follow_v here_o the_o transome_n if_o there_o be_v room_n enough_o for_o the_o measurer_n to_o go_v far_o enough_o back_o must_v be_v put_v low_a in_o the_o second_o distance_n 9_o if_o the_o sight_n be_v from_o the_o begin_n of_o the_o transverie_n parallel_v to_o the_o length_n to_o be_v measure_v as_o in_o the_o index_n the_o difference_n of_o the_o great_a segment_n be_v unto_o the_o lesser_a so_o be_v the_o difference_n of_o the_o second_o station_n unto_o the_o length_n this_o kind_n of_o geodaesy_n be_v somewhat_o more_o subtle_a than_o the_o former_a be_v the_o figure_n be_v thus_o in_o which_o let_v the_o first_o aim_v be_v from_o a_o the_o beginning_n of_o the_o transome_n and_o out_o of_o a_o i_o the_o length_n seek_v by_o o_o the_o end_n of_o the_o index_n unto_o e_o the_o top_n of_o the_o height_n and_o let_v the_o segment_n of_o the_o index_n be_v o_o u._fw-mi the_o second_o aim_v let_v it_o be_v from_o y_fw-mi the_o beginning_n of_o the_o transome_n out_o of_o a_o great_a distance_n by_o s_o the_o end_n of_o the_o index_n unto_o e_o the_o same_o note_n of_o the_o height_n and_o let_v the_o segment_n of_o the_o index_n be_v s_o r._n here_o the_o measure_v perform_v be_v the_o take_n of_o the_o difference_n between_o o_fw-fr u._fw-mi and_o s_o r._n the_o rest_n be_v feign_a only_o for_o demonstration_n sake_n therefore_o in_o the_o first_o station_n let_v a_o m_o l_o be_v from_o the_o begin_n of_o the_o transome_n be_v parallel_n to_o y_o e._n here_o first_o m_z u._fw-mi be_v equal_a to_o s_o r._n for_o the_o triangleâ—_n m_o u._fw-mi a_o and_o s_o r_o y_fw-fr be_v equal_a in_o their_o shank_n you_o a_o and_o r_o y_fw-fr by_o the_o grant_n because_o the_o transome_n stand_v still_o in_o his_o own_o place_n and_o the_o angle_n at_o m_o u._fw-mi a_o u_z a_o m_z be_v equal_a to_o the_o angle_n and_o all_o right_a angle_n be_v equal_a by_o the_o 14_o e_fw-la iij._o these_o be_v the_o outter_n and_o inner_a opposite_a one_o to_o another_o and_o such_o be_v equal_a by_o the_o 1._o e._n v_o therefore_o they_o be_v equilater_n by_o the_o 2_o e_fw-la seven_o and_o o_fw-it m_o be_v the_o difference_n of_o the_o segment_n of_o the_o index_n then_o as_o o_o m_o be_v to_z m_z u_z so_o be_v e_o l_o to_z l_o i_o as_o the_o equation_n of_o three_o degree_n do_v show_v for_o by_o the_o 12_o e_fw-la seven_o as_o o_o m_o be_v to_z m_z a_o so_o be_v e_o l_o to_o l_o a_o and_o as_o m_z a_o be_v to_o m_o u._fw-mi so_o be_v l_o a_o to_z l_o i._n therefore_o by_o right_a as_o o_o m_o be_v to_z m_z u._fw-mi so_o be_v e_o l_o to_z l_o i_o and_o by_o the_o 12_o e_o v_o j_o so_o be_v y_o a_o to_o a_o i_o as_o if_o the_o difference_n of_o the_o first_o segment_n be_v 36_o part_n the_o second_o segment_n be_v 72_o part_n the_o difference_n of_o the_o second_o station_n 40_o foot_n the_o length_n seek_v shall_v be_v 80_o foot_n and_o here_o indeed_o be_v no_o height_n definite_o give_v that_o may_v make_v any_o bind_v of_o the_o principal_a proportion_n notwithstanding_o the_o height_n although_o it_o be_v of_o a_o unknown_a measure_n be_v the_o bind_v of_o the_o length_n seek_v and_o therefore_o it_o be_v a_o help_n and_o mean_n to_o argue_v the_o question_n because_o it_o be_v conceive_v to_o stand_v plumbe_v upon_o the_o outmost_a end_n of_o the_o length_n therefore_o that_o three_o kind_n of_o measure_v of_o length_n be_v oftentimes_o necessary_a when_o by_o neither_o of_o the_o former_a wayeâ—_n the_o length_n may_v possible_o be_v take_v