Selected quad for the lemma: ground_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
ground_n	angle_n	line_n	right_a	2,182	5	9.6733	4	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 especial_o by_o petiscus_n who_o assume_v a_o radius_fw-la of_o twenty_o six_o place_n that_o according_a to_o his_o supputation_n the_o diameter_n of_o the_o earth_n be_v know_v and_o the_o globe_n thereof_o suppose_v to_o be_v perfect_o round_a one_o shall_v not_o fail_v in_o the_o dimension_n of_o its_o whole_a circuit_n the_o nine_o hundred_o thousand_o scantling_n of_o the_o million_o part_n of_o a_o inch_n and_o yet_o not_o be_v able_a for_o all_o that_o to_o measure_v it_o without_o amiss_o for_o so_o indivisible_a the_o truth_n of_o a_o thing_n be_v that_o come_v you_o never_o so_o near_o it_o unless_o you_o hit_v upon_o it_o just_a to_o a_o point_n there_o be_v a_o error_n still_o definition_n a_o cord_n or_o subtense_n be_v a_o right_a line_n draw_v from_o the_o one_o extremity_n to_o the_o other_o of_o a_o archippus_n 2._o a_o right_a sine_fw-la be_v the_o half_a cord_n of_o the_o double_a arch_n propose_v and_o from_o one_o extremity_n of_o the_o arch_n fall_v 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 subtendentine_n the_o cosubtendents_n the_o basall_n the_o basidion_n the_o cobases_n the_o vertical_a the_o verticaline_a the_o coverticall_n and_o cocathetopposites_n be_v call_v the_o first_o either_o subtendent_fw-la base_a topangle_n or_o cocathetopposite_a whither_o in_o the_o great_a triangle_n or_o the_o little_a or_o in_o the_o correctangle_n if_o they_o be_v ingredient_n of_o that_o rectangular_a whereof_o most_o part_n be_v know_v which_o part_n be_v always_o a_o subtendent_fw-la and_o a_o cathetopposite_a but_o if_o they_o be_v in_o the_o other_o triangle_n they_o be_v call_v the_o second_o subtendent_o base_n and_o so_o forth_o 9_o the_o external_a double_a vertical_a be_v include_v by_o the_o perpendicular_a and_o subtendentall_a and_o divide_v by_o the_o subtendentine_a the_o internal_a be_v include_v by_o cosubtendents_n and_o divide_v by_o the_o perpendicular_a apodicticks_n the_o angles_n make_v by_o a_o right_a line_n fall_v on_o another_o right_a line_n be_v equal_a 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

in_o every_o circle_n a_o angle_n from_o the_o centre_n be_v two_o in_o the_o limb_n both_o of_o they_o have_v one_o part_n of_o the_o circumference_n for_o base_a for_o be_v a_o external_a angle_n and_o consequent_o equal_a to_o both_o the_o intrinsical_v angle_n and_o therefore_o equal_a to_o one_o another_o because_o of_o their_o be_v subtend_v by_o equal_a base_n viz._n the_o semidiameters_a it_o must_v needs_o be_v the_o double_a of_o the_o foresay_a angle_n in_o the_o limb_n 8._o triangle_n stand_v between_o two_o parallel_n upon_o one_o and_o the_o fame_n base_a be_v equal_a for_o the_o identity_n of_o the_o base_a whereon_o they_o be_v seat_v together_o with_o the_o equidistance_n of_o the_o line_n within_o the_o which_o they_o be_v confine_v make_v they_o of_o such_o a_o nature_n that_o how_o long_o so_o ever_o the_o line_n parallel_n to_o the_o base_a be_v protract_v the_o diagonall_a cut_n of_o in_o one_o off_o the_o triangle_n as_o much_o of_o breadth_n as_o it_o gain_v of_o length_n the_o one_o loss_n accrue_v to_o the_o profit_n of_o the_o other_o quantify_v they_o both_o to_o a_o equality_n the_o thing_n we_o do_v intend_v to_o prove_v 9_o hence_o do_v we_o infer_v that_o triangle_n betwixt_o two_o parallel_n be_v in_o the_o same_o proportion_n with_o their_o base_n 10._o therefore_o if_o in_o a_o triangle_n be_v draw_v a_o parallel_n to_o any_o of_o the_o side_n it_o divide_v the_o other_o side_n through_o which_o it_o pass_v