whereon_o the_o head_n thereof_o you_o shall_v find_v the_o book_n it_o belong_v to_o and_o the_o proposition_n or_o use_n continue_v in_o their_o order_n eight_o book_n of_o euclid_n element_n with_o the_o use_v of_o each_o proposition_n the_o first_o book_n euclid_n design_n in_o this_o book_n be_v to_o give_v the_o first_o principle_n of_o geometry_n and_o to_o do_v the_o same_o methodical_o he_o begin_v with_o the_o definition_n and_o explication_n of_o the_o most_o ordinary_a term_n than_o he_o exhibit_v certain_a supposition_n and_o have_v propose_v those_o maxim_n which_o natural_a reason_n teach_v he_o pretend_v to_o put_v forward_o nothing_o without_o demonstration_n and_o to_o convince_v any_o one_o which_o will_v consent_v to_o nothing_o but_o what_o he_o shall_v be_v oblige_v to_o acknowledge_v in_o his_o first_o proposition_n he_o treat_v of_o line_n and_o of_o the_o several_a angle_n make_v by_o their_o intersect_v each_o other_o and_o have_v occasion_n to_o demonstrate_v their_o propriety_n and_o compare_v certain_a triangle_n he_o do_v the_o same_o in_o the_o first_o eight_o proposition_n then_o teach_v the_o practical_a way_n of_o divide_v a_o angle_n and_o a_o line_n into_o two_o equal_a part_n and_o to_o draw_v a_o perpendicular_a he_o pursue_v to_o the_o propriety_n of_o a_o triangle_n and_o have_v show_v those_o of_o parallel_n line_n he_o make_v a_o end_n of_o the_o explication_n of_o this_o first_o figure_n and_o pass_v forward_o to_o parallellogram_n give_v a_o way_n to_o reduce_v all_o sort_n of_o polygon_n into_o a_o more_o regular_a figure_n he_o end_v this_o book_n with_o that_o celebrate_a proposition_n of_o pythagoras_n and_o demonstrate_v that_o in_o a_o rectangular_a triangle_n the_o square_a of_o the_o base_a be_v equal_a to_o the_o sum_n of_o the_o square_n of_o the_o side_n include_v the_o right_a angle_n definition_n 1._o a_o point_n be_v that_o which_o have_v no_o part_n this_o definition_n be_v to_o be_v understand_v in_o this_o sense_n the_o quantity_n which_o we_o conceive_v without_o distinguish_v its_o part_n or_o without_o think_v that_o it_o have_v any_o be_v a_o mathematical_a point_n far_o differ_v from_o those_o of_o zeno_n which_o be_v altogether_o indivisible_a since_o one_o may_v doubt_v with_o a_o great_a deal_n of_o reason_n if_o those_o last_o be_v possible_a which_o yet_o we_o can_v of_o the_o first_o if_o we_o conceive_v they_o as_o we_o ought_v 2._o a_o line_n be_v a_o length_n without_o breadth_n the_o sense_n of_o this_o definition_n be_v the_o same_o with_o that_o of_o the_o forego_n the_o quantity_n which_o we_o consider_v have_v length_n without_o make_v any_o reflection_n on_o its_o breadth_n be_v that_o we_o understand_v by_o the_o word_n line_n although_o one_o can_v draw_v a_o real_a line_n which_o have_v not_o a_o determinate_a breadth_n it_o be_v general_o say_v that_o a_o line_n be_v produce_v by_o the_o motion_n of_o a_o point_n which_o we_o ought_v well_o to_o take_v notice_n of_o see_v that_o by_o a_o motion_n after_o that_o manner_n may_v be_v produce_v all_o sort_n of_o quantity_n imagine_v then_o that_o a_o point_n move_v and_o that_o it_o leave_v a_o trace_n in_o the_o middle_n of_o the_o way_n which_o it_o pass_v the_o trace_v be_v a_o line_n 3._o the_o two_o end_n of_o a_o line_n be_v point_n 4._o a_o straight_a line_n be_v that_o who_o point_n be_v place_v exact_o in_o the_o midst_n or_o if_o you_o will_v rather_o have_v it_o a_o straight_a line_n be_v the_o short_a of_o all_o the_o line_n which_o may_v be_v draw_v from_o one_o point_n to_o another_o 5._o a_o superficies_n be_v a_o quantity_n to_o which_o be_v give_v length_n and_o breadth_n without_o consider_v the_o thickness_n 6._o the_o extremity_n of_o a_o superficies_n be_v line_n 7._o a_o plain_a or_o straight_o superficies_n be_v that_o who_o line_n be_v place_v equal_o between_o the_o extremity_n or_o that_o to_o which_o a_o straight_a line_n may_v be_v apply_v any_o manner_n of_o way_n 1._o plate_n i._n fig._n 1._o i_o have_v already_o take_v notice_n that_o motion_n be_v capable_a of_o produce_v all_o sort_n of_o quantity_n whence_o we_o say_v that_o when_o a_o line_n pass_v over_o another_o it_o produce_v a_o superficies_n or_o a_o plain_n and_o that_o that_o motion_n have_v a_o likeness_n to_o arithmetical_a multiplication_n imagine_v that_o the_o line_n ab_fw-la move_v along_o the_o line_n bc_n keep_v the_o same_o situation_n without_o incline_v one_o way_n or_o the_o other_o the_o point_n a_o shall_v describe_v the_o line_n ad_fw-la the_o point_n b_o the_o line_n bc_n and_o the_o other_o point_n between_o other_o parallel_n line_n which_o shall_v compose_v the_o superficies_n abcd._n i_o add_v that_o this_o motion_n correspond_v with_o arithmetical_a multiplication_n for_o if_o i_o know_v the_o number_n of_o point_n which_o be_v in_o the_o line_n ab_fw-la bc_n multiply_v of_o they_o one_o by_o the_o other_o i_o shall_v have_v the_o number_n of_o point_n which_o compose_v the_o superficies_n abcd_v as_o if_o ab_fw-la contain_v four_o point_n and_o bc_n six_o say_v four_o time_n six_o be_v twenty_o four_o the_o superficies_n ab_fw-la cd_o shall_v be_v compose_v of_o twenty_o four_o point_n now_o i_o may_v take_v for_o a_o mathematical_a point_n any_o quantity_n whatsoever_o for_o example_n a_o foot_n provide_v i_o do_v not_o subdivide_v the_o same_o into_o part_n 8._o a_o plain_a angle_n be_v the_o open_n of_o two_o line_n which_o intersect_v each_o other_o and_o which_o compose_v not_o one_o single_a line_n 2._o fig._n 2._o as_o the_o opening_z d_o of_o the_o line_n ab_fw-la cb_n which_o be_v not_o part_n of_o the_o same_o line_n a_o right_a line_a angle_n be_v the_o open_n of_o two_o straight_a line_n it_o be_v principal_o of_o this_o sort_n of_o angle_n which_o i_o intend_v to_o treat_v of_o at_o present_a because_o experience_n do_v make_v i_o perceive_v that_o the_o most_o part_n of_o those_o who_o begin_v do_v mistake_v the_o measure_v the_o quantity_n of_o a_o angle_n by_o the_o length_n of_o the_o line_n which_o compose_v the_o same_o 4._o fig._n 3_o 4._o the_o most_o open_a angle_n be_v the_o great_a that_o be_v to_o say_v when_o the_o line_n include_v a_o angle_n be_v far_o asunder_o than_o those_o of_o another_o angle_n take_v they_o at_o the_o same_o distance_n from_o the_o point_n of_o intersection_n of_o their_o line_n the_o first_o be_v great_a than_o the_o second_o so_o the_o angle_n a_o be_v great_a than_o e_z because_o if_o we_o take_v the_o point_n b_o and_o d_o as_o far_o distant_a from_o the_o point_n a_o as_o the_o point_n g_o and_o l_o be_v from_o the_o point_n e_o the_o point_n b_o and_o d_o be_v far_o asunder_o than_o the_o point_n g_o and_o l_o from_o whence_o i_o conclude_v that_o if_o eglantine_n el_fw-es be_v continue_v the_o angle_n e_o will_v be_v of_o the_o same_o measure_n and_o less_o than_o the_o angle_n a._n we_o