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enter_v into_o and_o full_o understand_v without_o any_o strange_a help_n or_o succour_n only_o by_o diligent_a observation_n of_o the_o order_n and_o course_n of_o euclides_n write_n so_o that_o he_o which_o diligent_o have_v peruse_v and_o full_o understand_v the_o 9_o book_n go_v before_o and_o mark_v also_o earnest_o the_o principle_n and_o definition_n of_o this_o â—enth_fw-mi book_n he_o shall_v well_o perceive_v that_o euclid_n be_v of_o himself_o a_o sufficient_a teacher_n and_o instructor_n understand_v and_o need_v not_o the_o help_n of_o any_o other_o and_o shall_v soon_o see_v that_o this_o ten_o book_n be_v not_o of_o such_o hardness_n and_o obscurity_n as_o it_o have_v be_v hitherto_o think_v yea_o i_o doubt_v not_o but_o that_o by_o the_o travel_n and_o industry_n take_v in_o this_o translation_n and_o by_o addition_n and_o emendation_n get_v of_o other_o there_o shall_v appear_v in_o it_o no_o hardness_n at_o all_o but_o shall_v be_v as_o easy_a as_o the_o rest_n of_o his_o book_n be_v definition_n definition_n 1_o magnitude_n commensurable_a be_v suchwhich_v one_o and_o the_o self_n same_o measure_n do_v measure_n first_o he_o show_v what_o magnitude_n be_v commensurable_a one_o to_o a_o other_o to_o the_o better_a and_o more_o clear_a understanding_n of_o this_o definition_n note_v that_o that_o measure_v whereby_o any_o magnitude_n be_v measure_v be_v less_o than_o the_o magnitude_n which_o it_o measure_v or_o at_o least_o equal_a unto_o it_o for_o the_o great_a can_v by_o no_o mean_n measure_v the_o less_o far_o it_o behove_v that_o that_o measure_n if_o it_o be_v equal_a to_o that_o which_o be_v measure_v take_v once_o make_v the_o magnitude_n which_o be_v measure_v if_o it_o be_v less_o than_o oftentimes_o take_v and_o repeat_v it_o must_v precise_o render_v and_o make_v the_o magnitude_n which_o it_o measure_v which_o thing_n in_o number_n be_v easy_o see_v for_o that_o as_o be_v before_o say_v all_o number_n be_v commensurable_a one_o to_o a_o other_o and_o although_o euclid_n in_o this_o definition_n comprehend_v purpose_o only_o magnitude_n which_o be_v continual_a quantity_n as_o be_v line_n superficiece_n and_o body_n yet_o undoubted_o the_o explication_n of_o this_o and_o such_o like_a place_n be_v apt_o to_o be_v seek_v of_o number_n as_o well_o rational_a as_o irrational_a for_o that_o all_o quantity_n commensurable_a have_v that_o proportion_n the_o one_o to_o the_o other_o which_o number_n have_v to_o number_n in_o number_n therefore_o 9_o and_o 12_o be_v commensurable_a because_o there_o be_v one_o common_a measure_n which_o measure_v they_o both_o namely_o the_o number_n 3_n first_o it_o measure_v 12_o for_o it_o be_v less_o than_o 12._o and_o be_v take_v certain_a time_n namely_o 4_o time_n it_o make_v exact_o 12_o 3_o time_n 4_o be_v 12_o it_o also_o measure_v 9_o for_o it_o be_v less_o than_o 9_o and_o also_o take_v certain_a time_n namely_o 3_o time_n it_o make_v precise_o 9_o 3_o time_n 3_o be_v 9_o 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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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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 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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
that_o the_o line_n of_o be_v make_v equal_a to_o the_o line_n ad_fw-la which_o be_v the_o diameter_n of_o the_o square_n abcd_o of_o which_o square_v the_o line_n ab_fw-la be_v a_o side_n it_o be_v certain_a that_o the_o â—ide_v of_o a_o square_n be_v incommensurable_a in_o length_n to_o the_o diameter_n of_o the_o same_o square_n if_o there_o be_v yet_o find_v any_o one_o superficies_n which_o measure_v the_o two_o square_n abcd_o and_o efgh_a as_o here_o do_v the_o triangle_n abdella_n or_o the_o triangle_n acd_v note_v in_o the_o square_n abcd_o or_o any_o of_o the_o four_o triangle_n note_v in_o the_o square_a efgh_o as_o appear_v somewhat_o more_o manifest_o in_o the_o second_o example_n in_o the_o declaration_n of_o the_o last_o definition_n go_v before_o the_o line_n of_o be_v also_o a_o rational_a line_n note_v that_o these_o line_n which_o here_o be_v call_v rational_a line_n be_v not_o rational_a line_n of_o purpose_n or_o by_o supposition_n as_o be_v the_o first_o 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
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 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 â—â—_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
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 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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
virescit_fw-la vulnere_fw-la veritas_fw-la the_o element_n of_o geometry_n of_o the_o most_o ancient_a philosopher_n euclid_n of_o megara_n faithful_o now_o first_o translate_a into_o the_o english_a tongue_n by_o h._n billingsley_n citizen_n of_o london_n whereunto_o be_v annex_v certain_a scholics_n annotation_n and_o invention_n of_o the_o best_a mathematician_n both_o of_o time_n past_a and_o iâ—_z this_o our_o age_n with_o a_o very_a fruitful_a preface_n make_v by_o m._n i._n do_v you_o specify_v the_o chief_a mathematical_a science_n what_o they_o be_v and_o whereunto_o commodiousâ—_n where_o also_o be_v disclose_v certain_a new_a secret_n mathematical_a and_o mechanical_a until_o these_o our_o day_n great_o miss_v imprint_v at_o london_n by_o john_n day_n the_o translator_n to_o the_o reader_n there_o be_v gentle_a reader_n nothing_o the_o word_n of_o god_n only_o set_v apart_o which_o so_o much_o beautify_v and_o adorn_v the_o soul_n and_o mind_n of_o man_n as_o do_v the_o 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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 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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
namely_o the_o first_o ten_o proposition_n as_o they_o follow_v in_o order_n which_o be_v undoubted_o great_a pleasure_n to_o consider_v also_o great_a increase_n &_o furniture_n of_o knowledge_n who_o proposition_n be_v set_v orderly_a after_o the_o proposition_n of_o euclid_n every_o one_o of_o barlaam_n correspondent_a to_o the_o same_o of_o euclid_n and_o doubtless_o it_o be_v wonderful_a to_o see_v how_o these_o two_o contrary_a kind_n of_o quantity_n quantity_n discrete_a or_o number_n and_o quantity_n continual_a or_o magnitude_n which_o be_v the_o subject_n or_o matterâ—_n of_o arithmitique_a and_o geometry_n shall_v have_v in_o they_o one_o and_o the_o same_o propriety_n common_a to_o they_o both_o in_o very_a many_o point_n and_o affection_n although_o not_o in_o all_o for_o a_o line_n may_v in_o such_o sort_n be_v divide_v that_o what_o proportion_n the_o whole_a have_v to_o the_o great_a part_n the_o same_o shall_v the_o great_a part_n have_v to_o the_o less_o but_o that_o can_v not_o be_v in_o number_n for_o a_o number_n can_v not_o so_o be_v divide_v that_o the_o whole_a number_n to_o the_o great_a part_n thereof_o shall_v have_v that_o proportion_n which_o the_o great_a part_n have_v to_o the_o less_o as_o jordane_n very_o plain_o prove_v in_o his_o book_n of_o arithmetic_n which_o thing_n campane_n also_o as_o we_o shall_v afterward_o in_o the_o 9_o book_n after_o the_o 15._o proposition_n see_v prove_v and_o as_o touch_v these_o ten_o first_o proposition_n of_o the_o second_o book_n of_o euclid_n demonstrate_v by_o barlaam_n in_o number_n they_o be_v also_o demonstrate_v of_o campane_n after_o the_o 15._o proposition_n of_o the_o 9_o book_n who_o demonstration_n i_o mind_n by_o god_n help_n to_o set_v forth_o when_o i_o shall_v come_v to_o the_o place_n they_o be_v also_o demonstrate_v of_o jordane_n that_o excellet_fw-la learned_a author_n in_o the_o first_o book_n of_o his_o arithmetic_n in_o the_o mean_a time_n i_o think_v it_o not_o amiss_o here_o to_o set_v forth_o the_o demonstration_n of_o barlaam_n for_o that_o they_o geve_v great_a light_n to_o the_o second_o book_n of_o euclid_n beside_o the_o inestimable_a pleasure_n which_o they_o bring_v to_o the_o studious_a considerer_n and_o now_o to_o declare_v the_o first_o proposition_n by_o number_n i_o have_v put_v this_o example_n follow_v take_v two_o number_n the_o one_o undivided_a as_o 74._o the_o other_o divide_v into_o what_o part_n and_o how_o many_o you_o listen_v as_o 37._o divide_v into_o 20._o 10._o 5._o and_o 2d_o which_o altogether_o make_v the_o whole_a 37._o then_o if_o you_o multiply_v the_o number_n undivided_a namely_o 74_o into_o all_o the_o part_n of_o the_o number_n divide_v as_o into_o 20._o 10._o 5._o and_o 2._o you_o shall_v produce_v 1480._o 740._o 370._o 148._o which_o add_v together_o make_v 2738_o which_o self_n number_n be_v also_o produce_v if_o you_o multiply_v the_o two_o number_n first_o give_v the_o one_o into_o the_o other_o as_o you_o see_v in_o the_o example_n on_o the_o other_o side_n set_v so_o by_o the_o aid_n of_o this_o proposition_n be_v get_v a_o compendious_a way_n of_o multiplication_n by_o break_v of_o one_o of_o the_o number_n into_o his_o part_n which_o oftentimes_o serve_v to_o great_a use_n in_o working_n chiâ—â—ly_o in_o the_o rule_n of_o proportion_n the_o demonstration_n of_o which_o proposition_n follow_v in_o barlaam_n but_o â—irst_n be_v put_v of_o the_o author_n these_o principle_n follow_v barlaam_n ¶_o principles_n 1._o a_o number_n be_v sâ—yd_v to_o multiply_v a_o other_o number_n when_o the_o number_n multiply_v be_v so_o oftentimes_o add_v to_o itself_o as_o there_o be_v unity_n in_o the_o number_n which_o multiply_v whereby_o be_v produce_v a_o certain_a number_n which_o the_o number_n multiply_v measure_v by_o the_o unity_n which_o be_v in_o the_o number_n which_o multipli_v 2._o and_o the_o number_n produce_v of_o that_o a_o multiplication_n be_v call_v a_o plain_a or_o superficial_a number_n 3._o a_o square_a number_n be_v that_o which_o be_v produce_v of_o the_o multiplicatian_n of_o any_o number_n into_o itself_o 4._o 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 5._o number_n who_o one_o and_o the_o self_n same_o number_n measure_v equal_o that_o be_v by_o one_o and_o the_o self_n same_o number_n be_v equal_a the_o one_o to_o the_o other_o 6._o number_n that_o be_v equemultiplâ—ces_n to_o one_o and_o the_o self_n same_o number_n that_o be_v which_o contain_v one_o and_o the_o same_o number_n equal_o and_o alike_o be_v equal_a the_o one_o to_o the_o other_o the_o first_o proposition_n two_o number_n be_v give_v if_o the_o one_o of_o they_o be_v divide_v into_o any_o number_n how_o many_o soever_o barlaam_n the_o plain_n or_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o two_o number_n first_o give_v the_o one_o into_o the_o other_o shall_v be_v equal_a to_o the_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n not_o divide_v into_o every_o part_n of_o the_o number_n divide_v suppose_v that_o there_o be_v two_o number_n ab_fw-la and_o c._n and_o divide_v the_o number_n ab_fw-la into_o certain_a other_o number_n how_o many_o soever_o as_o into_o ad_fw-la de_fw-fr and_o ebb_v then_o i_o say_v that_o the_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n c_o into_o the_o number_n ab_fw-la be_v equal_a to_o the_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n c_o into_o the_o number_n ad_fw-la and_o of_o c_o into_o de_fw-fr and_o of_o c_o into_o ebb_n for_o let_v fletcher_n be_v the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n c_o into_o the_o number_n ab_fw-la and_o let_v gh_o be_v the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o c_o into_o ad_fw-la and_o let_v high_a be_v produce_v of_o the_o multiplication_n of_o c_o into_o de_fw-fr aâ—d_v final_o of_o the_o multiplication_n of_o c_o into_o ebb_n let_v there_o be_v produce_v the_o number_n ik_fw-mi now_o forasmuch_o as_o ab_fw-la multiply_v the_o number_n c_o produce_v the_o number_n f_o therefore_o the_o number_n c_o measure_v the_o number_n f_o by_o the_o unity_n which_o be_v in_o the_o number_n ab_fw-la and_o by_o the_o same_o reason_n may_v be_v prove_v that_o the_o number_n c_o do_v also_o measure_v the_o number_n gh_o by_o the_o unity_n which_o be_v in_o the_o number_n ad_fw-la and_o that_o it_o do_v measure_v the_o number_n high_a by_o the_o unity_n which_o be_v in_o the_o number_n df_o and_o final_o that_o it_o measure_v the_o number_n ik_fw-mi by_o the_o unity_n which_o be_v in_o the_o number_n ebb_n wherefore_o the_o number_n c_o measure_v the_o whole_a number_n gk_o by_o the_o unity_n which_o be_v in_o the_o number_n ab_fw-la but_o it_o before_o measure_v the_o number_n f_o by_o the_o unity_n which_o be_v in_o the_o number_n ab_fw-la wherefore_o either_o of_o these_o number_n fletcher_n and_o gk_o be_v equemultiplex_n to_o the_o number_n c._n but_o number_n which_o be_v equemultiplices_fw-la to_o one_o &_o the_o self_n same_o number_n be_v equal_a the_o one_o to_o the_o other_o by_o the_o 6._o definition_n wherefore_o the_o number_n fletcher_n be_v equal_a to_o the_o number_n gk_o but_o the_o number_n fletcher_n be_v the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n c_o into_o the_o number_n ab_fw-la and_o the_o number_n gk_o be_v compose_v of_o the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n c_o not_o divide_v into_o every_o one_o of_o the_o number_n ad_fw-la de_fw-fr and_o ebb_v if_o therefore_o there_o be_v two_o number_n give_v and_o the_o one_o of_o they_o be_v divide_v etc_n etc_n which_o be_v require_v to_o be_v prove_v the_o 2._o theorem_a the_o 2._o proposition_n if_o a_o right_a line_n be_v divide_v by_o chance_n the_o rectangle_v figure_n which_o be_v comprehend_v under_o the_o whole_a and_o every_o one_o of_o the_o part_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o whole_a svppose_v that_o the_o right_a line_n ab_fw-la be_v by_o chance_n deny_v in_o the_o point_n c._n then_o i_o say_v that_o the_o rectangle_n figure_n comprehend_v under_o ab_fw-la and_o bc_o together_o with_o the_o rectangle_n comprehend_v under_o ab_fw-la and_o ac_fw-la be_v equal_a unto_o the_o square_n make_v of_o ab_fw-la construction_n describe_v by_o the_o 46._o of_o the_o first_o upon_o ab_fw-la a_o square_a adeb_n and_o by_o the_o 31_o of_o the_o first_o by_o the_o point_n c_o draw_v a_o line_n parallel_n unto_o either_o of_o these_o line_n ad_fw-la aâ—d_fw-la be_v and_o let_v the_o same_o
