Selected quad for the lemma: cause_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
cause_n	bring_v	good_a	let_v	1,459	5	4.0417	3	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A00429	The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed; Elements. English	Euclid.; Dee, John, 1527-1608.; Candale, François de Foix, comte de, 1502-1594.; Billingsley, Henry, Sir, d. 1606.	1570 (1570)	STC 10560; ESTC S106699	1,020,889	884

again_o of_o thing_n be_v in_o like_a swiftness_n of_o move_v to_o think_v the_o near_a to_o move_v fast_o and_o the_o far_o much_o slow_a nay_o of_o two_o thing_n whereof_o the_o one_o incomparable_o do_v move_v swift_a than_o the_o other_o to_o deem_v the_o slow_a to_o move_v very_o swift_a &_o the_o other_o to_o stand_v what_o a_o error_n be_v this_o of_o our_o eye_n of_o the_o rainbow_n both_o of_o his_o colour_n of_o the_o order_n of_o the_o colour_n of_o the_o bigness_n of_o it_o the_o place_n and_o heith_n of_o it_o etc_o etc_o to_o know_v the_o cause_n demonstrative_a be_v it_o not_o pleasant_a be_v it_o not_o necessary_a of_o two_o or_o three_o son_n appear_v of_o blaze_a star_n and_o such_o like_a thing_n by_o natural_a cause_n bring_v to_o pass_v and_o yet_o nevertheless_o of_o far_a matter_n significative_a be_v it_o not_o commodious_a for_o man_n to_o know_v the_o very_a true_a cause_n &_o occasion_n natural_a yea_o rather_o be_v it_o not_o great_o against_o the_o sovereignty_n of_o man_n nature_n to_o be_v so_o overshoot_v and_o abuse_v with_o thing_n at_o hand_n before_o his_o eye_n as_o with_o a_o peacock_n tail_n and_o a_o dove_n neck_n or_o a_o whole_a ore_n in_o water_n hold_v to_o seem_v break_v thing_n far_o of_o to_o seem_v never_o and_o never_o to_o seem_v far_o of_o small_a thing_n to_o seem_v great_a and_o great_a to_o seem_v small_a one_o man_n to_o seem_v a_o army_n or_o a_o man_n to_o be_v curse_o afraid_a of_o his_o own_o shadow_n yea_o so_o much_o to_o fear_v that_o if_o you_o be_v alone_o near_o a_o certain_a glass_n and_o proffer_n with_o dagger_n or_o sword_n to_o foin_v at_o the_o glass_n you_o shall_v sudden_o be_v move_v to_o give_v back_o in_o manner_n by_o reason_n of_o a_o image_n ☞_o appear_v in_o the_o air_n between_o you_o &_o the_o glass_n with_o like_a hand_n sword_n or_o dagger_n &_o with_o like_a quickness_n foin_a at_o your_o very_a eye_n likewise_o as_o you_o do_v at_o the_o glass_n strange_a this_o be_v to_o hear_v of_o but_o more_o marvellous_a to_o behold_v than_o these_o my_o word_n can_v signify_v and_o nevertheless_o by_o demonstration_n optical_a the_o order_n and_o cause_n thereof_o be_v certify_v even_o so_o as_o the_o effect_n be_v consequent_a yea_o thus_o much_o more_o dare_v i_o take_v upon_o i_o towards_o the_o satisfy_n of_o the_o noble_a courage_n that_o long_v ardent_o for_o the_o wisdom_n of_o cause_n natural_a as_o to_o let_v he_o understand_v that_o in_o london_n he_o may_v wish_v his_o own_o eye_n have_v proof_n of_o that_o which_o i_o have_v say_v herein_o a_o gentleman_n which_o for_o his_o good_a service_n s.w.p._n do_v to_o his_o country_n be_v famous_a and_o honourable_a and_o for_o skill_n in_o the_o mathematical_a science_n and_o language_n be_v the_o odd_a man_n of_o this_o kind_n etc_n etc_n even_o he_o be_v able_a and_o i_o be_o sure_a will_v very_o willing_o let_v the_o glass_n and_o proof_n be_v see_v and_o so_o i_o here_o request_v he_o for_o the_o increase_n of_o wisdom_n in_o the_o honourable_a and_o for_o the_o stop_n of_o the_o mouth_n malicious_a and_o repress_v the_o arrogancy_n of_o the_o ignorant_a you_o may_v easy_o guess_v what_o i_o mean_v this_o art_n of_o perspective_n be_v of_o that_o excellency_n and_o may_v be_v lead_v to_o the_o certify_n and_o execute_v of_o such_o thing_n as_o no_o man_n will_v easy_o believe_v without_o actual_a proof_n perceive_v i_o speak_v nothing_o of_o natural_a philosophy_n which_o without_o perspective_n can_v not_o be_v full_o understand_v nor_o perfect_o attein_v unto_o nor_o of_o astronomy_n which_o without_o perspective_n can_v not_o well_o be_v ground_v nor_o astrology_n natural_o verify_v and_o avouch_v that_o part_n hereof_o which_o deal_v with_o glass_n which_o name_n glass_n be_v a_o general_a name_n in_o this_o art_n for_o any_o thing_n from_o which_o a_o beam_n rebound_v be_v call_v catoptrike_v and_o have_v so_o many_o use_n both_o marvellous_a and_o profitable_a that_o both_o it_o will_v hold_v i_o to_o long_o to_o no●●_n therein_o the_o