make_v use_v of_o three_o letter_n to_o express_v a_o angle_n and_o the_o second_o letter_n denote_v the_o angular_a point_n as_o the_o angle_n bad_a be_v the_o angle_n which_o the_o line_n ba_o ad_fw-la do_v form_n at_o the_o point_n a_o the_o angle_n bac_n be_v that_o which_o be_v form_v by_o the_o line_n ba_o ac_fw-la the_o angle_n god_n be_v comprehend_v under_o the_o line_n ca_n ad._n 3._o fig._n 3._o the_o arch_n of_o a_o circle_n be_v the_o measure_n of_o a_o angle_n thus_o design_v to_o measure_v the_o quantity_n of_o the_o angle_n bad_a i_o put_v one_o foot_n of_o the_o compass_n on_o the_o point_v a_o and_o with_o the_o other_o i_o describe_v a_o arch_n of_o a_o circle_n bcd_v the_o angle_n shall_v be_v the_o great_a by_o how_o much_o the_o arch_n bcd_v which_o be_v the_o measure_n thereof_o shall_v contain_v a_o great_a portion_n of_o a_o circle_n and_o because_o that_o common_o a_o arch_n of_o a_o circle_n be_v divide_v into_o three_o hundred_o and_o sixty_o equal_a part_n call_v degree_n it_o be_v say_v that_o a_o angle_n contain_v twenty_o thirty_o forty_o degree_n when_o the_o arch_n include_v betwixt_o its_o line_n contain_v twenty_o thirty_o forty_o degree_n so_o the_o angle_n be_v great_a which_o contain_v the_o great_a number_n of_o degree_n as_o the_o angle_n bad_a be_v great_a than_o gel_n the_o line_n ca_n divide_v the_o angle_n bad_a in_o the_o middle_n because_o the_o arch_n bc_n cd_o be_v equal_a and_o the_o angle_n bac_n be_v a_o part_n of_o bad_a because_o the_o arch_a bc_n be_v part_n of_o the_o arch_n bd._n 10._o when_o a_o line_n fall_v on_o another_o line_n make_v the_o angle_n on_o each_o side_n thereof_o equal_a those_o angles_n be_v right_a angle_n and_o the_o line_n so_o fall_v be_v a_o perpendicular_a 5._o fig._n 5._o as_o if_o the_o line_n ab_fw-la fall_v on_o cd_o
b._n by_o consequence_n the_o whole_a angle_n acd_v be_v equal_a to_o the_o two_o angel_n ace_n ecd_n of_o who_o it_o be_v compose_v it_o shall_v then_o be_v equal_a to_o a_o and_o b_o take_v together_o in_o the_o second_o place_n the_o angles_n acd_v acb_n be_v equal_a to_o two_o right_v by_o the_o 13_o and_o i_o have_v demonstrate_v that_o the_o angle_n acd_v be_v equal_a to_o the_o angle_n a_o and_o b_o take_v together_o therefore_o the_o angle_n acd_v be_v equal_a to_o a_o and_o b_o that_o be_v to_o say_v all_o the_o angle_n of_o the_o triangle_n abc_n be_v equal_a to_o two_o right_a or_o 180_o degree_n corollariy_n 1._o the_o three_o angle_n of_o one_o triangle_n be_v equal_a to_o the_o three_o angle_n of_o another_o triangle_n coral_n 2._o if_o two_o angle_n of_o a_o triangle_n be_v equal_a to_o two_o angle_n of_o another_o triangle_n the_o remain_a angle_n in_o the_o one_o shall_v be_v equal_a to_o the_o remain_a angle_n in_o the_o other_o coral_n 3._o if_o a_o triangle_n have_v one_o of_o its_o angle_n right_o the_o other_o two_o shall_v be_v acute_a and_o be_v take_v together_o shall_v be_v equal_a to_o one_o right_a angle_n coral_n 4._o from_o one_o and_o the_o same_o point_n of_o a_o line_n there_o can_v be_v draw_v but_o one_o perpendicular_a because_o a_o triangle_n can_v have_v two_o right_a angle_n coral_n 5._o the_o perpendicular_a be_v the_o short_a line_n which_o can_v be_v draw_v from_o a_o point_n to_o a_o line_n coral_n 6._o in_o a_o right_a angle_a triangle_n the_o great_a angle_n be_v a_o right_a angle_n and_o the_o long_a side_n be_v opposite_a thereto_o coral_n 7._o each_o angle_n of_o a_o equilateral_a triangle_n contain_v 60_o degree_n that_o be_v to_o say_v the_o three_o of_o 180._o use_v 32._o use_v 32._o this_o proposition_n be_v of_o use_n to_o we_o in_o astronomy_n to_o determine_v the_o parallax_n let_v the_o point_n a_o represent_v the_o centre_n of_o the_o earth_n and_o from_o the_o point_n b_o on_o the_o superficies_n of_o the_o earth_n let_v there_o be_v take_v by_o observation_n the_o angle_n dbc_n that_o be_v to_o say_v the_o apparent_a distance_n of_o a_o planet_n or_o comet_n from_o the_o zenith_n d._n i_o say_v if_o the_o earth_n be_v transparent_a this_o planet_n or_o comet_n view_v from_o the_o centre_n of_o the_o earth_n a_o will_v appear_v distant_a from_o the_o zenith_n d_o equal_a to_o the_o quantity_n of_o the_o angle_n god_n which_o be_v less_o than_o the_o angle_n cbd_v for_o the_o angle_n cbd_v be_v exterior_a in_o respect_n of_o the_o triangle_n abc_n be_v equal_a by_o the_o 32d._o to_o the_o opposite_a angle_n a_o and_o c._n whence_o the_o angle_n c_o shall_v be_v equal_a to_o the_o excess_n of_o the_o angle_n cbd_v above_o the_o angle_n a._n from_o whence_o i_o conclude_v that_o if_o i_o know_v by_o a_o astronomical_a table_n how_o far_o that_o planet_n or_o comet_n ought_v to_o appear_v distant_a from_o the_o zenith_n to_o one_o which_o shall_v be_v in_o the_o centre_n of_o the_o earth_n and_o if_o i_o observe_v at_o the_o same_o time_n the_o difference_n of_o those_o two_o angle_n shall_v be_v the_o parallax_n viz._n the_o angle_n bca_n proposition_n xxxiii_o theorem_fw-la if_o two_o equal_a and_o parallel_v line_n be_v join_v together_o with_o two_o other_o right_a line_n then_o be_v those_o line_n also_o equal_a and_o parallel_v let_v the_o line_n ab_fw-la gd_v be_v parallel_n and_o equal_a and_o let_v they_o be_v join_v with_o agnostus_n bd._n i_o say_v that_o the_o line_n agnostus_n bd_o be_v equal_a and_o parallel_v draw_v the_o diagonal_a bg_n demonstration_n see_v the_o line_n ab_fw-la gd_v be_v parallel_n the_o alternate_a angle_n abg_n bgd_v shall_v be_v equal_a by_o the_o 29_o and_o the_o side_n gb_n be_v common_a to_o both_o the_o triangle_n abg_n bgd_v and_o the_o side_n ab_fw-la gd_v equal_a with_o the_o angles_n abg_n bgd_v equal_a as_o before_o the_o base_n of_o those_o triangle_n agnostus_n bd_o shall_v be_v equal_a by_o the_o 4._o as_o also_o the_o angle_n dbg_n bga_n which_o because_o alternate_a the_o line_n agnostus_n bd_o shall_v be_v parallel_n by_o the_o 27_o use_v 33._o use_v 33._o this_o proposition_n be_v put_v in_o practice_n to_o measure_v as_o well_o the_o perpendicular_a height_n agnostus_n of_o mountain_n as_o their_o horizontal_n line_n c_o g_o which_o be_v hide_v in_o their_o thickness_n to_o effect_v which_o we_o make_v use_v of_o a_o very_a large_a square_a abdella_n put_v one_o end_n thereof_o in_o a_o in_o such_o manner_n that_o the_o other_o side_n thereof_o bd_o may_v be_v perpendicular_a to_o the_o horizon_n than_o we_o measure_v the_o side_n ad_fw-la bd_o than_o we_o do_v the_o like_a again_o at_o the_o point_n b_o and_o measure_n be_v aec_fw-la the_o side_n parallel_v to_o the_o horizon_n that_o be_v to_o say_v ad_fw-la be_v be_v add_v together_o gives_z the_o horizontal_n line_n cg_n and_o the_o perpendicular_a side_n bd_o aec_fw-la be_v add_v give_v the_o perpendicular_a height_n ag._n proposition_n xxxiv_o theorem_fw-la the_o side_n and_o the_o opposite_a angle_n in_o a_o parallelogram_n be_v equal_a