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arise_v the_o four_o rate_n or_o comparison_n 7._o if_o a_o right_a line_n be_v cut_v into_o two_o equal_a part_n and_o otherwise_o the_o oblong_a of_o the_o unequal_a segment_n with_o the_o quadrate_n of_o the_o segment_n between_o they_o be_v equal_a to_o the_o quadrate_n of_o the_o bisegment_n 5_o p_o ij_o the_o three_o section_n do_v follow_v from_o whence_o the_o five_o reason_n arise_v 8._o if_o a_o right_a line_n be_v cut_v into_o equal_a part_n and_o continue_v the_o oblong_v make_v of_o the_o continue_a and_o the_o continuation_n with_o the_o quadrate_n of_o the_o bisegment_n or_o half_n be_v equal_a to_o the_o quadrate_n of_o the_o line_n compound_v of_o the_o bisegment_n and_o continuation_n 6_o p_o ij_o from_o hence_o arise_v the_o mesographus_n or_o mesolabus_n of_o heron_n the_o mechanic_n so_o name_v of_o the_o invention_n of_o two_o line_n continual_o proportional_a between_o two_o line_n give_v whereupon_o arise_v the_o deliacke_a problem_n which_o trouble_v apollo_n 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right_a line_n give_v the_o difference_n of_o the_o right_a line_n from_o the_o midst_n of_o the_o conterminall_a side_n of_o the_o say_a quadrate_n make_v above_o the_o same_o half_a shall_v be_v the_o great_a segment_n of_o the_o line_n give_v proportional_o cut_v 11_o p_o ij_o or_o thus_o if_o a_o square_a be_v make_v of_o a_o right_a line_n give_v the_o difference_n of_o a_o right_a line_n draw_v from_o the_o angle_n of_o the_o square_n make_v unto_o the_o midst_n of_o the_o next_o side_n above_o the_o half_a of_o the_o side_n shall_v be_v the_o great_a segment_n of_o the_o line_n give_v be_v proportional_o cut_v h._n for_o of_o y_o a_o let_v the_o quadrate_n a_fw-fr y_fw-fr s_o r_o be_v make_v and_o let_v s_o r_o be_v continue_v unto_o l._n now_o by_o the_o 8_o e_z xiij_o the_o oblong_n of_o o_fw-fr y_fw-fr and_o a_o y_z with_o the_o quadrate_n of_o you_o a_o be_v equal_a to_o the_o quadrate_n of_o u._fw-mi y_fw-mi that_o be_v by_o the_o construction_n of_o u._fw-mi e_fw-es and_o therefore_o by_o the_o 9_o e_fw-la xij_o it_o be_v equal_a to_o the_o quadrate_n e_o a_o and_o a_o u._fw-mi take_v away_o from_o each_o side_n the_o common_a oblong_v a_o l_o and_o the_o quadrate_n y_fw-fr r_o shall_v be_v equal_a to_o the_o oblong_n r_o i._n therefore_o the_o three_o right_a line_n e_z a_o a_o r_o and_o r_o e_o by_o the_o 8_o e_fw-la xij_o be_v continual_a proportional_a and_o the_o right_a line_n a_o e_o be_v cut_v proportional_o therefore_o 4_o if_o a_o right_a line_n cut_v proportional_o be_v continue_v with_o the_o great_a segment_n the_o whole_a shall_v be_v cut_v proportional_o and_o the_o great_a segment_n shall_v be_v the_o line_n give_v 5_o p_o xiij_o as_o in_o the_o same_o example_n the_o right_a line_n o_o y_fw-fr be_v continue_v with_o the_o great_a segment_n and_o the_o oblong_a of_o the_o whole_a and_o the_o lesser_a segment_n be_v equal_a to_o the_o quadrate_n of_o the_o great_a and_o thus_o one_o may_v by_o infinite_o proportional_o cut_v increase_n 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as_o be_v a_o o_o the_o gnomon_n l_o m_z n_o shall_v be_v four_o time_n so_o much_o as_o be_v u._fw-mi a_o who_o quadruple_a also_o by_o the_o 14._o e_fw-la xij_o be_v a_o v_o therefore_o it_o be_v equal_a to_o the_o gnomon_n now_o a_o j_o be_v equal_a to_o a_o e_o therefore_o it_o be_v the_o double_a also_o of_o a_o o_o that_o be_v of_o a_o y_o and_o therefore_o by_o the_o 24._o e_fw-la x._o it_o be_v the_o double_a of_o a_o t_o and_o therefore_o it_o be_v equal_a to_o the_o compliment_n i_o y_fw-fr and_o y_z s_z therefore_o the_o other_o diagonall_a y_o r_o be_v equal_a to_o the_o other_o rectangle_n i_o v._n wherefore_o by_o the_o 8_o e_fw-la xij_o as_o e_z v_o that_o be_v a_o e_z be_v to_z y_z t_o that_o
be_v a_o i_o so_o be_v a_o i_o unto_z i_z e_z wherefore_o by_o the_o â—_o e_o a_o e_z be_v proportional_a cut_n and_o the_o great_a segment_n be_v a_o i_o the_o same_o remain_v the_o other_o propriety_n of_o the_o quintuple_a do_v follow_v 6_o the_o lesser_a segment_n continue_v to_o the_o half_a of_o the_o great_a be_v of_o power_n quintuple_a to_o the_o same_o half_a è_fw-mi 3_o p_o x_o iij._o the_o rate_n of_o the_o triple_a follow_v 7_o the_o whole_a line_n and_o the_o lesser_a segment_n be_v in_o power_n treble_a unto_o the_o great_a è_fw-it 4_o p_o xiij_o 8_o a_o obliquangled_a parallelogramme_n be_v either_o a_o rhombus_fw-la or_o a_o rhomboide_n 9_o a_o rhombus_fw-la be_v a_o obliquangled_a equilater_n parallelogramme_n 32_o dj_o it_o be_v otherwise_o of_o some_o call_v a_o diamond_n 10_o a_o rhomboide_n be_v a_o obliquangled_a parallelogramâ—e_n not_o equilater_n 33._o dj_o and_o a_o rhomboide_n be_v so_o oppose_v to_o a_o oblong_a as_o a_o rhombus_fw-la be_v to_o a_o quadrate_n and_o the_o rhomboide_n be_v so_o call_v as_o one_o will_v say_v rhombuslike_n although_o beside_o the_o inequality_n of_o the_o angle_v it_o have_v nothing_o like_o to_o a_o rhombus_fw-la a_o example_n of_o measure_v of_o a_o rhombus_fw-la be_v thus_o 11_o a_o trapezium_fw-la be_v a_o quadrangle_n not_o parallelogramme_n 34._o dj_o the_o example_n both_o of_o the_o figure_n and_o of_o the_o measure_n of_o the_o same_o let_v these_o be_v therefore_o triangulate_v quadrangle_v be_v of_o this_o sort_n 12_o a_o multangle_n be_v a_o figure_n that_o be_v comprehend_v of_o more_o than_o four_o right_a line_n 23._o dj_o by_o this_o general_a name_n all_o other_o sort_n of_o right_n line_v figure_n hereafter_o follow_v be_v by_o euclid_n comprehend_v as_o be_v the_o quinquangle_n sexangle_v septangle_n and_o such_o like_a innumerable_a take_v their_o name_n of_o the_o number_n of_o their_o angle_n in_o every_o kind_n of_o multangle_n there_o be_v one_o ordinate_a as_o we_o have_v 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there_o be_v consider_v no_o more_o but_o the_o motion_n the_o point_n in_o the_o end_n of_o the_o ray_n describe_v the_o periphery_a here_o be_v consider_v the_o motion_n of_o the_o whole_a ray_n make_v the_o whole_a plot_n contain_v within_o the_o periphery_n a_o circle_n of_o all_o plain_n be_v the_o most_o ordinate_a figure_n as_o be_v before_o teach_v at_o the_o 10_o e_fw-la iiij_o 2_o circle_n be_v as_o the_o quadrate_n or_o square_n make_v of_o their_o diameter_n 2_o p._n x_o ij_o therefore_o 3._o the_o diameter_n be_v as_o their_o periphery_n pappus_n 5_o l_o x_o j_o and_o 26_o the_o 18._o as_o here_o thou_o see_v in_o a_o e_fw-la and_o i_z o._n 