by_o reason_n of_o some_o impediment_n in_o the_o way_n to_o wit_n of_o a_o wall_n or_o tree_n or_o house_n or_o mountain_n whereby_o the_o end_n of_o the_o length_n may_v not_o be_v see_v which_o be_v the_o first_o way_n nor_o a_o height_n next_o adjoin_v to_o the_o end_n of_o the_o length_n be_v give_v which_o be_v the_o second_o way_n hitherto_o we_o have_v speak_v of_o the_o threefold_a measure_n of_o longitude_n the_o first_o and_o second_o out_o of_o a_o height_n give_v the_o three_o out_o of_o a_o double_a distance_n the_o measure_n of_o height_n follow_v next_o and_o that_o be_v also_o threefold_a now_o height_n be_v a_o perpendicular_a line_n fall_v from_o the_o top_n of_o the_o magnitude_n unto_o the_o ground_n or_o plain_a whereon_o the_o measurer_n do_v stand_v after_o which_o manner_n altitude_n on_o height_n be_v define_v at_o the_o 9_o e_fw-la iiij_o the_o first_o geodesy_n or_o manner_n of_o measure_v of_o heighth_n be_v thus_o 10._o if_o the_o sight_n be_v from_o the_o begin_n of_o the_o transome_a perpendicular_a unto_o the_o height_n to_o be_v measure_v 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 to_o the_o height_n let_v the_o segment_n of_o the_o transome_n be_v 60_o part_n the_o segment_n of_o the_o index_n 36_o the_o length_n give_v 120_o foot_n the_o height_n seek_v shall_v be_v by_o the_o golden_a rule_n 72_o foot_n 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 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 of_o a_o know_a part_n of_o a_o height_n from_o whence_o the_o rest_n may_v be_v know_v as_o in_o the_o figure_n the_o first_o and_o second_o kind_n of_o measure_v of_o height_n be_v thus_o the_o three_o follow_v 14_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 as_o in_o the_o index_n the_o difference_n of_o the_o segmeut_n be_v unto_o the_o difference_n of_o the_o distance_n or_o station_n so_o be_v the_o segment_n of_o the_o transome_n unto_o the_o height_n hitherto_o you_o must_v recall_v that_o subtlety_n which_o be_v use_v in_o the_o three_o manner_n of_o measure_v of_o length_n let_v the_o first_o aim_n be_v take_v from_o a_o the_o beginning_n of_o the_o index_n perpendicular_a unto_o the_o height_n to_o be_v measure_v and_o from_o a_o unknown_a length_n a_o i_o by_z o_o the_o end_n of_o the_o transome_n unto_o e_o the_o top_n of_o the_o height_n e_o i_o and_o let_v the_o segment_n of_o the_o index_n be_v u._fw-mi a._n the_o 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
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
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
word_n from_o hence_o have_v three_o line_n give_v be_v the_o invention_n of_o the_o four_o proportional_a and_o out_o of_o that_o have_v two_o line_n give_v arise_v the_o invention_n of_o the_o three_o proportional_a 2_o have_v three_o right_a line_n give_v if_o the_o first_o and_o the_o three_o make_v a_o angle_n and_o knit_v together_o with_o a_o base_a be_v continue_v the_o first_o equal_o to_o the_o second_o the_o three_o infinite_o a_o parallel_n from_o the_o end_n of_o the_o second_o unto_o the_o continuation_n of_o the_o three_o shall_v intercept_v the_o four_o proportional_a 12._o puj._n the_o diagramme_n and_o demonstration_n be_v the_o same_o with_o our_o 31._o e_z or_o 3_o c_o of_o ramus_n 3_o if_o two_o right_a line_n give_v make_v a_o angle_n and_o knit_v together_o with_o a_o base_a be_v continue_v the_o first_o equal_o to_o the_o second_o the_o second_o infinite_o a_o