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book_n definition_n plate_n use_v 2,262 5 11.5518 5 false
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ID Title Author Corrected Date of Publication (TCP Date of Publication) STC Words Pages
A38722 The elements of Euclid, explained and demonstrated in a new and most easie method with the uses of each proposition in all the parts of the mathematicks / by Claude Francois Milliet D'Chales, a Jesuit ; done out of French, corrected and augmented, and illustrated with nine copper plates, and the effigies of Euclid, by Reeve Williams ...; Huict livres des Eléments d'Euclide rendus plus faciles. English Dechales, Claude-François Milliet, 1621-1678.; Euclid. Elements.; Williams, Reeve, fl. 1682-1703. 1685 (1685) Wing E3399; ESTC R10241 136,603 430

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the_o quantity_n of_o a_o angle_n by_o the_o length_n of_o the_o line_n which_o compose_v the_o same_o 4._o fig._n 3_o 4._o the_o most_o open_a angle_n be_v the_o great_a that_o be_v to_o say_v when_o the_o line_n include_v a_o angle_n be_v far_o asunder_o than_o those_o of_o another_o angle_n take_v they_o at_o the_o same_o distance_n from_o the_o point_n of_o intersection_n of_o their_o line_n the_o first_o be_v great_a than_o the_o second_o so_o the_o angle_n a_o be_v great_a than_o e_z because_o if_o we_o take_v the_o point_n b_o and_o d_o as_o far_o distant_a from_o the_o point_n a_o as_o the_o point_n g_o and_o l_o be_v from_o the_o point_n e_o the_o point_n b_o and_o d_o be_v far_o asunder_o than_o the_o point_n g_o and_o l_o from_o whence_o i_o conclude_v that_o if_o eglantine_n el_fw-es be_v continue_v the_o angle_n e_o will_v be_v of_o the_o same_o measure_n and_o less_o than_o the_o angle_n a._n we_o make_v use_v of_o three_o letter_n to_o express_v a_o angle_n and_o the_o 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thirty_o forty_o degree_n when_o the_o arch_n include_v betwixt_o its_o line_n contain_v twenty_o thirty_o forty_o degree_n so_o the_o angle_n be_v great_a which_o contain_v the_o great_a number_n of_o degree_n as_o the_o angle_n bad_a be_v great_a than_o gel_n the_o line_n ca_n divide_v the_o angle_n bad_a in_o the_o middle_n because_o the_o arch_n bc_n cd_o be_v equal_a and_o the_o angle_n bac_n be_v a_o part_n of_o bad_a because_o the_o arch_a bc_n be_v part_n of_o the_o arch_n bd._n 10._o when_o a_o line_n fall_v on_o another_o line_n make_v the_o angle_n on_o each_o side_n thereof_o equal_a those_o angles_n be_v right_a angle_n and_o the_o line_n so_o fall_v be_v a_o perpendicular_a 5._o fig._n 5._o as_o if_o the_o line_n ab_fw-la fall_v on_o cd_o
under_o ab_fw-la and_o ac_fw-la shall_v be_v three_o time_n 8_o or_o 24_o the_o square_a of_o ac_fw-la 3_o be_v 9_o the_o rectangle_n comprehend_v under_o ac_fw-la 3_o and_o cb_n 5_o be_v 3_o time_n 5_o or_o 15._o it_o be_v evident_a that_o 15_o and_o 9_o be_v 24._o use_v at_fw-fr  _fw-fr 43_o c_z 40._o 3_o b_o  _fw-fr 3_o 120._o  _fw-fr 9_o 129_o  _fw-fr  _fw-fr this_o proposition_n serve_v likewise_o to_o demonstrate_v the_o ordinary_a practice_n of_o multiplication_n for_o example_n if_o one_o will_v multiply_v the_o number_n 43_o by_o 3_o have_v separate_v the_o number_n of_o 43_o into_o two_o part_n in_o 40_o and_o 3_o three_o time_n 43_o shall_v be_v as_o much_o as_o three_o time_n 3_o which_o be_v nine_o the_o square_a of_o three_o and_o three_o time_n forty_o which_o be_v 120_o for_o 129_o be_v three_o time_n 43._o those_o which_o be_v young_a beginner_n ought_v not_o to_o be_v discourage_v if_o they_o do_v not_o conceive_v immediate_o these_o proposition_n for_o they_o be_v not_o difficult_a but_o because_o they_o do_v imagine_v they_o contain_v some_o great_a mystery_n proposition_n iu_o theorem_fw-la if_o a_o line_n be_v divide_v into_o two_o part_n the_o square_a of_o the_o whole_a line_n shall_v be_v equal_a to_o the_o two_o square_n make_v of_o its_o part_n and_o to_o two_o rectangle_v comprehend_v under_o the_o same_o part_n let_v the_o line_n ab_fw-la be_v divide_v in_o c_o and_o let_v the_o square_a thereof_o abde_n be_v make_v let_v the_o diagonal_a ebb_n be_v draw_v and_o the_o perpendicular_a cf_n cut_v the_o same_o and_o through_o that_o point_n let_v there_o be_v draw_v gl_n parallel_n to_o ab_fw-la it_o be_v evident_a that_o the_o square_a abde_n be_v equal_a to_o the_o four_o rectangle_v gf_n cl_n cg_n lf_n the_o two_o first_o be_v the_o square_a of_o ac_fw-la and_o of_o cb_n the_o two_o compliment_n be_v comprehend_v under_o ac_fw-la cb._n demonstration_n the_o side_n ae_n ab_fw-la be_v equal_a thence_o the_o angle_n aeb_fw-mi abe_n be_v half_a right_n and_o because_o of_o the_o parallel_n gl_n ab_fw-la the_o angle_n of_o the_o triangle_n of_o the_o square_a ge_z by_o the_o 29_o shall_v be_v equal_a as_o also_o the_o side_n by_o the_o 6_o of_o the_o 1._o thence_o gf_n be_v the_o square_a of_o ac_fw-la in_o like_a manner_n the_o rectangle_n cl_n be_v the_o square_a of_o cb_n the_o rectangle_n gc_n be_v comprehend_v under_o ac_fw-la and_o agnostus_n equal_a to_o bl_v or_o bc_n the_o rectangle_n lf_o be_v comprehend_v under_o ld_n equal_a to_o ac_fw-la and_o under_o fd_n equal_a to_o bc._n coral_n if_o a_o diagonal_a be_v draw_v in_o a_o square_a the_o rectangle_v through_o which_o it_o pass_v be_v square_n use_v a_o 144_o b_o 22_o c_o 12_o this_o proposition_n give_v we_o the_o practical_a way_n of_o find_v or_o extract_v the_o square_a root_n of_o a_o number_n propound_v let_v the_o same_o be_v the_o number_n a_o 144_o represent_v by_o the_o square_a ad_fw-la and_o its_o root_n by_o the_o line_n ab_fw-la moreover_o i_o know_v that_o the_o line_n