proportional_o for_o beside_o that_o it_o make_v the_o four_o segment_n to_o be_v four_o base_n it_o become_v if_o two_o diagonall_a line_n be_v extend_v from_o the_o end_n thereof_o to_o the_o end_n of_o its_o parallel_n a_o common_a base_a to_o two_o equal_a triangle_n to_o which_o two_o the_o triangle_n of_o the_o first_o two_o segment_n have_v reference_n according_a to_o the_o difference_n of_o their_o base_n and_o these_o two_o be_v equal_a as_o it_o be_v to_o the_o one_o so_o must_v it_o be_v to_o the_o other_o and_o therefore_o the_o first_o base_a must_v be_v to_o the_o second_o which_o be_v the_o segment_n of_o one_o side_n of_o the_o triangle_n as_o the_o three_o to_o the_o four_o which_o be_v the_o segment_n of_o the_o second_o all_o which_o be_v to_o be_v demonstrate_v 11._o from_o hence_o do_v we_o collect_v that_o equiangle_v triangle_n have_v their_o side_n about_o the_o equal_a angle_n proportional_a to_o one_o another_o this_o say_v petiscus_n be_v the_o golden_a foundation_n and_o chief_a ground_n of_o trigonometry_n 12._o a_o angle_n in_o a_o semicircle_n be_v right_a because_o it_o be_v equal_a to_o both_o the_o angle_n at_o the_o base_a which_o by_o cut_v the_o diameter_n in_o two_o be_v perceivable_a to_o any_o 13._o of_o four_o proportional_a line_n the_o rectangled_a figure_n make_v of_o the_o two_o extreme_n be_v equal_a to_o the_o rectangular_a compose_v of_o the_o mean_n for_o as_o four_o and_o one_o be_v equal_a to_o two_o and_o three_o by_o a_o arithmetical_a proportion_n and_o the_o four_o term_n geometrical_o exceed_v or_o be_v less_o than_o the_o three_o as_o the_o second_o be_v more_o or_o less_o than_o the_o first_o what_o the_o four_o have_v or_o want_v from_o and_o above_o the_o three_o be_v supply_v or_o impair_v by_o the_o surplusage_n or_o deficiency_n of_o the_o first_o from_o and_o above_o the_o second_o these_o analogy_n be_v still_o take_v in_o a_o geometrical_a way_n make_v the_o oblong_a of_o the_o two_o middle_n equal_a to_o that_o of_o the_o extreme_n which_o be_v to_o be_v prove_v 14._o in_o all_o plain_a rectangle_v triangle_n the_o ambient_n be_v equal_a in_o power_n to_o the_o subtendent_fw-la for_o by_o demit_v from_o the_o right_a angle_n a_o perpendicular_a there_o will_v arise_v two_o correctangle_v from_o who_o equiangularity_n with_o the_o great_a rectangle_n will_v proceed_v such_o a_o proportion_n among_o the_o homologall_a side_n of_o all_o the_o three_o that_o if_o you_o set_v they_o right_o in_o the_o rule_n begin_v your_o analogy_n at_o the_o main_n subtendent_fw-la see_v the_o including_z side_n of_o the_o total_a rectangle_n prove_v subtendent_o in_o the_o partial_a correctangle_v and_o the_o base_n of_o those_o rectanglet_n the_o segment_n of_o the_o great_a subtendent_fw-la it_o will_v fall_v out_o that_o as_o the_o main_n subtendent_fw-la be_v to_o his_o base_a on_o either_o side_n for_o either_o of_o the_o leg_n of_o a_o rectangled_a triangle_n in_o reference_n to_o one_o another_o be_v both_o base_a and_o perpendicular_a so_o the_o same_o base_n which_o be_v subtendent_o in_o the_o lesser_a rectangle_v be_v to_o their_o base_n the_o segment_n of_o the_o prime_n subtendent_fw-la then_o by_o the_o golden_a rule_n we_o find_v that_o the_o multiply_a of_o the_o middle_a term_n which_o be_v nothing_o else_o but_o the_o square_n of_o the_o comprehend_v side_n of_o the_o prime_n rectangular_a afford_v two_o product_n equal_a to_o the_o oblong_v make_v of_o the_o great_a subtendent_fw-la and_o his_o respective_a segment_n the_o aggregat_fw-la whereof_o by_o equation_n be_v the_o same_o with_o the_o square_n of_o the_o chief_a subtendent_fw-la or_o hypotenusa_fw-la which_o be_v to_o be_v demonstrate_v 15._o in_o every_o total_a