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
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
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
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
there_o be_v two_o proportional_a number_n but_o how_o many_o number_n fall_v in_o continual_a proportion_n between_o c_o and_o d_o so_o many_o by_o the_o 8._o of_o the_o eight_o fall_n there_o between_o the_o number_n that_o have_v the_o same_o proportion_n with_o they_o wherefore_o between_o a_o and_o b_o there_o be_v two_o mean_v proportional_a number_n which_o let_v be_v e_o and_o f._n and_o forasmuch_o as_o there_o be_v four_o number_n in_o continual_a proportion_n namely_o a_o e_o f_o b_o and_o a_o be_v a_o cube_fw-la number_n therefore_o by_o the_o 2d_o of_o the_o eight_o b_o also_o be_v a_o cube_fw-la number_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o flussates_n flussates_n between_o a_o square_a number_n and_o a_o number_n that_o be_v not_o a_o square_a number_n fall_v not_o the_o proportion_n of_o one_o square_a number_n to_o a_o other_o for_o if_o the_o first_o be_v a_o square_a number_n the_o second_o also_o shall_v be_v a_o square_a number_n which_o be_v contrary_a to_o the_o supposition_n likewise_o between_o a_o cube_fw-la number_n and_o a_o number_n that_o be_v no_o cube_fw-la number_n fall_v not_o the_o proportion_n of_o one_o cube_fw-la number_n to_o a_o other_o for_o if_o the_o first_o be_v a_o cube_fw-la number_n the_o second_o also_o shall_v be_v a_o cube_fw-la number_n which_o be_v contrary_a to_o the_o supposition_n &_o therefore_o impossible_a ¶_o the_o 24._o theorem_a the_o 26._o proposition_n like_o plain_a number_n be_v in_o the_o same_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n be_v to_o a_o square_a number_n svppose_v that_o a_o and_o b_o be_v like_o plain_a number_n then_o i_o say_v that_o a_o be_v unto_o b_o in_o the_o same_o proportion_n that_o a_o square_a number_n be_v to_o a_o square_a number_n for_o forasmuch_o as_o a_o b_o be_v like_o plain_a number_n construction_n therefore_o between_o a_o and_o b_o there_o fall_v one_o mean_a proportional_a number_n by_o the_o 18._o of_o the_o eight_o let_v there_o fall_v such_o a_o number_n and_o let_v the_o same_o be_v c._n and_o by_o the_o 35._o of_o the_o seven_o take_v the_o three_o lest_o number_n that_o have_v one_o and_o the_o same_o proportion_n with_o a_o c_o b_o and_o let_v the_o same_o be_v d_o e_o f_o wherefore_o by_o the_o corollary_n of_o the_o 2d_o â—_z of_o the_o eight_o their_o 〈◊〉_d that_o be_v d_o f_o be_v square_a number_n and_o for_o that_o as_z d_z be_v to_o fletcher_n so_o be_v a_o to_o b_o by_o the_o 14._o of_o the_o seven_o and_o d_z f_o be_v square_a number_n therefore_o a_o be_v unto_o b_o in_o that_o proportion_n that_o a_o square_a number_n be_v unto_o a_o square_a numâ—er_n which_o be_v require_v to_o be_v prove_v the_o 25._o theorem_a the_o 27._o proposition_n like_o solid_a number_n be_v in_o the_o same_o proportion_n the_o one_o to_o the_o other_o that_o a_o cube_fw-la number_n be_v to_o a_o cube_fw-la number_n svppose_v that_o a_o a_o and_o b_o be_v like_o solid_a number_n then_o i_o say_v that_o a_o be_v unto_o b_o in_o the_o same_o proportion_n that_o a_o cube_fw-la numb_a be_v to_o to_o a_o cube_fw-la number_n for_o forasmuch_o as_o a_o b_o be_v like_o solid_a number_n construction_n therefore_o by_o the_o 19_o of_o the_o eight_o between_o a_o and_o b_o there_o fall_v two_o mean_v proportional_a number_n let_v there_o fall_v two_o such_o number_n and_o let_v the_o same_o be_v c_o and_o d._n and_o take_v by_o the_o 35._o of_o the_o seven_o the_o least_o number_n that_o have_v one_o and_o the_o same_o proportion_n with_o a_o c_o d_o b_o demonstration_n and_o equal_a also_o with_o they_o in_o multitude_n and_o let_v the_o same_o be_v e_o f_o g_o h._n wherefore_o by_o the_o corollary_n of_o the_o 2._o of_o the_o eight_o their_o extreme_n that_o be_v eh_o be_v cube_fw-la number_n but_o as_o e_o be_v to_o h_o so_o be_v a_o to_o b._n wherefore_o a_o be_v unto_o b_o in_o the_o same_o proportion_n that_o a_o cube_fw-la number_n be_v to_o a_o cube_fw-la number_n which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o corollary_n add_v by_o flussates_n if_o two_o nnmber_n be_v in_o the_o same_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n be_v to_o a_o square_a number_n those_o