how_o often_o therefore_o note_n these_o five_o sundry_a sort_n of_o operation_n do_v for_o the_o most_o part_n of_o their_o execution_n differre_fw-la from_o the_o five_o operation_n of_o like_a general_a property_n and_o name_n in_o our_o whole_a number_n practisable_v so_o often_o for_o a_o more_o distinct_a doctrine_n we_o vulgar_o account_v and_o name_v it_o a_o other_o kind_n of_o arithmetic_n and_o by_o this_o reason_n the_o consideration_n doctrine_n and_o work_v in_o whole_a number_n only_o where_o of_o a_o unit_fw-la be_v no_o less_o part_n to_o be_v allow_v be_v name_v as_o it_o be_v a_o arithmetic_n by_o itself_o and_o so_o of_o the_o arithmetic_n of_o fraction_n in_o like_o sort_n the_o necessary_a wonderful_a and_o secret_a doctrine_n of_o proportion_n and_o proportionalytie_n have_v purchase_v unto_o itself_o a_o peculiar_a manner_n of_o handle_v and_o work_n and_o so_o may_v seem_v a_o other_o form_n of_o arithmetic_n moreover_o the_o astronomer_n for_o speed_n and_o 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and_o mixtynge_n consider_v howâ—_n i_o be_v desirous_a to_o deliver_v the_o student_n from_o error_n and_o cavillation_n do_v give_v to_o this_o practice_n the_o name_n of_o the_o arithmetic_n of_o radical_a number_n not_o of_o irrational_a or_o fur_v numbersâ—_n which_o other_o while_n be_v rational_a though_o they_o have_v the_o sign_n of_o a_o rote_n before_o they_o which_o arithmetic_n of_o whole_a number_n most_o usual_a will_v say_v they_o have_v no_o such_o root_n and_o so_o account_v they_o fur_v number_n which_o general_o speak_v be_v untrue_a as_o euclides_n ten_o book_n may_v teach_v you_o therefore_o to_o call_v they_o general_o radical_a number_n by_o reason_n of_o the_o sign_n √_o prefix_a be_v a_o sure_a way_n and_o a_o sufficient_a general_a distinction_n from_o all_o other_o order_v and_o use_v of_o number_n and_o yet_o beside_o all_o this_o consider_v the_o infinite_a desire_n of_o knowledge_n and_o incredible_a power_n of_o man_n search_n and_o capacity_n how_o they_o joint_o have_v wade_v far_o by_o mixt_v of_o speculation_n and_o practice_n and_o have_v find_v out_o and_o attain_v to_o the_o very_a chief_a perfection_n almost_o of_o number_n practical_a use_n which_o thing_n be_v well_o to_o be_v perceive_v in_o that_o great_a arithmetical_a art_n of_o aequation_n common_o call_v the_o rule_n of_o coss._n or_o algebra_n the_o latin_n term_v it_o regulam_fw-la rei_fw-la &_o census_n that_o be_v the_o rule_n of_o the_o thing_n and_o his_o value_n with_o a_o apt_a name_n comprehend_v the_o first_o and_o last_o point_n of_o the_o work_n and_o the_o vulgar_a name_n both_o in_o italian_a french_a and_o spanish_a depend_v in_o name_a it_o upon_o the_o signification_n of_o the_o latin_a word_n res_n a_o thing_n unleast_o they_o use_v the_o name_n of_o algebra_n and_o therein_o common_o be_v a_o double_a error_n the_o one_o of_o they_o which_o think_v it_o to_o be_v of_o geber_n his_o invent_v the_o other_o of_o such_o as_o call_v it_o algebra_n for_o first_o though_o geber_n for_o his_o great_a skill_n in_o number_n geometry_n astronomy_n and_o other_o marvellous_a art_n may_v have_v seem_v able_a to_o have_v first_o devise_v the_o say_a rule_n and_o also_o the_o name_n carry_v with_o it_o a_o very_a near_a likeness_n of_o geber_n his_o name_n yet_o true_a it_o be_v that_o a_o greek_a philosopher_n and_o mathematicien_n name_v diophantus_n before_o geber_n his_o time_n write_v 13._o book_n thereof_o of_o which_o six_o be_v yet_o extant_a and_o i_o have_v they_o to_o 1550._o use_v of_o the_o famous_a mathematicien_fw-fr and_o my_o great_a friend_n petrus_n monâ—aureus_n and_o second_o the_o very_a name_n be_v algiebar_v and_o not_o algebra_n as_o by_o the_o arabien_fw-fr avicen_n may_v be_v prove_v who_o have_v these_o precise_a word_n in_o latin_a by_o andrea_n alpagus_n most_o perfect_a in_o the_o arabik_a tongue_n so_o translate_v scientia_fw-la faciendi_fw-la algiebar_n &_o almachabel_n i._o scientia_fw-la inveniendi_fw-la numerum_fw-la ignotum_fw-la per_fw-la additionem_fw-la numeri_fw-la &_o divisionem_fw-la &_o aequationem_fw-la which_o be_v to_o say_v the_o science_n of_o work_n algiebar_n and_o almachabel_n that_o be_v the_o science_n of_o find_v a_o unknown_a number_n by_o add_v of_o a_o number_n &_o division_n &_o aequation_n here_o have_v you_o the_o name_n and_o also_o the_o principal_a part_n of_o the_o rule_n touch_v to_o name_v it_o the_o rule_n or_o art_n of_o aequation_n do_v signify_v the_o middle_a part_n and_o the_o state_n of_o the_o rule_n this_o rule_n have_v his_o peculiar_a character_n and_o the_o principal_a part_n of_o arithmetic_n to_o it_o appertain_v do_v differe_o from_o the_o other_o arithmetical_a operation_n this_o arithmetic_n have_v number_n simple_a compound_v mix_v and_o fraction_n accordingly_z this_o rule_n and_o arithmetic_n of_o algiebar_n be_v so_o profound_a so_o general_a and_o so_o in_o manner_n contain_v the_o whole_a power_n of_o number_n application_n 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learn_v and_o exercise_v the_o excellent_a science_n of_o arithmetic_n and_o first_o who_o near_o at_o hand_n can_v be_v a_o better_a witness_n of_o the_o fruit_n receive_v by_o arithmetic_n than_o all_o kind_n of_o merchant_n though_o not_o all_o alike_o either_o need_n it_o or_o use_v it_o how_o can_v they_o forbear_v the_o use_n and_o help_v of_o the_o rule_n call_v the_o golden_a rule_n simple_n and_o compoundeâ—_n both_o forward_a and_o backward_o how_o may_v they_o miss_v arithmetical_a help_n in_o the_o rule_n of_o felowshypâ—_n either_o without_o time_n or_o with_o timer_n and_o between_o the_o merchant_n &_o his_o â—actorâ—_n the_o rulâ—â—_n of_o baâ—tering_v in_o ware_n only_o or_o part_n in_o ware_n and_o part_n in_o money_n will_v they_o glad_o want_v our_o merchant_n venture_n and_o travayler_n over_o sea_n how_o can_v they_o order_v their_o do_n just_o and_o without_o loss_n unleast_o certaâ—ne_n and_o general_a rule_n for_o exchange_n of_o money_n and_o rechaunge_n be_v
thereof_o be_v 〈◊〉_d this_o iâ—_z also_o to_o be_v note_v that_o of_o line_n some_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o and_o some_o be_v commensurable_a the_o one_o to_o the_o other_o in_o power_n of_o line_n commensurable_a in_o length_n the_o one_o to_o the_o other_o be_v give_v a_o example_n in_o the_o declaration_n of_o the_o first_o definition_n namely_o the_o line_n a_o and_o b_o which_o be_v commensurable_a in_o length_n one_o and_o the_o self_n measure_n namely_o the_o line_n c_o measure_v the_o length_n of_o either_o of_o they_o of_o the_o other_o kind_n be_v give_v this_o definition_n here_o set_v for_o the_o open_n of_o which_o take_v this_o example_n let_v there_o be_v a_o certain_a line_n namely_o the_o line_n bc_o and_o let_v the_o square_n of_o that_o line_n be_v the_o square_n bcde_v suppose_v also_o a_o other_o line_n namely_o the_o line_n fh_o &_o let_v the_o square_v thereof_o be_v the_o square_a fhik_n and_o let_v a_o certain_a 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here_o define_v to_o be_v absolute_o incommensurable_a these_o thing_n thus_o stand_v it_o may_v easy_o appear_v that_o if_o a_o line_n be_v assign_v and_o lay_v before_o we_o there_o may_v be_v innumerable_a other_o line_n commensurable_a unto_o it_o and_o other_o incommensurable_a unto_o it_o of_o commensurable_a line_n some_o be_v commensurable_a in_o length_n and_o power_n and_o some_o in_o power_n only_o 5_o and_o that_o right_a line_n so_o set_v forth_o be_v call_v a_o rational_a line_n thus_o may_v you_o see_v how_o to_o the_o suppose_a line_n first_o set_v may_v be_v compare_v infinite_a line_n line_n some_o commensurable_a both_o in_o length_n &_o power_n and_o some_o commensurable_a in_o power_n only_o and_o incommensurable_a in_o length_n and_o some_o incommensurable_a both_o in_o power_n &_o in_o length_n and_o this_o first_o line_n so_o set_v whereunto_o and_o to_o who_o square_n the_o other_o line_n and_o their_o square_n be_v compare_v be_v call_v a_o rational_a line_n common_o of_o the_o most_o part_n of_o writer_n line_n but_o some_o there_o be_v which_o mislike_n that_o it_o shall_v be_v call_v a_o rational_a line_n &_o that_o not_o without_o just_a cause_n in_o the_o greek_a copy_n it_o be_v call_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d rete_fw-la which_o signify_v a_o thing_n that_o may_v be_v speak_v &_o express_v by_o word_n a_o thing_n certain_a grant_v and_o appoint_v wherefore_o flussates_n a_o man_n which_o bestow_v great_a travel_n and_o diligence_n in_o restore_v of_o these_o element_n of_o euclid_n leave_v this_o word_n rational_a call_v this_o line_n suppose_v and_o first_o set_v a_o line_n certain_a because_o the_o part_n thereof_o into_o which_o it_o be_v divide_v be_v certain_a certain_a and_o know_v and_o may_v be_v express_v by_o voice_n and_o also_o be_v count_v by_o number_n other_o line_n be_v to_o this_o line_n incommensurable_a who_o part_n be_v not_o distinct_o know_v but_o be_v uncertain_a nor_o can_v be_v express_v by_o name_n nor_o assign_v by_o number_n which_o be_v of_o other_o man_n call_v irrational_a he_o call_v uncertain_a and_o surd_a line_n petrus_n montaureus_n although_o he_o do_v not_o very_o well_o like_a of_o the_o name_n yet_o he_o alter_v it_o not_o but_o use_v it_o in_o all_o his_o book_n likewise_o will_v we_o do_v here_o for_o that_o the_o word_n have_v be_v and_o be_v so_o universal_o receive_v and_o therefore_o will_v we_o use_v the_o same_o name_n and_o call_v it_o a_o rational_a line_n for_o it_o be_v not_o so_o great_a a_o matter_n what_o name_n we_o geve_v to_o thing_n so_o that_o we_o full_o understand_v the_o thing_n which_o the_o name_n signify_v this_o rational_a line_n thus_o here_o define_v be_v the_o ground_n and_o foundation_n of_o all_o the_o proposition_n almost_o of_o this_o whole_a ten_o book_n book_n and_o chief_o from_o the_o ten_o proposition_n forward_o so_o that_o unless_o you_o first_o place_n this_o rational_a line_n and_o have_v a_o special_a and_o continual_a 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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
quadrates_n but_o all_o other_o kind_n of_o rectiline_a figure_n plain_a plait_n &_o superficiese_n what_o so_o ever_o so_o that_o if_o any_o such_o figure_n be_v commensurable_a unto_o that_o rational_a squareâ—_n it_o be_v also_o rational_a as_o suppose_v that_o the_o square_a of_o the_o rational_a line_n which_o be_v also_o rational_a be_v abcd_o suppose_v 〈◊〉_d so_o some_o other_o square_a as_o the_o square_a efgh_o to_o be_v commensurable_a to_o the_o same_o they_o be_v the_o square_a efgh_o also_o rational_a so_o also_o if_o the_o rectiline_a figure_n klmn_n which_o be_v a_o figure_n on_o the_o one_o side_n long_o be_v commensurable_a unto_o the_o say_a square_a as_o be_v suppose_v in_o this_o exampleâ—_n it_o be_v also_o a_o rational_a superficies_n and_o so_o of_o all_o other_o superficiese_n 10_o such_o which_o be_v incommensurable_a unto_o it_o be_v irrational_a definition_n where_o it_o be_v say_v in_o this_o definition_n such_o which_o be_v incommensurable_a it_o be_v general_o to_o be_v take_v as_o be_v this_o word_n commensurable_a in_o the_o definition_n before_o for_o all_o superficiese_n whether_o they_o be_v square_n or_o figure_n on_o the_o one_o side_n long_o or_o otherwise_o what_o manner_n of_o right_n line_v figure_n so_o ever_o it_o be_v if_o they_o be_v incommensurable_a unto_o the_o rational_a square_n suppose_v they_o be_v they_o irrational_a as_o let_v thâ—_z square_v abcd_o be_v the_o square_n of_o the_o suppose_a rational_a line_n which_o square_n therefore_o be_v also_o rational_a suppose_v also_o also_o a_o other_o square_a namely_o the_o square_a e_o suppose_v also_o any_o other_o figure_n as_o for_o example_n sake_n a_o figure_n of_o one_o side_n long_o which_o let_v be_v f_o now_o if_o the_o square_a e_o and_o the_o figure_n fletcher_n be_v both_o incommensurable_a to_o the_o rational_a square_n abcd_o then_o be_v 〈◊〉_d of_o these_o figure_n e_o &_o fletcher_n irrational_a and_o so_o of_o other_o 11_o and_o these_o line_n who_o powere_n they_o be_v be_v irrational_a if_o they_o be_v square_n then_o be_v their_o side_n irrational_a if_o they_o be_v not_o square_n definition_n but_o some_o other_o rectiline_a figure_n then_o shall_v the_o line_n who_o square_n be_v equal_a to_o these_o rectiline_a figure_n be_v irrational_a suppose_v that_o the_o rational_a square_n be_v abcd._n suppose_v also_o a_o other_o square_a namely_o the_o square_a e_o which_o let_v be_v incommensurable_a to_o the_o rational_a square_n &_o therefore_o be_v it_o irrational_a and_o let_v the_o side_n or_o line_n which_o produce_v this_o square_n be_v the_o line_n fg_o then_o shall_v the_o line_n fg_o by_o this_o definition_n be_v a_o irrational_a line_n because_o it_o be_v the_o side_n of_o a_o irrational_a square_n let_v also_o the_o figure_n h_o be_v a_o figure_n on_o the_o one_o side_n long_o which_o may_v be_v any_o other_o rectiline_a figure_n rectangle_v or_o not_o rectangle_v triangle_n pentagone_fw-mi trapezite_fw-mi or_o what_o so_o ever_o else_o be_v incommensurable_a to_o the_o rational_a square_n abcd_o then_o because_o the_o figure_n h_o be_v not_o a_o square_a it_o have_v no_o side_n or_o root_n to_o produce_v it_o yet_o may_v there_o be_v a_o square_n make_v equal_a unto_o it_o for_o that_o all_o such_o figure_n may_v be_v reduce_v into_o triangle_n and_o so_o into_o square_n by_o the_o 14._o of_o the_o second_o suppose_v that_o the_o square_a queen_n be_v equal_a to_o the_o irrational_a figure_n h._n the_o side_n of_o which_o figure_n q_n let_v be_v the_o line_n kl_o then_o shall_v the_o line_n kl_o be_v also_o a_o irrational_a line_n because_o the_o power_n or_o square_v thereof_o be_v equal_a to_o the_o irrational_a figure_n h_o and_o thus_o conceive_v of_o other_o the_o like_a these_o irrational_a line_n and_o figure_n be_v the_o chief_a matter_n and_o subject_n which_o be_v entreat_v of_o in_o all_o this_o ten_o book_n the_o knowledge_n of_o which_o be_v deep_a and_o secret_a and_o pertain_v to_o the_o high_a and_o most_o worthy_a part_n of_o geometry_n wherein_o stand_v the_o pith_n and_o marry_v of_o the_o hole_n science_n the_o knowledge_n hereof_o bring_v light_n to_o all_o the_o book_n follow_v with_o out_o which_o they_o be_v hard_a and_o can_v be_v at_o all_o understand_v and_o for_o the_o more_o plainness_n you_o shall_v note_v that_o of_o irrational_a line_n there_o be_v diâ—ers_n sort_n and_o kind_n but_o they_o who_o name_n be_v set_v in_o a_o table_n here_o follow_v and_o be_v in_o number_n 13._o be_v the_o chief_a and_o in_o this_o ten_o book_n sufficient_o for_o euclides_n principal_a purpose_n discourse_v on_o a_o medial_a line_n a_o binomial_a line_n a_o first_o bimedial_a line_n a_o second_o bimedial_a line_n a_o great_a line_n a_o line_n contain_v in_o power_n a_o rational_a superficies_n and_o a_o medial_a superficies_n a_o line_n contain_v in_o power_n two_o medial_a superficiece_n a_o residual_a line_n a_o first_o medial_a residual_a line_n a_o second_o medial_a residual_a line_n a_o less_o line_n a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a of_o all_o which_o kind_n the_o definition_n together_o with_o there_o declaration_n shall_v be_v set_v here_o after_o in_o their_o due_a place_n ¶_o the_o 1._o theorem_a the_o 1._o proposition_n two_o unequal_a magnitude_n be_v give_v if_o from_o the_o great_a be_v take_v away_o more_o than_o the_o half_a and_o from_o the_o residue_n be_v again_o take_v away_o more_o than_o the_o half_a and_o so_o be_v do_v still_o continual_o there_o shall_v at_o length_n be_v leave_v a_o certain_a magnitude_n lesser_a than_o the_o less_o of_o the_o magnitude_n first_o give_v svppose_v that_o there_o be_v two_o unequal_a magnitude_n ab_fw-la and_o c_o of_o which_o let_v ab_fw-la be_v the_o great_a then_o i_o say_v that_o if_o from_o ab_fw-la be_v take_v away_o more_o than_o the_o half_a construction_n and_o from_o the_o residue_n be_v take_v again_o more_o than_o the_o half_a and_o so_o still_o continual_o there_o shall_v at_o the_o length_n be_v leave_v a_o certain_a magnitude_n lesser_a than_o the_o less_o magnitude_n give_v namely_o then_o c._n for_o forasmuch_o as_o c_z be_v the_o less_o magnitude_n therefore_o c_z may_v be_v so_o multiply_v that_o at_o the_o length_n it_o will_v be_v great_a than_o the_o magnitude_n ab_fw-la by_o the_o 5._o definition_n of_o the_o five_o book_n demonstration_n let_v it_o be_v so_o multiply_v and_o let_v the_o multiplex_n of_o c_z great_a than_o ab_fw-la be_v de._n and_o divide_v de_fw-fr into_o the_o part_n equal_a unto_o c_o which_o let_v be_v df_o fg_o and_o ge._n and_o from_o the_o magnitude_n ab_fw-la take_v away_o more_o than_o the_o half_a which_o let_v be_v bh_o and_o again_o from_o ah_o take_v away_o more_o than_o the_o half_a which_o let_v be_v hk_a and_o so_o do_v continual_o until_o the_o division_n which_o be_v in_o the_o magnitude_n ab_fw-la be_v equal_a in_o multitude_n unto_o the_o division_n which_o be_v in_o the_o magnitude_n de._n so_o that_o let_v the_o division_n ak_o kh_o and_o hb_o be_v equal_a in_o multitude_n unto_o the_o division_n df_o fg_o and_o ge._n and_o forasmuch_o as_o the_o magnitude_n de_fw-fr be_v great_a than_o the_o magnitude_n ab_fw-la and_o from_o de_fw-fr be_v take_v away_o less_o than_o the_o half_a that_o be_v eglantine_n which_o detraction_n or_o take_v away_o be_v understand_v to_o be_v do_v by_o the_o former_a division_n of_o the_o magnitude_n de_fw-fr into_o the_o part_n equal_a unto_o c_o for_o as_o a_o magnitude_n be_v by_o multiplication_n increase_v so_o be_v it_o by_o division_n diminish_v and_o from_o ab_fw-la be_v take_v away_o more_o than_o the_o half_a that_o be_v bh_o therefore_o the_o residue_n gd_o be_v great_a than_o the_o residue_n ha_o which_o thing_n be_v most_o true_a and_o most_o easy_a to_o conceive_v if_o we_o remember_v this_o principle_n that_o the_o residue_n of_o a_o great_a magnitude_n after_o the_o take_v away_o of_o the_o half_a or_o less_o than_o the_o half_a be_v ever_o great_a than_o the_o residue_n of_o a_o less_o magnitude_n after_o the_o take_v away_o of_o more_o than_o the_o half_a and_o forasmuch_o as_o the_o magnitude_n gd_o be_v great_a than_o the_o magnitude_n ha_o and_o from_o gd_o be_v take_v away_o the_o half_a that_o be_v gf_o and_o from_o ah_o be_v take_v away_o more_o than_o the_o half_a that_o be_v hk_o therefore_o the_o residue_n df_o be_v great_a than_o the_o residue_n ak_o by_o the_o foresay_a principle_n but_o the_o magnitude_n df_o be_v equal_a unto_o the_o magnitude_n c_o by_o supposition_n wherefore_o also_o the_o magnitude_n c_o be_v great_a than_o the_o magnitude_n ak_o wherefore_o the_o magnitude_n ak_o be_v less_o than_o the_o magnitude_n c._n wherefore_o of_o the_o magnitude_n
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
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
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 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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 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into_o part_n as_o many_o as_o you_o listen_v equal_a or_o unequal_a but_o as_o touch_v breadth_n it_o remain_v indivisible_a as_o the_o line_n ab_fw-la which_o be_v only_o draw_v in_o length_n may_v be_v divide_v in_o the_o point_n c_o equal_o or_o in_o the_o point_n d_o unequal_o and_o so_o into_o as_o many_o part_n as_o you_o listen_v there_o be_v also_o of_o diverse_a other_o give_v other_o definition_n of_o a_o line_n as_o these_o which_o follow_v line_n a_o line_n be_v the_o move_n of_o a_o point_n as_o the_o motion_n or_o draught_n of_o a_o pin_n or_o a_o pen_n to_o your_o sense_n make_v a_o line_n other_o again_o a_o line_n be_v a_o magnitude_n have_v one_o only_a space_n or_o dimension_n namely_o length_n want_v breadth_n and_o thickness_n line_n 3_o the_o end_n or_o limit_n of_o a_o line_n be_v point_n for_o a_o line_n have_v his_o beginning_n from_o a_o point_n and_o likewise_o end_v in_o a_o point_n so_o that_o by_o this_o also_o it_o be_v manifest_a that_o point_n for_o 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of_o the_o sun_n say_v astronomer_n then_o happen_v when_o the_o sun_n the_o moon_n &_o our_o eye_n be_v in_o one_o right_a line_n for_o the_o moon_n then_o be_v in_o the_o midst_n between_o we_o and_o the_o sun_n cause_v it_o to_o be_v darken_v diverse_a other_o define_v a_o right_a line_n diverse_o definition_n as_o follow_v a_o right_a line_n be_v that_o which_o stand_v firm_a between_o his_o extreme_n other_o again_o a_o right_a line_n be_v that_o which_o with_o a_o other_o line_n of_o like_a form_n can_v make_v a_o figure_n again_o other_o a_o right_a line_n be_v that_o which_o have_v not_o one_o part_n in_o a_o plain_a superficies_n and_o a_o other_o erect_v on_o high_a again_o other_o a_o right_a line_n be_v that_o all_o who_o part_n agree_v together_o with_o all_o his_o other_o part_n again_o other_o a_o right_a line_n be_v that_o who_o extreme_n abide_v can_v be_v alter_v euclid_n do_v not_o here_o define_v a_o crooked_a line_n line_n for_o it_o need_v not_o it_o may_v easy_o be_v understand_v by_o the_o definition_n of_o a_o right_a line_n for_o every_o contrary_n be_v well_o manifest_v &_o set_v forth_o by_o his_o contrary_n one_o crooked_a line_n may_v be_v more_o crooked_a than_o a_o other_o and_o from_o one_o point_n to_o a_o other_o may_v be_v draw_v infinite_a crooked_a line_n but_o one_o right_a line_n can_v be_v right_a than_o a_o other_o and_o therefore_o from_o one_o point_n to_o a_o other_o there_o may_v be_v draw_v but_o one_o tight_a line_n as_o by_o figure_n above_o set_n you_o may_v see_v 5_o a_o superficies_n be_v that_o which_o have_v only_a length_n and_o breadth_n superficies_n a_o superficies_n be_v the_o second_o kind_n of_o quantity_n and_o to_o it_o be_v attribute_v two_o dimension_n namely_o length_n and_o breadth_n way_n as_o in_o the_o
diameter_n be_v double_a to_o that_o square_n who_o diameter_n it_o be_v corollary_n the_o 34._o theorem_a the_o 48._o proposition_n if_o the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n of_o a_o triangle_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o two_o other_o side_n of_o the_o same_o triangle_n the_o angle_n comprehend_v under_o those_o two_o other_o side_n be_v a_o right_a angle_n svppose_v that_o abc_n be_v a_o triangle_n and_o let_v the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n there_o namely_o of_o the_o side_n bc_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o side_n basilius_n and_o ac_fw-la then_o i_o say_v that_o the_o angle_n bac_n be_v a_o right_a angle_n raise_v up_o by_o the_o 11._o proposition_n from_o the_o point_n a_o unto_o the_o right_a line_n ac_fw-la a_o perpendicular_a line_n ad._n and_o by_o the_o third_o proposition_n unto_o the_o line_n ab_fw-la put_v a_o equal_a line_n ad._n and_o by_o 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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 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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 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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
triangle_n it_o can_v not_o fall_v within_o nor_o in_o acuteangle_a triangle_n without_o for_o then_o the_o outward_a angle_n of_o a_o triangle_n shall_v be_v less_o than_o the_o inward_a and_o opposite_a angle_n which_o be_v contrary_a to_o the_o 16._o of_o the_o first_o triangle_n and_o this_o be_v to_o be_v note_v that_o although_o proper_o a_o acuteangle_n triangle_n by_o the_o definition_n thereof_o give_v in_fw-ge the_o first_o book_n be_v that_o triangle_n who_o angle_n be_v all_o acute_a yet_o forasmuch_o as_o there_o be_v no_o triangle_n but_o that_o it_o have_v a_o acute_a angle_n this_o proposition_n be_v to_o be_v understand_v &_o be_v true_a general_o in_o all_o kind_n of_o triangle_n whatsoever_o and_o may_v be_v declare_v by_o they_o as_o you_o may_v easy_o prove_v the_o 2._o problem_n the_o 14._o proposition_n unto_o a_o rectiline_a figure_n give_v to_o make_v a_o square_a equal_a svppose_v that_o the_o rectiline_a figure_n give_v be_v a._n it_o be_v require_v to_o make_v a_o square_a equal_a unto_o the_o rectiline_a figure_n a._n construction_n make_v by_o the_o 45._o of_o the_o first_o unto_o the_o rectiline_a figure_n a_o a_o equal_a rectangle_n parallelogram_n bcde_v now_o if_o the_o line_n be_v be_v equal_a unto_o the_o line_n ed_z then_z be_v the_o thing_n do_v which_o be_v require_v for_o unto_o the_o rectiline_a figure_n a_o be_v make_v a_o equal_a square_a bd._n but_o if_o not_o one_o of_o these_o line_n be_v &_o be_v ed_z the_o great_a let_v be_v be_v the_o great_a and_o let_v it_o be_v produce_v unto_o the_o point_fw-fr f._n and_o by_o the_o 3._o of_o the_o first_o put_v unto_o ed_z a_o equal_a line_n ef._n and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n bf_o into_o two_o equal_a part_n in_o the_o point_n g._n and_o make_v the_o centre_n the_o point_n g_o and_o the_o space_n gb_o or_o gf_o describe_v a_o semicircle_n bhf_n and_o by_o the_o 2._o petition_n extend_v the_o line_n de_fw-fr unto_o the_o point_n h._n demonstration_n and_o by_o the_o 1._o petition_n draw_v a_o line_n from_o g_z to_o h._n and_o forasmuch_o as_o the_o right_a line_n fb_o be_v divide_v into_o two_o equal_a part_n in_o the_o point_n g_o and_o into_o two_o unequal_a part_n in_o the_o point_n e_o therefore_o by_o the_o 5._o of_o the_o second_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n be_v and_o of_o together_o with_o the_o square_n which_o be_v make_v of_o the_o line_n eglantine_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n gf_o but_o the_o line_n gf_o be_v equal_a unto_o the_o line_n gh_o wherefore_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n be_v and_o of_o together_o with_o the_o square_n which_o be_v make_v of_o the_o line_n ge_z be_v equal_a to_o square_v which_o be_v make_v of_o the_o line_n gh_o but_o unto_o the_o square_n which_o be_v make_v of_o the_o line_n gh_o be_v equal_a the_o square_n which_o be_v make_v of_o the_o line_n he_o and_o ge_z by_o the_o 47._o of_o the_o first_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n be_v and_o of_o together_o with_o the_o square_v which_o be_v make_v of_o ge_z be_v equal_a to_o the_o square_n which_o be_v make_v of_o he_o and_o ge._n take_v away_o the_o square_a of_o the_o line_n eglantine_n common_a to_o they_o both_o wherefore_o the_o rectangle_n figure_n contain_v under_o the_o line_n be_v &_o of_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n eh_o but_o that_o which_o be_v contain_v under_o the_o line_n be_v and_o of_o be_v the_o parallelogram_n bd_o for_o the_o line_n of_o be_v equal_a unto_o the_o line_n ed._n wherefore_o the_o parallelogram_n bd_o be_v equal_a to_o the_o square_v which_o be_v make_v of_o the_o line_n he._n but_o the_o parallelogram_n bd_o be_v equal_a unto_o the_o rectiline_a figure_n a._n wherefore_o the_o rectiline_a figure_n a_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n he._n wherefore_o unto_o the_o rectiline_a figure_n give_v a_o be_v make_v a_o equal_a square_n describe_v of_o the_o line_n eh_o which_o be_v require_v to_o be_v do_v ¶_o the_o end_n of_o the_o second_o book_n of_o euclides_n element_n ¶_o the_o three_o book_n of_o euclides_n element_n book_n this_o three_o book_n of_o euclid_n entreat_v of_o the_o most_o perfect_a figure_n which_o be_v a_o circle_n wherefore_o it_o be_v much_o more_o to_o be_v esteem_v then_o the_o two_o book_n go_v before_o in_o which_o he_o do_v set_v forth_o the_o most_o simple_a propriety_n of_o rightline_v figure_n for_o science_n take_v their_o dignity_n of_o the_o worthiness_n of_o the_o matter_n that_o they_o entreat_v of_o but_o of_o all_o figure_n the_o circle_n be_v of_o most_o absolute_a perfection_n who_o propriety_n and_o passion_n be_v here_o set_v forth_o and_o most_o certain_o demonstrate_v here_o also_o be_v entreat_v of_o right_a line_n subtend_v to_o arke_n in_o circle_n also_o of_o angle_n set_v both_o at_o the_o circumference_n and_o at_o the_o centre_n of_o a_o circle_n and_o of_o the_o variety_n and_o difference_n of_o they_o wherefore_o the_o read_v of_o this_o book_n be_v very_o profitable_a to_o the_o attain_v to_o the_o knowledge_n of_o chord_n and_o arke_n it_o teach_v moreover_o which_o be_v circle_n contingent_a and_o which_o be_v cut_v the_o one_o the_o other_o and_o also_o that_o the_o angle_n of_o contingence_n be_v the_o least_o of_o all_o acute_a rightline_v angle_n and_o that_o the_o diameter_n in_o a_o circle_n be_v the_o long_a line_n that_o can_v be_v draw_v in_o a_o circle_n far_o in_o it_o may_v we_o learn_v how_o three_o point_n be_v give_v how_o soever_o so_o that_o they_o be_v not_o set_v in_o a_o right_a line_n may_v be_v draw_v a_o circle_n pass_v by_o they_o all_o three_o again_o how_o in_o a_o solid_a body_n as_o in_o a_o sphere_n cube_n or_o such_o like_a may_v be_v find_v the_o two_o opposite_a point_n which_o be_v a_o thing_n very_o necessary_a and_o commodious_a chief_o for_o those_o that_o shall_v make_v instrument_n serve_v to_o astronomy_n and_o other_o art_n definition_n definition_n equal_a circle_n be_v such_o who_o diameter_n be_v equal_a or_o who_o line_n draw_v from_o the_o centre_n be_v equal_a the_o circle_n a_o and_o b_o be_v equal_a if_o their_o diameter_n namely_o of_o and_o cd_o be_v equal_a or_o if_o their_o semidiameter_n which_o be_v line_n draw_v from_o the_o centre_n to_o the_o circumferenceâ—_n namely_o of_o and_o bd_o be_v equal_a by_o this_o also_o be_v know_v the_o definition_n of_o unequal_a circle_n circle_n circle_n who_o diameter_n or_o semidiameter_n be_v unequal_a be_v also_o unequal_a and_o that_o circled_a which_o have_v the_o great_a diameter_n or_o semidiameter_n be_v the_o great_a circle_n and_o that_o circle_v which_o have_v the_o less_o diameter_n or_o semidiameter_n be_v the_o less_o circle_n a_o right_a line_n be_v say_v to_o touch_v a_o circle_n definition_n which_o touch_v the_o circle_n and_o be_v produce_v cut_v it_o not_o as_o the_o right_a line_n of_o draw_v from_o the_o point_n e_o and_o pass_a by_o a_o point_n of_o the_o circle_n namely_o by_o the_o point_n g_o to_o the_o point_n fletcher_n only_o touch_v the_o circle_n gh_o and_o cut_v it_o not_o nor_o enter_v within_o it_o for_o a_o right_a line_n enter_v within_o a_o circle_n cut_v and_o divide_v the_o circle_n as_o the_o right_a line_n kl_o divide_v and_o cut_v the_o circle_n klm_n and_o enter_v within_o it_o and_o therefore_o touch_v it_o in_o two_o place_n but_o a_o right_a line_n touch_v a_o circle_n which_o be_v common_o call_v a_o contingent_a line_n line_n touch_v the_o circle_n only_o in_o one_o point_n circle_n be_v say_v to_o touch_v the_o one_o the_o other_o definition_n which_o touch_v the_o one_o the_o other_o cut_v not_o the_o one_o the_o other_o as_o the_o two_o circle_n ab_fw-la and_o bc_o touch_v the_o one_o the_o other_o for_o their_o circumference_n touch_v together_o in_o the_o point_n b._n but_o neither_o of_o they_o cut_v or_o divide_v the_o other_o neither_o do_v any_o part_n of_o the_o one_o enter_v within_o the_o other_o only_o and_o such_o a_o touch_n of_o circle_n be_v ever_o in_o one_o point_n only_o which_o point_n only_o be_v common_a to_o they_o both_o as_o the_o point_n b_o be_v in_o the_o conference_n of_o the_o circle_n ab_fw-la and_o also_o 〈…〉_o the_o â—â—â—â—â—ference_n of_o the_o circle_n bc._n way_n circle_n may_v touch_v together_o two_o manner_n of_o way_n either_o outward_o the_o one_o whole_o without_o the_o other_o or_o else_o the_o one_o be_v contain_v within_o the_o other_o as_o the_o circle_n de_fw-fr and_o df_o of_o which_o the_o one_o de_fw-fr contain_v the_o other_o namely_o df_o and_o touch_v the_o one_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
the_o point_n of_o the_o touch_n namely_o b_o be_v draw_v into_o the_o circle_n a_o certain_a right_a line_n bc_o therefore_o by_o the_o 32._o of_o the_o three_o the_o angle_n fbc_n be_v equal_a to_o the_o angle_n bac_n which_o be_v in_o the_o alternate_a segment_n but_o the_o angle_n fbc_n be_v equal_a to_o the_o angle_n d._n wherefore_o the_o angle_n bac_n which_o consist_v in_o the_o segment_n bac_n be_v equal_a to_o the_o angle_n d._n wherefore_o from_o the_o circle_n give_v abc_n be_v cut_v away_o a_o segment_n bac_n which_o contain_v a_o angle_n equal_a to_o the_o rectiline_a angle_n give_v which_o be_v require_v to_o be_v do_v the_o 29._o theorem_a the_o 35._o proposition_n if_o in_o a_o circle_n two_o right_a line_n do_v cut_v the_o one_o the_o other_o the_o rectangle_n parallelogram_n comprehend_v under_o the_o segment_n or_o part_n of_o the_o one_o line_n be_v equal_a to_o the_o rectangle_n parallelogram_n comprehend_v under_o the_o segment_n or_o part_n of_o the_o other_o line_n let_v the_o circle_n be_v abcd_o and_o in_o it_o let_v these_o two_o right_a line_n ac_fw-la and_o bd_o câ—t_o the_o one_o the_o other_o in_o the_o point_n e._n then_o i_o say_v that_o the_o rectangle_n parallelogram_n contain_v under_o the_o part_n ae_n and_o ec_o be_v equal_a to_o the_o rectangle_n parallelogram_n contain_v under_o the_o part_n de_fw-fr and_o ebb_v case_n for_o if_o the_o line_n ac_fw-la and_o bd_o be_v draw_v by_o the_o centre_n then_o be_v it_o manifest_a that_o for_o as_o much_o as_o the_o line_n ae_n and_o ec_o be_v equal_a to_o the_o line_n de_fw-fr and_o ebb_v by_o the_o definition_n of_o a_o circle_n demonstration_n the_o rectangle_n parallelogram_n also_o contain_v under_o the_o line_n ae_n and_o ec_o be_v equal_a to_o the_o rectangle_n parallelogram_n contain_v under_o the_o line_n de_fw-fr and_o ebb_v but_o now_o suppose_v that_o the_o line_n ac_fw-la and_o db_o be_v not_o extend_v by_o the_o centre_n câ—se_n 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 abcd_o and_o let_v the_o same_o be_v the_o point_n fletcher_n construction_n and_o from_o the_o point_n f_o draw_v to_o the_o right_a line_n ac_fw-la and_o db_o perpendicular_a line_n fg_o and_o fh_o by_o the_o 12._o of_o the_o first_o and_o draw_v these_o right_a line_n fb_o fc_o and_o fe_o and_o forasmuch_o as_o a_o certain_a right_a line_n fg_o draw_v by_o the_o centre_n demonstration_n cut_v a_o certain_a right_a line_n ac_fw-la not_o draw_v by_o the_o centre_n in_o such_o sort_n that_o it_o make_v right_a angle_n it_o therefore_o divide_v the_o line_n ac_fw-la into_o two_o equal_a part_n by_o the_o 3._o of_o the_o three_o wherefore_o the_o line_n agnostus_n be_v equal_a to_o the_o line_n gc_o and_o forasmuch_o as_o the_o right_a line_n ac_fw-la be_v divide_v into_o two_o equal_a part_n in_o the_o point_n g_o and_o into_o two_o unequal_a part_n in_o the_o point_n e_o therefore_o by_o the_o 5._o of_o the_o second_o the_o rectangle_n parallelogram_n contain_v under_o the_o line_n ae_n and_o ec_o together_o with_o the_o square_n of_o the_o line_n eglantine_n be_v equal_a to_o the_o square_n of_o the_o line_n gc_o put_v the_o square_n of_o the_o line_n gf_o common_a to_o they_o both_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ae_n &_o ec_o together_o with_o the_o square_n of_o the_o line_n eglantine_n and_o gf_o be_v equal_a to_o the_o square_n of_o the_o line_n gf_o &_o gc_o but_o unto_o the_o square_n of_o the_o line_n eglantine_n &_o gf_o be_v equal_a the_o square_a of_o the_o line_n fe_o by_o the_o 47._o of_o the_o first_o and_o to_o the_o square_n of_o the_o line_n gc_o and_o gf_o be_v equal_a the_o square_n of_o the_o line_n fc_o by_o the_o same_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ec_o together_o with_o the_o square_n of_o the_o line_n fe_o be_v equal_a to_o the_o square_n of_o the_o line_n fc_o but_o the_o line_n fc_o be_v equal_a to_o the_o line_n fb_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o eâ—_n together_o with_o the_o square_n of_o the_o line_n fe_o be_v equal_a to_o the_o square_n of_o the_o line_n fb_o and_o by_o the_o same_o demonstration_n that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o ebb_v together_o with_o the_o square_n of_o the_o line_n fe_o be_v equal_a to_o the_o square_n of_o the_o line_n fb_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ec_o together_o with_o the_o square_n of_o the_o line_n of_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o ebb_v together_o with_o the_o square_n of_o the_o line_n ef._n take_v away_o the_o square_a of_o the_o line_n fe_o which_o be_v common_a to_o they_o both_o wherefore_o the_o rectangle_n parallelogram_n remain_v which_o be_v contain_v under_o the_o line_n ae_n and_o ec_o be_v equal_a to_o the_o rectangle_n parallelogram_n remain_v which_o be_v contain_v under_o the_o line_n de_fw-fr and_o ebb_v if_o therefore_o in_o a_o circle_n two_o right_a line_n do_v cut_v the_o one_o the_o other_o the_o rectangle_n parallelogram_n comprehend_v under_o the_o segment_n or_o part_n of_o the_o one_o line_n be_v equal_a to_o the_o rectangle_n parallelogram_n comprehend_v under_o the_o segment_n or_o part_n of_o the_o other_o line_n which_o be_v require_v to_o be_v demonstrate_v in_o this_o proposition_n be_v three_o case_n proposition_n for_o either_o both_o the_o line_n pass_v by_o the_o centre_n or_o neither_o of_o they_o pass_v by_o the_o centre_n or_o the_o one_o pass_v by_o the_o centre_n and_o the_o other_o not_o the_o two_o first_o case_n be_v before_o demonstrate_v amongst_o all_o the_o proposition_n in_o this_o three_o book_n doubtless_o this_o be_v one_o of_o the_o chief_a for_o it_o set_v forth_o unto_o we_o the_o wonderful_a nature_n of_o a_o circle_n so_o that_o by_o it_o may_v be_v do_v many_o goodly_a conclusion_n in_o geometry_n as_o shall_v afterward_o be_v declare_v when_o occasion_n shall_v serve_v the_o 30._o theorem_a the_o 36._o proposition_n if_o without_o a_o circle_n be_v take_v a_o certain_a point_n and_o from_o that_o point_n be_v draw_v to_o the_o circle_n two_o right_a line_n so_o that_o the_o one_o of_o they_o do_v cut_v the_o circle_n and_o the_o other_o do_v touch_v the_o circle_n the_o rectangle_n parallelogram_n which_o be_v comprehend_v under_o the_o whole_a right_a line_n which_o cut_v the_o circle_n and_o that_o portion_n of_o the_o same_o line_n that_o lie_v between_o the_o point_n and_o the_o utter_a circumference_n of_o the_o circle_n be_v equal_a to_o the_o square_n make_v of_o the_o line_n that_o touch_v the_o circle_n svppose_v that_o the_o circle_n be_v abc_n and_o without_o the_o same_o circle_n take_v any_o point_n at_o all_o adventure_n and_o let_v the_o same_o be_v d._n construction_n and_o from_o the_o point_n d_o let_v there_o be_v draw_v to_o the_o circle_n two_o right_a line_n dca_n and_o db_o and_o let_v the_o right_a line_n dca_n cut_v the_o circle_n acb_o in_o the_o point_n c_o and_o let_v the_o right_a line_n bd_o touch_v the_o same_o then_o i_o say_v that_o the_o rectangle_n parallelogram_n contain_v under_o the_o line_n ad_fw-la and_o dc_o proposition_n be_v equal_a to_o the_o square_n of_o the_o line_n bd._n now_o the_o line_n dca_n be_v either_o draw_v by_o the_o centre_n or_o not_o first_o let_v it_o be_v draw_v by_o the_o centre_n case_n and_o by_o the_o first_o of_o the_o three_o let_v the_o point_n fletcher_n be_v the_o centre_n of_o the_o circle_n abc_n and_o draw_v a_o line_n from_o fletcher_n to_o b._n wherefore_o the_o angle_n fbd_v be_v a_o right_a angle_n demonstration_n and_o for_o asmuch_o as_o the_o right_a line_n ac_fw-la be_v divide_v into_o two_o equal_a part_n in_o the_o point_n fletcher_n and_o unto_o it_o be_v add_v direct_o a_o right_a line_n cd_o therefore_o by_o the_o 6._o of_o the_o second_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o dc_o together_o with_o the_o square_n of_o the_o line_n cf_o be_v equal_a to_o the_o square_n of_o the_o line_n fd._n but_o the_o line_n fc_o be_v equal_a to_o the_o line_n fb_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o dc_o together_o with_o the_o square_n of_o the_o line_n fb_o be_v equal_a to_o the_o square_n of_o the_o line_n fd._n but_o the_o square_a of_o the_o line_n fd_o be_v by_o the_o 47._o of_o the_o first_o equal_a to_o the_o square_n of_o the_o line_n fb_o and_o bd_o for_o the_o angle_n fbd_v be_v a_o right_a angle_n wherefore_o
