Selected quad for the lemma: reason_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
reason_n	doctrine_n	part_n	solution_n	4,929	5	15.6531	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A95751	The trissotetras: or, a most exquisite table for resolving all manner of triangles, whether plaine or sphericall, rectangular or obliquangular, with greater facility, then ever hitherto hath been practised: most necessary for all such as would attaine to the exact knowledge of fortification, dyaling, navigation, surveying, architecture, the art of shadowing, taking of heights, and distances, the use of both the globes, perspective, the skill of making the maps, the theory of the planets, the calculating of their motions, and of all other astronomicall computations whatsoever. Now lately invented, and perfected, explained, commented on, and with all possible brevity, and perspicuity, in the hiddest, and most re-searched mysteries, from the very first grounds of the science it selfe, proved, and convincingly demonstrated. / By Sir Thomas Urquhart of Cromartie Knight. Published for the benefit of those that are mathematically affected.	Urquhart, Thomas, Sir, 1611-1660.	1645 (1645)	Wing U140; Thomason E273_9; ESTC R212170	85,776	129

certain_a all_o magnitude_n be_v figure_n at_o least_o in_o power_n and_o all_o figure_n either_o triangle_n or_o triangle_v that_o the_o arithmetical_a solution_n of_o any_o geometrical_a question_n depend_v on_o the_o doctrine_n of_o triangle_n 5._o and_o though_o the_o proportion_n betwixt_o the_o part_n of_o a_o triangle_n can_v be_v without_o some_o error_n because_o of_o crooked_a line_n to_o right_a line_n and_o of_o crooked_a line_n among_o themselves_o the_o reason_n be_v inscrutable_a no_o man_n be_v able_a to_o find_v out_o the_o exact_a proportion_n of_o the_o diameter_n to_o the_o circumference_n yet_o both_o in_o plain_a triangle_n where_o the_o measure_n of_o the_o angles_n be_v of_o a_o different_a species_n from_o the_o side_n and_o in_o sphericall_n wherein_o both_o the_o angle_n and_o side_n be_v of_o a_o circular_a nature_n crooked_a line_n be_v in_o some_o measure_n reduce_v to_o right_a line_n by_o the_o definition_n of_o quantity_n which_o right_a line_n viz._n sines_n tangent_n and_o secant_v apply_v to_o a_o circle_n have_v in_o respect_n of_o the_o radius_fw-la o●_n half-diameter_n 6._o and_o therefore_o though_o the_o circle_n quadrature_n be_v not_o find_v out_o it_o be_v in_o our_o power_n to_o make_v the_o diameter_n or_o the_o semi-diameter_n which_o be_v the_o radius_fw-la of_o as_o many_o part_n as_o we_o please_v and_o be_v sure_o so_o much_o the_o more_o that_o the_o radius_fw-la be_v take_v the_o error_n will_v be_v the_o lesser_a for_o albeit_o the_o sin_n tangent_n and_o secant_v be_v irrational_a thereto_o for_o the_o most_o part_n and_o their_o proportion_n inexplicable_a by_o any_o number_n whatsoever_o whither_o whole_a or_o break_a yet_o if_o they_o be_v right_o make_v they_o will_v be_v such_o as_o that_o in_o they_o all_fw-mi no_o number_n will_v be_v different_a from_o the_o truth_n by_o a_o integer_fw-la or_o unity_n of_o those_o part_n whereof_o the_o radius_fw-la be_v take_v which_o be_v so_o exact_o do_v by_o some_o 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perpendicular_o on_o the_o radius_fw-la pass_v by_o the_o other_o end_n thereof_o 3._o a_o tangent_fw-la be_v a_o right_a line_n draw_v from_o the_o secant_fw-la by_o one_o end_n of_o the_o arch_n perpendicular_o on_o the_o extremity_n of_o the_o diameter_n pass_v by_o the_o other_o end_n of_o the_o say_v arch._n 4._o a_o secant_fw-la be_v the_o prolong_v radius_fw-la which_o pass_v by_o the_o upper_a extremity_n of_o the_o arch_n till_o it_o meet_v with_o the_o sine_fw-la tangent_fw-la of_o the_o say_v arch._n 5._o compliment_n be_v the_o difference_n betwixt_o the_o lesser_a arch_n and_o a_o quadrant_n or_o betwixt_o a_o right_a angle_n and_o a_o acute_a 6._o the_o compliment_n to_o a_o semicircle_n be_v the_o difference_n betwixt_o the_o half-circumference_n and_o any_o arch_n lesser_a or_o betwixt_o two_o right_a angle_n and_o a_o oblique_a angle_n whither_o