right_a line_n but_o many_o do_v fall_v out_o to_o be_v in_o a_o crooked_a line_n and_o in_o a_o sphere_n a_o cone_n &_o cylinderâ—_n a_o ruler_n may_v be_v apply_v but_o it_o must_v be_v a_o sphearicall_a conicall_a or_o cylindraceall_n but_o by_o the_o example_n of_o a_o right_a line_n do_v vitellio_n 2_o p_o i_o demand_n that_o between_o two_o line_n a_o surface_n may_v be_v extend_v and_o so_o may_v it_o seem_v in_o the_o element_n of_o many_o figure_n both_o plain_a and_o solid_n by_o euclid_n to_o be_v demand_v that_o a_o figure_n may_v be_v describe_v at_o the_o 7._o and_o 8._o e_fw-la ij_o item_n that_o a_o figure_n may_v be_v make_v up_o at_o the_o 8._o 14._o 16._o 23.28_o p._n uj_o which_o be_v of_o plain_n item_n at_o the_o 25._o 31._o 33._o 34._o 36._o 49._o p.xj._n which_o be_v of_o solid_n yet_o notwithstanding_o a_o plain_a surface_n and_o a_o plain_a body_n do_v measure_v their_o rectitude_n by_o a_o right_a line_n so_o that_o jus_o postulandi_fw-la this_o right_a of_o beg_v to_o have_v a_o thing_n grant_v may_v seem_v primary_o to_o be_v in_o a_o right_n plain_a line_n now_o the_o continuation_n of_o a_o right_a line_n be_v nothing_o else_o but_o the_o draw_v out_o far_o of_o a_o line_n now_o draw_v and_o that_o from_o a_o point_n unto_o a_o point_n as_o we_o may_v continue_v the_o right_a line_n a_o e._n unto_o i._o wherefore_o the_o first_o and_o second_o petition_n of_o euâ—lde_n do_v agree_v in_o one_o and_o 7._o to_o set_v at_o a_o point_n assign_v a_o right_a line_n equal_a to_o another_o right_a line_n give_v and_o from_o a_o great_a to_o cut_v off_o a_o part_n equal_a to_o a_o lesser_a 2._o and_o 3._o pj._n therefore_o 8._o one_o right_a line_n or_o two_o cut_v one_o another_o be_v in_o the_o same_o plain_a out_o of_o the_o 1._o and_o 2._o p_o xj_o one_o right_a line_n may_v be_v the_o common_a section_n of_o two_o plain_n yet_o all_o or_o the_o whole_a in_o the_o same_o plain_a be_v one_o and_o all_o the_o whole_a be_v in_o the_o same_o other_o and_o so_o the_o whole_a be_v the_o same_o plain_a two_o right_a line_n cut_v one_o another_o may_v be_v in_o two_o plain_n cut_v one_o of_o another_o but_o then_o a_o plainâ—_n may_v be_v draw_v by_o they_o therefore_o both_o of_o they_o shall_v be_v in_o the_o same_o plain_a and_o this_o plain_n be_v geometrical_o to_o be_v conceive_v because_o the_o same_o plain_a be_v not_o always_o make_v the_o ground_n whereupon_o one_o oblique_a line_n or_o two_o cut_v one_o another_o be_v draw_v when_o a_o periphery_n be_v in_o a_o sphearicall_a neither_o may_v all_o periphery_n cut_v one_o another_o be_v possible_o in_o one_o plain_a and_o 9_o with_o a_o right_a line_n give_v to_o describe_v a_o peripherie_n talus_fw-la the_o nephew_n of_o daedalus_n by_o his_o sister_n be_v say_v in_o the_o viij_o book_n of_o ovid_n metamorphosis_n to_o have_v be_v the_o inventour_n of_o this_o instrument_n for_o there_o he_o thus_o write_v of_o he_o and_o this_o matter_n et_fw-la ex_fw-la uno_fw-la duo_fw-la ferrea_fw-la brachia_fw-la nodo_fw-la junxit_fw-la ut_fw-la aequali_fw-la spatio_fw-la distantibus_fw-la ipsis_fw-la altera_fw-la pars_fw-la staret_fw-la pars_fw-la altera_fw-la duceâ—et_fw-la orbem_fw-la therefore_o 10._o the_o raiâ—s_n of_o the_o same_o or_o of_o a_o equal_a periphery_n be_v equal_a the_o reason_n be_v because_o the_o same_o right_a line_n be_v every_o where_o convert_v or_o turn_v about_o but_o here_o by_o the_o ray_n of_o the_o periphery_a must_v be_v understand_v the_o ray_n the_o figure_n contain_v within_o the_o periphery_n 11._o if_o two_o equal_a periphery_n from_o the_o end_n of_o equal_a shank_n of_o a_o assign_a rectilineall_a angle_n do_v meet_v before_o it_o a_o right_a line_n draw_v from_o the_o meeting_n of_o they_o unto_o the_o top_n or_o point_n of_o the_o angle_n shall_v cut_v it_o into_o two_o equal_a part_n 9_o pj._n hitherto_o we_o have_v speak_v of_o plain_a line_n their_o affection_n follow_v and_o first_o in_o the_o bisection_n or_o divide_v of_o a_o angle_n into_o two_o equal_a part_n 12._o if_o two_o equal_a periphery_n from_o the_o end_n of_o a_o right_a line_n give_v do_v meet_v on_o each_o side_n of_o the_o same_o a_o right_a line_n draw_v from_o those_o meeting_n shall_v divide_v the_o right_a line_n give_v into_o two_o equal_a part_n 10._o pj._n 13._o if_o a_o right_a line_n do_v stand_v perpendicular_a upon_o another_o right_a line_n it_o make_v on_o each_o side_n right_a angle_n and_o contrary_a wise_a the_o rular_a for_o the_o make_n of_o straight_a line_n on_o a_o plain_a be_v the_o first_o geometrical_a instrument_n the_o compass_n for_o the_o describe_v of_o a_o circle_n be_v the_o second_o the_o norma_n or_o square_n for_o the_o true_a erect_n of_o a_o right_a line_n in_o the_o same_o plain_a upon_o another_o right_a line_n and_o then_o of_o a_o surface_n and_o body_n upon_o a_o surface_n or_o body_n be_v the_o three_o the_o figure_n therefore_o be_v thus_o therefore_o 14._o if_o a_o right_a line_n do_v stand_v upon_o a_o right_a line_n it_o make_v the_o angle_n on_o each_o side_n equal_a to_o two_o right_a angle_n and_o contrariwise_o out_o of_o the_o 13._o and_o 14._o pj._n and_o 15._o if_o two_o right_a line_n do_v cut_v one_o another_o they_o do_v make_v the_o angle_n at_o the_o top_n equal_a and_o all_o equal_a to_o four_o right_a angle_n 15._o pj._n and_o 16._o if_o two_o right_a line_n cut_v with_o one_o right_a line_n do_v make_v the_o inner_a angle_n on_o the_o same_o side_n great_a than_o two_o right_a angle_n those_o on_o the_o other_o side_n against_o they_o shall_v be_v lesser_a than_o two_o right_a angle_n 17._o if_o from_o â—â—oint_n assign_v of_o a_o infinite_a right_a line_n give_v two_o equal_a part_n be_v on_o each_o side_n cut_v off_o and_o then_o from_o the_o point_n of_o those_o section_n two_o equal_a circle_n do_v meet_v a_o right_a line_n draw_v from_o their_o meeting_n unto_o the_o point_n assign_v shall_v be_v perpendicular_a unto_o the_o line_n give_v 11._o pj._n 18._o if_o a_o part_n of_o a_o infinite_a right_a line_n be_v by_o a_o periphery_a from_o a_o point_n give_v without_o cut_v off_o a_o right_a line_n from_o the_o say_a point_n cut_v in_o two_o the_o say_a part_n shall_v be_v perpendicular_a upon_o the_o line_n give_v 12._o pj._n 19_o if_o two_o right_a line_n draw_v at_o length_n in_o the_o same_o plain_a do_v never_o meet_v they_o be_v parallelly_n è_fw-it 35._o dj_o therefore_o 20._o if_o a_o infinite_a right_a line_n do_v cut_v one_o of_o the_o infinite_a right_a parallel_n line_n it_o shall_v also_o cut_v the_o other_o as_o in_o the_o same_o example_n u._fw-mi y._n cut_v a_o e._n it_o shall_v also_o cuâ—_n i_o o._n otherwise_o if_o it_o shall_v not_o cut_v it_o it_o shall_v be_v parallel_n unto_o it_o by_o the_o 18_o e._n and_o that_o against_o the_o grant_n 21._o if_o right_a line_n cut_v with_o a_o right_a line_n be_v pararellell_n they_o do_v make_v the_o inner_a angle_n on_o the_o same_o side_n equal_a to_o two_o right_a angle_n and_o also_o the_o alterne_a angle_n equal_a between_o themselves_o and_o the_o outter_n to_o the_o inner_a opposite_a to_o it_o and_o contrariwise_o 29,28,27_o p_o 1._o the_o cause_n of_o this_o threefold_a propriety_n be_v from_o the_o perpendicular_a or_o plumbline_n which_o fall_v upon_o the_o parallel_n breed_v and_o discover_v all_o this_o variety_n as_o here_o they_o be_v right_a angle_n which_o be_v the_o inner_a on_o the_o same_o part_n or_o side_n item_n the_o alterne_a angle_n item_n the_o inner_a and_o the_o outter_n and_o therefore_o they_o be_v equal_a both_o i_o mean_v the_o two_o inner_a to_o two_o right_a angle_n and_o the_o alterne_a angle_n between_o themselves_o and_o the_o outter_n to_o the_o inner_a opposite_a to_o it_o if_o so_o be_v that_o the_o cut_a line_n be_v oblique_a that_o be_v fall_v not_o upon_o they_o plumbe_v or_o perpendicular_o the_o same_o shall_v on_o the_o contrary_n befall_v the_o parallel_n for_o by_o that_o same_o obliquation_n or_o slant_v the_o right_a line_n remain_v and_o the_o angle_n unaltered_a in_o like_a manner_n both_o one_o of_o the_o inner_a to_o wit_n e_fw-it u._fw-mi y_fw-mi be_v make_v obtuse_a the_o other_o to_o wiâ—_n u._fw-mi y_fw-mi o_o be_v make_v acute_a and_o the_o alterne_a angle_n be_v make_v acute_a and_o obtuse_a as_o also_o the_o outter_n and_o inner_a opposite_a be_v likewise_o make_v acute_a and_o obtuse_a the_o same_o impossibility_n shall_v be_v conclude_v if_o they_o shall_v be_v say_v to_o be_v lesser_a than_o two_o right_a anglesâ—_n the_o second_o and_o three_o part_n may_v be_v conclude_v out_o of_o the_o first_o the_o second_o be_v thus_o twice_o two_o angle_n be_v equal_a to_o two_o right_a