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 those_o line_n be_v equal_a and_o that_o line_n be_v say_v to_o be_v more_o distant_a upon_o who_o fall_v the_o great_a perpendicular_a line_n as_o in_o the_o circle_n abcd_o who_o centre_n be_v e_o the_o two_o line_n ab_fw-la and_o cd_o have_v equal_a distance_n from_o the_o centre_n e_o because_o that_o the_o line_n of_o draw_v from_o the_o centre_n e_o perpendicular_o upon_o the_o line_n ab_fw-la and_o the_o line_n eglantine_n draw_v likewise_o perpendilar_o from_o the_o centre_n e_o upon_o the_o line_n cd_o be_v equal_a the_o one_o to_o the_o other_o but_o in_o the_o circle_n hklm_n who_o centre_n be_v no_o the_o line_n hk_o have_v great_a distance_n from_o the_o centre_n n_o than_o have_v the_o line_n lm_o for_o that_o the_o line_n on_o draw_v from_o the_o centre_n n_o perpendicular_o upon_o the_o line_n hk_o be_v great_a than_o the_o line_n np_v which_o be_v draw_v from_o the_o centre_n n_o perpendicular_o upon_o the_o line_n lm_o so_o likewise_o in_o the_o other_o figure_n the_o line_n ab_fw-la and_o dc_o in_o the_o circle_n abcd_o be_v equidistant_v from_o the_o centre_n gâ—_n â—_z because_o the_o line_n og_n and_o gp_n perpendicular_o draw_v from_o the_o centre_n g_o upon_o the_o say_a line_n ab_fw-la and_o dc_o be_v equal_a and_o the_o line_n ab_fw-la have_v great_a distance_n from_o the_o centre_n g_o than_o have_v the_o the_o line_n of_o because_o the_o line_n og_n perpendicular_o drowen_v from_o the_o centre_n g_o to_o the_o line_n ab_fw-la be_v greâ—ter_n than_o the_o line_n hg_o which_o be_v perpendicular_o draw_v from_o the_o câ—â—tre_n g_o to_o the_o line_n ef._n definition_n a_o section_n or_o segment_n of_o a_o circle_n be_v a_o figure_n comprehend_v under_o a_o right_a line_n and_o a_o portion_n of_o the_o circumference_n of_o a_o circle_n as_o the_o figure_n abc_n be_v a_o section_n of_o a_o circle_n because_o it_o be_v comprehend_v under_o the_o right_a line_n ac_fw-la and_o the_o circumference_n of_o a_o circle_n abc_n likewise_o the_o figure_n 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 under_o the_o circumference_n bdc_o and_o the_o right_a line_n bc._n and_o these_o angle_n be_v common_o call_v mix_v angle_n angle_n because_o they_o be_v contain_v under_o a_o right_a line_n and_o a_o crooked_a and_o these_o portion_n of_o circumference_n be_v common_o call_v arke_n arke_n and_o the_o right_a line_n be_v call_v chord_n chord_n or_o right_a line_n subtend_v and_o the_o great_a section_n have_v ever_o the_o great_a angle_n and_o the_o less_o section_n the_o less_o angle_n a_o angle_n be_v say_v to_o be_v in_o a_o section_n definition_n when_o in_o the_o circumference_n be_v take_v any_o point_n and_o from_o that_o point_n be_v draw_v right_a line_n to_o the_o end_n of_o the_o right_a line_n which_o be_v the_o base_a of_o the_o segment_n the_o angle_n which_o be_v contain_v under_o the_o right_a line_n draw_v from_o the_o point_n be_v i_o say_v say_v to_o be_v a_o angle_n in_o a_o section_n as_o the_o angle_n abc_n be_v a_o angle_n be_v the_o section_n abc_n because_o 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 section_n have_v in_o it_o the_o great_a angle_n but_o when_o the_o right_a line_n which_o comprehend_v the_o angle_n do_v receive_v any_o circumference_n of_o a_o circle_n definition_n than_o that_o angle_n be_v say_v to_o be_v correspondent_a and_o to_o pertain_v to_o that_o circumference_n as_o the_o right_a line_n basilius_n and_o bc_o which_o contain_v the_o angle_n ab_fw-la c_z and_o receive_v the_o circumference_n adc_o therefore_o the_o angle_n abc_n be_v say_v to_o subtend_v and_o to_o pertain_v to_o the_o circumference_n adc_o and_o if_o the_o right_a line_n which_o cause_n the_o angle_n concur_v in_o the_o centre_n of_o a_o circle_n then_o the_o angle_n be_v say_v to_o be_v in_o the_o centre_n of_o a_o circle_n as_o the_o angle_n efd_n be_v say_v to_o be_v in_o the_o centre_n of_o a_o circle_n for_o that_o it_o be_v comprehend_v of_o two_o right_a line_n fe_o and_o fd_v which_o concur_v and_o touch_v in_o the_o centre_n f._n and_o this_o angle_n likewise_o subtend_v the_o circumference_n egg_v which_o circumference_n also_o be_v the_o measure_n of_o the_o greatness_n of_o the_o angle_n efd_n a_o sector_n of_o a_o circle_n be_v a_o angle_n be_v set_v at_o the_o centre_n of_o a_o circle_n a_o figure_n contain_v under_o the_o right_a line_n which_o make_v that_o angle_n definition_n and_o the_o part_n of_o the_o circumference_n receive_v of_o they_o as_o the_o figure_n abc_n be_v a_o sector_n of_o a_o circle_n for_o that_o it_o have_v a_o angle_n at_o the_o centre_n namely_o the_o angle_n bac_n &_o be_v contain_v of_o the_o two_o right_a line_n ab_fw-la and_o ac_fw-la which_o contain_v that_o angle_n and_o the_o circumference_n receive_v by_o they_o definition_n like_a segment_n or_o section_n of_o a_o circle_n be_v those_o which_o have_v equal_a angle_n or_o in_o who_o be_v equal_a angle_n definition_n here_o be_v set_v two_o definition_n of_o like_a section_n of_o a_o circle_n the_o one_o pertain_v to_o the_o angle_n which_o be_v set_v in_o the_o 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
same_o reason_n also_o as_o e_o be_v to_o fletcher_n so_o be_v m_o to_o n._n but_o as_o a_o be_v to_o b_o so_o be_v e_o to_o f._n wherefore_o by_o yâ—_n 11._o of_o the_o five_o as_z g_z be_v to_o h_o so_o be_v m_o to_o n._n and_o forasmuch_o as_o as_z b_o be_v to_o c_o so_o be_v d_z to_o e_o and_o unto_o b_o &_o d_o be_v take_v equemultiplices_fw-la h_o &_o king_n and_o likewise_o unto_o c_z and_o e_o be_v take_v certain_a other_o equemultiplices_fw-la l_o and_o m_o therefore_o by_o the_o 4._o of_o the_o five_o as_o h_o be_v to_o l_o so_o be_v king_n to_o m_o and_o alternate_o also_o by_o the_o 16._o of_o the_o five_o as_o b_o be_v to_o d_z so_o be_v c_o to_o e._n and_o forasmuch_o as_o h_o and_o edward_n be_v the_o equemultiplices_fw-la of_o b_o and_o d_o but_o the_o part_n of_o equemultiplices_fw-la be_v in_o the_o same_o proportion_n that_o their_o equemultiplices_fw-la be_v by_o the_o 15._o of_o the_o five_o therefore_o as_o b_o be_v to_o d_z so_o be_v h_o to_o k._n but_o as_o b_o be_v to_o d_z so_o be_v c_z to_o e_o therefore_o by_o the_o 11._o of_o the_o first_o as_o h_o be_v to_o king_n so_o be_v c_o to_o e._n again_o forasmuch_o as_o l_o and_o m_o be_v the_o equemultiplices_fw-la of_o c_o and_o e_o therefore_o as_o c_z be_v to_o e_o so_o be_v l_o to_o m._n but_o as_o c_o be_v to_o e_o so_o be_v h_o to_o king_n therefore_o as_o h_o it_o to_o king_n so_o be_v l_o to_o m_o and_o alternate_o by_o the_o 16._o of_o the_o five_o as_o h_o be_v to_o l_o so_o be_v king_n to_o m._n *_o but_o as_o a_o be_v to_o b_o so_o be_v e_o to_o fletcher_n by_o supposition_n wherefore_o as_o g_z be_v to_o h_o so_o be_v e_o fletcher_n by_o the_o 11._o of_o the_o five_o again_o for_o as_o much_o as_o m_o and_o n_o be_v equemultiplices_fw-la unto_o e_o aâ—d_n fletcher_n therefore_o again_o by_o the_o 15._o of_o the_o five_o as_z â—_o be_v to_o â—_o so_fw-ge â—â—_o m_o to_o n._n but_o as_o e_o be_v to_o fletcher_n so_o have_v we_o prove_v be_v g_z to_o h_o wherefore_o as_o g_z be_v to_o h_o so_o be_v m_o to_o n_o by_o the_o 11._o of_o the_o five_o and_o for_o that_o â—â—_o â—_o be_v to_o â—_o so_o be_v d_z to_o e_o by_o supposition_n and_o unto_o b_o and_o d_o be_v take_v equemultiplices_fw-la h_o and_o edward_n and_o unto_o c_o and_o e_o be_v take_v certain_a other_o equemultiplices_fw-la l_o and_o m_o therefore_o as_o h_o be_v to_o l_o so_fw-mi be_v king_n to_o m_o by_o the_o 4_o of_o this_o book_n but_o it_o be_v prove_v that_o as_z g_z be_v to_o h_o so_o be_v m_o to_o n._n see_v therefore_o that_o there_o be_v in_o one_o orde_z threeâ—_n magnitude_n namely_o g_o h_o l_o and_o as_o many_o other_o magnitude_n in_o a_o other_o order_n namely_o king_n m_o n_o which_o be_v take_v two_o and_o two_o in_o each_o order_n be_v in_o one_o and_o the_o same_o proportion_n and_o their_o proportion_n be_v perturbate_n therefore_o of_o equality_n by_o the_o 21._o of_o the_o five_o if_o g_z exceed_v l_o then_o shall_v king_n exceed_v n_o and_o if_o it_o be_v equal_a it_o shall_v be_v equal_a and_o if_o it_o be_v less_o it_o shall_v be_v less_o but_o g_z and_o king_n be_v equemultiplices_fw-la unto_o a_o and_o d_o and_o l_o and_o n_o be_v certain_a other_o equemultiplices_fw-la unto_o c_o and_o f._n wherefore_o as_o a_o be_v to_o c_o so_o be_v d_z to_o fletcher_n by_o the_o 6._o definition_n of_o the_o five_o if_o therefore_o there_o be_v three_o magnitude_n in_o one_o order_n and_o as_o many_o other_o magnitude_n in_o a_o other_o order_n which_o be_v take_v two_o and_o two_o in_o each_o order_n be_v in_o one_o and_o the_o same_o proportion_n and_o if_o also_o their_o proportion_n be_v perturbate_n then_o of_o equality_n they_o shall_v be_v in_o one_o and_o the_o same_o proportion_n which_o be_v require_v to_o be_v prove_v from_o this_o mark_n *_o first_o to_o the_o same_o mark_n again_o you_o may_v if_o you_o will_v in_o stead_n of_o theâ—â—â—_n argument_n which_o seem_v somewhat_o intricate_a read_v those_o argument_n follow_v print_v with_o a_o other_o letter_n which_o be_v very_o perspicuous_a and_o brief_a and_o follow_v of_o the_o most_o interpreter_n and_o even_o as_o by_o the_o demonstration_n in_o three_o magnitude_n be_v take_v the_o proof_n in_o four_o magnitude_n by_o leave_v out_o one_o of_o note_n the_o mean_n so_o by_o the_o demonstration_n in_o four_o magnitude_n be_v take_v the_o proof_n in_o five_o magnitude_n by_o leave_v out_o two_o of_o the_o mean_n and_o by_o the_o demonstration_n in_o five_o the_o proof_n in_o six_o by_o leave_v out_o three_o mean_n and_o so_o forward_o continual_o which_o be_v also_o to_o be_v understand_v in_o the_o former_a kind_n of_o proportion_n of_o equality_n which_o be_v in_o ordinate_a proportion_n the_o 24._o theorem_a the_o 24._o proposition_n if_o the_o first_o have_v unto_o the_o second_o the_o same_o proportion_n that_o the_o three_o have_v to_o the_o four_o and_o if_o the_o five_o have_v unto_o the_o second_o the_o same_o proportion_n that_o the_o six_o have_v to_o the_o four_o then_o also_o the_o first_o and_o five_o compose_v together_o shall_v have_v unto_o the_o second_o the_o same_o proportion_n that_o the_o three_o and_o six_o compose_v together_o have_v unto_o the_o four_o svppose_v that_o there_o be_v six_o magnitude_n ab_fw-la c_o de_fw-fr fletcher_n bg_o &_o eh_o of_o which_o let_v ab_fw-la be_v the_o first_o c_o the_o second_o de_fw-fr the_o three_o fletcher_n the_o four_o bg_o the_o five_o and_o eh_o the_o six_o and_o suppose_v that_o ab_fw-la the_o first_o have_v unto_o c_o the_fw-fr second_o the_o same_o proportion_n that_o de_fw-fr the_o three_o have_v to_o fletcher_n the_o four_o magnitude_n and_o let_v bg_o the_o five_o have_v unto_o c_o the_fw-fr second_o the_o same_o proportion_n that_o eh_o the_o six_o have_v unto_o fletcher_n the_o four_o then_o i_o say_v that_o the_o first_o and_o five_o compose_v together_o namely_o agnostus_n have_v unto_o c_z the_o second_o the_o same_o proportion_n that_o the_o three_o and_o six_o compose_v together_o namely_o dh_o have_v unto_o fletcher_n the_o four_o for_o for_o that_o as_o bg_o be_v to_o c_o so_o be_v eh_o to_o fletcher_n then_o also_o by_o conversion_n by_o the_o corollary_n of_o the_o 4._o of_o the_o five_o as_o c_o be_v to_o bg_o so_o be_v fletcher_n to_o eh_o and_o for_o that_o as_o ab_fw-la be_v to_o c_o so_o be_v de_fw-fr to_o fletcher_n but_o as_o c_z be_v to_o gb_o so_o be_v fletcher_n to_o eh_o therefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o ab_fw-la be_v to_o bg_o so_o be_v de_fw-fr to_o eh_o and_o forasmuch_o as_o when_o magnitude_n divide_v be_v proportional_a they_o also_o compose_v be_v proportional_a by_o the_o 18._o of_o the_o five_o therefore_o as_o agnostus_n be_v to_o gb_o so_o be_v dh_o to_o he_o but_o as_o bg_o be_v to_o c_o so_o be_v eh_o to_o fletcher_n wherefore_o again_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o agnostus_n be_v to_o c_o so_o be_v dh_fw-mi to_o f._n if_o therefore_o the_o first_o have_v unto_o the_o second_o the_o same_o proportion_n that_o the_o three_o have_v to_o the_o four_o and_o if_o the_o five_o have_v unto_o the_o second_o the_o same_o proportion_n that_o the_o six_o have_v to_o the_o four_o then_o also_o the_o first_o and_o five_o compose_v together_o shall_v have_v unto_o the_o second_o the_o same_o proportion_n that_o the_o three_o and_o six_o compose_v together_o have_v unto_o the_o four_o which_o be_v require_v to_o be_v prove_v the_o 25._o theorem_a the_o 25._o proposition_n if_o there_o be_v four_o magnitude_n proportional_a the_o great_a and_o the_o least_o of_o they_o shall_v be_v great_a than_o the_o other_o remain_v svppose_v that_o there_o be_v four_o magnitude_n proportional_a ab_fw-la cd_o e_o and_o f._n so_o that_o as_o ab_fw-la be_v to_o cd_o so_o let_v e_o be_v to_o f._n and_o let_v the_o great_a of_o they_o be_v ab_fw-la &_o the_o lest_o of_o they_o be_v f._n then_o i_o say_v that_o these_o two_o magnitude_n ab_fw-la and_o f_o be_v great_a than_o the_o two_o magnitude_n cd_o &_o e._n forasmuch_o as_o ab_fw-la be_v suppose_v to_o be_v the_o great_a of_o all_o four_o therefore_o it_o be_v great_a than_o e._n therefore_o from_o the_o great_a ab_fw-la cut_n of_o by_o the_o 3._o of_o the_o first_o unto_o e_o a_fw-fr equal_a magnitude_n ag._n and_o likewise_o by_o the_o same_o from_o cd_o cut_n of_o unto_o fletcher_n a_o equal_a magnitude_n ch._n which_o may_v be_v do_v for_o that_o the_o magnitude_n cd_o be_v great_a than_o the_o magnitude_n e_o for_o that_o as_o ab_fw-la be_v to_o cd_o so_o be_v e_o to_o fletcher_n
