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
A48344	Nine geometricall exercises, for young sea-men and others that are studious in mathematicall practices: containing IX particular treatises, whose contents follow in the next pages. All which exercises are geometrically performed, by a line of chords and equal parts, by waies not usually known or practised. Unto which the analogies or proportions are added, whereby they may be applied to the chiliads of logarithms, and canons of artificiall sines and tangents. By William Leybourn, philomath.	Leybourn, William, 1626-1716.	1669 (1669)	Wing L1925; ESTC R213612	96,458	212

length_n of_o the_o perpendicular_a c_o a._n the_o analogy_n or_o proportion_n be_v as_o the_o sine_fw-la of_o the_o angle_n at_o c_z be_v to_o the_o log._n of_o a_o b_o so_o be_v the_o sine_fw-la of_o the_o angle_n b_o to_o the_o logar_n of_o c_o a._n or_o as_o the_o radius_fw-la be_v to_o the_o logar_n of_o a_o b_o so_o be_v the_o tangent_fw-la of_o b_o to_o the_o logar_n of_o c_o a._n case_n iu._n the_o hypotenuse_n c_o b_o 225_o the_o angle_n c_o 53_o degr_n 7_o min._n and_o the_o angle_n at_o b_o 36_o degr_n 53_o min._n give_v to_o find_v the_o base_a b_o a_o and_o the_o perpendicular_a c_o a._n draw_v a_o right_a line_n c_o b_o contain_v 225_o of_o your_o line_n of_o equal_a part_n then_o take_v 60_o deg_fw-la out_o of_o your_o line_n of_o chord_n set_v one_o foot_n of_o the_o compass_n in_o b_o and_o with_o the_o other_o describe_v the_o arch_n e_o d_o also_o the_o compass_n continue_v at_o the_o same_o distance_n place_v one_o foot_n in_o c_o and_o with_o the_o other_o describe_v the_o arch_n g_o f._n then_o from_o the_o point_n b_o and_o through_o the_o point_n d_o draw_v a_o right_a line_n also_o from_o the_o point_n c_o and_o through_o the_o point_n f_o draw_v another_o right_a line_n these_o two_o line_n will_v intersect_v or_o cross_v each_o other_o in_o the_o point_n a_o form_v the_o triangle_n c_o a_o b._n last_o take_v the_o line_n a_o b_o in_o your_o compass_n and_o apply_v it_o to_o your_o scale_n of_o equal_a part_n you_o shall_v find_v it_o to_o contain_v 180_o and_o that_o be_v the_o length_n of_o the_o base_a a_o b._n likewise_o a_o c_z be_v take_v in_o the_o compass_n and_o measure_v upon_o the_o line_n of_o equal_a part_n will_v be_v find_v to_o contain_v 135_o which_o be_v the_o length_n of_o the_o perpendicular_a c_o a._n the_o analogy_n or_o proportion_n be_v as_o the_o radius_fw-la be_v to_o the_o logarithm_n of_o c_o b_o so_o be_v the_o sine_fw-la of_o c_o to_o the_o logarithm_n of_o a_o b_o and_o the_o sine_fw-la of_o b_o to_o the_o logarithm_n of_o c_o a._n case_n v._o the_o hypotenuse_n c_o b_o 225_o and_o the_o base_a a_o b_o 180_o be_v give_v to_o find_v the_o perpendicular_a c_o a._n draw_v a_o right_a line_n a_o b_o contain_v 180_o of_o your_o scale_n of_o equal_a part_n and_o upon_o the_o end_n a_o erect_v a_o perpendicular_a a_o c._n then_o take_v out_o of_o your_o scale_n of_o equal_a part_n 225_o the_o length_n of_o your_o hypotenuse_n give_v and_o set_v one_o foot_n of_o the_o compass_n in_o b_o with_o the_o other_o describe_v the_o arch_n h_o k_o cut_v the_o perpendicular_a a_o c_o in_o c_o then_o