4._o circular_a geometry_n be_v either_o in_o line_n or_o in_o the_o segment_n of_o a_o circle_n this_o partition_n of_o the_o subject_a matter_n howsoever_o be_v take_v for_o the_o distinguish_n and_o sever_n with_o some_o light_n a_o matter_n somewhat_o confuse_v and_o indeed_o concern_v line_n the_o consideration_n of_o secant_v be_v 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the_o diameter_n be_v the_o chief_a for_o it_o show_v the_o centre_n and_o also_o the_o reason_n or_o proportion_n of_o all_o other_o inscript_n therefore_o the_o invention_n and_o make_n of_o the_o diameter_n of_o a_o circle_n be_v first_o to_o be_v teach_v 7._o if_o a_o inscript_n do_v cut_v into_o two_o equal_a part_n another_o inscript_n perpendicular_o it_o be_v the_o diameter_n of_o the_o circle_n and_o the_o midst_n of_o it_o be_v the_o centre_n 1_o p_o iij._o the_o cause_n be_v the_o same_o which_o be_v of_o the_o 5_o e_z x_o i_o because_o the_o inscript_n cut_v into_o half_n if_o for_o the_o side_n of_o the_o inscribe_v rectangle_n and_o it_o do_v subtend_v the_o periphery_n cut_v also_o into_o two_o part_n by_o the_o which_o both_o the_o inscript_n and_o periphery_a also_o be_v in_o like_a manner_n cut_v into_o two_o equal_a part_n therefore_o the_o right_a line_n thus_o half_v in_o the_o diameter_n of_o the_o rectangle_n but_o that_o the_o middle_n of_o the_o circle_n be_v the_o centre_n be_v manife_a out_o of_o the_o 7_o e_o v_o and_o 29_o e_fw-la iiij_o euclid_n think_v better_a of_o impossibile_fw-it than_o he_o do_v of_o the_o cause_n and_o thus_o he_o force_v it_o for_o if_o y_o be_v not_o the_o centre_n but_o s_o the_o part_n must_v be_v equal_a to_o the_o whole_a for_o the_o triangle_n a_o o_o s_o shall_v be_v equilater_n to_o the_o triangle_n e_o o_fw-fr s._n for_o a_o o_o oe_o be_v equal_a by_o the_o grant_n item_n be_v a_o and_o s_z e_z be_v the_o ray_n of_o the_o circle_n and_o s_o o_o be_v common_a to_o both_o the_o triangle_n therefore_o by_o the_o 1_o e_fw-la seven_o the_o angle_v no_o each_o side_n at_o o_o be_v equal_a and_o by_o the_o 13_o e_o v_o they_o be_v both_o right_a angle_n therefore_o s_o o_o e_o be_v a_o right_a angle_n it_o be_v therefore_o equal_a by_o the_o grant_n to_o the_o right_a angle_n y_fw-fr o_fw-fr e_fw-es that_o be_v the_o part_n be_v equal_a to_o the_o whole_a which_o be_v impossible_a wherefore_o y_o be_v not_o the_o centre_n the_o same_o will_v fall_v out_o of_o any_o other_o point_n whatsoever_o â—ut_v of_o y._n therefore_o 8._o if_o two_o râ—ght_a line_n do_v perpendicular_o half_a two_o inscript_n the_o meeting_n of_o these_o two_o bisecant_v shall_v be_v the_o centre_n of_o the_o circle_n è_fw-mi 25_o p_o iij._o and_o one_o may_n 9_o draw_v a_o periphery_a by_o three_o point_n which_o do_v not_o fall_v in_o a_o right_a line_n 10._o if_o a_o diameter_n do_v half_o a_o inscript_n that_o be_v nâ—t_v a_o diameter_n it_o do_v cut_v it_o perpendicular_o and_o contrariwise_o 3_o p_o iij._o 11._o if_o inscript_n which_o be_v not_o diameter_n do_v cut_v one_o another_o the_o segment_n shall_v be_v unequal_a 4_o p_o iij._o but_o rate_n have_v be_v hitherto_o in_o the_o part_n of_o inscript_n proportion_n in_o the_o same_o part_n follow_v 12_o if_o two_o inscript_n do_v cut_v one_o another_o the_o rectangle_n of_o the_o segment_n of_o the_o one_o be_v equal_a to_o the_o rectangle_n of_o the_o segment_n of_o the_o other_o 35_o p_o iij._o and_o this_o be_v the_o comparison_n of_o the_o part_n inscript_n the_o rate_n of_o whole_a inscript_n do_v follow_v the_o which_o whole_a one_o diameter_n do_v make_v 13_o inscript_n be_v equal_a distant_a from_o the_o centre_n unto_o which_o the_o perpendicular_o from_o the_o centre_n be_v equal_a 4_o d_o iij._o 14._o if_o inscript_n be_v equal_a they_o be_v equal_o distant_a from_o the_o centre_n and_o contrariwise_o 13_o p_o iij._o the_o diameter_n in_o the_o same_o circle_n by_o the_o 28_o e_o iiijâ—_n be_v equal_a and_o they_o be_v equal_o distant_a from_o the_o centre_n see_v they_o be_v by_o the_o centre_n or_o rather_o be_v no_o whit_n at_o all_o
in_o a_o periphery_n and_o do_v differ_v only_o in_o base_a 14_o the_o angle_n in_o opposite_a section_n be_v equal_a to_o two_o right_a angle_n 22._o p_o iij._o the_o reason_n or_o rate_n of_o a_o section_n be_v thus_o the_o similitude_n do_v follow_v 15_o if_o section_n do_v receive_v or_o contain_v equal_v angle_n they_o be_v alike_o e_fw-la 10._o d_o iij._o 16_o if_o like_a section_n be_v upon_o a_o equal_a base_a they_o be_v equal_a and_o contrariwise_o 23,24_o p_o iij._o in_o the_o first_o figure_n let_v the_o base_a be_v the_o same_o and_o if_o they_o shall_v be_v say_v to_o unequal_a section_n and_o one_o of_o they_o great_a than_o another_o the_o angle_n in_o that_o a_o o_o e_o shall_v be_v less_o than_o the_o angle_n a_o i_o e_o in_o the_o lesser_a section_n by_o the_o 16_o e_fw-la uj._o which_o notwithstanding_o by_o the_o grant_n be_v equal_a in_o the_o second_o figure_n if_o one_o section_n be_v put_v upon_o another_o it_o will_v agree_v with_o it_o otherwise_o against_o the_o first_o part_n like_o section_n upon_o the_o same_o base_a shall_v not_o be_v equal_a but_o congruency_n be_v here_o sufficient_a by_o the_o former_a two_o proposition_n and_o by_o the_o 9_o e_fw-la x_o v._n one_o may_v find_v a_o section_n like_a unto_o another_o assign_v or_o else_o from_o a_o circle_n give_v to_o cut_v off_o one_o like_a unto_o it_o 17_o a_o angle_n of_o a_o section_n be_v that_o which_o be_v comprehend_v of_o the_o bound_n of_o a_o section_n 18_o a_o section_n be_v either_o a_o semicircle_n or_o that_o which_o be_v unequal_a to_o a_o semicircle_n a_o section_n be_v two_o fold_n a_o semicircle_n to_o wit_n when_o it_o be_v cut_v by_o the_o diameter_n or_o unequal_a to_o a_o semicircle_n when_o it_o be_v cut_v by_o a_o line_n lesser_a than_o the_o diameter_n 19_o a_o semicircle_n be_v the_o half_a section_n of_o a_o circle_n or_o it_o be_v that_o which_o be_v make_v the_o diameter_n therefore_o 20_o a_o semicircle_n be_v comprehend_v of_o a_o periphery_a and_o the_o diameter_n 18_o dj_o 21_o the_o angle_n in_o a_o semicircle_n be_v a_o right_a angle_n the_o angle_n of_o a_o semicircle_n be_v lesser_a than_o a_o rectilineall_a right_a angle_n but_o great_a than_o any_o acute_a angle_n the_o angle_n in_o a_o great_a section_n be_v lesser_a than_o a_o right_a angle_n of_o a_o great_a it_o be_v a_o great_a in_o a_o lesser_a it_o be_v great_a of_o a_o lesser_a it_o be_v lesser_a ê_fw-la 31_o and_o 16._o p_o iij._o or_o thus_o the_o angle_n in_o