Selected quad for the lemma: power_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
power_n	line_n	number_n	square_a	3,745	5	14.0094	5	false

Text snippets containing the quad

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

flat_a and_o blunt_a angle_n occurse_v be_v a_o meeting_n together_o from_o occurro_fw-la occursum_fw-la oppobasall_n be_v say_v of_o those_o mood_n which_o have_v a_o catheteuretick_n concordance_n in_o their_o datas_fw-la of_o the_o same_o cathetopposite_a angle_n and_o the_o same_o base_n oppocathetall_n be_v say_v of_o those_o loxogonosphericals_n which_o have_v a_o datisterurgetick_a concordance_n in_o their_o datas_fw-la of_o the_o same_o angle_n at_o the_o base_a and_o the_o perpendicular_a oppoverticall_a be_v say_v of_o those_o mood_n which_o have_v a_o catheteuretick_n concordance_n in_o their_o datas_fw-la of_o the_o same_o cathetopposites_n and_o vertical_a angle_n orthogonosphericall_a be_v say_v of_o right_a angle_a sphericals_n of_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d rectus_fw-la 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d angulus_fw-la and_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d gobus_fw-la oxygonosphericall_a be_v say_v of_o acuteangled_n sphericals_n of_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d and_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d p._n parallelisme_n be_v a_o parallel_n equality_n of_o right_a line_n cut_v with_o a_o right_a line_n or_o of_o sphericals_n with_o a_o spherical_a from_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d equidistan_n of_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d and_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d parallelogram_n be_v a_o oblong_a long_o square_a rectangle_n or_o figure_n make_v of_o parallel_a line_n of_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d and_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d linea_fw-la partial_a be_v say_v of_o enodandas_fw-la depend_v on_o several_a axiom_n particularise_v specialise_v by_o some_o especial_a difference_n to_o contract_v the_o generality_n of_o a_o thing_n partition_n be_v say_v of_o the_o several_a operation_n of_o every_o loxogonosphericall_a mood_n and_o be_v divide_v in_o praenoscendall_n catheteuretick_n and_o hysterurgetick_n permutat_fw-la proportion_n or_o proportion_n by_o permutation_n or_o alternat_a proportion_n be_v when_o the_o antecedent_n be_v compare_v to_o the_o antecedent_n and_o the_o consequent_a to_o the_o consequent_a vide_fw-la perturbat_fw-la perpendicularity_n be_v the_o affection_n of_o the_o perpendicular_a or_o plumbline_n which_o come_v from_o perpendendo_fw-la id_fw-la est_fw-la explorando_fw-la altitudinem_fw-la perturbat_fw-la be_v the_o same_o as_o permutat_fw-la and_o call_v so_o because_o the_o order_n of_o the_o analogy_n be_v perturb_v planobliquangular_a be_v say_v of_o plain_a triangle_n wherein_o there_o be_v no_o right_a angle_n at_o all_o planorectangular_a be_v say_v of_o plain_a rightangled_n triangle_n planotriangular_a be_v say_v of_o plain_a triangle_n that_o be_v such_o as_o be_v not_o spherical_a pleuseotechnie_n the_o art_n of_o navigation_n of_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d navigatio_fw-la and_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d ars_fw-la plusminused_a be_v say_v of_o mood_n which_o admit_v of_o mensurator_n or_o who_o illatitious_a term_n be_v never_o the_o same_o but_o either_o more_o or_o less_o than_o the_o main_a quaehta_n poliechyrologie_n the_o art_n of_o fortify_v town_n and_o city_n of_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d vrbs_fw-la civet_n as_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d munio_fw-la firmo_fw-la and_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d