcf_n and_o ebkh_n be_v parallelogram_n about_o the_o diameter_n for_o that_o each_o of_o they_o have_v for_o his_o diameter_n a_o part_n of_o the_o diameter_n of_o the_o whole_a parallelogram_n as_o ck_v and_o kb_v the_o particular_a diameter_n be_v part_n of_o the_o line_n cb_o which_o be_v the_o diameter_n of_o the_o whole_a parallelogram_n and_o the_o two_o parallelogram_n aegk_n and_o khfd_n be_v supplement_n because_o they_o be_v without_o the_o diameter_n of_o the_o whole_a parallelogram_n namely_o cb._n now_o any_o one_o of_o those_o partial_a parallelogram_n
ae_n or_o ec_o and_o multiply_v 4._o into_o himself_o and_o there_o be_v produce_v 16_o which_o add_v unto_o 64_o and_o there_o shall_v be_v produce_v 80_o who_o root_n be_v â_o â_o 80_o which_o be_v the_o line_n ebb_n or_o the_o line_n of_o by_o the_o 47._o of_o the_o first_o and_o forasmuch_o as_o the_o line_n of_o be_v â_o â_o 80._o &_o the_o line_n ea_fw-la be_v 4._o therefore_o the_o line_n of_o be_v â_o â_o 80_o 4._o and_o so_o much_o shall_v the_o line_n ah_o be_v and_o the_o line_n bh_o shall_v be_v 8_o â_o â_o 80_o 4_o that_o be_v 12_o â_o â_o 80._o now_o they_o 12_o â_o â_o 80_o multiply_v into_o 8_o shall_v be_v as_o much_o as_o â_o â_o 80_o 4._o multiply_v into_o itself_o for_o of_o either_o of_o they_o be_v produce_v 96_o â_o 5120._o the_o 11._o theorem_a the_o 12._o proposition_n in_o obtuseangle_v triangle_n the_o square_n which_o be_v make_v of_o the_o side_n subtend_v the_o obtuse_a angle_n be_v great_a than_o the_o square_n which_o be_v make_v of_o the_o side_n which_o comprehend_v the_o obtuse_a angle_n by_o the_o rectangle_n figure_n which_o be_v comprehend_v twice_o under_o one_o of_o those_o side_n which_o be_v about_o the_o obtuse_a angle_n upon_o which_o be_v produce_v fall_v a_o perpendicular_a line_n and_o that_o which_o be_v outward_o take_v between_o the_o perpendicular_a line_n and_o the_o obtuse_a angle_n svppose_v that_o abc_n be_v a_o obtuseangle_n triangle_n have_v the_o angle_n bac_n obtuse_a and_o from_o the_o point_n b_o by_o the_o 12._o of_o the_o first_o draw_v a_o perpendicular_a line_n unto_o ca_n produce_v and_o let_v the_o same_o be_v bd._n then_o i_o say_v that_o the_o square_n which_o be_v make_v of_o the_o side_n bc_o be_v great_a than_o the_o square_n which_o be_v make_v of_o the_o side_n basilius_n and_o ac_fw-la by_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n ca_n and_o ad_fw-la twice_o demonstration_n for_o forasmuch_o as_o the_o right_a line_n cd_o be_v by_o chance_n divide_v in_o the_o point_n a_o therefore_o by_o the_o 4._o of_o the_o second_o the_o square_a which_o be_v make_v of_o cd_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o ca_n and_o ad_fw-la and_o unto_o the_o rectangle_n figure_n contain_v under_o ca_n and_o ad_fw-la twice_o put_v the_o square_n which_o be_v make_v of_o db_o common_a unto_o they_o both_o wherefore_o the_o square_n which_o be_v make_v of_o cd_o and_o db_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n ca_n ad_fw-la and_o db_o and_o unto_o the_o rectangle_n figure_n contain_v under_o the_o line_n ca_n and_o ad_fw-la twice_o but_o by_o the_o 47._o of_o the_o first_o the_o square_a which_o be_v make_v of_o cb_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n cd_o and_o db._n for_o the_o angle_n at_o the_o point_n d_o be_v a_o right_a angle_n and_o unto_o the_o square_n which_o be_v make_v of_o ad_fw-la and_o db_o by_o the_o self_n same_o be_v equal_a the_o square_n which_o be_v mâde_v of_o ab_fw-la wherefore_o the_o square_n which_o be_v make_v of_o cb_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o ca_n and_o ab_fw-la and_o unto_o the_o rectangle_n figure_n contain_v under_o the_o line_n ca_n and_o ad_fw-la twice_o wherefore_o the_o square_v which_o be_v make_v of_o cb_o be_v great_a than_o the_o square_n which_o be_v make_v of_o ca_n and_o ab_fw-la by_o the_o rectangle_n figure_n contain_v under_o the_o line_n ca_n and_o ad_fw-la twice_o in_o obtuseangle_v triangle_n therefore_o the_o square_n which_o be_v make_v of_o the_o side_n subtend_v the_o obtuse_a angle_n be_v great_a than_o the_o square_n which_o be_v make_v of_o the_o side_n which_o comprehend_v the_o obtuse_a angle_n by_o the_o rectangle_n figure_n which_o be_v comprehend_v twice_o under_o one_o of_o those_o side_n which_o be_v about_o the_o obtuse_a angle_n upon_o which_o be_v produce_v fall_v a_o perpendicular_a line_n and_o that_o which_o be_v outward_o take_v between_o the_o perpendicular_a line_n and_o the_o obtuse_a angle_n which_o be_v require_v to_o be_v demonstrate_v of_o what_o force_n this_o proposition_n and_o the_o proposition_n follow_v touch_v the_o measure_n of_o the_o obtuseangle_n triangle_n and_o the_o acuteangle_a triangle_n with_o the_o aid_n of_o the_o 47._o proposition_n of_o the_o first_o book_n touch_v the_o rightangle_n triangle_n he_o shall_v well_o perceive_v which_o shall_v at_o any_o time_n need_v the_o art_n of_o triangle_n in_o which_o by_o three_o thing_n know_v be_v ever_o search_v out_o three_o other_o thing_n unknown_a by_o help_n of_o the_o table_n of_o arke_n and_o cord_n the_o 12._o theorem_a the_o 13._o proposition_n in_o acuteangle_v triangle_n the_o square_n which_o be_v make_v of_o the_o side_n that_o subtend_v the_o acute_a angle_n be_v less_o than_o the_o square_n which_o be_v make_v of_o the_o side_n which_o comprehend_v the_o acute_a angle_n by_o the_o rectangle_n figure_n which_o be_v comprehend_v twice_o under_o one_o of_o those_o side_n which_o be_v about_o the_o acuteangle_n upon_o which_o fall_v a_o perpendicular_a line_n and_o that_o which_o be_v inward_o take_v between_o the_o perpendicular_a line_n and_o the_o acute_a angle_n svppose_v that_o abc_n be_v a_o acuteangle_n triangle_n have_v the_o angle_n at_o the_o point_fw-fr b_o acute_a &_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n a_o draw_v unto_o the_o line_n bc_o a_o perpendicular_a line_n ad._n then_o i_o say_v that_o the_o square_n which_o be_v make_v of_o the_o line_n ac_fw-la be_v less_o than_o the_o square_n which_o be_v make_v of_o the_o line_n cb_o and_o basilius_n by_o the_o rectangle_n figure_n contain_v under_o the_o line_n cb_o and_o bd_o twice_o demonstration_n for_o forasmuch_o as_o the_o right_a line_n bc_o be_v by_o chance_n divide_v in_o the_o point_n d_o therefore_o by_o the_o 7._o of_o the_o second_o the_o square_n which_o be_v make_v of_o the_o line_n cb_o and_o bd_o be_v equal_a to_o the_o rectangle_n figure_n contain_v under_o the_o line_n cb_o and_o db_o twice_o and_o unto_o the_o square_n which_o be_v make_v of_o line_n cd_o put_v the_o square_n which_o be_v make_v of_o the_o line_n dam_fw-ge common_a unto_o they_o both_o wherefore_o the_o square_n which_o be_v make_v of_o the_o line_n cb_o bd_o and_o da_z be_v equal_a unto_o the_o rectangle_n figure_n contain_v under_o the_o line_n cb_o and_o bd_o twice_o and_o unto_o the_o square_n which_o be_v make_v of_o ad_fw-la and_o dc_o but_o to_o the_o square_n which_o be_v make_v of_o the_o line_n bd_o and_o da_z be_v equal_a the_o square_v which_o be_v make_v of_o the_o line_n ab_fw-la for_o th'angle_v at_o the_o point_fw-fr d_o be_v a_o right_a angle_n and_o unto_o the_o square_n which_o be_v make_v of_o the_o line_n ad_fw-la and_o dc_o be_v equal_a the_o square_n which_o be_v make_v of_o the_o line_n ac_fw-la by_o the_o 47._o of_o the_o first_o wherefore_o the_o square_n which_o be_v make_v of_o the_o line_n cb_o and_o basilius_n 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 and_o to_o that_o which_o be_v contain_v under_o the_o line_n cb_o and_o bd_o twice_o wherefore_o the_o square_n which_o be_v make_v of_o the_o line_n ac_fw-la be_v take_v alone_o be_v less_o than_o the_o square_n which_o be_v make_v of_o the_o line_n cb_o and_o basilius_n by_o the_o rectangle_n figure_n which_o be_v contain_v under_o the_o line_n cb_o and_o bd_o twice_o in_o rectangle_n triangle_n therefore_o the_o square_n which_o be_v make_v of_o the_o side_n that_o subtend_v the_o acute_a angle_n be_v less_o than_o the_o square_n which_o be_v make_v of_o the_o side_n which_o comprehend_v the_o acute_a angle_n by_o the_o rectangle_n figure_n which_o be_v comprehend_v twice_o under_o one_o of_o those_o side_n which_o be_v about_o the_o acute_a angle_n upon_o which_o fall_v a_o perpendicular_a line_n and_o that_o which_o be_v inward_o take_v between_o the_o perpendicular_a line_n and_o the_o acute_a angle_n which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o orontius_n corollary_n hereby_o be_v easy_o gather_v that_o such_o a_o perpendicular_a line_n in_o rectangle_n triangle_n fall_v of_o necessity_n upon_o the_o side_n of_o the_o triangle_n that_o be_v neither_o within_o the_o triangle_n nor_o without_o but_o in_o obtuseangle_v triangle_n it_o fall_v without_o and_o in_o acuteangle_v triangle_n within_o for_o the_o perpendicular_a line_n in_o obtuseangle_a triangle_n and_o acuteangle_v triangle_n can_v not_o exact_o agree_v with_o the_o side_n of_o the_o triangle_n for_o then_o a_o obtuse_a &_o a_o acuteangle_n shall_v be_v equal_a to_o a_o right_a angle_n contrary_a to_o the_o eleven_o and_o twelve_o definition_n of_o the_o first_o book_n likewise_o in_o obtuseangle_a
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 ãâ¦ã_o the_o âââââference_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
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 centre_n be_v always_o great_a than_o that_o line_n which_o be_v more_o distant_a which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n after_o campane_n in_o the_o circle_n abcd_o who_o centre_n let_v be_v the_o point_n e_o draw_v these_o line_n ab_fw-la ac_fw-la ad_fw-la fg_o and_o hk_o of_o which_o let_v the_o line_n ad_fw-la be_v the_o diameter_n of_o the_o circle_n campane_n then_o i_o say_v that_o the_o line_n ad_fw-la be_v the_o great_a of_o all_o the_o line_n and_o the_o other_o line_n each_o of_o the_o one_o be_v so_o much_o great_a than_o each_o of_o the_o other_o how_o much_o nigh_o it_o be_v unto_o the_o centre_n join_v together_o the_o end_n of_o all_o these_o line_n with_o the_o centre_n by_o draw_v these_o right_a line_n ebb_v ec_o eglantine_n eke_o eh_o and_o ef._n and_o by_o the_o 20._o of_o the_o first_o the_o two_o side_n of_o and_o eglantine_n of_o the_o triangle_n efg_o shall_v be_v great_a than_o the_o three_o side_n fg._n and_o for_o asmuch_o as_o the_o say_a side_n of_o &_o eglantine_n be_v equal_a to_o the_o line_n ad_fw-la by_o the_o definition_n of_o a_o circle_n therefore_o the_o line_n ad_fw-la be_v great_a than_o the_o line_n fg._n and_o by_o the_o same_o reason_n it_o be_v great_a than_o every_o one_o of_o the_o rest_n of_o the_o line_n if_o they_o be_v put_v to_o be_v base_n of_o triangle_n for_o that_o every_o two_o side_n draw_v from_o the_o centre_n be_v equal_a to_o the_o line_n ad._n which_o be_v the_o first_o part_n of_o the_o proposition_n again_o for_o asmuch_o as_o the_o two_o side_n of_o and_o eglantine_n of_o the_o triangle_n efg_o be_v equal_a to_o the_o
5._o and_o 32._o of_o the_o first_o the_o angle_n bfc_n also_o shall_v be_v the_o three_o part_n of_o two_o right_a angle_n wherefore_o either_o of_o the_o two_o angle_n remain_v fbc_n and_o fcb_n for_o asmuch_o as_o they_o be_v equal_a by_o the_o 5._o of_o the_o first_o shall_v be_v two_o three_o part_n of_o two_o right_a angle_n by_o the_o 32._o of_o the_o same_o or_o by_o the_o 4._o of_o the_o first_o forasmuch_o as_o the_o angle_n bfc_n be_v equal_a to_o the_o angle_n fba_n and_o the_o two_o side_n fb_o and_o fc_o be_v equal_a to_o the_o two_o side_n ab_fw-la and_o bf_o the_o base_a bc_o shall_v be_v equal_a to_o the_o base_a bf_o and_o therefore_o be_v equal_a to_o the_o line_n fc_o wherefore_o the_o triangle_n fbc_n be_v equilater_n and_o equiangle_n last_o make_v the_o angle_n cfd_v equal_a to_o either_o of_o the_o angle_n at_o the_o point_n fletcher_n by_o draw_v the_o line_n fd._n and_o draw_v a_o line_n from_o c_o to_o d._n now_o then_o by_o the_o former_a reason_n the_o triangle_n fcd_n shall_v be_v equilater_n and_o equiangle_n and_o for_o asmuch_o as_o the_o three_o angle_n at_o the_o point_n f_o be_v equal_a to_o two_o right_a angle_n for_o each_o of_o they_o be_v the_o three_o part_n of_o two_o right_a angle_n therefore_o by_o the_o 14._o of_o the_o first_o ad_fw-la be_n one_o right_a line_n and_o for_o that_o cause_n be_v the_o diameter_n of_o the_o circle_n wherefore_o if_o the_o other_o semicircle_n afd_v be_v divide_v into_o so_o many_o equal_a part_n as_o the_o semicircle_n abcd_o be_v divide_v into_o it_o shall_v comprehend_v so_o many_o equal_a line_n subtend_v unto_o it_o wherefore_o the_o line_n ab_fw-la be_v the_o side_n of_o a_o equilater_n hexagon_n figure_n to_o be_v inscribe_v in_o the_o circle_n which_o hexagon_n figure_n also_o shall_v be_v equiangle_n for_o the_o half_a of_o the_o whole_a angle_n b_o be_v equal_a to_o the_o half_a of_o the_o whole_a angle_n c_o which_o be_v require_v to_o be_v do_v now_o than_o if_o we_o draw_v from_o the_o centre_n f_o a_o perpendicular_a line_n unto_o ad_fw-la which_o let_v be_v fe_o and_o draw_v also_o these_o right_a line_n be_v and_o ce_fw-fr there_o shall_v be_v describe_v a_o triangle_n bec_n who_o angle_n e_o which_o be_v at_o the_o top_n shall_v be_v the_o 6._o part_n of_o two_o right_a angle_n by_o the_o 20._o of_o the_o three_o for_o the_o angle_n bfc_n be_v double_a unto_o it_o and_o either_o of_o the_o two_o angle_n at_o the_o base_a namely_o the_o angle_n ebc_n and_o ecb_n be_v dupla_fw-la sesquialter_fw-la to_o the_o angle_n e_o that_o be_v either_o of_o they_o contain_v the_o angle_n e_o twice_o and_o half_o the_o angle_n e._n and_o by_o this_o reason_n be_v find_v out_o the_o side_n of_o a_o hexagon_n figure_n correlary_a hereby_o it_o be_v manifest_a that_o the_o side_n of_o a_o hexagon_n figure_n describe_v in_o a_o circle_n be_v equal_a to_o a_o right_a line_n draw_v from_o the_o centre_n of_o the_o say_a circle_n unto_o the_o circumference_n and_o if_o by_o the_o point_n a_o b_o c_o d_o e_o f_o be_v draw_v right_a line_n touch_v the_o circle_n then_o shall_v there_o be_v describe_v about_o the_o circle_n a_o hexagon_n figure_n equilater_n and_o equiangle_n which_o may_v be_v demonstrate_v by_o that_o which_o have_v be_v speak_v of_o the_o describe_v of_o a_o pentagon_n about_o a_o circle_n and_o moreover_o by_o those_o thing_n which_o have_v be_v speak_v of_o pentagons_n we_o may_v in_o a_o hexagon_n give_v either_o describe_v or_o circumscribe_v a_o circle_n which_o be_v require_v to_o be_v do_v the_o 16._o problem_n the_o 16._o proposition_n in_o a_o circle_n give_v to_o describe_v a_o quindecagon_n or_o figure_n of_o fifteen_o angle_n equilater_n and_o equiangle_n svppose_v that_o the_o circle_n give_v be_v abcd._n it_o be_v require_v in_o the_o circle_n abcd_o to_o describe_v a_o figure_n of_o fifteen_o angle_n consist_v of_o equal_a side_n and_o of_o equal_a angle_n describe_v in_o the_o circle_n abcd_o the_o side_n of_o a_o equilater_n triangle_n and_o let_v the_o same_o be_v ac_fw-la and_o in_o the_o ark_n ac_fw-la describe_v the_o side_n of_o a_o equilater_n pentagon_n and_o let_v the_o same_o be_v ab_fw-la construction_n now_o then_o of_o such_o equal_a part_n whereof_o the_o whole_a circle_n abcd_o contain_v fifteen_o of_o such_o part_n i_o say_v the_o circumference_n abc_n be_v the_o three_o part_n of_o the_o circle_n shall_v contain_v five_o and_o the_o circumference_n ab_fw-la be_v the_o five_o part_n of_o a_o circle_n shall_v contain_v three_o wherefore_o the_o residue_n bc_o shall_v contain_v two_o divide_v by_o the_o 30._o of_o the_o first_o the_o ark_n bc_o into_o two_o equal_a part_n in_o the_o point_n e._n demonstration_n wherefore_o either_o of_o these_o circumference_n be_v &_o ec_o be_v the_o fifteen_o part_n of_o the_o circle_n abcd._n if_o therefore_o there_o be_v draw_v right_a line_n from_o b_o to_o e_o and_o from_o e_o to_o c_z and_o then_o begin_v at_o the_o point_n b_o or_o at_o the_o point_n c_o there_o be_v apply_v into_o the_o circle_n abcd_o right_a line_n equal_a unto_o ebb_n or_o ec_o and_o so_o continue_v till_o you_o come_v to_o the_o point_n c_o if_o you_o begin_v at_o b_o or_o to_o the_o point_fw-fr b_o if_o you_o begin_v at_o c_o and_o there_o shall_v be_v describe_v in_o the_o circle_n abcd_o a_o figure_n of_o fifteen_o angle_n equilater_n and_o equiangle_n which_o be_v require_v to_o be_v do_v and_o in_o like_a sort_n as_o in_o a_o pentagon_n if_o by_o the_o point_n where_o the_o circle_n be_v divide_v be_v draw_v right_a line_n touch_v the_o circle_n in_o the_o say_a point_n there_o shall_v be_v describe_v about_o the_o circle_n a_o figure_n of_o fifteen_o angle_n equilater_n &_o equiangle_n and_o in_o like_a sort_n by_o the_o self_n same_o observation_n that_o be_v in_o pentagons_n we_o may_v in_o a_o figure_n of_o fifteen_o angle_n give_v be_v equilater_n and_o equiangle_n either_o inscribe_v or_o circumscribe_v a_o circle_n flussates_n ¶_o a_o addition_n of_o flussates_n to_o find_v out_o infinite_a figure_n of_o many_o angle_n if_o into_o a_o circle_n from_o one_o point_n be_v apply_v the_o side_n of_o two_o side_n poligonon_n figure_n the_o excess_n of_o the_o great_a ark_n above_o the_o less_o shall_v comprehend_v a_o ark_n contain_v so_o many_o side_n of_o the_o poligonon_n figure_n to_o be_v inscribe_v by_o how_o many_o unity_n the_o denomination_n of_o the_o poligonon_n figure_n of_o the_o less_o side_n exceed_v the_o denomination_n of_o the_o poligonon_n figure_n of_o the_o great_a side_n and_o the_o number_n of_o the_o side_n of_o the_o poligonon_n figure_n to_o be_v inscribe_v be_v produce_v of_o the_o multiplication_n of_o the_o denomination_n of_o the_o foresay_a poligonon_n figure_v the_o one_o into_o the_o other_o as_o for_o example_n suppose_v that_o into_o the_o circle_n abe_n be_v apply_v the_o side_n of_o a_o equilater_n and_o equiangle_n hexagon_n figure_n by_o the_o 15._o of_o this_o book_n which_o let_v be_v ab_fw-la and_o likewise_o the_o side_n of_o a_o pentagon_n by_o the_o 11._o of_o this_o book_n which_o let_v be_v ac_fw-la and_o the_o side_n of_o a_o square_n by_o the_o 6._o of_o this_o book_n which_o let_v be_v ad_fw-la and_o the_o side_n of_o a_o equilater_n triangle_n by_o the_o 2._o of_o this_o book_n which_o let_v be_v ae_n then_o i_o say_v that_o the_o excess_n of_o the_o ark_n ad_fw-la above_o the_o ark_n ab_fw-la which_o excess_n be_v the_o ark_n bd_o contain_v so_o many_o side_n of_o the_o poligonon_n figure_n to_o be_v inscribe_v of_o how_o many_o unity_n the_o denominator_fw-la of_o the_o hexagon_n ab_fw-la which_o be_v six_o exceed_v the_o denominator_fw-la of_o the_o square_a ad_fw-la which_o be_v four_o and_o forasmuch_o as_o that_o excess_n it_o two_o unity_n therefore_o in_o bd_o there_o shall_v be_v two_o side_n and_o the_o denominator_fw-la of_o the_o poligonon_n figure_n which_o be_v to_o be_v inscribe_v shall_v be_v produce_v of_o the_o multiplication_n of_o the_o denominator_n of_o the_o foresay_a poligonon_n figure_n namely_o of_o the_o multiplication_n of_o 6._o into_o 4._o which_o make_v 24._o which_o number_n be_v the_o denominator_fw-la of_o the_o poligonon_n figure_n who_o two_o side_n shall_v subtend_v the_o ark_n bd._n for_o of_o such_o equal_a part_n whereof_o the_o whole_a circumference_n contain_v 24_o of_o such_o part_n i_o say_v the_o circumference_n ab_fw-la contain_v 4_o and_o the_o circumference_n ad_fw-la contain_v 6._o wherefore_o if_o from_o ad_fw-la which_o subtend_v 6._o part_n be_v take_v away_o 4._o which_o ab_fw-la subtend_v there_o shall_v remain_v unto_o bd_o two_o of_o such_o part_n of_o which_o the_o whole_a contain_v 24._o wherefore_o of_o a_o hexagon_n and_o a_o square_n be_v make_v a_o
the_o denomination_n of_o they_o both_o the_o denomination_n of_o sextupla_fw-la proportion_n be_v 6_o the_o denomination_n of_o dupla_fw-la proportion_n be_v 2._o now_o divide_v 6._o the_o denomination_n of_o the_o one_o by_o 2._o the_o denomination_n of_o the_o other_o the_o quotient_n shall_v be_v 3_o which_o be_v the_o denomination_n of_o a_o new_a proportion_n namely_o tripla_fw-la so_o that_o when_o dupla_fw-la proportion_n be_v subtrahe_v from_o sextupla_fw-la there_o shall_v remain_v tripla_fw-la proportion_n and_o thus_o may_v you_o do_v in_o all_o other_o 6._o a_o parallelogram_n apply_v to_o a_o right_a line_n be_v say_v to_o want_v in_o form_n by_o a_o parallelogram_n like_a to_o one_o give_v when_o the_o parallelogram_n apply_v want_v to_o the_o fill_n of_o the_o whole_a line_n by_o a_o parallelogram_n like_a to_o one_o give_v definition_n and_o then_o be_v it_o say_v to_o exceed_v when_o it_o exceed_v the_o line_n by_o a_o parallelogram_n like_a to_o that_o which_o be_v give_v as_o let_v e_o be_v a_o parallelogramme_n give_v and_o let_v ab_fw-la be_v a_o right_a line_n to_o who_o be_v apply_v the_o parallelogram_n acdf_n now_o if_o it_o want_v of_o the_o fill_n of_o the_o line_n ab_fw-la by_o the_o parallelogram_n dfgb_n be_v like_a to_o the_o parallelogram_n give_v e_o then_o be_v the_o parallelogram_n say_v to_o want_v in_o form_n by_o a_o parallelogram_n like_a unto_o a_o parallelogram_n give_v likewise_o if_o it_o exceed_v as_o the_o parallelogram_n acgd_v apply_v to_o the_o linâ_n abâ_n if_o it_o exceed_v it_o by_o the_o parallelogram_n fgbd_v be_v like_o to_o the_o parallelogram_n fletcher_n which_o be_v give_v then_o be_v the_o parallelogram_n abgd_v say_v to_o exceed_v in_o form_n by_o a_o parallelogram_n like_a to_o a_o parallelogram_n give_v this_o definition_n be_v add_v by_o flussates_n as_o it_o seem_v it_o be_v not_o in_o any_o common_a greek_a book_n abroad_o nor_o in_o any_o commentary_n it_o be_v for_o many_o theorem_n follow_v very_o necessary_a the_o 1._o theorem_a the_o 1._o proposition_n triangle_n &_o parallelogram_n which_o be_v under_o one_o &_o the_o self_n same_o altitude_n be_v in_o proportion_n as_o the_o base_a of_o the_o one_o be_v to_o the_o base_a of_o the_o other_o and_o forasmuch_o as_o the_o line_n cb_o bg_o and_o gh_o be_v equal_a the_o one_o to_o the_o other_o part_n therefore_o the_o triangle_n also_o ahg_n agb_n and_o abc_n be_v by_o the_o 38._o of_o the_o first_o equal_a the_o one_o to_o the_o other_o wherefore_o how_o multiplex_n the_o base_a hc_n be_v to_o the_o base_a bc_o so_o multiplex_n also_o be_v the_o triangle_n ahc_n to_o the_o triangle_n abc_n and_o by_o the_o same_o reason_n also_o how_o multiplex_n the_o base_a lc_n be_v to_o the_o base_a dc_o so_o multiplex_n also_o be_v the_o triangle_n alc_n to_o the_o triangle_n adc_o wherefore_o if_o the_o base_a hc_n be_v equal_a unto_o the_o base_a cl_n then_o by_o the_o 38._o of_o the_o first_o the_o triangle_n ahc_n be_v equal_a unto_o the_o triangle_n acl_n and_o if_o the_o base_a hc_n exceed_v the_o base_a cl_n then_o also_o the_o triangle_n ahc_n exceed_v the_o triangle_n acl_n and_o if_o the_o base_a be_v less_o the_o triangle_n also_o shall_v be_v less_o now_o then_o there_o be_v four_o magnitude_n namely_o the_o two_o basâs_n bc_o and_o cd_o and_o the_o two_o triangle_n abc_n and_o acd_o and_o to_o the_o base_a bc_o and_o to_o the_o triangle_n abc_n namely_o to_o the_o first_o and_o the_o three_o be_v take_v equemulâiplices_n namely_o the_o base_a hc_n and_o the_o triangle_n ahc_n and_o likewise_o to_o the_o base_a cd_o and_o to_o the_o triangle_n adc_o namely_o to_o the_o second_o and_o the_o four_o be_v take_v certain_a other_o equemultiplices_fw-la that_o be_v the_o base_a cl_n and_o the_o triangle_n alc_n and_o it_o have_v be_v prove_v that_o if_o the_o multiplex_n of_o the_o first_o magnitude_n that_o be_v the_o base_a hc_n do_v exceed_v the_o multiplex_n of_o the_o second_o that_o be_v the_o base_a cl_n the_o multiplex_n also_o of_o the_o three_o that_o be_v the_o triangle_n ahc_n exceed_v the_o multiplex_n of_o the_o â_z that_o be_v the_o triangle_n alc_n and_o if_o the_o say_v base_a hc_n be_v equal_a to_o the_o say_a baââ_n cl_n the_o triangle_n also_o ahc_n be_v equal_a to_o the_o triangle_n alc_n and_o if_o it_o be_v less_o it_o iâ_z less_o wherefore_o by_o the_o six_o definition_n of_o the_o five_o as_o the_o first_o of_o the_o foresay_a magnitude_n be_v to_o the_o second_o so_o be_v the_o three_o to_o the_o four_o wherefore_o as_o the_o base_a bc_o be_v to_o the_o base_a cd_o so_o be_v the_o triangle_n abc_n to_o the_o triangle_n acd_o and_o because_o by_o the_o 41._o of_o the_o first_o the_o parallelogram_n ec_o be_v double_a to_o the_o triangle_n abc_n part_n and_o by_o the_o same_o the_o parallelogram_n fc_o be_v double_a to_o the_o triangle_n acd_o therefore_o the_o parallelogram_n ec_o and_o fc_fw-la be_v equemultiplices_fw-la unto_o the_o triangle_n abc_n and_o acd_o but_o the_o part_n of_o equemultiplices_fw-la by_o the_o 15._o of_o the_o five_o have_v one_o and_o the_o same_o proportion_n with_o theiâ_z equemultiplices_fw-la wherefore_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n acd_o so_o be_v the_o parallelogram_n ec_o to_o the_o parallelogram_n fc_o and_o forasmuch_o as_o it_o have_v be_v demonstrate_v that_o as_o the_o base_a bc_o be_v to_o the_o base_a cd_o so_o be_v the_o triangle_n abc_n to_o the_o triangle_n acd_o and_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n acd_v so_o be_v the_o parallelogram_n ec_o to_o the_o parallelogram_n fc_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o base_a bc_o be_v to_o the_o base_a cd_o so_o be_v the_o parallelogram_n ec_o to_o the_o parallelogram_n fc_o the_o parallelogram_n may_v also_o be_v demonstrate_v a_o part_n by_o themselves_o as_o the_o triangle_n be_v if_o we_o describe_v upon_o the_o base_n bg_o gh_o and_o dk_o &_o kl_o parallelogram_n under_o the_o self_n same_o altitude_n that_o the_o parallelogrammeâ_n give_v be_v wherefore_o triangle_n and_o parallelogram_n which_o be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o proportion_n as_o the_o base_a of_o the_o one_o be_v to_o the_o base_a of_o the_o other_o which_o be_v require_v to_o be_v demonstrate_v here_o flussates_n add_v this_o corollary_n if_o two_o right_a line_n be_v give_v the_o one_o of_o they_o be_v divide_v how_o so_o ever_o flussates_n the_o rectangle_n figure_v contain_v under_o the_o whole_a line_n undivided_a and_o each_o of_o the_o segment_n of_o the_o line_n divide_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o segment_n be_v the_o one_o to_o the_o other_o for_o imagine_v the_o figure_n basilius_n and_o ad_fw-la in_o the_o former_a description_n to_o be_v rectangle_v the_o rectangle_n figure_v contain_v under_o the_o whole_a right_a line_n ac_fw-la and_o the_o segment_n of_o the_o right_a line_n bd_o which_o be_v cuâ_n in_o the_o point_n c_o namely_o the_o parallelogram_n basilius_n and_o ad_fw-la be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o segmente_n bc_o and_o cd_o be_v the_o 2._o theorem_a the_o 2._o proposition_n if_o to_o any_o one_o of_o the_o side_n of_o a_o triangle_n be_v draw_v a_o parallel_n right_a line_n it_o shall_v cut_v the_o side_n of_o the_o same_o triangle_n proportional_o and_o if_o the_o side_n of_o a_o triangle_n be_v cut_v proportional_o a_o right_a line_n draw_v from_o section_n to_o section_n be_v a_o parallel_n to_o the_o other_o side_n of_o the_o triangle_n svppose_v that_o there_o be_v a_o triangle_n abc_n unto_o one_o of_o the_o side_n whereof_o namely_o unto_o bc_o let_v there_o be_v draw_v a_o parallel_a line_n de_fw-fr cut_v the_o side_n ac_fw-la and_o ab_fw-la in_o the_o point_n e_o and_o d._n then_o i_o say_v first_o that_o as_o bd_o be_v to_o da_z so_o be_v ce_fw-fr to_o ea_fw-la theorem_a draw_v a_o line_n from_o b_o to_o e_o &_o also_o from_o c_z to_o d._n wherefore_o by_o the_o 37._o of_o the_o first_o the_o triangle_n bde_v be_v equal_a unto_o the_o triangle_n cde_o for_o they_o be_v set_v upon_o one_o and_o the_o same_o base_a de_fw-fr and_o be_v contain_v within_o the_o self_n same_o parallel_n de_fw-fr and_o bc._n consider_v also_o a_o certain_a other_o triangle_n ade_n now_o thing_n equal_a by_o the_o 7._o of_o the_o five_o have_v to_o oneself_o thing_n one_o and_o the_o same_o proportion_n wherefore_o as_o the_o triangle_n bde_n be_v to_o the_o triangle_n ade_n so_o be_v the_o triangle_n cde_o to_o the_o triangle_n ade_n but_o as_o the_o triangle_n bde_n be_v to_o the_o triangle_n ade_n so_o be_v the_o base_a bd_o to_o the_o base_a dam_fw-la by_o the_o first_o of_o this_o book_n for_o they_o be_v under_o one_o
triangle_n unto_o the_o section_n divide_v the_o angle_n of_o the_o triangle_n into_o two_o equal_a part_n this_o construction_n be_v the_o half_a part_n of_o that_o gnomical_a figure_n describe_v in_o the_o 43._o proposition_n of_o the_o first_o book_n which_o gnomical_a figure_n be_v of_o great_a use_n in_o a_o manner_n in_o all_o geometrical_a demonstration_n the_o 4._o theorem_a the_o 4._o proposition_n in_o equiangle_n triangle_n the_o side_n which_o contain_v the_o equal_a angle_n be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n svppose_v that_o there_o be_v two_o equiangle_n triangle_n abc_n and_o dce_n and_o let_v the_o angle_n abc_n of_o the_o one_o triangle_n be_v equal_a unto_o the_o angle_v dce_n of_o the_o other_o triangle_n and_o the_o angle_n bac_n equal_a unto_o the_o angle_v cde_o and_o moreover_o the_o angle_n acb_o equal_a unto_o the_o angle_n dec_n then_o i_o say_v that_o those_o side_n of_o the_o triangle_n abc_n &_o dce_n which_o include_v the_o equal_a angle_n be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n construction_n for_o let_v two_o side_n of_o the_o say_a triangle_n namely_o two_o of_o those_o side_n which_o be_v subtend_v under_o equal_a angle_n as_o for_o example_n the_o side_n bc_o and_o ce_fw-fr be_v so_o set_v that_o they_o both_o make_v one_o right_a line_n and_o because_o the_o angle_n abc_n &_o acb_o be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o first_o but_o the_o angle_n acb_o be_v equal_a unto_o the_o angle_n dec_n therefore_o the_o angle_n abc_n &_o dec_n be_v less_o they_o two_o right_a angle_n wherefore_o the_o line_n basilius_n &_o ed_z be_v produce_v will_v at_o the_o length_n meet_v together_o let_v they_o meet_v and_o join_v together_o in_o the_o point_n f._n demonstration_n and_o because_o by_o supposition_n the_o angle_n dce_n be_v equal_a unto_o the_o angle_n abc_n therefore_o the_o line_n bf_o be_v by_o the_o 28._o of_o the_o first_o a_o parallel_n unto_o tâe_z line_n cd_o and_o forasmuch_o as_o by_o supposition_n the_o angle_n acb_o be_v equal_a unto_o the_o angle_n dec_n therefore_o again_o by_o the_o 28._o of_o the_o first_o the_o line_n ac_fw-la be_v a_o parallel_n unto_o the_o line_n fe_o wherefore_o fadc_n be_v a_o parallelogram_n wherefore_o the_o side_n favorina_n be_v equal_a unto_o the_o side_n dc_o and_o the_o side_n ac_fw-la unto_o the_o side_n fd_o by_o the_o 34._o of_o the_o first_o and_o because_o unto_o one_o of_o the_o side_n of_o the_o triangle_n bfe_n namely_o to_o fe_o be_v draw_v a_o parallel_a line_n ac_fw-la therefore_o as_o basilius_n be_v to_o of_o so_o be_v bc_o to_o ce_fw-fr by_o the_o 2._o of_o the_o six_o but_o of_o be_v equal_a unto_o cd_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o basilius_n be_v to_o cd_o so_o be_v bc_o to_o ce_fw-fr which_o be_v side_n subtend_v under_o equal_a angle_n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o ab_fw-la be_v to_o bc_o so_o be_v dc_o to_o ce._n again_o forasmuch_o as_o cd_o be_v a_o parallel_n unto_o bf_o therefore_o again_o by_o the_o 2._o of_o the_o six_o as_o bc_o be_v to_o ce_fw-fr so_o be_v fd_v to_o de._n but_o fd_o be_v equal_a unto_o ac_fw-la wherefore_o as_o bc_o be_v to_o ce_fw-fr so_o be_v ac_fw-la to_o de_fw-fr which_o be_v also_o side_n subtend_v under_o equal_a angle_n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o âs_v bc_o be_v to_o ca_n so_o be_v ce_fw-fr to_o edâ_n wherefore_o forasmuch_o as_o it_o have_v be_v demonstrate_v that_o as_o ab_fw-la be_v unto_o be_v so_o be_v dc_o unto_o ceâ_n but_o as_o dc_o be_v unto_o ca_n so_o be_v ce_fw-fr unto_o edâ_n it_o follow_v of_o equality_n by_o the_o 22._o of_o the_o five_o that_o âs_a basilius_n be_v unto_o ac_fw-la so_o be_v cd_o unto_o deâ_n wherefore_o in_o equiangle_n triangleâ_n the_o side_n which_o include_v the_o equal_a angle_n be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n âhich_z be_v require_v to_o be_v demonstrate_v the_o 5._o theorem_a the_o 5._o proposition_n if_o two_o triangle_n have_v their_o side_n proportional_a the_o triangââs_n be_v equiangle_n and_o those_o angle_n in_o they_o be_v equal_a under_o which_o be_v subtend_v side_n of_o like_a proportion_n svppose_v that_o there_o be_v two_o triangle_n abc_n &_o def_n have_v their_o side_n proportional_a as_o ab_fw-la be_v to_o bc_o so_o let_v de_fw-fr be_v to_o of_o proposition_n &_o as_o bc_o be_v to_o ac_fw-la so_o let_v of_o be_v to_o df_o and_o moreover_o as_o basilius_n be_v to_o ac_fw-la so_o let_v ed_z be_v to_o df._n then_o i_o say_v that_o the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n def_n and_o those_o angle_n in_o they_o be_v equal_a under_o which_o be_v subtend_v side_n of_o like_a proportion_n that_o be_v the_o angle_n abc_n be_v equal_a unto_o the_o angle_n def_n and_o the_o angle_n bca_o unto_o the_o angle_n efd_n and_o moreover_o the_o angle_n bac_n to_o the_o angle_n edf_o upon_o the_o right_a line_n of_o construction_n and_o unto_o the_o point_n in_o it_o e_o &_o f_o describe_v by_o the_o 23._o of_o the_o first_o angle_n equal_a unto_o the_o angle_n abc_n &_o acb_o which_o let_v be_v feg_fw-mi and_o efg_o namely_o let_v the_o angle_n feg_fw-mi be_v equal_a unto_o the_o angle_n abc_n and_o let_v the_o angle_n efg_o be_v equal_a to_o the_o angle_n acb_o demonstration_n and_o forasmuch_o as_o the_o angle_n abc_n and_o acb_o be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o first_o therefore_o also_o the_o angle_n feg_v and_o efg_o be_v less_o than_o two_o right_a angle_n wherefore_o by_o the_o 5._o petition_n of_o the_o first_o the_o right_a line_n eglantine_n &_o fg_o shall_v at_o the_o length_n concur_v let_v they_o concur_v in_o the_o point_n g._n wherefore_o efg_o be_v a_o triangle_n wherefore_o the_o angle_n remain_v bac_n be_v equal_a unto_o the_o angle_n remain_v egf_n by_o the_o first_o corollary_n of_o the_o 32._o of_o the_o first_o wherefore_o the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n gef_n wherefore_o in_o the_o triangle_n abc_n and_o egf_n the_o side_n which_o include_v the_o equal_a angle_n by_o the_o 4._o of_o the_o six_o be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n wherefore_o as_o ab_fw-la be_v to_o bc_o so_o be_v ge_z to_o ef._n but_o as_o ab_fw-la be_v to_o bc_o so_o by_o supposition_n be_v de_fw-fr to_o ef._n wherefore_o as_o de_fw-fr be_v to_o of_o so_o be_v ge_z to_o of_o by_o the_o 11._o of_o the_o five_o wherefore_o either_o of_o these_o de_fw-fr and_o eglantine_n have_v to_o of_o one_o and_o the_o same_o proportion_n wherefore_o by_o the_o 9_o of_o the_o five_o de_fw-fr be_v equal_a unto_o eglantine_n and_o by_o the_o same_o reason_n also_o df_o be_v equal_a unto_o fg._n now_o forasmuch_o as_o de_fw-fr be_v equal_a to_o eglantine_n and_o of_o be_v common_a unto_o they_o both_o therefore_o these_o two_o side_n de_fw-fr &_o of_o be_v equal_a unto_o these_o two_o side_n ge_z and_o of_o and_o the_o base_a df_o be_v equal_a unto_o the_o base_a fg._n wherefore_o the_o angle_n def_n by_o the_o 8._o of_o the_o first_o be_v equal_a unto_o the_o angle_n gef_n and_o the_o triangle_n def_n by_o the_o 4._o of_o the_o first_o be_v equal_a unto_o the_o triangle_n gef_n and_o the_o rest_n of_o the_o angle_n of_o the_o one_o triangle_n be_v equal_a unto_o the_o rest_n of_o the_o angle_n of_o the_o other_o triangle_n the_o one_o to_o the_o outstretch_o under_o which_o be_v subtend_v equal_a side_n wherefore_o the_o angle_n dfe_n be_v equal_a unto_o the_o angle_n gfe_n and_o the_o angle_n edf_o unto_o the_o angle_n egf_n and_o becauseâ_n the_o angle_n feed_v be_v equal_a unto_o the_o angle_n gef_n but_o the_o angle_n gef_n be_v equal_a unto_o the_o angle_n abc_n therefore_o the_o angle_n abc_n be_v also_o equal_a unto_o the_o angle_n feed_v and_o by_o the_o same_o reason_n the_o angle_n acb_o be_v equal_a unto_o the_o angle_v dfeâ_n and_o moreover_o the_o angle_n bac_n unto_o the_o angle_n edf_o wherefore_o the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n def_n if_o two_o triangle_n therefore_o have_v their_o side_n proportional_a the_o triangle_n shall_v be_v equiangle_n &_o those_o angle_n in_o they_o shall_v be_v equal_a under_o which_o be_v subtend_v side_n of_o like_a proportion_n which_o be_v require_v to_o be_v demonstrate_v the_o 6._o theorem_a the_o 6._o proposition_n if_o there_o be_v two_o triangle_n whereof_o the_o one_o have_v one_o angle_n equal_a to_o one_o angle_n of_o
to_o bg_o and_o draw_v a_o line_n from_o a_o to_o g._n now_o forasmuch_o as_o ab_fw-la be_v to_o bc_o as_o de_fw-fr be_v to_o of_o therefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o ab_fw-la be_v to_o de_fw-fr so_o be_v bc_o to_o ef._n demonstration_n but_o as_o bc_o be_v to_o of_o so_o be_v of_o to_o bg_o wherefore_o also_o by_o the_o 11._o of_o the_o five_o as_o ab_fw-la be_v to_o de_fw-fr so_o be_v of_o to_o bg_o wherefore_o the_o side_n of_o the_o triangle_n abg_o &_o def_n which_o include_v the_o equal_a angle_n be_v reciprocal_a proportional_a but_o if_o in_o triangle_n have_v one_o angle_n of_o the_o one_o equal_a to_o one_o angle_n of_o the_o outstretch_o the_o side_n which_o include_v the_o equal_a angle_n be_v reciprokal_n the_o triangle_n also_o by_o the_o 15._o the_o six_o shall_v be_v equal_a wherefore_o the_o triangle_n abg_o be_v equal_a unto_o the_o triangle_n def_n and_o for_o that_o as_z a'n_z line_n bc_o be_v to_o the_o line_n of_o so_o be_v the_o line_n of_o to_o the_o line_n bg_o but_o if_o there_o be_v three_o line_n in_o proportion_n the_o first_o shall_v have_v to_o the_o three_o double_a proportion_n that_o it_o have_v to_o the_o second_o by_o the_o 10._o definition_n of_o the_o five_o therefore_o the_o line_n bc_o have_v unto_o the_o line_n bg_o double_a proportion_n that_o it_o have_v to_o the_o line_n ef._n but_o as_o bc_o be_v to_o bg_o so_o by_o the_o 1._o of_o the_o six_o be_v the_o triangle_n abc_n to_o the_o triangle_n abg_o wherefore_o the_o tiangle_n abc_n be_v unto_o the_o triangle_n abg_o in_o double_a proportion_n that_o the_o side_n bc_o be_v to_o the_o side_n ef._n but_o the_o triangle_n abg_o be_v equal_a to_o the_o triangle_n def_n wherefore_o also_o the_o triangle_n abc_n be_v unto_o the_o triangle_n def_n in_o double_a proportion_n that_o the_o side_n bc_o be_v to_o the_o side_n ef._n wherefore_o like_a triangle_n be_v one_o to_o the_o other_o in_o double_a proportion_n that_o the_o side_n of_o like_a proportion_n be_v which_o be_v require_v to_o be_v prove_v corollary_n corollary_n hereby_o it_o be_v manifest_a that_o if_o there_o be_v three_o right_a line_n in_o proportion_n as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o triangle_n describe_v upon_o the_o first_o unto_o the_o triangle_n describe_v upon_o the_o second_o so_o that_o the_o say_v triangle_n be_v like_a and_o in_o like_a âort_n describe_v for_o it_o have_v be_v prove_v that_o as_o the_o line_n cb_o be_v to_o the_o line_n bg_o so_o be_v the_o triangle_n abc_n to_o the_o triangle_n def_n which_o be_v require_v to_o be_v demonstrate_v the_o 14._o theorem_a the_o 20._o proposition_n like_a poligonon_n figure_n be_v divide_v into_o like_a triangle_n and_o equal_a in_o number_n and_o of_o like_a proportion_n to_o the_o whole_a and_o the_o one_o poligonon_n figure_n be_v to_o the_o other_o poligonon_n figure_n in_o double_a proportion_n that_o one_o of_o the_o side_n of_o like_a proportion_n be_v to_o one_o of_o the_o side_n of_o like_a proportion_n svppose_v that_o the_o like_a poligonon_n figure_n be_v abcde_o &_o fghkl_o have_v the_o angle_n at_o the_o point_n f_o equal_a to_o the_o angle_n at_o the_o point_n a_o and_o the_o angle_n at_o the_o point_n â_o equal_a to_o the_o angle_n at_o the_o point_n b_o and_o the_o angle_n at_o the_o point_n h_o equal_a to_o you_o angle_v at_o the_o point_n c_o and_o so_o of_o the_o rest_n and_o moreover_o as_o the_o side_n ab_fw-la be_v to_o the_o side_n bc_o so_o let_v the_o side_n fg_o be_v to_o the_o side_n gh_o and_o as_o the_o side_n bc_o be_v to_o the_o side_n cd_o so_o let_v the_o side_n gh_o be_v to_o the_o side_n hk_o and_o so_o forth_o and_o let_v the_o side_n ab_fw-la &_o fg_o be_v side_n of_o like_a proportion_n then_o i_o say_v first_o theorem_a that_o these_o poligonon_n figure_v abcde_o &_o fghkl_o be_v divide_v into_o like_a triangle_n and_o equal_a in_o number_n for_o draw_v these_o right_a line_n ac_fw-la ad_fw-la fh_o &_o fkâ_n and_o forasmuch_o as_o by_o supposition_n that_o be_v by_o reason_n the_o figure_n abcde_o be_v like_a unto_o the_o figure_n fghkl_o the_o angle_n b_o be_v equal_a unto_o the_o angle_n g_o and_o as_z the_o side_n ab_fw-la be_v to_o the_o side_n bc_o so_o be_v the_o side_n fg_o to_o the_o side_n gh_o it_o follow_v that_o the_o two_o triangle_n abc_n and_o fgh_n have_v one_o angle_n of_o the_o one_o equal_a to_o one_o angle_n of_o the_o other_o and_o have_v also_o the_o side_n about_o the_o equal_a angle_n proportional_a wherefore_o by_o the_o 6._o of_o the_o six_o the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n fgh_n and_o those_o angle_n in_o they_o be_v equal_a under_o which_o be_v subtend_v side_n of_o like_a proportion_n namely_o the_o angle_n bac_n be_v equal_a to_o the_o angle_n gfh_v and_o the_o angle_n bca_o to_o the_o angle_n ghf_n wherefore_o by_o the_o 4._o of_o the_o six_o the_o side_n which_o be_v about_o the_o equal_a angle_n be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n wherefore_o as_o ac_fw-la be_v to_o bc_o so_o be_v fh_o to_o gh_o but_o by_o supposition_n as_o bc_o be_v to_o cd_o so_o be_v gh_o to_o hk_v wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o ac_fw-la be_v to_o cd_o so_o be_v fh_o to_o hk_v and_o forasmuch_o as_o by_o supposition_n the_o whole_a angle_n bcd_o be_v equal_a to_o the_o whole_a angle_n ghk_o and_o it_o be_v prove_v that_o the_o angle_n bca_o be_v eââall_a to_o the_o angle_n ghf_n therefore_o the_o angle_n remain_v acd_o be_v equal_a to_o the_o angle_n remain_v fhk_n by_o the_o 3._o common_a sentence_n wherefore_o the_o âriaâglâs_v acd_o and_o fhk_v have_v again_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 and_o the_o side_n which_o be_v about_o the_o equal_a side_n be_v proportional_a wherefore_o by_o the_o âame_n six_o of_o this_o book_n the_o triangle_n acd_o &_o fhk_v be_v equiangle_n and_o by_o the_o 4._o of_o this_o book_n the_o side_n which_o be_v about_o the_o equal_a angle_n ãâã_d proportional_a and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o triangle_n adâ_n be_v equiangle_n unto_o the_o triangle_n fkl_n and_o that_o the_o side_n which_o be_v about_o the_o equal_a angle_n be_v proportional_a wherefore_o the_o triangle_n abc_n be_v like_o to_o the_o triangle_n fgh_n and_o the_o triangle_n acd_o to_o the_o triangle_n fhk_n and_o also_o the_o triangle_n ade_n to_o the_o triangle_n fkl_n by_o the_o first_o definition_n of_o this_o six_o book_n wherefore_o the_o poligonon_n figure_v give_v abcde_o and_o fghkl_o be_v divide_v into_o triangle_n like_a and_o equal_a in_o number_n demonstrate_v i_o say_v moreover_o that_o the_o triangle_n be_v the_o one_o to_o the_o other_o and_o to_o the_o whole_a poligonon_n figure_v proportional_a that_o be_v as_o the_o triangle_n abc_n be_v to_o the_o triangle_n fgh_o so_o be_v the_o triangle_n acd_o to_o the_o triangle_n fhk_n and_o the_o triangle_n ade_n to_o the_o triangle_n fkl_n and_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n fgh_o so_o be_v the_o poligonon_n figure_n abcde_o to_o the_o poligonon_n figure_n fghkl_o for_o forasmuch_o as_o the_o triangle_n abc_n be_v like_a to_o the_o triangle_n fgh_o and_o ac_fw-la and_o fh_o be_v side_n of_o like_a proportion_n therefore_o the_o proportion_n of_o the_o triangle_n abc_n to_o the_o triangle_n fgh_n be_v double_a to_o the_o proportion_n of_o the_o side_n ac_fw-la to_o the_o ãâã_d fh_o by_o the_o former_a proposition_n and_o therefore_o also_o the_o proportion_n of_o the_o triangle_n acd_o to_o the_o triangle_n fkh_n be_v double_a to_o the_o proportion_n that_o the_o same_o side_n ac_fw-la have_v to_o the_o side_n fh_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n fgh_o so_o be_v the_o triangle_n acd_o to_o the_o triangle_n fhk_n again_o forasmuch_o as_o the_o triangle_n acd_o be_v like_a to_o the_o triangle_n fhk_n and_o the_o side_n ad_fw-la &_o fk_o be_v of_o like_a proportion_n therefore_o the_o proportion_n of_o the_o triangle_n acd_o to_o the_o triangle_n fhk_n be_v double_a to_o the_o proportion_n of_o the_o side_n ad_fw-la to_o the_o side_n fk_o by_o the_o foresay_a 19_o of_o the_o six_o and_o by_o the_o same_o reason_n the_o proportion_n of_o the_o triangle_n ade_n to_o the_o triangle_n fkl_n be_v double_a to_o the_o proportion_n of_o the_o same_o side_n ad_fw-la to_o the_o side_n fk_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o triangle_n acd_o be_v to_o the_o triangle_n fhk_n so_o be_v the_o triangle_n ade_n to_o the_o triangle_n fkl_n but_o
