two_o side_n eh_o and_o eke_o of_o the_o triangle_n ehk_n and_o the_o angle_n feg_n be_v great_a than_o the_o angle_n hek_n therefore_o by_o the_o 24._o of_o the_o first_o the_o base_a fg_o be_v great_a than_o the_o base_a hk_n and_o by_o the_o same_o reason_n may_v it_o be_v prove_v that_o the_o line_n ac_fw-la be_v great_a than_o the_o line_n ab_fw-la and_o so_o be_v manifest_v the_o whole_a proposition_n the_o 15._o theorem_a the_o 16._o proposition_n if_o from_o the_o end_n of_o the_o diameter_n of_o a_o circle_n be_v draw_v a_o right_a line_n make_v right_a angle_n it_o shall_v fall_v without_o the_o circle_n and_o between_o that_o right_a line_n and_o the_o circumference_n can_v not_o be_v draw_v a_o other_o right_a line_n and_o the_o angle_n of_o the_o semicircle_n be_v great_a than_o any_o acute_a angle_n make_v of_o right_a line_n but_o the_o other_o angle_n be_v less_o than_o any_o acute_a angle_n make_v of_o right_a line_n svppose_v that_o there_o be_v a_o circle_n abc_n who_o centre_n let_v be_v the_o point_n d_o and_o let_v the_o diameter_n thereof_o be_v ab_fw-la then_o i_o say_v that_o a_o right_a line_n draw_v from_o the_o point_n a_o make_v with_o the_o diameter_n ab_fw-la right_a angle_n shall_v fall_v without_o the_o circle_n theorem_a for_o if_o it_o do_v not_o then_o if_o it_o be_v possible_a let_v it_o fall_v within_o the_o circle_n as_o the_o line_n ac_fw-la do_v and_o draw_v a_o line_n from_o the_o point_n d_o to_o the_o point_n c._n absurdity_n and_o for_o asmuch_o as_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_n dam_n be_v equal_a to_o the_o line_n dc_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n therefore_o the_o angle_n dac_o be_v equal_a to_o the_o angle_n acd_o but_o the_o angle_n dac_o be_v by_o supposition_n a_o right_a angle_n wherefore_o also_o the_o angle_n acd_o be_v a_o right_a angle_n wherefore_o the_o angle_n dac_o and_o acd_o be_v equal_a to_o two_o right_a angle_n which_o by_o the_o 17._o of_o the_o first_o be_v impossible_a wherefore_o a_o right_a line_n draw_v from_o the_o point_n a_o make_v with_o the_o diameter_n ab_fw-la right_a angle_n shall_v not_o fall_v within_o the_o circle_z in_o like_a sort_n also_o may_v we_o prove_v that_o it_o fall_v not_o in_o the_o circumference_n wherefore_o it_o fall_v without_o as_o the_o line_n ae_n do_v i_o say_v also_o part_n that_o between_o the_o right_a line_n ae_n and_o the_o circumference_n acb_o can_v not_o be_v draw_v a_o other_o right_a line_n for_o if_o it_o be_v possible_a let_v the_o line_n of_o so_o be_v draw_v and_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n d_o draw_v unto_o the_o line_n favorina_n a_o perpendicular_a line_n dg_o and_o for_o asmuch_o as_o agd_n be_v a_o right_a angle_n but_o dag_n be_v less_o than_o a_o right_a angle_n therefore_o by_o the_o 19_o of_o the_o first_o the_o side_n ad_fw-la be_v great_a than_o the_o side_n dg_o but_o the_o line_n dam_n be_v equal_a to_o the_o line_n dh_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n wherefore_o the_o line_n dh_o be_v great_a than_o the_o line_n dg_o namely_o the_o less_o great_a than_o the_o great_a which_o be_v impossible_a wherefore_o between_o the_o right_a line_n ae_n and_o the_o circumference_n acb_o can_v not_o be_v draw_v a_o other_o right_a line_n i_o say_v moreover_o part_n that_o the_o angle_n of_o the_o semicircle_n contain_v under_o the_o right_a line_n ab_fw-la and_o the_o circumference_n cha_fw-mi be_v great_a than_o any_o acute_a angle_n make_v of_o right_a line_n and_o the_o angle_n remain_v contain_v under_o the_o circumference_n cha_fw-mi and_o the_o right_a line_n ae_n be_v less_o than_o any_o acute_a angle_n make_v of_o right_a line_n for_o if_o there_o be_v any_o angle_n make_v of_o right_a line_n great_a than_o that_o angle_n which_o be_v contain_v under_o the_o right_a line_n basilius_n and_o the_o circumference_n cha_fw-mi or_o less_o than_o that_o which_o be_v contain_v under_o the_o circumference_n cha_fw-mi and_o the_o right_a line_n ae_n then_o between_o the_o circumference_n cha_fw-mi and_o the_o right_a line_n ae_n there_o shall_v fall_v a_o right_a line_n which_o make_v the_o angle_n contain_v under_o the_o right_a line_n great_a than_o that_o angle_n which_o be_v contain_v under_o the_o right_a line_n basilius_n and_o the_o circumference_n cha_fw-mi and_o less_o than_o the_o angle_n which_o be_v contain_v under_o the_o circumference_n cha_fw-mi and_o the_o right_a line_n ae_n but_o there_o can_v fall_v no_o such_o line_n as_o it_o have_v before_o be_v prove_v wherefore_o no_o acute_a angle_n contain_v under_o right_a line_n be_v great_a than_o the_o angle_n contain_v under_o the_o right_a line_n basilius_n and_o the_o circumference_n cha_fw-mi nor_o also_o less_o than_o the_o angle_n contain_v under_o the_o circumference_n cha_fw-mi and_o the_o line_n ae_n correlary_a hereby_o it_o be_v manifest_a that_o a_o perpendicular_a line_n draw_v from_o the_o end_n of_o the_o diameter_n of_o a_o circle_n touch_v the_o circle_n and_o that_o a_o right_a line_n touch_v a_o circle_n in_o one_o point_n only_o for_o it_o be_v prove_v by_o the_o 2._o of_o the_o three_o that_o a_o right_a line_n draw_v from_o two_o point_n take_v in_o the_o circumference_n of_o a_o circle_n shall_v fall_v within_o the_o circle_n which_o be_v require_v to_o be_v demonstrate_v the_o 2._o problem_n the_o 17._o proposition_n from_o a_o point_n give_v to_o draw_v a_o right_a line_n which_o shall_v touch_v a_o circle_n give_v construction_n svppose_v that_o the_o point_n give_v be_v a_o and_o let_v the_o circle_n give_v be_v bcd_o it_o be_v require_v from_o the_o point_n a_o to_o draw_v a_o right_a line_n which_o shall_v touch_v the_o circle_n bcd_o take_v by_o the_o first_o of_o the_o three_o the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v e._n and_o by_o the_o first_o petition_n draw_v the_o right_a line_n ade_n and_o make_v the_o centre_n e_o and_o the_o space_n ae_n describe_v by_o the_o three_o petition_n a_o circle_n afg_n and_o from_o the_o point_n d_o raise_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o line_n ea_fw-la a_o perpendicular_a line_n df._n and_o by_o the_o first_o petition_n draw_v these_o line_n ebf_a and_o ab_fw-la then_o i_o say_v that_o from_o the_o point_n a_o be_v draw_v to_o the_o circle_n bcd_v a_o touch_n line_n ab_fw-la demonstration_n for_o for_o asmuch_o as_o the_o point_n e_o be_v the_o centre_n of_o the_o circle_n bcd_o and_o also_o of_o the_o circle_n afg_n therefore_o the_o line_n ea_fw-la be_v equal_a to_o the_o line_n of_o and_o the_o line_n ed_z to_o the_o line_n ebb_n for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n wherefore_o to_o these_o two_o line_n ae_n and_o ebb_n be_v equal_a these_o two_o line_n of_o &_o ed_z and_o the_o angle_n at_o the_o point_n e_o be_v common_a to_o they_o both_o wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a df_o be_v equal_a to_o the_o base_a ab_fw-la and_o the_o triangle_n def_n be_v equal_a to_o the_o triangle_n eba_n and_o the_o rest_n of_o the_o angle_n remain_v to_o the_o rest_n of_o the_o angle_n remain_v wherefore_o the_o angle_n edf_o be_v equal_a to_o the_o angle_n eba_n but_o the_o angle_n edf_o be_v a_o right_a angle_n wherefore_o also_o the_o angle_n eba_n be_v a_o right_a angle_n and_o the_o line_n ebb_n be_v draw_v from_o the_o centre_n but_o a_o perpendicular_a line_n draw_v from_o the_o end_n of_o the_o diameter_n of_o a_o circle_n touch_v the_o circle_n by_o the_o corellary_n of_o the_o 16._o of_o the_o three_o wherefore_o the_o line_n ab_fw-la touch_v the_o circle_n bcd_o wherefore_o from_o the_o point_n give_v namely_o a_o be_v draw_v unto_o the_o circle_n give_v bcd_o a_o touch_n line_n ab_fw-la which_o be_v require_v to_o be_v do_v ¶_o a_o addition_n of_o pelitarius_n pelitarius_n unto_o a_o right_a line_n which_o cut_v a_o circle_n to_o draw_v a_o parallel_n line_n which_o shall_v touch_v the_o circle_n suppose_v that_o the_o right_a line_n ab_fw-la do_v out_o the_o circle_n abc_n in_o the_o point_n a_o and_o b._n it_o be_v require_v to_o draw_v unto_o the_o line_n ab_fw-la a_o parallel_n line_n which_o shall_v touch_v the_o circle_n let_v the_o centre_n of_o the_o circle_n be_v the_o point_n d._n and_o divide_v the_o line_n ab_fw-la into_o two_o equal_a part_n in_o the_o point_n e._n and_o by_o the_o point_n e_o and_o by_o the_o centre_n d_o draw_v the_o diameter_n cdef_n and_o from_o the_o point_n fletcher_n which_o be_v the_o end_n of_o the_o diameter_n raise_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o diameter_n cf_o a_o perpendicular_a line_n gfh_v then_o i_o
the_o touch_n and_o cut_v the_o circle_n may_v either_o pass_v by_o the_o centre_n or_o not_o if_o it_o pass_v by_o the_o centre_n then_o be_v it_o manifest_a by_o the_o 18._o of_o this_o book_n that_o it_o fall_v perpendicular_o upon_o the_o touch_n line_n and_o divide_v the_o circle_n into_o two_o equal_a part_n so_o that_o all_o the_o angle_n in_o each_o semicircle_n be_v by_o the_o former_a proposition_n right_a angle_n and_o therefore_o equal_a to_o the_o alternate_a angle_n make_v by_o the_o say_v perpendicular_a line_n and_o the_o touch_n line_n if_o it_o pass_v not_o by_o the_o centre_n then_o follow_v the_o construction_n and_o demonstration_n before_o put_v the_o 5._o problem_n the_o 33._o proposition_n upon_o a_o right_a line_n give_v to_o describe_v a_o segment_n of_o a_o circle_n which_o shall_v contain_v a_o angle_n equal_a to_o a_o rectiline_a angle_n give_v svppose_v that_o the_o right_a line_n give_v be_v ab_fw-la and_o let_v the_o rectiline_a angle_n geâ—â—n_v be_v c._n it_o be_v require_v upon_o the_o right_a line_n give_v ab_fw-la to_o describe_v a_o segment_n of_o a_o circle_n which_o shall_v contain_v a_o angle_n equal_a to_o the_o angle_n c._n now_o the_o angle_n c_o be_v either_o a_o acute_a angle_n or_o a_o right_a angle_n or_o a_o obtuse_a angle_n case_n first_o let_v it_o be_v a_o acute_a angle_n as_o in_o the_o first_o description_n and_o by_o the_o 23_o of_o the_o first_o upon_o the_o right_a line_n ab_fw-la and_o to_o the_o point_n in_o it_o a_o describe_v a_o angle_n equal_a to_o the_o angle_n c_o construction_n and_o let_v the_o same_o be_v dab_n wherefore_o the_o angle_n dab_n be_v a_o acute_a angle_n from_o the_o point_n a_o raise_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o line_n ad_fw-la a_o perpendicular_a line_n af._n and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n ab_fw-la into_o two_o equal_a part_n in_o the_o point_n f._n and_o by_o the_o 11._o of_o the_o same_o from_o the_o point_n fletcher_n raise_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n fg_o and_o draw_v a_o line_n from_o g_z to_o b._n demonstration_n and_o forasmuch_o as_o the_o line_n of_o be_v equal_a to_o the_o line_n fb_o and_o the_o line_n fg_o be_v common_a to_o they_o both_o therefore_o these_o two_o line_n of_o and_o fg_o be_v equal_a to_o these_o two_o line_n fb_o and_o fg_o and_o the_o angle_n afg_n be_v by_o the_o 4._o petition_n equal_a to_o the_o angle_n gfb_n wherefore_o by_o the_o 4._o of_o the_o same_o the_o base_a agnostus_n be_v equal_a to_o the_o base_a gb_o wherefore_o make_v the_o centre_n g_o and_o the_o space_n ga_o describe_v by_o the_o 3._o petition_n a_o circle_n and_o it_o shall_v pass_v by_o the_o point_n b_o describe_v such_o a_o circle_n &_o let_v the_o same_o be_v abe_n and_o draw_v a_o line_n from_o e_o to_o b._n now_o forasmuch_o as_o from_o the_o end_n of_o the_o diameter_n ae_n namely_o from_o the_o point_n a_o be_v 〈◊〉_d a_o right_a line_n ad_fw-la make_v together_o with_o the_o right_a line_n ae_n a_o right_a angle_n therefore_o by_o the_o corollary_n of_o the_o 16._o of_o the_o three_o the_o line_n ad_fw-la touch_v the_o circle_n abe_n and_o forasmuch_o as_o a_o certain_a right_a line_n ad_fw-la touch_v the_o circle_n abe_n &_o from_o the_o point_n a_o where_o the_o touch_n be_v be_v draw_v into_o the_o circle_v a_o certain_a right_a line_n ab_fw-la therefore_o by_o the_o 32._o of_o the_o three_o the_o angle_n dab_n be_v equal_a to_o the_o angle_n aeb_fw-mi which_o be_v in_o the_o alternate_a segment_n of_o the_o circle_n but_o the_o angle_n dab_n be_v equal_a to_o the_o angle_n c_o wherefore_o the_o angle_n c_o be_v equal_a to_o the_o angle_n aeb_fw-mi wherefore_o upon_o the_o right_a line_n give_v ab_fw-la be_v describe_v a_o segment_n of_o a_o circle_n which_o contain_v the_o angle_n aeb_fw-mi which_o be_v equal_a to_o the_o angle_n give_v namely_o to_o c._n case_n but_o now_o suppose_v that_o the_o angle_n c_o be_v a_o right_a angle_n it_o be_v again_o require_v upon_o the_o right_a line_n ab_fw-la to_o describe_v a_o segment_n of_o a_o circle_n which_o shall_v contain_v a_o angle_n equal_a to_o the_o right_a angle_n c._n construction_n describe_v again_o upon_o the_o right_a line_n ab_fw-la and_o to_o the_o point_n in_o it_o a_o a_o angle_n bid_v equal_a to_o the_o rectiline_a angle_n give_v c_o by_o the_o 23._o of_o the_o first_o as_o it_o be_v set_v forth_o in_o the_o second_o description_n and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n ab_fw-la into_o two_o equal_a part_n in_o the_o point_n f._n and_o make_v the_o centre_n the_o point_n f_o and_o the_o space_n favorina_n or_o fb_o describe_v by_o the_o 3._o petition_n the_o circle_z aeb_fw-mi demonstration_n wherefore_o the_o right_a line_n ad_fw-la touch_v the_o circle_n aeb_fw-mi for_o that_o the_o angle_n bad_a be_v a_o right_a angle_n wherefore_o the_o angle_v bad_a be_v equal_a to_o the_o angle_n which_o be_v in_o the_o segment_n aeb_fw-mi for_o the_o angle_n which_o be_v in_o a_o semicircle_n be_v a_o right_a angle_n by_o the_o 31._o of_o the_o three_o but_o the_o angle_n bad_a be_v equal_a to_o the_o angle_n c._n wherefore_o there_o be_v again_o describe_v upon_o the_o line_n ab_fw-la a_o segment_n of_o a_o circle_n namely_o aeb_n which_o contain_v a_o angle_n equal_a to_o the_o angle_n give_v namely_o to_o c._n but_o now_o suppose_v that_o the_o angle_n c_o be_v a_o obtuse_a angle_n case_n upon_o the_o right_a line_n ab_fw-la and_o to_o the_o point_n in_o it_o a_o describe_v by_o the_o 23._o of_o the_o first_o a_o angle_n bid_v equal_a to_o the_o angle_n c_o as_o it_o be_v in_o the_o three_o description_n construction_n and_o from_o the_o point_n a_o raise_v up_o unto_o the_o line_n ad_fw-la a_o perpendicular_a line_n ae_n by_o the_o 11._o of_o the_o first_o and_o again_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n ab_fw-la into_o two_o equal_a part_n in_o the_o point_n f._n and_o from_o the_o point_n f._n raâ—se_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n fg_o by_o the_o 11._o of_o the_o same_o &_o draw_v a_o line_n from_o g_z to_o b._n demonstration_n and_o now_o forasmuch_o as_o the_o line_n of_o be_v equal_a to_o the_o line_n fb_o and_o the_o line_n fg_o be_v common_a to_o they_o both_o therefore_o these_o two_o line_n of_o and_o fg_o be_v equal_a to_o these_o two_o line_n bf_o and_o fg_o and_o the_o angle_n afg_n be_v by_o the_o 4._o petition_n equal_a to_o the_o angle_n bfg_n wherefore_o by_o the_o 4._o of_o the_o same_o the_o base_a agnostus_n be_v equal_a to_o the_o base_a gb_o wherefore_o make_v the_o centre_n g_o and_o the_o space_n ga_o describe_v by_o the_o 3._o petition_n a_o circle_n and_o it_o shall_v pass_v by_o the_o point_n b_o let_v it_o be_v describe_v as_o the_o circle_n aeb_fw-mi be_v and_o forasmuch_o as_o from_o the_o end_n of_o the_o diameter_n ae_n be_v draw_v a_o perpendicular_a line_n ad_fw-la therefore_o by_o the_o corollary_n of_o the_o 16._o of_o the_o three_o the_o line_n ad_fw-la touch_v the_o circle_n aeb_fw-mi &_o from_o the_o point_n of_o the_o touch_n namely_o a_o be_v extend_v the_o line_n ab_fw-la wherefore_o by_o the_o 32._o of_o the_o three_o the_o angle_n bad_a be_v equal_a to_o the_o angle_n ahb_n which_o be_v in_o the_o alternate_a segment_n of_o the_o circle_n but_o the_o angle_n bad_a be_v equal_a to_o the_o angle_n c._n wherefore_o the_o angle_n which_o be_v in_o the_o segment_n ahb_n be_v equal_a to_o the_o angle_n c._n wherefore_o upon_o the_o right_a line_n give_v ab_fw-la be_v describe_v a_o segment_n of_o a_o circle_n ahb_n which_o contain_v a_o angle_n equal_a to_o the_o angle_n give_v namely_o c_o which_o be_v require_v to_o be_v do_v the_o 6._o problem_n the_o 34._o proposition_n from_o a_o circle_n give_v to_o cut_v away_o a_o section_n which_o shall_v contain_v a_o angle_n equal_a to_o a_o rectiline_a angle_n give_v svppose_v that_o the_o circle_n give_v be_v ac_fw-la and_o let_v the_o rectiline_a angle_n give_v be_v d._n it_o be_v require_v from_o the_o circle_n abc_n to_o cut_v away_o a_o segment_n which_o shall_v contain_v a_o angle_n equal_a to_o the_o angle_n d._n construction_n draw_v by_o the_o 17._o of_o the_o three_o a_o line_n touch_v the_o circle_n and_o let_v the_o same_o be_v of_o and_o let_v it_o touch_v in_o the_o point_n b._n and_o by_o the_o 23._o of_o the_o first_o upon_o the_o right_a line_n of_o and_o to_o the_o point_n in_o it_o b_o describe_v the_o angle_n fbc_n equal_a to_o the_o angle_n d._n demonstration_n now_o forasmuch_o as_o a_o certain_a right_a line_n of_o touch_v the_o circle_n abc_n in_o the_o point_n b_o and_o frora_fw-la
proportion_n that_o the_o first_o of_o the_o three_o line_n put_v be_v to_o the_o 〈◊〉_d â—or_a tâ—e_z 〈◊〉_d line_n to_o the_o three_o namely_o the_o line_n ae_n to_o the_o line_n ebb_n be_v in_o double_a proportion_n that_o it_o be_v to_o the_o second_o by_o the_o 10._o definition_n of_o the_o fiâ—t_z ¶_o the_o second_o corollary_n hereby_o may_v we_o learn_v how_o from_o a_o rectiline_a â—igure_n give_v to_o take_v away_o a_o part_n appoint_v leaâ—ing_v the_o rest_n of_o the_o rectiline_a â—igure_n like_o unto_o the_o whole_a corollary_n for_o if_o from_o the_o right_a line_n ab_fw-la be_v cut_v of_o a_o part_n appoint_v namely_o ebb_n by_o the_o 9_o of_o this_o book_n as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o rectiline_a â—igure_n describe_v of_o the_o line_n of_o to_o the_o rectiline_a figure_n describe_v of_o the_o line_n fb_o the_o say_v figure_n being_n suppose_v to_o be_v like_o both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o rectiline_a â—igure_n describe_v of_o the_o line_n ab_fw-la and_o be_v also_o in_o like_a sort_n situate_a wherefore_o take_v away_o â—rom_o the_o rectiline_a â—igure_n describe_v of_o the_o line_n ab_fw-la the_o rectiline_a figure_n describe_v of_o the_o line_n fb_o the_o residue_n namely_o the_o rectiline_a figure_n describe_v of_o the_o line_n of_o shall_v be_v both_o like_a unto_o the_o whole_a rectiline_a â—igure_n give_v describe_v of_o the_o line_n ab_fw-la and_o in_o like_a sort_n situate_a ¶_o the_o three_o corollary_n to_o compose_v two_o like_o rectiline_a â—_z into_o one_o rectiline_a figure_n like_o and_o equal_a to_o the_o same_o figure_n let_v their_o side_n of_o like_a proportion_n be_v set_v so_o that_o they_o make_v a_o right_a angle_n corollary_n as_o the_o line_n of_o and_o fb_o be_v and_o upon_o the_o line_n subtend_v the_o say_a angle_n namely_o the_o line_n ab_fw-la describe_v a_o rectiline_a â—igure_n like_o unto_o the_o rectiline_a figure_n give_v and_o in_o like_a sort_n situate_a by_o the_o 18._o of_o this_o book_n and_o the_o same_o shall_v be_v equal_a to_o the_o two_o rectiline_a figure_n give_v by_o the_o 31._o of_o this_o book_n ¶_o the_o second_o proposition_n if_o two_o right_a line_n cut_v the_o one_o the_o other_o obtuseangle_v wise_a and_o from_o the_o end_n of_o the_o line_n which_o â—ut_z the_o one_o the_o other_o be_v draw_v perpendicular_a line_n to_o either_o line_n the_o line_n which_o be_v between_o the_o end_n and_o the_o perpendicular_a line_n be_v cut_v reciprocal_a suppose_v that_o there_o be_v two_o right_a line_n ab_fw-la and_o gd_o cut_v the_o one_o the_o other_o in_o the_o point_n e_o and_o make_v a_o obtuse_a angle_n in_o the_o section_n e._n and_o from_o the_o end_n of_o the_o line_n namely_o a_o and_o g_o let_v there_o be_v draw_v to_o either_o line_n perpendicular_a line_n namely_o from_o the_o point_n a_o to_o the_o line_n gd_o which_o let_v be_v ad_fw-la and_o from_o the_o point_n g_o to_o the_o right_a line_n ab_fw-la which_o let_v be_v gb_o then_o i_o say_v that_o the_o right_a line_n ab_fw-la and_o gd_o do_v between_o the_o end_n a_o and_o the_o perpendicular_a b_o and_o the_o end_n g_o and_o the_o perpendicular_a d_o cut_v the_o one_o the_o other_o reciprocal_a in_o the_o point_n e_o so_o that_o as_o the_o line_n ae_n be_v to_o the_o line_n ed_z so_o be_v the_o line_n ge_z to_o the_o line_n ebb_n proposition_n for_o forasmuch_o as_o the_o angle_n ade_n and_o gbe_n be_v right_a angle_n therefore_o they_o be_v equal_a but_o the_o angle_n aed_n and_o geb_fw-mi be_v also_o equal_a by_o the_o 15._o of_o the_o first_o wherefore_o the_o angle_n remain_v namely_o ead_n &_o egb_n be_v equal_a by_o the_o corollary_n of_o the_o 32._o of_o the_o first_o wherefore_o the_o triangle_n aed_n and_o geh_a be_v equiangle_n wherefore_o the_o side_n about_o the_o equal_a angle_n shall_v be_v proportional_a by_o the_o 4._o of_o the_o six_o wherefore_o as_o the_o line_n ae_n be_v to_o the_o line_n ed_z so_o be_v the_o line_n ge_z to_o the_o line_n ebb_n if_o therefore_o two_o right_a line_n cut_v the_o one_o the_o other_o obtuseangled_a wife_n etc_o etc_o which_o be_v require_v to_o be_v prove_v ¶_o the_o three_o proposition_n if_o two_o right_a line_n make_v a_o acute_a angle_n and_o from_o their_o end_n be_v draw_v to_o each_o line_n perpendicular_a line_n cut_v they_o the_o two_o right_a line_n give_v shall_v be_v reciprocal_a proportional_a as_o the_o segment_n which_o be_v about_o the_o angle_n suppose_v that_o there_o be_v two_o right_a line_n ab_fw-la and_o gb_o make_v a_o acute_a angle_n abg_o and_o from_o the_o point_n a_o and_o g_o let_v there_o be_v draw_v unto_o the_o line_n ab_fw-la and_o gb_o perpendicular_a line_n ac_fw-la and_o ge_z cut_v the_o line_n ab_fw-la and_o gb_o in_o the_o point_n e_o and_o â—_o then_o i_o say_v that_o the_o line_n namely_o ab_fw-la to_o gb_o be_v reciprocal_a proportional_a as_o the_o segment_n namely_o cb_o to_o ebb_v which_o be_v about_o the_o acute_a angle_n b._n proposition_n for_o forasmuch_o as_o thâ—_z right_o angle_n acb_o and_o gerard_n be_v equal_a and_o the_o angleâ—_n abg_o be_v common_a to_o the_o triangle_n abc_n and_o gbeâ—_n â—_z therefore_o the_o angle_n remain_v bac_n and_o egb_n be_v equal_a by_o the_o corollary_n of_o the_o 32._o of_o the_o first_o wherefore_o the_o triangle_n abc_n and_o gbe_n be_v equiangle_n wherefore_o the_o sideâ—_n about_o the_o equal_a angle_n be_v proportional_a by_o the_o 4._o of_o the_o six_o â—_o so_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n fc_o so_o be_v the_o line_n gb_o to_o the_o line_n be._n wherefore_o alternate_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n gb_o so_o be_v the_o line_n cb_o to_o the_o line_n be._n if_o therefore_o two_o right_a line_n makâ—_n aâ—â—câ—te_a angleâ—_n &_o câ—_n which_o be_v require_v to_o be_v prove_v â—_o the_o four_o proposition_n if_o in_o a_o circle_n be_v draw_v two_o right_a line_n cut_v the_o one_o the_o other_o the_o section_n of_o the_o one_o to_o the_o section_n of_o the_o other_o shall_v be_v reciprocal_a proportional_a in_o the_o circle_n agb_n let_v these_o two_o right_a line_n 〈…〉_o one_o the_o other_o in_o the_o point_n e._n proposition_n thâ—â—_n i_o say_v that_o reciprocal_a 〈◊〉_d â—hâ—_n line_n ae_n be_v to_o the_o line_n ed_z so_o be_v the_o line_n ge_z to_o the_o line_n ebb_n for_o forasmuch_o as_o by_o the_o 35._o of_o the_o three_o the_o rectangle_n figure_n contain_v under_o the_o line_n ae_n and_o ebb_n be_v equal_a to_o the_o rectangle_n figure_n contain_v under_o the_o line_n ge_z and_o ed_z but_o in_o equal_a rectangle_n parallelogram_n the_o side_n about_o the_o equal_a angle_n be_v reciprocal_a by_o the_o 14._o of_o the_o six_o therefore_o the_o line_n ae_n be_v to_o the_o line_n ed_z reciprocal_a as_o the_o line_n ge_z be_v to_o the_o line_n ebb_n by_o the_o second_o definition_n of_o the_o six_o if_o therefore_o in_o a_o circle_n be_v draw_v two_o right_a line_n etc_o etc_o which_o be_v require_v to_o be_v prove_v ¶_o the_o five_o proposition_n if_o from_o a_o point_n give_v be_v draw_v in_o a_o plain_a superficies_n two_o right_a line_n to_o the_o concave_a circumference_n of_o a_o circle_n they_o shall_v be_v reciprocal_a proportional_a with_o their_o part_n take_v without_o the_o circle_n and_o moreover_o a_o right_a line_n draw_v from_o the_o say_a point_n &_o touch_v the_o circle_n shall_v be_v a_o mean_a proportional_a between_o the_o whole_a line_n and_o the_o utter_a segment_n suppose_v that_o there_o be_v a_o circle_n abdella_n and_o without_o it_o take_v a_o certain_a point_n namely_o g._n and_o from_o the_o point_n g_o draw_v unto_o the_o concave_a circumference_n two_o right_a line_n gb_o and_o gd_o cut_v the_o circle_n in_o the_o point_n c_o and_o e._n and_o let_v the_o line_n ga_o touch_v the_o circle_n in_o the_o point_n a._n then_o i_o say_v that_o the_o line_n namely_o gb_o to_o gd_o be_v reciprocal_a as_o their_o part_n take_v without_o the_o circle_n namely_o as_o gc_o to_o ge._n proposition_n for_o forasmuch_o as_o by_o the_o corollary_n of_o the_o 36._o of_o the_o three_o the_o rectangle_n figure_n contain_v under_o the_o line_n gb_o and_o ge_z be_v equal_a to_o the_o rectangle_n figure_n contain_v under_o the_o line_n gd_o and_o gc_o therefore_o by_o the_o 14._o of_o the_o six_o reciprocal_a as_o the_o line_n gb_o be_v to_o the_o line_n gd_o so_o be_v the_o line_n gc_o to_o the_o line_n ge_z for_o they_o be_v side_n contain_v equal_a angle_n i_o say_v moreover_o that_o between_o the_o line_n gb_o and_o ge_z or_o between_o the_o line_n gd_o and_o gc_o the_o touch_n line_n ga_o be_v a_o mean_a proportional_a part_n for_o forasmuch_o as_o the_o rectangle_n
draw_v by_o the_o 11._o of_o the_o firsâ—_o a_o perpendicular_a line_n ab_fw-la and_o let_v ab_fw-la be_v a_o rational_a line_n and_o make_v perfectâ—_n the_o parallelogram_n bc._n wherefore_o bg_o be_v irrational_a by_o that_o which_o be_v declare_v and_o prove_v in_o manner_n of_o a_o assumpt_n in_o the_o end_n of_o the_o demonstration_n of_o the_o 38_o and_o the_o line_n that_o contain_v it_o iâ—_z power_n be_v also_o irrational_a let_v the_o line_n cd_o contain_v in_o power_n the_o superficies_n bc._n wherefore_o cd_o be_v irrational_a &_o not_o of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o for_o the_o square_n of_o the_o line_n cd_o apply_v to_o a_o rational_a line_n namely_o ab_fw-la make_v the_o breadth_n a_o medial_a line_n namely_o ac_fw-la but_o the_o square_n of_o none_o of_o the_o foresay_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n a_o medial_a line_n again_o make_v perfect_v the_o parallelogram_n ed._n wherefore_o the_o parallelogram_n ed_z be_v also_o irrational_a by_o the_o say_v assumpt_v in_o the_o end_n of_o the_o 98._o his_o demonstration_n brieâ—ly_o prove_v and_o the_o line_n which_o contain_v it_o in_o power_n be_v irrational_o let_v the_o line_n which_o contain_v it_o in_o power_n be_v df._n wherefore_o df_o be_v irrational_a and_o not_o of_o the_o self_n same_o kind_n with_o any_o of_o the_o foresay_a irrational_a line_n for_o the_o square_n of_o none_o of_o the_o foresay_a irrational_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n the_o line_n cd_o wherefore_o of_o a_o medial_a line_n be_v produce_v infinite_a irrational_a line_n of_o which_o none_o be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 92._o theorem_a the_o 116._o proposition_n now_o let_v we_o prove_v that_o in_o square_a figure_n the_o diameter_n be_v incommensurable_a in_o length_n to_o the_o side_n svppose_v that_o abcd_o be_v a_o square_a and_o let_v the_o diameter_n thereof_o be_v ac_fw-la then_o i_o say_v that_o the_o diameter_n ac_fw-la be_v incommensurable_a in_o length_n to_o the_o side_n ab_fw-la for_o if_o it_o be_v possible_a impossibility_n let_v it_o be_v commensurable_a in_o length_n i_o say_v that_o they_o this_o will_v follow_v that_o one_o and_o the_o self_n same_o number_n shall_v be_v both_o a_o even_a number_n &_o a_o odd_a number_n it_o be_v manifest_a by_o the_o 47._o of_o the_o first_o that_o the_o square_a of_o the_o line_n ac_fw-la be_v double_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o for_o that_o the_o line_n ac_fw-la be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la by_o supposition_n therefore_o the_o line_n ac_fw-la have_v unto_o the_o line_n ab_fw-la that_o proportion_n that_o a_o number_n have_v to_o a_o number_n by_o the_o 5._o of_o the_o ten_o let_v the_o line_n ac_fw-la have_v unto_o the_o line_n ab_fw-la that_o proportion_n that_o the_o number_n of_o have_v to_o the_o number_n g._n and_o let_v of_o and_o g_z be_v the_o least_o number_n that_o have_v one_o and_o the_o same_o proportion_n with_o they_o wherefore_o of_o be_v not_o unity_n for_o if_o of_o be_v unity_n and_o it_o have_v to_o the_o number_n g_o that_o proportion_n that_o the_o line_n ac_fw-la have_v to_o the_o line_n ab_fw-la and_o the_o line_n ac_fw-la be_v great_a than_o the_o line_n ab_fw-la wherefore_o unity_n of_o be_v great_a than_o the_o number_n g_o which_o be_v impossible_a wherefore_o fe_o be_v not_o unity_n wherefore_o it_o be_v a_o number_n and_o for_o that_o as_o the_o square_n of_o the_o line_n ac_fw-la be_v to_o the_o square_n of_o the_o line_n ab_fw-la so_o be_v the_o square_a number_n of_o the_o number_n of_o to_o the_o square_a number_n of_o the_o number_n g_o for_o in_o each_o be_v the_o proportion_n of_o their_o side_n double_v by_o the_o corollary_n of_o the_o 20_o of_o the_o six_o and_o 11._o of_o the_o eight_o and_o the_o proportion_n of_o the_o line_n ac_fw-la to_o the_o line_n ab_fw-la double_v be_v equal_a to_o the_o proportion_n of_o the_o number_n of_o to_o the_o number_n g_o double_a for_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n ab_fw-la so_o be_v the_o number_n of_o to_o the_o number_n g._n but_o the_o square_n of_o the_o line_n ac_fw-la be_v double_a to_o the_o square_n of_o the_o line_n ab_fw-la wherefore_o the_o square_a number_n produce_v of_o the_o number_n of_o be_v double_a to_o the_o square_a number_n produce_v of_o the_o number_n g._n wherefore_o the_o square_a number_n produce_v of_o of_o be_v a_o even_a number_n wherefore_o of_o be_v also_o a_o even_a number_n for_o if_o of_o be_v a_o odd_a number_n the_o square_a number_n also_o produce_v of_o it_o shall_v by_o the_o 23._o and_o 29._o of_o the_o nine_o be_v a_o odd_a number_n for_o if_o odd_a number_n how_o many_o soever_o be_v add_v together_o and_o if_o the_o multitude_n of_o they_o be_v odd_a the_o whole_a also_o shall_v be_v odd_a wherefore_o of_o be_v a_o even_a number_n divide_v the_o number_n of_o into_o two_o equal_a part_n in_o h._n and_o forasmuch_o as_o the_o number_n of_o and_o g_z be_v the_o least_a number_n in_o that_o proportion_n therefore_o by_o the_o 24._o of_o the_o seven_o they_o be_v prime_a number_n the_o one_o to_o the_o other_o and_o of_o be_v a_o even_a number_n wherefore_o g_z be_v a_o odd_a number_n for_o if_o g_z be_v a_o even_a number_n the_o number_n two_z shall_v measure_v both_o the_o number_n of_o and_o the_o number_n g_z for_o every_o even_a number_n have_v a_o half_a part_n by_o the_o definition_n but_o these_o number_n of_o &_o g_o be_v prime_a the_o one_o to_o the_o other_o wherefore_o it_o be_v impossible_a that_o they_o shall_v be_v measure_v by_o two_o or_o by_o any_o other_o number_n beside_o unity_n wherefore_o g_z be_v a_o odd_a number_n and_o forasmuch_o as_o the_o number_n of_o be_v double_a to_o the_o number_n eh_o therefore_o the_o square_a number_n produce_v of_o of_o be_v quadruple_a to_o the_o square_a number_n produce_v of_o eh_o and_o the_o square_a number_n produce_v of_o of_o be_v double_a to_o the_o square_a number_n produce_v of_o g._n wherefore_o the_o square_a number_n produce_v of_o g_o be_v double_a to_o the_o square_a number_n produce_v of_o eh_o wherefore_o the_o square_a number_n produce_v of_o g_o be_v a_o even_a number_n wherefore_o also_o by_o those_o thing_n which_o have_v be_v before_o speak_v the_o number_n g_z be_v a_o even_a number_n but_o it_o be_v prove_v that_o it_o be_v a_o odd_a number_n which_o be_v impossible_a wherefore_o the_o line_n ac_fw-la be_v not_o commensurable_a in_o length_n to_o the_o line_n ab_fw-la wherefore_o it_o be_v incommensurable_a an_o other_o demonstration_n we_o may_v by_o a_o other_o demonstration_n prove_v that_o the_o diameter_n of_o a_o square_n be_v incommensurable_a to_o the_o side_n thereof_o impossibility_n suppose_v that_o there_o be_v a_o square_a who_o diameter_n let_v be_v a_o and_o let_v the_o side_n thereof_o be_v b._n then_o i_o say_v that_o the_o line_n a_o be_v incommensurable_a in_o length_n to_o the_o line_n b._n for_o if_o it_o be_v possible_a let_v it_o be_v commensurable_a in_o length_n and_o again_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o let_v the_o number_n of_o be_v to_o the_o number_n g_o and_o let_v they_o be_v the_o least_o that_o have_v one_o and_o the_o same_o proportion_n with_o they_o wherefore_o the_o number_n of_o and_o g_z be_v prime_a the_o one_o to_o the_o other_o first_o i_o say_v that_o g_z be_v not_o unity_n for_o if_o it_o be_v possible_a let_v it_o be_v unity_n and_o for_o that_o the_o square_a of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o as_o the_o square_a number_n produce_v of_o of_o be_v to_o the_o square_a number_n produce_v of_o g_z as_o it_o be_v prove_v in_o the_o â—ormer_a demonstration_n but_o the_o square_n of_o the_o line_n a_o be_v double_a to_o the_o square_n of_o the_o line_n b._n wherefore_o the_o square_a number_n produce_v of_o of_o be_v double_a to_o the_o square_a number_n produce_v of_o g._n and_o by_o your_o supposition_n g_z be_v unity_n wherefore_o the_o square_a number_n produce_v of_o of_o be_v the_o number_n two_o which_o be_v impossible_a wherefore_o g_z be_v not_o unity_n wherefore_o it_o be_v a_o number_n and_o for_o that_o as_o the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o so_o be_v the_o square_a number_n produce_v of_o of_o to_o the_o square_a number_n produce_v of_o g._n wherefore_o the_o square_a number_n produce_v of_o of_o be_v double_a to_o the_o square_a number_n produce_v of_o g._n wherefore_o the_o square_a number_n produce_v of_o g._n
