other_o number_n de_fw-fr be_v of_o a_o other_o number_n fletcher_n and_o let_v ab_fw-la be_v less_o than_o de._n then_o i_o say_v that_o alternate_o also_o what_o part_n or_o part_n ab_fw-la be_v of_o de_fw-fr the_o self_n same_o part_n or_o part_n be_v c_o of_o f._n forasmuch_o as_o what_o part_v ab_fw-la be_v of_o c_o the_o self_n same_o part_n be_v de_fw-fr of_o fletcher_n construction_n therefore_o how_o many_o part_n of_o c_o there_o be_v in_o ab_fw-la so_o many_o part_n of_o fletcher_n also_o be_v there_o in_o de._n divide_v ab_fw-la into_o the_o part_n of_o c_o that_o be_v into_o ag_z and_o gb_o and_o likewise_o de_fw-fr into_o the_o part_n of_o fletcher_n that_o be_v dh_o and_o he._n now_o then_o the_o multitude_n of_o these_o agnostus_n and_o gb_o be_v equal_a unto_o the_o multitude_n of_o these_o dh_o and_o he._n demonstration_n and_o forasmuch_o as_o what_o part_n agnostus_n be_v of_o c_o the_o self_n same_o part_n be_v dh_o of_o fletcher_n therefore_o alternate_o also_o by_o the_o former_a what_o part_n or_o part_n agnostus_n be_v of_o dh_o the_o self_n same_o part_n or_o part_n be_v c_o of_o f._n and_o by_o the_o same_o reason_n also_o what_o part_n or_o part_n gb_o be_v of_o he_o the_o same_o part_n or_o part_n be_v c_o of_o f._n wherefore_o what_o part_n or_o part_n agnostus_n be_v of_o dh_o the_o self_n same_o part_n or_o part_n be_v ab_fw-la of_o de_fw-fr by_o the_o 6._o of_o the_o seven_o but_o what_o part_n or_o part_n agnostus_n be_v of_o dh_o the_o self_n same_o part_n or_o part_n be_v it_o prove_v that_o c_z be_v of_o f._n wherefore_o what_o part_n or_o part_n ab_fw-la be_v of_o d_o e_o the_o self_n same_o part_n or_o part_n be_v c_o of_o fletcher_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 9_o theorem_a the_o 11._o proposition_n if_o the_o whole_a be_v to_o the_o whole_a as_o a_o part_n take_v away_o be_v to_o a_o part_n take_v away_o then_o shall_v the_o residue_n be_v unto_o the_o residue_n as_o the_o whole_a be_v to_o the_o whole_a quantity_n svppose_v that_o the_o whole_a number_n ab_fw-la be_v unto_o the_o whole_a number_n cd_o as_o the_o part_n take_v away_o ae_n be_v to_o the_o part_n take_v away_o cf._n then_o i_o say_v that_o the_o residue_n ebb_n be_v to_o the_o residue_n fd_o as_o the_o whole_a ab_fw-la be_v to_o the_o whole_a cd_o for_o forasmuch_o as_o ab_fw-la be_v to_o cd_o as_o ae_n be_v to_o cf_o therefore_o what_o part_n or_o part_n ab_fw-la be_v of_o cd_o the_o self_n same_o part_n or_o part_n be_v ae_n of_o cf._n wherefore_o also_o the_o residue_n ebb_n be_v of_o the_o residue_n fd_o by_o the_o 8._o of_o the_o seven_o the_o self_n same_o part_n oâ—_n part_n that_o ab_fw-la be_v of_o cd_o demonstration_n wherefore_o also_o by_o the_o 21._o definition_n of_o this_o book_n as_o ebb_n be_v to_o fd_o so_o be_v ab_fw-la to_o cd_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 10._o theorem_a the_o 12._o proposition_n if_o there_o be_v a_o multitude_n of_o number_n how_o many_o soever_o proportional_a 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 svppose_v that_o there_o be_v a_o multitude_n of_o number_n how_o many_o soever_o proportional_a namely_o a_o b_o c_o d_o so_fw-mi that_o as_o a_o be_v to_o b_o so_o let_v c_o be_v to_o d._n quantity_n then_o i_o say_v that_o as_o one_o of_o the_o antecedentes_fw-la namely_o a_o be_v to_o one_o of_o the_o consequentes_fw-la namely_o to_o b_o or_o as_z c_z be_v to_o d_z so_o be_v all_o the_o antecedentes_fw-la namely_o a_o and_o c_o to_o all_o the_o consequentes_fw-la namely_o to_o b_o and_o d._n for_o forasmuch_o as_o by_o supposition_n as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z therefore_o what_o part_n or_o part_n a_o be_v of_o b_o the_o self_n same_o part_n or_o part_n be_v c_z of_o d_z by_o the_o 21._o definition_n of_o this_o book_n wherefore_o alternate_o what_o part_n or_o part_n a_o be_v of_o c_o the_o self_n same_o part_n or_o part_n be_v b_o of_o d_z by_o the_o nine_o and_o ten_o of_o the_o seven_o wherefore_o both_o these_o number_n add_v together_o demonstration_n a_o and_o c_o be_v of_o both_o these_o numbers_z b_o and_o d_z add_v together_o the_o self_n same_o part_n or_o part_n that_o a_o be_v of_o b_o by_o the_o 5._o and_o 6._o of_o the_o seven_o wherefore_o by_o the_o 21._o definition_n of_o the_o seven_o as_o one_o of_o the_o antecedent_n namely_o a_o be_v to_o one_o of_o the_o consequentes_fw-la namely_o to_o b_o so_o be_v all_o the_o antecedentes_fw-la a_fw-la and_o c_o to_o all_o the_o consequentes_fw-la b_o &_o d._n which_o be_v require_v to_o be_v prove_v ¶_o the_o 11._o theorem_a the_o 13._o proposition_n if_o there_o be_v four_o number_n proportional_a then_o alternate_o also_o they_o shall_v be_v proportional_a svppose_v that_o there_o be_v four_o number_n proportional_a quantity_n a_o b_o c_o d_o so_fw-mi that_o as_o a_o be_v to_o b_o so_o let_v c_o be_v to_o d._n then_o i_o say_v that_o alternate_o also_o they_o shall_v be_v proportional_a that_o be_v as_o a_o be_v to_o c_o so_o be_v b_o to_o d._n for_o forasmuch_o as_o by_o supposition_n as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z therefore_o by_o the_o 21._o definition_n of_o this_o book_n what_o part_n or_o part_n a_o be_v of_o b_o the_o self_n same_o part_n or_o part_n be_v c_o of_o d._n therefore_o alternate_o what_o part_n or_o part_n a_o be_v of_o c_o the_o self_n same_o part_n or_o part_n be_v b_o of_o d_z by_o the_o 9_o of_o the_o seven_o &_o also_o by_o the_o 10._o of_o the_o same_o wherefore_o as_o a_o be_v to_o c_o so_o be_v b_o to_o d_z by_o the_o 21._o definition_n of_o this_o book_n which_o be_v require_v to_o be_v prove_v here_o be_v to_o be_v note_v that_o although_o in_o the_o foresay_a example_n and_o demonstration_n the_o number_n a_o be_v suppose_v to_o be_v less_o than_o the_o number_n b_o and_o so_o the_o number_n c_o be_v less_o than_o the_o number_n d_o note_n yet_o will_v the_o same_o serve_v also_o though_o a_o be_v suppose_v to_o be_v great_a than_o b_o whereby_o also_o c_o shall_v be_v great_a than_o d_z as_z in_o thâ—s_n example_n here_o put_v for_o for_o that_o by_o supposition_n as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z and_o a_o be_v suppose_v to_o be_v great_a than_o b_o and_o c_z great_a than_o d_z therefore_o by_o the_o 21._o definition_n of_o this_o book_n how_o multiplex_n a_o be_v to_o b_o so_o multiplex_n be_v c_o to_o d_z and_o therefore_o what_o part_n or_o part_n b_o be_v of_o a_o the_o self_n same_o part_n or_o part_n be_v d_z of_o c._n wherefore_o alternate_o what_o part_n or_o part_n b_o be_v of_o d_o the_o self_n same_o part_n or_o part_n be_v a_o of_o c_o and_o therefore_o by_o the_o same_o definition_n b_o be_v to_o d_z as_z a_o be_v to_o c._n and_o so_o must_v you_o understand_v of_o the_o former_a proposition_n next_o go_v before_o ¶_o the_o 12._o theorem_a the_o 14._o proposition_n if_o there_o be_v a_o multitude_n of_o number_n how_o many_o soever_o and_o also_o other_o number_n equal_a unto_o they_o in_o multitude_n which_o be_v compare_v two_o and_o two_o be_v in_o one_o and_o the_o same_o proportion_n they_o shall_v also_o of_o equality_n be_v in_o one_o and_o the_o same_o proportion_n quantity_n svppose_v that_o there_o be_v a_o multitude_n of_o number_n how_o many_o soever_o namely_o a_o b_o c_o and_o let_v the_o other_o number_n equal_a unto_o they_o in_o multitude_n be_v d_o e_o f_o which_o be_v compare_v two_o and_o two_o let_v be_v in_o one_o and_o the_o same_o proportion_n that_o be_v as_z a_o to_o b_o so_o let_v d_o be_v to_o e_o and_o as_o b_o be_v to_o c_o so_o let_v e_o be_v to_o f._n then_o i_o say_v that_o of_o equality_n as_o a_o be_v to_o c_o so_o be_v d_z to_o f._n for_o forasmuch_o as_o by_o supposition_n as_o a_o be_v to_o b_o so_o be_v d_z to_o e_o therefore_o alternate_o also_o by_o the_o 13_o of_o the_o seven_o as_o a_o be_v to_o d_o so_o be_v b_o to_o e._n again_o for_o that_o as_z b_o be_v to_o c_o so_o be_v e_o to_o fletcher_n demonstration_n therefore_o alternate_o also_o by_o the_o self_n same_o as_z b_o be_v to_o e_o so_o be_v c_o to_o f._n but_o as_o b_o be_v to_o e_o so_o be_v a_o to_o d._n wherefore_o by_o the_o seven_o common_a sentence_n of_o the_o seven_o as_o a_o be_v to_o d_o so_o be_v c_o to_o f._n wherefore_o alternate_o by_o the_o 13._o of_o the_o seven_o as_o a_o be_v to_o c_o so_o be_v d_z to_o fletcher_n which_o
it_o comprehend_v wherefore_o the_o pyramid_n who_o base_a be_v the_o square_n abcd_o and_o altitude_n the_o self_n same_o that_o the_o cone_fw-mi have_v be_v great_a than_o the_o half_a of_o the_o cone_fw-it divide_v by_o the_o 30._o of_o the_o three_o every_o one_o of_o the_o circumference_n ab_fw-la bc_o cd_o and_o dam_n into_o two_o equal_a part_n in_o the_o point_n e_o f_o g_o and_o h_o and_o draw_v these_o right_a line_n ae_n ebb_n bf_o fc_o cg_o gd_o dh_o and_o ha._n wherefore_o every_o one_o of_o these_o triangle_n aeb_fw-mi bfc_n cgd_v and_o dha_n be_v great_a than_o the_o half_a part_n of_o the_o segment_n of_o the_o circle_n describe_v about_o it_o upon_o every_o one_o of_o these_o triangle_n aeb_fw-mi bfc_n cgd_v and_o dha_n describe_v a_o pyramid_n of_o equal_a altitude_n with_o the_o cone_n and_o after_o the_o same_o manner_n every_o one_o of_o those_o pyramid_n so_o describe_v be_v great_a than_o the_o half_a part_n of_o the_o segment_n of_o the_o cone_fw-it set_v upon_o the_o segment_n of_o the_o circle_n now_o therefore_o divide_v by_o the_o 30_o of_o the_o three_o the_o circumference_n remain_v into_o two_o equal_a part_n &_o draw_v right_a line_n &_o raise_v up_o upon_o every_o one_o of_o those_o triangle_n a_o pyramid_n of_o equal_a altitude_n with_o the_o cone_n and_o do_v this_o continual_o we_o shall_v at_o the_o length_n by_o the_o first_o of_o the_o ten_o leave_v certain_a segment_n of_o the_o cone_n which_o shall_v be_v less_o then_o the_o excess_n whereby_o the_o cone_n exceed_v the_o three_o part_n of_o the_o cylinder_n let_v those_o segment_n be_v ae_n ebb_n bf_o fc_o cg_o gd_o dh_o and_o ha._n wherefore_o the_o pyramid_n remain_v who_o base_a be_v the_o poligonon_n figure_n aebfcgdh_n and_o altitude_n the_o self_n same_o with_o the_o cone_n be_v great_a than_o the_o three_o part_n of_o the_o cylinder_n but_o the_o pyramid_n who_o base_a be_v the_o polygonon_n figure_n aebfcgdh_n and_o altitude_n the_o self_n same_o with_o the_o cone_n be_v the_o three_o part_n of_o the_o prism_n who_o base_a be_v the_o poligonon_n figure_n aebfcgdh_n and_o altitude_n the_o self_n same_o with_o the_o cylinder_n wherefore_o you_o the_o prism_n who_o base_a be_v the_o polygonon_n figure_n aebfcgdh_n and_o altitude_n the_o self_n same_o with_o the_o cylinder_n be_v great_a than_o the_o cylinder_n who_o base_a be_v the_o circle_n abcd._n but_o it_o be_v also_o less_o for_o it_o be_v contain_v of_o it_o which_o be_v impossible_a wherefore_o the_o cylinder_n be_v not_o in_o less_o proportion_n to_o the_o cone_n then_o in_o treble_a proportion_n and_o it_o be_v prove_v that_o it_o be_v not_o in_o great_a proportion_n to_o the_o cone_n then_o in_o treble_a proportion_n wherefore_o the_o cone_n be_v the_o three_o part_n of_o the_o cylinder_n wherefore_o every_o cone_n be_v the_o three_o part_n of_o a_o cylinder_n have_v one_o &_o the_o self_n same_o base_a and_o one_o and_o the_o self_n same_o altitude_n with_o it_o which_o be_v require_v to_o be_v demonstrate_v ¶_o add_v by_o m._n john_n dee_n ¶_o a_o theorem_a 1._o the_o superficies_n of_o every_o upright_o cylinder_n except_o his_o base_n be_v equal_a to_o that_o circle_n who_o semidiameter_n be_v middle_a proportional_a between_o the_o side_n of_o the_o cylinder_n and_o the_o diameter_n of_o his_o base_a ¶_o a_o theorem_a 2._o the_o superficies_n of_o every_o upright_o or_o isosceles_a cone_n except_o the_o base_a be_v equal_a to_o that_o circle_n who_o semidiameter_n be_v middle_a proportional_a between_o the_o side_n of_o that_o cone_n and_o the_o semidiameter_n of_o the_o circle_n which_o be_v the_o base_a of_o the_o cone_n my_o intent_n in_o addition_n be_v not_o to_o amend_v euclideâ—_n method_n which_o need_v little_a add_v or_o none_o at_o all_o but_o my_o desire_n be_v somewhat_o to_o furnish_v you_o towards_o a_o more_o general_a art_n mathematical_a them_z euclides_n element_n addition_n remain_v in_o the_o term_n in_o which_o they_o be_v write_v can_v sufficient_o help_v you_o unto_o and_o though_o euclides_n element_n with_o my_o addition_n run_v not_o in_o one_o methodical_a race_n towards_o my_o mark_n yet_o in_o the_o mean_a space_n my_o addition_n either_o geve_v light_a where_o they_o be_v annex_v to_o euclides_n matter_n or_o geve_v some_o ready_a aid_n and_o show_v the_o way_n to_o dilate_v your_o discourse_n mathematical_a or_o to_o invent_v and_o practice_v thing_n mechanical_o and_o in_o deed_n if_o more_o leysor_n have_v happen_v many_o more_o strange_a matter_n mathematical_a have_v according_a to_o my_o purpose_n general_a be_v present_o publish_v to_o your_o knowledge_n but_o want_v of_o due_a leasour_n cause_v you_o to_o want_v that_o which_o my_o good_a will_n towards_o you_o most_o hearty_o do_v wish_v you_o as_o concern_v the_o two_o theorem_n here_o annex_v their_o verity_n be_v by_o archimedes_n in_o his_o book_n of_o the_o sphere_n and_o cylinder_n manifest_o demonstrate_v and_o at_o large_a you_o may_v therefore_o bold_o trust_v to_o they_o and_o use_v they_o as_o supposition_n in_o any_o your_o purpose_n till_o you_o have_v also_o their_o demonstration_n but_o if_o you_o well_o remember_v my_o instruction_n upon_o the_o first_o proposition_n of_o this_o book_n and_o my_o other_o addition_n upon_o the_o second_o with_o the_o supposition_n how_o a_o cylinder_n and_o a_o cone_n be_v mathematical_o produce_v you_o will_v not_o need_v archimedes_n demonstration_n nor_o yet_o be_v utter_o ignorant_a of_o the_o solid_a quantity_n of_o this_o cylinder_n and_o cone_n here_o compare_v the_o diameter_n of_o their_o base_a and_o heith_n be_v know_v in_o any_o measure_n neither_o can_v their_o crooked_a superficies_n remain_v unmeasured_a whereof_o undoubted_o great_a pleasure_n and_o commodity_n may_v grow_v to_o the_o sincere_a student_n and_o precise_a practiser_n ¶_o the_o 11._o theorem_a the_o 11._o proposition_n cones_n and_o cylinder_n be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o that_o proportion_n the_o one_o other_o that_o their_o base_n be_v in_o like_a sort_n also_o may_v we_o proveâ—_v that_o as_o the_o circle_n efgh_o be_v to_o the_o circle_n abcd_o so_o be_v not_o the_o cone_n en_fw-fr to_o any_o solid_a less_o than_o the_o cone_n al._n now_o i_o say_v that_o as_o the_o circle_n abcd_o be_v to_o the_o circle_n efgh_o so_o be_v not_o the_o cone_n all_o to_o any_o solid_a great_a than_o the_o cone_fw-mi en_fw-fr for_o if_o it_o be_v possible_a let_v it_o be_v unto_o a_o great_a namely_o to_o the_o solid_a x._o wherefore_o by_o conversion_n as_o the_o circle_n efgh_o be_v to_o the_o circle_n abcd_o so_o be_v the_o solid_a x_o to_o the_o cone_n all_o but_o as_o the_o solid_a x_o be_v to_o the_o cone_n all_o so_o be_v the_o cone_n en_fw-fr to_o some_o solid_a less_o than_o the_o cone_n all_o as_o we_o may_v see_v by_o the_o assumpt_n put_v after_o thâ—_z second_o of_o this_o book_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o circle_n efgh_o be_v to_o the_o circle_n abcgâ—_n so_o be_v the_o cone_n en_fw-fr to_o some_o solid_a less_o than_o the_o cone_n all_o which_o we_o have_v prove_v to_o be_v impossible_a wherefore_o as_o the_o circle_n abcd_o be_v to_o the_o circle_n efgh_o so_o be_v not_o the_o cone_n all_o to_o any_o solid_a great_a than_o the_o cone_fw-mi en_fw-fr and_o it_o be_v also_o prove_v that_o it_o be_v not_o to_o any_o less_o wherefore_o as_o the_o circle_n abcd_o be_v to_o the_o circle_n efgh_o so_o be_v the_o cone_n all_o to_o the_o cone_n en_fw-fr but_o as_o the_o cone_n be_v to_o the_o cone_n so_o be_v the_o cylinder_n to_o the_o cylinder_n by_o the_o 15._o of_o the_o five_o for_o the_o one_o be_v in_o treble_a proportion_n to_o the_o other_o cylinder_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o circle_n abcd_o be_v to_o the_o circle_n efgh_o so_o be_v the_o cylinder_n which_o be_v set_v upon_o they_o the_o one_o to_o the_o other_o the_o say_a cylinder_n being_n under_o equal_a altitude_n with_o the_o cones_fw-la cones_n therefore_o and_o cylinder_n be_v under_o one_o &_o the_o self_n same_o altitude_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 12._o theorem_a the_o 12._o proposition_n like_a cones_n and_o cylinder_n be_v in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o their_o base_n be_v now_o also_o i_o say_v that_o the_o cone_fw-mi abcdl_n be_v not_o to_o any_o solid_a great_a than_o the_o cone_n efghn_n case_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bd_o be_v to_o the_o diameter_n fh_o for_o if_o it_o be_v possible_a let_v it_o be_v to_o a_o great_a namely_o to_o the_o solid_a x._o wherefore_o by_o conversion_n by_o the_o
construction_n two_o case_n in_o this_o proposition_n first_o caseâ—_n second_o case_n demonstration_n construction_n demonstration_n construction_n demonstration_n an_o other_o way_n after_o peliâ—arius_n construction_n demonstration_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n three_o case_n in_o this_o proposition_n the_o three_o case_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o proposition_n add_v by_o petarilius_n note_n construction_n demonstration_n an_o other_o way_n also_o after_o pelitarius_n construction_n demonstration_n an_o other_o way_n to_o do_v the_o samâ—_n after_o pelitarius_n demonstration_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n construction_n demonstration_n demonstration_n a_o â—ther_n way_n to_o do_v the_o same_o after_o orontius_n an_o other_o way_n after_o pelitarius_n construction_n demonstration_n a_o addition_n of_o flussates_n flussates_n a_o poligonon_n figure_n be_v a_o figure_n consist_v of_o many_o side_n the_o argument_n of_o this_o five_o book_n the_o first_o author_n of_o this_o book_n eudoxus_n the_o first_o definition_n a_o part_n take_v two_o manner_n of_o way_n the_o fiâ—st_a way_n the_o second_o way_n how_o a_o less_o quantity_n be_v say_v to_o measure_v a_o great_a in_o what_o signification_n euclid_n here_o take_v a_o part_n parâ—_n metienâ—_n or_o mensuranâ—_n pars_fw-la multiplicatiâ—a_fw-la pars_fw-la aliquota_fw-la this_o kind_n of_o part_n common_o use_v in_o arithmetic_n the_o other_o kind_n of_o part_n pars_fw-la constitâ—ens_fw-la or_o componens_fw-la pars_fw-la aliquanta_fw-la the_o second_o definition_n number_n very_o necessary_a for_o the_o understanding_n of_o this_o book_n and_o the_o other_o book_n follow_v the_o tâ—ird_o definition_n rational_a proportion_n divide_v â—â—to_o two_o kind_n proportion_n of_o equality_n proportion_n of_o inequality_n proportion_n of_o the_o great_a to_o the_o less_o multiplex_fw-la duple_n proportion_n triple_a quadruple_a quintuple_a superperticular_a sesquialtera_fw-la sesquitertia_fw-la sesquiquarta_fw-la superpartiens_fw-la superbipartiens_fw-la supertripartiens_fw-la superquadripartiens_fw-la superquintipartiens_fw-la multiplex_fw-la superperticular_a dupla_fw-la sesquialtera_fw-la dupla_fw-la sesquitertia_fw-la tripla_fw-la sesquialtera_fw-la multiplex_fw-la superpartiens_fw-la dupla_fw-la superbipartiens_fw-la dupla_fw-la supertripartiens_fw-la tripla_fw-la superbipartiens_fw-la tripla_fw-la superquadâ—ipartiens_fw-la how_o to_o knoâ—_n the_o denomination_n of_o any_o proportion_n proportion_n of_o the_o less_o in_o the_o great_a submultiplex_fw-la subsuperparticular_a subsuperpertient_fw-fr etc_n etc_n the_o four_o definition_n example_n of_o this_o definition_n in_o magnitude_n example_n thereof_o in_o number_n note_n the_o five_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n why_o euclid_n in_o define_v of_o proportion_n use_v multiplication_n the_o six_o definition_n a_o example_n of_o this_o definitiin_n in_o magnitude_n a_o example_n in_o number_n an_o other_o example_n in_o number_n an_o other_o example_n in_o number_n note_v this_o particle_n according_a to_o any_o multiplication_n a_o example_n where_o the_o equimultiplice_n of_o the_o first_o and_o three_o excee_v the_o equimultiplice_n of_o the_o second_o and_o four_o and_o yet_o the_o quantity_n give_v be_v not_o in_o one_o and_o the_o self_n same_o proportion_n a_o rule_n to_o produce_v equimultiplice_n of_o the_o first_o and_o three_o equal_a to_o the_o equimultiplice_n of_o the_o secondâ—_n and_o forth_o example_n thereof_o the_o seven_o definition_n 9_z 12_o 3_o 4_o proportionality_n of_o two_o sort_n continual_a and_o discontinuall_a a_o example_n of_o continual_a proportionality_n in_o number_n 16.8.4.2.1_o in_o coutinnall_a proportionality_n the_o quantity_n can_v be_v of_o one_o kind_n discontinuall_a propâ—rtionalitie_n example_n of_o discontinual_a proportionality_n in_o number_n in_o discontinual_a proportionality_n the_o proportion_n may_v be_v of_o diverse_a kind_n the_o eight_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n a_o example_n in_o number_n note_n the_o nine_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n example_n â—n_a number_n the_o ten_o definition_n a_o rule_n to_o add_v proportion_n to_o proportion_n 8._o 4._o 2._o 1._o 2_o 2_o 2_o 1_o 1_o 1_o the_o eleven_o definition_n example_n of_o this_o definition_n in_o magnitude_n example_n in_o number_n the_o twelfâ—h_o definition_n example_n of_o this_o definition_n in_o magnitud_n example_n in_o number_n the_o thirteen_o definition_n example_n of_o this_o definition_n in_o magnitud_n example_n in_o number_n the_o fourteen_o definition_n example_n of_o this_o definition_n in_o magnitud_n example_n in_o number_n the_o fiâ—tâ—ne_a definition_n this_o be_v the_o converse_n of_o the_o former_a definition_n example_n in_o magnitude_n example_n in_o number_n the_o sixteen_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n a_o example_n in_o number_n the_o sevententh_n definition_n a_o example_n of_o this_o definition_n in_o magnitude_n a_o example_n in_o number_n note_n the_o eighttenth_n definition_n a_o example_n of_o this_o definition_n in_o magnitude_n example_n in_o number_n the_o nineteen_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n example_n in_o number_n the_o 20._o definition_n the_o 2d_o definition_n these_o two_o last_o definition_n not_o find_v in_o the_o greek_a exampler_n construction_n demonstration_n demonstrationâ—_n construction_n demonstration_n construction_n demonstration_n alemmae_fw-la or_o a_o assumpt_n a_o corollary_n converse_v proportion_n construction_n demonstration_n two_o case_n in_o this_o propotion_n the_o second_o the_o second_o part_n demonstrate_v the_o first_o part_n of_o this_o proposition_n demonstrate_v the_o second_o part_n of_o the_o proposition_n demonstrate_v first_o differâ—câ—_n of_o the_o first_o part_n demonstratiâ—_n of_o tâ—e_z same_o first_o difference_n second_o difference_n three_o difference_n the_o second_o part_n â—f_o this_o proposition_n the_o first_o parâ—_n of_o this_o proposition_n demonstrate_v the_o second_o part_n prove_v the_o first_o part_n of_o this_o proposition_n prove_v the_o second_o part_n demonstrate_v construction_n demonstrationâ—_n constrâ—ction_n demonstrationâ—_n construction_n demonstration_n a_o addition_n of_o campane_n demonstration_n construction_n demonstration_n demonstration_n of_o alternate_a proportion_n construction_n demonstration_n demonstration_n of_o proportion_n by_o division_n construction_n demonstration_n demonstration_n of_o proportion_n by_o composition_n this_o proposition_n be_v the_o converse_n of_o the_o former_a demonstrationâ—eâ—aing_v to_o a_o impossibility_n that_o which_o the_o five_o of_o this_o book_n prove_v only_o touch_v multiplices_fw-la this_o prove_v general_o of_o all_o magnitude_n alemma_n a_o corollary_n conversion_n of_o proportion_n this_o proposition_n pertain_v to_o proportion_n of_o equality_n inordinate_a proportionality_n the_o second_o difference_n the_o three_o difference_n thâ—r_a proposition_n pertain_v to_o proportion_n of_o equality_n in_o perturbate_n proportionality_n the_o three_o difference_n proportion_n of_o equality_n in_o ordinate_a proportionality_n construction_n demonstration_n when_o there_o be_v more_o than_o three_o magnitude_n in_o either_o order_n aâ—cdeâ—gh_n proportion_n of_o equality_n in_o perturbate_n proprotionalitie_n construction_n demonstration_n note_n that_o which_o the_o second_o proposition_n of_o this_o book_n prove_v only_o touch_v multiplices_fw-la be_v here_o prove_v general_o touch_v magnitude_n an_o other_o demonstration_n of_o the_o same_o affirmative_o an_o other_o demonstration_n of_o the_o same_o affirmative_o an_o other_o demonstration_n of_o the_o same_o demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n of_o the_o same_o affirmative_o demonstration_n demonstrationâ—_n demonstration_n the_o argument_n of_o this_o six_o book_n this_o book_n necessary_a for_o the_o use_n of_o instrument_n of_o geometry_n the_o first_o definition_n the_o second_o definition_n reciprocal_a figure_n call_v mutual_a figure_n the_o three_o definition_n the_o four_o definition_n the_o five_o definition_n an_o other_o example_n of_o substraction_n of_o proportion_n the_o six_o definition_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n a_o corollary_n add_v by_o flussates_n the_o first_o part_n of_o this_o theorem_a demonstration_n of_o the_o second_o part_n a_o corollary_n add_v by_o flussates_n construction_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o construction_n demonstration_n this_o be_v the_o converse_n of_o the_o former_a proposition_n construction_n demonstration_n construction_n the_o first_o part_n of_o this_o proposition_n demonstration_n lead_v to_o a_o impossibility_n the_o second_o part_n of_o this_o proposition_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o corollary_n out_o of_o flussates_n by_o this_o and_o the_o former_a proposition_n may_v a_o right_a line_n be_v divide_v into_o what_o part_n soever_o you_o will._n construction_n demonstration_n an_o other_o way_n after_o pelitarius_n a_o otâ—eâ—_n way_n after_o pelitarius_n construction_n demonstration_n an_o other_o way_n after_o campane_n construction_n demonstration_n a_o proposition_n add_v by_o pelitarius_n the_o
first_o part_n of_o this_o proposition_n demonstration_n of_o the_o of_o the_o same_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o the_o first_o parâ—_n of_o this_o proposition_n demonstration_n of_o the_o same_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o demonstration_n of_o the_o first_o part_n the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o the_o first_o part_n of_o thâ—â—_n theorem_a the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o a_o corollary_n description_n of_o the_o rectiline_a figure_n require_v demonstration_n demonstration_n a_o corollary_n the_o first_o parâ—_n of_o this_o theorem_a the_o second_o part_n demonstrate_v the_o three_o part_n the_o first_o corollary_n the_o second_o corollary_n demonstration_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o first_o note_v that_o this_o be_v prove_v in_o the_o assumpt_n follow_v a_o assumpt_n an_o other_o demonstration_n of_o the_o second_o part_n after_o flussates_n an_o other_o demonstration_n after_o flussate_n demonstration_n of_o this_o proposition_n wherein_o be_v first_o prove_v that_o the_o parallegramme_n eglantine_n be_v like_a to_o the_o whole_a parallelogram_n abcd._n that_o the_o parallelogram_n kh_o be_v like_a to_o the_o whole_a parallelogram_n abcd_o that_o the_o parallelogram_n eglantine_n and_o kh_o be_v like_o the_o one_o to_o the_o other_o an_o other_o demonstration_n after_o flussates_n a_o addition_n of_o pelitarius_n another_o addition_n of_o pelitarius_n construction_n demonstration_n demonstration_n demonstration_n by_o the_o dimetient_fw-la be_v understand_v here_o the_o dimetient_fw-la which_o be_v draw_v from_o the_o angle_n which_o be_v common_a to_o they_o both_o to_o the_o opposite_a angle_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o way_n after_o flussates_n in_o this_o proposition_n be_v two_o case_n in_o the_o first_o the_o parallelogram_n compare_v to_o the_o parallelogram_n describe_v of_o the_o half_a line_n be_v describe_v upon_o a_o line_n great_a they_o the_o half_a line_n in_o the_o second_o upon_o a_o line_n less_o the_o first_o case_n where_o the_o parellelogramme_n compare_v namely_o of_o be_v describe_v upon_o the_o line_n ak_o which_o be_v great_a than_o the_o half_a line_n ac_fw-la demonstration_n of_o this_o case_n the_o second_o case_n where_o the_o parallelogram_n compare_v namely_o ae_n be_v describe_v upon_o the_o line_n ad_fw-la which_o be_v less_o than_o the_o line_n ac_fw-la demonstration_n of_o the_o second_o case_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n a_o corollary_n add_v by_o flussates_n and_o be_v put_v of_o theon_n as_o a_o assumpt_n beâ—ore_o the_o 17._o proposition_n of_o the_o ten_o book_n which_o â—or_a that_o it_o follow_v of_o this_o proposition_n i_o think_v it_o not_o amiss_o here_o to_o place_n construction_n demonstration_n construction_n demonstration_n an_o other_o way_n construction_n demonstration_n the_o converse_n of_o the_o former_a proposition_n demonstration_n that_o the_o angle_n at_o thâ—_z centre_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o circumference_n whereon_o they_o be_v that_o the_o angle_n at_o the_o circumference_n be_v so_o also_o that_o the_o sector_n be_v so_o also_o construction_n of_o the_o problem_n demonstartion_n of_o the_o same_o the_o first_o corollary_n the_o second_o corollary_n the_o three_o corollary_n demonstration_n of_o this_o proposition_n demonstration_n of_o this_o proposition_n demonstration_n of_o this_o proposition_n demonstration_n of_o the_o first_o part_n of_o this_o proposition_n demonstration_n of_o the_o second_o part_n why_o euclid_n in_o the_o midst_n of_o his_o work_n be_v compel_v to_o add_v these_o three_o book_n of_o number_n arithmetic_n of_o more_o excellency_n than_o geometry_n thing_n intellectual_a of_o more_o worthiness_n theâ—_z thing_n sensible_a arithmetic_n minister_v prinâ—ciples_n and_o ground_n in_o a_o manner_n to_o all_o science_n boetius_fw-la cap._n 2._o lib._n prim_fw-la arithmeti_n timaus_n the_o argument_n of_o the_o seven_o book_n the_o first_o definition_n without_o unity_n shall_v be_v confusion_n of_o thing_n â—oetius_fw-la in_o his_o book_n dâ—_z unitate_fw-la &_o uno_fw-la an_o other_o desinition_n of_o unity_n the_o second_o definition_n difference_n between_o a_o point_n and_o unity_n boetius_fw-la an_o other_o desinition_n of_o number_n jordane_n an_o other_o definition_n of_o number_n unity_n have_v in_o it_o the_o virtue_n and_o power_n of_o all_o number_n number_n consider_v three_o manner_n of_o wayâ—_n the_o three_o definition_n the_o four_o definition_n the_o five_o definition_n the_o six_o definition_n boetius_fw-la an_o other_o definition_n of_o a_o even_a number_n note_n pithagoraâ—_n an_o other_o definition_n an_o other_o definition_n an_o other_o definition_n the_o seven_o definition_n an_o other_o definition_n of_o a_o odd_a number_n an_o other_o definition_n the_o eight_o definition_n campane_n an_o other_o definition_n of_o a_o even_o even_o number_n flussates_n an_o other_o definition_n boetius_fw-la an_o other_o definition_n the_o nine_o definition_n campane_n an_o other_o definition_n flussates_n an_o other_o definition_n an_o other_o definition_n the_o ten_o definition_n this_o definition_n not_o find_v in_o the_o greek_n an_o other_o definition_n boeâ—ius_n definition_n of_o a_o number_n even_o even_o and_o even_o â—d_a the_o eleven_o definition_n flusâ—ates_n an_o other_o definition_n the_o twelve_o definition_n prime_n number_n call_v incomposed_a number_n the_o thirteen_o definition_n the_o fourteen_o definition_n the_o fifteen_o definition_n the_o sixteen_o definition_n two_o number_n require_v in_o multiplication_n the_o seventeen_o definition_n why_o they_o be_v call_v superficial_a number_n the_o eighteen_o definition_n why_o they_o be_v call_v solid_a number_n the_o nineteen_o definition_n why_o it_o be_v call_v a_o square_a number_n the_o twenty_o definition_n why_o it_o be_v call_v a_o cube_fw-la number_n the_o twenty_o one_o definition_n why_o the_o definition_n of_o proportional_a magnitude_n be_v unlike_a to_o the_o definitio_fw-la of_o proportional_a number_n the_o twenty_o two_o definition_n the_o twenty_o three_o definition_n perfect_a number_n rare_a &_o of_o great_a use_n in_o magic_a &_o in_o secret_a philosophy_n in_o what_o respect_n a_o number_n be_v perfect_a two_o kind_n of_o imperfect_a number_n a_o â—â—mber_n wanâ—â—ngâ—_n common_a sentence_n â—irst_n common_a sentence_n â—â—cond_v â—ommon_a sentence_n three_o common_a sentence_n forth_o common_a sentence_n â—iâ—th_o common_a sentence_n six_o common_a sentence_n seven_o comâ—mon_a sentence_n constrâ—ctioâ—_n demonstratiâ—â—_n lead_v to_o a_o absurdity_n the_o converse_n of_o â—his_n proposition_n how_o to_o â—now_v whether_o two_o number_n give_v be_v prime_a the_o one_o to_o the_o other_o two_o case_n in_o this_o problem_n the_o first_o case_n the_o second_o case_n demonstration_n of_o the_o second_o case_n that_o cf_o be_v a_o common_a measure_n to_o the_o number_n ab_fw-la and_o cd_o that_o cf_o be_v the_o great_a common_a measure_n to_o ab_fw-la and_o cd_o the_o second_o case_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n this_o proposition_n and_o the_o 6._o proposition_n in_o discrete_a quantity_n answer_v to_o the_o first_o of_o the_o five_o in_o continual_a quantity_n demonstration_n construction_n demonstration_n thiâ—_n proposition_n and_o the_o next_o follow_v in_o discreet_a quantity_n answer_v to_o the_o five_o proposition_n of_o the_o five_o book_n in_o continual_a quantity_n construction_n demonstration_n constuâ—ction_n demonstration_n an_o other_o demonstration_n after_o flussates_n construction_n demonstration_n construction_n demonstration_n this_o proposition_n iâ—_z discreet_a quantity_n answer_v to_o the_o nine_o proposition_n of_o the_o five_o book_n in_o continual_a quantity_n demonstration_n this_o in_o discreet_a quantity_n answer_v to_o the_o twelve_o proposition_n of_o the_o five_o in_o continual_a quantity_n demonstration_n this_o in_o discrete_a quantity_n answer_v to_o the_o sixteen_o proposition_n of_o the_o five_o book_n in_o continual_a quantity_n note_n this_o in_o discrete_a quantity_n answer_v to_o tâ—â—_n twenty_o one_o proposition_n oâ—_n the_o five_o book_n in_o continual_a quantity_n demonstration_n certain_a addition_n of_o campane_n the_o second_o case_n proportionality_n divide_v proportionality_o compose_v euerse_a proportionality_n the_o conversâ—_n of_o the_o same_o prâ—position_n demonstration_n a_o corollary_n follow_v thâ—se_a proposition_n adâ—ed_v by_o campane_n coâ—strâ—ctioâ—_n demonstration_n demonstration_n â—emonstraâ—ion_n a_o corollary_n add_v by_o flussâ—tes_n demonstration_n this_o proposition_n and_o the_o former_a may_v be_v extend_v to_o number_n how_o many_o soever_o the_o second_o part_n of_o this_o proposition_n which_o be_v the_o converse_n of_o the_o first_o demonstration_n a_o assumpt_v add_v by_o campane_n this_o proposition_n in_o number_n demonstrate_v that_o which_o the_o 17._o of_o the_o six_o demonstrate_v in_o line_n demonstration_n the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o demonstration_n demonstration_n lead_v to_o a_o impossibility_n this_o proposition_n in_o discreet_a quantity_n answer_v to_o the_o 23._o proposition_n of_o the_o five_o book_n in_o continual_a quantity_n this_o and_o the_o eleven_o proposition_n follow_v
declare_v the_o passion_n and_o property_n ofâ—_n prime_a number_n demonstration_n lead_v to_o a_o impossibility_n this_o be_v the_o converse_n of_o the_o former_a proposition_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n demonstration_n demonstration_n demonstration_n of_o the_o first_o part_n lead_v to_o a_o absurdity_n demonstration_n of_o the_o second_o part_n which_o be_v the_o conâ—câ—se_n of_o the_o first_o lean_v also_o to_o a_o absurdity_n demonstrasion_n lead_v to_o a_o absurdity_n demonstrasion_n a_o corollary_n â—â—ded_v by_o campave_n demonstration_n lâ—ading_v to_o a_o impossibility_n an_o other_o demonstration_n demonstration_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n add_v by_o campane_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o absurdity_n the_o second_o caseâ—_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n leaâ—iâ—g_v â—o_o a_o absurdity_n the_o second_o case_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n demonstration_n the_o converse_n of_o the_o former_a proposition_n demonstration_n construction_n demonstration_n leâ—ding_v to_o a_o absurdity_n a_o corollary_n adâ—ed_v by_o campane_n how_o to_o â—inde_v out_o the_o second_o least_o number_n and_o the_o three_o and_o so_o â—orth_o infinite_o how_o to_o siâ—â—_n out_o the_o least_o â—â—mâ—_n a_o conâ—ayâ—â—g_n â—â—e_z paâ—â—s_n of_o part_n the_o arguâ—â—â—â—_n of_o the_o eight_o book_n demonstration_n lead_v to_o a_o absurdity_n construction_n demonstration_n this_o proposition_n be_v the_o â—â—uerse_a of_o the_o first_o demonstrationâ—_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o absurdity_n the_o second_o case_n demonstration_n this_o proposition_n in_o number_n answer_v to_o the_o of_o the_o six_o touch_v parellelogramme_n construction_n demonstration_n an_o other_o demonstration_n after_o campane_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n a_o corollary_n add_v by_o flussates_n construction_n demonstration_n this_o proposition_n be_v the_o converse_n of_o the_o former_a construction_n demonstration_n the_o first_o part_n of_o this_o proposition_n demonstrate_v the_o second_o part_n demonstrate_v construction_n the_o first_o part_n of_o this_o prâ—position_n demonstrate_v the_o second_o part_n demonstrate_v construction_n demonstration_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o a_o negative_a proportion_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o a_o negative_a proposition_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o demonstration_n of_o the_o fiâ—st_a part_n of_o this_o proposition_n demonstration_n of_o the_o second_o part_n demonstration_n of_o the_o first_o part_n of_o this_o proposition_n the_o second_o part_n this_o proposition_n be_v the_o converse_n of_o the_o 18._o proposition_n construction_n demonstration_n this_o proposition_n be_v the_o converse_n of_o the_o 19_o proposition_n construction_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n a_o corollary_n add_v by_o flussates_n construction_n construction_n demonstration_n a_o corollary_n add_v by_o flussates_n another_o corollary_n add_v by_o flussates_n the_o argument_n of_o the_o niâ—th_o book_n demonstration_n this_o proposition_n be_v the_o converse_n oâ—_n tâ—e_z formâ—â—_n demonstration_n a_o corollary_n aâ—ded_v by_o campane_n demonstration_n demonstration_n demonstration_n a_o corollary_n add_v by_o campane_n demonstration_n demonstration_n demonstration_n of_o the_o first_o part_n the_o second_o part_n demonstrate_v demostration_n of_o the_o three_o part_n demostration_n of_o the_o first_o part_n of_o this_o proposition_n the_o second_o pâ—rt_n demonstrate_v demonstration_n of_o the_o first_o part_n leave_v to_o a_o absurdity_n demonstration_n of_o the_o â—â—cond_a pâ—â—â—_n lead_v alâ—o_o to_o a_o absurdity_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o demonstration_n aâ—ter_n flussates_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o demonstration_n after_o campane_n demonstration_n lead_v to_o a_o absurdity_n a_o proposition_n add_v by_o campane_n construction_n demonstration_n demonstration_n to_o prove_v that_o the_o number_n a_o and_o c_o be_v prime_a to_o b._n demonstratiou_n this_o proposition_n be_v the_o converse_n of_o the_o former_a demonstration_n this_o answer_v to_o the_o 2._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 3._o of_o the_o three_o demonstration_n this_o answer_v to_o thâ—_z 4._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 5._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 6._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 7._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 8._o of_o the_o second_o demonstratition_n this_o answer_v to_o thâ—_z 9_o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 10._o oâ—_n the_o second_o demonstration_n a_o negative_a proposition_n demonstration_n leaâ—ing_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o abjurditie_n three_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n the_o three_o case_n divert_v case_n â—n_v this_o proposition_n the_o first_o case_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n a_o proposition_n add_v by_o campane_n a_o other_o add_v by_o he_o demonstration_n lead_v to_o a_o absurdity_n demonstration_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o demonstration_n demonstration_n demonstration_n this_o proposition_n teachâ—th_v how_o to_o find_v out_o a_o perfect_a number_n construction_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n the_o argument_n of_o the_o ten_o book_n difference_n between_o number_n and_o magnitude_n a_o line_n be_v not_o make_v of_o point_n as_o number_n be_v make_v of_o unity_n this_o book_n the_o hard_a to_o understand_v of_o all_o the_o book_n of_o euclid_n in_o this_o book_n be_v entreat_v of_o a_o strange_a manner_n of_o matter_n then_o in_o the_o former_a many_o even_o of_o the_o well_o learn_v have_v think_v that_o this_o book_n can_v not_o well_o be_v understand_v without_o algebra_n the_o nine_o former_a book_n &_o the_o principle_n of_o this_o â—ooke_n well_o understand_v this_o book_n will_v not_o be_v hard_a to_o understand_v the_o fâ—rst_a definition_n the_o second_o definition_n contraries_n make_v manifest_a by_o the_o compare_v of_o the_o one_o to_o the_o other_o the_o third_o definition_n what_o the_o power_n of_o a_o line_n be_v the_o four_o definition_n unto_o the_o suppose_a line_n first_o set_v may_v be_v compare_v infinite_a line_n why_o some_o mislike_n that_o the_o line_n first_o set_v shall_v be_v call_v a_o rational_a line_n flussates_n call_v this_o line_n a_o line_n certain_a this_o rational_a line_n the_o ground_n in_o a_o manner_n of_o all_o the_o proposition_n in_o this_o ten_o book_n note_n the_o line_n rational_a of_o purpose_n the_o six_o definition_n campâ—nus_n â—ath_z cause_v much_o obscurity_n in_o this_o ten_o book_n the_o seven_o definition_n flussates_n in_o steed_n of_o this_o word_n irrational_a use_v this_o word_n uncertain_a why_o they_o be_v call_v irrational_a line_n the_o cause_n of_o the_o obscurity_n and_o confusedness_n in_o this_o book_n the_o eight_o definition_n the_o nine_o definition_n the_o ten_o definition_n the_o eleven_o definition_n construction_n demonstration_n a_o corollary_n construction_n demonstration_n this_o proposition_n teach_v that_o incontinuall_a quantity_n which_o the_o first_o of_o the_o seven_o teach_v in_o discrete_a quantity_n construction_n demonstration_n lead_v to_o a_o absurdity_n two_o case_n in_o this_o proposition_n the_o first_o case_n this_o proposition_n teach_v that_o in_o continual_a quantity_n which_o the_o 2._o of_o the_o sâ—â—ith_v teach_v in_o number_n the_o second_o case_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n this_o problem_n reduce_v to_o a_o theorem_a this_o proposition_n teach_v that_o in_o continual_a quantity_n which_o the_o 3._o of_o the_o second_o teach_v in_o number_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o absurdity_n the_o second_o case_n a_o leâ—ma_n necessary_a
second_o part_n of_o the_o first_o case_n the_o second_o case_n first_o part_n of_o the_o second_o case_n second_o part_n of_o the_o second_o case_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o first_o part_n of_o the_o first_o case_n 〈◊〉_d second_o 〈◊〉_d of_o the_o 〈◊〉_d case_n the_o second_o case_n a_o corollary_n the_o first_o senary_a by_o substraction_n demonstration_n an_o other_o demonstration_n after_o campane_n definition_n of_o the_o eight_o irrational_a line_n definition_n of_o 〈◊〉_d â—inth_o irrational_a line_n an_o other_o demonstration_n after_o campane_n construction_n demonstration_n definition_n of_o the_o ten_o irrational_a line_n definition_n of_o the_o eleven_o irrational_a line_n â—â—â—â—iâ—ition_n of_o the_o twelve_o irraâ—ionall_a line_n definition_n of_o the_o thirteen_o and_o last_o irrational_a line_n a_o assumpt_n of_o campane_n i_o dee_n though_o campane_n lemma_n be_v true_a yeâ—_z the_o manner_n of_o demonstrate_v it_o narrow_o consider_v be_v not_o artificial_a second_o senary_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o absurdity_n construction_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n 〈◊〉_d a_o abjurdâ—tâ—â—â—_n six_o kind_n of_o reâ—iduall_a line_n first_o definition_n second_o definition_n three_o definition_n four_o definition_n five_o definition_n six_o definition_n three_o senary_n construction_n demonstration_n construction_n demoâ—stratiâ—â—_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o other_o more_o ready_a way_n to_o find_v out_o the_o six_o residual_a line_n four_o senary_n the_o â—irst_n parâ—_n of_o the_o construction_n the_o first_o part_n of_o the_o demonstration_n note_n a_z and_o fk_o conclude_v rational_a parallelogram_n note_n dh_o and_o fk_o parallelogram_n medial_a second_o part_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n ln_o be_v the_o only_a liâ—e_z â—hat_n we_o seek_v &_o consider_v first_o part_n of_o the_o construction_n the_o first_o part_n of_o thâ—_z demonstration_n a_z and_o fk_o conclude_v parallelogram_n medial_a dh_o &_o eke_o rational_a the_o second_o part_n of_o the_o construction_n the_o second_o part_n of_o the_o demonstration_n demonstration_n analytical_o the_o proâ—e_n hereof_o follow_v amongâ—_n other_o thing_n the_o line_n ln_o find_v which_o be_v the_o principal_a drift_n of_o all_o the_o former_a discourse_n the_o first_o part_n of_o the_o construction_n the_o fiâ—st_a part_n of_o the_o demonstration_n note_n a_z and_o fk_o medial_a note_n dh_o and_o eke_o medial_a note_n ai_fw-fr incommensurable_a to_o eke_o second_o part_n of_o the_o construction_n the_o principal_a line_n ln_o find_v find_v because_o the_o line_n of_o and_o â—g_z be_v prove_v commensurable_a in_o length_n length_n by_o the_o first_o oâ—_n the_o six_o and_o ten_o of_o the_o ten_o the_o first_o part_n of_o the_o construction_n the_o first_o part_n of_o the_o demonstration_n note_n ak_o rational_a note_n dk_o medial_a a_z and_o fk_o incommensurable_a the_o second_o part_n of_o the_o construction_n the_o second_o part_n of_o the_o demonstration_n ln_o the_o chief_a line_n of_o this_o theorem_a find_v demonstration_n the_o line_n ln_o demonstration_n the_o five_o senary_a these_o six_o proposition_n follow_v be_v the_o conuâ—rses_n of_o the_o six_o former_a proposition_n construction_n demonstration_n demonstration_n by_o the_o 20._o of_o the_o ten_o ten_o by_o the_o 21._o of_o the_o ten_o ten_o by_o the_o 22._o of_o the_o ten_o â—f_o conclude_v a_o residual_a line_n construction_n demonstration_n cf_o conclude_v a_o residual_a line_n construction_n demonstration_n cf_o conclude_v a_o residual_a line_n construction_n demonstration_n cf_o prove_v a_o residual_a line_n cf_o prove_v a_o residual_a line_n construction_n demonstration_n cf_o â—roved_v â—_o residual_a the_o six_o senary_n construction_n demonstration_n cd_o conclude_v a_o residual_a line_n note_n construction_n demonstration_n cd_o prove_v a_o medial_a construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n seven_o senary_n constraction_n demonstration_n construction_n demonstration_n demonstration_n construction_n demonstration_n on_o lead_v to_o a_o impossibility_n a_o corollary_n the_o determination_n have_v sundry_a part_n orderly_a to_o be_v prove_v construction_n demonstration_n this_o be_v a_o assumpt_n problematical_a artificial_o use_v and_o demonstrate_v demonstrate_v therefore_o those_o three_o line_n be_v in_o continual_a proportion_n fe_o conclude_v a_o residual_a liâ—â—_n which_o be_v sumwhat_o prepâ—â—icroâ—sly_o in_o respect_n oâ—_n the_o â—â—der_n propound_v both_o in_o the_o proposition_n and_o also_o in_o the_o determination_n construction_n demonstration_n construction_n demonstration_n here_o be_v the_o â—ower_a part_n of_o the_o propositiâ—_n more_o orderly_a huddle_v theâ—_z in_o the_o former_a demöstration_n construction_n demonstration_n a_o assumpt_n an_o other_o demonstration_n after_o flussas_n construction_n demonstration_n this_o be_v in_o a_o manner_n the_o converse_n of_o both_o the_o former_a proposition_n joint_o construction_n demonstration_n construction_n demonstration_n demonstration_n an_o other_o demonstration_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n lead_v to_o a_o impossibility_n the_o argument_n of_o the_o eleven_o book_n a_o point_n the_o beginning_n of_o all_o quantity_n continual_a the_o method_n use_v by_o euclid_n in_o the_o ten_o â—â—â—mer_a boot_n â—irst_n boâ—â—e_n second_o â—â—oâ—e_n three_o booâ—e_n forth_o bâ—oâ—e_n â—iveth_v boâ—â—e_n six_o booâ—e_n seven_o bookâ—_n â—ight_a booâ—â—_n nine_o book_n ten_o booâ—e_n what_o be_v entreaâ—ea_n of_o in_o the_o fiâ—e_a boot_n follow_v 〈◊〉_d â—â—â—ular_a bodiesâ—_n the_o â—â—all_a end_n 〈…〉_o oâ—_n i_o uâ—â—â—â—es_v â—eomeâ—â—â—all_a element_n comparison_n â—â—_o the_o 〈◊〉_d â—â—oâ—e_n and_o 〈◊〉_d book_n 〈◊〉_d first_o definition_n a_o solid_a the_o most_o perfect_a quantity_n no_o science_n of_o thing_n infinite_a second_o definition_n three_o definition_n two_o difâ—initions_n include_v in_o this_o definition_n declaration_n of_o the_o first_o part_n declaration_n of_o the_o second_o part_n four_o definition_n five_o definition_n six_o definition_n seven_o defâ—inition_n eight_o definition_n nine_o dissilition_n ten_o definition_n eleven_o definition_n an_o other_o definition_n of_o a_o prism_n which_o be_v a_o special_a definition_n of_o a_o prism_n as_o it_o be_v common_o call_v and_o use_v this_o body_n call_v figura_fw-la serratilis_fw-la psellus_n twelve_o definition_n what_o be_v to_o be_v taâ—â—n_v heed_n of_o in_o the_o definition_n of_o a_o sphere_n give_v by_o johannes_n de_fw-fr sacro_fw-la busco_n theodosiuâ—_n definition_n of_o a_o sphere_n the_o circumference_n of_o a_o sphere_n galens_n definition_n 〈◊〉_d a_o sphâ—râ—_n the_o dignity_n of_o a_o sphere_n a_o sphere_n call_v a_o globe_n thirteen_o definition_n theodosius_n definition_n of_o the_o axe_n of_o a_o sphere_n fourtenth_n definition_n theodosius_n definition_n of_o the_o centre_n of_o a_o sphere_n flussas_n definition_n of_o the_o centre_n of_o a_o sphere_n fivetenth_fw-mi definition_n difference_n between_o the_o diameter_n &_o axe_n of_o a_o sphere_n sevententh_n definition_n first_o kind_n of_o cones_n a_o cone_n call_v of_o campane_n a_o roâ—â—de_a pyramid_n sevententh_n definition_n a_o conical_a superficies_n eightenth_n definition_n ninetenth_fw-mi definition_n a_o cillindricall_a superficies_n corollary_n a_o roundâ—_n column_n or_o sphere_n a_o corollary_n add_v by_o campane_n twenty_o 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the_o line_n ab_fw-la be_v long_o than_o the_o line_n cd_o &_o if_o you_o take_v from_o they_o two_o equal_a line_n as_o ebb_v and_o fd_o the_o part_n remain_v which_o be_v the_o line_n ae_n and_o cf_o shall_v be_v unequal_a the_o one_o to_o the_o other_o namely_o the_o line_n ae_n shall_v be_v great_a than_o the_o line_n cf_o which_o be_v ever_o true_a in_o all_o quantity_n whatsoever_o 5_o and_o if_o to_o unequal_a thing_n you_o add_v equal_a thing_n the_o whole_a shall_v be_v unequal_a as_o if_o you_o have_v two_o unequal_a line_n namely_o ae_n the_o great_a and_o cf_o the_o less_o &_o if_o you_o add_v unto_o they_o two_o equal_a line_n ebb_n &_o fd_o then_o may_v you_o conclude_v that_o the_o whole_a line_n compose_v be_v unequal_a namely_o that_o the_o whole_a line_n ab_fw-la be_v great_a than_o the_o whole_a line_n cd_o and_o so_o of_o all_o other_o quantity_n 6_o thing_n which_o be_v double_a to_o one_o and_o the_o self_n same_o thing_n be_v equal_a the_o one_o to_o the_o 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be_v do_v be_v a_o theorem_a be_v a_o proposition_n which_o require_v the_o search_a ouâ—_n and_o demonstration_n of_o some_o property_n or_o passion_n of_o some_o figure_n wherein_o be_v only_a speculation_n and_o contemplation_n of_o mind_n without_o do_v or_o work_v of_o any_o thing_n as_o the_o five_o proposition_n of_o the_o first_o book_n which_o be_v thus_o a_o isosceles_a or_o triangle_n of_o two_o equal_a side_n have_v his_o angle_n at_o thâ—_z base_a equal_a the_o one_o to_o the_o other_o etc_n etc_n be_v a_o theorem_a for_o in_o it_o be_v require_v only_o to_o be_v prove_v and_o make_v plain_a by_o reason_n and_o demonstration_n that_o these_o two_o angle_n be_v equal_a without_o further_a work_n or_o do_v and_o in_o the_o endâ—_n of_o â—uâ—ry_a theorem_a after_o the_o demonstration_n be_v conclude_v after_o this_o manner_n which_o thing_n be_v require_v to_o be_v demonstrate_v or_o prove_v the_o first_o problem_n the_o first_o proposition_n upon_o a_o right_a line_n give_v not_o be_v infinite_a to_o describe_v a_o equilater_n triangle_n or_o a_o triangle_n of_o three_o equal_a side_n svppose_v that_o the_o right_a line_n give_v be_v ab_fw-la it_o be_v require_v upon_o the_o line_n ab_fw-la to_o describe_v a_o equilater_n triangle_n namely_o a_o triangle_n of_o three_o equal_a side_n construction_n now_o therefore_o make_v the_o centre_n the_o point_n a_o and_o the_o space_n ab_fw-la describe_v by_o the_o three_o petition_n a_o circle_n bcd_o and_o again_o by_o the_o same_o make_v the_o centre_n the_o point_n b_o and_o the_o space_n bâ—_n describe_v a_o other_o circle_n ace_n and_o by_o the_o first_o petition_n from_o the_o point_n c_o wherein_o the_o circle_n cut_v the_o one_o the_o other_o draw_v one_o right_a line_n to_o the_o point_n a_o and_o a_o other_o right_a line_n to_o the_o point_n b._n and_o forasmuch_o as_o the_o point_n a_o be_v the_o centre_n of_o the_o circle_n cbd_v demonstration_n therefore_o by_o the_o 15._o definition_n the_o line_n ac_fw-la be_v equal_a to_o the_o line_n ab_fw-la again_o forasmuch_o as_o the_o point_n b_o be_v the_o centre_n of_o the_o circle_n caesar_n therefore_o by_o the_o same_o definition_n the_o line_n bc_o be_v equal_a to_o the_o line_n basilius_n and_o it_o be_v prove_v that_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n ab_fw-la wherefore_o either_o of_o these_o line_n ca_n and_o cb_o be_v equal_a to_o the_o line_n ab_fw-la but_o thing_n which_o be_v equal_a to_o one_o and_o the_o same_o thing_n be_v also_o equal_a the_o one_o to_o the_o other_o by_o the_o first_o common_a sentence_n wherefore_o the_o line_n ca_n also_o be_v equal_a to_o the_o line_n cb._n wherefore_o these_o three_o right_a line_n ca_n ab_fw-la and_o bc_o be_v equal_a the_o one_o to_o the_o other_o wherefore_o the_o triangle_n abc_n be_v equilater_n wherefore_o upon_o the_o line_n ab_fw-la be_v describe_v a_o equilater_n triangle_n abc_n wherefore_o upon_o a_o line_n give_v not_o be_v infinite_a there_o be_v describe_v a_o equilater_n triangle_n which_o be_v the_o thing_n which_o be_v require_v to_o be_v do_v a_o triangle_n or_o any_o other_o rectiline_v figure_n be_v then_o say_v to_o be_v set_v or_o describe_v upon_o a_o line_n when_o the_o line_n be_v one_o of_o the_o side_n of_o the_o figure_n this_o first_o proposition_n be_v a_o problem_n because_o it_o require_v act_n or_o do_v namely_o to_o describe_v a_o triangle_n and_o this_o be_v to_o be_v note_v that_o every_o proposition_n whether_o it_o be_v a_o problem_n or_o a_o theorem_a common_o contain_v in_o it_o a_o thing_n give_v and_o a_o thing_n require_v to_o be_v search_v out_o although_o it_o be_v not_o always_o so_o and_o the_o thing_n give_v give_v be_v ever_o