as_o the_o triangle_n acd_o be_v to_o the_o triangle_n fhk_n so_o be_v it_o prove_v that_o the_o triangle_n abc_n be_v to_o the_o triangle_n fgh_n wherefore_o also_o by_o the_o 11._o of_o the_o five_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n fgh_o so_o be_v the_o triangle_n ade_n to_o the_o triangle_n fkl_n wherefore_o the_o foresay_a triangle_n be_v proportional_a namely_o as_o abc_n be_v to_o fgh_v so_o be_v acd_o to_o fhk_v and_o ade_n to_o fkl_n wherefore_o by_o the_o 12._o of_o the_o five_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 wherefore_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n fgh_o so_o be_v the_o poligonon_n figure_n abcde_o to_o the_o poligonon_n figure_n fghkl_o wherefore_o the_o triangle_n be_v proportional_a both_o the_o one_o to_o the_o other_o &_o also_o to_o the_o whole_a poligonon_n figure_n last_o i_o say_v part_n that_o the_o poligonon_n figure_n abcde_o have_v to_o the_o poligonon_n figure_n fghkl_o a_o double_a proportion_n to_o that_o which_o the_o side_n ab_fw-la have_v to_o the_o side_n fg_o which_o be_v side_n of_o like_a proportion_n for_o it_o be_v prove_v that_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n fgh_o so_o be_v the_o poligonon_n figure_n abcde_o to_o the_o poligonon_n figure_n fghkl_o but_o the_o triangle_n abc_n have_v to_o the_o triangle_n fgh_v a_o double_a proportion_n to_o that_o which_o the_o side_n ab_fw-la have_v to_o the_o side_n fg_o by_o the_o former_a 19_o proposition_n of_o this_o book_n for_o it_o be_v prove_v that_o the_o triangle_n abc_n be_v like_a to_o the_o triangle_n fgh_n wherefore_o the_o proportion_n of_o the_o poligonon_n figure_n abcde_o to_o the_o poligonon_n figure_n fghkl_o be_v double_a to_o the_o proportion_n of_o the_o side_n ab_fw-la to_o the_o side_n fg_o which_o be_v side_n of_o like_a proportion_n wherefore_o like_a poligonon_n figure_n be_v divide_v etc_n etc_n as_o beforeâ_n which_o be_v require_v to_o be_v prove_v the_o first_o corollary_n hereby_o it_o be_v manifest_a that_o all_o like_a rectiline_a figure_n what_o soever_o be_v the_o one_o to_o the_o other_o in_o double_a proportion_n that_o the_o side_n of_o like_a proportion_n be_v corollary_n for_o any_o like_a rectiline_a figure_n whatsoever_o be_v by_o this_o proposition_n divide_v into_o like_a triangle_n and_o equal_a in_o number_n the_o second_o corollary_n hereby_o also_o it_o be_v manifest_a that_o if_o there_o be_v three_o right_a line_n proportional_a corollary_n as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o figure_n describe_v upon_o the_o first_o to_o the_o figure_n describe_v upon_o the_o second_o so_o that_o the_o say_v figure_n be_v like_a and_o in_o like_a sort_n describe_v for_o it_o be_v prove_v that_o the_o proportion_n of_o the_o poligonon_n figure_n abcde_o to_o the_o poligonon_n figure_n fghkl_o be_v double_a to_o the_o proportion_n of_o the_o side_n ab_fw-la to_o the_o side_n fg._n and_o if_o by_o the_o 11._o of_o the_o six_o unto_o the_o line_n ab_fw-la and_o fg_o we_o take_v a_o three_o line_n in_o proportion_n namely_o mn_v the_o first_o line_n namely_o ab_fw-la shall_v have_v unto_o the_o three_o line_n namely_o to_o mn_v double_a proportion_n that_o it_o have_v to_o the_o second_o line_n namely_o to_o fg_o by_o the_o 10._o definition_n of_o the_o five_o wherefore_o as_o the_o line_v ab_fw-la be_v to_o the_o line_n mn_o so_o be_v the_o rectiline_a figure_n abc_n to_o the_o rectiline_a figure_n fgh_o the_o say_v rectiline_a figure_n be_v like_o &_o in_o like_a sort_n describe_v the_o 15._o theorem_a the_o 21._o proposition_n rectiline_a figure_n which_o be_v like_a unto_o one_o and_o the_o same_o rectiline_a figure_n be_v also_o like_o the_o one_o to_o the_o other_o svppose_v there_o be_v two_o rectiline_a figure_n a_o and_o b_o like_o unto_o the_o rectiline_a figure_n c._n then_o i_o say_v that_o the_o figure_n a_o be_v also_o like_a unto_o the_o figure_n b._n for_o forasmuch_o as_o the_o figure_n a_o be_v like_a unto_o the_o figure_n c_o demonstration_n it_o be_v also_o equiangle_n unto_o it_o by_o the_o conversion_n of_o the_o first_o definition_n of_o the_o six_o &_o the_o side_n include_v the_o equal_a angle_n shall_v be_v proportional_a again_o forasmuch_o as_o the_o figure_n b_o be_v like_a unto_o the_o figure_n c_o it_o be_v also_o by_o the_o same_o definition_n equiangle_n unto_o it_o and_o the_o side_n about_o the_o equal_a angle_n be_v proportional_a wherefore_o both_o these_o figure_n a_o and_o b_o be_v equiangle_n unto_o the_o figure_n c_o and_o the_o side_n about_o the_o equal_a angle_n be_v proportional_a wherefore_o by_o the_o first_o common_a sentence_n the_o figure_n a_o be_v equiangle_n unto_o the_o figure_n b_o and_o the_o side_n about_o the_o equal_a angle_n be_v proportional_a wherefore_o the_o figure_n b_o be_v like_a unto_o the_o figure_n a_o which_o be_v require_v to_o be_v prove_v the_o 16._o theorem_a the_o 22._o proposition_n if_o there_o be_v four_o right_a line_n proportional_a the_o rectiline_a figure_n also_o describe_v upon_o they_o be_v like_a and_o in_o like_a sort_n situate_a shall_v be_v proportional_a and_o if_o the_o rectiline_a figure_n upon_o they_o describe_v be_v proportional_a those_o right_a line_n also_o shall_v be_v proportional_a svppose_v there_o be_v four_o right_a line_n ab_fw-la cd_o of_o and_o gh_o and_o as_o ab_fw-la be_v to_o cd_o so_o let_v of_o be_v to_o gh_o and_o upon_o the_o line_n ab_fw-la and_o cd_o by_o the_o 1â_o of_o the_o six_o let_v there_o be_v describe_v two_o rectiline_a figure_n kab_n and_o lcd_v like_o the_o one_o to_o the_o other_o and_o in_o like_a sort_n situate_a and_o upon_o the_o line_n of_o and_o gh_o by_o the_o same_o let_v there_o be_v describe_v also_o two_o rectiline_a figure_n mf_a and_o nh_v like_o the_o one_o to_o the_o other_o and_o in_o like_a sort_n situate_a proposition_n then_o i_o say_v that_o as_o the_o âigure_n kab_n be_v to_o the_o figure_n lcd_n so_o be_v the_o figure_n mf_n to_o the_o figure_n nh_o unto_o the_o line_n ab_fw-la and_o cd_o by_o the_o 11._o of_o the_o six_o make_v a_o three_o line_n in_o proportion_n namely_o oh_o and_o unto_o the_o line_n of_o and_o gh_o in_o like_a sort_n make_v a_o three_o line_n in_o a_o line_n proportion_n namely_o p._n and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o be_v the_o line_n of_o to_o the_o line_n gh_o but_o as_o the_o line_n cd_o be_v to_o the_o line_n oh_o so_o be_v the_o line_n gh_o to_o the_o line_n p._n wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o the_o line_n ab_fw-la be_v unto_o the_o line_n oh_o so_o be_v the_o line_n of_o to_o the_o line_n p._n but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n oh_o so_o be_v the_o figure_n kab_n to_o the_o figure_n lcd_v by_o the_o second_o corollary_n of_o the_o 20._o of_o the_o six_o and_o as_o the_o line_n of_o be_v to_o the_o line_n p_o so_o be_v the_o figure_n m_o fletcher_n to_o the_o figure_n nh_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o figure_n kab_n be_v to_o the_o figure_n lcd_n so_o be_v the_o figure_n m_o fletcher_n to_o the_o figure_n nh_o but_o now_o suppose_v that_o as_o the_o figure_n kab_n be_v to_o the_o figure_n lcd_n first_o so_o be_v the_o figure_n m_o fletcher_n to_o the_o figure_n nh_o than_o i_o say_v that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o be_v the_o line_n of_o to_o the_o line_n gh_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o by_o the_o 1â_o of_o the_o six_o let_v the_o line_n of_o be_v to_o the_o line_n qr_o and_o upon_o the_o line_n qr_o by_o the_o 18._o of_o the_o six_o describe_v unto_o either_o of_o these_o figure_n mf_a and_o nh_o a_o like_a figure_n and_o in_o like_a sort_n situate_a sir_n now_o forasmuch_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o be_v the_o line_n of_o to_o the_o line_n qr_o and_o upon_o the_o line_n ab_fw-la and_o cd_o be_v describe_v two_o figure_n like_o and_o in_o like_a sort_n situate_a kab_n and_o lcd_v and_o upon_o the_o line_n of_o and_o qr_o be_v describe_v also_o two_o figure_n like_o and_o in_o like_a sort_n situate_a mf_n and_o sir_n therefore_o as_o the_o figure_n kab_n be_v to_o the_o figure_n lcd_n so_o be_v the_o figure_n mf_n to_o the_o figure_n sir_n wherefore_o also_o by_o the_o 11._o of_o the_o five_o as_o the_o figure_n mf_n be_v to_o the_o figure_n sir_n so_o be_v the_o figure_n mf_n to_o the_o figure_n nh_o wherefore_o the_o figure_n m_o fletcher_n have_v to_o either_o of_o these_o figure_n nh_v and_o sir_n one_o
namely_o the_o circumference_n blakoc_n be_v equal_a unto_o the_o circumference_n remain_v of_o the_o self_n same_o circle_n abc_n namely_o to_o the_o circumference_n cpblak_n wherefore_o the_o angle_n bpc_n be_v equal_a unto_o the_o angle_n cok_n by_o the_o 27._o of_o the_o three_o wherefore_o by_o the_o 10._o definition_n of_o the_o three_o the_o segment_n bpc_n be_v like_a unto_o the_o segment_n cok_n and_o they_o be_v set_v upon_o equal_a right_a line_n bc_o and_o kc_o but_o like_o segment_n of_o circle_n which_o consist_v upon_o equal_a right_a line_n be_v also_o equal_a the_o one_o to_o the_o other_o by_o the_o 24._o of_o the_o three_o wherefore_o the_o segment_n bpc_n be_v equal_a unto_o the_o segment_n cok._n and_o the_o triangle_n gbc_n be_v equal_a unto_o the_o triangle_n gck._n wherefore_o the_o sector_n gbc_n be_v equal_a unto_o the_o sector_n gck._n and_o by_o the_o same_o reason_n also_o the_o sector_n gkl_n be_v equal_a unto_o either_o of_o the_o sector_n gbc_n and_o gck._n wherefore_o the_o three_o sector_n gbc_n and_o gck_n and_o gkl_n be_v equal_a the_o one_o to_o the_o outstretch_o and_o by_o the_o same_o reason_n also_o the_o sector_n hef_n and_o hfm_n and_o hmn_v be_v equal_a the_o one_o to_o the_o other_o wherefore_o how_o multiplex_n the_o circumference_n bl_o be_v to_o the_o circumference_n bc_o so_o multiplex_n be_v the_o sector_n glb_n to_o the_o sector_n gbc_n and_o by_o the_o same_o reason_n how_o multiplex_n the_o circumference_n ne_fw-fr be_v to_o the_o circumference_n of_o so_o multiplex_n be_v the_o sector_n hen_n to_o the_o sector_n hef_fw-fr if_o therefore_o the_o circumference_n bl_o be_v equal_a unto_o the_o circumference_n en_fw-fr the_o sector_n also_o bgl_n be_v equal_a unto_o the_o sector_n ehn._n and_o if_o the_o circumference_n bl_o exceed_v the_o circumference_n en_fw-fr the_o sector_n also_o bgl_n exceed_v the_o sector_n ehn._n and_o if_o the_o circumference_n be_v less_o the_o sector_n also_o be_v less_o now_o they_o there_o be_v four_o magnitude_n namely_o the_o two_o circumference_n bc_o and_o of_o and_o the_o two_o sector_n gbc_n &_o hef_n and_o to_o the_o circumference_n bc_o &_o to_o the_o sector_n gbc_n namely_o to_o the_o first_o and_o the_o three_o be_v take_v equemultiplices_fw-la that_o be_v the_o circumference_n bl_o and_o the_o sector_n gbl_n and_o likewise_o to_o the_o circumference_n of_o and_o to_o the_o sector_n hef_fw-fr namely_o to_o the_o second_o and_o four_o be_v take_v certain_a other_o equimultiplice_n namely_o the_o circumference_n en_fw-fr and_o the_o sector_n hen._n and_o it_o be_v prove_v that_o if_o the_o circumference_n bl_o exceed_v the_o circumference_n en_fw-fr the_o sector_n also_o bgl_n exceed_v the_o sector_n ehn._n and_o if_o the_o circumference_n be_v equal_a the_o segment_n also_o be_v equal_a and_o if_o the_o circumference_n be_v less_o the_o segment_n also_o be_v less_o wherefore_o by_o the_o conversion_n of_o the_o six_o definition_n of_o the_o five_o as_o the_o circumference_n bc_o be_v to_o the_o circumference_n of_o so_o be_v the_o sector_n gbc_n unto_o the_o sector_n hef_fw-fr which_o be_v all_o that_o be_v require_v to_o be_v prove_v corollary_n and_o hereby_o it_o be_v manifest_a that_o as_o the_o sector_n be_v to_o the_o sector_n so_o be_v angle_n to_o angle_v by_o the_o 11._o of_o the_o five_o flussates_n here_o add_v five_o proposition_n whereof_o one_o be_v a_o problem_n have_v three_o corollarye_n follow_v of_o it_o and_o the_o rest_n be_v theorem_n which_o for_o that_o they_o be_v both_o witty_a &_o also_o serve_v to_o great_a use_n as_o we_o shall_v afterward_o see_v i_o think_v not_o good_a to_o omit_v but_o have_v here_o place_v they_o but_o only_o that_o i_o have_v not_o put_v they_o to_o follow_v in_o order_n with_o the_o proposition_n of_o euclid_n as_o he_o have_v do_v ¶_o the_o first_o proposition_n add_v by_o flussates_n to_o describe_v two_o rectiline_a figure_n equal_a and_o like_a unto_o a_o rectiline_a figure_n give_v and_o in_o like_a sort_n situate_a which_o shall_v have_v also_o a_o proportion_n give_v suppose_v that_o the_o rectiline_a figure_n give_v be_v abh_n problem_n and_o let_v the_o proportion_n give_v be_v the_o proportion_n of_o the_o line_n gc_o and_o cd_o and_o by_o the_o 10._o of_o this_o book_n divide_v the_o line_n ab_fw-la like_a unto_o the_o line_n gd_o in_o the_o point_n e_o so_o that_o as_o the_o line_n gc_o be_v to_o the_o line_n cd_o so_o let_v the_o line_n ae_n be_v to_o the_o line_n ebb_n and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n afb_o and_o from_o the_o point_n e_o erect_v by_o the_o 11._o of_o the_o first_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n of_o cut_v the_o circumference_n in_o the_o point_n f._n and_o draw_v these_o line_n of_o and_o fb_o and_o upon_o either_o of_o these_o line_n describe_v rectiline_a figure_n like_o unto_o the_o rectiline_a figure_n ahb_n and_o in_o like_a sort_n situate_a by_o the_o 18._o of_o the_o six_o which_o let_v be_v akf_n &_o fib_n then_o i_o say_v that_o the_o rectiline_a figure_n akf_a and_o fib_n have_v the_o proportion_n give_v namely_o the_o proportion_n of_o the_o line_n gc_o to_o the_o line_n cd_o and_o be_v equal_a to_o the_o rectiline_a figure_n give_v abh_n unto_o which_o they_o be_v describe_v like_o and_o in_o like_a sort_n situate_a for_o forasmuch_o as_o afb_o be_v a_o semicircle_n therefore_o the_o angle_n afb_o be_v a_o right_a angle_n by_o the_o 31._o of_o the_o three_o and_o fe_o be_v a_o perpendicular_a line_n same_o wherefore_o by_o the_o 8._o of_o this_o book_n the_o triangle_n afe_a and_o fbe_n be_v like_a both_o to_o the_o whole_a triangle_n afb_o and_o also_o the_o one_o to_o the_o other_o wherefore_o by_o the_o 4._o of_o this_o book_n as_o the_o line_n of_o be_v to_o the_o line_n fb_o so_o be_v the_o line_n ae_n to_o the_o line_n of_o and_o the_o line_n of_o to_o the_o line_n ebb_n which_o be_v side_n contain_v equal_a angle_n wherefore_o by_o the_o 22._o of_o this_o book_n as_o the_o rectiline_a figure_n describe_v of_o the_o line_n of_o be_v to_o the_o rectiline_a figure_n describe_v of_o the_o line_n fb_o so_o be_v the_o rectiline_a figure_n describe_v of_o the_o line_n ae_n to_o the_o rectiline_a figure_n describe_v of_o the_o line_n of_o the_o say_v rectiline_a figure_n be_v like_o and_o in_o like_a sort_n situate_a but_o as_o the_o rectiline_a figure_n describe_v of_o the_o line_n ae_n be_v the_o âirst_n be_v to_o the_o rectiline_a âigure_n describe_v of_o the_o line_n of_o be_v the_o second_o so_o be_v the_o line_n ae_n the_o first_o 10._o the_o line_n âb_n the_o three_o by_o the_o 2._o corollary_n of_o the_o 20._o of_o this_o book_n wherefore_o the_o rectiline_a figure_n describe_v of_o the_o line_n of_o be_v to_o the_o rectiline_a figure_n describe_v of_o the_o line_n fb_o as_o the_o line_n aâ_z be_v to_o the_o line_n ebb_n but_o the_o line_n ae_n be_v to_o the_o line_n ebb_n by_o construction_n as_o the_o line_n gc_o be_v to_o the_o line_n cd_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o line_n gc_o be_v to_o the_o line_n cd_o so_o be_v the_o rectiline_a âigure_n describe_v of_o the_o line_n of_o to_o the_o rectiline_a âigure_n describe_v ãâã_d the_o line_n âb_n the_o say_v rectiline_a figure_n be_v like_o and_o in_o like_a sort_n describe_v but_o the_o ãâ¦ã_o describe_v oâ_n the_o line_n of_o and_o fb_o be_v equal_a to_o the_o rectiline_a âigure_n dâââââbed_v oâ_n the_o line_n ab_fw-la unto_o which_o they_o be_v by_o construction_n describe_v like_o and_o in_o like_a sort_n situate_a wherefore_o there_o be_v describe_v two_o rectiline_a figure_n akf_a and_o fib_n equal_a and_o like_a unto_o the_o rectiline_a figure_n give_v abh_n and_o in_o like_a sort_n situate_a and_o they_o haâe_z also_o the_o one_o to_o the_o other_o the_o proportion_n give_v namely_o the_o proportion_n of_o the_o line_n gc_o to_o the_o line_n cd_o which_o be_v require_v to_o be_v do_v ¶_o the_o first_o corollary_n to_o resolve_v a_o rectiline_a figure_n geve_v into_o two_o like_o rectiline_a â_z which_o shall_v hauâ_n also_o a_o proportiâ_n geâen_o for_o iâ_z there_o be_v put_v three_o right_a line_n in_o the_o proportion_n give_v and_o if_o the_o line_n ab_fw-la be_v cuâ_n in_o the_o same_o proportion_n that_o the_o first_o line_n be_v to_o the_o three_o corollary_n the_o rectiline_a â_z describe_v of_o the_o lineâ_z aâ_z and_o fb_o which_o figure_n have_v the_o same_o proportion_n that_o the_o line_n ae_n and_o ebb_n have_v shall_v be_v in_o double_a proportion_n to_o that_o which_o the_o line_n of_o and_o fb_o be_v by_o the_o âirsââorollary_n oâ_n the_o 20._o oâ_n this_o book_n wherefore_o the_o right_a line_n of_o and_o fb_o be_v the_o oâe_n to_o the_o other_o in_o the_o same_o
but_o not_o every_o pyramid_n a_o tetrahedron_n and_o in_o deed_n psellus_n in_o number_v of_o these_o five_o solid_n or_o body_n call_v a_o tetrahedron_n a_o pyramid_n in_o manifest_a word_n pyramid_n this_o i_o say_v may_v make_v flussas_n &_o other_o as_o i_o think_v it_o do_v to_o omit_v the_o definition_n of_o a_o tetrahedron_n in_o this_o place_n as_o sufficient_o comprehend_v within_o the_o definition_n of_o a_o pyramid_n give_v before_o but_o why_o then_o do_v he_o not_o count_v that_o definition_n of_o a_o pyramid_n faulty_a for_o that_o it_o extend_v itself_o to_o large_a and_o comprehend_v under_o it_o a_o tetrahedron_n which_o differ_v from_o a_o pyramid_n by_o that_o it_o be_v contain_v of_o equal_a triangle_n as_o he_o not_o so_o advise_o do_v before_o the_o definition_n of_o a_o prism_n definition_n 23_o a_o octohedron_n be_v a_o solid_a or_o bodily_a figure_n contain_v under_o eight_o equal_a and_o equilater_n triangle_n as_o a_o cube_n be_v a_o solid_a figure_n 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any_o line_n or_o section_n require_v to_o be_v draw_v in_o any_o of_o the_o say_v eight_o triangle_n which_o be_v the_o side_n of_o that_o body_n 24_o a_o dodecahedron_n be_v a_o solid_a or_o bodily_a figure_n contain_v under_o twelve_o equal_a equilater_n definition_n and_o equiangle_n pentagons_n as_o a_o cube_n a_o tetrahedron_n and_o a_o octohedron_n be_v contain_v under_o equal_a plain_a figure_n a_o cube_n under_o square_n the_o other_o two_o under_o triangle_n so_o be_v this_o solid_a figure_n contain_v under_o twelve_o equilater_n equiangle_n and_o equal_a pentagons_n or_o figure_n of_o five_o side_n as_o in_o these_o two_o figure_n here_o set_v you_o may_v perceive_v of_o which_o the_o first_o which_o thing_n also_o be_v before_o note_v of_o a_o cube_n a_o tetrahedron_n and_o a_o octohedron_n be_v the_o common_a description_n of_o it_o in_o a_o plain_a the_o other_o be_v the_o description_n of_o it_o by_o art_n upon_o a_o plain_a to_o make_v it_o to_o appear_v somewhat_o 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body_n of_o this_o condition_n and_o perfection_n but_o only_o these_o five_o wherefore_o they_o have_v ever_o of_o the_o ancient_a philosopher_n be_v have_v in_o great_a estimation_n and_o admiration_n and_o have_v be_v think_v worthy_a of_o much_o contemplation_n about_o which_o they_o have_v bestow_v most_o diligent_a study_n and_o endeavour_n to_o search_v out_o the_o nature_n &_o property_n of_o they_o they_o be_v as_o it_o be_v the_o end_n and_o perfection_n of_o all_o geometry_n for_o who_o sake_n be_v write_v whatsoever_o be_v write_v in_o geometry_n they_o be_v as_o man_n say_v first_o invent_v by_o the_o most_o witty_a pythagoras_n then_o afterward_o set_v forth_o by_o the_o divine_a plato_n and_o last_o of_o all_o marvellous_o teach_v and_o declare_v by_o the_o most_o excellent_a philosopher_n euclid_n in_o these_o book_n follow_v and_o ever_o since_o wonderful_o embrace_v of_o all_o learned_a philosopher_n body_n the_o knowledge_n of_o they_o contain_v infinite_a secret_n of_o nature_n pithagâras_n timeus_n and_o plato_n by_o they_o search_v out_o the_o composition_n of_o the_o world_n with_o the_o harmony_n and_o preservation_n thereof_o and_o apply_v these_o âive_a solid_n to_o the_o simple_a part_n thereof_o the_o pyramid_n or_o tetrahedron_n they_o ascribe_v to_o the_o âire_n fire_n for_o that_o it_o ascend_v upward_o according_a to_o the_o figure_n of_o the_o pyramid_n to_o the_o air_n they_o ascribe_v the_o octohedron_n air_n for_o that_o through_o the_o subtle_a moisture_n which_o it_o have_v it_o extend_v itself_o every_o way_n to_o the_o one_o side_n and_o to_o the_o other_o accord_v as_o that_o figure_n do_v unto_o the_o water_n they_o assign_v the_o ikosahedron_n for_o that_o it_o be_v continual_o flow_v and_o move_v water_n and_o as_o it_o be_v make_v angleâ_n ãâ¦ã_o âide_v according_a to_o that_o figure_n and_o to_o the_o earth_n they_o attribute_v a_o cube_n earth_n as_o to_o a_o thing_n stableâ_n ãâã_d and_o sure_a as_o the_o figure_n
angle_n bac_n god_n &_o dab_n be_v equal_a the_o one_o to_o the_o other_o then_o be_v it_o manifest_v that_o two_o of_o they_o which_o two_z so_o ever_o be_v take_v be_v great_a than_o the_o three_o but_o if_o not_o let_v the_o angle_n bac_n be_v the_o great_a of_o the_o three_o angle_n and_o unto_o the_o right_a line_n ab_fw-la and_o from_o the_o point_n a_o make_v in_o the_o plain_a superficies_n bac_n unto_o the_o angle_n dab_n a_o equal_a angle_n bae_o and_o by_o the_o 2._o of_o the_o first_o make_v the_o line_n ae_n equal_a to_o the_o line_n ad._n now_o a_o right_a line_n bec_n draw_v by_o the_o point_n e_o shall_v cut_v the_o right_a line_n ab_fw-la and_o ac_fw-la in_o the_o point_n b_o and_o c_o draw_v a_o right_a line_n from_o d_z to_o b_o and_o a_o outstretch_o from_o d_z to_o c._n and_o forasmuch_o as_o the_o line_n dam_n be_v equal_a to_o the_o line_n ae_n demonstration_n and_o the_o line_n ab_fw-la be_v common_a to_o they_o both_o therefore_o these_o two_o line_n dam_n and_o ab_fw-la be_v equal_a to_o these_o two_o line_n ab_fw-la and_o ae_n and_o the_o angle_n dab_n be_v equal_a to_o the_o angle_n bae_o wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a db_o be_v equal_a to_o the_o base_a be._n and_o forasmuch_o as_o these_o two_o line_n db_o and_o dc_o be_v great_a than_o the_o line_n bc_o of_o which_o the_o line_n db_o be_v prove_v to_o be_v equal_a to_o the_o line_n be._n wherefore_o the_o residue_n namely_o the_o line_n dc_o be_v great_a than_o the_o residue_n namely_o than_o the_o line_n ec_o and_o forasmuch_o as_o the_o line_n dam_n be_v equal_a to_o the_o line_n ae_n and_o the_o line_n ac_fw-la be_v common_a to_o they_o both_o and_o the_o base_a dc_o be_v great_a than_o the_o base_a ec_o therefore_o the_o angle_n dac_o be_v great_a than_o the_o angle_n eac_n and_o it_o be_v prove_v that_o the_o angle_n dab_n be_v equal_a to_o the_o angle_n bae_o wherefore_o the_o angle_n dab_n and_o dac_o be_v great_a than_o the_o angle_n bac_n if_o 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the_o same_o be_v b_o c_o d._n and_o draw_v these_o right_a line_n bc_o cd_o and_o db._n demonstration_n and_o forasmuch_o as_o the_o angle_n b_o be_v a_o solid_a angle_n for_o it_o be_v contain_v under_o three_o superficial_a angle_n that_o be_v under_o cba_o abdella_n and_o cbd_n therefore_o by_o the_o 20._o of_o the_o eleven_o two_o of_o they_o which_o two_z so_o ever_o be_v take_v be_v great_a than_o the_o three_o wherefore_o the_o angle_n cba_o and_o abdella_n be_v great_a than_o the_o angle_n cbd_v and_o by_o the_o same_o reason_n the_o angle_n bca_o and_o acd_o be_v great_a than_o the_o angle_n bcdâ_n and_o moreover_o the_o angle_n cda_n and_o adb_o be_v great_a than_o the_o angle_n cdb_n wherefore_o these_o six_o angle_n cba_o abdella_n bca_o acd_o cda_n and_o adb_o be_v great_a they_o these_o three_o angle_n namely_o cbd_v bcd_o &_o cdb_n but_o the_o three_o angle_n cbd_v bdc_o and_o bcd_o be_v equal_a to_o two_o right_a angle_n wherefore_o the_o six_o angle_n cba_o abdella_n 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i_o put_v for_o the_o better_a sight_n of_o the_o demonstration_n of_o the_o proposition_n next_o go_v before_o only_o here_o be_v not_o require_v the_o draught_n of_o the_o line_n ae_n although_o this_o demonstration_n of_o euclid_n be_v here_o put_v for_o solid_a angle_n contain_v under_o three_o superficial_a angle_n yet_o after_o the_o like_a manner_n may_v you_o proceed_v if_o the_o solid_a angle_n be_v contain_v under_o superficial_a angle_n how_o many_o so_o ever_o as_o for_o example_n if_o it_o be_v contain_v under_o four_o superficial_a angle_n if_o you_o follow_v the_o former_a construction_n the_o base_a will_v be_v a_o quadrangled_a figure_n who_o four_o angle_n be_v equal_a to_o four_o right_a angle_n but_o the_o 8._o angle_n at_o the_o base_n of_o the_o 4._o triangle_n set_v upon_o this_o quadrangled_a figure_n may_v by_o the_o 20._o proposition_n of_o this_o book_n be_v prove_v to_o be_v great_a than_o those_o 4._o angle_n of_o the_o quadrangled_a figure_n as_o we_o see_v by_o the_o discourse_n of_o the_o former_a demonstration_n wherefore_o those_o 8._o angle_n be_v great_a than_o four_o right_a angle_n but_o the_o 12._o angle_n of_o those_o four_o triangle_n be_v equal_a to_o 8._o right_a angle_n wherefore_o the_o four_o angle_n remain_v at_o the_o top_n which_o make_v the_o solid_a angle_n be_v less_o than_o four_o right_a angle_n and_o observe_v this_o course_n you_o may_v proceed_v infinite_o ¶_o the_o 20._o theorem_a the_o 22._o proposition_n if_o there_o be_v three_o superficial_a plain_a angle_n of_o which_o two_o how_o soever_o they_o be_v take_v be_v great_a than_o the_o three_o and_o if_o the_o right_a line_n also_o which_o contain_v those_o angle_n be_v equal_a then_o of_o the_o line_n couple_v those_o equal_a right_a line_n together_o it_o be_v possible_a to_o make_v a_o triangle_n svppose_v that_o there_o be_v three_o superficial_a angle_n abc_n def_n and_o ghk_v of_o which_o let_v two_o which_o two_o soever_o be_v take_v be_v great_a than_o the_o three_o that_o be_v let_v the_o angle_n abc_n and_o def_n be_v great_a than_o the_o angle_n ghk_o and_o let_v the_o angle_n def_n and_o ghk_o be_v great_a than_o the_o angle_n abc_n and_o moreover_o let_v the_o angle_n ghk_v and_o abc_n be_v great_a than_o the_o angle_n def_n and_o let_v the_o right_a line_n ab_fw-la bc_o de_fw-fr of_o gh_o and_o hk_o be_v equal_a the_o one_o to_o the_o other_o and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n c_o and_o a_o other_o from_o the_o point_n d_o to_o the_o point_n fletcher_n and_o moreover_o a_o other_o from_o the_o point_n g_o to_o the_o point_n k._n proposition_n then_o i_o say_v that_o it_o be_v possible_a of_o three_o right_a line_n equal_a to_o the_o line_n ac_fw-la df_o and_o gk_o to_o make_v a_o triangle_n that_o be_v that_o two_o of_o the_o right_a line_n ac_fw-la df_o and_o gk_o which_o then_o soever_o be_v take_v be_v great_a than_o the_o three_o now_o if_o the_o angle_n abc_n def_n case_n and_o ghk_v be_v equal_a the_o one_o to_o the_o other_o it_o be_v manifest_a that_o these_o right_a line_n ac_fw-la df_o and_o gk_o be_v also_o by_o the_o 4._o of_o the_o first_o equal_a the_o one_o to_o the_o other_o it_o be_v possible_a of_o three_o right_a line_n equal_a to_o the_o line_n ac_fw-la df_o and_o gk_o to_o make_v a_o triangle_n case_n but_o if_o they_o be_v not_o equal_a let_v they_o be_v unequal_a and_o by_o the_o 23._o of_o the_o first_o unto_o the_o right_a line_n hk_o construction_n and_o at_o the_o point_n in_o it_o h_o make_v unto_o the_o angle_n abc_n a_o equal_a angle_n khl._n and_o by_o the_o â_o of_o the_o first_o to_o one_o of_o the_o line_n
reform_v by_o m._n dee_n describe_v for_o it_o in_o the_o plain_n especial_o if_o you_o remember_v the_o form_n of_o the_o figure_n of_o the_o 29._o proposition_n of_o this_o book_n only_o that_o which_o there_o you_o conceive_v to_o be_v the_o base_a imagine_v here_o in_o both_o the_o figure_n of_o this_o second_o case_n to_o be_v the_o upper_a superficies_n opposite_a to_o the_o base_a and_o that_o which_o be_v there_o suppose_v to_o be_v the_o upper_a superficies_n conceive_v here_o to_o be_v the_o base_a you_o may_v describe_v they_o upon_o past_v paper_n for_o your_o better_a sight_n take_v heed_n you_o note_v the_o letter_n right_o according_a as_o the_o construction_n require_v flussas_n demonstrate_v this_o proposition_n a_o otherway_o take_v only_o the_o base_n of_o the_o solid_n and_o that_o after_o this_o manner_n take_v equal_a base_n which_o yet_o for_o the_o sure_a understanding_n let_v be_v utter_o unlike_a namely_o aebf_n and_o adch_n and_o let_v one_o of_o the_o side_n of_o each_o concur_v in_o one_o &_o the_o same_o right_a line_n aed_n &_o the_o base_n be_v upon_o one_o and_o the_o self_n same_o plain_n let_v there_o be_v suppose_v to_o be_v set_v upon_o they_o parallelipipedon_n under_o one_o &_o the_o self_n same_o altitude_n then_o i_o say_v that_o the_o solid_a set_v upon_o the_o base_a ab_fw-la be_v equal_a to_o the_o solid_a set_v upon_o the_o base_a ah_o by_o the_o point_n e_o draw_v unto_o the_o line_n ac_fw-la a_o parallel_n line_n eglantine_n which_o if_o it_o fall_v without_o the_o base_a ab_fw-la produce_v the_o right_a line_n hc_n to_o the_o point_n i_o now_o forasmuch_o as_o ab_fw-la and_o ah_o be_v parallelogrmae_n therefore_o by_o the_o 24._o of_o this_o book_n the_o triangle_n aci_n and_o egl_n shall_v be_v equaliter_fw-la the_o one_o to_o the_o other_o and_o by_o the_o 4._o of_o the_o first_o they_o shall_v be_v equiangle_n and_o equal_a and_o by_o the_o first_o definition_n of_o the_o six_o and_o four_o proposition_n of_o the_o same_o they_o shall_v be_v like_a wherefore_o prism_n erect_v upon_o those_o triangle_n and_o under_o the_o same_o altitude_n that_o the_o solid_n ab_fw-la and_o ah_o aâe_n shall_v be_v equal_a and_o like_a by_o the_o 8._o definition_n of_o this_o book_n for_o they_o be_v contain_v under_o like_o plain_a superficiece_n equal_a both_o in_o multitude_n and_o magnitude_n add_v the_o solid_a set_v upon_o the_o base_a acle_n common_a to_o they_o both_o wherefore_o the_o solid_a set_v upon_o the_o base_a aegc_n be_v equal_a to_o the_o solid_a set_v upon_o the_o base_a aeli_n and_o forasmuch_o as_o the_o superficiece_n aebf_n and_o adhc_n be_v equal_a by_o supposition_n and_o the_o part_n take_v away_o agnostus_n be_v equal_a to_o the_o part_n take_v away_o all_o therefore_o the_o residue_n by_o shall_v be_v equal_a to_o the_o residue_n gd_o wherefore_o as_o agnostus_n be_v to_o gd_a as_o all_o be_v to_o by_o namely_o equal_n to_o equal_n but_o as_o agnostus_n be_v to_o gd_o so_o iâ_z the_o solid_a set_v upon_o agnostus_n to_o the_o solid_a set_v upon_o gd_o by_o the_o 25._o of_o this_o book_n for_o it_o be_v cut_v by_o a_o plain_a superficies_n set_v upon_o the_o line_n ge_z which_o superficies_n be_v parallel_n to_o the_o opposite_a superficiece_n wherefore_o as_o all_o be_v to_o by_o so_o be_v the_o solid_a set_v upon_o all_o to_o the_o solid_a set_v upon_o by_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o solid_a set_v upon_o agnostus_n or_o upon_o all_o which_o be_v equal_a unto_o it_o be_v to_o the_o solid_a set_v upon_o gd_o so_o be_v the_o same_o solid_a set_v upon_o agnostus_n or_o all_o to_o the_o solid_a set_v upon_o by_o wherefore_o by_o the_o 2._o part_n of_o the_o 9_o of_o the_o five_o the_o solid_n set_v upon_o gd_o and_o by_o shall_v be_v equal_a unto_o which_o solid_n if_o you_o add_v equal_a solid_n namely_o the_o solid_a set_v upon_o agnostus_n to_o the_o solid_a set_v upon_o gd_o and_o the_o solid_a set_v upon_o all_o to_o the_o solid_a set_v upon_o by_o the_o whole_a solid_n set_v upon_o the_o base_a ah_o and_o upon_o the_o base_a ab_fw-la âhall_v be_v equal_a wherefore_o parallelipedon_n consist_v upon_o equal_a base_n and_o be_v under_o one_o and_o the_o self_n same_o altitude_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 27._o theorem_a the_o 32._o proposition_n parallelipipedon_n be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v svppose_v that_o these_o parallelipipedon_n ab_fw-la and_o cd_o be_v under_o one_o &_o the_o self_n same_o altitude_n then_o i_o say_v that_o those_o parallelipipedon_n ab_fw-la and_o cd_o be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v that_o be_v that_o as_o the_o base_a ae_n be_v to_o the_o base_a cf_o so_o be_v the_o parallelipipedon_n ab_fw-la to_o the_o parallelipipedon_n cd_o construction_n upon_o the_o line_n fg_o describe_v by_o the_o 45._o of_o the_o first_o the_o parallelogram_n fh_o equal_a to_o the_o parallelogram_n ae_n and_o equiangle_n with_o the_o parallelogram_n cf._n and_o upon_o the_o base_a fh_o describe_v a_o parallelipipedom_n of_o the_o self_n same_o altitude_n that_o the_o parallelipipedom_n cd_o be_v &_o let_v the_o same_o be_v gk_o demonstration_n now_o by_o the_o 31._o of_o the_o eleven_o the_o parallelipipedon_n ab_fw-la be_v equal_a to_o the_o parallelipipedon_n gk_o for_o they_o consist_v upon_o equal_a base_n namely_o ae_n and_o fh_o and_o be_v under_o one_o and_o the_o self_n same_o altitude_n and_o forasmuch_o as_o the_o parallelipipedon_n ck_o be_v cut_v by_o a_o plain_a superficies_n dg_o be_v parallel_n to_o either_o of_o the_o opposite_a plain_a superâicieces_n therefore_o by_o the_o 25._o of_o the_o eleven_o as_o the_o base_a hf_o be_v to_o the_o base_a fc_n so_o be_v the_o parallelipipedon_n gk_o to_o parallelipipedon_v cd_o but_o the_o base_a hf_o be_v equal_a to_o the_o base_a ae_n and_o the_o parallelipipedon_n gk_o be_v prove_v equal_a to_o the_o parallelipipedon_n ab_fw-la wherefore_o as_o the_o base_a aâe_n be_v to_o the_o base_a cf_o so_o be_v the_o parallelipedon_n ab_fw-la to_o the_o parallelipipedon_n cd_o wherefore_o parallelipipedon_n be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v which_o be_v require_v to_o be_v demonstrate_v i_o need_v not_o to_o put_v any_o other_o figure_n for_o the_o declaration_n of_o this_o demonstration_n for_o it_o be_v easy_a to_o see_v by_o the_o figure_n there_o describe_v howbeit_o you_o may_v for_o the_o more_o full_a sight_n thereof_o describe_v solid_n of_o past_v paper_n according_a to_o the_o construction_n there_o set_v forth_o which_o will_v not_o be_v hard_o for_o you_o to_o do_v if_o you_o remember_v the_o description_n of_o such_o body_n before_o teach_v a_o corollary_n add_v by_o flussas_n equal_a parallelipipedon_n contain_v under_o one_o and_o the_o self_n same_o altitude_n have_v also_o their_o base_n equal_a for_o if_o the_o base_n shall_v be_v unequal_a the_o parallelipipedon_n also_o shall_v be_v unequal_a by_o this_o 32_o proposition_n and_o equal_a parallelipipedon_n have_v equal_a base_n have_v also_o one_o and_o the_o self_n same_o altitude_n for_o if_o they_o shall_v have_v a_o great_a altitude_n they_o shall_v exceed_v the_o equal_a parallelipipedon_n which_o have_v the_o self_n same_o altitude_n but_o if_o they_o shall_v have_v a_o less_o they_o shall_v want_v so_o much_o of_o those_o self_n same_o equal_a parallelipipedon_n the_o 28._o theorem_a the_o 33._o proposition_n like_a parallelipipedon_n be_v in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o their_o side_n of_o like_a proportion_n be_v svppose_v that_o these_o parallelipipedon_n ab_fw-la and_o cd_o be_v like_a &_o let_v the_o side_n ae_n and_o cf_o be_v side_n of_o like_a proportion_n then_o i_o say_v the_o parallelipipedon_n ab_fw-la be_v unto_o the_o parallelipipedon_n cd_o in_o treble_a proportion_n of_o that_o in_o which_o the_o side_n ae_n be_v to_o the_o side_n cf._n extend_v the_o right_a line_n ae_n ge_z and_o he_o to_o the_o point_n king_n l_o m._n construction_n and_o by_o the_o 2._o of_o the_o first_o unto_o the_o line_n cf_o put_v the_o line_n eke_o equal_a and_o unto_o the_o line_n fn_o put_v the_o line_n el_n equal_a and_o moreover_o unto_o the_o line_n fr_z put_v the_o line_n em_n equal_a and_o make_v perfect_a the_o parallelogram_n kl_o and_o the_o parallelipipedon_n ko_o demonstration_n now_o forasmuch_o as_o these_o two_o line_n eke_o and_o el_n be_v equal_a to_o these_o two_o line_n cf_o and_o fn_o but_o the_o angle_n kel_n be_v equal_a to_o the_o angle_n cfn_n for_o the_o angle_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 parââ_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