and_o the_o diamter_n do_v divide_v the_o same_o into_o two_o equal_a part_n let_v the_o figure_n ab_fw-la cd_o be_v a_o parallelogram_n that_o be_v to_o say_v let_v the_o side_n ab_fw-la cd_o ac_fw-la bd_o be_v parallel_n i_o say_v that_o the_o opposite_a side_n ab_fw-la cd_o ac_fw-la bd_o be_v equal_a as_o also_o the_o angle_n a_o and_o d_o abdella_n acd_v and_o that_o the_o diameter_n bc_n do_v divide_v the_o figure_n into_o two_o equal_a part_n demonstration_n the_o line_n ab_fw-la cd_o be_v suppose_v parallel_n therefore_o the_o alternate_a angel_n abc_n bcd_v shall_v be_v equal_a by_o the_o 29_o likewise_o the_o side_n ac_fw-la bd_o be_v suppose_a parallel_n the_o alternate_a angle_n acb_n cbd_v be_v equal_a both_o which_o triangle_n have_v the_o same_o side_n bc_n and_o the_o angel_n abc_n bcd_v acb_n cbd_v equal_a they_o shall_v be_v equal_a in_o every_o respect_n by_o the_o 26_o therefore_o the_o side_n ab_fw-la cd_o ac_fw-la bd_o and_o the_o angle_n a_o and_o d_o be_v equal_a and_o the_o diameter_n divide_v the_o figure_n into_o two_o equal_a part_n and_o see_v the_o angel_n abc_n bcd_v acd_v cbd_a be_v equal_a add_v together_o abc_n cbd_v bcd_n acd_a we_o conclude_v that_o the_o opposite_a angle_n abdella_n acd_v shall_v be_v equal_a use_v 34._o use_v 34._o svrveyer_n have_v sometime_o occasion_n to_o make_v use_n of_o this_o proposition_n to_o part_v or_o divide_v land_n if_o the_o field_n be_v a_o parallelogram_n it_o may_v be_v divide_v into_o two_o equal_a part_n by_o the_o diameter_n ad._n but_o if_o one_o be_v oblige_v to_o divide_v the_o same_o into_o two_o equal_a part_n from_o the_o point_n e_o divide_v the_o diameter_n into_o two_o equal_a part_n in_o the_o point_n f_o and_o draw_v the_o line_n efg_v that_o line_n shall_v divide_v the_o field_n into_o two_o equal_a part_n for_o the_o triangle_n aef_n gfd_n which_o have_v their_o alternate_a angel_n eaf_n fdg_n aef_n fgd_v and_o the_o side_n of_o fd_n equal_a be_v equal_a by_o the_o 26_o and_o since_o the_o trapezium_fw-la befd_n together_o with_o the_o triangle_n afe_n that_o be_v to_o say_v the_o triangle_n adb_n be_v one_o half_a of_o the_o parallelogram_n by_o the_o 34â—h._n the_o same_o trapezium_fw-la efbd_v together_o with_o the_o triangle_n dfg_v shall_v be_v one_o half_a of_o the_o figure_n or_o field_n and_o the_o line_n eglantine_n divide_v the_o same_o into_o two_o equal_a part_n proposition_n xxxv_o the_o parallellogram_n be_v equal_a when_o have_v the_o same_o base_a they_o be_v between_o the_o same_o parallel_n lines_n let_v the_o parallelograms_n abec_n abdf_n have_v the_o same_o base_a ab_fw-la and_o be_v between_o the_o same_o parallel_n ab_fw-la cd_o i_o say_v they_o be_v equal_a demonstration_n the_o sides_n ab_fw-la ce_fw-fr be_v equal_a by_o the_o 34_o as_o also_o ab_fw-la fd_n wherefore_o ce_fw-fr fd_n be_v equal_a and_o add_v of_o to_o each_o the_o line_n cf_n ed_z shall_v be_v equal_a the_o triangle_n cfa_n ebb_v have_v the_o side_n ca_n ebb_n cf_n ed_z equal_a with_o the_o angles_n deb_n fca_n by_o the_o 29_o the_o one_o be_v exterior_a and_o the_o other_o interiour_n on_o the_o same_o side_n whence_o by_o the_o 4_o the_o triangle_n ace_n bed_n be_v equal_a and_o take_v away_o from_o both_o that_o which_o be_v common_a to_o both_o that_o be_v to_o say_v the_o little_a triangle_n egf_n the_o trapezium_fw-la fgbd_v shall_v be_v equal_a to_o the_o trapezium_fw-la cage_n and_o add_v to_o both_o the_o little_a triangle_n agb_n the_o parallellogram_n abec_n abdf_n shall_v be_v equal_a demonstration_n by_o the_o method_n of_o indivisibles_n this_o method_n be_v new_o invent_v by_o cavalerius_fw-la which_o be_v approve_v
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 let_v the_o angle_n a_o and_o d_o be_v in_o a_o segment_n bac_n less_o than_o a_o semicircle_n they_o shall_v notwithstanding_o be_v equal_a demonstration_n the_o angle_n of_o the_o triangle_n abe_n be_v equal_a to_o all_o the_o angle_n of_o the_o triangle_n dec_n the_o angles_n ecd_v abe_n be_v equal_a by_o the_o precede_a case_n since_o they_o be_v in_o the_o same_o segment_n abcd_n great_a than_o a_o semicircle_n the_o angle_n in_o e_o be_v likewise_o equal_a by_o the_o 15_o of_o the_o 1_o therefore_o the_o angle_n a_o and_o d_o shall_v be_v equal_a which_o angle_n have_v also_o the_o same_o arch_a bfc_n for_o base_a use_v xxi_o prop._n xxi_o it_o be_v prove_v in_o optic_n that_o the_o line_n bc_n shall_v appear_v equal_a be_v see_v from_o a_o and_o d_o since_o it_o always_o appear_v under_o equal_a angle_n we_o make_v use_v of_o this_o proposition_n to_o describe_v a_o great_a circle_n without_o have_v its_o centre_n for_o example_n when_o we_o will_v give_v a_o spherical_a 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 angle_n touch_v the_o circumference_n of_o the_o circle_n abcd_v i_o say_v that_o its_o opposite_a angle_n bad_a bcd_v be_v equal_a to_o two_o right_a draw_v the_o diagonal_n ac_fw-la bd._n demonstration_n all_o the_o angle_n of_o the_o triangle_n bad_a be_v equal_a to_o two_o right_a in_o the_o place_n of_o its_o angle_n abdella_n put_v the_o angle_n acd_v which_o be_v equal_a thereto_o by_o the_o 21_o as_o be_v in_o the_o same_o segment_n abcd_v and_o in_o the_o place_n of_o its_o angle_n adb_n put_v the_o angle_n acb_n which_o be_v in_o the_o same_o segment_n of_o the_o circle_n bcda_n so_o then_o the_o angle_n bad_a and_o the_o angle_n acd_v acb_n that_o be_v to_o say_v the_o whole_a angle_n bcd_n be_v equal_a to_o two_o right_a use_v ptolomy_n make_v use_v of_o this_o proposition_n to_o make_v the_o table_n of_o chord_n or_o subtendent_o i_o have_v also_o make_v use_v thereof_o in_o trigonometry_n in_o the_o three_o book_n to_o prove_v that_o the_o side_n of_o a_o obtuse_a angle_a triangle_n have_v the_o same_o reason_n among_o themselves_o as_o the_o sin_n of_o their_o opposite_a angle_n proposition_n xxiii_o theorem_fw-la two_o like_a segment_n of_o a_o circle_n describe_v on_o the_o same_o line_n be_v equal_a i_o call_v like_o segment_n of_o a_o circle_n those_o which_o contain_v equal_a angle_n and_o i_o say_v that_o if_o they_o be_v describe_v on_o the_o same_o line_n ab_fw-la they_o shall_v fall_v one_o on_o the_o other_o and_o shall_v not_o surpass_v each_o other_o in_o any_o part_n for_o if_o they_o do_v surpass_v each_o other_o as_o do_v the_o segment_n acb_n the_o segment_n adb_n they_o will_v not_o be_v like_a and_o to_o demonstrate_v it_o draw_v the_o line_n adc_a db_n and_o bc._n demonstration_n the_o angle_n adb_n be_v exterior_a in_o respect_n of_o the_o triangle_n bdc_n thence_o by_o the_o 21_o of_o the_o 1_o it_o be_v great_a than_o the_o angle_n acb_n and_o by_o consequence_n the_o segments_n adb_n acb_n contain_v unequal_a angle_n which_o i_o call_v unlike_a proposition_n xxiv_o theorem_fw-la two_o like_a segment_n of_o circle_n describe_v on_o equal_a line_n be_v equal_a if_o the_o