two_o number_n shall_v be_v like_o superficial_a number_n flussates_n and_o if_o they_o be_v in_o the_o same_o proportion_n the_o one_o to_o the_o other_o that_o a_o cube_fw-la number_n be_v to_o a_o cube_fw-la number_n they_o shall_v be_v like_o solid_a number_n first_o let_v the_o number_v a_o have_v unto_o the_o number_n b_o the_o same_o proportion_n that_o the_o square_a number_n c_o have_v to_o the_o square_a number_n d._n then_o i_o say_v that_o a_o and_o b_o be_v like_o superficial_a number_n for_o forasmuch_o as_o between_o the_o square_a number_n c_o and_o d_o there_o fall_v a_o mean_a proportional_a by_o the_o 11._o of_o this_o book_n there_o shall_v also_o between_o a_o and_o b_o which_o have_v one_o and_o the_o same_o proportion_n with_o c_o and_o d_o fall_v a_o mean_a proportional_a by_o the_o 8._o of_o this_o book_n wherefore_o a_o and_o b_o be_v like_o superficial_a number_n by_o the_o 20._o of_o this_o book_n but_o if_o a_o be_v unto_o b_o as_z the_o cube_fw-la number_n c_z be_v to_o the_o cube_fw-la number_n d._n then_o be_v a_o &_o b_o like_v solid_a number_n for_o forasmuch_o as_o c_z and_o d_z be_v cube_fw-la number_n there_o fall_v between_o they_o too_o mean_v proportional_a number_n by_o the_o 12._o of_o this_o book_n and_o therefore_o by_o the_o 8._o of_o the_o same_o between_o a_o and_o b_o which_o be_v in_o the_o same_o proportion_n that_o c_z be_v to_o d_z there_o fall_v also_o two_o mean_a proportional_a number_n wherefore_o by_o the_o 21._o of_o this_o book_n a_o and_o b_o be_v like_o solid_a number_n an_o other_o corollary_n add_v also_o by_o flussates_n if_o a_o number_n multiply_v a_o square_a number_n produce_v not_o a_o square_a number_n the_o say_a number_n multiply_v shall_v bâ—_z no_o square_a number_n flussates_n for_o if_o it_o shall_v be_v a_o square_a number_n than_o shall_v it_o and_o the_o number_n multiply_v be_v like_o superficial_a number_n by_o reason_n they_o be_v square_a number_n have_v a_o mean_a proportional_a by_o the_o 18._o of_o this_o book_n and_o the_o number_n produce_v of_o the_o say_v mean_a shall_v be_v equal_a to_o the_o number_n contain_v under_o the_o extreme_n which_o be_v square_a number_n by_o the_o 20._o of_o the_o seven_o wherefore_o the_o number_n produce_v of_o the_o extreme_n be_v equal_a to_o the_o square_a number_n produce_v of_o the_o mean_a shall_v be_v a_o square_a number_n but_o the_o say_a number_n by_o supposition_n be_v no_o square_a number_n wherefore_o neither_o be_v the_o number_n multiply_v the_o square_a number_n a_o square_a number_n the_o first_o part_n of_o the_o first_o corollary_n be_v the_o converse_n of_o the_o 26._o proposition_n of_o this_o book_n and_o have_v some_o use_n in_o the_o ten_o book_n the_o second_o part_n of_o the_o same_o also_o be_v the_o converse_n of_o the_o 27._o proposition_n of_o the_o same_o the_o end_n of_o the_o eight_o book_n of_o euclides_n element_n ¶_o the_o nine_o book_n of_o euclides_n element_n in_o this_o nine_o book_n euclid_n continue_v his_o purpose_n touch_v number_n partly_o prosecute_a thing_n more_o full_o book_n which_o be_v before_o somewhat_o speak_v of_o as_o of_o square_a and_o cube_fw-la number_n and_o partly_o set_v out_o the_o nature_n and_o propriety_n of_o such_o kind_n of_o number_n as_o have_v not_o yet_o be_v entreat_v of_o which_o yet_o be_v most_o necessary_a to_o be_v know_v as_o be_v number_n even_o and_o odd_a who_o passion_n and_o condition_n be_v in_o this_o book_n large_o teach_v with_o their_o composition_n and_o subduction_n of_o the_o one_o from_o the_o other_o with_o many_o other_o general_a and_o special_a thing_n to_o be_v note_v worthy_a the_o knowledge_n ¶_o the_o 1._o theorem_a the_o 1._o proposition_n if_o two_o like_o plain_a number_n multiply_v the_o one_o the_o other_o produce_v any_o number_n the_o number_n of_o they_o produce_v shall_v be_v a_o square_a number_n svppose_v that_o a_o and_o b_o be_v two_o like_o plain_a number_n and_o let_v a_o multiply_v b_o produce_v the_o number_n c._n then_o i_o say_v that_z c_z be_v a_o square_a number_n for_o let_v a_o multiply_v himself_o produce_v d._n wherefore_o d_z be_v a_o square_a number_n demonstration_n and_o forasmuch_o as_o a_o multiply_v himself_o produce_v d_z and_o multiply_v b_o produce_v c_z therefore_o by_o the_o 17._o of_o the_o seven_o as_o a_o be_v to_o b_o so_o be_v d_z to_o c._n and_o forasmuch_o as_o a_o b_o be_v like_o plain_a number_n therefore_o by_o the_o 18._o of_o
