chapter_n of_o his_o clavis_fw-la mathematicae_fw-la where_o he_o say_v that_o 43_o 7_o be_v the_o proportion_n of_o 31_o to_o 7_o for_o his_o meaning_n be_v that_o the_o proportion_n of_o 43_o 7_o to_o one_o be_v the_o proportion_n of_o 31_o to_o 7_o whereas_o if_o he_o mean_v as_o you_o do_v than_o 86_o 7_o shall_v be_v double_a the_o proportion_n of_o 31_o to_o 7._o partly_o also_o because_o you_o think_v as_o in_o the_o end_n of_o the_o twenty_o proposition_n that_o if_o the_o proportion_n of_o the_o numerator_n of_o these_o fraction_n to_o their_o denominator_n decrease_v eternal_o they_o shall_v so_o vanish_v at_o last_o as_o to_o leave_v the_o proportion_n of_o the_o sum_n of_o all_o the_o square_n to_o the_o sum_n of_o the_o great_a so_o often_o take_v that_o be_v a_o infinite_a number_n of_o time_n as_o one_o to_o three_o or_o the_o sum_n of_o the_o great_a to_o the_o sum_n of_o the_o increase_a square_n as_o three_o to_o one_o for_o which_o there_o be_v no_o more_o reason_n than_o for_o four_o to_o one_o or_o five_o to_o one_o or_o any_o other_o such_o proportion_n for_o if_o the_o proportion_n come_v eternal_o near_a and_o near_o to_o the_o subtriple_a they_o must_v needs_o also_o come_v near_o and_o near_o to_o subquadruple_a and_o you_o may_v as_o well_o conclude_v thence_o that_o the_o upper_a quantity_n shall_v be_v to_o the_o low_a quantity_n as_o one_o to_o four_o or_o as_o one_o to_o five_o etc._n etc._n as_o conclude_v they_o be_v as_o one_o to_o three_o you_o can_v see_v without_o admonition_n what_o effect_n this_o false_a ground_n of_o you_o will_v produce_v in_o the_o whole_a structure_n of_o your_o arithmetica_fw-la infinitorum_fw-la and_o how_o it_o make_v all_o that_o you_o have_v say_v unto_o the_o end_n of_o your_o thirty-eighth_a proposition_n undemonstrate_v and_o much_o of_o it_o false_a the_o thirty-nineth_a be_v this_o other_o lemma_n in_o a_o series_n of_o quantity_n begin_v with_o a_o point_n or_o cypher_n and_o proceed_v according_a to_o the_o series_n of_o the_o cubique_a number_n as_o 0._o 1._o 8._o 27._o 64_o etc._n etc._n to_o find_v the_o proportion_n of_o the_o sum_n of_o the_o cube_n to_o the_o sum_n of_o the_o great_a cube_n so_o many_o time_n take_v as_o there_o be_v term_n and_o you_o conclude_v that_o they_o have_v the_o proportion_n of_o 1_o to_o 4_o which_o be_v false_a let_v the_o first_o series_n be_v of_o three_o term_n subscribe_v with_o the_o great_a the_o sum_n of_o the_o cube_n be_v nine_o the_o sum_n of_o all_o the_o great_a be_v 24_o a_o quarter_n whereof_o be_v 6._o but_o 9_o be_v great_a than_o 6_o by_o three_o unity_n a_o unity_n be_v something_o let_v it_o be_v therefore_o a._n therefore_o the_o row_n of_o cube_n be_v great_a than_o a_o quarter_n of_o three_o time_n eight_o by_o three_o a._n again_o let_v the_o series_n have_v four_o term_n as_o the_o sum_n of_o the_o cube_n be_v 36_o a_o quarter_n of_o the_o sum_n of_o all_o the_o great_a be_v twenty-seven_o but_o thirty-six_a be_v great_a than_o twenty-seven_o by_o nine_a that_o be_v by_z 9_o a._n the_o excess_n therefore_o of_o the_o sum_n of_o the_o cube_n above_o the_o four_o part_n of_o the_o sum_n of_o all_o the_o great_a be_v increase_v by_o the_o increase_n of_o the_o number_n of_o term_n again_o let_v the_o term_n be_v five_o as_o the_o sum_n of_o the_o cube_n be_v one_o hundred_o the_o sum_n of_o all_o the_o great_a three_o hundred_o and_o twenty_o a_o quarter_n whereof_o be_v