require_v ab_fw-la must_v have_v two_o figure_n i_o therefore_o imagine_v that_o the_o line_n ab_fw-la be_v divide_v in_o c_o and_o that_o ac_fw-la represent_v the_o first_o figure_n and_o bc_n the_o second_o i_o seek_v the_o root_n of_o the_o first_o figure_n of_o the_o number_n 144_o which_o be_v 100_o and_o i_o find_v that_o it_o be_v 10_o and_o make_v its_o square_a 100_o represent_v by_o the_o square_a gf_n i_o subtract_v the_o same_o from_o 144_o and_o there_o remain_v 44_o for_o the_o rectangle_v gc_a fl_fw-mi and_o the_o square_a cl._n but_o because_o this_o gnomonicall_a figure_n be_v not_o proper_a i_o transport_v the_o rectangle_n fl_fw-mi in_o kg_v and_o so_o i_o have_v the_o rectangle_n kl_n contain_v 44._o i_o know_v also_o almost_o all_o the_o length_n of_o the_o side_n kb_n for_o ac_fw-la be_v 10_o therefore_o kc_n be_v 20_o i_o must_v then_o divide_v 44_o by_o 20_o that_o be_v to_o say_v to_o find_v the_o divisor_n i_o double_v the_o root_n find_v and_o i_o say_v how_o many_o time_n 20_o in_o 44_o i_o find_v it_o 2_o time_n for_o the_o side_n bl_n but_o because_o 20_o be_v not_o the_o whole_a side_n kb_n but_o only_a kc_n this_o 2_o which_o come_v in_o the_o quotient_n be_v to_o be_v add_v to_o the_o divisor_n which_o then_o will_v be_v 22._o so_o i_o find_v the_o same_o 2_o time_n precise_o in_o 44_o the_o square_a root_n then_o shall_v be_v 12._o you_o see_v that_o the_o square_a of_o 144_o be_v equal_a to_o the_o square_n of_o 10_o to_o the_o square_n of_o 2_o which_o be_v 4_o and_o to_o twice_o 20_o which_o be_v two_o rectangle_v comprehend_v under_o 2_o and_o under_o 10._o proposition_n v._o theorem_fw-la if_o a_o right_a line_n be_v cut_v into_o equal_a part_n and_o into_o unequal_a part_n the_o rectangle_n comprehend_v under_o the_o unequal_a part_n together_o with_o the_o square_n which_o be_v of_o the_o middle_a part_n or_o difference_n of_o the_o part_n be_v equal_a to_o the_o square_n of_o half_a the_o line_n if_o the_o line_n ab_fw-la be_v divide_v equal_o in_o c_z and_o unequal_o in_o d_o the_o rectangle_n ah_o comprehend_v under_o the_o segment_n ad_fw-la db_fw-la together_o with_o the_o square_n of_o cd_o shall_v be_v equal_a to_o the_o square_a cf_n that_o be_v of_o half_a of_o ab_fw-la viz._n cb._n make_v a_o end_n of_o the_o figure_n as_o you_o see_v it_o the_o rectangle_v lg_n diego_n shall_v be_v square_n by_o the_o coral_n of_o the_o 4_o i_o prove_v that_o the_o rectangle_n ah_o comprehend_v under_o ad_fw-la and_o dh_n equal_a to_o db_v with_o the_o square_a lg_n be_v equal_a to_o the_o square_a cf._n demonstration_n the_o rectangle_n all_o be_v equal_a to_o the_o rectangle_n df_n the_o one_o and_o the_o other_o be_v comprehend_v under_o half_a the_o line_n ab_fw-la and_o under_o bd_o or_o dh_n equal_a thereto_o add_v to_o both_o the_o rectangle_n ch_z the_o rectangle_n ah_o shall_v be_v equal_a to_o the_o gnomon_n lbg_n again_o to_o both_o add_v the_o square_a lg_n the_o rectangle_n ah_o with_o the_o square_a lg_n shall_v be_v equal_a to_o the_o square_a cf._n arithmetical_o let_v ab_fw-la be_v 10_o ac_fw-la be_v 5_o as_o also_o cb._n let_v cd_o be_v 2_o and_o db_n 3_o the_o rectangle_n comprehend_v under_o ad_fw-la 7_o and_o db_n 3_o that_o be_v to_o say_v 21_o with_o the_o square_n of_o cd_o 2_o which_o be_v 4_o shall_v be_v equal_a to_o the_o square_n of_o cb_n 5_o which_o be_v 25._o use_v this_o proposition_n be_v very_o useful_a in_o the_o three_o book_n we_o make_v use_v thereof_o in_o algebra_n to_o demonstrate_v the_o way_n of_o find_v the_o root_n of_o a_o affect_a square_a or_o equation_n proposition_n vi_o theorem_fw-la if_o one_o add_v a_o line_n to_o another_o which_o be_v divide_v into_o two_o equal_a part_n the_o rectangle_n comprehend_v under_o the_o line_n compound_v of_o both_o and_o under_o the_o line_n add_v together_o with_o the_o square_n of_o half_a the_o divide_a line_n be_v equal_a to_o the_o square_n of_o a_o line_n compound_v of_o half_a the_o divide_a line_n and_o the_o line_n add_v if_o one_o add_v the_o line_n bd_o to_o the_o line_n ab_fw-la which_o be_v equal_o divide_v in_o c_o the_o rectangle_n a_fw-la comprehend_v under_o ad_fw-la and_o under_o dn_n or_o db_n with_o the_o square_n of_o cb_n be_v equal_a to_o the_o square_n of_o cd_o make_v the_o square_n of_o cd_o and_o have_v draw_v the_o diagonal_a fd_n draw_v bg_n parallel_v to_o fc_n which_o cut_v fd_v in_o the_o point_n h_n through_o which_o pass_v hn_n parallel_v to_o ab_fw-la kg_n shall_v be_v the_o square_n of_o bc_n and_o bn_n that_o of_o bd._n demonstration_n the_o rectangle_v ak_v ch_z on_o equal_a base_n ac_fw-la bc_n be_v equal_a by_o the_o 38_o of_o the_o 1_o the_o compliment_n ch_z he_o be_v equal_a by_o the_o 43_o of_o the_o 1_o therefore_o the_o rectangle_v ak_v he_o be_v equal_a add_v to_o both_o the_o rectangle_n cn_fw-la and_o the_o square_a kg_n the_o rectangle_v ak_v cn_fw-la that_o be_v to_o say_v the_o rectangle_n a_fw-la with_o the_o square_a kg_n shall_v be_v equal_a to_o the_o rectangle_v cn_fw-la he_o and_o to_o the_o square_a kg_n that_o be_v to_o say_v to_o the_o square_a ce._n arithmetical_o or_o by_o number_n let_v ab_fw-la be_v 8_o ac_fw-la 4_o cb_n 4_o bd_o 3_o than_o ad_fw-la shall_v be_v 11._o it_o be_v evident_a that_o the_o rectangle_n a_fw-la three_o time_n 11_o that_o be_v to_o say_v 33_o with_o the_o square_n of_o kg_v 16_o which_o together_o be_v 49_o be_v equal_a to_o the_o square_n of_o cd_o 7_o which_o be_v 49_o for_o 7_o time_n 7_o be_v 49._o use_v 6._o fig._n 6._o maurolycus_n measure_v the_o whole_a earth_n by_o one_o single_a
