give_v 6̇91̇69̇_n first_o single_a root_n b_o =_o 2_o and_o bb_v =_o 4.0000_o which_o 4.0000_o be_v subtract_v from_o the_o number_n give_v 69169_o then_o there_o remain_v of_o the_o number_n give_v 291̇69̇_n remain_v of_o the_o number_n give_v 291̇69̇_n root_n decuplate_n b_o =_o 20_o divisor_n 2._o b_o 40.00_o the_o second_o single_a root_n c_o =_o 6_o 2._o bc_n 240.00_o cc_o 36.00_o  _fw-fr 276.00_o subtract_v 276.00_o remain_n of_o the_o number_n give_v 15669̇_n the_o root_n increase_v b_o =_o 26_o root_n increase_v and_o decuplate_v b_o =_o 260_o divisor_n be_v 2_o b_o =_o 520_o the_o three_o single_a root_n c_o =_o 3_o 2_o bc_n 1560_o cc_o 0009_o totall_n 1569_o subtract_v 1569_o remain_n of_o the_o number_n give_v 0000_o the_o root_n increase_v 263_o be_v therefore_o the_o true_a root_n as_o may_v be_v prove_v by_o recomposition_n or_o multiply_v 263_o by_o 263_o for_o the_o product_n will_v be_v 69169_o which_o be_v the_o number_n give_v the_o cipher_n which_o be_v put_v after_o in_o the_o divisor_n and_o subtract_n be_v only_o to_o fill_v up_o the_o number_n of_o place_n by_o which_o the_o number_n give_v or_o rather_o the_o remain_a point_n will_v else_o exceed_v for_o the_o like_a purpose_n be_v use_v the_o decuplation_n of_o the_o root_n as_o only_o to_o supply_v a_o place_n until_o another_o figure_n succeed_v in_o place_n of_o the_o cipher_n and_o in_o nothing_o else_o do_v this_o work_n differ_v from_o the_o ordinary_a extraction_n of_o the_o square_a root_n common_o teach_v and_o know_v the_o reason_n of_o it_o depend_v upon_o the_o 4_o prop._n of_o the_o second_o book_n of_o euclid_n where_o it_o be_v demonstrate_v that_o if_o a_o right_a line_n be_v divide_v by_o chance_n into_o two_o part_n the_o square_n make_v of_o the_o whole_a be_v equal_a to_o the_o square_n of_o the_o part_n and_o to_o the_o rectangle_n make_v of_o the_o part_n twice_o so_o it_o be_v here_o as_o follow_v the_o square_a of_o the_o great_a part_n that_o be_v of_o 260_o bb_v =_o 67600_o the_o square_a of_o the_o lesser_a part_n that_o be_v of_o 3._o cc_o =_o 00009_o the_o rectangle_n of_o the_o part_n that_o be_v 260_o into_o 3_o twice_o 2_o bc_n =_o 01560_o equal_a to_o the_o whole_a square_n 69169_o nor_o do_v these_o letter_n represent_v so_o natural_o the_o thing_n themselves_o in_o a_o divide_a superficies_n only_o but_o as_o proper_o and_o clear_o the_o part_n of_o solid_a body_n of_o which_o two_o or_o three_o example_n for_o satisfaction_n in_o which_o i_o admonish_v the_o reader_n to_o be_v intent_n to_o the_o several_a pointing_n of_o the_o quantity_n according_a to_o their_o due_a order_n as_o be_v before_o express_v and_o also_o to_o the_o place_n of_o the_o divisor_n and_o substract_v by_o cipher_n as_o before_o also_o be_v intimate_v for_o this_o to_o the_o ingenious_a be_v enough_o &_o a_o long_a verbosity_n to_o other_o will_n scarce_o be_v so_o of_o cubicall_a aequation_n let_v there_o be_v a_o cube_n aaa_o =_o fff_n or_o propose_v in_o number_n aaa_o =_o 41781923_o put_v as_o before_o b_o +_o c_o =_o a_o then_o their_o cube_n also_o shall_v