which_o be_v require_v to_o be_v prove_v ¶_o the_o 34._o theorem_a the_o 34._o proposition_n if_o a_o number_n be_v neither_o double_v from_o two_o nor_o have_v to_o his_o half_a part_n a_o odd_a number_n it_o shall_v be_v a_o number_n both_o even_o even_o and_o even_o odd_a svppose_v that_o the_o number_n a_o be_v a_o number_n neither_o double_v from_o the_o number_n two_o neither_o also_o let_v it_o have_v to_o his_o half_a part_n a_o odd_a number_n then_o i_o say_v that_o a_o be_v a_o number_n both_o even_o even_o and_o even_o odd_a demonstration_n that_o a_o be_v even_o even_o it_o be_v manifest_a for_o the_o half_a thereof_o be_v not_o odd_a and_o be_v measure_v by_o the_o number_n 2._o which_o be_v a_o even_a number_n now_o i_o say_v that_o it_o be_v even_o odd_a also_o for_o if_o we_o divide_v a_o into_o two_o equal_a part_n and_o so_o continue_v still_o we_o shall_v at_o the_o length_n light_n upon_o a_o certain_a odd_a number_n which_o shall_v measure_v a_o by_o a_o even_a number_n for_o if_o we_o shall_v not_o light_v upon_o such_o a_o odd_a number_n which_o measure_v a_o by_o a_o even_a number_n we_o shall_v at_o the_o length_n come_v unto_o the_o number_n two_o and_o so_o shall_v a_o be_v one_o of_o those_o number_n which_o be_v double_v from_o two_o upward_a which_o be_v contrary_a to_o the_o supposition_n wherefore_o a_o be_v even_o odd_a and_o it_o be_v prove_v that_o it_o be_v even_o even_o wherefore_o a_o be_v a_o number_n both_o even_o even_o and_o even_o odd_a which_o be_v require_v to_o be_v demonstrate_v this_o proposition_n and_o the_o two_o former_a manifest_o declare_v that_o which_o we_o note_v upon_o the_o ten_o definition_n of_o the_o seven_o book_n namely_o that_o campane_n and_o flussates_n and_o diverse_a other_o interpreter_n of_o euclid_n only_a theon_n except_v do_v not_o right_o understand_v the_o 8._o and_o 9_o definition_n of_o the_o same_o book_n concern_v a_o number_n even_o even_o and_o a_o number_n even_o odd_a for_o in_o the_o one_o definition_n they_o add_v unto_o euclides_n word_n extant_a in_o the_o greek_a this_o word_n only_o as_o we_o there_o note_v and_o in_o the_o other_o this_o word_n all_o so_o that_o after_o their_o definition_n a_o number_n can_v not_o be_v even_o even_o unless_o it_o be_v measure_v only_o by_o even_a number_n likewise_o a_o number_n can_v not_o be_v even_o odd_a unless_o all_o the_o even_a number_n which_o do_v measure_v it_o do_v measure_v it_o by_o a_o odd_a number_n the_o contrary_a whereof_o in_o this_o proposition_n we_o manifest_o see_v for_o here_o euclid_n prove_v that_o one_o number_n may_v be_v both_o even_o even_o and_o even_o odd_a and_o in_o the_o two_o former_a proposition_n he_o prove_v that_o some_o number_n be_v even_o even_o only_o and_o some_o even_o odd_a only_o which_o word_n only_o have_v be_v in_o vain_a of_o he_o add_v if_o no_o number_n even_o even_o can_v be_v measure_v by_o a_o odd_a number_n or_o if_o all_o the_o number_n that_o measure_v a_o number_n even_o odd_a must_v needs_o measure_v it_o by_o a_o odd_a number_n although_o campane_n and_o flussates_n to_o avoid_v this_o absurdity_n have_v wrest_v the_o 32._o proposition_n of_o this_o book_n from_o the_o true_a sense_n of_o the_o greek_a and_o as_o it_o be_v interpret_v of_o theon_n so_o also_o have_v flussates_n wrest_v the_o 33._o proposition_n for_o whereas_o euclid_n say_v 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 they_o say_v only_o the_o number_n produce_v by_o the_o double_n of_o two_o be_v even_o even_o likewise_o whereas_o euclid_n say_v a_o number_n who_o hafle_n part_n be_v odd_a be_v even_o odd_a only_o flussates_n say_v only_o a_o number_n who_o half_a part_n be_v odd_a be_v even_o odd_a which_o their_o interpretation_n be_v not_o true_a neither_o can_v be_v apply_v to_o the_o proposition_n as_o they_o be_v extant_a in_o the_o greek_a in_o deed_n the_o say_v 32._o and_o 33._o proposition_n as_o they_o put_v they_o be_v true_a touch_v those_o number_n which_o be_v even_o even_o only_o or_o even_o odd_a only_o for_o no_o number_n be_v even_o even_o only_o but_o those_o only_o which_o be_v double_v from_o two_o upward_o likewise_o no_o number_n be_v even_o odd_a only_o but_o those_o only_o who_o half_o be_v a_o odd_a number_n but_o this_o let_v not_o but_o that_o a_o number_n may_v