square_n the_o supplement_n about_o the_o partial_a and_o interior_n square_n be_v equal_a the_o one_o to_o the_o other_o for_o by_o draw_v a_o diagonall_a line_n the_o great_a square_a be_v divide_v into_o two_o equal_a triangle_n because_o of_o their_o stand_n on_o equal_a base_n betwixt_o two_o parallel_n by_o the_o nine_o apodictick_n it_o be_v evident_a that_o in_o either_o of_o these_o great_a triangle_n there_o be_v two_o partial_a one_o equal_a to_o the_o two_o of_o the_o other_o each_o to_o his_o own_o by_o the_o same_o reason_n of_o the_o nine_o if_o from_o equal_a thing_n viz._n the_o total_a triangle_n be_v take_v equal_a thing_n to_o wit_n the_o two_o pair_n of_o partial_a triangle_n equal_a thing_n must_v needs_o remain_v which_o be_v the_o foresay_a supplement_n who_o equality_n i_o undertake_v to_o prove_v 16._o if_o a_o right_a line_n cut_v into_o two_o equal_a part_n be_v increase_v the_o square_n make_v of_o the_o additonall_a line_n and_o one_o of_o the_o bisegment_n join_v in_o one_o less_o by_o the_o square_n of_o the_o half_a of_o the_o line_n bisect_v be_v equal_a to_o the_o oblong_v contain_v under_o the_o prolong_a line_n and_o the_o line_n of_o continuation_n for_o if_o annex_o to_o the_o long_a side_n of_o the_o propose_v oblong_o be_v describe_v the_o foresay_a square_n there_o will_v jet_v out_o beyond_o the_o quadrat_fw-la figure_n a_o space_n or_o rectangle_n which_o for_o be_v power_v by_o the_o bisegment_n and_o additionall_n line_n will_v be_v equal_a to_o the_o near_a supplement_n and_o consequent_o to_o the_o other_o the_o equality_n of_o supplement_n be_v prove_v by_o the_o last_o apodictick_n by_o virtue_n whereof_o a_o gnomon_n in_o the_o great_a square_n lack_v nothing_o of_o its_o whole_a area_n but_o the_o space_n of_o the_o square_n of_o the_o bisected_a line_n be_v apparent_a to_o equalise_v the_o parallelogram_n propose_v which_o be_v to_o be_v demonstrate_v 17._o from_o hence_o proceed_v this_o sequel_n that_o if_o from_o any_o point_n without_o a_o circle_n two_o line_n cut_v it_o be_v protract_v to_o the_o other_o extremity_n thereof_o make_v two_o cord_n the_o oblong_v contain_v under_o the_o total_a line_n and_o the_o excess_n of_o the_o subtense_n be_v equal_a one_o to_o another_o for_o whether_o any_o of_o the_o line_n pass_v through_o the_o centre_n or_o not_o if_o the_o subtense_n be_v bisect_v see_v all_o line_n from_o the_o centre_n fall_v perpendicular_o upon_o the_o chordall_n point_n of_o bisection_n because_o the_o two_o semidiameters_a and_o bisegment_v substern_v under_o equal_a angle_n in_o two_o triangle_n evince_v the_o equality_n of_o the_o three_o angle_n to_o the_o three_o by_o the_o five_o apodictick_n which_o two_o angle_n be_v make_v by_o the_o fall_n of_o one_o right_a line_n upon_o another_o must_v needs_o be_v right_a by_o the_o ten_o definition_n of_o the_o first_o of_o euchilde_n the_o bucarnon_n of_o pythagoras_n demonstrate_v in_o my_o fourteen_o apodictick_n will_v by_o quadrosubduction_n of_o ambient_n from_o one_o another_o and_o their_o quadrobiquadrequation●_n with_o the_o hypotenusa_fw-la together_o with_o other_o analogy_n of_o equation_n with_o the_o power_n of_o like_a rectangular_a triangle_n comprehend_v within_o the_o same_o circle_n manifest_v the_o equality_n of_o long_a square_n or_o oblong_v radical_o meet_v in_o a_o exterior_a point_n and_o make_v of_o the_o prolong_a subtense_n and_o the_o line_n of_o interception_n betwixt_o the_o limb_n of_o the_o circle_n and_o the_o point_n of_o concourse_n quod_fw-la probandum_fw-la fuit_fw-la 18._o now_o to_o look_v back_o on_o the_o eleaventh_fw-mi apodictick_n where_o according_a to_o petiscus_n

