Selected quad for the lemma: reason_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
reason_n	angle_n	equal_a	line_n	4,117	5	11.1250	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A47139	An essay for the discovery of some new geometrical problems (judged by some learned men, impracticable) concerning angular sections, beginning with the geometrical trisection of any right lined angle, by plain geometry of right lines and arches of circles, with rule and compass only, with out all conick sections, and cubick æquations. Whether the following praxis, and apparent demonstration thereof doth not only make it practicable, but easie to the understanding of a tiro, who but understands a little in true geometrical learning. Which layeth a foundation of a plain method how to sect any angle into any other number of parts required, even as 4. 6. 8. 10; or uneven, as 5. 7. 9. 11. &c. As also to divide a circle into any number even, or uneven of equal parts. All which have great uses in the improvement of the mathematical sciences, some of which are here specified. Proposed and submitted to the impartial tryal and examination of the right reason of such artises, to whose hands it may come. By G.K.	Keith, George, 1639?-1716.	1697 (1697)	Wing K160; ESTC R221663	9,043	17

a_o essay_n for_o the_o discovery_n of_o some_o new_a geometrical_a problem_n judge_v by_o some_o learned_a man_n impracticable_a concern_v angular_a section_n begin_v with_o the_o geometrical_a trisection_n of_o any_o right_a line_a angle_n by_o plain_a geometry_n of_o right_a line_n and_o arch_n of_o circle_n with_o rule_n and_o compass_n only_o without_o all_o conic_a section_n and_o cubick_a aequation_n whether_o the_o follow_a praxis_fw-la and_o apparent_a demonstration_n thereof_o do_v not_o only_o make_v it_o practicable_a but_o easy_a to_o the_o understanding_n of_o a_o tiro_n who_o but_o understand_v a_o little_a in_o true_a geometrical_a learning_n which_o lay_v a_o foundation_n of_o a_o plain_a method_n how_o to_o sect_n any_o angle_n into_o any_o other_o number_n of_o part_n require_v even_o as_o 4._o 6._o 8._o 10_o or_o vneven_a as_o 5._o 7._o 9_o 11._o etc._n etc._n as_o also_o to_o divide_v a_o circle_n into_o any_o number_n even_o or_o vneven_a of_o equal_a part_n all_o which_o have_v great_a uses_n in_o the_o improvement_n of_o the_o mathematical_a science_n some_o of_o which_o be_v here_o specify_v propose_v and_o submit_v to_o the_o impartial_a trial_n and_o examination_n of_o the_o right_a reason_n of_o such_o artist_n to_o who_o hand_n it_o may_v come_v by_o g._n k._n london_n print_v 1697._o and_o to_o be_v sell_v by_o the_o author_n at_o his_o house_n in_o pudding-lane_n at_o the_o sign_n of_o he_o golden-ball_n near_o the_o monument_n and_o by_o b._n aylmer_n at_o the_o three_o pigeon_n over_o against_o the_o royal-exchange_n some_o new_a geometrical_a problem_n etc._n etc._n although_o the_o trisection_n of_o a_o right_n line_v angle_n and_o also_o its_o section_n into_o any_o part_n require_v to_o a_o true_a mathematical_a exactness_n be_v deny_v to_o be_v practicable_a