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
demonstration_n construction_n demonstration_n an_o other_o way_n after_o peliâ—arius_n construction_n demonstration_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n three_o case_n in_o this_o proposition_n the_o three_o case_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o proposition_n add_v by_o petarilius_n note_n construction_n demonstration_n an_o other_o way_n also_o after_o pelitarius_n construction_n demonstration_n an_o other_o way_n to_o do_v the_o samâ—_n after_o pelitarius_n demonstration_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n construction_n demonstration_n demonstration_n a_o â—ther_n way_n to_o do_v the_o same_o after_o orontius_n an_o other_o way_n after_o pelitarius_n construction_n demonstration_n a_o addition_n of_o flussates_n flussates_n a_o poligonon_n figure_n be_v a_o figure_n consist_v of_o many_o side_n the_o argument_n of_o this_o five_o book_n the_o first_o author_n of_o this_o book_n eudoxus_n the_o first_o definition_n a_o part_n take_v two_o manner_n of_o way_n the_o fiâ—st_a way_n the_o second_o way_n how_o a_o less_o quantity_n be_v say_v to_o measure_v a_o great_a in_o what_o signification_n euclid_n here_o take_v a_o part_n parâ—_n metienâ—_n or_o mensuranâ—_n pars_fw-la multiplicatiâ—a_fw-la pars_fw-la aliquota_fw-la this_o kind_n of_o part_n common_o use_v in_o arithmetic_n the_o other_o kind_n of_o part_n pars_fw-la constitâ—ens_fw-la or_o componens_fw-la pars_fw-la aliquanta_fw-la the_o second_o definition_n number_n very_o necessary_a for_o the_o understanding_n of_o this_o book_n and_o the_o other_o book_n follow_v the_o tâ—ird_o definition_n rational_a proportion_n divide_v â—â—to_o two_o kind_n proportion_n of_o equality_n proportion_n of_o inequality_n proportion_n of_o the_o great_a to_o the_o less_o multiplex_fw-la duple_n proportion_n triple_a quadruple_a quintuple_a superperticular_a sesquialtera_fw-la sesquitertia_fw-la sesquiquarta_fw-la superpartiens_fw-la superbipartiens_fw-la supertripartiens_fw-la superquadripartiens_fw-la superquintipartiens_fw-la multiplex_fw-la superperticular_a dupla_fw-la sesquialtera_fw-la dupla_fw-la sesquitertia_fw-la tripla_fw-la sesquialtera_fw-la multiplex_fw-la superpartiens_fw-la dupla_fw-la superbipartiens_fw-la dupla_fw-la supertripartiens_fw-la tripla_fw-la superbipartiens_fw-la tripla_fw-la superquadâ—ipartiens_fw-la how_o to_o knoâ—_n the_o denomination_n of_o any_o proportion_n proportion_n of_o the_o less_o in_o the_o great_a submultiplex_fw-la subsuperparticular_a subsuperpertient_fw-fr etc_n etc_n the_o four_o definition_n example_n of_o this_o definition_n in_o magnitude_n example_n thereof_o in_o number_n note_n the_o five_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n why_o euclid_n in_o define_v of_o proportion_n use_v multiplication_n the_o six_o definition_n a_o example_n of_o this_o definitiin_n in_o magnitude_n a_o example_n in_o number_n an_o other_o example_n in_o number_n an_o other_o example_n in_o number_n note_v this_o particle_n according_a to_o any_o multiplication_n a_o example_n where_o the_o equimultiplice_n of_o the_o first_o and_o three_o excee_v the_o equimultiplice_n of_o the_o second_o and_o four_o and_o yet_o the_o quantity_n give_v be_v not_o in_o one_o and_o the_o self_n same_o proportion_n a_o rule_n to_o produce_v equimultiplice_n of_o the_o first_o and_o three_o equal_a to_o the_o equimultiplice_n of_o the_o secondâ—_n and_o forth_o example_n thereof_o the_o seven_o definition_n 9_z 12_o 3_o 4_o proportionality_n of_o two_o sort_n continual_a and_o discontinuall_a a_o example_n of_o continual_a proportionality_n in_o number_n 16.8.4.2.1_o in_o coutinnall_a proportionality_n the_o quantity_n can_v be_v of_o one_o kind_n discontinuall_a propâ—rtionalitie_n example_n of_o discontinual_a proportionality_n in_o number_n in_o discontinual_a proportionality_n the_o proportion_n may_v be_v of_o diverse_a kind_n the_o eight_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n a_o example_n in_o number_n note_n the_o nine_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n example_n â—n_a number_n the_o ten_o definition_n a_o rule_n to_o add_v proportion_n to_o proportion_n 8._o 4._o 2._o 1._o 2_o 2_o 2_o 1_o 1_o 1_o the_o eleven_o definition_n example_n of_o this_o definition_n in_o magnitude_n example_n in_o number_n the_o twelfâ—h_o definition_n example_n of_o this_o definition_n in_o magnitud_n example_n in_o number_n the_o thirteen_o definition_n example_n of_o this_o definition_n in_o magnitud_n example_n in_o number_n the_o fourteen_o definition_n example_n of_o this_o definition_n in_o magnitud_n example_n in_o number_n the_o fiâ—tâ—ne_a definition_n this_o be_v the_o converse_n of_o the_o former_a definition_n example_n in_o magnitude_n example_n in_o number_n the_o sixteen_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n a_o example_n in_o number_n the_o sevententh_n definition_n a_o example_n of_o this_o definition_n in_o magnitude_n a_o example_n in_o number_n note_n the_o eighttenth_n definition_n a_o example_n of_o this_o definition_n in_o magnitude_n example_n in_o number_n the_o nineteen_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n example_n in_o number_n the_o 20._o definition_n the_o 2d_o definition_n these_o two_o last_o definition_n not_o find_v in_o the_o greek_a exampler_n construction_n demonstration_n demonstrationâ—_n construction_n demonstration_n construction_n demonstration_n alemmae_fw-la or_o a_o assumpt_n a_o corollary_n converse_v proportion_n construction_n demonstration_n two_o case_n in_o this_o propotion_n the_o second_o the_o second_o part_n demonstrate_v the_o first_o part_n of_o this_o proposition_n demonstrate_v the_o second_o part_n of_o the_o