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 oâ—_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 amblygonosphericallâ—_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 quadrobiquadrequationâ—_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
give_v side_n from_o the_o sum_n of_o the_o middle_a term_n i_o mean_v the_o logarithm_n of_o the_o one_o and_o the_o other_o which_o be_v the_o total_a sine_fw-la and_o the_o leg_n propose_v we_o shall_v have_v the_o hypotenusa_fw-la require_v for_o it_o be_v as_o the_o sine_fw-la of_o the_o angle_n opposite_a to_o the_o side_n give_v to_o the_o foresay_a give_v side_n so_o the_o total_a sine_fw-la to_o the_o subtendent_fw-la require_v the_o reason_n of_o this_o proportion_n be_v ground_v on_o the_o second_o axiom_n eproso_n for_o k._n the_o three_o consonant_a of_o its_o directory_n give_v we_o to_o understand_v that_o it_o be_v one_o of_o the_o enodandas_n thereof_o the_o second_o mood_n of_o enave_n be_v egalem_fw-la which_o comprehend_v all_o those_o problem_n wherein_o one_o of_o the_o ambient_n and_o a_o oblique_a angle_n be_v give_v the_o other_o ambient_fw-la be_v require_v and_o by_o its_o resolver_n rad_n taxeg_v eglantine_n ☞_o your_o show_v that_o if_o we_o add_v the_o logarithm_n of_o the_o side_n give_v to_o the_o logarithm_n of_o the_o tangent_fw-la of_o the_o angle_n conterminall_a with_o that_o side_n and_o from_o the_o sum_n if_o we_o cut_v off_o the_o first_o digit_fw-la on_o the_o left_a hand_n which_o be_v equivalent_a to_o the_o subtract_v of_o the_o radius_fw-la whether_o double_v or_o single_a the_o remainder_n will_v afford_v we_o a_o logarithm_n so_o near_o as_o the_o irrationality_n of_o the_o term_n will_v admit_v in_o the_o table_n of_o equal_a part_n expressive_a of_o the_o side_n require_v for_o it_o be_v as_o the_o whole_a sine_fw-la to_o the_o tangent_fw-la of_o a_o angle_n incident_a on_o the_o give_v side_n so_o the_o side_n propose_v to_o the_o side_n require_v the_o reason_n hereof_o be_v ground_v on_o the_o second_o axiom_n for_o the_o four_o consonant_a of_o its_o directory_n show_v that_o egalem_fw-la be_v eprosos_n enodandum_fw-la the_o four_o figure_n be_v ereva_n who_o mood_n be_v echemun_n and_o etenar_n the_o first_o viz._n echemun_n comprehend_v all_o those_o problem_n wherein_o the_o two_o ambient_n be_v give_v the_o subtendent_fw-la be_v require_v and_o not_o be_v logarithmical_o resolvable_a in_o less_o than_o two_o operation_n have_v for_o its_o prae_fw-la and_o possubservient_n the_o mood_n of_o etenar_n and_o ekarul_a for_o a_o oblique_a angle_n by_o this_o method_n be_v to_o be_v search_v before_o the_o subtendent_fw-la can_v be_v find_v out_o and_o by_o reason_n of_o these_o several_a workâ—_n this_o mood_n be_v ground_v on_o the_o two_o first_o axiom_n and_o be_v a_o enodandum_n partial_o depend_v on_o eradetul_n and_o eproso_n yet_o if_o you_o will_v be_v please_v to_o be_v at_o the_o pain_n of_o extract_v the_o square_a root_n you_o may_v have_v the_o subtendent_fw-la at_o one_o work_n by_o a_o quadrobiquadraequation_n as_o the_o bucarnon_n do_v instruct_v we_o who_o demonstration_n you_o have_v plain_o set_v down_o in_o the_o fourteen_o of_o my_o apodictick_n the_o second_o mood_n of_o this_o figure_n be_v etenar_n which_o include_v all_o those_o question_n wherein_o the_o two_o contain_v side_n be_v give_v one_o of_o the_o oblique_n be_v require_v and_o by_o its_o resolver_n e_z ge_n rad_n ☞_o toge_z manifest_v that_o if_o from_o the_o sum_n of_o the_o radius_fw-la and_o logarithm_n of_o the_o side_n give_v we_o subtract_v the_o logarithm_n of_o the_o other_o propose_v side_n the_o remainder_n will_v afford_v the_o tangent_fw-la of_o the_o angle_n opposite_a to_o one_o of_o the_o give_v side_n the_o compliment_n of_o which_o angle_n to_o a_o right_a one_o be_v always_o the_o measure_n of_o the_o other_o angle_n by_o the_o five_o of_o my_o apodictick_n for_o it_o be_v as_o the_o one_o ambient_fw-la be_v to_o the_o other_o ambient_fw-la so_o the_o total_a sine_fw-la to_o the_o tangent_fw-la of_o a_o angle_n which_o find_v out_o be_v either_o the_o angle_n require_v or_o the_o compliment_n thereof_o to_o a_o right_a angle_n the_o reason_n of_o this_o analogy_n be_v ground_v on_o the_o second_o branch_n of_o the_o first_o axiom_n as_o by_o the_o characteristic_a of_o the_o directory_n be_v perceivable_a enough_o to_o any_o industrious_a reader_n of_o the_o planobliquangular_a triangle_n there_o be_v four_o figure_n alaheme_n emenarole_n enerolome_n and_o erelea_n the_o first_o and_o last_o of_o these_o four_o be_v monotropall_a figure_n and_o have_v but_o each_o one_o mood_n but_o the_o other_o two_o have_v a_o couple_n a_o piece_n so_o that_o for_o the_o planobliquangular_n all_o the_o four_o together_o afford_v we_o six_o datoquaeres_fw-la the_o mood_n of_o alaheme_n be_v danarele_n which_o comprehend_v all_o those_o problem_n wherein_o two_o angles_n be_v give_v and_o a_o side_n another_o side_n be_v demand_v and_o by_o its_o resolver_n sapeg_n eglantine_n sapyr_n ☞_o your_o show_v that_o if_o to_o the_o sum_n of_o the_o logarithm_n of_o the_o side_n give_v and_o of_o the_o sine_fw-la of_o the_o angle_n opposite_a to_o the_o side_n require_v we_o add_v the_o difference_n of_o the_o secant_fw-la compliment_n from_o the_o radius_fw-la by_o some_o call_v the_o arithmetical_a compliment_n of_o the_o sine_fw-la and_o in_o master_n speidel_n logarithmicall_a canon_n of_o sines_n tangent_n and_o secant_v with_o good_a reason_n term_v the_o secant_fw-la for_o though_o it_o do_v not_o cut_v any_o arch_n thereby_o more_o etymological_o to_o deserve_v the_o name_n of_o secant_fw-la yet_o work_v it_o the_o same_o effect_n that_o the_o prolong_v radius_fw-la do_v the_o operation_n will_v proceed_v so_o neat_o that_o if_o from_o these_o three_o logarithm_n thus_o sum_v up_o we_o only_o cut_v off_o a_o digit_fw-la at_o the_o left_a hand_n we_o will_v find_v as_o much_o by_o addition_n alone_o perform_v in_o this_o case_n as_o if_o from_o the_o propose_a sum_n the_o sine_fw-la of_o the_o angle_n have_v be_v abstract_v for_o the_o total_a sine_fw-la thus_o unradiuse_v be_v the_o logarithm_n of_o the_o side_n require_v but_o such_o as_o be_v not_o acquaint_v with_o this_o compendious_a manner_n of_o calculate_v or_o peradventure_o be_v not_o accommodate_v with_o a_o convenient_a canon_n for_o the_o purpose_n may_v in_o god_n name_n use_v their_o own_o way_n the_o resolver_n be_v of_o such_o amplitude_n that_o it_o extend_v itself_o to_o all_o sort_n of_o operation_n whereby_o the_o truth_n of_o the_o four_o ternary_n in_o this_o mood_n may_v be_v attain_v unto_o for_o it_o be_v analogise_v thus_o as_o the_o sine_fw-la of_o the_o angle_n opposite_a to_o the_o side_n give_v be_v to_o the_o same_o give_v side_n so_o the_o sine_fw-la of_o the_o angle_n opposite_a to_o the_o side_n require_v to_o the_o require_a side_n the_o reason_n of_o this_o proportion_n be_v ground_v on_o the_o second_o axiom_n the_o first_o determinater_n of_o who_o directory_n show_v that_o danarele_n be_v one_o of_o eprosos_n enodandas_n the_o second_o figure_n of_o the_o planobliquangular_n be_v emenarole_n who_o mood_n be_v therelabmo_n and_o zelemabne_n the_o first_o comprehend_v all_o those_o planobliquangular_a problem_n wherein_o two_o side_n be_v give_v with_o a_o interjacent_a angle_n a_o opposite_a angle_n be_v demand_v and_o by_o its_o resolver_n aggres_n ze_n talfagros_n ☞_o talzos_n show_v that_o if_o from_o the_o sum_n of_o the_o logarithm_n of_o the_o difference_n of_o the_o side_n and_o tangent_fw-la of_o half_a the_o sum_n of_o the_o opposite_a angle_n be_v subduce_v the_o aggregat_fw-la or_o sum_n of_o the_o logarithm_n of_o the_o two_o propose_a side_n the_o remainder_n thereof_o will_v prove_v the_o logarithm_n of_o the_o tangent_fw-la of_o half_a the_o difference_n of_o the_o opposite_a angle_n the_o which_o join_v to_o the_o one_o and_o abstract_v from_o the_o other_o afford_v we_o the_o measure_n of_o the_o angle_n we_o require_v for_o the_o theorem_a be_v as_o the_o aggregat_fw-la of_o the_o give_v side_n to_o the_o difference_n of_o thâ—se_a side_n so_o the_o tangent_fw-la of_o half_a the_o sum_n of_o the_o opposite_a angle_n to_o the_o tangent_fw-la of_o half_a the_o difference_n of_o those_o angles_n which_o without_o any_o more_o ado_n by_o simple_a addition_n and_o subtraction_n afford_v the_o angle_n we_o demand_v the_o three_o axiom_n and_o the_o theorem_fw-la of_o the_o resolver_n of_o this_o mood_n be_v but_o one_o and_o the_o same_o thing_n i_o must_v make_v bold_a to_o remit_v you_o to_o my_o apodictick_n for_o the_o reason_n of_o the_o analogy_n thereof_o the_o only_a determinater_n of_o who_o directory_n be_v θ._n point_v out_o the_o mood_n of_o therelabmo_n for_o the_o sole_a enodandum_n appropriate_v thereunto_o the_o second_o mood_n of_o this_o figure_n be_v zelemabue_n which_o involve_v all_o the_o planobliquangulary_a problem_n wherein_o two_o side_n be_v give_v with_o the_o angle_n between_o the_o three_o side_n be_v demand_v and_o not_o be_v calculable_a by_o the_o logarithmicall_a canon_n in_o less_o than_o two_o operation_n because_o it_o require_v the_o
the_o deify_n of_o the_o revealer_n of_o so_o great_a a_o mystery_n and_o true_o beside_o they_o a_o great_a many_o other_o learned_a man_n who_o for_o the_o laboriousnesse_n of_o long_a and_o various_a multiplication_n division_n and_o radical_a extraction_n of_o several_a sort_n be_v deter_v from_o the_o prosecute_n and_o divulge_v of_o their_o knowledge_n in_o the_o chief_a and_o most_o noble_a part_n of_o the_o mathematics_n will_v have_v leave_v behind_o they_o diverse_a exquisite_a volume_n of_o a_o incomprehensible_a value_n if_o the_o arithmetical_a equality_n of_o difference_n agreeable_a to_o every_o continue_a geometrical_a proportion_n have_v be_v make_v know_v unto_o they_o wherefore_o i_o be_o infallible_o persuade_v that_o in_o the_o estimation_n of_o scientifical_o dispose_v spirit_n the_o philosopher_n stone_n be_v but_o trash_n to_o this_o invention_n which_o will_v always_o in_o their_o judicious_a opinion_n be_v account_v of_o more_o worth_n to_o the_o mathematical_a world_n than_o be_v the_o find_v out_o of_o america_n to_o the_o king_n of_o spain_n or_o the_o discovery_n of_o the_o near_a way_n to_o the_o eastindies_a will_v be_v to_o the_o northerly_a occidental_a merchant_n what_o the_o merit_n then_o of_o the_o author_n be_v let_v the_o most_o envious_a judge_n for_o my_o own_o part_n i_o do_v not_o praise_v he_o so_o much_o for_o that_o he_o be_v my_o compatriot_n as_o i_o must_v extol_v the_o happiness_n of_o my_o country_n for_o have_v produce_v so_o brave_a a_o spark_n in_o who_o alone_o i_o may_v with_o confidence_n aver_v it_o be_v more_o glorious_a then_o if_o it_o have_v be_v the_o conquer_a kingdom_n of_o five_o hundred_o potent_a nation_n for_o by_o how_o much_o the_o gift_n of_o the_o mind_n be_v more_o excellent_a than_o those_o of_o either_o body_n or_o fortune_n by_o so_o much_o the_o divine_a effect_n of_o the_o faculty_n thereof_o be_v of_o great_a consequence_n than_o what_o be_v perform_v by_o mere_a force_n of_o arm_n or_o chance_n of_o war_n i_o may_v say_v more_o in_o commendation_n of_o this_o gallant_a man_n but_o that_o my_o discourse_n be_v direct_v to_o the_o reader_n he_o will_v possible_o expect_v to_o be_v entertain_v with_o some_o other_o purpose_n than_o encomiastics_n and_o therefore_o to_o undeceive_v he_o of_o those_o hope_n if_o any_o such_o there_o be_v i_o will_v assure_v he_o that_o to_o no_o other_o end_n i_o do_v require_v his_o observance_n here_o but_o to_o be_v inform_v by_o i_o of_o the_o laudable_a endowment_n of_o that_o honourable_a baron_n who_o eminency_n above_o other_o wher-eve_a he_o be_v speak_v of_o deserve_v such_o a_o ample_a elegy_n by_o itself_o that_o the_o paper_n grace_v with_o the_o receive_n of_o his_o name_n and_o character_n shall_v not_o be_v blur_v with_o the_o course_n impression_n of_o any_o other_o stuff_n however_o the_o reader_n ought_v not_o to_o conceive_v amiss_o for_o his_o be_v detain_v so_o long_o upon_o this_o eulogistick_a subject_n without_o the_o variety_n of_o any_o peculiar_a instruction_n bestow_v on_o he_o see_v i_o be_o certain_a there_o be_v nothing_o more_o advantageous_a to_o he_o or_o that_o more_o efficacious_o can_v tend_v to_o his_o improvement_n than_o the_o imitation_n of_o that_o admirable_a gentleman_n who_o immortal_a fame_n in_o spite_n of_o time_n will_v outlast_v all_o age_n and_o look_v eternity_n in_o the_o face_n the_o reader_n wellwisher_a t._n u._n a_o epaenetick_a and_o doxologetick_a express_a in_o commendation_n of_o this_o book_n and_o the_o author_n thereof_o to_o all_o philomathet_n see_v trigonometry_n which_o handsome_o unlock_v the_o choice_a and_o most_o intime_fw-la mystery_n of_o the_o mathematics_n have_v be_v hitherto_o expose_v to_o the_o world_n in_o a_o method_n who_o intricacy_n deter_v many_o from_o adventure_v on_o it_o we_o be_v all_o and_o every_o one_o of_o we_o by_o duty_n bind_v to_o acknowledge_v ourselves_o behold_v to_o the_o author_n of_o this_o treatise_n who_o by_o reduce_v all_o the_o secret_n of_o that_o noble_a science_n into_o a_o most_o exquisite_a order_n have_v so_o facilitated_a the_o way_n to_o the_o learner_n that_o in_o seven_o week_n at_o most_o he_o may_v attain_v to_o more_o knowledge_n therein_o then_o otherwise_o he_o can_v do_v for_o his_o heart_n in_o the_o space_n of_o a_o twelvemonth_n and_o who_o for_o the_o better_a encouragement_n of_o the_o studious_a have_v so_o gentile_o expatiate_v his_o spirit_n upon_o all_o its_o actiom_n principle_n analogy_n precept_n and_o whole_a subject_v matter_n that_o this_o mathematical_a tractate_n do_v no_o less_o bespeak_v he_o a_o good_a poet_n and_o good_a orator_n then_o by_o his_o elabour_v poem_n he_o have_v show_v himself_o already_o a_o good_a philosopher_n and_o mathematician_n thus_o do_v the_o various_a mixture_n of_o most_o excellent_a quality_n in_o he_o give_v such_o evidence_n of_o the_o transcendent_a faculty_n of_o his_o mind_n that_o as_o the_o muse_n never_o yet_o inspire_v sublime_a conception_n in_o a_o more_o refine_a stile_n then_o be_v to_o be_v find_v in_o the_o accurate_a strain_n of_o his_o most_o ingenious_a epigram_n so_o on_o the_o other_o part_n be_v the_o abstruse_a difficulty_n of_o this_o science_n by_o he_o so_o neat_o unfold_v and_o with_o such_o exactness_n have_v he_o resolve_v the_o hard_a and_o most_o entangle_a doubt_n thereof_o that_o i_o may_v just_o say_v what_o praise_n in_o his_o epistle_n or_o rather_o preface_n to_o the_o reader_n he_o have_v be_v please_v out_o of_o his_o ingenuity_n to_o confer_v on_o the_o learned_a and_o honourable_a neper_n do_v without_o any_o diminution_n in_o every_o jot_n as_o due_o belong_v unto_o himself_o for_o i_o be_o certain_o persuade_v he_o that_o use_v logarithm_n shall_v not_o gain_v so_o much_o time_n on_o the_o worker_n by_o the_o natural_a sin_n and_o tangent_n as_o by_o virtue_n of_o this_o succinct_a manner_n of_o calculation_n shall_v be_v get_v on_o he_o that_o know_v it_o not_o how_o compendious_o soever_o else_o with_o addition_n and_o subtraction_n or_o addition_n alone_o he_o frame_v his_o computation_n however_o he_o who_o together_o with_o that_o of_o the_o logarithm_n make_v use_n of_o this_o invention_n be_v in_o a_o way_n which_o will_v bring_v he_o so_o straight_o and_o ready_o to_o the_o perfect_a practice_n of_o trigonometry_n that_o compare_v with_o the_o old_a beat_a path_n tread_v upon_o by_o regiomontanus_n ptolemy_n and_o other_o ancient_a mathematician_n it_o be_v like_o the_o sea_n voyage_n in_o regard_n of_o that_o by_o land_n betwixt_o the_o two_o pillar_n of_o hercules_n common_o call_v the_o strait_o of_o gibraltar_n whereof_o the_o one_o be_v but_o of_o six_o hour_n sail_v at_o most_o and_o the_o other_o a_o journey_n of_o seven_o thousand_o long_a mile_n if_o we_o then_o consider_v how_o a_o great_a many_o despair_v ever_o to_o get_v out_o if_o once_o enter_v of_o the_o confuse_a obscurity_n wherein_o the_o doctrine_n of_o triangle_n have_v be_v from_o time_n to_o time_n involve_v have_v rather_o content_v themselves_o bare_o with_o scale_n and_o compass_n and_o other_o mechanic_n tool_n and_o instrument_n to_o prosecute_v their_o operation_n and_o in_o any_o reasonable_a measure_n to_o glance_v somewhat_o near_o the_o truth_n then_o through_o so_o many_o pesterment_n and_o harsh_a encumbrance_n to_o touch_v it_o to_o a_o point_n in_o its_o most_o indivisible_a and_o infallible_a reality_n and_o how_o other_o for_o all_o their_o being_n more_o industrious_a in_o prove_v their_o conclusion_n by_o the_o medium_n from_o which_o they_o be_v necessary_o infer_v be_v nevertheless_o even_o when_o they_o have_v bestow_v half_o a_o age_n in_o the_o trigonometricall_a practice_n oftentimes_o so_o far_o to_o seek_v that_o without_o a_o great_a deal_n of_o premeditation_n advisement_n and_o recollect_v of_o themselves_o they_o know_v not_o how_o to_o discuss_v some_o query_n corollary_n problem_n consectary_n proportion_n way_n of_o perpendicular_a fall_n and_o other_o such_o like_o occur_v debatable_a matter_n incident_a to_o the_o scientifick_a measure_n of_o triangle_n we_o can_v choose_v these_o thing_n be_v mature_o perpend_v but_o be_v much_o take_v with_o the_o pregnancy_n of_o this_o device_n whereby_o we_o shall_v soon_o hit_v to_o a_o minute_n upon_o the_o verity_n of_o a_o angle_n or_o side_n demand_v and_o trace_v it_o to_o the_o very_a source_n and_o original_n from_o whence_o it_o flow_v than_o another_o mechanical_o shall_v be_v able_a to_o come_v within_o three_o degree_n thereof_o although_o he_o can_v for_o the_o same_o little_a he_o do_v afford_v any_o reason_n at_o all_o and_o so_o sudden_o resolve_v any_o trigonometricall_a question_n without_o pain_n or_o labour_n how_o perplex_v soever_o it_o be_v with_o all_o the_o dependence_n thereto_o belong_v as_o if_o it_o be_v a_o knowledge_n mere_o infuse_v from_o above_o and_o reveal_v by_o the_o peculiar_a inspiration_n of_o some_o