ratio_fw-la possubservient_fw-fr be_v that_o which_o after_o another_o serve_v for_o the_o resolve_v of_o a_o question_n of_o post_n and_o subserviens_n of_o sub_fw-la and_o seruio_fw-la potentia_fw-la be_v that_o wherein_o the_o force_n and_o whole_a result_n of_o another_o thing_n lie_v power_n be_v the_o square_a quadrat_fw-la or_o product_v of_o a_o line_n extend_v upon_o itself_o or_o of_o a_o number_n in_o itself_o multiply_v power_v square_v quadrify_v precept_n document_n from_o praecipio_fw-la praeceptum_fw-la praeroscenda_fw-la be_v the_o term_n which_o must_v be_v know_v before_o we_o can_v attain_v to_o the_o knowledge_n of_o the_o main_a quaesitas_fw-la of_o prae_fw-la and_o nosco_fw-la praenoscendall_n be_v say_v of_o the_o concordance_n of_o those_o mood_n which_o agree_v in_o the_o same_o praenoscendas_fw-la praesection_n praesectionall_a be_v concern_v the_o digit_fw-la towards_o the_o left_a who_o cut_v off_o save_v the_o labour_n of_o subtract_v the_o double_a or_o single_a radius_fw-la praescinded_a problem_n be_v those_o speculative_a datoquaeres_fw-la which_o be_v not_o apply_v to_o any_o matter_n by_o way_n of_o practice_n praesubservient_fw-fr be_v say_v of_o those_o mood_n which_o in_o the_o first_o place_n we_o must_v make_v use_n of_o for_o the_o explanation_n of_o other_o of_o prae_fw-la and_o ●ub●ervio_fw-la prime_n be_v say_v of_o the_o further_a cathetopposite_a or_o angle_n at_o the_o base_a contain_v within_o the_o triangle_n to_o be_v resolve_v primifie_v the_o radius_fw-la be_v to_o put_v the_o radius_fw-la in_o the_o first_o place_n primumque_fw-la inter_fw-la terminos_fw-la collocare_fw-la proportionales_fw-la problem_n problemet_fw-la a_o question_n or_o datoquaere_fw-la from_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d unde_fw-la 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d propositum_fw-la objectaculum_fw-la product_v be_v the_o result_n factus_fw-la or_o operatum_fw-la of_o a_o multiplication_n from_o produco_fw-la productum_fw-la proportion_n proportionality_n be_v the_o same_o as_o analogy_n and_o analogisme_n the_o first_o be_v a_o likeness_n of_o term_n the_o other_o of_o proportion_n proposition_n a_o propose_a sentence_n whether_o theorem_fw-la or_o problem_n prosiliencie_n be_v a_o demission_n or_o fall_v of_o the_o perpendicular_a from_o prosilio_fw-la ex_fw-la pro_fw-la &_o salio_fw-la proturgetick_a be_v say_v of_o the_o first_o operation_n of_o every_o disergetick_a mood_n of_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d and_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d the_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d be_v attical_o contract_v into_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d q._n quadrant_n the_o four_o part_n of_o a_o circle_n quadrant_v the_o protract_n of_o a_o spherical_a side_n unto_o a_o quadrant_n quadrat_fw-la a_o square_a a_o forma_fw-la quadrae_fw-la the_o power_n or_o possibility_n of_o a_o line_n vide_fw-la power_n quadrobiquadraequation_n concern_v the_o square_a of_o the_o subtendent_fw-la side_n which_o be_v equal_a to_o the_o biquadrat_n or_o two_o square_n of_o the_o ambient_n quadrosubduction_n be_v concern_v the_o subtract_v of_o the_o square_n of_o one_o of_o the_o ambient_n from_o the_o square_n of_o the_o subtendent_fw-la quaesitas_fw-la the_o thing_n demand_v from_o quaero_fw-la quaesitum_fw-la quotient_a be_v the_o result_n of_o a_o division_n from_o quoties_fw-la how_o many_o time_n r._n radical_o meeting_n be_v say_v of_o those_o oblong_v or_o square_n who_o side_n do_v meet_v together_o radius_fw-la ray_n or_o beam_n be_v the_o semidiameter_n call_v so_o metaphorical_o from_o the_o speak_v of_o a_o wheel_n which_o be_v to_o the_o limb_n thereof_o as_o the_o semidiameter_n to_o the_o circumference_n of_o