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n_o and_o let_v the_o inteâ—â—â—ction_n of_o the_o two_o line_n a_o c_o and_o d_o o_fw-fr be_v in_o r_o which_o be_v do_v the_o triangle_n m_o n_o t_o r_o c_o t_o will_v be_v like_a and_o equal_a therefore_o m_z n_o and_o r_o c_o be_v equal_a and_o â—…sequently_o the_o strait_a line_n i_o m_o t_o u_o shall_v pass_v through_o c._n divide_v therefore_o a_o c_o in_o the_o midst_n at_o it_o and_o s_o n_o in_o the_o midst_n at_o l_o and_o join_v to_o n_o l_o l_o the_o line_n l_o l_o t_o n_o and_o c_o t_o produce_v with_o all_o meet_a i_o 〈◊〉_d one_o and_o the_o same_o point_n of_o b_o s_o produce_v suppose_v at_o q._n therefore_o the_o point_n q_o be_v give_v by_o the_o two_o know_a point_n t_o and_o i_o the_o line_n draw_v from_o q_o through_o equal_a part_n of_o the_o sine_fw-la of_o the_o arch_n b_o i_o for_o example_n through_o the_o point_n p_o q_o r_o of_o the_o sine_fw-la m_o i_o shall_v cut_v off_o equal_a arch_n as_o b_o l_o l_o n_o n_o o_o o_o i._n and_o this_o be_v enough_o to_o 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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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semidiameter_n of_o circle_n be_v proportional_a which_o you_o deny_v and_o therefore_o deny_v because_o you_o say_v it_o follow_v by_o the_o same_o ratiocination_n that_o circle_n also_o and_o sphere_n be_v proportional_a to_o their_o semidiameter_n for_o the_o same_o distance_n you_o say_v of_o the_o perimeter_n from_o the_o centre_n which_o determine_v the_o magnitude_n of_o the_o semidiameter_n determine_v also_o the_o magnitude_n both_o of_o the_o circle_n and_o of_o the_o sphere_n you_o acknowledge_v that_o perimeter_n and_o semidiameter_n have_v the_o cause_n of_o their_o determination_n such_o as_o in_o equal_a time_n make_v equal_a space_n suppose_v now_o a_o sphere_n generate_v by_o the_o semidiameter_n whilst_o the_o semicircle_n be_v turn_v about_o there_o be_v but_o one_o radius_fw-la of_o a_o infinite_a number_n of_o radii_fw-la which_o describe_v a_o great_a circle_n all_o the_o
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who_o they_o be_v nevertheless_o that_o other_o man_n may_v see_v the_o spirit_n of_o a_o ambitious_a part_n of_o the_o clergy_n i_o will_v tell_v you_o where_o i_o read_v it_o it_o be_v in_o the_o prolegomena_n of_o lalovera_fw-fr a_o jesuit_n to_o his_o quadrature_n of_o the_o circle_n pag._n 13_o &_o 14_o in_o these_o worâ—s_n quamvis_fw-la circuli_fw-la tetragonismus_fw-la fit_a 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d possibilis_fw-la a_o taman_n etiam_fw-la sit_fw-la 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d hoc_fw-la est_fw-la post_fw-la adae_fw-la lapsum_fw-la homo_fw-la ejus_fw-la scientiam_fw-la absque_fw-la speciali_fw-la diâ—inae_fw-la gratiae_fw-la auxilio_fw-la possit_fw-la comparare_fw-la jure_fw-la merito_fw-la inquirunt_fw-la theologi_fw-la pronunciantque_fw-la hanc_fw-la veritatem_fw-la tanta_fw-la esse_fw-la caligine_fw-la involutam_fw-la ut_fw-la illam_fw-la videre_fw-la nemo_fw-la possit_fw-la nisi_fw-la ignorantiae_fw-la ex_fw-la primi_fw-la parentis_fw-la praevaricatione_fw-la propagatas_fw-la tenebras_fw-la indebitus_fw-la divina_fw-la lucis_fw-la radius_fw-la dissipet_fw-la quod_fw-la verissimum_fw-la esse_fw-la sentio_fw-la wherein_o i_o observe_v that_o he_o suppose_v he_o have_v find_v that_o quadrature_n will_v have_v we_o believe_v it_o be_v not_o by_o the_o ordinary_a and_o natural_a help_n of_o god_n whereby_o one_o man_n reason_v judge_v and_o remember_v better_a than_o another_o but_o by_o a_o