how_o often_o therefore_o note_n these_o five_o sundry_a sort_n of_o operation_n do_v for_o the_o most_o part_n of_o their_o execution_n differre_fw-la from_o the_o five_o operation_n of_o like_a general_a property_n and_o name_n in_o our_o whole_a number_n practisable_v so_o often_o for_o a_o more_o distinct_a doctrine_n we_o vulgar_o account_v and_o name_v it_o a_o other_o kind_n of_o arithmetic_n and_o by_o this_o reason_n the_o consideration_n doctrine_n and_o work_v in_o whole_a number_n only_o where_o of_o a_o unit_fw-la be_v no_o less_o part_n to_o be_v allow_v be_v name_v as_o it_o be_v a_o arithmetic_n by_o itself_o and_o so_o of_o the_o arithmetic_n of_o fraction_n in_o like_o sort_n the_o necessary_a wonderful_a and_o secret_a doctrine_n of_o proportion_n and_o proportionalytie_n have_v purchase_v unto_o itself_o a_o peculiar_a manner_n of_o handle_v and_o work_n and_o so_o may_v seem_v a_o other_o form_n of_o arithmetic_n moreover_o the_o astronomer_n for_o speed_n and_o 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and_o mixtynge_n consider_v howâ—_n i_o be_v desirous_a to_o deliver_v the_o student_n from_o error_n and_o cavillation_n do_v give_v to_o this_o practice_n the_o name_n of_o the_o arithmetic_n of_o radical_a number_n not_o of_o irrational_a or_o fur_v numbersâ—_n which_o other_o while_n be_v rational_a though_o they_o have_v the_o sign_n of_o a_o rote_n before_o they_o which_o arithmetic_n of_o whole_a number_n most_o usual_a will_v say_v they_o have_v no_o such_o root_n and_o so_o account_v they_o fur_v number_n which_o general_o speak_v be_v untrue_a as_o euclides_n ten_o book_n may_v teach_v you_o therefore_o to_o call_v they_o general_o radical_a number_n by_o reason_n of_o the_o sign_n √_o prefix_a be_v a_o sure_a way_n and_o a_o sufficient_a general_a distinction_n from_o all_o other_o order_v and_o use_v of_o number_n and_o yet_o beside_o all_o this_o consider_v the_o infinite_a desire_n of_o knowledge_n and_o incredible_a power_n of_o man_n search_n and_o capacity_n how_o 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for_o first_o though_o geber_n for_o his_o great_a skill_n in_o number_n geometry_n astronomy_n and_o other_o marvellous_a art_n may_v have_v seem_v able_a to_o have_v first_o devise_v the_o say_a rule_n and_o also_o the_o name_n carry_v with_o it_o a_o very_a near_a likeness_n of_o geber_n his_o name_n yet_o true_a it_o be_v that_o a_o greek_a philosopher_n and_o mathematicien_n name_v diophantus_n before_o geber_n his_o time_n write_v 13._o book_n thereof_o of_o which_o six_o be_v yet_o extant_a and_o i_o have_v they_o to_o 1550._o use_v of_o the_o famous_a mathematicien_fw-fr and_o my_o great_a friend_n petrus_n monâ—aureus_n and_o second_o the_o very_a name_n be_v algiebar_v and_o not_o algebra_n as_o by_o the_o arabien_fw-fr avicen_n may_v be_v prove_v who_o have_v these_o precise_a word_n in_o latin_a by_o andrea_n alpagus_n most_o perfect_a in_o the_o arabik_a tongue_n so_o translate_v scientia_fw-la faciendi_fw-la algiebar_n &_o almachabel_n i._o scientia_fw-la inveniendi_fw-la numerum_fw-la ignotum_fw-la per_fw-la additionem_fw-la numeri_fw-la &_o divisionem_fw-la &_o aequationem_fw-la which_o be_v to_o say_v the_o science_n of_o work_n algiebar_n and_o almachabel_n that_o be_v the_o science_n of_o find_v a_o unknown_a number_n by_o add_v of_o a_o number_n &_o division_n &_o aequation_n here_o have_v you_o the_o name_n and_o also_o the_o principal_a part_n of_o the_o rule_n touch_v to_o name_v it_o the_o rule_n or_o art_n of_o aequation_n do_v signify_v the_o middle_a part_n and_o the_o state_n of_o the_o rule_n this_o rule_n have_v his_o peculiar_a character_n and_o the_o principal_a part_n of_o arithmetic_n to_o it_o appertain_v do_v differe_o from_o the_o other_o arithmetical_a operation_n this_o arithmetic_n have_v number_n simple_a compound_v mix_v and_o fraction_n accordingly_z this_o rule_n and_o arithmetic_n of_o algiebar_n be_v so_o profound_a so_o general_a and_o so_o in_o manner_n contain_v the_o whole_a power_n of_o number_n application_n 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learn_v and_o exercise_v the_o excellent_a science_n of_o arithmetic_n and_o first_o who_o near_o at_o hand_n can_v be_v a_o better_a witness_n of_o the_o fruit_n receive_v by_o arithmetic_n than_o all_o kind_n of_o merchant_n though_o not_o all_o alike_o either_o need_n it_o or_o use_v it_o how_o can_v they_o forbear_v the_o use_n and_o help_v of_o the_o rule_n call_v the_o golden_a rule_n simple_n and_o compoundeâ—_n both_o forward_a and_o backward_o how_o may_v they_o miss_v arithmetical_a help_n in_o the_o rule_n of_o felowshypâ—_n either_o without_o time_n or_o with_o timer_n and_o between_o the_o merchant_n &_o his_o â—actorâ—_n the_o rulâ—â—_n of_o baâ—tering_v in_o ware_n only_o or_o part_n in_o ware_n and_o part_n in_o money_n will_v they_o glad_o want_v our_o merchant_n venture_n and_o travayler_n over_o sea_n how_o can_v they_o order_v their_o do_n just_o and_o without_o loss_n unleast_o certaâ—ne_n and_o general_a rule_n for_o exchange_n of_o money_n and_o rechaunge_n be_v
thereof_o be_v 〈◊〉_d this_o iâ—_z also_o to_o be_v note_v that_o of_o line_n some_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o and_o some_o be_v commensurable_a the_o one_o to_o the_o other_o in_o power_n of_o line_n commensurable_a in_o length_n the_o one_o to_o the_o other_o be_v give_v a_o example_n in_o the_o declaration_n of_o the_o first_o definition_n namely_o the_o line_n a_o and_o b_o which_o be_v commensurable_a in_o length_n one_o and_o the_o self_n measure_n namely_o the_o line_n c_o measure_v the_o length_n of_o either_o of_o they_o of_o the_o other_o kind_n be_v give_v this_o definition_n here_o set_v for_o the_o open_n of_o which_o take_v this_o example_n let_v there_o be_v a_o certain_a line_n namely_o the_o line_n bc_o and_o let_v the_o square_n of_o that_o line_n be_v the_o square_n bcde_v suppose_v also_o a_o other_o line_n namely_o the_o line_n fh_o &_o let_v the_o square_v thereof_o be_v the_o square_a fhik_n and_o let_v a_o certain_a 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here_o define_v to_o be_v absolute_o incommensurable_a these_o thing_n thus_o stand_v it_o may_v easy_o appear_v that_o if_o a_o line_n be_v assign_v and_o lay_v before_o we_o there_o may_v be_v innumerable_a other_o line_n commensurable_a unto_o it_o and_o other_o incommensurable_a unto_o it_o of_o commensurable_a line_n some_o be_v commensurable_a in_o length_n and_o power_n and_o some_o in_o power_n only_o 5_o and_o that_o right_a line_n so_o set_v forth_o be_v call_v a_o rational_a line_n thus_o may_v you_o see_v how_o to_o the_o suppose_a line_n first_o set_v may_v be_v compare_v infinite_a line_n line_n some_o commensurable_a both_o in_o length_n &_o power_n and_o some_o commensurable_a in_o power_n only_o and_o incommensurable_a in_o length_n and_o some_o incommensurable_a both_o in_o power_n &_o in_o length_n and_o this_o first_o line_n so_o set_v whereunto_o and_o to_o who_o square_n the_o other_o line_n and_o their_o square_n be_v compare_v be_v call_v a_o rational_a line_n common_o of_o the_o most_o part_n of_o writer_n line_n but_o some_o there_o be_v which_o mislike_n that_o it_o shall_v be_v call_v a_o rational_a line_n &_o that_o not_o without_o just_a cause_n in_o the_o greek_a copy_n it_o be_v call_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d rete_fw-la which_o signify_v a_o thing_n that_o may_v be_v speak_v &_o express_v by_o word_n a_o thing_n certain_a grant_v and_o appoint_v wherefore_o flussates_n a_o man_n which_o bestow_v great_a travel_n and_o diligence_n in_o restore_v of_o these_o element_n of_o euclid_n leave_v this_o word_n rational_a call_v this_o line_n suppose_v and_o first_o set_v a_o line_n certain_a because_o the_o part_n thereof_o into_o which_o it_o be_v divide_v be_v certain_a certain_a and_o know_v and_o may_v be_v express_v by_o voice_n and_o also_o be_v count_v by_o number_n other_o line_n be_v to_o this_o line_n incommensurable_a who_o part_n be_v not_o distinct_o know_v but_o be_v uncertain_a nor_o can_v be_v express_v by_o name_n nor_o assign_v by_o number_n which_o be_v of_o other_o man_n call_v irrational_a he_o call_v uncertain_a and_o surd_a line_n petrus_n montaureus_n although_o he_o do_v not_o very_o well_o like_a of_o the_o name_n yet_o he_o alter_v it_o not_o but_o use_v it_o in_o all_o his_o book_n likewise_o will_v we_o do_v here_o for_o that_o the_o word_n have_v be_v and_o be_v so_o universal_o receive_v and_o therefore_o will_v we_o use_v the_o same_o name_n and_o call_v it_o a_o rational_a line_n for_o it_o be_v not_o so_o great_a a_o matter_n what_o name_n we_o geve_v to_o thing_n so_o that_o we_o full_o understand_v the_o thing_n which_o the_o name_n signify_v this_o rational_a line_n thus_o here_o define_v be_v the_o ground_n and_o foundation_n of_o all_o the_o proposition_n almost_o of_o this_o whole_a ten_o book_n book_n and_o chief_o from_o the_o ten_o proposition_n forward_o so_o that_o unless_o you_o first_o place_n this_o rational_a line_n and_o have_v a_o special_a and_o continual_a regard_n unto_o it_o before_o you_o begin_v any_o demonstration_n you_o shall_v not_o easy_o understand_v it_o for_o it_o be_v as_o it_o be_v the_o touch_n and_o trial_n of_o all_o other_o line_n by_o which_o it_o be_v know_v whether_o any_o of_o they_o be_v rational_a or_o not_o note_n and_o this_o may_v be_v call_v the_o first_o rational_a line_n purpose_n the_o line_n rational_a of_o purpose_n or_o a_o rational_a line_n set_v in_o the_o first_o place_n and_o so_o make_v distinct_a and_o sever_v from_o other_o rational_a line_n of_o which_o shall_v be_v speak_v afterward_o and_o this_o must_v you_o well_o commit_v to_o memory_n definition_n 6_o line_n which_o be_v commensurable_a to_o this_o line_n whether_o in_o length_n and_o power_n or_o in_o power_n only_o be_v also_o call_v rational_a this_o definition_n need_v no_o declaration_n at_o all_o but_o be_v easy_o perceive_v if_o the_o first_o definition_n be_v remember_v which_o show_v what_o magnitude_n be_v commensurable_a and_o the_o three_o which_o show_v what_o line_n be_v commensurable_a in_o power_n here_o notâ—_n how_o apt_o &_o natural_o euclid_n in_o this_o place_n use_v these_o word_n commensurable_a either_o in_o length_n and_o power_n or_o in_o power_n only_o because_o that_o all_o line_n which_o be_v commensurable_a in_o length_n be_v also_o commensurable_a in_o pour_v when_o he_o speak_v of_o line_n commensurable_a in_o length_n he_o ever_o add_v and_o in_o power_n but_o when_o he_o speak_v of_o line_n commensurable_a in_o power_n he_o add_v this_o word_n only_o and_o add_v not_o this_o word_n in_o length_n as_o he_o in_o the_o other_o add_v this_o word_n in_o power_n for_o not_o all_o line_n which_o be_v commensurable_a in_o power_n be_v straight_o way_n commensurable_a also_o in_o length_n of_o this_o definition_n take_v this_o example_n let_v the_o first_o line_n rational_a of_o purpose_n which_o be_v suppose_v and_o lay_v forth_o who_o part_n be_v certain_a &_o know_v and_o may_v be_v express_v name_v and_o number_v be_v ab_fw-la the_o quadrate_n whereof_o let_v be_v abcd_o then_o suppose_v again_o a_o other_o line_n namely_o the_o line_n of_o which_o let_v be_v commensurable_a both_o in_o length_n and_o in_o power_n to_o the_o first_o rational_a line_n that_o be_v as_o before_o be_v teach_v let_v one_o line_n measure_v the_o length_n of_o each_o line_n and_o also_o lâ—t_v one_o superficies_n measure_v the_o two_o square_n of_o the_o say_v two_o line_n as_o here_o in_o the_o example_n be_v suppose_v and_o also_o appear_v to_o the_o eye_n then_o be_v the_o line_n e_o fletcher_n also_o a_o rational_a line_n moreover_o if_o the_o line_n of_o be_v commensurable_a in_o power_n only_o to_o the_o rational_a line_n ab_fw-la first_o set_v and_o suppose_v so_o that_o no_o one_o line_n do_v measure_v the_o two_o line_n ab_fw-la and_o of_o as_o in_o example_n yâ—_n see_v to_o be_v for_o
what_o manly_a virtue_n in_o other_o noble_a man_n flourish_v before_o his_o eye_n he_o sthe_o aspire_v after_o what_o prowess_n he_o purpose_v and_o mean_v to_o achieve_v with_o what_o feat_n and_o art_n he_o begin_v to_o furnish_v and_o fraught_v himself_o for_o the_o better_a service_n of_o his_o king_n and_o country_n both_o in_o peace_n &_o war_n these_o i_o say_v his_o heroical_a meditation_n forecasâ—inges_n and_o determination_n no_o twain_o i_o think_v beside_o myself_o can_v so_o perfect_o and_o true_o report_v and_o therefore_o in_o conscience_n i_o count_v it_o my_o part_n for_o the_o honour_n preferment_n &_o procure_v of_o virtue_n thus_o brief_o to_o have_v put_v his_o name_n in_o the_o register_n of_o faâ—â—_n immortal_a to_o our_o purpose_n this_o john_n by_o one_o of_o his_o act_n beside_o many_o other_o both_o in_o england_n and_o france_n by_o i_o in_o he_o note_v do_v dislose_v his_o hearty_a love_n to_o virtuous_a science_n and_o his_o noble_a intent_n to_o excel_v in_o martial_a prowess_n when_o he_o with_o humble_a request_n and_o instant_n solicit_v get_v the_o best_a rule_n either_o in_o time_n pass_v by_o greek_a or_o roman_a or_o in_o our_o time_n use_v and_o new_a stratagem_n therein_o deâ—ised_v for_o order_v of_o all_o company_n sum_n and_o numberâ—_n of_o man_n many_n or_o few_o with_o one_o kind_n of_o weapon_n or_o more_o appoint_v with_o artillery_n or_o without_o on_o horseback_n or_o on_o foot_n to_o give_v or_o take_v onset_n to_o seem_v many_o be_v fewâ—_n to_o sâ—em_v few_o be_v many_o to_o march_v in_o battle_n or_o jornay_n with_o many_o such_o feat_n to_o fight_v fâ—â—ld_n skarmoush_n or_o ambush_n appartain_v and_o of_o all_o these_o lively_a desiâ—nementes_n most_o curious_o to_o be_v in_o velame_n parchment_n describe_v with_o note_n &_o peculiar_a mark_n as_o the_o art_n require_v and_o all_o these_o rulesâ—_n and_o description_n arithmetical_a somerset_n enclose_v in_o a_o rich_fw-fr case_n of_o gold_n he_o use_v to_o wear_v about_o his_o neckeâ—_n as_o his_o jewel_n most_o precious_a and_o counselor_n most_o trusty_a thus_o arithmetic_n of_o he_o be_v shrine_v in_o gold_n of_o number_n fruit_n he_o have_v good_a hope_n now_o number_n therefore_o innumerable_a in_o number_n praise_v his_o shâ—â—ne_n shall_v find_v what_o need_n i_o for_o far_a proof_n to_o you_o of_o the_o schoolmaster_n of_o justice_n to_o require_v testimony_n how_o needful_a how_o fruitful_a how_o skilful_a a_o thing_n arithmetic_n be_v i_o meaneâ—_n the_o lawyer_n of_o all_o sort_n undoubted_o the_o civilians_n can_v marvelous_o declare_v how_o neither_o the_o ancient_a roman_a law_n without_o good_a knowledge_n of_o number_n art_n can_v be_v perceive_v nor_o justice_n in_o infinite_a case_n without_o due_a proportion_n narrow_o consider_v be_v able_a to_o be_v executedâ—_n how_o just_o &_o with_o great_a knowledge_n of_o art_n do_v papâ—â—â—â—anus_n institute_v a_o law_n of_o partition_n and_o allowance_n between_o man_n and_o wife_n aâ—â—â—_n a_o â—â—â—orce_n but_o how_o accâ—rsius_n baldâ—s_n bartolus_n jason_n alexâ—nder_n and_o final_o alciatâ—â—_n be_v otherwise_o notable_o well_o learn_v do_v jumble_v guess_v and_o err_v from_o the_o â—quityâ—â—rt_n â—nd_v iâ—â—ent_v of_o the_o lawmaker_n arithmetic_n can_v detect_v and_o convince_v and_o clear_o make_v the_o truth_n to_o shine_v good_a bartolus_n tire_v in_o the_o examine_n &_o proportion_v of_o the_o matter_n â—nd_v with_o accursius_fw-la gloss_n much_o cumber_v burst_v out_o and_o say_v nulla_fw-la est_fw-la iâ—_z 〈◊〉_d librâ—_n 〈◊〉_d glossa_fw-la difficiliâ—r_fw-la evils_n câ—mputationem_fw-la nec_fw-la scholasâ—ici_fw-la â—ec_fw-la doctâ—res_fw-la intelligent_a etc_n etc_n that_o isâ—_n in_o the_o whole_a book_n there_o be_v no_o â—losse_n hard_a than_o thiâ—â—_n who_o account_n or_o reckon_n neither_o the_o scholar_n nor_o the_o doctor_n understand_v etc_n etc_n what_o can_v they_o say_v of_o iâ—lianus_n law_n si_fw-mi ita_fw-la sâ—riptâ—m_fw-la etc_n etc_n of_o the_o testator_n will_v just_o perform_v between_o the_o wife_n son_n and_o daughter_n how_o can_v they_o perceive_v the_o â—â—â—â—â—tie_n of_o aphricanus_fw-la arithmetical_a reckon_a where_o he_o treat_v of_o lex_n falcidâ—aâ—_n â—_z how_o â—â—n_fw-la they_o deliver_v he_o from_o his_o reprover_n and_o their_o maintainer_n as_o iâ—â—â—â—es_n accursius_fw-la hippolytus_n and_o alciatus_fw-la how_o â—ustly_o and_o artificial_o be_v africanâ—s_n reckon_v make_v proportionate_v to_o the_o sum_n bequeath_v the_o contribution_n of_o each_o part_n namely_o for_o the_o hundred_o present_o receive_v 17_o 1_o 7._o and_o for_o the_o hundred_o receive_v after_o ten_o month_n 12_o 6_o 7_o which_o make_v the_o 30_o which_o be_v to_o be_v contribute_v by_o the_o legatariâ—s_n to_o the_o heir_n for_o what_o proportion_n 100_o have_v to_o 75_o the_o same_o have_v 17_o 1_o 7_o to_o 12_o 6_o 7_o which_o be_v sesquitertia_fw-la that_o be_v as_z 4_o to_o 3._o which_o make_v 7._o wonderful_a many_o place_n in_o the_o civil_a law_n require_v a_o expert_a arithmeticien_fw-fr to_o understand_v the_o deep_a judgement_n &_o just_a determination_n of_o the_o ancient_a roman_a lawmaker_n but_o much_o more_o expert_a aught_o he_o to_o be_v who_o shall_v be_v able_a to_o decide_v with_o equity_n the_o infinite_a variety_n of_o case_n which_o do_v or_o may_v happen_v under_o every_o one_o of_o those_o law_n and_o ordinance_n civil_a hereby_o easy_o you_o may_v now_o conjecture_v that_o in_o the_o