e_o produce_v d_z wherefore_o a_o measure_v d_z but_o it_o also_o measure_v it_o not_o which_o be_v impossible_a wherefore_o it_o be_v impossible_a to_o find_v out_o a_o four_o number_n proportional_a with_o these_o number_n a_o b_o c_o whensoever_o a_o measure_v not_o d._n ¶_o the_o 20._o theorem_a the_o 20._o proposition_n prime_n number_n be_v give_v how_o many_o soever_o there_o may_v be_v give_v more_o prime_a number_n svppose_v that_o the_o prime_a number_n give_v be_v a_o b_o c._n proposition_n then_o i_o say_v that_o there_o be_v yet_o more_o prime_a number_n beside_o a_o b_o c._n take_v by_o the_o 38._o of_o the_o seven_o the_o lest_o number_v who_o these_o number_n a_o b_o c_o do_v measure_n and_o let_v the_o same_o be_v de._n and_o unto_o de_fw-fr add_v unity_n df._n now_o of_o be_v either_o a_o prime_a number_n or_o not_o first_o let_v it_o be_v a_o prime_a number_n case_n then_o be_v there_o find_v these_o prime_a number_n a_o b_o c_o and_o of_o more_o in_o multitude_n then_o the_o prime_a number_n â—irst_n give_v a_o b_o c._n but_o now_o suppose_v that_o of_o be_v not_o prime_a case_n wherefore_o some_o prime_a number_n measure_v it_o by_o the_o 24._o of_o the_o seven_o let_v a_o prime_a number_n measure_v it_o namely_o g._n then_o i_o say_v that_z g_z be_v none_o of_o these_o number_n a_o b_o c._n for_o if_o g_z be_v one_o and_o the_o same_o with_o any_o of_o these_o a_o b_o c._n but_o a_o b_o c_o measure_v the_o number_n de_fw-fr wherefore_o g_z also_o measure_v de_fw-fr and_o it_o also_o measure_v the_o whole_a ef._n wherefore_o g_o be_v a_o number_n shall_v measure_v the_o residue_n df_o be_v unity_n which_o be_v impossible_a wherefore_o g_z be_v not_o one_o and_o the_o same_o with_o any_o of_o these_o prime_a number_n a_o b_o c_o and_o it_o be_v also_o suppose_a to_o be_v a_o prime_a number_n wherefore_o there_o be_v â—ound_v these_o prime_a number_n a_o b_o c_o g_o be_v more_o in_o multitude_n then_o the_o prime_a number_n give_v a_o b_o c_o which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n by_o this_o proposition_n it_o be_v manifest_a that_o the_o multitude_n of_o prime_a number_n be_v infinite_a ¶_o the_o 21._o theorem_a the_o 21._o proposition_n if_o even_o number_n how_o many_o soever_o be_v add_v together_o the_o whole_a shall_v be_v even_o svppose_v that_o these_o even_a number_n ab_fw-la bc_o cd_o and_o de_fw-fr be_v add_v together_o then_o i_o say_v that_o the_o whole_a number_n namely_o ae_n be_v a_o even_a number_n demonstration_n for_o forasmuch_o as_o every_o one_o of_o these_o number_n ab_fw-la bc_o cd_o and_o de_fw-fr be_v a_o even_a number_n therefore_o every_o one_o of_o they_o have_v a_o half_a wherefore_o the_o whole_a ae_n also_o have_v a_o half_a but_o a_o even_a number_n by_o the_o definition_n be_v that_o which_o may_v be_v divide_v into_o two_o equal_a part_n wherefore_o ae_n be_v a_o even_a number_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 22._o theorem_a the_o 22._o proposition_n if_o odd_a number_n how_o many_o soever_o be_v add_v together_o &_o if_o their_o multitude_n be_v even_o the_o whole_a also_o shall_v be_v even_o svppose_v that_o these_o odd_a number_n ab_fw-la bc_o cd_o and_o de_fw-fr be_v even_o in_o multitude_n be_v add_v together_o then_o i_o say_v that_o the_o whole_a ae_n be_v a_o even_a number_n for_o forasmuch_o as_o every_o one_o of_o these_o number_n ab_fw-la bc_o cd_o and_o de_fw-fr be_v a_o odd_a number_n be_v you_o take_v away_o unity_n from_o every_o one_o of_o they_o demonstration_n that_o which_o remain_v oâ—_n every_o one_o of_o they_o be_v a_o even_a number_n wherefore_o they_o all_o add_v together_o be_v by_o the_o 21._o of_o the_o nine_o a_o even_a number_n and_o the_o multitude_n of_o the_o unity_n take_v away_o be_v even_o wherefore_o the_o whole_a ae_n be_v a_o even_a number_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 23._o theorem_a the_o 23._o proposition_n if_o odd_a number_n how_o many_o soever_o be_v add_v together_o and_o if_o the_o multitude_n of_o they_o be_v odd_a the_o whole_a also_o shall_v be_v odd_a svppose_v that_o these_o odd_a number_n ab_fw-la bc_o and_o cd_o be_v odd_a in_o multitude_n be_v add_v together_o then_o i_o say_v that_o the_o whole_a ad_fw-la be_v a_o odd_a number_n take_v away_o from_o cd_o unity_n de_fw-fr wherefore_o that_o which_o remain_v ce_fw-fr be_v a_o even_a number_n but_o ac_fw-la also_o by_o the_o 22._o of_o the_o nine_o be_v a_o even_a number_n demonstration_n wherefore_o the_o whole_a ae_n be_v a_o even_a number_n but_o de_fw-fr which_o be_v unity_n be_v add_v to_o the_o even_a number_n ae_n make_v the_o whole_a ad_fw-la aâ—_z odd_a number_n which_o be_v require_v to_o be_v provedâ—_n ¶_o the_o 24._o theorem_a the_o 24._o proposition_n if_o from_o a_o even_a number_n be_v take_v away_o a_o even_a number_n that_o which_o remain_v shall_v be_v a_o even_a number_n svppose_v that_o ab_fw-la be_v a_o even_a number_n and_o from_o it_o take_v away_o a_o even_a number_n cb._n demonstration_n then_o i_o say_v that_o that_o which_o remain_v namely_o ac_fw-la be_v a_o even_a number_n for_o forasmuch_o as_o ab_fw-la be_v a_o even_o even_a number_n it_o have_v a_o half_a and_o by_o the_o same_o reason_n also_o bc_o have_v a_o half_a wherefore_o the_o residue_n ca_n have_v a_o half_a wherefore_o ac_fw-la be_v a_o even_a number_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 25._o theorem_a the_o 25._o proposition_n if_o from_o a_o even_a number_n be_v take_v away_o a_o odd_a number_n that_o which_o remain_v shall_v be_v a_o odd_a number_n svppose_v that_o ab_fw-la be_v a_o even_a number_n and_o take_v away_o from_o it_o bc_o a_o odd_a number_n then_o i_o say_v that_o the_o residue_n ca_n be_v a_o odd_a number_n demonstration_n take_v away_o from_o bc_o unity_n cd_o wherefore_o db_o be_v a_o even_a number_n and_o ab_fw-la also_o be_v a_o even_a number_n wherefore_o the_o residue_n ad_fw-la be_v a_o even_a number_n by_o the_o â—ormer_a proposition_n but_o cd_o which_o be_v unity_n be_v take_v away_o from_o the_o even_a number_n ad_fw-la make_v the_o residue_n ac_fw-la a_o odd_a number_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 26._o theorem_a the_o 26._o proposition_n if_o from_o a_o odd_a number_n be_v take_v away_o a_o odd_a number_n that_o which_o remain_v shall_v be_v a_o even_a number_n svppose_v that_o ab_fw-la be_v a_o odd_a number_n and_o from_o it_o take_v away_o a_o odd_a number_n bc._n then_o i_o say_v that_o the_o residue_n ca_n be_v a_o even_a number_n demonstration_n for_o forasmuch_o as_o ab_fw-la be_v a_o odd_a number_n take_v away_o from_o it_o unity_n bd._n wherefore_o the_o residue_n ad_fw-la be_v even_o and_o by_o the_o same_o reason_n cd_o be_v a_o even_a number_n wherefore_o the_o residue_n ca_n be_v a_o even_a number_n by_o the_o 24._o of_o this_o book_n â—_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 27._o theorem_a the_o 27._o proposition_n if_o from_o a_o odd_a number_n be_v take_v a_o way_n a_o even_a number_n the_o residue_n shall_v be_v a_o odd_a number_n svppose_v that_o ab_fw-la be_v a_o odd_a number_n and_o from_o it_o take_v away_o a_o even_a number_n bc._n demonstration_n then_o i_o say_v that_o the_o residue_n ca_n be_v a_o odd_a number_n take_v away_o from_o ab_fw-la unity_n ad._n wherefore_o the_o residue_n db_o be_v a_o even_a number_n &_o bc_o be_v by_o supposition_n even_o wherefore_o the_o residue_n cd_o be_v a_o even_a number_n wherefore_o dam_n which_o be_v unity_n be_v add_v unto_o cd_o which_o be_v a_o even_a number_n make_v the_o whole_a ac_fw-la a_o â—dde_a number_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 28._o theorem_a the_o 28._o proposition_n if_o a_o odd_a number_n multiply_v a_o even_a number_n produce_v any_o number_n the_o number_n produce_v shall_v be_v a_o even_a number_n svppose_v that_o a_o be_v a_o odd_a number_n multiply_v b_o be_v a_o even_a number_n do_v produce_v the_o number_n c._n then_o i_o say_v that_o c_z be_v a_o even_a number_n demonstration_n for_o forasmuch_o as_o a_o multiply_v b_o produce_v c_z therefore_o c_z be_v compose_v of_o so_o many_o number_n equal_a unto_o b_o as_o there_o be_v in_o unity_n in_o a._n but_o b_o be_v a_o even_a number_n wherefore_o c_o be_v compose_v of_o so_o many_o even_a number_n as_o there_o be_v unity_n in_o a._n but_o if_o even_a number_n how_o many_o soever_o be_v add_v together_o the_o whole_a by_o the_o 21._o of_o the_o nine_o be_v a_o even_a number_n wherefore_o c_o be_v a_o even_a number_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 29._o theorem_a the_o 29._o proposition_n iâ—_z a_o odd_a number_n multiply_v a_o
28._o proposition_n to_o find_v out_o medial_a right_a line_n commensurable_a in_o power_n only_o contain_v a_o medial_a parallelogram_n let_v there_o be_v put_v three_o rational_a right_a line_n commensurable_a in_o power_n only_o namely_o a_o b_o and_o c_o construction_n and_o by_o the_o 13._o of_o the_o six_o take_v the_o mean_a proportional_a between_o the_o line_n a_o and_o b_o &_o let_v thâ—_z same_o be_v d._n and_o as_o the_o line_n b_o be_v to_o the_o line_n c_o so_o by_o the_o 12._o of_o the_o six_o let_v the_o line_n d_o be_v to_o the_o line_n e._n and_o forasmuch_o as_o the_o line_n a_o and_o b_o be_v rational_a commensurable_a in_o power_n only_o demonstration_n therefore_o by_o the_o 21._o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o that_o be_v the_o square_a of_o the_o line_n d_o be_v medial_a wherefore_o d_z be_v a_o medial_a line_n and_o forasmuch_o as_o the_o line_n b_o and_o c_o be_v commensurable_a in_o power_n only_o and_o as_o the_o line_n b_o be_v to_o the_o line_n c_o so_o be_v the_o line_n d_o to_o the_o line_n e_o wherefore_o the_o line_n d_z and_o e_o be_v commensurable_a in_o power_n only_o by_o the_o corollary_n of_o the_o ten_o of_o this_o book_n but_o d_o be_v a_o medial_a line_n wherefore_o e_o also_o be_v a_o medial_a line_n by_o the_o 23._o of_o this_o book_n wherefore_o d_z &_o e_o be_v medial_a line_n commensurable_a in_o power_n only_o i_o say_v also_o that_o they_o contain_v a_o medial_a parallelogram_n for_o for_o that_o as_o the_o line_n b_o be_v to_o the_o line_n c_o so_o be_v the_o line_n d_o to_o the_o line_n e_o therefore_o alternate_o by_o the_o 16_o of_o the_o five_o as_o the_o line_n b_o be_v to_o the_o line_n d_o so_o be_v the_o line_n c_o to_o the_o line_n e._n but_o as_o the_o line_n b_o be_v to_o the_o line_n d_o so_o be_v the_o line_n d_o to_o the_o line_n aâ—_z by_o converse_n proportion_n which_o be_v prove_v by_o the_o corollary_n of_o the_o four_o of_o the_o five_o wherefore_o as_o the_o line_n d_o be_v to_o the_o line_n a_o so_o be_v the_o line_n c_o to_o the_o line_n e._n wherefore_o that_o which_o be_v contain_v under_o the_o line_n a_o &_o c_o be_v by_o the_o 16._o of_o the_o sixâ—_o equal_a to_o that_o which_o be_v contain_v under_o the_o line_n d_o &_o e._n