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shall_v only_o give_v the_o reader_n to_o understand_v that_o i_o have_v faithful_o render_v this_o piece_n into_o english_a according_a to_o the_o sense_n of_o the_o author_n but_o here_o and_o there_o omit_v some_o small_a matter_n which_o i_o judge_v not_o so_o proper_o relate_v to_o the_o subject_n of_o this_o work_n and_o therein_o will_v make_v no_o want_n to_o the_o reader_n nor_o i_o hope_v be_v any_o offence_n to_o the_o ingenious_a author_n himself_o i_o have_v only_o one_o thing_n more_o to_o add_v and_o that_o upon_o the_o account_n of_o a_o objection_n i_o have_v meet_v with_o that_o here_o be_v not_o all_o the_o book_n of_o euclid_n and_o it_o be_v true_a here_o be_v not_o all_o here_o be_v only_o the_o first_o six_o book_n and_o the_o eleven_o and_o twelve_o the_o other_o be_v purposely_o omit_v by_o our_o learned_a author_n who_o judge_v the_o understanding_n of_o these_o to_o be_v sufficient_a for_o all_o the_o part_n of_o the_o mathematics_n 304._o mathematics_n see_v 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st._n michael_n alley_n in_o cornhill_n advertisement_n whereas_o in_o the_o french_a all_o the_o definition_n which_o need_v all_o the_o proposition_n as_o also_o all_o those_o use_n our_o author_n think_v fit_a to_o illustrate_v by_o scheme_n be_v do_v in_o the_o book_n in_o wood_n here_o they_o be_v in_o copper_n plate_n to_o be_v place_v at_o the_o end_n of_o the_o book_n the_o definition_n and_o use_n be_v in_o the_o plate_n mark_v with_o arithmetical_a charactars_n or_o zipher_n but_o the_o propsition_n in_o alphabetical_a zipher_n at_o the_o begin_n of_o the_o book_n or_o at_o least_o when_o you_o come_v to_o the_o first_o definition_n you_o be_v refer_v to_o the_o number_n of_o the_o plate_n in_o which_o you_o shall_v find_v the_o scheme_n proper_a thereto_o as_o also_o the_o def_n prop._n and_o use_v belong_v unto_o the_o book_n unless_o the_o plate_n can_v not_o contain_v they_o and_o then_o you_o be_v refer_v to_o the_o next_o plate_n
the_o square_n of_o the_o other_o two_o sides_n ab_fw-la ac_fw-la draw_v the_o line_n ah_o parallel_n to_o bd_o ce_fw-fr and_o draw_v also_o the_o line_n ad_fw-la ae_n fc_n bg_n i_o prove_v that_o the_o square_a of_o be_v equal_a to_o the_o right_o angle_a figure_n or_o long_a square_a bh_n and_o the_o square_a agnostus_n to_o the_o right_o angle_a figure_n ch_z and_o that_o so_o the_o square_n be_v be_v equal_a to_o the_o two_o square_n of_o ag._n demonstration_n the_o triangle_n fbc_n abdella_n have_v their_o sides_n ab_fw-la bf_n bd_o bc_n equal_a and_o the_o angel_n fbc_n abdella_n be_v equal_a see_v that_o each_o of_o they_o beside_o the_o right_a angle_n include_v the_o angle_n abc_n thence_o by_o the_o 4_o the_o triangle_n abdella_n fbc_n be_v equal_a now_o the_o square_n of_o be_v double_a to_o the_o triangle_n fbc_n by_o the_o 41_o because_o they_o have_v the_o same_o base_a bf_n and_o be_v between_o the_o same_o parallel_n bf_a ac_fw-la likewise_o the_o right_o line_a 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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
of_o to_o make_v all_o sort_n of_o figure_n great_a or_o lesser_a for_o it_o be_v more_o universal_a then_o the_o 47th_o of_o the_o one_a which_o notwithstanding_o be_v so_o useful_a that_o it_o seem_v that_o almost_o all_o geometry_n be_v establish_v on_o this_o principle_n the_o 32d_o proposition_n be_v unnecessary_a proposition_n xxxiii_o theorem_fw-la in_o equal_a circle_n the_o angle_n as_o well_o those_o at_o the_o centre_n as_o those_o at_o the_o circumference_n as_o also_o the_o sector_n be_v in_o the_o same_o ratio_fw-la as_o the_o ark_n which_o serve_v to_o they_o as_o base_a if_o the_o circle_n anc_n dof_n be_v equal_a there_o shall_v be_v the_o same_o ratio_fw-la of_o the_o angle_n abc_n to_o the_o angle_n def_n as_o of_o the_o ark_n ac_fw-la to_o the_o ark_n df._n let_v the_o ark_n agnostus_n gh_n hc_n he_o equal_a ark_n and_o consequent_o aliquot_fw-la part_n of_o the_o ark_n ac_fw-la and_o let_v be_v divide_v the_o ark_n df_n into_o so_o many_o equal_a to_o agnostus_n as_o may_v be_v find_v therein_o and_o let_v the_o line_n ei_o eke_o and_o the_o rest_n be_v draw_v demonstration_n all_o the_o angles_n abg_n gbh_n hbc_n dei_fw-la jek_n and_o the_o rest_n be_v equal_a by_o the_o 37th_o of_o the_o three_o so_o then_o agnostus_n a_o aliquot_fw-la part_n of_o ac_fw-la be_v find_v in_o the_o ark_n df_n as_o many_o time_n as_o the_o angle_n abg_n a_o aliquot_fw-la part_n of_o the_o angle_n abc_n be_v find_v in_o def_n there_o be_v therefore_o the_o same_o ratio_fw-la of_o the_o ark_n ac_fw-la to_o the_o ark_n df_n as_o of_o the_o angle_n abc_n to_o the_o angle_n def_n and_o because_o n_n and_o o_o be_v the_o half_n of_o the_o angel_n abc_n def_n they_o shall_v be_v in_o the_o same_o ratio_fw-la as_o be_v those_o angles_n there_o be_v therefore_o the_o same_o ratio_fw-la of_o the_o angle_n n_n to_o the_o angle_n o_o as_o of_o the_o ark_n ac_fw-la to_o the_o ark_n df._n it_o be_v the_o same_o with_o sector_n for_o if_o the_o line_n agnostus_n gh_n hc_n 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without_o know_v what_o this_o book_n teach_v we_o which_o make_v it_o very_o necessary_a in_o the_o great_a part_n of_o the_o mathematics_n in_o the_o first_o place_n theodosius_n spherics_n do_v whole_o suppose_v it_o spherical_a trigonometry_n the_o three_o part_n of_o practical_a geometry_n several_a proposition_n of_o staticks_n and_o of_o geography_n be_v establish_v on_o the_o principle_n of_o solid_a body_n gnomonic_n conical_a section_n and_o the_o treatise_n of_o stone-cutting_a be_v not_o difficult_a but_o because_o one_o be_v often_o oblige_v to_o represent_v on_o paper_n figure_n that_o be_v raise_v and_o which_o be_v comprise_v under_o several_a surface_n i_o leave_v out_o the_o seven_o the_o eight_o the_o nine_o and_o the_o ten_o book_n of_o euclid_n element_n because_o they_o be_v unnecessary_a in_o almost_o all_o the_o part_n of_o the_o mathematics_n i_o have_v often_o wonder_v that_o they_o have_v be_v put_v among_o the_o element_n see_v it_o be_v evident_a that_o 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plane_n ii_o fig._n ii_o as_o the_o line_n ab_fw-la shall_v be_v at_o right_a angle_n to_o the_o plane_n cd_o if_o it_o be_v perpendicular_a to_o the_o line_n cd_o fe_o which_o line_n be_v draw_v in_o the_o plane_n cd_o pass_v through_o the_o point_n b_o in_o such_o sort_n that_o the_o angel_n abc_n abdella_n abe_n abf_n be_v right_o 4._o a_o plane_n be_v perpendicular_a to_o another_o when_o the_o line_n perpendicular_a to_o the_o common_a section_n of_o the_o plane_n and_o draw_v in_o the_o one_o be_v also_o perpendicular_a to_o the_o other_o plane_n iii_o fig._n iii_o we_o call_v the_o common_a section_n of_o the_o plane_n a_o line_n which_o be_v in_o both_o plane_n as_o the_o line_n ab_fw-la which_o be_v also_o in_o the_o plane_n ac_fw-la as_o well_o as_o in_o the_o plane_n ad._n if_o then_o the_o line_n de_fw-fr draw_v in_o the_o plane_n ad_fw-la and_o perpendicular_a to_o ab_fw-la be_v also_o perpendicular_a to_o the_o plane_n ac_fw-la the_o plane_n ad_fw-la shall_v be_v right_o or_o perpendicular_a to_o the_o plane_n ac_fw-la 5._o iu._n fig._n iu._n if_o the_o line_n ab_fw-la be_v not_o perpendicular_a to_o the_o plane_n cd_o and_o if_o there_o be_v draw_v from_o the_o point_n a_o the_o line_n ae_n perpendicular_a to_o the_o plane_n cd_o and_o then_o the_o line_n be_v the_o angle_n abe_n be_v that_o of_o the_o inclination_n of_o the_o line_n ab_fw-la to_o the_o plane_n cd_o 6._o v._n fig._n v._n the_o inclination_n of_o one_o plane_n to_o another_o be_v the_o acute_a angle_n comprehend_v by_o the_o two_o perpendicular_o to_o the_o common_a section_n draw_v in_o each_o plane_n as_o the_o inclination_n of_o the_o plane_n ab_fw-la to_o the_o plane_n ad_fw-la be_v no_o other_o than_o the_o angle_n bcd_v comprehend_v by_o the_o line_n bccd_v drawnin_n both_o plane_n perpendicular_a to_o the_o common_a section_n ae_n 7._o plain_n shall_v be_v incline_v after_o the_o same_o manner_n if_o the_o angle_n of_o inclination_n be_v equal_a 8._o planes_n which_o be_v parallel_v be_v continue_v as_o far_o as_o you_o please_v be_v notwithstanding_o equidistant_a 9_o like_o solid_a figure_n be_v comprehend_v or_o terminate_v under_o like_a plane_n equal_a in_o number_n 10._o solid_n figure_n equal_a and_o like_a be_v comprehend_v or_o termine_v under_o like_a