side_n the_o rectiline_a angle_n mab_n either_o of_o which_o be_v equal_a to_o the_o rectiline_a angle_n give_v cde_o which_o be_v require_v to_o be_v do_v an_o other_o construction_n also_o and_o demonstration_n after_o pelitarâ—us_n and_o if_o the_o perpendicular_a line_n chance_n to_o fall_v without_o the_o angle_n give_v namely_o if_o the_o angle_n give_v be_v a_o acute_a angle_n the_o self_n same_o manner_n of_o demonstration_n will_v serve_v but_o only_o that_o in_o stead_n of_o the_o second_o common_a sentence_n must_v be_v use_v the_o 3._o common_a sentence_n appollonius_n put_v another_o construction_n &_o demonstration_n of_o this_o proposition_n which_o though_o the_o demonstration_n thereof_o depend_v of_o proposition_n put_v in_o the_o three_o book_n yet_o for_o that_o the_o construction_n be_v very_o good_a for_o he_o that_o will_v ready_o and_o mechanical_o without_o demonstration_n describe_v upon_o a_o line_n give_v and_o to_o a_o point_n in_o it_o give_v a_o rectiline_a angle_n equal_a to_o a_o rectiline_a angle_n give_v i_o think_v not_o amiss_o here_o to_o place_v it_o and_o it_o be_v thus_o proposition_n oenopides_n be_v the_o first_o inventor_n of_o this_o proposition_n as_o witness_v eudemius_fw-la the_o 15._o theorem_a the_o 24._o proposition_n if_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_a to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n and_o if_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o one_o be_v great_a than_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o other_o the_o base_a also_o of_o the_o same_o shall_v be_v great_a than_o the_o base_a of_o the_o other_o svppose_v that_o there_o be_v two_o triangle_n abc_n and_o def_n have_v two_o side_n of_o the_o one_o that_o be_v ab_fw-la and_o ac_fw-la equal_a to_o two_o side_n of_o the_o other_o that_o be_v to_o de_fw-fr and_o df_o each_o to_o his_o correspondent_a side_n that_o be_v the_o side_n ab_fw-la to_o the_o side_n de_fw-fr and_o the_o side_n ac_fw-la to_o the_o side_n df_o and_o suppose_v that_o the_o angle_n bac_n be_v great_a than_o the_o angle_n edf_o then_o i_o say_v that_o the_o base_a bc_o be_v great_a than_o the_o base_a ef._n for_o forasmuch_o as_o the_o angle_n bac_n be_v great_a than_o the_o angle_n edf_o construction_n make_v by_o the_o 23._o proposition_n upon_o the_o right_a line_n de_fw-fr and_o to_o the_o point_n in_o it_o give_v d_o a_o angle_n edge_v equal_a to_o the_o angle_n give_v bac_n and_o to_o one_o of_o these_o line_n that_o be_v either_o to_o ac_fw-la or_o df_o put_v a_o equal_a line_n dg_o and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n g_o to_o the_o point_n e_o and_o a_o other_o from_o the_o point_n fletcher_n demonstration_n to_o the_o point_n g._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n de_fw-fr and_o the_o line_n ac_fw-la to_o the_o line_n dg_o the_o one_o to_o the_o outstretch_o and_o the_o angle_n bac_n be_v by_o construction_n equal_a to_o the_o angle_n edg_n therefore_o by_o the_o 4._o proposition_n the_o base_a bc_o be_v equal_a to_o the_o base_a eglantine_n again_o for_o as_o much_o as_o the_o line_n dg_o be_v equal_a to_o the_o line_n df_o therâ—_n by_o the_o 5._o proposition_n the_o angle_n dgf_n be_v equal_a to_o the_o angle_n dfg_n wherefore_o the_o angle_n dfg_n be_v great_a than_o the_o angle_n egf_n wherefore_o the_o angle_n efg_o be_v much_o great_a than_o the_o angle_n egf_n and_o forasmuch_o as_o efg_o be_v a_o triangle_n have_v the_o angle_n efg_o great_a than_o the_o angle_n egf_n and_o by_o the_o 18._o prâ—position_n under_o the_o great_a angle_n be_v subtend_v the_o great_a side_n therefore_o the_o side_n eglantine_n be_v great_a than_o the_o side_n ef._n but_o the_o side_n eglantine_n be_v equal_a to_o the_o side_n bc_o wherefore_o the_o side_n bc_o be_v great_a than_o the_o side_n ef._n if_o therefore_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_a to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n and_o if_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o one_o be_v great_a than_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o other_o the_o base_a also_o of_o the_o same_o shall_v be_v great_a than_o the_o base_a of_o the_o other_o which_o be_v require_v to_o be_v prove_v in_o this_o theorem_a may_v be_v three_o case_n prâ—position_n for_o the_o angle_n edg_n be_v put_v equal_a to_o the_o angle_n bac_n and_o the_o line_n dg_o be_v put_v equal_a to_o the_o line_n ac_fw-la and_o a_o line_n be_v draw_v from_o e_o to_o g_z the_o line_n eglantine_n shall_v either_o fall_v above_o the_o line_n gf_o or_o upon_o it_o or_o under_o it_o euclides_n demonstration_n serve_v 2._o when_o the_o line_n ge_z fall_v above_o the_o line_n gf_o as_o we_o have_v before_o manifest_o see_v but_o now_o let_v the_o line_n eglantine_n case_n fall_v under_o the_o line_n e_o fletcher_n as_o in_o the_o figure_n here_o put_v and_o forasmuch_o as_o these_o two_o line_n ab_fw-la and_o ac_fw-la be_v equal_a to_o these_o two_o line_n de_fw-fr and_o dg_o the_o one_o to_o the_o other_o and_o they_o contain_v equal_a angle_n therefore_o by_o the_o 4._o proposition_n the_o base_a bc_o be_v equal_a to_o the_o base_a eglantine_n and_o forasmuch_o as_o within_o the_o triangle_n deg_n the_o two_o linne_n df_o and_o fe_o be_v set_v upon_o the_o side_n de_fw-fr therefore_o by_o the_o 21._o proposition_n the_o line_n df_o and_o fâ—_n be_v less_o than_o the_o outward_a line_n dg_o and_o ge_z but_o the_o line_n dg_o be_v equal_a to_o the_o line_n df._n wherefore_o the_o line_n ge_z be_v great_a than_o the_o line_n fe_o but_o ge_z be_v equal_a to_o bc._n wherefore_o the_o line_n bc_o be_v great_a the_o the_o line_n ef._n which_o be_v require_v to_o be_v prove_v it_o may_v peradventure_o semeâ—_n that_o euclid_n shall_v here_o in_o this_o proposition_n have_v prove_v that_o not_o only_o the_o base_n of_o the_o triangle_n be_v unequal_a but_o also_o that_o the_o area_n of_o the_o same_o be_v unequal_a for_o so_o in_o the_o four_o proposition_n after_o he_o have_v prove_v the_o base_a to_o be_v equal_a he_o prove_v also_o the_o area_n to_o be_v equal_a but_o hereto_o may_v be_v answer_v &c._a that_o in_o equal_a angle_n and_o base_n and_o unequal_a angle_n and_o base_n the_o consideration_n be_v not_o like_a for_o the_o angle_n and_o base_n be_v equal_a the_o triangle_n also_o shall_v of_o necessity_n be_v equal_a but_o the_o angle_n and_o base_n be_v unequal_a the_o area_n shall_v not_o of_o necessity_n be_v equal_a for_o the_o triangle_n may_v both_o be_v equal_a and_o unequal_a and_o that_o may_v be_v the_o great_a which_o have_v the_o great_a angle_n and_o the_o great_a base_n and_o it_o may_v also_o be_v the_o less_o and_o for_o that_o cause_n euclid_n make_v not_o mention_v of_o the_o comparison_n of_o the_o triangle_n whereof_o this_o also_o may_v be_v a_o cause_n for_o that_o to_o the_o demonstration_n thereof_o be_v require_v certain_a proposition_n concern_v parallel_a line_n which_o we_o be_v not_o as_o yet_o come_v unto_o etc_n howbeit_o after_o the_o 37â—_n proposition_n of_o his_o book_n you_o shall_v find_v the_o comparison_n of_o the_o area_n of_o triangle_n which_o have_v their_o side_n equal_a and_o their_o base_n and_o angle_n at_o the_o top_n unequal_a the_o 16._o theorem_a the_o 25._o proposition_n if_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_a to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n and_o if_o the_o base_a of_o the_o one_o be_v great_a than_o the_o base_a of_o the_o other_o the_o angle_n also_o of_o the_o same_o contain_v under_o the_o equal_a right_a linesâ—_n shall_v be_v great_a than_o the_o angle_n of_o the_o other_o svppose_v that_o there_o be_v two_o triangle_n a_o b_o c_o and_o def_n have_v two_o side_n of_o tb'one_n that_o be_v ab_fw-la and_o ac_fw-la equal_a to_o two_o side_n of_o the_o other_o that_o be_v to_o de_fw-fr and_o df_o each_o to_o his_o correspondent_a side_n namely_o the_o side_n ab_fw-la to_o the_o side_n df_o and_o the_o side_n ac_fw-la to_o the_o side_n df._n but_o let_v the_o base_a bc_o be_v great_a than_o the_o base_a ef._n then_o i_o say_v they_o the_o angle_n bac_n be_v great_a than_o the_o angle_n edf_o for_o if_o not_o absurdity_n then_o be_v it_o either_o equal_a unto_o it_o or_o less_o than_o it_o but_o the_o angle_n bac_n be_v not_o equal_a to_o the_o angle_n edf_o for_o if_o it_o be_v equal_a the_o base_a also_o bc_o shall_v by_o the_o 4._o proposition_n be_v equal_a to_o the_o base_a of_o but_o by_o supposition_n it_o be_v not_o wherefore_o
namely_o the_o first_o ten_o proposition_n as_o they_o follow_v in_o order_n which_o be_v undoubted_o great_a pleasure_n to_o consider_v also_o great_a increase_n &_o furniture_n of_o knowledge_n who_o proposition_n be_v set_v orderly_a after_o the_o proposition_n of_o euclid_n every_o one_o of_o barlaam_n correspondent_a to_o the_o same_o of_o euclid_n and_o doubtless_o it_o be_v wonderful_a to_o see_v how_o these_o two_o contrary_a kind_n of_o quantity_n quantity_n discrete_a or_o number_n and_o quantity_n continual_a or_o magnitude_n which_o be_v the_o subject_n or_o matterâ—_n of_o arithmitique_a and_o geometry_n shall_v have_v in_o they_o one_o and_o the_o same_o propriety_n common_a to_o they_o both_o in_o very_a many_o point_n and_o affection_n although_o not_o in_o all_o for_o a_o line_n may_v in_o such_o sort_n be_v divide_v that_o what_o proportion_n the_o whole_a have_v to_o the_o great_a part_n the_o same_o shall_v the_o great_a part_n have_v to_o the_o less_o but_o that_o can_v not_o be_v in_o number_n for_o a_o number_n can_v not_o so_o be_v divide_v that_o the_o whole_a number_n to_o the_o great_a part_n thereof_o shall_v have_v that_o proportion_n which_o the_o great_a part_n have_v to_o the_o less_o as_o jordane_n very_o plain_o prove_v in_o his_o book_n of_o arithmetic_n which_o thing_n campane_n also_o as_o we_o shall_v afterward_o in_o the_o 9_o book_n after_o the_o 15._o proposition_n see_v prove_v and_o as_o touch_v these_o ten_o first_o proposition_n of_o the_o second_o book_n of_o euclid_n demonstrate_v by_o barlaam_n in_o number_n they_o be_v also_o demonstrate_v of_o campane_n after_o the_o 15._o proposition_n of_o the_o 9_o book_n who_o demonstration_n i_o mind_n by_o god_n help_n to_o set_v forth_o when_o i_o shall_v come_v to_o the_o place_n they_o be_v also_o demonstrate_v of_o jordane_n that_o excellet_fw-la learned_a author_n in_o the_o first_o book_n of_o his_o arithmetic_n in_o the_o mean_a time_n i_o think_v it_o not_o amiss_o here_o to_o set_v forth_o the_o demonstration_n of_o barlaam_n for_o that_o they_o geve_v great_a light_n to_o the_o second_o book_n of_o euclid_n beside_o the_o inestimable_a pleasure_n which_o they_o bring_v to_o the_o studious_a considerer_n and_o now_o to_o declare_v the_o first_o proposition_n by_o number_n i_o have_v put_v this_o example_n follow_v take_v two_o number_n the_o one_o undivided_a as_o 74._o the_o other_o divide_v into_o what_o part_n and_o how_o many_o you_o listen_v as_o 37._o divide_v into_o 20._o 10._o 5._o and_o 2d_o which_o altogether_o make_v the_o whole_a 37._o then_o if_o you_o multiply_v the_o number_n undivided_a namely_o 74_o into_o all_o the_o part_n of_o the_o number_n divide_v as_o into_o 20._o 10._o 5._o and_o 2._o you_o shall_v produce_v 1480._o 740._o 370._o 148._o which_o add_v together_o make_v 2738_o which_o self_n number_n be_v also_o produce_v if_o you_o multiply_v the_o two_o number_n first_o give_v the_o one_o into_o the_o other_o as_o you_o see_v in_o the_o example_n on_o the_o other_o side_n set_v so_o by_o the_o aid_n of_o this_o proposition_n be_v get_v a_o compendious_a way_n of_o multiplication_n by_o break_v of_o one_o of_o the_o number_n into_o his_o part_n which_o oftentimes_o serve_v to_o great_a use_n in_o working_n chiâ—â—ly_o in_o the_o rule_n of_o proportion_n the_o demonstration_n of_o which_o proposition_n follow_v in_o barlaam_n but_o â—irst_n be_v put_v of_o the_o author_n these_o principle_n follow_v barlaam_n ¶_o principles_n 1._o a_o number_n be_v sâ—yd_v to_o multiply_v a_o other_o number_n when_o the_o number_n multiply_v be_v so_o oftentimes_o add_v to_o itself_o as_o there_o be_v unity_n in_o the_o number_n which_o multiply_v whereby_o be_v produce_v a_o certain_a number_n which_o the_o number_n multiply_v measure_v by_o the_o unity_n which_o be_v in_o the_o number_n which_o multipli_v 2._o and_o the_o number_n produce_v of_o that_o a_o multiplication_n be_v call_v a_o plain_a or_o superficial_a number_n 3._o a_o square_a number_n be_v that_o which_o be_v produce_v of_o the_o multiplicatian_n of_o any_o number_n into_o itself_o 4._o every_o less_o number_n compare_v to_o a_o great_a be_v say_v to_o be_v a_o part_n of_o the_o great_a whether_o the_o less_o measure_n the_o great_a or_o measure_v it_o not_o 5._o number_n who_o one_o and_o the_o self_n same_o number_n measure_v equal_o that_o be_v by_o one_o and_o the_o self_n same_o number_n be_v equal_a the_o one_o to_o the_o other_o 6._o number_n that_o be_v equemultiplâ—ces_n to_o one_o and_o the_o self_n same_o number_n that_o be_v which_o contain_v one_o and_o the_o same_o number_n equal_o and_o alike_o be_v equal_a the_o one_o to_o the_o other_o the_o first_o proposition_n two_o number_n be_v give_v if_o the_o one_o of_o they_o be_v divide_v into_o any_o number_n how_o many_o soever_o barlaam_n the_o plain_n or_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o two_o number_n first_o give_v the_o one_o into_o the_o other_o shall_v be_v equal_a to_o the_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n not_o divide_v into_o every_o part_n of_o the_o number_n divide_v suppose_v that_o there_o be_v two_o number_n ab_fw-la and_o c._n and_o divide_v the_o number_n ab_fw-la into_o certain_a other_o number_n how_o many_o soever_o as_o into_o ad_fw-la de_fw-fr and_o ebb_v then_o i_o say_v that_o the_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n c_o into_o the_o number_n ab_fw-la be_v equal_a to_o the_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n c_o into_o the_o number_n ad_fw-la and_o of_o c_o into_o de_fw-fr and_o of_o c_o into_o ebb_n for_o let_v fletcher_n be_v the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n c_o into_o the_o number_n ab_fw-la and_o let_v gh_o be_v the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o c_o into_o ad_fw-la and_o let_v high_a be_v produce_v of_o the_o multiplication_n of_o c_o into_o de_fw-fr aâ—d_v final_o of_o the_o multiplication_n of_o c_o into_o ebb_n let_v there_o be_v produce_v the_o number_n ik_fw-mi now_o forasmuch_o as_o ab_fw-la multiply_v the_o number_n c_o produce_v the_o number_n f_o therefore_o the_o number_n c_o measure_v the_o number_n f_o by_o the_o unity_n which_o be_v in_o the_o number_n ab_fw-la and_o by_o the_o same_o reason_n may_v be_v prove_v that_o the_o number_n c_o do_v also_o measure_v the_o number_n gh_o by_o the_o unity_n which_o be_v in_o the_o number_n ad_fw-la and_o that_o it_o do_v measure_v the_o number_n high_a by_o the_o unity_n which_o be_v in_o the_o number_n df_o and_o final_o that_o it_o measure_v the_o number_n ik_fw-mi by_o the_o unity_n which_o be_v in_o the_o number_n ebb_n wherefore_o the_o number_n c_o measure_v the_o whole_a number_n gk_o by_o the_o unity_n which_o be_v in_o the_o number_n ab_fw-la but_o it_o before_o measure_v the_o number_n f_o by_o the_o unity_n which_o be_v in_o the_o number_n ab_fw-la wherefore_o either_o of_o these_o number_n fletcher_n and_o gk_o be_v equemultiplex_n to_o the_o number_n c._n but_o number_n which_o be_v equemultiplices_fw-la to_o one_o &_o the_o self_n same_o number_n be_v equal_a the_o one_o to_o the_o other_o by_o the_o 6._o definition_n wherefore_o the_o number_n fletcher_n be_v equal_a to_o the_o number_n gk_o but_o the_o number_n fletcher_n be_v the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n c_o into_o the_o number_n ab_fw-la and_o the_o number_n gk_o be_v compose_v of_o the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n c_o not_o divide_v into_o every_o one_o of_o the_o number_n ad_fw-la de_fw-fr and_o ebb_v if_o therefore_o there_o be_v two_o number_n give_v and_o the_o one_o of_o they_o be_v divide_v etc_n etc_n which_o be_v require_v to_o be_v prove_v the_o 2._o theorem_a the_o 2._o proposition_n if_o a_o right_a line_n be_v divide_v by_o chance_n the_o rectangle_v figure_n which_o be_v comprehend_v under_o the_o whole_a and_o every_o one_o of_o the_o part_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o whole_a svppose_v that_o the_o right_a line_n ab_fw-la be_v by_o chance_n deny_v in_o the_o point_n c._n then_o i_o say_v that_o the_o rectangle_n figure_n comprehend_v under_o ab_fw-la and_o bc_o together_o with_o the_o rectangle_n comprehend_v under_o ab_fw-la and_o ac_fw-la be_v equal_a unto_o the_o square_n make_v of_o ab_fw-la construction_n describe_v by_o the_o 46._o of_o the_o first_o upon_o ab_fw-la a_o square_a adeb_n and_o by_o the_o 31_o of_o the_o first_o by_o the_o point_n c_o draw_v a_o line_n parallel_n unto_o either_o of_o these_o line_n ad_fw-la aâ—d_fw-la be_v and_o let_v the_o same_o
number_n of_o the_o number_n ad_fw-la and_o db_o be_v double_a to_o the_o square_a number_n of_o ac_fw-la and_o cd_o for_o forasmuch_o as_o the_o number_n ad_fw-la be_v divide_v into_o the_o number_n ab_fw-la and_o bd_o therefore_o the_o square_a number_n of_o the_o number_n ad_fw-la and_o db_o be_v equal_a to_o the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n ad_fw-la and_o db_o the_o on_o into_o the_o other_o twice_o together_o with_o the_o square_n of_o the_o number_n ab_fw-la by_o the_o 7_o proposition_n but_o the_o square_n of_o the_o number_n ab_fw-la be_v equal_a to_o four_o square_n of_o either_o of_o the_o number_n ac_fw-la or_o cb_o for_o ac_fw-la be_v equal_a to_o the_o number_n cb_o wherefore_o also_o the_o square_n of_o the_o number_n ad_fw-la and_o db_o be_v equal_a to_o the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n ad_fw-la and_o db_o the_o one_o into_o the_o other_o twice_o and_o to_o four_o square_n of_o the_o number_n bc_o or_o ca._n and_o forasmuch_o as_o the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n ad_fw-la and_o db_o the_o one_o into_o the_o other_o together_o with_o the_o square_n of_o the_o number_n cb_o be_v equal_a to_o square_v of_o the_o number_n cd_o by_o the_o 6_o proposition_n therefore_o the_o number_n produce_v of_o the_o multiplication_n of_o the_o number_n ad_fw-la and_o db_o the_o one_o into_o the_o other_o twice_o together_o with_o two_o square_n of_o the_o number_n cb_o be_v equal_a to_o two_o square_n of_o the_o number_n cd_o wherefore_o the_o square_n of_o the_o number_n ad_fw-la and_o db_o be_v equal_a to_o two_o square_n of_o the_o number_n cd_o and_o to_o two_o square_n of_o the_o number_n ac_fw-la wherefore_o they_o be_v double_a to_o the_o square_n of_o the_o number_n ac_fw-la and_o cd_o and_o the_o square_a of_o the_o number_n ad_fw-la be_v the_o square_n of_o the_o whole_a and_o of_o the_o number_n add_v and_o the_o square_a of_o db_o be_v the_o square_a of_o the_o number_n add_v the_o square_a also_o of_o the_o number_n cd_o be_v the_o square_a of_o the_o number_n compose_v of_o the_o half_a and_o of_o the_o number_n add_v if_o therefore_o a_o even_a number_n be_v divide_v etc_n etc_n which_o be_v require_v to_o be_v prove_v the_o 1._o problem_n the_o 11._o proposition_n to_o divide_v a_o right_a line_n give_v in_o such_o sort_n that_o the_o rectangle_n figure_n comprehend_v under_o the_o whole_a and_o one_o of_o the_o part_n shall_v be_v equal_a unto_o the_o square_n make_v of_o the_o other_o part_n svppose_v that_o the_o right_a line_n give_v be_v ab_fw-la now_o it_o be_v require_v to_o divide_v the_o line_n ab_fw-la in_o such_o sort_n that_o the_o rectangle_n figure_n contain_v under_o the_o whole_a and_o one_o of_o the_o part_n shall_v be_v equal_a unto_o the_o square_n which_o be_v make_v of_o the_o other_o part_n describe_v by_o the_o 46._o of_o the_o first_o upon_o ab_fw-la a_o square_a abcd._n construction_n and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n ac_fw-la into_o two_o equal_a part_n in_o the_o point_n e_o and_o draw_v a_o line_n from_o b_o to_o e._n and_o by_o the_o second_o petition_n extend_v ca_n unto_o the_o point_n f._n and_o by_o the_o 3._o of_o the_o first_o put_v the_o line_n of_o equal_a unto_o the_o line_n be._n and_o by_o the_o 46._o of_o the_o first_o upon_o the_o line_n of_o describe_v a_o square_a fgah_n and_o by_o the_o 2._o petition_n extend_v gh_o unto_o the_o point_n k._n then_o i_o say_v that_o the_o line_n ab_fw-la be_v divide_v in_o the_o point_n h_o in_o such_o sort_n that_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o ab_fw-la and_o bh_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o ah_o demonstration_n for_o forasmuch_o as_o the_o right_a line_n ac_fw-la be_v divide_v into_o two_o equal_a part_n in_o the_o point_n e_o and_o unto_o it_o be_v add_v a_o other_o right_a line_n af._n therefore_o by_o the_o 6._o of_o the_o second_o the_o rectangle_n figure_n contain_v under_o cf_o and_o favorina_n together_o with_o the_o square_n which_o be_v make_v of_o ae_n be_v equal_a to_o the_o square_v which_o be_v make_v of_o ef._n but_o of_o be_v equal_a unto_o ebb_n wherefore_o the_o rectangle_n figure_n contain_v under_o cf_o and_o favorina_n together_o with_o the_o square_n which_o be_v make_v of_o ea_fw-la be_v equal_a to_o the_o square_n which_o be_v make_v of_o ebb_n but_o by_o 47._o of_o the_o first_o unto_o the_o square_n which_o be_v make_v of_o ebb_n be_v equal_a the_o square_n which_o be_v make_v of_o basilius_n and_o ae_n for_o the_o angle_n at_o the_o point_n a_o be_v a_o right_a angle_n wherefore_o that_o which_o be_v contain_v under_o cf_o and_o favorina_n together_o with_o the_o square_n which_o be_v make_v of_o ae_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o basilius_n and_o ae_n take_v away_o the_o square_n which_o be_v make_v of_o ae_n which_o be_v common_a to_o they_o both_o wherefore_o the_o rectangle_n figure_n remain_v contain_v under_o cf_o and_o favorina_n be_v equal_a unto_o the_o square_n which_o be_v make_v of_o ab_fw-la and_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o favorina_n be_v the_o figure_n fk_o for_o the_o line_n favorina_n be_v equal_a unto_o the_o line_n fg._n and_o the_o square_n which_o be_v make_v of_o ab_fw-la be_v the_o figure_n ad._n wherefore_o the_o figure_n fk_o be_v equal_a unto_o the_o figure_n ad._n take_v away_o the_o figure_n ak_o which_o be_v common_a to_o they_o both_o wherefore_o the_o residue_n namely_o the_o figure_n fh_o be_v equal_a unto_o the_o residue_n namely_o unto_o the_o figure_n hd_v but_o the_o figure_n hd_v be_v that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bh_o for_o ab_fw-la be_v equal_a unto_o bd._n and_o the_o figure_n fh_o be_v the_o square_n which_o be_v make_v of_o ah_o wherefore_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n ab_fw-la and_o bh_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n ha._n wherefore_o the_o right_a line_n give_v ab_fw-la be_v divide_v in_o the_o point_n h_o in_o such_o sort_n that_o the_o rectangle_n figure_n contain_v under_o ab_fw-la and_o bh_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o ah_o which_o be_v require_v to_o be_v do_v this_o proposition_n have_v many_o singular_a use_n upon_o it_o depend_v the_o demonstration_n of_o that_o worthy_a problem_n the_o 10._o proposition_n of_o the_o 4._o book_n proposition_n which_o teach_v to_o describe_v a_o isosceles_a triangle_n in_o which_o either_o of_o the_o angle_n at_o the_o base_a shall_v be_v double_a to_o the_o angle_n at_o the_o top_n many_o and_o diverse_a use_n of_o a_o line_n so_o divide_v shall_v you_o find_v in_o the_o 13._o book_n of_o euclid_n this_o be_v to_o be_v note_v that_o this_o proposition_n can_v not_o as_o the_o former_a proposition_n of_o this_o second_o book_n be_v reduce_v unto_o number_n number_n for_o the_o line_n ebb_n have_v unto_o the_o line_n ae_n no_o proportion_n that_o can_v be_v name_v and_o therefore_o it_o can_v not_o be_v express_v by_o number_n for_o forasmuch_o as_o the_o square_n of_o ebb_n be_v equal_a to_o the_o two_o square_n of_o ab_fw-la and_o ae_n by_o the_o 47._o of_o the_o first_o and_o ae_n be_v the_o half_a of_o ab_fw-la therefore_o the_o line_n be_v be_v irrational_a for_o even_o as_o two_o equal_a square_a number_n join_v together_o can_v not_o make_v a_o square_a number_n so_o also_o two_o square_a number_n of_o which_o the_o one_o be_v the_o square_a of_o the_o half_a root_n of_o the_o other_o can_v not_o make_v a_o square_a number_n as_o by_o a_o example_n take_v the_o square_n of_o 8._o which_o be_v 64._o which_o double_v that_o be_v 128._o make_v not_o a_o square_a number_n so_o take_v the_o half_a of_o 8._o which_o be_v 4._o and_o the_o square_n of_o 8._o and_o 4._o which_o be_v 64._o and_o 16._o add_v together_o likewise_o make_v not_o a_o square_a number_n for_o they_o make_v 80._o who_o have_v no_o root_n square_v which_o thing_n must_v of_o necessity_n be_v if_o this_o problem_n shall_v have_v place_n in_o number_n but_o in_o irrational_a number_n it_o be_v true_a and_o may_v by_o this_o example_n be_v declare_v let_v 8._o be_v so_o divide_v that_o that_o which_o be_v produce_v of_o the_o whole_a into_o one_o of_o his_o part_n shall_v be_v equal_a to_o the_o square_a number_n produce_v of_o the_o other_o part_n multiply_v 8._o into_o himself_o and_o there_o shall_v be_v produce_v 64._o that_o be_v the_o square_a abcd._n divide_v 8._o into_o two_o equal_a part_n that_o be_v into_o 4_o and_o 4._o as_o the_o line_n
&_o m._n demonstrationâ—_n if_o therefore_o g_z exceed_v l_o then_z also_o h_o exceed_v m_o and_o if_o it_o be_v equal_a it_o be_v equal_a and_o if_o it_o be_v less_o it_o be_v less_o by_o the_o converse_n of_o the_o 6â—_o definition_n of_o the_o five_o again_o because_o that_o as_o c_o be_v to_o d_z so_o be_v e_o to_o fletcher_n and_o to_o c_o and_o e_o be_v take_v â—â—â—emâ—ltiplices_n h_o â—â—d_z king_n and_o likewise_o to_o d_z &_o f_o be_v take_v certain_a other_o equemultiplices_fw-la m_o &_o n._n if_o therefore_o h_o exceed_v m_o than_o also_o king_n exceed_v n_o and_o if_o it_o be_v equal_a it_o be_v equal_a and_o if_o it_o be_v less_o it_o be_v less_o by_o the_o same_o converse_n but_o if_o king_n exceed_v m_o than_o also_o g_z exceed_v l_o and_o if_o it_o be_v equal_a it_o be_v equal_a and_o if_o it_o be_v less_o it_o be_v less_o by_o the_o same_o converse_n wherefore_o if_o g_z exceed_v l_o then_z king_n also_o exceed_v n_o and_o if_o it_o be_v equal_a it_o be_v equal_a and_o if_o it_o be_v less_o it_o be_v less_o but_o g_z &_o king_n be_v equemultiplices_fw-la of_o a_o &_o e._n and_o l_o &_o n_o be_v certain_a other_o equemultiplices_fw-la of_o b_o &_o f._n wherefore_o by_o the_o 6._o definition_n as_o a_o be_v to_o b_o so_o be_v e_o to_o f._n proportion_n therefore_o which_o be_v one_o and_o the_o self_n same_o to_o any_o one_o proportion_n be_v also_o the_o self_n same_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v the_o 12._o theorem_a the_o 12._o proposition_n if_o there_o be_v a_o number_n of_o magnitude_n how_o many_o soeâ—â—r_n proportional_a 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 svppose_v that_o there_o be_v a_o number_n of_o magnitude_n how_o many_o soever_o namely_o a_o b_o c_o d_o e_o f_o in_o proportion_n so_o that_o as_o a_o be_v to_o b_o so_o let_v c_o be_v to_o d_o and_o e_o to_o f._n then_o i_o say_v that_o as_o a_o be_v to_o b_o so_o 〈◊〉_d a_o c_o e_o to_o b_o d_o f._n take_v equemultiplices_fw-la to_o a_o c_o and_o e._n constrâ—ction_n and_o let_v the_o same_o be_v g_o h_o k._n and_o likewise_o to_o b_o d_o and_o fletcher_n â—ake_z any_o other_o equemultiplices_fw-la which_o to_o be_v l_o m_o n._n and_o because_o that_o 〈◊〉_d a_o be_v to_o b_o so_o iâ—_z c_o to_o d_z and_o e_o to_o f._n demonstrationâ—_n and_o to_o a_o c_o e_o be_v take_v â—quemultiplices_n g_o h_o king_n and_o likewise_o to_o â—_o d_o f_o be_v take_v certain_a other_o equemâ—â—tipliâ—â—s_n l_o m_o n._n if_o therefore_o g_z exceed_v l_o h_o also_o exceed_v m_o and_o kn_n and_o if_o it_o be_v equal_a it_o be_v equal_a and_o if_o it_o be_v less_o it_o be_v less_o â—y_a the_o converse_n of_o the_o sixâ—_o definition_n of_o the_o five_o wherefore_o if_o g_z exceed_v l_o then_z g_z h_o edward_n also_o exceed_v l_o m_o n_o and_o if_o they_o be_v equal_a they_o be_v equal_a and_o if_o they_o be_v less_o they_o be_v less_o by_o the_o same_o but_o g_z and_o g_z h_o edward_n be_v equemultiplices_fw-la to_o the_o magnitude_n a_o and_o to_o the_o magnitude_n a_o c_o e._n for_o by_o the_o first_o of_o the_o five_o if_o there_o be_v a_o number_n of_o magnitude_n equemultiplices_fw-la to_o a_o like_a number_n of_o magnitude_n each_o to_o each_o how_o multiplex_n one_o magnitude_n be_v to_o one_o so_o multiplices_fw-la be_v all_o the_o magnitude_n to_o all_o and_o by_o the_o same_o reason_n also_o l_o and_o l_o m_o n_o be_v equemultiplices_fw-la to_o the_o magnitude_n b_o and_o to_o the_o magnitude_n b_o d_o f_o wherefore_o as_o a_o be_v to_o b_o so_o be_v a_o c_o e_o to_o b_o d_o f_o by_o the_o six_o definition_n of_o the_o five_o if_o therefore_o there_o be_v a_o number_n of_o magnitude_n how_o many_o soever_o proportional_a 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 which_o be_v require_v to_o be_v prove_v the_o 13._o theorem_a the_o 13._o proposition_n if_o the_o first_o have_v unto_o the_o second_o the_o self_n same_o proportion_n that_o the_o three_o have_v to_o the_o four_o and_o if_o the_o three_o have_v unto_o the_o four_o a_o great_a proportion_n they_o the_o five_o have_v to_o the_o six_o they_o shall_v the_o first_o also_o have_v unto_o the_o second_o a_o great_a proportion_n than_o have_v the_o five_o to_o the_o six_o svppose_v that_o there_o be_v six_o magnitude_n of_o which_o let_v a_o be_v the_o first_o b_o the_o second_o c_o the_o three_o d_z the_o four_o e_o the_o five_o and_o fletcher_n the_o six_o suppose_v that_o a_o the_o first_o have_v unto_o b_o the_o second_o the_o self_n same_o proportion_n that_o c_o the_o three_o have_v to_o d_o the_o four_o and_o let_v c_o the_fw-fr three_o have_v unto_o d_z the_o four_o a_o great_a proportion_n than_o have_v e_o the_o five_o to_o f_o the_o six_o then_o i_o say_v that_o a_o the_o first_o have_v to_o b_o the_o second_o a_o great_a proportion_n then_o have_v e_o the_o five_o to_o f_o the_o six_o construction_n for_o forasmuch_o as_o c_z have_v to_o d_z a_o great_a proportion_n than_o have_v e_o to_o fletcher_n therefore_o there_o be_v certain_a equemultiplices_fw-la to_o c_o and_o e_o and_o likewise_o any_o other_o equemultiplices_fw-la whatsoever_o to_o d_z and_o f_o which_o be_v compare_v together_o the_o multiplex_n to_o c_o shall_v exceed_v the_o multiplex_n to_o d_o but_o the_o multiplex_n to_o e_o shall_v not_o exceed_v the_o multiplex_n to_o fletcher_n by_o the_o converse_n of_o the_o eight_o definition_n of_o this_o book_n let_v those_o multiplices_fw-la be_v take_v and_o suppose_v that_o the_o equemultiplices_fw-la to_o c_o and_o e_o be_v g_z and_o h_o and_o likewise_o to_o d_z and_o f_o take_v any_o other_o equemultiplices_fw-la whatsoever_o and_o let_v the_o same_o be_v king_n and_o l_o so_fw-mi that_o let_v g_z exceed_v king_n but_o let_v not_o h_o exceed_v l._n and_o how_o multiplex_n g_z be_v to_o c_o so_o multiplex_n let_v m_o be_v to_o a._n and_o how_o multiplex_n king_n be_v to_o d_o so_o multiplex_n also_o let_v n_o be_v to_o b._n and_o because_o that_o as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z and_o to_o a_o and_o c_o be_v take_v equemultiplices_fw-la m_o and_o g._n demonstration_n and_o likewise_o to_o b_o and_o d_o be_v take_v certain_a other_o equemultiplices_fw-la n_o &_o king_n if_o therefore_o m_o exceed_v n_o g_o also_o exceed_v king_n and_o if_o it_o be_v equal_a it_o be_v equal_a and_o if_o it_o be_v less_o it_o be_v less_o by_o the_o conversion_n of_o the_o six_o definition_n of_o the_o five_o but_o by_o construction_n g_o excedetâ—_n edward_n wherefore_o m_o also_o exceed_v n_o but_o h_o exceed_v not_o l._n but_o m_o &_o h_o be_v equemultiplices_fw-la to_o a_o &_o e_o and_o n_o &_o l_o be_v certain_a other_o equemultiplicesâ—_n whatsoever_o to_o b_o and_o f._n wherefore_o a_o have_v unto_o b_o a_o great_a proportion_n than_o e_o have_v to_o fletcher_n by_o the_o 8._o definition_n if_o therefore_o the_o first_o have_v unto_o the_o second_o the_o self_n same_o proportion_n that_o the_o three_o have_v to_o the_o four_o and_o if_o the_o three_o have_v unto_o the_o four_o a_o great_a proportion_n than_o the_o five_o have_v to_o the_o six_o then_o shall_v the_o firsâ—_o also_o have_v unto_o the_o second_o a_o great_a proportion_n than_o have_v the_o 〈◊〉_d to_o the_o sixths_n which_o be_v require_v to_o be_v prove_v ¶_o a_o addition_n of_o campane_n if_o there_o be_v four_o quantity_n campane_n and_o if_o the_o first_o have_v unto_o the_o second_o a_o great_a proportion_n they_o have_v the_o three_o to_o the_o four_o then_o shall_v there_o be_v some_o equemultiplices_fw-la of_o the_o first_o and_o the_o three_o which_o be_v compare_v to_o some_o equemultiplices_fw-la of_o the_o second_o and_o the_o four_o the_o multiplex_n of_o the_o first_o shall_v be_v great_a than_o the_o multiplex_n of_o the_o second_o but_o the_o multiplex_n of_o the_o three_o shall_v not_o be_v great_a than_o the_o multiplex_n of_o the_o four_o although_o this_o proposition_n here_o put_v by_o campane_n need_v no_o demonstration_n for_o that_o it_o be_v but_o the_o converse_n of_o the_o 8._o definition_n of_o this_o book_n yet_o think_v i_o it_o not_o worthy_a to_o be_v omit_v for_o that_o it_o reach_v the_o way_n to_o find_v out_o such_o equemultiplices_fw-la that_o the_o multiplex_n of_o the_o first_o shall_v exceed_v the_o multiplex_n of_o the_o second_o but_o the_o multiplex_n of_o the_o three_o shall_v not_o exceed_v the_o multiplex_n of_o the_o four_o the_o 14._o theorem_a the_o 14
therefore_o alternate_o as_o ab_fw-la be_v to_o e_o so_o be_v cd_o to_o fletcher_n by_o the_o 16._o of_o the_o five_o but_o ab_fw-la be_v great_a than_o e_o wherefore_o also_o cd_o be_v great_a than_o fletcher_n which_o thing_n may_v also_o be_v prove_v by_o the_o 14._o of_o the_o same_o now_o for_o that_o as_o ab_fw-la be_v to_o cd_o so_o be_v e_o to_o fletcher_n but_o e_o be_v equal_a unto_o agnostus_n and_o fletcher_n be_v equal_a unto_o change_z therefore_o as_o ab_fw-la be_v to_o cd_o so_o be_v ag_z to_o change_z and_o forasmuch_o as_o the_o whole_a ab_fw-la be_v to_o the_o whole_a cd_o so_o be_v the_o part_n take_v away_o agnostus_n to_o the_o part_n take_v away_o change_z therefore_o the_o residue_n gb_o by_o the_o 1â—_o of_o the_o five_o be_v unto_o the_o residue_n hd_v as_o the_o whole_a ab_fw-la be_v to_o the_o whole_a cd_o but_o ab_fw-la the_o first_o be_v great_a than_o cd_o the_o three_o wherefore_o gb_o the_o second_o be_v great_a than_o hd_v the_o four_o by_o the_o 14._o of_o the_o five_o and_o forasmuch_o as_o agnostus_n be_v equal_a unto_o e_o &_o change_z be_v equal_a unto_o fletcher_n therefore_o agnostus_n and_o fletcher_n be_v equal_a unto_o ch_n and_o e._n and_o forasmuch_o as_o if_o unto_o thing_n unequal_a be_v add_v thing_n equal_a all_o shall_v be_v unequal_a by_o the_o four_o common_a sentence_n therefore_o see_v that_o gb_o and_o dh_o be_v unequal_a and_o gb_o be_v the_o great_a if_o unto_o gb_o be_v add_v agnostus_n and_o fletcher_n and_o likewise_o if_o unto_o hd_n be_v add_v change_z &_o e_o there_o shall_v be_v produce_v ab_fw-la and_o fletcher_n great_a than_o cd_o &_o e._n if_o therefore_o there_o be_v four_o magnitude_n proportional_a the_o great_a and_o the_o least_o of_o they_o shall_v be_v great_a they_o the_o other_o remain_v which_o be_v require_v to_o be_v demonstrate_v here_o follow_v certain_a proposition_n add_v by_o campane_n which_o be_v not_o to_o be_v contemn_v and_o be_v cite_v even_o of_o the_o best_a learned_a namely_o of_o johannes_n regio_fw-la montanus_fw-la in_o the_o epitome_n which_o he_o write_v upon_o ptolemy_n ¶_o the_o first_o proposition_n if_o there_o be_v four_o quantity_n and_o if_o the_o proportion_n of_o the_o first_o to_o the_o second_o be_v great_a than_o the_o proportion_n of_o the_o three_o to_o the_o four_o then_o contrariwise_o by_o conversion_n the_o proportion_n of_o the_o second_o to_o the_o first_o shall_v be_v less_o than_o the_o proportion_n of_o the_o four_o to_o the_o three_o affirmative_o it_o may_v also_o be_v demonstrate_v direct_o for_o let_v e_o be_v unto_o b_o as_z c_z be_v to_o d._n then_o converse_o b_o be_v to_o e_o as_z d_z be_v to_o c._n and_o forasmuch_o as_o a_o be_v great_a than_o e_o by_o the_o first_o part_n of_o the_o ten_o of_o this_o book_n therefore_o by_o the_o second_o part_n of_o the_o 8_o of_o the_o same_o b_o have_v unto_o a_o a_o less_o proportion_n than_o have_v b_o to_o e._n wherefore_o by_o the_o 13._o of_o the_o same_o b_o have_v unto_o aâ—_n â—_z less_o proportion_n than_o have_v d_z to_o c_z which_o be_v require_v to_o be_v prove_v ¶_o the_o second_o proposition_n if_o there_o be_v four_o quantity_n and_o if_o the_o proportion_n of_o the_o first_o to_o the_o second_o be_v great_a than_o the_o proportion_n of_o the_o three_o to_o the_o four_o then_o alternate_o the_o proportion_n of_o the_o first_o to_o the_o three_o shall_v be_v great_a than_o the_o proportion_n of_o the_o second_o to_o the_o four_o affirmative_o this_o may_v also_o be_v demonstrate_v affirmative_o let_v e_o be_v unto_o b_o as_z c_z be_v to_o d._n now_o they_o by_o the_o first_o part_n of_o the_o ten_o of_o this_o book_n e_o be_v less_o than_o a_o wherefore_o by_o the_o first_o part_n of_o the_o 8._o of_o the_o same_o the_o proportion_n of_o a_o to_o c_o be_v great_a than_o the_o proportion_n of_o e_o to_o c._n but_o alternate_o e_o be_v to_o c_o as_z b_o be_v to_o d._n wherefore_o by_o the_o 13._o of_o the_o same_o a_o have_v to_o c_o a_o 〈…〉_o ¶_o the_o three_o proposition_n if_o there_o be_v four_o quantity_n and_o if_o the_o proportion_n of_o the_o first_o tâ—_n the_o second_o be_v great_a than_o the_o proportion_n of_o the_o three_o to_o the_o four_o then_o by_o composition_n also_o the_o proportion_n of_o the_o fâ—â—th_n and_o second_v to_o the_o second_o shall_v be_v great_a than_o the_o proportion_n of_o the_o three_o and_o fourthâ—_o to_o the_o four_o this_o may_v also_o be_v demonstrate_v affirmative_o same_o forasmuch_o as_o the_o proportion_n of_o a_o to_o b_o be_v great_a than_o the_o proportion_n of_o c_o to_o d_z let_v e_o be_v unto_o b_o as_z c_z be_v to_o d._n and_o so_o by_o the_o first_o part_n of_o the_o 10._o of_o this_o book_n e_o shall_v be_v less_o than_o a._n and_o therefore_o by_o the_o common_a sentence_n ebb_n shall_v be_v less_o than_o ab_fw-la wherefore_o by_o the_o first_o part_n of_o the_o 8._o of_o the_o same_o ab_fw-la have_v unto_o b_o a_o great_a proportion_n than_o have_v ebb_n to_o b._n but_o by_o composition_n ebb_n be_v to_o b_o as_o cd_o be_v to_o d._n for_o by_o supposition_n e_o be_v unto_o b_o as_o be_v to_o d._n wherefore_o by_o the_o 12._o of_o this_o book_n ab_fw-la have_v to_o b_o a_o great_a proportion_n than_o have_v cd_o to_o d_o which_o be_v require_v to_o be_v prove_v ¶_o the_o four_o proposition_n if_o there_o be_v four_o quantity_n and_o if_o the_o proportion_n of_o the_o first_o and_o the_o second_o to_o the_o second_o be_v great_a than_o the_o proportion_n of_o the_o three_o and_o four_o to_o the_o four_o then_o by_o division_n also_o the_o proportion_n of_o the_o first_o to_o the_o second_o shall_v be_v great_a thâ—n_n the_o proportion_n of_o the_o third_o to_o the_o four_o suppose_v that_o the_o proportion_n of_o ab_fw-la to_o b_o be_v great_a than_o the_o proportion_n of_o cd_o to_o d._n impossibility_n then_o i_o say_v that_o by_o division_n also_o the_o proportion_n of_o a_o to_o b_o be_v great_a than_o the_o proportion_n of_o c_o to_o d._n for_o it_o can_v be_v the_o same_o for_o then_o by_o composition_n ab_fw-la shall_v be_v to_o b_o as_o cd_o be_v to_o d._n neither_o also_o can_v it_o be_v less_o for_o if_o the_o proportion_n of_o c_o to_o d_z be_v great_a than_o the_o proportion_n of_o a_o to_o b_o then_o by_o the_o former_a proposition_n the_o proportion_n of_o cd_o to_o d_z shall_v be_v great_a than_o the_o proportion_n of_o ab_fw-la to_o b_o which_o be_v contrary_a also_o to_o the_o supposition_n wherefore_o the_o proportion_n of_o a_o to_o b_o be_v neither_o one_o and_o the_o same_o with_o the_o proportion_n of_o c_o to_o d_z 〈…〉_o it_o wherefore_o it_o be_v great_a than_o it_o which_o be_v require_v to_o be_v prove_v affirmative_o the_o same_o may_v also_o be_v prove_v affirmative_o suppose_v that_o ebb_n be_v unto_o b_o as_o cd_o be_v to_o d._n now_o then_o by_o the_o first_o part_n of_o the_o 10._o of_o the_o five_o ebb_n shall_v be_v less_o than_o ab_fw-la and_o therefore_o by_o the_o common_a sentence_n e_o be_v less_o than_o a_o wherefore_o by_o the_o first_o part_n of_o the_o 8._o of_o this_o book_n the_o proportion_n of_o e_o to_o b_o be_v less_o than_o the_o proportion_n of_o a_o to_o b_o but_o as_z e_o be_v to_o b_o so_o be_v c_z to_o d_z wherefore_o the_o proportion_n of_o c_o to_o d_z be_v less_o than_o the_o proportion_n of_o a_o to_o b._n wherefore_o the_o proportion_n of_o a_o to_o b_o be_v great_a than_o the_o proportion_n of_o c_o to_o d_z which_o be_v require_v to_o be_v prove_v ¶_o the_o five_o proposition_n if_o there_o be_v four_o quantity_n and_o if_o the_o proportion_n of_o the_o first_o and_o the_o second_o to_o the_o second_o be_v great_a than_o the_o proportion_n of_o the_o three_o and_o the_o four_o to_o the_o four_o then_o by_o eversion_n the_o proportion_n of_o the_o first_o and_o second_o to_o the_o first_o shall_v be_v less_o than_o the_o proportion_n of_o the_o three_o and_o four_o to_o the_o three_o suppose_v that_o the_o proportion_n of_o ab_fw-la to_o b_o be_v great_a than_o the_o proportion_n of_o cd_o to_o d._n then_o i_o say_v that_o by_o eversion_n the_o proportion_n of_o ab_fw-la to_o a_o be_v less_o than_o the_o proportion_n of_o cd_o to_o hundred_o demonstration_n for_o by_o division_n by_o the_o former_a proposition_n the_o proportion_n of_o a_o to_o b_o be_v great_a than_o the_o proportion_n of_o c_o to_o d._n wherefore_o by_o the_o first_o of_o these_o proposition_n converse_o b_o have_v unto_o a_o a_o less_o proportion_n they_o have_v d_z to_o c._n wherefore_o by_o the_o 3._o of_o the_o same_o by_o composition_n the_o proportion_n of_o ab_fw-la to_o a_o be_v less_o they_o the_o
the_o part_n add_v together_o be_v not_o equal_a to_o the_o whole_a nor_o make_v the_o whole_a but_o make_v either_o more_o or_o less_o wherefore_o of_o imperfect_a number_n there_o be_v two_o kind_n number_n the_o one_o be_v call_v abundant_a or_o a_o building_n the_o other_o 〈◊〉_d or_o want_v a_o number_n abund_v be_v that_o who_o part_n be_v all_o add_v together_o make_v more_o than_o the_o wâ—â—ds_n number_n who_o part_n they_o be_v as_o 12._o be_v a_o abundant_a number_n for_o all_o the_o parteâ—_n of_o 12._o namely_o 6.4.3.2_o and_o 1._o add_v together_o make_v 16_o which_o be_v more_o than_o 12._o likewise_o 18._o be_v a_o number_n abund_v all_o his_o partâ—_n name_o 9.6.3.2_o and_o 1._o add_v together_o make_v 20._o which_o be_v more_o than_o 18_o and_o so_o of_o other_o wanâ—â—ngâ—_n a_o number_n diminute_a or_o want_v be_v that_o who_o part_n be_v all_o add_v together_o make_v less_o than_o the_o whole_a or_o number_n who_o part_n they_o be_v as_o 9_o be_v a_o diminute_a or_o want_v number_n for_o all_o his_o part_n name_o 3._o and_o 1._o more_o part_n he_o have_v not_o add_v together_o make_v only_o 4_o which_o be_v less_o than_o 9_o also_o 26._o be_v a_o diminute_a number_n all_o his_o part_n namely_o 13.2.1_o add_v together_o make_v only_o 16_o which_o be_v a_o number_n much_o less_o than_o 26._o and_o so_o of_o such_o like_a sentence_n campane_n and_o flussates_n here_o add_v certain_a common_a sentence_n some_o of_o which_o for_o that_o they_o be_v in_o these_o three_o book_n follow_v sometime_o allege_v i_o think_v good_a here_o to_o annex_v sentence_n 1_o the_o less_o part_n be_v that_o which_o have_v the_o great_a denomination_n and_o the_o great_a part_n be_v that_o which_o have_v the_o less_o denomination_n as_o the_o number_n 6._o and_o 8._o be_v either_o of_o they_o a_o part_n of_o the_o number_n 24_o 6._o be_v a_o four_o part_n 4._o time_n 6._o be_v 24_o and_o 8._o be_v a_o three_o part_n 3._o time_n 8._o be_v 24._o now_o forasmuch_o as_o 4_o which_o denominate_v what_o part_n 6._o be_v of_o 24_o be_v great_a than_o 3._o which_o denominate_v what_o part_n 8._o be_v of_o 24._o therefore_o be_v 6._o a_o less_o part_n of_o 24â—_n than_o be_v 8._o and_o so_o be_v 8._o a_o great_a part_n of_o 24._o they_o 6._o be_v and_o so_o in_o other_o sentence_n 2_o whatsoever_o number_n be_v equemultiplices_fw-la to_o one_o &_o the_o self_n same_o number_n or_o to_o equal_a number_n be_v also_o equal_a the_o one_o to_o the_o other_o as_o if_o unto_o the_o number_n 3_o be_v take_v two_o number_n contain_v the_o same_o number_n four_o time_n that_o be_v be_v equemultiplices_fw-la to_o the_o same_o number_n three_o the_o say_v two_o number_n shall_v be_v equal_a for_o 4._o time_n 3._o will_v ever_o be_v 12._o so_o also_o will_v it_o be_v if_o unto_o the_o two_o equal_a number_n 3._o &_o 3._o be_v take_v two_o number_n the_o one_o contain_v the_o one_o number_n 3._o four_o time_n the_o other_o contain_v the_o other_o number_n 3._o also_o four_o time_n that_o be_v be_v equemultiplices_fw-la to_o the_o equal_a number_n 3._o and_o 3._o sentence_n 3_o those_o number_n to_o who_o one_o and_o the_o self_n same_o number_n be_v equmultiplex_n or_o who_o equemultiplices_fw-la be_v equal_a be_v also_o equal_a the_o on_o to_o the_o other_o as_o if_o the_o number_n 18._o be_v equemultiplex_n to_o any_o two_o number_n that_o be_v contain_v any_o two_o number_n twice_o thrice_o four_o time_n &_o c_o as_o for_o example_n 3._o time_n then_o be_v the_o say_v two_o number_n equal_a for_o 18._o divide_v by_o 3._o will_v ever_o bring_v forth_o 6._o so_o that_o that_o division_n make_v twice_o will_v bring_v forth_o 6._o and_o 6._o two_o equal_a number_n so_o also_o will_v it_o follow_v if_o the_o two_o number_n have_v equal_a equemultiplices_fw-la namely_o if_o 18._o and_o 18._o which_o be_v equal_a number_n contain_v any_o two_o number_n 3._o time_n 4_o if_o a_o number_n measure_v the_o whole_a and_o a_o part_n take_v away_o it_o shall_v also_o measure_v the_o residue_n sentence_n as_o if_o from_o 24._o be_v take_v away_o 9_o there_o remain_v 15._o and_o for_o as_o much_o as_o the_o number_n 3_o measure_v the_o whole_a number_n 24_o &_o also_o the_o number_n take_v away_o namely_o 9_o it_o shall_v also_o measure_v the_o residue_n which_o be_v 15â—_n for_o 3._o measure_v 15_o by_o five_o five_o time_n 3._o be_v 15._o and_o so_o of_o other_o sentence_n 5_o if_o a_o number_n measure_v any_o number_n it_o also_o measure_v every_o number_n that_o the_o say_a number_n measure_v as_o the_o number_n 6._o measure_v the_o number_n 12._o shall_v also_o measure_v all_o the_o number_n that_o 1â—_o measurethâ—_n as_o the_o number_n 24.36.48.60_o and_o so_o forth_o which_o the_o number_n 12._o do_v measure_n by_o the_o numberâ—_n 2.3.4_o and_o 5._o and_o for_o as_o much_o as_o the_o number_n 12._o do_v measure_v the_o number_n 24.36.48_o and_o 60._o and_o the_o number_n 6_o do_v measure_v the_o number_n 12._o namely_o by_o 2._o it_o follow_v by_o this_o common_a sentence_n that_o the_o number_n 6._o measure_v each_o of_o thâ—se_a number_n 24._o 36.48_o and_o 60._o and_o so_o of_o other_o 6_o if_o a_o number_n measure_v two_o number_n sentence_n it_o shall_v also_o measure_v the_o number_n compose_v of_o they_o as_o the_o number_n 3_o measure_v these_o two_o number_n 6._o and_o 9â—_n it_o measure_v 6._o by_o 2d_o and_o 9_o by_o 3._o and_o therefore_o by_o this_o common_a sentence_n it_o measure_v the_o number_n 15._o which_o be_v compose_v of_o the_o number_n 6._o and_o 9_o namely_o it_o measure_v it_o by_o 5._o 7_o if_o in_o number_n there_o be_v proportion_n how_o manysoever_o equal_a or_o the_o self_n same_o to_o one_o proportion_n sentence_n they_o shall_v alâ—o_o be_v equal_a or_o the_o self_n same_o the_o one_o to_o the_o other_o as_o if_o the_o proportion_n of_o the_o number_n 6._o to_o the_o number_n 3._o be_v as_o the_o proportion_n of_o the_o number_n 8._o to_o the_o number_n 4_o if_o also_o the_o proportion_n of_o the_o number_n 10._o to_o the_o number_n 5._o be_v as_o the_o proportion_n of_o the_o number_n 8._o to_o the_o number_n 4_o then_o shall_v the_o proportion_n of_o the_o number_n 6._o to_o the_o number_n 3._o be_v as_o the_o proportion_n of_o the_o number_n 10._o be_v to_o the_o number_n 5_o namely_o each_o proportion_n be_v duple_n and_o so_o of_o other_o euclid_n in_o his_o â—_o book_n the_o 11._o proposition_n demonstrate_v this_o also_o in_o continual_a quantity_n which_o although_o as_o touch_v that_o kind_n of_o quantity_n it_o may_v have_v be_v put_v also_o as_o a_o principle_n as_o in_o number_n he_o take_v it_o yet_o for_o that_o in_o all_o magnitude_n their_o proportion_n can_v not_o be_v express_v as_o have_v before_o be_v note_v &_o shall_v be_v afterward_o in_o the_o ten_o book_n more_o at_o large_a make_v manifest_v therefore_o he_o demonstrate_v it_o there_o in_o that_o place_n and_o prove_v that_o it_o be_v true_a as_o touch_v all_o proportion_n general_o whither_o they_o be_v rational_a or_o irrational_a ¶_o the_o first_o proposition_n the_o first_o theorem_a if_o there_o be_v give_v two_o unequal_a number_n and_o if_o in_o take_v the_o less_o continual_o from_o the_o great_a the_o number_n remain_v do_v not_o measure_v the_o number_n go_v before_o until_o it_o shall_v come_v to_o unity_n then_o be_v those_o number_n which_o be_v at_o the_o beginning_n give_v prime_a the_o one_o to_o the_o other_o svppose_v that_o there_o be_v two_o unequal_a number_n ab_fw-la the_o great_a and_o cd_o the_o less_o and_o from_o ab_fw-la the_o great_a take_v away_o cd_o the_o less_o as_o oâ—ten_v as_o you_o can_v leave_v favorina_n constrâ—ctioâ—_n and_o from_o cd_o take_v away_o favorina_n as_o often_o as_o you_o can_v leave_v the_o number_n gc_o and_o from_o favorina_n take_v away_o gc_o as_o often_o as_o you_o can_v and_o so_o do_v continual_o till_o there_o remain_v only_a unity_n which_o let_v be_v ha._n absurdity_n then_o i_o say_v that_o no_o number_n measure_v the_o number_n ab_fw-la and_o cd_o for_o if_o it_o be_v possible_a let_v some_o number_n measure_v they_o and_o let_v the_o same_o be_v e._n now_o cd_o measure_n ab_fw-la leave_v a_o less_o number_n than_o itself_o which_o let_v be_v faâ—_n â—_z and_o favorina_n measure_v dc_o leave_v also_o a_o less_o than_o itself_o namely_o gc_o and_o gc_o measure_v favorina_n leave_v unity_n ha._n and_o forasmuch_o as_o the_o number_n e_o measure_v dc_o and_o the_o number_n cd_o measure_v the_o number_n bf_o therefore_o the_o number_n e_o also_o measure_v bf_o and_o it_o measure_v the_o whole_a number_n basilius_n wherefore_o it_o also_o measure_v that_o which_o remain_v namely_o the_o number_n favorina_n by_o the_o
the_o double_a of_o b_o be_v prime_a unto_o the_o double_a of_o b_o then_o the_o two_o number_n whereof_o the_o number_n be_v compose_v namely_o the_o number_n compose_v of_o a_o and_o c_o and_o the_o double_a of_o b_o shall_v be_v prime_a the_o one_o to_o the_o other_o by_o the_o 30_o of_o the_o seven_o and_o therefore_o the_o number_n compose_v of_o a_o and_o c_o shall_v be_v prime_a to_o b_o take_v once_o for_o if_o any_o number_n shall_v measure_v the_o two_o number_n namely_o the_o number_n compose_v of_o a_o and_o c_o and_o the_o number_n b_o it_o shall_v also_o measure_v the_o number_n compose_v of_o a_o and_o c_o and_o the_o double_a of_o b_o by_o the_o 5._o common_a sentence_n of_o the_o seven_o which_o be_v not_o possible_a for_o that_o they_o be_v prove_v to_o be_v prime_a number_n here_o have_v i_o add_v a_o other_o demonstration_n of_o the_o former_a proposition_n after_o campane_n which_o prove_v that_o in_o number_n how_o many_o soever_o which_o be_v there_o prove_v only_o touch_v three_o number_n and_o the_o demonstration_n seement_n somewhat_o more_o perspicuous_a than_o theon_n demonstration_n and_o thus_o he_o put_v the_o proposition_n if_o number_n how_o many_o soever_o be_v in_o continual_a proportion_n be_v the_o least_o that_o have_v one_o &_o the_o same_o proportion_n with_o they_o every_o one_o of_o they_o shall_v be_v to_o the_o number_n compose_v of_o the_o rest_n prime_a second_o i_o say_v that_o this_o be_v so_o in_o every_o one_o of_o they_o namely_o that_o c_z be_v a_o prime_a number_n to_o the_o number_n compose_v of_o a_o b_o d._n for_o if_o not_o then_o as_o before_o let_v e_o measure_v c_o and_o the_o number_n compose_v of_o a_o b_o d_o which_o e_o shall_v be_v a_o number_n not_o prime_a either_o to_o fletcher_n or_o to_o g_z by_o the_o former_a proposition_n add_v by_o campane_n wherefore_o let_v h_o measure_v they_o and_o forasmuch_o as_o h_o measure_v e_o it_o shall_v also_o measure_v the_o whole_a a_o b_o c_o d_o who_o e_o measure_v and_o forasmuch_o as_o h_o measure_v one_o of_o these_o number_n fletcher_n or_o g_o it_o shall_v measure_v one_o of_o the_o extreme_n a_o or_o d_o which_o be_v produce_v of_o fletcher_n or_o g_o by_o the_o second_o of_o the_o eight_o if_o they_o be_v multiply_v into_o the_o mean_n l_o or_o k._n and_o moreover_o the_o same_o h_o shall_v measure_v the_o meame_n bc_o by_o the_o 5._o common_a sentence_n of_o the_o seven_o when_o as_o by_o supposition_n it_o measure_v either_o fletcher_n or_o g._n which_o measure_n b_o c_o by_o the_o second_o of_o the_o eight_o but_o the_o same_o h_o measure_v the_o whole_a a_o b_o c_o d_o as_o we_o have_v prove_v for_o that_o it_o measure_v e._n wherefore_o it_o shall_v also_o measure_v the_o residue_n namely_o the_o number_n compose_v of_o the_o extreme_n a_o and_o d_z by_o the_o 4._o common_a sentence_n of_o the_o seven_o and_o it_o measure_v one_o of_o these_o a_o or_o d_z for_o it_o measure_v one_o of_o these_o fletcher_n or_o g_o which_o produce_v a_o and_o d_z wherefore_o the_o same_o h_o shall_v measure_v one_o of_o these_o a_o or_o d_o and_o also_o the_o other_o of_o they_o by_o the_o former_a common_a sentence_n which_o number_n a_o and_o d_o be_v by_o the_o 3._o of_o the_o eight_o prime_n the_o one_o to_o the_o other_o which_o be_v absurd_a this_o may_v also_o be_v prove_v in_o every_o one_o of_o these_o number_n a_o b_o c_o d._n wherefore_o no_o number_n shall_v measure_v one_o of_o these_o number_n a_o b_o c_o d_o and_o the_o number_n compose_v of_o the_o rest_n wherefore_o they_o be_v prime_a the_o one_o to_o the_o other_o if_o therefore_o number_n how_o many_o soever_o etc_o etc_o which_o be_v require_v to_o be_v prove_v here_o as_o i_o promise_v i_o have_v add_v campane_n demonstration_n of_o those_o proposition_n in_o number_n which_o euclâ—de_n in_o the_o second_o book_n demonstrate_v in_o line_n and_o that_o in_o this_o place_n so_o much_o the_o rather_o for_o that_o theon_n as_o we_o see_v in_o the_o demonstration_n of_o the_o 15._o proposition_n seem_v to_o allege_v the_o 3._o &_o 4._o proposition_n of_o the_o second_o book_n which_o although_o they_o concern_v line_n only_o yet_o as_o we_o there_o declare_v and_o prove_v be_v they_o true_a also_o in_o number_n ¶_o the_o first_o proposition_n add_v by_o campane_n that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o one_o number_n into_o number_n how_o many_o soever_o be_v equal_a to_o that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o same_o number_n into_o the_o number_n compose_v of_o they_o this_o prove_v that_o in_o number_n which_o the_o first_o of_o the_o second_o prove_v touch_v line_n suppoâ—â—_n that_o the_o number_n a_o be_v multiply_v into_o the_o number_n b_o and_o into_o the_o number_n c_o demonstratiou_n and_o into_o the_o number_n d_o do_v produce_v the_o number_n e_o f_o and_o g._n then_o i_o say_v that_o the_o number_n produce_v of_o a_o multiply_a into_o the_o number_n compose_v of_o b_o c_o and_o d_z be_v equal_a to_o the_o number_n compose_v of_o e_o f_o and_o g._n for_o by_o the_o converse_n of_o the_o definition_n of_o a_o number_n multiply_v what_o part_n unity_n be_v of_o a_o the_o self_n same_o part_n be_v b_o of_o e_o and_o c_o of_o fletcher_n and_o also_o d_z of_o g._n wherefore_o by_o the_o 5._o of_o the_o seven_o what_o part_n unity_n be_v of_o a_o the_o self_n same_o part_n be_v the_o number_n compose_v of_o b_o c_o and_o d_o of_o the_o number_n compose_v of_o e_o f_o and_o g._n wherefore_o by_o the_o definition_n that_o which_o be_v produce_v of_o a_o into_o the_o number_n compose_v of_o b_o c_o d_o be_v equal_a to_o the_o number_n compose_v of_o e_o f_o g_o which_o be_v require_v to_o be_v prove_v the_o second_o proposition_n that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o number_n how_o many_o soever_o into_o one_o number_n be_v equal_a to_o that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n compose_v of_o they_o into_o the_o same_o number_n this_o be_v the_o converse_n of_o the_o former_a as_o if_o the_o â—â—â—bers_n â—_o and_o g_z and_o d_o multiply_v into_o the_o number_n a_o do_v produce_v the_o number_n e_o and_o f_o and_o g._n former_a then_o the_o number_n compose_v of_o b_o c_o d._n multiply_v into_o the_o number_n a_o shall_v produce_v the_o number_n compose_v of_o the_o number_n e_o f_o g._n which_o thing_n be_v easy_o prove_v by_o the_o 16._o of_o the_o seven_o and_o by_o the_o former_a proposition_n ¶_o the_o three_o proposition_n that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o number_n how_o many_o soever_o into_o other_o number_n how_o many_o soever_o be_v equal_a to_o that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n compose_v of_o those_o first_o number_n into_o the_o number_n compose_v of_o these_o latter_a number_n as_o if_o the_o number_n a_o b_o c_o do_v multiply_v the_o number_n d_o e_o f_o each_o one_o each_o other_o and_o if_o the_o number_n produce_v be_v add_v together_o then_o i_o say_v that_o the_o number_n compose_v of_o the_o number_n produce_v be_v equal_a to_o the_o number_n produce_v of_o the_o number_n compose_v of_o the_o number_n a_o b_o demonstration_n c_o into_o the_o number_n compose_v of_o the_o number_n d_o e_o f._n for_o by_o the_o former_a proposition_n that_o which_o be_v produce_v of_o the_o number_n compose_v of_o a_o b_o c_o into_o d_z be_v equal_a to_o that_o which_o be_v produce_v of_o every_o one_o of_o the_o say_a number_n into_o d_o and_o by_o the_o same_o reason_n that_o which_o be_v produce_v of_o the_o number_n compose_v of_o a_o b_o c_o into_o e_o be_v equal_a to_o that_o which_o be_v produce_v of_o every_o one_o of_o the_o say_a number_n into_o e_o and_o so_o likewise_o that_o which_o be_v produce_v of_o the_o number_n compose_v of_o a_o b_o c_o into_o fletcher_n be_v equal_a to_o that_o which_o be_v produce_v of_o every_o one_o of_o the_o say_a number_n into_o f._n but_o by_o the_o first_o of_o these_o proposition_n thâ—â—_n which_o be_v produce_v of_o the_o number_n compose_v of_o these_o number_n a_o b_o c_o into_o every_o one_o of_o these_o number_n d_o e_o fletcher_n be_v equal_a to_o that_o which_o be_v produce_v of_o the_o number_n compose_v into_o the_o number_n compose_v wherefore_o that_o be_v manifest_v which_o be_v require_v to_o be_v prove_v ¶_o the_o four_o proposition_n if_o a_o number_n be_v deuâ—â—â—d_v into_o part_n how_o many_o soever_o that_o number_n which_o be_v produce_v of_o the_o whole_a into_o himself_o be_v equal_a to_o that_o number_n which_o be_v produce_v of_o