but_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o c_o be_v medial_a by_o the_o 21._o of_o the_o ten_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o be_v medial_a wherefore_o there_o be_v find_v out_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o medial_a superficies_n which_o be_v require_v to_o be_v do_v a_o assumpt_n to_o find_v out_o two_o square_a number_n which_o add_v together_o make_v a_o square_a number_n let_v there_o be_v put_v two_o like_o superficial_a number_n ab_fw-la and_o bc_o which_o how_o to_o find_v out_o have_v be_v teach_v after_o the_o 9_o proposition_n of_o this_o book_n and_o let_v they_o both_o be_v either_o even_a number_n or_o odd_a and_o let_v the_o great_a number_n be_v ab_fw-la and_o forasmuch_o as_o if_o from_o any_o even_a number_n be_v take_v away_o a_o even_a number_n or_o from_o a_o odd_a number_n be_v take_v away_o a_o odd_a number_n the_o residue_n shall_v be_v even_o by_o the_o 24._o and_o 26_o of_o the_o nine_o if_o therefore_o from_o ab_fw-la be_v a_o even_a number_n be_v take_v away_o bc_o a_o even_a number_n or_o from_o ab_fw-la be_v a_o odd_a number_n be_v take_v away_o bc_o be_v also_o odd_a the_o residue_n ac_fw-la shall_v be_v even_o divide_v the_o number_n ac_fw-la into_o two_o equal_a part_n in_o d_o wherefore_o the_o number_n which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o cd_o be_v by_o the_o six_o of_o the_o second_o as_o barlaam_n demonstrate_v it_o in_o number_n equal_a to_o the_o square_a number_n of_o bd._n but_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v a_o square_a number_n for_o it_o be_v prove_v by_o the_o first_o of_o the_o nine_o that_o 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 produce_v shall_v be_v a_o square_a number_n wherefore_o there_o be_v find_v out_o two_o square_a number_n the_o one_o be_v the_o square_a number_n which_o be_v produce_v of_o ab_fw-la into_o bc_o and_o the_o other_o the_o square_a number_n produce_v of_o cd_o which_o add_v together_o make_v a_o square_a number_n namely_o the_o square_a number_n produce_v of_o bd_o multiply_v into_o himself_o forasmuch_o as_o they_o be_v demonstrate_v equal_a to_o it_o corollary_n a_o corollary_n â—umber_n and_o hereby_o it_o be_v manifest_a that_o there_o be_v find_v out_o two_o square_a number_n namely_o the_o 〈◊〉_d the_o square_a number_n of_o bd_o and_o the_o other_o the_o square_a number_n of_o cd_o so_o that_o that_o number_n wherein_o the_o one_o exceed_v the_o other_o the_o number_n i_o say_v which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v also_o a_o square_a number_n namely_o when_o aâ—_z &_o bc_o be_v like_o plain_a number_n but_o when_o they_o be_v not_o like_o plain_a number_n then_o be_v there_o find_v out_o two_o square_a number_n the_o square_a number_n of_o bd_o and_o the_o square_a number_n of_o dc_o who_o excess_n that_o be_v the_o number_n whereby_o the_o great_a exceed_v the_o less_o namely_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v not_o a_o square_a number_n ¶_o a_o assumpt_n to_o find_v out_o two_o square_a number_n which_o add_v together_o make_v not_o a_o square_a number_n assumpt_n let_v ab_fw-la and_o bc_o be_v like_o plain_a number_n so_o that_o by_o the_o first_o of_o the_o nine_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v a_o square_a number_n and_o let_v ac_fw-la be_v a_o even_a number_n and_o divide_v câ—_n into_o two_o equal_a parâ—es_n in_o d._n now_o by_o that_o which_o have_v before_o be_v say_v in_o the_o former_a assumpt_n it_o be_v manifest_a that_o the_o square_a number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o cd_o be_v equal_a to_o the_o square_a number_n of_o bd._n take_v away_o from_o cd_o unity_n de._n wherefore_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_n of_o ce_fw-fr be_v less_o than_o the_o square_a number_n of_o bd._n now_o then_o i_o say_v that_o the_o square_a numâ—er_n produce_v of_o ab_fw-la into_o bc_o add_v to_o the_o square_a number_n of_o ce_fw-fr make_v not_o a_o square_a number_n for_o if_o they_o do_v make_v a_o square_a number_n than_o that_o square_a number_n which_o they_o make_v be_v either_o great_a they_o the_o square_a number_n of_o be_v or_o equal_a unto_o it_o or_o less_o than_o it_o first_o great_a it_n can_v be_v for_o it_o be_v already_o prove_v that_o the_o square_a number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v less_o than_o the_o square_a number_n of_o bd._n but_o between_o the_o square_a number_n of_o bd_o and_o the_o square_a number_n of_o be_v there_o be_v no_o mean_a square_a number_n for_o the_o number_n bd_o exceed_v the_o number_n be_v only_o by_o unity_n which_o unity_n can_v by_o no_o mean_n be_v divide_v into_o number_n or_o if_o the_o number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_n of_o the_o number_n ce_fw-fr shall_v be_v great_a than_o the_o square_a of_o the_o number_n be_v then_o shall_v the_o self_n same_o number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_n of_o the_o number_n ce_fw-fr be_v equal_a to_o the_o square_n of_o the_o number_n bd_o the_o contrary_a whereof_o be_v already_o prove_v wherefore_o if_o it_o be_v possible_a let_v that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o the_o number_n ce_fw-fr be_v equal_a to_o the_o square_a number_n of_o be._n and_o let_v ga_o be_v double_a to_o unity_n de_fw-fr that_o be_v let_v it_o be_v the_o number_n two_o now_o forasmuch_o as_o the_o whole_a number_n ac_fw-la be_v by_o supposition_n double_a to_o the_o whole_a number_n cd_o of_o which_o the_o number_n agnostus_n be_v double_a to_o unity_n de_fw-fr therefore_o by_o the_o 7._o of_o the_o seven_o the_o residue_n namely_o the_o number_n gc_o be_v double_a to_o the_o residue_n namely_o to_o the_o number_n ec_o wherefore_o the_o number_n gc_o be_v divide_v into_o two_o equal_a part_n in_o e._n wherefore_o that_o which_o be_v produce_v of_o gb_o into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v equal_a to_o the_o square_a number_n
in_o that_o sort_n of_o understanding_n there_o be_v no_o number_n but_o that_o it_o may_v be_v divide_v into_o two_o equal_a part_n as_o this_o number_n 7_o may_v be_v divide_v into_o 3_o part_n namely_o 3._o 1._o and_o 3._o of_o which_o two_o namely_o 3_o and_o 3_o be_v equal_a yet_o iâ—_z not_o 7_o a_o even_a number_n because_o 3_o and_o 3_o add_v together_o make_v not_o 7._o boetius_fw-la therefore_o in_o the_o first_o book_n of_o his_o arithmetic_n boetius_fw-la for_o the_o more_o plain_n after_o this_o manner_n define_v a_o even_a number_n numerun_v paâ—_n est_fw-la qui_fw-la potest_fw-la in_o â—quâ—lia_fw-la due_a dividi_fw-la una_fw-la mediâ—â—â—n_fw-la intercedenta_fw-la that_o be_v a_o even_a number_n be_v that_o which_o may_v be_v divide_v into_o two_o equal_a part_n without_o a_o unity_n come_v between_o they_o number_n as_o 8_o be_v divide_v into_o 4_o and_o 4_o two_o equal_a part_n without_o anâ—_z unity_n come_v between_o they_o which_o add_v together_o make_v 8_o so_o that_o the_o sense_n of_o this_o definition_n be_v that_o a_o even_a number_n be_v that_o which_o be_v divide_v into_o two_o such_o equal_a part_n which_o be_v his_o two_o half_a part_n here_o be_v also_o to_o be_v note_v note_n that_o a_o part_n be_v take_v in_o this_o definition_n and_o in_o certain_a definition_n follow_v not_o in_o the_o signification_n as_o it_o be_v before_o define_v namely_o for_o such_o a_o part_n asâ—_n measure_v the_o whole_a number_n but_o for_o any_o part_n which_o help_v to_o the_o make_n of_o the_o whole_a and_o into_o which_o the_o whole_a may_v be_v resolve_v so_o be_v 3_o and_o 5_o part_n of_o 8_o in_o this_o sense_n but_o not_o in_o the_o other_o sense_n for_o neither_o 3_o not_o 5_o measure_v â—_o pythagoras_n and_o his_o scholar_n give_v a_o other_o definition_n of_o a_o even_a number_n which_o definition_n 〈◊〉_d also_o have_v after_o this_o manner_n pithagoraâ—_n a_o even_a number_n be_v that_o which_o in_o one_o and_o the_o same_o division_n be_v divide_v into_o the_o great_a and_o into_o the_o least_o into_o the_o great_a as_o touch_v space_n and_o into_o the_o least_o as_o touch_v quantity_n definition_n as_o 10._o be_v divide_v into_o 5_o &_o 5._o which_o 〈◊〉_d his_o great_a part_n which_o greatness_n of_o part_n he_o call_v space_n and_o in_o the_o same_o division_n the_o number_n 10_o be_v divide_v but_o into_o two_o part_n but_o into_o less_o them_z two_o part_n nothing_o can_v be_v divide_v which_o thing_n he_o call_v quantity_n so_o that_o 10_o divide_v into_o 5_o and_o 5._o in_o that_o one_o division_n be_v divide_v into_o the_o great_a namely_o into_o his_o haluesâ—_n and_o into_o the_o least_o namely_o into_o two_o part_n and_o no_o more_o there_o be_v also_o another_o definition_n more_o ancient_a which_o be_v thus_o definition_n a_o even_a number_n be_v that_o which_o may_v be_v divide_v into_o two_o equal_a part_n and_o into_o two_o unequal_a part_n but_o in_o neither_o division_n to_o the_o constitution_n of_o the_o whole_a to_o the_o even_a part_n be_v add_v the_o odd_a neither_o to_o the_o odd_a be_v add_v the_o even_o as_o 8_o may_v be_v divide_v diverse_o partly_o into_o even_a part_n as_o into_o 4_o and_o 4._o likewise_o into_o 6_o and_o 2._o and_o partly_o into_o odd_a part_n as_o into_o 5_o and_o 3._o also_o into_o 7_o and_o 1._o in_o all_o which_o division_n you_o see_v no_o odd_a part_n join_v to_o a_o even_o nor_o a_o even_a part_n join_v to_o a_o odd_a but_o if_o the_o one_o be_v even_o the_o other_o be_v even_o and_o if_o the_o one_o be_v odd_a the_o other_o be_v odd_a in_o the_o two_o first_o division_n both_o part_n be_v even_o and_o in_o the_o two_o last_o both_o part_n be_v odd_a it_o be_v to_o be_v consider_v that_o the_o two_o part_n add_v together_o must_v make_v the_o whole_a this_o definition_n be_v general_a and_o common_a to_o all_o even_a number_n except_o to_o the_o number_n 2._o which_o can_v not_o be_v divide_v into_o two_o unequal_a part_n but_o only_o into_o two_o unity_n which_o be_v equal_a definition_n there_o be_v yet_o give_v a_o other_o definition_n of_o a_o even_a number_n namely_o thus_o a_o even_a number_n be_v that_o which_o only_o by_o a_o unity_n either_o above_o it_o or_o under_o it_o differ_v from_o a_o odd_a as_z 8_o be_v a_o even_a number_n differ_v from_o 9_o a_o odd_a number_n be_v above_o it_o but_o by_o one_o and_o also_o from_o 7._o under_o it_o it_o differ_v likewise_o by_o one_o and_o so_o oâ—_n other_o definition_n 7_o a_o odd_a number_n be_v that_o which_o can_v be_v divide_v into_o two_o equal_a part_n or_o that_o which_o only_o by_o a_o unity_n differ_v from_o a_o even_a number_n as_o the_o number_n 5_o can_v be_v by_o no_o mean_n divide_v into_o two_o equal_a part_n namely_o such_o two_o which_o add_v together_o shall_v make_v 5._o or_o by_o the_o second_o definition_n 5_o a_o odd_a number_n differ_v from_o 6_o a_o even_a number_n above_o â—t_a