line_n fb_o favorina_n &_o fe_o and_o for_o asmuch_o as_o the_o angle_n bcd_o be_v equal_a to_o the_o angle_n cde_o &_o the_o half_a of_o the_o angle_n bcd_o be_v the_o angle_n fcd_n and_o likewise_o the_o half_a of_o the_o angle_n cde_o be_v the_o angle_n cdf_n wherefore_o the_o angle_n fcd_n be_v equal_a to_o the_o angle_n fdc_n wherefore_o the_o side_n fc_o be_v equal_a to_o the_o side_n fd._n in_o like_a sort_n also_o may_v it_o be_v prove_v that_o every_o one_o of_o these_o line_n fb_o favorina_n and_o fe_o be_v equal_a to_o every_o one_o of_o these_o line_n fc_a and_o fd._n wherefore_o these_o five_o right_a line_n favorina_n fb_o fc_o fd_o and_o fe_o be_v equal_a the_o one_o to_o the_o other_o wherefore_o make_v the_o centre_n f_o and_o the_o space_n favorina_n or_o fb_o or_o fc_n or_o fd_o or_o fe_o describe_v a_o circle_n and_o it_o will_v pass_v by_o the_o point_n a_o b_o c_o d_o e_o and_o shall_v be_v describe_v â—bout_o the_o five_o angle_a figure_n abcde_o which_o be_v equiangle_n and_o equilater_n let_v this_o circle_n be_v describe_v and_o let_v the_o same_o be_v abcde_o wherefore_o about_o the_o pentagon_n give_v beâ—â—g_v both_o equiangle_n and_o equilater_n be_v describe_v a_o circle_n which_o be_v require_v to_o be_v â—one_n the_o 15._o problem_n the_o 15._o proposition_n in_o a_o circle_n give_v to_o describe_v a_o hexagon_n or_o figure_n of_o six_o angle_n equilater_n and_o equiangle_n svppose_v that_o the_o circle_n give_v be_v abcdef_n it_o be_v require_v in_o the_o circle_n give_v abcde_o to_o describe_v a_o figure_n of_o six_o angle_n of_o equal_a side_n and_o of_o equal_a angle_n dâ—aw_v the_o diameter_n of_o the_o circle_n abcdef_n and_o let_v the_o same_o be_v ad._n and_o by_o the_o first_o of_o the_o three_o take_v the_o câ—nâ—re_n of_o the_o circled_a and_o let_v the_o same_o be_v g._n and_o make_v the_o centre_n d_o and_o the_o space_n dg_o describe_v by_o the_o three_o petition_n a_o circle_n cgeh_a and_o draw_v right_a line_n from_o e_o to_o g_z and_o from_o g_z to_o c_z extend_v they_o to_o the_o point_n b_o and_o f_o of_o the_o circumference_n of_o the_o circle_n give_v and_o draw_v these_o right_a line_n ab_fw-la bc_o cd_o de_fw-fr of_o and_o fa._n then_o i_o say_v that_o abcdef_n be_v a_o hexagon_n figure_n of_o equal_a side_n and_o of_o equal_a angle_n demonstration_n for_o forasmuch_o as_o the_o point_n g_z be_v the_o centre_n of_o the_o circle_n abcdef_n therefore_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_n ge_z be_v equal_a unto_o the_o line_n gd_o again_o forasmuch_o as_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n cgeh_a therefore_o by_o the_o self_n same_o the_o line_n de_fw-fr be_v equal_a unto_o the_o line_n dg_o and_o it_o be_v prove_v that_o the_o line_n ge_z be_v equal_a unto_o the_o line_n gd_o wherefore_o the_o line_n ge_z be_v equal_a unto_o the_o line_n ed_z by_o the_o first_o common_a sentence_n wherefore_o the_o triangle_n egg_v be_v equilater_n and_o biâ—_n three_o angle_n namely_o egg_v gde_v deg_n be_v equal_a the_o one_o to_o the_o other_o and_o forasmuch_o as_o by_o the_o 5._o of_o the_o first_o in_o triangle_n of_o two_o equal_a side_n colmonell_n call_v isosceles_a the_o angle_n at_o the_o base_a be_v equal_a the_o one_o to_o the_o other_o and_o the_o three_o angle_n of_o a_o triangle_n be_v by_o the_o 30._o of_o the_o first_o equal_v â—nto_o two_o right_a angle_n therefore_o the_o angle_n egg_v be_v the_o three_o part_n of_o two_o right_a angle_n and_o in_o like_a sorâ—e_n may_v it_o be_v prove_v that_o the_o angle_n dgc_n be_v the_o three_o part_n of_o two_o right_a angle_n and_o forasmuch_o as_o the_o right_a line_n cg_o stand_v upon_o the_o right_a line_n ebb_n do_v by_o the_o 13._o of_o the_o fiâ—â—t_n make_v the_o two_o side_n angle_n â—gc_n and_o cgâ—_n equal_a to_o two_o right_a angle_n therefore_o the_o angle_n remain_v cgb_n be_v the_o three_o part_n of_o two_o right_a angle_n wherefore_o the_o angle_n egg_v dgc_n and_o cgb_n be_v equal_a the_o one_o to_o the_o other_o wherefore_o their_o head_n angle_n that_o be_v bga_n agf_n and_o fge_n be_v by_o the_o 15._o of_o the_o first_o equal_a to_o these_o angle_n egg_v dgc_n and_o cgb_n wherefore_o these_o six_o angle_n egg_v dgc_n cgb_n bga_n agf_n and_o fge_n be_v equal_a the_o one_o to_o the_o other_o but_o equal_a angle_n consist_v upon_o equal_a circumference_n by_o 〈…〉_o the_o three_o therefore_o these_o six_o circumference_n abâ—_n be_v cdâ—_n deâ—_n efâ—_n and_o â—a_o be_v equal_a the_o one_o to_o the_o other_o but_o under_o equal_a circumference_n be_v â—â—btâ—â—ded_v equal_a right_a line_n by_o the_o 29._o of_o the_o same_o wherefore_o these_o six_o right_a line_n ab_fw-la bc_o cd_o de_fw-fr of_o and_o favorina_n be_v equal_a the_o one_o to_o the_o other_o wherefore_o the_o hexagon_n abcdef_n be_v equilater_n i_o say_v also_o that_o it_o be_v equiangle_n for_o forasmuch_o as_o the_o circumference_n of_o be_v equal_a unto_o the_o circumference_n ed_z add_v the_o circumference_n abcd_o common_a to_o they_o both_o wherefore_o the_o whole_a circumference_n fabcd_n be_v equal_a to_o the_o whole_a circumference_n edcba_n and_o upon_o the_o circumference_n fabcd_n consist_v the_o angle_n feed_v and_o upon_o the_o circumference_n edcba_n consist_v the_o angle_n afe._n wherefore_o the_o angle_n afe_n be_v equal_a to_o the_o angle_n def_n in_o like_a sort_n also_o may_v it_o be_v prove_v that_o the_o rest_n of_o the_o angle_n of_o the_o hexagon_n abcdef_n that_o be_v every_o one_o of_o these_o angle_n fab._n abc_n bcd_o and_o cde_o be_v equal_a to_o every_o one_o of_o these_o angle_n afe_a and_o feed_v wherefore_o the_o hexagon_n figure_n abcdef_n be_v equiangle_n and_o it_o be_v prove_v that_o it_o be_v also_o equilater_n and_o it_o be_v describe_v in_o the_o circle_n abcdef_n wherefore_o in_o the_o circle_n give_v abcdef_n be_v describe_v a_o figure_n of_o six_o angle_n of_o equal_a side_n and_o of_o equal_a angle_n which_o be_v require_v to_o be_v doneâ—_n ¶_o a_o other_o way_n to_o do_v the_o same_o after_o orontius_n suppose_v that_o the_o circle_n give_v be_v abcdef_n in_o which_o first_o let_v there_o be_v describe_v a_o equilater_n and_o equiangle_n triangle_n ace_n by_o the_o second_o of_o this_o book_n orontius_n wherefore_o the_o arke_n abc_n cde_o efa_n be_v by_o the_o 28._o of_o the_o three_o equal_v the_o one_o to_o the_o other_o divide_v every_o one_o of_o those_o three_o arke_n into_o two_o equal_a part_n by_o the_o 30._o of_o the_o same_o in_o the_o point_n b_o d_o &_o f._n and_o draw_v these_o right_a line_n abâ—_n bc_o cd_o de_fw-fr of_o and_o fa._n now_o then_o by_o the_o 2._o definition_n of_o this_o book_n there_o shall_v be_v describe_v in_o the_o circle_n geve_v a_o hexagon_n figure_n abcdef_n which_o must_v needs_o be_v equilater_n for_o that_o every_o one_o of_o the_o arke_n which_o subtend_v the_o side_n thereof_o be_v equal_a the_o one_o to_o the_o other_o i_o say_v also_o that_o it_o be_v equiangle_n for_o every_o angle_n of_o the_o hexagon_n figure_n be_v set_v upon_o equal_a arke_n namely_o upon_o four_o such_o part_n of_o the_o circumference_n whereof_o the_o whole_a circumference_n contain_v six_o wherefore_o the_o angle_n of_o the_o hexagon_n figure_n be_v equal_a the_o one_o to_o the_o other_o by_o the_o 27._o of_o the_o three_o wherefore_o in_o the_o circle_n give_v abcdef_n be_v inscribe_v a_o equilater_n and_o equiangle_n hexagon_n figure_n which_o be_v require_v to_o be_v do_v ¶_o a_o other_o way_n to_o do_v the_o same_o after_o pelitarius_n suppose_v that_o the_o circle_n in_o which_o be_v to_o be_v inscribe_v 〈◊〉_d equilater_n &_o â—â—â—iangle_n hexagon_n figure_n be_v abcde_o pelitarius_n who_o câ—ntre_n let_v be_v f._n and_o from_o the_o centre_n draw_v the_o semidiameter_n fa._n and_o from_o the_o point_n a_o apply_v by_o the_o first_o oâ—â—hys_n book_n the_o line_n â—â—_o equal_a to_o the_o semidiameter_n which_o i_o say_v be_v the_o side_n of_o a_o equilater_n and_o equiangle_n hexagon_n figure_n to_o be_v inscribe_v in_o the_o circle_n abcde_o draw_v â—â—â—ght_n â—ine_o from_o e_o to_o bâ—_n â—_z and_o for_o asmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n faâ—_n â—_z &_o it_o be_v also_o equal_a to_o the_o line_n fb_o therefore_o the_o triangle_n afb_o be_v equilater_n and_o by_o the_o 5._o of_o the_o first_o equiangle_n now_o then_o upon_o the_o centre_n f_o describe_v the_o angle_n bfc_n equal_a to_o the_o angle_n afâ—_n or_o to_o the_o angle_n fba_n which_o be_v all_o one_o by_o draw_v the_o right_a line_n fc_o and_o draw_v a_o line_n from_o b_o to_o c._n and_o for_o asmuch_o as_o the_o angle_n afb_o be_v the_o three_o part_n of_o two_o right_a angle_n by_o the_o
poligonon_n figure_n of_o 24._o side_n likewise_o of_o the_o hexagon_n ab_fw-la and_o of_o the_o pentagon_n ac_fw-la shall_v be_v make_v a_o poligonon_n figure_n of_o 30._o side_n one_o of_o who_o side_n shall_v subtend_v the_o ark_n bc._n for_o the_o denomination_n of_o ab_fw-la which_o be_v 6._o exceed_v the_o denomination_n of_o ac_fw-la which_o be_v 5._o only_a by_o unity_n so_o also_o forasmuch_o as_o the_o denomination_n of_o ab_fw-la which_o be_v 6._o exceed_v the_o denomination_n of_o ae_n which_o be_v 3._o by_o 3._o therefore_o the_o ark_n be_v shall_v contain_v 3._o side_n of_o a_o poligonon_n figure_n of_o .18_o side_n and_o observe_v this_o self_n same_o method_n and_o order_n a_o man_n may_v find_v out_o infinite_a side_n of_o a_o poligonon_n figure_n the_o end_n of_o the_o four_o book_n of_o euclides_n element_n ¶_o the_o five_o book_n of_o euclides_n element_n this_o five_o book_n of_o euclid_n be_v of_o very_o great_a commodity_n and_o use_n in_o all_o geometry_n book_n and_o much_o diligence_n ought_v to_o be_v bestow_v therein_o it_o ought_v of_o all_o other_o to_o be_v thorough_o and_o most_o perfect_o and_o ready_o know_v for_o nothing_o in_o the_o book_n follow_v can_v be_v understand_v without_o it_o the_o knowledge_n of_o they_o all_o depend_v of_o it_o and_o not_o only_o they_o and_o other_o write_n of_o geometry_n but_o all_o other_o science_n also_o and_o art_n as_o music_n astronomy_n perspective_n arithmetic_n the_o art_n of_o account_n and_o reckon_n with_o other_o such_o like_a this_o book_n therefore_o be_v as_o it_o be_v a_o chief_a treasure_n and_o a_o peculiar_a jewel_n much_o to_o be_v account_v of_o it_o entreat_v of_o proportion_n and_o analogy_n or_o proportionality_n which_o pertain_v not_o only_o unto_o line_n figure_n and_o body_n in_o geometry_n but_o also_o unto_o sound_n &_o voice_n of_o which_o music_n entreat_v as_o witness_v boetius_fw-la and_o other_o which_o write_v of_o music_n also_o the_o whole_a art_n of_o astronomy_n teach_v to_o measure_v proportion_n of_o time_n and_o move_n archimedes_n and_o jordan_n with_o other_o write_v of_o weight_n affirm_v that_o there_o be_v proportion_n between_o weight_n and_o weight_n and_o also_o between_o place_n &_o place_n you_o see_v therefore_o how_o large_a be_v the_o use_n of_o this_o five_o book_n wherefore_o the_o definition_n also_o thereof_o be_v common_a although_o hereof_o euclid_n they_o be_v accommodate_v and_o apply_v only_o to_o geometry_n the_o first_o author_n of_o this_o book_n be_v as_o it_o be_v affirm_v of_o many_o one_o eudoxus_n who_o be_v plato_n scholar_n eudoxus_n but_o it_o be_v afterward_o frame_v and_o put_v in_o order_n by_o euclid_n definition_n definition_n a_o part_n be_v a_o less_o magnitude_n in_o respect_n of_o a_o great_a magnitude_n when_o the_o less_o measure_v the_o great_a as_o in_o the_o other_o book_n before_o so_o in_o this_o the_o author_n first_o set_v orderly_a the_o definition_n and_o declaration_n of_o such_o term_n and_o word_n which_o be_v necessary_o require_v to_o the_o entreaty_n of_o the_o subject_n and_o matter_n thereof_o which_o be_v proportion_n and_o comparison_n of_o proportion_n or_o proportionality_n and_o first_o he_o show_v what_o a_o part_n be_v here_o be_v to_o be_v consider_v that_o all_o the_o definition_n of_o this_o five_o book_n be_v general_a to_o geometry_n and_o arithmetic_n and_o be_v true_a in_o both_o art_n even_o as_o proportion_n and_o proportionality_n be_v common_a to_o they_o both_o and_o chief_o appertain_v to_o number_v neither_o can_v they_o apt_o be_v apply_v to_o matter_n of_o geometry_n but_o in_o respect_n of_o number_n and_o by_o number_n yet_o in_o this_o book_n and_o in_o these_o definition_n here_o set_v euclid_n seem_v to_o speak_v of_o they_o only_o geometrical_o as_o they_o be_v apply_v to_o quantity_n continual_a as_o to_o line_n superficiece_n and_o body_n for_o that_o he_o yet_o continue_v in_o geometry_n i_o will_v notwithstanding_o for_o facility_n and_o far_a help_n of_o the_o reader_n declare_v they_o both_o by_o example_n in_o number_n and_o also_o in_o line_n for_o the_o clear_a understanding_n of_o a_o part_n it_o be_v to_o be_v note_v way_n that_o a_o part_n be_v take_v in_o the_o mathematical_a science_n two_o manner_n of_o way_n way_n one_o way_n a_o part_n be_v a_o less_o quantity_n in_o respect_n of_o a_o great_a whether_o it_o measure_v the_o great_a oâ—_n no._n the_o second_o way_n way_n a_o part_n be_v only_o that_o less_o quantity_n in_o respect_n of_o the_o great_a which_o measure_v the_o great_a a_o less_o quantity_n be_v say_v to_o measure_n or_o number_v a_o great_a quantity_n great_a when_o it_o be_v oftentimes_o take_v make_v precise_o the_o great_a quantity_n without_o more_o or_o less_o or_o be_v as_o oftentimes_o take_v from_o the_o great_a as_o it_o may_v there_o remain_v nothing_o as_o suppose_v the_o line_n ab_fw-la to_o contain_v 3._o and_o the_o line_n cd_o to_o contain_v 9_o they_o do_v the_o line_n ab_fw-la measure_n the_o line_n cd_o for_o that_o if_o it_o be_v take_v certain_a time_n namely_o 3._o time_n it_o make_v precise_o the_o line_n cd_o that_o be_v 9_o without_o more_o or_o less_o again_o if_o the_o say_v less_o line_n ab_fw-la be_v take_v from_o the_o great_a cd_o as_o often_o as_o it_o may_v be_v namely_o 3._o time_n there_o shall_v remain_v nothing_o of_o the_o great_a so_o the_o number_n 3._o be_v say_v to_o measure_v 12._o for_o that_o be_v take_v certain_a time_n namely_o four_o time_n it_o make_v just_a 12._o the_o great_a quantity_n and_o also_o be_v take_v from_o 12._o as_o often_o as_o it_o may_v namely_o 4._o time_n there_o shall_v remain_v nothing_o and_o in_o this_o meaning_n and_o signification_n do_v euclid_n undoubted_o here_o in_o this_o define_v a_o part_n part_n say_v that_o it_o be_v a_o less_o magnitude_n in_o comparison_n of_o a_o great_a when_o the_o less_o measure_v the_o great_a as_o the_o line_n ab_fw-la before_o set_v contain_v 3_n be_v a_o less_o quantity_n in_o comparison_n of_o the_o line_n cd_o which_o contain_v 9_o and_o also_o measure_v it_o for_o it_o be_v certain_a time_n take_v namely_o 3._o time_n precise_o make_v it_o or_o take_v from_o it_o as_o often_o as_o it_o may_v there_o remain_v nothing_o wherefore_o by_o this_o definition_n the_o line_n ab_fw-la be_v a_o part_n of_o the_o line_n cd_o likewise_o in_o number_n the_o number_n 5._o be_v a_o part_n of_o the_o number_n 15._o for_o it_o be_v a_o less_o number_n or_o quantity_n compare_v to_o the_o great_a and_o also_o it_o measure_v the_o great_a for_o be_v take_v certain_a time_n namely_o 3._o time_n it_o make_v 15._o and_o this_o kind_n of_o part_n be_v call_v common_o pars_fw-la metiens_fw-la or_o mensurans_fw-la mensuranâ—_n that_o be_v a_o measure_a part_n some_o call_v it_o pars_fw-la multiplicatina_fw-la multiplicatiâ—a_fw-la and_o of_o the_o barbarous_a it_o be_v call_v pars_fw-la aliquota_fw-la aliquota_fw-la that_o be_v a_o aliquote_v part_n and_o this_o kind_n of_o part_n be_v common_o use_v in_o arithmetic_n arithmetic_n the_o other_o kind_n of_o a_o part_n part_n be_v any_o less_o quantity_n in_o comparison_n of_o a_o great_a whether_o it_o be_v in_o number_n or_o magnitude_n and_o whether_o it_o measure_v or_o no._n as_o suppose_v the_o line_n ab_fw-la to_o be_v 17._o and_o let_v it_o be_v divide_v into_o two_o part_n in_o the_o point_n c_o namely_o into_o the_o line_n ac_fw-la &_o the_o line_n cb_o and_o let_v the_o line_n ac_fw-la the_o great_a part_n contain_v 12._o and_o let_v the_o line_n bc_o the_o less_o part_n contain_v 5._o now_o either_o of_o these_o line_n by_o this_o definition_n be_v a_o part_n of_o the_o whole_a line_n ab_fw-la for_o either_o of_o they_o be_v a_o less_o magnitude_n or_o quantity_n in_o comparison_n of_o the_o whole_a line_n ab_fw-la but_o neither_o of_o they_o measure_v the_o whole_a line_n ab_fw-la for_o the_o less_o line_n cb_o contain_v 5._o take_v as_o often_o as_o you_o listen_v will_v never_o make_v precise_o ab_fw-la which_o contain_v 17._o if_o you_o take_v it_o 3._o time_n it_o make_v only_o 15._o so_o lack_v it_o 2._o of_o 17._o which_o be_v to_o little_a if_o you_o take_v it_o 4._o time_n so_o make_v it_o 20._o them_z be_v there_o three_o to_o much_o so_o it_o never_o make_v precise_o 17._o but_o either_o to_o much_o or_o to_o little_a likewise_o the_o other_o part_n ac_fw-la measure_v not_o the_o whole_a line_n ab_fw-la for_o take_v once_o it_o make_v but_o 12._o which_o be_v less_o than_o 17._o and_o take_v twice_o it_o make_v 24._o which_o be_v more_o than_o 17._o by_o â—_o so_o it_o never_o precise_o make_v by_o take_v thereof_o the_o whole_a ab_fw-la but_o either_o more_o or_o less_o and_o this_o kind_n of_o part_n they_o common_o call_v pars_fw-la
constituens_fw-la or_o componens_fw-la componens_fw-la because_o that_o it_o with_o some_o other_o part_n or_o part_n make_v the_o whole_a as_o the_o line_n cb_o together_o with_o the_o line_n ac_fw-la make_v the_o whole_a line_n ab_fw-la of_o the_o barbarous_a it_o be_v call_v pars_fw-la aliquanta_fw-la aliquanta_fw-la in_o this_o signification_n it_o be_v take_v in_o bâ—rlaâ—â—_n in_o the_o begin_n of_o his_o book_n in_o the_o definition_n of_o a_o part_n when_o he_o say_v every_o less_o number_n compare_v to_o a_o great_a be_v say_v to_o be_v a_o part_n of_o the_o great_a whether_o the_o less_o measure_n the_o great_a or_o measure_v it_o not_o multiplex_fw-la be_v a_o great_a magnitude_n in_o respect_n of_o the_o less_o when_o the_o less_o measure_v the_o great_a definition_n as_o the_o line_n cd_o before_o set_v in_o the_o first_o example_n be_v multiplex_n to_o the_o line_n ab_fw-la for_o that_o cd_o a_o line_n contain_v 9_o be_v the_o great_a magnitude_n and_o be_v compare_v to_o the_o less_o namely_o to_o the_o line_n ab_fw-la contain_v 3._o and_o also_o the_o less_o line_n ab_fw-la measure_v the_o great_a line_v cd_o for_o take_v 3._o time_n it_o make_v it_o as_o be_v above_o say_v so_o in_o number_n 12._o be_v multiplex_n to_o 3_o for_o 12_o be_v the_o great_a number_n and_o be_v compare_v to_o the_o less_o namely_o to_o 3._o which_o 3._o also_o measure_v it_o follow_v for_o 3_o take_v 4_o time_n make_v 12._o by_o this_o word_n multiplex_n which_o be_v a_o term_n proper_a to_o arithmetic_n and_o number_n it_o be_v easy_a to_o consider_v that_o there_o can_v be_v no_o exact_a knowledge_n of_o proportion_n and_o proportionality_n and_o so_o of_o this_o five_o book_n with_o all_o the_o other_o book_n follow_v without_o the_o aid_n and_o knowledge_n of_o number_n proportion_n be_v a_o certain_a respect_n of_o two_o magnitude_n of_o one_o kind_n definition_n according_a to_o quantity_n kind_n proportion_n rational_a be_v divide_v into_o two_o kind_n into_o proportion_n of_o equality_n and_o into_o proportion_n of_o inequality_n proportion_n of_o equality_n be_v when_o one_o quantity_n be_v refer_v to_o a_o other_o equal_a unto_o itself_o as_o if_o you_o compare_v 5_o to_o 5_o or_o 7_o to_o 7_o &_o so_o of_o other_o and_o this_o proportion_n have_v great_a use_n in_o the_o rule_n of_o cosse_n equality_n for_o in_o it_o all_o the_o rule_n of_o equation_n tend_v to_o none_o other_o end_n but_o to_o find_v out_o and_o bring_v forth_o a_o number_n equal_a to_o the_o number_n suppose_v which_o be_v to_o put_v the_o proportion_n of_o equality_n inequality_n proportion_n of_o inequality_n be_v when_o one_o unequal_a quantity_n be_v compare_v to_o a_o other_o as_o the_o great_a to_o the_o less_o as_o 8._o to_o 4_o or_o 9_o to_o 3_o or_o the_o less_o to_o the_o great_a as_o 4._o to_o 8_o or_o 3._o to_o 9_o less_o proportion_n of_o the_o great_a to_o the_o less_o have_v five_o kind_n namely_o multiplex_n superparticular_a superpartiens_fw-la multiplex_fw-la superperticular_a and_o multiplex_fw-la superpartiens_fw-la multiplex_fw-la be_v when_o the_o antecedent_n contain_v in_o itself_o the_o consequent_a certain_a time_n without_o more_o or_o less_o multiplex_fw-la as_o twice_o thrice_o four_o time_n and_o so_o far_o and_o this_o proportion_n have_v under_o it_o infinite_a kind_n for_o if_o the_o antecedent_n contain_v the_o consequent_a just_o twice_o it_o be_v call_v dupla_fw-la proportion_n proportion_n as_o 4_o to_o 2._o if_o thrice_o tripla_fw-la quintuple_a as_o 9_o to_o 3._o if_o 4._o time_n quadrupla_fw-la as_o 12._o to_o 3._o if_o 5._o time_n quintupla_fw-la as_o 15._o to_o 3._o and_o so_o infinite_o after_o the_o same_o manner_n superperticular_a superperticular_a be_v when_o the_o antecedent_n contain_v the_o consequent_a only_o once_o &_o moreover_o some_o one_o part_n thereof_o as_o a_o half_a a_o three_o or_o four_o etc_n etc_n this_o kind_n also_o have_v under_o it_o infinite_a kind_n for_o if_o the_o antecedent_n contain_v the_o consequent_a once_o and_o a_o half_a thereof_o it_o be_v call_v sesquialtera_fw-la sesquialtera_fw-la as_o 6._o to_o 4_o if_o once_o and_o a_o three_o part_n sesquitertia_fw-la sesquitertia_fw-la as_o 4._o to_o 3_o if_o once_o and_o a_o four_o part_n sesquiquarta_fw-la sesquiquarta_fw-la as_o 5._o to_o 4._o and_o so_o in_o like_a manner_n infinite_o superpartiens_fw-la superpartiens_fw-la be_v when_o the_o antecedent_n contain_v the_o consequent_a only_o once_o &_o moreover_o more_o part_n than_o one_o of_o the_o same_o as_o two_o three_o three_o fourthe_n four_o fifthe_n and_o so_o forth_o this_o also_o have_v infinite_a kind_n under_o it_o for_o if_o the_o antecedent_n contain_v above_o the_o consequent_a two_o part_n it_o be_v call_v superbipartiens_fw-la superbipartiens_fw-la as_o 7._o to_o 5._o if_o 3._o part_n supertripartiens_fw-la as_o 7._o to_o 4._o supertripartiens_fw-la if_o 4._o part_n superquadripartiens_n superquadripartiens_fw-la as_o 9_o to_o 5._o if_o 5._o part_n superquintipartiens_n as_o 11._o to_o 6._o superquintipartiens_fw-la and_o so_o forth_o infinite_o superperticular_a multiplex_fw-la superperticular_a be_v when_o the_o antecedent_n contain_v the_o consequent_a more_o than_o once_o and_o moreover_o only_o one_o part_n of_o the_o same_o this_o kind_n likewise_o have_v infinite_a kind_n under_o it_o for_o if_o the_o antecedent_n contain_v the_o consequent_a twice_o and_o half_o thereof_o it_o be_v call_v dupla_fw-la sesquialtera_fw-la sesquialtera_fw-la as_o 5._o to_o 2._o if_o twice_o and_o a_o three_o dupla_fw-la sesquitertia_fw-la as_o 7._o to_o 3._o sesquitertia_fw-la if_o thrice_o and_o a_o half_a tripla_fw-la sesquialtera_fw-la as_o 7._o to_o 2._o sesquialtera_fw-la if_o four_o time_n and_o a_o half_a quadrâ—pla_fw-la sesquialtera_fw-la as_o 9_o to_o 2._o and_o so_o go_v on_o infinite_o superpartiens_fw-la multiplex_fw-la superpartient_fw-fr be_v when_o the_o antecedent_n contain_v the_o consequent_a more_o than_o once_o and_o also_o more_o part_n than_o one_o of_o the_o consequent_a and_o this_o kind_n also_o have_v infinite_a kind_n under_o it_o for_o if_o the_o antecedent_n contain_v the_o consequent_a twice_o and_o two_o part_n over_o it_o be_v call_v dupla_fw-la superbipartiens_fw-la as_o 8._o to_o 3._o superbipartiens_fw-la if_o twice_o and_o three_o part_n dupla_fw-la supertripartiens_fw-la as_o 11._o to_o 4._o supertripartiens_fw-la if_o thrice_o and_o two_o part_n it_o be_v name_v tripla_fw-la superbipartiens_fw-la as_o 11._o to_o 3._o superbipartiens_fw-la if_o three_o time_n and_o four_o part_n treble_a superquadripartiens_n as_o 31._o to_o 9_o superquadâ—ipartiens_fw-la and_o so_o forth_o infinite_o here_o be_v to_o be_v note_v that_o the_o denomination_n of_o the_o proportion_n between_o any_o two_o number_n be_v have_v by_o devide_v of_o the_o great_a by_o the_o less_o for_o the_o quotient_a oâ—_n number_v produce_v of_o that_o division_n be_v even_o the_o denomination_n of_o the_o proportion_n proportion_n which_o in_o the_o first_o kind_n of_o proportion_n namely_o multiplex_n be_v ever_o a_o whole_a number_n and_o in_o all_o other_o kind_n of_o proportion_n it_o be_v a_o break_a number_n as_o if_o you_o will_v know_v the_o denomination_n of_o the_o proportion_n between_o 9_o and_o 3._o divide_v 9_o by_o 3._o so_o shall_v you_o have_v in_o the_o quotient_a 3._o which_o be_v a_o whole_a number_n and_o be_v the_o denomination_n of_o the_o proportion_n and_o show_v that_o the_o proportion_n between_o 9_o &_o 3._o be_v tripla_fw-la so_o the_o proportion_n between_o 12._o and_o 3._o be_v quadrupla_fw-la for_o that_o 12._o be_v divide_v by_o 3._o the_o quotient_n be_v 4._o and_o so_o of_o other_o in_o the_o kind_n of_o multiplex_n and_o although_o in_o this_o kind_a the_o quotient_n be_v ever_o a_o whole_a number_n yet_o proper_o it_o be_v refer_v to_o unity_n and_o so_o be_v represent_v in_o manner_n of_o a_o break_a number_n as_o 5_o â—_o and_o 4_o â—_o for_o unity_n be_v the_o denomination_n to_o a_o whole_a number_n likewise_o the_o denomination_n of_o the_o proportion_n between_o 4_o and_o 3_o be_v 1._o 1_o â—_o for_o that_o 4_o divide_v by_o 3._o have_v in_o the_o quotient_a 1_o 1_o â—_o one_o and_o a_o three_o part_n of_o which_o three_o part_n it_o be_v call_v sesquitercia_fw-la so_o the_o proportion_n between_o 7_o and_o 6._o be_v 1_o â…™_n one_o and_o a_o six_o of_o which_o fix_v part_v it_o be_v call_v sesquisexta_fw-la and_o so_o of_o other_o of_o that_o kind_n also_o between_o 7_o and_o 5_o the_o denomination_n of_o the_o proportion_n be_v 1_o â—_o â—_o one_o and_o two_o fifthe_n which_o denomination_n consist_v of_o two_o part_n namely_o of_o the_o munerator_n and_o denominator_fw-la of_o the_o quotient_a of_o 2._o and_o 5_o of_o which_o two_o fifthe_n it_o be_v call_v superbipartiens_fw-la quintas_fw-la for_o 2_o the_o numerator_n show_v the_o denomination_n of_o the_o number_n of_o the_o part_n and_o 5._o the_o denominator_fw-la show_v the_o denomination_n what_o part_n they_o be_v &_o so_o of_o other_o also_o the_o denomination_n between_o
proportion_n than_o have_v bc_o to_o ef._n wherefore_o a_o have_v to_o d_o a_o great_a proportion_n than_o have_v bc_o to_o ef._n wherefore_o alternate_o a_o have_v to_o bc_o a_o great_a proportion_n than_o have_v d_z to_o of_o wherefore_o by_o composition_n abc_n have_v to_o bc_o a_o great_a proportion_n than_o have_v def_n to_o ef._n wherefore_o again_o alternate_o abc_n have_v to_o def_n a_o great_a proportion_n than_o have_v bc_o to_o ef._n wherefore_o by_o the_o former_a proposition_n the_o proportion_n of_o a_o to_o d_z be_v great_a than_o the_o proportion_n of_o abc_n to_o def_n which_o be_v require_v to_o be_v prove_v the_o end_n of_o the_o five_o book_n of_o euclides_n element_n ¶_o the_o six_o book_n of_o euclides_n element_n this_o spaniard_o book_n be_v for_o use_v and_o practice_v book_n a_o most_o special_a book_n in_o it_o be_v teach_v the_o proportion_n of_o one_o figure_n to_o a_o other_o figure_n &_o of_o their_o side_n the_o one_o to_o the_o other_o and_o of_o the_o side_n of_o one_o to_o the_o side_n of_o a_o other_o likewise_o of_o the_o angle_n of_o the_o one_o to_o the_o angle_n of_o the_o other_o moreover_o it_o teach_v the_o description_n of_o figure_n like_o to_o â—_z give_v and_o marvelous_a application_n of_o figure_n to_o line_n even_o or_o with_o decrease_n or_o excess_n with_o many_o other_o theorem_n not_o only_o of_o the_o propoâ—tions_n of_o right_n line_v figure_n but_o also_o of_o sector_n of_o circle_n with_o their_o angle_n on_o the_o theorem_n and_o problem_n of_o this_o book_n depend_v for_o the_o most_o part_n the_o composition_n of_o all_o instrument_n of_o measure_v length_n breadth_n or_o deepen_n and_o also_o the_o reason_n of_o the_o use_n of_o the_o same_o instrument_n as_o of_o the_o geometrical_a â—quarâ—_n geometry_n the_o scale_n of_o the_o astrolabe_n the_o quadrant_a the_o staff_n and_o such_o other_o the_o use_n of_o which_o instrument_n beside_o all_o other_o mechanical_a instrument_n of_o raise_v up_o of_o move_v and_o draw_v huge_a thing_n incredible_a to_o the_o ignorant_a and_o infinite_a other_o begin_v which_o likewise_o have_v their_o ground_n out_o of_o this_o book_n be_v of_o wonderful_a and_o unspeakable_a profit_n beside_o the_o inestimable_a pleasure_n which_o be_v in_o they_o definition_n 1._o like_o rectiline_a figure_n be_v such_o definition_n who_o angle_n be_v equal_a the_o one_o to_o the_o other_o and_o who_o side_n about_o the_o equal_a angle_n be_v proportional_a as_o if_o you_o take_v any_o two_o rectiline_a figure_n as_o for_o example_n two_o triangle_n abc_n and_o def_n 〈…〉_o of_o the_o one_o triangle_n be_v equal_a to_o the_o angle_n of_o the_o other_o namely_o if_o the_o angle_n a_o be_v equal_a to_o the_o angle_n d_o and_o the_o angle_n b_o equal_v to_o the_o angle_n e_o &_o also_o the_o angle_n c_o equal_v to_o the_o angle_n f._n and_o moreover_o iâ—_z the_o side_n which_o contain_v the_o equal_a angle_n be_v proportional_a as_o if_o the_o side_n ab_fw-la have_v that_o proportion_n to_o the_o side_n bc_o whâ—ch_v the_o side_n de_fw-fr have_v to_o the_o side_n of_o and_o also_o if_o the_o side_n bc_o be_v unto_o the_o side_n ca_n as_o â—he_n side_n of_o be_v to_o the_o side_n fd_o and_o moreover_o if_o the_o side_n ca_n be_v to_o the_o side_n ab_fw-la as_o the_o side_n fd_o be_v to_o the_o side_n de_fw-fr then_o be_v these_o two_o triangle_n say_v to_o be_v like_o and_o so_o judge_v you_o of_o any_o other_o kind_n of_o figure_n as_o if_o in_o the_o parallelogram_n abcd_o and_o efgh_a the_o angle_n a_o be_v equal_a to_o the_o angle_n e_o and_o the_o angle_n b_o equal_v to_o the_o angle_n fletcher_n and_o the_o angle_n c_o equal_v to_o the_o angle_n g_o and_o the_o angle_n d_o equal_v to_o the_o angle_n h._n and_o farthermore_o if_o the_o side_n ac_fw-la have_v that_o proportion_n to_o the_o side_n cd_o which_o the_o side_n eglantine_n have_v to_o the_o side_n gh_o and_o if_o also_o the_o side_n cd_o be_v to_o the_o side_n db_o as_o the_o side_n 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
and_o the_o same_o proportion_n wherefore_o by_o the_o 9_o of_o the_o five_o the_o figure_n nh_o be_v equal_a unto_o the_o figure_n sir_n and_o it_o be_v unto_o it_o like_v and_o in_o like_a sort_n situate_a follow_v but_o in_o like_a and_o equal_a rectiline_a figure_n be_v in_o like_a sort_n situate_a the_o side_n of_o like_a proportion_n on_o which_o they_o be_v describe_v be_v equal_a wherefore_o the_o line_n gh_o be_v equal_a unto_o the_o line_n qr_o and_o because_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o be_v the_o line_n of_o to_o the_o line_n qr_o but_o the_o line_n qr_o be_v equal_a unto_o the_o line_n gh_o therefore_o as_o the_o line_n ab_fw-la be_v to_o he_o line_n cd_o so_o be_v the_o line_n of_o to_o the_o line_n gh_o if_o therefore_o there_o be_v four_o right_a line_n proportional_a the_o rectiline_a figure_n also_o describe_v upon_o they_o be_v like_a and_o in_o like_a sort_n situate_a shall_v be_v proportional_a and_o if_o the_o rectiline_a figure_n upon_o they_o describe_v be_v like_o and_o in_o like_a sort_n situate_v be_v proportional_a those_o right_a line_n also_o shall_v be_v proportional_a which_o be_v require_v to_o be_v prove_v a_o assumpt_n and_o now_o that_o in_o like_a and_o equal_a figure_n be_v in_o like_a sort_n situate_a the_o side_n of_o like_a proportion_n be_v also_o equal_a which_o thing_n be_v before_o in_o this_o proposition_n take_v as_o grant_v may_v thus_o be_v prove_v assumpt_n suppose_v that_o the_o rectiline_a figure_n nh_v and_o sir_n be_v equal_a and_o like_a and_o as_o hg_o be_v to_o gn_v so_o let_v rq_n be_v to_o q_n and_o let_v gh_o and_o qr_o be_v side_n of_o like_a proportion_n then_o i_o say_v that_o the_o side_n rq_n be_v equal_a unto_o the_o side_n gh_o for_o if_o they_o be_v unequal_a the_o one_o of_o they_o be_v great_a than_o the_o other_o let_v the_o side_n rq_n be_v great_a than_o the_o side_n hg_o and_o for_o that_o as_o the_o line_n rq_n be_v to_o the_o line_n q_n so_o be_v the_o line_n hg_o to_o the_o line_n gn_n and_o alternate_o also_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n rq_n be_v to_o the_o line_n hg_o so_o be_v the_o line_n q_n to_o the_o line_n gn_n but_o the_o line_n rq_n be_v great_a than_o the_o line_n hg_o wherefore_o also_o the_o line_n q_n be_v great_a than_o the_o line_n gn_n wherefore_o also_o the_o figure_n rs_n be_v great_a than_o the_o figure_n hn_o but_o by_o supposition_n it_o be_v equal_a unto_o it_o which_o be_v impossible_a wherefore_o the_o line_n qr_o be_v not_o great_a than_o the_o line_n gh_o in_o like_a sort_n also_o may_v we_o prove_v that_o it_o be_v not_o less_o than_o it_o wherefore_o it_o be_v equal_a unto_o it_o which_o be_v require_v to_o be_v prove_v flussates_n demonstrate_v this_o second_o part_n more_o brief_o flussates_n by_o the_o first_o corollary_n of_o the_o _o of_o this_o book_n thus_o forasmuch_o as_o the_o rectiline_a figure_n be_v by_o supposition_n in_o one_o and_o the_o same_o proportion_n and_o the_o same_o proportion_n be_v double_a to_o the_o proportion_n of_o the_o side_n ab_fw-la to_o cd_o and_o of_o to_o gh_o by_o the_o foresay_a corollary_n the_o proportion_n also_o of_o the_o side_n shall_v be_v one_o and_o the_o self_n same_o by_o the_o 7._o common_a sentence_n namely_o the_o line_n ab_fw-la shall_v be_v unto_o the_o line_n cd_o as_o the_o line_n of_o be_v to_o the_o line_n gh_o the_o 17._o theorem_a the_o 23._o proposition_n equiangle_n parallelogrammes_n have_v the_o one_o to_o the_o other_o that_o proportion_n which_o be_v compose_v of_o the_o side_n flussates_n demonstrate_v this_o theorem_a without_o take_v of_o these_o three_o line_n king_n l_o m_o after_o this_o manner_n flussate_n forasmuch_o as_o say_v he_o it_o have_v be_v declare_v upon_o the_o 10._o definition_n of_o the_o five_o book_n and_o â—ift_a definition_n of_o this_o book_n that_o the_o proportion_n of_o the_o extreme_n consist_v of_o the_o proportion_n of_o the_o mean_n let_v we_o suppose_v two_o equiangle_n parallelogram_n abgd_v and_o gezi_n and_o let_v the_o angle_n at_o the_o point_n g_o in_o either_o be_v equal_a and_o let_v the_o line_n bg_o and_o give_v be_v set_v direct_o that_o they_o both_o make_v one_o right_a line_n namely_o bgi._n wherefore_o egg_v also_o shall_v be_v one_o right_a line_n by_o the_o converse_n of_o the_o 15._o of_o the_o first_o make_v complete_a the_o parallelogram_n gt_fw-mi then_o i_o say_v that_o the_o proportion_n of_o the_o parallelogram_n agnostus_n &_o gz_n be_v compose_v of_o the_o proportion_n of_o the_o side_n bg_o to_o give_v and_o dg_o to_o ge._n for_o forasmuch_o as_o that_o there_o be_v three_o magnitude_n agnostus_n gt_n and_o gz_n and_o gt_n be_v the_o mean_a of_o the_o say_a magnitude_n and_o the_o proportion_n of_o the_o extreme_n agnostus_n to_o gz_n consist_v of_o the_o mean_a proportion_n by_o the_o 5._o definition_n of_o this_o book_n namely_o of_o the_o proportion_n of_o agnostus_n to_o gt_n and_o of_o the_o proportion_n gt_fw-mi to_o gz_n but_o the_o proportion_n of_o agnostus_n to_o gt_n be_v one_o and_o the_o self_n same_o with_o the_o proportion_n of_o the_o side_n bg_o to_o give_v by_o the_o first_o of_o this_o book_n and_o the_o proportion_n also_o of_o gt_n to_o gz_n be_v one_o and_o the_o self_n same_o with_o the_o proportion_n of_o the_o other_o side_n namely_o dg_o to_o ge_z by_o the_o same_o proposition_n wherefore_o the_o proportion_n of_o the_o parallelogram_n agnostus_n to_o gz_n consist_v of_o the_o proportion_n of_o the_o side_n bg_o to_o give_v and_o dc_o to_o ge._n wherefore_o equiangle_n parallelogram_n be_v the_o one_o to_o the_o other_o in_o that_o proportion_n which_o be_v compose_v of_o their_o side_n which_o be_v require_v to_o be_v prove_v the_o 18._o theorem_a the_o 24._o proposition_n in_o every_o parallelogram_n the_o parallelogram_n about_o the_o dimecient_a be_v like_a unto_o the_o whole_a and_o also_o like_o the_o one_o to_o the_o other_o abcd._n svppose_v that_o there_o be_v a_o parallelogram_n abcd_o and_o let_v the_o dimecient_a thereof_o be_v ac_fw-la and_o let_v the_o parallelogram_n about_o the_o dimecient_a ac_fw-la be_v eglantine_n and_o hk_o then_o i_o say_v that_o either_o of_o these_o parallelogram_n eglantine_n and_o hk_o be_v like_a unto_o the_o whole_a parallelogram_n abcd_o and_o also_o be_v like_o the_o one_o to_o the_o other_o for_o forasmuch_o as_o to_o one_o of_o the_o side_n of_o the_o triangle_n abc_n namely_o to_o bc_o be_v draw_v a_o parallel_a line_n of_o therefore_o as_o be_v be_v to_o ea_fw-la so_o by_o the_o 2._o of_o the_o six_o be_v cf_o to_o fa._n again_o forasmuch_o as_o to_o one_o of_o the_o side_n of_o the_o triangle_n adc_o namely_o to_o cd_o be_v draw_v a_o parallel_a line_n fâ—_n therefore_o by_o the_o same_o as_o cf_o be_v to_o favorina_n so_o be_v dg_o to_o ga._n but_o as_o cf_o be_v to_o favorina_n so_o be_v it_o prove_v that_o be_v be_v to_o ea_fw-la wherefore_o as_o be_v be_v to_o ea_fw-la so_o by_o the_o 11._o of_o the_o five_o â—_o be_v dg_fw-mi to_o ga._n wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o basilius_n be_v to_o aeâ—_n so_o be_v dam_n to_o ag._n and_o alternate_o by_o the_o 16._o of_o the_o five_o as_o basilius_n be_v to_o ad_fw-la so_o be_v ea_fw-la to_o ag._n wherefore_o in_o the_o parallelogrammesâ—_n abcd_o and_o eglantine_n the_o side_n which_o be_v about_o the_o common_a angle_n bad_a be_v proportional_a and_o because_o the_o line_n gf_o be_v a_o parallel_n unto_o the_o line_n dcâ—_n therefore_o the_o angle_n agf_n by_o the_o 29â—_n of_o theâ—_z first_o be_v equal_a unto_o the_o angle_v adc_o â—_o the_o angle_n gfa_n equal_a unto_o the_o angle_v dca_n and_o the_o angle_n dac_o be_v common_a to_o the_o two_o triangle_n adc_o and_o afg_n wherefore_o the_o triangle_n dac_o be_v equiangle_n unto_o the_o triangle_n agf_n and_o by_o the_o same_o reason_n the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n aef_n wherefore_o the_o whole_a parallelogram_n abcd_o be_v equiangle_n unto_o the_o parallelogram_n eglantine_n wherefore_o as_o ad_fw-la be_v in_o proportion_n to_o dc_o so_o by_o the_o 4._o of_o the_o six_o be_v agnostus_n to_o gf_o and_o as_o dc_o be_v to_o ca_n so_o be_v gf_o to_o fa._n and_o as_o ac_fw-la be_v to_o cb_o so_o be_v of_o to_o fe_o and_o moreover_o as_o cb_o be_v to_o ba_o so_o be_v fe_o to_o ea_fw-la and_o forasmuch_o as_o it_o be_v prove_v that_o as_o dâ—_n be_v to_o ca_n so_o be_v gf_o to_o favorina_n but_o as_o ac_fw-la be_v to_o câ—_n so_o be_v of_o to_o fe_o wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o dc_o be_v to_o cb_o so_o be_v gf_o to_o fe_o wherefore_o in_o the_o parallelogram_n abcd_o and_o