that_o the_o line_n of_o be_v make_v equal_a to_o the_o line_n ad_fw-la which_o be_v the_o diameter_n of_o the_o square_n abcd_o of_o which_o square_v the_o line_n ab_fw-la be_v a_o side_n it_o be_v certain_a that_o the_o â—ide_v of_o a_o square_n be_v incommensurable_a in_o length_n to_o the_o diameter_n of_o the_o same_o square_n if_o there_o be_v yet_o find_v any_o one_o superficies_n which_o measure_v the_o two_o square_n abcd_o and_o efgh_a as_o here_o do_v the_o triangle_n abdella_n or_o the_o triangle_n acd_v note_v in_o the_o square_n abcd_o or_o any_o of_o the_o four_o triangle_n note_v in_o the_o square_a efgh_o as_o appear_v somewhat_o more_o manifest_o in_o the_o second_o example_n in_o the_o declaration_n of_o the_o last_o definition_n go_v before_o the_o line_n of_o be_v also_o a_o rational_a line_n note_v that_o these_o line_n which_o here_o be_v call_v rational_a line_n be_v not_o rational_a line_n of_o purpose_n or_o by_o supposition_n as_o be_v the_o first_o rational_a line_n but_o be_v rational_a only_o by_o reason_n of_o relation_n and_o comparison_n which_o they_o have_v unto_o it_o because_o they_o be_v commensurable_a unto_o it_o either_o in_o length_n and_o power_n or_o in_o power_n only_o far_o here_o be_v to_o be_v note_v that_o these_o word_n length_n and_o power_n and_o power_n only_o be_v join_v only_o with_o these_o wordeâ—_n commensurable_a or_o incommensurable_a and_o be_v never_o join_v with_o these_o word_n rational_a or_o irrational_a so_o that_o no_o line_n can_v be_v call_v rational_a in_o length_n or_o in_o power_n nor_o like_o wise_a can_v they_o be_v call_v irrational_a in_o length_n or_o in_o power_n wherein_o undoubted_o campanus_n be_v deceive_v book_n who_o use_v those_o word_n &_o speech_n indifferent_o cause_v &_o bring_v in_o great_a obscurity_n to_o the_o proposition_n and_o demonstration_n of_o this_o book_n which_o he_o shall_v easy_o see_v which_o mark_v with_o diligence_n the_o demonstration_n of_o campanus_n in_o this_o book_n 7_o line_n which_o be_v incommensurable_a to_o the_o rational_a line_n be_v call_v irrational_a definition_n by_o line_n incommensurable_a to_o the_o rational_a line_n suppose_v in_o this_o place_n he_o understand_v such_o as_o be_v incommensurable_a unto_o it_o both_o in_o length_n and_o in_o power_n for_o there_o be_v no_o line_n incommensurable_a in_o power_n only_o for_o it_o can_v be_v that_o any_o line_n shall_v so_o be_v incommensurable_a in_o power_n only_o that_o they_o be_v not_o also_o incommensurable_a in_o length_n what_o so_o ever_o line_n be_v incommensurable_a in_o power_n the_o same_o be_v also_o incommensurable_a in_o length_n neither_o can_v euclid_n here_o in_o this_o place_n mean_a line_n incommensurable_a in_o length_n only_o for_o in_o the_o definition_n before_o he_o call_v they_o rational_a line_n neither_o may_v they_o be_v place_v amongst_o irrational_a line_n wherefore_o it_o remain_v that_o in_o this_o diffintion_n he_o speak_v only_o of_o those_o line_n which_o be_v incommensurable_a to_o the_o rational_a line_n first_o give_v and_o suppose_v both_o in_o length_n and_o in_o power_n which_o by_o all_o mean_n be_v incommensurable_a to_o the_o rational_a line_n &_o therefore_o most_o apt_o be_v they_o call_v irrational_a line_n this_o definition_n be_v easy_a to_o be_v understand_v by_o that_o which_o have_v be_v say_v before_o yet_o for_o the_o more_o plainness_n see_v this_o example_n let_v the_o â—â—rst_a rational_a line_n suppose_v be_v the_o line_n ab_fw-la who_o square_a or_o quadrate_n let_v be_v abcd._n and_o let_v there_o be_v give_v a_o other_o line_n of_o which_o lâ—t_v be_v to_o the_o rational_a line_n incommensurable_a in_o length_n and_o power_n so_o that_o let_v no_o one_o line_n measure_v the_o length_n of_o the_o two_o line_n ab_fw-la and_o of_o and_o let_v the_o square_n of_o the_o line_n of_o be_v efgh_a now_o if_o also_o there_o be_v no_o one_o superficies_n which_o measure_v the_o two_o square_n abcd_o and_o efgh_a as_o be_v suppose_v to_o be_v in_o this_o example_n they_o be_v the_o line_n of_o a_o irrational_a line_n which_o word_n irrational_a as_o before_o do_v this_o word_n rational_a mislike_v many_o learned_a in_o this_o knowledge_n of_o geometry_n flussates_n uncertain_a as_o he_o leave_v the_o word_n rational_a and_o in_o stead_n thereof_o use_v this_o word_n certain_a so_o here_o he_o leave_v the_o word_n irrational_a and_o use_v in_o place_n thereof_o this_o word_n uncertain_a and_o ever_o name_v these_o line_n uncertain_a line_n petrus_n montaureus_n also_o mislike_v the_o word_n irrational_a will_v rather_o have_v they_o to_o be_v call_v surd_a line_n yet_o because_o this_o word_n irrational_a have_v ever_o by_o custom_n and_o long_a use_n so_o general_o be_v receive_n he_o use_v continual_o the_o same_o in_o greek_a such_o line_n be_v call_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d alogoi_n which_o signify_v nameless_a unspeakable_a uncertain_a in_o determinate_a line_n and_o with_o out_o proportion_n not_o that_o these_o irrational_a line_n have_v no_o proportion_n at_o all_o either_o to_o the_o first_o rational_a line_n or_o between_o themselves_o but_o be_v so_o name_v for_o that_o their_o proportion_n to_o the_o rational_a line_n can_v be_v express_v in_o number_n that_o be_v undoubted_o very_o untrue_a which_o many_o write_v that_o their_o proportion_n be_v unknown_a both_o to_o we_o and_o to_o nature_n be_v it_o not_o think_v you_o a_o thing_n very_o absurd_a to_o say_v that_o there_o be_v any_o thing_n in_o nature_n and_o produce_v by_o nature_n to_o be_v hide_v from_o nature_n and_o not_o to_o be_v know_v of_o nature_n it_o can_v not_o be_v say_v that_o their_o proportion_n be_v utter_o hide_v and_o unknown_a to_o we_o much_o less_o unto_o nature_n although_o we_o can_v geve_v they_o their_o name_n and_o distinct_o express_v they_o by_o number_n otherwise_o shall_v euclid_n have_v take_v all_o this_o travel_n and_o wonderful_a diligence_n bestow_v in_o this_o bookeâ—_n in_o vain_a and_o to_o no_o useâ—_n in_o which_o he_o do_v nothing_o ellâ—_n but_o teach_v the_o propriety_n and_o passion_n of_o these_o irrational_a linesâ—_n and_o show_v the_o proportion_n which_o they_o have_v the_o one_o to_o the_o other_o here_o be_v also_o to_o be_v note_v which_o thing_n also_o tartalea_n have_v before_o diligent_o notedâ—_n that_o campanus_n and_o many_o other_o writer_n of_o geometryâ—_n over_o much_o â—â—â—ed_a and_o be_v deceive_v in_o that_o they_o write_v and_o teach_v that_o all_o these_o line_n who_o square_n be_v not_o signify_v and_o may_v be_v express_v by_o a_o square_a number_n although_o they_o may_v by_o any_o other_o number_n as_o by_o 11._o 12._o 14._o and_o such_o other_o not_o square_a number_n be_v irrational_a line_n which_o be_v manifest_o repugnant_a to_o the_o ground_n and_o principle_n of_o euclid_n who_o will_v that_o all_o line_n which_o be_v commensurable_a to_o the_o rational_a line_n whether_o it_o be_v in_o length_n and_o power_n or_o in_o power_n only_o shall_v be_v rational_a undoubted_o this_o have_v be_v one_o of_o the_o chief_a and_o great_a cause_n of_o the_o wonderful_a confusion_n and_o darkness_n of_o this_o book_n book_n which_o so_o have_v toss_v and_o turmoil_v the_o wit_n of_o all_o both_o writer_n and_o reader_n master_n and_o scholar_n and_o so_o overwhelm_v they_o that_o they_o can_v not_o with_o out_o infinite_a travel_n and_o sweat_v attain_v to_o the_o truth_n and_o perfect_a understanding_n thereof_o definition_n 8_o the_o square_n which_o be_v describe_v of_o the_o rational_a right_a line_n suppose_v be_v rational_a until_o this_o definition_n have_v euclid_n set_v forth_o the_o nature_n and_o propriety_n of_o the_o first_o kind_n of_o magnitude_n namely_o of_o line_n how_o they_o be_v rational_a or_o irrational_a now_o he_o begin_v to_o â—hew_v how_o the_o second_o kind_n of_o magnitude_n namely_o superficies_n be_v one_o to_o the_o other_o rational_a or_o irrational_a this_o definition_n be_v very_a plain_n suppose_v the_o line_n ab_fw-la to_o be_v the_o rational_a line_n have_v his_o part_n and_o division_n certain_o know_v the_o square_a of_o which_o line_n let_v be_v the_o square_a abcd._n now_o because_o it_o be_v the_o square_a of_o the_o rational_a line_n ab_fw-la it_o be_v also_o call_v rational_a and_o as_o the_o line_n ab_fw-la be_v the_o first_o rational_a line_n unto_o which_o other_o line_n compare_v be_v count_v rational_a or_o irrational_a so_o be_v the_o quadrat_fw-la or_o square_v thereof_o the_o â—irst_n rational_a superficies_n unto_o which_o all_o other_o square_n or_o figure_n compare_v be_v count_v and_o name_v rational_a or_o irrational_a 9_o such_o which_o be_v commensurable_a unto_o it_o be_v rational_a definition_n in_o this_o definition_n where_o it_o be_v say_v such_o as_o be_v commensurable_a to_o the_o square_n of_o the_o rational_a line_n be_v not_o understand_v only_o other_o square_n or_o
now_o forasmuch_o as_o d_z measure_v ab_fw-la and_o bc_o it_o also_o measure_v the_o whole_a magnitude_n ac_fw-la and_o it_o measure_v ab_fw-la wherefore_o d_z measure_v these_o magnitude_n ca_n and_o ab_fw-la wherefore_o ca_n &_o ab_fw-la be_v commensurable_a and_o they_o be_v suppose_v to_o be_v incommensurableâ—_n which_o be_v impossible_a wherefore_o no_o magnitude_n measure_v ab_fw-la and_o bc._n wherefore_o the_o magnitude_n ab_fw-la and_o bc_o be_v incommensurable_a and_o in_o like_a sort_n may_v they_o be_v prove_v to_o be_v incommensurable_a if_o the_o magnitude_n ac_fw-la be_v suppose_v to_o be_v incommensurable_a unto_o bc._n if_o therefore_o there_o be_v two_o magnitude_n incommensurable_a compose_v the_o whole_a also_o shall_v be_v incommensurable_a unto_o either_o of_o the_o two_o part_n component_a and_o if_o the_o whole_a be_v incommensurable_a to_o one_o of_o the_o part_n component_n those_o first_o magnitude_n shall_v be_v incommensurable_a which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o montaureus_n if_o a_o whole_a magnitude_n be_v incommensurable_a to_o one_o of_o the_o two_o magnitude_n which_o make_v the_o whole_a magnitude_n it_o shall_v also_o be_v incommensurable_a to_o the_o other_o of_o the_o two_o magnitude_n for_o if_o the_o whole_a magnitude_n ac_fw-la be_v incommensurable_a unto_o the_o magnitude_n bc_o then_o by_o the_o 2_o part_n of_o this_o 16._o theorem_o the_o magnitude_n ab_fw-la and_o bc_o shall_v be_v incommensurable_a wherefore_o by_o the_o first_o part_n of_o the_o same_o theorem_a the_o magnitude_n ac_fw-la shall_v be_v incommensurable_a to_o either_o of_o these_o magnitude_n ab_fw-la and_o bc._n this_o corollary_n 〈◊〉_d use_v in_o the_o demonstration_n of_o the_o â—3_o theorem_a &_o also_o of_o other_o proposition_n ¶_o a_o assumpt_n if_o upon_o a_o right_a line_n be_v apply_v a_o parallelogram_n want_v in_o figure_n by_o a_o square_n the_o parallelogram_n so_o apply_v be_v equal_a to_o that_o parallelogram_n which_o be_v contain_v under_o the_o segment_n of_o the_o right_a line_n which_o segment_n be_v make_v by_o reason_n of_o that_o application_n this_o assumpt_n i_o before_o add_v as_o a_o corollary_n out_o of_o flussates_n after_o the_o 28._o proposition_n of_o the_o six_o book_n ¶_o the_o 14._o theorem_a the_o 17._o proposition_n if_o there_o be_v two_o right_a line_n unequal_a and_o if_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o less_o line_n and_o want_v in_o figure_n by_o a_o square_a if_o also_o the_o parallelogram_n thus_o apply_v divide_v the_o line_n where_o upon_o it_o be_v apply_v into_o part_n commensurable_a in_o length_n then_o shall_v the_o great_a line_n be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o great_a and_o if_o the_o great_a be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o right_a line_n commensurable_a in_o length_n unto_o the_o great_a and_o if_o also_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o less_o line_n and_o want_v in_o figure_n by_o a_o square_n then_o shall_v it_o divide_v the_o great_a line_n into_o part_n commensurable_a but_o now_o suppose_v that_o the_o line_n bc_o be_v in_o power_n more_o than_o the_o line_n a_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n bc._n first_o and_o upon_o the_o line_n bc_o let_v there_o be_v apply_v a_o rectangle_n parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o line_n a_o and_o want_v in_o figure_n by_o a_o square_a and_o let_v the_o say_a parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n bd_o and_o dc_o then_o must_v we_o prove_v that_o the_o line_n bd_o be_v unto_o the_o line_n dc_o commensurable_a in_o length_n the_o same_o construction_n and_o supposition_n that_o be_v before_o remain_v we_o may_v in_o like_a sort_n prove_v that_o the_o line_n bc_o be_v in_o power_n more_o than_o the_o line_n a_o by_o the_o square_n of_o the_o line_n fd._n but_o by_o supposition_n the_o line_n bc_o be_v in_o power_n more_o they_o the_o line_n a_o by_o the_o square_n of_o a_o line_n commensurable_a unto_o it_o in_o length_n wherefore_o the_o line_n bc_o be_v unto_o the_o line_n fd_v commensurable_a in_o length_n wherefore_o the_o line_n compose_v of_o the_o two_o line_n bf_o and_o dc_o be_v commensurable_a in_o length_n unto_o the_o line_n fd_o by_o the_o second_o part_n of_o the_o 15._o of_o the_o ten_o wherefore_o by_o the_o 12._o of_o the_o ten_o or_o by_o the_o first_o part_n of_o the_o 15._o of_o the_o ten_o the_o line_n bc_o be_v commensurable_a in_o length_n to_o the_o line_n compose_v of_o bf_o and_o dc_o but_o the_o whole_a line_n conpose_v bf_o and_o dc_o be_v commensurable_a in_o length_n unto_o dc_o for_o bf_o as_o before_o have_v be_v prove_v be_v equal_a to_o dc_o wherefore_o the_o line_n bc_o be_v commensurable_a in_o length_n unto_o the_o line_n dc_o by_o the_o 12._o of_o the_o ten_o whâ—â—â—fore_o also_o the_o line_n bd_o be_v commensurable_a in_o length_n unto_o the_o line_n dc_o by_o the_o second_o part_n of_o thâ—_z 15._o of_o the_o teâ—th_n if_o therefore_o there_o be_v two_o right_a line_n unequal_a and_o if_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o less_o and_o want_v in_o figure_n by_o a_o square_a if_o also_o the_o parallelogram_n thus_o apply_v divide_v the_o line_n whereupon_o it_o be_v apply_v into_o part_n commensurable_a in_o length_n then_o shall_v the_o great_a line_n be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o great_a and_o if_o the_o great_a be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o great_a and_o if_o also_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n make_v of_o the_o less_o and_o want_v in_o figure_n by_o a_o square_n then_o shall_v it_o divide_v the_o great_a line_n into_o part_n commensurable_a in_o length_n which_o be_v require_v to_o be_v prove_v campanâ—_n after_o this_o proposition_n reach_v how_o we_o may_v ready_o apply_v upon_o the_o line_n bc_o a_o parallelogram_n equal_a to_o the_o four_o part_n of_o the_o square_n of_o half_n of_o the_o line_n a_o and_o want_v in_o figure_n by_o a_o square_a after_o this_o manner_n divide_v the_o line_n bc_o into_o two_o line_n in_o such_o sort_n that_o half_a of_o the_o line_n a_o shall_v be_v the_o mean_a proportional_a between_o those_o two_o line_n which_o be_v possible_a when_o as_o the_o line_n bc_o be_v suppose_v to_o be_v great_a than_o the_o line_n a_o and_o may_v thus_o be_v do_v proposition_n divide_v the_o line_n bc_o into_o two_o equal_a part_n in_o the_o point_n e_o and_o describe_v upon_o the_o line_n bc_o a_o semicircle_n bhc_n and_o unto_o the_o line_n bc_o and_o from_o the_o point_n c_o erect_v a_o perpâ—dicular_a line_n ck_o and_o put_v the_o line_n ck_o equal_a to_o half_o of_o the_o line_n aâ—_z and_o by_o the_o point_n king_n draw_v unto_o the_o line_n ec_o a_o parallel_n line_n kh_o cut_v the_o semicircle_n in_o the_o point_n h_o which_o it_o must_v needs_o cut_v forasmuch_o as_o the_o line_n bc_o be_v great_a than_o the_o line_n a_o and_o from_o the_o point_n h_o draw_v unto_o the_o line_n bc_o a_o perpendicular_a liâ—e_z hd_v which_o line_n hdâ—_n forasmuch_o as_o by_o the_o 34_o of_o the_o first_o it_o be_v equal_a unto_o the_o line_n kc_o shall_v also_o be_v equal_a to_o half_o of_o the_o line_n a_o draw_v the_o line_n bh_o and_o hc_n now_o then_o by_o the_o â—â—_o of_o the_o three_o the_o angle_n bhc_n be_v a_o right_a aâ—gle_n wherefore_o by_o the_o corollary_n of_o the_o eight_o of_o the_o six_o book_n the_o line_n hd_v be_v the_o mean_a proportional_a between_o the_o line_n bd_o and_o dc_o wherefore_o the_o half_a of_o the_o line_n a_o which_o be_v equal_a unto_o the_o line_n hd_v be_v the_o mean_a proportional_a between_o the_o line_n bd_o and_o dc_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n bd_o and_o dc_o be_v equal_a to_o the_o four_o part_n of_o the_o square_n of_o the_o line_n a._n and_o so_o if_o upon_o the_o line_n bd_o be_v describe_v a_o rectangle_n parallelogram_n have_v his_o other_o side_n equal_a to_o the_o line_n dc_o there_o shall_v be_v apply_v upon_o the_o line_n bc_o a_o rectangle_n parallelogram_n equal_a unto_o the_o square_n of_o half_n of_o the_o line_n a_o and_o want_v in_o figure_n by_o
the_o line_n ab_fw-la be_v rational_a by_o the_o definition_n wherefore_o by_o the_o definition_n also_o of_o rational_a figure_n the_o parallelogram_n cd_o shall_v be_v rational_a now_o rest_v a_o other_o caâ—e_n of_o the_o third_o kind_n of_o rational_a line_n commensurable_a in_o length_n the_o one_o to_o the_o other_o which_o be_v to_o the_o rational_a line_n ab_fw-la first_o set_v commensurable_a in_o power_n only_o and_o yet_o be_v therefore_o rational_a line_n and_o let_v the_o line_n ce_fw-fr and_o ed_z be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o length_n now_o then_o let_v the_o self_n same_o construction_n remain_v that_o be_v in_o the_o former_a so_o that_o let_v the_o line_n ce_fw-fr and_o ed_z be_v rational_a commensurable_a in_o power_n only_o unto_o the_o line_n ab_fw-la but_o let_v they_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o then_o i_o say_v that_o in_o this_o case_n also_o the_o parallelogram_n cd_o be_v rational_a 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shall_v be_v a_o rational_a line_n commensurable_a in_o length_n unto_o that_o line_n whereupon_o the_o rational_a parallelogram_n be_v apply_v which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o assumpt_n a_o line_n contain_v in_o power_n a_o irrational_a superficies_n be_v irrational_a assumpt_n suppose_v that_o the_o line_n ab_fw-la contain_v in_o power_n a_o irrational_a superficies_n that_o be_v let_v the_o square_v describe_v upon_o the_o line_n ab_fw-la be_v equal_a unto_o a_o irrational_a superficies_n then_o i_o say_v that_o the_o line_n ab_fw-la be_v irrational_a for_o if_o the_o line_n ab_fw-la be_v rational_a they_o shall_v the_o square_a of_o the_o line_n ab_fw-la be_v also_o rational_a for_o so_o be_v it_o put_v in_o the_o definition_n but_o by_o supposition_n it_o be_v not_o wherefore_o the_o line_n ab_fw-la be_v irrational_a a_o line_n therefore_o contain_v in_o power_n a_o irrational_a superficies_n be_v irrational_a ¶_o the_o 18._o theorem_a the_o 21._o proposition_n a_o rectangle_n figure_n comprehend_v under_o two_o rational_a right_a line_n commensurable_a in_o power_n only_o be_v irrational_a and_o the_o line_n which_o in_o power_n contain_v that_o rectangle_n figure_n be_v irrational_a &_o be_v call_v a_o medial_a line_n svppose_v that_o this_o rectangle_n figure_n ac_fw-la be_v comprehend_v under_o these_o rational_a right_a line_n ab_fw-la and_o bc_o commensurable_a in_o power_n only_o then_o i_o say_v that_o the_o superficies_n ac_fw-la be_v irrational_a and_o the_o line_n which_o contain_v it_o in_o power_n be_v irrational_a and_o be_v call_v a_o medial_a line_n construction_n describe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n ab_fw-la a_o square_a ad._n demonstration_n wherefore_o the_o square_a ad_fw-la be_v rational_a and_o forasmuch_o as_o the_o line_n ab_fw-la be_v unto_o the_o line_n bc_o incommensurable_a in_o length_n for_o they_o be_v suppose_v to_o be_v commensurable_a in_o power_n only_o and_o the_o line_n ab_fw-la be_v equal_a unto_o the_o line_n bd_o therefore_o also_o the_o lineâ—_z bd_o be_v unto_o the_o line_n bc_o incommensurable_a in_o length_n and_o 〈◊〉_d â—hâ—_n linâ—_n 〈…〉_o be_v to_o the_o line_n â—_o c_o so_fw-mi 〈◊〉_d the_o square_a ad_fw-la to_o the_o parallelogram_n ac_fw-la by_o the_o first_o of_o the_o fiuâ—_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a dam_n be_v unto_o the_o parallelogram_n ac_fw-la incommensurable_a but_o the_o square_a dam_n be_v rational_a wherefore_o the_o parallelogram_n ac_fw-la be_v irrational_a wherefore_o also_o the_o line_n that_o contain_v the_o superficies_n ac_fw-la in_o power_n that_o be_v who_o square_a be_v equal_a unto_o the_o parallelogram_n ac_fw-la be_v by_o the_o assumpt_n go_v before_o irrational_a and_o it_o be_v call_v a_o medial_a line_n for_o that_o the_o square_n which_o be_v make_v of_o it_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o and_o therefore_o it_o be_v by_o the_o second_o part_n of_o the_o 17._o of_o the_o six_o a_o mean_a proportional_a line_n between_o the_o line_n ab_fw-la and_o bc._n a_o rectangle_n figure_n therefore_o comprehend_v under_o rational_a right_a line_n which_o be_v commensurable_a in_o power_n only_o be_v irrational_a and_o the_o line_n which_o in_o power_n contain_v that_o rectangle_n figure_n be_v irrational_a and_o be_v call_v a_o medial_a line_n at_o this_o proposition_n do_v euclid_n first_o entreat_v of_o the_o generation_n and_o production_n of_o irrational_a line_n and_o here_o he_o search_v out_o the_o first_o kind_n of_o they_o which_o he_o call_v a_o medial_a line_n and_o the_o definition_n thereof_o be_v full_o gather_v and_o take_v out_o of_o this_o 21._o proposition_n which_o be_v this_o a_o medial_a line_n be_v a_o irrational_a line_n who_o square_a be_v equal_a to_o a_o rectangled_a figure_n contain_v of_o two_o rational_a line_n commensurable_a in_o power_n only_o line_n it_o be_v call_v a_o 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be_v irrational_a corollary_n for_o if_o the_o line_n ab_fw-la be_v rational_a and_o
by_o his_o motion_n describe_v the_o round_a conical_a superficies_n about_o the_o cone_n and_o as_o the_o circumference_n of_o the_o semicircle_n describe_v the_o round_a spherical_a superficies_n about_o the_o sphere_n in_o this_o example_n it_o be_v the_o superficies_n describe_v of_o the_o line_n dc_o corollary_n by_o this_o definition_n it_o be_v plain_a that_o the_o two_o circle_n or_o base_n of_o a_o cilind_a be_v ever_o equal_a and_o parallel_n for_o that_o the_o line_n move_v which_o produce_v they_o remain_v always_o equal_a and_o parallel_n also_o the_o axe_n of_o a_o cilind_a be_v ever_o a_o erect_a line_n unto_o either_o of_o the_o base_n for_o with_o all_o the_o line_n describe_v in_o the_o base_n and_o touch_v it_o it_o make_v right_a angle_n sphere_n campane_n vitellâ—o_o with_o other_o late_a writer_n call_v this_o solid_a or_o body_n a_o round_a columnâ—_n or_o pillar_n and_o campane_n add_v unto_o this_o definition_n this_o as_o a_o corollary_n that_o of_o a_o 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â—g_z of_o the_o cone_n abc_n be_v to_o to_o the_o axe_n ei_o of_o the_o cone_n def_n as_o the_o diameter_n ac_fw-la of_o the_o cone_n abc_n be_v to_o the_o diameter_n df_o of_o the_o cone_fw-it def_n then_o aâ—e_v the_o cones_fw-la abc_n and_o def_n like_o cones_fw-la likewise_o in_o the_o cilinder_n if_o the_o axe_n ln_o of_o the_o cilind_a lhmn_n have_v that_o proportion_n to_o the_o axe_n oq_fw-la of_o the_o cilind_a ropq_n which_o the_o diameter_n he_o have_v to_o the_o diameter_n rp_n then_o be_v the_o cilinder_n hlmn_n and_o ropq_n like_o cilinder_n and_o so_o of_o all_o other_o 21_o a_o cube_n be_v a_o solid_a or_o bodily_a figure_n contain_v under_o six_o equal_a square_n definition_n as_o be_v a_o die_v which_o have_v six_o side_n and_o each_o of_o they_o be_v a_o full_a and_o perfect_a square_n as_o limit_n or_o border_n under_o which_o it_o be_v contain_v and_o as_o you_o may_v conceive_v in_o a_o piece_n of_o timber_n contain_v a_o foot_n square_v every_o way_n or_o in_o 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section_n draw_v in_o any_o one_o of_o the_o six_o side_n which_o can_v not_o be_v so_o well_o see_v in_o the_o other_o figure_n describe_v upon_o a_o playnd_v and_o as_o touch_v the_o first_o figure_n which_o be_v set_v at_o the_o end_n of_o the_o other_o side_n of_o this_o leaf_n you_o see_v that_o there_o be_v six_o parallelogram_n which_o you_o must_v conceive_v to_o be_v both_o equilater_n and_o rectangle_n although_o in_o deed_n there_o can_v be_v in_o this_o description_n only_o two_o of_o they_o rectangle_n they_o may_v in_o deed_n be_v describe_v all_o equilater_n now_o if_o you_o imagine_v one_o of_o the_o six_o parallelogram_n as_o in_o this_o example_n the_o parallelogram_n abcd_o to_o be_v the_o base_a lie_v upon_o a_o ground_n plain_a superfice_n and_o so_o conceive_v the_o parallelogram_n efgh_o to_o be_v in_o the_o top_n over_o it_o in_o such_o sort_n that_o the_o line_n ae_n cg_o dh_o &_o bf_o may_v be_v erect_v perpendicular_o from_o the_o point_n a_o c_o b_o d_o to_o the_o ground_n plain_a superficies_n or_o square_a abcd._n for_o by_o this_o imagination_n this_o figure_n will_v show_v unto_o you_o bodilike_o and_o this_o imagination_n perfect_o have_v will_v make_v many_o of_o the_o proposition_n in_o these_o five_o book_n follow_v in_o which_o be_v require_v to_o be_v describe_v such_o like_a solid_n although_o not_o all_o cube_n to_o be_v more_o plain_o and_o easy_o conceive_v in_o many_o example_n of_o the_o greek_a and_o also_o of_o the_o latin_a there_o be_v in_o this_o place_n set_v the_o definition_n of_o a_o tetrahedron_n which_o be_v thus_o definition_n 22_o a_o tetrahedron_n be_v a_o solid_a which_o be_v contain_v under_o four_o triangle_n equal_a and_o equilater_n a_o form_n of_o this_o solid_a you_o may_v see_v in_o these_o two_o example_n here_o set_v whereof_o one_o be_v as_o it_o be_v common_o describe_v in_o a_o plain_n neither_o be_v it_o hard_o to_o conceive_v for_o as_o we_o before_o teach_v in_o a_o pyramid_n if_o you_o imagine_v the_o triangle_n bcd_v to_o lie_v upon_o a_o ground_n plain_a superficies_n and_o the_o point_n a_o to_o be_v pull_v up_o together_o with_o the_o line_n ab_fw-la ac_fw-la and_o ad_fw-la you_o shall_v perceive_v the_o form_n of_o the_o tetrahedron_n to_o be_v contain_v under_o 4._o triangle_n which_o you_o must_v imagine_v to_o be_v all_o four_o equilater_n and_o equiangle_n though_o they_o can_v not_o so_o be_v draw_v in_o a_o plain_a and_o a_o tetrahedron_n thus_o describe_v be_v of_o more_o use_n in_o these_o five_o book_n follow_v then_o be_v the_o other_o although_o the_o other_o appear_v in_o form_n to_o the_o eye_n more_o bodilike_o body_n why_o this_o definition_n be_v here_o leave_v out_o both_o of_o campane_n and_o of_o flussas_n i_o can_v not_o but_o marvel_n consider_v that_o a_o tetrahedron_n be_v of_o all_o philosopher_n count_v one_o of_o the_o five_o chief_a solid_n which_o be_v here_o define_v of_o euclid_n which_o be_v call_v common_o regular_a body_n without_o mention_v of_o which_o the_o entreaty_n of_o these_o shall_v seem_v much_o maim_v unless_o they_o think_v it_o sufficient_o define_v under_o the_o definition_n of_o a_o pyramid_n pyramid_n which_o plain_o and_o general_o take_v include_v in_o deed_n a_o tetrahedron_n although_o a_o tetrahedron_n proper_o much_o differ_v from_o a_o pyramid_n as_o a_o thing_n special_a or_o a_o particular_a from_o a_o more_o general_a for_o so_o take_v it_o every_o tetrahedron_n be_v a_o pyramid_n but_o not_o every_o pyramid_n be_v a_o tetrahedron_n by_o the_o general_a definition_n of_o a_o pyramid_n the_o superficiece_n of_o the_o side_n may_v be_v as_o many_o in_o number_n as_o you_o listen_v as_o 3.4_o 5.6_o or_o more_o according_a to_o the_o form_n of_o the_o base_a whereon_o it_o be_v set_v whereof_o before_o in_o the_o definition_n of_o a_o pyramid_n be_v example_n give_v but_o in_o a_o tetrahedron_n the_o superficiece_n erect_v can_v be_v but_o three_o in_o number_n according_a to_o the_o base_a thereof_o which_o be_v ever_o a_o triangle_n again_o by_o the_o general_a definition_n of_o a_o pyramiâ—_n the_o superficiece_n erect_v may_v ascend_v as_o high_a as_o you_o listen_v but_o in_o a_o tetrahedron_n they_o must_v all_o be_v equal_a to_o the_o base_a wherefore_o a_o pyramid_n may_v seem_v to_o be_v more_o general_a than_o a_o tetrahedron_n as_o before_o a_o prism_n seem_v to_o be_v more_o general_a than_o a_o parallelipipedon_n or_o a_o side_v column_n so_o that_o every_o parallelipipedon_n be_v a_o prism_n but_o not_o every_o prism_n be_v a_o parallelipipedon_n and_o every_o axe_n in_o a_o sphere_n be_v a_o diameter_n but_o not_o every_o diameter_n of_o a_o sphere_n be_v the_o axe_n thereof_o so_o also_o note_v well_o the_o definition_n of_o a_o pyramid_n every_o tetrahedron_n may_v be_v call_v a_o pyramid_n
signify_v last_o of_o all_o a_o dodecahedron_n heaven_n for_o that_o it_o be_v make_v of_o pâ—ntagoâ—_n who_o angle_n be_v more_o ample_a and_o large_a than_o the_o angle_n of_o the_o other_o body_n and_o by_o that_o â—eaâ—â—â—_n draw_v more_o â—â—_o roundness_n 〈◊〉_d &_o to_o the_o form_n and_o nature_n of_o a_o sphere_n they_o assign_v to_o a_o sphere_n namely_o 〈…〉_o who_o so_o will_v 〈…〉_o in_o his_o tineus_n shall_v â—ead_a of_o these_o figure_n and_o of_o their_o mutual_a proportionâ—â—â—raunge_a maâ—terâ—_n which_o hâ—re_o be_v not_o to_o be_v entreat_v of_o this_o which_o be_v say_v shall_v be_v sufficient_a for_o the_o 〈◊〉_d of_o they_o and_o for_o thâ—_z declaration_n of_o their_o definition_n after_o all_o these_o definition_n here_o set_v of_o euclid_n flussas_n have_v add_v a_o other_o definition_n which_o 〈◊〉_d of_o a_o parallelipipedon_n which_o because_o it_o have_v not_o hitherto_o of_o euclid_n in_o any_o place_n be_v define_v and_o because_o it_o be_v very_o good_a and_o necessary_a to_o be_v have_v i_o think_v good_a not_o to_o omit_v it_o thus_o it_o be_v a_o parallelipipedon_n be_v a_o solid_a figure_n comprehend_v under_o four_o plain_a quadrangle_n figure_n parallelipipedon_n of_o which_o those_o which_o be_v opposite_a be_v parallel_n because_o these_o five_o regular_a body_n here_o define_v be_v not_o by_o these_o figure_n here_o set_v so_o full_o and_o lively_o express_v that_o the_o studious_a beholder_n can_v thorough_o according_a to_o their_o definition_n conceive_v they_o i_o have_v here_o give_v of_o they_o other_o description_n draw_v in_o a_o plain_n by_o which_o you_o may_v easy_o attain_v to_o the_o knowledge_n of_o they_o for_o if_o you_o draw_v the_o like_a form_n in_o matter_n that_o will_v bow_v and_o geve_v place_n as_z most_o apt_o you_o may_v do_v in_o fine_a past_v paper_n such_o as_o pastwive_n make_v woman_n paste_n of_o &_o they_o with_o a_o knife_n cut_v every_o line_n fine_o not_o through_o but_o half_a way_n only_o if_o they_o you_o bow_v and_o bend_v they_o according_o you_o shall_v most_o plain_o and_o manifest_o see_v the_o form_n and_o shape_n of_o these_o body_n even_o as_o their_o definition_n show_v and_o it_o shall_v be_v very_o necessary_a for_o you_o to_o hadâ—â—tore_v of_o that_o past_v paper_n by_o you_o for_o so_o shall_v yoâ—_n upon_o it_o 〈…〉_o the_o form_n of_o other_o body_n as_o prism_n and_o parallelipopedon_n 〈…〉_o set_v forth_o in_o these_o five_o book_n follow_v and_o see_v the_o very_a 〈◊〉_d of_o thâ—se_a body_n there_o mention_v which_o will_v make_v these_o book_n concern_v body_n as_o easy_a unto_o you_o as_o be_v the_o other_o book_n who_o figure_n you_o may_v plain_o see_v upon_o a_o plain_a superficies_n describe_v thiâ—_z figurâ—_n which_o consi_v of_o twâ—luâ—â—quilâ—â—â—â—_n and_o equiangle_n pâ—ntâ—â—â—â—â—_n upoâ—_n the_o foresay_a matter_n and_o fine_o cut_v as_o before_o be_v â—â—ught_v tâ—â—â—lâ—uâ—n_v line_n contain_v within_o thâ—_z figurâ—_n and_o bow_n and_o fold_n the_o penâ—â—gonâ—_n according_o and_o they_o will_v so_o close_o together_o thaâ—_z thâ—y_n will_v â—â—kâ—_n thâ—_z very_a form_n of_o a_o dodecahedron_n dâ—dâ—â—â—â—edron_n icosaâ—edron_n if_o you_o describe_v this_o figure_n which_o consist_v of_o twenty_o equilater_n and_o equiangle_n triangle_n upon_o the_o foresay_a matter_n and_o fine_o cut_v as_o before_o be_v show_v the_o ninâ—tâ—ne_a line_n which_o be_v contain_v within_o the_o figure_n and_o then_o bow_n and_o fold_n they_o according_o they_o will_v in_o such_o sort_n close_o together_o that_o therâ—_n will_v be_v make_v a_o perfect_v form_n of_o a_o icosahedron_n because_o in_o these_o five_o book_n there_o be_v sometime_o require_v other_o body_n beside_o the_o foresay_a five_o regular_a body_n as_o pyramid_n of_o diverse_a form_n prism_n and_o other_o i_o have_v here_o set_v forth_o three_o figure_n of_o three_o sundry_a pyramid_n one_o have_v to_o his_o base_a a_o triangle_n a_o other_o a_o quadrangle_n figure_n the_o other_o â—_o pentagonâ—_n which_o if_o you_o describe_v upon_o the_o foresay_a matter_n &_o fine_o cut_v as_o it_o be_v before_o teach_v the_o line_n contain_v within_o each_o figure_n namely_o in_o the_o first_o three_o line_n in_o the_o second_o four_o line_n and_o in_o the_o three_o five_o line_n and_o so_o bend_v and_o fold_v they_o according_o they_o will_v so_o close_o together_o at_o the_o top_n that_o they_o will_v â—ake_v pyramid_n of_o that_o form_n that_o their_o base_n be_v of_o and_o if_o you_o conceive_v well_o the_o describe_v of_o these_o you_o may_v most_o easy_o describe_v the_o body_n of_o a_o pyramid_n of_o what_o form_n so_o ever_o you_o will._n because_o these_o five_o book_n follow_v be_v somewhat_o hard_o for_o young_a beginner_n by_o reason_n they_o must_v in_o the_o figure_n describe_v in_o a_o plain_a imagine_v line_n and_o superficiece_n to_o be_v elevate_v and_o erect_v the_o one_o to_o the_o other_o and_o also_o conceive_v solid_n or_o body_n which_o for_o that_o they_o have_v not_o hitherto_o be_v acquaint_v with_o will_v at_o the_o first_o sight_n be_v somewhat_o strange_a unto_o they_o i_o have_v for_o their_o more_o â—ase_a in_o this_o eleven_o book_n at_o the_o end_n of_o the_o demonstration_n of_o every_o proposition_n either_o set_v new_a figure_n if_o they_o concern_v the_o elevate_n or_o erect_v of_o line_n or_o superficiece_n or_o else_o if_o they_o concern_v body_n i_o have_v show_v how_o they_o shall_v describe_v body_n to_o be_v compare_v with_o the_o construction_n and_o demonstration_n of_o the_o proposition_n to_o they_o belong_v and_o if_o they_o diligent_o weigh_v the_o manner_n observe_v in_o this_o eleven_o book_n touch_v the_o description_n of_o new_a figure_n agree_v with_o the_o figure_n describe_v in_o the_o plain_a it_o shall_v not_o be_v hard_o for_o they_o of_o themselves_o to_o do_v the_o like_a in_o the_o other_o book_n follow_v when_o they_o come_v to_o a_o proposition_n which_o concern_v either_o the_o elevate_n or_o erect_v of_o line_n and_o superficiece_n or_o any_o kind_n of_o body_n to_o be_v imagine_v ¶_o the_o 1._o theorem_a the_o 1._o proposition_n that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n &_o part_v elevate_v upward_o be_v impossible_a for_o if_o it_o be_v possible_a let_v part_n of_o the_o right_a line_n abc_n namely_o the_o part_n ab_fw-la be_v in_o a_o ground_n plain_a superficies_n and_o the_o other_o part_n thereof_o namely_o bc_o be_v elevate_v upward_o and_o produce_v direct_o upon_o the_o ground_n plain_a superficies_n the_o right_a line_n ab_fw-la beyond_o the_o point_n b_o unto_o the_o point_n d._n wherefore_o unto_o two_o right_a line_n give_v abc_n and_o abdella_n the_o line_n ab_fw-la be_v a_o common_a section_n or_o part_n which_o be_v impossible_a impossibility_n for_o a_o right_a line_n can_v not_o touch_v a_o right_a line_n in_o 〈◊〉_d point_n then_o one_o uâ—lesse_o those_o right_n be_v exact_o agree_v and_o lay_v the_o one_o upon_o the_o other_o wherefore_o that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n and_o part_v elevate_v upward_o be_v impossible_a which_o be_v require_v to_o be_v prove_v this_o figure_n more_o plain_o set_v forth_o the_o foresay_a demonstration_n if_o you_o elevate_v the_o superficies_n wherein_o the_o line_n bc._n an_o other_o demonstration_n after_o flâ—sâ—â—s_n if_o it_o be_v possible_a let_v there_o be_v a_o right_a line_n abg_o who_o part_n ab_fw-la let_v be_v in_o the_o ground_n plain_a superficies_n aed_n flussas_n and_o let_v the_o rest_n thereof_o bg_o be_v elevate_v on_o high_a that_o be_v without_o the_o plain_n aed_n then_o i_o say_v that_o abg_o be_v not_o one_o right_a line_n for_o forasmuch_o as_o aed_n be_v a_o plain_a superficies_n produce_v direct_o &_o equal_o upon_o the_o say_a plain_n aed_n the_o right_a line_n ab_fw-la towards_o d_z which_o by_o the_o 4._o definition_n of_o the_o first_o shall_v be_v a_o right_a line_n and_o from_o some_o one_o point_n of_o the_o right_a line_n abdella_n namely_o from_o c_o draâ—_n unto_o the_o point_n g_o a_o right_a line_n cg_o wherefore_o in_o the_o triangle_n 〈…〉_o the_o outward_a angâ—â—_n abâ—_n be_v equal_a to_o the_o two_o inward_a and_o opposite_a angle_n by_o the_o 32._o of_o the_o first_o and_o therefore_o it_o be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o same_o wherefore_o the_o line_n abg_o forasmuch_o as_o it_o make_v a_o angle_n be_v not_o â—_o right_n line_n wherefore_o that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n and_o part_v elevate_v upward_o be_v impossible_a if_o you_o mark_v well_o the_o figure_n before_o add_v for_o the_o plain_a declaration_n of_o euclides_n demonstration_n iâ—_z will_v not_o be_v hard_o for_o you_o to_o coâ—â—â—â—e_v this_o figure_n which_o ulyss_n put_v for_o his_o demonstration_n â—_o wherein_o
a_o triangle_n and_o if_o the_o parallelogram_n be_v double_a to_o the_o triangle_n those_o prism_n be_v by_o the_o 40._o of_o the_o eleven_o equal_v the_o one_o to_o the_o other_o therefore_o the_o prism_n contain_v under_o the_o two_o triangle_n bkf_a and_o ehg_n and_o under_o the_o three_o parallelogram_n ebfg_n ebkh_n and_o khfg_n part_n be_v equal_a to_o the_o prism_n contain_v under_o the_o two_o triangle_n gfc_a and_o hkl_n and_o under_o the_o three_o parallelogram_n kfcl_n lcgh_n and_o hkfg_n and_o it_o be_v manifest_a that_o both_o these_o prism_n of_o which_o the_o base_a of_o one_o be_v the_o parallelogram_n ebfg_n pyramid_n and_o the_o opposite_a unto_o it_o the_o line_n kh_o and_o the_o base_a of_o the_o other_o be_v the_o triangle_n gfc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n klh_n be_v great_a than_o both_o these_o pyramid_n who_o base_n be_v the_o triangle_n age_n and_o hkl_n and_o top_n the_o point_n h_o &_o d._n for_o if_o we_o draw_v these_o right_a line_n of_o and_o eke_o the_o prism_n who_o base_a be_v the_o parallelogram_n ebfg_n and_o the_o opposite_a unto_o it_o the_o right_a line_n hk_o be_v great_a than_o the_o pyramid_n who_o base_a be_v the_o triangle_n ebf_n &_o top_n the_o point_n k._n but_o the_o pyramid_n who_o base_a be_v the_o triangle_n ebf_n and_o top_n the_o point_n king_n be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h_o for_o they_o be_v contain_v under_o equal_a and_o like_a plain_a superficiece_n wherefore_o also_o the_o prism_n who_o base_a be_v the_o parallelogram_n ebfg_n and_o the_o opposite_a unto_o it_o the_o right_a line_n hk_o be_v great_a than_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h._n but_o the_o prism_n who_o base_a be_v the_o parallelogram_n ebfg_n and_o the_o opposite_a unto_o it_o the_o right_a line_n hk_o be_v equal_a to_o the_o prism_n who_o base_a be_v the_o triangle_n gfc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n hkl_n and_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h_o be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n hkl_n and_o top_n the_o point_n d._n part_n wherefore_o the_o foresay_a two_o prism_n be_v great_a than_o the_o foresay_a two_o pyramid_n who_o base_n be_v the_o triangle_n aeg_n hkl_n and_o top_n the_o point_n h_o and_o d._n wherefore_o the_o whole_a pyramid_n who_o base_a be_v the_o triangle_n abc_n proposition_n and_o top_n the_o point_n d_o be_v divide_v into_o two_o pyramid_n equal_a and_o like_v the_o one_o to_o the_o other_o and_o like_v also_o unto_o the_o whole_a pyramid_n have_v also_o triangle_n to_o their_o base_n and_o into_o two_o equal_a prism_n and_o the_o two_o prism_n be_v great_a than_o half_a of_o the_o whole_a pyramid_n which_o be_v require_v to_o be_v demonstrate_v if_o you_o will_v with_o diligence_n read_v these_o four_o book_n follow_v of_o euclid_n which_o concern_v body_n and_o clear_o see_v the_o demonstration_n in_o they_o contain_v it_o shall_v be_v requisite_a for_o you_o when_o you_o come_v to_o any_o proposition_n which_o concern_v a_o body_n or_o body_n whether_o they_o be_v regular_a or_o not_o first_o to_o describe_v of_o pâ—sâ—ed_a paper_n according_a as_o i_o teach_v you_o in_o the_o end_n of_o the_o definition_n of_o the_o eleven_o book_n such_o a_o body_n or_o body_n as_o be_v there_o require_v and_o have_v your_o body_n or_o body_n thus_o describe_v when_o you_o have_v note_v it_o with_o letter_n according_a to_o the_o figure_n set_v forth_o upon_o a_o plain_a in_o the_o proposition_n follow_v the_o construction_n require_v in_o the_o proposition_n as_o for_o example_n in_o this_o three_o proposition_n it_o be_v say_v that_o every_o pyramid_n have_v a_o triangle_n to_o â—is_n base_a may_v be_v divide_v into_o two_o pyramid_n etc_n etc_n here_o first_o describe_v a_o pyramid_n of_o past_v paper_n haâ—ing_v his_o base_a triangle_v it_o skill_v not_o whether_o it_o be_v equilater_n or_o equiangle_v or_o not_o only_o in_o this_o proposition_n be_v require_v that_o the_o base_a be_v a_o triangle_n then_o for_o that_o the_o proposition_n suppose_v the_o base_a of_o the_o pyramid_n to_o be_v the_o triangle_n abc_n note_v the_o base_a of_o your_o pyramid_n which_o you_o have_v describe_v with_o the_o letter_n abc_n and_o the_o top_n of_o your_o pyramid_n with_o the_o letter_n d_o for_o so_o be_v require_v in_o the_o proposition_n and_o thus_o have_v you_o your_o body_n order_v ready_a to_o the_o construction_n now_o in_o the_o construction_n it_o be_v require_v that_o you_o divide_v the_o line_n ab_fw-la bc_o ca._n etc_o etc_o namely_o the_o six_o line_n which_o be_v the_o side_n of_o the_o four_o triangle_n contain_v the_o pyramid_n into_o two_o equal_a part_n in_o the_o poyntet_fw-la â—_o f_o g_o etc_n etc_n that_o be_v you_o must_v divide_v the_o line_n ab_fw-la of_o your_o pyramid_n into_o two_o equal_a part_n and_o note_v the_o point_n of_o the_o division_n with_o the_o letter_n e_o and_o so_o the_o line_n bc_o be_v divide_v into_o two_o equal_a part_n note_v the_o point_n of_o the_o division_n with_o the_o letter_n f._n and_o so_o the_o rest_n and_o this_o order_n follow_v you_o as_o touch_v the_o rest_n of_o the_o construction_n there_o put_v and_o when_o you_o have_v finish_v the_o construction_n compare_v your_o body_n thus_o describe_v with_o the_o demonstration_n and_o it_o will_v make_v it_o very_o plain_a and_o easy_a to_o be_v understand_v whereas_o without_o such_o a_o body_n describe_v of_o matter_n it_o be_v hard_o for_o young_a beginner_n unless_o they_o have_v a_o very_a deep_a imagination_n full_o to_o conceive_v the_o demonstration_n by_o the_o sigâ—e_n as_o it_o be_v describe_v in_o a_o plain_a here_o for_o the_o better_a declaration_n of_o that_o which_o i_o have_v say_v have_v i_o set_v a_o figure_n who_o form_n if_o you_o describe_v upon_o past_v paper_n note_v with_o the_o like_a letter_n and_o cut_v the_o line_n â—a_o dam_n dc_o and_o fold_v it_o according_o it_o will_v make_v a_o pyramid_n describe_v according_a to_o the_o construction_n require_v in_o the_o proposition_n and_o this_o order_n follow_v you_o as_o touch_v all_o other_o proposition_n which_o concern_v body_n ¶_o a_o other_o demonstration_n after_o campane_n of_o the_o 3._o proposition_n suppose_v that_o there_o be_v a_o pyramid_n abcd_o have_v to_o his_o base_a the_o triangle_n bcd_o and_o let_v his_o top_n be_v the_o solid_a angle_n a_o from_o which_o let_v there_o be_v draw_v three_o subtended_a line_n ab_fw-la ac_fw-la and_o ad_fw-la to_o the_o three_o angle_n of_o the_o base_a and_o divide_v all_o the_o side_n of_o the_o base_a into_o two_o equal_a part_n in_o the_o three_o point_n e_o f_o g_o divide_v also_o the_o three_o subtended_a line_n ab_fw-la ac_fw-la and_o ad_fw-la in_o two_o equal_a part_n in_o the_o three_o point_n h_o edward_n l._n and_o draw_v in_o the_o base_a these_o two_o line_n of_o and_o eglantine_n so_o shall_v the_o base_a of_o the_o pyramid_n be_v divide_v into_o three_o superficiece_n whereof_o two_o be_v the_o two_o triangle_n bef_n and_o egg_v which_o be_v like_o both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o whole_a base_a by_o the_o 2_o part_n of_o the_o second_o of_o the_o six_o &_o by_o the_o definition_n of_o like_a superâ—iciecâ—s_n &_o they_o be_v equal_v the_o one_o to_o the_o other_o by_o the_o 8._o of_o the_o first_o the_o three_o superficies_n be_v a_o quadrangled_a parallelogram_n namely_o efgc_n which_o be_v double_a to_o the_o triangle_n egg_v by_o the_o 40._o and_o 41._o of_o the_o first_o now_o then_o again_o from_o the_o point_n h_o draw_v unto_o the_o point_n e_o and_o f_o these_o two_o subtendent_fw-la line_n he_o and_o hf_o draw_v also_o a_o subtended_a line_n from_o the_o point_n king_n to_o the_o point_n g._n and_o draw_v these_o line_n hk_o kl_o and_o lh_o wherefore_o the_o whole_a pyramid_n abcd_o be_v divide_v into_o two_o pyramid_n which_o be_v hbef_n and_o ahkl_n and_o into_o two_o prism_n of_o which_o the_o one_o be_v ehfgkc_n and_o be_v set_v upon_o the_o quadrangled_a base_a cfge_n the_o other_o be_v egdhkl_n and_o have_v to_o his_o base_a the_o triangle_n egg_v now_o as_o touch_v the_o two_o pyramid_n hbef_a and_o ahkl_n that_o they_o be_v equal_v the_o one_o to_o the_o other_o and_o also_o that_o they_o be_v like_o both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o whole_a it_o be_v manifest_a by_o the_o definition_n of_o equal_a and_o like_a body_n and_o by_o the_o 10._o of_o the_o eleven_o and_o by_o 2._o part_n of_o the_o second_o of_o the_o six_o and_o that_o the_o two_o prism_n be_v equal_a it_o
be_v manifest_a by_o the_o last_o of_o the_o eleven_o and_o now_o that_o both_o the_o prism_n take_v together_o be_v great_a than_o the_o half_a of_o the_o whole_a pyramid_n hereby_o it_o be_v manifest_a for_o that_o either_o of_o they_o may_v be_v divide_v into_o two_o pyramid_n of_o which_o the_o one_o be_v a_o triangular_a pyramid_n equal_a to_o one_o of_o the_o two_o pyramid_n into_o which_o together_o with_o the_o two_o prism_n be_v divide_v the_o whole_a pyramid_n and_o the_o other_o be_v a_o quadrangled_a pyramid_n double_a to_o the_o other_o pyramid_n wherefore_o it_o be_v plain_a that_o the_o two_o prism_n take_v together_o be_v three_o quarter_n of_o the_o whole_a pyramid_n divide_v but_o if_o you_o be_v desirous_a to_o know_v the_o proportion_n between_o they_o read_v the_o â—_o of_o this_o book_n but_o now_o here_o to_o this_o purpose_n it_o shall_v be_v sufficient_a to_o know_v that_o the_o two_o prism_n take_v together_o do_v exceed_v in_o quantity_n the_o two_o partial_a pyramid_n take_v together_o into_o which_o together_o with_o the_o two_o prism_n the_o whole_a pyramid_n be_v divide_v ¶_o the_o 4._o theorem_a the_o 4._o proposition_n if_o there_o be_v two_o pyramid_n under_o equal_a altitude_n have_v triangle_n to_o their_o base_n and_o either_o of_o those_o pyramid_n be_v divide_v into_o two_o pyramid_n equal_v the_o one_o to_o the_o other_o and_o like_v unto_o the_o whole_a and_o into_o two_o squall_n prism_n and_o again_o if_o in_o either_o of_o the_o pyramid_n make_v of_o the_o two_o first_o pyramid_n be_v still_o observe_v the_o same_o order_n and_o manner_n then_o as_o the_o base_a of_o the_o one_o pyramid_n be_v to_o the_o base_a of_o the_o other_o pyramid_n so_o be_v all_o the_o prism_n which_o be_v in_o the_o one_o pyramid_n to_o all_o the_o prism_n which_o be_v in_o the_o other_o be_v equal_a in_o multitude_n with_o they_o svppose_v that_o there_o be_v two_o pyramid_n under_o equal_a altitude_n have_v triangle_n to_o their_o base_n namely_o abc_n and_o def_n and_o have_v to_o their_o top_n the_o point_n g_o and_o h._n and_o let_v either_o of_o these_o pyramid_n be_v divide_v into_o two_o pyramid_n equal_v the_o one_o to_o the_o other_o and_o like_v unto_o the_o whole_a and_o into_o two_o equal_a prism_n according_a to_o the_o method_n of_o the_o former_a proposition_n and_o again_o let_v either_o of_o those_o pyramid_n soâ—_n make_v of_o the_o two_o first_o pyramid_n be_v imagine_v to_o be_v after_o the_o same_o order_n divide_v and_o so_o do_v continual_o then_o i_o say_v that_o as_o the_o base_a abc_n be_v to_o the_o base_a def_n so_o be_v all_o the_o prism_n which_o be_v in_o the_o pyramid_n abcg_o to_o all_o the_o prism_n which_o be_v in_o the_o pyramid_n defh_o be_v equal_a in_o multitude_n with_o they_o for_o forasmuch_o as_o the_o line_n bx_n be_v equal_a to_o the_o line_n xc_o and_o the_o line_n all_o to_o the_o line_n lc_n for_o as_o we_o see_v in_o the_o construction_n pertain_v to_o the_o former_a proposition_n all_o the_o six_o side_n of_o the_o whole_a pyramid_n be_v each_o divide_v into_o two_o equal_a part_n the_o like_a of_o which_o construction_n be_v in_o this_o proposition_n also_o suppose_v therefore_o the_o line_n xl_o be_v a_o parallel_n to_o the_o line_n ab_fw-la &_o the_o triangle_n abc_n be_v like_a to_o the_o triangle_n lxc_n by_o the_o corollary_n of_o the_o second_o of_o the_o six_o and_o by_o the_o same_o reason_n the_o triangle_n def_n be_v like_a to_o the_o triangle_n rwf_n and_o forasmuch_o as_o the_o line_n bc_o be_v double_a to_o the_o line_n cx_o and_o the_o line_n fe_o to_o the_o line_n fw_o therefore_o as_o the_o line_n bc_o be_v to_o the_o line_n cx_o so_o be_v the_o line_n of_o to_o the_o line_n fw_o and_o upon_o the_o line_n bc_o and_o cx_o be_v describe_v rectiline_a figure_n like_a and_o in_o like_a sort_n set_v namely_o the_o triangle_n abc_n and_o lxc_n and_o upon_o the_o line_n of_o and_o fw_o be_v also_o describe_v rectiline_a figure_n like_a and_o in_o like_a sort_n set_v namely_o the_o triangle_n def_n &_o rwfâ—_n but_o if_o there_o be_v four_o right_a line_n proportional_a the_o rectiline_a figure_n describe_v of_o they_o be_v like_a and_o in_o like_a sort_n set_v shall_v also_o be_v proportional_a by_o the_o 22._o of_o the_o six_o wherefore_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n lxc_n so_o be_v the_o triangle_n def_n to_o the_o triangle_n rwf_n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n def_n so_o be_v the_o triangle_n lxc_n to_o the_o triangle_n rwf_n assumpt_n but_o as_o the_o triangle_n lxc_n be_v to_o the_o triangle_n rwf_n so_o be_v the_o prism_n who_o base_a be_v the_o triangle_n lxc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n omnes_n to_o the_o prism_n who_o base_a be_v the_o triangle_n rwf_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n stv_n by_o the_o corollary_n of_o the_o 40._o of_o the_o eleven_o for_o these_o prism_n be_v under_o one_o &_o the_o self_n same_o altitude_n namely_o under_o the_o half_a of_o the_o altitude_n of_o the_o whole_a pyramid_n which_o pyramid_n be_v suppose_v to_o be_v under_o one_o and_o the_o self_n same_o altitude_n this_o be_v also_o prove_v in_o the_o assumpt_n follow_v wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n def_n so_o be_v the_o prism_n who_o base_a be_v the_o triangle_n lxc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n omnes_n to_o the_o prism_n who_o base_a be_v the_o triangle_n rwf_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n stv_n and_o forasmuch_o as_o there_o be_v two_o prism_n in_o the_o pyramid_n abcg_o equal_a the_o one_o to_o the_o other_o &_o two_o prism_n also_o in_o the_o pyramid_n defh_o equal_a the_o one_o to_o the_o other_o therefore_o as_o the_o prism_n who_o base_a be_v the_o parallelogram_n bklx_fw-fr and_o the_o opposite_a side_n unto_o it_o the_o line_n more_n be_v to_o the_o prism_n who_o base_a be_v the_o triangle_n lxc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n omnes_n so_o be_v the_o prism_n who_o base_a be_v the_o parallelogram_n perw_n and_o the_o opposite_a unto_o it_o the_o line_n n-ab_n to_o the_o prism_n who_o base_a be_v the_o triangle_n rwf_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n stv_n wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o the_o prism_n kbxlmo_n &_o lxcmno_n be_v to_o the_o prism_n lxcmno_n so_o be_v the_o prism_n pewrst_a and_o rwfstv_n to_o the_o prism_n rwfstv_n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o two_o prism_n kbxlmo_n and_o lxcmno_n be_v to_o the_o two_o prism_n pewrst_a and_o rwfstv_n so_o be_v the_o prism_n lxcmno_n to_o the_o prism_n rwfstv_n but_o as_o the_o prism_n lxcmno_n be_v to_o the_o prism_n rwfstv_n so_o have_v we_o prove_v that_o the_o base_a lxc_n be_v to_o the_o base_a rwf_n and_o the_o base_a abc_n to_o the_o base_a def_n wherefore_o by_o the_o 16._o of_o the_o five_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n def_n so_o be_v both_o the_o prism_n which_o be_v in_o the_o pyramid_n abcg_o to_o both_o the_o prism_n which_o be_v in_o the_o pyramid_n defh_o and_o in_o like_a sort_n if_o we_o divide_v the_o other_o pyramid_n after_o the_o self_n same_o manner_n namely_o the_o pyramid_n omng_v and_o the_o pyramid_n stuh_o as_o the_o base_a omnes_n be_v to_o the_o base_a stv_n so_o shall_v both_o the_o prism_n that_o be_v in_o the_o pyramid_n omng_v be_v to_o both_o the_o prism_n which_o be_v in_o the_o pyramid_n stuh_o but_o as_o the_o base_a omnes_n be_v to_o the_o base_a stv_n so_o be_v the_o base_a abc_n to_o the_o base_a def_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o base_a abc_n be_v to_o the_o base_a def_n so_o be_v the_o two_o prism_n that_o be_v in_o the_o pyramid_n abcg_o to_o the_o two_o prism_n that_o be_v in_o the_o pyramid_n defh_o and_o the_o two_o prism_n that_o be_v in_o the_o pyramid_n omng_v to_o the_o two_o prism_n that_o be_v in_o the_o pyramid_n stuh_o and_o the_o four_o prism_n to_o the_o four_o prism_n and_o so_o also_o shall_v it_o follow_v in_o the_o prism_n make_v by_o divide_v the_o two_o pyramid_n aklo_n and_o dpr_n and_o of_o all_o the_o other_o pyramid_n in_o general_a be_v equal_a in_o multitude_n andrea_n that_o as_o the_o triangle_n lxc_n be_v to_o the_o triangle_n rwf_n so_o be_v the_o prism_n who_o base_a be_v the_o triangle_n lxc_n and_o the_o opposite_a side_n omnes_n assumpt_n to_o the_o prism_n who_o base_a be_v the_o triangle_n rwf_n and_o the_o
same_o be_v treble_a it_o be_v manifest_a by_o the_o 12._o of_o the_o five_o that_o all_o the_o prism_n be_v to_o all_o the_o pyramid_n treble_a wherefore_o parallelipipedon_n be_v treble_a to_o pyramid_n set_v upon_o the_o self_n same_o base_a with_o they_o and_o under_o the_o same_o altitude_n they_o for_o that_o they_o contain_v two_o prism_n three_o corollary_n if_o two_o prism_n be_v under_o one_o and_o the_o self_n same_o altitude_n have_v to_o their_o base_n either_o both_o triangle_n or_o both_o parallelogram_n the_o prism_n be_v the_o one_o to_o the_o other_o as_o their_o base_n be_v for_o forasmuch_o as_o those_o prism_n be_v equemultiqlice_n unto_o the_o pyramid_n upon_o the_o self_n same_o base_n and_o under_o the_o same_o altitude_n which_o pyramid_n be_v in_o proportion_n as_o their_o base_n it_o be_v manifest_a by_o the_o 15._o of_o the_o five_o that_o the_o prism_n be_v in_o the_o proportion_n of_o the_o base_n for_o by_o the_o former_a corollary_n the_o prism_n be_v treble_a to_o the_o pyramid_n sâ—t_v upon_o the_o triangular_a base_n four_o corollary_n prism_n be_v in_o sesquealtera_fw-la proportion_n to_o pyramid_n which_o have_v the_o self_n same_o quadrangled_a base_a that_o the_o prism_n have_v and_o be_v under_o the_o self_n same_o altitude_n for_o that_o pyramid_n contain_v two_o pyramid_n set_v upon_o a_o triangular_a base_a of_o the_o same_o prism_n for_o it_o be_v prove_v that_o that_o prism_n be_v treble_a to_o the_o pyramid_n which_o be_v set_v upon_o the_o half_a of_o his_o quadrangled_a base_a unto_o which_o the_o other_o which_o be_v set_v upon_o the_o whole_a base_a be_v double_a by_o the_o six_o of_o this_o book_n five_v corollary_n wherefore_o we_o may_v in_o like_a sort_n conclude_v that_o solid_n mention_v in_o the_o second_o corollary_n which_o solid_n campane_n call_v side_v column_n be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n which_o be_v poligonon_o figure_n for_o they_o be_v in_o the_o proportion_n of_o the_o pyramid_n or_o prism_n set_v upon_o the_o self_n same_o base_n and_o under_o the_o self_n same_o altitude_n that_o be_v they_o be_v in_o the_o proportion_n of_o the_o base_n of_o the_o say_a pyramid_n or_o prism_n for_o those_o solid_n may_v be_v divide_v into_o prism_n have_v the_o self_n same_o altitude_n when_o as_o their_o opposite_a base_n may_v be_v divide_v into_o triangle_n by_o the_o 20_o of_o the_o six_o upon_o which_o triangle_n prism_n be_v set_v be_v in_o proportion_n as_o their_o base_n by_o this_o 7._o proposition_n it_o plain_o appear_v that_o â—uâ—lide_a as_o it_o be_v before_o note_v in_o the_o diffinitionâ—â—_n under_o the_o definition_n of_o a_o prism_n comprehend_v also_o those_o kind_n of_o solid_n which_o campane_n call_v side_v column_n for_o in_o that_o he_o say_v every_o prism_n have_v a_o triangle_n to_o his_o base_a may_v be_v devidedâ—_n etc_n etc_n he_o need_v not_o take_v a_o prism_n in_o that_o sense_n which_o campane_n and_o most_o man_n take_v it_o to_o have_v add_v that_o particle_n have_v to_o his_o base_a a_o triangle_n for_o by_o their_o sense_n there_o be_v no_o prism_n but_o it_o may_v have_v to_o his_o base_a a_o triangleâ—_n and_o so_o it_o may_v seem_v that_o euclid_n ought_v without_o exception_n have_v say_v that_o every_o prism_n whatsoever_o may_v be_v divide_v into_o three_o pyramid_n equal_a the_o one_o to_o the_o other_o have_v also_o triangle_n to_o â—heir_a base_n for_o so_o do_v campane_n and_o flussas_n put_v the_o proposition_n leave_v out_o the_o former_a particle_n have_v to_o his_o base_a a_o triangle_n which_o yet_o be_v red_a in_o the_o greek_a copy_n &_o not_o leâ—t_v out_o by_o any_o other_o interpreter_n know_v abroad_o except_o by_o campane_n and_o flussas_n yea_o and_o the_o corollary_n follow_v of_o this_o proposition_n add_v by_o theon_n or_o euclid_n and_o amend_v by_o m._n dee_n seem_v to_o confirm_v this_o sense_n of_o this_o â—s_a 〈◊〉_d make_v manifest_a that_o every_o pyramid_n be_v the_o three_o part_n of_o the_o prism_n have_v the_o same_o base_a with_o it_o and_o equal_a altitude_n for_o and_o if_o the_o base_a of_o the_o prism_n have_v any_o other_o right_o line_v figure_n than_o a_o triangle_n and_o also_o the_o superficies_n opposite_a to_o the_o base_a the_o same_o figure_n that_o prism_n may_v be_v divide_v into_o prism_n have_v triangle_v base_n and_o the_o superficiece_n to_o those_o base_n opposite_a also_o triangle_v a_o â—â—ike_a and_o equal_o for_o there_o as_o we_o see_v be_v put_v these_o word_n â—or_a and_o if_o the_o base_a of_o the_o prism_n be_v any_o other_o right_o line_v figureâ—_n etc_n etc_n whereof_o a_o man_n may_v well_o infer_v that_o the_o base_a may_v be_v any_o other_o rectiline_a figure_n whatsoever_o &_o not_o only_o a_o triangle_n or_o a_o parallelogram_n and_o it_o be_v true_a also_o in_o that_o sense_n as_o it_o be_v plain_a to_o see_v by_o the_o second_o corollary_n add_v out_o of_o flussas_n which_o corollary_n as_o also_o the_o first_o of_o his_o corollary_n be_v in_o a_o manner_n all_o one_o with_o the_o corollary_n add_v by_o theon_n or_o euclid_n far_o theon_n in_o the_o demonstration_n of_o the_o 10._o proposition_n of_o this_o book_n as_o we_o shall_v afterwards_o see_v most_o plain_o call_v not_o only_o side_v column_n prism_n but_o also_o parallelipipedon_n and_o although_o the_o 40._o proposition_n of_o the_o eleven_o book_n may_v seem_v hereunto_o to_o be_v a_o lâ—t_n for_o that_o it_o can_v be_v understand_v of_o those_o prism_n only_o which_o have_v triangle_n to_o their_o like_a equal_a opposite_a and_o parallel_v side_n or_o but_o of_o some_o side_v column_n and_o not_o of_o all_o yet_o may_v that_o let_v be_v thus_o remove_v away_o to_o say_v that_o euclid_n in_o that_o proposition_n use_v genus_fw-la pro_fw-la specie_fw-la that_o be_v the_o general_a word_n for_o some_o special_a kind_n thereof_o which_o thing_n also_o be_v not_o rare_a not_o only_o with_o he_o but_o also_o with_o other_o learned_a philosopher_n thus_o much_o i_o think_v good_a by_o the_o way_n to_o note_v in_o far_a defence_n of_o euclid_n definition_n of_o a_o prism_n the_o 8._o theorem_a the_o 8._o proposition_n pyramid_n be_v like_o &_o have_a triangle_n to_o their_o base_n be_v in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o their_o side_n of_o like_a proportion_n be_v svppose_v that_o these_o pyramid_n who_o base_n be_v the_o triangle_n gbc_n and_o hef_n and_o top_n the_o point_n a_o and_o d_o be_v like_o and_o in_o like_a sort_n describe_v and_o let_v ab_fw-la and_o de_fw-fr be_v side_n of_o like_a proportion_n then_o i_o say_v that_o the_o pyramid_n abcg_o be_v to_o the_o pyramid_n defh_o in_o treble_a proportion_n of_o that_o in_o which_o the_o side_n ab_fw-la be_v to_o the_o side_n de._n make_v perfect_a the_o parallelipipedon_n namely_o the_o solid_n bckl_n &_o efxo_n and_o forasmuch_o as_o the_o pyramid_n abcg_o be_v like_a to_o the_o pyramid_n defh_o construction_n therefore_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n def_n demonstration_n &_o the_o angle_n gbc_n to_o the_o angle_n hef_fw-fr and_o moreover_o the_o angle_n abg_o to_o the_o angle_n deh_fw-it and_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n de_fw-fr so_o be_v the_o line_n bc_o to_o the_o line_n of_o and_o the_o line_n bg_o to_o the_o line_n eh_o and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n de_fw-fr so_o be_v the_o line_n bc_o to_o the_o line_n of_o and_o the_o side_n about_o the_o equal_a angle_n be_v proportional_a therefore_o the_o parallelogram_n bm_n be_v like_a to_o the_o parallelogram_n ep_n and_o by_o the_o same_o reason_n the_o parallelogram_n bn_o be_v like_a to_o the_o parallelogram_n er_fw-mi and_o the_o parellelogramme_n bk_o be_v like_a unto_o the_o parallelogram_n exit_fw-la wherefore_o the_o three_o parallelogram_n bm_n kb_o and_o bn_o be_v like_a to_o the_o three_o parallelogram_n ep_n exit_fw-la and_o er._n but_o the_o three_o parallelogram_n bm_n kb_o and_o bn_o be_v equal_a and_o like_a to_o the_o three_o opposite_a parallelogram_n and_o the_o three_o parallelogram_n ep_n exit_fw-la and_o ere_o be_v equal_a and_o like_a to_o the_o three_o opposite_a parallelogram_n wherefore_o the_o parallelipipedon_n bckl_n and_o efxo_n be_v comprehend_v under_o plain_a superficiece_n like_a and_o equal_a in_o multitude_n wherefore_o the_o solid_a bckl_n be_v like_a to_o the_o solid_a efxo_n but_o like_o parallelipipedon_n be_v by_o the_o 33._o of_o the_o eleven_o in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o side_n of_o like_a proportion_n be_v to_o side_n of_o like_a proportion_n wherefore_o the_o solid_a bckl_n be_v to_o the_o solid_a efxo_n in_o treble_a