favourable_a angel_n beside_o these_o advantage_n administer_v unto_o we_o by_o the_o mean_n of_o this_o exquisite_a book_n this_o main_a commodity_n accrue_v to_o the_o diligent_a peruser_n of_o it_o that_o instead_o of_o three_o quarter_n of_o a_o year_n usual_o by_o professor_n allow_v to_o their_o scholar_n for_o the_o right_a conceive_n of_o this_o science_n which_o notwithstanding_o through_o any_o little_a discontinuance_n be_v by_o they_o so_o apt_a to_o be_v forget_v that_o the_o expense_n of_o a_o week_n or_o two_o will_v hardly_o suffice_v to_o reseat_v it_o in_o their_o memory_n they_o shall_v not_o need_v by_o this_o method_n to_o bestow_v above_o a_o month_n and_o with_o such_o ease_n and_o facility_n for_o retention_n when_o they_o have_v learn_v and_o acquire_v it_o that_o if_o multiplicity_n of_o business_n or_o serious_a plod_n upon_o other_o study_n happen_v to_o blot_v it_o out_o of_o their_o mind_n they_o may_v as_o firm_o recover_v in_o one_o quarter_n of_o a_o hour_n the_o whole_a knowledge_n and_o remembrance_n thereof_o as_o when_o they_o have_v it_o best_o and_o be_v most_o punctual_o verse_v in_o it_o a_o secret_a in_o my_o opinion_n so_o precious_a that_o as_o the_o author_n speak_v of_o marchiston_n i_o may_v with_o the_o like_a pertinencie_n avouch_v of_o he_o that_o his_o country_n and_o kindred_n will_v not_o have_v be_v more_o honour_v by_o he_o have_v he_o purchase_v million_o of_o gold_n and_o several_a rich_a territory_n of_o a_o great_a and_o vast_a extent_n then_o for_o this_o subtle_a and_o divine_a invention_n which_o will_v outlast_v the_o continuance_n of_o any_o inheritance_n and_o remain_v fresh_a in_o the_o understanding_n of_o man_n of_o profound_a literature_n when_o house_n and_o possession_n will_v change_v their_o owner_n the_o wealthy_a become_v poor_a and_o the_o child_n of_o the_o needy_a enjoy_v the_o treasure_n of_o those_o who_o heir_n be_v impoverish_v therefore_o see_v for_o the_o manifold_n use_v thereof_o in_o divers_a art_n and_o science_n in_o speculation_n and_o practice_n peace_n and_o war_n sport_n and_o earnest_n with_o the_o admirable_a furtherance_n we_o reap_v by_o it_o in_o the_o knowledge_n of_o sea_n and_o land_n and_o heaven_n and_o earth_n it_o can_v be_v otherwise_o then_o permanent_a together_o with_o the_o author_n fame_n so_o long_o as_o any_o of_o those_o endure_v i_o will_v god_n will_v in_o the_o ruin_n of_o all_o these_o and_o when_o time_n itself_o be_v expire_v in_o testimony_n of_o my_o thankfulness_n in_o particular_a for_o so_o great_a a_o benefit_n if_o after_o the_o resurrection_n there_o be_v any_o complemental_a affability_n express_v myself_o then_o as_o i_o do_v now_o the_o author_n most_o affectionate_a and_o most_o humble_o devote_a servant_n j._n a._n the_o diatyposis_n of_o the_o whole_a doctrine_n of_o triangle_n the_o plane_n triangle_n have_v 13._o mood_n planorectangular_n 7._o 1_o upalem_n 2_o uberman_n 3_o uphener_n 4_o ekarul_a 5_o egalem_fw-la 6_o echemun_n 7_o etenaâ—_n planobliquangulars_n 6._o 1_o danarele_n 2_o thereâ—abmo_n 3_o zelemabne_n 4_o xemenoro_n 5_o shenerolem_n 6_o pserelema_n the_o sphericals_n have_n 28_o mood_n orthogonosphericals_n 16._o 1._o upalam_fw-la 2._o ubamen_fw-la 3._o uphanep_v 4._o ukelamb_n 5._o ugemon_n 6._o uchener_n 7._o eâ—alum_n 8._o edamon_n 9_o ethaner_n 10._o ezolum_fw-la 11._o exoman_n 12._o epsoner_n 13._o alamun_n 14._o amane_a 15._o enerul_a 16._o eâ—elam_fw-la of_o these_o 16._o mood_n 〈◊〉_d mood_n of_o vâ—â—â—gen_n â—re_n pround_v upon_o the_o axiâ—â—re_n of_o suproâ—â—â—_n the_o 8._o of_o pubkutetkepsaler_n on_o sbaprotea_n and_o the_o 5._o of_o uchedezexam_n on_o proso_n by_o these_o 16._o representative_n 1._o leâ—_n 2._o yet_o 3._o râ—c_fw-la 4._o cle._n 5._o lu._n 6._o tul._n 7._o tere._n 8._o tol._n 9_o leâ—_n 10._o at._n 11._o tul._n 12._o clet_n 13._o great_fw-mi 14._o tur._n 15._o tur._n 16._o le_fw-fr a._n signify_v a_o oblique_a angle_n e._n the_o perpendicular_a u._n the_o subtendent_fw-la c._n initial_a the_o compliment_n of_o a_o side_n to_o a_o quadrant_a â—_o final_a the_o side_n continue_v to_o the_o radius_fw-la or_o a_o quadrant_n i._o leave_v r._n right_a and_o t._n one_o of_o the_o top_n triangle_n of_o the_o scheme_n it_o be_v evidence_v in_o what_o part_n of_o the_o diagram_n the_o analogy_n of_o my_o of_o the_o 16._o mood_n begin_v which_o be_v once_o know_v the_o progressive_a sequence_n of_o the_o proportionable_a side_n &_o angle_n be_v easy_o discern_v out_o of_o the_o orderly_a in_o volution_n of_o the_o figure_n itself_o here_o it_o be_v to_o be_v observe_v that_o as_o the_o book_n explain_v the_o trissotetral_a table_n so_o this_o trigonodiatyposis_n unfold_v all_o the_o intricate_a difficulty_n of_o the_o book_n loxogonosphericals_n 12._o that_o the_o scheme_n and_o type_n of_o triangulary_a analogy_n be_v not_o seat_v in_o the_o room_n where_o they_o be_v treat_v of_o i_o purposely_o have_v do_v it_o to_o the_o end_n that_o be_v all_o perceive_v at_o one_o view_n their_o multiplicity_n which_o will_v appear_v confuse_v in_o their_o dispersed_a method_n may_v â—â—ot_v any_o way_n discourage_v 〈…〉_z der_fw-ge beside_o that_o this_o their_o â—eing_v together_o in_o their_o â—uâ—_n order_n and_o rank_v record_v to_o the_o exigence_n of_o the_o side_n or_o angle_n be_v such_o a_o furtherance_n to_o the_o memory_n and_o illustration_n to_o the_o judgement_n that_o it_o make_v trigonometry_n which_o of_o all_o science_n be_v account_v the_o abstruse_a to_o be_v in_o effect_n the_o most_o 〈◊〉_d and_o 〈◊〉_d monurgetick_n 4_o 1_o lamaneprep_n 2_o menerolo_n 3_o nerelema_n 4_o ralamâ—â—â—_n of_o the_o disergetick_n there_o be_v 8._o mood_n each_o whereof_o be_v divide_v into_o four_o case_n ahalebmane_n 1._o alamebna_n dasimforaug_n dadisfoâ—â—ug_n dadisgatin_n simomatin_n 2._o alamebne_n dasimforauxy_n dadiscracforeug_n dadiscramgatin_n simomatin_n ahamepnare_fw-la 〈…〉_z dadissepamforaur_n dadissexamforeur_fw-fr dasimatin_n simomatin_n 4._o ammaneprela_n dadissepamfor_n dadisseâ—amfor_n dasimin_n simomatin_n ehenabrole_fw-fr 〈…〉_z dacramfor_n damracfor_n dasimquzin_a simomatin_n 6._o eâ—neralâ—la_n dacforamb_v damforac_n daâ—imatâ—m_fw-la simomatin_n eherolabme_o 〈…〉_z dacracforeur_fw-fr dambracforeur_fw-fr dacrambatin_n simomatin_n 8._o eâ—relome_o dakyxamfor_n dambyxamfor_n dakypambin_n simomatin_n in_o eruditum_fw-la d._n thomae_fw-la vrquharti_fw-la equitis_fw-la trissotetrados_n librum_fw-la si_fw-it cupis_fw-la aetherios_fw-la tutò_fw-la peragrare_fw-la meatus_fw-la et_fw-la sulcare_fw-la audes_fw-ge si_fw-mi vada_fw-la salsa_fw-la maris_fw-la well_fw-mi tibi_fw-la si_fw-la cordi_fw-la est_fw-la terrae_fw-la spatia_fw-la ampla_fw-la metiri_fw-la huc_fw-la ades_fw-la hunc_fw-la doctum_fw-la percipe_fw-la ment_fw-la librum_fw-la hoc_fw-la sine_fw-la daedaleis_fw-la pennis_fw-la volitare_fw-la per_fw-la auras_fw-la et_fw-la sine_fw-la neptuno_fw-it near_fw-mi per_fw-la alta_fw-la vale_n hoc_fw-la deuce_fw-la jam_fw-la lybicos_fw-la poteris_fw-la superare_fw-la calores_fw-la atque_fw-la pati_fw-la scythici_n frigora_fw-la saeva_fw-la poli_fw-fr pergo_fw-la thoma_n tali_fw-la tandem_fw-la gaudebit_fw-la alumno_fw-la scotia_n quam_fw-la scriptis_fw-la tollis_fw-la in_o astra_fw-la tuis_fw-la al._n ross_n position_n every_o circle_n be_v divide_v into_o three_o hundred_o and_o sixty_o part_n call_v degree_n whereof_o each_o one_o be_v sexagesimate_v subsexagesimate_v resubsexagesimate_v and_o biresubsexagesimate_v in_o minute_n second_o three_o fourths_o and_o so_o far_o forth_o as_o any_o computist_n be_v please_v to_o proceed_v for_o the_o exactness_n of_o a_o research_n in_o the_o calculation_n of_o any_o orbiculary_a dimension_n 2._o as_o degree_n be_v the_o measure_n of_o arch_n so_o be_v they_o of_o angle_n but_o that_o those_o be_v call_v circumferentiall_a these_o angulary_a degree_n each_o whereof_o be_v the_o three_o hundred_o and_o sixty_v part_n of_o four_o right_a angle_n which_o be_v nothing_o else_o but_o the_o surface_n of_o a_o plain_a to_o any_o point_n circumjacent_a for_o any_o space_n whatsoever_o about_o a_o point_n be_v divide_v in_o 360._o part_n and_o the_o better_a to_o conceive_v the_o analogy_n that_o be_v betwixt_o these_o two_o sort_n of_o gradual_a measure_n we_o must_v know_v that_o there_o be_v the_o same_o proportion_n of_o any_o angle_n to_o 4._o right_a angle_n as_o of_o a_o arch_n of_o so_o many_o circumferential_a degree_n to_o the_o whole_a circumference_n 3._o hence_o be_v it_o that_o the_o same_o number_n serve_v the_o angle_n and_o the_o arch_n that_o vault_n it_o and_o that_o divers_a quantity_n be_v measure_v as_o it_o be_v with_o the_o same_o gradual_a measure_n angle_n and_o arch_n than_o be_v analogical_a and_o the_o same_o reason_n be_v of_o both_o 4._o see_v any_o give_a proportion_n may_v be_v find_v in_o number_n and_o that_o any_o two_o quantity_n have_v the_o same_o proportion_n that_o the_o two_o number_n have_v according_a to_o the_o which_o they_o be_v measure_v if_o for_o the_o measure_n of_o triangle_n there_o must_v be_v certain_a proportion_n of_o all_o the_o part_n of_o a_o triangle_n to_o one_o another_o know_v and_o those_o proportion_n explain_v in_o number_n it_o be_v most_o
practise_v for_o brevity_n sake_n likewise_o it_o be_v my_o main_a design_n in_o the_o frame_n of_o this_o table_n to_o make_v alcapable_a trigonometrically-affected_n student_n with_o much_o facility_n and_o little_a labour_n attain_v to_o the_o whole_a knowledge_n of_o the_o noble_a science_n of_o the_o doctrine_n of_o triangle_n i_o deem_v it_o expedient_a the_o more_o firm_o and_o ready_o to_o imprint_v the_o several_a datoquaeres_fw-la or_o praescinded_a problem_n thereof_o in_o their_o memory_n to_o accommodate_v they_o according_o with_o letter_n proper_a for_o the_o purpose_n which_o if_o the_o ingenious_a reader_n will_v be_v please_v to_o consider_v he_o will_v find_v by_o the_o very_a letter_n themselves_o the_o place_n and_o number_n of_o each_o datoquaere_fw-la this_o be_v the_o reason_n why_o my_o trissotetras_n conform_a to_o the_o etymology_n of_o its_o name_n be_v in_o so_o many_o divers_a ternary_n and_o quaternary_n divide_v and_o that_o the_o sharp_a mean_a blunt_a double_a and_o liquid_a consonant_n of_o the_o greek_a alphabet_n be_v so_o orderly_o bestow_v in_o their_o several_a room_n be_v all_o and_o every_o one_o of_o they_o seat_v according_a to_o the_o nature_n of_o the_o mood_n and_o figure_n who_o characteristic_n they_o be_v three_o the_o mood_n of_o the_o planotriangular_a table_n be_v in_o all_o thirteen_o whereof_o there_o be_v seven_o rectangular_a and_o six_o obliquangular_a be_v fit_o comprehend_v by_o the_o three_o blunt_a three_o mean_a three_o sharp_a and_o sour_a double_a consonant_n the_o hebrew_n shin_n be_v account_v for_o one_o of_o they_o four_o the_o sixteen_o mood_n of_o the_o orthogonosphericall_a trissotetras_n be_v contain_v under_o three_o sharp_a three_o mean_a three_o blunt_a three_o double_a and_o four_o liquid_n which_o four_o do_v orderly_o particularise_v the_o binary_n of_o the_o last_o two_o figure_n five_o the_o four_o monurgetick_a loxogonosphericals_n be_v deciphr_v by_o each_o its_o own_o liquid_a in_o front_n according_a to_o their_o literal_a order_n six_o the_o eight_o loxogonosphericall_a disergetick_n be_v also_o distinguish_v by_o the_o four_o liquid_n but_o with_o this_o difference_n from_o the_o monurgetick_n that_o the_o vowel_n of_o a_o and_o e_o precede_v they_o in_o the_o first_o syllable_n import_v thereby_o the_o datas_fw-la of_o a_o angle_n or_o a_o side_n now_o because_o these_o disergetick_n be_v eight_o in_o all_o there_o be_v allot_v to_o every_o liquid_a that_o characterise_v the_o figure_n the_o better_a to_o diversify_v the_o first_o and_o second_o datas_fw-la of_o each_o respective_a binarie_n from_o one_o another_o in_o so_o far_o as_o they_o have_v reference_n to_o each_o its_o own_o quaesitum_fw-la the_o figurative_a liquid_a be_v double_v when_o a_o side_n be_v require_v and_o remain_v single_a when_o a_o angle_n furthermore_o in_o the_o oblique_a sphericodisergetick_n so_o far_o as_o the_o sense_n of_o the_o resolver_n can_v bear_v it_o i_o do_v trinifie_v they_o with_o letter_n convenient_a for_o the_o purpose_n according_a to_o the_o several_a case_n of_o their_o datoquaeres_fw-la who_o diversity_n reach_v not_o above_o the_o extent_n of_o Ï€._fw-la β._fw-la φ._fw-la and_o Ï„._fw-la δ._fw-la θ._fw-la i_o have_v almost_o omit_v to_o tell_v you_o that_o for_o the_o more_o variety_n in_o the_o last_o two_o figure_n of_o the_o orthogonosphericals_n be_v set_v down_o the_o two_o letter_n of_o ch._n and_o shin_n the_o first_o a_o spanish_a and_o the_o second_o a_o hebrew_n letter_n now_o if_o to_o those_o help_n for_o the_o memory_n which_o in_o this_o table_n i_o have_v afford_v the_o reader_n both_o by_o the_o alphabetical_a order_n of_o some_o consonant_n and_o homogeneity_n of_o other_o in_o their_o affection_n of_o sharpness_n meanness_n obtusity_n and_o duplicity_n he_o join_v that_o artificial_a aid_n in_o have_v every_o part_n of_o thâ—â—_n chemâ—locally_o in_o his_o mind_n of_o all_o way_n both_o for_o facility_n in_o remember_v and_o steadfastness_n of_o retention_n without_o doubt_n the_o most_o expedite_a or_o otherwise_o place_v the_o representatives_n of_o word_n according_a to_o the_o method_n of_o the_o art_n of_o memory_n in_o the_o several_a corner_n of_o a_o house_n which_o in_o regard_n of_o their_o paucity_n be_v containable_a within_o a_o parlour_n or_o dine_a room_n at_o most_o he_o may_v with_o ease_n get_v they_o all_o by_o heart_n in_o less_o than_o the_o space_n of_o a_o hour_n which_o be_v no_o great_a expense_n of_o time_n though_o bestow_v on_o matter_n of_o mean_a consequence_n the_o commentary_n the_o axiom_n of_o plain_a triangle_n be_v four_o viz._n rulerst_n eproso_n grediftal_a and_o bagrediffus_n ruler_v that_o be_v to_o say_v the_o subtendent_fw-la in_o plain_a triangle_n may_v be_v either_o radius_fw-la or_o secant_fw-la and_o the_o ambient_n either_o radius_fw-la sines_n or_o tangent_n for_o it_o be_v a_o maxim_n in_o planangular_a triangle_n that_o any_o side_n may_v be_v put_v for_o radius_fw-la ground_v on_o this_o that_o from_o any_o point_n at_o any_o distance_n a_o circle_n may_v be_v describe_v therefore_o if_o any_o of_o the_o side_n of_o a_o plain_a triangle_n be_v give_v together_o with_o the_o angler_n each_o of_o the_o other_o two_o side_n be_v give_v by_o a_o threefold_a proportion_n that_o be_v whether_o you_o put_v that_o or_o this_o or_o the_o three_o side_n for_o the_o radius_fw-la which_o difference_n occasion_v both_o in_o plain_a and_o spherical_a triangle_n great_a variety_n in_o their_o calculation_n the_o branch_n of_o this_o axiom_n be_v vradesso_n and_o eradetul_n vradesso_fw-it that_o be_v when_o the_o hypotenusa_fw-la be_v radius_fw-la the_o side_n be_v sin_n of_o their_o opposite_a angle_n so_o that_o there_o be_v two_o arch_n describe_v with_o that_o hypotenusal_a identity_n of_o distance_n who_o centre_n be_v in_o the_o two_o extremity_n of_o the_o subtendent_fw-la for_o so_o the_o case_n will_v be_v make_v plain_a in_o both_o the_o leg_n which_o otherwise_o will_v not_o appear_v but_o in_o one_o eradetul_n when_o any_o of_o the_o side_n be_v radius_fw-la the_o other_o of_o they_o be_v a_o tangent_fw-la and_o the_o subtendent_fw-la a_o secant_fw-la the_o reason_n of_o this_o be_v find_v in_o the_o very_a definition_n of_o the_o sin_n tangent_n and_o secant_v to_o the_o which_o if_o the_o reader_n please_v he_o may_v have_v recourse_n for_o i_o have_v set_v they_o down_o among_o my_o definition_n hence_o it_o be_v according_a to_o mr._n speidel_n observation_n in_o his_o book_n of_o sphericals_n that_o the_o sine_fw-la of_o any_o arch_n be_v radius_fw-la that_o which_o be_v the_o total_a sine_fw-la become_v the_o secant_fw-la compliment_n of_o the_o say_a arch_n and_o that_o the_o tangent_fw-la of_o any_o arch_n be_v radius_fw-la what_o be_v radius_fw-la become_v tangent_fw-la compliment_n of_o that_o arch._n the_o directory_n of_o this_o axiom_n be_v vphechâ—t_o which_o show_v we_o that_o there_o be_v three_o planorectangular_a enodandas_n belong_v thereto_o viz._n vphener_n echemun_n and_o etenar_n as_o for_o pserelema_n which_o be_v the_o loxogonian_a one_o point_v at_o in_o my_o trissotetras_n because_o it_o be_v but_o a_o partial_a enodandum_fw-la i_o have_v purposely_o omit_v to_o mention_v it_o in_o the_o directory_n of_o eradetul_n the_o second_o axiom_n be_v eprosâ—_n that_o be_v the_o side_n be_v proportional_a to_o one_o another_o as_o the_o sin_n of_o their_o opposite_a angle_n for_o see_v about_o any_o triangle_n a_o circle_n may_v be_v circumscribe_v in_o which_o case_n each_o side_n be_v a_o cord_n or_o subtense_n the_o half_a whereof_o be_v the_o sine_fw-la of_o its_o opposite_a angle_n and_o there_o be_v always_o the_o same_o reason_n of_o the_o whole_a to_o the_o whole_a as_o of_o the_o half_a to_o the_o half_a the_o side_n must_v needs_o be_v proportional_a to_o one_o another_o as_o the_o sin_n of_o their_o opposite_a angle_n quod_fw-la probandum_fw-la erat_fw-la the_o directory_n of_o this_o second_o axiom_n be_v pubkegdaxesh_a which_o declare_v that_o there_o be_v seven_o enodandas_n ground_v on_o it_o to_o wit_n four_o rectangular_a upalem_n ubeman_n ekarul_a egalem_fw-la and_o three_o obliquangular_a danarele_n xemenoro_n and_o shenerolem_n the_o three_o axiom_n be_v grediftal_a that_o be_v in_o all_o plain_a triangle_n as_o the_o sum_n of_o the_o two_o side_n be_v to_o their_o difference_n so_o be_v the_o tangent_fw-la of_o the_o half_a sum_n of_o the_o opposite_a angle_n to_o the_o tangent_fw-la of_o half_a their_o difference_n for_o if_o a_o line_n be_v draw_v equal_a to_o the_o sum_n of_o the_o two_o side_n and_o if_o on_o the_o point_n of_o extension_n with_o the_o distance_n of_o the_o short_a side_n a_o semicircle_n be_v describe_v and_o that_o from_o the_o extremity_n of_o the_o protract_a line_n a_o diameter_n be_v draw_v through_o the_o circle_n where_o it_o touch_v the_o top_n of_o the_o triangle_n in_o question_n till_o it_o occur_v with_o a_o parallel_n to_o the_o three_o side_n there_o will_v
