proposition_n after_o prâ—â—lus_n a_o corollary_n take_v out_o of_o flussates_n demonstration_n lead_v to_o 〈◊〉_d absurdity_n a_o addition_n oâ—_n pelitarius_n demonstration_n three_o case_n in_o this_o proposition_n the_o first_o case_n construction_n demonstration_n three_o case_n in_o this_o proposition_n the_o first_o case_n every_o case_n may_v happen_v seven_o diverse_a way_n the_o like_a variety_n in_o each_o of_o the_o other_o two_o case_n euclides_n construction_n and_o demostration_n serve_v in_o all_o these_o case_n and_o in_o their_o varity_n also_o construction_n demonstration_n how_o triangle_n be_v say_v to_o be_v in_o the_o self_n same_o parallel_a line_n comparison_n of_o two_o triangle_n who_o side_n be_v equal_a their_o base_n and_o angle_n at_o the_o top_n be_v unequal_a when_o they_o be_v less_o than_o two_o right_a angle_n construction_n demonstration_n three_o case_n in_o this_o proposition_n each_o of_o these_o case_n also_o may_v be_v diverse_o note_n an_o other_o addition_n of_o pelitarius_n construction_n demonstration_n this_o theorem_a the_o converse_n of_o the_o 37._o proposition_n a_o addition_n of_o flâ—ssases_n a_o addition_n of_o campanus_n construction_n demonstration_n lead_v to_o a_o absurdity_n this_o proposition_n be_v the_o converse_n of_o the_o 38._o proposition_n demonstration_n two_o case_n in_o this_o proposition_n a_o corollary_n the_o self_n same_o demonstration_n will_v serve_v if_o the_o triangle_n &_o the_o parallelogram_n be_v upon_o equal_a base_n the_o converse_n of_o this_o proposition_n an_o other_o converse_n of_o the_o same_o proposition_n comparison_n of_o a_o triangle_n and_o a_o trapesium_n be_v upon_o one_o &_o the_o self_n same_o base_a and_o in_o the_o self_n same_o parallel_a line_n construction_n demonstration_n supplement_n &_o complement_n three_o case_n in_o this_o theorem_a the_o first_o case_n this_o proposition_n call_v gnomical_a and_o mystical_a the_o converse_n of_o this_o proposition_n 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 wonderful_a proposition_n in_o this_o book_n first_o definition_n what_o a_o parallelogram_n be_v four_o kind_n of_o parallelogram_n second_o definition_n a_o proposition_n add_v by_o campane_n after_o the_o last_o proposition_n of_o the_o first_o book_n construction_n demonstration_n barlaam_n barlaam_n construction_n demonstration_n barlaam_n construction_n demonstration_n barlaam_n construction_n demonstration_n a_o corollary_n barlaam_n construction_n demonstration_n constrâ—ction_n demonstration_n construction_n demonstration_n construction_n demonstration_n many_o and_o singular_a use_n of_o this_o proposition_n this_o proposition_n can_v not_o be_v reduce_v unto_o number_n demonstration_n demonstration_n a_o corollary_n this_o proposition_n true_a in_o all_o kind_n of_o triangle_n construction_n demonstration_n the_o argument_n of_o this_o book_n the_o first_o definition_n definition_n of_o unequal_a circle_n second_o definition_n a_o contigent_a line_n three_o definition_n 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 the_o first_o part_n of_o this_o proposition_n demonstration_n second_o part_n three_o part_n this_o demonstrate_v by_o a_o argument_n lead_v to_o a_o impossibilie_n an_o other_o demonstration_n of_o the_o latter_a part_n of_o the_o proposition_n lead_v also_o to_o a_o impossibility_n a_o corollary_n three_o part_n an_o other_o demonstration_n of_o the_o latter_a part_n lead_v also_o to_o a_o impossibility_n this_o proposion_n be_v common_o call_v caâ—dâ—_n panonis_fw-la a_o corollary_n construction_n demonstration_n an_o other_o demonstration_n of_o the_o same_o lead_v also_o to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n of_o the_o same_o lead_v also_o to_o a_o impossibility_n construction_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n of_o the_o same_o lead_v also_o to_o a_o impossibility_n the_o same_o again_o demonstrate_v by_o a_o argument_n lead_v to_o a_o absurdititie_n demonstratiâ—_n lead_v 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 circumscribe_v of_o figure_n in_o or_o abouâ—_n circle_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n after_o orontius_n demonstration_n lead_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n the_o one_o when_o the_o angle_n set_v at_o the_o circumference_n include_v the_o centre_n demonstration_n the_o other_o when_o the_o same_o angle_n set_v at_o the_o circumference_n include_v not_o the_o centre_n construction_n demonstration_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 construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n a_o addition_n of_o campane_n demonstrate_v by_o pelitariâ—s_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n construction_n three_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n the_o second_o case_n the_o three_o case_n a_o addition_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n the_o converse_n of_o the_o former_a proposition_n construction_n demonstration_n construction_n demonstration_n second_o part_n thirâ—_n part_n the_o five_o and_o last_o part_n an_o other_o demonstration_n to_o prove_v that_o the_o angâ—e_n in_o a_o semicircle_n be_v a_o right_a angle_n a_o corollary_n a_o addition_n of_o pâ—litarius_n demonstration_n leaâ—ing_v to_o a_o absurdity_n a_o addition_n of_o campane_n construction_n demonstration_n two_o case_n in_o this_o proposition_n three_o case_n in_o this_o proposition_n the_o first_o case_n construction_n demonstration_n the_o second_o case_n construction_n demonstration_n the_o three_o case_n construction_n demonstration_n construction_n demonstration_n two_o case_n in_o this_o proposition_n first_o case_n demonstration_n the_o second_o câ—se_n construction_n demonstration_n three_o case_n in_o this_o proposition_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n the_o second_o case_n construction_n demonstration_n first_o corollary_n second_o corollary_n three_o corollary_n this_o proposition_n be_v the_o converse_n of_o the_o former_a construction_n demonstration_n an_o other_o demonstration_n after_o pelitarius_n the_o argument_n of_o this_o book_n first_o definition_n second_o definition_n the_o inscriptition_n and_o circumscription_n of_o rectiline_a figure_n pertain_v only_o to_o regular_a figure_n the_o three_o definition_n the_o four_o definition_n the_o five_o definition_n the_o six_o devition_n seven_o definition_n construction_n two_o case_n in_o this_o proposition_n first_o caseâ—_n second_o case_n demonstration_n construction_n
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 first_o it_o may_v be_v prove_v as_o before_o that_o the_o parallelogram_n cd_o be_v commensurable_a to_o the_o square_a df._n wherefore_o by_o the_o 12._o of_o this_o book_n the_o parallelogram_n cd_o shall_v be_v commensurable_a to_o the_o square_n of_o the_o line_n abâ—_n but_o the_o square_n of_o the_o line_n ab_fw-la be_v rational_a case_n wherefore_o by_o the_o definition_n the_o parallelogram_n cd_o shall_v be_v also_o rational_a this_o case_n be_v well_o to_o be_v note_v for_o it_o serve_v to_o the_o demonstration_n and_o understanding_n of_o the_o 25._o proposition_n of_o this_o book_n ¶_o the_o 17._o theorem_a the_o 20._o proposition_n if_o upon_o a_o rational_a line_n be_v apply_v a_o rational_a rectangle_n parallelogram_n the_o other_o side_n that_o make_v the_o breadth_n thereof_o shall_v be_v a_o rational_a line_n and_o commensurable_a in_o length_n unto_o that_o line_n whereupon_o the_o rational_a parallelogram_n be_v apply_v svppose_v that_o this_o rational_a rectangle_n parallelogram_n ac_fw-la be_v apply_v upon_o the_o line_n ab_fw-la which_o let_v be_v rational_a according_a to_o any_o one_o of_o the_o foresay_a way_n whether_o it_o be_v the_o first_o rational_a line_n set_v proposition_n or_o any_o other_o line_n commensurable_a to_o the_o rational_a line_n first_o set_v and_o that_o in_o length_n and_o in_o power_n or_o in_o power_n only_o for_o one_o of_o these_o three_o way_n as_o be_v declare_v in_o the_o assumpt_n put_v before_o the_o 19_o proposition_n of_o this_o book_n be_v a_o line_n call_v rational_a and_o make_v in_o breadth_n the_o line_n bc._n then_o i_o say_v that_o the_o line_n bc_o be_v rational_a and_o commensurable_a in_o length_n unto_o the_o line_n basilius_n construction_n desrcribe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n basilius_n a_o square_a ad._n wherefore_o by_o the_o 9_o definition_n of_o the_o ten_o the_o square_a ad_fw-la be_v rational_a but_o the_o parallelogram_n ac_fw-la also_o be_v rational_a by_o supposition_n demonstration_n wherefore_o by_o the_o conversion_n of_o the_o definition_n of_o rational_a figure_n or_o by_o the_o 12._o of_o this_o book_n the_o square_a dam_n be_v commensurable_a unto_o the_o parallelogram_n ac_fw-la but_o as_o the_o square_a dam_n be_v to_o the_o parallelogram_n ac_fw-la so_o be_v the_o line_n db_o to_o the_o line_n bc_o by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o line_n db_o be_v commensurable_a unto_o the_o line_n bc._n but_o the_o line_n db_o be_v equal_a unto_o the_o line_n basilius_n wherefore_o the_o line_n ab_fw-la be_v commensurable_a unto_o the_o line_n bc._n but_o the_o line_n ab_fw-la be_v rational_a wherefore_o the_o line_n bc_o also_o be_v rational_a and_o commensurable_a in_o length_n unto_o the_o line_n basilius_n if_o therefore_o upon_o a_o rational_a line_n be_v apply_v a_o rational_a rectangle_n parallelogram_n the_o other_o side_n that_o make_v the_o breadth_n thereof_o 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 medial_a line_n as_o theon_n right_o say_v for_o two_o cause_n first_o for_o that_o the_o power_n or_o square_v which_o it_o producethâ—_n be_v equal_a to_o a_o medial_a superficies_n or_o parallelogram_n for_o as_o that_o line_n which_o produce_v a_o rational_a square_n be_v call_v a_o rational_a line_n and_o that_o line_n which_o produce_v a_o irrational_a square_n or_o a_o square_n equal_a to_o a_o irrational_a figure_n general_o be_v call_v a_o irrational_a line_n so_o iâ—_z thaâ—_z line_n which_o produce_v a_o medial_a square_n or_o a_o square_n equal_a to_o a_o medial_a superficies_n call_v by_o special_a name_n a_o medial_a line_n second_o it_o be_v call_v a_o medial_a line_n because_o it_o be_v a_o mean_a proportional_a between_o the_o two_o line_n commensurable_a in_o power_n only_o which_o comprehend_v the_o medial_a superficies_n ¶_o a_o corollary_n add_v by_o flussates_n a_o rectangle_n parallelogram_n contain_v under_o a_o rational_a line_n and_o a_o irrational_a line_n be_v irrational_a corollary_n for_o if_o the_o line_n ab_fw-la be_v rational_a and_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
