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	side_n	2,221	5	9.5367	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

right_a sine_fw-la of_o a_o arch_n and_o the_o secant_n of_o its_o compliment_n demonst._n in_o the_o precede_a diagram_n the_o triangle_n aef_o and_o ahg_o be_v like_a therefore_o af._n ae_o ∷_o ag._n ah_o 31_o as_o the_o sine_fw-la of_o a_o arch_n or_o angle_n be_v to_o rad._n so_o be_v the_o tangent_fw-la of_o the_o same_o arch_n to_o the_o secant_n thereof_o demonst._n in_o the_o precede_a diagram_n the_o triangle_n aef_o and_o agh_o be_v like_a therefore_o ef._n ae_o ∷_o hg_o ah_o 32_o as_o radius_fw-la be_v to_o the_o secant_n of_o a_o arch_n so_o be_v the_o cotangent_n of_o the_o same_o arch_n to_o the_o cosecant_v thereof_o demonst._n in_o the_o precede_a diagram_n the_o triangle_n alh_o and_o ack_o be_v like_a therefore_o lh_o ah_o ∷_o ck_o ak_o other_o more_o easy_a and_o expeditious_a way_n of_o make_v the_o tangent_n and_o secant_n you_o may_v see_v in_o the_o first_o chap._n of_o my_o trigonometria_fw-la britannica_fw-la but_o the_o canon_n be_v now_o already_o make_v these_o rule_n we_o deem_v sufficient_a the_o construction_n of_o the_o artificial_a sine_n and_o tangent_n we_o have_v purposely_o omit_v they_o be_v nothing_o else_o but_o the_o logarithm_n of_o the_o natural_a of_o which_o logarithm_n we_o have_v show_v the_o construction_n in_o a_o former_a institution_n by_o the_o extraction_n of_o root_n and_o in_o my_o trigonometria_fw-la britannica_fw-la by_o multiplication_n and_o therefore_o shall_v now_o proceed_v to_o the_o use_n of_o the_o canon_n of_o sine_n tangent_n and_o secant_n in_o the_o solution_n of_o all_o triangle_n whether_o plain_a or_o spherical_a chap._n iv._o of_o the_o calculation_n of_o plain_a triangle_n a_o plain_a triangle_n be_v contain_v under_o three_o right_a line_n and_o be_v either_o rightangle_v or_o oblique_a 2_o in_o all_o plain_a triangle_n two_o angle_n be_v give_v the_o three_o be_v also_o give_v and_o one_o angle_n be_v give_v the_o sum_n of_o the_o other_o two_o be_v give_v because_o the_o three_o angle_n together_o be_v equal_a to_o two_o right_a by_o the_o 9th_o of_o the_o second_o therefore_o in_o a_o plain_a right_n angle_v triang._n one_o of_o the_o acute_a angle_n be_v the_o compliment_n of_o the_o other_o 3_o in_o the_o resolution_n of_o plain_a triangle_n the_o angle_n only_o be_v give_v the_o side_n can_v be_v find_v but_o only_o the_o the_o reason_n of_o the_o side_n it_o be_v therefore_o necessary_a that_o one_o of_o the_o side_n be_v know_v 4_o in_o a_o rightangled_a triangle_n two_o term_n beside_o the_o right-angle_n will_v serve_v to_o find_v the_o three_o so_o the_o one_o of_o they_o be_v a_o side_n 5_o in_o oblique_a angle_a triangle_n there_o must_v be_v three_o thing_n give_v to_o find_v a_o four_o 6_o in_o rightangled_a plain_a triangle_n there_o be_v seven_o case_n and_o five_o