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 succour_v of_o number_n but_o the_o matter_n which_o remain_v to_o be_v teach_v and_o to_o be_v speak_v of_o in_o these_o his_o geometrical_a book_n which_o follow_v as_o in_o the_o ten_o eleven_o and_o so_o forth_o he_o can_v by_o no_o mean_n full_o and_o clear_o make_v plain_a &_o demonstrate_v without_o the_o help_n and_o aid_n of_o number_n in_o the_o ten_o be_v entreat_v of_o line_n irrational_a and_o uncertain_a and_o that_o of_o many_o &_o sundry_a kind_n and_o in_o the_o eleven_o &_o the_o other_o follow_v he_o teach_v the_o nature_n of_o body_n and_o compare_v their_o side_n and_o line_n together_o all_o which_o for_o the_o most_o part_n be_v also_o irrational_a and_o as_o rational_a quantity_n and_o the_o comparison_n and_o proportion_n of_o they_o can_v be_v know_v nor_o exact_o try_v but_o by_o the_o mean_a of_o number_n in_o which_o they_o be_v first_o see_v and_o perceive_v even_o so_o likewise_o can_v irrational_a quantity_n be_v know_v and_o find_v out_o without_o number_n as_o straightness_n be_v the_o trial_n of_o crookedness_n and_o inequality_n be_v try_v by_o equality_n so_o be_v quantity_n irrational_a perceive_v and_o know_v by_o quantity_n rational_a which_o be_v â—irst_n and_o chief_o find_v among_o number_n wherefore_o in_o these_o three_o book_n follow_v be_v as_o it_o be_v in_o the_o midst_n of_o his_o element_n he_o be_v compel_v of_o necessity_n to_o entreat_v of_o number_n number_n although_o not_o so_o full_o as_o the_o nature_n of_o number_n require_v yet_o so_o much_o as_o shall_v seem_v to_o be_v fit_a and_o sufficient_o to_o serve_v for_o his_o purpose_n whereby_o be_v see_v the_o necessity_n that_o the_o one_o art_n namely_o geometry_n have_v of_o the_o other_o namely_o of_o arithmetic_n and_o also_o of_o what_o excellency_n and_o worthiness_n arithmetic_n be_v above_o geometry_n geometry_n in_o that_o geometry_n borrow_v of_o it_o principle_n aid_n and_o succour_n and_o be_v as_o it_o be_v maim_v with_o out_o it_o whereas_o arithmetic_n be_v of_o itself_o sufficient_a and_o nead_v not_o at_o all_o any_o aid_n of_o geometry_n but_o be_v absolute_a and_o perfect_a in_o itself_o and_o may_v well_o be_v teach_v and_o attain_a unto_o without_o it_o again_o the_o matter_n or_o subject_n where_o about_o geometry_n be_v occupy_v which_o be_v line_n figure_n and_o body_n be_v such_o as_o offer_v themselves_o to_o the_o sense_n as_o triangle_n square_n circle_n cube_n and_o other_o be_v see_v &_o judge_v to_o be_v such_o as_o they_o be_v by_o the_o sight_n but_o number_n which_o be_v the_o subject_n and_o matter_n of_o arithmetic_n fall_v under_o no_o sense_n nor_o be_v represent_v by_o any_o shape_n form_n or_o figure_n and_o therefore_o can_v be_v judge_v by_o any_o sense_n but_o only_o by_o consideration_n of_o the_o mind_n and_o understanding_n now_o thing_n sensible_a be_v far_o under_o in_o degree_n then_o be_v thing_n intellectual_a sensible_a and_o be_v of_o nature_n much_o more_o gross_a than_o they_o 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&_o teach_v most_o pithy_o and_o pure_o the_o secret_a and_o hide_a knowledge_n of_o the_o nature_n and_o condition_n of_o all_o thing_n by_o number_n and_o by_o the_o propriety_n and_o passion_n of_o they_o of_o what_o force_n number_n be_v in_o perspective_n let_v he_o declare_v and_o judge_v who_o have_v any_o thing_n travel_v therein_o yea_o to_o be_v short_a what_o can_v be_v worthy_o and_o with_o praise_n practise_v in_o common_a life_n of_o any_o man_n of_o any_o condition_n without_o the_o knowledge_n of_o number_n yea_o it_o have_v be_v teach_v of_o the_o chief_a amongst_o philosopher_n that_o all_o natural_a thing_n be_v frame_v and_o have_v their_o constitution_n of_o number_n boetius_fw-la say_v hoc_fw-la fuit_fw-la principal_n in_o animâ—_n câ—â—ditoris_fw-la exemplar_n arithmeti_n number_n say_v he_o be_v the_o principal_a example_n and_o patron_n in_o the_o mind_n of_o the_o creator_n of_o the_o world_n do_v not_o that_o great_a philosop_n timaus_n in_o his_o book_n timaus_n &_o also_o plato_n in_o his_o timâ—â—_n follow_v he_o show_v how_o the_o soul_n be_v compose_v of_o harmonical_a number_n and_o consonantes_fw-la of_o music_n number_n compass_v all_o thing_n and_o be_v after_o these_o man_n the_o be_v and_o very_a essence_n of_o all_o thing_n and_o minister_v aid_n and_o help_v as_o to_o all_o other_o knowledge_n so_o also_o no_o small_a to_o geometry_n which_o thing_n cause_v euclidâ—_n in_o the_o midst_n of_o his_o book_n of_o geometry_n to_o insert_v and_o place_n these_o three_o book_n of_o arithmetic_n as_o without_o the_o aid_n of_o which_o he_o can_v not_o well_o pass_v any_o father_n in_o this_o seven_o book_n he_o â—irst_n place_v the_o general_a principle_n and_o first_o ground_n of_o arithmetic_n and_o set_v the_o definition_n of_o number_n or_o multitude_n and_o the_o kind_n thereof_o as_o in_o the_o first_o book_n he_o do_v of_o magnitude_n and_o the_o kind_n and_o part_n thereof_o book_n after_o that_o he_o entreat_v of_o 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 book_n for_o the_o like_a reason_n and_o cause_n be_v a_o point_n first_o define_v unity_n say_v euclid_n be_v that_o whereby_o every_o thing_n be_v say_v to_o be_v one_o that_o be_v unity_n be_v that_o whereby_o every_o thing_n be_v divide_v and_o separate_v from_o a_o other_o and_o remain_v on_o in_o itself_o pure_a and_o distinct_a from_o all_o other_o otherwise_o be_v not_o this_o unity_n thing_n whereby_o all_o thing_n be_v sejoin_v the_o on_o from_o the_o other_o all_o thing_n shall_v suffer_v mixtion_n and_o be_v in_o confusion_n and_o where_o confusion_n be_v there_o be_v no_o order_n nor_o any_o thing_n can_v be_v exact_o know_v either_o what_o it_o be_v or_o what_o be_v the_o nature_n and_o what_o be_v the_o property_n thereof_o unity_n therefore_o be_v that_o which_o make_v every_o thing_n to_o be_v that_o which_o it_o be_v boetius_fw-la say_v very_o apt_o
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 by_o 5_o a_o odd_a number_n in_o this_o definition_n also_o be_v find_v by_o all_o the_o expositor_n of_o euclid_n the_o same_o want_n that_o be_v find_v in_o the_o definition_n next_o before_o and_o for_o that_o it_o extend_v itself_o to_o large_a for_o there_o be_v infinite_a number_n which_o even_a number_n do_v measure_n by_o odd_a number_n which_o yet_o after_o their_o mind_n be_v not_o even_o odd_a number_n as_o for_o example_n 12._o for_o 4_o a_o even_a number_n measure_v 12_o by_o â—_o a_o odd_a numberâ—_n three_o time_n 4_o be_v 12._o yet_o be_v not_o 12_o as_z they_o think_v a_o even_o odd_a number_n wherefore_o campane_n amend_v it_o after_o his_o think_n campane_n by_o add_v of_o this_o word_n all_o as_o he_o do_v in_o the_o first_o and_o define_v it_o after_o this_o manner_n a_o number_n even_o odd_a be_v when_o all_o the_o even_a number_n which_o measure_v it_o do_v measure_n it_o by_o uneven_a time_n that_o be_v by_o a_o odd_a number_n as_o 10._o be_v a_o number_n even_o 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that_o which_o may_v be_v divide_v into_o two_o equal_a part_n but_o that_o part_n can_v again_o be_v divide_v into_o two_o equal_a part_n as_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 neither_o of_o they_o can_v be_v divide_v into_o two_o equal_a part_n for_o that_o 3._o be_v a_o odd_a number_n and_o suffer_v no_o such_o division_n definition_n 10_o a_o number_n odd_o even_o call_v in_o lattin_a in_o pariter_fw-la par_fw-fr be_v that_o which_o a_o odd_a number_n measure_v by_o a_o even_a number_n definition_n as_o the_o number_n 12_o for_o 3._o a_o odd_a number_n measure_v 12._o by_o 4._o which_o be_v a_o even_a number_n three_o time_n 4._o be_v 12._o this_o definition_n be_v not_o find_v in_o the_o greek_a neither_o be_v it_o doubtless_o ever_o in_o this_o manner_n write_v by_o euclid_n greek_n which_o thing_n the_o slenderness_n and_o the_o imperfection_n thereof_o and_o the_o absurdity_n follow_v also_o of_o the_o same_o declare_v most_o manifest_o the_o definition_n next_o before_o give_v be_v in_o substance_n all_o one_o with_o this_o for_o what_o number_n soever_o a_o evenâ—_n number_n do_v measure_n by_o a_o odd_a the_o self_n same_o number_n do_v a_o odd_a number_n measure_n by_o a_o even_o as_o 2._o a_o even_a number_n measure_v 6._o by_o 3._o a_o odd_a number_n wherefore_o 3._o a_o odd_a number_n do_v also_o measure_v the_o same_o number_n 6._o by_o 2._o a_o even_a number_n now_o if_o these_o two_o definition_n be_v definition_n of_o two_o distinct_a kind_n of_o number_n then_o be_v this_o number_n 6._o both_o even_o even_o and_o also_o even_o odd_a and_o so_o be_v contain_v under_o two_o diverse_a kind_n of_o number_n which_o be_v direct_o against_o the_o authority_n of_o euclid_n who_o plain_o pâ—ovoâ—h_v here_o after_o in_o the_o 9_o book_n that_o every_o number_n who_o half_a be_v a_o odd_a number_n be_v a_o number_n even_o odd_a only_o flussates_n have_v here_o very_a well_o note_v that_o these_o two_o 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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
no_o part_n to_o measure_v they_o but_o only_a unity_n definition_n 13_o number_n prime_a the_o one_o to_o the_o other_o be_v they_o which_o only_a unity_n do_v measure_n be_v a_o common_a measure_n to_o they_o as_o 15._o and_o 22._o be_v number_n prime_a the_o one_o to_o the_o other_o .15_o of_o itself_o be_v no_o prime_a number_n for_o not_o only_a unity_n do_v measure_v it_o but_o also_o the_o numbeâ—â—_n 5._o and_o 3_o for_o â—_o time_n 5._o be_v lx_o likewise_o 22._o be_v of_o itself_o no_o prime_a number_n for_o it_o be_v measure_v by_o 2._o and_o 11_o beside_o unity_n for_o 11._o twice_o or_o 2._o eleven_o time_n make_v 22._o so_o that_o although_o neither_o of_o these_o two_o number_n 15._o and_o 22._o be_v a_o prime_n or_o incomposed_a number_n but_o either_o have_v part_n of_o his_o own_o whereby_o it_o may_v be_v measure_v beside_o unity_n yet_o compare_v together_o they_o be_v prime_a the_o one_o to_o the_o other_o for_o no_o one_o number_n do_v as_o a_o common_a measure_n measure_n both_o of_o they_o but_o only_a unity_n which_o be_v a_o common_a measure_n to_o all_o number_n the_o number_n 5._o and_o 3._o which_o measure_n 15._o will_v not_o measure_v 22_o again_o the_o number_n 2._o and_o 11._o which_o measure_n 22_o do_v not_o measure_v 15._o 14_o a_o number_n compose_v be_v that_o which_o some_o one_o number_n measure_v definition_n a_o number_n compose_v be_v not_o measure_v only_o by_o unity_n as_o be_v a_o prime_a number_n but_o have_v some_o number_n which_o measure_v it_o as_o 15_o the_o number_n 3._o measure_v 15_o namely_o take_v 5._o time_n also_o the_o number_n 5._o measure_v 15_o namely_o take_v 3._o time_n 5._o time_n â—_o and_o 3._o time_n 5_o be_v 15._o likewise_o 18._o be_v a_o compose_a number_n it_o be_v measure_v by_o these_o number_n 6.3.9.2_o and_o so_o of_o other_o these_o number_n be_v also_o call_v common_o second_v number_n as_o contrary_a to_o prime_n or_o first_o number_n 15_o number_n compose_v the_o one_o to_o the_o other_o be_v they_o which_o some_o one_o number_n be_v a_o common_a measure_n to_o they_o both_o do_v measure_n definition_n as_o 12._o and_o 8._o be_v two_o compose_a number_n the_o one_o to_o the_o other_o for_o the_o number_n 4_o be_v a_o common_a measure_n to_o they_o both_o 4._o take_v three_o time_n make_v 12_o and_o the_o same_o 4._o take_v two_o time_n make_v 8._o so_o be_v 9_o and_o 15_o 3._o measure_v they_o both_o also_o 10._o and_o 25_o for_o 5._o measure_v both_o of_o they_o and_o so_o infinite_o of_o other_o in_o this_o do_v number_n compose_v the_o one_o to_o the_o other_o or_o second_o number_n differre_fw-la from_o number_n prime_a the_o one_o to_o the_o other_o for_o that_o two_o number_n be_v compose_v the_o one_o to_o the_o other_o each_o of_o they_o several_o be_v of_o necessity_n a_o compose_a number_n as_o in_o the_o example_n before_o 8._o and_o 12._o be_v compose_v number_n likewise_o 9_o and_o 15_o also_o 10_o and_o 25_o but_o if_o they_o be_v two_o number_n prime_a the_o one_o to_o the_o other_o â—it_n be_v not_o of_o necessity_n that_o each_o of_o they_o several_o be_v a_o prime_a number_n as_o 9_o and_o 22._o be_v two_o number_n prime_a the_o one_o to_o the_o other_o no_o one_o number_n measure_v both_o of_o they_o and_o yet_o neither_o of_o they_o in_o itself_o and_o in_o his_o own_o nature_n be_v a_o prime_a number_n but_o each_o of_o they_o be_v a_o compose_a number_n for_o 3._o measure_v 9_o and_o 11._o and_o 2._o measure_n 22._o 16_o a_o number_n be_v say_v to_o multiply_v a_o number_n when_o the_o number_n multiply_v definition_n be_v so_o oftentimes_o add_v to_o itself_o as_o there_o be_v in_o the_o number_n multiply_v unity_n and_o a_o other_o number_n be_v produce_v in_o multiplication_n be_v ever_o require_v two_o number_n the_o one_o be_v whereby_o you_o multiply_v common_o call_v the_o multiply_o or_o multiplicant_fw-la the_o other_o be_v that_o which_o be_v multiply_v multiplication_n the_o number_n by_o which_o a_o other_o be_v multiply_v namely_o the_o multiplyer_n be_v say_v to_o multiply_v as_o if_o you_o will_v multiply_v 4._o by_o 3_o then_o be_v three_o say_v to_o multiply_v 4_o therefore_o according_a to_o this_o definition_n because_o in_o 3._o there_o be_v three_o unity_n add_v 4,3_o time_n to_o itself_o say_v 3._o time_n 4_o so_o shall_v you_o bring_v forth_o a_o other_o number_n namely_o 12_o which_o be_v the_o sum_n produce_v of_o that_o multiplication_n and_o so_o of_o all_o other_o multiplication_n 17_o when_o two_o number_n multiply_v themselves_o the_o one_o the_o other_o produce_v a_o other_o definition_n the_o number_n produce_v be_v call_v a_o plain_a or_o superficial_a number_n and_o the_o number_n which_o muliply_v themselves_o the_o one_o by_o the_o other_o be_v the_o side_n of_o that_o number_n as_o let_v these_o two_o number_n 3._o and_o 6_o multiply_v the_o one_o the_o other_o say_v 3._o time_n 6_o or_o six_o time_n 3_o they_o shall_v produce_v 18._o this_o number_n 18._o thus_o produce_v be_v call_v a_o plain_a number_n or_o a_o superficial_a number_n and_o the_o two_o multiply_a number_n which_o produce_v 12_o namely_o 3._o and_o 6_o be_v the_o side_n of_o the_o same_o superficial_a or_o plain_a number_n that_o be_v the_o length_n and_o breadth_n thereof_o likewise_o if_o 9_o multiply_v 11_o or_o eleven_o nine_o there_o shall_v be_v produce_v 99_o a_o plain_a number_n who_o side_n be_v the_o two_o number_n 9_o and_o 11_o â—_o as_o the_o length_n and_o breadth_n of_o the_o same_o they_o be_v call_v plain_a and_o superficial_a number_n number_n because_o be_v describe_v by_o their_o unity_n on_o a_o plain_a superficies_n they_o represent_v some_o superficial_a form_n or_o figure_n geometrical_a have_v length_n and_o breadth_n as_o you_o see_v of_o this_o example_n and_o so_o of_o other_o and_o all_o such_o plain_a or_o superficial_a number_n do_v ever_o represent_v right_o angle_v figure_n as_o appear_v in_o the_o example_n definition_n 18_o when_o three_o number_n multiply_v together_o the_o one_o into_o the_o other_o produce_v any_o number_n the_o number_n produce_v be_v call_v a_o solid_a number_n and_o the_o number_n multiply_v themselves_o the_o one_o into_o the_o outstretch_o be_v the_o side_n thereof_o as_o take_v these_o three_o number_n 3.4_o 5._o multiply_v the_o one_o into_o the_o other_o first_o 4._o into_o 5._o say_n four_o time_n 5._o be_v 20_o then_o multiply_v that_o number_n produce_v name_o 20._o into_o 3_o which_o be_v the_o three_o number_n so_o shall_v you_o produce_v 60._o which_o be_v a_o solid_a number_n and_o the_o three_o number_n which_o produce_v the_o number_n namely_o 3.4_o and_o 5._o be_v the_o side_n of_o the_o same_o and_o they_o be_v call_v solid_a number_n number_n because_o be_v describe_v by_o their_o unity_n they_o represent_v solid_a and_o bodylicke_a figure_n of_o geometry_n which_o have_v length_n breadth_n and_o thickness_n as_o you_o see_v this_o number_n 60._o express_v here_o by_o his_o unity_n who_o length_n be_v his_o side_n 5_o his_o breadth_n be_v 3_o and_o thickness_n 4._o and_o thus_o may_v you_o do_v of_o all_o other_o three_o number_n multiply_v the_o one_o the_o other_o definition_n 19_o a_o square_a number_n be_v that_o which_o be_v equal_o equal_a or_o that_o which_o be_v contain_v under_o two_o equal_a number_n as_o multiply_v two_o equal_a number_n the_o one_o into_o the_o other_o as_o 9_o by_o 9_o you_o shall_v produce_v 81_o which_o be_v a_o square_a number_n euclid_n call_v it_o a_o number_n equal_o equal_a because_o it_o be_v produce_v of_o the_o multiplication_n of_o two_o equal_a number_n the_o one_o into_o the_o other_o which_o number_n be_v also_o say_v in_o the_o second_o definition_n to_o contain_v a_o square_a number_n as_o in_o the_o definition_n of_o the_o second_o book_n two_o line_n be_v say_v to_o contain_v a_o square_a or_o a_o parallelogram_n figure_n it_o be_v call_v a_o square_a number_n number_n because_o be_v describe_v by_o his_o unity_n it_o represent_v the_o figure_n of_o a_o square_a in_o geometry_n as_o you_o here_o see_v do_v the_o number_n 81._o who_o side_n that_o be_v to_o say_v who_o length_n and_o breadth_n be_v 9_o and_o 9_o equal_a number_n which_o also_o be_v say_v to_o contain_v the_o square_a number_n 81_o and_o so_o of_o other_o definition_n 20_o a_o cube_fw-la number_n be_v that_o which_o be_v equal_o equal_a equal_o or_o that_o which_o be_v contain_v under_o three_o equal_a number_n as_o multiply_v three_o equal_a number_n the_o one_o into_o the_o other_o as_z 9_z 9_z and_o 9_z first_o 9_o by_o 9_o so_o shall_v you_o have_v 81_o which_o again_o multiply_v by_o 9_o so_o shall_v you_o produce_v 729._o which_o be_v a_o cube_fw-la number_n and_o euclid_n call_v
there_o be_v two_o proportional_a number_n but_o how_o many_o number_n fall_v in_o continual_a proportion_n between_o c_o and_o d_o so_o many_o by_o the_o 8._o of_o the_o eight_o fall_n there_o between_o the_o number_n that_o have_v the_o same_o proportion_n with_o they_o wherefore_o between_o a_o and_o b_o there_o be_v two_o mean_v proportional_a number_n which_o let_v be_v e_o and_o f._n and_o forasmuch_o as_o there_o be_v four_o number_n in_o continual_a proportion_n namely_o a_o e_o f_o b_o and_o a_o be_v a_o cube_fw-la number_n therefore_o by_o the_o 2d_o of_o the_o eight_o b_o also_o be_v a_o cube_fw-la number_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o flussates_n flussates_n between_o a_o square_a number_n and_o a_o number_n that_o be_v not_o a_o square_a number_n fall_v not_o the_o proportion_n of_o one_o square_a number_n to_o a_o other_o for_o if_o the_o first_o be_v a_o square_a number_n the_o second_o also_o shall_v be_v a_o square_a number_n which_o be_v contrary_a to_o the_o supposition_n likewise_o between_o a_o cube_fw-la number_n and_o a_o number_n that_o be_v no_o cube_fw-la number_n fall_v not_o the_o proportion_n of_o one_o cube_fw-la number_n to_o a_o other_o for_o if_o the_o first_o be_v a_o cube_fw-la number_n the_o second_o also_o shall_v be_v a_o cube_fw-la number_n which_o be_v contrary_a to_o the_o supposition_n &_o therefore_o impossible_a ¶_o the_o 24._o theorem_a the_o 26._o proposition_n like_o plain_a number_n be_v in_o the_o same_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 svppose_v that_o a_o and_o b_o be_v like_o plain_a number_n then_o i_o say_v that_o a_o be_v unto_o b_o in_o the_o same_o proportion_n that_o a_o square_a number_n be_v to_o a_o square_a number_n for_o forasmuch_o as_o a_o b_o be_v like_o plain_a number_n construction_n therefore_o between_o a_o and_o b_o there_o fall_v one_o mean_a proportional_a number_n by_o the_o 18._o of_o the_o eight_o let_v there_o fall_v such_o a_o number_n and_o let_v the_o same_o be_v c._n and_o by_o the_o 35._o of_o the_o seven_o take_v the_o three_o lest_o number_n that_o have_v one_o and_o the_o same_o proportion_n with_o a_o c_o b_o and_o let_v the_o same_o be_v d_o e_o f_o wherefore_o by_o the_o corollary_n of_o the_o 2d_o â—_z of_o the_o eight_o their_o 〈◊〉_d that_o be_v d_o f_o be_v square_a number_n and_o for_o that_o as_z d_z be_v to_o fletcher_n so_o be_v a_o to_o b_o by_o the_o 14._o of_o the_o seven_o and_o d_z f_o be_v square_a number_n therefore_o a_o be_v unto_o b_o in_o that_o proportion_n that_o a_o square_a number_n be_v unto_o a_o square_a numâ—er_n which_o be_v require_v to_o be_v prove_v the_o 25._o theorem_a the_o 27._o proposition_n like_o solid_a number_n be_v in_o the_o same_o proportion_n the_o one_o to_o the_o other_o that_o a_o cube_fw-la number_n be_v to_o a_o cube_fw-la number_n svppose_v that_o a_o a_o and_o b_o be_v like_o solid_a number_n then_o i_o say_v that_o a_o be_v unto_o b_o in_o the_o same_o proportion_n that_o a_o cube_fw-la numb_a be_v to_o to_o a_o cube_fw-la number_n for_o forasmuch_o as_o a_o b_o be_v like_o solid_a number_n construction_n therefore_o by_o the_o 19_o of_o the_o eight_o between_o a_o and_o b_o there_o fall_v two_o mean_v proportional_a number_n let_v there_o fall_v two_o such_o number_n and_o let_v the_o same_o be_v c_o and_o d._n and_o take_v by_o the_o 35._o of_o the_o seven_o the_o least_o number_n that_o have_v one_o and_o the_o same_o proportion_n with_o a_o c_o d_o b_o demonstration_n and_o equal_a also_o with_o they_o in_o multitude_n and_o let_v the_o same_o be_v e_o f_o g_o h._n wherefore_o by_o the_o corollary_n of_o the_o 2._o of_o the_o eight_o their_o extreme_n that_o be_v eh_o be_v cube_fw-la number_n but_o as_o e_o be_v to_o h_o so_o be_v a_o to_o b._n wherefore_o a_o be_v unto_o b_o in_o the_o same_o proportion_n that_o a_o cube_fw-la number_n be_v to_o a_o cube_fw-la number_n which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o corollary_n add_v by_o flussates_n if_o two_o nnmber_n be_v in_o the_o same_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 those_o two_o number_n shall_v be_v like_o superficial_a number_n flussates_n and_o if_o they_o be_v in_o the_o same_o proportion_n the_o one_o to_o the_o other_o that_o a_o cube_fw-la number_n be_v to_o a_o cube_fw-la number_n they_o shall_v be_v like_o solid_a number_n first_o let_v the_o number_v a_o have_v unto_o the_o number_n b_o the_o same_o proportion_n that_o the_o square_a number_n c_o have_v to_o the_o square_a number_n d._n then_o i_o say_v that_o a_o and_o b_o be_v like_o superficial_a number_n for_o forasmuch_o as_o between_o the_o square_a number_n c_o and_o d_o there_o fall_v a_o mean_a proportional_a by_o the_o 11._o of_o this_o book_n there_o shall_v also_o between_o a_o and_o b_o which_o have_v one_o and_o the_o same_o proportion_n with_o c_o and_o d_o fall_v a_o mean_a proportional_a by_o the_o 8._o of_o this_o book_n wherefore_o a_o and_o b_o be_v like_o superficial_a number_n by_o the_o 20._o of_o this_o book_n but_o if_o a_o be_v unto_o b_o as_z the_o cube_fw-la number_n c_z be_v to_o the_o cube_fw-la number_n d._n then_o be_v a_o &_o b_o like_v solid_a number_n for_o forasmuch_o as_o c_z and_o d_z be_v cube_fw-la number_n there_o fall_v between_o they_o too_o mean_v proportional_a number_n by_o the_o 12._o of_o this_o book_n and_o therefore_o by_o the_o 8._o of_o the_o same_o between_o a_o and_o b_o which_o be_v in_o the_o same_o proportion_n that_o c_z be_v to_o d_z there_o fall_v also_o two_o mean_a proportional_a number_n wherefore_o by_o the_o 21._o of_o this_o book_n a_o and_o b_o be_v like_o solid_a number_n an_o other_o corollary_n add_v also_o by_o flussates_n if_o a_o number_n multiply_v a_o square_a number_n produce_v not_o a_o square_a number_n the_o say_a number_n multiply_v shall_v bâ—_z no_o square_a number_n flussates_n for_o if_o it_o shall_v be_v a_o square_a number_n than_o shall_v it_o and_o the_o number_n multiply_v be_v like_o superficial_a number_n by_o reason_n they_o be_v square_a number_n have_v a_o mean_a proportional_a by_o the_o 18._o of_o this_o book_n and_o the_o number_n produce_v of_o the_o say_v mean_a shall_v be_v equal_a to_o the_o number_n contain_v under_o the_o extreme_n which_o be_v square_a number_n by_o the_o 20._o of_o the_o seven_o wherefore_o the_o number_n produce_v of_o the_o extreme_n be_v equal_a to_o the_o square_a number_n produce_v of_o the_o mean_a shall_v be_v a_o square_a number_n but_o the_o say_a number_n by_o supposition_n be_v no_o square_a number_n wherefore_o neither_o be_v the_o number_n multiply_v the_o square_a number_n a_o square_a number_n the_o first_o part_n of_o the_o first_o corollary_n be_v the_o converse_n of_o the_o 26._o proposition_n of_o this_o book_n and_o have_v some_o use_n in_o the_o ten_o book_n the_o second_o part_n of_o the_o same_o also_o be_v the_o converse_n of_o the_o 27._o proposition_n of_o the_o same_o the_o end_n of_o the_o eight_o book_n of_o euclides_n element_n ¶_o the_o nine_o book_n of_o euclides_n element_n in_o this_o nine_o book_n euclid_n continue_v his_o purpose_n touch_v number_n partly_o prosecute_a thing_n more_o full_o book_n which_o be_v before_o somewhat_o speak_v of_o as_o of_o square_a and_o cube_fw-la number_n and_o partly_o set_v out_o the_o nature_n and_o propriety_n of_o such_o kind_n of_o number_n as_o have_v not_o yet_o be_v entreat_v of_o which_o yet_o be_v most_o necessary_a to_o be_v know_v as_o be_v number_n even_o and_o odd_a who_o passion_n and_o condition_n be_v in_o this_o book_n large_o teach_v with_o their_o composition_n and_o subduction_n of_o the_o one_o from_o the_o other_o with_o many_o other_o general_a and_o special_a thing_n to_o be_v note_v worthy_a the_o knowledge_n ¶_o the_o 1._o theorem_a the_o 1._o proposition_n if_o two_o like_o plain_a number_n multiply_v the_o one_o the_o other_o produce_v any_o number_n the_o number_n of_o they_o produce_v shall_v be_v a_o square_a number_n svppose_v that_o a_o and_o b_o be_v two_o like_o plain_a number_n and_o let_v a_o multiply_v b_o produce_v the_o number_n c._n then_o i_o say_v that_z c_z be_v a_o square_a number_n for_o let_v a_o multiply_v himself_o produce_v d._n wherefore_o d_z be_v a_o square_a number_n demonstration_n and_o forasmuch_o as_o a_o multiply_v himself_o produce_v d_z and_o multiply_v b_o produce_v c_z therefore_o by_o the_o 17._o of_o the_o seven_o as_o a_o be_v to_o b_o so_o be_v d_z to_o c._n and_o forasmuch_o as_o a_o b_o be_v like_o plain_a number_n therefore_o by_o the_o 18._o of_o
number_n be_v make_v of_o unity_n and_o therefore_o can_v a_o point_n be_v a_o common_a part_n of_o all_o line_n and_o measure_v they_o as_o unity_n be_v a_o common_a part_n of_o all_o number_n and_o measure_v they_o unity_n take_v certain_a time_n make_v any_o number_n for_o there_o be_v not_o in_o any_o number_n infinite_a unity_n but_o a_o point_n take_v certain_a time_n yea_o as_o often_o as_o you_o listen_v never_o make_v any_o line_n for_o that_o in_o every_o line_n there_o be_v infinite_a point_n wherefore_o line_n figure_n and_o body_n in_o geometry_n be_v oftentimes_o incommensurable_a and_o irrational_a now_o which_o be_v rational_a and_o which_o irrational_a which_o commensurable_a and_o which_o incommensurable_a how_o many_o and_o how_o sundry_a sort_n and_o kind_n there_o be_v of_o they_o what_o be_v their_o nature_n passion_n and_o property_n do_v euclid_n most_o manifest_o show_v in_o this_o book_n and_o demonstrate_v they_o most_o exact_o this_o ten_o book_n have_v 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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
first_o set_v they_o also_o must_v needs_o be_v at_o the_o least_o commensurable_a in_o power_n the_o one_o to_o the_o other_o for_o forasmuch_o as_o their_o square_n be_v rational_a they_o shall_v be_v commensurable_a to_o the_o square_n of_o the_o rational_a line_n first_o set_v wherefore_o by_o the_o 12._o of_o this_o book_n they_o be_v also_o commensurable_a the_o one_o to_o the_o other_o wherefore_o their_o line_n be_v at_o the_o least_o commensurable_a in_o power_n the_o one_o to_o the_o other_o and_o it_o be_v possible_a also_o that_o they_o may_v be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o for_o suppose_v that_o a_o be_v a_o rational_a liâ—e_z first_o set_v and_o let_v the_o line_n b_o be_v unto_o the_o same_o rational_a line_n a_o commensurable_a in_o power_n only_o that_o be_v incommensurable_a in_o length_n unto_o it_o let_v there_o be_v also_o a_o other_o line_n c_o commensurable_a in_o length_n to_o the_o line_n b_o which_o be_v possible_a by_o the_o principle_n of_o this_o book_n now_o by_o the_o 13._o of_o the_o ten_o it_o be_v manifest_a that_o the_o line_n c_o be_v incommensurable_a in_o length_n unto_o the_o line_n a._n but_o the_o square_n of_o the_o line_n a_o be_v commensurable_a to_o the_o square_n of_o the_o line_n b_o by_o supposition_n and_o the_o square_a of_o the_o line_n c_o be_v also_o commensurable_a to_o the_o square_n of_o the_o line_n b_o by_o supposition_n wherefore_o by_o the_o 12._o of_o this_o book_n the_o square_a of_o the_o line_n c_o be_v commensurable_a to_o the_o square_n of_o the_o line_n a._n wherefore_o by_o the_o definition_n the_o line_n c_o shall_v be_v rational_a commensurable_a in_o power_n only_o to_o the_o line_n a_o as_o also_o be_v the_o line_n b._n wherefore_o there_o be_v give_v two_o rational_a line_n commensurable_a in_o power_n only_o to_o the_o rational_a line_n first_o set_v and_o commensurable_a in_o length_n the_o one_o to_o the_o other_o here_o be_v to_o be_v note_v which_o thing_n also_o we_o before_o note_v in_o the_o definition_n that_o campane_n and_o other_o which_o follow_v he_o bring_v in_o these_o phrase_n of_o speech_n to_o call_v some_o line_n rational_a in_o power_n only_o cause_n and_o other_o some_o rational_a in_o length_n and_o in_o power_n which_o we_o can_v find_v that_o euclid_n ever_o use_v for_o these_o word_n in_o length_n and_o in_o power_n be_v never_o refer_v to_o rationality_n or_o irrationalitie_n but_o always_o to_o the_o commensurabilitie_n or_o incommensurablitie_n of_o line_n which_o pervert_v of_o word_n as_o be_v there_o declare_v have_v much_o increase_v the_o difficulty_n and_o obscureness_n of_o this_o book_n book_n and_o now_o i_o think_v it_o good_a again_o to_o put_v you_o in_o mind_n that_o in_o these_o proposition_n which_o follow_v we_o must_v ever_o have_v before_o our_o eye_n the_o rational_a line_n first_o set_v note_n unto_o which_o other_o line_n compare_v be_v either_o rational_a or_o irrational_a according_a to_o their_o commensurability_n or_o incommensurabilitie_n ¶_o the_o 16._o theorem_a the_o 19_o proposition_n a_o rectangle_n figure_n comprehend_v under_o right_a line_n commensurable_a in_o length_n be_v rational_a according_a to_o one_o of_o the_o foresay_a way_n be_v rational_a svppose_v that_o this_o rectangle_n figure_n ac_fw-la be_v comprehend_v under_o these_o right_a line_n ab_fw-la and_o bc_o be_v commensurable_a in_o length_n and_o rational_a according_a to_o one_o of_o the_o foresay_a way_n then_o i_o say_v that_o the_o superficies_n ac_fw-la be_v rational_a describe_v by_o the_o 46._o oâ—_n the_o first_o upon_o the_o line_n ab_fw-la a_o square_a ad._n construction_n wherefore_o that_o square_a ad_fw-la be_v rational_a by_o the_o definition_n demonstration_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v commensurable_a in_o length_n unto_o the_o line_n bc_o and_o the_o line_n ab_fw-la be_v equal_a unto_o the_o line_n bd_o therefore_o the_o line_n bd_o be_v commensurable_a in_o length_n unto_o the_o line_n bc._n and_o as_o the_o line_n bd_o be_v to_o the_o line_n bc_o so_o be_v the_o square_a dam_n to_o the_o superficies_n ac_fw-la by_o the_o first_o of_o the_o six_o but_o it_o be_v prove_v that_o the_o line_n bd_o be_v commensurable_a unto_o the_o line_n bc_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a dam_n be_v commensurable_a unto_o the_o rectangle_n superficies_n ac_fw-la but_o the_o square_a dam_n be_v rational_a wherefore_o the_o rectangle_n superficies_n ac_fw-la also_o be_v rational_a by_o the_o definition_n a_o rectangle_n figure_n therefore_o comprehend_v under_o right_a line_n commensurable_a in_o length_n be_v rational_a accord_v to_o one_o of_o the_o foresay_a way_n be_v rational_a which_o be_v require_v to_o be_v prove_v proposition_n where_o as_o in_o the_o former_a demonstration_n the_o square_n be_v describe_v upon_o the_o less_o line_n we_o may_v also_o demonstrate_v the_o proposition_n if_o we_o describe_v the_o square_n upon_o the_o great_a line_n and_o that_o after_o this_o manner_n suppose_v that_o the_o rectangle_n superficies_n bc_o be_v contain_v of_o these_o unequal_a line_n ab_fw-la and_o ac_fw-la which_o let_v be_v rational_a commensurable_a the_o one_o to_o the_o other_o in_o length_n and_o let_v the_o line_n ac_fw-la be_v the_o great_a case_n and_o upon_o the_o line_n ac_fw-la describe_v the_o square_a dc_o then_o i_o say_v that_o the_o parallelogram_n bc_o be_v rational_a length_n for_o the_o line_n ac_fw-la be_v commensurable_a in_o length_n unto_o the_o line_n ab_fw-la by_o supposition_n and_o the_o line_n dam_n be_v equal_a to_o the_o line_n ac_fw-la wherefore_o the_o line_n dam_n be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la but_o what_o proportion_n the_o line_n dam_n have_v to_o the_o line_n ab_fw-la the_o same_o have_v the_o square_a dc_o to_o the_o paraâ—lelogramme_n câ—_n by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10._o of_o this_o book_n the_o square_a dc_o be_v commensurable_a to_o the_o parallelogram_n cb._n but_o it_o be_v manifest_a that_o the_o square_a dc_o be_v rational_a for_o that_o it_o be_v the_o square_a of_o a_o rational_a line_n namely_o ac_fw-la wherefore_o by_o the_o definition_n the_o parallelogram_n also_o cb_o be_v rational_a moreover_o forasmuch_o as_o those_o two_o former_a demonstration_n seem_v to_o speak_v of_o that_o parallelogram_n which_o be_v make_v of_o two_o line_n of_o which_o any_o one_o may_v be_v the_o liâ—e_z first_o set_v which_o be_v call_v the_o first_o rational_a line_n from_o which_o we_o say_v ought_v to_o be_v take_v the_o measure_n of_o the_o other_o line_n compare_v unto_o it_o and_o the_o other_o be_v commensurable_a in_o length_n to_o the_o same_o first_o rational_a line_n length_n which_o be_v the_o first_o kind_n of_o rational_a line_n commensurable_a in_o length_n i_o think_v it_o good_a here_o to_o set_v a_o other_o case_n of_o the_o other_o kind_n of_o rational_a line_n of_o line_n i_o say_v rational_a commensurable_a in_o length_n compare_v to_o a_o other_o rational_a line_n first_o set_v to_o declare_v the_o general_a truth_n of_o this_o theorem_a and_o that_o we_o may_v see_v that_o this_o particle_n according_a to_o any_o of_o the_o foresay_a way_n be_v not_o here_o in_o vain_a put_v now_o then_o suppose_v first_o a_o rational_a line_n ab_fw-la let_v there_o be_v also_o a_o parallelogram_n cd_o contain_v under_o the_o line_n ce_fw-fr and_o ed_z which_o line_n let_v be_v rational_a that_o be_v commensurable_a in_o length_n to_o the_o â—irst_n rational_a line_n propound_v ab_fw-la howbeit_o let_v those_o two_o line_n ce_fw-fr and_o ed_z be_v diverse_a and_o unequal_a line_n unto_o the_o first_o rational_a line_n ab_fw-la then_o i_o say_v that_o the_o parallelogram_n cd_o be_v rational_a case_n describe_v the_o square_a of_o the_o line_n de_fw-fr which_o let_v be_v df._n first_o it_o be_v manifest_a by_o the_o 12._o of_o this_o book_n that_o the_o line_n ce_fw-fr &_o ed_z be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o for_o either_o of_o they_o be_v suppose_v to_o be_v commensurable_a in_o length_n unto_o the_o line_n ab_fw-la but_o the_o line_n ed_z be_v equal_a to_o the_o line_n ef._n wherefore_o the_o line_n ce_fw-fr be_v commensurable_a in_o length_n to_o the_o line_n bf_o but_o 〈◊〉_d the_o line_n ce_fw-fr be_v â—o_o the_o line_n â—_o f_o â—o_o be_v the_o parallelogram_n cd_o to_o the_o square_a df_o by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10._o of_o this_o book_n the_o parallelogram_n cd_o shall_v be_v commensurable_a to_o the_o square_a df._n but_o the_o square_a df_o be_v commensurable_a to_o the_o square_n of_o the_o line_n ab_fw-la which_o be_v the_o first_o rational_a line_n propound_v wherefore_o by_o the_o 12._o of_o this_o book_n the_o parallelogram_n cd_o be_v commensurable_a to_o the_o square_n of_o the_o line_n ab_fw-la but_o the_o square_n of_o
