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
in_o excess_n or_o in_o defect_n by_o less_o than_o ⅙_n of_o the_o whole_a circumference_n let_v it_o be_v x._o if_o therefore_o ½_o ±_o x_o be_v the_o single_a arch_n the_o double_a will_v be_v 1_o ±_o 2x_n and_o the_o subtense_n thereof_o whither_o great_a or_o less_o than_o one_o entire_a revolution_n will_v be_v the_o same_o with_o that_o of_o 2x_n and_o therefore_o x_o being_n less_o than_o ⅙_n 2x_n will_v require_v a_o less_o subtense_n than_o that_o of_o ½_o ±_o x_o that_o be_v less_o than_o the_o subtense_n of_o a_o trient_a but_o this_o great_a than_o it_o and_o the_o like_a be_v to_o be_v understand_v in_o other_o case_n of_o like_a nature_n xxviii_o suppose_v therefore_o as_o before_o or_o c_o must_v in_o this_o case_n be_v a_o negative_a quantity_n or_o if_o we_o put_v c_o affirmative_a then_o must_v z_o be_v negative_a or_o less_o than_o nothing_o for_o bq_n −_o zq_fw-fr where_o a_o great_a quantity_n be_v to_o be_v subtend_v from_o a_o less_o must_v needs_o be_v negative_a that_o be_v bq_n −_o zq_fw-fr =_o zc_a where_o zc_a be_v a_o negative_a either_o z_o or_o c_o must_v be_v so_o too_o or_o else_o put_v all_o affirmative_a zq_n −_o bq_fw-fr =_o zc_a and_o zq_n =_o bq+zc_a xxix_o which_o be_v evident_a also_o from_o the_o diagram_n where_o for_o this_o reason_n zz_v become_v diagonal_n and_o both_o bb_n and_o zc_n opposite_a side_n and_o therefore_o zq_a −_o bq_fw-fr =_o zc_a or_o zq_n =_o bq+zc_a and_o that_o be_v xxx_o the_o square_a of_o the_o subtense_n of_o a_o single_a arch_n great_a than_o a_o trient_a but_o xii_o less_o than_o two_o trient_n be_v equal_a to_o the_o square_n of_o the_o subtense_n of_o the_o double_a arch_n together_o with_o a_o rect-angle_n of_o the_o subtense_n of_o the_o single_a and_o triple_a arch._n and_o xxxi_o the_o square_a of_o the_o subtense_n of_o a_o single_a arch_n great_a than_o a_o trient_a but_o less_o than_o two_o trient_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 rectangle_n of_o the_o subtense_n of_o the_o single_a and_o triple_a arch._n and_o therefore_o xxxii_o if_o the_o square_a of_o the_o subtense_n of_o a_o single_a arch_n great_a than_o a_o trient_a but_o less_o than_o two_o trient_n want_v the_o square_a of_o the_o subtense_n of_o the_o double_a arch_n be_v divide_v by_o that_o of_o the_o single_a the_o result_n be_v the_o subtense_n of_o the_o triple_a arch_n or_o if_o divide_v by_o that_o of_o the_o triple_a the_o result_n be_v that_o of_o the_o single_a or_o xxxiii_o if_o from_o the_o subtense_n of_o a_o single_a arch_n great_a than_o a_o trient_a but_o less_o than_o two_o trient_n we_o subtract_v the_o square_n of_o the_o subtense_n of_o the_o double_a arch_n divide_v by_o that_o of_o the_o single_a the_o remainder_n be_v equal_a to_o the_o subtense_n of_o the_o triple_a xxxiv_o but_o because_o of_o or_o z_o zq_n −_o bq_fw-fr c_o and_o zq_n −_o bq_fw-fr =_o z+b_n into_o z_o −_o b_o we_o have_v thence_o this_o analogy_n that_o be_v xxxv_o as_o the_o subtense_n of_o a_o single_a arch_n great_a than_o a_o trient_a but_o less_o than_o two_o trient_n to_o the_o aggregate_v of_o the_o subtense_n of_o the_o single_a and_o double_a so_o be_v that_o of_o the_o single_a want_v that_o of_o the_o double_a to_o that_o of_o the_o triple_a xxxvi_o now_o because_o as_o we_o have_v show_v zq_n −_o bq_fw-fr =_o zc_a and_o by_o §_o 7_o 8_o chap._n prece_v therefore_o and_o that_o be_v xxxvii_o if_o from_o the_o cube_n of_o the_o subtense_n of_o a_o single_a arch_n great_a than_o a_o trient_a but_o less_o than_o two_o trient_n divide_v by_o the_o square_n of_o the_o radius_fw-la we_o subtract_v the_o triple_a of_o that_o subtense_n the_o remainder_n be_v equal_a to_o the_o subtense_n of_o the_o triple_a arch._n xxxviii_o if_o the_o arch_n to_o be_v tripled_a be_v great_a than_o two_o trient_n it_o be_v the_o same_o as_o if_o it_o be_v less_o than_o one_o trient_a for_o the_o residue_n of_o the_o whole_a to_o which_o it_o also_o subtend_v be_v then_o less_o than_o a_o trient_a and_o therefore_o the_o same_o chord_n suppose_v a_o or_o e_z subtend_v as_o well_o to_o a_o arch_n great_a than_o two_o trient_n as_o to_o one_o less_o than_o one_o trient_a thirty-nine_o if_o the_o arch_n to_o be_v tripled_a be_v equal_a