square_n of_o the_o line_n add_v be_v double_a to_o the_o square_n of_o half_a the_o line_n and_o to_o the_o square_n which_o be_v compose_v of_o the_o half_a line_n and_o the_o line_n add_v if_o one_o suppose_v ab_fw-la to_o be_v divide_v in_o the_o middle_n at_o the_o point_n c_o and_o if_o thereto_o be_v add_v the_o line_n bd_o the_o square_n of_o ab_fw-la and_o bd_o shall_v be_v double_a to_o the_o square_n of_o ac_fw-la and_o cd_o add_v together_o draw_v the_o perpendicular_n ce_fw-fr df_n equal_a to_o ac_fw-la then_o draw_v the_o line_n ae_n of_o agnostus_n ebg_n demonstration_n the_o line_n ac_fw-la ce_fw-fr cb_n be_v equal_a and_o the_o angle_n at_o the_o point_n c_o being_n right_n the_o angle_n aec_fw-la ceb_fw-mi cbe_n dbg_n dgb_n shall_v be_v half_a right_n and_o the_o line_n db_n dg_fw-mi and_o of_o fg_v cd_o shall_v be_v equal_a the_o square_a of_o ae_n be_v double_a to_o the_o square_n of_o ac_fw-la the_o square_a of_o eglantine_n be_v double_a to_o the_o square_n of_o of_o or_o cd_o by_o the_o 47_o of_o the_o 1_o now_o the_o square_n of_o agnostus_n be_v equal_a to_o the_o square_n of_o ae_n eglantine_n by_o the_o 47_o of_o the_o 1_o therefore_o the_o square_n of_o agnostus_n be_v double_a to_o the_o square_n of_o ac_fw-la cd_o the_o same_o agnostus_n by_o the_o 47_o of_o the_o 1_o be_v equal_a to_o the_o square_n of_o ad_fw-la bd_o or_o gd_v therefore_o the_o square_n of_o ad_fw-la bd_o be_v double_a the_o square_n of_o ac_fw-la cd_o arithmetical_o let_v ab_fw-la be_v 6_o part_n ac_fw-la 3_o cb_n 3_o bd_o 4_o the_o square_a of_o ad_fw-la 10_o be_v 100_o the_o square_a of_o bd_o 4_o be_v 16_o which_o be_v 116._o the_o square_a of_o ac_fw-la 3_o be_v 9_o the_o square_a of_o cd_o 7_o be_v 49._o now_o 49._o and_o 9_o be_v 58_o the_o half_a of_o 116._o proposition_n xi_o problem_n to_o divide_v a_o line_n so_o that_o the_o rectangle_n comprehend_v under_o the_o whole_a line_n and_o under_o one_o of_o its_o part_n shall_v be_v equal_a to_o the_o square_n of_o the_o other_o part_n it_o be_v propose_v to_o divide_v the_o line_n ab_fw-la so_o that_o the_o rectangle_n comprehend_v under_o the_o whole_a line_n ab_fw-la and_o under_o hb_n be_v equal_a to_o the_o square_n of_o ah_o make_v a_o square_a of_o ab_fw-la by_o the_o 46_o of_o the_o 1_o divide_v ad_fw-la in_o the_o middle_n in_o e_z then_o draw_v ebb_v and_o make_v of_o equal_a to_o ebb_v make_v the_o square_n of_o that_o be_v to_o say_v that_o of_o ah_o be_v equal_a i_o say_v that_o the_o square_a of_o ah_o shall_v be_v equal_a to_o the_o rectangle_n hc_n comprehend_v under_o hb_n and_o the_o line_n bc_n equal_a to_o ab_fw-la demonstration_n the_o line_n ad_fw-la be_v equal_o divide_v in_o e_o and_o there_o be_v add_v thereto_o the_o line_n favorina_n thence_o by_o the_o 6_o the_o rectangle_n dg_n comprehend_v under_o df_n and_o fg_n equal_a to_o of_o with_o the_o square_n of_o ae_n be_v equal_a to_o the_o square_n of_o of_o equal_a to_o ebb_v now_o the_o square_n of_o ebb_n be_v equal_a to_o the_o square_n of_o ab_fw-la ae_n by_o the_o 47_o of_o the_o 1_o therefore_o the_o square_n of_o ab_fw-la ae_n be_v equal_a to_o the_o rectangle_n dg_n and_o to_o the_o square_n of_o ae_n and_o take_v away_o from_o both_o the_o square_a of_o ae_n the_o square_a of_o ab_fw-la which_o be_v ac_fw-la shall_v be_v equal_a to_o the_o rectangle_n dg_n take_v also_o away_o the_o rectangle_n dh_n which_o be_v in_o both_o the_o rectangle_n hc_n shall_v be_v equal_a to_o the_o square_n of_o ag._n use_v this_o proposition_n serve_v to_o cut_v a_o line_n in_o extreme_a and_o mean_a proportion_n as_o shall_v be_v show_v in_o the_o six_o book_n it_o be_v use_v often_o in_o the_o 14_o of_o euclid_n element_n to_o find_v the_o side_n of_o regular_a body_n it_o serve_v for_o the_o 10_o of_o the_o four_o book_n to_o inscribe_v a_o pentagone_fw-mi in_o a_o circle_n as_o also_o a_o pentadecagone_fw-mi you_o shall_v see_v other_o use_n of_o a_o line_n thus_o divide_v in_o the_o 30_o of_o the_o six_o book_n proposition_n xii_o theorem_fw-la in_o a_o obtuse_a angle_a triangle_n the_o square_a of_o the_o side_n opposite_a to_o the_o obtuse_a angle_n be_v equal_a to_o the_o square_n of_o the_o other_o two_o side_n and_o to_o two_o rectangle_v comprehend_v under_o the_o side_n on_o which_o one_o draw_v a_o perpendicular_a and_o under_o the_o line_n which_o be_v between_o the_o triangle_n and_o that_o perpendicular_a let_v the_o angle_n acb_n of_o the_o triangle_n abc_n be_v obtuse_a and_o let_v ad_fw-la be_v draw_v perpendicular_a to_o bc._n the_o square_a of_o the_o side_n ab_fw-la be_v equal_a to_o the_o square_n of_o the_o side_n ac_fw-la cb_n and_o to_o two_o rectangle_v comprehend_v under_o the_o side_n bc_n and_o under_o dc_o demonstration_n the_o square_a of_o ab_fw-la be_v equal_a to_o the_o square_n of_o ad_fw-la db._n by_o the_o 47_o of_o the_o 1_o the_o square_a of_o db_n be_v equal_a to_o the_o square_n of_o dc_o and_o cb_n and_o to_o two_o rectangle_v comprehend_v under_o dc_o cb_n by_o the_o 4_o therefore_o the_o square_n of_o ab_fw-la be_v equal_a to_o the_o square_n of_o ad_fw-la dc_o cb_n and_o to_o two_o rectangle_v comprehend_v under_o dc_o cb_n in_o the_o place_n of_o the_o two_o last_v square_n ad_fw-la dc_o put_v the_o square_n of_o ac_fw-la which_o be_v equal_a to_o they_o by_o the_o 47_o of_o the_o 1_o the_o square_a of_o ab_fw-la shall_v be_v equal_a to_o the_o square_n of_o ac_fw-la and_o cb._n and_o to_o two_o rectangle_v comprehend_v under_o dc_o cb._n use_v this_o proposition_n be_v useful_a to_o measure_v the_o area_n of_o a_o triangle_n it_o be_v three_o side_n be_v know_v for_o example_n if_o the_o side_n ab_fw-la be_v twenty_o foot_n ac_fw-la 13_o bc_n 11_o the_o square_a of_o ab_fw-la will_v be_v four_o hundred_o the_o square_a of_o