principal_a conclusion_n all_o ready_a know_v and_o also_o perchance_o some_o thing_n may_v lack_v due_a credit_n with_o you_o and_o i_o thereby_o to_o lose_v my_o labor●_n and_o you_o to_o slip_v into_o light_a judgement_n ☞_o ●_o before_o you_o have_v learn_v sufficient_o the_o power_n of_o nature_n and_o arte._n now_o to_o proceed_v astronomy_n be_v a_o art_n mathematical_a which_o demonstrate_v the_o distance_n magnitude_n and_o all_o natural_a motion_n apparence_n and_o passion_n proper_a to_o the_o planet_n and_o fix_a start_n for_o any_o time_n past_a present_a and_o to_o come_v in_o respect_n of_o a_o certain_a horizon_n or_o without_o respect_n of_o any_o horizon_n by_o this_o art_n we_o be_v certify_v of_o the_o distance_n of_o the_o starry_a sky_n and_o of_o each_o planete_n from_o the_o centre_n of_o the_o earth_n and_o of_o the_o greatness_n of_o any_o fix_a star_n see_v or_o planete_n in_o respect_n of_o the_o earth_n greatness_n as_o we_o be_v sure_a by_o this_o art_n that_o the_o solidity_n massine_n and_o body_n of_o the_o son_n contain_v the_o quantity_n of_o the_o whole_a earth_n and_o sea_n a_o hundred_o three_o score_n and_o two_o time_n less_o by_o 1_o ●_o one_o eight_o part_n of_o the_o earth_n but_o the_o body_n of_o the_o whole_a earthly_a globe_n and_o sea_n be_v big_a than_o the_o body_n of_o the_o moon_n three_o and_o forty_o time_n less_o by_o 1_o ●_o of_o the_o moon_n wherefore_o the_o son_n be_v big_a than_o the_o moon_n 7000_o time_n less_o by_o 59_o ●●_o 164_o that_o be_v precise_o 6940_o ●5_o ●●_o big_a than_o the_o moon_n and_o yet_o the_o unskilful_a man_n will_v judge_v they_o a_o like_a big_a wherefore_o of_o necessity_n the_o one_o be_v much_o far_o from_o we_o than_o the_o other_o the_o son_n when_o he_o be_v far_o from_o the_o earth_n which_o now_o in_o our_o age_n be_v when_o he_o be_v in_o the_o 8._o degree_n of_o cancer_n be_v 1179_o semidiameter_n of_o the_o earth_n distante_fw-la and_o the_o moon_n when_o she_o be_v far_o from_o the_o earth_n be_v 68_o semidiameter_n of_o the_o earth_n and_o 1_o ●_o the_o near_o that_o the_o moon_n come_v to_o the_o earth_n be_v semidiameter_n 52_o ¼_n the_o distance_n of_o the_o starry_a sky_n be_v from_o we_o in_o semidiameter_n of_o the_o earth_n 20081_o 1_o ●_o twenty_o thousand_o fourscore_o one_o and_o almost_o a_o half_a subtract_v from_o this_o the_o moan_n near_o distance_n from_o the_o earth_n and_o thereof_o remain_v semidiameter_n of_o the_o earth_n 20029_o 1_o ●_o twenty_o thousand_o nine_o and_o twenty_o and_o a_o quarter_n note_n so_o thick_o be_v the_o heavenly_a palace_n that_o the_o planet_n have_v all_o their_o exercise_n in_o and_o most_o marvelous_o perform_v the_o commandment_n and_o charge_n to_o they_o give_v by_o the_o omnipotent_a majesty_n of_o the_o king_n of_o king_n this_o be_v that_o which_o in_o genesis_n be_v call_v ha_o rakia_n consider_v it_o well_o the_o 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psalm_n well_o well_o it_o be_v time_n for_o some_o to_o lay_v hold_v on_o wisdom_n and_o to_o judge_v true_o of_o thing_n and_o not_o so_o to_o expound_v the_o holy_a word_n all_o by_o allegory_n as_o to_o neglect_v the_o wisdom_n power_n and_o goodness_n of_o god_n in_o and_o by_o his_o creature_n and_o creation_n to_o be_v see_v and_o learn_v by_o parable_n and_o analogy_n of_o who_o nature_n and_o property_n the_o course_n of_o the_o holy_a scripture_n also_o declare_v to_o we_o very_o many_o mystery_n the_o whole_a frame_n of_o god_n creature_n which_o be_v the_o whole_a world_n be_v to_o we_o a_o bright_a glass_n from_o which_o by_o reflection_n rebound_v to_o our_o knowledge_n and_o perceiverance_n beam_n and_o radiations●_n represent_v the_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 forecas●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 fa●●_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 de●ised_v for_o order_v of_o all_o company_n sum_n and_o number●_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 few●_n to_o s●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 f●●ld_n skarmoush_n or_o ambush_n appartain_v and_o of_o all_o these_o lively_a desi●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 rules●_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 necke●_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 sh●●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 meane●_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 executed●_n how_o just_o &_o with_o great_a knowledge_n of_o art_n do_v pap●●●●anus_n institute_v a_o law_n of_o partition_n and_o allowance_n between_o man_n and_o wife_n a●●●_n a_o ●●●orce_n but_o how_o acc●rsius_n bald●s_n bartolus_n jason_n alex●nder_n and_o final_o alciat●●_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 ●quity●●rt_n ●nd_v i●●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 ●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 i●_z 〈◊〉_d libr●_n 〈◊〉_d glossa_fw-la difficili●r_fw-la evils_n c●mputationem_fw-la nec_fw-la scholas●ici_fw-la ●ec_fw-la doct●res_fw-la intelligent_a etc_n etc_n that_o is●_n in_o the_o whole_a book_n there_o be_v no_o ●losse_n hard_a than_o thi●●_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 i●lianus_n law_n si_fw-mi ita_fw-la s●ript●m_fw-la etc_n etc_n of_o the_o testator_n will_v 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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 d●●n_fw-la id_fw-la profecto_fw-la efficiet●_n deum_fw-la antem_fw-la aliquem_fw-la magis_fw-la quam_fw-la fortunam_fw-la ●d_a sa●●tem_fw-la nostram_fw-la hoc_fw-la m●nus_fw-la nobis_fw-la arbitror_fw-la contulisse_fw-la etc_n etc_n nam_fw-la ipsum_fw-la ●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 th●_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 expressed●_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

first_o set_v they_o also_o must_v needs_o be_v at_o the_o least_o commensurable_a in_o power_n the_o one_o to_o the_o other_o for_o forasmuch_o as_o their_o square_n be_v rational_a they_o shall_v be_v commensurable_a to_o the_o square_n of_o the_o rational_a line_n first_o set_v wherefore_o by_o the_o 12._o of_o this_o book_n they_o be_v also_o commensurable_a the_o one_o to_o the_o other_o wherefore_o their_o line_n be_v at_o the_o least_o commensurable_a in_o power_n the_o one_o to_o the_o other_o and_o it_o be_v possible_a also_o that_o they_o may_v be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o for_o suppose_v that_o a_o be_v a_o rational_a li●e_z first_o set_v and_o let_v the_o line_n b_o be_v unto_o the_o same_o rational_a line_n a_o commensurable_a in_o power_n only_o that_o be_v incommensurable_a in_o length_n unto_o it_o let_v there_o be_v also_o a_o other_o line_n c_o commensurable_a in_o length_n to_o the_o line_n b_o which_o be_v possible_a by_o the_o principle_n of_o this_o book_n now_o by_o the_o 13._o of_o the_o ten_o it_o be_v manifest_a that_o the_o line_n c_o be_v incommensurable_a in_o length_n unto_o the_o line_n a._n but_o the_o square_n of_o the_o line_n a_o be_v commensurable_a to_o the_o square_n of_o the_o line_n b_o by_o supposition_n and_o the_o square_a of_o the_o line_n c_o be_v also_o commensurable_a to_o the_o square_n of_o the_o line_n b_o by_o supposition_n wherefore_o by_o the_o 12._o of_o this_o book_n the_o square_a of_o the_o line_n c_o be_v commensurable_a to_o the_o square_n of_o the_o line_n a._n wherefore_o by_o the_o definition_n the_o line_n c_o shall_v be_v rational_a commensurable_a in_o power_n only_o to_o the_o line_n a_o as_o also_o be_v the_o line_n b._n wherefore_o there_o be_v give_v two_o rational_a line_n commensurable_a in_o power_n only_o to_o the_o rational_a line_n first_o set_v and_o commensurable_a in_o length_n the_o one_o to_o the_o other_o here_o be_v to_o be_v note_v which_o thing_n also_o we_o before_o note_v in_o the_o definition_n that_o campane_n and_o other_o which_o follow_v he_o bring_v in_o these_o phrase_n of_o speech_n to_o call_v some_o line_n rational_a in_o power_n only_o cause_n and_o other_o some_o rational_a in_o length_n and_o in_o power_n which_o we_o can_v find_v that_o euclid_n ever_o use_v for_o these_o word_n in_o length_n and_o in_o power_n be_v never_o refer_v to_o rationality_n or_o irrationalitie_n but_o always_o to_o the_o commensurabilitie_n or_o incommensurablitie_n of_o line_n which_o pervert_v of_o word_n as_o be_v there_o declare_v have_v much_o increase_v the_o difficulty_n and_o obscureness_n of_o this_o book_n book_n and_o now_o i_o think_v it_o good_a again_o to_o put_v you_o in_o mind_n that_o in_o these_o proposition_n which_o follow_v we_o must_v ever_o have_v before_o our_o eye_n the_o rational_a line_n first_o set_v note_n unto_o which_o other_o line_n compare_v be_v either_o rational_a or_o irrational_a according_a to_o their_o commensurability_n or_o incommensurabilitie_n ¶_o the_o 16._o theorem_a the_o 19_o proposition_n a_o rectangle_n figure_n