special_a which_o must_v be_v a_o superturall_a help_v of_o god_n that_o he_o have_v give_v to_o he_o of_o the_o order_n of_o jesus_n above_o other_o that_o have_v attempt_v the_o same_o in_o vain_a insinuate_v thereby_o as_o handsome_o as_o he_o can_v a_o special_a love_n of_o god_n towards_o the_o jesuit_n but_o you_o take_v no_o notice_n of_o the_o word_n special_a will_v have_v man_n think_v i_o hold_v that_o humane_a science_n may_v be_v acquire_v without_o any_o help_n of_o god_n and_o thereupon_o proceed_v in_o a_o great_a deal_n of_o ill_a language_n to_o the_o end_n of_o your_o objection_n to_o this_o chapter_n but_o i_o shall_v take_v notice_n of_o your_o manner_n for_o altogether_o in_o my_o next_o lesson_n at_o the_o nineteen_o chapter_n you_o see_v not_o you_o say_v the_o method_n like_a enough_o in_o this_o chapter_n i_o consider_v not_o the_o cause_n of_o reflection_n which_o consist_v in_o the_o resistance_n of_o body_n natural_a but_o i_o consider_v the_o consequence_n arise_v from_o the_o supposition_n of_o the_o equality_n of_o the_o angle_n of_o reflection_n to_o that_o of_o incidence_n leave_v the_o cause_n both_o of_o reflection_n and_o of_o refraction_n to_o be_v handle_v together_o in_o the_o 24_o chapter_n which_o method_n think_v what_o you_o will_v i_o still_o think_v best_a second_o you_o say_v i_o define_v not_o here_o but_o many_o chapter_n after_o what_o a_o angle_n of_o incidence_n and_o what_o a_o angle_n of_o reflection_n be_v have_v you_o not_o be_v more_o hasty_a than_o diligent_a reader_n you_o have_v find_v that_o those_o definition_n of_o the_o angle_n of_o incidence_n and_o of_o reflection_n be_v here_o set_v down_o in_o the_o first_o article_n and_o not_o defer_v to_o the_o 24._o let_v not_o therefore_o your_o own_o oversight_n be_v any_o more_o bring_v in_o for_o a_o objection_n three_o you_o say_v there_o be_v no_o great_a difficulty_n in_o the_o business_n of_o this_o chapter_n it_o may_v be_v so_o now_o it_o be_v down_o but_o before_o it_o be_v do_v i_o doubt_v not_o but_o you_o that_o be_v a_o professor_n will_v have_v do_v the_o same_o as_o well_o as_o you_o have_v do_v that_o of_o the_o angle_n of_o contact_n or_o the_o business_n of_o your_o arithmetica_fw-la infinitorum_fw-la but_o what_o a_o novice_n in_o geometry_n will_v have_v do_v i_o can_v tell_v to_o the_o three_o four_o and_o five_o article_n you_o object_v a_o want_n of_o determination_n and_o show_v it_o by_o instance_n as_o to_o the_o three_o article_n but_o what_o those_o determination_n shall_v be_v you_o
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
multiplication_n they_o may_v be_v make_v equal_a to_o the_o altitude_n of_o the_o whole_a figure_n may_v not_o the_o half_a the_o three_o the_o four_o or_o the_o firth_n part_n etc._n etc._n be_v make_v equal_a to_o the_o whole_a by_o multiplication_n why_o can_v you_o not_o have_v say_v plain_o so_o far_o forth_o as_o that_o every_o one_o of_o those_o infinite_o little_a altitude_n be_v not_o only_a something_o but_o a_o aliquoâ—part_n of_o the_o wholâ—_n â—o_o you_o will_v have_v a_o infinite_o little_a altitude_n that_o be_v to_o say_v a_o point_n to_o be_v both_o nothing_o and_o something_o and_o a_o aliquot_fw-la part_n and_o all_o this_o proceed_v from_o not_o understand_v the_o ground_n of_o your_o profession_n well_o the_o lemma_n be_v true_a let_v we_o see_v the_o theorem_n you_o draw_v from_o it_o the_o first_o be_v pag._n 3._o that_o a_o triangle_n to_o a_o parallelogram_n of_o equal_a base_a and_o altitude_n be_v as_o one_o to_o two_o the_o conclusion_n be_v true_a but_o how_o know_v you_o that_o because_o