line_n which_o subtend_v the_o angle_n zob_n to_o the_o three_o line_n which_o subtend_v the_o angle_n zkb_n but_o by_o construction_n boy_n be_v equal_a to_o bk_o therefore_o oz_n be_v equal_a to_o kz_o and_o the_o three_o alâ—o_o be_v equal_a to_o the_o three_o wherefore_o the_o point_n z_o in_o respect_n of_o the_o two_o triangle_n rectangle_v ozb_n and_o kzb_n determine_v one_o and_o the_o same_o magnitude_n iâ—_z the_o line_n bz_o which_o can_v not_o be_v if_o any_o other_o point_n in_o the_o line_n bz_o be_v assign_v near_o or_o far_o of_o from_o the_o point_n b._n one_o only_a point_n therefore_o be_v that_o at_o which_o the_o two_o perpendicular_n kz_o and_o oz_n fall_v but_o by_o construction_n oz_n fall_v at_o z_o the_o point_n and_o therefore_o at_o the_o same_o z_o do_v the_o perpendicular_a draw_v from_o king_n fall_v likewise_o which_o be_v require_v to_o be_v demonstrate_v although_o a_o brief_a monition_n may_v herein_o have_v serve_v for_o the_o pregnant_a or_o the_o humble_a learner_n yet_o for_o they_o that_o be_v well_o please_v to_o have_v thing_n make_v plain_a with_o many_o word_n and_o for_o the_o stiffnecked_a busy_a body_n it_o be_v necessary_a with_o my_o controlment_n of_o other_o to_o annex_v the_o cause_n &_o reason_n thereof_o both_o invincible_a and_o also_o evident_a a_o corollary_n 1._o hereby_o it_o be_v manifest_a that_o two_o equal_a circle_n cut_v one_o the_o other_o by_o the_o whole_a diameter_n if_o from_o one_o and_o the_o same_o end_n of_o their_o common_a diameter_n equal_a portion_n of_o their_o circumference_n be_v take_v and_o from_o the_o point_n end_v those_o equal_a portion_n two_o perpendicular_n be_v let_v down_o to_o their_o common_a diameter_n those_o perpendicular_n shall_v fall_v upon_o one_o and_o the_o same_o point_n of_o their_o common_a diameter_n 2._o second_o it_o follow_v that_o those_o perpendicular_n be_v equal_a ¶_o note_n from_o circle_n in_o our_o first_o supposition_n each_o to_o other_o perpendicular_o erect_v we_o proceed_v and_o infer_v now_o these_o corollary_n whether_o they_o be_v perpendicular_o erect_v or_o no_o by_o reasou_v the_o demonstration_n have_v a_o like_a force_n upon_o our_o supposition_n here_o use_v ¶_o the_o 16._o theorem_a the_o 18._o proposition_n sphere_n be_v in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o their_o diameter_n be_v svppose_v that_o there_o be_v two_o sphere_n abc_n and_o def_n and_o let_v their_o diameter_n be_v bc_o and_o ef._n then_o i_o say_v that_o the_o sphere_n abc_n be_v to_o the_o sphere_n def_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n ef._n proposition_n for_o if_o not_o than_o the_o sphere_n abc_n be_v in_o treble_a proportion_n of_o that_o in_o which_o bc_o be_v to_o of_o either_o to_o some_o sphere_n less_o than_o the_o sphere_n def_n or_o to_o some_o sphere_n great_a first_o let_v it_o be_v unto_o a_o less_o namely_o to_o ghk_v case_n and_o imagine_v that_o the_o sphere_n def_n and_o ghk_o be_v both_o about_o one_o and_o the_o self_n same_o centre_n impossibility_n and_o by_o the_o proposition_n next_o go_v before_o describe_v in_o the_o great_a sphere_n def_n a_o polihedron_n or_o a_o solid_a of_o many_o side_n not_o touch_v the_o superficies_n of_o the_o less_o sphere_n ghk_v and_o suppose_v also_o that_o in_o the_o sphere_n abc_n be_v inscribe_v a_o polihedron_n like_a to_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n wherefore_o by_o the_o corollary_n of_o the_o same_o the_o polihedron_n which_o be_v in_o the_o sphere_n abc_n be_v to_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n ef._n but_o by_o supposition_n the_o sphere_n abc_n be_v to_o the_o sphere_n ghk_v in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n ef._n wherefore_o as_o the_o sphere_n abc_n be_v to_o the_o sphere_n ghk_o so_o be_v the_o polihedron_n which_o be_v describe_v in_o the_o sphere_n abc_n to_o the_o polihedron_n which_o be_v describe_v in_o the_o sphere_n def_n by_o the_o 11._o of_o the_o five_o wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o sphere_n abc_n be_v to_o the_o polihedron_n which_o be_v describe_v in_o it_o so_o be_v the_o sphere_n ghk_v to_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n but_o the_o sphere_n abc_n be_v great_a than_o the_o polihedron_n which_o be_v describe_v in_o it_o wherefore_o also_o the_o sphere_n ghk_o be_v great_a than_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n by_o the_o 14._o of_o the_o five_o but_o it_o be_v also_o less_o for_o it_o be_v contain_v in_o it_o which_o impossible_a wherefore_o the_o sphere_n abc_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n of_o to_o any_o sphere_n less_o than_o the_o sphere_n def_n in_o like_a sort_n also_o may_v we_o prove_v that_o the_o sphere_n def_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o be_v to_o the_o diameter_n bc_o to_o any_o sphere_n less_o than_o the_o sphere_n abc_n case_n now_o i_o say_v that_o the_o sphere_n abc_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n of_o to_o any_o sphere_n great_a they_o the_o sphere_n def_n for_o if_o it_o be_v possible_a let_v it_o be_v to_o a_o great_a namely_o to_o lmn_o wherefore_o by_o conversion_n the_o sphere_n lmn_o be_v to_o the_o sphere_n abc_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o be_v to_o the_o diameter_n bc._n but_o as_o the_o sphere_n lmn_o be_v to_o the_o sphere_n abc_n so_o be_v the_o sphere_n def_n to_o some_o sphere_n less_o they_o the_o sphere_n abc_n booâ—â—_n as_o it_o have_v before_o be_v prove_v for_o the_o sphere_n lmn_o be_v great_a than_o the_o sphere_n def_n wherefore_o the_o sphere_n def_n be_v in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o be_v to_o the_o diameter_n bc_o to_o some_o sphere_n less_o they_o the_o sphere_n abc_n which_o be_v prove_v to_o be_v impossible_a wherefore_o the_o sphere_n abc_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o be_v be_v to_o of_o to_o any_o sphere_n great_a they_o the_o sphere_n def_n and_o it_o be_v also_o prove_v that_o it_o be_v not_o to_o any_o less_o wherefore_o the_o sphere_n abc_n be_v to_o the_o sphere_n def_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n of_o which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o flussas_n hereby_o it_o be_v manifest_a that_o sphere_n be_v the_o one_o to_o the_o other_o as_o like_o polihedrons_n and_o in_o like_a sort_n describe_v in_o they_o be_v namely_o each_o be_v in_o triple_a proportion_n of_o that_o in_o which_o the_o diameter_n a_o corollary_n add_v by_o mâ—_n dee_n it_o be_v then_o evident_a how_o to_o geve_v two_o right_a line_n have_v that_o proportion_n between_o they_o which_o any_o two_o sphere_n give_v have_v the_o one_o to_o the_o other_o for_o if_o to_o their_o diameter_n as_o to_o the_o first_o and_o second_o line_n of_o four_o in_o continual_a proportion_n you_o adjoin_v a_o three_o and_o a_o four_o line_n in_o continual_a proportion_n as_o i_o have_v teach_v before_o the_o first_o and_o four_o line_n shall_v answer_v the_o problem_n rule_n how_o general_a this_o rule_n be_v in_o any_o two_o like_a solid_n with_o their_o correspondent_a or_o omologall_n line_n i_o need_v not_o with_o more_o word_n declare_v ¶_o certain_a theorem_n and_o problem_n who_o use_n be_v manifold_a in_o sphere_n cones_n cylinder_n and_o other_o solid_n add_v by_o joh._n dee_n a_o theorem_a 1._o the_o whole_a superficies_n of_o any_o sphere_n be_v quadrupla_fw-la to_o the_o great_a circle_n in_o the_o same_o sphere_n contain_v it_o be_v needless_a to_o bring_v archimedes_n demonstration_n hereof_o into_o this_o place_n see_v his_o book_n of_o the_o sphere_n and_o cylinder_n with_o other_o his_o woâ—kes_n be_v every_o where_o to_o be_v have_v and_o the_o demonstration_n thereof_o easy_a a_o theorem_a 2._o every_o sphere_n be_v quadruplâ—_n to_o that_o cone_n who_o base_a be_v the_o great_a circle_n &_o height_n the_o semidiameter_n of_o the_o same_o sphere_n this_o be_v the_o 32._o proposition_n of_o archimedes_n fiâ—st_n book_n of_o the_o sphere_n and_o cylinder_n a_o problem_n 1._o a_o sphere_n be_v give_v to_o make_v a_o upright_a cone_n equal_a to_o the_o same_o or_o in_o any_o other_o proportion_n give_v between_o two_o right_a line_n and_o as_o concern_v the_o other_o part_n of_o
the_o square_a of_o ed_z to_o the_o square_n of_o dk_o but_o by_o construction_n ed_z be_v subquintuple_a to_o df._n wherefore_o the_o square_a of_o ed_z be_v subquintuple_a to_o the_o square_n of_o dk_o and_o therefore_o the_o square_a of_o dk_o be_v quintuple_a to_o the_o square_n of_o ed._n and_o ed_z be_v equal_a to_o ed_z by_o construction_n therefore_o the_o square_a of_o dk_o be_v quintuple_a to_o the_o square_n of_o e_o d._n wherefore_o the_o double_a of_o bd_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n who_o great_a segment_n be_v bkâ—_n â—_z by_o the_o second_o of_o this_o thirteâ—th_o but_o by_o construction_n ab_fw-la be_v the_o double_a of_o â—dâ—_n â—_z wherefore_o ab_fw-la be_v divide_v by_o extreme_a and_o mean_a proportion_n and_o his_o great_a segment_n be_v bk_o and_o thereby_o kâ—_n â—_z the_o point_n of_o the_o division_n we_o have_v therefore_o divide_v by_o extreme_a and_o mean_a proportion_n any_o right_a line_n give_v in_o length_n and_o position_n which_o be_v to_o be_v do_v noteâ—_n each_o of_o these_o way_n may_v well_o be_v execute_v but_o in_o the_o first_o you_o have_v this_o advantage_n that_o the_o diameter_n be_v take_v at_o pleasure_n which_o â—n_v the_o second_o way_n be_v ever_o just_a thrice_o so_o long_o as_o the_o line_n give_v to_o be_v divide_v john_n dee_n ¶_o the_o 4._o theorem_a the_o 4._o proposition_n if_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n the_o square_n make_v of_o the_o whole_a line_n and_o of_o the_o less_o segment_n be_v treble_a to_o the_o square_n make_v of_o the_o great_a segment_n svppose_v that_o the_o right_a line_n ab_fw-la be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n in_o the_o point_n c._n and_o let_v the_o great_a segment_n thereof_o be_v ac_fw-la then_o i_o say_v that_o the_o square_n make_v of_o the_o line_n ab_fw-la and_o bc_o be_v treble_a to_o the_o square_n of_o the_o line_n ac_fw-la describe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n ab_fw-la a_o square_a adeb_n and_o make_v perfect_a the_o figure_n now_o forasmuch_o as_o the_o line_n ab_fw-la demonstration_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c_o and_o the_o great_a segment_n thereof_o be_v the_o line_n ac_fw-la therefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o cb_o be_v the_o parallelogram_n ak_o and_o the_o square_a of_o the_o line_n ac_fw-la be_v the_o square_a fd._n wherefore_o the_o parallelogram_n ak_o be_v equal_a to_o the_o square_a fd._n and_o the_o parallelogram_n of_o be_v equal_a to_o the_o parallelogram_n fe_o put_v the_o square_a ck_n common_a to_o they_o both_o wherefore_o the_o whole_a parallelogram_n ak_o be_v equal_a to_o the_o whole_a parallelogram_n ce_fw-fr wherefore_o the_o parallelogram_n ce_fw-fr and_o ak_o be_v double_a to_o the_o parallelogram_n ak_o bâ—t_v the_o parallelogram_n ak_o and_o ce_fw-fr be_v the_o gnomon_n lmn_o and_o the_o square_a ck_n wherefore_o the_o gnomon_n lmn_o and_o the_o square_a ck_n be_v double_a to_o the_o parallelogram_n ak_o but_o it_o be_v prove_v that_o the_o parallelogram_n ak_o be_v equal_a to_o the_o square_a df._n wherefore_o the_o gnomon_n lmn_o and_o the_o square_a ck_n be_v double_a to_o the_o square_a df._n wherefore_o the_o gnomon_n lmn_o and_o the_o square_n ck_a and_o df_o be_v treble_a to_o the_o square_a df._n but_o the_o gnomon_n lmn_o and_o the_o square_n ck_a and_o df_o be_v the_o whole_a square_a ae_n together_o with_o the_o square_a ck_n which_o be_v the_o square_n of_o the_o line_n ab_fw-la and_o bc._n and_o df_o be_v the_o square_a of_o the_o line_n ac_fw-la wherefore_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v treble_a to_o the_o square_n of_o the_o line_n ac_fw-la if_o therefore_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n the_o square_n make_v of_o the_o whole_a line_n and_o of_o the_o less_o segment_n be_v treble_a to_o the_o square_n make_v of_o the_o great_a segment_n which_o be_v require_v to_o be_v prove_v look_v for_o a_o other_o demonstration_n of_o this_o proposition_n after_o the_o five_o proposition_n of_o this_o book_n ¶_o the_o 5._o theorem_a the_o 5._o proposition_n if_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o unto_o it_o be_v add_v a_o right_a â—ine_o proportion_n equal_a to_o the_o great_a segment_n the_o whole_a right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o the_o great_a segment_n thereof_o be_v the_o right_a line_n give_v at_o the_o beginning_n svppose_v that_o the_o right_a line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c_o and_o let_v the_o great_a segment_n thereof_o be_v ac_fw-la and_o unto_o the_o line_n ab_fw-la add_v the_o line_n ad_fw-la equal_a to_o the_o line_n ac_fw-la then_o i_o say_v that_o the_o line_n dâ—_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n a_o and_o the_o great_a segment_n thereof_o be_v the_o right_a line_n pâ—t_n at_o the_o beginning_n namely_o ab_fw-la describe_v by_o the_o 46._o of_o the_o first_o upon_o on_o the_o line_n ab_fw-la a_o square_a ae_n and_o make_v perfect_a the_o figure_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c_o therefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v the_o parallelogram_n ceâ—_n and_o the_o square_a of_o the_o â—â—ne_n ac_fw-la be_v the_o square_a ch._n wherefore_o the_o parallelogram_n ce_fw-fr be_v equal_a to_o the_o square_a ch._n but_o unto_o the_o square_a change_z be_v equal_a the_o square_a dh_o by_o the_o first_o of_o the_o six_o and_o unto_o the_o parallelogram_n ce_fw-fr be_v equal_a the_o parallelogram_n he._n wherefore_o the_o parallelogram_n dh_o be_v equal_a to_o the_o parallelogram_n he._n add_v the_o parallelogram_n hb_o common_a to_o they_o both_o wherefore_o the_o whole_a parallelogram_n dk_o be_v equal_a to_o the_o whole_a square_a ae_n and_o the_o parallelogram_n dk_o be_v that_o which_o be_v contain_v under_o the_o line_n bd_o and_o dam_fw-ge for_o the_o line_n ad_fw-la be_v equal_a to_o the_o line_n dl_o &_o the_o square_a ae_n be_v the_o square_a of_o the_o line_n ab_fw-la wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la wherefore_o as_o the_o line_n db_o be_v to_o the_o line_n basilius_n so_o be_v the_o line_n basilius_n to_o the_o line_n ad_fw-la by_o the_o 17._o of_o the_o six_o but_o the_o line_n db_o be_v great_a than_o the_o line_n basilius_n wherefore_o the_o line_n basilius_n be_v great_a than_o the_o line_n ad._n wherefore_o the_o line_n bd_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n a_o and_o his_o great_a segment_n be_v the_o line_n ab_fw-la if_o therefore_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o unto_o â—t_a be_v add_v a_o right_a line_n equal_a to_o the_o great_a segment_n the_o whole_a right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o the_o great_a segment_n thereof_o be_v the_o right_a line_n give_v at_o the_o beginning_n which_o be_v require_v to_o be_v demonstrate_v this_o proposition_n be_v again_o afterward_o demonstrate_v a_o corollary_n add_v by_o campane_n hereby_o it_o be_v 〈…〉_o from_o the_o greaâ—â—â—_n 〈◊〉_d of_o a_o line_n divide_v by_o a_o extreme_a &_o mean_a proportion_n proposition_n be_v 〈◊〉_d away_o 〈◊〉_d segment_n the_o say_v great_a a_o segment_n shall_v be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o the_o great_a segment_n thereof_o shall_v be_v the_o line_n take_v away_o as_o let_v the_o line_n â—â—_o be_v divide_v by_o a_o extreme_a and_o meanâ—_z proportion_n in_o the_o point_n c._n and_o leâ—_z the_o 〈…〉_o line_n 〈…〉_o a._n d._n i_o say_v that_o ac_fw-la be_v also_o divide_v by_o a_o extreme_a and_o meanâ—_z proportion_n in_o the_o point_n d_o and_o that_o his_o great_a portion_n be_v dc_o for_o by_o the_o definitino_fw-it of_o a_o line_n so_o divide_v ab_fw-la be_v to_o ac_fw-la as_o ac_fw-la be_v to_o cb._n but_o as_o ac_fw-la be_v to_o cb_o so_o be_v ac_fw-la to_o dc_o by_o the_o 7._o of_o the_o 〈…〉_o be_v equal_a to_o cb_o wherefore_o by_o the_o 11._o of_o the_o five_o as_z ab_fw-la be_v to_o ac_fw-la so_o be_v ac_fw-la to_o cd_o
the_o whole_a line_n mg_o to_o the_o whole_a line_n ea_fw-la by_o the_o 18._o of_o the_o five_o wherefore_o as_o mg_o the_o side_n of_o the_o cube_fw-la be_v to_o ea_fw-la the_o semidiameter_n so_o be_v the_o line_n fghim_n to_o the_o octohedron_n abkdlc_n inscribe_v in_o one_o &_o the_o self_n same_o sphere_n if_o therefore_o a_o cube_fw-la and_o a_o octohedron_n be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n they_o shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o semidiameter_n of_o the_o sphere_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n distinct_o to_o notify_v the_o power_n of_o the_o side_n of_o the_o five_o solid_n by_o the_o power_n of_o the_o diameter_n of_o the_o sphere_n the_o side_n of_o the_o tetrahedron_n and_o of_o the_o cube_fw-la do_v cut_v the_o power_n of_o the_o diameter_n of_o the_o sphere_n into_o two_o square_n which_o be_v in_o proportion_n double_a the_o one_o to_o the_o other_o the_o octohedron_n cut_v the_o power_n of_o the_o diameter_n into_o two_o equal_a square_n the_o icosahedron_n into_o two_o square_n who_o proportion_n be_v duple_n to_o the_o proportion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n who_o less_o segment_n be_v the_o side_n of_o the_o icosahedron_n and_o the_o dodecahedron_n into_o two_o square_n who_o proportion_n be_v quadruple_a to_o the_o proportion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n who_o less_o segment_n be_v the_o side_n of_o the_o dodecahedron_n for_o ad_fw-la the_o diameter_n of_o the_o sphere_n contain_v in_o power_n ab_fw-la the_o side_n of_o the_o tetrahedron_n and_o bd_o the_o side_n of_o the_o cube_fw-la which_o bd_o be_v in_o power_n half_a of_o the_o side_n ab_fw-la the_o diameter_n also_o of_o the_o sphere_n contain_v in_o power_n ac_fw-la and_o cd_o two_o equal_a side_n of_o the_o octohedron_n but_o the_o diameter_n contain_v in_o power_n the_o whole_a line_n ae_n and_o the_o great_a segment_n thereof_o ed_z which_o be_v the_o side_n of_o the_o icosahedron_n by_o the_o 15._o of_o this_o book_n wherefore_o their_o power_n be_v in_o duple_n proportion_n of_o that_o in_o which_o the_o side_n be_v by_o the_o first_o corollary_n of_o the_o 20._o of_o the_o six_o have_v their_o proportion_n duple_n to_o the_o proportion_n of_o a_o extreme_a &_o mean_a proportion_n far_o the_o diameter_n contain_v in_o power_n the_o whole_a line_n of_o and_o his_o less_o segment_n fd_o which_o be_v the_o side_n of_o the_o dodecahedron_n by_o the_o same_o 15._o of_o this_o book_n wherefore_o the_o whole_a have_v to_o the_o less_o â—_o double_a proportion_n of_o that_o which_o the_o extreme_a have_v to_o the_o mean_a namely_o of_o the_o whole_a to_o the_o great_a segment_n by_o the_o 10._o definition_n of_o the_o five_o it_o follow_v that_o the_o proportion_n of_o the_o power_n be_v double_a to_o the_o double_a proportion_n of_o the_o side_n by_o the_o same_o first_o corollary_n of_o the_o 20._o of_o the_o six_o that_o be_v be_v quadruple_a to_o the_o proportion_n of_o the_o extreme_a and_o of_o the_o mean_a by_o the_o definition_n of_o the_o six_o a_o advertisement_n add_v by_o flussas_n by_o this_o mean_n therefore_o the_o diameter_n of_o a_o sphere_n be_v give_v there_o shall_v be_v give_v the_o side_n of_o every_o one_o of_o the_o body_n inscribe_v and_o forasmuch_o as_o three_o of_o those_o body_n have_v their_o side_n commensurable_a in_o power_n only_o and_o not_o in_o length_n unto_o the_o diameter_n give_v for_o their_o power_n be_v in_o the_o proportion_n of_o a_o square_a number_n to_o a_o number_n not_o square_a wherefore_o they_o have_v not_o the_o proportion_n of_o a_o square_a number_n to_o a_o square_a number_n by_o the_o corollary_n of_o the_o 25._o of_o the_o eight_o wherefore_o also_o their_o side_n be_v incommensurabe_n in_o length_n by_o the_o 9_o of_o the_o ten_o therefore_o it_o be_v sufficient_a to_o compare_v the_o power_n and_o not_o the_o length_n of_o those_o side_n the_o one_o to_o the_o otherâ—_n which_o power_n be_v contain_v in_o the_o power_n of_o the_o diameter_n namely_o from_o the_o power_n of_o the_o diameter_n let_v there_o ble_v take_v away_o the_o power_n of_o the_o cube_fw-la and_o there_o shall_v remain_v the_o power_n of_o the_o tetrahedron_n and_o take_v away_o the_o power_n of_o the_o tetrahedron_n there_o remain_v the_o power_n of_o the_o cube_fw-la and_o take_v away_o from_o the_o power_n of_o the_o diameter_n half_o the_o power_n thereof_o there_o shall_v be_v leave_v the_o power_n of_o the_o side_n of_o the_o octohedron_n but_o forasmuch_o as_o the_o side_n of_o the_o dodecahedron_n and_o of_o the_o icosahedron_n be_v prove_v to_o be_v irrational_a for_o the_o side_n of_o the_o icosahedron_n be_v a_o less_o line_n by_o the_o 16._o of_o the_o thirteen_o and_o the_o side_n of_o the_o dedocahedron_n be_v a_o residual_a line_n by_o the_o 17._o of_o the_o same_o therefore_o those_o side_n be_v unto_o the_o diameter_n which_o be_v a_o rational_a line_n set_v incommensurable_a both_o in_o length_n and_o in_o power_n wherefore_o their_o comparison_n can_v not_o be_v define_v or_o describe_v by_o any_o proportion_n express_v by_o number_n by_o the_o 8._o of_o the_o ten_o neither_o can_v they_o be_v compare_v the_o one_o to_o the_o other_o for_o irrational_a line_n of_o diverse_a kind_n be_v incommensurable_a the_o one_o to_o the_o other_o for_o if_o they_o shall_v be_v commensurable_a they_o shall_v be_v of_o one_o and_o the_o self_n same_o kind_n by_o the_o 103._o and_o 105._o of_o the_o ten_o which_o be_v impossible_a wherefore_o we_o seek_v to_o compare_v they_o to_o the_o power_n of_o the_o diameter_n think_v they_o can_v not_o be_v more_o apt_o express_v then_o by_o such_o proportion_n which_o cut_v that_o rational_a power_n of_o the_o diameter_n according_a to_o their_o side_n namely_o divide_v the_o power_n of_o the_o diameter_n by_o line_n which_o have_v that_o proportion_n that_o the_o great_a segment_n have_v to_o the_o less_o to_o put_v the_o less_o segment_n to_o be_v the_o side_n of_o the_o icosahedron_n &_o devide_v the_o say_a power_n of_o the_o diameter_n by_o line_n have_v the_o proportion_n of_o the_o whole_a to_o the_o less_o segment_n to_o express_v the_o side_n of_o the_o dodecahedron_n by_o the_o less_o segment_n which_o thing_n may_v well_o be_v do_v between_o magnitude_n incommensurable_a the_o end_n of_o the_o fourteen_o book_n of_o euclides_n element_n after_o flussas_n ¶_o the_o fifteen_o book_n of_o euclides_n element_n this_o finetenth_n and_o last_o book_n of_o euclid_n or_o rather_o the_o second_o book_n of_o appollonius_n or_o hypsicles_n book_n teach_v the_o inscription_n and_o circumscription_n of_o the_o five_o regular_a body_n one_o within_o and_o about_o a_o other_o a_o thing_n undouted_o pleasant_a and_o delectable_a in_o mind_n to_o contemplate_v and_o also_o profitable_a and_o necessary_a in_o act_n to_o practice_v for_o without_o practice_n in_o act_n it_o be_v very_o hard_a to_o see_v and_o conceive_v the_o construction_n and_o demonstration_n of_o the_o proposition_n of_o this_o book_n unless_o a_o man_n have_v a_o very_a deep_a sharp_a &_o fine_a imagination_n wherefore_o i_o will_v wish_v the_o diligent_a student_n in_o this_o book_n to_o make_v the_o study_n thereof_o more_o pleasant_a unto_o he_o to_o have_v present_o before_o his_o eye_n the_o body_n form_v &_o frame_v of_o past_v paper_n as_o i_o teach_v after_o the_o definition_n of_o the_o eleven_o book_n and_o then_o to_o draw_v and_o describe_v the_o line_n and_o division_n and_o superficiece_n according_a to_o the_o construction_n of_o the_o proposition_n in_o which_o description_n if_o he_o be_v wary_a and_o diligent_a he_o shall_v find_v all_o thing_n in_o these_o solid_a matter_n as_o clear_v and_o as_o manifest_v unto_o the_o eye_n as_o be_v thing_n before_o teach_v only_o in_o plain_a or_o superficial_a figure_n and_o although_o i_o have_v before_o in_o the_o twelve_o book_n admonish_v the_o reader_n hereof_o yet_o because_o in_o this_o book_n chief_o that_o thing_n be_v require_v i_o think_v it_o shall_v not_o be_v irksome_a unto_o he_o again_o to_o be_v put_v in_o mind_n thereof_o far_o this_o be_v to_o be_v note_v that_o in_o the_o greek_a exemplar_n be_v find_v in_o this_o 15._o book_n only_o 5._o proposition_n which_o 5._o be_v also_o only_o touch_v and_o set_v forth_o by_o hypsicy_n unto_o which_o campane_n add_v 8._o and_o so_o make_v up_o the_o number_n of_o 13._o campane_n undoubted_o although_o he_o be_v very_o well_o learn_v and_o that_o general_o in_o all_o kind_n of_o learning_n yet_o assure_o be_v bring_v up_o in_o a_o time_n of_o rudeness_n when_o all_o good_a letter_n be_v darken_v &_o barberousnes_n have_v
side_n gd_v the_o angle_n m_o n_o under_o the_o side_n ab_fw-la the_o angle_n t_o s_o under_o the_o side_n bg_o the_o angle_n p_o oh_o and_o under_o the_o side_n agnostus_n the_o angle_n r_o q_o so_o there_o rest_v 4._o angle_n who_o true_a place_n we_o will_v now_o appoint_v forasmuch_o as_o a_o cube_fw-la contain_v in_o one_o and_o the_o self_n same_o sphere_n with_o a_o dodecahedron_n be_v inscribe_v in_o the_o same_o dodecahedron_n as_o it_o be_v manifest_a by_o the_o 17._o of_o the_o thirteen_o and_o 8._o of_o this_o book_n it_o follow_v that_o a_o cube_fw-la and_o a_o dodecahedron_n circumscribe_v about_o it_o be_v contain_v in_o one_o and_o the_o self_n same_o body_n for_o that_o their_o angle_n concur_v in_o one_o and_o the_o self_n same_o point_n and_o it_o be_v prove_v in_o the_o 18._o of_o this_o book_n that_o 4._o angle_n of_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n be_v set_v in_o the_o middle_a section_n of_o the_o perpendicularâ—_n which_o be_v draw_v from_o the_o solid_a angle_n of_o the_o pyramid_n to_o the_o opposite_a base_n wherefore_o the_o other_o 4._o angle_n of_o the_o dodecahedron_n be_v also_o as_o the_o angle_n of_o the_o cube_fw-la set_v in_o those_o middle_a section_n of_o the_o perpendicular_n namely_o the_o angle_n v_o be_v set_v in_o the_o midst_n of_o the_o perpendicular_a ahâ—_n the_o angle_n y_fw-fr in_o the_o midst_n of_o the_o perpendicular_a bf_o the_o angle_n x_o in_o the_o midst_n of_o the_o perpendicular_a ge_z and_o last_o the_o angle_n d_o in_o the_o midst_n of_o the_o perpendicular_a d_o which_o be_v draw_v from_o the_o top_n of_o the_o pyramid_n to_o the_o opposite_a base_a wherefore_o those_o 4._o angle_n of_o the_o dodecahedron_n may_v be_v say_v to_o be_v direct_o under_o the_o solid_a angle_n of_o the_o pyramid_n or_o they_o may_v be_v say_v to_o be_v set_v at_o the_o perpendicular_n wherefore_o the_o dodecahedron_n after_o this_o manner_n set_v be_v inscribe_v in_o the_o pyramid_n give_v by_o the_o first_o definition_n of_o this_o book_n for_o that_o upon_o every_o one_o of_o the_o base_n of_o the_o pyramid_n be_v set_v a_o angle_n of_o the_o dodecahedron_n inscribe_v wherefore_o in_o a_o trilater_n equilater_n pyramid_n be_v inscribe_v a_o dodecahedron_n the_o 21._o problem_n the_o 21._o proposition_n in_o every_o one_o of_o the_o regular_a solid_n to_o inscribe_v a_o sphere_n in_o the_o 13._o of_o thâ—_z thirteen_o and_o thâ—_z other_o 4._o proposition_n follow_v he_o iâ—_z be_v declare_v that_o â—he_n â—â—_o regular_a solidesâ—â—re_o so_o contain_v in_o a_o sphere_n that_o â—ight_a linâ—â—_n draw_v from_o the_o cenâ—â—â—_n oâ—_n the_o 〈…〉_o of_o 〈◊〉_d solid_a inscribe_v be_v equal_a which_o right_a line_n therefore_o make_v pyramid_n who_o â—oppes_n be_v the_o centre_n of_o the_o sphere_n or_o of_o the_o solid_a and_o the_o basâ—â—â—â—e_a cuâ—â—â—_n one_o of_o the_o base_n of_o those_o solid_n and_o 〈…〉_o solid_a squall_n and_o like_v the_o one_o to_o the_o other_o and_o describe_v in_o equal_a circle_n those_o circle_n shall_v cut_v the_o sphere_n for_o the_o angle_n which_o touch_v the_o circumference_n of_o the_o circle_n touch_v also_o the_o superficies_n of_o the_o sphere_n wherefore_o perpendiculars_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o base_n or_o to_o the_o plain_a superficiece_n of_o the_o equal_a circle_n be_v equal_a by_o the_o corollary_n of_o the_o assumpt_n of_o the_o 1â—_o of_o the_o twelve_o wherefore_o make_v the_o centre_n the_o 〈◊〉_d of_o the_o sphere_n which_o 〈◊〉_d the_o solid_a and_o thâ—_z space_n some_o one_o of_o the_o equal_a perpendicularâ—_n dâ—scribâ—_n a_o sphere_n and_o it_o shall_v touch_v every_o one_o of_o the_o base_n of_o 〈◊〉_d solid_a 〈…〉_o perficies_fw-la of_o the_o sphere_n pass_v beyond_o those_o base_n when_o as_o those_o pâ—â—peâ—diculars_n 〈…〉_o be_v draw_v from_o the_o centre_n to_o the_o base_n by_o the_o 3._o corollary_n of_o the_o saâ—â—â—â—â—umpt_n wherâ—fore_o â—e_v have_v iâ—_z every_o one_o of_o the_o regular_a body_n inscribe_v a_o sphere_n which_o regular_a boâ—â—â—_n be_v in_o number_n one_o iâ—_z 〈◊〉_d by_o the_o corollary_n of_o the_o 1â—_o of_o the_o 〈◊〉_d a_o corollary_n the_o regular_a figure_n inscribe_v in_o sphere_n and_o also_o the_o sphere_n circumscribe_v about_o they_o or_o contain_v they_o have_v one_o and_o the_o self_n same_o centre_n namely_o their_o pyramid_n the_o â—ngles_n of_o who_o base_n touch_v the_o superâ—â—â—â—â—â—_n of_o thâ—â—â—here_o do_v from_o those_o angle_n cause_v equal_a right_a line_n to_o be_v drawâ—â—_n to_o one_o and_o â—he_n self_n 〈◊〉_d poynâ—_n make_v the_o topâ—â—â—_n of_o the_o pyramid_n in_o the_o same_o point_n and_o therefore_o they_o 〈…〉_o thâ—_z câ—â—tres_n of_o the_o sphere_n in_o the_o self_n same_o top_n when_o 〈◊〉_d the_o right_a line_n draw_v from_o those_o angle_n to_o the_o croâ—â—ed_a superficies_n wherein_o be_v 〈◊〉_d the_o angle_n of_o the_o base_n of_o the_o pyramid_n be_v equal_o a_o advertisement_n of_o flussasâ—_n â—_z of_o these_o solid_n only_o the_o octohedron_n receive_v the_o other_o solid_n inscribe_v one_o with_o 〈…〉_o other_o for_o the_o octohedron_n contain_v the_o icosahedron_n inscribe_v in_o it_o and_o the_o same_o icosahedron_n contain_v the_o dodecahedron_n inscribe_v in_o the_o same_o icosahedron_n and_o the_o same_o dodecahedron_n contain_v the_o cube_fw-la inscribe_v in_o the_o same_o octohedron_n and_o 〈…〉_o â—â—râ—â—mscribeth_v the_o pyramid_n inscribe_v in_o the_o say_v octohedron_n but_o this_o happen_v not_o in_o the_o other_o solid_n the_o end_n of_o the_o fivetenth_fw-mi book_n of_o euclides_n elemenâ—â—â—_n after_o caâ—paâ—â—_n and_o 〈◊〉_d ¶_o the_o sixteen_o book_n of_o the_o element_n of_o geometry_n add_v by_o flussas_n in_o the_o former_a fivetenth_fw-mi book_n have_v be_v teach_v how_o to_o inscribe_v the_o five_o regular_a solid_n one_o with_o in_o a_o other_o now_o seem_v to_o rest_n to_o compare_v those_o solid_a so_o inscribe_v one_o to_o a_o other_o and_o to_o set_v forth_o their_o passion_n and_o propriety_n which_o thing_n flussas_n consider_v in_o this_o sixteen_o book_n add_v by_o he_o book_n have_v excellent_o well_o and_o most_o cunning_o perform_v for_o which_o undoubted_o he_o have_v of_o all_o they_o which_o have_v a_o love_n to_o the_o mathematicals_n deserve_v much_o praise_n and_o commendation_n both_o for_o the_o great_a traâ—ailes_n and_o paynâ—s_n which_o it_o be_v most_o likely_a he_o have_v taâ—â—n_v in_o invent_v such_o strange_a and_o wonderful_a proposition_n with_o their_o demonstration_n in_o this_o book_n contain_v as_o also_o for_o participate_v and_o communicate_v abroad_o the_o same_o to_o other_o which_o book_n also_o that_o the_o reader_n shall_v want_v nothing_o conduce_v to_o the_o perfection_n of_o euclides_n element_n i_o have_v with_o some_o travail_n translate_v &_o for_o the_o worthiness_n â—hereof_o have_v add_v it_o aâ—_z a_o sixteen_o book_n to_o the_o 15._o book_n of_o euclid_n vouchsafe_v therefore_o gentle_a reader_n diligent_o to_o read_v and_o poise_v it_o for_o in_o it_o shall_v you_o find_v noâ—_n only_a matter_n strange_a and_o delectable_a but_o also_o occasion_n of_o invention_n of_o great_a thing_n pertain_v to_o the_o nature_n of_o the_o five_o regular_a solidâ—sâ—_n ¶_o the_o 1._o proposition_n a_o dodecahedron_n and_o a_o cube_fw-la inscribe_v in_o it_o and_o a_o pyramid_n inscribe_v in_o the_o same_o cube_fw-la be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n for_o the_o angle_n of_o the_o pyramiâ—_n be_v seâ—_z in_o the_o angel_n of_o the_o cube_fw-la wherein_o it_o be_v inscribe_v by_o the_o first_o of_o the_o fivetenthâ—_n and_o all_o the_o angle_n of_o the_o cube_fw-la be_v set_v in_o the_o angle_n of_o the_o dodecahedâ—â—â—_n circumscribe_v 〈…〉_o 〈◊〉_d the_o 8._o of_o the_o fifteen_o and_o all_o the_o angle_n of_o the_o dodecahedron_n be_v set_v in_o the_o superficies_n of_o the_o sphere_n by_o the_o 17._o of_o the_o thirteen_o wherefore_o those_o three_o solid_n inscribe_v one_o within_o a_o other_o be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o first_o definition_n of_o the_o fifteen_o a_o dodecahedron_n therefore_o and_o a_o cube_fw-la inscribe_v in_o it_o and_o a_o pyramid_n inscribe_v in_o the_o same_o cube_fw-la be_v contain_v 〈…〉_o â—â—lfe_n same_o sphere_n 〈…〉_o these_o three_o solid_n livre_n 〈…〉_o elf_n same_o icosahedron_n or_o octohedron_n or_o pyramid_n 〈…〉_o i_o icosahedron_n by_o the_o 5.11_o &_o 12._o of_o the_o fifteen_o and_o they_o be_v 〈…〉_o ctohedron_n by_o the_o 4._o 6._o and_o 16._o of_o the_o same_o last_o they_o be_v inscribe_v in_o 〈…〉_o the_o first_o 18._o and_o 19_o of_o the_o same_o for_o the_o angle_n of_o all_o these_o solid_a 〈…〉_o the_o circumscribe_v icosahedron_n or_o octohedron_n or_o pyramid_n ¶_o the_o 〈…〉_o the_o proportion_n of_o a_o dodecahedron_n circumscribe_v about_o a_o cube_fw-la to_o a_o dodecahedron_n inscribe_v in_o the_o same_o cube_fw-la be_v
a_o right_a line_n couple_v their_o centre_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n the_o right_a line_n which_o couple_v the_o centre_n of_o the_o next_o base_n if_o by_o the_o centre_n of_o five_o base_n set_v upon_o one_o base_a be_v draw_v a_o plain_a superficies_n and_o by_o the_o centre_n of_o the_o base_n which_o be_v set_v upon_o the_o opposite_a base_a be_v draw_v also_o a_o plain_a superficies_n and_o then_o be_v draw_v a_o right_a line_n couple_v the_o centre_n of_o the_o opposite_a base_n that_o right_a line_n be_v so_o cut_v that_o each_o of_o his_o part_n set_v without_o the_o plain_a superficies_n be_v the_o great_a segment_n of_o that_o part_n which_o be_v contain_v between_o the_o plain_n the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o the_o dodecahedron_n to_o one_o of_o the_o base_n be_v in_o power_n quintuple_a to_o half_a the_o line_n which_o be_v between_o the_o plain_n and_o therefore_o the_o whole_a line_n which_o couple_v the_o centre_n of_o the_o opposite_a base_n be_v in_o power_n quintuple_a to_o the_o whole_a line_n which_o be_v between_o the_o say_a plain_n the_o line_n which_o subtâ—deth_v the_o angle_n of_o the_o base_a of_o the_o dodecahedron_n together_o with_o the_o side_n of_o the_o base_a be_v in_o power_n quintuple_a to_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n which_o contain_v the_o base_a to_o the_o circumference_n a_o section_n of_o a_o sphere_n contain_v three_o base_n of_o the_o dodecahedron_n take_v a_o three_o part_n of_o the_o diameter_n of_o the_o say_a sphere_n the_o side_n of_o the_o dodecahedron_n and_o the_o line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n be_v equal_a to_o the_o right_a line_n which_o couple_v the_o middle_a section_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n ¶_o the_o end_n of_o the_o element_n of_o geometry_n of_o the_o most_o ancient_a philosopher_n 〈◊〉_d of_o megara_n the_o intent_n of_o this_o preface_n number_n note_v the_o word_n unit_fw-la to_o express_v the_o greek_a monaâ—_n &_o not_o unity_n as_o we_o hauâ—_n all_o common_o till_o now_o use_v magnitude_n a_o point_n a_o line_n magnitude_n ano._n 1488._o ☞_o arithmetic_n note_n note_n anno._n 1550._o r._n b._n note_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d ☞_o this_o noble_a earl_n die_v anno._n 1554._o scarce_o of_o 24._o year_n of_o ageâ—_n have_v no_o issue_n by_o his_o wife_n daughter_n to_o the_o duke_n of_o somerset_n justice._n ☞_o ☞_o plato_n 7._o de_fw-fr rep._n ☞_o ☞_o note_n the_o difference_n between_o strataruhmetrie_n and_o tacticie_n i.d._n fâ—endâ—â—_n you_o will_v find_v it_o hard_o to_o perform_v my_o description_n of_o â—his_n fâ—ate_n but_o by_o châ—râ—graphieâ—_n you_o may_v help_v yourself_o some_o â—hat_n wherâ—_n thâ—_z figure_v know_v in_o sidâ—â—â—nd_a angle_n be_v not_o regular_a and_o â—_z resolution_n into_o triangle_n can_v sâ—â—uâ—_n etc_n etc_n and_o yet_o you_o will_v find_v it_o strange_a to_o deal_v thus_o general_o with_o arithmetical_a figure_n and_o that_o for_o battle_n â—ay_n their_o coâ—tentâ—â—â—_n differ_v so_o much_o from_o like_o geometrâ—call_a â—_z a_o marvelous_a glass_n ☞_o s.w.p._n ☞_o note_n 1._o 2._o 3._o 5._o 6._o 7._o 8._o i.d._n read_v in_o aristotle_n his_o 8._o book_n of_o politic_n the_o 5_o 6_o and_o 7._o chapter_n where_o you_o shall_v have_v some_o occasion_n far_o to_o think_v of_o music_n than_o common_o be_v think_v ☜_o ☜_o anno._n 1548_o and_o 1549._o in_o lovayn_n note_n i.d._n the_o cut_v of_o a_o sphere_n according_a to_o any_o proportion_n assign_v may_v by_o this_o proposition_n be_v do_v mechanical_o by_o temper_v liquour_n to_o a_o certain_a weight_n in_o respect_n of_o the_o weight_n of_o the_o sphere_n 〈◊〉_d swyâ—â—â—ng_n a_o common_a error_n note_v a_o paradox_n n._n t._n the_o wonderful_a use_n of_o these_o proposition_n the_o practice_n statical_a to_o know_v the_o proportion_n between_o the_o cube_n and_o the_o sphere_n i_o d._n d._n for_o so_o have_v you_o 256._o part_n of_o a_o grain_n grain_n the_o proportion_n of_o the_o square_n to_o the_o circle_n inscribe_v inscribe_v the_o square_n of_o the_o circle_v mechanical_o mechanical_o to_o any_o squirâ—_n gâ—uenâ—_n to_o 〈…〉_o note_v square_n of_o the_o circle_n without_o knowledge_n of_o the_o proportion_n between_o circumference_n and_o diameter_n to_o double_a the_o cube_n ready_o by_o art_n mechanical_a depend_v upon_o demonstration_n mathematical_a i_o d._n the_o 4._o side_n of_o this_o pyramiâ—_n must_v be_v 4._o isosceles_a triangle_n â—_o like_o and_o squall_v i_o d._n d._n in_o all_o workinger_n with_o this_o pyramid_n or_o cone_n let_v their_o situation_n be_v in_o all_o pointeâ—_n and_o condition_n a_o like_a oâ—_n all_o one_o while_o you_o be_v about_o â—ne_a work_n else_o you_o will_v 〈◊〉_d i_o d._n d._n consider_v well_o when_o you_o must_v put_v your_o wateâ—â—_n together_o and_o when_o you_o must_v empty_a youâ—_n first_o waterâ—_n out_o of_o your_o pyramiâ—_n or_o cone_n elâ—_n you_o will_v 〈◊〉_d 〈◊〉_d vitrwius_fw-la lib._n 9_o cap._n 3._o ☞_o god_n bâ—_z thank_v â—or_a this_o invention_n &_o the_o fruit_n ensue_v ensue_v note_n note_v as_o concern_v the_o spherical_a superficies_n of_o the_o water_n ☞_o ☞_o note_n note_v this_o abridgement_n of_o double_v 〈◊〉_d cube_n â—â—_o â—â—_o note_n *_o ☜_o to_o give_v cube_n one_o to_o the_o other_o in_o any_o proportion_n rational_a or_o irrational_a irrational_a empty_v the_o first_o the_o demonstration_n of_o this_o double_v of_o the_o cube_n and_o oâ—_n the_o rest_n i.d._n i.d._n here_o 〈…〉_o of_o the_o water_n water_n by_o the_o 33._o of_o the_o eleven_o book_n of_o euclid_n i.d._n i.d._n and_o your_o diligence_n in_o practice_n can_v â—o_o in_o weight_n of_o wateâ—â—_n perform_v it_o therefore_o now_o you_o arâ—_n 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de_fw-fr his_fw-la quae_fw-la mundo_fw-la mirabiliter_fw-la eveniunt_fw-la cap._n 8._o tusc._n â—_o â—_o ☞_o a_o digression_n apologetical_a apologetical_a a_o proverb_n fair_a fish_v and_o catch_v a_o frog_n ☞_o psal._n 140._o act._n 7._o c._n lib._n 30._o cap._n 1._o ☜_o r._n b._n ☞_o university_n ☜_o ☞_o the_o ground_n plat_z of_o this_o preface_n in_o a_o table_n the_o argumâ—â—â—_n of_o the_o first_o book_n definition_n of_o a_o point_n definition_n of_o a_o point_n after_o pythagoras_n definition_n of_o a_o liâ—â—_n an_o other_o definition_n of_o a_o line_n a_o other_o the_o end_n of_o a_o line_n difference_n of_o a_o point_n frâ—â—nity_n unity_n be_v a_o part_n of_o number_n a_o point_n be_v no_o part_n of_o quantity_n definition_n of_o a_o right_a line_n definition_n thereof_o after_o plato_n an_o other_o definition_n a_o other_o a_o other_o a_o other_o a_o other_o why_o euclid_n here_o define_v not_o a_o crooked_a line_n definition_n of_o a_o superficies_n a_o superficies_n 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definition_n of_o rectilined_a figure_n
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construction_n demonstration_n application_n of_o space_n with_o excess_n or_o want_v a_o ancient_a invention_n of_o pythagoras_n how_o a_o figure_n be_v say_v to_o be_v apply_v to_o a_o line_n three_o thing_n give_v in_o this_o proposition_n the_o converse_n of_o this_o proposition_n construction_n demonstration_n a_o addition_n of_o pelitarius_n to_o describe_v a_o square_n mechanical_o a_o addition_n of_o procâ—â—â—_n the_o converse_n thereof_o construction_n demonstration_n pythagoras_n the_o first_o inventor_n of_o this_o proposition_n a_o addition_n of_o pâ—lâ—tariâ—â—_n an_o other_o addition_n of_o pelitarius_n an_o other_o addition_n of_o pelitarius_n an_o other_o addition_n of_o pelitarius_n a_o corollary_n this_o proposition_n be_v the_o converse_n of_o the_o former_a the_o argument_n of_o the_o second_o book_n what_o be_v the_o power_n of_o a_o line_n many_o compendious_a rule_n of_o reckon_v gather_v one_o of_o this_o book_n and_o also_o many_o rule_n of_o algebra_n two_o 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the_o touch_n of_o circle_n be_v 〈◊〉_d in_o one_o poâ—â—â—_n only_o circle_n may_v touch_v together_o two_o maâ—â—â—_n of_o way_n four_o definition_n five_o definition_n six_o definition_n mix_v angle_n arke_n chord_n seven_o definition_n difference_n of_o a_o angle_n of_o a_o section_n and_o of_o a_o angle_n in_o a_o section_n eight_o definition_n nine_o definition_n ten_o definition_n two_o definition_n first_o second_o why_o euclid_n define_v not_o equal_a section_n constuction_n demonstration_n lead_v to_o a_o impossibility_n correlary_a demonstration_n lead_v to_o a_o impossibility_n the_o first_o para_fw-it of_o this_o proposition_n construction_n demonstration_n the_o second_o part_n converse_v of_o the_o first_o demonstration_n demonstration_n lead_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n construction_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibility_n two_o caseâ—_n in_o this_o proposition_n construction_n 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to_o a_o impossibility_n an_o other_o demonstration_n after_o pelitarius_n lead_v also_o to_o a_o absurdity_n of_o circle_n which_o touch_v the_o one_o the_o other_o inward_o of_o circle_n which_o touch_v the_o one_o the_o other_o outward_o an_o other_o demonstration_n after_o pelitarius_n &_o flussates_n of_o circle_n which_o touch_n the_o one_o the_o other_o outward_o of_o circle_n which_o touch_n the_o one_o the_o other_o inward_o the_o first_o part_n of_o this_o theorem_a construction_n demonstration_n demonstration_n the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o an_o other_o demonstration_n of_o the_o first_o part_n after_o campane_n construction_n demonstration_n an_o other_o demonstration_n after_o campane_n the_o first_o part_n of_o this_o theorem_a demonstration_n lead_v to_o a_o absurdity_n second_o part_n three_o part_n construction_n demonstration_n a_o addition_n of_o pelitarius_n this_o problem_n commodious_a for_o the_o inscribe_v and_o 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to_o be_v prâ—â—â—d_v beâ—oâ—e_n 〈◊〉_d â—all_n to_o the_o demonstration_n construction_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n this_o problem_n reduce_v to_o a_o theorem_a construction_n demonstration_n how_o magnitude_n be_v say_v to_o be_v in_o proportion_n the_o onâ—_n to_o the_o other_o as_o number_n be_v to_o number_v this_o proposition_n be_v the_o converse_n of_o forme_a construction_n demonstration_n a_o corollary_n construction_n demonstration_n construction_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n this_o be_v the_o 〈…〉_o demonstration_n the_o first_o part_n demonstrate_v an_o other_o demonstration_n of_o the_o first_o part_n an_o other_o demonstration_n oâ—_n the_o same_o first_o part_n after_o montaureus_n demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o former_a an_o other_o demonstration_n of_o the_o second_o part_n this_o assumpt_n follow_v as_o a_o corollary_n of_o the_o 25_o but_o so_o as_o it_o may_v also_o be_v here_o in_o method_n place_v you_o shall_v â—inde_v it_o after_o the_o 53._o of_o this_o book_n absolute_o demonstrate_v for_o there_o it_o serve_v to_o the_o 54._o his_o demonstration_n demonstration_n of_o the_o three_o part_n demonstration_n of_o the_o four_o part_n which_o be_v the_o converse_n of_o the_o â—_o conclusion_n of_o the_o whole_a proposition_n a_o corollary_n proâ—e_v of_o the_o first_o part_n of_o the_o corollary_n proof_n of_o the_o second_o part_n proof_n of_o the_o three_o pâ—rt_n proâ—e_a oâ—_n the_o four_o part_n certain_a annotation_n â—ut_z of_o montauâ—â—us_fw-la rule_v to_o know_v whether_o two_o superficial_a number_n be_v like_a or_o no._n this_o assumpt_n be_v the_o converse_n of_o the_o 26._o of_o the_o eight_o demonstration_n oâ—_n the_o first_o part_n demonstration_n of_o the_o second_o partâ—_n a_o corollary_n to_o find_v out_o the_o first_o line_n incommensurable_a in_o length_n only_o to_o the_o line_n give_v to_o find_v out_o the_o second_o line_n incommensurable_a both_o in_o length_n and_o in_o power_n to_o the_o line_n give_v construction_n demonstration_n tâ—is_n be_v wiâ—h_n zambert_n a_o aâ—â—â—mpt_n but_o uâ—â—eâ—ly_o improper_o â—lâ—ssateâ—_n mane_v iâ—_z a_o corollary_n but_o the_o greeâ—e_n and_o montaureus_n maâ—e_n it_o a_o proposition_n but_o every_o way_n a_o â—nfallible_a truth_n 〈…〉_o demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n a_o corollary_n demonstration_n an_o other_o way_n to_o prove_v that_o the_o line_n a_o e_o c_o f_o be_v proportional_a demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o paâ—t_n which_o be_v the_o converse_n of_o the_o first_o a_o corollary_n demonstration_n of_o the_o first_o part_n by_o a_o argument_n leadindg_fw-mi to_o a_o absurdity_n demonstration_n of_o the_o second_o paâ—t_n lead_v also_o to_o a_o impossibility_n and_o this_o second_o part_n be_v the_o converse_n of_o the_o first_o demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o how_o to_o divide_v the_o line_n bc_o ready_o in_o such_o sort_n as_o iâ—_z require_v in_o the_o proposition_n demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o tâ—e_z former_a a_o other_o demonstrationâ—y_v a_o argument_n lead_v to_o a_o absurdity_n a_o assumpt_n a_o corollary_n add_v by_o montaureuâ—_n cause_n cause_o of_o increase_v the_o difficulty_n of_o this_o book_n note_n construction_n demonstration_n diverse_a caâ—es_n in_o this_o proposition_n the_o second_o case_n the_o first_o kind_n of_o rational_a line_n commensurable_a in_o length_n this_o particle_n in_o the_o proposition_n according_a to_o any_o of_o the_o foresay_a way_n be_v not_o in_o vain_a put_v the_o second_o kind_n of_o rational_a line_n commensurable_a in_o length_n the_o three_o case_n the_o three_o kind_n of_o rational_a line_n commensurable_a in_o length_n the_o four_o case_n this_o proposition_n be_v the_o converse_n of_o the_o former_a proposition_n construction_n demonstration_n a_o assumpt_n construction_n demonstration_n definition_n of_o a_o medial_a line_n a_o corollary_n this_o assumpt_n be_v nothing_o else_o but_o a_o part_n of_o the_o first_o proposition_n of_o the_o six_o book_n 〈◊〉_d how_o a_o square_n be_v say_v to_o be_v apply_v upon_o a_o line_n construction_n demonstration_n construction_n demonstration_n note_n a_o corollary_n construction_n demonstration_n lead_v to_o a_o absurdity_n construction_n demonstration_n construction_n demonstration_n demonstration_n a_o corollary_n to_o find_v out_o two_o square_a number_n exceed_v the_o one_o the_o other_o by_o a_o square_a â—umber_n a_o assumpt_n construction_n demonstration_n montaureus_n make_v this_o a_o assumpt_n as_o the_o grecke_n text_n seem_v to_o do_v likewise_o but_o without_o a_o cause_n construction_n demonstration_n this_o assumpt_n set_v foâ—th_o nothing_o â—_z but_o that_o which_o the_o first_o oâ—_n the_o sâ—â—t_n jet_v â—orth_o and_o therefore_o in_o sâ—me_a examplar_n it_o be_v not_o find_v construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o corollary_n i_o dee_n dee_n the_o second_o corollary_n corollary_n therefore_o if_o you_o divide_v the_o square_n of_o the_o side_n ac_fw-la by_o the_o side_n bc_o the_o portion_n dc_o will_v be_v the_o product_n etc_n etc_n as_o in_o the_o former_a corollary_n i_o dâ—e_n dâ—e_n the_o third_o corollary_n corollary_n therefore_o if_o the_o parallelogram_n of_o basilius_n and_o ac_fw-la be_v divide_v by_o bc_o the_o product_n will_v geve_v the_o perpendicular_a d_o a._n these_o three_o corollarye_n in_o practice_n logistical_a and_o geometrical_a be_v profitable_a an_o other_o demonstration_n of_o this_o four_o part_n of_o the_o determination_n a_o assumpt_n construction_n demonstration_n the_o first_o part_n of_o the_o determination_n conclude_v the_o second_o part_n conclude_v the_o total_a conclusion_n construction_n demonstration_n the_o first_o part_n of_o the_o determination_n conclude_v the_o second_o part_n conclude_v the_o total_a conclusion_n construction_n demonstration_n the_o first_o part_n conclude_v the_o second_o part_n conclude_v the_o three_o part_n conclude_v the_o total_a conclusion_n the_o first_o senary_a by_o composition_n definition_n of_o a_o binomial_a line_n six_o kind_n of_o binomial_a line_n demonstration_n definition_n of_o a_o first_o bimedial_a line_n construction_n demonstration_n definition_n of_o a_o second_o bimedial_a line_n demonstration_n definition_n of_o a_o great_a line_n definition_n of_o a_o line_n who_o power_n be_v rational_a and_o medial_a definition_n of_o a_o liâ—e_z contain_v in_o power_n two_o medial_n a_o assumpt_n the_o second_o senary_a by_o composition_n demonstration_n lead_v to_o a_o impossibility_n a_o corollary_n demonstration_n lead_v to_o a_o impossibilâ—â—e_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibiâ—â—â—e_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n lead_v to_o a_o absurdity_n six_o kind_n of_o binomial_a line_n a_o binomial_a line_n consi_v of_o two_o paâ—tâ—s_n firsâ—_o definition_n seconâ—_o definition_n three_o â—â—â—â—â—â—ition_n four_o definition_n five_o definition_n six_o definition_n the_o three_o senary_a by_o composition_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o corollary_n add_v by_o flussates_n m._n do_v you_o his_o book_n call_v tyâ—â—câ—niâ—m_fw-la mathematicum_fw-la this_o assumpt_n as_o be_v before_o note_v follow_v most_o â—riâ—fly_o without_o far_a demonstration_n of_o the_o 25._o of_o this_o book_n demonstration_n a_o assumpt_n the_o four_o senary_a by_o composition_n construction_n demonstration_n the_o first_o part_n of_o this_o demonstration_n conclude_v the_o second_o part_n of_o the_o demonstration_n conclude_v the_o three_o part_n conclude_v the_o total_a conclusion_n demonstration_n the_o first_o part_n of_o this_o demonstration_n conclude_v the_o three_o part_n conclude_v the_o four_o part_n concludeâ—_n the_o five_o part_n conclude_v the_o total_a conclusion_n demonstration_n construction_n demonstration_n demonstration_n demonstration_n demonstration_n look_v after_o the_o assumpt_n conclude_v at_o this_o mark_n for_o plain_a open_n of_o this_o place_n the_o use_n of_o this_o assumpt_n be_v in_o the_o next_o proposition_n &_o other_o follow_v the_o five_o senary_a by_o composition_n construction_n demonstration_n conclude_v that_o dg_o be_v a_o binomial_a line_n construction_n demonstration_n conclude_v that_o dg_o be_v a_o binomial_a line_n construction_n demonstration_n †_o *_o ‡_o dg_o conclude_v a_o binomial_a line_n a_o corollary_n add_v by_o m._n dee_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n the_o six_o senary_n construction_n demonstration_n construction_n demonstration_n a_o corollary_n add_v by_o flussetes_n note_n construction_n demonstration_n an_o other_o demonstration_n after_o p._n montaureus_n an_o other_o demonstration_n after_o campane_n construction_n demonstration_n an_o other_o demonstration_n afâ—â—r_v campane_n construction_n demonstration_n a_o assumpt_n an_o other_o demonstration_n after_o campanâ—_n note_n
casâ—_n demonstration_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n note_v this_o manner_n of_o imagination_n mathematical_a demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n in_o tâ—is_fw-la â—ronoâ—â—â—oâ—â—â—_n must_v understand_v the_o proportional_a â—artes_n or_o sâ—â—â—ions_n to_o be_v thâ—se_a which_o be_v contain_v 〈◊〉_d the_o parallel_a superficies_n construction_n demonstration_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n construction_n demonstration_n two_o case_n in_o this_o proposition_n the_o first_o case_n second_o case_n construction_n demonstration_n an_o other_o demonstration_n construction_n demonstration_n construction_n three_o case_n in_o this_o proposition_n the_o first_o case_n a_o necessary_a thing_n to_o be_v prove_v before_o he_o procee_v any_o â—arther_a in_o the_o construction_n of_o the_o problem_n problem_n which_o how_o to_o find_v out_o be_v teach_v at_o the_o end_n of_o this_o demonstration_n and_o also_o be_v teach_v in_o the_o assumpt_n put_v before_o the_o 14._o proposition_n of_o the_o ten_o book_n demonstration_n of_o the_o first_o case_n second_o case_n three_o case_n an_o other_o demonstration_n to_o prove_v that_o thâ—_z line_n ab_fw-la be_v not_o less_o they_o the_o line_n lx._n this_o be_v before_o taâ—ght_n in_o the_o ten_o book_n in_o the_o assumpt_n put_v before_o the_o 14._o proposition_n proposition_n m._n dee_n to_o avoid_v cavillation_n add_v to_o euclides_n proposition_n this_o word_n six_o who_o i_o have_v follow_v according_o and_o not_o zamberts_n in_o this_o this_o kind_n of_o body_n mention_v in_o the_o proposition_n be_v call_v a_o parallelipipedom_n according_a to_o the_o definition_n before_o give_v thereof_o demonstration_n that_o the_o opposite_a side_n be_v parallelogram_n demonstration_n that_o the_o opposite_a superficies_n be_v equal_a equal_a ab_fw-la be_v equal_a to_o dc_o because_o the_o superficies_n ac_fw-la be_v prove_v a_o parallelogram_n and_o by_o the_o same_o reason_n be_v bh_o equal_a to_o cf_o because_o the_o superficies_n fb_o be_v prove_v a_o parallelogram_n therefore_o the_o 34._o of_o the_o first_o be_v our_o proof_n fâ—â—â—â—_n corollary_n second_o corolry_n these_o solid_n which_o he_o speak_v of_o in_o this_o corollary_n be_v of_o some_o call_v side_v column_n three_o corollary_n constrâ—ction_n demonstration_n demonstration_n look_v at_o the_o end_n of_o the_o demonstration_n what_o be_v understand_v by_o stand_a line_n john_n do_v you_o his_o figure_n by_o this_o figure_n it_o appear_v why_o â—uch_v prism_n be_v call_v pledge_n of_o 〈◊〉_d uâ—ry_a shape_n of_o a_o wedge_n as_o be_v the_o solid_a defgac_n etc_n etc_n stand_v line_n construction_n demonstration_n i_o dees_n figure_n two_o case_n in_o this_o proposition_n thâ—_z first_o case_n construction_n we_o be_v behold_v to_o m._n do_v you_o for_o invent_v this_o figure_n with_o other_o which_o till_o his_o reform_n be_v as_o much_o misshapen_a as_o this_o be_v and_o so_o both_o in_o the_o greek_a and_o latin_a copy_n remain_v demonstration_n demonstration_n note_v now_o how_o the_o base_a respective_o be_v takend_v for_o so_o may_v alteration_n of_o respect_n alter_v the_o name_n of_o the_o bound_n either_o of_o solid_n or_o plain_n second_o case_n case_n there_o you_o perceive_v how_o the_o base_a be_v diverse_o consider_v &_o choose_v as_o before_o we_o advertise_v you_o construction_n demonstration_n construction_n demonstration_n *_o *_o note_v this_o famous_a lemma_n the_o double_a oâ—â—he_n cube_n cube_n note_n note_n 〈…〉_o lemma_n note_v what_o iâ—_z yet_o lack_v requisite_a to_o the_o double_n of_o the_o cube_n the_o converse_n of_o both_o the_o part_n of_o the_o first_o case_n the_o converse_n of_o the_o second_o case_n the_o general_a conclusion_n construction_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n to_o find_v two_o middle_a proportional_n between_o two_o number_n give_v note_v the_o practice_n of_o approach_v to_o precieâ—es_n in_o cubik_fw-mi root_n root_n this_o be_v the_o way_n to_o apply_v any_o square_n give_v to_o a_o line_n also_o give_v sufficienty_fw-mi extend_v a_o problem_n worth_a the_o search_n for_o construction_n demonstration_n construction_n demonstration_n demonstration_n it_o be_v evident_a that_o those_o perpendicular_n be_v all_o one_o with_o the_o stand_a line_n of_o the_o solid_n if_o their_o solid_a angle_n be_v make_v of_o superâ—icâ—â—il_n right_a angle_n only_o double_v of_o the_o 〈◊〉_d etc_n etc_n etc_n dâ—mâ—â—â—â—ratioâ—â—f_a possibility_n in_o the_o problem_n an_o other_o argument_n to_o comâ—oâ—t_v the_o studious_a demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o part_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n construction_n demonstration_n demonstration_n which_o of_o some_o be_v call_v side_v column_n side_v column_n construction_n demonstration_n the_o square_n of_o the_o circle_n demonstration_n â—eading_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n the_o first_o case_n that_o a_o square_a within_o any_o circle_n describe_v be_v big_a than_o half_a the_o circle_n that_o the_o isosceles_a triangle_n without_o the_o square_n be_v great_a than_o half_a the_o segment_n wherein_o they_o be_v second_o case_n case_n this_o as_o 〈…〉_o afterwards_o at_o the_o end_n of_o the_o demâ—straâ—ion_n prove_v construction_n demonstration_n construction_n demonstration_n diâ—â—reence_n between_o the_o first_o problem_n and_o the_o second_o construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n note_v this_o well_o for_o it_o iâ—_z of_o great_a use_n an_o other_o way_n of_o demonstration_n of_o the_o fâ—â—st_a problem_n of_o thâ—â—_n addition_n note_v this_o properâ—ie_n of_o a_o triangle_n rectangle_n construction_n demonstration_n demonstration_n though_o i_o say_v without_o the_o square_n yet_o you_o must_v think_v that_o it_o may_v be_v also_o within_o the_o square_n &_o that_o diverse_o wherefore_o this_o problem_n may_v have_v diverse_a case_n so_o but_o brief_o to_o aside_o all_o may_v thus_o be_v say_v cut_v any_o side_n of_o that_o square_n into_o 3_n parte_v in_o the_o proportion_n of_o x_o to_o y._n note_v the_o manner_n of_o the_o drift_n in_o this_o demonstration_n and_o construction_n mix_o and_o with_o no_o determination_n to_o the_o construction_n as_o common_o iâ—_z in_o problemeâ—â—_n which_o be_v here_o of_o i_o so_o usedâ—_n for_o a_o example_n to_o young_a student_n of_o variety_n in_o art_n construction_n demonstration_n demonstration_n note_v and_o remember_v one_o â—eâ—th_n in_o these_o solid_n the_o conclusion_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n namely_o that_o it_o be_v divide_v moreover_o into_o two_o equal_a prism_n conclusion_n of_o the_o second_o part_n demonstration_n of_o the_o last_o part_n that_o the_o two_o prism_n be_v great_a than_o the_o half_a of_o the_o whole_a pyramid_n conclusion_n of_o the_o last_o part_n conclusion_n of_o the_o whole_a proposition_n proposition_n a_o assumpt_n a_o assumpt_n conclusion_n of_o the_o whole_a demonstration_n lead_v to_o a_o impossibility_n impossibility_n in_o the_o assuâ—pâ—â—â—llowinâ—_n the_o second_o proposition_n of_o this_o bâ—â—ke_n construction_n demonstration_n demonstration_n note_n side_v column_n sometime_o call_v prism_n be_v triple_a to_o pyramid_n have_v one_o base_a and_o equal_a heâ—th_n with_o they_o note_v â—arallelipipedons_n treble_v to_o pyramid_n of_o one_o base_a and_o heith_n with_o they_o construction_n demonstration_n a_o addition_n by_o campane_n and_o flussas_n demonstration_n of_o the_o first_o part_n demonstration_n â—f_n the_o second_o part_n which_o iâ—_z the_o converse_n of_o the_o first_o constr_n 〈◊〉_d 〈◊〉_d parallelipipedon_n call_v prism_n prism_n by_o this_o it_o be_v manifest_a that_o euclid_n comprehend_v side_v column_n also_o under_o the_o name_n of_o a_o prism_n prism_n a_o prism_n have_v for_o his_o base_a a_o polygonon_n figure_n as_o we_o have_v often_o before_o note_v unto_o you_o note_v m._n do_v you_o his_o chief_a purpose_n in_o his_o addition_n demonstration_n touch_v cylinder_n second_o case_n second_o parâ—_n which_o concern_v cillinder_n construction_n demonstration_n construction_n demonstration_n touch_v cylinder_n demonstration_n touch_v cones_n first_o part_n of_o the_o proposition_n demonstrate_v touch_v cones_n two_o case_n in_o this_o proposition_n the_o first_o case_n second_o case_n construction_n demonstration_n touch_v cylinder_n second_o part_n demonstrate_v construction_n construction_n note_v this_o lm_o because_o of_o kz_o in_o the_o next_o proposition_n and_o here_o the_o point_n m_o for_o the_o point_n z_o in_o the_o next_o demonstration_n i_o dee_n dee_n for_o that_o the_o section_n be_v make_v by_o the_o number_n two_o that_o be_v by_o take_v half_n and_o of_o the_o residue_n the_o halâ—eâ—_n and_o so_o to_o ld_n be_v a_o half_a and_o a_o residue_n which_o shall_v be_v a_o common_a measure_n back_o again_o to_o make_v side_n of_o the_o poligonon_n figure_n construction_n