be_v equal_a that_o be_v bbb_v +_o 3_o bbc_n +_o 3_o bcc_fw-la +_o ccc_o =_o 41781923_o the_o resolution_n the_o homogeneal_a number_n give_v 41̇781̇923̇_n the_o first_o single_a cubique_a root_n b_o =_o 3_o and_o bbb_v =_o 27.000000_o subtract_v 27.00000_o remain_n of_o the_o number_n give_v 14781̇923̇_n remain_v of_o the_o number_n give_v 14781̇923̇_n the_o first_o root_n decuplate_v b_o =_o 30_o divisor_n 2790.000_o second_v single_a root_n c_o =_o 4_o subtract_v 12304.000_o remain_n of_o the_o number_n give_v 02477923̇_n the_o root_n increase_v b_o =_o 34_o root_n increase_v decuplate_n b_o =_o 340_o the_o three_o single_a root_n c_o =_o 7_o subtract_v 2477923_o remain_n last_o of_o the_o number_n give_v 000_o the_o root_n increase_v b_o +_o c_o =_o 347_o which_o be_v the_o true_a root_n of_o the_o cube_n 41781923_o as_o may_v be_v prove_v by_o recomposition_n that_o be_v by_o multiply_v 347_o by_o 347_o and_o the_o product_n again_o by_o 347_o the_o last_o product_n shall_v be_v equal_a to_o the_o cube_n which_o be_v give_v to_o be_v resolve_v and_o as_o above_o in_o the_o square_a the_o canon_n of_o the_o resolution_n be_v the_o letter_n bb_v +_o 2_o bc_n +_o cc_o be_v the_o true_a square_n of_o b_o +_o c._n and_o those_o letter_n do_v answer_v exact_o to_o the_o part_n of_o the_o square_n divide_v alike_o in_o both_o the_o dimension_n so_o here_o also_o the_o canon_n of_o resolution_n or_o the_o letter_n bbb_v +_o 3_o bbc_n +_o 3_o bcc_fw-la +_o ccc_o do_v exact_o answer_v to_o the_o part_n or_o member_n of_o a_o cube_n divide_v into_o two_o part_n alike_o in_o all_o the_o three_o dimension_n as_o any_o one_o may_v prove_v upon_o a_o cube_n make_v of_o some_o slender_a matter_n and_o cut_v through_o all_o three_o way_n for_o he_o shall_v find_v the_o whole_a cube_n suppose_v equal_a to_o 41781923_o as_o before_o just_o make_v up_o of_o the_o two_o cube_n of_o the_o two_o segment_n that_o be_v bbb_n and_o ccc_o and_o three_o parallelepipedon_n who_o length_n and_o breadth_n be_v equal_a to_o b_o and_o their_o thickness_n to_o c_o those_o three_o be_v the_o 3_o bbc_n and_o last_o three_o other_o parallelepipedon_n who_o length_n and_o breadth_n be_v equal_a to_o c_o and_o their_o thickness_n to_o b_o such_o be_v the_o 3_o bcc_fw-la see_v the_o follow_a schematisme_n the_o cuhe_n of_o the_o great_a segment_n which_o be_v 340_o bbb_v 39304000_o the_o three_o great_a parallelepipedon_n 3_o bbc_n ·2427600_n the_o three_o lesser_a parallelepipedon_n or_o 3_o bcc_fw-la ···46980_n the_o cube_n of_o the_o lesser_a segment_n which_o be_v 7_o ccc_o ·····343_n the_o whole_a cube_n give_v 41781923_o note_n that_o the_o great_a segment_n be_v the_o aggregate_v of_o all_o the_o single_a root_n except_o the_o last_o be_v due_o value_v by_o a_o cipher_n as_o here_o it_o be_v 340_o but_o the_o lesser_a segment_n be_v the_o last_o single_a root_n only_o as_o here_o 7_o i_o have_v do_v this_o to_o let_v the_o reader_n see_v that_o he_o may_v be_v sure_o let_v the_o quantity_n to_o be_v resolve_v be_v great_a or_o little_o whatsoever_o if_o he_o be_v careful_a to_o