Selected quad for the lemma: book_n

Word A Word B Word C Word D Occurrence Frequency Band MI MI Band Prominent
book_n demonstration_n proposition_n quantity_n 1,827 5 13.9195 5 false
View all documents for the selected quad

Text snippets containing the quad

ID Title Author Corrected Date of Publication (TCP Date of Publication) STC Words Pages
A47183 A supplement to a late treatise, called An essay for the discovery of some new geometrical problems concerning angular sections, resolving what was there problematically proposed; and with some rectification made in the former essay, showing an easie method truly geometrical, without any conick section, or cubick æquation, to sect any angle or arch of a circle into 3. 5. 7. or any other uneven number of equal parts. By G. K. Keith, George, 1639?-1716. 1697 (1697) Wing K216A; ESTC R216625 4,362 7

There are 2 snippets containing the selected quad. | View original text

a_o supplement_n to_o a_o late_a treatise_n call_v a_o essay_n for_o the_o discovery_n of_o some_o new_a geometrical_a problem_n concern_v angular_a section_n resolve_v what_o be_v there_o problematical_o propose_v and_o with_o some_o rectification_n make_v in_o the_o former_a essay_n show_v a_o easy_a method_n true_o geometrical_a without_o any_o conic_a section_n or_o cubick_a aequation_n to_o sect_n any_o angle_n or_o arch_n of_o a_o circle_n into_o 3.5.7_o or_o any_o other_o uneven_a number_n of_o equal_a part_n by_o g.k._n whereas_o it_o be_v suppose_v in_o the_o former_a proposal_n that_o a_o straight_a line_n can_v be_v draw_v through_o the_o extreme_a point_n of_o three_o or_o more_o concentric_a arch_n at_o both_o end_n the_o arch_n beginning_n or_o end_v upon_o a_o straight_a line_n ●oming_v from_o the_o centre_n of_o those_o concentric_a arch_n have_v equal_a cord_n though_o not_o equal_a arch_n upon_o further_a consideration_n it_o be_v find_v that_o however_o seem_o such_o a_o line_n may_v appear_v to_o be_v straight_o in_o many_o case_n as_o when_o the_o radius_fw-la be_v short_a or_o the_o angle_n very_o acute_a yet_o in_o no_o case_n be_v such_o a_o line_n mathematical_o straight_o but_o be_v a_o regular_a curve_v and_o can_v be_v as_o regular_o draw_v and_o by_o as_o true_a geometrical_a art_n as_o any_o parabola_fw-la or_o other_o conic_a section_n can_v and_o with_o great_a facility_n and_o readiness_n and_o which_o any_o tiro_n who_o understand_v nothing_o of_o conic_a section_n and_o cubick_a equation_n may_v do_v the_o way_n of_o draw_v the_o say_v curve_v be_v this_o let_v a_o short_a cross-rule_n be_v set_v at_o right_a angle_n with_o another_o long_a rule_n and_o let_v the_o length_n of_o the_o cross_a rule_n be_v at_o pleasure_n 2_o or_o 3_o inch_n or_o more_o as_o 6_o or_o 7_o as_o you_o have_v a_o mind_n to_o make_v the_o length_n of_o the_o cord_n of_o each_o part_n of_o the_o section_n of_o your_o angle_n which_o as_o in_o the_o follow_a figure_n let_v be_v 3_o inch_n and_o let_v the_o just_a half_n of_o the_o cross-rule_n be_v on_o the_o left_a side_n of_o the_o long_a rule_n and_o let_v a_o small_a brass_n or_o steel-pin_n be_v fix_v on_o the_o right_a end_n of_o the_o say_v cross_a rule_n that_o as_o the_o rule_n be_v move_v may_v make_v a_o impression_n on_o the_o paper_n as_o the_o point_n