line_n be_v when_o the_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o the_o less_o part_n by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o it_o and_o neither_o part_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v as_o suppose_v the_o line_n ce_fw-fr to_o be_v a_o binomial_a line_n who_o part_n be_v join_v together_o in_o the_o point_n d_o and_o let_v the_o square_n of_o the_o line_n cd_o the_o great_a part_n exceed_v the_o square_n of_o the_o less_o part_n de_fw-fr by_o the_o square_n of_o the_o line_n fg_o and_o let_v the_o line_n fg_o be_v commensurable_a in_o length_n to_o the_o line_n cd_o the_o great_a part_n of_o the_o binomial_a moreover_o let_v neither_o the_o great_a part_v cd_o nor_o the_o less_o part_n de_fw-fr be_v commensurable_a in_o length_n to_o the_o rational_a line_n ab_fw-la then_o be_v the_o line_n ce_fw-fr by_o this_o definition_n a_o three_o binomial_a line_n a_o four_o binomial_a line_n be_v definition_n when_o the_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o the_o less_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o great_a part_n and_o the_o great_a be_v also_o commensurable_a in_o length_n to_o the_o rational_a line_n as_o let_v the_o line_n ce_fw-fr be_v a_o binomial_a line_n who_o part_n let_v be_v cd_o &_o de_fw-fr &_o let_v the_o square_n of_o the_o line_n cd_o the_o great_a part_n exceed_v the_o square_n of_o the_o line_n de_fw-fr the_o less_o by_o the_o square_n of_o the_o line_n fg._n and_o let_v the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n cd_o the_o great_a let_v also_o the_o line_n cd_o the_o great_a part_n be_v commensurable_a in_o length_n unto_o the_o rational_a line_n ab_fw-la then_o by_o this_o definition_n the_o line_n ce_fw-fr be_v a_o four_o binomial_a line_n a_o five_o binomial_a line_n be_v definition_n when_o the_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o the_o less_o part_n by_o the_o square_n of_o a_o line_n incommensurable_a unto_o it_o in_o length_n and_o the_o less_o part_n also_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v as_o suppose_v that_o ce_fw-fr be_v a_o binomial_a line_n who_o great_a part_n let_v be_v cd_o and_o let_v the_o less_o part_n be_v de._n and_o let_v the_o square_n of_o the_o line_n cd_o exceed_v the_o square_n of_o the_o line_n de_fw-fr by_o the_o square_n of_o the_o line_n fg_o which_o let_v be_v incommensurable_a in_o length_n unto_o the_o line_n cd_o the_o great_a part_n of_o the_o binomial_a line_n and_o let_v the_o line_n de_fw-fr the_o second_o part_n of_o the_o binomial_a line_n be_v commensurable_a in_o length_n unto_o the_o rational_a line_n ab_fw-la so_o be_v the_o line_n ce_fw-fr by_o this_o definition_n a_o five_o binomial_a line_n a_o six_o binomial_a line_n be_v definition_n when_o the_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o the_o less_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o it_o and_o neither_o part_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v as_o let_v the_o line_n ce_fw-fr be_v a_o binomial_a line_n divide_v into_o his_o name_n in_o the_o point_n d._n the_o square_n of_o who_o great_a part_n cd_o let_v exceed_v the_o square_n of_o the_o less_o part_n de_fw-fr by_o the_o square_n of_o the_o line_n fg_o and_o let_v the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n cd_o the_o great_a part_n of_o the_o binomial_a line_n let_v also_o neither_o cd_o the_o great_a part_n nor_o de_fw-fr the_o less_o part_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n ab_fw-la and_o so_o by_o this_o definition_n the_o line_n ce_fw-fr be_v a_o six_o binomial_a line_n so_o you_o see_v that_o by_o these_o definition_n &_o their_o example_n and_o declaration_n all_o the_o kind_n of_o binomial_a line_n be_v make_v very_o plain_a this_o be_v to_o be_v note_v that_o here_o be_v nothing_o speak_v of_o those_o line_n both_o who_o portion_n aâ—e_v commensurable_a in_o length_n unto_o the_o rational_a line_n first_o set_v for_o that_o such_o line_n can_v be_v binomial_a line_n â—or_a binomial_a line_n be_v compose_v of_o two_o rational_a line_n commensurable_a in_o power_n only_o by_o the_o 36._o of_o this_o book_n but_o line_n both_o who_o portion_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n first_o set_v be_v not_o binomial_a line_n for_o that_o the_o part_n of_o such_o line_n shall_v by_o the_o 12._o of_o this_o book_n be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o and_o so_o shall_v they_o not_o be_v such_o line_n as_o be_v require_v to_o the_o composition_n of_o a_o binomial_a line_n moreover_o such_o line_n shall_v not_o be_v irrational_a but_o rational_a for_o that_o they_o be_v commensurable_a tâ—â—ch_o of_o the_o part_n whereof_o they_o be_v compose_v by_o the_o 15._o oâ—_n this_o book_n and_o therefore_o they_o shall_v be_v rational_a for_o that_o the_o line_n which_o composâ—_n they_o be_v rational_a ¶_o the_o 13._o problem_n the_o 48._o proposition_n to_o find_v out_o a_o first_o binomial_a line_n composition_n take_v two_o number_n ac_fw-la and_o cb_o &_o let_v they_o be_v such_o that_o the_o number_n which_o be_v make_v of_o they_o both_o add_v together_o namely_o ab_fw-la have_v unto_o one_o of_o they_o 〈◊〉_d unto_o bc_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n â—ut_z unto_o the_o other_o namely_o unto_o ca_n let_v it_o not_o have_v that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n such_o as_o be_v every_o square_a number_n which_o may_v be_v divide_v into_o a_o square_a number_n and_o into_o a_o number_n not_o square_a construction_n take_v also_o a_o certain_a rational_a line_n and_o let_v the_o same_o be_v d._n and_o unto_o the_o line_n d_o let_v the_o line_n of_o be_v commensurable_a in_o length_n wherefore_o the_o line_n of_o be_v rational_a and_o as_o the_o number_n ab_fw-la be_v to_o the_o number_n ac_fw-la so_o let_v the_o square_n of_o the_o line_n of_o be_v to_o the_o square_n of_o a_o other_o â—iâ—e_n namely_o of_o fg_o by_o the_o corollary_n of_o the_o six_o of_o the_o ten_o wherefore_o the_o square_a of_o the_o line_n of_o have_v to_o the_o square_n of_o the_o line_n fg_o that_o proportion_n that_o number_n have_v to_o number_v wherefore_o the_o square_a of_o the_o line_n of_o be_v commensurable_a to_o the_o square_n of_o the_o line_n fg_o by_o the_o 6._o of_o this_o book_n and_o the_o line_n of_o be_v rational_a demonstration_n wherefore_o the_o line_n fg_o also_o be_v rational_a and_o forasmuch_o as_o the_o number_n ab_fw-la have_v not_o to_o the_o number_n ac_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n neither_o shall_v the_o square_a of_o the_o line_n of_o have_v to_o the_o square_n of_o the_o line_n fg_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n of_o be_v incommensurable_a in_o length_n to_o the_o line_n fg_o by_o the_o 9_o of_o this_o book_n wherefore_o the_o line_n of_o and_o fg_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n eglantine_n be_v a_o binomial_a line_n by_o the_o 36._o of_o the_o ten_o i_o say_v also_o that_o it_o be_v a_o â—irst_n binomial_a line_n for_o for_o that_o as_o the_o 〈◊〉_d basilius_n be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n â—g_z but_o the_o number_n basilius_n be_v great_a than_o the_o number_n ac_fw-la wherefore_o the_o square_a of_o the_o line_n â—f_n be_v also_o great_a than_o the_o square_a oâ—_n the_o line_n fg._n unto_o the_o square_n of_o the_o line_n of_o let_v the_o square_n of_o the_o line_n fg_o and_o h_o be_v equal_a which_o how_o to_o find_v out_o be_v teach_v in_o the_o assumpt_n put_v â—ftâ—r_v the_o 13._o of_o the_o tâ—nth_o and_o fâ—r_v that_o as_o thâ—_z number_n basilius_n be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n fg_o therefore_o by_o coâ—uersion_n or_o eâ—ersion_n of_o proportion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o as_o the_o number_n ab_fw-la be_v to_o the_o number_n bc_o so_o be_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n h._n but_o the_o number_n ab_fw-la have_v to_o the_o number_n bc_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o also_o the_o square_a of_o the_o line_n of_o have_v to_o the_o square_n of_o the_o line_n h_o
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 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superficies_n stqr_n unto_o the_o superficies_n mnop_n and_o so_o by_o this_o definition_n these_o two_o superficiece_n be_v say_v to_o be_v in_o like_a sort_n incline_v definition_n 6_o parallel_n plain_a superficiece_n be_v those_o which_o be_v produce_v or_o extend_v any_o way_n never_o touch_v or_o concur_v together_o neither_o need_v this_o definition_n any_o declaration_n but_o be_v very_o easy_a to_o be_v understand_v by_o the_o definition_n of_o parallel_a line_n â—or_a as_o they_o be_v draw_v on_o any_o part_n never_o touch_v or_o come_v together_o so_o parallel_v plain_a superâ—icieces_n be_v such_o which_o admit_v no_o touch_n that_o be_v be_v produce_v any_o way_n infinite_o never_o meet_v or_o come_v together_o defâ—inition_n 7_o like_o solid_a or_o bodily_a figure_n be_v such_o which_o be_v contain_v under_o like_o plain_a superficiece_n and_o equal_a in_o multitude_n what_o plain_a superâ—icieces_n be_v call_v like_o have_v in_o the_o begin_n of_o the_o six_o book_n be_v sufficient_o declare_v now_o when_o solid_a figure_n or_o body_n be_v contain_v under_o such_o like_a plain_a superficiece_n as_o there_o be_v define_v and_o equal_a in_o number_n that_o be_v that_o the_o one_o solid_a have_v as_o many_o in_o number_n as_o the_o other_o in_o their_o side_n and_o limit_n they_o be_v call_v like_o solid_a figure_n or_o like_a body_n definition_n 8_o equal_a and_o like_a solid_a or_o bodily_a figure_n be_v those_o which_o be_v contain_v under_o like_a superficiece_n and_o equal_a both_o in_o multitude_n and_o in_o magnitude_n in_o like_o solid_a figure_n it_o be_v sufficient_a that_o the_o superficiece_n which_o contain_v they_o be_v like_a and_o equal_a in_o number_n only_o but_o in_o like_a solid_a figure_n and_o equal_a it_o be_v necessary_a that_o the_o like_a superficiece_n contain_v they_o be_v also_o equal_a in_o magnitude_n so_o that_o beside_o the_o likeness_n between_o they_o they_o be_v each_o be_v compare_v to_o his_o correspondent_a superficies_n oâ—_n one_o greatness_n and_o that_o their_o area_n or_o field_n be_v equal_a when_o such_o superâ—icieces_n contain_v body_n or_o solid_n then_o be_v such_o body_n equal_a and_o like_a solid_n or_o body_n dissilition_n 9_o a_o solid_a or_o bodily_a angle_n be_v a_o inclination_n of_o more_o than_o two_o line_n to_o all_o the_o line_n which_o touch_v themselves_o mutual_o and_o be_v not_o in_o one_o and_o the_o self_n same_o superficies_n or_o else_o thus_o a_o solid_a or_o bodily_a angle_n be_v that_o which_o be_v contain_v under_o more_o than_o two_o plain_a angle_n not_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n but_o consist_v all_o at_o one_o point_n of_o a_o solid_a angle_n do_v euclid_n here_o geve_v two_o several_a definition_n the_o first_o be_v give_v by_o the_o concourse_n and_o touch_n of_o many_o line_n the_o second_o by_o the_o touch_n &_o concourse_n of_o many_o superficial_a angle_n and_o both_o these_o definition_n tend_v to_o one_o 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the_o second_o definition_n what_o superficial_a or_o plain_a angle_n be_v have_v be_v teach_v before_o in_o the_o first_o book_n namely_o that_o it_o be_v the_o touch_n of_o two_o right_a line_n and_o as_o a_o superâ—iciall_a or_o plain_a angle_n be_v cause_v &_o contain_v of_o right_a line_n so_o si_fw-mi a_o solid_a angle_n cause_v &_o contain_v of_o plain_a superficial_a angle_n two_o right_a line_n touch_v together_o make_v a_o plain_a angle_n but_o two_o plain_a angle_n join_v together_o can_v not_o make_v a_o solid_a angle_n but_o according_a to_o the_o definition_n they_o must_v be_v more_o they_o two_o as_z three_o â—oure_n â—ive_a or_o moâ—_n which_o also_o must_v not_o be_v in_o one_o &_o the_o self_n same_o superficies_n but_o must_v be_v in_o diverse_a superficiece_n â—eeting_v at_o one_o point_n this_o definition_n be_v not_o hard_o but_o may_v easy_o be_v conceive_v in_o a_o cube_fw-la or_o a_o die_n where_o you_o see_v three_o angle_n of_o any_o three_o superficiece_n 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triangle_n afc_n and_o upon_o the_o side_n bc_o the_o triangle_n bfc_n and_o so_o bow_v the_o triangle_n raise_v up_o that_o their_o top_n namely_o the_o point_n f_o meet_a and_o join_v together_o in_o one_o point_n you_o shall_v easy_o and_o plain_o see_v how_o these_o three_o superficial_a angle_n afbbfc_a cfa_n join_v and_o close_a together_o touch_v the_o one_o the_o other_o in_o the_o point_n fletcher_n and_o so_o make_v a_o solid_a angle_n 10_o a_o pyramid_n be_v a_o solid_a figure_n contain_v under_o many_o plain_a superficiece_n set_v upon_o one_o plain_a superficies_n and_o gather_v together_o to_o one_o point_n definition_n two_o superficiece_n raise_v upon_o any_o ground_n can_v not_o make_v a_o pyramid_n for_o that_o two_o superficial_a angle_n join_v together_o in_o the_o top_n can_v as_o before_o be_v say_v make_v a_o solid_a angle_n wherefore_o when_o three_o four_o five_o or_o more_o how_o many_o soever_o superficiece_n be_v raise_v up_o from_o one_o superficies_n be_v 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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
reform_v by_o m._n dee_n describe_v for_o it_o in_o the_o plain_n especial_o if_o you_o remember_v the_o form_n of_o the_o figure_n of_o the_o 29._o proposition_n of_o this_o book_n only_o that_o which_o there_o you_o conceive_v to_o be_v the_o base_a imagine_v here_o in_o both_o the_o figure_n of_o this_o second_o case_n to_o be_v the_o upper_a superficies_n opposite_a to_o the_o base_a and_o that_o which_o be_v there_o suppose_v to_o be_v the_o upper_a superficies_n conceive_v here_o to_o be_v the_o base_a you_o may_v describe_v they_o upon_o past_v paper_n for_o your_o better_a sight_n take_v heed_n you_o note_v the_o letter_n right_o according_a as_o the_o construction_n require_v flussas_n demonstrate_v this_o proposition_n a_o otherway_o take_v only_o the_o base_n of_o the_o solid_n and_o that_o after_o this_o manner_n take_v equal_a base_n which_o yet_o for_o the_o sure_a understanding_n let_v be_v utter_o unlike_a namely_o aebf_n and_o adch_n and_o let_v one_o of_o the_o side_n of_o each_o concur_v in_o one_o &_o the_o same_o right_a line_n aed_n &_o the_o base_n be_v upon_o one_o and_o the_o self_n same_o plain_n let_v there_o be_v suppose_v to_o be_v set_v upon_o they_o parallelipipedon_n under_o one_o &_o the_o self_n same_o altitude_n then_o i_o say_v that_o the_o solid_a set_v upon_o the_o base_a ab_fw-la be_v equal_a to_o the_o solid_a set_v upon_o the_o base_a ah_o by_o the_o point_n e_o draw_v unto_o the_o line_n ac_fw-la a_o parallel_n line_n eglantine_n which_o if_o it_o fall_v without_o the_o base_a ab_fw-la produce_v the_o right_a line_n hc_n to_o the_o point_n i_o now_o forasmuch_o as_o ab_fw-la and_o ah_o be_v parallelogrmae_n therefore_o by_o the_o 24._o of_o this_o book_n the_o triangle_n aci_n and_o egl_n shall_v be_v equaliter_fw-la the_o one_o to_o the_o other_o and_o by_o the_o 4._o of_o the_o first_o they_o shall_v be_v equiangle_n and_o equal_a and_o by_o the_o first_o definition_n of_o the_o six_o and_o four_o proposition_n of_o the_o same_o they_o shall_v be_v like_a wherefore_o prism_n erect_v upon_o those_o triangle_n and_o under_o the_o same_o altitude_n that_o the_o solid_n ab_fw-la and_o ah_o aâ—e_n shall_v be_v equal_a and_o like_a by_o the_o 8._o definition_n of_o this_o book_n for_o they_o be_v contain_v under_o like_o plain_a superficiece_n equal_a both_o in_o multitude_n and_o magnitude_n add_v the_o solid_a set_v upon_o the_o base_a acle_n common_a to_o they_o both_o wherefore_o the_o solid_a set_v upon_o the_o base_a aegc_n be_v equal_a to_o the_o solid_a set_v upon_o the_o base_a aeli_n and_o forasmuch_o as_o the_o superficiece_n aebf_n and_o adhc_n be_v equal_a by_o supposition_n and_o the_o part_n take_v away_o agnostus_n be_v equal_a to_o the_o part_n take_v away_o all_o therefore_o the_o residue_n by_o shall_v be_v equal_a to_o the_o residue_n gd_o wherefore_o as_o agnostus_n be_v to_o gd_a as_o all_o be_v to_o by_o namely_o equal_n to_o equal_n but_o as_o agnostus_n be_v to_o gd_o so_o iâ—_z the_o solid_a set_v upon_o agnostus_n to_o the_o solid_a set_v upon_o gd_o by_o the_o 25._o of_o this_o book_n for_o it_o be_v cut_v by_o a_o plain_a superficies_n set_v upon_o the_o line_n ge_z which_o superficies_n be_v parallel_n to_o the_o opposite_a superficiece_n wherefore_o as_o all_o be_v to_o by_o so_o be_v the_o solid_a set_v upon_o all_o to_o the_o solid_a set_v upon_o by_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o solid_a set_v upon_o agnostus_n or_o upon_o all_o which_o be_v equal_a unto_o it_o be_v to_o the_o solid_a set_v upon_o gd_o so_o be_v the_o same_o solid_a set_v upon_o agnostus_n or_o all_o to_o the_o solid_a set_v upon_o by_o wherefore_o by_o the_o 2._o part_n of_o the_o 9_o of_o the_o five_o the_o solid_n set_v upon_o gd_o and_o by_o shall_v be_v equal_a unto_o which_o solid_n if_o you_o add_v equal_a solid_n namely_o the_o solid_a set_v upon_o agnostus_n to_o the_o solid_a set_v upon_o gd_o and_o the_o solid_a set_v upon_o all_o to_o the_o solid_a set_v upon_o by_o the_o whole_a solid_n set_v upon_o the_o base_a ah_o and_o upon_o the_o base_a ab_fw-la â—hall_v be_v equal_a wherefore_o parallelipedon_n consist_v upon_o equal_a base_n and_o be_v under_o one_o and_o the_o self_n same_o altitude_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 27._o theorem_a the_o 32._o proposition_n parallelipipedon_n be_v under_o one_o and_o the_o self_n same_o altitude_n 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 svppose_v that_o these_o parallelipipedon_n ab_fw-la and_o cd_o be_v under_o one_o &_o the_o self_n same_o altitude_n then_o i_o say_v that_o those_o parallelipipedon_n ab_fw-la and_o cd_o 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 that_o be_v that_o as_o the_o base_a ae_n be_v to_o the_o base_a cf_o so_o be_v the_o parallelipipedon_n ab_fw-la to_o the_o parallelipipedon_n cd_o construction_n upon_o the_o line_n fg_o describe_v by_o the_o 45._o of_o the_o first_o the_o parallelogram_n fh_o equal_a to_o the_o parallelogram_n ae_n and_o equiangle_n with_o the_o parallelogram_n cf._n and_o upon_o the_o base_a fh_o describe_v a_o parallelipipedom_n of_o the_o self_n same_o altitude_n that_o the_o parallelipipedom_n cd_o be_v &_o let_v the_o same_o be_v gk_o demonstration_n now_o by_o the_o 31._o of_o the_o eleven_o the_o parallelipipedon_n ab_fw-la be_v equal_a to_o the_o parallelipipedon_n gk_o for_o they_o consist_v upon_o equal_a base_n namely_o ae_n and_o fh_o and_o be_v under_o one_o and_o the_o self_n same_o altitude_n and_o forasmuch_o as_o the_o parallelipipedon_n ck_o be_v cut_v by_o a_o plain_a superficies_n dg_o be_v parallel_n to_o either_o of_o the_o opposite_a plain_a superâ—icieces_n therefore_o by_o the_o 25._o of_o the_o eleven_o as_o the_o base_a hf_o be_v to_o the_o base_a fc_n so_o be_v the_o parallelipipedon_n gk_o to_o parallelipipedon_v cd_o but_o the_o base_a hf_o be_v equal_a to_o the_o base_a ae_n and_o the_o parallelipipedon_n gk_o be_v prove_v equal_a to_o the_o parallelipipedon_n ab_fw-la wherefore_o as_o the_o base_a aâ—e_n be_v to_o the_o base_a cf_o so_o be_v the_o parallelipedon_n ab_fw-la to_o the_o parallelipipedon_n cd_o wherefore_o parallelipipedon_n be_v under_o one_o and_o the_o self_n same_o altitude_n 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 which_o be_v require_v to_o be_v demonstrate_v i_o need_v not_o to_o put_v any_o other_o figure_n for_o the_o declaration_n of_o this_o demonstration_n for_o it_o be_v easy_a to_o see_v by_o the_o figure_n there_o describe_v howbeit_o you_o may_v for_o the_o more_o full_a sight_n thereof_o describe_v solid_n of_o past_v paper_n according_a to_o the_o construction_n there_o set_v forth_o which_o will_v not_o be_v hard_o for_o you_o to_o do_v if_o you_o remember_v the_o description_n of_o such_o body_n before_o teach_v a_o corollary_n add_v by_o flussas_n equal_a parallelipipedon_n contain_v under_o one_o and_o the_o self_n same_o altitude_n have_v also_o their_o base_n equal_a for_o if_o the_o base_n shall_v be_v unequal_a the_o parallelipipedon_n also_o shall_v be_v unequal_a by_o this_o 32_o proposition_n and_o equal_a parallelipipedon_n have_v equal_a base_n have_v also_o one_o and_o the_o self_n same_o altitude_n for_o if_o they_o shall_v have_v a_o great_a altitude_n they_o shall_v exceed_v the_o equal_a parallelipipedon_n which_o have_v the_o self_n same_o altitude_n but_o if_o they_o shall_v have_v a_o less_o they_o shall_v want_v so_o much_o of_o those_o self_n same_o equal_a parallelipipedon_n the_o 28._o theorem_a the_o 33._o proposition_n like_a parallelipipedon_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 parallelipipedon_n ab_fw-la and_o cd_o be_v like_a &_o let_v the_o side_n ae_n and_o cf_o be_v side_n of_o like_a proportion_n then_o i_o say_v the_o parallelipipedon_n ab_fw-la be_v unto_o the_o parallelipipedon_n cd_o in_o treble_a proportion_n of_o that_o in_o which_o the_o side_n ae_n be_v to_o the_o side_n cf._n extend_v the_o right_a line_n ae_n ge_z and_o he_o to_o the_o point_n king_n l_o m._n construction_n and_o by_o the_o 2._o of_o the_o first_o unto_o the_o line_n cf_o put_v the_o line_n eke_o equal_a and_o unto_o the_o line_n fn_o put_v the_o line_n el_n equal_a and_o moreover_o unto_o the_o line_n fr_z put_v the_o line_n em_n equal_a and_o make_v perfect_a the_o parallelogram_n kl_o and_o the_o parallelipipedon_n ko_o demonstration_n now_o forasmuch_o as_o these_o two_o line_n eke_o and_o el_n be_v equal_a to_o these_o two_o line_n cf_o and_o fn_o but_o the_o angle_n kel_n be_v equal_a to_o the_o angle_n cfn_n for_o the_o angle_n
at_o all_o adventure_n namely_o d_o five_o g_o s_o and_o a_o right_a line_n be_v draw_v from_o the_o point_n d_o to_o the_o point_n g_o and_o a_o other_o from_o the_o point_n five_o to_o the_o point_n s._n wherefore_o by_o the_o 7._o of_o the_o eleven_o the_o line_n dg_o and_o we_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n and_o forasmuch_o as_o the_o line_n de_fw-fr be_v a_o parallel_n to_o the_o line_n bg_o therefore_o by_o the_o 24._o of_o the_o first_o the_o angle_n edt_n be_v equal_a to_o the_o angle_n bgt_fw-mi for_o they_o be_v alternate_a angle_n and_o likewise_o the_o angle_n dtv_n be_v equal_a to_o the_o angle_n gts_fw-fr now_o then_o there_o be_v two_o triangle_n that_o be_v dtv_n and_o gts_fw-fr have_v two_o angle_n of_o the_o one_o equal_a to_o two_o angle_n of_o the_o other_o and_o one_o side_n of_o the_o one_o equal_a to_o one_o side_n of_o the_o other_o namely_o the_o side_n which_o subtend_v the_o equal_a angle_n that_o be_v the_o side_n dv_o to_o the_o side_n g_n for_o they_o be_v the_o half_n of_o the_o line_n de_fw-fr and_o bg_o wherefore_o the_o side_n remain_v be_v equal_a to_o the_o side_n remain_v wherefore_o the_o line_n dt_n be_v equal_a to_o the_o line_n tg_n and_o the_o line_n vt_fw-la to_o the_o line_n it_o be_v if_o therefore_o the_o opposite_a side_n of_o a_o parallelipipedon_n be_v divide_v into_o two_o equal_a part_n and_o by_o their_o section_n be_v extend_v plain_a superficiece_n the_o common_a section_n of_o those_o plain_a superficiece_n and_o the_o diameter_n of_o the_o parallelipipedon_n do_v divide_v the_o one_o the_o other_o into_o two_o equal_a part_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o flussas_n every_o plain_a superficies_n extend_v by_o the_o centre_n of_o a_o parallelipipedon_n divide_v that_o solid_a into_o two_o equal_a part_n and_o so_o do_v not_o any_o other_o plain_a superficies_n not_o extend_v by_o the_o centre_n for_o every_o plain_n extend_v by_o the_o centre_n cut_v the_o diameter_n of_o the_o parallelipipedon_n in_o the_o centre_n into_o two_o equal_a part_n for_o it_o be_v prove_v that_o plain_a superficiece_n which_o cut_v the_o solid_a into_o two_o equal_a part_n do_v cut_v the_o dimetient_fw-la into_o two_o equal_a part_n in_o the_o centre_n wherefore_o all_o the_o line_n draw_v by_o the_o centre_n in_o that_o plain_a superficies_n shall_v make_v angle_n with_o the_o dimetient_fw-la and_o forasmuch_o as_o the_o diameter_n fall_v upon_o the_o parallel_n right_a line_n of_o the_o solid_a which_o describe_v the_o opposite_a side_n of_o the_o say_v solid_a or_o upon_o the_o parallel_n plain_a superficiece_n of_o the_o solid_a which_o make_v angel_n at_o the_o end_n of_o the_o diameter_n the_o triangle_n contain_v under_o the_o diameter_n and_o the_o right_a line_n extend_v in_o that_o plain_n by_o the_o centre_n and_o the_o right_a line_n which_o be_v draw_v in_o the_o opposite_a superficiece_n of_o the_o solid_a join_v together_o the_o end_n of_o the_o foresay_a right_a line_n namely_o the_o end_n of_o the_o diameter_n and_o the_o end_n of_o the_o line_n draw_v by_o the_o centre_n in_o the_o superficies_n extend_v by_o the_o centre_n shall_v always_o be_v equal_a and_o equiangle_n by_o the_o 26._o of_o the_o first_o for_o the_o opposite_a right_a line_n draw_v by_o the_o opposite_a plain_a superficiece_n of_o the_o solid_a do_v make_v equal_a angle_n with_o the_o diameter_n forasmuch_o as_o they_o be_v parallel_a line_n by_o the_o 16._o of_o this_o book_n but_o the_o angle_n at_o the_o centre_n be_v equal_a by_o the_o 15._o of_o the_o first_o for_o they_o be_v head_n angle_n &_o one_o side_n be_v equal_a to_o one_o side_n namely_o half_o the_o dimetient_fw-la wherefore_o the_o triangle_n contain_v under_o every_o right_a line_n draw_v by_o the_o centre_n of_o the_o parallelipipedon_n in_o the_o superficies_n which_o be_v extend_v also_o by_o the_o say_a centre_n and_o the_o diameter_n thereof_o who_o end_n be_v the_o angle_n of_o the_o solid_a be_v equal_a equilater_n &_o equiangle_n by_o the_o 26._o of_o the_o first_o wherefore_o it_o follow_v that_o the_o plain_a superficies_n which_o cut_v the_o parallelipipedon_n do_v make_v the_o part_n of_o the_o base_n on_o the_o opposite_a side_n equal_a and_o equiangle_n and_o therefore_o like_a and_o equal_a both_o in_o multitude_n and_o in_o magnitude_n wherefore_o the_o two_o solid_a section_n of_o that_o solid_a shall_v be_v equal_a and_o like_a by_o the_o 8._o definition_n of_o this_o book_n and_o now_o that_o no_o other_o plain_a superficies_n beside_o that_o which_o be_v extend_v by_o the_o centre_n divide_v the_o parallelipipedon_n into_o two_o equal_a part_n it_o be_v manifest_a if_o unto_o the_o plain_a superficies_n which_o be_v not_o extend_v by_o the_o centre_n we_o extend_v by_o the_o centre_n a_o parallel_n plain_a superficies_n by_o the_o corollary_n of_o the_o 15._o of_o this_o book_n for_o forasmuch_o as_o that_o superficies_n which_o be_v extend_v by_o the_o centre_n do_v divide_v the_o parallelipipedom_n into_o two_o equal_a parâ—â—_n it_o be_v manifest_a that_o the_o other_o plain_a superficies_n which_o be_v parallel_n to_o the_o superficies_n which_o divide_v the_o solid_a into_o two_o equal_a part_n be_v in_o one_o of_o the_o equal_a part_n of_o the_o solid_a wherefore_o see_v that_o the_o whole_a be_v ever_o great_a than_o his_o part_n it_o must_v needs_o be_v that_o one_o of_o these_o section_n be_v less_o than_o the_o half_a of_o the_o solid_a and_o therefore_o the_o other_o be_v great_a for_o the_o better_a understanding_n of_o this_o former_a proposition_n &_o also_o of_o this_o corollary_n add_v by_o flussas_n it_o shall_v be_v very_o needful_a for_o you_o to_o describe_v of_o past_v paper_n or_o such_o like_a matter_n a_o parallelipipedom_n or_o a_o cube_n and_o to_o divide_v all_o the_o parallelogram_n thereof_o into_o two_o equal_a part_n by_o draw_v by_o the_o centre_n of_o the_o say_a parallelogram_n which_o centre_n be_v the_o point_n make_v by_o the_o cut_n of_o diagonal_a line_n draw_v from_o thâ—_z opposite_a angle_n of_o the_o say_a parallelogram_n line_n parallel_n to_o the_o side_n of_o the_o parallelogram_n as_o in_o the_o former_a figure_n describe_v in_o a_o plain_a you_o may_v see_v be_v the_o six_o parallelogram_n de_fw-fr eh_o ha_o ad_fw-la dh_o and_o cg_o who_o these_o parallel_a line_n draw_v by_o the_o centre_n of_o the_o say_a parallelogram_n namely_o xo_o or_o pr._n and_o px_n do_v divide_v into_o two_o equal_a part_n by_o which_o four_o line_n you_o must_v imagine_v a_o plain_a superficies_n to_o be_v extend_v also_o these_o parallel_a line_n kl_o ln_o nm_o and_o mâ—_n by_o which_o four_o line_n likewise_o yâ—_n must_v imagine_v a_o plain_a superficies_n to_o be_v extend_v you_o may_v if_o you_o will_v put_v within_o your_o body_n make_v thus_o of_o past_v paper_n two_o superficiece_n make_v also_o of_o the_o say_a paper_n have_v to_o their_o limit_n line_n equal_a to_o the_o foresay_a parallel_n line_n which_o superficiece_n must_v also_o be_v divide_v into_o two_o equal_a part_n by_o parallel_a line_n draw_v by_o their_o centre_n and_o must_v cut_v the_o one_o the_o other_o by_o these_o parallel_a line_n and_o for_o the_o diameter_n of_o this_o body_n extend_v a_o thread_n from_o one_o angle_n in_o the_o base_a of_o the_o solid_a to_o his_o opposite_a angle_n which_o shall_v pass_v by_o the_o centre_n of_o the_o parallelipipedon_n as_o do_v the_o line_n dg_o in_o the_o figure_n before_o describe_v in_o the_o plain_n and_o draw_v in_o the_o base_a and_o the_o opposite_a superficies_n unto_o it_o diagonal_a line_n from_o the_o angle_n from_o which_o be_v extend_v the_o diameter_n of_o the_o solid_a as_o in_o the_o former_a description_n be_v the_o line_n bg_o and_o de._n and_o when_o you_o have_v thus_o describe_v this_o body_n compare_v it_o with_o the_o former_a demonstration_n and_o it_o will_v make_v it_o very_o plain_a unto_o you_o so_o your_o letter_n agree_v with_o the_o letter_n of_o the_o figure_n describe_v in_o the_o book_n and_o this_o description_n will_v plain_o set_v forth_o unto_o you_o the_o corollary_n follow_v that_o proposition_n for_o where_o as_o to_o the_o understanding_n of_o the_o demonstration_n of_o the_o proposition_n the_o superficiece_n put_v within_o the_o body_n be_v extend_v by_o parallel_a line_n draw_v by_o the_o centre_n of_o the_o base_n of_o the_o parallelipipedon_n to_o the_o understanding_n of_o the_o say_a corollary_n you_o may_v extend_v a_o superficies_n by_o any_o other_o line_n draw_v in_o the_o say_a base_n so_o that_o yet_o it_o pass_v through_o the_o midst_n of_o the_o thread_n which_o be_v suppose_v to_o be_v the_o centre_n of_o the_o parallelipipedon_n ¶_o the_o 35._o theorem_a the_o 40._o proposition_n if_o there_o be_v
