Selected quad for the lemma: cause_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
cause_n	body_n	effect_n	part_n	1,666	5	4.5382	4	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

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three_o book_n in_o the_o four_o and_o six_o chapter_n may_v appear_v 6._o and_o what_o be_v the_o cause_n of_o the_o apt_a bond_n and_o friendly_a fellowship_n of_o the_o intellectual_a and_o mental_a part_n of_o we_o with_o our_o gross_a &_o corruptible_a body_n but_o a_o certain_a mean_a and_o harmonious_a spirituality_n with_o both_o participate_n &_o of_o both_o i●_z a_o manner_n 〈◊〉_d in_o the_o 7._o t●ne_n of_o man_n voice_n and_o also_o 8._o the_o sound_n of_o instrument_n what_o may_v be_v say_v of_o harmony_n no_o common_a musicien_n will_v light_o believe_v but_o of_o the_o sundry_a mixture_n as_o i_o may_v term_v it_o and_o concourse_n think_v diverse_a collation_n and_o application_n of_o these_o harmony_n as_o of_o three_o four_o five_o or_o more_o marvellous_a have_v the_o effect_n be_v and_o yet_o may_v be_v find_v and_o produce_v the_o like_a with_o some_o proportional_a consideration_n for_o our_o time_n and_o be_v in_o respect_n of_o the_o state_n of_o the_o thing_n 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and_o better_o know_v than_o i_o can_v express_v and_o so_o allow_v and_o like_v of_o as_o i_o will_v wish_v some_o other_o thing_n have_v the_o like_a hap_n i_o will_v spare_v to_o enlarge_v my_o line_n any_o far_o but_o consequent_o follow_v my_o purpose_n of_o cosmography_n i_o appoint_v brief_o in_o this_o place_n to_o geve_v you_o some_o intelligence_n cosmography_n be_v the_o whole_a and_o perfect_a description_n of_o the_o heavenly_a and_o also_o elemental_a part_n of_o the_o world_n and_o their_o homologal_a application_n and_o mutual_a collation_n necessary_a this_o art_n require_v astronomy_n geographie_n hydrography_n and_o music_n therefore_o it_o be_v no_o small_a art_n nor_o so_o simple_a as_o in_o common_a practice_n it_o be_v slight_o consider_v this_o match_v heaven_n and_o the_o earth_n in_o one_o frame_n and_o apt_o appli_v part_n correspondent_a so_o as_o the_o heavenly_a globe_n may_v in_o practice_n be_v due_o describe_v upon_o the_o geographical_a and_o hydrographical_a globe_n and_o there_o for_o we_o to_o consider_v a_o equinoctial_a circle_n a_o ecliptic_n line_n colour_n pole_n star_n in_o their_o true_a longitude_n latitude_n declination_n and_o verticalitie_n also_o clime_n and_o parallel_n and_o by_o a_o horizon_n annex_v and_o revolution_n of_o the_o earthly_a globe_n as_o the_o heaven_n is●_n by_o the_o pr●movan●_n carry_v about_o in_o 24●_n equal_a hour_n to_o learn_v the_o rise_n and_o settinge_n of_o star_n of_o virgil_n in_o his_o georgikes_fw-gr of_o hes●od_a of_o hypocrates_n in_o his_o medicinal_a sphere_n to_o perdic●●_n king_n of_o the_o macedonian_n of_o diocles_n to_o king_n antigonus_n and_o of_o other_o famous_a philosopher_n prescribe_v a_o thing_n necessary_a for_o due_a manure_v of_o the_o earth_n for_o navigation_n for_o the_o alteration_n of_o man_n body●_n be_v whole_a sick_a wound_a or_o bruise_a by_o the_o revolution_n also_o or_o move_v of_o the_o globe_n cosmographical_a the_o rise_n and_o set_v of_o the_o son_n

bring_v to_o the_o uttermost_a they_o may_v be_v stretch_v unto_o and_o apply_v sensible_o as_o for_o example_n the_o natural_a philosopher_n dispute_v and_o make_v goodly_a show_n of_o reason_n and_o the_o astronomer_n and_o the_o optical_a mechanicien_fw-fr put_v some_o thing_n in_o experience_n but_o neither_o all_o that_o they_o may_v nor_o yet_o sufficient_o and_o to_o the_o utmost_a those_o which_o they_o do_v there_o then_o the_o archemaster_n step_v in_o and_o lead_v forth_o on_o the_o experience_n by_o order_n of_o his_o doctrine_n experimental_a to_o the_o chief_a and_o final_a power_n of_o natural_a and_o mathematical_a artes._n of_o two_o or_o three_o man_n in_o who_o this_o description_n of_o archemastry_n be_v experimental_o verify_v