comprehend_v under_o right_a line_n commensurable_a in_o length_n be_v rational_a according_a to_o one_o of_o the_o foresay_a way_n be_v rational_a svppose_v that_o this_o rectangle_n figure_n ac_fw-la be_v comprehend_v under_o these_o right_a line_n ab_fw-la and_o bc_o be_v commensurable_a in_o length_n and_o rational_a according_a to_o one_o of_o the_o foresay_a way_n then_o i_o say_v that_o the_o superficies_n ac_fw-la be_v rational_a describe_v by_o the_o 46._o o●_n the_o first_o upon_o the_o line_n ab_fw-la a_o square_a ad._n construction_n wherefore_o that_o square_a ad_fw-la be_v rational_a by_o the_o definition_n demonstration_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v commensurable_a in_o length_n unto_o the_o line_n bc_o and_o the_o line_n ab_fw-la be_v equal_a unto_o the_o line_n bd_o therefore_o the_o line_n bd_o be_v commensurable_a in_o length_n unto_o the_o line_n bc._n and_o as_o the_o line_n bd_o be_v to_o the_o line_n bc_o so_o be_v the_o square_a dam_n to_o the_o superficies_n ac_fw-la by_o the_o first_o of_o the_o six_o but_o it_o be_v prove_v that_o the_o line_n bd_o be_v commensurable_a unto_o the_o line_n bc_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a dam_n be_v commensurable_a unto_o the_o rectangle_n superficies_n ac_fw-la but_o the_o square_a dam_n be_v rational_a wherefore_o the_o rectangle_n superficies_n ac_fw-la also_o be_v rational_a by_o the_o definition_n a_o rectangle_n figure_n therefore_o comprehend_v under_o right_a line_n commensurable_a in_o length_n be_v rational_a accord_v to_o one_o of_o the_o foresay_a way_n be_v rational_a which_o be_v require_v to_o be_v prove_v proposition_n where_o as_o in_o the_o former_a demonstration_n the_o square_n be_v describe_v upon_o the_o less_o line_n we_o may_v also_o demonstrate_v the_o proposition_n if_o we_o describe_v the_o square_n upon_o the_o great_a line_n and_o that_o after_o this_o manner_n suppose_v that_o the_o rectangle_n superficies_n bc_o be_v contain_v of_o these_o unequal_a line_n ab_fw-la and_o ac_fw-la which_o let_v be_v rational_a commensurable_a the_o one_o to_o the_o other_o in_o length_n and_o let_v the_o line_n ac_fw-la be_v the_o great_a case_n and_o upon_o the_o line_n ac_fw-la describe_v the_o square_a dc_o then_o i_o say_v that_o the_o parallelogram_n bc_o be_v rational_a length_n for_o the_o line_n ac_fw-la be_v commensurable_a in_o length_n unto_o the_o line_n ab_fw-la by_o supposition_n and_o the_o line_n dam_n be_v equal_a to_o the_o line_n ac_fw-la wherefore_o the_o line_n dam_n be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la but_o what_o proportion_n the_o line_n dam_n have_v to_o the_o line_n ab_fw-la the_o same_o have_v the_o square_a dc_o to_o the_o para●lelogramme_n c●_n by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10._o of_o this_o book_n the_o square_a dc_o be_v commensurable_a to_o the_o parallelogram_n cb._n but_o it_o be_v manifest_a that_o the_o square_a dc_o be_v rational_a for_o that_o it_o be_v the_o square_a of_o a_o rational_a line_n namely_o ac_fw-la wherefore_o by_o the_o definition_n the_o parallelogram_n also_o cb_o be_v rational_a moreover_o forasmuch_o as_o those_o two_o former_a demonstration_n seem_v to_o speak_v of_o that_o parallelogram_n which_o be_v make_v of_o two_o line_n of_o which_o any_o one_o may_v be_v the_o li●e_z first_o set_v which_o be_v call_v the_o first_o rational_a line_n from_o which_o we_o say_v ought_v to_o be_v take_v the_o measure_n of_o the_o other_o line_n compare_v unto_o it_o and_o the_o other_o be_v commensurable_a in_o length_n to_o the_o same_o first_o rational_a line_n length_n which_o be_v the_o first_o kind_n of_o rational_a line_n commensurable_a in_o length_n i_o think_v it_o good_a here_o to_o set_v a_o other_o case_n of_o the_o other_o kind_n of_o rational_a line_n of_o line_n i_o say_v rational_a commensurable_a in_o length_n compare_v to_o a_o other_o rational_a line_n first_o set_v to_o declare_v the_o general_a truth_n of_o this_o theorem_a and_o that_o we_o may_v see_v that_o this_o particle_n according_a to_o any_o of_o the_o foresay_a way_n be_v not_o here_o in_o vain_a put_v now_o then_o suppose_v first_o a_o rational_a line_n ab_fw-la let_v there_o be_v also_o a_o parallelogram_n cd_o contain_v under_o the_o line_n ce_fw-fr and_o ed_z which_o line_n let_v be_v rational_a that_o be_v commensurable_a in_o length_n to_o the_o ●irst_n rational_a line_n propound_v ab_fw-la howbeit_o let_v those_o