a_o circle_n reciprocal_a be_v say_v of_o proportionality_n or_o two_o row_n of_o proportional_n wherein_o the_o first_o of_o the_o first_o be_v to_o the_o first_o of_o the_o second_o as_o the_o last_o of_o the_o second_o be_v to_o the_o last_o of_o the_o first_o and_o contrary_o rectangular_a be_v say_v of_o those_o figure_n which_o have_v right_a angle_n refined_o be_v say_v when_o we_o go_v the_o short_a way_n to_o work_v by_o primify_v the_o radius_fw-la renvoy_fw-fr a_o remit_v from_o one_o place_n to_o another_o it_o come_v from_o the_o french_a word_n renvoyer_n representative_a be_v say_v of_o the_o letter_n which_o stand_v for_o whole_a word_n as_o e._n for_o side_n l._n for_o secant_fw-la u._n for_o subtendent_fw-la residuat_n be_v to_o leave_v a_o remainder_n nempe_fw-la id_fw-la quod_fw-la residet_fw-la &_o superest_fw-la resolver_n be_v that_o which_o lose_v and_o unti_v the_o knot_n of_o a_o difficulty_n of_o re_fw-mi and_o solvo_fw-la resolutory_a be_v say_v of_o the_o last_o partition_n of_o the_o loxogonosphericall_a operation_n result_v be_v the_o last_o effect_n of_o a_o work_n root_n be_v the_o side_n of_o a_o square_a cube_n or_o any_o cossick_n figure_n s._n scheme_n signify_v here_o the_o delineation_n of_o a_o geometrical_a figure_n with_o all_o part_n necessary_a for_o the_o illustrate_v of_o a_o demonstration_n from_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d habeo_fw-la sciography_n the_o art_n of_o shadow_v of_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d umbra_fw-la and_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d scribo_fw-la segment_n the_o portion_n of_o a_o thing_n cut_v off_o quasi_fw-la secamentum_fw-la quod_fw-la a_o re_fw-mi aliqua_fw-la secatur_fw-la sexagesimat_fw-la subsexagesimat_fw-la resubsexagesimat_fw-la and_o biresubsexagesimat_n be_v say_v of_o the_o division_n subdivision_n resubdivision_n and_o reer-resubdivision_a of_o degree_n into_o minute_n second_o three_o and_o fourth_n in_o 60._o of_o each_o other_o the_o devisor_n of_o the_o fore_a goer_n be_v successive_o the_o follow_a dividend_n and_o the_o quotient_a always_o sixty_o sharp_a be_v say_v of_o acute_a angle_n sindiforall_a be_v say_v of_o those_o mood_n the_o four_o term_n of_o who_o analogy_n be_v only_o illatitious_a to_o the_o main_a quaesitum_fw-la sindiforation_n be_v the_o affection_n of_o those_o foresay_a mood_n whereby_o the_o value_n of_o

certain_a all_o magnitude_n be_v figure_n at_o least_o in_o power_n and_o all_o figure_n either_o triangle_n or_o triangle_v that_o the_o arithmetical_a solution_n of_o any_o geometrical_a question_n depend_v on_o the_o doctrine_n of_o triangle_n 5._o and_o though_o the_o proportion_n betwixt_o the_o part_n of_o a_o triangle_n can_v be_v without_o some_o error_n because_o of_o crooked_a line_n to_o right_a line_n and_o of_o crooked_a line_n among_o themselves_o the_o reason_n be_v inscrutable_a no_o man_n be_v able_a to_o find_v out_o the_o exact_a proportion_n of_o the_o diameter_n to_o the_o circumference_n yet_o both_o in_o plain_a triangle_n where_o the_o measure_n of_o the_o angles_n be_v of_o a_o different_a species_n from_o the_o side_n and_o in_o sphericall_n wherein_o both_o the_o angle_n and_o side_n be_v of_o a_o circular_a nature_n crooked_a line_n be_v in_o some_o measure_n reduce_v to_o right_a line_n by_o the_o definition_n of_o quantity_n which_o right_a line_n viz._n sines_n tangent_n and_o secant_v apply_v to_o a_o circle_n have_v in_o respect_n of_o the_o radius_fw-la o●_n half-diameter_n 6._o and_o therefore_o though_o the_o circle_n quadrature_n be_v not_o find_v out_o it_o be_v in_o our_o power_n to_o make_v the_o diameter_n or_o the_o semi-diameter_n which_o be_v the_o radius_fw-la of_o as_o many_o part_n as_o we_o please_v and_o be_v sure_o so_o