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equal_a to_o lt_v therefore_o it_o shall_v fall_v on_o the_o point_n i_o which_o be_v the_o point_n be_v those_o circle_n cut_v each_o other_o but_o if_o you_o deny_v it_o and_o say_v it_o shall_v fall_v on_o some_o other_o point_n as_o at_o o_o than_o i_o say_v the_o line_n ho_o that_o be_v to_o say_v vt_fw-la will_v be_v great_a than_o he_o and_o the_o line_n go_v that_o be_v to_o say_v lt_v will_v be_v less_o than_o give_v which_o be_v contrary_a to_o the_o hypo._n from_o whence_o i_o conclude_v the_o triangle_n do_v agree_v in_o all_o their_o part_n and_o that_o therefore_o the_o angle_n gih_o be_v equal_a to_o the_o angle_n ltv_n this_o proposition_n be_v necessary_a for_o the_o establish_n the_o next_o follow_v moreover_o when_o we_o can_v measure_v a_o angle_n which_o be_v make_v in_o a_o solid_a body_n by_o reason_n we_o can_v place_v our_o instrument_n we_o take_v the_o three_o side_n of_o a_o triangle_n and_o make_v another_o on_o paper_n who_o angle_n we_o easy_o measure_v this_o be_v very_o useful_a in_o dial_v and_o in_o cut_v of_o stone_n and_o bevel_v of_o timber_n proposition_n ix_o problem_n to_o bisect_n or_o divide_v into_o two_o equal_a part_n a_o right-lined_n angle_v give_v sat._n take_v as_o equal_a to_o at_o and_o draw_v the_o line_n st_n upon_o which_o make_v a_o equilateral_a triangle_n sut_o draw_v the_o right_a line_n valerio_n it_o shall_v bisect_v the_o angle_n demonstration_n as_o be_v equal_a to_o at_o and_o the_o side_n ave_fw-la be_v common_a and_o the_o base_a sv_n equal_a to_o vt_fw-la therefore_o the_o angle_n sav_n be_v equal_a to_o tav_n which_o be_v to_o be_v do_v use_v this_o proposition_n be_v very_o useful_a to_o divide_v a_o quadrant_n into_o degree_n it_o be_v the_o same_o thing_n to_o divide_v a_o angle_n or_o to_o divide_v a_o arch_n into_o two_o equal_a part_n for_o the_o line_n ave_fw-la divide_v as_o well_o the_o arch_a st_n at_o the_o angle_n sit_v for_o if_o you_o put_v the_o semi-diameter_n on_o a_o quadrant_n you_o cut_v off_o a_o arch_n of_o 60_o degree_n and_o divide_v that_o arch_n into_o two_o equal_a part_n you_o have_v 30_o degree_n which_o be_v again_o divide_v into_o two_o equal_a part_n you_o have_v 15_o degree_n it_o be_v true_a to_o make_v a_o end_n of_o this_o division_n you_o must_v divide_v a_o arch_n in_o three_o which_o be_v not_o yet_o know_v to_o geomatrician_n our_o sea_n compass_n be_v divide_v into_o 32_o point_n by_o this_o proposition_n proposition_n x._o problem_n to_o divide_v a_o line_n give_v into_o two_o equal_a part_n let_v the_o line_n ab_fw-la be_v propose_v to_o be_v divide_v into_o two_o equal_a part_n erect_v a_o equilateral_a triangle_n abc_n
let_v the_o line_n ab_fw-la represent_v the_o plain_a and_o from_o the_o centre_n of_o the_o earth_n c_o let_v the_o line_n ca_n be_v draw_v perpendicular_a to_o the_o line_n ab_fw-la i_o say_v that_o a_o boul_a be_v place_v on_o the_o point_n b_o ought_v not_o to_o stand_v on_o that_o point_n because_o no_o heavy_a body_n will_v stand_v still_o while_o it_o may_v descend_v now_o the_o boul_a b_o going_z towards_o a_o be_v always_o descend_v and_o come_v near_a and_o near_o the_o centre_n of_o the_o earth_n c_o because_o in_o the_o triangle_n cab_n the_o perpendicular_a ca_n be_v short_a than_o bc._n we_o also_o prove_v in_o like_a manner_n that_o a_o liquid_a body_n ought_v to_o descend_v from_o b_o to_o a_o and_o that_o the_o surface_n ought_v to_o be_v â—ound_v proposition_n xx._n theorem_fw-la the_o two_o side_n of_o a_o triangle_n take_v together_o be_v great_a than_o the_o three_o i_o say_v that_o the_o two_o side_n tl_n lv_o of_o the_o triangle_n tlv_n be_v great_a than_o the_o side_n tu._n some_o prove_v this_o proposition_n by_o the_o definition_n of_o a_o straight_a line_n which_o be_v the_o short_a which_o can_v be_v draw_v between_o two_o point_n therefore_o the_o line_n tu_fw-la be_v less_o than_o the_o two_o line_n tl_n lv._o but_o we_o may_v demonstrate_v the_o same_o another_o way_n continue_v the_o side_n lv_o to_o r_n and_o let_v lr_n lt_n be_v equal_a then_o draw_v the_o line_n rt_n demonstration_n the_o side_n lt_v lr_n of_o the_o triangle_n ltr_n be_v equal_a therefore_o the_o angle_n r_n and_o rtl_n equal_a by_o the_o 5â—h_n but_o the_o angle_n rtv_n be_v great_a than_o the_o angle_n rtl_n therefore_o the_o angle_n rtv_n be_v great_a than_o the_o angle_n r_n and_o by_o the_o 17_o in_o the_o triangle_n rtv_n the_o side_n rv_n that_o be_v to_o say_v the_o sum_n of_o the_o side_n lt_n lv_o be_v great_a than_o the_o side_n tu._n proposition_n xxi_o theorem_fw-la if_o on_o the_o same_o base_a you_o draw_v a_o lesser_a triangle_n in_o a_o great_a the_o side_n of_o the_o lesser_a shall_v be_v short_a than_o the_o great_a but_o contain_v a_o great_a angle_n let_v the_o less_o triangle_n adb_n be_v draw_v within_o the_o great_a acb_n on_o the_o same_o base_a ab_fw-la i_o say_v in_o the_o first_o place_n that_o the_o side_n ac_fw-la bc_n be_v great_a than_o the_o side_n ad_fw-la bd._n continue_v the_o side_n ad_fw-la unto_o e._n demonstration_n in_o the_o triangle_n ace_n the_o side_n ac_fw-la ce_fw-fr take_v together_o be_v great_a than_o the_o side_n ae_n by_o the_o 20_o add_v thereto_o the_o side_n ebb_n the_o side_n ac_fw-la ceb_fw-mi shall_v be_v great_a than_o the_o side_n ae_n ebb_n likewise_o in_o the_o triangle_n dbe_n the_o two_o side_n be_v ed_z take_v together_o be_v great_a than_o bd_o and_o add_v thereto_o ad_fw-la the_o side_n ae_n ebb_n shall_v be_v great_a than_o ad_fw-la bd._n moreover_o i_o say_v that_o the_o angle_n adb_n be_v great_a than_o the_o angle_n acb_n for_o the_o angle_n adb_n be_v exterior_a in_o respect_n of_o the_o triangle_n deb_n it_o be_v therefore_o great_a than_o the_o interiour_n deb_n by_o the_o 16_o likewise_o the_o angle_n deb_n be_v exterior_a in_o respect_n of_o the_o triangle_n ace_n be_v great_a than_o the_o angle_n ace_n therefore_o the_o angle_n adb_n be_v great_a than_o the_o angle_n acb_n use_v xxi_o prop._n xxi_o we_o demonstrate_v in_o optic_n by_o this_o proposition_n that_o if_o from_o the_o point_n c_o one_o shall_v see_v the_o base_a ab_fw-la it_o will_v seem_v less_o than_o if_o one_o shall_v see_v the_o same_o from_o the_o point_n d_o according_a to_o this_o principle_n that_o quantity_n see_v under_o a_o great_a angle_n appear_v great_a for_o which_o reason_n vitruvius_n will_v that_o the_o top_n of_o very_a high_a pillar_n shall_v be_v make_v but_o little_o taper_v because_o that_o their_o top_n be_v at_o a_o good_a distance_n from_o the_o eye_n will_v of_o themselves_o appear_v very_o much_o diminish_v proposition_n xxii_o problem_n to_o make_v a_o triangle_n have_v its_o side_n equal_a to_o three_o right_a line_n give_v provide_v that_o two_o of_o they_o be_v great_a than_o the_o three_o let_v it_o be_v propose_v to_o make_v a_o triangle_n have_v its_o three_o side_n equal_a to_o the_o three_o give_v line_n ab_fw-la d_o e_o take_v with_o your_o compass_n the_o line_n d_o and_o put_v one_o foot_n thereof_o in_o the_o point_n b_o make_v a_o arch_n then_o take_v in_o your_o compass_n the_o line_n e_o and_o put_v one_o foot_n in_o the_o point_n a_o cross_a with_o the_o other_o foot_n the_o former_a arch_n in_o c_o draw_v the_o line_n ac_fw-la bc._n i_o say_v that_o the_o triangle_n abc_n be_v what_o you_o desire_v demonstration_n the_o side_n ac_fw-la be_v equal_a to_o the_o line_n e_o since_o it_o be_v the_o radius_fw-la of_o a_o arch_n draw_v on_o the_o centre_n a_o equal_a in_o length_n to_o the_o line_n e_o likewise_z the_o side_n bc_n be_v equal_a to_o the_o line_n d_o therefore_o the_o three_o sides_n ac_fw-la bc_n ad_fw-la be_v equal_a to_o the_o line_n e_o d_o ab_fw-la use_v we_o make_v use_v of_o this_o proposition_n to_o make_v a_o figure_n equal_a or_o like_v unto_o another_o for_o have_v divide_v the_o figure_n into_o triangle_n and_o make_v other_o triangle_n have_v their_o side_n equal_a to_o the_o side_n of_o the_o triangle_n in_o the_o propose_a figure_n we_o shall_v have_v a_o figure_n like_a unto_o the_o same_o in_o all_o respect_n but_o if_o we_o desire_v it_o shall_v be_v only_o like_o thereunto_o but_o lesser_a for_o example_n if_o we_o will_v have_v the_o form_n of_o any_o plain_a or_o piece_n of_o ground_n on_o paper_n have_v divide_v the_o same_o into_o triangle_n and_o measure_v all_o their_o side_n we_o make_v triangle_n like_a unto_o those_o of_o the_o plain_a by_o the_o help_n of_o a_o scale_n of_o equal_a part_n from_o which_o we_o take_v the_o number_n of_o part_n which_o their_o side_n contain_v whether_o foot_n rod_n or_o any_o other_o measure_n and_o apply_v they_o as_o be_v here_o teach_v proposition_n xxiii_o problem_n to_o make_v a_o angle_n equal_a to_o a_o angle_n give_v in_o any_o point_n of_o a_o line_n let_v it_o be_v propose_v to_o make_v a_o angle_n equal_a to_o edf_n at_o the_o point_n a_o of_o the_o line_n ab_fw-la at_o the_o point_n a_o and_o d_o as_o centre_n draw_v two_o arch_n bc_n of_o with_o the_o same_o extent_n of_o the_o compass_n then_o take_v the_o distance_n of_o between_o your_o compass_n put_v one_o foot_n in_o b_o and_o cut_v off_o bc_n and_o draw_v ac_fw-la i_o say_v that_o the_o angles_n bac_n edf_n be_v equal_a demonstration_n the_o triangle_n abc_n def_n have_v the_o side_n ab_fw-la ac_fw-la equal_a to_o the_o side_n de_fw-fr df_n since_o that_o the_o arch_n bc_n of_o be_v describe_v with_o the_o same_o extent_n of_o the_o compass_n they_o have_v also_o their_o base_n bc_n of_o equal_a therefore_o the_o angle_n bac_n def_n be_v equal_a by_o the_o 8_o use_v this_o problem_n be_v so_o necessary_a in_o survey_v fortification_n prospective_n dial_v and_o in_o all_o other_o part_n of_o the_o mathematics_n so_o that_o the_o great_a part_n of_o their_o practice_n will_v be_v impossible_a if_o one_o angle_n can_v not_o be_v make_v equal_a to_o another_o or_o of_o any_o number_n of_o degree_n require_v proposition_n xxiv_o theorem_fw-la if_o two_o triangle_n which_o have_v two_o side_n of_o the_o one_o equal_a to_o the_o two_o side_n of_o the_o other_o that_o which_o have_v the_o great_a angle_n have_v the_o great_a base_a let_v the_o side_n ab_fw-la de_fw-fr ac_fw-la df_n of_o the_o triangle_n abc_n def_n be_v equal_a and_o let_v the_o angle_n bac_n be_v great_a than_o the_o angle_n edf_n i_o say_v that_o the_o base_a bc_n be_v great_a than_o the_o base_a of_o make_v the_o angle_n edg_n equal_a to_o the_o angle_n bac_n by_o the_o 23d._o and_o the_o line_n dg_fw-mi equal_a to_o ac_fw-la then_o draw_v eglantine_n in_o the_o first_o place_n the_o triangle_n abc_n deg_n have_v the_o side_n ab_fw-la de_fw-fr ac_fw-la dg_n equal_a and_o the_o angle_n edg_n equal_a to_o bac_n their_o base_n bc_n eglantine_n be_v equal_a by_o the_o 4_o and_o the_o line_n dg_v df_n be_v equal_a to_o ac_fw-la they_o shall_v be_v equal_a among_o themselves_o demonstration_n in_o the_o triangle_n dgf_n the_o side_n dg_n df_n be_v equal_a the_o angles_n dgf_n dfg_n be_v equal_a by_o the_o 5_o but_o the_o angle_n egf_n be_v less_o than_o dgf_n and_o the_o angle_n efg_v be_v great_a than_o dfg_v therefore_o in_o the_o triangle_n efg_v the_o angle_n efg_v shall_v be_v great_a than_o the_o angle_n egf_n thence_o by_o the_o 18_o the_o line_n eglantine_n opposite_a to_o the_o great_a angle_n efg_v shall_v be_v