say_v you_o the_o triangle_n consist_v as_o it_o be_v as_o be_v be_v be_v no_o phrase_n of_o a_o geometrician_n of_o a_o infinite_a number_n of_o straâ—ght_a parallel_n line_n do_v it_o so_o then_o by_o your_o own_o doctrine_n which_o be_v that_o line_n have_v no_o breadth_n the_o altitude_n of_o your_o triangle_n consist_v of_o a_o infinite_a number_n of_o no_o altitude_n that_o be_v of_o a_o infinite_a number_n of_o nothing_o and_o consequent_o the_o area_n of_o your_o triangle_n have_v no_o quantity_n if_o you_o say_v that_o by_o the_o parallel_n you_o mean_v infinite_o little_a parallellogram_n you_o be_v never_o the_o better_a for_o if_o infinite_o little_a either_o they_o be_v nothing_o or_o if_o somewhat_o yet_o see_v that_o no_o two_o side_n of_o a_o triangle_n be_v parallel_n those_o parallel_n can_v be_v parallellogram_n i_o see_v they_o may_v be_v count_v for_o parallellogram_n by_o not_o consider_v the_o quantity_n of_o their_o altitude_n in_o the_o demonstration_n but_o you_o be_v bar_v of_o that_o plea_n by_o your_o spiteful_a argue_v against_o it_o in_o your_o elenchus_n therefore_o this_o three_o proposition_n and_o with_o it_o the_o four_o be_v undemonstrate_v your_o five_o proposition_n be_v the_o spirall_n line_n be_v equal_a to_o half_a the_o circle_n of_o the_o first_o revolution_n but_o what_o spirall_n line_n we_o shall_v understand_v that_o by_o your_o construction_n which_o be_v this_o the_o straight_a line_n ma_n in_o your_o figure_n which_o i_o have_v place_v at_o the_o end_n of_o the_o five_o lesson_n turn_v round_o the_o point_n m_o remain_v unmoved_a be_v suppose_v to_o describe_v with_o its_o point_n a_o the_o circle_n aoa_n whilst_o some_o point_n in_o the_o same_o ma_n whilst_o it_o go_v about_o be_v suppose_v to_o be_v move_v uniform_o from_o m_n to_o a_o describe_v the_o spirall_n line_n this_o therefore_o be_v the_o spirall_n line_n of_o archimedes_n and_o your_o proposition_n affirm_v it_o to_o be_v equal_a to_o the_o half_a of_o the_o circle_n aoa_n which_o you_o perceive_v not_o long_o after_o to_o be_v false_a but_o think_v it_o have_v be_v true_a you_o go_v about_o to_o prove_v it_o by_o inscribe_v in_o the_o circle_n a_o infinite_a multitude_n of_o equal_a angle_n and_o consequent_o a_o infinite_a number_n of_o sector_n who_o arch_n will_v therefore_o be_v in_o arithmetical_a proportion_n which_o be_v true_a and_o the_o aggregate_v of_o those_o arch_n equal_a to_o half_a the_o circumference_n aoa_n which_o be_v true_a also_o and_o thence_o you_o conclude_v that_o the_o spirall_n line_n be_v equal_a to_o half_a the_o circumference_n of_o the_o circle_n aoa_n which_o be_v false_a for_o the_o aggregate_v of_o that_o infinite_a number_n of_o infinite_o little_a arch_n be_v not_o the_o spirall_n line_n make_v by_o your_o construction_n see_v by_o your_o construction_n the_o line_n you_o make_v be_v manifest_o the_o spiral_n of_o archimedes_n whereas_o no_o number_n though_o infinite_a of_o arch_n of_o circle_n how_o little_a soever_o be_v any_o kind_n of_o spirall_n at_o all_o and_o though_o you_o call_v it_o a_o spirall_n that_o be_v but_o a_o patch_n to_o cover_v your_o fault_n and_o deceive_v no_o man_n but_o yourself_o beside_o you_o see_v not_o how_o absurd_a it_o be_v for_o you_o that_o hold_v a_o point_n to_o be_v absolute_o nothing_o to_o make_v a_o infinite_a number_n of_o equal_a angle_n the_o radius_fw-la increase_v as_o the_o number_n of_o angles_n increase_v and_o then_o to_o say_v that_o the_o arch_n of_o the_o sector_n who_o angle_n they_o be_v be_v as_o 0_o 1_o 2_o 3_o 4_o etc._n etc._n for_o you_o make_v the_o first_o angle_n 0_o