a_o semicircle_n be_v a_o right_a angle_n the_o angle_n of_o a_o semicircle_n be_v less_o than_o a_o right_n rightline_v angle_n but_o great_a than_o any_o acute_a angle_n the_o angle_n in_o the_o great_a section_n be_v less_o than_o a_o right_a angle_n the_o angle_n of_o the_o great_a section_n be_v great_a than_o a_o right_a angle_n the_o angle_n in_o the_o lesser_a section_n be_v great_a than_o a_o right_a angle_n the_o angle_n of_o the_o lesser_a section_n be_v lesser_a than_o a_o right_a angle_n h._n the_o second_o part_n that_o the_o angle_n of_o a_o semicircle_n be_v lesser_a than_o a_o right_a angle_n be_v manifest_a out_o of_o that_o because_o it_o be_v the_o part_n of_o a_o right_a angle_n for_o the_o angle_n of_o the_o semicircle_n a_o i_o e_o be_v a_o part_n of_o the_o rectilineall_a right_a angle_n a_o i_o u._n the_o three_o part_n that_o it_o be_v great_a than_o any_o acute_a angle_n be_v manifest_a out_o of_o the_o 23._o e_fw-la x_o v._n for_o otherwise_o a_o tangent_fw-la be_v not_o on_o the_o same_o part_n one_o only_a and_o no_o more_o the_o four_o part_n be_v thus_o make_v manifest_a the_o angle_n at_o i_o in_o the_o great_a section_n a_o e_fw-it i_fw-it be_v lesser_a than_o a_o right_a angle_n because_o it_o be_v in_o the_o same_o triangle_n a_o e_fw-it i_fw-it which_o at_o a_o be_v right_a angle_n and_o if_o neither_o of_o the_o shank_n be_v by_o the_o centre_n notwithstanding_o a_o angle_n may_v be_v make_v equal_a to_o the_o assign_a in_o the_o same_o section_n the_o five_o be_v thus_o the_o angle_n of_o the_o great_a section_n e_fw-la a_o i_o be_v great_a than_o a_o right_a angle_n because_o it_o contain_v a_o rightangle_n the_o six_o be_v thus_o the_o angle_n a_o o_o e_o in_o a_o lesser_a section_n be_v great_a than_o a_o right_a angle_n by_o the_o 14_o e_fw-la x_o five_o i_o because_o that_o which_o be_v in_o the_o opposite_a section_n be_v lesser_a than_o a_o right_a angle_n the_o seven_o be_v thus_o the_o angle_n e_o a_o o_o be_v lesser_a than_o a_o rightangle_n because_o it_o be_v part_n of_o a_o right_a angle_n to_o wit_n of_o the_o outter_n angle_n if_o i_o a_o be_v draw_v out_o at_o length_n and_o thus_o much_o of_o the_o angle_n of_o a_o circle_n of_o all_o which_o the_o most_o effectual_a and_o of_o great_a power_n and_o use_n be_v the_o angle_n in_o a_o semicircle_n and_o therefore_o it_o be_v not_o without_o cause_n so_o often_o mention_v of_o aristotle_n this_o geometry_n therefore_o of_o aristotle_n let_v we_o somewhat_o more_o full_o open_a and_o declare_v for_o from_o hence_o do_v arise_v many_o thing_n therefore_o 22_o if_o two_o right_a line_n joint_o bound_v with_o the_o diameter_n of_o a_o circle_n be_v joint_o bound_v in_o the_o periphery_n they_o do_v make_v a_o right_a angle_n or_o thus_o if_o two_o right_a line_n have_v the_o same_o term_n with_o the_o diameter_n be_v join_v together_o in_o one_o point_n of_o the_o circomference_n they_o make_v a_o right_a angle_n h._n this_o corollary_n be_v draw_v out_o of_o the_o first_o part_n of_o the_o former_a element_n where_o it_o be_v say_v that_o a_o angle_n in_o a_o semicircle_n be_v a_o right_a angle_n and_o 23_o if_o a_o infinite_a right_a line_n be_v cut_v of_o a_o periphery_a of_o a_o external_a centre_n in_o a_o point_n assign_v and_o contingent_a and_o the_o diameter_n be_v draw_v from_o the_o contingent_a point_n a_o right_a line_n from_o the_o point_n assign_v knit_v it_o with_o the_o diameter_n shall_v be_v perpendicular_a unto_o the_o infinite_a line_n give_v let_v the_o infinite_a right_a line_n be_v a_o e_fw-es from_o who_o point_n a_o a_o perpendicular_a be_v to_o be_v raise_v and_o 24_o if_o a_o right_a line_n from_o a_o point_n give_v make_v a_o acute_a angle_n with_o a_o infinite_a line_n be_v make_v the_o diameter_n of_o a_o periphery_a cut_v the_o infinite_a a_o right_a line_n from_o the_o point_n assign_v knit_v the_o segment_n shall_v be_v perpendicular_a upon_o the_o infinite_a line_n as_o in_o the_o same_o example_n have_v a_o external_a point_n give_v let_v a_o perpendicular_a unto_o the_o infinite_a right_a line_n a_o e_o be_v seek_v let_v the_o right_a line_n i_o o_o e_o be_v make_v the_o diameter_n of_o the_o peripherie_n and_o withal_o let_v it_o make_v with_o the_o infinite_a right_a line_n giyen_v a_o acute_a angle_n in_o e_o from_o who_o bisection_n for_o the_o centre_n let_v a_o periphery_n cut_v the_o infinite_a etc._n etc._n and_o 25_o if_o of_o two_o right_a line_n the_o great_a be_v make_v the_o diameter_n of_o a_o circle_n and_o the_o lesser_a joint_o bound_v with_o the_o great_a an_o inscribe_v be_v knit_v together_o the_o power_n of_o the_o great_a shall_v be_v more_o than_o the_o power_n of_o the_o lesser_a by_o the_o quadrate_n of_o that_o which_o knit_v they_o both_o together_o ad_fw-la 13_o p._n x._o 26_o if_o a_o right_a line_n continue_v or_o continual_o make_v of_o two_o right_a line_n give_v be_v make_v the_o diameter_n of_o a_o circle_n the_o perpendicular_a from_o the_o point_n of_o their_o continuation_n unto_o the_o periphery_n shall_v be_v the_o mean_a proportional_a between_o the_o two_o line_n give_v 13_o p_o uj._o so_o if_o the_o side_n of_o a_o quadrate_n of_o 10._o foot_n content_a be_v seek_v let_v the_o side_n 1_o foot_n and_o 10_o foot_n a_o oblong_a equal_a to_o that_o same_o quadrate_n be_v continue_v the_o mean_a proportional_a shall_v be_v the_o side_n of_o the_o quadrate_n that_o be_v the_o power_n of_o it_o shall_v be_v 10._o foot_n the_o reason_n of_o the_o angle_n in_o opposite_a section_n do_v follow_v 27_o the_o angle_n in_o opposite_a section_n be_v equal_a in_o the_o alterne_a angle_n make_v of_o the_o secant_fw-la and_o touch_v line_n 32._o p_o iij._o as_o let_v the_o unequal_a section_n be_v e_o i_o o_o and_o e_z a_o o_o the_o tangent_fw-la let_v it_o be_v u._fw-mi e_fw-es y_fw-es and_o the_o angle_n in_o the_o opposite_a section_n e_z a_o o_o and_o e_z i_z o._n i_o say_v they_o be_v equal_a in_o the_o alterne_a angle_n of_o the_o secant_fw-la and_o touch_v line_n o_fw-mi e_fw-es y_fw-es and_o o_o e_o u._n first_o that_o which_o be_v at_o a_o be_v equal_a to_o the_o