to_o a_o trient_a it_o be_v indifferent_a to_o whether_o of_o the_o two_o case_n it_o be_v refer_v that_o of_o the_o great_a or_o that_o of_o the_o lesser_a than_o a_o trient_a and_o the_o same_o happen_v if_o it_o be_v suppose_v equal_a to_o two_o or_o more_o trient_n or_o to_o one_o or_o more_o entire_a revolution_n xl._o if_o the_o arch_n to_o be_v tripled_a be_v great_a than_o one_o or_o more_o entire_a revolution_n its_o subtense_n be_v the_o same_o with_o that_o of_o its_o excess_n above_o those_o entire_a revolution_n and_o to_o be_v consider_v in_o like_a manner_n which_o thing_n be_v evident_a and_o need_v no_o further_o demonstration_n xli_o now_o what_o have_v be_v several_o deliver_v concern_v the_o triplication_n of_o a_o arch_n or_o angle_n less_o than_o a_o trient_a and_o of_o one_o great_a than_o a_o trient_a but_o less_o than_o two_o trient_n to_o one_o of_o which_o case_n every_o arch_n may_v be_v refer_v as_o be_v already_o show_v we_o may_v thus_o joint_o put_v together_o xlii_o the_o difference_n of_o the_o square_n of_o the_o subtense_n of_o the_o single_a and_o double_a arch_n xii_o whether_o soever_o of_o they_o be_v the_o great_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 triple_a by_o §_o 4_o and_o 31._o that_o be_v bq_n −_o aq_n =_o ac_fw-la and_o zq_n −_o bq_fw-fr =_o zc_a and_o therefore_o xliii_o if_o the_o difference_n of_o the_o square_n of_o the_o subtense_n of_o the_o single_a and_o double_a arch_n be_v divide_v by_o the_o subtense_n of_o the_o single_a it_o give_v that_o of_o the_o triple_a if_o by_o that_o of_o the_o triple_a it_o give_v that_o of_o the_o single_a by_o §_o 5_o 32._o that_o be_v and_o and_o and_o xliv_o as_o the_o subtense_n of_o the_o single_a arch_n to_o the_o aggregate_v of_o the_o subtense_n of_o the_o single_a and_o double_a so_o be_v the_o difference_n of_o these_o subtense_n to_o the_o subtense_n of_o the_o triple_a by_o §_o 7_o 35._o that_o be_v and_o xlv_o now_o for_o as_o much_o as_o by_o §_o 12_o 26._o the_o three_o arch_n a_o e_o z_o if_o tripled_a will_v have_v the_o same_o subtense_n of_o the_o triple_a arch_n c_o it_o be_v thence_o manifest_a that_o such_o equation_n as_o this_o which_o concern_v the_o trisection_n of_o a_o arch_n must_v have_v in_o all_o three_o root_n as_o a_o e_o z_o for_o every_o of_o these_o upon_o such_o triplication_n will_v have_v the_o subtense_n of_o the_o triple_a arch_n c_o yet_o so_o that_o where_o a_o and_o e_o be_v affirmative_a root_n z_o be_v a_o negative_a and_o contrariwise_o where_o this_o be_v affirmative_a those_o be_v negative_a that_o be_v in_o this_o equation_n the_o root_n be_v +_o a_o +_o e_o −_o z._n butler_n in_o this_o the_o root_n be_v −_o a_o −_o e_o +_o z._n and_o therefore_o and_o consequent_o 3arq_n −_o ac_fw-la =_o 3erq_n −_o aec_fw-la =_o crq_n =_o zc_a −_o 3zrq_n xlvi_o since_o therefore_o 3arq_n −_o ac_fw-la =_o zc_a −_o 3zrq_n and_o consequent_o by_o transposition_n 3zrq_n +_o 3arq_n =_o zc_a +_o ac_fw-la it_o be_v also_o divide_v both_o side_n by_o z_o +_o a_o for_o z_o +_o a_o into_o zq_n −_o za_n +_o aq_n be_v equal_a to_o zc_v +_o ac_fw-la as_o will_v appear_v by_o multiplication_n and_o contrariwise_o if_o this_o be_v divide_v by_o either_o of_o those_o the_o quotient_n will_v be_v the_o other_o of_o they_o as_o will_v be_v find_v by_o division_n and_o in_o like_a manner_n because_o also_o 3erq_n −_o aec_fw-la =_o zc_a −_o 3zrq_n therefore_o 3zrq_n +_o 3erq_n =_o zc_a +_o aec_fw-la and_o that_o be_v xlvii_o of_o two_o arch_n whereof_o the_o one_o exceed_v the_o other_o by_o a_o trient_a of_o the_o whole_a circumference_n or_o else_o whereof_o the_o one_o do_v as_o much_o exceed_v a_o trient_a as_o the_o other_o want_v of_o it_o the_o square_n of_o the_o subtense_n want_v a_o rect-angle_n of_o the_o same_o subtense_n be_v equal_a to_o the_o square_n of_o the_o subtense_n of_o a_o trient_a or_o three_o time_n the_o square_n of_o the_o radius_fw-la that_o be_v zq_n −_o za_n +_o aq_n =_o 3rq_n =_o tq_fw-fr =_o zq_fw-fr −_o see_fw-mi +_o eq._n xlviii_o now_o the_o angle_n contain_v by_o the_o leg_n za_n or_o see_fw-mi stand_v on_o the_o chord_n t_o be_v a_o angle_n of_o 60_o degree_n as_o be_v a_o angle_n in_o the_o circumference_n stand_v