segment_n of_o circle_n aeb_fw-mi cfd_n be_v like_a and_o if_o the_o line_n ab_fw-la cd_o be_v equal_a they_o shall_v be_v equal_a demonstration_n let_v it_o be_v imagine_v that_o the_o line_n cd_o be_v place_v on_o ab_fw-la they_o shall_v not_o surpass_v each_o other_o see_v they_o be_v suppose_v equal_a and_o then_o the_o segment_n aeb_fw-mi ce_v shall_v be_v describe_v on_o the_o same_o line_n and_o they_o shall_v then_o be_v equal_a by_o the_o precede_a proposition_n use_v 24._o use_v 24._o cvrve_v line_a figure_n be_v often_o reduce_v to_o right_o line_v by_o this_o proposition_n as_o if_o one_o shall_v describe_v two_o like_a segment_n of_o circle_n aec_fw-la adb_fw-la on_o the_o equal_a side_n ab_fw-la ac_fw-la of_o the_o triangle_n abc_n it_o be_v evident_a that_o transpose_v the_o segment_n aec_fw-la on_o adb_n the_o triangle_n abc_n be_v equal_a to_o the_o figure_n adbcea_n proposition_n xxv_o problem_n to_o complete_a a_o circle_n whereof_o we_o have_v but_o a_o part_n there_o be_v give_v the_o arch_a abc_n and_o we_o will_v complete_a the_o circle_n there_o need_v but_o to_o find_v its_o centre_n draw_v the_o line_n ab_fw-la bc_n and_o have_v divide_v they_o in_o the_o middle_n in_o d_o and_z e_z draw_v their_o perpendicular_o diego_n ei_o which_o shall_v meet_v each_o other_o in_o the_o point_n i_o the_o centre_n of_o the_o circle_n demonstration_n the_o centre_n be_v in_o the_o line_n diego_n by_o the_o coral_n of_o the_o 1_o it_o be_v also_o in_o ei_o it_o be_v then_o in_o the_o point_n 1._o use_v 25._o use_v 25._o this_o proposition_n come_v very_o frequent_o in_o use_n it_o may_v be_v propound_v another_o way_n as_o to_o inscribe_v a_o triangle_n in_o a_o circle_n or_o to_o make_v a_o circle_n pass_v through_o three_o give_v point_n provide_v they_o be_v not_o place_v in_o a_o straight_a line_n let_v be_v propose_v the_o point_n a_o b_o c_o put_v the_o point_n of_o the_o compass_n in_o c_o and_o at_o what_o open_v soever_o describe_v two_o ark_n f_o and_o e._n transport_v the_o foot_n of_o the_o compass_n to_o b_o and_o with_o the_o same_o open_n describe_z two_o arck_n to_o cut_v the_o former_a in_o e_z and_z f._n describe_v on_o b_o as_o centre_n at_o what_o open_v soever_o the_o arch_n h_n and_o g_o and_o at_o the_o same_o open_n of_o the_o compass_n describe_v on_o the_o centre_n a_o two_o ark_n to_o cut_v the_o same_o in_o g_z and_z h._n draw_v the_o line_n fe_o and_o gh_a which_o will_v cut_v each_o other_o in_o the_o point_n d_o the_o centre_n of_o the_o circle_n the_o demonstration_n be_v evident_a enough_o for_o if_o you_o have_v draw_v
a_o line_n which_o be_v found_v on_o this_o proposition_n for_o example_n to_o erect_v a_o perpendicular_a from_o the_o point_n a_o of_o the_o line_n ab_fw-la i_o put_v the_o foot_n of_o the_o compass_n on_o the_o point_n c_o take_v at_o discretion_n and_o extend_v the_o other_o to_o a_o i_o describe_v a_o circle_n which_o may_v cut_v the_o line_n ab_fw-la in_o the_o point_n b._n i_o draw_v the_o line_n bcd_n it_o be_v evident_a that_o the_o line_n dam_fw-ge shall_v be_v perpendicular_a to_o the_o line_n ab_fw-la see_v the_o angle_n bad_a be_v in_o the_o semicircle_n proposition_n xxxii_o theorem_fw-la the_o line_n which_o cut_v the_o circle_n at_o the_o point_n of_o touch_v make_v with_o the_o touch_n line_n the_o angle_n equal_a to_o those_o of_o the_o alternate_a segment_n let_v the_o line_n bd_o cut_v the_o circle_n in_o the_o point_n b_o which_o be_v the_o point_n where_o the_o line_n ab_fw-la do_v touch_v the_o same_o i_o say_v that_o the_o angle_n cbd_v which_o the_o line_n bd_o comprehend_v with_o the_o touch_n line_n abc_n be_v equal_a to_o the_o angle_n e_o which_o be_v that_o of_o the_o alternate_a segment_n bed_n and_o that_o the_o angle_n abdella_n be_v equal_a to_o the_o angle_n f_o of_o the_o segment_n bfd_n in_o the_o first_o place_n if_o the_o line_n pass_v through_o the_o centre_n as_o do_v the_o line_n be_v it_o will_v make_v with_o the_o touch_n line_n ab_fw-la two_o right_a angle_n by_o the_o 17_o and_o the_o angle_n of_o the_o semicircle_n will_v be_v also_o right_o by_o the_o precede_a so_o the_o proposition_n will_v be_v true_a if_o the_o line_n pass_v not_o through_o the_o centre_n as_o do_v the_o line_n bd_o draw_v the_o line_n be_v through_o the_o centre_n and_o join_v the_o line_n de_fw-fr demonstration_n the_o line_n be_v make_v two_o right_a angle_n with_o the_o touch_n line_n and_o all_o the_o angle_n of_o the_o triangle_n bde_n be_v equal_a to_o two_o right_v by_o the_o 32d_o of_o the_o one_a so_o take_v away_o the_o right_a angle_n abe_n and_o d_o which_o be_v in_o a_o semicircle_n and_o take_v again_o away_o the_o angle_n ebb_v which_o be_v common_a to_o both_o the_o angle_n cbd_v shall_v be_v equal_a to_o the_o angle_n bed_n three_o the_o angle_n abdella_n be_v equal_a to_o the_o angle_n f_o because_o in_o the_o quadrilateral_a bfde_v which_o be_v inscribe_v in_o a_o circle_n the_o opposite_a angle_n e_o and_o f_o be_v equal_a to_o two_o right_n by_o the_o 22d_o the_o angles_n abdella_n dbc_n be_v also_o equal_a to_o two_o right_v by_o the_o 13_o of_o the_o one_a and_o the_o angle_n dbc_n and_o e_o be_v equal_a as_o just_a now_o i_o do_v demonstrate_v therefore_o the_o angles_n abdella_n and_o bfd_n be_v equal_a use_v this_o proposition_n be_v necessary_a for_o that_o which_o follow_v proposition_n xxxiii_o problem_n to_o describe_v upon_o a_o line_n a_o segment_n of_o a_o circle_n which_o shall_v contain_v a_o give_v angle_n it_o be_v propose_v to_o describe_v on_o the_o line_n ab_fw-la a_o segment_n of_o a_o circle_n to_o contain_v the_o angle_n c._n make_v the_o angle_n bid_v equal_a to_o the_o angle_n c_o and_o draw_v to_o ad_fw-la the_o perpendicular_a ae_n make_v also_o the_o angle_n abf_n equal_a to_o the_o angle_n baf_n and_o last_o describe_v a_o circle_n on_o the_o point_n f_o as_o centre_n at_o the_o open_a bf_n or_o favorina_n the_o segment_n beaumont_n contain_v a_o angle_n equal_a to_o the_o angle_n c._n demonstration_n the_o angel_n baf_n abf_n be_v equal_a the_o line_n favorina_n fb_n be_v equal_a by_o the_o 6_o and_o the_o circle_n which_o be_v describe_v on_o the_o centre_n f_o pass_v through_o a_o and_o b_o now_o the_o angle_n dae_n be_v right_o the_o line_n da_fw-la touch_v the_o circle_n in_o the_o point_n a_o by_o the_o 16_o therefore_o the_o angle_n which_o the_o segment_n beaumont_n comprehend_v as_o the_o angle_n e_o be_v equal_a to_o the_o angle_n dab_n that_o be_v to_o say_v to_o the_o angle_n c._n but_o if_o the_o angle_n be_v obtuse_a we_o must_v take_v the_o acute_a angle_n which_o be_v its_o compliment_n to_o 180_o degree_n proposition_n xxxiv_o problem_n a_o circle_n be_v give_v to_o cut_v therefrom_o a_o segment_n to_o contain_v a_o assign_a angle_n to_o cut_v from_o the_o circle_n be_v a_o segment_n to_o contain_v the_o angle_n a._n draw_v by_o the_o 17_o the_o