arise_v two_o equicrural_a triangle_n one_o whereof_o have_v one_o angle_n common_a with_o the_o triangle_n propose_v and_o the_o three_o of_o the_o one_o be_v equal_a to_o the_o three_o of_o the_o other_o any_o one_o of_o the_o equal_a angle_n in_o the_o foresay_a isosceles_n must_v needs_o be_v the_o one_o half_a of_o the_o two_o unknown_a angle_n this_o be_v the_o first_o step_n to_o the_o obtain_n of_o what_o we_o demand_v then_o do_v we_o find_v that_o the_o three_o side_n cut_v the_o side_n of_o the_o great_a triangle_n according_a to_o the_o analogy_n require_v which_o be_v perceivable_a enough_o if_o with_o the_o distance_n of_o the_o outmost_a parallel_n from_o the_o low_a end_n thereof_o as_o centre_n be_v describe_v a_o new_a circle_n for_o than_o will_v the_o tangent_n be_v perspicuous_a and_o so_o much_o the_o more_o for_o their_o rectangularity_n the_o one_o with_o the_o radius_fw-la and_o the_o other_o with_o its_o parallel_n which_o be_v touch_v at_o a_o angle_n describe_v in_o a_o semicircle_n confirm_v the_o rectangularity_n of_o both_o by_o the_o parallel_n likewise_o be_v infer_v the_o equality_n of_o the_o alternate_a angle_n who_o addition_n and_o subduction_n to_o and_o from_o half_a the_o sum_n of_o the_o two_o unknown_a angle_n make_v up_o both_o the_o great_a and_o lesser_a angle_n hereby_o it_o be_v evident_a how_o the_o sum_n of_o the_o two_o side_n etc._n etc._n which_o be_v to_o be_v prove_v the_o directory_n of_o this_o three_o axiom_n be_v θ._o only_o for_o it_o have_v no_o enodandum_n but_o therelabmo_fw-la the_o four_o axiom_n be_v bagrediffiu_n that_o be_v as_o the_o base_a or_o great_a side_n be_v to_o the_o sum_n of_o the_o other_o side_n so_o the_o difference_n of_o the_o other_o side_n to_o the_o difference_n of_o the_o segment_n of_o the_o base_a for_o if_o upon_o the_o centre_n of_o the_o vertical_a angle_n with_o the_o distance_n of_o the_o short_a side_n be_v describe_v a_o circle_n it_o will_v so_o cut_v the_o two_o great_a side_n of_o the_o give_v triangle_n that_o find_v thereby_o two_o oblong_n of_o the_o nature_n of_o those_o who_o equality_n be_v demonstrate_v in_o my_o apodictick_n we_o may_v infer_v the_o oblong_o make_v of_o the_o sum_n of_o the_o side_n and_o difference_n of_o the_o side_n be_v equal_a to_o the_o oblong_o make_v of_o the_o base_a and_o the_o difference_n of_o its_o segment_n that_o their_o side_n be_v reciprocal_o proportional_a that_o be_v as_o the_o great_a side_n be_v to_o the_o sum_n of_o the_o other_o side_n so_o the_o difference_n of_o the_o other_o side_n to_o the_o difference_n of_o the_o segment_n of_o the_o base_a or_o great_a side_n the_o directory_n of_o this_o axiom_n be_v θ._n and_o its_o only_a enodandum_n though_o but_o a_o partial_a one_o pserelema_n the_o planorectangular_a table_n have_v four_o figure_n it_o be_v to_o be_v observe_v that_o figure_n here_o be_v not_o take_v geometrical_o but_o in_o the_o sense_n that_o it_o be_v use_v in_o the_o logick_n when_o a_o syllogism_n be_v say_v to_o be_v in_o the_o first_o second_o or_o three_o figure_n for_o as_o there_o by_o the_o various_a application_n of_o the_o medium_fw-la or_o mean_v term_v the_o figure_n be_v constitute_v diverse_a so_o do_v the_o difference_n of_o the_o datas_fw-la in_o a_o triangle_n distinguish_v these_o trissotetrall_a figure_n from_o one_o another_o and_o to_o continue_v yet_o further_o in_o the_o syllogistical_a analogy_n be_v according_a to_o the_o several_a demand_n when_o the_o datas_fw-la be_v the_o same_o subdivide_v into_o mood_n the_o first_o two_o vowel_n give_v notice_n of_o the_o data_n and_o the_o three_o of_o what_o be_v demand_v so_o that_o uale_n and_o euphonetical_o pronounce_v vale_n which_o be_v the_o first_o figure_n show_v that_o the_o subtendent_fw-la and_o one_o angle_n be_v give_v and_o that_o one_o of_o the_o contain_v side_n be_v require_v vemane_n be_v the_o second_o figure_n which_o point_v out_o all_o those_o problem_n wherein_o the_o hypotenusa_fw-la and_o one_o leg_n be_v give_v and_o a_o angle_n or_o the_o other_o leg_n be_v require_v the_o three_o figure_n be_v enave_n which_o comprehend_v all_o the_o problem_n wherein_o one_o of_o the_o ambient_n be_v give_v with_o a_o oblique_a angle_n and_o the_o subtendent_fw-la or_o other_o ambient_fw-la require_v the_o four_o and_o last_o of_o the_o rectangular_a figure_n be_v ereva_n which_o stand_v for_o those_o datoquaeres_fw-la wherein_o the_o including_z side_n be_v give_v and_o the_o subtendent_fw-la or_o a_o angle_n demand_v now_o let_v we_o come_v to_o the_o mood_n of_o those_o figure_n the_o first_o figure_n vale_n have_v but_o one_o mood_n and_o therefore_o of_o as_o great_a extent_n as_o itself_o which_o be_v upalem_n who_o nature_n be_v to_o let_v we_o know_v when_o a_o plane_n right_o angle_v triangle_n be_v give_v we_o to_o resolve_v who_o subtendent_fw-la and_o one_o of_o the_o oblique_n be_v propose_v and_o one_o of_o the_o ambient_n require_v that_o we_o must_v have_v recourse_n unto_o its_o resolver_n which_o be_v rad_n u_o sapy_a ☞_o your_o show_v that_o if_o we_o join_v the_o artificial_a sine_fw-la of_o the_o angle_n opposite_a to_o the_o side_n demand_v with_o the_o logarithm_n of_o the_o subtendent_fw-la the_o sum_n search_v in_o the_o canon_n of_o absolute_a number_n will_v afford_v we_o the_o logarithm_n of_o the_o side_n require_v the_o reason_n hereof_o be_v find_v in_o the_o second_o axiom_n the_o first_o consonant_a of_o who_o directory_n evidence_v that_o upalem_n be_v eprosos_n enodandum_fw-la for_o it_o be_v as_o the_o total_a sine_fw-la to_o the_o hypotenusa_fw-la so_o the_o sine_fw-la of_o the_o angle_n opposite_a to_o the_o side_n require_v be_v to_o the_o say_v require_v side_n according_a to_o the_o nature_n of_o the_o foresay_a axiom_n whereupon_o it_o be_v ground_v the_o second_o figure_n vemane_n have_v two_o mood_n ubeman_n and_o uphener_n the_o first_o whereof_o comprehend_v all_o those_o question_n wherein_o the_o subtendent_fw-la and_o a_o ambient_fw-la be_v give_v a_o oblique_a be_v require_v and_o by_o its_o resolver_n v._o rad_n eglantine_n ☞_o so._n thus_o satisfy_v our_o demand_n that_o if_o we_o subtract_v the_o logarithm_n of_o the_o subtendent_fw-la from_o the_o sum_n of_o the_o logarithm_n of_o the_o middle_a term_n we_o have_v the_o logarithm_n of_o the_o sine_fw-la of_o the_o opposite_a angle_n we_o seek_v for_o for_o it_o be_v as_o the_o subtendent_fw-la to_o the_o total_a sine_fw-la so_o the_o contain_v side_n give_v to_o the_o sine_fw-la of_o the_o opposite_a angle_n require_v the_o reason_n likewise_o of_o this_o analogy_n be_v find_v in_o the_o second_o axiom_n eproso_n upon_o the_o which_o this_o mood_n be_v ground_v as_o the_o second_o consonant_a of_o its_o directory_n give_v we_o to_o understand_v the_o second_o mood_n or_o datoquaere_fw-la of_o this_o figure_n be_v uphener_n which_o show_v that_o those_o question_n in_o plain_a triangle_n wherein_o the_o hypotenusa_fw-la and_o a_o leg_n be_v give_v the_o other_o leg_n be_v demand_v be_v to_o be_v calculate_v by_o its_o resolver_n which_o because_o the_o canon_n of_o logarithm_n can_v perform_v it_o at_o one_o operation_n there_o be_v a_o necessity_n to_o find_v one_o of_o the_o oblique_a angle_n before_o the_o four_o term_n can_v be_v bring_v into_o a_o analogy_n allow_v two_o subservient_n for_o the_o achievement_n thereof_o viz._n vbeman_n the_o first_o mood_n of_o the_o second_o figure_n for_o the_o find_v out_o of_o the_o angle_n and_o here_o because_o anterior_a in_o the_o work_n call_v praesubservient_fw-fr than_o vpalem_n the_o first_o mood_n of_o all_o for_o find_v out_o of_o the_o leg_n inquire_v and_o here_o call_v possubservient_fw-fr because_o of_o its_o posteriority_n in_o the_o operation_n yet_o be_v it_o not_o for_o the_o facility_n which_o addition_n and_o subtraction_n only_o afford_v we_o in_o this_o manner_n of_o calculation_n we_o may_v do_v it_o with_o one_o work_n alone_o by_o the_o bucarnon_n or_o pythagorases_n diodot_n which_o plain_o show_v we_o that_o by_o subduce_v the_o square_n of_o the_o leg_n give_v from_o the_o square_n of_o the_o subtendent_fw-la we_o have_v for_o the_o remainder_n another_o square_n who_o root_n be_v the_o side_n require_v the_o reason_n of_o this_o be_v in_o my_o apodictick_n but_o that_o of_o the_o former_a resolver_n by_o two_o operation_n be_v in_o the_o first_o axiom_n as_o by_o the_o first_o syllable_n of_o its_o directory_n be_v manifest_a the_o three_o figure_n be_v enave_n which_o have_v two_o mood_n ekarul_a and_o egalem_fw-la the_o first_o comprehend_v all_o those_o problem_n wherein_o one_o of_o the_o including_z side_n and_o a_o angle_n be_v give_v the_o subtendent_fw-la be_v require_v and_o by_o its_o resolver_n sapeg_n eglantine_n rad_n ☞_o ur_fw-la show_v that_o if_o we_o subtract_v the_o sine_fw-la of_o the_o angle_n opposite_a to_o the_o
find_v out_o of_o another_o angle_n before_o it_o can_v fix_v upon_o the_o side_n therelabmo_n be_v allow_v it_o for_o a_o praesubservient_fw-fr by_o virtue_n whereof_o a_o opposite_a angle_n be_v obtain_v and_o danarele_n for_o its_o possubservient_fw-fr and_o final_a resolver_n by_o who_o mean_n we_o get_v the_o side_n require_v the_o reason_n of_o the_o first_o operation_n be_v ground_v on_o the_o three_o axiom_n and_o of_o the_o second_o operation_n on_o the_o second_o but_o because_o this_o mood_n be_v mere_o a_o partial_a enodandum_fw-la neither_o of_o the_o foresay_a axiom_n afford_v any_o directory_n concern_v it_o otherwise_o then_o in_o the_o two_o subservient_n thereof_o the_o three_o figure_n be_v enerolom_n who_o two_o mood_n be_v xâ—monorâ—_n and_o shenerolem_n the_o first_o mood_n of_o this_o figure_n include_v all_o those_o planobliquangularie_a problem_n wherein_o two_o side_n be_v give_v with_o a_o opposite_a angle_n another_o opposite_a angle_n be_v demand_v and_o by_o its_o resolver_n e_z sog_n ge_n ☞_o so_o show_v that_o if_o from_o the_o aggregat_fw-la of_o the_o logarithm_n of_o one_o of_o the_o give_v side_n and_o that_o of_o the_o sine_fw-la of_o the_o opposite_a angle_n propose_v we_o subtract_v the_o logarithm_n of_o the_o other_o give_v side_n the_o residue_n will_v afford_v we_o the_o logarithm_n of_o the_o sine_fw-la of_o the_o opposite_a angle_n require_v for_o it_o be_v analogise_v thus_o as_o one_o of_o the_o side_n to_o the_o sine_fw-la of_o the_o opposite_a angle_n give_v so_o the_o other_o side_n propose_v to_o the_o sine_fw-la of_o the_o opposite_a angle_n require_v the_o reason_n of_o this_o proportion_n be_v from_o the_o second_o axiom_n the_o six_o characteristic_a of_o who_o directory_n import_v that_o xemenoro_n be_v one_o of_o eprosos_n enodandas_fw-la the_o second_o mood_n of_o this_o figure_n be_v shenerolem_a which_o contain_v all_o those_o planobliquangularie_a problem_n wherein_o two_o side_n be_v give_v with_o a_o opposite_a angle_n the_o three_o side_n be_v demand_v which_o not_o be_v findable_a by_o the_o logarithmicall_a table_n upon_o the_o foresay_a datas_fw-la in_o less_o than_o two_o operation_n because_o a_o angle_n must_v be_v obtain_v first_o before_o the_o side_n can_v be_v have_v xemenoro_n praesubserve_v it_o for_o a_o angle_n and_o danarele_n become_v its_o possubservient_fw-fr for_o the_o side_n require_v the_o reason_n of_o both_o these_o operation_n be_v found_v on_o the_o second_o axiom_n the_o last_o characteristick_n of_o who_o directory_n inroll_v shenerolem_a for_o one_o of_o eprosos_n enodandas_fw-la the_o four_o figure_n be_v erelea_n which_o be_v monotropall_a have_v no_o mood_n but_o pserelema_n this_o pserelema_n encompass_v all_o those_o planobliquangulary_a problem_n wherein_o the_o three_o side_n be_v propose_v a_o angle_n be_v require_v this_o datoquaere_fw-la not_o be_v resolvable_a by_o the_o logarithm_n in_o less_o than_o two_o operation_n because_o the_o segment_n of_o the_o base_a or_o sustain_a side_n must_v needs_o be_v find_v out_o that_o by_o demit_v of_o a_o perpendicular_a from_o the_o top_n angle_n we_o may_v hit_v upon_o the_o angle_n demand_v the_o resolver_n for_o the_o segment_n be_v basilius_n gres_n ze_v ☞_o zius_n whereby_o we_o learn_v that_o if_o from_o the_o logarithm_n of_o the_o sum_n of_o the_o side_n join_v to_o the_o logarithm_n of_o the_o difference_n of_o the_o side_n we_o subtract_v the_o logarithm_n of_o the_o base_a the_o remainder_n be_v the_o logarithm_n of_o the_o difference_n of_o the_o segment_n which_o difference_n be_v take_v from_o the_o whole_a base_a half_o the_o difference_n prove_v to_o be_v the_o lesser_a segment_n this_o theorem_fw-la be_v thus_o the_o praesubservient_fw-fr of_o this_o mood_n its_o possubservient_fw-fr be_v vbeman_n who_o general_a resolver_n v._o rad_n eglantine_n ☞_o sor_n be_v particularise_v for_o this_o case_n uxiug_v rad_n ing_n ☞_o sor_n which_o show_v that_o if_o from_o the_o sum_n of_o the_o logarithm_n of_o the_o total_a sine_fw-la and_o of_o one_o of_o the_o segment_v give_v we_o subduce_v the_o logarithm_n of_o the_o hypotenusa_fw-la conterminall_a with_o the_o segment_n propose_v the_o remainder_n will_v be_v the_o logarithm_n of_o the_o sine_fw-la of_o the_o opposite_a angle_n require_v for_o the_o demit_v of_o the_o perpendicular_a open_v a_o way_n to_o have_v the_o theorem_fw-la to_o be_v first_o in_o general_n propound_v thus_o as_o the_o subtendent_fw-la to_o the_o total_a sine_fw-la so_o the_o contain_v side_n give_v to_o the_o sine_fw-la of_o the_o angle_n require_v or_o in_o particular_a thus_o as_o the_o sine_fw-la of_o the_o cosubtendent_n adjoin_v the_o segment_n give_v be_v to_o the_o radius_fw-la so_o be_v the_o say_a segment_n propose_v to_o the_o sine_fw-la of_o the_o angle_n require_v thus_o far_o for_o the_o calculate_v of_o plain_a triangle_n both_o right_a and_o oblique_a now_o follow_v the_o sphericals_n there_o be_v three_o principal_a axiom_n upon_o which_o depend_v the_o resolve_v of_o spherical_a triangle_n to_o wit_n suprosca_n sbaprotca_n and_o seproso_n the_o first_o maxim_n or_o axiom_n suprosca_n show_v that_o of_o several_a rectangled_a sphericals_n which_o have_v one_o and_o the_o same_o acute_a angle_n at_o the_o base_a the_o sin_n of_o the_o hypotenusas_n be_v proportional_a to_o the_o sin_n of_o their_o perpendicular_o for_o from_o the_o same_o inclination_n every_o where_o of_o the_o one_o plain_a to_o the_o other_o there_o arise_v a_o equiangularity_n in_o the_o two_o rectangle_v out_o of_o which_o we_o may_v confident_o infer_v the_o homologall_a side_n which_o be_v the_o sin_n of_o the_o subtendent_o and_o of_o the_o perpendicular_o of_o the_o one_o and_o the_o other_o to_o be_v among_o themselves_o proportional_a it_o be_v directory_n be_v uphugen_o by_o the_o which_o we_o learn_v that_o uphanep_v ugemon_n and_o enarul_a be_v its_o three_o enodandas_fw-la the_o second_o axiom_n be_v sbaprotca_n whereby_o we_o learn_v that_o in_o all_o rectangled_a sphericals_n that_n have_v one_o and_o the_o same_o acute_a angle_n at_o the_o base_a the_o sin_n of_o the_o base_n be_v proportional_a to_o the_o tangent_n of_o their_o perpendicular_o which_o analogy_n proceed_v from_o the_o equiangularity_n of_o such_o rectangled_a sphericals_n by_o the_o semblable_a incline_n of_o the_o plain_a towards_o they_o both_o this_o proportion_n nevertheless_o will_v never_o hold_v betwixt_o the_o sin_n of_o the_o base_n and_o the_o sin_n of_o their_o perpendicular_o because_o if_o the_o sin_n of_o the_o base_n be_v proportional_a to_o the_o sin_n of_o the_o perpendicular_o the_o sin_n of_o the_o perpendicular_o be_v already_o demonstrate_v proportional_a to_o the_o sin_n of_o the_o subtendent_o either_o the_o sine_fw-la of_o the_o perpendicular_a or_o the_o sine_fw-la of_o the_o base_a will_v be_v the_o cord_n of_o the_o same_o arch_n whereof_o it_o be_v a_o sine_fw-la which_o be_v impossible_a by_o reason_n that_o nothing_o can_v be_v both_o a_o whole_a and_o a_o part_n in_o regard_n of_o one_o and_o the_o same_o thing_n and_o therefore_o do_v we_o only_o say_v that_o the_o sin_n of_o the_o base_n and_o tangent_n of_o the_o perpendicular_o and_o contrary_o be_v proportional_a it_o be_v directory_n be_v pubkutethepsale_a which_o show_v that_o upalam_n ubamen_fw-la vkelamb_n etalum_n ethaner_n epsoner_n alamun_n and_o erelam_n be_v the_o eight_o enodandas_n the_o reupon_o depend_v the_o three_o axiom_n be_v that_o the_o sin_n of_o the_o side_n be_v proportional_a to_o the_o sin_n of_o their_o opposite_a angle_n the_o truth_n whereof_o hold_v in_o all_o spherical_a triangle_n whatsoever_o which_o be_v prove_v partly_o out_o of_o the_o proportion_n betwixt_o the_o sin_n of_o the_o perpendicular_o substern_v under_o equal_a angle_n and_o the_o sin_n of_o the_o hypotenusas_n and_o partly_o by_o the_o analogy_n that_o be_v betwixt_o the_o sin_n of_o the_o angle_n sustain_v by_o several_a perpendicular_o demit_v from_o one_o point_n and_o the_o sin_n of_o the_o perpendicular_o themselves_o the_o directory_n of_o this_o axiom_n be_v vchedezexam_n whereby_o we_o know_v that_o uchener_n edamon_n ezolum_n exoman_n and_o amane_a be_v the_o five_o enodandas_n thereof_o the_o orthogonosphericall_a table_n consist_v of_o these_o six_o figure_n valamenep_v vemanore_n enarulome_n erolumane_a achave_v and_o esheva_n the_o first_o figure_n valamenep_v comprehend_v all_o those_o question_n wherein_o the_o subtendent_fw-la and_o a_o angle_n be_v give_v either_o another_o angle_n or_o one_o of_o the_o ambient_n be_v demand_v of_o this_o figure_n there_o be_v three_o mood_n viz._n upalam_n ubamen_fw-la and_o uphanep_v the_o first_o to_o wit_n upalam_n contain_v all_o those_o orthogonosphericall_a problem_n wherein_o the_o subtendent_fw-la and_o one_o oblique_a angle_n be_v give_v another_o oblique_a angle_n be_v require_v and_o by_o its_o resolver_n torb_n tag_n nu_n ☞_o mir_fw-fr show_v that_o the_o sum_n of_o the_o sine_fw-la compliment_n of_o the_o subtendent_fw-la side_n and_o tangent_fw-la of_o