same_o event_n inscribe_v xvi_o a_o quadrilater_n so_o as_o that_o a_o d_o and_o a_o b_o may_v be_v opposite_a side_n and_o b_o c_o diagonal_n for_o than_o bc_n −_o basilius_n =_o da._n and_o therefore_o into_o will_v equal_a da._n that_o be_v and_o as_o before_o but_o of_o this_o we_o shall_v say_v more_o at_o §_o 86._o etc._n etc._n xvii_o but_o for_o as_o much_o as_o the_o same_o d_o subtend_v not_o only_o the_o quadruple_a of_o xv._n the_o arch_a a_o but_o also_o the_o quadruple_a of_o the_o arch_n e_o which_o therefore_o together_o with_o the_o arch_a a_o will_v complete_v a_o quadrant_n of_o the_o whole_a circumference_n it_o may_v in_o like_a manner_n be_v show_v that_o and_o therefore_o xviii_o a_o arch_n less_o than_o a_o quadrant_n and_o the_o arch_n which_o this_o want_n of_o a_o quadrant_n have_v both_o the_o same_o subtense_n of_o the_o quadruple_a arch_n d._n and_o according_o ae_n be_v two_o affirmative_a root_n of_o that_o equation_n xix_o but_o there_o be_v yet_o two_o other_o root_n but_o both_o negative_a as_o will_v after_o xvii_o appear_v of_o the_o same_o equation_n which_o we_o will_v call_v p_o s_o whereof_o one_o subtend_v a_o quadrant_n increase_v by_o the_o arch_a a_o the_o other_o a_o quadrant_n increase_v by_o the_o arch_n e._n for_o it_o be_v manifest_a by_o what_o be_v say_v at_o §_o 23_o chap._n prece_v that_o these_o also_o must_v have_v the_o same_o subtense_n of_o the_o quadruple_a arch_n with_o a_o and_o e._n for_o four_o time_n ¼+a_n be_v 1+4â—_n and_o will_v therefore_o have_v the_o same_o subtense_n with_o 4a_n and_o the_o like_a of_o four_o time_n ¼+e_v which_o be_v 1+4e_n who_o subtense_n be_v the_o same_o with_o that_o of_o 4e_n and_o the_o like_a will_v follow_v in_o case_n two_o three_o or_o more_o quadrant_n be_v thus_o increase_v and_o consequent_o xx._n a_o arch_n great_a than_o a_o quadrant_n or_o than_o two_o three_o or_o more_o quadrant_n will_v require_v the_o same_o subtense_n of_o its_o quadruple_a arch_n with_o its_o excess_n above_o a_o quadrant_n or_o above_o these_o two_o three_o or_o more_o quadrant_n xxi_o but_o the_o same_o p_o subtend_v as_o well_o to_o a_o quadrant_n increase_v by_o the_o arch_n of_o a_o as_o to_o three_o quadrant_n want_v the_o say_a arch_n as_o also_o to_o a_o semicircumference_n or_o two_o quadrant_n increase_v by_o the_o arch_n of_o e_o or_o want_v that_o archippus_n as_o be_v manifest_a to_o view_v by_o the_o scheme_n and_o in_o like_a manner_n saint_n subtend_v as_o well_o to_o a_o quadrant_n increase_v by_o the_o arch_n of_o e_o as_o to_o three_o quadrant_n want_v that_o arch_n as_o also_o to_o a_o semicircumference_n or_o two_o quadrant_n increase_v by_o the_o arch_n of_o a_o or_o want_v that_o arch._n xxii_o now_o that_o p_o s_o be_v negative_a root_n will_v thus_o appear_v for_o suppose_v for_o instance_n and_o p_o a_o subtendent_fw-la of_o a_o arch_n great_a than_o a_o quadrant_n but_o less_o than_o three_o quadrant_n otherwise_o it_o be_v the_o same_o as_o if_o it_o be_v less_o than_o one_o quadrant_n for_o the_o same_o chord_n which_o subtend_v a_o arch_n great_a than_o three_o quadrant_n subtend_v also_o to_o less_o than_o one_o p_o will_v in_o this_o case_n be_v great_a than_o the_o subtense_n of_o a_o quadrant_n and_o therefore_o 2rq_n −_o pq_fw-fr a_o negative_a quantity_n because_o of_o great_a quantity_n subtract_v from_o a_o lesser_a and_o therefore_o also_o p_o must_v be_v negative_a that_o so_o 2rqp_n −_o pc_n compound_v by_o the_o multiplication_n of_o two_o negative_n may_v be_v a_o positive_a quantity_n and_o therefore_o the_o whole_a affirmative_a also_o and_o what_o be_v say_v of_o p_o hold_v in_o like_a manner_n of_o s._n xxiii_o but_o if_o we_o choose_v to_o make_v p_o affirmative_a then_o must_v be_v negative_a and_o therefore_o change_v the_o sign_n affirmative_a and_o the_o like_a of_o s._n but_o of_o this_o more_o afterward_o xxiv_o but_o for_o what_o reason_n the_o equation_n or_o have_v two_o affirmative_a root_n a_o e_o and_o two_o negative_n p_o s_o for_o the_o same_o reason_n will_v the_o equation_n or_o have_v two_o negative_n a_o e_o and_o p_o s_o affirmative_n xxv_o if_o now_o we_o consider_v the_o quadrilater_n who_o say_v opposite_a side_n be_v a_o a_o and_o e_z p_o then_o because_o the_o arch_n a_o e_o do_v together_o make_v up_o a_o quadrant_n the_o