a_o circle_n and_o therein_o inscribe_v the_o chord_n b_o and_o another_o on_o the_o middle_n hereof_o at_o right-angle_n which_o will_v therefore_o bisect_v that_o and_o be_v a_o diameter_n and_o from_o both_o end_n of_o this_o to_o either_o end_n of_o b_o draw_v the_o line_n a_o e_o as_o before_z and_o this_o construction_n be_v better_a than_o the_o former_a because_o of_o the_o uncertainty_n of_o the_o precise_a point_n of_o contact_n or_o section_n in_o case_n the_o section_n be_v somewhat_o oblique_a xxv_o now_o if_o it_o be_v desire_v in_o like_a manner_n to_o give_v a_o like_a construction_n in_o case_n of_o such_o biquadratick_a equation_n or_o quadraticks_n of_o a_o plain-root_n where_o the_o high_a power_n be_v affirmative_a though_o that_o be_v here_o a_o digression_n as_o in_o all_o the_o rest_n that_o follow_v to_z §_o 35._o it_o be_v thus_o suppose_v the_o equation_n aqq_n −_o vqaq_n =_o vqeq_fw-fr =_o pqq_fw-fr +_o vqpq_fw-fr who_o affirmative_a root_n be_v aq_n and_o pq_n and_o therefore_o vq_n vqeq_n and_o consequent_o eq_n be_v know_v quantity_n therefore_o by_o transposition_n aqq_n −_o vqaq_n +_o vqpq_fw-fr and_o divide_v by_o and_o therefore_o aq_n −_o vq_fw-fr =_o pq_fw-fr and_o pq_n +_o vq_fw-fr =_o aq_n and_o by_o multiplication_n aqq_n −_o vqaq_n =_o aqpq_fw-fr =_o pqq_fw-fr +_o vqpq_fw-fr =_o vqeq_fw-fr xxvi_o the_o equation_n therefore_o propose_v divide_v all_o by_o vq_n come_v to_o this_o that_o be_v who_o root_n be_v and_o namely_o and_o and_o these_o multiply_v into_o v._o a_o know_a quantity_n make_v aq_n and_o pq_n namely_o and_o and_o consequent_o a_o be_v a_o mean_a proportional_a between_o five_o and_o and_o p_o a_o mean_a proportional_a between_o five_o and_o therefore_o xxvii_o and_o equation_n be_v propose_v in_o one_o of_o these_o form_n aqq_fw-fr −_o vqaq_n v_o =_o vqeq_fw-fr =_o pqq_fw-fr +_o vqpq_fw-fr the_o absolute_a term_n vqeq_n be_v divide_v by_o the_o co-efficient_a of_o the_o middle_a term_n vq_n the_o quantity_n result_v be_v eq_n who_o square-root_n e_z set_v perpendicular_a on_o the_o end_n of_o a_o straight_a line_n equal_a to_o v_z the_o square-root_n of_o the_o co-efficient_a which_o we_o may_v suppose_v the_o diameter_n of_o a_o circle_n to_o which_o that_o perpendicular_a be_v a_o tangent_fw-la on_o the_o same_o centre_n with_o this_o circle_n and_o on_o the_o same_o diameter_n continue_v by_o the_o top_n of_o that_o perpendicular_a draw_v a_o second_o circle_n the_o diameter_n of_o this_o second_o circle_n be_v by_o that_o perpendicular_a e_o cut_v into_o two_o segment_n which_o be_v the_o root_n of_o these_o equation_n that_o be_v and_o xxviii_o or_o without_o draw_v that_o second_o circle_n from_o the_o top_n of_o that_o perpendicular_a in_o a_o straight_a line_n through_o the_o centre_n of_o the_o first_o which_o will_v cut_v the_o circumference_n in_o two_o point_n to_o the_o first_o section_n be_v to_o the_o second_o xxix_o these_o two_o root_n multiply_v one_o into_o the_o other_o become_v equal_a to_o the_o absolute_a quantity_n and_o multiply_v into_o v_n become_v aq_n pq_n or_o thus_o p_o be_v a_o mean_a proportional_a between_o five_o and_o u_o and_o a_o between_z v_z and_o v+u_n or_o thus_o p_o be_v a_o mean_a proportional_a between_o five_o and_o and_o because_o by_o §_o 25._o aq_n =_o pq_fw-fr +_o vq_fw-fr a_o be_v the_o hypothenuse_n in_o a_o rightangled_n triangle_n to_o the_o leg_n p_o v._o and_o this_o be_v no_o contemptible_a method_n for_o the_o resolve_v quadratick_a equation_n of_o a_o plain-root_n wherein_o the_o high_a term_n be_v affirmative_a the_o whole_a geometric_a construction_n be_v clear_a enough_o from_o the_o figure_n adjoin_v where_o yet_o the_o circle_n for_o the_o most_o part_n serve_v rather_o for_o the_o demonstration_n than_o the_o construction_n xxx_o again_o by_o the_o same_o §_o 25._o and_o therefore_o a_o and_o e_o be_v also_o the_o leg_n of_o a_o rightangled_n triangle_n who_o hypothenuse_n be_v five_o +_o u_o which_o by_o p_o a_o perpendicular_a on_o it_o from_o the_o right-angle_n be_v cut_v into_o those_o two_o segment_n xxxi_o from_o the_o same_o construction_n therefore_o we_o have_v also_o the_o geometrical_a construction_n of_o this_o problem_n in_o a_o rightangled_n triangle_n have_v one_o of_o the_o leg_n e_o with_o the_o far_a