great_a than_o of_o thence_o bc_n equal_a to_o eglantine_n be_v great_a than_o the_o base_a ef._n proposition_n xxv_o theorem_fw-la of_o two_o triangle_n which_o 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 that_o triangle_n which_o have_v the_o great_a base_n have_v also_o the_o great_a angle_n let_v the_o side_n ab_fw-la de_fw-fr ac_fw-la df_n of_o the_o triangle_n abc_n def_n be_v equal_a and_o let_v the_o base_a bc_n be_v great_a than_o the_o base_a ef._n i_o say_v that_o the_o angle_n a_o shall_v be_v great_a than_o the_o angle_n d._n demonstration_n if_o the_o angle_n a_o be_v not_o great_a than_o the_o angle_n d_o it_o will_v be_v equal_a or_o less_o if_o equal_a in_o this_o case_n the_o base_n bc_n will_v be_v equal_a by_o the_o 4_o if_o less_o than_o the_o base_a of_o will_v be_v great_a than_o the_o base_a bc_n by_o the_o 24_o both_o which_o be_v contrary_a to_o our_o hyp._n these_o proposition_n be_v of_o use_n to_o demonstrate_v those_o which_o follow_v proposition_n xxvi_o theorem_fw-la a_o triangle_n which_o have_v one_o side_n and_o two_o angle_n equal_a to_o those_o of_o a_o other_o shall_v be_v equal_a thereto_o in_o every_o respect_n let_v the_o angel_n abc_n def_n acd_v dfe_n of_o the_o triangle_n abc_n def_n be_v equal_a and_o let_v the_o side_n bc_n of_o which_o be_v between_o those_o angles_n be_v also_o equal_a to_o each_o other_o i_o say_v that_o their_o other_o side_n be_v equal_a for_o example_n ac_fw-la df_n but_o let_v it_o be_v imagine_v that_o the_o side_n df_n be_v great_a than_o ac_fw-la and_o that_o gf_n be_v equal_a to_o ac_fw-la and_o draw_v the_o line_n ge._n demonstration_n the_o triangle_n abc_n gef_n have_v the_o side_n of_o bc_n ac_fw-la gf_n equal_a the_o angle_n c_o be_v also_o suppose_v equal_a to_o the_o angle_n f_o thence_o by_o the_o 4_o the_o triangle_n abc_n gef_n be_v equal_a in_o every_o respect_n and_o the_o angel_n gef_n abc_n be_v equal_a but_o according_a to_o our_o first_o hyp._n the_o angel_n abc_n def_n be_v equal_a by_o this_o argument_n the_o angles_n def_n gef_n will_v be_v equal_a that_o be_v to_o say_v the_o whole_a equal_a to_o a_o part_n which_o be_v impossible_a therefore_o df_n can_v be_v great_a than_o ac_fw-la nor_o ac_fw-la great_a than_o df_n because_o the_o same_o demonstration_n may_v be_v make_v in_o the_o triangle_n abc_n second_o let_v we_o suppose_v that_o the_o angle_n a_o and_o d_o c_o and_o f_o be_v equal_a and_o that_o the_o side_n bc_n of_o which_o be_v opposite_a to_o the_o equal_a angle_n a_o and_o d_o be_v also_o equal_a to_o each_o other_o i_o say_v the_o other_o side_n be_v equal_a for_o if_o df_n be_v great_a than_o ac_fw-la make_v gf_n equal_a to_o ac_fw-la and_o draw_v the_o line_n ge._n demonstration_n the_o triangle_n abc_n gef_n have_v the_o side_n of_o bc_n fg_v ca_n equal_a they_o be_v then_o equal_a in_o every_o respect_n by_o the_o 4_o and_o the_o angles_n egf_n bac_n shall_v be_v equal_a but_o according_a to_o our_o hyp._n a_o and_o d_o be_v equal_a thus_o the_o angle_n d_o and_o egf_n shall_v be_v equal_a which_o be_v impossible_a since_o that_o the_o angle_n egf_n be_v exterior_a in_o respect_n of_o the_o triangle_n egg_v it_o ought_v to_o be_v great_a than_o the_o interiour_n angle_n d_o by_o the_o 16_o therefore_o the_o side_n df_n be_v not_o great_a than_o ac_fw-la use_v 26._o use_v 26._o thales_n make_v use_v of_o this_o proposition_n to_o measure_v inaccessible_a distance_n the_o distance_n ad_fw-la be_v propose_v from_o the_o point_n a_o he_o draw_v the_o line_n ac_fw-la perpendicular_a to_o ad_fw-la then_o place_v a_o semicircle_n at_o the_o point_n c_o he_o measure_v the_o angle_n acd_v than_o he_o take_v a_o angle_n equal_a thereto_o on_o the_o other_o side_n draw_v the_o line_n cb_n until_o it_o meet_v the_o line_n dam_fw-ge continue_v to_o the_o point_n b._n he_o demonstrate_v that_o the_o line_n ad_fw-la ab_fw-la be_v equal_a so_o measure_v actual_o the_o accessible_a line_n he_o may_v know_v by_o that_o mean_v the_o other_o for_o the_o two_o triangle_n adc_a abc_n have_v the_o right_a angle_n god_n cab_n equal_a both_o the_o angle_n acd_v acb_n be_v take_v equal_a to_o each_o other_o and_o the_o side_n ac_fw-la be_v common_a to_o both_o triangle_n therefore_o by_o the_o 26_o the_o side_n ad_fw-la ab_fw-la be_v equal_a lemma_n 26._o lem._n 26._o a_o line_n which_o be_v perpendicular_a to_o one_o parallel_n be_v also_o perpendicular_a to_o the_o other_o let_v the_o line_n ab_fw-la cd_o be_v parallel_n to_o each_o other_o and_o let_v of_o be_v perpendicular_a to_o cd_o i_o say_v that_o it_o be_v perpendicular_a to_o ab_fw-la cut_a off_o two_o equal_a line_n cf_n fd_n at_o the_o point_n c_o and_o d_o erect_v two_o perpendicular_o to_o the_o line_n cd_o which_o shall_v also_o be_v equal_a to_o fe_o by_o the_o definition_n of_o parallel_n and_o draw_v the_o line_n aec_fw-la ed._n demonstration_n the_o triangle_n cef_n feed_v have_v the_o side_n fe_o common_a the_o side_n fd_v fc_n be_v equal_a the_o angle_n at_o f_o be_v right_o and_o by_o consequence_n equal_a therefore_o by_o the_o 4_o the_o base_n aec_fw-la ed_z the_o angles_n feed_v fec_n fde_v fce_n be_v equal_a and_o those_o two_o last_o be_v take_v away_o from_o the_o right_a angel_n ace_n bdf_n leave_v the_o equal_a angel_n edb_n eca_n now_o the_o triangle_n caesar_n dbe_n shall_v by_o the_o 4_o have_v the_o angles_n deb_n cea_n equal_a which_o angle_n be_v add_v to_o the_o equal_a angel_n cef_n feed_v make_v equal_a angle_n feb_n fea_n therefore_o of_o be_v perpendicular_a to_o ab_fw-la proposition_n xxvii_o theorem_fw-la if_o a_o right_a line_n fall_v upon_o two_o right_a line_n make_v the_o alternate_a angle_n equal_a the_o one_o to_o the_o other_o then_o be_v the_o right_a line_n parallel_v let_v the_o line_n eh_n fall_v on_o the_o right_a line_n ab_fw-la cd_o make_v therewith_o the_o alternate_a angle_n afg_v fgd_v equal_a i_o say_v in_o the_o first_o place_n that_o the_o line_n ab_fw-la cd_o shall_v not_o meet_v although_o continue_v as_o far_o as_o one_o list_n for_o suppose_v they_o shall_v meet_v in_o ay_o and_o that_o fbi_n cdi_o be_v two_o straight_a line_n demonstration_n if_o fbi_n gdi_n be_v two_o straight_a line_n then_o fig_n be_v a_o triangle_n then_o by_o the_o 16_o the_o exterior_a angle_n afg_v shall_v be_v great_a than_o the_o interior_a fgi_n wherefore_o that_o the_o equality_n of_o the_o angle_n may_v subsist_v the_o line_n ab_fw-la cd_o must_v never_o meet_v each_o other_o but_o because_o we_o have_v example_n of_o some_o crooked_a line_n that_o never_o intersect_v which_o notwithstanding_o be_v not_o parallel_n but_o approach_v continual_o to_o prove_v the_o forego_n i_o make_v another_o demonstration_n as_o follow_v first_o i_o say_v that_o if_o the_o line_n eh_n fall_v on_o the_o line_n ab_fw-la cd_o make_v the_o alternate_a angle_n afg_v fgd_v equal_a the_o line_n ab_fw-la cd_o be_v parallel_n that_o be_v in_o every_o part_n equidistant_n from_o each_o other_o for_o which_o reason_n the_o perpendicular_o shall_v be_v equal_a to_o each_o other_o draw_z from_z g_z to_o the_o line_n ab_fw-la the_o perpendicular_a give_v and_o cd_o be_v take_v equal_a to_o of_o draw_v fd._n demonstration_n the_o triangle_n agf_n fgd_v have_v the_o side_n gf_n common_a the_o side_n gd_v be_v take_v equal_a to_o of_o it_o be_v suppose_v that_o the_o angel_n afg_n fgd_v be_v equal_a therefore_o by_o the_o 4_o ac_fw-la fd_n be_v equal_a and_o the_o angle_n gdf_n be_v equal_a to_o the_o right_a angle_n gaf_n thence_o fd_n be_v perpendicular_a furthermore_o that_o ab_fw-la be_v parallel_n to_o cd_o for_o the_o parallel_n to_o cd_o be_v to_o be_v draw_v from_o the_o point_n f_o and_o must_v pass_v through_o the_o point_n a_o according_a to_o our_o definition_n of_o parallel_n which_o be_v that_o the_o perpendicular_o agnostus_n fd_n be_v equal_a proposition_n xxviii_o theorem_fw-la if_o a_o right_a line_n fall_v upon_o two_o right_a line_n make_v the_o exterior_a angle_n equal_a to_o the_o interiour_n opposite_a angle_n of_o the_o other_o on_o the_o same_o side_n or_o the_o two_o interiour_n on_o the_o same_o side_n equal_a to_o two_o right_a then_o be_v the_o right_a line_n parallel_v have_v draw_v a_o figure_n like_a unto_o the_o former_a let_v the_o line_n eh_n fall_v on_o ab_fw-la cd_o make_v in_o the_o first_o place_n the_o exterior_a angle_n efb_n equal_a to_o the_o interiour_n opposite_a angle_n of_o the_o other_o on_o the_o same_o side_n fgd_v i_o say_v that_o the_o line_n ab_fw-la cd_o be_v parallel_n demonstration_n the_o angle_n efb_n be_v equal_a to_o the_o angle_n afg_v by_o the_o 15_o and_o it_o be_v suppose_v that_o efb_n be_v also_o equal_a to_o
the_o 1_o because_o the_o side_n bc_n bd_o be_v equal_a the_o angle_n abc_n shall_v be_v double_a of_o each_o the_o second_o case_n be_v when_o a_o angle_n enclose_v the_o other_o and_o the_o line_n make_v the_o same_o angle_n not_o meet_v each_o other_o as_o you_o see_v in_o the_o second_o figure_n the_o angle_n bid_v be_v in_o the_o centre_n and_o the_o angle_n bad_a be_v at_o the_o circumference_n draw_v the_o line_n aic_n through_o the_o centre_n demonstration_n the_o angle_n bic_n be_v double_a to_o the_o angle_n bac_n and_o cid_n double_a to_o the_o angle_n god_n by_o the_o precede_a case_n therefore_o the_o angle_n bid_v shall_v be_v double_a to_o the_o angle_n bad_a use_v there_o be_v give_v ordinary_o a_o practical_a way_n to_o describe_v a_o horizontal_n dial_n by_o a_o single_a open_n of_o the_o compass_n which_o be_v ground_v in_o part_n on_o this_o proposition_n second_o when_o we_o will_v determine_v the_o apogaeon_fw-mi of_o the_o sun_n and_o the_o excentricity_n of_o his_o circle_n by_o three_o observation_n we_o suppose_v that_o the_o angle_n at_o the_o centre_n be_v double_a to_o the_o angle_n at_o the_o circumference_n ptolemy_n make_v often_o use_v of_o this_o proposition_n to_o determine_v as_o well_o the_o excentricity_n of_o the_o sun_n as_o the_o moon_n be_v epicycle_n the_o first_o proposition_n of_o the_o three_o book_n of_o trigonometry_n be_v ground_v on_o this_o proposition_n xxi_o theorem_fw-la the_o angle_n which_o be_v in_o the_o same_o segment_n of_o a_o circle_n or_o that_o have_v the_o same_o arch_n for_o base_a be_v equal_a if_o the_o angles_n bac_n bdc_n be_v in_o the_o same_o segment_n of_o a_o circle_n great_a than_o a_o semicircle_n they_o shall_v be_v equal_a draw_v the_o line_n by_o ci._n demonstration_n the_o angles_n a_o and_o d_o be_v each_o of_o they_o half_o of_o the_o angle_n bic_n by_o the_o precede_a proposition_n therefore_o they_o be_v equal_a they_o have_v also_o the_o same_o arch_a bc_n for_o base_a second_o 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figure_n to_o brass_n cauldron_n to_o the_o end_n we_o may_v work_v thereon_o and_o to_o polish_v prospective_n or_o telescope_n glass_n for_o have_v make_v in_o iron_n a_o angle_n bac_n equal_a to_o that_o which_o the_o segment_n abc_n contain_v and_o have_v put_v in_o the_o point_n b_o and_o c_o two_o small_a pin_n of_o iron_n if_o the_o triangle_n bac_n be_v make_v to_o move_v after_o such_o a_o manner_n that_o the_o side_n ab_fw-la may_v always_o touch_v the_o pin_n b_o and_o the_o side_n ac_fw-la the_o pin_n c_o the_o point_v a_o shall_v be_v always_o in_o the_o circumference_n of_o the_o circle_n abcd._n this_o way_n of_o describe_v a_o circle_n may_v also_o serve_v to_o make_v large_a astrolabe_fw-la proposition_n xxii_o theorem_fw-la qvadrilateral_a figure_n describe_v in_o a_o circle_n have_v their_o opposite_a angle_n equal_a to_o two_o right_a let_v a_o quadrilateral_a or_o four_o side_v figure_n be_v describe_v in_o a_o circle_n in_o such_o manner_n that_o all_o its_o 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abc_n draw_v the_o touch_n line_n feed_v by_o the_o 17_o of_o the_o 3d._n and_o make_v at_o the_o point_n of_o touch_v e_z the_o angle_n deh_fw-it equal_a to_o the_o angle_n b_o and_o the_o angle_n feg_n equal_a to_o the_o angle_n c_o by_o the_o 23d_o of_o the_o one_a draw_v the_o line_n gh_o the_o triangle_n egh_n shall_v be_v equi_fw-la angle_v to_o abc_n demonstration_n the_o angle_n deh_n be_v equal_a to_o the_o angle_n egh_n of_o the_o alternate_a segment_n by_o the_o 32d_o of_o the_o 3d._n now_o the_o angle_n deh_n be_v make_v equal_a to_o the_o angle_n b_o and_o by_o consequence_n the_o angle_n b_o and_o g_o be_v equal_a the_o angle_n c_o and_o h_o be_v also_o equal_a for_o the_o same_o reason_n and_o by_o the_o coral_n 2._o of_o the_o 32d_o of_o the_o one_a the_o angle_n a_o and_o geh_a shall_v be_v equal_a therefore_o the_o triangle_n egh_n abc_n be_v equiangle_v proposition_n iii_o problem_n to_o describe_v about_o a_o circle_n a_o triangle_n equiangle_v to_o another_o if_o one_o will_v describe_v about_o a_o circle_n gkh_n a_o triangle_n equiangle_v to_o abc_n one_o of_o the_o side_n bc_n must_v be_v continue_v to_o d_o and_z e_z and_o make_v the_o angle_n gih_n equal_a to_o the_o angle_n abdella_n and_o hik_n equal_a to_o the_o angle_n ace_n then_o draw_v the_o tangent_n lgm_n lkn_n nhm_o through_o the_o point_n g_o k_o h._n the_o tangent_n shall_v meet_v each_o other_o for_o the_o angles_n ikl_n igl_n be_v right_o if_o one_o shall_v draw_v the_o line_n kg_v which_o be_v not_o draw_v the_o angles_n kgl_n gkl_n will_v be_v less_o than_o two_o right_n therefore_o by_o the_o 11_o axiom_n the_o line_n gl_n kl_n aught_o to_o concur_v demonstration_n all_o the_o angle_n of_o the_o quadrilateral_a gihm_n be_v equal_a to_o four_o right_n see_v it_o may_v be_v reduce_v into_o two_o triangle_n the_o angel_n igm_n ihm_fw-ge which_o be_v make_v by_o the_o tangent_n be_v right_o thence_o the_o angle_n m_o an_o i_o be_v equivalent_a to_o two_o right_a as_o well_o 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dbe_n dbf_n be_v also_o equal_a the_o angle_n abc_n have_v be_v divide_v into_o two_o equal_o the_o side_n db_n be_v common_a therefore_o by_o the_o 26_o of_o the_o one_a these_o triangle_n shall_v be_v equal_a in_o every_o respect_n and_o the_o side_n de_fw-fr df_n shall_v be_v equal_a one_o may_v demonstrate_v after_o the_o same_o manner_n that_o the_o side_n df_n dg_n be_v equal_a one_o may_v therefore_o describe_v a_o circle_n which_o shall_v pass_v through_o the_o point_n e_o f_o g_o and_o see_v the_o angle_n e_o f_o g_o be_v right_o the_o side_n ab_fw-la ac_fw-la bc_n touch_v the_o circle_n which_o shall_v by_o consequence_n be_v inscribe_v in_o the_o triangle_n proposition_n v._o problem_n to_o describe_v a_o circle_n about_o a_o triangle_n if_o you_o will_v describe_v a_o circle_n