alterne_n o_fw-fr e_fw-es y_fw-es because_o also_o three_o angle_n o_o e_o y_fw-es o_z e_z a_o and_o a_o e_z u._fw-mi be_v equal_a to_o two_o right_a angle_n by_o the_o 14_o e_fw-la v_o unto_o which_o also_o be_v equal_a the_o three_o angle_n in_o the_o triangle_n a_o e_o o_o by_z the_o 13_o e_z uj._o from_o three_o equal_n take_v away_o the_o two_o right_a angle_n a_o u._fw-mi e_fw-it and_o a_o o_o e_o for_o a_o o_o e_o be_v a_o right_a angle_n by_o the_o 21_o e_z because_o it_o be_v in_o a_o semicircle_n take_v away_o also_o the_o common_a angle_n a_o e_o o_o and_o the_o remainder_n e_o a_fw-fr o_o and_o o_o e_fw-it y_fw-es alterne_a angle_n shall_v be_v equal_a therefore_o 28_o if_o at_o the_o end_n of_o a_o right_a line_n give_v a_o right_n line_v angle_n be_v make_v equal_a to_o a_o angle_n give_v and_o from_o the_o top_n of_o the_o angle_n now_o make_v a_o perpendicular_a unto_o the_o other_o side_n do_v meet_v with_o a_o perpendicular_a draw_v from_o the_o midst_n of_o the_o line_n give_v the_o meeting_n shall_v be_v the_o centre_n of_o the_o circle_n describe_v by_o the_o equal_v angle_n in_o who_o opposite_a section_n the_o angle_n upon_o the_o line_n give_v shall_v be_v make_v equal_a to_o the_o assign_v è_fw-mi 33_o p_o iij._o and_o 29_o if_o the_o angle_n of_o the_o secant_fw-la and_o touch_v line_n be_v equal_a to_o a_o assign_a rectilineall_a angle_n the_o angle_n in_o the_o opposite_a section_n shall_v likewise_o be_v equal_a to_o the_o same_o 34._o piij._n of_o geometry_n the_o seventeen_o book_n of_o the_o adscription_n of_o a_o circle_n and_o triangle_n hitherto_o we_o have_v speak_v of_o the_o geometry_n of_o rectilineall_a plain_n and_o of_o a_o circle_n now_o follow_v the_o adscription_n of_o both_o this_o be_v general_o define_v in_o the_o first_o book_n 12_o e._n now_o the_o periphery_a of_o a_o circle_n be_v the_o bind_v thereof_o therefore_o a_o rectilineall_a be_v inscribe_v into_o a_o circle_n when_o the_o periphery_n do_v touch_v the_o angle_n of_o it_o 3_o d_o iiij_o it_o be_v circumscribe_v when_o it_o be_v touch_v of_o every_o side_n by_o the_o periphery_a 4_o d_o iij._o 1._o if_o a_o rectilineall_a ascribe_v unto_o a_o circle_n be_v a_o equilater_n it_o be_v equiangle_n of_o the_o circumscript_n it_o be_v likewise_o true_a if_o the_o circumscript_n be_v understand_v to_o be_v a_o circle_n for_o the_o perpendicular_o from_o the_o centre_n a_o unto_o the_o side_n of_o the_o circumscript_n by_o the_o 9e_n xij_o shall_v make_v triangle_n on_o each_o side_n equilater_n &_o equiangl_n by_o draw_v the_o semidiameter_n unto_o the_o corner_n as_o in_o the_o same_o example_n 2._o it_o be_v equal_a to_o a_o triangle_n of_o equal_a base_a to_o the_o perimeter_n but_o of_o height_n to_o the_o perpendicular_a from_o the_o centre_n to_o the_o side_n as_o here_o be_v manifest_a by_o the_o 8_o e_fw-la seven_o for_o there_o be_v in_o one_o triangle_n three_o triangle_n of_o equal_a height_n the_o same_o will_v fall_v out_o in_o a_o triangulate_a as_o here_o in_o a_o quadrate_n for_o here_o shall_v be_v make_v four_o triangle_n of_o equal_a height_n last_o every_o equilater_n rectilineall_a ascribe_v to_o a_o circle_n shall_v be_v equal_a to_o a_o triangle_n of_o base_a equal_a to_o the_o perimeter_n of_o the_o adscript_n because_o the_o perimeter_n contain_v the_o base_n of_o the_o triangle_n into_o the_o which_o the_o rectilineall_a be_v resolve_v 3._o like_a rectilineall_n inscribe_v into_o circle_n be_v one_o to_o another_o as_o the_o quadrate_n of_o their_o diameter_n 1_o p._n x_o i_o i_o in_o like_a triangulate_v see_v by_o the_o 4_o e_fw-la x_o they_o may_v be_v resolve_v into_o like_a triangle_n the_o same_o will_v fall_v out_o therefore_o 4._o if_o it_o be_v as_o the_o diameter_n of_o the_o circle_n be_v unto_o the_o side_n of_o rectilineall_a inscribe_v so_o the_o diameter_n of_o the_o second_o circle_n be_v unto_o the_o side_n of_o the_o second_o rectilineall_a inscribe_v and_o the_o several_a triangle_n of_o the_o inscript_n be_v alike_o and_o likely_a situate_a the_o rectilineall_n inscribe_v shall_v be_v alike_o and_o likely_a situate_a this_o euclid_n do_v thus_o assume_v at_o the_o 2_o p_o xij_o and_o indeed_o as_o it_o seem_v out_o of_o the_o 18_o p_o uj._o both_o which_o be_v contain_v in_o the_o 23_o e_fw-la iiij_o and_o therefore_o we_o also_o have_v assume_v it_o adscription_n of_o a_o circle_n be_v with_o any_o triangle_n but_o with_o a_o triangulate_v it_o be_v with_o that_o only_a which_o be_v ordinate_a and_o indeed_o adscription_n of_o a_o circle_n be_v common_a to_o all_o 5._o if_o two_o right_a line_n do_v cut_v into_o two_o equal_a part_n two_o angle_n of_o a_o assign_a rectilineall_a the_o circle_n of_o the_o ray_n from_o their_o meeting_n perpendicular_a unto_o the_o side_n shall_v be_v inscribe_v unto_o the_o assign_a rectilineall_a 4_o and_o 8._o p._n iiij_o the_o same_o argument_n shall_v serve_v in_o a_o triangulate_a 6._o if_o two_o right_a line_n do_v right_a anglewise_o cut_v into_o two_o equal_a part_n two_o side_n of_o a_o assign_a rectilineall_a the_o circle_n of_o the_o ray_n from_o their_o meeting_n unto_o the_o angle_n shall_v be_v circumscribe_v unto_o the_o assign_a rectilineall_a 5_o p_o iiij_o as_o in_o the_o former_a figure_n the_o demonstration_n be_v the_o same_o with_o the_o former_a for_o the_o three_o ray_n by_o the_o 2_o e_fw-la seven_o be_v equal_a and_o the_o meeting_n of_o they_o by_o the_o 17_o ex_fw-la be_v the_o centre_n and_o thus_o be_v the_o common_a adscription_n of_o a_o circle_n the_o adscription_n of_o a_o rectilineall_a follow_v and_o first_o of_o a_o triangle_n 7._o if_o two_o inscript_n from_o the_o touch_n point_n of_o a_o right_a line_n and_o a_o periphery_a do_v make_v two_o angle_n on_o each_o side_n equal_a to_o two_o angle_n of_o the_o triangle_n assign_v be_v knit_v together_o they_o shall_v inscribe_v a_o triangle_n into_o the_o circle_n give_v equiangular_a to_o the_o triangle_n give_fw-mi è_fw-mi 2_o p_o iiij_o the_o circumscription_n here_o be_v also_o special_a 8_o if_o two_o angle_n in_o the_o centre_n of_o a_o circle_n give_v be_v equal_a at_o a_o common_a ray_n to_o the_o outter_n angle_v of_o a_o triangle_n give_v right_a line_n touch_v a_o periphery_a in_o the_o shank_n of_o the_o angle_n shall_v circumscribe_v a_o triangle_n about_o the_o circle_n give_v like_o to_o the_o triangle_n give_v 3_o p_o iiij_o therefore_o 9_o if_o a_o triangle_n be_v a_o rectangle_n a_o obtusangle_n a_o acute_a angle_n the_o centre_n of_o the_o circumscribe_v triangle_n be_v in_o the_o side_n out_o of_o the_o side_n and_o within_o the_o side_n and_o contrariwise_o 5_o e_fw-la iiij_o as_o thou_o see_v in_o these_o three_o figure_n underneath_o the_o centre_n a._n of_o geometry_n the_o eighteen_o book_n of_o the_o adscription_n of_o a_o triangulate_a such_o be_v the_o adscription_n of_o a_o triangle_n the_o adscription_n of_o a_o ordinate_a triangulate_a be_v now_o to_o be_v teach_v and_o first_o the_o common_a