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 gh_o be_v to_o the_o side_n hf_o and_o moreover_o if_o the_o side_n db_o be_v to_o the_o side_n basilius_n as_o the_o side_n hf_o be_v to_o the_o side_n fe_o and_o final_o if_o the_o side_n basilius_n be_v to_o the_o side_n ac_fw-la as_o the_o side_n fe_o be_v to_o the_o side_n eglantine_n then_o be_v these_o parallelogram_n like_a definition_n 2._o reciprocal_a figure_n be_v those_o when_o the_o termeâ—_n of_o proportion_n be_v both_o antecedentes_fw-la and_o consequentes_fw-la in_o either_o figure_n as_o if_o you_o have_v two_o parallelogram_n abcd_o and_o efgh_o if_o the_o side_n ab_fw-la to_o the_o side_n of_o a_o antecedent_n of_o the_o first_o figure_n to_o a_o consequent_a of_o the_o second_o figure_n have_v mutual_o the_o same_o proportion_n which_o the_o side_n eglantine_n have_v to_o the_o side_n ac_fw-la a_o antecedent_n of_o the_o second_o figure_n to_o a_o consequent_a of_o the_o first_o figure_n then_o be_v these_o two_o figure_n reciprocal_a they_o be_v call_v of_o some_o figure_n of_o mutual_a side_n and_o that_o undoubted_o not_o amiss_o nor_o unapt_o figure_n and_o to_o make_v this_o definition_n more_o plain_a campane_n and_o pestitarius_n and_o othersâ—_n thus_o put_v it_o reciprocal_a figure_n be_v when_o the_o side_n of_o other_o 〈◊〉_d mutual_o proportional_a as_o in_o the_o example_n and_o declaration_n before_o give_v among_o the_o barbarous_a they_o be_v call_v mutekesia_n reserve_v still_o the_o arabike_a word_n definition_n 3._o a_o right_a line_n be_v say_v to_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n when_o the_o whole_a be_v to_o the_o great_a part_n as_o the_o great_a part_n be_v to_o the_o less_o as_o if_o the_o line_n ab_fw-la be_v so_o divide_v in_o the_o point_n c_o that_o the_o whole_a line_n ab_fw-la have_v the_o same_o proportion_n to_o the_o great_a part_n thereof_o namely_o to_o ac_fw-la which_o the_o same_o great_a part_n ac_fw-la have_v to_o the_o less_o part_n thereof_o namely_o to_o cb_o then_o be_v 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 to_o be_v contain_v within_o two_o equidistant_a line_n so_o that_o figure_v to_o have_v one_o altitude_n and_o to_o be_v contain_v within_o two_o equidistant_a line_n be_v all_o one_o so_o in_o all_o these_o example_n if_o from_o the_o high_a point_n of_o the_o figure_n you_o draw_v a_o equidistant_a line_n to_o the_o base_a thereof_o and_o then_o from_o that_o point_n draw_v a_o perpendicular_a to_o the_o same_o base_a that_o perpendicular_a be_v the_o altitude_n of_o the_o figure_n 5._o a_o proportion_n be_v say_v to_o be_v make_v of_o two_o proportion_n or_o more_o when_o the_o quantity_n of_o the_o proportion_n multiply_v the_o one_o into_o the_o other_o produce_v a_o other_o quantity_n definition_n an_o other_o example_n where_o the_o great_a inequality_n and_o the_o less_o inequality_n be_v mix_v together_o 6._o 4._o 2._o 3._o the_o denomination_n of_o the_o proportion_n of_o 6._o to_o 4_o be_v 1_o â—_o â—_o example_n of_o 4._o to_o 2_o be_v â—_o â—_o and_o of_o 2._o to_o 3_o 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
figure_n comprehend_v under_o the_o line_n gb_o and_o ge_z be_v equal_a to_o the_o square_n make_v of_o the_o line_n agnostus_n by_o the_o 36._o of_o the_o three_o it_o follow_v that_o the_o touch_n line_n ga_o be_v a_o mean_a proportional_a between_o the_o extreme_n gb_o and_o ge_z by_o the_o second_o part_n of_o the_o 17._o of_o the_o six_o for_o that_o by_o that_o proposition_n the_o line_n gb_o ga_o and_o ge_z be_v proportional_a and_o by_o the_o same_o reason_n may_v it_o be_v prove_v that_o the_o line_n ga_o be_v a_o mean_a proportional_a between_o the_o line_n gd_o and_o gc_o and_o so_o of_o all_o other_o if_o therefore_o from_o a_o point_n gevenâ—_n &_o câ—_n which_o be_v require_v to_o be_v demonstrate_v the_o end_n of_o the_o six_o book_n of_o euclides_n element_n ¶_o the_o seven_o book_n of_o euclides_n element_n hitherto_o in_o the_o six_o book_n before_o have_v euclid_n pass_v through_o and_o entreat_v of_o the_o element_n of_o geometry_n without_o the_o aid_n and_o 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number_n general_o and_o of_o their_o part_n and_o search_v and_o demonstrate_v in_o general_a the_o most_o common_a passion_n and_o propriety_n of_o the_o same_o and_o chief_o of_o number_n prime_a or_o incommensurable_a and_o of_o number_n compose_v or_o commensurable_a and_o of_o their_o propriety_n and_o partly_o also_o of_o the_o comparison_n oâ—_n proportion_n of_o one_o number_n to_o a_o other_o ¶_o definition_n definition_n 1_o unity_n be_v that_o whereby_o every_o thing_n that_o be_v be_v say_v to_o be_v on_o as_o a_o point_n in_o magnitude_n be_v the_o least_o thing_n in_o magnitude_n and_o no_o magnitude_n at_o all_o &_o yet_o the_o ground_n and_o beginning_n of_o all_o magnitude_n even_o so_o be_v unity_n in_o multitude_n or_o number_n the_o least_o thing_n in_o number_n and_o no_o number_n at_o all_o and_o yet_o the_o ground_n and_o beginning_n of_o all_o number_n and_o therefore_o it_o be_v here_o in_o this_o place_n of_o euclid_n first_o define_v as_o in_o the_o first_o 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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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same_o number_n 12._o by_o 3._o a_o odd_a number_n add_v therefore_o be_v 12._o a_o number_n even_o even_o and_o even_o odd_a and_o so_o of_o such_o other_o there_o be_v also_o a_o other_o definition_n give_v of_o this_o kind_n of_o number_n by_o boetius_fw-la and_o other_o common_o which_o be_v thus_o â—d_a a_o number_n eveâ—ly_o even_o and_o even_o odd_a be_v that_o which_o may_v be_v divide_v into_o two_o equal_a part_n and_o each_o of_o they_o may_v aâ—ayne_o be_v divide_v into_o two_o equal_a part_n and_o so_o forth_o but_o this_o division_n be_v at_o length_n stay_v and_o continue_v not_o till_o it_o come_v to_o unity_n as_o for_o example_n 48_o which_o may_v be_v divide_v into_o two_o equal_a part_n namely_o into_o 24._o and_o 24._o again_o 24._o which_o be_v on_o of_o the_o part_n may_v be_v divide_v into_o two_o equal_a part_n 12._o and_o 12._o again_o 12._o into_o 6._o and_o 6._o and_o again_o 6_o may_v be_v divide_v into_o two_o equal_a part_n into_o 3._o and_o 3_o but_o 3_n can_v be_v divide_v into_o two_o equal_a part_n wherefore_o the_o division_n there_o stay_v and_o continue_v not_o till_o it_o come_v to_o unity_n as_o it_o do_v in_o these_o number_n which_o be_v even_o even_o only_o definition_n 11_o a_o number_n odd_o odd_a be_v that_o which_o a_o odd_a number_n do_v measure_n by_o a_o odd_a number_n as_o 25_o which_o 5._o a_o odd_a number_n measure_v by_o a_o odd_a number_n namely_o by_o 5._o five_o time_n five_o be_v 25_o likewise_o 21._o who_o 7._o a_o odd_a number_n do_v measure_n by_o 3_o which_o be_v likewise_o a_o odd_a number_n three_o time_n 7._o be_v 21._o flusâ—ates_n flussatus_n give_v this_o definition_n follow_v of_o this_o kind_n of_o number_n which_o be_v all_o one_o in_o substance_n with_o the_o former_a definition_n a_o number_n odd_o odd_a it_o that_o which_o only_o a_o odd_a number_n do_v measure_n definition_n as_o 15._o for_o no_o number_n measure_v 15._o but_o only_o 5._o and_o 3_o also_o 25_o none_o measure_v it_o but_o only_o 5._o which_o be_v a_o odd_a number_n and_o so_o of_o other_o definition_n 12_o a_o prime_a or_o first_o number_n be_v that_o which_o only_a unity_n do_v measure_n as_o 5.7.11.13_o for_o no_o number_n measure_v 5_o but_o only_a unity_n for_o v_o unity_n make_v the_o number_n 5._o so_o no_o number_n measure_v 7_o but_o only_a unity_n .2_o take_v 3._o time_n make_v 6._o which_o be_v less_o than_o 7_o and_o 2._o take_v 4._o time_n be_v 8_o which_o be_v more_o than_o 7._o and_o so_o of_o 11.13_o and_o such_o other_o so_o that_o all_o prime_a number_n number_n which_o also_o be_v call_v first_o number_n and_o number_n uncomposed_a have_v
the_o double_a of_o b_o be_v prime_a unto_o the_o double_a of_o b_o then_o the_o two_o number_n whereof_o the_o number_n be_v compose_v namely_o the_o number_n compose_v of_o a_o and_o c_o and_o the_o double_a of_o b_o shall_v be_v prime_a the_o one_o to_o the_o other_o by_o the_o 30_o of_o the_o seven_o and_o therefore_o the_o number_n compose_v of_o a_o and_o c_o shall_v be_v prime_a to_o b_o take_v once_o for_o if_o any_o number_n shall_v measure_v the_o two_o number_n namely_o the_o number_n compose_v of_o a_o and_o c_o and_o the_o number_n b_o it_o shall_v also_o measure_v the_o number_n compose_v of_o a_o and_o c_o and_o the_o double_a of_o b_o by_o the_o 5._o common_a sentence_n of_o the_o seven_o which_o be_v not_o possible_a for_o that_o they_o be_v prove_v to_o be_v prime_a number_n here_o have_v i_o add_v a_o other_o demonstration_n of_o the_o former_a proposition_n after_o campane_n which_o prove_v that_o in_o number_n how_o many_o soever_o which_o be_v there_o prove_v only_o touch_v three_o number_n and_o the_o demonstration_n seement_n somewhat_o more_o perspicuous_a than_o theon_n demonstration_n and_o thus_o he_o put_v the_o proposition_n if_o number_n how_o many_o soever_o be_v in_o continual_a proportion_n be_v the_o least_o that_o have_v one_o &_o the_o same_o proportion_n with_o they_o every_o one_o of_o they_o shall_v be_v to_o the_o number_n compose_v of_o the_o rest_n prime_a second_o i_o say_v that_o this_o be_v so_o in_o every_o one_o of_o they_o namely_o that_o c_z be_v a_o prime_a number_n to_o the_o number_n compose_v of_o a_o b_o d._n for_o if_o not_o then_o as_o before_o let_v e_o measure_v c_o and_o the_o number_n compose_v of_o a_o b_o d_o which_o e_o shall_v be_v a_o number_n not_o prime_a either_o to_o fletcher_n or_o to_o g_z by_o the_o former_a proposition_n add_v by_o campane_n wherefore_o let_v h_o measure_v they_o and_o forasmuch_o as_o h_o measure_v e_o it_o shall_v also_o measure_v the_o whole_a a_o b_o c_o d_o who_o e_o measure_v and_o forasmuch_o as_o h_o measure_v one_o of_o these_o number_n fletcher_n or_o g_o it_o shall_v measure_v one_o of_o the_o extreme_n a_o or_o d_o which_o be_v produce_v of_o fletcher_n or_o g_o by_o the_o second_o of_o the_o eight_o if_o they_o be_v multiply_v into_o the_o mean_n l_o or_o k._n and_o moreover_o the_o same_o h_o shall_v measure_v the_o meame_n bc_o by_o the_o 5._o common_a sentence_n of_o the_o seven_o when_o as_o by_o supposition_n it_o measure_v either_o fletcher_n or_o g._n which_o measure_n b_o c_o by_o the_o second_o of_o the_o eight_o but_o the_o same_o h_o measure_v the_o whole_a a_o b_o c_o d_o as_o we_o have_v prove_v for_o that_o it_o measure_v e._n wherefore_o it_o shall_v also_o measure_v the_o residue_n namely_o the_o number_n compose_v of_o the_o extreme_n a_o and_o d_z by_o the_o 4._o common_a sentence_n of_o the_o seven_o and_o it_o measure_v one_o of_o these_o a_o or_o d_z for_o it_o measure_v one_o of_o these_o fletcher_n or_o g_o which_o produce_v a_o and_o d_z wherefore_o the_o same_o h_o shall_v measure_v one_o of_o these_o a_o or_o d_o and_o also_o the_o other_o of_o they_o by_o the_o former_a common_a sentence_n which_o number_n a_o and_o d_o be_v by_o the_o 3._o of_o the_o eight_o prime_n the_o one_o to_o the_o other_o which_o be_v absurd_a this_o may_v also_o be_v prove_v in_o every_o one_o of_o these_o number_n a_o b_o c_o d._n wherefore_o no_o number_n shall_v measure_v one_o of_o these_o number_n a_o b_o c_o d_o and_o the_o number_n compose_v of_o the_o rest_n wherefore_o they_o be_v prime_a the_o one_o to_o the_o other_o if_o therefore_o number_n how_o many_o soever_o etc_o etc_o which_o be_v require_v to_o be_v prove_v here_o as_o i_o promise_v i_o have_v add_v campane_n demonstration_n of_o those_o proposition_n in_o number_n which_o euclâ—de_n in_o the_o second_o book_n demonstrate_v in_o line_n and_o that_o in_o