demonstration_n demonstration_n the_o circle_n so_o make_v or_o so_o consider_v in_o the_o sphere_n be_v call_v the_o great_a circle_n all_o other_o not_o have_v the_o centre_n of_o the_o sphere_n to_o be_v their_o centre_n alsoâ—_n be_v call_v less_o circle_n note_v these_o description_n description_n an_o other_o corollary_n corollary_n an_o other_o corollary_n construction_n construction_n this_o be_v also_o prove_v in_o the_o assumpt_n before_o add_v out_o oâ—_n flussas_n note_v what_o a_o great_a or_o great_a circle_n in_o a_o spear_n be_v first_o part_n of_o the_o construction_n noteâ—_n noteâ—_n you_o know_v full_a well_o that_o in_o the_o superficies_n of_o the_o sphere_n â—â—â—ly_o the_o circumference_n of_o the_o circle_n be_v but_o by_o thâ—se_a circumference_n the_o limitation_n and_o assign_v of_o circle_n be_v use_v and_o so_o the_o circumference_n of_o a_o circle_n usual_o call_v a_o circle_n which_o in_o this_o place_n can_v not_o offend_v this_o figure_n be_v restore_v by_o m._n dee_n his_o diligence_n for_o in_o the_o greek_a and_o latin_a euclides_n the_o line_n gl_n the_o line_n agnostus_n and_o the_o line_n kz_o in_o which_o three_o line_n the_o chief_a pinch_n of_o both_o the_o demonstration_n do_v stand_v be_v untrue_o draw_v as_o by_o compare_v the_o studious_a may_v perceive_v note_n you_o must_v imagine_v 〈◊〉_d right_a line_n axe_n to_o be_v perpendicular_a upon_o the_o diameter_n bd_o and_o ce_fw-fr though_o here_o ac_fw-la the_o semidiater_n seem_v to_o be_v part_n of_o ax._n and_o so_o in_o other_o point_n in_o this_o figure_n and_o many_o other_o strengthen_v your_o imagination_n according_a to_o the_o tenor_n of_o construction_n though_o in_o the_o delineation_n in_o plain_a sense_n be_v not_o satisfy_v note_n boy_n equal_a to_o bk_o in_o respect_n of_o m._n dee_n his_o demonstration_n follow_v follow_v note_v â—his_fw-la point_n z_o that_o you_o may_v the_o better_a understand_v m._n dee_n his_o demonstration_n second_o part_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n demonstration_n which_o of_o necessity_n shall_v fall_v upon_o z_o as_o m._n dee_n prove_v it_o and_o his_o proof_n be_v set_v after_o at_o this_o mark_n ✚_o follow_v i_o dee_n dee_n but_o az_o be_v great_a them_z agnostus_n as_o in_o the_o former_a proposition_n km_o be_v evident_a to_o be_v great_a than_o kg_v so_o may_v it_o also_o be_v make_v manifest_a that_o kz_o do_v neither_o touch_n nor_o cut_v the_o circle_n fgâ—h_n a_o other_o prove_v that_o the_o line_n ay_o be_v great_a they_o the_o line_n ag._n ag._n this_o as_o a_o assumpt_n be_v present_o prove_v two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o impossibility_n second_o case_n case_n as_o it_o be_v â—asiâ—_n to_o gather_v by_o the_o assumpt_n put_v after_o the_o secoâ—â—_n of_o this_o booâ—â—_n note_v a_o general_a rule_n the_o second_o part_n of_o the_o problem_n two_o way_n execute_v a_o upright_a cone_n the_o second_o part_n of_o the_o problem_n the_o second_o â—aâ—â—_n oâ—_n the_o problem_n ☜_o ☜_o this_o may_v easy_o be_v demonstrate_v as_o in_o thâ—_z 17._o proposition_n the_o section_n of_o a_o sphere_n be_v prove_v to_o be_v a_o circle_n circle_n for_o take_v away_o all_o doubt_n this_o aâ—_z a_o lemma_n afterward_o be_v demonstrate_v a_o lemma_n as_o it_o be_v present_o demonstrate_v construction_n demonstration_n the_o second_o part_n of_o the_o problem_n *_o construction_n demonstration_n an_o other_o way_n of_o execute_v this_o problem_n the_o converse_n of_o the_o assumpt_n a_o great_a error_n common_o maintain_v between_o straight_o and_o crooked_a all_o manner_n of_o proportion_n may_v be_v give_v construction_n demonstration_n the_o definition_n of_o a_o circled_a â—â—apâ—â—d_n in_o a_o spâ—erâ—_n construction_n demonstration_n this_o be_v manifest_a if_o you_o consider_v the_o two_o triangle_n rectangle_v hom_o and_o hon_n and_o then_o with_o all_o use_v the_o 47._o of_o the_o first_o of_o euclid_n construction_n demonstration_n construction_n demonstration_n this_o in_o manner_n of_o a_o lemmâ—_n be_v present_o prove_v note_v here_o of_o axe_n base_a &_o solidity_n more_o than_o i_o need_n to_o bring_v any_o far_a proof_n for_o note_n note_n i_o say_v half_a a_o circular_a revolution_n for_o that_o suffice_v in_o the_o whole_a diameter_n n-ab_n to_o describe_v a_o circle_n by_o iâ—_z it_o be_v move_v â—â—out_v his_o centre_n q_o etc_n etc_n lib_fw-la 2_o prop_n 2._o de_fw-fr spheâ—a_n &_o cylindrâ—_n note_n note_n a_o rectangle_n parallelipipedon_n give_v equal_a to_o a_o sphere_n give_v to_o a_o sphere_n or_o to_o any_o part_n of_o a_o sphere_n assign_v as_o a_o three_o four_o five_o etc_o etc_o to_o geve_v a_o parallelipipedon_n equal_a side_v colume_n pyramid_n and_o prism_n to_o be_v give_v equal_a to_o a_o sphere_n or_o to_o any_o certain_a part_n thereof_o to_o a_o sphere_n or_o any_o segment_n or_o sector_n of_o the_o same_o to_o geve_v a_o cone_n or_o cylinder_n equal_a or_o in_o any_o proportion_n assign_v far_o use_v of_o spherical_a geometry_n the_o argument_n of_o the_o thirteen_o book_n construction_n demonstration_n demonstration_n the_o assumpt_v prove_v prove_v because_o ac_fw-la be_v suppose_v great_a than_o ad_fw-la therefore_o his_o residue_n be_v less_o than_o the_o residue_n of_o ad_fw-la by_o the_o common_a sentence_n wherefore_o by_o the_o supposition_n db_o be_v great_a than_o â—c_z the_o chieâ—e_a line_n in_o all_o euclides_n geometry_n what_o be_v mean_v here_o by_o a_o section_n in_o one_o only_a poiâ—t_n construction_n demonstration_n demonstration_n note_v how_o ce_fw-fr and_o the_o gnonom_n xop_n be_v prove_v equal_a for_o it_o serve_v in_o the_o converse_n demonstrate_v by_o m._n dee_n here_o next_o after_o this_o proposition_n â—the_v converse_v of_o the_o former_a former_a as_o we_o haâ—e_z note_v the_o place_n of_o the_o peculiar_a proâ—e_n there_o â—in_a the_o demonstration_n of_o the_o 3._o 3._o therefore_o by_o my_o second_o theorem_a add_v upon_o the_o second_o proposition_n dc_o be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n a._n and_o because_o ac_fw-la be_v big_a than_o cb_o therefore_o dam_n be_v great_a than_o ac_fw-la wherefore_o if_o a_o right_a line_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v demonstrate_v therefore_o by_o my_o second_o theorem_a add_v upon_o the_o second_o proposition_n dc_o be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n a._n and_o because_o ac_fw-la be_v big_a than_o cb_o therefore_o dam_n be_v great_a than_o ac_fw-la wherefore_o if_o a_o right_a line_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v construction_n construction_n though_o i_o say_v perpendicular_a yes_o you_o may_v perceive_v how_o infinite_a other_o position_n will_v serve_v so_o that_o diego_n and_o ad_fw-la make_v a_o angle_n for_o a_o triangle_n to_o have_v his_o side_n proportional_o cut_v etc_n etc_n demonstration_n demonstration_n i_o dee_n this_o be_v most_o evident_a of_o my_o second_o theorem_a add_v to_o the_o three_o proposition_n for_o to_o add_v to_o a_o whole_a line_n a_o line_n equal_a to_o the_o great_a segment_n &_o to_o add_v to_o the_o great_a segment_n a_o line_n equal_a to_o the_o whole_a line_n be_v all_o one_o thing_n in_o the_o line_n produce_v by_o the_o whole_a line_n i_o mean_v the_o line_n divide_v by_o extreme_a and_o mean_a proportion_n this_o be_v before_o demonstrate_v most_o evident_o and_o brief_o by_o m._n dee_n after_o the_o 3._o proposition_n note_n note_v 4._o proportional_a line_n note_v two_o middle_a proportional_n note_v 4._o way_n of_o progression_n in_o the_o proportion_n of_o a_o line_n divide_v by_o extreme_a and_o middle_a proportion_n what_o resolution_n and_o composition_n be_v have_v before_o be_v teach_v in_o the_o begin_n of_o the_o first_o book_n book_n proclus_n in_o the_o greek_a in_o the_o 58._o page_n construction_n demonstration_n two_o case_n in_o this_o proposition_n construction_n thâ—_z first_o case_n demonstration_n the_o second_o case_n construction_n demonstration_n construction_n demonstration_n this_o corollary_n be_v the_o 3._o proposition_n of_o theâ—4_n â—4_o book_n after_o campane_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n construction_n demonstration_n construstion_n demonstration_n constrâ—yction_n demonstration_n this_o corollary_n be_v the_o 11._o proposition_n of_o the_o 14._o book_n after_o campane_n this_o corollary_n be_v the_o 3._o corollary_n after_o the_o 17._o proposition_n of_o the_o 14_o book_n after_o campane_n campane_n by_o the_o name_n oâ—_n a_o pyramid_n both_o here_o &_o iâ—_z this_o book_n follow_v understand_v a_o tetrahedron_n an_o other_o construction_n and_o demonstration_n of_o the_o second_o part_n after_o fâ—ussas_n three_o part_n of_o the_o demonstration_n this_o corollary_n be_v the_o 15._o proposition_n of_o the_o 14._o book_n after_o campane_n this_o corollary_n campane_n put_v as_o a_o corollary_n after_o
the_o 17._o proposition_n of_o the_o 14._o book_n construction_n â—rist_n part_n of_o the_o demonstration_n demonstration_n for_o the_o 4._o angle_n at_o the_o point_n king_n be_v equal_a to_o four_o right_a angle_n by_o the_o corollary_n of_o tâ—e_z 15._o of_o the_o first_o and_o those_o 4._o angle_n be_v equal_a the_o one_o to_o the_o other_o by_o the_o â—_o of_o the_o â—irst_n and_o therefore_o each_o be_v a_o right_a angle_n second_o part_n of_o the_o demonstration_n demonstration_n for_o the_o square_n of_o the_o line_n ab_fw-la which_o be_v prove_v equal_a to_o the_o square_n of_o the_o line_n lm_o be_v double_a to_o the_o square_n of_o the_o line_n bd_o which_o be_v also_o equal_a to_o the_o square_n of_o the_o line_n le._n this_o corollary_n be_v the_o 16._o proposition_n of_o the_o 14._o book_n after_o campane_n first_o part_n of_o the_o demonstration_n second_o part_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n demonstration_n by_o the_o 2._o assumpt_v of_o the_o 13._o of_o this_o book_n three_o part_n of_o the_o demonstration_n second_o part_n of_o the_o demonstration_n for_o the_o line_n qw_o be_v equal_a to_o the_o line_n iz_n &_o the_o line_n zw_n be_v common_a to_o they_o both_o this_o part_n be_v again_o afterward_o demonstrate_v by_o flussas_n the_o pentagon_n vbwcr_n prove_v to_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n the_o pentagon_n vbwcz_n it_o prove_v equiangle_n equiangle_n look_v for_o a_o far_a construction_n after_o flussas_n at_o the_o end_n of_o the_o demonstration_n that_o the_o side_n of_o the_o dodecahedron_n be_v a_o residual_a line_n draw_v in_o the_o former_a figure_n these_o line_n cta_n ctl_n câ—d_v the_o side_n of_o a_o pyramid_n the_o side_n of_o a_o cube_fw-la the_o side_n of_o a_o dodecahedron_n comparison_n of_o the_o five_o side_n of_o the_o foresay_a body_n an_o other_o demommonstration_n to_o prove_v that_o the_o side_n of_o the_o icosahedron_n be_v great_a than_o the_o side_n of_o the_o dodecahedron_n that_o 3._o square_n of_o the_o line_n fb_o be_v great_a they_o 6._o square_n of_o the_o line_n nb._n that_o there_o can_v be_v no_o other_o solid_a beside_o these_o five_o contain_v under_o equilater_n and_o equiangle_n base_n that_o the_o angle_n of_o a_o equilater_n and_o equiangle_n pentagon_n be_v one_o right_a angle_n and_o a_o 〈◊〉_d part_n 〈◊〉_d which_o thing_n be_v also_o before_o prove_v in_o the_o 〈◊〉_d of_o the_o 32._o of_o the_o â—irst_n the_o side_n of_o the_o angle_n of_o the_o inclâ—â—â—tion_n of_o the_o 〈◊〉_d of_o the_o 〈◊〉_d be_v 〈◊〉_d rational_a the_o side_n of_o the_o angle_n of_o the_o inclination_n of_o the_o 〈◊〉_d â—f_n tâ—e_z 〈…〉_o that_o the_o plain_n of_o a_o octohedron_n be_v in_o liâ—e_z sort_n incline_v that_o the_o plain_n of_o a_o icosahedron_n be_v in_o like_a sort_n incline_v that_o the_o plain_n of_o a_o dâ—â—â—â—â—hedron_n be_v 〈◊〉_d like_o sort_n incline_v the_o side_n of_o the_o angle_n of_o the_o inclination_n of_o the_o supeâ—ficieces_n of_o the_o tetrahedron_n be_v prove_v rational_a the_o side_n of_o the_o angle_n of_o the_o inclination_n of_o the_o superficiece_n of_o the_o cube_fw-la prove_v rational_a the_o side_n of_o thâ—_z angle_n etc_n etc_n of_o the_o octohedron_n prove_v rational_a the_o side_n of_o the_o angle_n etc_n etc_n of_o the_o icosahedron_n prove_v irrational_a how_o to_o know_v whether_o the_o angle_n of_o the_o inclination_n be_v a_o right_a angle_n a_o acute_a angle_n or_o a_o oblique_a angle_n the_o argument_n of_o the_o fourteen_o book_n first_o proposition_n after_o flussas_n construction_n demonstration_n demonstration_n this_o be_v manifest_a by_o the_o 12._o proposition_n of_o the_o thirtenh_o book_n as_o campane_n well_o gather_v in_o a_o corollary_n of_o the_o same_o the_o 4._o pâ—posâ—tion_n after_o flussas_n flussas_n this_o be_v afterward_o prove_v in_o the_o 4._o proposition_n this_o assumpt_n be_v the_o 3._o proposition_n after_o flussas_n construction_n of_o the_o assumpt_n demonstration_n of_o the_o assumpt_n construction_n of_o the_o proposition_n demonstration_n of_o the_o proposition_n proposition_n thâ—_z 5._o proposition_n aâ—tâ—r_a 〈◊〉_d construction_n demonstration_n the_o 5._o proposition_n aâ—ter_a fâ—ussas_n demonstration_n demonstration_n this_o be_v the_o reason_n of_o the_o corollary_n follow_v a_o corollary_n which_o also_o flussas_n put_v as_o a_o corollary_n after_o the_o 5._o proposition_n in_o his_o order_n the_o 6._o pâ—â—positionâ—â—ter_a flussas_n construction_n demonstration_n demonstration_n this_o be_v not_o hard_a to_o prove_v by_o the_o 15._o 16._o and_o 19_o of_o the_o â—â—â—eth_v â—â—â—eth_v in_o the_o corollary_n of_o the_o 17._o of_o the_o tâ—irtenth_o tâ—irtenth_o 〈◊〉_d again_o be_v require_v the_o assumpt_n which_o be_v afterward_o prove_v in_o this_o 4_o proposition_n proposition_n but_o first_o the_o assumpt_n follow_v the_o construction_n whereof_o here_o begin_v be_v to_o be_v prove_v the_o assumpt_n which_o also_o flussas_n put_v as_o a_o assumpt_n aâ—ter_v the_o 6._o proposition_n demonstration_n of_o the_o assumpt_n construction_n pertain_v to_o the_o second_o demonstration_n of_o the_o 4._o proposition_n second_o demonstration_n oâ—_n the_o 4._o proposition_n the_o 7â—_n proposition_n after_o flussas_n construction_n demonstration_n demonstration_n here_o again_o be_v require_v the_o assumpt_n afterward_o prove_v in_o this_o 4._o proposition_n proposition_n as_o may_v by_o the_o assumpt_n afterward_o in_o this_o proposition_n be_v plain_o prove_v the_o 8._o prodition_n aâ—ter_a flussas_n flussas_n by_o the_o corollary_n add_v by_o flussas_n after_o have_v assumpt_v put_v after_o the_o 17._o proposition_n of_o the_o 12._o book_n corollary_n of_o the_o 8._o after_o flussas_n this_o assumpt_n be_v the_o 3._o proposition_n aâ—ter_v â—lussas_n and_o be_v it_o which_o 〈◊〉_d time_n have_v be_v take_v aâ—_z grant_v in_o this_o book_n and_o oâ—ce_n also_o in_o the_o last_o proposition_n of_o the_o 13._o book_n as_o we_o have_v beâ—ore_o note_v demonstration_n demonstration_n in_o the_o 4._o section_n â—f_o this_o proposition_n proposition_n in_o the_o 1._o and_o 3_o section_n of_o the_o same_o proposition_n proposition_n in_o the_o 5._o section_n of_o the_o same_o proposition_n a_o corollary_n the_o first_o proposition_n after_o campane_n construction_n demonstration_n the_o 2._o proposition_n after_o campane_n demonstration_n lead_v to_o a_o impossibility_n the_o 4._o proposition_n after_o campane_n construction_n demonstration_n this_o corollary_n campane_n also_o â—utteth_v after_o the_o 4._o proposition_n in_o his_o order_n the_o 5._o proposition_n after_o campane_n construction_n demonstration_n this_o be_v the_o 6._o and_o 7._o proposition_n after_o campane_n construction_n demonstration_n this_o corollary_n campane_n also_o add_v after_o the_o 7._o proposition_n iâ—_z his_o order_n the_o 5._o proposition_n aâ—ter_v campane_n construction_n demonstration_n this_o assumpt_a campane_n also_o have_v after_o the_o 8._o proposition_n in_o his_o order_n construction_n demonstration_n the_o 9_o proposition_n after_o campane_n construction_n demonstration_n this_o campane_n put_v aâ—_z a_o corollary_n in_o the_o 9_o proposition_n after_o his_o order_n this_o corollary_n be_v the_o 9_o proposition_n after_o campane_n the_o 12._o proposition_n after_o campane_n construction_n demonstration_n the_o 13._o proposition_n after_o campane_n the_o 14._o proposition_n after_o campane_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n the_o 17._o proposition_n after_o campane_n firâ—t_o part_n of_o the_o construction_n first_o part_n of_o the_o demonstration_n second_o parâ—_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n the_o 18._o proposition_n after_o campane_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n the_o corollary_n of_o the_o 8._o proposition_n after_o campane_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n the_o argument_n of_o the_o 15._o book_n book_n in_o this_o proposition_n as_o also_o in_o all_o the_o other_o follow_v by_o the_o name_n of_o a_o pyramid_n understand_v a_o tetrahedron_n as_o i_o have_v before_o admonish_v construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n that_o which_o here_o follow_v concern_v the_o inclination_n of_o the_o plain_n of_o the_o five_o solid_n be_v before_o teach_v â—hough_o not_o altogether_o after_o the_o same_o manner_n out_o of_o flussas_n in_o the_o latter_a â—nde_n of_o the_o 13_o book_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n first_o part_n of_o the_o construction_n first_o part_n of_o the_o demonstration_n second_o part_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n three_o part_n of_o the_o construction_n three_o part_n of_o the_o demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n produce_v in_o the_o figure_n the_o line_n tf_n to_o the_o point_n b._n construction_n demonstration_n this_o proposition_n campane_n have_v &_o be_v the_o last_o also_o in_o order_n of_o the_o 15._o book_n with_o he_o the_o argument_n of_o the_o 16._o book_n construction_n demonstration_n demonstration_n by_o a_o pyramid_n understand_v a_o tetrahedron_n throughout_o all_o this_o book_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n demonstration_n that_o be_v aâ—_z 18._o to_o 1._o demonstration_n demonstration_n that_o iâ—_z as_o 9_o to_o 2._o 2._o that_o be_v as_o 18._o to_o 2._o or_o 9_o to_o 1._o draw_v in_o the_o figure_n a_o line_n from_o b_o to_o h._n h._n what_o the_o duple_n of_o a_o extreme_a and_o mean_a proportion_n be_v construction_n demonstration_n demonstration_n constrution_n demonstration_n construction_n demonstration_n demonstration_n demonstration_n that_o be_v at_o 13._o 1_o â—_o be_v toâ—_n â—_z demonstration_n demonstration_n construction_n demonstration_n construction_n demonstration_n extend_v in_o the_o figure_n a_o line_n â—rom_o the_o point_n e_o to_o the_o point_n b._n extend_v in_o the_o figure_n a_o line_n from_o the_o point_n e_o to_o the_o point_n b._n demonstration_n construction_n demonstration_n second_o part_n of_o the_o demonstration_n icosidodecahedron_n exoctohedron_n that_o the_o exoctohedron_n be_v contain_v in_o a_o sphere_n that_o the_o exoctohedron_n be_v contain_v in_o the_o sphere_n give_v that_o the_o diaâ—â—â—ter_n of_o the_o sphere_n be_v doâ—ble_a to_o the_o side_n â—f_o the_o exoctohedron_n that_o the_o icosidodecahedron_n be_v contain_v in_o the_o sphere_n give_v give_v that_o be_v as_o 8._o 103â—_n â—_z fault_n escape_v â—cl_n â—ag_n line_n faultesâ—_n co_n 〈◊〉_d 〈◊〉_d  _o  _o  _o errata_fw-la lib._n 1._o  _o 1_o 2_o 41_o point_n b._n at_o campane_n point_n c_o aâ—_z campane_n 3_o 1_o 22_o aâ—l_a line_n draw_v all_o righâ—_n 〈…〉_o 3_o 1_o 28_o line_n draw_v to_o the_o superficies_n right_a line_n draw_v to_o the_o circumference_n 9_o 1_o 42_o liâ—es_n ab_fw-la and_o ac_fw-la line_n ab_fw-la and_o bc_o 15_o 1_o 35_o be_v equal_a be_v prove_v equal_a 20_o 2_o 28_o by_o the_o first_o by_o the_o four_o 21_o 1_o 39_o tâ—e_z centre_n c._n the_o centre_n e_o 2d_o 2_o â—_o iâ—â—ower_a right_n if_o two_o right_a 25_o 2_o 3_o fâ—â—â—_n petition_n five_o petition_n 49_o 2_o 7_o 14._o â—â—_o 32.64_o etc_n etc_n 4.8.16.32.64_o &_o 53_o 1_o 39_o the_o triangle_n ng_z the_o triangle_n kâ—_n 54_o 2_o 25_o by_o the_o 44_o by_o the_o 42_o 57_o 2_o 23_o and_o câ—g_v in_o the_o and_o cgb_n be_v thâ—_z  _o  _o  _o in_o stead_n of_o â—lussates_n through_o out_o 〈◊〉_d whole_a book_n read_v â—lusâ—as_v  _o  _o  _o errata_fw-la lib._n 2._o  _o 60_o 2_o 29_o gnomon_n fgeh_a gnomon_n ahkd_v  _o  _o 30_o gnomon_n ehfg_n gnomon_n â—ckd_v 69_o 1_o 18_o the_o whole_a line_n the_o whole_a â—igure_n 76_o 2_o 9_o the_o diameter_n cd_o the_o diameter_n ahf_n  _o  _o  _o errata_fw-la lib._n 3._o  _o 82_o 2_o 36_o angle_n equal_a to_o the_o angle_n 92_o 1_o last_o the_o line_n ac_fw-la be_v the_o line_n of_o 〈◊〉_d  _o  _o  _o errata_fw-la lib._n 4._o  _o 110_o 2_o 10_o cd_o touch_v the_o ed_z touch_v the_o  _o  _o 12_o side_n of_o the_o other_o angle_n of_o the_o other_o 115_o 1_o 21_o and_o hb_o and_o he_o 117_o 2_o 44_o the_o angle_n acd_o the_o angle_n acb_o 118_o 1_o 2_o into_o ten_o equal_a into_o two_o equal_a 121_o 1_o 3â—_o cd_o and_o ea_fw-la cd_o de_fw-fr &_o ea_fw-la â—â—â—_o 1_o 29_o the_o first_o the_o three_o  _o  _o  _o errata_fw-la lib._n 5._o  _o 126_o 1_o 43_o it_o make_v 12._o more_o than_o 17._o by_o 5._o it_o make_v 24._o more_o than_o 17._o by_o 7._o 129_o 1_n  _o in_o stead_n of_o the_o figure_n of_o the_o 6._o definition_n draw_v in_o the_o magâ—â—_n a_o figure_n like_o unto_o thâ—s_n 134_o 2_o 4_o as_o ab_fw-la be_v to_o a_o so_o be_v cd_o to_o c_o as_z ab_fw-la be_v to_o b_o so_o be_v cd_o to_o d_o 141_o 2_o last_v but_o if_o king_n exceed_v m_o but_o if_o h_o exceed_v m_o lief_a be_v death_n and_o death_n be_v lief_a aetatis_fw-la svae_fw-la xxxx_n at_o london_n print_v by_o john_n day_n dwell_v over_o aldersgate_n beneath_o saint_n martins_n ¶_o these_o book_n be_v to_o be_v sell_v at_o his_o shop_n under_o the_o gate_n 1570._o
the_o lengthe_n of_o day_n and_o night_n the_o hour_n and_o time_n both_o night_n and_o day_n be_v know_v with_o very_a many_o other_o pleasant_a and_o necessary_a use_n whereof_o some_o be_v know_v but_o better_a remain_v for_o such_o to_o know_v and_o use_v who_o of_o a_o spark_n of_o true_a fire_n can_v make_v a_o wonderful_a bonfire_n by_o apply_v of_o due_a matter_n due_o ☜_o of_o astrology_n here_o i_o make_v a_o arâ—e_n several_a from_o astronomieâ—_n â—_z not_o by_o new_a devise_n but_o by_o good_a reason_n and_o authority_n for_o astrology_n be_v a_o art_n mathematical_a which_o reasonable_o demonstrate_v the_o operation_n and_o effect_n of_o the_o natural_a beam_n of_o light_n and_o secret_a influence_n of_o the_o star_n and_o planet_n in_o every_o element_n and_o elemental_a body_n at_o all_o time_n in_o any_o horizon_n assign_v this_o art_n be_v furnish_v with_o many_o other_o great_a art_n and_o experience_n as_o with_o perfect_v perspective_n astronomy_n cosmography_n natural_a philosophy_n of_o the_o 4._o element_n the_o art_n of_o graduation_n and_o some_o good_a understanding_n in_o music_n and_o yet_o moreover_o with_o a_o other_o great_a art_n hereafter_o follow_v though_o i_o here_o set_v this_o before_o for_o some_o consideration_n i_o move_v sufficient_a you_o see_v be_v the_o stuff_n to_o make_v this_o rare_a and_o secret_a art_n of_o and_o hard_a enough_o to_o frame_v to_o the_o conclusion_n syllogistical_a yet_o both_o the_o manifold_a and_o continual_a travail_n of_o the_o most_o ancient_a and_o wise_a philosopher_n for_o the_o attain_n of_o this_o art_n and_o by_o example_n of_o effect_n to_o confirm_v the_o same_o have_v leave_v unto_o we_o sufficient_a proof_n and_o witness_v and_o we_o also_o daily_o may_v perceive_v that_o man_n body_n and_o all_o other_o elemental_a body_n be_v alter_v dispose_v order_a pleasure_v and_o displeasure_v by_o the_o influential_a work_n of_o the_o sun_n moon_n and_o the_o other_o star_n and_o planet_n and_o therefore_o say_v aristotle_n in_o the_o first_o of_o his_o meteorological_a book_n in_o the_o second_o chapter_n est_fw-la autem_fw-la necessariò_fw-la mundus_fw-la iste_fw-la supernis_fw-la lationibus_fw-la ferè_fw-la continuus_fw-la ut_fw-la inde_fw-la vis_fw-la eius_fw-la universa_fw-la regatur_fw-la ea_fw-la siquidem_fw-la causa_fw-la prima_fw-la putanda_fw-la omnibus_fw-la est_fw-la unde_fw-la motus_fw-la principium_fw-la existit_fw-la that_o be_v this_o elemental_a world_n be_v of_o necessity_n almost_o next_o adjoin_v to_o the_o heavenly_a motion_n that_o from_o thence_o all_o his_o virtue_n or_o force_n may_v be_v govern_v for_o that_o be_v to_o be_v think_v the_o first_o cause_n unto_o all_o from_o which_o the_o beginning_n of_o motion_n be_v and_o again_o in_o the_o ten_o chapter_n opâ—rtet_fw-la igitur_fw-la &_o horum_fw-la principia_fw-la sumamus_fw-la &_o causas_fw-la omnium_fw-la similiter_fw-la principium_fw-la igitur_fw-la vy_z movens_fw-la praecipuumque_fw-la &_o omnium_fw-la primum_fw-la circulus_fw-la ille_fw-la est_fw-la in_fw-la quo_fw-la manifest_a solis_fw-la latio_fw-la etc_n etc_n and_o so_o forth_o his_o meteorological_a book_n be_v full_a of_o argument_n and_o effectual_a demonstration_n of_o the_o virtue_n operation_n and_o power_n of_o the_o heavenly_a body_n in_o and_o upon_o the_o four_o element_n and_o other_o body_n of_o they_o either_o perfect_o or_o unperfect_o compose_v and_o in_o his_o second_o book_n de_fw-fr generatione_fw-la &_o corruption_n in_o the_o ten_o chapter_n quocirca_fw-la &_o prima_fw-la latiâ—_n orâ—us_n &_o interitus_fw-la causa_fw-la non_fw-la est_fw-la sed_fw-la obliqui_fw-la circuli_fw-la latio_fw-la ea_fw-la namque_fw-la &_o continua_fw-la est_fw-la &_o dvobus_fw-la motibus_fw-la fit_a in_o english_a thus_o wherefore_o the_o uppermost_a motion_n be_v not_o the_o cause_n of_o generation_n and_o corruption_n but_o the_o motion_n of_o the_o zodiac_n for_o that_o both_o be_v continual_a and_o be_v cause_v of_o two_o move_n and_o in_o his_o second_o book_n and_o second_o chapter_n of_o his_o physike_n homo_fw-la namque_fw-la generate_fw-la hominem_fw-la atque_fw-la sol._n for_o man_n say_v he_o and_o the_o son_n be_v cause_n of_o man_n generation_n authority_n may_v be_v bring_v very_o many_o both_o of_o 1000_o 2000_o yea_o and_o 3000._o year_n antiquity_n of_o great_a philosopher_n expert_a wise_a and_o godly_a man_n for_o that_o conclusion_n which_o daily_a and_o hourly_o we_o man_n may_v discern_v and_o perceive_v by_o sense_n and_o reason_n all_o beast_n do_v feel_v and_o simple_o show_v by_o their_o action_n and_o passion_n outward_a and_o inward_a all_o plant_n herb_n tree_n flower_n and_o fruit_n and_o final_o the_o element_n and_o all_o thing_n of_o the_o element_n compose_v do_v geve_v testimony_n as_o aristotle_n say_v that_o their_o whole_a disposition_n virtue_n and_o natural_a motion_n depend_v of_o the_o activity_n of_o the_o heavenly_a motion_n and_o influence_n whereby_o beside_o the_o specifical_a order_n and_o form_n due_a to_o every_o seed_n and_o beside_o the_o nature_n proper_a to_o the_o individual_a matrix_fw-la of_o the_o thing_n produce_v what_o shall_v be_v the_o heavenly_a impression_n the_o perfect_a and_o circumspect_a astrologien_n have_v to_o conclude_v not_o only_o by_o apotelesme_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d but_o by_o natural_a and_o mathematical_a demonstration_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d whereunto_o what_o science_n be_v requisite_a without_o exception_n i_o partly_o have_v here_o warn_v and_o in_o my_o bropâ—deâ—â—â—â—_n beside_o other_o matter_n there_o disclose_v i_o have_v mathematical_o furnish_v up_o the_o whole_a method_n to_o this_o our_o age_n not_o so_o careful_o handle_v by_o any_o that_o ever_o i_o see_v or_o hear_v of_o i_o be_v for_o lovayn_n â—1_n year_n ago_o by_o certain_a earnest_a disputation_n of_o the_o learned_a gâ—â—ardus_n mâ—rcâ—tâ—â—_n and_o 〈◊〉_d gogaâ—a_n and_o other_o thereto_o so_o provoke_v and_o by_o my_o constant_a and_o invincible_a zeal_n to_o the_o verity_n in_o observation_n of_o heavenly_a influence_n to_o the_o minâ—te_n of_o time_n than_o so_o diligent_a and_o chief_o by_o the_o supernatural_a influence_n from_o the_o star_n of_o jacob_n so_o direct_v that_o any_o modest_a and_o sober_a student_n careful_o and_o diligent_o selâ—ing_v for_o the_o truth_n will_v both_o find_v &_o confess_v therein_o to_o be_v the_o verity_n of_o these_o my_o word_n and_o also_o become_v a_o reasonable_a reformer_n of_o three_o sort_n of_o people_n about_o these_o influential_a operation_n great_o err_v from_o the_o truth_n whereof_o the_o one_o be_v light_a believer_n the_o other_o note_n light_a despiser_n and_o the_o three_o light_a practiser_n the_o first_o &_o most_o common_a sort_n think_v the_o heaven_n and_o star_n to_o be_v answerable_a to_o any_o their_o doubt_n or_o desire_n which_o be_v not_o so_o and_o in_o deed_n they_o to_o much_o over_o reach_n the_o second_o sort_n think_v no_o influential_a virtue_n from_o the_o heavenly_a body_n to_o bear_v any_o sway_n in_o generation_n and_o corruption_n in_o this_o elemental_a world_n and_o to_o the_o sun_n moon_n and_o star_n be_v so_o many_o so_o pure_a so_o bright_a so_o wonderful_a big_a so_o far_o in_o distance_n so_o manifold_a in_o their_o motion_n so_o constant_a in_o their_o periode_n etc_n etc_n they_o assign_v a_o sleight_n simple_a office_n or_o two_o and_o so_o allow_v unto_o they_o according_a to_o their_o capacity_n as_o much_o virtue_n and_o power_n influential_a as_o to_o the_o sign_n of_o the_o sun_n moon_n and_o seven_o star_n hang_v up_o for_o sign_n in_o london_n for_o distinction_n of_o house_n &_o such_o gross_a help_n in_o our_o worldly_a affair_n and_o they_o understand_v not_o or_o will_v not_o understand_v of_o the_o other_o work_n and_o virtue_n of_o the_o heavenly_a sun_n moon_n and_o star_n not_o so_o much_o as_o the_o mariner_n or_o husband_n manâ—_n no_o not_o so_o much_o as_o the_o elephant_n do_v as_o the_o cynocephalus_n as_o the_o porpentine_n doâ—h_v nor_o will_v allow_v these_o perfect_a and_o incorruptible_a mighty_a body_n so_o much_o virtual_a radiation_n &_o force_n as_o they_o see_v in_o a_o little_a piece_n of_o a_o magnes_n stone_n which_o at_o great_a distance_n show_v his_o operation_n and_o perchance_o they_o think_v the_o sea_n &_o rivers_n as_o the_o thames_n to_o be_v some_o quick_a thing_n and_o sâ—_n to_o ebb_v aâ—d_z slow_a run_v in_o and_o out_o of_o themselves_o at_o â—heiâ—_n own_o fantasy_n god_n help_v god_n help_v sure_o these_o man_n come_v to_o short_a and_o either_o be_v to_o dull_a or_o wilful_o blind_a or_o perhaps_o to_o malicious_a the_o three_o man_n be_v the_o common_a and_o vulgar_a astrologien_n or_o practiser_n who_o be_v not_o due_o artificial_o and_o perfect_o furnish_v yet_o either_o for_o vain_a glory_n or_o gain_v or_o like_o a_o simple_a dolt_n &_o blind_a bayardâ—_n both_o in_o matter_n and_o manner_n err_v to_o the_o discredit_n of_o the_o wary_a and_o modest_a astrologien_n and_o to_o the_o rob_n of_o those_o most_o noble_a corporal_a
be_v wood_n copper_n tin_n lead_v silver_n etc_n etc_n be_v as_o i_o say_v of_o like_a nature_n condition_n and_o like_o weight_n throughout_o and_o you_o may_v by_o say_v balance_n have_v prepare_v a_o great_a number_n of_o the_o small_a weight_n which_o by_o those_o balance_n can_v be_v discern_v or_o try_v and_o so_o have_v proceed_v to_o make_v you_o a_o perfect_a pyle_z company_n &_o number_n of_o weight_n to_o the_o weight_n of_o six_o eight_o or_o twelve_o pound_n weight_n most_o diligent_o try_v all_o and_o of_o every_o one_o the_o content_a know_v in_o your_o least_o weight_n that_o be_v wayable_a they_o that_o can_v not_o have_v these_o weight_n of_o preciseness_n may_v by_o sand_n uniform_a and_o well_o dust_v make_v they_o a_o number_n of_o weight_n somewhat_o never_o preciseness_n by_o half_v ever_o the_o sand_n they_o shall_v at_o length_n come_v to_o a_o least_o common_a weight_n therein_o i_o leave_v the_o far_a matter_n to_o their_o discretion_n who_o need_n shall_v pinch_n the_o venetian_n consideration_n of_o weight_n may_v seem_v precise_a enough_o d._n by_o eight_o descent_n progressionall_a grain_n half_v from_o a_o grain_n your_o cube_n sphere_n apt_a balance_n and_o convenient_a weight_n be_v ready_a fall_v to_o work_v â—_o first_o way_n your_o cube_n note_v the_o number_n of_o the_o weight_n way_n after_o that_o your_o sphere_n note_v likewise_o the_o number_n of_o the_o weight_n if_o you_o now_o find_v the_o weight_n of_o your_o cube_n to_o be_v to_o the_o weight_n of_o the_o sphere_n as_o 21._o be_v to_o 11_o then_o you_o see_v how_o the_o mechanicien_n and_o experimenter_n without_o geometry_n and_o demonstration_n be_v as_o near_o in_o effect_n teach_v the_o proportion_n of_o the_o cube_n to_o the_o sphere_n as_o i_o have_v demonstrate_v it_o in_o the_o end_n of_o the_o twelve_o book_n of_o euclid_n often_o try_v with_o the_o same_o cube_n and_o sphere_n then_o change_n your_o sphere_n and_o cube_n to_o a_o other_o matter_n or_o to_o a_o other_o bigness_n till_o you_o have_v make_v a_o perfect_a universal_a experience_n of_o it_o possible_a it_o be_v that_o you_o shall_v win_v to_o near_a term_n in_o the_o proportion_n when_o you_o have_v find_v this_o one_o certain_a drop_n of_o natural_a verity_n proceed_v on_o to_o infer_v and_o due_o to_o make_v assay_n of_o matter_n depend_v as_o because_o it_o be_v well_o demonstrate_v that_o a_o cylinder_n who_o heith_n and_o diameter_n of_o his_o base_a be_v equal_a to_o the_o diameter_n of_o the_o sphere_n be_v sesquialter_fw-la to_o the_o same_o sphere_n that_o be_v as_o 3._o to_o 2_o to_o the_o number_n of_o the_o weight_n of_o the_o sphere_n add_v half_o so_o much_o as_o it_o be_v and_o so_o have_v you_o the_o number_n of_o the_o weight_n of_o that_o cylinder_n which_o be_v also_o comprehend_v of_o our_o former_a cube_n so_o that_o the_o base_a of_o that_o cylinder_n be_v a_o circle_n describe_v in_o the_o square_a which_o be_v the_o base_a of_o our_o cube_n but_o the_o cube_n and_o the_o cylinder_n be_v both_o of_o one_o heith_n have_v their_o base_n in_o the_o same_o proportion_n in_o the_o which_o they_o be_v one_o to_o a_o other_o in_o their_o massine_n or_o solidity_n but_o before_o we_o have_v two_o number_n express_v their_o massine_n solidity_n and_o quantity_n by_o weight_n wherefore_o we_o have_v inscribe_v the_o proportion_n of_o the_o square_n to_o the_o circle_n inscribe_v in_o the_o same_o square_n and_o so_o be_v we_o fall_v into_o the_o knowledge_n sensible_a and_o experimental_a of_o archimedes_n great_a secret_n of_o he_o by_o great_a travail_n of_o mind_n seek_v and_o find_v wherefore_o to_o any_o circle_n give_v you_o can_v give_v a_o square_n equal_a mechanical_o as_o i_o have_v teach_v in_o my_o annotation_n upon_o the_o first_o proposition_n of_o the_o twelve_o book_n and_o likewise_o to_o any_o square_n give_v you_o may_v give_v a_o circle_n equal_a 〈…〉_o if_o you_o describe_v a_o circle_n which_o shall_v be_v in_o that_o proportion_n to_o your_o circle_n inscribe_v as_o the_o square_n be_v to_o the_o same_o circle_n this_o you_o may_v do_v by_o my_o annotation_n upon_o the_o second_o proposition_n of_o the_o twelve_o book_n of_o euclid_n in_o my_o three_o problem_n there_o your_o diligence_n may_v come_v to_o a_o proportion_n of_o the_o square_n to_o the_o circle_n inscribe_v near_o the_o truth_n then_o be_v the_o proportion_n of_o 14._o to_o 11._o and_o consider_v that_o you_o may_v begin_v at_o the_o circle_n and_o square_n and_o so_o come_v to_o conclude_v of_o the_o sphere_n &_o the_o cube_n what_o their_o proportion_n be_v as_o now_o you_o come_v from_o the_o sphere_n to_o the_o circle_n for_o of_o silver_n or_o gold_n or_o latton_n lamyns_n or_o plate_n through_o one_o hole_n draw_v as_o the_o manner_n be_v if_o you_o make_v a_o square_a figureâ—_n &_o way_n it_o and_o then_o describe_v thereon_o the_o circle_n inscribe_v &_o cut_v of_o &_o file_v away_o precise_o to_o the_o circle_n the_o overplus_n of_o the_o square_n you_o shall_v then_o weigh_v your_o circle_n see_v whether_o the_o weight_n of_o the_o square_n be_v to_o your_o circle_n as_o 14._o to_o 11._o as_o i_o have_v note_a in_o the_o begin_n of_o euclides_n twelve_o book_n etc_n etc_n after_o this_o resort_n to_o my_o last_o proposition_n upon_o the_o last_o of_o the_o twelve_o and_o there_o help_v yourself_o to_o the_o end_n and_o here_o note_v this_o diameter_n by_o the_o way_n that_o we_o may_v square_a the_o circle_n without_o have_v knowledge_n of_o the_o proportion_n of_o the_o circumference_n to_o the_o diameter_n as_o you_o have_v here_o perceive_v and_o otherways_o also_o i_o can_v demonstrate_v it_o so_o that_o many_o have_v cumberd_v themselves_o superfluous_o by_o travail_v in_o that_o point_n first_o which_o be_v not_o of_o necessity_n first_o and_o also_o very_o intricate_a and_o easy_o you_o may_v and_o that_o diverse_o come_v to_o the_o knowledge_n of_o the_o circumference_n the_o circle_n quantity_n be_v first_o know_v which_o thing_n i_o leave_v to_o your_o consideration_n make_v haste_n to_o dispatch_v a_o other_o magistral_a problem_n and_o to_o bring_v it_o near_o to_o your_o knowledge_n and_o ready_a deal_n with_o than_o the_o world_n before_o this_o day_n have_v it_o for_o you_o that_o i_o can_v tell_v of_o and_o that_o be_v a_o mechanical_a double_a of_o the_o cube_n etc_n etc_n mathematical_a which_o may_v thus_o be_v do_v make_v of_o copper_n plate_n or_o tyn_n plate_n a_o foursquare_n upright_o pyramid_n or_o a_o cone_n perfect_o fashion_v in_o the_o hollow_a within_o wherein_o let_v great_a diligence_n be_v use_v to_o approach_v as_o never_o as_o may_v be_v to_o the_o mathematical_a perfection_n of_o those_o figure_n at_o their_o base_n let_v they_o be_v all_o open_a every_o where_o else_o most_o close_o and_o just_a to_o from_o the_o vertex_fw-la to_o the_o circumference_n of_o the_o base_a of_o the_o cone_n &_o to_o the_o side_n of_o the_o base_a of_o the_o pyramid_n let_v 4._o straight_a line_n be_v draw_v in_o the_o inside_n of_o the_o cone_n and_o pyramid_n squall_v make_v at_o their_o fall_n on_o the_o perimeter_n of_o the_o base_n equal_a angle_n on_o both_o side_n themselves_o with_o the_o say_a perimeter_n these_o 4._o line_n in_o the_o pyramid_n and_o as_o many_o in_o the_o cone_n divide_v one_o in_o 12._o equal_a part_n and_o a_o other_o in_o 24._o a_o other_o in_o 60_o and_o a_o other_o in_o 100_o reckon_v up_o from_o the_o vertex_fw-la or_o use_v other_o number_n of_o division_n d._n as_o experience_n shall_v reach_v youâ—_n then_o 〈◊〉_d set_v your_o cone_n or_o pyramid_n with_o the_o vertex_fw-la downward_o perpendicular_o in_o respect_n of_o the_o base_a though_o it_o be_v otherways_o it_o hinder_v nothing_o so_o let_v they_o most_o steady_o be_v stay_v now_o if_o there_o be_v a_o cube_n which_o you_o will_v have_v double_v make_v you_o a_o pretty_a cube_n of_o copper_n silver_n lead_v tin_n wood_n stone_n or_o bone._n or_o else_o make_v a_o hollow_a cube_n or_o cubiâ—_n coffin_n of_o copper_n silver_n tin_n or_o wood_n etc_n etc_n these_o you_o may_v so_o proportion_v in_o respect_n of_o your_o pyramid_n or_o cone_n that_o the_o pyramid_n or_o cone_n will_v be_v able_a to_o contain_v the_o weight_n of_o they_o in_o waâ—â—_n 3._o or_o 4._o time_n at_o the_o least_o what_o stuff_n so_o ever_o they_o be_v make_v ofâ—_n let_v not_o your_o solid_a angle_n at_o the_o vertex_fw-la be_v to_o sharp_a but_o that_o the_o water_n may_v come_v with_o ease_n to_o the_o very_a vertex_fw-la of_o your_o hollow_a cone_n or_o pyramid_n put_v one_o of_o your_o solid_a cube_n in_o a_o balance_n apt_a d._n take_v the_o weight_n thereof_o exact_o in_o water_n pour_v that_o water_n without_o loss_n into_o the_o hollow_a pyramid_n or_o cone_n quiet_o
be_v câ—_n demonstration_n now_o be_v the_o parallelogram_n ae_n equal_a to_o the_o parallelogram_n of_o and_o ce_fw-fr by_o the_o first_o of_o this_o book_n but_o ae_n be_v the_o square_n make_v of_o ab_fw-la and_o of_o be_v the_o rectangle_n parallelogram_n comprehend_v under_o the_o line_n basilius_n and_o ac_fw-la for_o it_o be_v comprehend_v under_o the_o line_n dam_n and_o ac_fw-la but_o the_o line_n ad_fw-la be_v equal_a unto_o the_o line_n ab_fw-la and_o likewise_o the_o parrallelogramme_n ce_fw-fr be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o for_o the_o line_n be_v be_v equal_a unto_o the_o line_n ab_fw-la wherefore_o that_o which_o be_v contain_v under_o basilius_n and_o ac_fw-la together_o with_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o square_n make_v of_o the_o line_n ab_fw-la if_o therefore_o a_o right_a line_n be_v divide_v by_o chance_n the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o whole_a and_o every_o one_o of_o the_o part_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o whole_a which_o be_v require_v to_o be_v demonstrate_v an_o other_o demonstration_n of_o campane_n suppose_v that_o the_o line_n ab_fw-la be_v divide_v into_o the_o line_n ac_fw-la cd_o and_o db._n then_o i_o say_v that_o the_o square_a of_o the_o whole_a line_n ab_fw-la which_o let_v be_v aebf_n be_v equal_a to_o the_o rectangle_n figure_n which_o be_v contain_v under_o the_o whole_a and_o every_o one_o of_o the_o part_n for_o take_v the_o line_n king_n which_o let_v be_v equal_a to_o the_o line_n a_o b._n now_o then_o by_o the_o first_o proposition_n the_o rectangle_n figure_n contain_v under_o the_o line_n ab_fw-la and_o king_n be_v equal_a to_o the_o rectangle_n figure_v contain_v under_o the_o line_n king_n and_o all_o the_o part_n of_o the_o line_n ab_fw-la but_o that_o which_o be_v contain_v under_o the_o line_n king_n and_o ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o the_o rectangle_n figure_v contain_v under_o the_o line_n king_n and_o all_o the_o part_n of_o ab_fw-la be_v equal_a to_o the_o rectangle_n figure_v contain_v under_o the_o line_n ab_fw-la and_o all_o the_o part_n of_o the_o line_n ab_fw-la for_o the_o line_n ab_fw-la and_o king_n be_v equal_a wherefore_o that_o be_v manifest_v which_o be_v require_v to_o be_v prove_v a_o example_n of_o this_o proposition_n in_o number_n take_v a_o number_n as_o 11._o and_o divide_v it_o into_o two_o part_n namely_o 7._o and_o 4_o and_o multiply_v 11._o into_o 7_o and_o then_o into_o 4._o and_o there_o shall_v be_v produce_v 77._o and_o 44_o both_o which_o number_n add_v together_o make_v 121._o which_o be_v equal_a to_o the_o square_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n 11._o into_o himself_o as_o you_o see_v in_o the_o example_n the_o demonstration_n whereof_o follow_v in_o barlaam_n the_o second_o proposition_n barlaam_n if_o a_o number_n give_v be_v divide_v into_o two_o other_o number_n the_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o whole_a into_o either_o part_n add_v together_o be_v equal_a to_o the_o square_a number_n of_o the_o whole_a number_n give_v suppose_v that_o the_o number_n give_v by_o ab_fw-la and_o let_v it_o be_v divide_v into_o two_o other_o number_n ac_fw-la and_o cb._n then_o i_o say_v that_o the_o two_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o ab_fw-la into_o ac_fw-la and_o of_o ab_fw-la into_o bc_o those_o two_o superficial_a number_n i_o say_v be_v add_v together_o shall_v be_v equal_a to_o the_o square_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n ab_fw-la into_o itself_o for_o let_v the_o number_n ab_fw-la multiply_v itself_o produce_v the_o number_n d._n let_v the_o number_n ac_fw-la also_o multiply_v the_o number_n ab_fw-la produce_v the_o number_n of_o again_o let_v the_o number_n cb_o multipli_v the_o self_n same_o number_n ab_fw-la produce_v the_o number_n fg._n now_o forasmuch_o as_o the_o number_n ac_fw-la multiply_v the_o number_n ab_fw-la produce_v the_o number_n of_o therefore_o the_o number_n ab_fw-la measure_v the_o number_n of_o by_o the_o unity_n which_o be_v in_o ac_fw-la again_o forasmuch_o as_o the_o number_n cb_o multiply_v the_o number_n ab_fw-la and_o produce_v the_o number_n fg_o therefore_o the_o number_n ab_fw-la measure_v the_o number_n fg_o by_o the_o unity_n which_o be_v in_o the_o number_n cb._n but_o the_o same_o number_n ab_fw-la before_o measure_a the_o number_n of_o by_o the_o unity_n which_o be_v in_o the_o number_n ac_fw-la wherefore_o the_o number_n ab_fw-la measure_v the_o whole_a number_n â—g_z by_o the_o unity_n which_o be_v in_o ab_fw-la far_o forasmuch_o as_o the_o number_n ab_fw-la multiply_v itself_o produce_v the_o number_n d_o therefore_o the_o number_n ab_fw-la measure_v the_o number_n d_o by_o the_o unity_n which_o be_v in_o himself_o wherefore_o it_o measure_v either_o of_o these_o number_n namely_o the_o number_n d_o and_o the_o number_n eglantine_n by_o the_o unity_n which_o be_v in_o himself_o wherefore_o how_o multiplex_n the_o number_n d_o be_v to_o the_o number_n ab_fw-la so_o multiplex_n be_v the_o number_n eglantine_n to_o the_o same_o number_n ab_fw-la but_o number_n which_o be_v equemultiplices_fw-la to_o one_o and_o the_o self_n same_o numberâ—_n be_v equal_a the_o one_o to_o the_o other_o wherefore_o the_o number_n d_o be_v equal_a to_o the_o number_n eglantine_n and_o the_o number_n d_o be_v the_o square_a number_n make_v of_o the_o number_n ab_fw-la and_o the_o number_n eglantine_n be_v compose_v of_o the_o two_o superficial_a number_n produce_v of_o ab_fw-la into_o bc_o and_o of_o basilius_n into_o ac_fw-la wherefore_o the_o square_a number_n produce_v of_o the_o number_n ab_fw-la be_v equal_a to_o the_o superficial_a number_n produce_v of_o the_o number_n ab_fw-la into_o the_o number_n bc_o and_o of_o ab_fw-la into_o ac_fw-la add_v together_o if_o therefore_o a_o number_n be_v divide_v into_o two_o other_o number_n etc_n etc_n which_o be_v require_v to_o be_v prove_v the_o 3._o theorem_a the_o 3._o proposition_n if_o a_o right_a line_n be_v divide_v by_o chance_n the_o rectangle_n figure_n comprehend_v under_o the_o whole_a and_o one_o of_o the_o part_n be_v equal_a to_o the_o rectangle_n figure_n comprehend_v under_o the_o part_n &_o unto_o the_o square_n which_o be_v make_v of_o the_o foresay_a part_n svppose_v that_o the_o right_a line_n give_v ab_fw-la be_v divide_v by_o chance_n in_o the_o point_n c._n then_o i_o say_v that_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a unto_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n ac_fw-la and_o cb_o and_o also_o unto_o the_o square_n which_o be_v make_v of_o the_o line_n bc._n construction_n describe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n be_v square_a cdeb_n and_o by_o the_o second_o petition_n extend_v ed_z unto_o f._n and_o by_o the_o point_n a_o draw_v by_o the_o 31._o of_o the_o first_o a_o line_n parallel_n unto_o either_o of_o these_o line_n cd_o and_o be_v and_o let_v the_o same_o be_v af._n now_o the_o parallelogram_n ae_n be_v equal_a unto_o the_o parallelogram_n ad_fw-la and_o ce._n demonstration_n and_o ae_n be_v the_o rectangle_n figure_n comprehend_v under_o the_o line_n ab_fw-la and_o bc._n for_o it_o be_v comprehend_v under_o the_o line_n ab_fw-la and_o be_v which_o line_n be_v be_v equal_a unto_o the_o line_n bc._n and_o the_o parallelogram_n ad_fw-la be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o for_o the_o line_n dc_o be_v equal_a unto_o the_o line_n cb_o and_o db_o be_v the_o square_n which_o be_v make_v of_o the_o line_n cb._n wherefore_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n ac_fw-la and_o cb_o &_o also_o unto_o the_o square_n which_o be_v make_v of_o the_o line_n bc._n if_o therefore_o a_o right_a line_n be_v divide_v by_o chance_n the_o rectangle_n figure_n comprehend_v under_o the_o whole_a and_o one_o of_o the_o part_n be_v equal_a to_o the_o rectangle_n figure_n comprehend_v under_o the_o part_n and_o unto_o the_o square_n which_o be_v make_v of_o the_o foresay_a part_n which_o be_v require_v to_o be_v prove_v a_o example_n of_o this_o proposition_n in_o number_n suppose_v a_o number_n namely_o 14._o to_o be_v divide_v into_o two_o part_n 8._o and_o 6._o the_o whole_a number_n 14._o multiply_v into_o 8._o one_o of_o his_o part_n produce_v 112_o the_o part_n 8._o &_o 6._o multiply_v the_o one_o into_o the_o other_o produce_v 48_o which_o add_v to_o 64_o which_o be_v the_o square_a of_o 8._o the_o
poligonon_n figure_n of_o 24._o side_n likewise_o of_o the_o hexagon_n ab_fw-la and_o of_o the_o pentagon_n ac_fw-la shall_v be_v make_v a_o poligonon_n figure_n of_o 30._o side_n one_o of_o who_o side_n shall_v subtend_v the_o ark_n bc._n for_o the_o denomination_n of_o ab_fw-la which_o be_v 6._o exceed_v the_o denomination_n of_o ac_fw-la which_o be_v 5._o only_a by_o unity_n so_o also_o forasmuch_o as_o the_o denomination_n of_o ab_fw-la which_o be_v 6._o exceed_v the_o denomination_n of_o ae_n which_o be_v 3._o by_o 3._o therefore_o the_o ark_n be_v shall_v contain_v 3._o side_n of_o a_o poligonon_n figure_n of_o .18_o side_n and_o observe_v this_o self_n same_o method_n and_o order_n a_o man_n may_v find_v out_o infinite_a side_n of_o a_o poligonon_n figure_n the_o end_n of_o the_o four_o book_n of_o euclides_n element_n ¶_o the_o five_o book_n of_o euclides_n element_n this_o five_o book_n of_o euclid_n be_v of_o very_o great_a commodity_n and_o use_n in_o all_o geometry_n book_n 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proportion_n of_o equality_n and_o into_o proportion_n of_o inequality_n proportion_n of_o equality_n be_v when_o one_o quantity_n be_v refer_v to_o a_o other_o equal_a unto_o itself_o as_o if_o you_o compare_v 5_o to_o 5_o or_o 7_o to_o 7_o &_o so_o of_o other_o and_o this_o proportion_n have_v great_a use_n in_o the_o rule_n of_o cosse_n equality_n for_o in_o it_o all_o the_o rule_n of_o equation_n tend_v to_o none_o other_o end_n but_o to_o find_v out_o and_o bring_v forth_o a_o number_n equal_a to_o the_o number_n suppose_v which_o be_v to_o put_v the_o proportion_n of_o equality_n inequality_n proportion_n of_o inequality_n be_v when_o one_o unequal_a quantity_n be_v compare_v to_o a_o other_o as_o the_o great_a to_o the_o less_o as_o 8._o to_o 4_o or_o 9_o to_o 3_o or_o the_o less_o to_o the_o great_a as_o 4._o to_o 8_o or_o 3._o to_o 9_o less_o proportion_n of_o the_o great_a to_o the_o less_o have_v five_o kind_n namely_o multiplex_n superparticular_a superpartiens_fw-la multiplex_fw-la superperticular_a and_o multiplex_fw-la superpartiens_fw-la multiplex_fw-la be_v when_o the_o antecedent_n contain_v in_o itself_o the_o consequent_a certain_a time_n without_o more_o or_o less_o multiplex_fw-la as_o twice_o thrice_o four_o time_n and_o so_o far_o and_o this_o proportion_n have_v under_o it_o infinite_a kind_n for_o if_o the_o antecedent_n contain_v the_o consequent_a just_o twice_o it_o be_v call_v dupla_fw-la proportion_n proportion_n as_o 4_o to_o 2._o if_o thrice_o tripla_fw-la quintuple_a as_o 9_o to_o 3._o if_o 4._o time_n quadrupla_fw-la as_o 12._o to_o 3._o if_o 5._o time_n quintupla_fw-la as_o 15._o to_o 3._o and_o so_o infinite_o after_o the_o same_o manner_n superperticular_a superperticular_a be_v when_o the_o antecedent_n contain_v the_o consequent_a only_o once_o &_o moreover_o some_o one_o part_n thereof_o as_o a_o half_a a_o three_o or_o four_o etc_n etc_n this_o kind_n also_o have_v under_o it_o infinite_a kind_n for_o if_o the_o antecedent_n contain_v the_o consequent_a once_o and_o a_o half_a thereof_o it_o be_v call_v sesquialtera_fw-la sesquialtera_fw-la as_o 6._o to_o 4_o if_o once_o and_o a_o three_o part_n sesquitertia_fw-la sesquitertia_fw-la as_o 4._o to_o 3_o if_o once_o and_o a_o four_o part_n sesquiquarta_fw-la sesquiquarta_fw-la as_o 5._o to_o 4._o and_o so_o in_o like_a manner_n infinite_o superpartiens_fw-la superpartiens_fw-la be_v when_o the_o antecedent_n contain_v the_o consequent_a only_o once_o &_o moreover_o more_o part_n than_o one_o of_o the_o same_o as_o two_o three_o three_o fourthe_n four_o fifthe_n and_o so_o forth_o this_o also_o have_v infinite_a kind_n under_o it_o for_o if_o the_o antecedent_n contain_v above_o the_o consequent_a two_o part_n it_o be_v call_v superbipartiens_fw-la superbipartiens_fw-la as_o 7._o to_o 5._o if_o 3._o part_n supertripartiens_fw-la as_o 7._o to_o 4._o supertripartiens_fw-la if_o 4._o part_n superquadripartiens_n superquadripartiens_fw-la as_o 9_o to_o 5._o if_o 5._o part_n superquintipartiens_n as_o 11._o to_o 6._o superquintipartiens_fw-la and_o so_o forth_o infinite_o superperticular_a multiplex_fw-la superperticular_a be_v when_o the_o antecedent_n contain_v the_o consequent_a more_o than_o once_o and_o moreover_o only_o one_o part_n of_o the_o same_o this_o kind_n likewise_o have_v infinite_a kind_n under_o it_o for_o if_o the_o antecedent_n contain_v the_o consequent_a twice_o and_o half_o thereof_o it_o be_v call_v dupla_fw-la sesquialtera_fw-la sesquialtera_fw-la as_o 5._o to_o 2._o if_o twice_o and_o a_o three_o dupla_fw-la sesquitertia_fw-la as_o 7._o to_o 3._o sesquitertia_fw-la if_o thrice_o and_o a_o half_a tripla_fw-la sesquialtera_fw-la as_o 7._o to_o 2._o sesquialtera_fw-la if_o four_o time_n and_o a_o half_a quadrâ—pla_fw-la sesquialtera_fw-la as_o 9_o to_o 2._o and_o so_o go_v on_o infinite_o superpartiens_fw-la multiplex_fw-la superpartient_fw-fr be_v when_o the_o antecedent_n contain_v the_o consequent_a 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denomination_n of_o the_o proportion_n proportion_n which_o in_o the_o first_o kind_n of_o proportion_n namely_o multiplex_n be_v ever_o a_o whole_a number_n and_o in_o all_o other_o kind_n of_o proportion_n it_o be_v a_o break_a number_n as_o if_o you_o will_v know_v the_o denomination_n of_o the_o proportion_n between_o 9_o and_o 3._o divide_v 9_o by_o 3._o so_o shall_v you_o have_v in_o the_o quotient_a 3._o which_o be_v a_o whole_a number_n and_o be_v the_o denomination_n of_o the_o proportion_n and_o show_v that_o the_o proportion_n between_o 9_o &_o 3._o be_v tripla_fw-la so_o the_o proportion_n between_o 12._o and_o 3._o be_v quadrupla_fw-la for_o that_o 12._o be_v divide_v by_o 3._o the_o quotient_n be_v 4._o and_o so_o of_o other_o in_o the_o kind_n of_o multiplex_n and_o although_o in_o this_o kind_a the_o quotient_n be_v ever_o a_o whole_a number_n yet_o proper_o it_o be_v refer_v to_o unity_n and_o so_o be_v represent_v in_o manner_n of_o a_o break_a number_n as_o 5_o â—_o and_o 4_o â—_o for_o unity_n be_v the_o denomination_n to_o a_o whole_a number_n likewise_o the_o denomination_n of_o the_o proportion_n between_o 4_o and_o 3_o be_v 1._o 1_o â—_o for_o that_o 4_o divide_v by_o 3._o have_v in_o the_o quotient_a 1_o 1_o â—_o one_o and_o a_o three_o part_n of_o which_o three_o part_n it_o be_v call_v sesquitercia_fw-la so_o the_o proportion_n between_o 7_o and_o 6._o be_v 1_o â…™_n one_o and_o a_o six_o of_o which_o fix_v part_v it_o be_v call_v sesquisexta_fw-la and_o so_o of_o other_o of_o that_o kind_n also_o between_o 7_o and_o 5_o the_o denomination_n of_o the_o proportion_n be_v 1_o â—_o â—_o one_o and_o two_o fifthe_n which_o denomination_n consist_v of_o two_o part_n namely_o of_o the_o munerator_n and_o denominator_fw-la of_o the_o quotient_a of_o 2._o and_o 5_o of_o which_o two_o fifthe_n it_o be_v call_v superbipartiens_fw-la quintas_fw-la for_o 2_o the_o numerator_n show_v the_o denomination_n of_o the_o number_n of_o the_o part_n and_o 5._o the_o denominator_fw-la show_v the_o denomination_n what_o part_n they_o be_v &_o so_o of_o other_o also_o the_o denomination_n between_o