by_o 1._o and_o the_o same_o 5_o differ_v from_o 4_o a_o even_a number_n under_o it_o likewise_o by_o 1._o 〈…〉_o number_n after_o this_o manner_n a_o odd_a number_n be_v that_o which_o can_v be_v divide_v into_o two_o equal_a part_n but_o that_o a_o unity_n shall_v be_v between_o they_o as_o if_o you_o divide_v 5_o into_o 2_o and_o â—_o which_o be_v two_o equal_a part_n number_n there_o remain_v one_o or_o a_o unity_n between_o they_o to_o make_v the_o whole_a number_n 5._o there_o be_v yet_o a_o other_o definition_n of_o a_o odd_a number_n a_o odd_a number_n be_v that_o which_o be_v divide_v into_o two_o unequal_a part_n howsoever_o the_o one_o be_v ever_o even_o and_o the_o other_o odd_a as_o if_o 9_o be_v divide_v into_o two_o part_n which_o add_v together_o make_v the_o whole_a namely_o into_o 4_o and_o 5._o which_o be_v unequal_a definition_n you_o see_v the_o one_o be_v even_o namely_o 4_o and_o the_o other_o be_v odd_a namely_o 5._o so_o if_o you_o divide_v 9_o into_o 6_o and_o 3._o or_o into_o 8_o and_o 1._o the_o one_o part_n be_v ever_o even_o and_o the_o other_o odd_a definition_n 8_o a_o number_n even_o even_o call_v in_o latin_a pariter_fw-la par_fw-fr be_v that_o number_n which_o a_o even_a number_n measure_v by_o a_o even_a number_n as_o 8_o be_v a_o number_n even_o even_o for_o 4_o a_o even_a number_n measure_v 8_o by_o 2_o which_o be_v also_o a_o even_a number_n this_o definition_n have_v much_o trouble_v many_o and_o seem_v not_o a_o true_a definition_n for_o that_o there_o be_v many_o number_n which_o even_a number_n do_v measure_n and_o that_o by_o even_a number_n which_o yet_o be_v not_o even_o even_a number_n after_o most_o man_n mind_n as_o 24._o which_o 6_o a_o even_a number_n do_v measure_n by_o four_o which_o be_v also_o a_o even_a number_n and_o yet_o as_o they_o think_v be_v not_o 24_o a_o even_o even_a number_n for_o that_o 8_o a_o even_a number_n do_v measure_n also_o â—4_o by_o 3._o a_o odd_a number_n campane_n wherefore_o campane_n to_o make_v his_o sentence_n plain_a after_o this_o manner_n set_v forth_o this_o definition_n pariter_fw-la par_fw-fr est_fw-la quem_fw-la cuncti_fw-la pares_fw-la â—um_fw-la numerantes_fw-la paribus_fw-la vicibuâ—_n numerane_n that_o be_v number_n a_o even_o even_o number_n be_v when_o all_o the_o even_a number_n which_o measure_v it_o do_v measure_v it_o by_o even_a time_n that_o be_v by_o even_a number_n as_o 16._o all_o the_o even_a number_n whith_z measure_n 16_o as_o be_v 8._o 4._o and_o 2._o do_v measure_n it_o by_o even_a number_n as_o 8_o by_o 2_o twice_o 8_o be_v 16_o 4_o by_o 4_o four_o time_n 4_o be_v 16_o and_o 2_o by_o 8_o 8_o time_n 2_o be_v 16._o which_o particle_n all_o even_a number_n add_v by_o campane_n make_v 24_o to_o be_v no_o even_o even_o number_n for_o that_o some_o one_o even_o number_n measure_v it_o by_o a_o odd_a number_n as_o 8_o by_o 3._o flussates_n also_o be_v plain_o of_o this_o mind_n flussates_n that_o euclid_n give_v not_o this_o definition_n in_o such_o manner_n as_o it_o be_v by_o theon_n write_v for_o the_o largeness_n &_o generality_n thereof_o â—or_a that_o it_o extend_v it_o to_o infinite_a number_n which_o be_v not_o even_o even_o as_o he_o think_v for_o which_o cause_n in_o place_n thereof_o he_o give_v this_o definition_n definition_n a_o number_n even_o even_o be_v that_o which_o only_o even_o number_n do_v measure_n as_o 16_o be_v measure_v of_o none_o but_o of_o even_a number_n and_o therefore_o be_v even_o even_o there_o be_v also_o of_o boetius_fw-la give_v a_o other_o definition_n of_o more_o facility_n boetius_fw-la include_v in_o it_o no_o doubt_n at_o all_o which_o be_v most_o common_o use_v of_o all_o writer_n and_o be_v thus_o definition_n a_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 odd_a definition_n for_o no_o even_a number_n but_o only_o 2_o measure_v 10._o and_o that_o be_v by_o 5_o a_o odd_a number_n but_o not_o all_o the_o even_a number_n which_o measure_n 12._o do_v measure_n it_o by_o odd_a number_n for_o 6_o a_o even_a number_n measure_v 12_o by_o 2_o which_o be_v also_o even_o wherefore_o 12_o be_v not_o by_o this_o definition_n a_o number_n even_o odd_a flussates_n also_o offend_v with_o the_o over_o large_a generality_n of_o this_o definition_n to_o make_v the_o definition_n agree_v with_o the_o thing_n define_v put_v it_o after_o this_o manner_n flussates_n a_o number_n even_o odd_a be_v that_o which_o a_o odd_a number_n do_v measure_n only_o by_o a_o even_a number_n as_o 14._o which_o 7._o a_o odd_a number_n do_v measure_n only_o by_o 2._o which_o be_v a_o even_a number_n definition_n there_o be_v also_o a_o other_o definition_n of_o this_o kind_n of_o number_n common_o give_v of_o more_o plain_n which_o be_v this_o a_o number_n even_o odd_a in_o 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 even_o odd_a and_o odd_o even_o be_v take_v of_o euclid_n for_o on_o and_o the_o self_n same_o kind_n of_o number_n but_o the_o number_n which_o here_o ought_v to_o have_v be_v place_v be_v call_v of_o the_o best_a interpreter_n of_o euclid_n numerus_fw-la pariter_fw-la par_fw-fr &_o nupar_fw-la that_o be_v a_o number_n even_o even_o and_o even_o odd_a yeâ—_n and_o it_o be_v so_o call_v of_o euclid_n himself_o in_o the_o 34._o proposition_n of_o his_o 9_o book_n which_o kind_n of_o number_n campanus_n and_o flussates_n in_o stead_n of_o the_o insufficient_a and_o unapt_a definition_n before_o give_v assign_v this_o definition_n a_o number_n even_o even_o and_o even_o â—dde_v be_v that_o which_o a_o even_a number_n do_v measure_n sometime_o by_o a_o even_a number_n and_o sometime_o by_o a_o odd_a definition_n as_o the_o number_n 12_o for_o 2._o a_o even_a number_n measure_v 12._o by_o 6._o a_o even_a number_n two_o time_n 6._o iâ—_z 12._o also_o 4._o a_o even_a number_n measure_v the_o 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
odd_a number_n produce_v any_o number_n the_o number_n produce_v shall_v be_v a_o odd_a number_n svppose_v that_o a_o be_v a_o odd_a number_n multiply_v b_o be_v also_o a_o odd_a number_n do_v produce_v the_o number_n c._n then_o i_o say_v that_o c_z be_v a_o odd_a number_n for_o forasmuch_o as_o a_o multiply_a b_o produce_v c_z therefore_o c_z be_v compose_v of_o so_o many_o number_n equal_a unto_o b_o as_o there_o be_v unity_n in_o a._n demonstration_n but_o either_o of_o these_o number_n a_o and_o b_o be_v a_o odd_a number_n wherefore_o c_o be_v compose_v of_o odd_a number_n who_o multitude_n also_o be_v odd_a wherefore_o by_o the_o 23._o of_o the_o nine_o c_o be_v a_o odd_a number_n which_o be_v require_v to_o be_v demonstrate_v a_o proposition_n add_v by_o campane_n campane_n if_o a_o odd_a number_n measure_v a_o even_a number_n it_o shall_v measure_v it_o by_o a_o even_a number_n for_o if_o it_o shall_v measure_v it_o by_o a_o odd_a number_n then_o of_o a_o odd_a number_n multiply_v into_o a_o odd_a number_n shall_v be_v produce_v a_o odd_a number_n which_o by_o the_o former_a proposition_n be_v impossible_a an_o other_o proposition_n add_v by_o he_o he_o if_o a_o odd_a number_n measure_v a_o odd_a number_n it_o shall_v measure_v it_o by_o a_o odd_a number_n for_o if_o it_o shall_v measure_v it_o by_o a_o even_a number_n then_o of_o a_o odd_a number_n multiply_v into_o a_o even_a number_n shall_v be_v produce_v a_o odd_a number_n which_o by_o the_o 28._o of_o this_o book_n be_v impossible_a ¶_o the_o 30._o theorem_a the_o 30._o proposition_n if_o a_o odd_a number_n measure_v a_o even_a number_n it_o shall_v also_o measure_v the_o half_a thereof_o svppose_v that_o a_o be_v a_o odd_a number_n do_v measure_n b_o be_v a_o even_a number_n thâ—â—_n i_o say_v that_o it_o shall_v measure_v the_o half_a thereof_o for_o forasmuch_o as_o a_o measure_v b_o let_v iâ—_z measure_v it_o by_o c._n absurdity_n then_o i_o say_v that_o c_z be_v a_o even_a number_n for_o if_o not_o then_o if_o it_o be_v possible_a leâ—_z iâ—_z be_v odd_a and_o forasmuch_o as_o a_o measure_v b_o by_o c_o therefore_o a_o multiply_v c_o produce_v b._n wherefore_o b_o be_v compose_v of_o odd_a number_n who_o multitude_n also_o be_v odd_a wherefore_o b_o be_v a_o odd_a number_n by_o the_o 29._o of_o this_o book_n which_o be_v absurd_a for_o it_o be_v suppose_v to_o be_v even_o wherefore_o c_o be_v a_o even_a numâ—er_n wherefore_o a_o measure_v b_o by_o a_o even_a number_n and_o c_z measure_v b_o by_o a._n but_o either_o of_o these_o number_n c_o and_o b_o have_v a_o half_a part_n wherefore_o as_o c_o be_v to_o b_o so_o be_v the_o half_a to_o the_o half_a but_o c_o measure_v b_o by_o a._n wherefore_o the_o half_a of_o c_o measure_v the_o half_a of_o b_o by_o a_o wherefore_o a_o multiply_a the_o half_a of_o c_o produce_v the_o half_a of_o b._n wherefore_o a_o measure_v the_o half_a of_o b_o and_o it_o measure_v it_o by_o the_o half_a of_o c._n wherefore_o a_o measure_v the_o half_a of_o the_o number_n b_o which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 31._o theorem_a the_o 31._o proposition_n if_o a_o odd_a number_n be_v prime_a to_o any_o number_n it_o shall_v also_o be_v prime_a to_o the_o double_a thereof_o svppose_v that_o a_o be_v a_o odd_a number_n be_v prime_a unto_o the_o number_n b_o and_o let_v the_o double_a of_o b_o be_v c._n demonstration_n then_o i_o say_v that_o a_o be_v prime_a unto_o c._n for_o if_o a_o and_o c_o be_v not_o prime_a the_o one_o to_o the_o other_o some_o one_o number_n measure_v they_o both_o let_v there_o be_v such_o a_o number_n which_o measure_v they_o both_o and_o let_v the_o same_o be_v d._n but_o a_o be_v a_o odd_a number_n wherefore_o d_z also_o be_v a_o odd_a number_n for_o if_o d_z which_o measure_v a_o shall_v be_v a_o even_a number_n than_o shall_v a_o also_o be_v a_o even_a number_n by_o the_o 21._o of_o this_o book_n which_o be_v contrary_a to_o the_o supposition_n for_o a_o be_v suppose_v to_o be_v a_o odd_a number_n &_o therefore_o d_z also_o be_v a_o odd_a number_n and_o forasmuch_o as_o d_z be_v a_o odd_a number_n measure_v c_z but_o c_o be_v a_o even_a number_n for_o that_o it_o have_v a_o half_a namely_o b_o wherefore_o by_o the_o proposition_n next_o go_v before_o d_o measure_v the_o half_a of_o c._n but_o the_o half_a of_o c_o be_v b._n wherefore_o d_o measure_v b_o and_o it_o also_o measure_v a._n wherefore_o d_o measure_v a_o and_o b_o being_n prime_n the_o one_o to_o the_o other_o which_o be_v absurd_a wherefore_o no_o number_n measure_v the_o number_n a_o &_o c._n wherefore_o a_o be_v a_o prime_a number_n unto_o c._n wherefore_o these_o number_n a_o and_o c_o be_v prime_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 32._o theorem_a the_o 32._o proposition_n every_o number_n produce_v by_o the_o double_n of_o two_o upward_a be_v even_o even_o only_o svppose_v that_o a_o be_v the_o number_n two_o and_o from_o a_o upward_a double_a number_n how_o many_o soever_o as_z b_o c_o d._n then_o i_o say_v that_z b_o c_o d_o be_v