two_o line_n ce_fw-fr and_o ed_z be_v diverse_a and_o unequal_a line_n unto_o the_o first_o rational_a line_n ab_fw-la then_o i_o say_v that_o the_o parallelogram_n cd_o be_v rational_a case_n describe_v the_o square_a of_o the_o line_n de_fw-fr which_o let_v be_v df._n first_o it_o be_v manifest_a by_o the_o 12._o of_o this_o book_n that_o the_o line_n ce_fw-fr &_o ed_z be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o for_o either_o of_o they_o be_v suppose_v to_o be_v commensurable_a in_o length_n unto_o the_o line_n ab_fw-la but_o the_o line_n ed_z be_v equal_a to_o the_o line_n ef._n wherefore_o the_o line_n ce_fw-fr be_v commensurable_a in_o length_n to_o the_o line_n bf_o but_o 〈◊〉_d the_o line_n ce_fw-fr be_v ●o_o the_o line_n ●_o f_o ●o_o be_v the_o parallelogram_n cd_o to_o the_o square_a df_o by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10._o of_o this_o book_n the_o parallelogram_n cd_o shall_v be_v commensurable_a to_o the_o square_a df._n but_o the_o square_a df_o be_v commensurable_a to_o the_o square_n of_o the_o line_n ab_fw-la which_o be_v the_o first_o rational_a line_n propound_v wherefore_o by_o the_o 12._o of_o this_o book_n the_o parallelogram_n cd_o be_v commensurable_a to_o the_o square_n of_o the_o line_n ab_fw-la but_o the_o square_n of_o

of_o be._n but_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v suppose_v to_o be_v equal_a to_o the_o square_a number_n of_o be_v wherefore_o that_o which_o be_v produce_v of_o gb_o into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v equal_a to_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce._n wherefore_o take_v away_o the_o square_a number_n of_o ce_fw-fr which_o be_v common_a to_o they_o both_o the_o number_n ab_fw-la shall_v be_v equal_a to_o the_o number_n gb_o namely_o the_o great_a to_o the_o less_o which_o be_v impossible_a wherefore_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v not_o equal_a to_o the_o square_a number_n of_o be_v i_o say_v also_o that_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v not_o less_o than_o the_o square_a number_n of_o be._n for_o if_o it_o be_v possible_a they_o shall_v it_o be_v equal_a to_o some_o square_a number_n less_o than_o the_o square_a number_n of_o be._n wherefore_o let_v the_o number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_n of_o the_o number_n ce_fw-fr be_v equal_a to_o the_o square_a number_n of_o bf_o and_o let_v the_o number_n ha_o be_v double_a to_o the_o number_n df._n then_o also_o it_o follow_v that_o the_o number_n hc_n be_v double_a to_o the_o number_n cf_o so_o that_o hc_n also_o be_v divide_v into_o two_o equal_a part_n in_o fletcher_n and_o therefore_o also_o the_o number_n which_o be_v produce_v of_o hd_a into_o bc_o together_o with_o the_o square_a number_n of_o fc_n be_v equal_a to_o the_o square_a number_n of_o the_o number_n bf_o but_o by_o supposition_n the_o number_n which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v equal_a to_o the_o square_a number_n of_o bf_o wherefore_o it_o follow_v that_o the_o number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v equal_a to_o that_o which_o be_v produce_v of_o hb_o into_o bc_o together_o with_o the_o square_a number_n cf_o which_o be_v impossible_a for_o if_o it_o shall_v be_v equal_a then_o forasmuch_o as_o the_o square_n of_o cf_o be_v less_o than_o the_o square_a of_o ce_fw-fr the_o number_n produce_v of_o hb_o into_o bc_o shall_v be_v great_a than_o th●_z number_v produce_v of_o ab_fw-la into_o bc._n and_o so_o also_o shall_v the_o number_n hb_o be_v great_a than_o the_o number_n ab_fw-la when_o yet_o it_o be_v less_o than_o it_o wherefore_o the_o number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v not_o less_o than_o the_o square_a number_n of_o ●_o e._n and_o it_o be_v also_o prove_v that_o it_o can_v be_v equal_a to_o the_o square_a number_n of_o be_v neither_o great_a than_o it_o wherefore_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o add_v to_o the_o square_a number_n of_o ce_fw-fr make_v not_o a_o square_a number_n and_o although_o it_o be_v possible_a to_o demonstrate_v this_o many_o other_o way_n yet_o this_o seem_v to_o we_o sufficient_a lest_o the_o matter_n be_v over_o long_o shall_v seem_v to_o much_o tedious_a ¶_o the_o 6._o problem_n the_o 29._o proposition_n to_o find_v out_o two_o such_o rational_a right_a line_n commensurable_a in_o power_n only_o that_o the_o great_a shall_v be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o