two_o prism_n under_o equal_a altitude_n &_o the_o one_o have_v to_o his_o base_a a_o parallelogram_n and_o the_o other_o 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 equal_a the_o one_o to_o the_o other_o svppose_v that_o these_o two_o prism_n abcdef_n ghkmon_n be_v under_o equal_a altitude_n and_o let_v the_o one_o have_v to_o his_o base_a the_o parallelogram_n ac_fw-la and_o the_o other_o the_o triangle_n ghk_o and_o let_v the_o parallelogram_n ac_fw-la be_v double_a to_o the_o triangle_n ghk_o then_o i_o say_v that_o the_o prism_n abcdef_n be_v equal_a to_o the_o prism_n ghkmon_n construction_n make_v perfect_v the_o parallelipipedon_n axe_n &_o go_v and_o forasmuch_o as_o the_o parallelogram_n ac_fw-la be_v double_a to_o the_o triangle_n ghk_o demonstration_n but_o the_o parallelogram_n gh_o be_v also_o by_o the_o 41._o of_o the_o first_o double_a to_o the_o triangle_n ghk_o wherefore_o the_o parallelogram_n ac_fw-la be_v equal_a to_o the_o parallelogram_n gh_o but_o parallelipipedon_n consist_v upon_o equal_a base_n and_o under_o one_o and_o the_o self_n same_o altitude_n be_v equal_a the_o one_o to_o the_o other_o by_o the_o 31._o of_o the_o eleven_o wherefore_o the_o solid_a axe_n be_v equal_a to_o the_o solid_a go_v but_o the_o half_a of_o the_o solid_a axe_n be_v the_o prism_n abcdef_n and_o the_o half_a of_o the_o solid_a go_v be_v the_o prism_n ghkmon_n wherefore_o the_o prism_n abcdef_n be_v equal_a to_o the_o prism_n ghkmon_n if_o therefore_o there_o be_v two_o prism_n under_o equal_a altitude_n and_o the_o one_o have_v to_o his_o base_a a_o parallelogram_n &_o the_o other_o 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 equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v this_o proposition_n and_o the_o demonstration_n thereof_o be_v not_o hard_a to_o conceive_v by_o the_o former_a figure_n but_o you_o may_v for_o your_o full_a understanding_n of_o they_o take_v two_o equal_a parallelipipedon_n equilater_n and_o equiangle_n the_o one_o to_o the_o other_o describe_v of_o past_v paper_n or_o such_o like_a matter_n and_o in_o the_o base_a of_o the_o one_o parallelipipedon_n draw_v a_o diagonal_a line_n and_o draw_v a_o other_o diagonal_a line_n in_o the_o upper_a superficies_n opposite_a unto_o the_o say_a diagonal_a line_n draw_v in_o the_o base_a and_o in_o one_o of_o the_o parallelogram_n which_o be_v set_v upon_o the_o base_a of_o the_o other_o parallelipipedon_n draw_v a_o diagonal_a line_n and_o draw_v a_o other_o diagonal_a line_n in_o the_o parallelogram_n opposite_a to_o the_o same_o for_o so_o if_o you_o extend_v plain_a superficiece_n by_o those_o diagonal_a line_n there_o will_v be_v make_v two_o prism_n in_o each_o body_n you_o must_v take_v heed_n that_o you_o put_v for_o the_o base_n of_o each_o of_o these_o parallelipipedon_n equal_a parallelogram_n and_o then_o note_v they_o with_o letter_n according_a to_o the_o letter_n of_o the_o figure_n before_o describe_v in_o the_o plain_a and_o compare_v they_o with_o the_o demonstration_n and_o they_o will_v make_v both_o it_o and_o the_o proposition_n very_o clear_o unto_o you_o they_o will_v also_o geve_v great_a light_n to_o the_o corollary_n follow_v add_v by_o flussas_n a_o corollary_n add_v by_o flussas_n by_o this_o and_o the_o former_a proposition_n it_o be_v manifest_a that_o prism_n and_o solid_n column_n contain_v under_o two_o poligoâ—on_a figure_n equal_a like_a and_o parallel_n and_o the_o rest_n parallelogram_n may_v be_v compare_v the_o one_o to_o the_o other_o after_o the_o self_n same_o manner_n that_o parallelipipedon_n be_v for_o forasmuch_o as_o by_o this_o proposition_n and_o by_o the_o second_o corollary_n of_o the_o 2d_o of_o this_o book_n it_o be_v manifest_a that_o every_o parallelipipedon_n may_v be_v resolve_v into_o two_o like_a and_o equal_a prism_n of_o one_o and_o the_o same_o altitude_n who_o base_n shall_v be_v one_o and_o the_o self_n same_o with_o the_o base_a of_o the_o parallelipipedon_n or_o the_o half_a thereof_o which_o prisâ—es_v also_o shall_v be_v contâ—yned_v under_o the_o self_n same_o sideâ—_n with_o the_o parallelipipedom_n the_o say_a sideâ—_n be_v also_o side_n of_o like_a proportion_n i_o say_v that_o prismeâ—_n may_v be_v compare_v together_o after_o the_o like_a manner_n that_o their_o parallelipipedonâ—_n areâ—_n for_o if_o we_o will_v divide_v a_o prism_n like_a unto_o his_o foliâ—e_n by_o the_o 25._o of_o this_o book_n you_o shall_v find_v in_o the_o corollaryes_fw-mi of_o the_o 25._o proposition_n that_o that_o which_o be_v set_v forth_o touch_v a_o parallelipipedon_n follow_v not_o only_o in_o a_o prism_n but_o also_o in_o any_o side_v column_n who_o opposite_a base_n be_v equal_a and_o like_a and_o his_o side_n parallelogram_n if_o it_o be_v require_v by_o the_o 27._o proposition_n upon_o a_o right_a line_n give_v to_o describe_v a_o prism_n like_a and_o in_o like_a sort_n situate_a to_o a_o prism_n give_v describe_v â—â—â—st_n the_o whole_a parallelipipedon_n whereof_o the_o prism_n give_v be_v the_o half_n which_o thing_n you_o see_v by_o this_o 40._o proposition_n may_v be_v do_v and_o unto_o that_o parallelipipedom_n describe_v upon_o the_o right_a line_n give_v by_o the_o say_v 27._o proposition_n a_o other_o parallelipipedon_n like_o and_o the_o half_a thereof_o shall_v be_v the_o prism_n which_o you_o seek_v for_o namely_o shall_v be_v a_o prism_n describe_v upon_o the_o right_a line_n give_v and_o like_v unto_o the_o prism_n give_v in_o deed_n prism_n can_v not_o be_v cut_v according_a to_o the_o 28._o proposition_n for_o that_o in_o their_o opposite_a side_n can_v be_v draw_v no_o diagonal_a line_n howbeit_o by_o that_o 28._o proposition_n those_o prism_n be_v manifest_o confirm_v to_o be_v equal_a and_o like_a which_o be_v the_o half_n of_o one_o and_o the_o self_n same_o parallelipipedon_n and_o as_o touch_v the_o 29._o proposition_n and_o the_o three_o follow_v it_o which_o prove_v that_o parallelipipedon_n under_o one_o and_o the_o self_n same_o altitude_n and_o upon_o equal_a base_n or_o the_o self_n same_o base_n be_v equal_a or_o if_o they_o be_v under_o one_o and_o the_o self_n same_o altitude_n they_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n areâ—_n to_o apply_v these_o comparison_n unto_o 〈◊〉_d it_o be_v to_o 〈◊〉_d require_v that_o the_o base_n of_o the_o prism_n compare_v together_o be_v either_o all_o parallelogram_n or_o all_o triaâ—gles_n for_o so_o one_o and_o the_o self_n altitude_n remain_v the_o comparison_n of_o thing_n equal_a 〈◊〉_d one_o and_o thâ—â—_n self_n same_o and_o the_o half_n of_o the_o base_n be_v ever_o the_o one_o to_o the_o other_o in_o the_o same_o proportion_n that_o their_o whole_n be_v wherefore_o prism_n which_o be_v the_o half_n of_o the_o parallelipipedon_n and_o which_o have_v the_o same_o proportion_n the_o one_o to_o the_o other_o that_o the_o whole_a parallelipipedon_n have_v which_o be_v under_o one_o and_o thâ—_z selfâ—â—ame_v altitude_n must_v needs_o cause_v that_o their_o base_n be_v the_o half_n of_o the_o baseâ—_n of_o the_o parallelipâ—pâ—â—â—â—â—â—e_n in_o the_o same_o proportion_n the_o one_o to_o the_o other_o that_o their_o whole_a parallelipipâ—donâ—_n be_v if_o therefore_o the_o wâ—ole_a parallelipipedon_n be_v in_o the_o proportion_n of_o the_o whole_a base_n their_o hâ—lâ—â—â—_n also_o which_o be_v prism_n shall_v be_v in_o the_o proportion_n either_o of_o the_o whole_n if_o their_o base_n be_v parallelâ—grâ—mmâ—â—â—_n or_o of_o the_o halâ—â—â—â—f_n they_o be_v triangle_n which_o be_v ever_o all_o one_o by_o the_o 15._o of_o the_o five_o and_o forasmuch_o as_o by_o the_o 33._o proposition_n like_o parallelipipedon_n which_o be_v the_o double_n of_o their_o prism_n be_v in_o treble_a proportion_n the_o one_o to_o the_o other_o that_o their_o side_n of_o like_a proportion_n be_v it_o be_v manifest_a that_o prism_n be_v their_o half_n which_o have_v the_o one_o to_o the_o other_o the_o same_o proportion_n that_o their_o whole_n have_v by_o the_o 15_o of_o the_o five_o and_o have_v the_o self_n same_o side_n that_o theiâ—_z parallelipipedon_n have_v be_v the_o one_o to_o the_o other_o in_o treble_a proportion_n of_o that_o which_o the_o side_n of_o like_a proportion_n be_v and_o for_o that_o prism_n be_v the_o one_o to_o the_o other_o in_o the_o same_o proportion_n that_o their_o parallelipipedon_n be_v and_o the_o base_n of_o the_o prism_n be_v all_o either_o triangle_n or_o parallelogram_n be_v the_o one_o to_o the_o other_o in_o the_o same_o proportion_n that_o the_o base_n of_o the_o parallelipipedon_n be_v who_o altitude_n also_o be_v always_o equal_a we_o may_v by_o the_o 34._o proposition_n conclude_v that_o the_o base_n of_o the_o prism_n and_o the_o base_n of_o the_o parallelipipedon_n their_o double_n be_v each_o the_o one_o to_o the_o
other_o in_o one_o and_o the_o self_n same_o proportion_n be_v to_o the_o altitude_n in_o the_o same_o proportion_n that_o the_o base_n of_o the_o double_a solid_n namely_o of_o the_o parallelipipedon_n be_v for_o if_o the_o base_n of_o the_o equal_a parallelipipedon_n be_v reciprocal_a with_o their_o altitude_n than_o their_o half_n which_o be_v prism_n shall_v have_v their_o base_n reciprocal_a with_o their_o altitude_n by_o the_o 36._o proposition_n we_o may_v conclude_v that_o if_o there_o be_v three_o right_a line_n proportional_a the_o angle_n of_o a_o prism_n make_v of_o these_o three_o line_n be_v common_a with_o the_o angle_n of_o his_o parallelipipedon_n which_o be_v double_a do_v make_v a_o prism_n which_o be_v equal_a to_o the_o prism_n describe_v of_o the_o middle_a line_n and_o contain_v the_o like_a angle_n consist_v also_o of_o equal_a side_n for_o aâ—_z in_o the_o parallelipipedon_n so_o also_o in_o the_o prism_n this_o one_o thing_n be_v require_v namely_o that_o the_o three_o dimension_n of_o the_o proportional_a line_n do_v make_v a_o angle_n like_o unto_o the_o angle_n contain_v of_o the_o middle_a line_n take_v three_o time_n now_o than_o if_o the_o solid_a angle_n of_o the_o prism_n be_v make_v of_o those_o three_o right_a line_n there_o shall_v of_o they_o be_v make_v a_o angle_n like_o to_o the_o angle_n of_o the_o parallelipipedon_n which_o be_v double_a unto_o it_o wherefore_o it_o follow_v of_o necessity_n that_o the_o prism_n which_o be_v always_o the_o half_n of_o the_o parallelipipedon_n be_v equiangle_n the_o one_o to_o the_o other_o as_o also_o be_v their_o double_n although_o they_o be_v not_o equilater_n and_o therefore_o those_o half_n of_o equal_a solid_n be_v equal_a the_o one_o to_o the_o other_o namely_o that_o which_o be_v describe_v of_o the_o middle_n proportional_a line_n be_v equal_a to_o that_o which_o be_v describe_v of_o the_o three_o proportional_a line_n by_o the_o 37._o proposition_n also_o we_o may_v conclude_v the_o same_o touch_v prism_n which_o be_v conclude_v touch_a parallelipipedon_n for_o forasmuch_o as_o prism_n describe_v like_o &_o in_o like_a sort_n of_o the_o line_n give_v be_v the_o half_n of_o the_o parallelipipedon_n which_o be_v like_a and_o in_o like_a sort_n describe_v it_o follow_v that_o these_o prism_n have_v the_o one_o to_o the_o other_o the_o same_o proportion_n that_o the_o solid_n which_o be_v their_o double_n have_v and_o therefore_o if_o the_o line_n which_o describe_v they_o be_v porportionall_a they_o shall_v be_v proportional_a and_o so_o converse_o according_o to_o the_o rule_n of_o the_o say_v 37._o proposition_n but_o forasmuch_o as_o the_o 39_o proposition_n suppose_v the_o opposite_a superficial_a side_n of_o the_o solid_a to_o be_v parallelogram_n and_o the_o same_o solid_a to_o have_v one_o diameter_n which_o thing_n a_o prism_n can_v not_o have_v therefore_o this_o proposition_n can_v by_o â—o_o mean_v by_o apply_v to_o prism_n but_o as_o touch_v solid_n who_o base_n be_v two_o like_v equal_a and_o parallel_v poligonon_o figure_n and_o their_o side_n be_v parallelogram_n column_n forasmuch_o as_o by_o the_o second_o corollary_n of_o the_o 25._o of_o this_o book_n it_o have_v be_v declare_v that_o such_o solid_n be_v compose_v of_o prism_n it_o may_v easy_o be_v prove_v that_o their_o nature_n be_v such_o as_o be_v the_o nature_n of_o the_o prism_n whereof_o they_o be_v compose_v wherefore_o a_o parallelipipedon_n be_v by_o the_o 27._o proposition_n of_o this_o book_n describe_v there_o may_v also_o be_v describe_v the_o half_a thereof_o which_o be_v a_o prism_n and_o by_o the_o description_n of_o prism_n there_o may_v be_v compose_v a_o solid_a like_o unto_o a_o solid_a give_v compose_v of_o prism_n so_o that_o it_o be_v manifest_a that_o that_o which_o the_o 29._o 30._o 31._o 32._o 33._o 34._o and_o 37._o proposition_n set_v forth_o touch_v parallelipipedon_n may_v well_o be_v apply_v also_o to_o these_o kind_n of_o solid_n the_o end_n of_o the_o eleven_o book_n of_o euclides_n element_n ¶_o the_o twelve_o book_n of_o euclides_n element_n in_o this_o twelve_v book_n euclid_n set_v forth_o the_o passion_n and_o propriety_n of_o pyramid_n prism_n cones_n cylinder_n and_o sphere_n and_o compare_v pyramid_n first_o to_o pyramid_n then_o to_o prism_n so_o likewise_o do_v he_o cones_n and_o cylinder_n and_o last_o he_o compare_v sphere_n the_o one_o to_o the_o other_o but_o before_o he_o go_v to_o the_o treaty_n of_o those_o body_n he_o prove_v that_o like_o poligonon_n figure_v inscribe_v in_o circle_n and_o also_o the_o circle_n then selue_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o square_n of_o the_o diameter_n of_o those_o circle_n be_v because_o that_o be_v necessary_a to_o be_v prove_v for_o the_o confirmation_n of_o certain_a passion_n and_o propriety_n of_o those_o body_n ¶_o the_o 1._o theorem_a the_o 1._o proposition_n like_a poligonon_n figure_v describe_v in_o circle_n be_v in_o 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 be_v svppose_v that_o there_o be_v two_o circle_n abcde_o and_o fghkl_o and_o in_o they_o let_v there_o be_v describe_v like_o poligonon_n figure_n namely_o abcde_o and_o fghkl_o and_o let_v the_o diameter_n of_o the_o circle_n be_v bm_n and_o gn_n then_o i_o say_v that_o as_o the_o square_n of_o the_o line_n bm_n be_v to_o the_o square_n of_o the_o line_n gn_n so_o be_v the_o poligonon_n figure_n abcde_o to_o the_o poligonon_n figure_n fghkl_o construction_n draw_v these_o right_a line_n be_v be_o gl_n and_o fn_o and_o forasmuch_o as_o the_o poligonon_n figure_n abcde_o be_v like_a to_o the_o poligonon_n figure_n fghkl_o demonstration_n therefore_o the_o angle_n bae_o be_v equal_a to_o the_o angle_n gfl_fw-mi and_o as_o the_o line_n basilius_n be_v to_o the_o line_n ae_n so_o be_v the_o line_n gf_o to_o the_o line_n fl_fw-mi by_o the_o definition_n of_o like_a poligonon_n figure_v now_o therefore_o there_o be_v two_o triangle_n bae_o and_o gfl_fw-mi have_v one_o angle_n of_o the_o one_o equal_a to_o one_o angle_n of_o the_o other_o namely_o the_o angle_n bae_o equal_a to_o the_o angle_n gfl_fw-mi and_o the_o side_n about_o the_o equal_a angle_n be_v proportional_a wherefore_o by_o the_o first_o definition_n of_o the_o six_o the_o triangle_n abe_n be_v equiangle_n to_o the_o triangle_n fgl_n wherefore_o the_o angle_n aeb_fw-mi be_v equal_a to_o the_o angle_n flg_n but_o by_o the_o 21._o of_o the_o three_o the_o angle_n aeb_fw-mi be_v equal_a to_o the_o angle_n amb_n for_o they_o consist_v upon_o one_o and_o the_o self_n same_o circumference_n and_o by_o the_o same_o reason_n the_o angle_n flg_n be_v equal_a to_o the_o angle_n fng_v wherefore_o the_o angle_n amb_n be_v equal_a to_o the_o angle_n fng_v and_o the_o right_a angle_n bam_n be_v by_o the_o 4._o petition_n equal_a to_o the_o right_a angle_n gfn_n wherefore_o the_o angle_n remain_v be_v equal_a to_o the_o angle_n remain_v wherefore_o the_o triangle_n amb_n be_v equiangle_n to_o the_o triangle_n fng_v wherefore_o proportional_o as_o the_o line_n bm_n be_v to_o the_o line_n gn_n so_o be_v the_o line_n basilius_n to_o the_o line_n gf_o but_o the_o square_n of_o the_o line_n bm_n be_v to_o the_o square_n of_o the_o line_n gn_n in_o double_a proportion_n of_o that_o which_o the_o line_n bm_n be_v to_o the_o line_n gn_n by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o and_o the_o poligonon_n figure_n abcde_o be_v to_o the_o poligonon_n figure_n fghkl_o in_o double_a proportion_n of_o that_o which_o the_o line_n basilius_n be_v to_o the_o line_n gf_o by_o the_o _o of_o the_o six_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o square_n of_o the_o line_n bm_n be_v to_o the_o square_n of_o the_o line_n gn_n so_o be_v the_o poligonon_n figure_n abcde_o to_o the_o poligonon_n figure_n fghkl_o wherefore_o like_a poligonon_n figure_v describe_v in_o circle_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o the_o square_n of_o the_o diameter_n be_v which_o be_v require_v to_o be_v demonstrate_v ¶_o john_n dee_n his_o fruitful_a instruction_n with_o certain_a corollary_n and_o their_o great_a use_n who_o can_v not_o easy_o perceive_v what_o occasion_n and_o aid_n archimedes_n have_v by_o these_o first_o &_o second_o proposition_n to_o â—inde_v the_o near_a area_n or_o content_a of_o a_o circle_n between_o a_o poligonon_n figure_n within_o the_o circle_n and_o the_o like_a about_o the_o same_o circle_n describe_v who_o precise_a quantity_n be_v most_o easy_o know_v be_v comprehend_v of_o right_a line_n where_o also_o to_o avoid_v all_o occasion_n of_o error_n it_o be_v good_a in_o number_n not_o have_v precise_a square_a root_n to_o use_v the_o logistical_a process_n according_a to_o the_o rule_n with_o √_o â—_o 12_o √_o â—_o 19_o
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
same_o be_v treble_a it_o be_v manifest_a by_o the_o 12._o of_o the_o five_o that_o all_o the_o prism_n be_v to_o all_o the_o pyramid_n treble_a wherefore_o parallelipipedon_n be_v treble_a to_o pyramid_n set_v upon_o the_o self_n same_o base_a with_o they_o and_o under_o the_o same_o altitude_n they_o for_o that_o they_o contain_v two_o prism_n three_o corollary_n if_o two_o prism_n be_v under_o one_o and_o the_o self_n same_o altitude_n have_v to_o their_o base_n either_o both_o triangle_n or_o both_o parallelogram_n the_o prism_n be_v the_o one_o to_o the_o other_o as_o their_o base_n be_v for_o forasmuch_o as_o those_o prism_n be_v equemultiqlice_n unto_o the_o pyramid_n upon_o the_o self_n same_o base_n and_o under_o the_o same_o altitude_n which_o pyramid_n be_v in_o proportion_n as_o their_o base_n it_o be_v manifest_a by_o the_o 15._o of_o the_o five_o that_o the_o prism_n be_v in_o the_o proportion_n of_o the_o base_n for_o by_o the_o former_a corollary_n the_o prism_n be_v treble_a to_o the_o pyramid_n sâ—t_v upon_o the_o triangular_a base_n four_o corollary_n prism_n be_v in_o sesquealtera_fw-la proportion_n to_o pyramid_n which_o have_v the_o self_n same_o quadrangled_a base_a that_o the_o prism_n have_v and_o be_v under_o the_o self_n same_o altitude_n for_o that_o pyramid_n contain_v two_o pyramid_n set_v upon_o a_o triangular_a base_a of_o the_o same_o prism_n for_o it_o be_v prove_v that_o that_o prism_n be_v treble_a to_o the_o pyramid_n which_o be_v set_v upon_o the_o half_a of_o his_o quadrangled_a base_a unto_o which_o the_o other_o which_o be_v set_v upon_o the_o whole_a base_a be_v double_a by_o the_o six_o of_o this_o book_n five_v corollary_n wherefore_o we_o may_v in_o like_a sort_n conclude_v that_o solid_n mention_v in_o the_o second_o corollary_n which_o solid_n campane_n call_v side_v column_n be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n which_o be_v poligonon_o figure_n for_o they_o be_v in_o the_o proportion_n of_o the_o pyramid_n or_o prism_n set_v upon_o the_o self_n same_o base_n and_o under_o the_o self_n same_o altitude_n that_o be_v they_o be_v in_o the_o proportion_n of_o the_o base_n of_o the_o say_a pyramid_n or_o prism_n for_o those_o solid_n may_v be_v divide_v into_o prism_n have_v the_o self_n same_o altitude_n when_o as_o their_o opposite_a base_n may_v be_v divide_v into_o triangle_n by_o the_o 20_o of_o the_o six_o upon_o which_o triangle_n prism_n be_v set_v be_v in_o proportion_n as_o their_o base_n by_o this_o 7._o proposition_n it_o plain_o appear_v that_o â—uâ—lide_a as_o it_o be_v before_o note_v in_o the_o diffinitionâ—â—_n under_o the_o definition_n of_o a_o prism_n comprehend_v also_o those_o kind_n of_o solid_n which_o campane_n call_v side_v column_n for_o in_o that_o he_o say_v every_o prism_n have_v a_o triangle_n to_o his_o base_a may_v be_v devidedâ—_n etc_n etc_n he_o need_v not_o take_v a_o prism_n in_o that_o sense_n which_o campane_n and_o most_o man_n take_v it_o to_o have_v add_v that_o particle_n have_v to_o his_o base_a a_o triangle_n for_o by_o their_o sense_n there_o be_v no_o prism_n but_o it_o may_v have_v to_o his_o base_a a_o triangleâ—_n and_o so_o it_o may_v seem_v that_o euclid_n ought_v without_o exception_n have_v say_v that_o every_o prism_n whatsoever_o may_v be_v divide_v into_o three_o pyramid_n equal_a the_o one_o to_o the_o other_o have_v also_o triangle_n to_o â—heir_a base_n for_o so_o do_v campane_n and_o flussas_n put_v the_o proposition_n leave_v out_o the_o former_a particle_n have_v to_o his_o base_a a_o triangle_n which_o yet_o be_v red_a in_o the_o greek_a copy_n &_o not_o leâ—t_v out_o by_o any_o other_o interpreter_n know_v abroad_o except_o by_o campane_n and_o flussas_n yea_o and_o the_o corollary_n follow_v of_o this_o proposition_n add_v by_o theon_n or_o euclid_n and_o amend_v by_o m._n dee_n seem_v to_o confirm_v this_o sense_n of_o this_o â—s_a 〈◊〉_d make_v manifest_a that_o every_o pyramid_n be_v the_o three_o part_n of_o the_o prism_n have_v the_o same_o base_a with_o it_o and_o equal_a altitude_n for_o and_o if_o the_o base_a of_o the_o prism_n have_v any_o other_o right_o line_v figure_n than_o a_o triangle_n and_o also_o the_o superficies_n opposite_a to_o the_o base_a the_o same_o figure_n that_o prism_n may_v be_v divide_v into_o prism_n have_v triangle_v base_n and_o the_o superficiece_n to_o those_o base_n opposite_a also_o triangle_v a_o â—â—ike_a and_o equal_o for_o there_o as_o we_o see_v be_v put_v these_o word_n â—or_a and_o if_o the_o base_a of_o the_o prism_n be_v any_o other_o right_o line_v figureâ—_n etc_n etc_n whereof_o a_o man_n may_v well_o infer_v that_o the_o base_a may_v be_v any_o other_o rectiline_a figure_n whatsoever_o &_o not_o only_o a_o triangle_n or_o a_o parallelogram_n and_o it_o be_v true_a also_o in_o that_o sense_n as_o it_o be_v plain_a to_o see_v by_o the_o second_o corollary_n add_v out_o of_o flussas_n which_o corollary_n as_o also_o the_o first_o of_o his_o corollary_n be_v in_o a_o manner_n all_o one_o with_o the_o corollary_n add_v by_o theon_n or_o euclid_n far_o theon_n in_o the_o demonstration_n of_o the_o 10._o proposition_n of_o this_o book_n as_o we_o shall_v afterwards_o see_v most_o plain_o call_v not_o only_o side_v column_n prism_n but_o also_o parallelipipedon_n and_o although_o the_o 40._o proposition_n of_o the_o eleven_o book_n may_v seem_v hereunto_o to_o be_v a_o lâ—t_n for_o that_o it_o can_v be_v understand_v of_o those_o prism_n only_o which_o have_v triangle_n to_o their_o like_a equal_a opposite_a and_o parallel_v side_n or_o but_o of_o some_o side_v column_n and_o not_o of_o all_o yet_o may_v that_o let_v be_v thus_o remove_v away_o to_o say_v that_o euclid_n in_o that_o proposition_n use_v genus_fw-la pro_fw-la specie_fw-la that_o be_v the_o general_a word_n for_o some_o special_a kind_n thereof_o which_o thing_n also_o be_v not_o rare_a not_o only_o with_o he_o but_o also_o with_o other_o learned_a philosopher_n thus_o much_o i_o think_v good_a by_o the_o way_n to_o note_v in_o far_a defence_n of_o euclid_n definition_n of_o a_o prism_n the_o 8._o theorem_a the_o 8._o proposition_n pyramid_n be_v like_o &_o have_a triangle_n to_o their_o base_n be_v in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o their_o side_n of_o like_a proportion_n be_v svppose_v that_o these_o pyramid_n who_o base_n be_v the_o triangle_n gbc_n and_o hef_n and_o top_n the_o point_n a_o and_o d_o be_v like_o and_o in_o like_a sort_n describe_v and_o let_v ab_fw-la and_o de_fw-fr be_v side_n of_o like_a proportion_n then_o i_o say_v that_o the_o pyramid_n abcg_o be_v to_o the_o pyramid_n defh_o in_o treble_a proportion_n of_o that_o in_o which_o the_o side_n ab_fw-la be_v to_o the_o side_n de._n make_v perfect_a the_o parallelipipedon_n namely_o the_o solid_n bckl_n &_o efxo_n and_o forasmuch_o as_o the_o pyramid_n abcg_o be_v like_a to_o the_o pyramid_n defh_o construction_n therefore_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n def_n demonstration_n &_o the_o angle_n gbc_n to_o the_o angle_n hef_fw-fr and_o moreover_o the_o angle_n abg_o to_o the_o angle_n deh_fw-it and_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n de_fw-fr so_o be_v the_o line_n bc_o to_o the_o line_n of_o and_o the_o line_n bg_o to_o the_o line_n eh_o and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n de_fw-fr so_o be_v the_o line_n bc_o to_o the_o line_n of_o and_o the_o side_n about_o the_o equal_a angle_n be_v proportional_a therefore_o the_o parallelogram_n bm_n be_v like_a to_o the_o parallelogram_n ep_n and_o by_o the_o same_o reason_n the_o parallelogram_n bn_o be_v like_a to_o the_o parallelogram_n er_fw-mi and_o the_o parellelogramme_n bk_o be_v like_a unto_o the_o parallelogram_n exit_fw-la wherefore_o the_o three_o parallelogram_n bm_n kb_o and_o bn_o be_v like_a to_o the_o three_o parallelogram_n ep_n exit_fw-la and_o er._n but_o the_o three_o parallelogram_n bm_n kb_o and_o bn_o be_v equal_a and_o like_a to_o the_o three_o opposite_a parallelogram_n and_o the_o three_o parallelogram_n ep_n exit_fw-la and_o ere_o be_v equal_a and_o like_a to_o the_o three_o opposite_a parallelogram_n wherefore_o the_o parallelipipedon_n bckl_n and_o efxo_n be_v comprehend_v under_o plain_a superficiece_n like_a and_o equal_a in_o multitude_n wherefore_o the_o solid_a bckl_n be_v like_a to_o the_o solid_a efxo_n but_o like_o parallelipipedon_n be_v by_o the_o 33._o of_o the_o eleven_o in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o side_n of_o like_a proportion_n be_v to_o side_n of_o like_a proportion_n wherefore_o the_o solid_a bckl_n be_v to_o the_o solid_a efxo_n in_o treble_a
with_o the_o altitude_n of_o the_o say_a pyramid_n a_o and_o b_o shall_v be_v equal_a by_o the_o 6._o of_o this_o book_n wherefore_o by_o the_o first_o part_n of_o this_o proposition_n the_o base_n of_o the_o pyramid_n c_o to_o d_z be_v reciprocal_a with_o the_o altitude_n of_o d_o to_o c._n but_o in_o what_o proportion_n be_v the_o base_n c_o to_o d_z in_o the_o same_o be_v the_o base_n a_o to_o b_o forasmuch_o as_o they_o be_v equal_a and_o in_o what_o proportion_n be_v the_o altitude_n of_o d_z to_o c_z in_o the_o same_o be_v the_o altitude_n of_o b_o to_o a_o which_o altitude_n be_v likewise_o equal_a wherefore_o by_o the_o 11._o of_o the_o five_o in_o what_o proportion_n the_o base_n a_o to_o b_o be_v in_o the_o same_o reciprocal_a be_v the_o altitude_n of_o the_o pyramid_n b_o to_o a._n in_o like_a sort_n by_o the_o second_o part_n of_o this_o proposition_n may_v be_v prove_v the_o converse_n of_o this_o corollary_n the_o same_o thing_n follow_v also_o in_o a_o prism_n and_o in_o a_o side_v column_n as_o have_v before_o at_o large_a be_v declare_v in_o the_o corollary_n of_o the_o 40._o proposition_n of_o the_o 11._o book_n for_o those_o solid_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o pyramid_n or_o parallelipipedon_n for_o they_o be_v either_o part_n of_o equemultiplices_fw-la or_o equemultiplices_fw-la to_o part_n the_o 10._o theorem_a the_o 10._o proposition_n 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 and_o one_o and_o the_o self_n same_o altitude_n with_o it_o svppose_v that_o there_o be_v a_o cone_fw-mi have_v to_o his_o base_a the_o circle_n abcd_o and_o let_v there_o be_v a_o cilind_a have_v the_o self_n same_o base_a and_o also_o the_o same_o altitude_n that_o the_o cone_fw-mi have_v then_o i_o say_v that_o the_o cone_fw-mi be_v the_o three_o part_n of_o the_o cilind_a that_o be_v that_o the_o cilind_a be_v in_o treble_a proportion_n to_o the_o cone_n for_o if_o the_o cilind_a be_v not_o in_o treble_a proportion_n to_o the_o cone_n than_o the_o cilind_a be_v either_o in_o great_a proportion_n then_o triple_a to_o the_o cone_n or_o else_o in_o less_o first_o let_v it_o be_v in_o great_a than_o triple_a 〈◊〉_d and_o describe_v by_o the_o 6._o of_o the_o four_o in_o the_o circle_n abcd_o a_o square_a abcd._n now_o the_o square_n abcd_o be_v great_a than_o the_o half_a of_o the_o circle_n abcd_o for_o if_o about_o the_o circle_n abcd_o we_o describe_v a_o square_a the_o square_n describe_v in_o the_o circle_n abcd_o be_v the_o half_a of_o the_o square_n describe_v about_o the_o circle_n and_o let_v there_o be_v parallelipipedon_n prism_n describe_v upon_o those_o square_n prism_n equal_a in_o altitude_n with_o the_o cilind_a but_o prism_n 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 of_o the_o eleven_o and_o 5._o corollary_n of_o the_o 7._o of_o this_o book_n wherefore_o the_o prism_n describe_v upon_o the_o square_a abcd_o be_v the_o half_a of_o the_o prism_n describe_v upon_o the_o square_a that_o be_v describe_v about_o the_o circle_n now_o the_o clind_a be_v less_o than_o the_o prism_n which_o be_v make_v of_o the_o square_n describe_v abouâ—_n the_o circle_n abcd_o be_v equal_a in_o altitude_n with_o it_o for_o it_o contain_v it_o wherefore_o the_o prism_n describe_v upon_o the_o square_a abcd_o and_o be_v equal_a in_o altitude_n with_o the_o cylinder_n be_v great_a than_o half_a the_o cylinder_n divide_v by_o the_o 30._o of_o the_o three_o the_o circumference_n ab_fw-la bc_o cd_o and_o da_z into_o two_o equal_a part_n in_o the_o point_n e_o f_o g_o h_o and_o draw_v these_o right_a line_n ae_n ebb_n bf_o fc_o cg_o gd_o dh_o &_o ha._n wherefore_o every_o one_o of_o these_o triangle_n aeb_fw-mi bfc_n cgd_v and_o dha_n be_v great_a than_o half_a of_o that_o segment_n of_o the_o circle_n abcd_o which_o be_v describe_v about_o it_o as_o we_o have_v before_o in_o the_o 2._o proposition_n declare_v describe_v upon_o every_o one_o of_o these_o triangle_n aeb_fw-mi bfc_n cgd_v and_o dha_n a_o prism_n of_o equal_a altitude_n with_o the_o cylinder_n wherefore_o every_o one_o of_o these_o prism_n so_o describe_v be_v great_a than_o the_o half_a part_n of_o the_o segment_n of_o the_o cylinder_n that_o be_v set_v upon_o the_o say_a segment_n of_o the_o circle_n for_o if_o by_o the_o point_n e_o f_o g_o h_o be_v draw_v parallel_a line_n to_o the_o line_n ab_fw-la bc_o cd_o and_o da_z and_o then_o be_v make_v perfect_a the_o parallelogram_n make_v by_o those_o parallel_a line_n and_o moreover_o upon_o those_o parallelogram_n be_v erect_v parallelipipedon_n equal_v in_o altitude_n with_o the_o cylinder_n the_o prism_n which_o be_v describe_v upon_o each_o of_o the_o triangle_n aeb_fw-mi bfc_n cgd_v and_o dha_n be_v the_o half_n of_o every_o one_o of_o those_o parallelipipedon_n and_o the_o segment_n of_o the_o cylinder_n be_v less_o than_o those_o parallelipipedon_n so_o describe_v wherefore_o also_o every_o one_o of_o the_o prism_n which_o be_v describe_v upon_o the_o triangle_n aeb_fw-mi bfc_n cgd_v and_o dha_n be_v great_a than_o the_o half_a of_o the_o segment_n of_o the_o cylinder_n set_n upon_o the_o say_a segment_n now_o therefore_o devide_v every_o one_o of_o the_o circumference_n remain_v into_o two_o equal_a part_n and_o draw_v right_a line_n and_o raise_v up_o upon_o every_o one_o of_o these_o triangle_n prism_n equal_a in_o altitude_n with_o the_o cylinder_n and_o do_v this_o continual_o we_o shall_v at_o the_o length_n by_o the_o first_o of_o the_o ten_o leave_v certain_a segment_n of_o the_o cylinder_n which_o shall_v be_v less_o then_o the_o excess_n whereby_o the_o cylinder_n exceed_v the_o cone_n more_o than_o thrice_o let_v those_o segment_n be_v ae_n ebb_n bf_o fc_o cg_o gd_o dh_o and_o ha._n wherefore_o the_o prism_n remain_v who_o base_a be_v the_o polygonon_n â—igure_n aebfcgdh_n and_o altitude_n the_o self_n same_o that_o the_o cylinder_n have_v be_v great_a than_o the_o cone_n take_v three_o time_n prism_n but_o the_o prism_n who_o base_a be_v the_o polygonon_n figure_n aebfcgdh_n and_o altitude_n the_o self_n same_o that_o the_o cylinder_n have_v be_v treble_a to_o the_o pyramid_n who_o base_a be_v the_o polygonon_n figure_n aebfcgda_n and_o altitude_n the_o self_n same_o that_o the_o cone_fw-mi have_v by_o the_o corollary_n of_o the_o 3._o of_o this_o book_n wherefore_o also_o the_o pyramid_n who_o base_a be_v the_o polygonon_n figure_n aebfcgdh_n and_o top_n the_o self_n same_o that_o the_o cone_fw-mi have_v be_v great_a than_o the_o cone_n which_o have_v to_o his_o base_a the_o circle_n abcd._n but_o it_o be_v also_o less_o for_o it_o be_v contain_v of_o it_o which_o be_v impossible_a wherefore_o the_o cylinder_n be_v not_o in_o great_a proportion_n then_o triple_a to_o the_o cone_n i_o say_v moreover_o that_o the_o cylinder_n be_v not_o in_o less_o proportion_n then_o triple_a to_o the_o coneâ—_n for_o if_o it_o be_v possible_a let_v the_o cylinder_n be_v in_o less_o proportion_n then_o triple_a to_o the_o cone_n wherefore_o by_o conversion_n the_o cone_n be_v great_a than_o the_o three_o part_n of_o the_o cylinder_n describe_v now_o by_o the_o six_o of_o the_o four_o in_o the_o circle_n abcd_o a_o square_a abcd._n wherefore_o the_o square_a abcd_o be_v great_a than_o the_o half_a of_o the_o circle_n abcd_o upon_o the_o square_n abcd_o describe_v a_o pyramid_n have_v one_o &_o the_o self_n same_o altitude_n with_o the_o cone_n wherefore_o the_o pyramid_n so_o describe_v be_v great_a they_o half_n of_o the_o cone_fw-it for_o if_o as_o we_o have_v before_o declare_v we_o describe_v a_o square_a about_o the_o circle_n the_o square_a abcd_o be_v the_o half_a of_o the_o square_n describe_v about_o the_o circle_n and_o if_o upon_o the_o square_n be_v describe_v parallelipipedon_n equal_v in_o altitude_n with_o the_o cone_n which_o solid_n be_v also_o call_v prism_n the_o prism_n or_o parallelipipedon_n describe_v upon_o the_o square_a abcd_o be_v the_o half_a of_o the_o prism_n which_o be_v describe_v upon_o the_o square_n describe_v about_o the_o circle_n for_o they_o be_v the_o one_o to_o the_o other_o in_o that_o proportion_n that_o their_o base_n be_v by_o the_o 32._o of_o the_o eleven_o &_o 5._o corollary_n of_o the_o 7._o of_o this_o book_n wherefore_o also_o their_o three_o part_n be_v in_o the_o self_n same_o proportion_n by_o the_o 15._o of_o the_o five_o wherefore_o the_o pyramid_n who_o base_a be_v the_o square_a abcd_o be_v the_o half_a of_o the_o pyramid_n set_v upon_o the_o square_n describe_v about_o the_o circle_n but_o the_o pyramid_n set_v upon_o the_o square_n describe_v about_o the_o circle_n be_v great_a than_o the_o cone_n who_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
superficies_n or_o solidity_n in_o the_o hole_n or_o in_o partâ—_n such_o certain_a knowledge_n demonstrative_a may_v arise_v and_o such_o mechanical_a exercise_n thereby_o be_v devise_v that_o sure_a i_o be_o to_o the_o sincere_a &_o true_a student_n great_a light_n aid_n and_o comfortable_a courage_n far_a to_o wade_v will_v enter_v into_o his_o heart_n and_o to_o the_o mechanical_a witty_a and_o industrous_a deviser_n new_a manner_n of_o invention_n &_o execution_n in_o his_o work_n will_v with_o small_a travail_n for_o foot_n application_n come_v to_o his_o perceiveraunce_n and_o understanding_n therefore_o even_o a_o manifold_a speculation_n &_o practice_n may_v be_v have_v with_o the_o circle_n his_o quantity_n be_v not_o know_v in_o any_o kind_n of_o small_a certain_a measure_n so_o likewise_o of_o the_o sphere_n many_o problem_n may_v be_v execute_v and_o his_o precise_a quantity_n in_o certain_a measure_n not_o determine_v or_o know_v yet_o because_o both_o one_o of_o the_o first_o humane_a occasion_n of_o invent_v and_o stablish_v this_o art_n be_v measure_v of_o the_o earth_n and_o therefore_o call_v geometria_n that_o be_v earthmeasure_v and_o also_o the_o chief_a and_o general_a end_n in_o deed_n be_v measure_n and_o measure_n require_v a_o determination_n of_o quantity_n in_o a_o certain_a measure_n by_o number_n express_v it_o be_v needful_a for_o mechanical_a earthmeasure_n not_o to_o be_v ignorant_a of_o the_o measure_n and_o content_n of_o the_o circle_n neither_o of_o the_o sphere_n his_o measure_n and_o quantity_n as_o near_o as_o sense_n can_v imagine_v or_o wish_v and_o in_o very_a deed_n the_o quantity_n and_o measure_n of_o the_o circle_n be_v know_v make_v not_o only_o the_o cone_n and_o cylinder_n but_o also_o the_o sphere_n his_o quantity_n to_o be_v as_o precise_o know_v and_o certain_a therefore_o see_v in_o respect_n of_o the_o circle_n quantity_n by_o archimedes_n specify_v this_o theorem_a be_v note_v unto_o you_o i_o will_v by_o order_n upon_o that_o as_o a_o supposition_n infer_v the_o conclusion_n of_o this_o our_o theorem_n note_n 1._o wherefore_o if_o you_o divide_v the_o one_o side_n as_o tq_n of_o the_o cube_fw-la tx_n into_o 21._o equal_a part_n and_o where_o 11._o part_n do_v end_n reckon_v from_o t_o suppose_v the_o point_n p_o and_o by_o that_o point_n p_o imagine_v a_o plain_a pass_v parallel_n to_o the_o opposite_a base_n to_o cut_v the_o cube_fw-la tx_n and_o thereby_o the_o cube_fw-la tx_n to_o be_v divide_v into_o two_o rectangle_n parallelipipedon_n namely_o tn_n and_o px_n it_o be_v manifest_a give_v tn_n to_o be_v equal_a to_o the_o sphere_n a_o by_o construction_n and_o the_o 7._o of_o the_o five_o note_n 2._o second_o the_o whole_a quantity_n of_o the_o sphere_n a_o assign_v be_v contain_v in_o the_o rectangle_n parallelipipedon_n tn_n you_o may_v easy_o transform_v the_o same_o quantity_n into_o other_o parallelipipedon_n rectangle_v of_o what_o height_n and_o of_o what_o parallelogram_n base_a you_o listen_v by_o my_o first_o and_o second_o problem_n upon_o the_o 34._o of_o this_o book_n and_o the_o like_a may_v you_o do_v to_o any_o assign_a part_n of_o the_o sphere_n a_o by_o the_o like_a mean_n devide_v the_o parallelipipedon_n tn_n as_o the_o part_n assign_v do_v require_v as_o if_o a_o three_o four_o five_o or_o six_o part_n of_o the_o sphere_n a_o be_v to_o be_v have_v in_o a_o parallelipipedon_n of_o any_o parallelograâ—â—e_n base_a assign_v or_o of_o any_o heith_n assign_v then_o devide_v tp_n into_o so_o many_o part_n as_o into_o 4._o if_o a_o four_o part_n be_v to_o be_v transform_v or_o into_o five_o if_o a_o five_o part_n be_v to_o be_v transform_v etc_n etc_n and_o then_o proceed_v â—s_v you_o do_v with_o cut_v of_o tn_n from_o tx_n and_o that_o i_o say_v of_o parallelipipedon_n may_v in_o like_a sort_n by_o my_o â—â—yd_n two_o problem_n add_v to_o the_o 34._o of_o this_o book_n be_v do_v in_o any_o side_v column_n pyramid_n and_o prismeâ—_n so_o thâ—â—_n in_o pyramid_n and_o some_o prism_n you_o use_v the_o caution_n necessary_a in_o respect_n of_o their_o quan_fw-mi 〈…〉_o odyes_n have_v parallel_n equal_a and_o opposite_a base_n who_o part_n 〈…〉_o re_fw-mi in_o their_o proposition_n be_v by_o euclid_n demonstrate_v and_o final_o 〈…〉_o addition_n you_o have_v the_o way_n and_o order_n how_o to_o geve_v to_o a_o sphere_n or_o any_o segment_n oâ—_n the_o same_o cones_n or_o cylinder_n equal_a or_o in_o any_o proportion_n between_o two_o right_a line_n give_v with_o many_o other_o most_o necessary_a speculation_n and_o practice_n about_o the_o sphere_n i_o trust_v that_o i_o have_v sufficient_o â—raughted_v your_o imagination_n for_o your_o honest_a and_o profitable_a study_n herein_o and_o also_o give_v you_o reaâ—â—_n â—â—tter_n wheâ—â—_n with_o to_o sâ—â—p_v the_o mouth_n of_o the_o malicious_a ignorant_a and_o arrogant_a despiser_n of_o the_o most_o excellent_a discourse_n travail_n and_o invention_n mathematical_a sting_v aswell_o the_o heavenly_a sphere_n &_o star_n their_o spherical_a solidity_n geometry_n with_o their_o convene_n spherical_a superficies_n to_o the_o earth_n at_o all_o time_n respect_v and_o their_o distance_n from_o the_o earth_n as_o also_o the_o whole_a earthly_a sphere_n and_o globe_n itself_o and_o infinite_a other_o case_n concern_v sphere_n or_o globe_n may_v hereby_o with_o as_o much_o ease_n and_o certainty_n be_v determine_v of_o as_o of_o the_o quantity_n of_o any_o bowl_n ball_n or_o bullet_n which_o we_o may_v gripe_v in_o our_o hand_n reason_n and_o experience_n be_v our_o witness_n and_o without_o these_o aid_n such_o thing_n of_o importance_n never_o able_a of_o we_o certain_o to_o be_v know_v or_o attain_a unto_o here_o end_n m._n john_n do_v you_o his_o addition_n upon_o the_o last_o proposition_n of_o the_o twelve_o book_n a_o proposition_n add_v by_o flussas_n if_o a_o sphere_n touch_v a_o plain_a superficiesâ—_n a_o right_a line_n draw_v from_o the_o centre_n to_o the_o touch_n shall_v be_v erect_v perpendicular_o to_o the_o plain_a superficies_n suppose_v that_o there_o be_v a_o sphere_n bcdl_n who_o centre_n let_v be_v the_o point_n a._n and_o let_v the_o plain_a superficies_n gci_n touch_v the_o spear_n in_o the_o point_n c_o and_o extend_v a_o right_a line_n from_o the_o centre_n a_o to_o the_o point_n c._n then_o i_o say_v that_o the_o line_n ac_fw-la be_v erect_v perpendicular_o to_o tâ—e_z plain_a gic._n let_v the_o sphere_n be_v cut_v by_o plain_a superficiece_n pass_v by_o the_o right_a line_n lac_n which_o plain_n let_v be_v abcdl_n and_o acel_n which_o let_v cut_v the_o plain_n gci_n by_o the_o right_a line_n gch_o and_o kci_fw-la now_o it_o be_v manifest_a by_o the_o assumpt_n put_v before_o the_o 17._o of_o this_o book_n that_o the_o two_o section_n of_o the_o sphere_n shall_v be_v circle_n have_v to_o their_o diameter_n the_o line_n lac_n which_o be_v also_o the_o diameter_n of_o the_o sphere_n wherefore_o the_o right_a line_n gch_o and_o kci_fw-la which_o be_v draw_v in_o the_o plain_n gci_n do_v at_o the_o point_n c_o fall_n without_o the_o circle_n bcdl_n and_o ecl._n wherefore_o they_o touch_v the_o circle_n in_o the_o point_n c_o by_o the_o second_o definition_n of_o the_o three_o wherefore_o the_o right_a line_n lac_n make_v right_a angle_n with_o the_o line_n gch_v and_o kci_fw-la by_o the_o 16._o of_o the_o three_o wherefore_o by_o the_o 4._o of_o the_o eleven_o the_o right_a line_n ac_fw-la be_v erect_v perpendicular_o to_o to_o the_o plain_a superficies_n gci_n wherein_o be_v draw_v the_o line_n gch_v and_o kci_fw-la if_o therefore_o a_o sphere_n touch_v a_o plain_a superficies_n a_o right_a line_n draw_v from_o the_o centre_n to_o the_o touch_n shall_v be_v erect_v perpendicular_o to_o the_o plain_a superficies_n which_o be_v require_v to_o be_v prove_v the_o end_n of_o the_o twelve_o book_n of_o euclides_n element_n ¶_o the_o thirteen_o book_n of_o euclides_n element_n in_o this_o thirteen_o book_n be_v set_v forth_o certain_a most_o wonderful_a and_o excellent_a passion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n book_n a_o matter_n undoubted_o of_o great_a and_o infinite_a use_n in_o geometry_n as_o you_o shall_v both_o in_o this_o book_n and_o in_o the_o other_o book_n follow_v most_o evident_o perceive_v it_o teach_v moreover_o the_o composition_n of_o the_o five_o regular_a solid_n and_o how_o to_o inscribe_v they_o in_o a_o sphere_n give_v and_o also_o set_v forth_o certain_a comparison_n of_o the_o say_a body_n both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o sphere_n wherein_o they_o be_v describe_v the_o 1._o theorem_a the_o 1._o proposition_n if_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o to_o the_o great_a segment_n be_v add_v the_o half_a of_o the_o whole_a line_n the_o square_n make_v of_o those_o two_o