blunt_a or_o sharp_a 7._o the_o verse_v sine_fw-la be_v the_o remainder_n of_o the_o radius_fw-la the_o sine_fw-la compliment_n be_v subtract_v from_o it_o and_o though_o great_a use_n may_v be_v make_v of_o the_o verse_v sin_n for_o find_v out_o of_o the_o angle_n by_o the_o side_n and_o side_n by_o the_o angles_n yet_o in_o logarithmicall_a calculation_n they_o be_v altogether_o useless_a and_o therefore_o in_o my_o trissotetras_n there_o be_v no_o mention_n make_v of_o they_o 8._o in_o amblygonosphericall●_n which_o admit_v both_o of_o a_o extrinsecall_n and_o intrinsical_v demission_n of_o the_o perpendicular_a nineteen_o several_a part_n be_v to_o be_v consider_v viz._n the_o perpendicular_a the_o subtendentall_a the_o subtendentine_n two_o cosubtendents_n the_o basall_n the_o basidion_n the_o chief_a segment_n of_o the_o base_a two_o cobases_n the_o double_a vertical_a the_o vertical_a the_o verticaline_a two_o coverticall_n the_o next_o cathetopposite_a the_o prime_a cathetopposite_a and_o the_o two_o cocathetopposites_n fourteen_o whereof_o to_o wit_n the_o subtendentall_a the_o 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to_o two_o right_a angle_n because_o every_o angle_n be_v measure_v by_o a_o arch_n or_o part_n of_o a_o circumference_n and_o a_o right_a angle_n by_o ninety_o degree_n if_o upon_o the_o middle_n of_o the_o ground_n line_n as_o centre_n be_v describe_v a_o semicircle_n it_o will_v be_v the_o measure_n of_o the_o angle_n comprehend_v betwixt_o the_o fall_n and_o sustain_a line_n 2._o hence_o it_o be_v that_o the_o four_o opposite_a angle_n make_v by_o one_o line_n cross_v another_o be_v always_o each_o to_o its_o own_o opposite_a equal_a for_o if_o upon_o the_o point_n of_o intersection_n as_o centre_n be_v describe_v a_o circle_n every_o two_o of_o those_o angle_n will_v fill_v up_o the_o semicircle_n therefore_o the_o first_o and_o second_o will_v be_v equal_a to_o the_o second_o and_o three_o and_o consequent_o the_o second_o which_o be_v the_o common_a angle_n to_o both_o these_o couple_n be_v remove_v the_o first_o will_v remain_v equal_a to_o the_o three_o and_o by_o the_o same_o reason_n the_o second_o to_o the_o four_o which_o be_v to_o be_v demonstrate_v 3._o if_o a_o right_a line_n fall_v upon_o two_o other_o right_a line_n make_v the_o alternat_a angle_n equal_a these_o line_n must_v needs_o be_v parallel_n for_o if_o they_o do_v meet_v the_o alternat_a angle_n will_v not_o be_v equal_a because_o in_o all_o plain_a triangle_n the_o outward_a angle_n be_v great_a than_o any_o of_o the_o remote_a inward_a angle_n which_o be_v prove_v by_o the_o first_o 4._o if_o one_o of_o the_o side_n of_o a_o triangle_n be_v produce_v the_o outward_a angle_n be_v equal_a to_o both_o the_o inner_a and_o opposite_a angle_n together_o because_o according_a to_o the_o accline_a or_o decline_v of_o the_o conterminall_a side_n be_v leave_v a_o angulary_a space_n for_o the_o receive_n of_o a_o parallel_n to_o the_o opposite_a side_n in_o the_o point_n of_o who_o occourse_n at_o the_o base_a the_o exterior_a angle_n be_v divide_v into_o two_o which_o for_o their_o like_a and_o alternat_a situation_n with_o the_o two_o interior_n angle_n be_v equal_a each_o to_o its_o own_o conform_a to_o the_o nature_n of_o angles_n make_v by_o a_o right_a line_n cross_v divers_a parallel_n 5._o from_o hence_o we_o gather_v that_o the_o three_o angle_n of_o a_o plain_a triangle_n be_v equal_a to_o two_o right_n for_o the_o two_o inward_a be_v equal_a to_o the_o external_a one_o and_o there_o remain_v of_o the_o three_o but_o one_o which_o be_v prove_v in_o the_o first_o apodictick_n to_o be_v the_o external_a angle_n compliment_n to_o two_o right_n it_o must_v needs_o fall_v forth_o what_o be_v equal_a to_o a_o three_o be_v equal_a among_o themselves_o that_o the_o three_o angle_n of_o a_o plain_a triangle_n be_v equal_a to_o two_o right_a angle_n the_o which_o we_o undertake_v to_o prove_v 6._o by_o the_o same_o reason_n the_o two_o acute_a of_o a_o rectangle_v plain_a triangle_n be_v equal_a to_o one_o right_a angle_n and_o any_o one_o of_o they_o the_o other_o compliment_n thereto_o 7._o