great_a perfection_n than_o be_v a_o line_n but_o here_o in_o the_o definition_n of_o a_o solid_a or_o body_n euclid_n attribute_v unto_o it_o all_o the_o three_o dimension_n length_n breadth_n and_o thickness_n wherefore_o a_o solid_a be_v the_o most_o perfect_a quantity_n quantity_n which_o want_v no_o dimension_n at_o all_o pass_v a_o line_n by_o two_o dimension_n and_o pass_v a_o superficies_n by_o one_o this_o definition_n of_o a_o solid_a be_v without_o any_o designation_n of_o â—orme_n or_o figure_n easy_o understand_v only_o conceive_v in_o mind_n or_o behold_v with_o the_o eye_n a_o piece_n of_o timber_n or_o stone_n or_o what_o matter_n so_o ever_o else_o who_o dimension_n let_v be_v equal_a or_o unequal_a for_o example_n let_v the_o length_n thereof_o be_v 5._o inch_n the_o breadth_n 4._o and_o the_o thickness_n 2._o if_o the_o dimension_n be_v equal_a the_o reason_n be_v like_a and_o all_o one_o as_o it_o be_v in_o a_o sphere_n and_o in_o â—_o cube_fw-la for_o in_o that_o respect_n and_o consideration_n only_o that_o it_o be_v long_o broad_a and_o thick_a it_o bear_v the_o name_n of_o a_o solid_a or_o body_n â—nd_n have_v the_o nature_n and_o property_n thereof_o there_o be_v add_v to_o the_o endâ—_n of_o the_o definition_n of_o a_o solid_a that_o the_o term_n and_o limit_v of_o a_o solid_a â—s_n a_o superficies_n of_o thing_n infinity_n there_o iâ—_z no_o art_n or_o science_n all_o quantity_n therefore_o in_o this_o art_n entreat_v of_o be_v imagine_v to_o be_v finite_a infinite_a and_o to_o have_v their_o end_n and_o border_n as_o have_v be_v show_v in_o the_o first_o book_n that_o the_o limit_n and_o end_n of_o a_o line_n be_v point_n and_o the_o limit_n or_o border_n of_o a_o superficies_n be_v line_n so_o now_o he_o say_v thaâ—_z the_o end_n limit_n or_o border_n of_o a_o solideâ—_n be_v superficiece_n as_o the_o side_n of_o any_o square_a piece_n of_o timber_n or_o of_o a_o table_n or_o die_v or_o any_o other_o like_a be_v the_o term_n and_o limit_n of_o they_o 2_o a_o right_a line_n be_v then_o erect_v perpendicular_o to_o a_o pl_z 〈…〉_o erficies_n when_o the_o right_a line_n make_v right_a angle_n with_o all_o the_o line_n 〈…〉_o it_o definition_n and_o be_v draw_v upon_o the_o ground_n plain_a superficies_n suppose_v that_o upon_o the_o ground_n plain_a superficies_n cdef_n from_o the_o point_n b_o be_v erect_v a_o right_a line_n namely_o â—a_o so_o that_o let_v the_o point_v a_o be_v a_o loâ—e_n in_o the_o air_n draw_v also_o from_o the_o point_n â—_o in_o the_o plain_a superficies_n cdbf_n as_o many_o right_a line_n as_o you_o listen_v as_o the_o line_n bc_o bd_o â—â—_o bf_n bg_o hk_o bh_o and_o bl_n if_o the_o erect_a line_n basilius_n with_o all_o these_o line_n draw_v in_o the_o superficies_n cdef_n make_v a_o right_a angle_n so_o that_o all_o those_o â—ngles_v aâ—â—_n aâ—d_n aâ—e_n abfâ—_n aâ—g_v aâ—k_n abh_n abl_n and_o so_o of_o other_o be_v right_a angle_n then_o by_o this_o definition_n the_o line_n ab_fw-la iâ—_z a_o line_n â—â—â—cted_v upon_o the_o superficies_n cdef_n it_o be_v also_o call_v common_o a_o perpendicular_a line_n or_o a_o plumb_n line_n unto_o or_o upon_o a_o superficies_n definition_n 3_o a_o plain_a superficies_n be_v then_o upright_o or_o erect_v perpendicular_o to_o a_o plain_a superficies_n when_o all_o the_o right_a line_n draw_v in_o one_o of_o the_o plain_a superficiece_n unto_o the_o common_a section_n of_o those_o two_o plain_a superficiece_n make_v therewith_o right_a angle_n do_v also_o make_v right_a angle_n to_o the_o other_o plain_a superficies_n inclination_n or_o lean_v of_o a_o right_a line_n to_o a_o plain_a superficies_n be_v a_o acute_a angle_n contain_v under_o a_o right_a line_n fall_v from_o a_o point_n above_o to_o the_o plain_a superficies_n and_o under_o a_o other_o right_a line_n from_o the_o low_a end_n of_o the_o say_a line_n let_v down_o draw_v in_o the_o same_o plain_a superficies_n by_o a_o certain_a point_n assign_v where_o a_o right_a line_n from_o the_o first_o point_n above_o to_o the_o same_o plain_a superficies_n fall_v perpendicular_o touch_v in_o this_o three_o definition_n be_v include_v two_o definition_n the_o first_o be_v of_o a_o plain_a superficies_n erect_v perpendicular_o upon_o a_o plain_a superficies_n definition_n the_o second_o be_v of_o the_o inclination_n or_o lean_v of_o a_o right_a line_n unto_o a_o superficies_n of_o the_o first_o take_v this_o example_n suppose_v you_o have_v two_o superâ—icieces_n abcd_o and_o cdef_a of_o which_o let_v the_o superficies_n cdef_n be_v a_o ground_n plain_a superficies_n and_o let_v the_o superficies_n abcd_o be_v erect_v unto_o it_o and_o let_v the_o line_n cd_o be_v a_o common_a term_n or_o intersection_n to_o they_o both_o that_o be_v let_v it_o be_v the_o end_n or_o bind_v of_o either_o of_o they_o part_n &_o be_v draw_v in_o either_o of_o they_o in_o which_o line_n note_n at_o pleasure_n certain_a point_n as_o the_o point_n g_o h._n from_o which_o point_n unto_o the_o line_n cd_o draw_v perpendicular_a line_n in_o the_o superficies_n abcd_o which_o let_v be_v gl_n and_o hk_o which_o fall_v upon_o the_o superficies_n cdef_n if_o they_o cause_v right_a angle_n with_o it_o that_o be_v with_o line_n draw_v in_o it_o from_o the_o same_o point_n g_z and_o h_o as_o if_o the_o angle_n lgm_n or_o the_o angle_n lgn_v contain_v under_o the_o line_n â—g_z draw_v in_o the_o superficies_n erect_v and_o under_o the_o gm_n or_o gn_n draw_v in_o the_o ground_n superficies_n cdef_n lie_v flat_a be_v a_o right_a angle_n then_o by_o this_o definition_n the_o superficies_n abcd_o be_v upright_o or_o erect_v upon_o the_o superficies_n cdef_n it_o be_v also_o common_o call_v a_o superficies_n perpendicular_a upon_o or_o unto_o a_o superficies_n part_n for_o the_o second_o part_n of_o this_o definition_n which_o be_v of_o the_o inclination_n of_o a_o right_a line_n unto_o a_o plain_a superficies_n take_v this_o example_n let_v abcd_o be_v a_o ground_n plain_a superficies_n upon_o which_o from_o a_o point_n be_v a_o loft_n namely_o the_o point_n e_o suppose_v a_o right_a line_n to_o fall_v which_o let_v be_v the_o line_n eglantine_n touch_v the_o plain_a superficies_n abcd_o at_o the_o point_n g._n again_o from_o the_o point_n e_o be_v the_o top_n or_o high_a limit_n and_o end_n of_o the_o incline_a line_n eglantine_n let_v a_o perpendicular_a line_n fall_v unto_o the_o plain_a superficies_n abcd_o which_o let_v be_v the_o line_n of_o and_o let_v fletcher_n be_v the_o point_n where_o of_o touch_v the_o plain_a superficies_n abcd._n then_o from_o the_o point_n of_o the_o fall_n of_o the_o line_n incline_v upon_o the_o superficies_n unto_o the_o point_n of_o the_o fall_n of_o the_o perpendicular_a line_n upon_o the_o same_o superficies_n that_o be_v from_o the_o point_n g_o to_o the_o point_n fletcher_n draw_v a_o right_a line_n gf_o now_o by_o this_o definition_n the_o acute_a angle_n egf_n be_v the_o inclination_n of_o the_o line_n eglantine_n unto_o the_o superficies_n abcd._n because_o it_o be_v contain_v of_o the_o incline_a line_n and_o of_o the_o right_a line_n draw_v in_o the_o superficies_n from_o the_o point_n of_o the_o fall_n of_o the_o line_n incline_v to_o the_o point_n of_o the_o fall_n of_o the_o perpendicular_a line_n which_o angle_n must_v of_o necessity_n be_v a_o acute_a angle_n for_o the_o angle_n efg_o be_v by_o construction_n a_o right_a angle_n and_o three_o angle_n in_o a_o triangle_n be_v equâ—_fw-la 〈…〉_o â—ight_a angle_n wherefore_o the_o other_o two_o angle_n namely_o the_o angle_n egf_n and_o gef_n be_v equâ—_fw-la 〈…〉_o right_n angle_n wherefore_o either_o of_o they_o be_v less_o than_o a_o right_a angle_n wherefore_o the_o angle_n egf_n be_v a_o 〈…〉_o gle_a definition_n 4_o inclination_n of_o a_o plain_a superficies_n to_o a_o plain_a superficies_n be_v a_o acute_a angle_n contain_v under_o the_o right_a line_n which_o be_v draw_v in_o either_o of_o the_o plain_a superficiece_n to_o one_o &_o the_o self_n same_o point_n of_o the_o common_a section_n make_v with_o the_o section_n right_a angle_n suppose_v that_o there_o be_v two_o superficiece_n abcd_o &_o efgh_a and_o let_v the_o superficies_n abcd_o be_v suppose_v to_o be_v erect_v not_o perpendicular_o but_o somewhat_o lean_v and_o incline_v unto_o the_o plain_a superficies_n efgh_o as_o much_o or_o as_o little_a as_o you_o will_v the_o common_a term_n or_o section_n of_o which_o two_o superficiece_n let_v be_v the_o line_n cd_o from_o some_o one_o point_n aâ—_z from_o the_o point_n m_o assign_v in_o the_o common_a section_n of_o the_o two_o superficiece_n namely_o in_o the_o line_n cd_o draw_v a_o perpendicular_a line_n in_o either_o superficies_n in_o the_o ground_n superficies_n efgh_o draw_v the_o line_n mk_o and_o in_o the_o superficies_n abcd_o draw_v the_o line_n ml_o now_o
and_o one_o side_n of_o the_o one_o equal_a to_o one_o side_n of_o the_o other_o namely_o that_o side_n which_o subtend_v one_o of_o the_o equal_a angle_n that_o be_v the_o side_n ha_o be_v equal_a to_o the_o side_n dm_o by_o construction_n wherefore_o the_o side_n remain_v be_v by_o the_o 26._o of_o the_o first_o equal_a to_o the_o side_n remain_v wherefore_o the_o side_n ac_fw-la be_v equal_a to_o the_o side_n df._n in_o like_a sort_n may_v we_o prove_v that_o the_o side_n ab_fw-la be_v equal_a to_o the_o side_n de_fw-fr if_o you_o draw_v a_o right_a line_n from_o the_o point_n h_o to_o the_o point_n b_o and_o a_o other_o from_o the_o point_n m_o to_o the_o point_n e._n for_o forasmuch_o as_o the_o square_n of_o the_o line_n ah_o be_v by_o the_o 47._o of_o the_o first_o equal_v to_o the_o square_n of_o the_o line_n ak_o and_o kh_o and_o by_o the_o same_o unto_o the_o square_n of_o the_o line_n ak_o be_v equal_a the_o square_n of_o the_o line_n ab_fw-la and_o bk_o wherefore_o the_o square_n of_o the_o line_n ab_fw-la bk_o and_o kh_o be_v equal_a to_o the_o square_n of_o the_o line_n ah_o but_o unto_o the_o square_n of_o the_o line_n bk_o and_o kh_o be_v equal_a the_o square_n of_o the_o line_n bh_o by_o the_o 47._o of_o the_o first_o for_o the_o angle_n hkb_n be_v a_o right_a angle_n for_o that_o the_o line_n hk_o be_v erect_v perpendicular_o to_o the_o ground_n plain_a superficies_n wherefore_o the_o square_a of_o the_o line_n ah_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o bh_o wherefore_o by_o the_o 48._o of_o the_o first_o the_o angle_n abh_n be_v a_o right_a angle_n and_o by_o the_o same_o reason_n the_o angle_n dem_n be_v a_o right_a angle_n now_o the_o angle_n bah_o be_v equal_a to_o the_o angle_n edm_n for_o it_o be_v so_o suppose_a and_o the_o line_n ah_o be_v equal_a to_o the_o line_n dm_o wherefore_o by_o the_o 26._o of_o the_o first_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n de._n now_o forasmuch_o as_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n df_o and_o the_o line_n ab_fw-la to_o the_o line_n de_fw-fr therefore_o these_o two_o line_n ac_fw-la and_o ab_fw-la be_v equal_a to_o these_o two_o line_n fd_v and_o de._n but_o the_o angle_n also_o cab_n be_v by_o supposition_n equal_a to_o the_o angle_n fde_v wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a bc_o be_v equal_a to_o the_o base_a of_o and_o the_o triangle_n to_o the_o triangle_n and_o the_o rest_n of_o the_o angle_n to_o the_o rest_n of_o the_o angle_n wherefore_o the_o angle_n acb_o be_v equal_a to_o the_o angle_n dfe_n and_o the_o right_a angle_n ack_z be_v equal_a to_o the_o right_a angle_n dfn._n wherefore_o the_o angle_n remain_v namely_o bck_n be_v equal_a to_o the_o angle_n remain_v namely_o to_o efn_n and_o by_o the_o same_o reason_n also_o the_o angle_n cbk_n be_v equal_a to_o the_o angle_n fen_n wherefore_o there_o be_v two_o triangle_n bck_o &_o efn_n have_v two_o angle_n of_o the_o one_o equal_a to_o two_o angle_n of_o the_o other_o each_o to_o his_o correspondent_a angle_n and_o one_o side_n of_o the_o one_o equal_a to_o one_o side_n of_o the_o other_o namely_o that_o side_n that_o lie_v between_o the_o equal_a angle_n that_o be_v the_o side_n bc_o be_v equal_a to_o the_o side_n of_o wherefore_o by_o the_o 26._o of_o the_o first_o the_o side_n remain_v be_v equal_a to_o the_o side_n remain_v wherefore_o the_o side_n ck_o be_v equal_a to_o the_o side_n fn_o but_o the_o side_n ac_fw-la be_v equal_a to_o the_o side_n df._n wherefore_o these_o two_o side_n ac_fw-la and_o ck_o be_v equal_a to_o these_o two_o side_n df_o and_o fn_o and_o they_o contain_v equal_a angle_n wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a ak_n be_v equal_a to_o the_o base_a dn_o and_o forasmuch_o as_o the_o line_n ah_o be_v equal_a to_o the_o line_n dm_o therefore_o the_o square_a of_o the_o line_n ah_o be_v equal_a to_o the_o square_n of_o the_o line_n dm_o but_o unto_o the_o square_n of_o the_o line_n ah_o be_v equal_a the_o square_n of_o the_o line_n ak_o and_o kh_o by_o the_o 47._o of_o the_o first_o for_o the_o angle_n akh_n be_v a_o right_a angle_n and_o to_o the_o square_n of_o the_o line_n dm_o be_v equal_a the_o square_n of_o the_o line_n dn_o and_o nm_o for_o the_o angle_n dnm_n be_v a_o right_a angle_n wherefore_o the_o square_n of_o the_o line_n ak_o and_o kh_o be_v equal_a to_o the_o square_n of_o the_o line_n dn_o and_o nm_o of_o which_o two_o the_o square_a of_o the_o line_n ak_o be_v equal_a to_o the_o square_n of_o the_o line_n dn_o for_o the_o line_n ak_o be_v prove_v equal_a to_o the_o line_n a_fw-la wherefore_o the_o residue_n namely_o the_o square_a of_o the_o line_n kh_o be_v equal_a to_o the_o residue_n namely_o to_o the_o square_n of_o the_o line_n nm_o wherefore_o the_o line_n hk_o be_v equal_a to_o the_o line_n mn_v and_o forasmuch_o as_o these_o two_o line_n ha_o and_o ak_o be_v equal_a to_o these_o two_o line_n md_o and_o dn_o the_o one_o to_o the_o other_o and_o the_o base_a hk_n be_v equal_a to_o the_o base_a mn_n therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n hak_n be_v equal_a to_o the_o angle_n mdn._n if_o therefore_o there_o be_v two_o superficial_a angle_n equal_a and_o from_o the_o point_n of_o those_o angle_n be_v elevate_v on_o high_a right_a line_n comprehend_v together_o with_o those_o right_a line_n which_o be_v put_v at_o the_o beginning_n equal_a angle_n each_o to_o his_o corespondent_a angle_n and_o if_o in_o each_o of_o the_o erect_a line_n be_v take_v a_o point_n at_o all_o adventure_n and_o from_o those_o point_n be_v draw_v perpendicular_a line_n to_o the_o plain_a superficiece_n in_o which_o be_v the_o angle_n give_v at_o the_o beginning_n and_o frâ—â—_n the_o point_n which_o be_v by_o the_o perpendicular_a line_n make_v in_o the_o two_o plain_a superficiece_n be_v join_v right_a line_n to_o those_o angle_n which_o be_v put_v at_o the_o beginning_n those_o right_a line_n shall_v together_o with_o the_o line_n elevate_v on_o high_a make_v equal_a angle_n which_o be_v require_v to_o be_v prove_v because_o the_o figure_n of_o the_o former_a demonstration_n be_v somewhat_o hard_a to_o conceive_v as_o they_o be_v there_o draw_v in_o a_o plain_a by_o reason_n of_o the_o line_n that_o be_v imagine_v to_o be_v elevate_v on_o high_a i_o have_v here_o set_v other_o figure_n wherein_o you_o must_v erect_v perpendicular_o to_o the_o ground_n superficiece_v the_o two_o triangle_n bhk_v and_o emn_a and_o then_o elevate_v the_o triangle_n dfm_n &_o ache_n in_o such_o sort_n that_o the_o angle_n m_o and_o h_o of_o these_o triangle_n may_v concur_v with_o the_o angle_n m_o and_o h_o of_o the_o other_o erect_v triangle_n and_o then_o imagine_v only_o a_o line_n to_o be_v draw_v from_o the_o point_n g_o of_o the_o line_n agnostus_n to_o the_o point_n l_o in_o the_o ground_n superficies_n compare_v it_o with_o the_o former_a construction_n &_o demonstration_n and_o it_o will_v make_v it_o very_o easy_a to_o conceive_v ¶_o corollary_n by_o this_o it_o be_v manifest_a that_o if_o there_o be_v two_o rectiline_v superficial_a angle_n equal_a and_o upon_o those_o angle_n be_v elevate_v on_o high_a equal_a right_a line_n contain_v together_o with_o the_o right_a line_n put_v at_o the_o beginning_n equal_a angle_n perpendicular_a line_n draw_v from_o those_o elevate_v line_n to_o the_o ground_n plain_a superficiece_n wherein_o be_v the_o angle_n put_v at_o the_o beginning_n be_v equal_a the_o one_o to_o the_o other_o for_o it_o be_v manifest_a that_o the_o perpendicular_a line_n hk_o &_o mn_a which_o be_v draw_v from_o the_o end_n of_o the_o equal_a elevate_v line_n ah_o and_o dm_o to_o the_o ground_n superficiece_n be_v equal_a ¶_o the_o 31._o theorem_a the_o 36._o proposition_n if_o there_o be_v three_o right_a line_n proportional_a a_o parallelipipedon_n describe_v of_o those_o three_o right_a line_n be_v equal_a to_o the_o parallelipipedon_n describe_v of_o the_o middle_a line_n so_o that_o it_o consist_v of_o equal_a side_n and_o also_o be_v equiangle_n to_o the_o foresay_a parallelipipedon_n svppose_v that_o these_o three_o line_n a_o b_o c_o be_v proportional_a as_o a_o be_v to_o b_o so_o let_v b_o be_v to_o c._n then_o i_o say_v that_o the_o parallelipipedon_n make_v of_o the_o line_n a_o b_o c_o be_v equal_a to_o the_o parallelipipedon_n make_v of_o the_o line_n b_o so_fw-mi that_o the_o solid_a make_v of_o the_o line_n b_o consist_v of_o equal_a side_n and_o be_v also_o equiangle_n to_o the_o solid_a make_v of_o the_o line_n a_o b_o c._n describe_v by_o the_o 23._o of_o the_o
one_o of_o the_o circle_n ¶_o a_o assumpt_n add_v by_o flussas_n if_o a_o sphere_n be_v cut_v of_o a_o plain_a superficies_n the_o common_a section_n of_o the_o superficiece_n shall_v be_v the_o circumference_n of_o a_o circle_n suppose_v that_o the_o sphere_n abc_n be_v cut_v by_o the_o plain_a superficies_n aeb_n and_o let_v the_o centre_n of_o the_o sphere_n be_v the_o point_n d._n construction_n and_o from_o the_o point_n d_o let_v there_o be_v draw_v unto_o the_o plain_a superficies_n aeb_n a_o perpendicular_a line_n by_o the_o 11._o of_o the_o eleven_o which_o let_v be_v the_o line_n de._n and_o from_o the_o point_n e_o draw_v in_o the_o plain_a superficies_n aeb_n unto_o the_o common_a section_n of_o the_o say_a superficies_n and_o the_o sphere_n line_n how_o many_o so_o ever_o namely_o ea_fw-la and_o ebb_n and_o draw_v these_o line_n dam_fw-ge and_o db._n now_o forasmuch_o as_o the_o right_a angle_n dea_n and_o deb_n be_v equal_a for_o the_o line_n de_fw-fr be_v erect_v perpendicular_o to_o the_o plain_a superficies_n demonstration_n and_o the_o right_a line_n dam_n &_o db_o which_o subtend_v those_o angle_n be_v by_o the_o 12._o definition_n of_o the_o eleven_o equal_a which_o right_a line_n moreover_o by_o the_o 47._o of_o the_o first_o do_v contain_v in_o power_n the_o square_n of_o the_o line_n de_fw-fr ea_fw-la and_o de_fw-fr ebb_n if_o therefore_o from_o the_o square_n of_o the_o line_n de_fw-fr ea_fw-la and_o de_fw-fr ebb_v you_o take_v away_o the_o square_a of_o the_o line_n de_fw-fr which_o be_v common_a unto_o they_o the_o residue_n namely_o the_o square_n of_o the_o line_n ea_fw-la &_o ebb_n shall_v be_v equal_a wherefore_o also_o the_o line_n ea_fw-la and_o ebb_n be_v equal_a and_o by_o the_o same_o reason_n may_v we_o prove_v that_o all_o the_o right_a line_n draw_v from_o the_o point_n e_o to_o the_o line_n which_o be_v the_o common_a section_n of_o the_o superficies_n of_o sphere_n and_o of_o the_o plain_a superficies_n be_v equal_a wherefore_o that_o line_n shall_v be_v the_o circumference_n of_o a_o circle_n by_o the_o 15._o definition_n of_o the_o first_o circle_n but_o if_o it_o happen_v the_o plain_a superficies_n which_o cut_v the_o sphere_n to_o pass_v by_o the_o centre_n of_o the_o sphere_n the_o right_a line_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o their_o common_a section_n shalâ—_n be_v equal_a by_o the_o 12._o definition_n of_o the_o eleven_o for_o that_o common_a section_n be_v in_o the_o superficies_n of_o the_o sphere_n wherefore_o of_o necessity_n the_o plain_a superficies_n comprehend_v under_o that_o line_n of_o the_o common_a section_n shall_v be_v a_o circle_n and_o his_o centre_n shall_v be_v one_o and_o the_o same_o with_o the_o centre_n of_o the_o sphere_n john_n dee_n euclid_n have_v among_o the_o definition_n of_o solid_n omit_v certain_a which_o be_v easy_a to_o conceive_v by_o a_o kind_n of_o analogy_n as_o a_o segment_n of_o a_o sphere_n a_o sector_n of_o a_o sphere_n the_o vertex_fw-la or_o top_n of_o the_o segment_n of_o a_o sphere_n with_o such_o like_a but_o that_o if_o need_n be_v some_o fartheâ—_n light_n may_v be_v give_v in_fw-ge this_o figure_n next_o before_o description_n understand_v a_o segment_n of_o the_o sphere_n abc_n to_o be_v that_o part_n of_o the_o sphere_n contain_v between_o the_o circle_n ab_fw-la who_o centre_n be_v e_o and_o the_o spherical_a superficies_n afb_o to_o which_o be_v a_o less_o segment_n add_v the_o cone_n adb_o who_o base_a be_v the_o former_a circle_n and_o top_n the_o centre_n of_o the_o sphere_n and_o you_o have_v dafb_n a_o sector_n of_o a_o sphere_n or_o solid_a sector_n as_o i_o call_v it_o de_fw-fr extend_a to_o fletcher_n show_v the_o top_n or_o vertex_fw-la of_o the_o segment_n to_o be_v the_o point_n f_o and_o of_o be_v the_o altitude_n of_o the_o segment_n spherical_a of_o segment_n some_o be_v great_a they_o the_o half_a sphere_n some_o be_v less_o as_o before_o abf_n be_v less_o the_o remanent_fw-la abc_n be_v a_o segment_n great_a than_o the_o half_a sphere_n ¶_o a_o corollary_n add_v by_o the_o same_o flussas_n by_o the_o foresay_a assumpt_n it_o be_v manifest_a that_o if_o from_o the_o centre_n of_o a_o sphere_n the_o line_n draw_v perpendicular_o unto_o the_o circle_n which_o cut_v the_o sphere_n be_v equal_a those_o circle_n be_v equal_a and_o the_o perpendicular_a line_n so_o draw_v fall_n upon_o the_o centre_n of_o the_o same_o circle_n for_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n contain_v in_o power_n the_o power_n of_o the_o perpendicular_a line_n and_o the_o power_n of_o the_o line_n which_o join_v together_o the_o end_n of_o those_o line_n wherefore_o from_o that_o square_a or_o power_n of_o the_o line_n from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n or_o common_a section_n draw_v which_o be_v the_o semidiameter_n of_o the_o sphere_n take_v away_o the_o power_n of_o the_o perpendicular_a which_o be_v common_a to_o they_o â—ound_v about_o it_o follow_v that_o the_o residue_v how_o many_o so_o ever_o they_o be_v be_v equal_a power_n and_o therefore_o the_o line_n be_v equal_a the_o one_o to_o the_o other_o wherefore_o they_o will_v describe_v equal_a circle_n by_o the_o first_o definition_n of_o the_o three_o and_o upon_o their_o centre_n fall_v the_o perpendicular_a line_n by_o the_o 9_o of_o the_o three_o corollary_n and_o those_o circle_n upon_o which_o fall_v the_o great_a perpendicular_a line_n be_v the_o less_o circle_n for_o the_o power_n of_o the_o line_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n be_v always_o one_o and_o equal_a to_o the_o power_n of_o the_o perpendicular_a line_n and_o also_o to_o the_o power_n of_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n to_o their_o circumference_n the_o great_a that_o the_o power_n of_o the_o perpendicular_a line_n take_v away_o from_o the_o power_n contain_v they_o both_o be_v the_o less_o be_v the_o power_n and_o therefore_o the_o line_n remain_v which_o be_v the_o semidiameter_n of_o the_o circle_n and_o therefore_o the_o less_o be_v the_o circle_n which_o they_o describe_v wherefore_o if_o the_o circle_n be_v equal_a the_o perpendicular_a line_n fall_v from_o the_o centre_n of_o the_o sphere_n upon_o they_o shall_v also_o be_v equal_a for_o if_o they_o shall_v be_v great_a or_o less_o the_o circle_n shall_v be_v unequal_a as_o it_o be_v before_o manifest_a but_o we_o suppose_v the_o perpendicular_n to_o be_v equal_a corollary_n also_o the_o perpendicular_a line_n fall_v upon_o those_o base_n be_v the_o least_o of_o all_o that_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n for_o the_o other_o draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n of_o the_o circle_n be_v in_o power_n equal_a both_o to_o the_o power_n of_o the_o perpendicular_n and_o to_o the_o power_n of_o the_o line_n join_v these_o perpendicular_n and_o these_o subtendent_fw-la line_n together_o make_v triangle_n rectangleround_v about_o as_o most_o easy_o you_o may_v conceive_v of_o the_o figure_n here_o annex_v a_o the_o centre_n of_o the_o sphere_n ab_fw-la the_o line_n from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n of_o the_o circle_n make_v by_o the_o section_n bcb_n the_o diameter_n of_o circle_n make_v by_o the_o section_n ac_fw-la the_o perpendicular_n from_o the_o centre_n of_o the_o sphere_n to_o the_o circlesâ—_n who_o diameter_n bcb_n be_v one_o both_o side_n or_o in_o any_o situation_n else_o cb_o the_o semidiameter_n of_o the_o circle_n make_v by_o the_o section_n ao_o a_o perpendicular_a long_o than_o ac_fw-la and_o therefore_o the_o semidiameter_n ob_fw-la be_v less_o acb_o &_o aob_n triangle_n rectangle_n ¶_o the_o 2._o problem_n the_o 17._o proposition_n two_o sphere_n consist_v both_o about_o one_o &_o the_o self_n same_o centre_n be_v give_v to_o inscribe_v in_o the_o great_a sphere_n a_o solid_a of_o many_o side_n which_o be_v call_v a_o polyhedron_n which_o shall_v not_o touch_v the_o superficies_n of_o the_o less_o sphere_n svppose_v that_o there_o be_v two_o sphere_n about_o one_o &_o the_o self_n same_o centre_n namely_o about_o a._n it_o be_v require_v in_o the_o great_a sphere_n to_o inscribe_v a_o polyhedron_n or_o a_o solid_a of_o many_o side_n which_o shall_v not_o with_o his_o superficies_n touch_v the_o superficies_n of_o the_o less_o sphere_n let_v the_o sphere_n be_v cut_v by_o some_o one_o plain_a superficies_n pass_v by_o the_o centre_n a._n construction_n then_o shall_v their_o section_n be_v circle_n flussas_n for_o by_o the_o 12._o definition_n of_o the_o eleven_o the_o diameter_n remain_v fix_v and_o the_o semicircle_n be_v turn_v round_o about_o make_v a_o sphere_n wherefore_o in_o what_o position_n so_o ever_o you_o imagine_v the_o
semicircle_n to_o be_v the_o plain_a superficies_n which_o pass_v by_o it_o shall_v make_v in_o the_o superficies_n of_o the_o sphere_n a_o circle_n and_o it_o be_v manifest_a that_o be_v also_o a_o great_a circle_n for_o the_o diameter_n of_o the_o sphere_n which_o be_v also_o the_o diameter_n of_o the_o semicircle_n and_o therefore_o also_o of_o the_o circle_n be_v by_o the_o 15._o of_o the_o three_o great_a then_o all_o the_o right_a line_n draw_v in_o the_o circle_n or_o sphere_n which_o circle_n shall_v have_v both_o one_o centre_n be_v both_o also_o in_o that_o one_o plain_a superficies_n be_v by_o which_o the_o sphere_n be_v cut_v suppose_v that_o that_o section_n or_o circle_n in_o the_o great_a sphere_n be_v bcde_v and_o in_o the_o less_o be_v the_o circle_n fgh_n draw_v the_o diameter_n of_o those_o two_o circle_n in_o such_o sort_n that_o they_o make_v right_a angle_n and_o let_v those_o diameter_n be_v bd_o and_o ce._n construction_n and_o let_v the_o line_n agnostus_n be_v part_n of_o the_o line_n ab_fw-la be_v the_o semidiameter_n of_o the_o less_o sphere_n and_o circle_n as_z ab_fw-la be_v the_o semidiameter_n of_o the_o great_a sphere_n and_o great_a circle_n both_o the_o sphere_n and_o circle_n have_v one_o and_o the_o same_o centre_n now_o two_o circle_n that_o be_v bcde_v and_o fgh_a consist_v both_o about_o one_o and_o the_o self_n same_o centre_n be_v give_v let_v there_o be_v describe_v by_o the_o proposition_n next_o go_v before_o in_o the_o great_a circle_n bcde_v a_o polygonon_n figure_n consist_v of_o equal_a and_o even_a side_n not_o touch_v the_o less_o circle_n fgh_a and_o let_v the_o side_n of_o that_o figure_n in_o the_o four_o part_n of_o the_o circle_n namely_o in_o be_v be_v bk_o kl_o lm_o and_o me._n and_o draw_v a_o right_a line_n from_o the_o point_n king_n to_o the_o point_n a_o and_o extend_v it_o to_o the_o point_n n._n and_o by_o the_o 12._o of_o the_o eleven_o from_o the_o a_o raise_v up_o to_o the_o superficies_n of_o the_o circle_n bcde_v a_o perpendicular_a line_n axe_n and_o let_v it_o light_v upon_o the_o superficies_n of_o the_o great_a sphere_n in_o the_o point_n x._o noteâ—_n and_o by_o the_o line_n axe_n and_o by_o either_o of_o these_o line_n bd_o and_o kn_n extend_v plain_a superficiece_n now_o by_o that_o which_o be_v before_o speak_v those_o plain_a superficiece_n shall_v in_o the_o perceive_v superficies_n of_o the_o sphere_n make_v two_o great_a circle_n let_v their_o semicircle_n consist_v upon_o the_o diameter_n bd_o and_o kn_n be_v bxd_v and_o kxn_o and_o forasmuch_o satisfy_v as_o the_o line_n xa_n be_v erect_v perpendicular_o to_o the_o plain_a superficies_n of_o the_o circle_n bcde_v therefore_o all_o the_o plain_a superficiece_n which_o be_v draw_v by_o the_o line_n xa_n be_v erect_v perpendicular_o to_o the_o superficies_n of_o the_o circle_n bcde_v by_o the_o 18._o of_o the_o eleven_o wherefore_o the_o semicircle_n bxd_v and_o kxn_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n of_o the_o circle_n bcde_v and_o forasmuch_o as_o the_o semicircle_n bed_n bxd_v and_o kxn_o be_v equal_a for_o they_o consist_v upon_o equal_a diameter_n â—d_a and_o kn_n therefore_o also_o the_o four_o part_n or_o quarter_n of_o those_o circle_n namely_o be_v bx_n and_o kx_n be_v equal_a the_o one_o to_o the_o other_o wherefore_o how_o many_o side_n of_o a_o polygonon_n figure_n there_o be_v in_o the_o four_o part_n or_o quarter_n be_v so_o many_o also_o be_v there_o in_o the_o other_o four_o part_n or_o quarter_n bx_n and_o kx_n equal_a to_o the_o right_a line_n bk_o kl_o lm_o and_o me._n let_v those_o side_n be_v describe_v and_o let_v they_o be_v boy_n open_v pr._n rx_n k_n st_z tu_fw-la and_o vx_n follow_v and_o draw_v these_o right_a line_n so_o tâ—_n and_o vâ—_n and_o from_o the_o point_n oh_o and_o â—_o draw_v to_o the_o plain_a superficies_n of_o the_o circle_n bcde_v perpendicular_a line_n which_o perpendicular_a line_n will_v fall_v upon_o the_o common_a section_n of_o the_o plain_a superficiece_n namely_o upon_o the_o line_n bd_o &_o kn_n by_o the_o 38._o of_o the_o eleven_o for_o that_o the_o plain_a superficiece_n of_o the_o semicircle_n bxd_v kxn_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n of_o the_o circle_n bcde_v let_v those_o perpendicular_a line_n be_v gz_n demonstration_n and_o sweet_a and_o draw_v a_o right_a line_n from_o the_o point_n z_o to_o the_o point_n w._n and_o forasmuch_o as_o in_o the_o equal_a semicircle_n bxd_v and_o kxn_o the_o right_a line_n boy_n and_o k_n be_v equal_a from_o the_o end_n whereof_o be_v draw_v perpendicular_a line_n oz_n and_o sweet_a therefore_o by_o the_o corollary_n of_o the_o 35._o of_o the_o eleven_o the_o line_n oz_n be_v equal_a to_o the_o line_n sweet_a &_o the_o line_n bz_o be_v equal_a to_o the_o line_n kw_n flussas_n prove_v this_o a_o other_o way_n thus_o forasmuch_o as_o in_o the_o triangle_n swk_n and_o ozb_n the_o two_o angle_n swk_n and_o ozb_n be_v equal_a for_o that_o by_o construction_n they_o be_v right_a angle_n and_o by_o the_o 27._o of_o three_o the_o angle_n wk_n and_o zbo_n be_v equal_a for_o they_o subtend_v equal_a circumference_n sxn_v and_o oxd_v and_o the_o side_n sk_n be_v equal_a to_o the_o side_n ob_fw-la as_o it_o have_v before_o be_v prove_v wherefore_o by_o the_o 26._o of_o the_o first_o the_o other_o side_n &_o angle_n be_v equal_a namely_o the_o line_n oz_n to_o the_o line_n sweet_a and_o the_o line_n bz_o to_o the_o line_n kw_n but_o the_o whole_a line_n basilius_n be_v equal_a to_o the_o whole_a line_n ka_o by_o the_o definition_n of_o a_o circle_n wherefore_o the_o residue_n za_n be_v equal_a to_o the_o residue_n wa_n wherefore_o the_o line_n zw_n be_v a_o parallel_n to_o the_o line_n bk_o by_o the_o 2._o of_o the_o six_o and_o forasmuch_o as_o either_o of_o these_o line_n oz_n &_o sweet_a be_v erect_v perpendicular_o to_o the_o plain_a superficies_n of_o the_o circle_n bcde_v therefore_o the_o line_n oz_n be_v a_o parallel_n to_o the_o line_n sweet_a by_o the_o 6._o of_o the_o eleven_o and_o it_o be_v prove_v that_o it_o be_v also_o equal_a unto_o it_o wherefore_o the_o line_n wz_n and_o so_o be_v also_o equal_a and_o parallel_n by_o the_o 7_o of_o the_o eleven_o and_o the_o 33._o of_o the_o first_o and_o by_o the_o 3._o of_o the_o first_o and_o forasmuch_o as_o wz_n be_v a_o parallel_n to_o so_o but_o zw_n be_v a_o parallel_n be_v to_o kb_v wherefore_o so_o be_v also_o a_o parallel_n to_o kb_v by_o the_o 9_o of_o the_o eleven_o and_o the_o line_n boy_n and_o k_n do_v knit_v they_o together_o wherefore_o the_o four_o side_v figure_n book_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n for_o by_o the_o 7._o of_o the_o eleven_o if_o there_o be_v any_o two_o parallel_n right_a line_n and_o if_o in_o either_o of_o they_o be_v take_v a_o point_n at_o allaventure_n a_o right_a line_n draw_v by_o these_o point_n be_v in_o one_o &_o the_o self_n same_o plain_a superficies_n with_o the_o parallel_n and_o by_o the_o same_o reason_n also_o every_o one_o of_o the_o four_o side_v figure_n sopt_v and_o tpkv_n be_v in_o one_o and_o thâ—_z self_n same_o plain_a superficies_n and_o the_o triangle_n vrx_n be_v also_o in_o one_o and_o the_o self_n same_o plain_a superficies_n by_o the_o â—_o of_o the_o eleven_o now_o if_o we_o imagine_v right_a line_n draw_v from_o the_o point_n oh_o s_o p_o t_o r_o five_o to_o the_o point_n 〈◊〉_d there_o shall_v be_v describe_v a_o polyhedron_n or_o a_o solid_a figure_n of_o many_o side_n between_o the_o circumference_n bx_n and_o kx_n compose_v of_o pyramid_n who_o base_n be_v the_o four_o side_v figure_n bkos_n sopt_v tprv_n and_o the_o triangle_n vrx_n and_o top_n the_o point_n a._n and_o if_o in_o every_o one_o of_o the_o side_n kâ—l_o lm_o and_o i_o we_o use_v the_o self_n same_o construction_n that_o we_o do_v in_o bk_o and_o moreover_o in_o the_o other_o three_o quadrant_n or_o quarter_n and_o also_o in_o the_o other_o half_n of_o the_o sphere_n there_o shall_v then_o be_v make_v a_o polyhedron_n or_o solid_a figure_n consist_v of_o many_o side_n describe_v in_o the_o sphere_n which_o polyhedron_n be_v make_v of_o the_o pyramid_n who_o base_n be_v the_o foresay_a four_o side_v figure_n and_o the_o triangle_n vrk_n and_o other_o which_o be_v in_o the_o self_n same_o order_n with_o they_o and_o common_a top_n to_o they_o all_o in_o the_o point_n a._n now_o i_o say_v that_o the_o foresaid_a polihedron_n solid_a of_o many_o side_n touch_v not_o the_o superficies_n of_o the_o less_o sphere_n in_o which_o be_v the_o circle_n fgh_o construction_n draw_v by_o the_o 11._o of_o the_o eleven_o from_o the_o point_n a_o to_o the_o