i_o have_v read_v and_o hard_a and_o good_a record_n be_v of_o their_o such_o perfection_n so_o that_o this_o art_n be_v no_o fantastical_a imagination_n as_o some_o sophister_n might_n cum_fw-la suis_fw-la insolubilibus_fw-la make_v a_o 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ma_fw-fr 〈…〉_o here_o in_o these_o realm_n of_o england_n and_o ireland_n tha●_z 〈…〉_o &_o compass_n who_o with_o their_o own_o skill_n and_o expe_v 〈…〉_o ble_o by_o these_o good_a help_n and_o information_n to_o find_v 〈…〉_o s_o strange_a engine_n and_o instrument_n for_o sundry_a purp_v 〈…〉_o wealth_n or_o for_o private_a pleasure_n and_o for_o the_o better_a maintain_v 〈…〉_o own_o estate_n i_o will_v not_o therefore_o fight_v against_o my_o own_o shadow_n for_o no_o man_n i_o be_o s●re_v will_v open_v his_o mouth_n against_o this_o enterprise_n no_o man_n i_o say_v who_o either_o have_v charity_n towards_o his_o brother_n and_o will_v be_v glad_a of_o his_o furtherance_n in_o virtuous_a knowledge_n o●_n that_o have_v any_o care_n &_o zeal_n for_o the_o better_n of_o the_o common_a state_n of_o this_o realm_n neither_o any_o that_o make_v account_n what_o the_o wise_a sort_n of_o man_n sage_a and_o stay_v do_v think_v of_o they_o to_o ●one_n therefore_o will_v i_o make_v any_o 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5._o and_o 32._o of_o the_o first_o the_o angle_n bfc_n also_o shall_v be_v the_o three_o part_n of_o two_o right_a angle_n wherefore_o either_o of_o the_o two_o angle_n remain_v fbc_n and_o fcb_n for_o asmuch_o as_o they_o be_v equal_a by_o the_o 5._o of_o the_o first_o shall_v be_v two_o three_o part_n of_o two_o right_a angle_n by_o the_o 32._o of_o the_o same_o or_o by_o the_o 4._o of_o the_o first_o forasmuch_o as_o the_o angle_n bfc_n be_v equal_a to_o the_o angle_n fba_n and_o the_o two_o side_n fb_o and_o fc_o be_v equal_a to_o the_o two_o side_n ab_fw-la and_o bf_o the_o base_a bc_o shall_v be_v equal_a to_o the_o base_a bf_o and_o therefore_o be_v equal_a to_o the_o line_n fc_o wherefore_o the_o triangle_n fbc_n be_v equilater_n and_o equiangle_n last_o make_v the_o angle_n cfd_v equal_a to_o either_o of_o the_o angle_n at_o the_o point_n fletcher_n by_o draw_v the_o line_n fd._n and_o draw_v a_o line_n from_o c_o to_o d._n now_o then_o by_o the_o former_a reason_n the_o triangle_n fcd_n shall_v be_v equilater_n and_o equiangle_n and_o for_o asmuch_o as_o the_o three_o angle_n at_o the_o point_n f_o be_v equal_a to_o two_o right_a angle_n for_o each_o of_o they_o be_v the_o three_o part_n of_o two_o right_a angle_n therefore_o by_o the_o 14._o of_o the_o first_o ad_fw-la be_n one_o right_a line_n and_o for_o that_o cause_n be_v the_o diameter_n of_o the_o circle_n wherefore_o if_o the_o other_o semicircle_n afd_v be_v divide_v into_o so_o many_o equal_a part_n as_o the_o semicircle_n abcd_o be_v divide_v into_o it_o shall_v comprehend_v so_o many_o equal_a line_n subtend_v unto_o it_o wherefore_o the_o line_n ab_fw-la be_v the_o side_n of_o a_o equilater_n hexagon_n figure_n to_o be_v inscribe_v in_o the_o circle_n which_o hexagon_n figure_n also_o shall_v be_v equiangle_n for_o the_o half_a of_o the_o whole_a angle_n b_o be_v equal_a to_o the_o half_a of_o the_o whole_a angle_n c_o which_o be_v require_v to_o be_v do_v now_o than_o if_o we_o draw_v 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by_o the_o point_n a_o b_o c_o d_o e_o f_o be_v draw_v right_a line_n touch_v the_o circle_n then_o shall_v there_o be_v describe_v about_o the_o circle_n a_o hexagon_n figure_n equilater_n and_o equiangle_n which_o may_v be_v demonstrate_v by_o that_o which_o have_v be_v speak_v of_o the_o describe_v of_o a_o pentagon_n about_o a_o circle_n and_o moreover_o by_o those_o thing_n which_o have_v be_v speak_v of_o pentagons_n we_o may_v in_o a_o hexagon_n give_v either_o describe_v or_o circumscribe_v a_o circle_n which_o be_v require_v to_o be_v do_v the_o 16._o problem_n the_o 16._o proposition_n in_o a_o circle_n give_v to_o describe_v a_o quindecagon_n or_o figure_n