proposition_n demonstrate_v first_o differâ—câ—_n of_o the_o first_o part_n demonstratiâ—_n of_o tâ—e_z same_o first_o difference_n second_o difference_n three_o difference_n the_o second_o part_n â—f_o this_o proposition_n the_o first_o parâ—_n of_o this_o proposition_n demonstrate_v the_o second_o part_n prove_v the_o first_o part_n of_o this_o proposition_n prove_v the_o second_o part_n demonstrate_v construction_n demonstrationâ—_n constrâ—ction_n demonstrationâ—_n construction_n demonstration_n a_o addition_n of_o campane_n demonstration_n construction_n demonstration_n demonstration_n of_o alternate_a proportion_n construction_n demonstration_n demonstration_n of_o proportion_n by_o division_n construction_n demonstration_n demonstration_n of_o proportion_n by_o composition_n this_o proposition_n be_v the_o converse_n of_o the_o former_a demonstrationâ—eâ—aing_v to_o a_o impossibility_n that_o which_o the_o five_o of_o this_o book_n prove_v only_o touch_v multiplices_fw-la this_o prove_v general_o of_o all_o magnitude_n alemma_n a_o corollary_n conversion_n of_o proportion_n this_o proposition_n pertain_v to_o proportion_n of_o equality_n inordinate_a proportionality_n the_o second_o difference_n the_o three_o difference_n thâ—r_a proposition_n pertain_v to_o proportion_n of_o equality_n in_o perturbate_n proportionality_n the_o three_o difference_n proportion_n of_o equality_n in_o ordinate_a proportionality_n construction_n demonstration_n when_o there_o be_v more_o than_o three_o magnitude_n in_o either_o order_n aâ—cdeâ—gh_n proportion_n of_o equality_n in_o perturbate_n proprotionalitie_n construction_n demonstration_n note_n that_o which_o the_o second_o proposition_n of_o this_o book_n prove_v only_o touch_v multiplices_fw-la be_v here_o prove_v general_o touch_v magnitude_n an_o other_o demonstration_n of_o the_o same_o affirmative_o an_o other_o demonstration_n of_o the_o same_o affirmative_o an_o other_o demonstration_n of_o the_o same_o demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n of_o the_o same_o affirmative_o demonstration_n demonstrationâ—_n demonstration_n the_o argument_n of_o this_o six_o book_n this_o book_n necessary_a for_o the_o use_n of_o instrument_n of_o geometry_n the_o first_o definition_n the_o second_o definition_n reciprocal_a figure_n call_v mutual_a figure_n the_o three_o definition_n the_o four_o definition_n the_o five_o definition_n an_o other_o example_n of_o substraction_n of_o proportion_n the_o six_o definition_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n a_o corollary_n add_v by_o flussates_n the_o first_o part_n of_o this_o theorem_a demonstration_n of_o the_o second_o part_n a_o corollary_n add_v by_o flussates_n construction_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o construction_n demonstration_n this_o be_v the_o converse_n of_o the_o former_a proposition_n construction_n demonstration_n construction_n the_o first_o part_n of_o this_o proposition_n demonstration_n lead_v to_o a_o impossibility_n the_o second_o part_n of_o this_o proposition_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o corollary_n out_o of_o flussates_n by_o this_o and_o the_o former_a proposition_n may_v a_o right_a line_n be_v divide_v into_o what_o part_n soever_o you_o will._n construction_n demonstration_n an_o other_o way_n after_o pelitarius_n a_o otâ—eâ—_n way_n after_o pelitarius_n construction_n demonstration_n an_o other_o way_n after_o campane_n construction_n demonstration_n a_o proposition_n add_v by_o pelitarius_n the_o
first_o 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 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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 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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
side_n the_o rectiline_a angle_n mab_n either_o of_o which_o be_v equal_a to_o the_o rectiline_a angle_n give_v cde_o which_o be_v require_v to_o be_v do_v an_o other_o construction_n also_o and_o demonstration_n after_o pelitarâ—us_n and_o if_o the_o perpendicular_a line_n chance_n to_o fall_v without_o the_o angle_n give_v namely_o if_o the_o angle_n give_v be_v a_o acute_a angle_n the_o self_n same_o manner_n of_o demonstration_n will_v serve_v but_o only_o that_o in_o stead_n of_o the_o second_o common_a sentence_n must_v be_v use_v the_o 3._o common_a sentence_n appollonius_n put_v another_o construction_n &_o demonstration_n of_o this_o proposition_n which_o though_o the_o demonstration_n thereof_o depend_v of_o proposition_n put_v in_o the_o three_o book_n yet_o for_o that_o the_o construction_n be_v very_o good_a for_o he_o that_o will_v ready_o and_o mechanical_o without_o demonstration_n describe_v upon_o a_o line_n give_v and_o to_o a_o point_n in_o it_o give_v a_o rectiline_a angle_n equal_a to_o a_o rectiline_a angle_n give_v i_o think_v not_o amiss_o here_o to_o place_v it_o and_o it_o be_v thus_o proposition_n oenopides_n be_v the_o first_o inventor_n of_o this_o proposition_n as_o witness_v eudemius_fw-la the_o 15._o theorem_a the_o 24._o proposition_n if_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_a to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n and_o if_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o one_o be_v great_a than_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o other_o the_o base_a also_o of_o the_o same_o shall_v be_v great_a than_o the_o base_a of_o the_o other_o svppose_v that_o there_o be_v two_o triangle_n abc_n and_o def_n have_v two_o side_n of_o the_o one_o that_o be_v ab_fw-la and_o ac_fw-la equal_a to_o two_o side_n of_o the_o other_o that_o be_v to_o de_fw-fr and_o df_o each_o to_o his_o correspondent_a side_n that_o be_v the_o side_n ab_fw-la to_o the_o side_n de_fw-fr