right_a line_n commensurable_a in_o length_n unto_o the_o great_a let_v there_o be_v put_v a_o rational_a line_n ab_fw-la and_o take_v also_o two_o such_o square_a number_n cd_o and_o de_fw-fr construction_n that_o their_o excess_n ce_fw-fr be_v not_o a_o square_a number_n by_o the_o corollary_n of_o the_o first_o assumpt_n of_o the_o 28._o of_o the_o ten_o and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n afb_o and_o by_o the_o corollary_n of_o the_o 6._o of_o the_o ten_o as_o the_o number_n dc_o be_v to_o the_o number_n ce_fw-fr so_o let_v the_o square_n of_o the_o line_n basilius_n be_v to_o the_o square_n of_o the_o line_n af._n demonstration_n and_o draw_v a_o line_n from_o fletcher_n to_o b._n now_o for_o that_o as_o the_o square_n of_o the_o line_n basilius_n be_v to_o the_o square_n of_o the_o line_n of_o so_o be_v the_o number_n cd_o to_o the_o number_n ce_fw-fr therefore_o the_o square_a of_o the_o line_n basilius_n have_v to_o the_o square_n of_o the_o line_n of_o that_o proportion_n that_o the_o number_n cd_o have_v to_o the_o number_n ce._n wherefore_o the_o square_a of_o the_o line_n basilius_n be_v commensurable_a to_o the_o square_n of_o the_o line_n of_o by_o the_o 6._o of_o the_o ten_o but_o the_o square_n of_o the_o line_n ab_fw-la be_v rational_a wherefore_o also_o the_o square_a of_o the_o line_n of_o be_v rational_a wherefore_o also_o the_o line_n of_o be_v rational_a and_o forasmuch_o as_o the_o number_n cd_o have_v not_o unto_o the_o number_n ce_fw-fr that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o neither_o also_o have_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n of_o that_o proportion_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 the_o ten_o the_o line_n ab_fw-la be_v unto_o the_o line_n of_o incommensurable_a in_o length_n wherefore_o the_o line_n of_o and_o ab_fw-la be_v rational_a commensurable_a in_o power_n only_o and_o for_o that_o as_o the_o number_n dc_o be_v to_o the_o number_n ce_fw-fr so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n of_o therefore_o by_o conversion_n or_o everse_v proportion_n which_o be_v demonstrate_v by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o as_o the_o number_n cd_o be_v to_o the_o number_n de_fw-fr so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n bf_o which_o be_v the_o excess_n of_o the_o square_n of_o the_o line_n ab_fw-la above_o the_o square_n of_o the_o line_n of_o by_o the_o assumpt_n put_v before_o the_o 14._o of_o this_o book_n but_o the_o number_n cd_o have_v to_o the_o number_n de_fw-fr that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o square_a of_o the_o line_n ab_fw-la have_v to_o the_o square_n of_o the_o line_n bf_o that_o proportion_n that_o a_o square_a num●er_n have_v to_o a_o square_a number_n wherefore_o by_o the_o 9_o of_o the_o ten_o the_o line_n ab_fw-la be_v commensurable_a in_o length_n unto_o the_o line_n bf_o and_o by_o the_o 47._o of_o the_o first_o the_o square_a of_o the_o line_n ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n of_o and_o fb_o wherefore_o the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n of_o by_o the_o square_n of_o the_o line_n bf_o which_o be_v commensurable_a in_o length_n unto_o the_o line_n ab_fw-la wherefore_o there_o be_v find_v out_o two_o such_o rational_a line_n commensurable_a in_o power_n only_o namely_o ab_fw-la and_o of_o so_o that_o the_o great_a line_n ab_fw-la be_v in_o power_n more_o than_o the_o less_o line_n of_o by_o the_o square_n of_o the_o line_n fb_o which_o be_v commensurable_a in_o length_n unto_o the_o line_n ab_fw-la which_o be_v require_v to_o be_v do_v ¶_o the_o 7._o theorem_a the_o 30._o proposition_n to_o find_v out_o two_o such_o rational_a line_n commensurable_a in_o power_n only_o cause_n that_o the_o great_a shall_v be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o right_a line_n incommensurable_a in_o length_n to_o the_o great_a let_v there_o be_v put_v a_o rational_a line_n ab_fw-la and_o take_v also_o by_o the_o 2._o assumpt_n of_o the_o 28._o of_o the_o ten_o two_o square_a number_n ce_fw-fr and_o ed_z which_o be_v add_v together_o make_v not_o a_o square_a number_n and_o let_v the_o number_n ce_fw-fr and_o ed_z add_v together_o make_v the_o number_n cd_o construction_n and_o upon_o the_o line_n ab_fw-la describe_v a_o sencircle_v afb_o and_o by_o the_o corollary_n of_o the_o 6._o of_o the_o ten_o as_o the_o number_n dc_o be_v to_o the_o number_n ce_fw-fr so_o let_v the_o square_n of_o the_o line_n ab_fw-la be_v to_o the_o square_n of_o the_o line_n of_o and_o draw_v a_o line_n from_o fletcher_n to_o b._n and_o we_o may_v in_o like_a sort_n demonstration_n as_o we_o do_v in_o the_o former_a proposition_n prove_v that_o the_o line_n basilius_n and_o of_o be_v rational_a commensurable_a in_o power_n only_o and_o for_o that_o as_o the_o number_n dc_o be_v