this_o place_n so_o much_o the_o rather_o for_o that_o theon_n as_o we_o see_v in_o the_o demonstration_n of_o the_o 15._o proposition_n seem_v to_o allege_v the_o 3._o &_o 4._o proposition_n of_o the_o second_o book_n which_o although_o they_o concern_v line_n only_o yet_o as_o we_o there_o declare_v and_o prove_v be_v they_o true_a also_o in_o number_n ¶_o the_o first_o proposition_n add_v by_o campane_n that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o one_o number_n into_o number_n how_o many_o soever_o be_v equal_a to_o that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o same_o number_n into_o the_o number_n compose_v of_o they_o this_o prove_v that_o in_o number_n which_o the_o first_o of_o the_o second_o prove_v touch_v line_n suppoâ—â—_n that_o the_o number_n a_o be_v multiply_v into_o the_o number_n b_o and_o into_o the_o number_n c_o demonstratiou_n and_o into_o the_o number_n d_o do_v produce_v the_o number_n e_o f_o and_o g._n then_o i_o say_v that_o the_o number_n produce_v of_o a_o multiply_a into_o the_o number_n compose_v of_o b_o c_o and_o d_z be_v equal_a to_o the_o number_n compose_v of_o e_o f_o and_o g._n for_o by_o the_o converse_n of_o the_o definition_n of_o a_o number_n multiply_v what_o part_n unity_n be_v of_o a_o the_o self_n same_o part_n be_v b_o of_o e_o and_o c_o of_o fletcher_n and_o also_o d_z of_o g._n wherefore_o by_o the_o 5._o of_o the_o seven_o what_o part_n unity_n be_v of_o a_o the_o self_n same_o part_n be_v the_o number_n compose_v of_o b_o c_o and_o d_o of_o the_o number_n compose_v of_o e_o f_o and_o g._n wherefore_o by_o the_o definition_n that_o which_o be_v produce_v of_o a_o into_o the_o number_n compose_v of_o b_o c_o d_o be_v equal_a to_o the_o number_n compose_v of_o e_o f_o g_o which_o be_v require_v to_o be_v prove_v the_o second_o proposition_n that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o number_n how_o many_o soever_o into_o one_o number_n be_v equal_a to_o that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n compose_v of_o they_o into_o the_o same_o number_n this_o be_v the_o converse_n of_o the_o former_a as_o if_o the_o â—â—â—bers_n â—_o and_o g_z and_o d_o multiply_v into_o the_o number_n a_o do_v produce_v the_o number_n e_o and_o f_o and_o g._n former_a then_o the_o number_n compose_v of_o b_o c_o d._n multiply_v into_o the_o number_n a_o shall_v produce_v the_o number_n compose_v of_o the_o number_n e_o f_o g._n which_o thing_n be_v easy_o prove_v by_o the_o 16._o of_o the_o seven_o and_o by_o the_o former_a proposition_n ¶_o the_o three_o proposition_n that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o number_n how_o many_o soever_o into_o other_o number_n how_o many_o soever_o be_v equal_a to_o that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n compose_v of_o those_o first_o number_n into_o the_o number_n compose_v of_o these_o latter_a number_n as_o if_o the_o number_n a_o b_o c_o do_v multiply_v the_o number_n d_o e_o f_o each_o one_o each_o other_o and_o if_o the_o number_n produce_v be_v add_v together_o then_o i_o say_v that_o the_o number_n compose_v of_o the_o number_n produce_v be_v equal_a to_o the_o number_n produce_v of_o the_o number_n compose_v of_o the_o number_n a_o b_o demonstration_n c_o into_o the_o number_n compose_v of_o the_o number_n d_o e_o f._n for_o by_o the_o former_a proposition_n that_o which_o be_v produce_v of_o the_o number_n compose_v of_o a_o b_o c_o into_o d_z be_v equal_a to_o that_o which_o be_v produce_v of_o every_o one_o of_o the_o say_a number_n into_o d_o and_o by_o the_o same_o reason_n that_o which_o be_v produce_v of_o the_o number_n compose_v of_o a_o b_o c_o into_o e_o be_v equal_a to_o that_o which_o be_v produce_v of_o every_o one_o of_o the_o say_a number_n into_o e_o and_o so_o likewise_o that_o which_o be_v produce_v of_o the_o number_n compose_v of_o a_o b_o c_o into_o fletcher_n be_v equal_a to_o that_o which_o be_v produce_v of_o every_o one_o of_o the_o say_a number_n into_o f._n but_o by_o the_o first_o of_o these_o proposition_n thâ—â—_n which_o be_v produce_v of_o the_o number_n compose_v of_o these_o number_n a_o b_o c_o into_o every_o one_o of_o these_o number_n d_o e_o fletcher_n be_v equal_a to_o that_o which_o be_v produce_v of_o the_o number_n compose_v into_o the_o number_n compose_v wherefore_o that_o be_v manifest_v which_o be_v require_v to_o be_v prove_v ¶_o the_o four_o proposition_n if_o a_o number_n be_v deuâ—â—â—d_v into_o part_n how_o many_o soever_o that_o number_n which_o be_v produce_v of_o the_o whole_a into_o himself_o be_v equal_a to_o that_o number_n which_o be_v produce_v of_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
if_o the_o angle_n lmk_n be_v a_o acute_a angle_n then_o be_v that_o angle_n the_o inclination_n of_o the_o superficies_n abcd_o unto_o the_o superficies_n efgh_o by_o this_o definition_n because_o it_o be_v contain_v of_o perpendicular_a line_n draw_v in_o either_o of_o the_o superficiece_n to_o one_o and_o the_o self_n same_o point_n be_v the_o common_a section_n of_o they_o both_o 5_o plain_a superficiece_n be_v in_o like_a sort_n incline_v the_o onâ—_n 〈…〉_o she_o definition_n when_o the_o say_v angle_n of_o inclination_n be_v equal_a the_o one_o to_o the_o o_o 〈…〉_o this_o definition_n need_v no_o declaration_n at_o all_o but_o be_v most_o manifest_a by_o the_o definition_n last_o go_v before_o for_o in_o consider_v the_o inclination_n of_o diverse_a superficiece_n to_o other_o if_o the_o acute_a angle_n contain_v under_o the_o perpendicular_a line_n draw_v in_o they_o from_o one_o point_n assign_v in_o each_o of_o their_o common_a section_n be_v equal_a as_o if_o to_o the_o angle_n lmk_n in_o the_o former_a example_n be_v give_v a_o other_o angle_n in_o the_o inclination_n of_o two_o other_o superficiece_n equal_a then_o be_v the_o inclination_n of_o these_o superficiece_n like_o and_o be_v by_o this_o definition_n say_v in_o like_a sort_n to_o incline_v the_o one_o to_o the_o other_o now_o also_o let_v there_o be_v a_o other_o ground_n plain_a superficies_n namely_o the_o superficies_n mnop_n unto_o who_o also_o let_v lean_a and_o incline_v the_o superficies_n qâ—â—t_n and_o let_v the_o common_a section_n or_o segment_n of_o they_o be_v the_o line_n qr_o and_o draw_v in_o the_o superficies_n mnop_n to_o some_o one_o point_n of_o the_o common_a section_n as_o to_o the_o point_n x_o the_o line_n vx_n make_v with_o the_o common_a section_n right_a angle_n namely_o the_o angle_n vxr_n or_o the_o angle_n vxq_n also_o in_o the_o superficies_n stqr_n draw_v the_o right_a line_n yx_n to_o the_o same_o point_n x_o in_o the_o common_a section_n make_v therewith_o right_a angle_n as_o the_o angle_n yxâ—_n or_o the_o angle_n yxq._n now_o as_o say_v the_o definition_n if_o the_o angle_n contain_v under_o the_o right_a line_n draw_v in_o these_o superficiece_n &_o make_v right_a angle_n with_o the_o common_a section_n be_v in_o the_o point_n that_o be_v in_o the_o point_n of_o their_o meet_v in_o the_o common_a section_n equal_a then_o be_v the_o inclination_n of_o the_o superficiece_n equal_a as_o in_o this_o example_n if_o the_o angle_n lgh_n contain_v under_o the_o line_n lg_n be_v in_o the_o incline_a superficies_n â—kef_a and_o under_o the_o line_n hg_o be_v in_o the_o ground_n superficies_n abcd_o be_v equal_a to_o the_o angle_n yxv_n contain_v under_o the_o line_n vx_n be_v in_o the_o ground_n superficies_n mnop_n and_o under_o the_o line_n yx_n be_v in_o the_o incline_a superficies_n stqr_n then_o be_v the_o inclination_n of_o the_o superficies_n ikef_n unto_o the_o superficies_n abcd_o like_a unto_o the_o inclination_n of_o the_o 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of_o nature_n pithagâ—ras_n timeus_n and_o plato_n by_o they_o search_v out_o the_o composition_n of_o the_o world_n with_o the_o harmony_n and_o preservation_n thereof_o and_o apply_v these_o â—ive_a solid_n to_o the_o simple_a part_n thereof_o the_o pyramid_n or_o tetrahedron_n they_o ascribe_v to_o the_o â—ire_n fire_n for_o that_o it_o ascend_v upward_o according_a to_o the_o figure_n of_o the_o pyramid_n to_o the_o air_n they_o ascribe_v the_o octohedron_n air_n for_o that_o through_o the_o subtle_a moisture_n which_o it_o have_v it_o extend_v itself_o every_o way_n to_o the_o one_o side_n and_o to_o the_o other_o accord_v as_o that_o figure_n do_v unto_o the_o water_n they_o assign_v the_o ikosahedron_n for_o that_o it_o be_v continual_o flow_v and_o move_v water_n and_o as_o it_o be_v make_v angleâ—_n 〈…〉_o â—ide_v according_a to_o that_o figure_n and_o to_o the_o earth_n they_o attribute_v a_o cube_n earth_n as_o to_o a_o thing_n stableâ—_n 〈◊〉_d and_o sure_a as_o the_o figure_n
signify_v last_o of_o all_o a_o dodecahedron_n heaven_n for_o that_o it_o be_v make_v of_o pâ—ntagoâ—_n who_o angle_n be_v more_o ample_a and_o large_a than_o the_o angle_n of_o the_o other_o body_n and_o by_o that_o â—eaâ—â—â—_n draw_v more_o â—â—_o roundness_n 〈◊〉_d &_o to_o the_o form_n and_o nature_n of_o a_o sphere_n they_o assign_v to_o a_o sphere_n namely_o 〈…〉_o who_o so_o will_v 〈…〉_o in_o his_o tineus_n shall_v â—ead_a of_o these_o figure_n and_o of_o their_o mutual_a proportionâ—â—â—raunge_a maâ—terâ—_n which_o hâ—re_o be_v not_o to_o be_v entreat_v of_o this_o which_o be_v say_v shall_v be_v sufficient_a for_o the_o 〈◊〉_d of_o they_o and_o for_o thâ—_z declaration_n of_o their_o definition_n after_o all_o these_o definition_n here_o set_v of_o euclid_n flussas_n have_v add_v a_o other_o definition_n which_o 〈◊〉_d of_o a_o parallelipipedon_n which_o because_o it_o have_v not_o hitherto_o of_o euclid_n in_o any_o place_n be_v define_v and_o because_o it_o be_v very_o good_a and_o necessary_a to_o be_v have_v i_o think_v good_a not_o to_o omit_v it_o thus_o it_o be_v a_o parallelipipedon_n be_v a_o solid_a figure_n comprehend_v under_o four_o plain_a quadrangle_n figure_n parallelipipedon_n of_o which_o those_o which_o be_v opposite_a be_v parallel_n because_o these_o five_o regular_a body_n here_o define_v be_v not_o by_o these_o figure_n here_o set_v so_o full_o and_o lively_o express_v that_o the_o studious_a beholder_n can_v thorough_o according_a to_o their_o definition_n conceive_v they_o i_o have_v here_o give_v of_o they_o other_o description_n draw_v in_o a_o plain_n by_o which_o you_o may_v easy_o attain_v to_o the_o knowledge_n of_o they_o for_o if_o you_o draw_v the_o like_a form_n in_o matter_n that_o will_v bow_v and_o geve_v place_n as_z most_o apt_o you_o may_v do_v in_o fine_a past_v paper_n such_o as_o pastwive_n make_v woman_n paste_n of_o &_o they_o with_o a_o knife_n cut_v every_o line_n fine_o not_o through_o but_o half_a way_n only_o if_o they_o you_o bow_v and_o bend_v they_o according_o you_o shall_v most_o plain_o and_o manifest_o see_v the_o form_n and_o shape_n of_o these_o body_n even_o as_o their_o definition_n show_v and_o it_o shall_v be_v very_o necessary_a for_o you_o to_o hadâ—â—tore_v of_o that_o past_v paper_n by_o you_o for_o so_o shall_v yoâ—_n upon_o it_o 〈…〉_o the_o form_n of_o other_o body_n as_o prism_n and_o parallelipopedon_n 〈…〉_o set_v forth_o in_o these_o five_o book_n follow_v and_o see_v the_o very_a 〈◊〉_d of_o thâ—se_a body_n there_o mention_v which_o will_v make_v these_o book_n concern_v body_n as_o easy_a unto_o you_o as_o be_v the_o other_o book_n who_o figure_n you_o may_v plain_o see_v upon_o a_o plain_a superficies_n describe_v thiâ—_z figurâ—_n which_o consi_v of_o twâ—luâ—â—quilâ—â—â—â—_n and_o equiangle_n pâ—ntâ—â—â—â—â—_n upoâ—_n the_o foresay_a matter_n and_o fine_o cut_v as_o before_o be_v â—â—ught_v tâ—â—â—lâ—uâ—n_v line_n contain_v within_o thâ—_z figurâ—_n and_o bow_n and_o fold_n the_o penâ—â—gonâ—_n according_o and_o they_o will_v so_o close_o together_o thaâ—_z thâ—y_n will_v â—â—kâ—_n thâ—_z very_a form_n of_o a_o dodecahedron_n dâ—dâ—â—â—â—edron_n icosaâ—edron_n if_o you_o describe_v this_o figure_n which_o consist_v of_o twenty_o equilater_n and_o equiangle_n triangle_n upon_o the_o foresay_a matter_n and_o fine_o cut_v as_o before_o be_v show_v the_o ninâ—tâ—ne_a line_n which_o be_v contain_v within_o the_o figure_n and_o then_o bow_n and_o fold_n they_o according_o they_o will_v in_o such_o sort_n close_o together_o that_o therâ—_n will_v be_v make_v a_o perfect_v form_n of_o a_o icosahedron_n because_o in_o these_o five_o book_n there_o be_v sometime_o require_v other_o body_n beside_o the_o foresay_a five_o regular_a body_n as_o pyramid_n of_o diverse_a form_n prism_n and_o other_o i_o have_v here_o set_v forth_o three_o figure_n of_o three_o sundry_a pyramid_n one_o have_v to_o his_o base_a a_o triangle_n a_o other_o a_o quadrangle_n figure_n the_o other_o â—_o pentagonâ—_n which_o if_o you_o describe_v upon_o the_o foresay_a matter_n &_o fine_o cut_v as_o it_o be_v before_o teach_v the_o line_n contain_v within_o each_o figure_n namely_o in_o the_o first_o three_o line_n in_o the_o second_o four_o line_n and_o in_o the_o three_o five_o line_n and_o so_o bend_v and_o fold_v they_o according_o they_o will_v so_o close_o together_o at_o the_o top_n that_o they_o will_v â—ake_v pyramid_n of_o that_o form_n that_o their_o base_n be_v of_o and_o if_o you_o conceive_v well_o the_o describe_v of_o these_o you_o may_v most_o easy_o describe_v the_o body_n