number_n even_o even_o only_o that_o every_o one_o of_o they_o be_v even_o even_o it_o be_v manifest_a demonstration_n for_o every_o one_o of_o they_o be_v produce_v by_o the_o double_n of_o two_o i_o say_v also_o that_o every_o one_o of_o they_o be_v even_o even_o only_o take_v unity_n e._n and_o forasmuch_o as_o from_o unity_n be_v certain_a number_n in_o continual_a proportion_n &_o a_o which_o follow_v next_o after_o unity_n be_v a_o prime_a number_n therefore_o by_o the_o 13._o of_o the_o three_o no_o number_n measure_v d_z be_v the_o great_a number_n of_o these_o number_n a_o b_o c_o d_o beside_o the_o self_n same_o number_n in_o proportion_n but_o every_o one_o of_o these_o number_n a_o b_o c_o be_v even_o even_o wherefore_o d_z be_v even_o even_o only_o in_o like_a sort_n may_v we_o prove_v that_o every_o one_o of_o these_o number_n a_o b_o c_o be_v even_o even_o only_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 33._o theorem_a the_o 33._o proposition_n a_o number_n who_o half_a part_n be_v odd_a be_v even_o odd_a only_o svppose_v that_o a_o be_v a_o number_n who_o half_a part_n be_v odd_a then_o i_o say_v that_o a_o be_v even_o odd_a only_o that_o it_o be_v even_o odd_a it_o be_v manifest_a for_o his_o half_a be_v odd_a measure_v he_o by_o a_o even_a number_n namely_o by_o 2._o absurdity_n by_o the_o definition_n i_o say_v also_o that_o it_o be_v even_o odd_a only_o for_o if_o a_o be_v even_o even_o his_o half_a also_o be_v even_o for_o by_o the_o definition_n a_o even_a number_n measure_v he_o by_o a_o even_a number_n wherefore_o that_o even_a number_n which_o measure_v he_o by_o a_o even_a number_n shall_v also_o measure_v the_o half_a thereof_o be_v a_o odd_a number_n by_o the_o 4._o common_a sentence_n of_o the_o seven_o which_o be_v absurd_a wherefore_o a_o be_v a_o number_n even_o odd_a only_o which_o be_v require_v to_o be_v prove_v an_o other_o demonstration_n to_o prove_v the_o same_o suppose_v that_o the_o number_n a_o have_v to_o his_o half_n a_o odd_a number_n namely_o b._n then_o i_o say_v that_o a_o be_v even_o odd_a only_o that_o it_o be_v even_o odd_a need_v no_o proof_n forasmuch_o as_o the_o number_n 2._o a_o even_a number_n measure_v it_o by_o the_o half_a thereof_o which_o be_v a_o odd_a number_n demonstration_n let_v c_o be_v the_o number_n 2._o by_o which_o b_o measure_v a_o for_o that_o a_o be_v suppose_v to_o be_v double_a unto_o b_o and_o let_v a_o even_a number_n namely_o d_o measure_v a_o which_o be_v possible_a for_o that_o a_o be_v a_o even_a number_n by_o the_o definition_n by_o f._n and_o forasmuch_o as_o that_o which_o be_v produce_v of_o c_o into_o b_o be_v equal_a to_o that_o which_o be_v produce_v of_o d_o into_o fletcher_n therefore_o by_o the_o 19_o of_o the_o seven_o as_o c_o be_v to_o d_z so_o be_v b_o to_o f._n but_o c_o the_o number_n two_o measure_v d_z be_v a_o even_a number_n wherefore_o fletcher_n also_o measure_v b_o which_o be_v the_o half_a of_o a._n wherefore_o fletcher_n be_v a_o odd_a number_n for_o if_o fletcher_n be_v a_o even_a number_n than_o shall_v it_o in_o the_o b_o who_o it_o measure_v a_o odd_a number_n also_o by_o the_o 21._o of_o this_o book_n which_o be_v contrary_a to_o the_o supposition_n and_o in_o like_a manner_n may_v we_o prove_v that_o all_o the_o even_a number_n which_o measure_v the_o number_n aâ—_z do_v measure_v it_o by_o odd_a number_n wherefore_o a_o be_v a_o number_n even_o odd_a only_o
to_o the_o same_o and_o so_o the_o line_n bd_o be_v a_o six_o residual_a line_n and_o the_o line_n kh_o be_v a_o six_o binomial_a line_n wherefore_o kh_o be_v a_o binomial_a line_n who_o name_n kf_a and_o fh_o be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n bd_o namely_o to_o bc_o and_o cd_o and_o in_o the_o self_n same_o proportion_n and_o the_o binomial_a line_n kh_o be_v in_o the_o self_n same_o order_n of_o binomial_a line_n that_o the_o residual_a bd_o be_v of_o residual_a line_n wherefore_o the_o square_a of_o a_o rational_a line_n apply_v unto_o a_o residual_a line_n make_v the_o breadth_n or_o other_o side_n a_o binomial_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n and_o in_o the_o self_n same_o proportion_n and_o moreover_o the_o binomial_a line_n be_v in_o the_o self_n same_o order_n of_o binomial_a line_n that_o the_o residual_a line_n be_v of_o residual_a line_n which_o be_v require_v to_o be_v demonstrate_v the_o assumpt_v confirm_v now_o let_v we_o declare_v how_o as_o the_o line_n kh_o be_v to_o the_o line_n eh_o so_o to_o make_v the_o line_n hf_o to_o the_o line_n fe_o add_v unto_o the_o line_n kh_o direct_o a_o line_n equal_a to_o he_o and_o let_v the_o whole_a line_n be_v kl_o and_o by_o the_o ten_o of_o the_o six_o let_v the_o line_n he_o be_v divide_v as_o the_o whole_a line_n kl_o be_v divide_v in_o the_o point_n h_o let_v the_o line_n he_o be_v so_o divide_v in_o the_o point_n f._n wherefore_o as_o the_o line_n kh_o be_v to_o the_o line_n hl_o that_o be_v to_o the_o line_n he_o so_o be_v the_o line_n hf_o to_o the_o line_n fe_o an_o other_o demonstration_n after_o flussas_n suppose_v that_o a_o be_v a_o rational_a line_n and_o let_v bd_o be_v a_o residual_a line_n and_o upon_o the_o line_n bd_o apply_v the_o parallelogram_n dt_n equal_a to_o the_o square_n of_o the_o line_n a_o by_o the_o 45._o of_o the_o first_o make_v in_o breadth_n the_o line_n bt_n flussas_n then_o i_o say_v that_o bt_n be_v a_o binominall_a line_n such_o a_o one_o as_o be_v require_v in_o the_o proposition_n forasmuch_o as_o bd_o be_v a_o residual_a line_n let_v the_o line_n convenient_o join_v unto_o it_o be_v gd_o wherefore_o the_o line_n bg_o and_o gd_o be_v rational_a commensurable_a in_o power_n only_o construction_n upon_o the_o rational_a line_n bg_o apply_v the_o parallelogram_n by_o equal_a to_o the_o square_n of_o the_o line_n a_o and_o make_v in_o breadth_n the_o line_n be._n wherefore_o the_o line_n be_v be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n bg_o by_o the_o 20._o of_o the_o ten_o now_o forasmuch_o as_o the_o parallelogram_n by_o and_o td_v be_v equal_a for_o that_o they_o be_v each_o equal_a to_o the_o square_n of_o the_o line_n a_o demonstration_n therefore_o reciprocal_a by_o the_o 14._o of_o the_o six_o as_o the_o line_n bt_n be_v to_o the_o line_n be_v so_o be_v the_o line_n bg_o to_o the_o line_n bd._n wherefore_o by_o conversion_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 line_n bt_n be_v to_o the_o line_n te_fw-la so_o be_v the_o line_n bg_o to_o the_o line_n gd_o as_o the_o line_n bg_o be_v to_o the_o line_n gd_o so_o let_v the_o line_n tz_n be_v to_o the_o line_n see_fw-mi by_o the_o corollary_n of_o the_o 10._o of_o the_o six_o wherefore_o by_o the_o 11._o of_o the_o five_o the_o line_n bt_n be_v to_o the_o line_n te_fw-la as_o the_o line_n tz_n be_v to_o the_o line_n ze._n for_o either_o of_o they_o be_v as_o the_o line_n bg_o be_v to_o the_o line_n gd_o wherefore_o the_o residue_n bz_o be_v to_o the_o residue_n zt_n as_o the_o whole_a bt_n be_v to_o the_o whole_a te_fw-la by_o the_o 19_o of_o the_o five_o wherefore_o by_o the_o 11._o of_o the_o five_o the_o line_n bz_o be_v to_o the_o line_n zt_n as_o the_o line_n zt_n be_v to_o the_o line_n ze._n wherefore_o the_o line_n tz_n be_v the_o mean_a proportional_a between_o the_o line_n bz_o and_o ze._n wherefore_o the_o square_a of_o the_o first_o namely_o of_o the_o line_n bz_o be_v to_o the_o square_n of_o the_o second_o namely_o of_o the_o line_n zt_n as_o the_o first_o namely_o the_o line_n bz_o be_v to_o the_o three_o namely_o to_o the_o line_n see_fw-mi by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o and_o for_o that_o as_o the_o line_n bg_o be_v to_o the_o line_n gd_o so_o be_v the_o line_n tz_n to_o the_o line_n see_fw-mi but_o as_o the_o line_n tz_n be_v to_o the_o line_n see_fw-mi so_o be_v the_o line_n bz_o to_o the_o line_n zt_n wherefore_o as_o the_o line_n bg_o be_v to_o the_o line_n gd_o so_o be_v the_o line_n bz_o to_o the_o line_n zt_n by_o the_o 11._o of_o the_o five_o wherefore_o the_o line_n bz_o and_o zt_n be_v commensurable_a in_o power_n only_o as_o also_o be_v the_o line_n bg_o and_o gd_o which_o be_v the_o name_n of_o the_o residual_a line_n bd_o by_o the_o 10._o of_o this_o book_n wherefore_o the_o right_a line_n bz_o and_o see_fw-mi be_v commensurable_a in_o length_n for_o we_o have_v prove_v that_o they_o be_v in_o the_o same_o proportion_n that_o the_o square_n of_o the_o line_n bz_o and_o zt_n be_v and_o therefore_o by_o the_o corollary_n of_o the_o 15._o of_o this_o book_n the_o residue_n be_v which_o be_v a_o rational_a line_n be_v commensurable_a in_o length_n unto_o the_o same_o line_n bz_o wherefore_o also_o the_o line_n bg_o which_o be_v commensurable_a in_o length_n unto_o the_o line_n be_v shall_v also_o be_v commensurable_a in_o length_n unto_o the_o same_o line_n ez_o by_o the_o 12._o of_o the_o ten_o and_o it_o be_v prove_v that_o the_o line_n rz_n be_v to_o the_o line_n zt_n commensurable_a in_o power_n only_o wherefore_o the_o right_a line_n bz_o and_o zt_n be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n bt_n be_v a_o binomial_a line_n by_o the_o 36._o of_o this_o book_n and_o for_o that_o as_o the_o line_n bg_o be_v to_o the_o line_n gd_o so_o be_v the_o line_n bz_o to_o the_o line_n zt_n therefore_o alternate_o by_o the_o 16._o of_o the_o five_o the_o line_n bg_o be_v to_o the_o line_n bz_o as_o the_o line_n gd_o be_v to_o the_o line_n zt_n but_o the_o line_n bg_o be_v commensurable_a in_o length_n unto_o the_o line_n bz_o wherefore_o by_o the_o 10._o of_o this_o book_n the_o line_n gd_o be_v commensurable_a in_o length_n unto_o the_o line_n zt_n wherefore_o the_o name_n bg_o and_o gd_o of_o the_o residual_a line_n bd_o be_v commensurable_a in_o length_n unto_o the_o name_n bz_o and_o zt_v of_o the_o binomial_a line_n bt_n and_o the_o line_n bz_o be_v to_o the_o line_n zt_n in_o the_o same_o proportion_n that_o the_o line_n bg_o be_v to_o the_o line_n gd_o as_o before_o it_o be_v more_o manifest_a and_o that_o they_o be_v of_o one_o and_o the_o self_n same_o order_n be_v thus_o prove_v if_o the_o great_a or_o less_o name_n of_o the_o residual_a line_n namely_o the_o right_a line_n bg_o or_o gd_o be_v commensurable_a in_o length_n to_o any_o rational_a line_n put_v the_o great_a name_n also_o or_o less_o namely_o bz_o or_o zt_n shall_v be_v commensurable_a in_o length_n to_o the_o same_o rational_a line_n put_v by_o the_o 12._o of_o this_o book_n and_o if_o neither_o of_o the_o name_n of_o the_o residual_a line_n be_v commensurable_a in_o length_n unto_o the_o rational_a line_n put_v neither_o of_o the_o name_n of_o the_o binomial_a line_n shall_v be_v commensurable_a in_o length_n unto_o the_o same_o rational_a line_n put_v by_o the_o 13._o of_o the_o ten_o and_o if_o the_o great_a name_n bg_o be_v in_o power_n more_o than_o the_o less_o name_n by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n bg_o the_o great_a name_n also_o bz_o shall_v be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n bz_o and_o if_o the_o one_o be_v in_o power_n more_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n the_o other_o also_o shall_v be_v in_o power_n more_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n by_o the_o 14._o of_o this_o book_n the_o square_a therefore_o of_o a_o rational_a line_n etc_n etc_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 90._o theorem_a the_o 114._o proposition_n joint_o if_o a_o parallelogram_n be_v contain_v under_o a_o residual_a line_n &_o a_o binomial_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n and_o in_o the_o selâ—e_a same_o proportion_n the_o line_n which_o contain_v in_o power_n