much_o the_o more_o that_o the_o radius_fw-la be_v take_v the_o error_n will_v be_v the_o lesser_a for_o albeit_o the_o sin_n tangent_n and_o secant_v be_v irrational_a thereto_o for_o the_o most_o part_n and_o their_o proportion_n inexplicable_a by_o any_o number_n whatsoever_o whither_o whole_a or_o break_a yet_o if_o they_o be_v right_o make_v they_o will_v be_v such_o as_o that_o in_o they_o all_fw-mi no_o number_n will_v be_v different_a from_o the_o truth_n by_o a_o integer_fw-la or_o unity_n of_o those_o part_n whereof_o the_o radius_fw-la be_v take_v which_o be_v so_o exact_o do_v by_o some_o especial_o by_o petiscus_n who_o assume_v a_o radius_fw-la of_o twenty_o six_o place_n that_o according_a to_o his_o supputation_n the_o diameter_n of_o the_o earth_n be_v know_v and_o the_o globe_n thereof_o suppose_v to_o be_v perfect_o round_a one_o shall_v not_o fail_v in_o the_o dimension_n of_o its_o whole_a circuit_n the_o nine_o hundred_o thousand_o scantling_n of_o the_o million_o part_n of_o a_o inch_n and_o yet_o not_o be_v able_a for_o all_o that_o to_o measure_v it_o without_o amiss_o for_o so_o indivisible_a the_o truth_n of_o a_o thing_n be_v that_o come_v you_o never_o so_o near_o it_o unless_o you_o hit_v upon_o it_o just_a to_o a_o point_n there_o be_v a_o error_n still_o definition_n a_o cord_n or_o subtense_n be_v a_o right_a line_n draw_v from_o the_o one_o extremity_n to_o the_o other_o of_o a_o archippus_n 2._o a_o right_a sine_fw-la be_v the_o half_a cord_n of_o the_o double_a arch_n propose_v and_o from_o one_o extremity_n of_o the_o arch_n fall_v perpendicular_o on_o the_o radius_fw-la pass_v by_o the_o other_o end_n thereof_o 3._o a_o tangent_fw-la be_v a_o right_a line_n draw_v from_o the_o secant_fw-la by_o one_o end_n of_o the_o arch_n perpendicular_o on_o the_o extremity_n of_o the_o diameter_n pass_v by_o the_o other_o end_n of_o the_o say_v arch._n 4._o a_o secant_fw-la be_v the_o prolong_v radius_fw-la which_o pass_v by_o the_o upper_a extremity_n of_o the_o arch_n till_o it_o meet_v with_o the_o sine_fw-la tangent_fw-la of_o the_o say_v arch._n 5._o compliment_n be_v the_o difference_n betwixt_o the_o lesser_a arch_n and_o a_o quadrant_n or_o betwixt_o a_o right_a angle_n and_o a_o acute_a 6._o the_o compliment_n to_o a_o semicircle_n be_v the_o difference_n betwixt_o the_o half-circumference_n and_o any_o arch_n lesser_a or_o betwixt_o two_o right_a angle_n and_o a_o oblique_a angle_n whither_o blunt_a or_o sharp_a 7._o the_o verse_v sine_fw-la be_v the_o remainder_n of_o the_o radius_fw-la the_o sine_fw-la compliment_n be_v subtract_v from_o it_o and_o though_o great_a use_n may_v be_v make_v of_o the_o verse_v sin_n for_o find_v out_o of_o the_o angle_n by_o the_o side_n and_o side_n by_o the_o angles_n yet_o in_o logarithmicall_a calculation_n they_o be_v altogether_o useless_a and_o therefore_o in_o my_o trissotetras_n there_o be_v no_o mention_n make_v of_o they_o 8._o in_o amblygonosphericall●_n which_o admit_v both_o of_o a_o extrinsecall_n and_o intrinsical_v demission_n of_o the_o perpendicular_a nineteen_o several_a part_n be_v to_o be_v consider_v viz._n the_o perpendicular_a the_o subtendentall_a the_o subtendentine_n two_o cosubtendents_n the_o basall_n the_o basidion_n the_o chief_a segment_n of_o the_o base_a two_o cobases_n the_o double_a vertical_a the_o vertical_a the_o verticaline_a two_o coverticall_n the_o next_o cathetopposite_a the_o prime_a cathetopposite_a and_o the_o two_o cocathetopposites_n fourteen_o whereof_o to_o wit_n the_o subtendentall_a the_o subtendentine_n the_o cosubtendents_n the_o basall_n the_o basidion_n the_o cobases_n the_o vertical_a the_o verticaline_a the_o coverticall_n and_o cocathetopposites_n be_v call_v the_o first_o either_o subtendent_fw-la base_a topangle_n or_o