make_v his_o canon_n right_a the_o letter_n themselves_o will_v direct_v he_o how_o to_o frame_v his_o divisor_n and_o subtract_n in_o order_n to_o the_o final_a resolution_n especial_o in_o these_o unmix_a quantity_n where_o the_o point_n limit_v how_o far_o the_o subtract_v shall_v advance_v at_o every_o operation_n begin_v first_o at_o the_o point_n next_o the_o left_a hand_n not_o further_o and_o to_o the_o second_o point_n only_o at_o the_o second_o work_n and_o not_o otherwise_o in_o all_o that_o follow_v and_o in_o mix_a aequation_n if_o they_o be_v make_v up_o of_o cube_n with_o addition_n of_o certain_a square_n or_o certain_a root_n or_o both_o square_n and_o root_n or_o by_o subtraction_n of_o the_o same_o the_o canon_n of_o the_o resolution_n must_v ever_o be_v make_v by_o multiply_v the_o assume_v root_n b_o +_o c_o in_o the_o place_n of_o the_o quesititious_a root_n a_o quite_o through_o the_o aequation_n in_o all_o the_o degree_n thereof_o for_o so_o shall_v arise_v all_o the_o several_a parcel_n of_o which_o the_o several_a subtract_n be_v orderly_a to_o be_v make_v in_o a_o cubique_a aequation_n if_o all_o the_o quantity_n be_v present_a there_o be_v no_o need_n to_o point_v any_o but_o the_o cubiques_n and_o root_n yet_o i_o have_v here_o distinguish_v the_o place_n of_o the_o square_n also_o with_o little_a cross_n oblique_o which_o labour_n when_o the_o workman_n be_v intent_n upon_o his_o business_n may_v well_o enough_o be_v spare_v of_o the_o resolution_n of_o mix_a cubiques_n let_v the_o aequation_n aaa_o +_o daa_o −_o ffa_o =_o ggg_fw-mi be_v propose_v in_o number_n as_o let_v it_o be_v aaa_o +_o 32_o aa_o −_o 75_o a_o =_o 29282970_o therefore_o d_o =_o 32_o and_o ff_fw-mi =_o 75_o and_o ggg_fw-mi =_o 29282970_o put_v b_o +_o c_o =_o a_o and_o make_v the_o canon_n of_o resolution_n by_o substitute_v b_o +_o c_o in_o the_o place_n of_o a_o quite_o through_o the_o several_a quantity_n aaa_o +_o daa_o −_o ffa_o the_o canon_n right_o make_v will_v be_v +_o bbb_v +_o 3_o bbc_n +_o dbb_v −_o ffb_n +_o 3_o bcc_fw-la +_o 2_o dbc_n −_o ffc_fw-fr +_o ccc_o +_o dcc_o these_o several_a parcel_n of_o the_o canon_n be_v right_o subtract_v from_o the_o homogeneal_a number_n 29282970_o the_o number_n shall_v be_v thereby_o resolve_v and_o the_o root_n a_o find_v ☞_o note_v first_o that_o all_o the_o parcel_n in_o the_o canon_n which_o have_v not_o the_o secondary_a root_n c_o in_o they_o as_o +_o bbb_v +_o dbb_v
caa_o −_o da_fw-mi −_o f._n which_o be_v the_o same_o aequation_n which_o be_v to_o be_v resolve_v of_o which_o therefore_o gk_v be_v a_o true_a root_n in_o like_a sort_n may_v no_o be_v prove_v a_o true_a root_n which_o be_v to_o be_v demonstrate_v des_fw-mi cartes_n demonstrate_v of_o all_o this_o no_o more_o but_o the_o case_n where_o the_o aequation_n be_v aaaa_o =_o caa_o −_o da_fw-mi +_o f_o and_o leave_v the_o reader_n to_o please_v himself_o in_o find_v proof_n for_o the_o rest_n i_o have_v choose_v this_o case_n to_o demonstrate_v &_o have_v demonstrate_v the_o case_n of_o the_o two_o mean_n and_o trisection_n not_o only_o