segment_n of_o the_o hypothenuse_n v_n to_o find_v the_o other_o segment_n and_o so_o the_o whole_a v_o v_o u_o and_o the_o perpendicular_a p_o and_o the_o other_o leg_n a_o and_o the_o who_o triangle_n xxxii_o we_o have_v thence_o also_o this_o analogy_n and_o or_o thus_o and_o xxxiii_o if_o therefore_o we_o make_v five_o the_o radius_fw-la of_o a_o circle_n then_o be_v a_o the_o vi_o secant_fw-la p_o the_o tangent_fw-la e_z a_o parallel_n to_o the_o right-sine_a in_o contrary_a position_n from_o the_o end_n of_o the_o secant_fw-la to_o the_o diameter_n produce_v if_o we_o make_v a_o the_o radius_fw-la then_o be_v p_o the_o right-sine_a and_o e_z the_o tangent_fw-la of_o the_o same_o arch_n and_o v._o the_o sine_fw-la of_o the_o compliment_n or_o difference_n between_o the_o radius_fw-la and_o verse_v sine_fw-la from_o hence_o therefore_o xxxiv_o the_o tangent_fw-la e_z and_o sine_fw-la of_o the_o compliment_n v_n be_v give_v we_o have_v the_o right-sine_a p_o and_o the_o radius_fw-la a._n but_o §_o 25_o and_o all_o hitherto_o be_v a_o digression_n xxxv_o if_o in_o a_o semicircle_n on_o the_o diameter_n 2_o r_o we_o inscribe_v b_o the_o subtense_n vii_o of_o a_o double_a arch_n a_o perpendicular_a on_o the_o middle_a point_n hereof_o will_v cut_v the_o arch_n of_o that_o semicircle_n into_o two_o segment_n who_o subtense_n be_v a_o e_o either_o of_o which_o be_v a_o single_a arch_n to_o the_o double_a whereof_o b_o be_v a_o subtense_n this_o as_o to_o e_o be_v evident_a from_o 4_o è_fw-la 1_o and_o 28_o è_fw-la 3_o and_o as_o to_o a_o from_o §_o 15_o of_o this_o xxxvi_o but_o also_o by_o the_o same_o reason_n the_o arch_a β_n the_o difference_n of_o the_o arch_n a_o e_o and_o b_o the_o double_a of_o either_o will_v if_o double_v have_v the_o same_o subtense_n of_o their_o double_a arch._n that_o be_v the_o double_a of_o the_o double_a of_o either_o and_o the_o double_a of_o their_o difference_n will_v have_v the_o same_o subtense_n xxxvii_o if_o a_o arch_a to_o be_v double_v be_v just_o a_o three_o part_n of_o the_o circumference_n viii_o the_o subtense_n of_o the_o double_a be_v equal_a to_o that_o of_o the_o single_a archippus_n for_o the_o same_o subtense_n which_o on_o one_o side_n subtend_v two_o trient_n do_v on_o the_o other_o side_n subtend_v but_o one_o that_o be_v by_z §_o 7_o and_o therefore_o by_o transposition_n and_o 3rq_n =_o aq._n that_o be_v xxxviii_o the_o square_a of_o the_o subtense_n to_o a_o trient_a of_o the_o circumference_n or_o of_o the_o side_n of_o a_o equilater_n triangle_n inscribe_v be_v equal_a to_o three_o square_n of_o the_o radius_fw-la thirty-nine_o again_o the_o same_o be_v the_o subtense_n of_o the_o double_a trient_a and_o of_o the_o double_a sextant_a for_o a_o trient_a and_o a_o sextant_a complete_a the_o half_n ⅓_n +_o ⅙_n =_o ½_o the_o square_a of_o the_o subtense_n of_o a_o sextant_a eq_n be_v the_o difference_n of_o the_o square_n of_o that_o of_o the_o trient_a and_o the_o diameter_n or_o that_o of_o the_o semicircumference_n that_o be_v 4rq_n −_o aq_n =_o eq_fw-fr that_o be_v by_o §_o prece_v 4rq_n −_o 3rq_n =_o rq_n =_o eq_fw-fr and_o e_z =_o r._n that_o be_v xl._o the_o subtense_n of_o a_o sextant_a or_o side_n of_o the_o inscribe_v equilater_n hexagon_n be_v equal_a to_o the_o radius_fw-la chap._n ii_o of_o the_o triplication_n and_o trisection_n of_o a_o arch_n or_o angle_n i._o if_o in_o a_o circle_n be_v inscribe_v a_o quadrilater_n who_o three_o side_n be_v a_o a_o a_o ix_o subtense_n of_o a_o single_a arch_n and_o the_o four_o c_z the_o subtense_n of_o the_o triple_a arch_n the_o diagonal_n be_v b_o b_o the_o subtense_n of_o the_o double_a as_o be_v evident_a but_o it_o be_v evident_a also_o that_o in_o this_o case_n a_o be_v less_o than_o a_o trient_a of_o the_o whole_a circumference_n ii_o and_o therefore_o the_o rect-angle_n of_o the_o diagonal_n be_v equal_a to_o the_o two_o rectangle_n of_o the_o opposite_a side_n bq_n =_o aq+ac_n and_o therefore_o bq_a −_o aq_n =_o ac_fw-la and_o that_o be_v iii_o the_o square_a of_o the_o subtense_n of_o the_o double_a arch_n be_v equal_a to_o the_o square_n of_o the_o subtense_n of_o the_o single_a arch_n less_o than_o a_o trient_a of_o the_o circumference_n and_o the_o rectangle_n of_o the_o subtense_n of_o the_o sngle_a and_o treble_a arch._n and_o therefore_o iu._n the_o square_a of_o the_o subtense_n of_o the_o double_a arch_n want_v the_o square_a of_o the_o subtense_n of_o the_o single_a arch_n be_v