ac_fw-la one_o hundred_o sixty_o nine_o and_o the_o square_a of_o bc_n one_o hundred_o twenty_o one_o the_o sum_n of_o the_o two_o last_o be_v two_o hundred_o and_o ninety_o which_o be_v subtract_v from_o four_o hundred_o leave_v one_o hundred_o and_o ten_o for_o the_o two_o rectangle_v under_o bc_n cd_o the_o one_o half-fifty_a five_z shall_v be_v one_o of_o those_o rectangle_v which_o divide_v by_o bc_n 11_o we_o shall_v have_v five_o for_o the_o line_n cd_o who_o square_a be_v twenty_o five_o which_o be_v subtract_v from_o the_o square_n of_o ac_fw-la one_o hundred_o sixty_o nine_o there_o remain_v the_o square_a of_o ad_fw-la one_o hundred_o forty_o four_o and_o its_o root_n shall_v be_v the_o side_n ad_fw-la which_o be_v multiply_v by_o 5½_n the_o half_a of_o bc_n you_o have_v the_o area_n of_o the_o triangle_n abc_n contain_v 66_o square_a foot_n proposition_n xiii_o theorem_fw-la in_o any_o triangle_n whatever_o the_o square_a of_o the_o side_n opposite_a to_o a_o acute_a angle_n together_o with_o two_o rectangle_v comprehend_v under_o the_o side_n on_o which_o the_o perpendicular_a fall_v and_o under_o the_o line_n which_o be_v betwixt_o the_o perpendicular_a and_o that_o angle_n be_v equal_a to_o the_o square_n of_o the_o other_o side_n let_v the_o propose_a triangle_n be_v abc_n which_o have_v the_o angle_n c_o acute_a and_o if_o one_o draw_v ad_fw-la perpendicular_a to_o bc_n the_o square_a of_o the_o side_n ab_fw-la which_o be_v opposite_a to_o the_o acute_a angle_n c_o together_z with_o two_o rectangle_v comprehend_v under_o bc_n dc_o shall_v be_v equal_a to_o the_o square_n of_o ac_fw-la bc._n demonstration_n the_o line_n bc_n be_v divide_v in_o d_o whence_o by_o the_o 7_o the_o square_a of_o bc_n dc_o be_v equal_a to_o two_o rectangle_v under_o bc_n dc_o and_o to_o the_o square_n of_o bd_o add_v to_o both_o the_o square_n of_o ad_fw-la the_o square_a of_o bd_o dc_o ad_fw-la shall_v be_v equal_a to_o two_o rectangle_v under_o bc_n dc_o and_o to_o the_o square_n of_o bd_o ad_fw-la in_o the_o place_n of_o the_o square_n of_o cd_o ad_fw-la put_v the_o square_n of_o ac_fw-la which_o be_v equal_a to_o they_o by_o the_o 47_o of_o the_o 1_o and_o instead_o of_o the_o square_n of_o bd_o ab_fw-la substitute_v the_o square_n of_o ab_fw-la which_o be_v equal_a to_o they_o the_o square_n of_o bc_n ac_fw-la shall_v be_v equal_a to_o the_o square_n of_o ab_fw-la and_o to_o two_o rectangle_v comprehend_v under_o bc_n dc_o use_v these_o proposition_n be_v very_o necessary_a in_o trigonometry_n i_o make_v use_v thereof_o in_o the_o eight_o proposition_n of_o the_o three_o book_n to_o prove_v that_o in_o a_o triangle_n there_o be_v the_o same_o reason_n between_o the_o whole_a sine_fw-la and_o the_o sine_fw-la of_o a_o angle_n as_o be_v between_o the_o rectangle_n of_o the_o side_n comprehend_v that_o angle_n and_o
of_o 4_o foot_n and_o thrice_o in_o the_o line_n of_o 6_o the_o first_o to_o the_o second_o shall_v have_v the_o same_o reason_n as_o 2_o to_o 3._o irrational_a reason_n be_v between_o two_o magnitude_n of_o the_o same_o species_n which_o be_v incommensurable_a that_o be_v to_o say_v that_o have_v not_o a_o common_a measure_n as_o the_o reason_n of_o the_o side_n of_o a_o square_a to_o its_o diagonal_a for_o there_o can_v be_v find_v any_o measure_n although_o never_o so_o little_a which_o will_v measure_v both_o precise_o four_o magnitude_n shall_v be_v in_o the_o same_o reason_n or_o shall_v be_v proportional_n when_o the_o reason_n of_o the_o first_o to_o the_o second_o shall_v be_v the_o same_o or_o like_v to_o that_o of_o the_o three_o to_o the_o four_o wherefore_o to_o speak_v proper_o proportion_n be_v a_o similitude_n of_o reason_n but_o one_o find_v it_o difficult_a to_o understand_v in_o what_o consist_v this_o similitude_n of_o reason_n it_o be_v only_o to_o say_v that_o two_o habitude_n or_o relation_n be_v alike_o for_o euclid_n have_v not_o give_v a_o just_a definition_n and_o which_o may_v have_v explain_v its_o nature_n have_v content_v himself_o to_o give_v we_o a_o mark_n by_o which_o we_o may_v know_v if_o magnitude_n have_v the_o same_o reason_n and_o the_o obscurity_n of_o this_o definition_n have_v make_v this_o book_n difficult_a i_o will_v endeavour_v to_o supply_v this_o default_n 5._o euclid_n say_v that_o four_o magnitude_n have_v the_o same_o reason_n when_o have_v take_v the_o equimultiplices_a of_o the_o first_o and_o of_o the_o three_o and_o other_o equimultiplice_n of_o the_o second_o and_o of_o the_o four_o whatever_o combination_n be_v make_v when_o the_o multiplex_n of_o the_o first_o be_v great_a than_o the_o multiplex_n of_o the_o second_o the_o multiplex_n of_o the_o three_o shall_v be_v also_o great_a than_o the_o multiplex_n of_o the_o four_o and_o when_o the_o multiplex_n of_o the_o first_o be_v equal_a or_o less_o than_o the_o multiplex_n of_o the_o second_o the_o multiplex_n of_o the_o three_o be_v equal_a or_o less_o than_o the_o multiplex_n of_o the_o four_o that_o then_o there_o be_v the_o same_o reason_n between_o the_o first_o and_o second_o as_o there_o be_v between_o the_o three_o and_o four_o a_o b_o c_o d_o 2._o 4._o 3._o 6._o e_o f_o g_o h_o 10_o 8._o 15_o 12._o k_o l_o m_o n_o 8._o 8._o 12._o 12._o o_o p_o q_o r_o 6._o 16_o 9_o 24_o as_o if_o there_o be_v propose_v four_o magnitude_n a_o b_o c_o d._n have_v take_v the_o equimultiplexes_a of_o a_o and_o c_o which_o let_v be_v e_o and_o g_o quintuplex_fw-la f_o and_o h_n double_a to_o b_o and_z d._n in_o like_a manner_n take_v k_o and_o m_n quadruple_a to_o a_o and_o c_o l_o and_z n_z double_a to_o b_o and_z d._n take_v again_o o_n and_o q_n triple_a to_o a_o and_o c_o p_o and_z r_z