and_o all_o the_o rest_n equal_a to_o it_o and_o so_o make_v 0_o 0_o 0_o 0_o 0_o etc._n etc._n to_o be_v the_o same_o progression_n with_o 0_o 1_o 2_o 3_o 4_o etc._n etc._n the_o influence_n of_o this_o absurdity_n reach_v to_o the_o end_n of_o the_o eighteen_o proposition_n so_o many_o be_v therefore_o false_a or_o nothing_o worth_a and_o you_o need_v not_o to_o wonder_v that_o the_o doctrine_n contain_v in_o they_o be_v omit_v by_o archimedes_n who_o never_o be_v so_o senseless_a as_o to_o think_v a_o spiâ—â—ll_n line_n be_v compound_v of_o arch_n of_o circle_n your_o nineteen_o proposition_n be_v this_o other_o lemma_n in_o a_o series_n or_o a_o row_n of_o quantity_n begin_v from_o a_o point_n or_o cypher_n and_o proceed_v according_a to_o the_o order_n of_o the_o square_a number_n as_o 0_o 1_o 4_o 9_o 16_o etc._n etc._n to_o find_v what_o proportion_n the_o whole_a series_n have_v to_o so_o many_o time_n the_o great_a and_o you_o conclude_v the_o proportion_n to_o be_v that_o of_o 1_o to_o 3._o which_o be_v false_a as_o you_o shall_v present_o see_v first_o let_v the_o series_n of_o square_n with_o the_o prefix_a cypher_n and_o under_o every_o one_o of_o they_o the_o great_a 4_o be_v and_o you_o have_v for_o the_o sum_n of_o the_o square_n 5_o and_o for_o thrice_o the_o great_a 12_o the_o 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be_v 30_o the_o sum_n of_o all_o the_o great_a will_v be_v 80._o the_o three_o part_n whereof_o be_v 26_o â…”_n but_o 30_o be_v great_a than_o 26â…“_n by_o 3â…“_n that_o be_v by_o three_o twelfth_n of_o twelve_o and_o â…“_n of_o a_o twelve_o that_o be_v by_o 3â…“_n a._n likewise_o in_o the_o series_n continue_v to_o six_o place_n with_o the_o great_a six_o time_n subscribe_v as_o the_o sum_n of_o the_o square_n be_v 55_o and_o the_o sum_n of_o the_o great_a six_o time_n take_v be_v 150_o the_o three_o part_n whereof_o be_v 50._o but_o 55_o be_v great_a than_o 50_o by_o 5_o that_o be_v by_o five_o twelfth_n of_o 12_o that_o be_v by_o 5._o a._n and_o so_o continual_o as_o the_o row_n grow_v long_o the_o excess_n also_o of_o the_o aggregate_v of_o the_o square_n above_o the_o three_o part_n of_o the_o aggregate_v of_o so_o many_o time_n the_o great_a square_n grow_v great_a and_o consequent_o if_o the_o number_n of_o the_o square_n be_v infinite_a their_o sum_n will_v be_v so_o far_o from_o be_v equal_a to_o the_o three_o part_n of_o the_o aggregate_v of_o the_o great_a as_o often_o take_v as_o that_o it_o will_v be_v great_a than_o it_o by_o a_o quantity_n great_a than_o any_o that_o can_v be_v give_v or_o name_v that_o which_o deceive_v you_o be_v partly_o this_o that_o you_o think_v as_o you_o do_v in_o your_o elenchus_n that_o these_o fraction_n etc._n etc._n be_v proportion_n as_o if_o 1_o 12_o be_v the_o proportion_n of_o one_o to_o twelve_o and_o consequent_o 2_o 12_o double_a the_o proportion_n of_o one_o to_o twelve_o which_o be_v as_o unintelligible_a as_o school-divinity_n and_o i_o assure_v you_o far_o from_o the_o meaning_n of_o mr._n ougthr_v in_o the_o six_o
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 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determine_v not_o because_o you_o can_v not_o the_o word_n in_o the_o three_o article_n be_v first_o these_o if_o there_o fall_v two_o straight_a line_n parallel_v etc._n etc._n which_o be_v too_o general_a it_o shall_v be_v if_o there_o fall_v the_o same_o way_n two_o straight_a line_n parallel_v etc._n etc._n next_o these_o their_o reflect_v line_n produce_v inward_o shall_v make_v a_o angle_n etc._n etc._n this_o also_o be_v too_o general_a i_o shall_v have_v say_v their_o