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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 a_o right_a line_n and_o again_o decrease_v it_o the_o lesser_a segment_n of_o a_o right_a line_n proportional_o cut_v be_v the_o great_a segment_n of_o the_o great_a proportional_o cut_v and_o from_o hence_o a_o decrease_a may_v be_v make_v infinite_o 5_o the_o great_a segment_n continue_v to_o the_o half_a of_o the_o whole_a be_v of_o power_n quintuple_a unto_o the_o say_v half_a that_o be_v five_o time_n so_o great_a as_o it_o be_v and_o if_o the_o power_n of_o a_o right_a line_n be_v quintuple_a to_o his_o segment_n the_o remainder_n make_v the_o double_a of_o the_o former_a be_v cut_v proportional_o and_o the_o great_a segment_n be_v the_o same_o remainder_n 1._o and_o 2._o p_o x_o iij._o this_o be_v the_o fabric_n or_o manner_n of_o make_v a_o proportional_a section_n a_o threefold_a rate_n follow_v the_o first_o be_v of_o the_o great_a segment_n the_o converse_n be_v apparent_a in_o the_o same_o example_n for_o see_v that_o i_o o_o be_v of_o power_n five_o time_n so_o much_o 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
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 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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 tetraedrum_n 13_o p_o xiij_o for_o a_o solid_a to_o be_v comprehend_v of_o right_a line_n understand_v plain_n comprehend_v of_o right_a line_n as_o in_o other_o place_n follow_v the_o twenty_o three_o book_n of_o geometry_n of_o a_o prisma_fw-la 1_o a_o pyramidate_n be_v a_o plain_a solid_a comprehend_v of_o pyramid_n 2._o a_o pyramidate_n be_v a_o prisma_fw-la or_o a_o mingle_a polyedrum_fw-la 3._o a_o prisma_fw-la be_v a_o pyramidate_n who_o opposite_a plain_n be_v equal_a alike_o and_o parallel_v the_o rest_n parallelogramme_n 13_o dxj_o therefore_o 4._o the_o flatte_n of_o a_o prisma_fw-la be_v two_o more_o than_o be_v the_o angle_n in_o the_o base_a and_o indeed_o as_o the_o augmentation_n of_o a_o pyramid_n from_o a_o quaternary_a be_v infinite_a so_o be_v it_o of_o a_o prisma_fw-la from_o a_o quinary_a as_o if_o it_o be_v from_o a_o triangular_a quadrangular_a or_o quinquangular_a base_a you_o shall_v have_v a_o pentaedrum_fw-la hexaedrum_n heptaedrum_fw-la and_o so_o in_o infinite_a 5._o the_o plain_a of_o the_o base_a and_o height_n be_v the_o solidity_n of_o a_o right_a prisma_fw-la 6._o a_o prisma_fw-la be_v the_o triple_a of_o a_o pyramid_n of_o equal_a base_a and_o height_n è_fw-it 7_o p._n x_o i_o i_o if_o the_o base_a be_v triangular_a the_o prisma_fw-la may_v be_v resolve_v into_o prisma_n of_o triangular_a base_n and_o the_o theorem_a shall_v be_v conclude_v as_o afore_o therefore_o 7._o the_o plain_a make_v of_o the_o base_a and_o the_o three_o part_n of_o the_o height_n be_v the_o solidity_n of_o a_o pyramid_n of_o equal_a base_a and_o height_n so_o in_o the_o example_n follow_v let_v 36_o the_o quadrate_n of_o 6_o the_o ray_n be_v take_v out_o of_o 292_o 9_o 1156_o the_o quadrate_n of_o the_o side_n 17_o 3_o 34_o the_o side_n 16_o 3_o 34_o of_o 256_o 9_o 1156_o the_o remainder_n shall_v be_v the_o height_n who_o three_o part_n be_v 5_o 37_o 102_o the_o plain_a of_o which_o by_o the_o base_a 72_o ¼_n shall_v be_v 387_o 11_o 24_o for_o the_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 height_n as_o the_o base_n by_o the_o conclusion_n be_v equal_a the_o prisma_n be_v equal_a and_o therefore_o also_o their_o half_n a_o e_o i_o o_o u._fw-mi and_o y_z s_z n_o l_o r_o