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 point_n of_o the_o touch_n namely_o b_o be_v draw_v into_o the_o circle_n a_o certain_a right_a line_n bc_o therefore_o by_o the_o 32._o of_o the_o three_o the_o angle_n fbc_n be_v equal_a to_o the_o angle_n bac_n which_o be_v in_o the_o alternate_a segment_n but_o the_o angle_n fbc_n be_v equal_a to_o the_o angle_n d._n wherefore_o the_o angle_n bac_n which_o consist_v in_o the_o segment_n bac_n be_v equal_a to_o the_o angle_n d._n wherefore_o from_o the_o circle_n give_v abc_n be_v cut_v away_o a_o segment_n bac_n which_o contain_v a_o angle_n equal_a to_o the_o rectiline_a angle_n give_v which_o be_v require_v to_o be_v do_v the_o 29._o theorem_a the_o 35._o proposition_n if_o in_o a_o circle_n two_o right_a line_n do_v cut_v the_o one_o the_o other_o the_o rectangle_n parallelogram_n comprehend_v under_o the_o segment_n or_o part_n of_o the_o one_o line_n be_v equal_a to_o the_o rectangle_n parallelogram_n comprehend_v under_o the_o segment_n or_o part_n of_o the_o other_o line_n let_v the_o circle_n be_v abcd_o and_o in_o it_o let_v these_o two_o right_a line_n ac_fw-la and_o bd_o câ—t_o the_o one_o the_o other_o in_o the_o point_n e._n then_o i_o say_v that_o the_o rectangle_n parallelogram_n contain_v under_o the_o part_n ae_n and_o ec_o be_v equal_a to_o the_o rectangle_n parallelogram_n contain_v under_o the_o part_n de_fw-fr and_o ebb_v case_n for_o if_o the_o line_n ac_fw-la and_o bd_o be_v draw_v by_o the_o centre_n then_o be_v it_o manifest_a that_o for_o as_o much_o as_o the_o line_n ae_n and_o ec_o be_v equal_a to_o the_o line_n de_fw-fr and_o ebb_v by_o the_o definition_n of_o a_o circle_n demonstration_n the_o rectangle_n parallelogram_n also_o contain_v under_o the_o line_n ae_n and_o ec_o be_v equal_a to_o the_o rectangle_n parallelogram_n contain_v under_o the_o line_n de_fw-fr and_o ebb_v but_o now_o suppose_v that_o the_o line_n ac_fw-la and_o db_o be_v not_o extend_v by_o the_o centre_n câ—se_n and_o take_v by_o the_o 1._o of_o the_o three_o the_o centre_n of_o the_o circle_n abcd_o and_o let_v the_o same_o be_v the_o point_n fletcher_n construction_n and_o from_o the_o point_n f_o draw_v to_o the_o right_a line_n ac_fw-la and_o db_o perpendicular_a line_n fg_o and_o fh_o by_o the_o 12._o of_o the_o first_o and_o draw_v these_o right_a line_n fb_o fc_o and_o fe_o and_o forasmuch_o as_o a_o certain_a right_a line_n fg_o draw_v by_o the_o centre_n demonstration_n cut_v a_o certain_a right_a line_n ac_fw-la not_o draw_v by_o the_o centre_n in_o such_o sort_n that_o it_o make_v right_a angle_n it_o therefore_o divide_v the_o line_n ac_fw-la into_o two_o equal_a part_n by_o the_o 3._o of_o the_o three_o wherefore_o the_o line_n agnostus_n be_v equal_a to_o the_o line_n gc_o and_o forasmuch_o as_o the_o right_a line_n ac_fw-la be_v divide_v into_o two_o equal_a part_n in_o the_o point_n g_o and_o into_o two_o unequal_a part_n in_o the_o point_n e_o therefore_o by_o the_o 5._o of_o the_o second_o the_o rectangle_n parallelogram_n contain_v under_o the_o line_n ae_n and_o ec_o together_o with_o the_o square_n of_o the_o line_n eglantine_n be_v equal_a to_o the_o square_n of_o the_o line_n gc_o put_v the_o square_n of_o the_o line_n gf_o common_a to_o they_o both_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ae_n &_o ec_o together_o with_o the_o square_n of_o the_o line_n eglantine_n and_o gf_o be_v equal_a to_o the_o square_n of_o the_o line_n gf_o &_o gc_o but_o unto_o the_o square_n of_o the_o line_n eglantine_n &_o gf_o be_v equal_a the_o square_a of_o the_o line_n fe_o by_o the_o 47._o of_o the_o first_o and_o to_o the_o square_n of_o the_o line_n gc_o and_o gf_o be_v equal_a the_o square_n of_o the_o line_n fc_o by_o the_o same_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ec_o together_o with_o the_o square_n of_o the_o line_n fe_o be_v equal_a to_o the_o square_n of_o the_o line_n fc_o but_o the_o line_n fc_o be_v equal_a to_o the_o line_n fb_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o eâ—_n together_o with_o the_o square_n of_o the_o line_n fe_o be_v equal_a to_o the_o square_n of_o the_o line_n fb_o and_o by_o the_o same_o demonstration_n that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o ebb_v together_o with_o the_o square_n of_o the_o line_n fe_o be_v equal_a to_o the_o square_n of_o the_o line_n fb_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ec_o together_o with_o the_o square_n of_o the_o line_n of_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o ebb_v together_o with_o the_o square_n of_o the_o line_n ef._n take_v away_o the_o square_a of_o the_o line_n fe_o which_o be_v common_a to_o they_o both_o wherefore_o the_o rectangle_n parallelogram_n remain_v which_o be_v contain_v under_o the_o line_n ae_n and_o ec_o be_v equal_a to_o the_o rectangle_n parallelogram_n remain_v which_o be_v contain_v under_o the_o line_n de_fw-fr and_o ebb_v if_o therefore_o in_o a_o circle_n two_o right_a line_n do_v cut_v the_o one_o the_o other_o the_o rectangle_n parallelogram_n comprehend_v under_o the_o segment_n or_o part_n of_o the_o one_o line_n be_v equal_a to_o the_o rectangle_n parallelogram_n comprehend_v under_o the_o segment_n or_o part_n of_o the_o other_o line_n which_o be_v require_v to_o be_v demonstrate_v in_o this_o proposition_n be_v three_o case_n proposition_n for_o either_o both_o the_o line_n pass_v by_o the_o centre_n or_o neither_o of_o they_o pass_v by_o the_o centre_n or_o the_o one_o pass_v by_o the_o centre_n and_o the_o other_o not_o the_o two_o first_o case_n be_v before_o demonstrate_v amongst_o all_o the_o proposition_n in_o this_o three_o book_n doubtless_o this_o be_v one_o of_o the_o chief_a for_o it_o set_v forth_o unto_o we_o the_o wonderful_a nature_n of_o a_o circle_n so_o that_o by_o it_o may_v be_v do_v many_o goodly_a conclusion_n in_o geometry_n as_o shall_v afterward_o be_v declare_v when_o occasion_n shall_v serve_v the_o 30._o theorem_a the_o 36._o proposition_n if_o without_o a_o circle_n be_v take_v a_o certain_a point_n and_o from_o that_o point_n be_v draw_v to_o the_o circle_n two_o right_a line_n so_o that_o the_o one_o of_o they_o do_v cut_v the_o circle_n and_o the_o other_o do_v touch_v the_o circle_n the_o rectangle_n parallelogram_n which_o be_v comprehend_v under_o the_o whole_a right_a line_n which_o cut_v the_o circle_n and_o that_o portion_n of_o the_o same_o line_n that_o lie_v between_o the_o point_n and_o the_o utter_a circumference_n of_o the_o circle_n be_v equal_a to_o the_o square_n make_v of_o the_o line_n that_o touch_v the_o circle_n svppose_v that_o the_o circle_n be_v abc_n and_o without_o the_o same_o circle_n take_v any_o point_n at_o all_o adventure_n and_o let_v the_o same_o be_v d._n construction_n and_o from_o the_o point_n d_o let_v there_o be_v draw_v to_o the_o circle_n two_o right_a line_n dca_n and_o db_o and_o let_v the_o right_a line_n dca_n cut_v the_o circle_n acb_o in_o the_o point_n c_o and_o let_v the_o right_a line_n bd_o touch_v the_o same_o then_o i_o say_v that_o the_o rectangle_n parallelogram_n contain_v under_o the_o line_n ad_fw-la and_o dc_o proposition_n be_v equal_a to_o the_o square_n of_o the_o line_n bd._n now_o the_o line_n dca_n be_v either_o draw_v by_o the_o centre_n or_o not_o first_o let_v it_o be_v draw_v by_o the_o centre_n case_n and_o by_o the_o first_o of_o the_o three_o let_v the_o point_n fletcher_n be_v the_o centre_n of_o the_o circle_n abc_n and_o draw_v a_o line_n from_o fletcher_n to_o b._n wherefore_o the_o angle_n fbd_v be_v a_o right_a angle_n demonstration_n and_o for_o asmuch_o as_o the_o right_a line_n ac_fw-la be_v divide_v into_o two_o equal_a part_n in_o the_o point_n fletcher_n and_o unto_o it_o be_v add_v direct_o a_o right_a line_n cd_o therefore_o by_o the_o 6._o of_o the_o second_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o dc_o together_o with_o the_o square_n of_o the_o line_n cf_o be_v equal_a to_o the_o square_n of_o the_o line_n fd._n but_o the_o line_n fc_o be_v equal_a to_o the_o line_n fb_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o dc_o together_o with_o the_o square_n of_o the_o line_n fb_o be_v equal_a to_o the_o square_n of_o the_o line_n fd._n but_o the_o square_a of_o the_o line_n fd_o be_v by_o the_o 47._o of_o the_o first_o equal_a to_o the_o square_n of_o the_o line_n fb_o and_o bd_o for_o the_o angle_n fbd_v be_v a_o right_a angle_n wherefore_o
poligonon_n figure_n of_o 24._o side_n likewise_o of_o the_o hexagon_n ab_fw-la and_o of_o the_o pentagon_n ac_fw-la shall_v be_v make_v a_o poligonon_n figure_n of_o 30._o side_n one_o of_o who_o side_n shall_v subtend_v the_o ark_n bc._n for_o the_o denomination_n of_o ab_fw-la which_o be_v 6._o exceed_v the_o denomination_n of_o ac_fw-la which_o be_v 5._o only_a by_o unity_n so_o also_o forasmuch_o as_o the_o denomination_n of_o ab_fw-la which_o be_v 6._o exceed_v the_o denomination_n of_o ae_n which_o be_v 3._o by_o 3._o therefore_o the_o ark_n be_v shall_v contain_v 3._o side_n of_o a_o poligonon_n figure_n of_o .18_o side_n and_o observe_v this_o self_n same_o method_n and_o order_n a_o man_n may_v find_v out_o infinite_a side_n of_o a_o poligonon_n figure_n the_o end_n of_o the_o four_o book_n of_o euclides_n element_n ¶_o the_o five_o book_n of_o euclides_n element_n this_o five_o book_n of_o euclid_n be_v of_o very_o great_a commodity_n and_o use_n in_o all_o geometry_n book_n and_o much_o diligence_n ought_v to_o be_v bestow_v therein_o it_o ought_v of_o all_o other_o to_o be_v thorough_o and_o most_o perfect_o and_o ready_o know_v for_o nothing_o in_o the_o book_n follow_v can_v be_v understand_v without_o it_o the_o knowledge_n of_o they_o all_o depend_v of_o it_o and_o not_o only_o they_o and_o other_o write_n of_o geometry_n but_o all_o other_o science_n also_o and_o art_n as_o music_n astronomy_n perspective_n arithmetic_n the_o art_n of_o account_n and_o reckon_n with_o other_o such_o like_a this_o book_n therefore_o be_v as_o it_o be_v a_o chief_a treasure_n and_o a_o peculiar_a jewel_n much_o to_o be_v account_v of_o it_o entreat_v of_o proportion_n and_o analogy_n or_o proportionality_n which_o pertain_v not_o only_o unto_o line_n figure_n and_o body_n in_o geometry_n but_o also_o unto_o sound_n &_o voice_n of_o which_o music_n entreat_v as_o witness_v boetius_fw-la and_o other_o which_o write_v of_o music_n also_o the_o whole_a art_n of_o astronomy_n teach_v to_o measure_v proportion_n of_o time_n and_o move_n archimedes_n and_o jordan_n with_o other_o write_v of_o weight_n affirm_v that_o there_o be_v proportion_n between_o weight_n and_o weight_n and_o also_o between_o place_n &_o place_n you_o see_v therefore_o how_o large_a be_v the_o use_n of_o this_o five_o book_n wherefore_o the_o definition_n also_o thereof_o be_v common_a although_o hereof_o euclid_n they_o be_v