touch_n line_n bd_o and_o make_v the_o angle_n dbc_n equal_a to_o the_o angle_n a._n it_o be_v evident_a by_o the_o 32d_o that_o the_o segment_n bec_n contain_v a_o angle_n equal_a to_o dbc_n and_o by_o consequence_n to_o the_o angle_n a._n use_v i_o have_v make_v use_n of_o this_o proposition_n to_o find_v geometrical_o the_o excentricity_n of_o the_o annual_a circle_n of_o the_o sun_n and_o his_o apogeon_n three_o observation_n be_v give_v it_o be_v also_o make_v use_n of_o in_o optic_n two_o unequal_a line_n be_v propose_v to_o find_v a_o point_n where_o they_o shall_v appear_v equal_a or_o under_o equal_a angle_n make_v on_o each_o segment_v which_o may_v contain_v equal_a angle_n proposition_n xxxv_o theorem_fw-la if_o two_o line_n cut_v each_o other_o in_o a_o circle_n the_o rectangle_n comprehend_v under_o the_o part_n of_o the_o one_o be_v equal_a to_o the_o rectangle_n comprehend_v under_o the_o part_n of_o the_o other_o in_o the_o first_o place_n if_o two_o line_n cut_v each_o other_o in_o the_o centre_n of_o the_o circle_n they_o shall_v be_v equal_a and_o divide_v equal_o so_o than_o it_o be_v evident_a that_o the_o rectangle_n comprehend_v under_o the_o part_n of_o the_o one_o be_v equal_a to_o the_o rectangle_n comprehend_v under_o the_o part_n of_o the_o other_o second_o let_v one_o of_o the_o line_n pass_v through_o the_o centre_n f_o as_o ac_fw-la and_o divide_v the_o line_n bd_o in_o two_o equal_o in_o the_o point_n e_o i_o say_v that_o the_o rectangle_n comprehend_v under_o ae_n aec_fw-la be_v equal_a to_o the_o rectangle_n comprehend_v under_o be_v ed_z that_o be_v to_o say_v to_o the_o square_n of_o be._n the_o line_n ac_fw-la be_v perpendicular_a to_o bd_o by_o the_o three_o demonstration_n see_v that_o the_o line_n ac_fw-la be_v divide_v equal_o in_o f_o and_o unequal_o in_o f_o the_o rectangle_n comprehend_v under_o ae_n aec_fw-la with_o the_o square_n of_o fe_o be_v equal_a to_o the_o square_n of_o fc_n or_o fb_n by_o the_o 5_o of_o the_o 2d_o now_o the_o angle_n e_o being_n right_n the_o square_a of_o fb_n be_v equal_a to_o the_o square_n of_o be_v of_o therefore_o the_o rectangle_n comprehend_v under_o ae_n aec_fw-la with_o the_o square_n of_o of_o be_v equal_a to_o the_o square_n of_o be_v of_o and_o take_v away_o the_o square_n of_o of_o there_o remain_v that_o the_o square_a of_o be_v be_v equal_a to_o the_o rectangle_n under_o be_v ed._n three_o let_v the_o line_n ab_fw-la pass_v through_o the_o centre_n f_o and_o let_v it_o divide_v the_o line_n cd_o unequal_o in_o the_o point_n e_o draw_v fg_v perpendicular_o to_o cd_o and_o by_o the_o 3d._n the_o line_n cg_n gd_v shall_v be_v equal_a demonstration_n see_v the_o line_n ab_fw-la be_v divide_v equal_o in_o f_o and_o unequal_o in_o e_z the_o rectangle_n comprehend_v under_o ae_n ebb_n with_o the_o square_n of_o of_o be_v equal_a to_o the_o square_n of_o bf_n or_o fc_n by_o the_o 5_o of_o the_o 2d_o in_o the_o place_n of_o of_o put_v the_o square_n of_o fg_n ge_z which_o be_v equal_a thereto_o by_o the_o 47th_o of_o the_o one_a in_o like_a manner_n the_o line_n cd_o be_v equal_o divide_v in_o g_o and_o unequal_o in_o e_z the_o rectangle_n ce_v with_o the_o square_n of_o ge_z shall_v be_v equal_a to_o the_o square_n of_o gc_n add_v the_o square_n of_o gf_n the_o rectangle_n of_o ce_fw-fr ed_z with_o the_o square_n of_o ge_z fg_v shall_v be_v equal_a to_o the_o square_n of_o gc_n gf_n that_o be_v to_o say_v by_o the_o 47th_o of_o the_o one_a to_o the_o square_n of_o cf._n therefore_o the_o rectangle_n aeb_n with_o the_o square_n of_o ge_z gf_n and_o the_o rectangle_n of_o ce_fw-fr ed_z with_o the_o same_o square_n be_v equal_a and_o by_o consequence_n take_v away_o the_o same_o square_n the_o rectangle_n aeb_n be_v equal_a to_o the_o rectangle_n cfd_n four_o let_v the_o line_n cd_o he_o cut_v each_o other_o in_o the_o point_n e_o so_fw-mi that_o neither_o of_o they_o pass_v through_o the_o centre_n i_o say_v that_o the_o rectangle_n ce_v be_v equal_a to_o the_o rectangle_n hei_fw-la for_o draw_v the_o line_n afb_n the_o rectangle_v ce_v hei_fw-la be_v equal_a to_o the_o rectangle_n aeb_fw-mi by_o the_o precede_a case_n therefore_o they_o be_v equal_a use_v one_o might_n by_o this_o proposition_n have_v a_o practical_a way_n to_o find_v the_o four_o proportional_a to_o three_o give_v line_n or_o
the_o base_a bc_n let_v there_o be_v imagine_v another_o triangle_n def_n have_v a_o angle_n d_o equal_v to_o the_o angle_n a_o and_o the_o side_n de_fw-fr df_n equal_a to_o ab_fw-la ac_fw-la now_o since_o the_o side_n ab_fw-la ac_fw-la be_v equal_a the_o four_o line_n ab_fw-la ac_fw-la de_fw-fr df_n shall_v be_v equal_a demonstration_n because_o the_o side_n ab_fw-la de_fw-fr ac_fw-la df_n be_v equal_a as_o also_o the_o angle_n a_o and_o d_o if_o we_o put_v the_o triangle_n def_n on_o the_o triangle_n abc_n the_o line_n de_fw-fr shall_v fall_v upon_o ab_fw-la and_o df_n on_o ac_fw-la and_o fe_o on_o bc_n by_o the_o four_o therefore_o the_o angle_n def_n shall_v be_v equal_a to_o abc_n and_o since_o one_o part_n of_o the_o line_n de_fw-fr fall_n on_o ab_fw-la the_o whole_a line_n diego_n shall_v fall_v on_o agnostus_n otherwise_o two_o straight_a line_n will_v contain_v a_o space_n therefore_o the_o angle_n jef_n shall_v be_v equal_a to_o the_o angle_n gbc_n now_o if_o you_o shall_v turn_v the_o triangle_n def_n and_o make_v the_o line_n df_n to_o fall_v on_o ab_fw-la and_o de_fw-fr on_o ac_fw-la because_o the_o four_o line_n ab_fw-la df_n ac_fw-la de_fw-fr be_v equal_a as_o also_o the_o angle_n a_o and_o d_o the_o two_o triangle_n abc_n def_n shall_v lie_v exact_o on_o each_o other_o and_o the_o angles_n acb_n def_n hcb_n jef_n shall_v be_v equal_a now_o according_a to_o our_o first_o comparison_n the_o angle_n abc_n be_v equal_a to_o def_n and_o gbc_n to_o jef_n therefore_o the_o angel_n abc_n acb_n which_o be_v equal_a to_o the_o same_o def_n and_o gbc_n hcb_n which_o be_v also_o equal_a to_o the_o same_o jef_n be_v also_o equal_a among_o themselves_o i_o think_v fit_a not_o to_o make_v use_n of_o euclid_n demonstration_n because_o it_o be_v too_o difficult_a for_o young_a learner_n to_o apprehend_v they_o be_v often_o discourage_v to_o proceed_v proposition_n vi_o theorem_fw-la if_o two_o angle_n of_o a_o triangle_n be_v equal_a that_o triangle_n shall_v be_v a_o isosceles_a triangle_n let_v the_o angel_n abc_n acb_n of_o the_o triangle_n abc_n be_v equal_a i_o affirm_v that_o their_o opposite_a side_n ab_fw-la ac_fw-la be_v equal_a to_o each_o other_o let_v the_o triangle_n def_n have_v the_o base_a df_n equal_a to_o bc_n and_o the_o angle_n def_n equal_a to_o abc_n as_o also_o def_n equal_a to_o acb_n since_o than_o that_o we_o suppose_v that_o the_o angel_n abc_n acb_n be_v equal_a the_o four_o angel_n abc_n acb_n def_n dfe_n be_v equal_a now_o let_v we_o imagine_v the_o base_a of_o be_v so_o put_v on_o bc_n that_o the_o