alterne_n o_fw-fr e_fw-es y_fw-es because_o also_o three_o angle_n o_o e_o y_fw-es o_z e_z a_o and_o a_o e_z u._fw-mi be_v equal_a to_o two_o right_a angle_n by_o the_o 14_o e_fw-la v_o unto_o which_o also_o be_v equal_a the_o three_o angle_n in_o the_o triangle_n a_o e_o o_o by_z the_o 13_o e_z uj._o from_o three_o equal_n take_v away_o the_o two_o right_a angle_n a_o u._fw-mi e_fw-it and_o a_o o_o e_o for_o a_o o_o e_o be_v a_o right_a angle_n by_o the_o 21_o e_z because_o it_o be_v in_o a_o semicircle_n take_v away_o also_o the_o common_a angle_n a_o e_o o_o and_o the_o remainder_n e_o a_fw-fr o_o and_o o_o e_fw-it y_fw-es alterne_a angle_n shall_v be_v equal_a therefore_o 28_o if_o at_o the_o end_n of_o a_o right_a line_n give_v a_o right_n line_v angle_n be_v make_v equal_a to_o a_o angle_n give_v and_o from_o the_o top_n of_o the_o angle_n now_o make_v a_o perpendicular_a unto_o the_o other_o side_n do_v meet_v with_o a_o perpendicular_a draw_v from_o the_o midst_n of_o the_o line_n give_v the_o meeting_n shall_v be_v the_o centre_n of_o the_o circle_n describe_v by_o the_o equal_v angle_n in_o who_o opposite_a section_n the_o angle_n upon_o the_o line_n give_v shall_v be_v make_v equal_a to_o the_o assign_v è_fw-mi 33_o p_o iij._o and_o 29_o if_o the_o angle_n of_o the_o secant_fw-la and_o touch_v line_n be_v equal_a to_o a_o assign_a rectilineall_a angle_n the_o angle_n in_o the_o opposite_a section_n shall_v likewise_o be_v equal_a to_o the_o same_o 34._o piij._n of_o geometry_n the_o seventeen_o book_n of_o the_o adscription_n of_o a_o circle_n and_o triangle_n hitherto_o we_o have_v speak_v of_o the_o geometry_n of_o rectilineall_a plain_n and_o of_o a_o circle_n now_o follow_v the_o adscription_n of_o both_o this_o be_v general_o define_v in_o the_o first_o book_n 12_o e._n now_o the_o periphery_a of_o a_o circle_n be_v the_o bind_v thereof_o therefore_o a_o rectilineall_a be_v inscribe_v into_o a_o circle_n when_o the_o periphery_n do_v touch_v the_o angle_n of_o it_o 3_o d_o iiij_o it_o be_v circumscribe_v when_o it_o be_v touch_v of_o every_o side_n by_o the_o periphery_a 4_o d_o iij._o 1._o if_o a_o rectilineall_a ascribe_v unto_o a_o circle_n be_v a_o equilater_n it_o be_v equiangle_n of_o the_o circumscript_n it_o be_v likewise_o true_a if_o the_o circumscript_n be_v understand_v to_o be_v a_o circle_n for_o the_o perpendicular_o from_o the_o centre_n a_o unto_o the_o side_n of_o the_o circumscript_n by_o the_o 9e_n xij_o shall_v make_v triangle_n on_o each_o side_n equilater_n &_o equiangl_n by_o draw_v the_o semidiameter_n unto_o the_o corner_n as_o in_o the_o same_o example_n 2._o it_o be_v equal_a to_o a_o triangle_n of_o equal_a base_a to_o the_o perimeter_n but_o of_o height_n to_o the_o perpendicular_a from_o the_o centre_n to_o the_o side_n as_o here_o be_v manifest_a by_o the_o 8_o e_fw-la seven_o for_o there_o be_v in_o one_o triangle_n three_o triangle_n of_o equal_a height_n the_o same_o will_v fall_v out_o in_o a_o triangulate_a as_o here_o in_o a_o quadrate_n for_o here_o shall_v be_v make_v four_o triangle_n of_o equal_a height_n last_o every_o equilater_n rectilineall_a ascribe_v to_o a_o circle_n shall_v be_v equal_a to_o a_o triangle_n of_o base_a equal_a to_o the_o perimeter_n of_o the_o adscript_n because_o the_o perimeter_n contain_v the_o base_n of_o the_o triangle_n into_o the_o which_o the_o rectilineall_a be_v resolve_v 3._o like_a rectilineall_n inscribe_v into_o circle_n be_v one_o to_o another_o as_o the_o quadrate_n of_o their_o diameter_n 1_o p._n x_o i_o i_o in_o like_a triangulate_v see_v by_o the_o 4_o e_fw-la x_o they_o may_v be_v resolve_v into_o like_a triangle_n the_o same_o will_v fall_v out_o therefore_o 4._o if_o it_o be_v as_o the_o diameter_n of_o the_o circle_n be_v unto_o the_o side_n of_o rectilineall_a inscribe_v so_o the_o diameter_n of_o the_o second_o circle_n be_v unto_o the_o side_n of_o the_o second_o rectilineall_a inscribe_v and_o the_o several_a triangle_n of_o the_o inscript_n be_v alike_o and_o likely_a situate_a the_o rectilineall_n inscribe_v shall_v be_v alike_o and_o likely_a situate_a this_o euclid_n do_v thus_o assume_v at_o the_o 2_o p_o xij_o and_o indeed_o as_o it_o seem_v out_o of_o the_o 18_o p_o uj._o both_o which_o be_v contain_v in_o the_o 23_o e_fw-la iiij_o and_o therefore_o we_o also_o have_v assume_v it_o adscription_n of_o a_o circle_n be_v with_o any_o triangle_n but_o with_o a_o triangulate_v it_o be_v with_o that_o only_a which_o be_v ordinate_a and_o indeed_o adscription_n of_o a_o circle_n be_v common_a to_o all_o 5._o if_o two_o right_a line_n do_v cut_v into_o two_o equal_a part_n two_o angle_n of_o a_o assign_a rectilineall_a the_o circle_n of_o the_o ray_n from_o their_o meeting_n perpendicular_a unto_o the_o side_n shall_v be_v inscribe_v unto_o the_o assign_a rectilineall_a 4_o and_o 8._o p._n iiij_o the_o same_o argument_n shall_v serve_v in_o a_o triangulate_a 6._o if_o two_o right_a line_n do_v right_a anglewise_o cut_v into_o two_o equal_a part_n two_o side_n of_o a_o assign_a rectilineall_a the_o circle_n of_o the_o ray_n from_o their_o meeting_n unto_o the_o angle_n shall_v be_v circumscribe_v unto_o the_o assign_a rectilineall_a 5_o p_o iiij_o as_o in_o the_o former_a figure_n the_o demonstration_n be_v the_o same_o with_o the_o former_a for_o the_o three_o ray_n by_o the_o 2_o e_fw-la seven_o be_v equal_a and_o the_o meeting_n of_o they_o by_o the_o 17_o ex_fw-la be_v the_o centre_n and_o thus_o be_v the_o common_a adscription_n of_o a_o circle_n the_o adscription_n of_o a_o rectilineall_a follow_v and_o first_o of_o a_o triangle_n 7._o if_o two_o inscript_n from_o the_o touch_n point_n of_o a_o right_a line_n and_o a_o periphery_a do_v make_v two_o angle_n on_o each_o side_n equal_a to_o two_o angle_n of_o the_o triangle_n assign_v be_v knit_v together_o they_o shall_v inscribe_v a_o triangle_n into_o the_o circle_n give_v equiangular_a to_o the_o triangle_n give_fw-mi è_fw-mi 2_o p_o iiij_o the_o circumscription_n here_o be_v also_o special_a 8_o if_o two_o angle_n in_o the_o centre_n of_o a_o circle_n give_v be_v equal_a at_o a_o common_a ray_n to_o the_o outter_n angle_v of_o a_o triangle_n give_v right_a line_n touch_v a_o periphery_a in_o the_o shank_n of_o the_o angle_n shall_v circumscribe_v a_o triangle_n about_o the_o circle_n give_v like_o to_o the_o triangle_n give_v 3_o p_o iiij_o therefore_o 9_o if_o a_o triangle_n be_v a_o rectangle_n a_o obtusangle_n a_o acute_a angle_n the_o centre_n of_o the_o circumscribe_v triangle_n be_v in_o the_o side_n out_o of_o the_o side_n and_o within_o the_o side_n and_o contrariwise_o 5_o e_fw-la iiij_o as_o thou_o see_v in_o these_o three_o figure_n underneath_o the_o centre_n a._n of_o geometry_n the_o eighteen_o book_n of_o the_o adscription_n of_o a_o triangulate_a such_o be_v the_o adscription_n of_o a_o triangle_n the_o adscription_n of_o a_o ordinate_a triangulate_a be_v now_o to_o be_v teach_v and_o first_o the_o common_a adscription_n and_o yet_o out_o of_o the_o former_a adscription_n after_o this_o manner_n 1._o if_o right_a line_n do_v touch_v a_o periphery_a in_o the_o angle_n of_o the_o inscript_n ordinate_a triangulate_a they_o shall_v unto_o a_o circle_n circumscribe_v a_o triangulate_a homogeneal_a to_o the_o inscribe_v triangulate_v the_o example_n shall_v be_v lay_v down_o according_a as_o the_o species_n or_o several_a kind_n do_v come_v in_o order_n the_o special_a inscription_n therefore_o shall_v first_o be_v teach_v and_o that_o by_o one_o side_n which_o reiterated_a as_o oft_o as_o need_v shall_v require_v may_v fill_v up_o the_o whole_a periphery_n for_o that_o euclid_n do_v in_o the_o quindecangle_n one_o of_o the_o kind_n we_o will_v do_v it_o in_o all_o the_o rest_n 2._o if_o the_o diameter_n do_v cut_v one_o another_o right-anglewise_a a_o right_a line_n subtend_v or_o draw_v against_o the_o right_a angle_n shall_v be_v the_o side_n of_o the_o quadrate_n è_fw-it 6_o p_o iiij_o therefore_o 3._o a_o quadrate_n inscribe_v be_v the_o half_a of_o that_o which_o be_v circumscribe_v because_o the_o side_n of_o the_o circumscribe_v which_o here_o be_v equal_a to_o the_o diameter_n of_o the_o circle_n be_v of_o power_n double_a to_o the_o side_n of_o the_o inscript_n by_o the_o 9_o e_fw-la x_o i_o i_o anâ—_n 4._o it_o be_v great_a than_o the_o half_a of_o the_o circumscribe_v circle_n because_o the_o circumscribe_v quadrate_n which_o be_v his_o double_a be_v great_a than_o the_o whole_a circle_n for_o the_o inscribe_v of_o other_o multangled_a odde-sided_n figure_n we_o must_v needs_o use_v the_o help_n of_o a_o triangle_n each_o of_o who_o angle_n at_o the_o base_a be_v manifold_a to_o the_o other_o in_o a_o quinguangle_n first_o that_o which_o be_v double_a