accommodate_v and_o apply_v only_o to_o geometry_n the_o first_o author_n of_o this_o book_n be_v as_o it_o be_v affirm_v of_o many_o one_o eudoxus_n who_o be_v plato_n scholar_n eudoxus_n but_o it_o be_v afterward_o frame_v and_o put_v in_o order_n by_o euclid_n definition_n definition_n a_o part_n be_v a_o less_o magnitude_n in_o respect_n of_o a_o great_a magnitude_n when_o the_o less_o measure_v the_o great_a as_o in_o the_o other_o book_n before_o so_o in_o this_o the_o author_n first_o set_v orderly_a the_o definition_n and_o declaration_n of_o such_o term_n and_o word_n which_o be_v necessary_o require_v to_o the_o entreaty_n of_o the_o subject_n and_o matter_n thereof_o which_o be_v proportion_n and_o comparison_n of_o proportion_n or_o proportionality_n and_o first_o he_o show_v what_o a_o part_n be_v here_o be_v to_o be_v consider_v that_o all_o the_o definition_n of_o this_o five_o book_n be_v general_a to_o geometry_n and_o arithmetic_n and_o be_v true_a in_o both_o art_n even_o as_o proportion_n and_o proportionality_n be_v common_a to_o they_o both_o and_o chief_o appertain_v to_o number_v neither_o can_v they_o apt_o be_v apply_v to_o matter_n of_o geometry_n but_o in_o respect_n of_o number_n and_o by_o number_n yet_o in_o this_o book_n and_o in_o these_o definition_n here_o set_v euclid_n seem_v to_o speak_v of_o they_o only_o geometrical_o as_o they_o be_v apply_v to_o quantity_n continual_a as_o to_o line_n superficiece_n and_o body_n for_o that_o he_o yet_o continue_v in_o geometry_n i_o will_v notwithstanding_o for_o facility_n and_o far_a help_n of_o the_o reader_n declare_v they_o both_o by_o example_n in_o number_n and_o also_o in_o line_n for_o the_o clear_a understanding_n of_o a_o part_n it_o be_v to_o be_v note_v way_n that_o a_o part_n be_v take_v in_o the_o mathematical_a science_n two_o manner_n of_o way_n way_n one_o way_n a_o part_n be_v a_o less_o quantity_n in_o respect_n of_o a_o great_a whether_o it_o measure_v the_o great_a oâ—_n no._n the_o second_o way_n way_n a_o part_n be_v only_o that_o less_o quantity_n in_o respect_n of_o the_o great_a which_o measure_v the_o great_a a_o less_o quantity_n be_v say_v to_o measure_n or_o number_v a_o great_a quantity_n great_a when_o it_o be_v oftentimes_o take_v make_v precise_o the_o great_a quantity_n without_o more_o or_o less_o or_o be_v as_o oftentimes_o take_v from_o the_o great_a as_o it_o may_v there_o remain_v nothing_o as_o suppose_v the_o line_n ab_fw-la to_o contain_v 3._o and_o the_o line_n cd_o to_o contain_v 9_o they_o do_v the_o line_n ab_fw-la measure_n the_o line_n cd_o for_o that_o if_o it_o be_v take_v certain_a time_n namely_o 3._o time_n it_o make_v precise_o the_o line_n cd_o that_o be_v 9_o without_o more_o or_o less_o again_o if_o the_o say_v less_o line_n ab_fw-la be_v take_v from_o the_o great_a cd_o as_o often_o as_o it_o may_v be_v namely_o 3._o time_n there_o shall_v remain_v nothing_o of_o the_o great_a so_o the_o number_n 3._o be_v say_v to_o measure_v 12._o for_o that_o be_v take_v certain_a time_n namely_o four_o time_n it_o make_v just_a 12._o the_o great_a quantity_n and_o also_o be_v take_v from_o 12._o as_o often_o as_o it_o may_v namely_o 4._o time_n there_o shall_v remain_v nothing_o and_o in_o this_o meaning_n and_o signification_n do_v euclid_n undoubted_o here_o in_o this_o define_v a_o part_n part_n say_v that_o it_o be_v a_o less_o magnitude_n in_o comparison_n of_o a_o great_a when_o the_o less_o measure_v the_o great_a as_o the_o line_n ab_fw-la before_o set_v contain_v 3_n be_v a_o less_o quantity_n in_o comparison_n of_o the_o line_n cd_o which_o contain_v 9_o and_o also_o measure_v it_o for_o it_o be_v certain_a time_n take_v namely_o 3._o time_n precise_o make_v it_o or_o take_v from_o it_o as_o often_o as_o it_o may_v there_o remain_v nothing_o wherefore_o by_o this_o definition_n the_o line_n ab_fw-la be_v a_o part_n of_o the_o line_n cd_o likewise_o in_o number_n the_o number_n 5._o be_v a_o part_n of_o the_o number_n 15._o for_o it_o be_v a_o less_o number_n or_o quantity_n compare_v to_o the_o great_a and_o also_o it_o measure_v the_o great_a for_o be_v take_v certain_a time_n namely_o 3._o time_n it_o make_v 15._o and_o this_o kind_n of_o part_n be_v call_v common_o pars_fw-la metiens_fw-la or_o mensurans_fw-la mensuranâ—_n that_o be_v a_o measure_a part_n some_o call_v it_o pars_fw-la multiplicatina_fw-la multiplicatiâ—a_fw-la and_o of_o the_o barbarous_a it_o be_v call_v pars_fw-la aliquota_fw-la aliquota_fw-la that_o be_v a_o aliquote_v part_n and_o this_o kind_n of_o part_n be_v common_o use_v in_o arithmetic_n arithmetic_n the_o other_o kind_n of_o a_o part_n part_n be_v any_o less_o quantity_n in_o comparison_n of_o a_o great_a whether_o it_o be_v in_o number_n or_o magnitude_n and_o whether_o it_o measure_v or_o no._n as_o suppose_v the_o line_n ab_fw-la to_o be_v 17._o and_o let_v it_o be_v divide_v into_o two_o part_n in_o the_o point_n c_o namely_o into_o the_o line_n ac_fw-la &_o the_o line_n cb_o and_o let_v the_o line_n ac_fw-la the_o great_a part_n contain_v 12._o and_o let_v the_o line_n bc_o the_o less_o part_n contain_v 5._o now_o either_o of_o these_o line_n by_o this_o definition_n be_v a_o part_n of_o the_o whole_a line_n ab_fw-la for_o either_o of_o they_o be_v a_o less_o magnitude_n or_o quantity_n in_o comparison_n of_o the_o whole_a line_n ab_fw-la but_o neither_o of_o they_o measure_v the_o whole_a line_n ab_fw-la for_o the_o less_o line_n cb_o contain_v 5._o take_v as_o often_o as_o you_o listen_v will_v never_o make_v precise_o ab_fw-la which_o contain_v 17._o if_o you_o take_v it_o 3._o time_n it_o make_v only_o 15._o so_o lack_v it_o 2._o of_o 17._o which_o be_v to_o little_a if_o you_o take_v it_o 4._o time_n so_o make_v it_o 20._o them_z be_v there_o three_o to_o much_o so_o it_o never_o make_v precise_o 17._o but_o either_o to_o much_o or_o to_o little_a likewise_o the_o other_o part_n ac_fw-la measure_v not_o the_o whole_a line_n ab_fw-la for_o take_v once_o it_o make_v but_o 12._o which_o be_v less_o than_o 17._o and_o take_v twice_o it_o make_v 24._o which_o be_v more_o than_o 17._o by_o â—_o so_o it_o never_o precise_o make_v by_o take_v thereof_o the_o whole_a ab_fw-la but_o either_o more_o or_o less_o and_o this_o kind_n of_o part_n they_o common_o call_v pars_fw-la
quadrates_n but_o all_o other_o kind_n of_o rectiline_a figure_n plain_a plait_n &_o superficiese_n what_o so_o ever_o so_o that_o if_o any_o such_o figure_n be_v commensurable_a unto_o that_o rational_a squareâ—_n it_o be_v also_o rational_a as_o suppose_v that_o the_o square_a of_o the_o rational_a line_n which_o be_v also_o rational_a be_v abcd_o suppose_v 〈◊〉_d so_o some_o other_o square_a as_o the_o square_a efgh_o to_o be_v commensurable_a to_o the_o same_o they_o be_v the_o square_a efgh_o also_o rational_a so_o also_o if_o the_o rectiline_a figure_n klmn_n which_o be_v a_o figure_n on_o the_o one_o side_n long_o be_v commensurable_a unto_o the_o say_a square_a as_o be_v suppose_v in_o this_o exampleâ—_n it_o be_v also_o a_o rational_a superficies_n and_o so_o of_o all_o other_o superficiese_n 10_o such_o which_o be_v incommensurable_a unto_o it_o be_v irrational_a definition_n where_o it_o be_v say_v in_o this_o definition_n such_o which_o be_v incommensurable_a it_o be_v general_o to_o be_v take_v as_o be_v this_o word_n commensurable_a in_o the_o definition_n before_o for_o all_o superficiese_n whether_o they_o be_v square_n or_o figure_n on_o the_o one_o side_n long_o or_o otherwise_o what_o manner_n of_o right_n line_v figure_n so_o ever_o it_o be_v if_o they_o be_v incommensurable_a unto_o the_o rational_a square_n suppose_v they_o be_v they_o irrational_a as_o let_v thâ—_z square_v abcd_o be_v the_o square_n of_o the_o suppose_a rational_a line_n which_o square_n therefore_o be_v also_o rational_a suppose_v also_o also_o a_o other_o square_a namely_o the_o square_a e_o suppose_v also_o any_o other_o figure_n as_o for_o example_n sake_n a_o figure_n of_o one_o side_n long_o which_o let_v be_v f_o now_o if_o the_o square_a e_o and_o the_o figure_n fletcher_n be_v both_o incommensurable_a to_o the_o rational_a square_n abcd_o then_o be_v 〈◊〉_d of_o these_o figure_n e_o &_o fletcher_n irrational_a and_o so_o of_o other_o 11_o and_o these_o line_n who_o powere_n they_o be_v be_v irrational_a if_o they_o be_v square_n then_o be_v their_o side_n irrational_a if_o they_o be_v not_o square_n definition_n but_o some_o other_o rectiline_a figure_n then_o shall_v the_o line_n who_o square_n be_v equal_a to_o these_o rectiline_a figure_n be_v irrational_a suppose_v that_o the_o rational_a square_n be_v abcd._n suppose_v also_o a_o other_o square_a namely_o the_o square_a e_o which_o let_v be_v incommensurable_a to_o the_o rational_a square_n &_o therefore_o be_v it_o irrational_a and_o let_v the_o side_n or_o line_n which_o produce_v this_o square_n be_v the_o line_n fg_o then_o shall_v the_o line_n fg_o by_o this_o definition_n be_v a_o irrational_a line_n because_o it_o be_v the_o side_n of_o a_o irrational_a square_n let_v also_o the_o figure_n h_o be_v a_o figure_n on_o the_o one_o side_n long_o which_o may_v be_v any_o other_o rectiline_a figure_n rectangle_v or_o not_o rectangle_v triangle_n pentagone_fw-mi trapezite_fw-mi or_o what_o so_o ever_o else_o be_v incommensurable_a to_o the_o rational_a square_n abcd_o then_o because_o the_o figure_n h_o be_v not_o a_o square_a it_o have_v no_o side_n or_o root_n to_o produce_v it_o yet_o may_v there_o be_v a_o square_n make_v equal_a unto_o it_o for_o that_o all_o such_o figure_n may_v be_v reduce_v into_o triangle_n and_o so_o into_o square_n by_o the_o 14._o of_o the_o second_o suppose_v that_o the_o square_a queen_n be_v equal_a to_o the_o irrational_a figure_n h._n the_o side_n of_o which_o figure_n q_n let_v be_v the_o line_n kl_o then_o shall_v the_o line_n kl_o be_v also_o a_o irrational_a line_n because_o the_o power_n or_o square_v thereof_o be_v equal_a to_o the_o irrational_a figure_n h_o and_o thus_o conceive_v of_o other_o the_o like_a these_o irrational_a line_n and_o figure_n be_v the_o chief_a matter_n and_o subject_n which_o be_v entreat_v of_o in_o all_o this_o ten_o book_n the_o knowledge_n of_o which_o be_v deep_a and_o secret_a and_o pertain_v to_o the_o high_a and_o most_o worthy_a part_n of_o geometry_n wherein_o stand_v the_o pith_n and_o marry_v of_o the_o hole_n science_n the_o knowledge_n hereof_o bring_v light_n to_o all_o the_o book_n follow_v with_o out_o which_o they_o be_v hard_a and_o can_v be_v at_o all_o understand_v and_o for_o the_o more_o plainness_n you_o shall_v note_v that_o of_o irrational_a line_n there_o be_v diâ—ers_n sort_n and_o kind_n but_o they_o who_o name_n be_v set_v in_o a_o table_n here_o follow_v and_o be_v