that_o superficies_n be_v rational_a svppose_v that_o a_o parallelogram_n be_v contain_v under_o a_o residual_a line_n ab_fw-la and_o a_o binomial_a line_n cd_o and_o let_v the_o great_a name_n of_o the_o binomial_a line_n be_v ce_fw-fr and_o the_o less_o name_n be_v ed_z and_o let_v the_o name_n of_o the_o binomial_a line_n namely_o ce_fw-fr and_o ed_z be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n namely_o to_o of_o and_o fâ—_n and_o in_o the_o self_n same_o proportion_n and_o let_v the_o line_n which_o contain_v in_o power_n that_o parallelogram_n be_v g._n then_o i_o say_v that_o the_o line_n g_o be_v rational_a take_v a_o rational_a line_n namely_o h._n and_o unto_o the_o line_n cd_o apply_v a_o parallelogram_n equal_a to_o the_o square_v of_o the_o line_n h_o construction_n and_o make_v in_o breadth_n the_o line_n kl_o wherefore_o by_o the_o 112._o of_o the_o ten_o kl_o be_v a_o residual_a line_n demonstration_n who_o name_n let_v be_v km_o and_o ml_o which_o be_v by_o the_o same_o commensurable_a to_o the_o name_n of_o the_o binomial_a line_n that_o be_v to_o ce_fw-fr and_o ed_z and_o be_v in_o the_o self_n same_o proportion_n but_o by_o position_n the_o line_n ce_fw-fr and_o ed_z be_v commensurable_a to_o the_o line_n of_o and_o fb_o and_o be_v in_o the_o self_n same_o proportion_n wherefore_o by_o the_o 12._o of_o the_o ten_o as_o the_o line_n of_o be_v to_o the_o line_n fbâ—_n so_o be_v the_o line_n km_o to_o the_o line_n ml_o wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n of_o be_v to_o the_o line_n km_o so_o be_v the_o line_n bf_o to_o the_o line_n lm_o wherefore_o the_o residue_n ab_fw-la be_v to_o the_o residue_n kl_o as_o the_o whole_a of_o be_v to_o the_o whole_a km_o but_o the_o line_n of_o be_v commensurable_a to_o the_o line_n km_o for_o either_o of_o the_o line_n of_o and_o km_o be_v commensurable_a to_o the_o line_n ce._n wherefore_o also_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n kl_o and_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n kl_o so_o by_o the_o first_o of_o the_o six_o be_v the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o ab_fw-la to_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o kl_o wherefore_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o ab_fw-la be_v commensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o kl_o but_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o kl_o be_v equal_a to_o the_o square_n of_o the_o line_n h._n wherefore_o the_o parallelogram_n contain_v under_o the_o line_n cd_o &_o ab_fw-la be_v commensurable_a to_o the_o square_n of_o the_o line_n h._n but_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n g._n wherefore_o the_o square_a of_o the_o line_n h_o be_v commensurable_a to_o the_o square_n of_o the_o line_n g._n but_o the_o square_n of_o the_o line_n h_o be_v rational_a wherefore_o the_o square_a of_o the_o line_n g_o be_v also_o rational_a wherefore_o also_o the_o line_n g_o be_v rational_a and_o it_o contain_v in_o power_n the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o cd_o if_o therefore_o a_o parallelogram_n be_v contain_v under_o a_o residual_a line_n and_o a_o binomial_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n and_o in_o the_o self_n same_o proportion_n the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v rational_a which_o be_v require_v to_o be_v prove_v ¶_o corollary_n hereby_o it_o be_v manifest_a that_o a_o rational_a parallelogram_n may_v be_v contain_v under_o irrational_a line_n ¶_o a_o otâ—â—r_n 〈…〉_o flussas_n 〈…〉_o line_n â—d_a whosâ—_n name_v aâ—_z and_o â—d_a let_v be_v commensurable_a in_o length_n unto_o the_o name_n of_o the_o residual_a line_n aâ—_z which_o let_v be_v of_o and_o fb_o and_o let_v the_o liâ—e_z aeâ—_n be_v to_o the_o line_n edâ—_n in_o the_o same_o proportion_n that_o the_o line_n of_o be_v to_o the_o line_n fâ—_n and_o let_v the_o right_a line_n â—_o contain_v in_o power_n the_o superficies_n dâ—_n then_o i_o say_v thaâ—_z the_o liâ—e_z â—_o be_v a_o rational_a linâ—_n 〈…〉_o lâ—ne_n construction_n which_o lâ—â—_n bâ—â—_n and_o upon_o the_o line_n â—â—_o describe_v by_o the_o 4â—_o of_o the_o first_o a_o parallelogram_n equal_a to_o the_o square_n of_o the_o line_n â—â—_o and_o make_n in_o breadth_n the_o line_n dc_o wherefore_o by_o the_o â—12_n of_o this_o book_n cd_o be_v a_o residuâ—ll_a lineâ—_z who_o name_n which_o let_v be_v â—â—_o and_o odd_a shall_v be_v coâ—mensurablâ—_n in_o length_n unto_o the_o name_n aâ—_z and_o â—d_a and_o the_o line_n c_o o_fw-fr shall_v be_v unto_o the_o line_n odd_a demonstration_n in_o the_o same_o proportion_n that_o the_o line_n ae_n be_v to_o the_o line_n edâ—_n but_o as_o the_o line_n aâ—_z be_v to_o the_o line_n â—d_a so_o by_o supposition_n be_v the_o line_n of_o to_o the_o line_n fe_o wherefore_o as_o the_o line_n county_n be_v to_o the_o line_n odd_a so_o be_v the_o line_n of_o to_o the_o line_n fâ—â—_v wherefore_o the_o line_n county_n and_o odd_a be_v commensurable_a with_o the_o line_n aâ—_z and_o â—â—_o by_o the_o â—â—_o of_o this_o book_n wherefore_o the_o residue_n namely_o the_o line_n cd_o be_v to_o the_o residue_n namely_o to_o the_o line_n aâ—_z as_o the_o line_n county_n be_v to_o the_o line_n of_o by_o the_o 19_o of_o the_o five_o but_o it_o be_v prove_v that_o the_o line_n county_n be_v commensurable_a unto_o the_o line_n af._n wherefore_o the_o line_n cd_o be_v commensurable_a unto_o the_o line_n ab_fw-la wherefore_o by_o the_o first_o of_o the_o six_o the_o parallelogram_n ca_n be_v commensurable_a to_o the_o parallelogram_n dâ—_n but_o the_o parallelogram_n â—â—_o iâ—_z by_o construction_n rational_a for_o it_o be_v equal_a to_o the_o square_n of_o the_o rational_a line_n â—_o wherefore_o the_o parallelogram_n â—d_a â—s_n also_o rational_o wherâ—fore_o the_o line_n â—_o which_o by_o supposition_n contain_v in_o power_n the_o superficies_n â—dâ—_n be_v also_o rational_a if_o therefore_o a_o parallelogram_n be_v contain_v etc_o etc_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 91._o theorem_a the_o 115._o proposition_n of_o a_o medial_a line_n be_v produce_v infinite_a irrational_a line_n of_o which_o none_o be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o svppose_v that_o a_o be_v a_o medial_a line_n then_o i_o say_v that_o of_o the_o line_n a_o may_v be_v produce_v infinite_a irrational_a line_n of_o which_o none_o shall_v be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o take_v a_o rational_a line_n b._n and_o unto_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o let_v the_o square_n of_o the_o line_n c_o be_v equal_a by_o the_o 14._o of_o the_o second_o â—_o demonstration_n wherefore_o the_o line_n c_o be_v irrational_a for_o a_o superficies_n contain_v under_o a_o rational_a line_n and_o a_o irrational_a line_n be_v by_o the_o assumpt_n follow_v the_o 38._o of_o the_o ten_o irrational_a and_o the_o line_n which_o contain_v in_o power_n a_o irrational_a superficies_n be_v by_o the_o assumpt_n go_v before_o the_o 21._o of_o the_o ten_o irrational_a and_o it_o be_v not_o one_o and_o the_o self_n same_o with_o any_o of_o those_o thirteen_o that_o be_v before_o for_o none_o of_o the_o line_n that_o be_v before_o apply_v to_o a_o rational_a line_n make_v the_o breadth_n medial_a again_o unto_o that_o which_o be_v contain_v under_o the_o line_n b_o and_o c_o let_v the_o square_n of_o d_o be_v equal_a wherefore_o the_o square_a of_o d_o be_v irrational_a wherefore_o also_o the_o line_n d_o be_v irrational_a and_o not_o of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o for_o the_o square_n of_o none_o of_o the_o line_n which_o be_v before_o apply_v to_o a_o rational_a line_n make_v the_o breadth_n the_o line_n c._n in_o like_a sort_n also_o shall_v it_o so_o follow_v if_o a_o man_n proceed_v infinite_o wherefore_o it_o be_v manifest_a that_o of_o a_o medial_a line_n be_v produce_v infinite_a irrational_a line_n of_o which_o none_o be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o which_o be_v require_v to_o be_v prove_v an_o other_o demonstration_n demonstration_n suppose_v that_o ac_fw-la be_v a_o medial_a line_n then_o i_o say_v that_o of_o the_o line_n ac_fw-la may_v be_v produce_v infinite_a irrational_a line_n of_o which_o none_o shall_v be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o irrational_a line_n before_o name_v unto_o the_o line_n ac_fw-la and_o from_o the_o point_n a_o
of_o these_o triangle_n ekf_a flg_n gmh_o and_o hne_n be_v great_a than_o the_o half_a of_o the_o segment_n of_o the_o circle_n which_o be_v describe_v about_o it_o now_o then_o devide_v the_o circumference_n remain_v into_o two_o equal_a part_n and_o draw_v right_a line_n from_o the_o point_n where_o those_o division_n be_v make_v &_o so_o continual_o do_v this_o we_o shall_v at_o the_o length_n by_o the_o 1._o of_o the_o ten_o leave_v certain_a segment_n of_o the_o circle_n which_o shall_v be_v less_o than_o the_o excess_n whereby_o the_o circle_n efgh_o exceed_v the_o superficies_n s._n for_o it_o have_v be_v prove_v in_o the_o first_o proposition_n of_o the_o ten_o book_n that_o 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 likewise_o again_o from_o the_o residue_n more_o than_o the_o halâ—e_n and_o so_o continual_o there_o shall_v at_o the_o length_n be_v leave_v a_o certain_a magnitude_n which_o shall_v be_v less_o than_o the_o less_o magnitude_n give_v let_v there_o be_v such_o segment_n leave_v &_o let_v the_o segment_n of_o the_o circle_n efgh_o namely_o which_o be_v make_v by_o the_o line_n eke_o kf_n fl_fw-mi lg_n gm_n mh_o hn_o and_o ne_fw-fr be_v less_o than_o the_o excess_n whereby_o the_o circle_n efgh_o exceed_v the_o superficies_n s._n wherefore_o the_o residue_n namely_o the_o poligonon_n figure_n ekflgmhn_n be_v great_a than_o the_o superficies_n s._n inscribe_v in_o the_o circle_n abcd_o a_o poligonon_n figure_n like_o to_o the_o poligonon_n figure_n ekflgmhn_n and_o let_v the_o same_o be_v axbocpdr_n wherefore_o by_o the_o proposition_n next_o go_v before_o as_o the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o so_o be_v the_o poligonon_n figure_n