parallel_n to_o the_o base_a from_o the_o end_n of_o the_o first_o continuation_n unto_o the_o second_o shall_v intercept_v the_o three_o proportional_a 11._o p_o five_o i_o the_o diagramme_n here_o also_o and_o demonstration_n be_v in_o all_o respect_v the_o same_o with_o our_o 30_o e_o or_o 2_o c_o of_o ramus_n thus_o far_o ramus_n and_o here_o by_o the_o judgement_n of_o the_o learned_a finkius_n two_o element_n of_o ptolomey_n be_v to_o be_v adjoin_v 32_o if_o two_o right_a line_n cut_v one_o another_o be_v again_o cut_v with_o many_o parallel_n the_o parallel_n be_v proportional_a unto_o their_o next_o segment_n the_o same_o demonstation_n shall_v serve_v if_o the_o line_n do_v cross_a one_o another_o or_o do_v vertical_o cut_v one_o another_o as_o in_o the_o same_o diagramme_n appear_v for_o if_o the_o assign_v a_o i_o and_o u_z s_z do_v cut_v one_o another_o vertical_o in_o o_o let_v they_o be_v cut_v with_o the_o parallel_v a_o u._fw-mi and_o s_z i_z the_o precedent_a fabric_n or_o figure_n be_v make_v up_o it_o shall_v be_v by_o 28._o e._n as_o a_o u._fw-mi be_v unto_o a_o o_o the_o segment_n next_o unto_o it_o so_o a_o y_fw-mi that_o be_v i_z s_z shall_v be_v unto_o o_o ay_o his_o next_o segment_n the_o 28._o e_o teach_v how_o to_o find_v out_o the_o three_o and_o four_o proportional_a this_o afford_v we_o a_o mean_n how_o to_o find_v out_o the_o continual_o mean_v proportional_a single_a or_o double_a therefore_o 33._o if_o two_o right_a line_n give_v be_v continue_v into_o one_o a_o perpendicular_a from_o the_o point_n of_o continuation_n unto_o the_o angle_n of_o the_o squire_n include_v the_o continue_a line_n with_o the_o continuation_n be_v the_o mean_a proportional_a between_o the_o two_o right_a line_n give_v a_o squire_n norma_n gnomon_n or_o canon_n be_v a_o instrument_n consist_v of_o two_o shank_n include_v a_o right_a angle_n of_o this_o we_o hear_v before_o at_o the_o 13_o e_o by_o the_o mean_n of_o this_o a_o mean_a proportional_a unto_o two_o line_n give_v be_v easy_o find_v whereupon_o it_o may_v also_o be_v call_v a_o mesolabium_fw-la or_o mesographus_n simplex_fw-la or_o single_a mean_a finder_n and_o 34_o if_o two_o assign_v right_a line_n join_v together_o by_o their_o end_n right_a anglewise_o be_v continue_v vertical_o a_o square_n fall_v with_o one_o of_o his_o shank_n and_o another_o to_o it_o parallel_n and_o movable_a upon_o the_o end_n of_o the_o assign_v with_o the_o angle_n upon_o the_o continue_a line_n shall_v cut_v between_o they_o from_o the_o continue_v two_o mean_n continual_o proportional_a to_o the_o assign_a the_o former_a consectary_n be_v of_o a_o single_a mesolabium_fw-la this_o be_v of_o a_o double_a who_o use_n in_o make_v of_o solid_n to_o this_o or_o that_o bigness_n desire_v be_v notable_a and_o thus_o we_o have_v the_o composition_n and_o use_n both_o of_o the_o single_a and_o double_a mesolabium_fw-la 35._o if_o of_o four_o right_a line_n two_o do_v make_v a_o angle_n the_o other_o reflect_v or_o turn_v back_o upon_o themselves_o from_o the_o end_n of_o these_o do_v cut_v the_o former_a the_o reason_n of_o the_o one_o unto_o his_o own_o â—egment_n or_o of_o the_o segment_n between_o themselves_o be_v make_v of_o the_o reason_n of_o the_o so_o joint_o bound_v that_o the_o first_o of_o the_o maker_n be_v joint_o bound_v with_o the_o begin_n of_o the_o antecedent_n make_v the_o second_o of_o this_o consequent_a joint_o bound_v with_o the_o end_n do_v end_n in_o the_o end_n