in_o number_n 13._o be_v the_o chief_a and_o in_o this_o ten_o book_n sufficient_o for_o euclides_n principal_a purpose_n discourse_v on_o a_o medial_a line_n a_o binomial_a line_n a_o first_o bimedial_a line_n a_o second_o bimedial_a line_n a_o great_a line_n a_o line_n contain_v in_o power_n a_o rational_a superficies_n and_o a_o medial_a superficies_n a_o line_n contain_v in_o power_n two_o medial_a superficiece_n a_o residual_a line_n a_o first_o medial_a residual_a line_n a_o 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than_o the_o half_a construction_n and_o from_o the_o residue_n be_v take_v again_o more_o than_o the_o half_a and_o so_o still_o continual_o there_o shall_v at_o the_o length_n be_v leave_v a_o certain_a magnitude_n lesser_a than_o the_o less_o magnitude_n give_v namely_o then_o c._n for_o forasmuch_o as_o c_z be_v the_o less_o magnitude_n therefore_o c_z may_v be_v so_o multiply_v that_o at_o the_o length_n it_o will_v be_v great_a than_o the_o magnitude_n ab_fw-la by_o the_o 5._o definition_n of_o the_o five_o book_n demonstration_n let_v it_o be_v so_o multiply_v and_o let_v the_o multiplex_n of_o c_z great_a than_o ab_fw-la be_v de._n and_o divide_v de_fw-fr into_o the_o part_n equal_a unto_o c_o which_o let_v be_v df_o fg_o and_o ge._n and_o from_o the_o magnitude_n ab_fw-la take_v away_o more_o than_o the_o half_a which_o let_v be_v bh_o and_o again_o from_o ah_o take_v away_o more_o than_o the_o half_a which_o let_v be_v hk_a and_o so_o do_v continual_o until_o the_o division_n which_o be_v in_o the_o magnitude_n ab_fw-la be_v equal_a in_o multitude_n unto_o the_o division_n which_o be_v in_o the_o magnitude_n de._n so_o that_o let_v the_o division_n ak_o kh_o and_o hb_o be_v equal_a in_o multitude_n unto_o the_o division_n df_o fg_o and_o ge._n and_o forasmuch_o as_o the_o magnitude_n de_fw-fr be_v great_a than_o the_o magnitude_n ab_fw-la and_o from_o de_fw-fr be_v take_v away_o less_o than_o the_o half_a that_o be_v eglantine_n which_o detraction_n or_o take_v away_o be_v understand_v to_o be_v do_v by_o the_o former_a division_n of_o the_o magnitude_n de_fw-fr into_o the_o part_n equal_a unto_o c_o for_o as_o a_o magnitude_n be_v by_o multiplication_n increase_v so_o be_v it_o by_o division_n diminish_v and_o from_o ab_fw-la be_v take_v away_o more_o than_o the_o half_a that_o be_v bh_o therefore_o the_o residue_n gd_o be_v great_a than_o the_o residue_n ha_o which_o thing_n be_v most_o true_a and_o most_o easy_a to_o conceive_v if_o we_o remember_v this_o principle_n that_o the_o residue_n of_o a_o great_a magnitude_n after_o the_o take_v away_o of_o the_o half_a or_o less_o than_o the_o half_a be_v ever_o great_a than_o the_o residue_n of_o a_o less_o magnitude_n after_o the_o take_v away_o of_o more_o than_o the_o half_a and_o forasmuch_o as_o the_o magnitude_n gd_o be_v great_a than_o the_o magnitude_n ha_o and_o from_o gd_o be_v take_v away_o the_o half_a that_o be_v gf_o and_o from_o ah_o be_v take_v away_o more_o than_o the_o half_a that_o be_v hk_o therefore_o the_o residue_n df_o be_v great_a than_o the_o residue_n ak_o by_o the_o foresay_a principle_n but_o the_o magnitude_n df_o be_v equal_a unto_o the_o magnitude_n c_o by_o supposition_n wherefore_o also_o the_o magnitude_n c_o be_v great_a than_o the_o magnitude_n ak_o wherefore_o the_o magnitude_n ak_o be_v less_o than_o the_o magnitude_n c._n wherefore_o of_o the_o magnitude_n
first_o set_v they_o also_o must_v needs_o be_v at_o the_o least_o commensurable_a in_o power_n the_o one_o to_o the_o other_o for_o forasmuch_o as_o their_o square_n be_v rational_a they_o shall_v be_v commensurable_a to_o the_o square_n of_o the_o rational_a line_n first_o set_v wherefore_o by_o the_o 12._o of_o this_o book_n they_o be_v also_o commensurable_a the_o one_o to_o the_o other_o wherefore_o their_o line_n be_v at_o the_o least_o commensurable_a in_o power_n the_o one_o to_o the_o other_o and_o it_o be_v possible_a also_o that_o they_o may_v be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o for_o suppose_v that_o a_o be_v a_o rational_a liâ—e_z first_o set_v and_o let_v the_o line_n b_o be_v unto_o the_o same_o rational_a line_n a_o commensurable_a in_o power_n only_o that_o be_v incommensurable_a in_o length_n unto_o it_o let_v there_o be_v also_o a_o other_o line_n c_o commensurable_a in_o length_n to_o the_o line_n b_o which_o be_v possible_a by_o the_o principle_n of_o this_o book_n now_o by_o the_o 13._o of_o the_o ten_o it_o be_v manifest_a that_o the_o line_n c_o be_v incommensurable_a in_o length_n unto_o the_o line_n a._n but_o the_o square_n of_o the_o line_n a_o be_v commensurable_a to_o the_o square_n of_o the_o line_n b_o by_o supposition_n and_o the_o square_a of_o the_o line_n c_o be_v also_o commensurable_a to_o the_o square_n of_o the_o line_n b_o by_o supposition_n wherefore_o by_o the_o 12._o of_o this_o book_n the_o square_a of_o the_o line_n c_o be_v commensurable_a to_o the_o square_n of_o the_o line_n a._n wherefore_o by_o the_o definition_n the_o line_n c_o shall_v be_v rational_a commensurable_a in_o power_n only_o to_o the_o line_n a_o as_o also_o be_v the_o line_n b._n wherefore_o there_o be_v give_v two_o rational_a line_n commensurable_a in_o power_n only_o to_o the_o rational_a line_n first_o set_v and_o commensurable_a in_o length_n the_o one_o to_o the_o other_o here_o be_v to_o be_v note_v which_o thing_n also_o we_o before_o note_v in_o the_o definition_n that_o campane_n and_o other_o which_o follow_v he_o bring_v in_o these_o phrase_n of_o speech_n to_o call_v some_o line_n rational_a in_o power_n only_o cause_n and_o other_o some_o rational_a in_o length_n and_o in_o power_n which_o we_o can_v find_v that_o euclid_n ever_o use_v for_o these_o word_n in_o length_n and_o in_o power_n be_v never_o refer_v to_o rationality_n or_o irrationalitie_n but_o always_o to_o the_o commensurabilitie_n or_o incommensurablitie_n of_o line_n which_o pervert_v of_o word_n as_o be_v there_o declare_v have_v much_o increase_v the_o difficulty_n and_o obscureness_n of_o this_o book_n book_n and_o now_o i_o think_v it_o good_a again_o to_o put_v you_o in_o mind_n that_o in_o these_o proposition_n which_o follow_v we_o must_v ever_o have_v before_o our_o eye_n the_o rational_a line_n first_o set_v note_n unto_o which_o other_o line_n compare_v be_v either_o rational_a or_o irrational_a according_a to_o their_o commensurability_n or_o incommensurabilitie_n ¶_o the_o 16._o theorem_a the_o 19_o proposition_n a_o rectangle_n figure_n comprehend_v under_o right_a line_n commensurable_a in_o length_n be_v rational_a according_a to_o one_o of_o the_o foresay_a way_n be_v rational_a svppose_v that_o this_o rectangle_n figure_n ac_fw-la be_v comprehend_v under_o these_o right_a line_n ab_fw-la and_o bc_o be_v commensurable_a in_o length_n and_o rational_a according_a to_o one_o of_o the_o foresay_a way_n then_o i_o say_v that_o the_o superficies_n ac_fw-la be_v rational_a describe_v by_o the_o 46._o oâ—_n the_o first_o upon_o the_o line_n ab_fw-la a_o square_a ad._n construction_n wherefore_o that_o square_a ad_fw-la be_v rational_a by_o the_o definition_n demonstration_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v commensurable_a in_o length_n unto_o the_o line_n bc_o and_o the_o line_n ab_fw-la be_v equal_a unto_o the_o line_n bd_o therefore_o the_o line_n bd_o be_v commensurable_a in_o length_n unto_o the_o line_n bc._n and_o as_o the_o line_n bd_o be_v to_o the_o line_n bc_o so_o be_v the_o square_a dam_n to_o the_o superficies_n ac_fw-la by_o the_o first_o of_o the_o six_o but_o it_o be_v prove_v that_o the_o line_n bd_o be_v commensurable_a unto_o the_o line_n bc_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a dam_n be_v commensurable_a unto_o the_o rectangle_n superficies_n ac_fw-la but_o the_o square_a dam_n be_v rational_a wherefore_o the_o rectangle_n superficies_n ac_fw-la also_o be_v rational_a by_o the_o definition_n a_o rectangle_n figure_n therefore_o comprehend_v under_o right_a line_n commensurable_a in_o length_n be_v rational_a accord_v to_o one_o of_o the_o foresay_a way_n be_v rational_a which_o be_v require_v to_o be_v prove_v proposition_n where_o as_o in_o the_o former_a demonstration_n the_o square_n be_v describe_v upon_o the_o less_o line_n we_o may_v also_o demonstrate_v the_o proposition_n if_o we_o describe_v the_o square_n upon_o the_o great_a line_n and_o that_o after_o this_o manner_n suppose_v that_o the_o rectangle_n superficies_n bc_o be_v contain_v of_o these_o unequal_a line_n ab_fw-la and_o ac_fw-la which_o let_v be_v rational_a commensurable_a the_o one_o to_o the_o other_o in_o length_n and_o let_v the_o line_n ac_fw-la be_v the_o great_a case_n and_o upon_o the_o line_n ac_fw-la describe_v the_o square_a dc_o then_o i_o say_v that_o the_o parallelogram_n bc_o be_v rational_a length_n for_o the_o line_n ac_fw-la be_v commensurable_a in_o length_n unto_o the_o line_n ab_fw-la by_o supposition_n and_o the_o line_n dam_n be_v equal_a to_o the_o line_n ac_fw-la wherefore_o the_o line_n dam_n be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la but_o what_o proportion_n the_o line_n dam_n have_v to_o the_o line_n ab_fw-la the_o same_o have_v the_o square_a dc_o to_o the_o paraâ—lelogramme_n câ—_n by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10._o of_o this_o book_n the_o square_a dc_o be_v commensurable_a to_o the_o parallelogram_n cb._n but_o it_o be_v manifest_a that_o the_o square_a dc_o be_v rational_a for_o that_o it_o be_v the_o square_a of_o a_o rational_a line_n namely_o ac_fw-la wherefore_o by_o the_o definition_n the_o parallelogram_n also_o cb_o be_v rational_a moreover_o forasmuch_o as_o those_o two_o former_a demonstration_n seem_v to_o speak_v of_o that_o parallelogram_n which_o be_v make_v of_o two_o line_n of_o which_o any_o one_o may_v be_v the_o liâ—e_z first_o set_v which_o be_v call_v the_o first_o rational_a line_n from_o which_o we_o say_v ought_v to_o be_v take_v the_o measure_n of_o the_o other_o line_n compare_v unto_o it_o and_o the_o other_o be_v commensurable_a in_o length_n to_o the_o same_o first_o rational_a line_n length_n which_o be_v the_o first_o kind_n of_o rational_a line_n commensurable_a in_o length_n i_o think_v it_o good_a here_o to_o set_v a_o other_o case_n of_o the_o other_o kind_n of_o rational_a line_n of_o line_n i_o say_v rational_a commensurable_a in_o length_n compare_v to_o a_o other_o rational_a line_n first_o set_v to_o declare_v the_o general_a truth_n of_o this_o theorem_a and_o that_o we_o may_v see_v that_o this_o particle_n according_a to_o any_o of_o the_o foresay_a way_n be_v not_o here_o in_o vain_a put_v now_o then_o suppose_v first_o a_o rational_a line_n ab_fw-la let_v there_o be_v also_o a_o parallelogram_n cd_o contain_v under_o the_o line_n ce_fw-fr and_o ed_z which_o line_n let_v be_v rational_a that_o be_v commensurable_a in_o length_n to_o the_o â—irst_n rational_a line_n propound_v ab_fw-la howbeit_o let_v those_o two_o line_n ce_fw-fr and_o ed_z be_v diverse_a and_o unequal_a line_n unto_o the_o first_o rational_a line_n ab_fw-la then_o i_o say_v that_o the_o parallelogram_n cd_o be_v rational_a case_n describe_v the_o square_a of_o the_o line_n de_fw-fr which_o let_v be_v df._n first_o it_o be_v manifest_a by_o the_o 12._o of_o this_o book_n that_o the_o line_n ce_fw-fr &_o ed_z be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o for_o either_o of_o they_o be_v suppose_v to_o be_v commensurable_a in_o length_n unto_o the_o line_n ab_fw-la but_o the_o line_n ed_z be_v equal_a to_o the_o line_n ef._n wherefore_o the_o line_n ce_fw-fr be_v commensurable_a in_o length_n to_o the_o line_n bf_o but_o 〈◊〉_d the_o line_n ce_fw-fr be_v â—o_o the_o line_n â—_o f_o â—o_o be_v the_o parallelogram_n cd_o to_o the_o square_a df_o by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10._o of_o this_o book_n the_o parallelogram_n cd_o shall_v be_v commensurable_a to_o the_o square_a df._n but_o the_o square_a df_o be_v commensurable_a to_o the_o square_n of_o the_o line_n ab_fw-la which_o be_v the_o first_o rational_a line_n propound_v wherefore_o by_o the_o 12._o of_o this_o book_n the_o parallelogram_n cd_o be_v commensurable_a to_o the_o square_n of_o the_o line_n ab_fw-la but_o the_o square_n of_o