about_o a_o triangle_n abc_n divide_v the_o side_n ab_fw-la bc_n into_o two_o equal_o in_o d_o and_o e_o drawing_z the_o perpendicular_o df_n of_o which_o concur_v in_o the_o point_n f._n if_o you_o describe_v a_o circle_n on_o the_o centre_n f_o at_o the_o open_a fb_n it_o shall_v pass_v through_o a_o and_o c_o that_o be_v to_o say_v that_o the_o line_n favorina_n fb_n fc_fw-la be_v equal_a demonstration_n the_o triangle_n adf_n bdf_n have_v the_o side_n df_n common_a and_o the_o side_n ad_fw-la db_fw-la equal_a see_v the_o side_n ab_fw-la have_v be_v divide_v equal_o and_o the_o angle_n in_o d_o be_v equal_a be_v right_o thence_o by_o the_o four_o of_o the_o one_a the_o base_n of_o bf_n be_v equal_a and_o for_o the_o same_o reason_n the_o base_n bf_n cf._n use_v we_o have_v often_o need_v to_o inscribe_v a_o triangle_n in_o a_o circle_n as_o in_o the_o first_o proposition_n of_o the_o three_o book_n of_o trigonometry_n this_o practice_n be_v necessary_a for_o to_o measure_v the_o area_n of_o a_o triangle_n and_o upon_o several_a other_o occasion_n proposition_n vi_o problem_n to_o inscribe_v a_o square_a in_o a_o circle_n to_o inscribe_v a_o square_a in_o the_o circle_n abcd_v draw_v to_o the_o diameter_n ab_fw-la the_o perpendicular_a dc_o which_o may_v pass_v through_o the_o centre_n e._n draw_v also_o the_o line_n ac_fw-la cb_n bd_o ad_fw-la and_o you_o will_v have_v inscribe_v in_o the_o circle_n the_o square_n acbd_v demonstration_n the_o triangle_n aec_fw-la ceb_fw-mi have_v their_o side_n equal_a and_o the_o angle_n aec_fw-la ceb_fw-mi equal_a see_v they_o be_v right_o therefore_o the_o base_n ac_fw-la cb_n be_v equal_a by_o the_o four_o of_o the_o one_a moreover_o see_v the_o side_n ae_n ce_fw-fr be_v equal_a and_o the_o angle_n e_o being_n right_n they_o shall_v each_o of_o they_o be_v semi-right_a by_o the_o 32d_o of_o the_o one_a so_o then_o the_o angle_n ecb_n be_v semi-right_a and_o by_o consequence_n the_o angle_n acb_n shall_v be_v right_o it_o be_v the_o same_o of_o all_o the_o other_o angles_n therefore_o the_o figure_n acdb_n be_v a_o square_a proposition_n vii_o problem_n to_o describe_v a_o square_a about_o a_o circle_n have_v draw_v the_o two_o diameter_n ab_fw-la cd_o which_o cut_v each_o other_o perpendicular_o in_o the_o centre_n e_o draw_v the_o touch_n line_n fg_v gh_o he_o fi_n through_o the_o point_v a_o d_o b_o c_o and_o you_o will_v have_v describe_v a_o square_a fghi_n about_o the_o circle_n acbd_v demonstration_n the_o angle_n e_o and_o a_o be_v right_o thence_o by_o the_o 29_o of_o the_o one_a the_o line_n fg_v cd_o be_v parallel_n i_o prove_v after_o the_o same_o manner_n that_o cd_o he_o fi_n ab_fw-la ab_fw-la gh_n be_v parallel_n thence_o the_o figure_n fcdg_n be_v a_o parallelogram_n and_o by_o the_o 34th_o of_o the_o one_a the_o line_n fg_v cd_o be_v equal_a as_o also_o cd_o ih_o fi_n ab_fw-la ab_fw-la gh_n and_o by_o consequence_n the_o side_n of_o the_o figure_n fg_v gh_o he_o if_o be_v equal_a moreover_o see_v the_o line_n fg_v cd_o be_v parallel_n and_o that_o the_o angle_n fce_n be_v right_o the_o angle_n g_o shall_v be_v also_o right_o by_o the_o 29_o of_o the_o one_a i_o demonstrate_v after_o the_o same_o manner_n that_o the_o angle_n f_o h_o and_o i_o be_v right_o therefore_o the_o figure_n fghi_n be_v a_o square_a and_o its_o side_n touch_v the_o circle_n proposition_n viii_o problem_n to_o inscribe_v a_o circle_n in_o a_o square_a if_o you_o will_v inscribe_v a_o circle_n in_o the_o square_a fghi_n divide_v the_o side_n fg_v gh_o he_o fi_n in_o the_o middle_n in_o a_o d_o b_o c_o and_o draw_v the_o line_n ab_fw-la cd_o which_o cut_v each_o other_o in_o the_o point_n e._n i_o demonstrate_v that_o the_o line_n ea_fw-la ed_z aec_fw-la ebb_n be_v equal_a and_o that_o the_o angle_n in_o a_o b_o c_o d_o be_v right_o and_o that_o so_o you_o may_v describe_v a_o circle_n on_o the_o centre_n e_o which_o shall_v pass_v through_o a_o d_o b_o c_o and_o which_o touch_v the_o side_n of_o the_o square_n demonstration_n see_v the_o line_n ab_fw-la gh_n conjoyn_v the_o line_n agnostus_n bh_n which_o be_v parallel_v and_o equal_a they_o shall_v be_v also_o parallel_a and_o equal_a therefore_o the_o figure_n aghb_n be_v a_o parallelogram_n and_o the_o line_n ae_n gd_a agnostus_n ed_z be_v parallel_n and_o agnostus_n gd_v be_v equal_a ae_n ed_z shall_v be_v also_o equal_a it_o be_v the_o same_o with_o the_o other_o ae_n aec_fw-la ebb_n moreover_o agnostus_n ed_z be_v parallel_n and_o the_o angle_n g_z be_v right_o the_o angle_n d_o shall_v be_v also_o a_o right_a angle_n one_o may_v then_o on_o the_o centre_n e_o describe_v the_o circle_n adbc_n which_o shall_v pass_v through_o the_o point_v a_o d_o b_o c_o and_o which_o shall_v touch_v the_o side_n of_o the_o square_n proposition_n ix_o problem_n to_o describe_v a_o circle_n about_o a_o square_n
triangle_n abc_n the_o line_n de_fw-fr be_v parallel_n to_o the_o base_a bc_n the_o side_n ab_fw-la ac_fw-la shall_v be_v divide_v proportional_o that_o be_v to_o say_v that_o there_o shall_v be_v the_o same_o reason_n of_o ad_fw-la to_o db_n as_o of_o ae_n to_o aec_fw-la draw_v the_o line_n de_fw-fr be._n the_o triangle_n dbe_v dec_n which_o have_v the_o same_o base_a de_fw-fr and_o be_v between_o the_o same_o parallel_n de_fw-fr bc_n be_v equal_a by_o the_o 37th_o of_o the_o one_a demonstration_n the_o triangle_n ade_n dbe_n have_v the_o same_o point_n e_o for_o their_o vertical_a if_o we_o take_v ad_fw-la db_fw-la for_o their_o base_n and_o if_o one_o shall_v draw_v through_o the_o point_n e_o a_o parallel_n to_o ab_fw-la they_o will_v be_v both_o between_o the_o same_o parallel_n they_o shall_v have_v thence_o the_o same_o reason_n as_o their_o base_n by_o the_o one_a that_o be_v to_o say_v that_o there_o be_v the_o same_o reason_n of_o ad_fw-la to_o db_n as_o of_o the_o triangle_n ade_n to_o the_o triangle_n dbe_n or_o to_o its_o equal_a ce_v now_o there_o be_v the_o same_o reason_n of_o the_o triangle_n ade_n to_o the_o triangle_n ce_v as_o of_o the_o base_a ae_n to_o aec_fw-la there_o be_v therefore_o the_o same_o reason_n of_o ad_fw-la to_o db_n as_o of_o ae_n to_o aec_fw-la and_o if_o there_o be_v the_o same_o reason_n of_o ae_n to_o aec_fw-la as_o of_o ad_fw-la to_o db_n i_o say_v that_o the_o line_n de_fw-fr bc_n will_v then_o be_v parallel_n demonstration_n there_o be_v the_o same_o reason_n of_o ad_fw-la to_o db_n as_o of_o the_o triangle_n ade_n to_o the_o triangle_n dbe_n by_o the_o one_a there_o be_v also_o the_o same_o reason_n of_o ae_n to_o aec_fw-la as_o of_o the_o triangle_n ade_n to_o the_o triangle_n dec_n consequent_o there_o be_v the_o same_o reason_n of_o the_o triangle_n ade_n to_o the_o triangle_n bde_v as_o of_o the_o same_o triangle_n ade_n to_o the_o triangle_n ce_v so_o then_o by_o the_o seven_o of_o the_o 5_o the_o triangle_n bde_v ced_a be_v equal_a and_o by_o the_o 39th_o of_o the_o one_a they_o be_v between_o the_o same_o parallel_n use_v this_o proposition_n be_v absolute_o necessary_a in_o the_o follow_a proposition_n one_o may_v make_v use_n thereof_o in_o measure_v as_o in_o the_o follow_a figure_n if_o it_o be_v require_v to_o measure_v the_o height_n be_v have_v the_o length_n of_o the_o staff_n dam_fw-ge there_o be_v the_o same_o reason_n of_o cd_o to_o da_fw-la as_o of_o bc_n to_o be._n proposition_n iii_o theorem_fw-la that_o line_n which_o divide_v the_o angle_n of_o a_o triangle_n into_o two_o equal_a part_n divide_v its_o base_a in_o two_o part_n which_o be_v in_o the_o same_o reason_n to_o each_o other_o as_o be_v their_o side_n and_o if_o that_o line_n divide_v the_o base_a into_o part_n proportional_a to_o the_o side_n it_o shall_v divide_v the_o angle_n into_o two_o equal_o if_o the_o line_n ad_fw-la divide_v the_o angle_n bac_n into_o two_o equal_a part_n there_o shall_v be_v the_o same_o reason_n of_o ab_fw-la to_o ac_fw-la as_o of_o bd_o to_o dc_o continue_v the_o side_n ca_n and_o make_v ae_n equal_a to_o ab_fw-la then_o draw_v the_o line_n ebb_n demonstration_n the_o exterior_a angle_n cab_n be_v equal_a to_o the_o two_o interior_a angle_n aeb_n abe_n which_o be_v equal_a by_o the_o 5_o of_o the_o one_a see_v the_o side_n ae_n ab_fw-la be_v equal_a the_o angle_n bad_a the_o half_a of_o bac_n shall_v be_v equal_a to_o one_o of_o they_o that_o be_v to_o say_v to_o the_o angle_n abe_n thence_o by_o the_o 27_o of_o the_o one_a the_o line_n ad_fw-la ebb_n be_v parallel_n and_o by_o the_o 2d_o there_o be_v the_o same_o reason_n of_o ea_fw-la or_o ab_fw-la to_o ac_fw-la as_o of_o bd_o to_o dc_o second_o if_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_o ac_fw-la as_o of_o bd_o to_o dc_o the_o angle_n bac_n shall_v be_v divide_v into_o two_o equal_o demonstra_fw-la there_o be_v the_o same_o reason_n of_o ab_fw-la or_o ae_n to_o ac_fw-la as_o of_o bd_o to_o dc_o thence_o the_o line_n ebb_v ad_fw-la be_v parallel_n and_o by_o the_o 29_o of_o the_o one_a the_o alternate_a angle_n eba_n bad_a the_o internal_a bea_fw-mi and_o the_o external_a dac_n shall_v be_v equal_a and_o the_o angles_n eba_n aeb_n be_v equal_a the_o angel_n bad_a dac_n shall_v be_v so_o likewise_o wherefore_o the_o angle_n bac_n have_v be_v divide_v equal_o use_v we_o make_v use_v of_o this_o proposition_n to_o attain_v to_o the_o proportion_n of_o the_o side_n proposition_n iu_o theorem_fw-la equiangular_a triangle_n have_v their_o side_n proportional_a if_o the_o triangle_n abc_n dce_n be_v equiangular_a that_o be_v to_o say_v that_o the_o angel_n abc_n dce_n bac_n cde_v be_v equal_a there_o will_v be_v the_o same_o reason_n of_o basilius_n to_o bc_n as_o of_o cd_o to_o ce._n in_o like_a manner_n the_o reason_n of_o basilius_n to_o ac_fw-la shall_v be_v the_o same_o with_o that_o of_o cd_o to_o de._n join_v the_o triangle_n after_o such_o a_o manner_n that_o their_o base_n bc_n ce_fw-fr be_v on_o the_o same_o line_n and_o continue_v the_o side_n ed_z ba_z see_v the_o angle_n acb_n dec_n be_v equal_a the_o line_n ac_fw-la of_o be_v parallel_n and_o so_o cd_o bf_n by_o the_o 29_o of_o the_o one_a and_o of_o dc_o shall_v be_v a_o parallelogram_n demonstration_n in_o the_o triangle_n bfe_n ac_fw-la be_v parallel_n to_o the_o base_a fe_o thence_o by_o the_o 2d_o there_o shall_v be_v the_o same_o reason_n of_o basilius_n to_o of_o or_o cd_o as_o of_o bc_n to_o ce_fw-fr and_o by_o exchange_n there_o shall_v be_v the_o same_o reason_n of_o ab_fw-la to_o bc_n as_o of_o dc_o to_o ce._n in_o like_a manner_n in_o the_o same_o triangle_n cd_o be_v parallel_n to_o the_o base_a bf_n there_o shall_v be_v the_o same_o reason_n of_o fd_n or_o ac_fw-la to_o de_fw-fr as_o of_o bc_n to_o ge_z by_o the_o 2d_o and_o by_o exchange_n there_o shall_v be_v the_o same_o reason_n of_o ac_fw-la to_o bc_n as_o of_o de_fw-fr to_z ce._n use_v this_o proposition_n be_v of_o a_o great_a extent_n and_o may_v pass_v for_o a_o universal_a principle_n in_o all_o sort_n of_o measure_v for_o in_o the_o first_o place_n the_o ordinary_a practice_n in_o measure_v inaccessible_a line_n by_o make_v a_o little_a triangle_n like_o unto_o that_o which_o be_v make_v or_o imagine_v to_o be_v make_v on_o the_o ground_n be_v found_v on_o this_o proposition_n as_o also_o the_o great_a part_n of_o those_o instrument_n on_o which_o be_v make_v triangle_n like_v unto_o those_o that_o we_o will_v measure_v as_o the_o geometrical_a square_n sinical_a quadrant_n jacob_n staff_n and_o other_o moreover_o we_o can_v not_o take_v the_o plane_n of_o a_o place_n but_o by_o this_o proposition_n wherefore_o to_o explain_v its_o use_n we_o shall_v be_v force_v to_o bring_v in_o the_o first_o book_n of_o practical_a geometry_n proposition_n v._o theorem_fw-la triangle_n who_o side_n be_v proportional_a be_v equiangule_a if_o the_o triangle_n abc_n def_n have_v their_o side_n proportional_a that_o be_v to_o say_v if_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_o bc_n as_o of_o de_fw-fr to_o of_o as_o also_o if_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_o ac_fw-la as_o of_o de_fw-fr to_o df_n the_o angel_n abc_n def_n a_o and_o d_o c_z and_z f_o shall_v be_v equal_a make_v the_o angle_n feg_v equal_a to_o the_o angle_n b_o and_o efg_v equal_a to_o the_o angle_n c._n demonstration_n the_o triangle_n abc_n efg_v have_v two_o angle_n equal_a they_o be_v thence_o equiangle_v by_o the_o cor._n of_o the_o 32d_o of_o the_o one_a and_o by_o the_o four_o there_o be_v the_o same_o reason_n of_o de_fw-fr to_o of_o as_o of_o eglantine_n to_o ef._n now_o it_o be_v suppose_v that_o there_o be_v the_o same_o reason_n of_o de_fw-fr to_o of_o as_o of_o eglantine_n to_o ef._n thence_o by_o the_o seven_o of_o the_o 5_o de_fw-fr eglantine_n be_v equal_a in_o like_a manner_n df_n fg_n be_v also_o equal_a and_o by_o the_o 8_o of_o the_o one_a the_o triangle_n def_n gef_n be_v equiangular_a now_o the_o angle_n gef_n be_v make_v equal_a to_o the_o angle_n b_o thence_o def_n be_v equal_a to_o the_o angle_n b_o and_o the_o angle_n dfe_n to_o the_o angle_n c._n so_o that_o the_o triangle_n abc_n def_n be_v equiangular_a proposition_n vi_o theorem_fw-la triangle_n which_o have_v their_o side_n proportional_a which_o include_v a_o equal_a angle_n be_v equiangular_a if_o the_o angle_n b_o and_o e_o of_o the_o triangle_n abc_n def_n be_v equal_a there_o be_v the_o same_o reason_n of_o ab_fw-la to_o bc_n as_o of_o de_fw-fr to_o of_o the_o triangle_n abc_n def_n shall_v be_v equiangular_a make_v the_o angle_n feg_n equal_a to_o the_o angle_n b_o and_o