reflect_v line_n produce_v inward_o if_o they_o meet_v within_o shall_v make_v a_o angle_n etc._n etc._n which_o do_v both_o this_o article_n and_o the_o 4_o and_o 5_o be_v full_o demonstrate_v and_o without_o it_o a_o intelligent_a reader_n have_v be_v satisfy_v supply_v the_o want_n himself_o by_o the_o construction_n to_o the_o eight_o you_o object_n only_o the_o two_o great_a length_n and_o labour_n of_o it_o because_o you_o can_v do_v it_o a_o short_a way_n perhaps_o so_o now_o as_o be_v easy_a to_o shorten_v 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shall_v not_o have_v think_v fit_a to_o reprehend_v this_o fault_n upon_o this_o occasion_n in_o a_o english_a man_n nor_o to_o take_v notice_n of_o it_o but_o that_o i_o find_v you_o in_o some_o place_n nibble_v but_o causeless_o at_o my_o latin_a concern_v the_o twenty_o chapter_n before_o i_o answer_v to_o the_o objection_n against_o the_o proposition_n themselves_o i_o must_v answer_v to_o the_o exception_n you_o first_o take_v to_o these_o word_n of_o i_o quae_fw-la de_fw-fr dimensione_n circuli_fw-la &_o angulorum_fw-la pronuntiata_fw-la sunt_fw-la tanquam_fw-la exacté_fw-fr inuenta_fw-la accipiat_fw-la lector_fw-la â—â—nquam_fw-la dicta_fw-la problematice_fw-la to_o which_o you_o say_v thus_o we_o be_v wont_n in_o geometry_n to_o call_v some_o proposition_n theorem_n other_o problem_n etc._n etc._n of_o which_o 〈◊〉_d theorem_a be_v that_o wherein_o some_o assertion_n be_v propouâ—nded_v to_o be_v prove_v a_o problem_n that_o wherein_o something_o be_v command_v to_o be_v do_v do_v you_o mean_v to_o be_v do_v and_o not_o prove_v by_o your_o favour_n a_o problem_n in_o all_o ancient_a writer_n signify_v no_o more_o but_o a_o proposition_n utter_v to_o the_o end_n to_o have_v it_o by_o they_o to_o who_o it_o be_v utter_v examine_v whether_o it_o be_v true_a or_o not_o true_a faiâ—able_a or_o not_o faisable_v and_o differ_v not_o among_o geometrician_n from_o a_o theorem_a but_o in_o the_o manner_n of_o propound_v for_o this_o proposition_n to_o make_v a_o equilateral_a triangle_n so_o propound_v they_o call_v a_o problem_n but_o if_o propound_v thus_o if_o upon_o the_o end_n of_o a_o straight_a line_n give_v be_v describe_v two_o circle_n who_o radius_fw-la be_v the_o same_o straight_a line_n and_o there_o be_v draw_v from_o the_o intersection_n of_o the_o circle_n to_o their_o two_o centre_n two_o straight_a line_n there_o will_v be_v make_v a_o equilateral_a triangle_n than_o they_o call_v it_o a_o theorem_a and_o yet_o the_o proposition_n be_v the_o same_o therefore_o these_o word_n accipiat_fw-la lector_fw-la tanquam_fw-la dicta_fw-la problematiâ—ey_n signify_v plain_o this_o that_o i_o will_v have_v the_o reader_n take_v for_o propound_v to_o he_o to_o examine_v whether_o from_o my_o construction_n the_o quadrature_n of_o the_o circle_n can_v be_v true_o infer_v or_o not_o and_o this_o be_v not_o to_o bid_v he_o as_o you_o interpret_v it_o to_o square_v the_o circle_n and_o if_o you_o believe_v that_o problematicey_n signify_v probable_o you_o have_v be_v very_o negligent_a in_o observe_v the_o sense_n of_o the_o ancient_a gâ—eek_a philosopher_n in_o the_o word_n problem_n therefore_o your_o solemus_fw-la in_o geometria_n etc._n etc._n be_v nothing_o to_o the_o pu_fw-fr pose_n nor_o have_v it_o be_v though_o you_o have_v speak_v more_o proper_o and_o say_v solâ—nt_n leave_v out_o yourselves_o my_o first_o article_n have_v this_o title_n from_o a_o false_a supposition_n a_o false_a quadrature_n of_o the_o circle_n see_v therefore_o you_o be_v resolve_v to_o show_v where_o i_o err_v you_o shall_v have_v prove_v