because_o some_o reader_n may_v be_v as_o much_o please_v to_o have_v this_o do_v to_o their_o hand_n as_o left_a to_o do_v themselves_o but_o also_o that_o all_o may_v see_v that_o the_o general_a way_n of_o demonstrate_v all_o sort_n of_o case_n depend_v on_o these_o two_o thing_n first_o that_o the_o right_a parameter_n of_o the_o parabola_fw-la be_v always_o unity_n if_o any_o of_o the_o root_n be_v put_v equal_a to_o a_o the_o intercept_a diameter_n will_v be_v always_o aa_o second_o there_o may_v be_v ever_o find_v two_o square_n equal_a to_o two_o other_o square_n and_o either_o the_o first_o two_o or_o second_o two_o equal_a to_o the_o square_n of_o radius_fw-la by_o help_v of_o these_o two_o thing_n may_v any_o case_n hereof_o be_v prove_v i_o will_v conclude_v with_o a_o breviat_fw-la of_o such_o equation_n as_o be_v not_o resoluble_a by_o ruler_n and_o compass_n 1_o if_o there_o be_v as_o many_o vowel_n as_o consonant_n and_o the_o vowel_n unequal_a as_o ae_z −_o da_o =_o db_fw-ge ae_z +_o da_fw-mi =_o db_fw-ge or_o −_o ae_z +_o da_fw-mi =_o db_fw-ge 2_o though_o but_o of_o two_o dimension_n and_o in_o few_o term_n as_o ae_z =_o bb_v though_o such_o be_v solvable_a yet_o it_o may_v be_v by_o infinite_a way_n and_o therefore_o can_v be_v apply_v to_o any_o limit_a proposition_n 3_o if_o there_o be_v but_o one_o vowel_n but_o cubical_o multiply_v or_o high_o as_o aaa_o =_o bbc_n or_o aaaa_o =_o bbbc_n where_o the_o aequation_n be_v already_o in_o the_o least_o term_n and_o not_o to_o be_v bring_v down_o by_o any_o common_a divisor_n nor_o the_o homogeneal_a bbc_n reducible_a to_o any_o solid_a more_o regular_a as_o to_o fff_n it_o be_v irresoluble_a chap._n xv._o have_v say_v in_o the_o conclusion_n of_o the_o former_a chapter_n that_o the_o aequation_n ae_z +_o da_fw-mi =_o db_fw-ge be_v as_o by_o right_a line_n and_o circle_n only_o irresoluble_a i_o will_v here_o show_v a_o problem_n by_o resolve_v whereof_o the_o say_a aequation_n will_v be_v happen_v on_o which_o be_v this_o follow_v probl._n 1._o in_o any_o rectangle_n bdca_n give_v from_o a_o angle_n in_o it_o c_o to_o draw_v a_o right_a line_n cf_o cut_v one_o opposite_a side_n bd_a in_o o_o and_o concur_v with_o the_o other_o bq_fw-la produce_v in_o f_o so_o as_o the_o intercept_a line_n foe_n may_v be_v equal_a to_o z_o any_o other_o right_a line_n give_v put_v bd_v =_o b_o ed_z =_o d_o do_v =_o a_o and_o bf_n =_o e_o and_o because_o the_o triangle_n bof_o do_v be_v like_a therefore_o it_o be_v b_o −_o a′_n a″_n e′_n d″_n and_o db_n −_o da_fw-mi =_o ae_z that_o be_v ae_z +_o da_fw-mi =_o db_fw-ge so_o we_o be_v quick_o come_v to_o the_o aequation_n require_v which_o aequation_n have_v as_o many_o unknown_a quantity_n as_o a_o e_fw-la as_o know_v to_o wit_n b_o &_o d_o be_v hitherto_o useless_a that_o the_o problem_n therefore_o may_v be_v solve_v we_o must_v work_v another_o way_n and_o bring_v it_o to_o a_o solid_a aequation_n by_o make_v for_o more_o convenience_n cd_o =_o b._n foe_fw-mi =_o c._n and_o bd_v =_o d._n and_o bo_o =_o a._n then_z d_o −_o a′_n a″_n b′_n and_o