by_o plain_a geometry_n of_o right_a line_n and_o arch_n of_o circle_n with_o rule_n and_o compass_n only_o without_o all_o conic_a section_n and_o algebra_n equation_n by_o some_o learned_a artist_n yet_o other_o as_o learned_a be_v not_o so_o positive_a but_o acknowledge_v the_o practice_n of_o it_o be_v not_o as_o yet_o discover_v among_o who_o be_v the_o learned_a be_v barrow_n who_o have_v write_v thus_o in_o corol._n ad_fw-la 9_o 1._o elem_fw-la eucl._n methodus_fw-la vero_fw-la regula_fw-la &_o circino_n angulos_fw-la secandi_fw-la in_o aequales_fw-la quotcunque_fw-la hactenus_fw-la geometras_fw-la latuit_fw-la and_o in_o schol._n ad_fw-la 16._o 4._o elem._n coeterum_fw-la divisio_fw-la circumferentiae_fw-la in_o part_n datas_fw-la etiamnum_fw-la desideratur_fw-la the_o praxis_fw-la of_o the_o trisection_n of_o any_o give_a angle_n 1._o divide_v the_o cord_n bc_n into_o 3_o equal_a part_n as_o be_v =_o i_fw-mi =_o aec_fw-la by_o 10._o 6._o elem_fw-la euclid_n and_o draw_v the_o 2_o perpendicular_o ef._n ef._n 2._o with_o the_o extent_n of_o â…“_n of_o the_o cord_n as_o aec_fw-la measure_n on_o the_o arch_n from_o c_o to_o g_z and_o draw_v the_o line_n or_o cord_n gc_n =_o aec_fw-la 3._o with_o another_o radius_fw-la less_o than_o ac_fw-la viz._n ak_v draw_v a_o second_o arch_n as_o kl_n until_o it_o cut_v the_o perpendicular_a of_o and_o let_v the_o radius_fw-la ak_v be_v so_o long_o that_o the_o extent_n of_o the_o cord_n betwixt_o k_o and_o l_o be_n somewhat_o long_o than_o aec_fw-la and_o with_o the_o same_o extent_n aec_fw-la measure_n on_o this_o second_o arch_n kr_n =_o aec_fw-la 4._o from_o g_z to_o r_o draw_v a_o straight_a line_n by_o post_n 1._o 1._o elem_fw-la and_o produce_v the_o same_o by_o post_n 2._o until_o it_o cut_v the_o perpendicular_o at_o i_o and_o o._n 5._o draw_v the_o straight_a line_n a_z and_o ao_o and_o extend_v they_o to_o h_n on_o both_o side_n so_o that_o the_o one_o line_n shall_v be_v aih_o and_o the_o other_o aoh_n which_o two_o line_n shall_v sect_n the_o give_v angle_n bac_n into_o three_o part_n or_o angle_n viz._n bah_o hah_o hac_fw-la the_o question_n or_o problem_n to_o be_v resolve_v be_v whether_o these_o three_o angle_n be_v not_o equal_a and_o consequent_o that_o the_o angle_n bac_n be_v trisect_v into_o equal_a angle_n the_o apparent_a demonstration_n that_o be_v here_o propose_v to_o prove_v it_o follow_v in_o order_n to_o the_o apparent_a follow_a demonstration_n by_o way_n of_o preparation_n 1._o draw_v the_o line_n oi_o which_o shall_v be_v parallel_n to_o i_fw-mi by_o 28._o 1._o elem_fw-la and_o extend_v it_o on_o both_o side_n from_o i_o to_o n_o on_o the_o one_o side_n and_o from_o o_o to_o n_o on_o the_o other_o make_v in_o =_o on_o =_o aec_fw-la =_o oi_o 2._o with_o radius_fw-la in_o and_o on_o describe_v the_o arch_n on_o both_o side_n nm_n until_o these_o arch_n cut_v the_o line_n ac_fw-la and_o ab_fw-la so_o shall_v the_o line_n or_o cord_n draw_v betwixt_o i_o and_o m_o and_o o_o and_o m_o viz._n in_o and_o om_fw-la =_o aec_fw-la 3._o make_v the_o angle_n imp_v =_o angle_v min_n