of_o fifteen_o angle_n equilater_n and_o equiangle_n svppose_v that_o the_o circle_n give_v be_v abcd._n it_o be_v require_v in_o the_o circle_n abcd_o to_o describe_v a_o figure_n of_o fifteen_o angle_n consist_v of_o equal_a side_n and_o of_o equal_a angle_n describe_v in_o the_o circle_n abcd_o the_o side_n of_o a_o equilater_n triangle_n and_o let_v the_o same_o be_v ac_fw-la and_o in_o the_o ark_n ac_fw-la describe_v the_o side_n of_o a_o equilater_n pentagon_n and_o let_v the_o same_o be_v ab_fw-la construction_n now_o then_o of_o such_o equal_a part_n whereof_o the_o whole_a circle_n abcd_o contain_v fifteen_o of_o such_o part_n i_o say_v the_o circumference_n abc_n be_v the_o three_o part_n of_o the_o circle_n shall_v contain_v five_o and_o the_o circumference_n ab_fw-la be_v the_o five_o part_n of_o a_o circle_n shall_v contain_v three_o wherefore_o the_o residue_n bc_o shall_v contain_v two_o divide_v by_o the_o 30._o of_o the_o first_o the_o ark_n bc_o into_o two_o equal_a part_n in_o the_o point_n e._n demonstration_n wherefore_o either_o of_o these_o circumference_n be_v &_o ec_o be_v the_o fifteen_o part_n of_o the_o circle_n abcd._n if_o therefore_o there_o be_v draw_v right_a line_n from_o b_o to_o e_o and_o from_o e_o to_o c_z and_o then_o begin_v at_o the_o point_n b_o or_o at_o the_o point_n c_o there_o be_v apply_v into_o the_o circle_n abcd_o right_a line_n equal_a unto_o ebb_n or_o ec_o and_o so_o continue_v till_o you_o come_v to_o the_o point_n c_o if_o you_o begin_v at_o b_o or_o to_o the_o point_fw-fr b_o if_o you_o begin_v at_o c_o and_o there_o shall_v be_v describe_v in_o the_o circle_n abcd_o a_o figure_n of_o fifteen_o angle_n equilater_n and_o equiangle_n which_o be_v require_v to_o be_v do_v and_o in_o like_a sort_n as_o in_o a_o pentagon_n if_o by_o the_o point_n where_o the_o circle_n be_v divide_v be_v draw_v right_a line_n touch_v the_o circle_n in_o the_o say_a point_n there_o shall_v be_v describe_v about_o the_o circle_n a_o figure_n of_o fifteen_o angle_n equilater_n &_o equiangle_n and_o in_o like_a sort_n by_o the_o self_n same_o observation_n that_o be_v in_o pentagons_n we_o may_v in_o a_o figure_n of_o fifteen_o angle_n give_v be_v equilater_n and_o equiangle_n either_o inscribe_v or_o circumscribe_v a_o circle_n flussates_n ¶_o a_o addition_n of_o flussates_n to_o find_v out_o infinite_a figure_n of_o many_o angle_n if_o into_o a_o circle_n from_o one_o point_n be_v apply_v the_o side_n of_o two_o side_n poligonon_n figure_n the_o excess_n of_o the_o great_a ark_n above_o the_o less_o shall_v comprehend_v a_o ark_n contain_v so_o many_o side_n of_o the_o poligonon_n figure_n to_o be_v inscribe_v by_o how_o many_o unity_n the_o denomination_n of_o the_o poligonon_n figure_n of_o the_o less_o side_n exceed_v the_o denomination_n of_o the_o poligonon_n figure_n of_o the_o great_a side_n and_o the_o number_n of_o the_o side_n of_o the_o poligonon_n figure_n to_o be_v inscribe_v be_v produce_v of_o the_o multiplication_n of_o the_o denomination_n of_o the_o foresay_a poligonon_n figure_v the_o one_o into_o the_o other_o as_o for_o example_n suppose_v that_o into_o the_o circle_n abe_n be_v apply_v the_o side_n of_o a_o equilater_n and_o equiangle_n hexagon_n figure_n by_o the_o 15._o of_o this_o book_n which_o let_v be_v ab_fw-la and_o likewise_o the_o side_n of_o a_o pentagon_n by_o the_o 11._o of_o this_o book_n which_o let_v be_v ac_fw-la and_o the_o side_n of_o a_o square_n by_o the_o 6._o of_o this_o book_n which_o let_v be_v ad_fw-la and_o the_o side_n of_o a_o equilater_n triangle_n by_o the_o 2._o of_o this_o book_n which_o let_v be_v ae_n then_o i_o say_v that_o the_o excess_n of_o the_o ark_n ad_fw-la above_o the_o ark_n ab_fw-la which_o excess_n be_v the_o ark_n bd_o contain_v so_o many_o side_n of_o the_o poligonon_n figure_n to_o be_v inscribe_v of_o how_o many_o unity_n the_o denominator_fw-la of_o the_o hexagon_n ab_fw-la which_o be_v six_o exceed_v the_o denominator_fw-la of_o the_o square_a ad_fw-la which_o be_v four_o and_o