and_o the_o square_a thereof_o be_v equal_a to_o cc_o by_o the_o 47._o of_o the_o 1._o of_o euclid_n that_o be_v multiply_v all_o by_o the_o denominator_fw-la dd_v −_o 2_o da_fw-mi +_o aa_o it_o make_v ddaa_n −_o 2_o daaa_o +_o aaaa_o +_o bbaa_o =_o =_o ddcc_fw-la −_o 2_o dcca_fw-mi +_o ccaa_o that_o be_v aaaa_o −_o 2_o daaa_o +_o bbaa_o −_o ccaa_n +_o ddaa_o +_o 2_o dcca_fw-it =_o ddcc_fw-la make_v bb_v +_o dd_fw-mi −_o cc_o =_o ff_n than_o ff_n shall_v be_v sign_v +_o because_o hereby_o supposition_n it_o shall_v be_v bb_n +_o dd_fw-mi >_o cc._n and_o the_o aequation_n will_n c_o aaaa_o −_o 2_o daaa_o +_o ffaa_o +_o 2_o dcca_fw-it =_o ddcc_fw-la expunge_v the_o second_o term_n which_o be_v −_o 2_o daaa_n by_o the_o second_o rule_n of_o the_o 4._o chap._n and_o because_o the_o rule_n be_v not_o full_o exemplify_v there_o in_o the_o operosity_n thereof_o i_o will_v here_o work_v it_o at_o large_a because_o aaaa_n have_v four_o dimension_n therefore_o make_v 4′_n 1″_n 2_o d′_n ½_n d″_n again_o because_o the_o first_o and_o second_o term_n have_v different_a sign_n therefore_o put_v e_o +_o ½_n d_o =_o a_o chap._n 4._o rule_n 2._o the_o new_a aequation_n arise_v thereof_o will_v be_v the_o homogeneal_a −_o ddcc_fw-la be_v here_o put_v on_o the_o same_o side_n with_o the_o rest_n because_o for_o the_o present_a it_o seem_v better_a to_o stand_v so_o that_o it_o may_v be_v the_o last_o term_n in_o relation_n to_o that_o which_o be_v go_v before_o chap._n 4._o sect._n 4._o of_o the_o second_o rule_n in_o this_o last_o aequation_n it_o be_v manifest_a that_o the_o second_o term_n 2_o deee_v be_v through_o contradiction_n of_o +_o and_o −_o abolish_v as_o be_v require_v and_o now_o because_o the_o quesitition_n root_n e_o must_v be_v find_v by_o help_n of_o a_o parabola_fw-la as_o before_o in_o the_o like_a case_n be_v use_v it_o be_v necessary_a to_o reduce_v the_o aequation_n to_o some_o such_o form_n as_o have_v be_v show_v before_o in_o the_o note_n of_o the_o former_a chap._n first_o therefore_o to_o reduce_v the_o three_o term_n because_o d_o >_o f_o and_o +_o 6_o 4_o dd_o take_v from_o −_o 3_o dd_fw-mi rest_v −_o 6_o 4_o dd_o >_o ff_n make_v 6_o 4_o dd_o −_o ff_n =_o gg_fw-mi so_o all_o of_o the_o three_o term_n shall_v be_v −_o ggee_n likewise_o for_o the_o four_o term_n if_o +_o 4_o 8_o ddd_o −_o 6_o 4_o ddd_o +_o dff_n be_v sum_v up_o together_o the_o aggregate_v will_v be_v −_o ddd_fw-mi +_o ffd_a make_v dd_v −_o ff_n +_o 2_o cc_o =_o hh_o than_o all_o the_o four_o term_n will_v be_v +_o dhhe_fw-it now_o for_o the_o last_o term_n −_o ¼_n dddd_n −_o 1_o 16_o dddd_fw-mi =_o 3_o 16_o dddd_a and_o therefore_o make_v 3_o 16_o dd_fw-mi −_o ¼_n ff_n =_o will_v the_o aggregate_v of_o the_o last_o term_n be_v thereby_o −_o ddll_v for_o ddcc_fw-la be_v through_o contradiction_n of_o the_o sign_n annul_v and_o now_o the_o aequation_n be_v eeee_o −_o ggee_n +_o dhhe_fw-it −_o ddll_v =_o o_o make_v gg_n d_o =_o m_o and_o hh_v d_o =_o n_z and_o will_v d_o =_o p_o then_z the_o aequation_n will_v be_v eeee_o −_o dmee_fw-fr +_o ddne_fw-mi −_o dddp_n =_o o_o and_o