axbocpdr_n to_o the_o poligonon_n figure_n ekflgmhn_n but_o as_o the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fg_o so_o be_v the_o circle_n abcd_o suppose_v to_o be_v to_o the_o superficies_n s._n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o circle_n abcd_o be_v to_o the_o superficies_n s_o so_o be_v the_o poligonon_n figure_n axbocpdr_n to_o the_o poligonon_n figure_n ekflgmhn_n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o circle_n abcd_o be_v to_o the_o poligonon_n figure_n describe_v in_o it_o so_o be_v the_o superficies_n s_o to_o the_o poligonon_n figure_n ekflgmhn_n but_o the_o circle_n abcd_o be_v great_a than_o the_o poligonon_n figure_n describe_v in_o it_o wherefore_o also_o the_o superficies_n saint_n be_v great_a than_o the_o poligonon_n figure_n ekflghmn_a but_o it_o be_v also_o less_o which_o be_v impossible_a wherefore_o as_o the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o so_o be_v not_o the_o circle_n abcd_o to_o any_o superficies_n less_o than_o the_o circle_n efgh_o case_n in_o like_a sort_n also_o may_v wprove_v that_o as_o the_o square_n of_o the_o line_n fh_o be_v to_o the_o square_n of_o the_o line_n bd_o so_o be_v not_o the_o circle_n efgh_o to_o any_o superficies_n less_o than_o the_o circle_n abcd._n i_o say_v namely_o that_o as_o the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o so_o be_v not_o the_o circle_n abcd_o to_o any_o superficies_n great_a they_o than_o the_o circle_n efgh_o for_o if_o it_o be_v possible_a let_v it_o be_v to_o a_o great_a namely_o to_o the_o superficies_n s._n wherefore_o by_o conversion_n as_o the_o square_n of_o the_o line_n fh_o be_v to_o the_o square_n of_o the_o line_n bd_o so_o be_v the_o superficies_n s_o to_o the_o circle_n abcd._n prove_v but_o as_o the_o superficies_n saint_n be_v to_o the_o circle_n abcd_o so_o be_v the_o circle_n efgh_o to_o some_o superficies_n lâ—sse_v they_o the_o circle_n abcd._n 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 fh_o be_v to_o the_o square_n of_o the_o line_n bd_o so_o be_v the_o circle_n efgh_o to_o some_o superficies_n less_o than_o the_o circle_n abcd_o which_o be_v in_o the_o first_o case_n prove_v to_o be_v impossible_a wherefore_o as_o the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o so_o be_v not_o the_o circle_n abcd_o to_o any_o superficies_n great_a than_o the_o circle_n efgh_o and_o it_o be_v also_o prove_v that_o it_o be_v not_o to_o any_o less_o wherefore_o as_o the_o square_n of_o the_o lâ—ne_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o so_o be_v the_o circle_n abcd_o to_o the_o circle_n efgh_o wherefore_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 which_o be_v require_v to_o be_v prove_v ¶_o a_o assumpt_n i_o say_v now_o that_o the_o superficies_n saint_n be_v great_a than_o the_o circle_n efgh_o as_o the_o superficies_n saint_n be_v to_o the_o circle_n abcd_o so_o be_v the_o circle_n efgh_o to_o some_o superficies_n less_o than_o the_o circle_n abcd._n for_o as_o the_o superficies_n saint_n be_v to_o the_o circle_n abcd_o so_o let_v the_o circle_n efgh_o be_v to_o the_o superficies_n t._n now_o i_o say_v that_o the_o superficies_n t_n be_v less_o than_o the_o circle_n abcd._n for_o for_o that_o as_o the_o superficies_n saint_n be_v to_o the_o circle_n abcd_o so_o be_v the_o circle_n efgh_o to_o the_o superficies_n t_o therefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o superficies_n saint_n be_v to_o the_o circle_n efgh_o so_o be_v the_o circle_n abcd_o to_o the_o superficies_n t._n but_o the_o superficies_n saint_n be_v great_a than_o the_o circle_n efgh_o by_o supposition_n wherefore_o also_o the_o circle_n abcd_o be_v great_a than_o the_o superficies_n t_n by_o the_o 14._o of_o the_o five_o wherefore_o as_o the_o superficies_n saint_n be_v to_o the_o circle_n abcd_o so_o be_v the_o circle_n efgh_o to_o some_o superficies_n less_o than_o the_o circle_n abcd_o which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o corollary_n add_v by_o flussas_n circle_n have_v the_o one_o to_o the_o other_o that_o proportion_n that_o like_o poligonon_n figure_n and_o in_o like_a sort_n describe_v in_o they_o have_v for_o it_o be_v by_o the_o first_o proposition_n prove_v that_o the_o poligonon_n figure_n have_v that_o proportion_n the_o one_o to_o the_o other_o that_o the_o square_n of_o the_o diameter_n have_v which_o proportion_n likewise_o by_o this_o proposition_n the_o circle_n have_v ¶_o very_o needful_a problem_n and_o corollarye_n by_o master_n john_n dee_n invent_v who_o wonderful_a use_n also_o be_v partly_o declare_v a_o problem_n 1._o two_o circle_n be_v give_v to_o find_v two_o right_a line_n which_o have_v the_o same_o proportion_n one_o to_o the_o other_o that_o the_o give_v circle_n have_v oâ—e_v to_o the_o otherâ—_n suppose_v a_o and_o b_o to_o be_v the_o diameter_n of_o two_o circle_n give_v i_o say_v that_o two_o right_a line_n be_v to_o be_v find_v have_v that_o proportion_n that_o the_o circle_n of_o a_o have_v to_o the_o circle_n of_o b._n let_v to_o a_o &_o b_o by_o the_o 11_o of_o the_o six_o a_o three_o proportional_a line_n be_v find_v which_o suppose_v to_o be_v c._n construction_n i_o say_v now_o that_o a_o have_v to_o c_o that_o proportion_n which_o the_o circle_n of_o a_o have_v to_o the_o circle_n of_o b._n for_o forasmuch_o as_o a_o b_o and_o c_o be_v by_o construction_n three_o proportional_a line_n demonstration_n the_o square_a of_o a_o be_v to_o the_o square_n of_o b_o as_z a_o be_v to_o c_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o â—_o but_o as_o the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o so_o be_v the_o circle_n who_o diameter_n be_v the_o line_n a_o to_o the_o circle_n who_o diameter_n be_v the_o line_n b_o by_o this_o second_o of_o the_o eleven_o wherefore_o the_o circle_n of_o the_o lineâ—_z a_o and_o b_o be_v in_o the_o proportion_n of_o the_o right_a line_n a_o and_o c._n therefore_o two_o circle_n be_v give_v we_o have_v find_v two_o right_a line_n have_v the_o same_o proportion_n between_o they_o that_o the_o circle_n give_v have_v one_o to_o the_o other_o which_o ought_v to_o be_v do_v a_o problem_n 2._o two_o circle_n be_v give_v and_o a_o right_a line_n to_o find_v a_o other_o right_a line_n to_o which_o the_o line_n give_v shall_v have_v that_o proportion_n which_o the_o one_o circle_n have_v to_o the_o other_o suppose_v two_o circle_n give_v which_o let_v be_v a_o &_o b_o &_o a_o right_a line_n give_v which_o let_v be_v c_o i_o say_v that_o a_o other_o right_a line_n be_v to_o be_v â—ounde_v to_o which_o the_o line_n c_o shall_v have_v that_o proportion_n that_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
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
demonstration_n demonstration_n the_o circle_n so_o make_v or_o so_o consider_v in_o the_o sphere_n be_v call_v the_o great_a circle_n all_o other_o not_o have_v the_o centre_n of_o the_o sphere_n to_o be_v their_o centre_n alsoâ—_n be_v call_v less_o circle_n note_v these_o description_n description_n an_o other_o corollary_n corollary_n an_o other_o corollary_n construction_n construction_n this_o be_v also_o prove_v in_o the_o assumpt_n before_o add_v out_o oâ—_n flussas_n note_v what_o a_o great_a or_o great_a circle_n in_o a_o spear_n be_v first_o part_n of_o the_o construction_n noteâ—_n noteâ—_n you_o know_v full_a well_o that_o in_o the_o superficies_n of_o the_o sphere_n â—â—â—ly_o the_o circumference_n of_o the_o circle_n be_v but_o by_o thâ—se_a circumference_n the_o limitation_n and_o assign_v of_o circle_n be_v use_v and_o so_o the_o circumference_n of_o a_o circle_n usual_o call_v a_o circle_n which_o in_o this_o place_n can_v not_o offend_v this_o figure_n be_v restore_v by_o m._n dee_n his_o diligence_n for_o in_o the_o greek_a and_o latin_a euclides_n the_o line_n gl_n the_o line_n agnostus_n and_o the_o line_n kz_o in_o which_o three_o line_n the_o chief_a pinch_n of_o both_o the_o demonstration_n do_v stand_v be_v untrue_o draw_v as_o by_o compare_v the_o studious_a may_v perceive_v note_n you_o must_v imagine_v 〈◊〉_d right_a line_n axe_n to_o be_v perpendicular_a upon_o the_o diameter_n bd_o and_o ce_fw-fr though_o here_o ac_fw-la the_o semidiater_n seem_v to_o be_v part_n of_o ax._n and_o so_o in_o other_o point_n in_o this_o figure_n and_o many_o other_o strengthen_v your_o imagination_n according_a to_o the_o tenor_n of_o construction_n though_o in_o the_o delineation_n in_o plain_a sense_n be_v not_o satisfy_v note_n boy_n equal_a to_o bk_o in_o respect_n of_o m._n dee_n his_o demonstration_n follow_v follow_v note_v â—his_fw-la point_n z_o that_o you_o may_v the_o better_a understand_v m._n dee_n his_o demonstration_n second_o part_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n demonstration_n which_o of_o necessity_n shall_v fall_v upon_o z_o as_o m._n dee_n prove_v it_o and_o his_o proof_n be_v set_v after_o at_o this_o mark_n ✚_o follow_v i_o dee_n dee_n but_o az_o be_v great_a them_z agnostus_n as_o in_o the_o former_a proposition_n km_o be_v evident_a to_o be_v great_a than_o kg_v so_o may_v it_o also_o be_v make_v manifest_a that_o kz_o do_v neither_o touch_n nor_o cut_v the_o circle_n fgâ—h_n a_o other_o prove_v that_o the_o line_n ay_o be_v great_a they_o the_o line_n ag._n ag._n this_o as_o a_o assumpt_n be_v present_o prove_v two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o impossibility_n second_o case_n case_n as_o it_o be_v â—asiâ—_n to_o gather_v by_o the_o assumpt_n put_v after_o the_o secoâ—â—_n of_o this_o booâ—â—_n note_v a_o general_a rule_n the_o second_o part_n of_o the_o problem_n two_o way_n execute_v a_o upright_a cone_n the_o second_o part_n of_o the_o problem_n the_o second_o â—aâ—â—_n oâ—_n the_o problem_n ☜_o ☜_o this_o may_v easy_o be_v demonstrate_v as_o in_o thâ—_z 17._o proposition_n the_o section_n of_o a_o sphere_n be_v prove_v to_o be_v a_o circle_n circle_n for_o take_v away_o all_o doubt_n this_o aâ—_z a_o lemma_n afterward_o be_v demonstrate_v a_o lemma_n as_o it_o be_v present_o demonstrate_v construction_n demonstration_n the_o second_o part_n of_o the_o problem_n *_o construction_n demonstration_n an_o other_o way_n of_o execute_v this_o problem_n the_o converse_n of_o the_o assumpt_n a_o great_a error_n common_o maintain_v between_o straight_o and_o crooked_a all_o manner_n of_o proportion_n may_v be_v give_v construction_n demonstration_n the_o definition_n of_o a_o circled_a â—â—apâ—â—d_n in_o a_o spâ—erâ—_n construction_n demonstration_n this_o be_v manifest_a if_o you_o consider_v the_o two_o triangle_n rectangle_v hom_o and_o hon_n and_o then_o with_o all_o use_v the_o 47._o of_o the_o first_o of_o euclid_n construction_n demonstration_n construction_n demonstration_n this_o in_o manner_n of_o a_o lemmâ—_n be_v present_o prove_v note_v here_o of_o axe_n base_a &_o solidity_n more_o than_o i_o need_n to_o