line_n kg_v into_o two_o equal_a part_n in_o the_o point_n m._n fâ—ussas_n and_o draw_v a_o line_n from_o m_o to_o g._n and_o forasmuch_o as_o by_o construction_n the_o line_n kh_o be_v equal_a to_o the_o line_n ac_fw-la and_o the_o line_n hl_o to_o the_o line_n cb_o therefore_o the_o whole_a line_n ab_fw-la be_v equal_a to_o the_o whole_a line_n kl_o wherefore_o also_o the_o half_a of_o the_o line_n kl_o namely_o the_o line_n lm_o be_v equal_a to_o the_o semidiameter_n bn_o wherefore_o take_v away_o from_o those_o equal_a line_n equal_a part_n bc_o and_o lh_o the_o residue_v nc_n and_o mh_o shall_v be_v equal_a wherefore_o in_o the_o two_o triangle_n mhg_n and_o ncd_a the_o two_o side_n about_o the_o equal_a right_n angle_n dcn_n and_o ghm_n namely_o the_o side_n dc_o cn_fw-la and_o gh_o he_o be_v equal_a wherefore_o the_o base_n mg_o and_o nd_o be_v equal_a by_o the_o 4._o of_o the_o first_o and_o by_o the_o same_o reason_n may_v it_o be_v prove_v that_o right_a line_n draw_v from_o the_o point_n m_o to_o the_o point_n e_o and_o f_o be_v equal_a to_o the_o line_n nd_o but_o the_o right_a line_n nd_o be_v equal_a to_o the_o line_n a_fw-la which_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n wherefore_o the_o line_n mg_o be_v equal_a to_o the_o line_n mk_o &_o also_o to_o the_o line_n i_o mf_n and_o ml_o wherefore_o make_v the_o centre_n the_o point_n m_o and_o the_o space_n mk_o or_o mg_o describe_v a_o semicircle_n kgl_n and_o the_o diameter_n kl_o abide_v fix_v let_v the_o say_a semicircle_n kgl_n be_v move_v round_o about_o until_o it_o return_v to_o the_o same_o place_n from_o whence_o it_o begin_v to_o be_v move_v and_o there_o shall_v be_v describe_v a_o sphere_n about_o the_o centre_n m_o by_o the_o 12._o definition_n of_o the_o eleven_o touch_v every_o one_o of_o the_o angle_n of_o the_o pyramid_n which_o be_v at_o the_o point_n king_n e_o f_o g_o for_o those_o angle_n be_v equal_o distant_a from_o the_o centre_n of_o the_o sphere_n namely_o by_o the_o semidiameter_n of_o the_o say_a sphere_n as_o have_v before_o be_v prove_v wherefore_o in_o the_o sphere_n give_v who_o diameter_n be_v the_o line_n kl_o or_o the_o line_n ab_fw-la be_v inscribe_v a_o tetrahedron_n efgk_n now_o i_o say_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n sesquialtera_fw-la to_o the_o side_n of_o the_o pyramid_n demonstration_n for_o forasmuch_o as_o the_o line_n ac_fw-la be_v double_a to_o the_o line_n cb_o by_o construction_n therefore_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n bc._n wherefore_o by_o conversion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o the_o line_n ab_fw-la be_v sesquialtera_fw-la to_o the_o line_n ac_fw-la but_o as_o the_o line_n basilius_n be_v to_o the_o line_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n basilius_n to_o the_o square_n of_o the_o line_n ad._n for_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n d_o as_o the_o line_n bd_o be_v to_o the_o line_n ad_fw-la so_o be_v the_o same_o ad_fw-la to_o the_o line_n ac_fw-la by_o reason_n of_o the_o likeness_n of_o the_o ttriangle_n dab_n &_o dac_o by_o the_o 8._o of_o the_o six_o &_o by_o reason_n also_o that_o as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o square_a of_o the_o first_o to_o the_o square_n of_o the_o second_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n basilius_n be_v sesquialter_fw-la to_o the_o square_n of_o the_o line_n ad._n but_o the_o line_n basilius_n be_v equal_a to_o the_o diameter_n of_o the_o sphere_n give_v namely_o to_o the_o line_n kl_o as_o have_v be_v prove_v &_o the_o line_n ad_fw-la be_v equal_a to_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o sphere_n wherefore_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o the_o pyramid_n wherefore_o there_o be_v make_v a_o pyramid_n comprehend_v in_o a_o sphere_n give_v and_o the_o diameter_n of_o the_o sphere_n be_v sesquialtera_fw-la to_o the_o side_n of_o the_o pyramid_n which_o be_v require_v to_o be_v do_v and_o prove_v ¶_o a_o other_o demonstration_n to_o prove_v that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ad_fw-la to_o the_o square_n of_o the_o line_n dc_o let_v the_o description_n of_o the_o semicircle_n adb_o be_v as_o in_o the_o first_o description_n and_o upon_o the_o line_n ac_fw-la describe_v by_o the_o 46._o of_o the_o first_o a_o square_a ec_o and_o make_v perfect_v the_o parallelogram_n fb_o now_o forasmuch_o as_o the_o triangle_n dab_n be_v equiangle_n to_o the_o triangle_n dac_o by_o the_o 32._o of_o the_o six_o therefore_o as_o the_o line_n basilius_n be_v to_o the_o line_n ad_fw-la so_o be_v the_o line_n dam_fw-ge to_o the_o line_n ac_fw-la by_o the_o 4._o of_o the_o six_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n ad_fw-la by_o the_o 17._o of_o the_o six_o and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o parallelogram_n ebb_v to_o the_o parallelogram_n fb_o by_o the_o 1._o of_o the_o six_o and_o the_o parallelogram_n ebb_n be_v that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la for_o the_o line_n ea_fw-la be_v equal_a to_o the_o line_n ac_fw-la and_o the_o parallelogram_n bf_o be_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o bc._n wherefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb._n but_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n ad_fw-la by_o the_o corollary_n of_o the_o 8._o of_o the_o six_o and_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o be_v equal_a to_o the_o square_n of_o the_o line_n cd_o for_o the_o perpendicular_a line_n dc_o be_v the_o mean_a proportional_a between_o the_o segment_n of_o the_o base_a namely_o ac_fw-la and_o cb_o by_o the_o former_a corollary_n of_o the_o 8._o of_o the_o six_o for_o that_o the_o angle_n adb_o be_v a_o right_a angle_n wherefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ad_fw-la to_o the_o square_n of_o the_o line_n dc_o by_o the_o 11._o of_o the_o five_o which_o be_v require_v to_o be_v prove_v ¶_o two_o assumpte_n add_v by_o campane_n first_o assumpt_n suppose_v that_o upon_o the_o line_n ab_fw-la be_v erect_v perpendicular_o the_o line_n dc_o which_o line_n dc_o let_v be_v the_o mean_a proportional_a between_o the_o part_n of_o the_o line_n ab_fw-la namely_o ac_fw-la &_o cb_o so_o that_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cd_o so_o let_v the_o line_n cd_o be_v to_o the_o line_n cb._n and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n then_o i_o say_v that_o the_o circumference_n of_o that_o semicircle_n shall_v pass_v by_o the_o point_n d_o which_o be_v the_o end_n of_o the_o perpendicular_a line_n but_o if_o not_o than_o it_o shall_v either_o cut_v the_o line_n cd_o or_o it_o shall_v pass_v above_o it_o and_o include_v it_o not_o touch_v it_o first_o let_v it_o cut_v it_o in_o the_o point_n e._n and_o draw_v these_o right_a line_n ebb_v and_o ea_fw-la wherefore_o by_o the_o 31._o of_o the_o three_o the_o whole_a angle_n aeb_n be_v a_o right_a angle_n wherefore_o by_o the_o first_o part_n of_o the_o corollary_n of_o the_o 8._o of_o the_o six_o the_o line_n ac_fw-la be_v to_o the_o line_n â—c_z as_z the_o line_n ec_o be_v to_o the_o line_n cb._n but_o by_o the_o 8._o of_o the_o five_o the_o proportion_n of_o the_o line_n ac_fw-la to_o the_o line_n ec_o be_v great_a than_o the_o proportion_n of_o the_o same_o line_n ac_fw-la to_o the_o line_n cd_o for_o the_o line_n ce_fw-fr be_v less_o than_o the_o line_n cd_o now_o for_o that_o the_o line_n ce_fw-fr be_v to_o thâ—_z line_n cb_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n ce_fw-fr and_o the_o line_n cd_o be_v to_o the_o line_n cb_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cd_o therefore_o by_o the_o 13._o of_o the_o five_o the_o proportion_n of_o the_o line_n ec_o to_o the_o line_n cb_o be_v great_a than_o the_o proportion_n of_o the_o line_n cd_o to_o the_o line_n cb._n wherefore_o by_o the_o 10._o of_o the_o five_o the_o line_n ec_o be_v great_a than_o the_o line_n dc_o
right_a linâ—s_n now_o then_o multiply_v the_o 20._o triangle_n into_o the_o side_n of_o one_o of_o the_o triangle_n and_o so_o shall_v there_o be_v produce_v 6â—_o â—he_z half_a of_o which_o be_v 30._o and_o so_o many_o side_n have_v a_o icosahedron_n and_o in_o like_a sort_n in_o a_o dodecahedron_n forasmuch_o as_o 12._o pentagon_n make_v a_o dodecahedron_n and_o every_o pentagon_n contain_v â—_o right_a linesâ—_n multiply_v â—â—_o into_o 12._o and_o there_o shall_v be_v produce_v 60._o the_o half_a of_o which_o be_v 30._o and_o so_o many_o be_v the_o side_n of_o a_o dodecahedron_n and_o the_o reason_n why_o we_o take_v the_o half_a iâ—_z for_o that_o every_o side_n whether_o it_o be_v of_o a_o triangle_n or_o of_o a_o pentagon_n or_o of_o a_o square_a as_o in_o a_o cube_fw-la â—s_v take_v twice_o and_o by_o the_o same_o reason_n may_v you_o find_v out_o how_o many_o side_n be_v in_o a_o cube_fw-la and_o in_o a_o pyramid_n and_o in_o a_o octohedron_n but_o now_o again_o if_o you_o will_v find_v out_o the_o number_n of_o the_o angle_n of_o every_o one_o of_o the_o solid_a figure_n when_o you_o have_v do_v the_o same_o multiplication_n that_o you_o do_v before_o divide_v the_o same_o side_n by_o the_o number_n of_o the_o plain_a superficiece_n which_o comprehend_v one_o of_o the_o angle_n of_o the_o solid_n as_o for_o example_n forasmuch_o as_o 5._o triangle_n contain_v the_o solid_a angle_n of_o a_o icosahedron_n divide_v 60._o by_o 5._o and_o there_o will_v come_v forth_o 12._o and_o so_o many_o solid_a angle_n have_v a_o icosahedron_n in_o a_o dodecahedron_n forasmuch_o as_o three_o pentagon_n comprehend_v a_o angle_n divide_v 60._o by_o 3._o and_o there_o will_v come_v forth_o 20_o and_o so_o many_o be_v the_o angle_n of_o a_o dodecahedron_n and_o by_o the_o same_o reason_n may_v you_o find_v out_o how_o many_o angle_n be_v in_o each_o of_o the_o rest_n of_o the_o solid_a figure_n book_n if_o it_o be_v require_v to_o be_v know_v how_o one_o of_o the_o plain_n of_o any_o of_o the_o five_o solid_n be_v give_v there_o may_v be_v find_v out_o the_o inclination_n of_o the_o say_a plain_n the_o one_o to_o the_o other_o which_o contain_v each_o of_o the_o solid_n this_o as_o say_v isidorus_n our_o great_a master_n be_v foâ—â—d_v out_o after_o this_o manner_n it_o be_v manifest_a that_o in_o a_o cube_fw-la the_o plain_n which_o contain_v iâ—_z doâ—_n 〈◊〉_d the_o one_o the_o other_o by_o a_o right_a angle_n but_o in_o a_o tetrahedron_n one_o of_o the_o triangle_n be_v give_v let_v the_o end_n of_o one_o of_o the_o side_n of_o the_o say_a triangle_n be_v the_o centre_n and_o let_v the_o space_n be_v the_o perpendicular_a line_n draw_v from_o the_o top_n of_o the_o triangle_n to_o the_o base_a and_o describe_v circumferâ—nces_n of_o a_o circle_n which_o shall_v cut_v the_o one_o the_o other_o and_o from_o the_o intersection_n to_o the_o centre_n draw_v right_a line_n which_o shall_v contain_v the_o inclination_n of_o the_o plain_n contain_v the_o tetrahedron_n in_o a_o octoâ—edron_n take_v one_o of_o the_o side_n of_o the_o triangle_n thereof_o and_o upon_o it_o describe_v a_o square_a and_o draw_v the_o diagonal_a line_n and_o make_v the_o centre_n the_o end_n of_o the_o diagonal_a line_n and_o the_o space_n likewise_o the_o perpendicular_a line_n draw_v from_o the_o top_n of_o the_o triangle_n to_o the_o base_a describe_v circumference_n and_o again_o from_o the_o common_a section_n to_o the_o centre_n draw_v right_a line_n and_o they_o shall_v contain_v the_o inclination_n seek_v for_o in_o a_o icosahedron_n upon_o the_o side_n of_o one_o of_o the_o triangle_n thereof_o describe_v a_o pentagon_n and_o draw_v the_o line_n which_o subtend_v one_o of_o the_o angle_n of_o the_o say_a pentagon_n and_o make_v the_o centre_n the_o end_n of_o that_o line_n and_o the_o space_n the_o perpendicular_a line_n of_o the_o triangle_n describe_v circumference_n and_o draw_v from_o the_o common_a intersection_n of_o the_o circumference_n unto_o the_o centre_n right_a line_n and_o they_o shall_v contain_v likewise_o the_o inclination_n of_o the_o plain_n of_o the_o icosahedron_n in_o a_o dodecahedron_n take_v one_o of_o the_o pentagon_n and_o draw_v likewise_o the_o line_n which_o subtend_v one_o of_o the_o angle_n of_o the_o pentagon_n and_o make_v the_o centre_n the_o end_n of_o that_o line_n and_o the_o space_n the_o perpendicular_a line_n draw_v from_o the_o section_n into_o two_o equal_a part_n of_o that_o line_n to_o the_o side_n of_o the_o pentagon_n which_o be_v parallel_n unto_o it_o describe_v circumference_n and_o from_o the_o point_n of_o the_o intersection_n of_o the_o circumference_n draw_v unto_o the_o centre_n right_a line_n and_o they_o shall_v also_o contain_v the_o inclination_n of_o the_o plain_n of_o the_o dodecahedron_n thus_o do_v this_o most_o singular_a learned_a man_n reason_n think_v the_o demonstration_n in_o every_o one_o of_o they_o to_o be_v plain_a and_o clear_a but_o to_o make_v the_o demonstration_n of_o they_o manifest_a i_o think_v it_o good_a to_o declare_v and_o make_v open_a his_o wordesâ—_n and_o first_o in_o a_o tâ—trahedronâ—_n the_o end_n of_o the_o fivetenth_fw-mi book_n of_o euclides_n element_n after_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d ¶_o the_o 6._o proposition_n the_o 6._o problem_n in_o a_o octohedron_n give_v to_o inscribe_v a_o trilater_n equilater_n pyramid_n svppose_v thaâ—_z the_o octohedron_n whereâ—â—_n the_o tetrahedron_n be_v require_v to_o be_v insâ—riâ—â—â—_n be_v abgdei_n take_v 〈…〉_o base_n of_o the_o octoâ—â—dron_n that_o be_v construction_n 〈…〉_o close_o in_o the_o loweâ—â—_n triangle_n bgd_v namely_o aeâ—_n head_n igd_v and_o let_v the_o four_o be_v aib_n which_o be_v opposite_a to_o the_o low_a triangle_n before_o put_v namely_o to_o egg_v and_o take_v the_o centre_n of_o those_o four_o base_n which_o let_v be_v the_o point_n h_o c_o n_o â—_o and_o upon_o the_o triangle_n hcn_n erect_v a_o pyramid_n hcnl_n now_o forasmuch_o as_o these_o two_o base_n of_o the_o octohedron_n namely_o age_n and_o abi_n be_v set_v upon_o the_o right_a line_n eglantine_n and_o by_o which_o be_v opposite_a the_o one_o to_o the_o otherâ—_n in_o the_o square_a gebi_n of_o the_o octohedron_n from_o the_o poinâ—_n a_o draâ—e_v by_o the_o centre_n of_o the_o base_n namely_o by_o the_o centre_n h_o l_o perpendicular_a line_n ahf_n alk_v cut_v the_o line_n eglantine_n and_o by_o 〈◊〉_d two_o equal_a part_n in_o the_o point_n f_o edward_n by_o the_o corollary_n of_o the_o 1â—_o of_o the_o thirteen_o wherefore_o a_o right_a line_n draw_v from_o the_o point_n f_o to_o the_o point_n king_n demonstration_n shall_v be_v a_o parallel_n and_o equal_a to_o the_o side_n of_o the_o octohedron_n namely_o to_o â—â—_o and_o give_v by_o the_o 33._o of_o the_o first_o and_o the_o right_a line_n hl_o which_o cut_v the_o 〈…〉_o of_o ak_o proportional_o for_o ah_o and_o all_o be_v draw_v from_o the_o centre_n of_o equal_a circle_n to_o the_o circumference_n be_v a_o parallel_n to_o the_o right_a line_n fk_o by_o the_o 2._o of_o the_o six_o and_o also_o to_o the_o side_n of_o the_o octohedron_n namely_o to_o eâ—_n and_o ig_n by_o the_o 9_o of_o the_o eleven_o wherefore_o as_o the_o line_n of_o be_v to_o the_o line_n ah_o so_o be_v the_o line_n fk_o to_o the_o line_n hl_o by_o the_o 4._o of_o the_o six_o for_o the_o triangle_n afk_v and_o ahl_n be_v like_o by_o thâ—_z corollary_n of_o the_o 2._o of_o the_o six_o but_o the_o line_n of_o be_v in_o sesquialter_fw-la proportion_n to_o the_o line_n ah_o for_o the_o side_n eglantine_n make_v hf_o the_o half_a of_o the_o right_a line_n ah_o by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o fk_n or_o give_v the_o side_n of_o the_o octohedron_n be_v sesquialter_fw-la to_o the_o righâ—line_n hl._n and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o side_n of_o the_o octohedron_n be_v sesquialter_fw-la to_o the_o rest_n of_o the_o right_a line_n which_o make_v the_o pyramid_n hnci_n namely_o to_o the_o right_a lineâ—â—_n n_o nc_n ci_o ln_o and_o change_z wherefore_o those_o right_a line_n be_v equal_a and_o therefore_o the_o triangleâ—_n which_o be_v describe_v of_o they_o namely_o the_o triangle_n hcn_n hnl_n ncl_n and_o chl._n which_o make_v the_o pyramid_n hncl_n be_v equal_a and_o equilater_n and_o forasmuch_o as_o the_o angle_n of_o the_o same_o pyramid_n namely_o the_o angle_n h_o c_o n_o l_o do_v end_n in_o the_o centre_n of_o the_o base_n of_o the_o octohedron_n therefore_o it_o be_v inscribe_v â—o_o the_o same_o octohedron_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o octohedron_n â—even_v be_v inscribe_v a_o trilate_a equilaâ—â—â—â—â—â—amisâ—_n which_o be_v require_v to_o â—e_a donâ—_n a_o corollary_n the_o base_n of_o a_o pyramid_n inscribe_v in_o a_o octohedron_n be_v parallel_n
compare_v they_o all_o with_o triangle_n &_o also_o together_o the_o one_o with_o the_o other_o in_o it_o also_o be_v teach_v how_o a_o figure_n of_o any_o form_n may_v be_v change_v into_o a_o figure_n of_o a_o other_o form_n and_o for_o that_o it_o entreat_v of_o these_o most_o common_a and_o general_a thing_n this_o book_n be_v more_o universal_a than_o be_v the_o second_o three_o or_o any_o other_o and_o therefore_o just_o occupy_v the_o first_o place_n in_o order_n as_o that_o without_o which_o the_o other_o book_n of_o eâ—clide_n which_o follow_v and_o also_o the_o work_n of_o other_o which_o have_v write_v in_o geometry_n can_v be_v perceive_v nor_o understand_v and_o forasmuch_o â—s_v all_o the_o demonstration_n and_o proof_n of_o all_o the_o proposition_n in_o this_o whole_a book_n depend_v of_o these_o ground_n and_o principle_n follow_v which_o by_o reason_n of_o their_o plainness_n need_v no_o great_a declaration_n yet_o to_o remove_v all_o be_v it_o never_o so_o little_a obscurity_n there_o be_v here_o set_v certain_a short_a and_o manife_a exposition_n of_o they_o definition_n 1._o a_o sign_n or_o point_n be_v that_o which_o have_v no_o part_n point_n the_o better_a to_o understand_v what_o manâ—r_n of_o thing_n a_o sign_n or_o point_n be_v you_o must_v note_v that_o the_o nature_n and_o property_n of_o quantity_n whereof_o geometry_n entreat_v be_v to_o be_v divide_v so_o that_o whatsoever_o may_v be_v divide_v into_o sundâ—y_a part_n be_v call_v quantity_n but_o a_o point_n although_o it_o pertain_v to_o quantity_n and_o have_v his_o be_v in_o quantity_n yet_o be_v it_o no_o quantity_n for_o that_o it_o can_v be_v divide_v because_o as_o the_o definition_n say_v it_o have_v no_o part_n into_o which_o it_o shall_v be_v divide_v so_o that_o a_o point_n be_v the_o least_o thing_n that_o by_o mind_n and_o understanding_n can_v be_v imagine_v and_o conceive_v than_o which_o there_o can_v be_v nothing_o less_o as_o the_o point_n a_o in_o the_o margin_n a_o sign_n or_o point_n be_v of_o pythagoras_n scholar_n after_o this_o manner_n define_v pythagoras_n a_o point_n be_v a_o unity_n which_o have_v position_n number_n be_v conceive_v in_o mind_n without_o any_o form_n &_o figure_n and_o therefore_o without_o matter_n whereon_o to_o 〈◊〉_d figure_n &_o consequent_o without_o place_n and_o position_n wherefore_o unity_n be_v a_o part_n of_o number_n have_v no_o position_n or_o determinate_a place_n whereby_o it_o be_v manifest_a that_o â—umbâ—â—_n iâ—_z more_o simple_a and_o pure_a then_o be_v magnitude_n and_o also_o immaterial_a and_o so_o unity_n which_o iâ—_z the_o beginning_n of_o number_n be_v less_o material_a than_o a_o â—igne_n or_o poyâ—â—_n which_o be_v the_o begin_n of_o magnitude_n for_o a_o point_n be_v material_a and_o require_v position_n and_o place_n and_o â—â—â—rby_o differ_v from_o unity_n â—_o a_o line_n be_v length_n without_o breadth_n liâ—â—_n there_o pertain_v to_o quantiâ—â—e_v three_o dimension_n length_n breadth_n &_o thickness_n or_o depth_n and_o by_o these_o three_o be_v all_o quamtitieâ—_n measure_v &_o make_v know_v there_o be_v also_o according_a to_o these_o three_o dimension_n three_o kind_n of_o continual_a quantity_n a_o line_n a_o superficies_n or_o plain_a and_o a_o body_n the_o first_o kind_n namely_o a_o line_n be_v here_o define_v in_o these_o word_n a_o line_n be_v length_n without_o breadth_n a_o point_n for_o that_o it_o be_v no_o quantity_n nor_o have_v any_o part_n into_o which_o it_o may_v be_v divide_v but_o remain_v indivisible_a have_v not_o nor_o can_v have_v any_o of_o these_o three_o dimension_n it_o neither_o have_v length_n breadth_n nor_o thickness_n but_o to_o a_o line_n which_o be_v the_o first_o kind_n of_o quantity_n be_v attribute_v the_o first_o dimension_n namely_o length_n and_o only_o that_o for_o it_o have_v neither_o breadth_n nor_o thickness_n but_o be_v conceive_v to_o be_v draw_v in_o length_n only_o and_o by_o it_o it_o may_v be_v divide_v into_o part_n as_o many_o as_o you_o listen_v equal_a or_o unequal_a but_o as_o touch_v breadth_n it_o remain_v indivisible_a as_o the_o line_n ab_fw-la which_o be_v only_o draw_v in_o length_n may_v be_v divide_v in_o the_o point_n c_o equal_o or_o in_o the_o point_n d_o unequal_o and_o so_o into_o as_o many_o part_n as_o you_o listen_v there_o be_v also_o of_o diverse_a other_o give_v other_o definition_n of_o a_o line_n as_o these_o which_o follow_v line_n a_o line_n be_v the_o move_n of_o a_o point_n as_o the_o motion_n or_o draught_n of_o a_o pin_n or_o a_o pen_n to_o your_o sense_n make_v a_o line_n other_o again_o a_o line_n be_v a_o magnitude_n have_v one_o only_a space_n or_o dimension_n namely_o length_n want_v breadth_n and_o thickness_n line_n 3_o the_o end_n or_o limit_n of_o a_o line_n be_v point_n for_o a_o line_n have_v his_o beginning_n from_o a_o point_n and_o likewise_o end_v in_o a_o point_n so_o that_o by_o this_o also_o it_o be_v manifest_a that_o point_n for_o their_o simplicity_n and_o lack_n of_o composition_n be_v neither_o quantity_n nor_o part_n of_o quantity_n but_o only_o the_o term_n and_o end_n of_o quantity_n as_o the_o point_n a_o b_o be_v only_o the_o end_n of_o the_o line_n ab_fw-la and_o no_o part_n thereof_o and_o herein_o differ_v a_o point_n in_o quantity_n frâ—â—nity_n from_o unity_n in_o numberâ—_n for_o that_o although_o unity_n be_v the_o beginning_n of_o number_n and_o no_o number_n as_o a_o point_n be_v the_o begin_n of_o quantity_n and_o no_o quantity_n yet_o be_v unity_n a_o part_n of_o number_n number_n for_o number_n be_v nothing_o else_o but_o a_o collection_n of_o unity_n and_o therefore_o may_v be_v divide_v into_o they_o as_o into_o his_o part_n but_o a_o point_n be_v no_o part_n of_o quantity_n quantity_n or_o of_o a_o lyneâ—_n neither_o be_v a_o line_n compose_v of_o point_n as_o number_n be_v of_o unity_n for_o thing_n indivisible_a being_n never_o so_o many_o add_v together_o can_v never_o make_v a_o thing_n divisible_a as_o a_o instant_n in_o time_n be_v neither_o time_n nor_o part_n of_o time_n but_o only_o the_o beginning_n and_o end_n of_o time_n and_o couple_v &_o join_v part_n of_o time_n together_o line_n 4_o a_o right_a line_n be_v that_o which_o lie_v equal_o between_o his_o point_n as_o the_o whole_a line_n ab_fw-la lie_v straight_a and_o equal_o between_o the_o point_n ab_fw-la without_o any_o go_n up_o or_o come_v down_o on_o either_o side_n a_o right_a line_n be_v the_o short_a extension_n or_o draught_n that_o be_v or_o may_v be_v from_o one_o point_n to_o a_o other_o archimedes_n define_v it_o thus_o plato_n plato_n define_v a_o right_a line_n after_o this_o manner_n a_o right_a line_n be_v that_o who_o middle_a part_n shadow_v the_o exâ—remeâ—_n as_o if_o you_o put_v any_o thing_n in_o the_o middle_n of_o a_o right_a line_n you_o shall_v not_o see_v from_o the_o one_o end_n to_o the_o other_o which_o thing_n happen_v not_o in_o a_o crooked_a line_n the_o eclipse_n 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then_o by_o the_o common_a sentence_n the_o line_n bk_o shall_v be_v equal_v to_o the_o line_n all_o for_o by_o the_o definition_n of_o a_o circle_n the_o line_n bk_o be_v equal_a to_o the_o line_n bg_o and_o the_o line_n all_o be_v equal_a to_o the_o line_n agnostus_n which_o be_v equal_a to_o the_o line_n bg_o wherefore_o the_o line_n ak_o be_v great_a than_o the_o line_n bk_o and_o by_o the_o same_o reason_n may_v it_o be_v prove_v that_o the_o line_n bk_o be_v great_a than_o the_o line_n ab_fw-la wherefore_o the_o triangle_n abk_n consist_v of_o three_o unequal_a side_n and_o so_o have_v you_o upon_o the_o line_n give_v describe_v all_o the_o kind_n of_o triangle_n this_o be_v to_o be_v note_v that_o if_o a_o man_n will_v mechanical_o and_o ready_o not_o regard_v demonstration_n upon_o a_o line_n give_v describe_v a_o triangle_n of_o three_o equal_a side_n mechanical_o he_o need_v not_o to_o describe_v the_o whole_a foresay_a circle_n but_o only_o a_o little_a part_n of_o each_o namely_o where_o they_o cut_v the_o one_o the_o other_o and_o so_o from_o the_o point_n of_o the_o section_n to_o draw_v the_o line_n to_o the_o end_n of_o the_o line_n gevenâ—_n as_o in_o this_o figure_n here_o put_v and_o likewise_o if_o upon_o the_o say_a line_n he_o will_v describe_v a_o triangle_n of_o two_o equal_a side_n ready_o let_v he_o extend_v the_o compass_n according_a to_o the_o quantity_n that_o he_o will_v have_v the_o side_n to_o be_v whether_o long_o than_o the_o line_n give_v or_o short_a and_o so_o draw_v only_o a_o little_a part_n of_o each_o circle_n where_o they_o cut_v the_o one_o the_o other_o &_o from_o the_o point_n of_o the_o section_n draw_v the_o line_n to_o the_o end_n of_o the_o line_n give_v as_o in_o the_o figure_n here_o put_v note_v that_o in_o this_o the_o two_o side_n must_v be_v such_o that_o be_v join_v together_o they_o be_v long_o than_o the_o line_n give_v and_o so_o also_o if_o upon_o the_o say_a right_a line_n he_o will_v describe_v a_o triangle_n of_o three_o unequal_a side_n ready_o let_v he_o extend_v the_o compass_n first_o according_a to_o the_o quantity_n that_o he_o will_v have_v one_o of_o the_o unequal_a side_n to_o be_v and_o so_o draw_v a_o little_a part_n of_o the_o circle_n &_o then_o extend_v it_o according_a to_o the_o quantity_n that_o he_o will_v have_v the_o other_o unequal_a side_n to_o be_v and_o draw_v likewise_o a_o little_a part_n of_o the_o circle_n and_o that_o do_v from_o the_o point_n of_o the_o section_n draâ—_n the_o line_n to_o the_o end_n of_o the_o line_n give_v as_o in_o the_o figure_n here_o put_v note_v that_o in_o this_o the_o two_o side_n must_v be_v such_o that_o the_o circle_n describe_v according_a to_o their_o quantity_n may_v cut_v the_o one_o the_o other_o the_o second_o problem_n the_o second_o proposition_n fron_n a_o point_n give_v to_o draw_v a_o right_a line_n equal_a to_o a_o rightline_v give_v svppose_v that_o the_o point_n give_v be_v a_o &_o let_v the_o right_a line_n give_v be_v bc._n it_o be_v require_v from_o the_o point_n a_o to_o draw_v a_o right_a line_n equal_a to_o the_o line_n bc._n draw_v by_o the_o first_o petition_n from_o the_o point_n a_o to_o the_o point_n b_o a_o right_a line_n ab_fw-la construâ—tiâ—â—_n and_o upon_o the_o line_n ab_fw-la describe_v by_o the_o first_o proposition_n a_o equilater_n triangle_n and_o let_v the_o same_o be_v dab_n and_o extend_v by_o the_o second_o petition_n the_o right_a line_n dam_n &_o db_o to_o the_o point_n e_o and_o f_o &_o by_o the_o three_o petition_n make_v the_o centre_n b_o and_o the_o space_n bc_o describe_v a_o circle_n cgh_n &_o again_o by_o the_o same_o make_v the_o centre_n d_o and_o the_o space_n dg_o describe_v a_o circle_n gkl_n