by_o this_o mean_n 14_o if_o a_o right_a line_n equal_a to_o the_o axis_fw-la of_o the_o sphearicall_a and_o to_o it_o from_o the_o end_n of_o the_o perpendicular_a be_v knit_v unto_o the_o centre_n a_o right_a line_n draw_v from_o the_o cut_n of_o the_o periphery_a unto_o the_o say_a end_n shall_v be_v the_o side_n of_o the_o icosahedrum_fw-la 15_o of_o the_o five_o ordinate_a body_n inscribe_v into_o the_o same_o sphere_n the_o tetrahedrum_fw-la in_o respect_n of_o the_o greatness_n oâ—_n his_o side_n be_v first_o the_o octahedrum_fw-la the_o second_o the_o cube_n the_o three_o the_o icosahedrum_fw-la the_o four_o and_o the_o dodecahedrum_fw-la the_o five_o the_o latter_a euclid_n do_v demonstrate_v with_o a_o great_a circumstance_n therefore_o out_o of_o the_o former_a figure_n and_o demonstration_n let_v here_o be_v repeat_v the_o section_n of_o the_o axis_fw-la first_o into_o a_o double_a reason_n in_o we_o and_o the_o side_n of_o the_o sexangle_n r_o l_o and_o the_o side_n of_o 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â…”_n 126_o 4_o 63_o do_v make_v 1152_o 46_o 63._o again_o the_o lesser_a sectour_z by_o the_o next_o precedent_n be_v 410_o â…”_n and_o here_o the_o inner_a cone_n be_v 126_o 4_o 63_o and_o therefore_o 410_o 2_o â—_o 126_o 4_o 63_o that_o be_v 284_o 38_o 63_o be_v the_o lesser_a section_n finis_fw-la or_o thus_o or_o thus_o or_o thus_o
angle_n o_o y_fw-mi u._fw-mi and_o e_z u_z y_z by_o the_o former_a part_n item_n a_o u._fw-mi y_fw-mi and_o e_z u_z y_z by_o the_o 14_o e._n therefore_o they_o be_v equal_a between_o themselves_o now_o from_o the_o equal_a take_v away_o e_z u_z y_z the_o common_a angle_n and_o the_o remainder_n the_o alterne_a angle_n at_o u._fw-mi and_o y_z shall_v be_v least_o equal_a the_o three_o be_v thus_o the_o angle_n e_o u._fw-mi y_fw-mi and_o o_z y_z s_z be_v equal_a to_o the_o same_o u._fw-mi y_fw-mi i_fw-it by_o the_o second_o propriety_n and_o by_o the_o 15_o e._n therefore_o they_o be_v equal_a between_o themselves_o if_o they_o be_v oblique_a angle_n as_o here_o the_o line_n one_o slant_v or_o lique_o cross_v one_o another_o the_o angle_n on_o one_o side_n will_v grow_v less_o on_o the_o other_o side_n great_a therefore_o they_o will_v not_o be_v equal_a to_o two_o right_a angle_n against_o the_o grant_n from_o hence_o the_o second_o and_o three_o part_n may_v be_v conclude_v the_o 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altenate_n foot_n be_v parallel_n be_v equal_a h._n and_o 26_o if_o parallel_n do_v bind_v parallel_n the_o opposite_a line_n be_v equal_a è_fw-mi 34_o p.j._n or_o thus_o if_o parallel_n do_v enclose_v parallel_n the_o opposite_a parallel_n be_v equal_a h._n and_o 27._o if_o right_a line_n do_v joint_o bind_v on_o the_o same_o side_n equal_a and_o parallel_a line_n they_o be_v also_o equal_a and_o parallel_v on_o the_o same_o part_n or_o side_n it_o be_v say_v lest_o any_o man_n may_v understand_v right_a line_n knit_v together_o by_o opposite_a bound_n as_o here_o 28._o if_o right_a line_n be_v cut_v joint_o by_o many_o parallel_n right_a line_n the_o segment_n between_o those_o line_n shall_v be_v proportional_a one_o to_o another_o out_o of_o the_o 2_o p_o uj_o and_o 17_o p_o x_o i_o thus_o much_o of_o the_o perpendicle_n and_o parallel_a equality_n of_o plain_a right_a line_n their_o proportion_n be_v the_o last_o thing_n to_o be_v consider_v of_o they_o if_o the_o line_n cut_v be_v not_o parallel_n but_o do_v lean_v one_o towards_o another_o the_o portion_n cut_v or_o intercept_v between_o they_o will_v not_o be_v equal_a yet_o shall_v they_o be_v proportional_a one_o to_o another_o and_o look_v how_o much_o great_a the_o line_n thus_o cut_v be_v so_o much_o great_a shall_v the_o intersegment_n or_o portion_n intercept_v be_v and_o contrariwise_o look_v how_o much_o less_o so_o much_o lesser_a shall_v they_o be_v the_o three_o parallel_n in_o the_o top_n be_v not_o express_v yet_o must_v it_o be_v understand_v this_o element_n be_v very_o fruitful_a for_o from_o hence_o do_v arise_v and_o issue_n first_o the_o manner_n of_o cut_v a_o line_n according_a to_o any_o rate_n or_o proportion_n assign_v and_o then_o the_o invention_n or_o way_n to_o find_v out_o both_o the_o three_o and_o four_o proportional_o 29._o if_o a_o right_a line_n make_v a_o angle_n with_o another_o right_a line_n be_v cut_v according_a to_o any_o reason_n or_o proportion_n assign_v 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two_o part_n or_o three_o part_n or_o into_o as_o many_o patt_n as_o you_o shall_v think_v good_a or_o general_o after_o what_o manner_n of_o way_n soever_o thou_o shall_v command_v or_o desire_v a_o line_n to_o be_v cut_v or_o divide_v now_o 〈◊〉_d be_v cut_v into_o three_o parte_v 〈◊〉_d which_o the_o first_o let_v it_o be_v the_o half_a of_o the_o second_o and_o the_o second_o the_o half_a of_o the_o three_o and_o the_o conter_fw-la minall_a or_o right_a line_n make_v a_o angle_n with_o the_o say_v assign_v line_n let_v it_o be_v cut_v one_o part_v a_o o_o then_o double_a this_o in_o o_o u._fw-mi last_o let_v u._fw-mi i_o be_v take_v double_a to_o o_o u._fw-mi and_o let_v the_o whole_a diagramme_n be_v make_v up_o with_o three_o parallel_n yâ—_n and_o os_fw-la the_o four_o parallel_n in_o the_o top_n as_o a_o foresaid_a shall_v be_v understand_v therefore_o that_o section_n which_o be_v make_v in_o the_o conterminall_a line_n by_o the_o 28_o e_fw-la shall_v be_v in_o the_o assign_a line_n because_o the_o segment_n or_o portion_n intercept_v be_v between_o the_o parallel_n and_o 30._o if_o two_o right_a line_n give_v make_v a_o angle_n be_v continue_v the_o first_o equal_o to_o the_o second_o the_o second_o infinite_o parallel_v draw_v from_o the_o end_n of_o the_o first_o continuation_n unto_o the_o begin_n of_o the_o second_o and_o some_o contingent_a point_n in_o the_o same_o shall_v intercept_v between_o they_o the_o three_o proportional_a 11._o p_o five_o i_o and_o 31._o if_o of_o three_o right_a line_n give_v the_o first_o and_o the_o three_o make_v a_o angle_n be_v continue_v the_o first_o equal_o to_o the_o second_o and_o the_o three_o infinite_o parallel_n draw_v from_o the_o end_n of_o the_o first_o continuation_n unto_o the_o begin_n of_o the_o second_o and_o some_o contingent_a point_n the_o same_o shall_v intercept_v between_o they_o the_o four_o proportional_a 12._o p_o uj._o let_v the_o line_n give_v be_v these_o the_o first_o a_o e_o the_o second_o e_z i_z the_o third_z a_o o_o and_o let_v the_o whole_a diagramme_n be_v make_v up_o according_a to_o the_o prescript_n of_o the_o consectary_n here_o by_o 28._o e_fw-la as_o a_o e_z be_v to_z e_z i_z so_o be_v a_o o_o to_z o_o u._fw-mi thus_o far_o ramus_n lazarus_n schonerus_n who_o about_o some_o 25._o year_n since_o do_v revise_v and_o augment_v this_o work_n of_o our_o author_n have_v not_o only_o alter_v the_o form_n of_o these_o two_o next_o precedent_n consectary_n but_o he_o have_v also_o change_v their_o order_n and_o that_o which_o be_v here_o the_o second_o be_v in_o his_o edition_n the_o three_o and_o the_o three_o here_o be_v in_o he_o the_o second_o and_o to_o the_o former_a declaration_n of_o they_o he_o add_v these_o
angle_n namely_o the_o inward_a angle_n general_o be_v equal_a unto_o the_o even_a number_n from_o two_o forward_a but_o the_o outward_a angle_n be_v equal_a but_o to_o 4._o right_a angle_n h._n 5_o a_o rectilineall_a be_v either_o a_o triangle_n or_o a_o triangulate_a as_o before_o of_o a_o line_n be_v make_v a_o lineate_v so_o here_o in_o like_a manner_n of_o a_o triangle_n be_v make_v a_o triangulate_a 6_o a_o triangle_n be_v a_o rectilineall_a figure_n comprehend_v of_o three_o rightlines_n 21._o dj_o therefore_o 7_o a_o triangle_n be_v the_o prime_a figure_n of_o rectilineal_n a_o triangle_n or_o threeside_v figure_n be_v the_o prime_n or_o most_o simple_a figure_n of_o all_o rectilineal_n for_o among_o rectilineall_a figure_n there_o be_v none_o of_o two_o side_n for_o two_o right_a line_n can_v enclose_v a_o figure_n what_o be_v mean_v by_o a_o prime_a figure_n be_v teach_v at_o the_o 7._o e._n iiij_o and_o 8_o if_o a_o infinite_a right_a line_n do_v cut_v the_o angle_n of_o a_o triangle_n it_o do_v also_o cut_v the_o base_a of_o the_o same_o vitell._n 29._o to_o i_o 9_o any_o two_o side_n of_o a_o triangle_n be_v great_a than_o the_o other_o let_v the_o triangle_n be_v a_o e_o i_o i_o say_v the_o side_n a_o i_o be_v short_a than_o the_o two_o side_n a_o e_o and_o e_z i_z because_o by_o the_o 6._o e_fw-la ij_o a_o right_a line_n be_v between_o the_o same_o bound_n the_o short_a therefore_o 10_o if_o of_o three_o right_a line_n give_v any_o two_o of_o they_o be_v great_a than_o the_o other_o and_o periphery_n describe_v upon_o the_o end_n of_o the_o one_o at_o the_o distance_n of_o the_o other_o two_o shall_v meet_v the_o ray_n from_o that_o meeting_n unto_o the_o say_a end_n shall_v make_v a_o triangle_n of_o the_o line_n give_v and_o 11_o if_o two_o equal_a periphery_n from_o the_o end_n of_o a_o right_a line_n give_v and_o at_o his_o distance_n do_v meet_v liâ—es_v draw_v from_o the_o meeting_n unto_o the_o say_a end_n shall_v make_v a_o equilater_n triangle_n upon_o the_o line_n give_v 1_o p.j._n 12_o if_o a_o right_a line_n in_o a_o triangle_n be_v parallel_n to_o the_o base_a