draw_v the_o line_n c_o b_o so_o have_v you_o constitute_v the_o triangle_n c_o a_o b._n last_o take_v in_o your_o compass_n the_o length_n of_o the_o line_n a_o c_o and_o apply_v it_o to_o your_o line_n of_o equal_a part_n where_o you_o shall_v find_v that_o it_o will_v contain_v 135_o and_o that_o be_v the_o length_n of_o the_o perpendicular_a c_o a._n the_o analogy_n or_o proportion_n be_v 1._o operation_n as_o the_o logarithm_n of_o c_o b_o be_v to_o the_o radius_fw-la so_o be_v the_o logarithm_n of_o a_o b_o to_o the_o sine_fw-la of_o c._n 2._o operation_n as_o the_o radius_fw-la be_v to_o the_o logarithm_n of_o c_o b_o so_o be_v the_o sine_fw-la of_o b_o the_o compliment_n of_o c_o to_o the_o log._n of_o c_o a._n case_n vi_o the_o base_a a_o b_o 180_o the_o angle_n c_o 53_o degr_n 7_o min._n and_o the_o angle_n b_o 36_o degr_n 53_o min._n be_v give_v to_o find_v the_o hypotenuse_n c_o b._n draw_v a_o right_a line_n a_o b_o contain_v 180_o part_n of_o your_o scale_n for_o the_o base_a of_o your_o triangle_n and_o on_o the_o end_n a_o erect_v a_o perpendicular_a a_o c._n then_o take_v 60_o degr_n out_o of_o your_o line_n of_o chord_n and_o upon_o the_o point_n b_o with_o that_o distance_n describe_v the_o arch_n d_o e_z and_o because_o the_o angle_n at_o b_o be_v 36_o degr_n 53_o min._n take_v 36_o degr_n 53_o min._n from_o your_o line_n of_o chord_n and_o set_v it_o upon_o the_o arch_n from_o d_o to_z e._n then_o from_o b_o through_o the_o point_n e_o draw_v a_o right_a line_n till_o it_o meet_v with_o the_o perpendicular_a before_o draw_v which_o it_o will_v do_v in_o the_o point_n c._n and_o thus_o have_v you_o protract_v your_o triangle_n c_o a_o b._n last_o take_v in_o your_o compass_n the_o length_n of_o the_o hypotenuse_n c_o b_o and_o measure_v it_o upon_o your_o scale_n of_o equal_a part_n and_o you_o shall_v find_v it_o to_o contain_v 225._o the_o analogy_n or_o proportion_n be_v as_o the_o sine_fw-la of_o c_o be_v to_o the_o logarithm_n of_o a_o b_o so_o be_v the_o radius_fw-la to_o the_o logarithm_n of_o c_o b._n case_n vii_o the_o base_a a_o b_o 180_o and_o the_o perpendicular_a c_o at_fw-fr 135_o be_v give_v to_o find_v the_o hypotenuse_n c_o a._n draw_v a_o right_a line_n a_o b_o contain_v 180_o equal_a part_n and_o upon_o the_o end_n a_o erect_v the_o perpendicular_a a_o c_o and_o out_o of_o your_o scale_n of_o equal_a part_n take_v the_o length_n thereof_o 135_o which_o set_v from_o a_o to_o c_o and_o draw_v the_o line_n c_o b_o which_o constitute_v the_o triangle_n c_o a_o b._n last_o take_v the_o length_n of_o the_o hypotenuse_n c_o b_o in_o your_o compass_n and_o measure_v it_o upon_o your_o line_n of_o equal_a part_n you_o shall_v find_v it_o to_o contain_v 225._o the_o analogy_n or_o proportion_n be_v 1._o operation_n as_o the_o logarithm_n of_o a_o b_o be_v to_o the_o logarithm_n of_o c_o a_o so_o be_v the_o radius_fw-la to_o the_o tangent_fw-la of_o b._n 2._o operation_n as_o the_o sine_fw-la of_o b_o be_v to_o the_o logarithm_n of_o c_o a_o so_o be_v the_o radius_fw-la to_o the_o logarithm_n of_o c_o b._n these_o be_v the_o several_a variety_n or_o case_n that_o can_v at_o any_o time_n fall_v out_o in_o the_o solution_n of_o right-angled_n