and_o forasmuch_o as_o the_o point_n b_o be_v the_o centre_n of_o the_o circle_n cgh_n demonstration_n therefore_o by_o the_o 15._o definition_n the_o line_n bc_o be_v equal_a to_o the_o line_n bg_o and_o forasmuch_o as_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n gkl_n therefore_o by_o the_o same_o the_o line_n dl_o be_v equal_a to_o the_o line_n dg_o of_o which_o the_o line_n dam_n be_v equal_a to_o a_o line_n db_o by_o the_o proposition_n go_v before_o wherefore_o the_o residue_n namely_o the_o line_n all_o be_v equal_a to_o the_o residue_n namely_o to_o the_o line_n bg_o by_o the_o three_o common_a sentence_n and_o it_o be_v prove_v that_o the_o line_n bc_o be_v equal_a to_o the_o line_n bg_o wherefore_o either_o of_o these_o line_n all_o &_o bc_o be_v equal_a to_o the_o line_n bg_o but_o thing_n which_o be_v equal_a to_o one_o and_o the_o same_o thing_n be_v also_o equal_a the_o one_o to_o the_o other_o by_o the_o first_o common_a sentence_n wherefore_o the_o line_n all_o be_v equal_a to_o the_o line_n bc._n wherefore_o from_o the_o point_n give_v namely_o a_o be_v draw_v a_o right_a line_n all_o equal_a to_o the_o right_a line_n give_v bc_o which_o be_v require_v to_o be_v do_v of_o problem_n and_o theoremesâ—_n as_o we_o have_v before_o note_v some_o have_v no_o case_n at_o all_o which_o be_v those_o which_o have_v only_o one_o position_n and_o construction_n and_o other_o some_o have_v many_o and_o diverse_a case_n which_o be_v such_o proposition_n which_o have_v diverse_a description_n &_o construction_n and_o change_v their_o position_n of_o which_o sort_n be_v this_o second_o proposition_n which_o be_v also_o a_o problem_n this_o proposition_n have_v two_o thing_n give_v proposition_n namely_o a_o point_n and_o a_o line_n the_o thing_n require_v be_v that_o from_o the_o point_n give_v wheresoever_o it_o be_v put_v be_v draw_v a_o line_n equal_a to_o the_o line_n give_v now_o this_o point_n give_v may_v have_v diverse_a position_n for_o it_o may_v be_v place_v either_o without_o the_o right_a line_n give_v or_o in_o some_o point_n in_o it_o if_o it_o be_v without_o it_o either_o it_o be_v on_o the_o side_n of_o it_o so_o that_o the_o right_a line_n draw_v from_o it_o to_o the_o end_n of_o the_o right_a line_n give_v make_v a_o angleâ—_n or_o else_o it_o be_v put_v direct_o unto_o itâ—_n so_o that_o the_o right_a line_n give_v be_v produce_v shall_v fall_v upon_o the_o point_n give_v which_o be_v without_o but_o if_o it_o be_v in_o the_o line_n give_v then_o either_o it_o be_v in_o one_o of_o the_o end_n or_o extreme_n thereof_o or_o in_o some_o place_n between_o the_o extreme_n so_o be_v there_o four_o diverse_a position_n of_o the_o point_n in_o respect_n of_o the_o line_n whereupon_o follow_v diverse_a delineation_n and_o construction_n and_o consequent_o variety_n of_o case_n case_n to_o the_o second_o case_n the_o figure_n here_o on_o the_o side_n set_v belong_v and_o as_o touch_v the_o order_n both_o oâ—_n construction_n and_o of_o demonstration_n it_o be_v â—ll_a one_o with_o the_o first_o the_o four_o case_n as_o touch_v construction_n herein_o differ_v from_o the_o two_o first_o case_n for_o thâ—t_v whereas_o in_o they_o you_o be_v will_v to_o draw_v a_o right_a line_n from_o the_o point_n give_v namely_o a_o to_o the_o point_n b_o which_o be_v one_o of_o the_o end_n of_o the_o line_n give_v here_o you_o shall_v not_o need_v to_o draw_v that_o line_n for_o that_o it_o be_v already_o draw_v as_o touch_v the_o rest_n both_o in_o construction_n and_o demonstration_n you_o may_v proceed_v as_o in_o the_o two_o firsteâ—_n as_o it_o be_v manifest_a to_o see_v in_o this_o figure_n here_o on_o the_o side_n put_v the_o 3._o problem_n the_o 3._o proposition_n two_o unequal_a right_a line_n be_v give_v to_o cut_v of_o from_o the_o great_a a_o right_a line_n equal_a to_o the_o less_o svppose_v that_o the_o two_o unequal_a right_a line_n give_v be_v ab_fw-la &_o c_o of_o which_o lâ—t_v the_o line_n ab_fw-la be_v the_o great_a it_o be_v require_v from_o the_o line_n ab_fw-la be_v the_o great_a to_o cut_v of_o a_o right_a line_n equal_a to_o the_o right_a line_n c_o which_o be_v the_o less_o line-drawâ—_a on_o by_o the_o second_o proposition_n from_o the_o point_n a_o a_o right_a line_n equal_a to_o the_o line_n c_o and_o let_v the_o same_o be_v ad_fw-la and_o make_v the_o centre_n a_o and_o the_o space_n ad_fw-la describe_v by_o the_o three_o petition_n a_o circle_n def_n and_o forasmuch_o as_o the_o point_n a_o be_v the_o centre_n of_o the_o circle_n def_n therefore_o ae_n be_v equal_a to_o ad_fw-la but_o the_o line_n c_o be_v equal_a to_o the_o line_n ad._n wherefore_o either_o of_o these_o line_n ae_n and_o c_o be_v equal_a to_o ad_fw-la wherefore_o the_o line_n ae_n be_v equal_a to_o the_o line_n c_o wherefore_o two_o unequal_a right_a line_n be_v give_v namely_o ab_fw-la and_o c_o the_fw-mi be_v cut_v of_o from_o ab_fw-la being_n the_o great_a a_o right_a line_n ae_n equal_a to_o the_o less_o line_n namely_o to_o c_o which_o be_v require_v to_o be_v do_v this_o proposition_n which_o be_v a_o problem_n have_v two_o thing_n give_v 〈◊〉_d namely_o two_o unequal_a right_a line_n the_o thing_n require_v be_v from_o the_o great_a to_o cut_v of_o a_o line_n equal_a to_o the_o less_o it_o have_v also_o diverse_a case_n for_o the_o line_n give_v either_o be_v distinct_a the_o one_o from_o the_o other_o or_o be_v join_v together_o at_o one_o of_o their_o end_n or_o they_o cut_v the_o one_o the_o other_o or_o the_o one_o cut_v the_o other_o in_o one_o of_o the_o extreme_n which_o may_v be_v two_o way_n for_o either_o the_o great_a cut_v the_o less_o or_o the_o less_o the_o great_a if_o they_o cut_v the_o one_o the_o other_o either_o â—ch_n cut_v the_o other_a into_o equal_a part_n or_o into_o unequal_a part_n or_o the_o one_o into_o equal_a part_n and_o the_o other_o into_o unequal_a part_n which_o may_v hap_v in_o two_o sort_n for_o the_o great_a may_v be_v cut_v into_o equal_a part_n and_o the_o less_o into_o unequal_a part_n or_o contrariwise_o case_n when_o the_o unequal_a line_n give_v be_v distinct_a the_o one_o from_o the_o other_o the_o figure_n before_o put_v serve_v but_o if_o the_o one_o cut_v the_o other_o in_o one_o of_o the_o extreme_n case_n as_o for_o example_n suppose_v that_o the_o unequal_a right_a line_n give_v be_v ab_fw-la and_o cd_o of_o which_o let_v the_o line_n cd_o be_v the_o great_a and_o let_v the_o line_n cd_o cut_v the_o line_n ab_fw-la in_o his_o extreme_a c._n then_o make_v the_o centre_n a_o and_o the_o space_n ab_fw-la describe_v a_o circle_n be._n and_o upon_o the_o line_n ac_fw-la describe_v a_o equilater_n triangle_n by_o the_o â—irst_n which_o let_v bâ—_z aec_o &_o produce_v the_o line_n ea_fw-la and_o ec_o and_o again_o make_v the_o centre_n e_o and_o the_o space_n of_o describe_v a_o ciâ—â—â—e_n gf_o likewise_o make_v the_o centre_n c_o and_o the_o space_n cg_o describe_v a_o circle_n gl_n now_o forasmuch_o as_o the_o line_n of_o be_v equal_a to_o the_o line_n eglantine_n â—or_a e_o be_v the_o centre_n of_o which_o the_o line_n ea_fw-la be_v equal_a to_o the_o line_n eâ—_n therefore_o the_o residue_n of_o be_v equal_v to_o the_o residue_n cg_o â—ut_fw-la the_o line_n of_o be_v equal_a to_o the_o line_n ab_fw-la for_o a_o be_v the_o centre_n whereâ—oâ—e_v also_o the_o line_n cg_o be_v equal_a to_o the_o line_n ab_fw-la but_o the_o line_n cg_o be_v also_o equal_a â—o_o the_o line_n cl_n for_o the_o point_n c_o be_v the_o centre_n wherefore_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n cl._n wherefore_o from_o the_o line_n cd_o be_v cut_v of_o the_o line_n cl_n which_o be_v equal_a to_o the_o line_n ab_fw-la case_n or_o it_o be_v less_o than_o the_o half_a and_o then_o make_v the_o centre_n c_o &_o the_o space_n cd_o describe_v a_o circle_n which_o shall_v cut_v of_o from_o the_o line_n ab_fw-la a_o line_n equal_a to_o the_o line_n cd_o case_n or_o it_o be_v great_a than_o the_o half_a and_o they_o unto_o the_o point_n a_o put_v the_o line_n of_o equal_a to_o the_o line_n cd_o by_o the_o second_o and_o make_v the_o centre_n a_o &_o the_o space_n of_o describe_v a_o circle_n which_o shall_v cut_v of_o from_o the_o line_n ab_fw-la a_o line_n equal_a to_o the_o line_n of_o that_o be_v unto_o the_o line_n cd_o but_o if_o it_o be_v great_a than_o the_o half_a case_n then_o again_o unto_o the_o point_n a_o put_v the_o line_n of_o equal_a to_o the_o line_n cd_o by_o the_o second_o proposition_n &_o make_v the_o centre_n a_o and_o the_o space_n of_o describe_v a_o circle_n which_o shall_v cut_v of_o from_o the_o line_n
suppose_v that_o the_o angle_n at_o the_o base_a be_v equal_a which_o in_o the_o former_a proposition_n be_v the_o conclusion_n and_o the_o conclusion_n be_v that_o the_o two_o side_n subtend_v the_o two_o angle_n be_v equal_a which_o in_o the_o former_a proposition_n be_v the_o supposition_n this_o be_v the_o chief_a kind_n of_o conversion_n uniform_a and_o certain_a there_o be_v a_o other_o kind_n of_o conversion_n first_o but_o not_o so_o full_a a_o conversion_n nor_o so_o perfect_a as_o the_o first_o be_v which_o happen_v in_o compose_a proposition_n that_o be_v in_o such_o which_o have_v more_o supposition_n than_o one_o and_o pass_v from_o these_o supposition_n to_o one_o conclusion_n in_o the_o connuerse_n of_o such_o proposition_n you_o pass_v from_o the_o conclusion_n of_o the_o first_o proposition_n with_o one_o or_o more_o of_o the_o supposition_n of_o the_o same_o &_o conclude_v some_o other_o supposition_n of_o the_o self_n first_o proposition_n of_o this_o kind_n there_o be_v many_o in_o euclid_n thereof_o you_o may_v have_v a_o example_n in_o the_o 8._o proposition_n be_v the_o converse_n of_o the_o fourâ—h_n this_o conversion_n be_v not_o so_o uniform_a as_o the_o other_o but_o more_o diverse_a and_o uncertain_a according_a to_o the_o multitude_n of_o the_o thing_n give_v or_o supposition_n in_o the_o proposition_n but_o because_o in_o the_o five_o proposition_n there_o be_v two_o conclusion_n proposition_n the_o first_o that_o the_o two_o angle_n at_o the_o baâ—e_n be_v equal_a the_o second_o that_o the_o angle_n under_o the_o baâ—e_n be_v equal_a this_o be_v to_o be_v note_v that_o this_o six_o proposition_n be_v the_o converse_n of_o the_o â—ame_n five_o as_o touch_v the_o first_o conclusion_n only_o conclusion_n you_o may_v in_o like_a manner_n make_v a_o converse_n of_o the_o same_o proposition_n touch_v the_o second_o conclusion_n thereof_o and_o that_o after_o this_o manner_n the_o two_o side_n of_o a_o triangle_n be_v produce_v if_o the_o angle_n under_o the_o base_a be_v equal_a the_o say_a triangle_n shall_v be_v a_o isosceles_a triangle_n in_o which_o proposition_n the_o supposition_n be_v that_o the_o angle_n under_o the_o base_a be_v equal_a which_o in_o the_o five_o proposition_n be_v the_o conclusionâ—_n &_o the_o conclusion_n in_o this_o proposition_n be_v that_o the_o two_o side_n of_o the_o triangle_n be_v equal_a which_o in_o the_o five_o proposition_n be_v the_o supposition_n but_o now_o for_o proof_n of_o the_o say_a proposition_n for_o suppose_v that_o ac_fw-la be_v equal_a to_o ad_fw-la and_o produce_v the_o line_n ca_n to_o the_o point_n e_o and_o put_v the_o line_n ae_n equal_a to_o the_o line_n db_o by_o the_o three_o proposition_n wherefore_o the_o whole_a line_n ce_fw-fr be_v equal_a to_o the_o whole_a line_n ab_fw-la by_o the_o second_o common_a sentence_n draw_v a_o line_n from_o the_o point_n e_o to_o the_o point_n b._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n ec_o and_o the_o line_n bc_o be_v common_a to_o they_o both_o and_o the_o angle_n acb_o be_v suppose_v to_o be_v equal_a to_o the_o angle_n abc_n wherefore_o by_o the_o four_o proposition_n the_o triangle_n ebc_n be_v equal_a to_o the_o triangle_n abc_n name_o the_o whole_a to_o the_o part_n which_o be_v impossible_a the_o 4._o theorem_a the_o 7._o proposition_n if_o from_o the_o end_n of_o one_o line_n be_v draw_v two_o right_a line_n to_o any_o point_n there_o can_v not_o from_o the_o self_n same_o end_n on_o the_o same_o side_n be_v draw_v two_o other_o line_n equal_a to_o the_o two_o first_o line_n the_o one_o to_o the_o other_o unto_o any_o other_o point_n for_o if_o it_o be_v possible_a then_o from_o the_o end_n of_o one_o &_o the_o self_n same_o right_a line_n namely_o ab_fw-la from_o the_o point_n i_o say_v a_o and_o b_o let_v there_o be_v draw_v two_o right_a line_n ac_fw-la and_o cb_o to_o the_o point_n c_o and_o from_o the_o same_o end_n of_o the_o line_n ab_fw-la let_v there_o be_v draw_v two_o other_o right_n right_a line_n ad_fw-la and_o db_o equal_a to_o the_o line_n ac_fw-la and_o cb_o the_o one_o to_o the_o other_o be_v each_o to_o his_o correspondent_a line_n and_o on_o one_o and_o the_o same_o side_n and_o to_o a_o other_o point_n namely_o to_o d_o so_o that_o let_v ca_n be_v equal_a to_o dam_n be_v both_o draw_v from_o one_o end_n that_o be_v a_o &_o let_v cb_o be_v equal_a to_o db_o be_v both_o also_o draw_v from_o one_o end_n that_o be_v b._n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n c_o to_o the_o point_n d._n absurdity_n now_o forasmuch_o as_o ac_fw-la be_v equal_a to_o ad_fw-la the_o angle_n acd_v also_o be_v by_o the_o 5._o proposition_n equal_a to_o the_o angle_n adc_o wherefore_o the_o angle_n acd_o be_v less_o they_o the_o angle_n bdc_o wherefore_o the_o angle_n bcd_o be_v much_o less_o than_o the_o angle_n bdc_o again_o forasmuch_o as_o bc_o be_v equal_a to_o bd_o and_o therefore_o also_o the_o angle_n bcd_o be_v equal_a to_o the_o angle_n bdc_o and_o it_o be_v prove_v that_o it_o be_v much_o less_o than_o it_o which_o be_v impossible_a if_o therefore_o from_o the_o end_n of_o one_o line_n be_v draw_v too_o right_a line_n to_o any_o point_n there_o can_v not_o from_o the_o self_n same_o end_n on_o the_o same_o side_n be_v draw_v two_o other_o line_n equal_a to_o the_o two_o first_o line_n the_o one_o to_o the_o other_o unto_o any_o other_o point_n which_o be_v require_v to_o be_v demonstrate_v in_o this_o proposition_n the_o conclusion_n be_v a_o negation_n which_o very_o rare_o happen_v in_o the_o mathematical_a art_n art_n for_o they_o ever_o for_o the_o most_o part_n use_v to_o conclude_v affirmative_o &_o not_o negative_o for_o a_o proposition_n universal_a affirmative_a be_v most_o agreeable_a to_o science_n as_o say_v aristotle_n and_o be_v of_o itself_o strong_a and_o need_v no_o negative_a to_o his_o proof_n but_o a_o universal_a proposition_n negative_a must_v of_o necessity_n have_v to_o his_o proof_n a_o affirmative_a for_o of_o only_o negative_a proposition_n there_o can_v be_v no_o demonstration_n and_o therefore_o science_n use_v demonstration_n conclude_v affirmative_o and_o very_o seldom_o use_v negative_a conclusion_n an_o other_o demonstration_n after_o campanus_n suppose_v that_o there_o be_v a_o line_n ab_fw-la from_o who_o end_n a_o and_o b_o let_v there_o be_v draw_v two_o line_n ac_fw-la and_o bc_o on_o one_o side_n which_o let_v concur_v in_o the_o point_n c._n then_o i_o say_v that_o on_o the_o same_o side_n there_o can_v be_v draw_v two_o other_o line_n from_o the_o end_n of_o the_o line_n ab_fw-la which_o shall_v concur_v at_o any_o other_o point_n so_o that_o that_o which_o be_v draw_v from_o the_o point_n a_o shall_v be_v equal_a to_o the_o line_n ac_fw-la and_o that_o which_o be_v draw_v from_o the_o point_n b_o shall_v be_v equal_a to_o the_o line_n bc._n for_o if_o it_o be_v possible_a let_v there_o be_v draw_v two_o other_o line_n on_o the_o self_n same_o side_n which_o let_v concur_v in_o the_o point_n d_o and_o let_v the_o line_n ad_fw-la be_v equal_a to_o the_o line_n ac_fw-la &_o the_o line_n bd_o equal_a to_o the_o line_n bc._n demonstration_n wherefore_o the_o point_n d_o shall_v fall_v either_o within_o the_o triangle_n abc_n or_o without_o for_o it_o can_v fall_v in_o one_o of_o the_o side_n for_o then_o a_o part_n shall_v be_v equal_a to_o his_o whole_a if_o therefore_o it_o fall_v without_o then_o either_o one_o of_o the_o line_n ad_fw-la and_o db_o shall_v cut_v one_o of_o the_o line_n ac_fw-la and_o cb_o or_o else_o neither_o shall_v cut_v neither_o case_n first_o let_v one_o cut_v the_o other_o and_o draw_v a_o right_a line_n from_o c_o to_o d._n now_o forasmuch_o as_o in_o the_o triangle_n acd_o the_o two_o side_n ac_fw-la and_o ad_fw-la be_v equal_a therefore_o the_o angle_n acd_o be_v equal_a to_o the_o angle_n adc_o by_o the_o five_o proposition_n likewise_o forasmuch_o as_o in_o the_o triangle_n bcd_o the_o two_o side_n bc_o and_o bd_o be_v equal_a therefore_o by_o the_o same_o the_o angle_n bcd_o &_o bdc_o be_v also_o equal_a and_o forasmuch_o as_o the_o angle_n bdc_o be_v great_a they_o the_o angle_n adc_o it_o follow_v that_o the_o angle_n bcd_o be_v great_a than_o the_o angle_n acd_o namely_o the_o part_n great_a than_o the_o whole_a which_o be_v impossible_a but_o if_o the_o point_n d_o fall_n without_o the_o triangle_n abc_n case_n so_o that_o the_o line_n cut_v not_o the_o one_o the_o other_o draw_v a_o line_n from_o d_z to_o c._n and_o produce_v the_o line_n bd_o &_o bc_o beyond_o the_o base_a cd_o unto_o the_o point_n e_o &_o f._n and_o forasmuch_o as_o the_o line_n ac_fw-la and_o ad_fw-la be_v equal_a the_o angle_n
acd_o and_o adc_o shall_v also_o be_v equal_a by_o the_o five_o proposition_n likewise_o for_o asmuch_o as_o the_o line_n bc_o and_o bd_o be_v equal_a the_o angle_n under_o the_o base_a namely_o the_o angle_n fdc_n and_o ecd_n be_v equal_a by_o the_o second_o part_n of_o the_o same_o proposition_n and_o for_o as_o much_o as_o the_o angle_n ecd_n be_v less_o than_o the_o angle_n acd_o it_o follow_v that_o the_o angle_n fdc_n be_v less_o they_o the_o angle_n adc_o which_o be_v impossible_a for_o that_o the_o angle_n ad_fw-la c_z be_v a_o part_n of_o the_o angle_n fd_o c._n and_o the_o same_o inconvenience_n will_v follow_v if_o the_o point_n d_o fall_n within_o the_o triangle_n abc_n the_o five_o theorem_a the_o 8._o proposition_n if_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_n to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n &_o have_v also_o the_o base_a of_o the_o one_o equal_a to_o the_o base_a of_o the_o other_o they_o shall_v have_v also_o the_o angle_n contain_v under_o the_o equal_a right_a line_n of_o the_o one_o equal_a to_o the_o angle_n contain_v under_o the_o equal_a right_a line_n of_o the_o other_o svppose_v that_o there_o be_v two_o triangle_n abc_n and_o def_n &_o let_v these_o two_o side_n of_o the_o one_o ab_fw-la and_o ac_fw-la be_v equal_a to_o these_o two_o side_n of_o the_o other_o de_fw-fr and_o df_o each_o to_o his_o correspondent_a side_n that_o be_v ab_fw-la to_o d_o e_o and_o ac_fw-la to_o df_o &_o let_v the_o base_a of_o the_o one_o namely_o bc_o be_v equal_a to_o the_o base_a of_o the_o other_o namely_o to_o ef._n then_o i_o say_v that_o the_o angle_n bac_n be_v equal_a to_o the_o angle_n edf_o for_o the_o triangle_n abc_n exact_o agree_v with_o the_o triangle_n de_fw-fr fletcher_n impossibility_n and_o the_o point_n b_o be_v put_v upon_o the_o point_n e_o and_o the_o right_a line_n bc_o upon_o the_o right_a line_n of_o the_o point_n c_o shall_v exact_o agree_v with_o the_o point_n fletcher_n for_o the_o line_n bc_o be_v equal_a to_o the_o line_n of_o and_o bc_o exact_o agree_v with_o of_o the_o line_n also_o basilius_n and_o ac_fw-la shall_v exact_o agree_v with_o the_o line_n ed_z &_o df._n for_o if_o the_o base_a bc_o do_v exact_o agree_v with_o the_o base_a fe_o but_o the_o side_n basilius_n &_o ac_fw-la do_v not_o exact_o agree_v with_o the_o side_n ed_z &_o df_o but_o differ_v as_o fg_o &_o gf_o do_v they_o from_o the_o end_n of_o one_o line_n shall_v be_v draw_v two_o right_a line_n to_o a_o point_n &_o from_o the_o self_n same_o end_n on_o the_o same_o side_n shall_v be_v draw_v then_o other_o line_n equal_a to_o the_o two_o first_o line_n the_o one_o to_o the_o other_o and_o unto_o a_o other_o point_n but_o that_o be_v impossible_a by_o the_o seven_o proposition_n wherefore_o the_o base_a bc_o exact_o agree_v with_o the_o base_a of_o the_o side_n also_o basilius_n and_o ac_fw-la do_v exact_o agree_v with_o the_o side_n ed_z and_o df._n wherefore_o also_o the_o angle_n bac_n shall_v exact_o agree_v with_o the_o angle_n edf_o and_o therefore_o shall_v also_o be_v equal_a to_o it_o if_o therefore_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_a to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n and_o have_v also_o the_o base_a of_o the_o one_o equal_a to_o the_o base_a of_o the_o other_o they_o shall_v have_v also_o the_o angle_n contain_v under_o the_o equal_a right_a line_n of_o the_o one_o equal_a to_o the_o angle_n contain_v under_o the_o equal_a right_a line_n of_o the_o other_o which_o be_v require_v to_o be_v prove_v conversion_n this_o theorem_a be_v the_o converse_n of_o the_o four_o but_o it_o be_v not_o the_o chief_a and_o principal_a kind_a oâ—_n conversion_n for_o it_o turn_v not_o the_o whole_a supposition_n into_o the_o conclusion_n and_o the_o whole_a conclusion_n into_o the_o supposition_n for_o the_o four_o proposition_n who_o converse_n this_o be_v be_v a_o compound_n theorem_a have_v two_o thing_n give_v or_o suppose_v which_o be_v these_o the_o one_o that_o two_o side_n of_o the_o one_o triangle_n be_v equal_a to_o two_o side_n of_o the_o other_o triangle_n the_o other_a that_o the_o angle_n contain_v of_o the_o two_o side_n of_o the_o one_o be_v equal_a to_o the_o angle_n contain_v of_o the_o two_o side_n of_o the_o one_o but_o have_v amongst_o other_o one_o thing_n require_v which_o be_v that_o the_o base_a of_o the_o one_o be_v equal_a to_o the_o base_a of_o the_o other_o now_o in_o this_o 8._o proposition_n be_v the_o converse_n therofâ—_n that_o the_o base_a of_o the_o one_o be_v equal_a to_o the_o base_a of_o the_o other_a be_v the_o supposition_n or_o the_o thing_n give_v which_o in_o the_o former_a proposition_n be_v the_o conclusion_n and_o this_o that_o two_o side_n of_o the_o one_o be_v equal_a to_o two_o side_n of_o the_o other_o be_v in_o this_o proposition_n also_o a_o supposition_n like_a as_o it_o be_v in_o the_o former_a proposition_n so_o that_o it_o be_v a_o thing_n give_v in_o either_o proposition_n the_o conclusion_n of_o this_o proposition_n be_v that_o the_o angle_n enclose_v of_o the_o two_o equal_a side_n of_o the_o one_o triangle_n be_v equal_a to_o the_o angle_n enclose_v of_o the_o two_o equal_a side_n of_o the_o other_o triangle_n which_o in_o the_o former_a proposition_n be_v one_o of_o the_o thing_n give_v philo_n and_o his_o scholas_fw-la demonstrate_v this_o proposition_n without_o the_o help_n of_o the_o former_a proposition_n in_o this_o manner_n first_o let_v it_o fall_v direct_o case_n and_o forasmuch_o as_o the_o line_n de_fw-fr be_v equal_a to_o the_o line_n e_o g_o and_o dfg_n be_v one_o right_a line_n therefore_o deg_n be_v a_o isosceles_a triangle_n and_o so_o by_o the_o five_o proposition_n the_o angle_n at_o the_o point_n d_o be_v equal_a to_o the_o angle_n at_o the_o point_n g_o which_o be_v require_v to_o be_v prove_v the_o 4._o problem_n the_o 9_o proposition_n to_o divide_v a_o rectiline_a angle_n give_v into_o two_o equal_a part_n svppose_v that_o the_o rectiline_a angle_n give_v be_v bac_n it_o be_v require_v to_o divide_v the_o angle_n bac_n into_o two_o equal_a part_n construction_n in_o the_o line_n ab_fw-la take_v a_o point_n at_o all_o adventure_n &_o let_v the_o same_o be_v d._n and_o by_o the_o three_o proposition_n from_o the_o line_n ac_fw-la cut_v of_o the_o line_n ae_n equal_a to_o ad._n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n d_o to_o the_o point_n e._n and_o by_o the_o first_o proposition_n upon_o the_o line_n de_fw-fr describe_v a_o equilater_n triangle_n and_o let_v the_o same_o be_v dfe_n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n f._n then_o i_o say_v that_o the_o angle_n bac_n be_v by_o the_o line_n of_o divide_v into_o two_o equal_a part_n demonstration_n for_o forasmuch_o as_o ad_fw-la be_v equal_a to_o ae_n and_o of_o be_v common_a to_o they_o both_o therefore_o these_o two_o dam_n and_o of_o be_v equal_a to_o these_o two_o ea_fw-la and_o of_o the_o one_o to_o the_o other_o but_o by_o the_o first_o proposition_n the_o base_a df_o be_v equal_a to_o the_o base_a of_o wherefore_o by_o the_o 8._o proposition_n the_o angle_n daf_n be_v equal_a to_o the_o angle_n fae_z wherefore_o the_o rectiline_a angle_n give_v namely_o b_o ac_fw-la be_v divide_v into_o two_o equal_a part_n by_o the_o right_a line_n afâ—_n which_o be_v require_v to_o be_v do_v in_o this_o proposition_n be_v not_o teach_v to_o divide_v a_o right_n line_v angle_n into_o more_o part_n than_o two_o albeit_o to_o divide_v a_o angle_n so_o it_o be_v a_o right_a angle_n into_o three_o part_n it_o be_v not_o hard_a and_o it_o be_v teach_v of_o viâ—ellio_n in_o his_o first_o book_n of_o perspective_n nature_n the_o 28._o proposition_n â—or_a to_o divide_v a_o acute_a angle_n into_o three_o equal_a part_n be_v as_o say_v proclus_n impossible_a unless_o it_o be_v by_o the_o help_n of_o other_o line_n which_o be_v of_o a_o mix_a nature_n which_o thing_n nicomedes_n do_v by_o such_o line_n which_o be_v call_v concoideâ—_n linea_fw-la who_o first_o search_v out_o the_o invention_n nature_n &_o property_n of_o such_o line_n and_o other_o do_v it_o by_o other_o mean_v as_o by_o the_o help_n of_o quadrant_a line_n invent_v by_o hippias_n &_o nicomedes_n other_o by_o helices_n or_o spiral_n line_n invent_v of_o archimedes_n but_o these_o be_v thing_n of_o much_o difficulty_n and_o hardness_n and_o not_o here_o to_o be_v entreat_v of_o here_o against_o this_o proposition_n may_v of_o the_o adversary_n be_v bring_v a_o manifest_a instance_n for_o he_o may_v cavil_v that_o the_o
head_n of_o the_o equilater_n triangle_n shall_v not_o fall_v between_o the_o two_o right_a line_n but_o in_o one_o of_o they_o or_o without_o they_o both_o as_o for_o example_n there_o may_v also_o in_o this_o proposition_n be_v diverse_a casesâ—_n proposition_n â—f_n it_o so_o happen_v that_o there_o be_v no_o space_n under_o the_o base_a de_fw-fr to_o describe_v a_o equilater_n triangle_n but_o that_o of_o necessity_n you_o must_v describe_v it_o on_o the_o same_o side_n that_o the_o line_n ab_fw-la and_o ac_fw-la be_v for_o then_o the_o side_n of_o the_o equilater_n triangle_n either_o exact_o agree_v with_o the_o line_n ad_fw-la and_o ae_n if_o the_o say_a line_n ad_fw-la and_o ae_n be_v equal_a with_o the_o base_a de._n or_o they_o fall_v without_o they_o if_o the_o line_n ad_fw-la and_o ae_n be_v less_o than_o the_o base_a de._n or_o they_o fall_v within_o they_o if_o the_o say_a line_n be_v great_a theâ—_z the_o base_a de._n the_o 5._o problem_n the_o 10._o proposition_n to_o divide_v a_o right_a line_n give_v be_v finite_a into_o two_o equal_a part_n svppose_v that_o the_o right_a line_n give_v be_v ab_fw-la it_o be_v require_v to_o divide_v the_o line_n ab_fw-la into_o two_o equal_a part_n construction_n describe_v by_o the_o first_o proposition_n upon_o the_o line_n ab_fw-la a_o equilater_n triangle_n and_o let_v the_o same_o be_v abc_n and_o by_o the_o former_a proposition_n divide_v the_o angle_n acb_o into_o two_o equal_a part_n by_o the_o right_a line_n cd_o then_o i_o say_v that_o the_o right_a line_n ab_fw-la be_v divide_v into_o two_o equal_a part_n in_o the_o point_n d._n demonstration_n for_o forasmuch_o as_o by_o the_o first_o proposition_n ac_fw-la be_v equal_a to_o cb_o and_o cd_o be_v common_a to_o they_o both_o therefore_o these_o two_o line_n ac_fw-la &_o cd_o be_v equal_a to_o these_o two_o line_n bc_o &_o cd_o the_o one_o to_o the_o outstretch_o and_o the_o angle_n acd_o be_v equal_a to_o the_o angle_n bcd_o wherefore_o by_o the_o 4._o proposition_n the_o base_a ad_fw-la be_v equal_a to_o the_o base_a bd._n wherefore_o the_o right_a line_n give_v ab_fw-la be_v divide_v into_o two_o equal_a part_n in_o the_o point_n d_o which_o be_v require_v to_o be_v do_v apollonius_n reach_v to_o divide_v a_o right_a line_n be_v finite_a into_o two_o equal_a part_n after_o this_o manner_n apollonius_n the_o 6._o problem_n the_o 11._o proposition_n upon_o a_o right_a line_n give_v to_o raise_v up_o from_o a_o point_n give_v in_o the_o same_o line_n a_o perpendicular_a line_n svppose_v that_o the_o right_a line_n give_v be_v ab_fw-la &_o let_v the_o point_n in_o it_o give_v be_v c._n it_o be_v require_v from_o the_o point_n c_o to_o raise_z up_o unto_o the_o right_a line_n ab_fw-la a_o perpendicular_a line_n take_v in_o the_o line_n ac_fw-la a_o point_n at_o all_o adventure_n construction_n &_o let_v the_o same_o be_v d_o and_o by_o the_o 3._o proposition_n put_v unto_o dc_o a_o equal_a line_n ce._n and_o by_o the_o first_o proposition_n upon_o the_o line_n de_fw-fr describe_v a_o equilater_n triangle_n fde_v &_o draw_v a_o line_n from_o fletcher_n to_o c._n then_o i_o say_v that_o unto_o the_o right_a line_n give_v ab_fw-la and_o from_o the_o point_n in_o it_o give_v namely_o c_o be_v raise_v up_o a_o perpendicular_a line_n fc_o for_o forasmuch_o as_o dc_o be_v equal_a to_o ce_fw-fr demonstration_n &_o the_o line_n cf_o be_v common_a to_o they_o both_o therefore_o these_o two_o dc_o and_o cf_o be_v equal_a to_o these_o two_o ec_o &_o cf_o the_o one_o to_o the_o other_o and_o by_o the_o first_o proposition_n the_o base_a df_o be_v equal_a to_o the_o base_a of_o wherefore_o by_o the_o 8._o proposition_n the_o angle_n dcf_n be_v equal_a to_o the_o angle_n ecf_n and_o they_o be_v side_n angle_n but_o when_o a_o right_a line_n stand_v upon_o a_o right_a line_n do_v make_v the_o two_o side_n angle_n equal_a the_o one_o to_o the_o other_o either_o of_o those_o equal_a angle_n be_v by_o the._n 10._o definition_n a_o right_a angle_n &_o the_o line_n stand_v upon_o the_o right_a line_n be_v call_v a_o perpendicular_a line_n wherefore_o the_o angle_n dcf_n &_o thangle_v fce_n be_v right_a angle_n wherefore_o unto_o the_o right_a line_n give_v ab_fw-la &_o from_o the_o point_n in_o it_o c_o be_v raise_v up_o a_o perpendicular_a line_n cf_o which_o be_v require_v to_o be_v do_v although_o the_o point_n give_v shall_v be_v set_v in_o one_o of_o the_o end_n of_o the_o right_a line_n give_v it_o be_v easy_a so_o do_v it_o as_o it_o be_v before_o for_o produce_v the_o line_n in_o length_n from_o the_o point_n by_o the_o second_o petition_n you_o may_v work_v as_o you_o do_v before_o but_o if_o one_o require_v to_o erect_v a_o right_a line_n perpendicular_o from_o the_o point_n at_o the_o end_n of_o the_o line_n without_o produce_v the_o rightlyne_v that_o also_o may_v well_o be_v do_v after_o this_o manner_n appollonius_n teach_v to_o raise_v up_o unto_o a_o line_n give_v from_o a_o point_n in_o it_o give_v a_o perpendicular_a line_n after_o this_o manner_n construction_n the_o 7._o problem_n the_o 12._o proposition_n unto_o a_o right_a line_n give_v be_v infinite_a and_o from_o a_o point_n give_v not_o be_v in_o the_o same_o line_n to_o draw_v a_o perpendicular_a line_n let_v the_o right_a line_n give_v be_v infinite_a be_v ab_fw-la &_o let_v the_o point_n give_v not_o be_v in_o the_o say_a line_n ab_fw-la be_v c._n it_o be_v require_v from_o the_o point_n give_v namely_o c_o to_o draw_v unto_o the_o right_a line_n give_v ab_fw-la a_o perpendicular_a line_n construction_n take_v on_o the_o other_o side_n of_o the_o line_n ab_fw-la namely_o on_o that_o side_n wherein_o be_v not_o the_o point_n c_o a_o point_n at_o all_o adventure_n and_o let_v the_o same_o be_v d._n and_o make_v the_o centre_n c_o and_o the_o space_n cd_o describe_v by_o the_o three_o petition_n a_o circle_n and_o let_v the_o same_o be_v efg_o which_o let_v cut_v the_o line_n ab_fw-la in_o the_o point_n e_o and_o g._n and_o by_o the_o x._o proposition_n divide_v the_o line_n eglantine_n into_o two_o equal_a part_n in_o the_o point_n h._n and_o by_o the_o first_o petition_n draw_v these_o right_a line_n cg_o change_z and_o ce._n then_o i_o say_v that_o unto_o the_o right_a line_n give_v ab_fw-la &_o from_o the_o point_n give_v not_o be_v in_o it_o namely_o c_o be_v draw_v a_o perpendicular_a line_n ch._n demonstration_n for_o forasmuch_o as_o gh_o be_v equal_a to_o he_o and_o hc_n be_v common_a to_o they_o both_o therefore_o these_o two_o side_n gh_o and_o hc_n be_v equal_a to_o these_o two_o side_n eh_o &_o hc_n the_o one_o to_o the_o otherâ—_n and_o by_o the_o 15_o definition_n the_o base_a cg_o be_v equal_a to_o the_o base_a ce_fw-fr wherefore_o by_o the_o 8._o proposition_n the_o angle_n chg_n be_v equal_a to_o the_o angle_n i_o and_o they_o be_v side_n angle_n but_o when_o a_o right_a line_n stand_v upon_o a_o right_a line_n make_v the_o two_o side_n angle_n equal_a the_o one_o to_o the_o other_o either_o of_o those_o equal_a angle_n be_v by_o the_o 10._o definition_n a_o right_a angle_n and_o the_o line_n stand_v upon_o the_o say_a right_a line_n be_v call_v a_o perpendicular_a line_n wherefore_o unto_o the_o right_a line_n give_v ab_fw-la and_o from_o the_o point_n give_v c_o which_o be_v not_o in_o the_o line_n ab_fw-la be_v draw_v a_o perpendicular_a line_n change_z which_o be_v require_v to_o be_v do_v this_o problem_n do_v oenâ—pides_n first_o find_v out_o problem_n consider_v the_o necessary_a use_n thereof_o to_o the_o study_n of_o astronomy_n there_o arâ—_n two_o kind_n of_o perpendicular_a line_n solid_a whereof_o one_o be_v a_o plain_a perpendicular_a line_n the_o other_o be_v a_o solid_a a_o plain_a perpendicular_a line_n be_v when_o the_o point_n from_o whence_o the_o perpendiâ—uler_a line_n iâ—_z draw_v be_v in_o the_o same_o plain_a superficies_n with_o the_o line_n whereunto_o it_o be_v a_o perpendicular_a a_o solid_a perpendicular_a line_n be_v when_o the_o point_n from_o whence_o the_o perpendicular_a be_v draw_v be_v on_o high_a and_o without_o the_o plain_a superficies_n so_o that_o a_o plain_a perpendicular_a line_n be_v draw_v to_o a_o right_a line_n &_o a_o solid_a perpendicular_a line_n be_v draw_v to_o a_o superficies_n a_o plainâ—_n perpendicular_a line_n cause_v right_a angle_n with_o one_o only_a line_n namely_o with_o that_o upon_o who_o it_o fall_v but_o a_o solid_a perpendicular_a line_n cause_v right_a anâ—leâ—_n not_o only_o with_o one_o line_n but_o with_o as_o many_o line_n as_o may_v be_v draw_v in_o that_o superficies_n by_o the_o touch_n thereof_o line_n this_o proposition_n teach_v to_o draw_v a_o plain_a perpendicular_a line_n for_o it_o be_v draw_v to_o one_o line_n and_o suppose_v to_o