it_o do_v cut_v the_o shank_n proportional_o and_o contrariwise_o 2_o p_o five_o i_o as_o here_o in_o the_o triangle_n a_o e_o i_o let_v o_o u._fw-mi be_v parallel_n to_o the_o base_a and_o let_v a_o three_o parallel_n be_v understand_v to_o be_v in_o the_o top_n a_o therefore_o by_o the_o 28._o e.u._n the_o intersegment_n be_v proportional_a the_o converse_n be_v force_v out_o of_o the_o antecedent_n because_o otherwise_o the_o whole_a shall_v be_v less_o than_o the_o part_n for_o if_o o_fw-mi u._fw-mi be_v not_o parallel_v to_o the_o base_a e_o i_o then_z y_z u_z be_v here_o by_o the_o grant_n and_o by_o the_o antecedent_n see_v a_o o_o o_o e_o a_o y_z y_fw-es e_fw-es be_v proportional_a and_o the_o first_o a_o o_o be_v lesser_a than_o a_o y_o the_o three_o o_o e_o the_o second_o must_v be_v lesser_a than_o y_z e_z the_o four_o that_o be_v the_o whole_a than_o the_o part_n 13_o the_o three_o angle_n of_o a_o triangle_n be_v equal_a to_o two_o right_a angle_n 32._o p_o i_o therefore_o 14._o any_o two_o angle_n of_o a_o triangle_n be_v less_o than_o two_o right_a angle_n for_o if_o three_o angle_n be_v equal_a to_o two_o right_a angle_n then_o be_v two_o lesser_a than_o two_o right_a angle_n and_o 15_o the_o one_o side_n of_o any_o triangle_n be_v continue_v or_o draw_v out_o the_o outter_n angle_n shall_v be_v equal_a to_o the_o two_o inner_a opposite_a angle_n therefore_o 16_o the_o say_a outter_n angle_n be_v great_a than_o either_o of_o the_o inner_a opposite_a angle_n 16._o p_o i_o this_o be_v a_o consectary_n follow_v necessary_o upon_o the_o next_o former_a consectary_n 17_o if_o a_o triangle_n be_v equicrural_a the_o angle_n at_o the_o base_a be_v equal_a and_o contrariwise_o 5._o and_o 6._o p.j._n therefore_o 18_o if_o the_o equal_a shank_n of_o a_o triangle_n be_v continue_v or_o draw_v out_o the_o angle_n under_o the_o base_a shall_v be_v equal_a between_o themselves_o and_o 19_o if_o a_o triangle_n be_v a_o equilater_n it_o be_v also_o a_o equiangle_n and_o contrariwise_o and_o 20_o the_o angle_n of_o a_o equilater_n triangle_n do_v countervail_v two_o three_o part_n of_o a_o right_a angle_n regio_fw-la 23._o p_o i_o for_o see_v that_o 3._o angle_n be_v equal_a to_o 2._o 1._o must_v needs_o be_v equal_a to_o â…”_n and_o 21_o six_o equilater_n triangle_n do_v fill_v a_o place_n 22_o the_o great_a side_n of_o a_o triangle_n subtend_v the_o great_a angle_n and_o the_o great_a angle_n be_v subtend_v of_o the_o great_a side_n 19_o and_o 18._o p_o i_o the_o converse_n be_v manifest_a by_o the_o same_o figure_n as_o let_v the_o angle_v a_o e_o i_o be_v great_a than_o the_o angle_n a_o i_o e._n therefore_o by_o the_o same_o 9_o e_z iij._o it_o be_v great_a in_o base_a for_o what_o be_v there_o speak_v of_o angle_n in_o general_a be_v here_o assume_v special_o of_o the_o angle_n in_o a_o triangle_n 23_o if_o a_o right_a line_n in_o a_o triangle_n do_v cut_v the_o angle_n in_o two_o equal_a part_n it_o shall_v cut_v the_o base_a according_a to_o the_o reason_n of_o the_o shank_n and_o contrariwise_o 3._o p_o five_o i_o the_o mingle_a proportion_n of_o the_o side_n and_o angle_n do_v now_o remain_v to_o be_v handle_v in_o the_o last_o place_n the_o converse_n likewise_o be_v demonstrate_v in_o the_o same_o figure_n for_o as_o e_z a_o be_v to_o a_o i_o so_o be_v e_z o_o to_z o_o i_fw-it and_o so_o be_v e_z a_o to_o a_o u._fw-mi by_o the_o 12_o e_fw-la therefore_o a_o i_o and_o a_o u._fw-mi be_v equal_a item_n the_o angle_n e_o a_fw-fr o_o and_o o_o a_o i_o be_v equal_a to_o the_o angle_n at_o you_o and_o i_o by_o the_o 21._o e_o uâ—_n which_o be_v equal_a between_o themselves_o by_o the_o 17._o e._n of_o geometry_n the_o seven_o book_n of_o the_o comparison_n of_o triangle_n 1_o equilater_n triangle_n be_v equiangle_n 8._o p.j._n thus_o forre_v of_o the_o geometry_n or_o affection_n and_o reason_n of_o one_o triangle_n the_o comparison_n of_o two_o triangle_n one_o with_o another_o do_v follow_v and_o first_o of_o their_o rate_n or_o reason_n out_o of_o their_o side_n and_o angle_n whereupon_o triangle_n between_o themselves_o be_v say_v to_o be_v equilater_n and_o equiangle_n first_o out_o of_o the_o equality_n of_o the_o side_n be_v draw_v also_o the_o equality_n of_o the_o angle_n triangle_n therefore_o be_v here_o joint_o call_v equilater_n who_o side_n be_v several_o equal_a the_o first_o to_o the_o first_o the_o second_o to_o the_o second_o the_o three_o to_o the_o three_o although_o every_o several_a triangle_n be_v inequilaterall_a therefore_o the_o equality_n of_o the_o side_n do_v argue_v the_o equality_n of_o the_o angle_n by_o the_o 7._o e_fw-la iij._o as_o here_o 2_o if_o two_o triangle_n be_v equal_a in_o angle_n either_o the_o two_o equicrurals_n or_o two_o of_o equal_a either_o shank_n or_o base_a of_o two_o angle_n they_o be_v equilater_n 4._o and_o 26._o p_o i_o oh_o thus_o if_o two_o triangle_n be_v equal_a in_o their_o angle_n either_o in_o two_o angle_n contain_v under_o equal_a foot_n or_o in_o two_o angle_n who_o side_n or_o base_a of_o both_o be_v equal_a those_o angle_n be_v equilater_n h._n this_o element_n have_v three_o part_n or_o it_o do_v conclude_v two_o triangle_n to_o be_v equilater_n three_o way_n 1._o the_o first_o part_n be_v apparent_a thus_o let_v the_o two_o triangle_n be_v a_o e_o i_o and_o o_o u._fw-mi y_fw-mi because_o the_o equal_a angle_n at_o a_o and_o o_o be_v equicrural_a therefore_o they_o be_v equal_a in_o base_a by_o the_o 7._o e_fw-la iij._o 3_o the_o three_o part_n be_v thus_o force_v in_o the_o triangle_n a_o e_o i_o and_o o_o u._fw-mi y_fw-mi let_v the_o angle_n at_o e_o and_o i_o and_o u_z and_o y_z be_v equal_a as_o afore_o and_o a_o e._n the_o base_a of_o the_o angle_n at_o i_o be_v equal_a to_o o_fw-mi u._fw-mi the_o base_a of_o angle_n at_o y_o i_o say_v that_o the_o two_o triangle_n give_v be_v equilater_n for_o if_o the_o side_n e_o i_o be_v great_a than_o the_o side_n u._fw-mi y_fw-mi let_v e_o s_o be_v cut_v off_o equal_a to_o it_o and_o draw_v the_o right_a line_n a_o s._n therefore_o by_o the_o antecedent_n the_o two_o triangle_n a_o e_o s_o and_o o_o u._fw-mi y_fw-mi equal_a in_o the_o angle_n of_o their_o equal_a shank_n be_v equiangle_n and_o the_o angle_n a_o s_o e_o be_v equal_a to_o the_o angle_n o_o y_fw-fr u._fw-mi which_o be_v equal_a by_o the_o grant_n unto_o the_o angle_n a_o i_o e._n therefore_o a_o s_o e_o be_v equal_a to_o a_o i_o e_o
these_o latter_a day_n the_o german_n especial_o as_o regiomontanus_n werner_n schoner_n and_o appian_n have_v grace_v it_o but_o above_o all_o other_o the_o learned_a gemma_fw-la phrisius_n in_o a_o several_a work_n of_o that_o argument_n only_o have_v illustrate_v and_o teach_v the_o use_n of_o it_o plain_o and_o full_o the_o jacob_n staff_n therefore_o according_a to_o his_o own_o and_o those_o geometrical_a part_n shall_v here_o be_v describe_v the_o astronomical_a distribution_n we_o reserve_v to_o his_o time_n and_o place_n and_o that_o do_v the_o use_n of_o it_o shall_v be_v show_v in_o the_o measure_n of_o line_n 2_o the_o shank_n of_o the_o staff_n be_v the_o index_n and_o the_o transome_n 3_o the_o index_n be_v the_o double_a and_o one_o ten_o part_n of_o the_o transome_n or_o thus_o the_o index_n be_v to_o the_o transversary_n double_a and_o 1_o 10_o part_n thereof_o h._n as_o here_o thou_o see_v 4_o the_o transome_n be_v that_o which_o ride_v upon_o the_o index_n and_o be_v to_o be_v slide_v high_a or_o low_a at_o pleasure_n or_o the_o transversary_n be_v to_o be_v move_v upon_o the_o index_n sometime_o high_o sometime_o low_a h._n this_o proportion_n in_o define_v and_o make_v of_o the_o shank_n of_o the_o instrument_n be_v perpetual_o to_o be_v observe_v as_o if_o the_o transome_n be_v 10._o part_n the_o index_n must_v be_v 21._o if_o that_o be_v 189._o this_o shall_v be_v 90._o or_o if_o it_o be_v 2000_o this_o shall_v be_v 4200._o neither_o do_v it_o skill_n what_o the_o number_n be_v so_o this_o be_v their_o proportion_n more_o than_o this_o that_o the_o great_a the_o number_n be_v that_o be_v the_o lesser_a that_o the_o division_n be_v the_o better_a will_v it_o be_v in_o the_o use_n and_o because_o the_o index_n must_v bear_v and_o the_o transome_n be_v to_o be_v bear_v let_v the_o index_n be_v thick_a and_o the_o transome_a the_o thin_a but_o of_o what_o matter_n each_o part_n of_o the_o staff_n be_v make_v whether_o of_o brass_n or_o wood_n it_o 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demonstration_n be_v by_o the_o 12_o e_fw-la seven_o as_o afore_o but_o here_o be_v to_o be_v add_v the_o height_n of_o the_o measurer_n which_o if_o it_o be_v 4_o foot_n the_o whole_a height_n shall_v be_v 76_o foot_n therefore_o in_o a_o eversed_a altitude_n 11._o if_o the_o sight_n be_v from_o the_o begin_n of_o the_o index_n parallel_v to_o the_o height_n as_o the_o segment_n of_o the_o transome_n be_v unto_o the_o segment_n of_o the_o index_n so_o shall_v