a_o treatise_n of_o angular_a section_n by_o john_n wallis_n d._n d._n professor_n of_o geometry_n in_o the_o university_n of_o oxford_n and_o a_o member_n of_o the_o royal_a society_n london_n london_n print_v by_o john_n playford_n for_o richard_n davis_n bookseller_n in_o the_o university_n of_o oxford_n 1684._o a_o treatise_n of_o angular_a section_n chap._n 1._o of_o the_o duplication_n and_o bisection_n of_o a_o arch_n or_o angle_n i._o let_v the_o chord_n or_o subtense_n of_o a_o arch_n propose_v be_v call_v a_o or_o e_z of_o the_o double_a b_o of_o the_o treble_a c_o of_o the_o quadruple_a d_o of_o the_o quintruple_n f_o etc._n etc._n the_o radius_fw-la r_o the_o diameter_n 2r_n but_o sometime_o we_o shall_v give_v the_o name_n of_o the_o subtense_n a_o e_o etc._n etc._n to_o the_o arch_n who_o subtense_n it_o be_v yet_o with_o that_o care_n as_o not_o to_o be_v liable_a to_o a_o mistake_n ii_o where_o the_o subtense_n of_o a_o arch_n be_v a_o let_v the_o verse_v sine_fw-la be_v v_z where_o i._n that_o be_v e_o let_v this_o be_v u._n which_o draw_v into_o or_o multiply_v by_o the_o remainder_n of_o the_o diameter_n 2_o r_z −_o v_o makes_z 2_o r_z wq_n the_o square_a of_o the_o right-sine_a this_o sine_fw-la be_v a_o mean-proportional_a between_o the_o segment_n of_o the_o diameter_n on_o which_o it_o stand_v erect_v by_o 13_o â—_o 6._o that_o be_v q_o ½_o b_o the_o square_a of_o the_o right-sine_a or_o half_a the_o subtense_n of_o the_o double_a arch_n that_o be_v 2_o r_z v_o −_o vq_fw-fr =_o q_o ½_o b_o =_o ¼_n bq._n iii_o if_o to_o this_o we_o add_v vq_n the_o square_n of_o the_o versed-sine_a it_o make_v 2_o rv_n =_o ¼_n bq_n +_o vq_fw-fr =_o aq._n and_o by_o the_o same_o reason_n 2_o r_o u_o =_o eq._n that_o be_v iu._n the_o subtense_n of_o a_o arch_n be_v a_o mean_a proportional_a between_o the_o diameter_n and_o the_o versed-sine_a v._o again_o because_o 2_o r_o v_o v_o aq_n therefore_o divide_v both_o by_o 2_o r_o and_o the_o square_a thereof_o which_o subtract_v from_o aq_n leave_v the_o square_a of_o the_o right-sine_a and_o in_o like_a manner_n and_o and_o that_o be_v vi_o if_o from_o the_o square_n of_o the_o subtense_n we_o take_v its_o biquadrate_n divide_v by_o the_o square_n of_o the_o diameter_n the_o remainder_n be_v equal_a to_o the_o square_n of_o the_o right-sine_a and_o the_o square-root_n of_o that_o remainder_n to_o the_o sine_fw-la itself_o and_o the_o double_a of_o this_o to_o the_o subtense_n of_o the_o double_a arch._n vii_o according_o because_o therefore_o its_o quadruple_a and_o and_o in_o like_a manner_n that_o be_v viii_o if_o from_o fourtime_n the_o square_a of_o a_o subtense_n be_v take_v its_o biquadrate_n divide_v by_o the_o square_n of_o the_o radius_fw-la the_o remainder_n be_v the_o square_a of_o the_o subtense_n of_o the_o double_a arch_n and_o the_o quadratick_a root_n of_o that_o remainder_n be_v the_o subtense_n itself_o ix_o but_o that_o be_v x._o the_o rect-angle_n of_o the_o subtense_n of_o a_o arch_n and_o of_o its_o remainder_n to_o a_o semicircle_n divide_v by_o the_o radius_fw-la be_v equal_a to_o the_o subtense_n of_o the_o double_a arch._n xi_o because_o therefore_o ae_n =_o rb_fw-ge =_o 2r_o ×_o ½b_n and_o r._n a_o b._n b_o and_o therefore_o because_o a_o e_o contain_v a_o right-angle_n as_o be_v a_o angle_n in_o a_o semicircle_n xii_o in_o a_o rightangled_n triangle_n the_o rect-angle_n of_o the_o two_o leg_n contain_v the_o right-angle_n be_v equal_a to_o that_o of_o the_o hypothenuse_n and_o the_o perpendicular_a from_o the_o right-angle_n thereupon_o and_o xiii_o as_o the_o radius_fw-la to_o the_o subtense_n of_o a_o arch_n so_o the_o subtense_n of_o its_o remainder_n to_o a_o semicircle_n be_v to_o that_o of_o the_o double_a arch._n fourteen_o because_o b_o the_o subtense_n of_o a_o double_a arch_n do_v indifferent_o subtend_v the_o two_o segment_n which_o complete_a the_o whole_a circumference_n and_o consequent_o the_o half_a of_o either_o may_v be_v the_o single_a arch_n of_o this_o double_a it_o be_v therefore_o necessary_a that_o this_o equation_n have_v two_o affirmative_a root_n the_o great_a of_o which_o we_o