be_v in_o the_o self_n same_o plain_a superficies_n the_o 6._o theorem_a the_o 13._o proposition_n when_o a_o right_a line_n stand_v upon_o a_o right_a line_n make_v any_o angle_n those_o angle_n shall_v be_v either_o two_o right_a angle_n or_o equal_a to_o two_o right_a angle_n svppose_v that_o the_o right_a line_n ab_fw-la stand_v upon_o the_o right_a line_n cd_o do_v make_v these_o angle_n cba_o and_o abdella_n then_o i_o say_v that_o the_o angle_n cba_o and_o abdella_n be_v either_o two_o right_a angle_n or_o its_o squall_n to_o two_o right_a angle_n if_o the_o angle_n cba_o be_v equal_a to_o the_o angle_n abdella_n then_o be_v they_o two_o right_a angle_n by_o the_o ten_o definition_n construction_n but_o if_o not_o raise_v up_o by_o the_o 11._o proposition_n unto_o the_o right_a line_n cd_o and_o from_o the_o point_n give_v in_o it_o name_o demonstration_n b_o a_o perpendicular_a line_n beâ—_n wherefore_o by_o the_o x._o definition_n the_o angle_n cbe_n and_o ebb_v arâ—_n right_a angle_n now_o forasmuch_o as_o the_o angle_n cbe_n be_v equal_a to_o these_o two_o angle_n cba_o and_o abe_n put_v the_o angle_n ebb_v common_a to_o they_o bothâ—_n wherefore_o the_o angle_n cbe_n and_o ebb_v be_v equal_a to_o these_o three_o angle_n cba_o abe_n and_o ebb_v again_o forasmuch_o as_o the_o angle_n dba_n be_v equal_a unto_o these_o two_o angle_n dbe_n and_o eba_n put_v the_o angle_n abc_n common_a to_o they_o both_o wherefore_o the_o angle_n dba_n and_o abc_n be_v equal_a to_o these_o three_o angle_n dbe_n eba_n and_o abc_n and_o it_o be_v prove_v that_o the_o angle_n cbe_n and_o ebb_v be_v equal_a to_o the_o self_n same_o three_o angle_n but_o thing_n equal_a to_o one_o &_o the_o self_n same_o thing_n be_v also_o by_o the_o first_o common_a sentence_n equal_v the_o oâ—e_n to_o the_o oath_n wherefore_o the_o angle_n cbe_n and_o ebb_v be_v equal_a to_o the_o angle_n dba_n &_o abc_n but_o the_o angle_n cbe_n and_o ebb_v be_v two_o right_a angle_n wherefore_o also_o the_o angle_n dba_n and_o abc_n be_v equal_a to_o two_o right_a angle_n wherefore_o when_o a_o right_a line_n stand_v upon_o a_o right_a line_n make_v any_o angle_n those_o angle_n shall_v be_v either_o two_o right_a angle_n or_o equal_a to_o two_o right_a angle_n which_o be_v require_v to_o be_v demonstrate_v an_o other_o demonstration_n after_o pelitarius_n suppose_v that_o the_o right_a line_n ab_fw-la do_v stand_v upon_o the_o right_a line_n cd_o then_o i_o say_v that_o the_o two_o angle_n abc_n and_o abdella_n peliâ—â—riuâ—_n be_v either_o two_o right_a angle_n or_o equal_a to_o two_o right_a angle_n for_o if_o ab_fw-la be_v perpendicule_a unto_o cd_o they_o be_v it_o manifest_a that_o they_o be_v right_a angle_n by_o the_o conversion_n of_o the_o definition_n but_o if_o it_o incline_v towards_o the_o end_n c_o then_z by_o the_o 11._o proposition_n from_o the_o point_n b_o erect_v unto_o the_o line_n cd_o a_o perpendicular_a line_n be._n by_o which_o construction_n the_o proposition_n be_v very_o manifest_a for_o forasmuch_o as_o the_o angle_n abdella_n be_v greâ—ter_a than_o the_o right_a angle_n dbe_n by_o the_o angle_n abeâ—_n â—_z and_o the_o other_o angle_n abc_n be_v less_o than_o the_o right_a angle_n cbe_n by_o the_o self_n same_o angle_n abe_n if_o from_o the_o great_a bâ—e_n take_v away_o the_o excess_n and_o the_o same_o be_v add_v to_o the_o less_o they_o shall_v be_v make_v two_o right_a angle_n that_o be_v if_o from_o the_o obtuse_a angle_n abdella_n be_v take_v away_o the_o anglâ—_n abe_n there_o shall_v remain_v the_o right_a angle_n dbe_n and_o then_o if_o the_o same_o angle_n abe_n be_v add_v to_o the_o â—cute_a angle_n cba_o there_o shall_v be_v make_v the_o right_a angle_n cbe_n wherefore_o it_o be_v manifest_a that_o the_o two_o angle_n namely_o the_o obtuse_a angle_n abdella_n &_o the_o acute_a angle_n abc_n be_v equal_a to_o the_o two_o right_a angle_n cbe_v and_o dbe_n which_o be_v require_v to_o be_v prove_v the_o 7._o theorem_a the_o 14._o proposition_n if_o unto_o a_o right_a line_n and_o to_o a_o point_n in_o the_o same_o line_n be_v draw_v two_o right_a line_n noâ—_n both_o on_o one_o and_o the_o same_o side_n make_v the_o side_n angle_n equal_a to_o two_o right_a angle_n those_o two_o right_a line_n shall_v make_v direct_o one_o right_a line_n unto_o the_o right_a line_n ab_fw-la &_o to_o you_o point_fw-fr in_o it_o bâ—_n let_v therâ—_n be_v draw_v two_o right_a line_n bc_o and_o bd_o unto_o contrary_a side_n make_v the_o side_n angle_n namely_o abc_n &_o abdella_n equal_a to_o two_o right_a angle_n then_o i_o say_v that_z the_o right_a line_n bd_o and_o bc_o make_v both_o one_o right_a line_n absurdity_n for_o if_o cb_o and_o bd_o do_v not_o make_v both_o one_o right_a line_n let_v the_o right_a line_n be_v be_v so_o draw_v to_o bc_o that_o they_o both_o make_v one_o right_a line_n now_o forasmuch_o as_o the_o right_a line_n ab_fw-la stand_v upon_o the_o right_a line_n cbe_n therefore_o the_o angle_n abc_n and_o abe_n be_v equal_a to_o two_o right_a angle_n by_o the_o 13._o proposition_n but_o by_o supposition_n the_o angle_n abc_n and_o abdella_n be_v equal_a to_o two_o right_a angle_n wherefore_o the_o angle_n cba_o and_o abe_n be_v equal_a to_o the_o angle_n cba_o and_o abdella_n take_v away_o the_o angle_n abc_n which_o be_v common_a to_o they_o both_o wherefore_o the_o angle_n remain_v abe_n be_v equal_a to_o the_o angle_n remain_v abdella_n namely_o the_o less_o to_o the_o great_a which_o be_v impossible_a wherefore_o the_o line_n be_v be_v not_o so_o direct_o draw_v to_o bc_o that_o they_o both_o make_v one_o right_a line_n in_o like_a sort_n may_v we_o prove_v that_o no_o other_o line_n beside_o bd_o can_v so_o be_v drawnet_n wherefore_o the_o line_n cb_o and_o bd_o make_v both_o one_o right_a line_n if_o therefore_o unto_o a_o right_a line_n &_o to_o a_o point_n in_o the_o same_o line_n be_v draw_v two_o right_a line_n not_o both_o on_o one_o and_o the_o same_o side_n make_v the_o side_n angle_n equal_a to_o two_o right_a angle_n those_o two_o line_n shall_v make_v direct_o one_o right_a line_n which_o be_v require_v to_o be_v prove_v an_o other_o demonstration_n after_o pelitarius_n suppose_v that_o there_o be_v a_o right_a line_n ab_fw-la unto_o who_o point_n â—_o pelitarius_n let_v there_o be_v draw_v two_o right_a line_n cb_o and_o bd_o unto_o contrary_a side_n and_o let_v the_o two_o angle_n cba_o and_o dba_n be_v either_o two_o right_a angle_n or_o equal_a to_o two_o right_a angle_n then_o i_o say_v that_o the_o two_o line_n cb_o and_o bd_o do_v make_v direct_o one_o right_a line_n namely_o cd_o for_o if_o they_o do_v not_o them_z lâ—t_o â—e_z bâ—â—o_o draw_v unto_o cb_o that_o they_o both_o make_v direct_o one_o right_a line_n câ—â—_n which_o shall_v pass_v â—ither_o above_o the_o line_n bd_o or_o under_o it_o first_o lâ—t_v it_o pass_v above_o it_o and_o for_o aâ—_z much_o as_o the_o two_o angle_n cba_o and_o abe_n be_v by_o the_o former_a proposition_n equal_a to_o two_o right_a angle_n and_o be_v a_o part_n of_o the_o two_o angle_n cba_o and_o abdella_n but_o the_o angle_n cba_o and_o aâ—d_a aâ—e_n by_o supposition_n equal_v also_o to_o two_o right_a anglesâ—_n therefore_o the_o parâ—â—_n be_v equal_a to_o the_o whole_a which_o be_v impossible_a and_o the_o like_a absurdity_n will_v follow_v if_o cb_o e_o pass_n under_o the_o line_n bd_o namely_o that_o the_o whole_a shall_v be_v equal_v to_o the_o partâ—_n which_o be_v also_o impossible_a wherefore_o cd_o be_v one_o right_a line_n which_o be_v require_v to_o be_v prove_v the_o 8._o theorem_a the_o 15._o proposition_n if_o two_o right_a line_n cut_v the_o one_o the_o other_o the_o head_n angle_n shall_v be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o these_o two_o right_a line_n ab_fw-la and_o cd_o do_v cut_v the_o one_o the_o other_o in_o the_o point_n e._n then_o i_o say_v that_o the_o angle_n aec_o be_v equal_a to_o the_o angle_n deb_n demonstration_n for_o forasmuch_o as_o the_o right_a line_n ae_n stand_v upon_o the_o right_a line_n dc_o make_v these_o angle_n cea_n and_o aed_n therefore_o by_o the_o 13._o proposition_n the_o angle_n cea_n and_o aed_n be_v equal_a to_o two_o right_a angle_n again_o forasmuch_o as_o the_o right_a line_n de_fw-fr stand_v upon_o the_o right_a line_n ab_fw-la make_v these_o angle_n aed_n and_o deb_n therefore_o by_o the_o same_o proposition_n the_o angle_n aed_n and_o deb_n be_v equal_a to_o two_o right_a angle_n and_o it_o be_v prove_v that_o the_o angle_n cea_n and_o aed_n be_v also_o equal_a to_o two_o right_a angle_n wherefore_o the_o angle_n cea_n and_o aed_n be_v equal_a to_o the_o angle_n aed_n and_o deb_n take_v away_o the_o angle_n aed_n which_o
be_v require_v to_o be_v demonstrate_v this_o proposition_n may_v also_o be_v demonstrate_v without_o produce_v any_o of_o the_o side_n aâ—ter_v this_o manner_n the_o same_o may_v yet_o also_o be_v demonstrate_v a_o other_o way_n this_o proposition_n may_v yet_o moreover_o be_v demonstrate_v by_o a_o argument_n lead_v to_o a_o absurdity_n and_o that_o after_o this_o manner_n line_n a_o man_n may_v also_o more_o brief_o demonstrate_v this_o proposition_n by_o campanus_n definition_n of_o a_o right_a line_n which_o as_o we_o have_v before_o declare_v be_v thus_o a_o right_a line_n be_v the_o short_a extension_n or_o drawght_fw-mi that_fw-mi be_v or_o may_v be_v from_o one_o point_n to_o another_o wherefore_o any_o one_o side_n of_o a_o triangle_n for_o that_o it_o be_v a_o right_a line_n draw_v from_o some_o one_o point_n to_o some_o other_o one_o point_n be_v of_o necessity_n short_a than_o the_o other_o two_o side_n draw_v from_o and_o to_o the_o same_o point_n understanding_n epicurus_n and_o such_o as_o follow_v he_o deride_v this_o proposition_n not_o count_v it_o worthy_a to_o be_v add_v in_o the_o number_n of_o proposition_n of_o geometry_n for_o the_o easiness_n thereof_o for_o that_o it_o be_v manifest_a even_o to_o the_o sense_n but_o not_o all_o thing_n manifest_a to_o sense_n be_v straight_o way_n manifest_a to_o reason_n and_o understanding_n it_o pertain_v to_o one_o that_o be_v a_o teacher_n of_o science_n by_o proof_n and_o demonstration_n to_o render_v a_o certain_a and_o undoubted_a reason_n why_o it_o so_o appear_v to_o the_o senseâ—_n and_o in_o that_o only_o consist_v science_n the_o 14._o theorem_a the_o 21._o proposition_n if_o from_o the_o end_n of_o one_o of_o the_o side_n of_o a_o triangle_n be_v draw_v to_o any_o point_n within_o the_o say_a triangle_n two_o right_a line_n those_o right_a line_n so_o draw_v shall_v be_v less_o then_o the_o two_o other_o side_n of_o the_o triangle_n but_o shall_v contain_v the_o great_a angle_n svppose_v that_o abc_n be_v a_o triangle_n and_o from_o the_o end_n of_o the_o side_n bc_o namely_o from_o the_o point_n b_o and_o c_o let_v there_o be_v draw_v within_o the_o triangle_n two_o right_a line_n bd_o and_o cd_o to_o the_o point_n d._n then_o i_o say_v that_o the_o line_n bd_o and_o cd_o be_v less_o than_o the_o other_o side_n of_o the_o triangle_n namely_o than_o the_o side_n basilius_n and_o ac_fw-la and_o that_o the_o angle_n which_o they_o contain_v namely_o bdc_o be_v great_a than_o the_o angle_n bac_n extend_v by_o the_o second_o petition_n the_o line_n bd_o to_o the_o point_n e._n demonstration_n and_o forasmuch_o as_o by_o the_o 20._o proposition_n in_o every_o triangle_n the_o two_o side_n be_v great_a than_o the_o side_n remain_v therefore_o the_o two_o side_n of_o the_o triangle_n abe_n namely_o the_o side_n ab_fw-la and_o ae_n be_v great_a than_o the_o side_n ebb_n put_v the_o line_n ec_o common_a to_o they_o both_o wherefore_o the_o line_n basilius_n and_o ac_fw-la be_v great_a than_o the_o line_n be_v and_o ecâ—_n again_o forasmuch_o as_o by_o the_o same_o in_o the_o triangle_n ce_z the_o two_o side_n ce_fw-fr and_o ed_z be_v great_a than_o the_o side_n dc_o put_v the_o line_n db_o common_a to_o they_o bothâ—_n wherefore_o the_o line_n ce_fw-fr and_o ed_z be_v great_a than_o the_o line_n cd_o and_o db._n but_o it_o be_v prove_v that_o the_o line_n basilius_n and_o ac_fw-la be_v great_a than_o the_o line_n be_v and_o ec_o wherefore_o the_o line_n basilius_n and_o ac_fw-la be_v much_o great_a than_o the_o line_n bd_o and_o dc_o again_o forasmuch_o as_o by_o the_o 16._o proposition_n in_o every_o triangle_n the_o outward_a angle_n be_v great_a than_o the_o inward_a and_o opposite_a angle_n therefore_o the_o outward_a angle_n of_o the_o triangle_n cde_o namely_o bdc_o be_v great_a than_o the_o angle_n ce_z wherefore_o also_o by_o the_o same_o the_o outward_a angle_n of_o the_o triangle_n abe_n namely_o the_o angle_n ceb_n be_v great_a than_o the_o angle_n bac_n but_o it_o be_v prove_v that_o the_o angle_n bdc_o be_v great_a than_o the_o angle_n ceb_fw-mi wherefore_o the_o angle_n bdc_o be_v much_o great_a than_o the_o angle_n bac_n wherefore_o if_o from_o the_o end_n of_o one_o of_o the_o side_n of_o a_o triangle_n be_v draw_v to_o any_o point_n within_o the_o say_a triangle_n two_o right_a line_n those_o right_a line_n so_o draw_v shall_v be_v less_o than_o the_o two_o other_o side_n of_o the_o triangle_n but_o shall_v contain_v the_o great_a angle_n which_o be_v require_v to_o be_v demonstrate_v in_o this_o proposition_n be_v express_v that_o the_o two_o right_a line_n draw_v within_o the_o triangle_n have_v their_o beginning_n at_o the_o extreme_n of_o the_o side_n of_o the_o triangle_n for_o from_o the_o one_o extreme_a of_o the_o side_n of_o the_o triangle_n and_o from_o some_o one_o point_n of_o the_o same_o side_n may_v be_v draw_v two_o right_a line_n within_o the_o triangle_n which_o shall_v be_v long_o they_o the_o two_o outward_a line_n which_o be_v wonderful_a and_o seem_v strange_a that_o two_o right_a line_n draw_v upon_o a_o part_n of_o a_o line_n shall_v be_v great_a than_o two_o right_a line_n draw_v upon_o the_o whole_a line_n and_o again_o it_o be_v possible_a from_o the_o one_o extreme_a of_o the_o side_n of_o a_o triangle_n and_o from_o some_o one_o point_n of_o the_o same_o side_n to_o draw_v two_o right_a line_n within_o the_o triangle_n which_o shall_v contain_v a_o angle_n less_o than_o the_o angle_n contain_v under_o the_o two_o outward_a line_n as_o touch_v the_o first_o part_n the_o 8._o problem_n the_o 22._o proposition_n of_o three_o right_a line_n which_o be_v equal_a to_o three_o right_a line_n give_v to_o make_v a_o triangle_n but_o it_o behove_v two_o of_o those_o line_n which_o then_o soever_o be_v take_v to_o be_v great_a than_o the_o three_o for_o that_o in_o every_o triangle_n two_o side_n which_o two_o side_n soever_o be_v take_v be_v great_a than_o the_o side_n remain_v svppose_v that_o the_o three_o right_a line_n give_v be_v a_o b_o c_o of_o which_o let_v two_o of_o they_o which_o two_o soever_o be_v take_v be_v great_a than_o the_o three_o that_o be_v let_v the_o line_n a_o b_o be_v great_a than_o the_o line_n c_o and_o the_o line_n a_o c_o then_z the_o line_n b_o and_o the_o line_n b_o c_o then_z the_o line_n a._n it_o be_v require_v of_o three_o right_a line_n equal_a to_o the_o right_a line_n a_o b_o c_o to_o make_v a_o triangle_n construction_n take_v a_o right_a line_n have_v a_o appoint_a end_n on_o the_o side_n d_o and_o be_v infinite_a on_o the_o side_n e._n and_o by_o the_o 3._o proposition_n put_v unto_o the_o line_n a_o a_o equal_a line_n df_o and_o put_v unto_o the_o line_n b_o a_o equal_a line_n fg_o and_o unto_o the_o line_v c_o a_o equal_a line_n gh_o and_o make_v the_o centre_n fletcher_n and_o the_o space_n df_o describe_v by_o the_o 3._o petition_n a_o circle_n dkl_n again_o make_v the_o centre_n g_o and_o the_o space_n g_o h_o describe_v by_o the_o same_o a_o circle_n hkl_n and_o let_v the_o point_n of_o the_o intersection_n of_o the_o say_a circle_n be_v king_n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n king_n to_o the_o point_fw-fr fletcher_n &_o a_o other_o from_o the_o point_n king_n to_o the_o point_n g._n then_o i_o say_v that_o of_o three_o right_a line_n equal_a to_o the_o line_n a_o b_o c_o be_v make_v a_o triangle_n kfg_n demonstration_n for_o forasmuch_o as_o the_o point_n fletcher_n be_v the_o centre_n of_o the_o circle_n dkl_n therefore_o by_o the_o 15._o definition_n the_o line_n fd_o be_v equal_a to_o the_o line_n fk_o but_o the_o line_n a_o be_v equal_a to_o the_o line_n fd_o wherefore_o by_o the_o first_o common_a sentence_n the_o line_n fk_o be_v equal_a to_o the_o line_n a._n again_o forasmuch_o as_o the_o point_n g_o be_v the_o centre_n of_o the_o circle_n lkh_n therefore_o by_o the_o same_o definition_n the_o line_n gk_o be_v equal_a to_o the_o line_n ghâ—_n but_o the_o line_n c_o be_v equal_a to_o the_o line_n gh_o wherefore_o by_o the_o first_o common_a sentence_n the_o line_n kg_n be_v equal_a to_o the_o line_n c._n but_o the_o line_n fg_o be_v by_o supposition_n equal_a to_o the_o line_n b_o wherefore_o these_o three_o right_a line_n gf_o fk_o and_o kg_v be_v equal_a to_o these_o three_o right_a line_n a_o b_o c._n wherefore_o of_o three_o right_a line_n that_o be_v kf_n fg_o and_o gk_o which_o be_v equal_a to_o the_o three_o right_a line_n give_v that_o be_v to_o a_o b_o c_o be_v make_v a_o triangle_n kfg_n which_o be_v require_v to_o be_v do_v an_o other_o construction_n and_o demonstration_n after_o flussates_n suppose_v that_o the_o
square_n which_o be_v make_v of_o the_o line_n ab_fw-la and_o ac_fw-la construction_n describe_v by_o the_o 46._o proposition_n upon_o the_o line_n bc_o a_o square_a bdce_n and_o by_o the_o same_o upon_o the_o line_n basilius_n and_o ac_fw-la describe_v the_o square_n abfg_n and_o ackh_n and_o by_o the_o point_n a_o draw_v by_o the_o 31._o proposition_n to_o either_o of_o these_o line_n bd_o and_o ce_fw-fr a_o parallel_a line_n al._n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n d_o and_o a_o other_o from_o the_o point_n c_o to_o the_o point_n e._n demonstration_n and_o forasmuch_o as_o the_o angle_n bac_n and_o bag_n be_v right_a angle_n therefore_o unto_o a_o right_a line_n basilius_n and_o to_o a_o point_n in_o it_o give_v a_o be_v draw_v two_o right_a line_n ac_fw-la and_o ag_z not_o both_o on_o one_o and_o the_o same_o side_n make_v the_o two_o side_n angle_n equal_a to_o two_o right_a angle_n wherefore_o by_o the_o 14._o proposition_n the_o line_n ac_fw-la and_o ag_z make_v direct_o one_o right_a line_n and_o by_o the_o same_o reason_n the_o line_n basilius_n and_o ah_o make_v also_o direct_o one_o right_a line_n and_o forasmuch_o as_o the_o angle_n dbc_n be_v equal_a to_o the_o angle_n fba_n for_o either_o of_o they_o be_v a_o right_a angle_n put_v the_o angle_n abc_n common_a to_o they_o both_o wherefore_o the_o whole_a angle_n dba_n be_v equal_a to_o the_o whole_a angle_n fbc_n and_o forasmuch_o as_o these_o two_o line_n ab_fw-la and_o bd_o be_v equal_a to_o these_o two_o line_n bf_o and_o bc_o the_o one_o to_o the_o other_o and_o the_o angle_n dba_n be_v equal_a to_o the_o angle_n fbc_n therefore_o by_o the_o 4._o proposition_n the_o base_a ad_fw-la be_v equal_a to_o the_o base_a fc_n and_o the_o triangle_n abdella_n be_v equal_a to_o the_o triangle_n fbc_n but_o by_o the_o 31._o proposition_n the_o parallelogram_n bl_o be_v double_a to_o the_o triangle_n abdella_n for_o they_o have_v both_o one_o and_o the_o same_o base_a namely_o bd_o and_o be_v in_o the_o self_n same_o parallel_a line_n that_o be_v bd_o and_o all_o and_o by_o the_o same_o the_o square_a gb_o be_v double_a to_o the_o triangle_n fbc_n for_o they_o have_v both_o one_o and_o the_o self_n same_o base_a that_o be_v bf_o and_o be_v in_o the_o self_n same_o parallel_a line_n that_o be_v fb_o and_o gc_o but_o the_o double_n of_o thing_n equal_a be_v by_o the_o six_o common_a sentence_n equal_v the_o one_o to_o the_o other_o wherefore_o the_o parallelogram_n bl_o be_v equal_a to_o the_o square_a gb_o and_o in_o like_a sort_n if_o by_o the_o first_o petition_n there_o be_v draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n e_o and_o a_o other_o from_o the_o point_n b_o to_o the_o point_n king_n we_o may_v prove_v that_o the_o parallelogram_n cl_n be_v equal_a to_o the_o square_a hc_n wherefore_o the_o whole_a square_a bdec_n be_v equal_a to_o the_o two_o square_n gb_o and_o hc_n but_o the_o square_a bdec_n be_v describe_v upon_o the_o line_n bc_o and_o the_o square_n gb_o and_o hc_n be_v describe_v upon_o the_o line_n basilius_n &_o ac_fw-la wherefore_o the_o square_a of_o the_o side_n bc_o be_v equal_a to_o the_o square_n of_o the_o side_n basilius_n and_o ac_fw-la wherefore_o in_o rectangle_n triangle_n the_o square_n which_o be_v make_v of_o the_o side_n that_o subtend_v the_o right_a angle_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o side_n contain_v the_o right_a angle_n which_o be_v require_v to_o be_v demonstrate_v this_o most_o excellent_a and_o notable_a theorem_a be_v first_o invent_v of_o the_o great_a philosopher_n pythagoras_n proposition_n who_o for_o the_o exceed_a joy_n conceive_v of_o the_o invention_n thereof_o offer_v in_o sacrifice_n a_o ox_n as_o record_n hiâ—rone_n proclus_n lycius_n &_o vitruutus_fw-la and_o it_o have_v be_v common_o call_v of_o barbarous_a writer_n of_o the_o latter_a time_n dulcarnon_n pâ—lâ—tariâ—â—_n a_o addition_n of_o pelitarius_n to_o reduce_v two_o unequal_a square_n to_o two_o equal_a square_n suppose_v that_o the_o square_n of_o the_o line_n ab_fw-la and_o ac_fw-la be_v unequal_a it_o be_v require_v to_o reduce_v they_o to_o two_o equal_a square_n join_v the_o two_o line_n ab_fw-la and_o ac_fw-la at_o their_o end_n in_o such_o sort_n that_o they_o make_v a_o right_a angle_n bac_n and_o draw_v a_o line_n from_o b_o to_o c._n then_o upon_o the_o two_o end_n b_o and_o c_o make_v two_o angle_n each_o of_o which_o may_v be_v equal_a to_o half_a a_o right_a angle_n this_o be_v do_v by_o erect_v upon_o the_o line_n bc_o perpendicule_a line_n from_o the_o point_n b_o and_o c_o and_o so_o by_o the_o 9_o proposition_n devide_v â—che_fw-mi of_o the_o right_a angle_n into_o two_o equal_a part_n and_o let_v the_o angle_n bcd_o and_o cbd_v be_v either_o of_o they_o half_o of_o a_o right_a angle_n and_o let_v the_o line_n bd_o and_o cd_o concur_v in_o the_o point_n d._n then_o i_o say_v that_o the_o two_o square_n of_o the_o side_n bd_o and_o cd_o be_v equal_a to_o the_o two_o square_n of_o the_o side_n ab_fw-la and_o ac_fw-la for_o by_o the_o 6_o proposition_n the_o two_o side_n db_o and_o dc_o be_v equal_a and_o the_o angle_n at_o the_o point_n d_o be_v by_o the_o 32._o proposition_n a_o right_a angle_n wherefore_o the_o square_a of_o the_o side_n bc_o be_v equal_a to_o the_o square_n of_o the_o two_o side_n db_o and_o dc_o by_o the_o 47._o proposition_n but_o it_o be_v also_o equal_a to_o the_o square_n of_o the_o two_o side_n ab_fw-la and_o ac_fw-la by_o the_o self_n same_o proposition_n wherefore_o by_o the_o common_a sentence_n the_o square_n of_o the_o two_o side_n bd_o and_o dc_o be_v equal_a to_o the_o square_n of_o the_o two_o side_n ab_fw-la and_o ac_fw-la which_o be_v require_v to_o be_v do_v an_o other_o addition_n of_o pelitarius_n pelitarius_n if_o two_o right_a angle_a triangle_n have_v equal_a base_n the_o square_n of_o the_o two_o side_n of_o the_o one_o be_v equal_a to_o the_o square_n of_o the_o two_o side_n of_o the_o other_o this_o be_v manifest_a by_o the_o former_a construction_n and_o demonstration_n an_o other_o addition_n of_o pelitarius_n pelitarius_n two_o unequal_a line_n be_v give_v to_o know_v how_o much_o the_o square_n of_o the_o one_o be_v great_a than_o the_o square_a of_o the_o other_o suppose_v that_o there_o be_v two_o unequal_a line_n ab_fw-la and_o bc_o of_o which_o let_v ab_fw-la be_v the_o great_a it_o be_v require_v to_o search_v out_o how_o much_o the_o square_n of_o ab_fw-la exceed_v the_o square_a of_o bc._n that_o be_v i_o will_v find_v out_o the_o square_n which_o with_o the_o square_n of_o the_o line_n bc_o shall_v be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la put_v the_o line_n a_o b_o and_o bc_o direct_o that_o they_o make_v both_o one_o right_a line_n and_o make_v the_o centre_n the_o point_n b_o and_o the_o space_n basilius_n describe_v a_o circle_n ade_n and_o produce_v the_o line_n ac_fw-la to_o the_o circumference_n and_o let_v it_o concur_v with_o it_o in_o the_o point_n e._n and_o upon_o the_o line_n ae_n and_o from_o the_o point_n c_o erect_v by_o the_o 11._o proposition_n a_o perpendicular_a line_n cd_o which_o produce_v till_o it_o concur_v with_o the_o circumference_n in_o the_o point_n d_o &_o draw_v a_o line_n from_o b_o to_o d._n then_o i_o say_v that_o the_o square_n of_o the_o line_n cd_o be_v the_o excess_n of_o the_o square_n of_o the_o line_n ab_fw-la above_o the_o square_n of_o the_o line_n bc._n for_o forasmuch_o as_o in_o the_o triangle_n bcd_o the_o angle_n at_o the_o point_n c_o be_v a_o right_a angle_n the_o square_a of_o the_o base_a bd_o be_v equal_a to_o the_o square_n of_o the_o two_o side_n bc_o and_o cd_o by_o this_o 47._o proposition_n wherefore_o also_o the_o square_a of_o the_o line_n ab_fw-la be_v equal_a to_o the_o self_n same_o square_n of_o the_o line_n bc_o and_o cd_o wherefore_o the_o square_a of_o the_o line_n bc_o be_v so_o much_o less_o than_o the_o square_a of_o the_o line_n ab_fw-la as_o be_v the_o square_a of_o the_o line_n cd_o which_o be_v require_v to_o search_v out_o an_o other_o addition_n of_o pelitarius_n pelitarius_n the_o diameter_n of_o a_o square_n be_v give_v to_o geve_v the_o square_a thereof_o this_o be_v easy_a to_o be_v do_v for_o if_o upon_o the_o two_o end_n of_o the_o line_n be_v draw_v two_o half_a right_a angle_n and_o so_o be_v make_v perfect_a the_o triangle_n then_o shall_v be_v describe_v half_a of_o the_o square_n the_o other_o half_o whereof_o also_o be_v after_o the_o same_o manner_n easy_a to_o be_v describe_v hereby_o it_o be_v manifest_a that_o the_o square_n of_o the_o
the_o other_o in_o the_o point_n d_o and_o that_o only_a point_n be_v common_a to_o they_o both_o neither_o do_v the_o one_o enter_v into_o the_o other_o if_o any_o part_n of_o the_o one_o enter_v into_o any_o part_n of_o the_o other_o than_o the_o one_o cut_v and_o divide_v the_o other_o and_o touch_v the_o one_o the_o other_o not_o in_o one_o point_n only_o as_o in_o the_o other_o before_o but_o in_o two_o pointâ—s_n and_o have_v also_o a_o superficies_n common_a to_o they_o both_o as_o the_o circle_n ghk_v and_o hlk_v cut_v the_o one_o the_o other_o in_o two_o point_n h_o and_o edward_n and_o the_o one_o enter_v into_o the_o other_o also_o the_o superficies_n hk_o be_v common_a to_o they_o both_o for_o it_o be_v a_o part_n of_o the_o circle_n ghk_o and_o also_o it_o be_v a_o part_n of_o the_o circle_n hlk._n definition_n right_o line_n in_o a_o circle_n be_v say_v to_o be_v equal_o distant_a from_o the_o centre_n when_o perpendicular_a line_n draw_v from_o the_o centre_n unto_o those_o line_n be_v equal_a and_o that_o line_n be_v say_v to_o be_v more_o distant_a upon_o who_o fall_v the_o great_a perpendicular_a line_n as_o in_o the_o circle_n abcd_o who_o centre_n be_v e_o the_o two_o line_n ab_fw-la and_o cd_o have_v equal_a distance_n from_o the_o centre_n e_o because_o that_o the_o line_n of_o draw_v from_o the_o centre_n e_o perpendicular_o upon_o the_o line_n ab_fw-la and_o the_o line_n eglantine_n draw_v likewise_o perpendilar_o from_o the_o centre_n e_o upon_o the_o line_n cd_o be_v equal_a the_o one_o to_o the_o other_o but_o in_o the_o circle_n hklm_n who_o centre_n be_v no_o the_o line_n hk_o have_v great_a distance_n from_o the_o centre_n n_o than_o have_v the_o line_n lm_o for_o that_o the_o line_n on_o draw_v from_o the_o centre_n n_o perpendicular_o upon_o the_o line_n hk_o be_v great_a than_o the_o line_n np_v which_o be_v draw_v from_o the_o centre_n n_o perpendicular_o upon_o the_o line_n lm_o so_o likewise_o in_o the_o other_o figure_n the_o line_n ab_fw-la and_o dc_o in_o the_o circle_n abcd_o be_v equidistant_v from_o the_o centre_n gâ—_n â—_z because_o the_o line_n og_n and_o gp_n perpendicular_o draw_v from_o the_o centre_n g_o upon_o the_o say_a line_n ab_fw-la and_o dc_o be_v equal_a and_o the_o line_n ab_fw-la have_v great_a distance_n from_o the_o centre_n g_o than_o have_v the_o the_o line_n of_o because_o the_o line_n og_n perpendicular_o drowen_v from_o the_o centre_n g_o to_o the_o line_n ab_fw-la be_v greâ—ter_n than_o the_o line_n hg_o which_o be_v perpendicular_o draw_v from_o the_o câ—â—tre_n g_o to_o the_o line_n ef._n definition_n a_o section_n or_o segment_n of_o a_o circle_n be_v a_o figure_n comprehend_v under_o a_o right_a line_n and_o a_o portion_n of_o the_o circumference_n of_o a_o circle_n as_o the_o figure_n abc_n be_v a_o section_n of_o a_o circle_n because_o it_o be_v comprehend_v under_o the_o right_a line_n ac_fw-la and_o the_o circumference_n of_o a_o circle_n abc_n likewise_o the_o figure_n def_n be_v a_o section_n of_o a_o circle_n for_o that_o it_o be_v comprehend_v under_o the_o right_a line_n df_o and_o the_o circumference_n def_n and_o the_o figure_n abc_n for_o that_o it_o contain_v within_o it_o the_o centre_n of_o the_o circle_n be_v call_v the_o great_a section_n of_o a_o circle_n and_o the_o figure_n def_n be_v the_o less_o section_n of_o a_o circle_n because_o it_o be_v whole_o without_o the_o centre_n of_o the_o circle_n as_o it_o be_v note_v in_o the_o 16._o definition_n of_o the_o first_o book_n a_o angle_n of_o a_o section_n or_o segment_n be_v that_o angle_n which_o be_v contain_v under_o a_o right_a line_n and_o the_o circumference_n of_o the_o circle_n definition_n as_o the_o angle_n abc_n in_o the_o section_n abc_n be_v a_o angle_n of_o a_o section_n because_o it_o be_v contain_v of_o the_o circumference_n bac_n and_o the_o right_a line_n bc._n likewise_o the_o angle_n cbd_v be_v a_o angle_n of_o the_o section_n bdc_o because_o it_o be_v contain_v under_o the_o circumference_n bdc_o and_o the_o right_a line_n bc._n and_o these_o angle_n be_v common_o call_v mix_v angle_n angle_n because_o they_o be_v contain_v under_o a_o right_a line_n and_o a_o crooked_a and_o these_o portion_n of_o circumference_n be_v common_o call_v arke_n arke_n and_o the_o right_a line_n be_v call_v chord_n chord_n or_o right_a line_n subtend_v and_o the_o great_a section_n have_v ever_o the_o great_a angle_n and_o the_o less_o section_n the_o less_o angle_n a_o angle_n be_v say_v to_o be_v in_o a_o section_n definition_n when_o in_o the_o circumference_n be_v take_v any_o point_n and_o from_o that_o point_n be_v draw_v right_a line_n to_o the_o end_n of_o the_o right_a line_n which_o be_v the_o base_a of_o the_o segment_n the_o angle_n which_o be_v contain_v under_o the_o right_a line_n draw_v from_o the_o point_n be_v i_o say_v say_v to_o be_v a_o angle_n in_o a_o section_n as_o the_o angle_n abc_n be_v a_o angle_n be_v the_o section_n abc_n because_o from_o the_o point_n b_o be_v a_o point_n in_o the_o circumference_n abc_n be_v draw_v two_o right_a line_n bc_o and_o basilius_n to_o the_o end_n of_o the_o line_n ac_fw-la which_o be_v the_o base_a of_o the_o section_n abc_n likewise_o the_o angle_n adc_o be_v a_o angle_n in_o the_o section_n adc_o because_o from_o the_o point_n d_o be_v in_o the_o circumference_n adc_o be_v draw_v two_o right_a line_n namely_o dc_o &_o da_z to_o the_o end_n of_o the_o right_a line_n ac_fw-la which_o be_v also_o the_o base_a to_o the_o say_a section_n adc_o so_o you_o see_v it_o be_v not_o all_o one_o to_o say_v section_n a_o angle_n of_o a_o section_n and_o a_o angle_n in_o a_o section_n a_o angle_n of_o a_o section_n consist_v of_o the_o touch_n of_o a_o right_a line_n and_o a_o crooked_a and_o a_o angle_n in_o a_o section_n be_v place_v on_o the_o circumference_n and_o be_v contain_v of_o two_o right_a line_n also_o the_o great_a section_n have_v in_o it_o the_o less_o angle_n and_o the_o less_o section_n have_v in_o it_o the_o great_a angle_n but_o when_o the_o right_a line_n which_o comprehend_v the_o angle_n do_v receive_v any_o circumference_n of_o a_o circle_n definition_n than_o that_o angle_n be_v say_v to_o be_v correspondent_a and_o to_o pertain_v to_o that_o circumference_n as_o the_o right_a line_n basilius_n and_o bc_o which_o contain_v the_o angle_n ab_fw-la c_z and_o receive_v the_o circumference_n adc_o therefore_o the_o angle_n abc_n be_v say_v to_o subtend_v and_o to_o pertain_v to_o the_o circumference_n adc_o and_o if_o the_o right_a line_n which_o cause_n the_o angle_n concur_v in_o the_o centre_n of_o a_o circle_n then_o the_o angle_n be_v say_v to_o be_v in_o the_o centre_n of_o a_o circle_n as_o the_o angle_n efd_n be_v say_v to_o be_v in_o the_o centre_n of_o a_o circle_n for_o that_o it_o be_v comprehend_v of_o two_o right_a line_n fe_o and_o fd_v which_o concur_v and_o touch_v in_o the_o centre_n f._n and_o this_o angle_n likewise_o subtend_v the_o 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centre_n of_o the_o circle_n and_o receive_v the_o circumference_n of_o the_o say_a section_n first_o the_o other_o pertain_v to_o the_o angle_n in_o the_o section_n which_o as_o before_o be_v say_v be_v ever_o in_o the_o circumference_n as_o if_o the_o angle_n bac_n be_v in_o the_o centre_n a_o and_o receive_v of_o the_o circumference_n blc_n be_v equal_a to_o the_o angle_n feg_n be_v also_o in_o the_o centre_n e_o and_o receive_v of_o the_o circumference_n fkg_v then_o be_v the_o two_o section_n bcl_n and_o fgk_n like_v by_o the_o first_o definition_n by_o the_o same_o definition_n also_o be_v the_o other_o two_o section_n like_a namely_o bcd_o and_o fgh_a for_o that_o the_o angle_n bac_n be_v equal_a to_o the_o