the_o angle_n efg_v equal_a to_o the_o angle_n c._n demonstra_fw-la the_o triangle_n abc_n egf_n be_v equiangular_a by_o the_o cor._n of_o the_o 32d_o of_o the_o one_a there_o be_v thence_o the_o same_o reason_n of_o ab_fw-la to_o bc_n as_o of_o eglantine_n to_o of_o by_o the_o four_o now_o as_o ab_fw-la to_o bc_n so_o be_v de_fw-fr to_o of_o there_o be_v then_o the_o same_o reason_n of_o de_fw-fr to_o of_o as_o of_o ge_z to_o ef._n so_o then_o by_o the_o seven_o of_o the_o 5_o de_fw-fr eglantine_n be_v equal_a and_o the_o triangle_n def_n gef_n which_o have_v their_o angle_n def_n gef_n each_o of_o they_o equal_a to_o the_o angle_n b_o and_o the_o side_n de_fw-fr eglantine_n equal_a with_o the_o side_n of_o common_a they_o shall_v be_v equal_a in_o every_o respect_n by_o the_o four_o of_o the_o one_a they_o be_v thence_o equiangular_a and_o the_o triangle_n egf_n be_v equal_a to_o the_o triangle_n abc_n the_o triangle_n abc_n def_n be_v equiangular_a the_o seven_o proposition_n be_v unnecessary_a proposition_n viii_o theorem_fw-la a_o perpendicular_a be_v draw_v from_o the_o right_a angle_n of_o a_o right_a angle_a triangle_n to_o the_o opposite_a side_n divide_v the_o same_o into_o two_o triangle_n which_o be_v a_o like_a thereto_o if_o from_o the_o right_a angle_n abc_n be_v draw_v a_o perpendicular_a bd_o to_o the_o opposite_a side_n ac_fw-la it_o will_v divide_v the_o right_o angle_a triangle_n abc_n into_o two_o triangle_n adb_n bdc_fw-la which_o shall_v be_v like_a or_o equiangular_a to_o the_o triangle_n abc_n demonstration_n the_o triangle_n abc_n adb_n have_v the_o same_o angle_n a_o the_o angle_n adb_n abc_n be_v right_a they_o be_v thence_o equiangular_a by_o the_o cor._n 2._o of_o the_o 32d_o of_o the_o one_a in_o like_a manner_n the_o triangle_n bdc_a abc_n have_v the_o angle_n c_o common_n and_o the_o angel_n abc_n bdc_n be_v right_a they_o be_v also_o equal_a thence_o the_o triangle_n abc_n dbc_n be_v like_a use_v we_o measure_v inaccessible_a 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great_a into_o a_o lesser_a wherefore_o this_o proposition_n extend_v almost_o to_o all_o art_n in_o which_o it_o be_v necessary_a to_o take_v a_o design_n or_o model_n proposition_n xix_o theorem_fw-la like_a triangle_n be_v in_o duplicate_v ratio_fw-la to_o their_o homologous_n side_n if_o the_o triangle_n abc_n def_n be_v like_a or_o equiangular_a they_o shall_v be_v in_o duplicate_v ratio_fw-la of_o their_o homologous_n side_n bc_n of_o that_o be_v to_o say_v that_o the_o ratio_fw-la of_o the_o triangle_n abc_n to_o the_o triangle_n def_n shall_v be_v in_o duplicate_v ratio_fw-la of_o bc_n to_o of_o wherefore_o by_o seek_v the_o three_o proportional_a he_o to_o the_o line_n bc_n of_o or_o so_o make_v it_o that_o there_o may_v be_v the_o same_o ratio_fw-la of_o bc_n to_o of_o as_o of_o of_o to_o he_o the_o triangle_n abc_n shall_v have_v the_o same_o ratio_fw-la to_o the_o triangle_n def_n as_o the_o line_n bc_n have_v to_o the_o line_n he_o which_o be_v call_v a_o duplicate_v 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gil_n shall_v be_v equal_a thence_o the_o triangle_n acd_v gil_n shall_v be_v like_o by_o the_o 15_o so_o that_o it_o be_v easy_a after_o the_o same_o manner_n to_o go_v round_o about_o the_o angle_n of_o the_o polygon_n and_o to_o prove_v they_o be_v circular_a or_o alike_o i_o further_o add_v that_o the_o triangle_n be_v in_o the_o same_o ratio_fw-la as_o be_v the_o polygon_n demonstration_n see_v all_o the_o triangle_n be_v like_a their_o side_n shall_v be_v in_o the_o same_o ratio_fw-la by_o the_o four_o now_o each_o triangle_n be_v to_o its_o like_a in_o duplicate_v ratio_fw-la of_o its_o homologous_n side_n by_o the_o 19_o thence_o each_o triangle_n of_o one_o polygon_n to_o each_o triangle_n of_o the_o other_o polygon_n be_v in_o duplicate_v ratio_fw-la to_o the_o side_n which_o be_v the_o same_o there_o shall_v be_v the_o same_o ratio_fw-la of_o each_o triangle_n to_o its_o like_a as_o of_o all_o the_o triangle_n of_o one_o polygon_n to_o all_o the_o triangle_n of_o the_o 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by_o equality_n the_o same_o ratio_fw-la of_o bc_n to_o g_o as_o of_o ht_a to_o p_o and_o this_o ratio_fw-la shall_v be_v duplicate_v or_o double_v of_o the_o ratio_fw-la of_o bc_n to_o of_o or_o of_o ht_a to_o mn_v demonstration_n the_o polygon_n abc_n to_o the_o polygon_n def_n be_v in_o duplicate_v or_o double_v ratio_fw-la of_o the_o ratio_fw-la of_o bc_n to_o of_o by_o the_o 21_o that_o be_v to_o say_v as_o bc_n be_v to_o g_o and_o the_o polygon_n hl_n to_o more_n have_v the_o same_o ratio_fw-la as_o ht_n to_o p._n there_o be_v therefore_o the_o same_o ratio_fw-la of_o abc_n to_o def_n as_o of_o hl_n to_o mo._n and_o if_o like_v polygon_n be_v proportional_a
aed_n by_o draw_v the_o line_n ad_fw-la bd_o and_o the_o perpendicular_o cg_n bf_n ei._n for_o multiply_v the_o half_a of_o bd_o by_o cg_n and_o the_o half_a of_o ad_fw-la by_o bf_n and_o by_o ei_o we_o have_v the_o area_n of_o all_o those_o triangle_n which_o add_v together_o be_v equal_a to_o the_o content_a of_o the_o right_o line_v figurâ—_n abcde_v 39_o use_v 39_o we_o find_v the_o area_n of_o regular_a polygon_n by_o multiply_v one_o half_a of_o their_o periphery_n by_o the_o perpendicular_a draw_v from_o their_o centre_n to_o one_o of_o their_o side_n for_o multiply_v ig_a by_o agnostus_n we_o have_v the_o rectangle_n hklm_n equal_a to_o the_o triangle_n aib_n and_o do_v the_o same_o for_o every_o triangle_n take_v always_o the_o semi_a base_a we_o have_v the_o rectangle_n hkon_v who_o side_n quoth_fw-mi compose_v of_o the_o semi-base_n and_o consequent_o equal_a to_o the_o semi-periphery_a and_o the_o side_n hk_v equal_a to_o the_o perpendicular_a ig_n according_a to_o this_o principle_n archimedes_n have_v demonstrate_v that_o a_o circle_n be_v equal_a to_o a_o rectangle_n comprehend_v under_o the_o semi-diameter_n and_o a_o line_n equal_a to_o the_o semi-circumference_n proposition_n xlii_o problem_n to_o make_v a_o parallelogram_n equal_a to_o a_o triangle_n give_v in_o a_o angle_n equal_a to_o a_o right_o line_v angle_n give_v it_o be_v require_v to_o make_v a_o parallelogram_n equal_a to_o the_o triangle_n abc_n have_v a_o angle_n equal_a to_o the_o angle_n e._n divide_v the_o base_a bc_n into_o two_o equal_a part_n in_o d._n draw_v agnostus_n parallel_v to_o bc_n by_o the_o 31_o make_v also_o the_o angle_n cdf_n equal_a to_o the_o angle_n e_o by_o the_o 23d._o then_o draw_v the_o parallel_n cg_n the_o figure_n fdcg_n be_v a_o parallelogram_n see_v the_o line_n fg_v dc_o df_n cg_n be_v parallel_n the_o angle_n cdf_n be_v equal_a to_o the_o angle_n e._n i_o say_v that_o the_o parallelogram_n be_v equal_a to_o the_o triangle_n abc_n demonstration_n the_o 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one_o of_o its_o angle_n equal_a to_o the_o right_o line_a angle_n e_o which_o must_v be_v equal_a to_o the_o triangle_n abc_n make_v by_o the_o precede_a problem_n the_o parallelogram_n bfgh_o have_v the_o angle_n hbf_o equal_a to_o the_o angle_n e_o and_o which_o may_v be_v equal_a to_o the_o triangle_n abc_n continue_v the_o side_n gh_o cf_n and_o make_v he_o equal_a to_o the_o line_n d_o then_o draw_v the_o line_n ibn_n and_o by_o the_o point_n i_o and_o n_n draw_z il_fw-fr parallel_n to_o gn_n and_o nl_n parallel_n to_o give_v and_o continue_v hb_a to_o m_o and_o fb_n to_o k._n the_o parallelogram_n klbm_n be_v the_o parallelogram_n require_v demonstration_n the_o angle_n hbf_o equal_a to_o the_o give_v angle_n e_o be_v also_o equal_a to_o the_o angle_n kbm_n by_o the_o 15_o and_o the_o side_n kb_n be_v equal_a to_o the_o line_n he_o or_o d._n last_o the_o parallelogram_n mk_n be_v equal_a by_o the_o forego_n to_o the_o parallelogram_n gfbh_n and_o this_o be_v make_v equal_a to_o the_o triangle_n abc_n therefore_o the_o parallelogram_n mk_n be_v equal_a to_o the_o triangle_n abc_n use_v 44._o use_v 44._o in_o this_o proposition_n be_v contain_v a_o sort_n of_o geometrical_a division_n for_o in_o arithmetical_a division_n there_o be_v propose_v a_o number_n which_o may_v be_v imagine_v to_o be_v like_o a_o rectangle_n for_o example_n the_o rectangle_n ab_fw-la contain_v twelve_o square_a foot_n which_o be_v to_o be_v divide_v by_o another_o number_n as_o by_o two_o that_o be_v to_o say_v that_o there_o must_v be_v make_v another_o rectangle_n equal_a to_o the_o rectangle_n ab_fw-la which_o may_v have_v bd_o too_o for_o one_o of_o its_o side_n and_o to_o find_v how_o many_o foot_n the_o other_o side_n ought_v to_o be_v that_o be_v to_o say_v the_o quotient_a one_o may_v attain_v it_o geometrical_o by_o a_o rule_n and_o compass_n take_v bd_o of_o the_o length_n of_o two_o foot_n and_o draw_v the_o diagonal_a def_n the_o line_n of_o be_v that_o which_o you_o seek_v for_o for_o have_v make_v the_o rectangle_n dcfg_v the_o compliment_n eglantine_n aec_fw-la be_v equal_a by_o the_o 43d._o and_o eglantine_n have_v for_o one_o of_o its_o side_n eh_n equal_a to_o bd_o two_o foot_n and_o ei_o equal_a to_o af._n this_o way_n of_o division_n be_v call_v application_n because_o the_o rectangle_n ab_fw-la be_v apply_v to_o the_o line_n bd_o or_o eh_fw-it and_o this_o be_v the_o reason_n why_o division_n be_v call_v application_n for_o the_o ancient_a geometrician_n do_v choose_v rather_o to_o make_v use_n of_o a_o ruler_n and_o compass_n than_o arithmetic_n proposition_n xlv_o problem_n upon_o a_o right-line_n to_o make_v a_o parallelogram_n equal_a to_o a_o right-lined_n figure_n give_v at_o a_o right-lined_n angle_v give_v there_o be_v propose_v or_o give_v the_o right_o line_a figure_n abcd_v and_o it_o be_v require_v to_o make_v a_o parallelogram_n equal_a thereto_o have_v a_o angle_n equal_a to_o the_o angle_n e._n reduce_v the_o right_o line_a figure_n into_o triangle_n by_o draw_v the_o right_a line_n bd_o and_o make_v by_o the_o 42_o a_o parallelogram_n fghi_n have_v the_o angle_n fgh_o equal_a to_o the_o angle_n e_o and_o be_v in_o content_a equal_a to_o the_o triangle_n abdella_n make_v also_o by_o the_o 44_o a_o parallelogram_n ihkl_n equal_a to_o the_o triangle_n bcd_v and_o have_v a_o line_n equal_a to_o ih_o and_o the_o angle_n ihk_v equal_a to_o the_o angle_n e._n the_o parallelogram_n fgkl_n shall_v be_v equal_a to_o the_o right_o line_a figure_n abcd._n demonstration_n it_o remain_v to_o prove_v that_o the_o parallellogram_n fghi_n hkli_n be_v but_o one_o that_o be_v to_o say_v that_o gh_a hk_v make_v but_o one_o straight_a line_n the_o angles_n ghi_n and_o k_o be_v equal_a to_o the_o angle_n e_o the_o angle_n k_n and_o khi_n be_v equal_a to_o two_o right_a see_v we_o have_v make_v a_o parallelogram_n khil_fw-la whereof_o the_o angle_n ghi_n khi_n be_v equal_a to_o two_o right_a and_o by_o the_o 14_o gh_v hk_n be_v one_o straight_a line_n use_v this_o proposition_n be_v in_o the_o practice_n much_o like_o to_o the_o former_a and_o serve_v to_o measure_v the_o content_a of_o any_o figure_n whatsoever_o by_o reduce_v it_o first_o into_o triangle_n then_o make_v a_o right_a angle_a parallelogram_n equal_a thereto_o we_o may_v also_o make_v a_o right_a angle_a parallelogram_n on_o a_o determine_a or_o give_v line_n and_o which_o shall_v be_v equal_a to_o several_a irregular_a figure_n likewise_o have_v several_a figure_n we_o may_v make_v another_o equal_a to_o their_o difference_n proposition_n xlvi_o problem_n upon_o a_o right_a line_n to_o describe_v a_o square_n to_o describe_v a_o square_a on_o the_o right_a line_n ab_fw-la erect_v two_o perpendicular_n ac_fw-la bd_o equal_a to_o ab_fw-la and_o draw_v the_o line_n bd._n demonstration_n the_o angle_n at_o a_o and_z b_o be_v right_o the_o line_n ac_fw-la bd_o be_v parallel_n by_o the_o 28_o they_o be_v also_o equal_a therefore_o the_o line_n ab_fw-la cd_o be_v parallel_n and_o equal_a by_o the_o 33d._o and_o the_o angle_n at_o a_o and_o c_o b_o and_o d_o equal_a to_o two_o right_v by_o the_o 29_o and_o see_v a_o and_o b_o be_v right_a angle_n the_o angle_n c_o and_o d_o shall_v be_v also_o right_o thence_o the_o figure_n ad_fw-la have_v all_o its_o side_n equal_a and_o all_o its_o angle_n right_o and_o consequent_o be_v a_o square_a proposition_n xlvii_o theorem_fw-la the_o square_n of_o the_o base_a of_o a_o right_a angle_a triangle_n be_v equal_a to_o the_o sum_n of_o the_o square_n of_o the_o other_o two_o side_n it_o be_v suppose_v that_o the_o angle_n bac_n be_v a_o right_a angle_n and_o that_o there_o be_v describe_v square_n on_o the_o several_a side_n bc_n ab_fw-la ac_fw-la the_o square_a of_o the_o base_a bc_n shall_v be_v equal_a to_o