be_v equal_a the_o measure_n of_o a_o pentaedrall_n prisma_fw-la be_v even_o now_o general_o teach_v the_o matter_n in_o special_a may_v be_v conceive_v in_o these_o two_o example_n follow_v the_o plain_a of_o 18._o the_o perimeter_n of_o the_o triangular_a base_a and_o 12_o the_o height_n be_v 216._o this_o add_v to_o the_o triangular_a base_a 15_o 18_o 3â—_n or_o 15_o â…—_n almost_o twice_o take_v that_o be_v 31_o â…•_n do_v make_v 247_o â…•_n for_o the_o sum_n of_o the_o whole_a surface_n but_o the_o plain_a of_o the_o same_o base_a 15_o â…–_n and_o the_o height_n 12._o be_v 187_o â…•_n for_o the_o whole_a solidity_n so_o in_o the_o pentaedrum_fw-la the_o second_o prisma_fw-la which_o be_v call_v cuneus_n a_o wedge_n of_o the_o sharpness_n and_o which_o also_o more_o proper_o of_o cut_v be_v call_v a_o 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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side_n of_o a_o octaedrum_fw-la be_v double_v the_o side_n of_o the_o double_a shall_v be_v the_o diagony_n as_o in_o the_o figure_n follow_v the_o side_n be_v 6._o the_o quadrate_n 36._o the_o double_a be_v 72._o who_o side_n 8_o 8_o 17._o be_v the_o diagony_n 13_o a_o icosaedrum_fw-la be_v a_o ordinate_a polyedrum_fw-la comprehend_v of_o 20_o triangle_n 29_o d_o xj_o therefore_o 14_o the_o side_n of_o a_o icosaedrum_fw-la be_v 30._o the_o plain_a angle_n 60._o the_o solid_a 12._o 15_o if_o twenty_o ordinate_a and_o equal_a triangle_n be_v join_v with_o solid_a angle_n they_o shall_v comprehend_v a_o icosaedrum_fw-la 16_o if_o ordinate_a figure_n to_o wit_n a_o double_a quinquangle_n and_o one_o decangle_v be_v so_o inscribe_v into_o the_o same_o circle_n that_o the_o side_n of_o both_o the_o quinquangle_n do_v subtend_v two_o side_n of_o the_o decangle_n six_o right_a line_n perpendicular_a to_o the_o circle_n and_o equal_v to_o his_o ray_n five_o from_o the_o angle_n of_o one_o of_o 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
side_n 28_o p_o xj_o it_o answer_v to_o the_o 34._o pj._n and_o 16_o if_o it_o be_v half_v by_o two_o plain_n half_v the_o opposite_a side_n the_o common_a bisection_n and_o diagony_n do_v half_a one_o another_o 39_o p_o xj_o 17_o if_o three_o line_n be_v proportional_a the_o parallelepipedum_fw-la of_o mean_a shall_v be_v equal_a to_o the_o equiangle_v pâ—rallelepipedum_n of_o all_o they_o è_fw-it 36._o p_o x_o i_o it_o be_v a_o consectary_n out_o of_o the_o 8_o e._n 18_o eight_o rectangle_v parallelepiped_n do_v fill_v a_o solid_a place_n 19_o the_o figurate_a of_o a_o rectangle_v parallelepipedum_fw-la be_v call_v a_o solid_a make_v of_o three_o number_n 17._o d_o seven_o as_o if_o thou_o shall_v multiply_v 1,2,3_o continual_o thou_o shall_v make_v the_o solid_a 6._o item_n if_o thou_o shall_v in_o like_a manner_n multiply_v 2,3,4_o thou_o shall_v make_v the_o solid_a 24._o and_o the_o side_n of_o that_o solid_a 6_o solid_a shall_v be_v 1,2,3_o of_o 24_o they_o shall_v be_v 2,3,4_o therefore_o 20_o if_o two_o solid_n be_v alike_o they_o have_v their_o side_n proportional_o and_o two_o mean_a proportional_o 21_o d_o seven_o 19_o 21._o p_o viij_o it_o be_v a_o consectary_n out_o of_o the_o 5_o e_fw-la xxij_o but_o the_o mean_a proportional_o be_v make_v of_o the_o side_n of_o the_o like_a solid_n to_o wit_n of_o the_o second_o three_o and_o four_o item_n of_o the_o three_o four_o and_o five_o as_o here_o thoâ—_n see_v of_o geometry_n the_o twenty_o four_o book_n of_o a_o cube_n 1_o a_o rightangle_v parallelepipedum_fw-la be_v either_o a_o cube_n