three_o proposition_n we_o will_v use_v the_o same_o supposition_n and_o construction_n there_o specify_v so_o far_o as_o they_o shall_v serve_v our_o purpose_n begin_v therefore_o at_o the_o conclusion_n we_o must_v infer_v the_o part_n of_o the_o proposition_n before_o grant_v it_o be_v conclude_v that_o the_o square_n of_o the_o line_n db_o be_v quintuple_a to_o the_o square_n of_o the_o line_n dc_o his_o own_o segment_n therefore_o dn_o the_o square_n of_o db_o be_v quintuple_a to_o gf_o the_o square_n of_o dc_o but_o the_o squaâ—e_n of_o ac_fw-la the_o double_a of_o dc_o which_o be_v r_n be_v quadruple_a to_o gf_o by_o the_o second_o corollary_n of_o the_o 20._o of_o the_o six_o and_o therefore_o r_n with_o gf_o be_v quintuple_a to_o gf_o and_o so_o it_o be_v evident_a that_o the_o square_a dn_o be_v equal_a to_o the_o square_a r_n together_o with_o the_o square_a gf_o wherefore_o from_o those_o two_o equal_n take_v the_o square_a gf_o common_a to_o they_o both_o remain_v the_o square_a r_n equal_a to_o the_o gnomon_n xop_n but_o to_o the_o gnomon_n xop_n the_o parallelogram_n 3._o ce_fw-fr be_v equal_a wherefore_o the_o square_a of_o the_o line_n ac_fw-la which_o be_v r_n be_v equal_a to_o the_o parallelogram_n câ—_n which_o parallelogamme_n be_v contain_v under_o be_v equal_a to_o ab_fw-la and_o cb_o the_o part_n remain_v of_o the_o first_o line_n gruen_o which_o be_v db._n and_o the_o line_n ab_fw-la be_v make_v of_o the_o double_a of_o the_o segment_n dc_o and_o of_o cbâ—_n the_o other_o part_n of_o the_o line_n db_o first_o goven_v wherefore_o the_o double_a of_o the_o segment_n dc_o with_o cb_o the_o part_n remain_v which_o altogether_o be_v the_o whole_a line_n ab_fw-la be_v to_o ac_fw-la the_o double_a of_o the_o segment_n dc_o as_o that_o same_o ac_fw-la be_v to_o cb_o by_o the_o second_o part_n of_o the_o 16._o of_o the_o six_o therefore_o by_o the_o 3._o definition_n of_o the_o six_o book_n the_o whole_a line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n &_o ac_fw-la the_o double_a of_o the_o segment_n dc_o be_v middle_a proportional_a be_v the_o great_a part_n thereof_o wherefore_o if_o a_o right_a line_n be_v quintuple_a in_o power_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v or_o thus_o it_o may_v be_v demonstrate_v forasmuch_o as_o the_o square_n dn_o be_v quintuple_a to_o the_o square_a gf_o i_o mean_v the_o square_n of_o db_o the_o line_n give_v too_o the_o square_a oâ—_n dc_o the_o segment_n and_o the_o same_o square_a dn_o be_v equal_a to_o the_o parallelogram_n under_o ab_fw-la cb_o with_o the_o square_n make_v of_o the_o line_n dc_o by_o the_o six_o of_o the_o second_o for_o unto_o the_o line_n ac_fw-la equal_o divide_v the_o line_n cb_o be_v as_o it_o be_v adjoin_v wherefore_o the_o parallelogram_n under_o ab_fw-la cb_o together_o with_o the_o square_n of_o dc_o which_o be_v gf_o be_v quintuple_a to_o the_o square_a gf_o make_v oâ—_n thâ—_z line_n dc_o take_v then_o that_o square_a gf_o â—rom_o the_o parallelogram_n under_o ab_fw-la cb_o that_o parallelogram_n under_o ab_fw-la cb_o remain_v alone_o be_v but_o quadruple_a to_o the_o say_v square_a of_o the_o line_n dc_o but_o by_o the_o 4._o of_o the_o second_o or_o the_o second_o corollary_n of_o the_o 20._o of_o the_o six_o r_n â—he_z square_a of_o the_o line_n ac_fw-la be_v quadrupla_fw-la to_o the_o same_o square_a gfâ—_n wherefore_o by_o the_o 7._o of_o the_o five_o the_o square_a of_o the_o line_n ac_fw-la be_v equal_a to_o the_o parallelogram_n under_o ab_fw-la cb_o and_o so_o by_o the_o second_o part_n of_o the_o 16._o of_o the_o six_o ab_fw-la ac_fw-la and_o cb_o be_v three_o line_n in_o continual_a proportion_n and_o see_v ab_fw-la be_v great_a they_o ac_fw-la the_o same_o ac_fw-la the_o double_a of_o the_o line_n dc_o shall_v be_v great_a than_o the_o part_n bc_o remain_v wherefore_o by_o the_o 3._o definition_n of_o the_o six_o ab_fw-la compose_v or_o make_v of_o the_o double_a of_o dc_o and_o the_o other_o part_n of_o db_o remain_v be_v divide_v by_o a_o extreme_a and_o middle_n proportion_n and_o also_o his_o great_a segment_n be_v ac_fw-la the_o double_a of_o the_o segment_n dc_o wherefore_o if_o a_o right_a line_n be_v quintuple_a in_o power_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v a_o theorem_a 2._o if_o a_o right_a line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n be_v give_v and_o to_o the_o great_a segment_n â—herof_o he_o direct_o adjoin_v a_o line_n equal_a to_o the_o whole_a line_n give_v that_o adjoin_v line_n and_o the_o say_v great_a segment_n do_v make_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n who_o great_a segment_n be_v the_o line_n adjoin_v suppose_v the_o line_n give_v divide_v by_o extreme_a and_o mean_a proportion_n to_o be_v ab_fw-la divide_v in_o the_o point_n c_o and_o his_o great_a segment_n let_v be_v ac_fw-la unto_fw-la ac_fw-la direct_o adjoin_v a_o line_n equal_a to_o ab_fw-la let_v that_o be_v ad_fw-la i_o say_v that_o ad_fw-la together_o with_o ac_fw-la that_o be_v dc_o be_v a_o divide_v by_o extreme_a and_o middle_n proportion_n who_o great_a segment_n be_v ad_fw-la the_o line_n adjoin_v divide_v ad_fw-la equal_o in_o the_o point_n e._n now_o forasmuch_o as_o ae_n be_v the_o half_a of_o ad_fw-la by_o construction_n it_o be_v also_o the_o half_a of_o ab_fw-la equal_a to_o ad_fw-la by_o construction_n wherefore_o by_o the_o 1._o of_o the_o thirteen_o the_o square_a of_o the_o line_n compose_v of_o ac_fw-la and_o ae_n which_o â—ne_n be_v ec_o be_v quintuple_a to_o the_o square_n of_o the_o line_n ae_n wherefore_o the_o double_a of_o ae_n and_o the_o line_n ac_fw-la compose_v as_o in_o one_o right_a line_n be_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n by_o the_o converse_n of_o this_o three_o by_o i_o demonstrate_v and_o the_o double_a of_o ae_n be_v the_o great_a segment_n but_o dc_o be_v the_o line_n compose_v of_o the_o double_a of_o ae_n &_o the_o line_n ac_fw-la and_o with_o all_o ad_fw-la be_v the_o double_a of_o ae_n wherefore_o dc_o be_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n and_o ad_fw-la iâ—_z hiâ—_z great_a segment_n if_o a_o right_a line_n therefore_o divide_v by_o extreme_a and_o mean_a proportion_n be_v give_v and_o to_o the_o great_a segment_n thereof_o be_v direct_o adjoin_v a_o line_n equal_a to_o the_o whole_a line_n give_v that_o adjoin_v line_n and_o the_o say_v great_a segment_n do_v make_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n who_o great_a segment_n be_v the_o line_n adjoin_v which_o be_v require_v to_o be_v demonstrate_v two_o other_o brief_a demonstration_n of_o the_o same_o forasmuch_o as_o ad_fw-la be_v to_o ac_fw-la as_o ab_fw-la be_v to_o ac_fw-la because_o ad_fw-la be_v equal_a to_o ab_fw-la by_o construction_n but_o as_o ab_fw-la be_v to_o ac_fw-la so_o be_v ac_fw-la to_o cb_o by_o supposition_n therefore_o by_o the_o 11._o of_o the_o five_o as_z ac_fw-la be_v to_o cb_o so_o be_v ad_fw-la to_o ac_fw-la demonstrate_v wherefore_o as_o ac_fw-la and_o cb_o which_o be_v ab_fw-la be_v to_o cb_o so_o be_v ad_fw-la and_o ac_fw-la which_o be_v dc_o to_o ac_fw-la therefore_o everse_o as_o ab_fw-la be_v to_o ac_fw-la so_o be_v dc_o to_o ad._n and_o it_o be_v prove_v ad_fw-la to_o be_v to_o ac_fw-la as_o ac_fw-la be_v to_o cb._n wherefore_o as_o ab_fw-la be_v to_o ac_fw-la and_o ac_fw-la to_o cb_o so_o be_v dc_o to_o ad_fw-la and_o ad_fw-la to_o ac_fw-la but_o ab_fw-la ac_fw-la and_o cb_o be_v in_o continual_a proportion_n by_o supposition_n wherefore_o dc_o ad_fw-la and_o ac_fw-la be_v in_o continual_a proportion_n wherefore_o by_o the_o 3._o definition_n of_o the_o six_o book_n dc_o be_v divide_v by_o extreme_a and_o middle_a proportion_n and_o his_o great_a segment_n be_v ad._n which_o be_v to_o be_v demonstrate_v note_v from_o the_o mark_n demonstrate_v how_o this_o have_v two_o demonstration_n one_o i_o have_v set_v in_o the_o margin_n by_o ¶_o a_o corollary_n 1._o upon_o euclides_n three_o proposition_n demonstrate_v it_o be_v make_v evident_a that_o of_o a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n if_o you_o produce_v the_o less_o segment_n equal_o to_o the_o length_n of_o the_o great_a the_o line_n thereby_o adjoin_v together_o with_o the_o say_v less_o segment_n make_v a_o new_a line_n divide_v by_o extreme_a and_o middle_a proportion_n who_o less_o segment_n be_v the_o line_n adjoin_v for_o if_o ab_fw-la be_v divide_v by_o extreme_a and_o middle_a proportion_n in_o the_o point_n c_o ac_fw-la be_v the_o great_a segment_n and_o cb_o be_v produce_v from_o the_o point_n b_o make_v a_o line_n with_o cb_o equal_a to_o ac_fw-la which_o let_v be_v cq_n and_o the_o
place_n from_o whence_o first_o it_o begin_v to_o be_v move_v it_o shall_v pass_v by_o the_o point_n f_o g_o h_o and_o the_o octohedron_n shall_v be_v comprehend_v in_o a_o sphere_n i_o say_v also_o that_o it_o be_v comprehend_v in_o the_o sphere_n give_v for_o forasmuch_o as_o the_o line_n lk_n be_v equal_a to_o the_o line_n km_o by_o position_n and_o the_o line_n ke_o be_v common_a to_o they_o both_o and_o they_o contain_v right_a angle_n by_o the_o 3._o definition_n of_o the_o eleven_o therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a le_fw-fr be_v equal_a to_o the_o base_a em_n and_o forasmuch_o as_o the_o angle_n lem_n be_v a_o right_a angle_n by_o the_o 31._o of_o the_o three_o for_o it_o be_v in_o a_o semicircle_n as_o have_v be_v prove_v therefore_o the_o square_a of_o the_o line_n lm_o be_v double_a to_o the_o square_n of_o the_o line_n le_fw-fr by_o the_o 47._o of_o the_o first_o again_o forasmuch_o as_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n bc_o therefore_o the_o line_n ab_fw-la be_v double_a to_o the_o line_n bc_o by_o the_o definition_n of_o a_o circle_n but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n bd_o by_o the_o corollary_n of_o the_o 8._o and_o _o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v double_a to_o the_o square_n of_o the_o line_n bd._n and_o it_o be_v prove_v that_o the_o square_a of_o the_o line_n lm_o be_v double_a to_o the_o square_n of_o the_o line_n le._n wherefore_o the_o square_a of_o the_o line_n bd_o be_v equal_a to_o the_o square_n of_o the_o line_n le._n for_o the_o line_n eh_o which_o be_v equal_a to_o the_o line_n lf_n be_v put_v to_o be_v equal_a to_o the_o line_n db._n wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n lm_o wherefore_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n lm_o and_o the_o line_n ab_fw-la be_v the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o line_n lm_o be_v equal_a to_o the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o octoedron_n be_v contain_v in_o the_o sphere_n give_v le._n and_o it_o be_v also_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n wherefore_o there_o be_v make_v a_o octohedron_n and_o it_o be_v comprehend_v in_o the_o sphere_n give_v wherein_o be_v comprehend_v the_o pyramid_n and_o it_o be_v prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedrn_n which_o be_v require_v to_o be_v do_v and_o to_o be_v prove_v certain_a corollary_n add_v by_o flussas_n first_o corollary_n the_o side_n of_o a_o pyramid_n be_v in_o power_n sesquitertia_fw-la to_o the_o side_n of_o a_o octahedron_n inscribe_v in_o the_o same_o sphere_n for_o forasmch_v as_o the_o diameter_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n therefore_o of_o what_o part_n the_o diameter_n contain_v in_o power_n 6._o of_o the_o same_o the_o side_n of_o the_o octohedron_n contain_v in_o power_n 3._o but_o of_o what_o part_n the_o diameter_n contain_v 6._o of_o the_o same_o the_o side_n of_o the_o pyramid_n contain_v 4._o by_o the_o 13._o of_o this_o book_n wherefore_o of_o what_o part_n the_o side_n of_o the_o pyramid_n contain_v 4._o of_o the_o same_o the_o side_n of_o the_o octohedron_n contain_v 3._o second_o corollary_n a_o octohedron_n be_v divide_v into_o two_o equal_a and_o like_a pyramid_n the_o common_a base_n of_o these_o pyramid_n be_v set_v upon_o every_o square_n contain_v of_o the_o side_n of_o the_o octohedron_n upon_o which_o square_a be_v set_v the_o â—â—_o triangle_n of_o the_o octohedron_n campane_n which_o pyramid_n be_v by_o the_o â—_o definition_n of_o the_o eleven_o equal_a and_o like_a and_o the_o foresay_a square_n common_a to_o those_o pyramid_n be_v the_o half_a of_o the_o square_n of_o the_o diameter_n of_o the_o sphere_n for_o it_o be_v the_o square_a of_o the_o side_n of_o the_o octohedron_n three_o corollary_n the_o three_o diameter_n of_o the_o octohedron_n do_v cut_v the_o one_o the_o other_o perpendicular_o into_o two_o equal_a part_n in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o say_a octohedron_n as_o it_o be_v manifest_a by_o the_o three_o diameter_n eglantine_n fh_o and_o lm_o which_o cut_v the_o one_o the_o other_o in_o the_o centre_n king_n equal_o and_o perpendicular_o ¶_o the_o 3._o problem_n the_o 15._o proposition_n to_o make_v a_o solid_a call_v a_o cube_fw-la and_o to_o comprehend_v it_o in_o the_o sphere_n give_v namely_o that_o sphere_n wherein_o the_o former_a two_o solid_n be_v comprehendâ—dâ—_n and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n treble_a to_o the_o side_n of_o the_o cube_fw-la take_v the_o diameter_n of_o the_o sphere_n give_v namely_o ab_fw-la and_o divide_v it_o in_o the_o point_n câ—_n so_o that_o let_v the_o line_n ac_fw-la be_v double_a to_o the_o line_n bc_o by_o the_o 9_o of_o the_o six_o and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n adb_o and_o by_o the_o 11._o of_o the_o first_o from_o the_o pâ—ynt_n c_o râ—yse_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n cd_o and_o draw_v a_o right_a linâ—_n db._n and_o describe_v a_o square_a efgh_o have_v every_o one_o of_o his_o side_n equal_a to_o the_o line_n db_o and_o from_o the_o point_n e_o f_o g_o h_o raise_v up_o by_o the_o 12._o of_o the_o eleven_o unto_o the_o plain_a superficies_n of_o the_o square_a efgh_o perpendicular_a line_n eke_o fl_fw-mi gm_n and_o hn_o and_o let_v every_o one_o of_o the_o line_n eke_o fl_fw-mi gm_n and_o hn_o be_v put_v equal_a to_o one_o of_o the_o line_n of_o fg_o gh_o or_o he_o which_o be_v the_o side_n of_o the_o square_n and_o draw_v these_o right_a line_n kl_o lm_o mn_a and_o nk_v demonstration_n wherefore_o there_o be_v make_v a_o cube_fw-la namely_o fn_o which_o be_v contain_v under_o six_o equal_a square_n construction_n now_o it_o be_v require_v to_o comprehend_v the_o same_o cube_fw-la in_o the_o sphere_n give_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n ble_o to_o the_o side_n of_o the_o cube_fw-la demonstration_n draw_v these_o right_a line_n kg_v and_o eglantine_n and_o forasmuch_o as_o the_o angle_n keg_n be_v a_o right_a angle_n for_o that_o the_o line_n ke_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n eâ—_n and_o therefore_o also_o to_o the_o right_a line_n eglantine_n by_o the_o 2._o definition_n of_o the_o eleven_o wherefore_o a_o semicircle_n describe_v upon_o the_o line_n kg_v shall_v book_n pass_v by_o the_o point_n e._n again_o forasmuch_o as_o the_o line_n fg_o be_v erect_v perpendicular_o to_o either_o of_o these_o line_n fl_fw-mi and_o fe_o by_o the_o definition_n of_o a_o square_a &_o by_o the_o 2._o definition_n of_o the_o eleven_o therefore_o the_o line_n fg_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n fk_o by_o the_o 4._o of_o the_o eleven_o wherefore_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n f_o to_o the_o point_n king_n the_o line_n gf_o shall_v be_v erect_v perpendicular_o to_o the_o line_n kf_o by_o the_o 2._o definition_n of_o the_o eleven_o and_o by_o the_o same_o reason_n again_o a_o semicircle_n describe_v upon_o the_o line_n gk_o shall_v pass_v also_o by_o the_o point_n f._n and_o likewise_o shall_v it_o pass_v by_o the_o rest_n of_o the_o point_n of_o the_o angle_n of_o that_o cube_fw-la if_o now_o the_o diameter_n kg_v abide_v fix_v the_o semicircle_n be_v turn_v round_o about_o until_o it_o return_v into_o the_o self_n same_o place_n from_o whence_o it_o begin_v first_o to_o be_v move_v the_o cube_fw-la shall_v be_v comprehend_v in_o a_o sphere_n i_o say_v also_o that_o it_o be_v comprehend_v in_o the_o sphere_n give_v demonstration_n for_o forasmuch_o as_o the_o line_n gf_o be_v equal_a to_o the_o linâ—â—e_n and_o the_o angle_n fletcher_n be_v a_o right_a angle_n therefore_o the_o square_a of_o the_o line_n eglantine_n be_v by_o the_o 47._o of_o the_o first_o double_a to_o the_o square_n of_o the_o line_n â—f_n but_o the_o line_n of_o be_v equal_a to_o the_o line_n eke_o wherefore_o the_o square_a of_o the_o line_n eglantine_n be_v double_a to_o the_o square_n of_o the_o line_n eke_o wherefore_o the_o square_n of_o eglantine_n and_o eke_o that_o be_v the_o square_a of_o the_o line_n gk_o by_o the_o 47._o of_o the_o first_o be_v treble_a to_o the_o square_n of_o the_o line_n eke_o and_o forasmuch_o as_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n bc_o but_o
double_a to_o the_o side_n of_o the_o octohedron_n &_o the_o side_n be_v in_o power_n sequitertia_fw-la to_o the_o perpendiclar_a line_n by_o the_o 12._o of_o this_o book_n wherefore_o the_o diameter_n thereof_o be_v in_o power_n duple_fw-fr superbipartiens_fw-la tertias_fw-la to_o the_o perpendicular_a line_n wherefore_o also_o the_o diameter_n and_o the_o perpendicular_a line_n be_v rational_a and_o commensurable_a by_o the_o 6._o of_o the_o ten_o as_o touch_v a_o icosahedron_n it_o be_v prove_v in_o the_o 16._o of_o this_o book_n that_o the_o side_n thereof_o be_v a_o less_o line_n when_o the_o diameter_n of_o the_o sphere_n be_v rational_a and_o forasmuch_o as_o the_o angle_n of_o the_o inclination_n of_o the_o base_n thereof_o be_v contain_v of_o the_o perpendicular_a line_n of_o the_o triangle_n and_o subtend_v of_o the_o right_a line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n which_o contain_v five_o side_n of_o the_o icosahedron_n and_o unto_o the_o perpendicular_a line_n the_o side_n be_v commensurable_a namely_o be_v in_o power_n sesquitertia_fw-la unto_o they_o by_o the_o corollary_n of_o the_o 12._o of_o this_o book_n therefore_o the_o perpendicular_a line_n which_o contain_v the_o angle_n be_v irrational_a line_n namely_o less_o line_n by_o the_o 105._o of_o the_o ten_o book_n and_o forasmuch_o as_o the_o diameter_n contain_v in_o power_n both_o the_o side_n of_o the_o icosahedron_n and_o the_o line_n which_o subtend_v the_o foresay_a angle_n if_o from_o the_o power_n of_o the_o diameter_n which_o be_v rational_a be_v take_v away_o the_o power_n of_o the_o side_n of_o the_o icosahedron_n which_o be_v irrational_a it_o be_v manifest_a that_o the_o residue_n which_o be_v the_o power_n of_o the_o subtend_a line_n shall_v be_v irrational_a for_o if_o it_o shall_v be_v rational_a the_o number_n which_o measure_v the_o whole_a power_n of_o the_o diameter_n and_o the_o part_n take_v away_o of_o the_o subtend_a line_n shall_v also_o by_o the_o 4._o common_a sentence_n of_o the_o seven_o measure_n the_o residue_n namely_o the_o power_n of_o the_o side_n irrational_a which_o be_v irrational_a for_o that_o it_o be_v a_o less_o line_n which_o be_v absurd_a wherefore_o it_o be_v manifest_a that_o the_o right_a line_n which_o compose_v the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o the_o icosahedron_n be_v irrational_a line_n for_o the_o subtend_a line_n have_v to_o the_o line_n contain_v a_o great_a proportion_n than_o the_o whole_a have_v to_o the_o great_a segment_n the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o a_o dodecahedron_n be_v contain_v under_o two_o perpendicular_n of_o the_o base_n of_o the_o dodecahedron_n and_o be_v subtend_v of_o that_o right_a line_n who_o great_a segment_n be_v the_o side_n of_o a_o cube_n inscribe_v in_o the_o dodecahedron_n which_o right_a line_n be_v equal_a to_o the_o line_n which_o couple_v the_o section_n into_o two_o equal_a part_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n and_o this_o couple_a line_n we_o say_v be_v a_o irrational_a line_n for_o that_o the_o diameter_n of_o the_o sphere_n contain_v in_o power_n both_o the_o couple_a line_n and_o the_o side_n of_o the_o dodecahedron_n but_o the_o side_n of_o the_o dodecahedron_n be_v a_o irrational_a line_n namely_o a_o residual_a line_n by_o the_o 17._o of_o this_o book_n wherefore_o the_o residue_n namely_o the_o couple_a line_n be_v a_o irrational_a line_n as_o it_o be_v â—asy_a to_o prove_v by_o the_o 4._o common_a sentence_n of_o the_o seven_o and_o that_o the_o perpendicular_a line_n which_o contain_v the_o angle_n of_o the_o inclination_n be_v irrational_a be_v thus_o prove_v by_o the_o proportion_n of_o the_o subtend_a line_n of_o the_o foresay_a angle_n of_o inclination_n to_o the_o line_n which_o contain_v the_o angle_n be_v find_v out_o the_o obliquity_n of_o the_o angle_n angle_n for_o if_o the_o subtend_a line_n be_v in_o power_n double_a to_o the_o line_n which_o contain_v the_o angle_n then_o be_v the_o angle_n a_o right_a angle_n by_o the_o 48._o of_o the_o first_o but_o if_o it_o be_v in_o power_n less_o than_o the_o double_a it_o be_v a_o acute_a angle_n by_o the_o 23._o of_o the_o second_o but_o if_o it_o be_v in_o power_n more_o than_o the_o double_a or_o have_v a_o great_a proportion_n than_o the_o whole_a have_v to_o the_o great_a segmentâ—_n the_o angle_n shall_v be_v a_o obtuse_a angle_n by_o the_o 12._o of_o the_o second_o and_o 4._o of_o the_o thirteen_o by_o which_o may_v be_v prove_v that_o the_o square_a of_o the_o whole_a be_v great_a than_o the_o double_a of_o the_o square_n of_o the_o great_a segment_n this_o be_v to_o be_v note_v that_o that_o which_o flussas_n have_v here_o teach_v touch_v the_o inclination_n of_o the_o base_n of_o the_o â—ive_a regular_a body_n hypsicles_n teach_v after_o the_o 5_o proposition_n of_o the_o 15._o book_n where_o he_o confess_v that_o he_o receive_v it_o of_o one_o isidorus_n and_o seek_v to_o make_v the_o matter_n more_o clear_a he_o endeavour_v himself_o to_o declare_v that_o the_o angle_n of_o the_o inclination_n of_o the_o solid_n be_v give_v and_o that_o they_o be_v either_o acute_a or_o obtuse_a according_a to_o the_o nature_n of_o the_o solid_a although_o euclid_v in_o all_o his_o 15._o book_n have_v not_o yet_o show_v what_o a_o thing_n give_v be_v wherefore_o flussas_n frame_v his_o demonstration_n upon_o a_o other_o ground_n proceed_v after_o a_o other_o manner_n which_o seem_v more_o plain_a and_o more_o apt_o hereto_o be_v place_v then_o there_o albeit_o the_o reader_n in_o that_o place_n shall_v not_o be_v frustrate_a of_o his_o also_o the_o end_n of_o the_o thirteen_o book_n of_o euclides_n element_n ¶_o the_o fourteen_o book_n of_o euclides_n element_n in_o this_o book_n which_o be_v common_o account_v the_o 14._o book_n of_o euclid_n be_v more_o at_o large_a entreat_v of_o our_o principal_a purpose_n book_n namely_o of_o the_o comparison_n and_o proportion_n of_o the_o five_o regular_a body_n customable_o call_v the_o 5._o figure_n or_o form_n of_o pythagoras_n the_o one_o to_o the_o other_o and_o also_o of_o their_o side_n together_o each_o to_o other_o which_o thing_n be_v of_o most_o secret_a use_n and_o inestimable_a pleasure_n and_o commodity_n to_o such_o as_o diligent_o search_v for_o they_o and_o attain_v unto_o they_o which_o thing_n also_o undoubted_o for_o the_o worthiness_n and_o hardness_n thereof_o for_o thing_n of_o most_o price_n be_v most_o hard_a be_v first_o search_v and_o find_v out_o of_o philosopher_n not_o of_o the_o inferior_a or_o mean_a sort_n but_o of_o the_o deep_a and_o most_o ground_a philosopher_n and_o best_a exercise_v in_o geometry_n and_o albeit_o this_o book_n with_o the_o book_n follow_v namely_o the_o 15._o book_n have_v be_v hitherto_o of_o all_o man_n for_o the_o most_o part_n and_o be_v also_o at_o this_o day_n number_v and_o account_v among_o euclides_n book_n and_o suppose_v to_o be_v two_o of_o he_o namely_o the_o 14._o and_o 15._o in_o order_n as_o all_o exemplar_n not_o only_o new_a and_o late_o set_v abroad_o but_o also_o old_a monument_n write_v by_o hand_n do_v manifest_o witness_v yet_o it_o be_v think_v by_o the_o best_a learned_a in_o these_o day_n that_o these_o two_o book_n be_v none_o of_o euclides_n but_o of_o some_o other_o author_n no_o less_o worthy_a nor_o of_o less_o estimation_n and_o authority_n notwithstanding_o than_o euclid_n apollonius_n a_o man_n of_o deep_a knowledge_n a_o great_a philosopher_n and_o in_o geometry_n marvellous_a who_o wonderful_a book_n write_v of_o the_o section_n of_o cones_fw-la which_o exercise_n &_o occupy_v thewitte_v of_o the_o wise_a and_o best_a learned_a be_v yet_o remain_v be_v think_v and_o that_o not_o without_o just_a cause_n to_o be_v the_o author_n of_o they_o or_o as_o some_o think_v hypsicles_n himself_o for_o what_o can_v be_v more_o plain_o then_o that_o which_o he_o himself_o witness_v in_o the_o preface_n of_o this_o book_n basilides_n of_o tire_n say_v hypsicles_n and_o my_o father_n together_o scan_v and_o peyse_v a_o writing_n or_o book_n of_o apollonius_n which_o be_v of_o the_o comparison_n of_o a_o dodecahedron_n to_o a_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n and_o what_o proportion_n these_o figure_n have_v the_o one_o to_o the_o other_o find_v that_o apollonius_n have_v fail_v in_o this_o matter_n but_o afterward_o say_v he_o i_o find_v a_o other_o copy_n or_o book_n of_o apollonius_n wherein_o the_o demonstration_n of_o that_o matter_n be_v full_a and_o perfect_a and_o show_v it_o unto_o they_o whereat_o they_o much_o rejoice_v by_o which_o word_n it_o seem_v to_o be_v manifest_a that_o apollonius_n be_v the_o first_o author_n of_o this_o book_n which_o be_v afterward_o set_v forth_o by_o hypsicles_n for_o so_o his_o own_o word_n after_o in_o
22._o of_o the_o six_o wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o both_o the_o line_n ab_fw-la &_o bc_o add_v the_o one_o to_o the_o other_o together_o with_o the_o line_n ac_fw-la that_o be_v as_z two_z such_o line_n as_o ab_fw-la be_v be_v to_o the_o line_n ac_fw-la so_o be_v both_o the_o line_n de_fw-fr and_o of_o add_v the_o one_o to_o the_o other_o together_o with_o the_o line_n df_o that_o be_v two_o such_o line_n as_o de_fw-fr be_v to_o the_o line_n df._n and_o in_o the_o same_o proportion_n be_v the_o half_n of_o the_o antecedent_n by_o the_o 15._o of_o the_o five_o wherefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n ac_fw-la so_o be_v the_o line_n de_fw-fr to_o the_o line_n df._n and_o therefore_o by_o the_o 19_o of_o the_o five_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o line_n df_o to_o the_o line_n fe_o wherefore_o also_o by_o division_n by_o the_o 17._o of_o the_o five_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cb_o so_o be_v the_o line_n df_o to_o the_o line_n de_fw-fr now_o that_o we_o have_v prove_v proposition_n that_o any_o right_a line_n whatsoever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n what_o proportion_n the_o line_n contain_v in_o power_n the_o square_n make_v of_o the_o whole_a line_n and_o of_o the_o great_a segment_n add_v together_o have_v to_o the_o line_n contain_v in_o power_n the_o square_n make_v of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o the_o same_o proportion_n have_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n now_o also_o that_o we_o have_v prove_v proposition_n that_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n so_o be_v the_o superficies_n of_o the_o dodecahedron_n to_o the_o superficies_n of_o the_o icosahedron_n be_v both_o describe_v in_o one_o and_o the_o self_n same_o sphere_n and_o moreover_o see_v that_o we_o have_v prove_v proposition_n that_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v the_o dodecahedron_n to_o the_o icosahedron_n for_o that_o both_o the_o pentagon_n of_o the_o dodecahedron_n and_o the_o triangle_n of_o the_o icosahedron_n be_v comprehend_v in_o one_o and_o the_o self_n same_o circle_n corollary_n all_o these_o thing_n i_o say_v be_v prove_v it_o be_v manifest_a that_o if_o in_o one_o and_o the_o self_n same_o sphere_n be_v describe_v a_o dodecahedron_n and_o a_o icosahedron_n they_o shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_z a_o right_a line_n whatsoever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o great_a segment_n add_v together_o be_v to_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o for_o for_o that_o as_o the_o dodecahedron_n be_v to_o the_o icosahedron_n so_o be_v the_o superficies_n of_o the_o dodecahedron_n to_o the_o superficies_n of_o the_o icosahedron_n that_o be_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n but_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n so_o any_o right_a line_n what_o so_o ever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n be_v the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o great_a segment_n add_v together_o to_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o wherefore_o as_o a_o dodecahedron_n be_v to_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n so_o any_o right_a line_n what_o so_o ever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n be_v the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n &_o of_o the_o great_a segment_n add_v together_o to_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o the_o end_n of_o the_o fourteen_o book_n of_o euclides_n element_n after_o hypsicles_n ¶_o the_o fourteen_o book_n of_o euclides_n element_n after_o flussas_n for_o that_o the_o fouretenth_fw-mi book_n as_o it_o be_v set_v forth_o by_o flussas_n contain_v in_o it_o more_o proposition_n than_o be_v find_v in_o hypsicles_n &_o also_o some_o of_o those_o proposition_n which_o hypsicles_n have_v be_v by_o he_o somewhat_o otherwise_o demonstrate_v i_o think_v my_o labour_n well_o bestow_v for_o the_o reader_n sake_n to_o turn_v it_o also_o all_o whole_a notwithstanding_o my_o travail_n before_o take_v in_o turn_v the_o same_o book_n after_o hypsicles_n where_o note_v you_o that_o here_o in_o this_o 14._o book_n after_o flussas_n and_o in_o the_o other_o book_n follow_v namely_o the_o 15._o and_o 16._o i_o have_v in_o allege_v of_o the_o proposition_n of_o the_o same_o 14._o book_n follow_v the_o order_n and_o number_n of_o the_o proposition_n as_o flussas_n have_v place_v they_o ¶_o the_o first_o proposition_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 inscribe_v in_o the_o same_o circle_n campane_n be_v the_o half_a of_o these_o two_o line_n take_v together_o namely_o 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 inscribe_v in_o the_o same_o circle_n take_v a_o circle_n abc_n and_o inscribe_v in_o it_o the_o side_n of_o a_o pentagon_n which_o let_v be_v bc_o and_o take_v the_o centre_n of_o the_o circle_n which_o let_v be_v the_o point_n d_o construction_n and_o from_o it_o draw_v unto_o the_o side_n bc_o a_o perpendicular_a line_n de_fw-fr which_o produce_v to_o the_o point_n â—_o and_o unto_o the_o line_n e_o fletcher_n put_v the_o line_n eglantine_n equal_a and_o draw_v these_o right_a line_n cg_o cd_o and_o cf._n then_o i_o say_v that_o the_o right_a 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 â—he_n sideâ—_n of_o the_o decagon_n and_o hexagon_n take_v together_o demonstration_n forasmuch_o as_o the_o line_n de_fw-fr be_v a_o perpendicular_a â—nto_n the_o line_n bc_o therefore_o the_o section_n be_v and_o ec_o shall_v be_v equal_a by_o the_o 3._o of_o the_o three_o and_o the_o line_n of_o be_v common_a unto_o they_o both_o and_o the_o angle_n fec_n and_o feb_n be_v right_a angle_n by_o supposition_n wherefore_o the_o base_n bf_o and_o fc_o be_v equal_a by_o the_o 4._o of_o the_o first_o but_o the_o line_n bc_o be_v the_o side_n of_o a_o pentagon_n by_o construction_n wherefore_o fc_o which_o subtend_v the_o half_a of_o the_o side_n of_o the_o pentagon_n be_v the_o side_n of_o the_o decagon_n inscribe_v in_o the_o circle_n abc_n but_o unto_o the_o line_n fc_o be_v by_o the_o 4._o of_o the_o first_o equal_a the_o line_n cg_o for_o they_o subtend_v right_a angle_n ceg_v and_o cef_n which_o be_v contain_v under_o equal_a side_n wherefore_o also_o the_o angle_n cge_n and_o cfe_n of_o the_o triangle_n cfg_n be_v equal_a by_o the_o 5._o of_o the_o first_o and_o forasmuch_o as_o the_o ark_n fc_o be_v subtend_v of_o the_o side_n of_o a_o decagon_n the_o ark_n ca_n shall_v be_v quadruple_a to_o the_o ark_n cf_o wherefore_o also_o the_o angle_n cda_n shall_v be_v quadruple_a to_o the_o angle_n cdf_n by_o the_o last_o of_o the_o sixâ—_o and_o forasmuch_o as_o the_o same_o angle_n cda_n which_o be_v set_v at_o the_o centre_n be_v double_a to_o the_o angle_n cfa_n which_o be_v set_v at_o the_o circumference_n by_o the_o 20._o of_o the_o three_o therefore_o the_o angle_n cfa_n or_o cfd_n be_v double_a to_o the_o angle_n cdâ—_n namely_o the_o half_a of_o quadruple_a but_o unto_o the_o angle_n cfd_n or_o cfg_n be_v prove_v equal_a the_o angle_n cgf_n wherefore_o the_o outward_a angle_n cgf_n be_v double_a to_o the_o angle_n cdf_n wherefore_o the_o angle_n cdg_n and_o dcg_n shall_v be_v equal_a for_o unto_o those_o two_o angle_n the_o angle_n cgf_n be_v equal_a by_o the_o 32._o of_o the_o first_o wherefore_o the_o side_n gc_o and_o gd_o be_v equal_a by_o the_o 6._o of_o the_o first_o wherefore_o also_o the_o line_n gd_o be_v equal_a to_o the_o line_n fc_o which_o be_v the_o side_n of_o the_o decagon_n but_o unto_o the_o right_a line_n fe_o be_v equal_a the_o line_n eglantine_n by_o construction_n wherefore_o the_o whole_a line_n de_fw-fr be_v equal_a to_o the_o two_o line_n câ—_n and_o fe_o wherefore_o those_o line_n take_v together_o namely_o the_o line_n df_o and_o fc_o shall_v