bc_o be_v to_o ca_n so_o be_v ca_n to_o cd_o the_o 1._o problem_n the_o 9_o proposition_n a_o right_a line_n be_v give_v to_o cut_v of_o from_o it_o any_o part_n appoint_v let_v the_o right_a line_n give_v be_v ab_fw-la it_o be_v require_v that_o from_o the_o same_o line_n ab_fw-la be_v cut_v of_o any_o part_n appoint_v suppose_v that_o a_o thyrd_o part_n be_v appoint_v to_o be_v cut_v of_o construction_n from_o the_o point_fw-fr a_o draw_v a_o right_a line_n ac_fw-la make_v with_o the_o line_n ab_fw-la a_o angle_n and_o in_o the_o line_n ac_fw-la take_v a_o point_n at_o all_o adventure_n which_o let_v be_v d._n and_o beginning_n at_o d_o put_v unto_o ad_fw-la two_o equal_a line_n de_fw-fr &_o ec_o by_o the_o 2._o of_o the_o first_o and_o draw_v a_o right_a line_n from_o b_o to_o c_z and_o by_o the_o point_n d_o by_o the_o â—1_n of_o the_o first_o draw_v unto_o bc_o a_o parallel_a line_n df._n demonstration_n now_o forasmuch_o as_o unto_o one_o of_o y_fw-fr side_n of_o the_o triangle_n abc_n namely_o unto_o the_o side_n bc_o be_v draw_v a_o parallel_a line_n fd_o it_o follow_v by_o the_o 2._o of_o this_o book_n that_o as_o cd_o be_v in_o proportion_n unto_o dam_n so_o be_v bf_o to_o fa._n but_o by_o construction_n cd_o be_v double_a to_o da._n wherefore_o y_fw-fr line_n bf_n be_v also_o double_a to_o the_o line_n fa._n wherefore_o the_o line_n basilius_n be_v treble_a unto_o the_o line_n af._n wherefore_o from_o the_o right_a lineâ—_z give_v ab_fw-la be_v cut_v of_o a_o three_o part_n appoint_v namely_o of_o which_o be_v require_v to_o be_v do_v the_o 2._o problem_n the_o 10._o proposition_n to_o divide_v a_o right_a line_n give_v not_o divide_v like_a unto_o a_o right_a line_n give_v be_v divide_v svppose_v that_o the_o right_a line_n give_v not_o divide_v be_v ab_fw-la and_o the_o right_a line_n give_v be_v divide_v let_v be_v ac_fw-la it_o be_v require_v to_o divide_v the_o line_n ab_fw-la which_o be_v not_o divide_v like_o unto_o the_o line_n ac_fw-la which_o be_v divide_v construction_n suppose_v the_o line_n ac_fw-la be_v divide_v in_o the_o point_n d_o and_o e_o &_o let_v you_o line_n ab_fw-la &_o ac_fw-la so_o be_v put_v that_o they_o make_v a_o angle_n at_o all_o adventure_n and_o draw_v a_o line_n from_o b_o to_o c_z and_o by_o the_o point_n d_o and_o e_o draw_v unto_o the_o line_n bc_o by_o the_o 31._o of_o the_o first_o two_z parallel_v line_n df_o and_o eglantine_n and_o by_o the_o point_n d_o unto_o the_o line_n ab_fw-la by_o the_o same_o draw_v a_o parallel_n line_n dhk_n wherefore_o either_o of_o these_o figureâ—_n fh_o and_o hb_o be_v parallelogram_n wherefore_o the_o line_n dh_o be_v equal_a unto_o the_o line_n fg_o and_o the_o line_n hk_o be_v equal_a unto_o the_o line_n gb_o demonstration_n and_o because_o to_o one_o of_o the_o side_n of_o the_o triangle_n dkc_n namely_o to_o the_o side_n kc_o be_v draw_v a_o parallel_a line_n he_o therefore_o the_o line_n ce_fw-fr by_o thâ—_z 2._o of_o the_o six_o be_v in_o proportion_n unto_o the_o line_n ed_z as_o the_o line_n kh_o be_v to_o the_o line_n hd_v but_o the_o line_n kh_o be_v equal_a unto_o the_o line_n bg_o and_o the_o line_n hd_v be_v equal_a unto_o the_o line_n gf_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o ce_fw-fr be_v unto_o ed_z so_o be_v bg_o to_o gf_o again_o because_o to_o one_o of_o the_o side_n of_o the_o triangle_n age_n namely_o to_o ge_z be_v draw_v a_o parallel_a line_n fd_o therefore_o the_o line_n ed_z by_o the_o 2._o of_o the_o six_o be_v in_o proportion_n unto_o the_o line_n dam_fw-ge as_o the_o line_n gf_o be_v to_o the_o line_n fa._n and_o it_o be_v already_o prove_v that_o as_o ce_fw-fr be_v to_o ed_z so_o be_v bg_o to_o gf_o wherefore_o as_o ce_fw-fr be_v to_o ed_z so_o be_v bg_o to_o gf_o and_o as_z ed_z be_v to_o da_z so_o be_v gf_o to_o fa._n wherefore_o the_o right_a line_n give_v not_o divide_v namely_o ab_fw-la be_v divide_v like_o unto_o the_o right_a line_n give_v be_v divide_v which_o be_v ac_fw-la which_o be_v require_v to_o be_v do_v ¶_o a_o corollary_n out_o of_o flussates_n flussates_n by_o this_o proposition_n we_o may_v divide_v any_o right_a line_n give_v accord_v to_o the_o proportion_n of_o any_o right_a line_n give_v for_o let_v those_o right_a line_n have_v proportion_n be_v join_v together_o direct_o that_o they_o may_v make_v all_o one_o right_a line_n and_o then_o join_v they_o to_o the_o line_n give_v anglewise_o and_o so_o proceed_v as_o in_o the_o proposition_n where_o you_o see_v that_o the_o right_a line_n give_v ab_fw-la be_v divide_v into_o the_o right_a line_n of_o fg_o and_o gb_o which_o have_v the_o self_n same_o proportion_n that_o the_o right_a line_n ad_fw-la de_fw-fr and_o ec_o have_v by_o this_o and_o the_o former_a proposition_n also_o may_v a_o right_a line_n give_v be_v easy_o divide_v into_o what_o part_n so_o ever_o you_o will_v name_v will._n as_o if_o you_o will_v divide_v the_o line_n ab_fw-la into_o three_o equal_a part_n let_v the_o line_n de_fw-fr be_v make_v equal_a to_o the_o line_n ad_fw-la and_o the_o line_n ec_o make_v equal_a to_o the_o same_o by_o the_o three_o of_o the_o first_o and_o then_o use_v the_o self_n same_o manner_n of_o construction_n that_o be_v before_o the_o line_n ab_fw-la shall_v be_v divide_v into_o three_o equal_a part_n and_o so_o of_o any_o kind_n of_o part_n whatsoever_o the_o 3._o problem_n the_o 11._o proposition_n unto_o two_o right_a line_n give_v to_o find_v a_o three_o in_o proportion_n with_o they_o svppose_v that_o there_o be_v two_o right_a line_n give_v basilius_n and_o ac_fw-la and_o let_v they_o be_v so_o put_v that_o they_o comprehend_v a_o angle_n howsoever_o it_o be_v it_o be_v require_v to_o find_v unto_o basilius_n and_o unto_o ac_fw-la a_o three_o line_n in_o proportion_n construction_n produce_v the_o line_n ab_fw-la and_o ac_fw-la unto_o the_o point_n d_o and_o e._n and_o unto_o the_o line_n ac_fw-la by_o the_o 2._o of_o the_o first_o put_v a_o equal_a line_n bd_o and_o draw_v a_o line_n from_o b_o to_o c._n and_o by_o the_o point_n d_o by_o the_o 31._o of_o the_o first_o draw_v unto_o the_o line_n bc_o a_o parallel_a line_n de_fw-fr which_o let_v concur_v with_o the_o line_n ac_fw-la in_o the_o point_n e._n demonstration_n now_o forasmuch_o as_o unto_o one_o of_o the_o side_n of_o the_o triangle_n ade_n namely_o to_o de_fw-fr be_v draw_v a_o parallel_a line_n bc_o therefore_o as_o ab_fw-la be_v in_o proportion_n unto_o bd_o so_o by_o the_o 2._o of_o the_o six_o be_v ac_fw-la unto_o ce._n but_o the_o line_n bd_o be_v equal_a unto_o the_o line_n ac_fw-la wherefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n ac_fw-la so_o be_v the_o line_n ac_fw-la to_o the_o line_n ce._n wherefore_o unto_o the_o two_o right_a line_n give_v ab_fw-la and_o ac_fw-la be_v find_v a_o three_o line_n ce_fw-fr in_o proportion_n with_o they_o which_o be_v require_v to_o be_v do_v ¶_o a_o other_o way_n after_o pelitarius_n let_v the_o line_n ab_fw-la and_o bc_o be_v set_v direct_o in_o such_o sort_n that_o they_o both_o make_v one_o right_a line_n pelitarius_n then_o from_o the_o point_n a_o erect_v the_o line_n ad_fw-la make_v with_o the_o line_n ab_fw-la a_o angle_n at_o all_o adventure_n and_o put_v the_o line_n ad_fw-la equal_a to_o the_o line_n bc._n and_o draw_v a_o right_a line_n from_o d_z to_o b_o which_o produce_v beyond_o the_o point_n b_o unto_o the_o point_n e._n and_o by_o the_o point_n c_o draw_v unto_o the_o line_n da_z a_o parallel_a line_n ce_fw-fr concur_v with_o the_o line_n de_fw-fr in_o the_o point_n e._n then_o i_o say_v that_o the_o line_n ce_fw-fr be_v the_o three_o line_n proportional_a with_o the_o line_n ab_fw-la and_o bc._n for_o forâ—asmuch_o as_o by_o the_o 15._o of_o the_o first_o the_o angle_n b_o of_o the_o triangle_n abdella_n be_v equal_a to_o the_o angle_n b_o of_o the_o triangle_n cbe_n and_o by_o the_o 29._o of_o the_o same_o the_o angle_n a_o be_v equal_a to_o the_o angle_n c_o and_o the_o angle_n d_o to_o the_o angle_n e_o therefore_o by_o the_o 4._o of_o this_o book_n ab_fw-la be_v to_o dam_n as_o bc_o be_v to_o ce._n wherefore_o by_o the_o 11._o of_o the_o five_o ab_fw-la be_v to_o bc_o as_o bc_o be_v to_o ce_fw-fr which_o be_v require_v to_o be_v do_v ¶_o a_o other_o way_n also_o after_o pelitarius_n let_v the_o line_n ab_fw-la and_o bc_o be_v so_o join_v together_o pelitarius_n that_o they_o may_v make_v a_o right_a angle_n namely_o abc_n and_o draw_v a_o line_n from_o a_o to_o c_o and_o from_o the_o point_n c_o draw_v unto_o the_o line_n ac_fw-la a_o perpendicular_a cd_o by_o the_o 11._o of_o the_o first_o and_o
it_o a_o number_n equal_o equal_a equal_o because_o it_o be_v produce_v of_o the_o multiplication_n of_o three_o equal_a number_n the_o one_o into_o the_o other_o which_o three_o number_n be_v say_v in_o the_o second_o definition_n wherein_o he_o speak_v more_o apply_v to_o geometry_n to_o contain_v the_o cube_fw-la number_n it_o be_v therefore_o call_v a_o cube_fw-la number_n number_n because_o be_v describe_v by_o his_o unity_n it_o represent_v the_o form_n of_o a_o cube_fw-la in_o geometry_n who_o side_n that_o be_v to_o say_v who_o length_n breadth_n and_o thickness_n be_v the_o three_o equal_a number_n 9_o 9_z and_o 9_z of_o which_o he_o be_v produce_v which_o three_o side_n also_o be_v say_v to_o contain_v the_o cube_fw-la number_n 729_o behold_v here_o the_o description_n thereof_o definition_n 21_o number_n proportional_a be_v when_o the_o first_o be_v to_o the_o second_o equemultiplex_n as_o the_o three_o be_v to_o the_o four_o or_o the_o self_n same_o part_n or_o the_o self_n same_o part_n here_o he_o define_v which_o number_n be_v call_v proportional_a that_o be_v what_o number_n have_v one_o and_o the_o self_n same_o proportion_n for_o example_n 6._o to_o 3_o and_o 4._o to_o 2_o be_v number_n proportional_a and_o have_v one_o and_o thâ—_z self_n same_o proportion_n for_o 6._o the_o first_o be_v to_o 3._o the_o second_o equemultiplex_n as_o 4._o the_o three_o be_v to_o 2._o the_o four_o â—_o be_v double_a to_o 3_o and_o so_o be_v 4._o double_a to_o 2._o likewise_o these_o four_o number_n be_v in_o like_a proportion_n 3_o 9.4.1â—_n for_o what_o part_n 3._o be_v of_o 9_o such_o part_n be_v 4._o of_o 12_o â—3_o of_o 9_o be_v a_o three_o part_n so_o be_v also_o 4._o of_o 12._o a_o three_o part_n so_o be_v these_o four_o number_n also_o in_o proportion_n â—_o 5_o 4._o 10_o what_o part_n 2._o be_v of_o 5_o such_o part_n be_v 4._o of_o 10_o 2_o of_o 5_o be_v two_o â—ift_a part_n likewise_o 4._o of_o 10_o be_v two_o five_o part_n moreover_o these_o number_n 8.6_o 1â—_o 9_o be_v in_o proportion_n for_o what_o and_o how_o many_o part_n 8._o be_v of_o 6_o such_o &_o so_o many_o part_n be_v 12._o of_o 9_o 8._o of_o 6_o be_v four_o three_o part_n for_o one_o three_o part_n of_o 6._o be_v 2_o which_o take_v four_o time_n make_v 8_o so_o 12._o of_o 9_o be_v also_o four_o thyrd_o part_n for_o one_o three_o part_n of_o 9_o be_v 3_o which_o take_v four_o time_n make_v 1â—_o and_o so_o conceive_v you_o of_o all_o other_o proportional_a number_n in_o the_o 〈◊〉_d definition_n of_o the_o v_o book_n euclid_n give_v a_o â—arre_n other_o definition_n of_o magnitude_n proportional_a and_o much_o unlike_a to_o this_o which_o he_o here_o give_v of_o number_n proportional_a number_n the_o reason_n be_v as_o there_o also_o be_v partly_o note_v for_o that_o there_o he_o give_v a_o definition_n common_a to_o all_o quantity_n discrete_a and_o continual_a rational_a and_o irrational_a and_o therefore_o be_v constrain_v to_o geve_v the_o definition_n by_o the_o excess_n equality_n or_o want_n of_o their_o equemultiplices_fw-la and_o that_o general_o only_a for_o that_o irrational_a quantity_n have_v no_o certain_a part_n or_o common_a measure_n to_o be_v measure_v by_o or_o know_v neither_o can_v they_o be_v express_v by_o any_o certain_a number_n but_o here_o in_o this_o place_n because_o in_o number_n there_o be_v no_o irrational_a quantity_n but_o all_o be_v certain_o know_v so_o that_o both_o they_o and_o the_o proportion_n between_o they_o may_v be_v express_v by_o number_n certain_a and_o know_v by_o reason_n of_o their_o part_n certain_a and_o for_o that_o they_o have_v some_o common_a measure_n to_o measure_v they_o at_o the_o least_o unity_n which_o be_v a_o common_a measure_n to_o all_o number_n he_o give_v here_o this_o definition_n of_o proportional_a number_n by_o that_o the_o one_o be_v like_o equemultiplex_n to_o the_o other_o or_o the_o same_o part_n or_o the_o same_o part_n which_o definition_n be_v much_o easy_a than_o be_v the_o other_o and_o be_v not_o so_o large_a as_o be_v the_o other_o neither_o extend_v itself_o general_o to_o all_o kind_n of_o quantity_n rational_a and_o irrational_a but_o contain_v itself_o within_o the_o limit_n and_o bond_n of_o rational_a quantity_n and_o number_n 22_o like_o plain_a number_n and_o like_o solid_a number_n be_v such_o definition_n which_o have_v their_o side_n proportional_a before_o he_o show_v that_o a_o plain_a number_n have_v two_o side_n and_o a_o solid_a number_n three_o side_n now_o he_o show_v by_o this_o definition_n which_o plain_a number_n and_o which_o solid_a number_n be_v like_o the_o one_o to_o the_o other_o the_o likeness_n of_o which_o number_n depend_v altogether_o of_o the_o proportion_n of_o the_o side_n of_o these_o number_n so_o that_o if_o the_o two_o side_n of_o one_o plain_a number_n have_v the_o same_o proportion_n the_o one_o to_o the_o other_o that_o the_o two_o side_n of_o the_o other_o plain_a number_n have_v the_o one_o to_o the_o other_o then_o be_v such_o two_o plain_a number_n like_o for_o a_o example_n 6_o and_o 24_o be_v two_o plain_a number_n the_o side_n of_o 6_o be_v 2_o and_o 3_o two_o time_n 3_o make_v 6_o the_o side_n of_o 24_o be_v 4_o and_o 6_o four_o time_n 6_o make_v 24._o again_o the_o same_o proportion_n that_o be_v between_o 3_o and_o 2_o the_o side_n of_o 6._o be_v also_o between_o 6_o and_o 4_o the_o side_n of_o 24._o wherefore_o 24_o and_o 6_o be_v two_o like_o plain_a and_o superficial_a number_n and_o so_o of_o other_o plain_a number_n after_o the_o same_o manner_n be_v it_o in_o solid_a number_n if_o three_o side_n of_o the_o one_o be_v in_o like_a proportion_n together_o as_o be_v the_o three_o side_n of_o the_o other_o then_o be_v the_o one_o solid_a number_n like_o to_o the_o other_o as_o 24_o and_o 192_o be_v solid_a number_n the_o side_n of_o 24._o be_v 2._o 3._o and_o 4_o two_o time_n there_o take_v 4_o time_n be_v 24._o the_o side_n of_o 192_o be_v 4.6_o and_o 8_o for_o four_o time_n 6._o 8_o time_n make_v 192._o again_o the_o proportion_n of_o 4_o to_o 3_o be_v sesquitercia_fw-la the_o proportion_n of_o 3_o to_o 2_o be_v sesquialtera_fw-la which_o be_v the_o proportion_n of_o the_o side_n of_o the_o one_o solid_a number_n namely_o of_o 24_o the_o proportion_n between_o 8_o and_o 6_o be_v sesquâ—â—ercia_n the_o proportion_n between_o 6_o and_o 4_o be_v sesquialtera_fw-la which_o be_v the_o proportion_n of_o the_o side_n of_o the_o other_o solid_a number_n namely_o of_o 191._o and_o they_o be_v one_o and_o the_o same_o with_o the_o proportione_n of_o the_o 〈◊〉_d of_o the_o other_o wherefore_o thâ—se_a two_z solid_a number_n 24_o and_o 192_o be_v like_a and_o so_o of_o other_o solid_a number_n 23_o a_o perfect_a number_n be_v that_o which_o be_v equal_a to_o all_o his_o part_n definition_n as_o the_o part_n of_o 6_o be_v 1._o 2._o 3._o three_o be_v the_o half_a of_o 6_o two_o the_o three_o part_n and_o 1._o the_o six_o part_n and_o more_o port_n 6_o have_v not_o which_o thrâ—â—_n payte_v 1._o â—_o 3_o add_v together_o make_v 6_o the_o whole_a number_n who_o part_n they_o be_v wherefore_o 6_o be_v a_o perfect_a number_n so_o likewise_o be_v 28_o a_o perfect_a number_n the_o part_n whereof_o be_v these_o number_n 14._o 7._o 2._o and_o 1_o 14_o be_v the_o half_a thereof_o 7_o be_v the_o quarter_n 4_o be_v the_o seven_o part_n 2_o be_v a_o fourteen_o part_n and_o 1_o a_o 28_o part_n and_o these_o be_v all_o the_o part_n of_o 28._o all_o which_o namely_o 1_o 2_o 3_o 4_o 7_o and_o 14_o add_v together_o make_v just_o without_o more_o or_o lâ—sse_v 28._o wherefore_o 〈…〉_o a_o perfect_a number_n and_o so_o of_o other_o the_o like_a this_o kind_n of_o number_n be_v very_o rare_a and_o seldom_o find_v philosophy_n from_o 1_o to_o 10_o there_o be_v but_o one_o perfect_a number_n namely_o 6._o from_o 10_o to_o a_o 100_o there_o be_v also_o but_o one_o that_o be_v 28._o also_o from_o 100_o to_o 1000_o there_o be_v but_o one_o which_o be_v 496._o from_o 1000_o to_o 10000_o likewise_o but_o one_o so_o that_o between_o every_o sâ—ay_n in_o number_v which_o be_v ever_o in_o the_o ten_o place_n there_o be_v find_v but_o one_o perfect_a number_n and_o for_o their_o rareness_n and_o great_a perfection_n they_o be_v of_o marvellous_a use_n in_o magic_a and_o in_o the_o secret_a part_n of_o philosophy_n this_o kind_n of_o number_n be_v call_v perfect_a perfect_a in_o respect_n 〈…〉_o number_n which_o be_v imperfect_a for_o as_o the_o nature_n of_o a_o perfect_a number_n stand_v in_o this_o that_o all_o part_n add_v together_o be_v equal_a to_o the_o whole_a and_o make_v the_o whole_a so_o in_o a_o imperfect_a number_n all_o
measure_v b_o by_o the_o unity_n which_o be_v in_o b._n wherefore_o unity_n e_o so_o many_o time_n measure_v the_o number_n b_o as_o a_o measure_v d._n but_o unity_n e_o so_fw-mi many_o time_n measure_v the_o number_n b_o as_o a_o measure_v c._n wherefore_o a_o measure_v either_o of_o these_o number_n c_o and_o d_o a_o like_a wherefore_o by_o the_o 3._o common_a sentence_n of_o this_o book_n c_z be_v equal_a unto_o d_o which_o be_v require_v to_o be_v demonstrate_v the_o 15._o theorem_a the_o 17._o proposition_n if_o one_o number_n multiply_v two_o number_n and_o produce_v other_o number_n the_o number_n produce_v of_o they_o shall_v be_v in_o the_o self_n same_o proportion_n that_o the_o number_n multiply_v be_v svppose_v that_o the_o number_n a_o multiply_v two_o number_n b_o and_o c_o do_v produce_v the_o number_n d_o and_o e._n then_o i_o say_v that_o as_z b_o be_v to_o c_o so_o be_v d_z to_o e._n take_v unity_n namely_o f._n and_o forasmuch_o as_o a_o multiply_v b_o produce_v d_z therefore_o b_o measure_v d_z by_o those_o unity_n that_o be_v in_o a._n and_o unity_n fletcher_n measure_v a_o by_o thâ—sâ—_n uâ—itiâ—â—_fw-fr whih_o be_v in_o a._n â—emonstraâ—ion_n wherefore_o unity_n fletcher_n so_o many_o time_n measure_v the_o number_n a_o as_z b_o measure_v d._n wherefore_o as_o unity_n i_o be_v to_o the_o number_n a._n so_o be_v the_o number_n b_o to_o the_o number_n d_o by_o the_o 21_o definition_n of_o this_o book_n and_o by_o the_o same_o reason_n as_o unity_n fletcher_n be_v to_o the_o number_n a_o so_o be_v the_o number_n c_o to_o the_o number_n e_o wherefore_o also_o by_o the_o 7._o common_a sentence_n of_o this_o book_n as_o b_o be_v to_o d_z so_o be_v c_o to_o e._n wherefore_o alternate_o by_o the_o 15._o of_o the_o seven_o as_o b_o be_v to_o c_o so_o be_v d_z to_o e._n if_o therefore_o one_o number_n multiply_v two_o numbersâ—_n and_o produce_v other_o number_n the_o number_n produce_v of_o they_o shall_v be_v in_o the_o self_n same_o proportion_n that_o the_o number_n multiply_v be_v which_o be_v require_v to_o be_v prove_v here_o fluâ—â—tes_n add_v thiâ—_z corollary_n if_o two_o numberâ—_n have_v one_o and_o the_o samâ—_n proportion_n with_o two_o other_o number_n do_v multiply_v thâ—_z oâ—e_v the_o other_o alternate_o flussâ—tes_n and_o produce_v any_o number_n the_o number_n produce_v of_o they_o shall_v be_v equal_a the_o one_o to_o the_o other_o suppose_v that_o there_o be_v two_o numberâ—_n â—_o and_o b_o and_o also_o two_o other_o number_n c_o and_o d_o have_v thâ—_z same_o proportion_n that_o the_o number_n a_o and_o b_o have_v and_o let_v the_o number_n a_o and_o b_o multiply_v the_o numberâ—_n c_o &_o d_o alternate_o that_o be_v let_v a_o multiply_v d_o produce_v f_o and_o let_v b_o multiply_v c_o produce_v e._n then_o i_o say_v that_o the_o number_n produce_v namely_o e_o &_o f_o be_v equal_a let_v a_o and_o b_o multiply_v the_o one_o the_o other_o in_o such_o sort_n that_o let_v a_o multiply_v b_o produce_v g_o and_o let_v b_o multiply_v a_o produce_v h_o now_o then_o the_o number_n g_o and_o h_o be_v equal_a by_o the_o 16._o of_o this_o bookeâ—_n and_o forasmuch_o as_o a_o multipliâ—ng_a the_o two_o number_n b_o and_o d_z produce_v the_o number_n g_o and_o f_o therefore_o g_z be_v to_o â—_o as_z b_o be_v to_o d_z by_o this_o proposition_n so_o likewise_o b_o multiply_v the_o two_o number_n a_o and_o c_o produce_v the_o two_o number_n h_o and_o e._n wherefore_o by_o the_o same_o h_o be_v to_o e_o as_o a_o be_v to_o c._n but_o alternate_o by_o the_o 13._o of_o this_o book_n a_o be_v to_o c_o as_z b_o be_v to_o d_z but_o as_o a_o be_v to_o c_o so_o be_v h_z to_o e_o and_o as_z â—_o be_v to_o d_z so_o be_v g_z to_o â—_o wherefore_o by_o the_o seven_o common_a sentence_n as_o h_o be_v to_o eâ—_n â—_z so_o be_v g_z to_o f._n wherefore_o alternate_o by_o the_o 13._o of_o this_o book_n h_o be_v to_o g_z as_z e_o be_v to_o f._n but_o it_o be_v prove_v that_o g_z &_o h_o be_v equal_a wherefore_o e_o and_o fletcher_n which_o have_v the_o same_o proportion_n that_o a_o and_o b_o have_v be_v equal_a if_o therefore_o there_o be_v two_o number_n etc_n etc_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 16._o theorem_a the_o 18._o proposition_n if_o two_o number_n multiply_v any_o number_n &_o produce_v other_o number_n the_o number_n of_o they_o produce_v shall_v be_v in_o the_o same_o proportion_n that_o the_o number_n multiply_v be_v svppose_v that_o two_o number_n a_o and_o b_o multiply_v the_o number_n c_o do_v produce_v the_o number_n d_o and_o e._n demonstration_n then_o i_o say_v that_o as_o a_o be_v to_o b_o so_o be_v d_z to_o e._n for_o forasmuch_o as_o a_o multiply_v c_o produce_v d_z therefore_o c_o multiply_v a_o produce_v also_o d_z by_o the_o 16._o of_o this_o book_n and_o by_o the_o same_o reason_n c_o multiply_v b_o produce_v e._n now_o then_o one_o number_n c_z multiply_v two_o number_n a_o and_o b_o produce_v the_o number_n d_z and_o e._n wherefore_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 e_o which_o be_v require_v to_o be_v demonstrate_v this_o proposition_n and_o the_o former_a touch_v two_o number_n may_v be_v extend_v to_o number_n how_o many_o soever_o soever_o so_o that_o if_o one_o number_n multiply_v number_n how_o many_o soever_o and_o produce_v any_o number_n the_o proportion_n of_o the_o number_n produce_v and_o of_o the_o number_n multiply_v shall_v be_v one_o and_o the_o self_n same_o likewise_o if_o number_n how_o many_o soever_o multiply_v one_o number_n and_o produce_v any_o number_n the_o proportion_n of_o the_o number_n producedâ—_n and_o of_o the_o number_n multiply_v shall_v be_v one_o and_o the_o self_n same_o which_o thing_n by_o this_o and_o the_o former_a proposition_n repeat_v as_o often_o as_o be_v needful_a be_v not_o hard_a to_o prove_v ¶_o the_o 17._o theorem_a the_o 19_o proposition_n if_o there_o be_v four_o number_n in_o proportion_n the_o number_n produce_v of_o the_o first_o and_o the_o four_o be_v equal_a to_o that_o number_n which_o be_v produce_v of_o the_o second_o and_o the_o three_o and_o if_o the_o number_n which_o be_v produce_v of_o the_o first_o and_o the_o four_o be_v equal_a to_o that_o which_o be_v produce_v of_o the_o second_o &_o the_o three_o those_o four_o number_n shall_v be_v in_o proportion_n but_o now_o again_o suppose_v that_o e_o be_v equal_a unto_o f._n then_o i_o say_v that_o as_o a_o be_v to_o b_o so_o be_v c_o to_o d._n first_o for_o the_o same_o order_n of_o construction_n remain_v still_o forasmuch_o as_o a_o multiply_v c_o &_o d_o produce_v g_z and_o e_o therefore_o by_o the_o 17._o of_o the_o seven_o as_o c_o be_v to_o d_z so_o be_v g_z to_o e_o but_o e_o be_v equal_a unto_o fletcher_n but_o if_o two_o number_n be_v equal_a one_o number_n shall_v have_v unto_o they_o onâ—_n and_o the_o same_o proportion_n wherefore_o as_o g_z be_v to_o e_o so_o be_v g_z to_o f._n but_o as_o g_z be_v to_o e_o so_o be_v c_o to_o d._n wherefore_o as_o c_o be_v to_o d_z so_o be_v g_z to_o fletcher_n but_o as_o g_z be_v to_o fletcher_n so_o be_v a_o to_o b_o by_o the_o 18._o of_o the_o seven_o wherefore_o as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z demonstration_n which_o be_v require_v to_o be_v prove_v campane_n here_o campane_n add_v that_o it_o be_v needless_a to_o demonstrate_v that_o if_o one_o number_n have_v to_o two_o number_n one_o and_o the_o same_o proportion_n the_o say_v two_o number_n shall_v be_v equal_a or_o that_o if_o they_o be_v equal_a one_o number_n have_v to_o they_o one_o and_o the_o same_o proportion_n for_o say_v he_o if_o g_o have_v unto_o e_o and_o f_o one_o and_o the_o same_o proportion_n they_o either_o what_o part_n or_o part_n g_z be_v to_o e_o the_o same_o part_n or_o part_n be_v g_z also_o of_o fletcher_n or_o how_o multiplex_n g_z be_v to_o e_o so_o multiplex_n be_v g_z to_o fletcher_n by_o the_o 21._o definition_n and_o therefore_o by_o the_o 2_o and_o 3_o common_a sentence_n the_o say_a number_n shall_v be_v equal_a and_o so_o converse_o if_o the_o two_o number_n e_o and_o f_o be_v equal_a then_o shall_v the_o number_n e_o and_o f_o be_v either_o the_o self_n same_o part_n or_o part_n of_o the_o number_n g_o or_o they_o shall_v be_v equemultiplices_fw-la unto_o it_o and_o therefore_o by_o the_o same_o definition_n the_o number_n g_o shall_v have_v to_o the_o number_n e_o and_o f_o one_o and_o the_o same_o proportion_n ¶_o the_o 18._o theorem_a the_o 20._o proposition_n if_o there_o be_v three_o number_n in_o proportion_n the_o number_n
wherefore_o a_o be_v a_o plain_a number_n and_o the_o side_n thereof_o be_v d_z and_o fletcher_n by_o the_o 17._o definition_n of_o the_o seven_o again_o forasmuch_o as_o d_z and_o e_o be_v the_o least_a number_n that_o have_v one_o &_o the_o same_o proportion_n with_o c_o b_o therefore_o by_o the_o 21._o of_o the_o seven_o how_o many_o time_n d_z measure_v c_o so_fw-mi many_o time_n do_v e_o measure_n b._n how_o often_o e_o measure_v b_o so_fw-mi many_o unity_n let_v there_o be_v in_o g._n wherefore_o e_o measure_v b_o by_o those_o unity_n which_o be_v in_o g_o wherefore_o g_z multiply_a e_o produce_v b_o wherefore_o b_o be_v a_o plain_a number_n by_o the_o 17._o definition_n of_o the_o seven_o demonstration_n and_o the_o side_n thereof_o be_v e_o and_o g._n wherefore_o those_o two_o number_n a_o and_o b_o be_v two_o plain_a number_n i_o say_v moreover_o that_o they_o be_v like_a for_o forasmuch_o as_o fletcher_n multiply_v e_o produce_v c_z and_o g_z multiply_v e_o produce_v b_o therefore_o by_o the_o 17._o of_o the_o seven_o as_o fletcher_n be_v to_o g_o so_o be_v c_z to_o b_o but_o as_z c_z be_v to_o b_o so_o be_v d_z to_o e_o wherefore_o as_o d_z be_v to_o e_o so_o be_v fletcher_n to_o g._n wherefore_o a_o and_o b_o be_v like_o plain_a number_n for_o their_o side_n be_v proportional_a which_o be_v require_v to_o be_v prove_v ¶_o the_o 19_o theorem_a the_o 21._o proposition_n if_o between_o two_o number_n there_o be_v two_o mean_v proportional_a number_n those_o number_n be_v like_o solid_a number_n proposition_n svppose_v that_o between_o two_o number_n a_o and_o b_o there_o be_v two_o mean_v proportional_a number_n c_o d._n then_o i_o say_v that_o a_o and_o b_o be_v like_o solid_a number_n take_v by_o the_o 3â—_o of_o the_o seven_o or_o 2._o of_o the_o eight_o three_o of_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 and_o let_v the_o same_o be_v e_o f_o g._n wherefore_o by_o the_o 3._o of_o the_o eight_o their_o extreme_n e_o g_o be_v prime_a the_o one_o to_o the_o other_o and_o forasmuch_o as_o between_o the_o number_n e_o and_o g_o there_o be_v one_o mean_v proportional_a number_n construction_n therefore_o by_o the_o 20_o of_o the_o eight_o they_o be_v like_o plain_a number_n suppose_v that_o the_o side_n of_o e_o be_v h_o and_o k._n and_o let_v the_o side_n of_o g_o be_v l_o and_o m._n now_o it_o be_v manifest_a that_o these_o number_n e_o f_o g_o be_v in_o continual_a proportion_n demonstration_n and_o in_o the_o same_o proportion_n that_o h_o be_v to_o l_o and_o that_o king_n be_v to_o m._n and_o forasmuch_o aâ—_z e_o f_o g_o be_v 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 therefore_o of_o equality_n by_o the_o 14._o of_o the_o seven_o as_o e_o be_v to_o g_z so_o be_v a_o to_o d._n but_o e_o g_o be_v by_o the_o 3._o of_o the_o eight_o prime_a number_n yea_o they_o be_v prime_a and_o the_o least_o but_o the_o least_o number_n by_o the_o 21._o of_o the_o seven_o measure_v those_o number_n that_o have_v one_o &_o the_o same_o proportion_n with_o they_o equal_o the_o great_a the_o great_a and_o the_o less_o the_o less_o that_o be_v the_o antecedent_n the_o antecedent_n &_o the_o consequent_a the_o consequent_a therefore_o how_o many_o timeâ—_n e_o measure_v a_o so_o many_o time_n g_z measure_v d._n how_o many_o time_n e_o measure_v a_o so_o many_o unity_n let_v there_o be_v in_o n._n wherefore_o n_o multiply_v e_o produce_v a._n but_o e_o be_v produce_v of_o the_o number_n h_o k._n wherefore_o n_o multiply_v that_o which_o be_v produce_v of_o h_o edward_n produce_v a._n wherefore_o a_o be_v a_o solid_a number_n and_o the_o side_n thereof_o be_v h_o edward_n n._n again_o forasmuch_o as_o e_o f_o g_o be_v the_o least_o number_n that_o have_v one_o and_o the_o same_o proportion_n with_o c_o d_o b_o therefore_o how_o many_o time_n e_o measure_v c_o so_fw-mi many_o time_n g_z measure_v b._n how_o oftentimes_o g_z measure_v b_o so_fw-mi many_o unity_n let_v there_o be_v in_o x._o wherefore_o g_o measure_v b_o by_o those_o unity_n which_o be_v in_o x._o wherefore_o x_o multiply_v g_o produce_v b._n but_o g_o be_v produce_v of_o the_o number_n l_o m._n wherefore_o x_o multiply_v that_o number_n which_o be_v produce_v of_o l_o and_o m_o produce_v b._n wherefore_o b_o be_v a_o solid_a number_n and_o the_o side_n thereof_o be_v l_o m_o x._o wherefore_o a_o b_o be_v solid_a number_n i_o say_v moreover_o that_o they_o be_v like_o solid_a number_n for_o forasmuch_o as_o n_o and_o x_o multiply_v e_o produce_v a_o and_o c_o therefore_o by_o the_o 18._o of_o the_o seven_o as_o n_o be_v to_o x_o so_o iâ—_z a_o to_o c_o that_o be_v e_o â—o_z f._n but_o as_o e_o be_v to_o fletcher_n so_o be_v h_o to_o l_o and_o king_n to_o m_o therefore_o as_o h_o be_v to_o l_o so_o be_v king_n to_o m_o and_o n_o to_o x._o and_o h_o edward_n n_o be_v the_o side_n of_o a_o and_o likewise_o l_o m_o x_o aâ—â—_n thâ—_z side_n of_o bâ—_n wherefore_o a_o bâ—_n be_v like_o solid_a number_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 20._o theorem_a the_o 22._o proposition_n if_o three_o number_n be_v in_o continual_a proportion_n and_o if_o the_o first_o be_v a_o square_a number_n the_o three_o also_o shall_v be_v a_o square_a number_n svppose_v that_o there_o be_v three_o number_n in_o continual_a proportion_n a_o b_o c_o and_o let_v the_o first_o be_v a_o square_a number_n demonstration_n then_o i_o say_v that_o the_o three_o be_v also_o a_o square_a number_n for_o forasmuch_o as_o between_o a_o and_o c_o there_o be_v one_o mean_v proportional_a numberâ—_n namely_o b_o therefore_o by_o the_o 20._o of_o the_o eight_o a_o and_o c_o be_v like_o plain_a number_n but_o a_o be_v a_o square_a number_n wherefore_o c_z also_o be_v a_o square_a number_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 21._o theorem_a the_o 23._o proposition_n if_o four_o number_n be_v in_o continual_a proportion_n and_o if_o the_o first_o be_v a_o cube_fw-la number_n the_o four_o also_o shall_v be_v a_o cube_fw-la number_n svppose_v that_o there_o be_v four_o number_n in_o continual_a proportion_n a_o b_o c_o d._n demonstration_n and_o let_v a_o be_v a_o cube_fw-la number_n then_o i_o say_v that_o d_o also_o be_v a_o cube_fw-la number_n for_o forasmuch_o as_o between_o a_o and_o d_o there_o be_v two_o mean_v proportional_a number_n bâ—_n c._n therefore_o a_o d_o be_v like_o solid_a number_n by_o the_o 21._o of_o this_o book_n but_o a_o be_v a_o cube_fw-la number_n wherefore_o d_z also_o be_v a_o cube_fw-la numberâ—_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 22._o theorem_a the_o 24._o proposition_n if_o two_o number_n be_v 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 and_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 svppose_v that_o two_o number_n a_o and_o b_o be_v in_o the_o same_o proportion_n that_o the_o square_a number_n c_o be_v unto_o the_o squâ—â—â—_n number_n d._n and_o let_v a_o be_v a_o square_a number_n demonstration_n then_o i_o say_v that_o b_o also_o be_v a_o square_a number_n for_o forasmuch_o as_o c_z and_o d_o be_v square_a number_n therefore_o g_z and_o d_o be_v like_o plain_a number_n wherefore_o by_o the_o 18._o of_o the_o eight_o between_o c_o and_o d_o there_o be_v one_o mean_v proportional_a number_n but_o as_o c_o be_v to_o d_z so_o be_v a_o to_o b._n wherefore_o between_o a_o and_o b_o there_o be_v one_o mean_v proportional_a number_n by_o the_o 8._o of_o the_o eight_o but_o a_o be_v a_o square_a number_n wherefore_o by_o the_o 22._o of_o the_o eight_o b_o also_o be_v a_o square_a number_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 23._o theorem_a the_o 25._o proposition_n if_o two_o 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 and_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 svppose_v that_o two_o number_n a_o and_o b_o be_v in_o the_o same_o proportion_n the_o one_o to_o the_o other_o that_o the_o cube_fw-la number_n c_z be_v unto_o the_o cube_fw-la number_n d._n and_o let_v a_o be_v a_o cube_fw-la number_n then_o i_o say_v that_o b_o also_o be_v a_o cube_fw-la number_n for_o forasmuch_o as_o c_o d_o be_v cube_fw-la number_n therefore_o c_z demonstration_n d_o be_v like_o solid_a number_n wherefore_o by_o the_o 19_o of_o the_o eight_o between_o c_o and_o d_o
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
number_n be_v make_v of_o unity_n and_o therefore_o can_v a_o point_n be_v a_o common_a part_n of_o all_o line_n and_o measure_v they_o as_o unity_n be_v a_o common_a part_n of_o all_o number_n and_o measure_v they_o unity_n take_v certain_a time_n make_v any_o number_n for_o there_o be_v not_o in_o any_o number_n infinite_a unity_n but_o a_o point_n take_v certain_a time_n yea_o as_o often_o as_o you_o listen_v never_o make_v any_o line_n for_o that_o in_o every_o line_n there_o be_v infinite_a point_n wherefore_o line_n figure_n and_o body_n in_o geometry_n be_v oftentimes_o incommensurable_a and_o irrational_a now_o which_o be_v rational_a and_o which_o irrational_a which_o commensurable_a and_o which_o incommensurable_a how_o many_o and_o how_o sundry_a sort_n and_o kind_n there_o be_v of_o they_o what_o be_v their_o nature_n passion_n and_o property_n do_v euclid_n most_o manifest_o show_v in_o this_o book_n and_o demonstrate_v they_o most_o exact_o this_o ten_o book_n have_v ever_o hitherto_o of_o all_o man_n and_o be_v yet_o think_v &_o account_v to_o be_v the_o hard_a book_n to_o understand_v of_o all_o the_o book_n of_o euclid_n euclid_n which_o common_a receive_a opinion_n have_v cause_v many_o to_o shrink_v and_o have_v as_o it_o be_v deter_v they_o from_o the_o handle_n and_o treaty_n thereof_o there_o have_v be_v in_o deed_n in_o time_n past_a and_o be_v present_o in_o these_o our_o day_n many_o which_o have_v deal_v and_o have_v take_v great_a and_o good_a diligence_n in_o comment_v amend_v and_o restore_a of_o the_o six_o first_o book_n of_o euclid_n and_o there_o have_v stay_v themselves_o and_o go_v no_o far_o be_v deter_v and_o make_v afraid_a as_o it_o seem_v by_o the_o opinion_n of_o the_o hardness_n of_o this_o book_n to_o pass_v forth_o to_o the_o book_n follow_v truth_n it_o be_v that_o this_o book_n have_v in_o it_o somewhat_o a_o other_o &_o strange_a manner_n of_o matter_n entreat_v of_o former_a them_z the_o other_o book_n before_o have_v and_o the_o demonstration_n also_o thereof_o &_o the_o order_n seem_v likewise_o at_o the_o first_o somewhat_o strange_a and_o unaccustomed_a which_o thing_n may_v seem_v also_o to_o cause_v the_o obscurity_n thereof_o and_o to_o fear_v away_o many_o from_o the_o read_n and_o diligent_a study_n of_o the_o same_o so_o much_o that_o many_o of_o the_o well_o learned_a have_v much_o complain_v of_o the_o darkness_n and_o difficulty_n thereof_o and_o have_v think_v it_o a_o very_a hard_a thing_n and_o in_o manner_n impossible_a to_o attain_v to_o the_o right_n and_o full_a understanding_n of_o this_o book_n algebra_n without_o the_o aid_n and_o help_v of_o some_o other_o knowledge_n and_o learning_n and_o chief_o without_o the_o knowledge_n of_o that_o more_o secret_a and_o subtle_a part_n of_o arithmetic_n common_o call_v algebra_n which_o undoubted_o first_o well_o have_v and_o know_v will_v geve_v great_a light_n thereunto_o yet_o certain_o may_v this_o book_n very_o well_o be_v enter_v into_o and_o full_o understand_v without_o any_o strange_a help_n or_o succour_n only_o by_o diligent_a observation_n of_o the_o order_n and_o course_n of_o euclides_n write_n so_o that_o he_o which_o diligent_o have_v peruse_v and_o full_o understand_v the_o 9_o book_n go_v before_o and_o mark_v also_o earnest_o the_o principle_n and_o definition_n of_o this_o â—enth_fw-mi book_n he_o shall_v well_o perceive_v that_o euclid_n be_v of_o himself_o a_o sufficient_a teacher_n and_o instructor_n understand_v and_o need_v not_o the_o help_n of_o any_o other_o and_o shall_v soon_o see_v that_o this_o ten_o book_n be_v not_o of_o such_o hardness_n and_o obscurity_n as_o it_o have_v be_v hitherto_o think_v yea_o i_o doubt_v not_o but_o that_o by_o the_o travel_n and_o industry_n take_v in_o this_o translation_n and_o by_o addition_n and_o emendation_n get_v of_o other_o there_o shall_v appear_v in_o it_o no_o hardness_n at_o all_o but_o shall_v be_v as_o easy_a as_o the_o rest_n of_o his_o book_n be_v definition_n definition_n 1_o magnitude_n commensurable_a be_v suchwhich_v one_o and_o the_o self_n same_o measure_n do_v measure_n first_o he_o show_v what_o magnitude_n be_v commensurable_a one_o to_o a_o other_o to_o the_o better_a and_o more_o clear_a understanding_n of_o this_o definition_n note_v that_o that_o measure_v whereby_o any_o magnitude_n be_v measure_v be_v less_o than_o the_o magnitude_n which_o it_o measure_v or_o at_o least_o equal_a unto_o it_o for_o the_o great_a can_v by_o no_o mean_n measure_v the_o less_o far_o it_o behove_v that_o that_o measure_n if_o it_o be_v equal_a to_o that_o which_o be_v measure_v take_v once_o make_v the_o magnitude_n which_o be_v measure_v if_o it_o be_v less_o than_o oftentimes_o take_v and_o repeat_v it_o must_v precise_o render_v and_o make_v the_o magnitude_n which_o it_o measure_v which_o thing_n in_o number_n be_v easy_o see_v for_o that_o as_o be_v before_o say_v all_o number_n be_v commensurable_a one_o to_o a_o other_o and_o although_o euclid_n in_o this_o definition_n comprehend_v purpose_o only_o magnitude_n which_o be_v continual_a quantity_n as_o be_v line_n superficiece_n and_o body_n yet_o undoubted_o the_o explication_n of_o this_o and_o such_o like_a place_n be_v apt_o to_o be_v seek_v of_o number_n as_o well_o rational_a as_o irrational_a for_o that_o all_o quantity_n commensurable_a have_v that_o proportion_n the_o one_o to_o the_o other_o which_o number_n have_v to_o number_n in_o number_n therefore_o 9_o and_o 12_o be_v commensurable_a because_o there_o be_v one_o common_a measure_n which_o measure_v they_o both_o namely_o the_o number_n 3_n first_o it_o measure_v 12_o for_o it_o be_v less_o than_o 12._o and_o be_v take_v certain_a time_n namely_o 4_o time_n it_o make_v exact_o 12_o 3_o time_n 4_o be_v 12_o it_o also_o measure_v 9_o for_o it_o be_v less_o than_o 9_o and_o also_o take_v certain_a time_n namely_o 3_o time_n it_o make_v precise_o 9_o 3_o time_n 3_o be_v 9_o likewise_o be_v it_o in_o magnitude_n if_o one_o magnitude_n measure_v two_o other_o magnitude_n those_o two_o magnitude_n so_o measure_v be_v say_v to_o be_v commensurable_a as_o for_o example_n if_o the_o line_n c_o be_v double_v do_v make_v the_o line_n b_o and_o the_o same_o line_n c_o triple_a do_v make_v the_o line_n a_o then_o be_v the_o two_o line_n a_o and_o b_o line_n or_o magnitude_n commensurable_a for_o that_o one_o measure_n namely_o the_o line_n c_o measure_v they_o both_o first_o the_o line_n c_o be_v less_o they_o the_o line_n a_o and_o alsolesse_v they_o the_o line_n b_o also_o the_o line_n c_o take_v or_o repeat_v certain_a time_n namely_o 3_o time_n make_v precise_o the_o line_n a_o and_o the_o same_o line_n c_o take_v also_o certain_a time_n namely_o two_o time_n make_v precise_o the_o line_n b._n so_o that_o the_o line_n c_o be_v a_o common_a measure_n to_o they_o both_o and_o do_v measure_v they_o both_o and_o therefore_o be_v the_o two_o line_n a_o and_o b_o line_n commensurable_a and_o so_o imagine_v you_o of_o magnitude_n of_o other_o kind_n as_o of_o superficial_a figure_n and_o also_o of_o body_n 2_o incommensurable_a magnitude_n be_v such_o which_o no_o one_o common_a measure_n do_v measure_n definition_n this_o definition_n nead_v no_o explanation_n at_o all_o other_o it_o be_v easy_o understand_v by_o the_o definition_n go_v before_o of_o line_n commensurable_a for_o contrary_n be_v make_v manifest_a by_o compare_v of_o the_o one_o to_o the_o other_o as_o if_o the_o line_n c_o or_o any_o other_o line_n oftentimes_o iterate_v do_v not_o render_v precise_o the_o line_n a_o nor_o the_o line_n b_o them_z be_v the_o line_n a_o and_o b_o incommensurable_a also_o if_o the_o line_n c_o or_o any_o other_o line_n certain_a time_n repeat_v do_v exact_o render_v the_o line_n a_o and_o do_v not_o measure_v the_o line_n b_o or_o if_o it_o measure_v the_o line_n b_o and_o measure_v not_o also_o the_o line_n a_o the_o line_n a_o and_o b_o be_v yet_o line_n incommensurable_a &_o so_o of_o other_o magnitude_n as_o of_o superficiece_n and_o body_n 3_o right_a line_n commensurable_a in_o power_n be_v such_o who_o square_n one_o and_o the_o self_n same_o superficies_n area_n or_o plat_n do_v measure_n definition_n to_o the_o declaration_n of_o this_o definition_n we_o must_v first_o call_v to_o mind_n what_o be_v understand_v &_o mean_v by_o the_o power_n of_o a_o line_n which_o as_o we_o have_v before_o in_o the_o former_a book_n note_v be_v nothing_o else_o but_o the_o square_a thereof_o or_o any_o other_o plain_a figure_n equal_a to_o the_o square_a thereof_o and_o so_o great_a power_n &_o hability_n â—s_v a_o line_n say_v to_o have_v as_o be_v the_o quantity_n of_o the_o square_n which_o it_o be_v able_a to_o describe_v or_o a_o figure_n superficial_a equal_a to_o the_o square_a
which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o by_o a_o rational_a superficies_n for_o either_o of_o they_o be_v rational_a wherefore_o also_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o exceed_v the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o by_o a_o rational_a superficies_n when_o yet_o each_o of_o they_o be_v a_o medial_a superficies_n which_o be_v impossible_a wherefore_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a be_v in_o one_o point_n only_o divide_v into_o his_o name_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 35._o theorem_a the_o 47._o proposition_n a_o line_n contain_v in_o power_n two_o medial_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n svppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n be_v divide_v into_o his_o name_n in_o the_o point_n c_o so_fw-mi that_o let_v the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la &_o cb_o medial_a and_o that_o also_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o medial_a and_o moreover_o incommensurable_a â—o_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb._n then_o i_o say_v construction_n that_o the_o line_n ab_fw-la can_v in_o no_o other_o point_n be_v divide_v into_o his_o name_n but_o only_o in_o the_o point_n c._n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n d_o so_fw-mi that_o let_v not_o the_o line_n ac_fw-la be_v one_o and_o the_o same_o that_o be_v equal_a with_o the_o line_n db_o but_o by_o supposition_n let_v the_o line_n ac_fw-la be_v the_o great_a and_o take_v a_o rational_a line_n ef._n and_o by_o the_o 43._o of_o the_o first_o upon_o the_o line_n of_o apply_v a_o rectangle_n parallelogram_n eglantine_n equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o and_o likewise_o upon_o the_o line_n hc_n which_o be_v equal_a to_o the_o line_n of_o apply_v the_o parallelogram_n hk_o equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o wherefore_o the_o whole_a parallelogram_n eke_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la absurdity_n again_o upon_o the_o same_o line_n of_o describe_v the_o parallelogram_n el_n equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o db._n wherefore_o the_o residue_n namely_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o be_v equal_a to_o the_o parallelogram_n remain_v namely_o to_o mk_o and_o forasmuch_o as_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v by_o supposition_n medial_a therefore_o the_o parallelogram_n eglantine_n which_o be_v equal_a unto_o it_o be_v also_o medial_a and_o it_o be_v apply_v upon_o the_o rational_a line_n ef._n wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n he_o be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n ef._n and_o by_o the_o same_o reason_n also_o the_o line_n hn_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o same_o line_n ef._n and_o forasmuch_o as_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o for_o it_o be_v suppose_v to_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o once_o therefore_o the_o parallelogram_n eglantine_n be_v incommensurable_a to_o the_o parallelogram_n h_o â—_o wherefore_o the_o line_n eh_o also_o be_v incommensurable_a in_o length_n to_o the_o line_n hn_o and_o they_o be_v rational_a line_n wherefore_o the_o line_n eh_o and_o hn_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n en_fw-fr be_v a_o binomial_a line_n and_o be_v divide_v into_o his_o name_n in_o the_o point_n h._n and_o in_o like_a sort_n may_v we_o prove_v that_o the_o same_o binomial_a line_n en_fw-fr be_v divide_v into_o his_o name_n in_o the_o point_n m_o and_o that_o the_o line_n eh_o be_v not_o one_o and_o the_o same_o that_o be_v equal_a with_o the_o line_n mn_o as_o it_o be_v prove_v in_o the_o end_n of_o the_o demonstration_n of_o the_o 44._o of_o this_o book_n wherefore_o a_o binomial_a line_n be_v divide_v into_o his_o name_n in_o two_o sundry_a point_n which_o be_v impossible_a by_o the_o 42._o of_o the_o ten_o wherefore_o a_o line_n contain_v in_o power_n two_o medial_n be_v not_o in_o sundry_a point_n divide_v into_o his_o name_n wherefore_o it_o be_v divide_v in_o one_o point_n only_o which_o be_v require_v to_o be_v demonstrate_v ¶_o second_a definition_n it_o be_v show_v before_o that_o of_o binomial_a line_n there_o be_v six_o kind_n line_n the_o definition_n of_o all_o which_o be_v here_o now_o set_v and_o be_v call_v second_o definition_n all_o binomial_a line_n as_o all_o other_o kind_n of_o irrational_a line_n be_v conceive_v consider_v and_o perfect_o understand_v only_o in_o respect_n of_o a_o rational_a line_n who_o part_n as_o before_o be_v teach_v be_v certain_a and_o know_v and_o may_v be_v distinct_o express_v by_o number_n unto_o which_o line_n they_o be_v compare_v this_o rationalâ—_fw-la line_n must_v you_o ever_o have_v before_o your_o eye_n in_o all_o these_o definition_n so_o shall_v they_o all_o be_v â—asie_a enough_o paâ—tâ—s_n a_o binomial_a line_n you_o know_v be_v make_v of_o two_o part_n or_o name_n whereof_o the_o one_o be_v great_a than_o the_o other_o wherefore_o the_o power_n or_o square_v also_o of_o the_o one_o be_v great_a than_o the_o power_n or_o square_a oâ—_n the_o other_o the_o three_o first_o kind_n of_o binomial_a line_n namely_o the_o first_o the_o seconâ—_o &_o the_o three_o be_v produce_v when_o the_o square_a of_o the_o great_a name_n or_o part_n of_o a_o binomâ—all_n eâ—cedeth_v the_o square_a of_o the_o less_o name_n or_o part_n by_o the_o square_n of_o a_o line_n which_o be_v commensurable_a in_o length_n to_o it_o namely_o to_o the_o great_a the_o three_o last_o kind_n namely_o the_o four_o the_o â—iâ—t_n and_o the_o six_o be_v produce_v when_o the_o square_a of_o the_o great_a name_n or_o part_n â—â—â—â—edeth_v the_o square_a of_o the_o less_o name_n or_o part_n by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o it_o that_o be_v to_o the_o great_a part_n definition_n a_o first_o binomial_a line_n be_v who_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o tâ—e_z less_o part_n â—y_a the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o great_a part_n and_o the_o great_a part_n be_v also_o commensurable_a in_o length_n to_o tâ—e_z rational_a line_n first_o set_v as_o lâ—t_v the_o rational_a line_n first_o set_v be_v ab_fw-la who_o part_n be_v distinct_o know_v suppose_v also_o that_o the_o line_n ce_fw-fr be_v a_o binomial_a line_n who_o name_n or_o part_n let_v be_v cd_o and_o de._n and_o let_v the_o square_n of_o the_o line_n cd_o the_o great_a part_n exceed_v the_o square_n of_o the_o line_n de_fw-fr the_o less_o part_n by_o the_o square_n of_o the_o line_n fg_o which_o line_n fg_o let_v bâ—e_o commensurable_a in_o length_n to_o the_o line_n cd_o which_o be_v the_o great_a part_n of_o the_o binomial_a line_n and_o moreover_o let_v the_o line_n cd_o the_o great_a paâ—t_n be_v commensurble_n in_o length_n to_o the_o rational_a line_n first_o set_v namely_o to_o ab_fw-la so_o by_o this_o dâ—â—inition_n the_o binomial_a line_n ce_fw-fr be_v a_o first_o binomial_a line_n definition_n a_o second_o binomial_a line_n be_v when_o the_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o the_o less_o part_n by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o it_o and_o the_o less_o part_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n first_o set_v as_o suppose_v ever_o the_o rational_a line_n let_v ce_fw-fr be_v a_o binomial_a line_n divide_v in_o the_o point_n d._n the_o square_n of_o who_o great_a part_n cd_o let_v exceed_v the_o square_n of_o the_o less_o part_n de_fw-fr by_o the_o square_n of_o the_o line_n fg_o which_o line_n â—g_z let_v be_v commensurable_a in_o length_n unto_o the_o line_n cd_o tâ—e_z great_a pâ—â—â—_n oâ—_n the_o binomial_a line_n and_o let_v also_o the_o line_n de_fw-fr the_o less_o part_n of_o the_o binomial_a line_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n first_o set_v ab_fw-la so_o by_o this_o definition_n the_o binomial_a line_n ce_fw-fr be_v a_o second_o binomial_a line_n â—â—â—â—â—â—ition_n a_o three_o binomial_a
shall_v in_o like_a sort_n by_o the_o 14._o of_o the_o ten_o be_v in_o power_n more_o than_o the_o line_n df_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n cf_o and_o then_o if_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n the_o line_n cf_o shall_v also_o in_o like_a sort_n be_v commensurable_a in_o length_n to_o the_o same_o rational_a line_n and_o so_o either_o of_o the_o line_n ab_fw-la and_o cd_o be_v a_o four_o residual_a line_n and_o if_o the_o line_n be_v be_v commensurable_a in_o length_n to_o the_o rational_a line_n the_o line_n df_o shall_v also_o be_v commensurable_a in_o length_n to_o the_o same_o line_n and_o so_o either_o of_o the_o line_n ab_fw-la &_o cd_o be_v a_o â—iâ—t_o residual_a line_n and_o if_o neither_o of_o the_o line_n ae_n nor_o be_v be_v commensurable_a in_o length_n to_o the_o rational_a line_n in_o like_a sort_n neither_o of_o the_o line_n cf_o nor_o df_o shall_v be_v commensurable_a in_o length_n to_o the_o same_o rational_a line_n and_o so_o either_o of_o the_o line_n ab_fw-la &_o cd_o be_v a_o six_o residual_a line_n wherefore_o the_o line_n cd_o be_v a_o residual_a line_n of_o the_o self_n same_o order_n that_o the_o line_n ab_fw-la be_v a_o line_n therefore_o commensurable_a in_o length_n to_o a_o residual_a line_n be_v itself_o also_o a_o residual_a line_n of_o the_o self_n same_o order_n which_o be_v require_v to_o be_v prove_v as_o before_o touch_v binomial_a line_n so_o also_o touch_v residual_a line_n this_o be_v to_o be_v note_v that_o a_o line_n commensurable_a in_o length_n to_o a_o residual_a line_n be_v always_o a_o residual_a line_n of_o the_o self_n same_o order_n that_o the_o residual_a line_n be_v unto_o who_o it_o be_v commensurable_a note_n as_o have_v before_o in_o this_o 103._o proposition_n be_v prove_v but_o if_o a_o line_n be_v commensurable_a in_o power_n only_o to_o a_o residual_a lineâ—_z than_o follow_v it_o not_o yea_o it_o be_v impossible_a that_o that_o line_n shall_v be_v a_o residual_a of_o the_o self_n same_o order_n that_o the_o residual_a line_n be_v unto_o who_o it_o be_v commensurable_a in_o power_n only_o howbeit_o those_o two_o line_n shall_v of_o necessity_n be_v both_o either_o of_o the_o three_o first_o order_n of_o residâ—â—ll_a line_n or_o of_o the_o three_o last_o order_n which_o be_v not_o hard_a to_o prove_v if_o you_o mark_v diligent_o the_o former_a demonstration_n and_o that_o which_o be_v speak_v of_o binomial_a line_n as_o touch_v this_o matter_n ¶_o the_o 80._o theorem_a the_o 104._o proposition_n a_o line_n commensurable_a to_o a_o medial_a residual_a line_n be_v itself_o also_o a_o medial_a residual_a line_n and_o of_o the_o self_n same_o order_n svppose_v that_o ab_fw-la be_v a_o medial_a residual_a line_n unto_o who_o let_v the_o line_n cd_o be_v commensurable_a in_o length_n and_o in_o power_n or_o in_o power_n only_o then_o i_o say_v that_o cd_o be_v also_o a_o medial_a residual_a line_n and_o of_o the_o self_n same_o order_n for_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o medial_a residual_a line_n construction_n let_v the_o line_n convenient_o join_v unto_o iâ—_z 〈◊〉_d be_v wherefore_o the_o line_n ae_n and_o be_v be_v medial_a commensurable_a in_o power_n only_o as_o ab_fw-la be_v to_o cd_o so_o by_o the_o 22._o of_o the_o six_o let_v be_v be_v to_o df._n and_o in_o like_a sort_n as_o in_o the_o former_a so_o also_o in_o this_o may_v we_o prove_v demonstration_n that_o the_o line_n ae_n be_v commensurable_a in_o length_n and_o in_o power_n or_o in_o power_n only_o unto_o the_o line_n cf_o &_o the_o line_n be_v 〈◊〉_d the_o line_n df._n wherefore_o by_o the_o 23._o of_o the_o ten_o 〈◊〉_d line_n cf_o be_v a_o medial_a line_n and_o the_o line_n df_o be_v also_o a_o medial_a line_n for_o that_o it_o be_v commensurable_a to_o the_o medial_a line_n be._n and_o in_o like_a sort_n the_o line_n cf_o and_o df_o be_v commensurable_a in_o power_n only_o for_o that_o they_o have_v the_o self_n same_o proportion_n the_o one_o to_o the_o other_o that_o the_o line_n ae_n and_o ebb_n have_v which_o be_v commensurable_a in_o power_n only_o wherefore_o the_o line_n cd_o be_v a_o medial_a residual_a line_n medial_a i_o say_v moreover_o that_o it_o be_v of_o the_o self_n same_o order_n that_o the_o line_n ab_fw-la be_v for_o for_o that_o as_o the_o line_n ae_n be_v to_o the_o line_n be_v so_o be_v the_o line_n cf_o to_o the_o line_n df._n but_o as_o the_o line_n ae_n be_v to_o the_o line_n be_v so_o be_v the_o square_a of_o the_o line_n ae_n to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o be_v by_o the_o first_o of_o the_o six_o and_o as_o the_o line_n cf_o be_v to_o the_o line_n df_o so_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o df._n wherefore_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o be_v so_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o df._n wherefore_o alternate_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o square_n of_o the_o line_n cf_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o be_v to_o the_o parallelogram_n contain_v under_o the_o â—ines_n cf_o and_o df._n but_o the_o square_n of_o the_o line_n ae_n be_v commensurable_a to_o the_o square_n of_o the_o line_n cf_o for_o the_o line_n ae_n be_v commensurable_a to_o the_o line_n cf_o wherefore_o also_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o be_v be_v commensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o df._n wherefore_o if_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n be_v rational_a the_o parallelogram_n also_o contain_v under_o the_o line_n cf_o and_o fd_o shall_v be_v rational_a and_o then_o either_o of_o the_o line_n ab_fw-la and_o cd_o be_v a_o first_o medial_a residual_a line_n but_o if_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o be_v be_v medial_a the_o parallelogram_n also_o contain_v under_o the_o line_n cf_o and_o fd_o shall_v be_v also_o medial_a by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o and_o so_o either_o of_o the_o line_n ab_fw-la and_o cd_o be_v a_o second_o medial_a residual_a line_n wherefore_o the_o line_n cd_o be_v a_o medial_a residual_a line_n of_o the_o self_n same_o order_n that_o the_o line_n ab_fw-la be_v a_o line_n therefore_o commensurable_a to_o a_o medial_a residual_a line_n be_v itself_o also_o a_o medial_a residual_a line_n of_o the_o self_n same_o order_n which_o be_v require_v to_o be_v demonstrate_v this_o theorem_a be_v understand_v general_o that_o whether_o a_o line_n be_v commensurable_a in_o length_n &_o in_o power_n or_o in_o power_n only_o to_o a_o medial_a residual_a line_n it_o be_v itself_o also_o a_o medial_a residual_a line_n and_o of_o the_o self_n same_o order_n which_o thing_n also_o be_v to_o be_v understand_v of_o the_o three_o theorem_n which_o follow_v an_o other_o demonstration_n after_o campane_n suppose_v that_o a_o be_v a_o medial_a residual_a line_n unto_o who_o let_v the_o line_n b_o be_v commensurable_a in_o length_n or_o in_o power_n only_o and_o take_v a_o rational_a line_n cd_o unto_o which_o apply_v the_o parallelogram_n ce_fw-fr equal_a to_o the_o square_n of_o the_o line_n a_o construction_n and_o unto_o the_o line_n fe_o which_o be_v equal_a to_o the_o line_n cd_o apply_v the_o parallelogram_n fâ—_n equal_a to_o the_o square_n of_o the_o line_n b._n now_o then_o the_o parallelogram_n ce_fw-fr and_o fg_o shall_v be_v commensurable_a demonstration_n for_o that_o the_o line_n a_o b_o be_v commensurable_a in_o power_n wherefore_o by_o the_o 1._o of_o the_o six_o and_o 10._o of_o this_o book_n thâ—_z line_n de_fw-fr and_o fg_o be_v commensurable_a in_o length_n now_o than_o if_o a_o be_v a_o first_o medial_a residual_a line_n then_o be_v the_o line_n de_fw-fr a_fw-fr second_v residual_a line_n by_o the_o 98._o of_o this_o book_n and_o if_o the_o line_n a_o be_v a_o sâ—cond_o medial_a residual_a line_n then_o be_v the_o line_n â—â—_o â—_z a_o three_o residual_a line_n by_o the_o 99_o of_o this_o book_n but_o if_o de_fw-fr be_fw-mi a_o second_o residual_a line_n gâ—_n also_o shall_v be_v a_o second_o residual_a line_n by_o the_o â—03_n of_o this_o book_n and_o if_o de_fw-fr be_fw-mi a_o three_o residual_a line_n ge_z also_o shall_v by_o the_o same_o be_v also_o a_o three_o residual_a line_n wherefore_o it_o follow_v by_o the_o 9â—_n and_o 93._o of_o this_o book_n that_o b_o be_v either_o a_o first_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