the_o angle_n give_v the_o logarithm_n of_o these_o be_v always_o to_o be_v understand_v a_o digit_fw-la be_v prescind_v from_o the_o left_a be_v equal_a to_o the_o tangent_fw-la compliment_n of_o the_o angle_n require_v for_o the_o proposition_n go_v thus_o as_o the_o radius_fw-la to_o the_o tangent_fw-la of_o the_o angle_n give_v so_o the_o sine_fw-la compliment_n of_o the_o subtendent_fw-la side_n to_o the_o tangent_fw-la compliment_n of_o the_o angle_n require_v and_o because_o tangent_n and_o tangent_fw-la compliment_n be_v reciprocal_o proportional_a instead_o of_o to_o tag_n nu_n ☞_o mir_fw-fr or_o to_z lu._n madge_n ☞_o tir_v which_o for_o that_o the_o radius_fw-la be_v a_o mean_a proportional_a betwixt_o the_o l._n and_o n._n the_o t._n and_o m_n be_v all_o one_o for_o infer_v of_o the_o same_o four_o proportional_a or_o foresay_a quaesitum_fw-la we_o may_v say_v madge_n nu_n to_o ☞_o mir_fw-fr that_o be_v as_o the_o tangent_fw-la compliment_n of_o the_o give_v angle_n to_o the_o cousin_n of_o the_o subtendent_fw-la so_o the_o total_a sine_fw-la to_o the_o antitangent_fw-la of_o the_o angle_n demand_v for_o the_o total_a sine_fw-la be_v as_o i_o have_v tell_v you_o a_o mean_a proportional_a betwixt_o the_o tangent_n and_o cotangent_n the_o subtract_v of_o the_o cotangent_fw-la or_o tangent_fw-la compliment_n from_o the_o sum_n of_o the_o radius_fw-la and_o antisine_n residuat_v a_o logarithm_n equal_a to_o that_o of_o the_o remainder_n by_o abstract_v the_o radius_fw-la from_o the_o sum_n of_o the_o cousin_n of_o the_o subtendent_fw-la and_o tangent_fw-la of_o the_o angle_n give_v either_o of_o which_o will_v fall_v out_o to_o be_v the_o antitangent_fw-la of_o the_o require_a angle_n notandum_fw-la here_o always_o be_v to_o be_v observe_v that_o the_o subtract_v of_o logarithm_n may_v be_v avoid_v by_o substitute_v the_o arithmetical_a compliment_n thereof_o to_o be_v add_v to_o the_o logarithm_n of_o the_o two_o middle_a proportional_n which_o arithmetical_a compliment_n according_a to_o gellibrand_n be_v nothing_o else_o but_o the_o difference_n between_o the_o logarithm_n to_o be_v subtract_v and_o another_o consist_v of_o a_o unit_fw-la or_o binarie_a with_o the_o addition_n of_o cypher_n that_o be_v the_o single_a or_o double_a radius_fw-la for_o so_o the_o sum_n of_o the_o three_o logarithm_n cut_v off_o a_o unit_fw-la or_o binarie_a towards_o the_o left_a hand_n will_v still_o be_v the_o logarithm_n of_o the_o four_o proportional_a require_v for_o the_o great_a ease_n therefore_o in_o trigonometricall_a computation_n such_o a_o logarithmicall_a canon_n be_v to_o be_v wish_v for_o wherein_o the_o radius_fw-la be_v leave_v out_o of_o all_o the_o secant_v and_o all_o the_o tangent_n of_o major_a arch_n according_a to_o the_o method_n prescribe_v by_o mr._n speidel_n who_o be_v willing_a to_o take_v the_o pain_n to_o make_v such_o a_o new_a canon_n better_a than_o any_o that_o ever_o hitherto_o have_v be_v make_v use_n of_o so_o that_o the_o public_a who_o it_o most_o concern_v or_o some_o potent_a man_n well_o mind_v towards_o the_o mathematics_n will_v be_v so_o generous_a as_o to_o relieve_v he_o of_o the_o charge_n it_o must_v needs_o cost_v he_o which_o consider_v his_o great_a affection_n to_o and_o ability_n in_o those_o science_n will_v certain_o be_v as_o small_a a_o sum_n as_o possible_o he_o can_v bring_v it_o to_o this_o parenthesis_n though_o somewhat_o with_o the_o long_a will_v not_o i_o hope_v be_v displease_v to_o the_o studious_a reader_n the_o second_o mood_n of_o the_o first_o figure_n be_v ubamen_n which_o comprehend_v all_o those_o problem_n wherein_o the_o subtendent_fw-la and_o one_o oblique_a angle_n be_v give_v the_o ambient_fw-la adjoin_v the_o angle_n give_v be_v require_v and_o by_o its_o resolver_n nag_n mu_o torp_n ☞_o myr_n show_v that_o if_o to_o the_o sum_n of_o the_o logarithm_n of_o the_o two_o middle_a proportional_n we_o add_v the_o arithmetical_a compliment_n of_o the_o first_o the_o cut_n off_o the_o index_n from_o the_o aggregat_fw-la of_o the_o three_o will_v residuat_n the_o tangent_fw-la compliment_n of_o the_o side_n require_v and_o therefore_o with_o the_o total_a sine_fw-la in_o the_o first_o place_n it_o may_v be_v thus_o propound_v torp_n mu_o lag_a ☞_o myr_n for_o the_o first_o theorem_n be_v as_o the_o sine_fw-la compliment_n of_o the_o angle_n give_v to_o the_o tangent_fw-la compliment_n of_o the_o subtendent_fw-la side_n so_o the_o total_a sine_fw-la to_o the_o tangent_fw-la compliment_n of_o the_o side_n require_v just_o so_o the_o second_o theorem_fw-la which_o be_v that_o refine_a be_v as_o the_o total_a sine_fw-la to_o the_o tangent_fw-la compliment_n of_o the_o subtendent_fw-la so_o the_o secant_fw-la of_o the_o give_a angle_n to_o the_o tangent_fw-la compliment_n of_o the_o demand_v side_n here_o you_o must_v consider_v as_o i_o have_v tell_v you_o already_o that_o of_o the_o whole_a secant_fw-la ay_o take_v but_o its_o excess_n above_o the_o radius_fw-la as_o i_o do_v of_o all_o tangent_n above_o 45._o degree_n because_o the_o cut_n off_o the_o first_o digit_fw-la on_o the_o left_a supply_v the_o subtraction_n requisite_a for_o the_o find_v out_o of_o the_o four_o proportional_a so_o that_o by_o addition_n only_o the_o whole_a operation_n may_v be_v perform_v of_o all_o way_n the_o most_o succinct_a and_o ready_a otherwise_o because_o of_o the_o total_a sin_n mean_v proportionality_n betwixt_o the_o sine_fw-la compliment_n and_o the_o secant_fw-la and_o betwixt_o the_o tangent_fw-la and_o tangent_fw-la compliment_n it_o may_v be_v regulate_v thus_o to_o tu_fw-la nag_n ☞_o tire_v that_o be_v as_o the_o radius_fw-la to_o the_o tangent_fw-la of_o the_o subtendent_fw-la so_o the_o sine_fw-la compliment_n of_o the_o angle_n give_v to_o the_o tangent_fw-la of_o the_o side_n require_v the_o reason_n of_o the_o resolution_n both_o of_o this_o and_o of_o the_o former_a datoquaere_fw-la be_v ground_v on_o the_o second_o axiom_n and_o the_o proportion_n that_o in_o several_a rectangled_a sphericals_n which_o have_v the_o same_o acute_a angle_n at_o the_o base_a be_v find_v betwixt_o the_o sin_n of_o their_o perpendicular_o and_o tangent_n of_o their_o base_n as_o be_v show_v you_o by_o the_o two_o first_o consonant_n of_o the_o directory_n of_o sbaprotca_n the_o three_o and_o last_o mood_n of_o the_o first_o figure_n be_v uphane_a which_o comprehend_v all_o those_o problem_n wherein_o the_o hypotenusa_fw-la and_o one_o of_o the_o oblique_n be_v give_v the_o opposite_a ambient_fw-la be_v require_v and_o by_o its_o resolver_n tol_o sag_fw-ge sum_z ☞_o sir_n show_v that_o if_o we_o add_v the_o logarithm_n of_o the_o sine_fw-la of_o the_o angle_n and_o sine_fw-la of_o the_o subtendent_fw-la cut_v off_o the_o left_a supernumerarie_a digit_fw-la from_o the_o sum_n it_o give_v we_o the_o logarithm_n of_o the_o sine_fw-la of_o the_o side_n demand_v for_o it_o be_v as_o the_o total_a sine_fw-la to_o the_o sine_fw-la of_o the_o angle_n give_v so_o the_o sine_fw-la of_o the_o subtendent_fw-la side_n to_o the_o sine_fw-la of_o the_o side_n require_v and_o because_o by_o the_o axiom_n of_o rulerst_n it_o be_v prove_v that_o when_o the_o sine_fw-la of_o any_o arch_n be_v make_v radius_fw-la what_o be_v then_o the_o total_a sine_fw-la become_v a_o secant_fw-la and_o therefore_o secant_fw-la compliment_n of_o that_o arch_n instead_o of_o tol_o sag_fw-ge sum_z ☞_o sir_n we_o may_v say_v to_z ru_n rag_n ☞_o ryr_n that_o be_v as_o the_o total_a sine_fw-la be_v to_o the_o secant_fw-la compliment_n of_o the_o subtendent_fw-la so_o the_o secant_fw-la compliment_n of_o the_o angle_n give_v to_o the_o secant_fw-la compliment_n of_o the_o side_n demand_v the_o resolution_n of_o this_o datoquaere_fw-la by_o sines_n be_v ground_v on_o the_o first_o axiom_n of_o sphericals_n which_o elucidat_n the_o proportion_n betwixt_o the_o sin_n of_o the_o hypotenusas_n and_o perpendicular_o as_o it_o be_v declare_v to_o we_o by_o the_o first_o syllable_n of_o suproscas_n directory_n the_o second_o figure_n be_v vemanore_n which_o contain_v all_o those_o orthogonosphericall_a question_n wherein_o the_o subtendent_fw-la and_o a_o ambient_fw-la be_v propose_v either_o of_o the_o oblique_n or_o the_o other_o ambient_fw-la be_v require_v and_o have_v three_o mood_n viz._n ukelamb_n ugemon_n and_o uchener_n the_o first_o mood_n ukelamb_n comprehend_v all_o those_o orthogonosphericall_a problem_n wherein_o the_o subtendent_fw-la and_o one_o include_v side_n be_v give_v the_o interjacent_a angle_n be_v demand_v and_o by_o its_o resolver_n meg_n torp_n mu_o ☞_o nir_fw-fr or_o because_o of_o the_o total_a sin_n mean_v proportion_n betwixt_o the_o tangent_fw-la and_o tangent_fw-la compliment_n torp-teg_n mu_o ☞_o nir_fw-fr which_o be_v the_o same_o in_o effect_n show_v that_o if_o from_o the_o sum_n of_o the_o logarithm_n of_o the_o middle_a term_n which_o in_o the_o first_o analogy_n be_v the_o radius_fw-la and_o tangent_fw-la compliment_n of_o the_o subtendent_fw-la we_o subtract_v the_o tangent_fw-la
first_o leave_v digit_fw-la from_o the_o sum_n of_o the_o tangent_fw-la compliment_n of_o the_o ambient_fw-la propose_v and_o the_o sine_fw-la compliment_n of_o the_o give_v angle_n afford_v we_o the_o tangent_fw-la compliment_n of_o the_o subtendent_fw-la require_v for_o the_o theorem_fw-la go_v thus_o as_o the_o total_a sine_fw-la to_o the_o tangent_fw-la compliment_n of_o the_o give_v side_n so_o the_o sine_fw-la compliment_n of_o the_o angle_n give_v to_o the_o tangent_fw-la compliment_n of_o the_o hypotenusa_fw-la require_v and_o because_o the_o total_a sine_fw-la have_v the_o same_o proportion_n to_o the_o tangent_fw-la compliment_n which_o the_o sine_fw-la have_v to_o the_o sine_fw-la compliment_n we_o may_v as_o well_o say_v to-meg-sa_a ☞_o nur_v that_o be_v as_o the_o radius_fw-la to_o the_o tangent_fw-la compliment_n of_o the_o ambient_fw-la side_n so_o the_o sine_fw-la of_o the_o angle_n give_v to_z the_o sine_z compliment_n of_o the_o subtendent_fw-la require_v the_o progress_n of_o this_o mood_n depend_v on_o the_o axiom_n of_o sbaprotca_n as_o you_o may_v perceive_v by_o the_o four_o consonant_n of_o its_o directory_n pubkutethepsaler_n the_o second_o mood_n of_o the_o three_o figure_n be_v edamon_n which_o comprehend_v all_o those_o orthogonosphericall_a problem_n wherein_o a_o ambient_fw-la and_o a_o adjacent_a angle_n be_v give_v the_o opposite_a oblique_a viz._n the_o angle_n under_o which_o the_o ambient_fw-la be_v subtend_v be_v require_v and_o by_o its_o resolver_n to-neg-sa_a ☞_o nir_fw-fr show_v that_o the_o addition_n of_o the_o cousin_n of_o the_o ambient_fw-la and_o of_o the_o sine_fw-la of_o the_o angle_n propose_v afford_v we_o if_o we_o omit_v not_o the_o usual_a presection_n the_o cousin_n of_o the_o angle_n we_o seek_v for_o for_o it_o be_v as_o the_o radius_fw-la to_o the_o cousin_n or_o sine_fw-la compliment_n of_o the_o give_v side_n so_o the_o sine_fw-la of_o the_o angle_n propose_v to_o the_o antisin_n or_o sine_fw-la compliment_n of_o the_o angle_n demand_v now_o the_o radius_fw-la be_v always_o a_o mean_a proportional_a betwixt_o the_o sine_fw-la compliment_n and_o the_o secant_fw-la we_o may_v for_o to_o neg_fw-mi sa_o ☞_o nir_fw-fr say_v to_z leg_n ra_o ☞_o lir_n or_o to_o rag_n le_fw-fr ☞_o lir_n that_o be_v as_o the_o total_a sine_fw-la to_o the_o secant_fw-la or_o cutter_n of_o the_o side_n give_v or_o to_o the_o cosecant_v or_o secant_fw-la compliment_n of_o the_o give_v angle_n so_o be_v the_o secant_fw-la compliment_n of_o the_o angle_n or_o secant_fw-la of_o the_o side_n to_o the_o secant_fw-la or_o cutter_n of_o the_o angle_n require_v the_o reason_n of_o all_o this_o be_v ground_v on_o seproso_n because_o it_o run_v upon_o the_o proportion_n betwixt_o the_o sin_n of_o the_o side_n and_o the_o sin_n of_o their_o opposite_a angle_n as_o be_v perspicuous_a to_o any_o by_o the_o second_o syllable_n of_o the_o directory_n of_o that_o axiom_n the_o last_o mood_n of_o the_o three_o figure_n be_v ethane_a which_o comprehend_v all_o those_o orthogonosphericall_a problem_n wherein_o a_o ambient_fw-la with_o a_o oblique_a annex_v thereto_o be_v give_v and_o the_o other_o arch_n about_o the_o right_a angle_n be_v require_v and_o by_o its_o resolver_n torb_n tag_n se_fw-la ☞_o tire_v show_v that_o if_o we_o join_v the_o logarithm_n of_o the_o two_o middle_a proportional_n which_o be_v the_o tangent_fw-la of_o the_o give_a angle_n and_o the_o sine_fw-la of_o the_o side_n the_o usual_a prefection_n be_v observe_v we_o shall_v thereby_o have_v the_o tangent_fw-la or_o toucher_n of_o the_o ambient_fw-la side_n desire_a for_o it_o be_v as_o the_o radius_fw-la to_o the_o tangent_fw-la of_o the_o angle_n give_v so_o the_o sine_fw-la of_o the_o contain_v side_n propose_v to_o the_o side_n require_v and_o because_o the_o tangent_fw-la compliment_n and_o tangent_fw-la be_v reciprocal_o proportional_a the_o sine_fw-la likewise_o and_o secant_fw-la compliment_n for_o to_o tag_n se_fw-la ☞_o tire_v we_o may_v say_v keep_v the_o same_o proportion_n to_o reg_n ma_n ☞_o myr_n that_o be_v as_o the_o radius_fw-la to_o the_o secant_fw-la compliment_n of_o the_o give_v side_n so_o the_o tangent_fw-la compliment_n of_o the_o angle_n propose_v to_o the_o tangent_fw-la compliment_n of_o the_o side_n require_v the_o truth_n of_o all_o these_o operation_n depend_v on_o sbaprotca_n the_o second_o axiom_n of_o the_o sphericals_n as_o be_v evidence_v by_o θ._n the_o five_o characteristic_a of_o its_o directory_n pubkutethepsaler_n the_o four_o figure_n be_v erollumane_a which_o include_v all_o orthogonosphericall_a question_n wherein_o a_o ambient_fw-la and_o a_o opposite_a oblique_a be_v give_v the_o subtendent_fw-la the_o other_o oblique_a or_o the_o other_o ambient_fw-la be_v demand_v it_o have_v likewise_o conform_v to_o the_o three_o former_a figure_n three_o mood_n belong_v to_o it_o the_o first_o whereof_o be_v ezolum_n this_o ezolum_n comprehend_v all_o those_o orthogonosphericall_a problem_n wherein_o one_o of_o the_o leg_n with_o a_o opposite_a angle_n be_v give_v the_o subtendent_fw-la be_v require_v and_o by_o its_o resolver_n sag_fw-ge sep_n rad_n ☞_o sur_fw-fr or_o by_o put_v the_o radius_fw-la in_o the_o first_o place_n to_z serag_n ☞_o sur_fw-fr sheweth_z that_o the_o abstract_n of_o the_o radius_fw-la from_o the_o sum_n of_o the_o sine_fw-la of_o the_o side_n and_o secant_fw-la compliment_n of_o the_o angle_n give_v residuat_v the_o sine_fw-la of_o the_o hypotenusa_fw-la require_v for_o it_o be_v as_o the_o sine_fw-la of_o the_o angle_n give_v to_o the_o sine_fw-la of_o the_o opposite_a side_n so_o the_o radius_fw-la to_o the_o sine_fw-la of_o the_o subtendent_fw-la or_o more_o refined_o as_o the_o total_a sine_fw-la to_o the_o sine_fw-la of_o the_o side_n so_o the_o secant_fw-la compliment_n of_o the_o angle_n give_v to_o the_o sine_fw-la of_o the_o subtendent_fw-la side_n and_o because_o of_o the_o sin_n and_o antisecant_v or_o secant_fw-la compliment_v reciprocal_a proportionality_n to_o sag_fw-ge re_fw-mi ☞_o ru_n that_o be_v as_o the_o radius_fw-la to_o the_o sine_fw-la of_o the_o angle_n give_v so_o the_o secant_fw-la compliment_n of_o the_o propose_a side_n to_o the_o secant_fw-la compliment_n of_o the_o subtendent_fw-la require_v the_o reason_n of_o all_o this_o be_v ground_v on_o the_o three_o axiom_n seproso_n as_o be_v make_v manifest_a by_o the_o three_o syllable_n of_o its_o directory_n the_o second_o mood_n of_o this_o figure_n be_v exoman_n which_o comprehend_v all_o those_o problem_n wherein_o a_o contain_v side_n and_o a_o opposite_a oblique_a be_v give_v the_o adjacent_a oblique_a be_v require_v and_o by_o its_o resolver_n ne_o to_o nag_n ☞_o sir_n or_o more_o refined_o to_o le_fw-fr nag_n ☞_o sir_n sheweth_z that_o the_o sum_n of_o the_o sine_fw-la of_o the_o angle_n together_o with_o the_o arithmetical_a compliment_n of_o the_o antisine_n of_o the_o leg_n which_o in_o the_o table_n i_o have_v so_o much_o recommend_v unto_o the_o reader_n be_v set_v down_o for_o a_o secant_fw-la the_o usual_a prefection_n be_v observe_v afford_v we_o the_o sine_fw-la of_o the_o angle_n require_v and_o because_o of_o the_o reciprocal_a proportion_n betwixt_o the_o sine_fw-la compliment_n and_o secant_fw-la and_o betwixt_o the_o sine_fw-la and_o secant_fw-la compliment_n the_o theorem_fw-la may_v be_v compose_v thus_o to_o neg_fw-mi la_o ☞_o rir_fw-it that_o be_v as_o the_o radius_fw-la to_o the_o sine_fw-la compliment_n of_o the_o give_v side_n so_o the_o secant_fw-la of_o the_o angle_n propose_v to_o the_o secant_fw-la compliment_n of_o the_o angle_n demand_v the_o reason_n of_o this_o be_v likewise_o ground_v on_o seproso_n as_o you_o may_v perceive_v by_o the_o four_o characteristic_a of_o its_o directory_n the_o last_o mood_n of_o this_o figure_n be_v epsoner_n which_o contain_v all_o those_o orthogonosphericall_a problem_n wherein_o a_o ambient_fw-la and_o a_o opposite_a oblique_a be_v give_v the_o other_o ambient_fw-la be_v demand_v and_o by_o its_o resolver_n tag_n tolb_n te_fw-la ☞_o sir_n or_o more_o elaboured_o tolb_n madge_n te_fw-la ☞_o sir_n show_v that_o the_o praescind_n of_o the_o radius_fw-la from_o the_o sum_n of_o the_o tangent_fw-la of_o the_o side_n and_o antitangent_fw-la of_o the_o give_a angle_n residuat_v the_o sine_fw-la of_o the_o side_n require_v for_o it_o be_v as_o the_o tangent_fw-la of_o the_o angle_n propose_v to_o the_o total_a sine_fw-la so_o the_o tangent_fw-la of_o the_o give_a side_n to_o the_o sine_fw-la of_o the_o side_n demand_v or_o as_o the_o radius_fw-la to_o the_o tangent_fw-la compliment_n of_o the_o angle_n give_v so_o the_o tangent_fw-la of_o the_o give_a side_n to_o the_o sine_fw-la of_o the_o side_n require_v and_o because_o of_o the_o reciprocal_a analogy_n betwixt_o the_o tangent_n and_o co-tangents_a and_o betwixt_o the_o sin_n and_o cosecants_a we_o may_v with_o the_o same_o confidence_n as_o former_o set_v it_o thus_o in_o the_o rule_n to_o meg_n ta_o ☞_o ryr_n and_o it_o will_v find_v out_o the_o same_o quaesitum_fw-la the_o reason_n of_o the_o
