Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	arch_n	line_n	right_a	1,244	5	9.1480	5	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A52262	Geometrical trigonometry, or, The explanation of such geometrical problems as are most useful & necessary, either for the construction of the canons of triangles, or for the solution of them together with the proportions themselves suteable unto every case both in plain and spherical triangles ... / by J. Newton ...	Newton, John, 1622-1678.	1659 (1659)	Wing N1059; ESTC R17311	30,729	123

bcq_n by_o the_o 19_o hereof_o 22_o if_o a_o plain_a triangle_n be_v inscrihe_v in_o a_o circle_n the_o angle_n opposite_a to_o the_o circumference_n be_v half_a as_o much_o as_o that_o part_n of_o the_o circumference_n which_o be_v opposite_a to_o the_o angle_n demonst._n in_o the_o triangle_n ebb_o ang._n edb_o ebb_o by_o the_o second_o hereof_o and_o ang._n aeb_o equal_a to_o both_o by_o the_o 9th_o hereof_o the_o arch_n ab_o be_v the_o measure_n of_o the_o angle_n aeb_o by_o the_o 25_o of_o the_o first_o therefore_o the_o arch_n ab_o be_v the_o double_a measure_n of_o the_o angle_n adb_o as_o be_v to_o be_v prove_v 1_o consectary_n if_o the_o side_n of_o a_o plain_a triangle_n inscribe_v in_o a_o circle_n be_v the_o diameter_n the_o angle_n opposite_a to_o that_o side_n be_v a_o right_a angle_n as_o the_o angle_n abd_o opposite_a to_o the_o diameter_n ad_fw-la 2_o consectary_n if_o divers_a right_n line_a triangle_n be_v inscribe_v in_o the_o same_o segment_n of_o a_o circle_n upon_o one_o base_a the_o angle_n in_o the_o circumference_n be_v equal_a as_o the_o triang._n abd_a and_o acd_o be_v inscribe_v in_o the_o same_o segment_n of_o the_o circle_n abcd_o and_o upon_o the_o same_o base_a ad_fw-la have_v their_o angle_n at_o b_o and_o d_o fall_v in_o the_o circumference_n equal_a 23_o if_o a_o quadrilateral_a figure_n be_v inscribe_v in_o a_o circle_n the_o angle_n thereof_o which_o be_v opposite_a to_o one_o another_o be_v together_o equal_a to_o two_o right_a angle_n demonst._n ang._n cdb_o cab_o and_o bda_o bca_o by_o the_o last_o aforegoing_a therefore_o ang._n cda_o bca_o bac_o and_o abc_o bac_o bca_o 2_o r._n ang._n by_o the_o 9th_o hereof_o and_o therefore_o ang._n abc_o adc_o 2_o r._n ang._n as_o be_v to_o be_v prove_v 24_o if_o in_o a_o quadrilateral_a figure_n inscribe_v in_o a_o circle_n there_o be_v draw_v two_o diagonal_a line_n the_o rectangle_n under_o the_o diagonal_o be_v equal_a to_o the_o two_o rectangle_n under_o the_o opposite_a side_n demonst._n let_v ang._n de_fw-la cab_o by_o construction_n then_o shall_v ang._n dac_o eab_o and_o ang._n acd_o abe_o because_o the_o arch_a a_o d_o be_v the_o double_a measure_n to_o they_o both_o and_o therefore_o the_o triangle_n adc_o and_o aeb_o be_v like_a again_o ang._n adb_o acb_o because_o the_o arch_n ab_o be_v the_o double_a measure_n to_o they_o both_o and_o ang._n de_fw-la cab_o by_o construction_n and_o the_o triang._n aed_o and_o abc_o like_o therefore_o ac_o cb_o ∷_o ad_fw-la de_o and_o ac_o cd_o ∷_o ab_o be._n therefore_o ac_o ×_o de_o cb_o ×_o ad_fw-la and_o also_o ac_o ×_o be_v cd_o ×_o ab_o and_o ac_o ×_o de_o ac_o ×_o be_v ac_o ×_o db._n therefore_o ac_o ×_o db_o cb_o ×_o ad_fw-la cd_o ×_o ab_o as_o be_v to_o be_v prove_v chap._n iii_o of_o the_o construction_n of_o the_o canon_n of_o triangle_n that_o the_o proportion_n which_o the_o part_n of_o a_o triangle_n have_v one_o to_o another_o may_v be_v certain_a the_o arch_n of_o circle_n by_o which_o the_o angle_n of_o all_o triangle_n and_o of_o spherical_a triangle_n the_o side_n be_v also_o measure_v must_v be_v first_o reduce_v into_o right_a line_n by_o define_v the_o quantity_n of_o right_a line_n as_o they_o be_v apply_v to_o the_o arch_n of_o a_o circle_n 2_o right_a line_n be_v apply_v to_o the_o arch_n of_o a_o circle_n three_o way_n viz._n either_o as_o they_o be_v draw_v within_o the_o circle_n without_o the_o circle_n or_o as_o they_o be_v draw_v through_o it_o 3_o right_a line_n within_o the_o circle_n be_v chord_n and_o sine_n 4_o a_o chord_n or_o subtense_n be_v a_o right_a line_n inscribe_v in_o a_o ci●cle_n divide_v the_o whole_a circle_n into_o two_o segment_n and_o in_o like_o manner_n subtend_v both_o the_o segment_n as_o the_o right_a line_n ck_o divide_v the_o circle_n gedk_o into_o the_o two_o segment_n cegk_n and_o cdk_o and_o subtend_v both_o the_o segment_n that_o be_v the_o right_a line_n ck_o be_v the_o chord_n of_o the_o arch_n cgk_o and_o also_o the_o chord_n of_o the_o arch_a cdk_n 5_o a_o sine_fw-la be_v a_o right_a line_n in_o a_o semicircle_n fall_v perpendicular_a from_o the_o term_n of_o a_o arch_n 6_o a_o sine_fw-la be_v either_o right_a or_o verse_v 7_o a_o right_a sine_fw-la be_v a_o right_a line_n in_o a_o semicircle_n which_o from_o the_o term_n of_o a_o arch_n be_v perpendicular_a to_o the_o diameter_n divide_v the_o semicircle_n into_o two_o segment_n and_o in_o like_o manner_n refer_v to_o both_o thus_o the_o right_a line_n ca_o be_v the_o sine_fw-la of_o the_o arch_n cd_o less_o than_o a_o quadrant_a and_o also_o the_o sine_fw-la of_o the_o arch_n ceg_o great_a than_o a_o quadrant_a and_o hence_o instead_o of_o the_o obtuse_a angle_n gbc_o we_o take_v the_o acute_a angle_n cba_o the_o compliment_n thereof_o to_o a_o semicircle_n and_o so_o our_o canon_n of_o triangle_n do_v never_o exceed_v 90_o deg._n 8_o a_o right_a sine_fw-la be_v either_o sinus_fw-la totus_fw-la that_o be_v the_o radius_fw-la or_o whole_a sine_fw-la as_o the_o right_a line_n eb_o or_o sinus_fw-la simpliciter_fw-la the_o first_o sine_fw-la or_o a_o sine_fw-la less_o than_o radius_fw-la as_o ac_o or_o ab_o the_o one_o whereof_o be_v always_o the_o compliment_n of_o the_o other_o to_o 90_o degree_n we_o usual_o call_v they_o sine_fw-la and_o cousin_n 10_o right_a line_n without_o the_o circle_n who_o quantity_n we_o be_v to_o define_v be_v such_o as_o touch_v the_o circle_n and_o be_v call_v tangent_n 11_o a_o tangent_fw-la be_v a_o right_a line_n which_o touch_v the_o circle_n without_o be_v perpendicular_a from_o the_o end_n of_o the_o diameter_n to_o the_o radius_fw-la continue_v through_o