the_o word_n of_o psellus_n in_o his_o epitome_n of_o geometry_n where_o he_o entreat_v of_o the_o production_n and_o constitution_n of_o these_o body_n his_o word_n be_v these_o psellus_n all_o rectiline_v figure_n be_v erect_v upon_o their_o plain_n or_o base_n by_o right_a angle_n make_v prism_n who_o perceive_v not_o but_o that_o a_o pentagon_n erect_v upon_o his_o base_a of_o â—ive_a side_n make_v by_o his_o motion_n a_o side_v column_n of_o five_o side_n likewise_o a_o hexagon_n erect_v at_o right_a angle_n produce_v a_o column_n have_v six_o side_n and_o so_o of_o all_o other_o rectillne_v figure_n all_o which_o solid_n or_o body_n so_o produce_v whether_o they_o be_v side_v column_n or_o parallelipipedon_n be_v here_o in_o most_o plain_a word_n of_o this_o excellent_a and_o ancient_a greek_a author_n psellus_n call_v prism_n wherefore_o if_o the_o definition_n of_o a_o prism_n give_v of_o euclid_n shall_v extend_v itself_o so_o large_o as_o flussas_n imagine_v and_o shall_v include_v such_o figure_n or_o body_n as_o he_o note_v he_o ought_v not_o yet_o for_o all_o that_o so_o much_o to_o be_v offend_v and_o so_o narrow_o to_o have_v seek_v fault_n for_o euclid_n in_o so_o define_v may_v have_v that_o meaning_n &_o sense_n of_o a_o prism_n which_o psellus_n have_v so_o you_o see_v that_o euclid_n may_v be_v defend_v either_o of_o these_o two_o way_n either_o by_o that_o that_o the_o definition_n extend_v not_o to_o these_o figure_n and_o so_o not_o to_o be_v over_o general_n nor_o stretch_v far_o than_o it_o ought_v or_o else_o by_o that_o that_o if_o it_o shall_v stretch_v so_o far_o it_o be_v not_o so_o heinous_a for_o that_o as_o you_o see_v many_o have_v takâ—_n it_o in_o that_o sense_n in_o deed_n common_o a_o prism_n be_v take_v in_o that_o signification_n and_o meaning_n in_o which_o campaâ—â—â—_n flussas_n and_o other_o take_v it_o in_o which_o sense_n it_o seem_v also_o that_o in_o diverse_a proposition_n in_o these_o book_n follow_v it_o ought_v of_o necessity_n to_o be_v take_v 12_o a_o sphere_n be_v a_o figure_n which_o be_v make_v definition_n when_o the_o diameter_n of_o a_o semicircle_n abide_v fix_v the_o semicircle_n be_v turn_v round_o about_o until_o it_o return_v unto_o the_o self_n same_o place_n from_o whence_o it_o begin_v to_o be_v move_v to_o the_o end_n we_o may_v full_o and_o perfect_o understand_v this_o definition_n how_o a_o sphere_n be_v produce_v of_o the_o motion_n of_o a_o semicircle_n it_o shall_v be_v expedient_a to_o consider_v how_o quantity_n mathematical_o be_v by_o imagination_n conceive_v to_o be_v produce_v by_o flow_v and_o motion_n as_o be_v somewhat_o touch_v in_o the_o begin_n of_o the_o first_o book_n ever_o the_o less_o quantity_n by_o his_o motion_n bring_v forâ—h_v the_o quantity_n next_o above_o it_o as_o a_o point_n move_v flow_a or_o glide_v bring_v forth_o a_o line_n which_o be_v the_o first_o quantity_n and_o next_o to_o a_o point_n a_o line_n move_v produce_v a_o superficies_n 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johannes_n de_fw-fr sacro_fw-la busco_n in_o his_o book_n of_o the_o sphere_n busco_n of_o this_o definition_n which_o he_o take_v out_o of_o euclid_n do_v well_o collect_n but_o it_o be_v to_o be_v note_v and_o take_v heed_n of_o that_o none_o be_v deceive_v by_o the_o definition_n of_o a_o sphere_n give_v by_o johannes_n de_fw-fr sacro_fw-la busco_n a_o sphere_n say_v he_o be_v the_o passage_n or_o move_v of_o the_o circumference_n of_o a_o semicircle_n till_o it_o return_v unto_o the_o place_n where_o it_o begin_v which_o agree_v not_o with_o euclid_n euclid_n plain_o say_v that_o a_o sphere_n be_v the_o passage_n or_o motion_n of_o a_o semicircle_n and_o not_o the_o passage_n or_o motion_n of_o the_o circumference_n of_o a_o semicircle_n neither_o can_v it_o be_v true_a that_o the_o circumference_n of_o a_o semicircle_n which_o be_v a_o line_n shall_v describe_v a_o body_n it_o be_v before_o note_v that_o every_o quantity_n move_v describe_v and_o produce_v the_o quantity_n next_o unto_o it_o wherefore_o a_o line_n move_v can_v not_o bring_v forth_o a_o body_n but_o a_o superficies_n only_o as_o if_o you_o imagine_v a_o right_a line_n fasten_v at_o one_o of_o his_o end_n to_o move_v about_o from_o some_o one_o point_n till_o it_o return_v to_o the_o same_o again_o it_o shall_v describe_v a_o plain_a superficies_n namely_o a_o circle_n so_o also_o if_o you_o likewise_o conceive_v of_o a_o crooked_a line_n such_o as_o be_v the_o circumference_n of_o a_o semicircle_n that_o his_o diameter_n fasten_v on_o both_o the_o end_n it_o shall_v move_v from_o a_o point_n assign_v till_o it_o return_v to_o the_o same_o again_o it_o shall_v describe_v &_o produce_v a_o â—ound_a superficies_n only_o which_o be_v the_o superficies_n and_o limit_n of_o the_o sphere_n and_o shall_v not_o produce_v the_o body_n and_o solidity_n of_o the_o sphere_n but_o the_o whole_a semicircle_n which_o be_v a_o superficies_n by_o his_o motion_n as_o be_v before_o say_v produce_v a_o body_n that_o 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of_o not_o fall_v into_o error_n sphere_n theodosius_n in_o his_o book_n de_fw-fr sphericis_n a_o book_n very_o necessary_a for_o all_o those_o which_o will_v see_v the_o ground_n and_o principle_n of_o geometry_n and_o astronomy_n which_o also_o i_o have_v translate_v into_o our_o vulgar_a tongue_n ready_a to_o the_o press_n define_v a_o sphere_n after_o this_o manner_n a_o sphere_n be_v a_o solid_a or_o body_n contain_v under_o one_o superficies_n in_o the_o middle_n whereof_o there_o be_v a_o point_n from_o which_o all_o line_n draw_v to_o the_o circumference_n be_v equal_a this_o definition_n of_o theodosius_n be_v more_o essential_a and_o natural_a then_o be_v the_o other_o give_v by_o euclid_n the_o other_o do_v not_o so_o much_o declare_v the_o inward_a nature_n and_o substance_n of_o a_o sphere_n as_o it_o show_v the_o industry_n and_o knowledge_n of_o the_o produce_v of_o a_o sphere_n and_o therefore_o be_v a_o causal_n definition_n give_v by_o the_o cause_n efficient_a or_o rather_o a_o description_n then_o a_o definition_n but_o this_o definition_n be_v very_o essential_a declare_v the_o nature_n and_o substance_n of_o a_o sphere_n as_o if_o a_o circle_n shall_v be_v thus_o define_v as_o it_o well_o may_v a_o circle_n be_v the_o passage_n or_o move_v of_o a_o line_n from_o a_o point_n till_o it_o return_v to_o the_o same_o point_n against_o it_o be_v a_o causal_n definition_n show_v the_o efficient_a cause_n whereof_o a_o circle_n be_v produce_v namely_o of_o the_o motion_n of_o a_o line_n and_o it_o be_v a_o very_a good_a description_n full_o show_v what_o a_o circle_n be_v such_o like_a description_n be_v the_o definition_n of_o a_o sphere_n give_v oâ—_n euclideâ—_n â—_z by_o the_o motion_n of_o a_o semicircle_n but_o when_o a_o circle_n be_v define_v to_o be_v a_o plain_a superficies_n
at_o all_o adventure_n namely_o d_o five_o g_o s_o and_o a_o right_a line_n be_v draw_v from_o the_o point_n d_o to_o the_o point_n g_o and_o a_o other_o from_o the_o point_n five_o to_o the_o point_n s._n wherefore_o by_o the_o 7._o of_o the_o eleven_o the_o line_n dg_o and_o we_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n and_o forasmuch_o as_o the_o line_n de_fw-fr be_v a_o parallel_n to_o the_o line_n bg_o therefore_o by_o the_o 24._o of_o the_o first_o the_o angle_n edt_n be_v equal_a to_o the_o angle_n bgt_fw-mi for_o they_o be_v alternate_a angle_n and_o likewise_o the_o angle_n dtv_n be_v equal_a to_o the_o angle_n gts_fw-fr now_o then_o there_o be_v two_o triangle_n that_o be_v dtv_n and_o gts_fw-fr have_v two_o angle_n of_o the_o one_o equal_a to_o two_o angle_n of_o the_o other_o and_o one_o side_n of_o the_o one_o equal_a to_o one_o side_n of_o the_o other_o namely_o the_o side_n which_o subtend_v the_o equal_a angle_n that_o be_v the_o side_n dv_o to_o the_o side_n g_n for_o they_o be_v the_o half_n of_o the_o line_n de_fw-fr and_o bg_o wherefore_o the_o side_n remain_v be_v equal_a to_o the_o side_n remain_v wherefore_o the_o line_n dt_n be_v equal_a to_o the_o line_n tg_n and_o the_o line_n vt_fw-la to_o the_o line_n it_o be_v if_o therefore_o the_o opposite_a side_n of_o a_o parallelipipedon_n be_v divide_v into_o two_o equal_a part_n and_o by_o their_o section_n be_v extend_v plain_a superficiece_n the_o common_a section_n of_o those_o plain_a superficiece_n and_o the_o diameter_n of_o the_o parallelipipedon_n do_v divide_v the_o one_o the_o other_o into_o two_o equal_a part_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o flussas_n every_o plain_a superficies_n extend_v by_o the_o centre_n of_o a_o parallelipipedon_n divide_v that_o solid_a into_o two_o equal_a part_n and_o so_o do_v not_o any_o other_o plain_a superficies_n not_o extend_v by_o the_o centre_n for_o every_o plain_n extend_v by_o the_o centre_n cut_v the_o diameter_n of_o the_o parallelipipedon_n in_o the_o centre_n into_o two_o equal_a part_n for_o it_o be_v prove_v that_o plain_a superficiece_n which_o cut_v the_o solid_a into_o two_o equal_a part_n do_v cut_v the_o dimetient_fw-la into_o two_o equal_a part_n in_o the_o centre_n wherefore_o all_o the_o line_n draw_v by_o the_o centre_n in_o that_o plain_a superficies_n shall_v make_v angle_n with_o the_o dimetient_fw-la and_o forasmuch_o as_o the_o diameter_n fall_v upon_o the_o parallel_n right_a line_n of_o the_o solid_a which_o describe_v the_o opposite_a side_n of_o the_o say_v solid_a or_o upon_o the_o parallel_n plain_a superficiece_n of_o the_o solid_a which_o make_v angel_n at_o the_o end_n of_o the_o diameter_n the_o triangle_n contain_v under_o the_o diameter_n and_o the_o right_a line_n extend_v in_o that_o plain_n by_o the_o centre_n and_o the_o right_a line_n which_o be_v draw_v in_o the_o opposite_a superficiece_n of_o the_o solid_a join_v together_o the_o end_n of_o the_o foresay_a right_a line_n namely_o the_o end_n of_o the_o diameter_n and_o the_o end_n of_o the_o line_n draw_v by_o the_o centre_n in_o the_o superficies_n extend_v by_o the_o centre_n shall_v always_o be_v equal_a and_o equiangle_n by_o the_o 26._o of_o the_o first_o for_o the_o opposite_a right_a line_n draw_v by_o the_o opposite_a plain_a superficiece_n of_o the_o solid_a do_v make_v equal_a angle_n with_o the_o diameter_n forasmuch_o as_o they_o be_v parallel_a line_n by_o the_o 16._o of_o this_o book_n but_o the_o angle_n at_o the_o centre_n be_v equal_a by_o the_o 15._o of_o the_o first_o for_o they_o be_v head_n angle_n &_o one_o side_n be_v equal_a to_o one_o side_n namely_o half_o the_o dimetient_fw-la wherefore_o the_o triangle_n contain_v under_o every_o right_a line_n draw_v by_o the_o centre_n of_o the_o parallelipipedon_n in_o the_o superficies_n which_o be_v extend_v also_o by_o the_o say_a centre_n and_o the_o diameter_n thereof_o who_o end_n be_v the_o angle_n of_o the_o solid_a be_v equal_a equilater_n &_o equiangle_n by_o the_o 26._o of_o the_o first_o wherefore_o it_o follow_v that_o the_o plain_a superficies_n which_o cut_v the_o parallelipipedon_n do_v make_v the_o part_n of_o the_o base_n on_o the_o opposite_a side_n equal_a and_o equiangle_n and_o therefore_o like_a and_o equal_a both_o in_o multitude_n and_o in_o magnitude_n wherefore_o the_o two_o solid_a section_n of_o that_o solid_a shall_v be_v equal_a and_o like_a by_o the_o 8._o definition_n of_o this_o book_n and_o now_o that_o no_o other_o plain_a superficies_n beside_o that_o which_o be_v extend_v by_o the_o centre_n divide_v the_o parallelipipedon_n into_o two_o equal_a part_n it_o be_v manifest_a if_o unto_o the_o plain_a superficies_n which_o be_v not_o extend_v by_o the_o centre_n we_o extend_v by_o the_o centre_n a_o parallel_n plain_a superficies_n by_o the_o corollary_n of_o the_o 15._o of_o this_o book_n for_o forasmuch_o as_o that_o superficies_n which_o be_v extend_v by_o the_o centre_n do_v divide_v the_o parallelipipedom_n into_o two_o equal_a parâ—â—_n it_o be_v manifest_a that_o the_o other_o plain_a superficies_n which_o be_v parallel_n to_o the_o superficies_n which_o divide_v the_o solid_a into_o two_o equal_a part_n be_v in_o one_o of_o the_o equal_a part_n of_o the_o solid_a wherefore_o see_v that_o the_o whole_a be_v ever_o great_a than_o his_o part_n it_o must_v needs_o be_v that_o one_o of_o these_o section_n be_v less_o than_o the_o half_a of_o the_o solid_a and_o therefore_o the_o other_o be_v great_a for_o the_o better_a understanding_n of_o this_o former_a proposition_n &_o also_o of_o this_o corollary_n add_v by_o flussas_n it_o shall_v be_v very_o needful_a for_o you_o to_o describe_v of_o past_v paper_n or_o such_o like_a matter_n a_o parallelipipedom_n or_o a_o cube_n and_o to_o divide_v all_o the_o parallelogram_n thereof_o into_o two_o equal_a part_n by_o draw_v by_o the_o centre_n of_o the_o say_a parallelogram_n which_o centre_n be_v the_o point_n make_v by_o the_o cut_n of_o diagonal_a line_n draw_v from_o thâ—_z opposite_a angle_n of_o the_o say_a parallelogram_n line_n parallel_n to_o the_o side_n of_o the_o parallelogram_n as_o in_o the_o former_a figure_n describe_v in_o a_o plain_a you_o may_v see_v be_v the_o six_o parallelogram_n de_fw-fr eh_o ha_o ad_fw-la dh_o and_o cg_o who_o these_o parallel_a line_n draw_v by_o the_o centre_n of_o the_o say_a parallelogram_n namely_o xo_o or_o pr._n and_o px_n do_v divide_v into_o two_o equal_a part_n by_o which_o four_o line_n you_o must_v imagine_v a_o plain_a superficies_n to_o be_v extend_v also_o these_o parallel_a line_n kl_o ln_o nm_o and_o mâ—_n by_o which_o four_o line_n likewise_o yâ—_n must_v imagine_v a_o plain_a superficies_n to_o be_v extend_v you_o may_v if_o you_o will_v put_v within_o your_o body_n make_v thus_o of_o past_v paper_n two_o superficiece_n make_v also_o of_o the_o say_a paper_n have_v to_o their_o limit_n line_n equal_a to_o the_o foresay_a parallel_n line_n which_o superficiece_n must_v also_o be_v divide_v into_o two_o equal_a part_n by_o parallel_a line_n draw_v by_o their_o centre_n and_o must_v cut_v the_o one_o the_o other_o by_o these_o parallel_a line_n and_o for_o the_o diameter_n of_o this_o body_n extend_v a_o thread_n from_o one_o angle_n in_o the_o base_a of_o the_o solid_a to_o his_o opposite_a angle_n which_o shall_v pass_v by_o the_o centre_n of_o the_o parallelipipedon_n as_o do_v the_o line_n dg_o in_o the_o figure_n before_o describe_v in_o the_o plain_n and_o draw_v in_o the_o base_a and_o the_o opposite_a superficies_n unto_o it_o diagonal_a line_n from_o the_o angle_n from_o which_o be_v extend_v the_o diameter_n of_o the_o solid_a as_o in_o the_o former_a description_n be_v the_o line_n bg_o and_o de._n and_o when_o you_o have_v thus_o describe_v this_o body_n compare_v it_o with_o the_o former_a demonstration_n and_o it_o will_v make_v it_o very_o plain_a unto_o you_o so_o your_o letter_n agree_v with_o the_o letter_n of_o the_o figure_n describe_v in_o the_o book_n and_o this_o description_n will_v plain_o set_v forth_o unto_o you_o the_o corollary_n follow_v that_o proposition_n for_o where_o as_o to_o the_o understanding_n of_o the_o demonstration_n of_o the_o proposition_n the_o superficiece_n put_v within_o the_o body_n be_v extend_v by_o parallel_a line_n draw_v by_o the_o centre_n of_o the_o base_n of_o the_o parallelipipedon_n to_o the_o understanding_n of_o the_o say_a corollary_n you_o may_v extend_v a_o superficies_n by_o any_o other_o line_n draw_v in_o the_o say_a base_n so_o that_o yet_o it_o pass_v through_o the_o midst_n of_o the_o thread_n which_o be_v suppose_v to_o be_v the_o centre_n of_o the_o parallelipipedon_n ¶_o the_o 35._o theorem_a the_o 40._o proposition_n if_o there_o be_v
last_o proposition_n in_o the_o second_o book_n and_o also_o of_o the_o 31._o in_o the_o six_o book_n ¶_o a_o problem_n 5._o two_o unequal_a circle_n be_v give_v to_o find_v a_o circle_n equal_a to_o the_o excess_n of_o the_o great_a to_o the_o less_o suppose_v the_o two_o unequal_a circle_n give_v to_o be_v abc_n &_o def_n &_o let_v abc_n be_v the_o great_a construction_n who_o diameter_n suppose_v to_o be_v ac_fw-la &_o the_o diameter_n of_o def_n suppose_v to_o be_v df._n i_o say_v a_o circle_n must_v be_v find_v equal_a to_o that_o excess_n in_o magnitude_n by_o which_o abc_n be_v great_a thâ—_z def_n by_o the_o first_o of_o the_o four_o in_o the_o circle_n abc_n apply_v a_o right_a line_n equal_a to_o df_o who_o one_o end_n let_v be_v at_o c_o and_o the_o other_o let_v be_v at_o b._n fron_n b_o to_o a_o draw_v a_o right_a line_n by_o the_o 30._o of_o the_o three_o it_o may_v appear_v demonstration_n that_o abc_n be_v a_o right_a angle_n and_o thereby_o abc_n the_o triangle_n be_v rectangle_v wherefore_o by_o the_o first_o of_o the_o two_o corollary_n here_o before_o the_o circle_z abc_n be_v equal_a to_o the_o circle_n def_n for_o bc_o by_o construction_n be_v equal_a to_o df_o and_o more_o over_o to_o the_o circle_n who_o diameter_n be_v ab_fw-la that_o circle_n therefore_o who_o diameter_n be_v ab_fw-la be_v the_o circle_n contain_v the_o magnitude_n by_o which_o abc_n be_v great_a than_o def_n wherefore_o two_o unequal_a circle_n be_v give_v we_o have_v find_v a_o circle_n equal_a to_o the_o excess_n of_o the_o great_a to_o the_o less_o which_o ought_v to_o be_v do_v a_o problem_n 6._o a_o circle_n be_v give_v to_o find_v two_o circle_n equal_a to_o the_o same_o which_o find_v circle_n shall_v have_v the_o one_o to_o the_o other_o any_o proportion_n give_v in_o two_o right_a line_n suppose_v abc_n a_o circle_n give_v and_o the_o proportion_n give_v let_v it_o be_v that_o which_o be_v between_o the_o two_o right_a line_n d_z and_o e._n i_o say_v that_o two_o circle_n be_v to_o be_v find_v equal_a to_o abc_n and_o with_o all_o one_o to_o the_o other_o construction_n in_o the_o proportion_n of_o d_o to_o e._n let_v the_o diameter_n of_o abc_n be_v ac_fw-la as_z d_z be_v to_o e_o so_o let_v ac_fw-la be_v divide_v by_o the_o 10._o of_o the_o six_o in_o the_o point_n f._n at_o fletcher_n to_o the_o line_n ac_fw-la let_v a_o perpendicular_a be_v draw_v fb_o and_o let_v it_o mete_v the_o circumference_n at_o the_o point_n b._n from_o the_o point_n b_o to_o the_o point_v a_o and_o c_o let_v right_a line_n be_v draw_v ba_o and_o bc._n i_o say_v that_o the_o circle_n who_o diameter_n be_v the_o line_n basilius_n and_o bc_o be_v equal_a to_o the_o circle_n abc_n and_o that_o those_o circle_n have_v to_o their_o diameter_n the_o line_n basilius_n and_o bc_o be_v one_o to_o the_o other_o in_o the_o proportion_n of_o the_o line_n d_o to_o the_o line_n e._n for_o first_o that_o they_o be_v equal_a it_o be_v evident_a by_o reason_n that_o abc_n be_v a_o triangle_n rectangle_n wherefore_o by_o the_o 47._o of_o the_o first_o the_o square_n of_o basilius_n and_o bc_o be_v equal_a to_o the_o square_n of_o ac_fw-la and_o so_o by_o this_o second_o it_o be_v manife_a the_o two_o circle_n to_o be_v equal_a to_o the_o circle_n abc_n second_o as_o d_o be_v to_o â—_o so_o be_v of_o to_o fc_n by_o construction_n and_o as_o the_o line_n of_o be_v to_o the_o line_n fc_o so_o be_v the_o square_a of_o the_o line_n â—a_o to_o the_o square_n of_o the_o line_n bc._n which_o thing_n we_o will_v brief_o prove_v thus_o the_o parallelogram_n contain_v under_o ac_fw-la and_o of_o use_n be_v equal_a to_o the_o square_n of_o basilius_n by_o the_o lemma_n after_o the_o 32._o of_o the_o ten_o book_n and_o by_o the_o same_o lemma_n or_o assumpt_n the_o parallelogram_n contain_v under_o ac_fw-la and_o â—c_z be_v equal_a to_o the_o square_n of_o the_o line_n bc._n wherefore_o as_o the_o first_o parallelogram_n have_v itself_o to_o the_o secondâ—_n so_o have_v the_o square_a of_o basilius_n equal_a to_o the_o first_o parallelogram_n itself_o to_o the_o square_n of_o bc_o equal_a to_o the_o second_o parallelogram_n but_o both_o the_o parallelogram_n have_v one_o height_n namely_o the_o line_n ac_fw-la and_o base_n the_o line_n of_o and_o fc_n wherefore_o as_o of_o be_v to_o fc_n so_o be_v the_o parallelogâ—amme_n contain_v under_o ac_fw-la of_o to_o the_o parallelogram_n contain_v under_o ac_fw-la fc_fw-la by_o the_o fiâ—st_n of_o the_o six_o and_o therefore_o as_o of_o be_v to_o fc_n so_o be_v the_o square_a of_o basilius_n to_o the_o square_n of_o bc._n and_o as_o the_o square_n of_o basilius_n be_v to_o the_o square_n of_o bc_o so_o be_v the_o circle_n who_o diameter_n be_v basilius_n to_o the_o circle_n who_o diameter_n be_v bc_o by_o this_o second_o of_o the_o twelve_o wherefore_o the_o circle_n who_o diameter_n be_v basilius_n be_v to_o the_o circle_n who_o diameter_n be_v bc_o as_o d_z be_v to_o e._n and_o before_o we_o prove_v they_o equal_a to_o the_o circle_n abc_n wherâ—fore_o a_o circle_n be_v give_v we_o have_v find_v two_o circle_n equal_a to_o the_o same_o which_o have_v the_o one_o to_o the_o other_o any_o proportion_n give_v in_o two_o right_a line_n which_o ought_v to_o be_v do_v note_n addition_n heâ—e_o may_v you_o perceive_v a_o other_o way_n how_o to_o execute_v my_o first_o problem_n for_o if_o you_o make_v a_o right_a angle_n contain_v of_o the_o diameter_n give_v as_o in_o this_o figure_n suppose_v they_o basilius_n and_o bc_o and_o then_o subtend_v the_o right_a angle_n with_o the_o line_n ac_fw-la and_o from_o the_o right_a angle_n let_v fall_v a_o line_n perpendicular_a to_o the_o base_a ac_fw-la that_o perpendicular_a at_o the_o point_n of_o his_o fall_n divide_v ac_fw-la into_o of_o and_o fc_n of_o the_o proportion_n require_v a_o corollary_n it_o follow_v of_o thing_n manifest_o prove_v in_o the_o demonstration_n of_o this_o problem_n that_o in_o a_o triangle_n rectangle_n if_o from_o the_o right_a angle_n to_o the_o base_a a_o perpendicular_a be_v let_v fall_v the_o same_o perpendicular_a cut_v the_o base_a into_o two_o part_n rectangle_n in_o that_o proportion_n one_o to_o the_o other_o that_o the_o square_n of_o the_o righâ—_n line_n contain_v the_o right_a angle_n be_v in_o one_o to_o the_o other_o those_o on_o the_o one_o side_n the_o perpendicular_a be_v compare_v to_o those_o on_o the_o other_o both_o square_a and_o segment_n a_o problem_n 7._o between_o two_o circle_n give_v to_o find_v a_o circle_n middle_a proportional_a let_v the_o two_o circle_n give_v be_v acd_o and_o bef_a i_o say_v that_o a_o circle_n be_v to_o be_v find_v which_o between_o acd_a and_o bef_n be_v middle_a proportional_a construction_n let_v the_o diameter_n of_o acd_a be_v ad_fw-la and_o of_o bef_n let_v bâ—_n be_v the_o diameter_n between_o ad_fw-la and_o bf_o find_v a_o line_n middle_a proportional_a by_o the_o 13._o of_o the_o six_o which_o let_v be_v hk_a i_o say_v that_o a_o circle_n who_o diameter_n be_v hk_n be_v middle_a proportional_a between_o acd_a and_o bef_a to_o ad_fw-la hk_o and_o bf_o three_o right_a line_n in_o continual_a proportion_n by_o construction_n let_v a_o four_o line_n be_v find_v to_o which_o bf_o shall_v have_v that_o proportion_n that_o ad_fw-la have_v to_o hk_n by_o the_o 12._o of_o the_o six_o &_o let_v that_o line_n be_v â—_o demonstration_n it_o be_v manifest_a that_o the_o â—ower_a line_n ad_fw-la hk_o bf_o and_o l_o be_v in_o continual_a proportion_n for_o by_o construction_n as_o ad_fw-la be_v to_o hk_n so_o be_v bâ—_n to_o l._n and_o by_o construction_n on_o as_z ad_fw-la be_v to_o hk_n so_o be_v hk_v to_o bf_o wherefore_o hk_n be_v to_o bf_o as_o bf_o be_v to_o l_o by_o the_o 11._o of_o the_o five_o wherefore_o the_o 4._o line_n be_v in_o continual_a proportion_n wherefore_o as_o the_o first_o be_v to_o the_o three_o that_o be_v ad_fw-la to_o bf_o so_o be_v the_o square_a of_o the_o first_o to_o the_o square_n of_o the_o second_o that_o be_v the_o square_a of_o ad_fw-la to_o the_o square_n of_o hk_n by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o and_o by_o the_o same_o corollary_n as_o hk_n be_v to_o l_o so_o be_v the_o square_a of_o hk_n to_o the_o square_n of_o bf_o but_o by_o alternate_a proportion_n the_o line_n ad_fw-la be_v to_o bf_o as_o hk_n be_v to_o l_o wherefore_o the_o square_a of_o ad_fw-la be_v to_o the_o square_n of_o hk_n as_o the_o square_n of_o hk_n be_v to_o the_o square_n of_o bf_o wherefore_o the_o square_a of_o hk_n be_v middle_a proportional_a between_o the_o square_n of_o ad_fw-la and_o the_o square_n of_o bf_o but_o as_o the_o square_n be_v
triangle_n abc_n acd_o &_o cde_o and_o likewise_o the_o base_a fghkl_o into_o these_o triangle_n fgh_o fhl_n and_o hkl_n and_o imagine_v that_o upon_o every_o one_o of_o those_o triangle_n be_v set_v a_o pyramid_n of_o equal_a altitude_n with_o the_o two_o pyramid_n put_v at_o the_o beginning_n demonstration_n and_o for_o that_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n adc_o so_o be_v the_o pyramid_n abcm_n to_o the_o pyramid_n adcm_n by_o the_o 5._o of_o this_o book_n wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o the_o four_o side_v figure_n abcd_o be_v to_o the_o triangle_n acd_o so_o be_v the_o pyramid_n abcdm_n to_o the_o pyramid_n acdm_n but_o as_o the_o triangle_n acd_o be_v to_o the_o triangle_n cde_o so_o be_v the_o pyramid_n acdm_n to_o the_o pyramid_n cdem_fw-la wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o the_o base_a abcd_o be_v to_o thâ—_z base_a cde_o so_o be_v the_o pyramid_n abcdm_n to_o the_o pyramid_n cdem_fw-la wherefore_o again_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o the_o base_a abcde_o be_v to_o the_o base_a cde_o so_o be_v the_o pyramid_n abcedm_n to_o the_o pyramid_n cdem_fw-la and_o by_o the_o same_o reason_n also_o as_o the_o base_a fghkl_o be_v to_o the_o base_a hkl_n so_o be_v the_o pyramid_n fghkln_v to_o the_o pyramid_n hkln_n and_o forasmuch_o as_o there_o be_v two_o pyramid_n cdem_n and_o hkln_n have_v triangle_n to_o their_o base_n and_o be_v under_o one_o and_o the_o self_n same_o altitude_n therefore_o by_o the_o 5._o of_o the_o twelve_o as_o the_o base_a cde_o be_v to_o the_o base_a hkl_n so_o be_v the_o pyramid_n cdem_n to_o the_o pyramid_n hkln_n now_o for_o that_o as_o the_o base_a abce_v be_v to_o the_o base_a cde_o so_o be_v the_o pyramid_n abcedm_n to_o the_o pyramid_n cdem_fw-la but_o as_o the_o base_a cde_o be_v to_o the_o base_a hkl_n so_o be_v the_o pyramid_n cdem_n to_o the_o pyramid_n hkln_n wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o the_o base_a abce_v be_v to_o the_o base_a hkl_n so_o be_v the_o pyramid_n abcedm_n to_o the_o pyramid_n hkln_n but_o also_o as_o the_o base_a hkl_n be_v the_o base_a fghkl_o so_o be_v the_o pyramid_n hkln_v to_o to_o the_o pyramid_n fghkln_n wherefore_o again_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o the_o base_a abce_v be_v to_o the_o base_a fghkl_o so_o be_v the_o pyramid_n abcedm_n to_o the_o pyramid_n fghkln_n wherefore_o pyramid_n consist_v under_o one_o and_o the_o self_n same_o altitude_n and_o have_v polygonon_n figure_n to_o their_o base_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v which_o be_v require_v to_o be_v prove_v the_o 7._o theorem_a the_o 7._o proposition_n every_o prism_n have_v a_o triangle_n to_o his_o base_a may_v be_v divide_v into_o three_o pyramid_n equal_a the_o one_o to_o the_o other_o have_v also_o triangle_n to_o their_o base_n svppose_v that_o abcdef_n be_v a_o prism_n have_v to_o his_o base_a the_o triangle_n abc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n def_n then_o i_o say_v that_o the_o prism_n abcdef_n may_v be_v divide_v into_o three_o pyramid_n equal_a the_o one_o to_o the_o other_o and_o have_v triangle_n to_o their_o base_n demonstration_n draw_v these_o right_a line_n bd_o ec_o and_o cd_o and_o forasmuch_o as_o abed_o be_v a_o parallelogram_n and_o his_o diameter_n be_v the_o line_n bd_o therefore_o the_o triangle_n abdella_n be_v equal_a to_o the_o triangle_n edb_n wherefore_o also_o the_o pyramid_n who_o base_a be_v the_o triangle_n abdella_n and_o top_n the_o point_n c_o be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n edb_n &_o top_n the_o point_n c_o by_o the_o 5._o of_o this_o book_n but_o the_o pyramid_n who_o base_a be_v the_o triangle_n edb_n and_o top_n the_o point_n c_o be_v one_o and_o the_o same_o which_o the_o pyramid_n who_o base_a be_v the_o triangle_n ebc_n and_o top_n the_o point_n d_o for_o they_o be_v comprehend_v of_o the_o self_n same_o plain_a superficiece_n namely_o of_o the_o triangle_n bdedec_n dbc_n and_o ebc_n wherefore_o also_o the_o pyramid_n who_o base_a be_v the_o triangle_n abdella_n and_o top_n the_o point_n c_o be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n ebc_n and_o top_n the_o point_n d._n again_o forasmuch_o as_o bcfe_n be_v a_o parallelogram_n and_o the_o diameter_n thereof_o be_v ec_o therefore_o the_o triangle_n ecf_n be_v equal_a to_o the_o triangle_n cbe_n wherefore_o also_o the_o pyramid_n who_o base_a be_v the_o triangle_n ebc_n and_o top_n the_o point_n d_o be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n ecf_n and_o top_n the_o point_n d_o by_o the_o 5._o of_o this_o book_n but_o the_o pyramid_n who_o base_a be_v the_o triangle_n bec_n and_o top_n the_o point_n d_o be_v prove_v to_o be_v equal_a to_o the_o pyramid_n who_o base_a iâ—_z the_o triangle_n abdella_n and_o top_n the_o point_n c._n wherefore_o also_o the_o pyramid_n who_o base_a be_v the_o triangle_n cef_n and_o top_n the_o point_n d_o be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n abdella_n &_o top_n the_o point_n c._n wherefore_o the_o prism_n abdef_n be_v divide_v into_o three_o equal_a pyramid_n have_v triangle_n to_o their_o base_n and_o forasmuch_o as_o the_o pyramid_n who_o base_a be_v the_o triangle_n abdella_n and_o top_n the_o point_n c_o be_v one_o &_o the_o self_n same_o with_o the_o pyramid_n who_o base_a be_v the_o triangle_n cab_n &_o top_n the_o point_n d_o for_o they_o be_v contain_v under_o the_o self_n same_o plain_a superficiece_n but_o it_o have_v be_v prove_v that_o the_o pyramid_n who_o base_a be_v the_o triangle_n abdella_n and_o top_n the_o point_n c_o be_v the_o three_o pyramid_n of_o the_o prism_n who_o base_a be_v the_o triangle_n abc_n aâ—d_z the_o opposite_a side_n unto_o it_o the_o triangle_n def_n wherefore_o the_o pyramid_n who_o base_a be_v the_o triangle_n abc_n and_o top_n the_o point_n d_o be_v the_o three_o pyramid_n of_o the_o prism_n who_o base_a be_v the_o triangle_n abc_n and_o opposite_a side_n the_o triangle_n def_n wherefore_o every_o prism_n have_v a_o triangle_n to_o his_o base_a may_v be_v divide_v into_o three_o pyramid_n equal_a the_o one_o to_o the_o other_o have_v also_o triangle_n to_o their_o base_n 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 every_o pyramid_n be_v the_o three_o part_n of_o a_o prism_n have_v one_o and_o the_o same_o base_a with_o it_o and_o also_o be_v under_o the_o self_n same_o altitude_n with_o it_o for_o if_o the_o base_a of_o the_o prism_n be_v any_o other_o rectiline_a figure_n they_o a_o triangle_n that_o also_o may_v be_v divide_v into_o prism_n which_o shall_v have_v triangle_n to_o their_o base_n here_o campane_n and_o flussas_n add_v certain_a corollarye_n first_o corollary_n every_o prism_n be_v treble_a to_o the_o pyramid_n which_o have_v the_o self_n same_o triangle_n to_o his_o base_a that_o the_o prism_n have_v and_o the_o self_n same_o altitude_n as_o it_o be_v manifest_a by_o this_o proposition_n where_o the_o prism_n be_v divide_v into_o three_o equal_a pyramid_n of_o which_o two_o be_v upon_o one_o and_o the_o self_n same_o base_a and_o under_o one_o and_o the_o self_n same_o altitude_n but_o if_o the_o prism_n have_v to_o his_o â—ase_n a_o parallelogram_n and_o if_o the_o pyramid_n have_v to_o his_o base_a the_o half_a of_o the_o same_o parallelogram_n and_o their_o altitude_n be_v equal_a then_o again_o the_o pyramid_n shall_v be_v the_o three_o part_n of_o the_o prism_n for_o it_o be_v manifest_a by_o the_o 40._o of_o the_o cleveth_n that_o prism_n be_v under_o equal_a altitude_n and_o the_o one_o have_v to_o his_o base_a a_o triangle_n and_o the_o other_o a_o parallelogram_n double_a to_o the_o same_o triangle_n be_v equal_a the_o one_o to_o the_o other_o whereof_o follow_v the_o former_a conclusion_n second_o corollary_n if_o there_o be_v many_o prism_n under_o one_o and_o the_o same_o altitude_n and_o have_v triangle_n to_o their_o base_n they_o and_o if_o the_o triangular_a base_n be_v so_o join_v together_o upon_o one_o and_o the_o same_o plain_n that_o they_o compose_v a_o poligonon_n figure_n a_o pyramid_n set_v upon_o that_o base_a be_v a_o poligonon_n figure_n and_o under_o the_o same_o altitude_n be_v the_o three_o part_n of_o that_o solid_a which_o be_v compose_v of_o all_o the_o prism_n add_v together_o for_o forasmuch_o as_o euâ—ry_v one_o of_o the_o prism_n which_o have_v to_o his_o base_a a_o triangle_n to_o every_o one_o of_o the_o pyramid_n set_v upon_o the_o same_o base_a the_o altitude_n be_v always_o one_o and_o the_o
fd._n wherefore_o cones_fw-la &_o cylinder_n consist_v upon_o equal_a base_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o altitude_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 15._o theorem_a the_o 15._o proposition_n in_o equal_a cones_n and_o cylinder_n the_o base_n be_v reciprocal_a to_o their_o altitude_n and_o cones_fw-la and_o cylinder_n who_o base_n be_v reciprocal_a to_o their_o altitude_n be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o these_o cones_fw-la acl_n egn_n or_o these_o cylinder_n axe_n eo_fw-la who_o base_n be_v the_o circle_n abcd_o efgh_o and_o axe_n kl_o and_o mn_v which_o axe_n be_v also_o the_o altitude_n of_o the_o cones_fw-la &_o cylinder_n be_v equal_a the_o one_o to_o the_o other_o cones_n then_o i_o say_v that_o the_o base_n of_o the_o cylinder_n xa_n &_o eo_fw-la be_v reciprokal_n to_o their_o altitude_n that_o be_v that_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o proposition_n so_o the_o altitude_n mn_v to_o the_o altitude_n kl_o for_o the_o altitude_n kl_o be_v either_o equal_a to_o the_o altitude_n mn_a or_o not_o first_o let_v it_o be_v equal_a case_n but_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eq_n but_o cones_fw-la and_o cylinder_n consist_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n be_v by_o the_o 11._o of_o the_o twelve_o wherefore_o the_o base_a abcd_o be_v equal_a to_o the_o base_a efgh_o wherefore_o also_o they_o be_v reciprokal_a as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o but_o now_o suppose_v that_o the_o altitude_n lk_n be_v not_o equal_a to_o the_o altitude_n m._n n_o construction_n but_o let_v the_o altitude_n mn_v be_v great_a and_o by_o the_o 3._o of_o the_o first_o from_o the_o altitude_n mn_v take_v away_o pm_v equal_a to_o the_o altitude_n kl_o so_o that_o let_v the_o line_n pm_n be_v put_v equal_a to_o the_o line_n kl_o and_o by_o the_o point_n p_o let_v there_o be_v extend_v a_o plain_a superficies_n tus_z which_o let_v cut_v the_o cylinder_n eo_fw-la and_o be_v a_o parallel_n to_o the_o two_o opposite_a plain_a superâ—icieces_n that_o be_v to_o the_o circle_n efgh_o and_o ro._n cylinder_n and_o make_v the_o base_a the_o circle_n efgh_o &_o the_o altitude_n mp_n imagine_v a_o cylinder_n es._n and_o for_o that_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eo_fw-la and_o there_o be_v a_o other_o cylinder_n es_fw-ge therefore_o by_o the_o 7._o of_o the_o five_o as_o the_o cylinder_n axe_n be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o cylinder_n eo_fw-la to_o the_o cylinder_n es._n but_o as_o the_o cylinder_n axe_n be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o base_a abcd_o to_o the_o base_a efgh_o for_o the_o cylinder_n axe_n and_o es_fw-ge be_v under_o one_o and_o the_o self_n same_o altitude_n and_o as_o the_o cylinder_n eo_fw-la be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o altitude_n mn_v to_o the_o altitude_n mp_n for_o cylinder_n consist_v upon_o equal_a base_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o altitude_n wherefore_o as_o the_o base_a abcd_o be_v â—o_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n mp_n but_o the_o altitude_n pm_n be_v equal_a to_o the_o altitude_n kl_o wherefore_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o wherefore_o in_o the_o equal_a cylinder_n axe_n and_o eo_fw-la the_o base_n be_v reciprocal_a to_o their_o altitude_n but_o now_o suppose_v that_o the_o base_n of_o the_o cylinder_n axe_n and_o eo_fw-la be_v reciprokal_n to_o their_o altitude_n that_o be_v as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o demonstrate_v then_o i_o say_v that_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eo_fw-la for_o the_o self_n same_o order_n of_o construction_n remain_v for_o that_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o but_o the_o altitude_n kl_o be_v equal_a to_o the_o altitude_n pm_n wherefore_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n pm_n but_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o cylinder_n axe_n to_o the_o cylinder_n es_fw-ge for_o they_o be_v under_o equal_a altitude_n and_o as_o the_o altitude_n mn_o be_v to_o the_o altitude_n pm_n so_o be_v the_o cylinder_n eo_fw-la to_o the_o cylinder_n es_fw-ge by_o the_o 14._o of_o the_o twelve_o wherefore_o also_o as_o the_o cylinder_n axe_n be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o cylinder_n eo_fw-la to_o the_o cylinder_n es._n wherefore_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eo_fw-la by_o the_o 9_o of_o the_o five_o and_o so_o also_o be_v it_o in_o the_o cones_fw-la which_o haâ—â—_n the_o self_n same_o base_n and_o altitude_n with_o the_o cylinder_n wherefore_o in_o equal_a cones_fw-la and_o cylinder_n the_o base_n be_v reciprocal_a to_o their_o altitude_n etc_n etc_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o campane_n and_o flussas_n hitherto_o have_v be_v show_v the_o passion_n and_o propriety_n of_o cones_fw-la and_o cylinder_n who_o altitude_n fall_v perpendicular_o upon_o the_o base_n now_o will_v we_o declare_v that_o cones_fw-la and_o cilinder_n who_o altitude_n fall_v oblique_o upon_o their_o base_n have_v also_o the_o self_n same_o passion_n and_o propriety_n which_o the_o foresay_a cones_fw-la and_o cilinder_n have_v forasmuch_o as_o in_o the_o ten_o of_o this_o book_n it_o be_v say_v that_o every_o cone_n be_v the_o three_o part_n of_o a_o cilind_a have_v one_o and_o the_o self_n same_o base_a &_o one_o &_o the_o self_n same_o altitude_n with_o it_o which_o thing_n be_v demonstrate_v by_o a_o cilind_a give_v who_o base_a be_v cut_v by_o a_o square_n inscribe_v in_o it_o and_o upon_o the_o side_n of_o the_o square_n be_v describe_v isosceles_a triangle_n make_v a_o polygonon_n figure_n and_o again_o upon_o the_o side_n of_o this_o polygonon_n figure_n be_v infinite_o after_o the_o same_o manner_n describe_v other_o isosceles_a triangle_n take_v away_o more_o they_o the_o half_a as_o have_v oftentimes_o be_v declare_v therefore_o it_o be_v manifest_a that_o the_o solid_n set_v upon_o these_o base_n be_v under_o the_o same_o altitude_n that_o the_o cilind_a incline_v be_v and_o be_v also_o include_v in_o the_o same_o cilind_a do_v take_v away_o more_o than_o the_o half_a of_o the_o cilind_a and_o also_o more_o they_o the_o half_a of_o the_o residue_n as_o it_o have_v be_v prove_v in_o erect_a cylinder_n for_o these_o incline_n solid_n be_v under_o equal_a altitude_n and_o upon_o equal_a base_n with_o the_o erect_a solid_n be_v equal_a to_o the_o erect_a solid_n by_o the_o corollary_n of_o the_o _o of_o the_o eleven_o wherefore_o they_o also_o in_o like_a sort_n as_o the_o erect_a take_v away_o more_o than_o the_o half_a if_o therefore_o we_o compare_v the_o incline_v cilind_a to_o a_o cone_n set_v upon_o the_o self_n same_o base_a and_o have_v his_o altitude_n erect_v and_o reason_n by_o a_o argument_n lead_v to_o a_o impossibility_n by_o the_o demonstration_n of_o the_o ten_o of_o this_o book_n we_o may_v prove_v that_o the_o side_v solid_a include_v in_o the_o incline_v cylinder_n be_v great_a than_o the_o triple_a of_o his_o pyramid_n and_o it_o be_v also_o equal_a to_o the_o same_o which_o be_v impossible_a and_o this_o be_v the_o first_o case_n wherein_o it_o be_v prove_v that_o the_o cilind_a not_o be_v equal_a to_o the_o triple_a of_o the_o cone_n be_v not_o great_a than_o the_o triple_a of_o the_o same_o and_o as_o touch_v the_o second_o case_n we_o may_v after_o the_o same_o manner_n conclude_v that_o that_o â—ided_v solid_a contain_v in_o the_o cylinder_n be_v great_a than_o the_o cylinder_n which_o be_v very_o absurd_a wherefore_o if_o the_o cylinder_n be_v neither_o great_a than_o the_o triple_a of_o the_o cone_n nor_o less_o it_o must_v needs_o be_v equal_a to_o the_o same_o the_o demonstration_n of_o these_o incline_v cylinder_n most_v plain_o follow_v the_o demonstration_n of_o the_o erect_a cylinder_n for_o it_o have_v already_o be_v prove_v that_o pyramid_n and_o side_v solid_n which_o be_v also_o call_v general_o prism_n be_v set_v upon_o equal_a base_n and_o under_o one_o and_o the_o self_n same_o altitude_n whether_o the_o altitude_n be_v erect_v or_o incline_v be_v equal_a the_o one_o to_o the_o other_o namely_o be_v in_o proportion_n as_o their_o base_n be_v by_o
the_o 7._o of_o the_o five_o our_o conclusion_n be_v infer_v the_o superficies_n spherical_a of_o the_o segment_n caesar_n to_o be_v to_o the_o superficies_n spherical_a of_o the_o segment_n fgh_o as_o ad_fw-la be_v to_o give_v a_o theorem_a 6._o to_o any_o solid_a sector_n of_o a_o sphere_n that_o upright_o câ—â—e_n be_v equal_a who_o base_a be_v equal_a to_o the_o connex_v spherical_a superficies_n of_o that_o sector_n and_o heith_n equal_a to_o the_o semidiameter_n of_o the_o same_o sphere_n hereof_o the_o demonstration_n in_o respect_n of_o the_o premise_n and_o the_o common_a argument_n of_o inscription_n and_o circumscription_n of_o figure_n be_v easy_a and_o nevertheless_o if_o your_o own_o write_n will_v not_o help_v you_o sufficient_o you_o may_v take_v help_n at_o archimedes_n hand_n in_o his_o first_o book_n &_o last_o proposition_n of_o the_o sphere_n and_o cylinder_n whether_o if_o you_o have_v recourse_n you_o shalâ—_n perceive_v how_o your_o theorem_a here_o amend_v the_o common_a translation_n there_o and_o also_o our_o delineation_n give_v more_o sâ—uâ—y_a show_n of_o the_o chief_a circumstance_n necessary_a to_o the_o construction_n than_o there_o you_o shall_v find_v of_o the_o sphere_n here_o imagine_v to_o be_v a_o we_o note_v a_o solid_a sector_n by_o the_o letterâ—_n pqro._n so_o that_o pqr_n do_v signify_v the_o spherical_a superficies_n to_o that_o solid_a sector_n belong_v note_n which_o be_v also_o common_a to_o the_o segment_n of_o the_o same_o sphere_n prq_n and_o therefore_o a_o line_n draw_v from_o the_o top_n of_o that_o segmentâ—_n which_o top_n suppose_v to_o be_v queen_n be_v the_o semidiameter_n of_o the_o circle_n which_o be_v equal_a to_o the_o spherical_a superficies_n of_o the_o say_v solid_a sector_n or_o segmentâ—_n as_o before_o be_v teach_v let_v that_o line_n be_v qp_n by_o q_n draw_v a_o line_n contingent_a which_o let_v be_v sqt._n at_o the_o point_n q_o from_o the_o line_n q_n cut_v a_o line_n equal_a to_o pq_n which_o let_v be_v sq_n and_o unto_o sq_n make_v qt_o equal_a then_o draw_v the_o right_a line_n osoâ—_n and_o oq_fw-la about_o which_o oq_fw-la as_o a_o axe_n faâ—â—ened_v if_o you_o imagine_v the_o triangle_n ost_n to_o make_v a_o etc_n half_a circular_a revolution_n you_o shall_v have_v the_o upright_a cone_fw-mi ost_n who_o heith_n be_v oq_fw-la the_o semidiameter_n of_o the_o sphere_n and_o base_a the_o circle_n who_o diameter_n be_v n-ab_n equal_a to_o the_o solid_a sector_n pqro._n a_o theorem_a 7._o to_o any_o segment_n or_o portion_n of_o a_o sphere_n that_o cone_fw-mi iâ—_z equal_a which_o have_v that_o circle_n to_o his_o base_a which_o be_v the_o bâ—se_a of_o the_o segment_n and_o heith_n a_o right_a line_n which_o unto_o the_o heith_n of_o the_o segment_n have_v that_o proportion_n which_o the_o semidiameter_n of_o the_o sphere_n together_o with_o the_o heith_n of_o the_o other_o segment_n remayâ—â—â—g_o have_v to_o the_o heith_n of_o the_o same_o other_o segment_n remaynâ—ng_v this_o be_v well_o demonstrate_v by_o archiâ—â—des_n &_o therefore_o need_v no_o invention_n of_o i_o cylindrâ—_n to_o confirm_v the_o same_o and_o for_o that_o the_o say_a demonstration_n be_v over_o long_o here_o to_o be_v add_v i_o will_v refer_v you_o thither_o for_o the_o demonstration_n and_o here_o supply_v that_o which_o to_o archimedes_n demonstration_n shall_v geve_v light_a and_o to_o your_o far_a speculation_n and_o practice_n shall_v be_v a_o great_a aid_n and_o direction_n suppose_v king_n to_o be_v a_o sphere_n &_o the_o great_a circle_n king_n in_o contain_v let_v be_v abce_n and_o his_o diameter_n be_v &_o centre_n d._n let_v the_o sphere_n king_n be_v cut_v by_o a_o plain_a superficies_n perpendicular_o erect_v upon_o the_o say_v great_a circle_n abce_n let_v the_o section_n be_v the_o circle_n about_o ac_fw-la and_o let_v the_o segment_n of_o the_o sphere_n be_v the_o one_o that_o wherein_o be_v abc_n who_o â—oppe_n be_v bâ—_n and_o the_o other_o let_v be_v that_o wherein_o be_v aec_o and_o his_o top_n let_v be_v e_o i_o say_v that_o a_o cone_fw-mi which_o have_v his_o base_a the_o circle_n about_o ac_fw-la &_o hold_v a_o line_n which_o to_o bf_o the_o heith_n of_o the_o segment_n who_o top_n be_v b_o have_v that_o proportion_n that_o a_o line_n compoâ—ed_v of_o de_fw-fr the_o semidiameter_n of_o the_o sphere_n and_o of_o the_o heith_n of_o the_o other_o remain_v segment_n who_o top_n be_v e_o have_v to_o of_o the_o heith_n of_o that_o other_o segment_n remain_v be_v equal_a to_o the_o segment_n of_o the_o sphere_n king_n who_o top_n be_v b._n to_o make_v this_o cone_fw-mi take_v my_o easy_a order_n thus_o frame_v your_o work_n for_o the_o find_n of_o the_o four_o proportional_a lineâ—_z by_o make_v of_o the_o first_o and_o a_o line_n compose_v of_o de_fw-fr and_o of_o the_o secondâ—_n and_o the_o three_o let_v be_v bf_o then_o by_o the_o 12._o of_o the_o sixâ—h_o let_v the_o four_o proportional_a line_n be_v find_v which_o let_v be_v fgâ—_n upon_o fletcher_n the_o centre_n of_o the_o base_a of_o the_o segment_n who_o top_n be_v b_o erect_v a_o line_n perpendicular_a equal_a to_o fg_o find_v and_o draw_v the_o line_n ga_o and_o gc_o and_o so_o make_v perfect_a the_o cone_n gac_n i_o say_v that_o the_o cone_fw-mi gac_n be_v equal_a to_o the_o segment_n of_o the_o sphere_n king_n who_o top_n be_v b._n in_o like_a manner_n for_o the_o other_o segment_n who_o top_n be_v e_o to_o find_v the_o heith_n due_a for_o a_o cone_fw-mi equal_a to_o it_o by_o the_o order_n of_o the_o theorem_a you_o must_v thus_o frame_v your_o line_n let_v the_o first_o be_v bf_o the_o second_o db_o and_o bf_o compose_v in_o one_o right_a line_n and_o the_o three_o must_v be_v of_o where_o by_o the_o 12._o of_o the_o six_o find_v the_o four_o it_o shall_v be_v the_o heith_n to_o rear_v upon_o the_o base_a the_o circle_n about_o ac_fw-la to_o make_v a_o upright_a cone_fw-mi equal_a to_o the_o segment_n who_o top_n be_v e._n ¶_o logistical_o ¶_o the_o logistical_a find_v hereof_o be_v most_o easy_a the_o diameter_n of_o the_o sphere_n be_v give_v and_o the_o portion_n of_o the_o diameter_n in_o the_o segment_n contain_v or_o axe_n of_o the_o segment_n be_v know_v then_o order_n your_o number_n in_o the_o rule_n of_o proportion_n as_o i_o here_o have_v make_v most_o plain_a in_o order_v of_o the_o line_n for_o the_o â—ought_v heith_n will_v be_v the_o producte_v a_o corollary_n 1._o hereby_o and_o other_o the_o premise_n it_o be_v evident_a that_o to_o any_o segment_n of_o a_o sphere_n note_n who_o whole_a diameter_n be_v know_v and_o the_o axe_n of_o the_o segment_n give_v a_o upright_a cone_n may_v be_v make_v equal_a or_o in_o any_o proportion_n between_o two_o right_a line_n assignedâ—_n and_o therefore_o also_o a_o cylinder_n may_v to_o the_o say_a segment_n of_o the_o sphere_n be_v make_v equal_a â—r_n in_o any_o proportion_n give_v between_o two_o right_a line_n a_o corollary_n 2._o manifest_o also_o of_o the_o former_a theorem_a it_o may_v be_v infer_v that_o a_o sphere_n and_o his_o diameter_n be_v divide_v by_o one_o and_o the_o same_o plain_a superficies_n to_o which_o the_o say_a diameter_n be_v perpendicularâ—_n the_o two_o segment_n of_o the_o sphere_n be_v one_o to_o the_o other_o in_o that_o proportion_n in_o which_o a_o rectangle_n parallelipipedon_n have_v for_o his_o base_a the_o square_n of_o the_o great_a part_n of_o the_o diameter_n and_o his_o heith_n a_o line_n compose_v of_o the_o less_o portion_n of_o the_o diameter_n and_o the_o semidiameter_n to_o the_o rectangle_n parallelipâ—pedon_n have_v for_o his_o base_a the_o square_n of_o the_o less_o portion_n of_o the_o diameter_n &_o his_o heith_n a_o line_n compose_v of_o the_o semidiameter_n &_o the_o great_a part_n of_o the_o diameter_n a_o theorem_a 8._o every_o sphere_n to_o the_o cube_fw-la make_v of_o his_o diameter_n be_v in_o manner_n as_o 11._o to_o 21._o as_o upon_o the_o first_o and_o second_o proposition_n of_o this_o book_n i_o begin_v my_o addition_n with_o the_o circle_n be_v the_o chief_a among_o plain_a figure_n and_o therein_o bring_v manifold_a consideration_n about_o circle_n as_o of_o the_o proportion_n between_o their_o circumference_n and_o their_o diameter_n of_o the_o content_a or_o area_n of_o circle_n of_o the_o proportion_n of_o circle_n to_o the_o square_n describe_v of_o their_o diameter_n &_o of_o circle_n to_o be_v give_v in_o all_o proâ—portions_n to_o other_o circle_n with_o diverse_a other_o most_o necessary_a problem_n who_o use_n be_v partly_o there_o specify_v so_o have_v i_o in_o the_o end_n of_o this_o book_n add_v some_o such_o problem_n &_o theoremesâ—_n about_o the_o sphere_n be_v among_o solid_n the_o chief_a as_o of_o the_o same_o either_o in_o itself_o consider_v or_o to_o cone_n and_o cylinder_n compare_v by_o reason_n of_o
double_a to_o the_o side_n of_o the_o octohedron_n &_o the_o side_n be_v in_o power_n sequitertia_fw-la to_o the_o perpendiclar_a line_n by_o the_o 12._o of_o this_o book_n wherefore_o the_o diameter_n thereof_o be_v in_o power_n duple_fw-fr superbipartiens_fw-la tertias_fw-la to_o the_o perpendicular_a line_n wherefore_o also_o the_o diameter_n and_o the_o perpendicular_a line_n be_v rational_a and_o commensurable_a by_o the_o 6._o of_o the_o ten_o as_o touch_v a_o icosahedron_n it_o be_v prove_v in_o the_o 16._o of_o this_o book_n that_o the_o side_n thereof_o be_v a_o less_o line_n when_o the_o diameter_n of_o the_o sphere_n be_v rational_a and_o forasmuch_o as_o the_o angle_n of_o the_o inclination_n of_o the_o base_n thereof_o be_v contain_v of_o the_o perpendicular_a line_n of_o the_o triangle_n and_o subtend_v of_o the_o right_a line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n which_o contain_v five_o side_n of_o the_o icosahedron_n and_o unto_o the_o perpendicular_a line_n the_o side_n be_v commensurable_a namely_o be_v in_o power_n sesquitertia_fw-la unto_o they_o by_o the_o corollary_n of_o the_o 12._o of_o this_o book_n therefore_o the_o perpendicular_a line_n which_o contain_v the_o angle_n be_v irrational_a line_n namely_o less_o line_n by_o the_o 105._o of_o the_o ten_o book_n and_o forasmuch_o as_o the_o diameter_n contain_v in_o power_n both_o the_o side_n of_o the_o icosahedron_n and_o the_o line_n which_o subtend_v the_o foresay_a angle_n if_o from_o the_o power_n of_o the_o diameter_n which_o be_v rational_a be_v take_v away_o the_o power_n of_o the_o side_n of_o the_o icosahedron_n which_o be_v irrational_a it_o be_v manifest_a that_o the_o residue_n which_o be_v the_o power_n of_o the_o subtend_a line_n shall_v be_v irrational_a for_o if_o it_o shall_v be_v rational_a the_o number_n which_o measure_v the_o whole_a power_n of_o the_o diameter_n and_o the_o part_n take_v away_o of_o the_o subtend_a line_n shall_v also_o by_o the_o 4._o common_a sentence_n of_o the_o seven_o measure_n the_o residue_n namely_o the_o power_n of_o the_o side_n irrational_a which_o be_v irrational_a for_o that_o it_o be_v a_o less_o line_n which_o be_v absurd_a wherefore_o it_o be_v manifest_a that_o the_o right_a line_n which_o compose_v the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o the_o icosahedron_n be_v irrational_a line_n for_o the_o subtend_a line_n have_v to_o the_o line_n contain_v a_o great_a proportion_n than_o the_o whole_a have_v to_o the_o great_a segment_n the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o a_o dodecahedron_n be_v contain_v under_o two_o perpendicular_n of_o the_o base_n of_o the_o dodecahedron_n and_o be_v subtend_v of_o that_o right_a line_n who_o great_a segment_n be_v the_o side_n of_o a_o cube_n inscribe_v in_o the_o dodecahedron_n which_o right_a line_n be_v equal_a to_o the_o line_n which_o couple_v the_o section_n into_o two_o equal_a part_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n and_o this_o couple_a line_n we_o say_v be_v a_o irrational_a line_n for_o that_o the_o diameter_n of_o the_o sphere_n contain_v in_o power_n both_o the_o couple_a line_n and_o the_o side_n of_o the_o dodecahedron_n but_o the_o side_n of_o the_o dodecahedron_n be_v a_o irrational_a line_n namely_o a_o residual_a line_n by_o the_o 17._o of_o this_o book_n wherefore_o the_o residue_n namely_o the_o couple_a line_n be_v a_o irrational_a line_n as_o it_o be_v â—asy_a to_o prove_v by_o the_o 4._o common_a sentence_n of_o the_o seven_o and_o that_o the_o perpendicular_a line_n which_o contain_v the_o angle_n of_o the_o inclination_n be_v irrational_a be_v thus_o prove_v by_o the_o proportion_n of_o the_o subtend_a line_n of_o the_o foresay_a angle_n of_o inclination_n to_o the_o line_n which_o contain_v the_o angle_n be_v find_v out_o the_o obliquity_n of_o the_o angle_n angle_n for_o if_o the_o subtend_a line_n be_v in_o power_n double_a to_o the_o line_n which_o contain_v the_o angle_n then_o be_v the_o angle_n a_o right_a angle_n by_o the_o 48._o of_o the_o first_o but_o if_o it_o be_v in_o power_n less_o than_o the_o double_a it_o be_v a_o acute_a angle_n by_o the_o 23._o of_o the_o second_o but_o if_o it_o be_v in_o power_n more_o than_o the_o double_a or_o have_v a_o great_a proportion_n than_o the_o whole_a have_v to_o the_o great_a segmentâ—_n the_o angle_n shall_v be_v a_o obtuse_a angle_n by_o the_o 12._o of_o the_o second_o and_o 4._o of_o the_o thirteen_o by_o which_o may_v be_v prove_v that_o the_o square_a of_o the_o whole_a be_v great_a than_o the_o double_a of_o the_o square_n of_o the_o great_a segment_n this_o be_v to_o be_v note_v that_o that_o which_o flussas_n have_v here_o teach_v touch_v the_o inclination_n of_o the_o base_n of_o the_o â—ive_a regular_a body_n hypsicles_n teach_v after_o the_o 5_o proposition_n of_o the_o 15._o book_n where_o he_o confess_v that_o he_o receive_v it_o of_o one_o isidorus_n and_o seek_v to_o make_v the_o matter_n more_o clear_a he_o endeavour_v himself_o to_o declare_v that_o the_o angle_n of_o the_o inclination_n of_o the_o solid_n be_v give_v and_o that_o they_o be_v either_o acute_a or_o obtuse_a according_a to_o the_o nature_n of_o the_o solid_a although_o euclid_v in_o all_o his_o 15._o book_n have_v not_o yet_o show_v what_o a_o thing_n give_v be_v wherefore_o flussas_n frame_v his_o demonstration_n upon_o a_o other_o ground_n proceed_v after_o a_o other_o manner_n which_o seem_v more_o plain_a and_o more_o apt_o hereto_o be_v place_v then_o there_o albeit_o the_o reader_n in_o that_o place_n shall_v not_o be_v frustrate_a of_o his_o also_o the_o end_n of_o the_o thirteen_o book_n of_o euclides_n element_n ¶_o the_o fourteen_o book_n of_o euclides_n element_n in_o this_o book_n which_o be_v common_o account_v the_o 14._o book_n of_o euclid_n be_v more_o at_o large_a entreat_v of_o our_o principal_a purpose_n book_n namely_o of_o the_o comparison_n and_o proportion_n of_o the_o five_o regular_a body_n customable_o call_v the_o 5._o figure_n or_o form_n of_o pythagoras_n the_o one_o to_o the_o other_o and_o also_o of_o their_o side_n together_o each_o to_o other_o which_o thing_n be_v of_o most_o secret_a use_n and_o inestimable_a pleasure_n and_o commodity_n to_o such_o as_o diligent_o search_v for_o they_o and_o attain_v unto_o they_o which_o thing_n also_o undoubted_o for_o the_o worthiness_n and_o hardness_n thereof_o for_o thing_n of_o most_o price_n be_v most_o hard_a be_v first_o search_v and_o find_v out_o of_o philosopher_n not_o of_o the_o inferior_a or_o mean_a sort_n but_o of_o the_o deep_a and_o most_o ground_a philosopher_n and_o best_a exercise_v in_o geometry_n and_o albeit_o this_o book_n with_o the_o book_n follow_v namely_o the_o 15._o book_n have_v be_v hitherto_o of_o all_o man_n for_o the_o most_o part_n and_o be_v also_o at_o this_o day_n number_v and_o account_v among_o euclides_n book_n and_o suppose_v to_o be_v two_o of_o he_o namely_o the_o 14._o and_o 15._o in_o order_n as_o all_o exemplar_n not_o only_o new_a and_o late_o set_v abroad_o but_o also_o old_a monument_n write_v by_o hand_n do_v manifest_o witness_v yet_o it_o be_v think_v by_o the_o best_a learned_a in_o these_o day_n that_o these_o two_o book_n be_v none_o of_o euclides_n but_o of_o some_o other_o author_n no_o less_o worthy_a nor_o of_o less_o estimation_n and_o authority_n notwithstanding_o than_o euclid_n apollonius_n a_o man_n of_o deep_a knowledge_n a_o great_a philosopher_n and_o in_o geometry_n marvellous_a who_o wonderful_a book_n write_v of_o the_o section_n of_o cones_fw-la which_o exercise_n &_o occupy_v thewitte_v of_o the_o wise_a and_o best_a learned_a be_v yet_o remain_v be_v think_v and_o that_o not_o without_o just_a cause_n to_o be_v the_o author_n of_o they_o or_o as_o some_o think_v hypsicles_n himself_o for_o what_o can_v be_v more_o plain_o then_o that_o which_o he_o himself_o witness_v in_o the_o preface_n of_o this_o book_n basilides_n of_o tire_n say_v hypsicles_n and_o my_o father_n together_o scan_v and_o peyse_v a_o writing_n or_o book_n of_o apollonius_n which_o be_v of_o the_o comparison_n of_o a_o dodecahedron_n to_o a_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n and_o what_o proportion_n these_o figure_n have_v the_o one_o to_o the_o other_o find_v that_o apollonius_n have_v fail_v in_o this_o matter_n but_o afterward_o say_v he_o i_o find_v a_o other_o copy_n or_o book_n of_o apollonius_n wherein_o the_o demonstration_n of_o that_o matter_n be_v full_a and_o perfect_a and_o show_v it_o unto_o they_o whereat_o they_o much_o rejoice_v by_o which_o word_n it_o seem_v to_o be_v manifest_a that_o apollonius_n be_v the_o first_o author_n of_o this_o book_n which_o be_v afterward_o set_v forth_o by_o hypsicles_n for_o so_o his_o own_o word_n after_o in_o