by_o 23._o 1._o elem_fw-la and_o omp_n =_o mon_fw-fr and_o extend_v the_o line_n mp_n until_o it_o cut_v the_o line_n ai_fw-fr at_o p_o and_o ao_o at_z p._n 4._o draw_v the_o line_n open_v parallel_n to_o in_o until_o it_o cut_v mp_n whence_o the_o perfect_a rhombus_fw-la ompi_fw-la shall_v be_v form_v have_v the_o opposite_a side_n parallel_n and_o equal_a for_o pm_n by_o construction_n be_v parallel_n with_o oi_o and_o open_v with_o in_o and_o that_o the_o line_n open_v draw_v parallel_n to_o in_o can_v cut_v the_o line_n pm_n no_o where_o but_o at_o p_o on_o the_o line_n ai_fw-fr be_v prove_v thus_o suppose_v it_o cut_v the_o line_n pm_n any_o where_o else_o than_o on_o the_o line_n ai_fw-fr on_o either_o side_n it_o shall_v make_v pm_v either_o long_o or_o short_a than_o its_o opposite_a side_n for_o the_o arch_n draw_v by_o the_o radius_fw-la mi_fw-mi and_o cut_v the_o line_n ai_fw-fr at_o p_o prove_v mp_n =_o mi_fw-mi by_o def_n 15._o 1._o elem_fw-la as_o also_o open_a shall_v be_v long_o or_o short_a than_o in_o as_o the_o like_a arch_n conceive_v to_o be_v draw_v by_o radius_fw-la oi_o and_o cut_v a_z at_o p_o prove_v oi_o =_o op_n by_o the_o same_o def_n 15._o 1._o elem_fw-la but_o both_o these_o consequence_n be_v absurd_a make_v the_o opposite_a side_n of_o a_o paralellogram_n to_o be_v unequal_a contrary_a to_o 34._o 1._o elem_fw-la because_o the_o figure_n opmi_n be_v prove_v to_o be_v a_o perfect_a rhombus_fw-la have_v all_o the_o side_n equal_a whereof_o the_o line_n pi_n be_v the_o diagonal_a it_o be_v prove_v that_o open_a and_o pm_n terminate_v on_o a_z therefore_o the_o angle_n oip_n =_o oia_n =_o pim_n =_o aim_n by_o 8._o and_o 4._o 1._o elem_fw-la last_o these_o two_o triangle_n oai_n and_o jam_fw-la shall_v be_v according_a to_o the_o four_o prop._n 1._o elem_fw-la and_o so_o shall_v oai_n and_o oam_n for_o oi_o =_o in_o as_o above_o prove_v and_o ai_fw-fr be_v common_a to_o both_o and_o the_o angle_n oia_n =_o aim_v as_o be_v above_o prove_v therefore_o by_o 4._o 1._o elem_fw-la be_o =_o ao_o =_o ai_fw-fr and_o consequent_o by_o 8._o 1._o the_o angle_n jam_fw-la =_o oai_fw-gr =_o oam_v therefore_o the_o give_v angle_n bac_n be_v geometrical_o trisect_v by_o the_o line_n ao_o and_o ai_fw-fr extend_v to_o h._n q._n e._n d._n here_o note_v let_v the_o radius_fw-la be_v ever_o so_o much_o change_v betwixt_o m_n and_o c_o take_v it_o at_o any_o extent_n from_o a_o and_o let_v ever_o so_o many_o concentric_a arch_n be_v draw_v from_o the_o centre_n a_o betwixt_o in_o and_o gc_n their_o cord_n terminate_n on_o the_o straight_a line_n mc_n on_o the_o one_o side_n and_o ig_a on_o the_o other_o shall_v all_o be_v equal_a one_o to_o another_o and_o to_o the_o cord_n gc_fw-mi =_o abc_n as_o the_o demonstration_n above_o give_v prove_v for_o the_o reason_n that_o prove_v the_o cord_n of_o any_o 3_o concentric_a arch_n terminate_n on_o two_o straight_a line_n to_o be_v equal_a within_o the_o limit_n mi_fw-mi and_o gc_n be_v a_o trapezia_n prove_v any_o other_o 3_o to_o be_v equal_a within_o the_o same_o limit_n but_o if_o we_o draw_v any_o