this_o problem_n execute_v it_o be_v now_o easy_a to_o execute_v and_o that_o two_o way_n i_o mean_v to_o a_o the_o sphere_n give_v to_o make_v a_o upright_a cone_fw-mi in_o any_o proportion_n give_v between_o two_o right_a line_n for_o let_v the_o proportion_n give_v be_v that_o which_o be_v between_o x_o and_o y._n by_o the_o order_n of_o my_o addition_n upon_o the_o 2._o of_o this_o twelve_o book_n to_o the_o circle_n ekg_n make_v a_o other_o circle_n in_o that_o proportion_n that_o x_o be_v to_o you_o which_o let_v be_v z._n upon_o the_o centre_n of_o z_o rear_v a_o line_n perpendicular_a and_o equal_a to_o fl._n i_o say_v that_o the_o cone_fw-mi who_o base_a be_v z_o and_o the_o height_n equal_a to_o fl_fw-mi be_v to_o a_o in_o the_o proportion_n of_o x_o to_o y._n for_o the_o cone_n upon_o z_o by_o construction_n have_v height_n equal_a to_o the_o height_n of_o the_o cone_fw-it lekg_v and_o z_o by_o construction_n be_v to_o ekg●_n as_o x_o be_v to_o you_o wherefore_o by_o the_o 11._o of_o this_o twelve_o the_o cone_n upon_o z_o be_v to_o the_o cone_n lekg_v as_o x_o be_v to_o y._n but_o the_o cone_n lekg_n be_v prove_v equal_a to_o the_o sphere_n a._n wherefore_o the_o cone_n upon_o z_o be_v to_o a_o as_o x_o be_v to_o you_o by_o the_o 7._o of_o the_o five_o to_o a_o sphere_n give_v therefore_o we_o have_v make_v a_o cone_n in_o any_o proportion_n give_v between_o two_o right_a line_n second_o as_o x_o be_v to_o you_o so_o to_o fl_fw-mi let_v there_o be_v a_o four_o line_n by_o the_o 12._o of_o the_o six_o and_o suppose_v it_o to_o be_v w._n i_o say_v that_o a_o cone_fw-mi who_o base_a be_v equal_a to_o ekg_v and_o height_n the_o line_n w_n be_v to_o a_o as_o x_o be_v to_o y._n for_o by_o the_o 14._o of_o this_o twelve_o cones_fw-la be_v set_v on_o equal_a base_n be_v one_o to_o the_o other_o as_o their_o height_n be_v but_o by_o construction_n the_o height_n w_n be_v to_o the_o height_n fl_fw-mi a●_z x_o be_v to_o y._n wherefore_o the_o cone_n which_o have_v his_o base_a equal_a to_o ekg_v and_o height_n the_o line_n w_n be_v to_o the_o cone_n lekg_v as_o x_o be_v to_o y._n and_o it_o be_v prove_v that_o to_o the_o cone_n lekg_v the_o sphere_n a_o be_v equal_a wherefore_o by_o the_o 7._o of_o the_o five_o the_o cone_n who_o base_a be_v equal_a to_o ekg_v and_o height_n the_o line_n w_n be_v to_o a_o as_o x_o be_v to_o y._n therefore_o a_o sphere_n be_v give_v we_o have_v make_v a_o upright_a cone_fw-mi cone_n in_o any_o proportion_n give_v between_o two_o right_a line_n and_o before_o we_o make_v a_o upright_a cone_fw-mi equal_a to_o the_o sphere_n give_v wherefore_o a_o sphere_n be_v give_v we_o have_v make_v a_o upright_a cone_fw-mi equal_a to_o the_o same_o or_o in_o any_o other_o proportion_n give_v between_o two_o right_a line_n i_o call_v that_o a_o upright_a cone_fw-mi who_o axe_n be_v perpendicular_a to_o his_o base_a ¶_o a_o corollary_n of_o the_o first_o part_n of_o the_o demonstration_n it_o be_v evident_a a_o sphere_n be_v propound_v that_o a_o cone_n who_o base_a have_v his_o semidiameter_n equal_a to_o the_o diameter_n thereof_o and_o height_n equal_a to_o the_o semidiameter_n of_o the_o same_o sphere_n be_v equal_a to_o that_o sphere_n propound_v ¶_o a_o problem_n 2._o a_o sphere_n be_v give_v and_o a_o circle_n to_o rear_v a_o upright_a cone_n upon_o that_o circle_n as_o a_o base_a equal_a to_o the_o sphere_n give_v or_o in_o any_o proportion_n between_o two_o right_a line_n assign_v and_o the_o second_o part_n of_o this_o problem_n be_v thus_o perform_v suppose_v the_o proportion_n give_v to_o be_v that_o which_o be_v between_o x_o &_o y._n problem_n then_o as_o x_o be_v to_o you_o so_o let_v a_o other_o right_a line_n find_v be_v to_o the_o h●ight_n of_o fletcher_n which_o line_n let_v be_v g._n for_o this_o g_o the_o find_a height_n by_o construction_n be_v to_o the_o height_n of_o fletcher_n as_o x_o be_v to_o you_o do_v cause_n this_o cone_n which_o let_v be_v m_o upon_o c_o the_o circle_n give_v or_o a_o other_o to_o it_o equal_a due_o rear_v to_o be_v unto_o the_o cone_n fletcher_n