will_v call_v a_o and_o the_o lesser_a e_o and_o therefore_o that_o be_v xv._n any_o arch_n and_o its_o remainder_n to_o a_o semicircumference_n as_o also_o its_o excess_n above_o a_o semicircumference_n and_o either_o of_o they_o increase_v by_o one_o or_o more_o semicircumference_n will_v have_v the_o same_o subtense_n of_o the_o double_a archippus_n for_o in_o all_o these_o case_n the_o subtense_n of_o the_o single_a arch_n will_v be_v either_o a_o or_o e._n xvi_o because_o and_o therefore_o and_o 4aq_n rq_n −_o aqq_fw-fr =_o bq_fw-fr rq_n =_o 4eq_fw-fr rq_n −_o eqq_n therefore_o by_o transposition_n 4aq_n rq_n −_o 4eq_n rq_n =_o aqq_fw-fr −_o eqq_n and_o divide_v both_o by_o that_o be_v xvii_o the_o square_a of_o the_o diameter_n be_v equal_a to_o the_o difference_n of_o the_o biquadrate_n of_o the_o subtense_n of_o two_o arch_n which_o together_o complete_a a_o semicircumference_n divide_v by_o the_o difference_n of_o their_o square_n and_o this_o also_o equal_a to_o the_o sum_n of_o the_o square_n of_o those_o subtense_n that_o be_v because_o a_o e_o contain_v a_o right-angle_n xviii_o in_o a_o rightangled_n triangle_n the_o square_a of_o the_o hypothenuse_n 4rq_n be_v equal_a to_o the_o square_n of_o the_o side_n contain_v the_o right-angle_n aq+eq_fw-la xix_o or_o thus_o because_o b_o be_v the_o common_a subtense_n to_o two_o segment_n which_o ii_o together_o complete_a the_o whole_a circumference_n and_o therefore_o the_o half_a of_o both_o complete_a the_o semicircumference_n if_o therefore_o in_o a_o circle_n according_a to_o ptolemy_n lemma_n a_o trapezium_fw-la be_v inscribe_v who_o opposite_a side_n be_v a_o a_o and_o e_z e_o the_o diagonal_n will_v be_v diameter_n that_o be_v 2r_n and_o consequent_o 4rq_n =_o aq_n +_o eq_fw-fr as_o before_o xx._n hence_o therefore_o the_o radius_fw-la r_o with_o the_o subtense_n of_o a_o arch_n a_o or_o e_z be_v give_v we_o have_v thence_o the_o subtense_n of_o the_o double_a arch_n b_o which_o be_v the_o duplication_n of_o a_o arch_n or_o angle_n for_o r_o a_o be_v give_v we_o have_v or_o r_o e_o be_v give_v we_o have_v a_o =_o 4rq_n −_o eq_n and_o have_v r_o a_o e_o we_o have_v by_o §_o 9_o xxi_o the_o radius_fw-la r_o with_o b_o the_o subtense_n of_o the_o double_a arch_n be_v give_v we_o have_v thence_o the_o subtense_n of_o the_o single_a arch_n a_o or_o e._n which_o be_v the_o bisection_n of_o a_o arch_n or_o angle_n for_o by_o §_o 14_o and_o therefore_o 4rq_n aq_n −_o aqq_fw-fr =_o rq_n bq_fw-la =_o 4rq_n eq_fw-fr −_o eqq._n and_o the_o root_n of_o this_o equation_n or_o eq._n and_o the_o quadratick_a root_n of_o this_o be_v a_o or_o e._n xxii_o hence_o also_o we_o have_v a_o easy_a method_n for_o a_o geometrical_a construction_n for_o iii_o the_o resolution_n of_o such_o biquadratick_a equation_n or_o quadratick_a equation_n of_o a_o plain_a root_n wherein_o the_o high_a power_n be_v negative_a understand_v it_o in_o mr._n oughtred_n language_n who_o put_v the_o absolute_a quantity_n affirmative_a and_o by_o itself_o and_o the_o rest_n of_o the_o equation_n all_o on_o the_o other_o side_n suppose_v rq_n bq_fw-fr =_o 4rq_n aq_n −_o aqq_n or_o put_v p_o =_o ½_o b_o 4rq_n pq_n =_o 4rq_n aq_n −_o aqq._n for_n divide_v the_o absolute_a term_n rq_n bq_n or_o 4rq_n pq_n by_o the_o co-efficient_a of_o the_o middle_a term_n 4rq_n the_o result_n be_v ¼bq_n or_o pq_n and_o its_o root_n ½_o b_o or_o p._n which_o be_v set_v perpendicular_a on_o a_o diameter_n equal_a to_o 2_o r_o the_o square_a root_n of_o that_o co-efficient_a a_o straight_a line_n from_o the_o top_n of_o it_o parallel_n to_o that_o diameter_n will_v if_o the_o equation_n be_v not_o impossible_a cut_v the_o circle_n or_o at_o least_o touch_v it_o from_o which_o point_n of_o section_n or_o contact_n two_o streight-line_n draw_v to_o the_o end_n of_o the_o diameter_n be_v a_o and_o e_z the_o two_o root_n of_o that_o ambiguous_a biquadratick_a equation_n or_o if_o we_o call_v it_o a_o quadratick_a of_o a_o plain-root_n the_o root_n of_o the_o plain-root_n of_o such_o quadratick_a equation_n xxiii_o and_o this_o construction_n be_v the_o same_o with_o the_o resolution_n of_o this_o problem_n in_o a_o rightangled_n triangle_n the_o hypothenuse_n be_v give_v and_o a_o perpendicular_a from_o the_o right-angle_n thereupon_o to_o find_v the_o other_o side_n and_o if_o need_v be_v the_o angles_n the_o segment_n of_o the_o hypothenuse_n and_o the_o area_n of_o the_o triangle_n ½_o r_o b_o or_o p_o r._n xxiv_o or_o thus_o have_v r_o and_o b_o as_o at_o §_o 22._o with_o the_o radius_fw-la r_o describe_v
