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
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
or_o e_z less_o than_o a_o quadrant_n then_o a_o b_o and_o a_o d_o will_v be_v opposite_a side_n and_o b_o c_o diagonal_n and_o therefore_o cb_n −_o ab_fw-la =_o ad._n and_o consequent_o into_o equal_a to_o ad._n that_o be_v and_o as_o before_o and_o for_o the_o same_o reason_n and_o lxxxviii_o if_o the_o subtense_n of_o the_o single_a arch_n be_v p_o or_o saint_n great_a than_o a_o xxi_o quadrant_n and_o even_o great_a than_o a_o trient_a but_o less_o than_o two_o trient_n then_o b_o c_o and_o b_o p_o or_o b_o s_o will_v be_v opposite_a side_n and_o d_o p_o or_o d_o s_o diagonal_n and_o therefore_o bc+bp_v =_o pd_v or_o bc+bs_n =_o sd_z and_o consequent_o into_o equal_a to_o pd_v that_o be_v and_o as_o before_o and_o by_o the_o same_o reason_n bc+bs_n =_o sd_z if_o saint_n also_o be_v great_a than_o a_o trient_a and_o lxxxix_o but_o if_o the_o single_a arch_n be_v that_o of_o saint_n great_a than_o a_o quadrant_n but_o xxii_o less_o than_o a_o trient_a or_o p_o great_a than_o two_o trient_n but_o less_o than_o three_o quadrant_n then_o b_o c_o and_o d_o s_o be_v opposite_a side_n and_o b_o s_o diagonal_n and_o therefore_o b_n −_o bc_n =_o ds._n and_o consequent_o into_o that_o be_v and_o as_o before_o and_o in_o like_a manner_n bp_o −_o bc_n =_o pd_v if_o the_o arch_n of_o p_o be_v great_a than_o two_o trient_n which_o be_v the_o same_o as_o if_o less_o than_o one_o and_o xc_o from_o all_o which_o arise_v this_o general_n theorem_n the_o rect-angle_n of_o the_o subtense_n of_o the_o single_a and_o of_o the_o quadruple_a arch_n be_v equal_a to_o the_o subtense_n of_o the_o double_a multiply_v into_o the_o excess_n of_o the_o subtense_n of_o the_o triple_a above_o that_o of_o the_o single_a in_o case_n this_o be_v less_o than_o a_o quadrant_n or_o more_o than_o three_o quadrant_n or_o into_o the_o excess_n of_o the_o subtense_n of_o the_o single_a above_o that_o of_o the_o triple_a in_o case_n the_o single_a be_v more_o than_o a_o quadrant_n but_o less_o than_o a_o trient_a or_o more_o than_o two_o trient_n but_o less_o than_o three_o quadrant_n or_o last_o into_o the_o sum_n of_o the_o subtense_n of_o the_o triple_a and_o single_a in_o case_n this_o be_v more_o than_o a_o trient_a but_o less_o than_o two_o trient_n that_o be_v ad_fw-la =_o b_o into_o c_o −_o a_o if_o the_o arch_n of_o a_o be_v less_o than_o a_o quadrant_n or_o great_a than_o three_o quadrant_n a_o −_o c_o if_o it_o be_v great_a than_o a_o quadrant_n but_o less_o than_o a_o trient_a or_o great_a than_o two_o trient_n but_o less_o than_o three_o quadrant_n a+c_a if_o it_o be_v great_a than_o a_o trient_a but_o less_o than_o two_o trient_n xci_o and_o universal_o that_o be_v if_o the_o difference_n of_o 2rqa_n and_o a_o c_o whereof_o that_o be_v the_o great_a if_o the_o single_a arch_n be_v less_o than_o a_o quadrant_n or_o great_a than_o three_o quadrant_n but_o this_o if_o contrariwise_o divide_v by_o rc_n be_v multiply_v into_o product_v be_v equal_a to_o d._n xcii_o and_o therefore_o that_o be_v xciii_o as_o the_o cube_n of_o the_o radius_fw-la to_o the_o solid_a of_o the_o subtense_n of_o the_o single_a arch_n into_o the_o difference_n of_o the_o square_n of_o itself_o and_o of_o the_o double_a square_n of_o the_o radius_fw-la so_o be_v the_o subtense_n of_o the_o difference_n of_o that_o single_a arch_n from_o a_o semicircumference_n to_o the_o subtense_n of_o the_o quadruple_a arch._n xciv_o now_o what_o be_v before_o say_v at_o §_o 15_o chap._n 29._o that_o the_o subtense_n i._n of_o a_o arch_n with_o that_o of_o its_o remainder_n to_o a_o semicircumference_n or_o of_o its_o excess_n above_o a_o semicircumference_n will_v require_v the_o same_o subtense_n of_o the_o double_a arch_n be_v the_o same_o as_o to_o say_v that_o from_o any_o point_n of_o circumference_n two_o subtense_n draw_v to_o the_o two_o end_n of_o any_o inscribe_v diameter_n as_o a_o e_o will_v require_v the_o same_o subtense_n b_o of_o the_o double_a arch._n xcv_o and_o what_o be_v say_v at_o §_o 12_o 26_o chap._n prece_v that_o the_o subtense_n xi_o of_o a_o arch_n less_o than_o a_o trient_a and_o of_o its_o residue_n to_o a_o trient_a as_o a_o e_o and_o of_o a_o trient_a