a_o triangle_n and_o if_o the_o parallelogram_n be_v double_a to_o the_o triangle_n those_o prism_n be_v by_o the_o 40._o of_o the_o eleven_o equal_v the_o one_o to_o the_o other_o therefore_o the_o prism_n contain_v under_o the_o two_o triangle_n bkf_a and_o ehg_n and_o under_o the_o three_o parallelogram_n ebfg_n ebkh_n and_o khfg_n part_n be_v equal_a to_o the_o prism_n contain_v under_o the_o two_o triangle_n gfc_a and_o hkl_n and_o under_o the_o three_o parallelogram_n kfcl_n lcgh_n and_o hkfg_n and_o it_o be_v manifest_a that_o both_o these_o prism_n of_o which_o the_o base_a of_o one_o be_v the_o parallelogram_n ebfg_n pyramid_n and_o the_o opposite_a unto_o it_o the_o line_n kh_o and_o the_o base_a of_o the_o other_o be_v the_o triangle_n gfc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n klh_n be_v great_a than_o both_o these_o pyramid_n who_o base_n be_v the_o triangle_n age_n and_o hkl_n and_o top_n the_o point_n h_o &_o d._n for_o if_o we_o draw_v these_o right_a line_n of_o and_o eke_o the_o prism_n who_o base_a be_v the_o parallelogram_n ebfg_n and_o the_o opposite_a unto_o it_o the_o right_a line_n hk_o be_v great_a than_o the_o pyramid_n who_o base_a be_v the_o triangle_n ebf_n &_o top_n the_o point_n k._n but_o the_o pyramid_n who_o base_a be_v the_o triangle_n ebf_n and_o top_n the_o point_n king_n be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h_o for_o they_o be_v contain_v under_o equal_a and_o like_a plain_a superficiece_n wherefore_o also_o the_o prism_n who_o base_a be_v the_o parallelogram_n ebfg_n and_o the_o opposite_a unto_o it_o the_o right_a line_n hk_o be_v great_a than_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h._n but_o the_o prism_n who_o base_a be_v the_o parallelogram_n ebfg_n and_o the_o opposite_a unto_o it_o the_o right_a line_n hk_o be_v equal_a to_o the_o prism_n who_o base_a be_v the_o triangle_n gfc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n hkl_n and_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h_o be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n hkl_n and_o top_n the_o point_n d._n part_n wherefore_o the_o foresay_a two_o prism_n be_v great_a than_o the_o foresay_a two_o pyramid_n who_o base_n be_v the_o triangle_n aeg_n hkl_n and_o top_n the_o point_n h_o and_o d._n wherefore_o the_o whole_a pyramid_n who_o base_a be_v the_o triangle_n abc_n proposition_n and_o top_n the_o point_n d_o be_v divide_v into_o two_o pyramid_n equal_a and_o like_v the_o one_o to_o the_o other_o and_o like_v also_o unto_o the_o whole_a pyramid_n have_v also_o triangle_n to_o their_o base_n and_o into_o two_o equal_a prism_n and_o the_o two_o prism_n be_v great_a than_o half_a of_o the_o whole_a pyramid_n which_o be_v require_v to_o be_v demonstrate_v if_o you_o will_v with_o diligence_n read_v these_o four_o book_n follow_v of_o euclid_n which_o concern_v body_n and_o clear_o see_v the_o demonstration_n in_o they_o contain_v it_o shall_v be_v requisite_a for_o you_o when_o you_o come_v to_o any_o proposition_n which_o concern_v a_o body_n or_o body_n whether_o they_o be_v regular_a or_o not_o first_o to_o describe_v of_o pâsâed_a paper_n according_a as_o i_o teach_v you_o in_o the_o end_n of_o the_o definition_n of_o the_o eleven_o book_n such_o a_o body_n or_o body_n as_o be_v there_o require_v and_o have_v your_o body_n or_o body_n thus_o describe_v when_o you_o have_v note_v it_o with_o letter_n according_a to_o the_o figure_n set_v forth_o upon_o a_o plain_a in_o the_o proposition_n follow_v the_o construction_n require_v in_o the_o proposition_n as_o for_o example_n in_o this_o three_o proposition_n it_o be_v say_v that_o every_o pyramid_n have_v a_o triangle_n to_o âis_n base_a may_v be_v divide_v into_o two_o pyramid_n etc_n etc_n here_o first_o describe_v a_o pyramid_n of_o past_v paper_n haâing_v his_o base_a triangle_v it_o skill_v not_o whether_o it_o be_v equilater_n or_o equiangle_v or_o not_o only_o in_o this_o proposition_n be_v require_v that_o the_o base_a be_v a_o triangle_n then_o for_o that_o the_o proposition_n suppose_v the_o base_a of_o the_o pyramid_n to_o be_v the_o triangle_n abc_n note_v the_o base_a of_o your_o pyramid_n which_o you_o have_v describe_v with_o the_o letter_n abc_n and_o the_o top_n of_o your_o pyramid_n with_o the_o letter_n d_o for_o so_o be_v require_v in_o the_o proposition_n and_o thus_o have_v you_o your_o body_n order_v ready_a to_o the_o construction_n now_o in_o the_o construction_n it_o be_v require_v that_o you_o divide_v the_o line_n ab_fw-la bc_o ca._n etc_o etc_o namely_o the_o six_o line_n which_o be_v the_o side_n of_o the_o four_o triangle_n contain_v the_o pyramid_n into_o two_o equal_a part_n in_o the_o poyntet_fw-la â_o f_o g_o etc_n etc_n that_o be_v you_o must_v divide_v the_o line_n ab_fw-la of_o your_o pyramid_n into_o two_o equal_a part_n and_o note_v the_o point_n of_o the_o division_n with_o the_o letter_n e_o and_o so_o the_o line_n bc_o be_v divide_v into_o two_o equal_a part_n note_v the_o point_n of_o the_o division_n with_o the_o letter_n f._n and_o so_o the_o rest_n and_o this_o order_n follow_v you_o as_o touch_v the_o rest_n of_o the_o construction_n there_o put_v and_o when_o you_o have_v finish_v the_o construction_n compare_v your_o body_n thus_o describe_v with_o the_o demonstration_n and_o it_o will_v make_v it_o very_o plain_a and_o easy_a to_o be_v understand_v whereas_o without_o such_o a_o body_n describe_v of_o matter_n it_o be_v hard_o for_o young_a beginner_n unless_o they_o have_v a_o very_a deep_a imagination_n full_o to_o conceive_v the_o demonstration_n by_o the_o sigâe_n as_o it_o be_v describe_v in_o a_o plain_a here_o for_o the_o better_a declaration_n of_o that_o which_o i_o have_v say_v have_v i_o set_v a_o figure_n who_o form_n if_o you_o describe_v upon_o past_v paper_n note_v with_o the_o like_a letter_n and_o cut_v the_o line_n âa_o dam_n dc_o and_o fold_v it_o according_o it_o will_v make_v a_o pyramid_n describe_v according_a to_o the_o construction_n require_v in_o the_o proposition_n and_o this_o order_n follow_v you_o as_o touch_v all_o other_o proposition_n which_o concern_v body_n ¶_o a_o other_o demonstration_n after_o campane_n of_o the_o 3._o proposition_n suppose_v that_o there_o be_v a_o pyramid_n abcd_o have_v to_o his_o base_a the_o triangle_n bcd_o and_o let_v his_o top_n be_v the_o solid_a angle_n a_o from_o which_o let_v there_o be_v draw_v three_o subtended_a line_n ab_fw-la ac_fw-la and_o ad_fw-la to_o the_o three_o angle_n of_o the_o base_a and_o divide_v all_o the_o side_n of_o the_o base_a into_o two_o equal_a part_n in_o the_o three_o point_n e_o f_o g_o divide_v also_o the_o three_o subtended_a line_n ab_fw-la ac_fw-la and_o ad_fw-la in_o two_o equal_a part_n in_o the_o three_o point_n h_o edward_n l._n and_o draw_v in_o the_o base_a these_o two_o line_n of_o and_o eglantine_n so_o shall_v the_o base_a of_o the_o pyramid_n be_v divide_v into_o three_o superficiece_n whereof_o two_o be_v the_o two_o triangle_n bef_n and_o egg_v which_o be_v like_o both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o whole_a base_a by_o the_o 2_o part_n of_o the_o second_o of_o the_o six_o &_o by_o the_o definition_n of_o like_a superâiciecâs_n &_o they_o be_v equal_v the_o one_o to_o the_o other_o by_o the_o 8._o of_o the_o first_o the_o three_o superficies_n be_v a_o quadrangled_a parallelogram_n namely_o efgc_n which_o be_v double_a to_o the_o triangle_n egg_v by_o the_o 40._o and_o 41._o of_o the_o first_o now_o then_o again_o from_o the_o point_n h_o draw_v unto_o the_o point_n e_o and_o f_o these_o two_o subtendent_fw-la line_n he_o and_o hf_o draw_v also_o a_o subtended_a line_n from_o the_o point_n king_n to_o the_o point_n g._n and_o draw_v these_o line_n hk_o kl_o and_o lh_o wherefore_o the_o whole_a pyramid_n abcd_o be_v divide_v into_o two_o pyramid_n which_o be_v hbef_n and_o ahkl_n and_o into_o two_o prism_n of_o which_o the_o one_o be_v ehfgkc_n and_o be_v set_v upon_o the_o quadrangled_a base_a cfge_n the_o other_o be_v egdhkl_n and_o have_v to_o his_o base_a the_o triangle_n egg_v now_o as_o touch_v the_o two_o pyramid_n hbef_a and_o ahkl_n that_o they_o be_v equal_v the_o one_o to_o the_o other_o and_o also_o that_o they_o be_v like_o both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o whole_a it_o be_v manifest_a by_o the_o definition_n of_o equal_a and_o like_a body_n and_o by_o the_o 10._o of_o the_o eleven_o and_o by_o 2._o part_n of_o the_o second_o of_o the_o six_o and_o that_o the_o two_o prism_n be_v equal_a it_o
be_v manifest_a by_o the_o last_o of_o the_o eleven_o and_o now_o that_o both_o the_o prism_n take_v together_o be_v great_a than_o the_o half_a of_o the_o whole_a pyramid_n hereby_o it_o be_v manifest_a for_o that_o either_o of_o they_o may_v be_v divide_v into_o two_o pyramid_n of_o which_o the_o one_o be_v a_o triangular_a pyramid_n equal_a to_o one_o of_o the_o two_o pyramid_n into_o which_o together_o with_o the_o two_o prism_n be_v divide_v the_o whole_a pyramid_n and_o the_o other_o be_v a_o quadrangled_a pyramid_n double_a to_o the_o other_o pyramid_n wherefore_o it_o be_v plain_a that_o the_o two_o prism_n take_v together_o be_v three_o quarter_n of_o the_o whole_a pyramid_n divide_v but_o if_o you_o be_v desirous_a to_o know_v the_o proportion_n between_o they_o read_v the_o â_o of_o this_o book_n but_o now_o here_o to_o this_o purpose_n it_o shall_v be_v sufficient_a to_o know_v that_o the_o two_o prism_n take_v together_o do_v exceed_v in_o quantity_n the_o two_o partial_a pyramid_n take_v together_o into_o which_o together_o with_o the_o two_o prism_n the_o whole_a pyramid_n be_v divide_v ¶_o the_o 4._o theorem_a the_o 4._o proposition_n if_o there_o be_v two_o pyramid_n under_o equal_a altitude_n have_v triangle_n to_o their_o base_n and_o either_o of_o those_o pyramid_n be_v divide_v into_o two_o pyramid_n equal_v the_o one_o to_o the_o other_o and_o like_v unto_o the_o whole_a and_o into_o two_o squall_n prism_n and_o again_o if_o in_o either_o of_o the_o pyramid_n make_v of_o the_o two_o first_o pyramid_n be_v still_o observe_v the_o same_o order_n and_o manner_n then_o as_o the_o base_a of_o the_o one_o pyramid_n be_v to_o the_o base_a of_o the_o other_o pyramid_n so_o be_v all_o the_o prism_n which_o be_v in_o the_o one_o pyramid_n to_o all_o the_o prism_n which_o be_v in_o the_o other_o be_v equal_a in_o multitude_n with_o they_o svppose_v that_o there_o be_v two_o pyramid_n under_o equal_a altitude_n have_v triangle_n to_o their_o base_n namely_o abc_n and_o def_n and_o have_v to_o their_o top_n the_o point_n g_o and_o h._n and_o let_v either_o of_o these_o pyramid_n be_v divide_v into_o two_o pyramid_n equal_v the_o one_o to_o the_o other_o and_o like_v unto_o the_o whole_a and_o into_o two_o equal_a prism_n according_a to_o the_o method_n of_o the_o former_a proposition_n and_o again_o let_v either_o of_o those_o pyramid_n soâ_n make_v of_o the_o two_o first_o pyramid_n be_v imagine_v to_o be_v after_o the_o same_o order_n divide_v and_o so_o do_v continual_o then_o i_o say_v that_o as_o the_o base_a abc_n be_v to_o the_o base_a def_n so_o be_v all_o the_o prism_n which_o be_v in_o the_o pyramid_n abcg_o to_o all_o the_o prism_n which_o be_v in_o the_o pyramid_n defh_o be_v equal_a in_o multitude_n with_o they_o for_o forasmuch_o as_o the_o line_n bx_n be_v equal_a to_o the_o line_n xc_o and_o the_o line_n all_o to_o the_o line_n lc_n for_o as_o we_o see_v in_o the_o construction_n pertain_v to_o the_o former_a proposition_n all_o the_o six_o side_n of_o the_o whole_a pyramid_n be_v each_o divide_v into_o two_o equal_a part_n the_o like_a of_o which_o construction_n be_v in_o this_o proposition_n also_o suppose_v therefore_o the_o line_n xl_o be_v a_o parallel_n to_o the_o line_n ab_fw-la &_o the_o triangle_n abc_n be_v like_a to_o the_o triangle_n lxc_n by_o the_o corollary_n of_o the_o second_o of_o the_o six_o and_o by_o the_o same_o reason_n the_o triangle_n def_n be_v like_a to_o the_o triangle_n rwf_n and_o forasmuch_o as_o the_o line_n bc_o be_v double_a to_o the_o line_n cx_o and_o the_o line_n fe_o to_o the_o line_n fw_o therefore_o as_o the_o line_n bc_o be_v to_o the_o line_n cx_o so_o be_v the_o line_n of_o to_o the_o line_n fw_o and_o upon_o the_o line_n bc_o and_o cx_o be_v describe_v rectiline_a figure_n like_a and_o in_o like_a sort_n set_v namely_o the_o triangle_n abc_n and_o lxc_n and_o upon_o the_o line_n of_o and_o fw_o be_v also_o describe_v rectiline_a figure_n like_a and_o in_o like_a sort_n set_v namely_o the_o triangle_n def_n &_o rwfâ_n but_o if_o there_o be_v four_o right_a line_n proportional_a the_o rectiline_a figure_n describe_v of_o they_o be_v like_a and_o in_o like_a sort_n set_v shall_v also_o be_v proportional_a by_o the_o 22._o of_o the_o six_o wherefore_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n lxc_n so_o be_v the_o triangle_n def_n to_o the_o triangle_n rwf_n wherefore_o alternate_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 triangle_n def_n so_o be_v the_o triangle_n lxc_n to_o the_o triangle_n rwf_n assumpt_n but_o as_o the_o triangle_n lxc_n be_v to_o the_o triangle_n rwf_n so_o be_v the_o prism_n who_o base_a be_v the_o triangle_n lxc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n omnes_n to_o the_o prism_n who_o base_a be_v the_o triangle_n rwf_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n stv_n by_o the_o corollary_n of_o the_o 40._o of_o the_o eleven_o for_o these_o prism_n be_v under_o one_o &_o the_o self_n same_o altitude_n namely_o under_o the_o half_a of_o the_o altitude_n of_o the_o whole_a pyramid_n which_o pyramid_n be_v suppose_v to_o be_v under_o one_o and_o the_o self_n same_o altitude_n this_o be_v also_o prove_v in_o the_o assumpt_n follow_v wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n def_n so_o be_v the_o prism_n who_o base_a be_v the_o triangle_n lxc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n omnes_n to_o the_o prism_n who_o base_a be_v the_o triangle_n rwf_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n stv_n and_o forasmuch_o as_o there_o be_v two_o prism_n in_o the_o pyramid_n abcg_o equal_a the_o one_o to_o the_o other_o &_o two_o prism_n also_o in_o the_o pyramid_n defh_o equal_a the_o one_o to_o the_o other_o therefore_o as_o the_o prism_n who_o base_a be_v the_o parallelogram_n bklx_fw-fr and_o the_o opposite_a side_n unto_o it_o the_o line_n more_n be_v to_o the_o prism_n who_o base_a be_v the_o triangle_n lxc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n omnes_n so_o be_v the_o prism_n who_o base_a be_v the_o parallelogram_n perw_n and_o the_o opposite_a unto_o it_o the_o line_n n-ab_n to_o the_o prism_n who_o base_a be_v the_o triangle_n rwf_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n stv_n wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o the_o prism_n kbxlmo_n &_o lxcmno_n be_v to_o the_o prism_n lxcmno_n so_o be_v the_o prism_n pewrst_a and_o rwfstv_n to_o the_o prism_n rwfstv_n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o two_o prism_n kbxlmo_n and_o lxcmno_n be_v to_o the_o two_o prism_n pewrst_a and_o rwfstv_n so_o be_v the_o prism_n lxcmno_n to_o the_o prism_n rwfstv_n but_o as_o the_o prism_n lxcmno_n be_v to_o the_o prism_n rwfstv_n so_o have_v we_o prove_v that_o the_o base_a lxc_n be_v to_o the_o base_a rwf_n and_o the_o base_a abc_n to_o the_o base_a def_n wherefore_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 triangle_n def_n so_o be_v both_o the_o prism_n which_o be_v in_o the_o pyramid_n abcg_o to_o both_o the_o prism_n which_o be_v in_o the_o pyramid_n defh_o and_o in_o like_a sort_n if_o we_o divide_v the_o other_o pyramid_n after_o the_o self_n same_o manner_n namely_o the_o pyramid_n omng_v and_o the_o pyramid_n stuh_o as_o the_o base_a omnes_n be_v to_o the_o base_a stv_n so_o shall_v both_o the_o prism_n that_o be_v in_o the_o pyramid_n omng_v be_v to_o both_o the_o prism_n which_o be_v in_o the_o pyramid_n stuh_o but_o as_o the_o base_a omnes_n be_v to_o the_o base_a stv_n so_o be_v the_o base_a abc_n to_o the_o base_a def_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o base_a abc_n be_v to_o the_o base_a def_n so_o be_v the_o two_o prism_n that_o be_v in_o the_o pyramid_n abcg_o to_o the_o two_o prism_n that_o be_v in_o the_o pyramid_n defh_o and_o the_o two_o prism_n that_o be_v in_o the_o pyramid_n omng_v to_o the_o two_o prism_n that_o be_v in_o the_o pyramid_n stuh_o and_o the_o four_o prism_n to_o the_o four_o prism_n and_o so_o also_o shall_v it_o follow_v in_o the_o prism_n make_v by_o divide_v the_o two_o pyramid_n aklo_n and_o dpr_n and_o of_o all_o the_o other_o pyramid_n in_o general_a be_v equal_a in_o multitude_n andrea_n that_o as_o the_o triangle_n lxc_n be_v to_o the_o triangle_n rwf_n so_o be_v the_o prism_n who_o base_a be_v the_o triangle_n lxc_n and_o the_o opposite_a side_n omnes_n assumpt_n to_o the_o prism_n who_o base_a be_v the_o triangle_n rwf_n and_o the_o
with_o the_o altitude_n of_o the_o say_a pyramid_n a_o and_o b_o shall_v be_v equal_a by_o the_o 6._o of_o this_o book_n wherefore_o by_o the_o first_o part_n of_o this_o proposition_n the_o base_n of_o the_o pyramid_n c_o to_o d_z be_v reciprocal_a with_o the_o altitude_n of_o d_o to_o c._n but_o in_o what_o proportion_n be_v the_o base_n c_o to_o d_z in_o the_o same_o be_v the_o base_n a_o to_o b_o forasmuch_o as_o they_o be_v equal_a and_o in_o what_o proportion_n be_v the_o altitude_n of_o d_z to_o c_z in_o the_o same_o be_v the_o altitude_n of_o b_o to_o a_o which_o altitude_n be_v likewise_o equal_a wherefore_o by_o the_o 11._o of_o the_o five_o in_o what_o proportion_n the_o base_n a_o to_o b_o be_v in_o the_o same_o reciprocal_a be_v the_o altitude_n of_o the_o pyramid_n b_o to_o a._n in_o like_a sort_n by_o the_o second_o part_n of_o this_o proposition_n may_v be_v prove_v the_o converse_n of_o this_o corollary_n the_o same_o thing_n follow_v also_o in_o a_o prism_n and_o in_o a_o side_v column_n as_o have_v before_o at_o large_a be_v declare_v in_o the_o corollary_n of_o the_o 40._o proposition_n of_o the_o 11._o book_n for_o those_o solid_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o pyramid_n or_o parallelipipedon_n for_o they_o be_v either_o part_n of_o equemultiplices_fw-la or_o equemultiplices_fw-la to_o part_n the_o 10._o theorem_a the_o 10._o proposition_n every_o cone_n be_v the_o three_o part_n of_o a_o cilind_a have_v one_o and_o the_o self_n same_o base_a and_o one_o and_o the_o self_n same_o altitude_n with_o it_o svppose_v that_o there_o be_v a_o cone_fw-mi have_v to_o his_o base_a the_o circle_n abcd_o and_o let_v there_o be_v a_o cilind_a have_v the_o self_n same_o base_a and_o also_o the_o same_o altitude_n that_o the_o cone_fw-mi have_v then_o i_o say_v that_o the_o cone_fw-mi be_v the_o three_o part_n of_o the_o cilind_a that_o be_v that_o the_o cilind_a be_v in_o treble_a proportion_n to_o the_o cone_n for_o if_o the_o cilind_a be_v not_o in_o treble_a proportion_n to_o the_o cone_n than_o the_o cilind_a be_v either_o in_o great_a proportion_n then_o triple_a to_o the_o cone_n or_o else_o in_o less_o first_o let_v it_o be_v in_o great_a than_o triple_a ãâã_d and_o describe_v by_o the_o 6._o of_o the_o four_o in_o the_o circle_n abcd_o a_o square_a abcd._n now_o the_o square_n abcd_o be_v great_a than_o the_o half_a of_o the_o circle_n abcd_o for_o if_o about_o the_o circle_n abcd_o we_o describe_v a_o square_a the_o square_n describe_v in_o the_o circle_n abcd_o be_v the_o half_a of_o the_o square_n describe_v about_o the_o circle_n and_o let_v there_o be_v parallelipipedon_n prism_n describe_v upon_o those_o square_n prism_n equal_a in_o altitude_n with_o the_o cilind_a but_o prism_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v by_o the_o 32._o of_o the_o eleven_o and_o 5._o corollary_n of_o the_o 7._o of_o this_o book_n wherefore_o the_o prism_n describe_v upon_o the_o square_a abcd_o be_v the_o half_a of_o the_o prism_n describe_v upon_o the_o square_a that_o be_v describe_v about_o the_o circle_n now_o the_o clind_a be_v less_o than_o the_o prism_n which_o be_v make_v of_o the_o square_n describe_v abouâ_n the_o circle_n abcd_o be_v equal_a in_o altitude_n with_o it_o for_o it_o contain_v it_o wherefore_o the_o prism_n describe_v upon_o the_o square_a abcd_o and_o be_v equal_a in_o altitude_n with_o the_o cylinder_n be_v great_a than_o half_a the_o cylinder_n divide_v by_o the_o 30._o of_o the_o three_o the_o circumference_n ab_fw-la bc_o cd_o and_o da_z into_o two_o equal_a part_n in_o the_o point_n e_o f_o g_o h_o and_o draw_v these_o right_a line_n ae_n ebb_n bf_o fc_o cg_o gd_o dh_o &_o ha._n wherefore_o every_o one_o of_o these_o triangle_n aeb_fw-mi bfc_n cgd_v and_o dha_n be_v great_a than_o half_a of_o that_o segment_n of_o the_o circle_n abcd_o which_o be_v describe_v about_o it_o as_o we_o have_v before_o in_o the_o 2._o proposition_n declare_v describe_v upon_o every_o one_o of_o these_o triangle_n aeb_fw-mi bfc_n cgd_v and_o dha_n a_o prism_n of_o equal_a altitude_n with_o the_o cylinder_n wherefore_o every_o one_o of_o these_o prism_n so_o describe_v be_v great_a than_o the_o half_a part_n of_o the_o segment_n of_o the_o cylinder_n that_o be_v set_v upon_o the_o say_a segment_n of_o the_o circle_n for_o if_o by_o the_o point_n e_o f_o g_o h_o be_v draw_v parallel_a line_n to_o the_o line_n ab_fw-la bc_o cd_o and_o da_z and_o then_o be_v make_v perfect_a the_o parallelogram_n make_v by_o those_o parallel_a line_n and_o moreover_o upon_o those_o parallelogram_n be_v erect_v parallelipipedon_n equal_v in_o altitude_n with_o the_o cylinder_n the_o prism_n which_o be_v describe_v upon_o each_o of_o the_o triangle_n aeb_fw-mi bfc_n cgd_v and_o dha_n be_v the_o half_n of_o every_o one_o of_o those_o parallelipipedon_n and_o the_o segment_n of_o the_o cylinder_n be_v less_o than_o those_o parallelipipedon_n so_o describe_v wherefore_o also_o every_o one_o of_o the_o prism_n which_o be_v describe_v upon_o the_o triangle_n aeb_fw-mi bfc_n cgd_v and_o dha_n be_v great_a than_o the_o half_a of_o the_o segment_n of_o the_o cylinder_n set_n upon_o the_o say_a segment_n now_o therefore_o devide_v every_o one_o of_o the_o circumference_n remain_v into_o two_o equal_a part_n and_o draw_v right_a line_n and_o raise_v up_o upon_o every_o one_o of_o these_o triangle_n prism_n equal_a in_o altitude_n with_o the_o cylinder_n and_o do_v this_o continual_o we_o shall_v at_o the_o length_n by_o the_o first_o of_o the_o ten_o leave_v certain_a segment_n of_o the_o cylinder_n which_o shall_v be_v less_o then_o the_o excess_n whereby_o the_o cylinder_n exceed_v the_o cone_n more_o than_o thrice_o let_v those_o segment_n be_v ae_n ebb_n bf_o fc_o cg_o gd_o dh_o and_o ha._n wherefore_o the_o prism_n remain_v who_o base_a be_v the_o polygonon_n âigure_n aebfcgdh_n and_o altitude_n the_o self_n same_o that_o the_o cylinder_n have_v be_v great_a than_o the_o cone_n take_v three_o time_n prism_n but_o the_o prism_n who_o base_a be_v the_o polygonon_n figure_n aebfcgdh_n and_o altitude_n the_o self_n same_o that_o the_o cylinder_n have_v be_v treble_a to_o the_o pyramid_n who_o base_a be_v the_o polygonon_n figure_n aebfcgda_n and_o altitude_n the_o self_n same_o that_o the_o cone_fw-mi have_v by_o the_o corollary_n of_o the_o 3._o of_o this_o book_n wherefore_o also_o the_o pyramid_n who_o base_a be_v the_o polygonon_n figure_n aebfcgdh_n and_o top_n the_o self_n same_o that_o the_o cone_fw-mi have_v be_v great_a than_o the_o cone_n which_o have_v to_o his_o base_a the_o circle_n abcd._n but_o it_o be_v also_o less_o for_o it_o be_v contain_v of_o it_o which_o be_v impossible_a wherefore_o the_o cylinder_n be_v not_o in_o great_a proportion_n then_o triple_a to_o the_o cone_n i_o say_v moreover_o that_o the_o cylinder_n be_v not_o in_o less_o proportion_n then_o triple_a to_o the_o coneâ_n for_o if_o it_o be_v possible_a let_v the_o cylinder_n be_v in_o less_o proportion_n then_o triple_a to_o the_o cone_n wherefore_o by_o conversion_n the_o cone_n be_v great_a than_o the_o three_o part_n of_o the_o cylinder_n describe_v now_o by_o the_o six_o of_o the_o four_o in_o the_o circle_n abcd_o a_o square_a abcd._n wherefore_o the_o square_a abcd_o be_v great_a than_o the_o half_a of_o the_o circle_n abcd_o upon_o the_o square_n abcd_o describe_v a_o pyramid_n have_v one_o &_o the_o self_n same_o altitude_n with_o the_o cone_n wherefore_o the_o pyramid_n so_o describe_v be_v great_a they_o half_n of_o the_o cone_fw-it for_o if_o as_o we_o have_v before_o declare_v we_o describe_v a_o square_a about_o the_o circle_n the_o square_a abcd_o be_v the_o half_a of_o the_o square_n describe_v about_o the_o circle_n and_o if_o upon_o the_o square_n be_v describe_v parallelipipedon_n equal_v in_o altitude_n with_o the_o cone_n which_o solid_n be_v also_o call_v prism_n the_o prism_n or_o parallelipipedon_n describe_v upon_o the_o square_a abcd_o be_v the_o half_a of_o the_o prism_n which_o be_v describe_v upon_o the_o square_n describe_v about_o the_o circle_n for_o they_o be_v the_o one_o to_o the_o other_o in_o that_o proportion_n that_o their_o base_n be_v by_o the_o 32._o of_o the_o eleven_o &_o 5._o corollary_n of_o the_o 7._o of_o this_o book_n wherefore_o also_o their_o three_o part_n be_v in_o the_o self_n same_o proportion_n by_o the_o 15._o of_o the_o five_o wherefore_o the_o pyramid_n who_o base_a be_v the_o square_a abcd_o be_v the_o half_a of_o the_o pyramid_n set_v upon_o the_o square_n describe_v about_o the_o circle_n but_o the_o pyramid_n set_v upon_o the_o square_n describe_v about_o the_o circle_n be_v great_a than_o the_o cone_n who_o
place_n from_o whence_o first_o it_o begin_v to_o be_v move_v it_o shall_v pass_v by_o the_o point_n f_o g_o h_o and_o the_o octohedron_n shall_v be_v comprehend_v in_o a_o sphere_n i_o say_v also_o that_o it_o be_v comprehend_v in_o the_o sphere_n give_v for_o forasmuch_o as_o the_o line_n lk_n be_v equal_a to_o the_o line_n km_o by_o position_n and_o the_o line_n ke_o be_v common_a to_o they_o both_o and_o they_o contain_v right_a angle_n by_o the_o 3._o definition_n of_o the_o eleven_o therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a le_fw-fr be_v equal_a to_o the_o base_a em_n and_o forasmuch_o as_o the_o angle_n lem_n be_v a_o right_a angle_n by_o the_o 31._o of_o the_o three_o for_o it_o be_v in_o a_o semicircle_n as_o have_v be_v prove_v therefore_o the_o square_a of_o the_o line_n lm_o be_v double_a to_o the_o square_n of_o the_o line_n le_fw-fr by_o the_o 47._o of_o the_o first_o again_o forasmuch_o as_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n bc_o therefore_o the_o line_n ab_fw-la be_v double_a to_o the_o line_n bc_o by_o the_o definition_n of_o a_o circle_n but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n bd_o by_o the_o corollary_n of_o the_o 8._o and_o _o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v double_a to_o the_o square_n of_o the_o line_n bd._n and_o it_o be_v prove_v that_o the_o square_a of_o the_o line_n lm_o be_v double_a to_o the_o square_n of_o the_o line_n le._n wherefore_o the_o square_a of_o the_o line_n bd_o be_v equal_a to_o the_o square_n of_o the_o line_n le._n for_o the_o line_n eh_o which_o be_v equal_a to_o the_o line_n lf_n be_v put_v to_o be_v equal_a to_o the_o line_n db._n wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n lm_o wherefore_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n lm_o and_o the_o line_n ab_fw-la be_v the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o line_n lm_o be_v equal_a to_o the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o octoedron_n be_v contain_v in_o the_o sphere_n give_v le._n and_o it_o be_v also_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n wherefore_o there_o be_v make_v a_o octohedron_n and_o it_o be_v comprehend_v in_o the_o sphere_n give_v wherein_o be_v comprehend_v the_o pyramid_n and_o it_o be_v prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedrn_n which_o be_v require_v to_o be_v do_v and_o to_o be_v prove_v certain_a corollary_n add_v by_o flussas_n first_o corollary_n the_o side_n of_o a_o pyramid_n be_v in_o power_n sesquitertia_fw-la to_o the_o side_n of_o a_o octahedron_n inscribe_v in_o the_o same_o sphere_n for_o forasmch_v as_o the_o diameter_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n therefore_o of_o what_o part_n the_o diameter_n contain_v in_o power_n 6._o of_o the_o same_o the_o side_n of_o the_o octohedron_n contain_v in_o power_n 3._o but_o of_o what_o part_n the_o diameter_n contain_v 6._o of_o the_o same_o the_o side_n of_o the_o pyramid_n contain_v 4._o by_o the_o 13._o of_o this_o book_n wherefore_o of_o what_o part_n the_o side_n of_o the_o pyramid_n contain_v 4._o of_o the_o same_o the_o side_n of_o the_o octohedron_n contain_v 3._o second_o corollary_n a_o octohedron_n be_v divide_v into_o two_o equal_a and_o like_a pyramid_n the_o common_a base_n of_o these_o pyramid_n be_v set_v upon_o every_o square_n contain_v of_o the_o side_n of_o the_o octohedron_n upon_o which_o square_a be_v set_v the_o ââ_o triangle_n of_o the_o octohedron_n campane_n which_o pyramid_n be_v by_o the_o â_o definition_n of_o the_o eleven_o equal_a and_o like_a and_o the_o foresay_a square_n common_a to_o those_o pyramid_n be_v the_o half_a of_o the_o square_n of_o the_o diameter_n of_o the_o sphere_n for_o it_o be_v the_o square_a of_o the_o side_n of_o the_o octohedron_n three_o corollary_n the_o three_o diameter_n of_o the_o octohedron_n do_v cut_v the_o one_o the_o other_o perpendicular_o into_o two_o equal_a part_n in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o say_a octohedron_n as_o it_o be_v manifest_a by_o the_o three_o diameter_n eglantine_n fh_o and_o lm_o which_o cut_v the_o one_o the_o other_o in_o the_o centre_n king_n equal_o and_o perpendicular_o ¶_o the_o 3._o problem_n the_o 15._o proposition_n to_o make_v a_o solid_a call_v a_o cube_fw-la and_o to_o comprehend_v it_o in_o the_o sphere_n give_v namely_o that_o sphere_n wherein_o the_o former_a two_o solid_n be_v comprehendâdâ_n and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n treble_a to_o the_o side_n of_o the_o cube_fw-la take_v the_o diameter_n of_o the_o sphere_n give_v namely_o ab_fw-la and_o divide_v it_o in_o the_o point_n câ_n so_o that_o let_v the_o line_n ac_fw-la be_v double_a to_o the_o line_n bc_o by_o the_o 9_o of_o the_o six_o and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n adb_o and_o by_o the_o 11._o of_o the_o first_o from_o the_o pâynt_n c_o râyse_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n cd_o and_o draw_v a_o right_a linâ_n db._n and_o describe_v a_o square_a efgh_o have_v every_o one_o of_o his_o side_n equal_a to_o the_o line_n db_o and_o from_o the_o point_n e_o f_o g_o h_o raise_v up_o by_o the_o 12._o of_o the_o eleven_o unto_o the_o plain_a superficies_n of_o the_o square_a efgh_o perpendicular_a line_n eke_o fl_fw-mi gm_n and_o hn_o and_o let_v every_o one_o of_o the_o line_n eke_o fl_fw-mi gm_n and_o hn_o be_v put_v equal_a to_o one_o of_o the_o line_n of_o fg_o gh_o or_o he_o which_o be_v the_o side_n of_o the_o square_n and_o draw_v these_o right_a line_n kl_o lm_o mn_a and_o nk_v demonstration_n wherefore_o there_o be_v make_v a_o cube_fw-la namely_o fn_o which_o be_v contain_v under_o six_o equal_a square_n construction_n now_o it_o be_v require_v to_o comprehend_v the_o same_o cube_fw-la in_o the_o sphere_n give_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n ble_o to_o the_o side_n of_o the_o cube_fw-la demonstration_n draw_v these_o right_a line_n kg_v and_o eglantine_n and_o forasmuch_o as_o the_o angle_n keg_n be_v a_o right_a angle_n for_o that_o the_o line_n ke_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n eâ_n and_o therefore_o also_o to_o the_o right_a line_n eglantine_n by_o the_o 2._o definition_n of_o the_o eleven_o wherefore_o a_o semicircle_n describe_v upon_o the_o line_n kg_v shall_v book_n pass_v by_o the_o point_n e._n again_o forasmuch_o as_o the_o line_n fg_o be_v erect_v perpendicular_o to_o either_o of_o these_o line_n fl_fw-mi and_o fe_o by_o the_o definition_n of_o a_o square_a &_o by_o the_o 2._o definition_n of_o the_o eleven_o therefore_o the_o line_n fg_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n fk_o by_o the_o 4._o of_o the_o eleven_o wherefore_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n f_o to_o the_o point_n king_n the_o line_n gf_o shall_v be_v erect_v perpendicular_o to_o the_o line_n kf_o by_o the_o 2._o definition_n of_o the_o eleven_o and_o by_o the_o same_o reason_n again_o a_o semicircle_n describe_v upon_o the_o line_n gk_o shall_v pass_v also_o by_o the_o point_n f._n and_o likewise_o shall_v it_o pass_v by_o the_o rest_n of_o the_o point_n of_o the_o angle_n of_o that_o cube_fw-la if_o now_o the_o diameter_n kg_v abide_v fix_v the_o semicircle_n be_v turn_v round_o about_o until_o it_o return_v into_o the_o self_n same_o place_n from_o whence_o it_o begin_v first_o to_o be_v move_v the_o cube_fw-la shall_v be_v comprehend_v in_o a_o sphere_n i_o say_v also_o that_o it_o be_v comprehend_v in_o the_o sphere_n give_v demonstration_n for_o forasmuch_o as_o the_o line_n gf_o be_v equal_a to_o the_o linââe_n and_o the_o angle_n fletcher_n be_v a_o right_a angle_n therefore_o the_o square_a of_o the_o line_n eglantine_n be_v by_o the_o 47._o of_o the_o first_o double_a to_o the_o square_n of_o the_o line_n âf_n but_o the_o line_n of_o be_v equal_a to_o the_o line_n eke_o wherefore_o the_o square_a of_o the_o line_n eglantine_n be_v double_a to_o the_o square_n of_o the_o line_n eke_o wherefore_o the_o square_n of_o eglantine_n and_o eke_o that_o be_v the_o square_a of_o the_o line_n gk_o by_o the_o 47._o of_o the_o first_o be_v treble_a to_o the_o square_n of_o the_o line_n eke_o and_o forasmuch_o as_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n bc_o but_o
as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n bd_o by_o the_o corollary_n of_o the_o 8._o and_o 20._o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v treble_a to_o the_o square_n of_o the_o line_n bd._n and_o it_o be_v prove_v that_o the_o square_a of_o the_o line_n gk_o be_v treble_a to_o the_o square_n of_o the_o line_n ke_o and_o the_o line_n ke_o be_v put_v equal_a to_o the_o line_n bd._n wherefore_o the_o line_n kg_n be_v also_o equal_a to_o the_o line_n ab_fw-la and_o the_o line_n ab_fw-la be_v the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o line_n kg_n be_v equal_a to_o the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o cube_fw-la be_v comprehend_v in_o the_o sphere_n give_v and_o it_o be_v also_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n treble_a to_o the_o side_n of_o the_o cube_fw-la which_o be_v require_v tââe_o do_v and_o to_o be_v prove_v an_o other_o demonstration_n after_o flussas_n suppose_v that_o the_o diameter_n of_o the_o sphere_n give_v in_o the_o former_a proposition_n be_v the_o line_n aâ_z and_o let_v the_o centre_n be_v the_o point_n c_o upon_o which_o describe_v a_o semicircle_n adb_o and_o from_o the_o diameter_n ab_fw-la cut_v of_o a_o three_o part_n bg_o by_o the_o 9_o of_o the_o six_o and_o from_o the_o point_n g_o raise_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n dg_o by_o the_o 11._o of_o the_o first_o and_o draw_v these_o right_a line_n dam_n dc_o and_o db._n and_o unto_o the_o right_a line_n db_o put_v a_o equal_a right_a line_n zi_n and_o upon_o the_o line_n zi_n describe_v a_o square_a ezit_n and_o from_o the_o point_n e_o z_o i_o t_o erect_v unto_o the_o superficies_n ezit_n perpendicular_a line_n eke_o zh_n in_o tn_n by_o the_o 12._o of_o the_o eleven_o and_o put_v every_o one_o of_o those_o perpendicular_a line_n equal_a to_o the_o line_n zi_n and_o draw_v these_o right_a line_n kh_o hm_n mn_o and_o nk_v each_o of_o which_o shall_v be_v equal_a and_o parallel_n to_o the_o line_n zi_n and_o to_o the_o rest_n of_o the_o line_n of_o the_o square_n by_o the_o 33._o of_o the_o first_o and_o moreover_o they_o shall_v contain_v equal_a angle_n by_o the_o 10._o of_o the_o eleven_o and_o therefore_o the_o angle_n be_v right_a angle_n for_o that_o ezit_n be_v a_o square_n wherefore_o the_o rest_n of_o the_o base_n shall_v be_v square_n wherefore_o the_o solid_a ezitkhmn_n be_v contain_v under_o 6._o equal_a square_n be_v a_o cube_fw-la by_o the_o 21._o definition_n of_o the_o eleven_o extend_v by_o the_o opposite_a side_n ke_o and_o mi_n of_o the_o cube_fw-la a_o plain_a keim_n and_o again_o by_o the_o other_o opposite_a side_n nt_v and_o hz_n extend_v a_o other_o plain_a hztn_n now_o forasmuch_o as_o each_o of_o these_o plain_n divide_v the_o solid_a into_o two_o equal_a part_n namely_o into_o two_o prism_n equal_a and_o like_a by_o the_o 8._o definition_n of_o the_o eleven_o therefore_o those_o plain_n shall_v cut_v the_o cube_fw-la by_o the_o centre_n by_o the_o corollary_n of_o the_o 39_o of_o the_o eleven_o wherefore_o the_o common_a section_n of_o those_o plain_n shall_v pass_v by_o the_o centre_n let_v that_o common_a section_n be_v the_o line_n lf_n and_o forasmuch_o as_o the_o side_n hn_o and_o km_o of_o the_o superficiece_n keim_n and_o hztn_n do_v divide_v the_o one_o the_o other_o into_o two_o equal_a part_n by_o the_o corollary_n of_o the_o 34._o of_o the_o first_o and_o so_o likewise_o do_v the_o side_n zt_v and_o ei_o therefore_o the_o common_a section_n lf_o be_v draw_v by_o these_o section_n and_o divide_v the_o plain_n keim_n and_o hztn_n into_o two_o equal_a part_n by_o the_o first_o of_o the_o six_o for_o their_o base_n be_v equal_a and_o the_o altitude_n be_v one_o and_o the_o âame_n namely_o the_o altitude_n of_o the_o cube_fw-la wherefore_o the_o line_n lf_n shall_v divide_v into_o two_o equal_a part_n the_o diameter_n of_o his_o plain_n namely_o the_o right_a line_n ki_v em_n zn_n and_o nt_v which_o be_v the_o diameter_n of_o the_o cube_fw-la wherefore_o those_o diameter_n shall_v concur_v and_o cut_v one_o the_o other_o in_o one_o and_o the_o self_n same_o point_n let_v the_o same_o be_v o._n wherefore_o the_o right_a line_n ok_n oe_n oi_o om_o oh_o oz_n ot_fw-mi and_o on_o shall_v be_v equal_a the_o onâ_n to_o the_o other_o for_o that_o they_o be_v the_o half_n of_o the_o diameter_n of_o equal_a and_o like_a rectangle_n parallelogram_n wherefore_o make_v the_o centre_n the_o point_n oh_o and_o the_o space_n any_o of_o these_o line_n oe_z or_o ok_n etc_n etc_n a_o sphere_n describe_v shall_v pass_v by_o every_o one_o of_o the_o angle_n of_o the_o cube_fw-la namely_o which_o be_v at_o the_o point_n e_o z_o i_o t_o edward_n h_o m_o n_o by_o the_o 12._o definition_n of_o the_o eleven_o for_o that_o all_o the_o line_n draw_v from_o the_o point_n oh_o to_o the_o angle_n of_o the_o cube_fw-la be_v equal_a but_o the_o right_a line_n ei_o contain_v in_o power_n the_o two_o equal_a right_a line_n ez_o and_o zi_n by_o the_o 47._o of_o the_o first_o wherefore_o the_o square_a of_o the_o line_n ei_o be_v double_a to_o the_o square_n of_o the_o line_n zi_n and_o forasmuch_o as_o the_o right_a line_n ki_v subtend_v the_o right_a angle_n kei_n for_o that_o the_o right_a line_n keâ_n be_v erect_v perpendicular_o to_o the_o plaiâe_a superficies_n of_o the_o right_a line_n ez_o and_o zt_n by_o the_o 4._o of_o the_o eleven_o â_o therefore_o the_o square_a of_o the_o line_n ki_v be_v equal_a to_o the_o square_n of_o the_o line_n ei_o and_o eke_o but_o the_o square_n of_o the_o line_n ei_o be_v double_a to_o the_o square_n of_o the_o line_n eke_o for_o it_o be_v double_a to_o the_o square_n of_o the_o line_n zi_n as_o have_v be_v prove_v and_o the_o base_n of_o the_o cube_fw-la be_v equal_a square_n wherefore_o the_o square_a of_o the_o line_n ki_v be_v triple_a to_o the_o square_n of_o the_o line_n ke_o that_o be_v to_o the_o square_n of_o the_o line_n zi_n but_o the_o right_a line_n zi_n be_v equal_a to_o thâ_z right_a line_n db_o by_o position_n unto_o who_o square_a the_o square_n of_o the_o diameter_n ab_fw-la be_v triple_a by_o that_o which_o be_v demonstrate_v in_o the_o 13._o proposition_n of_o this_o book_n wherefore_o the_o diameter_n ki_v &_o db_o be_v equal_a wherefore_o there_o be_v describe_v a_o cube_fw-la king_n and_o it_o be_v comprehend_v in_o the_o sphere_n give_v wherein_o the_o other_o solid_n be_v contain_v the_o diameter_n of_o which_o sphere_n be_v the_o line_n ab_fw-la and_o the_o diameter_n ki_v or_o ab_fw-la of_o the_o same_o sphere_n be_v prove_v to_o be_v in_o power_n triple_a to_o the_o side_n of_o the_o cube_fw-la namely_o to_o the_o line_n db_o or_o zi_n ¶_o corollaryes_fw-mi add_v by_o flussas_n first_o corollary_n hereby_o it_o be_v manifest_a that_o the_o diameter_n of_o a_o sphere_n contain_v in_o power_n the_o side_n both_o of_o a_o pyramid_n and_o of_o a_o cube_fw-la inscribe_v in_o it_o for_o the_o power_n of_o the_o side_n of_o the_o pyramid_n be_v two_o three_o of_o the_o power_n of_o the_o diameter_n by_o the_o 13._o of_o this_o book_n and_o the_o power_n of_o the_o side_n of_o the_o cube_fw-la be_v by_o this_o proposition_n one_o three_o of_o the_o power_n of_o the_o say_a diameter_n wherefore_o the_o diameter_n of_o the_o sphere_n contain_v in_o power_n the_o side_n of_o the_o pyramid_n and_o of_o the_o cube_fw-la .._o ¶_o second_v corollary_n all_o the_o diameter_n of_o a_o cube_fw-la cut_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o cube_fw-la and_o moreover_o those_o diameter_n do_v in_o the_o self_n same_o point_n cut_v into_o two_o equal_a part_n the_o right_a line_n which_o join_v together_o the_o centre_n of_o the_o opposite_a base_n as_o it_o be_v manifest_a to_o see_v by_o the_o right_a line_n lof_o for_o the_o angle_n lko_n and_o fio_n be_v equal_a by_o the_o 29._o of_o the_o first_o and_o it_o be_v prove_v that_o they_o be_v contain_v under_o equal_a side_n wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_n lo_o and_o foe_n be_v equal_a in_o like_a sort_n may_v be_v prove_v that_o the_o rest_n of_o the_o right_a line_n which_o join_v together_o the_o centre_n of_o the_o opposite_a base_n do_v cut_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o the_o centre_n o._n ¶_o the_o 4._o problem_n the_o 16._o proposition_n to_o make_v a_o icosahedron_n and_o to_o comprehend_v it_o in_o the_o sphere_n