of_o the_o consequent_a make_v let_v therefore_o the_o two_o right_a line_n be_v â—_o e_o and_o a_o i_o and_o from_o the_o end_n of_o these_o other_o two_o reflect_v be_v i_o u._fw-mi and_o e_z o_o cut_v themselves_o in_o y_o and_o the_o two_o former_a in_o you_o and_o o._n the_o reason_n of_o the_o particular_a right_a line_n make_v shall_v be_v as_o the_o draught_n follow_v do_v manifest_a in_o which_o the_o antecedent_n of_o the_o maker_n be_v in_o the_o upper_a place_n the_o consequent_n be_v set_v under_o neathe_v their_o own_o antecedent_n the_o business_n be_v the_o same_o in_o the_o two_o other_o whether_o you_o do_v cross_v the_o bound_n or_o invert_v they_o here_o for_o demonstration_n sake_n we_o crave_v no_o more_o but_o that_o from_o the_o begin_n of_o a_o antecedent_n make_v a_o parallel_n be_v draw_v to_o the_o second_o consequent_a of_o the_o maker_n unto_o one_o of_o the_o assign_v infinite_o continue_v then_o the_o multiply_a proportion_n shall_v be_v the_o antecedent_n the_o consequent_a the_o antecedent_n the_o consequent_a of_o the_o second_o of_o the_o maker_n every_o way_n the_o reason_n or_o rate_n be_v of_o equality_n the_o antecedent_n the_o consequent_a of_o the_o first_o of_o the_o maker_n the_o parallel_n the_o antecedent_n of_o the_o second_o of_o the_o maker_n by_o the_o 32._o e._n therefore_o by_o multiplication_n of_o proportion_n the_o reason_n of_o the_o parallel_n unto_o the_o consequent_a of_o the_o second_o of_o the_o maker_n that_o be_v by_o the_o fabric_n or_o construction_n and_o the_o 32._o e._n the_o reason_n of_o the_o antecedent_n of_o the_o product_n unto_o the_o consequent_a be_v make_v of_o the_o reason_n etc._n etc._n after_o the_o manner_n above_o write_v again_o i_o say_v that_o the_o reason_n of_o e_o y_fw-fr unto_z y_z o_o be_v compound_v of_o the_o reason_n of_o e_o u._fw-mi unto_z u_z a_o and_o of_o a_o i_o unto_o i_o â—_o theon_n here_o draw_v a_o parallel_n from_o o_o unto_o u._fw-mi i._o by_o the_o general_a fabric_n it_o may_v be_v draw_v out_o of_o e_o unto_o o_o i._n therefore_o the_o reason_n of_o e_z n_o unto_z i_z o_o that_o be_v of_o e_o y_fw-fr unto_z y_z o_o shall_v be_v make_v of_o the_o foresay_a reason_n of_o the_o segment_n of_o divers_a right_a linesâ—_n the_o arabian_n have_v much_o under_o the_o name_n of_o the_o rule_n of_o six_o quantity_n and_o the_o theorem_n of_o althinâ—us_n concern_v this_o matter_n be_v in_o many_o man_n hand_n and_o regiomontanus_n in_o his_o algorithmus_fw-la and_o maurolycus_n upon_o the_o 1_o piij._n of_o menelaus_n do_v make_v mention_n of_o they_o but_o they_o contain_v nothing_o which_o may_v not_o by_o any_o man_n skilful_a in_o arithmetic_n be_v perform_v by_o the_o multiplication_n of_o proportion_n for_o all_o those_o way_n of_o they_o be_v no_o more_o but_o special_a example_n of_o that_o kind_n of_o multiplication_n of_o geometry_n the_o six_o book_n of_o a_o triangle_n 1_o like_a plain_n have_v a_o double_a reason_n of_o their_o homâ—logall_a side_n and_o one_o proportional_a mean_a out_o of_o 20_o p_o uj._o and_o xj_o and_o 18._o p_o viij_o or_o thus_o like_a plain_n have_v the_o proportion_n of_o their_o corespondent_a proportional_a side_n double_v &_o one_o mean_a proportional_a hitherto_o we_o have_v speak_v of_o plain_a line_n and_o their_o affection_n plain_a figure_n and_o their_o kind_n do_v follow_v in_o the_o next_o place_n and_o first_o there_o be_v premise_v a_o common_a corollary_n draw_v out_o of_o