plain_n equal_a both_o in_o multitude_n and_o magnitude_n 11._o vi_o fig._n vi_o a_o solid_a angle_n be_v the_o concourse_n or_o inclination_n of_o many_o line_n which_o be_v in_o divers_a plane_n as_o the_o concourse_n of_o the_o line_n ab_fw-la ac_fw-la ad_fw-la which_o be_v in_o divers_a plane_n 12._o vi_o fig._n vi_o a_o pyramid_n be_v a_o solid_a figure_n comprehend_v under_o divers_a plane_n set_v upon_o one_o plane_n and_o gather_v together_o to_o one_o point_n as_o the_o figure_n abcd._n 13._o vii_o fig._n vii_o a_o prism_n be_v a_o solid_a figure_n contain_v under_o planes_n whereof_o the_o two_o opposite_a be_v equal_a like_a and_o parallel_n but_o the_o other_o be_v parallellogram_n as_o the_o figure_n ab_fw-la its_o opposite_a plain_n may_v be_v polygon_n 14._o a_o sphere_n be_v a_o solid_a figure_n terminate_v by_o a_o single_a superficies_n from_o which_o draw_v several_a line_n to_o a_o point_n take_v in_o the_o middle_n of_o the_o figure_n they_o shall_v be_v all_o equal_a some_o define_v a_o sphere_n by_o the_o motion_n of_o a_o semicircle_n turn_v about_o its_o diameter_n the_o diameter_n remain_v immovable_a 15._o the_o axis_n of_o a_o sphere_n be_v that_o unmoveable_a line_n about_o which_o the_o semia-circle_n turn_v 16._o the_o centre_n of_o the_o sphere_n be_v the_o same_o with_o that_o of_o the_o semicircle_n which_o turn_v 17._o the_o diameter_n of_o a_o sphere_n be_v any_o line_n whatever_o which_o pass_v through_o the_o centre_n of_o the_o sphere_n and_o end_v in_o its_o
for_o pentagon_n be_v the_o most_o ordinary_a you_o must_v also_o take_v notice_n that_o these_o way_n of_o describe_v a_o pentagon_n about_o a_o circle_n may_v be_v apply_v to_o the_o other_o polygon_n i_o have_v give_v another_o way_n to_o inscribe_v a_o regular_a pentagon_n in_o a_o circle_n in_o military_a architecture_n proposition_n xv._o problem_n to_o inscribe_v a_o regular_a hexagon_n in_o a_o circle_n to_o inscribe_v a_o regular_a hexagon_n in_o the_o circle_n abcdef_n draw_v the_o diameter_n ad_fw-la and_o put_v the_o foot_n of_o the_o compass_n in_o the_o point_n d_o describe_v a_o circle_n at_o the_o open_a dg_n which_o shall_v intersect_v the_o circle_n in_o the_o point_n aec_fw-la then_o draw_v the_o diameter_n egb_n cgf_n and_o the_o line_n ab_fw-la of_o and_o the_o other_o demonstration_n it_o be_v evident_a that_o the_o triangle_n cdg_n dge_n be_v equilateral_a wherefore_o the_o angle_n cgd_v dge_n and_o their_o opposite_n bga_n agf_n be_v each_o of_o they_o the_o three_o part_n of_o two_o right_n and_o that_o be_v 60_o degree_n now_o all_o the_o angle_n which_o can_v be_v make_v about_o one_o point_n be_v equal_a to_o four_o right_n that_o be_v to_o say_v 360._o so_o take_v away_o four_o time_n 60_o that_o be_v 240_o from_o 360_o there_o remain_v 120_o degree_n for_o bgc_n and_o fge_n whence_o they_o shall_v each_o be_v 60_o degree_n so_o all_o the_o angle_n at_o the_o centre_n be_v equal_a all_o the_o ark_n and_o all_o the_o side_n shall_v be_v equal_a and_o each_o angle_n a_o b_o c_o etc._n etc._n shall_v be_v compose_v of_o two_o angle_n of_o sixty_o that_o be_v to_o say_v one_o hundred_o and_o twenty_o degree_n they_o shall_v therefore_o be_v equal_a coral_n the_o side_n of_o a_o hexagon_n be_v equal_a to_o the_o semi_a diameter_n use_v because_o that_o the_o side_n of_o a_o hexagon_n be_v the_o base_a of_o a_o ark_n of_o sixty_o degree_n and_o that_o be_v equal_a to_o the_o semi-diameter_n its_o half_n be_v the_o sine_fw-la of_o thirty_o and_o it_o be_v with_o this_o sine_fw-la we_o begin_v the_o table_n of_o sines_n euclid_n treat_v of_o hexagon_n in_o the_o last_o book_n of_o his_o element_n proposition_n xvi_o problem_n to_o inscribe_v a_o regular_a pentadecagon_n in_o a_o circle_n inscribe_v in_o a_o circle_n a_o equilateral_a triangle_n abc_n by_o the_o 2d_o and_o a_o regular_a pentagon_n by_o the_o 11_o in_o such_o sort_n that_o the_o angle_n meet_v in_o the_o point_n a._n the_o line_n bf_n by_o je_n shall_v be_v the_o side_n of_o the_o pentadecagon_n and_o by_o inscribe_v in_o the_o other_o ark_n line_n equal_a to_o bf_n by_o you_o may_v complete_a this_o polygon_n demonstration_n see_v the_o line_n ab_fw-la be_v the_o side_n of_o the_o equilateral_a triangle_n the_o ark_n aeb_fw-mi shall_v be_v the_o three_o of_o the_o whole_a circle_n or_o 5_o fifteenth_n and_o the_o ark_n ae_n be_v the_o five_o part_n it_o shall_v contain_v 3_o 15_o thence_o ebb_n contain_v two_o and_o if_o you_o divide_v it_o in_o the_o middle_n in_o i_o each_o part_n shall_v be_v a_o fifteen_o use_v this_o proposition_n serve_v only_o to_o open_v the_o way_n for_o other_o polygon_n we_o have_v in_o the_o compass_n of_o proportion_n very_o easy_a method_n to_o inscribe_v all_o the_o ordinary_a polygon_n but_o they_o be_v ground_v on_o this_o for_o one_o can_v not_o put_v polygon_n on_o that_o instrument_n if_o one_o do_v not_o find_v their_o side_n by_o this_o proposition_n or_o such_o like_a the_o end_n of_o the_o four_o book_n the_o five_o book_n of_o euclid_n element_n this_o five_o book_n be_v absolute_o necessary_a to_o demonstrate_v the_o propoposition_n of_o the_o six_o book_n it_o contain_v a_o most_o universal_a doctrine_n and_o a_o way_n of_o argue_v by_o proportion_n which_o be_v most_o subtle_a solid_a and_o brief_a so_o that_o all_o treatise_n which_o be_v found_v on_o proportion_n can_v be_v without_o this_o mathematical_a logic_n geometry_n arithmetic_n music_n astronomy_n staticks_n and_o to_o say_v in_o one_o word_n all_o the_o treatise_n of_o the_o science_n be_v demonstrate_v by_o the_o proposition_n of_o this_o book_n the_o great_a part_n of_o measure_v be_v do_v by_o proportion_n and_o in_o practisal_n geometry_n and_o one_o may_v demonstrate_v all_o the_o rule_n of_o arithmetic_n by_o the_o theorem_n hereof_o wherefore_o it_o be_v not_o necessary_a to_o have_v recourse_n to_o the_o seven_o eight_o or_o nine_o book_n the_o music_n of_o the_o ancient_n be_v scarce_o any_o thing_n else_o but_o the_o doctrine_n of_o proportion_n apply_v to_o the_o sense_n it_o be_v the_o same_o in_o staticks_n which_o consider_v the_o proportion_n of_o weight_n in_o fine_a one_o may_v affirm_v that_o if_o one_o shall_v take_v away_o from_o mathematician_n the_o knowledge_n of_o the_o proposition_n that_o this_o book_n give_v we_o the_o remainder_n will_v be_v of_o little_a use_n definition_n the_o whole_a correspond_v to_o its_o part_n and_o this_o shall_v be_v the_o great_a quantity_n compare_v with_o the_o lesser_a whether_o it_o contain_v the_o same_o in_o effect_n or_o that_o it_o do_v not_o contain_v the_o same_o part_n or_o quantity_n take_v in_o general_n be_v divide_v ordinary_o into_o aliquot_fw-la part_n and_o aliquant_a part_n 1._o an_fw-mi aliquot_fw-la part_n which_o euclid_n define_v in_o this_o book_n be_v a_o magnitude_n of_o a_o magnitude_n the_o lesser_a of_o the_o great_a when_o it_o measure_v it_o exact_o that_o be_v to_o say_v that_o it_o be_v a_o lesser_a quantity_n compare_v with_o a_o great_a which_o it_o measure_v precise_o as_o the_o line_n of_o two_o foot_n take_v three_o time_n be_v equal_a to_o a_o line_n of_o six_o foot_n 2._o multiplex_fw-la be_v a_o magnitude_n of_o a_o magnitude_n the_o great_a of_o the_o lesser_a when_o the_o lesser_a measure_v the_o great_a that_o be_v to_o say_v that_o multiplex_n be_v a_o great_a quantity_n compare_v with_o a_o lesser_a which_o it_o contain_v precise_o some_o number_n of_o time_n for_o example_n the_o 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to_o efg_v demonstration_n we_o have_v already_o demonstrate_v that_o there_o be_v a_o great_a reason_n of_o a_o b_o c_o to_z e_z f_o g_o than_o of_o the_o part_n bc_n to_o the_o part_n fg_v there_o shall_v thence_o be_v a_o great_a reason_n of_o a_o to_o e_o than_o of_o a_o b_o c_o to_z e_z f_o g_o by_o the_o 32d_o the_o end_n of_o the_o five_o book_n the_o sixth_z book_n of_o euclid_n element_n this_o book_n explain_v and_o begin_v to_o apply_v particular_a matter_n of_o the_o doctrine_n of_o proportion_n which_o the_o precede_a book_n explain_v but_o in_o general_n it_o begin_v with_o the_o most_o easy_a figure_n that_o be_v to_o say_v triangle_n give_v rule_n to_o determine_v not_o only_o the_o proportion_n of_o their_o side_n but_o also_o that_o of_o their_o capacity_n area_n or_o superficies_n then_o it_o teach_v to_o find_v proportional_a line_n and_o to_o augment_v or_o diminish_v any_o figure_n whatever_o according_a to_o a_o give_v ratio_fw-la it_o demonstrate_v