canon_n law_n and_o in_o the_o law_n of_o the_o realm_n which_o with_o we_o bear_v the_o chief_a authority_n justice_n and_o equity_n may_v be_v great_o prefer_v and_o skilful_o execute_v through_o due_a skill_n of_o arithmetic_n and_o proportion_n appertain_v the_o worthy_a philosopher_n and_o prudent_a lawmaker_n who_o have_v write_v many_o book_n de_fw-fr republica_n how_o the_o best_a state_n of_o common_a wealth_n may_v be_v procure_v and_o maintain_v have_v very_o well_o determine_v of_o justice_n which_o not_o only_o be_v the_o base_a and_o foundation_n of_o common_a weal_n but_o also_o the_o total_a perfection_n of_o all_o our_o work_n word_n and_o thought_n define_v it_o to_o be_v that_o virtue_n justice._n by_o which_o to_o every_o one_o be_v render_v that_o to_o he_o appertain_v god_n challenge_v this_o at_o our_o hand_n to_o be_v honour_v as_o god_n to_o belove_a as_o a_o father_n to_o be_v fear_v as_o a_o lord_n &_o master_n our_o neighbour_n proportion_n be_v also_o prescribe_v of_o the_o almighty_a lawmaker_n which_o be_v to_o do_v to_o other_o even_o as_o we_o will_v be_v do_v unto_o these_o proportion_n be_v in_o justice_n necessary_a in_o duty_n commendable_a and_o of_o common_a wealth_n the_o life_n strength_n stay_v and_o flourish_a aristotle_n in_o his_o ethic_n to_o fatch_z the_o seed_n of_o justice_n and_o light_n of_o direction_n to_o use_v and_o execute_v the_o samâ—_n be_v fain_o to_o fly_v to_o the_o perfection_n and_o power_n of_o number_n for_o proportion_n arithmetical_a and_o geometrical_a plato_n in_o his_o book_n call_v epinomis_n which_o book_n be_v the_o threasury_n of_o all_o his_o doctrine_n where_o his_o purpose_n be_v to_o seek_v a_o science_n which_o when_o a_o man_n have_v it_o perfect_o he_o may_v seem_v and_o so_o be_v in_o deed_n wise._n he_o brief_o of_o other_o science_n discourse_v find_v they_o not_o able_a to_o bring_v it_o to_o pass_v but_o of_o the_o science_n of_o number_n he_o say_v illa_fw-la qua_fw-la numerum_fw-la mortalium_fw-la generi_fw-la dâ—â—n_fw-la id_fw-la profecto_fw-la efficietâ—_n deum_fw-la antem_fw-la aliquem_fw-la magis_fw-la quam_fw-la fortunam_fw-la â—d_a saâ—â—tem_fw-la nostram_fw-la hoc_fw-la mâ—nus_fw-la nobis_fw-la arbitror_fw-la contulisse_fw-la etc_n etc_n nam_fw-la ipsum_fw-la â—onorum_fw-la omnium_fw-la authorem_fw-la cur_n non_fw-la maximi_fw-la boni_fw-la prudentiae_fw-la dico_fw-la causam_fw-la arbitramur_fw-la that_o science_n very_o which_o have_v teach_v mankind_n number_n shall_v be_v able_a to_o bring_v it_o to_o pass_v and_o i_o think_v a_o certain_a god_n rather_o than_o fortune_n to_o have_v give_v we_o this_o gift_n for_o our_o bliss_n for_o why_o shall_v we_o not_o judge_v he_o who_o be_v the_o author_n of_o all_o good_a thing_n to_o be_v also_o the_o cause_n of_o thâ—_z great_a good_a thing_n namely_o wisdom_n there_o at_o length_n he_o prove_v wisdom_n to_o be_v attain_v by_o good_a skill_n of_o number_n with_o which_o great_a testimony_n and_o the_o manifold_a profess_v and_o reason_n before_o expressedâ—_n you_o may_v be_v sufficient_o and_o full_o persuade_v of_o the_o perfect_a science_n of_o arithmetic_n to_o make_v this_o account_v that_o of_o all_o science_n ☞_o next_o to_o theologie_n it_o be_v most_o divine_a most_o pure_a most_o ample_a and_o general_a most_o profound_a most_o subtle_a most_o commodious_a and_o most_o necessary_a who_o be_v next_o sister_n be_v the_o absolute_a science_n of_o magnitude_n of_o which_o by_o the_o direction_n and_o aid_n of_o he_o who_o magnitude_n be_v infinite_a and_o of_o we_o incomprehensible_a i_o now_o
the_o lengthe_n of_o day_n and_o night_n the_o hour_n and_o time_n both_o night_n and_o day_n be_v know_v with_o very_a many_o other_o pleasant_a and_o necessary_a use_n whereof_o some_o be_v know_v but_o better_a remain_v for_o such_o to_o know_v and_o use_v who_o of_o a_o spark_n of_o true_a fire_n can_v make_v a_o wonderful_a bonfire_n by_o apply_v of_o due_a matter_n due_o ☜_o of_o astrology_n here_o i_o make_v a_o arâ—e_n several_a from_o astronomieâ—_n â—_z not_o by_o new_a devise_n but_o by_o good_a reason_n and_o authority_n for_o astrology_n be_v a_o art_n mathematical_a which_o reasonable_o demonstrate_v the_o operation_n and_o effect_n of_o the_o natural_a beam_n of_o light_n and_o secret_a influence_n of_o the_o star_n and_o planet_n in_o every_o element_n and_o elemental_a body_n at_o all_o time_n in_o any_o horizon_n assign_v this_o art_n be_v furnish_v with_o many_o other_o great_a art_n and_o experience_n as_o with_o perfect_v perspective_n astronomy_n cosmography_n natural_a philosophy_n of_o the_o 4._o element_n the_o art_n of_o graduation_n and_o some_o good_a understanding_n in_o music_n and_o yet_o moreover_o with_o a_o other_o great_a art_n hereafter_o follow_v though_o i_o here_o set_v this_o before_o for_o some_o consideration_n i_o move_v sufficient_a you_o see_v be_v the_o stuff_n to_o make_v this_o rare_a and_o secret_a art_n of_o and_o hard_a enough_o to_o frame_v to_o the_o conclusion_n syllogistical_a yet_o both_o the_o manifold_a and_o continual_a travail_n of_o the_o most_o ancient_a and_o wise_a philosopher_n for_o the_o attain_n of_o this_o art_n and_o by_o example_n of_o effect_n to_o confirm_v the_o same_o have_v leave_v unto_o we_o sufficient_a proof_n and_o witness_v and_o we_o also_o daily_o may_v perceive_v that_o man_n body_n and_o all_o other_o elemental_a body_n be_v alter_v dispose_v order_a pleasure_v and_o displeasure_v by_o the_o influential_a work_n of_o the_o sun_n moon_n and_o the_o other_o star_n and_o planet_n and_o therefore_o say_v aristotle_n in_o the_o first_o of_o his_o meteorological_a book_n in_o the_o second_o chapter_n est_fw-la autem_fw-la necessariò_fw-la mundus_fw-la iste_fw-la supernis_fw-la lationibus_fw-la ferè_fw-la continuus_fw-la ut_fw-la inde_fw-la vis_fw-la eius_fw-la universa_fw-la regatur_fw-la ea_fw-la siquidem_fw-la causa_fw-la prima_fw-la putanda_fw-la omnibus_fw-la est_fw-la unde_fw-la motus_fw-la principium_fw-la existit_fw-la that_o be_v this_o elemental_a world_n be_v of_o necessity_n almost_o next_o adjoin_v to_o the_o heavenly_a motion_n that_o from_o thence_o all_o his_o virtue_n or_o force_n may_v be_v govern_v for_o that_o be_v to_o be_v think_v the_o first_o cause_n unto_o all_o from_o which_o the_o beginning_n of_o motion_n be_v and_o again_o in_o the_o ten_o chapter_n opâ—rtet_fw-la igitur_fw-la &_o horum_fw-la principia_fw-la sumamus_fw-la &_o causas_fw-la omnium_fw-la similiter_fw-la principium_fw-la igitur_fw-la vy_z movens_fw-la praecipuumque_fw-la &_o omnium_fw-la primum_fw-la circulus_fw-la ille_fw-la est_fw-la in_fw-la quo_fw-la manifest_a solis_fw-la latio_fw-la etc_n etc_n and_o so_o forth_o his_o meteorological_a book_n be_v full_a of_o argument_n and_o effectual_a demonstration_n of_o the_o virtue_n operation_n and_o power_n of_o the_o heavenly_a body_n in_o and_o upon_o the_o four_o element_n and_o other_o body_n of_o they_o either_o perfect_o or_o unperfect_o compose_v and_o in_o his_o second_o book_n de_fw-fr generatione_fw-la &_o corruption_n in_o the_o ten_o chapter_n quocirca_fw-la &_o prima_fw-la latiâ—_n orâ—us_n &_o interitus_fw-la causa_fw-la non_fw-la est_fw-la sed_fw-la obliqui_fw-la circuli_fw-la latio_fw-la ea_fw-la namque_fw-la &_o continua_fw-la est_fw-la &_o dvobus_fw-la motibus_fw-la fit_a in_o english_a thus_o wherefore_o the_o uppermost_a motion_n be_v not_o the_o cause_n of_o generation_n and_o corruption_n but_o the_o motion_n of_o the_o zodiac_n for_o that_o both_o be_v continual_a and_o be_v cause_v of_o two_o move_n and_o in_o his_o second_o book_n and_o second_o chapter_n of_o his_o physike_n homo_fw-la namque_fw-la generate_fw-la hominem_fw-la atque_fw-la sol._n for_o man_n say_v he_o and_o the_o son_n be_v cause_n of_o man_n generation_n authority_n may_v be_v bring_v very_o many_o both_o of_o 1000_o 2000_o yea_o and_o 3000._o year_n antiquity_n of_o great_a philosopher_n expert_a wise_a and_o godly_a man_n for_o that_o conclusion_n which_o daily_a and_o hourly_o we_o man_n may_v discern_v and_o perceive_v by_o sense_n and_o reason_n all_o beast_n do_v feel_v and_o simple_o show_v by_o their_o action_n and_o passion_n outward_a and_o inward_a all_o plant_n herb_n tree_n flower_n and_o fruit_n and_o final_o the_o element_n and_o all_o thing_n of_o the_o element_n compose_v do_v geve_v testimony_n as_o aristotle_n say_v that_o their_o whole_a disposition_n virtue_n and_o natural_a motion_n depend_v of_o the_o activity_n of_o the_o heavenly_a motion_n and_o influence_n whereby_o beside_o the_o specifical_a order_n and_o form_n due_a to_o every_o seed_n and_o beside_o the_o nature_n proper_a to_o the_o individual_a matrix_fw-la of_o the_o thing_n produce_v what_o shall_v be_v the_o heavenly_a impression_n the_o perfect_a and_o circumspect_a astrologien_n have_v to_o conclude_v not_o only_o by_o apotelesme_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d but_o by_o natural_a and_o mathematical_a demonstration_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d whereunto_o what_o science_n be_v requisite_a without_o exception_n i_o partly_o have_v here_o warn_v and_o in_o my_o bropâ—deâ—â—â—â—_n beside_o other_o matter_n there_o disclose_v i_o have_v mathematical_o furnish_v up_o the_o whole_a method_n to_o this_o our_o age_n not_o so_o careful_o handle_v by_o any_o that_o ever_o i_o see_v or_o hear_v of_o i_o be_v for_o lovayn_n â—1_n year_n ago_o by_o certain_a earnest_a disputation_n of_o the_o learned_a gâ—â—ardus_n mâ—rcâ—tâ—â—_n and_o 〈◊〉_d gogaâ—a_n and_o other_o thereto_o so_o provoke_v and_o by_o my_o constant_a and_o invincible_a zeal_n to_o the_o verity_n in_o observation_n of_o heavenly_a influence_n to_o the_o minâ—te_n of_o time_n than_o so_o diligent_a and_o chief_o by_o the_o supernatural_a influence_n from_o the_o star_n of_o jacob_n so_o direct_v that_o any_o modest_a and_o sober_a student_n careful_o and_o diligent_o selâ—ing_v for_o the_o truth_n will_v both_o find_v &_o confess_v therein_o to_o be_v the_o verity_n of_o these_o my_o word_n and_o also_o become_v a_o reasonable_a reformer_n of_o three_o sort_n of_o people_n about_o these_o influential_a operation_n great_o err_v from_o the_o truth_n whereof_o the_o one_o be_v light_a believer_n the_o other_o note_n light_a despiser_n and_o the_o three_o light_a practiser_n the_o first_o &_o most_o common_a sort_n think_v the_o heaven_n and_o star_n to_o be_v answerable_a to_o any_o their_o doubt_n or_o desire_n which_o be_v not_o so_o and_o in_o deed_n they_o to_o much_o over_o reach_n the_o second_o sort_n think_v no_o influential_a virtue_n from_o the_o heavenly_a body_n to_o bear_v any_o sway_n in_o generation_n and_o corruption_n in_o this_o elemental_a world_n and_o to_o the_o sun_n moon_n and_o star_n be_v so_o many_o so_o pure_a so_o bright_a so_o wonderful_a big_a so_o far_o in_o distance_n so_o manifold_a in_o their_o motion_n so_o constant_a in_o their_o periode_n etc_n etc_n they_o assign_v a_o sleight_n simple_a office_n or_o two_o and_o so_o allow_v unto_o they_o according_a to_o their_o capacity_n as_o much_o virtue_n and_o power_n influential_a as_o to_o the_o sign_n of_o the_o sun_n moon_n and_o seven_o star_n hang_v up_o for_o sign_n in_o london_n for_o distinction_n of_o house_n &_o such_o gross_a help_n in_o our_o worldly_a affair_n and_o they_o understand_v not_o or_o will_v not_o understand_v of_o the_o other_o work_n and_o virtue_n of_o the_o heavenly_a sun_n moon_n and_o star_n not_o so_o much_o as_o the_o mariner_n or_o husband_n manâ—_n no_o not_o so_o much_o as_o the_o elephant_n do_v as_o the_o cynocephalus_n as_o the_o porpentine_n doâ—h_v nor_o will_v allow_v these_o perfect_a and_o incorruptible_a mighty_a body_n so_o much_o virtual_a radiation_n &_o force_n as_o they_o see_v in_o a_o little_a piece_n of_o a_o magnes_n stone_n which_o at_o great_a distance_n show_v his_o operation_n and_o perchance_o they_o think_v the_o sea_n &_o rivers_n as_o the_o thames_n to_o be_v some_o quick_a thing_n and_o sâ—_n to_o ebb_v aâ—d_z slow_a run_v in_o and_o out_o of_o themselves_o at_o â—heiâ—_n own_o fantasy_n god_n help_v god_n help_v sure_o these_o man_n come_v to_o short_a and_o either_o be_v to_o dull_a or_o wilful_o blind_a or_o perhaps_o to_o malicious_a the_o three_o man_n be_v the_o common_a and_o vulgar_a astrologien_n or_o practiser_n who_o be_v not_o due_o artificial_o and_o perfect_o furnish_v yet_o either_o for_o vain_a glory_n or_o gain_v or_o like_o a_o simple_a dolt_n &_o blind_a bayardâ—_n both_o in_o matter_n and_o manner_n err_v to_o the_o discredit_n of_o the_o wary_a and_o modest_a astrologien_n and_o to_o the_o rob_n of_o those_o most_o noble_a corporal_a
the_o other_o in_o the_o point_n d_o and_o that_o only_a point_n be_v common_a to_o they_o both_o neither_o do_v the_o one_o enter_v into_o the_o other_o if_o any_o part_n of_o the_o one_o enter_v into_o any_o part_n of_o the_o other_o than_o the_o one_o cut_v and_o divide_v the_o other_o and_o touch_v the_o one_o the_o other_o not_o in_o one_o point_n only_o as_o in_o the_o other_o before_o but_o in_o two_o pointâ—s_n and_o have_v also_o a_o superficies_n common_a to_o they_o both_o as_o the_o circle_n ghk_v and_o hlk_v cut_v the_o one_o the_o other_o in_o two_o point_n h_o and_o edward_n and_o the_o one_o enter_v into_o the_o other_o also_o the_o superficies_n hk_o be_v common_a to_o they_o both_o for_o it_o be_v a_o part_n of_o the_o circle_n ghk_o and_o also_o it_o be_v a_o part_n of_o the_o circle_n hlk._n definition_n right_o line_n in_o a_o circle_n be_v say_v to_o be_v equal_o distant_a from_o the_o centre_n when_o perpendicular_a line_n draw_v from_o the_o centre_n unto_o 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def_n be_v a_o section_n of_o a_o circle_n for_o that_o it_o be_v comprehend_v under_o the_o right_a line_n df_o and_o the_o circumference_n def_n and_o the_o figure_n abc_n for_o that_o it_o contain_v within_o it_o the_o centre_n of_o the_o circle_n be_v call_v the_o great_a section_n of_o a_o circle_n and_o the_o figure_n def_n be_v the_o less_o section_n of_o a_o circle_n because_o it_o be_v whole_o without_o the_o centre_n of_o the_o circle_n as_o it_o be_v note_v in_o the_o 16._o definition_n of_o the_o first_o book_n a_o angle_n of_o a_o section_n or_o segment_n be_v that_o angle_n which_o be_v contain_v under_o a_o right_a line_n and_o the_o circumference_n of_o the_o circle_n definition_n as_o the_o angle_n abc_n in_o the_o section_n abc_n be_v a_o angle_n of_o a_o section_n because_o it_o be_v contain_v of_o the_o circumference_n bac_n and_o the_o right_a line_n bc._n likewise_o the_o angle_n cbd_v be_v a_o angle_n of_o the_o section_n bdc_o because_o it_o be_v contain_v 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from_o the_o point_n b_o be_v a_o point_n in_o the_o circumference_n abc_n be_v draw_v two_o right_a line_n bc_o and_o basilius_n to_o the_o end_n of_o the_o line_n ac_fw-la which_o be_v the_o base_a of_o the_o section_n abc_n likewise_o the_o angle_n adc_o be_v a_o angle_n in_o the_o section_n adc_o because_o from_o the_o point_n d_o be_v in_o the_o circumference_n adc_o be_v draw_v two_o right_a line_n namely_o dc_o &_o da_z to_o the_o end_n of_o the_o right_a line_n ac_fw-la which_o be_v also_o the_o base_a to_o the_o say_a section_n adc_o so_o you_o see_v it_o be_v not_o all_o one_o to_o say_v section_n a_o angle_n of_o a_o section_n and_o a_o angle_n in_o a_o section_n a_o angle_n of_o a_o section_n consist_v of_o the_o touch_n of_o a_o right_a line_n and_o a_o crooked_a and_o a_o angle_n in_o a_o section_n be_v place_v on_o the_o circumference_n and_o be_v contain_v of_o two_o right_a line_n also_o the_o great_a section_n have_v in_o it_o the_o less_o angle_n and_o the_o less_o 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centre_n of_o the_o circle_n and_o receive_v the_o circumference_n of_o the_o say_a section_n first_o the_o other_o pertain_v to_o the_o angle_n in_o the_o section_n which_o as_o before_o be_v say_v be_v ever_o in_o the_o circumference_n as_o if_o the_o angle_n bac_n be_v in_o the_o centre_n a_o and_o receive_v of_o the_o circumference_n blc_n be_v equal_a to_o the_o angle_n feg_n be_v also_o in_o the_o centre_n e_o and_o receive_v of_o the_o circumference_n fkg_v then_o be_v the_o two_o section_n bcl_n and_o fgk_n like_v by_o the_o first_o definition_n by_o the_o same_o definition_n also_o be_v the_o other_o two_o section_n like_a namely_o bcd_o and_o fgh_a for_o that_o the_o angle_n bac_n be_v equal_a to_o the_o