of_o the_o compass_n do_v in_o draw_v a_o circle_n the_o length_n of_o the_o long_a rule_n be_v to_o be_v as_o occasion_n require_v as_o double_a or_o triple_a the_o length_n of_o the_o other_o have_v thus_o prepare_v your_o two_o rule_n the_o one_o cut_v the_o other_o at_o right_a angle_n and_o the_o cross-rule_n fix_v to_o it_o though_o it_o may_v be_v make_v also_o movable_a on_o it_o suppose_v the_o angle_n give_v to_o be_v trisect_v be_v bac_n measure_v by_o the_o arch_a bmc_n in_o order_n to_o draw_v the_o curve_v line_n with_o one_o draught_n of_o the_o hand_n set_v the_o left_a end_n of_o the_o cross-rule_n on_o the_o point_n b_o and_o from_o b_o let_v it_o run_v or_o slide_v along_o the_o line_n basilius_n and_o as_o it_o run_v along_o the_o say_a line_n let_v the_o left_a side_n of_o the_o long_a rule_n still_o run_v through_o the_o centre_n or_o vertical_a point_n a_o which_o be_v most_o easy_o do_v and_o let_v it_o run_v or_o slide_v along_o from_o b_o towards_o a_o until_o the_o other_o end_n of_o the_o cross-rule_n reach_v at_o least_o to_o the_o line_n ac_fw-la or_o further_o as_o one_o please_v and_o the_o brass_n pin_n on_o the_o other_o end_n of_o the_o cross_a rule_n shall_v describe_v the_o regular_a curve_v fig_n have_v thus_o draw_v the_o curve_v line_n at_o the_o distance_n of_o one_o half_a of_o the_o cross_a rule_n draw_v the_o straight_a line_n de_fw-fr parallel_n to_o ac_fw-la and_o where_o the_o curve_v line_n cut_v the_o straight_a line_n de_fw-fr as_o at_o i_o a_o line_n draw_v from_o the_o centre_n a_o to_o i_o shall_v by_o true_a geometry_n trisect_v the_o give_v angle_n bac_n the_o demonstration_n see_v it_o be_v the_o property_n of_o these_o two_o rule_n cross_v each_o other_o at_o right_a angle_n where_o ever_o the_o two_o end_n of_o the_o cross_a rule_n terminate_v to_o make_v a_o isosceles_a triangle_n make_v always_o two_o right_a angle_n triangle_n who_o base_n be_v equal_a and_o the_o perpendicular_a common_a to_o both_o therefore_o by_o the_o 4._o 1._o el._n eucl._n the_o hypotenusal_n be_v equal_a therefore_o with_o radius_fw-la ai_fw-fr describe_v the_o arch_n hlis_n draw_v the_o cord_n he_o and_o from_o i_o let_v fall_v a_o perpendicular_a on_o the_o line_n ac_fw-la as_o ik_fw-mi make_v he_o =_o li_n =_o ik_fw-mi therefore_o the_o arch_n of_o those_o equal_a sin_n be_v equal_a as_o hl_n =_o li_n =_o be_v q.e.d._n the_o same_o or_o any_o other_o angle_n obtruse_a or_o acute_a may_v be_v trisect_v into_o 3_o equal_a part_n without_o the_o curve_v line_n or_o any_o part_n of_o it_o by_o find_v the_o point_n i_o which_o can_v be_v find_v without_o the_o curve_v line_n by_o let_v the_o cross_a rule_n slide_v or_o run_v along_o the_o line_n ab_fw-la while_o the_o left_a side_n of_o the_o long_a rule_n still_o run_v through_o the_o centre_n a_o either_o upward_o or_o downward_o until_o the_o right_a side_n of_o the_o cross_a rule_n touch_v the_o straight_a line_n de_fw-fr which_o shall_v be_v at_o i._o and_o thus_o without_o any_o need_n of_o noticing_z or_o regard_v the_o curve_v line_n the_o point_n 1_o be_v find_v where_o the_o two_o straight_a line_n he_o and_o de_fw-fr meet_v together_o and_o as_o thus_o any_o angle_n may_v be_v trisect_v without_o draw_v any_o curve_v line_n so_o it_o may_v