and_o the_o side_n ac_fw-la to_o the_o side_n df_o and_o suppose_v that_o the_o angle_n bac_n be_v great_a than_o the_o angle_n edf_o then_o i_o say_v that_o the_o base_a bc_o be_v great_a than_o the_o base_a ef._n for_o forasmuch_o as_o the_o angle_n bac_n be_v great_a than_o the_o angle_n edf_o construction_n make_v by_o the_o 23._o proposition_n upon_o the_o right_a line_n de_fw-fr and_o to_o the_o point_n in_o it_o give_v d_o a_o angle_n edge_v equal_a to_o the_o angle_n give_v bac_n and_o to_o one_o of_o these_o line_n that_o be_v either_o to_o ac_fw-la or_o df_o put_v a_o equal_a line_n dg_o and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n g_o to_o the_o point_n e_o and_o a_o other_o from_o the_o point_n fletcher_n demonstration_n to_o the_o point_n g._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n de_fw-fr and_o the_o line_n ac_fw-la to_o the_o line_n dg_o the_o one_o to_o the_o outstretch_o and_o the_o angle_n bac_n be_v by_o construction_n equal_a to_o the_o angle_n edg_n therefore_o by_o the_o 4._o proposition_n the_o base_a bc_o be_v equal_a to_o the_o base_a eglantine_n again_o for_o as_o much_o as_o the_o line_n dg_o be_v equal_a to_o the_o line_n df_o therâ—_n by_o the_o 5._o proposition_n the_o angle_n dgf_n be_v equal_a to_o the_o angle_n dfg_n wherefore_o the_o angle_n dfg_n be_v great_a than_o the_o angle_n egf_n wherefore_o the_o angle_n efg_o be_v much_o great_a than_o the_o angle_n egf_n and_o forasmuch_o as_o efg_o be_v a_o triangle_n have_v the_o angle_n efg_o great_a than_o the_o angle_n egf_n and_o by_o the_o 18._o prâ—position_n under_o the_o great_a angle_n be_v subtend_v the_o great_a side_n therefore_o the_o side_n eglantine_n be_v great_a than_o the_o side_n ef._n but_o the_o side_n eglantine_n be_v equal_a to_o the_o side_n bc_o wherefore_o the_o side_n bc_o be_v great_a than_o the_o side_n ef._n if_o therefore_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_a to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n and_o if_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o one_o be_v great_a than_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o other_o the_o base_a also_o of_o the_o same_o shall_v be_v great_a than_o the_o base_a of_o the_o other_o which_o be_v require_v to_o be_v prove_v in_o this_o theorem_a may_v be_v three_o case_n prâ—position_n for_o the_o angle_n edg_n be_v put_v equal_a to_o the_o angle_n bac_n and_o the_o line_n dg_o be_v put_v equal_a to_o the_o line_n ac_fw-la and_o a_o line_n be_v draw_v from_o e_o to_o g_z the_o line_n eglantine_n shall_v either_o fall_v above_o the_o line_n gf_o or_o upon_o it_o or_o under_o it_o euclides_n demonstration_n serve_v 2._o when_o the_o line_n ge_z fall_v above_o the_o line_n gf_o as_o we_o have_v before_o manifest_o see_v but_o now_o let_v the_o line_n eglantine_n case_n fall_v under_o the_o line_n e_o fletcher_n as_o in_o the_o figure_n here_o put_v and_o forasmuch_o as_o these_o two_o line_n ab_fw-la and_o ac_fw-la be_v equal_a to_o these_o two_o line_n de_fw-fr and_o dg_o the_o one_o to_o the_o other_o and_o they_o contain_v equal_a angle_n therefore_o by_o the_o 4._o proposition_n the_o base_a bc_o be_v equal_a to_o the_o base_a eglantine_n and_o forasmuch_o as_o within_o the_o triangle_n deg_n the_o two_o linne_n df_o and_o fe_o be_v set_v upon_o the_o side_n de_fw-fr therefore_o by_o the_o 21._o proposition_n the_o line_n df_o and_o fâ—_n be_v less_o than_o the_o outward_a line_n dg_o and_o ge_z but_o the_o line_n dg_o be_v equal_a to_o the_o line_n df._n wherefore_o the_o line_n ge_z be_v great_a than_o the_o line_n fe_o but_o ge_z be_v equal_a to_o bc._n wherefore_o the_o line_n bc_o be_v great_a the_o the_o line_n ef._n which_o be_v require_v to_o be_v prove_v it_o may_v peradventure_o semeâ—_n that_o euclid_n shall_v here_o in_o this_o proposition_n have_v prove_v that_o not_o only_o the_o base_n of_o the_o triangle_n be_v unequal_a but_o also_o that_o the_o area_n of_o the_o same_o be_v unequal_a for_o so_o in_o the_o four_o proposition_n after_o he_o have_v prove_v the_o base_a to_o be_v equal_a he_o prove_v also_o the_o area_n to_o be_v equal_a but_o hereto_o may_v be_v answer_v &c._a that_o in_o equal_a angle_n and_o base_n and_o unequal_a angle_n and_o base_n the_o consideration_n be_v not_o like_a for_o the_o angle_n and_o base_n be_v equal_a the_o triangle_n also_o shall_v of_o necessity_n be_v equal_a but_o the_o angle_n and_o base_n be_v unequal_a the_o area_n shall_v not_o of_o necessity_n be_v equal_a for_o the_o triangle_n may_v both_o be_v equal_a and_o unequal_a and_o that_o may_v be_v the_o great_a which_o have_v the_o great_a angle_n and_o the_o great_a base_n and_o it_o may_v also_o be_v the_o less_o and_o for_o that_o cause_n euclid_n make_v not_o mention_v of_o the_o comparison_n of_o the_o triangle_n whereof_o this_o also_o may_v be_v a_o cause_n for_o that_o to_o the_o demonstration_n thereof_o be_v require_v certain_a proposition_n concern_v parallel_a line_n which_o we_o be_v not_o as_o yet_o come_v unto_o etc_n howbeit_o after_o the_o 37â—_n proposition_n of_o his_o book_n you_o shall_v find_v the_o comparison_n of_o the_o area_n of_o triangle_n which_o have_v their_o side_n equal_a and_o their_o base_n and_o angle_n at_o the_o top_n unequal_a the_o 16._o theorem_a the_o 25._o proposition_n if_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_a to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n and_o if_o the_o base_a of_o the_o one_o be_v great_a than_o the_o base_a of_o the_o other_o the_o angle_n also_o of_o the_o same_o contain_v under_o the_o equal_a right_a linesâ—_n shall_v be_v great_a than_o the_o angle_n of_o the_o other_o svppose_v that_o there_o be_v two_o triangle_n a_o b_o c_o and_o def_n have_v two_o side_n of_o tb'one_n that_o be_v ab_fw-la and_o ac_fw-la equal_a to_o two_o side_n of_o the_o other_o that_o be_v to_o de_fw-fr and_o df_o each_o to_o his_o correspondent_a side_n namely_o the_o side_n ab_fw-la to_o the_o side_n df_o and_o the_o side_n ac_fw-la to_o the_o side_n df._n but_o let_v the_o base_a bc_o be_v great_a than_o the_o base_a ef._n then_o i_o say_v they_o the_o angle_n bac_n be_v great_a than_o the_o angle_n edf_o for_o if_o not_o absurdity_n then_o be_v it_o either_o equal_a unto_o it_o or_o less_o than_o it_o but_o the_o angle_n bac_n be_v not_o equal_a to_o the_o angle_n edf_o for_o if_o it_o be_v equal_a the_o base_a also_o bc_o shall_v by_o the_o 4._o proposition_n be_v equal_a to_o the_o base_a of_o but_o by_o supposition_n it_o be_v not_o wherefore_o