give_v wherein_o be_v contain_v the_o former_a solid_n and_o to_o prove_v that_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n demonstration_n now_o forasmuch_o as_o the_o line_n qp_v qv_n qt_n q_n and_o qr_o do_v each_o subtend_v right_a angle_n contain_v under_o the_o side_n of_o a_o equilater_n hexagon_n &_o of_o a_o equilater_n decagon_fw-mi inscribe_v in_o the_o circle_n prstv_n or_o in_o the_o circle_n efghk_n which_o two_a circle_n be_v equal_a therefore_o the_o say_a line_n be_v each_o equal_a to_o the_o side_n of_o the_o pentagon_n inscribe_v in_o the_o foresay_a circle_n by_o the_o 10._o of_o this_o book_n and_o be_v equal_a the_o one_o to_o the_o other_o by_o the_o 4._o of_o the_o first_o for_o all_o the_o angle_n at_o the_o point_n w_n which_o they_o subtend_v be_v right_a angle_n wherefore_o the_o five_o triangle_n qpv_o qpr_fw-la qr_n qst_fw-la and_o qtv_n which_o be_v contain_v under_o the_o say_a line_n qv_n qp_n qr_o q_n qt_n and_o under_o the_o side_n of_o the_o pentagon_n vpr_v be_v equilater_n and_o equal_a to_o the_o ten_o former_a triangle_n and_o by_o the_o same_o reason_n the_o five_o triangle_n opposite_a unto_o they_o namely_o the_o triangle_n yml_n ymn_n ynx_n yxo_n and_o yol_n be_v equilater_n and_o equal_a to_o the_o say_v ten_o triangle_n for_o the_o line_n ill_o ym_fw-fr yn_n yx_n and_o you_o do_v subtend_v right_a angle_n contain_v under_o the_o side_n of_o a_o equilater_n hexagon_n and_o of_o a_o equilater_n decagon_fw-mi inscribe_v in_o the_o circle_n efghk_n which_o be_v equal_a to_o the_o circle_n prstv_n wherefore_o there_o be_v describe_v a_o solid_a contain_v under_o 20._o equilater_n triangle_n wherefore_o by_o the_o last_o definition_n of_o the_o eleven_o there_o be_v describe_v a_o icosahedron_n now_o it_o be_v require_v to_o comprehend_v it_o in_o the_o sphere_n give_v and_o to_o prove_v that_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n forasmuch_o as_o the_o line_n zw_n be_v the_o side_n of_o a_o hexagon_n &_o the_o line_n wq_n be_v the_o side_n of_o a_o decagon_n therefore_o the_o line_n zq_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n w_n and_o his_o great_a segment_n be_v zw_n by_o the_o 9_o of_o the_o thirteen_o wherefore_o as_o the_o line_n qz_n be_v to_o the_o line_n zw_n so_o be_v the_o line_n zw_n to_o the_o line_n wq_n but_o the_o zw_n be_v equal_a to_o the_o line_n zl_n by_o construction_n and_o the_o line_n wq_n to_o the_o line_n zy_n by_o construction_n also_o wherefore_o as_o the_o line_n qz_n be_v to_o the_o line_n zl_n so_o be_v the_o line_n zl_n to_o the_o line_n zy_n and_o the_o angle_n qzlâ_n and_o lzy_n be_v right_a angle_n by_o the_o 2._o definition_n of_o the_o eleven_o if_o therefore_o we_o draw_v a_o right_a line_n from_o the_o point_n l_o to_o the_o point_n qui_fw-fr the_o angle_n ylq_n shall_v be_v a_o right_a angle_n by_o reason_n of_o the_o likeness_n of_o the_o triangle_n ylq_n and_o zlq_n by_o the_o 8._o of_o the_o six_o wherefore_o a_o semicircle_n describe_v upon_o the_o line_n qy_n shall_v pass_v also_o by_o the_o point_n l_o by_o the_o assumpt_n add_v by_o campane_n after_o the_o 13._o of_o this_o book_n and_o by_o the_o same_o reason_n also_o for_o that_o as_o the_o line_n qz_n be_v the_o line_n zw_n so_o be_v the_o line_n zw_n to_o the_o line_n wq_n flussas_n but_o the_o line_n zq_n be_v equal_a to_o the_o line_n yw_n and_o the_o line_n zw_n to_o the_o line_n pw_o wherefore_o as_o the_o line_n yw_n be_v to_o the_o line_n wp_n so_o be_v the_o line_n pw_o to_o the_o line_n wq_n and_o therefore_o again_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n p_o to_o the_o point_n y_fw-fr the_o angle_n ypq_n shall_v be_v a_o right_a angle_n wherefore_o a_o semicircle_n describe_v upon_o the_o line_n qy_n shall_v pass_v also_o by_o the_o point_n p_o by_o the_o former_a assumpt_n &_o if_o the_o diameter_n qy_o abide_v fix_v the_o semicircle_n be_v turn_v round_o about_o until_o it_o come_v to_o the_o self_n same_o place_n from_o whence_o it_o begin_v first_o to_o be_v move_v it_o shall_v pass_v both_o by_o the_o point_n p_o and_o also_o by_o the_o rest_n of_o the_o point_n of_o the_o angle_n of_o the_o icosahedron_n and_o the_o icosahedron_n shall_v be_v comprehend_v in_o a_o sphere_n i_o say_v also_o that_o it_o be_v contain_v in_o the_o sphere_n give_v divide_v by_o the_o 10._o of_o the_o first_o the_o line_n zw_n into_o two_o equal_a part_n in_o the_o point_n a._n and_o forasmuch_o as_o the_o right_a line_n zq_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n w_n and_o his_o less_o segment_n be_v qw_o therefore_o the_o segment_n qw_o have_v add_v unto_o it_o the_o half_a of_o the_o great_a segment_n namely_o the_o line_n wa_n be_v by_o the_o 3._o of_o this_o book_n in_o power_n quintuple_a to_o the_o square_n make_v of_o the_o half_a of_o the_o great_a segment_n wherefore_o the_o square_a of_o the_o line_n qa_n be_v quintuple_a to_o the_o square_n of_o the_o line_n âw_o but_o unto_o the_o square_n of_o the_o qa_n the_o square_a of_o the_o line_n qy_n be_v quadruple_a by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o for_o the_o line_n qy_n be_v double_a to_o the_o line_n qa_n and_o by_o the_o same_o reason_n unto_o the_o square_n of_o the_o wa_n the_o square_a of_o the_o line_n zw_n be_v quadruple_a wherefore_o the_o square_a of_o the_o line_n qy_n be_v quintuple_a to_o the_o square_n of_o the_o line_n zw_n by_o the_o 15._o of_o the_o five_o and_o forasmuch_o as_o the_o line_n ac_fw-la be_v quadruple_a to_o the_o line_n cb_o therefore_o the_o line_n ab_fw-la be_v quintuple_a to_o the_o line_n cb._n but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n bd_o by_o the_o 8_o of_o the_o six_o and_o corollary_n of_o the_o 20._o of_o the_o same_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v quintuple_a to_o the_o square_n of_o the_o line_n bd_o and_o it_o be_v be_v prove_v that_o the_o square_a of_o the_o line_n qy_n be_v quintuple_a to_o the_o square_n of_o the_o line_n zw_n and_o the_o line_n bd_o be_v equal_a to_o the_o line_n zw_n for_o either_o of_o they_o be_v by_o position_n equal_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 efghk_v to_o the_o circumference_n wherefore_o the_o line_n ab_fw-la be_v equal_a to_o the_o yq_n but_o the_o line_n ab_fw-la be_v the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o line_n yq_n which_o be_v prove_v to_o be_v the_o diameter_n of_o the_o sphere_n contain_v the_o icosahedron_n be_v equal_a to_o the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o icosahedron_n be_v contain_v in_o the_o sphere_n give_v now_o i_o say_v that_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n for_o forasmuch_o as_o the_o diameter_n of_o the_o sphere_n be_v rational_a and_o be_v in_o power_n quintuple_a to_o the_o square_n of_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n olmnx_n wherefore_o also_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n olmnx_n be_v rational_a wherefore_o the_o diameter_n also_o be_v commensurable_a to_o the_o same_o line_n by_o the_o 6._o of_o the_o ten_o be_v rational_a but_o if_o in_o a_o circle_n have_v a_o rational_a line_n to_o his_o diameter_n be_v describe_v a_o equilater_n pentagon_n the_o side_n of_o the_o pentagon_n be_v by_o the_o 11._o of_o this_o book_n a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n but_o the_o side_n of_o the_o pentagon_n olmnx_n be_v also_o the_o side_n of_o the_o icosahedron_n describe_v as_o have_v before_o be_v prove_v wherefore_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n wherefore_o there_o be_v describe_v a_o icosahedron_n and_o it_o be_v contain_v in_o the_o sphere_n give_v and_o it_o be_v prove_v that_o the_o side_n of_o thou_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n which_o be_v require_v to_o be_v do_v and_o to_o be_v prove_v a_o corollary_n hereby_o it_o be_v manifest_a that_o the_o diameter_n of_o the_o sphere_n 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 to_o
line_n ai._n wherefore_o the_o square_a of_o the_o line_n bw_o be_v equal_a to_o the_o square_n of_o the_o line_n ai_fw-fr wherefore_o also_o the_o line_n bt_n be_v equal_a to_o the_o line_n ai._n and_o by_o the_o same_o reason_n be_v the_o line_n id_fw-la and_o wc_n equal_a to_o the_o same_o line_n now_o forasmuch_o as_o the_o line_n ai_fw-fr and_o id_fw-la and_o the_o line_n all_o and_o ld_n be_v equal_a and_o the_o base_a il_fw-fr be_v common_a to_o they_o both_o the_o angle_n ali_n and_o dli_o shall_v be_v equal_a by_o the_o 8._o of_o the_o first_o and_o therefore_o they_o be_v right_a angle_n by_o the_o 10._o definition_n of_o the_o first_o and_o by_o the_o same_o reason_n be_v the_o angle_n whb_n and_o whc_n right_a angle_n and_o forasmuch_o as_o the_o two_o line_n ht_v and_o tw_n be_v equal_a to_o the_o two_o line_n lct_n and_o cti_n and_o they_o contain_v equal_a angle_n that_o be_v right_a angle_n by_o supposition_n therefore_o the_o angle_n wht_v and_o ilct_n be_v equal_a by_o the_o 4._o of_o the_o first_o wherefore_o the_o plain_a superficies_n aid_n be_v in_o like_a sort_n incline_v to_o the_o plain_a superficies_n abcd_o as_o the_o plain_a superficies_n bwc_n be_v incline_v to_o the_o same_o plain_n abcd_o by_o the_o 4._o definition_n of_o the_o eleven_o in_o like_a sort_n may_v we_o prove_v that_o the_o plain_n wcdi_n be_v in_o like_a sort_n incline_v to_o the_o plain_n abcd_o as_o the_o plain_a buzc_n be_v to_o the_o plain_n ebcf._n for_o that_o in_o the_o triangle_n yoh_n and_o ctpk_n which_o consist_v of_o equal_a side_n each_o to_o his_o correspondent_a side_n the_o angle_n yho_n and_o ctkp_n which_o be_v the_o angle_n of_o the_o inclination_n be_v equal_a and_o now_o if_o the_o right_a line_n ctk_n be_v extend_v to_o the_o point_n a_o and_o the_o pentagon_n cwida_n be_v make_v perfect_a we_o may_v by_o the_o same_o reason_n prove_v that_o that_o plain_n be_v equiangle_n and_o equilater_n that_o we_o prove_v the_o pentagon_n buzcw_o to_o be_v equaliter_fw-la and_o equiangle_n and_o likewise_o if_o the_o other_o plain_n bwia_fw-la and_o aid_v be_v make_v perfect_a they_o may_v be_v prove_v to_o be_v equal_a and_o like_a pentagon_n and_o in_o like_a sort_n situate_a and_o they_o be_v set_v upon_o these_o common_a right_a line_n bw_o wc_n wi_n ai_fw-fr and_o id_fw-la and_o observe_v this_o method_n there_o shall_v upon_o every_o one_o of_o the_o 12._o âidâs_n of_o the_o cube_fw-la be_v set_v every_o one_o of_o the_o 12._o pentagon_n which_o compose_v the_o dodecahedron_n ¶_o certain_a corollarye_n add_v by_o flussas_n first_o corollary_n the_o side_n of_o a_o cube_fw-la be_v equal_a to_o the_o right_a line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n of_o a_o dodecahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n with_o the_o cube_fw-la for_o the_o angle_n bwc_n and_o aid_n be_v subtend_v of_o the_o line_n bc_o and_o ad._n which_o be_v side_n of_o the_o cubeâ_n ¶_o second_v corollary_n in_o a_o dodecahedron_n there_o be_v six_o side_n every_o two_o of_o which_o be_v parallel_n and_o opposite_a who_o section_n into_o two_o equal_a part_n be_v couple_v by_o three_o right_a line_n which_o in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o dodecahedron_n divide_v into_o two_o equal_a part_n and_o perpendicular_o both_o themselves_o and_o also_o the_o side_n for_o upon_o the_o six_o base_n of_o the_o cube_fw-la be_v set_v six_o side_n of_o the_o dodecahedron_n as_o it_o have_v be_v prove_v by_o the_o line_n zv_o wi_n &c._a &c._a which_o be_v cut_v into_o two_o equal_a part_n by_o right_a line_n which_o join_v together_o the_o centre_n of_o the_o base_n of_o the_o cube_fw-la as_o the_o line_n the_fw-mi produce_v and_o the_o other_o like_a which_o line_n couple_v together_o the_o centre_n of_o the_o base_n be_v three_o in_o number_n cut_v the_o one_o the_o other_o perpendicular_o for_o they_o be_v parallel_n to_o the_o side_n of_o the_o cube_fw-la and_o they_o cut_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o cube_fw-la by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n and_o unto_o these_o equal_a line_n join_v together_o the_o centre_n of_o the_o base_n of_o the_o cube_fw-la be_v without_o the_o base_n add_v equal_a part_n oy_o p_o ct_v and_o the_o other_o like_a which_o by_o supposition_n be_v equal_a to_o half_n of_o the_o side_n of_o the_o dodecahedron_n wherefore_o the_o whole_a line_n which_o join_v together_o the_o sectoin_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n be_v equal_a and_o they_o cut_v those_o side_n into_o two_o equal_a part_n and_o perpendicular_o three_o corollary_n a_o right_a line_n join_v together_o the_o point_n of_o the_o section_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n into_o two_o equal_a part_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n the_o great_a segment_n thereof_o shall_v be_v the_o side_n of_o the_o cube_fw-la and_o the_o less_o segment_n the_o side_n of_o the_o dodecahedron_n contain_v in_o the_o self_n same_o sphere_n for_o it_o be_v prove_v that_o the_o right_a line_n yq_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n oh_o and_o that_o his_o great_a segment_n oq_fw-la be_v half_a the_o side_n of_o the_o cube_fw-la and_o his_o less_o segment_n oy_o be_v half_o of_o the_o side_n we_o which_o be_v the_o side_n of_o the_o dodecahedron_n wherefore_o it_o follow_v by_o the_o 15._o of_o the_o five_o that_o their_o double_n be_v in_o the_o same_o proportion_n wherefore_o the_o double_a of_o the_o line_n yq_n which_o join_v the_o point_n opposite_a unto_o the_o line_n y_fw-fr be_v the_o whole_a and_o the_o great_a segment_n be_v the_o double_a of_o the_o line_n oq_fw-la which_o be_v the_o side_n of_o the_o cube_fw-la &_o the_o less_o segment_n be_v the_o double_a of_o the_o line_n the_fw-mi which_o be_v equal_a to_o the_o side_n of_o the_o dodecahedron_n namely_o to_o the_o side_n we_o ¶_o the_o 6._o problem_n the_o 18._o proposition_n to_o find_v out_o the_o side_n of_o the_o foresay_a five_o body_n and_o to_o compare_v they_o together_o take_v the_o diameter_n of_o the_o sphere_n give_v and_o let_v the_o same_o be_v ab_fw-la and_o divide_v it_o in_o the_o point_n c_o so_fw-mi that_o let_v the_o line_n ac_fw-la be_v equal_a to_o cb_o by_o the_o 10._o of_o the_o first_o and_o in_o the_o point_n d_o so_fw-mi that_o let_v ad_fw-la be_v double_a to_o db_o by_o the_o 9_o of_o the_o six_o and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n aeb_fw-mi and_o from_o the_o point_n c_o and_o d_o raise_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o line_n ab_fw-la perpendicular_a line_n ce_fw-fr and_o df._n and_o draw_v these_o right_a line_n of_o fb_o and_o be._n now_o forasmuch_o as_o the_o line_n ad_fw-la be_v double_a to_o the_o line_n db_o therefore_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n db._n wherefore_o the_o line_n basilius_n be_v sesquialter_fw-la to_o the_o line_n ad_fw-la for_o it_o be_v as_o 3._o to_o 2._o but_o as_o the_o line_n basilius_n be_v to_o the_o line_n ad_fw-la so_o be_v the_o square_a of_o the_o line_n basilius_n to_o the_o square_n of_o the_o line_n of_o by_o the_o 6._o of_o the_o six_o or_o by_o the_o corollary_n of_o the_o same_o and_o by_o the_o corollary_n of_o the_o 20._o of_o the_o same_o for_o the_o triangle_n afb_o be_v equiangle_n to_o the_o triangle_n afd_v wherefore_o the_o square_a of_o the_o line_n basilius_n be_v sesquialter_fw-la to_o the_o square_n of_o the_o line_n af._n but_o the_o diameter_n of_o a_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o sidâ_n of_o the_o pyramid_n pyramid_n by_o the_o 13._o of_o this_o book_n and_o the_o line_n ab_fw-la be_v the_o diameter_n of_o the_o sphere_n wherefore_o the_o line_n of_o be_v equal_a to_o the_o side_n of_o the_o pyramid_n again_o forasmuch_o as_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n bd_o but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bd_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n fb_o by_o the_o corollary_n of_o the_o 8._o and_o 20._o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v treble_a to_o the_o square_n of_o the_o line_n fb_o but_o the_o diameter_n oâ_n a_o sphere_n be_v in_o power_n treble_a to_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o this_o book_n and_o the_o diameter_n of_o the_o sphere_n be_v the_o line_n ab_fw-la wherefore_o the_o line_n bf_o be_v the_o side_n of_o the_o cube_fw-la cube_fw-la and_o forasmuch_o as_o the_o line_n fb_o be_v the_o
double_a to_o the_o side_n of_o the_o octohedron_n &_o the_o side_n be_v in_o power_n sequitertia_fw-la to_o the_o perpendiclar_a line_n by_o the_o 12._o of_o this_o book_n wherefore_o the_o diameter_n thereof_o be_v in_o power_n duple_fw-fr superbipartiens_fw-la tertias_fw-la to_o the_o perpendicular_a line_n wherefore_o also_o the_o diameter_n and_o the_o perpendicular_a line_n be_v rational_a and_o commensurable_a by_o the_o 6._o of_o the_o ten_o as_o touch_v a_o icosahedron_n it_o be_v prove_v in_o the_o 16._o of_o this_o book_n that_o the_o side_n thereof_o be_v a_o less_o line_n when_o the_o diameter_n of_o the_o sphere_n be_v rational_a and_o forasmuch_o as_o the_o angle_n of_o the_o inclination_n of_o the_o base_n thereof_o be_v contain_v of_o the_o perpendicular_a line_n of_o the_o triangle_n and_o subtend_v of_o the_o right_a line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n which_o contain_v five_o side_n of_o the_o icosahedron_n and_o unto_o the_o perpendicular_a line_n the_o side_n be_v commensurable_a namely_o be_v in_o power_n sesquitertia_fw-la unto_o they_o by_o the_o corollary_n of_o the_o 12._o of_o this_o book_n therefore_o the_o perpendicular_a line_n which_o contain_v the_o angle_n be_v irrational_a line_n namely_o less_o line_n by_o the_o 105._o of_o the_o ten_o book_n and_o forasmuch_o as_o the_o diameter_n contain_v in_o power_n both_o the_o side_n of_o the_o icosahedron_n and_o the_o line_n which_o subtend_v the_o foresay_a angle_n if_o from_o the_o power_n of_o the_o diameter_n which_o be_v rational_a be_v take_v away_o the_o power_n of_o the_o side_n of_o the_o icosahedron_n which_o be_v irrational_a it_o be_v manifest_a that_o the_o residue_n which_o be_v the_o power_n of_o the_o subtend_a line_n shall_v be_v irrational_a for_o if_o it_o shall_v be_v rational_a the_o number_n which_o measure_v the_o whole_a power_n of_o the_o diameter_n and_o the_o part_n take_v away_o of_o the_o subtend_a line_n shall_v also_o by_o the_o 4._o common_a sentence_n of_o the_o seven_o measure_n the_o residue_n namely_o the_o power_n of_o the_o side_n irrational_a which_o be_v irrational_a for_o that_o it_o be_v a_o less_o line_n which_o be_v absurd_a wherefore_o it_o be_v manifest_a that_o the_o right_a line_n which_o compose_v the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o the_o icosahedron_n be_v irrational_a line_n for_o the_o subtend_a line_n have_v to_o the_o line_n contain_v a_o great_a proportion_n than_o the_o whole_a have_v to_o the_o great_a segment_n the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o a_o dodecahedron_n be_v contain_v under_o two_o perpendicular_n of_o the_o base_n of_o the_o dodecahedron_n and_o be_v subtend_v of_o that_o right_a line_n who_o great_a segment_n be_v the_o side_n of_o a_o cube_n inscribe_v in_o the_o dodecahedron_n which_o right_a line_n be_v equal_a to_o the_o line_n which_o couple_v the_o section_n into_o two_o equal_a part_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n and_o this_o couple_a line_n we_o say_v be_v a_o irrational_a line_n for_o that_o the_o diameter_n of_o the_o sphere_n contain_v in_o power_n both_o the_o couple_a line_n and_o the_o side_n of_o the_o dodecahedron_n but_o the_o side_n of_o the_o dodecahedron_n be_v a_o irrational_a line_n namely_o a_o residual_a line_n by_o the_o 17._o of_o this_o book_n wherefore_o the_o residue_n namely_o the_o couple_a line_n be_v a_o irrational_a line_n as_o it_o be_v âasy_a to_o prove_v by_o the_o 4._o common_a sentence_n of_o the_o seven_o and_o that_o the_o perpendicular_a line_n which_o contain_v the_o angle_n of_o the_o inclination_n be_v irrational_a be_v thus_o prove_v by_o the_o proportion_n of_o the_o subtend_a line_n of_o the_o foresay_a angle_n of_o inclination_n to_o the_o line_n which_o contain_v the_o angle_n be_v find_v out_o the_o obliquity_n of_o the_o angle_n angle_n for_o if_o the_o subtend_a line_n be_v in_o power_n double_a to_o the_o line_n which_o contain_v the_o angle_n then_o be_v the_o angle_n a_o right_a angle_n by_o the_o 48._o of_o the_o first_o but_o if_o it_o be_v in_o power_n less_o than_o the_o double_a it_o be_v a_o acute_a angle_n by_o the_o 23._o of_o the_o second_o but_o if_o it_o be_v in_o power_n more_o than_o the_o double_a or_o have_v a_o great_a proportion_n than_o the_o whole_a have_v to_o the_o great_a segmentâ_n the_o angle_n shall_v be_v a_o obtuse_a angle_n by_o the_o 12._o of_o the_o second_o and_o 4._o of_o the_o thirteen_o by_o which_o may_v be_v prove_v that_o the_o square_a of_o the_o whole_a be_v great_a than_o the_o double_a of_o the_o square_n of_o the_o great_a segment_n this_o be_v to_o be_v note_v that_o that_o which_o flussas_n have_v here_o teach_v touch_v the_o inclination_n of_o the_o base_n of_o the_o âive_a regular_a body_n hypsicles_n teach_v after_o the_o 5_o proposition_n of_o the_o 15._o book_n where_o he_o confess_v that_o he_o receive_v it_o of_o one_o isidorus_n and_o seek_v to_o make_v the_o matter_n more_o clear_a he_o endeavour_v himself_o to_o declare_v that_o the_o angle_n of_o the_o inclination_n of_o the_o solid_n be_v give_v and_o that_o they_o be_v either_o acute_a or_o obtuse_a according_a to_o the_o nature_n of_o the_o solid_a although_o euclid_v in_o all_o his_o 15._o book_n have_v not_o yet_o show_v what_o a_o thing_n give_v be_v wherefore_o flussas_n frame_v his_o demonstration_n upon_o a_o other_o ground_n proceed_v after_o a_o other_o manner_n which_o seem_v more_o plain_a and_o more_o apt_o hereto_o be_v place_v then_o there_o albeit_o the_o reader_n in_o that_o place_n shall_v not_o be_v frustrate_a of_o his_o also_o the_o end_n of_o the_o thirteen_o book_n of_o euclides_n element_n ¶_o the_o fourteen_o book_n of_o euclides_n element_n in_o this_o book_n which_o be_v common_o account_v the_o 14._o book_n of_o euclid_n be_v more_o at_o large_a entreat_v of_o our_o principal_a purpose_n book_n namely_o of_o the_o comparison_n and_o proportion_n of_o the_o five_o regular_a body_n customable_o call_v the_o 5._o figure_n or_o form_n of_o pythagoras_n the_o one_o to_o the_o other_o and_o also_o of_o their_o side_n together_o each_o to_o other_o which_o thing_n be_v of_o most_o secret_a use_n and_o inestimable_a pleasure_n and_o commodity_n to_o such_o as_o diligent_o search_v for_o they_o and_o attain_v unto_o they_o which_o thing_n also_o undoubted_o for_o the_o worthiness_n and_o hardness_n thereof_o for_o thing_n of_o most_o price_n be_v most_o hard_a be_v first_o search_v and_o find_v out_o of_o philosopher_n not_o of_o the_o inferior_a or_o mean_a sort_n but_o of_o the_o deep_a and_o most_o ground_a philosopher_n and_o best_a exercise_v in_o geometry_n and_o albeit_o this_o book_n with_o the_o book_n follow_v namely_o the_o 15._o book_n have_v be_v hitherto_o of_o all_o man_n for_o the_o most_o part_n and_o be_v also_o at_o this_o day_n number_v and_o account_v among_o euclides_n book_n and_o suppose_v to_o be_v two_o of_o he_o namely_o the_o 14._o and_o 15._o in_o order_n as_o all_o exemplar_n not_o only_o new_a and_o late_o set_v abroad_o but_o also_o old_a monument_n write_v by_o hand_n do_v manifest_o witness_v yet_o it_o be_v think_v by_o the_o best_a learned_a in_o these_o day_n that_o these_o two_o book_n be_v none_o of_o euclides_n but_o of_o some_o other_o author_n no_o less_o worthy_a nor_o of_o less_o estimation_n and_o authority_n notwithstanding_o than_o euclid_n apollonius_n a_o man_n of_o deep_a knowledge_n a_o great_a philosopher_n and_o in_o geometry_n marvellous_a who_o wonderful_a book_n write_v of_o the_o section_n of_o cones_fw-la which_o exercise_n &_o occupy_v thewitte_v of_o the_o wise_a and_o best_a learned_a be_v yet_o remain_v be_v think_v and_o that_o not_o without_o just_a cause_n to_o be_v the_o author_n of_o they_o or_o as_o some_o think_v hypsicles_n himself_o for_o what_o can_v be_v more_o plain_o then_o that_o which_o he_o himself_o witness_v in_o the_o preface_n of_o this_o book_n basilides_n of_o tire_n say_v hypsicles_n and_o my_o father_n together_o scan_v and_o peyse_v a_o writing_n or_o book_n of_o apollonius_n which_o be_v of_o the_o comparison_n of_o a_o dodecahedron_n to_o a_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n and_o what_o proportion_n these_o figure_n have_v the_o one_o to_o the_o other_o find_v that_o apollonius_n have_v fail_v in_o this_o matter_n but_o afterward_o say_v he_o i_o find_v a_o other_o copy_n or_o book_n of_o apollonius_n wherein_o the_o demonstration_n of_o that_o matter_n be_v full_a and_o perfect_a and_o show_v it_o unto_o they_o whereat_o they_o much_o rejoice_v by_o which_o word_n it_o seem_v to_o be_v manifest_a that_o apollonius_n be_v the_o first_o author_n of_o this_o book_n which_o be_v afterward_o set_v forth_o by_o hypsicles_n for_o so_o his_o own_o word_n after_o in_o
the_o same_o preface_n seem_v to_o import_v the_o preface_n of_o hypsicles_n before_o the_o fourteen_o book_n friend_n protarchus_n when_o that_o basilides_n of_o tire_n come_v into_o alexandria_n have_v familiar_a friendship_n with_o my_o father_n by_o reason_n of_o his_o knowledge_n in_o the_o mathematical_a science_n he_o remain_v with_o he_o a_o long_a time_n yea_o even_o all_o the_o time_n of_o the_o pestilence_n and_o sometime_o reason_v between_o themselves_o of_o that_o which_o apollonius_n have_v write_v touch_v the_o comparison_n of_o a_o dodecahedron_n and_o of_o a_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n what_o proportion_n such_o body_n have_v the_o one_o to_o the_o other_o they_o judge_v that_o apollonius_n have_v somewhat_o err_v therein_o wherefore_o they_o as_o my_o father_n declare_v unto_o i_o diligent_o weigh_v it_o write_v it_o perfect_o howbeit_o afterward_o i_o happen_v to_o find_v a_o other_o book_n write_v of_o apollonius_n which_o contain_v in_o it_o the_o right_a demonstration_n of_o that_o which_o they_o seek_v for_o which_o when_o they_o see_v they_o much_o rejoice_v as_o for_o that_o which_o apollonius_n write_v may_v be_v see_v of_o all_o man_n for_o that_o it_o be_v in_o âuery_n man_n hand_n and_o that_o which_o be_v of_o we_o more_o diligent_o afterward_o write_v again_o i_o think_v good_a to_o send_v and_o dedicate_v unto_o you_o as_z to_o one_o who_o i_o think_v worthy_a commendation_n both_o for_o that_o deep_a knowledge_n which_o i_o know_v you_o have_v in_o all_o kind_n of_o learning_n and_o chief_o in_o geometry_n so_o that_o you_o be_v able_a ready_o to_o judge_v of_o those_o thing_n which_o be_v speak_v and_o also_o for_o the_o great_a love_n and_o good_a will_n which_o you_o bear_v towards_o my_o father_n and_o i_o wherefore_o vouchsafe_v gentle_o to_o accept_v this_o which_o i_o send_v unto_o you_o but_o now_o be_v it_o time_n to_o end_v our_o preface_n and_o to_o begin_v the_o matter_n ¶_o the_o 1._o theorem_a the_o 1._o proposition_n flussas_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 describe_v in_o the_o same_o circle_n be_v the_o half_a of_o these_o two_o line_n namely_o of_o the_o side_n of_o a_o hexagon_n figure_n and_o of_o the_o side_n of_o a_o decagon_n figure_n be_v both_o describe_v in_o the_o self_n same_o circle_n svppose_v that_o the_o circle_n be_v abc_n construction_n and_o let_v the_o side_n of_o a_o equilater_n pentagon_n describe_v in_o the_o circle_n abc_n be_v bc._n and_o by_o the_o 1._o of_o the_o three_o take_v the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v d._n and_o 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 bc_o a_o perpendicular_a line_n de._n and_o extend_v the_o right_a line_n de_fw-fr direct_o to_o the_o point_n f._n then_o i_o say_v that_o the_o 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 the_o side_n of_o a_o hexagon_n and_o of_o a_o decagon_n take_v together_o and_o describe_v in_o the_o same_o circle_n draw_v these_o right_a line_n dc_o and_o cf._n and_o unto_o the_o line_n of_o put_v a_o equal_a line_n ge._n and_o draw_v a_o right_a line_n from_o the_o point_n g_o to_o the_o point_n c._n demonstration_n now_o forasmuch_o as_o the_o circumference_n of_o the_o whole_a circle_n be_v quintuple_a to_o the_o circumference_n bfc_n which_o be_v subtend_v of_o the_o side_n of_o the_o pentagon_n and_o the_o circumference_n acf_n be_v the_o half_a of_o the_o circumference_n of_o the_o whole_a circle_n and_o the_o circumference_n cf_o which_o be_v subtend_v of_o the_o side_n of_o the_o decagon_n be_v the_o half_a of_o the_o circumference_n bcf_n therefore_o the_o circumference_n acf_n be_v quintuple_a to_o the_o circumference_n cf_o by_o the_o 15._o of_o the_o âiât_n wherefore_o the_o circumference_n ac_fw-la be_v qradruple_a to_o the_o circumference_n fc_o but_o as_o the_o circumference_n ac_fw-la be_v to_o the_o circumference_n fc_o so_o be_v the_o angle_n adc_o to_o the_o angle_n fdc_n by_o the_o last_o of_o the_o six_o wherefore_o the_o angle_n adc_o be_v quadruple_a to_o the_o angle_n fdc_n but_o the_o angle_n adc_o be_v double_a to_o the_o angle_n efc_n by_o the_o 20._o of_o the_o three_o wherefore_o the_o angle_n efc_n be_v double_a to_o the_o angle_n gdc_n but_o the_o angle_n efc_n be_v equal_a to_o the_o angle_n egc_n by_o the_o 4._o of_o the_o first_o wherefore_o the_o angle_n egc_n be_v double_a to_o the_o angle_n edc_n wherefore_o the_o line_n dg_o be_v equal_a to_o the_o line_n gc_o by_o the_o 32._o and_o 6._o of_o the_o first_o but_o the_o line_n gc_o be_v equal_a to_o the_o line_n cf_o by_o the_o 4._o of_o the_o first_o wherefore_o the_o line_n dg_o be_v equal_a to_o the_o line_n cf._n and_o the_o line_n ge_z be_v equal_a to_o the_o line_n of_o by_o construction_n wherefore_o the_o line_n de_fw-fr be_v equal_a to_o the_o line_n of_o and_o fc_n add_v together_o unto_o the_o line_n of_o and_o fc_n add_v the_o line_n de._n wherefore_o the_o line_n df_o and_o fc_o add_v together_o be_v double_a to_o the_o line_n de._n but_o the_o line_n df_o be_v equal_a to_o the_o side_n of_o the_o hexagon_n and_o fc_a to_o the_o side_n of_o the_o decagon_n wherefore_o the_o line_n de_fw-fr be_v the_o half_a 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 be_v both_o add_v together_o and_o describe_v in_o one_o and_o the_o self_n same_o circle_n it_o be_v manifest_a same_o by_o the_o proposition_n of_o the_o thirteen_o book_n that_o 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 equilater_n triangle_n describe_v in_o the_o same_o circle_n be_v half_a of_o the_o semidiameter_n of_o the_o circle_n wherefore_o by_o this_o proposition_n a_o perpendicular_a draw_v from_o the_o cântre_n of_o a_o circle_n to_o the_o side_n of_o a_o pentagon_n be_v equal_a to_o the_o perpendicular_a draw_v from_o the_o centre_n to_o the_o side_n of_o the_o triangle_n ând_v to_o half_a of_o the_o side_n of_o the_o decagon_n describe_v in_o the_o same_o circle_n ¶_o the_o 2._o theorem_a the_o 2._o proposition_n one_o and_o the_o self_n same_o circle_n comprehend_v both_o the_o pentagon_n of_o a_o dodecahedron_n and_o the_o triangle_n of_o a_o icosahedron_n flussas_n describe_v in_o one_o and_o the_o self_n same_o sphere_n this_o theorem_a be_v describe_v of_o aristeus_n in_o that_o book_n who_o title_n be_v the_o comparison_n of_o the_o five_o figure_n and_o be_v describe_v of_o apollonius_n in_o his_o second_o edition_n of_o the_o comparison_n of_o a_o dodecahedron_n to_o a_o icosahedron_n which_o be_v proposition_n that_o as_o the_o superficies_n of_o a_o dodecahedron_n be_v to_o the_o superficies_n of_o a_o icosahedron_n so_o be_v the_o dodecahedron_n to_o a_o icosahedron_n for_o that_o a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o a_o sphere_n to_o the_o pentagon_n of_o a_o dodecahedron_n and_o to_o the_o triangle_n of_o a_o icosahedron_n be_v one_o and_o the_o self_n same_o now_o must_v we_o also_o prove_v that_o one_o and_o the_o self_n same_o circle_n comprehend_v both_o the_o pentagon_n of_o a_o dodecahedron_n and_o also_o the_o triangle_n of_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n first_o this_o be_v prove_v flussas_n if_o in_o a_o circle_n be_v describe_v a_o equilater_n pentagon_n the_o square_n which_o be_v make_v of_o the_o side_n of_o the_o pentagon_n and_o of_o that_o right_a line_n which_o be_v subtend_v under_o two_o side_n of_o the_o pentagon_n be_v quintuple_a to_o the_o square_n of_o the_o semidiameter_n oâ_n the_o circle_n suppose_v that_o abc_n be_v a_o circle_n assumpt_n and_o let_v the_o side_n of_o a_o pentagon_n in_o the_o circle_n abc_n be_v ac_fw-la and_o take_v by_o the_o 1._o of_o the_o three_o the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v 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 line_n ac_fw-la a_o perpendicular_a line_n df._n and_o extend_v the_o line_n df_o on_o either_o side_n to_o the_o point_n b_o and_o e._n and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n b._n now_o i_o say_v that_o the_o square_n of_o the_o line_n basilius_n and_o ac_fw-la be_v quintuple_a to_o the_o square_n of_o the_o line_n de._n draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n e._n wherefore_o the_o line_n ae_n be_v the_o side_n of_o a_o decagon_n figure_n and_o forasmuch_o as_o the_o line_n be_v be_v double_a to_o thâ_z line_n de_fw-fr assumpt_n therefore_o the_o square_n
triangle_n wherefore_o six_o such_o triangle_n as_o dbc_n be_v be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o bc_o thrice_o but_o six_o sâch_a triangle_n as_o dbc_n be_v be_v equal_a to_o two_o such_o triangle_n as_o abc_n be_v wherefore_o that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o bc_o thrice_o be_v equal_a to_o two_o such_o triangle_n as_o abc_n be_v but_o two_o of_o those_o triangle_n take_v ten_o time_n contain_v the_o whole_a icosahedron_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n de_fw-fr &_o bc_o thirty_o time_n be_v equal_a to_o twenty_o such_o triangle_n as_o the_o triangle_n abc_n be_v that_o be_v to_o the_o whole_a superficies_n of_o the_o icosahedron_n follow_v wherefore_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v that_o which_o be_v contain_v under_o the_o line_n cd_o and_o fg_o to_o that_o which_o be_v contain_v under_o the_o line_n bc_o and_o de._n ¶_o corollary_n by_o this_o it_o be_v manifest_a that_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n order_n so_o be_v that_o which_o be_v contain_v under_o the_o side_n of_o the_o pentagon_n and_o the_o perpendicular_a line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n describe_v about_o the_o pentagon_n to_o the_o same_o side_n to_o that_o which_o be_v contain_v under_o the_o side_n of_o the_o icosahedron_n and_o the_o perpendicular_a line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n describe_v about_o the_o triangle_n to_o the_o same_o side_n so_o that_o the_o icosahedron_n and_o dodecahedron_n be_v both_o describe_v in_o one_o and_o the_o self_n same_o sphere_n ¶_o the_o 4._o theorem_a the_o 4._o proposition_n flussas_n this_o be_v do_v now_o be_v to_o be_v prove_v that_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n construction_n take_v by_o the_o 2._o theorem_a of_o this_o book_n a_o circle_n contain_v both_o the_o pentagon_n of_o a_o dodecahedron_n and_o the_o triangle_n of_o a_o icosahedron_n be_v both_o describe_v in_o one_o and_o the_o self_n same_o sphere_n and_o let_v the_o same_o circle_n be_v dbc_n and_o in_o the_o circle_n dbc_n describe_v the_o side_n of_o a_o equilater_n triangle_n namely_o cd_o and_o the_o side_n of_o a_o equilater_n pentagon_n namely_o ac_fw-la and_o take_v by_o the_o 1._o of_o the_o three_o the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v e._n and_o from_o the_o point_n e_o draw_v unto_o the_o line_n dc_o and_o ac_fw-la perpendicular_a line_n of_o and_o eglantine_n and_o extend_v the_o line_n eglantine_n direct_o to_o the_o point_n b._n and_o draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n c._n and_o let_v the_o side_n of_o the_o cube_fw-la be_v the_o line_n h._n now_o i_o say_v that_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v the_o line_n h_o to_o the_o line_n cd_o demonstration_n forasmuch_o as_o the_o line_n make_v of_o the_o line_n ebb_v and_o bc_o add_v together_o namely_o of_o the_o side_n of_o the_o hexagon_n and_o of_o the_o side_n of_o a_o decagon_n be_v by_o the_o 9_o of_o the_o thirteen_o divide_a by_o a_o extreme_a and_o mean_a proportion_n and_o his_o great_a segment_n be_v the_o line_n be_v and_o the_o line_n eglantine_n be_v also_o by_o the_o 1._o of_o the_o foâretenth_n the_o half_a of_o the_o same_o line_n and_o the_o line_n of_o be_v the_o half_a of_o the_o line_n be_v by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o the_o line_n eglantine_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n âââeth_v his_o great_a segment_n shall_v be_v the_o line_n ef._n and_o the_o line_n h_o also_o be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n his_o great_a segment_n be_v the_o line_n ca_n as_o it_o be_v prove_v tâirtenth_o in_o the_o dodecahedron_n proposition_n wherefore_o as_o the_o line_n h_o be_v to_o the_o line_n ca_n so_o be_v the_o line_n eglantine_n to_o the_o line_n ef._n wherefore_o by_o the_o 16._o of_o the_o six_o that_o which_o be_v contain_v under_o the_o line_n h_o and_o of_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ca_n and_o eglantine_n and_o for_o that_o as_o the_o line_n h_o be_v to_o the_o line_n cd_o so_o be_v that_o which_o be_v contain_v under_o the_o line_n h_o and_o of_o to_o that_o which_o be_v contain_v under_o the_o line_n cd_o and_o of_o by_o the_o 1._o of_o the_o six_o but_o unto_o that_o which_o be_v contain_v under_o the_o line_n h_o and_o of_o be_v equal_a that_o which_o be_v contain_v under_o the_o line_n ca_n and_o eglantine_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o line_n h_o be_v to_o the_o line_n cd_o so_o be_v that_o which_o be_v contain_v under_o the_o line_n ca_n and_o eglantine_n to_o that_o which_o be_v contain_v under_o the_o line_n cd_o and_o of_o that_o be_v by_o the_o corollary_n next_o go_v before_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v the_o line_n h_o to_o the_o line_n cd_o an_o other_o demonstration_n to_o prove_v that_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n prove_v let_v there_o be_v a_o circle_n abc_n and_o in_o it_o describe_v two_o side_n of_o a_o equilater_n pentagon_n by_o the_o 11._o of_o the_o five_o namely_o ab_fw-la and_o ac_fw-la and_o draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n c._n and_o by_o the_o 1._o of_o the_o three_o take_v the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v d._n and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n d_o and_o extend_v it_o direct_o to_o the_o point_n e_o and_o let_v it_o cut_v the_o line_n bc_o in_o the_o point_n g._n and_o let_v the_o line_n df_o be_v half_a to_o the_o line_n dam_fw-ge and_o let_v the_o line_n gc_o be_v treble_a to_o the_o line_n hc_n by_o the_o 9_o of_o the_o six_o proposition_n now_o i_o say_v that_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o bh_o be_v equal_a to_o the_o pentagon_n inscribe_v in_o the_o circle_n abc_n draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n d._n now_o forasmuch_o as_o the_o line_n ad_fw-la be_v double_a to_o the_o line_n df_o therefore_o the_o line_n of_o be_v sesquialter_fw-la to_o the_o line_n ad._n again_o assumpt_n forasmuch_o as_o the_o line_n gc_o be_v treble_a to_o the_o line_n change_z therefore_o the_o line_n gh_o be_v double_a to_o the_o line_n ch._n wherefore_o the_o line_n gc_o be_v sesquialter_fw-la to_o the_o line_n hg_o wherefore_o as_o the_o line_n favorina_n be_v to_o the_o line_n ad_fw-la so_o be_v the_o line_n gc_o to_o the_o line_n gh_o wherefore_o by_o the_o 16._o of_o the_o six_o that_o which_o be_v contain_v under_o the_o line_n of_o &_o hg_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n dam_n and_o gc_o but_o the_o line_n gc_o be_v equal_a to_o the_o line_n bg_o by_o the_o 3._o of_o the_o three_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o bg_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o gh_o but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o bg_o be_v equal_a to_o two_o such_o triangle_n as_o the_o triangle_n abdella_n be_v by_o the_o 41._o of_o the_o first_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o gh_o be_v equal_a to_o two_o such_o triangle_n as_o the_o triangle_n abdella_n be_v wherefore_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o gh_o âive_a time_n be_v equal_a to_o ten_o triangle_n but_o ten_o triangle_n be_v two_o pentagon_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o gh_o five_o time_n be_v equal_a to_o two_o pentagon_n and_o forasmuch_o as_o the_o line_n gh_o be_v double_a to_o the_o line_n hc_n therefore_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o gh_o be_v double_a to_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o hc_n by_o the_o 1._o of_o the_o six_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o change_z twice_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n