the_o rule_n 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the_o base_a gc_n to_o the_o base_a em_n as_o of_o the_o triangle_n agc_n to_o the_o triangle_n dem_n coral_n parallellogram_n draw_v on_o the_o same_o base_n and_o that_o be_v between_o the_o same_o parallel_n line_n be_v double_a to_o the_o triangle_n by_o the_o 41st_o they_o be_v thence_o in_o the_o same_o reason_n as_o triangle_n that_o be_v to_o say_v in_o the_o same_o reason_n as_o their_o base_n use_v i._n fig._n i._n this_o proposition_n be_v not_o only_o necessary_a to_o demonstrate_v those_o which_o follow_v but_o we_o may_v make_v use_n thereof_o in_o divide_v of_o land_n let_v there_o be_v propose_v a_o trapezium_fw-la abcd_n which_o have_v its_o side_n ad_fw-la bc_n parallel_n and_o admit_v one_o will_v cut_v of_o a_o three_o part_n let_v ce_fw-fr be_v make_v equal_a to_o ad_fw-la and_o bg_n the_o three_o part_n of_o be._n draw_v ag._n i_o say_v that_o the_o triangle_n abg_n be_v the_o three_o of_o the_o trapezium_fw-la abcd._n demonstration_n the_o triangle_n adf_n fce_n 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square_n of_o the_o line_n add_v be_v double_a to_o the_o square_n of_o half_a the_o line_n and_o to_o the_o square_n which_o be_v compose_v of_o the_o half_a line_n and_o the_o line_n add_v if_o one_o suppose_v ab_fw-la to_o be_v divide_v in_o the_o middle_n at_o the_o point_n c_o and_o if_o thereto_o be_v add_v the_o line_n bd_o the_o square_n of_o ab_fw-la and_o bd_o shall_v be_v double_a to_o the_o square_n of_o ac_fw-la and_o cd_o add_v together_o draw_v the_o perpendicular_n ce_fw-fr df_n equal_a to_o ac_fw-la then_o draw_v the_o line_n ae_n of_o agnostus_n ebg_n demonstration_n the_o line_n ac_fw-la ce_fw-fr cb_n be_v equal_a and_o the_o angle_n at_o the_o point_n c_o being_n right_n the_o angle_n aec_fw-la ceb_fw-mi cbe_n dbg_n dgb_n shall_v be_v half_a right_n and_o the_o line_n db_n dg_fw-mi and_o of_o fg_v cd_o shall_v be_v equal_a the_o square_a of_o ae_n be_v double_a to_o the_o square_n of_o ac_fw-la the_o square_a of_o eglantine_n be_v double_a to_o the_o square_n of_o 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under_o dc_o cb_n by_o the_o 4_o therefore_o the_o square_n of_o ab_fw-la be_v equal_a to_o the_o square_n of_o ad_fw-la dc_o cb_n and_o to_o two_o rectangle_v comprehend_v under_o dc_o cb_n in_o the_o place_n of_o the_o two_o last_v square_n ad_fw-la dc_o put_v the_o square_n of_o ac_fw-la which_o be_v equal_a to_o they_o by_o the_o 47_o of_o the_o 1_o the_o square_a of_o ab_fw-la shall_v be_v equal_a to_o the_o square_n of_o ac_fw-la and_o cb._n and_o to_o two_o rectangle_v comprehend_v under_o dc_o cb._n use_v this_o proposition_n be_v useful_a to_o measure_v the_o area_n of_o a_o triangle_n it_o be_v three_o side_n be_v know_v for_o example_n if_o the_o side_n ab_fw-la be_v twenty_o foot_n ac_fw-la 13_o bc_n 11_o the_o square_a of_o ab_fw-la will_v be_v four_o hundred_o the_o square_a of_o ac_fw-la one_o hundred_o sixty_o nine_o and_o the_o square_a of_o bc_n one_o hundred_o twenty_o one_o the_o sum_n of_o the_o two_o last_o be_v two_o hundred_o and_o ninety_o which_o be_v subtract_v from_o four_o hundred_o leave_v one_o hundred_o and_o ten_o for_o the_o two_o rectangle_v under_o bc_n cd_o the_o one_o half-fifty_a five_z shall_v be_v one_o of_o those_o rectangle_v which_o divide_v by_o bc_n 11_o we_o shall_v have_v five_o for_o the_o line_n cd_o who_o square_a be_v twenty_o five_o which_o be_v subtract_v from_o the_o square_n of_o ac_fw-la one_o hundred_o sixty_o nine_o there_o remain_v the_o square_a of_o ad_fw-la one_o hundred_o forty_o four_o and_o its_o root_n shall_v be_v the_o side_n ad_fw-la which_o be_v multiply_v by_o 5½_n the_o half_a of_o bc_n you_o have_v the_o area_n of_o the_o triangle_n abc_n contain_v 66_o square_a foot_n proposition_n xiii_o theorem_fw-la in_o any_o triangle_n whatever_o the_o square_a of_o the_o side_n opposite_a to_o a_o acute_a angle_n together_o with_o two_o rectangle_v comprehend_v under_o the_o side_n on_o which_o the_o perpendicular_a fall_v and_o under_o the_o line_n which_o be_v betwixt_o the_o perpendicular_a and_o that_o angle_n be_v equal_a to_o the_o square_n of_o the_o other_o side_n let_v the_o propose_a triangle_n be_v abc_n which_o have_v the_o angle_n c_o acute_a and_o if_o one_o draw_v ad_fw-la perpendicular_a to_o bc_n the_o square_a of_o the_o side_n ab_fw-la which_o be_v opposite_a to_o the_o acute_a angle_n c_o together_z with_o two_o rectangle_v comprehend_v under_o bc_n dc_o shall_v be_v equal_a to_o the_o square_n of_o ac_fw-la bc._n demonstration_n the_o line_n bc_n be_v divide_v in_o d_o whence_o by_o the_o 7_o the_o square_a of_o bc_n dc_o be_v equal_a to_o two_o rectangle_v under_o bc_n dc_o and_o to_o the_o square_n of_o bd_o add_v to_o both_o the_o square_n of_o ad_fw-la the_o square_a of_o bd_o dc_o ad_fw-la shall_v be_v equal_a to_o two_o rectangle_v under_o bc_n dc_o and_o to_o the_o square_n of_o bd_o ad_fw-la in_o the_o place_n of_o the_o square_n of_o cd_o ad_fw-la put_v the_o square_n of_o ac_fw-la which_o be_v equal_a to_o they_o by_o the_o 47_o of_o the_o 1_o and_o instead_o of_o the_o square_n of_o bd_o ab_fw-la substitute_v the_o square_n of_o ab_fw-la which_o be_v equal_a to_o they_o the_o square_n of_o bc_n ac_fw-la shall_v be_v equal_a to_o the_o square_n of_o ab_fw-la and_o to_o two_o rectangle_v comprehend_v under_o bc_n dc_o use_v these_o proposition_n be_v very_o necessary_a in_o trigonometry_n i_o make_v use_v thereof_o in_o the_o eight_o proposition_n of_o the_o three_o book_n to_o prove_v that_o in_o a_o triangle_n there_o be_v the_o same_o reason_n between_o the_o whole_a sine_fw-la and_o the_o sine_fw-la of_o a_o angle_n as_o be_v between_o the_o rectangle_n of_o the_o side_n comprehend_v that_o angle_n and_o
the_o three_o proportional_a to_o two_o line_n proposition_n xxxvi_o theorem_fw-la if_o from_o a_o point_n take_v without_o a_o circle_n one_o draw_v a_o touch_n line_n and_o another_o line_n to_o cut_v the_o circle_n the_o square_a of_o the_o tangent_fw-la line_n shall_v be_v equal_a to_o the_o rectangle_n comprehend_v under_o the_o whole_a secant_fw-la and_o under_o the_o exterior_a line_n let_v from_o the_o point_n a_o take_v without_o the_o circle_n be_v draw_v a_o line_n ab_fw-la to_o touch_v the_o same_o in_o b_o another_z ac_fw-la or_o ah_o to_o cut_v the_o circle_n the_o square_a of_o ab_fw-la shall_v be_v equal_a to_o the_o rectangle_n comprehend_v under_o ac_fw-la ao_o as_o also_o to_o the_o rectangle_n comprehend_v under_o ah_o af._n if_o the_o secant_fw-la pass_v through_o the_o centre_n as_o ac_fw-la draw_v the_o line_n ebb_n demonstration_n see_v the_o line_n oc_fw-fr be_v divide_v in_o the_o middle_n in_o e_z and_o that_o thereto_o be_v add_v the_o line_n ao_o the_o rectangle_n comprehend_v under_o ao_o ac_fw-la with_o the_o square_n of_o oe_n or_o ebb_v shall_v be_v equal_a to_o the_o square_n of_o ae_n by_o the_o 6_o of_o the_o 2d_o now_o the_o line_n ab_fw-la touch_v the_o circle_n in_o the_o point_n b_o so_o by_o the_o 17_o the_o angle_n b_o be_v right_o and_o by_o the_o 47th_o of_o the_o one_a the_o square_a of_o ae_n be_v equal_a to_o the_o square_n of_o ab_fw-la ebb_n therefore_o the_o rectangle_n under_o ac_fw-la ao_o with_o the_o square_n of_o ebb_n be_v equal_a to_o the_o square_n of_o ab_fw-la ebb_n and_o take_v away_o the_o square_n of_o ebb_n the_o rectangle_n under_o ac_fw-la ao_o shall_v be_v equal_a to_o the_o square_n of_o ab_fw-la second_o let_v the_o secant_fw-la ah_o not_o pass_v through_o the_o centre_n draw_v the_o line_n ah_o the_o perpendicular_a eglantine_n which_o shall_v divide_v in_o the_o middle_n in_o g_z the_o line_n fh_v draw_v also_o the_o line_n ef._n demonstration_n the_o line_n fh_v be_v divide_v equal_o in_o the_o point_n g_o and_o the_o line_n of_o be_v add_v thereto_o the_o rectangle_n comprehend_v under_o ah_o of_o with_o the_o square_n of_o fg_v shall_v be_v equal_a to_o the_o square_n of_o ag._n add_v to_o both_o the_o square_n of_o eglantine_n the_o rectangle_n ah_o of_o with_o the_o square_n of_o fg_v ge_z that_o be_v to_o say_v by_o the_o 47th_o of_o the_o one_a with_o the_o square_n of_o fe_o or_o ebb_v shall_v be_v equal_a to_o the_o square_n of_o agnostus_n ge_z 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 ae_n moreover_o the_o square_a of_o ae_n by_o the_o same_o be_v equal_a to_o the_o square_n of_o ebb_n ab_fw-la therefore_o the_o rectangle_n compprehend_v under_o ah_o of_o with_o the_o