diagonal_o q_o q_o be_v subtense_n of_o a_o quadrant_n or_o side_n of_o a_o inscribe_v square_a and_o therefore_o by_o §_o 68_o chap._n prece_v qq_fw-fr =_o 2rq_n and_o xxvi_o and_o therefore_o qq_a −_o aq_n =_o 2rq_n −_o aq_n =_o ep_n and_o consequent_o xvii_o and_o xxvii_o but_o the_o same_o p_o do_v also_o subtend_v a_o semicircumference_n want_v the_o arch_n of_o e_o and_o therefore_o and_o xxviii_o and_o by_o the_o same_o reason_n take_v a_o quadrilater_n who_o opposite_a side_n be_v e_o e_o and_o a_o s_o we_o have_v the_o diagonal_o and_o qq_fw-fr −_o eq_fw-fr =_o 2rq_n −_o eq_fw-fr =_o as_o and_o consequent_o because_o the_o arch_n a_o and_o s_o do_v complete_a the_o semicircumference_n and_o xxix_o now_o because_o as_o at_o §_o 27._o therefore_o and_o therefore_o 4rqeq_a −_o eqq_fw-fr =_o pqeq_fw-fr =_o 4rqq_fw-fr −_o 4rqaq_n +_o aqq_fw-fr and_o aqq_n +_o eqq_fw-fr =_o 4aqrq_n +_o 4eqrq_n −_o 4rqq_n and_o in_o like_a manner_n because_o therefore_o and_o 4rqaq_n −_o aqq_fw-fr =_o sqaq_n =_o rqq_fw-fr =_o 4rqq_fw-fr −_o 4rqeq_fw-fr +_o eqq_fw-fr and_o 4aqrq_n +_o 4eqrq_n −_o 4rqq_fw-fr =_o aqq_fw-fr +_o eqq._n xxx_o now_o the_o leg_n a_o e_o contain_z a_o sesquiquadrantal_n angle_n or_o of_o 135_o degree_n as_o be_v a_o angle_n in_o the_o peripherie_n stand_v on_o a_o arch_n of_o three_o quadrant_n and_o therefore_o xxxi_o in_o a_o rightlined_n triangle_n who_o angle_n at_o the_o top_n be_v 135_o degree_n if_o the_o double_a of_o the_o aggregate_v of_o the_o square_n of_o the_o leg_n contain_v it_o 2aq_n +_o 2eq_fw-fr want_v the_o square_a of_o the_o base_a qq_n =_o 2rq_n be_v multiply_v into_o the_o square_n of_o the_o base_a 2rq_n the_o product_n 4aqrq_n +_o 4eqrq_n −_o 4rqq_fw-fr =_o 2aq_n +_o 2eq_fw-fr −_o 2rq_n into_o 2rq_n be_v equal_a to_o the_o biquadrate_n of_o the_o leg_n aqq_n +_o eqq._n so_o that_o xxxii_o from_o hence_o appear_v a_o convenient_a method_n for_o add_v of_o biquadrates_n xxxiii_o the_o subduction_n of_o biquadrates_n may_v with_o a_o little_a alteration_n be_v perform_v almost_o in_o the_o same_o manner_n but_o it_o be_v more_o convenient_o do_v by_o multiply_v the_o sum_n of_o the_o square_n by_o the_o difference_n of_o they_o for_o aq_n +_o eq_fw-fr into_o aq_n −_o eq_fw-fr be_v equal_a to_o aqq_n −_o eqq._n but_o that_o be_v a_o speculation_n not_o of_o this_o place_n xxxiv_o again_o in_o such_o triangle_n who_o angle_n at_o the_o top_n be_v of_o 135_o degree_n if_o the_o double_a of_o the_o aggregate_v of_o the_o square_n of_o the_o leg_n be_v multiply_v into_o the_o square_n of_o the_o base_a the_o product_n be_v equal_a to_o the_o aggregate_v of_o the_o biquadrate_n of_o all_o the_o side_n for_o since_o 4aqrq_n +_o 4eqrq_n −_o qqq_fw-fr =_o 4rqq_fw-fr =_o aqq_fw-fr +_o eqq_fw-fr therefore_o aqq_fw-fr +_o eqq_fw-fr +_o qqq_fw-fr =_o 4aqrq_n +_o 4eqrq_n =_o 2aq_n +_o 2eq_fw-fr into_o 2rq_n =_o qq._n xxxv_o again_o because_o as_o at_o §_o 27_o 28._o xvii_o and_o therefore_o and_o 4rqq_n −_o 4rqaq_n +_o aqq_fw-fr =_o pqeq_fw-fr =_o 4pqrq_n −_o pqq._n therefore_o pqq_n +_o aqq_fw-fr =_o 4pqrq_n +_o 4aqrq_n −_o 4rqq_n and_o in_o like_a manner_n because_o and_o therefore_o and_o 4rqq_n −_o 4eqrq_n +_o eqq_fw-fr =_o sqaq_n =_o 4sqrq_n −_o sqq_n therefore_o sqq_fw-fr +_o eqq_fw-fr =_o 4sqrq_n +_o 4eqrq_n −_o 4rqq_n xxxvi_o but_o both_o a_o p_o and_o also_o e_z s_o contain_v a_o semiquadrantal_n angle_n or_o of_o 45_o degree_n as_o be_v a_o angle_n in_o the_o periphery_a stand_n on_o a_o quadrantal_a arch_n and_o one_o of_o the_o angle_n at_o the_o base_a obtuse_a and_o therefore_o xxxvii_o in_o a_o rightlined_n triangle_n who_o angle_n at_o the_o top_n be_v of_o 45_o degree_n or_o half_o a_o right-angle_n one_o of_o the_o other_o be_v obtuse_a if_o the_o double_a of_o the_o aggregate_v of_o the_o square_n of_o the_o leg_n as_o 2pq_n +_o 2aq_n want_v the_o square_a of_o the_o base_a qq_n =_o 2rq_n be_v multiply_v into_o the_o square_n of_o the_o base_a 2rq_n the_o product_n 4pqrq_n +_o 4aqrq_n −_o 4rqq_fw-fr =_o 2pq_fw-fr +_o 2aq_n −_o 2rq_n into_o 2rq_n be_v equal_a to_o the_o biquadrate_n of_o the_o leg_n pqq_n +_o aqq_fw-fr in_o like_a manner_n 2sq_n +_o 2eq_fw-fr −_o 2rq_n into_o 2rq_n =_o 4sqrq_n +_o 4eqrq_n −_o 4rqq_fw-fr =_o sqq_fw-fr +_o eqq._n so_o that_o xxxviii_o here_o be_v another_o method_n
