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do_v in_o figurative_a arithmetic_n most_o common_o by_o apply_v some_o line_n of_o separation_n between_o the_o dividend_n and_o the_o divisor_n so_o a_o b_o be_v a_o divide_v by_o b_o and_o abc_n f_o signify_v that_o abc_n be_v divide_v by_o f._n but_o yet_o if_o the_o letter_n f_o have_v be_v find_v in_o the_o dividend_n the_o application_n of_o this_o line_n have_v not_o be_v necessary_a for_o it_o may_v have_v be_v better_o do_v by_o take_v away_o that_o letter_n out_o of_o the_o dividend_n so_o afc_a divide_v by_o f_o quotient_a be_v ac_fw-la and_o ffcc_fw-la divide_v by_o fc_a quotient_n be_v fc_a by_o ff_n quotient_a be_v cc_o by_o cc_o quotient_a be_v ff_n by_o ffc_a quotient_n be_v c_o by_z fcc_fw-la quotient_a be_v f_z by_z f_z quotient_a be_v fcc_fw-la by_z c_o quotient_fw-fr be_v ffc_a and_o the_o like_a may_v easy_o be_v understand_v of_o all_o the_o rest_n symbol_n of_o majority_n >_o minority_n <_o> equality_n =_o addition_n +_o substraction_n −_o root_n of_o a_o quantity_n √_o proportionality_n continue_a ′_o ″_o ‴_o '_o '_o '_o '_o proportionality_n disjunct_n ′_o ″_o ′_o ″_o so_o b_o >_o c_o signify_v b_o great_a than_o c_o b_o <_o> c_o .._o b_o less_o than_o c_o b_o =_o c_o ......._o b_o equal_a to_o c_o b_o +_o c_o ......._o c_o add_v to_o b_o b_o −_o c_o .._o c_o take_v from_o b_o √_o 72_o signify_v the_o square_a root_n of_o 72_o etc._n etc._n and_o b′_n c″_n d‴_n f_o '_o '_o '_o '_o signify_v that_o as_o b_o be_v to_z c_o so_o be_v c_o to_o d_o and_z so_z d_o to_z f._n likewise_o b′_n c″_n f′_n g″_n signify_v that_o as_o b_o be_v to_z c_o so_z be_v f_z to_z g._n these_o thing_n before_o express_v be_v almost_o general_o receive_v and_o
remote_a from_o the_o say_a centre_n note_n this_o aequation_n −_o aaa_n +_o 3_o a_o −_o b_o =_o 0_o be_v natural_o without_o the_o second_o term_n aa_o which_o be_v the_o cause_n that_o it_o have_v the_o false_a root_n not_o discern_v by_o twice_o +_o or_o twice_o −_o succeed_a as_o have_v be_v speak_v of_o chap._n 4._o if_o therefore_o one_o will_v have_v it_o so_o he_o must_v fill_v up_o the_o second_o term_n by_o augment_v the_o root_n never_o so_o little_a putting_z e_z −_o obligatory_n =_o a._n the_o demonstration_n of_o this_o problem_n be_v as_o follow_v it_o be_v to_o be_v prove_v that_o kg_n in_o the_o section_n be_v equal_a to_o be_v the_o subtense_n of_o the_o three_o part_n of_o the_o angle_n give_v put_v kg_a =_o y._n then_o because_o of_o the_o section_n ag_fw-mi =_o yy_a from_o the_o centre_n m_o draw_z mk_n and_o man_fw-mi which_o be_v equal_a because_o of_o the_o circle_n and_o draw_v kn_v parallel_n to_o ae_z and_o produce_v i_o to_o it_o in_o n._n then_o it_o be_v kn_v =_o ge_fw-mi =_o 2_o −_o yy_a the_o square_a therefore_o of_o kn_n be_v 4_o −_o 4_o yy_a +_o yyyy_a and_o mn_v be_v equal_a to_o y_fw-fr plus_fw-fr half_a the_o subtense_n bg_n call_v bg_v by_o the_o single_a letter_n b_o as_o before_z then_o mn_v =_o y_fw-es +_o ½_n b_o the_o square_a of_o which_o be_v yy_a +_o by_o +_o ¼_n bb_v add_v to_o it_o the_o former_a square_n of_o kn_n that_o be_v 4_o −_o 4_o yy_a +_o yyyy_a it_o make_v +_o 4_o −_o 4_o yy_a +_o yyyy_a +_o yy_a +_o by_o +_o ¼_n bb_v equal_a to_o the_o square_n of_o the_o hypothenusal_a mk_n again_o the_o square_a of_o ae_z be_v 4_o and_o