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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 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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 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
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 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
superficies_n 18._o a_o cone_n be_v a_o figure_n make_v when_o one_o side_n of_o a_o right_a angle_a triangle_n viz._n one_o of_o those_o that_o contain_v the_o right_a angle_n remain_v fix_v the_o triangle_n be_v turn_v round_o about_o till_o it_o return_v to_o the_o place_n from_o whence_o it_o first_o move_v and_o if_o the_o fix_a right_a line_n be_v equal_a to_o the_o other_o which_o contain_v the_o right_a angle_n than_o the_o cone_n be_v a_o rectangled_a cone_n but_o if_o it_o be_v less_o it_o be_v a_o obtuse_a angle_a cone_n if_o great_a a_o acute_a angle_a cone_n 19_o the_o axis_n of_o a_o cone_n be_v that_o fix_a line_n about_o which_o the_o triangle_n be_v move_v 20._o a_o cylinder_n be_v a_o figure_n make_v by_o the_o move_a round_n of_o a_o right_a angle_a parallelogram_n one_o of_o the_o side_n thereof_o namely_o which_o contain_v the_o right_a angle_n abide_v fix_v till_o the_o parallelogram_n be_v turn_v about_o to_o the_o same_o place_n whence_o it_o begin_v to_o move_v 21._o like_a cones_n and_o cylinder_n be_v those_o who_o axe_n and_o diameter_n of_o their_o base_n be_v proportional_a cones_n be_v right_a when_o the_o axis_n be_v perpendicular_a to_o the_o plain_a of_o the_o base_a and_o they_o be_v say_v to_o be_v scalene_n when_o the_o axis_n be_v incline_v to_o the_o base_a and_o the_o diameter_n of_o their_o base_n be_v in_o the_o same_o ratio_fw-la we_o add_v that_o incline_v cones_n to_o be_v like_o their_o axe_n must_v have_v the_o same_o inclination_n to_o the_o plane_n of_o their_o base_n proposition_n i._o theorem_fw-la i._n plate_n vii_o prop._n i._n a_o straight_a line_n can_v have_v one_o of_o its_o part_n in_o a_o plane_n and_o the_o other_o without_o it_o if_o the_o line_n ab_fw-la be_v in_o the_o plane_n ad_fw-la it_o be_v continue_v shall_v not_o go_v without_o but_o all_o its_o part_n shall_v be_v in_o the_o same_o plane_n for_o if_o it_o can_v be_v that_o bc_n be_v a_o part_n of_o ab_fw-la continue_v draw_v in_o the_o plane_n cd_o the_o line_n bd_o perpendicular_a to_o ab_fw-la draw_v also_o in_o the_o same_o plane_n be_v perpendicular_a to_o bd._n demonstration_n the_o angle_n abdella_n bde_n be_v both_o right_a angle_n thence_o by_o the_o 14_o of_o the_o first_o ab_fw-la be_v do_v make_v but_o one_o line_n and_o consequent_o bc_n be_v not_o a_o part_n of_o the_o line_n ab_fw-la continue_v otherwise_o two_o strait_a line_n cb_n ebb_n will_v have_v the_o same_o part_n ab_fw-la that_o be_v ab_fw-la will_v be_v part_n of_o both_o which_o we_o have_v reject_v as_o false_a in_o the_o thirteen_o maxim_n of_o the_o first_o book_n use_v we_o establish_v on_o this_o proposition_n a_o principle_n in_o gnomonic_n to_o prove_v that_o the_o shadow_n of_o the_o stile_n fall_v not_o without_o the_o plane_n of_o a_o great_a circle_n in_o which_o the_o sun_n be_v see_v that_o the_o end_n or_o top_n of_o the_o stile_n be_v take_v for_o the_o centre_n of_o the_o heaven_n and_o consequent_o for_o the_o centre_n of_o all_o 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