triangle_n unto_o the_o section_n divide_v the_o angle_n of_o the_o triangle_n into_o two_o equal_a part_n this_o construction_n be_v the_o half_a part_n of_o that_o gnomical_a figure_n describe_v in_o the_o 43._o proposition_n of_o the_o first_o book_n which_o gnomical_a figure_n be_v of_o great_a use_n in_o a_o manner_n in_o all_o geometrical_a demonstration_n the_o 4._o theorem_a the_o 4._o proposition_n in_o equiangle_n triangle_n the_o side_n which_o contain_v the_o equal_a angle_n be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n svppose_v that_o there_o be_v two_o equiangle_n triangle_n abc_n and_o dce_n and_o let_v the_o angle_n abc_n of_o the_o one_o triangle_n be_v equal_a unto_o the_o angle_v dce_n of_o the_o other_o triangle_n and_o the_o angle_n bac_n equal_a unto_o the_o angle_v cde_o and_o moreover_o the_o angle_n acb_o equal_a unto_o the_o angle_n dec_n then_o i_o say_v that_o those_o side_n of_o the_o triangle_n abc_n &_o dce_n which_o include_v the_o equal_a angle_n be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n construction_n for_o let_v two_o side_n of_o the_o say_a triangle_n namely_o two_o of_o those_o side_n which_o be_v subtend_v under_o equal_a angle_n as_o for_o example_n the_o side_n bc_o and_o ce_fw-fr be_v so_o set_v that_o they_o both_o make_v one_o right_a line_n and_o because_o the_o angle_n abc_n &_o acb_o be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o first_o but_o the_o angle_n acb_o be_v equal_a unto_o the_o angle_n dec_n therefore_o the_o angle_n abc_n &_o dec_n be_v less_o they_o two_o right_a angle_n wherefore_o the_o line_n basilius_n &_o ed_z be_v produce_v will_v at_o the_o length_n meet_v together_o let_v they_o meet_v and_o join_v together_o in_o the_o point_n f._n demonstration_n and_o because_o by_o supposition_n the_o angle_n dce_n be_v equal_a unto_o the_o angle_n abc_n therefore_o the_o line_n bf_o be_v by_o the_o 28._o of_o the_o first_o a_o parallel_n unto_o tâ—e_z line_n cd_o and_o forasmuch_o as_o by_o supposition_n the_o angle_n acb_o be_v equal_a unto_o the_o angle_n dec_n therefore_o again_o by_o the_o 28._o of_o the_o first_o the_o line_n ac_fw-la be_v a_o parallel_n unto_o the_o line_n fe_o wherefore_o fadc_n be_v a_o parallelogram_n wherefore_o the_o side_n favorina_n be_v equal_a unto_o the_o side_n dc_o and_o the_o side_n ac_fw-la unto_o the_o side_n fd_o by_o the_o 34._o of_o the_o first_o and_o because_o unto_o one_o of_o the_o side_n of_o the_o triangle_n bfe_n namely_o to_o fe_o be_v draw_v a_o parallel_a line_n ac_fw-la therefore_o as_o basilius_n be_v to_o of_o so_o be_v bc_o to_o ce_fw-fr by_o the_o 2._o of_o the_o six_o but_o of_o be_v equal_a unto_o cd_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o basilius_n be_v to_o cd_o so_o be_v bc_o to_o ce_fw-fr which_o be_v side_n subtend_v under_o equal_a angle_n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o ab_fw-la be_v to_o bc_o so_o be_v dc_o to_o ce._n again_o forasmuch_o as_o cd_o be_v a_o parallel_n unto_o bf_o therefore_o again_o by_o the_o 2._o of_o the_o six_o as_o bc_o be_v to_o ce_fw-fr so_o be_v fd_v to_o de._n but_o fd_o be_v equal_a unto_o ac_fw-la wherefore_o as_o bc_o be_v to_o ce_fw-fr so_o be_v ac_fw-la to_o de_fw-fr which_o be_v also_o side_n subtend_v under_o equal_a angle_n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o â—s_v bc_o be_v to_o ca_n so_o be_v ce_fw-fr to_o edâ—_n wherefore_o forasmuch_o as_o it_o have_v be_v demonstrate_v that_o as_o ab_fw-la be_v unto_o be_v so_o be_v dc_o unto_o ceâ—_n but_o as_o dc_o be_v unto_o ca_n so_o be_v ce_fw-fr unto_o edâ—_n it_o follow_v of_o equality_n by_o the_o 22._o of_o the_o five_o that_o â—s_a basilius_n be_v unto_o ac_fw-la so_o be_v cd_o unto_o deâ—_n wherefore_o in_o equiangle_n triangleâ—_n the_o side_n which_o include_v the_o equal_a angle_n be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n â—hich_z be_v require_v to_o be_v demonstrate_v the_o 5._o theorem_a the_o 5._o proposition_n if_o two_o triangle_n have_v their_o side_n proportional_a the_o triangâ—â—s_n be_v equiangle_n and_o those_o angle_n in_o they_o be_v equal_a under_o which_o be_v subtend_v side_n of_o like_a proportion_n svppose_v that_o there_o be_v two_o triangle_n abc_n &_o def_n have_v their_o side_n proportional_a as_o ab_fw-la be_v to_o bc_o so_o let_v de_fw-fr be_v to_o of_o proposition_n &_o as_o bc_o be_v to_o ac_fw-la so_o let_v of_o be_v to_o df_o and_o moreover_o as_o basilius_n be_v to_o ac_fw-la so_o let_v ed_z be_v to_o df._n then_o i_o say_v that_o the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n def_n and_o those_o angle_n in_o they_o be_v equal_a under_o which_o be_v subtend_v side_n of_o like_a proportion_n that_o be_v the_o angle_n abc_n be_v equal_a unto_o the_o angle_n def_n and_o the_o angle_n bca_o unto_o the_o angle_n efd_n and_o moreover_o the_o angle_n bac_n to_o the_o angle_n edf_o upon_o the_o right_a line_n of_o construction_n and_o unto_o the_o point_n in_o it_o e_o &_o f_o describe_v by_o the_o 23._o of_o