two_o other_o proposition_n go_v next_o before_o it_o so_o far_o misplace_v that_o where_o they_o be_v word_n for_o word_n before_o duâ—ly_o place_v be_v the_o 105._o and_o 106._o yet_o here_o after_o the_o book_n end_v they_o be_v repeat_v with_o the_o number_n of_o 116._o and_o 117._o proposition_n zambert_n therein_o be_v more_o faithful_a to_o follow_v as_o he_o find_v in_o his_o greek_n example_n than_o he_o be_v skilful_a or_o careful_a to_o do_v what_o be_v necessary_a yea_o and_o some_o greek_n write_v ancient_a copy_n have_v they_o not_o so_o though_o in_o deed_n they_o be_v well_o demonstrate_v yet_o truth_n disord_v be_v half_a disgracedâ—_n especial_o where_o the_o pattern_n of_o good_a order_n by_o profession_n be_v avouch_v to_o be_v but_o through_o ignorance_n arrogancy_n and_o â—emerltie_n of_o unskilful_a method_n master_n many_o thing_n remain_v yet_o in_o these_o geometrical_a element_n undue_o tumble_v in_o though_o true_a yet_o with_o disgrace_n which_o by_o help_n of_o so_o many_o wit_n and_o hability_n of_o such_o as_o now_o may_v have_v good_a cause_n to_o be_v skilful_a herein_o will_v i_o hope_v ere_o long_o be_v take_v away_o and_o thing_n of_o importance_n want_v supply_v the_o end_n of_o the_o ten_o book_n of_o euclides_n element_n ¶_o the_o eleven_o book_n of_o euclides_n element_n book_n hitherto_o have_v â—vclidâ—_n in_o thâ—sâ—_n former_a book_n with_o a_o wonderful_a method_n and_o order_n entreat_v of_o such_o kind_n of_o figure_n superficial_a which_o be_v or_o may_v be_v describe_v in_o a_o superficies_n or_o plain_a and_o have_v teach_v and_o set_v forth_o their_o property_n nature_n generation_n and_o production_n even_o from_o the_o first_o root_n ground_n and_o beginning_n of_o they_o namely_o from_o a_o point_n which_o although_o it_o be_v indivisible_a yet_o be_v it_o the_o begin_n of_o all_o quantity_n continual_a and_o of_o it_o and_o of_o the_o motion_n and_o slow_v thereof_o be_v produce_v a_o line_n and_o consequent_o all_o quantity_n continual_a as_o all_o figure_n plain_a and_o solid_a what_o so_o ever_o euclid_n therefore_o in_o his_o first_o book_n begin_v with_o it_o boot_n and_o from_o thence_o go_v he_o to_o a_o line_n as_o to_o a_o thing_n most_o simple_a next_o unto_o a_o point_n then_o to_o a_o superficies_n and_o to_o angle_n and_o so_o through_o the_o whole_a first_o book_n boâ—â—e_n he_o entreat_v of_o these_o most_o simple_a and_o plain_a ground_n in_o the_o second_o book_n he_o entreat_v further_o â—â—oâ—e_n and_o go_v unto_o more_o hard_a matter_n and_o teach_v of_o division_n of_o line_n and_o of_o the_o multiplication_n of_o line_n and_o of_o their_o part_n and_o of_o their_o passion_n and_o property_n and_o for_o that_o rightline_v â—_z be_v far_o distant_a in_o nature_n and_o property_n from_o round_a and_o circular_a figure_n in_o the_o three_o book_n he_o instruct_v the_o reader_n of_o the_o nature_n and_o condition_n of_o circle_n booâ—e_n in_o the_o four_o book_n he_o compare_v figure_n of_o right_a line_n and_o circle_n together_o bâ—oâ—e_n and_o teach_v how_o to_o describe_v a_o figure_n of_o right_a line_n with_o in_o or_o about_o a_o circle_n and_o contrariwise_o a_o circle_n with_o in_o or_o about_o a_o rectiline_a figure_n in_o the_o five_o book_n he_o search_v out_o the_o nature_n of_o proportion_n a_o matter_n of_o wonderful_a use_n and_o deep_a consideration_n boâ—â—e_n for_o that_o otherwise_o he_o can_v not_o compare_v â—igure_n with_o figure_n or_o the_o side_n of_o figure_n together_o for_o whatsoever_o be_v compare_v to_o any_o other_o thing_n be_v compare_v unto_o it_o undoubted_o under_o some_o kind_n of_o proportion_n wherefore_o in_o the_o six_o book_n he_o compare_v figure_n together_o booâ—e_n one_o to_o a_o other_o likewise_o their_o side_n and_o for_o that_o the_o nature_n of_o proportion_n can_v not_o be_v full_o and_o clear_o see_v without_o the_o knowledge_n of_o number_n wherein_o it_o be_v first_o and_o chief_o find_v in_o the_o seven_o bookâ—_n eight_o booâ—â—_n and_o nine_o book_n book_n he_o entreatâ—th_v of_o number_n &_o of_o the_o kind_n and_o property_n thereof_o and_o because_o that_o the_o side_n of_o solid_a body_n for_o the_o most_o part_n be_v of_o such_o sort_n that_o compare_v together_o they_o have_v such_o proportion_n the_o one_o to_o the_o other_o booâ—e_n which_o can_v not_o be_v express_v by_o any_o number_n certain_a and_o therefore_o be_v call_v irrational_a line_n he_o in_o the_o ten_o book_n have_v write_v &_o teach_v which_o lineâ—_z be_v commensurable_a or_o incommensurable_a the_o one_o to_o the_o other_o and_o of_o the_o diversity_n of_o kind_n of_o irrational_a line_n with_o all_o the_o condition_n &_o propriety_n of_o they_o and_o thus_o have_v euclid_n in_o these_o ten_o foresay_a book_n full_o &_o most_o plenteous_o in_o a_o marvelous_a order_n teach_v whatsoever_o seem_v necessary_a and_o requisite_a to_o the_o knowledge_n of_o all_o superficial_a figure_n of_o what_o sort_n &_o form_n so_o ever_o they_o be_v now_o in_o these_o book_n follow_v he_o entreat_v of_o figure_n of_o a_o other_o kind_n namely_o of_o bodily_a figure_n follow_v as_o of_o cube_n pyramid_n cones_n column_n cilinder_n parallelipipedon_n sphere_n and_o such_o othersâ—_n and_o show_v the_o diversity_n of_o they_o the_o generation_n and_o production_n of_o they_o and_o demonstrate_v with_o great_a and_o wonderful_a art_n their_o propriety_n and_o passion_n with_o all_o their_o nature_n and_o condition_n he_o also_o compare_v one_o oâ—_n they_o to_o a_o other_o whereby_o to_o know_v the_o reason_n and_o proportion_n of_o the_o one_o to_o the_o other_o chief_o of_o the_o five_o body_n which_o be_v call_v regular_a body_n element_n and_o these_o be_v the_o thing_n of_o all_o other_o entreat_v of_o in_o geometry_n most_o worthy_a and_o of_o great_a dignity_n and_o as_o it_o be_v the_o end_n and_o final_a intent_n of_o the_o whole_a be_v of_o geometry_n and_o for_o who_o cause_n have_v be_v write_v and_o speak_v whatsoever_o have_v hitherto_o in_o the_o former_a book_n be_v say_v or_o write_v as_o the_o first_o book_n be_v a_o ground_n and_o a_o necessary_a entry_n to_o all_o the_o râ—st_o â—ollowing_v so_o be_v this_o eleven_o book_n a_o necessary_a entry_n and_o ground_n to_o the_o rest_n which_o follow_v 〈◊〉_d and_o as_o that_o contain_v the_o declaration_n of_o word_n and_o definition_n of_o thinger_n requisite_a to_o the_o knowledge_n of_o superficial_a figure_n and_o entreat_v of_o line_n and_o of_o their_o division_n and_o section_n which_o be_v the_o term_n and_o limit_n of_o superficial_a figure_n so_o in_o this_o book_n be_v set_v forth_o the_o declaration_n of_o word_n and_o definition_n of_o thing_n pertain_v to_o solid_a and_o corporal_a figure_n and_o also_o of_o superficiece_n which_o be_v the_o term_n &_o limit_n of_o solid_n moreover_o of_o the_o division_n and_o intersection_n of_o they_o and_o diverse_a other_o thing_n without_o which_o the_o 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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 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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
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
and_o it_o be_v in_o the_o segment_n abc_n which_o be_v great_a than_o the_o semicircle_n part_n and_o forasmuch_o as_o in_o the_o circle_n there_o be_v a_o figure_n of_o four_o side_n namely_o abcd._n but_o if_o within_o a_o circle_n be_v describe_v a_o figure_n of_o four_o side_n the_o angle_n thereof_o which_o be_v opposite_a the_o one_o to_o the_o other_o be_v equal_a to_o two_o right_a angle_n by_o the_o 22._o of_o the_o three_o wherefore_o by_o the_o same_o the_o angle_n abc_n and_o adc_o be_v equal_a to_o two_o right_a angle_n but_o the_o angle_n abc_n be_v less_o than_o a_o right_a angle_n wherefore_o the_o angle_n remain_v adc_o be_v great_a than_o a_o right_a angle_n and_o it_o be_v in_o a_o segment_n which_o be_v less_o than_o the_o semicircle_n again_o forasmuch_o as_o the_o angle_n comprehend_v under_o the_o right_a line_n ac_fw-la and_o of_o be_v a_o right_a angle_n part_n therefore_o the_o angle_n comprehend_v under_o the_o right_a line_n ca_n and_o the_o circumference_n adc_o be_v less_o than_o a_o right_a angle_n wherefore_o in_o a_o circle_n a_o angle_n make_v in_o the_o semicircle_n be_v a_o right_a angle_n but_o a_o angle_n make_v in_o the_o segment_n great_a than_o the_o semicircle_n be_v less_o than_o a_o right_a angle_n and_o a_o angle_n make_v in_o the_o segment_n less_o than_o the_o semicircle_n be_v great_a than_o a_o right_a angle_n and_o moreover_o the_o angle_n of_o the_o great_a segment_n be_v great_a than_o a_o right_a angle_n &_o the_o angle_n of_o the_o less_o segment_n be_v less_o than_o a_o right_a angle_n which_o be_v require_v to_o be_v demonstrate_v an_o other_o demonstration_n to_o prove_v that_o the_o angle_n bac_n be_v a_o right_a angle_n angle_n forasmuch_o as_o the_o angle_n aec_o be_v double_a to_o the_o angle_n bae_o by_o the_o 32._o of_o the_o first_o for_o it_o be_v equal_a to_o the_o two_o inward_a angle_n which_o be_v opposite_a but_o the_o inward_a angle_n be_v by_o the_o 5._o of_o the_o first_o equal_a the_o one_o to_o the_o other_o and_o the_o angle_n aeb_fw-mi be_v double_a to_o