a_o circle_n be_v take_v any_o point_n which_o be_v not_o the_o centre_n of_o the_o circle_n and_o from_o that_o point_n be_v draw_v unto_o the_o circumference_n certain_a right_a line_n the_o great_a of_o those_o line_n shall_v be_v that_o line_n wherein_o be_v the_o centre_n and_o the_o lest_o shall_v be_v the_o residue_n of_o the_o same_o line_n and_o of_o all_o the_o other_o line_n that_o which_o be_v nigh_o to_o the_o line_n which_o pass_v by_o the_o centre_n be_v great_a than_o that_o which_o be_v more_o distant_a and_o from_o that_o point_n can_v fall_v within_o the_o circle_n on_o each_o side_n of_o the_o least_o line_n only_o two_o equal_a right_a line_n svppose_v that_o there_o be_v a_o circle_n abcd_o and_o let_v the_o diameter_n thereof_o be_v ad._n and_o take_v in_o it_o any_o point_n beside_o the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v f._n construction_n and_o let_v the_o centre_n of_o the_o circle_n by_o the_o 1._o of_o the_o three_o be_v the_o point_n e._n and_o from_o the_o point_n f_o let_v there_o be_v draw_v unto_o the_o circumference_n abcd_o these_o right_a line_n fd_v fc_o and_o fg._n then_o i_o say_v that_o the_o line_n favorina_n be_v the_o great_a and_o the_o line_n fd_o be_v the_o jest_n and_o of_o the_o other_o line_n the_o line_n fb_o be_v great_a than_o the_o line_n fc_o and_o the_o line_n fc_o be_v great_a than_o the_o line_n fg._n proposition_n draw_v by_o the_o first_o petition_n these_o right_a line_n be_v ce_fw-fr and_o ge._n and_o for_o asmuch_o as_o by_o the_o 20._o of_o the_o first_o in_o every_o triangle_n two_o side_n be_v great_a than_o the_o three_o demonstration_n therefore_o the_o line_n ebb_v and_o of_o be_v great_a than_o the_o residue_n namely_o then_o the_o line_n fb_o but_o the_o line_n ae_n be_v equal_a unto_o the_o line_n be_v by_o the_o 15._o definition_n of_o the_o first_o wherefore_o the_o line_n be_v and_o of_o be_v equal_a unto_o the_o line_n af._n wherefore_o the_o line_n of_o be_v great_a than_o then_o the_o line_n bf_o again_o for_o asmuch_o as_o the_o line_n be_v be_v equal_a unto_o ce_fw-fr by_o the_o 15._o definition_n of_o the_o first_o and_o the_o line_n fe_o be_v common_a unto_o they_o both_o therefore_o these_o two_o line_n be_v and_o of_o be_v equal_a unto_o these_o two_o ce_fw-fr and_o ef._n but_o the_o angle_n bef_n be_v great_a than_o the_o angle_n cef_n wherefore_o by_o the_o 24._o of_o the_o first_o the_o base_a bf_o be_v great_a than_o the_o base_a cf_o and_o by_o the_o same_o reason_n the_o line_n cf_o be_v great_a than_o the_o line_n fg._n again_o for_o asmuch_o as_o the_o line_n gf_o and_o fe_o be_v great_a than_o the_o line_n eglantine_n by_o the_o 20._o of_o the_o first_o part_n but_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_n eglantine_n be_v equal_a unto_o the_o line_n ed_z wherefore_o the_o line_n gf_o and_o fe_o be_v great_a than_o the_o line_n ed_z take_v away_o of_o which_o be_v common_a to_o they_o both_o wherefore_o the_o residue_n gf_o be_v great_a than_o the_o residue_n fd._n wherefore_o the_o line_n favorina_n be_v the_o great_a and_o the_o line_n fd_o be_v the_o jest_n and_o the_o line_n fb_o be_v great_a than_o the_o line_n fc_o and_o the_o line_n fc_o be_v great_a than_o the_o line_n fg._n part_n now_o also_o i_o say_v that_o from_o the_o point_n f_o there_o can_v be_v draw_v only_o two_o equal_a right_a line_n into_o the_o circle_n abcd_o on_o each_o side_n of_o the_o least_o line_n namely_o fd._n for_n by_o the_o 23._o of_o the_o first_o upon_o the_o right_a line_n give_v of_o and_o to_o the_o point_n in_o it_o namely_o e_o make_v unto_o the_o angle_n gef_n a_o equal_a angle_n feh_n and_o by_o the_o first_o petition_n draw_v a_o line_n from_o fletcher_n to_o h._n now_o for_o asmuch_o as_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_n eglantine_n be_v equal_a unto_o the_o line_n eh_o and_o the_o line_n of_o be_v common_a unto_o they_o both_o therefore_o these_o two_o line_n ge_z and_o of_o be_v equal_a unto_o these_o two_o line_n he_o and_o of_o and_o by_o construction_n the_o angle_n gef_n be_v equal_a unto_o the_o angle_n hef_fw-fr wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a fg_o be_v equal_a unto_o the_o base_a fh_o impossibilie_n i_o say_v moreover_o that_o from_o the_o point_n f_o can_v be_v draw_v into_o the_o circle_n no_o other_o right_a line_n equal_a unto_o the_o line_n fg._n for_o if_o it_o possible_a let_v the_o lineâ—_z fk_o be_v equal_a unto_o the_o line_n fg._n and_o for_o asmuch_o as_o fk_n be_v equal_a unto_o fg._n but_o the_o line_n fh_o be_v equal_a unto_o the_o line_n fg_o therefore_o the_o line_n fk_o be_v equal_a unto_o the_o line_n fh_o wherefore_o the_o line_n which_o be_v nigh_o to_o the_o line_n which_o pass_v by_o the_o centre_n be_v equal_a to_o that_o which_o be_v far_o of_o which_o we_o have_v before_o prove_v to_o be_v impossible_a impossibility_n or_o else_o it_o may_v thus_o be_v demonstrate_v draw_v by_o the_o first_o petition_n a_o line_n from_o e_o to_o king_n and_o for_o asmuch_o as_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_v ge_z be_v equal_a unto_o the_o line_n eke_o and_o the_o line_n fe_o be_v common_a to_o they_o both_o and_o the_o base_a gf_o be_v equal_a unto_o the_o base_a fk_n therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n gef_n be_v equal_a to_o the_o angle_n kef_n but_o the_o angle_n gef_n be_v equal_a to_o the_o angle_n hef_fw-fr wherefore_o by_o the_o first_o common_a sentence_n the_o angle_n hef_fw-fr be_v equal_a to_o the_o angle_n kef_n the_o less_o unto_o the_o great_a which_o be_v impossible_a wherefore_o from_o the_o point_n fletcher_n there_o can_v be_v draw_v into_o the_o circle_n no_o other_o right_a line_n equal_a unto_o the_o line_n gf_o wherefore_o but_o one_o only_a if_o therefore_o in_o the_o diameter_n of_o a_o circle_n be_v take_v any_o point_n which_o be_v not_o the_o centre_n of_o the_o circle_n and_o from_o that_o point_n be_v draw_v unto_o the_o circumference_n certain_a right_a line_n the_o great_a of_o those_o right_a line_n shall_v be_v that_o wherein_o be_v the_o centre_n and_o the_o least_o shall_v be_v the_o residue_n and_o of_o all_o the_o other_o line_n that_o which_o be_v nigh_o to_o the_o line_n which_o pass_v by_o the_o centre_n be_v great_a than_o that_o which_o be_v more_o distant_a and_o from_o that_o point_n can_v fall_v within_o the_o circle_n on_o each_o side_n of_o the_o least_o line_n only_o two_o equal_a right_a line_n which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n corollary_n hereby_o it_o be_v manifest_a that_o two_o right_a line_n be_v draw_v from_o any_o one_o point_n of_o the_o diameter_n the_o one_o of_o one_o side_n and_o the_o other_o of_o the_o other_o side_n if_o with_o the_o diameter_n they_o make_v equal_a angle_n the_o say_v two_o right_a line_n be_v equal_a as_o in_o this_o place_n be_v the_o two_o line_n fg_o and_o fh_o the_o 7._o theorem_a the_o 8._o proposition_n if_o without_o a_o circle_n be_v take_v any_o point_n and_o from_o that_o point_n be_v draw_v into_o the_o circle_n unto_o the_o circumference_n certain_a right_a line_n of_o which_o let_v one_o be_v draw_v by_o the_o centre_n and_o let_v the_o rest_n be_v draw_v at_o all_o adventure_n the_o great_a of_o those_o line_n which_o fall_v in_o the_o concavity_n or_o hollowness_n of_o the_o circumference_n of_o the_o circle_n be_v that_o which_o pass_v by_o the_o centre_n and_o of_o all_o the_o other_o line_n that_o line_n which_o be_v nigh_o to_o the_o line_n which_o pass_v by_o the_o centre_n be_v great_a than_o that_o which_o be_v more_o distant_a but_o of_o those_o right_a line_n which_o end_n in_o the_o convexe_n part_n of_o the_o circumference_n that_o be_v the_o least_o which_o be_v draw_v from_o the_o point_n to_o the_o diameter_n and_o of_o the_o other_o line_n that_o which_o be_v nigh_o to_o the_o least_o be_v always_o less_o than_o that_o which_o be_v more_o distant_a and_o from_o that_o point_n can_v be_v draw_v unto_o the_o circumference_n on_o each_o side_n of_o the_o least_o only_o two_o equal_a right_a line_n part_n now_o also_o i_o say_v that_o from_o the_o point_n d_o can_v be_v draw_v unto_o the_o circumference_n on_o each_o side_n of_o dg_o the_o least_o only_o two_o equal_a right_a line_n upon_o the_o right_a line_n md_o and_o unto_o the_o point_n in_o it_o m_o make_v by_o the_o 23._o of_o the_o first_o unto_o the_o angle_n kmd_v a_o equal_a angle_n dmb._n and_o by_o the_o first_o petition_n draw_v a_o line_n from_o d_z to_o b._n and_o for_o asmuch_o as_o by_o the_o 15._o
definition_n of_o the_o first_o the_o line_n mb_n be_v equal_a unto_o the_o line_n mk_o put_v the_o line_n md_o common_a to_o the_o both_o wherefore_o these_o two_o line_n mk_o and_o md_o be_v equal_a to_o these_o two_o line_n bm_n and_o md_o the_o one_o to_o the_o other_o and_o the_o angle_n kmd_v be_v by_o the_o 23._o of_o the_o first_o equal_a to_o the_o angle_n bmd_v wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a dk_o be_v equal_a to_o the_o base_a db._n or_o it_o may_v thus_o be_v demonstrate_v impossibility_n draw_v by_o the_o first_o petition_n a_o line_n from_o m_o to_o n._n and_o for_o asmuch_o as_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_n km_o be_v equal_a unto_o the_o line_n mn_o and_o the_o line_n md_o be_v common_a to_o they_o both_o and_o the_o base_a kd_v be_v equal_a to_o the_o base_a dn_o by_o supposition_n therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n kmd_v be_v equal_a to_o the_o angle_n dmn_n but_o the_o angle_n kmd_v be_v equal_a to_o the_o angle_n bmd_v wherefore_o the_o angle_n bmd_v be_v equal_a to_o the_o angle_n nmd_v the_o less_o unto_o the_o great_a which_o be_v impossible_a wherefore_o from_o the_o point_n d_o can_v not_o be_v draw_v unto_o the_o circumference_n abc_n on_o each_o side_n of_o dg_o the_o jest_n more_o than_o two_o equal_a right_a line_n if_o therefore_o without_o a_o circle_n be_v take_v any_o point_n and_o from_o that_o point_n be_v draw_v into_o the_o circle_n unto_o the_o circumference_n certain_a right_a line_n of_o which_o let_v one_o be_v draw_v by_o the_o centre_n and_o let_v the_o rest_n be_v draw_v at_o all_o adventure_n the_o great_a of_o those_o right_a line_n which_o fall_v in_o the_o concavity_n or_o hollowness_n of_o the_o circumference_n of_o the_o circle_n be_v that_o which_o pass_v by_o the_o centre_n and_o of_o all_o the_o other_o line_n that_o line_n which_o be_v nigh_o to_o the_o line_n which_o pass_v by_o the_o centre_n be_v great_a than_o that_o which_o be_v more_o distant_a but_o of_o those_o right_a line_n which_o end_n in_o the_o convexe_n part_n of_o the_o circumference_n that_o line_n be_v the_o lest_o which_o be_v draw_v from_o the_o point_n to_o the_o dimetient_fw-la and_o of_o the_o other_o line_n that_o which_o be_v nigh_o to_o the_o least_o be_v always_o less_o than_o that_o which_o be_v more_o distant_a and_o from_o that_o point_n can_v be_v draw_v unto_o the_o circumference_n on_o each_o side_n of_o the_o lest_o only_a two_o equal_a right_a line_n which_o be_v require_v to_o be_v prove_v this_o proposition_n be_v call_v common_o in_o old_a book_n amongst_o the_o barbarous_a caâ—dâ—_n panonis_n panonis_fw-la that_o be_v the_o peacock_n tail_n ¶_o a_o corollary_n hereby_o it_o be_v manifest_a corollary_n that_o the_o right_a line_n which_o be_v draw_v from_o the_o point_n give_v without_o the_o circle_n and_o fall_v within_o the_o circle_n be_v equal_o distant_a from_o the_o least_o or_o from_o the_o great_a which_o be_v draw_v by_o the_o centre_n be_v equal_a the_o one_o to_o the_o other_o but_o contrariwise_o if_o they_o be_v unequal_o distant_a whether_o they_o light_v upon_o the_o concave_n or_o convexe_a circumference_n of_o the_o circle_n they_o be_v unequal_a the_o 8._o theorem_a the_o 9_o proposition_n if_o within_o a_o circle_n be_v take_v a_o point_n and_o from_o that_o point_n be_v draw_v unto_o the_o circumference_n more_o than_o two_o equal_a right_a line_n the_o point_n take_v be_v the_o centre_n of_o the_o circle_n svppose_v that_o the_o circle_n be_v abc_n and_o within_o it_o let_v there_o be_v take_v the_o point_n d._n and_o from_o d_z let_v there_o be_v draw_v unto_o the_o circumference_n abc_n more_o than_o two_o equal_a right_a line_n construction_n that_o be_v da_z db_o and_o dc_o then_o i_o say_v that_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n abc_n draw_v by_o the_o first_o petition_n these_o right_a line_n ab_fw-la and_o bc_o demonstration_n and_o by_o the_o 10._o of_o the_o first_o divide_v they_o into_o two_o equal_a part_n in_o the_o point_n e_o and_o f_o namely_o the_o line_n ab_fw-la in_o the_o point_n e_o and_o the_o line_n bc_o in_o the_o point_n f._n and_o draw_v the_o line_n ed_z and_o fd_o and_o by_o the_o second_o petition_n extend_v the_o line_n ed_z and_o fd_v on_o each_o side_n to_o the_o point_n king_n g_o and_o h_o l._n and_o for_o asmuch_o as_o the_o line_n ae_n be_v equal_a unto_o the_o line_n ebb_n and_o the_o line_n ed_z be_v common_a to_o they_o both_o therefore_o these_o two_o side_n ae_n and_o ed_z be_v equal_a unto_o these_o two_o side_n be_v and_o ed_z and_o by_o supposition_n the_o base_a dam_n be_v equal_a to_o the_o base_a db._n wherefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n aed_n be_v equal_a to_o the_o angle_n bed_n wherefore_o either_o of_o these_o angle_n aed_n and_o bed_n be_v a_o right_a angle_n wherefore_o the_o line_n gk_o divide_v the_o line_n ab_fw-la into_o two_o equal_a part_n and_o make_v right_a angle_n and_o for_o asmuch_o as_z if_o in_o a_o circle_n a_o right_a line_n divide_v a_o other_o right_a line_n into_o two_o equal_a part_n in_o such_o sort_n that_o it_o make_v also_o right_a angle_n in_o the_o line_n that_o divide_v be_v the_o centre_n of_o the_o circle_n by_o the_o corollary_n of_o the_o first_o of_o the_o three_o therefore_o by_o the_o same_o corollary_n in_o the_o line_n gk_o be_v the_o centre_n of_o the_o circle_n abc_n and_o by_o the_o same_o reason_n may_v we_o prove_v that_o in_o the_o line_n hl_o be_v the_o centre_n of_o the_o circle_n abc_n and_o the_o right_a line_n gk_o and_o hl_o have_v no_o other_o point_n common_a to_o they_o both_o beside_o the_o point_n d._n wherefore_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n abc_n if_o therefore_o within_o a_o circle_n be_v take_v a_o point_n and_o from_o that_o point_n be_v draw_v unto_o the_o circumference_n more_o than_o two_o equal_a right_a line_n the_o point_n take_v be_v the_o centre_n of_o the_o circle_n which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n let_v there_o be_v take_v within_o the_o circle_n abc_n the_o point_n d._n impossibility_n and_o from_o the_o point_n d_o let_v there_o be_v draw_v unto_o the_o circumference_n more_o than_o two_o equal_a right_a line_n namely_o dam_n db_o and_o dc_o then_o i_o say_v that_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n for_o if_o not_o then_o if_o it_o be_v possible_a let_v the_o point_n e_o be_v the_o centre_n and_o draw_v a_o line_n from_o d_z to_o e_o and_o extend_v de_fw-fr to_o the_o point_n fletcher_n and_o g._n wherefore_o the_o line_n fg_o be_v the_o diameter_n of_o the_o circle_n abc_n and_o for_o asmuch_o as_o in_o fg_o the_o diameter_n of_o the_o circle_n abc_n be_v take_v a_o point_n namely_o d_z which_o be_v not_o the_o centre_n of_o that_o circle_n therefore_o by_o the_o 7._o of_o the_o three_o the_o line_n dg_o be_v the_o great_a and_o the_o line_n dc_o be_v great_a than_o the_o line_n db_o and_o the_o line_n db_o be_v greateâ—_n than_o the_o line_n da._n but_o the_o line_n dc_o db_o dam_n be_v also_o equal_a by_o supposition_n which_o be_v impossible_a wherefore_o the_o point_n e_o be_v not_o the_o centre_n of_o the_o circle_n abc_n and_o in_o like_a sort_n may_v we_o prove_v that_o no_o other_o point_n beside_o d._n wherefore_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n abc_n which_o be_v require_v to_o be_v prove_v the_o 9_o theorem_a the_o 10._o proposition_n a_o circle_n cut_v not_o a_o circle_n in_o more_o point_n than_o two_o for_o if_o it_o be_v possible_a let_v the_o circle_n abc_n cut_v the_o circle_n def_n in_o more_o point_n than_o two_o impossibility_n that_o be_v in_o b_o g_o h_o &_o f._n and_o draw_v line_n from_o b_o to_o g_z and_o from_o b_o to_o h._n and_o by_o the_o 10._o of_o the_o first_o divide_v either_o of_o the_o line_n bg_o &_o bh_o into_o two_o equal_a part_n in_o the_o point_n king_n and_o l._n and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n king_n raise_v up_o unto_o the_o line_n bh_o a_o perpendicular_a line_n kc_o and_o likewise_o from_o the_o point_n l_o raise_v up_o unto_o the_o line_n bg_o a_o perpendicular_a line_n lm_o and_o extend_v the_o line_n ck_o to_o the_o point_n a_o and_o lnm_n to_o the_o point_n x_o and_o e._n and_o for_o asmuch_o as_o in_o the_o circle_n abc_n the_o right_a line_n ac_fw-la divide_v the_o right_a line_n bh_o into_o two_o equal_a part_n and_o make_v right_a angle_n therefore_o by_o the_o 3._o of_o the_o three_o in_o the_o line_n ac_fw-la be_v the_o centre_n of_o
when_o perpendicular_a line_n draw_v from_o the_o centre_n to_o those_o line_n be_v equal_a by_o the_o 4._o definition_n of_o the_o three_o wherefore_o the_o line_n ab_fw-la and_o cd_o be_v equal_o distant_a from_o the_o centre_n but_o now_o suppose_v that_o the_o right_a line_n ab_fw-la and_o cd_o be_v equal_o distant_a from_o the_o centre_n first_o that_o be_v let_v the_o perpendicular_a line_n of_o be_v equal_a to_o the_o perpendicular_a line_n eglantine_n then_o i_o say_v that_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n cd_o for_o the_o same_o order_n of_o construction_n remain_v we_o may_v in_o like_a sort_n prove_v that_o the_o line_n ab_fw-la be_v double_a to_o the_o line_n of_o and_o that_o the_o line_n cd_o be_v double_a to_o the_o line_n cg_o and_o for_o asmuch_o as_o the_o line_n ae_n be_v equal_a to_o the_o line_n ce_fw-fr for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n therefore_o the_o square_a of_o the_o line_n ae_n be_v equal_a to_o the_o square_n of_o the_o line_n ce._n but_o by_o the_o 47._o of_o the_o first_o to_o the_o square_n of_o the_o line_n ae_n be_v equal_a the_o square_n of_o the_o line_n of_o and_o fa._n and_o by_o the_o self_n same_o to_o the_o square_n of_o the_o line_n ce_fw-fr be_v equal_v the_o square_n of_o the_o line_n eglantine_n and_o gc_o wherefore_o the_o square_n of_o the_o line_n of_o and_o favorina_n be_v equal_a to_o the_o square_n of_o the_o line_n eglantine_n and_o gc_o of_o which_o the_o square_a of_o the_o line_n eglantine_n be_v equal_a to_o the_o square_n of_o the_o line_n of_o for_o the_o line_n of_o be_v equal_a to_o the_o line_n eglantine_n wherefore_o by_o the_o three_o common_a sentence_n the_o square_a remain_v namely_o the_o square_a of_o the_o line_n of_o be_v equal_a to_o the_o square_n of_o the_o line_n cg_o wherefore_o the_o line_n ac_fw-la be_v equal_a unto_o the_o line_n cg_o but_o the_o line_n ab_fw-la be_v double_a to_o the_o line_n of_o and_o the_o line_n cd_o be_v double_a to_o the_o line_n cg_o wherefore_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n cd_o wherefore_o in_o a_o circle_n equal_a right_a line_n be_v equal_o distant_a from_o the_o centre_n and_o line_n equal_o distant_a from_o the_o centre_n be_v equal_a the_o one_o to_o the_o outstretch_o which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n for_o the_o first_o part_n after_o campane_n campane_n suppose_v that_o there_o be_v a_o circle_n abdc_n who_o centre_n let_v be_v the_o point_n e._n and_o draw_v in_o it_o two_o equal_a line_n ab_fw-la and_o cd_o then_o i_o say_v that_o they_o be_v equal_o distant_a from_o the_o centre_n draw_v from_o the_o centre_n unto_o the_o line_n ab_fw-la and_o cd_o these_o perpendicular_a line_n of_o and_o eglantine_n and_o by_o the_o 2._o part_n of_o the_o 3._o of_o this_o book_n the_o line_n ab_fw-la shall_v be_v equal_o divide_v in_o the_o point_n f._n and_o the_o line_n cd_o shall_v be_v equal_o divide_v in_o the_o point_n g._n and_o draw_v these_o right_a line_n ea_fw-la ebb_n ec_o and_o ed._n and_o for_o asmuch_o as_o in_o the_o triangle_n aeb_fw-mi the_o two_o side_n ab_fw-la and_o ae_n be_v equal_a to_o the_o two_o side_n cd_o and_o ce_fw-fr of_o the_o triangle_n ce_z &_o the_o base_a ebb_n be_v equal_a to_o the_o base_a ed._n therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n at_o the_o point_n a_o shall_v be_v equal_a to_o the_o angle_n at_o the_o point_n c._n and_o for_o asmuch_o as_o in_o the_o triangle_n aef_n the_o two_o side_n ae_n and_o of_o be_v equal_a to_o the_o two_o side_n ce_fw-fr and_o cg_o of_o the_o triangle_n ceg_n and_o the_o angle_n eaf_n be_v equal_a to_o the_o angle_n ceg_n therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a of_o iâ—_z equal_a to_o the_o base_a eglantine_n which_o for_o asmuch_o as_o they_o be_v perpendicular_a line_n therefore_o the_o line_n ab_fw-la &_o cd_o be_v equal_o distant_a from_o the_o centre_n by_o the_o 4._o definition_n of_o this_o book_n the_o 14._o theorem_a the_o 15._o proposition_n in_o a_o circle_n the_o great_a line_n be_v the_o diameter_n and_o of_o all_o other_o line_n that_o line_n which_o be_v nigh_o to_o the_o centre_n be_v always_o great_a than_o that_o line_n which_o be_v more_o distant_a svppose_v that_o there_o be_v a_o circle_n abcd_o and_o let_v the_o diameter_n thereof_o be_v the_o line_n ad_fw-la and_o let_v the_o centre_n thereof_o be_v the_o point_n e._n and_o unto_o the_o diameter_n ad_fw-la let_v the_o line_n bc_o be_v nigh_a than_o the_o line_n fg._n then_o i_o say_v that_o the_o line_n ad_fw-la be_v the_o great_a and_o the_o line_n bc_o be_v great_a than_o the_o line_n fg._n draw_v by_o the_o 12._o of_o the_o first_o from_o the_o centre_n e_o to_o the_o line_n bc_o and_o fg_o perpendicular_a line_n eh_o and_o eke_o construction_n and_o for_o asmuch_o as_o the_o line_n bc_o be_v nigh_a unto_o the_o centre_n than_o the_o line_n fg_o therefore_o by_o the_o 4._o definition_n of_o the_o three_o the_o line_n eke_o be_v great_a than_o the_o line_n eh_o and_o by_o the_o three_o of_o the_o first_o put_v unto_o the_o line_n eh_o a_o equal_a line_n el._n and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n l_o raise_v up_o unto_o the_o line_n eke_o a_o perpendicular_a line_n lm_o and_o extend_v the_o line_n lm_o to_o the_o point_n n._n and_o by_o the_o first_o petition_n draw_v these_o right_a line_n em_n en_fw-fr of_o and_o eglantine_n and_o for_o asmuch_o as_o the_o line_n eh_o be_v equal_a to_o the_o line_n el_fw-es therefore_o by_o the_o 14._o of_o the_o three_o demonstration_n and_o by_o the_o 4._o definition_n of_o the_o same_o the_o line_n bc_o be_v equal_a to_o the_o line_n mn_v again_o for_o asmuch_o as_o the_o line_n ae_n be_v equal_a to_o the_o line_n em_n and_o the_o line_n ed_z to_o the_o line_n en_fw-fr therefore_o the_o line_n ad_fw-la be_v equal_a to_o the_o line_n menander_n and_o en_fw-fr but_o the_o line_n i_o and_o en_fw-fr be_v by_o the_o 20._o of_o the_o first_o great_a then_o the_o line_n mn_v wherefore_o the_o line_n ad_fw-la be_v great_a than_o the_o line_n mn_v and_o for_o asmuch_o as_o these_o two_o line_n menander_n and_o en_fw-fr be_v equal_v to_o these_o two_o line_n fe_o and_o eglantine_n by_o the_o 15._o definition_n of_o the_o first_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n and_o the_o angle_n men_n be_v great_a than_o the_o angle_n feg_n therefore_o by_o the_o 24._o of_o the_o first_o the_o base_a mn_n be_v great_a than_o the_o base_a fg._n but_o it_o be_v prove_v that_o the_o line_n mn_o be_v equal_a to_o the_o line_n bc_o wherefore_o the_o line_n bc_o also_o be_v great_a than_o the_o line_n fg._n wherefore_o the_o diameter_n ad_fw-la be_v the_o great_a and_o the_o line_n bc_o be_v great_a than_o the_o line_n fg._n wherefore_o in_o a_o circle_n the_o great_a line_n be_v the_o diameter_n and_o of_o all_o the_o other_o line_n that_o line_n which_o be_v nigh_o to_o the_o centre_n be_v always_o great_a than_o that_o line_n which_o be_v more_o distant_a which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n after_o campane_n in_o the_o circle_n abcd_o who_o centre_n let_v be_v the_o point_n e_o draw_v these_o line_n ab_fw-la ac_fw-la ad_fw-la fg_o and_o hk_o of_o which_o let_v the_o line_n ad_fw-la be_v the_o diameter_n of_o the_o circle_n campane_n then_o i_o say_v that_o the_o line_n ad_fw-la be_v the_o great_a of_o all_o the_o line_n and_o the_o other_o line_n each_o of_o the_o one_o be_v so_o much_o great_a than_o each_o of_o the_o other_o how_o much_o nigh_o it_o be_v unto_o the_o centre_n join_v together_o the_o end_n of_o all_o these_o line_n with_o the_o centre_n by_o draw_v these_o right_a line_n ebb_v ec_o eglantine_n eke_o eh_o and_o ef._n and_o by_o the_o 20._o of_o the_o first_o the_o two_o side_n of_o and_o eglantine_n of_o the_o triangle_n efg_o shall_v be_v great_a than_o the_o three_o side_n fg._n and_o for_o asmuch_o as_o the_o say_a side_n of_o &_o eglantine_n be_v equal_a to_o the_o line_n ad_fw-la by_o the_o definition_n of_o a_o circle_n therefore_o the_o line_n ad_fw-la be_v great_a than_o the_o line_n fg._n and_o by_o the_o same_o reason_n it_o be_v great_a than_o every_o one_o of_o the_o rest_n of_o the_o line_n if_o they_o be_v put_v to_o be_v base_n of_o triangle_n for_o that_o every_o two_o side_n draw_v from_o the_o centre_n be_v equal_a to_o the_o line_n ad._n which_o be_v the_o first_o part_n of_o the_o proposition_n again_o for_o asmuch_o as_o the_o two_o side_n of_o and_o eglantine_n of_o the_o triangle_n efg_o be_v equal_a to_o the_o
touch_v the_o circle_n abc_n and_o let_v the_o same_o be_v de._n construction_n and_o by_o the_o first_o of_o the_o same_o let_v the_o point_n fletcher_n be_v the_o centre_n of_o the_o circle_n abc_n and_o draw_v these_o right_a line_n fe_o fb_o and_o fd._n wherefore_o the_o angle_n feed_v be_v a_o right_a angle_n demonstration_n and_o for_o asmuch_o as_o the_o right_a line_n de_fw-fr touch_v the_o circle_n abc_n and_o the_o right_a line_n dca_n cut_v the_o same_o therefore_o by_o the_o proposition_n go_v before_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o dc_o be_v equal_a to_o the_o square_n of_o the_o line_n de._n but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o dc_o be_v suppose_v to_o be_v equal_a to_o the_o square_n of_o the_o line_n db._n wherefore_o the_o square_a of_o the_o line_n de_fw-fr be_v equal_a to_o the_o square_n of_o the_o line_n db._n wherefore_o also_o the_o line_n de_fw-fr be_v equal_a to_o the_o line_n db._n and_o the_o line_n fe_o be_v equal_a to_o the_o line_n fb_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n now_o therefore_o these_o two_o line_n de_fw-fr and_o of_o be_v equal_a to_o these_o two_o line_n db_o and_o bf_o and_o fd_o be_v a_o common_a base_a to_o they_o both_o wherefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n def_n be_v equal_a to_o the_o angle_n dbf_n but_o the_o angle_n def_n be_v a_o right_a angle_n wherefore_o also_o the_o angle_n dbf_n be_v a_o right_a angle_n and_o the_o line_n fb_o be_v produce_v shall_v be_v the_o diameter_n of_o the_o circle_n but_o if_o from_o the_o end_n of_o the_o diameter_n of_o a_o circle_n be_v draw_v a_o right_a line_n make_v right_a angle_n the_o right_a line_n so_o draw_v touch_v the_o circle_n by_o the_o corollary_n of_o the_o 16._o of_o the_o three_o wherefore_o the_o right_a line_n db_o touch_v the_o circle_n abc_n and_o the_o like_a demonstration_n will_v serve_v if_o the_o centre_n be_v in_o the_o line_n ac_fw-la if_o therefore_o without_o a_o circle_n be_v take_v a_o certain_a point_n and_o from_o that_o point_n be_v draw_v to_o the_o circle_n two_o right_a line_n of_o which_o the_o one_o do_v cut_v the_o circle_n and_o the_o other_o fall_v upon_o the_o circle_n and_o that_o in_o such_o sort_n that_o the_o rectangle_n parallelogram_n which_o be_v contain_v under_o the_o whole_a right_a line_n which_o cut_v the_o circle_n and_o that_o portion_n of_o the_o same_o line_n that_o lie_v between_o the_o point_n and_o the_o utter_a circumference_n of_o the_o circle_n be_v equal_a to_o the_o square_n make_v of_o the_o line_n that_o fall_v upon_o the_o circle_n then_o the_o line_n that_o so_o fall_v upon_o the_o circle_n shall_v touch_v the_o circle_n which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n after_o pelitarius_n suppose_v that_o there_o be_v a_o circle_n bcd_o who_o centre_n let_v be_v e_o pelitarius_n and_o take_v a_o point_n without_o it_o namely_o a_o and_o from_o the_o point_n a_o draw_v two_o right_a line_n abdella_n and_o ac_fw-la of_o which_o let_v abdella_n cut_v the_o circle_n in_o the_o point_n b_o &_o let_v the_o other_o fall_n upon_o it_o and_o let_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la then_o i_o say_v that_o the_o line_n ac_fw-la touch_v the_o circle_n for_o first_o if_o the_o line_n abdella_n do_v pass_v by_o the_o centre_n draw_v the_o right_a line_n ce._n and_o by_o the_o 6._o of_o the_o second_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o ab_fw-la together_o with_o the_o square_n of_o the_o line_n ebb_n that_o be_v with_o the_o square_n of_o the_o line_n ec_o for_o the_o line_n ebb_v and_o ec_o be_v equal_a be_v equal_a to_o the_o square_n of_o the_o line_n ae_n but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o ab_fw-la be_v suppose_v to_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la wherefore_o the_o square_a of_o the_o line_n ac_fw-la together_o with_o the_o square_n of_o the_o line_n ce_fw-fr be_v equal_a to_o the_o square_n of_o the_o line_n ae_n wherefore_o by_o the_o last_o of_o the_o first_o the_o angle_n at_o the_o point_n c_o be_v a_o right_a angle_n wherefore_o by_o the_o 18._o of_o this_o book_n the_o line_n ac_fw-la touch_v the_o circle_n but_o if_o the_o line_n abdella_n do_v not_o pass_v by_o the_o centre_n draw_v from_o the_o point_n a_o the_o line_n ad_fw-la in_o which_o let_v be_v the_o centre_n e._n and_o forasmuch_o as_o that_o which_o be_v contain_v under_o this_o whole_a line_n and_o his_o outward_a part_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o ab_fw-la by_o the_o first_o corollary_n before_o put_v therefore_o the_o same_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la wherefore_o the_o angle_n eca_n be_v a_o right_a angle_n as_o have_v before_o be_v prove_v in_o the_o first_o part_n of_o this_o proposition_n and_o therefore_o the_o line_n ac_fw-la touch_v the_o circle_n which_o be_v require_v to_o be_v prove_v the_o end_n of_o the_o three_o book_n of_o euclides_n element_n ¶_o the_o four_o book_n of_o euclides_n element_n this_o four_o book_n intreat_v of_o the_o inscription_n &_o circumscription_n of_o rectiline_a figure_n book_n how_o one_o right_a line_v figure_n may_v be_v inscribe_v within_o a_o other_o right_o line_v figure_n and_o how_o a_o right_a line_v figure_n may_v be_v circumscribe_v about_o a_o other_o right_o line_v figure_n in_o such_o as_o may_v be_v inscribe_v and_o circumscribe_v within_o or_o about_o the_o other_o for_o all_o right_n line_v figure_n can_v so_o be_v inscribe_v or_o circumscribe_v within_o or_o about_o the_o other_o also_o it_o teach_v how_o a_o triangle_n a_o square_a and_o certain_a other_o rectiline_a figure_n be_v regular_a may_v be_v inscribe_v within_o a_o circle_n also_o how_o they_o may_v be_v circumscribe_v about_o a_o circle_n likewise_o how_o a_o circle_n may_v be_v inscribe_v within_o they_o and_o how_o it_o may_v be_v circumscribe_v about_o they_o and_o because_o the_o manner_n of_o entreaty_n in_o this_o book_n be_v diverse_a from_o the_o entreaty_n of_o the_o former_a book_n he_o use_v in_o this_o other_o word_n and_o term_n than_o he_o use_v in_o they_o the_o definition_n of_o which_o in_o order_n here_o after_o follow_v definition_n a_o rectiline_a figure_n be_v say_v to_o be_v inscribe_v in_o a_o rectiline_a figure_n definition_n when_o every_o one_o of_o the_o angle_n of_o the_o inscribe_v figure_n touch_v every_o one_o of_o the_o side_n of_o the_o figure_n wherein_o it_o be_v inscribe_v as_o the_o triangle_n abc_n be_v inscribe_v in_o the_o triangle_n def_n because_o that_o every_o angle_n of_o the_o triangle_n inscribe_v namely_o the_o triangle_n abc_n touch_v every_o side_n of_o the_o triangle_n within_o which_o it_o be_v describe_v namely_o of_o the_o triangle_n def_n as_o the_o angle_n cab_n touch_v the_o side_n ed_z the_o angle_n abc_n touch_v the_o side_n df_o and_o the_o angle_n acb_o touch_v the_o side_n ef._n so_o likewise_o the_o square_a abcd_o be_v say_v to_o be_v inscribe_v within_o the_o square_a efgh_o for_o every_o angle_n of_o it_o touch_v some_o one_o side_n of_o the_o other_o so_o also_o the_o pentagon_n or_o five_o angle_a figure_n abcde_o be_v inscribe_v within_o the_o pentagon_n or_o five_o angle_a figure_n fghikâ—_n â—_z as_o you_o see_v in_o the_o figureâ—_n likewise_o a_o rectiline_a figure_n be_v say_v to_o be_v circumscribe_v about_o a_o rectiline_a figure_n definition_n when_o every_o one_o of_o the_o side_n of_o the_o figure_n circumscribe_v touch_v every_o one_o of_o the_o angle_n of_o the_o figure_n about_o which_o it_o be_v circumscribe_v as_o in_o the_o former_a description_n the_o triangle_n def_n be_v say_v to_o be_v circumscribe_v about_o the_o triangle_n abc_n for_o that_o every_o side_n of_o the_o figure_n circumscribe_v namely_o of_o the_o triangle_n def_n touch_v every_o angle_n of_o the_o figure_n wherabout_o it_o be_v circumscribe_v as_o the_o side_n df_o of_o the_o triangle_n def_n circumscribe_v touch_v the_o angle_n abc_n of_o the_o triangle_n abc_n about_o which_o it_o be_v circumscribe_v and_o the_o side_n of_o touch_v the_o angle_n bca_o and_o the_o side_n cd_o touch_v the_o angle_n cab_n likewise_o understand_v you_o of_o the_o square_a efgh_o which_o be_v circumscribe_v about_o the_o square_n abcd_o for_o every_o side_n of_o the_o one_o touch_v some_o one_o side_n of_o the_o other_o even_o so_o by_o the_o same_o reason_n the_o pentagon_n fghik_n be_v circumscribe_v about_o the_o pentagon_n abcde_o as_o you_o see_v in_o the_o figure_n on_o the_o other_o side_n and_o thus_o may_v you_o