point_n e_o fall_n on_o b_o it_o be_v evident_a that_o since_o they_o be_v equal_a the_o one_o shall_v not_o exceed_v the_o other_o anywise_o moreover_o the_o angle_n e_o being_n equal_v to_o the_o angle_n b_o and_o the_o angle_n f_o to_o the_o angle_n c_o the_o line_n ebb_n shall_v fall_v on_o ba_o and_o fd_v on_o ca_n therefore_o the_o line_n ed_z fb_n shall_v meet_v each_o other_o in_o the_o point_n a_o from_o whence_o i_o conclude_v that_o the_o line_n ed_z be_v equal_a to_o basilius_n again_o let_v we_o turn_v over_o the_o triangle_n def_n place_v the_o point_n e_o on_o c_o and_z f_o on_o b_o which_o must_v so_o happen_v because_o bc_n be_v suppose_v equal_a to_o of_o and_o because_o the_o angles_n f_o and_o b_o e_o and_o c_o be_v suppose_v equal_a the_o side_n fb_n shall_v fall_v on_o ba_o and_z ed_z on_o ca_n and_o the_o point_n d_o on_o a_o wherefore_o the_o line_n ac_fw-la de_fw-fr shall_v be_v equal_a from_o whence_o i_o conclude_v that_o the_o side_n ab_fw-la ac_fw-la be_v equal_a to_o each_o other_o since_o they_o be_v equal_a to_o the_o same_o side_n de._n use_v 6._o use_v 6._o this_o proposition_n be_v make_v use_n of_o in_o measure_v all_o inaccessible_a line_n it_o be_v say_v that_o thales_n be_v the_o first_o that_o measure_v the_o height_n of_o the_o egyptian_a pyramid_n by_o their_o shadow_n it_o may_v easy_o be_v do_v by_o this_o proposition_n for_o if_o you_o will_v measure_v the_o height_n of_o the_o pyramid_n ab_fw-la you_o must_v wait_v till_o the_o sun_n be_v elevate_v 45_o degree_n above_o the_o horizon_n that_o be_v to_o say_v until_o the_o angle_n agb_n be_v 45_o degree_n now_o by_o the_o proposition_n the_o shadow_n bc_n be_v equal_a to_o the_o pyramid_n ab_fw-la for_o see_v the_o angle_n abc_n be_v a_o right_a angle_n and_o that_o the_o angle_n acb_n be_v half_a right_n or_o 45_o degree_n the_o angle_n cab_n shall_v also_o be_v half_a a_o right_a angle_n as_o shall_v be_v prove_v hereafter_o therefore_o the_o angle_n bca_n bac_n be_v equal_a and_o by_o the_o six_o the_o side_n ab_fw-la bc_n be_v also_o equal_a i_o may_v measure_v the_o same_o without_o make_v use_n of_o the_o shadow_n by_o go_v backward_o from_o b_o until_o the_o angle_n acb_n be_v half_o a_o right_a angle_n which_o i_o may_v know_v by_o a_o quadrant_n those_o proposition_n be_v make_v use_n of_o in_o trigonometry_n and_o several_a other_o treatise_n i_o omit_v the_o seven_o who_o use_n be_v only_o to_o demonstrate_v the_o eight_o proposition_n viii_o theorem_n if_o two_o triangle_n be_v equal_o side_v they_o shall_v also_o have_v their_o opposite_a angle_n equal_a let_v the_o side_n give_v lt_n he_o vt_fw-la gh_o lv_o be_v equal_a i_o say_v that_o the_o angle_n gih_n shall_v be_v equal_a to_o the_o angle_n ltv_n igh_a to_o the_o angle_n l_o ihg_v to_o the_o angle_n v._o from_o the_o centre_n h_n at_o the_o distance_n he_o let_v the_o circle_n ig_n be_v describe_v and_o from_o the_o centre_n g_o with_o the_o extent_n give_v let_v the_o circle_n he_o be_v describe_v demonstration_n if_o the_o base_a lv_o be_v lay_v on_o the_o base_a hg_n they_o will_v agree_v because_o they_o be_v equal_a i_o further_o add_v that_o the_o point_n t_o shall_v fall_v precise_o at_o the_o circumference_n of_o the_o circle_n ig_n since_o we_o suppose_v the_o line_n hl_n vt_fw-la be_v equal_a it_o must_v likewise_o fall_v at_o the_o circumference_n ih_v see_v give_v be_v 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
under_o ab_fw-la and_o ac_fw-la shall_v be_v three_o time_n 8_o or_o 24_o the_o square_a of_o ac_fw-la 3_o be_v 9_o the_o rectangle_n comprehend_v under_o ac_fw-la 3_o and_o cb_n 5_o be_v 3_o time_n 5_o or_o 15._o it_o be_v evident_a that_o 15_o and_o 9_o be_v 24._o use_v at_fw-fr  _fw-fr 43_o c_z 40._o 3_o b_o  _fw-fr 3_o 120._o  _fw-fr 9_o 129_o  _fw-fr  _fw-fr this_o proposition_n serve_v likewise_o to_o demonstrate_v the_o ordinary_a practice_n of_o multiplication_n for_o example_n if_o one_o will_v multiply_v the_o number_n 43_o by_o 3_o have_v separate_v the_o number_n of_o 43_o into_o two_o part_n in_o 40_o and_o 3_o three_o time_n 43_o shall_v be_v as_o much_o as_o three_o time_n 3_o which_o be_v nine_o the_o square_a of_o three_o and_o three_o time_n forty_o which_o be_v 120_o for_o 129_o be_v three_o time_n 43._o those_o which_o be_v young_a beginner_n ought_v not_o to_o be_v discourage_v if_o they_o do_v not_o conceive_v immediate_o these_o proposition_n for_o they_o be_v not_o difficult_a but_o because_o they_o do_v imagine_v they_o contain_v some_o great_a mystery_n proposition_n iu_o theorem_fw-la if_o a_o line_n be_v divide_v into_o two_o part_n the_o square_a of_o the_o whole_a line_n shall_v be_v equal_a to_o the_o two_o square_n make_v of_o its_o part_n and_o to_o two_o rectangle_v comprehend_v under_o the_o same_o part_n let_v the_o line_n ab_fw-la be_v divide_v in_o c_o and_o let_v the_o square_a thereof_o abde_n be_v make_v let_v the_o diagonal_a ebb_n be_v draw_v and_o the_o perpendicular_a cf_n cut_v the_o same_o and_o through_o that_o point_n let_v there_o be_v draw_v gl_n parallel_n to_o ab_fw-la it_o be_v evident_a that_o the_o square_a abde_n be_v equal_a to_o the_o four_o rectangle_v gf_n cl_n cg_n lf_n the_o two_o first_o be_v the_o square_a of_o ac_fw-la and_o of_o cb_n the_o two_o compliment_n be_v comprehend_v under_o ac_fw-la cb._n demonstration_n the_o side_n ae_n ab_fw-la be_v equal_a thence_o the_o angle_n aeb_fw-mi abe_n be_v half_a right_n and_o because_o of_o the_o parallel_n gl_n ab_fw-la the_o angle_n of_o the_o triangle_n of_o the_o square_a ge_z by_o the_o 29_o shall_v be_v equal_a as_o also_o the_o side_n by_o the_o 6_o of_o the_o 1._o thence_o gf_n be_v the_o square_a of_o ac_fw-la in_o like_a manner_n the_o rectangle_n cl_n be_v the_o square_a of_o cb_n the_o rectangle_n gc_n be_v comprehend_v under_o ac_fw-la and_o agnostus_n equal_a to_o bl_v or_o bc_n the_o rectangle_n lf_o be_v comprehend_v under_o ld_n equal_a to_o ac_fw-la and_o under_o fd_n equal_a to_o bc._n coral_n if_o a_o diagonal_a be_v draw_v in_o a_o square_a the_o rectangle_v through_o which_o it_o pass_v be_v square_n use_v a_o 144_o b_o 22_o c_o 12_o this_o proposition_n give_v we_o the_o practical_a way_n of_o find_v or_o extract_v the_o square_a root_n of_o a_o number_n propound_v let_v the_o same_o be_v the_o number_n a_o 144_o represent_v by_o the_o square_a ad_fw-la and_o its_o root_n by_o the_o line_n ab_fw-la moreover_o i_o know_v that_o the_o line_n require_v ab_fw-la must_v have_v two_o figure_n i_o therefore_o imagine_v that_o the_o line_n ab_fw-la be_v divide_v in_o c_o and_o that_o ac_fw-la represent_v the_o first_o figure_n and_o bc_n the_o second_o i_o seek_v the_o root_n of_o the_o first_o figure_n of_o the_o number_n 144_o which_o be_v 100_o and_o i_o find_v that_o it_o be_v 10_o and_o make_v its_o square_a 100_o represent_v by_o the_o square_a gf_n i_o subtract_v the_o