be_v even_o even_o although_o it_o be_v not_o double_v from_o two_o upward_a &_o also_o that_o a_o number_n may_v be_v even_o odd_a although_o it_o have_v not_o to_o his_o half_n a_o odd_a number_n as_o in_o this_o 34._o proposition_n euclid_n have_v plain_o prove_v which_o thing_n can_v by_o no_o mean_n be_v true_a if_o the_o foresay_a 32._o &_o 33._o propositon_n of_o this_o book_n shall_v have_v that_o sense_n and_o meaning_n wherein_o they_o take_v it_o ¶_o the_o 35._o theorem_a the_o 35._o proposition_n if_o there_o be_v number_n in_o continual_a proportion_n how_o many_o soever_o and_o if_o from_o the_o second_o and_o last_o be_v take_v away_o number_n equal_a unto_o the_o first_o as_o the_o excess_n of_o the_o second_o be_v to_o the_o first_o so_o be_v the_o excess_n of_o the_o last_o to_o all_o the_o number_n go_v before_o the_o last_o svppose_v that_o these_o number_n a_o bc_o d_o and_o of_o be_v in_o continual_a proportion_n beginning_n at_o a_o the_o least_o and_o from_o bc_o which_o be_v the_o second_o take_v away_o cg_o equal_a unto_o the_o first_o namely_o to_o a_o and_o likewise_o from_o of_o the_o last_o take_v away_o fh_o equal_a also_o unto_o the_o first_o namely_o to_o a._n then_o i_o say_v that_o as_o the_o excess_n bg_o be_v to_o a_o the_o first_o so_o be_v he_o the_o excess_n to_o all_o the_o number_n d_o bc_o and_o a_o which_o go_v before_o the_o last_o number_n namely_o ef._n demonstration_n forasmuch_o as_o of_o be_v the_o great_a for_o the_o second_o be_v suppose_v great_a than_o the_o first_o put_v the_o number_n fl_fw-mi equal_v to_o the_o number_n d_o and_o likewise_o the_o number_n fk_o equal_a to_o the_o number_n bc._n and_o forasmuch_o as_o fk_n be_v equal_a unto_o cb_o of_o which_o fh_o be_v equal_a unto_o gc_o therefore_o the_o residue_n hk_o be_v equal_a unto_o the_o residue_n gb_o and_o for_o that_o as_o the_o whole_a fâ—_n be_v to_o the_o whole_a fl_fw-mi so_o be_v the_o part_n take_v away_o fl_fw-mi to_o the_o part_n take_v away_o fk_a therefore_o the_o residue_n le_fw-fr be_v to_o the_o residue_n kl_o as_o the_o whole_a â—e_n be_v to_o the_o whole_a fl_fw-mi by_o the_o 11._o of_o the_o seven_o so_o likewise_o for_o that_o fl_fw-mi be_v to_o fk_v as_o fk_n be_v to_o fh_v kl_o shall_v be_v to_o hk_v as_o the_o whole_a fl_fw-mi be_v to_o the_o whole_a fk_n by_o the_o same_o proposition_n but_o as_o fe_o be_v to_o fl_fw-mi and_o as_o fl_fw-mi be_v to_o fk_v and_o fk_a to_o fh_v so_o be_v fe_o to_o d_z and_o d_z to_o bc_o and_o bc_o 〈◊〉_d a._n wherefore_o as_o le_fw-fr be_v to_o kl_o and_o as_o kl_o be_v to_o hk_v so_o be_v d_z to_o bc._n wherefore_o alternate_o by_o the_o 23._o of_o the_o seven_o as_z le_fw-fr be_v to_o d_z so_o be_v kl_o to_o be_v bc_o and_o as_o kl_o be_v to_o bc_o so_o be_v hk_v to_o a._n wherefore_o also_o as_o one_o of_o the_o antecedentes_fw-la be_v to_o one_o of_o the_o consequentes_fw-la so_o be_v all_o the_o antecedentes_fw-la to_o all_o the_o consequentes_fw-la wherefore_o as_o kh_o be_v to_o a_o so_o be_v hk_v kl_o and_o le_fw-fr to_o d_z bc_o and_o a_o by_o the_o 12._o of_o the_o seven_o but_o it_o be_v prove_v that_o kh_o be_v equal_a unto_o bg_o wherefore_o as_o bg_o which_o be_v the_o excess_n of_o the_o second_o be_v to_o a_o so_o be_v eh_o the_o excess_n of_o the_o last_o unto_o the_o number_n go_v before_o d_o bc_o and_o a._n wherefore_o as_o the_o excess_n of_o the_o second_o be_v unto_o the_o first_o so_o be_v the_o excess_n of_o the_o last_o to_o all_o the_o number_n go_v before_o the_o last_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 36._o theorem_a the_o 36._o proposition_n if_o from_o unity_n be_v take_v number_n how_o many_o soever_o in_o double_a proportion_n continual_o until_o the_o whole_a add_v together_o be_v a_o prime_a number_n and_o if_o the_o whole_a multiply_v the_o last_o produce_v any_o number_n that_o which_o be_v produce_v be_v a_o perfect_v number_n number_n svppose_v that_o from_o unity_n be_v take_v these_o number_n a_o b_o c_o d_o in_o double_a proportion_n continual_o so_o that_o all_o those_o number_n a_o b_o c_o d_o &_o unity_n add_v together_o make_v a_o prime_a number_n and_o let_v e_o be_v the_o number_n compose_v of_o