plain_a triangle_n wherefore_o we_o will_v now_o proceed_v to_o the_o solution_n of_o oblique_a plain_a triangle_n ii_o of_o oblique-angled_n plain_a triangle_n the_o triangle_n which_o i_o shall_v make_v use_n of_o in_o the_o solution_n of_o the_o several_a case_n appertain_v to_o a_o oblique-angled_n plain_a triangle_n shall_v be_v this_o follow_v c_o d_o b_o in_o which_o  _fw-fr  _fw-fr part_n  _fw-fr d_o b_o the_o base_a contain_v 335_o  _fw-fr c_o b_o the_o long_a side_n 271_o  _fw-fr d_o c_o the_o short_a side_n 100_o  _fw-fr and_o  _fw-fr deg_fw-la m._n c_o the_o obtuse_a angle_n contain_v 122_o 00_o d_o the_o 2_o acute_a angle_n 43_o 20_o b_o the_o 2_o acute_a angle_n 14_o 40_o case_n i_o two_o side_n as_o the_o base_a d_o b_o 335_o and_o the_o side_n c_o b_o 271_o and_o the_o angle_n d_o 43_o degr_n 20_o min._n opposite_a to_o c_o b_o to_o find_v the_o angle_n at_o c_o opposite_a to_o the_o base_a d_o b._n draw_v a_o right_a line_n d_o b_o represent_v the_o base_a of_o your_o triangle_n which_o by_o help_n of_o your_o scale_n of_o equal_a part_n make_v to_o contain_v 335._o then_o upon_o the_o point_n d_o with_o the_o distance_n of_o 60_o degr_n of_o your_o line_n of_o chord_n describe_v the_o arch_n k_o l_o and_o from_o your_o chord_n take_v 43_o degr_n 20_o min._n the_o quantity_n of_o the_o angle_n at_o d_o and_o set_v it_o upon_o the_o arch-line_n from_o l_o to_o k_n draw_v the_o line_n c_o d._n and_o because_o your_o other_o give_v side_n b_o c_o contain_v 271_o part_n take_v 271_o out_o of_o your_o line_n of_o equal_a part_n and_o set_v one_o foot_n in_o b_o with_o the_o other_o describe_v the_o arch_n m_o n_z cross_v the_o former_a arch_n k_o l_o in_o the_o point_n c_o then_o draw_v the_o line_n c_o b._n so_o shall_v you_o have_v constitute_v the_o triangle_n c_o d_o b._n last_o because_o it_o be_v the_o angle_n at_o c_o that_o be_v require_v take_v 60_o degr_n of_o your_o chord_n and_o upon_o c_o describe_v the_o arch_n g_o h_o and_o take_v the_o distance_n between_o g_o and_z h_z apply_v it_o to_o your_o line_n of_o chord_n and_o you_o shall_v find_v it_o to_o reach_v from_o the_o begin_n thereof_o beyond_o the_o end_n of_o the_o line_n wherefore_o take_v 90_o degr_n the_o whole_a line_n and_o set_v that_o distance_n from_o g_o to_z o_fw-mi then_o take_v the_o remainder_n of_o the_o arch_n o_o h_o and_o measure_n that_o upon_o your_o chord_n and_o you_o shall_v find_v it_o to_o contain_v 32_o degr_n which_o add_v to_o 90_o degr_n make_v 122_o degr_n and_o that_o be_v the_o quantity_n of_o the_o angle_n at_o c_o which_o be_v require_v the_o analogy_n or_o proportion_n be_v as_o

and_o k_o h_n both_o parallel_n to_o the_o line_n b_o a._n then_o take_v in_o your_o compass_n the_o distance_n a_o 6_o set_v one_o foot_n of_o they_o in_o c_z and_o with_o the_o other_o describe_v the_o arch_n z_o and_o draw_v the_o line_n a_o x_o so_o that_o it_o only_o touch_v the_o arch_n z._n then_o take_v in_o your_o compass_n the_o distance_n a_o h_n set_v one_o foot_n upon_o the_o line_n a_o c_o move_v it_o along_o till_o the_o other_o foot_n be_v turn_v about_o will_v only_o touch_v the_o line_n a_o z_o and_o where_o the_o point_n of_o the_o compass_n rest_v upon_o