quadruple_a to_o b_o and_z d._n now_o because_o e_z be_v great_a than_o f_o g_z be_v great_a than_o h_n and_o k_o be_v equal_a to_o l_o m_o be_v equal_a to_o n_n in_o fine_a o_o be_v lesser_o than_o p_o q_o be_v lesser_a than_o r._n then_o a_o shall_v have_v the_o same_o reason_n to_o b_o as_o c_z to_z d._n i_o believe_v that_o euclid_n ought_v to_o have_v demonstrate_v this_o proposition_n see_v it_o be_v so_o entangle_v that_o it_o can_v pass_v for_o a_o maxim_n to_o explain_v well_o what_o proportion_n be_v it_o be_v to_o say_v that_o four_o magnitude_n have_v the_o same_o ratio_fw-la although_o one_o may_v say_v in_o general_a that_o to_o that_o end_n the_o first_o must_v be_v alike_o part_n or_o a_o like_a whole_a in_o respect_n of_o the_o second_o as_o be_v the_o three_o compare_v to_o the_o four_o notwithstanding_o because_o this_o definition_n do_v not_o convene_v with_o the_o reason_n of_o equality_n there_o must_v be_v give_v a_o more_o general_a and_o to_o make_v it_o intelligible_a it_o must_v be_v explain_v what_o be_v mean_v by_o a_o like_a aliquot_fw-la part_n like_o aliquot_fw-la part_n be_v those_o which_o be_v as_o many_o time_n in_o their_o whole_a as_o three_z in_o respect_n of_o nine_o two_o in_o respect_n of_o six_o be_v alike_o aliquot_fw-la part_n because_o each_o be_v find_v three_o time_n in_o their_o whole_n the_o first_o quantity_n will_v have_v the_o same_o reason_n to_o the_o second_o as_o the_o three_o have_v to_o the_o four_o if_o the_o first_o contain_v as_o many_o time_n any_o aliquot_fw-la part_n of_o the_o second_o whatever_n as_o the_o 3d._n contain_v alike_o aliquot_fw-la part_n of_o the_o four_o a_o b_o c_o d._n as_o if_o a_o contain_v as_o many_o time_n a_o hundreth_o a_o thousand_o a_o millionth_n part_n of_o b_o as_o c_z contain_v a_o hundreth_o a_o thousand_o a_o millionth_n part_n of_o d_o and_o so_o of_o any_o other_o aliquot_fw-la part_n imaginable_a there_o will_v be_v the_o same_o reason_n of_o a_o to_o b_o as_o of_o c_z to_z d._n to_o make_v this_o definition_n yet_o clear_a i_o will_v in_o the_o first_o place_n prove_v that_o if_o there_o be_v the_o same_o reason_n of_o a_o to_o b_o as_o there_o be_v of_o c_o to_o d_o a_o will_v contain_v as_o many_o time_n the_o aliquot_fw-la part_n of_o b_o as_o c_z do_v of_o d._n and_o i_o will_v afterward_o prove_v that_o if_o a_o contain_v as_o many_o time_n the_o aliquot_fw-la part_n of_o b_o as_o c_z do_v of_o d_o there_o will_v be_v the_o same_o reason_n of_o a_o to_o b_o as_o of_o c_z to_z d._n the_o first_o point_n seem_v evident_a enough_o provide_v one_o do_v conceive_v the_o term_n for_o if_o a_o contain_v one_o hundred_o and_o one_o time_n the_o ten_o part_n of_o b_o and_z c_z only_a one_o hundred_o time_n the_o ten_o part_n of_o d_o the_o magnitude_n a_o compare_v with_o b_o will_v be_v a_o great_a whole_a than_z c_o compare_v with_o d_o so_o that_o it_o can_v not_o be_v compare_v after_o the_o same_o manner_n that_o be_v to_o say_v the_o habitude_n or_o relation_n will_v not_o be_v the_o same_o the_o second_o point_n seem_v more_o difficult_a to_o wit_n whether_o if_o this_o propriety_n be_v so_o find_v the_o reason_n shall_v be_v the_o same_o that_o be_v to_o say_v if_o ab_fw-la contain_v as_o many_o time_n any_o aliquot_fw-la part_n whatever_a of_o cd_o as_o e_z contain_v like_o aliquot_fw-la part_n of_o f_o there_o shall_v be_v the_o same_o reason_n of_o ab_fw-la to_o cd_o as_o of_o e_z to_z f._n for_o i_o will_v prove_v that_o if_o there_o be_v not_o the_o same_o reason_n a_o will_v contain_v more_o time_n any_o aliquot_fw-la part_n of_o b_o than_z c_o contain_v alike_o aliquot_fw-la part_n of_o d_o which_o will_v be_v contrary_a to_o what_o we_o have_v suppose_v demonstration_n see_v there_o be_v the_o same_o reason_n of_o agnostus_n to_o cd_o as_o of_o e_o to_o f_o agnostus_n will_v contain_v as_o many_o time_n kd_v a_o aliquot_fw-la part_n of_o cd_o as_o e_z will_v contain_v a_o like_a aliquot_fw-la part_n of_o f._n now_o ab_fw-la contain_v kd_v once_o more_o than_o agnostus_n thence_o ab_fw-la will_v contain_v once_o more_o kd_v a_o aliquot_fw-la part_n of_o cd_o than_o e_z do_v contain_v a_o like_o aliquot_fw-la part_n of_o f_o which_o will_v be_v contrary_a to_o the_o supposition_n 6._o there_o will_v be_v a_o great_a reason_n of_o the_o first_o quantity_n to_o the_o second_o than_o of_o the_o three_o to_o the_o four_o if_o the_o first_o contain_v more_o time_n any_o aliquot_fw-la part_n of_o the_o second_o than_o the_o three_o do_v contain_v a_o like_o aliquot_fw-la part_n of_o the_o four_o as_o 101_o have_v a_o great_a reason_n to_o 10_o then_o 200_o to_o 20_o because_o that_o 101_o contain_v one_o hundred_o and_o one_o time_n the_o ten_o part_n of_o 10_o and_o 200_o contain_v only_o one_o hundred_o time_n the_o ten_o part_n of_o 20_o which_o be_v 2._o 7._o the_o magnitude_n or_o quantity_n which_o be_v in_o the_o same_o reason_n be_v call_v proportional_n 8._o a_o proportion_n or_o analogy_n be_v a_o similitude_n of_o reason_n or_o habitude_n 9_o a_o proportion_n ought_v to_o have_v at_o least_o three_o term_n for_o to_o the_o end_n there_o be_v similitude_n of_o reason_n there_o must_v be_v two_o reason_n now_o each_o reason_n have_v two_o term_n the_o antecedent_n and_o the_o consequent_a it_o seem_v there_o aught_o to_o be_v four_o as_o when_o we_o say_v that_o there_o be_v the_o same_o reason_n of_o a_o to_o b_o as_o of_o c_z to_z d_o but_o because_o the_o consequent_a of_o the_o first_o reason_n may_v be_v take_v for_o antecedent_n in_o the_o second_o three_o term_n may_v suffice_v as_o when_o i_o say_v that_o there_o be_v the_o same_o reason_n of_o a_o to_o b_o as_o of_o b_o to_z c._n 10._o magnitude_n be_v in_o continue_a proportion_n