or_o a_o oblong_a 2_o a_o cube_n be_v a_o right_n angle_v parallelepipedum_fw-la of_o equal_a flatte_n 25._o d_o xj_o as_o here_o thou_o see_v in_o these_o two_o figure_n therefore_o 3_o the_o side_n of_o a_o cube_fw-la be_v 12._o the_o plain_a angle_n 24._o the_o solid_a 8._o therefore_o 4_o if_o six_o equal_a quadrate_n be_v join_v with_o solid_a angle_n they_o shall_v comprehend_v a_o cube_fw-la as_o here_o in_o these_o two_o example_n therefore_o 5_o if_o from_o the_o angle_n of_o a_o quadrate_n perpendâ—culars_n equal_a to_o the_o side_n be_v tie_v together_o aloft_o they_o shall_v comprehend_v a_o cube_n è_fw-it 15_o p_o xj_o it_o be_v a_o consectary_n follow_v upon_o the_o former_a consectary_n for_o then_o shall_v six_o equal_a quadrate_n be_v knit_v together_o 6_o the_o diagony_n of_o a_o cube_n be_v of_o treble_a power_n unto_o the_o side_n for_o the_o diagony_n of_o a_o quadrate_n be_v of_o double_a power_n to_o the_o side_n by_z the_o 12_o e_z xij_o and_o the_o diagony_n of_o a_o cube_n be_v of_o as_o much_o power_n as_o the_o side_n the_o diagony_n of_o the_o quadrate_n by_o the_o same_o e._n therefore_o it_o be_v of_o treble_a power_n to_o the_o side_n 7_o if_o of_o four_o right_a line_n continual_o proportional_o the_o first_o be_v the_o half_a of_o the_o four_o the_o cube_fw-la of_o the_o first_o shall_v be_v the_o half_a of_o the_o cube_n of_o the_o second_o è_fw-la 33_o p_o xj_o it_o be_v a_o consectary_n out_o of_o the_o 25_o e_fw-la iiij_o from_o hence_o hypocrates_n first_o find_v how_o to_o answer_v apollo_n problem_n 8_o the_o solid_a plain_n of_o a_o cube_fw-la be_v call_v a_o cube_n to_o wit_n a_o solid_a of_o equal_a side_n 19_o d_o seven_o therefore_o 9_o it_o be_v make_v of_o a_o number_n multiply_v into_o his_o own_o quadrate_n so_o be_v a_o cube_n make_v by_o multiply_v a_o number_n by_o itself_o and_o the_o product_n again_o by_o the_o first_o such_o be_v these_o nine_o first_o cube_n make_v of_o the_o nine_o first_o arithmetical_a figure_n this_o be_v the_o general_a invention_n of_o a_o cube_n both_o geometrical_a and_o arithmetical_a 10_o if_o a_o right_a line_n be_v cut_v into_o two_o segment_n the_o cube_n of_o the_o whole_a shall_v be_v equal_a to_o the_o cube_n of_o the_o segment_n and_o a_o double_a solid_a thrice_o comprehend_v of_o the_o quadrate_n of_o his_o own_o segment_n and_o the_o other_o segment_n as_o for_o example_n the_o side_n 12_o let_v it_o be_v cut_v into_o two_o segment_n 10_o and_o 2._o the_o cube_fw-la of_o 12._o the_o whole_a which_o be_v 1728_o shall_v be_v equal_a to_o two_o cube_n 1000_o and_o 8_o make_v of_o the_o segment_v 10._o and_o 2._o and_o a_o double_a solid_a of_o which_o the_o first_o 600._o be_v thrice_o comprehend_v of_o 100_o the_o quadrate_n of_o his_o segment_n 10._o and_o of_o 2._o the_o other_o segment_n the_o second_o 120._o be_v thrice_o comprehend_v of_o 4_o the_o quadrate_n of_o his_o own_o segment_n and_o of_o 10._o the_o other_o segment_n now_o 1000_o 600_o 120._o 8_o be_v equal_a to_o 1728_o and_o therefore_o a_o right_a etc._n etc._n therefore_o 11._o the_o side_n of_o the_o first_o several_a cube_fw-la be_v the_o other_o side_n of_o the_o second_o solid_a and_o the_o quadrate_n of_o the_o same_o side_n be_v the_o other_o side_n of_o the_o first_o solid_a who_o other_o side_n be_v the_o side_n of_o the_o second_o cube_fw-la and_o the_o quadrate_n of_o the_o same_o other_o side_n be_v the_o other_o side_n of_o the_o second_o solid_a in_o that_o equation_n therefore_o of_o four_o solid_n with_o one_o