the_o length_n give_v be_v unto_o the_o height_n seek_v eversa_n altitudo_fw-la a_o eversed_a altitude_n reverse_v h_n be_v that_o which_o we_o call_v depth_n which_o indeed_o be_v nothing_o else_o in_o the_o geometer_n sense_n but_o height_n turn_v topsy_n turvie_o as_o we_o say_v or_o with_o the_o heel_n upward_o for_o out_o of_o the_o height_n conclude_v by_o subduct_v that_o which_o be_v above_o ground_n the_o height_n or_o depth_n of_o a_o well_o shall_v remain_v 12._o if_o the_o sight_n be_v from_o the_o begin_n of_o the_o index_n perpendicular_a to_o the_o height_n to_o be_v measure_v as_o the_o segment_n of_o the_o index_n be_v unto_o the_o segment_n of_o the_o transome_n so_o shall_v the_o length_n give_v be_v to_o the_o height_n therefore_o 13._o if_o the_o sight_n be_v from_o the_o begin_n of_o the_o index_n perpendicular_a to_o the_o magnitude_n to_o be_v measure_v by_o the_o name_n of_o the_o transome_n unto_o the_o end_n of_o some_o know_a part_n of_o the_o height_n as_o the_o distance_n of_o the_o name_n be_v unto_o the_o rest_n of_o the_o transome_n above_o they_o so_o shall_v the_o know_a part_n be_v unto_o the_o part_n seek_v or_o thus_o if_o the_o sight_n pass_v from_o the_o begin_n of_o the_o index_n be_v right_a by_o the_o vane_n of_o the_o transversary_n to_o the_o term_n of_o some_o part_n as_o the_o distance_n of_o the_o vane_n be_v unto_o the_o rest_n of_o the_o transversary_n above_o the_o index_n so_o be_v the_o part_n know_v unto_o the_o remainder_n h._n this_o be_v a_o consectary_n 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second_o aim_n let_v it_o be_v take_v from_o y_o the_o beginning_n of_o the_o same_o index_n and_o out_o of_o a_o great_a distance_n by_o s_o the_o end_n of_o the_o transome_n unto_o the_o same_o top_n e._n and_o the_o segment_n of_o the_o index_n let_v it_o be_v r_o l._n here_o as_o afore_o the_o measure_n be_v perform_v and_o do_v by_o the_o take_n of_o the_o difference_n of_o the_o say_v y_o r_o above_o a_o u._fw-mi now_o the_o demonstration_n be_v conclude_v as_o in_o the_o former_a be_v teach_v let_v the_o parallel_n l_o s_o m_o be_v erect_v against_o a_o o_o e._n here_o
be_v a_o i_o so_o be_v a_o i_o unto_z i_z e_z wherefore_o by_o the_o â—_o e_o a_o e_z be_v proportional_a cut_n and_o the_o great_a segment_n be_v a_o i_o the_o same_o remain_v the_o other_o propriety_n of_o the_o quintuple_a do_v follow_v 6_o the_o lesser_a segment_n continue_v to_o the_o half_a of_o the_o great_a be_v of_o power_n quintuple_a to_o the_o same_o half_a è_fw-mi 3_o p_o x_o iij._o the_o rate_n of_o the_o triple_a follow_v 7_o the_o whole_a line_n and_o the_o lesser_a segment_n be_v in_o power_n treble_a unto_o the_o great_a è_fw-it 4_o p_o xiij_o 8_o a_o obliquangled_a parallelogramme_n be_v either_o a_o rhombus_fw-la or_o a_o rhomboide_n 9_o a_o rhombus_fw-la be_v a_o obliquangled_a equilater_n parallelogramme_n 32_o dj_o it_o be_v otherwise_o of_o some_o call_v a_o diamond_n 10_o a_o rhomboide_n be_v a_o obliquangled_a parallelogramâ—e_n not_o equilater_n 33._o dj_o and_o a_o rhomboide_n be_v so_o oppose_v to_o a_o oblong_a as_o a_o rhombus_fw-la be_v to_o a_o quadrate_n and_o the_o rhomboide_n be_v so_o call_v as_o one_o will_v say_v rhombuslike_n although_o beside_o the_o inequality_n of_o the_o angle_v it_o have_v nothing_o like_o to_o a_o rhombus_fw-la a_o example_n of_o measure_v of_o a_o rhombus_fw-la be_v thus_o 11_o a_o trapezium_fw-la be_v a_o quadrangle_n not_o parallelogramme_n 34._o dj_o the_o example_n both_o of_o the_o figure_n and_o of_o the_o measure_n of_o the_o same_o let_v these_o be_v therefore_o triangulate_v quadrangle_v be_v of_o this_o sort_n 12_o a_o multangle_n be_v a_o figure_n that_o be_v comprehend_v of_o more_o than_o four_o right_a line_n 23._o dj_o by_o this_o general_a name_n all_o other_o sort_n of_o right_n line_v figure_n hereafter_o follow_v be_v by_o euclid_n comprehend_v as_o be_v the_o quinquangle_n sexangle_v septangle_n and_o such_o like_a innumerable_a take_v their_o name_n of_o the_o number_n of_o their_o angle_n in_o every_o kind_n of_o multangle_n there_o be_v one_o ordinate_a as_o we_o have_v in_o the_o former_a signify_v of_o which_o in_o this_o place_n we_o will_v say_v nothing_o but_o this_o one_o thing_n of_o the_o quinquangle_n the_o rest_n shall_v be_v reserve_v until_o we_o come_v to_o adscription_n 13_o multangle_v triangulate_v do_v take_v their_o measure_n also_o from_o their_o triangle_n 14_o if_o a_o equilater_n quinquangle_v have_v three_o side_n equal_a it_o be_v equiangle_v 7_o p_o 13._o this_o of_o some_o from_o the_o greek_a be_v call_v a_o pentagon_n of_o other_o a_o pentangle_v by_o a_o name_n partly_o greek_a partly_o latin_a the_o fifteen_o book_n of_o geometry_n of_o the_o line_n in_o a_o circle_n as_o yet_o we_o have_v have_v the_o geometry_n of_o rectilineal_n the_o geometry_n of_o curvilineal_n of_o which_o the_o circle_n be_v the_o chief_a do_v follow_v 1._o a_o circle_n be_v a_o round_a plain_n â—_o 15_o dj_o the_o mean_n to_o describe_v a_o circle_n be_v the_o same_o which_o be_v to_o make_v a_o periphery_a but_o with_o some_o difference_n for_o there_o be_v consider_v no_o more_o but_o the_o motion_n the_o point_n in_o the_o end_n of_o the_o ray_n describe_v the_o periphery_a here_o be_v consider_v the_o motion_n of_o the_o whole_a ray_n make_v the_o whole_a plot_n contain_v within_o the_o periphery_n a_o circle_n of_o all_o plain_n be_v the_o most_o ordinate_a figure_n as_o be_v before_o teach_v at_o the_o 10_o e_fw-la iiij_o 2_o circle_n be_v as_o the_o quadrate_n or_o square_n make_v of_o their_o diameter_n 2_o p._n x_o ij_o therefore_o 3._o the_o diameter_n be_v as_o their_o periphery_n pappus_n 5_o l_o x_o j_o and_o 26_o the_o 18._o as_o here_o thou_o see_v in_o a_o e_fw-la and_o i_z o._n 4._o circular_a geometry_n be_v either_o in_o line_n or_o in_o the_o segment_n of_o a_o circle_n this_o partition_n of_o the_o subject_a matter_n howsoever_o be_v take_v for_o the_o distinguish_n and_o sever_n with_o some_o light_n a_o matter_n somewhat_o confuse_v and_o indeed_o concern_v line_n the_o consideration_n of_o secant_v be_v here_o the_o foremost_a and_o first_o of_o inscript_n 5._o if_o a_o right_a line_n be_v bound_v by_o two_o point_n in_o the_o periphery_n it_o shall_v fall_v within_o the_o circle_n 2_o p_o iij._o from_o hence_o do_v follow_v the_o infinite_a section_n of_o which_o we_o speak_v at_o the_o 6_o e_fw-la i_o this_o proposition_n teach_v how_o a_o rightline_n be_v to_o be_v inscribe_v in_o a_o circle_n to_o wit_n by_o take_v of_o two_o point_n in_o the_o periphery_a 6._o if_o from_o the_o end_n of_o the_o diameter_n and_o with_o a_o ray_n of_o it_o equal_a to_o the_o right_a line_n give_v a_o periphery_n be_v describe_v a_o right_a line_n draw_v from_o the_o say_a end_n unto_o the_o meeting_n of_o the_o periphery_n shall_v be_v inscribe_v into_o the_o circle_n equal_a to_o the_o right_a line_n give_v 1_o p_o iiij_o and_o this_o proposition_n teach_v how_o a_o right_a line_n give_v be_v to_o be_v inscribe_v into_o a_o circle_n equal_a to_o a_o line_n give_v moreover_o of_o all_o inscript_n the_o diameter_n be_v the_o chief_a for_o it_o show_v the_o centre_n and_o also_o the_o reason_n or_o proportion_n of_o all_o other_o inscript_n therefore_o the_o invention_n and_o make_n of_o the_o diameter_n of_o a_o circle_n be_v first_o to_o be_v teach_v 7._o if_o a_o inscript_n do_v cut_v into_o two_o equal_a part_n another_o inscript_n perpendicular_o it_o be_v the_o diameter_n of_o the_o circle_n and_o the_o midst_n of_o it_o be_v the_o centre_n 1_o p_o iij._o the_o cause_n be_v the_o same_o which_o be_v of_o the_o 5_o e_z x_o i_o because_o the_o inscript_n cut_v into_o half_n if_o for_o the_o side_n of_o the_o inscribe_v rectangle_n and_o it_o do_v subtend_v the_o periphery_n cut_v also_o into_o two_o part_n by_o the_o which_o both_o the_o inscript_n and_o periphery_a also_o be_v in_o like_a manner_n cut_v into_o two_o equal_a part_n therefore_o the_o right_a line_n thus_o half_v in_o the_o diameter_n of_o the_o rectangle_n but_o that_o the_o middle_n of_o the_o circle_n be_v the_o centre_n be_v manife_a out_o of_o the_o 7_o e_o v_o and_o 29_o e_fw-la iiij_o euclid_n think_v better_a of_o impossibile_fw-it than_o he_o do_v of_o the_o cause_n and_o thus_o he_o force_v it_o for_o if_o y_o be_v not_o the_o centre_n but_o s_o the_o part_n must_v be_v equal_a to_o the_o whole_a for_o the_o triangle_n a_o o_o s_o shall_v be_v equilater_n to_o the_o triangle_n e_o o_fw-fr s._n for_o a_o o_o oe_o be_v equal_a by_o the_o grant_n item_n be_v a_o and_o s_z e_z be_v the_o ray_n of_o the_o circle_n and_o s_o o_o be_v common_a to_o both_o the_o triangle_n therefore_o by_o the_o 1_o e_fw-la seven_o the_o angle_v no_o each_o side_n at_o o_o be_v equal_a and_o by_o the_o 13_o e_o v_o they_o be_v both_o right_a angle_n therefore_o s_o o_o e_o be_v a_o right_a angle_n it_o