operation_n of_o this_o mood_n because_o of_o the_o ingrediencie_n of_o tangent_n depend_v on_o sbaprotca_n as_o be_v perceivable_a by_o the_o six_o determinater_n of_o its_o directory_n pubkutethepsaler_n the_o five_o figure_n of_o the_o orthogonosphericals_n be_v achave_v which_o contain_v all_o those_o problem_n wherein_o the_o angles_n be_v give_v the_o subtendent_fw-la or_o a_o ambient_fw-la be_v desire_v and_o have_v two_o mood_n alamun_n and_o amane_a alamun_n comprehend_v all_o those_o problem_n wherein_o the_o angles_n be_v propose_v the_o hypotenusa_fw-la be_v require_v and_o by_o its_o resolver_n tag_n torb_n ma_n ☞_o nurse_n or_o more_o compendious_o torb_n madge_n ma_n ☞_o nurse_n show_v that_o the_o sum_n of_o the_o co-tangents_a not_o exceed_v the_o place_n of_o the_o radius_fw-la be_v the_o sine_fw-la compliment_n of_o the_o subtendent_fw-la require_v for_o it_o be_v as_o the_o tangent_fw-la of_o one_o of_o the_o angle_n to_o the_o radius_fw-la so_o the_o tangent_fw-la compliment_n of_o the_o other_o angle_n to_o the_o sine_fw-la compliment_n of_o the_o hypotenusa_fw-la demand_v or_o as_o the_o total_a sine_fw-la to_o the_o tangent_fw-la compliment_n of_o one_o of_o the_o angles_n so_o the_o tangent_fw-la compliment_n of_o the_o other_o angle_n to_o the_o sine_fw-la compliment_n of_o the_o subtendent_fw-la we_o seek_v for_o the_o run_v of_o this_o mood_n upon_o tangent_n notifi_v its_o dependence_n on_o sbaprotca_n as_o be_v evident_a by_o the_o seven_o determinater_n of_o the_o directory_n thereof_o the_o second_o mood_n of_o this_o figure_n be_v amane_a which_o comprehend_v all_o those_o orthogonosphericall_a problem_n wherein_o the_o angles_n be_v give_v a_o ambient_fw-la be_v demand_v and_o by_o its_o resolver_n say_v nag_n tω_fw-gr ☞_o nyr_n or_o more_o perspicuous_o tω_fw-gr noy_n ray_n ☞_o nyr_n show_v that_o the_o sum_n of_o the_o logarithm_n of_o the_o antisine_n of_o the_o angle_n opposite_a to_o the_o side_n require_v and_o the_o arithmetical_a compliment_n of_o the_o sine_fw-la of_o the_o angle_n adjoin_v the_o say_a side_n which_o we_o call_v its_o secant_fw-la compliment_n with_o the_o usual_a presection_n be_v equal_a to_o the_o sine_fw-la compliment_n of_o the_o same_o side_n demand_v for_o it_o be_v as_o the_o sine_fw-la of_o the_o angle_n adjoin_v the_o side_n require_v to_o the_o antisine_n of_o the_o other_o angle_n so_o the_o total_a sine_fw-la to_o the_o antisine_n of_o the_o side_n demand_v or_o as_o the_o radius_fw-la to_o the_o antisine_n of_o the_o angle_n opposite_a to_o the_o demand_v side_n so_o the_o antisecant_v of_o the_o angle_n conterminat_fw-la with_o that_o side_n to_o the_o antisine_n of_o the_o side_n require_v and_o because_o of_o the_o analogy_n betwixt_o the_o antisines_n and_o secant_v and_o likewise_o betwixt_o the_o antisecant_v and_o sines_n we_o may_v express_v it_o to_z say_v la_fw-fr ☞_o lyr_n that_o be_v as_o the_o radius_fw-la to_o the_o sine_fw-la of_o the_o angle_n incident_a on_o the_o require_a side_n so_o the_o secant_fw-la of_o the_o other_o give_v angle_n to_o the_o secant_fw-la of_o the_o side_n that_o be_v demand_v here_o the_o angulary_a intermixture_n of_o proportion_n give_v we_o to_o understand_v that_o this_o mood_n depend_v on_o seproso_n as_o be_v manifest_v by_o the_o last_o characteristic_a of_o uchedezexam_n the_o directory_n of_o this_o axiom_n the_o six_o and_o last_o figure_n be_v escheva_n which_o comprehend_v all_o those_o problem_n wherein_o the_o two_o contain_v side_n be_v give_v either_o the_o subtendent_fw-la or_o a_o angle_n be_v demand_v it_o have_v two_o mood_n enerul_a and_o erelam_n the_o first_o mood_n thereof_o enerul_a contain_v all_o such_o problem_n as_o have_v the_o ambient_n give_v require_v the_o subtendent_fw-la and_o by_o its_o resolver_n tun_n neg_fw-mi ne_fw-la ☞_o nurse_n show_v that_o the_o sum_n of_o the_o logarithm_n of_o the_o cosines_n of_o the_o two_o leg_n unradiate_v be_v the_o logarithm_n of_o the_o cousin_a of_o the_o subtendent_fw-la for_o it_o be_v as_o the_o total_a sine_fw-la to_o the_o cousin_a of_o one_o of_o the_o ambient_n so_o the_o cousin_a of_o the_o other_o include_v leg_n give_v to_o the_o cousin_a of_o the_o require_v subtendent_fw-la and_o because_o of_o the_o co-sinal_a and_o secantine_a proportion_n we_o may_v safe_o say_v to_z leg_n le_fw-fr ☞_o lur._n that_o be_v as_o the_o radius_fw-la to_o the_o secant_fw-la of_o one_o shank_n or_o leg_n so_o the_o secant_fw-la of_o the_o other_o shank_n or_o leg_n to_o the_o secant_fw-la of_o the_o hypotenusa_fw-la demand_v the_o course_v thus_o upon_o sines_n and_o their_o proportional_n evidence_v that_o this_o mood_n depend_v on_o suprosca_n the_o first_o of_o the_o spherical_a axiom_n which_o be_v point_v at_o by_o the_o three_o and_o last_o characteristic_a of_o vphugen_n the_o directory_n thereof_o the_o second_o mood_n of_o the_o last_o figure_n and_o consequent_o the_o last_o mood_n of_o all_o the_o orthogonosphericals_n be_v erelam_n which_o comprehend_v all_o those_o orthogonosphericall_a problem_n wherein_o the_o two_o contain_v side_n be_v propose_v a_o angle_n be_v demand_v and_o by_o its_o resolver_n sei_fw-it teg_n torb_n ☞_o tir_v or_o by_o primify_v the_o radius_fw-la torb_n tepi-rexi_a ☞_o tir_v give_v we_o to_o understand_v that_o the_o cut_n off_o the_o radius_fw-la from_o the_o sum_n of_o the_o tangent_fw-la of_o the_o side_n opposite_a to_o the_o angle_n demand_v and_o the_o cosecant_v of_o the_o side_n conterminat_fw-la with_o the_o say_a angle_n residuat_v the_o touch-line_n of_o the_o angle_n in_o question_n for_o it_o be_v as_o the_o sine_fw-la of_o the_o side_n adjoin_v the_o angle_n require_v to_o the_o tangent_fw-la of_o the_o other_o give_v side_n so_o the_o radius_fw-la to_o the_o tangent_fw-la of_o the_o angle_n demand_v or_o as_o the_o total_a sine_fw-la to_o the_o tangent_fw-la of_o the_o ambient_fw-la opposite_a to_o the_o angle_n seek_v so_o the_o antisecant_v of_o the_o leg_n adjacent_a to_o the_o say_v ask_v angle_n to_o the_o tangent_fw-la or_o toucher_n thereof_o and_o because_o sines_n have_v the_o same_o proportion_n to_o cosecant_v which_o tangent_n have_v to_o cotangent_n we_o may_v say_v to_z sei_fw-it i_o ☞_o mir_fw-ge that_o be_v as_o the_o radius_fw-la to_o the_o sine_fw-la of_o the_o side_n conterminat_fw-la with_o the_o angle_n require_v so_o the_o cotangent_fw-la of_o the_o other_o leg_n to_o the_o cotangent_fw-la of_o the_o angle_n search_v after_o or_o yet_o more_o profound_o by_o a_o alternat_a proportion_n change_v the_o relation_n of_o the_o four_o proportional_a although_o the_o same_o former_o require_v angle_n thus_o to_o rei_fw-la i_o ☞_o mor_fw-mi that_o be_v as_o the_o radius_fw-la to_o the_o antisecant_v of_o the_o side_n adjacent_a to_o the_o angle_n seek_v for_o so_o the_o antitangent_fw-la of_o the_o other_o side_n to_o the_o antitangent_fw-la of_o that_o side_n opposite_a angle_n which_o be_v the_o angle_n demand_v the_o reason_n hereof_o be_v ground_v on_o sbaprotca_n for_o the_o tangentine_a proportion_n of_o the_o term_n of_o this_o mood_n specifi_v its_o dependence_n on_o the_o second_o axiom_n which_o be_v show_v unto_o we_o by_o the_o eight_o and_o last_o characteristic_a of_o its_o directory_n pubkutethepsaler_n here_o end_v the_o doctrine_n of_o the_o rightangled_n sphericall_n the_o whole_a diatyposis_n whereof_o be_v in_o the_o equisolea_n or_o hippocrepidian_a diagram_n who_o most_o intricate_a amfractuosity_n renvoy_n various_a mixture_n of_o analogy_n and_o perturbat_fw-la situation_n of_o proportional_a term_n can_v choose_v but_o be_v pervious_a to_o the_o understanding_n of_o any_o judicious_a reader_n that_o have_v peruse_v this_o comment_n aright_o and_o therefore_o let_v he_o give_v i_o leave_v if_o he_o think_v fit_a for_o his_o memory_n sake_n to_o remit_v he_o to_o it_o before_o he_o proceed_v any_o further_a the_o loxogonosphericall_a triangle_n whether_o amblygonosphericall_a or_o oxygonosphericall_a be_v either_o monurgetick_a or_o disergetick_a the_o monurgetick_n have_v two_o figure_n datamista_n and_o datapura_n datamista_n be_v of_o all_o those_o loxogonospherical_a monurgetick_a problem_n wherein_o the_o angle_n and_o side_n be_v intermix_o give_v and_o therefore_o one_o of_o they_o be_v always_o of_o another_o kind_n from_o the_o other_o two_o either_o a_o angle_n or_o a_o side_n be_v demand_v it_o have_v two_o mood_n lamaneprep_n and_o menerolo_n the_o first_o mood_n lamaneprep_n comprehend_v all_o those_o loxogonosphericall_a problem_n wherein_o two_o angle_n be_v give_v and_o a_o opposite_a side_n another_o opposite_a side_n be_v demand_v and_o by_o its_o resolver_n sapeg_n see_v sapy_a ☞_o sir_n show_v that_o if_o to_o the_o logarithm_n of_o the_o sine_fw-la of_o the_o side_n give_v and_o sine_z of_o the_o angle_n opposite_a to_o the_o side_n require_v we_o join_v the_o arithmetical_a compliment_n of_o the_o sine_fw-la of_o the_o angle_n opposite_a to_o the_o propose_a side_n which_o be_v the_o refine_a antisecant_v we_o will_v thereby_o attain_v to_o the_o
knowledge_n of_o the_o sine_fw-la of_o the_o side_n demand_v the_o reason_n of_o this_o be_v ground_v on_o the_o three_o axiom_n seprosa_n as_o you_o may_v perceive_v by_o the_o first_o syllable_n of_o the_o obliquangularie_a directory_n lame_a the_o second_o mood_n of_o this_o figure_n be_v menerolo_n which_o comprehend_v all_o those_o amblygonosphericall_a problem_n wherein_o two_o side_n be_v give_v with_o a_o opposite_a angle_n another_o opposite_a angle_n be_v demand_v and_o by_o its_o resolver_n sepag_n sa_o sepi_fw-la ☞_o sir_n sheweth_z that_o if_o to_o the_o sum_n of_o the_o logarithm_n of_o the_o sine_fw-la of_o the_o give_v angle_n and_o sine_fw-la of_o the_o side_n opposite_a to_o the_o angle_n require_v we_o join_v the_o arithmetical_a compliment_n of_o the_o sine_fw-la of_o the_o side_n opposite_a to_o the_o give_v angle_n which_o be_v the_o refine_a cosecant_v of_o the_o say_a angle_n it_o will_v afford_v we_o the_o sine_fw-la of_o the_o angle_n require_v the_o reason_n of_o this_o operation_n be_v ground_v on_o the_o three_o axiom_n of_o sphericall_n a_o progress_n in_o sin_n show_v clear_o how_o that_o both_o this_o and_o the_o former_a do_v total_o depend_v on_o the_o axiom_n of_o seprosa_n as_o be_v evident_a by_o the_o second_o syllable_n of_o its_o directory_n lame_a the_o second_o figure_n of_o the_o monurgetick_a loxogonosphericall_n treat_v of_o all_o those_o question_n wherein_o the_o datas_fw-la be_v either_o side_n alone_o or_o angle_n alone_o a_o angle_n or_o a_o side_n be_v demand_v this_o figure_n of_o datapura_n be_v divide_v into_o two_o mood_n viz._n nerelema_n and_o ralamane_a which_o be_v of_o such_o affinity_n that_o upon_o one_o and_o the_o same_o theorem_fw-la depend_v the_o analogy_n that_o resolve_v both_o the_o first_o mood_n thereof_o nerelema_n comprehend_v all_o those_o problem_n wherein_o the_o three_o side_n be_v give_v a_o angle_n be_v demand_v and_o be_v the_o three_o of_o the_o monurgetick_n as_o by_o its_o characteristic_a the_o three_o liquid_a be_v perceivable_a the_o courteous_a reader_n may_v be_v please_v to_o take_v notice_n that_o in_o both_o the_o mood_n of_o the_o datapurall_a figure_n i_o be_o in_o some_o measure_n necessitate_v for_o the_o better_a order_n sake_n to_o couch_v two_o precept_n or_o document_n for_o the_o faciendas_fw-la thereof_o and_o to_o premise_v that_o one_o concern_v the_o three_o leg_n give_v before_o i_o make_v any_o mention_n of_o the_o main_a resolver_n whereupon_o both_o the_o foresay_a mood_n be_v found_v to_o which_o resolver_n because_o of_o both_o their_o dependence_n on_o it_o i_o have_v allow_v here_o in_o the_o gloss_n the_o same_o middle_a place_n which_o it_o possess_v in_o the_o table_n of_o my_o trissotetras_n the_o precept_n of_o nerelema_n be_v halbasalze_n *_o ad*_n ab*_n sadsabreregalsbis_n ir_z that_o be_v to_o say_v for_o the_o find_v out_o of_o a_o angle_n when_o the_o three_o leg_n be_v give_v as_o soon_o as_o we_o have_v constitute_v the_o sustentative_a leg_n of_o that_o angle_n a_o base_a the_o half_a thereof_o must_v be_v take_v and_o to_o that_o half_a we_o must_v add_v half_o the_o difference_n of_o the_o other_o two_o leg_n and_o likewise_o from_o that_o half_a subtract_v the_o half_a difference_n of_o the_o foresay_a two_o leg_n than_o the_o sum_n and_o the_o residue_n be_v two_o arch_n we_o must_v to_o the_o logarithm_n of_o the_o sine_fw-la of_o the_o sum_n and_o sine_fw-la of_o the_o remainder_n join_v the_o logarithm_n of_o the_o arithmetical_a compliment_n of_o the_o sin_n of_o the_o side_n which_o be_v the_o refine_a antisecant_v of_o the_o say_a leg_n and_o half_o that_o sum_n will_v afford_v we_o the_o logarithm_n of_o the_o sine_fw-la of_o a_o arch_n which_o double_v be_v the_o vertical_a angle_n we_o demand_v for_o out_o of_o its_o resolver_n parse_n powto_n parsadsab_n ☞_o powsalvertir_fw-fr be_v the_o analogy_n of_o the_o former_a work_n make_v clear_a the_o theorem_fw-la be_v as_o the_o oblong_a or_o parallelogram_n contain_v under_o the_o sin_n of_o the_o leg_n to_o the_o square_n power_n or_o quadrat_fw-la of_o the_o total_a sine_fw-la so_o the_o rectangle_n or_o oblong_o make_v of_o the_o right_a sin_n of_o the_o sum_n and_o difference_n of_o the_o half_a base_a and_o difference_n of_o the_o leg_n to_o the_o square_n of_o the_o right_n sine_fw-la of_o half_a the_o vertical_a angle_n the_o reason_n hereof_o will_v be_v manifest_a enough_o to_o the_o industrious_a reader_n if_o when_o by_o a_o peculiar_a diagram_n of_o who_o equiangular_a triangle_n the_o foresay_a sin_n and_o difference_n be_v make_v the_o constitutive_a side_n he_o have_v evince_v their_o analogy_n to_o one_o another_o he_o be_v then_o please_v to_o perpend_v how_o in_o two_o row_n of_o proportional_a number_n the_o product_n arise_v of_o the_o homologall_a root_n be_v in_o the_o same_o proportion_n among_o themselves_o that_o the_o say_v root_n towards_o one_o another_o be_v wherewithal_o if_o he_o do_v consider_v how_o the_o half_n must_v needs_o keep_v the_o same_o proportion_n that_o their_o whole_n and_o then_o in_o the_o work_n itself_o of_o collation_v several_a order_n of_o proportional_a term_n both_o single_a and_o compound_v be_v careful_a to_o dash_v out_o a_o divider_n against_o a_o multiplyer_n and_o afterward_o proceed_v in_o all_o the_o rest_n according_a to_o the_o ordinary_a rule_n of_o aequation_n and_o analogy_n he_o can_v choose_v but_o extricat_fw-la himself_o with_o ease_n forth_o of_o all_o the_o wind_n of_o this_o elabour_v proposition_n upon_o this_o theorem_fw-la as_o i_o have_v tell_v you_o depend_v likewise_o the_o document_n for_o the_o faciendum_fw-la of_o ralamane_a which_o be_v the_o second_o mood_n of_o datapura_n and_o the_o last_o of_o the_o monurgetick_a loxogonosphericals_n as_o be_v point_v at_o by_o nera_fw-mi the_o directory_n thereof_o this_o mood_n ralamane_a comprehend_v all_o those_o loxogonosphericall_a problem_n wherein_o the_o three_o angle_n be_v give_v a_o side_n be_v demand_v and_o by_o its_o resolver_n parse_n powto_n parsadsab_n ☞_o powsalvertir_fw-fr according_a to_o the_o peculiar_a precept_n of_o this_o mood_n kourbfasines_n erele_v koufbraxypopyx_n show_v that_o if_o we_o take_v the_o compliment_n to_o a_o semicircle_n of_o the_o angle_n opposite_a to_o the_o side_n require_v which_o for_o distinction_n sake_n we_o do_v here_o call_v the_o base_a and_o frame_v of_o the_o foresay_a compliment_n to_o a_o semicircle_n a_o second_o base_a for_o the_o fabric_n of_o a_o new_a triangle_n who_o other_o two_o side_n have_v the_o gradual_a measure_n of_o the_o former_a triangle_n other_o two_o angle_n and_o so_o the_o three_o angle_n be_v convert_v into_o side_n the_o compliment_n to_o a_o semicircle_n of_o the_o new_a vertical_a or_o angle_v opposite_a to_o the_o new_a base_a will_v be_v the_o measure_n of_o the_o true_a base_a or_o leg_n require_v and_o the_o angle_n incident_a on_o the_o right_a end_n of_o the_o new_a base_a in_o the_o second_o triangle_n fall_v to_o be_v the_o side_n conterminall_a with_o the_o left_a end_n of_o the_o true_a base_a in_o the_o first_o triangle_n and_o the_o angle_n adjoin_v the_o left_a end_n of_o the_o false_a base_a in_o the_o second_o triangle_n become_v the_o side_n adjacent_a to_o the_o right_a end_n of_o the_o old_a base_a in_o the_o first_o triangle_n so_o that_o thus_o by_o the_o angle_n all_o andeach_n of_o the_o side_n be_v find_v out_o all_o which_o work_n be_v to_o be_v perform_v by_o the_o precede_a mood_n upon_o the_o theorem_fw-la whereof_o the_o reason_n of_o all_o these_o operation_n do_v depend_v the_o disergetick_a loxogonosphericals_n be_v ground_v on_o four_o axiom_n viz._n 1._o nabadprosver_n 2._o naverprorte_n siubprortab_v and_o niubprodnesver_n the_o four_o directory_n whereof_o each_o in_o order_n to_o its_o own_o axiom_n be_v alama_n allera_fw-la ammena_n and_o ennerra_fw-la the_o first_o axiom_n be_v nabadprosver_n that_o be_v in_o obliquangular_a sphericals_n if_o a_o perpendicular_a be_v demit_v from_o the_o vertical_a angle_n to_o the_o opposite_a side_n continued_z if_o need_v be_v the_o sin_n compliment_n of_o the_o angle_n at_o the_o base_a will_v be_v direct_o proportional_a to_o the_o sin_n of_o the_o vertical_a angle_n and_o contrary_a the_o reason_n hereof_o be_v infer_v out_o of_o the_o proportion_n which_o the_o sin_n of_o angle_n substern_v by_o perpendicular_o have_v to_o the_o sin_n of_o the_o say_a perpendicular_o so_o that_o they_o belong_v to_o the_o arch_n of_o great_a circle_n concur_v in_o the_o same_o point_n and_o that_o from_o some_o point_n of_o the_o one_o they_o be_v let_v fall_n on_o the_o other_o arch_n which_o proportion_n of_o the_o sin_n of_o the_o say_a perpendicular_o to_o the_o sin_n of_o the_o angle_n subtend_v by_o they_o stow_v immediate_o from_o the_o proportion_n which_o in_o several_a orthogonosphericals_n have_v the_o same_o acute_a angle_n at_o the_o base_a be_v betwixt_o