to_o the_o square_n of_o the_o line_n cb_o &_o bd._n but_o unto_o the_o square_n of_o the_o line_n f_o be_v equal_a the_o square_n of_o the_o line_n bc_o &_o cd_o for_o the_o side_n of_o a_o pentagon_n contain_v in_o power_n both_o the_o side_n of_o a_o six_o angle_a figure_n and_o the_o side_n of_o a_o ten_o angle_a figure_n by_o the_o 10._o of_o the_o thirteen_o wherefore_o as_o the_o square_n of_o the_o line_n g_o be_v to_o the_o square_n of_o the_o line_n e_o so_fw-mi be_v the_o square_n of_o the_o line_n bc_o and_o cd_o to_o the_o square_n of_o the_o line_n cb_o and_o bd._n but_o as_o the_o square_n of_o the_o line_n cb_o and_o cd_o be_v to_o the_o square_n of_o the_o line_n cb_o &_o bd_o prove_v so_o any_o right_a line_n what_o so_o ever_o it_o be_v 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 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 wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o square_n of_o the_o line_n g_z the_o side_z of_o the_o cube_fw-la be_v to_o the_o square_n of_o the_o line_n e_o so_o any_o right_a line_n 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 make_v of_o the_o whole_a line_n and_o of_o the_o great_a segment_n 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 but_o the_o line_n g_o be_v the_o side_n of_o the_o cube_n and_o the_o line_n e_o of_o the_o icosahedron_n by_o supposition_n if_o therefore_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n as_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 great_a segment_n 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 so_o be_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n be_v both_o describe_v in_o one_o and_o the_o self_n same_o sphere_n now_o will_v we_o prove_v that_o as_o the_o side_n of_o the_o cube_n be_v to_o the_o side_n of_o the_o icosahedron_n flussas_n so_o be_v the_o solid_a of_o the_o dodecahedron_n to_o the_o solid_a of_o the_o icosahedron_n forasmuch_o as_o equal_a circle_n comprehend_v both_o the_o pentagon_n of_o a_o dodecahedron_n and_o the_o triangle_n of_o a_o icosahedron_n be_v both_o describe_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o 2._o of_o this_o book_n but_o in_o a_o sphere_n equal_a circle_n be_v equal_o distant_a from_o the_o centre_n for_o the_o perpendicular_a line_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o plain_a superficiece_n of_o the_o circle_n be_v equal_a and_o do_v fall_v upon_o the_o centre_n of_o the_o circle_n book_n wherefore_o perpendicular_a line_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o centre_n of_o the_o circle_n comprehend_v both_o the_o triangle_n of_o a_o icosahedron_n and_o the_o pentagon_n of_o a_o dodecahedron_n be_v equal_a wherefore_o the_o pyramid_n who_o base_n be_v the_o pentagon_n of_o the_o dodecahedron_n be_v of_o equal_a altitude_n with_o the_o pyramid_n who_o base_n be_v the_o triangle_n of_o the_o icosahedron_n but_o pyramid_n of_o equal_a altitude_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v by_o the_o 5._o of_o the_o twelve_o wherefore_o as_o the_o pentagon_n be_v to_o the_o triangle_n so_o be_v the_o pyramid_n who_o base_a be_v the_o pentagon_n of_o the_o dodecahedron_n and_o top_n the_o centre_n of_o the_o sphere_n to_o the_o pyramid_n who_o base_a be_v the_o triangle_n and_o top_n the_o centre_n also_o of_o the_o sphere_n wherefore_o by_o the_o 15._o of_o the_o five_o as_o 12._o pentagon_n be_v to_o 20._o triangle_n so_o be_v 12._o pyramid_n have_v pentagon_n to_o their_o base_n to_o 20._o pyramid_n have_v triangle_n to_o their_o base_n but_o 12._o pentagon_n be_v the_o superficies_n of_o the_o dâdecahedron_n and_o 20._o triangle_n be_v the_o superficies_n of_o the_o icosahedron_n wherefore_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v 12._o pyramid_n have_v pentagon_n to_o their_o base_n to_o 20._o pyramid_n have_v triangle_n to_o their_o base_n but_o 12._o pyramid_n have_v pentagon_n to_o their_o base_n be_v the_o solid_a of_o the_o dodecahedron_n and_o 20._o pyramid_n have_v triangle_n to_o their_o base_n be_v the_o solid_a oâ_n the_o icosahedron_n flussas_n wherefore_o by_o the_o 11._o of_o the_o fifthe_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 âo_o be_v the_o solid_a of_o the_o dodecahedron_n to_o the_o solid_a of_o the_o icosahedron_n but_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 have_v we_o prove_v that_o the_o side_n oâ_n the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n wherefore_o by_o the_o 11._o of_o the_o five_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 solid_a of_o the_o dodecahedron_n to_o the_o solid_a of_o the_o icosahedron_n note_v if_o two_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n they_o shall_v every_o way_n be_v in_o like_a proportion_n which_o thing_n be_v thus_o demonstrate_v let_v the_o line_n ab_fw-la be_v by_o the_o 30._o of_o the_o six_o divide_a by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c_o and_o let_v the_o great_a segment_n thereof_o be_v the_o line_n ca._n and_o likewise_o also_o let_v the_o line_n 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 fletcher_n and_o let_v the_o great_a segment_n thereof_o be_v the_o line_n df._n then_o i_o say_v that_o as_o the_o whole_a line_n ab_fw-la be_v to_o the_o great_a segment_n thereof_o ac_fw-la so_o be_v the_o whole_a line_n de_fw-fr to_o the_o great_a segment_n thereof_o df._n for_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 by_o the_o definition_n of_o a_o line_n divide_v be_v a_o extreme_a and_o mean_a proportion_n demonstration_n and_o that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o of_o be_v also_o equal_a to_o the_o square_n of_o the_o line_n df_o by_o the_o same_o definition_n therefore_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 to_o the_o square_n of_o the_o line_n ac_fw-la so_o be_v that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o of_o to_o the_o square_n of_o the_o line_n df._n for_o in_o each_o be_v the_o proportion_n of_o equality_n wherefore_o as_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o four_o time_n be_v to_o the_o square_n of_o the_o line_n ac_fw-la so_o be_v that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o of_o four_o time_n to_o the_o square_n of_o the_o line_n df_o by_o the_o 15._o of_o the_o five_o wherefore_o by_o composition_n by_o the_o 18._o of_o the_o âiâth_n as_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o four_o time_n together_o with_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 ac_fw-la so_o be_v that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o of_o four_o time_n together_o with_o the_o square_n of_o the_o line_n df_o to_o the_o square_n of_o the_o line_n df._n wherefore_o as_o the_o square_n which_o be_v make_v of_o the_o line_n ab_fw-la and_o bc_o add_v together_o and_o make_v one_o line_n which_o square_a by_o the_o 8._o of_o the_o second_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o four_o time_n together_o with_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 ac_fw-la so_o be_v the_o square_n make_v of_o the_o line_n de_fw-fr &_o of_o add_v together_o and_o make_v one_o line_n which_o square_a be_v also_o by_o the_o same_o equal_a to_o that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o of_o four_o time_n together_o with_o the_o square_n of_o the_o line_n df_o to_o the_o square_n of_o the_o line_n df._n wherefore_o also_o as_o the_o line_n ab_fw-la &_o bc_o add_v together_o be_v to_o the_o line_n ac_fw-la so_o be_v the_o line_n de_fw-fr &_o of_o add_v together_o to_o the_o line_n df_o by_o the_o
spherâ_n contain_v the_o dodecahedron_n of_o this_o pentagon_n and_o the_o icosahedron_n of_o this_o triangle_n by_o the_o 4._o of_o this_o book_n â_o and_o the_o line_n cl_n fall_v perpendicular_o upon_o the_o side_n of_o the_o icosahedron_n and_o the_o line_n ci_o upon_o the_o side_n of_o the_o dodecahedron_n that_o which_o be_v 30._o time_n contain_v under_o the_o side_n and_o the_o perpendicular_a line_n fall_v upon_o it_o be_v equal_a to_o the_o superficies_n of_o that_o solid_a upon_o who_o side_n the_o perpendicular_a fall_v if_o therefore_o in_o a_o circle_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n order_n the_o superficiece_n of_o a_o dodecahedron_n and_o of_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n be_v the_o one_o to_o the_o other_o as_o that_o which_o be_v contain_v under_o the_o side_n of_o the_o one_o and_o the_o perpendicular_a line_n draw_v unto_o it_o from_o the_o centre_n of_o his_o base_a to_o that_o which_o be_v contain_v under_o the_o side_n of_o the_o other_o and_o the_o perpendicular_a line_n draw_v to_o it_o from_o the_o centre_n of_o his_o base_a for_o aâ_z thirtyâ_n timâs_o be_v to_o thirty_o time_n so_o be_v once_o to_o once_o by_o the_o 15._o of_o thâ_z five_o the_o 6._o proposition_n the_o superficies_n of_o a_o dodecahedron_n be_v to_o the_o superficies_n of_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n campane_n in_o that_o proportion_n that_o the_o side_n of_o the_o cube_n be_v to_o the_o side_n of_o the_o icosahedron_n contain_v in_o the_o self_n same_o sphere_n construction_n svppose_v that_o there_o be_v a_o circle_n abg_o &_o in_o it_o by_o the_o 4._o of_o this_o book_n let_v there_o be_v inscribe_v the_o sideâ_n of_o a_o dodecahedron_n and_o of_o a_o icosahedron_n contain_v in_o onâ_n and_o the_o self_n same_o sphere_n and_o let_v the_o side_n oâ_n the_o dodecahedron_n be_v agnostus_n and_o the_o side_n of_o the_o icosahedron_n be_v dg_o and_o let_v the_o centre_n be_v the_o point_n e_o from_o which_o draw_v unto_o those_o sâdes_n perpendicular_a line_n ei_o and_o ez_o and_o produce_v the_o line_n ei_o to_o the_o point_n b_o and_o draw_v the_o linâ_n bg_o and_o let_v the_o side_n of_o the_o cube_fw-la contain_v in_o the_o self_n same_o sphere_n be_v gc_o then_o i_o say_v that_o the_o superficies_n of_o the_o dodecahedron_n iâ_z to_o the_o superficies_n of_o the_o icosahedron_n as_o the_o line_n âg_z iâ_z to_o the_o liââ_n gd_o for_o forasmuch_o as_o the_o line_n ei_o beinâ_n divide_v by_o a_o extreme_a and_o mean_a proportion_n the_o great_a segment_n thereof_o shall_v be_v the_o linâ_n ez_o by_o the_o corollary_n of_o the_o first_o of_o this_o book_n demonstration_n and_o the_o line_n cg_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n his_o great_a segment_n be_v the_o line_n agnostus_n by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o wherefore_o the_o right_a line_n ei_o and_o cg_o ârâ_n cut_v proportional_o by_o the_o second_o of_o this_o bâoke_n whârâfore_o as_o the_o line_n cg_o be_v to_o the_o line_n agnostus_n so_o be_v the_o line_n ei_o to_o the_o line_n ez_fw-fr wherâfore_o that_o which_o it_o contain_v under_o the_o extreme_n cg_o and_o ez_o be_v squall_n to_o that_o which_o iâ_z contaynâd_v under_o the_o mean_n agnostus_n and_o ei._n by_o the_o 16._o of_o the_o six_o but_o as_o that_o which_o iâ_z contain_v under_o the_o linââ_n cg_o and_o âz_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n dg_o and_o ez_o so_o by_o the_o first_o of_o the_o six_o iâ_z the_o linâ_n cg_o to_o the_o line_n dg_o for_o both_o those_o parallelogram_n have_v oââ_n and_o the_o self_n same_o altitude_n namely_o the_o line_n ez_fw-fr wherefore_o as_o that_o which_o be_v contain_v under_o the_o line_n ei_o and_o agnostus_n which_o iâ_z prove_v equal_a to_o that_o which_o be_v contain_v under_o the_o lineâ_z cg_o and_o ez_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n dg_o and_o ez_o so_o be_v the_o line_n cg_o to_o the_o liââ_n dg_o but_o as_o that_o which_o be_v contain_v under_o the_o line_n ei_o and_o agnostus_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n dg_o and_o ez_o so_o by_o the_o corollary_n of_o the_o former_a proposition_n be_v the_o superficies_n of_o the_o dodecahedron_n to_o the_o superficies_n of_o the_o icosahedron_n wherefore_o as_o the_o superficies_n ââ_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v cg_o the_o side_n of_o the_o cube_fw-la to_o gd_v the_o side_n of_o the_o icosahedron_n the_o superficies_n therefore_o of_o a_o dodecahedron_n be_v to_o the_o superficiesâ_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v require_v to_o be_v prove_v a_o assumpt_n the_o pentagon_n of_o a_o dodecahedron_n order_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o perpendicular_a line_n which_o fall_v upon_o the_o base_a of_o the_o triangle_n of_o the_o icosahedron_n and_o five_o six_o part_n of_o the_o side_n of_o the_o cube_fw-la the_o say_v three_o solid_n be_v describe_v in_o one_o and_o the_o self_n same_o sphere_n suppose_v that_o in_o the_o circle_n abeg_n the_o pentagon_n of_o a_o dodecahedron_n be_v aâcig_a and_o let_v two_o side_n thereof_o ab_fw-la and_o agnostus_n be_v subtend_v of_o the_o right_a line_n bg_o and_o let_v the_o triangle_n of_o the_o icosahedron_n inscribe_v in_o the_o self_n same_o sphere_n construction_n by_o the_o 4._o of_o this_o book_n be_v afh_n and_o let_v the_o centre_n of_o the_o circle_n be_v the_o point_n d_o and_o let_v the_o diameter_n be_v ade_n cut_v fh_o the_o side_n of_o the_o triangle_n in_o the_o point_n z_o and_o cut_v the_o line_n bg_o in_o the_o point_n k._n and_o draw_v the_o right_a line_n bd._n and_o from_o the_o right_a line_n kg_v cut_v of_o a_o three_o part_n tg_n by_o the_o 9_o of_o the_o six_o now_o than_o the_o line_n bg_o subtend_v two_o side_n of_o the_o dodecahedron_n shall_v be_v the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o same_o sphere_n demonstration_n by_o the_o 17._o of_o the_o thirteen_o and_o the_o triangle_n of_o the_o icosahedron_n of_o the_o same_o sphere_n shall_v be_v aâh_a by_o the_o 4._o of_o this_o book_n and_o the_o line_n az_o which_o pass_v by_o the_o centre_n d_o shall_v fall_v perpendicular_o upon_o the_o side_n of_o the_o triangle_n for_o forasmuch_o as_o the_o angle_n gay_a &_o bae_o be_v equal_a by_o the_o 27._o of_o the_o thirdâ_o for_o they_o be_v see_v upon_o equal_a circumference_n therefore_o the_o âases_v bk_o and_o kg_n be_v by_o the_o â_o of_o the_o first_o equal_a wherefore_o the_o line_n bt_n contain_v 5._o six_o part_n of_o the_o line_n bg_o then_o i_o say_v that_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o bt_n be_v equal_a to_o the_o pentagon_n aâcâg_n for_o forasmuch_o as_o the_o line_n âz_n be_v sesquialter_n to_o the_o line_n ad_fw-la for_o the_o line_n dâ_n be_v divide_v into_o two_o equal_a part_n in_o the_o point_n z_o by_o the_o corollary_n of_o the_o â2â_n of_o the_o thirteen_o likewise_o by_o construction_n the_o line_n kg_n be_v sesquialter_fw-la to_o the_o line_n kt_n therefore_o as_o the_o line_n az_o be_v to_o the_o line_n ad_fw-la so_o be_v the_o line_n kg_v to_o the_o ãâã_d ât_a wherefore_o that_o which_o be_v contain_v undeâ_n the_o ãâã_d az_o and_o kt_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o mean_n ad_fw-la and_o kg_n by_o the_o 16._o of_o the_o six_o but_o unto_o the_o line_n kg_n be_v the_o line_n âk_v âroved_v equal_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o bk_o but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o bk_o be_v by_o the_o 41._o of_o the_o first_o double_a to_o the_o triangle_n abdella_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n be_v double_a to_o the_o same_o triangle_n abdella_n and_o forasmuch_o as_o the_o pentagon_n abcig_n contaynethâ_n ãâ¦ã_o equal_a âo_o the_o triangle_n abdella_n and_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n contain_v two_o such_o triangle_n therefore_o the_o pentagon_n abcig_n be_v duple_fw-fr sesquialter_fw-la to_o the_o rectangle_n parallelogram_n contain_v under_o the_o line_n az_o and_o kt_n and_o ãâ¦ã_o 1._o of_o the_o six_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o bt_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n as_o the_o base_a bt_n be_v to_o the_o base_a ââtâ_n therefore_o that_o which_o be_v contain_v under_o the_o line_n az_o
and_o bt_n be_v duple_fw-fr sesquialter_fw-la to_o that_o which_o be_v contain_v unââr_o the_o line_n az_o &_o kt_n but_o unto_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n the_o pentagon_n abcig_n be_v prove_v duple_n sesquialter_fw-la wherefore_o the_o pentagon_n abcig_n of_o the_o dodecahedron_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o perpendicular_a line_n az_o and_o under_o the_o line_n bt_v which_o be_v five_o six_o part_n of_o the_o line_n bg_o ¶_o the_o 7._o proposition_n campane_n a_o right_a line_n 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 whole_a line_n and_o the_o great_a segment_n have_v to_o the_o line_n contain_v in_o power_n the_o whole_a and_o the_o less_o segment_n the_o same_o 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 contain_v in_o one_o and_o the_o same_o sphere_n construction_n take_v a_o circle_n abe_n and_o in_o it_o by_o the_o 11._o of_o the_o four_o inscribe_v a_o equilater_n pentagon_n bzech_n and_o by_o the_o second_o of_o the_o same_o a_o equilater_n triangle_n abi_n and_o let_v the_o centre_n thereof_o be_v the_o point_n g._n and_o draw_v a_o line_n from_o g_z to_o b._n and_o divide_v the_o line_n gb_o by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n d_o by_o the_o 30._o of_o the_o six_o and_o let_v the_o line_n ml_o contain_v in_o power_n both_o the_o whole_a line_n gb_o and_o his_o less_o segment_n bd_o by_o the_o corollary_n of_o the_o 13._o of_o the_o ten_o and_o draw_v the_o right_a line_n bâ_n subtend_v the_o angle_n of_o the_o pentagon_n which_o shall_v be_v the_o side_n of_o the_o cube_fw-la by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o â_o and_o the_o line_n by_o shall_v be_v the_o side_n of_o the_o icosahedron_n and_o the_o line_n âz_n the_o side_n of_o the_o dodecahedron_n by_o the_o 4._o of_o this_o book_n then_o i_o say_v that_o be_v the_o side_n of_o the_o cube_fw-la be_v to_o by_o the_o side_n of_o the_o icosahedron_n as_o the_o line_n contain_v in_o power_n the_o line_n bg_o &_o gd_o be_v to_o the_o line_n contain_v in_o power_n the_o line_n gb_o and_o bd._n demonstration_n for_o forasmuch_o as_o by_o the_o 12._o of_o the_o thirteen_o the_o line_n by_o be_v in_o power_n triple_a to_o the_o line_n bg_o and_o by_o the_o 4._o of_o the_o same_o the_o square_n of_o the_o line_n gb_o &_o bd_o be_v triple_a to_o the_o square_n of_o the_o line_n gd_o wherefore_o by_o the_o 15._o of_o the_o five_o the_o square_a of_o the_o line_n by_o be_v to_o the_o square_n of_o the_o line_n gb_o &_o bd_o namely_o triple_a to_o treble_v as_o the_o square_n of_o the_o line_n bâ_n be_v to_o the_o square_n of_o the_o line_n gd_v namely_o as_o one_o be_v to_o one_o but_o as_o the_o square_n of_o the_o line_n bg_o be_v to_o the_o square_n of_o the_o gd_o so_o be_v the_o square_a of_o the_o line_n be_v to_o the_o square_n of_o the_o line_n bz_o for_o the_o line_n bg_o gd_o and_o be_v bz_o be_v in_o one_o and_o the_o same_o proportion_n by_o the_o second_o of_o this_o book_n for_o bz_o be_v the_o great_a segment_n of_o the_o line_n be_v by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o wherefore_o the_o square_a of_o the_o line_n be_v be_v to_o the_o square_n of_o the_o line_n bz_o as_o the_o square_n of_o the_o line_n by_o be_v to_o the_o square_n of_o the_o line_n bg_o and_o bd._n wherefore_o alternate_o the_o square_a of_o the_o line_n be_v be_v to_o the_o square_n of_o the_o line_n by_o as_o the_o square_n of_o the_o line_n bz_o be_v to_o the_o square_n of_o the_o line_n gb_o and_o bd._n but_o the_o square_n of_o the_o line_n bz_o be_v equal_a to_o the_o square_n of_o the_o line_n bg_o and_o gd_o by_o the_o 10._o of_o the_o thirteen_o for_o the_o line_n bg_o be_v equal_a to_o the_o side_n of_o the_o hexagon_n and_o the_o line_n gd_o to_o the_o side_n of_o the_o decagon_n by_o the_o corollary_n of_o the_o 9_o of_o the_o same_o wherefore_o the_o square_n of_o the_o line_n bg_o and_o gd_o be_v to_o the_o square_n of_o the_o line_n gâ_n and_o bd_o as_o the_o square_n of_o the_o line_n be_v be_v to_o the_o square_n of_o the_o line_n by_o but_o the_o line_n zb_n contain_v in_o power_n the_o line_n bg_o and_o gd_o and_o the_o line_n ml_o contain_v in_o power_n the_o line_n gb_o and_o bd_o by_o construction_n wherefore_o as_o the_o line_n zb_n which_o contain_v in_o power_n the_o whole_a line_n bg_o and_o the_o great_a segment_n gd_o be_v to_o the_o line_n ml_o which_o contain_v in_o in_o power_n the_o whole_a line_n gb_o and_o the_o less_o segment_n bd_o so_o be_v be_v the_o side_n of_o the_o cube_fw-la to_o by_o the_o side_n of_o the_o icosahedron_n by_o the_o 22._o of_o the_o six_o wherefore_o a_o right_a line_n 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 whole_a line_n and_o the_o great_a segment_n have_v to_o the_o line_n contain_v in_o power_n the_o whole_a line_n and_o the_o less_o segment_n the_o same_o 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 contain_v in_o one_o and_o the_o same_o sphere_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 8._o proposition_n order_n the_o solid_a of_o a_o dodecahedron_n be_v to_o the_o solid_a of_o a_o icosahedron_n as_o the_o side_n of_o a_o cube_n be_v to_o the_o side_n of_o a_o icosahedron_n all_o those_o solid_n be_v describe_v in_o one_o and_o the_o self_n same_o sphere_n forasmuch_o as_o in_o the_o 4._o of_o this_o book_n it_o have_v be_v prove_v that_o one_o and_o the_o self_n same_o circle_n contain_v both_o the_o triangle_n of_o a_o icosahedron_n and_o the_o pentagon_n of_o a_o dodecahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n wherefore_o the_o circle_n which_o contain_v those_o base_n be_v equal_a the_o perpendicular_n also_o which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o those_o circle_n shall_v be_v equal_a by_o the_o corollary_n of_o the_o assumpt_n of_o the_o 16_o of_o the_o twelve_o and_o therefore_o the_o pyramid_n set_v upon_o the_o base_n of_o those_o solid_n have_v one_o and_o the_o self_n same_o altitude_n for_o the_o altitude_n of_o those_o pyramid_n concurrâ_n in_o the_o centre_n wherefore_o they_o be_v in_o proportion_n as_o their_o base_n be_v by_o the_o 5._o and_o 6._o of_o the_o twelve_o and_o therefore_o the_o pyramid_n which_o compose_v the_o dodecahedron_n arâ_n to_o the_o pyramid_n which_o compose_v the_o icosahedron_n as_o the_o base_n be_v which_o base_n be_v the_o superficiece_n of_o those_o solid_n wherefore_o their_o solid_n be_v the_o one_o to_o the_o other_o as_o their_o superficiece_n be_v but_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 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 by_o the_o 6._o of_o this_o book_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o solid_a of_o the_o dodecahedron_n be_v to_o the_o solid_a of_o the_o icosahedron_n so_o be_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n all_o the_o say_v solid_n be_v inscribe_v in_o one_o and_o the_o self_n same_o sphere_n wherefore_o the_o solid_a of_o a_o dodecahedron_n be_v to_o the_o solid_a of_o a_o icosahedron_n as_o the_o side_n of_o a_o cube_fw-la be_v to_o the_o side_n of_o a_o icosahedron_n all_o those_o solid_n be_v describe_v in_o one_o and_o the_o self_n same_o sphere_n which_o be_v require_v to_o be_v prove_v a_o corollary_n the_o solid_a of_o a_o dodecahedron_n be_v to_o the_o solid_a of_o a_o icosahedron_n campane_n as_o the_o superficiece_n of_o the_o one_o be_v to_o the_o superficiece_n of_o the_o other_o be_v describe_v in_o one_o and_o the_o self_n same_o sphere_n namely_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 as_o be_v before_o manifest_v for_o they_o be_v resolve_v into_o pyramid_n of_o one_o and_o the_o self_n same_o altitude_n ¶_o the_o 9_o proposition_n if_o the_o side_n of_o a_o equilater_n triangle_n be_v rational_a the_o superficies_n shall_v be_v irrational_a campane_n of_o that_o kind_n which_o be_v call_v medial_a svppose_v that_o abg_o be_v a_o equilater_n triangle_n and_o from_o the_o point_n a_o draw_v unto_o the_o side_n bg_o a_o perpendicular_a line_n ad_fw-la and_o let_v the_o line_n ab_fw-la be_v rational_a then_o i_o say_v that_o the_o superficies_n abg_o be_v medial_a construction_n forasmuch_o as_o the_o line_n ab_fw-la be_v in_o power_n
and_o have_v the_o same_o altitude_n with_o it_o name_o the_o altitude_n of_o the_o parallel_a base_n as_o it_o be_v manifest_a by_o the_o former_a be_v equal_a to_o three_o of_o those_o pyramid_n of_o the_o octohedron_n by_o the_o first_o corollary_n of_o the_o seven_o of_o the_o twelve_o wherefore_o that_o prism_n shall_v have_v to_o the_o other_o prism_n under_o the_o same_o altitude_n compose_v of_o the_o 4._o pyramid_n of_o the_o whole_a octohedron_n the_o proportion_n of_o the_o triangular_a base_n by_o the_o 3._o corollary_n of_o the_o same_o and_o forasmuch_o as_o 4._o pyramid_n be_v unto_o 3._o pyramid_n in_o sesquitercia_fw-la proportion_n therefore_o the_o triangule_a base_a of_o the_o prism_n which_o contain_v 4._o pyramid_n be_v in_o sesquitertia_fw-la proportion_n to_o the_o base_a of_o the_o prism_n which_o contain_v three_o pyramid_n of_o the_o same_o octohedron_n and_o be_v set_v upon_o the_o base_a of_o the_o octohedron_n and_o under_o the_o altitude_n thereof_o that_o be_v in_o sesquitercia_fw-la proportion_n to_o the_o base_a of_o the_o octohedron_n but_o the_o base_a of_o the_o same_o octohedron_n be_v in_o sesquitertia_fw-la proportion_n to_o the_o base_a of_o the_o pyramid_n by_o the_o âenth_n of_o this_o book_n wherefore_o the_o triangular_a base_n namely_o of_o the_o prism_n which_o contain_v four_o pyramid_n of_o the_o octohedron_n and_o be_v under_o the_o altitude_n thereof_o be_v equal_a to_o the_o triangular_a base_n of_o the_o prism_n which_o contain_v three_o pyramid_n under_o the_o altitude_n of_o the_o pyramid_n efgh_o but_o the_o prism_n of_o the_o octohedron_n be_v equal_a to_o the_o octohedron_n and_o the_o prism_n of_o the_o pyramid_n efgh_o be_v prove_v triple_a to_o the_o same_o pyramid_n efgh_o now_o than_o the_o prism_n set_v upon_o equal_a base_n be_v the_o one_o to_o the_o other_o as_o their_o altitude_n be_v by_o the_o corollary_n of_o the_o 25._o of_o the_o eleven_o namely_o as_o be_v the_o parallelipidedon_n their_o double_n by_o the_o corollary_n of_o the_o 31._o of_o the_o eleven_o but_o the_o altitude_n of_o the_o octohedron_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la contain_v in_o the_o same_o sphere_n by_o the_o corollary_n of_o the_o 13._o of_o this_o book_n and_o the_o side_n of_o the_o cube_fw-la be_v in_o power_n to_o the_o altitude_n of_o the_o tetrahedon_n in_o that_o proportion_n that_o 12._o be_v to_o 16_o by_o the_o 18._o of_o the_o thirteen_o and_o the_o side_n of_o the_o octohedron_n be_v to_o the_o side_n of_o the_o pyramid_n in_o that_o proportion_n that_o 18._o be_v to_o 24._o by_o the_o same_o 18._o of_o the_o thirteen_o which_o proportion_n be_v one_o &_o the_o self_n same_o with_o the_o proportion_n of_o 12._o to_o 16._o wherefore_o that_o prism_n which_o be_v equal_a to_o the_o octohedron_n be_v to_o the_o prism_n which_o be_v triple_a to_o the_o tetrahedron_n in_o that_o proportion_n that_o the_o altitude_n or_o that_o the_o side_n be_v wherefore_o a_o octohedron_n be_v to_o the_o triple_a of_o a_o tetrahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n in_o that_o proportion_n that_o their_o side_n be_v which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n the_o side_n of_o a_o tetrahedron_n &_o of_o a_o octohedron_n be_v proportional_a with_o their_o altitude_n for_o the_o side_n &_o altitude_n be_v in_o power_n sesquitercia_fw-la moreover_o the_o diameter_n of_o the_o sphere_n be_v to_o the_o side_n of_o the_o tetrahedron_n as_o the_o side_n of_o the_o octohedron_n be_v to_o the_o ââde_n of_o the_o cubeâ_n namely_o the_o power_n of_o each_o be_v in_o sesquialter_fw-la proportion_n by_o the_o 18._o of_o the_o thirteen_o the_o 15._o proposition_n if_o a_o rational_a line_n contain_v in_o power_n two_o line_n make_v the_o whole_a and_o the_o great_a segment_n and_o again_o contain_v in_o power_n two_o line_n make_v the_o whole_a and_o the_o less_o segment_n the_o great_a segment_n shall_v be_v the_o side_n of_o the_o icosahedron_n and_o the_o less_o segment_n shall_v be_v the_o side_n of_o the_o dodecahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n svppose_v that_o agnostus_n be_v the_o diameter_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n abgc_n and_o let_v bg_o subtend_v the_o side_n of_o the_o pentagon_n describe_v of_o the_o side_n of_o the_o icosahedron_n by_o the_o 16._o of_o the_o thirteen_o moreover_o upon_o the_o same_o diameter_n agnostus_n or_o df_o equal_a unto_o it_o construction_n let_v there_o be_v describe_v a_o dodecahedron_n defh_o by_o the_o 1â_o of_o the_o thirteen_o who_o opposite_n side_n ed_z and_o fh_o let_v be_v cut_v into_o two_o equal_a part_n in_o the_o point_n i_o and_o king_n and_o draw_v a_o line_n from_o i_o to_o k._n and_o let_v the_o line_n of_o couple_n two_o of_o the_o opposite_a angle_n of_o the_o base_n which_o be_v join_v together_o then_o i_o say_v that_o ab_fw-la the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n which_o the_o diameter_n agnostus_n contain_v in_o power_n together_o with_o the_o whole_a line_n and_o line_n ed_z be_v the_o less_o segment_n which_o the_o same_o diameter_n agnostus_n or_o df_o contain_v in_o power_n together_o with_o the_o whole_a demonstration_n for_o forasmuch_o as_o the_o opposite_a side_n ab_fw-la and_o gc_o of_o the_o icosahedron_n be_v couple_v by_o the_o diameter_n agnostus_n and_o bc_o be_v equal_a &_o parallel_n by_o the_o 2._o corollary_n of_o the_o 16._o of_o the_o thirteen_o the_o right_a line_n bg_o &_o ac_fw-la which_o couple_v they_o together_o be_v equal_a &_o parallel_n by_o the_o 33._o of_o the_o first_o moreover_o the_o angle_n bac_n &_o abg_o be_v subtend_v of_o equal_a diamâters_n shall_v by_o the_o 8._o of_o the_o first_o be_v equal_a &_o by_o the_o 29_o of_o the_o ãâã_d they_o shall_v be_v right_a angle_n wherefore_o the_o right_a line_n agnostus_n ãâã_d in_o power_n the_o too_o line_n ab_fw-la and_o bg_o by_o the_o 47._o of_o ãâ¦ã_o and_o forasmuch_o as_o the_o line_n bg_o subtend_v the_o angle_n of_o the_o pentagon_n compose_v of_o the_o side_n of_o the_o icosahedron_n the_o great_a segment_n of_o the_o right_a line_n bg_o shall_v be_v the_o right_a line_n ab_fw-la by_o the_o â_o of_o the_o thirteen_o which_o line_n ab_fw-la together_o with_o the_o whole_a line_n bg_o the_o line_n agnostus_n contain_v in_o power_n and_o forasmuch_o aâ_z the_o line_n ik_fw-mi couple_v the_o opposite_a and_o parallel_a side_n ed_z and_o fh_o of_o the_o dodecahedron_n make_v at_o those_o point_n right_a angle_n by_o the_o 3._o corollary_n of_o the_o 17._o of_o tâe_z thirteen_o the_o right_a line_n of_o which_o couple_v together_o equal_a and_o parallel_a line_n ei_o &_o fk_v shall_v be_v equal_a to_o the_o same_o line_n ik_fw-mi by_o the_o 33._o of_o the_o first_o wherefore_o the_o angle_n def_n shall_v be_v â_o right_n angle_n by_o the_o 29._o of_o the_o first_o wherefore_o the_o diameter_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o ef._n but_o the_o less_o segment_n of_o the_o line_n ik_fw-mi be_v ed_z the_o side_n of_o the_o dodecahedron_n by_o the_o 4._o corollary_n of_o the_o 17â_n of_o the_o thirteen_o wherefore_o the_o same_o line_n ed_z be_v also_o the_o less_o segment_n of_o the_o line_n of_o which_o be_v equal_a unto_o the_o line_n ik_fw-mi wherefore_o the_o diameter_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o of_o by_o the_o 47._o of_o the_o first_o contain_v in_o powârâ_n ed_z the_o side_n of_o the_o dodecahedron_n the_o less_o segment_n together_o with_o the_o whole_a if_o therefore_o a_o rational_a line_n agnostus_n or_o df_o contain_v in_o power_n two_o line_n ab_fw-la and_o bg_o do_v make_v the_o whole_a line_n and_o the_o great_a segment_n and_o again_o contain_v in_o power_n two_o line_n of_o and_o ed_z do_v make_v the_o whole_a line_n and_o the_o less_o segment_n the_o great_a segment_n ab_fw-la shall_v be_v the_o side_n of_o the_o icosahedron_n and_o the_o less_o segment_n ed_z shall_v be_v the_o side_n of_o the_o dodecahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n the_o 16._o proposition_n if_o the_o power_n of_o the_o side_n of_o a_o octohedron_n be_v express_v by_o two_o right_a lineâ_z join_v together_o by_o a_o extreme_a and_o meâne_a proportion_n the_o side_n of_o the_o icosahedron_n contain_v in_o the_o same_o sphere_n shall_v be_v duple_v to_o the_o less_o segment_n let_v ab_fw-la the_o side_n of_o the_o octohedron_n abg_o contain_v in_o power_n the_o two_o line_n c_o and_o h_o which_o let_v have_v that_o proportion_n that_o the_o whole_a have_v to_o the_o great_a segment_n by_o the_o corollarye_a of_o the_o first_o proposition_n add_v by_o flussas_n after_o the_o last_o proposition_n of_o the_o six_o book_n construction_n and_o let_v the_o icosahedron_n contain_v in_o the_o same_o sphere_n be_v
def_n who_o side_n let_v be_v de_fw-fr and_o let_v the_o right_a line_n subtend_v the_o angle_n of_o the_o pentagon_n make_v of_o the_o side_n of_o the_o icosahedron_n be_v the_o line_n ef._n then_o i_o say_v that_o the_o side_n ed_z be_v in_o power_n double_a to_o the_o line_n h_o the_o less_o of_o those_o segment_n forasmuch_o as_o by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n demonstration_n it_o be_v manifest_a that_o ed_z the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n of_o the_o line_n efâ_n and_o that_o the_o diameter_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o of_o namely_o the_o whole_a and_o the_o great_a segment_n but_o by_o supposition_n the_o side_n ab_fw-la contain_v in_o power_n the_o two_o line_n c_o &_o h_o join_v together_o in_o the_o self_n same_o proportion_n wherefore_o the_o line_n of_o be_v to_o the_o line_n ed_z as_z the_o line_n c_o be_v to_o the_o line_n h_o by_o the_o â_o oâ_n this_o bokeâ_n and_o alternally_n by_o the_o 16._o of_o the_o five_o the_o line_n of_o be_v to_o the_o line_n c_o as_o the_o line_n ed_z be_v to_o the_o line_n h._n and_o forasmuch_o as_o the_o line_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o of_o and_o the_o line_n ab_fw-la contain_v in_o power_n the_o two_o line_n c_o and_o h_o therefore_o the_o square_n of_o the_o line_n of_o and_o ed_z be_v to_o the_o square_n of_o the_o line_n df_o as_o the_o square_n of_o the_o line_n c_o and_o h_o to_o the_o square_a ab_fw-la and_o alternate_o the_o square_n of_o the_o line_n of_o and_o âd_a be_v to_o the_o square_n of_o the_o line_n c_o and_o h_o as_o the_o square_n of_o the_o line_n df_o be_v to_o the_o square_n of_o the_o line_n abâ_n but_o df_o the_o diameter_n be_v by_o the_o 14._o of_o the_o thirtenâh_o iâ_z power_n double_a to_o ab_fw-la the_o side_n of_o the_o octohedron_n inscribe_v by_o supposition_n in_o the_o same_o sphere_n wherefore_o the_o square_n of_o the_o line_n of_o and_o ed_z be_v double_a to_o the_o square_n of_o the_o line_n c_o and_o h._n and_o therefore_o one_o square_a of_o the_o line_n ed_z be_v double_a to_o one_o square_a of_o the_o line_n h_o by_o the_o 12._o of_o the_o five_o wherefore_o ed_z the_o side_n of_o the_o icosahedron_n be_v in_o power_n duple_n to_o the_o line_n h_o which_o be_v the_o less_o segment_n if_o therefore_o the_o poweâ_n of_o the_o side_n of_o a_o octohedron_n be_v express_v by_o two_o right_a line_n join_v together_o by_o a_o extreme_a and_o mean_a proportion_n the_o side_n of_o the_o icosahedron_n contain_v in_o the_o same_o sphere_n shall_v be_v duple_v to_o the_o less_o segment_n the_o 17._o proposition_n if_o the_o side_n of_o a_o dodecahedron_n and_o the_o right_a line_n of_o who_o the_o say_a side_n be_v the_o less_o segment_n be_v so_o set_v that_o they_o make_v a_o right_a angle_n the_o right_a line_n which_o contain_v in_o power_n half_o the_o line_n subtend_v the_o angle_n be_v the_o side_n of_o a_o octohedron_n contain_v in_o the_o self_n same_o sphere_n svppose_v that_o ab_fw-la be_v the_o side_n of_o a_o dodecahedron_n and_o let_v the_o right_a line_n of_o which_o that_o side_n be_v the_o less_o segment_n be_v agnostus_n namely_o which_o couple_v the_o opposite_a side_n of_o the_o dodecahedron_n by_o the_o 4._o corollary_n of_o the_o 17._o of_o the_o thirteen_o construction_n and_o let_v those_o line_n be_v so_o set_v that_o they_o make_v a_o right_a angle_n at_o the_o point_n a._n and_o draw_v the_o right_a line_n bg_o and_o let_v the_o line_n d_o contain_n in_o power_n half_o the_o line_n bg_o by_o the_o first_o proposition_n add_v by_o flussas_n after_o the_o last_o of_o the_o six_o then_o i_o say_v that_o the_o line_n d_o be_v the_o side_n of_o a_o octohedron_n contain_v in_o the_o same_o sphere_n forasmuch_o as_o the_o line_n agnostus_n make_v the_o great_a segment_n gc_o the_o side_n of_o the_o cube_fw-la contain_v in_o the_o same_o sphere_n by_o the_o same_o 4._o corollary_n of_o the_o 17._o of_o the_o thirteen_o demonstration_n and_o the_o square_n of_o the_o whole_a line_n ag._n and_o of_o the_o less_o segment_n ab_fw-la be_v triple_a to_o the_o square_n of_o the_o great_a segment_n gc_o by_o the_o 4._o of_o the_o thirteen_o moreover_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n triple_a to_o the_o same_o line_n gc_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o the_o thirteen_o wherefore_o the_o line_n bg_o be_v equal_a to_o the_o ãâã_d for_o it_o contain_v in_o power_n the_o two_o line_n ab_fw-la and_o ag_z by_o the_o 47._o of_o the_o first_o and_o therefore_o it_o contain_v in_o power_n the_o triple_a of_o the_o line_n gc_o but_o the_o side_n of_o the_o octohedron_n contain_v in_o the_o same_o sphere_n be_v in_o power_n triple_a to_o half_a the_o diameter_n of_o the_o sphere_n by_o the_o 14._o of_o the_o thirteen_o and_o by_o supposition_n the_o line_n d_o contaiââââ_n in_o powââ_n the_o half_a of_o the_o line_n bg_o wherefore_o the_o line_n d_o contain_v in_o power_n the_o half_a of_o the_o same_o diameter_n be_v the_o side_n of_o a_o octohedron_n if_o therefore_o the_o side_n of_o a_o dodecahedron_n and_o the_o right_a line_n of_o who_o the_o say_a side_n be_v the_o less_o segment_n be_v so_o set_v that_o they_o make_v a_o right_a angle_n the_o right_a line_n which_o contain_v in_o power_n half_o the_o line_n subtend_v the_o angle_n be_v the_o side_n of_o a_o ocâââedron_n contain_v in_o the_o self_n same_o sphere_n which_o be_v require_v to_o be_v prove_v a_o corollary_n unto_o what_o right_a line_n the_o side_n of_o the_o octoâedron_n be_v in_o power_n sesquialter_fw-la unto_o the_o same_o line_n the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o sphere_n be_v the_o great_a segment_n for_o the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o the_o segment_n cg_o unto_o which_o d_z the_o side_n of_o the_o octohedron_n be_v in_o power_n sesquialter_n that_o be_v be_v half_a of_o the_o power_n of_o the_o line_n bg_o which_o be_v triple_a unto_o the_o line_n cg_o ¶_o the_o 18._o proposition_n if_o the_o side_n of_o a_o tetrahedron_n contain_v in_o power_n two_o right_a line_n join_v together_o by_o a_o extreme_a and_o mean_a proportion_n the_o side_n of_o a_o icosahedron_n describe_v in_o the_o self_n same_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o less_o right_a line_n svppose_v that_o abc_n be_v a_o tetrahedron_n and_o let_v his_o side_n be_v ab_fw-la construction_n who_o power_n let_v be_v divide_v into_o the_o line_n agnostus_n and_o gb_o join_v together_o by_o a_o extreme_a and_o mean_a proportion_n namely_o let_v it_o be_v divide_v into_o agnostus_n the_o whole_a line_n and_o gb_o the_o great_a seâment_n by_o the_o corollary_n of_o the_o first_o proposition_n add_v by_o flussas_n after_o the_o last_o of_o the_o six_o and_o let_v ed_z be_v the_o side_n of_o the_o icosahedron_n edf_o contain_v in_o the_o self_n same_o sphere_n and_o let_v the_o line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n describe_v of_o the_o side_n of_o the_o icosahedron_n be_v ef._n then_o i_o say_v that_z ed_z the_o side_n of_o the_o icosahedron_n be_v in_o power_n sesquialter_fw-la to_o the_o less_o line_n gb_o demonstration_n forasmuch_o as_o by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n the_o side_n ed_z be_v the_o greâter_a segment_n of_o the_o line_n of_o which_o subtend_v the_o angle_n of_o the_o pentagon_n but_o as_o the_o whole_a line_n of_o be_v to_o the_o great_a segment_n ed_z so_o be_v the_o same_o grââter_a segment_n to_o the_o less_o by_o the_o 30._o of_o the_o six_o and_o by_o supposition_n agnostus_n be_v the_o whole_a line_n and_o gâ_n the_o great_a segment_n wherefore_o as_o of_o be_v to_o ed_z so_o be_v agnostus_n to_o gâ_n by_o the_o second_o of_o the_o fourteen_o and_o alternate_o the_o line_n of_o be_v to_o the_o line_n agnostus_n as_o the_o line_n ed_z be_v to_o the_o line_n gb_o and_o forasmuch_o as_o by_o supposition_n the_o line_n ab_fw-la contain_v in_o power_n the_o two_o line_n agnostus_n and_o gb_o therefore_o by_o the_o 4â_o of_o the_o first_o the_o angle_n agb_n be_v a_o right_a angle_n but_o the_o angle_n def_n be_v a_o right_a angle_n by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n wherefore_o the_o triangle_n agâ_n and_o feed_v be_v equiangle_n by_o the_o â_o of_o the_o six_o wherefore_o their_o side_n be_v proportional_a namely_o as_o the_o line_n ed_z be_v to_o the_o line_n gb_o so_o be_v the_o line_n fd_o to_o the_o line_n ab_fw-la by_o the_o 4._o of_o the_o six_o but_o by_o that_o which_o have_v before_o