the_o 24._o e_fw-la iiij_o because_o in_o plain_n there_o be_v but_o two_o dimension_n 2_o a_o plain_a surface_n be_v either_o rectilineall_a or_o obliquelineall_a or_o rightline_n or_o crookedline_v h._n straightness_n and_o crookedness_n be_v the_o difference_n of_o line_n at_o the_o 4._o e_fw-la i_o i_o from_o thence_o be_v it_o here_o repeat_v and_o attribute_v to_o a_o surface_n which_o be_v geometrical_o make_v of_o line_n that_o make_v of_o right_a line_n be_v rectileniall_n that_o which_o be_v make_v of_o crooked_a line_n be_v obliquilineall_a 3_o a_o rectilineall_a surface_n be_v that_o which_o be_v comprehend_v of_o right_a line_n 4_o a_o rightilineall_a do_v make_v all_o his_o angle_n equal_a to_o right_a angle_n the_o inner_a one_o general_o to_o pair_n from_o two_o forward_a the_o outter_n always_o to_o four_o or_o thus_o a_o right_n line_v plain_n make_v his_o angle_n equal_a unto_o right_a
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 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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 skill_v not_o so_o it_o be_v firm_a and_o will_v not_o cast_v or_o warp_v notwithstanding_o the_o transome_n will_v more_o convenient_o be_v move_v up_o and_o down_o by_o brazen_a pipe_n both_o by_o itself_o and_o upon_o the_o index_n high_o or_o low_a right_o angle_v wise_a so_o touch_v one_o another_o that_o the_o alterne_a mouth_n of_o the_o one_o may_v touch_v the_o side_n of_o the_o other_o the_o three_o pipe_n be_v to_o be_v move_v or_o slide_v up_o and_o down_o from_o one_o end_n of_o the_o transome_n to_o the_o other_o and_o therefore_o it_o may_v be_v call_v the_o cursor_n the_o four_o and_o five_o pipe_n fix_v and_o immovable_a be_v set_v upon_o the_o end_n of_o the_o transome_n be_v unto_o the_o three_o and_o second_o of_o equal_a height_n with_o â—innes_n to_o restrain_v when_o need_n be_v the_o optic_a line_n and_o as_o it_o be_v with_o certain_a point_n to_o define_v it_o in_o the_o transome_n the_o three_o first_o pipe_n may_v as_o occasion_n shall_v require_v be_v 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the_o station_n be_v from_o the_o midst_n of_o the_o foot_n 5_o if_o the_o sight_n do_v pass_v from_o the_o begin_n of_o one_o shank_n it_o pass_v by_o the_o end_n of_o the_o other_o and_o the_o one_o shank_n be_v perpendicular_a unto_o the_o magnitude_n to_o be_v measure_v the_o other_o parallel_n these_o common_a and_o general_a thing_n be_v premise_v that_o the_o sight_n be_v from_o the_o begin_n of_o the_o index_n by_o the_o end_n of_o the_o transome_n or_o contrariwise_o from_o the_o beginning_n of_o the_o transome_n unto_o the_o end_n of_o the_o index_n and_o that_o the_o index_n be_v right_a that_o be_v perpendicular_a to_o the_o line_n to_o be_v measure_v the_o transome_a parallel_n or_o contrariwise_o now_o the_o perpendicularity_n of_o the_o index_n in_o measuring_n of_o lengtht_n may_v be_v try_v by_o a_o plummet_n of_o lead_n appendentâ—_n but_o in_o height_n and_o breadth_n the_o eye_n must_v be_v trust_v although_o a_o little_a vary_v of_o the_o plummet_n can_v make_v 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be_v constrain_v to_o change_v his_o place_n and_o make_v a_o double_a stand_n here_o observe_v that_o length_n and_o height_n may_v be_v joint_o measure_v both_o with_o one_o and_o with_o a_o double_a station_n but_o breadth_n may_v not_o be_v measure_v otherwise_o than_o with_o two_o 7_o if_o the_o sight_n be_v