i_o say_v that_o upon_o the_o mutual_a proportion_n of_o the_o side_n of_o equiangle_v triangle_n be_v found_v the_o whole_a science_n of_o trigonometry_n i_o do_v here_o respeak_v it_o and_o with_o confidence_n maintain_v the_o truth_n thereof_o because_o beside_o many_o other_o it_o be_v the_o ground_n of_o these_o subsequent_a theorem_n 1._o the_o right_a sine_fw-la of_o a_o arch_n be_v to_o its_o cousin_a as_o the_o radius_fw-la to_o the_o co-tangent_a of_o the_o say_v arch._n 2._o the_o cousin_a of_o a_o arch_n be_v to_o its_o sine_fw-la as_o the_o radius_fw-la to_o the_o tangent_fw-la of_o the_o say_v arch._n 3._o the_o sin_n and_o cosecants_a the_o secant_v and_o co-sines_n and_o the_o tangent_n and_o co-tangents_a be_v reciprocal_o proportional_a 4._o the_o radius_fw-la be_v a_o mean_a proportional_a betwixt_o the_o sine_fw-la and_o co-secant_a the_o secant_fw-la and_o cousin_a and_o the_o tangent_fw-la and_o co-tangent_a the_o verity_n of_o all_o these_o if_o a_o quadrant_n be_v describe_v and_o upon_o the_o two_o radiuse_n two_o tangent_n and_o two_o or_o three_o sin_n be_v erect_v which_o in_o respect_n of_o other_o arch_n will_v be_v co-sines_n and_o co-tangents_a and_o two_o secant_v draw_v which_o be_v likewise_o cosecants_a from_o the_o centre_n to_o the_o top_n of_o the_o tangent_n will_v appear_v by_o the_o foresay_a reason_n out_o of_o my_o eleaventh_fw-mi apodictick_n the_o trissotetras_n plain_a spherical_a plain_a trissotetras_n axiom_n four_o 1._o rulerst_n vradesso_n directory_n enodandas_n eradetul_n vphechet_n 3._o orth._n 1_o obl._n 2._o eproso_n directory_n enodandas_n 3_o ax._n grediftal_a dir._n ●_o pubkegdaxesh_a 4._o orth._n 4._o ax._n bagrediffiu_o dir._n ●_o 3._o obl._n the_o planorectangular_a table_n figure_n four_a 1._o va*_n le_fw-fr datas_fw-la quaesitas_fw-la resolver_n vp*_n al§em_fw-la rad_n five_o sapy_n ☞_o yr._n 2._o ve*_n mane_n vb*_n they_o §an_n v_o rad_n eglantine_n ☞_o so._n  _fw-fr praesubserv_v possubserv_v vph*_n en_fw-fr §er_fw-fr vb*_n they_o §an_n vp*_n al§em_o or_o eg*_n al§em_fw-la 3._o ena*_n ve_fw-la ek*_n be_v §ul_a sapeg_n eglantine_n rad_n ☞_o ur_fw-la eg*_n all_o §em_n rad_n taxeg_v eglantine_n ☞_o yr._n  _fw-fr  _fw-fr praesubserv_v possubserv_v 4._o ere*_n va_fw-fr ech*_n they_o §un_n et*_n en_fw-fr §ar_fw-fr ek*_n be_v §ul_a et*_n en_fw-fr §ar_fw-fr e_o ge_n rad_n ☞_o toge_z the_o planobliquangular_a table_n figures_n four_o 1._o alahe_fw-la *_o i_o da*na*re_o §le_a sapeg_n eglantine_n sapyr_n ☞_o yr._n 2._o emena*role_v the*re*_n lab_n §mo_fw-la aggre_n ze_n talfagros_n ☞_o talzo_n  _fw-fr praesubserv_v possubserv_v ze*le*mab_v §ne_n the*re*_n lab_n §mo_fw-la da*na*re_v §le_n 3._o enero*lome_n xe*_n me*_n no_o §ro_n e_o so_o ge_n ☞_o so._n  _fw-fr  _fw-fr praesubserv_v possubserv_v she*ne*_n ro_o §lem_n xe*_n me*_n no_o §ro_n da*na*re_fw-fr §le_fw-fr  _fw-fr  _fw-fr praesubserv_v possubserv_v 4._o erele*_n a_o pse*_n re*_n le_fw-fr §ma_fw-la bagreziu_fw-la vb*em_fw-la §an_n final_a resolver_n vxi●q_n rad_n 〈◊〉_d ☞_o sor._n the_o spherical_a trissotetras_n axiom_n three_o 1._o suprosca_n dir._n uphugen_o 2._o sbaprotca_fw-la pubkutethepsaler_n 3._o seproso_n uchedezexam_fw-la the_o orthogonospherical_a table_n figure_n 6._o datoquaeres_fw-la 16._o  _fw-fr dat._n quae_n resolver_n 1._o valam*menep_v vp*al§am_n torb_n tag_n nu_n ☞_o mir._n vb*am§en_n nag_n mu_o torp_n ☞_o myr._n or_o  _fw-fr torp_n mu_o lag_a ☞_o myr._n vph*an_v §ep_v tol_o sag_fw-ge sum_z ☞_o syr._n 2._o veman*nore_v vk*el§amb_n meg_n torp_n mu_o ☞_o nir._n or_o  _fw-fr torp_n teg_n mu_o ☞_o nir._n ug*em_fw-la §on_n sum_z seg_n tom_n ☞_o sir_n or_o  _fw-fr tom_n seg_n ru_n ☞_o sir_n uch*en_o §er_n neg_fw-mi to_o nu_n ☞_o nyr_n or_o  _fw-fr to_o le_fw-fr nu_n ☞_o nyr_n 3._o enar*rulome_n et*al§um_fw-la torp_n