forasmuch_o as_o that_o excess_n it_o two_o unity_n therefore_o in_o bd_o there_o shall_v be_v two_o side_n and_o the_o denominator_fw-la of_o the_o poligonon_n figure_n which_o be_v to_o be_v inscribe_v shall_v be_v produce_v of_o the_o multiplication_n of_o the_o denominator_n of_o the_o foresay_a poligonon_n figure_n namely_o of_o the_o multiplication_n of_o 6._o into_o 4._o which_o make_v 24._o which_o number_n be_v the_o denominator_fw-la of_o the_o poligonon_n figure_n who_o two_o side_n shall_v subtend_v the_o ark_n bd._n for_o of_o such_o equal_a part_n whereof_o the_o whole_a circumference_n contain_v 24_o of_o such_o part_n i_o say_v the_o circumference_n ab_fw-la contain_v 4_o and_o the_o circumference_n ad_fw-la contain_v 6._o wherefore_o if_o from_o ad_fw-la which_o subtend_v 6._o part_n be_v take_v away_o 4._o which_o ab_fw-la subtend_v there_o shall_v remain_v unto_o bd_o two_o of_o such_o part_n of_o which_o the_o whole_a contain_v 24._o wherefore_o of_o a_o hexagon_n and_o a_o square_n be_v make_v a_o

the_o ten_o the_o line_n ab_fw-la be_v incommensurable_a in_o length_n unto_o the_o line_n be._n wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o be_v by_o the_o first_o of_o the_o six_o and_o 10._o of_o this_o book_n but_o unto_o the_o square_n of_o the_o line_n ab_fw-la be_v equal_a the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o by_o the_o 47._o of_o the_o first_o and_o unto_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o be_v be_v equal_a that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd_v that_o be_v which_o be_v contain_v under_o ad_fw-la and_o db._n for_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o fd_o be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n ad_fw-la and_o db_o by_o the_o last_o part_n of_o the_o first_o assumpt_n go_v before_o the_o 33._o of_o this_o ten_o book_n conclude_v wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o db._n wherefore_o there_o be_v find_v out_o two_o right_a line_n ad_fw-la and_o db_o incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o medial_a superficies_n conclusion_n and_o the_o parallelogram_n contain_v under_o they_o make_v also_o a_o medial_a superficies_n which_o parallelogram_n moreover_o be_v incommensurable_a to_o the_o superficies_n compose_v of_o the_o square_n of_o those_o line_n add_v together_o which_o be_v require_v to_o be_v do_v the_o beginning_n of_o the_o senaries_n by_o composition_n ¶_o the_o 2d_o theorem_a the_o 36._o proposition_n if_o two_o rational_a line_n commensurable_a in_o power_n only_o be_v add_v together_o composition_n the_o whole_a line_n be_v irrational_a and_o be_v call_v a_o binomium_n or_o a_o binomial_a line_n 〈…〉_o b_o and_o bc_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n bc._n but_o unto_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v commensurable_a the_o parallelogram_n contain_v under_o ab_fw-la and_o bc_o twice_o by_o the_o 6._o of_o the_o ten_o wherefore_o that_o which_o be_v contain_v under_o ab_fw-la and_o bc_o twice_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n bc_o by_o the_o 13_o of_o the_o ten_o but_o unto_o the_o square_n of_o the_o line_n bc_o be_v commensurable_a that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o by_o the_o 15._o of_o the_o ten_o for_o by_o supposition_n the_o line_n ab_fw-la and_o bc_o be_v commensurable_a in_o power_n only_o wherefore_o by_o the_o 13._o of_o the_o ten_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o wherefore_o by_o the_o 16._o of_o the_o ten_o that_o which_o be_v contain_v under_o ab_fw-la and_o bc_o twice_o together_o with_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o which_o by_o the_o 4._o of_o the_o second_o be_v equal_a to_o the_o square_n of_o the_o whole_a line_n ac_fw-la be_v incommensurable_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o ab_fw-la and_o bc_o add_v together_o but_o that_o which_o be_v compose_v of_o the_o square_n of_o ab_fw-la and_o bc_o add_v together_o be_v rational_a for_o it_o be_v