of_o add_v biquadrate_n thirty-nine_o and_o likewise_o in_o such_o triangle_n who_o angle_n at_o the_o top_n be_v of_o 45_o degree_n and_o one_o of_o the_o other_o obtuse_a if_o the_o double_a of_o the_o aggregate_v of_o the_o square_n of_o the_o leg_n be_v multiply_v into_o the_o square_n of_o the_o base_a the_o product_n be_v equal_a to_o the_o biquadrate_n of_o all_o the_o side_n for_o because_o 4pqrq_n +_o 4aqrq_n −_o 4rqq_fw-fr =_o pqq_fw-fr +_o aqq_fw-fr therefore_o pqq_fw-fr +_o aqq_fw-fr +_o 4rqq_fw-fr =_o qqq_fw-fr =_o 4pqrq_n +_o 4aqrq_n =_o 2pq_fw-fr +_o 2aq_n into_o 2rq_n =_o qq._n and_o because_o 4sqrq_n +_o 4eqrq_n −_o 4rqq_fw-fr =_o sqq_fw-fr +_o eqq_fw-fr therefore_o sqq_n +_o eqq_fw-fr +_o 4rqq_fw-fr =_o qqq_fw-fr =_o 4sqrq_n +_o 4eqrq_n =_o 2sq_fw-la +_o 2eq_fw-fr into_o 2rq_n =_o qq._n xl._o furthermore_o if_o in_o a_o circle_n be_v inscribe_v a_o quadrilater_n who_o opposite_a xviii_o side_n be_v saint_n a_o and_o q_o q_o and_o the_o diagonal_n p_o p_o as_o in_o the_o scheme_n then_o pq_a −_o qq_fw-fr =_o 2rq_n =_o sa_o and_o therefore_o and_o and_o therefore_o and_o and_o consequent_o pqq_n −_o 4pqrq_n +_o 4rqq_fw-fr =_o aqsq_fw-la =_o 4rqaq_n −_o aqq_fw-fr =_o 4rqsq_fw-la −_o sqq._n xli_o and_o by_o the_o same_o reason_n if_o a_o quadrilater_n be_v inscribe_v who_o opposite_a side_n be_v e_o p_o and_o q_o q_o and_o the_o diagonal_n s_o s_o then_o sq_n −_o qq_fw-fr =_o 2rq_n =_o ep_n and_o therefore_o and_o and_o therefore_o and_o and_o consequent_o sqq_fw-fr −_o 4sqrq_n +_o 4rqq_fw-fr =_o pqeq_fw-fr =_o 4pqrq_n −_o pqq_fw-fr =_o 4eqrq_n −_o eqq._n xlii_o and_o either_o way_n we_o may_v conclude_v sqq_fw-fr +_o pqq_fw-fr =_o 4sqrq_n +_o 4pqrq_n −_o 4rqq_n xliii_o but_o p_o s_o contain_v half_o a_o right-angle_n or_o angle_n of_o 45_o degree_n as_o xviii_o be_v a_o angle_n in_o the_o periphery_a stand_n on_o a_o quadrantal_a arch_n and_o both_o the_o other_o angle_n acute_a and_o therefore_o xliv_o in_o a_o rightlined_n triangle_n who_o angle_n at_o the_o top_n be_v 45_o degree_n or_o half_o a_o right_a angle_n and_o both_o at_o the_o base_a acute_a if_o the_o double_a of_o the_o aggregate_v of_o the_o square_n of_o the_o leg_n as_o 2sq_n +_o 2pq_fw-fr want_v the_o square_a of_o the_o base_a qq_n =_o 2rq_n be_v multiply_v into_o the_o square_n of_o the_o base_a 2rq_n the_o product_n 4sqrq_n +_o 4pqrq_n −_o 4qq_fw-fr =_o 2sq_fw-la +_o 2pq_fw-fr −_o 2rq_n into_o 2rq_n be_v equal_a to_o the_o biquadrate_n of_o the_o leg_n sqq_fw-la +_o pqq._n xlv_o and_o this_o be_v a_o three_o method_n of_o add_v biquadrate_n xlvi_o and_o likewise_o in_o such_o triangle_n who_o angle_n at_o the_o top_n be_v of_o 45_o degree_n and_o both_o the_o other_o acute_a if_o the_o double_a of_o the_o aggregate_v of_o the_o square_n of_o the_o leg_n be_v multiply_v into_o the_o square_n of_o the_o base_a the_o product_n be_v equal_a to_o the_o biquadrate_n of_o all_o the_o side_n for_o since_o sqq_n +_o pqq_fw-fr =_o 4sqrq_n +_o 4pqrq_n −_o 4rqq_n therefore_o sqq_n +_o pqq_fw-fr +_o 4rqq_fw-fr =_o qqq_fw-fr =_o 4sqrq_n +_o 4pqrq_n =_o 2sq_fw-la +_o 2pq_fw-fr into_o 2rq_n xlvii_o these_o theorem_n thus_o demonstrate_v several_o whether_o the_o angle_n at_o the_o top_n be_v of_o 135_o degree_n or_o of_o 45_o degree_n and_o this_o whether_o the_o triangle_n be_v acuteangled_n or_o obtuse-angled_n to_o either_o of_o which_o we_o may_v refer_v the_o rectangle_v may_v be_v thus_o reduce_v to_o these_o general_n xlviii_o in_o a_o rightlined_n triangle_n who_o angle_n at_o the_o vertex_fw-la be_v either_o of_o 135_o degree_n or_o of_o 4â—_n degree_n the_o double_a aggregate_v of_o the_o square_n of_o the_o leg_n contain_v it_o want_v the_o square_a of_o the_o base_a multiply_v into_o the_o square_n of_o the_o base_a be_v equal_a to_o the_o biquadrate_n of_o the_o two_o leg_n which_o be_v the_o addition_n of_o biquadrates_n by_o §_o 31_o 37_o 44._o xlix_o and_o that_o double_v aggregate_v of_o the_o square_n of_o the_o leg_n multiply_v into_o the_o square_n of_o the_o base_a be_v equal_a to_o the_o biquadrate_n of_o all_o the_o three_o side_n by_o §_o 34_o 39_o 46._o l._n now_o the_o equation_n at_o §_o 29._o 4aqrq_n +_o 4eqrq_n −_o 4rqq_fw-fr =_o aqq_fw-fr +_o eqq_fw-fr and_o the_o other_o like_a to_o it_o at_o §_o 35_o 42._o be_v