i_o nag_n ☞_o mur._n ed*am§on_n to_o neg_fw-mi sa_o ☞_o nir._n eth*an§er_n torb_n tag_n se_fw-la ☞_o tyr._n 4._o erol*lumane_a ez*●l§um_n sag_fw-ge sep_n rad_n ☞_o sur._n or_o  _fw-fr rad_n seg_n rag_n ☞_o sur._n ex*●●_n §an_n ne_n to_o nag_n ☞_o sir_n or_o  _fw-fr to_o le_fw-fr nag_n ☞_o sir_n eps*_n on_o §er_n tag_n tolb_n te_fw-la ☞_o syr._n or_o  _fw-fr tolb_n madge_n te_fw-la ☞_o syr._n 5._o acha*_n ve_fw-la al*_n be_o §un_v tag_n torb_n ma_n ☞_o nur._n or_o  _fw-fr torb_n madge_n ma_n ☞_o nur._n am*_n a_o §er_n say_v nag_n t●_n ☞_o nyr_n or_o  _fw-fr tω_fw-gr noy_n ray_n ☞_o nyr_n 6._o eshe*va_n en*er_v §ul_a tun_n neg_fw-mi ne_fw-la ☞_o nur._n er*_n el_fw-es §am_fw-la sei_fw-it teg_n torb_n ☞_o tir._n or_o  _fw-fr torb_n tepi_n rexi_fw-la ☞_o tir._n the_o loxogonospherical_a trissotetras_n monurgetick_a disergetick_n the_o monurgetick_a loxogonospherical_a table_n axiom_n two_o 1._o seproso_n dir._n lame_a figure_n two_o 2._o parses_n dir._n nera_fw-mi mood_n four_o figure_n datas_fw-la quaes_n resolver_n 1._o datamista_n lam*an*ep_v §_o rep_v sapeg_n see_v sapy_a ☞_o syr._n me*ne*ro_fw-it §_o lo._n sepag_n sa_o sepi_fw-la ☞_o sir_n  _fw-fr  _fw-fr  _fw-fr add_v 2._o datapura_n ne*re*le_a §_o ma._n hal_n basaldileg_n sad_a sab_n re_n regals_n bis*ir_n  _fw-fr  _fw-fr ab_fw-la  _fw-fr  _fw-fr parse_n powto_n parsadsab_n ☞_o powsalvertir_fw-fr ra*la*ma_fw-la §_o ne_o kour_n bfasines_n erele_v kouf_n br*axypopyx_n the_o loxogonospherical_a disergeticks_n axiom_n four_o 1._o no_z bad_a prosver_n dir._n alama_n 2._o naverpr_fw-la or_o tes._n allera_fw-la 3._o siubpror_n tab._n ammena_n 4._o niub_v prodesver_o errenna_n figure_n 4._o mood_n 8._o  _fw-fr fig._n m._n sub_fw-la res_fw-la dat._n praen_n cathetothesis_n final_a resolver_n 1._o ab_fw-la a_o  _fw-fr  _fw-fr  _fw-fr  _fw-fr cafregpiq_fw-la  _fw-fr  _fw-fr la_fw-fr up_o tag_fw-mi ut_fw-mi *_o open_v §_o at_o dasimforaug_fw-mi sat-nop-seud_a †_o nob_n kir_n a_o meb_n all_o nu_n ud_a *_o ob_fw-la §_o and_o dadisforeug_v saud-nob-sat_a †_o nop_n ir._n  _fw-fr na._n am_n mirabel_n uth*_n oph_n §_o auth_n dadisgatin_a sauth-noph-seuth_a †_o nop_n ir._n leb_n 2._o sub._n res_fw-la dat._n p●ae●_n cathetothesis_n final_a resolver_n  _fw-fr al_n  _fw-fr  _fw-fr  _fw-fr  _fw-fr cafyxegeq_fw-fr  _fw-fr ma_fw-fr la_fw-fr up_o tag_fw-mi ut_fw-mi *_o open_fw-mi §_o at_o dasimforauxy_n nat-mut-naud_a †_o mwd_fw-mi  _fw-fr meb_fw-mi all_o nu_n ud_a *_o ob_n §_o and_o dadiscracforeug_v naud-mud-nat_a †_o mwt_fw-mi ne_fw-fr ne_fw-fr am_n mirabel_n we_fw-mi *_o oph_o §_o auth_v dadiscramgatin_n nauth-muth-neuth_a †_o mwth_n fig._n m._n  _fw-fr  _fw-fr  _fw-fr cathetothesis_n plus_fw-fr minus_fw-la a_o a_o sub._n re._n dat._n pr._n cafriq_fw-la final_a resolver_n sindifora_n  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr at_o  _fw-fr ma_fw-fr up_z tag_fw-mi ut*op_n §_o at_o dadissepamforaur_n nop-sat-nob_n ☞_o seudfr_n autir_v ha_o nep_n all_o nu_n ud*ob_n §_o and_o dadissexamforeur_fw-fr nob-saud-nop_a ☞_o satfr_n eutir_v  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr and_o  _fw-fr ramires_n am_n mirabel_n uth*_n oph_n §_o auth_n dasimatin_n noph-seuth-nops_a ☞_o soethj_fw-la authir_n mep_v 4._o  _fw-fr  _fw-fr  _fw-fr cathetothesis_n plus_fw-fr minus_fw-la  _fw-fr am_n sub._n re._n dat._n pr._n cafregpagiq_fw-la final_a resolver_n sindiforiu_n  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr aet_fw-la na_fw-mi ma_fw-fr ub_z mu_o ut*_n open_v §_o aet_fw-la dadissepamfor_n tob-top-saet_a ☞_o soedfr_n dyr_n  _fw-fr nep_n am_n lag_a ud*_n ob_fw-la §_o aed_n dadissexamfor_n top-tob-saed_n ☞_o soetfr_n dyr_n  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr aed_n re_fw-mi reb_n en_fw-fr myr_n uth*_n oph_n §_o a_v