the_o sin_n of_o the_o hypotenusas_n and_o the_o sin_n of_o the_o perpendicular_o the_o demonstration_n whereof_o be_v plain_o set_v down_o in_o my_o gloss_n on_o suprosca_n the_o first_o general_a axiom_n of_o the_o sphericals_n of_o which_o this_o axiom_n of_o nabadprosver_n be_v a_o consectary_n the_o directory_n of_o this_o axiom_n be_v alama_n which_o show_v that_o the_o mood_n of_o alamebna_n and_o amanepra_n be_v ground_v on_o it_o the_o second_o disergetick_a axiom_n be_v naverprortes_n that_o be_v to_o say_v the_o sin_n compliment_n of_o the_o vertical_a angle_n in_o obliquangular_a triangle_n a_o perpendicular_a be_v let_v fall_v from_o the_o double_a vertical_a on_o the_o opposite_a side_n be_v reciprocal_o proportional_a to_o the_o tangent_n of_o the_o side_n the_o reason_n hereof_o proceed_v from_o sbaprotca_n the_o second_o general_a axiom_n of_o the_o sphericals_n according_a to_o which_o if_o we_o do_v but_o regulate_v after_o the_o customary_a analogical_a manner_n two_o quaternary_n of_o proportional_o of_o the_o former_a sin_n compliment_n and_o tangent_n propose_v we_o will_v find_v by_o the_o extreme_n alone_a exclude_v all_o the_o intermediate_v term_n that_o the_o sin_n compliment_n of_o the_o vertical_a angle_n both_o forward_o and_o inverse_o be_v reciprocal_o proportion_v to_o the_o tangent_n of_o the_o side_n and_o contrariwise_o from_o the_o tangent_n to_o the_o sin_n the_o directory_n of_o this_o axiom_n be_v allera_fw-la which_o evidence_v that_o the_o mood_n of_o allamebne_n and_o erelomab_v depend_v upon_o it_o the_o three_o disergetick_a axiom_n be_v sinbprortab_v that_o be_v to_o say_v that_o in_o obliquangular_a sphericals_n if_o a_o perpendicular_a be_v draw_v from_o the_o vertical_a angle_n unto_o the_o opposite_a side_n continued_z if_o need_v be_v the_o sin_n of_o the_o segment_n of_o the_o base_a be_v reciprocal_o proportional_a to_o the_o tangent_n of_o the_o angle_n conterminate_v at_o the_o base_a and_o contrary_a the_o proof_n of_o this_o as_o well_o as_o that_o of_o the_o former_a confectary_n depend_v on_o sbaprotca_n the_o second_o general_a axiom_n of_o the_o sphericals_n according_a to_o which_o if_o we_o so_o diagrammatise_v a_o ambly_n gonosphericall_a triangle_n by_o quadrant_v the_o perpendicular_a and_o all_o the_o side_n and_o describe_v from_o the_o basangulary_a point_n two_o quadrantall_n arch_n till_o we_o hit_v upon_o two_o row_n of_o proportional_a sin_n of_o base_n to_o tangent_n of_o perpendicular_o then_o shall_v we_o be_v sure_a if_o we_o exclude_v the_o intermediate_v term_n to_o fall_v upon_o a_o reciprocal_a analogy_n of_o sines_n and_o tangent_n which_o alternat_o change_v will_v afford_v the_o reciprocal_a proportion_n of_o the_o sin_n of_o the_o segment_n of_o the_o base_a to_o the_o tangent_n of_o the_o angle_n conterminat_fw-la thereat_o the_o thing_n require_v the_o directory_n of_o this_o axiom_n be_v ammena_n which_o certify_v that_o ammanepreb_n and_o enerablo_n be_v found_v thereon_o the_o four_o and_o last_o disergetick_a axiom_n be_v niubprodnesver_n that_o be_v to_o say_v that_o in_o all_o loxogonosphericals_n where_o the_o cathetus_n be_v regular_o demit_v the_o sin_n compliment_n of_o the_o segment_n of_o the_o base_a be_v direct_o proportional_a to_o the_o sin_n compliment_n of_o the_o side_n of_o the_o vertical_a angle_n and_o contrary_a the_o reason_n hereof_o be_v make_v manifest_a by_o the_o proportion_n that_o be_v betwixt_o the_o sin_n of_o angle_n subtend_v by_o perpendicular_o and_o the_o sin_n of_o these_o perpendicular_o out_o of_o which_o we_o collation_n several_a proportion_n till_o both_o forward_o and_o inverse_o we_o pitch_v at_o last_o upon_o the_o direct_a proportion_n require_v the_o directory_n of_o this_o axiom_n be_v ennerra_n which_o declare_v that_o ennerable_a and_o errelome_o be_v its_o dependent_n of_o the_o disergetick_a loxogonosphericals_n there_o be_v in_o all_o four_o figure_n two_o angulary_a and_o two_o lateral_a the_o two_o angulary_a be_v ahalebmane_a and_o ahamepnare_n the_o two_o lateral_a be_v ehenabrole_v and_o eheromabne_n the_o first_o angulary_a disergetick_n loxogonosphericall_a figure_n ahalebmane_a comprehend_v all_o those_o problem_n wherein_o two_o angles_n be_v give_v with_o a_o side_n between_o either_o the_o three_o angle_n or_o a_o opposite_a side_n be_v demand_v and_o according_o have_v two_o mood_n the_o first_o whereof_o be_v alamebna_n and_o the_o second_o allamebne_n alamebna_n concern_v all_o those_o loxogonosphericall_a disergetick_a problem_n wherein_o two_o angles_n be_v propose_v with_o a_o interjacent_a side_n the_o three_o angle_n be_v require_v which_o angle_n according_a to_o the_o several_a case_n of_o this_o mood_n be_v always_o one_o of_o the_o angle_n at_o the_o base_a that_o be_v to_o say_v in_o the_o term_n of_o my_o trissotetr_n as_o a_o prime_a or_o next_o opposite_a or_o at_o least_o one_o of_o the_o co-opposites_a to_o the_o perpendicular_a to_o be_v demit_v and_o therefore_o conform_v to_o the_o nature_n of_o the_o case_n of_o the_o datoquaere_fw-la in_o hand_n and_o that_o it_o may_v the_o more_o convenient_o fall_v within_o the_o compass_n of_o the_o axiom_n of_o nabadprosver_n a_o angle_n by_o the_o first_o operation_n of_o this_o disergetick_n be_v to_o be_v find_v out_o which_o must_v either_o be_v a_o double_a vertical_a a_o vertical_a in_o the_o little_a rectangle_n or_o a_o vertical_a or_o coverticall_a as_o sometime_o i_o call_v it_o in_o one_o of_o the_o correctangle_v thus_o much_o i_o have_v think_v fit_a to_o premise_v of_o the_o praenoscendum_fw-la of_o this_o mood_n before_o i_o come_v to_o its_o cathetothesis_n because_o in_o my_o trissotetrall_a table_n to_o avoid_v the_o confusion_n of_o homogeneal_a term_n though_o the_o order_n of_o doctrine_n will_v seem_v to_o require_v another_o method_n the_o first_o and_o prime_n orthogonosphericall_a work_n be_v total_o unfold_v before_o i_o speak_v any_o thing_n of_o the_o variety_n of_o the_o perpendicular_o demission_n to_o which_o owe_v its_o rectangularity_n it_o thereby_o obtain_v a_o infallible_a progress_n to_o the_o quaesitum_fw-la but_o see_v in_o the_o gloss_n i_o be_o not_o to_o astrict_n myself_o to_o so_o little_a bound_n as_o in_o my_o table_n i_o will_v observe_v the_o order_n that_o be_v most_o expedient_a and_o before_o the_o resolution_n of_o any_o operation_n in_o this_o mood_n deduce_v the_o diversity_n of_o the_o perpendicular_o prosiliencie_n in_o the_o several_a case_n thereof_o let_v the_o reader_n then_o be_v please_v to_o consider_v that_o the_o general_a maxim_n for_o the_o cathetothesis_n of_o this_o mood_n be_v cafregpiq_n the_o meaning_n whereof_o be_v that_o whether_o the_o side_n whereon_o the_o perpendicular_a be_v demit_v be_v increase_v or_o not_o that_o be_v to_o say_v whether_o the_o perpendicular_a fall_n outward_o or_o inward_o it_o must_v fall_v from_o the_o extremity_n of_o the_o give_v side_n and_o opposite_a to_o the_o angle_n require_v however_o it_o be_v to_o be_v remark_v that_o in_o this_o mood_n whatever_o be_v the_o affection_n of_o the_o angles_n unless_o they_o be_v all_o three_o alike_o the_o perpendicular_a may_v fall_v out_o ward_o the_o general_a maxim_n for_o the_o cathetothesis_n of_o this_o mood_n as_o well_o as_o for_o that_o of_o all_o the_o rest_n be_v divide_v into_o four_o tenet_n according_a to_o the_o number_n of_o the_o case_n of_o every_o mood_n here_o must_v i_o admonish_v the_o reader_n that_o he_o startle_v not_o at_o the_o mention_v of_o four_o especial_a cathetothetick_a tenet_n and_o four_o several_a case_n belong_v to_o each_o disergetick_a mood_n see_v to_o the_o most_o observant_a eye_n there_o be_v but_o three_o of_o either_o perceptible_a in_o my_o trissotetr_n as_o for_o the_o four_o both_o tenet_n and_o case_n be_v the_o same_o by_o way_n of_o expression_n in_o all_o the_o mood_n and_o be_v full_o resolve_v by_o the_o three_o case_n of_o every_o mood_n it_o shall_v suffice_v to_o speak_v thereof_o here_o once_o for_o all_o the_o tenet_n of_o this_o common_a case_n be_v simomatin_n that_o be_v to_o say_v when_o all_o the_o three_o angle_n in_o any_o of_o those_o disergetick_n be_v of_o the_o same_o affection_n either_o all_o acute_a or_o all_o oblique_a the_o perpendicular_a fall_v inward_o whether_o the_o double_a vertical_a be_v a_o angle_n give_v a_o angle_n demand_v or_o neither_o yet_o here_o it_o be_v to_o be_v consider_v that_o see_v triangle_n may_v be_v either_o calculate_v by_o their_o real_a and_o natural_a or_o by_o their_o circular_a part_n or_o by_o both_o together_o and_o that_o for_o the_o more_o facility_n we_o oftentimes_o instead_o of_o the_o propose_a triangle_n resolve_v its_o opposite_a it_o be_v not_o the_o real_a and_o give_v triangle_n that_o in_o this_o case_n we_o so_o much_o take_v notice_n of_o as_o of_o its_o resolvable_a and_o equivalent_a the_o opposite_a triangle_n as_o for_o example_n if_o a_o spherical_a triangle_n with_o two_o obtuse_a angle_n and_o one_o acute_a be_v give_v
you_o to_o resolve_v it_o will_v fall_v within_o the_o compass_n of_o simomatin_n because_o its_o opposite_a spherical_a be_v simple_o acute_a angle_v and_o also_o if_o you_o be_v desire_v to_o calculate_v a_o spherical_a triangle_n with_o two_o acute_a angle_n and_o one_o obtuse_a it_o will_v likewise_o fall_v within_o the_o reach_n of_o the_o same_o case_n because_o its_o opposite_a spherical_a be_v simple_o obtusangle_v the_o reason_n of_o both_o the_o premise_n be_v from_o the_o equality_n of_o the_o opposite_a angle_n of_o concur_v quadrant_n which_o that_o they_o be_v equal_a no_o man_n needs_o to_o doubt_v that_o will_v take_v the_o pain_n to_o let_v fall_v a_o perpendicular_a from_o the_o middle_n of_o the_o one_o quadrant_n upon_o the_o other_o for_o so_o there_o will_v be_v two_o triangle_n make_v equilateral_a and_o see_v it_o be_v a_o universal_o receive_v truth_n that_o equal_a side_n sustain_v equal_a angle_n the_o identity_n of_o the_o perpendicular_a in_o both_o the_o foresay_a triangle_n must_v needs_o manifest_v the_o equality_n of_o the_o two_o opposite_a angle_n i_o have_v be_v the_o ample_a in_o the_o elucidating_a of_o this_o case_n that_o it_o overrun_v all_o the_o mood_n of_o the_o disergetick_a loxogonosphericals_n the_o reader_n in_o what_o mood_n or_o datoquare_n soever_o he_o please_v to_o resolve_v this_o foresay_a case_n may_v for_o that_o purpose_n to_o this_o place_n have_v recourse_n to_o the_o which_o without_o any_o further_a intend_a reiteration_n of_o this_o tenet_n i_o do_v hearty_o remit_v he_o the_o first_o especial_a tenet_n of_o the_o general_a maxim_n of_o the_o cathetothesis_n of_o this_o mood_n be_v dasimforaug_fw-mi that_o be_v when_o the_o give_v angle_n be_v of_o the_o same_o nature_n but_o different_a from_o the_o require_v the_o perpendicular_a fall_v outward_o and_o the_o first_o vertical_a be_v a_o give_v angle_n the_o second_o tenet_n belong_v to_o the_o second_o case_n of_o this_o mood_n be_v dadisforeug_v that_o be_v when_o the_o propose_a angle_n be_v of_o different_a affection_n the_o perpendicular_a be_v external_o demit_v and_o one_o of_o the_o give_v angle_n be_v a_o second_o vertical_a yet_o this_o discrepance_n be_v to_o be_v observe_v between_o the_o external_a prosiliencie_n of_o the_o perpendicular_a arch_n in_o this_o case_n and_o that_o other_o of_o the_o former_a that_o in_o the_o former_a it_o be_v no_o matter_n from_o which_o of_o the_o end_n of_o the_o propose_a side_n the_o perpendicular_a be_v let_v fall_n upon_o one_o of_o the_o comprehend_v leg_n of_o the_o angle_n require_v which_o leg_n must_v be_v increase_v for_o it_o be_v a_o general_a notandum_fw-la that_o the_o sustentative_a leg_n of_o a_o perpendicular_o exterior_a demission_n must_v always_o be_v continue_v but_o in_o this_o case_n the_o outward_a fall_n of_o the_o perpendicular_a be_v only_o from_o one_o extremity_n of_o the_o give_v side_n for_o if_o it_o be_v demit_v likewise_o from_o the_o other_o end_n it_o fall_v then_o inward_o and_o so_o produce_v the_o three_o tenet_n of_o this_o mood_n which_o be_v dadisgatin_n that_o be_v if_o the_o give_v angle_n be_v of_o a_o different_a quality_n and_o that_o the_o perpendicular_a be_v internal_o demit_v the_o double_a vertical_a be_v one_o of_o the_o propose_a angle_n the_o nature_n of_o the_o perpendicular_o demission_n in_o all_o the_o case_n of_o this_o mood_n be_v thus_o to_o the_o full_a explain_v we_o may_v without_o impediment_n proceed_v to_o the_o performance_n of_o all_o the_o orthogonosphericall_a operation_n each_o in_o its_o own_o order_n thereto_o belong_v to_o begin_v therefore_o at_o the_o first_o who_o quasitum_fw-la as_o i_o have_v tell_v you_o already_o be_v a_o vertical_a angle_n we_o must_v know_v see_v the_o work_n be_v orthogonospherical_o to_o be_v perform_v that_o the_o foremention_v praenoscendum_fw-la can_v be_v obtain_v without_o the_o help_n of_o one_o of_o the_o sixteen_o datoquaeres_fw-la and_o therefore_o in_o my_o trissotetras_n consider_v the_o nature_n of_o what_o be_v give_v and_o ask_v in_o the_o case_n of_o this_o mood_n i_o have_v appoint_v upalam_n to_o be_v the_o subservient_fw-fr of_o its_o praenoscendum_fw-la for_o by_o the_o resolver_n thereof_o to_o tag_n nu_n ☞_o mir_fw-fr the_o subtendent_fw-la and_o a_o angle_n be_v give_v for_o one_o of_o the_o give_v side_n of_o every_o loxogonosphericall_a if_o the_o perpendicular_a be_v right_o demit_v become_v a_o subtendent_fw-la and_o sometime_o two_o give_v side_n be_v subtendent_o both_o we_o frame_v these_o three_o peculiar_a problem_n for_o the_o three_o praenoscendas_fw-la to_o wit_n utopat_o for_o the_o double_a vertical_a by_o the_o mean_n of_o the_o great_a subtendent_fw-la side_n and_o the_o prime_n opposite_a angle_n second_o udobaud_n for_o obtain_v of_o the_o first_o vertical_a in_o the_o little_a rectangle_n by_o virtue_n of_o the_o lesser_a subtendent_fw-la in_o the_o same_o rectangle_n and_o the_o next_o opposite_a angle_n last_o uthophauth_n for_o the_o first_o coverticall_a by_o mean_n of_o the_o first_o cosubtendent_a and_o first_o coopposite_a angle_n all_o which_o be_v at_o large_a set_v down_o in_o the_o first_o partition_n of_o alamebna_n in_o my_o table_n the_o first_o and_o chief_a operation_n be_v thus_o perfect_v the_o vertical_a angle_n so_o find_v out_o must_v concur_v with_o each_o its_o correspondent_a opposite_a for_o the_o obtain_n of_o the_o perpendicular_a necessary_a for_o the_o accomplishment_n of_o the_o second_o operation_n in_o every_o one_o of_o the_o case_n of_o the_o foresay_a mood_n to_o which_o effect_n amane_a be_v make_v the_o subservient_fw-fr by_o who_o resolver_n say_v nag_n tw_n ☞_o nyr_n these_o three_o datoquaeres_fw-la opatca_fw-la obautca_n and_o ophauthca_n come_v to_o light_n and_o be_v manifest_o show_v how_o by_o any_o pair_n of_o three_o several_a couple_n of_o different_a angle_n the_o perpendicular_a be_v acquirable_a now_o though_o of_o this_o work_n as_o it_o be_v a_o single_a one_o no_o more_o than_o of_o the_o other_o succeed_v it_o in_o the_o same_o mood_n nor_o of_o the_o last_o two_o in_o any_o of_o the_o disergetick_n in_o their_o full_a analogy_n i_o do_v not_o make_v any_o mention_n at_o all_o in_o my_o table_n but_o after_o the_o couch_n of_o the_o first_o operation_n for_o the_o praenoscendas_fw-la supply_v the_o room_n of_o the_o other_o two_o with_o a_o equivalent_a row_n of_o proportional_n out_o of_o they_o specify_v for_o attain_v to_o the_o knowledge_n of_o the_o main_a quaesitum_fw-la yet_o in_o this_o comment_n upon_o that_o table_n for_o the_o more_o perspicuity_n sake_n and_o that_o the_o reader_n may_v as_o well_o know_v what_o way_n the_o rule_n be_v make_v as_o how_o thereby_o a_o demand_n be_v to_o be_v find_v out_o i_o have_v think_v fit_a to_o expatiat_fw-la myself_o for_o his_o satisfaction_n on_o each_o operation_n apart_o and_o analytical_o to_o display_v in_o the_o gloss_n what_o be_v compound_v in_o the_o trissotetras_n and_o therefore_o according_a to_o that_o prescribe_a method_n to_o proceed_v in_o this_o mood_n the_o perpendicular_a by_o the_o second_o operation_n be_v already_o obtain_v it_o be_v requisite_a for_o the_o promove_v of_o a_o three_o work_n that_o the_o say_v perpendicular_a be_v make_v to_o join_v with_o the_o second_o verticaline_a the_o double_a vertical_a and_o second_v coverticall_a conform_v to_o the_o quality_n of_o the_o three_o case_n thereby_o to_o obtain_v the_o angle_n at_o the_o base_a for_o the_o which_o all_o these_o operation_n have_v be_v set_v on_o foot_n to_o wit_n the_o next_o cathetopposite_a who_o compliment_n to_o a_o semicircle_n be_v always_o the_o angle_n require_v the_o prime_a cathetopposite_a and_o the_o second_o cocathetopposite_a for_o the_o prosecute_n of_o this_o last_o work_n edoman_n be_v the_o subservient_fw-fr by_o who_o resolver_n to_o neg_fw-mi sa_o ☞_o nir_fw-fr we_o be_v instruct_v how_o to_o regulate_v the_o problemet_n of_o catheudob_n cathatop_n and_o catheuthop_n now_o these_o two_o last_o operation_n be_v thus_o make_v patent_n in_o their_o several_a structure_n it_o be_v not_o amiss_o that_o we_o ponder_v how_o apposite_o they_o may_v be_v conflate_v into_o one_o to_o the_o end_n that_o the_o verity_n of_o all_o the_o final_a resolver_n of_o the_o disergetick_n in_o my_o trissotetras_n which_o be_v all_o and_o each_o of_o they_o compose_v of_o the_o ingredient_n term_n of_o two_o different_a work_n may_v be_v the_o more_o evident_o know_v and_o obvious_a to_o the_o reach_n of_o any_o ordinary_a capacity_n for_o the_o performance_n hereof_o the_o resolver_n of_o these_o two_o operation_n be_v to_o be_v lay_v before_o we_o say_v nag_n ta_o ☞_o nyr_n and_o to_o neg_fw-mi sa_o ☞_o nir_fw-fr and_o see_v out_o of_o both_o these_o order_n of_o proportional_n there_o must_v result_v but_o one_o it_o be_v to_o be_v consider_v which_o be_v the_o four_o ejectitious_a term_n and_o which_o those_o four_o we_o shall_v reserve_v for_o