eighty_o but_o one_o hundred_o be_v great_a than_o eighty_o by_o twenty_o that_o be_v by_o 20_o a._n so_o you_o see_v that_o this_o lemma_n also_o be_v false_a and_o yet_o there_o be_v ground_v upon_o it_o all_o that_o which_o you_o have_v of_o compare_v parabolas_fw-la and_o paraboloeides_n with_o the_o parallellogram_n wherein_o they_o be_v accommodate_v and_o therefore_o though_o it_o be_v true_a that_o the_o parabola_fw-la be_v â…”_n and_o the_o cubicall_a paraboloeides_n ¾_n of_o their_o parallellogram_n respective_o '_o yet_o it_o be_v more_o than_o you_o be_v certain_a of_o when_o you_o refer_v i_o for_o the_o learning_n of_o geometry_n to_o this_o book_n of_o you_o beside_o any_o man_n may_v perceive_v that_o without_o these_o two_o lemma_n which_o be_v mingle_v with_o all_o your_o compound_a series_n with_o their_o excess_n there_o be_v nothing_o demonstrate_v to_o the_o end_n of_o your_o book_n which_o to_o prosecute_v particular_o be_v but_o a_o vain_a expense_n of_o time_n true_o be_v it_o not_o that_o i_o must_v defend_v my_o reputation_n i_o shall_v not_o have_v show_v the_o world_n how_o little_a there_o be_v of_o sound_a doctrine_n in_o any_o of_o your_o book_n for_o when_o i_o think_v how_o deject_a you_o will_v be_v for_o the_o future_a and_o how_o the_o grief_n of_o so_o much_o time_n irrecoverable_o lose_v together_o with_o the_o conscience_n of_o take_v so_o great_a a_o stipend_n for_o mis-teaching_a the_o young_a man_n of_o the_o university_n &_o the_o consideration_n of_o how_o much_o your_o friend_n will_v be_v ashamed_a of_o you_o will_v accompany_v you_o for_o the_o rest_n of_o your_o life_n i_o have_v more_o compassion_n for_o you_o then_o you_o have_v deserve_v your_o treatise_n of_o the_o angle_n of_o contact_n i_o have_v before_o confute_v in_o a_o very_a few_o leaf_n and_o for_o that_o of_o your_o conique_a section_n it_o be_v so_o cover_v over_o with_o the_o scab_n of_o symbol_n that_o i_o have_v not_o the_o patience_n to_o examine_v whether_o it_o be_v well_o or_o ill_o demonstrate_v yet_o i_o observe_v thus_o much_o that_o you_o find_v a_o tangent_fw-la to_o a_o point_n give_v in_o the_o section_n by_o a_o diameter_n give_v and_o in_o the_o next_o chapter_n after_o you_o teach_v the_o find_n of_o a_o diameter_n which_o be_v not_o artificial_o do_v i_o observe_v also_o that_o you_o call_v the_o parameter_n a_o imaginary_a line_n as_o if_o the_o place_n thereof_o be_v less_o determine_v than_o the_o diameter_n itself_o and_o than_o you_o take_v a_o mean_a proportional_a between_o the_o intercept_a diameter_n and_o its_o contiguous_a ordinate_a line_n to_o find_v it_o and_o it_o be_v true_a you_o find_v itâ—_n but_o the_o parameter_n have_v a_o determine_a quantity_n to_o be_v find_v without_o take_v a_o mean_a proportional_a for_o the_o diameter_n and_o half_a the_o section_n be_v give_v draw_v a_o tangent_fw-la through_o the_o vertex_fw-la and_o divide_v the_o angle_n in_o the_o midst_n which_o be_v make_v by_o the_o diameter_n and_o tangent_fw-la the_o line_n that_o so_o divide_v the_o angle_n will_v cut_v the_o crooked_a line_n from_o 〈◊〉_d intersection_n draw_v a_o line_n if_o it_o be_v a_o parabola_fw-la parallel_n to_o the_o diameter_n and_o that_o line_n shall_v cut_v off_o in_o the_o tangent_fw-la from_o the_o vertex_fw-la the_o parameter_n seek_v but_o if_o the_o section_n be_v a_o eâ—lipsis_n or_o a_o hyperbole_n you_o may_v use_v the_o same_o method_n save_v