to_o efg_v demonstration_n we_o have_v already_o demonstrate_v that_o there_o be_v a_o great_a reason_n of_o a_o b_o c_o to_z e_z f_o g_o than_o of_o the_o part_n bc_n to_o the_o part_n fg_v there_o shall_v thence_o be_v a_o great_a reason_n of_o a_o to_o e_o than_o of_o a_o b_o c_o to_z e_z f_o g_o by_o the_o 32d_o the_o end_n of_o the_o five_o book_n the_o sixth_z book_n of_o euclid_n element_n this_o book_n explain_v and_o begin_v to_o apply_v particular_a matter_n of_o the_o doctrine_n of_o proportion_n which_o the_o precede_a book_n explain_v but_o in_o general_n it_o begin_v with_o the_o most_o easy_a figure_n that_o be_v to_o say_v triangle_n give_v rule_n to_o determine_v not_o only_o the_o proportion_n of_o their_o side_n but_o also_o that_o of_o their_o capacity_n area_n or_o superficies_n then_o it_o teach_v to_o find_v proportional_a line_n and_o to_o augment_v or_o diminish_v any_o figure_n whatever_o according_a to_o a_o give_v ratio_fw-la it_o demonstrate_v the_o rule_n of_o three_o it_o appli_v the_o forty_o seven_o of_o the_o first_o to_o all_o sort_n of_o figure_n in_o fine_a it_o give_v we_o the_o most_o easy_a and_o most_o certain_a principle_n to_o conduct_v we_o in_o all_o sort_n of_o measure_v the_o definition_n 1._o 6._o def._n 1._o fig._n i._o plate_n 6._o right_o line_v figure_n be_v like_a when_o that_o they_o have_v all_o their_o angle_n equal_a and_o the_o side_n which_o form_v those_o angel_n proportional_a i._n fig._n i._n as_o the_o triangle_n abc_n def_n shall_v be_v like_a if_o the_o angle_n a_o and_o d_o b_o and_z e_z c_z and_z f_o be_v equal_a and_o if_o there_o be_v the_o same_o ratio_fw-la of_o ab_fw-la to_o ac_fw-la as_o of_o de_fw-fr to_o df_n and_o of_o ab_fw-la to_o cb_n as_o of_o de_fw-fr to_o ef._n 2._o ii_o fig._n ii_o figure_n be_v reciprocal_a when_o they_o may_v be_v compare_v after_o such_o sort_n that_o the_o antecedent_n of_o one_o reason_n and_o the_o consequent_a of_o the_o other_o be_v find_v in_o the_o same_o figure_n that_o be_v when_o the_o analogy_n begin_v in_o one_o figure_n and_o end_v in_o the_o same_o as_o if_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_o cd_o as_o of_o de_fw-fr to_o bf_n 3._o iii_o fig._n iii_o a_o line_n be_v divide_v into_o extreme_a and_o mean_a proportion_n or_o reason_n when_o there_o be_v the_o same_o reason_n of_o the_o whole_a line_n to_o its_o great_a part_n as_o of_o its_o great_a part_n to_o its_o lesser_a part_n as_o if_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_o ac_fw-la as_o of_o ac_fw-la to_o cb_n the_o line_n ab_fw-la shall_v be_v divide_v in_o the_o point_n c_o in_o extreme_a and_o mean_a proportion_n 4._o the_o altitude_n or_o height_n of_o a_o figure_n be_v the_o length_n of_o the_o perpendicular_a draw_v from_o the_o top_n thereof_o to_o its_o base_a iu._n fig._n iu._n as_o in_o the_o triangle_n abc_n efg_v the_o perpendicular_a eh_n ad_fw-la whether_o it_o fall_v without_o or_o within_o the_o triangle_n be_v their_o height_n or_o altitude_n triangle_n and_o parallellogram_n who_o height_n be_v equal_a may_v be_v place_v between_o the_o same_o parallel_n for_o have_v place_v their_o base_n on_o the_o same_o line_n hc_n if_o the_o perpendicular_o da_fw-la he_o be_v equal_a the_o line_n ea_fw-la hc_fw-la shall_v be_v parallel_n 5._o a_o reason_n be_v compose_v of_o several_a reason_n when_o the_o quantity_n of_o those_o reason_n be_v multiply_v make_v a_o three_o reason_n it_o be_v to_o be_v take_v notice_n of_o that_o a_o reason_n at_o least_o the_o rational_a have_v its_o name_n take_v from_o some_o number_n which_o specifi_v the_o reason_n or_o habitude_n of_o the_o antecedent_n of_o that_o reason_n to_o its_o consequent_a as_o when_o there_o be_v propose_v two_o magnitude_n or_o quantity_n the_o one_o of_o twelve_o foot_n and_o the_o other_o of_o six_o we_o say_v that_o the_o reason_n of_o twelve_o to_o six_o be_v double_a in_o like_a manner_n when_o there_o be_v propose_v two_o magnitude_n four_o and_o twelve_o we_o will_v say_v it_o be_v a_o subtriple_a reason_n and_o â…“_n be_v the_o denominator_fw-la thereof_o which_o specifi_v that_o there_o be_v the_o same_o reason_n of_o four_o to_o twelve_o as_o of_o â…“_n to_o one_o or_o as_o of_o one_o to_o three_o one_o may_v call_v this_o denominaotr_n the_o quantity_n of_o the_o reason_n let_v there_o be_v propose_v three_o term_n twelve_o six_o two_o the_o first_o reason_n of_o twelve_o to_o six_o be_v double_a its_o denominator_fw-la be_v two_o the_o reason_n of_o six_o to_o two_o be_v triple_a its_o denominator_fw-la be_v three_o the_o reason_n of_o twelve_o to_o two_o be_v compound_v of_o the_o reason_n of_o twelve_o to_o six_o and_o of_o that_o of_o six_o to_o two_o to_o have_v thâ—…_n denominator_fw-la or_o the_o reason_n of_o twelve_o to_o two_o which_o be_v compound_v of_o double_a and_o of_o triple_a multiply_v three_o by_o two_o and_o you_o shall_v have_v six_o thence_o the_o reason_n of_o twelve_o to_o two_o be_v sextuple_a this_o be_v that_o which_o mathematician_n understand_v by_o composition_n of_o reason_n although_o it_o ought_v to_o be_v call_v multiplication_n of_o reason_n proposition_n i._o theorem_fw-la parallellogram_n and_o triangle_n of_o equal_a height_n have_v the_o same_o reason_n as_o their_o base_n vi._n plate_n vi._n let_v there_o be_v propose_v the_o triangle_n acg_n dem_n of_o equal_a height_n in_o such_o sort_n that_o they_o may_v be_v place_v between_o the_o same_o parallel_n ad_fw-la gm_n i_o say_v that_o there_o shall_v be_v the_o same_o reason_n of_o the_o base_a gc_n to_o the_o base_a em_n as_o of_o the_o triangle_n agc_n to_o the_o triangle_n dem_n let_v the_o base_a em_n be_v divide_v into_o as_o many_o equal_a part_n as_o you_o please_v and_o let_v there_o be_v draw_v to_o each_o division_n the_o line_n df_n dh_n etc._n etc._n let_v also_o the_o line_n gc_fw-mi be_v divide_v into_o part_n equal_a to_o those_o of_o em_n and_o let_v be_v draw_v line_n to_o each_o division_n from_o the_o top_n a_o all_o those_o little_a triangle_n which_o be_v make_v in_o the_o two_o great_a triangle_n be_v between_o the_o same_o parallel_n and_o they_o have_v equal_a base_n they_o be_v thence_o equal_a by_o the_o 28_o demonstration_n the_o base_a gc_n contain_v as_o many_o aliquot_fw-la part_n of_o the_o line_n em_n as_o can_v be_v find_v part_n equal_a to_o of_o now_o as_o many_o time_n as_o there_o be_v in_o the_o base_a gc_a part_n equal_a to_o of_o so_o many_o time_n the_o triangle_n agc_n contain_v the_o little_a triangle_n equal_a to_o those_o which_o be_v in_o the_o triangle_n dem_n which_o be_v equal_a among_o themselves_o be_v its_o aliquot_fw-la part_n thence_o as_o many_o time_n as_o the_o base_a gc_n contain_v aliquot_fw-la part_n of_o em_n so_o many_o time_n the_o triangle_n agc_n contain_v aliquot_fw-la part_n of_o the_o triangle_n dem_n which_o will_v happen_v in_o all_o manner_n of_o division_n there_o be_v thence_o the_o same_o reason_n of_o the_o base_a gc_n to_o the_o base_a em_n as_o of_o the_o triangle_n agc_n to_o the_o triangle_n dem_n coral_n parallellogram_n draw_v on_o the_o same_o base_n and_o that_o be_v between_o the_o same_o parallel_n line_n be_v double_a to_o the_o triangle_n by_o the_o 41st_o they_o be_v thence_o in_o the_o same_o reason_n as_o triangle_n that_o be_v to_o say_v in_o the_o same_o reason_n as_o their_o base_n use_v i._n fig._n i._n this_o proposition_n be_v not_o only_o necessary_a to_o demonstrate_v those_o which_o follow_v but_o we_o may_v make_v use_n thereof_o in_o divide_v of_o land_n let_v there_o be_v propose_v a_o trapezium_fw-la abcd_n which_o have_v its_o side_n ad_fw-la bc_n parallel_n and_o admit_v one_o will_v cut_v of_o a_o three_o part_n let_v ce_fw-fr be_v make_v equal_a to_o ad_fw-la and_o bg_n the_o three_o part_n of_o be._n draw_v ag._n i_o say_v that_o the_o triangle_n abg_n be_v the_o three_o of_o the_o trapezium_fw-la abcd._n demonstration_n the_o triangle_n adf_n fce_n be_v equiangle_v because_o of_o the_o parallel_n ad_fw-la ce_fw-fr and_o they_o have_v their_o sides_n ad_fw-la ce_fw-fr equal_a they_o be_v thence_o equal_a by_o the_o 26_o of_o the_o one_a and_o by_o consequence_n the_o triangle_n abe_n be_v equal_a to_o the_o trapezium_fw-la now_o the_o triangle_n abg_n be_v the_o three_o part_n of_o the_o triangle_n abe_n by_o the_o precede_a proposition_n thence_o the_o triangle_n abg_n be_v the_o three_o part_n of_o the_o trapezium_fw-la abcd._n proposition_n ii_o theorem_fw-la a_o line_n be_v draw_v in_o a_o triangle_n parallel_n to_o its_o base_a divide_v its_o side_n proportional_o and_o if_o a_o line_n divide_v proportional_o the_o side_n of_o a_o triangle_n it_o shall_v be_v parallel_n to_o its_o base_a if_o in_o the_o
equal_a they_o may_v be_v join_v after_o such_o manner_n that_o their_o join_a side_n cd_o de_fw-fr be_v on_o one_o straight_a line_n by_o the_o 15_o of_o the_o one_a continue_v the_o side_n ab_fw-la ge_z you_o will_v have_v complete_v the_o parallelogram_n bdeh_n demonstration_n see_v the_o parallelogram_n l_o and_o m_o be_v equal_a they_o shall_v have_v the_o same_o reason_n to_o the_o parallelogram_n bdeh_n now_o the_o reason_n of_o the_o parallelogram_n l_o to_o the_o parallelogram_n bdeh_n be_v the_o same_o with_o that_o of_o the_o base_a cd_o to_o the_o base_a de_fw-fr by_o the_o one_a and_o that_o of_o the_o parallelogram_n m_o or_o dfge_n be_v the_o same_o with_o that_o of_o the_o base_a fd_v to_o the_o base_a bd._n thence_o there_o be_v the_o same_o reason_n of_o cd_o to_o de_fw-fr as_o of_o fd_n to_o bd._n second_o if_o the_o equiangular_a parallellogram_n l_o and_o m_o have_v their_o side_n reciprocal_a they_o shall_v be_v equal_a demonstration_n the_o side_n of_o the_o parallellogram_n be_v reciprocal_a that_o be_v to_o say_v that_o there_o be_v the_o same_o reason_n of_o cd_o to_o de_fw-fr as_o of_o fd_a to_o bd_o now_o as_o the_o base_a cd_o be_v to_o de_fw-fr so_o be_v the_o parallelogram_n l_o to_o the_o parallelogram_n bdeh_n by_o the_o first_o and_o as_o fd_n be_v to_o db_v so_o be_v the_o parallelogram_n m_o to_o bedh_n there_o be_v thence_o the_o same_o reason_n of_o l_o to_o bdeh_n as_o of_o m_n to_o the_o same_o bdeh_n so_o then_o by_o the_o seven_o of_o the_o 5_o the_o parallellogram_n l_o and_o m_o be_v equal_a proposition_n xv._o theorem_fw-la equal_a triangle_n which_o have_v one_o angle_n equal_a have_v the_o side_n which_o form_n that_o angle_v reciprocal_a and_o if_o their_o side_n be_v reciprocal_a they_o shall_v be_v equal_a if_o the_o triangle_n f_o and_o g_o be_v equal_a have_v their_o angle_n acb_n ecd_n equal_a their_o side_n about_o that_o angle_n shall_v be_v reciprocal_a that_o be_v to_o say_v that_o there_o shall_v be_v the_o same_o ratio_fw-la of_o bc_n to_o ce_fw-fr as_o of_o cd_o to_o ca._n dispose_v the_o triangle_n after_o such_o a_o manner_n that_o the_o side_n cd_o ca_n be_v one_o straight_a line_n see_v the_o angle_n acb_n ecd_n be_v suppose_v equal_a the_o line_n bc_n ce_fw-fr shall_v be_v also_o a_o straight_a line_n by_o the_o 14_o of_o the_o one_a draw_v the_o line_n ae_n demonstration_n there_o be_v the_o same_o ratio_fw-la of_o the_o triangle_n abc_n to_o the_o triangle_n ace_n as_o of_o the_o triangle_n ecd_v equal_a to_o the_o first_o to_o the_o same_o triangle_n ace_n by_o the_o seven_o of_o the_o 5_o now_o as_o abc_n be_v to_o ace_n so_o be_v the_o base_a bc_n to_o the_o base_a ce_fw-fr by_o the_o one_a see_v they_o have_v the_o same_o vertical_a a_o and_o as_o fcd_n be_v to_o ace_n so_o be_v the_o base_a cd_o to_o ca._n now_o if_o it_o be_v suppose_v that_o the_o side_n be_v reciprocal_a that_o be_v to_o say_v that_o there_o be_v the_o same_o ratio_fw-la of_o bc_n to_o ce_fw-fr as_o of_o cd_o to_o ca_n the_o triangle_n abc_n cde_v shall_v be_v equal_a because_o they_o will_v then_o have_v the_o same_o ratio_fw-la to_o the_o triangle_n ace_n proposition_n xvi_o theorem_fw-la if_o four_o line_n be_v proportional_a the_o rectangle_n comprehend_v under_o the_o first_o and_o four_o be_v equal_a to_o the_o rectangle_n comprehend_v under_o the_o second_o and_o three_o and_o if_o the_o rectangle_n comprehend_v under_o the_o extremes_n be_v equal_a to_o the_o rectangle_n comprehend_v under_o the_o mean_n then_o be_v the_o four_o line_n proportional_a if_o the_o line_n ab_fw-la cd_o be_v proportional_a that_o be_v to_o say_v that_o there_o be_v the_o same_o ratio_fw-la of_o a_o to_z b_o as_o of_o c_z to_z d_o the_o rectangle_n comprehend_v under_o the_o first_o a_o and_z the_o fourth_z d_o shall_v be_v equal_a to_o the_o rectangle_n comprend_v under_o b_o and_z c._n demonstration_n the_o rectangle_v have_v their_o angle_n equal_a see_v it_o be_v right_o they_o have_v also_o their_o side_n reciprocal_a they_o be_v thence_o equal_a by_o the_o 16_o in_o like_a manner_n if_o