the_o square_a of_o i_o be_v ¼_n bb_v which_o two_o square_n be_v equal_a also_o to_o the_o square_n of_o mk_n because_o mk_v =_o ma._n therefore_o 4_o −_o 4_o yy_a +_o yyyy_a +_o yy_a +_o by_o +_o ¼_n bb_a =_o =_o 4_o +_o ¼_n bb_v that_o be_v −_o 3_o yy_a +_o yyyy_a +_o by_o =_o 0._o that_o be_v by_o add_v on_o each_o part_n 3_o yy_a and_o subtract_v yyyy_a +_o 3_o yy_a −_o yyyy_a =_o by_o or_o last_o divide_v all_o by_o y_fw-mi +_o 3_o y_z −_o yyy_a =_o b._n but_o this_o aequation_n be_v alike_o graduate_v and_o like_a affect_v as_o the_o first_o aequation_n +_o 3_o a_o −_o aaa_o =_o b._n wherefore_o you_o =_o a._n but_o a_o =_o be_v and_o y_fw-mi =_o kg._n and_o therefore_o kg_a =_o be_v which_o be_v to_o be_v prove_v in_o like_a sort_n it_o may_v be_v prove_v that_o fd_a be_v a_o true_a root_n of_o the_o aequation_n 3_o a_o −_o aaa_o =_o b_o in_o the_o first_o figure_n and_o the_o subtense_n of_o the_o three_o part_n of_o the_o compliment_n of_o the_o angle_n give_v bag_n to_o a_o circle_n and_o by_o such_o work_a one_o may_v find_v it_o evident_a that_o when_o a_o circle_n cut_v a_o parabola_fw-la in_o point_n how_o many_o soever_o the_o vertex_fw-la except_v perpendicular_n let_v fall_v from_o all_o those_o point_n to_o the_o axis_fw-la be_v all_o the_o several_a root_n of_o one_o and_o the_o same_o aequation_n nor_o have_v that_o aequation_n any_o more_o root_n then_o those_o perpendicular_o to_o the_o axis_fw-la note_n 1._o in_o the_o aequation_n +_o aaa_o −_o bca_fw-mi =_o bbd_v the_o construction_n differ_v somewhat_o from_o the_o former_a for_o b_o be_v repute_a unity_n if_o c_o as_o here_o be_v sign_v with_o −_o the_o axis_fw-la of_o the_o parabola_fw-la must_v be_v produce_v from_o the_o point_n c_o in_o the_o axis_fw-la within_o the_o section_n distant_a from_o a_o by_o ½_n beyond_o the_o vertex_fw-la till_o the_o continuation_n be_v equal_a to_o ½_n c_o and_o at_o the_o end_n thereof_o raise_v a_o perpendicular_a equal_a to_o ½_n d_o at_o the_o end_n of_o that_o be_v the_o centre_n of_o the_o circle_n desire_v and_o according_a to_o this_o method_n may_v any_o aequation_n not_o above_o biquadratical_a be_v resolve_v after_o by_o take_v away_o the_o second_o term_n if_o there_o be_v any_o by_o the_o second_o rule_n of_o chapter_n the_o four_o it_o be_v reduce_v to_o such_o a_o form_n as_o this_o aaa_o *_o bca_fw-la =_o bbd_v if_o the_o quantity_n unknown_a have_v but_o three_o dimension_n or_o if_o it_o have_v four_o then_o thus_o aaaa_o *_o bcaa_o *_o bbda_fw-es =_o =_o bbbf_n or_o else_o taking_z b_o for_o unity_n than_o thus_o aaa_o *_o caa_o =_o d_o and_o thus_o aaaa_o *_o caa_n *_o da_fw-mi =_o f_o the_o sign_n +_o and_o −_o be_v hear_v omit_v for_o they_o must_v be_v supply_v as_o the_o nature_n of_o the_o aequation_n require_v note_n 2._o note_v that_o in_o this_o breviate_v the_o line_n b_o be_v that_o which_o be_v ba_o in_o the_o example_n of_o trisection_n and_o that_o which_o be_v r_o or_o unity_n in_o the_o example_n of_o two_o mean_n also_o the_o line_n c_o be_v that_o which_o in_o the_o former_a example_n of_o trisection_n be_v 2_o ce_fw-fr or_o 3._o and_o if_o this_o quantity_n be_v nothing_o than_o the_o perpendicular_a equal_a to_o half_o d_o be_v to_o be_v erect_v at_o the_o end_n of_o half_a b_o or_o ½_n set_v off_o from_o the_o vertex_fw-la upon_o the_o