of_o a_o pyramid_n of_o what_o form_n so_o ever_o you_o will._n because_o these_o five_o book_n follow_v be_v somewhat_o hard_o for_o young_a beginner_n by_o reason_n they_o must_v in_o the_o figure_n describe_v in_o a_o plain_a imagine_v line_n and_o superficiece_n to_o be_v elevate_v and_o erect_v the_o one_o to_o the_o other_o and_o also_o conceive_v solid_n or_o body_n which_o for_o that_o they_o have_v not_o hitherto_o be_v acquaint_v with_o will_v at_o the_o first_o sight_n be_v somewhat_o strange_a unto_o they_o i_o have_v for_o their_o more_o â—ase_a in_o this_o eleven_o book_n at_o the_o end_n of_o the_o demonstration_n of_o every_o proposition_n either_o set_v new_a figure_n if_o they_o concern_v the_o elevate_n or_o erect_v of_o line_n or_o superficiece_n or_o else_o if_o they_o concern_v body_n i_o have_v show_v how_o they_o shall_v describe_v body_n to_o be_v compare_v with_o the_o construction_n and_o demonstration_n of_o the_o proposition_n to_o they_o belong_v and_o if_o they_o diligent_o weigh_v the_o manner_n observe_v in_o this_o eleven_o book_n touch_v the_o description_n of_o new_a figure_n agree_v with_o the_o figure_n describe_v in_o the_o plain_a it_o shall_v not_o be_v hard_o for_o they_o of_o themselves_o to_o do_v the_o like_a in_o the_o other_o book_n follow_v when_o they_o come_v to_o a_o proposition_n which_o concern_v either_o the_o elevate_n or_o erect_v of_o line_n and_o superficiece_n or_o any_o kind_n of_o body_n to_o be_v imagine_v ¶_o the_o 1._o theorem_a the_o 1._o proposition_n that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n &_o part_v elevate_v upward_o be_v impossible_a for_o if_o it_o be_v possible_a let_v part_n of_o the_o right_a line_n abc_n namely_o the_o part_n ab_fw-la be_v in_o a_o ground_n plain_a superficies_n and_o the_o other_o part_n thereof_o namely_o bc_o be_v elevate_v upward_o and_o produce_v direct_o upon_o the_o ground_n plain_a superficies_n the_o right_a line_n ab_fw-la beyond_o the_o point_n b_o unto_o the_o point_n d._n wherefore_o unto_o two_o right_a line_n give_v abc_n and_o abdella_n the_o line_n ab_fw-la be_v a_o common_a section_n or_o part_n which_o be_v impossible_a impossibility_n for_o a_o right_a line_n can_v not_o touch_v a_o right_a line_n in_o 〈◊〉_d point_n then_o one_o uâ—lesse_o those_o right_n be_v exact_o agree_v and_o lay_v the_o one_o upon_o the_o other_o wherefore_o that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n and_o part_v elevate_v upward_o be_v impossible_a which_o be_v require_v to_o be_v prove_v this_o figure_n more_o plain_o set_v forth_o the_o foresay_a demonstration_n if_o you_o elevate_v the_o superficies_n wherein_o the_o line_n bc._n an_o other_o demonstration_n after_o flâ—sâ—â—s_n if_o it_o be_v possible_a let_v there_o be_v a_o right_a line_n abg_o who_o part_n ab_fw-la let_v be_v in_o the_o ground_n plain_a superficies_n aed_n flussas_n and_o let_v the_o rest_n thereof_o bg_o be_v elevate_v on_o high_a that_o be_v without_o the_o plain_n aed_n then_o i_o say_v that_o abg_o be_v not_o one_o right_a line_n for_o forasmuch_o as_o aed_n be_v a_o plain_a superficies_n produce_v direct_o &_o equal_o upon_o the_o say_a plain_n aed_n the_o right_a line_n ab_fw-la towards_o d_z which_o by_o the_o 4._o definition_n of_o the_o first_o shall_v be_v a_o right_a line_n and_o from_o some_o one_o point_n of_o the_o right_a line_n abdella_n namely_o from_o c_o draâ—_n unto_o the_o point_n g_o a_o right_a line_n cg_o wherefore_o in_o the_o triangle_n 〈…〉_o the_o outward_a angâ—â—_n abâ—_n be_v equal_a to_o the_o two_o inward_a and_o opposite_a angle_n by_o the_o 32._o of_o the_o first_o and_o therefore_o it_o be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o same_o wherefore_o the_o line_n abg_o forasmuch_o as_o it_o make_v a_o angle_n be_v not_o â—_o right_n line_n wherefore_o that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n and_o part_v elevate_v upward_o be_v impossible_a if_o you_o mark_v well_o the_o figure_n before_o add_v for_o the_o plain_a declaration_n of_o euclides_n demonstration_n iâ—_z will_v not_o be_v hard_o for_o you_o to_o coâ—â—â—â—e_v this_o figure_n which_o ulyss_n put_v for_o his_o demonstration_n â—_o wherein_o
angle_n bac_n god_n &_o dab_n be_v equal_a the_o one_o to_o the_o other_o then_o be_v it_o manifest_v that_o two_o of_o they_o which_o two_z so_o ever_o be_v take_v be_v great_a than_o the_o three_o but_o if_o not_o let_v the_o angle_n bac_n be_v the_o great_a of_o the_o three_o angle_n and_o unto_o the_o right_a line_n ab_fw-la and_o from_o the_o point_n a_o make_v in_o the_o plain_a superficies_n bac_n unto_o the_o angle_n dab_n a_o equal_a angle_n bae_o and_o by_o the_o 2._o of_o the_o first_o make_v the_o line_n ae_n equal_a to_o the_o line_n ad._n now_o a_o right_a line_n bec_n draw_v by_o the_o point_n e_o shall_v cut_v the_o right_a line_n ab_fw-la and_o ac_fw-la in_o the_o point_n b_o and_o c_o draw_v a_o right_a line_n from_o d_z to_o b_o and_o a_o outstretch_o from_o d_z to_o c._n and_o forasmuch_o as_o the_o line_n dam_n be_v equal_a to_o the_o line_n ae_n demonstration_n and_o the_o line_n ab_fw-la be_v common_a to_o they_o both_o therefore_o these_o two_o line_n dam_n and_o ab_fw-la be_v equal_a to_o these_o two_o line_n ab_fw-la and_o ae_n and_o the_o angle_n dab_n be_v equal_a to_o the_o angle_n bae_o wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a db_o be_v equal_a to_o the_o base_a be._n and_o forasmuch_o as_o these_o two_o line_n db_o and_o dc_o be_v great_a than_o the_o line_n bc_o of_o which_o the_o line_n db_o be_v prove_v to_o be_v equal_a to_o the_o line_n be._n wherefore_o the_o residue_n namely_o the_o line_n dc_o be_v great_a than_o the_o residue_n namely_o than_o the_o line_n ec_o and_o forasmuch_o as_o the_o line_n dam_n be_v equal_a to_o the_o line_n ae_n and_o the_o line_n ac_fw-la be_v common_a to_o they_o both_o and_o the_o base_a dc_o be_v great_a than_o the_o base_a ec_o therefore_o the_o angle_n dac_o be_v great_a than_o the_o angle_n eac_n and_o it_o be_v prove_v that_o the_o angle_n dab_n be_v equal_a to_o the_o angle_n bae_o wherefore_o the_o angle_n dab_n and_o dac_o be_v great_a than_o the_o angle_n bac_n if_o therefore_o a_o solid_a angle_n be_v contain_v under_o three_o plain_a superficial_a angle_n every_o two_o of_o those_o three_o angle_n which_o then_o so_o ever_o be_v take_v be_v great_a than_o the_o three_o which_o be_v require_v to_o be_v prove_v in_o this_o figure_n you_o may_v plain_o behold_v the_o former_a demonstration_n if_o you_o elevate_v the_o three_o triangle_n abdella_n aâ—c_n and_o acd_o in_o such_o â—orâ—that_o they_o may_v all_o meet_v together_o in_o the_o point_n a._n the_o 19_o theorem_a the_o 21._o proposition_n every_o solid_a angle_n be_v comprehend_v under_o plain_a angle_n less_o than_o four_o right_a angle_n svppose_v that_o a_o be_v a_o solid_a angle_n contain_v under_o these_o superficial_a angle_n bac_n dac_o and_o dab_n then_o i_o say_v that_o the_o angle_n bac_n dac_o and_o dab_n be_v less_o than_o four_o right_a angle_n construction_n take_v in_o every_o one_o of_o these_o right_a line_n acab_n and_o ad_fw-la a_o point_n at_o all_o adventure_n and_o let_v the_o same_o be_v b_o c_o d._n and_o draw_v these_o right_a line_n bc_o cd_o and_o db._n demonstration_n and_o forasmuch_o as_o the_o angle_n b_o be_v a_o solid_a angle_n for_o it_o be_v contain_v under_o three_o superficial_a angle_n that_o be_v under_o cba_o abdella_n and_o cbd_n therefore_o by_o the_o 20._o of_o the_o eleven_o two_o of_o they_o which_o two_z so_o ever_o be_v take_v be_v great_a than_o the_o three_o wherefore_o the_o angle_n cba_o and_o abdella_n be_v great_a than_o the_o angle_n cbd_v and_o by_o the_o same_o reason_n the_o angle_n bca_o and_o acd_o be_v great_a than_o the_o angle_n bcdâ—_n and_o moreover_o the_o angle_n cda_n and_o adb_o be_v great_a than_o the_o angle_n cdb_n wherefore_o these_o six_o angle_n cba_o abdella_n bca_o acd_o cda_n and_o adb_o be_v great_a they_o these_o three_o angle_n namely_o cbd_v bcd_o &_o cdb_n but_o the_o three_o angle_n cbd_v bdc_o and_o bcd_o be_v equal_a to_o two_o right_a angle_n wherefore_o the_o six_o angle_n cba_o abdella_n bca_o acd_o cda_n and_o adb_o be_v great_a they_o two_o right_a angle_n and_o forasmuch_o as_o in_o every_o one_o of_o these_o triangle_n abc_n and_o abdella_n and_o acd_o three_o angle_n be_v equal_a two_o right_a angle_n by_o the_o 32._o of_o the_o first_o wherefore_o the_o nine_o angle_n of_o the_o three_o triangle_n that_o be_v the_o angle_n cba_o acb_o bac_n acd_o dac_o cda_n adb_o dba_n and_o bad_a be_v equal_a to_o six_o right_a angle_n of_o which_o angle_n the_o six_o angle_n abc_n bca_o acd_o cda_n adb_o and_o dba_n be_v great_a than_o two_o right_a angle_n wherefore_o the_o angle_n remain_v namely_o the_o angle_n bac_n god_n and_o dab_n which_o contain_v the_o solid_a angle_n be_v less_o than_o sour_a right_a angle_n wherefore_o every_o solid_a angle_n be_v comprehend_v under_o plain_a angle_n less_o than_o four_o right_a angle_n which_o be_v require_v to_o be_v prove_v if_o you_o will_v more_o full_o see_v this_o demonstration_n compare_v it_o with_o the_o figure_n which_o i_o put_v for_o the_o better_a sight_n of_o the_o demonstration_n of_o the_o proposition_n next_o go_v before_o only_o here_o be_v not_o require_v the_o draught_n of_o the_o line_n ae_n although_o this_o demonstration_n of_o euclid_n be_v here_o put_v for_o solid_a angle_n contain_v under_o three_o superficial_a angle_n yet_o after_o the_o like_a manner_n may_v you_o proceed_v if_o the_o solid_a angle_n be_v contain_v under_o superficial_a angle_n how_o many_o so_o ever_o as_o for_o example_n if_o it_o be_v contain_v under_o four_o superficial_a angle_n if_o you_o follow_v the_o former_a construction_n the_o base_a will_v be_v a_o quadrangled_a figure_n who_o four_o angle_n be_v equal_a to_o four_o right_a angle_n but_o the_o 8._o angle_n at_o the_o base_n of_o the_o 4._o triangle_n set_v upon_o this_o quadrangled_a figure_n may_v by_o the_o 20._o proposition_n of_o this_o book_n be_v prove_v to_o be_v great_a than_o those_o 4._o angle_n of_o the_o quadrangled_a figure_n as_o we_o see_v by_o the_o discourse_n of_o the_o former_a demonstration_n wherefore_o those_o 8._o angle_n be_v great_a than_o four_o right_a angle_n but_o the_o 12._o angle_n of_o those_o four_o triangle_n be_v equal_a to_o 8._o right_a angle_n wherefore_o the_o four_o angle_n remain_v at_o the_o top_n which_o make_v the_o solid_a angle_n be_v less_o than_o four_o right_a angle_n and_o observe_v this_o course_n you_o may_v proceed_v infinite_o ¶_o the_o 20._o theorem_a the_o 22._o proposition_n if_o there_o be_v three_o superficial_a plain_a angle_n of_o which_o two_o how_o soever_o they_o be_v take_v be_v great_a than_o the_o three_o and_o if_o the_o right_a line_n also_o which_o contain_v those_o angle_n be_v equal_a then_o of_o the_o line_n couple_v those_o equal_a right_a line_n together_o it_o be_v possible_a to_o make_v a_o triangle_n svppose_v that_o there_o be_v three_o superficial_a angle_n abc_n def_n and_o ghk_v of_o which_o let_v two_o which_o two_o soever_o be_v take_v be_v great_a than_o the_o three_o that_o be_v let_v the_o angle_n abc_n and_o def_n be_v great_a than_o the_o angle_n ghk_o and_o let_v the_o angle_n def_n and_o ghk_o be_v great_a than_o the_o angle_n abc_n and_o moreover_o let_v the_o angle_n ghk_v and_o abc_n be_v great_a than_o the_o angle_n def_n and_o let_v the_o right_a line_n ab_fw-la bc_o de_fw-fr of_o gh_o and_o hk_o be_v equal_a the_o one_o to_o the_o other_o and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n c_o and_o a_o other_o from_o the_o point_n d_o to_o the_o point_n fletcher_n and_o moreover_o a_o other_o from_o the_o point_n g_o to_o the_o point_n k._n proposition_n then_o i_o say_v that_o it_o be_v possible_a of_o three_o right_a line_n equal_a to_o the_o line_n ac_fw-la df_o and_o gk_o to_o make_v a_o triangle_n that_o be_v that_o two_o of_o the_o right_a line_n ac_fw-la df_o and_o gk_o which_o then_o soever_o be_v take_v be_v great_a than_o the_o three_o now_o if_o the_o angle_n abc_n def_n case_n and_o ghk_v be_v equal_a the_o one_o to_o the_o other_o it_o be_v manifest_a that_o these_o right_a line_n ac_fw-la df_o and_o gk_o be_v also_o by_o the_o 4._o of_o the_o first_o equal_a the_o one_o to_o the_o other_o it_o be_v possible_a of_o three_o right_a line_n equal_a to_o the_o line_n ac_fw-la df_o and_o gk_o to_o make_v a_o triangle_n case_n but_o if_o they_o be_v not_o equal_a let_v they_o be_v unequal_a and_o by_o the_o 23._o of_o the_o first_o unto_o the_o right_a line_n hk_o construction_n and_o at_o the_o point_n in_o it_o h_o make_v unto_o the_o angle_n abc_n a_o equal_a angle_n khl._n and_o by_o the_o â—_o of_o the_o first_o to_o one_o of_o the_o line_n