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 one_o to_o the_o other_o the_o octohedron_n cut_v the_o power_n of_o the_o diameter_n into_o two_o equal_a square_n the_o icosahedron_n into_o two_o square_n who_o proportion_n be_v duple_n to_o the_o proportion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n who_o less_o segment_n be_v the_o side_n of_o the_o icosahedron_n and_o the_o dodecahedron_n into_o two_o square_n who_o proportion_n be_v quadruple_a to_o the_o proportion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n who_o less_o segment_n be_v the_o side_n of_o the_o dodecahedron_n for_o ad_fw-la the_o diameter_n of_o the_o sphere_n contain_v in_o power_n ab_fw-la the_o side_n of_o the_o tetrahedron_n and_o bd_o the_o side_n of_o the_o cube_fw-la which_o bd_o be_v in_o power_n half_a of_o the_o side_n ab_fw-la the_o diameter_n also_o of_o the_o sphere_n contain_v in_o power_n ac_fw-la and_o cd_o two_o equal_a side_n of_o the_o octohedron_n but_o the_o diameter_n contain_v in_o power_n the_o whole_a line_n ae_n and_o the_o great_a segment_n thereof_o ed_z which_o be_v the_o side_n of_o the_o icosahedron_n by_o the_o 15._o of_o this_o book_n wherefore_o their_o power_n be_v in_o duple_n proportion_n of_o that_o in_o which_o the_o side_n be_v by_o the_o first_o corollary_n of_o the_o 20._o of_o the_o six_o have_v their_o proportion_n duple_n to_o the_o proportion_n of_o a_o extreme_a &_o mean_a proportion_n far_o the_o diameter_n contain_v in_o power_n the_o whole_a line_n of_o and_o his_o less_o segment_n fd_o which_o be_v the_o side_n of_o the_o dodecahedron_n by_o the_o same_o 15._o of_o this_o book_n wherefore_o the_o whole_a have_v to_o the_o less_o â—_o double_a proportion_n of_o that_o which_o the_o extreme_a have_v to_o the_o mean_a namely_o of_o the_o whole_a to_o the_o great_a segment_n by_o the_o 10._o definition_n of_o the_o five_o it_o follow_v that_o the_o proportion_n of_o the_o power_n be_v double_a to_o the_o double_a proportion_n of_o the_o side_n by_o the_o same_o first_o corollary_n of_o the_o 20._o of_o the_o six_o that_o be_v be_v quadruple_a to_o the_o proportion_n of_o the_o extreme_a and_o of_o the_o mean_a by_o the_o definition_n of_o the_o six_o a_o advertisement_n add_v by_o flussas_n by_o this_o mean_n therefore_o the_o diameter_n of_o a_o sphere_n be_v give_v there_o shall_v be_v give_v the_o side_n of_o every_o one_o of_o the_o body_n inscribe_v and_o forasmuch_o as_o three_o of_o those_o body_n have_v their_o side_n commensurable_a in_o power_n only_o and_o not_o in_o length_n unto_o the_o diameter_n give_v for_o their_o power_n be_v in_o the_o proportion_n of_o a_o square_a number_n to_o a_o number_n not_o square_a wherefore_o they_o have_v not_o the_o proportion_n of_o a_o square_a number_n to_o a_o square_a number_n by_o the_o corollary_n of_o the_o 25._o of_o the_o eight_o wherefore_o also_o their_o side_n be_v incommensurabe_n in_o length_n by_o the_o 9_o of_o the_o ten_o therefore_o it_o be_v sufficient_a to_o compare_v the_o power_n and_o not_o the_o length_n of_o those_o side_n the_o one_o to_o the_o otherâ—_n which_o power_n be_v contain_v in_o the_o power_n of_o the_o diameter_n namely_o from_o the_o power_n of_o the_o diameter_n let_v there_o ble_v take_v away_o the_o power_n of_o the_o cube_fw-la and_o there_o shall_v remain_v the_o power_n of_o the_o tetrahedron_n and_o take_v away_o the_o power_n of_o the_o tetrahedron_n there_o remain_v the_o power_n of_o the_o cube_fw-la and_o take_v away_o from_o the_o power_n of_o the_o diameter_n half_o the_o power_n thereof_o there_o shall_v be_v leave_v the_o power_n of_o the_o side_n of_o the_o octohedron_n but_o forasmuch_o as_o the_o side_n of_o the_o dodecahedron_n and_o of_o the_o icosahedron_n be_v prove_v to_o be_v irrational_a for_o the_o side_n of_o the_o icosahedron_n be_v a_o less_o line_n by_o the_o 16._o of_o the_o thirteen_o and_o the_o side_n of_o the_o dedocahedron_n be_v a_o residual_a line_n by_o the_o 17._o of_o the_o same_o therefore_o those_o side_n be_v unto_o the_o diameter_n which_o be_v a_o rational_a line_n set_v incommensurable_a both_o in_o length_n and_o in_o power_n wherefore_o their_o comparison_n can_v not_o be_v define_v or_o describe_v by_o any_o proportion_n express_v by_o number_n by_o the_o 8._o of_o the_o ten_o neither_o can_v they_o be_v compare_v the_o one_o to_o the_o other_o for_o irrational_a line_n of_o diverse_a kind_n be_v incommensurable_a the_o one_o to_o the_o other_o for_o if_o they_o shall_v be_v commensurable_a they_o shall_v be_v of_o one_o and_o the_o self_n same_o kind_n by_o the_o 103._o and_o 105._o of_o the_o ten_o which_o be_v impossible_a wherefore_o we_o seek_v to_o compare_v they_o to_o the_o power_n of_o the_o diameter_n think_v they_o can_v not_o be_v more_o apt_o express_v then_o by_o such_o proportion_n which_o cut_v that_o rational_a power_n of_o the_o diameter_n according_a to_o their_o side_n namely_o divide_v the_o power_n of_o the_o diameter_n by_o line_n which_o have_v that_o proportion_n that_o the_o great_a segment_n have_v to_o the_o less_o to_o put_v the_o less_o segment_n to_o be_v the_o side_n of_o the_o icosahedron_n &_o devide_v the_o say_a power_n of_o the_o diameter_n by_o line_n have_v the_o proportion_n of_o the_o whole_a to_o the_o less_o segment_n to_o express_v the_o side_n of_o the_o dodecahedron_n by_o the_o less_o segment_n which_o thing_n may_v well_o be_v do_v between_o magnitude_n incommensurable_a the_o end_n of_o the_o fourteen_o book_n of_o euclides_n element_n after_o flussas_n ¶_o the_o fifteen_o book_n of_o euclides_n element_n this_o finetenth_n and_o last_o book_n of_o euclid_n or_o rather_o the_o second_o book_n of_o appollonius_n or_o hypsicles_n book_n teach_v the_o inscription_n and_o circumscription_n of_o the_o five_o regular_a body_n one_o within_o and_o about_o a_o other_o a_o thing_n undouted_o pleasant_a and_o delectable_a in_o mind_n to_o contemplate_v and_o also_o profitable_a and_o necessary_a in_o act_n to_o practice_v for_o without_o practice_n in_o act_n it_o be_v very_o hard_a to_o see_v and_o conceive_v the_o construction_n and_o demonstration_n of_o the_o proposition_n of_o this_o book_n unless_o a_o man_n have_v a_o very_a deep_a sharp_a &_o fine_a imagination_n wherefore_o i_o will_v wish_v the_o diligent_a student_n in_o this_o book_n to_o make_v the_o study_n thereof_o more_o pleasant_a unto_o he_o to_o have_v present_o before_o his_o eye_n the_o body_n form_v &_o frame_v of_o past_v paper_n as_o i_o teach_v after_o the_o definition_n of_o the_o eleven_o book_n and_o then_o to_o draw_v and_o describe_v the_o line_n and_o division_n and_o superficiece_n according_a to_o the_o construction_n of_o the_o proposition_n in_o which_o description_n if_o he_o be_v wary_a and_o diligent_a he_o shall_v find_v all_o thing_n in_o these_o solid_a matter_n as_o clear_v and_o as_o manifest_v unto_o the_o eye_n as_o be_v thing_n before_o teach_v only_o in_o plain_a or_o superficial_a figure_n and_o although_o i_o have_v before_o in_o the_o twelve_o book_n admonish_v the_o reader_n hereof_o yet_o because_o in_o this_o book_n chief_o that_o thing_n be_v require_v i_o think_v it_o shall_v not_o be_v irksome_a unto_o he_o again_o to_o be_v put_v in_o mind_n thereof_o far_o this_o be_v to_o be_v note_v that_o in_o the_o greek_a exemplar_n be_v find_v in_o this_o 15._o book_n only_o 5._o proposition_n which_o 5._o be_v also_o only_o touch_v and_o set_v forth_o by_o hypsicy_n unto_o which_o campane_n add_v 8._o and_o so_o make_v up_o the_o number_n of_o 13._o campane_n undoubted_o although_o he_o be_v very_o well_o learn_v and_o that_o general_o in_o all_o kind_n of_o learning_n yet_o assure_o be_v bring_v up_o in_o a_o time_n of_o rudeness_n when_o all_o good_a letter_n be_v darken_v &_o barberousnes_n have_v
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
by_o the_o 13._o of_o the_o first_o for_o the_o right_a line_n ti_fw-mi and_o ik_fw-mi be_v set_v upon_o the_o line_n mo._n and_o by_o the_o same_o reason_n may_v the_o rest_n of_o the_o angle_n namely_o ikl_n klt_n lti_fw-la be_v prove_v right_a angle_n and_o they_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n namely_o mnxo_n by_o the_o 7._o of_o the_o eleven_o wherefore_o the_o right_a line_n which_o join_v together_o the_o centre_n of_o the_o plain_a superficial_a triangle_n which_o make_v the_o solid_a angle_n a_o do_v make_v the_o square_a itkl_n and_o by_o the_o same_o reason_n may_v be_v prove_v that_o the_o plain_a superficial_a triangle_n of_o the_o rest_n of_o the_o five_o solid_a angle_n of_o the_o octohedron_n set_v at_o the_o point_n b_o g_o z_o d_o e_o do_v in_o the_o centre_n of_o their_o base_n receive_v square_n so_o that_o there_o be_v in_o number_n six_o square_n for_o every_o octohedron_n have_v six_o solid_a angle_n and_o those_o square_n be_v equal_a for_o their_o side_n do_v contain_v equal_a angle_n of_o inclination_n contain_v under_o equal_a side_n namely_o under_o those_o side_n which_o be_v draw_v from_o the_o centre_n to_o the_o side_n of_o the_o equal_a triangle_n by_o the_o 2._o corollary_n of_o the_o 18._o of_o the_o thirteen_o wherefore_o itklrpus_n be_v a_o cube_fw-la by_o the_o 21._o definition_n of_o the_o eleven_o and_o have_v his_o angle_n in_o the_o centre_n of_o the_o base_n of_o the_o octohedron_n and_o therefore_o be_v inscribe_v in_o it_o by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o octohedron_n give_v be_v describe_v a_o cube_fw-la which_o be_v require_v to_o be_v do_v the_o 5._o proposition_n the_o 5._o problem_n in_o a_o icosahedron_n give_v to_o describe_v a_o dodecahedron_n take_v a_o icosahedron_n one_o of_o who_o solid_a angle_n let_v be_v z._n construction_n now_o forasmuch_o as_o by_o those_o thing_n which_o have_v be_v prove_v in_o the_o 16._o of_o the_o thirteen_o the_o base_n of_o the_o triangle_n which_o contain_v the_o angle_n of_o the_o icosahedron_n do_v make_v a_o pentagon_n inscribe_v in_o a_o circle_n let_v that_o pentagon_n be_v abgde_v which_o be_v make_v of_o the_o five_o base_n of_o the_o triangle_n who_o plain_a superficial_a angle_n remain_v make_v the_o solid_a angle_n give_v namely_o z._n and_o take_v the_o centre_n of_o the_o circle_n which_o contain_v the_o foresay_a triangle_n which_o centre_n let_v be_v the_o point_n i_o t_o edward_n m_o l_o and_o draw_v these_o right_a line_n it_o tk_n km_o ml_o li._n demonstration_n now_o then_o a_o perpendicular_a line_n draw_v from_o the_o point_n z_o to_o the_o plain_a superficies_n of_o the_o pentagon_n abgde_v shall_v fall_v upon_o the_o centre_n of_o the_o circle_n which_o contain_v the_o pentagon_n abgde_v by_o those_o thing_n which_o have_v be_v prove_v in_o the_o self_n same_o 16._o of_o the_o thirteen_o moreover_o perpendicular_a line_n draw_v from_o the_o centre_n to_o the_o side_n of_o the_o pentagon_n abgde_v shall_v in_o the_o point_n c_o n_o oh_o where_o they_o fall_v cut_v the_o right_a line_n ab_fw-la bg_o gd_o into_o two_o equal_a part_n by_o the_o 3._o of_o the_o three_o draw_v these_o right_a line_n cn_fw-la and_o no_o and_o forasmuch_o as_o the_o angle_n cbn_n and_o ngo_n be_v equal_a and_o be_v contain_v under_o equal_a side_n therefore_o the_o base_a cn_fw-la be_v equal_a to_o the_o base_a no_o by_o the_o 4._o of_o the_o first_o moreover_o perpendicular_a line_n drâ—â—â—e_v from_o the_o point_n z_o to_o the_o bâ—sâ—â—_n of_o the_o pentagon_n abgde_v shall_v likewise_o cut_v the_o base_n into_o two_o equal_a part_n by_o thâ—â—_n of_o the_o three_o for_o the_o perpendicular_n pass_v by_o the_o centre_n by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o thâ—se_a perpendicular_a line_n shall_v fall_v upon_o the_o point_n c_o n_o o._n and_o now_o forasmuch_o as_o the_o right_a line_n zi_n ig_n be_v equal_a to_o the_o right_a line_n zt_v tn_n &_o also_o to_o the_o right_a line_n zk_v ko_o by_o reason_n of_o the_o likeness_n of_o the_o equal_a triangle_n therefore_o the_o line_n it_o be_v a_o parallel_n to_o the_o line_n cn_fw-la and_o so_o also_o be_v the_o line_n tk_n to_o the_o line_n no_o by_o the_o 2._o of_o the_o six_o wherefore_o the_o angle_n itk_n and_o cno_n be_v equal_a by_o the_o 11._o of_o the_o eleven_o again_o forasmuch_o as_o the_o triangle_n cbn_n and_o ngo_n be_v isoscel_n triangle_n therefore_o the_o angle_n bcn_n and_o bnc_n be_v equal_a by_o the_o 5._o of_o the_o first_o and_o by_o the_o same_o reason_n the_o angle_n gno_fw-la and_o gon_n be_v equal_a and_o moreover_o the_o angle_n bcn_n and_o bnc_n be_v equal_a to_o the_o angle_n gno_fw-la and_o go_v for_o that_o the_o triangle_n cbn_n and_o ngo_n be_v equal_a and_o like_a bâ—â—_n the_o three_o angle_n bnc_fw-la cno_n ong_n be_v equal_a to_o two_o right_a angle_n by_o the_o 13._o of_o the_o first_o for_o that_o upon_o the_o right_a line_n bâ—_n be_v set_v the_o right_a line_n cn_fw-la &_o on_o and_o the_o three_o angle_n of_o the_o triangle_n cbn_n namely_o the_o angle_n bnc_fw-la bcn_n or_o gno_n for_o the_o angle_n gno_n be_v equal_a to_o the_o angle_n bcn_n as_o it_o have_v be_v prove_v and_o nbc_n be_v also_o equal_a to_o two_o right_a angle_n by_o the_o 32._o of_o the_o first_o wherefore_o take_v away_o the_o angle_n bnc_n &_o gno_fw-la the_o angle_n remain_v namely_o cno_n be_v equal_a to_o the_o angle_n remain_v namely_o to_o cbn_n wherefore_o also_o the_o angle_n itk_n which_o be_v prove_v to_o be_v equal_a to_o the_o angle_n cno_n be_v equal_a to_o the_o angle_n cbn_n wherefore_o itk_n be_v the_o angle_n of_o a_o pentagon_n and_o by_o the_o same_o reason_n may_v be_v proâ—ed_v that_o the_o rest_n of_o the_o angle_n name_o the_o angle_n till_o ilm_n lmk_n mkt_v be_v equal_a to_o the_o rest_n of_o the_o angle_n namely_o to_o bae_o aed_n edg_n dgb_n wherefore_o itkml_n be_v a_o equilater_n and_o equiangle_n pentagon_n by_o the_o 4._o of_o the_o first_o for_o the_o equal_a base_n of_o the_o pentagon_n itkml_n do_v subtend_v equal_a angle_n set_v at_o the_o point_n z_o and_o comprehend_v under_o equal_a side_n moreover_o it_o be_v manifest_a that_o the_o pentagon_n i_o tkml_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n for_o forasmuch_o as_o the_o angle_n onc_n and_o ncp_n be_v in_o one_o and_o the_o self_n sâ—me_v plain_a superficies_n namely_o in_o the_o superficies_n abgde_v but_o unto_o the_o same_o plain_a superficies_n the_o plain_a superficiece_n of_o the_o angle_n kti_fw-la and_o until_o be_v parallel_n by_o the_o 15._o of_o the_o eleven_o and_o the_o triangle_n kti_n and_o until_o concur_v wherefore_o they_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n by_o the_o corollary_n of_o the_o 16._o of_o the_o eleven_o and_o by_o the_o same_o reason_n so_o may_v we_o prove_v that_o the_o triangle_n ilm_n lmk_n mkt_v be_v in_o the_o self_n same_o plain_a superficies_n wherein_o be_v the_o triangle_n kti_n and_o till_o wherefore_o the_o pentagon_n itkml_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n wherefore_o the_o solid_a angle_n of_o the_o icosahedron_n namely_o the_o solid_a angle_n at_o the_o point_n z_o subtend_v a_o equilater_n and_o equiangle_n pentagon_n plain_a superficies_n which_o pentagon_n have_v his_o plain_a superficial_a angle_n in_o the_o centre_n of_o the_o triangle_n which_o make_v the_o solid_a angle_n z._n and_o in_o like_a sort_n may_v we_o prove_v that_o the_o other_o eleven_o solid_a angle_n of_o the_o icosahedron_n each_o of_o which_o eleven_o solid_a angle_n be_v equal_a and_o like_a to_o the_o solid_a angle_n z_o by_o the_o 16._o of_o the_o thirteen_o be_v subtend_v unto_o pentagon_n equal_a and_o like_a and_o in_o like_a sort_n set_v to_o the_o pentagon_n itkml_n and_o forasmuch_o as_o in_o those_o pentagon_n the_o right_a line_n which_o join_v together_o the_o centre_n of_o the_o base_n be_v common_a side_n it_o follow_v that_o those_o 12._o pentagon_n include_v a_o solid_a which_o solid_a be_v therefore_o a_o dodecahedron_n by_o the_o 24._o definition_n of_o the_o eleven_o and_o be_v by_o the_o first_o definition_n of_o this_o book_n describe_v in_o the_o icosahedron_n five_o side_n whereof_o 〈◊〉_d set_v upon_o the_o pentagon_n abgde_v wherefore_o in_o a_o icosahedron_n give_v iâ—_z inscribe_v a_o dodecahedron_n which_o be_v require_v to_o be_v do_v a_o annotation_n of_o hypsiâ—les_n this_o be_v to_o be_v note_v that_o if_o a_o man_n shall_v demand_v 〈◊〉_d many_o side_n a_o icosahedron_n have_v we_o may_v thus_o answer_v it_o be_v manifest_a that_o a_o icosahâ—râ—n_n be_v contain_v under_o 20._o triangle_n and_o that_o every_o triangle_n consist_v of_o three_o
that_o be_v the_o right_a line_n agnostus_n gb_o bd_o dam_fw-ge ca_n cg_o cb_o cd_o and_o ja_z ig_n ib_n id_fw-la be_v equal_a the_o one_o to_o the_o other_o wherefore_o also_o the_o 8._o triangle_n cag_n cgb_n cbd_v cda_n jag_n igb_n ibd_v &_o ida_n be_v equal_a and_o equilater_n and_o therefore_o agbdci_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 say_v octahedron_n be_v include_v in_o the_o dodecahedron_n by_o the_o first_o definition_n of_o this_o book_n for_o that_o all_o the_o angle_n thereof_o do_v at_o one_o time_n touch_v the_o side_n of_o the_o dodecahedron_n wherefore_o in_o the_o dodecahedron_n give_v be_v include_v a_o octohedron_n which_o be_v require_v to_o be_v do_v ¶_o the_o 10._o proposition_n the_o 10._o problem_n in_o a_o dodecahedron_n give_v to_o inscribe_v a_o equilater_n trilater_n pyramid_n svppose_v that_o the_o dodecahedron_n give_v be_v abcd_o of_o which_o dodecahedron_n take_v three_o base_n meet_v at_o the_o point_n s_o namely_o these_o three_o base_n alsik_n dnsle_n and_o sibrn_n construction_n and_o of_o those_o three_o base_n take_v the_o three_o angle_n at_o the_o point_n a_o b_o d_o and_o draw_v these_o right_a line_n ab_fw-la bd_o and_o dam_fw-ge and_o let_v the_o diameter_n of_o the_o sphere_n contain_v the_o dodecahedron_n be_v so_o and_o then_o draw_v these_o right_a line_n ao_o boy_n and_o do_v now_o forasmuch_o as_o by_o the_o 17._o of_o the_o thirteen_o the_o angle_n of_o the_o dodecahedron_n be_v set_v in_o the_o superficies_n of_o the_o sphere_n describe_v about_o the_o dodecahedronâ—_n demonstration_n therefore_o if_o upon_o the_o diameter_n so_o and_o by_o the_o point_n a_o be_v describe_v a_o semicircle_n it_o shall_v make_v the_o angle_n sao_n a_o right_a angle_n by_o the_o 31._o of_o the_o three_o and_o likewise_o if_o the_o same_o semicircle_n be_v draw_v by_o the_o point_n d_o and_o b_o it_o shall_v also_o make_v the_o angle_n sbo_n and_o sdo_v right_a angle_n wherefore_o the_o diameter_n so_o contain_v in_o power_n both_o the_o line_n sa_o ao_o or_o the_o line_n sb_n boy_n or_o else_o sd_z do_v but_o the_o line_n sa_o sd_z sb_n be_v equal_a the_o one_o to_o the_o other_o for_o they_o each_o subtend_v one_o of_o the_o angle_n of_o equal_a pentagon_n wherefore_o the_o other_o line_n remain_v namely_o ao_o boy_n do_v be_v equal_a the_o one_o to_o the_o other_o and_o by_o the_o same_o reason_n may_v be_v prove_v that_o the_o diameter_n hd_v which_o subtend_v the_o two_o right_a line_n ha_o ad_fw-la contain_v in_o power_n both_o the_o say_v two_o right_a line_n and_o also_o contain_v in_o power_n both_o the_o right_a line_n hb_o and_o bd_o which_o two_o right_a line_n it_o also_o suhtend_v and_o moreover_o by_o the_o same_o reason_n the_o diameter_n ac_fw-la which_o subtend_v the_o right_a line_n cb_o and_o basilius_n contain_v in_o power_n both_o the_o say_v right_a line_n câ—_n and_o basilius_n but_o the_o right_a line_n ha_o hb_o and_o cb_o be_v equal_a the_o one_o to_o the_o other_o for_o that_o each_o of_o they_o also_o subtend_v one_o of_o the_o angle_n of_o equal_a pentagonsâ—_n wherefore_o the_o right_a line_n remain_v namely_o ad_fw-la bd_o and_o basilius_n be_v equal_a the_o one_o to_o the_o other_o and_o by_o the_o same_o reason_n may_v be_v prove_v that_o each_o of_o those_o right_a line_n ad_fw-la bd_o and_o basilius_n be_v equal_a to_o each_o of_o the_o right_a line_n ao_o boy_n and_o do_v wherefore_o the_o six_o right_a line_n ab_fw-la bd_o dam_fw-ge ao_o boy_n &_o do_v be_v equal_a the_o one_o to_o the_o other_o and_o therefore_o the_o triangle_n which_o be_v make_v of_o they_o namely_o the_o triangle_n abdella_n aob_n aod_n and_o bod_fw-mi be_v equal_a and_o equilater_n which_o triangle_n therefore_o do_v make_v a_o pyramid_n abdo_n who_o base_a be_v abdella_n and_o top_n the_o point_n o._n each_o of_o the_o angle_n of_o which_o pyramid_n namely_o the_o angle_n at_o the_o point_n a_o b_o d_o oh_o do_v in_o the_o self_n same_o point_n touch_v the_o angle_n of_o the_o dodecahedron_n wherefore_o the_o say_a pyramid_n be_v inscribe_v in_o the_o dodecahedron_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o dodecahedron_n give_v be_v inscribe_v a_o trilater_n equilater_n pyramid_n which_o be_v require_v to_o be_v do_v ¶_o the_o 11._o proposition_n the_o 11._o problem_n in_o a_o icosahedron_n give_v to_o inscribe_v a_o cube_fw-la it_o be_v manifest_a by_o the_o 7._o of_o this_o book_n that_o the_o angle_n of_o a_o dodecahedron_n be_v set_v in_o the_o centre_n of_o the_o base_n of_o the_o icosahedron_n and_o by_o the_o 8._o of_o this_o book_n it_o be_v prove_v that_o the_o angle_n of_o a_o cube_fw-la be_v set_v in_o the_o angle_n of_o a_o dodecahedron_n wherefore_o the_o self_n same_o angle_n of_o the_o cube_fw-la shall_v of_o necessity_n be_v set_v in_o the_o centre_n of_o the_o base_n of_o icosahedron_n wherefore_o the_o cube_fw-la shall_v be_v inscribe_v in_o the_o icosahedron_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o icosahedron_n give_v be_v include_v a_o cube_fw-la which_o be_v require_v to_o be_v do_v ¶_o the_o 12._o proposition_n the_o 12._o problem_n in_o a_o icosahedron_n give_v to_o inscribe_v a_o trilater_n equilater_n pyramid_n by_o the_o former_a proposition_n it_o be_v manifest_a that_o the_o angle_n of_o a_o cube_fw-la be_v set_v in_o the_o centre_n of_o the_o base_n of_o the_o icosâ—hedron_n and_o by_o the_o first_o of_o this_o book_n it_o be_v plain_a that_o the_o four_o angle_n of_o a_o pyramid_n be_v set_v in_o four_o angle_n of_o a_o cube_fw-la wherefore_o it_o be_v evident_a by_o the_o first_o definition_n of_o this_o book_n that_o a_o pyramid_n describe_v of_o right_a line_n join_v together_o these_o four_o centre_n of_o the_o base_n of_o the_o icosahedron_n shall_v be_v inscribe_v in_o the_o same_o icosahedron_n wherefore_o in_o a_o icosadron_n give_v be_v inscribe_v a_o equilater_n trilater_n pyramid_n which_o be_v require_v to_o be_v do_v ¶_o the_o 13._o problem_n the_o 13._o proposition_n in_o a_o cube_n give_v to_o inscribe_v a_o dodecahedron_n take_v a_o cube_n adfl_fw-mi construction_n and_o divide_v every_o one_o of_o the_o side_n thereof_o into_o two_o equal_a part_n in_o the_o point_n t_o h_o edward_n p_o g_o l_o m_o f_o and_o pkq_n and_o draw_v these_o right_a line_n tk_n gf_o pq_n hk_n ps_n and_o lm_o which_o line_n again_o divide_v into_o two_o equal_a part_n in_o the_o point_n n_o five_o a'n_z ay_o z_o x._o and_o draw_v these_o right_a line_n nigh_o vx_n and_o iz_n now_o the_o three_o line_n nigh_o vx_n and_o iz_n together_o with_o the_o diameter_n of_o the_o cube_fw-la shall_v cut_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o the_o centre_n of_o the_o cube_fw-la by_o the_o 3â—_o of_o the_o eleven_o let_v that_o centre_n be_v the_o point_n o._n and_o not_o to_o stand_v long_o about_o the_o demonstration_n understand_v all_o these_o right_a line_n to_o be_v equal_a and_o parallel_n to_o the_o side_n of_o the_o cube_fw-la and_o to_o cut_v the_o one_o the_o other_o right_o angle_a wise_a by_o the_o 29._o of_o the_o first_o let_v their_o half_n namely_o fv_n gv_n high_a and_o king_n and_o the_o rest_n such_o like_a be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n by_o the_o 30._o of_o the_o six_o who_o great_a segment_n let_v be_v the_o line_n f_n gb_o hc_n and_o ke_o etc_n etc_n and_o draw_v these_o right_a line_n give_v ge_z bc_o and_o be._n now_o forasmuch_o as_o the_o line_n give_v be_v equal_a to_o the_o whole_a line_n gv_n which_o be_v the_o half_a of_o the_o side_n of_o the_o cube_fw-la demonstration_n and_o the_o line_n je_n be_v equal_a to_o the_o line_n bv_n that_o be_v to_o the_o less_o segment_n therefore_o the_o square_n of_o the_o line_n give_v and_o je_n be_v triple_a to_o the_o square_n of_o the_o line_n gb_o by_o the_o 4._o of_o the_o thirteen_o but_o unto_o the_o square_n of_o the_o line_n give_v and_o je_n the_o square_a of_o the_o line_n ge_z be_v equal_a by_o the_o 47._o of_o the_o firstâ—_o for_o the_o angle_n gie_n be_v a_o right_a angle_n wherefore_o the_o square_a of_o the_o line_n ge_z be_v triple_a to_o the_o square_n of_o the_o line_n gb_o and_o forasmuch_o as_o the_o line_n fg_o be_v erect_v perpendicular_o to_o the_o plain_a agkl_n by_o the_o 4._o of_o the_o eleven_o for_o it_o be_v erect_v perpendicular_o to_o the_o two_o line_n agnostus_n and_o give_v therefore_o the_o angle_n bge_n be_v a_o right_a angle_n for_o the_o line_n ge_z be_v draw_v in_o the_o plain_a agkl_n wherefore_o the_o line_n be_v contain_v in_o power_n the_o two_o line_n bg_o and_o ge_z by_o the_o 47._o of_o the_o first_o be_v in_o power_n quadruple_a to_o the_o
subtend_v angle_n of_o triangle_n like_v unto_o the_o triangle_n who_o angle_n the_o line_n ac_fw-la bf_o and_o pl_z subtend_n be_v cut_v into_o two_o equal_a part_n in_o the_o point_n z_o an_o i_o by_o the_o 4._o of_o the_o six_o so_o also_o be_v the_o other_o line_n nv_n xm_o d_n qt_fw-la which_o be_v equal_a unto_o the_o line_n hk_o &_o ge_z cut_v in_o like_a sort_n and_o they_o shall_v cut_v the_o line_n ac_fw-la bf_o and_o pl_z like_z wherefore_o the_o line_n ko_o which_o be_v equal_a to_o rz_n shall_v make_v the_o great_a segment_n the_o line_n ro_n which_o be_v equal_a to_o the_o line_n zk_n for_o the_o great_a segment_n of_o the_o rz_n be_v the_o line_n zk_n and_o therefore_o the_o line_n oi_o shall_v be_v the_o less_o segment_n when_o as_o the_o whole_a line_n ri_n be_v equal_a to_o the_o whole_a line_n rz_n wherefore_o the_o square_n of_o the_o whole_a line_n ko_o and_o of_o the_o less_o segment_n oi_o be_v triple_a to_o the_o square_n of_o the_o great_a segment_n ro_n by_o the_o 4._o of_o the_o thirteen_o wherefore_o the_o line_n ki_v which_o contain_v in_o power_n the_o two_o line_n ko_o and_o oi_o be_v in_o power_n triple_a to_o the_o line_n ro_n by_o the_o 47._o of_o the_o first_o for_o the_o angle_n koi_n be_v a_o right_a angle_n and_o forasmuch_o as_o the_o line_n fe_o and_o fg_o which_o be_v the_o less_o segment_n of_o the_o side_n of_o the_o octohedron_n be_v equal_a and_o the_o line_n fk_o be_v common_a to_o they_o both_o and_o the_o angle_n kfg_n and_o kfe_n of_o the_o triangle_n of_o the_o octohedron_n be_v equal_a the_o base_n kg_v and_o ke_o shall_v by_o the_o 4_o of_o the_o first_o be_v equal_a and_o therefore_o the_o angle_n kie_fw-mi and_o kig_n which_o they_o subtend_v be_v equal_a by_o the_o 8._o of_o the_o first_o wherefore_o they_o be_v right_a angle_n by_o the_o 13._o of_o the_o first_o wherefore_o the_o right_a line_n ke_o which_o contain_v in_o power_n the_o two_o line_n ki_v and_o â—e_n by_o the_o 47._o of_o the_o first_o be_v in_o power_n quadruple_a to_o the_o line_n ro_n or_o je_n for_o the_o line_n ri_n be_v prove_v to_o be_v in_o power_n triple_a to_o the_o same_o line_n ro_n but_o the_o line_n ge_z be_v double_a to_o the_o line_n je_n wherefore_o the_o line_n ge_z be_v also_o in_o power_n 〈…〉_o pf_n and_o by_o the_o same_o reason_n may_v be_v prove_v that_o the_o â—est_n of_o the_o eleven_o solid_a angle_n of_o the_o 〈◊〉_d be_v 〈…〉_o the_o section_n of_o every_o one_o of_o the_o side_n of_o the_o octohedron_n namely_o in_o the_o point_n e_o n_o five_o h_o â—_o m_o â—_o d_o s_o q_o t._n wherefore_o there_o be_v 12._o angle_n of_o the_o icosahedron_n moreover_o forasmuch_o as_o every_o one_o of_o the_o base_n of_o the_o octohedron_n do_v each_o contain_v triangle_n of_o the_o icosahedron_n 〈…〉_o pyramid_n abcâ—fp_n which_o be_v the_o half_a of_o the_o octohedron_n the_o triangle_n fcp_n receive_v in_o thâ—_z section_n of_o his_o side_n the_o â—_o triangle_n gm_n and_o the_o triangle_n cpb_n contain_v the_o triangle_n nxs_n and_o thâ—_z triangle_n â—ap_n contain_v the_o triangle_n hnd_n and_o moreover_o the_o triangle_n apf_n contain_v the_o triangle_n edge_n and_o the_o same_o may_v be_v prove_v in_o the_o opposite_a pyramid_n abcfl_fw-mi wherefore_o there_o shall_v be_v eight_o triangleâ—_n and_o forasmuch_o as_o beside_o these_o triangle_n to_o every_o one_o of_o the_o solid_a angle_n of_o the_o octohedron_n 〈◊〉_d subtend_v two_o triangle_n as_o the_o triangle_n keg_n and_o meg_n to_o the_o angle_n f_o and_o the_o triangle_n hnv_n and_o xnv_n to_o the_o angle_n b_o also_o the_o triangle_n nd_n and_o â—ds_n to_o the_o angle_n p_o likewise_o the_o triangleâ—_n dhk_n and_o qhk_v to_o the_o angle_n a_o moreover_o the_o triangle_n eqt_a and_o vqt_a to_o the_o angle_n l_o and_o final_o the_o triangle_n sxm_n and_o txm_n to_o the_o angle_n c_o these_o 12._o triangle_n be_v add_v to_o thâ—â—_n for_o 〈◊〉_d triangle_n shall_v produce_v _o triangle_n equal_a and_o equilater_v couple_v together_o which_o shall_v male_a a_o icosahedron_n by_o the_o 25._o definition_n of_o the_o eleven_o and_o it_o shall_v be_v inscribe_v in_o the_o octohedron_n give_v abcâ—â—l_o by_o the_o first_o definition_n of_o this_o book_n for_o the_o 1â—_o angle_n thereof_o be_v set_v in_o 1â—_o like_a section_n of_o the_o side_n of_o the_o octohedron_n wherefore_o in_o a_o octohedron_n give_v be_v inscribe_v a_o icosahedron_n ¶_o first_n corollary_n the_o side_n of_o a_o equilater_n triangle_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n a_o right_a line_n subtend_v within_o the_o triangle_n the_o angle_n which_o be_v contain_v under_o the_o great_a segment_n and_o the_o less_o be_v in_o power_n duple_n to_o the_o less_o segment_n of_o the_o same_o side_n for_o the_o line_n ke_o which_o subtend_v the_o angle_n kfe_n of_o the_o triangle_n afl_fw-mi which_o angle_n kfe_n be_v contain_v under_o the_o two_o segment_n kf_a &_o fe_o be_v prove_v equal_a 〈◊〉_d the_o line_n hk_o which_o contain_v in_o power_n the_o two_o less_o segment_n ha_o and_o ak_o by_o the_o 47._o of_o the_o â—â—rst_a foâ—_n 〈◊〉_d angle_n hak_n be_v 〈…〉_o second_o corollary_n the_o base_n of_o the_o icosahedron_n be_v concentrical_a that_o be_v have_v one_o and_o the_o self_n same_o centre_n with_o the_o base_n of_o the_o octohedron_n which_o contain_v it_o for_o suppose_v that_o 〈…〉_o octohedron_n 〈◊〉_d ecd_v the_o base_a of_o a_o icosahedron_n and_o let_v the_o centre_n of_o the_o base_a abg_o be_v the_o point_n f._n and_o draw_v these_o right_a line_n favorina_n fb_o fc_o and_o fe_o now_o than_o the_o 〈…〉_o to_o the_o two_o line_n fb_o and_o bc_o for_o they_o be_v line_n draw_v from_o the_o centre_n and_o be_v also_o less_o segment_n and_o they_o contain_v the_o 〈…〉_o ¶_o the_o 17._o problem_n the_o 17._o proposition_n in_o a_o octohedron_n give_v to_o inscribe_v a_o dodecahedron_n construction_n svppose_v that_o the_o octohedron_n give_v be_v abgdec_n who_o 12._o â—ides_n let_v be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n as_o in_o the_o former_a proposition_n it_o be_v manifest_a that_o of_o the_o right_a line_n which_o couple_v thâ—se_a section_n be_v make_v 20._o triangle_n of_o which_o 8._o be_v concentrical_a with_o the_o base_n of_o the_o octohedron_n by_o the_o second_o corollary_n of_o the_o former_a proposition_n if_o therefore_o in_o every_o one_o of_o the_o centre_n of_o the_o 20._o triangle_n be_v inscribe_v by_o the_o 1._o of_o this_o book_n every_o one_o of_o the_o â—â—_o â—â—gles_n of_o the_o dodecahedron_n demonstration_n we_o shall_v find_v that_o â—_o angle_n of_o the_o dodecahedron_n be_v set_v in_o the_o 8._o centre_n of_o the_o base_n of_o the_o octohedron_n namely_o these_o angle_n i_o u._fw-mi ct_n oh_o m_o a_o p_o and_o x_o and_o of_o the_o other_o 12._o solid_a angle_n there_o be_v two_o in_o the_o centre_n of_o the_o two_o triangle_n which_o have_v one_o side_n common_a under_o every_o one_o of_o the_o solid_a angle_n of_o the_o octohedron_n namely_o under_o the_o solid_a angle_n a_o the_o two_o solid_a angle_n king_n z_o under_o the_o solid_a angle_n b_o the_o two_o solid_a angle_n h_o t_o under_o the_o solid_a angle_n g_o the_o two_o solid_a angle_n in_o v_z under_o the_o solid_a angle_n d_o the_o two_o solid_a angle_n f_o l_o under_o the_o solid_a angle_n e_o the_o two_o solid_a angle_n s_o n_o under_o the_o solid_a angle_n c_o the_o two_o solid_a angle_n queen_n r_o and_o forasmuch_o as_o in_o the_o octohedron_n be_v six_o solid_a angle_n under_o they_o shall_v be_v subtend_v 12._o solid_a angle_n of_o the_o dodâ—cahedron_n and_o so_o be_v mâ—de_v 20._o solid_a angle_n compose_v of_o 12._o equal_a and_o equilater_v superficial_a pentagon_n as_o it_o be_v 〈◊〉_d by_o the_o 5._o of_o this_o book_n which_o therefore_o contain_v a_o dodecahedron_n by_o the_o 24._o definition_n of_o the_o eleven_o and_o it_o be_v inscribe_v in_o the_o octohedron_n by_o the_o 1._o definition_n of_o this_o book_n for_o that_o every_o one_o of_o the_o base_n of_o the_o octohedron_n do_v receive_v angle_n thereof_o wherefore_o in_o a_o octohedron_n give_v be_v inscribe_v a_o dodecahedron_n ¶_o the_o 18._o problem_n the_o 18._o proposition_n in_o a_o trilater_n and_o equilater_n pyramid_n to_o inscribe_v a_o cube_n construction_n svppose_v that_o there_o be_v a_o trilater_n equilater_n pyramid_n who_o base_a let_v be_v abc_n and_o â—oppe_v the_o point_n d._n and_o let_v it_o be_v comprehend_v in_o a_o sphereâ—_n by_o the_o 13._o of_o the_o 〈◊〉_d and_o lâ—â—_n the_o centre_n of_o that_o sphere_n be_v the_o point_n e._n and_o from_o the_o solid_a angle_n a_o b_o c_o d_o draw_v right_a line_n pass_v by_o the_o centre_n e_o unto_o the_o opposite_a base_n of_o
side_n gd_v the_o angle_n m_o n_o under_o the_o side_n ab_fw-la the_o angle_n t_o s_o under_o the_o side_n bg_o the_o angle_n p_o oh_o and_o under_o the_o side_n agnostus_n the_o angle_n r_o q_o so_o there_o rest_v 4._o angle_n who_o true_a place_n we_o will_v now_o appoint_v forasmuch_o as_o a_o cube_fw-la contain_v in_o one_o and_o the_o self_n same_o sphere_n with_o a_o dodecahedron_n be_v inscribe_v in_o the_o same_o dodecahedron_n as_o it_o be_v manifest_a by_o the_o 17._o of_o the_o thirteen_o and_o 8._o of_o this_o book_n it_o follow_v that_o a_o cube_fw-la and_o a_o dodecahedron_n circumscribe_v about_o it_o be_v contain_v in_o one_o and_o the_o self_n same_o body_n for_o that_o their_o angle_n concur_v in_o one_o and_o the_o self_n same_o point_n and_o it_o be_v prove_v in_o the_o 18._o of_o this_o book_n that_o 4._o angle_n of_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n be_v set_v in_o the_o middle_a section_n of_o the_o perpendicularâ—_n which_o be_v draw_v from_o the_o solid_a angle_n of_o the_o pyramid_n to_o the_o opposite_a base_n wherefore_o the_o other_o 4._o angle_n of_o the_o dodecahedron_n be_v also_o as_o the_o angle_n of_o the_o cube_fw-la set_v in_o those_o middle_a section_n of_o the_o perpendicular_n namely_o the_o angle_n v_o be_v set_v in_o the_o midst_n of_o the_o perpendicular_a ahâ—_n the_o angle_n y_fw-fr in_o the_o midst_n of_o the_o perpendicular_a bf_o the_o angle_n x_o in_o the_o midst_n of_o the_o perpendicular_a ge_z and_o last_o the_o angle_n d_o in_o the_o midst_n of_o the_o perpendicular_a d_o which_o be_v draw_v from_o the_o top_n of_o the_o pyramid_n to_o the_o opposite_a base_a wherefore_o those_o 4._o angle_n of_o the_o dodecahedron_n may_v be_v say_v to_o be_v direct_o under_o the_o solid_a angle_n of_o the_o pyramid_n or_o they_o may_v be_v say_v to_o be_v set_v at_o the_o perpendicular_n wherefore_o the_o dodecahedron_n after_o this_o manner_n set_v be_v inscribe_v in_o the_o pyramid_n give_v by_o the_o first_o definition_n of_o this_o book_n for_o that_o upon_o every_o one_o of_o the_o base_n of_o the_o pyramid_n be_v set_v a_o angle_n of_o the_o dodecahedron_n inscribe_v wherefore_o in_o a_o trilater_n equilater_n pyramid_n be_v inscribe_v a_o dodecahedron_n the_o 21._o problem_n the_o 21._o proposition_n in_o every_o one_o of_o the_o regular_a solid_n to_o inscribe_v a_o sphere_n in_o the_o 13._o of_o thâ—_z thirteen_o and_o thâ—_z other_o 4._o proposition_n follow_v he_o iâ—_z be_v declare_v that_o â—he_n â—â—_o regular_a solidesâ—â—re_o so_o contain_v in_o a_o sphere_n that_o â—ight_a linâ—â—_n draw_v from_o the_o cenâ—â—â—_n oâ—_n the_o 〈…〉_o of_o 〈◊〉_d solid_a inscribe_v be_v equal_a which_o right_a line_n therefore_o make_v pyramid_n who_o â—oppes_n be_v the_o centre_n of_o the_o sphere_n or_o of_o the_o solid_a and_o the_o basâ—â—â—â—e_a cuâ—â—â—_n one_o of_o the_o base_n of_o those_o solid_n and_o 〈…〉_o solid_a squall_n and_o like_v the_o one_o to_o the_o other_o and_o describe_v in_o equal_a circle_n those_o circle_n shall_v cut_v the_o sphere_n for_o the_o angle_n which_o touch_v the_o circumference_n of_o the_o circle_n touch_v also_o the_o superficies_n of_o the_o sphere_n wherefore_o perpendiculars_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o base_n or_o to_o the_o plain_a superficiece_n of_o the_o equal_a circle_n be_v equal_a by_o the_o corollary_n of_o the_o assumpt_n of_o the_o 1â—_o of_o the_o twelve_o wherefore_o make_v the_o centre_n the_o 〈◊〉_d of_o the_o sphere_n which_o 〈◊〉_d the_o solid_a and_o thâ—_z space_n some_o one_o of_o the_o equal_a perpendicularâ—_n dâ—scribâ—_n a_o sphere_n and_o it_o shall_v touch_v every_o one_o of_o the_o base_n of_o 〈◊〉_d solid_a 〈…〉_o perficies_fw-la of_o the_o sphere_n pass_v beyond_o those_o base_n when_o as_o those_o pâ—â—peâ—diculars_n 〈…〉_o be_v draw_v from_o the_o centre_n to_o the_o base_n by_o the_o 3._o corollary_n of_o the_o saâ—â—â—â—â—umpt_n wherâ—fore_o â—e_v have_v iâ—_z every_o one_o of_o the_o regular_a body_n inscribe_v a_o sphere_n which_o regular_a boâ—â—â—_n be_v in_o number_n one_o iâ—_z 〈◊〉_d by_o the_o corollary_n of_o the_o 1â—_o of_o the_o 〈◊〉_d a_o corollary_n the_o regular_a figure_n inscribe_v in_o sphere_n and_o also_o the_o sphere_n circumscribe_v about_o they_o or_o contain_v they_o have_v one_o and_o the_o self_n same_o centre_n namely_o their_o pyramid_n the_o â—ngles_n of_o who_o base_n touch_v the_o superâ—â—â—â—â—â—_n of_o thâ—â—â—here_o do_v from_o those_o angle_n cause_v equal_a right_a line_n to_o be_v drawâ—â—_n to_o one_o and_o â—he_n self_n 〈◊〉_d poynâ—_n make_v the_o topâ—â—â—_n of_o the_o pyramid_n in_o the_o same_o point_n and_o therefore_o they_o 〈…〉_o thâ—_z câ—â—tres_n of_o the_o sphere_n in_o the_o self_n same_o top_n when_o 〈◊〉_d the_o right_a line_n draw_v from_o those_o angle_n to_o the_o croâ—â—ed_a superficies_n wherein_o be_v 〈◊〉_d the_o angle_n of_o the_o base_n of_o the_o pyramid_n be_v equal_o a_o advertisement_n of_o flussasâ—_n â—_z of_o these_o solid_n only_o the_o octohedron_n receive_v the_o other_o solid_n inscribe_v one_o with_o 〈…〉_o other_o for_o the_o octohedron_n contain_v the_o icosahedron_n inscribe_v in_o it_o and_o the_o same_o icosahedron_n contain_v the_o dodecahedron_n inscribe_v in_o the_o same_o icosahedron_n and_o the_o same_o dodecahedron_n contain_v the_o cube_fw-la inscribe_v in_o the_o same_o octohedron_n and_o 〈…〉_o â—â—râ—â—mscribeth_v the_o pyramid_n inscribe_v in_o the_o say_v octohedron_n but_o this_o happen_v not_o in_o the_o other_o solid_n the_o end_n of_o the_o fivetenth_fw-mi book_n of_o euclides_n elemenâ—â—â—_n after_o caâ—paâ—â—_n and_o 〈◊〉_d ¶_o the_o sixteen_o book_n of_o the_o element_n of_o geometry_n add_v by_o flussas_n in_o the_o former_a fivetenth_fw-mi book_n have_v be_v teach_v how_o to_o inscribe_v the_o five_o regular_a solid_n one_o with_o in_o a_o other_o now_o seem_v to_o rest_n to_o compare_v those_o solid_a so_o inscribe_v one_o to_o a_o other_o and_o to_o set_v forth_o their_o passion_n and_o propriety_n which_o thing_n flussas_n consider_v in_o this_o sixteen_o book_n add_v by_o he_o book_n have_v excellent_o well_o and_o most_o cunning_o perform_v for_o which_o undoubted_o he_o have_v of_o all_o they_o which_o have_v a_o love_n to_o the_o mathematicals_n deserve_v much_o praise_n and_o commendation_n both_o for_o the_o great_a traâ—ailes_n and_o paynâ—s_n which_o it_o be_v most_o likely_a he_o have_v taâ—â—n_v in_o invent_v such_o strange_a and_o wonderful_a proposition_n with_o their_o demonstration_n in_o this_o book_n contain_v as_o also_o for_o participate_v and_o communicate_v abroad_o the_o same_o to_o other_o which_o book_n also_o that_o the_o reader_n shall_v want_v nothing_o conduce_v to_o the_o perfection_n of_o euclides_n element_n i_o have_v with_o some_o travail_n translate_v &_o for_o the_o worthiness_n â—hereof_o have_v add_v it_o aâ—_z a_o sixteen_o book_n to_o the_o 15._o book_n of_o euclid_n vouchsafe_v therefore_o gentle_a reader_n diligent_o to_o read_v and_o poise_v it_o for_o in_o it_o shall_v you_o find_v noâ—_n only_a matter_n strange_a and_o delectable_a but_o also_o occasion_n of_o invention_n of_o great_a thing_n pertain_v to_o the_o nature_n of_o the_o five_o regular_a solidâ—sâ—_n ¶_o the_o 1._o proposition_n a_o dodecahedron_n and_o a_o cube_fw-la inscribe_v in_o it_o and_o a_o pyramid_n inscribe_v in_o the_o same_o cube_fw-la be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n for_o the_o angle_n of_o the_o pyramiâ—_n be_v seâ—_z in_o the_o angel_n of_o the_o cube_fw-la wherein_o it_o be_v inscribe_v by_o the_o first_o of_o the_o fivetenthâ—_n and_o all_o the_o angle_n of_o the_o cube_fw-la be_v set_v in_o the_o angle_n of_o the_o dodecahedâ—â—â—_n circumscribe_v 〈…〉_o 〈◊〉_d the_o 8._o of_o the_o fifteen_o and_o all_o the_o angle_n of_o the_o dodecahedron_n be_v set_v in_o the_o superficies_n of_o the_o sphere_n by_o the_o 17._o of_o the_o thirteen_o wherefore_o those_o three_o solid_n inscribe_v one_o within_o a_o other_o be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o first_o definition_n of_o the_o fifteen_o a_o dodecahedron_n therefore_o and_o a_o cube_fw-la inscribe_v in_o it_o and_o a_o pyramid_n inscribe_v in_o the_o same_o cube_fw-la be_v contain_v 〈…〉_o â—â—lfe_n same_o sphere_n 〈…〉_o these_o three_o solid_n livre_n 〈…〉_o elf_n same_o icosahedron_n or_o octohedron_n or_o pyramid_n 〈…〉_o i_o icosahedron_n by_o the_o 5.11_o &_o 12._o of_o the_o fifteen_o and_o they_o be_v 〈…〉_o ctohedron_n by_o the_o 4._o 6._o and_o 16._o of_o the_o same_o last_o they_o be_v inscribe_v in_o 〈…〉_o the_o first_o 18._o and_o 19_o of_o the_o same_o for_o the_o angle_n of_o all_o these_o solid_a 〈…〉_o the_o circumscribe_v icosahedron_n or_o octohedron_n or_o pyramid_n ¶_o the_o 〈…〉_o the_o proportion_n of_o a_o dodecahedron_n circumscribe_v about_o a_o cube_fw-la to_o a_o dodecahedron_n inscribe_v in_o the_o same_o cube_fw-la be_v