overthrow_v and_o overwhelm_v the_o whole_a world_n he_o be_v utter_o rude_a and_o ignorant_a in_o the_o greek_a tongue_n so_o that_o certain_o he_o never_o read_v euclid_n in_o the_o greek_a nor_o of_o like_a translate_v out_o of_o the_o greek_a but_o have_v it_o translate_v out_o of_o the_o arabike_a tongue_n the_o arabian_n be_v man_n of_o great_a study_n and_o industry_n and_o common_o great_a philosopher_n notable_a physician_n and_o in_o mathematical_a art_n most_o expert_a so_o that_o all_o kind_n of_o good_a learning_n flourish_v and_o reign_v amongst_o they_o in_o a_o manner_n only_o these_o man_n turn_v whatsoever_o good_a author_n be_v in_o the_o greek_a tongue_n of_o what_o art_n and_o knowledge_n so_o ever_o it_o be_v into_o the_o arabike_a tongue_n and_o from_o thence_o be_v many_o of_o they_o turn_v into_o the_o latin_a and_o by_o that_o mean_v many_o greek_a author_n come_v to_o the_o hand_n of_o the_o latin_n and_o not_o from_o the_o first_o fountain_n the_o greek_a tongue_n wherein_o they_o be_v first_o write_v as_o appear_v by_o many_o word_n of_o the_o arabike_a tongue_n yet_o remain_v in_o such_o book_n as_o be_v zenith_n nadir_n helmuayn_fw-mi helmuariphe_n and_o infinite_a such_o other_o which_o arabian_n also_o in_o translate_n such_o greek_a work_n be_v accustom_v to_o add_v as_o they_o think_v good_a &_o for_o the_o full_a understanding_n of_o the_o author_n many_o thing_n as_o be_v to_o be_v see_v in_o diverse_a author_n as_o namely_o in_o theodosius_n de_fw-la sphera_fw-la where_o you_o see_v in_o the_o old_a translation_n which_o be_v undoubteldy_a out_o of_o the_o arabike_a many_o proposition_n almost_o every_o three_o or_o four_o leaf_n some_o such_o copy_n of_o euclid_n most_o likely_a do_v campanus_n follow_v wherein_o he_o find_v those_o proposition_n which_o he_o have_v more_o &_o above_o those_o which_o be_v find_v in_o the_o greek_a set_v out_o by_o hypsicles_n and_o that_o not_o only_o in_o this_o 15._o book_n but_o also_o in_o the_o 14._o book_n wherein_o also_o you_o find_v many_o proposition_n more_o they_o be_v find_v in_o the_o greek_a set_v out_o also_o by_o hypsicles_n likewise_o in_o the_o book_n before_o you_o shall_v find_v many_o proposition_n add_v and_o many_o invert_v and_o set_v out_o of_o order_n far_o otherwise_o than_o they_o be_v place_v in_o the_o greek_a examplar_n flussas_n also_o a_o diligent_a restorer_n of_o euclid_n a_o man_n also_o which_o have_v well_o deserve_v of_o the_o whole_a art_n of_o geometry_n have_v add_v moreover_o in_o this_o book_n as_o also_o in_o the_o former_a 14._o book_n he_o add_v 8._o proposition_n 9_o proposition_n of_o his_o own_o touch_v the_o inscription_n and_o circumscription_n ãâ¦ã_o body_n very_o singular_a ândoubtedly_o and_o witty_a all_o which_o for_o that_o nothing_o shall_v want_v to_o the_o desirous_a lover_n of_o knowledge_n i_o have_v faithful_o with_o no_o small_a pain_n turn_v and_o whereas_o flâssââ_n in_o the_o begin_n of_o the_o eleven_o book_n namely_o in_o the_o end_n of_o the_o definition_n there_o âeâ_n put_v two_o definition_n of_o the_o inscription_n and_o circumscription_n of_o solid_n or_o corporal_a figure_n within_o or_o about_o the_o one_o the_o other_o which_o certain_o be_v not_o to_o be_v reject_v yet_o for_o that_o until_o this_o present_a 15._o book_n there_o be_v no_o mention_n make_v of_o the_o inscription_n or_o circumscription_n of_o these_o body_n i_o think_v it_o not_o so_o convenient_a thârâ_n to_o place_v they_o but_o to_o refer_v they_o to_o the_o begin_n of_o this_o 15._o book_n where_o they_o be_v in_o manner_n of_o necessity_n require_v to_o the_o elucidation_n of_o the_o proposition_n and_o dâmonstrationâ_n of_o the_o same_o the_o definition_n be_v these_o definition_n 1._o a_o solid_a figure_n be_v then_o âaid_v to_o be_v inscribe_v in_o a_o solid_a figure_n when_o the_o angle_n of_o the_o figure_n inscribe_v touch_n together_o at_o one_o time_n either_o the_o angle_n of_o the_o figure_n circumscribe_v or_o the_o superficiece_n or_o the_o side_n definition_n 2._o a_o solid_a figure_n be_v then_o say_v to_o be_v circumscribe_v about_o a_o solid_a figure_n when_o together_o at_o one_o time_n either_o the_o angle_n or_o the_o superficiece_n or_o the_o side_n of_o the_o figure_n circumscribe_v âouch_v the_o angle_n of_o the_o figure_n inscribe_v in_o the_o fourââ_n book_n in_o the_o definition_n of_o the_o inscription_n or_o circumscription_n of_o plain_a rectiline_n figure_v one_o with_o in_o or_o about_o a_o other_o be_v require_v that_o all_o the_o angle_n of_o the_o figuââ_n inscribe_v shall_v at_o one_o time_n touch_v all_o the_o side_n of_o the_o figure_n circumscribe_v but_o in_o the_o five_o regular_a solid_n âo_o who_o chief_o these_o two_o definition_n pertain_v for_o that_o the_o number_n of_o their_o angle_n superficiece_n &_o side_n be_v not_o equal_a one_o compare_v to_o a_o other_o it_o be_v not_o of_o necessity_n that_o all_o the_o angle_n of_o the_o solid_a inscribe_v shall_v together_o at_o one_o time_n touch_v either_o all_o the_o angle_n or_o all_o the_o superficiece_n or_o all_o the_o side_n of_o the_o solid_a circumscribe_v but_o it_o be_v sufficient_a that_o those_o angle_n of_o the_o inscribe_v solid_a which_o touch_n do_v at_o one_o time_n together_o each_o touch_n some_o one_o angle_n of_o the_o figure_n circumscribe_v or_o some_o one_o base_a or_o some_o one_o side_n so_o that_o if_o the_o angle_n of_o the_o inscribe_v figure_n do_v at_o one_o time_n touch_v the_o angle_n of_o the_o figure_n circumscribe_v none_o of_o they_o may_v at_o the_o same_o time_n touch_v either_o the_o base_n or_o the_o side_n of_o the_o same_o circumscribe_v figure_n and_o so_o if_o they_o touch_v the_o base_n they_o may_v touch_v neither_o angle_n nor_o side_n and_o likewise_o if_o they_o touch_v the_o side_n they_o may_v touch_v neither_o angle_n nor_o base_n and_o although_o sometime_o all_o the_o angle_n of_o the_o figure_n inscribe_v can_v not_o touch_v either_o the_o angle_n or_o the_o base_n or_o the_o side_n of_o the_o figure_n circumscribe_v by_o reason_n the_o number_n of_o the_o angle_n base_n or_o side_n of_o the_o say_a figure_n circumscribe_v want_v of_o the_o number_n of_o the_o angle_n of_o the_o âigure_n inscribe_v yet_o shall_v those_o angle_n of_o the_o inscribe_v figure_n which_o touch_n so_o touch_v that_o the_o void_a place_n leave_v between_o the_o inscribe_v and_o circumscribe_v figure_n shall_v on_o every_o side_n be_v equal_a and_o like_a as_o you_o may_v afterward_o in_o this_o fifteen_o book_n most_o plain_o perceive_v ¶_o the_o 1._o proposition_n the_o 1._o problem_n in_o a_o cube_n give_v to_o describe_v admonish_v a_o trilater_n equilater_n pyramid_n svppose_v that_o the_o cube_fw-la give_v be_v abcdefgh_o in_o the_o same_o cube_fw-la it_o be_v require_v to_o inscribe_v a_o tetrahedron_n draw_v these_o right_a line_n ac_fw-la construction_n ce_fw-fr ae_n ah_o eh_o hc_n demonstration_n now_o it_o be_v manifest_a that_o the_o triangle_n aec_o ahe_o ahc_n and_o i_o be_v equilater_n for_o their_o side_n be_v the_o diameter_n of_o equal_a square_n wherefore_o aech_n be_v a_o trilater_n equilater_n pyramid_n or_o tetrahedron_n &_o it_o be_v inscribe_v in_o the_o cube_fw-la give_v by_o the_o first_o definition_n of_o this_o book_n which_o be_v require_v to_o be_v do_v ¶_o the_o 2._o proposition_n the_o 2._o problem_n in_o a_o trilater_n equilater_n pyramid_n give_v to_o describe_v a_o octohedron_n svppose_v that_o the_o trilater_n equilater_n pyramid_n give_v be_fw-mi abcd_o who_o side_n let_v be_v divide_v into_o two_o equal_a part_n in_o the_o point_n e_o z_o i_o king_n l_o t._n construction_n and_o draw_v these_o 12._o right_a line_n ez_o zi_n je_n kl_o lt_n tk_n eke_o kz_o zl_n li_n it_o and_o te_fw-la which_o 12._o right_a line_n be_v demonstration_n by_o the_o 4._o of_o the_o first_o equal_a for_o they_o subtend_v equal_a plain_a angle_n of_o the_o base_n of_o the_o pyramid_n and_o those_o equal_a angle_n be_v contain_v under_o equal_a side_n namely_o under_o the_o half_n of_o the_o side_n of_o the_o pyramid_n wherefore_o the_o triangle_n tkl_n tli_n tie_v tek_v zkl_n zli_n zib_n zek_n be_v equilater_n and_o they_o limitate_v and_o contain_v the_o solid_a tklezi_n wherefore_o the_o solid_a tklezi_n be_v a_o octohedron_n by_o the_o 23._o definition_n of_o the_o eleven_o and_o the_o angle_n of_o the_o same_o octohedron_n do_v touch_v the_o side_n of_o the_o pyramid_n abcd_o in_o the_o point_n e_o z_o i_o t_o edward_n l._n wherefore_o the_o octohedron_n be_v inscribe_v in_o the_o pyramid_n by_o the_o 1._o definition_n of_o this_o book_n wherefore_o in_o the_o trilater_n equilater_n pyramid_n give_v be_v inscribe_v a_o octohedron_n which_o be_v require_v to_o be_v do_v a_o corollary_n add_v by_o flussas_n hereby_o it_o be_v manifest_a that_o a_o pyramid_n be_v cut_v into_o two_o
right_a linâs_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 âhe_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 ââ_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 âs_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 foââd_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 ãâã_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 circumferânces_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 octoâedron_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 tâtrahedronâ_n the_o end_n of_o the_o fivetenth_fw-mi book_n of_o euclides_n element_n after_o ãâã_d ãâã_d ãâã_d ãâã_d ãâã_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 whereââ_n the_o tetrahedron_n be_v require_v to_o be_v insâriâââ_n be_v abgdei_n take_v ãâ¦ã_o base_n of_o the_o octoââdron_n that_o be_v construction_n ãâ¦ã_o close_o in_o the_o loweââ_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 draâe_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 ãâã_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 ââ_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 ãâ¦ã_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 righâline_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 lineââ_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 âo_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 âeven_v be_v inscribe_v a_o trilate_a equilaââââââamisâ_n which_o be_v require_v to_o âe_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
to_o the_o base_n of_o the_o octohedron_n for_o forasmuch_o as_o the_o side_n of_o the_o base_n of_o the_o pyramid_n touch_v the_o one_o the_o other_o be_v parallel_n to_o the_o side_n of_o the_o octohedron_n which_o also_o touch_v the_o one_o the_o other_o as_o for_o example_n hl_o be_v prove_v to_o be_v a_o parallel_n to_o give_v and_o lc_a to_o diego_n therefore_o by_o the_o 15._o of_o the_o eleven_o the_o plain_a superficies_n which_o be_v draw_v by_o the_o line_n hl_o and_o lc_n be_v a_o parallel_n to_o the_o plain_a superficies_n draw_v by_o the_o line_n give_v and_o di._n and_o so_o likewise_o of_o the_o rest_n second_o corollary_n a_o right_a line_n join_v together_o the_o centre_n of_o the_o opposite_a base_n of_o the_o octohedron_n be_v sesquialter_fw-la to_o the_o perpendicular_a line_n draw_v from_o the_o angle_n of_o the_o inscribe_v pyramid_n to_o the_o base_a thereof_o for_o forasmuch_o as_o the_o pyramid_n and_o the_o cube_fw-la which_o contain_v it_o do_v in_o the_o self_n same_o point_n end_v their_o angle_n by_o the_o 1._o of_o this_o book_n therefore_o they_o shall_v both_o be_v enclose_v in_o one_o and_o the_o self_n same_o octohedron_n by_o the_o 4._o of_o this_o book_n but_o the_o diameter_n of_o the_o cube_fw-la join_v together_o the_o centre_n of_o the_o opposite_a base_n of_o the_o octohedron_n and_o therefore_o be_v the_o diameter_n of_o the_o sphere_n which_o contain_v the_o cube_fw-la and_o the_o pyramid_n inscribe_v in_o the_o cube_fw-la by_o the_o 13._o and_o 14._o of_o the_o thirteen_o which_o diameter_n be_v sesquialter_fw-la to_o the_o perpendicular_a which_o be_v draw_v from_o the_o angle_n of_o the_o pyramid_n to_o the_o base_a thereof_o for_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o base_a of_o the_o pyramid_n be_v the_o six_o part_n of_o the_o diameter_n by_o the_o 3._o corollary_n of_o the_o 13._o of_o the_o thirteen_o wherefore_o of_o what_o part_n the_o diameter_n contain_v six_o of_o the_o same_o part_n the_o perpendicular_a contain_v four_o ¶_o the_o 7._o proposition_n the_o 7._o problem_n in_o a_o dodecahedron_n give_v to_o inscribe_v a_o icosahedron_n svppose_v that_o the_o dodecahedron_n give_v be_v abcde_o and_o let_v the_o centre_n of_o the_o circle_n which_o contain_v six_o base_n of_o the_o same_o dodecahedron_n be_v the_o polnes_n l_o m_o n_o p_o q_o o._n construction_n and_o draw_v these_o right_a line_n ol_n om_o on_o open_v oq_fw-la and_o moreover_o these_o right_a line_n lm_o mn_a np_n pq_n ql_n and_o now_o forasmuch_o as_o equal_v and_o equilater_n pentagon_n be_v contain_v in_o equal_a circle_n therefore_o perpendicular_a line_n draw_v from_o their_o centre_n to_o the_o side_n shall_v be_v equal_a by_o the_o 14._o of_o the_o three_o and_o shall_v divide_v the_o side_n of_o the_o dodecahedron_n into_o two_o equal_a part_n by_o the_o 3._o of_o the_o same_o wherefore_o the_o foresay_a perpendicular_a line_n shall_v coâoutre_v in_o the_o point_n of_o the_o section_n demonstration_n wherein_o the_o side_n be_v divide_v into_o two_o equal_a part_n as_o lf_a and_o mf_o do_v and_o they_o also_o contain_v equal_a angle_n namely_o the_o inclination_n of_o the_o base_n of_o the_o dodecahedron_n by_o the_o 2._o corollary_n of_o the_o 18._o of_o the_o thirteen_o wherefore_o the_o right_a line_n lm_o mn_a np_n pq_n ql_n and_o the_o rest_n of_o the_o right_a line_n which_o join_v together_o two_o centre_n of_o the_o base_n and_o which_o subtende_v the_o equal_a angle_n contain_v under_o the_o say_a equal_a perpendicular_a line_n be_v equal_a the_o one_o to_o the_o other_o by_o the_o 4._o of_o the_o first_o wherefore_o the_o triangle_n olm_n omnes_n onp_n opq_n oql_n and_o the_o rest_n of_o the_o triangle_n which_o be_v set_v at_o the_o centre_n of_o the_o pentagon_n be_v equilater_n and_o equal_a now_o forasmuch_o as_o the_o 12._o pentagon_n of_o a_o dodecahedron_n contain_v 60._o plain_a superficial_a angle_n of_o which_o 60._o every_n âhre_o make_v one_o solid_a angle_n of_o the_o dodecahedron_n it_o follow_v that_o a_o dodecahedron_n have_v 20._o solid_a angle_n but_o each_o of_o those_o solid_a angle_n be_v subtend_v of_o each_o of_o the_o triangle_n of_o the_o icosahedron_n namely_o of_o each_o of_o those_o triangle_n which_o join_v together_o the_o centre_n of_o the_o pentagon_n which_o make_v the_o solid_a angle_n as_o we_o have_v before_o prove_v wherefore_o the_o 20_o equal_a and_o equilater_n triangle_n which_o subtende_v the_o 20._o solid_a angle_n of_o the_o dodecahedron_n and_o have_v their_o side_n which_o be_v draw_v from_o the_o centre_n of_o the_o pentagon_n common_a do_v make_v a_o icosahedron_n by_o the_o 25._o definition_n of_o the_o eleven_o and_o it_o be_v inscribe_v in_o the_o dodecahedron_n give_v by_o the_o first_o definition_n of_o this_o book_n for_o that_o the_o angle_n thereof_o do_v all_o at_o one_o time_n touch_v the_o base_n of_o the_o dodecahedron_n wherefore_o in_o a_o dodecahedron_n gevenâ_n iâ_z inscribe_v a_o icosahedron_n which_o be_v require_v to_o be_v do_v ¶_o the_o 8._o proposition_n the_o 8._o problem_n in_o a_o dodecahedron_n give_v to_o include_v a_o cube_fw-la describe_v by_o the_o 17._o of_o the_o thirteen_o a_o dodecahedron_n construction_n and_o by_o the_o same_o take_v the_o 12._o side_n of_o the_o cube_fw-la each_o of_o which_o subtend_v one_o angle_n of_o each_o of_o the_o 12._o base_n of_o the_o dodecahedron_n for_o the_o side_n of_o the_o cube_fw-la subtend_v the_o angle_n of_o the_o pentagon_n of_o the_o dodecahedron_n by_o the_o 2._o corollary_n of_o the_o 17._o of_o the_o thirteen_o if_o therefore_o in_o the_o dodecahedron_n describe_v by_o the_o self_n same_o 17._o proposition_n we_o draw_v the_o 12._o right_a line_n subtend_v under_o the_o foresay_a 12._o angle_n and_o end_v in_o 8._o angle_n of_o the_o dodecahedron_n and_o concur_v together_o in_o such_o sort_n that_o they_o be_v in_o like_a sort_n situate_a as_o it_o be_v plain_o prove_v in_o that_o proposition_n then_o shall_v it_o be_v manifest_a that_o the_o right_a line_n draw_v in_o this_o dodecahedron_n from_o the_o foresay_a 8._o angle_n thereof_o do_v make_v the_o foresay_a cube_fw-la which_o therefore_o be_v include_v in_o the_o dodecahedron_n for_o that_o the_o side_n of_o the_o cube_fw-la be_v draw_v in_o the_o side_n of_o the_o dodecahedron_n and_o the_o angle_n of_o the_o same_o cube_fw-la be_v set_v in_o the_o angle_n of_o the_o say_a dodecahedron_n as_o for_o example_n take_v 4._o pentagon_n of_o a_o dodecahedron_n demonstration_n namely_o agibo_n bhcno_n ckedn_n and_o dfaon_n and_o draw_v these_o right_a line_n ab_fw-la bc_o cd_o da._n which_o four_o right_a line_n make_v a_o square_n for_o that_o each_o of_o those_o right_a line_n do_v subtend_v equal_a angle_n of_o equal_a pentagon_n &_o the_o angle_n which_o those_o 4._o right_a line_n contain_v be_v right_a angle_n as_o we_o prove_v in_o the_o construction_n of_o the_o dodecahedron_n in_o the_o 17._o proposition_n before_o allege_a wherefore_o the_o six_o base_n be_v square_n do_v make_v a_o cube_fw-la by_o the_o 21._o definition_n of_o the_o eleven_o and_o for_o that_o the_o 8._o angle_n of_o the_o say_v cube_fw-la be_v set_v in_o 8._o angle_n of_o the_o dodecaheeron_n therefore_o be_v the_o say_v cube_fw-la inscribe_v in_o the_o dodecahedron_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o dodecahedron_n be_v inscribe_v a_o cube_fw-la which_o be_v require_v to_o be_v do_v ¶_o the_o 9_o proposition_n the_o 9_o problem_n in_o a_o dodecahedron_n give_v to_o include_v a_o octohedron_n svppose_v that_o the_o dodecahedron_n give_v be_v abgd_v construction_n now_o by_o the_o 3._o corollary_n of_o the_o 17._o of_o the_o thirteen_o take_v the_o 6._o side_n which_o be_v opposite_a the_o one_o to_o the_o other_o those_o 6._o side_n i_o say_v who_o section_n wherein_o they_o be_v divide_v into_o two_o equal_a part_n be_v couple_v by_o three_o right_a line_n which_o in_o the_o centre_n of_o the_o sphere_n wherein_o the_o dodecahedron_n be_v contain_v do_v cut_v the_o one_o the_o other_o perpendicular_o and_o let_v the_o point_n wherein_o the_o foresay_a side_n be_v cut_v into_o two_o equal_a part_n be_v a_o b_o g_o d_o c_o i._n and_o let_v the_o foresay_a three_o right_a line_n join_v together_o the_o say_a section_n be_v ab_fw-la gd_o and_o ci._n and_o let_v the_o centre_n of_o the_o sphere_n be_v e._n demonstration_n now_o forasmuch_o as_o by_o the_o foresay_a corollary_n those_o three_o right_a line_n be_v equal_a it_o follow_v by_o the_o 4._o of_o the_o first_o that_o the_o right_a line_n subtend_v the_o right_a angle_n which_o they_o make_v at_o the_o centre_n of_o the_o sphere_n which_o right_a angle_n be_v contain_v under_o the_o half_n of_o the_o say_v three_o right_a line_n be_v equal_a the_o one_o to_o the_o other_o
subtend_v angle_n of_o triangle_n like_v unto_o the_o triangle_n who_o angle_n the_o line_n ac_fw-la bf_o and_o pl_z subtend_n be_v cut_v into_o two_o equal_a part_n in_o the_o point_n z_o an_o i_o by_o the_o 4._o of_o the_o six_o so_o also_o be_v the_o other_o line_n nv_n xm_o d_n qt_fw-la which_o be_v equal_a unto_o the_o line_n hk_o &_o ge_z cut_v in_o like_a sort_n and_o they_o shall_v cut_v the_o line_n ac_fw-la bf_o and_o pl_z like_z wherefore_o the_o line_n ko_o which_o be_v equal_a to_o rz_n shall_v make_v the_o great_a segment_n the_o line_n ro_n which_o be_v equal_a to_o the_o line_n zk_n for_o the_o great_a segment_n of_o the_o rz_n be_v the_o line_n zk_n and_o therefore_o the_o line_n oi_o shall_v be_v the_o less_o segment_n when_o as_o the_o whole_a line_n ri_n be_v equal_a to_o the_o whole_a line_n rz_n wherefore_o the_o square_n of_o the_o whole_a line_n ko_o and_o of_o the_o less_o segment_n oi_o be_v triple_a to_o the_o square_n of_o the_o great_a segment_n ro_n by_o the_o 4._o of_o the_o thirteen_o wherefore_o the_o line_n ki_v which_o contain_v in_o power_n the_o two_o line_n ko_o and_o oi_o be_v in_o power_n triple_a to_o the_o line_n ro_n by_o the_o 47._o of_o the_o first_o for_o the_o angle_n koi_n be_v a_o right_a angle_n and_o forasmuch_o as_o the_o line_n fe_o and_o fg_o which_o be_v the_o less_o segment_n of_o the_o side_n of_o the_o octohedron_n be_v equal_a and_o the_o line_n fk_o be_v common_a to_o they_o both_o and_o the_o angle_n kfg_n and_o kfe_n of_o the_o triangle_n of_o the_o octohedron_n be_v equal_a the_o base_n kg_v and_o ke_o shall_v by_o the_o 4_o of_o the_o first_o be_v equal_a and_o therefore_o the_o angle_n kie_fw-mi and_o kig_n which_o they_o subtend_v be_v equal_a by_o the_o 8._o of_o the_o first_o wherefore_o they_o be_v right_a angle_n by_o the_o 13._o of_o the_o first_o wherefore_o the_o right_a line_n ke_o which_o contain_v in_o power_n the_o two_o line_n ki_v and_o âe_n by_o the_o 47._o of_o the_o first_o be_v in_o power_n quadruple_a to_o the_o line_n ro_n or_o je_n for_o the_o line_n ri_n be_v prove_v to_o be_v in_o power_n triple_a to_o the_o same_o line_n ro_n but_o the_o line_n ge_z be_v double_a to_o the_o line_n je_n wherefore_o the_o line_n ge_z be_v also_o in_o power_n ãâ¦ã_o pf_n and_o by_o the_o same_o reason_n may_v be_v prove_v that_o the_o âest_n of_o the_o eleven_o solid_a angle_n of_o the_o ãâã_d be_v ãâ¦ã_o the_o section_n of_o every_o one_o of_o the_o side_n of_o the_o octohedron_n namely_o in_o the_o point_n e_o n_o five_o h_o â_o m_o â_o d_o s_o q_o t._n wherefore_o there_o be_v 12._o angle_n of_o the_o icosahedron_n moreover_o forasmuch_o as_o every_o one_o of_o the_o base_n of_o the_o octohedron_n do_v each_o contain_v triangle_n of_o the_o icosahedron_n ãâ¦ã_o pyramid_n abcâfp_n which_o be_v the_o half_a of_o the_o octohedron_n the_o triangle_n fcp_n receive_v in_o thâ_z section_n of_o his_o side_n the_o â_o triangle_n gm_n and_o the_o triangle_n cpb_n contain_v the_o triangle_n nxs_n and_o thâ_z triangle_n âap_n contain_v the_o triangle_n hnd_n and_o moreover_o the_o triangle_n apf_n contain_v the_o triangle_n edge_n and_o the_o same_o may_v be_v prove_v in_o the_o opposite_a pyramid_n abcfl_fw-mi wherefore_o there_o shall_v be_v eight_o triangleâ_n and_o forasmuch_o as_o beside_o these_o triangle_n to_o every_o one_o of_o the_o solid_a angle_n of_o the_o octohedron_n ãâã_d subtend_v two_o triangle_n as_o the_o triangle_n keg_n and_o meg_n to_o the_o angle_n f_o and_o the_o triangle_n hnv_n and_o xnv_n to_o the_o angle_n b_o also_o the_o triangle_n nd_n and_o âds_n to_o the_o angle_n p_o likewise_o the_o triangleâ_n dhk_n and_o qhk_v to_o the_o angle_n a_o moreover_o the_o triangle_n eqt_a and_o vqt_a to_o the_o angle_n l_o and_o final_o the_o triangle_n sxm_n and_o txm_n to_o the_o angle_n c_o these_o 12._o triangle_n be_v add_v to_o thââ_n for_o ãâã_d triangle_n shall_v produce_v _o triangle_n equal_a and_o equilater_v couple_v together_o which_o shall_v male_a a_o icosahedron_n by_o the_o 25._o definition_n of_o the_o eleven_o and_o it_o shall_v be_v inscribe_v in_o the_o octohedron_n give_v abcââl_o by_o the_o first_o definition_n of_o this_o book_n for_o the_o 1â_o angle_n thereof_o be_v set_v in_o 1â_o like_a section_n of_o the_o side_n of_o the_o octohedron_n wherefore_o in_o a_o octohedron_n give_v be_v inscribe_v a_o icosahedron_n ¶_o first_n corollary_n the_o side_n of_o a_o equilater_n triangle_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n a_o right_a line_n subtend_v within_o the_o triangle_n the_o angle_n which_o be_v contain_v under_o the_o great_a segment_n and_o the_o less_o be_v in_o power_n duple_n to_o the_o less_o segment_n of_o the_o same_o side_n for_o the_o line_n ke_o which_o subtend_v the_o angle_n kfe_n of_o the_o triangle_n afl_fw-mi which_o angle_n kfe_n be_v contain_v under_o the_o two_o segment_n kf_a &_o fe_o be_v prove_v equal_a ãâã_d the_o line_n hk_o which_o contain_v in_o power_n the_o two_o less_o segment_n ha_o and_o ak_o by_o the_o 47._o of_o the_o âârst_a foâ_n ãâã_d angle_n hak_n be_v ãâ¦ã_o second_o corollary_n the_o base_n of_o the_o icosahedron_n be_v concentrical_a that_o be_v have_v one_o and_o the_o self_n same_o centre_n with_o the_o base_n of_o the_o octohedron_n which_o contain_v it_o for_o suppose_v that_o ãâ¦ã_o octohedron_n ãâã_d ecd_v the_o base_a of_o a_o icosahedron_n and_o let_v the_o centre_n of_o the_o base_a abg_o be_v the_o point_n f._n and_o draw_v these_o right_a line_n favorina_n fb_o fc_o and_o fe_o now_o than_o the_o ãâ¦ã_o to_o the_o two_o line_n fb_o and_o bc_o for_o they_o be_v line_n draw_v from_o the_o centre_n and_o be_v also_o less_o segment_n and_o they_o contain_v the_o ãâ¦ã_o ¶_o the_o 17._o problem_n the_o 17._o proposition_n in_o a_o octohedron_n give_v to_o inscribe_v a_o dodecahedron_n construction_n svppose_v that_o the_o octohedron_n give_v be_v abgdec_n who_o 12._o âides_n let_v be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n as_o in_o the_o former_a proposition_n it_o be_v manifest_a that_o of_o the_o right_a line_n which_o couple_v thâse_a section_n be_v make_v 20._o triangle_n of_o which_o 8._o be_v concentrical_a with_o the_o base_n of_o the_o octohedron_n by_o the_o second_o corollary_n of_o the_o former_a proposition_n if_o therefore_o in_o every_o one_o of_o the_o centre_n of_o the_o 20._o triangle_n be_v inscribe_v by_o the_o 1._o of_o this_o book_n every_o one_o of_o the_o ââ_o ââgles_n of_o the_o dodecahedron_n demonstration_n we_o shall_v find_v that_o â_o angle_n of_o the_o dodecahedron_n be_v set_v in_o the_o 8._o centre_n of_o the_o base_n of_o the_o octohedron_n namely_o these_o angle_n i_o u._fw-mi ct_n oh_o m_o a_o p_o and_o x_o and_o of_o the_o other_o 12._o solid_a angle_n there_o be_v two_o in_o the_o centre_n of_o the_o two_o triangle_n which_o have_v one_o side_n common_a under_o every_o one_o of_o the_o solid_a angle_n of_o the_o octohedron_n namely_o under_o the_o solid_a angle_n a_o the_o two_o solid_a angle_n king_n z_o under_o the_o solid_a angle_n b_o the_o two_o solid_a angle_n h_o t_o under_o the_o solid_a angle_n g_o the_o two_o solid_a angle_n in_o v_z under_o the_o solid_a angle_n d_o the_o two_o solid_a angle_n f_o l_o under_o the_o solid_a angle_n e_o the_o two_o solid_a angle_n s_o n_o under_o the_o solid_a angle_n c_o the_o two_o solid_a angle_n queen_n r_o and_o forasmuch_o as_o in_o the_o octohedron_n be_v six_o solid_a angle_n under_o they_o shall_v be_v subtend_v 12._o solid_a angle_n of_o the_o dodâcahedron_n and_o so_o be_v mâde_v 20._o solid_a angle_n compose_v of_o 12._o equal_a and_o equilater_v superficial_a pentagon_n as_o it_o be_v ãâã_d by_o the_o 5._o of_o this_o book_n which_o therefore_o contain_v a_o dodecahedron_n by_o the_o 24._o definition_n of_o the_o eleven_o and_o it_o be_v inscribe_v in_o the_o octohedron_n by_o the_o 1._o definition_n of_o this_o book_n for_o that_o every_o one_o of_o the_o base_n of_o the_o octohedron_n do_v receive_v angle_n thereof_o wherefore_o in_o a_o octohedron_n give_v be_v inscribe_v a_o dodecahedron_n ¶_o the_o 18._o problem_n the_o 18._o proposition_n in_o a_o trilater_n and_o equilater_n pyramid_n to_o inscribe_v a_o cube_n construction_n svppose_v that_o there_o be_v a_o trilater_n equilater_n pyramid_n who_o base_a let_v be_v abc_n and_o âoppe_v the_o point_n d._n and_o let_v it_o be_v comprehend_v in_o a_o sphereâ_n by_o the_o 13._o of_o the_o ãâã_d and_o lââ_n the_o centre_n of_o that_o sphere_n be_v the_o point_n e._n and_o from_o the_o solid_a angle_n a_o b_o c_o d_o draw_v right_a line_n pass_v by_o the_o centre_n e_o unto_o the_o opposite_a base_n of_o
side_n gd_v the_o angle_n m_o n_o under_o the_o side_n ab_fw-la the_o angle_n t_o s_o under_o the_o side_n bg_o the_o angle_n p_o oh_o and_o under_o the_o side_n agnostus_n the_o angle_n r_o q_o so_o there_o rest_v 4._o angle_n who_o true_a place_n we_o will_v now_o appoint_v forasmuch_o as_o a_o cube_fw-la contain_v in_o one_o and_o the_o self_n same_o sphere_n with_o a_o dodecahedron_n be_v inscribe_v in_o the_o same_o dodecahedron_n as_o it_o be_v manifest_a by_o the_o 17._o of_o the_o thirteen_o and_o 8._o of_o this_o book_n it_o follow_v that_o a_o cube_fw-la and_o a_o dodecahedron_n circumscribe_v about_o it_o be_v contain_v in_o one_o and_o the_o self_n same_o body_n for_o that_o their_o angle_n concur_v in_o one_o and_o the_o self_n same_o point_n and_o it_o be_v prove_v in_o the_o 18._o of_o this_o book_n that_o 4._o angle_n of_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n be_v set_v in_o the_o middle_a section_n of_o the_o perpendicularâ_n which_o be_v draw_v from_o the_o solid_a angle_n of_o the_o pyramid_n to_o the_o opposite_a base_n wherefore_o the_o other_o 4._o angle_n of_o the_o dodecahedron_n be_v also_o as_o the_o angle_n of_o the_o cube_fw-la set_v in_o those_o middle_a section_n of_o the_o perpendicular_n namely_o the_o angle_n v_o be_v set_v in_o the_o midst_n of_o the_o perpendicular_a ahâ_n the_o angle_n y_fw-fr in_o the_o midst_n of_o the_o perpendicular_a bf_o the_o angle_n x_o in_o the_o midst_n of_o the_o perpendicular_a ge_z and_o last_o the_o angle_n d_o in_o the_o midst_n of_o the_o perpendicular_a d_o which_o be_v draw_v from_o the_o top_n of_o the_o pyramid_n to_o the_o opposite_a base_a wherefore_o those_o 4._o angle_n of_o the_o dodecahedron_n may_v be_v say_v to_o be_v direct_o under_o the_o solid_a angle_n of_o the_o pyramid_n or_o they_o may_v be_v say_v to_o be_v set_v at_o the_o perpendicular_n wherefore_o the_o dodecahedron_n after_o this_o manner_n set_v be_v inscribe_v in_o the_o pyramid_n give_v by_o the_o first_o definition_n of_o this_o book_n for_o that_o upon_o every_o one_o of_o the_o base_n of_o the_o pyramid_n be_v set_v a_o angle_n of_o the_o dodecahedron_n inscribe_v wherefore_o in_o a_o trilater_n equilater_n pyramid_n be_v inscribe_v a_o dodecahedron_n the_o 21._o problem_n the_o 21._o proposition_n in_o every_o one_o of_o the_o regular_a solid_n to_o inscribe_v a_o sphere_n in_o the_o 13._o of_o thâ_z thirteen_o and_o thâ_z other_o 4._o proposition_n follow_v he_o iâ_z be_v declare_v that_o âhe_n ââ_o regular_a solidesââre_o so_o contain_v in_o a_o sphere_n that_o âight_a linââ_n draw_v from_o the_o cenâââ_n oâ_n the_o ãâ¦ã_o of_o ãâã_d solid_a inscribe_v be_v equal_a which_o right_a line_n therefore_o make_v pyramid_n who_o âoppes_n be_v the_o centre_n of_o the_o sphere_n or_o of_o the_o solid_a and_o the_o basââââe_a cuâââ_n one_o of_o the_o base_n of_o those_o solid_n and_o ãâ¦ã_o solid_a squall_n and_o like_v the_o one_o to_o the_o other_o and_o describe_v in_o equal_a circle_n those_o circle_n shall_v cut_v the_o sphere_n for_o the_o angle_n which_o touch_v the_o circumference_n of_o the_o circle_n touch_v also_o the_o superficies_n of_o the_o sphere_n wherefore_o perpendiculars_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o base_n or_o to_o the_o plain_a superficiece_n of_o the_o equal_a circle_n be_v equal_a by_o the_o corollary_n of_o the_o assumpt_n of_o the_o 1â_o of_o the_o twelve_o wherefore_o make_v the_o centre_n the_o ãâã_d of_o the_o sphere_n which_o ãâã_d the_o solid_a and_o thâ_z space_n some_o one_o of_o the_o equal_a perpendicularâ_n dâscribâ_n a_o sphere_n and_o it_o shall_v touch_v every_o one_o of_o the_o base_n of_o ãâã_d solid_a ãâ¦ã_o perficies_fw-la of_o the_o sphere_n pass_v beyond_o those_o base_n when_o as_o those_o pââpeâdiculars_n ãâ¦ã_o be_v draw_v from_o the_o centre_n to_o the_o base_n by_o the_o 3._o corollary_n of_o the_o saâââââumpt_n wherâfore_o âe_v have_v iâ_z every_o one_o of_o the_o regular_a body_n inscribe_v a_o sphere_n which_o regular_a boâââ_n be_v in_o number_n one_o iâ_z ãâã_d by_o the_o corollary_n of_o the_o 1â_o of_o the_o ãâã_d a_o corollary_n the_o regular_a figure_n inscribe_v in_o sphere_n and_o also_o the_o sphere_n circumscribe_v about_o they_o or_o contain_v they_o have_v one_o and_o the_o self_n same_o centre_n namely_o their_o pyramid_n the_o ângles_n of_o who_o base_n touch_v the_o superââââââ_n of_o thâââhere_o do_v from_o those_o angle_n cause_v equal_a right_a line_n to_o be_v drawââ_n to_o one_o and_o âhe_n self_n ãâã_d poynâ_n make_v the_o topâââ_n of_o the_o pyramid_n in_o the_o same_o point_n and_o therefore_o they_o ãâ¦ã_o thâ_z cââtres_n of_o the_o sphere_n in_o the_o self_n same_o top_n when_o ãâã_d the_o right_a line_n draw_v from_o those_o angle_n to_o the_o croââed_a superficies_n wherein_o be_v ãâã_d the_o angle_n of_o the_o base_n of_o the_o pyramid_n be_v equal_o a_o advertisement_n of_o flussasâ_n â_z of_o these_o solid_n only_o the_o octohedron_n receive_v the_o other_o solid_n inscribe_v one_o with_o ãâ¦ã_o other_o for_o the_o octohedron_n contain_v the_o icosahedron_n inscribe_v in_o it_o and_o the_o same_o icosahedron_n contain_v the_o dodecahedron_n inscribe_v in_o the_o same_o icosahedron_n and_o the_o same_o dodecahedron_n contain_v the_o cube_fw-la inscribe_v in_o the_o same_o octohedron_n and_o ãâ¦ã_o âârââmscribeth_v the_o pyramid_n inscribe_v in_o the_o say_v octohedron_n but_o this_o happen_v not_o in_o the_o other_o solid_n the_o end_n of_o the_o fivetenth_fw-mi book_n of_o euclides_n elemenâââ_n after_o caâpaââ_n and_o ãâã_d ¶_o the_o sixteen_o book_n of_o the_o element_n of_o geometry_n add_v by_o flussas_n in_o the_o former_a fivetenth_fw-mi book_n have_v be_v teach_v how_o to_o inscribe_v the_o five_o regular_a solid_n one_o with_o in_o a_o other_o now_o seem_v to_o rest_n to_o compare_v those_o solid_a so_o inscribe_v one_o to_o a_o other_o and_o to_o set_v forth_o their_o passion_n and_o propriety_n which_o thing_n flussas_n consider_v in_o this_o sixteen_o book_n add_v by_o he_o book_n have_v excellent_o well_o and_o most_o cunning_o perform_v for_o which_o undoubted_o he_o have_v of_o all_o they_o which_o have_v a_o love_n to_o the_o mathematicals_n deserve_v much_o praise_n and_o commendation_n both_o for_o the_o great_a traâailes_n and_o paynâs_n which_o it_o be_v most_o likely_a he_o have_v taâân_v in_o invent_v such_o strange_a and_o wonderful_a proposition_n with_o their_o demonstration_n in_o this_o book_n contain_v as_o also_o for_o participate_v and_o communicate_v abroad_o the_o same_o to_o other_o which_o book_n also_o that_o the_o reader_n shall_v want_v nothing_o conduce_v to_o the_o perfection_n of_o euclides_n element_n i_o have_v with_o some_o travail_n translate_v &_o for_o the_o worthiness_n âhereof_o have_v add_v it_o aâ_z a_o sixteen_o book_n to_o the_o 15._o book_n of_o euclid_n vouchsafe_v therefore_o gentle_a reader_n diligent_o to_o read_v and_o poise_v it_o for_o in_o it_o shall_v you_o find_v noâ_n only_a matter_n strange_a and_o delectable_a but_o also_o occasion_n of_o invention_n of_o great_a thing_n pertain_v to_o the_o nature_n of_o the_o five_o regular_a solidâsâ_n ¶_o the_o 1._o proposition_n a_o dodecahedron_n and_o a_o cube_fw-la inscribe_v in_o it_o and_o a_o pyramid_n inscribe_v in_o the_o same_o cube_fw-la be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n for_o the_o angle_n of_o the_o pyramiâ_n be_v seâ_z in_o the_o angel_n of_o the_o cube_fw-la wherein_o it_o be_v inscribe_v by_o the_o first_o of_o the_o fivetenthâ_n and_o all_o the_o angle_n of_o the_o cube_fw-la be_v set_v in_o the_o angle_n of_o the_o dodecahedâââ_n circumscribe_v ãâ¦ã_o ãâã_d the_o 8._o of_o the_o fifteen_o and_o all_o the_o angle_n of_o the_o dodecahedron_n be_v set_v in_o the_o superficies_n of_o the_o sphere_n by_o the_o 17._o of_o the_o thirteen_o wherefore_o those_o three_o solid_n inscribe_v one_o within_o a_o other_o be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o first_o definition_n of_o the_o fifteen_o a_o dodecahedron_n therefore_o and_o a_o cube_fw-la inscribe_v in_o it_o and_o a_o pyramid_n inscribe_v in_o the_o same_o cube_fw-la be_v contain_v ãâ¦ã_o ââlfe_n same_o sphere_n ãâ¦ã_o these_o three_o solid_n livre_n ãâ¦ã_o elf_n same_o icosahedron_n or_o octohedron_n or_o pyramid_n ãâ¦ã_o i_o icosahedron_n by_o the_o 5.11_o &_o 12._o of_o the_o fifteen_o and_o they_o be_v ãâ¦ã_o ctohedron_n by_o the_o 4._o 6._o and_o 16._o of_o the_o same_o last_o they_o be_v inscribe_v in_o ãâ¦ã_o the_o first_o 18._o and_o 19_o of_o the_o same_o for_o the_o angle_n of_o all_o these_o solid_a ãâ¦ã_o the_o circumscribe_v icosahedron_n or_o octohedron_n or_o pyramid_n ¶_o the_o ãâ¦ã_o the_o proportion_n of_o a_o dodecahedron_n circumscribe_v about_o a_o cube_fw-la to_o a_o dodecahedron_n inscribe_v in_o the_o same_o cube_fw-la be_v
triple_a to_o a_o extreme_a &_o mean_a propartion_n forasmuch_o as_o in_o the_o ââ_o corollary_n of_o the_o 13._o of_o the_o fiuââenth_n it_o be_v prove_v that_o the_o side_n of_o a_o dodecahedron_n inscribe_v in_o a_o cube_fw-la be_v the_o less_o segment_n of_o the_o side_n of_o that_o cube_fw-la divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o the_o side_n of_o the_o dodecahedron_n circumscribe_v about_o the_o same_o cube_fw-la be_v the_o great_a segment_n of_o the_o side_n of_o the_o same_o cube_fw-la which_o thing_n also_o be_v teach_v in_o the_o 13._o of_o the_o fifteen_o the_o side_n of_o the_o dodecahedron_n circumscribe_v shall_v be_v to_o the_o side_n of_o the_o dodecahedron_n inscribe_v as_o the_o great_a segment_n of_o a_o right_a line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n be_v to_o the_o less_o segment_n of_o the_o same_o which_o proportion_n be_v call_v a_o extreme_a and_o mean_a proportion_n by_o the_o definition_n and_o by_o the_o 30._o of_o six_o but_o the_o proportion_n of_o like_a solid_a prolihedron_n be_v triple_a to_o the_o proportion_n of_o the_o sideâ_n of_o like_a proportion_n by_o the_o corollary_n of_o the_o 17._o of_o the_o twelve_o wherefore_o the_o proportion_n of_o the_o dodecahedron_n circumscribe_v about_o the_o cube_fw-la be_v to_o the_o dodecahedron_n inscribe_v in_o the_o same_o cube_fw-la in_o triple_a proportion_n of_o the_o side_n join_v together_o by_o a_o extreme_a and_o mean_a proportion_n the_o proportion_n therefore_o of_o â_o dodecahedron_n circumscribe_v about_o a_o cube_fw-la to_o a_o dodecahedron_n inscribe_v in_o the_o same_o cube_fw-la be_v triple_a to_o a_o extreme_a and_o mean_a proportion_n the_o 3._o proposition_n in_o every_o equiangle_n and_o equilater_n pentagon_n a_o perpendicular_a draw_v from_o one_o of_o the_o angle_n to_o the_o base_a be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n by_o a_o right_a line_n subtend_v the_o same_o angle_n svppose_v that_o abcdf_n be_v aâ_z equiangle_n and_o equilater_n pentagon_n construction_n and_o from_o one_o of_o the_o angle_n namely_o from_o a_o let_v there_o be_v draw_v to_o the_o base_a cd_o a_o perpendicular_a agnostus_n and_o let_v the_o line_n bf_o subtend_v the_o angle_n bap_n demonstration_n which_o line_n bf_o let_v the_o line_n ad_fw-la cut_v in_o the_o point_n i_o then_o i_o say_v that_o the_o line_n bf_o cut_v the_o line_n agnostus_n by_o a_o extreme_a and_o mean_a proportion_n for_o forasmuch_o as_o the_o angle_n give_v and_o gab_n be_v equal_a by_o the_o 27._o of_o the_o three_o and_o the_o angle_n aâf_n and_o afb_o be_v equal_a by_o the_o 5._o of_o the_o firstâ_o therefore_o the_o ââgles_n remain_v at_o the_o point_n e_o of_o the_o triangle_n aeb_fw-mi and_o aef_n be_v equal_a for_o that_o they_o be_v the_o residue_v of_o two_o right_a angle_n by_o the_o corollary_n of_o the_o 32._o of_o the_o first_o but_o the_o angle_n egc_n be_v by_o construction_n a_o right_a angleâ_n wherefore_o the_o line_n bf_o &_o cd_o be_v parallel_n by_o the_o 28._o of_o the_o first_o wherefore_o as_o the_o line_n diego_n be_v to_o the_o line_n ja_z so_o iâ_z the_o line_n ge_z to_o the_o line_n ea_fw-la by_o the_o 2._o of_o the_o six_o but_o the_o line_n dam_n be_v in_o the_o poinâ_n i_o divide_v by_o a_o extreme_a and_o mean_a proportion_n by_o the_o 8._o of_o the_o thirteen_o wherefore_o the_o line_n ga_o be_v in_o the_o point_n e_o divide_v by_o a_o extreme_a and_o mean_a proportion_n by_o the_o â_o of_o the_o fourteen_o wherefore_o in_o every_o equiangle_n and_o equilater_n pentagon_n a_o perpendicular_a draw_v from_o one_o of_o the_o angle_n to_o the_o base_a be_v divide_v by_o a_o exâreme_a and_o mean_a proportion_n by_o a_o right_a line_n subtend_v the_o same_o angleâ_n ¶_o a_o corollary_n the_o line_n which_o subtend_v the_o angle_n of_o a_o pentagon_n be_v a_o parallel_n to_o the_o side_n opposite_a unto_o the_o angle_n as_o it_o be_v manifest_a in_o the_o line_n âf_a and_o cd_o the_o 4._o proposition_n if_o from_o the_o angle_n of_o the_o base_a of_o a_o book_n pyramid_n be_v draw_v to_o the_o opposite_a side_n right_a line_n cut_v the_o say_a side_n by_o a_o extreme_a and_o mean_a proportion_n they_o shall_v contain_v the_o bise_n of_o the_o icosahedron_n inscribe_v in_o the_o pyramid_n which_o base_n shall_v be_v inscribe_v in_o a_o equilater_n triangle_n who_o angle_n cut_v the_o side_n of_o the_o base_a of_o the_o pyramid_n by_o a_o extreme_a and_o mean_a proportion_n construction_n svppose_v that_o abg_o be_v the_o base_a of_o a_o pyramid_n in_o which_o let_v be_v inscribe_v a_o equilater_n triangle_n fkh_n which_o be_v do_v by_o devide_v the_o side_n into_o two_o equal_a part_n and_o in_o âhis_n triangle_n let_v there_o be_v inscribe_v the_o base_a of_o the_o icosahedron_n inscribe_v in_o the_o pyramid_n which_o be_v describe_v by_o devide_v the_o side_n fk_o kh_o hf_o by_o a_o extreme_a &_o mean_a proportion_n in_o the_o point_n c_o d_o e_o by_o the_o 19_o of_o the_o fivetenth_n again_o let_v the_o side_n of_o the_o pyramid_n namely_o ab_fw-la bg_o and_o ga_o be_v divide_v by_o a_o extreme_a and_o meâne_a proportion_n in_o the_o point_n i_o m_o l_o by_o the_o 30._o of_o the_o six_o and_o draw_v these_o right_a line_n be_o bl_o give_v demonstration_n then_o i_o say_v that_o those_o line_n describe_v the_o triangle_n cde_o of_o the_o icosahedron_n for_o forasmuch_o as_o the_o line_n bg_o and_o fh_o be_v parallel_n by_o the_o 2._o of_o the_o six_o by_o the_o point_n d_o let_v the_o line_n odn_n be_v draw_v parallel_n to_o either_o of_o the_o line_n bg_o &_o fh_o wherefore_o the_o triangle_n hdn_n shall_v be_v like_v to_o the_o triangle_n hkg_v by_o the_o corollary_n of_o the_o 2._o of_o the_o six_o wherefore_o either_o of_o these_o line_n dn_o and_o nh_v shall_v be_v equal_a to_o the_o line_n dh_o the_o great_a segment_n of_o the_o line_n kh_o or_o fh_o and_o forasmuch_o as_o the_o line_n foyes_n be_v a_o parallel_n to_o the_o line_n hk_o and_o the_o line_n odd_a to_o the_o line_n fhâ_n the_o line_n odd_a shall_v be_v equal_a to_o the_o whole_a line_n fh_o in_o the_o parallelogram_n fodh_n by_o the_o 34._o of_o the_o âirst_n wherefore_o as_o the_o whole_a line_n fh_o be_v to_o the_o greaâer_a segment_n fe_o so_o shall_v the_o line_n equal_a to_o they_o be_v namely_o the_o line_n odd_a to_o the_o line_n dn_o by_o the_o 7._o of_o the_o five_o wherefore_o the_o line_n on_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n d_o by_o the_o 2._o of_o the_o fourteen_o but_o the_o triangle_n aod_n afe_n and_o abm_n be_v like_o the_o one_o to_o the_o other_o and_o so_o also_o be_v the_o triangle_n adn_n aeh_n and_o amg_n by_o the_o corollary_n of_o the_o second_o of_o the_o sixths_n wherefore_o as_z fe_o be_v to_o eh_o so_o be_v odd_a to_o dn_o and_o bm_n to_o mg_o wherefore_o the_o line_n be_o cut_v the_o line_n fh_o and_o on_o like_v unto_o the_o line_n bg_o in_o the_o point_n e_o d_o m_o describe_v ed_z the_o side_n of_o the_o triangle_n of_o the_o icosahedron_n ecd_v which_o be_v describe_v in_o the_o section_n e_o c_o d_o by_o supposition_n and_o by_o the_o same_o reason_n the_o line_n bl_o and_o give_v shall_v describe_v the_o other_o side_n ec_o and_o cd_o of_o the_o same_o triangle_n by_o the_o point_n e_o let_v there_o be_v draw_v to_o give_v a_o parallel_a line_n peq_n now_o forasmuch_o as_o the_o line_n bm_n and_o fe_o be_v parallel_n the_o line_n be_o be_v in_o the_o point_n e_o cut_v like_a to_o the_o line_n ab_fw-la in_o the_o point_n fletcher_n by_o the_o 2._o of_o the_o six_o wherefore_o the_o line_n ae_n be_v equal_a to_o the_o line_n em_n and_o unto_o the_o line_n em_n also_o be_v equal_a either_o of_o the_o line_n gd_o and_o diego_n which_o âre_n cut_v lâke_v unto_o the_o foresaid_a line_n again_o forasmuch_o as_o in_o the_o triangle_n adi_n the_o line_n diego_n and_o ep_n be_v parallel_n as_o the_o line_n diego_n be_v to_o the_o line_n ep_n so_o be_v the_o line_n ad_fw-la to_o the_o line_n ae_n but_o as_o the_o line_n ad_fw-la be_v to_o the_o line_n ae_n so_o be_v the_o line_n dg_o to_o the_o line_n eq_n by_o the_o 2._o of_o the_o six_o wherefore_o as_o the_o line_n diego_n be_v to_o the_o line_n ep_n so_o be_v the_o line_n dg_o to_o the_o line_n eq_n and_o alternate_o as_o the_o line_n diego_n be_v to_o the_o line_n dg_o so_o be_v the_o line_n ep_n to_o the_o line_n eq_n but_o the_o line_n diego_n and_o ig_n be_v equal_a wherefore_o also_o the_o line_n ep_n and_o eq_n be_v equal_a and_o forasmuch_o as_o the_o line_n ah_o be_v equal_a to_o the_o line_n fh_o who_o great_a segment_n be_v the_o line_n hnâ_n therefore_o the_o whole_a