same_o from_o 144_o and_o there_o remain_v 44_o for_o the_o rectangle_v gc_a fl_fw-mi and_o the_o square_a cl._n but_o because_o this_o gnomonicall_a figure_n be_v not_o proper_a i_o transport_v the_o rectangle_n fl_fw-mi in_o kg_v and_o so_o i_o have_v the_o rectangle_n kl_n contain_v 44._o i_o know_v also_o almost_o all_o the_o length_n of_o the_o side_n kb_n for_o ac_fw-la be_v 10_o therefore_o kc_n be_v 20_o i_o must_v then_o divide_v 44_o by_o 20_o that_o be_v to_o say_v to_o find_v the_o divisor_n i_o double_v the_o root_n find_v and_o i_o say_v how_o many_o time_n 20_o in_o 44_o i_o find_v it_o 2_o time_n for_o the_o side_n bl_n but_o because_o 20_o be_v not_o the_o whole_a side_n kb_n but_o only_a kc_n this_o 2_o which_o come_v in_o the_o quotient_n be_v to_o be_v add_v to_o the_o divisor_n which_o then_o will_v be_v 22._o so_o i_o find_v the_o same_o 2_o time_n precise_o in_o 44_o the_o square_a root_n then_o shall_v be_v 12._o you_o see_v that_o the_o square_a of_o 144_o be_v equal_a to_o the_o square_n of_o 10_o to_o the_o square_n of_o 2_o which_o be_v 4_o and_o to_o twice_o 20_o which_o be_v two_o rectangle_v comprehend_v under_o 2_o and_o under_o 10._o proposition_n v._o theorem_fw-la if_o a_o right_a line_n be_v cut_v into_o equal_a part_n and_o into_o unequal_a part_n the_o rectangle_n comprehend_v under_o the_o unequal_a part_n together_o with_o the_o square_n which_o be_v of_o the_o middle_a part_n or_o difference_n of_o the_o part_n be_v equal_a to_o the_o square_n of_o half_a the_o line_n if_o the_o line_n ab_fw-la be_v divide_v equal_o in_o c_z and_o unequal_o in_o d_o the_o rectangle_n ah_o comprehend_v under_o the_o segment_n ad_fw-la db_fw-la together_o with_o the_o square_n of_o cd_o shall_v be_v equal_a to_o the_o square_a cf_n that_o be_v of_o half_a of_o ab_fw-la viz._n cb._n make_v a_o end_n of_o the_o figure_n as_o you_o see_v it_o the_o rectangle_v lg_n diego_n shall_v be_v square_n by_o the_o coral_n of_o the_o 4_o i_o prove_v that_o the_o rectangle_n ah_o comprehend_v under_o ad_fw-la and_o dh_n equal_a to_o db_v with_o the_o square_a lg_n be_v equal_a to_o the_o square_a cf._n demonstration_n the_o rectangle_n all_o be_v equal_a to_o the_o rectangle_n df_n the_o one_o and_o the_o other_o be_v comprehend_v under_o half_a the_o line_n ab_fw-la and_o under_o bd_o or_o dh_n equal_a thereto_o add_v to_o both_o the_o rectangle_n ch_z the_o rectangle_n ah_o shall_v be_v equal_a to_o the_o gnomon_n lbg_n again_o to_o both_o add_v the_o square_a lg_n the_o rectangle_n ah_o with_o the_o square_a lg_n shall_v be_v equal_a to_o the_o square_a cf._n arithmetical_o let_v ab_fw-la be_v 10_o ac_fw-la be_v 5_o as_o also_o cb._n let_v cd_o be_v 2_o and_o db_n 3_o the_o rectangle_n comprehend_v under_o ad_fw-la 7_o and_o db_n 3_o that_o be_v to_o say_v 21_o with_o the_o square_n of_o cd_o 2_o which_o be_v 4_o shall_v be_v equal_a to_o the_o square_n of_o cb_n 5_o which_o be_v 25._o use_v this_o proposition_n be_v very_o useful_a in_o the_o three_o book_n we_o make_v use_v thereof_o in_o algebra_n to_o demonstrate_v the_o way_n of_o find_v the_o root_n of_o a_o affect_a square_a or_o equation_n proposition_n vi_o theorem_fw-la if_o one_o add_v a_o line_n to_o another_o which_o be_v divide_v into_o two_o equal_a part_n the_o rectangle_n comprehend_v under_o the_o line_n compound_v of_o both_o and_o under_o the_o line_n add_v together_o with_o the_o square_n of_o half_a the_o divide_a line_n be_v equal_a to_o the_o square_n of_o a_o line_n compound_v of_o half_a the_o divide_a line_n and_o the_o line_n add_v if_o one_o add_v the_o line_n bd_o to_o the_o line_n ab_fw-la which_o be_v equal_o divide_v in_o c_o the_o rectangle_n a_fw-la comprehend_v under_o ad_fw-la and_o under_o dn_n or_o db_n with_o the_o square_n of_o cb_n be_v equal_a to_o the_o square_n of_o cd_o make_v the_o square_n of_o cd_o and_o have_v draw_v the_o diagonal_a fd_n draw_v bg_n parallel_v to_o fc_n which_o cut_v fd_v in_o the_o point_n h_n through_o which_o pass_v hn_n parallel_v to_o ab_fw-la kg_n shall_v be_v the_o square_n of_o bc_n and_o bn_n that_o of_o bd._n demonstration_n the_o rectangle_v ak_v ch_z on_o equal_a base_n ac_fw-la bc_n be_v equal_a by_o the_o 38_o of_o the_o 1_o the_o compliment_n ch_z he_o be_v equal_a by_o the_o 43_o of_o the_o 1_o therefore_o the_o rectangle_v ak_v he_o be_v equal_a add_v to_o both_o the_o rectangle_n cn_fw-la and_o the_o square_a kg_n the_o rectangle_v ak_v cn_fw-la that_o be_v to_o say_v the_o rectangle_n a_fw-la with_o the_o square_a kg_n shall_v be_v equal_a to_o the_o rectangle_v cn_fw-la he_o and_o to_o the_o square_a kg_n that_o be_v to_o say_v to_o the_o square_a ce._n arithmetical_o or_o by_o number_n let_v ab_fw-la be_v 8_o ac_fw-la 4_o cb_n 4_o bd_o 3_o than_o ad_fw-la shall_v be_v 11._o it_o be_v evident_a that_o the_o rectangle_n a_fw-la three_o time_n 11_o that_o be_v to_o say_v 33_o with_o the_o square_n of_o kg_v 16_o which_o together_o be_v 49_o be_v equal_a to_o the_o square_n of_o cd_o 7_o which_o be_v 49_o for_o 7_o time_n 7_o be_v 49._o use_v 6._o fig._n 6._o maurolycus_n measure_v the_o whole_a earth_n by_o one_o single_a
the_o line_n ab_fw-la bc_n you_o will_v have_v divide_v they_o equal_o and_o perpendicular_o by_o so_o do_v this_o be_v very_o necessary_a to_o describe_v astrolabe_fw-la and_o to_o complete_a circle_n of_o which_o we_o have_v but_o a_o part_n that_o astronomical_a proposition_n which_o teach_v to_o find_v the_o apogeum_n and_o the_o excentricity_n of_o the_o sun_n circle_n require_v this_o proposition_n we_o often_o make_v use_n there_o of_o in_o the_o treatise_n of_o cut_v of_o stone_n proposition_n xxvi_o theorem_fw-la the_o equal_a angle_n which_o be_v at_o the_o centre_n or_o at_o the_o circumference_n of_o equal_a circle_n have_v for_o base_a equal_a arks._n if_o the_o equal_a angle_n d_o and_o i_o be_v in_o the_o centre_n of_o equal_a circle_n abc_n efg_n the_o ark_n bc_n fg_v shall_v be_v equal_a for_o if_o the_o ark_n bc_n be_v great_a or_o lesser_a than_o the_o ark_n fg_v see_v that_o the_o ark_n be_v the_o measure_n of_o the_o angle_n the_o angle_n d_o will_v be_v great_a or_o lesser_a than_o the_o angle_n 1._o and_o if_o the_o equal_a angle_n a_o and_o e_o be_v in_o the_o circumference_n of_o the_o equal_a circle_n the_o angel_n d_o and_o i_o which_o be_v the_o double_a of_o the_o angle_n a_o and_o e_o be_v also_o equal_a the_o ark_n bc_n fg_v shall_v be_v also_o equal_a proposition_n xxvii_o theorem_fw-la the_o angle_n which_o be_v either_o in_o the_o centre_n or_o in_o the_o circumference_n of_o equal_a circle_n and_o which_o have_v equal_a ark_n for_o base_a be_v also_o equal_a if_o the_o angle_n d_o an_o i_o be_v in_o the_o centre_n of_o equal_a circle_n and_o if_o they_o have_v for_o base_a equal_a arcks_n bc_n fg_v they_o shall_v be_v equal_a because_o that_o their_o measure_n bc_n fg_v be_v equal_a and_o if_o