commensurable_a to_o either_o of_o the_o square_n of_o the_o line_n ab_fw-la and_o bg_o of_o which_o either_o of_o they_o be_v rational_a by_o supposition_n wherefore_o the_o square_a of_o the_o line_n ac_fw-la be_v by_o the_o 10._o definition_n of_o the_o ten_o irrational_a wherefore_o the_o line_n ac_fw-la also_o be_v irrational_a and_o be_v call_v a_o binomial_a line_n line_n this_o proposition_n show_v the_o generation_n and_o production_n of_o the_o second_o kind_n of_o irrational_a line_n which_o be_v call_v a_o binomium_n or_o a_o binomial_a line_n the_o definition_n whereof_o be_v full_o gather_v out_o of_o this_o proposition_n and_o that_o thus_o a_o binomium_n or_o a_o binomial_a line_n be_v a_o irrational_a line_n compose_v of_o two_o rational_a line_n commensurable_a the_o one_o to_o the_o other_o in_o power_n only_o and_o it_o be_v call_v a_o binomium_n that_o be_v have_v two_o name_n because_o it_o be_v make_v of_o two_o such_o line_n as_o of_o his_o part_n which_o be_v only_o commensurable_a in_o power_n and_o not_o in_o length_n and_o therefore_o each_o part_n or_o line_n or_o at_o the_o least_o the_o one_o of_o they_o as_o touch_v length_n be_v uncertain_a and_o unknown_a wherefore_o be_v join_v together_o their_o quantity_n can_v be_v express_v by_o any_o one_o number_n or_o name_n but_o each_o part_n remain_v to_o be_v several_o name_v in_o such_o sort_n as_o it_o may_v and_o of_o these_o binomial_a line_n there_o be_v six_o several_a kind_n line_n the_o first_o binomiall_n the_o second_o the_o three_o the_o four_o the_o five_o and_o the_o six_o of_o what_o nature_n and_o condition_n each_o of_o these_o be_v shall_v be_v know_v by_o their_o definitious_a which_o be_v afterward_o set_v in_o their_o due_a place_n ¶_o the_o 25._o theorem_a the_o 37._o proposition_n if_o two_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o rational_a superficies_n be_v add_v together_o the_o whole_a line_n be_v irrational_a and_o be_v call_v a_o first_o bimedial_a line_n let_v these_o two_o medial_a line_n ab_fw-la and_o bc_o be_v commensurable_a in_o power_n only_o and_o contain_v a_o rational_a superficies_n the_o 27._o of_o the_o ten_o teach_v to_o find_v out_o two_o such_o line_n be_v compose_v then_o i_o say_v that_o the_o whole_a line_n ac_fw-la be_v irrational_a for_o as_o 〈◊〉_d say_v in_o the_o proposition_n next_o go_v before_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o 〈◊〉_d ab_fw-la and_o bc_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twis●_n wherefore_o by_o the_o 16._o of_o the_o ten_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o demonstration_n that_o be_v the_o square_a of_o the_o line_n ac_fw-la be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o i●_z commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o once_o by_o the_o 6._o of_o the_o ten_o wherefore_o the_o square_a of_o the_o whole_a line_n ac_fw-la be_v by_o the_o 13●_n of_o the_o ten_o incommensurable_a ●o_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o once_o but_o by_o supposition_n the_o line_n ab_fw-la and_o bc_o comprehend_v a_o rational_a superficies_n wherefore_o the_o square_a of_o the_o whole_a line_n ac_fw-la be_v irrational_a wherefore_o also_o the_o line_n ac_fw-la be_v irrational_a and_o it_o be_v call_v a_o first_o bimedial_a line_n the_o three_o irrational_a line_n which_o be_v call_v a_o first_o bimedial_a line_n be_v show_v by_o this_o proposition_n and_o the_o definition_n thereof_o be_v by_o it_o make_v manifest_a line_n which_o be_v this_o a_o first_o bimedial_a line_n be_v a_o irrational_a line_n which_o be_v compose_v of_o two_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o rational_a parallelogram_n it_o be_v call_v a_o first_o bimedial_a line_n by_o cause_n the_o two_o medial_a line_n or_o part_n whereof_o it_o be_v compose_v contain_v a_o rational_a superficies_n which_o be_v prefer_v before_o a_o irrational_a ¶_o the_o 26._o theorem_a the_o 38._o proposition_n if_o two_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o medial_a superficies_n