cube_n for_o forasmuch_o as_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o cube_fw-la subtend_v two_o side_n of_o the_o cube_fw-la which_o contain_v a_o right_a angle_n by_o the_o 1._o of_o the_o fifteen_o it_o be_v manifest_a by_o the_o 47._o of_o the_o first_o that_o the_o side_n of_o the_o pyramid_n subtend_v the_o say_a side_n be_v in_o power_n duple_n to_o the_o side_n of_o the_o cube_fw-la wherefore_o also_o the_o square_a of_o the_o side_n of_o the_o cube_fw-la be_v the_o half_a of_o the_o square_n of_o the_o side_n of_o the_o pyramid_n the_o side_n therefore_o of_o a_o cube_fw-la contain_v in_o power_n half_o the_o side_n of_o a_o equilater_n triangular_a pyramid_n inscribe_v in_o the_o say_v cube_fw-la ¶_o the_o 7._o proposition_n the_o side_n of_o a_o pyramid_n be_v duple_n to_o the_o side_n of_o a_o octohedron_n inscribe_v in_o it_o forasmuch_o as_o by_o the_o 2._o of_o the_o fivetenth_fw-mi it_o be_v prove_v that_o the_o side_n of_o the_o octohedron_n inscribe_v in_o a_o pyramid_n couple_v the_o middle_a section_n of_o the_o side_n of_o the_o pyramid_n wherefore_o the_o side_n of_o the_o pyramid_n and_o of_o the_o octohedron_n be_v parallel_n by_o the_o corollary_n of_o the_o 39_o of_o the_o first_o and_o therefore_o by_o the_o corollary_n of_o the_o 2._o of_o the_o six_o they_o subtend_v like_o triangle_n wherefore_o by_o the_o 4._o of_o the_o six_o the_o side_n of_o the_o pyramid_n be_v double_a to_o the_o side_n of_o the_o octohedron_n namely_o in_o the_o proportion_n of_o the_o side_n the_o side_n therefore_o of_o a_o pyramid_n be_v duple_n to_o the_o side_n of_o a_o octohedron_n inscribe_v in_o it_o ¶_o the_o 8._o proposition_n the_o side_n of_o a_o cube_n be_v in_o power_n duple_n to_o the_o side_n of_o a_o octohedron_n inscribe_v in_o it_o it_o be_v prove_v in_o the_o 3._o of_o the_o fifteen_o that_o the_o diameter_n of_o the_o octohedron_n inscribe_v in_o the_o cube_fw-la couple_v the_o centre_n of_o the_o opposite_a base_n of_o the_o cube_fw-la wherefore_o the_o say_a diameter_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la but_o the_o same_o be_v also_o the_o diameter_n of_o the_o square_n make_v of_o the_o side_n of_o the_o octohedron_n namely_o be_v the_o diameter_n of_o the_o sphere_n which_o contain_v it_o by_o the_o 14._o of_o the_o thirteen_o wherefore_o that_o diameter_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la be_v in_o power_n double_a to_o the_o side_n of_o that_o square_n or_o to_o the_o side_n of_o the_o octohedron_n inscribe_v in_o it_o by_o the_o 47._o of_o the_o first_o the_o side_n therefore_o of_o a_o cube_n be_v in_o power_n duple_n to_o the_o side_n of_o a_o octohedron_n inscribe_v in_o it_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 9_o proposition_n the_o side_n of_o a_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n which_o contain_v in_o power_n half_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o say_v dodecahedron_n svppose_v that_o of_o the_o dodecahedron_n abgd_v the_o side_n be_v ab_fw-la and_o let_v the_o base_a of_o the_o cube_fw-la inscribe_v in_o the_o dodecahedron_n be_v ecfh_n by_o the_o ââ_o of_o the_o fifteen_o and_o let_v the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o cube_fw-la be_fw-mi change_z by_o the_o 1._o of_o the_o fifteen_o construction_n wherefore_o the_o same_o pyramid_n be_v inscribe_v in_o the_o dodecahedron_n by_o the_o 10._o of_o the_o fifteen_o then_o i_o say_v that_o ab_fw-la the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n which_o contain_v in_o power_n half_o the_o line_n change_z which_o be_v the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o dodecahedron_n demonstration_n for_o forasmuch_o as_o ec_o the_o side_n of_o the_o cube_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n the_o line_n ab_fw-la the_o side_n of_o the_o dodecahedron_n by_o the_o âârst_a corollary_n of_o the_o 17._o of_o the_o thirteen_o for_o they_o be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o first_o of_o this_o book_n and_o the_o line_n ec_o the_o side_n of_o the_o cube_fw-la contain_v in_o power_n the_o half_a of_o the_o side_n change_z by_o the_o 6._o of_o this_o book_n wherefore_o ab_fw-la the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n ec_o which_o contain_v in_o power_n the_o half_a of_o the_o line_n change_z which_o be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o pyramid_n the_o side_n therefore_o of_o a_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n which_o contain_v in_o power_n half_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o say_v dodecahedron_n ¶_o the_o 10._o proposition_n the_o side_n of_o a_o icosahedron_n be_v the_o mean_a proportional_a between_o the_o side_n of_o the_o cube_n circumscribe_v about_o the_o icosahedron_n and_o the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o cube_n svppose_v that_o there_o be_v a_o cube_fw-la abfd_n in_o which_o let_v there_o be_v inscribe_v a_o icosahedron_n cligor_fw-la by_o the_o 14._o of_o the_o fifteen_o construction_n let_v also_o the_o dodecahedron_n inscribe_v in_o the_o same_o be_v edmnp_n by_o the_o 13._o of_o the_o same_o now_o forasmuch_o as_o cl_n the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n of_o ab_fw-la the_o side_n of_o the_o cube_fw-la circumscribe_v about_o it_o by_o the_o 3._o corollary_n of_o the_o 14._o of_o the_o fifteen_o demonstration_n and_o the_o side_n ed_z of_o the_o dodecahedron_n inscribe_v in_o thesame_o cube_fw-la be_v the_o less_o segment_n of_o the_o same_o side_n ab_fw-la of_o the_o cube_fw-la by_o the_o 2._o corollary_n of_o the_o 13._o of_o the_o fifteen_o it_o follow_v that_o ab_fw-la the_o side_n of_o the_o cube_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n cl_n the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o and_o the_o less_o segment_n ed_z the_o side_n of_o the_o dodecahedron_n likewise_o inscribe_v in_o it_o wherefore_o as_o the_o whole_a line_n ab_fw-la the_o side_n of_o the_o cube_fw-la be_v to_o the_o great_a segment_n cl_n the_o side_n of_o the_o icosahedron_n so_o be_v the_o great_a segment_n cl_n the_o side_n of_o the_o icosahedron_n to_o the_o less_o segment_n edâ_n the_o side_n of_o the_o dodecahedron_n by_o the_o three_o definition_n of_o the_o six_o wherefore_o the_o side_n of_o a_o icosahedron_n be_v the_o mean_a proportional_a between_o the_o side_n of_o the_o cube_fw-la circumscribe_v about_o the_o icosahedron_n and_o the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o cube_fw-la ¶_o the_o 11._o proposition_n the_o side_n of_o a_o pyramid_n be_v in_o power_n 1._o octodecuple_n to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o it_o for_o by_o that_o which_o be_v demonstrate_v in_o the_o 18._o of_o the_o fifteen_o the_o side_n of_o the_o pyramid_n be_v triple_a to_o the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la inscribe_v in_o it_o demonstration_n and_o therefore_o it_o be_v in_o power_n nonecuple_n to_o the_o same_o diameter_n by_o the_o 20._o of_o the_o six_o but_o the_o diamer_n be_v in_o power_n double_a to_o the_o side_n of_o the_o cube_fw-la by_o the_o 47._o of_o the_o first_o and_o the_o double_a of_o nonecuple_n make_v octodecuple_n wherefore_o the_o side_n of_o the_o pyramid_n be_v in_o power_n octodecuple_n to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o it_o ¶_o the_o 12._o proposition_n the_o side_n of_o a_o pyramid_n be_v in_o power_n octodecuple_n to_o that_o right_a line_n who_o great_a segment_n be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o pyramid_n forasmuch_o as_o the_o dodecahedron_n and_o the_o cube_fw-la inscribe_v in_o it_o be_v set_v in_o one_o and_o the_o sâlfâ_n same_o pyramid_n by_o the_o corollary_n of_o the_o first_o of_o this_o book_n and_o the_o side_n of_o the_o pyramid_n circumscribe_v about_o the_o cube_fw-la be_v in_o power_n octodecuple_n to_o the_o side_n of_o the_o cube_fw-la inscribe_v by_o the_o former_a proposition_n but_o the_o great_a segment_n of_o the_o self_n same_o side_n of_o the_o cube_fw-la be_v the_o side_n of_o the_o dodecahedron_n which_o contain_v the_o cube_fw-la by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o wherefore_o the_o side_n of_o the_o pyramid_n be_v in_o power_n octodecuple_n to_o that_o right_a line_n namely_o to_o the_o side_n of_o the_o cube_fw-la who_o great_a segment_n be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o pyramid_n ¶_o the_o 13._o proposition_n the_o side_n of_o a_o icosahedron_n inscribe_v in_o a_o octohedron_n be_v in_o power_n duple_n to_o the_o less_o segment_n of_o the_o side_n of_o the_o same_o
proportion_n which_o be_v in_o power_n duple_n to_o of_o the_o side_n of_o the_o octohedron_n inscribe_v in_o the_o dodecahedron_n draw_v the_o diameter_n el_n and_o fk_o of_o the_o octohedron_n now_o they_o couple_v the_o middle_a section_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n ab_fw-la and_o gd_o by_o the_o 9_o of_o the_o fifteen_o &_o 3._o corollary_n of_o the_o 17._o of_o the_o thirteen_o &_o every_o one_o of_o those_o diameter_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n do_v make_v the_o less_o segment_n the_o side_n of_o the_o dodecahedron_n by_o the_o 4._o corollary_n of_o the_o same_o wherefore_o the_o side_n ab_fw-la be_v the_o less_o segment_n of_o the_o line_n fk_o but_o the_o line_n fk_o contain_v in_o power_n the_o two_o equal_a line_n of_o &_o eke_o by_o the_o 47._o of_o the_o first_o for_o the_o angle_n fek_n be_v a_o right_a angle_n of_o the_o square_a fekl_n of_o the_o octohedron_n wherefore_o the_o line_n fk_o be_v in_o power_n duple_n to_o the_o line_n ef._n wherefore_o the_o line_n ab_fw-la the_o side_n of_o the_o dodecahedron_n be_v the_o less_o segment_n of_o the_o line_n fk_o which_o be_v in_o power_n duple_n to_o of_o the_o sidâ_n of_o the_o octohedron_n the_o side_n therefore_o of_o a_o dodecahedron_n iâ_z the_o less_o segment_n of_o that_o right_a line_n which_o be_v in_o power_n duple_n to_o the_o side_n of_o the_o octohedron_n inscribe_v in_o the_o same_o dodâcahedron_n ¶_o the_o 22._o proposition_n the_o diameter_n of_o a_o icosahedron_n be_v in_o power_n sesquitertia_fw-la to_o the_o side_n of_o the_o same_o icosahedron_n and_o also_o be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o icosahedron_n for_o forasmuch_o as_o it_o have_v be_v prove_v by_o the_o 10._o of_o this_o book_n that_o if_o from_o the_o power_n of_o the_o diameter_n of_o the_o icosahedron_n be_v take_v away_o the_o triple_a of_o the_o power_n of_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o it_o there_o shall_v be_v leave_v a_o square_a sesquitertia_fw-la to_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n but_o the_o power_n of_o the_o side_n of_o the_o cube_fw-la triple_a be_v the_o diameter_n of_o the_o same_o cube_fw-la by_o the_o 15._o of_o the_o thirteen_o and_o the_o cube_fw-la &_o the_o pyramid_n inscribe_v in_o it_o be_v contain_v in_o one_o &_o the_o self_n same_o sphere_n by_o the_o first_o of_o this_o book_n and_o in_o one_o &_o the_o self_n same_o icosahedron_n by_o the_o corollary_n of_o the_o same_o wherefore_o one_o and_o the_o self_n same_o diameter_n of_o the_o cube_fw-la or_o of_o the_o sphere_n which_o contain_v the_o cube_fw-la and_o the_o pyramid_n be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o the_o pyramid_n by_o the_o 13._o of_o the_o thirteen_o wherefore_o it_o follow_v that_o if_o from_o the_o diameter_n of_o the_o icosahedron_n be_v take_v away_o the_o triple_a power_n of_o the_o side_n of_o the_o cube_fw-la or_o the_o sesquialter_fw-la power_n of_o the_o side_n of_o the_o pyramid_n which_o be_v the_o power_n of_o one_o and_o the_o self_n same_o diameter_n there_o shall_v be_v leave_v the_o sesquitertia_fw-la power_n of_o the_o side_n of_o the_o icosahedron_n the_o diameter_n therefore_o of_o a_o icosahedron_n be_v in_o power_n sesquitertia_fw-la to_o the_o side_n of_o the_o same_o icosahedron_n and_o also_o be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o icosahedron_n the_o 23._o proposition_n the_o side_n of_o a_o dodecahedron_n be_v to_o the_o side_n of_o a_o icosahedron_n inscribe_v in_o it_o as_o the_o less_o segment_n of_o the_o perpendicular_a of_o the_o pentagon_n be_v to_o that_o line_n which_o be_v draw_v from_o the_o centre_n to_o the_o side_n of_o the_o same_o pentagon_n constrution_n let_v there_o be_v take_v a_o dodecahedron_n abgdfso_n who_o side_n let_v be_v as_o or_o so_o and_o let_v the_o icosahedron_n inscribe_v in_o it_o be_v klnmne_a who_o side_n let_v be_v kl_o from_o the_o two_o angle_n of_o the_o pentagonâ_n bas_n and_o fas_fw-la of_o the_o dodecahedron_n namely_o from_o the_o angleââ_n and_o fletcher_n let_v there_o be_v draw_v to_o the_o common_a base_a as_o perpendicular_a line_n bc_o &_o fc_n which_o shall_v pass_v by_o the_o centre_n king_n &_o l_o of_o the_o say_v pentagon_n by_o the_o corollary_n of_o the_o 10._o of_o the_o thirteen_o draw_v the_o line_n bf_o and_o ro._n now_o forasmuch_o as_o the_o line_n ro_n subtend_v the_o angle_n ofr_n of_o thâ_z pentagon_n of_o the_o dodecahedron_n it_o shall_v cut_v the_o line_n fc_o by_o a_o extreme_a and_o mean_a proportion_n by_o the_o 3._o of_o this_o book_n let_v it_o cut_v it_o in_o the_o point_n i_o and_o forasmuch_o as_o the_o line_n kl_o be_v the_o side_n of_o the_o icosahedron_n inscribe_v in_o the_o dodecahedron_n it_o couple_v the_o centre_n of_o the_o base_n of_o the_o dodecahedron_n for_o the_o angle_n of_o the_o icosahedron_n be_v set_v in_o the_o centre_n of_o the_o base_n of_o the_o dodecahedron_n by_o the_o 7._o of_o the_o fifteen_o now_o i_o say_v that_o so_o the_o side_n of_o the_o dodecahedron_n be_v to_o kl_o the_o side_n of_o the_o icosahedron_n as_o the_o less_o segment_n if_o of_o the_o perpendicular_a line_n cf_o be_v to_o the_o line_n lc_n which_o be_v draw_v from_o the_o centre_n l_o to_o as_o the_o side_n of_o the_o pentagon_n for_o forasmuch_o as_o in_o the_o triangle_n bcf_n the_o two_o side_n cb_o and_o cf_o be_v in_o the_o centre_n l_o and_o king_n cut_v like_o proportional_o demonstration_n the_o line_n bf_o and_o kl_o shall_v be_v parellel_n by_o the_o 2._o of_o the_o six_o wherefore_o the_o triangle_n bcf_n and_o kcl_n shall_v be_v equiangle_n by_o the_o corollary_n of_o the_o same_o wherefore_o as_o the_o line_n cl_n be_v to_o the_o line_n klâ_n so_o be_v the_o line_n cf_o to_o the_o line_n bf_o by_o the_o 4._o of_o the_o six_o but_o cf_o make_v the_o less_o segment_n the_o line_n if_o by_o the_o 3._o of_o this_o book_n and_o the_o linâ_n bf_o make_v the_o less_o segment_n the_o line_n so_o namely_o the_o side_n of_o the_o dodecahedron_n by_o the_o 2._o corollary_n of_o the_o 13._o of_o the_o fifteen_o for_o the_o line_n bf_o which_o couple_v the_o angle_n b_o and_o f_o of_o the_o base_n of_o the_o dodecahedron_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la which_o contain_v the_o dodecahedron_n by_o the_o .13_o of_o the_o fifteen_o wherefore_o as_o the_o whole_a line_n câ_n be_v to_o the_o whole_a line_n bf_o so_o be_v the_o less_o segment_n if_o to_o the_o less_o segment_n so_o by_o the_o 2._o of_o the_o 14_o but_o as_o the_o line_n cf_o be_v to_o the_o line_n bf_o so_o be_v the_o line_n cl_n prove_v to_o be_v to_o the_o line_n kl_o wherefore_o as_o the_o line_n if_o be_v to_o the_o line_n so_o so_o be_v the_o line_n cl_n to_o the_o line_n kl_o wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n if_o the_o less_o segment_n of_o the_o perpendicular_a of_o the_o pentagon_n fas_fw-la be_v to_o the_o line_n lc_n which_o be_v draw_v from_o the_o centre_n of_o the_o pentagon_n to_o the_o base_a so_o be_v the_o line_n so_o the_o side_n of_o the_o dodecahedron_n to_o thâ_z line_n kl_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o the_o side_n therefore_o of_o a_o dodecahedron_n be_v to_o the_o side_n of_o a_o icosahedron_n inscribe_v in_o it_o as_o the_o less_o segment_n of_o the_o perpendicular_a of_o the_o pentagon_n be_v to_o that_o line_n which_o be_v draw_v from_o the_o cenâre_n to_o the_o side_n of_o the_o same_o pentagon_n ¶_o the_o 24._o proposition_n if_o half_a of_o the_o side_n of_o a_o icosahedron_n be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n and_o if_o the_o less_o segment_n thereof_o be_v take_v away_o from_o the_o whole_a side_n and_o again_o from_o the_o residue_n be_v take_v away_o the_o three_o part_n that_o which_o remain_v shall_v be_v equal_a to_o the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o icosahedron_n svppose_v that_o abgdf_n be_v a_o pentagon_n construction_n contain_v five_o side_n of_o the_o icosahedron_n by_o the_o 16._o of_o the_o thirteen_o and_o let_v it_o be_v inscribe_v in_o a_o circle_n who_o centre_n let_v be_v the_o point_n e._n and_o upon_o the_o side_n of_o the_o pentagon_n let_v there_o be_v rear_v up_o triangle_n make_v a_o solid_a angle_n of_o the_o icosahedron_n at_o the_o point_n i_o by_o the_o 16._o of_o the_o thirteen_o and_o in_o the_o circle_n abdella_n inscribe_v a_o equilater_n triangle_n ahk_n from_o the_o centre_n e_o draw_v to_o hk_v the_o side_n of_o the_o triangle_n and_o gd_o the_o side_n of_o the_o pentagon_n a_o perpendicular_a line_n which_o
same_o the_o cube_fw-la contain_v 12_o namely_o be_v sesquialter_fw-la to_o the_o pyramid_n wherefore_o of_o what_o part_n the_o cube_fw-la contain_v 12_o of_o the_o same_o the_o whole_a octohedron_n which_o be_v double_a to_o the_o pyramid_n abdfc_n contain_v 54._o which_o 54._o have_v to_o 12._o quadruple_a sesquialter_fw-la proportion_n wherefore_o the_o whole_a octohedron_n be_v to_o the_o cube_fw-la inscribe_v in_o it_o in_o quadruple_a sesquialter_fw-la proportion_n wherefore_o we_o have_v prove_v that_o a_o octohedron_n give_v be_v quadruple_a sesquialter_fw-la to_o a_o cube_fw-la inscribe_v in_o it_o ¶_o a_o corollary_n a_o octohedron_n be_v to_o a_o cube_fw-la inscribe_v in_o it_o in_o that_o proportion_n that_o the_o square_n of_o their_o side_n be_v for_o by_o the_o 14._o of_o this_o book_n the_o side_n of_o the_o octohedron_n be_v in_o power_n quadruple_a sesquialter_fw-la to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o it_o ¶_o the_o 29._o proposition_n to_o prove_v that_o a_o octohedron_n give_v be_v â_z tredecuple_n sesquialter_fw-la to_o a_o trilater_n equilater_n pyramid_n inscribe_v in_o it_o let_v the_o octohedron_n geâen_o be_v ab_fw-la in_o which_o let_v there_o be_v inscribe_v a_o cube_fw-la fced_fw-la by_o the_o 4._o of_o the_o fifteen_o and_o in_o the_o cube_fw-la let_v there_o be_v inscribe_v a_o pyramid_n fegd_v by_o the_o â_o of_o the_o fifteen_o and_o forasmuch_o as_o the_o angle_n of_o the_o pyramid_n be_v by_o the_o same_o first_o of_o the_o fifteen_o set_v in_o the_o angle_n of_o the_o cube_fw-la and_o the_o angle_n of_o the_o cube_fw-la be_v set_v in_o the_o centre_n of_o the_o base_n of_o the_o octohedron_n namely_o in_o the_o point_n f_o e_o c_o d_o g_o by_o the_o 4._o of_o the_o fifteen_o wherefore_o the_o angle_n of_o the_o pyramid_n be_v set_v in_o the_o centre_n f_o c_o e_o d_o of_o the_o octohedron_n wherefore_o the_o pyramid_n fedg_n be_v inscribe_v in_o the_o octohedron_n by_o the_o 6._o of_o the_o fifteen_o and_o forasmuch_o as_o the_o octohedron_n ab_fw-la be_v to_o the_o cube_fw-la fced_fw-la inscribe_v in_o it_o quadruple_a sesquialter_fw-la by_o the_o former_a proposition_n and_o the_o cube_fw-la cdef_n be_v to_o the_o pyramid_n fedg_v inscribe_v in_o it_o triple_a by_o the_o 25._o of_o book_n wherefore_o three_o magnitude_n be_v give_v namely_o the_o octohedron_n the_o cube_fw-la and_o the_o pyramid_n the_o proportion_n of_o the_o extreme_n namely_o of_o the_o octohedron_n to_o the_o pyramid_n be_v make_v of_o the_o proportion_n of_o the_o mean_n namely_o of_o the_o octohedron_n to_o the_o cube_fw-la and_o of_o the_o cube_fw-la to_o the_o pyramid_n as_o it_o be_v easy_a to_o see_v by_o the_o declaration_n upon_o the_o 10._o definition_n of_o the_o five_o now_o then_o multiply_v the_o quantity_n or_o denomination_n of_o the_o proportion_n namely_o of_o the_o octohedron_n to_o the_o cube_fw-la which_o be_v 4_o 1_o â_o and_o of_o the_o cube_fw-la to_o the_o pyramid_n which_o be_v 3_o as_o be_v teach_v in_o the_o definition_n of_o the_o six_o there_o shall_v be_v produce_v 13_o 1_o â_o namely_o the_o proportion_n of_o the_o octohedron_n to_o the_o pyramid_n inscribe_v in_o it_o for_o 4_o ½_n multiply_v by_o 3._o produce_v 13_o ½_n wherefore_o the_o octohedron_n be_v to_o the_o pyramid_n inscribe_v in_o it_o in_o tredecuple_n sesquialter_fw-la proportion_n wherefore_o we_o have_v prove_v that_o a_o octohedron_n be_v to_o a_o trilater_n equilater_n pyramid_n inscribe_v in_o it_o in_o tredecuple_n sesquialter_fw-la proportion_n ¶_o the_o 30._o proposition_n to_o prove_v that_o a_o trilater_n equilater_n pyramid_n be_v noncuple_n to_o a_o cube_fw-la inscribe_v in_o it_o svppose_v that_o the_o pyramid_n give_v be_v abcd_o who_o two_o base_n let_v be_v abc_n and_o dbc_n and_o let_v their_o centre_n be_v the_o point_n g_o and_o i._n and_o from_o the_o angle_n a_o draw_v unto_o the_o base_a bc_o a_o perpendicular_a ae_n likewise_o from_o the_o angle_n d_o draw_v unto_o the_o same_o base_a bc_o a_o perpendicular_a de_fw-fr and_o they_o shall_v concur_v in_o the_o section_n e_o by_o the_o 3._o of_o the_o three_o and_o in_o they_o shall_v be_v the_o centre_n g_o an_o i_o by_o the_o corollary_n of_o the_o first_o of_o the_o three_o and_o forasmuch_o as_o the_o line_n ad_fw-la be_v the_o side_n of_o the_o pyramid_n the_o same_o ad_fw-la shall_v be_v the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la which_o contain_v the_o pyramid_n by_o the_o 1_o of_o the_o fivetenth_n demonstration_n draw_v the_o line_n give_v and_o forasmuch_o as_o the_o line_n give_v couple_v the_o centreâ_n of_o the_o base_n of_o the_o pyramid_n the_o say_a line_n give_v shall_v be_v the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n by_o the_o 18._o of_o the_o fifteen_o and_o forasmuch_o as_o the_o line_n agnostus_n be_v double_a to_o the_o line_n ge_z by_o the_o corollarye_a of_o the_o twelve_o of_o the_o thirteen_o the_o whole_a line_n ae_n shall_v be_v triple_a to_o the_o line_n ge_z and_o so_o be_v also_o the_o line_n de_fw-fr to_o the_o line_n je_n wherefore_o the_o line_n ad_fw-la and_o give_v be_v parallel_n by_o the_o 2._o of_o the_o six_o and_o therefore_o the_o triangle_n aed_n and_o gei_n be_v likeâ_n by_o the_o corollary_n of_o the_o same_o and_o forasmuch_o as_o the_o triangle_n aed_n and_o gei_n be_v like_a the_o line_n adâ_n shall_v be_v treble_v to_o the_o line_n give_v by_o the_o 4._o of_o the_o six_o but_o the_o line_n ad_fw-la be_v the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la circumscribe_v about_o the_o pyramid_n abcd_o and_o the_o line_n give_v be_v the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n abcd_o but_o the_o diameter_n of_o the_o base_n be_v equemultiplices_fw-la to_o the_o side_n namely_o be_v in_o power_n duple_n wherefore_o the_o side_n of_o the_o cube_fw-la circumscribe_v about_o the_o pyramid_n abcd_o be_v triple_a to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o same_o pyramid_n by_o the_o 15._o of_o the_o five_o but_o like_o cube_n be_v in_o triple_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o their_o side_n be_v by_o the_o 33._o of_o the_o eleven_o and_o the_o side_n be_v in_o triple_a proportion_n the_o one_o to_o the_o other_o wherefore_o triple_a take_v three_o time_n bring_v forth_o twenty_o sevencuple_n which_o be_v 27._o to_o 1_o for_o the_o 4._o term_n 27.9.3.1_o be_v set_v in_o triple_a proportion_n the_o proportion_n of_o the_o first_o to_o the_o four_o namely_o of_o 27._o to_o 1._o shall_v be_v treble_v to_o the_o proportion_n of_o the_o first_o to_o the_o second_o namely_o of_o 27._o to_o 9_o by_o the_o 10._o definition_n of_o the_o five_o which_o proportion_n of_o 27._o to_o 1._o be_v the_o proportion_n of_o the_o side_n triple_a which_o proportion_n also_o be_v find_v in_o like_o solides_fw-la wherefore_o of_o what_o part_n the_o cube_fw-la circumscribe_v contain_v 27._o of_o the_o same_o the_o cube_fw-la inscribe_v contain_v one_o but_o of_o what_o part_n the_o cube_fw-la circumscribe_v contain_v 27._o of_o the_o same_o the_o pyramid_n inscribe_v in_o it_o contain_v 9_o by_o the_o 25._o of_o this_o book_n wherefore_o of_o what_o part_n the_o pyramid_n ab_fw-la cd_o contain_v 9_o of_o the_o same_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n contain_v one_o wherefore_o we_o have_v prove_v that_o a_o trilater_n and_o equilater_n pyramid_n be_v nonâcuple_n to_o a_o cube_fw-la inscribe_v in_o it_o ¶_o the_o 31._o proposition_n a_o octohedron_n have_v to_o a_o icosohedron_n inscribe_v in_o it_o that_o proportion_n which_o two_o base_n of_o the_o octohedron_n have_v to_o five_o base_n of_o the_o icosahedron_n svppose_v that_o the_o octohedron_n give_v be_v abcd_o and_o let_v the_o icosahedron_n inscribe_v in_o it_o be_v fghmklio_n then_o i_o say_v that_o the_o octohedron_n be_v to_o the_o icosahedron_n as_o two_o base_n of_o the_o octohedron_n be_v to_o five_o base_n of_o the_o icosahedron_n for_o forasmuch_o as_o the_o solid_a of_o the_o octohedron_n consist_v of_o eight_o pyramid_n set_v upon_o the_o base_n of_o the_o octohedron_n demonstration_n and_o have_v to_o their_o altitude_n a_o perpendicular_a line_n draw_v from_o the_o centre_n to_o the_o base_a let_v that_o perpendicular_a be_v er_fw-mi or_o es_fw-ge be_v draw_v from_o the_o centre_n e_o which_o centre_n be_v common_a to_o either_o of_o the_o solid_n by_o the_o corollary_n of_o the_o 21._o of_o the_o fifteen_o to_o the_o centre_n of_o the_o base_n namely_o to_o the_o point_n r_o and_o s._n wherefore_o for_o that_o three_o pyramid_n be_v equal_a and_o like_a they_o shall_v be_v equal_a to_o a_o prism_n set_v upon_o the_o self_n same_o base_a and_o under_o the_o self_n same_o altitude_n by_o the_o corollary_n of_o the_o seven_o of_o the_o twelve_o but_o
in_o the_o point_n e._n and_o unto_o the_o line_n cg_o put_v the_o line_n cl_n equal_a now_o forasmuch_o as_o the_o line_n agnostus_n and_o gc_o be_v the_o great_a segâââtes_n of_o half_a the_o line_n ab_fw-la for_o âche_fw-mi of_o they_o be_v the_o half_a of_o the_o great_a segment_n of_o the_o whole_a line_n ab_fw-la the_o line_n ebb_v and_o ec_o shall_v be_v the_o less_o segment_n of_o half_a the_o line_n ab_fw-la wherefore_o the_o whole_a line_n câ_n be_v the_o great_a segment_n and_o the_o line_n ce_fw-fr be_v the_o less_o segment_n but_o as_o the_o line_n cl_n be_v to_o the_o line_n ce_fw-fr so_o be_v the_o line_n ce_fw-fr to_o the_o residue_n el._n wherefore_o the_o line_n el_n be_v the_o great_a segment_n of_o the_o line_n ce_fw-fr or_o of_o the_o line_n ebb_v which_o be_v equal_a unto_o it_o wherefore_o the_o residue_n lb_n be_v the_o less_o segment_n of_o the_o same_o ebb_n which_o be_v the_o lesâe_n segment_n of_o halfa_n the_o side_n of_o the_o cube_fw-la but_o the_o line_n agnostus_n gc_o and_o cl_n be_v three_o great_a segment_n of_o the_o half_a of_o the_o whole_a line_n ab_fw-la which_o three_o great_a segment_n make_v the_o altitude_n of_o the_o foresay_a solid_a wherefore_o the_o altitude_n of_o the_o say_v solid_a want_v of_o ab_fw-la the_o side_n of_o the_o cube_fw-la by_o the_o line_n lb_n which_o be_v the_o less_o segment_n of_o the_o line_n be._n which_o line_n be_v again_o be_v the_o less_o segment_n of_o half_a the_o side_n ab_fw-la of_o the_o cube_fw-la wherefore_o the_o foresay_a solid_a consist_v of_o the_o six_o solid_n whereby_o the_o dodecahedron_n exceed_v the_o cube_fw-la inscribe_v in_o it_o be_v set_v upon_o a_o base_a which_o want_v of_o the_o base_a of_o the_o cube_fw-la by_o a_o three_o part_n of_o the_o less_o segment_n and_o be_v under_o a_o altitude_n want_v of_o the_o side_n of_o the_o cube_fw-la by_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o side_n of_o the_o cube_fw-la the_o solid_a therefore_o of_o a_o dodecahedron_n exceed_v the_o solid_a of_o a_o cube_fw-la inscribe_v in_o it_o by_o a_o parallelipipedon_n who_o base_a want_v of_o the_o base_a of_o the_o cube_fw-la by_o a_o three_o part_n of_o the_o less_o segment_n and_o who_o altitude_n want_v of_o the_o altitude_n of_o the_o cube_fw-la by_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o side_n of_o the_o cube_fw-la ¶_o a_o corollary_n a_o dodecahedron_n be_v double_a to_o a_o cube_n inscribe_v in_o it_o take_v away_o the_o three_o part_n of_o the_o less_o segment_n of_o the_o cube_fw-la and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a of_o that_o excess_n for_o if_o there_o be_v give_v a_o cube_fw-la from_o which_o be_v cut_v of_o a_o solid_a set_v upon_o a_o three_o part_n of_o the_o less_o segment_n of_o the_o base_a and_o under_o one_o and_o the_o same_o altitude_n with_o the_o cube_fw-la that_o solid_a take_v away_o have_v to_o the_o whole_a solid_a the_o proportion_n of_o the_o section_n of_o the_o base_a to_o the_o base_a by_o the_o 32._o of_o the_o eleven_o wherefore_o from_o the_o cube_fw-la be_v take_v away_o a_o three_o âart_n of_o the_o less_o segment_n far_o forasmuch_o as_o the_o residue_n want_v of_o the_o altitude_n of_o the_o cube_fw-la by_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o altitude_n or_o side_n and_o that_o residue_n be_v a_o parallelipipedon_n if_o it_o be_v cut_v by_o a_o plain_a superficies_n parallel_v to_o the_o opposite_a plain_a superficiece_n cut_v the_o altitude_n of_o the_o cube_fw-la by_o a_o point_n it_o shall_v take_v away_o from_o that_o parallelipipedon_n a_o solid_a have_v to_o the_o whole_a the_o proportion_n of_o the_o section_n to_o the_o altitude_n by_o the_o 3._o corollary_n of_o the_o 25._o of_o the_o eleven_o wherefore_o the_o excess_n want_v of_o the_o same_o cube_fw-la by_o the_o thiâd_o part_n of_o the_o less_o segment_n and_o moreover_o by_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a of_o that_o excess_n ¶_o the_o 34._o proposition_n the_o proportion_n of_o the_o solid_a of_o a_o dodecahedron_n to_o the_o solid_a of_o a_o icosahedron_n inscribe_v in_o it_o consist_v of_o the_o proportion_n triple_a of_o the_o diameter_n to_o that_o line_n which_o couple_v the_o opposite_a base_n of_o the_o dodecahedron_n and_o of_o the_o proportion_n of_o the_o side_n of_o the_o cube_n to_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n svppose_v that_o ahbck_n be_v a_o dodecahedronâ_n who_o diameter_n let_v be_v ab_fw-la and_o let_v the_o line_n which_o couple_v the_o centre_n of_o the_o opposite_a base_n be_v khâ_n and_o let_v the_o icosahedron_n inscribe_v in_o the_o dodecahedron_n abc_n be_v do_v you_o who_o diameter_n let_v be_v de._n now_o forasmuch_o aâ_z oâe_n and_o the_o self_n same_o circle_n contain_v the_o pentagon_n of_o a_o dodecahedron_n &_o the_o triangle_n of_o a_o icosahedroâ_n describe_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o 14._o of_o the_o fourteen_o let_v that_o circle_n be_v igo._n wherefore_o io_o be_v the_o side_n of_o the_o cube_fw-la and_o ig_v the_o side_n of_o the_o icosahedron_n by_o the_o same_o then_o i_o say_v that_o the_o proportion_n of_o the_o dodecahedron_n ahbck_n to_o the_o icosahedron_n def_n inscribe_v in_o it_o consist_v of_o the_o proportion_n triple_a of_o the_o line_n ab_fw-la to_o the_o line_n kh_o and_o of_o the_o proportion_n of_o the_o line_n io_o to_o the_o line_n ig_n for_o forasmuch_o as_o the_o icosahedron_n def_n be_v inscribe_v in_o the_o dodecahedron_n abc_n demonstration_n by_o supposition_n the_o diameter_n de_fw-fr shall_v be_v equal_a to_o the_o line_n kh_o by_o the_o 7._o of_o the_o fifteen_o wherefore_o the_o dodecahedron_n set_v upon_o the_o diameter_n kh_o shall_v be_v inscribe_v in_o the_o same_o sphere_n wherein_o the_o icosahedron_n def_n be_v inscribe_v but_o the_o dodecahedron_n ahbck_n be_v to_o the_o dodecahedron_n upon_o the_o diameter_n kh_o in_o triple_a proportion_n of_o that_o in_o which_o the_o diameter_n ab_fw-la be_v to_o the_o diameter_n kh_o by_o the_o corollary_n of_o the_o 17._o of_o the_o twelve_o and_o the_o same_o dodecahedron_n which_o be_v set_v upon_o the_o diameter_n kh_o have_v to_o the_o icosahedron_n def_n which_o be_v set_v upon_o the_o same_o diameter_n or_o upon_o a_o diameter_n equal_a unto_o it_o namely_o de_fw-fr that_o proportion_n which_o io_o the_o side_n of_o the_o cube_fw-la have_v toâ_n ig_n the_o side_n of_o the_o icosahedron_n inscribe_v in_o one_o &_o the_o self_n same_o sphere_n by_o the_o 8_o of_o the_o fourteen_o wherefore_o the_o proportion_n of_o the_o dodecahedron_n ahbck_n to_o the_o icosahedron_n def_n inscribe_v in_o it_o consist_v of_o the_o proportion_n triple_a of_o the_o diameter_n ab_fw-la to_o the_o line_n kh_o which_o couple_v the_o centre_n of_o the_o opposite_a base_n of_o the_o dodecahedron_n which_o proportion_n be_v that_o which_o the_o dodecahedron_n ahbck_n have_v to_o the_o dodecahedron_n set_v upon_o the_o diameter_n kh_o and_o of_o the_o proportion_n of_o io_o the_o side_n of_o the_o cube_fw-la to_o ig_o the_o side_n of_o the_o icosahedron_n which_o be_v the_o proportion_n of_o the_o dodecahedron_n set_v upon_o the_o diameter_n kh_o to_o the_o icosahedron_n def_n describe_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o 5._o definition_n of_o the_o six_o the_o proportion_n therefore_o of_o the_o solid_a of_o a_o dodecahedron_n to_o the_o solid_a of_o a_o icosahedron_n inscribe_v in_o it_o consi_v of_o the_o proportion_n triple_a of_o the_o diameter_n to_o that_o line_n which_o couple_v the_o opposite_a base_n of_o the_o dodecahedron_n and_o of_o the_o proportion_n of_o the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n the_o 35._o proposition_n the_o solid_a of_o a_o dodecahedron_n contain_v of_o a_o pyramid_n circumscribe_v about_o it_o two_o nine_o part_n take_v away_o a_o three_o part_n of_o one_o nine_o part_n of_o the_o less_o segment_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o residue_n it_o have_v be_v prove_v that_o the_o dodecahedron_n together_o with_o the_o cube_fw-la inscribe_v in_o it_o be_v contain_v in_o one_o and_o the_o self_n same_o pyramid_n by_o the_o corollary_n of_o the_o first_o of_o this_o book_n and_o by_o the_o corollary_n of_o the_o 33._o of_o this_o book_n it_o be_v manifest_a that_o the_o dodecahedron_n be_v double_a to_o the_o same_o cube_fw-la take_v away_o the_o three_o part_n of_o the_o less_o segment_n and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a
the_o residue_n or_o of_o this_o excess_n but_o a_o pyramid_n be_v to_o the_o same_o cube_fw-la inscribe_v in_o it_o nonecuple_n by_o the_o 30._o of_o this_o book_n wherefore_o the_o dodecahedron_n inscribe_v in_o the_o pyramid_n and_o contain_v the_o same_o cube_fw-la twice_o take_v away_o the_o self_n same_o three_o of_o the_o less_o segment_n and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o residue_n shall_v contain_v two_o nine_o part_n of_o the_o solid_a of_o the_o pyramid_n of_o which_o nine_o part_v each_o be_v equal_a unto_o the_o cube_fw-la take_v away_o this_o self_n same_o excess_n the_o solid_a therefore_o of_o a_o dodecahedron_n contain_v of_o a_o pyramid_n circumscribe_v about_o it_o two_o nine_o part_n take_v away_o a_o three_o part_n of_o one_o nine_o part_n of_o the_o less_o segment_n of_o a_o line_n divide_v by_o a_o extmere_n and_o mean_a proportion_n and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o residue_n ¶_o the_o 36._o proposition_n a_o octohedron_n exceed_v a_o icosahedron_n inscribe_v in_o it_o by_o a_o parallelipipedon_n set_v upon_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n and_o have_v to_o his_o altitude_n the_o line_n which_o be_v the_o great_a segment_n of_o half_a the_o semidiameter_n of_o the_o octohedron_n svppose_v that_o there_o be_v a_o octohedron_n abcfpl_n construction_n in_o which_o let_v there_o be_v inscribe_v a_o icosahedron_n hkegmxnudsqtâ_n by_o the_o â6_o of_o the_o fifteen_o and_o draw_v the_o diameter_n azrcbroif_n and_o the_o perpendicular_a ko_o parallel_n to_o the_o line_n azr_n then_o i_o say_v that_o the_o octohedron_n abcfpl_n be_v great_a thân_n the_o icosahedron_n inscribe_v in_o it_o by_o a_o parallelipipedon_n set_v upon_o the_o square_n of_o the_o side_n hk_o or_o ge_z and_o have_v to_o his_o altitude_n the_o line_n ko_o or_o rz_n which_o be_v the_o great_a segment_n of_o the_o semidiameter_n ar._n forasmuch_o as_o in_o the_o same_o 16._o it_o have_v be_v prove_v that_o the_o triangle_n kdg_n and_o keq_n be_v describe_v in_o the_o base_n apf_n and_o alf_n of_o the_o octohedron_n demonstration_n therefore_o about_o the_o solid_a angle_n there_o remain_v upon_o the_o base_a feg_fw-mi three_o triangle_n keg_n kfe_n and_o kfg_n which_o contain_v a_o pyramid_n kefg_n unto_o which_o pyramid_n shall_v be_v equal_a and_o like_v the_o opposite_a pyramid_n mefg_n set_v upon_o the_o same_o base_a feg_n by_o the_o 8._o definition_n of_o the_o eleven_o and_o by_o the_o âame_n reason_n shall_v there_o at_o every_o solid_a angle_n of_o the_o octohedron_n remain_v two_o pyramid_n equal_a and_o like_a namely_o two_o upon_o the_o base_a ahk_n two_o upon_o the_o base_a bnv_n two_o upon_o the_o base_a dp_n and_o moreover_o two_o upon_o the_o base_a qlt._n now_o they_o there_o shall_v be_v make_v twelve_o pyramid_n set_v upon_o a_o base_a contain_v of_o the_o side_n of_o the_o icosahedron_n and_o under_o two_o leââe_a segment_n of_o the_o side_n of_o the_o octohedron_n contain_v a_o right_a angle_n as_o for_o example_n the_o base_a gef_n and_o forasmuch_o as_o the_o side_n ge_z subtend_v a_o right_a angle_n be_v by_o the_o 47._o of_o the_o âirst_n in_o power_n duple_n to_o either_o of_o the_o line_n of_o and_o fg_o and_o so_o the_o ââdeâ_n kh_o be_v in_o power_n duple_n to_o either_o of_o the_o side_n ah_o and_o ak_o and_o either_o of_o the_o line_n ah_o ak_o or_o of_o fg_o be_v in_o power_n duple_n to_o either_o of_o the_o line_n az_o or_o zk_v which_o contain_v a_o right_a angle_n make_v in_o the_o triangle_n or_o base_a ahk_n by_o the_o perpendicular_a az_o wherefore_o it_o follow_v that_o the_o side_n ge_z or_o hk_o be_v in_o power_n quadruple_a to_o the_o triangle_n efg_o or_o ahk_n but_o the_o pyramid_n kefg_n have_v his_o base_a efg_o in_o the_o plain_a flbp_n of_o the_o octohedron_n shall_v have_v to_o his_o altitude_n the_o perpendicular_a ko_o by_o the_o 4._o definition_n of_o the_o six_o which_o be_v the_o great_a segment_n of_o the_o semidiameter_n of_o the_o octohedron_n by_o the_o 16._o of_o the_o fifteen_o wherefore_o three_o pyramid_n set_v under_o the_o same_o altitude_n and_o upon_o equal_a base_n shall_v be_v equal_a to_o one_o prism_n set_v upon_o the_o same_o base_a and_o under_o the_o same_o altitude_n by_o the_o 1._o corollary_n of_o the_o 7._o of_o the_o twelve_o wherefore_o 4._o prism_n set_v upon_o the_o base_a gef_n quadruple_v which_o be_v equal_a to_o the_o square_n of_o the_o side_n ge_z and_o under_o the_o altitude_n ko_o or_o rz_n the_o great_a segment_n which_o be_v equal_a to_o ko_o shall_v contain_v a_o solid_a equal_a to_o the_o twelve_o pyramid_n which_o twelve_o pyramid_n make_v the_o excess_n of_o the_o octohedron_n above_o the_o icosahedron_n inscribe_v in_o it_o a_o octohedron_n therefore_o exceed_v a_o icosahedron_n inscribe_v in_o it_o by_o a_o parallelipipedon_n set_v upon_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n and_o have_v to_o his_o altitude_n the_o line_n which_o be_v the_o great_a segment_n of_o half_a the_o semidiameter_n of_o the_o octohedron_n ¶_o a_o corollary_n a_o pyramid_n exceed_v the_o double_a of_o a_o icosahedron_n inscribe_v in_o it_o by_o a_o solid_a set_v upon_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o and_o have_v to_o his_o altitude_n that_o whole_a line_n of_o which_o the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n for_o it_o be_v manifest_a by_o the_o 19_o of_o the_o fivetenth_n that_o a_o octohedron_n &_o a_o icosahedron_n inscribe_v in_o it_o be_v inscribe_v in_o one_o &_o the_o self_n same_o pyramid_n it_o have_v moreover_o be_v prove_v in_o the_o 26._o of_o this_o book_n that_o a_o pyramid_n be_v double_a to_o a_o octohedron_n inscribe_v in_o it_o wherefore_o the_o two_o excess_n of_o the_o two_o octohedron_n unto_o which_o the_o pyramid_n be_v equal_a above_o the_o two_o icosahedrons_n inscribe_v in_o the_o say_v two_o octohedron_n be_v bring_v into_o a_o solid_a the_o say_v solid_a shall_v be_v set_v upon_o the_o self_n same_o square_n of_o the_o side_n of_o the_o icosahedron_n and_o shall_v have_v to_o his_o altitude_n the_o perpendicular_a ko_o double_a who_o double_a couple_v the_o opposite_a side_n hk_o and_o xm_o make_v the_o great_a segment_n the_o same_o side_n of_o the_o icosahedron_n by_o the_o first_o and_o second_o corollary_n of_o the_o 14._o of_o the_o fiuââenâh_n the_o 37._o proposition_n if_o in_o a_o triangle_n have_v to_o his_o base_a a_o rational_a line_n set_v the_o side_n be_v commensurable_a in_o power_n to_o the_o base_a and_o from_o the_o top_n be_v draw_v to_o the_o base_a a_o perpendicular_a line_n cut_v the_o base_a the_o section_n of_o the_o base_a shall_v be_v commensurable_a in_o length_n to_o the_o whole_a base_a and_o the_o perpendicular_a shall_v be_v commensurable_a in_o power_n to_o the_o say_v whole_a base_a and_o now_o that_o the_o perpendicular_a ap_n be_v commensurable_a in_o power_n to_o the_o base_a bg_o demonstration_n iâ_z thus_o prove_v forasmuch_o as_o the_o square_n of_o ab_fw-la be_v by_o supposition_n commensurable_a to_o the_o square_n of_o bg_o and_o unto_o the_o rational_a square_n of_o ab_fw-la be_v commensurable_a the_o rational_a square_n of_o bp_o by_o the_o 12._o of_o the_o eleven_o wherefore_o the_o residue_n namely_o the_o square_a of_o pa_o be_v commensurable_a to_o the_o same_o square_n of_o bp_o by_o the_o 2._o part_n of_o the_o 15._o of_o the_o eleven_o wherefore_o by_o the_o 12._o of_o the_o ten_o the_o square_a of_o pa_o be_v commensurable_a to_o the_o whole_a square_n of_o bg_o wherefore_o the_o perpendicular_a ap_n be_v commensurable_a in_o power_n to_o the_o base_a bg_o by_o the_o 3._o definition_n of_o the_o ten_o which_o be_v require_v to_o be_v prove_v in_o demonstrate_v of_o this_o we_o make_v no_o mention_n at_o all_o of_o the_o length_n of_o the_o side_n ab_fw-la and_o ag_z but_o only_o of_o the_o length_n of_o the_o base_a bg_o for_o that_o the_o line_n bg_o be_v the_o rational_a line_n first_o set_v and_o the_o other_o line_n ab_fw-la and_o ag_z be_v suppose_v to_o be_v commensurable_a in_o power_n only_o to_o the_o line_n bg_o wherefore_o if_o that_o be_v plain_o demonstrate_v when_o the_o side_n be_v commensurable_a in_o power_n only_o to_o the_o base_a much_o more_o easy_o will_v it_o follow_v if_o the_o same_o side_n be_v suppose_v to_o be_v commensurable_a both_o in_o length_n and_o in_o power_n to_o the_o base_a that_o be_v if_o their_o lengthe_n be_v express_v by_o the_o root_n of_o square_a number_n ¶_o a_o corollary_n 1._o by_o the_o former_a thing_n demonstrate_v it_o be_v manifest_a that_o if_o from_o the_o power_n of_o the_o base_a and_o of_o one_o of_o the_o side_n be_v take_v away_o the_o