make_n d_o =_o 1_o than_o the_o aequation_n full_o redude_v and_o right_o prepare_v be_v +_o eeee_o −_o i_o +_o ne_o −_o p_o =_o o._n in_o reduce_v this_o or_o the_o like_a consider_v chap._n 5._o note_n 2._o or_o eeee_o =_o i_o −_o ne_fw-fr +_o p_o which_o be_v altogether_o the_o same_o with_o that_o in_o the_o former_a chapter_n and_o the_o work_n of_o it_o be_v there_o show_v except_o only_o because_o there_o the_o quantity_n f_o be_v sign_v −_o and_o here_o the_o like_a quantity_n p_o be_v sign_v +_o i_o shall_v although_o this_o case_n only_o be_v demonstrate_v in_o des_n cartes_n here_o demonstrate_v it_o thus_o describe_v the_o parabolaf_fw-mi ap_fw-mi according_a to_o the_o parameter_n d_o that_o be_v as_o &_o let_v ae_z be_v the_o axis_fw-la &_o make_v ac_fw-la =_o ½_n d_o ce_fw-fr =_o ½_n m_o and_o at_o right_a angle_n at_o e_o make_v i_o =_o ½_n x_o and_o draw_v the_o line_n mass_fw-mi make_v as_o =_o d_o and_z ax_z =_o p_o and_o upon_o x_n as_o a_o diameter_n describe_v the_o semicircle_n xh_n and_o from_o a_o to_o h_z raise_v the_o perpendicular_a ah_o cut_v the_o circle_n in_o h_z and_o with_o radius_fw-la mh_o describe_v the_o arch_a hky_n cut_v the_o section_n in_o k_n and_o from_o k_n let_v fall_v a_o perpendicular_a to_o i_o produce_v in_o q_o and_o draw_v the_o line_n mk_v and_o mh_o demonstration_n to_o prove_v gk_fw-mi =_o e_o suppose_v it_o do_v and_o because_o kg_v =_o qe_o =_o e_o and_o i_o =_o ½_n n_z therefore_o mq_fw-la =_o ½_n n_z +_o e_o and_o the_o square_a of_o it_o be_v ¼_n nn_n +_o ne_o +_o ee_fw-mi and_o because_o ae_z =_o ½_n d_o +_o
various_a analogy_n therein_o which_o i_o leave_v to_o the_o invention_n of_o the_o reader_n now_o if_o it_o be_v require_v to_o draw_v a_o line_n from_o a_o point_n give_v without_o a_o circle_n give_v through_o the_o circle_n so_o as_o to_o cut_v off_o a_o arch_n equal_a to_o a_o arch_n give_v that_o may_v very_o easy_o be_v do_v in_o this_o manner_n from_o b_o draw_v the_o tangent_fw-la br_fw-la and_o make_v br_a =_o c_o cd_o =_o b_o db_fw-ge =_o a_o than_o it_o will_v be_v b_o +_o a′_n c″_n a′″_n euclid_n 3.36_o therefore_o aa_o +_o ba_o =_o cc_o euclid_n 6.17_o wherefore_o a_o may_v he_o find_v by_o the_o first_o rule_n for_o plain_a aequation_n chap._n 2._o chap._n x._o the_o superficies_n of_o a_o ellipsis_n may_v be_v easy_o find_v as_o near_o the_o truth_n as_o that_o of_o a_o circle_n because_o it_o have_v be_v prove_v by_o diverse_a to_o be_v a_o mean_a proportional_a between_o the_o two_o circle_n describe_v several_o upon_o the_o diameter_n of_o the_o ellipsis_n and_o it_o be_v almost_o axiomatical_o evident_a by_o mere_a inspection_n of_o the_o figure_n follow_v and_o therefore_o it_o be_v as_o easy_a to_o give_v a_o ellipsis_n in_o any_o proportion_n to_o another_o ellipsis_n as_o to_o describe_v any_o ellipsis_n at_o all_o as_o for_o example_n let_v the_o great_a diameter_n of_o the_o semiellipsis_n adc_n be_v ac_fw-la =_o 28._o then_o the_o semicircle_n describe_v thereon_o shall_v be_v abc_n =_o 88_o 2_o and_o let_v the_o lesser_a diameter_n of_o the_o say_a ellipsis_n