they_o be_v equal_a their_o side_n be_v reciprocal_a that_o be_v to_o say_v that_o there_o be_v the_o same_o ratio_fw-la of_o a_o to_z b_o as_o of_o c_z to_z d._n proposition_n xvii_o theorem_fw-la if_o three_o line_n be_v proportional_a the_o rectangle_n comprehend_v under_o the_o first_o and_o three_o be_v equal_a to_o the_o square_n of_o the_o mean_a and_o if_o the_o square_a of_o the_o mean_v be_v equal_a to_o the_o rectangle_n of_o the_o extremes_n the_o three_o line_n be_v then_o proportional_a if_o the_o three_o line_n a_o b_o d_o be_v proportional_a the_o rectangle_n comprehend_v under_o a_o and_z under_z d_o shall_v be_v equal_a to_o the_o square_n of_o b._n take_v c_o equal_v to_o b_o there_o shall_v be_v the_o same_o ratio_fw-la of_o a_o to_z b_o as_o of_o c_z to_z d_o thence_o the_o four_o line_n a_o b_o c_o d_o be_v proportional_a demonstration_n the_o rectangle_n under_o a_o and_o d_o shall_v be_v equal_a to_o the_o rectangle_n under_o b_o and_z c_z by_o the_o forego_n now_o this_o last_o rectangle_n be_v a_o square_a see_v the_o line_n b_o and_z c_o be_v equal_a thence_o the_o rectangle_n under_o a_o and_o d_o be_v equal_a to_o the_o square_n of_o b._n in_o like_a manner_n if_o the_o rectangle_n under_o a_o and_o d_o be_v equal_a to_o the_o square_n of_o b_o there_o shall_v be_v the_o same_o ratio_fw-la of_o a_o to_z b_o as_o of_o c_z to_z d_o and_o see_v that_o b_o and_o c_o be_v equal_a there_o shall_v be_v the_o same_o ratio_fw-la of_o a_o to_z b_o as_o of_o b_o to_z d._n use_v those_o four_o proposition_n demonstrate_v that_o rule_n of_o arithmetic_n which_o we_o common_o call_v the_o rule_n of_o three_o and_o consequent_o the_o rule_n of_o fellowship_n false_a position_n and_o all_o other_o which_o be_v perform_v by_o proportion_n for_o example_n let_v there_o be_v propose_v these_o three_o number_n a_o eight_o b_o six_z c_z four_z and_o it_o be_v require_v to_o find_v the_o four_o proportional_a suppose_v it_o to_o be_v find_v and_o let_v it_o be_v d._n the_o rectangle_n comprehend_v under_o a_o and_o d_o be_v equal_a to_o to_o the_o rectangle_n comprehend_v under_o b_o and_z c._n now_o i_o can_v have_v this_o rectangle_n by_o multiply_v b_o by_o c_z that_o be_v to_o say_v six_o by_o four_o and_o i_o shall_v have_v twenty_o four_o thence_o the_o rectangle_n comprehend_v under_o a_o and_o d_o be_v twenty_o four_o wherefore_o divide_v the_o same_o by_o a_o eight_o the_o quotient_a three_o be_v the_o number_n i_o look_v for_o proposition_n xviii_o theorem_fw-la to_o describe_v a_o poligon_n like_o to_o another_o on_o a_o line_n give_v there_o be_v propose_v the_o line_n ab_fw-la on_o which_o one_o will_v describe_v a_o poligon_n like_o unto_o the_o poligon_n cfde_v have_v divide_v the_o poligon_n cfde_v into_o triangle_n make_v on_o the_o line_n ab_fw-la a_o triangle_n abh_n like_a unto_o the_o triangle_n cfe_n that_o be_v to_o say_v make_v the_o angle_n abh_n equal_a to_o the_o angle_n cfe_n and_o bah_o equal_a to_o fce_n so_o then_o the_o triangle_n abh_n cfe_n shall_v be_v equiangle_v by_o the_o 32d_o of_o the_o first_o make_v also_o on_o bh_n a_o triangle_n equiangle_v to_o fde_a demonstration_n see_v the_o triangle_n which_o be_v part_n of_o the_o poligon_n be_v equiangular_a the_o two_o poligon_n be_v equiangular_a moreover_o see_v the_o triangle_n abh_n cfe_n be_v equiangular_a there_o be_v the_o same_o ratio_fw-la of_o ab_fw-la to_o bh_n as_o of_o cf_n to_o fe_o by_o the_o four_o in_o like_a manner_n the_o triangle_n hbg_n efd_n be_v equiangular_a there_o shall_v be_v the_o same_o ratio_fw-la of_o bh_n to_o bg_n as_o of_o fe_o to_o fd_n and_o by_o equality_n there_o shall_v be_v the_o same_o ratio_fw-la of_o ab_fw-la to_o bg_n as_o of_o cf_n to_o fd._n and_o so_o of_o the_o rest_n of_o the_o side_n thence_o by_o the_o first_o definition_n the_o poligon_n be_v like_a to_o each_o other_o use_v it_o be_v on_o this_o proposition_n we_o establish_v the_o great_a part_n of_o the_o practical_a way_n to_o take_v the_o plane_n of_o a_o place_n of_o a_o edifice_n of_o a_o field_n of_o a_o forest_n or_o of_o a_o whole_a country_n for_o make_v use_n of_o the_o equal_a part_n of_o a_o line_n for_o foot_n or_o for_o chain_n we_o describe_v a_o figure_n like_o unto_o the_o prototype_n but_o lesser_a in_o which_o we_o may_v see_v the_o proportion_n of_o all_o its_o line_n and_o because_o it_o be_v easy_a on_o paper_n than_o on_o the_o ground_n we_o may_v comprehend_v in_o this_o proposition_n all_o geodesâ—…_n all_o chorography_n all_o geographical_a chart_n and_o way_n of_o reduce_v of_o the_o
bed_n dec_n by_o the_o 4_o i_o may_v prove_v by_o the_o same_o manner_n of_o argument_n that_o bf_n cf_n be_v equal_a demonstration_n the_o angel_n bed_n dec_n be_v equal_a it_o be_v also_o suppose_a that_o the_o angel_n bed_n aef_n be_v equal_a thence_o the_o opposite_a angle_n dec_n aef_n shall_v be_v equal_a and_o by_o the_o cor._n of_o the_o 15_o prop._n aec_fw-la be_v a_o straight_a line_n by_o consequence_n afc_n be_v a_o triangle_n in_o which_o the_o side_n of_o fc_n be_v great_a than_o aec_fw-la that_o be_v to_o say_v ae_n ebb_v for_o the_o line_n ae_n fc_n be_v equal_a to_o of_o ebb_n therefore_o the_o line_n of_o fb_n be_v great_a than_o ae_n be_v and_o since_o natural_a cause_n do_v act_n by_o the_o most_o short_a line_n therefore_o all_o reflection_n be_v make_v after_o this_o sort_n that_o the_o angle_n of_o reflection_n and_o of_o incidence_n be_v equal_a xv._o prop._n xv._o moreover_o because_o we_o can_v easy_o prove_v that_o all_o the_o angle_n which_o be_v make_v upon_o a_o plain_a by_o the_o meeting_n of_o never_o so_o many_o line_n in_o the_o same_o point_n be_v equal_a to_o four_o right_a angle_n since_o that_o in_o the_o first_o figure_n of_o this_o proposition_n the_o angle_n aec_fw-la aed_n be_v equal_a to_o two_o right_a as_o also_o bed_n bec_n we_o make_v this_o general_a rule_n to_o determine_v the_o number_n of_o polygon_n which_o may_v be_v join_v together_o to_o pave_v a_o a_o floor_n so_o we_o say_v that_o four_o square_n six_o triangle_n three_o hexigones_n may_v pave_v the_o same_o and_o that_o it_o be_v for_o this_o reason_n the_o bee_n make_v their_o cellules_a hexagonal_a proposition_n xvi_o theorem_fw-la the_o exterior_a angle_n of_o a_o triangle_n be_v great_a than_o either_o of_o the_o inward_a and_o opposite_a angle_n continue_v the_o side_n bc_n of_o the_o triangle_n abc_n i_o say_v that_o the_o exterior_a angle_n acd_v be_v great_a than_o the_o interior_a opposite_a angle_n abc_n or_o bac_n imagine_v that_o the_o triangle_n abc_n move_v along_o bd_o and_o that_o it_o be_v transport_v to_o ced_a demonstration_n it_o be_v impossible_a that_o the_o triangle_n abc_n shall_v be_v thus_o move_v and_o that_o the_o point_v a_o shall_v not_o alter_v its_o place_n move_v towards_o e_o now_o if_o it_o move_v towards_o e_o the_o angle_z ecd_v that_o be_v to_o say_v abc_n be_v less_o than_o the_o angle_n acd_v therefore_o the_o interiour_n angle_n abc_n be_v less_o than_o the_o exterior_a acd_v it_o be_v easy_a to_o prove_v that_o the_o angle_n a_o be_v also_o less_o than_o the_o exterior_a acd_v for_o have_v continue_v the_o side_n ac_fw-la to_o f_o the_o opposite_a angle_n bcf_n acd_v be_v equal_a by_o the_o 15_o and_o make_v the_o triangle_n abc_n to_o slide_v along_o the_o line_n acf_n it_o be_v demonstrable_a that_o the_o angle_n bcf_n be_v great_a than_o the_o angle_n a._n use_v 16._o use_v 16._o we_o draw_v from_o this_o proposition_n several_a very_o useful_a conclusion_n the_o first_o that_o from_o a_o give_v point_n there_o can_v be_v draw_v no_o more_o than_o one_o perpendicular_a to_o the_o same_o line_n example_n let_v the_o line_n ab_fw-la be_v perpendicular_a to_o the_o line_n bc_n i_o say_v ac_fw-la shall_v not_o be_v perpendicular_a thereto_o because_o the_o right_a angle_n abc_n shall_v be_v great_a than_o the_o interiour_n acb_n therefore_o acb_n shall_v not_o be_v a_o right_a angle_n neither_o ac_fw-la a_o perpendicular_a second_o that_o from_o the_o same_o point_v a_o there_o can_v only_o be_v draw_v two_o equal_a line_n for_o example_n ac_fw-la ad_fw-la and_o that_o if_o you_o draw_v a_o three_o ae_n it_o shall_v not_o be_v equal_a for_o since_o ac_fw-la ad_fw-la be_v equal_a the_o angles_n acd_v adc_a be_v equal_a by_o the_o 5_o now_o in_o the_o triangle_n aec_fw-la the_o exterior_a angle_n acb_n be_v great_a than_o the_o interiour_n aec_fw-la and_o thus_o be_v the_o angle_n ade_n great_a than_o aed_n therefore_o the_o line_n ae_n great_a than_o ad_fw-la and_o by_o consequence_n ac_fw-la ae_n be_v not_o equal_a three_o if_o the_o line_n ac_fw-la make_v the_o angle_n acb_n acute_a and_o acf_n obtuse_a the_o perpendicular_a draw_v from_o a_o shall_v fall_v on_o the_o same_o side_n which_o the_o acute_a angle_n be_v of_o for_o if_o one_o shall_v say_v that_o ae_n be_v a_o perpendicular_a and_o the_o angle_n aef_n be_v right_o then_o the_o right_a angle_n aef_n will_v be_v great_a than_o the_o obtuse_a angle_n ace_n those_o conclusion_n we_o make_v use_v of_o to_o measure_v all_o parallellogram_n triangle_n and_o trapeziams_n and_o to_o reduce_v they_o into_o rectangular_a figure_n proposition_n xvii_o theorem_fw-la two_o angle_n of_o any_o triangle_n be_v less_o than_o two_o right_a angle_n let_v abc_n be_v the_o propose_a triangle_n i_o say_v that_o two_o of_o the_o angle_n take_v together_o bac_n bca_n be_v less_o than_o two_o right_a angle_n continue_v the_o side_n ca_n to_o d._n demonstration_n the_o interiour_n angle_n c_o be_v less_o than_o the_o exterior_a bad_a by_o the_o 16_o add_v to_o both_o the_o angle_n bac_n the_o angles_n bac_n bca_n shall_v be_v less_o than_o the_o angles_n bac_n bad_a which_o latter_a be_v equal_a to_o two_o right_v by_o the_o 13_o therefore_o the_o angle_n bac_n bca_n be_v less_o than_o two_o right_n i_o may_v demonstrate_v after_o the_o same_o manner_n that_o the_o angel_n abc_n acb_n be_v less_o than_o two_o right_n by_o continue_v the_o side_n bc._n coral_n if_o one_o angle_n of_o a_o triangle_n be_v either_o right_a or_o obtuse_a the_o other_o two_o shall_v be_v acute_a this_o proposition_n be_v necessary_a to_o demonstrate_v those_o which_o follow_v proposition_n xviii_o theorem_fw-la the_o great_a side_n of_o every_o triangle_n subtend_v the_o great_a angle_n let_v the_o side_n bc_n of_o the_o triangle_n abc_n be_v great_a than_o the_o side_n ac_fw-la i_o say_v that_o the_o angle_n bac_n which_o be_v opposite_a to_o the_o side_n bc_n be_v great_a than_o the_o angle_n b_o which_o be_v opposite_a to_o the_o side_n ac_fw-la cut_a off_o from_o bc_n the_o line_n cd_o equal_a to_o ac_fw-la and_o draw_v ad._n demonstration_n see_v that_o the_o side_n ac_fw-la cd_o be_v equal_a the_o triangle_n acb_n be_v a_o isosceles_a triangle_n by_o the_o 5_o the_o angles_n cda_n god_n be_v therefore_o equal_a now_o the_o whole_a angle_n bac_n be_v great_a than_o the_o angle_n god_n thence_o the_o angle_n bac_n be_v great_a than_o the_o angle_n cda_n which_o be_v exterior_a in_o regard_n of_o the_o triangle_n abdella_n be_v great_a than_o the_o interior_a b_o by_o the_o 16_o therefore_o the_o angle_n bac_n be_v great_a than_o the_o angle_n b._n proposition_n xix_o theorem_fw-la in_o every_o triangle_n the_o great_a angle_n be_v opposite_a to_o the_o great_a side_n let_v the_o angle_n a_o of_o the_o triangle_n bac_n be_v great_a than_o the_o angle_n abc_n i_o say_v that_o the_o side_n bc_n opposite_a to_o the_o angle_n a_o be_v great_a than_o the_o side_n ac_fw-la opposite_a to_o the_o angle_n b._n demonstration_n if_o the_o side_n bc_n be_v equal_a to_o the_o side_n ac_fw-la in_o this_o case_n the_o angle_n a_o and_z b_o will_v be_v equal_a by_o the_o 5_o which_o be_v contrary_a to_o the_o hypothesis_n if_o the_o side_n bc_n be_v less_o than_o ac_fw-la than_o the_o angle_n b_o will_v be_v great_a than_o a_o which_o be_v also_o contrary_a to_o the_o hypothesis_n wherefore_o i_o conclude_v that_o the_o side_n bc_n be_v great_a than_o ac_fw-la use_v 19_o use_v 19_o we_o prove_v by_o these_o proposition_n not_o only_o that_o from_o the_o same_o point_n to_o a_o line_n give_v there_o can_v be_v but_o one_o perpendicular_a draw_v but_o also_o that_o that_o perpendicular_a be_v the_o short_a line_n of_o all_o those_o line_n which_o may_v be_v draw_v to_o the_o say_a line_n as_o for_o instance_n if_o the_o line_n rv_n be_v perpendicular_a to_o st_n it_o shall_v be_v short_a than_o r_n because_o the_o angle_n rus_n be_v right_o the_o angle_n rsv_n shall_v be_v acute_a by_o the_o cor._n of_o the_o 17_o and_o the_o line_n rv_n shall_v be_v short_a than_o r_n by_o the_o precede_a for_o this_o reason_n geomatrician_n always_o make_v use_n of_o the_o perpendicular_o in_o their_o measure_n and_o reduce_v irregular_a figure_n into_o those_o who_o angle_n be_v right_o i_o further_o add_v that_o see_v there_o can_v only_o be_v draw_v three_o perpendicular_o to_o one_o and_o the_o same_o point_n it_o can_v be_v imagine_v that_o there_o be_v more_o than_o three_o species_n of_o quantity_n viz._n a_o line_n a_o surface_n and_o a_o solid_a we_o also_o prove_v by_o these_o proposition_n that_o a_o boul_a which_o be_v exact_o round_a be_v put_v on_o a_o plain_a can_v stand_v but_o on_o one_o determinate_a point_n as_o for_o example_n