cocathetopposite_a whither_o in_o the_o great_a triangle_n or_o the_o little_a or_o in_o the_o correctangle_n if_o they_o be_v ingredient_n of_o that_o rectangular_a whereof_o most_o part_n be_v know_v which_o part_n be_v always_o a_o subtendent_fw-la and_o a_o cathetopposite_a but_o if_o they_o be_v in_o the_o other_o triangle_n they_o be_v call_v the_o second_o subtendent_o base_n and_o so_o forth_o 9_o the_o external_a double_a vertical_a be_v include_v by_o the_o perpendicular_a and_o subtendentall_a and_o divide_v by_o the_o subtendentine_a the_o internal_a be_v include_v by_o cosubtendents_n and_o divide_v by_o the_o perpendicular_a apodicticks_n the_o angles_n make_v by_o a_o right_a line_n fall_v on_o another_o right_a line_n be_v equal_a to_o two_o right_a angle_n because_o every_o angle_n be_v measure_v by_o a_o arch_n or_o part_n of_o a_o circumference_n and_o a_o right_a angle_n by_o ninety_o degree_n if_o upon_o the_o middle_n of_o the_o ground_n line_n as_o centre_n be_v describe_v a_o semicircle_n it_o will_v be_v the_o measure_n of_o the_o angle_n comprehend_v betwixt_o the_o fall_n and_o sustain_a line_n 2._o hence_o it_o be_v that_o the_o four_o opposite_a angle_n make_v by_o one_o line_n cross_v another_o be_v always_o each_o to_o its_o own_o opposite_a equal_a for_o if_o upon_o the_o point_n of_o intersection_n as_o centre_n be_v describe_v a_o circle_n every_o two_o of_o those_o angle_n will_v fill_v up_o the_o semicircle_n therefore_o the_o first_o and_o second_o will_v be_v equal_a to_o the_o second_o and_o three_o and_o consequent_o the_o second_o which_o be_v the_o common_a angle_n to_o both_o these_o couple_n be_v remove_v the_o first_o will_v remain_v equal_a to_o the_o three_o and_o by_o the_o same_o reason_n the_o second_o to_o the_o four_o which_o be_v to_o be_v demonstrate_v 3._o if_o a_o right_a line_n fall_v upon_o two_o other_o right_a line_n make_v the_o alternat_a angle_n equal_a these_o line_n must_v needs_o be_v parallel_n for_o if_o they_o do_v meet_v the_o alternat_a angle_n will_v not_o be_v equal_a because_o in_o all_o plain_a triangle_n the_o outward_a angle_n be_v great_a than_o any_o of_o the_o remote_a inward_a angle_n which_o be_v prove_v by_o the_o first_o 4._o if_o one_o of_o the_o side_n of_o a_o triangle_n be_v produce_v the_o outward_a angle_n be_v equal_a to_o both_o the_o inner_a and_o opposite_a angle_n together_o because_o according_a to_o the_o accline_a or_o decline_v of_o the_o conterminall_a side_n be_v leave_v a_o angulary_a space_n for_o the_o receive_n of_o a_o parallel_n to_o the_o opposite_a side_n in_o the_o point_n of_o who_o occourse_n at_o the_o base_a the_o exterior_a angle_n be_v divide_v into_o two_o which_o for_o their_o like_a and_o alternat_a situation_n with_o the_o two_o interior_n angle_n be_v equal_a each_o to_o its_o own_o conform_a to_o the_o nature_n of_o angles_n make_v by_o a_o right_a line_n cross_v divers_a parallel_n 5._o from_o hence_o we_o gather_v that_o the_o three_o angle_n of_o a_o plain_a triangle_n be_v equal_a to_o two_o right_n for_o the_o two_o inward_a be_v equal_a to_o the_o external_a one_o and_o there_o remain_v of_o the_o three_o but_o one_o which_o be_v prove_v in_o the_o first_o apodictick_n to_o be_v the_o external_a angle_n compliment_n to_o two_o right_n it_o must_v needs_o fall_v forth_o what_o be_v equal_a to_o a_o three_o be_v equal_a among_o themselves_o that_o the_o three_o angle_n of_o a_o plain_a triangle_n be_v equal_a to_o two_o right_a angle_n the_o which_o we_o undertake_v to_o prove_v 6._o by_o the_o same_o reason_n the_o two_o acute_a of_o a_o rectangle_v plain_a triangle_n be_v equal_a to_o one_o right_a angle_n and_o any_o one_o of_o they_o the_o other_o compliment_n thereto_o 7._o

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