definition_n of_o the_o first_o the_o line_n mb_n be_v equal_a unto_o the_o line_n mk_o put_v the_o line_n md_o common_a to_o the_o both_o wherefore_o these_o two_o line_n mk_o and_o md_o be_v equal_a to_o these_o two_o line_n bm_n and_o md_o the_o one_o to_o the_o other_o and_o the_o angle_n kmd_v be_v by_o the_o 23._o of_o the_o first_o equal_a to_o the_o angle_n bmd_v wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a dk_o be_v equal_a to_o the_o base_a db._n or_o it_o may_v thus_o be_v demonstrate_v impossibility_n draw_v by_o the_o first_o petition_n a_o line_n from_o m_o to_o n._n and_o for_o asmuch_o as_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_n km_o be_v equal_a unto_o the_o line_n mn_o and_o the_o line_n md_o be_v common_a to_o they_o both_o and_o the_o base_a kd_v be_v equal_a to_o the_o base_a dn_o by_o supposition_n therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n kmd_v be_v equal_a to_o the_o angle_n dmn_n but_o the_o angle_n kmd_v be_v equal_a to_o the_o angle_n bmd_v wherefore_o the_o angle_n bmd_v be_v equal_a to_o the_o angle_n nmd_v the_o less_o unto_o the_o great_a which_o be_v impossible_a wherefore_o from_o the_o point_n d_o can_v not_o be_v draw_v unto_o the_o circumference_n abc_n on_o each_o side_n of_o dg_o the_o jest_n more_o than_o two_o equal_a right_a line_n if_o therefore_o without_o a_o circle_n be_v take_v any_o point_n and_o from_o that_o point_n be_v draw_v into_o the_o circle_n unto_o the_o circumference_n certain_a right_a line_n of_o which_o let_v one_o be_v draw_v by_o the_o centre_n and_o let_v the_o rest_n be_v draw_v at_o all_o adventure_n the_o great_a of_o those_o right_a line_n which_o fall_v in_o the_o concavity_n or_o hollowness_n of_o the_o circumference_n of_o the_o circle_n be_v that_o which_o pass_v by_o the_o centre_n and_o of_o all_o the_o other_o line_n that_o line_n which_o be_v nigh_o to_o the_o line_n which_o pass_v by_o the_o centre_n be_v great_a than_o that_o which_o be_v more_o distant_a but_o of_o those_o right_a line_n which_o end_n in_o the_o convexe_n part_n of_o the_o circumference_n that_o line_n be_v the_o lest_o which_o be_v draw_v from_o the_o point_n to_o the_o dimetient_fw-la and_o of_o the_o other_o line_n that_o which_o be_v nigh_o to_o the_o least_o be_v always_o less_o than_o that_o which_o be_v more_o distant_a and_o from_o that_o point_n can_v be_v draw_v unto_o the_o circumference_n on_o each_o side_n of_o the_o lest_o only_a two_o equal_a right_a line_n which_o be_v require_v to_o be_v prove_v this_o proposition_n be_v call_v common_o in_o old_a book_n amongst_o the_o barbarous_a caâ—dâ—_n panonis_n panonis_fw-la that_o be_v the_o peacock_n tail_n ¶_o a_o corollary_n hereby_o it_o be_v manifest_a corollary_n that_o the_o right_a line_n which_o be_v draw_v from_o the_o point_n give_v without_o the_o circle_n and_o fall_v within_o the_o circle_n be_v equal_o distant_a from_o the_o least_o or_o from_o the_o great_a which_o be_v draw_v by_o the_o centre_n be_v equal_a the_o one_o to_o the_o other_o but_o contrariwise_o if_o they_o be_v unequal_o distant_a whether_o they_o light_v upon_o the_o concave_n or_o convexe_a circumference_n of_o the_o circle_n they_o be_v unequal_a the_o 8._o theorem_a the_o 9_o proposition_n if_o within_o a_o circle_n be_v take_v a_o point_n and_o from_o that_o point_n be_v draw_v unto_o the_o circumference_n more_o than_o two_o equal_a right_a line_n the_o point_n take_v be_v the_o centre_n of_o the_o circle_n svppose_v that_o the_o circle_n be_v abc_n and_o within_o it_o let_v there_o be_v take_v the_o point_n d._n and_o from_o d_z let_v there_o be_v draw_v unto_o the_o circumference_n abc_n more_o than_o two_o equal_a right_a line_n construction_n that_o be_v da_z db_o and_o dc_o then_o i_o say_v that_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n abc_n draw_v by_o the_o first_o petition_n these_o right_a line_n ab_fw-la and_o bc_o demonstration_n and_o by_o the_o 10._o of_o the_o first_o divide_v they_o into_o two_o equal_a part_n in_o the_o point_n e_o and_o f_o namely_o the_o line_n ab_fw-la in_o the_o point_n e_o and_o the_o line_n bc_o in_o the_o point_n f._n and_o draw_v the_o line_n ed_z and_o fd_o and_o by_o the_o second_o petition_n extend_v the_o line_n ed_z and_o fd_v on_o each_o side_n to_o the_o point_n king_n g_o and_o h_o l._n and_o for_o asmuch_o as_o the_o line_n ae_n be_v equal_a unto_o the_o line_n ebb_n and_o the_o line_n ed_z be_v common_a to_o they_o both_o therefore_o these_o two_o side_n ae_n and_o ed_z be_v equal_a unto_o these_o two_o side_n be_v and_o ed_z and_o by_o supposition_n the_o base_a dam_n be_v equal_a to_o the_o base_a db._n wherefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n aed_n be_v equal_a to_o the_o angle_n bed_n wherefore_o either_o of_o these_o angle_n aed_n and_o bed_n be_v a_o right_a angle_n wherefore_o the_o line_n gk_o divide_v the_o line_n ab_fw-la into_o two_o equal_a part_n and_o make_v right_a angle_n and_o for_o asmuch_o as_z if_o in_o a_o circle_n a_o right_a line_n divide_v a_o other_o right_a line_n into_o two_o equal_a part_n in_o such_o sort_n that_o it_o make_v also_o right_a angle_n in_o the_o line_n that_o divide_v be_v the_o centre_n of_o the_o circle_n by_o the_o corollary_n of_o the_o first_o of_o the_o three_o therefore_o by_o the_o same_o corollary_n in_o the_o line_n gk_o be_v the_o centre_n of_o the_o circle_n abc_n and_o by_o the_o same_o reason_n may_v we_o prove_v that_o in_o the_o line_n hl_o be_v the_o centre_n of_o the_o circle_n abc_n and_o the_o right_a line_n gk_o and_o hl_o have_v no_o other_o point_n common_a to_o they_o both_o beside_o the_o point_n d._n wherefore_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n abc_n if_o therefore_o within_o a_o circle_n be_v take_v a_o point_n and_o from_o that_o point_n be_v draw_v unto_o the_o circumference_n more_o than_o two_o equal_a right_a line_n the_o point_n take_v be_v the_o centre_n of_o the_o circle_n which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n let_v there_o be_v take_v within_o the_o circle_n abc_n the_o point_n d._n impossibility_n and_o from_o the_o point_n d_o let_v there_o be_v draw_v unto_o the_o circumference_n more_o than_o two_o equal_a right_a line_n namely_o dam_n db_o and_o dc_o then_o i_o say_v that_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n for_o if_o not_o then_o if_o it_o be_v possible_a let_v the_o point_n e_o be_v the_o centre_n and_o draw_v a_o line_n from_o d_z to_o e_o and_o extend_v de_fw-fr to_o the_o point_n fletcher_n and_o g._n wherefore_o the_o line_n fg_o be_v the_o diameter_n of_o the_o circle_n abc_n and_o for_o asmuch_o as_o in_o fg_o the_o diameter_n of_o the_o circle_n abc_n be_v take_v a_o point_n namely_o d_z which_o be_v not_o the_o centre_n of_o that_o circle_n therefore_o by_o the_o 7._o of_o the_o three_o the_o line_n dg_o be_v the_o great_a and_o the_o line_n dc_o be_v great_a than_o the_o line_n db_o and_o the_o line_n db_o be_v greateâ—_n than_o the_o line_n da._n but_o the_o line_n dc_o db_o dam_n be_v also_o equal_a by_o supposition_n which_o be_v impossible_a wherefore_o the_o point_n e_o be_v not_o the_o centre_n of_o the_o circle_n abc_n and_o in_o like_a sort_n may_v we_o prove_v that_o no_o other_o point_n beside_o d._n wherefore_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n abc_n which_o be_v require_v to_o be_v prove_v the_o 9_o theorem_a the_o 10._o proposition_n a_o circle_n cut_v not_o a_o circle_n in_o more_o point_n than_o two_o for_o if_o it_o be_v possible_a let_v the_o circle_n abc_n cut_v the_o circle_n def_n in_o more_o point_n than_o two_o impossibility_n that_o be_v in_o b_o g_o h_o &_o f._n and_o draw_v line_n from_o b_o to_o g_z and_o from_o b_o to_o h._n and_o by_o the_o 10._o of_o the_o first_o divide_v either_o of_o the_o line_n bg_o &_o bh_o into_o two_o equal_a part_n in_o the_o point_n king_n and_o l._n and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n king_n raise_v up_o unto_o the_o line_n bh_o a_o perpendicular_a line_n kc_o and_o likewise_o from_o the_o point_n l_o raise_v up_o unto_o the_o line_n bg_o a_o perpendicular_a line_n lm_o and_o extend_v the_o line_n ck_o to_o the_o point_n a_o and_o lnm_n to_o the_o point_n x_o and_o e._n and_o for_o asmuch_o as_o in_o the_o circle_n abc_n the_o right_a line_n ac_fw-la divide_v the_o right_a line_n bh_o into_o two_o equal_a part_n and_o make_v right_a angle_n therefore_o by_o the_o 3._o of_o the_o three_o in_o the_o line_n ac_fw-la be_v the_o centre_n of_o
constituens_fw-la or_o componens_fw-la componens_fw-la because_o that_o it_o with_o some_o other_o part_n or_o part_n make_v the_o whole_a as_o the_o line_n cb_o together_o with_o the_o line_n ac_fw-la make_v the_o whole_a line_n ab_fw-la of_o the_o barbarous_a it_o be_v call_v pars_fw-la aliquanta_fw-la aliquanta_fw-la in_o this_o signification_n it_o be_v take_v in_o bâ—rlaâ—â—_n in_o the_o begin_n of_o his_o book_n in_o the_o definition_n of_o a_o part_n when_o he_o say_v every_o less_o number_n compare_v to_o a_o great_a be_v say_v to_o be_v a_o part_n of_o the_o great_a whether_o the_o less_o measure_n the_o great_a or_o measure_v it_o not_o multiplex_fw-la be_v a_o great_a magnitude_n in_o respect_n of_o the_o less_o when_o the_o less_o measure_v the_o great_a definition_n as_o the_o line_n cd_o before_o set_v in_o the_o first_o example_n be_v multiplex_n to_o the_o line_n ab_fw-la for_o that_o cd_o a_o line_n contain_v 9_o be_v the_o great_a magnitude_n and_o be_v compare_v to_o the_o less_o namely_o to_o the_o line_n ab_fw-la contain_v 3._o and_o also_o the_o less_o line_n ab_fw-la measure_v the_o great_a line_v cd_o for_o take_v 3._o time_n it_o make_v it_o as_o be_v above_o say_v so_o in_o number_n 12._o be_v multiplex_n to_o 3_o for_o 12_o be_v the_o great_a number_n and_o be_v compare_v to_o the_o less_o namely_o to_o 3._o which_o 3._o also_o measure_v it_o follow_v for_o 3_o take_v 4_o time_n make_v 12._o by_o this_o word_n multiplex_n which_o be_v a_o term_n proper_a to_o arithmetic_n and_o number_n it_o be_v easy_a to_o consider_v that_o there_o can_v be_v no_o exact_a knowledge_n of_o proportion_n and_o proportionality_n and_o so_o of_o this_o five_o book_n with_o all_o the_o other_o book_n follow_v without_o the_o aid_n and_o knowledge_n of_o number_n proportion_n be_v a_o certain_a respect_n of_o two_o magnitude_n of_o one_o kind_n definition_n according_a to_o quantity_n kind_n proportion_n rational_a be_v divide_v into_o two_o kind_n into_o proportion_n of_o equality_n and_o into_o proportion_n of_o inequality_n proportion_n of_o equality_n be_v when_o one_o quantity_n be_v refer_v to_o a_o other_o equal_a unto_o itself_o as_o if_o you_o compare_v 5_o to_o 5_o or_o 7_o to_o 7_o &_o so_o of_o other_o and_o this_o proportion_n have_v great_a use_n in_o the_o rule_n of_o cosse_n equality_n for_o in_o it_o all_o the_o rule_n of_o equation_n tend_v to_o none_o other_o end_n but_o to_o find_v out_o and_o bring_v forth_o a_o number_n equal_a to_o the_o number_n suppose_v which_o be_v to_o put_v the_o proportion_n of_o equality_n inequality_n proportion_n of_o inequality_n be_v when_o one_o unequal_a quantity_n be_v compare_v to_o a_o other_o as_o the_o great_a to_o the_o less_o as_o 8._o to_o 4_o or_o 9_o to_o 3_o or_o the_o less_o to_o the_o great_a as_o 4._o to_o 8_o or_o 3._o to_o 9_o less_o proportion_n of_o the_o great_a to_o the_o less_o have_v five_o kind_n namely_o multiplex_n superparticular_a superpartiens_fw-la multiplex_fw-la superperticular_a and_o multiplex_fw-la superpartiens_fw-la multiplex_fw-la be_v when_o the_o antecedent_n contain_v in_o itself_o the_o consequent_a certain_a time_n without_o more_o or_o less_o multiplex_fw-la as_o twice_o thrice_o four_o time_n and_o so_o far_o and_o this_o proportion_n have_v under_o it_o infinite_a kind_n for_o if_o the_o antecedent_n contain_v the_o consequent_a just_o twice_o it_o be_v call_v dupla_fw-la proportion_n proportion_n as_o 4_o to_o 2._o if_o thrice_o tripla_fw-la quintuple_a as_o 9_o to_o 3._o if_o 4._o time_n quadrupla_fw-la as_o 12._o to_o 3._o if_o 5._o time_n quintupla_fw-la as_o 15._o to_o 3._o and_o so_o infinite_o after_o the_o same_o manner_n superperticular_a superperticular_a be_v when_o the_o antecedent_n contain_v the_o consequent_a only_o once_o &_o moreover_o some_o one_o part_n thereof_o as_o a_o half_a a_o three_o or_o four_o etc_n etc_n this_o kind_n also_o have_v under_o it_o infinite_a kind_n for_o if_o the_o antecedent_n contain_v the_o consequent_a once_o and_o a_o half_a thereof_o it_o be_v call_v sesquialtera_fw-la sesquialtera_fw-la as_o 6._o to_o 4_o if_o once_o and_o a_o three_o part_n sesquitertia_fw-la sesquitertia_fw-la as_o 4._o to_o 3_o if_o once_o and_o a_o four_o part_n sesquiquarta_fw-la sesquiquarta_fw-la as_o 5._o to_o 4._o and_o so_o in_o like_a manner_n infinite_o superpartiens_fw-la superpartiens_fw-la be_v when_o the_o antecedent_n contain_v the_o consequent_a only_o once_o &_o moreover_o more_o part_n than_o one_o of_o the_o same_o as_o two_o three_o three_o fourthe_n four_o fifthe_n and_o so_o forth_o this_o also_o have_v infinite_a kind_n under_o it_o for_o if_o the_o antecedent_n contain_v above_o the_o consequent_a two_o part_n it_o be_v call_v superbipartiens_fw-la superbipartiens_fw-la as_o 7._o to_o 5._o if_o 3._o part_n supertripartiens_fw-la as_o 7._o to_o 4._o supertripartiens_fw-la if_o 4._o part_n superquadripartiens_n superquadripartiens_fw-la as_o 9_o to_o 5._o if_o 5._o part_n superquintipartiens_n as_o 11._o to_o 6._o superquintipartiens_fw-la and_o so_o forth_o infinite_o superperticular_a multiplex_fw-la superperticular_a be_v when_o the_o antecedent_n contain_v the_o consequent_a more_o than_o once_o and_o moreover_o only_o one_o part_n of_o the_o same_o this_o kind_n likewise_o have_v infinite_a kind_n under_o it_o for_o if_o the_o antecedent_n contain_v the_o consequent_a twice_o and_o half_o thereof_o it_o be_v call_v dupla_fw-la sesquialtera_fw-la sesquialtera_fw-la as_o 5._o to_o 2._o if_o twice_o and_o a_o three_o dupla_fw-la sesquitertia_fw-la as_o 7._o to_o 3._o sesquitertia_fw-la if_o thrice_o and_o a_o half_a tripla_fw-la sesquialtera_fw-la as_o 7._o to_o 2._o sesquialtera_fw-la if_o four_o time_n and_o a_o half_a quadrâ—pla_fw-la sesquialtera_fw-la as_o 9_o to_o 2._o and_o so_o go_v on_o infinite_o superpartiens_fw-la multiplex_fw-la superpartient_fw-fr be_v when_o the_o antecedent_n contain_v the_o consequent_a more_o than_o once_o and_o also_o more_o part_n than_o one_o of_o the_o consequent_a and_o this_o kind_n also_o have_v infinite_a kind_n under_o it_o for_o if_o the_o antecedent_n contain_v the_o consequent_a twice_o and_o two_o part_n over_o it_o be_v call_v dupla_fw-la superbipartiens_fw-la as_o 8._o to_o 3._o superbipartiens_fw-la if_o twice_o and_o three_o part_n dupla_fw-la supertripartiens_fw-la as_o 11._o to_o 4._o supertripartiens_fw-la if_o thrice_o and_o two_o part_n it_o be_v name_v tripla_fw-la superbipartiens_fw-la as_o 11._o to_o 3._o superbipartiens_fw-la if_o three_o time_n and_o four_o part_n treble_a superquadripartiens_n as_o 31._o to_o 9_o superquadâ—ipartiens_fw-la and_o so_o forth_o infinite_o here_o be_v to_o be_v note_v that_o the_o denomination_n of_o the_o proportion_n between_o any_o two_o number_n be_v have_v by_o devide_v of_o the_o great_a by_o the_o less_o for_o the_o quotient_a oâ—_n number_v produce_v of_o that_o division_n be_v even_o the_o denomination_n of_o the_o proportion_n proportion_n which_o in_o the_o first_o kind_n of_o proportion_n namely_o multiplex_n be_v ever_o a_o whole_a number_n and_o in_o all_o other_o kind_n of_o proportion_n it_o be_v a_o break_a number_n as_o if_o you_o will_v know_v the_o denomination_n of_o the_o proportion_n between_o 9_o and_o 3._o divide_v 9_o by_o 3._o so_o shall_v you_o have_v in_o the_o quotient_a 3._o which_o be_v a_o whole_a number_n and_o be_v the_o denomination_n of_o the_o proportion_n and_o show_v that_o the_o proportion_n between_o 9_o &_o 3._o be_v tripla_fw-la so_o the_o proportion_n between_o 12._o and_o 3._o be_v quadrupla_fw-la for_o that_o 12._o be_v divide_v by_o 3._o the_o quotient_n be_v 4._o and_o so_o of_o other_o in_o the_o kind_n of_o multiplex_n and_o although_o in_o this_o kind_a the_o quotient_n be_v ever_o a_o whole_a number_n yet_o proper_o it_o be_v refer_v to_o unity_n and_o so_o be_v represent_v in_o manner_n of_o a_o break_a number_n as_o 5_o â—_o and_o 4_o â—_o for_o unity_n be_v the_o denomination_n to_o a_o whole_a number_n likewise_o the_o denomination_n of_o the_o proportion_n between_o 4_o and_o 3_o be_v 1._o 1_o â—_o for_o that_o 4_o divide_v by_o 3._o have_v in_o the_o quotient_a 1_o 1_o â—_o one_o and_o a_o three_o part_n of_o which_o three_o part_n it_o be_v call_v sesquitercia_fw-la so_o the_o proportion_n between_o 7_o and_o 6._o be_v 1_o â…™_n one_o and_o a_o six_o of_o which_o fix_v part_v it_o be_v call_v sesquisexta_fw-la and_o so_o of_o other_o of_o that_o kind_n also_o between_o 7_o and_o 5_o the_o denomination_n of_o the_o proportion_n be_v 1_o â—_o â—_o one_o and_o two_o fifthe_n which_o denomination_n consist_v of_o two_o part_n namely_o of_o the_o munerator_n and_o denominator_fw-la of_o the_o quotient_a of_o 2._o and_o 5_o of_o which_o two_o fifthe_n it_o be_v call_v superbipartiens_fw-la quintas_fw-la for_o 2_o the_o numerator_n show_v the_o denomination_n of_o the_o number_n of_o the_o part_n and_o 5._o the_o denominator_fw-la show_v the_o denomination_n what_o part_n they_o be_v &_o so_o of_o other_o also_o the_o denomination_n between_o