be_v therefore_o equal_a by_o the_o grant_n to_o the_o right_a angle_n y_fw-fr o_fw-fr e_fw-es that_o be_v the_o part_n be_v equal_a to_o the_o whole_a which_o be_v impossible_a wherefore_o y_o be_v not_o the_o centre_n the_o same_o will_v fall_v out_o of_o any_o other_o point_n whatsoever_o â—ut_v of_o y._n therefore_o 8._o if_o two_o râ—ght_a line_n do_v perpendicular_o half_a two_o inscript_n the_o meeting_n of_o these_o two_o bisecant_v shall_v be_v the_o centre_n of_o the_o circle_n è_fw-mi 25_o p_o iij._o and_o one_o may_n 9_o draw_v a_o periphery_a by_o three_o point_n which_o do_v not_o fall_v in_o a_o right_a line_n 10._o if_o a_o diameter_n do_v half_o a_o inscript_n that_o be_v nâ—t_v a_o diameter_n it_o do_v cut_v it_o perpendicular_o and_o contrariwise_o 3_o p_o iij._o 11._o if_o inscript_n which_o be_v not_o diameter_n do_v cut_v one_o another_o the_o segment_n shall_v be_v unequal_a 4_o p_o iij._o but_o rate_n have_v be_v hitherto_o in_o the_o part_n of_o inscript_n proportion_n in_o the_o same_o part_n follow_v 12_o if_o two_o inscript_n do_v cut_v one_o another_o the_o rectangle_n of_o the_o segment_n of_o the_o one_o be_v equal_a to_o the_o rectangle_n of_o the_o segment_n of_o the_o other_o 35_o p_o iij._o and_o this_o be_v the_o comparison_n of_o the_o part_n inscript_n the_o rate_n of_o whole_a inscript_n do_v follow_v the_o which_o whole_a one_o diameter_n do_v make_v 13_o inscript_n be_v equal_a distant_a from_o the_o centre_n unto_o which_o the_o perpendicular_o from_o the_o centre_n be_v equal_a 4_o d_o iij._o 14._o if_o inscript_n be_v equal_a they_o be_v equal_o distant_a from_o the_o centre_n and_o contrariwise_o 13_o p_o iij._o the_o diameter_n in_o the_o same_o circle_n by_o the_o 28_o e_o iiijâ—_n be_v equal_a and_o they_o be_v equal_o distant_a from_o the_o centre_n see_v they_o be_v by_o the_o centre_n or_o rather_o be_v no_o whit_n at_o all_o
distant_a from_o it_o other_o inscript_n be_v judge_v to_o be_v equal_a great_a or_o lesser_a one_o than_o another_o by_o the_o diameter_n or_o by_o the_o diameter_n centre_n euclid_n do_v demonstrate_v this_o proposition_n thus_o let_v first_o a_o e_o and_o i_z o_o be_v equal_a i_o say_v they_o be_v equidistant_a from_o the_o centre_n for_o let_v u._fw-mi y_fw-mi and_o u_z y_z be_v perpendicular_o they_o shall_v cut_v the_o assign_v a_o e_o &_o i_o o_o into_o half_n by_o the_o 5_o e_fw-la xj_o and_o y_o a_o and_o s_o i_o aâ—e_fw-fr equal_a because_o they_o be_v the_o half_n of_o equal_n now_o let_v the_o ray_n of_o the_o circle_n be_v u._fw-mi a_o aund_v u._fw-mi i_fw-it their_o quadrate_n by_o the_o 9_o e_fw-la xij_o be_v equal_a to_o the_o pair_n of_o quadrate_n of_o the_o shank_n which_o pair_n be_v therefore_o equal_a between_o themselves_o take_v from_o equal_n the_o quadrates_n y_o a_o and_o s_z i_z there_o shall_v remain_v y_fw-mi u._fw-mi and_o u._fw-mi s_o equal_n and_o therefore_o the_o side_n be_v equal_a by_o the_o 4_o e_fw-la 12._o the_o converse_n likewise_o be_v manifest_a for_o the_o perpendicular_o give_v do_v half_a they_o and_o the_o half_n as_o before_o be_v equal_a 15_o of_o unequal_a inscript_n the_o diameter_n be_v the_o great_a and_o that_o which_o be_v next_o to_o the_o diameter_n be_v great_a than_o that_o which_o be_v far_o off_o from_o it_o that_o which_o be_v far_a off_o from_o it_o be_v the_o least_o and_o that_o which_o be_v next_o to_o the_o least_o be_v lesser_a than_o that_o which_o be_v far_o off_o and_o those_o two_o only_a which_o be_v on_o each_o side_n of_o the_o diameter_n be_v equal_a è_fw-mi 15_o e_fw-la iij._o this_o proposition_n consist_v of_o five_o member_n the_o first_o be_v the_o diameter_n be_v the_o great_a iuscript_n the_o second_o that_o which_o be_v next_o to_o the_o diameter_n be_v great_a than_o that_o which_o be_v far_o off_o the_o three_o that_o which_o be_v far_a off_o from_o the_o diameter_n be_v the_o least_o the_o four_o that_o next_o to_o the_o least_o be_v lesser_a than_o that_o far_o off_o the_o five_o that_o two_o only_a on_o each_o side_n of_o the_o diameter_n be_v equal_a between_o themselves_o all_o which_o be_v manifest_a out_o of_o that_o same_o argument_n of_o equality_n that_o be_v the_o centre_n the_o beginning_n of_o decrease_v and_o the_o end_n of_o increase_v for_o look_v how_o much_o far_o off_o you_o go_v from_o the_o centre_n or_o how_o much_o near_o you_o come_v unto_o it_o so_o much_o lesâ—er_n or_o great_a do_v you_o make_v the_o inscript_n but_o euclides_n conclusion_n be_v by_o triangle_n of_o two_o side_n great_a than_o the_o other_o and_o of_o the_o great_a angle_n the_o first_o part_n be_v plain_a thus_o because_o the_o diameter_n a_o e_fw-es be_v equal_a to_o i_o l_o and_o l_o o_o viz._n to_o the_o ray_n and_o to_o those_o which_o be_v great_a than_o i_o o_o the_o base_a by_o the_o 9_o e_o v_o j_o etc._n etc._n the_o second_o part_n of_o the_o near_o be_v manifest_a by_o the_o 5_o e_fw-la seven_o because_o of_o the_o triangle_n i_o l_o o_o equicrural_a to_o the_o triangle_n u._fw-mi l_o y_fw-fr be_v great_a in_o angle_n and_o therefore_o it_o be_v also_o great_a in_o base_a the_o three_o and_o four_o be_v consectary_n of_o the_o first_o the_o five_o part_n be_v manifest_a by_o the_o second_o for_o if_o beside_o i_o o_o and_o s_z r_o there_o be_v suppose_v a_o three_o equal_a the_o same_o also_o shall_v be_v unequal_a because_o it_o shall_v be_v both_o near_o and_o far_o off_o from_o the_o diameter_n 16_o of_o right_a line_n draw_v from_o a_o point_n in_o the_o diameter_n which_o be_v not_o the_o centre_n unto_o the_o periphery_n that_o which_o pass_v by_o the_o centre_n be_v the_o great_a and_o that_o which_o be_v near_o to_o the_o great_a be_v great_a than_o that_o which_o be_v far_o off_o the_o other_o part_n of_o the_o great_a be_v the_o jest_n and_o that_o which_o be_v near_a to_o the_o least_o be_v lesser_a than_o that_o which_o be_v far_o off_o and_o two_o on_o each_o side_n of_o the_o great_a or_o least_o be_v only_o equal_a 7_o p_o iij._o the_o three_o that_o a_o y_fw-mi be_v lesser_a than_o a_o u._fw-mi because_o his_o y_o which_o be_v equal_a to_o we_o u._fw-mi be_v lesser_a than_o the_o right_a line_n be_v a_o and_o a_o u._fw-mi by_o the_o 9_o e_o v_o j_o and_o the_o common_a s_o a_o be_v take_v away_o a_o y_z shall_v be_v leave_v lesser_a than_o a_o u._n the_o four_o part_n follow_v of_o the_o three_o the_o five_o let_v it_o be_v thus_o s_o r_o make_v the_o angle_n a_o s_o r_o equal_a to_o the_o angle_n a_o s_o u._fw-mi the_o base_n a_o u._fw-mi and_o a_o r_o shall_v be_v equal_a by_o the_o 2_o e_fw-la five_o ij_o to_o these_o if_o the_o three_o be_v suppose_v to_o be_v equal_a as_o a_o l_o it_o will_v follow_v by_o the_o 1_o e_fw-la five_o ij_o that_o the_o whole_a angle_n s_o a_o shall_v be_v equal_a to_o r_o s_o a_o the_o particular_a angle_n which_o be_v impossible_a and_o out_o of_o this_o five_o part_n issue_v this_o consectary_n therefore_o 17_o if_o a_o point_n in_o a_o circle_n be_v the_o bind_v of_o three_o equal_a right_a line_n determine_v in_o the_o periphery_n it_o be_v the_o centre_n of_o the_o circle_n 9_o p_o iij._o let_v the_o point_n a_o in_o a_o circle_n be_v the_o common_a bind_v of_o three_o right_a line_n end_v in_o the_o periphery_a and_o equal_a between_o themselves_o be_v a_o e_fw-es a_o i_z a_o v_o i_o say_v this_o point_n be_v the_o centre_n of_o the_o circle_n 18_o of_o right_a line_n draw_v from_o a_o point_n assign_v without_o the_o periphery_n unto_o the_o concavity_n or_o hollow_a of_o the_o same_o that_o which_o be_v by_o the_o centre_n be_v the_o great_a and_o that_o next_o to_o the_o great_a be_v great_a than_o that_o which_o be_v far_o off_o but_o of_o those_o which_o fall_v upon_o the_o convexitiâ—_n of_o the_o circumference_n the_o segment_n of_o the_o great_a be_v leastâ—_n and_o that_o which_o be_v next_o unto_o the_o least_o be_v lesser_a than_o that_o be_v far_o off_o and_o two_o on_o each_o side_n of_o the_o great_a or_o least_o be_v only_o equal_a 8_o piij._n 19_o if_o a_o right_a line_n be_v perpendicular_a unto_o the_o end_n of_o the_o diameter_n it_o do_v touch_v the_o periphery_a and_o contrariwise_o è_fw-mi 16_o p_o iij._o as_o for_o example_n let_v the_o circle_n give_v a_o e_o be_v perpendicular_a to_o the_o end_n of_o the_o diameter_n or_o the_o end_n of_o the_o ray_n in_o the_o end_n a_o as_o suppose_v the_o ray_n be_v i_o a_o i_o say_v that_z e_z a_o do_v touch_v not_o cut_v the_o periphery_a in_o the_o common_a bind_v a._n therefore_o 20_o if_o a_o right_a line_n do_v pass_v by_o the_o centre_n and_o touch-point_n it_o be_v perpendicular_a to_o the_o tangent_fw-la or_o touch-line_n 18_o p_o iij._o and_o or_o thus_o as_o schoner_n amend_v it_o if_o a_o right_a line_n be_v the_o diameter_n by_o the_o touch_n point_n it_o be_v perpendicular_a to_o the_o tangent_fw-la 21_o if_o a_o right_a line_n be_v perpendicular_a unto_o the_o tangent_fw-la it_o do_v pass_v