the_o first_o angle_n equal_a unto_o the_o angle_n abc_n &_o acb_o which_o let_v be_v feg_fw-mi and_o efg_o namely_o let_v the_o angle_n feg_fw-mi be_v equal_a unto_o the_o angle_n abc_n and_o let_v the_o angle_n efg_o be_v equal_a to_o the_o angle_n acb_o demonstration_n and_o forasmuch_o as_o the_o angle_n abc_n and_o acb_o be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o first_o therefore_o also_o the_o angle_n feg_v and_o efg_o be_v less_o than_o two_o right_a angle_n wherefore_o by_o the_o 5._o petition_n of_o the_o first_o the_o right_a line_n eglantine_n &_o fg_o shall_v at_o the_o length_n concur_v let_v they_o concur_v in_o the_o point_n g._n wherefore_o efg_o be_v a_o triangle_n wherefore_o the_o angle_n remain_v bac_n be_v equal_a unto_o the_o angle_n remain_v egf_n by_o the_o first_o corollary_n of_o the_o 32._o of_o the_o first_o wherefore_o the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n gef_n wherefore_o in_o the_o triangle_n abc_n and_o egf_n the_o side_n which_o include_v the_o equal_a angle_n by_o the_o 4._o of_o the_o six_o be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n wherefore_o as_o ab_fw-la be_v to_o bc_o so_o be_v ge_z to_o ef._n but_o as_o ab_fw-la be_v to_o bc_o so_o by_o supposition_n be_v de_fw-fr to_o ef._n wherefore_o as_o de_fw-fr be_v to_o of_o so_o be_v ge_z to_o of_o by_o the_o 11._o of_o the_o five_o wherefore_o either_o of_o these_o de_fw-fr and_o eglantine_n have_v to_o of_o one_o and_o the_o same_o proportion_n wherefore_o by_o the_o 9_o of_o the_o five_o de_fw-fr be_v equal_a unto_o eglantine_n and_o by_o the_o same_o reason_n also_o df_o be_v equal_a unto_o fg._n now_o forasmuch_o as_o de_fw-fr be_v equal_a to_o eglantine_n and_o of_o be_v common_a unto_o they_o both_o therefore_o these_o two_o side_n de_fw-fr &_o of_o be_v equal_a unto_o these_o two_o side_n ge_z and_o of_o and_o the_o base_a df_o be_v equal_a unto_o the_o base_a fg._n wherefore_o the_o angle_n def_n by_o the_o 8._o of_o the_o first_o be_v equal_a unto_o the_o angle_n gef_n and_o the_o triangle_n def_n by_o the_o 4._o of_o the_o first_o be_v equal_a unto_o the_o triangle_n gef_n and_o the_o rest_n of_o the_o angle_n of_o the_o one_o triangle_n be_v equal_a unto_o the_o rest_n of_o the_o angle_n of_o the_o other_o triangle_n the_o one_o to_o the_o outstretch_o under_o which_o be_v subtend_v equal_a side_n wherefore_o the_o angle_n dfe_n be_v equal_a unto_o the_o angle_n gfe_n and_o the_o angle_n edf_o unto_o the_o angle_n egf_n and_o becauseâ—_n the_o angle_n feed_v be_v equal_a unto_o the_o angle_n gef_n but_o the_o angle_n gef_n be_v equal_a unto_o the_o angle_n abc_n therefore_o the_o angle_n abc_n be_v also_o equal_a unto_o the_o angle_n feed_v and_o by_o the_o same_o reason_n the_o angle_n acb_o be_v equal_a unto_o the_o angle_v dfeâ—_n and_o moreover_o the_o angle_n bac_n unto_o the_o angle_n edf_o wherefore_o the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n def_n if_o two_o triangle_n therefore_o have_v their_o side_n proportional_a the_o triangle_n shall_v be_v equiangle_n &_o those_o angle_n in_o they_o shall_v be_v equal_a under_o which_o be_v subtend_v side_n of_o like_a proportion_n which_o be_v require_v to_o be_v demonstrate_v the_o 6._o theorem_a the_o 6._o proposition_n if_o there_o be_v two_o triangle_n whereof_o the_o one_o have_v one_o angle_n equal_a to_o one_o angle_n of_o
that_o the_o line_n of_o be_v make_v equal_a to_o the_o line_n ad_fw-la which_o be_v the_o diameter_n of_o the_o square_n abcd_o of_o which_o square_v the_o line_n ab_fw-la be_v a_o side_n it_o be_v certain_a that_o the_o â—ide_v of_o a_o square_n be_v incommensurable_a in_o length_n to_o the_o diameter_n of_o the_o same_o square_n if_o there_o be_v yet_o find_v any_o one_o superficies_n which_o measure_v the_o two_o square_n abcd_o and_o efgh_a as_o here_o do_v the_o triangle_n abdella_n or_o the_o triangle_n acd_v note_v in_o the_o square_n abcd_o or_o any_o of_o the_o four_o triangle_n note_v in_o the_o square_a efgh_o as_o appear_v somewhat_o more_o manifest_o in_o the_o second_o example_n in_o the_o declaration_n of_o the_o last_o definition_n go_v before_o the_o line_n of_o be_v also_o a_o rational_a line_n note_v that_o these_o line_n which_o here_o be_v call_v rational_a line_n be_v not_o rational_a line_n of_o purpose_n or_o by_o supposition_n as_o be_v the_o first_o rational_a line_n but_o be_v rational_a only_o by_o reason_n of_o relation_n and_o comparison_n which_o they_o have_v unto_o it_o because_o they_o be_v commensurable_a unto_o it_o either_o in_o length_n and_o power_n or_o in_o power_n only_o far_o here_o be_v to_o be_v note_v that_o these_o word_n length_n and_o power_n and_o power_n only_o be_v join_v only_o with_o these_o wordeâ—_n commensurable_a or_o incommensurable_a and_o be_v never_o join_v with_o these_o word_n rational_a or_o irrational_a so_o that_o no_o line_n can_v be_v call_v rational_a in_o length_n or_o in_o power_n nor_o like_o wise_a can_v they_o be_v call_v irrational_a in_o length_n or_o in_o power_n wherein_o undoubted_o campanus_n be_v deceive_v book_n who_o use_v those_o word_n &_o speech_n indifferent_o cause_v &_o bring_v in_o great_a obscurity_n to_o the_o proposition_n and_o demonstration_n of_o this_o book_n which_o he_o shall_v easy_o see_v which_o mark_v with_o diligence_n the_o demonstration_n of_o campanus_n in_o this_o book_n 7_o line_n which_o be_v incommensurable_a to_o the_o rational_a line_n be_v call_v irrational_a definition_n by_o line_n incommensurable_a to_o the_o rational_a line_n suppose_v in_o this_o place_n he_o understand_v such_o as_o be_v incommensurable_a unto_o it_o both_o in_o length_n and_o in_o power_n for_o there_o be_v no_o line_n incommensurable_a in_o power_n only_o for_o it_o can_v be_v that_o