from_o the_o begin_n of_o the_o index_n râ—ght_v or_o plumbe_v unto_o the_o length_n and_o unto_o the_o father_n end_n of_o the_o same_o 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 be_v the_o height_n of_o the_o measurer_n unto_o the_o length_n the_o same_o manner_n of_o measure_v shall_v be_v use_v form_n a_o high_a place_n as_o out_o of_o y_o the_o segment_n of_o the_o index_n be_v 5._o part_n the_o segment_n of_o the_o transome_a 6_o and_o then_o the_o height_n be_v 10_o foot_n the_o same_o length_n shall_v be_v find_v to_o be_v 12_o foot_n neither_o be_v it_o any_o matter_n at_o all_o whether_o the_o length_n in_o a_o plain_a or_o level_v underneath_o or_o in_o a_o ascent_n or_o descent_n of_o a_o mountain_n as_o in_o the_o figure_n under_o write_v thus_o may_v thou_o measure_v the_o breadth_n of_o river_n valley_n and_o ditch_n for_o the_o length_n be_v always_o after_o this_o manner_n so_o that_o one_o may_v measure_v the_o distance_n of_o ship_n on_o the_o sea_n as_o also_o thales_n milesius_n in_o proclus_n at_o the_o 26_o pj_fw-la do_v measure_v they_o a_o example_n thou_o have_v here_o hereafter_o in_o the_o measure_n of_o longitude_n and_o altitude_n fight_n be_v unto_o the_o top_n of_o the_o height_n which_o here_o i_o do_v now_o forewarn_v thou_o of_o lest_o afterward_o it_o shall_v in_o vain_a be_v reiter_v often_o the_o second_o manner_n of_o measure_v a_o length_n be_v thus_o 8._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 length_n to_o be_v measure_v 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 height_n give_v be_v to_o the_o length_n as_o if_o the_o segment_n of_o the_o transome_n be_v 120_o part_n the_o height_n give_v 400-foote_n the_o segment_n of_o the_o index_n 210_o part_n the_o length_n by_o the_o golden_a rule_n shall_v be_v 700_o foot_n the_o figure_n be_v thus_o and_o the_o demonstration_n be_v like_a unto_o the_o former_a or_o indeed_o more_o easy_a for_o the_o triangle_n be_v equiangle_n as_o afore_o therefore_o as_o o_fw-mi u._fw-mi be_v to_o u._fw-mi a_o so_o be_v e_z i_z to_z i_z a._n this_o be_v the_o first_o and_o second_o kind_n of_o measure_v of_o a_o longitude_n by_o one_o single_a distance_n or_o station_n the_o three_o which_o be_v by_o a_o double_a distance_n do_v now_o
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 the_o decangle_v a_o r_o inscribe_v into_o the_o same_o circle_n circumscribe_v the_o quinquangle_n of_o a_o icosahedrum_fw-la and_o the_o perpendicular_o i_o s_o and_o u_z l._n here_o the_o two_o triangle_n a_o i_o e_o and_o i_o e_o s_o be_v by_o the_o 8_o e_z viij_o alike_o and_o as_o s_z e_z be_v unto_o e_fw-it i_fw-it so_o be_v i_o e_o unto_z e_z a_o and_o by_o 25_o e_fw-la iiij_o as_o s_z e_z be_v to_z e_z a_o so_o be_v the_o quadrate_n of_o s_o e_o to_o the_o quadrate_n of_o e_o i_o and_o invers_o or_o backward_o as_o a_o e_o be_v to_z s_z e_z so_o be_v the_o quadrate_n of_o i_o e_o to_o the_o quadrate_n of_o s_o e._n but_o a_o e_o be_v the_o triple_a of_o s_o e._n therefore_o the_o quadrate_n of_o i_o e_o be_v the_o triple_a of_o s_o e._n but_o the_o quadrate_n of_o a_o s_o by_o the_o grant_n and_o 14_o e_fw-la xij_o be_v the_o quadruple_a of_o the_o quadrate_n of_o s_o e._n therefore_o also_o it_o be_v great_a than_o the_o quadrate_n of_o i_o e_o and_o the_o right_a line_n a_o 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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 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