proportion_n than_o have_v bc_o to_o ef._n wherefore_o a_o have_v to_o d_o a_o great_a proportion_n than_o have_v bc_o to_o ef._n wherefore_o alternate_o a_o have_v to_o bc_o a_o great_a proportion_n than_o have_v d_z to_o of_o wherefore_o by_o composition_n abc_n have_v to_o bc_o a_o great_a proportion_n than_o have_v def_n to_o ef._n wherefore_o again_o alternate_o abc_n have_v to_o def_n a_o great_a proportion_n than_o have_v bc_o to_o ef._n wherefore_o by_o the_o former_a proposition_n the_o proportion_n of_o a_o to_o d_z be_v great_a than_o the_o proportion_n of_o abc_n to_o def_n which_o be_v require_v to_o be_v prove_v the_o end_n of_o the_o five_o book_n of_o euclides_n element_n ¶_o the_o six_o book_n of_o euclides_n element_n this_o spaniard_o book_n be_v for_o use_v and_o practice_v book_n a_o most_o special_a book_n in_o it_o be_v teach_v the_o proportion_n of_o one_o figure_n to_o a_o other_o figure_n &_o of_o their_o side_n the_o one_o to_o the_o other_o and_o of_o the_o side_n of_o one_o to_o the_o side_n of_o a_o other_o likewise_o of_o the_o angle_n of_o the_o one_o to_o the_o angle_n of_o the_o other_o moreover_o it_o teach_v the_o description_n of_o figure_n like_o to_o â—_z give_v and_o marvelous_a application_n of_o figure_n to_o line_n even_o or_o with_o decrease_n or_o excess_n with_o many_o other_o theorem_n not_o only_o of_o the_o propoâ—tions_n of_o right_n line_v figure_n but_o also_o of_o sector_n of_o circle_n with_o their_o angle_n on_o the_o theorem_n and_o problem_n of_o this_o book_n depend_v for_o the_o most_o part_n the_o composition_n of_o all_o instrument_n of_o measure_v length_n breadth_n or_o deepen_n and_o also_o the_o reason_n of_o the_o use_n of_o the_o same_o instrument_n as_o of_o the_o geometrical_a â—quarâ—_n geometry_n the_o scale_n of_o the_o astrolabe_n the_o quadrant_a the_o staff_n and_o such_o other_o the_o use_n of_o which_o instrument_n beside_o all_o other_o mechanical_a instrument_n of_o raise_v up_o of_o move_v and_o draw_v huge_a thing_n incredible_a to_o the_o ignorant_a and_o infinite_a other_o begin_v which_o likewise_o have_v their_o ground_n out_o of_o this_o book_n be_v of_o wonderful_a and_o unspeakable_a profit_n beside_o the_o inestimable_a pleasure_n which_o be_v in_o they_o definition_n 1._o like_o rectiline_a figure_n be_v such_o definition_n who_o angle_n be_v equal_a the_o one_o to_o the_o other_o and_o who_o side_n about_o the_o equal_a angle_n be_v proportional_a as_o if_o you_o take_v any_o two_o rectiline_a figure_n as_o for_o example_n two_o triangle_n abc_n and_o def_n 〈…〉_o of_o the_o one_o triangle_n be_v equal_a to_o the_o angle_n of_o the_o other_o namely_o if_o the_o angle_n a_o be_v equal_a to_o the_o angle_n d_o and_o the_o angle_n b_o equal_v to_o the_o angle_n e_o &_o also_o the_o angle_n c_o equal_v to_o the_o angle_n f._n and_o moreover_o iâ—_z the_o side_n which_o contain_v the_o equal_a angle_n be_v proportional_a as_o if_o the_o side_n ab_fw-la have_v that_o proportion_n to_o the_o side_n bc_o whâ—ch_v the_o side_n de_fw-fr have_v to_o the_o side_n of_o and_o also_o if_o the_o side_n bc_o be_v unto_o the_o side_n ca_n as_o â—he_n side_n of_o be_v to_o the_o side_n fd_o and_o moreover_o if_o the_o side_n ca_n be_v to_o the_o side_n ab_fw-la as_o the_o side_n fd_o be_v to_o the_o side_n de_fw-fr then_o be_v these_o two_o triangle_n say_v to_o be_v like_o and_o so_o judge_v you_o of_o any_o other_o kind_n of_o figure_n as_o if_o in_o the_o parallelogram_n abcd_o and_o efgh_a the_o angle_n a_o be_v equal_a to_o the_o angle_n e_o and_o the_o angle_n b_o equal_v to_o the_o angle_n fletcher_n and_o the_o angle_n c_o equal_v to_o the_o angle_n g_o and_o the_o angle_n d_o equal_v to_o the_o angle_n h._n and_o farthermore_o if_o the_o side_n ac_fw-la have_v that_o proportion_n to_o the_o side_n cd_o which_o the_o side_n eglantine_n have_v to_o the_o side_n gh_o and_o if_o also_o the_o side_n cd_o be_v to_o the_o side_n db_o as_o the_o side_n 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the_o line_n ab_fw-la divide_v by_o a_o extreme_a and_o mean_a proportion_n common_o it_o be_v call_v a_o line_n divide_v by_o proportion_n haâ—ing_v a_o mean_a and_o two_o extreme_n how_o to_o divide_v a_o line_n in_o such_o sort_n be_v teach_v in_o the_o 11._o proposition_n of_o the_o second_o book_n but_o not_o under_o this_o form_n of_o proportion_n 4._o the_o alitude_n of_o a_o figure_n be_v a_o perpendicular_a line_n draw_v from_o the_o top_n to_o the_o base_a definition_n as_o the_o altitude_n or_o height_n of_o the_o triangle_n abc_n be_v the_o line_n ad_fw-la be_v draw_v perpendicular_o from_o the_o point_n a_o be_v the_o top_n or_o high_a part_n of_o the_o triangle_n to_o the_o base_a thereof_o bc._n so_o likewise_o in_o other_o figure_n as_o you_o see_v in_o the_o example_n here_o set_v that_o which_o here_o â—ee_n call_v the_o altitude_n or_o height_n of_o a_o figure_n in_o the_o first_o book_n in_o the_o 35._o proposition_n and_o certain_a other_o follow_v he_o teach_v 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be_v â—_o â—_o now_o if_o you_o multiply_v as_o you_o ought_v all_o these_o denomination_n together_o you_o shall_v produce_v 12._o to_o 6_o namely_o dupla_fw-la proportion_n forasmuch_o as_o so_o much_o have_v hitherto_o be_v speak_v of_o addition_n of_o proportion_n it_o shall_v not_o be_v unnecessary_a somewhat_o also_o to_o say_v of_o substraction_n of_o they_o proportion_n where_o it_o be_v to_o be_v note_v that_o as_o addition_n of_o they_o be_v make_v by_o multiplication_n of_o their_o denomination_n the_o one_o into_o the_o other_o so_o be_v the_o substraction_n of_o the_o one_o from_o the_o other_o do_v by_o division_n of_o the_o denomination_n of_o the_o one_o by_o the_o denomination_n of_o the_o other_o as_o if_o you_o will_v from_o sextupla_fw-la proportion_n subtrahe_fw-la dupla_fw-la proportion_n take_v
number_n even_o even_o be_v that_o which_o may_v be_v divide_v into_o two_o even_a part_n and_o that_o part_n again_o into_o two_o even_a part_n and_o so_o continual_o devide_v without_o stay_n 〈◊〉_d come_v to_o unity_n as_o by_o exampleâ—_n 64._o may_v be_v divide_v into_o 31_o and._n 32._o and_o either_o of_o these_o part_n may_v be_v divide_v into_o two_o even_a part_n for_o 32_o may_v be_v divide_v into_o 16_o and_o 16._o again_o 16_o may_v be_v divide_v into_o 8_o and_o 8_o which_o be_v even_o part_n and_o 8_o into_o 4_o and_o 4._o again_o 4_o into_o â—_o and_o â—_o and_o last_o of_o all_o may_v â—_o be_v divide_v into_o one_o and_o one_o 9_o a_o number_n even_o odd_a call_v in_o latin_a pariter_fw-la impar_fw-la be_v that_o which_o a_o even_a number_n measure_v by_o a_o odd_a number_n definition_n as_o the_o number_n 6_o which_o 2_o a_o even_a number_n measure_v by_o 3_o a_o odd_a number_n three_o time_n 2_o be_v 6._o likewise_o 10._o which_o 2._o a_o even_a number_n measure_v 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rational_a line_n but_o be_v rational_a only_o by_o reason_n of_o relation_n and_o comparison_n which_o they_o have_v unto_o it_o because_o they_o be_v commensurable_a unto_o it_o either_o in_o length_n and_o power_n or_o in_o power_n only_o far_o here_o be_v to_o be_v note_v that_o these_o word_n length_n and_o power_n and_o power_n only_o be_v join_v only_o with_o these_o wordeâ—_n commensurable_a or_o incommensurable_a and_o be_v never_o join_v with_o these_o word_n rational_a or_o irrational_a so_o that_o no_o line_n can_v be_v call_v rational_a in_o length_n or_o in_o power_n nor_o like_o wise_a can_v they_o be_v call_v irrational_a in_o length_n or_o in_o power_n wherein_o undoubted_o campanus_n be_v deceive_v book_n who_o use_v those_o word_n &_o speech_n indifferent_o cause_v &_o bring_v in_o great_a obscurity_n to_o the_o proposition_n and_o demonstration_n of_o this_o book_n which_o he_o shall_v easy_o see_v which_o mark_v with_o diligence_n the_o demonstration_n of_o campanus_n in_o this_o book_n 7_o line_n which_o be_v incommensurable_a to_o the_o rational_a line_n be_v call_v irrational_a definition_n by_o line_n incommensurable_a to_o the_o rational_a line_n suppose_v in_o this_o place_n he_o understand_v such_o as_o be_v incommensurable_a unto_o it_o both_o in_o length_n and_o in_o power_n for_o there_o be_v no_o line_n incommensurable_a in_o power_n only_o for_o it_o can_v be_v that_o any_o line_n shall_v so_o be_v incommensurable_a in_o power_n only_o that_o they_o be_v not_o also_o incommensurable_a in_o length_n what_o so_o ever_o line_n be_v incommensurable_a in_o power_n the_o same_o be_v also_o incommensurable_a in_o length_n neither_o can_v euclid_n here_o in_o this_o place_n mean_a line_n incommensurable_a in_o length_n only_o for_o in_o the_o definition_n before_o he_o call_v they_o rational_a line_n neither_o may_v they_o be_v place_v amongst_o irrational_a line_n wherefore_o it_o remain_v that_o in_o this_o diffintion_n he_o speak_v only_o of_o those_o line_n which_o be_v incommensurable_a to_o the_o rational_a line_n first_o give_v and_o suppose_v both_o in_o length_n and_o in_o power_n which_o by_o all_o mean_n be_v incommensurable_a to_o the_o rational_a line_n &_o therefore_o most_o apt_o be_v they_o call_v irrational_a line_n this_o definition_n be_v easy_a to_o be_v understand_v by_o that_o which_o have_v be_v say_v before_o yet_o for_o the_o more_o plainness_n see_v this_o example_n let_v the_o â—â—rst_a rational_a line_n suppose_v be_v the_o line_n ab_fw-la who_o square_a or_o quadrate_n let_v be_v abcd._n and_o let_v there_o be_v give_v a_o other_o line_n of_o which_o lâ—t_v be_v to_o the_o rational_a line_n incommensurable_a in_o length_n and_o power_n so_o that_o let_v no_o one_o line_n measure_v the_o length_n of_o the_o two_o line_n ab_fw-la and_o of_o and_o let_v the_o square_n of_o the_o line_n of_o be_v efgh_a now_o if_o also_o there_o be_v no_o one_o superficies_n which_o measure_v the_o two_o square_n abcd_o and_o efgh_a as_o be_v suppose_v to_o be_v in_o this_o example_n they_o be_v the_o line_n of_o a_o irrational_a line_n which_o word_n irrational_a as_o before_o do_v this_o word_n rational_a mislike_v many_o learned_a in_o this_o knowledge_n of_o geometry_n flussates_n uncertain_a as_o he_o leave_v the_o word_n rational_a and_o in_o stead_n thereof_o use_v this_o word_n certain_a so_o here_o he_o leave_v the_o word_n irrational_a and_o use_v in_o place_n thereof_o this_o word_n uncertain_a and_o ever_o name_v these_o line_n uncertain_a line_n petrus_n montaureus_n also_o mislike_v the_o word_n irrational_a will_v rather_o have_v they_o to_o be_v call_v surd_a line_n yet_o because_o this_o word_n irrational_a have_v ever_o by_o custom_n and_o long_a use_n so_o general_o be_v receive_n he_o use_v continual_o the_o same_o in_o greek_a such_o line_n be_v call_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d alogoi_n which_o signify_v nameless_a unspeakable_a uncertain_a in_o determinate_a line_n and_o with_o out_o proportion_n not_o that_o these_o irrational_a line_n have_v no_o proportion_n at_o all_o either_o to_o the_o first_o rational_a line_n or_o between_o themselves_o but_o be_v so_o name_v for_o that_o their_o proportion_n to_o the_o rational_a line_n can_v be_v express_v in_o number_n that_o be_v undoubted_o very_o untrue_a which_o many_o write_v that_o their_o proportion_n be_v unknown_a both_o to_o we_o and_o to_o nature_n be_v it_o not_o think_v you_o a_o thing_n very_o absurd_a to_o say_v that_o there_o be_v any_o thing_n in_o nature_n and_o produce_v by_o nature_n to_o be_v hide_v from_o nature_n and_o not_o to_o be_v know_v of_o nature_n it_o can_v not_o be_v say_v that_o their_o proportion_n be_v utter_o hide_v and_o unknown_a to_o we_o much_o less_o unto_o nature_n although_o we_o can_v geve_v they_o their_o name_n and_o distinct_o express_v they_o by_o number_n otherwise_o shall_v euclid_n have_v take_v all_o this_o travel_n and_o wonderful_a diligence_n bestow_v in_o this_o bookeâ—_n in_o vain_a and_o to_o no_o useâ—_n in_o which_o he_o do_v nothing_o ellâ—_n but_o teach_v the_o propriety_n and_o passion_n of_o these_o irrational_a linesâ—_n and_o show_v the_o proportion_n which_o they_o have_v the_o one_o to_o the_o other_o here_o be_v also_o to_o be_v note_v which_o thing_n also_o tartalea_n have_v before_o diligent_o notedâ—_n that_o campanus_n and_o many_o other_o writer_n of_o geometryâ—_n over_o much_o â—â—â—ed_a and_o be_v deceive_v in_o that_o they_o write_v and_o teach_v that_o all_o these_o line_n who_o square_n be_v not_o signify_v and_o may_v be_v express_v by_o a_o square_a number_n although_o they_o may_v by_o any_o other_o number_n as_o by_o 11._o 12._o 14._o and_o such_o other_o not_o square_a number_n be_v irrational_a line_n which_o be_v manifest_o repugnant_a to_o the_o ground_n and_o principle_n of_o euclid_n who_o will_v that_o all_o line_n which_o be_v commensurable_a to_o the_o rational_a line_n whether_o it_o be_v in_o length_n and_o power_n or_o in_o power_n only_o shall_v be_v rational_a undoubted_o this_o have_v be_v one_o of_o the_o chief_a and_o great_a cause_n of_o the_o wonderful_a confusion_n and_o darkness_n of_o this_o book_n book_n which_o so_o have_v toss_v and_o turmoil_v the_o wit_n of_o all_o both_o writer_n and_o reader_n master_n and_o scholar_n and_o so_o overwhelm_v they_o that_o they_o can_v not_o with_o out_o infinite_a travel_n and_o sweat_v attain_v to_o the_o truth_n and_o perfect_a understanding_n thereof_o definition_n 8_o the_o square_n which_o be_v describe_v of_o the_o rational_a right_a line_n suppose_v be_v rational_a until_o this_o definition_n have_v euclid_n set_v forth_o the_o nature_n and_o propriety_n of_o the_o first_o kind_n of_o magnitude_n namely_o of_o line_n how_o they_o be_v rational_a or_o irrational_a now_o he_o begin_v to_o â—hew_v how_o the_o second_o kind_n of_o magnitude_n namely_o superficies_n be_v one_o to_o the_o other_o rational_a or_o irrational_a this_o definition_n be_v very_a plain_n suppose_v the_o line_n ab_fw-la to_o be_v the_o rational_a line_n have_v his_o part_n and_o division_n certain_o know_v the_o square_a of_o which_o line_n let_v be_v the_o square_a abcd._n now_o because_o it_o be_v the_o square_a of_o the_o rational_a line_n ab_fw-la it_o be_v also_o call_v rational_a and_o as_o the_o line_n ab_fw-la be_v the_o first_o rational_a line_n unto_o which_o other_o line_n compare_v be_v count_v rational_a or_o irrational_a so_o be_v the_o quadrat_fw-la or_o square_v thereof_o the_o â—irst_n rational_a superficies_n unto_o which_o all_o other_o square_n or_o figure_n compare_v be_v count_v and_o name_v rational_a or_o irrational_a 9_o such_o which_o be_v commensurable_a unto_o it_o be_v rational_a definition_n in_o this_o definition_n where_o it_o be_v say_v such_o as_o be_v commensurable_a to_o the_o square_n of_o the_o rational_a line_n be_v not_o understand_v only_o other_o square_n or_o
a_o assumpt_n forasmuch_o as_o in_o the_o eight_o book_n in_o the_o 26._o proposition_n it_o be_v prove_v that_o like_o plain_a number_n have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o likewise_o in_o the_o 24._o of_o the_o same_o book_n it_o be_v prove_v that_o if_o two_o number_n have_v that_o proportion_n the_o one_o to_o the_o other_o eight_o that_o a_o square_a number_n have_v to_o a_o square_a number_n those_o number_n be_v like_o plain_a number_n hereby_o it_o be_v manifest_a that_o unlike_o plain_a number_n that_o be_v who_o side_n be_v not_o proportional_a have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n for_o if_o they_o have_v than_o shall_v they_o be_v like_o plain_a number_n which_o be_v contrary_a to_o the_o supposition_n wherefore_o unlike_o plain_a number_n have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o therefore_o square_n which_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o unlike_o plain_a number_n have_v shall_v have_v their_o side_n incommensurable_a in_o length_n by_o the_o last_o part_n of_o the_o former_a proposition_n for_o that_o those_o square_n have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n ¶_o the_o 8._o theorem_a the_o 10._o proposition_n if_o four_o magnitude_n be_v proportional_a and_o if_o the_o first_o be_v commensurable_a unto_o the_o second_o the_o three_o also_o shall_v be_v commensurable_a unto_o the_o four_o and_o if_o the_o first_o be_v incommensurable_a unto_o the_o second_o the_o three_o shall_v also_o be_v incommensurable_a unto_o the_o four_o svppose_v that_o these_o four_o magnitude_n a_o b_o c_o d_o be_v proportional_a as_o a_o be_v to_o b_o so_o let_v c_o be_v to_o d_o and_o let_v a_o be_v commensurable_a unto_o b._n then_o i_o say_v that_o c_z be_v also_o commensurable_a unto_o d._n part_n for_o forasmuch_o as_o a_o be_v commensurable_a unto_o b_o it_o have_v by_o the_o five_o of_o the_o ten_o that_o proportion_n that_o number_n have_v to_o number_v but_o as_o a_o be_v to_o b_o so_o be_v c_o to_o d._n wherefore_o c_z also_o have_v unto_o d_z that_o proportion_n that_o number_n have_v to_o number_v wherefore_o c_o be_v commensurable_a unto_o d_z by_o the_o 6._o of_o the_o ten_o but_o now_o suppose_v that_o the_o magnitude_n a_o be_v incommensurable_a unto_o the_o magnitude_n b._n partâ—_n then_o i_o say_v that_o the_o magnitude_n c_o also_o be_v incommensurable_a unto_o the_o magnitude_n d._n for_o forasmuch_o as_o a_o be_v incommensurable_a unto_o b_o therefore_o by_o the_o 7._o of_o this_o book_n a_o have_v not_o unto_o b_o such_o proportion_n as_o number_n have_v to_o number_v but_o as_o a_o be_v to_o b_o so_o be_v c_o to_o d._n wherefore_o c_o have_v not_o unto_o d_z such_o proportion_n as_o number_n have_v to_o number_v wherefore_o by_o the_o 8._o of_o the_o ten_o c_o be_v incommensurable_a unto_o d._n if_o therefore_o there_o be_v four_o magnitude_n proportional_a and_o if_o the_o first_o be_v commensurable_a unto_o the_o second_o the_o three_o also_o shall_v be_v commensurable_a unto_o the_o four_o and_o if_o the_o first_o be_v incommensurable_a unto_o the_o second_o the_o three_o shall_v also_o be_v incommensurable_a unto_o the_o four_o which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o montaureus_n if_o there_o be_v four_o line_n proportional_a and_o if_o the_o two_o first_o or_o the_o two_o last_o be_v commensurable_a in_o power_n only_o the_o other_o two_o also_o shall_v be_v commensurable_a in_o power_n only_o corollary_n this_o be_v