less_o than_o a_o trient_a be_v equal_a to_o the_o rect-angle_n of_o the_o subtense_n of_o the_o single_a and_o treble_a arch._n and_o consequent_o v._o if_o the_o square_a of_o the_o subtense_n of_o the_o double_a arch_n want_v the_o square_a of_o the_o subtense_n of_o the_o single_a arch_n less_o than_o a_o trient_a be_v divide_v by_o the_o subtense_n of_o the_o single_a arch_n the_o result_n be_v the_o subtense_n of_o the_o triple_a arch._n vi_o because_o that_o by_o §_o 2._o and_o that_o b+a_n into_o b_o −_o at_fw-fr be_v equal_a to_o bq_n −_o aq_n as_o will_v appear_v by_o multiplication_n therefore_o that_o be_v vii_o as_o the_o subtense_n of_o a_o single_a arch_n less_o than_o a_o trient_a to_o the_o sum_n of_o the_o subtense_n of_o the_o single_a and_o double_a arch_n so_o be_v the_o excess_n of_o that_o of_o the_o double_a above_o that_o of_o the_o single_a to_o the_o subtense_n of_o the_o triple_a viii_o again_o because_o by_o §_o 7_o of_o the_o precedent_a chapter_n therefore_o and_o therefore_o that_o be_v ix_o the_o triple_a of_o the_o subtense_n of_o a_o arch_n less_o than_o a_o trient_a want_v the_o cube_n thereof_o divide_v by_o the_o square_n of_o the_o radius_fw-la be_v equal_a to_o the_o subtense_n of_o the_o triple_a arch._n x._o but_o because_o the_o same_o subtense_n c_o subtend_v also_o to_o another_o segment_n of_o the_o same_o circle_n the_o subtense_n of_o who_o trient_a we_o shall_v call_v e_z therefore_o xi_o and_o because_o the_o three_o arch_n a_o a_o a_o and_o the_o three_o arch_n e_o e_o e_o complete_a the_o whole_a circumference_n as_o be_v evident_a therefore_o once_o a_o and_o once_z e_z complete_a a_o trient_a or_o three_o part_n thereof_o therefore_o xii_o a_o arch_n less_o than_o the_o trient_a of_o a_o circumference_n and_o the_o residue_n of_o that_o ix_o trient_a a_o and_o e_z have_v the_o same_o subtense_n of_o their_o triple_a arch._n xiii_o again_o because_o as_o be_v show_v already_o and_o therefore_o 3rqa_o −_o ac_fw-la =_o 3rqe_o −_o aec_fw-la and_o 3rqa_n −_o 3rqe_fw-fr =_o ac_fw-la −_o aec_fw-la therefore_o divide_v both_o by_o a_o −_o e_o as_o will_v appear_v upon_o divide_v ac_fw-la −_o aec_fw-la by_o a_o −_o e_o or_o multiply_v a_o −_o e_o into_o aq+ae+eq_n fourteen_o but_o by_o §_o 37_o 38_o chap._n prece_v 3rq_n be_v the_o square_a of_o the_o subtense_n of_o a_o trient_a that_o be_v by_o §_o 11_o of_o this_o of_o the_o sum_n of_o the_o arch_n a_o and_o e._n therefore_o xv._n the_o square_a of_o the_o subtense_n of_o the_o trient_a of_o the_o circumference_n of_o a_o circle_n or_o three_o square_n of_o the_o radius_fw-la be_v equal_a to_o the_o square_n of_o the_o subtense_n of_o any_o two_o arch_n complete_n that_o trient_a and_o the_o rect-angle_n of_o they_o that_o be_v put_v t_n for_o the_o subtense_n of_o a_o trient_a tq_n =_o 3rq_n =_o aq+ae+aq_n xvi_o but_o the_o angle_n which_o ae_n contain_v as_o be_v a_o angle_n in_o the_o trient_a of_o a_o circle_n or_o insist_v on_o two_o trient_n be_v a_o angle_n of_o 120_o degree_n and_o therefore_o by_o §_o 15._o xvii_o in_o a_o rightlined_n triangle_n one_o of_o who_o angle_n be_v 120_o degree_n the_o square_a of_o the_o subtense_n to_o that_o angle_n be_v equal_a to_o the_o two_o square_n of_o the_o side_n contain_v it_o and_o a_o rect-angle_n of_o those_o side_n for_o if_o such_o triangle_n be_v inscribe_v in_o a_o circle_n the_o base_a of_o that_o triangle_n will_v be_v the_o subtendent_fw-la of_o a_o trient_a in_o such_o circle_n or_o rq._n xviii_o if_o a_o quadrilater_n be_v inscribe_v in_o a_o circle_n three_o of_o who_o side_n be_v x._o a_o e_o a_o or_o e_z a_o e_o and_o the_o four_o z_o each_o of_o the_o diagonal_n by_o §_o 11._o be_v t_n the_o subtense_n of_o a_o trient_a and_o therefore_o by_o §_o 13_o 14_o 15_o ze+aq_n =_o za+eq_fw-fr =_o tq_fw-fr =_o 3rq_n =_o aq+ae+eq_fw-fr and_o consequent_o see_fw-mi =_o ae+eq_fw-fr and_o za_n =_o aq+ae_fw-la and_o therefore_o z_o =_o a+e_o and_o therefore_o xix_o if_o to_o the_o aggregate_v of_o two_o arch_n a_o e_o complete_n a_o