any_o line_n shall_v so_o be_v incommensurable_a in_o power_n only_o that_o they_o be_v not_o also_o incommensurable_a in_o length_n what_o so_o ever_o line_n be_v incommensurable_a in_o power_n the_o same_o be_v also_o incommensurable_a in_o length_n neither_o can_v euclid_n here_o in_o this_o place_n mean_a line_n incommensurable_a in_o length_n only_o for_o in_o the_o definition_n before_o he_o call_v they_o rational_a line_n neither_o may_v they_o be_v place_v amongst_o irrational_a line_n wherefore_o it_o remain_v that_o in_o this_o diffintion_n he_o speak_v only_o of_o those_o line_n which_o be_v incommensurable_a to_o the_o rational_a line_n first_o give_v and_o suppose_v both_o in_o length_n and_o in_o power_n which_o by_o all_o mean_n be_v incommensurable_a to_o the_o rational_a line_n &_o therefore_o most_o apt_o be_v they_o call_v irrational_a line_n this_o definition_n be_v easy_a to_o be_v understand_v by_o that_o which_o have_v be_v say_v before_o yet_o for_o the_o more_o plainness_n see_v this_o example_n let_v the_o â—â—rst_a rational_a line_n suppose_v be_v the_o line_n ab_fw-la who_o square_a or_o quadrate_n let_v be_v abcd._n and_o let_v there_o be_v give_v a_o other_o line_n of_o which_o lâ—t_v be_v to_o the_o rational_a line_n incommensurable_a in_o length_n and_o power_n so_o that_o let_v no_o one_o line_n measure_v the_o length_n of_o the_o two_o line_n ab_fw-la and_o of_o and_o let_v the_o square_n of_o the_o line_n of_o be_v efgh_a now_o if_o also_o there_o be_v no_o one_o superficies_n which_o measure_v the_o two_o square_n abcd_o and_o efgh_a as_o be_v suppose_v to_o be_v in_o this_o example_n they_o be_v the_o line_n of_o a_o irrational_a line_n which_o word_n irrational_a as_o before_o do_v this_o word_n rational_a mislike_v many_o learned_a in_o this_o knowledge_n of_o geometry_n flussates_n uncertain_a as_o he_o leave_v the_o word_n rational_a and_o in_o stead_n thereof_o use_v this_o word_n certain_a so_o here_o he_o leave_v the_o word_n irrational_a and_o use_v in_o place_n thereof_o this_o word_n uncertain_a and_o ever_o name_v these_o line_n uncertain_a line_n petrus_n montaureus_n also_o mislike_v the_o word_n irrational_a will_v rather_o have_v they_o to_o be_v call_v surd_a line_n yet_o because_o this_o word_n irrational_a have_v ever_o by_o custom_n and_o long_a use_n so_o general_o be_v receive_n he_o use_v continual_o the_o same_o in_o greek_a such_o line_n be_v call_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d alogoi_n which_o signify_v nameless_a unspeakable_a uncertain_a in_o determinate_a line_n and_o with_o out_o proportion_n not_o that_o these_o irrational_a line_n have_v no_o proportion_n at_o all_o either_o to_o the_o first_o rational_a line_n or_o between_o themselves_o but_o be_v so_o name_v for_o that_o their_o proportion_n to_o the_o rational_a line_n can_v be_v express_v in_o number_n that_o be_v undoubted_o very_o untrue_a which_o many_o write_v that_o their_o proportion_n be_v unknown_a both_o to_o we_o and_o to_o nature_n be_v it_o not_o think_v you_o a_o thing_n very_o absurd_a to_o say_v that_o there_o be_v any_o thing_n in_o nature_n and_o produce_v by_o nature_n to_o be_v hide_v from_o nature_n and_o not_o to_o be_v know_v of_o nature_n it_o can_v not_o be_v say_v that_o their_o proportion_n be_v utter_o hide_v and_o unknown_a to_o we_o much_o less_o unto_o nature_n although_o we_o can_v geve_v they_o their_o name_n and_o distinct_o express_v they_o by_o number_n otherwise_o shall_v euclid_n have_v take_v all_o this_o travel_n and_o wonderful_a diligence_n bestow_v in_o this_o bookeâ—_n in_o vain_a and_o to_o no_o useâ—_n in_o which_o he_o do_v nothing_o ellâ—_n but_o teach_v the_o propriety_n and_o passion_n of_o these_o irrational_a linesâ—_n and_o show_v the_o proportion_n which_o they_o have_v the_o one_o to_o the_o other_o here_o be_v also_o to_o be_v note_v which_o thing_n also_o tartalea_n have_v before_o diligent_o notedâ—_n that_o campanus_n and_o many_o other_o writer_n of_o geometryâ—_n over_o much_o â—â—â—ed_a and_o be_v deceive_v in_o that_o they_o write_v and_o teach_v that_o all_o these_o line_n who_o square_n be_v not_o signify_v and_o may_v be_v express_v by_o a_o square_a number_n although_o they_o may_v by_o any_o other_o number_n as_o by_o 11._o 12._o 14._o and_o such_o other_o not_o square_a number_n be_v irrational_a line_n which_o be_v manifest_o repugnant_a to_o the_o ground_n and_o principle_n of_o euclid_n who_o will_v that_o all_o line_n which_o be_v commensurable_a to_o the_o rational_a line_n whether_o it_o be_v in_o length_n and_o power_n or_o in_o power_n only_o shall_v be_v rational_a undoubted_o this_o have_v be_v one_o of_o the_o chief_a and_o great_a cause_n of_o the_o wonderful_a confusion_n and_o darkness_n of_o this_o book_n book_n which_o so_o have_v toss_v and_o turmoil_v the_o wit_n of_o all_o both_o writer_n and_o reader_n master_n and_o scholar_n and_o so_o overwhelm_v they_o that_o they_o can_v not_o with_o out_o infinite_a travel_n and_o sweat_v attain_v to_o the_o truth_n and_o perfect_a understanding_n thereof_o definition_n 8_o the_o square_n which_o be_v describe_v of_o the_o rational_a right_a line_n suppose_v be_v rational_a until_o this_o definition_n have_v euclid_n set_v forth_o the_o nature_n and_o propriety_n of_o the_o first_o kind_n of_o magnitude_n namely_o of_o line_n how_o they_o be_v rational_a or_o irrational_a now_o he_o begin_v to_o â—hew_v how_o the_o second_o kind_n of_o magnitude_n namely_o superficies_n be_v one_o to_o the_o other_o rational_a or_o irrational_a this_o definition_n be_v very_a plain_n suppose_v the_o line_n ab_fw-la to_o be_v the_o rational_a line_n have_v his_o part_n and_o division_n certain_o know_v the_o square_a of_o which_o line_n let_v be_v the_o square_a abcd._n now_o because_o it_o be_v the_o square_a of_o the_o rational_a line_n ab_fw-la it_o be_v also_o call_v rational_a and_o as_o the_o line_n ab_fw-la be_v the_o first_o rational_a line_n unto_o which_o other_o line_n compare_v be_v count_v rational_a or_o irrational_a so_o be_v the_o quadrat_fw-la or_o square_v thereof_o the_o â—irst_n rational_a superficies_n unto_o which_o all_o other_o square_n or_o figure_n compare_v be_v count_v and_o name_v rational_a or_o irrational_a 9_o such_o which_o be_v commensurable_a unto_o it_o be_v rational_a definition_n in_o this_o definition_n where_o it_o be_v say_v such_o as_o be_v commensurable_a to_o the_o square_n of_o the_o rational_a line_n be_v not_o understand_v only_o other_o square_n or_o