of_o the_o first_o either_o of_o these_o line_n ab_fw-la and_o ad_fw-la into_o two_o equal_a part_n in_o the_o point_n e_o and_o f._n and_o by_o the_o point_n e_o by_o the_o 31._o of_o the_o first_o draw_v a_o line_n eh_o parallel_v unto_o either_o of_o these_o line_n ab_fw-la and_o dc_o demonstration_n and_o by_o the_o same_o by_o the_o point_n fletcher_n draw_v a_o line_n fk_a parallel_n unto_o either_o of_o these_o line_n ad_fw-la and_o bc._n wherefore_o every_o one_o of_o these_o figure_n ak_o kb_o ah_o hd_n agnostus_n gc_o bg_o and_o gd_o be_v a_o parallelogram_n and_o the_o side_n which_o be_v opposite_a the_o one_o to_o the_o other_o be_v by_o the_o 34._o of_o the_o first_o equal_a the_o one_o to_o the_o other_o and_o forasmuch_o as_o the_o line_n ad_fw-la be_v equal_a unto_o the_o line_n ab_fw-la and_o the_o half_a of_o the_o line_n ad_fw-la be_v the_o line_n ae_n and_o the_o half_a of_o the_o line_n ab_fw-la be_v the_o line_n of_o therefore_o the_o line_n ae_n be_v equal_a unto_o the_o line_n of_o wherefore_o by_o the_o same_o the_o side_n which_o be_v opposite_a be_v equal_a wherefore_o the_o line_n fg_o be_v equal_a unto_o the_o line_n eglantine_n in_o like_a sort_n may_v we_o prove_v that_o either_o of_o these_o line_n gh_o and_o gk_o be_v equal_a to_o either_o of_o these_o line_n fg_o and_o ge._n wherefore_o by_o the_o first_o common_a sentence_n these_o four_o line_n ge_z gf_o gh_o and_o gk_o be_v equal_a the_o one_o to_o the_o other_o wherefore_o make_v the_o centre_n g_o and_o the_o space_n either_o ge_z or_o gf_o gh_o or_o gk_o describe_v a_o circle_n and_o it_o will_v pass_v by_o the_o point_n e_o f_o h_o edward_n and_o will_v touch_v the_o right_a line_n ab_fw-la bc_o cd_o and_o da._n for_o the_o angle_n at_o the_o point_n e_o f_o h_o edward_n be_v right_a angle_n for_o if_o the_o circle_n do_v cut_v the_o right_a line_n ab_fw-la bc_o cd_o and_o dam_n than_o the_o line_n which_o be_v draw_v by_o the_o end_n of_o the_o diameter_n of_o the_o circle_n make_v right_a angle_n shall_v fall_v within_o the_o circle_n which_o be_v impossible_a by_o the_o 16._o of_o the_o three_o wherefore_o the_o centre_n be_v the_o point_n g_o and_o the_o space_n be_v ge_z or_o gf_o or_o gh_o or_o gk_o if_o a_o circle_n be_v describe_v it_o shall_v not_o cut_v the_o rigâ—t_a line_n ab_fw-la bc_o cd_o and_o da._n wherefore_o it_o shall_v touch_v they_o and_o it_o be_v describe_v in_o the_o square_n abcd_o wherefore_o in_o a_o square_n give_v be_v describe_v a_o circle_n which_o be_v require_v to_o be_v do_v the_o 9_o problem_n the_o 9_o proposition_n about_o a_o square_n give_v to_o describe_v a_o circle_n svppose_v that_o the_o square_n give_v be_v ab_fw-la cd_o it_o be_v require_v about_o the_o square_n abcd_o to_o describe_v a_o circle_n draw_v right_a line_n from_o a_o to_o c_o and_o from_o d_z to_o b_o &_o let_v they_o cut_v the_o one_o the_o other_o in_o the_o point_n e._n construction_n and_o forasmuch_o as_o the_o line_n dam_n be_v equal_a unto_o the_o line_n ab_fw-la demonstration_n and_o the_o line_n ac_fw-la be_v common_a unto_o they_o both_o therefore_o these_o two_o line_n dam_n and_o ac_fw-la be_v equal_a unto_o these_o two_o line_n basilius_n and_o ac_fw-la the_o one_o to_o the_o other_o and_o the_o base_a dc_o be_v equal_a unto_o the_o base_a bc._n wherefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n dac_o be_v equal_a unto_o the_o angle_n bac_n wherefore_o the_o angle_n dab_n be_v divide_v into_o two_o equal_a part_n by_o the_o line_n ac_fw-la and_o in_o like_a sorâ—_n may_v we_o prove_v that_o every_o one_o of_o these_o angle_n abc_n bcd_o and_o cda_n be_v divide_v into_o two_o equal_a part_n by_o the_o right_a line_n ac_fw-la and_o db._n and_o forasmuch_o as_o the_o angle_n dab_n be_v equal_a unto_o the_o angle_n abc_n and_o of_o the_o angle_n dab_n the_o angle_n eab_n be_v the_o half_a and_o of_o the_o angle_n abc_n the_o angle_n eba_n be_v the_o half_a therefore_o the_o angle_n eab_n be_v equal_a unto_o the_o angle_n eba_n wherefore_o by_o the_o 6._o of_o the_o first_o the_o side_n ea_fw-la be_v equal_a unto_o the_o side_n ebb_n in_o like_a sort_n may_v we_o prove_v that_o either_o of_o these_o right_a line_n ea_fw-la and_o ebb_n be_v equal_a unto_o either_o of_o these_o line_n ec_o and_o ed._n wherefore_o these_o four_o line_n ea_fw-la ebb_n ec_o and_o ed_z be_v equal_a the_o one_o to_o the_o other_o wherefore_o make_v the_o centre_n e_o and_o the_o space_n any_o of_o these_o line_n ea_fw-la ebb_n ec_o or_o ed._n describe_v a_o circle_n and_o it_o will_v pass_v by_o the_o point_n a_o b_o c_o d_o and_o shall_v be_v describe_v about_o the_o square_n abcd_o as_o it_o be_v evident_a in_o the_o figure_n abcd._n wherefore_o about_o a_o square_a give_v be_v describe_v a_o circle_n which_o be_v require_v to_o be_v do_v ¶_o a_o proposition_n add_v by_o pelitarius_n a_o square_n circumscribe_v about_o a_o circle_n be_v double_a to_o the_o square_n inscribe_v in_o the_o same_o circle_n this_o may_v also_o be_v demonstrate_v by_o the_o equality_n of_o the_o triangle_n and_o square_n contain_v in_o the_o great_a square_n the_o 10._o problem_n the_o 10._o proposition_n to_o make_v a_o triangle_n of_o two_o equal_a side_n call_v isosceles_a which_o shall_v have_v either_o of_o the_o angle_n at_o the_o base_a double_a to_o the_o other_o angle_n take_v a_o right_a line_n at_o all_o adventure_n which_o let_v be_v ab_fw-la &_o by_o the_o 11._o of_o the_o second_o let_v it_o be_v so_o divide_v in_o the_o point_n c_z construction_n that_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a unto_o the_o square_v which_o be_v make_v of_o the_o line_n ac_fw-la and_o make_v the_o centre_n the_o point_n a_o &_o the_o space_n ab_fw-la describe_v by_o the_o 3._o petition_n a_o circle_n bde_v and_o by_o the_o 1._o of_o the_o four_o into_o the_o circle_n bde_v apply_v a_o right_a line_n bd_o equal_a to_o the_o right_a line_n ac_fw-la which_o be_v not_o great_a than_o the_o diameter_n of_o the_o circle_n bde_v and_o draw_v line_n from_o a_o to_o d_z and_o from_o d_z to_o c._n and_o by_o the_o 5._o of_o the_o fourth_z about_o the_o triangle_n acd_v describe_v a_o circle_n acdf_n demonstration_n and_o forasmuch_o as_o the_o rectangle_n figure_n contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n ac_fw-la for_o that_o be_v by_o supposition_n but_o the_o line_n ac_fw-la be_v equal_a unto_o the_o line_n bd._n wherefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n bd._n and_o forasmuch_o as_o without_o the_o circle_n acdf_n be_v take_v a_o point_n b_o and_o from_o b_o unto_o the_o circle_n acdf_n be_v draw_v two_o right_a line_n bca_o and_o bd_o in_o such_o sort_n that_o the_o one_o of_o they_o cut_v the_o circle_n and_o the_o other_o end_v at_o the_o circumference_n and_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n bd_o therefore_o by_o the_o 17._o of_o the_o three_o the_o line_n bd_o touch_v the_o circle_n acdf_n and_o forasmuch_o as_o the_o line_n bd_o touch_v in_o the_o point_n d_o and_o from_o d_o where_o the_o touch_n be_v be_v draw_v a_o right_a line_n dc_o therefore_o by_o the_o 32._o of_o the_o same_o the_o angle_n bdc_o be_v equal_a unto_o the_o angle_n dac_o which_o be_v in_o the_o alternate_a segment_n of_o the_o circle_n and_o forasmuch_o as_o the_o angle_n bdc_o be_v equal_a unto_o the_o angle_v dac_o put_v the_o angle_n cda_n common_a unto_o they_o both_o wherefore_o the_o whole_a angle_n bda_n be_v equal_a to_o these_o two_o angle_n cda_n &_o dac_o but_o unto_o the_o angle_n cda_n &_o dac_o be_v equal_a the_o outward_a angle_n bcd_o by_o the_o 32._o of_o the_o 1._o wherefore_o the_o angle_n bda_n be_v equal_a unto_o the_o angle_v bcd_o but_o the_o angle_v bda_n be_v by_o the_o 5._o of_o the_o first_o equal_a unto_o the_o angle_n cbd_v for_o by_o the_o 15._o definition_n of_o y_fw-es first_o the_o side_n ad_fw-la be_v equal_a unto_o the_o side_n ab_fw-la wherefore_o by_o the_o 1._o common_a sentence_n the_o angle_n dba_n be_v equal_a unto_o the_o angle_n bcd_o wherefore_o these_o three_o angle_n bda_n dba_n and_o bcd_o be_v equal_v the_o one_o to_o the_o other_o and_o forasmuch_o as_o the_o angle_n dbc_n be_v equal_a unto_o the_o angle_n bcd_o the_o side_n therefore_o bd_o be_v equal_a unto_o the_o side_n dc_o but_o the_o line_n bd_o be_v by_o supposition_n
in_o the_o midst_n whereof_o be_v a_o point_n from_o which_o all_o line_n draw_v to_o the_o circumference_n thereof_o be_v equal_a this_o definition_n be_v essential_a and_o formal_a and_o declare_v the_o very_a nature_n of_o a_o circle_n and_o unto_o this_o definition_n of_o a_o circle_n be_v correspondent_a the_o definition_n of_o a_o sphere_n give_v by_o theodosius_n say_v that_o it_o be_v a_o solid_a oâ—_n body_n in_o the_o midst_n whereof_o there_o be_v a_o point_n from_o which_o all_o the_o line_n draw_v to_o the_o circumference_n be_v equal_a so_o see_v you_o the_o affinity_n between_o a_o circle_n and_o a_o sphere_n for_o what_o a_o circle_n be_v in_o a_o plain_a that_o be_v a_o sphere_n in_o a_o solid_a the_o fullness_n and_o content_n of_o a_o circle_n be_v describe_v by_o the_o motion_n of_o a_o line_n move_v about_o but_o the_o circumference_n thereof_o which_o be_v the_o limit_n and_o border_n thereof_o be_v describe_v of_o the_o end_n and_o point_n of_o the_o same_o line_n move_v about_o so_o the_o fullness_n content_n and_o body_n of_o a_o sphere_n or_o globe_n be_v describe_v of_o a_o semicircle_n move_v about_o but_o the_o spherical_a superficies_n which_o be_v the_o limit_n and_o border_n of_o a_o sphere_n sphere_n be_v describe_v of_o the_o circumference_n of_o the_o same_o semicircle_n move_v about_o and_o this_o be_v the_o superficies_n mean_v in_o the_o definition_n when_o it_o be_v say_v that_o it_o be_v contain_v under_o one_o superficies_n which_o superficies_n be_v call_v of_o johannes_n de_fw-fr â—acro_fw-la busco_n &_o other_o the_o circumference_n of_o the_o sphere_n sphâ—râ—_n galene_n in_o his_o book_n de_fw-fr diffinitionibus_fw-la mediciâ—â—_n â—_z give_v yet_o a_o other_o definition_n of_o a_o sphere_n by_o his_o property_n or_o common_a accidence_n of_o move_v which_o be_v thus_o a_o sphere_n be_v a_o figure_n most_o apt_a to_o all_o motion_n as_o have_v no_o base_a whereon_o the_o stay_n this_o be_v a_o very_a plain_a and_o 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semicircle_n be_v move_v definition_n as_o in_o the_o example_n before_o give_v in_o the_o definition_n of_o a_o sphere_n the_o line_n ab_fw-la about_o which_o his_o end_n be_v fix_v the_o semicircle_n be_v move_v which_o line_n also_o yet_o remain_v after_o the_o motion_n end_v be_v the_o axe_n of_o the_o sphere_n describe_v of_o that_o semicircle_n theodosius_n define_v the_o axe_n of_o a_o sphere_n after_o this_o manner_n sphere_n the_o axe_n of_o a_o sphere_n be_v a_o certain_a right_a line_n draw_v by_o the_o centre_n end_v on_o either_o side_n in_o the_o superficies_n of_o the_o sphere_n about_o which_o be_v fix_v the_o sphere_n be_v turn_v as_o the_o line_n ab_fw-la in_o the_o former_a example_n there_o need_v to_o this_o definition_n no_o other_o declaration_n but_o only_o to_o consider_v that_o the_o whole_a sphere_n turn_v upon_o that_o line_n ab_fw-la which_o pass_v by_o the_o centre_n d_o and_o be_v extend_v one_o either_o side_n to_o the_o superficies_n of_o the_o sphere_n wherefore_o by_o this_o definition_n of_o theodosius_n it_o be_v the_o axe_n of_o the_o sphere_n 14_o the_o centre_n of_o a_o sphere_n be_v that_o point_n which_o be_v also_o the_o centre_n of_o the_o semicircle_n definition_n this_o definition_n of_o the_o centre_n of_o a_o sphere_n be_v give_v as_o be_v the_o other_o definition_n of_o the_o axe_n namely_o have_v a_o relation_n to_o the_o definition_n of_o a_o sphere_n here_o give_v of_o euclid_n where_o it_o be_v say_v that_o a_o sphere_n be_v make_v by_o the_o revolution_n of_o a_o semicircle_n who_o diameter_n abide_v fix_v the_o diameter_n of_o a_o circle_n and_o of_o a_o semicrcle_n be_v all_o one_o and_o in_o the_o diameter_n either_o of_o a_o circle_n or_o of_o a_o semicircle_n be_v contain_v the_o centre_n of_o either_o of_o they_o for_o that_o they_o diameter_n of_o each_o ever_o pass_v by_o the_o centre_n now_o say_v euclid_n the_o point_n which_o be_v the_o centre_n of_o the_o semicircle_n by_o who_o motion_n the_o sphere_n be_v describe_v be_v also_o the_o centre_n of_o the_o sphere_n as_o in_o the_o example_n there_o give_v the_o point_n d_o be_v the_o centre_n both_o of_o the_o semicircle_n &_o also_o of_o the_o sphere_n theodosius_n give_v as_o other_o definition_n of_o the_o centre_n of_o a_o sphere_n which_o be_v thus_o sphere_n the_o centre_n of_o a_o sphere_n be_v a_o point_n with_o in_o the_o sphere_n from_o which_o all_o line_n draw_v to_o the_o superficies_n of_o the_o sphere_n be_v equal_a as_o in_o a_o circle_n be_v a_o plain_a figure_n there_o be_v a_o point_n in_o the_o midst_n from_o which_o all_o line_n draw_v to_o the_o circumfrence_n be_v equal_a which_o be_v the_o centre_n of_o the_o circle_n so_o in_o like_a manner_n with_o in_o a_o sphere_n which_o be_v a_o solid_a and_o bodily_a figure_n there_o must_v be_v conceive_v a_o point_n in_o the_o midst_n thereof_o from_o which_o all_o line_n draw_v to_o the_o superficies_n thereof_o be_v equal_a and_o this_o point_n be_v the_o centre_n of_o the_o sphere_n by_o this_o definition_n of_o theodosius_n flussas_n in_o define_v the_o centre_n of_o a_o sphere_n comprehend_v both_o those_o definition_n in_o one_o after_o this_o sort_n the_o centre_n of_o a_o sphere_n be_v a_o point_n assign_v in_o a_o sphere_n from_o which_o all_o the_o line_n draw_v to_o the_o superficies_n be_v equal_a and_o it_o be_v the_o same_o which_o be_v also_o the_o centre_n of_o the_o semicircle_n which_o describe_v the_o sphere_n sphere_n this_o definition_n be_v superfluous_a and_o contain_v more_o they_o need_v for_o either_o part_n thereof_o be_v a_o full_a and_o sufficient_a definition_n as_o before_o have_v be_v show_v or_o else_o have_v euclid_n be_v insufficient_a for_o leave_v out_o the_o one_o part_n or_o theodosius_n for_o leave_v out_o the_o other_o peradventure_o flussas_n do_v it_o for_o the_o more_o explication_n of_o either_o that_o the_o one_o part_n may_v open_v the_o other_o 15_o the_o diameter_n of_o a_o sphere_n be_v a_o certain_a right_a line_n draw_v by_o the_o centre_n and_o one_o each_o side_n end_v at_o the_o superficies_n of_o the_o same_o sphere_n definition_n this_o definition_n also_o be_v not_o hard_o but_o may_v easy_o be_v couceave_v by_o the_o definition_n of_o the_o diameter_n of_o a_o circle_n for_o as_o the_o diameter_n of_o a_o circle_n be_v a_o right_a line_n draw_v from_o one_o side_n of_o the_o circumfrence_n of_o a_o circle_n to_o the_o other_o pass_v by_o the_o centre_n of_o the_o circle_n so_o imagine_v you_o a_o right_a line_n to_o be_v draw_v from_o one_o side_n of_o the_o superficies_n of_o a_o sphere_n to_o the_o other_o pass_v by_o the_o centre_n of_o the_o sphere_n sphere_n and_o that_o line_n be_v the_o diameter_n of_o the_o sphere_n so_o it_o be_v not_o all_o one_o to_o say_v the_o axe_n of_o a_o sphere_n and_o the_o diameter_n of_o a_o sphere_n any_o line_n in_o a_o sphere_n draw_v from_o side_n to_o side_n by_o the_o centre_n be_v a_o diameter_n but_o not_o every_o 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same_o superficies_n wherefore_o these_o right_a line_n ab_fw-la bd_o and_o dc_o be_v in_o one_o and_o the_o self_n same_o superficies_n and_o either_o of_o these_o angle_n abdella_n and_o bdc_o be_v a_o right_a angle_n by_o supposition_n wherefore_o by_o the_o 28._o of_o the_o first_o the_o line_n ab_fw-la be_v a_o parallel_n to_o the_o line_n cd_o if_o therefore_o two_o right_a line_n be_v erect_v perpendicular_o to_o one_o and_o the_o self_n same_o plain_a superficies_n those_o right_a line_n be_v parallel_n the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v here_o for_o the_o better_a understanding_n of_o this_o 6._o proposition_n i_o have_v describe_v a_o other_o figure_n as_o touch_v which_o if_o you_o erect_v the_o superficies_n abdella_n perpendicular_o to_o the_o superficies_n bde_v and_o imagine_v only_o a_o line_n to_o be_v draw_v from_o the_o point_n a_o to_o the_o point_n e_o if_o you_o will_v you_o may_v extend_v a_o thread_n from_o the_o say_a 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wherefore_o by_o the_o three_o definition_n of_o this_o book_n qr_o be_v perpendicular_o erect_v to_o or_o upon_o open_a which_o be_v require_v to_o be_v prove_v io._n do_v you_o his_o advice_n upon_o the_o assumpt_n of_o the_o 6._o as_o concern_v the_o make_n of_o the_o line_n de_fw-fr equal_a to_o the_o right_a line_n ab_fw-la very_o the_o second_o of_o the_o first_o without_o some_o far_a consideration_n be_v not_o proper_o enough_o allege_v and_o no_o wonder_n it_o be_v for_o that_o in_o the_o former_a bookeâ—_n whatsoeâ—â—â—â—aâ—h_v of_o line_n be_v speak_v the_o same_o have_v always_o be_v imagine_v to_o be_v in_o one_o only_a plain_a superficies_n consider_v or_o execute_v but_o here_o the_o perpendicular_a line_n ab_fw-la be_v not_o in_o the_o same_o plaint_n superficies_n that_o the_o right_a line_n db_o be_v therefore_o some_o other_o help_n must_v be_v put_v into_o the_o hand_n of_o young_a beginner_n how_o to_o bring_v this_o problem_n to_o execution_n which_o be_v this_o most_o plain_a and_o brief_a understand_v that_o bd_o the_o right_n line_n be_v the_o common_a section_n of_o the_o plain_a superficies_n wherein_o the_o perpendicular_n ab_fw-la and_o cd_o be_v &_o of_o the_o other_o plain_a superficies_n to_o which_o they_o be_v perpendicular_n the_o first_o of_o these_o in_o my_o former_a demonstration_n of_o the_o 6_o â—_o i_o note_v by_o the_o plain_a superficies_n qr_o and_o the_o other_o i_o note_v by_o the_o plain_a superficies_n open_v wherefore_o bd_o be_v a_o right_a line_n common_a to_o both_o the_o plain_a supâ—rficieces_n qr_o &_o op_n thereby_o the_o ponit_n b_o and_o d_o be_v common_a to_o the_o plain_n qr_o and_o op_n now_o from_o bd_o sufficient_o extend_v cut_v a_o right_a line_n equal_a to_o ab_fw-la which_o suppose_v to_o be_v bf_o by_o the_o three_o of_o the_o first_o and_o orderly_a to_o bf_o make_v de_fw-fr equal_a by_o the_o 3._o oâ—_n the_o first_o if_o de_fw-fr be_fw-mi great_a than_o bf_o which_o always_o you_o may_v cause_v so_o to_o be_v by_o produce_v of_o de_fw-fr sufficient_o now_o forasmuch_o as_o bf_o by_o construction_n be_v cut_v equal_a to_o ab_fw-la and_o de_fw-fr also_o by_o construction_n put_v equal_a to_o bf_o therefore_o by_o the_o 1._o common_a sentence_n de_fw-fr be_v put_v equal_a to_o ab_fw-la which_o be_v require_v to_o be_v do_v in_o like_a sort_n if_o de_fw-fr be_v a_o line_n give_v to_o who_o ab_fw-la be_v to_o be_v cut_v and_o make_v equal_a first_o out_o of_o the_o line_n db_o sufficient_o produce_v cut_v of_o dg_o equal_a to_o de_fw-fr by_o the_o three_o of_o the_o first_o and_o by_o the_o same_o 3._o cut_v from_o basilius_n sufficient_o produce_v basilius_n equal_a to_o dg_v then_o be_v it_o evident_a that_o to_o the_o right_a line_n de_fw-fr the_o perpendicular_a line_n ab_fw-la be_v put_v equal_a and_o though_o this_o be_v easy_a to_o conceive_v yet_o i_o have_v design_v the_o figure_n according_o whereby_o you_o may_v instruct_v your_o imagination_n many_o such_o help_n be_v in_o this_o book_n requisite_a as_o well_o to_o inform_v the_o young_a student_n therewith_o as_o also_o to_o master_n the_o froward_a gaynesayer_n of_o our_o conclusion_n or_o interrupter_n of_o our_o demonstration_n course_n ¶_o the_o 7._o theorem_a the_o 7._o proposition_n if_o there_o be_v two_o parallel_n right_a line_n and_o in_o either_o of_o they_o be_v take_v a_o point_n at_o all_o adventure_n a_o right_a line_n draw_v by_o the_o say_a point_n be_v in_o the_o self_n same_o superficies_n with_o the_o parallel_n right_a line_n svppose_v that_o these_o two_o right_a line_n ab_fw-la and_o cd_o be_v parallel_n and_o in_o either_o of_o they_o take_v a_o point_n at_o all_o adventure_n namely_o e_o and_o f._n then_o i_o say_v that_o a_o right_a line_n draw_v from_o the_o point_n e_o to_o the_o point_n fletcher_n be_v in_o the_o self_n same_o plain_a superficies_n that_o the_o
at_o all_o adventure_n namely_o d_o five_o g_o s_o and_o a_o right_a line_n be_v draw_v from_o the_o point_n d_o to_o the_o point_n g_o and_o a_o other_o from_o the_o point_n five_o to_o the_o point_n s._n wherefore_o by_o the_o 7._o of_o the_o eleven_o the_o line_n dg_o and_o we_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n and_o forasmuch_o as_o the_o line_n de_fw-fr be_v a_o parallel_n to_o the_o line_n bg_o therefore_o by_o the_o 24._o of_o the_o first_o the_o angle_n edt_n be_v equal_a to_o the_o angle_n bgt_fw-mi for_o they_o be_v alternate_a angle_n and_o likewise_o the_o angle_n dtv_n be_v equal_a to_o the_o angle_n gts_fw-fr now_o then_o there_o be_v two_o triangle_n that_o be_v dtv_n and_o gts_fw-fr have_v two_o angle_n of_o the_o one_o equal_a to_o two_o angle_n of_o the_o other_o and_o one_o side_n of_o the_o one_o equal_a to_o one_o side_n of_o the_o other_o namely_o the_o side_n which_o subtend_v the_o equal_a angle_n that_o be_v the_o side_n dv_o to_o the_o side_n g_n for_o they_o be_v the_o half_n of_o the_o line_n de_fw-fr and_o bg_o wherefore_o the_o side_n remain_v be_v equal_a to_o the_o side_n remain_v wherefore_o the_o line_n dt_n be_v equal_a to_o the_o line_n tg_n and_o the_o line_n vt_fw-la to_o the_o line_n it_o be_v if_o therefore_o the_o opposite_a side_n of_o a_o parallelipipedon_n be_v divide_v into_o two_o equal_a part_n and_o by_o their_o section_n be_v extend_v plain_a superficiece_n the_o common_a section_n of_o those_o plain_a superficiece_n and_o the_o diameter_n of_o the_o parallelipipedon_n do_v divide_v the_o one_o the_o other_o into_o two_o equal_a part_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o flussas_n every_o plain_a superficies_n extend_v by_o the_o centre_n of_o a_o parallelipipedon_n divide_v that_o solid_a into_o two_o equal_a part_n and_o so_o do_v not_o any_o other_o plain_a superficies_n not_o extend_v by_o the_o centre_n for_o every_o plain_n extend_v by_o the_o centre_n cut_v the_o diameter_n of_o the_o parallelipipedon_n in_o the_o centre_n into_o two_o equal_a part_n for_o it_o be_v prove_v that_o plain_a superficiece_n which_o cut_v the_o solid_a into_o two_o equal_a part_n do_v cut_v the_o dimetient_fw-la into_o two_o equal_a part_n in_o the_o centre_n wherefore_o all_o the_o line_n draw_v by_o the_o centre_n in_o that_o plain_a superficies_n shall_v make_v angle_n with_o the_o dimetient_fw-la and_o forasmuch_o as_o the_o diameter_n fall_v upon_o the_o parallel_n right_a line_n of_o the_o solid_a which_o describe_v the_o opposite_a side_n of_o the_o say_v solid_a or_o upon_o the_o parallel_n plain_a superficiece_n of_o the_o solid_a which_o make_v angel_n at_o the_o end_n of_o the_o diameter_n the_o triangle_n contain_v under_o the_o diameter_n and_o the_o right_a line_n extend_v in_o that_o plain_n by_o the_o centre_n and_o the_o right_a line_n which_o be_v draw_v in_o the_o opposite_a superficiece_n of_o the_o solid_a join_v together_o the_o end_n of_o the_o foresay_a right_a line_n namely_o the_o end_n of_o the_o diameter_n and_o the_o end_n of_o the_o line_n draw_v by_o the_o centre_n in_o the_o superficies_n extend_v by_o the_o centre_n shall_v always_o be_v equal_a and_o equiangle_n by_o the_o 26._o of_o the_o first_o for_o the_o opposite_a right_a line_n draw_v by_o the_o opposite_a plain_a superficiece_n of_o the_o solid_a do_v make_v equal_a angle_n with_o the_o diameter_n forasmuch_o as_o they_o be_v parallel_a line_n by_o the_o 16._o of_o this_o book_n but_o the_o angle_n at_o the_o centre_n be_v equal_a by_o the_o 15._o of_o the_o first_o for_o they_o be_v head_n angle_n &_o one_o side_n be_v equal_a to_o one_o side_n namely_o half_o the_o dimetient_fw-la wherefore_o the_o triangle_n contain_v under_o every_o right_a line_n draw_v by_o the_o centre_n of_o the_o parallelipipedon_n in_o the_o superficies_n which_o be_v extend_v also_o by_o the_o say_a centre_n and_o the_o diameter_n thereof_o who_o end_n be_v the_o angle_n of_o the_o solid_a be_v equal_a equilater_n &_o equiangle_n by_o the_o 26._o of_o the_o first_o wherefore_o it_o follow_v that_o the_o plain_a superficies_n which_o cut_v the_o parallelipipedon_n do_v make_v the_o part_n of_o the_o base_n on_o the_o opposite_a side_n equal_a and_o equiangle_n and_o therefore_o like_a and_o equal_a both_o in_o multitude_n and_o in_o magnitude_n wherefore_o the_o two_o solid_a section_n of_o that_o solid_a shall_v be_v equal_a and_o like_a by_o the_o 8._o definition_n of_o this_o book_n and_o now_o that_o no_o other_o plain_a superficies_n beside_o that_o which_o be_v extend_v by_o the_o centre_n divide_v the_o parallelipipedon_n into_o two_o equal_a part_n it_o be_v manifest_a if_o unto_o the_o plain_a superficies_n which_o be_v not_o extend_v by_o the_o centre_n we_o extend_v by_o the_o centre_n a_o parallel_n plain_a superficies_n by_o the_o corollary_n of_o the_o 15._o of_o this_o book_n for_o forasmuch_o as_o that_o superficies_n which_o be_v extend_v by_o the_o centre_n do_v divide_v the_o parallelipipedom_n into_o two_o equal_a parâ—â—_n it_o be_v manifest_a that_o the_o other_o plain_a superficies_n which_o be_v parallel_n to_o the_o superficies_n which_o divide_v the_o solid_a into_o two_o equal_a part_n be_v in_o one_o of_o the_o equal_a part_n of_o the_o solid_a wherefore_o see_v that_o the_o whole_a be_v ever_o great_a than_o his_o part_n it_o must_v needs_o be_v that_o one_o of_o these_o section_n be_v less_o than_o the_o half_a of_o the_o solid_a and_o therefore_o the_o other_o be_v great_a for_o the_o better_a understanding_n of_o this_o former_a proposition_n &_o also_o of_o this_o corollary_n add_v by_o flussas_n it_o shall_v be_v very_o needful_a for_o you_o to_o describe_v of_o past_v paper_n or_o such_o like_a matter_n a_o parallelipipedom_n or_o a_o cube_n and_o to_o divide_v all_o the_o parallelogram_n thereof_o into_o two_o equal_a part_n by_o draw_v by_o the_o centre_n of_o the_o say_a parallelogram_n which_o centre_n be_v the_o point_n make_v by_o the_o cut_n of_o diagonal_a line_n draw_v from_o thâ—_z opposite_a angle_n of_o the_o say_a parallelogram_n line_n parallel_n to_o the_o side_n of_o the_o parallelogram_n as_o in_o the_o former_a figure_n describe_v in_o a_o plain_a you_o may_v see_v be_v the_o six_o parallelogram_n de_fw-fr eh_o ha_o ad_fw-la dh_o and_o cg_o who_o these_o parallel_a line_n draw_v by_o the_o centre_n of_o the_o say_a parallelogram_n namely_o xo_o or_o pr._n and_o px_n do_v divide_v into_o two_o equal_a part_n by_o which_o four_o line_n you_o must_v imagine_v a_o plain_a superficies_n to_o be_v extend_v also_o these_o parallel_a line_n kl_o ln_o nm_o and_o mâ—_n by_o which_o four_o line_n likewise_o yâ—_n must_v imagine_v a_o plain_a superficies_n to_o be_v extend_v you_o may_v if_o you_o will_v put_v within_o your_o body_n make_v thus_o of_o past_v paper_n two_o superficiece_n make_v also_o of_o the_o say_a paper_n have_v to_o their_o limit_n line_n equal_a to_o the_o foresay_a parallel_n line_n which_o superficiece_n must_v also_o be_v divide_v into_o two_o equal_a part_n by_o parallel_a line_n draw_v by_o their_o centre_n and_o must_v cut_v the_o one_o the_o other_o by_o these_o parallel_a line_n and_o for_o the_o diameter_n of_o this_o body_n extend_v a_o thread_n from_o one_o angle_n in_o the_o base_a of_o the_o solid_a to_o his_o opposite_a angle_n which_o shall_v pass_v by_o the_o centre_n of_o the_o parallelipipedon_n as_o do_v the_o line_n dg_o in_o the_o figure_n before_o describe_v in_o the_o plain_n and_o draw_v in_o the_o base_a and_o the_o opposite_a superficies_n unto_o it_o diagonal_a line_n from_o the_o angle_n from_o which_o be_v extend_v the_o diameter_n of_o the_o solid_a as_o in_o the_o former_a description_n be_v the_o line_n bg_o and_o de._n and_o when_o you_o have_v thus_o describe_v this_o body_n compare_v it_o with_o the_o former_a demonstration_n and_o it_o will_v make_v it_o very_o plain_a unto_o you_o so_o your_o letter_n agree_v with_o the_o letter_n of_o the_o figure_n describe_v in_o the_o book_n and_o this_o description_n will_v plain_o set_v forth_o unto_o you_o the_o corollary_n follow_v that_o proposition_n for_o where_o as_o to_o the_o understanding_n of_o the_o demonstration_n of_o the_o proposition_n the_o superficiece_n put_v within_o the_o body_n be_v extend_v by_o parallel_a line_n draw_v by_o the_o centre_n of_o the_o base_n of_o the_o parallelipipedon_n to_o the_o understanding_n of_o the_o say_a corollary_n you_o may_v extend_v a_o superficies_n by_o any_o other_o line_n draw_v in_o the_o say_a base_n so_o that_o yet_o it_o pass_v through_o the_o midst_n of_o the_o thread_n which_o be_v suppose_v to_o be_v the_o centre_n of_o the_o parallelipipedon_n ¶_o the_o 35._o theorem_a the_o 40._o proposition_n if_o there_o be_v