by_o the_o centre_n and_o touch-point_n 19_o piij._n or_o thus_o if_o it_o be_v perpendicular_a to_o the_o tangent_fw-la it_o be_v a_o diameter_n by_o the_o touch_n point_n schoner_n for_o a_o right_a line_n either_o from_o the_o centre_n unto_o the_o touch-point_n or_o from_o the_o touch_n point_n unto_o the_o centre_n be_v radius_fw-la or_o semidiameter_n and_o 22_o the_o touch-point_n be_v that_o into_o which_o the_o perpendicular_a from_o the_o centre_n do_v fall_v upon_o the_o touch_n line_n 23_o a_o tangent_fw-la on_o the_o same_o side_n be_v only_o one_o or_o touch_v line_n be_v but_o one_o upon_o one_o and_o the_o same_o side_n h._n or._n a_o tangent_fw-la be_v but_o one_o only_a in_o that_o point_n of_o the_o periphery_a schoner_n euclid_n propound_v this_o more_o special_o thus_o that_o no_o other_o right_a line_n may_v possible_o fall_v between_o the_o periphery_a and_o the_o tangent_fw-la and_o 24_o a_o touch-angle_n be_v lesser_a than_o any_o rectilineall_a aâ—ute_a angle_n è_fw-mi 16_o p_o ij_o angulus_n contractus_fw-la a_o touch_n angle_n be_v a_o angle_n of_o a_o straight_a touch-line_n and_o a_o periphery_n it_o be_v common_o call_v angulus_n contingentiae_fw-la of_o proclus_n it_o be_v name_v cornicularis_fw-la a_o horne-like_a corner_n because_o it_o be_v make_v of_o a_o right_a line_n and_o periphery_a like_a unto_o a_o horn_n it_o be_v less_o therefore_o than_o any_o acute_a or_o sharp_a rightlined_n angle_n because_o if_o it_o be_v not_o lesser_a a_o right_a line_n may_v fall_v between_o the_o periphery_a and_o the_o
unto_o the_o remainder_n which_o be_v thus_o find_v 5._o if_o a_o right_a line_n be_v cut_v proportional_o the_o base_a of_o that_o triangle_n who_o shank_n shall_v be_v equal_a to_o the_o whole_a line_n cut_v and_o the_o base_a to_o the_o great_a segment_n of_o the_o same_o shall_v have_v each_o of_o the_o angle_n at_o the_o base_a double_a to_o the_o remainder_n and_o the_o base_a shall_v be_v the_o side_n of_o the_o quinquangle_v inscribe_v with_o the_o triangle_n into_o a_o circle_n 10_o and_o 11._o p_o i_o i_o i_o i_o 6_o if_o two_o right_a line_n do_v subtend_v on_o each_o side_n two_o angle_n of_o a_o inscript_a quinquangle_n they_o be_v cut_v proportional_o and_o the_o great_a segment_n be_v the_o side_n of_o the_o say_a inscript_n è_fw-mi 8_o p_o x_o iij._o and_o from_o hence_o the_o fabric_n or_o construction_n of_o a_o ordinate_a quinquangle_n upon_o a_o right_a line_n give_v be_v manifest_a therefore_o 7_o if_o a_o right_a line_n give_v cut_v proportional_a be_v continue_v at_o each_o end_n with_o the_o great_a segment_n and_o six_o periphery_n at_o the_o distance_n of_o the_o line_n give_v shall_v meet_v two_o on_o each_o side_n from_o the_o end_n of_o the_o line_n give_v and_o the_o continue_a two_o other_o from_o their_o meeting_n right_a line_n draw_v from_o their_o meeting_n &_o the_o end_n of_o the_o assign_a shall_v make_v a_o ordinate_a quinquangle_n upon_o the_o assign_a 8_o if_o the_o diameter_n of_o a_o circle_n circumscribe_v about_o a_o quinquangle_n be_v rational_a it_o be_v irrational_a unto_o the_o side_n of_o the_o inscribe_v quinquangle_n è_fw-it 11._o p_o xiij_o so_o before_o the_o segment_n of_o a_o right_a line_n proportional_o cut_v be_v irrational_a the_o other_o triangulate_v hereafter_o multiply_v from_o the_o ternary_a quaternary_a or_o quinary_a of_o the_o side_n may_v be_v inscribe_v into_o a_o circle_n by_o a_o inscript_a triangle_n quadrate_n or_o quinquangle_v therefore_o by_o a_o triangle_n there_o may_v be_v inscribe_v a_o triangulate_a of_o 6._o 12,24,46_o angle_n by_o a_o quadrate_n a_o triangulate_a of_o 8._o 16,32,64_o angle_n by_o a_o quinquangle_n a_o triangulate_a of_o 10_o 20._o 40,80_o angle_n etc._n etc._n 9_o the_o ray_n of_o a_o circle_n be_v the_o side_n of_o the_o inscript_n sexangle_v è_fw-mi 15_o p_o iiij_o therefore_o 10_o three_o ordinate_a sexangle_n do_v fill_v up_o a_o place_n furthermore_o also_o no_o one_o figure_n among_o the_o plain_n do_v fill_v up_o a_o place_n a_o quinquangle_n do_v not_o for_o three_o angle_n a_o quinquangle_n may_v make_v only_o 3_o â—_o 5_o angle_n which_o be_v too_o little_a and_o four_o will_v make_v 4_o â—_o 5._o which_o be_v as_o much_o too_o great_a the_o angle_n of_o a_o septangle_n will_v make_v only_o two_o rightangle_v and_o 6_o 7_o of_o one_o three_o will_v make_v 3_o and_o 9_o 7_o that_o be_v in_o the_o whole_a 4._o 2_o 7_o which_o be_v too_o much_o etc._n etc._n to_o he_o that_o by_o induction_n shall_v thus_o make_v trial_n it_o will_v appear_v that_o a_o plain_a place_n may_v be_v fill_v up_o by_o three_o sort_n of_o ordinate_a plain_n only_o and_o 11_o if_o right_a line_n from_o one_o angle_n of_o a_o inscript_n sexangle_v unto_o the_o three_o angle_n on_o each_o side_n be_v knit_v together_o they_o shall_v inscribe_v a_o equilater_n triangle_n into_o the_o circle_n give_v 12_o the_o side_n of_o a_o inscribe_v equilater_n triangle_n have_v a_o treble_a power_n unto_o the_o ray_n of_o the_o circle_n 12._o p_o xiij_o 13_o if_o the_o side_n of_o a_o sexangle_n be_v cut_v proportional_o the_o great_a segment_n shall_v be_v the_o side_n of_o the_o decangle_n therefore_o 14_o if_o a_o decangle_n and_o a_o sexangle_v be_v inscribe_v in_o the_o same_o circle_n a_o right_a line_n continue_v and_o make_v of_o both_o side_n shall_v be_v cut_v proportional_o and_o the_o great_a segment_n shall_v be_v the_o side_n of_o a_o sexangle_n and_o if_o the_o great_a segment_n of_o a_o right_a line_n cut_v proportional_o be_v the_o side_n of_o a_o hexagon_n the_o rest_n shall_v be_v the_o side_n of_o a_o decagon_n 9_o p_o xiij_o the_o comparison_n of_o the_o decangle_n and_o sexangle_v with_o the_o quinangle_n follow_v 15_o if_o a_o decangle_n a_o sexangle_n and_o a_o pentangle_v be_v inscribe_v into_o the_o same_o circle_n the_o side_n of_o the_o pentangle_v shall_v in_o power_n countervail_v the_o side_n of_o the_o other_o and_o if_o a_o right_a line_n inscribe_v do_v countervail_v the_o side_n of_o the_o sexangle_n and_o decangle_v it_o be_v the_o side_n of_o the_o pentangle_v 10._o p_o fourteen_o let_v the_o proportion_n of_o this_o syllogism_n be_v demonstrate_v for_o this_o part_n only_o remain_v doubtful_a therefore_o two_o triangle_n a_o e_o i_o and_o y_fw-fr e_fw-it i_fw-it be_v equiangle_n have_v one_o common_a angle_n at_o e_o and_o also_o two_o equal_a one_o a_o e_o i_o and_o e_z i_z y_z the_o half_n to_o wit_n of_o the_o same_o e_o i_o s_o because_o that_o be_v by_o the_o 17_o e_fw-la uj_o one_o of_o the_o two_o equal_n unto_o the_o which_o e_o ay_o s_o the_o out_z angle_n be_v equal_a by_o the_o 15_o e._n uj._o and_o this_o do_v insist_v upon_o a_o half_a periphery_n for_o the_o half_a periphery_a a_o l_o s_o be_v equal_a to_o the_o half_a periphery_a a_o r_o s_o and_o also_o a_o l_o be_v equal_a to_o a_o r._n therefore_o the_o remnant_n l_o s_o be_v equal_a to_o the_o remnant_n r_o s_o and_o the_o whole_a r_o l_o be_v the_o double_a of_o the_o same_o r_o s_o and_o therefore_o e_o r_o be_v the_o double_a of_o e_o o_o and_o r_o s_o the_o double_a of_o o_o u._fw-mi for_o the_o bisegment_n be_v manifest_a by_o the_o 10_o e_z xv_o and_o the_o 11_o e_z xuj_o therefore_o the_o periphery_n e_o r_o s_o be_v the_o double_a of_o the_o periphery_n e_o o_fw-fr u._fw-mi and_o therefore_o the_o angle_n e_fw-it i_fw-it u._fw-mi be_v the_o half_a of_o the_o angle_n e_o i_o s_o by_z the_o 7_o e_z xuj_o therefore_o two_o angle_n of_o two_o triangle_n be_v equal_a wherefore_o the_o remainder_n by_o the_o 4_o e_fw-la seven_o be_v equal_a to_o the_o remainder_n wherefore_o by_o the_o 12_o e_z seven_o as_o the_o side_n a_o e_o be_v to_z e_o i_o so_o be_v e_z i_z to_z e_o y._n therefore_o by_o the_o 8_o e_fw-la xij_o the_o oblong_a of_o the_o extreme_n be_v equal_a to_o the_o quadrate_n of_o the_o mean_a now_o let_v o_o y_fw-es be_v knit_v together_o with_o a_o straight_o here_o again_o the_o two_o triangle_n a_o o_o e_o and_o a_o o_o y_fw-fr be_v equiangle_n have_v one_o common_a angle_n at_o a_o and_o a_o o_o y_fw-fr and_o o_z e_z a_o therefore_o also_o equal_a because_o both_o be_v equal_a to_o the_o angle_n at_o a_o that_o by_o the_o 17_o e_fw-la uj_o this_o by_o the_o 2_o e_z seven_o because_o the_o perpendicular_a half_v the_o side_n of_o the_o decangle_n do_v make_v two_o triangle_n equicrural_a and_o equal_a by_o the_o right_a angle_n of_o their_o shank_n and_o therefore_o they_o be_v equiangle_n therefore_o as_o e_z a_o be_v to_o a_o o_o so_o be_v e_z a_o to_o a_o y._n wherefore_o by_o the_o 8_o e_z xij_o the_o oblong_a of_o the_o two_o extreme_n be_v equal_a to_o the_o quadrate_n of_o the_o mean_a and_o the_o proposition_n of_o the_o syllogism_n which_o be_v to_o be_v demonstrate_v the_o converse_n from_o hence_o as_o manifest_v euclid_n do_v use_v at_o the_o 16_o p_o xiij_o 16._o if_o a_o triangle_n and_o a_o quinquangle_v be_v inscribe_v into_o the_o same_o circle_n at_o the_o same_o point_n the_o right_a line_n inscribe_v between_o the_o base_n of_o the_o both_o opposite_a to_o the_o say_a point_n shall_v be_v the_o side_n of_o the_o inscribe_v quindecangle_n 16._o p._n iiij_o therefore_o 17._o if_o a_o quinquangle_n and_o a_o sexangle_v be_v inscribe_v into_o the_o same_o circle_n at_o the_o same_o point_n the_o periphery_a intercept_v between_o both_o their_o side_n shall_v be_v the_o thirty_o part_n of_o the_o whole_a periphery_n of_o geometry_n the_o ninteenth_fw-mi book_n of_o the_o measure_v of_o ordinate_a multangle_n and_o of_o a_o circle_n out_o of_o the_o adscription_n of_o a_o circle_n and_o a_o rectilineall_a be_v draw_v the_o geodesy_n of_o ordinate_a multangle_v and_o first_o of_o the_o circle_n itself_o for_o the_o meeting_n of_o two_o right_a line_n equal_o divide_v two_o angle_n be_v the_o centre_n of_o the_o circumscribe_v circle_n from_o the_o centre_n unto_o the_o angle_n be_v the_o ray_n and_o then_o if_o the_o quadrate_n of_o half_a the_o side_n be_v take_v out_o of_o the_o quadrate_n of_o the_o ray_n the_o side_n of_o the_o remainder_n shall_v be_v the_o perpendicular_a by_o the_o 9_o e_fw-la xij_o therefore_o a_o special_a theorem_a be_v here_o thus_o make_v 1._o a_o plain_a make_v of_o the_o