square_n of_o be_v be_v equal_a to_o the_o square_n of_o be_v ab_fw-la and_o take_v from_o both_o the_o square_n of_o be_v the_o rectangle_n comprehend_v under_o ah_o of_o shall_v be_v equal_a to_o the_o square_n of_o ab_fw-la coral_n 1._o if_o you_o draw_v several_a secant_v ac_fw-la ah_o the_o rectangle_v ac_fw-la ao_o and_o ah_o of_o shall_v be_v equal_a among_o themselves_o see_v that_o the_o one_o and_o the_o other_o be_v equal_a to_o the_o square_n of_o ab_fw-la coral_n 2._o if_o there_o be_v draw_v two_o tangent_n ab_fw-la ai_fw-fr they_o shall_v be_v equal_a because_o their_o square_n be_v equal_a to_o the_o same_o rectangle_n ac_fw-la ao_o and_o by_o consequence_n among_o themselves_o as_o also_o the_o line_n proposition_n xxxvii_o theorem_fw-la if_o the_o rectangle_n comprehend_v under_o the_o secant_fw-la and_o under_o the_o exterior_a line_n be_v equal_a to_o a_o line_n which_o fall_v on_o the_o circle_n that_o line_n touch_v the_o same_o let_v be_v draw_v the_o secant_fw-la ac_fw-la or_o ah_o and_o let_v the_o rectangle_n ac_fw-la ao_o or_o the_o rectangle_n ah_o of_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la that_o line_n shall_v touch_v the_o circle_n draw_v the_o touch_n line_n ai_fw-fr by_o the_o 17_o and_o the_o line_n je_n demonstration_n see_v the_o line_n ai_fw-fr touch_v the_o circle_n the_o rectangle_n ac_fw-la ao_o or_o ah_o of_o shall_v be_v equal_a to_o the_o square_n of_o ai._n now_o the_o square_n of_o ab_fw-la be_v suppose_v equal_a to_o each_o rectangle_n therefore_o the_o square_n of_o ai_fw-fr and_o of_o ab_fw-la be_v equal_a and_o by_o consequence_n the_o line_n ai_fw-fr ab_fw-la so_o then_o the_o triangle_n abe_n aei_n which_o have_v all_o their_o side_n equal_a to_o each_o other_o shall_v be_v equiangle_v by_o the_o 8_o of_o the_o one_a and_o see_v the_o angle_n aye_o be_v right_o by_o the_o 17_o the_o line_n a_z be_v a_o touch_n line_n the_o angle_n abe_n shall_v be_v right_o and_o the_o line_n ab_fw-la a_o touch_n line_n also_o by_o the_o 16_o use_v xxxvii_o prop._n xxxvii_o maurolicus_n make_v use_v of_o this_o proposition_n to_o find_v the_o diameter_n of_o the_o earth_n for_o look_v from_o the_o top_n of_o a_o mountain_n oa_n to_o the_o extremity_n of_o the_o earth_n along_o the_o line_n basilius_n he_o observe_v the_o angle_n oab_n which_o the_o line_n ab_fw-la make_v with_o a_o plumb_n line_n ac_fw-la and_o he_o conclude_v the_o length_n of_o the_o line_n ab_fw-la by_o a_o trigonometrical_a calculation_n he_o multiply_v ab_fw-la by_o ab_fw-la to_o have_v its_o square_n which_o he_o divide_v by_o ao_o the_o height_n of_o the_o mountain_n from_o which_o have_v take_v away_o ao_o there_o remain_v oc_n the_o diameter_n of_o the_o earth_n this_o proposition_n serve_v also_o to_o prove_v the_o five_o proposition_n of_o the_o three_o book_n of_o trigonometry_n the_o end_n of_o the_o three_o book_n the_o four_o book_n of_o euclid_n element_n this_o book_n be_v very_o useful_a in_o trigonometry_n see_v that_o by_o inscribe_v of_o polygon_n in_o a_o circle_n we_o have_v the_o practice_n of_o make_v the_o table_n of_o subtendants_n of_o sines_n of_o tangent_n and_o of_o secant_v which_o be_v very_o necessary_a in_o all_o sort_n of_o measure_v second_o in_o inscribe_v of_o polygon_n in_o a_o circle_n we_o have_v the_o diversity_n of_o the_o aspect_n of_o the_o planet_n which_o take_v their_o name_n from_o the_o same_o polygon_n three_o in_o the_o practice_n hereof_o we_o have_v a_o method_n which_o give_v the_o quadrature_n of_o the_o circle_n as_o near_o the_o truth_n as_o shall_v be_v needful_a we_o demonstrate_v also_o that_o a_o circle_n be_v in_o duplicate_v reason_n to_o its_o diameter_n four_o military_a architecture_n have_v need_n of_o inscribe_v polygon_n into_o circle_n in_o the_o design_v or_o draw_v of_o regular_a fortification_n the_o definition_n 1._o v._n fig._n i._o plate_n v._n a_o right_a line_a figure_n be_v inscribe_v in_o a_o circle_n or_o the_o circle_n be_v describe_v about_o a_o figure_n when_o all_o its_o angle_n be_v in_o the_o circumference_n of_o the_o same_o circle_n as_o the_o triangle_n abc_n be_v inscribe_v in_o a_o circle_n and_o the_o circle_n be_v describe_v about_o the_o triangle_n because_o the_o angle_n a_o b_o c_o do_v touch_v the_o circumference_n the_o triangle_n def_n be_v not_o inscribe_v in_o the_o circle_n because_o the_o angle_n d_o do_v not_o touch_v the_o circumference_n of_o the_o circle_n 2._o i._n fig._n i._n a_o right_a line_a figure_n be_v describe_v about_o a_o circle_n and_o the_o circle_n be_v inscribe_v within_o the_o same_o figure_n when_o all_o the_o side_n of_o the_o figure_n touch_v the_o circumference_n of_o the_o circle_n as_o the_o triangle_n ghi_n be_v describe_v about_o the_o circle_n klm_n because_o its_o side_n touch_v the_o circle_n in_o k_o l_o m._n 3._o a_o line_n be_v fit_v or_o inscribe_v in_o a_o circle_n when_o it_o be_v terminate_v at_o both_o â—…s_n by_o the_o circumference_n of_o the_o circle_n as_o the_o line_n no_o the_o line_n rp_n be_v not_o inscribe_v in_o the_o circle_n proposition_n i._o problem_n to_o inscribe_v in_o a_o circle_n a_o line_n which_o surpass_v not_o its_o diameter_n it_o be_v propose_v to_o inscribe_v in_o a_o circle_n aebd_a a_o line_n which_o surpass_v not_o its_o diameter_n take_v the_o length_n thereof_o on_o the_o diameter_n and_o let_v it_o be_v for_o example_n bc._n put_v the_o foot_n of_o the_o compass_n in_o b_o and_o describe_v a_o circle_n at_o the_o extent_n of_o bc_n which_o cut_v the_o circle_n aebd_a in_o d_o and_o e._n draw_v the_o line_n bd_o or_o be._n it_o be_v evident_a they_o be_v equal_a to_o bc_n by_o the_o definition_n of_o a_o circle_n use_v this_o proposition_n be_v necessary_a for_o the_o practice_n of_o those_o which_o follow_v proposition_n ii_o problem_n to_o inscribe_v in_o a_o circle_n a_o triangle_n equiangle_v to_o another_o triangle_n the_o circle_n egh_n be_v propose_v in_o which_o one_o will_v inscribe_v a_o triangle_n equiangle_v to_o the_o triangle_n
superficies_n 18._o a_o cone_n be_v a_o figure_n make_v when_o one_o side_n of_o a_o right_a angle_a triangle_n viz._n one_o of_o those_o that_o contain_v the_o right_a angle_n remain_v fix_v the_o triangle_n be_v turn_v round_o about_o till_o it_o return_v to_o the_o place_n from_o whence_o it_o first_o move_v and_o if_o the_o fix_a right_a line_n be_v equal_a to_o the_o other_o which_o contain_v the_o right_a angle_n than_o the_o cone_n be_v a_o rectangled_a cone_n but_o if_o it_o be_v less_o it_o be_v a_o obtuse_a angle_a cone_n if_o great_a a_o acute_a angle_a cone_n 19_o the_o axis_n of_o a_o cone_n be_v that_o fix_a line_n about_o which_o the_o triangle_n be_v move_v 20._o a_o cylinder_n be_v a_o figure_n make_v by_o the_o move_a round_n of_o a_o right_a angle_a parallelogram_n one_o of_o the_o side_n thereof_o namely_o which_o contain_v the_o right_a angle_n abide_v fix_v till_o the_o parallelogram_n be_v turn_v about_o to_o the_o same_o place_n whence_o it_o begin_v to_o move_v 21._o like_a cones_n and_o cylinder_n be_v those_o who_o axe_n and_o diameter_n of_o their_o base_n be_v proportional_a cones_n be_v right_a when_o the_o axis_n be_v perpendicular_a to_o the_o plain_a of_o the_o base_a and_o they_o be_v say_v to_o be_v scalene_n when_o the_o axis_n be_v incline_v to_o the_o base_a and_o the_o diameter_n of_o their_o base_n be_v in_o the_o same_o ratio_fw-la we_o add_v that_o incline_v cones_n to_o be_v like_o their_o axe_n must_v have_v the_o same_o inclination_n to_o the_o plane_n of_o their_o base_n proposition_n i._o theorem_fw-la i._n plate_n vii_o prop._n i._n a_o straight_a line_n can_v have_v one_o of_o its_o part_n in_o a_o plane_n and_o the_o other_o without_o it_o if_o the_o line_n ab_fw-la be_v in_o the_o plane_n ad_fw-la it_o be_v continue_v shall_v not_o go_v without_o but_o all_o its_o part_n shall_v be_v in_o the_o same_o plane_n for_o if_o it_o can_v be_v that_o bc_n be_v a_o part_n of_o ab_fw-la continue_v draw_v in_o the_o plane_n cd_o the_o line_n bd_o perpendicular_a to_o ab_fw-la draw_v also_o in_o the_o same_o plane_n be_v perpendicular_a to_o bd._n demonstration_n the_o angle_n abdella_n bde_n be_v both_o right_a angle_n thence_o by_o the_o 14_o of_o the_o first_o ab_fw-la be_v do_v make_v but_o one_o line_n and_o consequent_o bc_n be_v not_o a_o part_n of_o the_o line_n ab_fw-la continue_v otherwise_o two_o strait_a line_n cb_n ebb_n will_v have_v the_o same_o part_n ab_fw-la that_o be_v ab_fw-la will_v be_v part_n of_o