a_o triangle_n and_o if_o the_o parallelogram_n be_v double_a to_o the_o triangle_n those_o prism_n be_v by_o the_o 40._o of_o the_o eleven_o equal_v the_o one_o to_o the_o other_o therefore_o the_o prism_n contain_v under_o the_o two_o triangle_n bkf_a and_o ehg_n and_o under_o the_o three_o parallelogram_n ebfg_n ebkh_n and_o khfg_n part_n be_v equal_a to_o the_o prism_n contain_v under_o the_o two_o triangle_n gfc_a and_o hkl_n and_o under_o the_o three_o parallelogram_n kfcl_n lcgh_n and_o hkfg_n and_o it_o be_v manifest_a that_o both_o these_o prism_n of_o which_o the_o base_a of_o one_o be_v the_o parallelogram_n ebfg_n pyramid_n and_o the_o opposite_a unto_o it_o the_o line_n kh_o and_o the_o base_a of_o the_o other_o be_v the_o triangle_n gfc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n klh_n be_v great_a than_o both_o these_o pyramid_n who_o base_n be_v the_o triangle_n age_n and_o hkl_n and_o top_n the_o point_n h_o &_o d._n for_o if_o we_o draw_v these_o right_a line_n of_o and_o eke_o the_o prism_n who_o base_a be_v the_o parallelogram_n ebfg_n and_o the_o opposite_a unto_o it_o the_o right_a line_n hk_o be_v great_a than_o the_o pyramid_n who_o base_a be_v the_o triangle_n ebf_n &_o top_n the_o point_n k._n but_o the_o pyramid_n who_o base_a be_v the_o triangle_n ebf_n and_o top_n the_o point_n king_n be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h_o for_o they_o be_v contain_v under_o equal_a and_o like_a plain_a superficiece_n wherefore_o also_o the_o prism_n who_o base_a be_v the_o parallelogram_n ebfg_n and_o the_o opposite_a unto_o it_o the_o right_a line_n hk_o be_v great_a than_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h._n but_o the_o prism_n who_o base_a be_v the_o parallelogram_n ebfg_n and_o the_o opposite_a unto_o it_o the_o right_a line_n hk_o be_v equal_a to_o the_o prism_n who_o base_a be_v the_o triangle_n gfc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n hkl_n and_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h_o be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n hkl_n and_o top_n the_o point_n d._n part_n wherefore_o the_o foresay_a two_o prism_n be_v great_a than_o the_o foresay_a two_o pyramid_n who_o base_n be_v the_o triangle_n aeg_n hkl_n and_o top_n the_o point_n h_o and_o d._n wherefore_o the_o whole_a pyramid_n who_o base_a be_v the_o triangle_n abc_n proposition_n and_o top_n the_o point_n d_o be_v divide_v into_o two_o pyramid_n equal_a and_o like_v the_o one_o to_o the_o other_o and_o like_v also_o unto_o the_o whole_a pyramid_n have_v also_o triangle_n to_o their_o base_n and_o into_o two_o equal_a prism_n and_o the_o two_o prism_n be_v great_a than_o half_a of_o the_o whole_a pyramid_n which_o be_v require_v to_o be_v demonstrate_v if_o you_o will_v with_o diligence_n read_v these_o four_o book_n follow_v of_o euclid_n which_o concern_v body_n and_o clear_o see_v the_o demonstration_n in_o they_o contain_v it_o shall_v be_v requisite_a for_o you_o when_o you_o come_v to_o any_o proposition_n which_o concern_v a_o body_n or_o body_n whether_o they_o be_v regular_a or_o not_o first_o to_o describe_v of_o pâ—sâ—ed_a paper_n according_a as_o i_o teach_v you_o in_o the_o end_n of_o the_o definition_n of_o the_o eleven_o book_n such_o a_o body_n or_o body_n as_o be_v there_o require_v and_o have_v your_o body_n or_o body_n thus_o describe_v when_o you_o have_v note_v it_o with_o letter_n according_a to_o the_o figure_n set_v forth_o upon_o a_o plain_a in_o the_o proposition_n follow_v the_o construction_n require_v in_o the_o proposition_n as_o for_o example_n in_o this_o three_o proposition_n it_o be_v say_v that_o every_o pyramid_n have_v a_o triangle_n to_o â—is_n base_a may_v be_v divide_v into_o two_o pyramid_n etc_n etc_n here_o first_o describe_v a_o pyramid_n of_o past_v paper_n haâ—ing_v his_o base_a triangle_v it_o skill_v not_o whether_o it_o be_v equilater_n or_o equiangle_v or_o not_o only_o in_o this_o proposition_n be_v require_v that_o the_o base_a be_v a_o triangle_n then_o for_o that_o the_o proposition_n suppose_v the_o base_a of_o the_o pyramid_n to_o be_v the_o triangle_n abc_n note_v the_o base_a of_o your_o pyramid_n which_o you_o have_v describe_v with_o the_o letter_n abc_n and_o the_o top_n of_o your_o pyramid_n with_o the_o letter_n d_o for_o so_o be_v require_v in_o the_o proposition_n and_o thus_o have_v you_o your_o body_n order_v ready_a to_o the_o construction_n now_o in_o the_o construction_n it_o be_v require_v that_o you_o divide_v the_o line_n ab_fw-la bc_o ca._n etc_o etc_o namely_o the_o six_o line_n which_o be_v the_o side_n of_o the_o four_o triangle_n contain_v the_o pyramid_n into_o two_o equal_a part_n in_o the_o poyntet_fw-la â—_o f_o g_o etc_n etc_n that_o be_v you_o must_v divide_v the_o line_n ab_fw-la of_o your_o pyramid_n into_o two_o equal_a part_n and_o note_v the_o point_n of_o the_o division_n with_o the_o letter_n e_o and_o so_o the_o line_n bc_o be_v divide_v into_o two_o equal_a part_n note_v the_o point_n of_o the_o division_n with_o the_o letter_n f._n and_o so_o the_o rest_n and_o this_o order_n follow_v you_o as_o touch_v the_o rest_n of_o the_o construction_n there_o put_v and_o when_o you_o have_v finish_v the_o construction_n compare_v your_o body_n thus_o describe_v with_o the_o demonstration_n and_o it_o will_v make_v it_o very_o plain_a and_o easy_a to_o be_v understand_v whereas_o without_o such_o a_o body_n describe_v of_o matter_n it_o be_v hard_o for_o young_a beginner_n unless_o they_o have_v a_o very_a deep_a imagination_n full_o to_o conceive_v the_o demonstration_n by_o the_o sigâ—e_n as_o it_o be_v describe_v in_o a_o plain_a here_o for_o the_o better_a declaration_n of_o that_o which_o i_o have_v say_v have_v i_o set_v a_o figure_n who_o form_n if_o you_o describe_v upon_o past_v paper_n note_v with_o the_o like_a letter_n and_o cut_v the_o line_n â—a_o dam_n dc_o and_o fold_v it_o according_o it_o will_v make_v a_o pyramid_n describe_v according_a to_o the_o construction_n require_v in_o the_o proposition_n and_o this_o order_n follow_v you_o as_o touch_v all_o other_o proposition_n which_o concern_v body_n ¶_o a_o other_o demonstration_n after_o campane_n of_o the_o 3._o proposition_n suppose_v that_o there_o be_v a_o pyramid_n abcd_o have_v to_o his_o base_a the_o triangle_n bcd_o and_o let_v his_o top_n be_v the_o solid_a angle_n a_o from_o which_o let_v there_o be_v draw_v three_o subtended_a line_n ab_fw-la ac_fw-la and_o ad_fw-la to_o the_o three_o angle_n of_o the_o base_a and_o divide_v all_o the_o side_n of_o the_o base_a into_o two_o equal_a part_n in_o the_o three_o point_n e_o f_o g_o divide_v also_o the_o three_o subtended_a line_n ab_fw-la ac_fw-la and_o ad_fw-la in_o two_o equal_a part_n in_o the_o three_o point_n h_o edward_n l._n and_o draw_v in_o the_o base_a these_o two_o line_n of_o and_o eglantine_n so_o shall_v the_o base_a of_o the_o pyramid_n be_v divide_v into_o three_o superficiece_n whereof_o two_o be_v the_o two_o triangle_n bef_n and_o egg_v which_o be_v like_o both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o whole_a base_a by_o the_o 2_o part_n of_o the_o second_o of_o the_o six_o &_o by_o the_o definition_n of_o like_a superâ—iciecâ—s_n &_o they_o be_v equal_v the_o one_o to_o the_o other_o by_o the_o 8._o of_o the_o first_o the_o three_o superficies_n be_v a_o quadrangled_a parallelogram_n namely_o efgc_n which_o be_v double_a to_o the_o triangle_n egg_v by_o the_o 40._o and_o 41._o of_o the_o first_o now_o then_o again_o from_o the_o point_n h_o draw_v unto_o the_o point_n e_o and_o f_o these_o two_o subtendent_fw-la line_n he_o and_o hf_o draw_v also_o a_o subtended_a line_n from_o the_o point_n king_n to_o the_o point_n g._n and_o draw_v these_o line_n hk_o kl_o and_o lh_o wherefore_o the_o whole_a pyramid_n abcd_o be_v divide_v into_o two_o pyramid_n which_o be_v hbef_n and_o ahkl_n and_o into_o two_o prism_n of_o which_o the_o one_o be_v ehfgkc_n and_o be_v set_v upon_o the_o quadrangled_a base_a cfge_n the_o other_o be_v egdhkl_n and_o have_v to_o his_o base_a the_o triangle_n egg_v now_o as_o touch_v the_o two_o pyramid_n hbef_a and_o ahkl_n that_o they_o be_v equal_v the_o one_o to_o the_o other_o and_o also_o that_o they_o be_v like_o both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o whole_a it_o be_v manifest_a by_o the_o definition_n of_o equal_a and_o like_a body_n and_o by_o the_o 10._o of_o the_o eleven_o and_o by_o 2._o part_n of_o the_o second_o of_o the_o six_o and_o that_o the_o two_o prism_n be_v equal_a it_o
the_o same_o preface_n seem_v to_o import_v the_o preface_n of_o hypsicles_n before_o the_o fourteen_o book_n friend_n protarchus_n when_o that_o basilides_n of_o tire_n come_v into_o alexandria_n have_v familiar_a friendship_n with_o my_o father_n by_o reason_n of_o his_o knowledge_n in_o the_o mathematical_a science_n he_o remain_v with_o he_o a_o long_a time_n yea_o even_o all_o the_o time_n of_o the_o pestilence_n and_o sometime_o reason_v between_o themselves_o of_o that_o which_o apollonius_n have_v write_v touch_v the_o comparison_n of_o a_o dodecahedron_n and_o of_o a_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n what_o proportion_n such_o body_n have_v the_o one_o to_o the_o other_o they_o judge_v that_o apollonius_n have_v somewhat_o err_v therein_o wherefore_o they_o as_o my_o father_n declare_v unto_o i_o diligent_o weigh_v it_o write_v it_o perfect_o howbeit_o afterward_o i_o happen_v to_o find_v a_o other_o book_n write_v of_o apollonius_n which_o contain_v in_o it_o the_o right_a demonstration_n of_o that_o which_o they_o seek_v for_o which_o when_o they_o see_v they_o much_o rejoice_v as_o for_o that_o which_o apollonius_n write_v may_v be_v see_v of_o all_o man_n for_o that_o it_o be_v in_o â—uery_n man_n hand_n and_o that_o which_o be_v of_o we_o more_o diligent_o afterward_o write_v again_o i_o think_v good_a to_o send_v and_o dedicate_v unto_o you_o as_z to_o one_o who_o i_o think_v worthy_a commendation_n both_o for_o that_o deep_a knowledge_n which_o i_o know_v you_o have_v in_o all_o kind_n of_o learning_n and_o chief_o in_o geometry_n so_o that_o you_o be_v able_a ready_o to_o judge_v of_o those_o thing_n which_o be_v speak_v and_o also_o for_o the_o great_a love_n and_o good_a will_n which_o you_o bear_v towards_o my_o father_n and_o i_o wherefore_o vouchsafe_v gentle_o to_o accept_v this_o which_o i_o send_v unto_o you_o but_o now_o be_v it_o time_n to_o end_v our_o preface_n and_o to_o begin_v the_o matter_n ¶_o the_o 1._o theorem_a the_o 1._o proposition_n flussas_n a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o a_o circle_n to_o the_o side_n of_o a_o pentagon_n describe_v in_o the_o same_o circle_n be_v the_o half_a of_o these_o two_o line_n namely_o of_o the_o side_n of_o a_o hexagon_n figure_n and_o of_o the_o side_n of_o a_o decagon_n figure_n be_v both_o describe_v in_o the_o self_n same_o circle_n svppose_v that_o the_o circle_n be_v abc_n construction_n and_o let_v the_o side_n of_o a_o equilater_n pentagon_n describe_v in_o the_o circle_n abc_n be_v bc._n and_o by_o the_o 1._o of_o the_o three_o take_v the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v d._n and_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n d_o draw_v unto_o the_o line_n bc_o a_o perpendicular_a line_n de._n and_o extend_v the_o right_a line_n de_fw-fr direct_o to_o the_o point_n f._n then_o i_o say_v that_o the_o line_n de_fw-fr which_o be_v draw_v from_o the_o centre_n to_o bc_o the_o side_n of_o the_o pentagon_n be_v the_o half_a of_o the_o side_n of_o a_o hexagon_n and_o of_o a_o decagon_n take_v together_o and_o describe_v in_o the_o same_o circle_n draw_v these_o right_a line_n dc_o and_o cf._n and_o unto_o the_o line_n of_o put_v a_o equal_a line_n ge._n and_o draw_v a_o right_a line_n from_o the_o point_n g_o to_o the_o point_n c._n demonstration_n now_o forasmuch_o as_o the_o circumference_n of_o the_o whole_a circle_n be_v quintuple_a to_o the_o circumference_n bfc_n which_o be_v subtend_v of_o the_o side_n of_o the_o pentagon_n and_o the_o circumference_n acf_n be_v the_o half_a of_o the_o circumference_n of_o the_o whole_a circle_n and_o the_o circumference_n cf_o which_o be_v subtend_v of_o the_o side_n of_o the_o decagon_n be_v the_o half_a of_o the_o circumference_n bcf_n therefore_o the_o circumference_n acf_n be_v quintuple_a to_o the_o circumference_n cf_o by_o the_o 15._o of_o the_o â—iâ—t_n wherefore_o the_o circumference_n ac_fw-la be_v qradruple_a to_o the_o circumference_n fc_o but_o as_o the_o circumference_n ac_fw-la be_v to_o the_o circumference_n fc_o so_o be_v the_o angle_n adc_o to_o the_o angle_n fdc_n by_o the_o last_o of_o the_o six_o wherefore_o the_o angle_n adc_o be_v quadruple_a to_o the_o angle_n fdc_n but_o the_o angle_n adc_o be_v double_a to_o the_o angle_n efc_n by_o the_o 20._o of_o the_o three_o wherefore_o the_o angle_n efc_n be_v double_a to_o the_o angle_n gdc_n but_o the_o angle_n efc_n be_v equal_a to_o the_o angle_n egc_n by_o the_o 4._o of_o the_o first_o wherefore_o the_o angle_n egc_n be_v double_a to_o the_o angle_n edc_n wherefore_o the_o line_n dg_o be_v equal_a to_o the_o line_n gc_o by_o the_o 32._o and_o 6._o of_o the_o first_o but_o the_o line_n gc_o be_v equal_a to_o the_o line_n cf_o by_o the_o 4._o of_o the_o first_o wherefore_o the_o line_n dg_o be_v equal_a to_o the_o line_n cf._n and_o the_o line_n ge_z be_v equal_a to_o the_o line_n of_o by_o construction_n wherefore_o the_o line_n de_fw-fr be_v equal_a to_o the_o line_n of_o and_o fc_n add_v together_o unto_o the_o line_n of_o and_o fc_n add_v the_o line_n de._n wherefore_o the_o line_n df_o and_o fc_o add_v together_o be_v double_a to_o the_o line_n de._n but_o the_o line_n df_o be_v equal_a to_o the_o side_n