a_o quadratick_a equation_n of_o a_o plain_a root_n whereof_o the_o root_n be_v 2rq_n the_o co-efficient_a of_o the_o middle_a term_n 2aq_n +_o 2eq_fw-fr which_o be_v therefore_o equal_a to_o the_o sum_n of_o two_o quantity_n who_o rect-angle_n be_v equal_a to_o the_o absolute_a quantity_n aqq_fw-fr +_o eqq._n li._n if_o we_o therefore_o order_v this_o according_a to_o the_o rule_n of_o other_o equation_n of_o the_o same_o form_n and_o according_o from_o aqq_n +_o 2aqeq_fw-fr +_o eqq_fw-fr the_o square_n of_o half_a the_o co-efficient_a aq_n +_o eq_fw-fr we_o subtract_v the_o absolute_a quantity_n aqq_n +_o eqq_fw-fr the_o remainder_n be_v 2aqeq_n and_o the_o square_a root_n of_o this_o add_v to_o or_o subduct_v from_o half_o the_o co-efficient_a aq_n +_o eq_fw-fr give_v the_o root_n of_o that_o equation_n lii_o but_o of_o this_o ambiguous_a equation_n it_o be_v evident_a that_o we_o be_v to_o make_v choice_n of_o the_o great_a root_n in_o the_o case_n of_o §_o 29_o because_o the_o angle_n at_o the_o vertex_fw-la 135_o degree_n be_v great_a than_o a_o right-angle_n and_o therefore_o the_o square_n of_o the_o base_a qq_n be_v to_o be_v great_a than_o aq_n +_o eq_fw-fr the_o square_n of_o the_o two_o side_n contain_v it_o and_o therefore_o that_o be_v liii_o if_o to_o the_o square_n of_o the_o leg_n contain_v a_o angle_n of_o 135_o degree_n or_o three_o half_n of_o a_o right-angle_n we_o add_v the_o rect-angle_n of_o those_o leg_n multiply_v by_o the_o aggregate_v be_v equal_a to_o the_o square_n of_o the_o base_a liv._o in_o the_o same_o manner_n may_v be_v show_v that_o the_o equation_n of_o §_o 35._o pqq_fw-fr xviii_o +_o aqq_fw-fr =_o 4pqrq_n +_o 4aqrq_n −_o 4rqq_n or_o sqq_n +_o eqq_fw-fr =_o 4sqrq_n +_o 4eqrq_n −_o 4rqq_n and_o of_o §_o 42._o sqq_fw-fr +_o pqq_fw-fr =_o 4sqrq_n +_o 4pqrq_n −_o 4rqq_n be_v quadratic_a equation_n of_o a_o plain_a root_n 2rq_n but_o in_o all_o these_o it_o be_v manifest_a the_o lesser_a root_n be_v to_o be_v choose_v because_o the_o angle_n at_o the_o vertex_fw-la be_v of_o 45_o degree_n be_v less_o than_o a_o right_a angle_n and_o therefore_o the_o square_n of_o the_o base_a less_o than_o the_o two_o square_n of_o the_o leg_n and_o therefore_o the_o root_n and_o and_o that_o be_v lv._o if_o from_o the_o square_n of_o the_o leg_n contain_v a_o angle_n of_o 45_o degree_n or_o half_o a_o right-angle_n we_o subtract_v the_o rect-angle_n of_o these_o leg_n multiply_v by_o the_o remainder_n be_v equal_a to_o the_o square_n of_o the_o base_a lvi_o or_o we_o may_v put_v both_o together_o thus_o if_o to_o the_o square_n of_o the_o leg_n bè_fw-fr add_v if_o they_o contain_v a_o angle_n of_o 135_o degree_n or_o subtract_v thence_o if_o they_o contain_v a_o angle_n of_o 45_o degree_n a_o rect-angle_n of_o those_o leg_n multiply_v into_o the_o result_n be_v equal_a to_o the_o square_n of_o the_o base_a by_o §_o 53_o 55._o lvii_o we_o be_v next_o to_o note_n that_o the_o subtense_n e_o and_o p_o as_o also_o a_o and_o s_o xix_o who_o two_o arch_n do_v together_o make_v up_o a_o semicircumference_n do_v by_o §_o 9_o chap._n 29._o require_v the_o same_o subtense_n of_o the_o double_a arch_n and_o therefore_o much_o more_o the_o same_o subtense_n of_o the_o quadruple_a that_o be_v be_v the_o subtense_n of_o the_o double_a arch_n both_o of_o e_z and_o of_o p_o and_o of_o the_o double_a arch_n of_o a_o and_o of_o s._n lviii_o the_o subtense_n therefore_o of_o the_o triple_a arch_n of_o e_o less_o than_o a_o quadrant_n and_o therefore_o much_o more_o less_o than_o a_o trient_a be_v by_o §_o 2_o 5_o chap._n prece_v as_o be_v 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 eq_n divide_v by_o the_o subtense_n of_o the_o single_a arch_n e._n lix_n but_o the_o same_o subtense_n of_o that_o triple_a arch_n by_o §_o 8_o 9_o chap._n prece_v be_v lx._n therefore_o and_o the_o aggregate_v of_o the_o subtense_n of_o the_o triple_a and_o single_a lxi_o which_o may_v also_o be_v thus_o prove_v because_o pq_n +_o eq_fw-fr =_o 4rq_n as_o be_v in_o a_o semicircle_n and_o therefore_o pq_n =_o 4rq_n −_o eq_n and_o pqe_v =_o 4rqe_n −_o aec_fw-la or_o pqe_v −_o rqe_fw-fr =_o 3rqe_n −_o ec._n therefore_o be_v the_o subtense_n of_o the_o trible_a and_o the_o aggregate_v of_o the_o