easy_o and_o true_o be_v do_v without_o either_o scale_n or_o compass_n other_o than_o what_o the_o two_o cross_a rule_n be_v as_o any_o artist_n may_v easy_o perceive_v if_o any_o object_n against_o this_o method_n as_o mechanical_a and_o not_o mathematical_a and_o true_o geometrical_a because_o perform_v by_o a_o instrument_n i_o shall_v refer_v they_o to_o two_o great_a geometrician_n for_o its_o vindication_n to_o wit_n des_fw-mi cartes_n in_o his_o second_o book_n of_o geometry_n and_o franciscus_n a_o schoten_n in_o his_o commentary_n on_o he_o argum_fw-la lib._n 2._o both_o which_o do_v prove_v that_o what_o be_v perform_v by_o instrument_n geometrical_o make_v be_v geometrical_a otherwise_o the_o plain_a geometry_n must_v be_v reject_v because_o its_o figure_n be_v draw_v by_o rule_n and_o compass_n both_o which_o be_v instrument_n and_o not_o only_a parabolas_fw-la and_o other_o conic_a section_n which_o be_v curve_n but_o divers_a other_o curve_n yea_o all_o such_o that_o can_v be_v draw_v by_o art_n with_o the_o help_n of_o instrument_n such_o as_o they_o have_v devise_v they_o contend_v to_o be_v true_o geometrical_a and_o both_o of_o they_o in_o their_o geometrical_a treatise_n use_v divers_a instrument_n for_o describe_v curve_n geometrical_o much_o more_o difficult_a to_o be_v make_v and_o with_o more_o difficulty_n to_o be_v use_v than_o what_o be_v here_o propose_v of_o two_o simple_a rule_n cut_v one_o another_o at_o right_a angle_n and_o see_v it_o have_v no_o dependence_n on_o solid_n or_o algebra_n equation_n and_o may_v be_v do_v without_o any_o curve_v line_n as_o be_v above_o show_v and_o who_o demonstration_n whole_o depend_v on_o a_o few_o easy_a proposition_n of_o the_o first_o book_n of_o euclid_n i_o see_v not_o why_o it_o may_v not_o be_v call_v plain_a geometry_n and_o as_o the_o word_n mechanical_a be_v use_v to_o signify_v a_o thing_n not_o mathematical_o exact_v but_o come_v near_o to_o it_o by_o approximation_n in_o this_o sense_n it_o be_v not_o mechanical_a but_o mathematical_a and_o pure_o geometrical_a be_v ground_v on_o as_o good_a demonstration_n as_o any_o proposition_n in_o euclid_n and_o be_v but_o a_o corrolary_n from_o some_o of_o they_o the_o next_o thing_n to_o be_v show_v be_v the_o quinquisection_n where_o to_o make_v one_o figure_n serve_v to_o both_o i_o make_v the_o cross_a rule_n only_o one_o inch_n and_o one_o four_o part_n from_o the_o middle_a line_n be_o set_v off_o on_o both_o side_n one_o half_a of_o the_o length_n of_o the_o cross_a rule_n draw_v the_o parallel_n lines_n ad_fw-la and_o be_v then_o let_v the_o cross_a rule_n side_n along_o the_o line_n ab_fw-la as_o in_o the_o trisection_n while_o the_o left_a side_n of_o the_o long_a rule_n slide_v through_o the_o centre_n a_o the_o other_o end_n of_o the_o cross_a rule_n shall_v describe_v a_o curve_v a_o part_n of_o which_o shall_v be_v g_o h_o that_o may_v be_v continue_v at_o pleasure_n again_o set_v the_o right_a end_n of_o the_o cross_a rule_n one_o the_o point_n d_o let_v it_o slide_v or_o move_v along_o the_o line_n da_fw-mi