that_o the_o line_n draw_v from_o the_o intersection_n must_v not_o be_v parallel_n but_o must_v pass_v through_o the_o end_n of_o the_o transverse_a diameter_n and_o then_o also_o it_o shall_v cut_v off_o a_o part_n of_o the_o tangent_fw-la which_o measure_v from_o the_o vertex_fw-la be_v the_o parameter_n so_o that_o there_o be_v no_o more_o reason_n to_o call_v the_o parameter_n a_o imaginary_a line_n than_o the_o diameter_n last_o i_o observe_v that_o in_o all_o this_o your_o new_a method_n of_o conique_n you_o show_v not_o how_o to_o find_v the_o burn_a point_n which_o writer_n call_v the_o foci_fw-la and_o umbilici_fw-la of_o the_o section_n which_o be_v of_o all_o other_o thing_n belong_v to_o the_o conique_n most_o useful_a in_o philosophy_n why_o therefore_o be_v they_o not_o as_o worthy_a of_o your_o pain_n as_o the_o rest_n for_o the_o rest_n also_o have_v already_o be_v demonstrate_v by_o other_o you_o know_v the_o focus_n of_o the_o parabola_fw-la be_v in_o the_o axis_n distant_a from_o the_o vertex_fw-la a_o quarter_n of_o the_o parameter_n know_v also_o that_o the_o focus_n of_o a_o hyperbole_n be_v in_o the_o axis_n distant_a from_o the_o vertex_fw-la as_o much_o as_o the_o hypotenusall_n of_o a_o rectangled_a triangle_n who_o one_o side_n be_v half_a the_o transverse_n axis_n the_o other_o side_n half_o the_o mean_a proportional_a between_o the_o whole_a transverse_n axis_n and_o the_o parameter_n be_v great_a than_o half_a the_o transverse_n axis_n the_o cause_n why_o you_o have_v perform_v nothing_o in_o any_o of_o your_o book_n save_v that_o in_o your_o elenâ—…hus_o you_o have_v spy_v a_o few_o negligence_n of_o i_o which_o i_o need_v not_o be_v ashamed_a of_o be_v this_o that_o you_o understand_v not_o what_o be_v quantity_n line_n superâ—…ies_n angle_n and_o proportion_n without_o which_o you_o can_v have_v the_o science_n
suppose_v that_o the_o crock_v line_n ab_fw-la in_o the_o seven_o figure_n be_v not_o a_o arch_n of_o a_o circle_n do_v you_o think_v that_o the_o angle_n which_o it_o make_v with_o the_o subtense_n ab_fw-la at_o the_o point_n a_o and_o b_o must_v needs_o be_v equal_a or_o if_o they_o be_v not_o do_v the_o excess_n of_o the_o superficies_n of_o the_o circle_n upon_o ad_fw-la above_o the_o superficies_n of_o the_o cone_n or_o the_o exceis_n of_o the_o superficies_n of_o the_o portion_n of_o the_o conoeides_n above_o the_o superficies_n of_o the_o same_o cone_n consist_v in_o the_o angle_n dab_n oâ—_n rather_o in_o the_o â—…tude_n of_o the_o two_o unequal_a angle_n dab_n and_o aba_n you_o shall_v have_v draw_v some_o other_o crooked_a line_n and_o make_v tangent_n to_o it_o through_o a_o and_o b_o and_o you_o will_v present_o have_v see_v your_o error_n see_v how_o you_o can_v answer_v this_o for_o if_o this_o demonstration_n of_o minestand_v firm_a i_o may_v be_v bold_a to_o say_v though_o the_o same_o be_v well_o demonstrate_v by_o arâ—…_n that_o this_o way_n of_o i_o be_v more_o natural_a as_o proceed_v immediate_o from_o the_o natural_a efficient_a cause_n of_o the_o effect_n contain_v in_o the_o conclusion_n and_o beside_o more_o brief_a and_o more_o easy_a to_o be_v follow_v by_o the_o fancy_n of_o the_o reader_n to_o the_o fourteen_o article_n you_o say_v that_o i_o commit_v a_o circle_n in_o that_o i_o require_v in_o the_o four_o article_n the_o find_n of_o two_o mean_v proportional_n and_o come_v not_o till_o now_o to_o show_v how_o it_o be_v to_o