prove_v by_o the_o 22._o of_o the_o six_o and_o by_o this_o ten_o proposition_n and_o this_o corollary_n euclid_n use_v in_o the_o 27._o and_o 28._o proposition_n of_o this_o book_n and_o in_o other_o proposition_n also_o ¶_o the_o 3._o problem_n the_o 11._o proposition_n unto_o a_o right_a line_n first_o set_v and_o give_v which_o be_v call_v a_o rational_a line_n to_o find_v out_o two_o right_a line_n incommensurable_a the_o one_o in_o length_n only_o and_o the_o other_o in_o length_n and_o also_o in_o power_n svppose_v that_o the_o right_a line_n first_o set_v and_o give_v which_o be_v call_v a_o rational_a line_n of_o purpose_n be_v a._n it_o be_v require_v unto_o the_o say_a line_n a_o to_o find_v out_o two_o right_a line_n incommensurable_a the_o one_o in_o length_n only_o the_o other_o both_o in_o length_n and_o in_o power_n give_v take_v by_o that_o which_o be_v add_v after_o the_o 9_o proposition_n of_o this_o book_n two_o number_n b_o and_o c_o not_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n that_o be_v let_v they_o not_o be_v like_o plain_a number_n for_o like_o plain_a number_n by_o the_o 26._o of_o the_o eight_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o as_o the_o number_n b_o be_v to_o the_o number_n c_o so_fw-mi let_v the_o square_n of_o the_o line_n a_o be_v unto_o the_o square_n of_o a_o other_o line_n namely_o of_o d_z how_o to_o do_v this_o be_v teach_v in_o the_o assumpt_n put_v before_o the_o 6._o proposition_n of_o this_o book_n wherefore_o the_o square_a of_o the_o line_n a_o be_v unto_o the_o square_n of_o the_o line_n d_o commensurable_a by_o the_o six_o of_o the_o ten_o and_o forasmuch_o as_o the_o number_n b_o have_v not_o unto_o the_o number_n c_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o the_o square_a of_o the_o line_n a_o have_v not_o unto_o the_o square_n of_o the_o line_n d_z that_o proportion_n that_o a_o square_a number_n have_v to_o a_o number_n wherefore_o by_o the_o 9_o of_o the_o ten_o the_o line_n a_o be_v unto_o the_o line_n d_o incommensurable_a in_o length_n only_o and_o so_o be_v find_v out_o the_o first_o line_n namely_o d_z incommensurable_a in_o length_n only_o to_o the_o line_n give_v a._n give_v again_o take_v by_o the_o 13._o of_o the_o six_o the_o mean_a proportional_a between_o the_o line_n a_o and_o d_o and_o let_v the_o same_o be_v e._n wherefore_o as_o the_o line_n a_o be_v to_o the_o line_n d_o so_o be_v the_o square_a of_o the_o line_n a_o to_o the_o square_n of_o the_o line_n e_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o but_o the_o line_n a_o be_v unto_o the_o line_n d_o incommensurable_a in_o length_n wherefore_o also_o the_o square_a of_o the_o line_n a_o be_v unto_o the_o square_n of_o the_o line_n e_o incommensurable_a by_o the_o second_o part_n of_o the_o former_a proposition_n now_o forasmuch_o as_o the_o square_n of_o the_o line_n a_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n e_o it_o follow_v by_o the_o definition_n of_o incommensurable_a line_n that_o the_o line_n a_o be_v incommensurable_a in_o power_n to_o the_o line_n e._n wherefore_o unto_o the_o right_a line_n give_v and_o first_o set_v a_o which_o be_v a_o rational_a line_n and_o which_o be_v suppose_v to_o have_v such_o division_n and_o so_o many_o part_n as_o you_o listen_v to_o conceive_v in_o mind_n as_o in_o this_o example_n 11_z whereunto_o as_o be_v declare_v in_o the_o 5._o definition_n of_o this_o book_n may_v be_v compare_v infinite_a other_o line_n either_o commensurable_a or_o incommensurable_a be_v find_v out_o the_o line_n d_o incommensurable_a in_o length_n only_o wherefore_o the_o line_n d_o be_v rational_a by_o the_o six_o definition_n of_o this_o book_n for_o that_o it_o be_v incommensurable_a in_o length_n only_o to_o the_o line_n a_o which_o be_v the_o first_o line_n set_v and_o be_v by_o supposition_n rational_a there_o be_v also_o find_v out_o the_o line_n e_o which_o be_v unto_o the_o same_o line_n a_o incommensurable_a not_o only_o in_o length_n but_o also_o in_o power_n which_o line_n e_o compare_v to_o the_o rational_a line_n a_o be_v by_o the_o definition_n irrational_a for_o euclid_n always_o call_v those_o line_n irrational_a which_o be_v incommensurable_a both_o in_o length_n and_o in_o power_n to_o the_o line_n first_o set_v and_o by_o supposition_n rational_a ¶_o the_o 9_o theorem_a the_o 12._o proposition_n magnitude_n commensurable_a to_o one_o and_o the_o self_n same_o magnitude_n be_v also_o commensurable_a the_o one_o to_o the_o other_o svppose_v that_o either_o of_o these_o magnitude_n a_o and_o b_o be_v commensurable_a unto_o the_o magnitude_n c_o then_o i_o say_v that_o the_o magnitude_n a_o be_v commensurable_a unto_o the_o magnitude_n b._n construction_n for_o forasmuch_o as_o the_o
measure_v the_o square_a number_n produce_v of_o ef._n wherefore_o also_o by_o the_o 14._o of_o the_o eight_o the_o number_n g_z measure_v the_o number_n of_o and_o the_o number_n g_o also_o measure_v itself_o wherefore_o the_o number_n g_z measure_v these_o number_n of_o and_o g_z when_o yet_o they_o be_v prime_a the_o one_o to_o the_o other_o which_o be_v impossible_a wherefore_o the_o diameter_n a_o be_v not_o commensurable_a in_o length_n to_o the_o side_n b._n wherefore_o it_o be_v incommensurable_a which_o be_v require_v to_o be_v demonstrate_v an_o other_o demonstration_n after_o flussas_n suppose_v that_o upon_o the_o line_n ab_fw-la be_v describe_v a_o square_n who_o diameter_n let_v be_v the_o line_n ac_fw-la then_o i_o say_v that_o the_o side_n ab_fw-la be_v incommensurable_a in_o length_n unto_o the_o diameter_n ac_fw-la forasmuch_o as_o the_o line_n ab_fw-la and_o bc_o be_v equal_a therefore_o the_o square_a of_o the_o line_n ac_fw-la be_v double_a to_o the_o square_n of_o the_o line_n ab_fw-la by_o the_o 47._o of_o the_o first_o take_v by_o the_o 2._o of_o the_o eight_o number_n how_o many_o soever_o in_o continual_a proportion_n from_o unity_n and_o in_o the_o proportion_n of_o the_o square_n of_o the_o line_n ab_fw-la and_o ac_fw-la which_o let_v be_v the_o number_n d_o e_o f_o g._n and_o forasmuch_o as_o the_o first_o from_o unity_n namely_o e_o be_v no_o square_a number_n for_o that_o it_o be_v a_o prime_a number_n neither_o be_v also_o any_o other_o of_o the_o say_a number_n a_o square_a number_n except_o the_o three_o from_o unity_n and_o so_o all_o the_o rest_n leving_fw-mi one_o between_o by_o the_o 10._o of_o the_o nine_o wherefore_o d_z be_v to_o e_o or_o e_o to_o fletcher_n or_o fletcher_n to_o g_z in_o that_o proportion_n that_o a_o square_a number_n be_v to_o a_o number_n not_o square_a wherefore_o by_o the_o corollary_n of_o the_o 25._o of_o the_o eight_o they_o be_v not_o in_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n be_v to_o a_o square_a number_n wherefore_o neither_o also_o have_v the_o square_n of_o the_o line_n ab_fw-la and_o ac_fw-la which_o be_v in_o the_o same_o proportion_n that_o porportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o by_o the_o 9_o of_o this_o book_n their_o side_n namely_o the_o side_n ab_fw-la and_o the_o diameter_n ac_fw-la be_v incommensurable_a in_o length_n the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v this_o demonstration_n i_o think_v good_a to_o add_v for_o that_o the_o former_a demonstration_n seem_v not_o so_o full_a and_o they_o be_v think_v of_o some_o to_o be_v none_o of_o theon_n as_o also_o the_o proposition_n to_o be_v none_o of_o euclides_n here_o follow_v a_o instruction_n by_o some_o studious_a and_o skilful_a grecian_a perchance_o theon_n which_o teach_v we_o of_o far_a use_n and_o fruit_n of_o these_o irrational_a line_n see_v that_o there_o be_v find_v out_o right_a line_n incommensurable_a in_o length_n the_o one_o to_o the_o â—ther_n as_o the_o line_n a_o and_o b_o there_o may_v also_o be_v find_v out_o many_o other_o magnitude_n have_v length_n and_o breadth_n such_o as_o be_v plain_a superficiece_n which_o shall_v be_v incommensurable_a the_o one_o to_o the_o other_o for_o if_o by_o the_o 13._o of_o the_o six_o between_o the_o line_n a_o and_o b_o there_o be_v take_v the_o mean_a proportional_a line_n namely_o c_z then_o by_o the_o second_o corollary_n of_o the_o 20._o of_o the_o six_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o figure_n describe_v upon_o the_o line_n a_o to_o the_o figure_n describe_v upon_o the_o line_n c_o be_v both_o like_a and_o in_o like_a sort_n describe_v that_o be_v whether_o they_o be_v square_n which_o be_v always_o like_o the_o one_o to_o the_o other_o or_o whether_o they_o be_v any_o other_o like_o rectiline_a figure_n or_o whether_o they_o be_v circle_n about_o the_o diameter_n a_o and_o c._n for_o circle_n have_v that_o proportion_n the_o one_o to_o the_o other_o that_o the_o square_n of_o their_o diameter_n have_v by_o the_o 2._o of_o the_o twelve_o wherefore_o by_o the_o second_o part_n of_o the_o 10._o of_o the_o ten_o the_o â—_z describe_v upon_o the_o line_n a_o and_o c_o being_n like_o and_o in_o like_a sort_n describe_v be_v incommensurable_a the_o one_o to_o the_o other_o wherefore_o by_o this_o mean_n there_o be_v find_v out_o superficieces_o incommensurable_a the_o one_o to_o the_o other_o in_o like_a sort_n there_o may_v be_v find_v out_o figure_n commensurable_a the_o one_o to_o the_o other_o if_o you_o put_v the_o line_n a_o and_o b_o to_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o and_o see_v that_o it_o be_v so_o now_o let_v we_o also_o prove_v that_o even_o in_o soliâ—es_n also_o or_o body_n there_o be_v some_o commensurable_a the_o one_o to_o the_o other_o and_o other_o some_o incommensurable_a the_o one_o to_o the_o other_o for_o if_o from_o each_o of_o the_o square_n of_o the_o line_n a_o and_o b_o or_o from_o any_o other_o rectiline_a figure_n equal_a to_o these_o square_n be_v erect_v solid_n of_o equal_a altitude_n whether_o those_o solid_n be_v compose_v of_o equidistant_a supersiciece_n or_o whether_o they_o be_v pyramid_n or_o prism_n thosâ—_n solid_n sâ—_n erected_a shall_v be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v by_o the_o 32._o oâ—_n the_o eleven_o and_o 5._o and_o 6._o of_o the_o twelve_o howbeit_o there_o be_v no_o such_o proposition_n concern_v prism_n and_o so_o if_o the_o base_n of_o the_o solid_n bâ—_z commensurable_a the_o one_o to_o the_o other_o the_o solid_n also_o shall_v be_v commensurable_a the_o one_o to_o the_o other_o and_o if_o the_o base_n be_v incommensurable_a the_o one_o to_o the_o other_o the_o solid_n also_o shall_v be_v incommensurable_a the_o one_o to_o the_o other_o by_o the_o 10._o of_o the_o ten_o and_o if_o there_o be_v two_o circle_n a_o and_o b_o and_o upon_o each_o of_o the_o circle_n be_v erect_v cones_n or_o cilinder_n of_o equal_a altitude_n those_o cones_n &_o cilinder_n sâ—all_v be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o the_o circle_n be_v which_o be_v their_o base_n by_o the_o 11._o of_o the_o twelve_o and_o so_o if_o the_o circle_n be_v commensurable_a the_o one_o to_o the_o other_o the_o cones_n and_o cilinder_n also_o shall_v be_v commensurable_a the_o one_o to_o the_o other_o but_o if_o the_o circle_n be_v incommensurable_a the_o one_o to_o the_o other_o the_o cones_n also_o and_o cilinder_n shall_v be_v incommensurable_a the_o one_o to_o the_o other_o by_o the_o 10._o of_o the_o ten_o wherefore_o it_o be_v manifest_a that_o not_o only_o in_o line_n and_o superâ—icieces_n but_o also_o in_o solid_n or_o body_n be_v find_v commensurabilitie_n or_o incommensurability_n a_o advertisement_n by_o john_n dee_n although_o this_o proposition_n be_v by_o euclid_n to_o this_o book_n allot_v as_o by_o the_o ancient_a graecian_n publish_v under_o the_o name_n of_o aristoteles_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d it_o will_v seem_v to_o be_v and_o also_o the_o property_n of_o the_o same_o agreeable_a to_o the_o matter_n of_o this_o book_n and_o the_o proposition_n itself_o so_o famous_a in_o philosophy_n and_o logic_n as_o it_o be_v will_v in_o manner_n crave_v his_o elemental_a place_n in_o this_o ten_o book_n yet_o the_o dignity_n &_o perfectionâ—_n of_o mathematical_a method_n can_v not_o allow_v it_o here_o as_o in_o due_a order_n follow_v but_o most_o apt_o after_o the_o 9_o proposition_n of_o this_o book_n as_o a_o corollary_n of_o the_o last_o part_n thereof_o and_o undoubted_o the_o proposition_n have_v for_o this_o 2000_o year_n be_v notable_o regard_v among_o the_o greek_n philosopher_n and_o before_o aristotle_n time_n be_v conclude_v with_o the_o very_a same_o inconvenience_n to_o the_o gaynesayer_n that_o the_o first_o demonstration_n here_o induce_v namely_o odd_a number_n to_o be_v equal_a to_o even_o as_o may_v appearâ—_n in_o aristotle_n work_n name_v analitica_fw-la prima_fw-la the_o first_o book_n and_o 40._o chapter_n but_o else_o in_o very_a many_o place_n of_o his_o work_n he_o make_v mention_n of_o the_o proposition_n evident_a also_o it_o be_v that_o euclid_n be_v about_o aristotle_n time_n and_o in_o that_o age_n the_o most_o excellent_a geometrician_n among_o the_o greek_n wherefore_o see_v it_o be_v so_o public_a in_o his_o time_n so_o famous_a and_o so_o appertain_v to_o the_o property_n of_o this_o book_n it_o be_v most_o likely_a both_o to_o be_v know_v to_o euclid_n and_o also_o to_o have_v be_v by_o he_o in_o apt_a order_n place_v but_o of_o the_o disorder_v of_o it_o can_v remain_v no_o doubt_n if_o you_o consider_v in_o zamberts_n translation_n
two_o other_o proposition_n go_v next_o before_o it_o so_o far_o misplace_v that_o where_o they_o be_v word_n for_o word_n before_o duâ—ly_o place_v be_v the_o 105._o and_o 106._o yet_o here_o after_o the_o book_n end_v they_o be_v repeat_v with_o the_o number_n of_o 116._o and_o 117._o proposition_n zambert_n therein_o be_v more_o faithful_a to_o follow_v as_o he_o find_v in_o his_o greek_n example_n than_o he_o be_v skilful_a or_o careful_a to_o do_v what_o be_v necessary_a yea_o and_o some_o greek_n write_v ancient_a copy_n have_v they_o not_o so_o though_o in_o deed_n they_o be_v well_o demonstrate_v yet_o truth_n disord_v be_v half_a disgracedâ—_n especial_o where_o the_o pattern_n of_o good_a order_n by_o profession_n be_v avouch_v to_o be_v but_o through_o ignorance_n arrogancy_n and_o â—emerltie_n of_o unskilful_a method_n master_n many_o thing_n remain_v yet_o in_o these_o geometrical_a element_n undue_o tumble_v in_o though_o true_a yet_o with_o disgrace_n which_o by_o help_n of_o so_o many_o wit_n and_o hability_n of_o such_o as_o now_o may_v have_v good_a cause_n to_o be_v skilful_a herein_o will_v i_o hope_v ere_o long_o be_v take_v away_o and_o thing_n of_o importance_n want_v supply_v the_o end_n of_o the_o ten_o book_n of_o euclides_n element_n ¶_o the_o eleven_o book_n of_o euclides_n element_n book_n hitherto_o have_v â—vclidâ—_n in_o thâ—sâ—_n former_a book_n with_o a_o wonderful_a method_n and_o order_n entreat_v of_o such_o kind_n of_o figure_n superficial_a which_o be_v or_o may_v be_v describe_v in_o a_o superficies_n or_o plain_a and_o have_v teach_v and_o set_v forth_o their_o property_n nature_n generation_n and_o production_n even_o from_o the_o first_o root_n ground_n and_o beginning_n of_o they_o namely_o from_o a_o point_n which_o although_o it_o be_v indivisible_a yet_o be_v it_o the_o begin_n of_o all_o quantity_n continual_a and_o of_o it_o and_o of_o the_o motion_n and_o slow_v thereof_o be_v produce_v a_o line_n and_o consequent_o all_o quantity_n continual_a as_o all_o figure_n plain_a and_o solid_a what_o so_o ever_o euclid_n therefore_o in_o his_o first_o book_n begin_v with_o it_o boot_n and_o from_o thence_o go_v he_o to_o a_o line_n as_o to_o a_o thing_n most_o simple_a next_o unto_o a_o point_n then_o to_o a_o superficies_n and_o to_o angle_n and_o so_o through_o the_o whole_a first_o book_n boâ—â—e_n he_o entreat_v of_o these_o most_o simple_a and_o plain_a ground_n in_o the_o second_o book_n he_o entreat_v further_o â—â—oâ—e_n and_o go_v unto_o more_o hard_a matter_n and_o teach_v of_o division_n of_o line_n and_o of_o the_o multiplication_n of_o line_n and_o of_o their_o part_n and_o of_o their_o passion_n and_o property_n and_o for_o that_o rightline_v â—_z be_v far_o distant_a in_o nature_n and_o property_n from_o round_a and_o circular_a figure_n in_o the_o three_o book_n he_o instruct_v the_o reader_n of_o the_o nature_n and_o condition_n of_o circle_n booâ—e_n in_o the_o four_o book_n he_o compare_v figure_n of_o right_a line_n and_o circle_n together_o bâ—oâ—e_n and_o teach_v how_o to_o describe_v a_o figure_n of_o right_a line_n with_o in_o or_o about_o a_o circle_n and_o contrariwise_o a_o circle_n with_o in_o or_o about_o a_o rectiline_a figure_n in_o the_o five_o book_n he_o search_v out_o the_o nature_n of_o proportion_n a_o matter_n of_o wonderful_a use_n and_o deep_a consideration_n boâ—â—e_n for_o that_o otherwise_o he_o can_v not_o compare_v â—igure_n with_o figure_n or_o the_o side_n of_o figure_n together_o for_o whatsoever_o be_v compare_v to_o any_o other_o thing_n be_v compare_v unto_o it_o undoubted_o under_o some_o kind_n of_o proportion_n wherefore_o in_o the_o six_o book_n he_o compare_v figure_n together_o booâ—e_n one_o to_o a_o other_o likewise_o their_o side_n and_o for_o that_o the_o nature_n of_o proportion_n can_v not_o be_v full_o and_o clear_o see_v without_o the_o knowledge_n of_o number_n wherein_o it_o be_v first_o and_o chief_o find_v in_o the_o seven_o bookâ—_n eight_o booâ—â—_n and_o nine_o book_n book_n he_o entreatâ—th_v of_o number_n &_o