adscription_n and_o yet_o out_o of_o the_o former_a adscription_n after_o this_o manner_n 1._o if_o right_a line_n do_v touch_v a_o periphery_a in_o the_o angle_n of_o the_o inscript_n ordinate_a triangulate_a they_o shall_v unto_o a_o circle_n circumscribe_v a_o triangulate_a homogeneal_a to_o the_o inscribe_v triangulate_v the_o example_n shall_v be_v lay_v down_o according_a as_o the_o species_n or_o several_a kind_n do_v come_v in_o order_n the_o special_a inscription_n therefore_o shall_v first_o be_v teach_v and_o that_o by_o one_o side_n which_o reiterated_a as_o oft_o as_o need_v shall_v require_v may_v fill_v up_o the_o whole_a periphery_n for_o that_o euclid_n do_v in_o the_o quindecangle_n one_o of_o the_o kind_n we_o will_v do_v it_o in_o all_o the_o rest_n 2._o if_o the_o diameter_n do_v cut_v one_o another_o right-anglewise_a a_o right_a line_n subtend_v or_o draw_v against_o the_o right_a angle_n shall_v be_v the_o side_n of_o the_o quadrate_n è_fw-it 6_o p_o iiij_o therefore_o 3._o a_o quadrate_n inscribe_v be_v the_o half_a of_o that_o which_o be_v circumscribe_v because_o the_o side_n of_o the_o circumscribe_v which_o here_o be_v equal_a to_o the_o diameter_n of_o the_o circle_n be_v of_o power_n double_a to_o the_o side_n of_o the_o inscript_n by_o the_o 9_o e_fw-la x_o i_o i_o anâ—_n 4._o it_o be_v great_a than_o the_o half_a of_o the_o circumscribe_v circle_n because_o the_o circumscribe_v quadrate_n which_o be_v his_o double_a be_v great_a than_o the_o whole_a circle_n for_o the_o inscribe_v of_o other_o multangled_a odde-sided_n figure_n we_o must_v needs_o use_v the_o help_n of_o a_o triangle_n each_o of_o who_o angle_n at_o the_o base_a be_v manifold_a to_o the_o other_o in_o a_o quinguangle_n first_o that_o which_o be_v double_a
unto_o the_o remainder_n which_o be_v thus_o find_v 5._o if_o a_o right_a line_n be_v cut_v proportional_o the_o base_a of_o that_o triangle_n who_o shank_n shall_v be_v equal_a to_o the_o whole_a line_n cut_v and_o the_o base_a to_o the_o great_a segment_n of_o the_o same_o shall_v have_v each_o of_o the_o angle_n at_o the_o base_a double_a to_o the_o remainder_n and_o the_o base_a shall_v be_v the_o side_n of_o the_o quinquangle_v inscribe_v with_o the_o triangle_n into_o a_o circle_n 10_o and_o 11._o p_o i_o i_o i_o i_o 6_o if_o two_o right_a line_n do_v subtend_v on_o each_o side_n two_o angle_n of_o a_o inscript_a quinquangle_n they_o be_v cut_v proportional_o and_o the_o great_a segment_n be_v the_o side_n of_o the_o say_a inscript_n è_fw-mi 8_o p_o x_o iij._o and_o from_o hence_o the_o fabric_n or_o construction_n of_o a_o ordinate_a quinquangle_n upon_o a_o right_a line_n give_v be_v manifest_a therefore_o 7_o if_o a_o right_a line_n give_v cut_v proportional_a be_v continue_v at_o each_o end_n with_o the_o great_a segment_n and_o six_o periphery_n at_o the_o distance_n of_o the_o line_n give_v shall_v meet_v two_o on_o each_o side_n from_o the_o end_n of_o the_o line_n give_v and_o the_o continue_a two_o other_o from_o their_o meeting_n right_a line_n draw_v from_o their_o meeting_n &_o the_o end_n of_o the_o assign_a shall_v make_v a_o ordinate_a quinquangle_n upon_o the_o assign_a 8_o if_o the_o diameter_n of_o a_o circle_n circumscribe_v about_o a_o quinquangle_n be_v rational_a it_o be_v irrational_a unto_o the_o side_n of_o the_o inscribe_v quinquangle_n è_fw-it 11._o p_o xiij_o so_o before_o the_o segment_n of_o a_o right_a line_n proportional_o cut_v be_v irrational_a the_o other_o triangulate_v hereafter_o multiply_v from_o the_o ternary_a quaternary_a or_o quinary_a of_o the_o side_n may_v be_v inscribe_v into_o a_o circle_n by_o a_o inscript_a triangle_n quadrate_n or_o quinquangle_v therefore_o by_o a_o triangle_n there_o may_v be_v inscribe_v a_o triangulate_a of_o 6._o 12,24,46_o angle_n by_o a_o quadrate_n a_o triangulate_a of_o 8._o 16,32,64_o angle_n by_o a_o quinquangle_n a_o triangulate_a of_o 10_o 20._o 40,80_o angle_n etc._n etc._n 9_o the_o ray_n of_o a_o circle_n be_v the_o side_n of_o the_o inscript_n sexangle_v è_fw-mi 15_o p_o iiij_o therefore_o 10_o three_o ordinate_a sexangle_n do_v fill_v up_o a_o place_n furthermore_o also_o no_o one_o figure_n among_o the_o plain_n do_v fill_v up_o a_o place_n a_o quinquangle_n do_v not_o for_o three_o angle_n a_o quinquangle_n may_v make_v only_o 3_o â—_o 5_o angle_n which_o be_v too_o little_a and_o four_o will_v make_v 4_o â—_o 5._o which_o be_v as_o much_o too_o great_a the_o angle_n of_o a_o septangle_n will_v make_v only_o two_o rightangle_v and_o 6_o 7_o of_o one_o three_o will_v make_v 3_o and_o 9_o 7_o that_o be_v in_o the_o whole_a 4._o 2_o 7_o which_o be_v too_o much_o etc._n etc._n to_o he_o that_o by_o induction_n shall_v thus_o make_v trial_n it_o will_v appear_v that_o a_o plain_a place_n may_v be_v fill_v up_o by_o three_o sort_n of_o ordinate_a plain_n only_o and_o 11_o if_o right_a line_n from_o one_o angle_n of_o a_o inscript_n sexangle_v unto_o the_o three_o angle_n on_o each_o side_n be_v knit_v together_o they_o shall_v inscribe_v a_o equilater_n triangle_n into_o the_o circle_n give_v 12_o the_o side_n of_o a_o inscribe_v equilater_n triangle_n have_v a_o treble_a power_n unto_o the_o ray_n of_o the_o circle_n 12._o p_o xiij_o 13_o if_o the_o side_n of_o a_o sexangle_n be_v cut_v proportional_o the_o great_a segment_n shall_v be_v the_o side_n of_o the_o decangle_n therefore_o 14_o if_o a_o decangle_n and_o a_o sexangle_v be_v inscribe_v in_o the_o same_o circle_n a_o right_a line_n continue_v and_o make_v of_o both_o side_n shall_v be_v cut_v proportional_o and_o the_o great_a segment_n shall_v be_v the_o side_n of_o a_o sexangle_n and_o if_o the_o great_a segment_n of_o a_o right_a line_n cut_v proportional_o be_v the_o side_n of_o a_o hexagon_n the_o rest_n shall_v be_v the_o side_n of_o a_o decagon_n 9_o p_o xiij_o the_o comparison_n of_o the_o decangle_n and_o sexangle_v with_o the_o quinangle_n follow_v 15_o if_o a_o decangle_n a_o sexangle_n and_o a_o pentangle_v be_v inscribe_v into_o the_o same_o circle_n the_o side_n of_o the_o pentangle_v shall_v in_o power_n countervail_v the_o side_n of_o the_o other_o and_o if_o a_o right_a line_n inscribe_v do_v countervail_v the_o side_n of_o the_o sexangle_n and_o decangle_v it_o be_v the_o side_n of_o the_o pentangle_v 10._o p_o fourteen_o let_v the_o proportion_n of_o this_o syllogism_n be_v demonstrate_v for_o this_o part_n only_o remain_v doubtful_a therefore_o two_o triangle_n a_o e_o i_o and_o y_fw-fr e_fw-it i_fw-it be_v equiangle_n have_v one_o common_a