the_o term_n of_o that_o arch_n of_o which_o it_o be_v the_o tangent_fw-la thus_o the_o right_a line_n fd_o be_v the_o tangent_fw-la of_o the_o arch_n cd_o 12_o right_a line_n draw_v through_o the_o circle_n who_o quantity_n we_o be_v to_o define_v be_v such_o as_o cut_v the_o circle_n and_o be_v call_v secant_n 13_o the_o secant_n of_o a_o arch_n be_v a_o right_a line_n draw_v through_o the_o term_n of_o a_o arch_n to_o the_o tangent_fw-la line_n of_o the_o same_o arch_n and_o thus_o the_o right_a line_n bf_o be_v the_o secant_n of_o the_o arch_n cd_o as_o also_o of_o the_o arch_n ceg_o the_o compliment_n thereof_o to_o a_o semicircle_n 14_o a_o canon_n of_o triangle_n than_o be_v that_o which_o contain_v the_o sine_n tangent_n and_o secant_n of_o all_o degree_n and_o part_n of_o degree_n in_o a_o quadrant_a according_a to_o a_o certain_a diameter_n or_o measure_n of_o a_o circle_n assume_v the_o construction_n whereof_o follow_v and_o first_o of_o the_o sine_n 15_o the_o right_a sine_n as_o they_o be_v to_o be_v consider_v in_o order_n to_o their_o construction_n be_v either_o primary_a or_o secondary_a 16_o the_o primary_a sine_n be_v those_o by_o which_o the_o rest_n be_v find_v and_o thus_o the_o radius_fw-la or_o whole_a sine_fw-la be_v the_o first_o primary_a sine_fw-la and_o be_v equal_a to_o the_o side_n of_o a_o six-angled_a figure_n inscribe_v in_o a_o circle_n consectary_n the_o radius_fw-la of_o a_o circle_n be_v give_v the_o sine_fw-la of_o 30_o deg._n be_v also_o give_v for_o by_o this_o proposition_n the_o radius_fw-la of_o a_o circle_n be_v the_o subtense_n of_o 60_o deg._n and_o the_o half_a thereof_o be_v the_o sine_fw-la of_o 30_o and_o therefore_o the_o radius_fw-la ab_o or_o bc_o be_v 1000.0000_o the_o sine_fw-la of_o 30_o deg._n be_v 500.0000_o 17_o the_o other_o primary_a sine_n be_v the_o sine_n of_o 60.18_o and_o 12_o deg._n be_v the_o half_a of_o the_o subtense_n of_o 120._o 36_o and_o 24_o degr._n and_o may_v be_v find_v by_o the_o problem_n follow_v 18_o the_o right_a sine_fw-la of_o a_o arch_n and_o the_o right_a sine_fw-la of_o its_o compliment_n be_v in_o power_n equal_a to_o radius_fw-la demonst._n in_o the_o first_o diagram_n of_o this_o chapter_n ac_o be_v the_o sine_fw-la of_o cd_o and_o ab_o the_o sine_fw-la of_o ce_fw-fr the_o compliment_n thereof_o which_o with_o the_o radius_fw-la bc_o make_v the_o rightangled_a triangle_n abc_o therefore_o abq_fw-la acq_n bcq_n by_o the_o 19_o of_o the_o second_o as_o be_v to_o be_v prove_v and_o hence_o the_o sine_fw-la of_o 60_o deg._n may_v thus_o be_v find_v let_v the_o sine_fw-la of_o 30_o deg._n ac_o be_v 500.0000_o the_o square_a whereof_o 250.00000_o be_v subtract_v from_o the_o square_a of_o bc_o radius_fw-la the_o remainder_n be_v 750.00000_o the_o square_a of_o ab_o who_o square_a root_n be_v 8660254_o the_o sine_fw-la of_o 60_o deg._n 19_o the_o subtense_n of_o 36_o deg._n be_v the_o side_n of_o a_o dec-angle_n inscribe_v in_o a_o circle_n or_o the_o great_a segment_n of_o a_o hexagon_n divide_v into_o extreme_a and_o mean_a proportion_n corsectary_n