when_o the_o term_n between_o they_o be_v take_v twice_o that_o be_v to_o say_v as_o antecedent_n and_o as_o consequent_a as_o if_o there_o be_v the_o same_o reason_n of_o a_o to_o b_o as_o of_o b_o to_z c_z and_z of_o c_z to_z d._n 11._o then_o a_o to_o c_o shall_v be_v in_o duplicate_v ratio_fw-la of_o a_o to_z b_o and_o the_o ratio_fw-la of_o a_o to_z d_o shall_v be_v in_o triplicate_v ratio_fw-la to_o that_o of_o a_o to_o b._n it_o be_v to_o be_v take_v notice_n that_o there_o be_v a_o great_a deal_n of_o difference_n between_o double_a ratio_fw-la and_o duplicate_v ratio_fw-la we_o say_v that_o the_o ratio_fw-la of_o four_o to_o two_o be_v double_a that_o be_v to_o say_v four_z be_v the_o double_a of_o two_o whence_o it_o follow_v that_o the_o number_n two_o be_v that_o which_o give_v the_o name_n to_o this_o ratio_fw-la or_o rather_o to_o the_o antecedent_n of_o this_o ratio_fw-la so_o we_o we_o say_v double_a triple_a quadruple_a quintuple_a which_o be_v denomination_n take_v from_o those_o number_n duo_fw-la tres_fw-la quatuor_fw-la quinque_fw-la compare_v with_o unity_n for_o we_o better_o conceive_v a_o reason_n when_o its_o term_n be_v small_a but_o as_o i_o have_v already_o take_v notice_n those_o denomination_n fall_v rather_o on_o the_o antecedent_n than_o on_o the_o reason_n itself_o we_o call_v that_o double_a triple_a reason_n or_o ratio_fw-la when_o the_o antecedent_n be_v double_a or_o triple_a to_o the_o consequent_a but_o when_o we_o say_v the_o reason_n be_v duplicate_v we_o mean_v a_o reason_n compound_v of_o two_o like_a reason_n as_o if_o there_o be_v the_o same_o reason_n of_o two_o to_o four_o as_o of_o four_o to_o eight_o the_o reason_n of_o two_o and_o eight_o be_v compound_v of_o the_o reason_n of_o two_o and_o four_o and_o of_o that_o of_o four_o and_o eight_o which_o be_v alike_o and_o as_o equal_v the_o reason_n or_o ratio_fw-la of_o two_o to_o eight_o shall_v be_v duplicate_v by_o each_o three_o to_o twenty_o seven_o be_v a_o duplicate_v reason_n of_o that_o of_o three_o to_o nine_o the_o reason_n of_o two_o to_o four_o be_v call_v subduple_n that_o be_v to_o say_v two_o be_v the_o half_a of_o four_o but_o the_o reason_n of_o two_o to_o eight_o be_v duple_n of_o the_o sub-duple_a that_o be_v to_o say_v that_o two_o be_v the_o half_a of_o the_o half_a of_o eight_o as_o three_o be_v the_o three_o of_o the_o three_o of_o twenty_o seven_o where_o you_o see_v there_o be_v take_v twice_o the_o denominator_fw-la ½_n and_o ⅓_n in_o like_a manner_n eight_o to_o two_o be_v a_o duplicate_v reason_n of_o eight_o to_o four_o because_o eight_o be_v double_a to_o four_o but_o eight_o be_v the_o double_a of_o the_o double_a of_o two_o if_o there_o be_v four_o term_n in_o the_o same_o continue_a reason_n that_o of_o the_o first_o and_o last_o be_v triple_a to_o that_o of_o the_o first_o and_o second_o as_o if_o one_o put_v these_o four_o number_n two_o four_o eight_o sixteen_o the_o reason_n of_o two_o to_o sixteen_o be_v triple_a of_o two_o to_o four_o for_o two_o be_v the_o half_a of_o the_o half_a of_o the_o half_a of_o sixteen_o as_o the_o reason_n of_o sixteen_o to_o two_o be_v triple_a of_o sixteen_o to_o eight_o for_o sixteen_o be_v the_o double_a of_o eight_o and_o it_o be_v the_o double_a of_o the_o double_a of_o the_o double_a of_o two_o 12._o magnitude_n be_v homologous_a the_o antecedent_n to_o the_o antecedent_n and_o the_o consequent_n to_o the_o consequent_n as_o if_o there_o be_v the_o same_o reason_n of_o a_o to_o b_o as_o of_o c_z to_z d_o a_o and_z c_o be_v homologous_a or_o magnitude_n of_o a_o like_o ratio_fw-la the_o follow_a definition_n be_v way_n of_o argue_v by_o proportion_n and_o it_o be_v principal_o to_o demonstrate_v the_o same_o that_o this_o book_n be_v compose_v 13._o alternate_a reason_n or_o by_o permutation_n or_o exchange_n be_v when_o we_o compare_v the_o antecedent_n one_o with_o the_o other_o as_o also_o the_o consequent_n for_o example_n if_o because_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o c_z to_z d_o i_o conclude_v there_o be_v the_o same_o reason_n of_o a_o to_o c_o as_o of_o b_o to_z d_o this_o way_n of_o reason_v can_v take_v place_n but_o when_o the_o four_o term_n be_v of_o the_o same_o specie_fw-la that_o be_v to_o say_v either_o all_o four_o line_n or_o superficies_n or_o solid_n proposition_n 16._o 14._o converse_v or_o inverse_v reason_n be_v a_o comparison_n of_o the_o consequent_n to_o the_o antecedent_n as_o if_o because_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o c_z to_z d_o i_o conclude_v there_o be_v the_o same_o reason_n of_o b_o to_o a_o as_o there_o be_v of_o d_o to_z c._n proposition_n 16._o 15._o composition_n of_o reason_n be_v a_o comparison_n of_o the_o antecedent_n and_o consequent_a take_v together_o to_o the_o consequent_a alone_o as_o if_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o c_z to_z d_o i_o conclude_v also_o that_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_o b_o as_o of_o cd_o to_o d._n prop._n 18._o 16._o division_n of_o reason_n be_v a_o comparison_n of_o the_o excess_n of_o the_o antecedent_n above_o the_o consequent_a to_o the_o same_o consequent_a as_o if_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_z b_o as_o of_o cd_o to_o d_o i_o conclude_v that_o there_o be_v the_o same_o reason_n of_o a_o to_o b_o as_o of_o cd_o prop._n 17._o 17._o conversion_n of_o reason_n be_v the_o comparison_n of_o the_o antecedent_n to_o the_o difference_n of_o the_o term_n as_o if_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_z b_o as_o of_o cd_o to_o d_o i_o conclude_v that_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_o