the_o angle_n eac_n wherefore_o the_o angle_n aeb_fw-mi and_o aec_o be_v double_a to_o the_o angle_n bac_n but_o the_o angle_n aeb_fw-mi and_o aec_o be_v equal_a to_o two_o right_a angle_n wherefore_o the_o angle_n bac_n be_v a_o right_a angle_n which_o be_v require_v to_o be_v demonstrate_v correlary_a hereby_o it_o be_v manifest_a corollary_n that_o if_o in_o a_o triangle_n one_o angle_n be_v equal_a to_o the_o two_o other_o angle_n remain_v the_o same_o angle_n be_v a_o right_a angle_n for_o that_o the_o side_n angle_n to_o that_o one_o angle_n namely_o the_o angle_n which_o be_v make_v of_o the_o side_n produce_v without_o the_o triangle_n be_v equal_a to_o the_o same_o angle_n but_o when_o the_o side_n angle_n be_v equal_a the_o one_o to_o the_o other_o they_o be_v also_o right_a angle_n ¶_o a_o addition_n of_o pelitarius_n if_o in_o a_o circle_n be_v inscribe_v a_o rectangle_n triangle_n the_o side_n opposite_a unto_o the_o right_a angle_n shall_v be_v the_o diameter_n of_o the_o circle_n pâ—litarius_n suppose_v that_o in_o the_o circle_n abc_n be_v inscribe_v a_o rectangle_n triangle_n abc_n who_o angle_n at_o the_o point_n b_o let_v be_v a_o right_a angle_n then_o i_o say_v that_o the_o side_n ac_fw-la be_v the_o diameter_n of_o the_o circle_n for_o if_o not_o then_o shall_v the_o centre_n be_v without_o the_o line_n ac_fw-la as_o in_o the_o point_n e._n absurdity_n and_o draw_v a_o line_n from_o the_o point_n a_o to_o the_o point_n e_o &_o produce_v it_o to_o the_o circumference_n to_o the_o point_n d_o and_o let_v aed_n be_v the_o diameter_n and_o draw_v a_o line_n from_o the_o point_n b_o to_o the_o point_n d._n now_o by_o this_o 31._o proposition_n the_o angle_n abdella_n shall_v be_v a_o right_a angle_n and_o therefore_o shall_v be_v equal_a to_o the_o right_a angle_n abc_n namely_o the_o part_n to_o the_o whole_a which_o be_v absurd_a even_o so_o may_v we_o prove_v that_o the_o centre_n be_v in_o no_o other_o where_o but_o in_o the_o line_n ac_fw-la wherefore_o ac_fw-la be_v the_o diameter_n of_o the_o circle_n which_o be_v require_v to_o be_v prove_v ¶_o a_o addition_n of_o campane_n campane_n by_o this_o 31._o proposition_n and_o by_o the_o 16._o proposition_n of_o this_o book_n it_o be_v manifest_a that_o although_o in_o mix_a angle_n which_o be_v contain_v under_o a_o right_a line_n and_o the_o circumference_n of_o a_o circle_n there_o may_v be_v give_v a_o angle_n less_o &_o great_a than_o a_o right_a angle_n yet_o can_v there_o never_o be_v give_v a_o angle_n equal_a to_o a_o right_a angle_n for_o every_o section_n of_o a_o circle_n be_v either_o a_o semicircle_n or_o great_a than_o a_o semicircle_n or_o less_o but_o the_o angle_n of_o a_o semicircle_n be_v by_o the_o 16._o of_o this_o book_n less_o than_o a_o right_a angle_n and_o so_o also_o be_v the_o angle_n of_o a_o less_o section_n by_o this_o 31._o proposition_n likewise_o the_o angle_n of_o a_o great_a section_n be_v great_a than_o a_o right_a angle_n as_o it_o have_v in_o this_o proposition_n be_v prove_v the_o 28._o theorem_a the_o 32._o proposition_n if_o a_o right_a line_n touch_v a_o circle_n and_o from_o the_o touch_n be_v draw_v a_o right_a line_n cut_v the_o circle_n the_o angle_n which_o that_o line_n and_o the_o touch_n line_n make_v be_v equal_a to_o the_o angle_n which_o consist_v in_o the_o alternate_a segment_n of_o the_o circle_n svppose_v that_o the_o right_a line_n of_o do_v touch_v the_o circle_n abcd_o in_o the_o point_n b_o and_o from_o the_o point_n b_o let_v there_o be_v draw_v into_o the_o circle_n abcd_o a_o right_a line_n cut_v the_o circle_n and_o let_v the_o same_o be_v bd._n then_o i_o say_v that_o the_o angle_n which_o the_o line_n bd_o together_o with_o the_o touch_n line_n of_o do_v make_v be_v equal_a to_o the_o angle_n which_o be_v in_o the_o alternate_a segment_n of_o the_o circle_n that_o be_v the_o angle_n fbd_v be_v equal_a to_o the_o angle_n which_o consist_v in_o the_o segment_n bad_a and_o the_o angle_n ebb_v be_v equal_a to_o the_o angle_n which_o consist_v in_o the_o segment_n bcd_o construction_n raise_v up_o by_o the_o 11._o of_o the_o first_o from_o the_o point_fw-fr b_o unto_o the_o right_a line_n of_o a_o perpendicular_a line_n basilius_n and_o in_o the_o circumference_n bd_o take_v a_o point_n at_o all_o adventure_n and_o let_v the_o same_o be_v c._n and_o draw_v these_o right_a line_n ad_fw-la dc_o and_o cb._n demonstration_n and_o for_o asmuch_o as_o a_o certain_a right_a line_n of_o touch_v the_o circle_n abc_n in_o the_o point_n b_o and_o from_o the_o point_n b_o where_o the_o touch_n be_v be_v raise_v up_o unto_o the_o touch_n line_v a_o perpendicular_a basilius_n therefore_o by_o the_o 19_o of_o the_o three_o in_o the_o line_n basilius_n be_v the_o centre_n of_o the_o circle_n abcd._n wherefore_o the_o angle_n adb_o be_v in_o the_o semicircle_n be_v by_o the_o 31._o of_o the_o three_o a_o right_a angle_n wherefore_o the_o angle_n remain_v bad_a and_o abdella_n be_v equal_a to_o one_o right_a angle_n but_o the_o angle_n abf_n be_v a_o right_a angle_n wherefore_o the_o angle_n abf_n be_v equal_a to_o the_o angle_n bad_a and_o abdella_n take_v away_o the_o angle_n abdella_n which_o be_v common_a to_o they_o both_o wherefore_o the_o angle_n remain_v dbf_n be_v equal_a to_o the_o angle_n remain_v bad_a which_o be_v in_o the_o alternate_a segment_n of_o the_o circle_n and_o for_o asmuch_o as_o in_o the_o circle_n be_v a_o figure_n of_o four_o side_n namely_o abcd_o therefore_o by_o the_o 22._o of_o the_o three_o the_o angle_n which_o be_v opposite_a the_o one_o to_o the_o other_o be_v equal_a to_o two_o right_a angle_n wherefore_o the_o angle_n bad_a and_o bcd_o be_v equal_a to_o two_o right_a angle_n but_o the_o angle_n dbf_n and_o dbe_n be_v also_o equal_a to_o two_o right_a angle_n wherefore_o the_o angle_n dbf_n and_o dbe_n be_v equal_a to_o the_o angle_n bad_a and_o bcd_o of_o which_o we_o have_v prove_v that_o the_o angle_n bad_a be_v equal_a to_o the_o angle_n dbf_n wherefore_o the_o angle_n remain_v dbe_n be_v equal_a to_o the_o angle_n remain_v dcb_n which_o be_v in_o the_o alternate_a segment_n of_o the_o circle_n namely_o in_o the_o segment_n dcb_n if_o therefore_o a_o right_a line_n touch_v a_o circle_n and_o from_o the_o touch_n be_v draw_v a_o right_a line_n cut_v the_o circle_n the_o angle_n which_o that_o line_n and_o the_o touch_n line_n make_v be_v equal_a to_o the_o angle_n which_o consist_v in_o the_o alternate_a segment_n of_o the_o circle_n which_o be_v require_v to_o be_v prove_v in_o this_o proposition_n may_v be_v two_o case_n proposition_n for_o the_o line_n draw_v from_o
the_o touch_n and_o cut_v the_o circle_n may_v either_o pass_v by_o the_o centre_n or_o not_o if_o it_o pass_v by_o the_o centre_n then_o be_v it_o manifest_a by_o the_o 18._o of_o this_o book_n that_o it_o fall_v perpendicular_o upon_o the_o touch_n line_n and_o divide_v the_o circle_n into_o two_o equal_a part_n so_o that_o all_o the_o angle_n in_o each_o semicircle_n be_v by_o the_o former_a proposition_n right_a angle_n and_o therefore_o equal_a to_o the_o alternate_a angle_n make_v by_o the_o say_v perpendicular_a line_n and_o the_o touch_n line_n if_o it_o pass_v not_o by_o the_o centre_n then_o follow_v the_o construction_n and_o demonstration_n before_o put_v the_o 5._o problem_n the_o 33._o proposition_n upon_o a_o right_a line_n give_v to_o describe_v a_o segment_n of_o a_o circle_n which_o shall_v contain_v a_o angle_n equal_a to_o a_o rectiline_a angle_n give_v svppose_v that_o the_o right_a line_n give_v be_v ab_fw-la and_o let_v the_o rectiline_a angle_n geâ—â—n_v be_v c._n it_o be_v require_v upon_o the_o right_a line_n give_v ab_fw-la to_o describe_v a_o segment_n of_o a_o circle_n which_o shall_v contain_v a_o angle_n equal_a to_o the_o angle_n c._n now_o the_o angle_n c_o be_v either_o a_o acute_a angle_n or_o a_o right_a angle_n or_o a_o obtuse_a angle_n case_n first_o let_v it_o be_v a_o acute_a angle_n as_o in_o the_o first_o description_n and_o by_o the_o 23_o of_o the_o first_o upon_o the_o right_a line_n ab_fw-la and_o to_o the_o point_n in_o it_o a_o describe_v a_o angle_n equal_a to_o the_o angle_n c_o construction_n and_o let_v the_o same_o be_v dab_n wherefore_o the_o angle_n dab_n be_v a_o acute_a angle_n from_o the_o point_n a_o raise_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o line_n ad_fw-la a_o perpendicular_a line_n af._n and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n ab_fw-la into_o two_o equal_a part_n in_o the_o point_n f._n and_o by_o the_o 11._o of_o the_o same_o from_o the_o point_n fletcher_n raise_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n fg_o and_o draw_v a_o line_n from_o g_z to_o b._n demonstration_n and_o forasmuch_o as_o the_o line_n of_o be_v equal_a to_o the_o line_n fb_o and_o the_o line_n fg_o be_v common_a to_o they_o both_o therefore_o these_o two_o line_n of_o and_o fg_o be_v equal_a to_o these_o two_o line_n fb_o and_o fg_o and_o the_o angle_n afg_n be_v by_o the_o 4._o petition_n equal_a to_o the_o angle_n gfb_n wherefore_o by_o the_o 4._o of_o the_o same_o the_o base_a agnostus_n be_v equal_a to_o the_o base_a gb_o wherefore_o make_v the_o centre_n g_o and_o the_o space_n ga_o describe_v by_o the_o 3._o petition_n a_o circle_n and_o it_o shall_v pass_v by_o the_o point_n b_o describe_v such_o a_o circle_n &_o let_v the_o same_o be_v abe_n and_o draw_v a_o line_n from_o e_o to_o b._n now_o forasmuch_o as_o from_o the_o end_n of_o the_o diameter_n ae_n namely_o from_o the_o point_n a_o be_v 〈◊〉_d a_o right_a line_n ad_fw-la make_v together_o with_o the_o right_a line_n ae_n a_o right_a angle_n therefore_o by_o the_o corollary_n of_o the_o 16._o of_o the_o three_o the_o line_n ad_fw-la touch_v the_o circle_n abe_n and_o forasmuch_o as_o a_o certain_a right_a line_n ad_fw-la touch_v the_o circle_n abe_n &_o from_o the_o point_n a_o where_o the_o touch_n be_v be_v draw_v into_o the_o circle_v a_o certain_a right_a line_n ab_fw-la therefore_o by_o the_o 32._o of_o the_o three_o the_o angle_n dab_n be_v equal_a to_o the_o angle_n aeb_fw-mi which_o be_v in_o the_o alternate_a segment_n of_o the_o circle_n but_o the_o angle_n dab_n be_v equal_a to_o the_o angle_n c_o wherefore_o the_o angle_n c_o be_v equal_a to_o the_o angle_n aeb_fw-mi wherefore_o upon_o the_o right_a line_n give_v ab_fw-la be_v describe_v a_o segment_n of_o a_o circle_n which_o contain_v the_o angle_n aeb_fw-mi which_o be_v equal_a to_o the_o angle_n give_v namely_o to_o c._n case_n but_o now_o suppose_v that_o the_o angle_n c_o be_v a_o right_a angle_n it_o be_v again_o require_v upon_o the_o right_a line_n ab_fw-la to_o describe_v a_o segment_n of_o a_o circle_n which_o shall_v contain_v a_o angle_n equal_a to_o the_o right_a angle_n c._n construction_n describe_v again_o