either_o that_o the_o supposition_n be_v true_a and_o the_o conclusion_n false_o infer_v or_o contrary_o that_o though_o the_o supposition_n be_v false_a yet_o the_o conclusion_n be_v true_a for_o else_o you_o object_v nothing_o to_o my_o geometry_n but_o only_o to_o my_o judgement_n in_o think_v fit_a to_o publish_v it_o which_o nevertheless_o you_o can_v just_o do_v see_v it_o be_v likely_a to_o give_v occasion_n to_o ingenuous_a man_n they_o practise_v of_o it_o be_v so_o accurate_a to_o sense_n to_o inquire_v wherein_o the_o fallacy_n do_v consist_v and_o for_o the_o problem_n as_o it_o be_v first_o print_v but_o never_o publish_v and_o consequent_o aught_o to_o have_v pass_v for_o a_o private_a paper_n steal_v out_o of_o my_o study_n your_o public_a object_v against_o it_o in_o the_o opinion_n of_o all_o man_n that_o have_v converse_v so_o much_o with_o honest_a company_n as_o to_o know_v what_o belong_v to_o civil_a conversation_n be_v sufficient_o barbarous_a in_o divine_n and_o see_v you_o know_v i_o have_v reject_v that_o proposition_n it_o be_v but_o a_o poor_a ambition_n to_o take_v wing_n as_o you_o think_v to_o do_v like_a beetle_n from_o my_o egestion_n but_o let_v that_o be_v as_o it_o will_v you_o will_v think_v strange_a now_o i_o shall_v resume_v and_o make_v good_a at_o least_o against_o your_o objection_n that_o very_o same_o proposition_n so_o much_o of_o the_o figure_n as_o be_v needful_a you_o will_v find_v note_v with_o the_o same_o letter_n and_o place_v at_o the_o end_n of_o this_o 5_o lesson_n wherein_o let_v b_o ay_o be_v a_o arch_n not_o great_a than_o the_o radius_fw-la of_o the_o circle_n and_o divide_v into_o four_o equal_a part_n in_o l_o n_o o._n draw_v s_o n_o the_o sine_fw-la of_o the_o arch_n b_o n_o and_o produce_v it_o to_o t_o so_o as_o saint_n t_o be_v double_a to_o s_o n_o that_o be_v equal_a to_o the_o chord_n b_o i._n draw_v likewise_o a_o l_o the_o sine_fw-la of_o the_o arch_n b_o l_o and_o produce_v it_o to_o c_o so_o as_o a_o c_o be_v quadruple_a to_o a_o l_o that_o be_v equal_a to_o the_o two_o chord_n b_o n_o n_o i._n upon_o the_o centre_n n_o with_o the_o radius_fw-la n_o i_o draw_v the_o aâ—ch_n i_o d_o cut_v b_o u_o the_o tangent_fw-la in_o d._n then_o will_n b_o n_o produce_v cut_v the_o arch_n i_o d_o in_o the_o midst_n at_o o._n in_o the_o line_n b_o s_o produce_v take_v saint_n b_o equal_a to_o b_o saint_n then_o draw_v and_o produce_v b_o n_o and_o it_o will_v fall_v on_o the_o point_n d._n and_o b_o d_o s_o t_o will_v be_v equal_a and_o d_o t_o join_v and_o produce_v will_v fall_v upon_o o_o the_o midst_n of_o the_o arch_n i_o d._n join_v i_o t_n and_o produce_v it_o to_o the_o tangent_fw-la b_o u_o in_o u._n i_o say_v that_o the_o strait_a line_n i_o t_o u_o shall_v pass_v through_o c._n for_o see_v b_o saint_n s_o h_o be_v equal_a and_o the_o angle_n at_o saint_n a_o right_a angle_n the_o straight_a line_n b_o n_o and_o b_o n_o be_v also_o equal_a and_o the_o triangle_n b_o n_o b_o d_o n_o o_o like_a and_o equal_a and_o the_o line_n d_o t_n t_o o_o equal_a draw_z o_o ay_o parallel_n to_o d_o u_o cut_v i_o u_o in_o i_o and_o the_o triangle_n d_o t_o v_o o_o t_o i_o will_v also_o be_v like_a and_o equal_a produce_v saint_n t_o to_o the_o arch_n d_o o_fw-fr i_o in_o e_o and_o produce_v it_o further_o to_o f_o so_o that_o the_o line_n e_o f_o be_v equal_a to_o t_o c_o and_o then_o saint_n f_o will_v be_v equal_a to_o a_o c._n therefore_o f_z c_o join_v will_v be_v parallel_n to_o b_o s_o in_o c_o f_o produce_v take_v f_z g_o equal_a to_z c_o f_o and_o draw_v g_o m_o parallel_n to_o d_o u_o cut_v i_o u_o in_o m_o and_o d_o o_z in_o