a_o as_o of_o cd_o to_o c._n proposition_n 18._o 18._o proportion_n of_o equality_n be_v a_o comparison_n of_o the_o extreme_a quantity_n in_o leave_v out_o those_o in_o the_o middle_n a_o b_o c_o d_o e_o f_o g_o h._n as_o if_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o e_z to_o f_o and_o of_o b_o to_z c_z as_o of_o f_o to_o g_z and_z of_o c_z to_z d_o as_o of_o g_z to_z h._n i_o draw_v this_o consequence_n that_o there_o be_v therefore_o the_o same_o reason_n of_o a_o to_z d_o as_o of_o e_z to_z h._n 19_o proportion_n of_o equality_n well_o rank_v be_v that_o in_o which_o one_o compare_v the_o term_n in_o the_o same_o manner_n of_o order_n as_o in_o the_o precede_a example_n prop._n 22._o 20._o proportion_n of_o equality_n ill_o rank_v be_v that_o in_o which_o one_o compare_v the_o term_n with_o a_o different_a order_n as_o if_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o g_z to_o h_n and_z of_o b_o to_z c_z as_o of_o f_o to_o g_z and_z of_o c_z to_z d_o as_o of_o e_z to_z f._n i_o draw_v this_o conclusion_n that_o there_o be_v the_o same_o reason_n of_o a_o to_z d_o as_o of_o e_z to_z h._n prop._n 28._o here_o be_v all_o the_o way_n of_o argue_v by_o proportion_n there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o c_z to_z d_o therefore_o by_o alternate_a reason_n there_o be_v the_o same_o reason_n of_o a_o to_o c_o as_o of_o b_o to_z d_o and_o by_o inversed_a reason_n there_o be_v the_o same_o reason_n of_o b_o to_o a_o as_o of_o d_o to_z c_z and_o by_o composition_n there_o be_v the_o same_o reason_n of_o ab_fw-la to_o b_o as_o of_o cd_o to_o d._n by_o division_n of_o reason_n if_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_o b_o as_o of_o cd_o to_o d_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o c_z to_z d_o and_o by_o conversion_n there_o be_v the_o same_o reason_n of_o ab_fw-la to_o a_o as_o of_o cd_o to_o c._n by_o reason_n of_o equality_n well_o rank_v if_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o c_z to_z d_o and_o also_o the_o same_o reason_n of_o b_o to_z e_z as_o of_o d_o to_o f_o there_o will_v be_v the_o same_o reason_n of_o a_o to_z e_z as_o of_o c_z to_z f._n by_o reason_n of_o equality_n ill_o rank_v if_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o d_o to_o f_o and_o also_o the_o same_o reason_n of_o b_o to_z e_z as_o of_o c_z to_z d_o there_o will_v be_v the_o same_o reason_n of_o a_o to_z e_z as_o of_o c_z to_z f._n this_o book_n contain_v twenty_o five_o proposition_n of_o euclid_n to_o which_o there_o have_v be_v add_v ten_o which_o be_v receive_v the_o first_o six_o of_o this_o book_n be_v useful_a only_o to_o prove_v the_o follow_a proposition_n by_o the_o method_n
superficies_n 18._o a_o cone_n be_v a_o figure_n make_v when_o one_o side_n of_o a_o right_a angle_a triangle_n viz._n one_o of_o those_o that_o contain_v the_o right_a angle_n remain_v fix_v the_o triangle_n be_v turn_v round_o about_o till_o it_o return_v to_o the_o place_n from_o whence_o it_o first_o move_v and_o if_o the_o fix_a right_a line_n be_v equal_a to_o the_o other_o which_o contain_v the_o right_a angle_n than_o the_o cone_n be_v a_o rectangled_a cone_n but_o if_o it_o be_v less_o it_o be_v a_o obtuse_a angle_a cone_n if_o great_a a_o acute_a angle_a cone_n 19_o the_o axis_n of_o a_o cone_n be_v that_o fix_a line_n about_o which_o the_o triangle_n be_v move_v 20._o a_o cylinder_n be_v a_o figure_n make_v by_o the_o move_a round_n of_o a_o right_a angle_a parallelogram_n one_o of_o the_o side_n thereof_o namely_o which_o contain_v the_o right_a angle_n abide_v fix_v till_o the_o parallelogram_n be_v turn_v about_o to_o the_o same_o place_n whence_o it_o begin_v to_o move_v 21._o like_a cones_n and_o cylinder_n be_v those_o who_o axe_n and_o diameter_n of_o their_o base_n be_v proportional_a cones_n be_v right_a when_o the_o axis_n be_v perpendicular_a to_o the_o plain_a of_o the_o base_a and_o they_o be_v say_v to_o be_v scalene_n when_o the_o axis_n be_v incline_v to_o the_o base_a and_o the_o diameter_n of_o their_o base_n be_v in_o the_o same_o ratio_fw-la we_o add_v that_o incline_v cones_n to_o be_v like_o their_o axe_n must_v have_v the_o same_o inclination_n to_o the_o plane_n of_o their_o base_n proposition_n i._o theorem_fw-la i._n plate_n vii_o prop._n i._n a_o straight_a line_n can_v have_v one_o of_o its_o part_n in_o a_o plane_n and_o the_o other_o without_o it_o if_o the_o line_n ab_fw-la be_v in_o the_o plane_n ad_fw-la it_o be_v continue_v shall_v not_o go_v without_o but_o all_o its_o part_n shall_v be_v in_o the_o same_o plane_n for_o if_o it_o can_v be_v that_o bc_n be_v a_o part_n of_o ab_fw-la continue_v draw_v in_o the_o plane_n cd_o the_o line_n bd_o perpendicular_a to_o ab_fw-la draw_v also_o in_o the_o same_o plane_n be_v perpendicular_a to_o bd._n demonstration_n the_o angle_n abdella_n bde_n be_v both_o right_a angle_n thence_o by_o the_o 14_o of_o the_o first_o ab_fw-la be_v do_v make_v but_o one_o line_n and_o consequent_o bc_n be_v not_o a_o part_n of_o the_o line_n ab_fw-la continue_v otherwise_o two_o strait_a line_n cb_n ebb_n will_v have_v the_o same_o part_n ab_fw-la that_o be_v ab_fw-la will_v be_v part_n of_o both_o which_o we_o have_v reject_v as_o false_a in_o the_o thirteen_o maxim_n of_o the_o first_o book_n use_v we_o establish_v on_o this_o proposition_n a_o principle_n in_o gnomonic_n to_o prove_v that_o the_o shadow_n of_o the_o stile_n fall_v not_o without_o the_o plane_n of_o a_o great_a circle_n in_o which_o the_o sun_n be_v see_v that_o the_o end_n or_o top_n of_o the_o stile_n be_v take_v for_o the_o centre_n of_o the_o heaven_n and_o consequent_o for_o