dasimin_n tops-toph-saeth_a ☞_o soethj_fw-la a_v syr._n fig._n m._n cathetothesis_n ebb_n en_fw-fr sub._n re._n dat._n pr._n cafregpigeq_n final_a resolver_n  _fw-fr er_fw-mi ub_z mu_o ut*_n open_v §_o aet_fw-la dacramfor_n soed-top-saet_a ☞_o tob._n kir_n en_fw-fr ab_fw-la am_n lag_a ud*_n ob_fw-la §_o ad_fw-la damracfor_a soet-tob-saed_n ☞_o top._n ir._n  _fw-fr lo_o en_fw-fr myr_n uth*_n oph_n §_o a_v dasimquaein_o soeth-toph-saeth_a ☞_o top_n ir._n ab_fw-la 6._o cathetothesis_n  _fw-fr en_fw-fr sub._n re._n dat._n q._n pr._n cafregpiq_fw-la final_a resolver_n ro_n ne_n ub_z mu_o ut*_n open_v §_o aet_fw-la dacforamb_v naet-nut-noed_n ☞_o nwd_fw-mi yr._n  _fw-fr rab_n am_n lag_a ud*_n ob_fw-la §_o ae_v damforac_n naed-nud-noet_a ☞_o nwt_fw-mi yr._n le_fw-fr le_fw-fr en_fw-fr myr_n uth*_n oph_n §_o a_v dakinatam_fw-la naeth-nuth-noeth_a ☞_o nwth_n yr._n fig._n m._n  _fw-fr  _fw-fr  _fw-fr  _fw-fr cathteothesis_n plus_fw-fr minus_fw-la ebb_n e_o sub._n re._n dat._n pr._n cafriq_fw-la final_a resolver_n sindifora_n  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr at_o  _fw-fr re_fw-mi up_o tag_fw-mi ut*_n open_v §_o at_o dacracforaur_n mut-nat-mwd_a ☞_o neudfr_n autir_v er_fw-mi lo_o all_o nu_n ud*_n ob_fw-la §_o and_o dambracforeur_fw-fr mud-naud-mwt_a ☞_o natfr_n autir_v  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr and_o  _fw-fr mab_n am_n mirabel_n uth*_n oph_n §_o auth_v dacrambatin_n muth-nauth-mwth_a ☞_o neuthj_fw-la authir_n om_fw-la 8._o  _fw-fr  _fw-fr  _fw-fr  _fw-fr cathetothesis_n plus_fw-fr minus_fw-la  _fw-fr er_fw-mi sub._n re._n dat._n pr._n cacurgyq_n

much_o worth_n and_o virtue_n of_o aequus_fw-la and_o valeo_fw-la erect_v be_v say_v of_o perpendicular_o which_o be_v set_v or_o raise_v upright_o upon_o a_o base_a from_o erigere_fw-la to_o raise_v up_o or_o set_v aloft_o external_a extrinsecall_v exterior_a outward_a or_o outer_a be_v say_v oft_a of_o angle_n which_o be_v without_o the_o area_n of_o a_o triangle_n be_v comprehend_v by_o two_o of_o its_o shank_n meet_v or_o cut_v one_o another_o according_o as_o one_o or_o both_o of_o they_o be_v protract_v beyond_o the_o extent_n of_o the_o figure_n f._n faciendas_fw-la be_v the_o thing_n which_o be_v to_o be_v do_v faciendum_fw-la be_v the_o gerund_n of_o facere_fw-la figurative_a be_v the_o same_o thing_n as_o characteristick_n and_o be_v apply_v to_o those_o letter_n which_o do_v figure_n and_o point_v we_o out_o a_o resemblance_n and_o distinction_n in_o the_o mood_n figure_n be_v take_v here_o for_o those_o partition_n of_o trigonometry_n which_o be_v divide_v into_o mood_n flat_a be_v say_v of_o obtuse_a or_o blunt_a angle_n forward_o be_v say_v of_o analogy_n progressive_a from_o the_o first_o term_n to_o the_o last_o fundamental_a be_v say_v of_o reason_n take_v from_o the_o first_o ground_n and_o principle_n of_o a_o science_n g._n geodesie_n the_o art_n of_o survey_v of_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d or_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d terra_fw-la and_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d divido_fw-la partior_fw-la geography_n the_o science_n of_o the_o terrestrial_a globe_n of_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d terra_fw-la and_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d describo_fw-la gloss_n signify_v a_o commentary_n or_o explication_n it_o come_v from_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d gnomon_n be_v a_o figure_n less_o than_o the_o total_a square_n by_o the_o square_n of_o a_o segment_n or_o according_a to_o ramus_n a_o figure_n compose_v of_o the_o two_o supplement_n and_o one_o of_o the_o diagonall_a square_n of_o a_o quadrat_fw-la gnomonick_a the_o art_n of_o dyall_v