trient_a be_v add_v a_o three_o equal_a to_o either_o of_o they_o z_o the_o subtense_n of_o the_o aggregate_v of_o all_o the_o three_o be_v equal_a to_o the_o sum_n of_o the_o subtense_n of_o those_o two_o that_o be_v z_o =_o a+e_o xx._n but_o the_o same_o chord_n z_o do_v subtend_v on_o the_o one_o side_n to_o a_o trient_a increase_v by_o the_o arch_a a_o and_o on_o the_o other_o side_n to_o a_o trient_a increase_v by_o the_o arch_n e_o as_o be_v evident_a that_o be_v to_o a_o arch_n which_o do_v as_o much_o exceed_v a_o trient_a or_o want_v of_o two_o trient_n as_o the_o arch_a a_o or_o e_o want_n of_o a_o trient_a therefore_o xxi_o the_o aggregate_v of_o the_o subtense_n of_o two_o arch_n which_o together_o make_v up_o a_o trient_a be_v equal_a to_o the_o subtense_n of_o another_o arch_n which_o do_v as_o much_o exceed_v a_o trient_a or_o want_v of_o two_o trient_n as_o either_o of_o those_o two_o want_n of_o a_o trient_a xxii_o the_o same_o will_n in_o like_a manner_n be_v infer_v if_o we_o inscribe_v a_o quadrilater_n xi_o who_o opposite_a side_n be_v a_o t_o and_o e_o t_o and_o the_o diagonal_n tz_n for_o than_o ta+te_a =_o tz_n and_o therefore_o a+e_o =_o z_o as_o before_z xxiii_o but_o if_o either_o of_o the_o arch_n to_o which_o z_o subtend_v great_a than_o a_o xi_o trient_a and_o less_o than_o two_o trient_n be_v tripled_a the_o subtense_n of_o this_o triple_a be_v the_o same_o with_o that_o of_o the_o triple_a of_o a_o or_o e._n for_o the_o triple_a of_o a_o arch_n great_a than_o a_o trient_a be_v equal_a to_o one_o whole_a circumference_n with_o the_o triple_a of_o that_o excess_n for_o the_o triple_a of_o â…“+a_n be_v 1+3a_n now_o because_o when_o we_o have_v once_o go_v round_o the_o whole_a circumference_n we_o be_v just_a there_o where_o at_o first_o we_o begin_v this_o therefore_o as_o to_o this_o point_n be_v as_o nothing_o and_o the_o whole_a distance_n to_o be_v acquire_v be_v but_o the_o triple_a of_o such_o excess_n and_o just_o the_o same_o as_o if_o only_a this_o excess_n have_v be_v thrice_o take_v xxiv_o as_o for_o example_n if_o the_o arch_n subtend_v by_o z_o be_v β_n γ_n δ_n that_o be_v a_o xii_o trient_a increase_v by_o the_o arch_n e_o and_o to_o this_o we_o add_v a_o second_o equal_a to_o it_o δ_n ζ_n θ_o the_o aggregate_v β_n γ_n δ_n ζ_n θ_o be_v the_o double_a arch_n and_o the_o subtense_n thereof_o be_v b_o or_o β_n θ_o which_o be_v also_o the_o subtense_n of_o the_o difference_n of_o the_o arch_n a_o e_o and_o if_o to_o these_o two_o we_o add_v a_o three_o equal_a to_o either_o of_o they_o θ_o γ_n χ_n then_o be_v β_n γ_n δ_n ζ_n θ_o γ_n χ_n the_o triple_a of_o the_o arch_n first_o propose_v and_o the_o subtense_n hereof_o that_o be_v the_o streight-line_n which_o join_v the_o beginning_n and_o the_o end_n of_o this_o triple_a arch_n be_v β_n χ_n =_o c_o the_o very_a same_o which_o subtend_v the_o triple_a of_o e._n xxv_o and_o just_o the_o same_o will_v come_v to_o pass_v if_o for_o the_o first_o arch_n we_o take_v β_n θ_o ζ_n δ_n that_o be_v a_o trient_a increase_v by_o a_o to_o which_o z_o be_v a_o subtendent_fw-la likewise_o for_o take_v a_o second_o equal_a to_o it_o δ_n χ_n γ_n β_n θ_o the_o aggregate_v β_n θ_o ζ_n δ_n χ_n γ_n β_n θ_o more_o than_o one_o entire_a circumference_n be_v the_o double_a arch_n and_o the_o subtense_n thereof_o b_o as_o before_o and_o if_o to_o these_o two_o we_o add_v a_o three_o equal_a to_o either_o θ_o ζ_n δ_n χ_n the_o triple_a arch_n be_v β_n θ_o ζ_n δ_n χ_n γ_n β_n θ_o ζ_n δ_n χ_n and_o the_o subtense_n hereof_o as_o before_o β_n χ_n or_o c_o the_o same_o with_o the_o subtendent_fw-la of_o the_o triple_a of_o a._n and_o therefore_o xxvi_o the_o triple_a of_o a_o arch_n great_a than_o a_o trient_a have_v the_o same_o subtense_n with_o the_o triple_a of_o its_o excess_n above_o a_o trient_a and_o the_o same_o for_o the_o same_o reason_n hold_v in_o arch_n great_a than_o 2_o 3_o or_o more_o trients_n xxvii_o but_o note_v here_o that_o in_o this_o case_n that_o be_v if_o the_o arch_n to_o be_v tripled_a be_v great_a than_o a_o trient_a but_o less_o than_o two_o trient_n for_o if_o more_o than_o two_o trient_n but_o less_o than_o the_o whole_a circumference_n it_o be_v the_o same_o as_o if_o it_o be_v less_o than_o a_o triâ—â—â—â—â—_n the_o subtense_n of_o the_o double_a be_v less_o than_o that_o of_o the_o single_a for_o in_o such_o case_n the_o arch_n will_v differ_v from_o that_o of_o a_o semicircle_n either_o