angle_n at_o e_o and_o also_o two_o equal_a one_o a_o e_o i_o and_o e_z i_z y_z the_o half_n to_o wit_n of_o the_o same_o e_o i_o s_o because_o that_o be_v by_o the_o 17_o e_fw-la uj_o one_o of_o the_o two_o equal_n unto_o the_o which_o e_o ay_o s_o the_o out_z angle_n be_v equal_a by_o the_o 15_o e._n uj._o and_o this_o do_v insist_v upon_o a_o half_a periphery_n for_o the_o half_a periphery_a a_o l_o s_o be_v equal_a to_o the_o half_a periphery_a a_o r_o s_o and_o also_o a_o l_o be_v equal_a to_o a_o r._n therefore_o the_o remnant_n l_o s_o be_v equal_a to_o the_o remnant_n r_o s_o and_o the_o whole_a r_o l_o be_v the_o double_a of_o the_o same_o r_o s_o and_o therefore_o e_o r_o be_v the_o double_a of_o e_o o_o and_o r_o s_o the_o double_a of_o o_o u._fw-mi for_o the_o bisegment_n be_v manifest_a by_o the_o 10_o e_z xv_o and_o the_o 11_o e_z xuj_o therefore_o the_o periphery_n e_o r_o s_o be_v the_o double_a of_o the_o periphery_n e_o o_fw-fr u._fw-mi and_o therefore_o the_o angle_n e_fw-it i_fw-it u._fw-mi be_v the_o half_a of_o the_o angle_n e_o i_o s_o by_z the_o 7_o e_z xuj_o therefore_o two_o angle_n of_o two_o triangle_n be_v equal_a wherefore_o the_o remainder_n by_o the_o 4_o e_fw-la seven_o be_v equal_a to_o the_o remainder_n wherefore_o by_o the_o 12_o e_z seven_o as_o the_o side_n a_o e_o be_v to_z e_o i_o so_o be_v e_z i_z to_z e_o y._n therefore_o by_o the_o 8_o e_fw-la xij_o the_o oblong_a of_o the_o extreme_n be_v equal_a to_o the_o quadrate_n of_o the_o mean_a now_o let_v o_o y_fw-es be_v knit_v together_o with_o a_o straight_o here_o again_o the_o two_o triangle_n a_o o_o e_o and_o a_o o_o y_fw-fr be_v equiangle_n have_v one_o common_a angle_n at_o a_o and_o a_o o_o y_fw-fr and_o o_z e_z a_o therefore_o also_o equal_a because_o both_o be_v equal_a to_o the_o angle_n at_o a_o that_o by_o the_o 17_o e_fw-la uj_o this_o by_o the_o 2_o e_z seven_o because_o the_o perpendicular_a half_v the_o side_n of_o the_o decangle_n do_v make_v two_o triangle_n equicrural_a and_o equal_a by_o the_o right_a angle_n of_o their_o shank_n and_o therefore_o they_o be_v equiangle_n therefore_o as_o e_z a_o be_v to_o a_o o_o so_o be_v e_z a_o to_o a_o y._n wherefore_o by_o the_o 8_o e_z xij_o the_o oblong_a of_o the_o two_o extreme_n be_v equal_a to_o the_o quadrate_n of_o the_o mean_a and_o the_o proposition_n of_o the_o syllogism_n which_o be_v to_o be_v demonstrate_v the_o converse_n from_o hence_o as_o manifest_v euclid_n do_v use_v at_o the_o 16_o p_o xiij_o 16._o if_o a_o triangle_n and_o a_o quinquangle_v be_v inscribe_v into_o the_o same_o circle_n at_o the_o same_o point_n the_o right_a line_n inscribe_v between_o the_o base_n of_o the_o both_o opposite_a to_o the_o say_a point_n shall_v be_v the_o side_n of_o the_o inscribe_v quindecangle_n 16._o p._n iiij_o therefore_o 17._o if_o a_o quinquangle_n and_o a_o sexangle_v be_v inscribe_v into_o the_o same_o circle_n at_o the_o same_o point_n the_o periphery_a intercept_v between_o both_o their_o side_n shall_v be_v the_o thirty_o part_n of_o the_o whole_a periphery_n of_o geometry_n the_o ninteenth_fw-mi book_n of_o the_o measure_v of_o ordinate_a multangle_n and_o of_o a_o circle_n out_o of_o the_o adscription_n of_o a_o circle_n and_o a_o rectilineall_a be_v draw_v the_o geodesy_n of_o ordinate_a multangle_v and_o first_o of_o the_o circle_n itself_o for_o the_o meeting_n of_o two_o right_a line_n equal_o divide_v two_o angle_n be_v the_o centre_n of_o the_o circumscribe_v circle_n from_o the_o centre_n unto_o the_o angle_n be_v the_o ray_n and_o then_o if_o the_o quadrate_n of_o half_a the_o side_n be_v take_v out_o of_o the_o quadrate_n of_o the_o ray_n the_o side_n of_o the_o remainder_n shall_v be_v the_o perpendicular_a by_o the_o 9_o e_fw-la xij_o therefore_o a_o special_a theorem_a be_v here_o thus_o make_v 1._o a_o plain_a make_v of_o the_o
great_a than_o the_o base_a i_o u._n therefore_o by_o the_o 5_o e_fw-la seven_o the_o angle_n o_o e_o i_o be_v great_a than_o the_o angle_n i_o e_o u._fw-mi therefore_o two_o angle_n a_o e_o o_o and_o o_o e_fw-it i_fw-it be_v great_a than_o a_o e_o i._n 10_o a_o plain_a solid_a be_v a_o pyramid_n or_o a_o pyramidate_n 11_o a_o pyramid_n be_v a_o plain_a solid_a from_o a_o rectilineall_a base_a equal_o decrease_v as_o here_o thou_o conceive_v from_o the_o triangular_a base_a a_o e_o i_o unto_o the_o top_n o_o the_o triangle_n a_o o_o e_o a_o o_o ay_o and_o e_z o_o ay_o to_o be_v rear_v up_o therefore_o 12_o the_o side_n of_o a_o pyramid_n be_v one_o more_o than_o be_v the_o base_a the_o side_n be_v here_o name_v hedrae_fw-la and_o 13_o a_o pyramid_n be_v the_o first_o figure_n of_o solid_n for_o a_o pyramid_n in_o solid_n be_v as_o a_o triangle_n be_v in_o plain_n for_o a_o pyramid_n may_v be_v resolve_v into_o other_o solid_a figure_n but_o it_o can_v be_v resolve_v into_o any_o one_o more_o simple_a than_o itself_o and_o which_o consist_v of_o few_o side_n than_o it_o do_v therefore_o 14_o pyramid_n of_o equal_a height_n be_v as_o their_o base_n be_v 5_o e_z and_o 6._o p_o xij_o and_o 15_o those_o which_o be_v reciprocal_a in_o base_a and_o height_n be_v equal_a 9_o p_o xij_o these_o consectary_n be_v draw_v out_o of_o the_o 16_o 18_o e._n iiij_o 16_o a_o tetraedrum_n be_v a_o ordinate_a pyramid_n comprehend_v of_o four_o triangle_n 26._o d_o xj_o therefore_o 17_o the_o edge_n of_o a_o tetraedrum_n be_v six_o the_o plain_a angle_n twelve_o the_o solid_a angle_n four_o for_o a_o tetraedrum_n be_v comprehend_v of_o four_o triangle_n each_o of_o they_o have_v three_o side_n and_o three_o corner_n a_o piece_n and_o every_o side_n be_v twice_o take_v therefore_o the_o number_n of_o edge_n be_v but_o half_a so_o many_o and_o 18_o twelve_o tetraedra_n doe_n fill_v up_o a_o solid_a place_n because_o 8._o solid_a right_a angle_n fill_v a_o place_n and_o 12._o angle_n of_o the_o tetraedrum_n be_v equal_a between_o themselves_o see_v that_o both_o of_o they_o be_v comprehend_v of_o 24._o plain_a rightangle_n for_o a_o solid_a right_a angle_n be_v comprehend_v of_o three_o plain_a right_a angle_n and_o therefore_o 8._o be_v comprehend_v of_o 24._o in_o like_a manner_n the_o angle_n of_o a_o tetraedrum_n be_v comprehend_v of_o three_o plain_a equilater_n that_o be_v of_o six_o three_o of_o one_o right_a angle_n and_o therefore_o of_o two_o right_a angle_n therefore_o 12_o be_v comprehend_v of_o 24._o and_o 19_o if_o four_o ordinate_a and_o equal_a triangle_n be_v join_v together_o in_o solid_a angle_n they_o shall_v comprehend_v a_o tetraedrum_n 20._o if_o a_o right_a