be_v do_v nor_o now_o neither_o but_o in_o the_o mean_a time_n you_o commit_v two_o mistake_n in_o say_v so_o the_o place_n cite_v by_o you_o in_o the_o four_o article_n be_v in_o the_o latin_a pag._n 149._o lin_n 9_o in_o the_o english_a pag._n 188._o lin._n 3._o let_v any_o reader_n judge_n whether_o that_o be_v a_o require_v it_o or_o a_o suppose_v it_o to_o be_v do_v this_o be_v your_o first_o mistake_n the_o second_o be_v that_o in_o this_o place_n the_o preposition_n itself_o which_o be_v if_o those_o deficient_a figure_n can_v be_v describe_v in_o a_o parallelogram_n exquisite_o there_o may_v be_v find_v thereby_o between_o any_o two_o line_n give_v as_o many_o mean_a proportional_n as_o one_o will_v be_v a_o theorem_a upon_o supposition_n of_o these_o crooked_a line_n exquisite_o draw_v but_o you_o take_v it_o for_o a_o problem_n and_o proceed_v in_o that_o error_n you_o undertake_v the_o invention_n of_o two_o mean_v proportional_n use_v therein_o my_o first_o figure_n which_o be_v of_o the_o same_o construction_n with_o the_o eight_o that_o belong_v to_o this_o fourteen_o article_n your_o construction_n be_v let_v there_o be_v take_v in_o the_o diameter_n ca_n figure_n 1_o the_o two_o give_v line_n or_o two_o other_o proportional_a to_o they_o as_o ch_z cg_n and_o their_o â—â—â—nate_a line_n hf_n ge_n which_o by_z construction_n be_v in_o subtriplicate_v proportion_n of_o the_o intercept_a diameter_n these_o line_n will_v show_v the_o proportion_n which_o those_o four_o proportional_n be_v to_o 〈◊〉_d but_o how_o will_v you_o find_v the_o length_n of_o hf_n or_o ge_n the_o ordinate_a line_n will_v you_o not_o do_v it_o by_o so_o draw_v the_o crooked_a line_n cfe_n as_o it_o may_v paâ—s_v through_o both_o the_o point_n f_o and_o e_o you_o may_v make_v it_o pass_v through_o one_o of_o they_o but_o to_o make_v it_o pass_v through_o the_o other_o you_o must_v find_v two_o mean_a proportional_n between_o gk_n and_o gl_n or_o between_o he_o and_o hp_n which_o you_o can_v do_v unless_o the_o crooked_a line_n be_v exact_o draw_v which_o it_o can_v be_v by_o the_o geometry_n of_o plunes_n go_v show_v this_o demonstration_n of_o you_o to_o orontius_n and_o see_v what_o he_o will_v say_v to_o it_o i_o be_o now_o come_v to_o a_o end_n of_o your_o objection_n to_o the_o seventeen_o chapter_n where_o you_o have_v a_o epiphonema_v not_o to_o be_v pass_v over_o in_o silence_n but_o because_o you_o pretend_v to_o the_o dâ—…tration_n of_o some_o of_o these_o proposition_n by_o another_o method_n in_o your_o arithmetica_fw-la infinitorum_fw-la i_o shall_v first_o try_v whether_o you_o be_v able_a to_o defend_v those_o demonstration_n as_o well_o as_o i_o have_v do_v they_o of_o i_o by_o the_o method_n of_o motion_n the_o first_o proposition_n of_o your_o arithmetica_fw-la infinitorum_fw-la be_v this_o lemma_n in_o a_o series_n or_o row_n of_o quantity_n arith_n â—tically_o proportional_a begin_v at_o a_o poiâ—t_n or_o cypâ—â—r_n as_o 0_o 1_o 2_o 3_o 4_o etc._n etc._n to_o find_v the_o proportion_n of_o the_o aggregate_v of_o they_o all_o to_z the_o aggregate_v of_o so_o many_o time_n the_o great_a as_o there_o be_v term_n this_o be_v to_o be_v do_v by_o multiply_v the_o great_a into_o half_n the_o number_n of_o the_o term_n the_o demonstration_n be_v easy_a but_o how_o do_v you_o demonstrate_v the_o same_o the_o most_o simple_a way_n say_v you_o of_o find_v this_o and_o some_o other_o problem_n it_o to_o do_v