of_o the_o kind_n and_o property_n thereof_o and_o because_o that_o the_o side_n of_o solid_a body_n for_o the_o most_o part_n be_v of_o such_o sort_n that_o compare_v together_o they_o have_v such_o proportion_n the_o one_o to_o the_o other_o booâ—e_n which_o can_v not_o be_v express_v by_o any_o number_n certain_a and_o therefore_o be_v call_v irrational_a line_n he_o in_o the_o ten_o book_n have_v write_v &_o teach_v which_o lineâ—_z be_v commensurable_a or_o incommensurable_a the_o one_o to_o the_o other_o and_o of_o the_o diversity_n of_o kind_n of_o irrational_a line_n with_o all_o the_o condition_n &_o propriety_n of_o they_o and_o thus_o have_v euclid_n in_o these_o ten_o foresay_a book_n full_o &_o most_o plenteous_o in_o a_o marvelous_a order_n teach_v whatsoever_o seem_v necessary_a and_o requisite_a to_o the_o knowledge_n of_o all_o superficial_a figure_n of_o what_o sort_n &_o form_n so_o ever_o they_o be_v now_o in_o these_o book_n follow_v he_o entreat_v of_o figure_n of_o a_o other_o kind_n namely_o of_o bodily_a figure_n follow_v as_o of_o cube_n pyramid_n cones_n column_n cilinder_n parallelipipedon_n sphere_n and_o such_o othersâ—_n and_o show_v the_o diversity_n of_o they_o the_o generation_n and_o production_n of_o they_o and_o demonstrate_v with_o great_a and_o wonderful_a art_n their_o propriety_n and_o passion_n with_o all_o their_o nature_n and_o condition_n he_o also_o compare_v one_o oâ—_n they_o to_o a_o other_o whereby_o to_o know_v the_o reason_n and_o proportion_n of_o the_o one_o to_o the_o other_o chief_o of_o the_o five_o body_n which_o be_v call_v regular_a body_n element_n and_o these_o be_v the_o thing_n of_o all_o other_o entreat_v of_o in_o geometry_n most_o worthy_a and_o of_o great_a dignity_n and_o as_o it_o be_v the_o end_n and_o final_a intent_n of_o the_o whole_a be_v of_o geometry_n and_o for_o who_o cause_n have_v be_v write_v and_o speak_v whatsoever_o have_v hitherto_o in_o the_o former_a book_n be_v say_v or_o write_v as_o the_o first_o book_n be_v a_o ground_n and_o a_o necessary_a entry_n to_o all_o the_o râ—st_o â—ollowing_v so_o be_v this_o eleven_o book_n a_o necessary_a entry_n and_o ground_n to_o the_o rest_n which_o follow_v 〈◊〉_d and_o as_o that_o contain_v the_o declaration_n of_o word_n and_o definition_n of_o thinger_n requisite_a to_o the_o knowledge_n of_o superficial_a figure_n and_o entreat_v of_o line_n and_o of_o their_o division_n and_o section_n which_o be_v the_o term_n and_o limit_n of_o superficial_a figure_n so_o in_o this_o book_n be_v set_v forth_o the_o declaration_n of_o word_n and_o definition_n of_o thing_n pertain_v to_o solid_a and_o corporal_a figure_n and_o also_o of_o superficiece_n which_o be_v the_o term_n &_o limit_n of_o solid_n moreover_o of_o the_o division_n and_o intersection_n of_o they_o and_o diverse_a other_o thing_n without_o which_o the_o 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rare_a thing_n in_o all_o learn_n chief_o in_o the_o mathematical_n to_o have_v one_o thing_n more_o general_a than_o a_o other_o be_v it_o not_o true_a that_o every_o isosceles_a be_v a_o triangle_n but_o not_o every_o triangle_n be_v a_o isosceles_a and_o why_o may_v not_o likewise_o a_o prism_n be_v more_o general_a than_o a_o parallelepipedon_n or_o a_o column_n have_v side_n and_o contain_v they_o under_o it_o as_o a_o triangle_n contain_v under_o it_o a_o isosceles_a and_o other_o kind_n of_o triangle_n so_o that_o every_o prallelipipedon_n or_o every_o side_v column_n be_v a_o prism_n but_o not_o every_o prism_n a_o parallelipipedom_n or_o a_o side_v column_n this_o ought_v not_o to_o be_v so_o much_o offensive_a and_o indeed_o it_o seem_v manifest_o of_o many_o yea_o &_o of_o the_o learned_a so_o to_o be_v take_v as_o clear_o appear_v by_o
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semicircle_n be_v move_v definition_n as_o in_o the_o example_n before_o give_v in_o the_o definition_n of_o a_o sphere_n the_o line_n ab_fw-la about_o which_o his_o end_n be_v fix_v the_o semicircle_n be_v move_v which_o line_n also_o yet_o remain_v after_o the_o motion_n end_v be_v the_o axe_n of_o the_o sphere_n describe_v of_o that_o semicircle_n theodosius_n define_v the_o axe_n of_o a_o sphere_n after_o this_o manner_n sphere_n the_o axe_n of_o a_o sphere_n be_v a_o certain_a right_a line_n draw_v by_o the_o centre_n end_v on_o either_o side_n in_o the_o superficies_n of_o the_o sphere_n about_o which_o be_v fix_v the_o sphere_n be_v turn_v as_o the_o line_n ab_fw-la in_o the_o former_a example_n there_o need_v to_o this_o definition_n no_o other_o declaration_n but_o only_o to_o consider_v that_o the_o whole_a sphere_n turn_v upon_o that_o line_n ab_fw-la which_o pass_v by_o the_o centre_n d_o and_o be_v extend_v one_o either_o side_n to_o the_o superficies_n of_o the_o sphere_n wherefore_o by_o this_o definition_n of_o theodosius_n it_o be_v the_o axe_n of_o the_o sphere_n 14_o the_o centre_n of_o a_o sphere_n be_v that_o point_n which_o be_v also_o the_o centre_n of_o the_o semicircle_n definition_n this_o definition_n of_o the_o centre_n of_o a_o sphere_n be_v give_v as_o be_v the_o other_o definition_n of_o the_o axe_n namely_o have_v a_o relation_n to_o the_o definition_n of_o a_o sphere_n here_o give_v of_o euclid_n where_o it_o be_v say_v that_o a_o sphere_n be_v make_v by_o the_o revolution_n of_o a_o semicircle_n who_o diameter_n abide_v fix_v the_o diameter_n of_o a_o circle_n and_o of_o a_o semicrcle_n be_v all_o one_o and_o in_o the_o diameter_n either_o of_o a_o circle_n or_o of_o a_o semicircle_n be_v contain_v the_o centre_n of_o either_o of_o they_o for_o that_o they_o diameter_n of_o each_o ever_o pass_v by_o the_o centre_n now_o say_v euclid_n the_o point_n which_o be_v the_o centre_n of_o the_o semicircle_n by_o who_o motion_n the_o sphere_n be_v describe_v be_v also_o the_o centre_n of_o the_o sphere_n as_o in_o the_o example_n there_o give_v the_o point_n d_o be_v the_o centre_n both_o of_o the_o semicircle_n &_o also_o of_o the_o sphere_n theodosius_n give_v as_o other_o definition_n of_o the_o centre_n of_o a_o sphere_n which_o be_v thus_o sphere_n the_o centre_n of_o a_o sphere_n be_v a_o point_n with_o in_o the_o sphere_n from_o which_o all_o line_n draw_v to_o the_o superficies_n of_o the_o sphere_n be_v equal_a as_o in_o a_o circle_n be_v a_o plain_a figure_n there_o be_v a_o point_n in_o the_o midst_n from_o which_o all_o line_n draw_v to_o the_o circumfrence_n be_v equal_a which_o be_v the_o centre_n of_o the_o circle_n so_o in_o like_a manner_n with_o in_o a_o sphere_n which_o be_v a_o solid_a and_o bodily_a figure_n there_o must_v be_v conceive_v a_o point_n in_o the_o midst_n thereof_o from_o which_o all_o line_n draw_v to_o the_o superficies_n thereof_o be_v equal_a and_o this_o point_n be_v the_o centre_n of_o the_o sphere_n by_o this_o definition_n of_o theodosius_n flussas_n in_o define_v the_o centre_n of_o a_o sphere_n comprehend_v both_o those_o definition_n in_o one_o after_o this_o sort_n the_o centre_n of_o a_o sphere_n be_v a_o point_n assign_v in_o a_o sphere_n from_o which_o all_o the_o line_n draw_v to_o the_o superficies_n be_v equal_a and_o it_o be_v the_o same_o which_o be_v also_o the_o centre_n of_o the_o semicircle_n which_o describe_v the_o sphere_n sphere_n this_o definition_n be_v superfluous_a and_o contain_v more_o they_o need_v for_o either_o part_n thereof_o be_v a_o full_a and_o sufficient_a definition_n as_o before_o have_v be_v show_v or_o else_o have_v euclid_n be_v insufficient_a for_o leave_v out_o the_o one_o part_n or_o theodosius_n for_o leave_v out_o the_o other_o peradventure_o flussas_n do_v it_o for_o the_o more_o explication_n of_o either_o that_o the_o one_o part_n may_v open_v the_o other_o 15_o the_o diameter_n of_o a_o sphere_n be_v a_o certain_a right_a line_n draw_v by_o the_o centre_n and_o one_o each_o side_n end_v at_o the_o superficies_n of_o the_o same_o sphere_n definition_n this_o definition_n also_o be_v not_o hard_o but_o may_v easy_o be_v couceave_v by_o the_o definition_n of_o the_o diameter_n of_o a_o circle_n for_o as_o the_o diameter_n of_o a_o circle_n be_v a_o right_a line_n draw_v from_o one_o side_n of_o the_o circumfrence_n of_o a_o circle_n to_o the_o other_o pass_v by_o the_o centre_n of_o the_o circle_n so_o imagine_v you_o a_o right_a line_n to_o be_v draw_v from_o one_o side_n of_o the_o superficies_n of_o a_o sphere_n to_o the_o other_o pass_v by_o the_o centre_n of_o the_o sphere_n sphere_n and_o that_o line_n be_v the_o diameter_n of_o the_o sphere_n so_o it_o be_v not_o all_o one_o to_o say_v the_o axe_n of_o a_o sphere_n and_o the_o diameter_n of_o a_o sphere_n any_o line_n in_o a_o sphere_n draw_v from_o side_n to_o side_n by_o the_o centre_n be_v a_o diameter_n but_o not_o every_o line_n so_o draw_v by_o the_o centre_n be_v the_o axe_n of_o the_o sphere_n but_o only_o one_o right_a line_n about_o which_o the_o sphere_n be_v imagine_v to_o be_v movedâ—_n so_o that_o the_o name_n of_o a_o diameter_n of_o a_o sphere_n be_v more_o general_a then_o be_v the_o name_n of_o a_o axe_n for_o every_o axe_n in_o a_o sphere_n be_v a_o diameter_n of_o the_o same_o but_o not_o every_o diameter_n of_o a_o sphere_n be_v a_o axe_n of_o the_o same_o and_o therefore_o flussas_n set_v a_o diameter_n in_o the_o definition_n of_o a_o axe_n as_o a_o more_o general_a word_n â—n_o this_o manner_n the_o axe_n of_o a_o sphere_n be_v that_o fix_a diameter_n above_o which_o the_o sphere_n be_v move_v a_o sphere_n as_o also_o a_o circle_n may_v have_v infinite_a diameter_n but_o it_o can_v have_v but_o
same_o be_v treble_a it_o be_v manifest_a by_o the_o 12._o of_o the_o five_o that_o all_o the_o prism_n be_v to_o all_o the_o pyramid_n treble_a wherefore_o parallelipipedon_n be_v treble_a to_o pyramid_n set_v upon_o the_o self_n same_o base_a with_o they_o and_o under_o the_o same_o altitude_n they_o for_o that_o they_o contain_v two_o prism_n three_o corollary_n if_o two_o prism_n be_v under_o one_o and_o the_o self_n same_o altitude_n have_v to_o their_o base_n either_o both_o triangle_n or_o both_o parallelogram_n the_o prism_n be_v the_o one_o to_o the_o other_o as_o their_o base_n be_v for_o forasmuch_o as_o those_o prism_n be_v equemultiqlice_n unto_o the_o pyramid_n upon_o the_o self_n same_o base_n and_o under_o the_o same_o altitude_n which_o pyramid_n be_v in_o proportion_n as_o their_o base_n it_o be_v manifest_a by_o the_o 15._o of_o the_o five_o that_o the_o prism_n be_v in_o the_o proportion_n of_o the_o base_n for_o by_o the_o former_a corollary_n the_o prism_n be_v treble_a to_o the_o pyramid_n sâ—t_v upon_o the_o triangular_a base_n four_o corollary_n prism_n be_v in_o sesquealtera_fw-la proportion_n to_o pyramid_n which_o have_v the_o self_n same_o quadrangled_a base_a that_o the_o prism_n have_v and_o be_v under_o the_o self_n same_o altitude_n for_o that_o pyramid_n contain_v two_o pyramid_n set_v upon_o a_o triangular_a base_a of_o the_o same_o prism_n for_o it_o be_v prove_v that_o that_o prism_n be_v treble_a to_o the_o pyramid_n which_o be_v set_v upon_o the_o half_a of_o his_o quadrangled_a base_a unto_o which_o the_o other_o which_o be_v set_v upon_o the_o whole_a base_a be_v double_a by_o the_o six_o of_o this_o book_n five_v corollary_n wherefore_o we_o may_v in_o like_a sort_n conclude_v that_o solid_n mention_v in_o the_o second_o corollary_n which_o solid_n campane_n call_v side_v column_n be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n which_o be_v poligonon_o figure_n for_o they_o be_v in_o the_o proportion_n of_o the_o pyramid_n or_o prism_n set_v upon_o the_o self_n same_o base_n and_o under_o the_o self_n same_o altitude_n that_o be_v they_o be_v in_o the_o proportion_n of_o the_o base_n of_o the_o say_a pyramid_n or_o prism_n for_o those_o solid_n may_v be_v divide_v into_o prism_n have_v the_o self_n same_o altitude_n when_o as_o their_o opposite_a base_n may_v be_v divide_v into_o triangle_n by_o the_o 20_o of_o the_o six_o upon_o which_o triangle_n prism_n be_v set_v be_v in_o proportion_n as_o their_o base_n by_o this_o 7._o proposition_n it_o plain_o appear_v that_o â—uâ—lide_a as_o it_o be_v before_o note_v in_o the_o diffinitionâ—â—_n under_o the_o definition_n of_o a_o prism_n comprehend_v also_o those_o kind_n of_o solid_n which_o campane_n call_v side_v column_n for_o in_o that_o he_o say_v every_o prism_n have_v a_o triangle_n to_o his_o base_a may_v be_v devidedâ—_n etc_n etc_n he_o need_v not_o take_v a_o prism_n in_o that_o sense_n which_o campane_n and_o most_o man_n take_v it_o to_o have_v add_v that_o particle_n have_v to_o his_o base_a a_o triangle_n for_o by_o their_o sense_n there_o be_v no_o prism_n but_o it_o may_v have_v to_o his_o base_a a_o triangleâ—_n and_o so_o it_o may_v seem_v that_o euclid_n ought_v without_o exception_n have_v say_v that_o every_o prism_n whatsoever_o may_v be_v divide_v into_o three_o pyramid_n equal_a the_o one_o to_o the_o other_o have_v also_o triangle_n to_o â—heir_a base_n for_o so_o do_v campane_n and_o flussas_n put_v the_o proposition_n leave_v out_o the_o former_a particle_n have_v to_o his_o base_a a_o triangle_n which_o yet_o be_v red_a in_o the_o greek_a copy_n &_o not_o leâ—t_v out_o by_o any_o other_o interpreter_n know_v abroad_o except_o by_o campane_n and_o flussas_n yea_o and_o the_o corollary_n follow_v of_o this_o proposition_n add_v by_o theon_n or_o euclid_n and_o amend_v by_o m._n dee_n seem_v to_o confirm_v this_o sense_n of_o this_o â—s_a 〈◊〉_d make_v manifest_a that_o every_o pyramid_n be_v the_o three_o part_n of_o the_o prism_n have_v the_o same_o base_a with_o it_o and_o equal_a altitude_n for_o and_o if_o the_o base_a of_o the_o prism_n have_v any_o other_o right_o line_v figure_n than_o a_o triangle_n and_o also_o the_o superficies_n opposite_a to_o the_o base_a the_o same_o figure_n that_o prism_n may_v be_v divide_v into_o prism_n have_v triangle_v base_n and_o the_o superficiece_n to_o those_o base_n opposite_a also_o triangle_v a_o â—â—ike_a and_o equal_o for_o there_o as_o we_o see_v be_v put_v these_o word_n â—or_a and_o if_o the_o base_a of_o the_o prism_n be_v any_o other_o right_o line_v figureâ—_n etc_n etc_n whereof_o a_o man_n may_v well_o infer_v that_o the_o base_a may_v be_v any_o other_o rectiline_a figure_n whatsoever_o &_o not_o only_o a_o triangle_n or_o a_o parallelogram_n and_o it_o be_v true_a also_o in_o that_o sense_n as_o it_o be_v plain_a to_o see_v by_o the_o second_o corollary_n add_v out_o of_o flussas_n which_o corollary_n as_o also_o the_o first_o of_o his_o corollary_n be_v in_o a_o manner_n all_o one_o with_o the_o corollary_n add_v by_o theon_n or_o euclid_n far_o theon_n in_o the_o demonstration_n of_o the_o 10._o proposition_n of_o this_o book_n as_o we_o shall_v afterwards_o see_v most_o plain_o call_v not_o only_o side_v column_n prism_n but_o also_o parallelipipedon_n and_o although_o the_o 40._o proposition_n of_o the_o eleven_o book_n may_v seem_v hereunto_o to_o be_v a_o lâ—t_n for_o that_o it_o can_v be_v understand_v of_o those_o prism_n only_o which_o have_v triangle_n to_o their_o like_a equal_a opposite_a and_o parallel_v side_n or_o but_o of_o some_o side_v column_n and_o not_o of_o all_o yet_o may_v that_o let_v be_v thus_o remove_v away_o to_o say_v that_o euclid_n in_o that_o proposition_n use_v genus_fw-la pro_fw-la specie_fw-la that_o be_v the_o general_a word_n for_o some_o special_a kind_n thereof_o which_o thing_n also_o be_v not_o rare_a not_o only_o with_o he_o but_o also_o with_o other_o learned_a philosopher_n thus_o much_o i_o think_v good_a by_o the_o way_n to_o note_v in_o far_a defence_n of_o euclid_n definition_n of_o a_o prism_n the_o 8._o theorem_a the_o 8._o proposition_n pyramid_n be_v like_o &_o have_a triangle_n to_o their_o base_n be_v in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o their_o side_n of_o like_a proportion_n be_v svppose_v that_o these_o pyramid_n who_o base_n be_v the_o triangle_n gbc_n and_o hef_n and_o top_n the_o point_n a_o and_o d_o be_v like_o and_o in_o like_a sort_n describe_v and_o let_v ab_fw-la and_o de_fw-fr be_v side_n of_o like_a proportion_n then_o i_o say_v that_o the_o pyramid_n abcg_o be_v to_o the_o pyramid_n defh_o in_o treble_a proportion_n of_o that_o in_o which_o the_o side_n ab_fw-la be_v to_o the_o side_n de._n make_v perfect_a the_o parallelipipedon_n namely_o the_o solid_n bckl_n &_o efxo_n and_o forasmuch_o as_o the_o pyramid_n abcg_o be_v like_a to_o the_o pyramid_n defh_o construction_n therefore_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n def_n demonstration_n &_o the_o angle_n gbc_n to_o the_o angle_n hef_fw-fr and_o moreover_o the_o angle_n abg_o to_o the_o angle_n deh_fw-it and_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n de_fw-fr so_o be_v the_o line_n bc_o to_o the_o line_n of_o and_o the_o line_n bg_o to_o the_o line_n eh_o and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n de_fw-fr so_o be_v the_o line_n bc_o to_o the_o line_n of_o and_o the_o side_n about_o the_o equal_a angle_n be_v proportional_a therefore_o the_o parallelogram_n bm_n be_v like_a to_o the_o parallelogram_n ep_n and_o by_o the_o same_o reason_n the_o parallelogram_n bn_o be_v like_a to_o the_o parallelogram_n er_fw-mi and_o the_o parellelogramme_n bk_o be_v like_a unto_o the_o parallelogram_n exit_fw-la wherefore_o the_o three_o parallelogram_n bm_n kb_o and_o bn_o be_v like_a to_o the_o three_o parallelogram_n ep_n exit_fw-la and_o er._n but_o the_o three_o parallelogram_n bm_n kb_o and_o bn_o be_v equal_a and_o like_a to_o the_o three_o opposite_a parallelogram_n and_o the_o three_o parallelogram_n ep_n exit_fw-la and_o ere_o be_v equal_a and_o like_a to_o the_o three_o opposite_a parallelogram_n wherefore_o the_o parallelipipedon_n bckl_n and_o efxo_n be_v comprehend_v under_o plain_a superficiece_n like_a and_o equal_a in_o multitude_n wherefore_o the_o solid_a bckl_n be_v like_a to_o the_o solid_a efxo_n but_o like_o parallelipipedon_n be_v by_o the_o 33._o of_o the_o eleven_o in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o side_n of_o like_a proportion_n be_v to_o side_n of_o like_a proportion_n wherefore_o the_o solid_a bckl_n be_v to_o the_o solid_a efxo_n in_o treble_a