be_v 2_o do_v or_o fg_a =_o 14._o last_o let_v it_o be_v require_v to_o describe_v a_o ellipsis_n which_o shall_v be_v to_o the_o ellipsis_n adc_n as_o 1_o to_o 4._o upon_o the_o line_n ac_fw-la from_o o_fw-la both_o way_n set_v off_o foe_n and_o go_v each_o of_o they_o equal_a to_o do_v and_o divide_v do_v into_o two_o equal_a part_n in_o h_z then_o describe_v the_o ellipsis_n which_o shall_v pass_v by_o the_o three_o point_n f_o h_o g_o i_o say_v that_o the_o ellipsis_n fhg_n be_v to_o the_o ellipsis_n adc_n as_o 1_o to_o 4._o for_o see_v the_o circle_n abc_n be_v to_o the_o circle_n fdg_fw-mi in_o diameter_n double_a therefore_o abc_n =_o 4_o fdg_n and_o of_o what_o part_n soever_o abc_n be_v 16_o of_o those_o fdg_n shall_v be_v 4._o and_o see_v the_o ellipsis_n adc_n be_v a_o mean_a betwixt_o they_o the_o say_a ellipsis_n be_v 8_o of_o the_o same_o part_n again_o by_o the_o same_o reason_n the_o circle_n fdg_n be_v quadruple_a to_o the_o circle_n nhk_fw-mi therefore_o of_o what_o part_n soever_o fdg_n be_v 4_o of_o those_o nhk_n shall_v be_v 1._o and_o see_v the_o ellipsis_n fhg_n be_v a_o mean_a betwixt_o they_o the_o say_a ellipsis_n be_v 2_o of_o the_o same_o part_n but_o the_o ellipsis_n give_v adc_n be_v 8._o and_o 2′_n 8″_n 1′_n 4″_n which_o be_v to_o be_v do_v in_o like_a sort_n have_v due_o proportion_v the_o diameter_n of_o circle_n may_v be_v make_v ellipse_n in_o any_o proportion_n one_o to_o another_o or_o in_o any_o proportion_n to_o a_o circle_n give_v and_o the_o work_n may_v be_v prove_v by_o induction_n as_o this_o also_o may_v have_v be_v for_o see_v the_o circle_n abc_n =_o 616_o the_o circle_n fdg_fw-mi =_o 154_o the_o ellipsis_n adc_n a_o mean_a betwixt_o they_o must_v be_v =_o 308._o again_o because_o the_o circle_n fdg_fw-mi =_o 154._o and_o the_o circle_n nhk_fw-mi =_o 038½_n the_o ellipsis_n fhg_v be_v a_o mean_a betwixt_o they_o must_v be_v =_o 77._o but_o 77′_n 308″_n 1′_n 4″_n etc._n etc._n note_n 1_o herein_o i_o make_v use_n of_o that_o proportion_n which_o be_v betwixt_o 22_o and_o 7_o for_o the_o circle_n to_o the_o diameter_n for_o easiness_n in_o account_n small_a and_o whole_a number_n be_v also_o better_o attend_v and_o understand_v soon_o by_o the_o reader_n and_o for_o no_o other_o cause_n the_o more_o exact_a proportion_n be_v as_o 355_o to_o 113_o or_o which_o be_v more_o use_v as_o 360_o to_o 114_o 5915492_o 10000000._o note_n 2_o hence_o it_o be_v manifest_a that_o the_o content_a of_o the_o lunula_n adbc_n comprehend_v by_o the_o circle_n abc_n and_o the_o ellipsis_n adc_n be_v according_o to_o this_o account_n half_o the_o circle_n abc_n that_o be_v 308._o as_o also_o the_o mix_a figure_n adf_a and_o cdg_n be_v here_o the_o residue_n of_o the_o semicircle_n fdg_fw-mi to_o the_o semiellipsis_n adc_fw-mi may_v be_v find_v out_o as_o exact_o as_o the_o superficies_n of_o a_o circle_n with_o which_o until_o a_o further_a discovery_n we_o must_v be_v content_a and_o i_o have_v here_o note_v it_o to_o show_v that_o investigation_n be_v not_o yet_o to_o be_v contemn_v as_o if_o the_o thing_n seek_v be_v