upright_o cylinder_n be_v cuâ—_n by_o a_o plain_a parallel_n to_o his_o base_n by_o construction_n therefore_o as_o the_o cylinder_n dc_o be_v to_o the_o cylinder_n be_v so_o be_v the_o axe_n of_o dc_o to_o the_o axe_n of_o be_v by_o the_o 13._o of_o this_o twelve_o demonstration_n wherefore_o as_o the_o axe_n be_v to_o the_o axe_n so_o be_v cylinder_n to_o cylinder_n but_o axe_n be_v to_o axe_n as_o side_n to_o side_n namely_o ce_fw-fr to_o qe_n because_o the_o axe_n be_v parallel_n to_o any_o side_n of_o a_o upright_o cylinder_n by_o the_o definition_n of_o a_o cylinder_n and_o the_o circle_n of_o the_o section_n be_v parallel_n to_o the_o base_n by_o construction_n wherefore_o in_o the_o parallelogram_n make_v of_o the_o axe_n and_o of_o two_o semidiameter_n on_o one_o side_n parallel_n one_o to_o the_o other_o be_v couple_v together_o by_o a_o line_n draw_v between_o their_o end_n in_o their_o circumference_n which_o line_n be_v the_o side_n qc_n it_o be_v evident_a that_o the_o axe_n of_o bc_o be_v cut_v in_o like_a proportion_n that_o the_o side_n qc_n be_v cut_v wherefore_o the_o cylinder_n dc_o be_v to_o the_o cylinder_n be_v as_o ec_o be_v to_o qe_n wherefore_o by_o composition_n the_o cylinder_n dc_o and_o be_v that_o be_v whole_a bc_o be_v to_o the_o cylinder_n be_v as_o ce_fw-fr and_o qe_n the_o whole_a right_a line_n qc_n be_v to_o qe_n but_o by_o construction_n qc_n be_v of_o 3._o such_o part_n as_o qe_n contain_v 2._o wherefore_o the_o cylinder_n bc_o be_v of_o 3._o such_o part_n as_o be_v contain_v 2._o wherefore_o bc_o the_o cylinder_n be_v to_o be_v as_o 3._o to_o 2_o which_o be_v sesquialtera_fw-la proportion_n but_o by_o the_o former_a theorem_a bc_o be_v sesquialtera_fw-la to_o the_o sphere_n a_o wherefore_o by_o the_o 7._o of_o the_o five_o be_v be_v equal_a to_o a._n therefore_o to_o a_o sphere_n give_v we_o have_v make_v a_o cylinder_n equal_a or_o thus_o more_o brief_o omit_v all_o cut_n of_o the_o cylinder_n forasmuch_o as_o bc_o be_v a_o upright_o cylinder_n his_o side_n be_v equal_a to_o his_o axe_n or_o heith_n therefore_o the_o two_o cylinder_n whereof_o one_o have_v the_o heith_n qc_n and_o the_o other_o the_o heith_n qe_n have_v both_o their_o base_n the_o great_a circle_n in_o the_o sphere_n a_o be_v one_o to_o the_o other_o as_o qc_n be_v to_o qe_n by_o the_o 14._o of_o this_o twelve_o but_o qc_n be_v to_o qe_n as_o 3._o to_o 2_o by_o construction_n and_o 3._o to_o 2._o be_v in_o sesquialtera_fw-la proportion_n therefore_o the_o cylinder_n bc_o have_v his_o heith_n qc_n &_o his_o base_a the_o great_a circle_n in_o a_o contain_v be_v sesquialtera_fw-la to_o the_o cylinder_n which_o have_v his_o base_a the_o great_a circle_n in_o a_o contain_v and_o heith_n the_o line_n qe_n but_o by_o the_o former_a theorem_a bc_o be_v also_o sesquialtera_fw-la to_o a_o wherefore_o the_o cylinder_n have_v the_o base_a bq_n which_o by_o supposition_n problem_n be_v equal_a to_o the_o great_a circle_n in_o a_o contain_v and_o heith_n qe_n be_v equal_a to_o the_o sphere_n a_o by_o the_o 7._o of_o the_o five_o and_o now_o it_o can_v not_o be_v hard_a to_o geve_v a_o cylinder_n to_o a_o in_o that_o proportion_n which_o be_v between_o x_o and_o y._n for_o let_v the_o side_n qe_n be_v to_o qp_v as_o you_o be_v to_o x_o by_o the_o 12._o of_o the_o six_o therefore_o backward_o qp_n be_v to_o qe_n as_o x_o to_o y._n wherefore_o the_o cylinder_n have_v the_o base_a the_o great_a circle_n in_o a_o and_o heith_n the_o line_n qp_n *_o be_v to_o the_o cylinder_n have_v the_o same_o base_a and_o heith_n the_o line_n qe_n as_o x_o be_v to_o you_o by_o the_o 14._o of_o this_o twelve_o but_o the_o cylinder_n have_v the_o heith_n qe_n &_o his_o base_a the_o great_a circle_n in_o a_o contain_v be_v prove_v equal_a to_o the_o sphere_n a._n wherefore_o by_o the_o 7._o of_o the_o five_o the_o cylinder_n who_o heith_n be_v qp_n and_o base_a the_o great_a circle_n in_o a_o contain_v be_v to_o the_o sphere_n a_o as_o x_o to_o y._n therefore_o to_o a_o sphere_n give_v we_o have_v make_v a_o cylinder_n in_o any_o proportion_n give_v between_o two_o right_a line_n and_o also_o before_o we_o have_v to_o a_o sphere_n give_v make_v a_o cylinder_n equal_a therefore_o to_o a_o sphere_n give_v we_o have_v make_v a_o cylinder_n equal_a or_o in_o any_o proportion_n give_v between_o two_o right_a line_n a_o problem_n 5._o a_o sphere_n be_v give_v and_o a_o circle_n upon_o that_o circle_n as_o a_o base_a to_o rear_v a_o cylinder_n equal_a to_o the_o sphere_n give_v or_o in_o any_o proportion_n give_v between_o two_o right_a line_n a_o problem_n 6._o a_o sphere_n be_v give_v and_o a_o right_a line_n to_o make_v a_o cylinder_n equal_a to_o the_o sphere_n give_v or_o in_o any_o other_o proportion_n between_o two_o right_a line_n give_v in_o this_o 5._o and_o 6._o problem_n first_o make_v a_o cylinder_n equal_a to_o the_o sphere_n give_v by_o the_o 4._o problem_n and_o then_o by_o the_o order_n of_o the_o 2._o and_o 3._o problem_n in_o cones_fw-la execute_v these_o accordingly_z in_o cylinder_n a_o problem_n 7._o two_o unlike_o cones_n or_o cylinder_n be_v give_v to_o find_v two_o right_a line_n which_o have_v the_o same_o proportion_n one_o to_o the_o other_o that_o the_o two_o give_v cones_fw-la or_o cylinder_n have_v one_o to_o the_o other_o upon_o one_o of_o their_o base_n rear_v a_o cone_n if_o cones_fw-la be_v compare_v or_o a_o cylinder_n if_o cylinder_n be_v compare_v equal_a to_o the_o other_o by_o the_o order_n of_o the_o second_o and_o three_o problem_n and_o the_o heith_n of_o the_o cone_n or_o cylinder_n on_o who_o base_a you_o rear_v a_o equal_a cone_fw-mi or_o cylinder_n with_o the_o new_a heith_n find_v have_v that_o proportion_n which_o the_o cones_fw-la or_o cylinder_n have_v one_o to_o the_o other_o by_o the_o 14._o of_o this_o twelve_o book_n a_o problem_n 8._o a_o upright_a cone_n and_o cylinder_n be_v give_v to_o find_v two_o right_a line_n have_v that_o proportion_n the_o one_o to_o the_o other_o which_o the_o cone_n and_o cylinder_n have_v one_o to_o the_o other_o suppose_v qek_v a_o upright_a cone_fw-mi and_o ab_fw-la a_o upright_o cylinder_n give_v i_o say_v two_o right_a line_n be_v to_o be_v give_v which_o shall_v have_v that_o proportion_n one_o to_o the_o other_o construction_n which_o qek_n and_o ab_fw-la have_v one_o to_o the_o other_o upon_o the_o base_a bh_o erect_v a_o cone_n equal_a to_o qek_v by_o the_o order_n of_o the_o second_o problem_n which_o let_v be_v obh_n and_o his_o heith_n let_v be_v oc_n demonstration_n and_o let_v the_o heith_n of_o ab_fw-la the_o cylinder_n be_v c_n produce_v c_n to_o p_o so_o that_o cp_n be_v tripla_fw-la to_o c_n &_o make_v perfect_a the_o cone_n pbh_n i_o say_v that_o fc_n and_o oc_n have_v that_o proportion_n which_o ab_fw-la have_v to_o qek_n for_o by_o constrâ—ction_n obh_n be_v equal_a to_o qek_v and_o pbh_n be_v equal_a to_o ab_fw-la as_o we_o will_v prove_v assumpt_v wise_a and_o pbh_n and_o obh_n be_v upon_o one_o base_a namely_o bh_o wherefore_o by_o the_o 14._o of_o this_o twelve_o â—bh_n and_o obh_n be_v one_o to_o the_o other_o as_o their_o heithe_n pc_n and_o oc_n be_v one_o to_o the_o other_o wherefore_o the_o cylinder_n and_o cone_n equal_a to_o pbh_n and_o obh_n be_v as_o pc_n be_v to_o oc_n by_o the_o 7._o of_o the_o five_o but_o ab_fw-la the_o cylinder_n qek_n the_o cone_n be_v equal_a to_o pbh_n and_o obh_n by_o construction_n wherefore_o ab_fw-la the_o cylinder_n be_v to_o qek_v the_o cone_n as_o pc_n be_v to_o oc_n wherefore_o we_o have_v find_v two_o right_a line_n have_v that_o proportion_n that_o a_o cone_fw-mi and_o a_o cylinder_n give_v have_v one_o to_o the_o other_o which_o thing_n we_o may_v execute_v upon_o the_o base_a of_o the_o cone_fw-it give_v as_o we_o do_v upon_o the_o base_a of_o the_o cylinder_n give_v on_o this_o manner_n upon_o the_o base_a of_o the_o cone_fw-it qek_n problem_n which_o base_a let_v be_v eke_o erect_v a_o cylinder_n equal_a to_o ab_fw-la by_o the_o order_n of_o my_o second_o problem_n which_o cylinder_n let_v be_v ed_z and_o gt_v his_o heith_n and_o let_v the_o heith_n of_o the_o cone_fw-it qek_n be_v qg_n take_v the_o line_n graccus_n the_o â—hird_n part_n oâ—_n qg_v by_o the_o 9_o of_o the_o six_o and_o with_o a_o plain_n pass_v by_o r_o parallel_n to_o eke_o cut_v of_o the_o cylinder_n eâ—_n which_o â—hall_v be_v equal_a to_o the_o cone_n qek_n by_o the_o assumpt_n follow_v i_o say_v now_o that_o ab_fw-la the_o cylinder_n be_v to_o qek_v the_o cone_n as_o gt_n be_v to_o gr._n for_o the_o cylinder_n ed_z be_v to_o the_o cylinder_n of_o as_o gt_n be_v
superficies_n or_o solidity_n in_o the_o hole_n or_o in_o partâ—_n such_o certain_a knowledge_n demonstrative_a may_v arise_v and_o such_o mechanical_a exercise_n thereby_o be_v devise_v that_o sure_a i_o be_o to_o the_o sincere_a &_o true_a student_n great_a light_n aid_n and_o comfortable_a courage_n far_a to_o wade_v will_v enter_v into_o his_o heart_n and_o to_o the_o mechanical_a witty_a and_o industrous_a deviser_n new_a manner_n of_o invention_n &_o execution_n in_o his_o work_n will_v with_o small_a travail_n for_o foot_n application_n come_v to_o his_o perceiveraunce_n and_o understanding_n therefore_o even_o a_o manifold_a speculation_n &_o practice_n may_v be_v have_v with_o the_o circle_n his_o quantity_n be_v not_o know_v in_o any_o kind_n of_o small_a certain_a measure_n so_o likewise_o of_o the_o sphere_n many_o problem_n may_v be_v execute_v and_o his_o precise_a quantity_n in_o certain_a measure_n not_o determine_v or_o know_v yet_o because_o both_o one_o of_o the_o first_o humane_a occasion_n of_o invent_v and_o stablish_v this_o art_n be_v measure_v of_o the_o earth_n and_o therefore_o call_v geometria_n that_o be_v earthmeasure_v and_o also_o the_o chief_a and_o general_a end_n in_o deed_n be_v measure_n and_o measure_n require_v a_o determination_n of_o quantity_n in_o a_o certain_a measure_n by_o number_n express_v it_o be_v needful_a for_o mechanical_a earthmeasure_n not_o to_o be_v ignorant_a of_o the_o measure_n and_o content_n of_o the_o circle_n neither_o of_o the_o sphere_n his_o measure_n and_o quantity_n as_o near_o as_o sense_n can_v imagine_v or_o wish_v and_o in_o very_a deed_n the_o quantity_n and_o measure_n of_o the_o circle_n be_v know_v make_v not_o only_o the_o cone_n and_o cylinder_n but_o also_o the_o sphere_n his_o quantity_n to_o be_v as_o precise_o know_v and_o certain_a therefore_o see_v in_o respect_n of_o the_o circle_n quantity_n by_o archimedes_n specify_v this_o theorem_a be_v note_v unto_o you_o i_o will_v by_o order_n upon_o that_o as_o a_o supposition_n infer_v the_o conclusion_n of_o this_o our_o theorem_n note_n 1._o wherefore_o if_o you_o divide_v the_o one_o side_n as_o tq_n of_o the_o cube_fw-la tx_n into_o 21._o equal_a part_n and_o where_o 11._o part_n do_v end_n reckon_v from_o t_o suppose_v the_o point_n p_o and_o by_o that_o point_n p_o imagine_v a_o plain_a pass_v parallel_n to_o the_o opposite_a base_n to_o cut_v the_o cube_fw-la tx_n and_o thereby_o the_o cube_fw-la tx_n to_o be_v divide_v into_o two_o rectangle_n parallelipipedon_n namely_o tn_n and_o px_n it_o be_v manifest_a give_v tn_n to_o be_v equal_a to_o the_o sphere_n a_o by_o construction_n and_o the_o 7._o of_o the_o five_o note_n 2._o second_o the_o whole_a quantity_n of_o the_o sphere_n a_o assign_v be_v contain_v in_o the_o rectangle_n parallelipipedon_n tn_n you_o may_v easy_o transform_v the_o same_o quantity_n into_o other_o parallelipipedon_n rectangle_v of_o what_o height_n and_o of_o what_o parallelogram_n base_a you_o listen_v by_o my_o first_o and_o second_o problem_n upon_o the_o 34._o of_o this_o book_n and_o the_o like_a may_v you_o do_v to_o any_o assign_a part_n of_o the_o sphere_n a_o by_o the_o like_a mean_n devide_v the_o parallelipipedon_n tn_n as_o the_o part_n assign_v do_v require_v as_o if_o a_o three_o four_o five_o or_o six_o part_n of_o the_o sphere_n a_o be_v to_o be_v have_v in_o a_o parallelipipedon_n of_o any_o parallelograâ—â—e_n base_a assign_v or_o of_o any_o heith_n assign_v then_o devide_v tp_n into_o so_o many_o part_n as_o into_o 4._o if_o a_o four_o part_n be_v to_o be_v transform_v or_o into_o five_o if_o a_o five_o part_n be_v to_o be_v transform_v etc_n etc_n and_o then_o proceed_v â—s_v you_o do_v with_o cut_v of_o tn_n from_o tx_n and_o that_o i_o say_v of_o parallelipipedon_n may_v in_o like_a sort_n by_o my_o â—â—yd_n two_o problem_n add_v to_o the_o 34._o of_o this_o book_n be_v do_v in_o any_o side_v column_n pyramid_n and_o prismeâ—_n so_o thâ—â—_n in_o pyramid_n and_o some_o prism_n you_o use_v the_o caution_n necessary_a in_o respect_n of_o their_o quan_fw-mi 〈…〉_o odyes_n have_v parallel_n equal_a and_o opposite_a base_n who_o part_n 〈…〉_o re_fw-mi in_o their_o proposition_n be_v by_o euclid_n demonstrate_v and_o final_o 〈…〉_o addition_n you_o have_v the_o way_n and_o order_n how_o to_o geve_v to_o a_o sphere_n or_o any_o segment_n oâ—_n the_o same_o cones_n or_o cylinder_n equal_a or_o in_o any_o proportion_n between_o two_o right_a line_n give_v with_o many_o other_o most_o necessary_a speculation_n and_o practice_n about_o the_o sphere_n i_o trust_v that_o i_o have_v sufficient_o â—raughted_v your_o imagination_n for_o your_o honest_a and_o profitable_a study_n herein_o and_o also_o give_v you_o reaâ—â—_n â—â—tter_n wheâ—â—_n with_o to_o sâ—â—p_v the_o mouth_n of_o the_o malicious_a ignorant_a and_o arrogant_a despiser_n of_o the_o most_o excellent_a discourse_n travail_n and_o invention_n mathematical_a sting_v aswell_o the_o heavenly_a sphere_n &_o star_n their_o spherical_a solidity_n geometry_n with_o their_o convene_n spherical_a superficies_n to_o the_o earth_n at_o all_o time_n respect_v and_o their_o distance_n from_o the_o earth_n as_o also_o the_o whole_a earthly_a sphere_n and_o globe_n itself_o and_o infinite_a other_o case_n concern_v sphere_n or_o globe_n may_v hereby_o with_o as_o much_o ease_n and_o certainty_n be_v determine_v of_o as_o of_o the_o quantity_n of_o any_o bowl_n ball_n or_o bullet_n which_o we_o may_v gripe_v in_o our_o hand_n reason_n and_o experience_n be_v our_o witness_n and_o without_o these_o aid_n such_o thing_n of_o importance_n never_o able_a of_o we_o certain_o to_o be_v know_v or_o attain_a unto_o here_o end_n m._n john_n do_v you_o his_o addition_n upon_o the_o last_o proposition_n of_o the_o twelve_o book_n a_o proposition_n add_v by_o flussas_n if_o a_o sphere_n touch_v a_o plain_a superficiesâ—_n a_o right_a line_n draw_v from_o the_o centre_n to_o the_o touch_n shall_v be_v erect_v perpendicular_o to_o the_o plain_a superficies_n suppose_v that_o there_o be_v a_o sphere_n bcdl_n who_o centre_n let_v be_v the_o point_n a._n and_o let_v the_o plain_a superficies_n gci_n touch_v the_o spear_n in_o the_o point_n c_o and_o extend_v a_o right_a line_n from_o the_o centre_n a_o to_o the_o point_n c._n then_o i_o say_v that_o the_o line_n ac_fw-la be_v erect_v perpendicular_o to_o tâ—e_z plain_a gic._n let_v the_o sphere_n be_v cut_v by_o plain_a superficiece_n pass_v by_o the_o right_a line_n lac_n which_o plain_n let_v be_v abcdl_n and_o acel_n which_o let_v cut_v the_o plain_n gci_n by_o the_o right_a line_n gch_o and_o kci_fw-la now_o it_o be_v manifest_a by_o the_o assumpt_n put_v before_o the_o 17._o of_o this_o book_n that_o the_o two_o section_n of_o the_o sphere_n shall_v be_v circle_n have_v to_o their_o diameter_n the_o line_n lac_n which_o be_v also_o the_o diameter_n of_o the_o sphere_n wherefore_o the_o right_a line_n gch_o and_o kci_fw-la which_o be_v draw_v in_o the_o plain_n gci_n do_v at_o the_o point_n c_o fall_n without_o the_o circle_n bcdl_n and_o ecl._n wherefore_o they_o touch_v the_o circle_n in_o the_o point_n c_o by_o the_o second_o definition_n of_o the_o three_o wherefore_o the_o right_a line_n lac_n make_v right_a angle_n with_o the_o line_n gch_v and_o kci_fw-la by_o the_o 16._o of_o the_o three_o wherefore_o by_o the_o 4._o of_o the_o eleven_o the_o right_a line_n ac_fw-la be_v erect_v perpendicular_o to_o to_o the_o plain_a superficies_n gci_n wherein_o be_v draw_v the_o line_n gch_v and_o kci_fw-la if_o therefore_o a_o sphere_n touch_v a_o plain_a superficies_n a_o right_a line_n draw_v from_o the_o centre_n to_o the_o touch_n shall_v be_v erect_v perpendicular_o to_o the_o plain_a superficies_n which_o be_v require_v to_o be_v prove_v the_o end_n of_o the_o twelve_o book_n of_o euclides_n element_n ¶_o the_o thirteen_o book_n of_o euclides_n element_n in_o this_o thirteen_o book_n be_v set_v forth_o certain_a most_o wonderful_a and_o excellent_a passion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n book_n a_o matter_n undoubted_o of_o great_a and_o infinite_a use_n in_o geometry_n as_o you_o shall_v both_o in_o this_o book_n and_o in_o the_o other_o book_n follow_v most_o evident_o perceive_v it_o teach_v moreover_o the_o composition_n of_o the_o five_o regular_a solid_n and_o how_o to_o inscribe_v they_o in_o a_o sphere_n give_v and_o also_o set_v forth_o certain_a comparison_n of_o the_o say_a body_n both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o sphere_n wherein_o they_o be_v describe_v the_o 1._o theorem_a the_o 1._o proposition_n if_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o to_o the_o great_a segment_n be_v add_v the_o half_a of_o the_o whole_a line_n the_o square_n make_v of_o those_o two_o
line_n thereby_o adjoin_v let_v be_v bq_n i_o say_v that_o cq_n be_v a_o line_n also_o divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n b_o and_o that_o bq_n the_o line_n adjoin_v be_v the_o less_o segment_n for_o by_o the_o third_o it_o be_v prove_v that_o half_a ac_fw-la which_o let_v be_v cd_o with_o cb_o as_o one_o line_n compose_v have_v his_o power_n or_o square_v quintuple_a to_o the_o power_n of_o the_o segment_n cd_o wherefore_o by_o the_o second_o of_o this_o book_n the_o double_a of_o c_o d_o be_v divide_v by_o extreme_a and_o middle_a proportioned_a and_o the_o great_a segment_n thereof_o shall_v be_v cb._n but_o by_o construction_n cq_n be_v the_o double_a of_o cd_o for_o it_o be_v equal_a to_o ac_fw-la wherefore_o cq_n be_v divide_v by_o extreme_a and_o middle_a proportion_n in_o the_o point_n b_o and_o the_o great_a segment_n thereof_o shall_v be_v cb._n wherefore_o bq_n be_v the_o less_o segment_n which_o be_v the_o line_n adjoin_v therefore_o a_o line_n be_v divide_v by_o extreme_a and_o middle_a proportion_n if_o the_o less_o segment_n be_v produce_v equal_o to_o the_o length_n of_o the_o great_a segment_n the_o line_n thereby_o adjoin_v together_o with_o the_o say_v less_o segment_n make_v a_o new_a line_n divide_v by_o extreme_a &_o mean_a proportion_n whoâ—e_v less_o segment_n be_v the_o line_n adjoin_v which_o be_v to_o be_v demonstrate_v ¶_o a_o corollary_n 2._o ifâ—_n from_o the_o great_a segment_n of_o a_o line_n divide_v by_o extreme_a and_o middle_a proportion_n a_o line_n equal_a to_o the_o less_o segment_n be_v cut_v of_o the_o great_a segment_n thereby_o be_v also_o divide_v by_o extreme_a and_o mean_a proportion_n who_o great_a segmentâ—_n shall_v be_v 〈◊〉_d that_o part_n of_o it_o which_o be_v cut_v of_o for_o take_v from_o ac_fw-la a_o line_n equal_a to_o cb_o let_v be_v remain_v i_o say_v that_o ac_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n r_o and_o that_o cr_n the_o line_n cut_v of_o be_v the_o great_a segment_n for_o it_o be_v prove_v in_o the_o former_a corollary_n that_o cq_n be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n b._n but_o ac_fw-la be_v equal_a to_o cq_n by_o construction_n and_o cr_n be_v equal_a to_o cb_o by_o construction_n wherefore_o the_o residue_n ar_n be_v equal_a to_o bq_n the_o residue_n see_v therefore_o the_o whole_a ac_fw-la be_v equal_a to_o the_o whole_a cq_n and_o the_o great_a part_n of_o ac_fw-la which_o be_v cr_n be_v equal_a to_o cb_o the_o great_a part_n of_o cq_n and_o the_o less_o segment_n also_o equal_a to_o the_o less_o and_o withal_o see_v cq_n be_v prove_v to_o be_v divide_v by_o extreme_a &_o mean_a proportion_n in_o the_o point_n b_o it_o follow_v of_o necessity_n that_o ac_fw-la be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n r._n and_o see_v cb_o be_v the_o great_a segment_n of_o cq_n cr_n shall_v be_v the_o great_a segment_n of_o ac_fw-la which_o be_v to_o be_v demonstrate_v a_o corollary_n 3._o it_o be_v evident_a thereby_o a_o line_n be_v divide_v by_o extreme_a and_o mean_a proportion_n that_o the_o line_n whereby_o the_o great_a segment_n exceed_v the_o less_o together_o with_o the_o less_o segment_n do_v make_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n who_o less_o segment_n be_v the_o say_a line_n of_o exceesse_n or_o difference_n between_o the_o segment_n john_n dee_n ¶_o two_o new_a way_n to_o divide_v any_o right_a line_n give_v by_o a_o extreme_a and_o mean_a proportion_n demonstrate_v and_o add_v by_o m._n dee_n a_o problem_n to_o divide_v by_o a_o extreme_a and_o mean_a proportion_n any_o right_a line_n give_v in_o length_n and_o position_n suppose_v a_o line_n give_v in_o length_n and_o position_n to_o be_v ab_fw-la i_o say_v that_o ab_fw-la be_v to_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n construction_n divide_v ab_fw-la into_o two_o equal_a part_n as_o in_o the_o point_n c._n produce_v ab_fw-la direct_o from_o the_o point_n b_o to_o the_o point_n d_o make_v bd_o equal_a to_o bc._n to_o the_o line_n ad_fw-la and_o at_o the_o point_n d_o let_v a_o line_n be_v draw_v etc_n perpendicular_a by_o the_o 11._o of_o the_o first_o which_o let_v be_v df_o of_o what_o length_n you_o will_v from_o df_o and_o at_o the_o point_n d_o cut_v of_o the_o six_o part_n of_o df_o by_o the_o 9_o of_o the_o six_o and_o let_v that_o six_o part_n be_v the_o line_n dg_o upon_o df_o as_o a_o diameter_n describe_v a_o semicircle_n which_o let_v be_v dhf_n from_o the_o point_n g_o rear_v a_o line_n perpendicular_a to_o df_o which_o suppose_v to_o be_v gh_o and_o let_v it_o come_v to_o the_o circumference_n of_o dhf_n in_o the_o point_n h._n draw_v right_a line_n hd_v and_o hf._n produce_v dh_o from_o the_o point_n h_o so_o long_o till_o a_o line_n adjoin_v with_o dh_o be_v equal_a to_o hf_o which_o let_v be_v di_fw-it equal_a to_o hf._n from_o the_o point_n h_o to_o the_o point_n b_o the_o one_o end_n of_o our_o line_n give_v let_v a_o right_a line_n be_v draw_v as_o hb_o from_o the_o point_n i_o let_v a_o line_n be_v draw_v to_o the_o line_n ab_fw-la so_o that_o it_o be_v also_o parallel_a to_o the_o line_n hb_o which_o parallel_n line_n suppose_v to_o be_v ik_fw-mi cut_v the_o line_n ab_fw-la at_o the_o point_n k._n i_o say_v that_o ab_fw-la be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n in_o the_o point_n k._n for_o the_o triangle_n dki_n have_v hb_o parallel_n to_o ik_fw-mi have_v his_o side_n dk_o and_o diego_n demonstration_n cut_v proportional_o by_o the_o 2._o of_o the_o six_o wherefore_o as_o ih_v be_v to_o hd_v so_o be_v kb_o to_o bd._n and_o therefore_o compound_o by_o the_o 18._o of_o the_o five_o as_o diego_n be_v to_o dh_o so_o be_v dk_o to_o db._n but_o by_o construction_n diego_n be_v equal_a to_o hf_o wherefore_o by_o the_o 7._o of_o the_o five_o di_z be_v to_o dh_o as_o hf_o be_v to_o dh_o wherefore_o by_o the_o 11._o of_o the_o five_o dk_o be_v to_o db_o as_o hf_o be_v to_o dh_o wherefore_o the_o square_a of_o dk_o be_v to_o the_o square_n of_o db_o as_o the_o square_n of_o hf_o be_v to_o the_o square_n of_o dh_o by_o the_o 22._o of_o the_o six_o but_o the_o square_n of_o hf_o be_v to_o the_o square_n of_o dh_o as_o the_o line_n gf_o be_v to_o the_o line_n gdâ—_n by_o my_o corollary_n upon_o the_o 5_o problem_n of_o my_o addition_n to_o the_o second_o proposition_n of_o the_o twelve_o wherefore_o by_o the_o 11._o of_o the_o five_o the_o square_a of_o dk_o be_v to_o the_o square_n of_o db_o as_o the_o line_n gf_o be_v to_o the_o line_n gd_o but_o by_o construction_n gf_o be_v quintuple_a to_o gd_o wherefore_o the_o square_a of_o dk_o be_v quintuple_a to_o the_o square_n of_o db_o and_o therefore_o the_o double_a of_o db_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o bk_o be_v the_o great_a segment_n thereof_o by_o the_o 2._o of_o this_o thirteen_o wherefore_o see_v ab_fw-la be_v the_o double_a of_o db_o by_o construction_n the_o line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o his_o great_a segment_n be_v the_o line_n bk_o wherefore_o ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n k._n we_o have_v therefore_o divide_v by_o extreme_a and_o mean_a proportion_n any_o line_n give_v in_o length_n and_o position_n which_o be_v requisite_a to_o be_v do_v the_o second_o way_n to_o execute_v this_o problem_n suppose_v the_o line_n give_v to_o be_v ab_fw-la divide_v aâ—_z into_o two_o equal_a part_n as_o suppose_v it_o to_o be_v do_v in_o the_o point_n c._n produce_v ab_fw-la from_o the_o point_n b_o adjoin_v a_o line_n equal_a to_o bc_o which_o let_v be_v bd._n to_o the_o right_a line_n ad_fw-la and_o at_o the_o point_n d_o erect_v a_o perpendicular_a line_n equal_a to_o bd_o let_v that_o be_v de._n produce_v ed_z from_o the_o point_n d_o to_o the_o point_n f_o make_v df_o to_o contain_v five_o such_o equal_a part_n as_o de_fw-fr be_v one_o now_o upon_o of_o as_o a_o diameter_n describe_v a_o semicircle_n which_o let_v 〈◊〉_d ekf_n and_o let_v the_o point_n where_o the_o circumference_n of_o ekf_n do_v cut_v the_o line_n ab_fw-la be_v the_o point_n k._n i_o say_v that_o ab_fw-la be_v divide_v in_o the_o point_n king_n by_o a_o extreme_a and_o mean_a proportion_n for_o by_o the_o 13._o of_o the_o six_o ed_z dk_o &_o df_o be_v three_o line_n in_o continual_a proportion_n dk_o be_v the_o middle_n proportional_a â—_o wherefore_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o as_z ed_z be_v to_o df_o so_o be_v
the_o circumference_n on_o which_o the_o icosahedron_n be_v describe_v and_o that_o the_o diameter_n of_o the_o sphere_n be_v compose_v of_o the_o side_n of_o a_o hexagon_n and_o of_o two_o side_n of_o a_o decagon_n describe_v in_o one_o and_o the_o self_n same_o circle_n flussas_n prove_v the_o icosahedron_n describe_v to_o be_v contain_v in_o a_o sphere_n by_o draw_v right_a line_n from_o the_o point_n a_o to_o the_o point_n p_o and_o g_o after_o this_o manner_n forasmuch_o as_o the_o line_n zw_n wp_n be_v put_v equal_a to_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumferenceâ—_n and_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n be_v double_a to_o the_o line_n awe_n by_o construction_n therefore_o the_o line_n wp_n be_v also_o double_a to_o the_o same_o line_n awe_n wherefore_o the_o square_a of_o the_o line_n wp_n be_v quadruple_a to_o the_o square_n of_o the_o line_n awe_n by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o and_o those_o line_n pw_o and_o wa_n contain_v a_o right_a angle_n pwa_n as_o have_v before_o be_v prove_v be_v subtend_v of_o the_o right_a line_n ap_fw-mi wherefore_o by_o the_o 47._o of_o the_o first_o the_o line_n ap_fw-mi contain_v in_o power_n the_o line_n pw_o and_o wa._n wherefore_o the_o right_a line_n ap_fw-mi be_v in_o power_n quintuple_a to_o the_o line_n wa._n wherefore_o the_o right_a line_n ap_fw-mi and_o aq_n be_v quintuple_a to_o one_o and_o the_o same_o line_n wa_n be_v by_o the_o 9_o of_o the_o flu_v equal_a in_o like_a sort_n also_o may_v we_o prove_v that_o unto_o those_o line_n ap_fw-mi and_o aq_n be_v equal_a the_o rest_n of_o the_o line_n draw_v from_o the_o point_n a_o to_o the_o rest_n of_o the_o angle_n r_o s_o t_o v._o for_o they_o subtend_v right_a angle_n contain_v of_o the_o line_n wa_n and_o of_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n and_o forasmuch_o as_o unto_o the_o line_n wa_n be_v equal_a the_o line_n valerio_n which_o be_v likewise_o erect_v perpendicular_o unto_o the_o other_o plain_a superficies_n olmnx_n therefore_o line_n draw_v from_o the_o point_n a_o to_o the_o angle_n oh_o l_o m_o n_o x_o and_o subtend_v right_a angle_n at_o the_o point_n z_o contain_v under_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n and_o under_o the_o line_n az_n be_v equal_a not_o only_o the_o one_o to_o the_o other_o but_o also_o to_o the_o line_n draw_v from_o the_o say_a point_n a_o to_o the_o former_a angle_n at_o the_o point_n p_o r_o s_o t_o vâ—_n for_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n of_o each_o circle_n be_v equal_a &_o the_o line_n awe_n be_v equal_a to_o the_o line_n az._n but_o the_o line_n ap_fw-mi be_v prove_v equal_a to_o the_o line_n aq_n which_o be_v the_o half_a of_o the_o whole_a qy_n therefore_o the_o residue_n ay_o be_v equal_a to_o the_o foresay_a line_n ap_fw-mi aq_n etc_n etc_n wherefore_o make_v the_o centre_n the_o point_n a_o and_o the_o space_n one_o of_o those_o line_n aq_n ap_fw-mi etc_n etc_n extend_v the_o superficies_n of_o a_o sphere_n &_o it_o shall_v touch_v the_o 12._o angle_n of_o the_o icosahedron_n which_o be_v at_o the_o point_n oh_o l_o m._n n_o x_o pea_n king_n s_o t_o five_o q_o y_fw-fr which_o sphere_n be_v describe_v if_o upon_o the_o diameter_n qy_n be_v draw_v a_o semicircle_n and_o the_o say_a semicircle_n be_v move_v about_o till_o it_o return_v unto_o the_o same_o place_n from_o whence_o it_o begin_v first_o to_o be_v move_v ¶_o a_o corollary_n add_v by_o flussas_n the_o opposite_a side_n of_o a_o icosahedron_n be_v parallel_n for_o the_o diameter_n of_o the_o sphere_n do_v fall_v upon_o the_o opposite_a angle_n of_o the_o icosahedron_n as_o it_o be_v manifest_a by_o the_o right_a line_n qy_n if_o therefore_o there_o be_v imagine_v to_o be_v draw_v the_o two_o diameter_n pn_n and_o om_o they_o shall_v concur_v in_o the_o point_n f_o wherefore_o the_o right_a line_n which_o join_v they_o together_o pv_n and_o ln_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n by_o the_o 2._o of_o the_o eleven_o and_o forasmuch_o as_o the_o alternate_a angle_n at_o the_o end_n of_o the_o diameter_n be_v equal_a by_o the_o 8._o of_o the_o first_o for_o the_o triangle_n contain_v under_o equal_a semidiameteâ—s_n and_o the_o side_n of_o the_o icosahedron_n be_v equiangle_n therefore_o by_o the_o 28._o of_o the_o first_o the_o line_n pv_n and_o ln_o be_v paralle_n ¶_o the_o 5._o problem_n the_o 17._o proposition_n to_o make_v a_o dodecahedron_n and_o to_o comprehend_v it_o in_o the_o sphere_n give_v wherein_o be_v comprehend_v the_o foresay_a solid_n and_o to_o prove_v that_o the_o side_n of_o the_o dodecahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o residual_a line_n superficies_n now_o i_o say_v that_o it_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n forasmuch_o as_o the_o line_n zv_n be_v a_o parallel_n to_o the_o line_n sir_n as_o be_v before_o prove_v but_o unto_o the_o same_o line_n sir_n be_v the_o line_n cb_o a_o parallel_n by_o the_o 28._o of_o the_o first_o wherefore_o by_o the_o 9_o of_o the_o eleven_o the_o line_n we_o be_v a_o parallel_n to_o the_o line_n cb._n wherefore_o by_o the_o seven_o of_o the_o eleven_o the_o right_a line_n which_o join_v they_o together_o be_v in_o the_o self_n same_o plain_n wherein_o be_v the_o parallel_a line_n wherefore_o the_o trapesium_n buzc_n be_v in_o one_o plain_n and_o the_o triangle_n bwc_n be_v in_o one_o plain_n by_o the_o 2._o of_o the_o eleven_o now_o to_o prove_v that_o the_o trapesium_n buzc_n &_o the_o triangle_n bwc_n be_v in_o one_o and_o the_o self_n same_o plain_a we_o must_v prove_v that_o the_o right_a line_n yh_n and_o hw_o be_v make_v direct_o one_o right_a line_n which_o thing_n be_v thus_o prove_v forasmuch_o as_o the_o line_n hp_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n t_o and_o his_o great_a segment_n be_v the_o line_n pt_v therefore_o as_o the_o line_n hp_n be_v to_o the_o line_n pt_v so_o be_v the_o line_n pt_v to_o the_o line_n th._n but_o the_o line_n hp_n be_v equal_a to_o the_o line_n ho_o and_o the_o line_n pt_v to_o either_o of_o these_o line_n tw_n and_o oy_o wherefore_o as_o the_o line_n ho_o be_v to_o the_o line_n oy_fw-it so_o be_v the_o line_n with_o to_o the_o line_n th._n but_o the_o line_n ho_o and_o tw_n be_v side_n of_o like_a proportion_n be_v parallel_n by_o the_o 6._o of_o the_o eleven_o for_o either_o of_o they_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n bd_o â—_o and_o the_o line_n th._n and_o oy_o be_v parallel_n which_o be_v also_o side_n of_o like_a proportion_n by_o the_o same_o 6._o of_o the_o eleven_o for_o either_o of_o they_o be_v also_o erect_v perpendicular_o to_o the_o plain_a superficies_n bf_o but_o when_o there_o be_v two_o triangle_n have_v two_o side_n proportional_a to_o two_o side_n so_o set_v upon_o one_o angle_n that_o their_o side_n of_o like_a proportion_n be_v also_o parallel_n as_o the_o triangle_n yoh_n and_o htw_n be_v â—_o who_o two_o side_n oh_o &_o ht_o be_v in_o the_o two_o base_n of_o the_o cube_fw-la make_v a_o angle_n at_o the_o point_n h_o the_o side_n remain_v of_o those_o triangle_n shall_v by_o the_o 32._o of_o the_o six_o be_v in_o one_o right_a line_n wherefore_o the_o line_n yh_n &_o hw_o make_v both_o one_o right_a line_n but_o every_o right_a line_n be_v by_o the_o 3._o of_o the_o eleven_o in_o one_o &_o the_o self_n same_o plain_a superficies_n wherefore_o if_o you_o draw_v a_o right_a line_n from_o b_o to_o a'n_z there_o shall_v be_v make_v a_o triangle_n bye_o which_o shall_v be_v in_o one_o and_o the_o self_n same_o plain_a by_o the_o 2._o of_o the_o eleven_o and_o therefore_o the_o whole_a pentagon_n figure_n vbwcz_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n now_o also_o i_o say_v that_o it_o be_v equiangle_n equiangle_n for_o forasmuch_o as_o the_o right_a line_n no_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n r_o and_o his_o great_a segment_n be_v or_o therefore_o as_o both_o the_o line_n no_o and_o or_o add_v together_o be_v to_o the_o line_n on_o so_o by_o the_o 5._o of_o this_o book_n be_v the_o line_n on_o to_o the_o line_n or_o but_o the_o line_n or_o be_v equal_a to_o the_o line_n os_fw-mi wherefore_o as_o the_o line_n sn_a be_v to_o the_o line_n no_o so_o be_v the_o line_n no_o to_o the_o line_n os_fw-mi wherefore_o the_o line_n sn_a be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n oh_o