spherâ—_n contain_v the_o dodecahedron_n of_o this_o pentagon_n and_o the_o icosahedron_n of_o this_o triangle_n by_o the_o 4._o of_o this_o book_n â—_o and_o the_o line_n cl_n fall_v perpendicular_o upon_o the_o side_n of_o the_o icosahedron_n and_o the_o line_n ci_o upon_o the_o side_n of_o the_o dodecahedron_n that_o which_o be_v 30._o time_n contain_v under_o the_o side_n and_o the_o perpendicular_a line_n fall_v upon_o it_o be_v equal_a to_o the_o superficies_n of_o that_o solid_a upon_o who_o side_n the_o perpendicular_a fall_v if_o therefore_o in_o a_o circle_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n order_n the_o superficiece_n of_o a_o dodecahedron_n and_o of_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n be_v the_o one_o to_o the_o other_o as_o that_o which_o be_v contain_v under_o the_o side_n of_o the_o one_o and_o the_o perpendicular_a line_n draw_v unto_o it_o from_o the_o centre_n of_o his_o base_a to_o that_o which_o be_v contain_v under_o the_o side_n of_o the_o other_o and_o the_o perpendicular_a line_n draw_v to_o it_o from_o the_o centre_n of_o his_o base_a for_o aâ—_z thirtyâ—_n timâ—s_o be_v to_o thirty_o time_n so_o be_v once_o to_o once_o by_o the_o 15._o of_o thâ—_z five_o the_o 6._o proposition_n the_o superficies_n of_o a_o dodecahedron_n be_v to_o the_o superficies_n of_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n campane_n in_o that_o proportion_n that_o the_o side_n of_o the_o cube_n be_v to_o the_o side_n of_o the_o icosahedron_n contain_v in_o the_o self_n same_o sphere_n construction_n svppose_v that_o there_o be_v a_o circle_n abg_o &_o in_o it_o by_o the_o 4._o of_o this_o book_n let_v there_o be_v inscribe_v the_o sideâ—_n of_o a_o dodecahedron_n and_o of_o a_o icosahedron_n contain_v in_o onâ—_n and_o the_o self_n same_o sphere_n and_o let_v the_o side_n oâ—_n the_o dodecahedron_n be_v agnostus_n and_o the_o side_n of_o the_o icosahedron_n be_v dg_o and_o let_v the_o centre_n be_v the_o point_n e_o from_o which_o draw_v unto_o those_o sâ—des_n perpendicular_a line_n ei_o and_o ez_o and_o produce_v the_o line_n ei_o to_o the_o point_n b_o and_o draw_v the_o linâ—_n bg_o and_o let_v the_o side_n of_o the_o cube_fw-la contain_v in_o the_o self_n same_o sphere_n be_v gc_o then_o i_o say_v that_o the_o superficies_n of_o the_o dodecahedron_n iâ—_z to_o the_o superficies_n of_o the_o icosahedron_n as_o the_o line_n â—g_z iâ—_z to_o the_o liâ—â—_n gd_o for_o forasmuch_o as_o the_o line_n ei_o beinâ—_n divide_v by_o a_o extreme_a and_o mean_a proportion_n the_o great_a segment_n thereof_o shall_v be_v the_o linâ—_n ez_o by_o the_o corollary_n of_o the_o first_o of_o this_o book_n demonstration_n and_o the_o line_n cg_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n his_o great_a segment_n be_v the_o line_n agnostus_n by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o wherefore_o the_o right_a line_n ei_o and_o cg_o â—râ—_n cut_v proportional_o by_o the_o second_o of_o this_o bâ—oke_n whâ—râ—fore_o as_o the_o line_n cg_o be_v to_o the_o line_n agnostus_n so_o be_v the_o line_n ei_o to_o the_o line_n ez_fw-fr wherâ—fore_o that_o which_o it_o contain_v under_o the_o extreme_n cg_o and_o ez_o be_v squall_n to_o that_o which_o iâ—_z contaynâ—d_v under_o the_o mean_n agnostus_n and_o ei._n by_o the_o 16._o of_o the_o six_o but_o as_o that_o which_o iâ—_z contain_v under_o the_o linâ—â—_n cg_o and_o â—z_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n dg_o and_o ez_o so_o by_o the_o first_o of_o the_o six_o iâ—_z the_o linâ—_n cg_o to_o the_o line_n dg_o for_o both_o those_o parallelogram_n have_v oâ—â—_n and_o the_o self_n same_o altitude_n namely_o the_o line_n ez_fw-fr wherefore_o as_o that_o which_o be_v contain_v under_o the_o line_n ei_o and_o agnostus_n which_o iâ—_z prove_v equal_a to_o that_o which_o be_v contain_v under_o the_o lineâ—_z cg_o and_o ez_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n dg_o and_o ez_o so_o be_v the_o line_n cg_o to_o the_o liâ—â—_n dg_o but_o as_o that_o which_o be_v contain_v under_o the_o line_n ei_o and_o agnostus_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n dg_o and_o ez_o so_o by_o the_o corollary_n of_o the_o former_a proposition_n be_v the_o superficies_n of_o the_o dodecahedron_n to_o the_o superficies_n of_o the_o icosahedron_n wherefore_o as_o the_o superficies_n â—â—_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v cg_o the_o side_n of_o the_o cube_fw-la to_o gd_v the_o side_n of_o the_o icosahedron_n the_o superficies_n therefore_o of_o a_o dodecahedron_n be_v to_o the_o superficiesâ—_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v require_v to_o be_v prove_v a_o assumpt_n the_o pentagon_n of_o a_o dodecahedron_n order_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o perpendicular_a line_n which_o fall_v upon_o the_o base_a of_o the_o triangle_n of_o the_o icosahedron_n and_o five_o six_o part_n of_o the_o side_n of_o the_o cube_fw-la the_o say_v three_o solid_n be_v describe_v in_o one_o and_o the_o self_n same_o sphere_n suppose_v that_o in_o the_o circle_n abeg_n the_o pentagon_n of_o a_o dodecahedron_n be_v aâ—cig_a and_o let_v two_o side_n thereof_o ab_fw-la and_o agnostus_n be_v subtend_v of_o the_o right_a line_n bg_o and_o let_v the_o triangle_n of_o the_o icosahedron_n inscribe_v in_o the_o self_n same_o sphere_n construction_n by_o the_o 4._o of_o this_o book_n be_v afh_n and_o let_v the_o centre_n of_o the_o circle_n be_v the_o point_n d_o and_o let_v the_o diameter_n be_v ade_n cut_v fh_o the_o side_n of_o the_o triangle_n in_o the_o point_n z_o and_o cut_v the_o line_n bg_o in_o the_o point_n k._n and_o draw_v the_o right_a line_n bd._n and_o from_o the_o right_a line_n kg_v cut_v of_o a_o three_o part_n tg_n by_o the_o 9_o of_o the_o six_o now_o than_o the_o line_n bg_o subtend_v two_o side_n of_o the_o dodecahedron_n shall_v be_v the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o same_o sphere_n demonstration_n by_o the_o 17._o of_o the_o thirteen_o and_o the_o triangle_n of_o the_o icosahedron_n of_o the_o same_o sphere_n shall_v be_v aâ—h_a by_o the_o 4._o of_o this_o book_n and_o the_o line_n az_o which_o pass_v by_o the_o centre_n d_o shall_v fall_v perpendicular_o upon_o the_o side_n of_o the_o triangle_n for_o forasmuch_o as_o the_o angle_n gay_a &_o bae_o be_v equal_a by_o the_o 27._o of_o the_o thirdâ—_o for_o they_o be_v see_v upon_o equal_a circumference_n therefore_o the_o â—ases_v bk_o and_o kg_n be_v by_o the_o â—_o of_o the_o first_o equal_a wherefore_o the_o line_n bt_n contain_v 5._o six_o part_n of_o the_o line_n bg_o then_o i_o say_v that_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o bt_n be_v equal_a to_o the_o pentagon_n aâ—câ—g_n for_o forasmuch_o as_o the_o line_n â—z_n be_v sesquialter_n to_o the_o line_n ad_fw-la for_o the_o line_n dâ—_n be_v divide_v into_o two_o equal_a part_n in_o the_o point_n z_o by_o the_o corollary_n of_o the_o â—2â—_n of_o the_o thirteen_o likewise_o by_o construction_n the_o line_n kg_n be_v sesquialter_fw-la to_o the_o line_n kt_n therefore_o as_o the_o line_n az_o be_v to_o the_o line_n ad_fw-la so_o be_v the_o line_n kg_v to_o the_o 〈◊〉_d â—t_a wherefore_o that_o which_o be_v contain_v undeâ—_n the_o 〈◊〉_d az_o and_o kt_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o mean_n ad_fw-la and_o kg_n by_o the_o 16._o of_o the_o six_o but_o unto_o the_o line_n kg_n be_v the_o line_n â—k_v â—roved_v equal_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o bk_o but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o bk_o be_v by_o the_o 41._o of_o the_o first_o double_a to_o the_o triangle_n abdella_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n be_v double_a to_o the_o same_o triangle_n abdella_n and_o forasmuch_o as_o the_o pentagon_n abcig_n contaynethâ—_n 〈…〉_o equal_a â—o_o the_o triangle_n abdella_n and_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n contain_v two_o such_o triangle_n therefore_o the_o pentagon_n abcig_n be_v duple_fw-fr sesquialter_fw-la to_o the_o rectangle_n parallelogram_n contain_v under_o the_o line_n az_o and_o kt_n and_o 〈…〉_o 1._o of_o the_o six_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o bt_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n as_o the_o base_a bt_n be_v to_o the_o base_a â—â—tâ—_n therefore_o that_o which_o be_v contain_v under_o the_o line_n az_o
def_n who_o side_n let_v be_v de_fw-fr and_o let_v the_o right_a line_n subtend_v the_o angle_n of_o the_o pentagon_n make_v of_o the_o side_n of_o the_o icosahedron_n be_v the_o line_n ef._n then_o i_o say_v that_o the_o side_n ed_z be_v in_o power_n double_a to_o the_o line_n h_o the_o less_o of_o those_o segment_n forasmuch_o as_o by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n demonstration_n it_o be_v manifest_a that_o ed_z the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n of_o the_o line_n efâ—_n and_o that_o the_o diameter_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o of_o namely_o the_o whole_a and_o the_o great_a segment_n but_o by_o supposition_n the_o side_n ab_fw-la contain_v in_o power_n the_o two_o line_n c_o &_o h_o join_v together_o in_o the_o self_n same_o proportion_n wherefore_o the_o line_n of_o be_v to_o the_o line_n ed_z as_z the_o line_n c_o be_v to_o the_o line_n h_o by_o the_o â—_o oâ—_n this_o bokeâ—_n and_o alternally_n by_o the_o 16._o of_o the_o five_o the_o line_n of_o be_v to_o the_o line_n c_o as_o the_o line_n ed_z be_v to_o the_o line_n h._n and_o forasmuch_o as_o the_o line_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o of_o and_o the_o line_n ab_fw-la contain_v in_o power_n the_o two_o line_n c_o and_o h_o therefore_o the_o square_n of_o the_o line_n of_o and_o ed_z be_v to_o the_o square_n of_o the_o line_n df_o as_o the_o square_n of_o the_o line_n c_o and_o h_o to_o the_o square_a ab_fw-la and_o alternate_o the_o square_n of_o the_o line_n of_o and_o â—d_a be_v to_o the_o square_n of_o the_o line_n c_o and_o h_o as_o the_o square_n of_o the_o line_n df_o be_v to_o the_o square_n of_o the_o line_n abâ—_n but_o df_o the_o diameter_n be_v by_o the_o 14._o of_o the_o thirtenâ—h_o iâ—_z power_n double_a to_o ab_fw-la the_o side_n of_o the_o octohedron_n inscribe_v by_o supposition_n in_o the_o same_o sphere_n wherefore_o the_o square_n of_o the_o line_n of_o and_o ed_z be_v double_a to_o the_o square_n of_o the_o line_n c_o and_o h._n and_o therefore_o one_o square_a of_o the_o line_n ed_z be_v double_a to_o one_o square_a of_o the_o line_n h_o by_o the_o 12._o of_o the_o five_o wherefore_o ed_z the_o side_n of_o the_o icosahedron_n be_v in_o power_n duple_n to_o the_o line_n h_o which_o be_v the_o less_o segment_n if_o therefore_o the_o poweâ—_n of_o the_o side_n of_o a_o octohedron_n be_v express_v by_o two_o right_a line_n join_v together_o by_o a_o extreme_a and_o mean_a proportion_n the_o side_n of_o the_o icosahedron_n contain_v in_o the_o same_o sphere_n shall_v be_v duple_v to_o the_o less_o segment_n the_o 17._o proposition_n if_o the_o side_n of_o a_o dodecahedron_n and_o the_o right_a line_n of_o who_o the_o say_a side_n be_v the_o less_o segment_n be_v so_o set_v that_o they_o make_v a_o right_a angle_n the_o right_a line_n which_o contain_v in_o power_n half_o the_o line_n subtend_v the_o angle_n be_v the_o side_n of_o a_o octohedron_n contain_v in_o the_o self_n same_o sphere_n svppose_v that_o ab_fw-la be_v the_o side_n of_o a_o dodecahedron_n and_o let_v the_o right_a line_n of_o which_o that_o side_n be_v the_o less_o segment_n be_v agnostus_n namely_o which_o couple_v the_o opposite_a side_n of_o the_o dodecahedron_n by_o the_o 4._o corollary_n of_o the_o 17._o of_o the_o thirteen_o construction_n and_o let_v those_o line_n be_v so_o set_v that_o they_o make_v a_o right_a angle_n at_o the_o point_n a._n and_o draw_v the_o right_a line_n bg_o and_o let_v the_o line_n d_o contain_n in_o power_n half_o the_o line_n bg_o by_o the_o first_o proposition_n add_v by_o flussas_n after_o the_o last_o of_o the_o six_o then_o i_o say_v that_o the_o line_n d_o be_v the_o side_n of_o a_o octohedron_n contain_v in_o the_o same_o sphere_n forasmuch_o as_o the_o line_n agnostus_n make_v the_o great_a segment_n gc_o the_o side_n of_o the_o cube_fw-la contain_v in_o the_o same_o sphere_n by_o the_o same_o 4._o corollary_n of_o the_o 17._o of_o the_o thirteen_o demonstration_n and_o the_o square_n of_o the_o whole_a line_n ag._n and_o of_o the_o less_o segment_n ab_fw-la be_v triple_a to_o the_o square_n of_o the_o great_a segment_n gc_o by_o the_o 4._o of_o the_o thirteen_o moreover_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n triple_a to_o the_o same_o line_n gc_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o the_o thirteen_o wherefore_o the_o line_n bg_o be_v equal_a to_o the_o 〈◊〉_d for_o it_o contain_v in_o power_n the_o two_o line_n ab_fw-la and_o ag_z by_o the_o 47._o of_o the_o first_o and_o therefore_o it_o contain_v in_o power_n the_o triple_a of_o the_o line_n gc_o but_o the_o side_n of_o the_o octohedron_n contain_v in_o the_o same_o sphere_n be_v in_o power_n triple_a to_o half_a the_o diameter_n of_o the_o sphere_n by_o the_o 14._o of_o the_o thirteen_o and_o by_o supposition_n the_o line_n d_o contaiâ—â—â—â—_n in_o powâ—â—_n the_o half_a of_o the_o line_n bg_o wherefore_o the_o line_n d_o contain_v in_o power_n the_o half_a of_o the_o same_o diameter_n be_v the_o side_n of_o a_o octohedron_n if_o therefore_o the_o side_n of_o a_o dodecahedron_n and_o the_o right_a line_n of_o who_o the_o say_a side_n be_v the_o less_o segment_n be_v so_o set_v that_o they_o make_v a_o right_a angle_n the_o right_a line_n which_o contain_v in_o power_n half_o the_o line_n subtend_v the_o angle_n be_v the_o side_n of_o a_o ocâ—â—â—edron_n contain_v in_o the_o self_n same_o sphere_n which_o be_v require_v to_o be_v prove_v a_o corollary_n unto_o what_o right_a line_n the_o side_n of_o the_o octoâ—edron_n be_v in_o power_n sesquialter_fw-la unto_o the_o same_o line_n the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o sphere_n be_v the_o great_a segment_n for_o the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o the_o segment_n cg_o unto_o which_o d_z the_o side_n of_o the_o octohedron_n be_v in_o power_n sesquialter_n that_o be_v be_v half_a of_o the_o power_n of_o the_o line_n bg_o which_o be_v triple_a unto_o the_o line_n cg_o ¶_o the_o 18._o proposition_n if_o the_o side_n of_o a_o tetrahedron_n contain_v in_o power_n two_o right_a line_n join_v together_o by_o a_o extreme_a and_o mean_a proportion_n the_o side_n of_o a_o icosahedron_n describe_v in_o the_o self_n same_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o less_o right_a line_n svppose_v that_o abc_n be_v a_o tetrahedron_n and_o let_v his_o side_n be_v ab_fw-la construction_n who_o power_n let_v be_v divide_v into_o the_o line_n agnostus_n and_o gb_o join_v together_o by_o a_o extreme_a and_o mean_a proportion_n namely_o let_v it_o be_v divide_v into_o agnostus_n the_o whole_a line_n and_o gb_o the_o great_a seâ—ment_n by_o the_o corollary_n of_o the_o first_o proposition_n add_v by_o flussas_n after_o the_o last_o of_o the_o six_o and_o let_v ed_z be_v the_o side_n of_o the_o icosahedron_n edf_o contain_v in_o the_o self_n same_o sphere_n and_o let_v the_o line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n describe_v of_o the_o side_n of_o the_o icosahedron_n be_v ef._n then_o i_o say_v that_z ed_z the_o side_n of_o the_o icosahedron_n be_v in_o power_n sesquialter_fw-la to_o the_o less_o line_n gb_o demonstration_n forasmuch_o as_o by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n the_o side_n ed_z be_v the_o greâ—ter_a segment_n of_o the_o line_n of_o which_o subtend_v the_o angle_n of_o the_o pentagon_n but_o as_o the_o whole_a line_n of_o be_v to_o the_o great_a segment_n ed_z so_o be_v the_o same_o grâ—â—ter_a segment_n to_o the_o less_o by_o the_o 30._o of_o the_o six_o and_o by_o supposition_n agnostus_n be_v the_o whole_a line_n and_o gâ—_n the_o great_a segment_n wherefore_o as_o of_o be_v to_o ed_z so_o be_v agnostus_n to_o gâ—_n by_o the_o second_o of_o the_o fourteen_o and_o alternate_o the_o line_n of_o be_v to_o the_o line_n agnostus_n as_o the_o line_n ed_z be_v to_o the_o line_n gb_o and_o forasmuch_o as_o by_o supposition_n the_o line_n ab_fw-la contain_v in_o power_n the_o two_o line_n agnostus_n and_o gb_o therefore_o by_o the_o 4â—_o of_o the_o first_o the_o angle_n agb_n be_v a_o right_a angle_n but_o the_o angle_n def_n be_v a_o right_a angle_n by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n wherefore_o the_o triangle_n agâ—_n and_o feed_v be_v equiangle_n by_o the_o â—_o of_o the_o six_o wherefore_o their_o side_n be_v proportional_a namely_o as_o the_o line_n ed_z be_v to_o the_o line_n gb_o so_o be_v the_o line_n fd_o to_o the_o line_n ab_fw-la by_o the_o 4._o of_o the_o six_o but_o by_o that_o which_o have_v before_o
overthrow_v and_o overwhelm_v the_o whole_a world_n he_o be_v utter_o rude_a and_o ignorant_a in_o the_o greek_a tongue_n so_o that_o certain_o he_o never_o read_v euclid_n in_o the_o greek_a nor_o of_o like_a translate_v out_o of_o the_o greek_a but_o have_v it_o translate_v out_o of_o the_o arabike_a tongue_n the_o arabian_n be_v man_n of_o great_a study_n and_o industry_n and_o common_o great_a philosopher_n notable_a physician_n and_o in_o mathematical_a art_n most_o expert_a so_o that_o all_o kind_n of_o good_a learning_n flourish_v and_o reign_v amongst_o they_o in_o a_o manner_n only_o these_o man_n turn_v whatsoever_o good_a author_n be_v in_o the_o greek_a tongue_n of_o what_o art_n and_o knowledge_n so_o ever_o it_o be_v into_o the_o arabike_a tongue_n and_o from_o thence_o be_v many_o of_o they_o turn_v into_o the_o latin_a and_o by_o that_o mean_v many_o greek_a author_n come_v to_o the_o hand_n of_o the_o latin_n and_o not_o from_o the_o first_o fountain_n the_o greek_a tongue_n wherein_o they_o be_v first_o write_v as_o appear_v by_o many_o word_n of_o the_o arabike_a tongue_n yet_o remain_v in_o such_o book_n as_o be_v zenith_n nadir_n helmuayn_fw-mi helmuariphe_n and_o infinite_a such_o other_o which_o arabian_n also_o in_o translate_n such_o greek_a work_n be_v accustom_v to_o add_v as_o they_o think_v good_a &_o for_o the_o full_a understanding_n of_o the_o author_n many_o thing_n as_o be_v to_o be_v see_v in_o diverse_a author_n as_o namely_o in_o theodosius_n de_fw-la sphera_fw-la where_o you_o see_v in_o the_o old_a translation_n which_o be_v undoubteldy_a out_o of_o the_o arabike_a many_o proposition_n almost_o every_o three_o or_o four_o leaf_n some_o such_o copy_n of_o euclid_n most_o likely_a do_v campanus_n follow_v wherein_o he_o find_v those_o proposition_n which_o he_o have_v more_o &_o above_o those_o which_o be_v find_v in_o the_o greek_a set_v out_o by_o hypsicles_n and_o that_o not_o only_o in_o this_o 15._o book_n but_o also_o in_o the_o 14._o book_n wherein_o also_o you_o find_v many_o proposition_n more_o they_o be_v find_v in_o the_o greek_a set_v out_o also_o by_o hypsicles_n likewise_o in_o the_o book_n before_o you_o shall_v find_v many_o proposition_n add_v and_o many_o invert_v and_o set_v out_o of_o order_n far_o otherwise_o than_o they_o be_v place_v in_o the_o greek_a examplar_n flussas_n also_o a_o diligent_a restorer_n of_o euclid_n a_o man_n also_o which_o have_v well_o deserve_v of_o the_o whole_a art_n of_o geometry_n have_v add_v moreover_o in_o this_o book_n as_o also_o in_o the_o former_a 14._o book_n he_o add_v 8._o proposition_n 9_o proposition_n of_o his_o own_o touch_v the_o inscription_n and_o circumscription_n 〈…〉_o body_n very_o singular_a â—ndoubtedly_o and_o witty_a all_o which_o for_o that_o nothing_o shall_v want_v to_o the_o desirous_a lover_n of_o knowledge_n i_o have_v faithful_o with_o no_o small_a pain_n turn_v and_o whereas_o flâ—ssâ—â—_n in_o the_o begin_n of_o the_o eleven_o book_n namely_o in_o the_o end_n of_o the_o definition_n there_o â—eâ—_n put_v two_o definition_n of_o the_o inscription_n and_o circumscription_n of_o solid_n or_o corporal_a figure_n within_o or_o about_o the_o one_o the_o other_o which_o certain_o be_v not_o to_o be_v reject_v yet_o for_o that_o until_o this_o present_a 15._o book_n there_o be_v no_o mention_n make_v of_o the_o inscription_n or_o circumscription_n of_o these_o body_n i_o think_v it_o not_o so_o convenient_a thâ—râ—_n to_o place_v they_o but_o to_o refer_v they_o to_o the_o begin_n of_o this_o 15._o book_n where_o they_o be_v in_o manner_n of_o necessity_n require_v to_o the_o elucidation_n of_o the_o proposition_n and_o dâ—monstrationâ—_n of_o the_o same_o the_o definition_n be_v these_o definition_n 1._o a_o solid_a figure_n be_v then_o â—aid_v to_o be_v inscribe_v in_o a_o solid_a figure_n when_o the_o angle_n of_o the_o figure_n inscribe_v touch_n together_o at_o one_o time_n either_o the_o angle_n of_o the_o figure_n circumscribe_v or_o the_o superficiece_n or_o the_o side_n definition_n 2._o a_o solid_a figure_n be_v then_o say_v to_o be_v circumscribe_v about_o a_o solid_a figure_n when_o together_o at_o one_o time_n either_o the_o angle_n or_o the_o superficiece_n or_o the_o side_n of_o the_o figure_n circumscribe_v â—ouch_v the_o angle_n of_o the_o figure_n inscribe_v in_o the_o fourâ—â—_n book_n in_o the_o definition_n of_o the_o inscription_n or_o circumscription_n of_o plain_a rectiline_n figure_v one_o with_o in_o or_o about_o a_o other_o be_v require_v that_o all_o the_o angle_n of_o the_o figuâ—â—_n inscribe_v shall_v at_o one_o time_n touch_v all_o the_o side_n of_o the_o figure_n circumscribe_v but_o in_o the_o five_o regular_a solid_n â—o_o who_o chief_o these_o two_o definition_n pertain_v for_o that_o the_o number_n of_o their_o angle_n superficiece_n &_o side_n be_v not_o equal_a one_o compare_v to_o a_o other_o it_o be_v not_o of_o necessity_n that_o all_o the_o angle_n of_o the_o solid_a inscribe_v shall_v together_o at_o one_o time_n touch_v either_o all_o the_o angle_n or_o all_o the_o superficiece_n or_o all_o the_o side_n of_o the_o solid_a circumscribe_v but_o it_o be_v sufficient_a that_o those_o angle_n of_o the_o inscribe_v solid_a which_o touch_n do_v at_o one_o time_n together_o each_o touch_n some_o one_o angle_n of_o the_o figure_n circumscribe_v or_o some_o one_o base_a or_o some_o one_o side_n so_o that_o if_o the_o angle_n of_o the_o inscribe_v figure_n do_v at_o one_o time_n touch_v the_o angle_n of_o the_o figure_n circumscribe_v none_o of_o they_o may_v at_o the_o same_o time_n touch_v either_o the_o base_n or_o the_o side_n of_o the_o same_o circumscribe_v figure_n and_o so_o if_o they_o touch_v the_o base_n they_o may_v touch_v neither_o angle_n nor_o side_n and_o likewise_o if_o they_o touch_v the_o side_n they_o may_v touch_v neither_o angle_n nor_o base_n and_o although_o sometime_o all_o the_o angle_n of_o the_o figure_n inscribe_v can_v not_o touch_v either_o the_o angle_n or_o the_o base_n or_o the_o side_n of_o the_o figure_n circumscribe_v by_o reason_n the_o number_n of_o the_o angle_n base_n or_o side_n of_o the_o say_a figure_n circumscribe_v want_v of_o the_o number_n of_o the_o angle_n of_o the_o â—igure_n inscribe_v yet_o shall_v those_o angle_n of_o the_o inscribe_v figure_n which_o touch_n so_o touch_v that_o the_o void_a place_n leave_v between_o the_o inscribe_v and_o circumscribe_v figure_n shall_v on_o every_o side_n be_v equal_a and_o like_a as_o you_o may_v afterward_o in_o this_o fifteen_o book_n most_o plain_o perceive_v ¶_o the_o 1._o proposition_n the_o 1._o problem_n in_o a_o cube_n give_v to_o describe_v admonish_v a_o trilater_n equilater_n pyramid_n svppose_v that_o the_o cube_fw-la give_v be_v abcdefgh_o in_o the_o same_o cube_fw-la it_o be_v require_v to_o inscribe_v a_o tetrahedron_n draw_v these_o right_a line_n ac_fw-la construction_n ce_fw-fr ae_n ah_o eh_o hc_n demonstration_n now_o it_o be_v manifest_a that_o the_o triangle_n aec_o ahe_o ahc_n and_o i_o be_v equilater_n for_o their_o side_n be_v the_o diameter_n of_o equal_a square_n wherefore_o aech_n be_v a_o trilater_n equilater_n pyramid_n or_o tetrahedron_n &_o it_o be_v inscribe_v in_o the_o cube_fw-la give_v by_o the_o first_o definition_n of_o this_o book_n which_o be_v require_v to_o be_v do_v ¶_o the_o 2._o proposition_n the_o 2._o problem_n in_o a_o trilater_n equilater_n pyramid_n give_v to_o describe_v a_o octohedron_n svppose_v that_o the_o trilater_n equilater_n pyramid_n give_v be_fw-mi abcd_o who_o side_n let_v be_v divide_v into_o two_o equal_a part_n in_o the_o point_n e_o z_o i_o king_n l_o t._n construction_n and_o draw_v these_o 12._o right_a line_n ez_o zi_n je_n kl_o lt_n tk_n eke_o kz_o zl_n li_n it_o and_o te_fw-la which_o 12._o right_a line_n be_v demonstration_n by_o the_o 4._o of_o the_o first_o equal_a for_o they_o subtend_v equal_a plain_a angle_n of_o the_o base_n of_o the_o pyramid_n and_o those_o equal_a angle_n be_v contain_v under_o equal_a side_n namely_o under_o the_o half_n of_o the_o side_n of_o the_o pyramid_n wherefore_o the_o triangle_n tkl_n tli_n tie_v tek_v zkl_n zli_n zib_n zek_n be_v equilater_n and_o they_o limitate_v and_o contain_v the_o solid_a tklezi_n wherefore_o the_o solid_a tklezi_n be_v a_o octohedron_n by_o the_o 23._o definition_n of_o the_o eleven_o and_o the_o angle_n of_o the_o same_o octohedron_n do_v touch_v the_o side_n of_o the_o pyramid_n abcd_o in_o the_o point_n e_o z_o i_o t_o edward_n l._n wherefore_o the_o octohedron_n be_v inscribe_v in_o the_o pyramid_n by_o the_o 1._o definition_n of_o this_o book_n wherefore_o in_o the_o trilater_n equilater_n pyramid_n give_v be_v inscribe_v a_o octohedron_n which_o be_v require_v to_o be_v do_v a_o corollary_n add_v by_o flussas_n hereby_o it_o be_v manifest_a that_o a_o pyramid_n be_v cut_v into_o two_o
right_a linâ—s_n now_o then_o multiply_v the_o 20._o triangle_n into_o the_o side_n of_o one_o of_o the_o triangle_n and_o so_o shall_v there_o be_v produce_v 6â—_o â—he_z half_a of_o which_o be_v 30._o and_o so_o many_o side_n have_v a_o icosahedron_n and_o in_o like_a sort_n in_o a_o dodecahedron_n forasmuch_o as_o 12._o pentagon_n make_v a_o dodecahedron_n and_o every_o pentagon_n contain_v â—_o right_a linesâ—_n multiply_v â—â—_o into_o 12._o and_o there_o shall_v be_v produce_v 60._o the_o half_a of_o which_o be_v 30._o and_o so_o many_o be_v the_o side_n of_o a_o dodecahedron_n and_o the_o reason_n why_o we_o take_v the_o half_a iâ—_z for_o that_o every_o side_n whether_o it_o be_v of_o a_o triangle_n or_o of_o a_o pentagon_n or_o of_o a_o square_a as_o in_o a_o cube_fw-la â—s_v take_v twice_o and_o by_o the_o same_o reason_n may_v you_o find_v out_o how_o many_o side_n be_v in_o a_o cube_fw-la and_o in_o a_o pyramid_n and_o in_o a_o octohedron_n but_o now_o again_o if_o you_o will_v find_v out_o the_o number_n of_o the_o angle_n of_o every_o one_o of_o the_o solid_a figure_n when_o you_o have_v do_v the_o same_o multiplication_n that_o you_o do_v before_o divide_v the_o same_o side_n by_o the_o number_n of_o the_o plain_a superficiece_n which_o comprehend_v one_o of_o the_o angle_n of_o the_o solid_n as_o for_o example_n forasmuch_o as_o 5._o triangle_n contain_v the_o solid_a angle_n of_o a_o icosahedron_n divide_v 60._o by_o 5._o and_o there_o will_v come_v forth_o 12._o and_o so_o many_o solid_a angle_n have_v a_o icosahedron_n in_o a_o dodecahedron_n forasmuch_o as_o three_o pentagon_n comprehend_v a_o angle_n divide_v 60._o by_o 3._o and_o there_o will_v come_v forth_o 20_o and_o so_o many_o be_v the_o angle_n of_o a_o dodecahedron_n and_o by_o the_o same_o reason_n may_v you_o find_v out_o how_o many_o angle_n be_v in_o each_o of_o the_o rest_n of_o the_o solid_a figure_n book_n if_o it_o be_v require_v to_o be_v know_v how_o one_o of_o the_o plain_n of_o any_o of_o the_o five_o solid_n be_v give_v there_o may_v be_v find_v out_o the_o inclination_n of_o the_o say_a plain_n the_o one_o to_o the_o other_o which_o contain_v each_o of_o the_o solid_n this_o as_o say_v isidorus_n our_o great_a master_n be_v foâ—â—d_v out_o after_o this_o manner_n it_o be_v manifest_a that_o in_o a_o cube_fw-la the_o plain_n which_o contain_v iâ—_z doâ—_n 〈◊〉_d the_o one_o the_o other_o by_o a_o right_a angle_n but_o in_o a_o tetrahedron_n one_o of_o the_o triangle_n be_v give_v let_v the_o end_n of_o one_o of_o the_o side_n of_o the_o say_a triangle_n be_v the_o centre_n and_o let_v the_o space_n be_v the_o perpendicular_a line_n draw_v from_o the_o top_n of_o the_o triangle_n to_o the_o base_a and_o describe_v circumferâ—nces_n of_o a_o circle_n which_o shall_v cut_v the_o one_o the_o other_o and_o from_o the_o intersection_n to_o the_o centre_n draw_v right_a line_n which_o shall_v contain_v the_o inclination_n of_o the_o plain_n contain_v the_o tetrahedron_n in_o a_o octoâ—edron_n take_v one_o of_o the_o side_n of_o the_o triangle_n thereof_o and_o upon_o it_o describe_v a_o square_a and_o draw_v the_o diagonal_a line_n and_o make_v the_o centre_n the_o end_n of_o the_o diagonal_a line_n and_o the_o space_n likewise_o the_o perpendicular_a line_n draw_v from_o the_o top_n of_o the_o triangle_n to_o the_o base_a describe_v circumference_n and_o again_o from_o the_o common_a section_n to_o the_o centre_n draw_v right_a line_n and_o they_o shall_v contain_v the_o inclination_n seek_v for_o in_o a_o icosahedron_n upon_o the_o side_n of_o one_o of_o the_o triangle_n thereof_o describe_v a_o pentagon_n and_o draw_v the_o line_n which_o subtend_v one_o of_o the_o angle_n of_o the_o say_a pentagon_n and_o make_v the_o centre_n the_o end_n of_o that_o line_n and_o the_o space_n the_o perpendicular_a line_n of_o the_o triangle_n describe_v circumference_n and_o draw_v from_o the_o common_a intersection_n of_o the_o circumference_n unto_o the_o centre_n right_a line_n and_o they_o shall_v contain_v likewise_o the_o inclination_n of_o the_o plain_n of_o the_o icosahedron_n in_o a_o dodecahedron_n take_v one_o of_o the_o pentagon_n and_o draw_v likewise_o the_o line_n which_o subtend_v one_o of_o the_o angle_n of_o the_o pentagon_n and_o make_v the_o centre_n the_o end_n of_o that_o line_n and_o the_o space_n the_o perpendicular_a line_n draw_v from_o the_o section_n into_o two_o equal_a part_n of_o that_o line_n to_o the_o side_n of_o the_o pentagon_n which_o be_v parallel_n unto_o it_o describe_v circumference_n and_o from_o the_o point_n of_o the_o intersection_n of_o the_o circumference_n draw_v unto_o the_o centre_n right_a line_n and_o they_o shall_v also_o contain_v the_o inclination_n of_o the_o plain_n of_o the_o dodecahedron_n thus_o do_v this_o most_o singular_a learned_a man_n reason_n think_v the_o demonstration_n in_o every_o one_o of_o they_o to_o be_v plain_a and_o clear_a but_o to_o make_v the_o demonstration_n of_o they_o manifest_a i_o think_v it_o good_a to_o declare_v and_o make_v open_a his_o wordesâ—_n and_o first_o in_o a_o tâ—trahedronâ—_n the_o end_n of_o the_o fivetenth_fw-mi book_n of_o euclides_n element_n after_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d ¶_o the_o 6._o proposition_n the_o 6._o problem_n in_o a_o octohedron_n give_v to_o inscribe_v a_o trilater_n equilater_n pyramid_n svppose_v thaâ—_z the_o octohedron_n whereâ—â—_n the_o tetrahedron_n be_v require_v to_o be_v insâ—riâ—â—â—_n be_v abgdei_n take_v 〈…〉_o base_n of_o the_o octoâ—â—dron_n that_o be_v construction_n 〈…〉_o close_o in_o the_o loweâ—â—_n triangle_n bgd_v namely_o aeâ—_n head_n igd_v and_o let_v the_o four_o be_v aib_n which_o be_v opposite_a to_o the_o low_a triangle_n before_o put_v namely_o to_o egg_v and_o take_v the_o centre_n of_o those_o four_o base_n which_o let_v be_v the_o point_n h_o c_o n_o â—_o and_o upon_o the_o triangle_n hcn_n erect_v a_o pyramid_n hcnl_n now_o forasmuch_o as_o these_o two_o base_n of_o the_o octohedron_n namely_o age_n and_o abi_n be_v set_v upon_o the_o right_a line_n eglantine_n and_o by_o which_o be_v opposite_a the_o one_o to_o the_o otherâ—_n in_o the_o square_a gebi_n of_o the_o octohedron_n from_o the_o poinâ—_n a_o draâ—e_v by_o the_o centre_n of_o the_o base_n namely_o by_o the_o centre_n h_o l_o perpendicular_a line_n ahf_n alk_v cut_v the_o line_n eglantine_n and_o by_o 〈◊〉_d two_o equal_a part_n in_o the_o point_n f_o edward_n by_o the_o corollary_n of_o the_o 1â—_o of_o the_o thirteen_o wherefore_o a_o right_a line_n draw_v from_o the_o point_n f_o to_o the_o point_n king_n demonstration_n shall_v be_v a_o parallel_n and_o equal_a to_o the_o side_n of_o the_o octohedron_n namely_o to_o â—â—_o and_o give_v by_o the_o 33._o of_o the_o first_o and_o the_o right_a line_n hl_o which_o cut_v the_o 〈…〉_o of_o ak_o proportional_o for_o ah_o and_o all_o be_v draw_v from_o the_o centre_n of_o equal_a circle_n to_o the_o circumference_n be_v a_o parallel_n to_o the_o right_a line_n fk_o by_o the_o 2._o of_o the_o six_o and_o also_o to_o the_o side_n of_o the_o octohedron_n namely_o to_o eâ—_n and_o ig_n by_o the_o 9_o of_o the_o eleven_o wherefore_o as_o the_o line_n of_o be_v to_o the_o line_n ah_o so_o be_v the_o line_n fk_o to_o the_o line_n hl_o by_o the_o 4._o of_o the_o six_o for_o the_o triangle_n afk_v and_o ahl_n be_v like_o by_o thâ—_z corollary_n of_o the_o 2._o of_o the_o six_o but_o the_o line_n of_o be_v in_o sesquialter_fw-la proportion_n to_o the_o line_n ah_o for_o the_o side_n eglantine_n make_v hf_o the_o half_a of_o the_o right_a line_n ah_o by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o fk_n or_o give_v the_o side_n of_o the_o octohedron_n be_v sesquialter_fw-la to_o the_o righâ—line_n hl._n and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o side_n of_o the_o octohedron_n be_v sesquialter_fw-la to_o the_o rest_n of_o the_o right_a line_n which_o make_v the_o pyramid_n hnci_n namely_o to_o the_o right_a lineâ—â—_n n_o nc_n ci_o ln_o and_o change_z wherefore_o those_o right_a line_n be_v equal_a and_o therefore_o the_o triangleâ—_n which_o be_v describe_v of_o they_o namely_o the_o triangle_n hcn_n hnl_n ncl_n and_o chl._n which_o make_v the_o pyramid_n hncl_n be_v equal_a and_o equilater_n and_o forasmuch_o as_o the_o angle_n of_o the_o same_o pyramid_n namely_o the_o angle_n h_o c_o n_o l_o do_v end_n in_o the_o centre_n of_o the_o base_n of_o the_o octohedron_n therefore_o it_o be_v inscribe_v â—o_o the_o same_o octohedron_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o octohedron_n â—even_v be_v inscribe_v a_o trilate_a equilaâ—â—â—â—â—â—amisâ—_n which_o be_v require_v to_o â—e_a donâ—_n a_o corollary_n the_o base_n of_o a_o pyramid_n inscribe_v in_o a_o octohedron_n be_v parallel_n
the_o square_a of_o the_o line_n a_o have_v not_o unto_o the_o square_n of_o the_o line_n b_o the_o same_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n then_o i_o say_v that_o the_o line_n a_o and_o b_o be_v incommensurable_a in_o length_n for_o if_o the_o line_n a_o and_o b_o be_v commensurable_a in_o length_n than_o the_o square_a of_o the_o line_n a_o shall_v have_v unto_o the_o square_n of_o the_o line_n b_o the_o same_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n by_o the_o first_o part_n of_o this_o proposition_n but_o by_o supposition_n it_o have_v not_o wherefore_o the_o line_n a_o and_o b_o be_v not_o commensurable_a in_o length_n proposition_n wherefore_o they_o be_v incomensurable_a in_o length_n wherefore_o square_n make_v of_o right_a line_n commensura_fw-la in_o length_n have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o square_n which_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o squaâ—e_a number_n shall_v also_o have_v the_o side_n commensurable_a in_o length_n but_o square_n describe_v of_o right_a line_n incommensurable_a in_o length_n have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o square_n which_o have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n have_v not_o also_o their_o side_n commensurable_a in_o length_n which_o be_v all_o that_o be_v require_v to_o be_v prove_v ¶_o corollary_n hereby_o it_o be_v manifest_a that_o right_a line_n commensurable_a in_o length_n be_v also_o ever_o commensurable_a in_o power_n but_o right_a line_n commensurable_a in_o power_n be_v not_o always_o commensurable_a in_o length_n corollary_n and_o right_a line_n incommensurable_a in_o length_n be_v not_o always_o incommensurable_a in_o power_n but_o right_a line_n incommensurable_a in_o power_n be_v ever_o also_o incommensurable_a in_o length_n for_o forasmuch_o as_o square_n make_v of_o right_a line_n commensurable_a in_o length_n have_v that_o proportion_n the_o one_o to_o the_o other_o corollary_n that_o a_o square_a number_n have_v to_o a_o square_a number_n by_o the_o first_o part_n of_o this_o proposition_n but_o magnitude_n which_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o number_n simple_o have_v to_o number_n be_v by_o the_o six_o of_o the_o ten_o commensurable_a wherefore_o right_a line_n commensurable_a in_o length_n be_v commensurable_a not_o only_o in_o length_n but_o also_o in_o power_n part_n again_o forasmuch_o as_o there_o be_v certain_a square_n which_o have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n but_o yet_o have_v that_o proportion_n the_o one_o to_o the_o other_o which_o number_n simple_o have_v to_o number_v their_o side_n in_o deed_n be_v in_o power_n commensurable_a for_o that_o they_o describe_v square_n which_o have_v that_o proportion_n which_o number_n simple_o have_v to_o number_n which_o square_n be_v therefore_o commensurable_a by_o the_o 6._o of_o this_o book_n but_o the_o say_v side_n be_v incommensurable_a in_o length_n by_o the_o latter_a part_n of_o this_o proposition_n wherâ—fore_o it_o be_v true_a that_o line_n commensurable_a in_o power_n be_v not_o straight_o way_n commensurable_a in_o length_n also_o pâ—rt_n and_o by_o the_o selâ—e_a same_o reason_n be_v prove_v also_o that_o three_o part_n of_o the_o corollary_n that_o line_n incommensurable_a in_o length_n be_v not_o always_o incommensurable_a in_o power_n for_o they_o may_v be_v incommensurable_a in_o length_n but_o yet_o commensurable_a in_o power_n as_o in_o those_o square_n which_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n but_o not_o as_o a_o square_a number_n be_v to_o a_o square_a number_n part_n but_o right_a line_n incommensurable_a in_o power_n be_v always_o also_o incommensurable_a in_o length_n for_o iâ—_z they_o be_v commensurable_a in_o length_n they_o shall_v also_o be_v commensurable_a in_o power_n by_o the_o first_o part_n of_o this_o corollary_n but_o they_o be_v suppose_v to_o be_v incommensurable_a in_o length_n which_o be_v absurd_a wherefore_o right_a line_n incommensurable_a in_o power_n be_v ever_o incommensurable_a in_o lengthy_a for_o the_o better_a understanding_n of_o this_o proposition_n and_o the_o other_o follow_v i_o have_v here_o add_v certain_a annotation_n take_v out_o of_o montaureus_n montauâ—â—us_fw-la and_o first_o as_o touch_v the_o signification_n oâ—_n word_n and_o term_n herein_o use_v whâ—ch_a arâ—_n such_o that_o unless_o they_o be_v well_o mark_v and_o poise_v the_o matter_n will_v be_v obscure_a and_o hard_a and_o in_o a_o manner_n inexplicable_a first_o this_o you_o must_v note_v that_o line_n to_o be_v commensurable_a in_o length_n and_o line_n to_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n be_v all_o one_o so_o that_o whatsoever_o line_n be_v commensurable_a in_o length_n be_v also_o in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_v and_o converse_o what_o so_o ever_o line_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n be_v also_o commensurable_a in_o length_n as_o it_o be_v manifest_a by_o the_o 5_o and_o 6_o of_o this_o book_n likewise_o line_n to_o be_v incommensurable_a in_o length_n and_o not_o to_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n be_v all_o one_o as_o it_o be_v manifest_a by_o the_o 7._o and_o 8._o of_o this_o book_n wherefore_o that_o which_o be_v say_v in_o this_o theorem_a aught_o to_o be_v understand_v of_o line_n commensurable_a in_o length_n and_o incommensurable_a in_o length_n this_o moreover_o be_v to_o be_v note_v that_o it_o be_v not_o all_o one_o numbers_z to_o be_v square_a number_n and_o to_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n for_o although_o square_a number_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n yet_o be_v not_o all_o those_o number_n which_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n square_a number_n for_o they_o may_v be_v like_o superficial_a number_n and_o yet_o not_o square_a number_n which_o yet_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n although_o all_o square_a number_n be_v like_o superficial_a number_n for_o between_o two_o square_a number_n there_o fall_v one_o mean_a proportional_a number_n by_o the_o 11._o of_o the_o eight_o but_o if_o between_o two_o number_n there_o fall_v one_o mean_v proportional_a number_n those_o two_o number_n be_v like_o superficial_a number_n by_o the_o 20._o of_o the_o eight_o so_o also_o if_o two_o number_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n they_o shall_v be_v like_o superficial_a number_n by_o the_o first_o corollary_n add_v after_o the_o last_o proposition_n of_o the_o eight_o book_n and_o now_o to_o know_v whether_o two_o superficial_a number_n give_v be_v like_o superficial_a number_n or_o no_o no._n it_o be_v thus_o find_v out_o first_o if_o between_o the_o two_o number_n give_v there_o fall_v no_o mean_a proportional_a then_o be_v not_o these_o two_o number_n like_o superficial_a number_n by_o the_o 18._o of_o the_o eight_o but_o if_o there_o do_v fall_v between_o they_o a_o mean_v proportional_a then_o be_v they_o like_v superâ—iciall_a number_n by_o the_o 20._o of_o the_o eight_o moreover_o two_o like_o superficial_a number_n multiply_v the_o one_o into_o the_o other_o do_v produce_v a_o square_a number_n by_o the_o firsâ—_o of_o the_o nine_o wherefore_o if_o they_o do_v not_o produce_v a_o square_a number_n then_o be_v they_o not_o like_o superficial_a number_n and_o if_o the_o one_o be_v multiply_v into_o the_o other_o they_o produce_v a_o square_a number_n then_o be_v they_o like_v superficial_a by_o the_o 2._o of_o the_o nine_o moreover_o if_o the_o say_v two_o superficial_a number_n be_v in_o superperticular_a or_o superbipartient_fw-fr proportion_n then_o be_v they_o not_o like_o superficial_a number_n for_o if_o they_o shall_v be_v like_a then_o shall_v there_o be_v a_o mean_a proportional_a between_o they_o by_o the_o 20._o of_o the_o eight_o but_o that_o be_v contrary_a to_o the_o corollary_n of_o the_o 20._o of_o the_o eight_o and_o the_o easy_o to_o conceive_v the_o demonstration_n follow_v take_v this_o example_n of_o that_o which_o we_o have_v say_v ¶_o
a_o assumpt_n forasmuch_o as_o in_o the_o eight_o book_n in_o the_o 26._o proposition_n it_o be_v prove_v that_o like_o plain_a number_n have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o likewise_o in_o the_o 24._o of_o the_o same_o book_n it_o be_v prove_v that_o if_o two_o number_n have_v that_o proportion_n the_o one_o to_o the_o other_o eight_o that_o a_o square_a number_n have_v to_o a_o square_a number_n those_o number_n be_v like_o plain_a number_n hereby_o it_o be_v manifest_a that_o unlike_o plain_a number_n that_o be_v who_o side_n be_v not_o proportional_a have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n for_o if_o they_o have_v than_o shall_v they_o be_v like_o plain_a number_n which_o be_v contrary_a to_o the_o supposition_n wherefore_o unlike_o plain_a number_n have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o therefore_o square_n which_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o unlike_o plain_a number_n have_v shall_v have_v their_o side_n incommensurable_a in_o length_n by_o the_o last_o part_n of_o the_o former_a proposition_n for_o that_o those_o square_n have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n ¶_o the_o 8._o theorem_a the_o 10._o proposition_n if_o four_o magnitude_n be_v proportional_a and_o if_o the_o first_o be_v commensurable_a unto_o the_o second_o the_o three_o also_o shall_v be_v commensurable_a unto_o the_o four_o and_o if_o the_o first_o be_v incommensurable_a unto_o the_o second_o the_o three_o shall_v also_o be_v incommensurable_a unto_o the_o four_o svppose_v that_o these_o four_o magnitude_n a_o b_o c_o d_o be_v proportional_a as_o a_o be_v to_o b_o so_o let_v c_o be_v to_o d_o and_o let_v a_o be_v commensurable_a unto_o b._n then_o i_o say_v that_o c_z be_v also_o commensurable_a unto_o d._n part_n for_o forasmuch_o as_o a_o be_v commensurable_a unto_o b_o it_o have_v by_o the_o five_o of_o the_o ten_o that_o proportion_n that_o number_n have_v to_o number_v but_o as_o a_o be_v to_o b_o so_o be_v c_o to_o d._n wherefore_o c_z also_o have_v unto_o d_z that_o proportion_n that_o number_n have_v to_o number_v wherefore_o c_o be_v commensurable_a unto_o d_z by_o the_o 6._o of_o the_o ten_o but_o now_o suppose_v that_o the_o magnitude_n a_o be_v incommensurable_a unto_o the_o magnitude_n b._n partâ—_n then_o i_o say_v that_o the_o magnitude_n c_o also_o be_v incommensurable_a unto_o the_o magnitude_n d._n for_o forasmuch_o as_o a_o be_v incommensurable_a unto_o b_o therefore_o by_o the_o 7._o of_o this_o book_n a_o have_v not_o unto_o b_o such_o proportion_n as_o number_n have_v to_o number_v but_o as_o a_o be_v to_o b_o so_o be_v c_o to_o d._n wherefore_o c_o have_v not_o unto_o d_z such_o proportion_n as_o number_n have_v to_o number_v wherefore_o by_o the_o 8._o of_o the_o ten_o c_o be_v incommensurable_a unto_o d._n if_o therefore_o there_o be_v four_o magnitude_n proportional_a and_o if_o the_o first_o be_v commensurable_a unto_o the_o second_o the_o three_o also_o shall_v be_v commensurable_a unto_o the_o four_o and_o if_o the_o first_o be_v incommensurable_a unto_o the_o second_o the_o three_o shall_v also_o be_v incommensurable_a unto_o the_o four_o which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o montaureus_n if_o there_o be_v four_o line_n proportional_a and_o if_o the_o two_o first_o or_o the_o two_o last_o be_v commensurable_a in_o power_n only_o the_o other_o two_o also_o shall_v be_v commensurable_a in_o power_n only_o corollary_n this_o be_v prove_v by_o the_o 22._o of_o the_o six_o and_o by_o this_o ten_o proposition_n and_o this_o corollary_n euclid_n use_v in_o the_o 27._o and_o 28._o proposition_n of_o this_o book_n and_o in_o other_o proposition_n also_o ¶_o the_o 3._o problem_n the_o 11._o proposition_n unto_o a_o right_a line_n first_o set_v and_o give_v which_o be_v call_v a_o rational_a line_n to_o find_v out_o two_o right_a line_n incommensurable_a the_o one_o in_o length_n only_o and_o the_o other_o in_o length_n and_o also_o in_o power_n svppose_v that_o the_o right_a line_n first_o set_v and_o give_v which_o be_v call_v a_o rational_a line_n of_o purpose_n be_v a._n it_o be_v require_v unto_o the_o say_a line_n a_o to_o find_v out_o two_o right_a line_n incommensurable_a the_o one_o in_o length_n only_o the_o other_o both_o in_o length_n and_o in_o power_n give_v take_v by_o that_o which_o be_v add_v after_o the_o 9_o proposition_n of_o this_o book_n two_o number_n b_o and_o c_o not_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n that_o be_v let_v they_o not_o be_v like_o plain_a number_n for_o like_o plain_a number_n by_o the_o 26._o of_o the_o eight_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o as_o the_o number_n b_o be_v to_o the_o number_n c_o so_fw-mi let_v the_o square_n of_o the_o line_n a_o be_v unto_o the_o square_n of_o a_o other_o line_n namely_o of_o d_z how_o to_o do_v this_o be_v teach_v in_o the_o assumpt_n put_v before_o the_o 6._o proposition_n of_o this_o book_n wherefore_o the_o square_a of_o the_o line_n a_o be_v unto_o the_o square_n of_o the_o line_n d_o commensurable_a by_o the_o six_o of_o the_o ten_o and_o forasmuch_o as_o the_o number_n b_o have_v not_o unto_o the_o number_n c_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o the_o square_a of_o the_o line_n a_o have_v not_o unto_o the_o square_n of_o the_o line_n d_z that_o proportion_n that_o a_o square_a number_n have_v to_o a_o number_n wherefore_o by_o the_o 9_o of_o the_o ten_o the_o line_n a_o be_v unto_o the_o line_n d_o incommensurable_a in_o length_n only_o and_o so_o be_v find_v out_o the_o first_o line_n namely_o d_z incommensurable_a in_o length_n only_o to_o the_o line_n give_v a._n give_v again_o take_v by_o the_o 13._o of_o the_o six_o the_o mean_a proportional_a between_o the_o line_n a_o and_o d_o and_o let_v the_o same_o be_v e._n wherefore_o as_o the_o line_n a_o be_v to_o the_o line_n d_o so_o be_v the_o square_a of_o the_o line_n a_o to_o the_o square_n of_o the_o line_n e_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o but_o the_o line_n a_o be_v unto_o the_o line_n d_o incommensurable_a in_o length_n wherefore_o also_o the_o square_a of_o the_o line_n a_o be_v unto_o the_o square_n of_o the_o line_n e_o incommensurable_a by_o the_o second_o part_n of_o the_o former_a proposition_n now_o forasmuch_o as_o the_o square_n of_o the_o line_n a_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n e_o it_o follow_v by_o the_o definition_n of_o incommensurable_a line_n that_o the_o line_n a_o be_v incommensurable_a in_o power_n to_o the_o line_n e._n wherefore_o unto_o the_o right_a line_n give_v and_o first_o set_v a_o which_o be_v a_o rational_a line_n and_o which_o be_v suppose_v to_o have_v such_o division_n and_o so_o many_o part_n as_o you_o listen_v to_o conceive_v in_o mind_n as_o in_o this_o example_n 11_z whereunto_o as_o be_v declare_v in_o the_o 5._o definition_n of_o this_o book_n may_v be_v compare_v infinite_a other_o line_n either_o commensurable_a or_o incommensurable_a be_v find_v out_o the_o line_n d_o incommensurable_a in_o length_n only_o wherefore_o the_o line_n d_o be_v rational_a by_o the_o six_o definition_n of_o this_o book_n for_o that_o it_o be_v incommensurable_a in_o length_n only_o to_o the_o line_n a_o which_o be_v the_o first_o line_n set_v and_o be_v by_o supposition_n rational_a there_o be_v also_o find_v out_o the_o line_n e_o which_o be_v unto_o the_o same_o line_n a_o incommensurable_a not_o only_o in_o length_n but_o also_o in_o power_n which_o line_n e_o compare_v to_o the_o rational_a line_n a_o be_v by_o the_o definition_n irrational_a for_o euclid_n always_o call_v those_o line_n irrational_a which_o be_v incommensurable_a both_o in_o length_n and_o in_o power_n to_o the_o line_n first_o set_v and_o by_o supposition_n rational_a ¶_o the_o 9_o theorem_a the_o 12._o proposition_n magnitude_n commensurable_a to_o one_o and_o the_o self_n same_o magnitude_n be_v also_o commensurable_a the_o one_o to_o the_o other_o svppose_v that_o either_o of_o these_o magnitude_n a_o and_o b_o be_v commensurable_a unto_o the_o magnitude_n c_o then_o i_o say_v that_o the_o magnitude_n a_o be_v commensurable_a unto_o the_o magnitude_n b._n construction_n for_o forasmuch_o as_o the_o
the_o circle_n a_o construction_n have_v to_o the_o circle_n b._n as_o the_o diameter_n of_o the_o circle_n a_o be_v to_o the_o diameter_n of_o the_o circle_n b_o so_fw-mi let_v the_o line_n c_o be_v to_o a_o four_o line_n by_o the_o 12._o of_o the_o 〈…〉_o line_n be_v d._n and_o by_o the_o 11._o of_o the_o six_o let_v a_o third_o line_n proportional_a be_v find_v to_o the_o line_n c_o &_o d_o which_o let_v be_v eâ—_n i_o say_v now_o that_o the_o line_n c_o have_v too_o the_o line_n e_o that_o proportion_n which_o the_o circle_n a_o have_v to_o the_o circle_n b._n for_o by_o construction_n the_o lineâ—_z c_o d_o and_o e_o demonstration_n be_v proportional_a therefore_o the_o square_n of_o câ—_n be_v to_o the_o square_n of_o d_o as_z c_z be_v to_o e_o by_o the_o corollary_n of_o the_o 10._o of_o the_o six_o but_o by_o construction_n as_o the_o diameter_n of_o the_o circle_n a_o be_v to_o the_o diameter_n of_o the_o circle_n b_o so_o be_v c_o to_o d_z wherefore_o as_o the_o square_n of_o the_o diameter_n of_o the_o circle_n a_o be_v to_o the_o square_n of_o the_o diameter_n of_o the_o circle_n b_o so_o be_v the_o square_a of_o the_o line_n c_o to_o the_o square_n of_o the_o line_n d_o by_o the_o 22._o of_o the_o six_o but_o as_o the_o square_n of_o the_o diameter_n of_o a_o the_o circle_n be_v to_o the_o square_n of_o the_o diameter_n of_o the_o circle_n d_o so_o be_v the_o circle_n a_o to_o the_o circle_n b_o by_o the_o second_o of_o the_o twelve_o wherefore_o by_o 11._o of_o the_o five_o as_o the_o circle_n a_o be_v to_o the_o circle_n b_o so_o be_v the_o square_a of_o the_o line_n c_o to_o the_o square_n of_o the_o liâ—e_z d._n but_o it_o be_v prove_v that_o â—s_v the_o square_a of_o the_o line_n c_o be_v to_o the_o square_n of_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 wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o circle_n a_o be_v to_o the_o circle_n b_o so_o be_v the_o line_n c_o to_o the_o line_n e._n two_o circle_n be_v give_v therefore_o and_o a_o right_a line_n we_o have_v find_v a_o right_a line_n to_o which_o the_o right_a line_n give_v have_v that_o proportion_n which_o the_o one_o circle_n have_v to_o the_o other_o which_o ought_v to_o be_v do_v note_n the_o difference_n between_o this_o problem_n and_o that_o next_o before_o be_v this_o there_o although_o we_o have_v two_o circle_n give_v second_o and_o two_o line_n be_v find_v in_o that_o proportion_n the_o one_o to_o the_o other_o in_o which_o the_o give_v circle_n be_v and_o here_o likewise_o be_v two_o circle_n give_v and_o two_o lineâ—_z also_o be_v have_v in_o the_o same_o proportion_n that_o the_o give_v circle_n be_v yet_o there_o we_o take_v at_o pleasure_n the_o first_o of_o the_o two_o line_n whereunto_o we_o frame_v the_o second_o proportional_o to_o the_o circle_n give_v but_o here_o the_o first_o of_o the_o two_o line_n be_v assign_v point_a aâ—d_z determine_v to_o we_o and_o not_o our_o choice_n to_o be_v have_v therein_o as_o be_v in_o the_o former_a problem_n a_o problem_n 3._o a_o circle_n be_v give_v to_o find_v a_o other_o circle_n to_o which_o the_o give_v câ—rcle_n be_v in_o any_o proportion_n give_v in_fw-ge two_o right_a line_n suppose_v the_o circle_n aâ—c_n give_v and_o therefore_o his_o semidiameter_n be_v give_v whereby_o his_o diameter_n also_o be_v give_v which_o diameter_n let_v be_v ac_fw-la let_v the_o proportion_n give_v be_v that_o which_o be_v between_o â—_o f_o two_o right_a line_n i_o say_v a_o circle_n be_v to_o be_v find_v unto_o which_o aâ—c_n have_v that_o proportion_n that_o â—_o have_v to_o â—_o as_o â—_o be_v to_o â—_o construction_n so_o let_v ac_fw-la the_o diameter_n be_v to_o a_o other_o right_a line_n by_o the_o 12._o of_o the_o six_o which_o line_n suppose_v to_o be_v 11._o between_o ac_fw-la and_o â—â—_o find_v a_o middle_n proportional_a line_n by_o the_o 13._o of_o the_o six_o which_o let_v be_v ln_o by_o the_o 10._o of_o the_o first_o divide_v â—n_n into_o two_o equal_a payte_n and_o let_v that_o be_v do_v in_o the_o point_n â—_o now_o upon_o â—â—_o o_o being_n make_v the_o centre_n describe_v a_o circle_n which_o let_v be_v â—nâ—_n demonstration_n i_o say_v that_o abc_n be_v to_o lmn_o as_o â—_o be_v to_o f._n for_o see_v that_o ac_fw-la ln_o and_o h_o be_v three_o right_a line_n in_o continual_a proportion_n by_o construction_n therefore_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o as_o ac_fw-la be_v to_o h_o so_o be_v the_o square_a of_o aâ—_z to_o the_o square_n of_o ln_o but_o ac_fw-la be_v to_o h_o as_z â—_o be_v to_o â—_o by_o construction_n wherefore_o the_o square_a of_o ac_fw-la be_v to_o the_o square_n of_o ln_o as_o e_o be_v fletcher_n but_o as_o the_o square_n of_o the_o diameter_n ac_fw-la be_v to_o the_o square_n of_o the_o diameter_n ln_o so_o be_v the_o circle_n abc_n to_o the_o circle_n â—mn_v by_o this_o 2._o of_o the_o twelve_o wherefore_o by_o the_o 11._o of_o the_o five_o the_o circle_n abc_n be_v to_o the_o circle_n lmn_o as_o â—_o be_v to_o f._n a_o circle_n be_v give_v therefore_o a_o other_o circle_n be_v find_v to_o which_o the_o give_v circle_n be_v in_o any_o proportion_n give_v between_o two_o right_a line_n which_o ought_v to_o be_v do_v a_o problem_n 4._o two_o circle_n be_v give_v to_o find_v one_o circle_n equal_a to_o they_o both_o suppose_v the_o two_o circle_n give_v have_v their_o diameter_n aâ—_z &_o cd_o i_o say_v that_o a_o circle_n must_v be_v â—ound_v equal_a to_o the_o two_o circle_n who_o diameter_n be_v aâ—_z and_o cd_o unto_o the_o line_n aâ—_z construction_n at_o the_o point_n a_o erect_v a_o perpendicular_a line_n aâ—_z from_o which_o sufficient_o produce_v cut_v a_o line_n equal_a to_o cd_o which_o let_v be_v af._n by_o the_o first_o petition_n draw_v from_o fletcher_n to_o â—_o a_o right_n line_n so_o be_v faâ—_n make_v a_o triangle_n rectangle_n i_o say_v now_o that_o a_o circle_n who_o diameter_n be_v fâ—_n be_v equal_a to_o the_o two_o circle_n who_o diameter_n be_v aâ—_z and_o â—d_a demonstration_n for_o by_o the_o 47._o of_o the_o first_o the_o square_a of_o fâ—_n be_v equal_a to_o the_o square_n of_o aâ—_z &_o af._n which_o of_o be_v by_o construction_n equal_a to_o cdâ—_n wherefore_o the_o square_a of_o fâ—_n be_v equal_a to_o the_o square_n of_o ab_fw-la and_o cd_o but_o circle_n be_v one_o to_o a_o other_o as_o the_o square_n of_o their_o diameter_n be_v one_o to_o the_o other_o by_o this_o second_o of_o the_o twelve_o therefore_o the_o circle_n who_o diameter_n be_v â—â—_o be_v equal_a to_o the_o circle_n who_o diameter_n be_v aâ—_z and_o cd_o therefore_o two_o circle_n be_v give_v we_o have_v find_v a_o circle_n equal_a to_o they_o both_o which_o be_v require_v to_o be_v do_v a_o corollary_n 1._o hereby_o it_o be_v make_v evident_a that_o in_o all_o triangle_n rectangle_n the_o circle_n semicircle_n quadrant_n oâ—_n any_o other_o portion_n of_o circle_n describe_v upon_o the_o subtendent_fw-la line_n be_v equal_a to_o the_o two_o circle_n semicircle_n quadrant_n or_o any_o two_o other_o like_a portion_n of_o circle_n describe_v on_o the_o two_o line_n comprehend_v the_o right_a angle_n like_v to_o like_v be_v compare_v for_o like_a part_n have_v that_o proportion_n between_o themselves_o that_o their_o whole_a magnitude_n have_v of_o which_o they_o be_v like_o part_n by_o the_o 15._o of_o the_o five_o but_o of_o the_o whole_a circle_n in_o the_o former_a problem_n it_o be_v evident_a and_o therefore_o in_o the_o forname_v like_o portion_n of_o circle_n it_o be_v a_o true_a consequent_a a_o corollary_n 2._o by_o the_o former_a problem_n it_o be_v also_o manifest_a unto_o circle_n three_o four_o five_o or_o to_o how_o many_o so_o ever_o one_o will_v geve_v one_o circle_n may_v be_v give_v equal_a for_o if_o first_o to_o any_o two_o by_o the_o former_a problem_n you_o find_v one_o equal_a and_o then_o unto_o your_o find_a circle_n and_o the_o three_o of_o the_o give_v circle_n as_o two_o give_v circle_n find_v one_o other_o circle_n equal_a and_o then_o to_o that_o second_o find_v circle_n and_o to_o the_o four_o of_o the_o first_o give_v circlesâ—_n as_o two_o circle_n one_o new_a circle_n be_v find_v equal_a and_o so_o proceed_v till_o you_o have_v once_o couple_v orderly_a every_o one_o of_o your_o propound_a circle_n except_o the_o first_o and_o second_o already_o do_v with_o the_o new_a circle_n thus_o find_v for_o so_o the_o last_o find_v circle_n be_v equal_a to_o all_o the_o first_o give_v circle_n if_o you_o doubt_v or_o sufficient_o understand_v i_o not_o help_v yourself_o by_o the_o discourse_n and_o demonstration_n of_o the_o