not_o only_o impossible_a but_o useless_a when_o so_o many_o neat_a proposition_n may_v thereby_o be_v start_v as_o will_v although_o not_o so_o absolute_o necessary_a for_o present_a use_n yet_o delight_v the_o modest_a eye_n with_o the_o novelty_n note_n 3_o moreover_o if_o the_o say_v lunula_n adbc_n be_v compose_v of_o two_o circle_n there_o may_v be_v a_o rectiline_a figure_n give_v equal_a to_o the_o superficies_n thereof_o that_o be_v if_o the_o superficies_n of_o a_o circle_n adc_fw-la be_v double_a to_o the_o superficies_n of_o the_o circle_n abc_n the_o line_n ab_fw-la bc_n be_v draw_v the_o triangle_n abc_n will_v be_v equal_a to_o the_o lunula_n adbc_n as_o may_v be_v prove_v if_o it_o be_v not_o easy_a and_o well_o enough_o know_v already_o so_o that_o some_o figure_n of_o crooked_a line_n either_o differ_v in_o kind_n or_o in_o quantity_n may_v be_v equal_v with_o rectiline_a figure_n or_o number_n and_o yet_o where_o cirle_n of_o equal_a quantity_n include_v any_o lunula_n or_o other_o figure_n this_o can_v yet_o be_v do_v so_o thin_a be_v that_o curtain_n which_o be_v draw_v between_o we_o and_o our_o desire_n note_v 4._o whereas_o in_o the_o former_a figure_n the_o make_n of_o the_o ellipse_n adc_n fhg_n be_v not_o show_v this_o may_v be_v here_o useful_a to_o some_o and_o it_o be_v as_o follow_v the_o great_a axis_n of_o the_o ellipsis_n be_v equal_a to_o the_o diameter_n of_o a_o circle_n abc_n namely_o to_o the_o right_a line_n ac_fw-la the_o other_o axis_n to_o be_v take_v at_o pleasure_n according_a to_o the_o occasion_n have_v here_o assign_v the_o line_n do_v for_o the_o half_a of_o the_o lesser_a axis_n draw_v from_o the_o circle_n to_o the_o diameter_n ac_fw-la perpendicular_n as_o many_o as_o you_o please_v then_o last_o divide_v each_o perpendicular_a into_o two_o part_n proportional_a with_o bd_a and_o do_v in_o certain_a point_n if_o by_o those_o point_n of_o which_o the_o more_o the_o better_a a_o line_n be_v draw_v with_o a_o even_a hand_n that_o line_n shall_v pass_v also_o by_o the_o point_n d_o and_o be_v the_o ellipsis_n require_v otherwise_o and_o more_o for_o mcchanick_n use_v have_v choose_v the_o two_o axe_n ac_fw-la and_o 2_o do_v and_o make_v they_o cut_v one_o another_o into_o two_o equal_a part_n and_o at_o right_a angle_n in_o the_o point_n o_o take_v the_o half_a of_o ac_fw-la 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more_o than_o name_v in_o any_o english_a book_n that_o i_o have_v see_v i_o will_v write_v a_o little_a here_o of_o a_o cone_n and_o all_o the_o section_n thereof_o comprehend_v in_o one_o figure_n and_o after_o take_v some_o principal_a definition_n and_o one_o or_o two_o way_n of_o describe_v the_o section_n and_o draw_v tangent_n to_o they_o and_o some_o few_o other_o problem_n out_o of_o claudius_n midorgius_n not_o word_n for_o word_n but_o as_o it_o shall_v seem_v convenient_a here_o chap._n xi_o definition_n general_a of_o a_o cone_n 1._o now_o let_v this_o triangle_n abc_n represent_v half_o the_o cone_n as_o aforesaid_a and_o then_o if_o a_o plain_a as_o eboaz_v touch_v