upon_o the_o right_a line_n ab_fw-la and_o to_o the_o point_n in_o it_o a_o a_o angle_n bid_v equal_a to_o the_o rectiline_a angle_n give_v c_o by_o the_o 23._o of_o the_o first_o as_o it_o be_v set_v forth_o in_o the_o second_o description_n and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n ab_fw-la into_o two_o equal_a part_n in_o the_o point_n f._n and_o make_v the_o centre_n the_o point_n f_o and_o the_o space_n favorina_n or_o fb_o describe_v by_o the_o 3._o petition_n the_o circle_z aeb_fw-mi demonstration_n wherefore_o the_o right_a line_n ad_fw-la touch_v the_o circle_n aeb_fw-mi for_o that_o the_o angle_n bad_a be_v a_o right_a angle_n wherefore_o the_o angle_v bad_a be_v equal_a to_o the_o angle_n which_o be_v in_o the_o segment_n aeb_fw-mi for_o the_o angle_n which_o be_v in_o a_o semicircle_n be_v a_o right_a angle_n by_o the_o 31._o of_o the_o three_o but_o the_o angle_n bad_a be_v equal_a to_o the_o angle_n c._n wherefore_o there_o be_v again_o describe_v upon_o the_o line_n ab_fw-la a_o segment_n of_o a_o circle_n namely_o aeb_n which_o contain_v a_o angle_n equal_a to_o the_o angle_n give_v namely_o to_o c._n but_o now_o suppose_v that_o the_o angle_n c_o be_v a_o obtuse_a angle_n case_n upon_o the_o right_a line_n ab_fw-la and_o to_o the_o point_n in_o it_o a_o describe_v by_o the_o 23._o of_o the_o first_o a_o angle_n bid_v equal_a to_o the_o angle_n c_o as_o it_o be_v in_o the_o three_o description_n construction_n and_o from_o the_o point_n a_o raise_v up_o unto_o the_o line_n ad_fw-la a_o perpendicular_a line_n ae_n by_o the_o 11._o of_o the_o first_o and_o again_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n ab_fw-la into_o two_o equal_a part_n in_o the_o point_n f._n and_o from_o the_o point_n f._n raâ—se_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n fg_o by_o the_o 11._o of_o the_o same_o &_o draw_v a_o line_n from_o g_z to_o b._n demonstration_n and_o now_o forasmuch_o as_o the_o line_n of_o be_v equal_a to_o the_o line_n fb_o and_o the_o line_n fg_o be_v common_a to_o they_o both_o therefore_o these_o two_o line_n of_o and_o fg_o be_v equal_a to_o these_o two_o line_n bf_o and_o fg_o and_o the_o angle_n afg_n be_v by_o the_o 4._o petition_n equal_a to_o the_o angle_n bfg_n wherefore_o by_o the_o 4._o of_o the_o same_o the_o base_a agnostus_n be_v equal_a to_o the_o base_a gb_o wherefore_o make_v the_o centre_n g_o and_o the_o space_n ga_o describe_v by_o the_o 3._o petition_n a_o circle_n and_o it_o shall_v pass_v by_o the_o point_n b_o let_v it_o be_v describe_v as_o the_o circle_n aeb_fw-mi be_v and_o forasmuch_o as_o from_o the_o end_n of_o the_o diameter_n ae_n be_v draw_v a_o perpendicular_a line_n ad_fw-la therefore_o by_o the_o corollary_n of_o the_o 16._o of_o the_o three_o the_o line_n ad_fw-la touch_v the_o circle_n aeb_fw-mi &_o from_o the_o point_n of_o the_o touch_n namely_o a_o be_v extend_v the_o line_n ab_fw-la wherefore_o by_o the_o 32._o of_o the_o three_o the_o angle_n bad_a be_v equal_a to_o the_o angle_n ahb_n which_o be_v in_o the_o alternate_a segment_n of_o the_o circle_n but_o the_o angle_n bad_a be_v equal_a to_o the_o angle_n c._n wherefore_o the_o angle_n which_o be_v in_o the_o segment_n ahb_n be_v equal_a to_o the_o angle_n c._n wherefore_o upon_o the_o right_a line_n give_v ab_fw-la be_v describe_v a_o segment_n of_o a_o circle_n ahb_n which_o contain_v a_o angle_n equal_a to_o the_o angle_n give_v namely_o c_o which_o be_v require_v to_o be_v do_v the_o 6._o problem_n the_o 34._o proposition_n from_o a_o circle_n give_v to_o cut_v away_o a_o section_n which_o shall_v contain_v a_o angle_n equal_a to_o a_o rectiline_a angle_n give_v svppose_v that_o the_o circle_n give_v be_v ac_fw-la and_o let_v the_o rectiline_a angle_n give_v be_v d._n it_o be_v require_v from_o the_o circle_n abc_n to_o cut_v away_o a_o segment_n which_o shall_v contain_v a_o angle_n equal_a to_o the_o angle_n d._n construction_n draw_v by_o the_o 17._o of_o the_o three_o a_o line_n touch_v the_o circle_n and_o let_v the_o same_o be_v of_o and_o let_v it_o touch_v in_o the_o point_n b._n and_o by_o the_o 23._o of_o the_o first_o upon_o the_o right_a line_n of_o and_o to_o the_o point_n in_o it_o b_o describe_v the_o angle_n fbc_n equal_a to_o the_o angle_n d._n demonstration_n now_o forasmuch_o as_o a_o certain_a right_a line_n of_o touch_v the_o circle_n abc_n in_o the_o point_n b_o and_o frora_fw-la
side_n wherefore_o the_o whole_a parallelipipedon_n ae_n be_v equal_a and_o like_a to_o the_o whole_a parallelipipedon_n yu._n extend_v by_o the_o second_o petition_n the_o line_n doctor_n and_o wv_n until_o they_o concur_v and_o let_v they_o concur_v in_o the_o point_n q._n and_o by_o the_o 31._o of_o the_o first_o by_o the_o point_n t_o draw_v unto_o the_o line_n rq_n a_o parallel_a line_n t_o 4_o and_o extend_v due_o the_o line_n ta_o and_o do_v until_o they_o concur_v and_o let_v they_o concur_v in_o the_o point_n ✚_o and_o make_v perfect_v the_o solid_n qy_n and_o ri._n now_o the_o solid_a qy_n who_o plain_n base_a be_v the_o parallelogram_n ry_o and_o the_o opposite_a side_n unto_o the_o base_a the_o parallelogram_n qb_n be_v equal_a to_o the_o solid_a yv_n who_o base_a be_v the_o parallelogram_n ry_o and_o the_o opposite_a side_n unto_o the_o base_a the_o parallelogram_n we_o for_o they_o consist_v upon_o one_o and_o the_o self_n fame_n base_a namely_o ry_o and_o be_v under_o one_o and_o the_o self_n same_o altitude_n and_o their_o stand_a line_n namely_o rq_n rv_n ta_o tw_n sn_a sd_v yb_n and_o yz_n be_v in_o the_o self_n same_o right_a line_n namely_o qw_o and_o nz_n but_o the_o solid_a yv_n be_v prove_v equal_a to_o the_o solid_a ae_n wherefore_o also_o the_o solid_a yq_n be_v equal_a to_o the_o solid_a ae_n now_o forasmuch_o as_o the_o parallelogram_n ruwt_fw-mi be_v equal_a to_o the_o parallelogram_n qt_n by_o the_o 35._o of_o the_o first_o and_o the_o parallelogram_n ab_fw-la be_v equal_a to_o the_o parallelogram_n rw_n therefore_o the_o parallelogram_n qt_o be_v equal_a to_o the_o parallelogram_n ab_fw-la and_o the_o parallelogram_n cd_o be_v equal_a to_o the_o parallelogram_n ab_fw-la by_o supposition_n wherefore_o the_o parallelogram_n cd_o be_v equal_a to_o the_o parallelogram_n qt_n and_o there_o be_v a_o certain_a other_o superficies_n namely_o dt_n wherefore_o by_o the_o 7._o of_o the_o five_o as_o the_o base_a cd_o be_v to_o the_o base_a dt_n so_o be_v the_o base_a qt_n to_o the_o base_a dt_n and_o forasmuch_o as_o the_o whole_a parallelipipedon_n ci_o be_v cut_v by_o the_o plain_a superficies_n rf_n which_o be_v a_o parallel_n to_o either_o of_o the_o opposite_a plain_a superficiece_n therefore_o as_o the_o base_a cd_o be_v to_o the_o base_a dt_n so_o be_v the_o solid_a cf_o to_o the_o solid_a ri_n by_o the_o 25._o of_o the_o eleven_o and_o by_o the_o same_o reason_n also_o forasmuch_o as_o the_o whole_a parallelipipedon_n qi_n be_v cut_v by_o the_o plain_a superficies_n ry_o which_o be_v a_o parallel_n to_o either_o of_o the_o opposite_a plain_a superficiece_n therefore_o as_o the_o base_a qt_n be_v to_o the_o base_a dt_n so_o be_v the_o solid_a qy_n to_o the_o solid_a ri._n but_o as_o the_o base_a cd_o be_v to_o the_o base_a dt_n so_o be_v the_o base_a qt_n to_o the_o base_a td_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o solid_a cf_o be_v to_o the_o solid_a ri_n so_o be_v the_o solid_a qy_n to_o the_o solid_a ri._n wherefore_o either_o of_o these_o solid_n cf_o and_o qy_n have_v to_o the_o solid_a ri_n one_o and_o the_o same_o proportion_n wherefore_o the_o solid_a cf_o be_v equal_a to_o the_o solid_a qy_n but_o it_o be_v prove_v that_o the_o solid_a qy_n be_v equal_a to_o the_o solid_a ae_n wherefore_o also_o the_o solid_a cf_o be_v equal_a to_o the_o solid_a ae_n but_o now_o suppose_v that_o the_o stand_a line_n namely_o agnostus_n hk_o case_n be_v lm_o cx_o open_v df_o and_o r_n be_v not_o erect_v perpendicular_o to_o the_o base_n ab_fw-la and_o cd_o then_o also_o i_o say_v that_o the_o solid_a ae_n be_v equal_a to_o the_o solid_a cf._n draw_v by_o the_o 11._o of_o the_o eleven_o unto_o the_o ground_n plain_a superficiece_n ab_fw-la and_o cd_o from_o these_o point_n king_n e_o c_o m_o p_o f_o x_o saint_n these_o perpendicular_a line_n kn_n et_fw-la gv_n mz_n pw_n fie_o xq_n and_o si_fw-it and_o draw_v these_o right_a line_n nt_v nv_n zv_n zt_n why_o wq_n iq_fw-la and_o jy_n now_o by_o that_o which_o have_v before_o be_v prove_v in_o this_o 31._o proposition_n the_o solid_a kz_o be_v equal_a to_o the_o solid_a piperollo_n for_o they_o consist_v upon_o equal_a you_o base_n namely_o km_o and_o ps_n and_o be_v under_o one_o and_o the_o self_n same_o altitude_n who_o stand_a line_n also_o be_v erect_v perpendicular_o to_o the_o base_n but_o the_o solid_a kz_o be_v equal_a to_o the_o solid_a ae_n by_o the_o 29._o of_o the_o eleven_o and_o the_o solid_a piperollo_n be_v by_o the_o same_o equal_a to_o the_o solid_a cf_o for_o they_o consist_v upon_o one_o and_o the_o self_n same_o base_a and_o be_v under_o one_o &_o the_o self_n same_o altitude_n who_o stand_a line_n be_v upon_o the_o self_n same_o right_a line_n wherefore_o also_o the_o solid_a ae_n be_v equal_a to_o the_o solid_a ce._n wherefore_o parallelipipedon_n consist_v upon_o equal_a base_n and_o be_v under_o one_o and_o the_o self_n same_o altitude_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v demonstrate_v the_o demonstration_n of_o the_o first_o case_n of_o this_o proposition_n be_v very_o hard_a to_o conceive_v by_o the_o figure_n describe_v for_o it_o in_o a_o plain_n and_o yet_o before_o m._n do_v you_o invent_v that_o figure_n which_o we_o have_v there_o place_v for_o it_o it_o be_v much_o hard_o for_o both_o in_o the_o greek_a and_o latin_a euclid_n it_o be_v very_o ill_o make_v and_o it_o be_v in_o a_o manner_n impossible_a to_o conceive_v by_o it_o the_o construction_n and_o demonstration_n thereto_o appertain_v wherefore_o i_o have_v here_o describe_v other_o figure_n which_o first_o describe_v upon_o past_v paper_n or_o such_o like_a matter_n and_o then_o order_v they_o in_o manner_n follow_v as_o touch_v the_o solid_a ae_n in_o the_o first_o case_n i_o need_v not_o to_o make_v any_o new_a description_n for_o it_o be_v plain_a enough_o to_o conceive_v as_o it_o be_v there_o draw_v although_o you_o may_v for_o your_o more_o ease_n of_o imagination_n describe_v of_o past_v paper_n a_o parallelipipedoâ—_n have_v his_o side_n equal_a with_o the_o side_n of_o the_o parallelipipedon_n ae_n before_o describe_v and_o have_v also_o the_o six_o parallelogram_n thereof_o contain_v under_o those_o side_n equiangle_n with_o the_o six_o parallelogram_n of_o that_o figure_n each_o side_n and_o each_o angle_n equal_a to_o his_o correspondent_a side_n and_o to_o his_o correspondent_a angle_n but_o