Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	draw_v	line_n	perpendicular_a	3,095	5	14.0786	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A60107	A Short treatise of the description of the sector wherein is also shown the great use of that excellent instrument, in the solution of several mathematical problems.	Worgan, John, surveyor.	1697 (1697)	Wing S3636; ESTC R41684	18,763	47

short_a treatise_n of_o the_o description_n of_o the_o sector_n wherein_o be_v also_o show_v the_o great_a use_n of_o that_o excellent_a instrument_n in_o the_o solution_n of_o several_a mathematical_a problem_n london_n print_v for_o and_o sell_v by_o john_n worgan_n mathematical_a instrument-maker_n at_o his_o shop_n under_o st._n dunstan_n church_n fleetstreet_n 1697._o where_o all_o sort_n of_o mathematic_a instrument_n both_o for_o sea_n and_o land_n be_v make_v and_o sold._n to_o the_o reader_n reader_n thou_o be_v here_o present_v with_o a_o small_a treatise_n of_o the_o use_n of_o the_o sector_n in_o which_o i_o have_v explain_v the_o true_a nature_n and_o property_n of_o the_o construction_n of_o this_o admirable_a instrument_n and_o have_v geometrical_o demonstrate_v the_o ground_n upon_o which_o all_o the_o operation_n depend_v i_o have_v also_o show_v the_o amendment_n that_o have_v be_v late_o make_v in_o dispose_v the_o line_n hereon_o place_v and_o the_o advantage_n accrue_v thereby_o in_o the_o next_o place_n i_o have_v exhibit_v its_o general_a use_n and_o declare_v wherein_o the_o great_a benefit_n of_o this_o noble_a contrivance_n lie_v and_o the_o universal_a usefulness_n of_o the_o same_o last_o i_o have_v a_o little_a illustrated_n its_o excellency_n in_o the_o solution_n of_o some_o few_o mathematic_a problem_n particular_o in_o arithmetic_n geometry_n and_o trigonometry_n which_o indeed_o be_v the_o foundation_n of_o all_o the_o rest_n and_o sufficient_a for_o the_o young_a tyro_n light_n to_o the_o solution_n of_o many_o more_o for_o to_o show_v 〈◊〉_d the_o 〈◊〉_d of_o 〈◊〉_d ●●●●●ment_n be_v little_o less_o than_o to_o write_v a_o body_n of_o the_o mathematics_n which_o be_v not_o my_o design_n at_o present_a my_o principal_a intent_n here_o be_v only_o to_o show_v how_o this_o instrument_n supply_v the_o place_n of_o almost_o all_o other_o particular_o of_o all_o kind_n of_o scale_n and_o rule_n which_o be_v make_v but_o to_o assign_a or_o particular_a radiuse_n which_o be_v the_o main_a end_n and_o sole_a design_n of_o this_o instrument_n if_o herein_o the_o public_a good_a be_v any_o way_n advance_v i_o shall_v be_v very_o glad_a of_o it_o vale._n john_n worgan_n londini_fw-la fecit_fw-la chap._n i._n a_o description_n of_o chord_n sin_n tangent_n etc._n etc._n because_o in_fw-la the_o measure_n of_o part_n of_o a_o circle_n which_o be_v often_o require_v there_o be_v no_o other_o way_n but_o by_o reduce_v they_o to_o strait_a line_n therefore_o do_v the_o ancient_n apply_v certain_a strait_a line_n to_o a_o circle_n which_o come_v in_o competition_n to_o arch-line_n and_o that_o several_a way_n viz._n as_o they_o be_v draw_v within_o a_o circle_n through_o it_o or_o without_o a_o circle_n line_n within_o a_o circle_n be_v chord_n and_o sin_n a_o chord_n of_o a_o arch_n be_v a_o strait_a line_n draw_v from_o one_o end_n of_o a_o arch_n to_o the_o other_o as_o in_o fig._n 1._o the_o line_n a_o b_o be_v the_o chord_n of_o the_o arch_n a_o c_o b._n it_o be_v also_o the_o chord_n of_o the_o arch_n a_o d_o b_o for_o it_o be_v common_a to_o both_o part_n of_o the_o circle_n so_o that_o from_o 15_o prop._n of_o the_o 3_o d_o book_n of_o euclid_n it_o be_v evident_a the_o great_a chord_n that_o can_v be_v draw_v be_v the_o diameter_n of_o the_o circle_n or_o chord_n of_o 180_o deg._n or_o half_a the_o circle_n and_o consequent_o that_o all_o chord_n of_o arch_n great_a than_o a_o semicircle_n be_v less_o than_o the_o diameter_n sin_n be_v either_o right_a or_o verse_v a_o right_a sine_fw-la of_o a_o arch_n be_v a_o line_n draw_v from_o one_o end_n of_o that_o arch_a perpendicular_a to_o a_o diameter_n draw_v from_o the_o other_o end_n of_o that_o arch_n and_o belong_v to_o both_o part_n of_o a_o semicircle_n as_o in_o fig._n 1._o ebb_n be_v the_o right_a sine_fw-la of_o the_o arch_n c_o b_o and_o also_o of_o the_o arch_n b_o d._n and_o here_o you_o may_v observe_v from_o the_o aforecited_a proposition_n of_o euclid_n that_o the_o great_a sine_fw-la be_v that_o of_o 90_o deg._n or_o a_o quarter_n of_o a_o circle_n and_o therefore_o all_o sin_n of_o arch_n great_a