be_v add_v together_o the_o whole_a line_n be_v irrational_a and_o be_v call_v a_o second_o bimedial_a line_n let_v these_o two_o medial_a line_n ab_fw-la and_o bc_o be_v commensurable_a in_o power_n only_o and_o contain_v a_o medial_a superficies_n the_o 28._o of_o the_o ten_o teach_v to_o find●_n out_o two_o such_o line_n be_v add_v together_o construction_n then_o i_o say_v that_o the_o whole_a line_n ac_fw-la be_v irrational_a take_v a_o rational_a line_n de._n and_o by_o the_o 44._o of_o the_o first_o upon_o the_o line_n de_fw-fr apply_v the_o parallelogram_n df_o equal_a to_o the_o square_n of_o the_o line_n ac_fw-la who_o be_v other_o side_n let_v be_v the_o line_n dg_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la be_v by_o the_o 4._o of_o the_o second_o equal_a to_o that_o which_o be_v compose_v of_o

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 du●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 disgraced●_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 ●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 ●vclid●_n in_o th●s●_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 bo●●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 ●●o●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 ●_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 boo●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 b●o●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 bo●●e_n for_o that_o otherwise_o he_o can_v not_o compare_v ●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 boo●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 book●_n eight_o boo●●_n and_o nine_o book_n book_n he_o entreat●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 boo●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 line●_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 others●_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 o●_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 r●st_o ●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 knowledge_n of_o bodily_a and_o solid_a form_n can_v not_o be_v attain_a unto_o and_o first_o be_v set_v the_o definition_n as_o follow_v definition_n a_o solid_a or_o body_n be_v that_o which_o have_v length_n breadth_n and_o thickness_n definition_n and_o the_o term_n or_o limit_v of_o a_o solid_a be_v a_o superficies_n there_o be_v three_o kind_n of_o continual_a quantity_n a_o line_n a_o superficies_n and_o a_o solid_a or_o body_n the_o beginning_n of_o all_o which_o as_o before_o have_v be_v say_v be_v a_o point_n which_o be_v indivisible_a two_o of_o these_o quantity_n namely_o a_o line_n and_o a_o superficies_n be_v define_v of_o euclid_n before_o in_o his_o first_o book_n but_o the_o three_o kind_n namely_o a_o solid_a or_o body_n he_o there_o define_v not_o as_o a_o thing_n which_o pertain_v not_o then_o to_o his_o purpose_n but_o here_o in_o this_o place_n he_o set_v the_o definition_n thereof_o as_o that_o which_o chief_o now_o pertain_v to_o his_o purpose_n and_o without_o which_o nothing_o in_o these_o thing_n can_v profitable_o be_v teach_v a_o solid_a say_v he_o be_v that_o which_o have_v length_n breadth_n and_o thickness_n or_o depth_n there_o be_v as_o before_o have_v be_v teach_v three_o reason_n or_o mean_n of_o measure_v which_o be_v call_v common_o dimension_n namely_o length_n breadth_n and_o thickness_n these_o dimension_n be_v ascribe_v unto_o quantity_n only_o by_o these_o be_v all_o kind_n of_o quantity_n define_v ●●_o be_v count_v perfect_a or_o imperfect_a according_a as_o they_o be_v partaker_n of_o few_o or_o more_o of_o they_o as_o euclid_n define_v a_o line_n ascribe_v unto_o it_o only_o one_o of_o these_o dimension_n namely_o length_n wherefore_o a_o line_n be_v the_o imperfect_a kind_n of_o quantity_n in_o define_v of_o a_o superficies_n he_o ascribe_v unto_o it_o two_o dimension_n namely_o length_n and_o breadth_n whereby_o a_o superficies_n be_v a_o quantity_n of_o