increase_v by_o either_o of_o those_o as_o z_o will_v have_v the_o same_o subtense_n of_o the_o triple_a arch_n be_v the_o same_o in_o effect_n with_o this_o that_o from_o any_o point_n of_o the_o circumference_n three_o subtense_n draw_v to_o the_o three_o angle_n of_o any_o inscribe_v regular_n trigone_n as_o a_o e_o z_o will_v have_v the_o same_o subtense_n c_o of_o the_o triple_a arch._n xcvi_o and_o what_o be_v say_v here_o at_o §_o 18_o 20._o that_o the_o subtense_n of_o a_o xxiii_o arch_n less_o than_o a_o quadrant_n and_o of_o its_o residue_n to_o a_o quadrant_n as_o a_o e_o and_o of_o a_o quadrant_n increase_v by_o either_o of_o these_o as_o p_o s_o will_v have_v the_o same_o subtense_n of_o the_o quadruple_a arch_n be_v the_o same_o with_o this_o that_o from_o any_o point_n of_o the_o circumference_n four_o subtense_n draw_v to_o the_o four_o angle_n of_o any_o inscribe_v regular_n tetragone_fw-mi as_o a_o e_o p_o s_o will_v have_v the_o same_o subtense_n d_o of_o the_o quadruple_a arch._n xcvii_o but_o the_o same_o hold_v respective_o in_o other_o multiplication_n of_o arch_n as_o five_o subtense_n from_o the_o same_o point_n to_o the_o five_o angle_n of_o a_o inscribe_v regular_n pentagon_n and_o six_o to_o the_o six_o angle_n of_o a_o hexagon_n etc._n etc._n will_v have_v the_o same_o subtense_n of_o the_o arch_n quintuple_a sextuple_a etc._n etc._n for_o they_o all_o depend_v on_o the_o same_o common_a principle_n that_o a_o semicircumference_n doubled_n a_o trient_a tripled_a a_o quadrant_n quadruple_v a_o quintant_a quintuple_v a_o sextant_a sextuple_v etc._n etc._n make_v one_o entire_a revolution_n which_o as_o to_o this_o business_n be_v the_o same_o as_o nothing_o and_o therefore_o universal_o xcviii_o from_o any_o point_n of_o the_o circumference_n two_o three_o four_o five_o six_o or_o more_o subtense_n draw_v to_o so_o many_o end_n of_o the_o diameter_n or_o angle_n of_o a_o regular_n polygone_a of_o so_o many_o angle_n however_o inscribe_v will_v have_v the_o same_o subtense_n of_o the_o arch_n multiply_v by_o the_o number_n of_o such_o end_n or_o angle_n and_o therefore_o cxix_o a_o equation_n belong_v to_o such_o multiplication_n or_o section_n of_o a_o arch_n or_o angle_n must_v have_v so_o many_o root_n affirmative_a or_o negative_a as_o be_v the_o exponent_fw-la of_o such_o multiplication_n or_o section_n as_o two_o for_o the_o bisection_n three_o for_o the_o trisection_n four_o for_o the_o quadrisection_n five_o for_o the_o quinquisection_n and_o so_o forth_o c._n and_o consequent_o such_o equation_n may_v according_o be_v resolve_v by_o such_o section_n of_o a_o angle_n as_o be_v before_o note_v at_o §_o 61_o chap._n prece_v of_o the_o trisection_n of_o a_o angle_n chap._n iu._n of_o the_o quintuplation_n and_o quinquisection_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 side_n a_o f_o the_o subtense_n xxiv_o of_o the_o single_a arch_n and_o the_o quintuple_a be_v parallel_n b_o b_o subtense_n of_o the_o double_a opposite_a the_o diagonal_n will_v be_v c_o c_o the_o subtense_n of_o the_o triple_a as_o be_v evident_a from_o the_o figure_n but_o it_o be_v evident_a also_o that_o in_o this_o case_n the_o single_a arch_n must_v be_v less_o than_o a_o quintant_a or_o fifth_z part_n 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 cq_n −_o bq_fw-fr =_o af._n and_o by_o the_o same_o reason_n cq_a −_o bq_fw-fr =_o ef._n that_o be_v iii_o the_o square_a of_o the_o subtense_n of_o the_o triple_a arch_n want_v 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 rect-angle_n of_o the_o subtense_n of_o the_o single_a and_o of_o the_o quintuple_a the_o single_a arch_n be_v less_o than_o a_o five_o part_n of_o the_o whole_a circumference_n iu._n and_o therefore_o if_o it_o be_v divide_v by_o one_o of_o they_o it_o give_v the_o other_o that_o be_v and_o and_o in_o like_a manner_n and_o v._o but_o c+b_n into_o c_o −_o b_o be_v equal_a to_o cq_n −_o bq._n and_o therefore_o that_o be_v vi_o as_o the_o subtense_n of_o the_o single_a arch_n less_o than_o a_o five_o part_n of_o the_o whole_a circumference_n to_o the_o aggregate_v of_o the_o subtense_n of_o the_o triple_a and_o double_a so_o be_v the_o excess_n of_o the_o subtense_n of_o the_o triple_a above_o that_o of_o the_o double_a to_o that_o of_o