from_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d the_o cock_n of_o a_o dial_n great_a circle_n vide_fw-la circle_n h._n homogeneal_a and_o homogeneity_n be_v say_v of_o angle_n of_o the_o same_o kind_n nature_n quality_n or_o affection_n from_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d communio_fw-la generis_fw-la homologall_n be_v say_v of_o side_n congruall_a correspondent_a and_o agreeable_a viz._n such_o as_o have_v the_o same_o reason_n or_o proportion_n from_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d similis_fw-la ratio_fw-la hypobasall_n be_v say_v of_o the_o concordance_n of_o those_o loxogonosphericall_a mood_n which_o when_o the_o perpendicular_a be_v demit_v have_v for_o the_o datas_fw-la of_o their_o second_o operation_n the_o same_o subtendent_fw-la and_o base_a hypocathetall_n be_v say_v of_o those_o which_o for_o the_o datas_fw-la of_o their_o three_o operation_n have_v the_o same_o subtendent_fw-la and_o perpendicular_a hypotenusall_n be_v say_v of_o subtendent_fw-la side_n from_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d and_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d hypotyposis_n a_o lay_v down_o of_o several_a thing_n before_o our_o eye_n at_o one_o time_n from_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d oculis_fw-la subjicio_fw-la delineo_fw-la &_o repraesento_fw-la vide_fw-la 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d hypoverticall_a be_v say_v of_o mood_n agree_v in_o the_o same_o catheteuretick_n datas_fw-la of_o subtendent_fw-la and_o vertical_a as_o the_o analysis_n of_o the_o word_n do_v show_v i_o identity_n a_o sameness_n from_o idem_fw-la the_o same_o illatitious_a or_o illative_a be_v say_v of_o the_o term_n which_o bring_v in_o the_o quaesitum_fw-la from_fw-la infero_fw-la illatum_fw-la inchase_v coagulate_v fix_v in_o compact_a or_o conflate_v be_v say_v of_o the_o last_o two_o loxogonosphericall_a operation_n put_v into_o one_o vide_fw-la compact_v conflated_a and_o coalescencie_n including_n side_n be_v the_o contain_v side_n of_o a_o angle_n of_o what_o affection_n soever_o it_o be_v vide_fw-la ambient_n leg_n etc._n etc._n individuate_v bring_v to_o the_o low_a division_n vide_fw-la specialise_v and_o specification_n endow_v be_v say_v of_o the_o term_n of_o a_o analogy_n whether_o side_n or_o angle_n as_o they_o stand_v affect_v with_o sines_n tangent_n secant_v or_o their_o compliment_n vide_fw-la invest_v ingredient_n be_v that_o which_o enter_v into_o the_o composition_n of_o a_o triangle_n or_o the_o progress_n of_o a_o operation_n from_o ingredior_fw-la of_o in_o and_o gradior_fw-la initial_a that_o which_o belong_v to_o the_o beginning_n from_o initium_fw-la ab_fw-la ineo_fw-la significante_fw-la incipio_fw-la incident_a be_v say_v of_o angle_n from_o insideo_fw-la vide_fw-la adjacent_a or_o conterminall_a interjacent_a lie_v betwixt_o of_o inter_fw-la and_o jaceo_fw-la it_o be_v say_v of_o the_o side_n or_o angle_n between_o intermediat_a be_v say_v of_o the_o middle_a term_n of_o a_o proportion_n inversionall_a be_v say_v of_o the_o concordance_n of_o those_o mood_n which_o agree_v in_o the_o manner_n of_o their_o inversion_n that_o be_v in_o place_v the_o second_o and_o four_o term_n of_o the_o analogy_n together_o with_o their_o endowment_n in_o the_o room_n of_o the_o first_o and_o three_o and_o contrariwise_o invest_v be_v the_o same_o as_o endow_v from_o investio_fw-la investire_fw-la irrational_a be_v those_o which_o be_v common_o call_v surd_a number_n and_o be_v inexplicable_a by_o any_o number_n whatsoever_o whether_o whole_a or_o break_v isosceles_a be_v the_o greek_a word_n of_o equicrural_a of_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d