the_o centre_n of_o all_o the_o great_a circle_n the_o shadow_n be_v always_o in_o a_o straight_a line_n with_o the_o ray_n draw_v from_o the_o sun_n to_o the_o opaque_fw-fr body_n this_o ray_n be_v in_o any_o great_a circle_n the_o shadow_n must_v also_o be_v therein_o proposition_n ii_o theorem_fw-la line_n which_o cut_v one_o another_o be_v in_o the_o same_o plane_n as_o well_o as_o all_o the_o part_n of_o a_o triangle_n if_o the_o two_o line_n be_v cd_o cut_v one_o another_o in_o the_o point_n a_o and_o if_o there_o be_v make_v a_o triangle_n by_o draw_v the_o base_a bc_n i_o say_v that_o all_o the_o part_n of_o the_o triangle_n abc_n be_v in_o the_o same_o plane_n and_o that_o the_o line_n be_v cd_o be_v likewise_o therein_o demonstration_n it_o can_v be_v say_v that_o any_o one_o part_n of_o the_o triangle_n abc_n be_v in_o a_o plane_n and_o that_o the_o other_o part_n be_v without_o without_o say_v that_o one_o part_n of_o a_o line_n be_v in_o one_o plane_n and_o that_o the_o other_o part_n of_o the_o same_o line_n be_v not_o therein_o which_o be_v contrary_a to_o the_o first_o proposition_n and_o see_v that_o the_o side_n of_o the_o triangle_n be_v in_o the_o same_o plane_n wherein_o the_o triangle_n be_v the_o line_n be_v cd_o shall_v be_v in_o the_o same_o plane_n use_v this_o proposition_n do_v sufficient_o determine_v a_o plane_n by_o two_o straight_a line_n mutual_o intersect_v each_o other_o or_o by_o a_o triangle_n i_o have_v make_v use_n thereof_o in_o optic_n to_o prove_v that_o the_o objective_a parallel_n line_n which_o fall_v on_o the_o tablet_n aught_o to_o be_v represent_v by_o line_n which_o concur_v in_o a_o point_n proposition_n iii_o theorem_fw-la the_o common_a section_n of_o two_o place_n be_v a_o straight_a line_n if_o two_o planes_n ab_fw-la cd_o cut_v one_o another_o their_o common_a section_n of_o shall_v be_v a_o straight_a line_n for_o if_o it_o be_v not_o take_v two_o point_n common_a to_o both_o plane_n which_o let_v be_v e_o and_o f_o and_o draw_v a_o strait_a line_n from_o the_o point_n e_o to_o the_o point_n f_o in_o the_o plane_n ab_fw-la which_o let_v be_v ehf_n draw_v also_o in_o the_o plane_n cd_o a_o straight_a line_n from_o e_o to_o f_o if_o it_o be_v not_o the_o same_o with_o the_o former_a let_v it_o be_v egf_n demonstration_n those_o line_n draw_v in_o the_o two_o plane_n be_v two_o different_a line_n and_o they_o comprehend_v a_o space_n which_o be_v contrary_a to_o the_o twelve_o maxim_n thence_o they_o be_v but_o one_o line_n which_o be_v in_o both_o plane_n shall_v be_v their_o common_a section_n use_v this_o proposition_n be_v fundamental_a we_o do_v suppose_v it_o in_o gnomonic_n when_o we_o represent_v in_o a_o dial_n the_o circle_n of_o the_o hour_n mark_v only_o the_o common_a section_n of_o their_o plane_n and_o that_o of_o the_o wall_n proposition_n iu_o theorem_fw-la if_o a_o line_n be_v perpendicular_a to_o two_o other_o line_n which_o cut_v one_o another_o it_o shall_v be_v also_o perpendicular_a to_o the_o plane_n of_o those_o line_n if_o the_o line_n ab_fw-la be_v perpendicular_a to_o the_o line_n cd_o of_o which_o cut_v one_o another_o in_o the_o point_n b_o in_o such_o manner_n that_o the_o angel_n abc_n abdella_n abe_n abf_n be_v right_a which_o a_o flat_a figure_n can_v represent_v it_o shall_v be_v perpendicular_a to_o the_o plane_n cd_o of_o that_o be_v to_o say_v that_o it_o shall_v be_v perpendicular_a to_o all_o the_o line_n that_o be_v draw_v in_o that_o plane_n through_o the_o point_n b_o as_o to_o the_o line_n gbh_n let_v equal_a line_n be_v cut_v bc_n bd_o be_v bf_n and_o let_v be_v draw_v the_o line_n aec_fw-la df_n ac_fw-la ad_fw-la ae_n of_o agnostus_n and_z ah_o demonstration_n the_o four_o triangle_n abc_n abdella_n abe_n abf_n have_v their_o angle_n right_o in_o the_o point_n b_o and_o the_o side_n bc_n bd_o be_v bf_n equal_a with_o the_o side_n ab_fw-la common_a to_o they_o all_o therefore_o their_o base_n ac_fw-la ad_fw-la ae_n of_o be_v equal_a by_o the_o four_o of_o the_o one_a 2._o the_o triangle_n ebc_n dbf_n shall_v be_v equal_a in_o every_o respect_n have_v the_o side_n bc_n bd_o be_v bf_n equal_a and_o the_o angel_n cbe_n dbf_n opposite_a at_o the_o vertex_fw-la be_v equal_a so_o than_o the_o angle_n be_v bdf_n bec_n bfd_n shall_v be_v equal_a by_o the_o four_o of_o the_o first_o and_o their_o base_n aec_fw-la df_n equal_a 3._o the_o triangle_n gbc_n dbh_n have_v their_o opposite_a angle_n cbg_n dbh_n equal_a as_o also_o the_o angle_n bdh_fw-mi bcg_n and_o the_o side_n bc_n bd_o they_o shall_v then_o have_v by_o the_o 26_o of_o the_o one_a their_o side_n bg_n bh_n cg_n dh_n equal_a 4._o the_o triangle_n ace_n afd_v have_v their_o side_n ac_fw-la ad_fw-la ae_n of_o equal_a and_o the_o base_n aec_fw-la df_n equal_a they_o shall_v have_v by_o the_o 8_o of_o the_o one_a the_o angles_n adf_n ace_n equal_a 5._o the_o triangle_n acg_n adh_n have_v the_o side_n ac_fw-la ad_fw-la cg_n dh_n equal_a with_o the_o angles_n adh_n agc_n thence_o they_o shall_v have_v their_o base_n agnostus_n ah_o equal_a last_o the_o triangle_n abh_n abg_n have_v all_o their_o side_n equal_a thence_o by_o the_o 27_o of_o the_o one_a the_o angles_n abg_n abh_n shall_v be_v equal_a and_o the_o line_n ab_fw-la perpendicular_a to_o gh_v so_o then_o the_o line_n ab_fw-la shall_v be_v perpendicular_a to_o any_o line_n which_o may_v be_v draw_v through_o the_o point_n b_o in_o the_o plane_n of_o the_o line_n cd_o of_o which_o i_o call_v perpendicular_a to_o the_o plane_n use_v this_o proposition_n come_v often_o in_o use_n in_o the_o first_o book_n of_o theodosius_n for_o example_n to_o demonstrate_v that_o the_o axis_n of_o the_o world_n be_v