the_o square_n of_o the_o other_o two_o sides_n ab_fw-la ac_fw-la draw_v the_o line_n ah_o parallel_n to_o bd_o ce_fw-fr and_o draw_v also_o the_o line_n ad_fw-la ae_n fc_n bg_n i_o prove_v that_o the_o square_a of_o be_v equal_a to_o the_o right_o angle_a figure_n or_o long_a square_a bh_n and_o the_o square_a agnostus_n to_o the_o right_o angle_a figure_n ch_z and_o that_o so_o the_o square_n be_v be_v equal_a to_o the_o two_o square_n of_o ag._n demonstration_n the_o triangle_n fbc_n abdella_n have_v their_o sides_n ab_fw-la bf_n bd_o bc_n equal_a and_o the_o angel_n fbc_n abdella_n be_v equal_a see_v that_o each_o of_o they_o beside_o the_o right_a angle_n include_v the_o angle_n abc_n thence_o by_o the_o 4_o the_o triangle_n abdella_n fbc_n be_v equal_a now_o the_o square_n of_o be_v double_a to_o the_o triangle_n fbc_n by_o the_o 41_o because_o they_o have_v the_o same_o base_a bf_n and_o be_v between_o the_o same_o parallel_n bf_a ac_fw-la likewise_o the_o right_o line_a figure_n bh_n be_v double_a to_o the_o triangle_n abdella_n see_v they_o have_v the_o same_o base_a bd_o and_o be_v between_o the_o same_o parallel_n bd_o ah_o therefore_o the_o square_n of_o be_v equal_a to_o the_o right_o line_a figure_n bh_n after_o the_o same_o manner_n the_o triangle_n ace_n gcb_n be_v equal_a by_o the_o 4_o the_o square_a agnostus_n be_v double_a the_o triangle_n bcg_n and_o the_o right_o line_a figure_n ch_n be_v double_a the_o triangle_n ace_n by_o the_o 41_o thence_o the_o square_a agnostus_n be_v equal_a to_o the_o right_o line_a figure_n ch_z and_o by_o consequence_n the_o sum_n of_o the_o square_n of_o agnostus_n be_v equal_a to_o the_o square_a bdec_n use_v 47._o use_v 47._o it_o be_v say_v that_o pythagoras_n have_v find_v this_o proposition_n sacrifice_v one_o hundred_o ox_n in_o thanks_o to_o the_o muse_n it_o be_v not_o without_o reason_n see_v this_o proposition_n serve_v for_o a_o foundation_n to_o a_o great_a part_n of_o the_o mathematics_n for_o in_o the_o first_o place_n trigonometry_n can_v be_v without_o it_o because_o it_o be_v necessary_a to_o make_v the_o table_n of_o all_o the_o line_n that_o can_v be_v draw_v within_o a_o circle_n that_o be_v to_o say_v of_o chord_n of_o sines_n also_o tangent_n and_o secant_v which_o i_o shall_v here_o show_v by_o one_o example_n let_v it_o be_v suppose_v that_o the_o semi-diameter_n ab_fw-la be_v divide_v into_o 10000_o part_n and_o that_o the_o arch_a bc_n be_v 30_o degree_n see_v the_o chord_n or_o subtendent_fw-la of_o 60_o degree_n be_v equal_a to_o the_o semi-diameter_n ac_fw-la bd_o the_o sine_fw-la of_o 30_o degree_n shall_v be_v equal_a to_o the_o half_a of_o ac_fw-la it_o shall_v therefore_o be_v 5000_o in_o the_o right_o angle_a triangle_n adb_n the_o square_a of_o ab_fw-la be_v equal_a to_o the_o square_n of_o bd_o and_o ad_fw-la make_v then_o the_o square_n of_o ab_fw-la by_o multiply_v 10000_o by_o 10000_o and_o from_o that_o product_n subtract_v the_o square_n of_o bd_o 5000_o there_o remain_v the_o square_a of_o ad_fw-la or_o bf_n the_o sine_fw-la of_o the_o compliment_n and_o extract_v the_o square_a root_n there_o be_v find_v the_o line_n fb_n then_o if_o by_o the_o rule_n of_o three_o you_o say_v as_o ad_fw-la be_v to_o bd_o so_o be_v ac_fw-la to_o ce_fw-fr you_o shall_v have_v the_o tangent_fw-la ce_fw-fr and_o add_v together_o the_o square_n of_o ac_fw-la ce_fw-fr you_o shall_v have_v by_o the_o 47_o the_o square_a of_o ae_n and_o by_o extract_v the_o root_n thereof_o you_o shall_v have_v the_o length_n of_o the_o line_n ae_n the_o secant_fw-la use_v 47._o we_o augment_v figure_n as_o much_o as_o we_o please_v by_o this_o proposition_n example_n to_o double_v the_o square_n abcd_v continue_v the_o side_n cd_o and_o make_v de_fw-fr equal_a to_o ad_fw-la the_o square_a of_o ae_n shall_v be_v the_o double_a of_o the_o square_n of_o abcd_n see_v that_o by_o the_o 47_o it_o be_v equal_a to_o the_o square_n of_o ad_fw-la and_o de._n and_o make_v a_o right_a angle_n aef_n and_o take_v of_o equal_a to_o ab_fw-la the_o square_a of_o of_o shall_v be_v triple_a to_o abcd._n and_o make_v again_o the_o right_a angle_n afg_v and_o fg_v equal_a to_o ab_fw-la the_o square_a of_o agnostus_n shall_v be_v quadruple_a to_o to_o abcd._n what_o i_o here_o say_v of_o a_o square_a be_v to_o be_v understand_v of_o all_o figure_n which_o be_v alike_o that_o be_v to_o say_v of_o the_o same_o species_n proposition_n xlviii_o theorem_fw-la if_o the_o two_o square_n make_v upon_o the_o side_n of_o a_o triangle_n be_v equal_a to_o the_o square_n make_v on_o the_o other_o side_n than_o the_o angle_n comprehend_v under_o the_o two_o other_o side_n of_o the_o triangle_n be_v a_o right_a angle_n if_o the_o square_a of_o the_o side_n np_n be_v equal_a to_o the_o square_n of_o the_o sides_n nl_n lp_n take_v together_o the_o angle_n nlp_n shall_v be_v a_o right_a angle_n draw_v lr_n perpendicular_a to_o nl_n and_o equal_a to_o lp_v then_o draw_v the_o line_n nr_n demonstration_n in_o the_o right_o angle_a triangle_n nlr_n the_o square_a of_o nr_n be_v equal_a to_o the_o square_n of_o nl_n and_o of_o lr_n or_o lp_v by_o the_o 47_o now_o the_o square_n of_o np_n be_v equal_a to_o the_o same_o square_n of_o nl_n lp_v therefore_o the_o square_n of_o nr_n be_v equal_a to_o that_o of_o np_n and_o by_o consequence_n the_o line_n nr_n np_n be_v equal_a and_o because_o the_o triangle_n nlr_n nlp_n have_v each_o of_o they_o the_o side_n nl_n common_a and_o that_o their_o base_n rn_v np_n be_v also_o equal_a the_o angel_n nlp_n nlr_n shall_v be_v equal_a by_o the_o 8_o and_o the_o angle_n nlr_n be_v a_o right_a angle_n the_o angle_z nlp_n shall_v be_v also_o a_o right_a angle_n the_o end_n of_o the_o first_o book_n the_o second_o book_n of_o euclid_n element_n euclid_n treat_v in_o this_o book_n of_o the_o power_n of_o straight_a line_n that_o be_v to_o say_v of_o their_o square_n compare_v the_o divers_a rectangle_v which_o be_v make_v on_o a_o line_n divide_v as_o well_o with_o the_o square_a as_o with_o the_o rectangle_n of_o the_o whole_a line_n this_o part_n be_v very_o useful_a see_v it_o serve_v for_o a_o foundation_n to_o the_o practical_a principle_n of_o algebra_n the_o three_o first_o proposition_n demonstrate_v the_o three_o rule_n of_o arithmetic_n the_o four_o teach_v we_o to_o find_v the_o square_a root_n of_o any_o number_n whatsoever_o those_o which_o follow_v unto_o the_o eight_o serve_v in_o several_a accident_n happen_v in_o algebra_n the_o remain_a proposition_n to_o the_o end_n of_o this_o book_n be_v conversant_a in_o trigonometry_n this_o book_n appear_v at_o the_o first_o sight_n very_o difficult_a because_o one_o do_v imagine_v that_o it_o contain_v mysterious_a or_o intricate_a matter_n notwithstanding_o the_o great_a part_n of_o the_o demonstration_n be_v found_v on_o a_o very_a evident_a principle_n viz._n that_o the_o whole_a be_v equal_a to_o all_o its_o part_n take_v together_o therefore_o one_o ought_v not_o to_o be_v discourage_v although_o one_o do_v not_o apprehend_v the_o demonstration_n of_o this_o book_n at_o the_o first_o read_v definition_n boook_v def._n 1._o of_o the_o second_o boook_v a_o rectangular_a parallelogram_n be_v comprehend_v under_o two_o right_a line_n which_o at_o their_o intersection_n contain_v a_o right_a angle_n it_o be_v to_o be_v note_v henceforward_o that_o we_o call_v that_o figure_n a_o rectangular_a parallelogram_n which_o have_v all_o its_o angle_n right_o and_o that_o the_o same_o shall_v be_v distinguish_v as_o much_o at_o be_v requisite_a if_o we_o give_v thereto_o length_n and_o breadth_n name_v only_o two_o of_o its_o line_n which_o comprehend_v any_o one_o angle_n as_o the_o line_n ab_fw-la bc_n for_o the_o rectangular_a parallelogram_n abcd_v be_v comprehend_v under_o the_o line_n ab_fw-la bc_n have_v bc_n for_o its_o length_n and_o ab_fw-la for_o its_o breadth_n whence_o it_o be_v not_o necessary_a to_o mention_v the_o other_o line_n because_o they_o be_v equal_a to_o those_o already_o speak_v of_o i_o have_v already_o take_v notice_n that_o the_o line_n ab_fw-la be_v in_o a_o perpendicular_a position_n in_o respect_n of_o bc_n produce_v the_o rectangle_n abcd_n if_o move_v along_o the_o line_n bc_n and_o that_o this_o motion_n represent_v arithmetical_a multiplication_n in_o this_o manner_n as_o the_o line_n ab_fw-la move_v along_o the_o line_n bc_n that_o be_v to_o say_v take_v as_o many_o time_n as_o there_o be_v point_n in_o bc_n compose_v the_o rectangle_n abcd_v wherefore_o multiply_v ab_fw-la by_o bc_n i_o shall_v have_v the_o rectangle_n abcd._n as_o suppose_v i_o know_v the_o number_n of_o mathematical_a point_n there_o be_v in_o the_o line_n ab_fw-la for_o example_n let_v there_o be_v 40_o and_o that_o in_o bc_n
for_o pentagon_n be_v the_o most_o ordinary_a you_o must_v also_o take_v notice_n that_o these_o way_n of_o describe_v a_o pentagon_n about_o a_o circle_n may_v be_v apply_v to_o the_o other_o polygon_n i_o have_v give_v another_o way_n to_o inscribe_v a_o regular_a pentagon_n in_o a_o circle_n in_o military_a architecture_n proposition_n xv._o problem_n to_o inscribe_v a_o regular_a hexagon_n in_o a_o circle_n to_o inscribe_v a_o regular_a hexagon_n in_o the_o circle_n abcdef_n draw_v the_o diameter_n ad_fw-la and_o put_v the_o foot_n of_o the_o compass_n in_o the_o point_n d_o describe_v a_o circle_n at_o the_o open_a dg_n which_o shall_v intersect_v the_o circle_n in_o the_o point_n aec_fw-la then_o draw_v the_o diameter_n egb_n cgf_n and_o the_o line_n ab_fw-la of_o and_o the_o other_o demonstration_n it_o be_v evident_a that_o the_o triangle_n cdg_n dge_n be_v equilateral_a wherefore_o the_o angle_n cgd_v dge_n and_o their_o opposite_n bga_n agf_n be_v each_o of_o they_o the_o three_o part_n of_o two_o right_n and_o that_o be_v 60_o degree_n now_o all_o the_o angle_n which_o can_v be_v make_v about_o one_o point_n be_v equal_a to_o four_o right_n that_o be_v to_o say_v 360._o so_o take_v away_o four_o time_n 60_o that_o be_v 240_o from_o 360_o there_o remain_v 120_o degree_n for_o bgc_n and_o fge_n whence_o they_o shall_v each_o be_v 60_o degree_n so_o all_o the_o angle_n at_o the_o centre_n be_v equal_a all_o the_o ark_n and_o all_o the_o side_n shall_v be_v equal_a and_o each_o angle_n a_o b_o c_o etc._n etc._n shall_v be_v compose_v of_o two_o angle_n of_o sixty_o that_o be_v to_o say_v one_o hundred_o and_o twenty_o degree_n they_o shall_v therefore_o be_v equal_a coral_n the_o side_n of_o a_o hexagon_n be_v equal_a to_o the_o semi_a diameter_n use_v because_o that_o the_o side_n of_o a_o hexagon_n be_v the_o base_a of_o a_o ark_n of_o sixty_o degree_n and_o that_o be_v equal_a to_o the_o semi-diameter_n its_o half_n be_v the_o sine_fw-la of_o thirty_o and_o it_o be_v with_o this_o sine_fw-la we_o begin_v the_o table_n of_o sines_n euclid_n treat_v of_o hexagon_n in_o the_o last_o book_n of_o his_o element_n proposition_n xvi_o problem_n to_o inscribe_v a_o regular_a pentadecagon_n in_o a_o circle_n inscribe_v in_o a_o circle_n a_o equilateral_a triangle_n abc_n by_o the_o 2d_o and_o a_o regular_a pentagon_n by_o the_o 11_o in_o such_o sort_n that_o the_o angle_n meet_v in_o the_o point_n a._n the_o line_n bf_n by_o je_n shall_v be_v the_o side_n of_o the_o pentadecagon_n and_o by_o inscribe_v in_o the_o other_o ark_n line_n equal_a to_o bf_n by_o you_o may_v complete_a this_o polygon_n demonstration_n see_v the_o line_n ab_fw-la be_v the_o side_n of_o the_o equilateral_a triangle_n the_o ark_n aeb_fw-mi shall_v be_v the_o three_o of_o the_o whole_a circle_n or_o 5_o fifteenth_n and_o the_o ark_n ae_n be_v the_o five_o part_n it_o shall_v contain_v 3_o 15_o thence_o ebb_n contain_v two_o and_o if_o you_o divide_v it_o in_o the_o middle_n in_o i_o each_o part_n shall_v be_v a_o fifteen_o use_v this_o proposition_n serve_v only_o to_o open_v the_o way_n for_o other_o polygon_n we_o have_v in_o the_o compass_n of_o proportion_n very_o easy_a method_n to_o inscribe_v all_o the_o ordinary_a polygon_n but_o they_o be_v ground_v on_o this_o for_o one_o can_v not_o put_v polygon_n on_o that_o instrument_n if_o one_o do_v not_o find_v their_o side_n by_o this_o proposition_n or_o such_o like_a the_o end_n of_o the_o four_o book_n the_o five_o book_n of_o euclid_n element_n this_o five_o book_n be_v absolute_o necessary_a to_o demonstrate_v the_o propoposition_n of_o the_o six_o book_n it_o contain_v a_o most_o universal_a doctrine_n and_o a_o way_n of_o argue_v by_o proportion_n which_o be_v most_o subtle_a solid_a and_o brief_a so_o that_o all_o treatise_n which_o be_v found_v on_o proportion_n can_v be_v without_o this_o mathematical_a