the_o line_n a_o c_o which_o it_o will_v do_v at_o e_o through_o that_o point_n draw_v the_o line_n o_o e_o parallel_n to_o b_o a._n so_o shall_v b_o o_o be_v measure_v upon_o your_o line_n of_o chord_n give_v you_o 69_o degr_n 22_o min._n the_o quantity_n of_o the_o inquire_v angle_n at_o b._n case_n viii_o the_o base_a a_o c_o 27_o degr_n 54_o min._n and_o the_o hypotenuse_n a_fw-it b_o 30_o degr_n be_v give_v to_o find_v the_o perpendicular_a b_o c._o the_o analogy_n or_o proportion_n be_v as_o the_o cousin_a of_o the_o base_a a_o c_o 62_o degr_n 6_o min._n be_v to_o the_o radius_fw-la so_o be_v the_o cousin_a of_o the_o hypotenuse_n 60_o degr_n to_o the_o cousin_a of_o the_o perpendicular_a b_o c._o have_v draw_v your_o quadrant_n a_o b_o c_o take_v out_o of_o your_o line_n of_o chord_n 62_o degr_n 6_o min._n the_o cousin_a of_o the_o base_a a_o c_o and_o set_v they_o from_o b_o to_z s._n also_o take_v from_o the_o chord_n 60_o degr_n the_o cousin_a of_o the_o hypotenuse_n and_o set_v they_o from_o b_o to_z m_z and_o draw_v the_o line_n s_o t_n and_o m_o n_o both_o parallel_n to_o b_o a._n then_o take_v the_o distance_n a_o t_o in_o your_o compass_n set_v one_o foot_n in_o c_o and_o with_o the_o other_o describe_v the_o arch_n v_n and_o draw_v the_o line_n a_o w_n so_o that_o it_o may_v only_o touch_v the_o arch_n v._o then_o take_v a_o n_z in_o your_o compass_n move_v one_o foot_n thereof_o gentle_o along_o the_o line_n c_o a_o till_o the_o other_o be_v turn_v about_o do_v only_o touch_v the_o line_n a_o w_n and_o where_o the_o point_n rest_v upon_o the_o line_n c_o a_o which_o you_o will_v find_v to_o be_v at_o c_o the_fw-mi make_v a_o mark_n and_o draw_v the_o line_n c_o a_o parallel_n to_o b_o a._n last_o take_v the_o distance_n from_o b_o to_o a_o and_o measure_v it_o upon_o your_o line_n of_o chord_n where_o you_o shall_v find_v it_o to_o contain_v 78_o degr_n 30_o min._n the_o compliment_n of_o the_o perpendicular_a or_o c_o a_o measure_a upon_o the_o chord_n will_v give_v you_o 11_o d._n 30_o min._n the_o perpendicular_a itself_o these_o five_o last_o be_v all_o the_o case_n in_o a_o right-angled_n spherical_a triangle_n that_o be_v resolvable_a by_o sin_n alone_o those_o which_o follow_v be_v to_o be_v resolve_v by_o sines_n and_o tangent_n joint_o and_o so_o will_v require_v another_o manner_n of_o operation_n than_o the_o former_a case_n ix_o the_o hypotenuse_n a_o b_o 30_o degr_n and_o the_o angle_n at_o the_o base_a a_o 23_o d._n 30_o min._n be_v give_v to_o find_v the_o angle_n at_o the_o perpendicular_a b._n the_o analogy_n or_o proportion_n be_v as_o the_o radius_fw-la be_v to_o the_o cousin_a of_o the_o hypotenuse_n a_fw-it b_o 60_o deg_fw-la so_o be_v the_o tangent_fw-la of_o the_o angle_n at_o the_o base_a a_o 23_o degr_n 30_o min._n to_o the_o co-tangent_a of_o the_o angle_n at_o the_o perpendicular_a b._n first_o draw_v a_o right_a line_n as_o c_o h_n and_o upon_o one_o end_n thereof_o as_o at_o c_o erect_a the_o perpendicular_a c_o a_n and_o with_o the_o distance_n of_o 60_o degr_n of_o your_o line_n of_o chord_n upon_o the_o centre_n c_o describe_v the_o quadrant_n c_o a_o d._n also_o upon_o the_o point_n a_o with_o 60_o degr_n of_o your_o