from_o a_o cube_fw-la and_o a_o octohedron_n compose_v into_o one_o solid_a there_o shall_v be_v leave_v a_o exocthedron_n moreover_o the_o solid_a angle_n take_v away_o from_o two_o pyramid_n compose_v into_o one_o solid_a there_o shall_v be_v leave_v a_o octohedron_n flussas_n after_o this_o set_v forth_o certain_a passion_n and_o property_n of_o the_o five_o simple_a regular_a body_n which_o although_o he_o demonstrate_v not_o yet_o be_v they_o not_o hard_a to_o be_v demonstrate_v if_o we_o well_o pease_n and_o conceive_v that_o which_o in_o the_o former_a book_n have_v be_v teach_v touch_v those_o solid_n of_o the_o nature_n of_o a_o trilater_n and_o equilater_n pyramid_n a_o trilater_n equilater_n pyramid_n be_v divide_v into_o two_o equal_a part_n by_o three_o equal_a square_n which_o in_o the_o centre_n of_o the_o pyramid_n cut_v the_o one_o the_o other_o into_o two_o equal_a part_n and_o perpendicular_o and_o who_o angle_n be_v set_v in_o the_o middle_a section_n of_o the_o side_n of_o the_o pyramid_n from_o a_o pyramid_n be_v take_v away_o 4._o pyramid_n like_a unto_o the_o whole_a which_o utter_o take_v away_o the_o side_n of_o the_o pyramid_n and_o that_o which_o be_v leave_v be_v a_o octohedron_n inscribe_v in_o the_o pyramy_n in_o which_o all_o the_o solid_n inscribe_v in_o the_o pyramid_n be_v contain_v a_o perpendicular_a draw_v from_o the_o angle_n of_o the_o pyramid_n to_o the_o base_a be_v double_a to_o the_o diameter_n of_o the_o cube_fw-la inscribe_v in_o it_o and_o a_o right_a line_n couple_v the_o middle_a section_n of_o the_o opposite_a side_n of_o the_o pyramid_n be_v triple_a to_o the_o side_n of_o the_o self_n same_o cube_fw-la the_o side_n also_o of_o the_o pyramid_n be_v triple_a to_o the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la wherefore_o the_o same_o side_n of_o the_o pyramid_n be_v in_o power_n duple_n 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 and_o it_o be_v in_o power_n sesquialter_fw-la to_o the_o perpendicular_a which_o be_v draw_v from_o the_o angle_n to_o the_o base_a wherefore_o the_o perpendicular_a be_v in_o power_n sesquitertia_fw-la to_o the_o line_n which_o couple_v the_o middle_a section_n of_o the_o opposite_a side_n a_o pyramid_n and_o a_o octohedron_n inscribe_v in_o it_o also_o a_o icosahedron_n inscribe_v in_o the_o same_o octohedron_n do_v contain_v one_o and_o the_o self_n same_o sphere_n of_o the_o nature_n of_o a_o octohedron_n four_o perpendicular_n of_o a_o octohedron_n draw_v in_o 4._o base_n thereof_o from_o two_o opposite_a angle_n of_o the_o say_v octohedron_n and_o couple_v together_o by_o those_o 4._o base_n describe_v a_o rhombus_fw-la or_o diamond_n figure_n one_o of_o who_o diameter_n be_v in_o power_n duple_n to_o the_o other_o diameter_n for_o it_o have_v the_o same_o proportion_n that_o the_o diameter_n of_o the_o octohedron_n have_v to_o the_o side_n of_o the_o octohedron_n a_o octohedron_n &_o a_o 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construction_n demonstration_n application_n of_o space_n with_o excess_n or_o want_v a_o ancient_a invention_n of_o pythagoras_n how_o a_o figure_n be_v say_v to_o be_v apply_v to_o a_o line_n three_o thing_n give_v in_o this_o proposition_n the_o converse_n of_o this_o proposition_n construction_n demonstration_n a_o addition_n of_o pelitarius_n to_o describe_v a_o square_n mechanical_o a_o addition_n of_o procâââ_n the_o converse_n thereof_o construction_n demonstration_n pythagoras_n the_o first_o inventor_n of_o this_o proposition_n a_o addition_n of_o pâlâtariââ_n an_o other_o addition_n of_o pelitarius_n an_o other_o addition_n of_o pelitarius_n an_o other_o addition_n of_o pelitarius_n a_o corollary_n this_o proposition_n be_v the_o converse_n of_o the_o former_a the_o argument_n of_o the_o second_o book_n what_o be_v the_o power_n of_o a_o line_n many_o compendious_a rule_n of_o reckon_v gather_v one_o of_o this_o book_n and_o also_o many_o rule_n of_o algebra_n two_o 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touch_n of_o circle_n be_v ãâã_d in_o one_o poâââ_n only_o circle_n may_v touch_v together_o two_o maâââ_n of_o way_n four_o definition_n five_o definition_n six_o definition_n mix_v angle_n arke_n chord_n seven_o definition_n difference_n of_o a_o angle_n of_o a_o section_n and_o of_o a_o angle_n in_o a_o section_n eight_o definition_n nine_o definition_n ten_o definition_n two_o definition_n first_o second_o why_o euclid_n define_v not_o equal_a section_n constuction_n demonstration_n lead_v to_o a_o impossibility_n correlary_a demonstration_n lead_v to_o a_o impossibility_n the_o first_o para_fw-it of_o this_o proposition_n construction_n demonstration_n the_o second_o part_n converse_v of_o the_o first_o demonstration_n demonstration_n lead_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n construction_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibility_n two_o caseâ_n in_o this_o proposition_n construction_n the_o first_o 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impossibility_n an_o other_o demonstration_n after_o pelitarius_n lead_v also_o to_o a_o absurdity_n of_o circle_n which_o touch_v the_o one_o the_o other_o inward_o of_o circle_n which_o touch_v the_o one_o the_o other_o outward_o an_o other_o demonstration_n after_o pelitarius_n &_o flussates_n of_o circle_n which_o touch_n the_o one_o the_o other_o outward_o of_o circle_n which_o touch_n the_o one_o the_o other_o inward_o the_o first_o part_n of_o this_o theorem_a construction_n demonstration_n demonstration_n the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o an_o other_o demonstration_n of_o the_o first_o part_n after_o campane_n construction_n demonstration_n an_o other_o demonstration_n after_o campane_n the_o first_o part_n of_o this_o theorem_a demonstration_n lead_v to_o a_o absurdity_n second_o part_n three_o part_n construction_n demonstration_n a_o addition_n of_o pelitarius_n this_o problem_n commodious_a for_o the_o inscribe_v and_o circumscribe_v of_o figure_n in_o or_o abouâ_n circle_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n after_o orontius_n demonstration_n lead_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n the_o one_o when_o the_o angle_n set_v at_o the_o circumference_n include_v the_o centre_n demonstration_n the_o other_o when_o the_o same_o angle_n set_v at_o the_o circumference_n include_v not_o the_o centre_n construction_n demonstration_n three_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n the_o three_o case_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n a_o addition_n of_o campane_n demonstrate_v by_o pelitariâs_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n construction_n three_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n the_o second_o case_n the_o three_o case_n a_o addition_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n the_o converse_n of_o the_o former_a proposition_n construction_n demonstration_n construction_n demonstration_n second_o part_n thirâ_n part_n the_o five_o and_o last_o part_n an_o other_o demonstration_n to_o prove_v that_o the_o angâe_n in_o a_o semicircle_n be_v a_o right_a angle_n a_o corollary_n a_o addition_n of_o pâlitarius_n demonstration_n leaâing_v to_o a_o absurdity_n a_o addition_n of_o campane_n construction_n demonstration_n two_o case_n in_o this_o proposition_n three_o case_n in_o this_o proposition_n the_o first_o case_n construction_n demonstration_n the_o second_o case_n construction_n demonstration_n the_o three_o case_n construction_n demonstration_n construction_n demonstration_n two_o case_n in_o this_o proposition_n first_o case_n demonstration_n the_o second_o câse_n construction_n demonstration_n three_o case_n in_o this_o proposition_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n the_o second_o case_n construction_n demonstration_n first_o corollary_n second_o corollary_n three_o corollary_n this_o proposition_n be_v the_o converse_n of_o the_o former_a construction_n demonstration_n an_o other_o demonstration_n after_o pelitarius_n the_o argument_n of_o this_o book_n first_o definition_n second_o definition_n the_o inscriptition_n and_o circumscription_n of_o rectiline_a figure_n pertain_v only_o to_o regular_a figure_n the_o three_o definition_n the_o four_o definition_n the_o five_o definition_n the_o six_o devition_n seven_o definition_n
demonstration_n demonstration_n the_o circle_n so_o make_v or_o so_o consider_v in_o the_o sphere_n be_v call_v the_o great_a circle_n all_o other_o not_o have_v the_o centre_n of_o the_o sphere_n to_o be_v their_o centre_n alsoâ_n be_v call_v less_o circle_n note_v these_o description_n description_n an_o other_o corollary_n corollary_n an_o other_o corollary_n construction_n construction_n this_o be_v also_o prove_v in_o the_o assumpt_n before_o add_v out_o oâ_n flussas_n note_v what_o a_o great_a or_o great_a circle_n in_o a_o spear_n be_v first_o part_n of_o the_o construction_n noteâ_n noteâ_n you_o know_v full_a well_o that_o in_o the_o superficies_n of_o the_o sphere_n âââly_o the_o circumference_n of_o the_o circle_n be_v but_o by_o thâse_a circumference_n the_o limitation_n and_o assign_v of_o circle_n be_v use_v and_o so_o the_o circumference_n of_o a_o circle_n usual_o call_v a_o circle_n which_o in_o this_o place_n can_v not_o offend_v this_o figure_n be_v restore_v by_o m._n dee_n his_o diligence_n for_o in_o the_o greek_a and_o latin_a euclides_n the_o line_n gl_n the_o line_n agnostus_n and_o the_o line_n kz_o in_o which_o three_o line_n the_o chief_a pinch_n of_o both_o the_o demonstration_n do_v stand_v be_v untrue_o draw_v as_o by_o compare_v the_o studious_a may_v perceive_v note_n you_o must_v imagine_v ãâã_d right_a line_n axe_n to_o be_v perpendicular_a upon_o the_o diameter_n bd_o and_o ce_fw-fr though_o here_o ac_fw-la the_o semidiater_n seem_v to_o be_v part_n of_o ax._n and_o so_o in_o other_o point_n in_o this_o figure_n and_o many_o other_o strengthen_v your_o imagination_n according_a to_o the_o tenor_n of_o construction_n though_o in_o the_o delineation_n in_o plain_a sense_n be_v not_o satisfy_v note_n boy_n equal_a to_o bk_o in_o respect_n of_o m._n dee_n his_o demonstration_n follow_v follow_v note_v âhis_fw-la point_n z_o that_o you_o may_v the_o better_a understand_v m._n dee_n his_o demonstration_n second_o part_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n demonstration_n which_o of_o necessity_n shall_v fall_v upon_o z_o as_o m._n dee_n prove_v it_o and_o his_o proof_n be_v set_v after_o at_o this_o mark_n â_o follow_v i_o dee_n dee_n but_o az_o be_v great_a them_z agnostus_n as_o in_o the_o former_a proposition_n km_o be_v evident_a to_o be_v great_a than_o kg_v so_o may_v it_o also_o be_v make_v manifest_a that_o kz_o do_v neither_o touch_n nor_o cut_v the_o circle_n fgâh_n a_o other_o prove_v that_o the_o line_n ay_o be_v great_a they_o the_o line_n ag._n ag._n this_o as_o a_o assumpt_n be_v present_o prove_v two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o impossibility_n second_o case_n case_n as_o it_o be_v âasiâ_n to_o gather_v by_o the_o assumpt_n put_v after_o the_o secoââ_n of_o this_o booââ_n note_v a_o general_a rule_n the_o second_o part_n of_o the_o problem_n two_o way_n execute_v a_o upright_a cone_n the_o second_o part_n of_o the_o problem_n the_o second_o âaââ_n oâ_n the_o problem_n â_o â_o this_o may_v easy_o be_v demonstrate_v as_o in_o thâ_z 17._o proposition_n the_o section_n of_o a_o sphere_n be_v prove_v to_o be_v a_o circle_n circle_n for_o take_v away_o all_o doubt_n this_o aâ_z a_o lemma_n afterward_o be_v demonstrate_v a_o lemma_n as_o it_o be_v present_o demonstrate_v construction_n demonstration_n the_o second_o part_n of_o the_o problem_n *_o construction_n demonstration_n an_o other_o way_n of_o execute_v this_o problem_n the_o converse_n of_o the_o assumpt_n a_o great_a error_n common_o maintain_v between_o straight_o and_o crooked_a all_o manner_n of_o proportion_n may_v be_v give_v construction_n demonstration_n the_o definition_n of_o a_o circled_a ââapââd_n in_o a_o spâerâ_n construction_n demonstration_n this_o be_v manifest_a if_o you_o consider_v the_o two_o triangle_n rectangle_v hom_o and_o hon_n and_o then_o with_o all_o use_v the_o 47._o of_o the_o first_o of_o euclid_n construction_n demonstration_n construction_n demonstration_n this_o in_o manner_n of_o a_o lemmâ_n be_v present_o prove_v note_v here_o of_o axe_n base_a &_o solidity_n more_o than_o i_o need_n to_o bring_v any_o far_a proof_n for_o note_n note_n i_o say_v half_a a_o circular_a revolution_n for_o that_o suffice_v in_o the_o whole_a diameter_n n-ab_n to_o describe_v a_o circle_n by_o iâ_z it_o be_v move_v ââout_v his_o centre_n q_o etc_n etc_n lib_fw-la 2_o prop_n 2._o de_fw-fr spheâa_n &_o cylindrâ_n note_n note_n a_o rectangle_n parallelipipedon_n give_v equal_a to_o a_o sphere_n give_v to_o a_o sphere_n or_o to_o any_o part_n of_o a_o sphere_n assign_v as_o a_o three_o four_o five_o etc_o etc_o to_o geve_v a_o parallelipipedon_n equal_a side_v colume_n pyramid_n and_o prism_n to_o be_v give_v equal_a to_o a_o sphere_n or_o to_o any_o certain_a part_n thereof_o to_o a_o sphere_n or_o any_o segment_n or_o sector_n of_o the_o same_o to_o geve_v a_o cone_n or_o cylinder_n equal_a or_o in_o any_o proportion_n assign_v far_o use_v of_o spherical_a geometry_n the_o argument_n of_o the_o thirteen_o book_n construction_n demonstration_n demonstration_n the_o assumpt_v prove_v prove_v because_o ac_fw-la be_v suppose_v great_a than_o ad_fw-la therefore_o his_o residue_n be_v less_o than_o the_o residue_n of_o ad_fw-la by_o the_o common_a sentence_n wherefore_o by_o the_o supposition_n db_o be_v great_a than_o âc_z the_o chieâe_a line_n in_o all_o euclides_n geometry_n what_o be_v mean_v here_o by_o a_o section_n in_o one_o only_a poiât_n construction_n demonstration_n demonstration_n note_v how_o ce_fw-fr and_o the_o gnonom_n xop_n be_v prove_v equal_a for_o it_o serve_v in_o the_o converse_n demonstrate_v by_o m._n dee_n here_o next_o after_o this_o proposition_n âthe_v converse_v of_o the_o former_a former_a as_o we_o haâe_z note_v the_o place_n of_o the_o peculiar_a proâe_n there_o âin_a the_o demonstration_n of_o the_o 3._o 3._o therefore_o by_o my_o second_o theorem_a add_v upon_o the_o second_o proposition_n dc_o be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n a._n and_o because_o ac_fw-la be_v big_a than_o cb_o therefore_o dam_n be_v great_a than_o ac_fw-la wherefore_o if_o a_o right_a line_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v demonstrate_v therefore_o by_o my_o second_o theorem_a add_v upon_o the_o second_o proposition_n dc_o be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n a._n and_o because_o ac_fw-la be_v big_a than_o cb_o therefore_o dam_n be_v great_a than_o ac_fw-la wherefore_o if_o a_o right_a line_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v construction_n construction_n though_o i_o say_v perpendicular_a yes_o you_o may_v perceive_v how_o infinite_a other_o position_n will_v serve_v so_o that_o diego_n and_o ad_fw-la make_v a_o angle_n for_o a_o triangle_n to_o have_v his_o side_n proportional_o cut_v etc_n etc_n demonstration_n demonstration_n i_o dee_n this_o be_v most_o evident_a of_o my_o second_o theorem_a add_v to_o the_o three_o proposition_n for_o to_o add_v to_o a_o whole_a line_n a_o line_n equal_a to_o the_o great_a segment_n &_o to_o add_v to_o the_o great_a segment_n a_o line_n equal_a to_o the_o whole_a line_n be_v all_o one_o thing_n in_o the_o line_n produce_v by_o the_o whole_a line_n i_o mean_v the_o line_n divide_v by_o extreme_a and_o mean_a proportion_n this_o be_v before_o demonstrate_v most_o evident_o and_o brief_o by_o m._n dee_n after_o the_o 3._o proposition_n note_n note_v 4._o proportional_a line_n note_v two_o middle_a proportional_n note_v 4._o way_n of_o progression_n in_o the_o proportion_n of_o a_o line_n divide_v by_o extreme_a and_o middle_a proportion_n what_o resolution_n and_o composition_n be_v have_v before_o be_v teach_v in_o the_o begin_n of_o the_o first_o book_n book_n proclus_n in_o the_o greek_a in_o the_o 58._o page_n construction_n demonstration_n two_o case_n in_o this_o proposition_n construction_n thâ_z first_o case_n demonstration_n the_o second_o case_n construction_n demonstration_n construction_n demonstration_n this_o corollary_n be_v the_o 3._o proposition_n of_o theâ4_n â4_o book_n after_o campane_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n construction_n demonstration_n construstion_n demonstration_n constrâyction_n demonstration_n this_o corollary_n be_v the_o 11._o proposition_n of_o the_o 14._o book_n after_o campane_n this_o corollary_n be_v the_o 3._o corollary_n after_o the_o 17._o proposition_n of_o the_o 14_o book_n after_o campane_n campane_n by_o the_o name_n oâ_n a_o pyramid_n both_o here_o &_o iâ_z this_o book_n follow_v understand_v a_o tetrahedron_n an_o other_o construction_n and_o demonstration_n of_o the_o second_o part_n after_o fâussas_n three_o part_n of_o the_o demonstration_n this_o corollary_n be_v the_o 15._o proposition_n of_o the_o 14._o book_n after_o campane_n this_o corollary_n campane_n put_v as_o a_o corollary_n after_o
the_o 17._o proposition_n of_o the_o 14._o book_n construction_n ârist_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 tâe_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 âirst_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 câd_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 ãâã_d part_n ãâã_d which_o thing_n be_v also_o before_o prove_v in_o the_o ãâã_d of_o the_o 32._o of_o the_o âirst_n the_o side_n of_o the_o angle_n of_o the_o inclâââtion_n of_o the_o ãâã_d of_o the_o ãâã_d be_v ãâã_d rational_a the_o side_n of_o the_o angle_n of_o the_o inclination_n of_o the_o ãâã_d âf_n tâe_z ãâ¦ã_o that_o the_o plain_n of_o a_o octohedron_n be_v in_o liâe_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 dâââââhedron_n be_v ãâã_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 supeâficieces_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 pâposâtion_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 aâtâr_a ãâã_d construction_n demonstration_n the_o 5._o proposition_n aâter_a fâussas_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 pââpositionââter_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 âââeth_v âââeth_v in_o the_o corollary_n of_o the_o 17._o of_o the_o tâirtenth_o tâirtenth_o ãâã_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 aâter_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 aâter_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 aâter_v âlussas_n and_o be_v it_o which_o ãâã_d time_n have_v be_v take_v aâ_z grant_v in_o this_o book_n and_o oâce_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 beâore_o note_v demonstration_n demonstration_n in_o the_o 4._o section_n âf_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 âutteth_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 aâter_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 firât_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 âhough_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 ânde_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 ârom_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 diaâââter_n of_o the_o sphere_n be_v doâble_a to_o the_o side_n âf_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 âcl_n âag_n line_n faultesâ_n co_n ãâã_d ãâã_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 aâl_a line_n draw_v all_o righâ_n ãâ¦ã_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 liâes_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 tâe_z centre_n c._n the_o centre_n e_o 2d_o 2_o â_o iââower_a right_n if_o two_o right_a 25_o 2_o 3_o fâââ_n petition_n five_o petition_n 49_o 2_o 7_o 14._o ââ_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 câg_v in_o the_o and_o cgb_n be_v thâ_z  _o  _o  _o in_o stead_n of_o âlussates_n through_o out_o ãâã_d whole_a book_n read_v âlusâas_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 âckd_v 69_o 1_o 18_o the_o whole_a line_n the_o whole_a âigure_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 ãâã_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 âââ_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 magââ_n a_o figure_n like_o unto_o thâs_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
fd._n wherefore_o cones_fw-la &_o cylinder_n consist_v upon_o equal_a base_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o altitude_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 15._o theorem_a the_o 15._o proposition_n in_o equal_a cones_n and_o cylinder_n the_o base_n be_v reciprocal_a to_o their_o altitude_n and_o cones_fw-la and_o cylinder_n who_o base_n be_v reciprocal_a to_o their_o altitude_n be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o these_o cones_fw-la acl_n egn_n or_o these_o cylinder_n axe_n eo_fw-la who_o base_n be_v the_o circle_n abcd_o efgh_o and_o axe_n kl_o and_o mn_v which_o axe_n be_v also_o the_o altitude_n of_o the_o cones_fw-la &_o cylinder_n be_v equal_a the_o one_o to_o the_o other_o cones_n then_o i_o say_v that_o the_o base_n of_o the_o cylinder_n xa_n &_o eo_fw-la be_v reciprokal_n to_o their_o altitude_n that_o be_v that_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o proposition_n so_o the_o altitude_n mn_v to_o the_o altitude_n kl_o for_o the_o altitude_n kl_o be_v either_o equal_a to_o the_o altitude_n mn_a or_o not_o first_o let_v it_o be_v equal_a case_n but_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eq_n but_o cones_fw-la and_o cylinder_n consist_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n be_v by_o the_o 11._o of_o the_o twelve_o wherefore_o the_o base_a abcd_o be_v equal_a to_o the_o base_a efgh_o wherefore_o also_o they_o be_v reciprokal_a as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o but_o now_o suppose_v that_o the_o altitude_n lk_n be_v not_o equal_a to_o the_o altitude_n m._n n_o construction_n but_o let_v the_o altitude_n mn_v be_v great_a and_o by_o the_o 3._o of_o the_o first_o from_o the_o altitude_n mn_v take_v away_o pm_v equal_a to_o the_o altitude_n kl_o so_o that_o let_v the_o line_n pm_n be_v put_v equal_a to_o the_o line_n kl_o and_o by_o the_o point_n p_o let_v there_o be_v extend_v a_o plain_a superficies_n tus_z which_o let_v cut_v the_o cylinder_n eo_fw-la and_o be_v a_o parallel_n to_o the_o two_o opposite_a plain_a superâicieces_n that_o be_v to_o the_o circle_n efgh_o and_o ro._n cylinder_n and_o make_v the_o base_a the_o circle_n efgh_o &_o the_o altitude_n mp_n imagine_v a_o cylinder_n es._n and_o for_o that_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eo_fw-la and_o there_o be_v a_o other_o cylinder_n es_fw-ge therefore_o by_o the_o 7._o of_o the_o five_o as_o the_o cylinder_n axe_n be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o cylinder_n eo_fw-la to_o the_o cylinder_n es._n but_o as_o the_o cylinder_n axe_n be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o base_a abcd_o to_o the_o base_a efgh_o for_o the_o cylinder_n axe_n and_o es_fw-ge be_v under_o one_o and_o the_o self_n same_o altitude_n and_o as_o the_o cylinder_n eo_fw-la be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o altitude_n mn_v to_o the_o altitude_n mp_n for_o cylinder_n consist_v upon_o equal_a base_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o altitude_n wherefore_o as_o the_o base_a abcd_o be_v âo_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n mp_n but_o the_o altitude_n pm_n be_v equal_a to_o the_o altitude_n kl_o wherefore_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o wherefore_o in_o the_o equal_a cylinder_n axe_n and_o eo_fw-la the_o base_n be_v reciprocal_a to_o their_o altitude_n but_o now_o suppose_v that_o the_o base_n of_o the_o cylinder_n axe_n and_o eo_fw-la be_v reciprokal_n to_o their_o altitude_n that_o be_v as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o demonstrate_v then_o i_o say_v that_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eo_fw-la for_o the_o self_n same_o order_n of_o construction_n remain_v for_o that_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o but_o the_o altitude_n kl_o be_v equal_a to_o the_o altitude_n pm_n wherefore_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n pm_n but_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o cylinder_n axe_n to_o the_o cylinder_n es_fw-ge for_o they_o be_v under_o equal_a altitude_n and_o as_o the_o altitude_n mn_o be_v to_o the_o altitude_n pm_n so_o be_v the_o cylinder_n eo_fw-la to_o the_o cylinder_n es_fw-ge by_o the_o 14._o of_o the_o twelve_o wherefore_o also_o as_o the_o cylinder_n axe_n be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o cylinder_n eo_fw-la to_o the_o cylinder_n es._n wherefore_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eo_fw-la by_o the_o 9_o of_o the_o five_o and_o so_o also_o be_v it_o in_o the_o cones_fw-la which_o haââ_n the_o self_n same_o base_n and_o altitude_n with_o the_o cylinder_n wherefore_o in_o equal_a cones_fw-la and_o cylinder_n the_o base_n be_v reciprocal_a to_o their_o altitude_n etc_n etc_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o campane_n and_o flussas_n hitherto_o have_v be_v show_v the_o passion_n and_o propriety_n of_o cones_fw-la and_o cylinder_n who_o altitude_n fall_v perpendicular_o upon_o the_o base_n now_o will_v we_o declare_v that_o cones_fw-la and_o cilinder_n who_o altitude_n fall_v oblique_o upon_o their_o base_n have_v also_o the_o self_n same_o passion_n and_o propriety_n which_o the_o foresay_a cones_fw-la and_o cilinder_n have_v forasmuch_o as_o in_o the_o ten_o of_o this_o book_n it_o be_v say_v that_o every_o cone_n be_v the_o three_o part_n of_o a_o cilind_a have_v one_o and_o the_o self_n same_o base_a &_o one_o &_o the_o self_n same_o altitude_n with_o it_o which_o thing_n be_v demonstrate_v by_o a_o cilind_a give_v who_o base_a be_v cut_v by_o a_o square_n inscribe_v in_o it_o and_o upon_o the_o side_n of_o the_o square_n be_v describe_v isosceles_a triangle_n make_v a_o polygonon_n figure_n and_o again_o upon_o the_o side_n of_o this_o polygonon_n figure_n be_v infinite_o after_o the_o same_o manner_n describe_v other_o isosceles_a triangle_n take_v away_o more_o they_o the_o half_a as_o have_v oftentimes_o be_v declare_v therefore_o it_o be_v manifest_a that_o the_o solid_n set_v upon_o these_o base_n be_v under_o the_o same_o altitude_n that_o the_o cilind_a incline_v be_v and_o be_v also_o include_v in_o the_o same_o cilind_a do_v take_v away_o more_o than_o the_o half_a of_o the_o cilind_a and_o also_o more_o they_o the_o half_a of_o the_o residue_n as_o it_o have_v be_v prove_v in_o erect_a cylinder_n for_o these_o incline_n solid_n be_v under_o equal_a altitude_n and_o upon_o equal_a base_n with_o the_o erect_a solid_n be_v equal_a to_o the_o erect_a solid_n by_o the_o corollary_n of_o the_o _o of_o the_o eleven_o wherefore_o they_o also_o in_o like_a sort_n as_o the_o erect_a take_v away_o more_o than_o the_o half_a if_o therefore_o we_o compare_v the_o incline_v cilind_a to_o a_o cone_n set_v upon_o the_o self_n same_o base_a and_o have_v his_o altitude_n erect_v and_o reason_n by_o a_o argument_n lead_v to_o a_o impossibility_n by_o the_o demonstration_n of_o the_o ten_o of_o this_o book_n we_o may_v prove_v that_o the_o side_v solid_a include_v in_o the_o incline_v cylinder_n be_v great_a than_o the_o triple_a of_o his_o pyramid_n and_o it_o be_v also_o equal_a to_o the_o same_o which_o be_v impossible_a and_o this_o be_v the_o first_o case_n wherein_o it_o be_v prove_v that_o the_o cilind_a not_o be_v equal_a to_o the_o triple_a of_o the_o cone_n be_v not_o great_a than_o the_o triple_a of_o the_o same_o and_o as_o touch_v the_o second_o case_n we_o may_v after_o the_o same_o manner_n conclude_v that_o that_o âided_v solid_a contain_v in_o the_o cylinder_n be_v great_a than_o the_o cylinder_n which_o be_v very_o absurd_a wherefore_o if_o the_o cylinder_n be_v neither_o great_a than_o the_o triple_a of_o the_o cone_n nor_o less_o it_o must_v needs_o be_v equal_a to_o the_o same_o the_o demonstration_n of_o these_o incline_v cylinder_n most_v plain_o follow_v the_o demonstration_n of_o the_o erect_a cylinder_n for_o it_o have_v already_o be_v prove_v that_o pyramid_n and_o side_v solid_n which_o be_v also_o call_v general_o prism_n be_v set_v upon_o equal_a base_n and_o under_o one_o and_o the_o self_n same_o altitude_n whether_o the_o altitude_n be_v erect_v or_o incline_v be_v equal_a the_o one_o to_o the_o other_o namely_o be_v in_o proportion_n as_o their_o base_n be_v by_o
the_o â_o of_o this_o book_n wherefore_o a_o cylinder_n incline_n shall_v be_v triple_a to_o every_o cone_n although_o also_o the_o cone_n be_v erect_v set_v upon_o one_o and_o the_o same_o base_a with_o it_o and_o be_v under_o the_o same_o altitude_n but_o the_o cilind_a erect_v be_v the_o triple_a of_o the_o same_o cone_fw-mi by_o the_o ten_o of_o this_o book_n wherefore_o the_o cilind_a incline_v be_v equal_a to_o the_o cilind_a erect_v being_n both_o set_v upon_o one_o and_o the_o self_n same_o base_a and_o have_v one_o and_o the_o self_n same_o altitude_n the_o same_o also_o come_v to_o pass_v in_o cones_fw-la which_o be_v the_o three_o part_n of_o equal_a cilinder_n &_o therefore_o be_v equal_a the_o one_o to_o the_o other_o wherefore_o according_a to_o the_o eleven_o of_o this_o book_n it_o follow_v that_o cylinder_n and_o cones_fw-la incline_v or_o erect_v be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n be_v for_o forasmuch_o as_o the_o erect_a be_v in_o proportion_n as_o their_o base_n be_v and_o to_o the_o erect_a cilinder_n the_o incline_n be_v equal_a therefore_o they_o also_o shall_v be_v in_o proportion_n as_o their_o base_n be_v and_o therefore_o by_o the_o 12._o of_o this_o book_n like_o cones_fw-la and_o cylinder_n be_v incline_v be_v in_o triple_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o the_o base_n be_v for_o forasmuch_o as_o they_o be_v equal_a to_o the_o erect_a which_o have_v the_o proportion_n by_o the_o 12._o of_o this_o book_n and_o their_o base_n also_o be_v equal_a with_o the_o base_n of_o the_o erect_v therefore_o they_o also_o shall_v have_v the_o same_o proportion_n wherefore_o it_o follow_v by_o the_o 13._o of_o this_o book_n thaâ_z cylinder_n incline_v be_v cut_v by_o a_o plain_a superficies_n parallel_v to_o the_o opposite_a plain_a superficiece_n thereof_o shall_v be_v cut_v according_a to_o the_o proportion_n of_o the_o axe_n for_o suppose_v that_o upon_o one_o and_o the_o self_n same_o base_a âe_n set_v a_o erect_a cylinder_n and_o a_o incline_v cylinder_n be_v both_o under_o one_o and_o the_o self_n same_o altitude_n which_o ãâ¦ã_o a_o plain_a superficies_n parallel_v to_o the_o opposite_a base_n now_o it_o be_v manifest_a that_o the_o section_n of_o the_o one_o cylinder_n be_v equal_a to_o the_o section_n of_o the_o other_o cylinder_n for_o they_o be_v set_v upon_o equal_a base_n and_o under_o one_o and_o the_o self_n same_o altitude_n namely_o between_o the_o parallel_n plain_a superficiece_n and_o their_o axe_n also_o be_v by_o those_o parallel_a plain_n superficiâ_fw-la ãâã_d proportional_o by_o the_o 1st_a of_o âhe_n âlevenâh_o wherefore_o the_o incline_v cylinder_n be_v equal_a to_o the_o erect_a cylinder_n shall_v have_v the_o proportion_n of_o theiâ_z axe_n aâ_z also_o have_v the_o erect_v for_o in_o echâ_n the_o proportion_n of_o the_o axe_n be_v one_o and_o the_o same_o wherefore_o incline_v cones_n and_o cylinder_n be_v set_v upon_o equal_a base_n shall_v by_o the_o 14._o of_o this_o book_n be_v in_o ãâã_d as_o their_o altitude_n ãâ¦ã_o forasmuch_o aâ_z the_o incline_v be_v equal_a to_o the_o erect_a which_o have_v the_o self_n same_o base_n and_o altitude_n and_o the_o erect_v be_v iâ_z proportion_n as_o their_o altitude_n therefore_o the_o incline_n shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_o âhe_z self_n same_o altitude_n which_o be_v common_a to_o each_o namely_o to_o the_o incline_v and_o to_o the_o erect_v and_o therefore_o in_o equal_a cones_fw-la and_o cylinder_n whether_o they_o be_v incline_v or_o erect_v the_o base_n shall_v be_v reciprocal_a proportional_a with_o the_o altitude_n and_o contrariwise_o by_o the_o 15._o of_o this_o book_n for_o forasmuch_o as_o we_o have_v oftentimes_o show_v that_o the_o incline_v cones_fw-la and_o cylinder_n be_v equal_a to_o the_o erect_a have_v the_o self_n same_o base_n and_o altitude_n with_o they_o and_o the_o erect_v unto_o who_o the_o incline_v be_v equal_a haâe_z their_o base_n reciprocal_a proportional_o with_o their_o altitude_n therefore_o it_o follow_v that_o the_o incline_v be_v equal_a to_o the_o erect_a have_v also_o their_o base_n and_o altitude_n which_o be_v common_a to_o each_o reciprocal_a proportional_a likewise_o if_o theiâ_z altitude_n &_o base_n be_v reciprocal_a proportional_a they_o themselves_o also_o shall_v be_v equal_a for_o that_o they_o be_v equal_a to_o the_o erect_a cylinder_n and_o cones_fw-la set_v upon_o the_o same_o base_n and_o be_v under_o the_o same_o altitude_n which_o erect_a cylinder_n be_v equal_a the_o one_o to_o the_o other_o by_o the_o same_o 15._o of_o this_o book_n wherefore_o we_o may_v conclude_v that_o those_o passion_n &_o propriety_n which_o in_o this_o twelve_o book_n have_v be_v prove_v to_o be_v in_o cones_fw-la and_o cylinder_n who_o altitude_n be_v erect_v perpendicular_o to_o the_o ãâ¦ã_o set_v oblique_o upoâ_n their_o base_n howbeit_o this_o be_v to_o be_v note_v that_o such_o incline_v cones_fw-la or_o cylinder_n be_v not_o perfect_a round_o as_o be_v the_o erect_v so_o that_o if_o they_o be_v cut_v by_o a_o plain_a superficies_n pass_v at_o right_a angle_n with_o their_o altitude_n this_o section_n be_v a_o conical_a section_n which_o be_v call_v ellipsis_n and_o shall_v not_o describe_v in_o their_o superficies_n a_o circle_n as_o it_o do_v in_o erect_a cylinder_n &_o cones_fw-la but_o a_o certain_a figure_n who_o less_o diameter_n be_v in_o cylinder_n equal_a to_o the_o dimetient_fw-la of_o the_o base_a that_o be_v be_v one_o and_o the_o same_o with_o it_o and_o the_o same_o thing_n happen_v also_o in_o cones_fw-la incline_v be_v cut_v after_o the_o same_o manner_n the_o 1._o problem_n the_o 16._o proposition_n two_o circle_n have_v both_o one_o and_o the_o self_n same_o centre_n be_v give_v to_o inscribe_v in_o the_o great_a circle_n a_o polygonon_n figure_n which_o shall_v consist_v of_o equal_a and_o even_a side_n and_o shall_v not_o touch_v the_o superficies_n of_o the_o less_o circle_n svppose_v that_o there_o be_v two_o circle_n abcd_o and_o efgh_a have_v one_o &_o the_o self_n same_o centre_n namely_o k._n it_o be_v require_v in_o the_o great_a circle_n which_o let_v be_v abcd_o to_o inscribe_v a_o polygonon_n figure_n which_o shall_v be_v of_o equal_a and_o even_a side_n and_o not_o touch_v the_o circle_n efgh_o draw_v by_o the_o centre_n king_n a_o right_a line_n bd._n construction_n and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n g_o raise_v up_o unto_o the_o right_a line_n bd_o a_o perpendicular_a line_n agnostus_n and_o extend_v it_o to_o the_o point_n c._n wherefore_o the_o line_n ac_fw-la touch_v the_o circle_n efgh_o by_o the_o 15._o of_o the_o three_o now_o therefore_o if_o by_o the_o 30._o of_o the_o three_o we_o divide_v the_o circumference_n bad_a into_o two_o equal_a part_n and_o again_o the_o half_a of_o that_o into_o two_o equal_a part_n and_o thus_o do_v continual_o we_o shall_v by_o the_o corollary_n of_o the_o 1._o of_o the_o ten_o at_o the_o length_n leave_v a_o certain_a circumference_n less_o than_o the_o circumference_n ad._n let_v the_o circumference_n leave_v be_v ld_n and_o from_o the_o point_n l._n draw_v by_o the_o 12._o of_o the_o first_o unto_o the_o line_n bd_o a_o perpendiculare_a line_n demonstration_n lm_o dee_n and_o extend_v it_o to_o the_o point_n n._n and_o draw_v these_o right_a line_n ld_n and_o dn_o and_o forasmuch_o as_o the_o angle_n dml_n and_o dmn_v be_v right_a angle_n therefore_o by_o the_o 3._o of_o the_o three_o the_o right_a line_n bd_o divide_v the_o right_a line_n ln_o into_o two_o equal_a part_n in_o the_o point_n m._n wherefore_o by_o the_o 4._o of_o the_o first_o the_o rest_n of_o the_o side_n of_o the_o triangle_n dml_n and_o dmn_v namely_o the_o line_n dl_o and_o dn_o shall_v be_v equal_v and_o forasmuch_o as_o the_o line_n ac_fw-la be_v a_o parallel_n to_o the_o ln_o by_o the_o 28._o of_o the_o first_o but_o ac_fw-la touch_v the_o circle_n efgh_o wherefore_o the_o line_n lm_o touch_v not_o the_o circle_n efgh_o and_o much_o less_o do_v the_o line_n ld_n and_o dn_o touch_v the_o circle_n efgh_o if_o therefore_o there_o be_v apply_v right_a line_n equal_a to_o the_o line_n ld_n continual_o into_o the_o circle_n abcd_o by_o the_o 1._o of_o the_o four_o there_o shall_v be_v describe_v in_o the_o circle_n abcd_o a_o polygonon_n figure_n which_o shall_v be_v of_o equal_a and_o figure_n even_o side_n and_o shall_v not_o touch_v the_o less_o circle_n namely_o efgh_o by_o the_o 14._o of_o the_o three_o or_o by_o the_o 29._o which_o be_v require_v to_o be_v do_v ¶_o corollary_n hereby_o it_o be_v manifest_a that_o a_o perpendicular_a line_n draw_v from_o the_o point_n l_o to_o the_o line_n bd_o touch_v not_o