logic_n geometry_n arithmetic_n music_n astronomy_n staticks_n and_o to_o say_v in_o one_o word_n all_o the_o treatise_n of_o the_o science_n be_v demonstrate_v by_o the_o proposition_n of_o this_o book_n the_o great_a part_n of_o measure_v be_v do_v by_o proportion_n and_o in_o practisal_n geometry_n and_o one_o may_v demonstrate_v all_o the_o rule_n of_o arithmetic_n by_o the_o theorem_n hereof_o wherefore_o it_o be_v not_o necessary_a to_o have_v recourse_n to_o the_o seven_o eight_o or_o nine_o book_n the_o music_n of_o the_o ancient_n be_v scarce_o any_o thing_n else_o but_o the_o doctrine_n of_o proportion_n apply_v to_o the_o sense_n it_o be_v the_o same_o in_o staticks_n which_o consider_v the_o proportion_n of_o weight_n in_o fine_a one_o may_v affirm_v that_o if_o one_o shall_v take_v away_o from_o mathematician_n the_o knowledge_n of_o the_o proposition_n that_o this_o book_n give_v we_o the_o remainder_n will_v be_v of_o little_a use_n definition_n the_o whole_a correspond_v to_o its_o part_n and_o this_o shall_v be_v the_o great_a quantity_n compare_v with_o the_o lesser_a whether_o it_o contain_v the_o same_o in_o effect_n or_o that_o it_o do_v not_o contain_v the_o same_o part_n or_o quantity_n take_v in_o general_n be_v divide_v ordinary_o into_o aliquot_fw-la part_n and_o aliquant_a part_n 1._o an_fw-mi aliquot_fw-la part_n which_o euclid_n define_v in_o this_o book_n be_v a_o magnitude_n of_o a_o magnitude_n the_o lesser_a of_o the_o great_a when_o it_o measure_v it_o exact_o that_o be_v to_o say_v that_o it_o be_v a_o lesser_a quantity_n compare_v with_o a_o great_a which_o it_o measure_v precise_o as_o the_o line_n of_o two_o foot_n take_v three_o time_n be_v equal_a to_o a_o line_n of_o six_o foot_n 2._o multiplex_fw-la be_v a_o magnitude_n of_o a_o magnitude_n the_o great_a of_o the_o lesser_a when_o the_o lesser_a measure_v the_o great_a that_o be_v to_o say_v that_o multiplex_n be_v a_o great_a quantity_n compare_v with_o a_o lesser_a which_o it_o contain_v precise_o some_o number_n of_o time_n for_o example_n the_o line_n of_o six_o foot_n be_v treble_a to_o a_o line_n of_o two_o foot_n aliquant_a part_n be_v a_o lesser_a quantity_n compare_v with_o a_o great_a which_o it_o measure_v not_o exact_o so_o a_o line_n of_o 4_o foot_n be_v a_o aliquant_a part_n of_o a_o line_n of_o 10_o foot_n equimultiplex_v be_v magnitude_n which_o contain_v equal_o their_o aliquot_fw-la part_n that_o be_v to_o say_v the_o same_o number_n of_o time_n 12._o 4._o 6._o 2._o a_o b_o c_o d_o for_a example_n if_o a_o contain_v as_o many_o time_n b_o as_o c_z contain_v d_o a_o and_z c_o shall_v be_v equal_a multiplex_v of_o b_o and_z d._n 3._o reason_n or_o ratio_fw-la be_v a_o mutual_a habitude_n or_o respect_n of_o one_o magnitude_n to_o another_o of_o the_o same_o species_n i_o have_v add_v of_o the_o same_o species_n 4._o for_o euclid_n say_v that_o magnitude_n have_v the_o same_o reason_n when_o be_v multiply_v they_o may_v surpass_v each_o other_o to_o do_v which_o they_o must_v be_v of_o the_o same_o species_n in_o effect_n a_o line_n have_v no_o manner_n of_o reason_n with_o a_o surface_n because_o a_o line_n take_v mathematical_o be_v consider_v without_o any_o breadth_n so_o that_o if_o it_o be_v multiply_v as_o many_o time_n as_o you_o please_v it_o give_v no_o breadth_n and_o notwithstanding_o a_o surface_n have_v breadth_n see_v that_o reason_n be_v a_o mutal_a habitude_n or_o respect_n of_o a_o magnitude_n to_o another_o it_o ought_v to_o have_v two_o term_n that_o which_o the_o philosopher_n will_v call_v foundation_n be_v name_v by_o the_o mathematician_n antecedent_n and_o the_o term_n be_v call_v consequent_a as_o if_o we_o compare_v the_o magnitude_n a_o to_o the_o magnitude_n i_o this_o habitude_n or_o reason_n shall_v have_v for_o antecedent_n the_o quantity_n a_o and_o for_o consequent_a the_o quantity_n b._n as_o on_o the_o contrary_a if_o we_o compare_v the_o magnitude_n b_o with_o a_o this_o reason_n of_o b_o to_o a_o shall_v have_v for_o antecedent_n the_o magnitude_n b_o and_o for_o consequent_a the_o magnitude_n a._n the_o reason_n or_o habitude_n of_o one_o magnitude_n to_o another_o be_v divide_v into_o rational_a reason_n and_o irrational_a reason_n rational_a reason_n be_v a_o habitude_n of_o one_o magnitude_n to_o another_o which_o be_v commensurable_a thereto_o that_o be_v to_o say_v that_o those_o magnitude_n have_v a_o common_a measure_n which_o measure_v both_o exact_o as_o the_o reason_n of_o a_o line_n of_o 4_o foot_n to_o a_o line_n of_o 6_o be_v rational_a because_o a_o line_n of_o two_o foot_n measure_v both_o exact_o and_o when_o this_o happen_v those_o magnitude_n have_v the_o same_o reason_n as_o one_o number_n have_v to_o another_o for_o example_n because_o that_o the_o line_n of_o two_o foot_n which_o be_v the_o common_a measure_n be_v find_v twice_o in_o the_o line_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
four_o of_o bd_o as_o a_o contain_v the_o four_o of_o b_o and_o what_o i_o say_v of_o the_o four_o verifi_v itself_o of_o all_o other_o aliquot_fw-la part_n there_o be_v therefore_o the_o same_o reason_n of_o a_o to_o b_o as_o of_o ac_fw-la to_o bd._n proposition_n xiii_o theorem_fw-la if_o of_o two_o equal_a reason_n the_o one_o be_v great_a than_o a_o three_o the_o other_o shall_v be_v so_o likewise_o ab_fw-la cd_o ef._n 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 and_o that_o there_o be_v a_o great_a reason_n of_o a_o to_o b_o than_z of_o e_z to_o f_o i_o say_v that_o there_o shall_v be_v a_o great_a reason_n of_o c_o to_o d_o than_z of_o e_z to_z f._n demonstration_n see_v there_o be_v a_o great_a reason_n of_o a_o to_o b_o than_z of_o e_z to_o f_o a_o shall_v contain_v more_o time_n any_o aliquot_fw-la part_n of_o b_o than_z e_z contain_v a_o like_a aliquot_fw-la part_n of_o f_o by_o the_o 6_o definition_n now_o c_z contain_v a_o like_a aliquot_fw-la part_n of_o d_o as_o many_o time_n as_o a_o contain_v that_o of_o b_o see_v 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 so_o then_o c_z contain_v a_o aliquot_fw-la part_n of_o d_o more_o time_n than_o e_z contain_v a_o like_a aliquot_fw-la part_n of_o f_o thence_o there_o be_v a_o great_a reason_n of_o c_o to_o d_o than_z of_o e_z to_z f._n proposition_n fourteen_o theorem_fw-la if_o there_o be_v the_o same_o reason_n of_o the_o first_o magnitude_n to_o the_o second_o as_o of_o the_o three_o to_o the_o four_o if_o the_o first_o be_v great_a equal_a or_o less_o than_o the_o three_o the_o second_o shall_v be_v great_a equal_a or_o less_o than_o the_o four_o a_o b_o c_o d_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 i_o say_v in_o the_o first_o place_n that_o if_o a_o be_v great_a than_o c_o b_o shall_v also_o be_v great_a than_o d._n demonstration_n see_v that_o a_o be_v great_a than_o c_o there_o will_v be_v by_o the_o 9th_o a_o great_a reason_n of_o a_o to_o b_o than_z of_o c_z to_z b._n now_o as_o a_o be_v to_o b_o so_o be_v c_z to_z d_o therefore_o there_o shall_v be_v a_o great_a reason_n of_o c_o to_o d_o than_z of_o c_z to_z b_o and_o by_o consequence_n according_a to_o the_o ten_o b_o shall_v be_v great_a than_o d._n i_o say_v in_o the_o second_o place_n that_o if_o a_o be_v equal_a to_o c_o b_o shall_v be_v also_o equal_a to_o d._n demonstration_n see_v that_o a_o and_o c_o be_v equal_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 b_o by_o the_o seven_o now_o as_o a_o be_v to_o b_o so_o be_v c_z to_z d_o thence_o there_o be_v the_o same_o reason_n of_o c_o to_o b_o as_o of_o c_z to_z d_o and_o by_o consequence_n b_o and_o d_o be_v equal_a by_o the_o nine_o and_o in_o the_o three_o place_n if_o a_o be_v less_o than_o c_o b_o shall_v be_v also_o less_o than_o d._n demonstration_n see_v that_o a_o be_v less_o than_o c_o the_fw-mi will_v be_v a_o lesser_a reason_n of_o a_o to_o b_o than_z of_o c_z to_z b_o by_o the_o 8_o now_o as_o a_o be_v to_o b_o so_o be_v c_z to_z d_o thence_o there_o will_v be_v a_o lesser_a reason_n of_o c_o to_o d_o than_z of_o c_z to_z b_o and_o by_o consequence_n according_a to_o the_o 10_o b_o shall_v be_v less_o than_o d._n proposition_n xv._o theorem_fw-la the_o equimultiplice_n and_o like_v aliquot_fw-la part_n be_v in_o the_o same_o reason_n a_o b_o c_o d._n 2._o 3._o 6._o 9_o e_o 2._o h_o 3._o f_o 2._o ay_o 3._o g_o 2._o k_o 3._o if_o the_o magnitude_n c_o and_o d_o are_z equimultiplice_n of_o a_o and_o b_o their_z aliquot_fw-la part_n there_o shall_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_o let_v the_o magnitude_n c_o be_v divide_v into_o part_n equal_a to_o a_o which_o shall_v be_v e_z f_o g_o let_v the_o magnitude_n d_o be_v divide_v into_o part_n equal_a to_o b_o seeing_z that_z c_o and_o d_o be_v equimultiplice_n of_o a_o and_o b_o there_o will_v be_v as_o many_o part_n in_o the_o one_o as_o in_o the_o other_o demonstration_n there_o be_v the_o same_o reason_n of_o e_o to_o h_n of_o f_o to_o i_o of_o g_z to_o k_o as_o of_o a_o to_o b_o seeing_z they_o be_v equal_a thence_o by_o the_o 12_o there_o shall_v be_v the_o same_o reason_n of_o e_o f_o g_o to_o hik_n that_o be_v to_o say_v of_o c_z to_z d_o as_o of_o a_o to_o b._n coral_n the_o same_o number_n of_o aliquot_fw-la part_n of_o two_o magnitude_n be_v in_o the_o same_o reason_n with_o those_o magnitude_n for_o see_v there_o be_v the_o same_o reason_n of_o e_o to_o h_n as_o of_o c_z to_z d_o and_o f_o to_o i_o there_o will_v be_v the_o same_o reason_n of_o of_o to_o he_o as_o of_o c_o to_z d._n proposition_n xvi_o theorem_fw-la alternate_a reason_n if_o four_o magnitude_n of_o the_o same_o species_n be_v proportional_a they_o shall_v also_o be_v proportional_a alternate_o a_o b_o c_o d._n 12._o 8._o 9_o 6._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 and_o if_o the_o four_o magnitude_n be_v of_o the_o same_o species_n that_o be_v to_o say_v if_o all_o four_o be_v line_n or_o all_o four_o superficies_n or_o all_o four_o solid_n there_o will_v 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._n now_o suppose_v that_o there_o be_v a_o great_a reason_n of_o a_o to_o c_o than_o of_o b_o to_z d._n demonstration_n see_v one_o will_v have_v it_o that_o there_o be_v a_o great_a reason_n of_o a_o to_o c_o than_z of_o b_o to_z d_o the_o magnitude_n a_o will_v contain_v a_o aliquot_fw-la part_n of_o c_o for_o example_n the_o three_o more_o time_n than_o b_o contain_v the_o three_o of_o d._n let_v a_o contain_v the_o three_o of_o c_o four_o time_n and_o b_o the_o three_o of_o d_o only_a three_o time_n have_v divide_v a_o into_o four_o part_n the_o three_o of_o c_o shall_v be_v once_o in_o each_o have_v also_o divide_v b_o into_o four_o part_n the_o three_o of_o d_o shall_v not_o be_v in_o each_o thence_o the_o three_o fourths_o of_o a_o shall_v contain_v the_o three_o three_o of_o c_o that_o be_v to_o say_v the_o magnitude_n c_o and_o the_o three_o fourths_o of_o b_o will_v not_o contain_v the_o three_o three_o of_o d_o that_o be_v to_o say_v the_o magnitude_n d._n again_o see_v 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 there_o will_v be_v also_o the_o same_o reason_n of_o the_o three_o fourths_o of_o a_o to_o the_o three_o fourths_o of_o b_o as_o of_o c_z to_z d_o by_o the_o coral_n of_o the_o 15_o and_o by_o the_o 14_o if_o the_o three_o fourths_o of_o a_o be_v great_a than_o c_o the_o three_o fourth_n of_o b_o shall_v be_v great_a than_o d_o although_o we_o have_v demonstrate_v the_o contrary_a lemma_n if_o there_o be_v the_o same_o reason_n of_o the_o first_o magnitude_n to_o the_o second_o as_o of_o the_o three_o to_o the_o four_o a_o aliquot_fw-la part_n of_o the_o first_o shall_v have_v the_o same_o reason_n to_o the_o second_o as_o a_o like_o aliquot_fw-la part_n of_o the_o three_o have_v to_o the_o four_o 16_o 3._o 32._o 6._o a_o b_o c_o d._n e_o  _fw-fr f_o  _fw-fr 4._o  _fw-fr 8._o  _fw-fr 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 that_o e_o be_v a_o aliquot_fw-la part_n of_o a_o and_o f_o a_o like_a aliquot_fw-la part_n of_o c_o i_o say_v there_o be_v the_o same_o reason_n of_o e_z to_z b_o as_o of_o f_o to_o d._n demonstration_n if_o there_o be_v a_o great_a reason_n of_o e_z to_z b_o than_o of_o f_o to_o d_o e_z will_v contain_v a_o aliquot_fw-la part_n of_o b_o more_o time_n than_o f_o contain_v a_o like_a aliquot_fw-la part_n of_o d._n then_o e_z take_v twice_o thrice_o or_o four_o time_n will_v contain_v a_o aliquot_fw-la part_n of_o b_o more_o time_n than_o f_o take_v twice_o thrice_o or_o four_o time_n will_v contain_v a_o aliquot_fw-la part_n of_o d._n now_o e_z taken_z four_o time_n be_v equal_a to_o a_o as_o f_o be_v to_o c._n so_o a_o will_v contain_v a_o aliquot_fw-la part_n of_o b_o more_o time_n than_o c_o contain_v a_o like_a aliquot_fw-la part_n of_o d_o thence_o there_o will_v be_v a_o great_a reason_n of_o a_o to_z b_o than_z of_o c_z to_z d_o which_o be_v contrary_a to_o the_o supposition_n corollary_n which_o in_o euclid_n be_v after_o the_o four_o proposition_n