the_o angle_n a_o and_o e_o be_v in_o the_o circumference_n of_o equal_a circle_n have_v for_o base_a equal_a ark_n bc_n eglantine_n the_o angle_n in_o the_o centre_n shall_v be_v equal_a and_o they_o be_v their_o half_n by_o the_o 20_o shall_v be_v also_o equal_a proposition_n xxviii_o theorem_fw-la equal_a line_n in_o equal_a circle_n correspond_v to_o equal_a arks._n if_o the_o line_n bc_n of_o be_v apply_v in_o equal_a circle_n abc_n def_n they_o shall_v be_v chord_n of_o equal_a ark_n bc_n ef._n draw_v the_o line_n ab_fw-la ac_fw-la de_fw-fr ef._n demonstration_n in_o the_o triangle_n abc_n def_n the_o side_n ab_fw-la ac_fw-la de_fw-fr of_o be_v equal_a be_v the_o semidiameters_a of_o equal_a circle_n the_o base_n bc_n of_o be_v suppose_v equal_a thence_o by_o the_o 8_o of_o the_o 1_o the_o angle_n a_o and_o d_o shall_v be_v equal_a and_o by_o the_o 16_o the_o ark_n bc_n of_o shall_v be_v also_o equal_a proposition_n xxix_o theorem_fw-la line_n which_o subtend_v equal_a arck_n in_o equal_a circle_n be_v equal_a if_o the_o line_n bc_n of_o subtend_v equal_a ark_n bc_n of_o in_o equal_a circle_n those_o line_n be_v equal_a demonstration_n the_o ark_n bc_n of_o be_v equal_a and_o part_n of_o equal_a circle_n therefore_o by_o the_o 27_o the_o angle_n a_o and_o d_o shall_v be_v equal_a so_o then_o in_o the_o triangle_n cab_n edf_n the_o side_n ab_fw-la ac_fw-la de_fw-fr df_n be_v equal_a as_o also_o the_o angle_n a_o and_o d_o the_o base_n bc_n of_o shall_v be_v equal_a by_o the_o 4_o of_o the_o 1_o use_v theodosius_n demonstrate_v by_o the_o 28_o and_o 29_o that_o the_o ark_n of_o the_o circle_n of_o the_o italian_a and_o babylonian_a hour_n comprehend_v between_o two_o parallel_n be_v equal_a we_o demonstrate_v also_o after_o the_o same_o manner_n that_o the_o ark_n of_o circle_n of_o astronomical_a hour_n comprehend_v between_o two_o parallel_n to_o the_o equator_fw-la be_v equal_a these_o proposition_n come_v almost_o continual_o in_o use_n in_o spherical_a trigonometry_n as_o also_o in_o gnomonic_n proposition_n xxx_o problem_n to_o divide_v a_o ark_n of_o a_o circle_n into_o two_o equal_a part_n it_o be_v propose_v to_o divide_v the_o ark_n aeb_fw-mi into_o two_o equal_a part_n put_v the_o foot_n of_o the_o compass_n in_o the_o point_n a_o make_v two_o ark_n f_o and_o g_o then_o transport_v the_o compass_n without_o open_v or_o shut_v it_o to_o the_o point_n b_o describe_v two_o ark_n cut_v the_o former_a in_o f_o and_o g_o the_o line_n gf_n will_v cut_v the_o ark_n ab_fw-la equal_o in_o the_o point_n e._n draw_v the_o line_n ab_fw-la demonstration_n you_o divide_v the_o line_n ab_fw-la equal_o by_o the_o construction_n for_o imagine_v the_o line_n of_o bf_n agnostus_n bg_n which_o i_o have_v not_o draw_v lest_o i_o shall_v imbroil_v the_o figure_n the_o triangle_n fga_n fgb_n have_v all_o their_o side_n equal_a so_o then_o by_o the_o 8_o of_o the_o 1_o the_o angles_n afd_v bfd_n be_v equal_a moreover_o the_o triangle_n dfa_n dfb_n have_v the_o sides_n df_n common_a the_o side_n of_o bf_n equal_a and_o the_o angles_n dfa_n dfb_n equal_a whence_o by_o the_o 4_o of_o the_o 1_o the_o base_n ad_fw-la db_fw-la be_v equal_a and_o the_o angles_n adf_n bdf_n be_v equal_a we_o have_v then_o divide_v the_o line_n ab_fw-la equal_o and_o perpendicular_o in_o the_o point_n d._n so_o then_o by_o the_o 1_o the_o centre_n of_o the_o circle_n be_v in_o the_o line_n eglantine_n let_v it_o be_v the_o point_n c_o and_o let_v be_v draw_v the_o line_n ca_n cb_n all_o the_o side_n of_o the_o triangle_n acd_v bcd_a be_v equal_a thence_o the_o angle_n acd_v bcd_a be_v equal_a by_o the_o 8_o of_o the_o 1_o and_o by_o the_o 27_o the_o ark_n ae_n ebb_n be_v equal_a use_v as_o we_o have_v often_o need_v to_o divide_v a_o ark_n in_o the_o middle_n the_o practice_n of_o this_o proposition_n be_v very_o ordinary_o in_o use_n it_o be_v by_o this_o mean_v we_o divide_v the_o mariner_n compass_n into_o 32_o rumb_n for_o have_v draw_v two_o diameter_n which_o cut_v each_o other_o at_o right_a angle_n we_o divide_v the_o circle_n in_o four_o and_o sub-dividing_a each_o quarter_n in_o the_o middle_n we_o have_v eight_o part_n and_o sub-dividing_a each_o part_n twice_o we_o come_v to_o thirty_o two_o part_n we_o have_v also_o occasion_n of_o the_o same_o practice_n to_o divide_v a_o semicircle_n into_o 180_o degree_n and_o because_o for_o the_o perform_v the_o same_o division_n throughout_o we_o be_v oblige_v to_o divide_v a_o ark_n into_o three_o all_o the_o ancient_a geometrician_n have_v endeavour_v to_o find_v a_o method_n to_o divide_v a_o angle_n or_o a_o ark_n into_o three_o equal_a part_n but_o it_o be_v not_o yet_o find_v proposition_n xxxi_o theorem_fw-la the_o angle_n which_o be_v in_o a_o semicircle_n be_v right_o that_o which_o be_v comprehend_v in_o a_o great_a segment_n be_v acute_a and_o that_o in_o a_o lesser_a segment_n be_v obtuse_a if_o the_o angle_n bac_n be_v in_o a_o semicircle_n i_o demonstrate_v that_o it_o be_v right_o draw_v the_o line_n da._n demonstration_n the_o angle_n adb_n exterior_a in_o respect_n of_o the_o triangle_n dac_n be_v equal_a by_o the_o 32d_o of_o the_o one_a to_o the_o two_o interiour_o dac_n dca_n and_o those_o be_v equal_a by_o the_o 5_o of_o the_o one_a see_v the_o side_n dam_fw-ge dc_o be_v equal_a it_o shall_v be_v double_a to_o the_o angle_n dac_n in_o like_a manner_n the_o angle_n adc_n be_v double_a to_o the_o angle_n dab_n therefore_o the_o two_o angel_n adb_n adc_fw-la which_o be_v equal_a to_o two_o right_a be_v double_a to_o the_o angle_n bac_n and_o by_o consequence_n the_o angle_n bac_n be_v a_o right_a angle_n second_o the_o angle_n aec_fw-la which_o be_v in_o the_o segment_n aec_fw-la be_v obtuse_a for_o in_o the_o quadrilateral_a abce_n the_o opposite_a angle_n e_z and_o b_o be_v equal_a to_o two_o right_v by_o the_o 22d_o the_o angle_n b_o be_v acute_a therefore_o the_o angle_n e_o shall_v be_v obtuse_a three_o the_o angle_n b_o which_o be_v in_o the_o segment_n abc_n great_a than_o a_o semicircle_n be_v acute_a see_v that_o in_o the_o triangle_n abc_n the_o angle_n bac_n be_v a_o right_a angle_n use_v 31._o use_v 31._o the_o workman_n have_v draw_v from_o this_o proposition_n the_o way_n of_o try_v if_o their_o square_n be_v exact_a for_o have_v draw_v a_o semicircle_n bad_a they_o apply_v the_o point_n a_o of_o their_o square_a bad_a on_o the_o circumference_n of_o the_o circle_n and_o one_o of_o its_o side_n ab_fw-la on_o the_o point_n b_o of_o the_o diameter_n the_o other_o side_n ad_fw-la must_v touch_v the_o other_o point_v d_o which_o be_v the_o other_o end_n of_o the_o diameter_n ptolemy_n make_v use_v of_o this_o proposition_n to_o make_v the_o table_n of_o subtendants_n or_o chord_n of_o which_o he_o have_v occasion_n in_o trigonometry_n 31._o use_v 31._o we_o have_v also_o a_o practical_a way_n to_o erect_v a_o perpendicular_a on_o the_o end_n of_o