axis_n within_o but_o if_o c_o have_v any_o length_n then_o at_o the_o distance_n of_o ½_n c_o from_o that_o end_n upon_o the_o axis_fw-la and_o this_o which_o have_v be_v say_v be_v enough_o for_o all_o cubiques_n prob._n 3._o but_o where_o the_o equation_n be_v aaaa_o −_o caa_n +_o da_fw-mi =_o f_o so_o place_v as_o here_o if_o there_o be_v +_o f_o and_o the_o problem_n be_v to_o find_v the_o value_n of_o the_o root_n a_o then_o produce_v ma_fw-fr towards_o a_o make_v as_o equal_a to_o the_o right_a parameter_n of_o the_o section_n and_o make_v axe_n =_o f_o and_o upon_o the_o diameter_n x_v describe_v the_o circle_n xh_v cut_v by_o a_o perpendicular_a to_o ma_fw-fr namely_o ah_o in_o h_z then_o make_v the_o centre_n m_o and_o the_o space_n mh_o describe_v the_o circle_n desire_v but_o if_o it_o be_v −_o f_o as_o in_o this_o example_n i_o put_v it_o then_o after_o ah_o be_v find_v as_o before_o upon_o the_o diameter_n be_o describe_v a_o circle_n and_o in_o it_o from_o a_o apply_v a_o line_n ai_fw-fr =_o ah_o and_o make_v the_o centre_n m_o and_o the_o space_n mi_fw-mi describe_v the_o circle_n fik_fw-mi which_o be_v the_o circle_n seek_v for_o now_o this_o circle_n fik_fw-mi may_v cut_v or_o touch_v the_o parabola_fw-la in_o 1_o 2_o 3_o or_o 4_o points_z from_o all_o which_o perpendicular_o let_v fall_v to_o the_o axis_fw-la give_v all_o the_o root_n of_o the_o aequation_n as_o well_o the_o true_a as_o false_a one_o namely_o if_o the_o quantity_n d_o be_v mark_v −_o than_o those_o perpendicular_o which_o be_v on_o that_o side_n the_o parabola_fw-la where_o the_o centre_n m_o be_v be_v the_o true_a root_n but_o if_o it_o be_v +_o d_o as_o here_o the_o true_a root_n be_v those_o of_o the_o other_o side_n as_o gk_v and_o no_o and_o those_o of_o the_o centre_n side_n as_o fz_n pq_fw-fr be_v the_o false_a demonstration_n put_v ce_fw-fr =_o c_o 2_o and_o draw_v i_o perpendicular_a to_o ag_v and_o gl_o equal_a and_o parallel_v to_o it_o last_o put_v gk_v =_o a_o then_o ag_v =_o aa_o and_o take_v from_o it_o ae_z that_o be_v ½_n c_o +_o ½_n then_o ge_o =_o aa_o −_o −_o ½_n c_o −_o ½_n who_o square_a be_v aaaa_o −_o caa_n −_o −_o aa_o +_o ¼_n cc_o +_o ½_n c_o +_o ¼_n and_o because_o by_o construction_n gl_o =_o ½_n d_o therefore_o kl_v =_o a_o +_o ½_n d_o and_o the_o square_a of_o it_o be_v aa_o +_o da_fw-mi +_o ¼_n dd_v which_o add_v to_o the_o former_a square_n of_o ge_fw-mi it_o give_v the_o square_a of_o km_n that_o be_v a4_n −_o caa_n +_o ¼_n cc_o +_o da_fw-mi +_o ¼_n dd_fw-mi +_o ½_n c_o +_o ¼_n again_o the_o square_a of_o ae_z be_v ¼_n cc_o +_o ½_n c_o +_o ¼_n to_o which_o add_v the_o square_a of_o i_o that_o be_v ¼_n dd_v the_o whole_a be_v the_o spuare_fw-la of_o ma_fw-fr to_o wit_n ¼_n cc_o +_o ¼_n dd_fw-mi +_o ½_n c_o +_o ¼_n but_o the_o square_n of_o ah_o that_o be_v ai_fw-fr be_v equal_a to_o f_o because_o sa_o =_o 1_o and_o xa_fw-la =_o f_o between_o which_z ah_o or_o ai_fw-fr be_v a_o mean_v therefore_o the_o square_a of_o mi_fw-mi be_v ¼_n cc_o +_o ¼_n dd_fw-mi +_o ½_n c_o +_o ¼_n −_o f_o but_o mi_fw-mi =_o mk_v therefore_o their_o square_n be_v equal_a that_o be_v aaaa_o −_o caa_n +_o ¼_n cc_o +_o ¼_n dd_fw-mi +_o da_fw-mi +_o ½_n c_o +_o ¼_n =_o ¼_n cc_o +_o ¼_n dd_fw-mi +_o ½_n c_o +_o ¼_n −_o f._n that_o be_v aaaa_o −_o caa_n +_o da_fw-mi =_o −_o f._n or_o else_o aaaa_o =_o