first_o demonstration_n demonstration_n lead_v to_o a_o impossibility_n this_o proposition_n in_o discreet_a quantity_n answer_v to_o the_o 23._o proposition_n of_o the_o five_o book_n in_o continual_a quantity_n this_o and_o the_o eleven_o proposition_n follow_v declare_v the_o passion_n and_o property_n ofâ—_n prime_a number_n demonstration_n lead_v to_o a_o impossibility_n this_o be_v the_o converse_n of_o the_o former_a proposition_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n demonstration_n demonstration_n demonstration_n of_o the_o first_o part_n lead_v to_o a_o absurdity_n demonstration_n of_o the_o second_o part_n which_o be_v the_o conâ—câ—se_n of_o the_o first_o lean_v also_o to_o a_o absurdity_n demonstrasion_n lead_v to_o a_o absurdity_n demonstrasion_n a_o corollary_n â—â—ded_v by_o campave_n demonstration_n lâ—ading_v to_o a_o impossibility_n an_o other_o demonstration_n demonstration_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n add_v by_o campane_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o absurdity_n the_o second_o caseâ—_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n leaâ—iâ—g_v â—o_o a_o absurdity_n the_o second_o case_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n demonstration_n the_o converse_n of_o the_o former_a proposition_n demonstration_n construction_n demonstration_n leâ—ding_v to_o a_o absurdity_n a_o corollary_n adâ—ed_v by_o campane_n how_o to_o â—inde_v out_o the_o second_o least_o number_n and_o the_o three_o and_o so_o â—orth_o infinite_o how_o to_o siâ—â—_n out_o the_o least_o â—â—mâ—_n a_o conâ—ayâ—â—g_n â—â—e_z paâ—â—s_n of_o part_n the_o arguâ—â—â—â—_n of_o the_o eight_o book_n demonstration_n lead_v to_o a_o absurdity_n construction_n demonstration_n this_o proposition_n be_v the_o â—â—uerse_a of_o the_o first_o demonstrationâ—_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o absurdity_n the_o second_o case_n demonstration_n this_o proposition_n in_o number_n answer_v to_o the_o of_o the_o six_o touch_v parellelogramme_n construction_n demonstration_n an_o other_o demonstration_n after_o campane_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n a_o corollary_n add_v by_o flussates_n construction_n demonstration_n this_o proposition_n be_v the_o converse_n of_o the_o former_a construction_n demonstration_n the_o first_o part_n of_o this_o proposition_n demonstrate_v the_o second_o part_n demonstrate_v construction_n the_o first_o part_n of_o this_o prâ—position_n demonstrate_v the_o second_o part_n demonstrate_v construction_n demonstration_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o a_o negative_a proportion_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o a_o negative_a proposition_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o demonstration_n of_o the_o fiâ—st_a part_n of_o this_o proposition_n demonstration_n of_o the_o second_o part_n demonstration_n of_o the_o first_o part_n of_o this_o proposition_n the_o second_o part_n this_o proposition_n be_v the_o converse_n of_o the_o 18._o proposition_n construction_n demonstration_n this_o proposition_n be_v the_o converse_n of_o the_o 19_o proposition_n construction_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n a_o corollary_n add_v by_o flussates_n construction_n construction_n demonstration_n a_o corollary_n add_v by_o flussates_n another_o corollary_n add_v by_o flussates_n the_o argument_n of_o the_o niâ—th_o book_n demonstration_n this_o proposition_n be_v the_o converse_n oâ—_n tâ—e_z formâ—â—_n demonstration_n a_o corollary_n aâ—ded_v by_o campane_n demonstration_n demonstration_n demonstration_n a_o corollary_n add_v by_o campane_n demonstration_n demonstration_n demonstration_n of_o the_o first_o part_n the_o second_o part_n demonstrate_v demostration_n of_o the_o three_o part_n demostration_n of_o the_o first_o part_n of_o this_o proposition_n the_o second_o pâ—rt_n demonstrate_v demonstration_n of_o the_o first_o part_n leave_v to_o a_o absurdity_n demonstration_n of_o the_o â—â—cond_a pâ—â—â—_n lead_v alâ—o_o to_o a_o absurdity_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o demonstration_n aâ—ter_n flussates_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o demonstration_n after_o campane_n demonstration_n lead_v to_o a_o absurdity_n a_o proposition_n add_v by_o campane_n construction_n demonstration_n demonstration_n to_o prove_v that_o the_o number_n a_o and_o c_o be_v prime_a to_o b._n demonstratiou_n this_o proposition_n be_v the_o converse_n of_o the_o former_a demonstration_n this_o answer_v to_o the_o 2._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 3._o of_o the_o three_o demonstration_n this_o answer_v to_o thâ—_z 4._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 5._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 6._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 7._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 8._o of_o the_o second_o demonstratition_n this_o answer_v to_o thâ—_z 9_o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 10._o oâ—_n the_o second_o demonstration_n a_o negative_a proposition_n demonstration_n leaâ—ing_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o abjurditie_n three_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n the_o three_o case_n divert_v case_n â—n_v this_o proposition_n the_o first_o case_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n a_o proposition_n add_v by_o campane_n a_o other_o add_v by_o he_o demonstration_n lead_v to_o a_o absurdity_n demonstration_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o demonstration_n demonstration_n demonstration_n this_o proposition_n teachâ—th_v how_o to_o find_v out_o a_o perfect_a number_n construction_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n the_o argument_n of_o the_o ten_o book_n difference_n between_o number_n and_o magnitude_n a_o line_n be_v not_o make_v of_o point_n as_o number_n be_v make_v of_o unity_n this_o book_n the_o hard_a to_o understand_v of_o all_o the_o book_n of_o euclid_n in_o this_o book_n be_v entreat_v of_o a_o strange_a manner_n of_o matter_n then_o in_o the_o former_a many_o even_o of_o the_o well_o learn_v have_v think_v that_o this_o book_n can_v not_o well_o be_v understand_v without_o algebra_n the_o nine_o former_a book_n &_o the_o principle_n of_o this_o â—ooke_n well_o understand_v this_o book_n will_v not_o be_v hard_a to_o understand_v the_o fâ—rst_a definition_n the_o second_o definition_n contraries_n make_v manifest_a by_o the_o compare_v of_o the_o one_o to_o the_o other_o the_o third_o definition_n what_o the_o power_n of_o a_o line_n be_v the_o four_o definition_n unto_o the_o suppose_a line_n first_o set_v may_v be_v compare_v infinite_a line_n why_o some_o mislike_n that_o the_o line_n first_o set_v shall_v be_v call_v a_o rational_a line_n flussates_n call_v this_o line_n a_o line_n certain_a this_o rational_a line_n the_o ground_n in_o a_o manner_n of_o all_o the_o proposition_n in_o this_o ten_o book_n note_n the_o line_n rational_a of_o purpose_n the_o six_o definition_n campâ—nus_n â—ath_z cause_v much_o obscurity_n in_o this_o ten_o book_n the_o seven_o definition_n flussates_n in_o steed_n of_o this_o word_n irrational_a use_v this_o word_n uncertain_a why_o they_o be_v call_v irrational_a line_n the_o cause_n of_o the_o obscurity_n and_o confusedness_n in_o this_o book_n the_o eight_o definition_n the_o nine_o definition_n the_o ten_o definition_n the_o eleven_o definition_n construction_n demonstration_n a_o corollary_n construction_n demonstration_n this_o proposition_n teach_v that_o incontinuall_a quantity_n which_o the_o first_o of_o the_o seven_o teach_v in_o discrete_a quantity_n construction_n demonstration_n lead_v to_o a_o absurdity_n two_o case_n in_o this_o proposition_n the_o first_o case_n this_o proposition_n teach_v that_o in_o continual_a quantity_n which_o the_o 2._o of_o the_o sâ—â—ith_v teach_v in_o number_n the_o second_o case_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n this_o problem_n reduce_v to_o a_o theorem_a this_o proposition_n teach_v that_o in_o continual_a quantity_n which_o the_o 3._o of_o the_o second_o teach_v in_o number_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o absurdity_n the_o second_o case_n a_o leâ—ma_n necessary_a
to_o be_v prâ—â—â—d_v beâ—oâ—e_n 〈◊〉_d â—all_n to_o the_o demonstration_n construction_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n this_o problem_n reduce_v to_o a_o theorem_a construction_n demonstration_n how_o magnitude_n be_v say_v to_o be_v in_o proportion_n the_o onâ—_n to_o the_o other_o as_o number_n be_v to_o number_v this_o proposition_n be_v the_o converse_n of_o forme_a construction_n demonstration_n a_o corollary_n construction_n demonstration_n construction_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n this_o be_v the_o 〈…〉_o demonstration_n the_o first_o part_n demonstrate_v an_o other_o demonstration_n of_o the_o first_o part_n an_o other_o demonstration_n oâ—_n the_o same_o first_o part_n after_o montaureus_n demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o former_a an_o other_o demonstration_n of_o the_o second_o part_n this_o assumpt_n follow_v as_o a_o corollary_n of_o the_o 25_o but_o so_o as_o it_o may_v also_o be_v here_o in_o method_n place_v you_o shall_v â—inde_v it_o after_o the_o 53._o of_o this_o book_n absolute_o demonstrate_v for_o there_o it_o serve_v to_o the_o 54._o his_o demonstration_n demonstration_n of_o the_o three_o part_n demonstration_n of_o the_o four_o part_n which_o be_v the_o converse_n of_o the_o â—_o conclusion_n of_o the_o whole_a proposition_n a_o corollary_n proâ—e_v of_o the_o first_o part_n of_o the_o corollary_n proof_n of_o the_o second_o part_n proof_n of_o the_o three_o pâ—rt_n proâ—e_a oâ—_n the_o four_o part_n certain_a annotation_n â—ut_z of_o montauâ—â—us_fw-la rule_v to_o know_v whether_o two_o superficial_a number_n be_v like_a or_o no._n this_o assumpt_n be_v the_o converse_n of_o the_o 26._o of_o the_o eight_o demonstration_n oâ—_n the_o first_o part_n demonstration_n of_o the_o second_o partâ—_n a_o corollary_n to_o find_v out_o the_o first_o line_n incommensurable_a in_o length_n only_o to_o the_o line_n give_v to_o find_v out_o the_o second_o line_n incommensurable_a both_o in_o length_n and_o in_o power_n to_o the_o line_n give_v construction_n demonstration_n tâ—is_n be_v wiâ—h_n zambert_n a_o aâ—â—â—mpt_n but_o uâ—â—eâ—ly_o improper_o â—lâ—ssateâ—_n mane_v iâ—_z a_o corollary_n but_o the_o greeâ—e_n and_o montaureus_n maâ—e_n it_o a_o proposition_n but_o every_o way_n a_o â—nfallible_a truth_n 〈…〉_o demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n a_o corollary_n demonstration_n an_o other_o way_n to_o prove_v that_o the_o line_n a_o e_o c_o f_o be_v proportional_a demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o paâ—t_n which_o be_v the_o converse_n of_o the_o first_o a_o corollary_n demonstration_n of_o the_o first_o part_n by_o a_o argument_n leadindg_fw-mi to_o a_o absurdity_n demonstration_n of_o the_o second_o paâ—t_n lead_v also_o to_o a_o impossibility_n and_o this_o second_o part_n be_v the_o converse_n of_o the_o first_o demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o how_o to_o divide_v the_o line_n bc_o ready_o in_o such_o sort_n as_o iâ—_z require_v in_o the_o proposition_n demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o tâ—e_z former_a a_o other_o demonstrationâ—y_v a_o argument_n lead_v to_o a_o absurdity_n a_o assumpt_n a_o corollary_n add_v by_o montaureuâ—_n cause_n cause_o of_o increase_v the_o difficulty_n of_o this_o book_n note_n construction_n demonstration_n diverse_a caâ—es_n in_o this_o proposition_n the_o second_o case_n the_o first_o kind_n of_o rational_a line_n commensurable_a in_o length_n this_o particle_n in_o the_o proposition_n according_a to_o any_o of_o the_o foresay_a way_n be_v not_o in_o vain_a put_v the_o second_o kind_n of_o rational_a line_n commensurable_a in_o length_n the_o three_o case_n the_o three_o kind_n of_o rational_a line_n commensurable_a in_o length_n the_o four_o case_n this_o proposition_n be_v the_o converse_n of_o the_o former_a proposition_n construction_n demonstration_n a_o assumpt_n construction_n demonstration_n definition_n of_o a_o medial_a line_n a_o corollary_n this_o assumpt_n be_v nothing_o else_o but_o a_o part_n of_o the_o first_o proposition_n of_o the_o six_o book_n 〈◊〉_d how_o a_o square_n be_v say_v to_o be_v apply_v upon_o a_o line_n construction_n demonstration_n construction_n demonstration_n note_n a_o corollary_n construction_n demonstration_n lead_v to_o a_o absurdity_n construction_n demonstration_n construction_n demonstration_n demonstration_n a_o corollary_n to_o find_v out_o two_o square_a number_n exceed_v the_o one_o the_o other_o by_o a_o square_a â—umber_n a_o assumpt_n construction_n demonstration_n montaureus_n make_v this_o a_o assumpt_n as_o the_o grecke_n text_n seem_v to_o do_v likewise_o but_o without_o a_o cause_n construction_n demonstration_n this_o assumpt_n set_v foâ—th_o nothing_o â—_z but_o that_o which_o the_o first_o oâ—_n the_o sâ—â—t_n jet_v â—orth_o and_o therefore_o in_o sâ—me_a examplar_n it_o be_v not_o find_v construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o corollary_n i_o dee_n dee_n the_o second_o corollary_n corollary_n therefore_o if_o you_o divide_v the_o square_n of_o the_o side_n ac_fw-la by_o the_o side_n bc_o the_o portion_n dc_o will_v be_v the_o product_n etc_n etc_n as_o in_o the_o former_a corollary_n i_o dâ—e_n dâ—e_n the_o third_o corollary_n corollary_n therefore_o if_o the_o parallelogram_n of_o basilius_n and_o ac_fw-la be_v divide_v by_o bc_o the_o product_n will_v geve_v the_o perpendicular_a d_o a._n these_o three_o corollarye_n in_o practice_n logistical_a and_o geometrical_a be_v profitable_a an_o other_o demonstration_n of_o this_o four_o part_n of_o the_o determination_n a_o assumpt_n construction_n demonstration_n the_o first_o part_n of_o the_o determination_n conclude_v the_o second_o part_n conclude_v the_o total_a conclusion_n construction_n demonstration_n the_o first_o part_n of_o the_o determination_n conclude_v the_o second_o part_n conclude_v the_o total_a conclusion_n construction_n demonstration_n the_o first_o part_n conclude_v the_o second_o part_n conclude_v the_o three_o part_n conclude_v the_o total_a conclusion_n the_o first_o senary_a by_o composition_n definition_n of_o a_o binomial_a line_n six_o kind_n of_o binomial_a line_n demonstration_n definition_n of_o a_o first_o bimedial_a line_n construction_n demonstration_n definition_n of_o a_o second_o bimedial_a line_n demonstration_n definition_n of_o a_o great_a line_n definition_n of_o a_o line_n who_o power_n be_v rational_a and_o medial_a definition_n of_o a_o liâ—e_z contain_v in_o power_n two_o medial_n a_o assumpt_n the_o second_o senary_a by_o composition_n demonstration_n lead_v to_o a_o impossibility_n a_o corollary_n demonstration_n lead_v to_o a_o impossibilâ—â—e_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibiâ—â—â—e_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n lead_v to_o a_o absurdity_n six_o kind_n of_o binomial_a line_n a_o binomial_a line_n consi_v of_o two_o paâ—tâ—s_n firsâ—_o definition_n seconâ—_o definition_n three_o â—â—â—â—â—â—ition_n four_o definition_n five_o definition_n six_o definition_n the_o three_o senary_a by_o composition_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o corollary_n add_v by_o flussates_n m._n do_v you_o his_o book_n call_v tyâ—â—câ—niâ—m_fw-la mathematicum_fw-la this_o assumpt_n as_o be_v before_o note_v follow_v most_o â—riâ—fly_o without_o far_a demonstration_n of_o the_o 25._o of_o this_o book_n demonstration_n a_o assumpt_n the_o four_o senary_a by_o composition_n construction_n demonstration_n the_o first_o part_n of_o this_o demonstration_n conclude_v the_o second_o part_n of_o the_o demonstration_n conclude_v the_o three_o part_n conclude_v the_o total_a conclusion_n demonstration_n the_o first_o part_n of_o this_o demonstration_n conclude_v the_o three_o part_n conclude_v the_o four_o part_n concludeâ—_n the_o five_o part_n conclude_v the_o total_a conclusion_n demonstration_n construction_n demonstration_n demonstration_n demonstration_n demonstration_n look_v after_o the_o assumpt_n conclude_v at_o this_o mark_n for_o plain_a open_n of_o this_o place_n the_o use_n of_o this_o assumpt_n be_v in_o the_o next_o proposition_n &_o other_o follow_v the_o five_o senary_a by_o composition_n construction_n demonstration_n conclude_v that_o dg_o be_v a_o binomial_a line_n construction_n demonstration_n conclude_v that_o dg_o be_v a_o binomial_a line_n construction_n demonstration_n †_o *_o ‡_o dg_o conclude_v a_o binomial_a line_n a_o corollary_n add_v by_o m._n dee_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n the_o six_o senary_n construction_n demonstration_n construction_n demonstration_n a_o corollary_n add_v by_o flussetes_n note_n construction_n demonstration_n an_o other_o demonstration_n after_o p._n montaureus_n an_o other_o demonstration_n after_o campane_n construction_n demonstration_n an_o other_o demonstration_n afâ—â—r_v campane_n construction_n demonstration_n a_o assumpt_n an_o other_o demonstration_n after_o campanâ—_n note_n