of_o the_o hexagon_n and_o fc_a to_o the_o side_n of_o the_o decagon_n wherefore_o the_o line_n de_fw-fr be_v the_o half_a of_o the_o side_n of_o the_o hexagon_n and_o of_o the_o side_n of_o the_o decagon_n be_v both_o add_v together_o and_o describe_v in_o one_o and_o the_o self_n same_o circle_n it_o be_v manifest_a same_o by_o the_o proposition_n of_o the_o thirteen_o book_n that_o a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o a_o circle_n to_o the_o side_n of_o a_o equilater_n triangle_n describe_v in_o the_o same_o circle_n be_v half_a of_o the_o semidiameter_n of_o the_o circle_n wherefore_o by_o this_o proposition_n a_o perpendicular_a draw_v from_o the_o câ—ntre_n of_o a_o circle_n to_o the_o side_n of_o a_o pentagon_n be_v equal_a to_o the_o perpendicular_a draw_v from_o the_o centre_n to_o the_o side_n of_o the_o triangle_n â—nd_v to_o half_a of_o the_o side_n of_o the_o decagon_n describe_v in_o the_o same_o circle_n ¶_o the_o 2._o theorem_a the_o 2._o proposition_n one_o and_o the_o self_n same_o circle_n comprehend_v both_o the_o pentagon_n of_o a_o dodecahedron_n and_o the_o triangle_n of_o a_o icosahedron_n flussas_n describe_v in_o one_o and_o the_o self_n same_o sphere_n this_o theorem_a be_v describe_v of_o aristeus_n in_o that_o book_n who_o title_n be_v the_o comparison_n of_o the_o five_o figure_n and_o be_v describe_v of_o apollonius_n in_o his_o second_o edition_n of_o the_o comparison_n of_o a_o dodecahedron_n to_o a_o icosahedron_n which_o be_v proposition_n that_o as_o the_o superficies_n of_o a_o dodecahedron_n be_v to_o the_o superficies_n of_o a_o icosahedron_n so_o be_v the_o dodecahedron_n to_o a_o icosahedron_n for_o that_o a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o a_o sphere_n to_o the_o pentagon_n of_o a_o dodecahedron_n and_o to_o the_o triangle_n of_o a_o icosahedron_n be_v one_o and_o the_o self_n same_o now_o must_v we_o also_o prove_v that_o one_o and_o the_o self_n same_o circle_n comprehend_v both_o the_o pentagon_n of_o a_o dodecahedron_n and_o also_o the_o triangle_n of_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n first_o this_o be_v prove_v flussas_n if_o in_o a_o circle_n be_v describe_v a_o equilater_n pentagon_n the_o square_n which_o be_v make_v of_o the_o side_n of_o the_o pentagon_n and_o of_o that_o right_a line_n which_o be_v subtend_v under_o two_o side_n of_o the_o pentagon_n be_v quintuple_a to_o the_o square_n of_o the_o semidiameter_n oâ—_n the_o circle_n suppose_v that_o abc_n be_v a_o circle_n assumpt_n and_o let_v the_o side_n of_o a_o pentagon_n in_o the_o circle_n abc_n be_v ac_fw-la and_o take_v by_o the_o 1._o of_o the_o three_o the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v d._n and_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n d_o draw_v unto_o the_o line_n ac_fw-la a_o perpendicular_a line_n df._n and_o extend_v the_o line_n df_o on_o either_o side_n to_o the_o point_n b_o and_o e._n and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n b._n now_o i_o say_v that_o the_o square_n of_o the_o line_n basilius_n and_o ac_fw-la be_v quintuple_a to_o the_o square_n of_o the_o line_n de._n draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n e._n wherefore_o the_o line_n ae_n be_v the_o side_n of_o a_o decagon_n figure_n and_o forasmuch_o as_o the_o line_n be_v be_v double_a to_o thâ—_z line_n de_fw-fr assumpt_n therefore_o the_o square_n
spherâ—_n contain_v the_o dodecahedron_n of_o this_o pentagon_n and_o the_o icosahedron_n of_o this_o triangle_n by_o the_o 4._o of_o this_o book_n â—_o and_o the_o line_n cl_n fall_v perpendicular_o upon_o the_o side_n of_o the_o icosahedron_n and_o the_o line_n ci_o upon_o the_o side_n of_o the_o dodecahedron_n that_o which_o be_v 30._o time_n contain_v under_o the_o side_n and_o the_o perpendicular_a line_n fall_v upon_o it_o be_v equal_a to_o the_o superficies_n of_o that_o solid_a upon_o who_o side_n the_o perpendicular_a fall_v if_o therefore_o in_o a_o circle_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n order_n the_o superficiece_n of_o a_o dodecahedron_n and_o of_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n be_v the_o one_o to_o the_o other_o as_o that_o which_o be_v contain_v under_o the_o side_n of_o the_o one_o and_o the_o perpendicular_a line_n draw_v unto_o it_o from_o the_o centre_n of_o his_o base_a to_o that_o which_o be_v contain_v under_o the_o side_n of_o the_o other_o and_o the_o perpendicular_a line_n draw_v to_o it_o from_o the_o centre_n of_o his_o base_a for_o aâ—_z thirtyâ—_n timâ—s_o be_v to_o thirty_o time_n so_o be_v once_o to_o once_o by_o the_o 15._o of_o thâ—_z five_o the_o 6._o proposition_n the_o superficies_n of_o a_o dodecahedron_n be_v to_o the_o superficies_n of_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n campane_n in_o that_o proportion_n that_o the_o side_n of_o the_o cube_n be_v to_o the_o side_n of_o the_o icosahedron_n contain_v in_o the_o self_n same_o sphere_n construction_n svppose_v that_o there_o be_v a_o circle_n abg_o &_o in_o it_o by_o the_o 4._o of_o this_o book_n let_v there_o be_v inscribe_v the_o sideâ—_n of_o a_o dodecahedron_n and_o of_o a_o icosahedron_n contain_v in_o onâ—_n and_o the_o self_n same_o sphere_n and_o let_v the_o side_n oâ—_n the_o dodecahedron_n be_v agnostus_n and_o the_o side_n of_o the_o icosahedron_n be_v dg_o and_o let_v the_o centre_n be_v the_o point_n e_o from_o which_o draw_v unto_o those_o sâ—des_n perpendicular_a line_n ei_o and_o ez_o and_o produce_v the_o line_n ei_o to_o the_o point_n b_o and_o draw_v the_o linâ—_n bg_o and_o let_v the_o side_n of_o the_o cube_fw-la contain_v in_o the_o self_n same_o sphere_n be_v gc_o then_o i_o say_v that_o the_o superficies_n of_o the_o dodecahedron_n iâ—_z to_o the_o superficies_n of_o the_o icosahedron_n as_o the_o line_n â—g_z iâ—_z to_o the_o liâ—â—_n gd_o for_o forasmuch_o as_o the_o line_n ei_o beinâ—_n divide_v by_o a_o extreme_a and_o mean_a proportion_n the_o great_a segment_n thereof_o shall_v be_v the_o linâ—_n ez_o by_o the_o corollary_n of_o the_o first_o of_o this_o book_n demonstration_n and_o the_o line_n cg_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n his_o great_a segment_n be_v the_o line_n agnostus_n by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o wherefore_o the_o right_a line_n ei_o and_o cg_o â—râ—_n cut_v proportional_o by_o the_o second_o of_o this_o bâ—oke_n whâ—râ—fore_o as_o the_o line_n cg_o be_v to_o the_o line_n agnostus_n so_o be_v the_o line_n ei_o to_o the_o line_n ez_fw-fr wherâ—fore_o that_o which_o it_o contain_v under_o the_o extreme_n cg_o and_o ez_o be_v squall_n to_o that_o which_o iâ—_z contaynâ—d_v under_o the_o mean_n agnostus_n and_o ei._n by_o the_o 16._o of_o the_o six_o but_o as_o that_o which_o iâ—_z contain_v under_o the_o linâ—â—_n cg_o and_o â—z_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n dg_o and_o ez_o so_o by_o the_o first_o of_o the_o six_o iâ—_z the_o linâ—_n cg_o to_o the_o line_n dg_o for_o both_o those_o parallelogram_n have_v oâ—â—_n and_o the_o self_n same_o altitude_n namely_o the_o line_n ez_fw-fr wherefore_o as_o that_o which_o be_v contain_v under_o the_o line_n ei_o and_o agnostus_n which_o iâ—_z prove_v equal_a to_o that_o which_o be_v contain_v under_o the_o lineâ—_z cg_o and_o ez_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n dg_o and_o ez_o so_o be_v the_o line_n cg_o to_o the_o liâ—â—_n dg_o but_o as_o that_o which_o be_v contain_v under_o the_o line_n ei_o and_o agnostus_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n dg_o and_o ez_o so_o by_o the_o corollary_n of_o the_o former_a proposition_n be_v the_o superficies_n of_o the_o dodecahedron_n to_o the_o superficies_n of_o the_o icosahedron_n wherefore_o as_o the_o superficies_n â—â—_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v cg_o the_o side_n of_o the_o cube_fw-la to_o gd_v the_o side_n of_o the_o icosahedron_n the_o superficies_n therefore_o of_o a_o dodecahedron_n be_v to_o the_o superficiesâ—_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v require_v to_o be_v prove_v a_o assumpt_n the_o pentagon_n of_o a_o dodecahedron_n order_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o perpendicular_a line_n which_o fall_v upon_o the_o base_a of_o the_o triangle_n of_o the_o icosahedron_n and_o five_o six_o part_n of_o the_o side_n of_o the_o cube_fw-la the_o say_v three_o solid_n be_v describe_v in_o one_o and_o the_o self_n same_o sphere_n suppose_v that_o in_o the_o circle_n abeg_n the_o pentagon_n of_o a_o dodecahedron_n be_v aâ—cig_a and_o let_v two_o side_n thereof_o ab_fw-la and_o agnostus_n be_v subtend_v of_o the_o right_a line_n bg_o and_o let_v the_o triangle_n of_o the_o icosahedron_n inscribe_v in_o the_o self_n same_o sphere_n construction_n by_o the_o 4._o of_o this_o book_n be_v afh_n and_o let_v the_o centre_n of_o the_o circle_n be_v the_o point_n d_o and_o let_v the_o diameter_n be_v ade_n cut_v fh_o the_o side_n of_o the_o triangle_n in_o the_o point_n z_o and_o cut_v the_o line_n bg_o in_o the_o point_n k._n and_o draw_v the_o right_a line_n bd._n and_o from_o the_o right_a line_n kg_v cut_v of_o a_o three_o part_n tg_n by_o the_o 9_o of_o the_o six_o now_o than_o the_o line_n bg_o subtend_v two_o side_n of_o the_o dodecahedron_n shall_v be_v the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o same_o sphere_n demonstration_n by_o the_o 17._o of_o the_o thirteen_o and_o the_o triangle_n of_o the_o icosahedron_n of_o the_o same_o sphere_n shall_v be_v aâ—h_a by_o the_o 4._o of_o this_o book_n and_o the_o line_n az_o which_o pass_v by_o the_o centre_n d_o shall_v fall_v perpendicular_o upon_o the_o side_n of_o the_o triangle_n for_o forasmuch_o as_o the_o angle_n gay_a &_o bae_o be_v equal_a by_o the_o 27._o of_o the_o thirdâ—_o for_o they_o be_v see_v upon_o equal_a circumference_n therefore_o the_o â—ases_v bk_o and_o kg_n be_v by_o the_o â—_o of_o the_o first_o equal_a wherefore_o the_o line_n bt_n contain_v 5._o six_o part_n of_o the_o line_n bg_o then_o i_o say_v that_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o bt_n be_v equal_a to_o the_o pentagon_n aâ—câ—g_n for_o forasmuch_o as_o the_o line_n â—z_n be_v sesquialter_n to_o the_o line_n ad_fw-la for_o the_o line_n dâ—_n be_v divide_v into_o two_o equal_a part_n in_o the_o point_n z_o by_o the_o corollary_n of_o the_o â—2â—_n of_o the_o thirteen_o likewise_o by_o construction_n the_o line_n kg_n be_v sesquialter_fw-la to_o the_o line_n kt_n therefore_o as_o the_o line_n az_o be_v to_o the_o line_n ad_fw-la so_o be_v the_o line_n kg_v to_o the_o 〈◊〉_d â—t_a wherefore_o that_o which_o be_v contain_v undeâ—_n the_o 〈◊〉_d az_o and_o kt_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o mean_n ad_fw-la and_o kg_n by_o the_o 16._o of_o the_o six_o but_o unto_o the_o line_n kg_n be_v the_o line_n â—k_v â—roved_v equal_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o bk_o but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o bk_o be_v by_o the_o 41._o of_o the_o first_o double_a to_o the_o triangle_n abdella_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n be_v double_a to_o the_o same_o triangle_n abdella_n and_o forasmuch_o as_o the_o pentagon_n abcig_n contaynethâ—_n 〈…〉_o equal_a â—o_o the_o triangle_n abdella_n and_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n contain_v two_o such_o triangle_n therefore_o the_o pentagon_n abcig_n be_v duple_fw-fr sesquialter_fw-la to_o the_o rectangle_n parallelogram_n contain_v under_o the_o line_n az_o and_o kt_n and_o 〈…〉_o 1._o of_o the_o six_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o bt_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n as_o the_o base_a bt_n be_v to_o the_o base_a â—â—tâ—_n therefore_o that_o which_o be_v contain_v under_o the_o line_n az_o