the_o analogy_n require_v all_o which_o that_o it_o may_v be_v the_o better_o understand_v by_o the_o industrious_a reader_n i_o will_v interpret_v the_o resolver_n so_o far_o forth_o as_o be_v requisite_a and_o therefore_o say_v nag_n to_o ☞_o nyr_n be_v as_o the_o sine_fw-la of_o one_o of_o the_o angle_n at_o the_o base_a or_o cathetopposite_a be_v to_o the_o sine_fw-la compliment_n of_o a_o vertical_a so_o the_o radius_fw-la to_o the_o sine_fw-la compliment_n of_o the_o perpendicular_a and_o the_o other_o to_z neg_fw-mi sa_o ☞_o nir_fw-fr be_v as_o the_o radius_fw-la to_o the_o sine_fw-la compliment_n of_o the_o perpendicular_a so_o the_o sine_fw-la of_o a_o vertical_a to_o the_o sine_fw-la compliment_n of_o a_o cathetopposite_a or_o angle_n at_o the_o base_a it_o be_v perceivable_a enough_o how_o both_o the_o radius_fw-la and_o the_o perpendicular_a be_v in_o both_o the_o row_n nor_o can_v it_o well_o escape_v the_o knowledge_n of_o one_o never_o so_o little_o verse_v in_o the_o element_n of_o arithmetic_n that_o the_o perpendicular_o be_v the_o four_o term_n in_o the_o first_o order_n of_o proportional_n be_v nothing_o else_o but_o that_o it_o be_v the_o quotient_a of_o the_o product_n of_o the_o middle_a term_n divide_v by_o the_o first_o or_o logarithmical_o the_o remainder_n of_o the_o first_o term_n abstraction_n from_o the_o sum_n of_o the_o middle_n two_o so_o that_o the_o whole_a power_n thereof_o be_v enclose_v in_o these_o three_o term_n whereby_o it_o be_v most_o evident_a that_o with_o what_o term_n soever_o the_o foresay_a perpendicular_a be_v employ_v to_o concur_v in_o operation_n the_o same_o effect_n will_v be_v produce_v by_o the_o concurrence_n of_o its_o ingredient_n with_o the_o say_a term_n and_o therefore_o in_o the_o second_o row_n of_o proportional_n where_o it_o be_v make_v use_n of_o for_o a_o fellow_n multiply_o with_o the_o three_o term_n to_o produce_v a_o factus_fw-la which_o divide_v may_v quote_v the_o main_a quaesitum_fw-la or_o logarithmical_o to_o join_v with_o the_o three_o term_n for_o the_o sum_n of_o a_o aggregat_fw-la from_o which_o the_o first_o term_n be_v abstract_v may_v residuat_n the_o term_n demand_v it_o be_v all_o one_o whether_o the_o work_n be_v perform_v by_o itself_o or_o by_o its_o equivalent_a viz._n the_o three_o first_o term_n of_o the_o first_o order_n of_o proportional_n in_o who_o potentia_fw-la it_o be_v whereupon_o the_o four_o term_n in_o the_o second_o row_n be_v that_o for_o the_o obtain_n whereof_o both_o the_o analogy_n be_v make_v we_o need_v not_o waste_v any_o labour_n about_o the_o find_v out_o of_o the_o perpendicular_a though_o a_o subservient_fw-fr to_o the_o chief_a quaesitum_fw-la but_o leave_v room_n for_o it_o in_o both_o the_o row_n that_o the_o equipollencie_n of_o its_o conflater_n may_v the_o better_o appear_v go_v on_o in_o work_n without_o it_o and_o by_o the_o mean_n of_o its_o constructive_a part_n with_o as_o much_o certainty_n effectuate_v the_o same_o design_n thus_o may_v you_o see_v then_o how_o the_o eight_o term_n of_o the_o forementioned_a resolver_n be_v reduce_v unto_o six_o but_o there_o remain_v yet_o two_o more_o to_o be_v eject_v that_o both_o the_o order_n may_v be_v bring_v unto_o a_o compound_n row_n of_o four_o proportional_n let_v we_o consider_v the_o radius_fw-la which_o be_v in_o both_o the_o row_n as_o i_o have_v once_o tell_v you_o already_o may_v peradventure_o without_o any_o prejudice_n to_o the_o work_n be_v spare_v out_o of_o both_o thus_o much_o thereof_o to_o any_o be_v perceivable_a that_o in_o the_o first_o resolver_n it_o be_v the_o three_o proportional_a and_o in_o the_o second_o the_o first_o and_o consequent_o a_o multiplyer_n in_o the_o one_o and_o in_o the_o other_o a_o divider_n or_o logarithmical_o in_o the_o second_o a_o subtractor_n and_o in_o the_o first_o a_o adder_n now_o it_o be_v well_o know_v that_o division_n overthrow_v the_o structure_n of_o multiplication_n and_o that_o what_o be_v make_v up_o by_o addition_n be_v by_o subtraction_n cast_v down_o we_o need_v not_o undergo_v the_o laboriousnesse_n of_o such_o a_o penelopaean_a task_n and_o by_o the_o division_n and_o abstraction_n of_o what_o we_o do_v add_v and_o multiply_v weave_v and_o unweave_v build_v up_o and_o throw_v down_o the_o self_n same_o thing_n but_o choose_v rather_o see_v the_o radius_fw-la undo_v in_o the_o one_o what_o it_o do_v in_o the_o other_o which_o ineffect_n be_v to_o do_v nothing_o at_o all_o to_o dash_v the_o one_o against_o the_o other_o and_o race_n it_o out_o of_o both_o then_o idle_o to_o expend_v time_n and_o have_v the_o proportion_n pester_v with_o unnecessary_a term_n thus_o from_o those_o two_o resolver_n four_o term_n be_v with_o reason_n eject_v we_o must_v for_o the_o find_v out_o of_o the_o last_o in_o the_o second_o resolver_n effectuate_v as_o much_o by_o three_o as_o former_o be_v on_o seven_o incumbent_n which_o three_o be_v the_o first_o and_o second_o term_n in_o the_o first_o row_n of_o proportional_o and_o the_o three_o in_o the_o second_o the_o two_o resolver_n say_v nag_n to_o ☞_o nyr_n &_o to_o neg_fw-mi sa_o ☞_o nir_fw-fr be_v comprehend_v by_o this_o one_o say_v nag_n sa_o ☞_o nir_fw-fr that_o be_v as_o the_o sine_fw-la of_o one_o vertical_a to_o the_o antisine_n of_o a_o opposite_a so_o the_o sine_fw-la of_o another_o vertical_a to_o the_o antisine_n of_o another_o opposite_a and_o though_o the_o second_o resolver_n do_v import_v that_o this_o other_o opposite_a be_v to_o be_v find_v out_o by_o the_o antisine_n of_o the_o perpendicular_a and_o sine_fw-la of_o a_o secondary_a vertical_a yet_o do_v it_o in_o nothing_o evince_v the_o coincidence_n of_o the_o two_o operation_n in_o one_o because_o the_o first_o two_o term_n of_o the_o resultative_n analogy_n do_v adaequat_o stand_v for_o the_o perpendicular_a which_o i_o have_v prove_v already_o and_o therefore_o these_o two_o in_o their_o proper_a place_n coworking_a with_o the_o three_o term_n according_a to_o the_o rule_n of_o proportion_n have_v the_o self_n same_o influence_n that_o the_o perpendicular_a so_o second_v have_v upon_o the_o operatum_fw-la now_o to_o contract_v the_o generality_n of_o this_o final_a resolver_n say_v nag_n sa_o ☞_o nir_fw-fr to_o all_o the_o particular_a case_n of_o this_o mood_n we_o must_v say_v when_o the_o give_v angle_n be_v of_o the_o same_o affection_n and_o the_o require_v diverse_a as_o in_o dasimforaug_n the_o first_o case_n sit_v nop_n feud_n ☞_o nob*_n kir_n that_o be_v as_o the_o sine_fw-la of_o the_o double_a vertical_a to_o the_o antisine_n of_o the_o prime_n cathetopposite_a so_o the_o sine_fw-la of_o the_o second_o verticaline_a or_o vertical_a in_o the_o lesser_a rectangle_n to_o the_o antisine_n of_o the_o next_o cathetopposite_a who_o compliment_n to_o a_o semicircle_n be_v the_o angle_n require_v but_o when_o the_o affection_n of_o the_o give_v angle_n be_v different_a the_o perpendicular_a be_v make_v to_o fall_v without_o as_o in_o dadisforeug_n the_o second_o case_n of_o this_o mood_n the_o resolver_n thereof_o be_v particularise_v thus_o save_v nob_n sit_v ☞_o nop_n *_o ir_z that_o be_v as_o the_o sine_fw-la of_o the_o first_o verticaline_a or_o vertical_a in_o the_o rectanglet_n to_o the_o sine_fw-la compliment_n of_o the_o near_a cathetopposite_a so_o the_o sine_fw-la of_o the_o double_a vertical_a to_o the_o sine_fw-la compliment_n of_o the_o prime_n cathetopposite_a which_o be_v the_o angle_n require_v and_o last_o if_o with_o the_o different_a quality_n of_o the_o give_v and_o demand_v angle_n the_o perpendicular_a be_v let_v fall_v within_o as_o in_o dadisgatin_n the_o three_o case_n of_o this_o mood_n then_o be_v the_o final_a resolver_n to_o be_v determine_v thus_o sauth_n noph_n seuth_n ☞_o nops_n *_o ir_z that_o be_v as_o the_o sine_fw-la of_o the_o first_o coverticall_a to_o the_o cousin_a of_o the_o first_o co-opposite_a so_o be_v the_o sine_fw-la of_o the_o second_o coverticall_a to_o the_o cousin_a of_o the_o second_o co-opposite_a which_o be_v the_o angle_n require_v the_o original_a reason_n of_o all_o these_o operation_n be_v ground_v on_o the_o axiom_n of_o nabadprosver_n as_o the_o first_o syllable_n of_o its_o directory_n alama_n give_v we_o to_o understand_v which_o we_o may_v easy_o perceive_v by_o the_o analogy_n that_o be_v only_o among_o the_o angle_n without_o any_o intermixture_n of_o side_n in_o the_o term_n of_o the_o proportion_n the_o second_o mood_n of_o the_o first_o angulary_a figure_n that_o be_v to_o say_v the_o first_o two_o term_n of_o who_o datas_fw-la be_v angle_n be_v allamebne_n which_o comprehend_v all_o those_o disergetick_a question_n wherein_o two_o angles_n be_v give_v and_o a_o side_n between_o one_o of_o the_o other_o side_n be_v demand_v which_o side_n the_o perpendicular_a be_v let_v fall_v be_v always_o one_o of_o the_o second_o subtendent_o viz._n in_o the_o first_o
for_o the_o perfect_a of_o this_o operation_n erelam_n be_v the_o subservient_fw-fr by_o who_o resolver_n sei_fw-it teg_n to_o ☞_o tir_v we_o be_v instruct_v how_o to_o unfold_v its_o peculiar_a problemet_n oedcathob_v oetcathop_n and_o oethcathop_n all_o the_o three_o operation_n be_v thus_o single_o accomplish_v according_a to_o our_o wont_a manner_n the_o last_o two_o must_v be_v inchace_v into_o one_o and_o therefore_o their_o resolver_n to_o tag_n se_fw-la ☞_o tire_v and_o sei_fw-it teg_n to_o ☞_o tir_v must_v be_v unterm_v of_o some_o of_o their_o proportional_o the_o which_o that_o we_o may_v perform_v the_o more_o judicions_o let_v we_o consider_v what_o they_o signify_v apart_o the_o first_o import_v as_o in_o the_o last_o mood_n i_o tell_v you_o that_o as_o the_o radius_fw-la be_v to_o the_o tangent_fw-la of_o one_o of_o the_o opposite_a angle_n so_o the_o sine_fw-la of_o one_o of_o the_o first_o base_n to_o the_o tangent_fw-la of_o the_o perpendicular_a the_o second_o sound_v as_o the_o sine_fw-la of_o one_o of_o the_o second_o base_n to_o the_o tangent_fw-la of_o the_o perpendicular_a so_o the_o radius_fw-la to_o the_o tangent_fw-la of_o a_o angle_n which_o either_o usher_n of_o be_v the_o angle_n require_v hereby_o it_o be_v evident_a how_o the_o radius_fw-la be_v a_o multiply_o in_o the_o one_o and_o a_o divider_n in_o the_o other_o and_o that_o the_o perpendicular_a which_o with_o the_o radius_fw-la be_v a_o multiply_o in_o the_o second_o row_n be_v in_o the_o power_n of_o the_o three_o first_o term_n of_o the_o first_o row_n whereof_o the_o radius_fw-la be_v one_o by_o virtue_n of_o all_o which_o we_o must_v proceed_v just_a so_o with_o these_o last_o two_o operation_n here_o as_o we_o have_v already_o do_v with_o the_o two_o last_o of_o the_o mood_n of_o alamebna_n allamebne_n and_o ammanepreb_n and_o eject_v the_o radius_fw-la and_o perpendicular_a out_o of_o both_o instead_o of_o to_o tag_n se_fw-la ☞_o tire_v and_o sei_fw-it teg_n to_o ☞_o tir_v set_v down_o sei_fw-it tag_fw-mi se_fw-la ☞_o tir_v that_o be_v as_o the_o sine_fw-la of_o one_o of_o the_o second_o base_n to_o the_o tangent_fw-la of_o one_o of_o the_o cathetopposites_n so_o be_v the_o sine_fw-la of_o one_o of_o the_o first_o base_n to_o the_o tangent_fw-la of_o one_o of_o the_o other_o cathetopposites_n which_o proposition_n comprehend_v to_o the_o full_a the_o last_o two_o operation_n and_o according_a to_o the_o three_o several_a case_n of_o this_o mood_n be_v to_o be_v individuate_v into_o so_o many_o final_a resolver_n the_o first_o thereof_o for_o dacramfor_n be_v soe_v top_n saet_fw-la ☞_o tob_z *_o kir_n that_o be_v as_o the_o sine_fw-la of_o the_o second_o basidion_n or_o little_a base_a be_v to_o the_o tangent_fw-la of_o the_o prime_n cathetopposite_a so_o the_o sine_fw-la of_o the_o first_o and_o great_a base_a to_o the_o tangent_fw-la of_o the_o next_o cathetopposite_a who_o compliment_n to_o a_o semicircle_n be_v the_o angle_n require_v the_o second_o final_a resolver_n be_v for_o damracfor_a the_o tenet_n of_o the_o second_o case_n and_o be_v soet_n tob_z say_z ☞_o top_n *_o ir_z that_o be_v to_o say_v as_o the_o sine_fw-la of_o the_o second_o and_o great_a base_a to_o the_o tangent_fw-la of_o the_o next_o cathetopposite_a so_o the_o sine_fw-la of_o the_o first_o basidion_n to_o the_o tangent_fw-la of_o the_o prime_n opposite_a which_o be_v the_o angle_n require_v the_o three_o and_o last_o final_a resolver_n be_v for_o the_o three_o case_n dasimquaein_o and_o be_v couch_v thus_o soeth_o toph_n saeth_n ☞_o top_n *_o ir_z that_o be_v as_o the_o sine_fw-la of_o the_o second_o cobase_a be_v to_o the_o tangent_fw-la of_o the_o first_o cocathetopposite_a so_o be_v the_o sine_fw-la of_o the_o first_o cobase_a to_o the_o tangent_fw-la of_o the_o second_o cocathetopposite_a which_o be_v the_o angle_n require_v the_o fundamental_a reason_n of_o all_o this_o be_v from_o the_o three_o disergetick_a axiom_n siubprortab_v the_o second_o determinater_n of_o who_o directory_n ammena_n show_v that_o the_o mood_n of_o enerablo_n in_o all_o the_o final_a resolver_n thereof_o owe_v the_o truth_n of_o its_o analogy_n to_o the_o maxim_n of_o siubprortab_n because_o of_o the_o reciprocal_a proportion_n thaâ—_n be_v among_o its_o term_n to_o be_v find_v betwixt_o the_o sin_n of_o the_o basall_a segment_n and_o the_o tangent_n of_o the_o cathetopposite_a angle_n the_o second_o mood_n of_o ehenabrole_n be_v ennerable_a which_o comprehend_v all_o those_o obliquangulary_a problem_n wherein_o two_o side_n be_v give_v with_o a_o angle_n intercept_v therein_o the_o three_o side_n demand_v which_o side_n be_v always_o one_o of_o the_o second_o subtendentâ—_n that_o be_v either_o the_o second_o subtendentine_n the_o second_o subtendentall_a 〈◊〉_d the_o second_o cosubtendent_a to_o the_o notice_n of_o all_o which_o that_o we_o may_v the_o more_o easy_o attain_v let_v we_o perpend_v the_o general_a maxim_n of_o the_o cathetothesis_n of_o this_o mood_n cafregpaq_n the_o meaning_n whereof_o be_v that_o in_o this_o mood_n whatever_o the_o case_n be_v the_o perpendicular_a may_v fall_v from_o the_o extremity_n of_o either_o of_o the_o give_v side_n but_o must_v fall_v from_o one_o of_o they_o opposite_a to_o the_o angle_n propose_v and_o upon_o the_o other_o give_v side_n continued_z if_o need_v be_v here_o may_v the_o reader_n be_v please_v to_o observe_v that_o the_o clause_n of_o the_o perpendicular_o fall_v opposite_a to_o the_o propose_a angle_n though_o it_o be_v only_o mention_v in_o this_o place_n may_v have_v as_o well_o be_v speak_v of_o in_o any_o one_o of_o the_o rest_n of_o the_o cathetothetick_a comment_n because_o it_o be_v a_o general_a tie_n incumbent_a on_o the_o demission_n of_o perpendicular_o in_o all_o loxogonosphericall_a disergetick_a figure_n whether_o amblygonian_a or_o oxygonian_a that_o it_o fall_v always_o opposite_a to_o a_o know_a angle_n and_o from_o the_o extremity_n of_o a_o know_a side_n of_o this_o general_a maxim_n cafregpaq_n according_a to_o the_o variety_n of_o the_o second_o subtendent_fw-la which_o be_v the_o side_n require_v there_o be_v these_o three_o especial_a tenet_n dacforamb_v damforac_n and_o dakinatam_fw-la dacforamb_v the_o tenet_n of_o the_o first_o case_n give_v we_o to_o understand_v that_o if_o the_o give_v angle_n be_v acute_a and_o that_o one_o only_a of_o the_o other_o two_o be_v a_o obtuse_a angle_n the_o perpendicular_a fall_v outward_o damforac_n the_o tenet_n of_o the_o second_o case_n signify_v that_o if_o the_o give_v angle_n be_v obtuse_a and_o the_o other_o two_o acute_a that_o the_o demission_n of_o the_o perpendicular_a be_v external_a as_o in_o the_o first_o three_o dakinatam_fw-la the_o tenet_n of_o the_o three_o case_n and_o variator_n of_o the_o first_o sheweth_z that_o if_o the_o propose_a angle_n be_v of_o the_o same_o affection_n with_o one_o of_o the_o other_o angle_n of_o the_o triangle_n as_o in_o the_o first_o case_n the_o perpendicular_a may_v fall_v inward_o the_o cathetology_n of_o this_o mood_n be_v thus_o expede_v the_o pranoscendas_fw-la thereof_o come_v next_o in_o hand_n to_o be_v discuss_v which_o be_v the_o first_o base_n who_o subservient_fw-fr be_v vbamen_n and_o its_o resolver_n torp_n mu_o lag_a ☞_o myr_n upon_o which_o depend_v the_o three_o subdatoquaera_n of_o vtopaet_n vdobaed_n and_o vthophaeth_n thus_o much_o i_o believe_v be_v express_v in_o the_o very_a table_n of_o my_o trissotetras_n and_o though_o a_o large_a explication_n may_v be_v with_o reason_n expect_v in_o this_o place_n of_o what_o be_v but_o summary_o mention_v there_o yet_o because_o what_o concern_v this_o matter_n have_v be_v already_o treat_v of_o in_o the_o last_o two_o mood_n of_o enerablo_n and_o 〈◊〉_d the_o whole_a discourse_n whereof_o may_v be_v as_o convenient_o peruse_v as_o if_o it_o be_v couch_v here_o i_o will_v not_o dull_a the_o reader_n with_o tedious_a rehearsal_n of_o one_o and_o the_o same_o thing_n but_o let_v pass_v the_o progress_n of_o this_o first_o work_n with_o the_o manner_n of_o which_o by_o my_o former_a instruction_n i_o suppose_v he_o sufficient_o well_o acquaint_v will_v proceed_v to_o the_o cathetouretick_n operation_n of_o this_o mood_n and_o perpend_v by_o what_o datas_fw-la the_o perpendicular_a be_v to_o be_v find_v out_o to_o this_o effect_n the_o praenoscendas_fw-la of_o ennerable_a to_o wit_n the_o first_o basal_n the_o first_o basidion_n and_o the_o first_o cobase_a be_v by_o the_o last_o work_n already_o obtain_v must_v concur_v with_o each_o its_o correspondent_a first_o subtendent_fw-la viz._n the_o first_o subtendentall_a the_o first_o subtendentine_n and_o the_o first_o cosubtendent_a discernible_a in_o their_o several_a nature_n by_o the_o figuratives_n of_o Ï„_n δ_n θ._n for_o the_o perfect_a of_o this_o second_o operation_n the_o subservient_fw-fr of_o this_o work_n be_v uchâ—ner_n by_o who_o resolver_n neg_fw-mi to_o nu_n ☞_o nyr_n the_o three_o subproblem_n utaeta_n vtadca_n