fd._n wherefore_o cones_fw-la &_o cylinder_n consist_v upon_o equal_a base_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o altitude_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 15._o theorem_a the_o 15._o proposition_n in_o equal_a cones_n and_o cylinder_n the_o base_n be_v reciprocal_a to_o their_o altitude_n and_o cones_fw-la and_o cylinder_n who_o base_n be_v reciprocal_a to_o their_o altitude_n be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o these_o cones_fw-la acl_n egn_n or_o these_o cylinder_n axe_n eo_fw-la who_o base_n be_v the_o circle_n abcd_o efgh_o and_o axe_n kl_o and_o mn_v which_o axe_n be_v also_o the_o altitude_n of_o the_o cones_fw-la &_o cylinder_n be_v equal_a the_o one_o to_o the_o other_o cones_n then_o i_o say_v that_o the_o base_n of_o the_o cylinder_n xa_n &_o eo_fw-la be_v reciprokal_n to_o their_o altitude_n that_o be_v that_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o proposition_n so_o the_o altitude_n mn_v to_o the_o altitude_n kl_o for_o the_o altitude_n kl_o be_v either_o equal_a to_o the_o altitude_n mn_a or_o not_o first_o let_v it_o be_v equal_a case_n but_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eq_n but_o cones_fw-la and_o cylinder_n consist_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n be_v by_o the_o 11._o of_o the_o twelve_o wherefore_o the_o base_a abcd_o be_v equal_a to_o the_o base_a efgh_o wherefore_o also_o they_o be_v reciprokal_a as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o but_o now_o suppose_v that_o the_o altitude_n lk_n be_v not_o equal_a to_o the_o altitude_n m._n n_o construction_n but_o let_v the_o altitude_n mn_v be_v great_a and_o by_o the_o 3._o of_o the_o first_o from_o the_o altitude_n mn_v take_v away_o pm_v equal_a to_o the_o altitude_n kl_o so_o that_o let_v the_o line_n pm_n be_v put_v equal_a to_o the_o line_n kl_o and_o by_o the_o point_n p_o let_v there_o be_v extend_v a_o plain_a superficies_n tus_z which_o let_v cut_v the_o cylinder_n eo_fw-la and_o be_v a_o parallel_n to_o the_o two_o opposite_a plain_a superâ—icieces_n that_o be_v to_o the_o circle_n efgh_o and_o ro._n cylinder_n and_o make_v the_o base_a the_o circle_n efgh_o &_o the_o altitude_n mp_n imagine_v a_o cylinder_n es._n and_o for_o that_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eo_fw-la and_o there_o be_v a_o other_o cylinder_n es_fw-ge therefore_o by_o the_o 7._o of_o the_o five_o as_o the_o cylinder_n axe_n be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o cylinder_n eo_fw-la to_o the_o cylinder_n es._n but_o as_o the_o cylinder_n axe_n be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o base_a abcd_o to_o the_o base_a efgh_o for_o the_o cylinder_n axe_n and_o es_fw-ge be_v under_o one_o and_o the_o self_n same_o altitude_n and_o as_o the_o cylinder_n eo_fw-la be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o altitude_n mn_v to_o the_o altitude_n mp_n for_o cylinder_n consist_v upon_o equal_a base_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o altitude_n wherefore_o as_o the_o base_a abcd_o be_v â—o_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n mp_n but_o the_o altitude_n pm_n be_v equal_a to_o the_o altitude_n kl_o wherefore_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o wherefore_o in_o the_o equal_a cylinder_n axe_n and_o eo_fw-la the_o base_n be_v reciprocal_a to_o their_o altitude_n but_o now_o suppose_v that_o the_o base_n of_o the_o cylinder_n axe_n and_o eo_fw-la be_v reciprokal_n to_o their_o altitude_n that_o be_v as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o demonstrate_v then_o i_o say_v that_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eo_fw-la for_o the_o self_n same_o order_n of_o construction_n remain_v for_o that_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o but_o the_o altitude_n kl_o be_v equal_a to_o the_o altitude_n pm_n wherefore_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n pm_n but_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o cylinder_n axe_n to_o the_o cylinder_n es_fw-ge for_o they_o be_v under_o equal_a altitude_n and_o as_o the_o altitude_n mn_o be_v to_o the_o altitude_n pm_n so_o be_v the_o cylinder_n eo_fw-la to_o the_o cylinder_n es_fw-ge by_o the_o 14._o of_o the_o twelve_o wherefore_o also_o as_o the_o cylinder_n axe_n be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o cylinder_n eo_fw-la to_o the_o cylinder_n es._n wherefore_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eo_fw-la by_o the_o 9_o of_o the_o five_o and_o so_o also_o be_v it_o in_o the_o cones_fw-la which_o haâ—â—_n the_o self_n same_o base_n and_o altitude_n with_o the_o cylinder_n wherefore_o in_o equal_a cones_fw-la and_o cylinder_n the_o base_n be_v reciprocal_a to_o their_o altitude_n etc_n etc_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o campane_n and_o flussas_n hitherto_o have_v be_v show_v the_o passion_n and_o propriety_n of_o cones_fw-la and_o cylinder_n who_o altitude_n fall_v perpendicular_o upon_o the_o base_n now_o will_v we_o declare_v that_o cones_fw-la and_o cilinder_n who_o altitude_n fall_v oblique_o upon_o their_o base_n have_v also_o the_o self_n same_o passion_n and_o propriety_n which_o the_o foresay_a cones_fw-la and_o cilinder_n have_v forasmuch_o as_o in_o the_o ten_o of_o this_o book_n it_o be_v say_v that_o every_o cone_n be_v the_o three_o part_n of_o a_o cilind_a have_v one_o and_o the_o self_n same_o base_a &_o one_o &_o the_o self_n same_o altitude_n with_o it_o which_o thing_n be_v demonstrate_v by_o a_o cilind_a give_v who_o base_a be_v cut_v by_o a_o square_n inscribe_v in_o it_o and_o upon_o the_o side_n of_o the_o square_n be_v describe_v isosceles_a triangle_n make_v a_o polygonon_n figure_n and_o again_o upon_o the_o side_n of_o this_o polygonon_n figure_n be_v infinite_o after_o the_o same_o manner_n describe_v other_o isosceles_a triangle_n take_v away_o more_o they_o the_o half_a as_o have_v oftentimes_o be_v declare_v therefore_o it_o be_v manifest_a that_o the_o solid_n set_v upon_o these_o base_n be_v under_o the_o same_o altitude_n that_o the_o cilind_a incline_v be_v and_o be_v also_o include_v in_o the_o same_o cilind_a do_v take_v away_o more_o than_o the_o half_a of_o the_o cilind_a and_o also_o more_o they_o the_o half_a of_o the_o residue_n as_o it_o have_v be_v prove_v in_o erect_a cylinder_n for_o these_o incline_n solid_n be_v under_o equal_a altitude_n and_o upon_o equal_a base_n with_o the_o erect_a solid_n be_v equal_a to_o the_o erect_a solid_n by_o the_o corollary_n of_o the_o _o of_o the_o eleven_o wherefore_o they_o also_o in_o like_a sort_n as_o the_o erect_a take_v away_o more_o than_o the_o half_a if_o therefore_o we_o compare_v the_o incline_v cilind_a to_o a_o cone_n set_v upon_o the_o self_n same_o base_a and_o have_v his_o altitude_n erect_v and_o reason_n by_o a_o argument_n lead_v to_o a_o impossibility_n by_o the_o demonstration_n of_o the_o ten_o of_o this_o book_n we_o may_v prove_v that_o the_o side_v solid_a include_v in_o the_o incline_v cylinder_n be_v great_a than_o the_o triple_a of_o his_o pyramid_n and_o it_o be_v also_o equal_a to_o the_o same_o which_o be_v impossible_a and_o this_o be_v the_o first_o case_n wherein_o it_o be_v prove_v that_o the_o cilind_a not_o be_v equal_a to_o the_o triple_a of_o the_o cone_n be_v not_o great_a than_o the_o triple_a of_o the_o same_o and_o as_o touch_v the_o second_o case_n we_o may_v after_o the_o same_o manner_n conclude_v that_o that_o â—ided_v solid_a contain_v in_o the_o cylinder_n be_v great_a than_o the_o cylinder_n which_o be_v very_o absurd_a wherefore_o if_o the_o cylinder_n be_v neither_o great_a than_o the_o triple_a of_o the_o cone_n nor_o less_o it_o must_v needs_o be_v equal_a to_o the_o same_o the_o demonstration_n of_o these_o incline_v cylinder_n most_v plain_o follow_v the_o demonstration_n of_o the_o erect_a cylinder_n for_o it_o have_v already_o be_v prove_v that_o pyramid_n and_o side_v solid_n which_o be_v also_o call_v general_o prism_n be_v set_v upon_o equal_a base_n and_o under_o one_o and_o the_o self_n same_o altitude_n whether_o the_o altitude_n be_v erect_v or_o incline_v be_v equal_a the_o one_o to_o the_o other_o namely_o be_v in_o proportion_n as_o their_o base_n be_v by_o
the_o â—_o of_o this_o book_n wherefore_o a_o cylinder_n incline_n shall_v be_v triple_a to_o every_o cone_n although_o also_o the_o cone_n be_v erect_v set_v upon_o one_o and_o the_o same_o base_a with_o it_o and_o be_v under_o the_o same_o altitude_n but_o the_o cilind_a erect_v be_v the_o triple_a of_o the_o same_o cone_fw-mi by_o the_o ten_o of_o this_o book_n wherefore_o the_o cilind_a incline_v be_v equal_a to_o the_o cilind_a erect_v being_n both_o set_v upon_o one_o and_o the_o self_n same_o base_a and_o have_v one_o and_o the_o self_n same_o altitude_n the_o same_o also_o come_v to_o pass_v in_o cones_fw-la which_o be_v the_o three_o part_n of_o equal_a cilinder_n &_o therefore_o be_v equal_a the_o one_o to_o the_o other_o wherefore_o according_a to_o the_o eleven_o of_o this_o book_n it_o follow_v that_o cylinder_n and_o cones_fw-la incline_v or_o erect_v be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n be_v for_o forasmuch_o as_o the_o erect_a be_v in_o proportion_n as_o their_o base_n be_v and_o to_o the_o erect_a cilinder_n the_o incline_n be_v equal_a therefore_o they_o also_o shall_v be_v in_o proportion_n as_o their_o base_n be_v and_o therefore_o by_o the_o 12._o of_o this_o book_n like_o cones_fw-la and_o cylinder_n be_v incline_v be_v in_o triple_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o the_o base_n be_v for_o forasmuch_o as_o they_o be_v equal_a to_o the_o erect_a which_o have_v the_o proportion_n by_o the_o 12._o of_o this_o book_n and_o their_o base_n also_o be_v equal_a with_o the_o base_n of_o the_o erect_v therefore_o they_o also_o shall_v have_v the_o same_o proportion_n wherefore_o it_o follow_v by_o the_o 13._o of_o this_o book_n thaâ—_z cylinder_n incline_v be_v cut_v by_o a_o plain_a superficies_n parallel_v to_o the_o opposite_a plain_a superficiece_n thereof_o shall_v be_v cut_v according_a to_o the_o proportion_n of_o the_o axe_n for_o suppose_v that_o upon_o one_o and_o the_o self_n same_o base_a â—e_n set_v a_o erect_a cylinder_n and_o a_o incline_v cylinder_n be_v both_o under_o one_o and_o the_o self_n same_o altitude_n which_o 〈…〉_o a_o plain_a superficies_n parallel_v to_o the_o opposite_a base_n now_o it_o be_v manifest_a that_o the_o section_n of_o the_o one_o cylinder_n be_v equal_a to_o the_o section_n of_o the_o other_o cylinder_n for_o they_o be_v set_v upon_o equal_a base_n and_o under_o one_o and_o the_o self_n same_o altitude_n namely_o between_o the_o parallel_n plain_a superficiece_n and_o their_o axe_n also_o be_v by_o those_o parallel_a plain_n superficiâ—_fw-la 〈◊〉_d proportional_o by_o the_o 1st_a of_o â—he_n â—levenâ—h_o wherefore_o the_o incline_v cylinder_n be_v equal_a to_o the_o erect_a cylinder_n shall_v have_v the_o proportion_n of_o theiâ—_z axe_n aâ—_z also_o have_v the_o erect_v for_o in_o echâ—_n the_o proportion_n of_o the_o axe_n be_v one_o and_o the_o same_o wherefore_o incline_v cones_n and_o cylinder_n be_v set_v upon_o equal_a base_n shall_v by_o the_o 14._o of_o this_o book_n be_v in_o 〈◊〉_d as_o their_o altitude_n 〈…〉_o forasmuch_o aâ—_z the_o incline_v be_v equal_a to_o the_o erect_a which_o have_v the_o self_n same_o base_n and_o altitude_n and_o the_o erect_v be_v iâ—_z proportion_n as_o their_o altitude_n therefore_o the_o incline_n shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_o â—he_z self_n same_o altitude_n which_o be_v common_a to_o each_o namely_o to_o the_o incline_v and_o to_o the_o erect_v and_o therefore_o in_o equal_a cones_fw-la and_o cylinder_n whether_o they_o be_v incline_v or_o erect_v the_o base_n shall_v be_v reciprocal_a proportional_a with_o the_o altitude_n and_o contrariwise_o by_o the_o 15._o of_o this_o book_n for_o forasmuch_o as_o we_o have_v oftentimes_o show_v that_o the_o incline_v cones_fw-la and_o cylinder_n be_v equal_a to_o the_o erect_a have_v the_o self_n same_o base_n and_o altitude_n with_o they_o and_o the_o erect_v unto_o who_o the_o incline_v be_v equal_a haâ—e_z their_o base_n reciprocal_a proportional_o with_o their_o altitude_n therefore_o it_o follow_v that_o the_o incline_v be_v equal_a to_o the_o erect_a have_v also_o their_o base_n and_o altitude_n which_o be_v common_a to_o each_o reciprocal_a proportional_a likewise_o if_o theiâ—_z altitude_n &_o base_n be_v reciprocal_a proportional_a they_o themselves_o also_o shall_v be_v equal_a for_o that_o they_o be_v equal_a to_o the_o erect_a cylinder_n and_o cones_fw-la set_v upon_o the_o same_o base_n and_o be_v under_o the_o same_o altitude_n which_o erect_a cylinder_n be_v equal_a the_o one_o to_o the_o other_o by_o the_o same_o 15._o of_o this_o book_n wherefore_o we_o may_v conclude_v that_o those_o passion_n &_o propriety_n which_o in_o this_o twelve_o book_n have_v be_v prove_v to_o be_v in_o cones_fw-la and_o cylinder_n who_o altitude_n be_v erect_v perpendicular_o to_o the_o 〈…〉_o set_v oblique_o upoâ—_n their_o base_n howbeit_o this_o be_v to_o be_v note_v that_o such_o incline_v cones_fw-la or_o cylinder_n be_v not_o perfect_a round_o as_o be_v the_o erect_v so_o that_o if_o they_o be_v cut_v by_o a_o plain_a superficies_n pass_v at_o right_a angle_n with_o their_o altitude_n this_o section_n be_v a_o conical_a section_n which_o be_v call_v ellipsis_n and_o shall_v not_o describe_v in_o their_o superficies_n a_o circle_n as_o it_o do_v in_o erect_a cylinder_n &_o cones_fw-la but_o a_o certain_a figure_n who_o less_o diameter_n be_v in_o cylinder_n equal_a to_o the_o dimetient_fw-la of_o the_o base_a that_o be_v be_v one_o and_o the_o same_o with_o it_o and_o the_o same_o thing_n happen_v also_o in_o cones_fw-la incline_v be_v cut_v after_o the_o same_o manner_n the_o 1._o problem_n the_o 16._o proposition_n two_o circle_n have_v both_o one_o and_o the_o self_n same_o centre_n be_v give_v to_o inscribe_v in_o the_o great_a circle_n a_o polygonon_n figure_n which_o shall_v consist_v of_o equal_a and_o even_a side_n and_o shall_v not_o touch_v the_o superficies_n of_o the_o less_o circle_n svppose_v that_o there_o be_v two_o circle_n abcd_o and_o efgh_a have_v one_o &_o the_o self_n same_o centre_n namely_o k._n it_o be_v require_v in_o the_o great_a circle_n which_o let_v be_v abcd_o to_o inscribe_v a_o polygonon_n figure_n which_o shall_v be_v of_o equal_a and_o even_a side_n and_o not_o touch_v the_o circle_n efgh_o draw_v by_o the_o centre_n king_n a_o right_a line_n bd._n construction_n and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n g_o raise_v up_o unto_o the_o right_a line_n bd_o a_o perpendicular_a line_n agnostus_n and_o extend_v it_o to_o the_o point_n c._n wherefore_o the_o line_n ac_fw-la touch_v the_o circle_n efgh_o by_o the_o 15._o of_o the_o three_o now_o therefore_o if_o by_o the_o 30._o of_o the_o three_o we_o divide_v the_o circumference_n bad_a into_o two_o equal_a part_n and_o again_o the_o half_a of_o that_o into_o two_o equal_a part_n and_o thus_o do_v continual_o we_o shall_v by_o the_o corollary_n of_o the_o 1._o of_o the_o ten_o at_o the_o length_n leave_v a_o certain_a circumference_n less_o than_o the_o circumference_n ad._n let_v the_o circumference_n leave_v be_v ld_n and_o from_o the_o point_n l._n draw_v by_o the_o 12._o of_o the_o first_o unto_o the_o line_n bd_o a_o perpendiculare_a line_n demonstration_n lm_o dee_n and_o extend_v it_o to_o the_o point_n n._n and_o draw_v these_o right_a line_n ld_n and_o dn_o and_o forasmuch_o as_o the_o angle_n dml_n and_o dmn_v be_v right_a angle_n therefore_o by_o the_o 3._o of_o the_o three_o the_o right_a line_n bd_o divide_v the_o right_a line_n ln_o into_o two_o equal_a part_n in_o the_o point_n m._n wherefore_o by_o the_o 4._o of_o the_o first_o the_o rest_n of_o the_o side_n of_o the_o triangle_n dml_n and_o dmn_v namely_o the_o line_n dl_o and_o dn_o shall_v be_v equal_v and_o forasmuch_o as_o the_o line_n ac_fw-la be_v a_o parallel_n to_o the_o ln_o by_o the_o 28._o of_o the_o first_o but_o ac_fw-la touch_v the_o circle_n efgh_o wherefore_o the_o line_n lm_o touch_v not_o the_o circle_n efgh_o and_o much_o less_o do_v the_o line_n ld_n and_o dn_o touch_v the_o circle_n efgh_o if_o therefore_o there_o be_v apply_v right_a line_n equal_a to_o the_o line_n ld_n continual_o into_o the_o circle_n abcd_o by_o the_o 1._o of_o the_o four_o there_o shall_v be_v describe_v in_o the_o circle_n abcd_o a_o polygonon_n figure_n which_o shall_v be_v of_o equal_a and_o figure_n even_o side_n and_o shall_v not_o touch_v the_o less_o circle_n namely_o efgh_o by_o the_o 14._o of_o the_o three_o or_o by_o the_o 29._o which_o be_v require_v to_o be_v do_v ¶_o corollary_n hereby_o it_o be_v manifest_a that_o a_o perpendicular_a line_n draw_v from_o the_o point_n l_o to_o the_o line_n bd_o touch_v not_o