together_o and_o a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o any_o base_a of_o the_o cube_fw-la be_v equal_a to_o half_a the_o side_n of_o the_o cube_fw-la which_o be_v require_v to_o be_v prouâ—d_v ¶_o a_o corollary_n if_o two_o three_o of_o the_o power_n of_o the_o diameter_n of_o the_o sphere_n be_v multiply_v into_o the_o perpendicular_a line_n equal_a to_o half_a the_o side_n of_o the_o cube_fw-la campane_n there_o shall_v be_v produce_v a_o solid_a equal_a to_o the_o solid_a of_o the_o cube_fw-la for_o it_o be_v before_o manifest_v that_o two_o three_o part_n of_o the_o power_n of_o the_o diameter_n of_o the_o sphere_n be_v equal_a to_o two_o base_n of_o the_o cube_fw-la if_o therefore_o unto_o each_o of_o those_o two_o three_o be_v apply_v half_o the_o altitude_n of_o the_o cube_fw-la they_o shall_v make_v each_o of_o those_o solid_n equal_a to_o half_o of_o the_o cube_fw-la by_o the_o 31._o of_o the_o eleven_o for_o they_o have_v equal_a base_n wherefore_o two_o of_o those_o solid_n be_v equal_a to_o the_o whole_a cube_fw-la you_o shall_v understand_v gentle_a reader_n that_o campane_n in_o his_o 14._o book_n of_o euclides_n element_n have_v 18._o proposition_n with_o diverse_a corollary_n follow_v of_o they_o some_o of_o which_o proposition_n and_o corollary_n i_o have_v before_o in_o the_o twelve_o and_o thirteen_o book_n add_v out_o of_o flussas_n as_o corollary_n which_o thing_n also_o i_o have_v note_v on_o the_o side_n of_o those_o corollary_n namely_o with_o what_o proposition_n or_o corollary_n of_o campane_n 14._o book_n they_o do_v agree_v the_o rest_n of_o his_o 18._o proposition_n and_o corollary_n be_v contain_v in_o the_o twelve_o former_a proposition_n and_o corollary_n of_o this_o 14._o book_n after_o flussas_n where_o you_o may_v see_v on_o the_o side_n of_o each_o proposition_n and_o corollary_n with_o what_o proposition_n and_o corollary_n of_o campane_n they_o agree_v but_o the_o eight_o proposition_n follow_v together_o with_o their_o corollary_n flussas_n have_v add_v of_o himself_o as_o he_o himself_o affirm_v the_o 13._o proposition_n one_o and_o the_o self_n same_o circle_n contain_v both_o the_o square_n of_o a_o cube_fw-la and_o the_o triangle_n of_o a_o octohedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n svppose_v that_o there_o be_v a_o cube_fw-la abg_o and_o a_o octohedron_n def_n describe_v in_o one_o and_o the_o self_n same_o sphere_n who_o diameter_n let_v be_v ab_fw-la or_o dh_o and_o let_v the_o line_n draw_v from_o the_o centre_n that_o be_v the_o semidiameter_n of_o the_o circle_n which_o ctonaine_v the_o base_n of_o those_o solid_n â—_o be_v ca_n and_o id_fw-la then_o i_o say_v that_o the_o line_n ca_n and_o id_fw-la be_v equal_a forasmuch_o as_o ab_fw-la the_o diameter_n of_o the_o sphere_n which_o contain_v the_o cube_fw-la demonstration_n be_v in_o power_n triple_a to_o bg_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o the_o thirteen_o unto_o which_o side_n ag_z the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la be_v in_o power_n double_a by_o the_o 47._o of_o the_o first_o which_o line_n agnostus_n be_v also_o the_o diameter_n of_o the_o circle_n which_o contain_v the_o base_a by_o the_o 9_o of_o the_o four_o therefore_o ab_fw-la the_o diameter_n of_o the_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o line_n agnostus_n namely_o of_o what_o part_n the_o line_n ab_fw-la contain_v in_o power_n 12._o of_o the_o same_o the_o line_n agnostus_n shall_v contain_v in_o power_n 8._o and_o therefore_o the_o right_a line_n ac_fw-la which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n to_o the_o circumference_n contain_v in_o power_n of_o the_o same_o part_n 2._o wherefore_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n sextuple_a to_o the_o line_n which_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o square_a of_o 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which_o contain_v the_o square_a of_o the_o cube_fw-la be_v also_o sextuple_a to_o the_o line_n id_fw-la draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o triangle_n of_o the_o octohedron_n wherefore_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n to_o the_o circumference_n which_o contain_v the_o base_n of_o the_o cube_fw-la and_o of_o the_o octohedron_n be_v equal_a and_o therefore_o the_o circle_n be_v equal_a by_o the_o first_o definition_n of_o the_o three_o wherefore_o one_o and_o the_o self_n same_o circle_n contain_v &c._a &c._a as_o in_o the_o proposition_n which_o be_v require_v to_o be_v prove_v a_o corollary_n hereby_o it_o be_v manifest_a that_o perpendicular_n couple_v together_o in_o a_o sphere_n the_o centre_n of_o the_o circle_n which_o contain_v the_o opposite_a base_n of_o the_o cube_fw-la and_o of_o the_o octohedron_n be_v equal_a for_o the_o circle_n be_v equal_a by_o the_o second_o corollary_n of_o 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the_o whole_a line_n mg_o to_o the_o whole_a line_n ea_fw-la by_o the_o 18._o of_o the_o five_o wherefore_o as_o mg_o the_o side_n of_o the_o cube_fw-la be_v to_o ea_fw-la the_o semidiameter_n so_o be_v the_o line_n fghim_n to_o the_o octohedron_n abkdlc_n inscribe_v in_o one_o &_o the_o self_n same_o sphere_n if_o therefore_o a_o cube_fw-la and_o a_o octohedron_n be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n they_o shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o semidiameter_n of_o the_o sphere_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n distinct_o to_o notify_v the_o power_n of_o the_o side_n of_o the_o five_o solid_n by_o the_o power_n of_o the_o diameter_n of_o the_o sphere_n the_o side_n of_o the_o tetrahedron_n and_o of_o the_o cube_fw-la do_v cut_v the_o power_n of_o the_o diameter_n of_o the_o sphere_n into_o two_o square_n which_o be_v in_o proportion_n double_a the_o 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overthrow_v and_o overwhelm_v the_o whole_a world_n he_o be_v utter_o rude_a and_o ignorant_a in_o the_o greek_a tongue_n so_o that_o certain_o he_o never_o read_v euclid_n in_o the_o greek_a nor_o of_o like_a translate_v out_o of_o the_o greek_a but_o have_v it_o translate_v out_o of_o the_o arabike_a tongue_n the_o arabian_n be_v man_n of_o great_a study_n and_o industry_n and_o common_o great_a philosopher_n notable_a physician_n and_o in_o mathematical_a art_n most_o expert_a so_o that_o all_o kind_n of_o good_a learning_n flourish_v and_o reign_v amongst_o they_o in_o a_o manner_n only_o these_o man_n turn_v whatsoever_o good_a author_n be_v in_o the_o greek_a tongue_n of_o what_o art_n and_o knowledge_n so_o ever_o it_o be_v into_o the_o arabike_a tongue_n and_o from_o thence_o be_v many_o of_o they_o turn_v into_o the_o latin_a and_o by_o that_o mean_v many_o greek_a author_n come_v to_o the_o hand_n of_o the_o latin_n and_o not_o from_o the_o first_o fountain_n the_o greek_a tongue_n wherein_o they_o be_v first_o write_v as_o appear_v by_o many_o word_n of_o the_o arabike_a tongue_n yet_o remain_v in_o such_o book_n as_o be_v zenith_n nadir_n helmuayn_fw-mi helmuariphe_n and_o infinite_a such_o other_o which_o arabian_n also_o in_o translate_n such_o greek_a work_n be_v accustom_v to_o add_v as_o they_o think_v good_a &_o for_o the_o full_a understanding_n of_o the_o author_n many_o thing_n as_o be_v to_o be_v see_v in_o diverse_a author_n as_o namely_o in_o theodosius_n de_fw-la sphera_fw-la where_o you_o see_v in_o the_o old_a translation_n which_o be_v undoubteldy_a out_o of_o the_o arabike_a many_o proposition_n almost_o every_o three_o or_o four_o leaf_n some_o such_o copy_n of_o euclid_n most_o likely_a do_v campanus_n follow_v wherein_o he_o find_v those_o proposition_n which_o he_o have_v more_o &_o above_o those_o which_o be_v find_v in_o the_o greek_a set_v out_o by_o hypsicles_n and_o that_o not_o only_o in_o this_o 15._o book_n but_o also_o in_o the_o 14._o book_n wherein_o also_o you_o find_v many_o proposition_n more_o they_o be_v find_v in_o the_o greek_a set_v out_o also_o by_o hypsicles_n likewise_o in_o the_o book_n before_o you_o shall_v find_v many_o proposition_n add_v and_o many_o invert_v and_o set_v out_o of_o order_n far_o otherwise_o than_o they_o be_v place_v in_o the_o greek_a examplar_n flussas_n also_o a_o diligent_a restorer_n of_o euclid_n a_o man_n also_o which_o have_v well_o deserve_v of_o the_o whole_a art_n of_o geometry_n have_v add_v moreover_o in_o this_o book_n as_o also_o in_o the_o former_a 14._o book_n he_o add_v 8._o proposition_n 9_o proposition_n of_o his_o own_o touch_v the_o inscription_n and_o circumscription_n 〈…〉_o body_n very_o singular_a â—ndoubtedly_o and_o witty_a all_o which_o for_o that_o nothing_o shall_v want_v to_o the_o desirous_a lover_n of_o knowledge_n i_o have_v faithful_o with_o no_o small_a pain_n turn_v and_o whereas_o flâ—ssâ—â—_n in_o the_o begin_n of_o the_o eleven_o book_n namely_o in_o the_o end_n of_o the_o definition_n there_o â—eâ—_n put_v two_o definition_n of_o the_o inscription_n and_o circumscription_n of_o solid_n or_o corporal_a figure_n within_o or_o about_o the_o one_o the_o other_o which_o certain_o be_v not_o to_o be_v reject_v yet_o for_o that_o until_o this_o present_a 15._o book_n there_o be_v no_o mention_n make_v of_o the_o inscription_n or_o circumscription_n of_o these_o body_n i_o think_v it_o not_o so_o convenient_a thâ—râ—_n to_o place_v they_o but_o to_o refer_v they_o to_o the_o begin_n of_o this_o 15._o book_n where_o they_o be_v in_o manner_n of_o necessity_n require_v to_o the_o elucidation_n of_o the_o proposition_n and_o dâ—monstrationâ—_n of_o the_o same_o the_o definition_n be_v these_o definition_n 1._o a_o solid_a figure_n be_v then_o â—aid_v to_o be_v inscribe_v in_o a_o solid_a figure_n when_o the_o angle_n of_o the_o figure_n inscribe_v touch_n together_o at_o one_o time_n either_o the_o angle_n of_o the_o figure_n circumscribe_v or_o the_o superficiece_n or_o the_o side_n definition_n 2._o a_o solid_a figure_n be_v then_o say_v to_o be_v circumscribe_v about_o a_o solid_a figure_n when_o together_o at_o one_o time_n either_o the_o angle_n or_o the_o superficiece_n or_o the_o side_n of_o the_o figure_n circumscribe_v â—ouch_v the_o angle_n of_o the_o figure_n inscribe_v in_o the_o fourâ—â—_n book_n in_o the_o definition_n of_o the_o inscription_n or_o circumscription_n of_o plain_a rectiline_n figure_v one_o with_o in_o or_o about_o a_o other_o be_v require_v that_o all_o the_o angle_n of_o the_o figuâ—â—_n inscribe_v shall_v at_o one_o time_n touch_v all_o the_o side_n of_o the_o figure_n circumscribe_v but_o in_o the_o five_o regular_a solid_n â—o_o who_o chief_o these_o two_o definition_n pertain_v for_o that_o the_o number_n of_o their_o angle_n superficiece_n &_o side_n be_v not_o equal_a one_o compare_v to_o a_o other_o it_o be_v not_o of_o necessity_n that_o all_o the_o angle_n of_o the_o solid_a inscribe_v shall_v together_o at_o one_o time_n touch_v either_o all_o the_o angle_n or_o all_o the_o superficiece_n or_o all_o the_o side_n of_o the_o solid_a circumscribe_v but_o it_o be_v sufficient_a that_o those_o angle_n of_o the_o inscribe_v solid_a which_o touch_n do_v at_o one_o time_n together_o each_o touch_n some_o one_o angle_n of_o the_o figure_n circumscribe_v or_o some_o one_o base_a or_o some_o one_o side_n so_o that_o if_o the_o angle_n of_o the_o inscribe_v figure_n do_v at_o one_o time_n touch_v the_o angle_n of_o the_o figure_n circumscribe_v none_o of_o they_o may_v at_o the_o same_o time_n touch_v either_o the_o base_n or_o the_o side_n of_o the_o same_o circumscribe_v figure_n and_o so_o if_o they_o touch_v the_o base_n they_o may_v touch_v neither_o angle_n nor_o side_n and_o likewise_o if_o they_o touch_v the_o side_n they_o may_v touch_v neither_o angle_n nor_o base_n and_o although_o sometime_o all_o the_o angle_n of_o the_o figure_n inscribe_v can_v not_o touch_v either_o the_o angle_n or_o the_o base_n or_o the_o side_n of_o the_o figure_n circumscribe_v by_o reason_n the_o number_n of_o the_o angle_n base_n or_o side_n of_o the_o say_a figure_n circumscribe_v want_v of_o the_o number_n of_o the_o angle_n of_o the_o â—igure_n inscribe_v yet_o shall_v those_o angle_n of_o the_o inscribe_v figure_n which_o touch_n so_o touch_v that_o the_o void_a place_n leave_v between_o the_o inscribe_v and_o circumscribe_v figure_n shall_v on_o every_o side_n be_v equal_a and_o like_a as_o you_o may_v afterward_o in_o this_o fifteen_o book_n most_o plain_o perceive_v ¶_o the_o 1._o proposition_n the_o 1._o problem_n in_o a_o cube_n give_v to_o describe_v admonish_v a_o trilater_n equilater_n pyramid_n svppose_v that_o the_o cube_fw-la give_v be_v abcdefgh_o in_o the_o same_o cube_fw-la it_o be_v require_v to_o inscribe_v a_o tetrahedron_n draw_v these_o right_a line_n ac_fw-la construction_n ce_fw-fr ae_n ah_o eh_o hc_n demonstration_n now_o it_o be_v manifest_a that_o the_o triangle_n aec_o ahe_o ahc_n and_o i_o be_v equilater_n for_o their_o side_n be_v the_o diameter_n of_o equal_a square_n wherefore_o aech_n be_v a_o trilater_n equilater_n pyramid_n or_o tetrahedron_n &_o it_o be_v inscribe_v in_o the_o cube_fw-la give_v by_o the_o first_o definition_n of_o this_o book_n which_o be_v require_v to_o be_v do_v ¶_o the_o 2._o proposition_n the_o 2._o problem_n in_o a_o trilater_n equilater_n pyramid_n give_v to_o describe_v a_o octohedron_n svppose_v that_o the_o trilater_n equilater_n pyramid_n give_v be_fw-mi abcd_o who_o side_n let_v be_v divide_v into_o two_o equal_a part_n in_o the_o point_n e_o z_o i_o king_n l_o t._n construction_n and_o draw_v these_o 12._o right_a line_n ez_o zi_n je_n kl_o lt_n tk_n eke_o kz_o zl_n li_n it_o and_o te_fw-la which_o 12._o right_a line_n be_v demonstration_n by_o the_o 4._o of_o the_o first_o equal_a for_o they_o subtend_v equal_a plain_a angle_n of_o the_o base_n of_o the_o pyramid_n and_o those_o equal_a angle_n be_v contain_v under_o equal_a side_n namely_o under_o the_o half_n of_o the_o side_n of_o the_o pyramid_n wherefore_o the_o triangle_n tkl_n tli_n tie_v tek_v zkl_n zli_n zib_n zek_n be_v equilater_n and_o they_o limitate_v and_o contain_v the_o solid_a tklezi_n wherefore_o the_o solid_a tklezi_n be_v a_o octohedron_n by_o the_o 23._o definition_n of_o the_o eleven_o and_o the_o angle_n of_o the_o same_o octohedron_n do_v touch_v the_o side_n of_o the_o pyramid_n abcd_o in_o the_o point_n e_o z_o i_o t_o edward_n l._n wherefore_o the_o octohedron_n be_v inscribe_v in_o the_o pyramid_n by_o the_o 1._o definition_n of_o this_o book_n wherefore_o in_o the_o trilater_n equilater_n pyramid_n give_v be_v inscribe_v a_o octohedron_n which_o be_v require_v to_o be_v do_v a_o corollary_n add_v by_o flussas_n hereby_o it_o be_v manifest_a that_o a_o pyramid_n be_v cut_v into_o two_o
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