as_o to_o l_o n_o all_o acute_a therefore_o as_o at_o §_o 114_o 115._o cxxii_o the_o difference_n of_o the_o quadricubes_n of_o the_o leg_n contain_v a_o angle_n of_o 72_o degree_n divide_v by_o the_o difference_n of_o those_o leg_n be_v equal_a to_o ten_o biquadrate_n of_o the_o radius_fw-la of_o the_o circumscribe_v circle_n and_o cxxiii_o the_o biquadrate_n of_o the_o leg_n contain_v a_o angle_n of_o 72_o degree_n together_o with_o the_o three_o mean_n proportional_a between_o those_o biquadrate_n be_v equal_a to_o ten_o biquadrate_n of_o the_o radius_fw-la of_o the_o circumscribe_v circle_n cxxiv_o but_o here_o the_o base_a of_o this_o triangle_n subtend_v to_o that_o angle_n of_o 72_o degree_n be_v the_o subtense_n of_o a_o biquintant_a or_o triquintant_a that_o be_v of_o ⅖_n =_o 144_o degree_n or_o of_o ⅗_n =_o 216_o degree_n which_o be_v by_o §_o 54._o and_o the_o square_a of_o this_o and_o its_o biquadrate_n which_o be_v to_o 10rqq_n as_o to_o 10_o or_o as_o to_o 4._o therefore_o cxxv_o the_o difference_n of_o the_o quadricubes_n of_o the_o leg_n contanine_v a_o angle_n of_o 72_o degree_n divide_v by_o the_o difference_n of_o those_o leg_n or_o the_o biquadrate_n of_o the_o leg_n contain_v such_o angle_n together_o with_o the_o three_o mean_n proportional_a between_o those_o biquadrate_n be_v to_o the_o biquadrate_n of_o the_o base_a subtend_v that_o angle_n as_o 4_o to_z cxxvi_o again_o because_o by_o §_o 109_o 5rqqa_o −_o 5rqac_n +_o aqc_a =_o rqqf_n =_o −_o 5rqql_o +_o 5rqlc_n −_o lqc_fw-fr l_o be_v great_a than_o a_o therefore_o by_o transposition_n 5rqql_v +_o 5rqqa_o =_o 5rqlc_n +_o 5rqac_n −_o lqc_fw-fr −_o aqc._n and_o divide_v all_o by_o l_o +_o a_o cxxvii_o but_o by_o §_o 46_o 47_o chap._n 30._o and_o therefore_o therefore_o that_o be_v cxxviii_o and_o again_o because_o as_o will_v appear_v by_o division_n xxix_o therefore_o lqq_n −_o lca_n +_o lqaq_n −_o lac_n +_o aqq_fw-fr =_o 10rqq_fw-fr cxxix_o but_o the_o angle_n contain_v by_o l_o a_o be_v of_o 36_o degree_n as_o be_v a_o angle_n at_o the_o circumfererence_n insist_v on_o a_o arch_n of_o 72_o degree_n or_o ⅗_n of_o the_o whole_a and_o one_o of_o the_o other_o obtuse_a cxxx_o and_o the_o same_o be_v to_o be_v say_v for_o the_o same_o reason_n of_o n_o e_o as_o of_o l_o a._n cxxxi_o and_o also_o because_o in_o like_a manner_n by_o §_o 109_o 5rqqm_o −_o 5rqmc_fw-fr +_o mqc_a =_o rqqf_n =_o −_o 5rqqn_v +_o 5rqnc_fw-la −_o nqc_n m_n be_v great_a than_o n_o therefore_o by_o the_o same_o method_n and_o the_o angle_n contain_v by_o m_n n_o be_v of_o 36_o degree_n and_o one_o of_o the_o other_o obtuse_a cxxxii_o and_o just_o the_o same_o for_o the_o same_o reason_n of_o m_n l_o save_v that_o here_o the_o angle_n be_v all_o acute_a cxxxiii_o and_o these_o be_v all_o the_o case_n that_o can_v happen_v the_o angle_n at_o the_o vertex_fw-la being_n 36_o degree_n for_o that_o of_o the_o leg_n v_o x_o be_v to_o be_v reduce_v to_o that_o of_o a_o l_o and_o that_o of_o x_o x_o to_o that_o of_o l_o m_o and_o the_o like_a be_v to_o be_v understand_v of_o other_o like_a case_n where_o a_o be_v extend_v to_o the_o whole_a quintant_a and_o e_o vanish_v into_o nothing_o therefore_o cxxxiv_o the_o sum_n of_o the_o quadricubes_n of_o the_o leg_n contain_v a_o angle_n of_o 36_o degree_n divide_v by_o the_o sum_n of_o those_o leg_n be_v equal_a to_o ten_o biquadrate_n of_o the_o radius_fw-la of_o the_o circumscribe_v circle_n by_o §_o 127_o 130_o 131_o 132._o and_o cxxxv_o the_o biquadrate_n of_o the_o leg_n contain_v a_o angle_n of_o 36_o degree_n with_o a_o mean_a proportional_a between_o those_o biquadrate_n want_v the_o first_o and_o three_o of_o three_o mean_n proportional_a between_o they_o be_v equal_a to_o ten_o biquadrate_n of_o the_o radius_fw-la of_o the_o circumscribe_v circle_n cxxxvi_o but_o the_o base_a subtend_v to_o this_o angle_n of_o 36_o degree_n be_v the_o side_n of_o a_o inscribe_v equilater_n pentagon_n as_o at_o §_o 116_o the_o biquadrate_n hereof_o be_v to_o 10rqq_n as_o to_o 4._o and_o therefore_o cxxxvii_o the_o sum_n of_o the_o quadricubes_n of_o the_o leg_n contain_v a_o angle_n of_o 36_o degree_n divide_v by_o the_o sum_n of_o those_o leg_n or_o the_o biquadrate_n of_o the_o leg_n contain_v such_o angle_n with_o a_o mean_a proportional_a between_o those_o biquadrate_n want_v the_o first_o