concentric_a arch_n without_o the_o trapezia_n igcm_n as_o with_o a_o less_o radius_fw-la than_o be_o or_o with_o a_o great_a radius_fw-la than_o ac_fw-la the_o case_n be_v alter_v and_o the_o equal_a cord_n will_v not_o terminate_v on_o the_o straight_a line_n ig_n but_o diviate_v or_o depart_v therefrom_o as_o both_o true_a reason_n do_v prove_v and_o even_o ocular_a inspection_n do_v show_v for_o though_o the_o eye_n be_v not_o able_a to_o judge_v of_o a_o straight_a line_n yet_o when_o a_o line_n be_v visible_o apparent_a to_o make_v a_o angle_n with_o another_o line_n these_o two_o line_n can_v be_v a_o straight_a line_n i_o call_v the_o figure_n imcg_n a_o trapezia_n for_o it_o be_v not_o any_o paralellogram_n because_o the_o

bfc_n and_o ao_o to_o x._o again_o with_o the_o extent_n ai_fw-fr on_o the_o centre_n i_o describe_v the_o circle_n asht._n and_o with_o the_o same_o radius_fw-la describe_v the_o arch_n klon_n again_o draw_v ih_o parallel_n to_o ak_v extend_v ih_v to_o the_o circumference_n of_o the_o circle_n asht._n again_o draw_v as_o parallel_n and_o equal_a to_o oi_o which_o shall_v cut_v the_o circle_n at_o saint_n because_o oi_o =_o as_o cut_v the_o arch_a kion_n have_v the_o same_o radius_fw-la again_o from_o the_o point_n s_o on_o the_o circumference_n draw_v the_o line_n shall_z parallel_v to_o ai_fw-fr which_o shall_v necessary_o cut_v the_o line_n ih_v on_o the_o circumference_n at_o the_o point_n h_n because_o by_o construction_n ih_o be_v parallel_n to_o ak_v and_o kh_n to_o ai_fw-fr which_o make_v the_o parallogram_n akhi_n who_o opposite_a side_n be_v equal_a but_o if_o shall_z and_o ih_v do_v meet_v any_o where_o else_o than_o on_o the_o circumference_n as_o either_o within_o or_o without_o it_o the_o opposite_a side_n of_o the_o parallogram_n shall_v not_o be_v equal_a contrary_n to_o 34._o 1._o eucl._n elem_fw-la again_o draw_v the_o line_n si_fw-it and_o extend_v the_o line_n oi_o from_o i_o to_o k_o until_o it_o cut_v shall_z at_o k_o which_o shall_v form_v another_o parallogram_n aski_n which_o be_v divide_v by_o the_o diagonal_a line_n si_fw-it shall_v give_v two_o equal_a isoscey_n triangle_n ais_fw-mi =_o oai_fw-gr isk._n the_o demonstration_n in_o the_o isoceles_a triangle_n ish_n who_o side_n be_v =_o ih_o by_o 15._o def_n 1._o elem_fw-la the_o angle_n ih_v =_o ish_n by_o 5._o 1._o elem_fw-la and_o ih_n =_o jak_n by_o 8._o 1._o elem_fw-la therefore_o jak_a =_o be_v but_o be_v =_o ais_fw-mi =_o oai_fw-gr be_v alternate_a angle_n betwixt_o parallel_n by_o 29._o 1._o elem_fw-la and_o therefore_o lasty_a oai_fw-fr =_o jak_a and_o by_o the_o like_a method_n oai_n be_v prove_v =_o oan_a draw_v the_o like_a parallel_n line_n on_o the_o other_o side_n of_o the_o triangle_n oai_n and_o therefore_o the_o angle_n bac_n be_v trisect_v by_o 3_o equal_a angle_n nao_n =_o oai_fw-gr =_o jak_a =_o bax_n =_o xad_v =_o dac_n q._n e._n d._n the_o praxis_fw-la of_o the_o section_n of_o a_o right_o line_v angle_n into_o six_o equal_a part_n because_o of_o the_o affinity_n of_o the_o figure_n of_o the_o trisection_n with_o the_o figure_n