line_n who_o power_n be_v sesquialter_fw-la unto_o the_o side_n of_o a_o equilater_n triangle_n be_v cut_v after_o a_o double_a reason_n the_o double_a segment_n perpendicular_a to_o the_o centre_n of_o the_o triangle_n knit_v together_o with_o the_o angle_n thereof_o shall_v comprehend_v a_o 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40._o p_o xj_o the_o cause_n be_v manifest_a and_o brief_a because_o they_o be_v the_o half_n of_o the_o same_o prisma_fw-la as_o here_o thou_o may_v perccive_v in_o a_o prisma_fw-la cut_v into_o two_o half_n by_o the_o diagoni'_v of_o the_o opposite_a side_n euclid_n do_v demonstrate_v it_o thus_o let_v the_o pentaedra_n a_o e_o i_o o_o u._fw-mi and_o y_z s_o r_o l_o m_o be_v of_o equal_a height_n the_o first_o of_o a_o triangular_a base_a e_o i_o o_o the_o second_o of_o a_o parallelogramme_n base_a s_o l_o double_a unto_o the_o triangular_a now_o let_v both_o of_o they_o be_v double_a and_o make_v up_o so_o that_o first_o be_v nâ—_n the_o second_o y_o s_o r_o l_o v_o f._n now_o again_o by_o the_o grant_n the_o base_a s_o l_o be_v the_o double_a of_o the_o base_a e_o i_o o_o who_o double_a be_v thâ—_n base_a e_o o_o by_o the_o 12_o e_fw-la x._o therefore_o the_o base_n s_o l_o and_o e_o o_o be_v equal_a and_o therefore_o see_v the_o prisma_n by_o the_o grant_n here_o be_v of_o equal_a 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prisma_fw-la the_o whole_a surface_n be_v 150_o and_o the_o solidity_n 90._o 13_o a_o prisma_fw-la compound_v of_o pentaâ—dra's_n be_v either_o a_o hexaedrum_n or_o polyedrum_fw-la and_o the_o hexaedrum_n be_v either_o a_o parallelepipedum_fw-la or_o a_o trapezium_fw-la 14_o a_o parallelepipedum_fw-la be_v that_o who_o opposite_a plain_n be_v parallelogramme_n ê_a 24._o p_o xj_o therefore_o a_o parallelepipedum_fw-la in_o solid_n answer_v to_o a_o parallelogramme_n in_o plain_n for_o here_o the_o opposite_a hedrae_fw-la or_o flatte_n be_v parallel_v there_o the_o opposite_a side_n be_v parallel_v therefore_o 15_o it_o be_v cut_v into_o two_o half_n with_o a_o plain_a by_o the_o diagony_n of_o the_o opposite_a
compound_v pyramid_n be_v find_v by_o the_o ray_n of_o the_o circle_n circumscribe_v about_o the_o base_a and_o by_o the_o semidiagony_n of_o the_o polyedrum_fw-la the_o base_a of_o the_o pyramid_n appear_v to_o the_o eye_n the_o height_n lie_v hide_v within_o but_o it_o be_v discover_v by_o a_o right_a angle_n triangle_n who_o base_a be_v the_o semidiagony_n or_o half_a diagony_n the_o shank_n the_o ray_n of_o the_o circle_n and_o the_o perpendicular_a of_o the_o height_n therefore_o subtract_v the_o quadrate_n of_o the_o ray_n from_o the_o quadrate_n of_o the_o halfa_n diagony_n the_o side_n of_o the_o remainder_n by_o the_o 9_o e_fw-la x_o ij_o shall_v be_v the_o height_n but_o the_o ray_n of_o the_o circle_n shall_v have_v a_o special_a invention_n according_a to_o the_o kind_n of_o the_o base_a first_o of_o a_o triangular_a and_o then_o next_o of_o a_o quinquangular_a 3_o a_o mingle_a ordinate_a polyedrum_fw-la have_v either_o a_o triangular_a or_o a_o quinquangular_a base_a the_o 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octoedrum_fw-la be_v 12._o the_o plain_a angle_n 24_o and_o the_o solid_a 6._o and_o 8_o nine_o octoedra_n doe_n fill_v a_o solid_a place_n for_o four_o angle_n of_o a_o tetraedrum_n be_v equal_a to_o three_o angle_n of_o the_o octoedrum_fw-la and_o therefore_o 12._o be_v equal_a to_o nine_o therefore_o nine_o angle_n of_o a_o octaedrum_fw-la do_v countervail_v eight_o solid_a right_a angle_n and_o 9_o if_o eight_o triangle_n equilater_n and_o equal_a be_v join_v together_o by_o their_o edge_n they_o shall_v comprehend_v a_o octaedrum_fw-la 10_o if_o a_o right_a line_n on_o each_o side_n perpendicular_a to_o the_o centre_n of_o a_o quadrate_n and_o equal_a to_o the_o half_a diagony_n be_v tie_v together_o with_o the_o angle_n it_o shall_v comprehend_v a_o octaedrum_fw-la 14._o d_o xiij_o therefore_o 11_o the_o diagony_n of_o a_o octaedrum_fw-la be_v of_o double_a power_n to_o the_o side_n as_o be_v manifest_a by_o the_o 9_o e_fw-la xij_o and_o 12_o if_o the_o quadrate_n of_o the_o 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the_o quinquangle_n knit_v together_o both_o between_o themselves_o and_o with_o the_o angle_n of_o the_o other_o quinquangle_n the_o six_o from_o the_o centre_n on_o each_o side_n continue_v with_o the_o side_n of_o the_o decangle_n and_o knit_v therewith_o the_o five_o perpendicular_o here_o with_o the_o angle_n of_o the_o second_o quinquangle_n they_o shall_v comprehend_v a_o icosaedrum_fw-la è_fw-it 15_o p_o xiij_o in_o like_a manner_n also_o shall_v it_o be_v prove_v of_o the_o five_o upper_a triangle_n by_o draw_v the_o right_a line_n d_o y_fw-fr and_o c_o n_o which_o as_o afore_o because_o they_o knit_v together_o equal_a parallel_n to_o wit_n d_o c_o and_o y_z n_o they_o shall_v be_v equal_a but_o d_o y_o be_v the_o side_n of_o a_o sexangle_n therefore_o c_o n_o shall_v be_v also_o the_o side_n of_o a_o sexangle_n and_o c_o g_o be_v the_o side_n of_o a_o decangle_n therefore_o a_o n_o who_o power_n be_v equal_a to_o both_o they_o by_o the_o 9_o e_fw-la xij_o shall_v by_o the_o converse_n of_o the_o 15_o e_fw-la xviij_o be_v the_o side_n of_o a_o quinquangle_n and_o in_o like_a manner_n g_o t_o shall_v be_v conclude_v to_o be_v the_o side_n of_o a_o quinquangle_n wherefore_o n_o g_o t_o be_v a_o equilater_n and_o the_o four_o other_o shall_v likewise_o be_v equilater_n the_o other_o five_o triangle_n beneath_o shall_v after_o the_o like_a manner_n be_v conclude_v to_o be_v equilater_n therefore_o one_o shall_v be_v for_o all_o to_o wit_n i_o b_o e_o by_o draw_v the_o ray_n d_o i_o and_o d_o e._n for_o i_o b_o who_o power_n as_o afore_o be_v asmuch_o as_o the_o side_n of_o the_o sexangle_n and_o decangle_v shall_v be_v the_o side_n of_o the_o quinquangle_n and_o in_o like_a sort_n b_o e_o be_v of_o equal_a power_n with_o d_o e_z and_o d_o o_o the_o side_n of_o the_o sexangle_n and_o decangle_v shall_v be_v the_o side_n of_o the_o quinquangle_n wherefore_o the_o triangle_n e_o b_o i_o be_v a_o equilater_n and_o the_o four_o other_o in_o like_a manner_n may_v be_v show_v to_o be_v equilater_n therefore_o all_o the_o side_n of_o the_o twenty_o triangle_n see_v they_o be_v equal_a they_o shall_v be_v equilater_n triangle_n and_o by_o the_o 8_o e_z seven_o equal_a 17_o the_o diagony_n of_o a_o icosaedrum_fw-la be_v irrational_a unto_o the_o side_n this_o be_v the_o four_o example_n of_o irrationality_n or_o