while_o the_o left_a side_n of_o the_o long_a rule_n run_v through_o the_o centre_n a_o the_o left_a end_n of_o the_o cross_a rule_n shall_v describe_v a_o part_n of_o another_o curve_v meet_v at_o h_o the_o other_o curve_v and_o have_v find_v the_o point_n h_o with_o radius_fw-la a_o h_o describe_v the_o arch_n v_o h_o z_o x_o y_o n_o which_o shall_v give_v five_o h_o =_o one_o five_o of_o the_o whole_a arch_n as_o be_v evident_a from_o the_o forego_n demonstration_n the_o quinquisection_n also_o may_v be_v make_v without_o any_o curve_v if_o two_o long_a rule_n be_v joint_v together_o like_o a_o sector_n and_o each_o have_v a_o movable_a cross_a rule_n to_o move_v on_o they_o at_o right_a angle_n with_o the_o long_a rule_n for_o let_v the_o centre_n of_o the_o two_o long_a rule_n be_v fix_v on_o the_o centre_n a_o and_o let_v the_o 2_o cross_a rule_n be_v move_v together_o from_o b_o and_o d_o until_o the_o left_a end_n of_o the_o one_o still_o touch_v the_o right_a end_n of_o the_o other_o the_o right_a end_n of_o the_o cross_n near_o to_o the_o line_n a_o d_o touch_n upon_o some_o point_n of_o it_o as_o at_o w_n the_o point_n at_o w_n shall_v give_v the_o quinquisection_n as_o above_o and_o thus_o a_o true_a geometrical_a line_n of_o cord_n may_v be_v make_v by_o any_o tiro_n without_o any_o conic_a section_n or_o algebra_n equation_n and_o without_o any_o table_n of_o natural_a sin_n or_o arithmetical_a operation_n for_o whereas_o euclid_n 11.4_o have_v teach_v how_o to_o find_v the_o cord_n of_o 36_o degr_n and_o also_o it_o be_v find_v by_o quinquisect_v the_o half_a circle_n as_o be_v above_o show_v it_o remain_v only_o to_o trisect_v the_o arch_n of_o 120_o degr_n which_o give_v the_o cord_n of_o 40_o and_o 36_o take_v from_o 40_o leave_v 4_o degr_n which_o bisect_v give_v two_o and_o that_o bisected_a give_v 1_o which_o be_v the_o one_o 360th_o part_n of_o the_o circle_n and_o one_o 90th_o of_o the_o quadrant_n and_o this_o be_v more_o methodical_a than_o to_o teach_v a_o beginner_n to_o make_v his_o line_n of_o cord_n for_o project_v of_o angle_n by_o send_v he_o to_o conic_a section_n and_o algebra_n equation_n or_o the_o table_n of_o natural_a sin_n which_o he_o be_v not_o capable_a at_o his_o entry_n nor_o after_o he_o have_v make_v some_o good_a progress_n to_o understand_v it_o be_v to_o teach_v ignotum_fw-la per_fw-la ignotius_fw-la a_o unknown_a thing_n by_o a_o more_o unknown_a quite_o contrary_a to_o all_o good_a method_n of_o true_a science_n such_o as_o geometry_n be_v the_o method_n of_o the_o quinquisection_n here_o deliver_v sufficient_o show_v without_o example_n any_o other_o section_n desire_v the_o corollary_n mention_v in_o the_o former_a treatise_n with_o the_o rectification_n here_o make_v be_v all_o valid_a some_o of_o the_o chief_a of_o they_o i_o shall_v here_o mention_v 1._o one_o great_a use_n be_v to_o teach_v a_o beginner_n how_o to_o make_v a_o true_a line_n of_o cord_n as_o be_v above_o show_v and_o how_o to_o divide_v a_o circle_n into_o any_o part_n require_v 2._o another_o great_a use_n descartes_n show_v in_o his_o three_o book_n of_o geometry_n for_o the_o resolve_v any_o such_o equation_n in_o algebra_n as_o z_o 3_o =_o +_o p_o zq_fw-fr where_o the_o root_n z_o be_v a_o unknown_a quantity_n and_o can_v be_v find_v by_o the_o trisection_n of_o a_o angle_n 3._o a_o three_o great_a use_n be_v to_o give_v some_o new_a promblem_n in_o practical_a geometry_n one_o whereof_o i_o shall_v