which_o be_v ever_o equal_a to_o χ+δ±μ_n however_o these_o part_n be_v intermingle_v which_o where_o it_o be_v +μ_n be_v common_o more_o obvious_a to_o the_o eye_n but_o where_o it_o be_v −_o μ_n be_v more_o perplex_a and_o will_v need_v more_o consideration_n to_o discern_v but_o it_o be_v equal_o true_a in_o both_o case_n the_o square_a of_o the_o base_a of_o a_o angle_n of_o 135_o degree_n be_v equal_a to_o the_o square_n of_o the_o leg_n with_o a_o rectangle_n of_o they_o multiply_v into_o vi_o if_o a_o be_v 45_o degree_n it_o will_v in_o like_a manner_n be_v show_v that_o because_o of_o b_o =_o χ+δ_n −_o μ._n into_o χ+δ_n −_o μ_n =_o b_o =_o bq._n that_o be_v the_o square_a of_o the_o base_a of_o a_o angle_n of_o 45_o degree_n be_v equal_a to_o the_o square_n of_o the_o leg_n want_v a_o rectangle_n of_o they_o multiply_v into_o vii_o and_o universal_o what_o ever_o be_v the_o angle_n a_o it_o will_v by_o like_a process_n be_v show_v that_o that_o be_v the_o square_a of_o the_o base_a whatever_o be_v the_o angle_n at_o the_o vertex_fw-la be_v equal_a to_o the_o square_n of_o the_o leg_n together_o with_o if_o it_o be_v great_a than_o a_o right-angle_n or_o want_v if_o less_o than_o such_o a_o plain_a which_o shall_v be_v to_o the_o rect-angle_n of_o the_o leg_n as_o a_o portion_n in_o the_o base-line_n intercept_v between_o two_o line_n from_o the_o vertex_fw-la make_v at_o the_o base_a a_o like_a angle_n with_o that_o of_o the_o vertex_fw-la to_o one_o of_o those_o two_o line_n so_o draw_v viii_o of_o this_o we_o be_v to_o give_v great_a variety_n of_o example_n in_o the_o follow_a chapter_n where_o this_o general_n theorem_n be_v apply_v to_o particular_a case_n and_o which_o be_v further_o improve_v by_o these_o two_o ensue_a proposition_n ix_o the_o radius_fw-la of_o a_o circle_n with_o the_o subtense_n of_o two_o arch_n be_v give_v the_o subtense_n of_o their_o aggregate_v be_v also_o give_v for_o suppose_v the_o subtense_n of_o the_o give_v arch_n to_o be_v a_o e_o the_o subtense_n of_o their_o remainder_n to_o a_o semicircle_n be_v also_o have_v suppose_v and_o and_o therefore_o inscribe_v a_o quadrilater_n who_o opposite_a side_n be_v a_o ε_n and_o e_z α_n one_o of_o the_o diagonal_o be_v the_o diameter_n =_o 2r_o the_o other_o the_o subtense_n of_o the_o sum_n or_o aggregate_v of_o those_o arch_n suppose_v x._o the_o same_o be_v give_v the_o subtense_n of_o the_o difference_n of_o those_o arch_n be_v also_o give_v for_o have_v as_o before_o a_o α_n e_z ε_n 2r_v we_o have_v by_o a_o quadrilater_n due_o inscribe_v the_o subtense_n of_o the_o difference_n xi_o it_o be_v manifest_a also_o from_o what_o be_v before_o deliver_v that_o the_o same_o triangle_n gγμ_n do_v indifferent_o serve_v for_o the_o angle_n of_o 120_o degree_n and_o of_o 60_o degree_n and_o in_o like_a manner_n for_o 135_o and_o 45_o and_o so_o for_o any_o two_o arch_n whereof_o one_o do_v as_o much_o exceed_v as_o the_o other_o want_v of_o a_o quadrant_n for_o the_o angle_n v_n be_v in_o both_o the_o same_o and_o the_o angle_n at_o the_o base_a differ_v only_o in_o this_o that_o in_o one_o the_o external_n angle_n in_o the_o other_o the_o internal_a which_o be_v the_o other_o compliment_n to_o two_o right-angle_n be_v equal_a to_o the_o angle_n of_o cd_o at_o the_o vertex_fw-la xii_o hence_o it_o follow_v that_o of_o two_o angle_n where_o the_o leg_n of_o the_o one_o be_v respective_o equal_a to_o those_o of_o the_o other_o the_o one_o as_o much_o exceed_v a_o right-angle_n as_o the_o other_o want_v of_o it_o the_o square_a of_o the_o base_a in_o the_o one_o do_v as_o much_o exceed_v the_o two_o square_n of_o the_o leg_n as_o in_o the_o other_o it_o want_v thereof_o xiii_o and_o consequent_o in_o any_o rightlined_n triangle_n however_o incline_v the_o square_n of_o the_o axis_n or_o diameter_n and_o of_o the_o half_a base_n twice_o take_v be_v equal_a to_o the_o square_n of_o the_o leg_n for_o suppose_v c_o c_o the_o two_o half_n of_o the_o base_a and_o b_o the_o diameter_n or_o axis_n of_o the_o triangle_n mean_v thereby_o a_o straight_a line_n from_o the_o vertex_fw-la to_o the_o middle_n of_o the_o base_a and_o b_o β_n