measure_v the_o square_a number_n produce_v of_o ef._n wherefore_o also_o by_o the_o 14._o of_o the_o eight_o the_o number_n g_z measure_v the_o number_n of_o and_o the_o number_n g_o also_o measure_v itself_o wherefore_o the_o number_n g_z measure_v these_o number_n of_o and_o g_z when_o yet_o they_o be_v prime_a the_o one_o to_o the_o other_o which_o be_v impossible_a wherefore_o the_o diameter_n a_o be_v not_o commensurable_a in_o length_n to_o the_o side_n b._n wherefore_o it_o be_v incommensurable_a which_o be_v require_v to_o be_v demonstrate_v an_o other_o demonstration_n after_o flussas_n suppose_v that_o upon_o the_o line_n ab_fw-la be_v describe_v a_o square_n who_o diameter_n let_v be_v the_o line_n ac_fw-la then_o i_o say_v that_o the_o side_n ab_fw-la be_v incommensurable_a in_o length_n unto_o the_o diameter_n ac_fw-la forasmuch_o as_o the_o line_n ab_fw-la and_o bc_o be_v equal_a therefore_o the_o square_a of_o the_o line_n ac_fw-la be_v double_a to_o the_o square_n of_o the_o line_n ab_fw-la by_o the_o 47._o of_o the_o first_o take_v by_o the_o 2._o of_o the_o eight_o number_n how_o many_o soever_o in_o continual_a proportion_n from_o unity_n and_o in_o the_o proportion_n of_o the_o square_n of_o the_o line_n ab_fw-la and_o ac_fw-la which_o let_v be_v the_o number_n d_o e_o f_o g._n and_o forasmuch_o as_o the_o first_o from_o unity_n namely_o e_o be_v no_o square_a number_n for_o that_o it_o be_v a_o prime_a number_n neither_o be_v also_o any_o other_o of_o the_o say_a number_n a_o square_a number_n except_o the_o three_o from_o unity_n and_o so_o all_o the_o rest_n leving_fw-mi one_o between_o by_o the_o 10._o of_o the_o nine_o wherefore_o d_z be_v to_o e_o or_o e_o to_o fletcher_n or_o fletcher_n to_o g_z in_o that_o proportion_n that_o a_o square_a number_n be_v to_o a_o number_n not_o square_a wherefore_o by_o the_o corollary_n of_o the_o 25._o of_o the_o eight_o they_o be_v not_o in_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n be_v to_o a_o square_a number_n wherefore_o neither_o also_o have_v the_o square_n of_o the_o line_n ab_fw-la and_o ac_fw-la which_o be_v in_o the_o same_o proportion_n that_o porportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o by_o the_o 9_o of_o this_o book_n their_o side_n namely_o the_o side_n ab_fw-la and_o the_o diameter_n ac_fw-la be_v incommensurable_a in_o length_n the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v this_o demonstration_n i_o think_v good_a to_o add_v for_o that_o the_o former_a demonstration_n seem_v not_o so_o full_a and_o they_o be_v think_v of_o some_o to_o be_v none_o of_o theon_n as_o also_o the_o proposition_n to_o be_v none_o of_o euclides_n here_o follow_v a_o instruction_n by_o some_o studious_a and_o skilful_a grecian_a perchance_o theon_n which_o teach_v we_o of_o far_a use_n and_o fruit_n of_o these_o irrational_a line_n see_v that_o there_o be_v find_v out_o right_a line_n incommensurable_a in_o length_n the_o one_o to_o the_o â—ther_n as_o the_o line_n a_o and_o b_o there_o may_v also_o be_v find_v out_o many_o other_o magnitude_n have_v length_n and_o breadth_n such_o as_o be_v plain_a superficiece_n which_o shall_v be_v incommensurable_a the_o one_o to_o the_o other_o for_o if_o by_o the_o 13._o of_o the_o six_o between_o the_o line_n a_o and_o b_o there_o be_v take_v the_o mean_a proportional_a line_n namely_o c_z then_o by_o the_o second_o corollary_n of_o the_o 20._o of_o the_o six_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o figure_n describe_v upon_o the_o line_n a_o to_o the_o figure_n describe_v upon_o the_o line_n c_o be_v both_o like_a and_o in_o like_a sort_n describe_v that_o be_v whether_o they_o be_v square_n which_o be_v always_o like_o the_o one_o to_o the_o other_o or_o whether_o they_o be_v any_o other_o like_o rectiline_a figure_n or_o whether_o they_o be_v circle_n about_o the_o diameter_n a_o and_o c._n for_o circle_n have_v that_o proportion_n the_o one_o to_o the_o other_o that_o the_o square_n of_o their_o diameter_n have_v by_o the_o 2._o of_o the_o twelve_o wherefore_o by_o the_o second_o part_n of_o the_o 10._o of_o the_o ten_o the_o â—_z describe_v upon_o the_o line_n a_o and_o c_o being_n like_o and_o in_o like_a sort_n describe_v be_v incommensurable_a the_o one_o to_o the_o other_o wherefore_o by_o this_o mean_n there_o be_v find_v out_o superficieces_o incommensurable_a the_o one_o to_o the_o other_o in_o like_a sort_n there_o may_v be_v find_v out_o figure_n commensurable_a the_o one_o to_o the_o other_o if_o you_o put_v the_o line_n a_o and_o b_o