solid_a thou_o shall_v consider_v a_o peculiar_a make_n and_o composition_n first_o that_o the_o last_o cube_fw-la be_v make_v of_o the_o last_o segment_n 2d_o then_o that_o the_o second_o solid_a of_o 4_o the_o quadrate_n of_o his_o own_o segment_n and_o of_o 10_o the_o other_o segment_n be_v thrice_o comprehend_v last_o that_o the_o first_o solid_a of_o 100_o the_o square_a of_o his_o own_o segment_n 10_o and_o the_o other_o segment_n 2_o be_v also_o thrice_o comprehend_v last_o that_o the_o cube_n 1000_o be_v make_v of_o the_o great_a segment_n 10._o out_o of_o this_o make_n etc._n etc._n the_o plain_a of_o the_o perimeter_n of_o the_o base_a 20_o and_o the_o altitude_n 5_o be_v 100_o this_o add_v to_o 25_o and_o 25_o both_o the_o base_n that_o be_v to_o 50_o make_v 150_o for_o the_o whole_a surface_n now_o the_o plain_a of_o 25_o the_o base_a and_o the_o height_n 5_o be_v 125_o for_o the_o whole_a solidity_n so_o in_o the_o oblong_a the_o plain_a of_o the_o base_n perimeter_n 20_o and_o the_o height_n 11_o be_v 220_o which_o add_v to_o the_o base_n 24_o and_o 24_o that_o be_v 48_o make_v 268_o for_o the_o whole_a surface_n but_o the_o plain_a of_o the_o base_a 24_o and_o the_o height_n 11_o be_v 264_o for_o the_o solidity_n thus_o be_v such_o kind_n of_o wall_n whether_o of_o mud_n brick_n or_o stone_n of_o most_o large_a house_n to_o be_v measure_v the_o same_o manner_n of_o geodesy_n be_v also_o to_o be_v use_v in_o the_o measure_n of_o a_o rhombe_n rhomboide_n trapezium_fw-la or_o mensall_n and_o any_o kind_n of_o multangled_a body_n the_o base_a be_v first_o to_o be_v measure_v as_o in_o the_o former_a then_o out_o of_o that_o and_o the_o height_n the_o solidity_n shall_v be_v manifest_v as_o in_o the_o rhombe_n the_o base_a be_v 24_o the_o height_n 4._o therefore_o the_o solidity_n be_v 96._o in_o the_o rhomboide_n the_o base_a be_v 64_o 3â—_n 12â—_n the_o hey_z 11._o therefore_o the_o solidity_n be_v 1028_o 44_o 1â—9_n the_o same_o be_v the_o geodesy_n of_o a_o trapezium_n as_o in_o these_o example_n the_o surface_n of_o the_o first_o be_v 198_o the_o solidity_n 192_o ½_o the_o surface_n of_o the_o second_o be_v 158_o 3_o 49_o the_o solidity_n be_v 91_o 29_o 4â—_n and_o from_o hence_o also_o may_v the_o capacity_n or_o content_n of_o vessel_n or_o measure_n make_v after_o any_o manner_n of_o plain_a solid_a be_v esteem_v and_o judge_v of_o as_o here_o thou_o see_v for_o here_o the_o plain_a of_o the_o sexangular_a base_a be_v 41_o 1_o 7_o for_o the_o ray_n by_o the_o 9_o e_fw-la xviij_o be_v the_o fide_fw-la and_o the_o height_n 5_o shall_v be_v 205_o 5_o 7._o therefore_o if_o a_o cubicall_a foot_n do_v contain_v 4_o quarter_n as_o we_o common_o call_v they_o then_o shall_v the_o vessel_n contain_v 822_o 6_o 7_o quarte_n that_o be_v almost_o 823_o quarte_n of_o geometry_n the_o twenty_o five_o book_n of_o mingle_a ordinate_a polyedra's_n 1._o a_o mingle_a ordinate_a polyedrum_fw-la be_v a_o pyramidate_n compound_v of_o pyramid_n with_o their_o top_n meet_v in_o the_o centre_n and_o their_o base_n only_o outward_o appear_v see_v therefore_o a_o mingle_a ordinate_a pyramidate_n be_v thus_o make_v or_o compound_v of_o pyramid_n the_o geodesy_n of_o it_o shall_v be_v have_v from_o the_o geodesy_n of_o the_o pyramid_n compound_v it_o and_o one_o base_a multiply_v by_o the_o number_n of_o all_o the_o base_n shall_v make_v the_o surface_n of_o the_o body_n and_o one_o pyramid_n by_o the_o number_n of_o all_o the_o pyramid_n shall_v make_v the_o solidity_n 2_o the_o height_n of_o the_o