the_o two_o leg_n it_o be_v manifest_a that_o of_o the_o two_o angle_n at_o the_o base_a which_o be_v each_o other_o compliment_n to_o two_o right-angle_n the_o one_o do_v as_o much_o exceed_v as_o the_o other_o want_v of_o a_o right-angle_n and_o therefore_o the_o square_n of_o one_o of_o the_o leg_n as_o bq_n do_v as_o much_o exceed_v as_o the_o other_o βq_n do_v come_v short_a of_o dq+cq_n and_o therefore_o both_o together_o bq+βq_fw-fr =_o 2dq+2cq_n fourteen_o and_o therefore_o the_o base_a and_o axis_n or_o diameter_n of_o a_o triangle_n remain_v the_o same_o however_o different_o incline_v the_o aggregate_v of_o the_o square_n of_o the_o two_o leg_n remain_v the_o same_o xv._n and_o the_o same_o be_v to_o be_v understand_v of_o the_o square_n of_o tangent_n of_o a_o parabola_fw-la hyperbola_fw-la elipsis_n or_o other_o curve_v line_n have_v diameter_n and_o ordinates_n from_o the_o two_o end_n of_o a_o inscribe_v ordinate_a to_o the_o point_n of_o the_o diameter_n produce_v if_o need_v be_v wherein_o those_o tangent_n meet_v xvi_o the_o same_o may_v be_v likewise_o accommodate_v to_o the_o segment_n of_o such_o leg_n or_o tangent_n cut_v off_o by_o line_n parallel_v to_o the_o base_a namely_o the_o square_n of_o such_o segment_n intercept_v by_o those_o parallel_n together_o take_v the_o axe_n of_o such_o trapezium_fw-la remain_v the_o same_o be_v the_o same_o whether_o such_o trapezium_fw-la be_v erect_v or_o however_o incline_v for_o such_o segment_n be_v still_o proportional_a to_o their_o whole_n chap._n vii_o application_n thereof_o to_o particular_a case_n i._o if_o a_o be_v a_o right-angle_n or_o of_o 90_o degree_n gγ_n be_v co-incident_a and_o μ_n =_o 0._o and_o therefore_o and_o consequent_o by_o §_o 7_o chap._n prece_v ii_o if_o a_o =_o 120_o degree_n then_o be_v five_o that_o be_v the_o angle_n contain_v of_o gγ_n =_o 60_o degree_n as_o be_v always_o the_o difference_n of_o 2_o a_o from_o two_o right-angle_n and_o consequent_o gγμ_n a_o equilater_n triangle_n for_o such_o also_o be_v the_o angle_n at_o the_o base_a each_o of_o which_o be_v the_o compliment_n of_o a_o to_o two_o right-angle_n and_o therefore_o μ_n =_o g_o and_o bq_fw-fr =_o cq+dq+cd_a iii_o if_o a_o =_o 60_o degree_n then_o also_o be_v five_o =_o 60_o degree_n and_o μ_n =_o g_o as_o before_z and_o therefore_o bq_fw-fr =_o cq+dq_n −_o cd_o iv._o if_o a_o =_o 135._o then_o five_o =_o 90_o and_o therefore_o by_o §_o 1._o μq_fw-fr =_o gq+γq_fw-fr that_o be_v because_o g_o =_o γ_n μq_n =_o 2gq_fw-fr and_o and_o therefore_o v._o if_o a_o =_o 45._o then_o also_o v._o =_o 90_o and_o therefore_o as_o before_o and_o consequent_o vi_o if_o a_o =_o 150_o then_o five_o =_o 120._o and_o therefore_o by_o §_o 2._o μq_n =_o gq+γq+gγ_n that_o be_v because_o g_o =_o γ_n μq_n =_o 3gq_fw-fr and_o and_o vii_o if_o a_o =_o 30_o then_o five_o =_o 120._o and_o therefore_o by_o §_o 2._o μq_n =_o gq+γq+gγ_n that_o be_v because_o g_o =_o γ_n μq_n =_o 3gq_fw-fr and_o and_o viii_o if_o a_o =_o 157½_n than_o v._o =_o 135._o and_o by_o §_o 4._o and_o therefore_o ix_o if_o a_o =_o 22½_n than_o v._o =_o 135._o and_o by_o §_o 4._o and_o therefore_o x._o if_o a_o =_o 112½_n than_o v._o =_o 45._o and_o by_o §_o 5._o and_o therefore_o xi_o if_o a_o =_o 6_o −_o ½_o than_o v._o =_o 45._o and_o by_o §_o 5._o and_o therefore_o xii_o if_o a_o =_o 165_o then_o five_o =_o 150._o and_o by_o §_o 6._o and_o therefore_o xiii_o if_o a_o =_o 15_o then_o five_o =_o 150._o and_o by_o §_o 6._o and_o therefore_o fourteen_o if_o a_o =_o 105_o then_o five_o =_o 30._o and_o by_o §_o 7._o and_o therefore_o xv._n if_o a_o =_o 75_o then_o five_o =_o 30._o and_o by_o §_o 7._o and_o therefore_o xvi_o if_o a_o =_o 172½_n than_o v._o =_o 165._o and_o by_o §_o 12._o and_o therefore_o xvii_o if_o a_o =_o 7½_n than_o v._o =_o 165._o and_o by_o §_o 12._o and_o therefore_o xviii_o if_o a_o =_o 97½_n than_o v._o =_o 15._o and_o by_o §_o 13._o and_o xix_o if_o a_o =_o 82½_n than_o v._o =_o 15._o and_o by_o §_o 13._o and_o xx._n if_o a_o =_o 142½_n than_o v._o =_o 105._o and_o by_o §_o 14._o and_o xxi_o if_o a_o =_o 37½_n than_o v._o =_o 105._o and_o by_o §_o 14._o and_o xxii_o if_o a_o =_o 127½_n than_o v._o =_o 75._o and_o by_o §_o 15._o and_o xxiii_o if_o a_o =_o 52½_n than_o v._o =_o 75._o and_o by_o §_o 15._o and_o and_o in_o