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
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
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