chord_n describe_v the_o quadrant_n a_o b_o c._n be_v thus_o prepare_v first_o take_v 60_o deg_fw-la the_o cousin_a of_o the_o hypotenuse_n a_fw-it b_o and_o set_v they_o from_o a_o to_o o_o also_o take_v 23_o d._n 30_o m._n the_o quantity_n of_o the_o angle_n at_o a_o and_o set_v they_o from_o c_z to_o r_o and_o draw_v the_o line_n a_o m_o f_o and_o the_o line_n o_o g_o parallel_n to_o a_o c._n then_o take_v in_o your_o compass_n the_o distance_n between_o c_o and_z g_z and_o set_v one_o foot_n in_o d_o with_o the_o other_o describe_v the_o arch_a k_o and_o draw_v the_o line_n a_o l_o so_o that_o it_o only_o touch_v the_o arch_n k._n then_o place_v one_o foot_n of_o the_o compass_n in_o f_o take_v the_o least_o distance_n you_o can_v to_o the_o line_n c_o l_o which_o set_v from_o c_o to_o e._n last_o draw_v the_o line_n a_o e_o cut_v the_o quadrant_n c_o b_o in_o n._n so_o the_o distance_n c_o n_n measure_v upon_o your_o chord_n shall_v give_v you_o 20_o degr_n 38_o min._n the_o compliment_n of_o the_o angle_n at_o b_o which_o be_v require_v or_o the_o distance_n b_o n_n will_v give_v you_o 69_o degr_n 22_o min._n the_o angle_n itself_o case_n x._o the_o hypotenuse_n a_o b_o 30_o degr_n and_o the_o angle_n at_o the_o base_a a_o 23_o d._n 30_o min._n be_v give_v to_o find_v the_o base_a a_o c._n the_o analogy_n or_o proportion_n be_v as_o the_o radius_fw-la be_v to_o the_o cousin_a of_o the_o angle_n at_o a_o 66_o d._n 30_o m._n so_o be_v the_o tangent_fw-la of_o a_o b_o the_o hypotenuse_n 30_o degr_n to_o the_o tangent_fw-la of_o the_o base_a a_o c._n have_v prepare_v your_o quadrant_n out_o of_o your_o line_n of_o chord_n take_v 66_o degr_n 30_o min._n the_o compliment_n of_o the_o angle_n at_o a_o and_o set_v they_o from_o a_o to_o a_o and_o draw_v the_o line_n a_o b_o parallel_n to_o a_o c_o also_o take_v 30_o degr_n the_o hypotenuse_n from_o your_o chord_n and_o set_v they_o from_o c_z to_z c_o and_o draw_v the_o line_n a_o c_o prolong_v it_o to_o d._n this_o do_v take_v in_o your_o compass_n the_o distance_n from_o c_o to_o d_o and_o set_v one_o foot_n in_o d_o with_o the_o other_o describe_v the_o arch_n p_o and_o draw_v the_o line_n c_o q_n so_o that_o it_o may_v only_o touch_v the_o arch_n p._n then_o set_v one_o foot_n of_o the_o compass_n in_o b_o take_v the_o least_o distance_n between_o b_o and_o the_o line_n c_o q_n which_o distance_n will_v reach_v from_o c_o to_z e._n last_o draw_v the_o line_n a_o e_o cut_v the_o quadrant_n a_o c_o b_o in_o the_o point_n m._n so_o shall_v the_o distance_n c_o m_o be_v take_v in_o the_o compass_n and_o measure_v upon_o the_o line_n of_o chord_n contain_v 27_o degr_n 54_o m._n which_o be_v the_o quantity_n of_o the_o base_a require_v case_n xi_o the_o base_a a_o c_o 27_o degr_n 54_o min._n and_o the_o angle_n at_o the_o base_a a_o 23_o degr_n 30_o min._n be_v give_v to_o find_v the_o perpendicular_a b_o c._o the_o analogy_n or_o proportion_n be_v as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o the_o base_a a_o c_o 27_o degr_n 54_o min._n so_o be_v the_o tangent_fw-la of_o the_o angle_n at_o a_o 23_o degr_n 30_o min._n to_o the_o tangent_fw-la of_o the_o 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