and_o three_o of_o three_o mean_v proportional_n between_o they_o be_v to_o the_o biquadrate_n of_o the_o base_a subtend_v that_o angle_n as_o 4_o to_z cxxxviii_o again_o because_o by_o §_o 109._o 5rqqa_o −_o 5rqac_n +_o aqc_a =_o rqqf_n =_o −_o 5rqqn_v +_o 5rqnc_fw-la −_o nqc_n n_o be_v great_a than_o a_o therefore_o as_o at_o §_o 126_o etc._n etc._n cxxxix_o and_o in_o like_a manner_n because_o 5rqqe_v −_o 5rqec_fw-fr +_o eqc_a =_o rqqf_n =_o −_o 5rqql_o +_o 5rqlc_n −_o lqc_n l_o be_v great_a than_o e_z therefore_o cxl_o but_o the_o angles_n contain_v by_o n_o a_o or_o by_o l_o e_z be_v angle_n of_o 108_o xxix_o degree_n as_o be_v angle_n at_o the_o circumference_n insist_v on_o a_o arch_n of_o 216_o degree_n or_o ⅗_n of_o the_o whole_a therefore_o cxli_o the_o sum_n of_o the_o quadricubes_n of_o the_o leg_n contain_v a_o angle_n of_o 108_o degree_n divide_v by_o the_o sum_n of_o those_o leg_n be_v equal_a to_o ten_o biquadrate_n of_o the_o radius_fw-la of_o a_o circumscribe_v circle_n and_o cxlii_o the_o biquadrate_n of_o the_o leg_n contain_v a_o angle_n of_o 108_o degree_n with_o a_o mean_a proportional_a between_o those_o biquadrate_n want_v the_o first_o and_o three_o of_o three_o mean_n proportional_a between_o they_o be_v equal_a to_o ten_o biquadrate_n of_o the_o radius_fw-la of_o a_o circumscribe_v circle_n cxliii_o but_o the_o base_a subtend_v to_o this_o angle_n of_o 108_o degree_n be_v the_o subtense_n of_o a_o biquintant_a or_o which_o be_v the_o same_o of_o a_o triquintant_a that_o be_v of_o ⅖_n or_o ⅗_n of_o the_o whole_a circumference_n and_o therefore_o as_o at_o §_o 124_o be_v to_o 10rqq_n as_o to_o 4._o therefore_o cxliv_o the_o sum_n of_o the_o quadricubes_n of_o leg_n contain_v a_o angle_n of_o 108_o degree_n divide_v by_o the_o sum_n of_o those_o leg_n or_o the_o biquadrate_n of_o the_o leg_n contain_v such_o angle_n with_o a_o mean_a proportional_a between_o those_o biquadrate_n want_v the_o first_o and_o three_o of_o three_o mean_n proportional_a between_o they_o be_v to_o the_o biquadrate_n of_o the_o base_a subtend_v that_o angle_n as_o 4_o to_z cxlv_o now_o these_o several_a theorem_n thus_o deliver_v in_o particular_a may_v be_v collect_v into_o these_o general_n follow_v namely_o cxlvi_o the_o difference_n of_o the_o quadricubes_n of_o leg_n contain_v a_o angle_n of_o 144_o or_o of_o 72_o degree_n divide_v by_o the_o difference_n of_o those_o leg_n or_o the_o sum_n of_o the_o quadricubes_n of_o leg_n contain_v a_o angle_n of_o 36_o degree_n or_o of_o 108_o degree_n divide_v by_o the_o sum_n of_o those_o leg_n or_o which_o be_v equivalent_a to_o those_o the_o biquadrate_n of_o the_o leg_n in_o the_o former_a case_n with_o the_o three_o mean_n proportional_a between_o they_o or_o in_o the_o latter_a case_n the_o biquadrate_n of_o the_o leg_n with_o a_o mean_a proportional_a between_o they_o want_v the_o first_o and_o three_o of_o three_o mean_n proportional_n be_v equal_a to_o ten_o biquadrate_n of_o the_o radius_fw-la of_o a_o circumscribe_v circle_n and_o these_z to_o the_o biquadrate_n of_o their_o respective_a base_n subtend_v such_o angle_n of_o 144_o degree_n or_o of_o 36_o degree_n be_v as_o 4_o to_o but_o of_o the_o base_n subtend_v such_o angle_n of_o 72_o degree_n or_o of_o 108_o degree_n as_o 4_o to_o or_o as_o 8_o to_o and_o 8_o to_z that_o be_v in_o the_o duplicate_v proportion_v to_o and_o of_o to_o cxlvii_o and_o those_o side_n contain_v these_o follow_a angle_n sides_n deg._n a_o e._n 144_o m_n a._n m_o e._n l_o n._n 72_o l_o a._n n_o e._n m_o l._n m_o n._n 36_o n_o a._n l_o e._n 108._o whereof_o the_o four_o first_o couple_n be_v side_n of_o like_a sign_n the_o six_o latter_a of_o unlike_a cxlviii_o the_o same_o equation_n may_v be_v thus_o also_o consider_v because_o by_o §_o 110_o 5rqqa_o −_o 5rqqe_fw-fr =_o 5rqac_n −_o 5rqec_fw-fr −_o aqc_fw-fr +_o eqc_a therefore_o divide_v all_o by_o a_o −_o e_o and_o again_o by_o 5rq_n and_o by_o transposition_n aq_n +_o ae_n +_o eq_fw-fr −_o rq_n into_o 5rq_n =_o aqq_fw-fr +_o ace_n +_o aqeq_fw-fr +_o aec_fw-la +_o eqq._n cxlix_o and_o in_o like_a manner_n because_o m_n a_o and_o m_o e_o and_o l_o n_o have_v xxix_o also_o like_o sign_n mq+ma+aq_n −_o rq_n into_o 5rq_n =_o mqq+mca+mqaq+mac+aqq_fw-fr mq+me+eq_fw-fr −_o rq_n into_o 5rq_n =_o