to_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o and_o see_v that_o it_o be_v so_o now_o let_v we_o also_o prove_v that_o even_o in_o soliâ—es_n also_o or_o body_n there_o be_v some_o commensurable_a the_o one_o to_o the_o other_o and_o other_o some_o incommensurable_a the_o one_o to_o the_o other_o for_o if_o from_o each_o of_o the_o square_n of_o the_o line_n a_o and_o b_o or_o from_o any_o other_o rectiline_a figure_n equal_a to_o these_o square_n be_v erect_v solid_n of_o equal_a altitude_n whether_o those_o solid_n be_v compose_v of_o equidistant_a supersiciece_n or_o whether_o they_o be_v pyramid_n or_o prism_n thosâ—_n solid_n sâ—_n erected_a shall_v be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v by_o the_o 32._o oâ—_n the_o eleven_o and_o 5._o and_o 6._o of_o the_o twelve_o howbeit_o there_o be_v no_o such_o proposition_n concern_v prism_n and_o so_o if_o the_o base_n of_o the_o solid_n bâ—_z commensurable_a the_o one_o to_o the_o other_o the_o solid_n also_o shall_v be_v commensurable_a the_o one_o to_o the_o other_o and_o if_o the_o base_n be_v incommensurable_a the_o one_o to_o the_o other_o the_o solid_n also_o shall_v be_v incommensurable_a the_o one_o to_o the_o other_o by_o the_o 10._o of_o the_o ten_o and_o if_o there_o be_v two_o circle_n a_o and_o b_o and_o upon_o each_o of_o the_o circle_n be_v erect_v cones_n or_o cilinder_n of_o equal_a altitude_n those_o cones_n &_o cilinder_n sâ—all_v be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o the_o circle_n be_v which_o be_v their_o base_n by_o the_o 11._o of_o the_o twelve_o and_o so_o if_o the_o circle_n be_v commensurable_a the_o one_o to_o the_o other_o the_o cones_n and_o cilinder_n also_o shall_v be_v commensurable_a the_o one_o to_o the_o other_o but_o if_o the_o circle_n be_v incommensurable_a the_o one_o to_o the_o other_o the_o cones_n also_o and_o cilinder_n shall_v be_v incommensurable_a the_o one_o to_o the_o other_o by_o the_o 10._o of_o the_o ten_o wherefore_o it_o be_v manifest_a that_o not_o only_o in_o line_n and_o superâ—icieces_n but_o also_o in_o solid_n or_o body_n be_v find_v commensurabilitie_n or_o incommensurability_n a_o advertisement_n by_o john_n dee_n although_o this_o proposition_n be_v by_o euclid_n to_o this_o book_n allot_v as_o by_o the_o ancient_a graecian_n publish_v under_o the_o name_n of_o aristoteles_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d it_o will_v seem_v to_o be_v and_o also_o the_o property_n of_o the_o same_o agreeable_a to_o the_o matter_n of_o this_o book_n and_o the_o proposition_n itself_o so_o famous_a in_o philosophy_n and_o logic_n as_o it_o be_v will_v in_o manner_n crave_v his_o elemental_a place_n in_o this_o ten_o book_n yet_o the_o dignity_n &_o perfectionâ—_n of_o mathematical_a method_n can_v not_o allow_v it_o here_o as_o in_o due_a order_n follow_v but_o most_o apt_o after_o the_o 9_o proposition_n of_o this_o book_n as_o a_o corollary_n of_o the_o last_o part_n thereof_o and_o undoubted_o the_o proposition_n have_v for_o this_o 2000_o year_n be_v notable_o regard_v among_o the_o greek_n philosopher_n and_o before_o aristotle_n time_n be_v conclude_v with_o the_o very_a same_o inconvenience_n to_o the_o gaynesayer_n that_o the_o first_o demonstration_n here_o induce_v namely_o odd_a number_n to_o be_v equal_a to_o even_o as_o may_v appearâ—_n in_o aristotle_n work_n name_v analitica_fw-la prima_fw-la the_o first_o book_n and_o 40._o chapter_n but_o else_o in_o very_a many_o place_n of_o his_o work_n he_o make_v mention_n of_o the_o proposition_n evident_a also_o it_o be_v that_o euclid_n be_v about_o aristotle_n time_n and_o in_o that_o age_n the_o most_o excellent_a geometrician_n among_o the_o greek_n wherefore_o see_v it_o be_v so_o public_a in_o his_o time_n so_o famous_a and_o so_o appertain_v to_o the_o property_n of_o this_o book_n it_o be_v most_o likely_a both_o to_o be_v know_v to_o euclid_n and_o also_o to_o have_v be_v by_o he_o in_o apt_a order_n place_v but_o of_o the_o disorder_v of_o it_o can_v remain_v no_o doubt_n if_o you_o consider_v in_o zamberts_n translation_n