both_o which_o we_o have_v reject_v as_o false_a in_o the_o thirteen_o maxim_n of_o the_o first_o book_n use_v we_o establish_v on_o this_o proposition_n a_o principle_n in_o gnomonic_n to_o prove_v that_o the_o shadow_n of_o the_o stile_n fall_v not_o without_o the_o plane_n of_o a_o great_a circle_n in_o which_o the_o sun_n be_v see_v that_o the_o end_n or_o top_n of_o the_o stile_n be_v take_v for_o the_o centre_n of_o the_o heaven_n and_o consequent_o for_o the_o centre_n of_o all_o the_o great_a circle_n the_o shadow_n be_v always_o in_o a_o straight_a line_n with_o the_o ray_n draw_v from_o the_o sun_n to_o the_o opaque_fw-fr body_n this_o ray_n be_v in_o any_o great_a circle_n the_o shadow_n must_v also_o be_v therein_o proposition_n ii_o theorem_fw-la line_n which_o cut_v one_o another_o be_v in_o the_o same_o plane_n as_o well_o as_o all_o the_o part_n of_o a_o triangle_n if_o the_o two_o line_n be_v cd_o cut_v one_o another_o in_o the_o point_n a_o and_o if_o there_o be_v make_v a_o triangle_n by_o draw_v the_o base_a bc_n i_o say_v that_o all_o the_o part_n of_o the_o triangle_n abc_n be_v in_o the_o same_o plane_n and_o that_o the_o line_n be_v cd_o be_v likewise_o therein_o demonstration_n it_o can_v be_v say_v that_o any_o one_o part_n of_o the_o triangle_n abc_n be_v in_o a_o plane_n and_o that_o the_o other_o part_n be_v without_o without_o say_v that_o one_o part_n of_o a_o line_n be_v in_o one_o plane_n and_o that_o the_o other_o part_n of_o the_o same_o line_n be_v not_o therein_o which_o be_v contrary_a to_o the_o first_o proposition_n and_o see_v that_o the_o side_n of_o the_o triangle_n be_v in_o the_o same_o plane_n wherein_o the_o triangle_n be_v the_o line_n be_v cd_o shall_v be_v in_o the_o same_o plane_n use_v this_o proposition_n do_v sufficient_o determine_v a_o plane_n by_o two_o straight_a line_n mutual_o intersect_v each_o other_o or_o by_o a_o triangle_n i_o have_v make_v use_n thereof_o in_o optic_n to_o prove_v that_o the_o objective_a parallel_n line_n which_o fall_v on_o the_o tablet_n aught_o to_o be_v represent_v by_o line_n which_o concur_v in_o a_o point_n proposition_n iii_o theorem_fw-la the_o common_a section_n of_o two_o place_n be_v a_o straight_a line_n if_o two_o planes_n ab_fw-la cd_o cut_v one_o another_o their_o common_a section_n of_o shall_v be_v a_o straight_a line_n for_o if_o it_o be_v not_o take_v two_o point_n common_a to_o both_o plane_n which_o let_v be_v e_o and_o f_o and_o draw_v a_o 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shall_v have_v their_o base_n agnostus_n ah_o equal_a last_o the_o triangle_n abh_n abg_n have_v all_o their_o side_n equal_a thence_o by_o the_o 27_o of_o the_o one_a the_o angles_n abg_n abh_n shall_v be_v equal_a and_o the_o line_n ab_fw-la perpendicular_a to_o gh_v so_o then_o the_o line_n ab_fw-la shall_v be_v perpendicular_a to_o any_o line_n which_o may_v be_v draw_v through_o the_o point_n b_o in_o the_o plane_n of_o the_o line_n cd_o of_o which_o i_o call_v perpendicular_a to_o the_o plane_n use_v this_o proposition_n come_v often_o in_o use_n in_o the_o first_o book_n of_o theodosius_n for_o example_n to_o demonstrate_v that_o the_o axis_n of_o the_o world_n be_v
the_o diameter_n basilius_n shall_v divide_v each_o other_o equal_o in_o o._n draw_v the_o line_n bg_n gk_n ah_o lh_n i_o prove_v in_o the_o first_o place_n that_o they_o make_v but_o one_o strait_a line_n for_o the_o triangle_n dbg_n kmg_n have_v their_o side_n bd_o km_n equal_a see_v they_o be_v the_o half_n of_o equal_a side_n as_o also_o gd_v gm_n moreover_o db_n km_fw-la be_v parallel_n the_o alternate_a angel_n bdg_n gmk_n shall_v be_v equal_a by_o the_o 29_o of_o the_o first_o so_o then_o by_o the_o four_o of_o the_o first_o the_o triangle_n dbg_n kgm_n shall_v be_v equal_a in_o every_o respect_n and_o consequent_o the_o angle_n bgd_v kgm_n now_o by_o the_o 16_o of_o the_o one_a bg_n gk_n be_v but_o one_o line_n as_o also_o lh_n ha_o thence_o all_o bk_n be_v but_o one_o plane_n in_o which_o be_v find_v the_o diameter_n ab_fw-la and_o gh_n the_o common_a section_n of_o the_o plane_n the_o plane_n all_o bk_n cut_v the_o parallel_n planes_n a_o cd_o shall_v have_v its_o common_a 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circle_n be_v like_a they_o shall_v be_v in_o the_o same_o ratio_fw-la as_o be_v the_o square_n of_o the_o diameter_n be_o fn_n draw_v the_o line_n bm_n gn_n fh_o ac_fw-la demonstration_n it_o be_v suppose_v that_o the_o polygon_n be_v like_a that_o be_v to_o say_v that_o the_o angle_n b_o and_z g_z be_v equal_a and_o that_o there_o be_v the_o same_o ratio_fw-la of_o ab_fw-la to_o bc_n as_o of_o fg_a to_o gh_v whence_o i_o conclude_v by_o the_o 6_o of_o the_o 6_o that_o the_o triangle_n abc_n fgh_a be_v equiangular_a and_o that_o the_o angles_n acb_n fhg_n be_v equal_a so_o then_o by_o the_o 21_o of_o the_o 3d._n the_o angel_n ambiguity_n fng_v be_v equal_a now_o the_o angel_n abm_n fgn_n be_v in_o a_o semicircle_n be_v right_o by_o the_o 31st_o of_o the_o 3d._n and_o consequent_o the_o triangle_n abm_n fgn_n be_v equiangular_a therefore_o by_o the_o 5_o of_o the_o 6_o there_o shall_v be_v the_o same_o ratio_fw-la of_o ab_fw-la to_o fg_n as_o of_o be_o to_o fn_n and_o by_o the_o 22d_o of_o the_o 11_o if_o one_o describe_v two_o similar_v polygon_n on_o ab_fw-la fg_v which_o be_v those_o propose_v and_o also_o two_o other_o like_a polygon_n be_o and_o fn_n which_o shall_v be_v their_o square_n there_o shall_v be_v the_o same_o ratio_fw-la of_o the_o polygon_n abcde_v to_o the_o polygon_n fgnkl_n as_o of_o the_o square_n of_o be_o to_o the_o square_n of_o fn_n this_o proposition_n be_v necessary_a to_o demonstrate_v the_o follow_a proposition_n lemma_n if_o a_o quantity_n be_v less_o than_o a_o circle_n there_o may_v be_v inscribe_v in_o the_o circle_n a_o regular_a polygon_n great_a than_o that_o quantity_n ii_o lemma_n fig._n ii_o let_v the_o figure_n a_o be_v less_o than_o the_o circle_n b_o there_o may_v be_v inscribe_v in_o the_o circle_n a_o regular_a polygon_n great_a than_o the_o figure_n a._n let_v the_o figure_n g_o be_v the_o difference_n of_o the_o figure_n a_o and_o the_o circle_n in_o such_o sort_n that_o the_o figure_n a_o and_o g_o take_v together_o be_v equal_a to_o the_o circle_n b._n inscribe_v in_o the_o circle_n b_o the_o square_n cdef_n by_o the_o 6_o of_o the_o four_o if_o that_o square_n be_v great_a than_o the_o figure_n a_o we_o shall_v have_v what_o we_o pretend_v to_o if_o it_o be_v less_o divide_v the_o quadrant_n of_o the_o circle_n cd_o de_fw-fr of_o fc_n into_o two_o equal_o in_o the_o point_v h_n i_o k_o l_o in_o such_o sort_n as_o you_o may_v have_v a_o octogon_n and_o if_o that_o octogon_n be_v less_o than_o the_o figure_n a_o subdivide_v the_o ark_n and_o you_o shall_v have_v a_o polygon_n of_o sixteen_o side_n than_o thirty_o two_o of_o sixty_o four_o etc._n etc._n i_o say_v in_o fine_a you_o shall_v have_v a_o polygon_n great_a than_o the_o figure_n a_o that_o be_v to_o say_v a_o polygon_n have_v less_o difference_n from_o the_o circle_n than_o the_o figure_n a_o in_o such_o sort_n that_o the_o difference_n shall_v be_v less_o than_o the_o figure_n g._n demonstration_n the_o square_n inscribe_v be_v more_o than_o the_o one_o half_a of_o the_o circle_n it_o be_v the_o half_a of_o the_o square_n describe_v about_o the_o circle_n and_o in_o describe_v the_o octagon_n you_o take_v away_o more_o than_o the_o half_a of_o the_o remainder_n that_o be_v to_o say_v the_o four_o segment_n chd_v die_v ekf_n clf_n for_o the_o triangle_n chd_n be_v the_o half_a of_o the_o rectangle_n co_n by_o the_o 34th_o of_o the_o one_a it_o be_v then_o more_o than_o the_o half_a of_o the_o segment_n chd_v it_o be_v the_o same_o of_o the_o other_o arks._n in_o like_a manner_n in_o describe_v a_o polygon_n of_o sixteen_o side_n you_o take_v away_o more_o than_o one_o half_a of_o that_o which_o remain_v in_o the_o circle_n and_o so_o of_o the_o rest_n you_o