first_o the_o triangle_n o_o u._fw-mi a_o &_o s_o r_o l_o be_v equilater_n by_o the_o 2_o e_fw-la seven_o see_v that_o the_o angle_n at_o a_o and_o l_o the_o external_a and_o internal_a be_v equal_a in_o base_n o_fw-mi u._fw-mi and_o s_o r_o for_o the_o segment_n in_o each_o distance_n be_v the_o same_o still_o therefore_o u_z a_o be_v equal_a to_o r_o l._n now_o the_o rest_n be_v conclude_v by_o a_o sorite_n of_o four_o degree_n as_o y_o r_o be_v unto_o y_fw-mi u._fw-mi so_o by_o the_o 12._o e_fw-la seven_o be_v his_o r_o that_o be_v o_o u._fw-mi unto_o e_fw-it i_fw-it and_o as_o o_fw-mi u._fw-mi be_v unto_o e_fw-it i_fw-it so_o be_v a_o u._fw-mi that_o be_v l_o r_o unto_o a_o i._o therefore_o the_o remainder_n y_fw-fr l_o unto_o the_o remainder_n y_o a_o shall_v be_v as_o y_o r_o be_v unto_o the_o whole_a y_fw-mi i_o and_o therefore_o from_o the_o first_o unto_o the_o last_o as_o s_z r_o be_v to_o e_o i._n therefore_o let_v the_o difference_n of_o the_o index_n be_v 23_o parte_v the_o difference_n of_o the_o distance_n 30._o foot_n the_o segment_n of_o the_o transome_n 23._o part_n the_o height_n shall_v be_v 57_o 9_o 23._o or_o foot_n therefore_o 15_o out_o of_o the_o geodesy_n of_o height_n the_o difference_n of_o two_o height_n be_v manifest_a or_o thus_o by_o the_o measure_n of_o one_o altitude_n we_o may_v know_v the_o difference_n of_o two_o altitude_n h._n for_o when_o thou_o have_v take_v or_o find_v both_o of_o they_o by_o some_o one_o of_o the_o former_a way_n take_v the_o lesser_a out_o of_o the_o great_a and_o the_o remain_n shall_v be_v the_o height_n desire_v from_o hence_o therefore_o by_o one_o of_o the_o tower_n of_o unequal_a height_n you_o may_v measure_v the_o height_n of_o the_o other_o first_o out_o of_o the_o lesser_a let_v the_o length_n be_v take_v by_o the_o first_o way_n because_o the_o height_n of_o the_o lesser_a wherein_o thou_o be_v be_v easy_a to_o be_v take_v either_o by_o a_o plumbe-line_n let_v fall_n from_o the_o top_n to_o the_o bottom_n or_o by_o some_o one_o of_o the_o former_a way_n then_o measure_v the_o height_n which_o be_v above_o the_o lesser_a and_o add_v that_o to_o the_o lesser_a and_o thou_o shall_v have_v the_o whole_a height_n by_o the_o first_o or_o second_o way_n the_o figure_n be_v thus_o and_o the_o demonstration_n be_v out_o of_o the_o 12._o e_fw-la seven_o for_o as_o a_o e_z be_v to_z e_z i_z so_o be_v a_o o_o to_z o_o u._fw-mi contrariwise_o out_o of_o a_o high_a tower_n one_o may_v measure_v a_o lesser_a 16_o if_o the_o sight_n be_v first_o from_o the_o top_n than_o again_o from_o the_o base_a or_o middle_a place_n of_o the_o great_a by_o the_o vane_n of_o the_o transome_n unto_o the_o top_n of_o the_o lesser_a height_n as_o the_o say_a part_n of_o the_o yard_n be_v unto_o the_o part_n of_o the_o first_o yard_n so_o the_o height_n between_o the_o station_n shall_v be_v unto_o his_o excess_n above_o the_o height_n desire_v for_o let_v the_o part_n of_o the_o yard_n be_v 12._o and_o 6._o and_o the_o sum_n of_o they_o 18._o now_o as_o 18._o be_v 12._o so_o be_v the_o whole_a altitude_n u._fw-mi y_fw-mi 190._o foot_n unto_o the_o excess_n 126⅔_n foot_n the_o remainder_n therefore_o 63⅓_n foot_n shall_v be_v a_o s_o the_o lesser_a height_n seek_v the_o second_o station_n may_v have_v be_v in_o o_o the_o end_n of_o the_o perpendicular_a from_o a._n but_o by_o take_v the_o aim_n out_o of_o the_o top_n of_o the_o lesser_a altitude_n the_o demonstration_n shall_v be_v yet_o again_o more_o easy_a and_o short_a by_o the_o two_o triangle_n at_o the_o yard_n a_o e_o i_o and_o a_o e_o f_o resemble_v the_o two_o whole_a triangle_n a_o o_o u._fw-mi and_o a_o o_fw-fr y_fw-fr in_o like_a situation_n the_o part_n of_o the_o shank_n cut_v be_v on_o each_o side_n the_o segment_n of_o the_o transome_n one_o may_v again_o also_o out_o of_o the_o top_n of_o a_o turret_n measure_v the_o distance_n of_o two_o turret_n one_o from_o another_o for_o it_o be_v the_o first_o manner_n of_o measure_v of_o longitude_n neither_o do_v it_o here_o differ_v any_o whit_n from_o it_o more_o than_o the_o yard_n be_v hang_v without_o the_o height_n give_v the_o figure_n be_v thus_o and_o the_o demonstration_n be_v by_o the_o 12._o e_fw-la seven_o for_o as_o a_o e_z the_o segment_n of_o the_o yard_n be_v unto_o e_fw-it i_fw-it the_o segment_n of_o the_o transome_n so_o be_v the_o assign_a altitude_n a_o o_o unto_o the_o length_n o_o u._n the_o geodesy_n or_o measure_n of_o altitude_n be_v thus_o where_o either_o the_o length_n or_o some_o part_n of_o the_o length_n be_v give_v as_o in_o the_o first_o and_o second_o way_n or_o where_o the_o distance_n be_v double_a as_o in_o the_o three_o 17_o if_o the_o sight_n be_v from_o the_o begin_n of_o the_o yard_n be_v right_a or_o perpendicular_a by_o the_o vane_n of_o the_o transome_n unto_o the_o end_n of_o the_o breadth_n as_o in_o the_o yard_n the_o difference_n of_o the_o segment_n be_v unto_o the_o difference_n of_o the_o distance_n so_o be_v the_o distance_n of_o the_o vane_n unto_o the_o breadth_n the_o measure_n of_o breadth_n that_o be_v of_o a_o thwart_a or_o cross_a line_n remain_v the_o figure_n and_o demonstration_n be_v thus_o the_o first_o aim_v let_v it_o be_v a_o e_o i_o by_z o_o and_o u_z the_o vane_n of_o the_o transome_n o_o u._fw-mi the_o second_o let_v it_o be_v y_fw-mi e_fw-it i_fw-it by_z s_z and_o r_o the_o vane_n of_o the_o transome_n s_o r._n then_o by_o the_o point_n s_o let_v the_o parallel_n l_o s_o m_o be_v draw_v against_o a_o o_o e._n here_o first_o the_o triangle_n o_o u._fw-mi a_o and_o s_o i_o l_o be_v equilater_n by_o the_o 2_o e_fw-la seven_o because_o the_o angle_n at_o n_o and_o j_o be_v right_a angle_n and_o u._fw-mi a_o o_o and_o j_o l_o s_z the_o outter_n and_o inner_a be_v equal_a in_o their_o base_n o_o u._fw-mi and_o s_z j_o by_o the_o grant_n because_o here_o the_o segment_n of_o the_o transome_n remain_v the_o same_o therefore_o u._fw-mi a_o be_v equal_a to_o j_o l._n 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be_v a_o rectilineall_a figure_n compound_v of_o triangle_n as_o before_o for_o the_o dichotomy_n sake_n of_o a_o line_n be_v make_v a_o lineate_v to_o signify_v the_o genus_fw-la of_o a_o surface_n and_o a_o body_n so_o now_o be_v for_o the_o same_o cause_n of_o a_o triangle_n make_v a_o triangulate_a to_o declare_v and_o express_v the_o genus_fw-la of_o a_o quadrilater_n and_o multilater_n and_o indeed_o more_o just_o then_o before_o in_o a_o lineate_v for_o triangle_n do_v compound_v and_o make_v the_o triangulate_a but_o line_n do_v not_o make_v the_o lineate_v therefore_o 2._o the_o side_n of_o a_o triangulate_a be_v two_o more_o than_o be_v the_o triangle_n of_o which_o it_o be_v make_v and_o 3._o homogeneal_a triangulate_v be_v cut_v into_o a_o equal_a number_n of_o triangle_n è_fw-la 20_o p_o uj._o for_o if_o they_o be_v quadrangle_v they_o be_v cut_v into_o two_o triangle_n if_o quinquangle_v into_o 3._o if_o hexangle_v into_o 4_o and_o so_o forth_o 4._o like_a triangulate_v be_v cut_v into_o triangle_n alike_o one_o to_o another_o and_o homologall_a to_o the_o whole_a è_fw-mi 20_o p_o uj._o or_o thus_o like_a triangulate_v be_v divide_v into_o triangle_n like_o one_o unto_o another_o and_o in_o porportion_n correspondent_a unto_o the_o whole_a h._n as_o in_o these_o two_o quinqualge_v first_o the_o particular_a triangle_n be_v like_a between_o themselves_o for_o the_o shank_n of_o a_o e_fw-it u._fw-mi and_o y_z s_z m_z equal_a angle_n be_v proportional_a by_o the_o grant_n therefore_o the_o triangle_n themselves_o be_v equiangle_n by_o 14_o e_fw-la seven_o and_o therefore_o alike_o by_o the_o 12_o