of_o the_o sextisection_n i_o shall_v begin_v with_o the_o sextisection_n and_o then_o proceed_v to_o the_o quinquisection_n in_o the_o 3d_o figure_n let_v the_o give_v angle_n be_v bac_n as_o above_o who_o cord_n be_v bc_n and_o its_o arch_a befgc_n 1._o divide_v the_o cord_n bc_n into_o 6_o equal_a part_n which_o division_n be_v mark_v with_o *****_o and_o on_o the_o middle_a division_n draw_v the_o line_n adi_n to_o the_o arch_a befgc_n 2._o with_o the_o extent_n of_o one_o of_o the_o division_n on_o the_o give_v arch_a measure_n from_o c_o to_o f_o and_o with_o the_o same_o extent_n from_o f_o to_o e_z and_o from_o e_z to_z g_z mark_v the_o point_n e._n f._n g._n and_o with_o the_o same_o extent_n measure_n from_o b_o to_z e_z from_o e_z to_o f_o and_o from_o f_o to_o g_z mark_v the_o point_n efg_v 3._o with_o a_o less_o radius_fw-la than_o ac_fw-la as_o ak_v draw_v a_o second_o arch_n as_o klhlk_v which_o radius_fw-la must_v be_v so_o long_o that_o the_o extent_n of_o the_o 1_o 6_o of_o the_o cord_n thrice_o repeat_v may_v fall_v short_a of_o reach_v the_o line_n adi_n 4._o with_o the_o same_o extent_n on_o this_o 2d_o arch_n measure_n from_o ac_fw-la and_o from_o ab_fw-la to_o k_o and_o from_o k_o to_o l_o and_o from_o l_o to_o h_n by_o a_o threefold_a repartition_n of_o the_o same_o extent_n 5._o from_o the_o point_n g_o on_o the_o first_o arch_n to_o the_o point_n h_n on_o the_o second_o arch_n draw_v a_o straight_a line_n and_o produce_v it_o until_o it_o cut_v the_o line_n adi_n at_o the_o point_n 1._o 6._o with_o the_o radius_fw-la ai_fw-fr draw_v a_o 3d._n concentric_a arch_n betwixt_o the_o line_n ab_fw-la and_o ac_fw-la i_o say_v the_o extent_n of_o the_o 16_o part_n of_o the_o cord_n bc_n =_o mc_n be_v 6_o time_n repeat_v shall_v cut_v the_o say_v 3d_o arch_n into_o six_o equal_a part_n to_o every_o one_o of_o which_o straight_a line_n draw_v from_o the_o centre_n a_o and_o produce_v to_o the_o outmost_a arch_n shall_v divide_v the_o give_v angle_n bac_n into_o six_o equal_a part_n orangle_v mark_v with_o 1._o 2._o 3._o 4._o 5._o 6._o in_o order_n to_o the_o demonstration_n of_o the_o praxis_fw-la of_o this_o section_n of_o a_o angle_n into_o 6_o equal_a part_n on_o the_o point_n 5_o make_v it_o a_o centre_n with_o radius_fw-la 5a_fw-la describe_v the_o circle_n aopqr_n 2._o from_o the_o point_n 5_o draw_v the_o line_n 5r_fw-fr parallel_n to_o ac_fw-la 3._o from_o r_o draw_v the_o line_n rt_a parallel_n to_o 5a_fw-la 4._o draw_v the_o line_n at_o from_o a_o to_o t_o 5._o draw_v the_o line_n t5_n and_o draw_v 5v_a parallel_n to_o at_o the_o demonstration_n in_o the_o triangle_n t5r_n be_v a_o isosceles_a the_o angle_n 5tr_n =_o 5rt_v by_o 5._o 1._o elem_fw-la and_o 5rt_v =_o 5r6_n 5a6_n therefore_o 5t6_n =_o 5a6_n again_o 5_o t6_n =_o 5tv_n a5t_v and_o a5t_v =_o 5a4_n by_o 34.1_o and_o therefore_o last_o 5_o a4_n =_o 5_o au._n and_o as_o these_o 2_o angle_n be_v prove_v equal_a by_o the_o like_a reason_n all_o the_o other_o angle_n can_v be_v prove_v equal_a by_o change_v the_o centre_n of_o the_o circle_n from_o 5_o to_o 4_o and_o from_o 