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
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
that_o superficies_n be_v rational_a svppose_v that_o a_o parallelogram_n be_v contain_v under_o a_o residual_a line_n ab_fw-la and_o a_o binomial_a line_n cd_o and_o let_v the_o great_a name_n of_o the_o binomial_a line_n be_v ce_fw-fr and_o the_o less_o name_n be_v ed_z and_o let_v the_o name_n of_o the_o binomial_a line_n namely_o ce_fw-fr and_o ed_z be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n namely_o to_o of_o and_o fâ—_n and_o in_o the_o self_n same_o proportion_n and_o let_v the_o line_n which_o contain_v in_o power_n that_o parallelogram_n be_v g._n then_o i_o say_v that_o the_o line_n g_o be_v rational_a take_v a_o rational_a line_n namely_o h._n and_o unto_o the_o line_n cd_o apply_v a_o parallelogram_n equal_a to_o the_o square_v of_o the_o line_n h_o construction_n and_o make_v in_o breadth_n the_o line_n kl_o wherefore_o by_o the_o 112._o of_o the_o ten_o kl_o be_v a_o residual_a line_n demonstration_n who_o name_n let_v be_v km_o and_o ml_o which_o be_v by_o the_o same_o commensurable_a to_o the_o name_n of_o the_o binomial_a line_n that_o be_v to_o ce_fw-fr and_o ed_z and_o be_v in_o the_o self_n same_o proportion_n but_o by_o position_n the_o line_n ce_fw-fr and_o ed_z be_v commensurable_a to_o the_o line_n of_o and_o fb_o and_o be_v in_o the_o self_n same_o proportion_n wherefore_o by_o the_o 12._o of_o the_o ten_o as_o the_o line_n of_o be_v to_o the_o line_n fbâ—_n so_o be_v the_o line_n km_o to_o the_o line_n ml_o wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n of_o be_v to_o the_o line_n km_o so_o be_v the_o line_n bf_o to_o the_o line_n lm_o wherefore_o the_o residue_n ab_fw-la be_v to_o the_o residue_n kl_o as_o the_o whole_a of_o be_v to_o the_o whole_a km_o but_o the_o line_n of_o be_v commensurable_a to_o the_o line_n km_o for_o either_o of_o the_o line_n of_o and_o km_o be_v commensurable_a to_o the_o line_n ce._n wherefore_o also_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n kl_o and_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n kl_o so_o by_o the_o first_o of_o the_o six_o be_v the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o ab_fw-la to_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o kl_o wherefore_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o ab_fw-la be_v commensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o kl_o but_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o kl_o be_v equal_a to_o the_o square_n of_o the_o line_n h._n wherefore_o the_o parallelogram_n contain_v under_o the_o line_n cd_o &_o ab_fw-la be_v commensurable_a to_o the_o square_n of_o the_o line_n h._n but_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n g._n wherefore_o the_o square_a of_o the_o line_n h_o be_v commensurable_a to_o the_o square_n of_o the_o line_n g._n but_o the_o square_n of_o the_o line_n h_o be_v rational_a wherefore_o the_o square_a of_o the_o line_n g_o be_v also_o rational_a wherefore_o also_o the_o line_n g_o be_v rational_a and_o it_o contain_v in_o power_n the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o cd_o if_o therefore_o a_o parallelogram_n be_v contain_v under_o a_o residual_a line_n and_o a_o binomial_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n and_o in_o the_o self_n same_o proportion_n the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v rational_a which_o be_v require_v to_o be_v prove_v ¶_o corollary_n hereby_o it_o be_v manifest_a that_o a_o rational_a parallelogram_n may_v be_v contain_v under_o irrational_a line_n ¶_o a_o otâ—â—r_n 〈…〉_o flussas_n 〈…〉_o line_n â—d_a whosâ—_n name_v aâ—_z and_o â—d_a let_v be_v commensurable_a in_o length_n unto_o the_o name_n of_o the_o residual_a line_n aâ—_z which_o let_v be_v of_o and_o fb_o and_o let_v the_o liâ—e_z aeâ—_n be_v to_o the_o line_n edâ—_n in_o the_o same_o proportion_n that_o the_o line_n of_o be_v to_o the_o line_n fâ—_n and_o let_v the_o right_a line_n â—_o contain_v in_o power_n the_o superficies_n dâ—_n then_o i_o say_v thaâ—_z the_o liâ—e_z â—_o be_v a_o rational_a linâ—_n 〈…〉_o lâ—ne_n construction_n which_o lâ—â—_n bâ—â—_n and_o upon_o the_o line_n â—â—_o describe_v by_o the_o 4â—_o of_o the_o first_o a_o parallelogram_n equal_a to_o the_o square_n of_o the_o line_n â—â—_o and_o make_n in_o breadth_n the_o line_n dc_o wherefore_o by_o the_o â—12_n of_o this_o book_n cd_o be_v a_o residuâ—ll_a lineâ—_z who_o name_n which_o let_v be_v â—â—_o and_o odd_a shall_v be_v coâ—mensurablâ—_n in_o length_n unto_o the_o name_n aâ—_z and_o â—d_a and_o the_o line_n c_o o_fw-fr shall_v be_v unto_o the_o line_n odd_a demonstration_n in_o the_o same_o proportion_n that_o the_o line_n ae_n be_v to_o the_o line_n edâ—_n but_o as_o the_o line_n aâ—_z be_v to_o the_o line_n â—d_a so_o by_o supposition_n be_v the_o line_n of_o to_o the_o line_n fe_o wherefore_o as_o the_o line_n county_n be_v to_o the_o line_n odd_a so_o be_v the_o line_n of_o to_o the_o line_n