than_o a_o quadrant_n be_v less_o than_o the_o sine_fw-la of_o 90_o deg._n which_o be_v half_a the_o diameter_n or_o that_o which_o in_o all_o proportion_n we_o call_v radius_fw-la the_o verse_v sine_fw-la of_o a_o arch_n be_v that_o part_n of_o the_o diameter_n betwixt_o the_o right_n sine_fw-la and_o the_o end_n of_o that_o arch_n it_o be_v the_o verse_v sine_fw-la off_o as_o in_o fig._n 1._o e_o c_o be_v the_o verse_v sine_fw-la of_o the_o arch_n c_o b_o and_o e_o d_o the_o verse_v sine_fw-la of_o the_o arch_n b_o d._n from_o hence_o it_o be_v also_o evident_a that_o the_o great_a verse_v sine_fw-la be_v that_o of_o 180_o deg._n line_n draw_v through_o a_o circle_n be_v secant_v a_o secant_fw-la be_v a_o strait_a line_n draw_v from_o the_o centre_n of_o a_o circle_n through_o one_o end_n of_o that_o arch_n which_o it_o be_v the_o secant_fw-la off_o till_o it_o meet_v with_o the_o tangent_fw-la which_o bound_n it_o as_o in_o fig._n 2._o ei_o in_o the_o secant_fw-la of_o the_o arch_n bh_n line_n draw_v without_o a_o circle_n be_v tangent_n a_o tangent_fw-la be_v a_o strait_a line_n that_o touch_v a_o circle_n and_o be_v erect_v perpendicular_a to_o a_o diameter_n draw_v from_o the_o touch-point_n be_v limit_v by_o the_o secant_fw-la which_o pass_v through_o the_o other_o end_n of_o the_o arch_n it_o be_v the_o tangent_fw-la off_o as_o in_o fig._n 2._o the_o strait_a line_n he_o be_v the_o tangent_fw-la of_o the_o arch_n b_o h._n hence_o it_o be_v evident_a there_o can_v be_v no_o secant_fw-la nor_o tangent_fw-la of_o 90_o degree_n for_o the_o secant_fw-la of_o 90_o degree_n be_v parallel_v to_o the_o tangent_fw-la line_n and_o therefore_o if_o infinite_o produce_v will_v never_o meet_v with_o it_o the_o half_a tangent_fw-la be_v only_a the_o tangent_fw-la of_o half_a the_o arch_n as_o the_o tangant_fw-la of_o 90_o degree_n be_v infinite_a but_o the_o half_n tangent_fw-la of_o 90_o deg._n be_v the_o tangent_fw-la of_o 45_o deg._n as_o in_o fig._n 2._o fe_o be_v the_o half_n tangent_fw-la of_o the_o arch_n bh_n which_o be_v but_o half_a the_o angle_n b_o a_o h_o for_o the_o angle_n at_o the_o centre_n be_v double_a to_o the_o angle_n at_o the_o circumference_n by_o 20_o prop._n of_o the_o three_o book_n of_o euclid_n chap._n ii_o how_o the_o sin_n of_o chord_n natural_a sines_n tangent_n secant_v etc._n etc._n be_v project_v and_o put_v on_o the_o common_a scale_n first_o with_o any_o distance_n describe_v the_o circle_n a_o b_o c_o f_o and_o draw_v the_o diameter_n a_o c_o which_o cross_n at_o right_a angle_n with_o the_o line_n d_o b_o produce_v in_o fig._n 3._o and_o draw_v a_o b_o then_o divide_v the_o arch_n a_o b_o into_o 9_o equal_a part_n set_v thereto_o the_o figure_n 10_o 20_o 30_o etc._n etc._n to_o 90_o and_o each_o of_o those_o 9_o division_n again_o into_o 10_o equal_a part_n one_o of_o which_o be_v a_o degree_n or_o the_o 360_o of_o the_o whole_a circle_n this_o do_v with_o one_o foot_n of_o the_o compass_n in_o a_o transfer_v these_o division_n of_o the_o arch_n to_o the_o strait_a line_n a_o b_o placing_z thereto_o the_o number_n 10_o 20_o etc._n etc._n as_o be_v set_v to_o the_o arch._n the_o line_n a_o b_o thus_o divide_v be_v a_o line_n of_o chord_n and_o may_v be_v transfer_v from_o the_o paper_n to_o a_o silver_n brass_n or_o box-scale_n second_o for_o the_o sin_n divide_v the_o arch_n b_o c_o as_o before_o viz._n into_o 90_o equal_a part_n note_v the_o 9_o grand_a division_n with_o figure_n as_o in_o the_o last_o from_o these_o point_n in_o the_o arch_n b_o c_o let_n fall_v perpendicular_o to_o the_o diameter_n a_o c_o these_o shall_v divide_v the_o semi-diameter_n d_o c_o into_o a_o line_n of_o right_a sin_n which_o may_v from_o hence_o be_v transfer_v to_o any_o scale_n or_o rule_n three_o for_o the_o verse_v sin_n which_o be_v only_o a_o double_a scale_n of_o right_a sin_n let_v fall_v perpendicular_o as_o before_o to_o divide_v the_o sin_n from_o every_o degree_n in_o the_o whole_a semicircle_n a_o b_o c_o these_o perpendicular_o will_v divide_v the_o diameter_n a_o c_o into_o a_o line_n of_o verse_v sin_n which_o be_v number_v according_a to_o the_o figure_n under_o the_o diameter_n but_o be_v seldom_o put_v on_o any_o scale_n except_o require_v four_o from_o c_o raise_z a_o perpendicular_a to_o a_o c_o and_o from_o the_o centre_n d_o through_o each_o division_n of_o the_o quadrant_n draw_v line_n produce_v they_o till_o they_o meet_v the_o line_n c_o e_o which_o line_n shall_v divide_v the_o say_a line_n into_o a_o scale_n of_o tangent_n to_o which_o