as_o x_o be_v to_o you_o by_o the_o 14._o of_o this_o twelve_o but_o fletcher_n be_v prove_v equal_a to_o the_o sphere_n give_v wherefore_o m_o be_v to_o the_o sphere_n give_v as_o x_o be_v to_o y._n and_o m_o be_v ●eared_v upon_o the_o circle_n give_v or_o his_o equal_a wherefore_o a_o sphere_n be_v give_v &_o a_o circle_n we_o have_v rear_v a_o up●ight_a cone_fw-mi upon_o that_o give_v circle_n as_o a_o base_a equal_a to_o the_o sphere_n give_v or_o in_o any_o proportion_n between_o two_o right_a line_n assign_v which_o be_v require_v to_o be_v do_v ¶_o a_o problem_n 3._o a_o sphere_n be_v give_v and_o a_o right_a line_n to_o make_v a_o upright_a cone_fw-mi equal_a to_o the_o sphere_n give_v or_o in_o any_o other_o proportion_n give_v between_o two_o right_a line_n which_o make_v cone_n shall_v have_v his_o height_n equal_a to_o the_o right_a line_n give_v problem_n for_o the_o second_o part_n sinde_z a_o circle_n which_o shall_v have_v to_o the_o base_a of_o l_o any_o proportion_n appoint_v in_o ●ight_a line_n as_o the_o proportion_n of_o x_o to_o you_o which_o by_o my_o addition_n upon_o the_o second_o of_o this_o book_n you_o have_v learned_a to_o do_v then_o with_o the_o height_n equal_a to_o the_o height_n of_o l_o rear_v upon_o this_o last_o find_v circle_n which_o l●t_v be_v t_o as_o a_o base_a you_o shall_v satisfy_v the_o problem_n l●t_v that_o cone_n be_v v._o for_o this_o last_o cone_n v._o be_v to_o l_o as_o his_o base_a be_v to_o the_o base_a of_o l_o by_o the_o 11._o of_o this_o twelve_o but_o l_o be_v prove_v equal_a to_o the_o sphere_n give_v wherefore_o by_o the_o 7._o of_o the_o five_o this_o l●●t_a cone_n v_o have_v to_o r_o the_o sphere_n give_v that_o proportion_n which_o be_v between_o x_o and_o you_o assign_v and_o forasmuch_o as_o the_o height_n of_o this_o cone_n v_o be_v equal_a to_o the_o height_n of_o l_o and_o the_o height_n of_o l_o equal_a to_o saint_n the_o right_a line_n give_v by_o construction_n it_o be_v evident_a that_o a_o sphere_n be_v give_v &_o a_o right_a line_n we_o have_v make_v a_o upright_a cone_fw-mi equal_a to_o the_o sphere_n give_v or_o in_o any_o other_o proportion_n give_v between_o two_o right_a line_n which_o make_v cones_fw-la have_v their_o height_n equal_a to_o the_o right_a line_n give_v which_o ought_v to_o be_v do_v vuwilling_a i_o be_o to_o use_v thus_o many_o word_n in_o matter_n so_o plain●_n and_o ease_n ☜_o but_o this_o i_o think_v can_v not_o hinder_v they_o that_o by_o nature_n be_v not_o so_o quick_a of_o invention_n as_o to_o lead_v every_o thing_n general_o speak_v to_o a_o particular_a execution_n ¶_o a_o theorem_a 3._o every_o cylinder_n which_o have_v his_o base_a the_o great_a circle_n in_o a_o sp●er●_n &_o heith_n equal_a to_o the_o diameter_n of_o that_o sphere_n be_v sesquialtera_fw-la to_o that_o sphere_n also_o the_o superficies_n of_o that_o cylinder_n with_o his_o two_o base_n be_v sesquilatera_n to_o the_o superficies_n of_o the_o sphere_n and_o without_o his_o two_o base_n be_v equal_a to_o the_o superficies_n of_o that_o spher●_n suppose_v a_o sphere_n to_o be_v signify_v by_o a_o and_o a_o upright_o cylinder_n have_v to_o his_o base_a a_o circle_n equal_a to_o the_o great_a circle_n in_o a_o contain_v and_o his_o heith_n equal_a to_o the_o diameter_n of_o a_o let_v be_v signify_v by_o fg._n i_o say_v that_o fg_o be_v sesquialter_fw-la to_o a_o second_o i_o say_v that_o the_o crooked_a cylindrical_a superficies_n of_o fg_o together_o with_o the_o superfici●ces_n of_o his_o two_o opposite_a base_n be_v sesquialtera_fw-la to_o the_o whole_a superficies_n spherical_a of_o a._n three_o i_o say_v that_o the_o cylindrical_a superficies_n of_o fg_o omit_v his_o two_o opposite_a base_n be_v equal_a to_o the_o superficies_n of_o the_o spear_n a._n let_v the_o base_a of_o fg_o be_v the_o circle_n flb_n who_o centre_n suppose_v m_o and_o diameter_n fb_o and_o the_o axe_n of_o the_o same_o fg_o let_v be_v mh_o which_o be_v his_o heith_n for_o we_o suppose_v the_o cylinder_n to_o be_v upright_o and_o suppose_v h_o to_o be_v his_o top_n or_o vertex_fw-la forasmuch_o as_o by_o supposition_n mh_o be_v equal_a to_o the_o diameter_n of_o a._n let_v mh_o be_v divide_v into_o two_o equal_a part_n in_o the_o point_n n_o by_o a_o plain_a superficies_n pass_v by_o the_o point_n n_o and_o be_v parallel_n to_o the_o opposite_a base_n of_o fg._n by_o the_o thirteen_o of_o this_o twelve_o book_n it_o than_o follow_v that_o the_o cylinder_n fg_o be_v