the_o thing_n itself_o a_o little_a way_n and_o to_o observe_v and_o compare_v the_o appear_a proportion_n and_o then_o by_o induction_n to_o conclude_v it_o universal_o egregious_a logician_n and_o geometrician_n that_o think_v a_o induction_n without_o a_o numeration_n of_o all_o the_o particular_n sufficient_a to_o infer_v a_o conclusion_n universal_a and_o fit_a to_o be_v receive_v for_o a_o geometrical_a demonstration_n but_o why_o do_v you_o limit_v it_o to_o the_o natural_a consecution_n of_o the_o number_n 0_o 1_o 2_o 3_o 4_o &_o c_o be_v it_o not_o also_o true_a in_o these_o number_n 0_o 2_o 4_o 6_o etc._n etc._n or_o in_o these_o 0_o 7_o 14_o 21_o &_o c_o or_o in_o any_o number_n where_o the_o difference_n of_o nothing_o and_o the_o first_o number_n be_v equal_a to_o the_o difference_n between_o the_o first_o and_o second_o and_o between_o the_o second_o and_o three_o &_o c_o again_o be_v not_o these_o quantity_n 1_o 3_o 5_o 7_o etc._n etc._n in_o continual_a proportion_n arithmaticall_a and_o if_o you_o put_v before_o they_o a_o cypher_n thus_o 0_o 1_o 3_o 5_o 7_o do_v you_o think_v that_o the_o sum_n of_o they_o be_v equal_a to_o the_o half_a of_o five_o time_n seven_o therefore_o though_o your_o lemma_n be_v true_a and_o by_o i_o chap._n 13._o art_n 5._o demonstrate_v yet_o you_o do_v not_o know_v why_o it_o be_v true_a which_o also_o appear_v most_o evident_o in_o the_o first_o proposition_n of_o your_o conique-section_n where_o first_o you_o have_v this_o that_o a_o parallelogram_n who_o altitude_n be_v infinite_o little_a that_o be_v to_o say_v none_o be_v scarce_o any_o thing_n else_o but_o a_o line_n be_v this_o the_o language_n of_o geometry_n how_o do_v you_o determine_v this_o word_n scarce_o the_o least_o altitude_n be_v somewhat_o or_o nothing_o if_o somewhat_o than_o the_o first_o character_n of_o your_o arithmetical_a progression_n must_v not_o be_v a_o cypher_n and_o consequent_o the_o first_o eighteen_o proposition_n of_o this_o your_o arithmetica_fw-la infinitorum_fw-la be_v all_o naught_o if_o nothing_o than_o your_o whole_a figure_n be_v without_o altitude_n and_o consequent_o your_o understanding_n naught_o again_o in_o the_o same_o proposition_n you_o say_v thus_o we_o will_v sometime_o call_v those_o parallellogram_n rather_o by_o the_o name_n of_o line_n then_o of_o parallellogram_n at_o least_o when_o there_o be_v no_o consideration_n of_o a_o determinate_a altitude_n but_o where_o there_o be_v a_o consideration_n of_o a_o determinate_a altitude_n which_o will_v happen_v sometime_o there_o that_o little_a altitude_n shall_v be_v so_o far_o consider_v as_o that_o be_v infinite_o multiply_v it_o may_v be_v equal_a to_o the_o altitude_n of_o the_o whole_a figure_n see_v here_o in_o what_o a_o confusion_n you_o be_v when_o you_o resist_v the_o truth_n when_o you_o consider_v no_o determinate_a altitude_n that_o be_v no_o quantity_n of_o altitude_n than_o you_o say_v your_o parallelogram_n shall_v be_v call_v a_o line_n but_o when_o the_o altitude_n be_v determine_v that_o be_v when_o it_o be_v quantity_n than_o you_o will_v call_v it_o a_o parallelogram_n be_v not_o this_o the_o very_a same_o doctrine_n which_o you_o so_o much_o wonder_n at_o and_o reprehend_v in_o i_o in_o your_o objection_n to_o my_o eight_o chapter_n and_o your_o word_n consider_v use_v as_o i_o use_v it_o it_o be_v very_o ugly_a in_o one_o that_o so_o bitter_o reprehend_v a_o doctrine_n in_o another_o to_o be_v drive_v upon_o the_o same_o himself_o by_o the_o force_n of_o truth_n when_o he_o think_v not_o on_o it_o again_o see_v you_o admit_v in_o any_o case_n those_o infinite_o little_a altitude_n to_o be_v quantity_n what_o need_v you_o this_o limitation_n of_o you_o so_o far_o forth_o as_o that_o by_o