in_o oblique_a for_o the_o solution_n of_o which_o the_o four_o axiom_n follow_v be_v sufficient_a 1_o axiom_n in_o a_o rightangled_a plain_a triangle_n the_o rectangle_n make_v of_o radius_fw-la and_o one_o of_o the_o side_n contain_v the_o right-angle_n be_v equal_a to_o the_o rect-angle_n make_v of_o the_o other_o contain_v side_n and_o the_o tangent_fw-la of_o the_o angle_n thereunto_o adjacent_a dem._n in_o the_o right_a angle_a plain_n triang._n bed_n draw_v the_o periphery_n fe_o then_o be_v be_v radius_fw-la et_fw-la de_o the_o tangent_fw-la of_o the_o angle_n at_o b_o make_v ca_o parallel_n to_o de_o then_o be_v the_o triangle_n abc_o and_o ebb_o like_o because_o of_o their_o rightangle_v at_o a_o and_o e_o and_o their_o common_a angle_n at_o b_o therefore_o ba_o be_v ∷_o ac_o ed._n and_o ba_o ×_o ed_o be_v ×_o ac_o that_o be_v ba_o ×_o tb_n rad._n ×_o ac_o as_o be_v to_o be_v prove_v 2_o axiom_n in_o all_o plain_a triangle_n the_o side_n be_v proportional_a to_o the_o sine_n of_o their_o opposite_a angle_n demonst._n in_o the_o ●plain_n triangle_n bcd_o extend_v bc_o to_o f_o make_v bf_o dc_o and_o draw_v the_o arch_n fg_o and_o ch_o then_o be_v the_o perpendicular_n fe_o and_o ca_o the_o sine_n of_o the_o angle_n at_o b_o and_o d_o by_o the_o seven_o of_o the_o three_o and_o the_o triangle_n bef_a and_o bac_o be_v like_a because_o of_o their_o right_a angle_n at_o e_o and_o a_o and_o their_o common_a angle_n at_o b._n therefore_o c._n c_o a_o ∷_o bf_o fe_o that_o be_v bc._n sine_fw-la d_o ∷_o dc_o bf_o sine_fw-la b._n as_o be_v to_o be_v prove_v 3_o axiom_n in_o all_o plain_a triangle_n as_o the_o half_a sum_n of_o the_o side_n be_v to_o their_o half_a difference_n so_o be_v the_o tangent_fw-la of_o the_o half_a sum_n of_o their_o opposite_a angle_n to_o the_o tangent_fw-la if_o their_o half_a difference_n demonst._n in_o the_o triangle_n bcd_o let_v the_o side_n be_v cb_o and_o cd_o and_o cg_o cb._n wherefore_o ½_o z_o crur._n eg_o and_o ½_o ×_o crur._n aec_o draw_v ch_o bisect_v bg_o at_o right_a angle_n and_o make_v the_o angle_n gci_o d_o then_o will_v the_o angle_n gch_o ½_o z_o angle_n b_o and_o d_o who_o tangent_fw-la be_v hg_o and_o the_o angle_n ich_o ½_o ×_o ang._n b_o and_o d_o who_o tangent_fw-la be_v hippolito_o but_o eg_o aec_o ∷_o hg_o hippolito_o that_o be_v ½_o z_o crur._n ½_o ×_o crur._n ∷_o t_o ½_o z_o ang_n it_o ½_o x_o ang._n 4_o axiom_n in_o all_o plain_a triangle_n as_o the_o base_a be_v to_o the_o sum_n of_o the_o other_o side_n so_o be_v the_o difference_n of_o those_o side_n to_o the_o difference_n of_o the_o segment_n of_o the_o base_a db._n bf_o ∷_o hb_o ga_o that_o be_v db._n z_o crur._n ∷_o ×_o crur._n ×_o seg_fw-mi base_a these_o thing_n premise_v we_o will_v now_o set_v down_o the_o several_a problem_n or_o case_n in_o all_o plain_a triangle_n rightangle_v and_o oblique_a with_o the_o proportion_n by_o which_o they_o may_v be_v solve_v and_o