incommensurability_n the_o first_o be_v of_o the_o diagony_n and_o side_n of_o a_o square_a or_o quadrate_n the_o second_o be_v of_o the_o segment_n of_o a_o line_n proportional_o cut_v the_o three_o of_o the_o diameter_n of_o a_o circle_n and_o the_o side_n of_o a_o quinquangle_n and_o 18_o the_o power_n of_o the_o diagony_n of_o a_o icosaedrum_fw-la be_v five_a time_n as_o much_o as_o the_o ray_n of_o the_o circle_n 19_o a_o mingle_a ordinate_a polyedrum_fw-la of_o a_o quinquangular_a base_a be_v that_o which_o be_v comprehend_v of_o 12_o quinquangle_n and_o it_o be_v call_v a_o dodecaedrum_fw-la therefore_o 20._o the_o side_n of_o a_o dodecaedrum_fw-la be_v 30_o the_o plain_a angle_n 60._o the_o solid_a 20._o and_o 21._o if_o 12_o ordinate_a equal_a quinquangle_n be_v join_v with_o solid_a angle_n they_o shall_v comprehend_v a_o dodecaedrum_fw-la as_o here_o thou_o see_v 22._o if_o the_o side_n of_o a_o cube_fw-la be_v with_o right_a line_n cut_v into_o two_o equal_a part_n and_o three_o bisegment_n of_o the_o bisecant_v in_o the_o abbute_v plain_n neither_o meet_v one_o the_o other_o nor_o parallel_v one_o unto_o another_o two_o of_o one_o the_o three_o of_o that_o next_o unto_o the_o remainder_n be_v so_o proportional_o cut_v that_o the_o lesser_a segment_n do_v bind_v the_o bisecant_v three_o line_n without_o the_o cube_fw-la perpendicular_a unto_o the_o say_a plain_n from_o the_o point_n of_o the_o proportional_a section_n equal_a to_o the_o great_a segment_n knit_v together_o two_o of_o the_o same_o bisecant_v between_o themselves_o and_o with_o the_o next_o angle_n of_o cube_fw-la the_o three_o with_o the_o same_o angle_n they_o shall_v comprehend_v a_o dodecaedrum_fw-la 17_o p_o xiij_o i_o say_v also_o that_o it_o be_v a_o plain_a quinquangle_n for_o it_o may_v be_v say_v to_o be_v a_o oblique_a quinquangle_n and_o to_o be_v seat_v in_o two_o plain_n let_v therefore_o f_z h_z be_v parallel_n to_o d_o b_o and_o c_o p_o and_o be_v equal_a unto_o they_o and_o let_v h_o z_o be_v draw_v this_o h_o z_o shall_v be_v cut_v one_o line_n by_o the_o 14_o e_fw-la seven_o for_o as_o the_o whole_a tr_fw-la that_o be_v r_o f_o be_v unto_o the_o great_a segment_n that_o be_v to_o f_o h_o so_o f_o h_o that_o be_v z_o g_o be_v unto_o g_o r._n and_o two_o pair_n of_o shank_n f_z h_z g_o r_o f_o c_o g_o z_o by_o the_o 6_o e_fw-la xxj_o be_v alterne_o or_o crossewise_a parallel_n therefore_o their_o base_n
be_v continual_a hitherto_o it_o have_v be_v prove_v that_o the_o quinquangle_v make_v be_v a_o equilater_n and_o plain_a it_o remain_v that_o it_o be_v prove_v to_o be_v equiangle_v let_v therefore_o the_o right_a line_n e_o p_o and_o e_o c_o be_v draw_v i_o say_v that_o the_o angle_n p_o b_o e_o and_o e_o z_o i_o be_v equal_a because_o they_o have_v by_o the_o construction_n the_o base_n of_o equal_a shank_n equal_a be_v to_o wit_n in_o value_n the_o quadruple_a of_o l_o e._n for_o the_o right_a line_n l_o f_o cut_v proportional_o and_o increase_v with_o the_o great_a segment_n d_o f_o that_o be_v f_o c_o be_v cut_v also_o proportional_o by_o the_o 4_o e_fw-la fourteen_o and_o by_o the_o 7_o e_fw-la fourteen_o the_o whole_a line_n proportional_o cut_v and_o the_o lesser_a segment_n that_o be_v c_o p_o be_v of_o treble_a value_n to_o the_o great_a f_o l_o that_o be_v of_o the_o say_v l_o e._n therefore_o e_o l_o and_o l_o c_o that_o be_v e_o c_o and_o c_o p_o that_o be_v e_o p_o be_v of_o quadruple_a power_n to_o e_o l_o and_o therefore_o by_o the_o 14_o e_fw-la xij_o it_o be_v the_o double_a of_o it_o and_o e_fw-it i_fw-it itself_o in_o like_a manner_n by_o the_o fabric_n or_o construction_n be_v the_o double_a of_o the_o same_o therefore_o the_o base_n be_v equal_a and_o after_o the_o same_o manner_n by_o draw_v the_o right_a line_n i_o d_o and_o i_z b_o the_o three_o angle_n b_o p_o i_o shall_v be_v conclude_v to_o be_v equal_a to_o the_o angle_n e_o z_o i._n therefore_o by_o the_o 13_o e_fw-la fourteen_o five_o angle_n be_v equal_a 23._o the_o diagony_n be_v irrational_a unto_o the_o side_n of_o the_o dodecahedrum_fw-la this_o be_v the_o five_o example_n of_o irrationality_n and_o incommensurability_n the_o first_o be_v of_o the_o diagony_n and_o side_n of_o a_o quadrate_n or_o square_n the_o second_o be_v of_o a_o line_n proportional_o cut_v and_o his_o segment_n the_o three_o be_v of_o the_o diameter_n of_o a_o circle_n and_o the_o side_n of_o a_o inscribe_v 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five_o ordinate_a solid_a plain_n this_o appear_v plain_o out_o of_o the_o nature_n of_o a_o solid_a angle_n by_o the_o kind_n of_o plain_a figure_n of_o two_o plain_a angle_n a_o solid_a angle_n can_v be_v comprehend_v of_o three_o angle_n of_o a_o ordinate_a triangle_n be_v the_o angle_n of_o a_o tetrahedrum_fw-la comprehend_v of_o four_o a_o octahedrum_fw-la of_o five_o a_o icosahedrum_fw-la of_o six_o none_o can_v be_v comprehend_v for_o six_o such_o like_a plain_a angle_n be_v equal_a to_o 12_o three_o of_o one_o right_a angle_n that_o be_v to_o four_o right_a angle_n but_o plain_a angle_n make_v a_o solid_a angle_n be_v lesser_a than_o four_o right_a angle_n by_o the_o 8_o e_fw-la xxij_o of_o seven_o therefore_o and_o of_o more_o it_o be_v much_o less_o possible_a of_o three_o quadrate_n angle_v the_o angle_n of_o a_o cube_fw-la be_v comprehend_v of_o 4._o such_o angle_v none_o may_v be_v comprehend_v for_o the_o same_o cause_n of_o three_o angle_n of_o a_o ordinate_a 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surface_n 2._o and_o it_o be_v either_o a_o sphere_n or_o a_o mingle_a form_n 3._o a_o sphere_n be_v a_o round_a imbossement_fw-fr therefore_o 4._o a_o sphere_n be_v make_v by_o the_o conversion_n of_o a_o semicircle_n the_o diameter_n stand_v still_o 14_o d_o xj_o as_o here_o thou_o see_v 5._o the_o great_a circle_n of_o a_o sphere_n be_v that_o which_o cut_v the_o sphere_n into_o two_o equal_a part_n therefore_o 6._o that_o circle_n which_o be_v near_a to_o the_o great_a be_v great_a than_o that_o which_o be_v far_o off_o and_o 7._o those_o which_o be_v equal_o distant_a from_o the_o great_a be_v equal_a as_o in_o the_o example_n above_o write_v 8._o the_o plain_a of_o the_o diameter_n and_o six_o part_n of_o the_o sphearicall_a be_v the_o solidity_n of_o the_o sphere_n therefore_o 9_o as_o 21_o be_v unto_o 11_o so_o be_v the_o cube_fw-la of_o the_o diameter_n unto_o the_o sphere_n as_o here_o the_o cube_n of_o 14_o be_v 2744._o for_o it_o be_v a_o easy_a matter_n for_o he_o that_o 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