the_o second_o case_n and_o be_v express_v thus_o mud_n naud_n mwt_n ☞_o natfr*aud*eut*ir_n that_o be_v as_o the_o tangent_fw-la compliment_n of_o the_o first_o subtendentine_n to_o the_o sine_fw-la compliment_n of_o the_o first_o verticalin_n so_o the_o tangent_fw-la compliment_n of_o the_o second_o subtendentall_a to_o the_o sine_fw-la compliment_n of_o the_o double_a vertical_a from_o which_o if_o you_o deduce_v the_o first_o verticalin_n there_o will_v remain_v the_o second_o vertical_a for_o the_o angle_n require_v the_o last_o final_a resolver_n be_v for_o the_o three_o case_n dacrambatin_n and_o be_v couch_v thus_o muth_o nauth_n mwth_n ☞_o neuth*jauth*ir_n that_o be_v as_o the_o tangent_fw-la compliment_n of_o the_o first_o cosubtendent_a to_o the_o sine_fw-la compliment_n of_o the_o first_o coverticall_a so_o the_o tangent_fw-la compliment_n of_o the_o second_o cosubtendent_a to_o the_o sine_fw-la compliment_n of_o the_o second_o coverticall_a which_o join_v to_o the_o first_o coverticall_a afford_v the_o angle_n require_v the_o proof_n of_o the_o verity_n of_o all_o these_o analogy_n be_v take_v out_o of_o the_o second_o disergetick_n amblygonosphericall_a axiom_n naverprortes_n the_o second_o determinater_n of_o who_o directory_n show_v that_o this_o mood_n be_v one_o of_o its_o dependent_n and_o with_o reason_n because_o of_o the_o reciprocal_a analogy_n that_o among_o its_o term_n be_v perceivable_a betwixt_o the_o tangent_n of_o the_o vertical_a side_n which_o in_o this_o mood_n be_v always_o first_o subtendent_o and_o the_o sine-complement_n of_o the_o vertical_a angle_n that_o be_v tosay_v according_a to_o the_o literal_a meaning_n of_o my_o final_a resolver_n of_o this_o mood_n the_o direct_a proportion_n that_o be_v betwixt_o the_o tangent-complement_n of_o the_o vertical_a side_n or_o rear_v subtendent_o &_o the_o sine-complement_n of_o the_o vertical_a angle_n for_o the_o proportion_n be_v the_o same_o with_o that_o whereof_o i_o have_v tell_v you_o somewhat_o already_o in_o the_o mood_n of_o allamebme_n the_o fellow_n dependent_a of_o erelomab_n the_o second_o mood_n of_o eherolabme_n four_o of_o the_o lateral_n eight_o of_o the_o sphericobliquangularie_a disergetick_n twelve_o of_o the_o loxogonosphericall_n eight_o and_o twenty_o of_o the_o sphericals_n and_o one_o and_o fourti_v or_o last_v of_o the_o triangular_n be_v errelome_o which_o comprehend_v all_o those_o obliquangularie_a problem_n wherein_o two_o side_n be_v give_v with_o a_o opposite_a angle_n the_o three_o side_n be_v require_v which_o side_n be_v always_o either_o one_o of_o the_o segment_n of_o the_o base_a or_o the_o base_a itself_o to_o the_o knowledge_n of_o all_o which_o that_o we_o may_v reach_v with_o ease_n we_o must_v perpend_v the_o general_a maxim_n of_o the_o cathetothesis_n of_o this_o mood_n which_o be_v cacurgyq_n that_o be_v to_o say_v the_o perpendicular_o demission_n in_o all_o the_o case_n of_o errelome_o must_v be_v from_o the_o concourse_n of_o the_o give_v side_n upon_o the_o side_n require_v continue_a if_o need_v be_v the_o plus-minus_a of_o this_o mood_n be_v sindiforiu_n which_o import_v that_o if_o the_o perpendicular_a fall_n internal_o the_o sum_n of_o the_o segment_n of_o the_o base_a or_o the_o total_a base_a be_v the_o side_n demand_v and_o if_o it_o fall_v without_o the_o difference_n of_o the_o base_n the_o little_a base_a be_v always_o but_o a_o segment_n of_o the_o great_a be_v the_o main_a quaesitum_fw-la the_o mood_n of_o ammanepreb_n be_v sindiforate_v in_o the_o same_o manner_n as_o this_o be_v because_o the_o main_a quaesitas_fw-la and_o four_o proportional_n of_o both_o do_v in_o nothing_o differ_v but_o that_o those_o be_v sinused_a and_o these_o run_v upon_o sine-complement_n the_o prosiliencie_n of_o the_o perpendicular_a in_o all_o spherical_a disergetick_n be_v so_o necessary_a to_o be_v know_v as_o i_o have_v often_o tell_v you_o because_o of_o the_o facility_n thereby_o to_o reduce_v they_o to_o rectangulary_a operation_n it_o fall_v out_o most_o convenient_o here_o according_a to_o the_o method_n propose_v to_o myself_o to_o speak_v somewhat_o of_o the_o three_o several_a tenet_n of_o the_o cathetothesis_n of_o this_o mood_n and_o what_o be_v understand_v by_o dakyxamfor_n dambyxamfor_n and_o dakypambin_n dakyxamfor_n which_o be_v the_o tenet_n of_o the_o first_o case_n declare_v that_o if_o the_o propose_a angle_n be_v acute_a and_o the_o side_n require_v conterminate_a with_o a_o obtuse_a angle_n the_o demission_n of_o the_o perpendicular_a be_v extrinsecall_v dambyxamfor_n the_o tenet_n of_o the_o second_o case_n import_v that_o if_o the_o give_v angle_n be_v obtuse_a and_o that_o the_o side_n require_v be_v annex_v thereto_o the_o perpendicular_a must_v as_o in_o the_o last_o fall_v outward_o three_o dakypambin_n the_o tenet_n of_o the_o last_o case_n signify_v that_o if_o the_o angle_n propose_v be_v sharp_a &_o that_o the_o demand_v side_n be_v subjacent_a to_o a_o obtuse_a or_o blunt_a angle_n the_o perpendicular_a fall_v inward_o have_v thus_o proceed_v in_o the_o enumeration_n of_o the_o cathetothetick_a tenet_n of_o this_o mood_n according_a to_o the_o manner_n by_o i_o observe_v in_o those_o of_o all_o the_o former_a disergetick_n save_o the_o first_o i_o be_o confident_a the_o reader_n if_o he_o have_v peruse_v all_o the_o tractat_fw-la until_o this_o place_n will_v not_o think_v strange_a why_o dakypambin_n be_v but_o the_o three_o i_o shall_v call_v it_o the_o tenet_n of_o the_o last_o case_n of_o this_o mood_n for_o though_o in_o alamebna_n i_o speak_v somewhat_o of_o every_o amblygonosphericall_a disergetick_a mood_n general_a cathetothetick_a maxim_n division_n into_o four_o especial_a tenet_n appropriable_a to_o so_o many_o several_a case_n yet_o the_o four_o case_n viz._n that_o wherein_o all_o the_o angle_n be_v homogeneal_a whether_o blunt_a or_o sharp_a not_o be_v limit_v to_o any_o one_o mood_n but_o adaequat_o extend_v to_o all_o the_o eight_o it_o seem_v to_o i_o more_o expedient_a to_o let_v its_o generality_n be_v know_v by_o mention_v it_o once_o or_o twice_o then_o by_o do_v no_o more_o in_o effect_n to_o make_v superfluous_a repetition_n and_o as_o in_o the_o first_o disergetick_a case_n for_o the_o reader_n instruction_n i_o do_v under_o the_o name_n of_o simomatin_n explicate_v the_o nature_n thereof_o so_o for_o his_o better_a remembrance_n have_v i_o choose_v rather_o to_o shut_v up_o my_o cathetothetick_a comment_n with_o the_o same_o discourse_n wherewith_o i_o do_v begin_v it_o then_o unnecessary_o to_o weary_v he_o with_o frequent_a reiteration_n and_o a_o tedious_a rehearsal_n of_o one_o and_o the_o same_o thing_n in_o all_o the_o six_o several_a intermediat_a mood_n it_o be_v not_o amiss_o now_o that_o the_o perpendicularity_n of_o this_o mood_n be_v discuss_v to_o consider_v what_o the_o praenoscendas_fw-la thereof_o be_v or_o the_o quaesitas_fw-la of_o the_o first_o operation_n but_o as_o i_o say_v in_o the_o last_o mood_n that_o there_o be_v no_o need_n to_o insist_v so_o long_o upon_o the_o explication_n of_o those_o praenoscendas_fw-la whereof_o ample_a relation_n have_v be_v already_o make_v in_o some_o of_o my_o proturgetick_a comment_n as_o upon_o those_o other_o which_o for_o be_v altogether_o different_a from_o such_o as_o have_v be_v former_o mention_v claim_v by_o the_o law_n of_o parity_n in_o their_o imparity_n right_a to_o a_o large_a discourse_n apart_o i_o will_v confine_v my_o pen_n upon_o this_o subject_n within_o those_o prescribe_a bound_n and_o see_v the_o first_o basal_n the_o first_o basidion_n and_o first_o cobase_a together_o with_o the_o datas_fw-la whereby_o they_o be_v find_v out_o viz._n for_o the_o side_n the_o first_o subtendentall_a the_o first_o subtendentine_n and_o the_o first_o co_fw-la subtendent_fw-la and_o for_o the_o angle_n the_o prime_n cathetopposite_a the_o next_o cathetopposite_a and_o the_o first_o cocathetopposite_a the_o datas_fw-la be_v both_o for_o side_n and_o angle_v the_o same_o here_o that_o they_o be_v in_o the_o former_a mood_n than_o the_o subservient_fw-fr ubamen_fw-la and_o its_o resolver_n torp_n mu_o lag_a ☞_o myr_n with_o the_o three_o peculiar_a problemet_n thereto_o belong_v utopat_o vdobaed_n and_o vthophaeth_n be_v all_o and_o every_o one_o of_o they_o the_o same_o in_o this_o mood_n of_o errelome_o that_o they_o be_v in_o the_o three_o forego_v mood_n of_o ammanepreb_n enerablo_n and_o ennerable_a these_o be_v the_o only_a four_o mood_n which_o have_v a_o lateral_a praenoscendal_n identity_n the_o reader_n will_v not_o in_o my_o opinion_n be_v so_o prodigal_a of_o his_o own_o labour_n nor_o covetous_a of_o i_o that_o either_o he_o will_v put_v himself_o or_o i_o to_o any_o further_a pain_n then_o have_v be_v already_o bestow_v upon_o this_o matter_n by_o myself_o for_o his_o instruction_n and_o therefore_o leave_v it_o for_o a_o suppose_a certainty_n that_o the_o praenoscendas_fw-la or_o first_o base_n according_a to_o the_o nature_n of_o the_o case_n can_v escape_v the_o reader_n knowledge_n by_o what_o
have_v be_v by_o i_o deliver_v of_o they_o i_o purpose_v here_o to_o give_v he_o notice_n that_o these_o foresay_a first_o base_n must_v concur_v with_o each_o its_o correspondent_a first_o subtendent_fw-la to_o wit_n the_o first_o subtendentall_a the_o first_o subtendentine_n and_o first_o cosubtendent_a dignoscible_a by_o the_o characteristic_n of_o τ._n δ._n θ_o for_o obtain_v of_o the_o perpendicular_a of_o which_o operation_n vchener_n be_v the_o subservient_fw-fr by_o who_o resolver_n neg_fw-mi to_o nu_n ☞_o nyr_n the_o problemet_n of_o utaetca_n udaedca_n and_o uthaethca_n be_v make_v manifest_a as_o to_o the_o same_o effect_n it_o remain_v couch_v in_o my_o comment_n upon_o ennerable_a which_o be_v the_o only_a mood_n that_o with_o this_o of_o errelome_o have_v a_o subtendentine_n and_o basal_a catheteuretick_n identity_n the_o second_o work_n be_v thus_o perfect_v the_o perpendicular_a thereby_o find_v out_o be_v to_o assist_v one_o of_o the_o rear_n subtendent_o in_o obtain_v the_o illatitious_a term_n of_o the_o main_a quasitum_fw-la correspondent_a thereto_o discernible_a by_o the_o characteristic_n or_o figurative_n of_o δ._n τ._n θ_o or_o more_o plain_o to_o express_v it_o the_o perpendicular_a must_v concur_v according_a as_o the_o case_n require_v it_o with_o the_o second_o subtendentine_n the_o second_o subtendentall_a and_o second_v cosubtendent_a as_o you_o may_v see_v in_o the_o last_o mood_n the_o datas_fw-la of_o the_o resolutory_a partition_n whereof_o be_v the_o same_o as_o here_o to_o find_v out_o three_o base_n which_o by_o abstract_v the_o first_o from_o another_o base_a then_o by_o abstract_v another_o base_a from_o the_o second_o and_o last_o by_o add_v the_o three_o base_a to_o another_o afford_v the_o sum_n and_o difference_n which_o be_v the_o require_v base_n for_o the_o performance_n of_o this_o operation_n the_o same_o subservient_fw-fr and_o resolver_n suffice_v which_o serve_v for_o the_o last_o so_o that_o uchener_n subserve_v it_o by_o who_o resolver_n neg_fw-mi to_o nu_n ☞_o nyr_n we_o be_v instruct_v how_o to_o explicate_v the_o subdatoquaeres_fw-la of_o wdcathoed_a wtcathoet_n and_o wthcatho_v or_o more_o orderly_o cathwdoe_v cathwtoet_fw-mi and_o cathwtho_v all_o the_o three_o work_n be_v thus_o accomplish_v the_o manner_n of_o conflate_v the_o last_o two_o in_o one_o rest_n to_o be_v treat_v of_o for_o the_o better_a perfect_a of_o which_o design_n the_o two_o resolver_n or_o the_o same_o in_o its_o great_a generality_n double_v viz._n neg_fw-mi to_o nu_n ☞_o nyr_n and_o neg_fw-mi to_o nu_n ☞_o nyr_n must_v be_v interpret_v the_o truth_n be_v both_o of_o they_o as_o they_o sound_v in_o their_o vast_a extent_n of_o signification_n express_v the_o same_o analogy_n without_o any_o difference_n which_o be_v as_o the_o sine_fw-la compliment_n of_o a_o give_v side_n to_o the_o radius_fw-la so_o the_o sine_fw-la compliment_n of_o a_o subtendent_fw-la to_o the_o sine_fw-la compliment_n of_o another_o side_n but_o when_o more_o contract_o according_a to_o the_o specification_n of_o the_o side_n they_o do_v suppone_v several_o they_o shall_v be_v thus_o expound_v the_o first_o as_o the_o sine_fw-la compliment_n of_o a_o first_o base_a to_o the_o total_a sine_fw-la so_o the_o sine_fw-la compliment_n of_o a_o first_o subtendent_fw-la to_o the_o sine_fw-la compliment_n of_o the_o perpendicular_a and_o the_o second_o as_o the_o sine_fw-la compliment_n of_o the_o perpendicular_a to_o the_o total_a sine_fw-la so_o the_o sine_fw-la compliment_n of_o a_o second_o subtendent_fw-la to_o the_o sine_fw-la compliment_n of_o a_o second_o base_a which_o usher_v the_o main_a quaesitum_fw-la now_o the_o perpendicular_a and_o radius_fw-la be_v both_o to_o be_v expel_v these_o two_o foresay_a order_n of_o proportional_a term_n for_o the_o reason_n which_o in_o the_o last_o precede_a mood_n and_o some_o other_o before_o it_o i_o have_v already_o mention_v and_o which_o to_o repeat_v further_o then_o that_o the_o sympathy_n of_o this_o place_n with_o that_o may_v be_v manifest_v in_o the_o tranf-seating_a of_o multiplyer_n and_o divider_n occasion_v by_o the_o four_o term_n in_o the_o first_o row_n be_v first_o in_o the_o second_o be_v altogether_o unnecessary_a in_o lieu_n of_o neg_fw-mi to_o nu_n ☞_o nyr_n and_o neg_fw-mi to_o nu_n ☞_o nyr_n we_o may_v say_v with_o as_o much_o truth_n power_n and_o efficacy_n and_o far_o more_o compendious_o nu_n ne_n nu_n ☞_o nyr_n that_o be_v as_o the_o sine_fw-la compliment_n of_o a_o subtendent_fw-la to_o the_o sine_fw-la compliment_n of_o a_o side_n so_o the_o sine_fw-la compliment_n of_o another_o subtendent_fw-la to_o the_o sine_fw-la compliment_n of_o another_o side_n or_o more_o particular_o and_o appliable_o to_o the_o present_a analogy_n as_o the_o sine_fw-la compliment_n of_o a_o first_o subtendent_fw-la to_o the_o sine_fw-la compliment_n of_o a_o first_o base_a so_o the_o sine_fw-la compliment_n of_o a_o second_o subtendent_fw-la to_o the_o sine_fw-la compliment_n of_o a_o second_o base_a illative_a to_o the_o quaesitum_fw-la this_o theorem_fw-la or_o proposition_n comprehend_v in_o every_o point_n all_o that_o be_v in_o the_o two_o last_o operation_n and_o not_o transcend_v the_o number_n of_o the_o case_n of_o this_o mood_n be_v divide_v into_o so_o many_o final_a resolver_n the_o first_o whereof_o for_o the_o first_o case_n dakyxamfor_n be_v nut_n naet_fw-la nwd_n ☞_o noedfr*aet*_n dyr_n that_o be_v as_o the_o sine_fw-la compliment_n of_o the_o first_o subtendent_fw-la all_o to_o the_o sine_fw-la compliment_n of_o the_o first_o basall_n so_o the_o sine_fw-la compliment_n of_o the_o second_o subtendentine_n to_o the_o sine_fw-la compliment_n of_o the_o second_o base_a which_o subduct_v from_o the_o first_o basal_n residuat_v the_o segment_n that_o be_v the_o side_n require_v the_o second_o final_a resolver_n of_o this_o mood_n and_o that_o which_o be_v for_o the_o second_o case_n thereof_o dambyxamfor_n be_v nud_a naed_n nwt_o ☞_o noetfr*aed*_n dyr_n that_o be_v as_o the_o sine_fw-la compliment_n of_o the_o first_o subtendentine_n to_o the_o sine_fw-la compliment_n of_o the_o first_o basidion_n so_o the_o sine_fw-la compliment_n of_o the_o second_o subtendentall_a to_o the_o sine_fw-la compliment_n of_o the_o second_o basall_n which_o the_o first_o basidion_n be_v subtract_v from_o it_o leaf_n for_o remainder_n or_o difference_n that_o segment_n of_o the_o base_a which_o be_v the_o side_n demand_v the_o last_o final_a resolver_n of_o this_o mood_n belong_v to_o the_o three_o case_n dakypambin_n as_o also_o to_o the_o four_o simomatin_n if_o what_o we_o have_v already_o speak_v of_o that_o matter_n will_v permit_v we_o to_o call_v it_o the_o four_o for_o simomatin_n together_o with_o the_o three_o case_n of_o every_o mood_n be_v still_o resolve_v by_o the_o last_o final_a resolver_n thereof_o be_v nuth_n naeth_n nwth_n ☞_o noethj*aeth*_n sir_n that_o be_v as_o the_o sine_fw-la compliment_n of_o the_o first_o cosubtendent_a to_o the_o sine_fw-la compliment_n of_o the_o first_o cobase_a so_o the_o sine_fw-la compliment_n of_o the_o second_o cosubtendent_a or_o alterne_a subtendent_fw-la to_o the_o sine_fw-la compliment_n of_o the_o second_o cobase_a or_o alterne_a base_a which_o add_v to_o the_o first_o cobase_a sum_v a_o aggregat_fw-la of_o subjacent_a side_n which_o be_v the_o total_a base_a or_o side_n require_v the_o fundamental_a ground_n of_o the_o truth_n of_o these_o analogy_n be_v in_o the_o four_o and_o last_o amblygonosphericall_a axiom_n niubprodnesver_n as_o we_o be_v make_v to_o understand_v by_o the_o second_o determinater_n of_o its_o directory_n ennerra_n for_o by_o the_o direct_a proportion_n that_o among_o the_o term_n thereof_o be_v visible_a viz._n betwixt_o the_o sin_n compliment_n of_o the_o subtendent_o or_o side_n of_o the_o vertical_a angle_n and_o the_o segment_n of_o the_o base_n and_o inverse_o it_o be_v apparent_a that_o this_o mood_n do_v no_o less_o firm_o depend_v upon_o it_o then_o that_o of_o ennerable_a former_o explain_v now_o with_o reason_n do_v i_o conjecture_v that_o without_o disappoint_v the_o reader_n of_o his_o expectation_n i_o may_v here_o secure_o make_v a_o end_n of_o this_o trigonometricall_a treatise_n because_o of_o that_o trissotetrall_a table_n which_o comprehend_v all_o the_o mystery_n axiom_n principle_n analogy_n and_o precept_n of_o the_o science_n of_o triangular_a calculation_n i_o have_v omit_v no_o material_a point_n unexplain_v yet_o see_v for_o avoid_v of_o prolixity_n i_o be_v please_v in_o my_o comment_n upon_o the_o eight_o loxogonospherical_a disergetick_n bare_o to_o express_v in_o their_o final_a resolver_n the_o analogy_n of_o the_o term_n without_o put_v myself_o to_o the_o pain_n i_o take_v in_o my_o sphericorectangular_n how_o to_o order_v the_o logarithm_n and_o antilogarithm_n of_o the_o proportional_o for_o obtain_v of_o the_o main_a quaesitas_fw-la and_o that_o by_o have_v to_o the_o full_a explicate_v the_o variety_n of_o the_o proportion_n of_o the_o foresay_a mood_n and_o upon_o what_o several_a axiom_n they_o do_v depend_v
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