manner_n of_o solve_v they_o both_o by_o natural_a and_o artificial_a number_n of_o right_o angle_v plain_a triangle_n in_o right_o angle_v plain_a triangle_n the_o side_n comprehend_v the_o right-angle_n we_o call_v the_o leg_n and_o the_o side_n subtend_v the_o right_a angle_n we_o call_v the_o hypotenuse_n 1_o probl._n the_o leg_n give_v to_o find_v a_o angle_n the_o give_a leg_n ab_o 230_o ac_o 143.72_o ac_o rad_n ∷_o ab_o ta_fw-fr cb_o by_o the_o 1_o axiom_n that_o the_o quantity_n of_o this_o angle_n or_o any_o other_o term_n require_v may_v be_v express_v in_o number_n if_o the_o solution_n be_v to_o be_v make_v in_o natural_a number_n multiply_v the_o second_o term_n give_v by_o the_o three_o and_o their_o product_n divide_v by_o the_o first_o the_o quotient_n be_v the_o four_o proportional_a seek_v but_o if_o the_o solution_n be_v to_o be_v make_v in_o artificial_a number_n from_o the_o sum_n of_o the_o logarithm_n of_o the_o second_o and_o three_o term_n give_v subtract_v the_o logarithm_n of_o the_o first_o the_o remainder_n shall_v be_v the_o logarithm_n of_o the_o four_o proportional_a require_v illustration_n by_o natural_a number_n as_o the_o leg_n ac_o 143.72_o be_v to_o the_o radius_fw-la ac_o 10000000_o so_o be_v the_o leg._n ab_o 235_o to_o the_o tang_n of_o acb_o grad._n 58.55_o the_o product_n of_o the_o second_o and_o three_o term_n be_v 2350000000_o which_o be_v divide_v by_o 143.72_o the_o first_o term_v give_v the_o quotient_n be_v 16351238_o the_o tangent_fw-la of_o the_o angle_n acb_o which_o be_v seek_v in_o the_o table_n the_o near_o less_o be_v 16350527_o and_o the_o arch_n answer_v thereto_o be_v grad._n 58._o 55_o part_n illustration_n by_o logarithm_n  _fw-fr logarithm_n as_o the_o leg_n ac_o 143.72_o 2.157517_o be_v to_o the_o rad._n i_o 10.0000_o so_o be_v the_o leg_n ab_o 235_o 2.371068_o to_o the_o tang_n of_o c_o gr._n 58.55_o 10.213551_o 2_o probl._n the_o angle_n and_o one_o leg_n give_v to_o find_v the_o other_o leg._n in_o the_o right_n angle_v plain_a triangle_n abc_o the_o leg_n ac_o be_v inquire_v the_o give_v angle_n abc_o leg._n ab_o rad._n ab_o ∷_o t_o abc_o ac_o by_o the_o first_o axiom_n 3_o prob._n the_o hypotenuse_n and_o a_o leg_n give_v to_o find_v a_o angle_n in_o the_o right_n angle_v plain_a triangle_n abc_o the_o angle_n acb_o be_v inquire_v the_o give_v hypoth_n bc._n leg_n ab_o bc._n rad_n ∷_o ab_o s._n acb_o by_o 2_o ax._n 4_o probl._n the_o hypotenuse_n and_o angle_n give_v to_o find_v either_o leg._n in_o the_o right_n angle_v plain_a triangle_n abc_o the_o leg._n ab_o be_v inquire_v the_o give_v hypot._n bc._n angle_z acb_o rad._n bc_o ∷_o s._n acb_o ab_o by_o 2_o ax._n 5_o probl._n the_o angle_n and_o a_o leg_n give_v to_o find_v the_o hypotenuse_n in_o the_o rightangled_a plain_a triangle_n abc_o the_o hypotenuse_n bc_o be_v inquire_v the_o give_v angle_n abc_o the_o give_a leg_n ac_o s._n abc_o ac_o ∷_o rad._n bc._n by_o