be_v term_v dial_v ancient_a be_v the_o use_n and_o more_o ancient_a be_v the_o invention_n the_o use_n do_v well_o appear_v to_o have_v be_v at_o the_o least_o above_o two_o thousand_o and_o three_o hundred_o year_n ago_o in_o 20._o king_n achaâ—_n dial_n then_o by_o the_o sun_n show_v the_o distinction_n of_o time_n by_o sun_n moon_n and_o star_n this_o dial_n may_v be_v perform_v and_o the_o precise_a time_n of_o day_n or_o night_n know_v but_o the_o demonstrative_a delineation_n of_o these_o dial_n of_o all_o sort_n require_v good_a skill_n both_o of_o astronomy_n and_o geometry_n elemental_a spherical_a phaenomenall_a and_o conikall_n then_o to_o use_v the_o ground_n of_o the_o art_n for_o any_o regular_a superficies_n in_o any_o place_n offer_v and_o in_o any_o possible_a apt_a position_n thereof_o thâ—ron_n to_o describe_v all_o manner_n of_o way_n how_o usual_a hour_n may_v be_v by_o the_o sun_n shadow_n true_o determine_v will_v be_v find_v no_o sleight_n painter_n work_v so_o to_o paint_n and_o prescribe_v the_o sun_n motion_n to_o the_o breadth_n of_o a_o hair_n in_o this_o feat_n in_o my_o youth_n i_o invent_v a_o way_n how_o in_o any_o horizontal_a mural_a or_o equinoctial_a dial_n etc_n etc_n at_o all_o hour_n the_o sun_n shine_v the_o sign_n and_o degree_n ascendent_a may_v be_v know_v which_o be_v a_o thing_n very_o necessary_a for_o the_o rise_n of_o those_o fix_a star_n who_o operation_n in_o the_o air_n be_v of_o great_a might_n evident_o i_o speak_v no_o further_o of_o the_o use_n hereof_o but_o forasmuch_o as_o man_n affair_n require_v knowledge_n of_o time_n &_o moment_n when_o neither_o sun_n moon_n or_o star_n can_v be_v see_v therefore_o by_o industry_n mechanical_a be_v invent_v first_o how_o by_o water_n run_v orderly_a the_o time_n and_o hour_n may_v be_v know_v whereof_o the_o famous_a ctesibius_n be_v inventor_n a_o man_n of_o vitrwius_fw-la to_o the_o sky_n just_o extol_v then_o after_o that_o by_o sand_n run_v be_v hour_n measure_v then_o by_o trochilic_a with_o weight_n and_o of_o late_a time_n by_o trochilic_a with_o spring_n without_o weight_n all_o these_o by_o sun_n or_o star_n direction_n in_o certain_a time_n require_v oversight_n and_o reformation_n according_a to_o the_o heavenly_a equinoctial_a motion_n beside_o the_o inequality_n of_o their_o own_o operation_n there_o remain_v without_o parabolical_a meaning_n herein_o among_o the_o philosopher_n motion_n a_o more_o excellent_a more_o commodious_a and_o more_o marvelous_a way_n than_o all_o these_o of_o have_v the_o motion_n of_o the_o primovant_n or_o first_o equinoctial_a motion_n by_o nature_n and_o arteâ—_n imitate_v which_o you_o shall_v by_o further_a search_n in_o weighty_a study_n hereafter_o understand_v more_o of_o and_o so_o it_o be_v time_n to_o finish_v this_o annotation_n of_o time_n distinction_n use_v in_o our_o common_a and_o private_a affair_n the_o commodity_n whereof_o no_o man_n will_v want_v that_o can_v tell_v how_o to_o bestow_v his_o tyme._n zographie_n be_v a_o art_n mathematical_a which_o teach_v and_o demonstrate_v how_o the_o intersection_n of_o all_o visual_a pyramid_n make_v by_o any_o plain_n assign_v the_o centre_n distance_n and_o light_n be_v determine_v may_v be_v by_o line_n and_o due_a proper_a colour_n represent_v a_o notable_a art_n be_v this_o and_o will_v require_v a_o whole_a volume_n to_o declare_v the_o property_n thereof_o and_o the_o commodity_n ensue_v great_a skill_n of_o geometry_n arithmetic_n perspective_n and_o anthropographie_n with_o many_o other_o particular_a artâ—s_n have_v the_o zographer_n need_n of_o for_o his_o perfection_n for_o the_o most_o excellent_a painter_n who_o be_v but_o the_o propre_fw-fr mechanicien_fw-fr &_o imitator_n sensible_a of_o the_o zographer_n have_v attein_v to_o such_o perfection_n that_o sense_n of_o man_n and_o beast_n have_v judge_v thing_n paint_v to_o be_v thing_n natural_a and_o not_o artificial_a alive_a and_o not_o dead_a this_o mechanical_a zographer_n common_o call_v the_o painter_n be_v marvellous_a in_o his_o skill_n and_o seem_v to_o have_v a_o certain_a divine_a power_n as_o of_o friend_n absent_a to_o make_v a_o friendly_a present_a comfort_n yea_o and_o of_o friend_n dead_a to_o give_v a_o continual_a silent_a presence_n not_o only_o with_o we_o but_o with_o our_o posterity_n for_o many_o age_n and_o so_o proceed_v consider_v how_o in_o winter_n he_o can_v show_v you_o the_o lively_a view_n of_o summer_n joy_n and_o riches_n and_o in_o summer_n exhibit_v the_o countenance_n of_o winter_n doleful_a state_n and_o nakedness_n city_n town_n fort_n wood_n army_n yea_o whole_a kingdom_n be_v they_o never_o so_o far_o or_o great_a can_v