here_o show_v let_v a_o straight_a line_n a_o f_o be_v give_v see_v the_o second_o figure_n and_o it_o be_v require_v on_o the_o point_n a_o to_o erect_v a_o isosceles_a abc_n who_o side_n bc_n produce_v shall_v terminate_v on_o a_o limit_a point_n d_o under_o the_o give_v straight_a line_n af._n the_o construction_n be_v thus_o draw_v a_o straight_a line_n from_o d_o to_o a_o as_o d_o a_o next_o make_v the_o right_a angle_z fae_z divide_v the_o angle_n ead_n into_o three_o equal_a part_n and_o with_o radius_fw-la ad_fw-la describe_v the_o semicircle_n gfe_n from_o c_z to_z b_o set_v off_o gb_v =_o ed._n then_o draw_v the_o line_n ab_fw-la and_o from_o b_o draw_v the_o line_n bcd_n which_o shall_v form_v the_o isosceles_a triangle_n abc_n who_o side_n bc_n be_v produce_v shall_v terminate_v on_o d._n q.e.f._n the_o use_n of_o this_o be_v obvious_a in_o architecture_n 4._o a_o four_o great_a use_n be_v to_o give_v we_o some_o new_a problem_n in_o geography_n and_o navigation_n example_n there_o be_v four_o place_n a._n b_o c._n d_o so_o situate_v a_o be_v distant_a from_o d_o 100_o league_n and_o bear_v south-easter_o from_o it_o 70_o degr_n b_o be_v distant_a from_o a_o 100_o league_n south-wester_o c_o be_v distant_a from_o b_o 100_o league_n north-wester_o c_z and_o a_o be_v in_o the_o same_o latitude_n and_o so_o that_o these_o three_o place_n b._n c._n d_o lie_v in_o a_o straight_a line_n one_o from_o another_o q._n what_o be_v the_o distance_n betwixt_o these_o two_o place_n b_o and_o d_o and_o the_o course_n on_o the_o rhumb_n line_n betwixt_o a_o and_o b_o and_o the_o distance_n betwixt_o a_o and_o c._n the_o resolution_n divide_v the_o angle_n ead_n into_o three_o equal_a part_n and_o make_v cab_n =_o one_o three_o part_n of_o ead_n and_o draw_v the_o line_n bad_a thus_o the_o four_o place_n a._n b._n c._n d_o shall_v be_v due_o situate_v and_o a_o isosceles_a triangle_n shall_v be_v form_v abdella_n who_o side_n ab_fw-la =_o ad_fw-la =_o 100_o league_n and_o the_o angle_n bid_v =_o 86_o degr_n 40._o consequent_o by_o plain_a trigonometry_n the_o angle_n of_o the_o course_n gab_n be_v find_v which_o be_v 23_o degr_n 30_o min._n the_o angle_n abdella_n its_o double_a be_v 46_o degr_n 40_o =_o adb_fw-ge by_o the_o rule_n of_o opposite_n as_o the_o sine_fw-la of_o 46_o d._n 40_o to_o ad_fw-la 100_o league_n so_o the_o sine_fw-la of_o 86._o 40_o log._n 9.861757_o 2.000000_o 9.999265_o 11.999265_o 9.861757_o to_o bd_o 137._o ●_o fere_n 2.137508_o from_o which_o substracting_a bc_n 100_o league_n there_o remain_v cd_o 37._o ●_o as_o be_v require_v a_o five_o great_a use_n of_o the_o trisection_n and_o other_o section_n be_v have_v the_o ratio_fw-la of_o any_o 2_o angle_n give_v in_o any_o plain_a triangle_n to_o find_v the_o quantity_n of_o they_o if_o the_o quantity_n of_o the_o 3d_o angle_n be_v give_v without_o any_o regard_n to_o their_o side_n what_o other_o use_v these_o angular_a section_n may_v have_v be_v leave_v to_o the_o search_n of_o industrious_a artist_n london_n print_v for_o the_o author_n and_o be_v to_o be_v have_v at_o the_o three_o pigeon_n over_o against_o the_o exchange_n and_o at_o his_o house_n in_o pudding-lane_n at_o the_o sign_n of_o the_o golden_a ball_n where_o he_o teach_v the_o mathematical_a arts._n