line_n which_o subtend_v the_o angle_n zob_n to_o the_o three_o line_n which_o subtend_v the_o angle_n zkb_n but_o by_o construction_n boy_n be_v equal_a to_o bk_o therefore_o oz_n be_v equal_a to_o kz_o and_o the_o three_o al●o_o be_v equal_a to_o the_o three_o wherefore_o the_o point_n z_o in_o respect_n of_o the_o two_o triangle_n rectangle_v ozb_n and_o kzb_n determine_v one_o and_o the_o same_o magnitude_n i●_z the_o line_n bz_o which_o can_v not_o be_v if_o any_o other_o point_n in_o the_o line_n bz_o be_v assign_v near_o or_o far_o of_o from_o the_o point_n b._n one_o only_a point_n therefore_o be_v that_o at_o which_o the_o two_o perpendicular_n kz_o and_o oz_n fall_v but_o by_o construction_n oz_n fall_v at_o z_o the_o point_n and_o therefore_o at_o the_o same_o z_o do_v the_o perpendicular_a draw_v from_o king_n fall_v likewise_o which_o be_v require_v to_o be_v demonstrate_v although_o a_o brief_a monition_n may_v herein_o have_v serve_v for_o the_o pregnant_a or_o the_o humble_a learner_n yet_o for_o they_o that_o be_v well_o please_v to_o have_v thing_n make_v plain_a with_o many_o word_n and_o for_o the_o stiffnecked_a busy_a body_n it_o be_v necessary_a with_o my_o controlment_n of_o other_o to_o annex_v the_o cause_n &_o reason_n thereof_o both_o invincible_a and_o also_o evident_a a_o corollary_n 1._o hereby_o it_o be_v manifest_a that_o two_o equal_a circle_n cut_v one_o the_o other_o by_o the_o whole_a diameter_n if_o from_o one_o and_o the_o same_o end_n of_o their_o common_a diameter_n equal_a portion_n of_o their_o circumference_n be_v take_v and_o from_o the_o point_n end_v those_o equal_a portion_n two_o perpendicular_n be_v let_v down_o to_o their_o common_a diameter_n those_o perpendicular_n shall_v fall_v upon_o one_o and_o the_o same_o point_n of_o their_o common_a diameter_n 2._o second_o it_o follow_v that_o those_o perpendicular_n be_v equal_a ¶_o note_n from_o circle_n in_o our_o first_o supposition_n each_o to_o other_o perpendicular_o erect_v we_o proceed_v and_o infer_v now_o these_o corollary_n whether_o they_o be_v perpendicular_o erect_v or_o no_o by_o reasou_v the_o demonstration_n have_v a_o like_a force_n upon_o our_o supposition_n here_o use_v ¶_o the_o 16._o theorem_a the_o 18._o proposition_n sphere_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 diameter_n be_v svppose_v that_o there_o be_v two_o sphere_n abc_n and_o def_n and_o let_v their_o diameter_n be_v bc_o and_o ef._n then_o i_o say_v that_o the_o sphere_n abc_n be_v to_o the_o sphere_n def_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n ef._n proposition_n for_o if_o not_o than_o the_o sphere_n abc_n be_v in_o treble_a proportion_n of_o that_o in_o which_o bc_o be_v to_o of_o either_o to_o some_o sphere_n less_o than_o the_o sphere_n def_n or_o to_o some_o sphere_n great_a first_o let_v it_o be_v unto_o a_o less_o namely_o to_o ghk_v case_n and_o imagine_v that_o the_o sphere_n def_n and_o ghk_o be_v both_o about_o one_o and_o the_o self_n same_o centre_n impossibility_n and_o by_o the_o proposition_n next_o go_v before_o describe_v in_o the_o great_a sphere_n def_n a_o polihedron_n or_o a_o solid_a of_o many_o side_n not_o touch_v the_o superficies_n of_o the_o less_o sphere_n ghk_v and_o suppose_v also_o that_o in_o the_o sphere_n abc_n be_v inscribe_v a_o polihedron_n like_a to_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n wherefore_o by_o the_o corollary_n of_o the_o same_o the_o polihedron_n which_o be_v in_o the_o sphere_n abc_n be_v to_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n ef._n but_o by_o supposition_n the_o sphere_n abc_n be_v to_o the_o sphere_n ghk_v in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n ef._n wherefore_o as_o the_o sphere_n abc_n be_v to_o the_o sphere_n ghk_o so_o be_v the_o polihedron_n which_o be_v describe_v in_o the_o sphere_n abc_n to_o the_o polihedron_n which_o be_v describe_v in_o the_o sphere_n def_n by_o the_o 11._o of_o the_o five_o wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o sphere_n abc_n be_v to_o the_o polihedron_n which_o be_v describe_v in_o it_o so_o be_v the_o sphere_n ghk_v