inscribe_v by_o the_o little_a rectangle_v through_o which_o the_o circumference_n of_o the_o circle_n pass_v and_o all_o those_o rectangle_v take_v together_o be_v equal_a to_o the_o rectangle_n al._n imagine_v that_o the_o semicircle_n be_v make_v to_o roll_n about_o the_o diameter_n ebb_n the_o semicircle_n shall_v describe_v a_o hemi-sphere_n and_o the_o inscribe_v rectangle_v will_v describe_v inscribe_v cylinder_n in_o the_o semi-sphere_n and_o the_o circumscribe_v will_v describe_v other_o cylinder_n demonstration_n the_o circumscribe_v cylinder_n surpass_v more_o the_o inscribe_v than_o do_v the_o hemi-sphere_n surpass_v the_o same_o inscribe_v cylinder_n see_v that_o they_o be_v comprehend_v within_o the_o circumscribe_v cylinder_n now_o the_o circumscribe_v surpass_v the_o inscribe_v by_o so_o much_o as_o be_v the_o cylinder_n all_o therefore_o the_o hemi-sphere_n shall_v surpass_v by_o less_o the_o inscribe_v cylinder_n than_o do_v the_o cylinder_n make_v by_o the_o rectangle_n al._n the_o cylinder_n all_o be_v less_o than_o the_o cylinder_n mp_n for_o there_o be_v the_o same_o ratio_fw-la of_o a_o great_a circle_n of_o the_o sphere_n which_o serve_v for_o base_a to_o the_o cylinder_n all_o as_o of_o mp_n to_o r_n so_o then_o by_o the_o forego_n a_o cylinder_n which_o have_v for_o base_a a_o great_a circle_n of_o the_o sphere_n and_o the_o altitude_n r_n will_v be_v equal_a to_o the_o cylinder_n mp_n consequent_o the_o hemi-sphere_n which_o surpass_v the_o quantity_n d_o by_o the_o cylinder_n mp_n and_o the_o inscribe_v cylinder_n by_o a_o quantity_n less_o than_o all_o surpass_v the_o inscribe_v cylinder_n by_o less_o than_o the_o quantity_n d._n therefore_o the_o quantity_n d_o be_v less_o than_o the_o inscribe_v cylinder_n what_o i_o have_v say_v of_o a_o hemi_a sphere_n may_v be_v say_v of_o a_o whole_a sphere_n lemma_n ii_o like_a cylinder_n inscribe_v in_o two_o sphere_n be_v in_o triple_a ratio_fw-la of_o the_o diameter_n of_o the_o sphere_n ii_o lemma_n fig._n ii_o if_o the_o two_o like_a cylinder_n cd_o of_o be_v inscribe_v in_o the_o sphere_n a_o b_o they_o shall_v be_v in_o triple_a ratio_fw-la of_o the_o diameter_n lm_o no_o draw_v the_o line_n gd_v if_o demonstration_n the_o like_a right_n cylinder_v cd_o of_o be_v like_a so_o then_o there_o be_v the_o same_o ratio_fw-la of_o hd_a to_o dr_n as_o of_o qf_n to_o f_n as_o also_o the_o same_o ratio_fw-la of_o kd_a to_o kg_v as_o of_o pf_n to_o pi._n and_o consequent_o the_o triangle_n gdk_v ifp_n be_v like_o by_o the_o 6_o of_o the_o 6_o so_o there_o shall_v be_v the_o same_o ratio_fw-la of_o kd_v to_o pf_n as_o of_o gd_a to_o if_o or_o of_o lm_o to_o on_o now_o the_o like_a cylinder_n cd_o of_o be_v in_o triple_a ratio_fw-la of_o kd_a and_o pf_n the_o diameter_n of_o their_o base_n by_o the_o 12_o therefore_o the_o like_a cylinder_n cd_o of_o inscribe_v in_o the_o sphere_n a_o and_o b_o be_v in_o triple_a ratio_fw-la of_o the_o diameter_n of_o the_o sphere_n proposition_n xviii_o theorem_fw-la sphere_n be_v in_o triple_a ratio_fw-la of_o their_o diameter_n the_o sphere_n a_o and_o b_o be_v in_o triple_a ratio_fw-la of_o their_o diameter_n cd_o ef._n for_o if_o they_o be_v not_o in_o triple_a ratio_fw-la one_o of_o the_o sphere_n as_o a_o shall_v be_v in_o a_o great_a ratio_fw-la then_o triple_a of_o that_o of_o cd_o to_o of_o therefore_o a_o quantity_n g_z less_o than_o the_o sphere_n a_o shall_v be_v in_o triple_a ratio_fw-la of_o that_o of_o cd_o to_o of_o and_o so_o one_o might_n according_a to_o the_o first_o lemma_n inscribe_v in_o the_o sphere_n a_o cylinder_v of_o the_o same_o height_n great_a than_o the_o quantity_n g._n let_v there_o be_v inscribe_v in_o the_o sphere_n b_o as_o many_o like_a cylinder_n as_o those_o of_o the_o cylinder_n a._n demonstration_n the_o cylinder_n of_o the_o sphere_n a_o to_o those_o of_o the_o sphere_n b_o shall_v be_v in_o triple_a ratio_fw-la of_o that_o of_o cd_o to_o of_o by_o the_o precede_a now_o the_o quantity_n g_o in_o respect_n of_o the_o sphere_n b_o be_v in_o triple_a ratio_fw-la of_o cd_o to_o of_o there_o be_v then_o the_o same_o ratio_fw-la of_o the_o cylinder_n of_o the_o sphere_n a_o to_o the_o like_a cylinder_n of_o the_o sphere_n b._n so_o then_o the_o cylinder_n of_o a_o be_v great_a than_o the_o quantity_n g_o the_o cylinder_n b_o that_o be_v to_o say_v inscribe_v in_o the_o sphere_n b_o will_v be_v great_a than_o the_o sphere_n b_o which_o be_v impossible_a therefore_o the_o sphere_n a_o and_o b_o be_v in_o triple_a ratio_fw-la of_o that_o of_o their_o diameter_n coral_n sphere_n be_v in_o the_o same_o ratio_fw-la as_o be_v the_o cube_n of_o their_o diameter_n see_v that_o the_o cube_n be_v like_o solid_n be_v in_o triple_a ratio_fw-la of_o that_o of_o their_o side_n finis_fw-la errata_fw-la page_n 14._o line_n 21._o read_v afd_v p._n 23._o l._n 17._o for_o df_n r._n ef._n l._n 16._o r._n dfe_n l._n 27._o r._n fd._n p._n 24._o l._n 1_o and_o 9_o for_o fb_n r._n fd._n p._n 26._o l._n 8._o r._n he_o p._n 52._o l._n 14._o r._n acb_n p._n 55._o l._n 22._o r._n acf_n p._n 71._o l._n 3._o r._n acb_n p._n 82._o l._n 19_o r._n abc_n p._n 117._o l._n 13._o r._n gfe_n p._n 125._o l._n 9_o r._n ad._n p._n 178._o l._n 16._o r._n cfd_n p._n 194._o l._n ult_n r._n af._n p._n 197._o l._n 2._o r._n cda_n p._n 218._o l._n 7._o r._n ⅓_n and_o ⅓_n p._n 268._o l._n 24._o r._n ce._n p._n 281._o l._n 3._o r._n ecd_n p._n 289._o l._n 4._o r._n be_v simular_a p._n 294._o l._n 13._o r._n eight_o ¾_n p._n 309._o l._n 15._o r._n dbe_n advertisement_n of_o globe_n book_n map_n etc._n etc._n make_v and_o sell_v by_o philip_n lea_n at_o the_o sign_n of_o the_o atlas_n and_o hercules_n in_o the_o poultry_n near_o cheapside_n london_n 1._o a_o new_a size_n of_o globe_n about_o 15_o inch_n diameter_n make_v according_a to_o the_o more_o accurate_a observation_n and_o discovery_n of_o our_o modern_a astronomer_n and_o geographer_n and_o much_o different_a from_o all_o that_o ever_o be_v yet_o extant_a all_o the_o southern_a constellation_n according_a to_o mr._n hally_n observation_n in_o the_o island_n of_o st._n helena_n and_o many_o of_o the_o northern_a price_n four_o pound_n 2._o a_o size_n of_o globe_n of_o about_o ten_o inch_n diameter_n very_o much_o correct_v price_n 50_o s._n 3._o concave_a hemisphere_n three_o inch_n diameter_n which_o serve_v as_o a_o case_n for_o a_o terrestrial_a globe_n and_o may_v be_v carry_v in_o the_o pocket_n or_o fit_v up_o in_o frames_n price_n 15_o or_o 20_o s._n 4._o another_o globe_n of_o about_o four_o inch_n diameter_n fit_v to_o move_v in_o circular_a line_n of_o brass_n for_o demonstrate_v the_o reason_n of_o dyall_v or_o be_v erect_v upon_o a_o small_a pedestal_n and_o fetch_v north_n and_o south_n will_v show_v the_o hour_n of_o the_o day_n by_o its_o own_o shadow_n or_o by_o the_o help_n of_o a_o move_a meridian_n will_v show_v the_o hour_n of_o the_o day_n in_o all_o part_n of_o the_o world_n etc._n etc._n 14._o there_o be_v in_o the_o press_n a_o particular_a description_n of_o the_o general_a use_n of_o quadrant_n for_o the_o easy_a resolve_v astronomical_a geometrical_a and_o gnomonical_a problem_n and_o find_v the_o hour_n and_o azimuth_n universal_o etc._n etc._n whereunto_o be_v add_v the_o use_n of_o the_o nocturnal_a and_o equinoctial_a dial._n 15._o a_o new_a map_n of_o england_n scotland_n and_o ireland_n with_o the_o road_n and_o a_o delineation_n of_o the_o genealogy_n of_o the_o king_n thereof_o from_o william_n the_o conqueror_n to_o this_o present_a time_n with_o a_o alphabetical_a table_n for_o the_o ready_a find_n of_o the_o place_n price_n 18_o d._n 16._o the_o element_n of_o euclid_n explain_v and_o demonstrate_v after_o a_o new_a and_o most_o easy_a method_n with_o the_o use_n of_o each_o proposition_n in_o all_o the_o part_n of_o the_o mathematics_n by_o claude_n francois_n millet_n dechales_n a_o jesuit_n 17._o new_a map_n of_o the_o world_n four_o quarter_n and_o of_o all_o the_o country_n and_o of_o all_o size_n make_v according_a to_o the_o late_a discovery_n extant_a may_v be_v have_v past_v upon_o cloth_n and_o colour_a also_o sea_n plait_n mathematical_a projection_n book_n and_o instrument_n whatsoever_o be_v make_v and_o sell_v by_o philip_n lea._n finis_fw-la euclid_n element_n with_o the_o use_v euclid_n london_n print_v for_o philip_n lea_n globe_n maker_n at_o the_o atlas_n and_o hercules_n in_o cheapside_n near_o friday_n street_n there_o all_o sort_n of_o globe_n sphere_n map_n sea-plat_n mathematical_a book_n and_o instrument_n be_v make_v and_o sell_v plate_n 2._o proposition_n and_o use_v of_o the_o first_o book_n see_v plate_n 3_o plate_n 1._o definition_n of_o the_o first_o book_n proposition_n and_o use_v of_o the_o first_o book_n see_v plate_n 2._o plate_n 3._o proposition_n &_o use_v of_o the_o first_o book_n definition_n proposition_n &_o use_v of_o the_o second_o book_n plate_n 4_o definition_n of_o the_o three_o book_n proposition_n proposition_n and_o use_v plate_n 5._o definition_n propsition_n of_o the_o fouth_z book_n plate_n 6._o definition_n of_o the_o six_o book_n proposition_n &_o use_v plate_n 7._o definition_n of_o the_o eleven_o book_n proposition_n plate_n 8._o proposi_fw-la of_o the_o twelve_o book_n