and_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d crus_fw-la l._n lateral_a belong_v to_o the_o side_n of_o a_o triangle_n from_o latus_fw-la lateris_fw-la leg_n be_v one_o of_o the_o including_z side_n of_o a_o angle_n two_o side_n of_o every_o triangle_n be_v call_v the_o leg_n and_o the_o three_o the_o base_a the_o leg_n therefore_o or_o shank_n of_o a_o angle_n be_v the_o bound_n insist_v or_o stand_v upon_o the_o base_a of_o the_o angle_n line_n of_o interception_n be_v the_o difference_n betwixt_o the_o secant_fw-la and_o the_o radius_fw-la and_o be_v common_o call_v the_o residuum_fw-la logarithm_n be_v those_o artificial_a number_n by_o which_o with_o addition_n and_o subtraction_n only_o we_o work_v the_o same_o effect_n as_o by_o other_o number_n with_o multiplication_n and_o division_n of_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d ratio_fw-la proportio_fw-la and_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d numerus_fw-la logarithmication_n be_v the_o work_n of_o a_o analogy_n by_o logarithm_n without_o have_v regard_n to_o the_o old_a laborious_a way_n of_o the_o natural_a sin_n and_o tangent_n we_o say_v likewise_o logarithmicall_a and_o logarithmical_o for_o logarithmeticall_a and_o logarithmetical_o for_o by_o the_o syncopise_n of_o et_fw-la the_o pronunciation_n of_o those_o word_n be_v make_v to_o the_o ear_n more_o pleasant_a a_o privilege_n warrant_v by_o all_o the_o dialect_v of_o the_o greek_a and_o other_o the_o most_o refine_a language_n in_o the_o world_n loxogonosphericall_a be_v say_v of_o oblique_a sphericals_n of_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d obliquus_fw-la and_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d ad_fw-la sphaeram_fw-la pertinens_fw-la from_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d globus_fw-la m._n mayor_z and_o minor_a arch_n vide_fw-la arch._n maxim_n a_o axiom_n or_o principle_n call_v so_o from_o maximus_fw-la because_o it_o be_v of_o great_a account_n in_o a_o art_n or_o science_n and_o the_o principal_a thing_n we_o ought_v to_o know_v mean_o or_o middle_a proportion_n be_v that_o the_o square_a whereof_o be_v equal_a to_o the_o plane_n of_o the_o extreme_n and_o call_v so_o because_o of_o its_o situation_n in_o the_o analogy_n mensurator_n be_v that_o whereby_o the_o illatitious_a term_n be_v compare_v or_o measure_v with_o the_o main_n quaesitum_fw-la monotropall_a be_v say_v of_o figure_n which_o have_v one_o only_a mood_n of_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d and_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d from_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d monurgetick_n be_v say_v of_o those_o mood_n the_o main_a quaesitas_fw-la whereof_o be_v obtain_v by_o one_o operation_n of_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d and_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d mood_n aetermine_v unto_o we_o the_o several_a manner_n of_o triangle_n from_o modus_fw-la a_o way_n or_o manner_n n._n natural_a the_o natural_a part_n of_o a_o triangle_n be_v those_o of_o which_o it_o be_v compound_v and_o the_o circular_a those_o whereby_o the_o main_a quaesitum_fw-la be_v find_v out_o near_o or_o next_o be_v say_v of_o that_o cathetopposite_a angle_n which_o be_v immediate_o opposite_a to_o the_o perpendicular_a notandum_fw-la be_v set_v down_o for_o a_o admonition_n to_o the_o reader_n of_o some_o remarkable_a thing_n to_o follow_v and_o be_v the_o gerund_n of_o noto_n notare_fw-la o._n oblique_a and_o obliquangulary_a be_v say_v of_o all_o angle_n that_o be_v not_o right_a oblong_o be_v a_o parallelogram_n or_o square_v more_o long_o they_o large_a from_o oblongus_fw-la very_o long_o obtuse_a and_o obtuse_a angle_v be_v say_v of_o