to_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n but_o the_o sphere_n abc_n be_v great_a than_o the_o polihedron_n which_o be_v describe_v in_o it_o wherefore_o also_o the_o sphere_n ghk_o be_v great_a than_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n by_o the_o 14._o of_o the_o five_o but_o it_o be_v also_o less_o for_o it_o be_v contain_v in_o it_o which_o impossible_a wherefore_o the_o sphere_n abc_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n of_o to_o any_o sphere_n less_o than_o the_o sphere_n def_n in_o like_a sort_n also_o may_v we_o prove_v that_o the_o sphere_n def_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o be_v to_o the_o diameter_n bc_o to_o any_o sphere_n less_o than_o the_o sphere_n abc_n case_n now_o i_o say_v that_o the_o sphere_n abc_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n of_o to_o any_o sphere_n great_a they_o the_o sphere_n def_n for_o if_o it_o be_v possible_a let_v it_o be_v to_o a_o great_a namely_o to_o lmn_o wherefore_o by_o conversion_n the_o sphere_n lmn_o be_v to_o the_o sphere_n abc_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o be_v to_o the_o diameter_n bc._n but_o as_o the_o sphere_n lmn_o be_v to_o the_o sphere_n abc_n so_o be_v the_o sphere_n def_n to_o some_o sphere_n less_o they_o the_o sphere_n abc_n boo●●_n as_o it_o have_v before_o be_v prove_v for_o the_o sphere_n lmn_o be_v great_a than_o the_o sphere_n def_n wherefore_o the_o sphere_n def_n be_v in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o be_v to_o the_o diameter_n bc_o to_o some_o sphere_n less_o they_o the_o sphere_n abc_n which_o be_v prove_v to_o be_v impossible_a wherefore_o the_o sphere_n abc_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o be_v be_v to_o of_o to_o any_o sphere_n great_a they_o the_o sphere_n def_n and_o it_o be_v also_o prove_v that_o it_o be_v not_o to_o any_o less_o wherefore_o the_o sphere_n abc_n be_v to_o the_o sphere_n def_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n of_o which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o flussas_n hereby_o it_o be_v manifest_a that_o sphere_n be_v the_o one_o to_o the_o other_o as_o like_o polihedrons_n and_o in_o like_a sort_n describe_v in_o they_o be_v namely_o each_o be_v in_o triple_a proportion_n of_o that_o in_o which_o the_o diameter_n a_o corollary_n add_v by_o m●_n dee_n it_o be_v then_o evident_a how_o to_o geve_v two_o right_a line_n have_v that_o proportion_n between_o they_o which_o any_o two_o sphere_n give_v have_v the_o one_o to_o the_o other_o for_o if_o to_o their_o diameter_n as_o to_o the_o first_o and_o second_o line_n of_o four_o in_o continual_a proportion_n you_o adjoin_v a_o three_o and_o a_o four_o line_n in_o continual_a proportion_n as_o i_o have_v teach_v before_o the_o first_o and_o four_o line_n shall_v answer_v the_o problem_n rule_n how_o general_a this_o rule_n be_v in_o any_o two_o like_a solid_n with_o their_o correspondent_a or_o omologall_n line_n i_o need_v not_o with_o more_o word_n declare_v ¶_o certain_a theorem_n and_o problem_n who_o use_n be_v manifold_a in_o sphere_n cones_n cylinder_n and_o other_o solid_n add_v by_o joh._n dee_n a_o theorem_a 1._o the_o whole_a superficies_n of_o any_o sphere_n be_v quadrupla_fw-la to_o the_o great_a circle_n in_o the_o same_o sphere_n contain_v it_o be_v needless_a to_o bring_v archimedes_n demonstration_n hereof_o into_o this_o place_n see_v his_o book_n of_o the_o sphere_n and_o cylinder_n with_o other_o his_o wo●kes_n be_v every_o where_o to_o be_v have_v and_o the_o demonstration_n thereof_o easy_a a_o theorem_a 2._o every_o sphere_n be_v quadrupl●_n to_o that_o cone_n who_o base_a be_v the_o great_a circle_n &_o height_n the_o semidiameter_n of_o the_o same_o sphere_n this_o be_v the_o 32._o proposition_n of_o archimedes_n fi●st_n book_n of_o the_o sphere_n and_o cylinder_n a_o problem_n 1._o a_o sphere_n be_v give_v to_o make_v a_o upright_a cone_n 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 and_o as_o concern_v the_o other_o part_n of_o