proportion_n which_o be_v in_o power_n duple_n to_o of_o the_o side_n of_o the_o octohedron_n inscribe_v in_o the_o dodecahedron_n draw_v the_o diameter_n el_n and_o fk_o of_o the_o octohedron_n now_o they_o couple_v the_o middle_a section_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n ab_fw-la and_o gd_o by_o the_o 9_o of_o the_o fifteen_o &_o 3._o corollary_n of_o the_o 17._o of_o the_o thirteen_o &_o every_o one_o of_o those_o diameter_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n do_v make_v the_o less_o segment_n the_o side_n of_o the_o dodecahedron_n by_o the_o 4._o corollary_n of_o the_o same_o wherefore_o the_o side_n ab_fw-la be_v the_o less_o segment_n of_o the_o line_n fk_o but_o the_o line_n fk_o contain_v in_o power_n the_o two_o equal_a line_n of_o &_o eke_o by_o the_o 47._o of_o the_o first_o for_o the_o angle_n fek_n be_v a_o right_a angle_n of_o the_o square_a fekl_n of_o the_o octohedron_n wherefore_o the_o line_n fk_o be_v in_o power_n duple_n to_o the_o line_n ef._n wherefore_o the_o line_n ab_fw-la the_o side_n of_o the_o dodecahedron_n be_v the_o less_o segment_n of_o the_o line_n fk_o which_o be_v in_o power_n duple_n to_o of_o the_o sidâ—_n of_o the_o octohedron_n the_o side_n therefore_o of_o a_o dodecahedron_n iâ—_z the_o less_o segment_n of_o that_o right_a line_n which_o be_v in_o power_n duple_n to_o the_o side_n of_o the_o octohedron_n inscribe_v in_o the_o same_o dodâ—cahedron_n ¶_o the_o 22._o proposition_n the_o diameter_n of_o a_o icosahedron_n be_v in_o power_n sesquitertia_fw-la to_o the_o side_n of_o the_o same_o icosahedron_n and_o also_o be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o icosahedron_n for_o forasmuch_o as_o it_o have_v be_v prove_v by_o the_o 10._o of_o this_o book_n that_o if_o from_o the_o power_n of_o the_o diameter_n of_o the_o icosahedron_n be_v take_v away_o the_o triple_a of_o the_o power_n of_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o it_o there_o shall_v be_v leave_v a_o square_a sesquitertia_fw-la to_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n but_o the_o power_n of_o the_o side_n of_o the_o cube_fw-la triple_a be_v the_o diameter_n of_o the_o same_o cube_fw-la by_o the_o 15._o of_o the_o thirteen_o and_o the_o cube_fw-la &_o the_o pyramid_n inscribe_v in_o it_o be_v contain_v in_o one_o &_o the_o self_n same_o sphere_n by_o the_o first_o of_o this_o book_n and_o in_o one_o &_o the_o self_n same_o icosahedron_n by_o the_o corollary_n of_o the_o same_o wherefore_o one_o and_o the_o self_n same_o diameter_n of_o the_o cube_fw-la or_o of_o the_o sphere_n which_o contain_v the_o cube_fw-la and_o the_o pyramid_n be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o the_o pyramid_n by_o the_o 13._o of_o the_o thirteen_o wherefore_o it_o follow_v that_o if_o from_o the_o diameter_n of_o the_o icosahedron_n be_v take_v away_o the_o triple_a power_n of_o the_o side_n of_o the_o cube_fw-la or_o the_o sesquialter_fw-la power_n of_o the_o side_n of_o the_o pyramid_n which_o be_v the_o power_n of_o one_o and_o the_o self_n same_o diameter_n there_o shall_v be_v leave_v the_o sesquitertia_fw-la power_n of_o the_o side_n of_o the_o icosahedron_n the_o diameter_n therefore_o of_o a_o icosahedron_n be_v in_o power_n sesquitertia_fw-la to_o the_o side_n of_o the_o same_o icosahedron_n and_o also_o be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o icosahedron_n the_o 23._o proposition_n the_o side_n of_o a_o dodecahedron_n be_v to_o the_o side_n of_o a_o icosahedron_n inscribe_v in_o it_o as_o the_o less_o segment_n of_o the_o perpendicular_a of_o the_o pentagon_n be_v to_o that_o line_n which_o be_v draw_v from_o the_o centre_n to_o the_o side_n of_o the_o same_o pentagon_n constrution_n let_v there_o be_v take_v a_o dodecahedron_n abgdfso_n who_o side_n let_v be_v as_o or_o so_o and_o let_v the_o icosahedron_n inscribe_v in_o it_o be_v klnmne_a who_o side_n let_v be_v kl_o from_o the_o two_o angle_n of_o the_o pentagonâ—_n bas_n and_o fas_fw-la of_o the_o dodecahedron_n namely_o from_o the_o angleâ—â—_n and_o fletcher_n let_v there_o be_v draw_v to_o the_o common_a base_a as_o perpendicular_a line_n bc_o &_o fc_n which_o shall_v pass_v by_o the_o centre_n king_n &_o l_o of_o the_o say_v pentagon_n by_o the_o corollary_n of_o the_o 10._o of_o the_o thirteen_o draw_v the_o line_n bf_o and_o ro._n now_o forasmuch_o as_o the_o line_n ro_n subtend_v the_o angle_n ofr_n of_o thâ—_z pentagon_n of_o the_o dodecahedron_n it_o shall_v cut_v the_o line_n fc_o by_o a_o extreme_a and_o mean_a proportion_n by_o the_o 3._o of_o this_o book_n let_v it_o cut_v it_o in_o the_o point_n i_o and_o forasmuch_o as_o the_o line_n kl_o be_v the_o side_n of_o the_o icosahedron_n inscribe_v in_o the_o dodecahedron_n it_o couple_v the_o centre_n of_o the_o base_n of_o the_o dodecahedron_n for_o the_o angle_n of_o the_o icosahedron_n be_v set_v in_o the_o centre_n of_o the_o base_n of_o the_o dodecahedron_n by_o the_o 7._o of_o the_o fifteen_o now_o i_o say_v that_o so_o the_o side_n of_o the_o dodecahedron_n be_v to_o kl_o the_o side_n of_o the_o icosahedron_n as_o the_o less_o segment_n if_o of_o the_o perpendicular_a line_n cf_o be_v to_o the_o line_n lc_n which_o be_v draw_v from_o the_o centre_n l_o to_o as_o the_o side_n of_o the_o pentagon_n for_o forasmuch_o as_o in_o the_o triangle_n bcf_n the_o two_o side_n cb_o and_o cf_o be_v in_o the_o centre_n l_o and_o king_n cut_v like_o proportional_o demonstration_n the_o line_n bf_o and_o kl_o shall_v be_v parellel_n by_o the_o 2._o of_o the_o six_o wherefore_o the_o triangle_n bcf_n and_o kcl_n shall_v be_v equiangle_n by_o the_o corollary_n of_o the_o same_o wherefore_o as_o the_o line_n cl_n be_v to_o the_o line_n klâ—_n so_o be_v the_o line_n cf_o to_o the_o line_n bf_o by_o the_o 4._o of_o the_o six_o but_o cf_o make_v the_o less_o segment_n the_o line_n if_o by_o the_o 3._o of_o this_o book_n and_o the_o linâ—_n bf_o make_v the_o less_o segment_n the_o line_n so_o namely_o the_o side_n of_o the_o dodecahedron_n by_o the_o 2._o corollary_n of_o the_o 13._o of_o the_o fifteen_o for_o the_o line_n bf_o which_o couple_v the_o angle_n b_o and_o f_o of_o the_o base_n of_o the_o dodecahedron_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la which_o contain_v the_o dodecahedron_n by_o the_o .13_o of_o the_o fifteen_o wherefore_o as_o the_o whole_a line_n câ—_n be_v to_o the_o whole_a line_n bf_o so_o be_v the_o less_o segment_n if_o to_o the_o less_o segment_n so_o by_o the_o 2._o of_o the_o 14_o but_o as_o the_o line_n cf_o be_v to_o the_o line_n bf_o so_o be_v the_o line_n cl_n prove_v to_o be_v to_o the_o line_n kl_o wherefore_o as_o the_o line_n if_o be_v to_o the_o line_n so_o so_o be_v the_o line_n cl_n to_o the_o line_n kl_o wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n if_o the_o less_o segment_n of_o the_o perpendicular_a of_o the_o pentagon_n fas_fw-la be_v to_o the_o line_n lc_n which_o be_v draw_v from_o the_o centre_n of_o the_o pentagon_n to_o the_o base_a so_o be_v the_o line_n so_o the_o side_n of_o the_o dodecahedron_n to_o thâ—_z line_n kl_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o the_o side_n therefore_o of_o a_o dodecahedron_n be_v to_o the_o side_n of_o a_o icosahedron_n inscribe_v in_o it_o as_o the_o less_o segment_n of_o the_o perpendicular_a of_o the_o pentagon_n be_v to_o that_o line_n which_o be_v draw_v from_o the_o cenâ—re_n to_o the_o side_n of_o the_o same_o pentagon_n ¶_o the_o 24._o proposition_n if_o half_a of_o the_o side_n of_o a_o icosahedron_n be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n and_o if_o the_o less_o segment_n thereof_o be_v take_v away_o from_o the_o whole_a side_n and_o again_o from_o the_o residue_n be_v take_v away_o the_o three_o part_n that_o which_o remain_v shall_v be_v equal_a to_o the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o icosahedron_n svppose_v that_o abgdf_n be_v a_o pentagon_n construction_n contain_v five_o side_n of_o the_o icosahedron_n by_o the_o 16._o of_o the_o thirteen_o and_o let_v it_o be_v inscribe_v in_o a_o circle_n who_o centre_n let_v be_v the_o point_n e._n and_o upon_o the_o side_n of_o the_o pentagon_n let_v there_o be_v rear_v up_o triangle_n make_v a_o solid_a angle_n of_o the_o icosahedron_n at_o the_o point_n i_o by_o the_o 16._o of_o the_o thirteen_o and_o in_o the_o circle_n abdella_n inscribe_v a_o equilater_n triangle_n ahk_n from_o the_o centre_n e_o draw_v to_o hk_v the_o side_n of_o the_o triangle_n and_o gd_o the_o side_n of_o the_o pentagon_n a_o perpendicular_a line_n which_o
the_o residue_n or_o of_o this_o excess_n but_o a_o pyramid_n be_v to_o the_o same_o cube_fw-la inscribe_v in_o it_o nonecuple_n by_o the_o 30._o of_o this_o book_n wherefore_o the_o dodecahedron_n inscribe_v in_o the_o pyramid_n and_o contain_v the_o same_o cube_fw-la twice_o take_v away_o the_o self_n same_o three_o of_o the_o less_o segment_n and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o residue_n shall_v contain_v two_o nine_o part_n of_o the_o solid_a of_o the_o pyramid_n of_o which_o nine_o part_v each_o be_v equal_a unto_o the_o cube_fw-la take_v away_o this_o self_n same_o excess_n the_o solid_a therefore_o of_o a_o dodecahedron_n contain_v of_o a_o pyramid_n circumscribe_v about_o it_o two_o nine_o part_n take_v away_o a_o three_o part_n of_o one_o nine_o part_n of_o the_o less_o segment_n of_o a_o line_n divide_v by_o a_o extmere_n and_o mean_a proportion_n and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o residue_n ¶_o the_o 36._o proposition_n a_o octohedron_n exceed_v a_o icosahedron_n inscribe_v in_o it_o by_o a_o parallelipipedon_n set_v upon_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n and_o have_v to_o his_o altitude_n the_o line_n which_o be_v the_o great_a segment_n of_o half_a the_o semidiameter_n of_o the_o octohedron_n svppose_v that_o there_o be_v a_o octohedron_n abcfpl_n construction_n in_o which_o let_v there_o be_v inscribe_v a_o icosahedron_n hkegmxnudsqtâ—_n by_o the_o â—6_o of_o the_o fifteen_o and_o draw_v the_o diameter_n azrcbroif_n and_o the_o perpendicular_a ko_o parallel_n to_o the_o line_n azr_n then_o i_o say_v that_o the_o octohedron_n abcfpl_n be_v great_a thâ—n_n the_o icosahedron_n inscribe_v in_o it_o by_o a_o parallelipipedon_n set_v upon_o the_o square_n of_o the_o side_n hk_o or_o ge_z and_o have_v to_o his_o altitude_n the_o line_n ko_o or_o rz_n which_o be_v the_o great_a segment_n of_o the_o semidiameter_n ar._n forasmuch_o as_o in_o the_o same_o 16._o it_o have_v be_v prove_v that_o the_o triangle_n kdg_n and_o keq_n be_v describe_v in_o the_o base_n apf_n and_o alf_n of_o the_o octohedron_n demonstration_n therefore_o about_o the_o solid_a angle_n there_o remain_v upon_o the_o base_a feg_fw-mi three_o triangle_n keg_n kfe_n and_o kfg_n which_o contain_v a_o pyramid_n kefg_n unto_o which_o pyramid_n shall_v be_v equal_a and_o like_v the_o opposite_a pyramid_n mefg_n set_v upon_o the_o same_o base_a feg_n by_o the_o 8._o definition_n of_o the_o eleven_o and_o by_o the_o â—ame_n reason_n shall_v there_o at_o every_o solid_a angle_n of_o the_o octohedron_n remain_v two_o pyramid_n equal_a and_o like_a namely_o two_o upon_o the_o base_a ahk_n two_o upon_o the_o base_a bnv_n two_o upon_o the_o base_a dp_n and_o moreover_o two_o upon_o the_o base_a qlt._n now_o they_o there_o shall_v be_v make_v twelve_o pyramid_n set_v upon_o a_o base_a contain_v of_o the_o side_n of_o the_o icosahedron_n and_o under_o two_o leâ—â—e_a segment_n of_o the_o side_n of_o the_o octohedron_n contain_v a_o right_a angle_n as_o for_o example_n the_o base_a gef_n and_o forasmuch_o as_o the_o side_n ge_z subtend_v a_o right_a angle_n be_v by_o the_o 47._o of_o the_o â—irst_n in_o power_n duple_n to_o either_o of_o the_o line_n of_o and_o fg_o and_o so_o the_o â—â—deâ—_n kh_o be_v in_o power_n duple_n to_o either_o of_o the_o side_n ah_o and_o ak_o and_o either_o of_o the_o line_n ah_o ak_o or_o of_o fg_o be_v in_o power_n duple_n to_o either_o of_o the_o line_n az_o or_o zk_v which_o contain_v a_o right_a angle_n make_v in_o the_o triangle_n or_o base_a ahk_n by_o the_o perpendicular_a az_o wherefore_o it_o follow_v that_o the_o side_n ge_z or_o hk_o be_v in_o power_n quadruple_a to_o the_o triangle_n efg_o or_o ahk_n but_o the_o pyramid_n kefg_n have_v his_o base_a efg_o in_o the_o plain_a flbp_n of_o the_o octohedron_n shall_v have_v to_o his_o altitude_n the_o perpendicular_a ko_o by_o the_o 4._o definition_n of_o the_o six_o which_o be_v the_o great_a segment_n of_o the_o semidiameter_n of_o the_o octohedron_n by_o the_o 16._o of_o the_o fifteen_o wherefore_o three_o pyramid_n set_v under_o the_o same_o altitude_n and_o upon_o equal_a base_n shall_v be_v equal_a to_o one_o prism_n set_v upon_o the_o same_o base_a and_o under_o the_o same_o altitude_n by_o the_o 1._o corollary_n of_o the_o 7._o of_o the_o twelve_o wherefore_o 4._o prism_n set_v upon_o the_o base_a gef_n quadruple_v which_o be_v equal_a to_o the_o square_n of_o the_o side_n ge_z and_o under_o the_o altitude_n ko_o or_o rz_n the_o great_a segment_n which_o be_v equal_a to_o ko_o shall_v contain_v a_o solid_a equal_a to_o the_o twelve_o pyramid_n which_o twelve_o pyramid_n make_v the_o excess_n of_o the_o octohedron_n above_o the_o icosahedron_n inscribe_v in_o it_o a_o octohedron_n therefore_o exceed_v a_o icosahedron_n inscribe_v in_o it_o by_o a_o parallelipipedon_n set_v upon_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n and_o have_v to_o his_o altitude_n the_o line_n which_o be_v the_o great_a segment_n of_o half_a the_o semidiameter_n of_o the_o octohedron_n ¶_o a_o corollary_n a_o pyramid_n exceed_v the_o double_a of_o a_o icosahedron_n inscribe_v in_o it_o by_o a_o solid_a set_v upon_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o and_o have_v to_o his_o altitude_n that_o whole_a line_n of_o which_o the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n for_o it_o be_v manifest_a by_o the_o 19_o of_o the_o fivetenth_n that_o a_o octohedron_n &_o a_o icosahedron_n inscribe_v in_o it_o be_v inscribe_v in_o one_o &_o the_o self_n same_o pyramid_n it_o have_v moreover_o be_v prove_v in_o the_o 26._o of_o this_o book_n that_o a_o pyramid_n be_v double_a to_o a_o octohedron_n inscribe_v in_o it_o wherefore_o the_o two_o excess_n of_o the_o two_o octohedron_n unto_o which_o the_o pyramid_n be_v equal_a above_o the_o two_o icosahedrons_n inscribe_v in_o the_o say_v two_o octohedron_n be_v bring_v into_o a_o solid_a the_o say_v solid_a shall_v be_v set_v upon_o the_o self_n same_o square_n of_o the_o side_n of_o the_o icosahedron_n and_o shall_v have_v to_o his_o altitude_n the_o perpendicular_a ko_o double_a who_o double_a couple_v the_o opposite_a side_n hk_o and_o xm_o make_v the_o great_a segment_n the_o same_o side_n of_o the_o icosahedron_n by_o the_o first_o and_o second_o corollary_n of_o the_o 14._o of_o the_o fiuâ—â—enâ—h_n the_o 37._o proposition_n if_o in_o a_o triangle_n have_v to_o his_o base_a a_o rational_a line_n set_v the_o side_n be_v commensurable_a in_o power_n to_o the_o base_a and_o from_o the_o top_n be_v draw_v to_o the_o base_a a_o perpendicular_a line_n cut_v the_o base_a the_o section_n of_o the_o base_a shall_v be_v commensurable_a in_o length_n to_o the_o whole_a base_a and_o the_o perpendicular_a shall_v be_v commensurable_a in_o power_n to_o the_o say_v whole_a base_a and_o now_o that_o the_o perpendicular_a ap_n be_v commensurable_a in_o power_n to_o the_o base_a bg_o demonstration_n iâ—_z thus_o prove_v forasmuch_o as_o the_o square_n of_o ab_fw-la be_v by_o supposition_n commensurable_a to_o the_o square_n of_o bg_o and_o unto_o the_o rational_a square_n of_o ab_fw-la be_v commensurable_a the_o rational_a square_n of_o bp_o by_o the_o 12._o of_o the_o eleven_o wherefore_o the_o residue_n namely_o the_o square_a of_o pa_o be_v commensurable_a to_o the_o same_o square_n of_o bp_o by_o the_o 2._o part_n of_o the_o 15._o of_o the_o eleven_o wherefore_o by_o the_o 12._o of_o the_o ten_o the_o square_a of_o pa_o be_v commensurable_a to_o the_o whole_a square_n of_o bg_o wherefore_o the_o perpendicular_a ap_n be_v commensurable_a in_o power_n to_o the_o base_a bg_o by_o the_o 3._o definition_n of_o the_o ten_o which_o be_v require_v to_o be_v prove_v in_o demonstrate_v of_o this_o we_o make_v no_o mention_n at_o all_o of_o the_o length_n of_o the_o side_n ab_fw-la and_o ag_z but_o only_o of_o the_o length_n of_o the_o base_a bg_o for_o that_o the_o line_n bg_o be_v the_o rational_a line_n first_o set_v and_o the_o other_o line_n ab_fw-la and_o ag_z be_v suppose_v to_o be_v commensurable_a in_o power_n only_o to_o the_o line_n bg_o wherefore_o if_o that_o be_v plain_o demonstrate_v when_o the_o side_n be_v commensurable_a in_o power_n only_o to_o the_o base_a much_o more_o easy_o will_v it_o follow_v if_o the_o same_o side_n be_v suppose_v to_o be_v commensurable_a both_o in_o length_n and_o in_o power_n to_o the_o base_a that_o be_v if_o their_o lengthe_n be_v express_v by_o the_o root_n of_o square_a number_n ¶_o a_o corollary_n 1._o by_o the_o former_a thing_n demonstrate_v it_o be_v manifest_a that_o if_o from_o the_o power_n of_o the_o base_a and_o of_o one_o of_o the_o side_n be_v take_v away_o the_o
from_o a_o cube_fw-la and_o a_o octohedron_n compose_v into_o one_o solid_a there_o shall_v be_v leave_v a_o exocthedron_n moreover_o the_o solid_a angle_n take_v away_o from_o two_o pyramid_n compose_v into_o one_o solid_a there_o shall_v be_v leave_v a_o octohedron_n flussas_n after_o this_o set_v forth_o certain_a passion_n and_o property_n of_o the_o five_o simple_a regular_a body_n which_o although_o he_o demonstrate_v not_o yet_o be_v they_o not_o hard_a to_o be_v demonstrate_v if_o we_o well_o pease_n and_o conceive_v that_o which_o in_o the_o former_a book_n have_v be_v teach_v touch_v those_o solid_n of_o the_o nature_n of_o a_o trilater_n and_o equilater_n pyramid_n a_o trilater_n equilater_n pyramid_n be_v divide_v into_o two_o equal_a part_n by_o three_o equal_a square_n which_o in_o the_o centre_n of_o the_o pyramid_n cut_v the_o one_o the_o other_o into_o two_o equal_a part_n and_o perpendicular_o and_o who_o angle_n be_v set_v in_o the_o middle_a section_n of_o the_o side_n of_o the_o pyramid_n from_o a_o pyramid_n be_v take_v away_o 4._o pyramid_n like_a unto_o the_o whole_a which_o utter_o take_v away_o the_o side_n of_o the_o pyramid_n and_o that_o which_o be_v leave_v be_v a_o octohedron_n inscribe_v in_o the_o pyramy_n in_o which_o all_o the_o solid_n inscribe_v in_o the_o pyramid_n be_v contain_v a_o perpendicular_a draw_v from_o the_o angle_n of_o the_o pyramid_n to_o the_o base_a be_v double_a to_o the_o diameter_n of_o the_o cube_fw-la inscribe_v in_o it_o and_o a_o right_a line_n couple_v the_o middle_a section_n of_o the_o opposite_a side_n of_o the_o pyramid_n be_v triple_a to_o the_o side_n of_o the_o self_n same_o cube_fw-la the_o side_n also_o of_o the_o pyramid_n be_v triple_a to_o the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la wherefore_o the_o same_o side_n of_o the_o pyramid_n be_v in_o power_n duple_n to_o the_o right_a line_n which_o couple_v the_o middle_a section_n of_o the_o opposite_a side_n and_o it_o be_v in_o power_n sesquialter_fw-la to_o the_o perpendicular_a which_o be_v draw_v from_o the_o angle_n to_o the_o base_a wherefore_o the_o perpendicular_a be_v in_o power_n sesquitertia_fw-la to_o the_o line_n which_o couple_v the_o middle_a section_n of_o the_o opposite_a side_n a_o pyramid_n and_o a_o octohedron_n inscribe_v in_o it_o also_o a_o icosahedron_n inscribe_v in_o the_o same_o octohedron_n do_v contain_v one_o and_o the_o self_n same_o sphere_n of_o the_o nature_n of_o a_o octohedron_n four_o perpendicular_n of_o a_o octohedron_n draw_v in_o 4._o base_n thereof_o from_o two_o opposite_a angle_n of_o the_o say_v octohedron_n and_o couple_v together_o by_o those_o 4._o base_n describe_v a_o rhombus_fw-la or_o diamond_n figure_n one_o of_o who_o diameter_n be_v in_o power_n duple_n to_o the_o other_o diameter_n for_o it_o have_v the_o same_o proportion_n that_o the_o diameter_n of_o the_o octohedron_n have_v to_o the_o side_n of_o the_o octohedron_n a_o octohedron_n &_o a_o icosahedron_n inscribe_v in_o it_o do_v contain_v one_o and_o the_o self_n same_o sphere_n the_o diameter_n of_o the_o solid_a of_o the_o octohedron_n be_v in_o power_n sesquialter_fw-la to_o the_o diameter_n of_o the_o circle_n which_o contain_v the_o base_a and_o be_v in_o power_n triple_a to_o the_o right_a line_n which_o couple_v the_o centre_n of_o the_o opposite_a base_n and_o be_v in_o power_n â—_z duple_fw-fr superbipartiens_fw-la tercias_fw-la to_o the_o perpendicular_a or_o side_n of_o the_o foresay_a rhombus_fw-la and_o moreover_o be_v in_o length_n triple_a to_o the_o line_n which_o couple_v the_o centre_n of_o the_o next_o base_n the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o the_o octohedron_n do_v with_o the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o the_o pyramid_n make_v angle_n equal_a to_o two_o right_a angle_n of_o the_o nature_n of_o a_o cube_n the_o diameter_n of_o a_o cube_fw-la be_v in_o power_n sesquialter_fw-la to_o the_o diameter_n of_o his_o base_a and_o be_v in_o power_n triple_a to_o his_o side_n and_o unto_o the_o line_n which_o couple_v the_o centre_n of_o the_o next_o base_n it_o be_v in_o power_n sextuple_a moreover_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o as_o the_o whole_a be_v to_o the_o great_a segment_n unto_o the_o side_n of_o the_o dodecahedron_n it_o be_v as_o the_o whole_a be_v to_o the_o less_o segment_n unto_o the_o side_n of_o the_o octohedron_n it_o be_v in_o power_n duple_n and_o unto_o the_o side_n of_o the_o pyramid_n it_o be_v in_o power_n subduple_n moreover_o the_o cube_fw-la be_v triple_a to_o the_o pyramid_n but_o to_o the_o cube_fw-la the_o dodecahedron_n be_v in_o a_o manner_n duple_n wherefore_o the_o same_o dodecahedron_n be_v in_o a_o manner_n sextuple_a to_o the_o say_a pyramid_n of_o the_o nature_n of_o a_o icosahedron_n five_o triangle_n of_o a_o icosahedron_n do_v make_v a_o solid_a angle_n the_o base_n of_o which_o triangle_n make_v a_o pentagon_n if_o therefore_o from_o the_o opposite_a base_n of_o the_o icosahedron_n be_v take_v the_o other_o pentagon_n by_o they_o describe_v these_o pentagon_n shall_v in_o such_o sort_n cut_v the_o diameter_n of_o the_o icosahedron_n which_o couple_v the_o foresaid_a opposite_a angle_n that_o that_o part_n which_o be_v contain_v between_o the_o plain_n of_o those_o two_o pentagon_n shall_v be_v the_o great_a segment_n and_o the_o residue_n which_o be_v draw_v from_o the_o plain_a to_o the_o angle_n shall_v be_v the_o less_o segment_n if_o the_o opposite_a angle_n of_o two_o base_n join_v together_o be_v couple_v by_o a_o right_a line_n the_o great_a segment_n of_o that_o right_a line_n be_v the_o side_n of_o the_o icosahedron_n a_o line_n draw_v from_o the_o centre_n of_o the_o icosahedron_n to_o the_o angle_n be_v in_o power_n quintuple_a to_o half_a that_o line_n which_o be_v take_v between_o the_o pentagon_n or_o of_o the_o half_a of_o that_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n which_o contain_v the_o foresay_a pentagon_n which_o two_o line_n be_v therefore_o equal_a the_o side_n of_o the_o icosahedron_n contain_v in_o power_n either_o of_o they_o and_o also_o the_o less_o segment_n namely_o the_o line_n which_o fall_v from_o the_o solid_a angle_n to_o the_o pentagon_n the_o diameter_n of_o the_o icosahedron_n contain_v in_o power_n the_o whole_a line_n which_o couple_v the_o opposite_a angle_n of_o the_o base_n join_v together_o and_o the_o great_a segment_n thereof_o namely_o the_o side_n of_o the_o icosahedron_n the_o diameter_n also_o be_v in_o power_n quintuple_a to_o the_o line_n which_o be_v take_v between_o the_o pentagon_n or_o 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 pentagon_n compose_v of_o the_o side_n of_o the_o icosahedron_n the_o dimetient_fw-la contain_v in_o power_n the_o right_a line_n which_o couple_v the_o centre_n of_o the_o opposite_a base_n of_o the_o icosahedron_n and_o the_o diameter_n of_o the_o circle_n which_o contain_v the_o base_a moreover_o the_o say_v dimetient_fw-la contain_v in_o power_n the_o diameter_n of_o the_o circle_n which_o contain_v the_o pentagon_n and_o also_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o same_o circle_n to_o the_o circumference_n that_o be_v it_o be_v quintuple_a to_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n the_o line_n which_o couple_v the_o centre_n of_o the_o opposite_a base_n contain_v in_o power_n the_o line_n which_o couple_v the_o centre_n of_o the_o next_o base_n and_o also_o the_o rest_n of_o that_o line_n of_o which_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o icosahedron_n be_v the_o great_a segment_n the_o line_n which_o couple_v the_o middle_a section_n of_o the_o opposite_a side_n be_v triple_a to_o the_o side_n of_o the_o dodecahedron_n inscribe_v in_o it_o wherefore_o if_o the_o side_n of_o the_o icosahedron_n and_o the_o great_a segment_n thereof_o be_v make_v one_o line_n the_o three_o part_n of_o the_o whole_a be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o icosahedron_n of_o the_o nature_n of_o a_o dodecahedron_n the_o diameter_n of_o a_o dodecahedron_n contain_v in_o power_n the_o side_n of_o the_o dodecahedron_n and_o also_o that_o right_a line_n unto_o which_o the_o side_n of_o the_o dodecahedron_n be_v the_o less_o segment_n and_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o it_o be_v the_o great_a segment_n which_o line_n be_v that_o which_o subtend_v the_o angle_n of_o the_o inclination_n of_o the_o base_n contain_v under_o two_o perpendicular_n of_o the_o base_n of_o the_o dodecahedron_n if_o there_o be_v take_v two_o base_n of_o the_o dodecahedron_n distant_a the_o one_o from_o the_o other_o by_o the_o length_n of_o one_o of_o the_o side_n
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
about_o the_o diameter_n together_o with_o the_o two_o supplement_n make_v a_o gnomon_n as_o the_o parallelogram_n ebkh_n with_o the_o two_o supplement_n aegk_n and_o khfd_n make_v the_o gnomon_n fgeh_a likewise_o the_o parallelogram_n gkcf_n with_o the_o same_o two_o supplement_n make_v the_o gnomon_n ehfg_n and_o this_o definition_n of_o a_o gnomon_n extend_v itself_o and_o be_v general_a to_o all_o kind_n of_o parallelogram_n whether_o they_o be_v square_n or_o figure_n of_o one_o side_n long_o or_o rhombus_fw-la or_o romboide_n to_o be_v short_a if_o you_o take_v away_o from_o the_o whole_a parallelogram_n one_o of_o the_o partial_a parallelogram_n which_o be_v about_o the_o diameter_n whether_o you_o will_v the_o rest_n of_o the_o figure_n be_v a_o gnomon_n campane_v after_o the_o last_o proposition_n of_o the_o first_o book_n add_v this_o proposition_n book_n two_o square_n be_v give_v to_o adjoin_v to_o one_o of_o they_o a_o gnomon_n equal_a to_o the_o other_o square_n which_o for_o that_o as_o than_o it_o be_v not_o teach_v what_o a_o gnomon_n be_v i_o there_o omit_v think_v that_o it_o may_v more_o apt_o be_v place_v here_o the_o do_v and_o demonstration_n whereof_o be_v thus_o suppose_v that_o there_o be_v two_o square_n ab_fw-la and_o cd_o unto_o one_o of_o which_o namely_o unto_o ab_fw-la it_o be_v require_v to_o add_v a_o gnomon_n equal_a to_o the_o other_o square_n namely_o to_o cd_o produce_v the_o side_n bf_o of_o the_o square_a ab_fw-la direct_o to_o the_o point_n e._n and_o put_v the_o line_n fe_o equal_a to_o the_o side_n of_o the_o square_a cd_o and_o draw_v a_o line_n from_o e_o to_o a._n now_o then_o forasmuch_o as_o efa_n be_v a_o rectangle_n triangle_n therefore_o by_o the_o 47._o of_o the_o first_o the_o square_a of_o the_o line_n ea_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n of_o &_o fa._n but_o the_o square_n of_o the_o line_n of_o be_v equal_a to_o the_o square_a cd_o &_o the_o square_a of_o the_o side_n favorina_n be_v the_o square_a ab_fw-la wherefore_o the_o square_a of_o the_o line_n ae_n be_v equal_a to_o the_o two_o square_n cd_o and_o ab_fw-la but_o the_o side_n of_o and_o favorina_n be_v by_o the_o 21._o of_o the_o first_o long_o than_o the_o side_n ae_n and_o the_o side_n favorina_n be_v equal_a to_o the_o side_n fb_o wherefore_o the_o side_n of_o and_o fb_o be_v long_o they_o the_o side_n ae_n wherefore_o the_o whole_a line_n be_v be_v long_o than_o the_o line_n ae_n from_o the_o line_n be_v cut_v of_o a_o line_n equal_a to_o the_o line_n ae_n which_o let_v be_v bc._n and_o by_o the_o 46._o proposition_n upon_o the_o line_n bc_o describe_v a_o square_a which_o let_v be_v bcgh_n which_o shall_v be_v equal_a to_o the_o square_n of_o the_o line_n ae_n but_o the_o square_n of_o the_o line_n ae_n be_v equal_a to_o the_o two_o square_n ab_fw-la and_o dc_o wherefore_o the_o square_a bcgh_n be_v equal_a to_o the_o same_o square_n wherefore_o forasmuch_o as_o the_o square_a bcgh_n be_v compose_v of_o the_o square_a ab_fw-la and_o of_o the_o gnomon_n fgah_n the_o say_a gnomon_n shall_v be_v equal_a unto_o the_o square_a cd_o which_o be_v require_v to_o be_v do_v a_o other_o more_o ready_a way_n after_o pelitarius_n suppose_v that_o there_o be_v two_o square_n who_o side_n let_v be_v ab_fw-la and_o bc._n it_o be_v require_v unto_o the_o square_n of_o the_o line_n ab_fw-la to_o add_v a_o gnomon_n equal_a to_o the_o square_n of_o the_o line_n bc._n set_v the_o line_n ab_fw-la and_o bc_o in_o such_o sort_n that_o they_o make_v a_o right_a angle_n abc_n and_o draw_v a_o line_n from_o a_o to_o c._n and_o upon_o the_o line_n ab_fw-la describe_v a_o square_n which_o let_v be_v abde_n and_o produce_v the_o line_n basilius_n to_o the_o point_n fletcher_n and_o put_v the_o line_n bf_o equal_a to_o the_o line_n ac_fw-la and_o upon_o the_o line_n bf_o describe_v a_o square_n which_o let_v be_v bfgh_o which_o shall_v be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la when_o as_o the_o line_n bf_o and_o ac_fw-la be_v equal_a and_o therefore_o it_o be_v equal_a to_o the_o square_n of_o the_o two_o line_n ab_fw-la and_o bc._n now_o forasmuch_o as_o the_o square_a bfgh_o be_v make_v complete_a by_o the_o square_a abde_n and_o by_o the_o gnomon_n fegd_v the_o gnomon_n fegd_v shall_v be_v equal_a to_o the_o square_n of_o the_o line_n bc_o which_o be_v require_v to_o be_v do_v the_o 1._o theorem_a the_o 1._o proposition_n if_o there_o be_v two_o right_a line_n and_o if_o the_o one_o of_o they_o be_v divide_v into_o part_n how_o many_o soever_o the_o rectangle_n figure_n comprehend_v under_o the_o two_o right_a line_n be_v equal_a to_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n undivided_a and_o under_o every_o one_o of_o the_o part_n of_o the_o other_o line_n svppose_v that_o there_o be_v two_o right_a line_n a_o and_o bc_o and_o let_v one_o of_o they_o namely_o bc_o be_v divide_v at_o all_o adventure_n in_o the_o point_n d_o and_o e._n then_o i_o say_v that_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n a_o and_o bc_o be_v equal_a unto_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n a_o and_o bd_o &_o unto_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n a_o and_o de_fw-fr and_o also_o unto_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n a_o and_o ec_o construction_n for_o from_o the_o point_n brayse_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o right_a line_n bc_o a_o perpendicular_a line_n bf_o &_o unto_o the_o line_n a_o by_o the_o three_o of_o the_o first_o put_v the_o line_n bg_o equal_a and_o by_o the_o point_n g_z by_o the_o 31._o of_o the_o first_o draw_v a_o parallel_n line_n unto_o the_o right_a line_n bc_o and_o let_v the_o same_o be_v gm_n and_o by_o the_o self_n same_o by_o the_o poor_n d_z e_o and_o c_o draw_v unto_o the_o line_n bg_o these_o parallel_a line_n dk_o demonstration_n el_n and_o ch._n now_o then_o the_o parallelogram_n bh_o be_v equal_a to_o these_o parallelogram_n bk_o dl_o and_o eh_o but_o the_o parallelogram_n bh_o be_v equal_a unto_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o bc._n for_o it_o be_v comprehend_v under_o the_o line_n gb_o &_o bc_o and_o the_o line_n gb_o be_v equal_a unto_o the_o line_n a_o and_o the_o parallelogram_n bk_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o bd_o for_o it_o be_v comprehend_v under_o the_o line_n gb_o and_o bd_o and_o bg_o be_v equal_a unto_o a_o and_o the_o parallelogram_n dl_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o de_fw-fr for_o the_o line_n dk_o that_o be_v bg_o be_v equal_a unto_o a_o and_o moreover_o likewise_o the_o parallelogram_n eh_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n a_o &_o ec_o wherefore_o that_o which_o be_v comprehend_v under_o the_o line_n a_o &_o bc_o be_v equal_a to_o that_o which_o be_v comprehend_v under_o the_o line_n a_o &_o bd_o &_o unto_o that_o which_o be_v comprehend_v under_o the_o line_n a_o and_o de_fw-fr and_o moreover_o unto_o that_o which_o be_v comprehend_v under_o the_o line_n a_o and_o ec_o if_o therefore_o there_o be_v two_o right_a line_n and_o if_o the_o one_o of_o they_o be_v divide_v into_o part_n how_o many_o soever_o the_o rectangle_n figure_n comprehend_v under_o the_o two_o right_a line_n be_v equal_a to_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n undivided_a and_o under_o every_o one_o of_o the_o part_n of_o the_o other_o line_n which_o be_v require_v to_o be_v demonstrate_v because_o that_o all_o the_o proposition_n of_o this_o second_o book_n for_o the_o most_o part_n be_v true_a both_o in_o line_n and_o in_o number_n and_o may_v be_v declare_v by_o both_o therefore_o have_v i_o have_v add_v to_o every_o proposition_n convenient_a number_n for_o the_o manifestation_n of_o the_o same_o and_o to_o the_o end_n the_o studious_a and_o diligent_a reader_n may_v the_o more_o full_o perceive_v and_o understand_v the_o agreement_n of_o this_o art_n of_o geometry_n with_o the_o science_n of_o arithmetic_n and_o how_o never_o &_o dear_a sister_n they_o be_v together_o so_o that_o the_o one_o can_v without_o great_a blemish_n be_v without_o the_o other_o i_o have_v here_o also_o join_v a_o little_a book_n of_o arithmetic_n write_v by_o one_o barlaam_n a_o greek_a author_n a_o man_n of_o great_a knowledge_n in_o which_o book_n be_v by_o the_o author_n demonstrate_v many_o of_o the_o self_n same_o propriety_n and_o passion_n in_o number_n which_o euclid_n in_o this_o his_o second_o book_n have_v demonstrate_v in_o magnitude_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