when_o the_o term_n between_o they_o be_v take_v twice_o that_o be_v to_o say_v as_o antecedent_n and_o as_o consequent_a as_o if_o there_o be_v the_o same_o reason_n of_o a_o to_o b_o as_o of_o b_o to_z c_z and_z of_o c_z to_z d._n 11._o then_o a_o to_o c_o shall_v be_v in_o duplicate_v ratio_fw-la of_o a_o to_z b_o and_o the_o ratio_fw-la of_o a_o to_z d_o shall_v be_v in_o triplicate_v ratio_fw-la to_o that_o of_o a_o to_o b._n it_o be_v to_o be_v take_v notice_n that_o there_o be_v a_o great_a deal_n of_o difference_n between_o double_a ratio_fw-la and_o duplicate_v ratio_fw-la we_o say_v that_o the_o ratio_fw-la of_o four_o to_o two_o be_v double_a that_o be_v to_o say_v four_z be_v the_o double_a of_o two_o whence_o it_o follow_v that_o the_o number_n two_o be_v that_o which_o give_v the_o name_n to_o this_o ratio_fw-la or_o rather_o to_o the_o antecedent_n of_o this_o ratio_fw-la so_o we_o we_o say_v double_a triple_a quadruple_a quintuple_a which_o be_v denomination_n take_v from_o those_o number_n duo_fw-la tres_fw-la quatuor_fw-la quinque_fw-la compare_v with_o unity_n for_o we_o better_o conceive_v a_o reason_n when_o its_o term_n be_v small_a but_o as_o i_o have_v already_o take_v notice_n those_o denomination_n fall_v rather_o on_o the_o antecedent_n than_o on_o the_o reason_n itself_o we_o call_v that_o double_a triple_a reason_n or_o ratio_fw-la when_o the_o antecedent_n be_v double_a or_o triple_a to_o the_o consequent_a but_o when_o we_o say_v the_o reason_n be_v duplicate_v we_o mean_v a_o reason_n compound_v of_o two_o like_a reason_n as_o if_o there_o be_v the_o same_o reason_n of_o two_o to_o four_o as_o of_o four_o to_o eight_o the_o reason_n of_o two_o and_o eight_o be_v compound_v of_o the_o reason_n of_o two_o and_o four_o and_o of_o that_o of_o four_o and_o eight_o which_o be_v alike_o and_o as_o equal_v the_o reason_n or_o ratio_fw-la of_o two_o to_o eight_o shall_v be_v duplicate_v by_o each_o three_o to_o twenty_o seven_o be_v a_o duplicate_v reason_n of_o that_o of_o three_o to_o nine_o the_o reason_n of_o two_o to_o four_o be_v call_v subduple_n that_o be_v to_o say_v two_o be_v the_o half_a of_o four_o but_o the_o reason_n of_o two_o to_o eight_o be_v duple_n of_o the_o sub-duple_a that_o be_v to_o say_v that_o two_o be_v the_o half_a of_o the_o half_a of_o eight_o as_o three_o be_v the_o three_o of_o the_o three_o of_o twenty_o seven_o where_o you_o see_v there_o be_v take_v twice_o the_o denominator_fw-la ½_n and_o ⅓_n in_o like_a manner_n eight_o to_o two_o be_v a_o duplicate_v reason_n of_o eight_o to_o four_o because_o eight_o be_v double_a to_o four_o but_o eight_o be_v the_o double_a of_o the_o double_a of_o two_o if_o there_o be_v four_o term_n in_o the_o same_o continue_a reason_n that_o of_o the_o first_o and_o last_o be_v triple_a to_o that_o of_o the_o first_o and_o second_o as_o if_o one_o put_v these_o four_o number_n two_o four_o eight_o sixteen_o the_o reason_n of_o two_o to_o sixteen_o be_v triple_a of_o two_o to_o four_o for_o two_o be_v the_o half_a of_o the_o half_a of_o the_o half_a of_o sixteen_o as_o the_o reason_n of_o sixteen_o to_o two_o be_v triple_a of_o sixteen_o to_o eight_o for_o sixteen_o be_v the_o double_a of_o eight_o and_o it_o be_v the_o double_a of_o the_o double_a of_o the_o double_a of_o two_o 12._o magnitude_n be_v homologous_a the_o antecedent_n to_o the_o antecedent_n and_o the_o consequent_n to_o the_o consequent_n as_o if_o there_o be_v the_o same_o reason_n of_o a_o to_o b_o as_o of_o c_z to_z d_o a_o and_z c_o be_v homologous_a or_o magnitude_n of_o a_o like_o ratio_fw-la the_o follow_a definition_n be_v way_n of_o argue_v by_o proportion_n and_o it_o be_v principal_o to_o demonstrate_v the_o same_o that_o this_o book_n be_v compose_v 13._o alternate_a reason_n or_o by_o permutation_n or_o exchange_n be_v when_o we_o compare_v the_o antecedent_n one_o with_o the_o other_o as_o also_o the_o consequent_n for_o example_n if_o because_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o c_z to_z d_o i_o conclude_v there_o be_v the_o same_o reason_n of_o a_o to_o c_o as_o of_o b_o to_z d_o this_o way_n of_o reason_v can_v take_v place_n but_o when_o the_o four_o term_n be_v of_o the_o same_o specie_fw-la that_o be_v to_o say_v either_o all_o four_o line_n or_o superficies_n or_o solid_n proposition_n 16._o 14._o converse_v or_o inverse_v reason_n be_v a_o comparison_n of_o the_o consequent_n to_o the_o antecedent_n as_o if_o because_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o c_z to_z d_o i_o conclude_v there_o be_v the_o same_o reason_n of_o b_o to_o a_o as_o there_o be_v of_o d_o to_z c._n proposition_n 16._o 15._o composition_n of_o reason_n be_v a_o comparison_n of_o the_o antecedent_n and_o consequent_a take_v together_o to_o the_o consequent_a alone_o as_o if_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o c_z to_z d_o i_o conclude_v also_o that_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_o b_o as_o of_o cd_o to_o d._n prop._n 18._o 16._o division_n of_o reason_n be_v a_o comparison_n of_o the_o excess_n of_o the_o antecedent_n above_o the_o consequent_a to_o the_o same_o consequent_a as_o if_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_z b_o as_o of_o cd_o to_o d_o i_o conclude_v that_o there_o be_v the_o same_o reason_n of_o a_o to_o b_o as_o of_o cd_o prop._n 17._o 17._o conversion_n of_o reason_n be_v the_o comparison_n of_o the_o antecedent_n to_o the_o difference_n of_o the_o term_n as_o if_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_z b_o as_o of_o cd_o to_o d_o i_o conclude_v that_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_o a_o as_o of_o cd_o to_o c._n proposition_n 18._o 18._o proportion_n of_o equality_n be_v a_o comparison_n of_o the_o extreme_a quantity_n in_o leave_v out_o those_o in_o the_o middle_n a_o b_o c_o d_o e_o f_o g_o h._n as_o if_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o e_z to_o f_o and_o of_o b_o to_z c_z as_o of_o f_o to_o g_z and_z of_o c_z to_z d_o as_o of_o g_z to_z h._n i_o draw_v this_o consequence_n that_o there_o be_v therefore_o the_o same_o reason_n of_o a_o to_z d_o as_o of_o e_z to_z h._n 19_o proportion_n of_o equality_n well_o rank_v be_v that_o in_o which_o one_o compare_v the_o term_n in_o the_o same_o manner_n of_o order_n as_o in_o the_o precede_a example_n prop._n 22._o 20._o proportion_n of_o equality_n ill_o rank_v be_v that_o in_o which_o one_o compare_v the_o term_n with_o a_o different_a order_n as_o if_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o g_z to_o h_n and_z of_o b_o to_z c_z as_o of_o f_o to_o g_z and_z of_o c_z to_z d_o as_o of_o e_z to_z f._n i_o draw_v this_o conclusion_n that_o there_o be_v the_o same_o reason_n of_o a_o to_z d_o as_o of_o e_z to_z h._n prop._n 28._o here_o be_v all_o the_o way_n of_o argue_v by_o proportion_n there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o c_z to_z d_o therefore_o by_o alternate_a reason_n there_o be_v the_o same_o reason_n of_o a_o to_o c_o as_o of_o b_o to_z d_o and_o by_o inversed_a reason_n there_o be_v the_o same_o reason_n of_o b_o to_o a_o as_o of_o d_o to_z c_z and_o by_o composition_n there_o be_v the_o same_o reason_n of_o ab_fw-la to_o b_o as_o of_o cd_o to_o d._n by_o division_n of_o reason_n if_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_o b_o as_o of_o cd_o to_o d_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o c_z to_z d_o and_o by_o conversion_n there_o be_v the_o same_o reason_n of_o ab_fw-la to_o a_o as_o of_o cd_o to_o c._n by_o reason_n of_o equality_n well_o rank_v if_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o c_z to_z d_o and_o also_o the_o same_o reason_n of_o b_o to_z e_z as_o of_o d_o to_o f_o there_o will_v be_v the_o same_o reason_n of_o a_o to_z e_z as_o of_o c_z to_z f._n by_o reason_n of_o equality_n ill_o rank_v if_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o d_o to_o f_o and_o also_o the_o same_o reason_n of_o b_o to_z e_z as_o of_o c_z to_z d_o there_o will_v be_v the_o same_o reason_n of_o a_o to_z e_z as_o of_o c_z to_z f._n this_o book_n contain_v twenty_o five_o proposition_n of_o euclid_n to_o which_o there_o have_v be_v add_v ten_o which_o be_v receive_v the_o first_o six_o of_o this_o book_n be_v useful_a only_o to_o prove_v the_o follow_a proposition_n by_o the_o method_n