be_v incongruous_a or_o a_o crooked_a and_o a_o straight_a line_n touch_v one_o another_o the_o contact_n be_v not_o in_o a_o line_n but_o only_o in_o one_o point_n and_o then_o your_o instance_n of_o a_o circle_n and_o a_o parabola_fw-la be_v a_o wilful_a cavil_n not_o befit_v a_o doctor_n if_o you_o either_o read_v they_o not_o or_o unstood_v they_o not_o it_o be_v your_o own_o fault_n in_o the_o rest_n that_o follow_v upon_o this_o article_n with_o your_o diagram_n there_o be_v nothing_o against_o i_o nor_o any_o thing_n of_o use_n novelty_n subtlety_n or_o learning_n at_o the_o seven_o article_n where_o i_o define_v both_o a_o angle_n simple_o so_o call_v and_o a_o angle_n of_o contingence_n by_o their_o several_a generation_n namely_o that_o the_o former_a be_v generate_v when_o two_o straight_a line_n be_v coincident_a and_o one_o of_o they_o be_v move_v and_o distract_v from_o the_o other_o by_o circular_a motion_n upon_o one_o common_a point_n rest_v etc._n etc._n you_o ask_v 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another_o yet_o they_o form_v no_o angle_n any_o where_o but_o in_o the_o very_a point_n of_o concourse_n you_o that_o deny_v a_o point_n to_o be_v any_o thing_n tell_v i_o how_o two_o nothing_o can_v form_v a_o angle_n or_o if_o the_o angle_n be_v not_o form_v neither_o before_o the_o concurrent_a line_n meet_v not_o in_o the_o point_n of_o concourse_n how_o can_v you_o apprehend_v that_o any_o angle_n can_v possible_o be_v frame_v but_o i_o wonder_v not_o at_o this_o absurdity_n because_o this_o whole_a treatise_n of_o you_o be_v but_o one_o absurdity_n continue_v from_o the_o begin_n to_o the_o end_n as_o shall_v then_o appear_v when_o i_o come_v to_o answer_v your_o objection_n to_o that_o which_o i_o have_v brief_o and_o full_o say_v of_o that_o subject_a in_o my_o 14._o chapter_n at_o the_o twelve_o article_n i_o confess_v your_o exception_n to_o my_o universal_a definition_n of_o parallel_n to_o be_v just_a though_o insolent_o set_v down_o for_o it_o be_v no_o fault_n of_o 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your_o geometry_n the_o same_o be_v observe_v also_o upon_o this_o place_n by_o one_o of_o the_o prime_a geometrician_n of_o paris_n and_o note_v in_o a_o letter_n to_o his_o friend_n in_o these_o word_n chap._n 14._o art_n 12._o the_o definition_n of_o parallel_n want_v somewhat_o to_o be_v supply_v and_o of_o the_o consectary_n he_o say_v it_o conclude_v not_o because_o it_o be_v ground_v on_o the_o definition_n of_o parallel_n true_o and_o severe_o enough_o though_o without_o any_o such_o word_n as_o savour_v of_o arrogance_n or_o of_o malice_n or_o of_o the_o clown_n at_o the_o thirteen_o article_n you_o recite_v the_o demonstration_n by_o which_o i_o prove_v the_o perimeter_n of_o two_o circle_n to_o be_v proportional_a to_o their_o semidiameter_n and_o with_o esto_fw-la fortasse_fw-la recte_fw-la omnino_fw-la noddy_v to_o the_o several_a part_n thereof_o you_o come_v at_o length_n to_o my_o last_o inference_n therefore_o by_o chap._n 13._o art_n 6._o the_o perimeter_n and_o 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