4_o to_z 3_o and_o from_o 3_o to_o 2_o and_o from_o 2_o to_o 1._o this_o demonstration_n have_v such_o affinity_n with_o the_o former_a of_o trisection_n and_o which_o can_v as_o easy_o be_v demonstrate_v both_o way_n as_o that_o of_o trisection_n i_o shall_v not_o enlarge_v upon_o it_o for_o he_o who_o understand_v and_o assent_v to_o the_o verity_n of_o the_o former_a by_o the_o same_o evidence_n will_v assent_v to_o the_o late_a the_o praxis_fw-la of_o the_o quinquisection_n 1._o according_a to_o the_o former_a method_n divide_v the_o cord_n of_o the_o give_v angle_n be_v into_o 5_o =_o part_n 2._o on_o the_o middle_a part_n mark_v with_o cd_o draw_v the_o line_n ch_n and_o dh_n cut_v the_o cord_n at_o right_a angle_n and_o parallel_v one_o to_o another_o 3._o on_o the_o arch_n of_o the_o give_v angle_n bfg_fw-mi take_v the_o â…•_n of_o the_o cord_n =_o cd_o and_o set_v it_o from_o e_o to_o f_o and_o from_o f_o to_o g_z also_o do_v the_o like_a from_o b_o to_o f_o and_o from_o f_o to_o g._n 4._o draw_v a_o second_o arch_n with_o a_o less_o radius_fw-la as_o ak_v so_o as_o the_o radius_fw-la may_v be_v so_o long_o that_o the_o extent_n of_o the_o â…•_n of_o the_o cord_n twice_o take_v on_o that_o second_o arch_n may_v not_o reach_v to_o the_o perpendicular_a dh_n 5._o with_o the_o same_o extent_n twice_o take_v as_o from_o k_o to_o *_o from_o *_o to_o *_o measure_n on_o the_o second_o arch_n to_o the_o second_o *_o 6._o from_o g_z to_o *_o draw_v a_o straight_a line_n until_o it_o cut_v the_o perpendicular_a line_n dh_n at_o h_n last_o with_o the_o radius_fw-la ah_o describe_v the_o three_o arch_n ihhi_n and_o draw_v the_o line_n ah_o ah_o which_o shall_v make_v the_o angle_n hah_o =_o â…•bae_fw-la the_o demonstration_n of_o this_o or_o any_o other_o section_n be_v after_o the_o same_o method_n with_o the_o former_a as_o also_o the_o praxis_fw-la it_o be_v superfluous_a to_o enlarge_v upon_o it_o or_o to_o add_v any_o new_a problem_n show_v how_o to_o sect_n any_o angle_n into_o any_o other_o part_v give_v as_o 7._o 8._o 9_o 10._o 11._o etc._n etc._n for_o he_o who_o understand_v by_o the_o forego_n method_n and_o praxis_fw-la to_o sect_n any_o angle_n into_o 3._o 5._o 6._o as_o be_v above_o show_v will_v by_o the_o like_a method_n and_o praxis_fw-la be_v able_a to_o sect_n any_o angle_n into_o 7._o 8._o 9_o 10._o etc._n etc._n equal_a part_n and_o also_o to_o demonstrate_v the_o same_o wherefore_o in_o the_o next_o place_n i_o shall_v proceed_v to_o show_v how_o a_o semicircle_n may_v be_v sect_v into_o any_o number_n of_o equal_a part_n even_o as_o 4._o 6._o 8._o 10_o etc._n etc._n or_o uneven_a as_o 5._o 7._o 9_o 11._o the_o praxis_fw-la of_o a_o section_n of_o a_o semicircle_n into_o 9_o equal_a part_n let_v the_o semicircle_n give_v be_v brsc_n who_o diameter_n be_v bc._n and_o who_o centre_n be_v a._n 1._o divide_v the_o diameter_n bc_n into_o 9_o equal_a part_n 2._o from_o the_o centre_n a_o set_a off_o ao_o and_o a_o make_v no_o =_o 1_o 9_o of_o the_o diameter_n and_o draw_v the_o perpendicular_o nr_n os_fw-la 3._o with_o the_o extent_n no_o