of_o equimultiplices_a and_o see_v i_o do_v not_o make_v use_n of_o that_o method_n i_o begin_v at_o the_o seven_o without_o change_v the_o order_n or_o number_n of_o the_o proposition_n the_o demand_n or_o proposition_n three_o magnitude_n abc_n be_v propose_v it_o be_v require_v that_o it_o be_v agree_v to_o or_o grant_v that_o there_o be_v a_o four_o magnitude_n in_o possibility_n to_o which_o the_o magnitude_n c_o have_v the_o same_o reason_n as_o the_o magnitude_n a_o have_v to_o the_o magnitude_n b._n proposition_n vii_o theorem_fw-la the_o magnitude_n which_o be_v equal_a have_v the_o same_o reason_n to_o a_o three_o magnitude_n and_o one_o and_o the_o same_o magnitude_n have_v the_o same_o reason_n to_o equal_a magnitude_n a_o 8._o c_o 4._o b_o 8._o if_o the_o magnitude_n a_o and_o b_o be_v equal_a they_o shall_v have_v the_o same_o reason_n to_o a_o three_o magnitude_n c._n demonstration_n if_o one_o of_o the_o two_o for_o example_n a_o shall_v have_v a_o great_a reason_n to_o the_o magnitude_n c_o than_z b_o a_o will_v contain_v more_o time_n a_o certain_a aliquot_fw-la part_n of_o c_o than_z b_o will_v contain_v thence_o a_o will_v be_v great_a than_o b_o contrary_a to_o what_o we_o have_v suppose_v second_o i_o say_v if_o a_o and_o b_o be_v equal_a the_o magnitude_n c_o shall_v have_v the_o same_o reason_n to_o the_o magnitude_n a_o as_o it_o have_v to_o the_o magnitude_n b._n demonstration_n if_o the_o magnitude_n c_o shall_v have_v a_o great_a reason_n to_o the_o magnitude_n a_o than_o it_o have_v to_o the_o magnitude_n b_o it_o to_o aught_o to_o contain_v more_o time_n a_o aliquot_fw-la of_o the_o magnitude_n a_o than_o it_o do_v contain_v a_o like_o aliquot_fw-la part_n of_o the_o magnitude_n b._n so_o that_o that_o part_n of_o a_o aught_o to_o be_v less_o than_o a_o like_o aliquot_fw-la part_n of_o b._n thence_o a_o shall_v be_v less_o than_o b_o which_o be_v contrary_a to_o the_o supposition_n proposition_n viii_o theorem_fw-la the_o great_a of_o two_o magnitude_n have_v a_o great_a reason_n to_o the_o same_o three_o than_o the_o lesser_a and_o the_o same_o three_o have_v a_o lesser_a reason_n to_o the_o great_a magnitude_n than_o it_o have_v to_o the_o lesser_a demonstration_n ad_fw-la and_o c_o be_v equal_a thence_o there_o be_v the_o same_o reason_n of_o ad_fw-la to_z of_o as_o there_o be_v of_o c_o to_z of_o by_o the_o seven_o and_o by_o the_o four_o definition_n ad_fw-la shall_v contain_v as_o many_o time_n gf_n a_o aliquot_fw-la part_n of_o of_o as_o c_z contain_v the_o same_o now_o ab_fw-la contain_v the_o same_o once_o more_o see_v db_n be_v great_a than_o gf_n whence_o by_o the_o 5_o definition_n the_o reason_n of_o ab_fw-la to_z of_o be_v great_a than_o that_o of_o c_o to_o the_o same_o three_o ef._n second_o i_o say_v that_o of_o have_v a_o lesser_a reason_n to_o ab_fw-la than_o to_o the_o magnitude_n c._n let_v there_o be_v take_v any_o aliquot_fw-la part_n of_o c_o for_o example_n the_o one_o four_o as_o many_o time_n as_o it_o can_v be_v take_v in_o ef._n let_v we_o suppose_v that_o it_o be_v find_v therein_o five_o time_n either_o it_o will_v leave_v something_o of_o the_o magnitude_n of_o after_o have_v be_v take_v five_o time_n or_o will_v leave_v nothing_o that_o be_v to_o say_v it_o will_v measure_v exact_o of_o it_o be_v evident_a that_o five_o fourth_n of_o the_o magnitude_n ab_fw-la will_v be_v a_o great_a line_n than_o the_o four_o of_o c_o take_v five_o time_n so_o that_o it_o can_v be_v find_v to_o be_v five_o time_n in_o ef._n and_o if_o the_o four_o of_o c_o take_v five_o time_n fall_v short_a as_o in_o g_o either_o the_o four_o of_o ab_fw-la take_v five_o time_n will_v reach_v to_o f_o or_o fall_v short_a in_o i._o if_o it_o reach_v to_o f_o there_o will_v be_v the_o same_o reason_n of_o of_o to_o ab_fw-la as_o of_o eglantine_n to_o c_o by_o the_o precede_a argument_n of_o to_o c_o have_v a_o great_a reason_n to_o c_o than_o to_o ab_fw-la that_o if_o the_o quarter_n of_o ab_fw-la take_v five_o time_n reach_v to_o i_o there_o will_v the_o same_o reason_n of_o ei_o to_o ab_fw-la as_o of_o eglantine_n and_o c._n now_o ei_o or_o of_o have_v a_o great_a reason_n than_o eglantine_n to_o c._n therefore_o of_o to_o c_o have_v a_o great_a reason_n than_o the_o same_o of_o to_o ab_fw-la proposition_n ix_o theorem_fw-la magnitude_n be_v equal_a when_o they_o have_v the_o same_o reason_n to_o a_o three_o magnitude_n a_o b_o c_o 12._o 12._o 16._o if_o the_o magnitude_n a_o and_o b_o have_v the_o same_o reason_n to_o a_o three_o magnitude_n c_o i_o say_v a_o and_o b_o be_v equal_a demonstration_n if_o one_o of_o the_o two_o for_o example_n a_o be_v great_a than_o b_o it_o will_v have_v a_o great_a reason_n to_o the_o magnitude_n c_o by_o the_o 8_o which_o will_v be_v contrary_a to_o the_o hypothesis_n second_o if_o the_o magnitude_n c_o have_v the_o same_o reason_n to_o the_o magnitude_n a_o as_o it_o have_v to_o the_o magnitude_n b_o i_o say_v that_o a_o and_o b_o be_v equal_a for_o if_o a_o be_v great_a than_o b_o c_z will_v have_v a_o great_a reason_n to_o the_o magnitude_n b_o than_o to_o the_o magnitude_n a_o which_o will_v be_v also_o contrary_a to_o the_o supposition_n proposition_n x._o theorem_fw-la the_o magnitude_n which_o have_v the_o great_a reason_n to_o the_o same_o magnitude_n be_v the_o great_a and_o that_o be_v the_o least_o to_o which_o the_o same_o have_v the_o great_a reason_n a_o b_o c._n if_o there_o be_v a_o great_a reason_n between_o a_o and_o c_o than_z between_z b_o and_o c_o i_o say_v that_o a_o be_v great_a than_o b_o for_o if_o a_o and_o b_o be_v equal_a they_o will_v have_v the_o same_o reason_n to_o c_o if_o a_o be_v less_o than_o b_o there_o will_v be_v a_o great_a reason_n between_o b_o and_o c_o than_o between_o a_o and_o c._n the_o one_o and_o the_o other_o be_v contrary_a to_o the_o hypothesis_n second_o if_o there_o be_v a_o lesser_a reason_n between_o c_o and_z a_o than_z between_z c_z and_z b._n i_o say_v that_o a_o shall_v be_v great_a than_o b._n for_o if_o a_o and_o b_o be_v equal_a c_o will_v have_v the_o same_o reason_n to_o both_o by_o the_o 8_o if_o a_o be_v less_o than_o b_o c_o will_v have_v less_o reason_n to_o b_o than_o to_o a_o the_o one_o and_o the_o other_o be_v contrary_a to_o what_o we_o have_v suppose_v proposition_n xi_o theorem_fw-la the_o reason_n which_o be_v equal_a to_o the_o same_o three_o be_v also_o the_o same_o one_o to_o another_o a_o b_o c_o d_o e_o f._n 4._o 2._o 8._o 4._o 6._o 3._o if_o there_o be_v the_o same_o reason_n between_o a_o and_o b_o as_o between_z c_z and_o d_o and_o if_o there_o be_v also_o the_o same_o reason_n between_o c_o and_o d_o as_o between_z e_z and_o f_o i_o say_v there_o will_v be_v the_o same_o reason_n between_o a_o and_o b_o as_o between_z e_z and_z f._n demonstration_n see_v there_o be_v the_o same_o reason_n between_o a_o and_o b_o as_o there_o be_v between_o c_o and_o d_o a_o shall_v contain_v as_o many_o time_n any_o aliquot_fw-la part_n whatever_n of_o b_o as_o c_z contain_v alike_o aliquot_fw-la part_n of_o d_o by_o the_o 5_o definition_n and_o in_o like_a manner_n as_o many_o time_n as_o c_z contain_v that_o aliquot_fw-la part_n of_o d_o e_z contain_v a_o like_a aliquot_fw-la part_n of_o f._n so_o then_o as_o many_o tim_n as_o a_o contain_v any_o aliquot_fw-la part_n whatever_n of_o b_o e_z shall_v contain_v also_o a_o like_o aliquot_fw-la part_n of_o f._n therefore_o there_o will_v be_v the_o same_o reason_n between_o a_o and_o b_o as_o there_o be_v between_o e_z and_z f._n proposition_n xii_o theorem_fw-la if_o several_a magnitude_n be_v proportional_n there_o shall_v be_v the_o same_o reason_n of_o one_o antecedent_n to_o its_o consequent_a as_o of_o all_o the_o antecedent_n take_v together_o to_o their_o consequent_n a_o b_o 3_o 12_o c_o d_o 2._o 8._o if_o there_o be_v the_o same_o reason_n of_o a_o to_o b_o as_o there_o be_v of_o c_o to_o d_o i_o say_v that_o there_o be_v the_o same_o reason_n of_o a_o and_o c_o taken_z together_z to_z b_o and_o d_o as_o of_o a_o to_o b._n demonstration_n see_v there_o be_v the_o same_o reason_n of_o a_o to_o b_o as_o of_o c_z to_z d_o the_o magnitude_n a_o shall_v contain_v as_o many_o time_n any_o aliquot_fw-la part_n whatever_n of_o b_o as_o c_z contain_v a_o like_a aliquot_fw-la part_n of_o d_o for_o example_n the_o four_o by_o the_o 5_o definition_n now_o the_o four_o of_o b_o and_o the_o four_o of_o d_o be_v one_o four_o of_o bd_o so_o than_o ac_fw-la shall_v contain_v as_o many_o time_n one_o