bring_v any_o far_a proof_n for_o note_n note_n i_o say_v half_a a_o circular_a revolution_n for_o that_o suffice_v in_o the_o whole_a diameter_n n-ab_n to_o describe_v a_o circle_n by_o iâ—_z it_o be_v move_v â—â—out_v his_o centre_n q_o etc_n etc_n lib_fw-la 2_o prop_n 2._o de_fw-fr spheâ—a_n &_o cylindrâ—_n note_n note_n a_o rectangle_n parallelipipedon_n give_v equal_a to_o a_o sphere_n give_v to_o a_o sphere_n or_o to_o any_o part_n of_o a_o sphere_n assign_v as_o a_o three_o four_o five_o etc_o etc_o to_o geve_v a_o parallelipipedon_n equal_a side_v colume_n pyramid_n and_o prism_n to_o be_v give_v equal_a to_o a_o sphere_n or_o to_o any_o certain_a part_n thereof_o to_o a_o sphere_n or_o any_o segment_n or_o sector_n of_o the_o same_o to_o geve_v a_o cone_n or_o cylinder_n equal_a or_o in_o any_o proportion_n assign_v far_o use_v of_o spherical_a geometry_n the_o argument_n of_o the_o thirteen_o book_n construction_n demonstration_n demonstration_n the_o assumpt_v prove_v prove_v because_o ac_fw-la be_v suppose_v great_a than_o ad_fw-la therefore_o his_o residue_n be_v less_o than_o the_o residue_n of_o ad_fw-la by_o the_o common_a sentence_n wherefore_o by_o the_o supposition_n db_o be_v great_a than_o â—c_z the_o chieâ—e_a line_n in_o all_o euclides_n geometry_n what_o be_v mean_v here_o by_o a_o section_n in_o one_o only_a poiâ—t_n construction_n demonstration_n demonstration_n note_v how_o ce_fw-fr and_o the_o gnonom_n xop_n be_v prove_v equal_a for_o it_o serve_v in_o the_o converse_n demonstrate_v by_o m._n dee_n here_o next_o after_o this_o proposition_n â—the_v converse_v of_o the_o former_a former_a as_o we_o haâ—e_z note_v the_o place_n of_o the_o peculiar_a proâ—e_n there_o â—in_a the_o demonstration_n of_o the_o 3._o 3._o therefore_o by_o my_o second_o theorem_a add_v upon_o the_o second_o proposition_n dc_o be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n a._n and_o because_o ac_fw-la be_v big_a than_o cb_o therefore_o dam_n be_v great_a than_o ac_fw-la wherefore_o if_o a_o right_a line_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v demonstrate_v therefore_o by_o my_o second_o theorem_a add_v upon_o the_o second_o proposition_n dc_o be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n a._n and_o because_o ac_fw-la be_v big_a than_o cb_o therefore_o dam_n be_v great_a than_o ac_fw-la wherefore_o if_o a_o right_a line_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v construction_n construction_n though_o i_o say_v perpendicular_a yes_o you_o may_v perceive_v how_o infinite_a other_o position_n will_v serve_v so_o that_o diego_n and_o ad_fw-la make_v a_o angle_n for_o a_o triangle_n to_o have_v his_o side_n proportional_o cut_v etc_n etc_n demonstration_n demonstration_n i_o dee_n this_o be_v most_o evident_a of_o my_o second_o theorem_a add_v to_o the_o three_o proposition_n for_o to_o add_v to_o a_o whole_a line_n a_o line_n equal_a to_o the_o great_a segment_n &_o to_o add_v to_o the_o great_a segment_n a_o line_n equal_a to_o the_o whole_a line_n be_v all_o one_o thing_n in_o the_o line_n produce_v by_o the_o whole_a line_n i_o mean_v the_o line_n divide_v by_o extreme_a and_o mean_a proportion_n this_o be_v before_o demonstrate_v most_o evident_o and_o brief_o by_o m._n dee_n after_o the_o 3._o proposition_n note_n note_v 4._o proportional_a line_n note_v two_o middle_a proportional_n note_v 4._o way_n of_o progression_n in_o the_o proportion_n of_o a_o line_n divide_v by_o extreme_a and_o middle_a proportion_n what_o resolution_n and_o composition_n be_v have_v before_o be_v teach_v in_o the_o begin_n of_o the_o first_o book_n book_n proclus_n in_o the_o greek_a in_o the_o 58._o page_n construction_n demonstration_n two_o case_n in_o this_o proposition_n construction_n thâ—_z first_o case_n demonstration_n the_o second_o case_n construction_n demonstration_n construction_n demonstration_n this_o corollary_n be_v the_o 3._o proposition_n of_o theâ—4_n â—4_o book_n after_o campane_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n construction_n demonstration_n construstion_n demonstration_n constrâ—yction_n demonstration_n this_o corollary_n be_v the_o 11._o proposition_n of_o the_o 14._o book_n after_o campane_n this_o corollary_n be_v the_o 3._o corollary_n after_o the_o 17._o proposition_n of_o the_o 14_o book_n after_o campane_n campane_n by_o the_o name_n oâ—_n a_o pyramid_n both_o here_o &_o iâ—_z this_o book_n follow_v understand_v a_o tetrahedron_n an_o other_o construction_n and_o demonstration_n of_o the_o second_o part_n after_o fâ—ussas_n three_o part_n of_o the_o demonstration_n this_o corollary_n be_v the_o 15._o proposition_n of_o the_o 14._o book_n after_o campane_n this_o corollary_n campane_n put_v as_o a_o corollary_n after_o
uno_fw-la vnum_fw-la quodque_fw-la idea_n est_fw-la quia_fw-la unum_fw-la numero_fw-la est_fw-la that_o be_v every_o thing_n therefore_o be_v that_o be_v therefore_o have_v his_o be_v in_o nature_n and_o be_v that_o it_o be_v for_o that_o it_o be_v on_o in_o number_n according_a whereunto_o jordane_n in_o that_o most_o excellent_a and_o absolute_a work_n of_o arithmetic_n which_o he_o write_v define_v unity_n after_o this_o manner_n vnitas_fw-la est_fw-la res_fw-la per_fw-la se_fw-la discretio_fw-la that_o be_v unity_n be_v proper_o and_o of_o itself_o the_o difference_n of_o any_o thing_n that_o be_v unity_n unity_n be_v that_o whereby_o every_o thing_n do_v proper_o and_o essential_o differ_v and_o be_v a_o other_o thing_n from_o all_o other_o certain_o a_o very_a apt_a definition_n and_o it_o make_v plain_a the_o definition_n here_o set_v of_o euclid_n definition_n 2_o number_n be_v a_o multitude_n compose_v of_o unity_n as_o the_o number_n of_o three_o be_v a_o multitude_n compose_v and_o make_v of_o three_o unity_n likewise_o the_o number_n of_o five_o be_v nothing_o else_o but_o the_o composition_n &_o put_v together_o of_o five_o unity_n although_o as_o be_v before_o say_v between_o a_o point_n in_o magnitude_n and_o unity_n in_o multitude_n there_o be_v great_a agreement_n and_o many_o thiâ—â—â—â—_n be_v common_a to_o they_o both_o for_o as_o a_o point_n be_v the_o begin_n of_o magnitude_n so_o be_v unity_n the_o beginning_n of_o number_n and_o as_o a_o point_n in_o magnitude_n be_v indivisible_a so_o be_v also_o unity_n in_o number_n indivisible_a yet_o in_o this_o they_o differ_v and_o disagree_v unity_n there_o be_v no_o line_n or_o magnitude_n make_v of_o point_n as_o of_o his_o part_n so_o that_o although_o a_o point_n be_v the_o beginning_n of_o a_o line_n yet_o be_v it_o no_o part_n thereof_o but_o unity_n as_o it_o be_v the_o begin_n of_o number_n so_o be_v it_o also_o a_o part_n thereof_o which_o be_v somewhat_o more_o manifest_o set_v of_o boetius_fw-la in_o a_o other_o definition_n of_o number_n which_o he_o give_v in_o his_o arithmetic_n boetius_fw-la which_o be_v thus_o numerus_fw-la est_fw-la quantitatâ—_fw-la acernus_fw-la ex_fw-la unitatibus_fw-la profusus_fw-la that_o be_v number_n be_v a_o mass_n or_o heap_n of_o quantity_n produce_v of_o unity_n which_o definition_n in_o substance_n be_v all_o one_o with_o the_o first_o wherein_o be_v say_v most_o plain_o that_o the_o heap_n or_o mass_n number_n that_o be_v the_o whole_a substance_n of_o the_o quantity_n of_o number_n be_v produce_v &_o make_v of_o unity_n so_o that_o unity_n be_v as_o it_o be_v the_o very_a matter_n of_o number_n as_o four_o unity_n add_v together_o be_v the_o matter_n whereof_o the_o number_n 4._o be_v make_v &_o each_o of_o these_o unity_n be_v a_o part_n of_o the_o number_n four_o namely_o a_o four_o part_n or_o a_o quarter_n unto_o this_o definition_n agree_v also_o the_o definition_n give_v of_o jordane_n jordane_n which_o be_v thus_o number_n be_v a_o quantity_n which_o gather_v together_o thing_n sever_v a_o sunder_o number_n as_o five_o man_n be_v in_o themselves_o sever_v and_o distinct_a be_v by_o the_o number_n five_o bring_v together_o as_o it_o be_v into_o one_o mass_n and_o so_o of_o other_o and_o although_o unity_n be_v no_o number_n yet_o it_o contain_v in_o it_o the_o virtue_n and_o power_n of_o all_o number_n and_o be_v set_v and_o take_v for_o they_o number_n in_o this_o place_n for_o the_o far_o elucidation_n of_o thing_n partly_o before_o set_v and_o chief_o hereafter_o to_o be_v set_v because_o euclid_n here_o do_v make_v mention_n of_o diverse_a kind_n of_o number_n and_o also_o define_v the_o same_o be_v to_o be_v note_v that_o number_n may_v be_v consider_v three_o manner_n of_o way_n first_o number_n may_v be_v consider_v absolute_o without_o compare_v it_o to_o any_o other_o number_n wayâ—_n or_o without_o apply_v it_o to_o any_o other_o thing_n only_o view_v â—nd_n payse_v what_o it_o be_v in_o itself_o and_o in_o his_o own_o nature_n only_o and_o what_o part_n it_o have_v and_o what_o propriety_n and_o passion_n as_o this_o number_n six_o may_v be_v consider_v absolute_o in_o his_o own_o nature_n that_o it_o be_v a_o even_a number_n and_o that_o it_o be_v a_o perfect_a number_n and_o have_v many_o more_o condition_n and_o propriety_n and_o so_o conceive_v you_o of_o all_o other_o number_n whatsoever_o of_o 9_o 12._o and_o so_o forth_o an_o other_o way_n number_n may_v be_v consider_v by_o way_n of_o comparison_n and_o in_o respect_n of_o some_o other_o number_n either_o as_o equal_a to_o itself_o or_o as_o great_a they_o itself_o or_o as_o less_o they_o itself_o as_o 12._o may_v be_v consider_v as_o compare_v to_o 12._o which_o be_v equal_a unto_o it_o or_o as_o to_o 24._o which_o be_v great_a than_o it_o for_o 12_o be_v the_o half_a thereof_o of_o as_o to_o 6._o which_o be_v less_o than_o it_o as_o be_v the_o double_a thereof_o and_o of_o this_o consideration_n of_o number_n arise_v and_o spring_v all_o kind_n and_o variety_n of_o proportion_n as_o have_v before_o be_v declare_v in_o the_o explanation_n of_o the_o principle_n of_o the_o five_o book_n so_o that_o of_o that_o matter_n it_o be_v needless_a any_o more_o to_o be_v say_v in_o this_o place_n thus_o much_o of_o this_o for_o the_o declaration_n of_o the_o thing_n follow_v 3_o a_o part_n be_v a_o less_o number_n in_o comparison_n to_o the_o great_a when_o the_o less_o measure_v the_o great_a definition_n as_o the_o number_n 3_o compare_v to_o the_o number_n 12._o be_v a_o part_n for_o 3_o be_v a_o less_o number_n than_o be_v 12._o and_o moreover_o it_o measure_v 12_o the_o great_a number_n for_o 3_o take_v or_o add_v to_o itself_o certain_a time_n namely_o 4_o time_n make_v 12._o for_o 3_o four_o time_n be_v 12._o likewise_o be_v â—_o a_o part_n of_o 8_o 2_o be_n less_o than_o 8_o and_o take_v 4_o time_n it_o make_v 8._o for_o the_o better_a understanding_n of_o this_o definition_n and_o how_o this_o word_n parte_fw-la be_v diverse_o take_v in_o arithmetic_n and_o in_o geometry_n read_v the_o declaration_n of_o the_o first_o definition_n of_o the_o 5._o book_n 