concern_v the_o other_o solid_a when_o you_o have_v describe_v these_o three_o figure_n upon_o past_v paper_n where_o note_n for_o the_o proportion_n of_o each_o line_n to_o make_v your_o figure_n of_o past_v paper_n equal_a with_o the_o figure_n before_o describe_v upon_o the_o plain_n let_v your_o line_n open_v cx_o r_n df_o etc_n etc_n namely_o the_o rest_n of_o the_o stand_a line_n of_o these_o figure_n be_v equal_a to_o the_o stand_a line_n open_v cx_o r_n df_o etc_n etc_n of_o that_o figure_n likewise_o let_v the_o line_n oc_fw-fr cr_n rd_o do_v etc_n etc_n namely_o the_o side_n which_o contain_v the_o base_n of_o these_o figure_n be_v equal_a to_o the_o line_n oc_fw-fr cr_n rd_o do_v etc_n etc_n namely_o to_o the_o side_n which_o contain_v the_o base_n of_o that_o figure_n moreover_o let_v the_o line_n px_n xâ—_n sf_n fp_n etc_n etc_n namely_o the_o rest_n of_o the_o line_n which_o contain_v the_o upper_a superficiece_n of_o these_o figure_n be_v equal_a to_o the_o line_n px_n xs_n sf_n fâ—_n etc_n etc_n namely_o to_o the_o rest_n of_o the_o line_n which_o contain_v the_o upper_a superficiece_n of_o that_o figure_n to_o have_v describe_v all_o those_o foresay_a line_n of_o these_o figure_n equal_a to_o all_o the_o line_n of_o that_o figure_n will_v have_v require_v much_o more_o space_n than_o here_o can_v be_v spare_v i_o have_v make_v they_o equal_v only_o to_o the_o half_n of_o those_o line_n but_o by_o the_o example_n of_o these_o you_o may_v if_o you_o will_v describe_v the_o like_a figure_n have_v their_o line_n equal_a to_o the_o whole_a line_n of_o the_o figure_n in_o the_o plain_n each_o line_n to_o his_o correspondent_a line_n when_o i_o say_v you_o have_v as_o before_o be_v teach_v describe_v these_o three_o figure_n cut_v fine_o the_o line_n xc_o sir_n fd_v of_o the_o first_o figure_n and_o the_o line_n sir_n it_o and_o i_o ✚_o of_o the_o second_o figure_n likewise_o the_o line_n â—r_o nq_n zw_n and_o it_o of_o the_o three_o figure_n and_o fold_v these_o figure_n according_o which_o you_o can_v not_o choose_v but_o do_v if_o you_o mark_v well_o the_o letter_n of_o every_o line_n as_o touch_v the_o second_o case_n you_o need_v no_o new_a figure_n for_o it_o be_v plain_a to_o see_v by_o the_o figure_n now_o
reform_v by_o m._n dee_n describe_v for_o it_o in_o the_o plain_n especial_o if_o you_o remember_v the_o form_n of_o the_o figure_n of_o the_o 29._o proposition_n of_o this_o book_n only_o that_o which_o there_o you_o conceive_v to_o be_v the_o base_a imagine_v here_o in_o both_o the_o figure_n of_o this_o second_o case_n to_o be_v the_o upper_a superficies_n opposite_a to_o the_o base_a and_o that_o which_o be_v there_o suppose_v to_o be_v the_o upper_a superficies_n conceive_v here_o to_o be_v the_o base_a you_o may_v describe_v they_o upon_o past_v paper_n for_o your_o better_a sight_n take_v heed_n you_o note_v the_o letter_n right_o according_a as_o the_o construction_n require_v flussas_n demonstrate_v this_o proposition_n a_o otherway_o take_v only_o the_o base_n of_o the_o solid_n and_o that_o after_o this_o manner_n take_v equal_a base_n which_o yet_o for_o the_o sure_a understanding_n let_v be_v utter_o unlike_a namely_o aebf_n and_o adch_n and_o let_v one_o of_o the_o side_n of_o each_o concur_v in_o one_o &_o the_o same_o right_a line_n aed_n &_o the_o base_n be_v upon_o one_o and_o the_o self_n same_o plain_n let_v there_o be_v suppose_v to_o be_v set_v upon_o they_o parallelipipedon_n under_o one_o &_o the_o self_n same_o altitude_n then_o i_o say_v that_o the_o solid_a set_v upon_o the_o base_a ab_fw-la be_v equal_a to_o the_o solid_a set_v upon_o the_o base_a ah_o by_o the_o point_n e_o draw_v unto_o the_o line_n ac_fw-la a_o parallel_n line_n eglantine_n which_o if_o it_o fall_v without_o the_o base_a ab_fw-la produce_v the_o right_a line_n hc_n to_o the_o point_n i_o now_o forasmuch_o as_o ab_fw-la and_o ah_o be_v parallelogrmae_n therefore_o by_o the_o 24._o of_o this_o book_n the_o triangle_n aci_n and_o egl_n shall_v be_v equaliter_fw-la the_o one_o to_o the_o other_o and_o by_o the_o 4._o of_o the_o first_o they_o shall_v be_v equiangle_n and_o equal_a and_o by_o the_o first_o definition_n of_o the_o six_o and_o four_o proposition_n of_o the_o same_o they_o shall_v be_v like_a wherefore_o prism_n erect_v upon_o those_o triangle_n and_o under_o the_o same_o altitude_n that_o the_o solid_n ab_fw-la and_o ah_o aâ—e_n shall_v be_v equal_a and_o like_a by_o the_o 8._o definition_n of_o this_o book_n for_o they_o be_v contain_v under_o like_o plain_a superficiece_n equal_a both_o in_o multitude_n and_o magnitude_n add_v the_o solid_a set_v upon_o the_o base_a acle_n common_a to_o they_o both_o wherefore_o the_o solid_a set_v upon_o the_o base_a aegc_n be_v equal_a to_o the_o solid_a set_v upon_o the_o base_a aeli_n and_o forasmuch_o as_o the_o superficiece_n aebf_n and_o adhc_n be_v equal_a by_o supposition_n and_o the_o part_n take_v away_o agnostus_n be_v equal_a to_o the_o part_n take_v away_o all_o therefore_o the_o residue_n by_o shall_v be_v equal_a to_o the_o residue_n gd_o wherefore_o as_o agnostus_n be_v to_o gd_a as_o all_o be_v to_o by_o namely_o equal_n to_o equal_n but_o as_o agnostus_n be_v to_o gd_o so_o iâ—_z the_o solid_a set_v upon_o agnostus_n to_o the_o solid_a set_v upon_o gd_o by_o the_o 25._o of_o this_o book_n for_o it_o be_v cut_v by_o a_o plain_a superficies_n set_v upon_o the_o line_n ge_z which_o superficies_n be_v parallel_n to_o the_o opposite_a superficiece_n wherefore_o as_o all_o be_v to_o by_o so_o be_v the_o solid_a set_v upon_o all_o to_o the_o solid_a set_v upon_o by_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o solid_a set_v upon_o agnostus_n or_o upon_o all_o which_o be_v equal_a unto_o it_o be_v to_o the_o solid_a set_v upon_o gd_o so_o be_v the_o same_o solid_a set_v upon_o agnostus_n or_o all_o to_o the_o solid_a set_v upon_o by_o wherefore_o by_o the_o 2._o part_n of_o the_o 9_o of_o the_o five_o the_o solid_n set_v upon_o gd_o and_o by_o shall_v be_v equal_a unto_o which_o solid_n if_o you_o add_v equal_a solid_n namely_o the_o solid_a set_v upon_o agnostus_n to_o the_o solid_a set_v upon_o gd_o and_o the_o solid_a set_v upon_o all_o to_o the_o solid_a set_v upon_o by_o the_o whole_a solid_n set_v upon_o the_o base_a ah_o and_o upon_o the_o base_a ab_fw-la â—hall_v be_v equal_a wherefore_o parallelipedon_n consist_v upon_o equal_a base_n and_o be_v under_o one_o and_o the_o self_n same_o altitude_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 27._o theorem_a the_o 32._o proposition_n parallelipipedon_n be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v svppose_v that_o these_o parallelipipedon_n ab_fw-la and_o cd_o be_v under_o one_o &_o the_o self_n same_o altitude_n then_o i_o say_v that_o those_o parallelipipedon_n ab_fw-la and_o cd_o be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v that_o be_v that_o as_o the_o base_a ae_n be_v to_o the_o base_a cf_o so_o be_v the_o parallelipipedon_n ab_fw-la to_o the_o parallelipipedon_n cd_o construction_n upon_o the_o line_n fg_o describe_v by_o the_o 45._o of_o the_o first_o the_o parallelogram_n fh_o equal_a to_o the_o parallelogram_n ae_n and_o equiangle_n with_o the_o parallelogram_n cf._n and_o upon_o the_o base_a fh_o describe_v a_o parallelipipedom_n of_o the_o self_n same_o altitude_n that_o the_o parallelipipedom_n cd_o be_v &_o let_v the_o same_o be_v gk_o demonstration_n now_o by_o the_o 31._o of_o the_o eleven_o the_o parallelipipedon_n ab_fw-la be_v equal_a to_o the_o parallelipipedon_n gk_o for_o they_o consist_v upon_o equal_a base_n namely_o ae_n and_o fh_o and_o be_v under_o one_o and_o the_o self_n same_o altitude_n and_o forasmuch_o as_o the_o parallelipipedon_n ck_o be_v cut_v by_o a_o plain_a superficies_n dg_o be_v parallel_n to_o either_o of_o the_o opposite_a plain_a superâ—icieces_n therefore_o by_o the_o 25._o of_o the_o eleven_o as_o the_o base_a hf_o be_v to_o the_o base_a fc_n so_o be_v the_o parallelipipedon_n gk_o to_o parallelipipedon_v cd_o but_o the_o base_a hf_o be_v equal_a to_o the_o base_a ae_n and_o the_o parallelipipedon_n gk_o be_v prove_v equal_a to_o the_o parallelipipedon_n ab_fw-la wherefore_o as_o the_o base_a aâ—e_n be_v to_o the_o base_a cf_o so_o be_v the_o parallelipedon_n ab_fw-la to_o the_o parallelipipedon_n cd_o wherefore_o parallelipipedon_n be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v which_o be_v require_v to_o be_v demonstrate_v i_o need_v not_o to_o put_v any_o other_o figure_n for_o the_o declaration_n of_o this_o demonstration_n for_o it_o be_v easy_a to_o see_v by_o the_o figure_n there_o describe_v howbeit_o you_o may_v for_o the_o more_o full_a sight_n thereof_o describe_v solid_n of_o past_v paper_n according_a to_o the_o construction_n there_o set_v forth_o which_o will_v not_o be_v hard_o for_o you_o to_o do_v if_o you_o remember_v the_o description_n of_o such_o body_n before_o teach_v a_o corollary_n add_v by_o flussas_n equal_a parallelipipedon_n contain_v under_o one_o and_o the_o self_n same_o altitude_n have_v also_o their_o base_n equal_a for_o if_o the_o base_n shall_v be_v unequal_a the_o parallelipipedon_n also_o shall_v be_v unequal_a by_o this_o 32_o proposition_n and_o equal_a parallelipipedon_n have_v equal_a base_n have_v also_o one_o and_o the_o self_n same_o altitude_n for_o if_o they_o shall_v have_v a_o great_a altitude_n they_o shall_v exceed_v the_o equal_a parallelipipedon_n which_o have_v the_o self_n same_o altitude_n but_o if_o they_o shall_v have_v a_o less_o they_o shall_v want_v so_o much_o of_o those_o self_n same_o equal_a parallelipipedon_n the_o 28._o theorem_a the_o 33._o proposition_n like_a parallelipipedon_n be_v in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o their_o side_n of_o like_a proportion_n be_v svppose_v that_o these_o parallelipipedon_n ab_fw-la and_o cd_o be_v like_a &_o let_v the_o side_n ae_n and_o cf_o be_v side_n of_o like_a proportion_n then_o i_o say_v the_o parallelipipedon_n ab_fw-la be_v unto_o the_o parallelipipedon_n cd_o in_o treble_a proportion_n of_o that_o in_o which_o the_o side_n ae_n be_v to_o the_o side_n cf._n extend_v the_o right_a line_n ae_n ge_z and_o he_o to_o the_o point_n king_n l_o m._n construction_n and_o by_o the_o 2._o of_o the_o first_o unto_o the_o line_n cf_o put_v the_o line_n eke_o equal_a and_o unto_o the_o line_n fn_o put_v the_o line_n el_n equal_a and_o moreover_o unto_o the_o line_n fr_z put_v the_o line_n em_n equal_a and_o make_v perfect_a the_o parallelogram_n kl_o and_o the_o parallelipipedon_n ko_o demonstration_n now_o forasmuch_o as_o these_o two_o line_n eke_o and_o el_n be_v equal_a to_o these_o two_o line_n cf_o and_o fn_o but_o the_o angle_n kel_n be_v equal_a to_o the_o angle_n cfn_n for_o the_o angle_n