he_o with_o ease_n bring_v with_o he_o home_n to_o any_o man_n judgement_n as_o pattern_n lively_a of_o the_o thing_n rehearse_v in_o one_o little_a house_n can_v he_o enclose_v with_o great_a pleasure_n of_o the_o beholder_n the_o portraiture_n lively_a of_o all_o visible_a creature_n either_o on_o earth_n or_o in_o the_o earth_n live_v or_o in_o the_o water_n lie_v creep_v slide_v or_o swim_v or_o of_o any_o â—oule_n or_o fly_v in_o the_o air_n fly_v nay_o in_o respect_n of_o the_o star_n the_o sky_n the_o cloud_n yea_o in_o the_o show_n of_o the_o very_a light_n itself_o that_o divine_a creature_n can_v he_o match_v our_o eye_n judgement_n most_o near_o what_o a_o thing_n be_v this_o thing_n not_o yet_o be_v he_o can_v represent_v so_o as_z at_o their_o be_v the_o picture_n shall_v seam_n in_o manner_n to_o have_v create_v they_o to_o what_o artificer_n be_v not_o picture_n a_o great_a pleasure_n and_o commodity_n which_o of_o they_o all_o will_v refuse_v the_o direction_n and_o aid_n of_o picture_n the_o architect_n the_o goldsmith_n and_o the_o arras_n weaver_n of_o picture_n make_v great_a account_n our_o lively_a herbal_n our_o portraiture_n of_o bird_n beast_n and_o fish_n and_o our_o curious_a anatomy_n which_o way_n be_v they_o most_o perfect_o make_v or_o with_o most_o pleasure_n of_o we_o beholden_a be_v it_o not_o by_o picture_n only_o and_o if_o picture_n by_o the_o industry_n of_o the_o painter_n be_v thus_o commodious_a and_o marvellous_a what_o shall_v be_v think_v of_o zographie_n the_o schoolmaster_n of_o picture_n and_o chief_a governor_n though_o i_o mention_v not_o sculpture_n in_o my_o table_n of_o art_n mathematical_a yet_o may_v all_o man_n perceive_v how_o that_o picture_n and_o sculpture_n be_v sister_n germane_a and_o both_o right_a profitable_a in_o a_o common_a wealth_n and_o of_o sculpture_n aswell_o as_o of_o picture_n excellent_a artificer_n have_v write_v great_a book_n in_o commendation_n witness_v i_o take_v of_o georgio_n vasari_n pittore_fw-la aretino_n of_o pomponius_n gauricusâ—_n â—_z and_o other_o to_o these_o two_o art_n with_o other_o be_v a_o certain_a odd_a art_n call_v althalmasat_n much_o behold_v more_o than_o the_o common_a sculptor_n entayler_n carver_n cutâ—er_n graver_n founder_n or_o painter_n etc_o etc_o know_v their_o art_n to_o be_v commodious_a architecture_n to_o many_o may_v seem_v not_o worthy_a or_o not_o mete_v objection_n to_o be_v reckon_v among_o the_o art_n mathematical_o â—_z to_o who_o i_o think_v good_a to_o give_v some_o account_n of_o my_o so_o do_v not_o worthy_a will_v they_o say_v because_o it_o be_v but_o for_o building_n of_o a_o house_n palace_n church_n forte_n or_o such_o like_a gross_a work_n and_o you_o also_o define_v the_o art_n mathematical_a to_o be_v such_o as_o deal_v with_o no_o material_a or_o corruptible_a thing_n and_o alâ—o_o do_v demonstrative_o proceed_v in_o their_o faculty_n by_o number_n or_o magnitude_n first_o you_o see_v that_o i_o count_v here_o architecture_n among_o those_o art_n mathematical_a answer_n which_o be_v derive_v from_o the_o principal_n and_o you_o know_v that_o such_o may_v deal_v with_o natural_a thing_n and_o sensibâ—â—â—aâ—â—er_n of_o which_o some_o draw_v near_o to_o the_o simple_a and_o absolute_a mathematical_a speculation_n than_o other_o do_v and_o though_o the_o architect_n ☜_o procure_v inform_v &_o direct_v the_o mechanicien_n to_o handworke_n &_o the_o build_n actual_a of_o house_n castle_n or_o palace_n and_o be_v chief_a judge_n of_o the_o same_o yet_o with_o himself_o as_o chief_a master_n and_o architect_n remain_v the_o demonstrative_a reason_n and_o cause_n of_o the_o mechaniciens_fw-la work_n in_o line_n plain_a and_o solid_a by_o geometrical_a arithmetical_a optical_a musical_a astronomical_a cosmographical_a &_o to_o be_v brief_a by_o all_o the_o former_a derive_v art_n mathematical_a and_o other_o natural_a art_n able_a to_o be_v confirm_v and_o establish_v if_o this_o be_v soâ—then_a may_v you_o think_v that_o architecture_n have_v good_a and_o due_a allowance_n in_o this_o honest_a company_n of_o art_n mathematical_a derivative_n i_o will_v herein_o crave_v judgement_n of_o two_o most_o perfect_a architectâ—s_n the_o one_o be_v vitrwius_fw-la the_o roman_a who_o do_v write_v ten_o
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
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