4_o part_n be_v a_o less_o number_n in_o respect_n of_o the_o great_a when_o the_o less_o measure_v not_o the_o great_a definition_n as_o the_o number_n 3_o compare_v to_o 5_o be_v part_n of_o 5_o and_o not_o a_o part_n for_o the_o number_n 3_o be_v less_o than_o the_o number_n 5_o and_o do_v not_o measure_v 5._o for_o take_v once_o it_o make_v but_o 3._o once_o 3_o be_v 3_o which_o be_v less_o than_o 5._o and_o 3_o take_v twice_o make_v 6_o which_o be_v more_o than_o 5._o wherefore_o it_o be_v no_o part_n of_o 5_o but_o part_n namely_o three_o five_o part_n of_o 5._o for_o in_o the_o number_n 3_o there_o be_v 3_o unity_n and_o every_o unity_n be_v the_o five_o part_n of_o 5._o wherefore_o 3_o be_v three_o five_o part_n of_o 5_o and_o so_o of_o other_o 5_o multiplex_n be_v a_o great_a number_n in_o comparison_n of_o the_o less_o when_o the_o less_o measure_v the_o great_a definition_n as_o 9_o compare_v to_o 3_o be_v multiplex_n the_o number_n 9_o be_v great_a than_o the_o number_n 3_n and_o moreover_o 3_o the_o less_o number_n measure_v 9_o the_o great_a number_n for_o 3_o take_v certain_a time_n namely_o 3_o time_n make_v 9_o three_o time_n three_o be_v 9_o for_o the_o more_o ample_a and_o full_a knowledge_n of_o this_o definition_n read_v what_o be_v say_v in_o the_o explanation_n of_o the_o second_o definition_n of_o the_o 5_o book_n where_o multiplex_n be_v sufficient_o entreat_v of_o with_o all_o his_o kind_n 6_o a_o even_a number_n be_v that_o which_o may_v be_v divide_v into_o two_o equal_a part_n definition_n as_o the_o number_n 6_o may_v be_v divide_v into_o 3_o and_o 3_o which_o be_v his_o part_n and_o they_o be_v equal_a the_o one_o not_o exceed_v the_o other_o this_o definition_n of_o euclid_n be_v to_o be_v understand_v of_o two_o such_o equal_a part_n which_o join_v together_o make_v the_o whole_a number_n as_o 3_o and_o 3_o the_o equal_a part_n of_o 6_o join_v together_o make_v 6_o for_o otherwise_o many_o number_n both_o even_o and_o odd_a may_v be_v divide_v into_o many_o equal_a part_n as_o into_o 4._o 5._o 6â—_o or_o more_o and_o therefore_o into_o 2._o as_o 9_o may_v be_v divide_v into_o 3_o and_o 3_o which_o be_v his_o part_n and_o be_v also_o equal_a for_o the_o one_o of_o they_o exceed_v not_o the_o other_o yet_o be_v not_o therefore_o this_o number_n â—_o a_o even_o number_n for_o that_o 3_o and_o 3_o these_o equal_a part_n of_o 9_o add_v together_o make_v not_o 9_o but_o only_o 6._o likewise_o take_v the_o definition_n so_o general_o every_o number_n whatsoever_o shall_v be_v a_o even_a numberâ—_n for_o
in_o the_o point_n e._n and_o unto_o the_o line_n cg_o put_v the_o line_n cl_n equal_a now_o forasmuch_o as_o the_o line_n agnostus_n and_o gc_o be_v the_o great_a segâ—â—â—tes_n of_o half_a the_o line_n ab_fw-la for_o â—che_fw-mi of_o they_o be_v the_o half_a of_o the_o great_a segment_n of_o the_o whole_a line_n ab_fw-la the_o line_n ebb_v and_o ec_o shall_v be_v the_o less_o segment_n of_o half_a the_o line_n ab_fw-la wherefore_o the_o whole_a line_n câ—_n be_v the_o great_a segment_n and_o the_o line_n ce_fw-fr be_v the_o less_o segment_n but_o as_o the_o line_n cl_n be_v to_o the_o line_n ce_fw-fr so_o be_v the_o line_n ce_fw-fr to_o the_o residue_n el._n wherefore_o the_o line_n el_n be_v the_o great_a segment_n of_o the_o line_n ce_fw-fr or_o of_o the_o line_n ebb_v which_o be_v equal_a unto_o it_o wherefore_o the_o residue_n lb_n be_v the_o less_o segment_n of_o the_o same_o ebb_n which_o be_v the_o lesâ—e_n segment_n of_o halfa_n the_o side_n of_o the_o cube_fw-la but_o the_o line_n agnostus_n gc_o and_o cl_n be_v three_o great_a segment_n of_o the_o half_a of_o the_o whole_a line_n ab_fw-la which_o three_o great_a segment_n make_v the_o altitude_n of_o the_o foresay_a solid_a wherefore_o the_o altitude_n of_o the_o say_v solid_a want_v of_o ab_fw-la the_o side_n of_o the_o cube_fw-la by_o the_o line_n lb_n which_o be_v the_o less_o segment_n of_o the_o line_n be._n which_o line_n be_v again_o be_v the_o less_o segment_n of_o half_a the_o side_n ab_fw-la of_o the_o cube_fw-la wherefore_o the_o foresay_a solid_a consist_v of_o the_o six_o solid_n whereby_o the_o dodecahedron_n exceed_v the_o cube_fw-la inscribe_v in_o it_o be_v set_v upon_o a_o base_a which_o want_v of_o the_o base_a of_o the_o cube_fw-la by_o a_o three_o part_n of_o the_o less_o segment_n and_o be_v under_o a_o altitude_n want_v of_o the_o side_n of_o the_o cube_fw-la by_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o side_n of_o the_o cube_fw-la the_o solid_a therefore_o of_o a_o dodecahedron_n exceed_v the_o solid_a of_o a_o cube_fw-la inscribe_v in_o it_o by_o a_o parallelipipedon_n who_o base_a want_v of_o the_o base_a of_o the_o cube_fw-la by_o a_o three_o part_n of_o the_o less_o segment_n and_o who_o altitude_n want_v of_o the_o altitude_n of_o the_o cube_fw-la by_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o side_n of_o the_o cube_fw-la ¶_o a_o corollary_n a_o dodecahedron_n be_v double_a to_o a_o cube_n inscribe_v in_o it_o take_v away_o the_o three_o part_n of_o the_o less_o segment_n of_o the_o cube_fw-la and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a of_o that_o excess_n for_o if_o there_o be_v give_v a_o cube_fw-la from_o which_o be_v cut_v of_o a_o solid_a set_v upon_o a_o three_o part_n of_o the_o less_o segment_n of_o the_o base_a and_o under_o one_o and_o the_o same_o altitude_n with_o the_o cube_fw-la that_o solid_a take_v away_o have_v to_o the_o whole_a solid_a the_o proportion_n of_o the_o section_n of_o the_o base_a to_o the_o base_a by_o the_o 32._o of_o the_o eleven_o wherefore_o from_o the_o cube_fw-la be_v take_v away_o a_o three_o â—art_n of_o the_o less_o segment_n far_o forasmuch_o as_o the_o residue_n want_v of_o the_o altitude_n of_o the_o cube_fw-la by_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o altitude_n or_o side_n and_o that_o residue_n be_v a_o parallelipipedon_n if_o it_o be_v cut_v by_o a_o plain_a superficies_n parallel_v to_o the_o opposite_a plain_a superficiece_n cut_v the_o altitude_n of_o the_o cube_fw-la by_o a_o point_n it_o shall_v take_v away_o from_o that_o parallelipipedon_n a_o solid_a have_v to_o the_o whole_a the_o proportion_n of_o the_o section_n to_o the_o altitude_n by_o the_o 3._o corollary_n of_o the_o 25._o of_o the_o eleven_o wherefore_o the_o excess_n want_v of_o the_o same_o cube_fw-la by_o the_o thiâ—d_o part_n of_o the_o less_o segment_n and_o moreover_o by_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a of_o that_o excess_n ¶_o the_o 34._o proposition_n the_o proportion_n of_o the_o solid_a of_o a_o dodecahedron_n to_o the_o solid_a of_o a_o icosahedron_n inscribe_v in_o it_o consist_v of_o the_o proportion_n triple_a of_o the_o diameter_n to_o that_o line_n which_o couple_v the_o opposite_a base_n of_o the_o dodecahedron_n and_o of_o the_o proportion_n of_o the_o side_n of_o the_o cube_n to_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n svppose_v that_o ahbck_n be_v a_o dodecahedronâ—_n who_o diameter_n let_v be_v ab_fw-la and_o let_v the_o line_n which_o couple_v the_o centre_n of_o the_o opposite_a base_n be_v khâ—_n and_o let_v the_o icosahedron_n inscribe_v in_o the_o dodecahedron_n abc_n be_v do_v you_o who_o diameter_n let_v be_v de._n now_o forasmuch_o aâ—_z oâ—e_n and_o the_o self_n same_o circle_n contain_v the_o pentagon_n of_o a_o dodecahedron_n &_o the_o triangle_n of_o a_o icosahedroâ—_n describe_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o 14._o of_o the_o fourteen_o let_v that_o circle_n be_v igo._n wherefore_o io_o be_v the_o side_n of_o the_o cube_fw-la and_o ig_v the_o side_n of_o the_o icosahedron_n by_o the_o same_o then_o i_o say_v that_o the_o proportion_n of_o the_o dodecahedron_n ahbck_n to_o the_o icosahedron_n def_n inscribe_v in_o it_o consist_v of_o the_o proportion_n triple_a of_o the_o line_n ab_fw-la to_o the_o line_n kh_o and_o of_o the_o proportion_n of_o the_o line_n io_o to_o the_o line_n ig_n for_o forasmuch_o as_o the_o icosahedron_n def_n be_v inscribe_v in_o the_o dodecahedron_n abc_n demonstration_n by_o supposition_n the_o diameter_n de_fw-fr shall_v be_v equal_a to_o the_o line_n kh_o by_o the_o 7._o of_o the_o fifteen_o wherefore_o the_o dodecahedron_n set_v upon_o the_o diameter_n kh_o shall_v be_v inscribe_v in_o the_o same_o sphere_n wherein_o the_o icosahedron_n def_n be_v inscribe_v but_o the_o dodecahedron_n ahbck_n be_v to_o the_o dodecahedron_n upon_o the_o diameter_n kh_o in_o triple_a proportion_n of_o that_o in_o which_o the_o diameter_n ab_fw-la be_v to_o the_o diameter_n kh_o by_o the_o corollary_n of_o the_o 17._o of_o the_o twelve_o and_o the_o same_o dodecahedron_n which_o be_v set_v upon_o the_o diameter_n kh_o have_v to_o the_o icosahedron_n def_n which_o be_v set_v upon_o the_o same_o diameter_n or_o upon_o a_o diameter_n equal_a unto_o it_o namely_o de_fw-fr that_o proportion_n which_o io_o the_o side_n of_o the_o cube_fw-la have_v toâ—_n ig_n the_o side_n of_o the_o icosahedron_n inscribe_v in_o one_o &_o the_o self_n same_o sphere_n by_o the_o 8_o of_o the_o fourteen_o wherefore_o the_o proportion_n of_o the_o dodecahedron_n ahbck_n to_o the_o icosahedron_n def_n inscribe_v in_o it_o consist_v of_o the_o proportion_n triple_a of_o the_o diameter_n ab_fw-la to_o the_o line_n kh_o which_o couple_v the_o centre_n of_o the_o opposite_a base_n of_o the_o dodecahedron_n which_o proportion_n be_v that_o which_o the_o dodecahedron_n ahbck_n have_v to_o the_o dodecahedron_n set_v upon_o the_o diameter_n kh_o and_o of_o the_o proportion_n of_o io_o the_o side_n of_o the_o cube_fw-la to_o ig_o the_o side_n of_o the_o icosahedron_n which_o be_v the_o proportion_n of_o the_o dodecahedron_n set_v upon_o the_o diameter_n kh_o to_o the_o icosahedron_n def_n describe_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o 5._o definition_n of_o the_o six_o the_o proportion_n therefore_o of_o the_o solid_a of_o a_o dodecahedron_n to_o the_o solid_a of_o a_o icosahedron_n inscribe_v in_o it_o consi_v of_o the_o proportion_n triple_a of_o the_o diameter_n to_o that_o line_n which_o couple_v the_o opposite_a base_n of_o the_o dodecahedron_n and_o of_o the_o proportion_n of_o the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n the_o 35._o proposition_n the_o solid_a 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 extreme_a 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 it_o have_v be_v prove_v that_o the_o dodecahedron_n together_o with_o the_o cube_fw-la inscribe_v in_o it_o be_v contain_v in_o one_o and_o the_o self_n same_o pyramid_n by_o the_o corollary_n of_o the_o first_o of_o this_o book_n and_o by_o the_o corollary_n of_o the_o 33._o of_o this_o book_n it_o be_v manifest_a that_o the_o dodecahedron_n be_v double_a to_o the_o same_o cube_fw-la take_v away_o the_o three_o part_n 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