fâ—â—_v wherefore_o the_o line_n county_n and_o odd_a be_v commensurable_a with_o the_o line_n aâ—_z and_o â—â—_o by_o the_o â—â—_o of_o this_o book_n wherefore_o the_o residue_n namely_o the_o line_n cd_o be_v to_o the_o residue_n namely_o to_o the_o line_n aâ—_z as_o the_o line_n county_n be_v to_o the_o line_n of_o by_o the_o 19_o of_o the_o five_o but_o it_o be_v prove_v that_o the_o line_n county_n be_v commensurable_a unto_o the_o line_n af._n wherefore_o the_o line_n cd_o be_v commensurable_a unto_o the_o line_n ab_fw-la wherefore_o by_o the_o first_o of_o the_o six_o the_o parallelogram_n ca_n be_v commensurable_a to_o the_o parallelogram_n dâ—_n but_o the_o parallelogram_n â—â—_o iâ—_z by_o construction_n rational_a for_o it_o be_v equal_a to_o the_o square_n of_o the_o rational_a line_n â—_o wherefore_o the_o parallelogram_n â—d_a â—s_n also_o rational_o wherâ—fore_o the_o line_n â—_o which_o by_o supposition_n contain_v in_o power_n the_o superficies_n â—dâ—_n be_v also_o rational_a if_o therefore_o a_o parallelogram_n be_v contain_v etc_o etc_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 91._o theorem_a the_o 115._o proposition_n of_o a_o medial_a line_n be_v produce_v infinite_a irrational_a line_n of_o which_o none_o be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o svppose_v that_o a_o be_v a_o medial_a line_n then_o i_o say_v that_o of_o the_o line_n a_o may_v be_v produce_v infinite_a irrational_a line_n of_o which_o none_o shall_v be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o take_v a_o rational_a line_n b._n and_o unto_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o let_v the_o square_n of_o the_o line_n c_o be_v equal_a by_o the_o 14._o of_o the_o second_o â—_o demonstration_n wherefore_o the_o line_n c_o be_v irrational_a for_o a_o superficies_n contain_v under_o a_o rational_a line_n and_o a_o irrational_a line_n be_v by_o the_o assumpt_n follow_v the_o 38._o of_o the_o ten_o irrational_a and_o the_o line_n which_o contain_v in_o power_n a_o irrational_a superficies_n be_v by_o the_o assumpt_n go_v before_o the_o 21._o of_o the_o ten_o irrational_a and_o it_o be_v not_o one_o and_o the_o self_n same_o with_o any_o of_o those_o thirteen_o that_o be_v before_o for_o none_o of_o the_o line_n that_o be_v before_o apply_v to_o a_o rational_a line_n make_v the_o breadth_n medial_a again_o unto_o that_o which_o be_v contain_v under_o the_o line_n b_o and_o c_o let_v the_o square_n of_o d_o be_v equal_a wherefore_o the_o square_a of_o d_o be_v irrational_a wherefore_o also_o the_o line_n d_o be_v irrational_a and_o not_o of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o for_o the_o square_n of_o none_o of_o the_o line_n which_o be_v before_o apply_v to_o a_o rational_a line_n make_v the_o breadth_n the_o line_n c._n in_o like_a sort_n also_o shall_v it_o so_o follow_v if_o a_o man_n proceed_v infinite_o wherefore_o it_o be_v manifest_a that_o of_o a_o medial_a line_n be_v produce_v infinite_a irrational_a line_n of_o which_o none_o be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o which_o be_v require_v to_o be_v prove_v an_o other_o demonstration_n demonstration_n suppose_v that_o ac_fw-la be_v a_o medial_a line_n then_o i_o say_v that_o of_o the_o line_n ac_fw-la may_v be_v produce_v infinite_a irrational_a line_n of_o which_o none_o shall_v be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o irrational_a line_n before_o name_v unto_o the_o line_n ac_fw-la and_o from_o the_o point_n a_o
signify_v last_o of_o all_o a_o dodecahedron_n heaven_n for_o that_o it_o be_v make_v of_o pâ—ntagoâ—_n who_o angle_n be_v more_o ample_a and_o large_a than_o the_o angle_n of_o the_o other_o body_n and_o by_o that_o â—eaâ—â—â—_n draw_v more_o â—â—_o roundness_n 〈◊〉_d &_o to_o the_o form_n and_o nature_n of_o a_o sphere_n they_o assign_v to_o a_o sphere_n namely_o 〈…〉_o who_o so_o will_v 〈…〉_o in_o his_o tineus_n shall_v â—ead_a of_o these_o figure_n and_o of_o their_o mutual_a proportionâ—â—â—raunge_a maâ—terâ—_n which_o hâ—re_o be_v not_o to_o be_v entreat_v of_o this_o which_o be_v say_v shall_v be_v sufficient_a for_o the_o 〈◊〉_d of_o they_o and_o for_o thâ—_z declaration_n of_o their_o definition_n after_o all_o these_o definition_n here_o set_v of_o euclid_n flussas_n have_v add_v a_o other_o definition_n which_o 〈◊〉_d of_o a_o parallelipipedon_n which_o because_o it_o have_v not_o hitherto_o of_o euclid_n in_o any_o place_n be_v define_v and_o because_o it_o be_v very_o good_a and_o necessary_a to_o be_v have_v i_o think_v good_a not_o to_o omit_v it_o thus_o it_o be_v a_o parallelipipedon_n be_v a_o solid_a figure_n comprehend_v under_o four_o plain_a quadrangle_n figure_n parallelipipedon_n of_o which_o those_o which_o be_v opposite_a be_v parallel_n because_o these_o five_o regular_a body_n here_o define_v be_v not_o by_o these_o figure_n here_o set_v so_o full_o and_o lively_o express_v that_o the_o studious_a beholder_n can_v thorough_o according_a to_o their_o definition_n conceive_v they_o i_o have_v here_o give_v of_o they_o other_o description_n draw_v in_o a_o plain_n by_o which_o you_o may_v easy_o attain_v to_o the_o knowledge_n of_o they_o for_o if_o you_o draw_v the_o like_a form_n in_o matter_n that_o will_v bow_v and_o geve_v place_n as_z most_o apt_o you_o may_v do_v in_o fine_a past_v paper_n such_o as_o pastwive_n make_v woman_n paste_n of_o &_o they_o with_o a_o knife_n cut_v every_o line_n fine_o not_o through_o but_o half_a way_n only_o if_o they_o you_o bow_v and_o bend_v they_o according_o you_o shall_v most_o plain_o and_o manifest_o see_v the_o form_n and_o shape_n of_o these_o body_n even_o as_o their_o definition_n show_v and_o it_o shall_v be_v very_o necessary_a for_o you_o to_o hadâ—â—tore_v of_o that_o past_v paper_n by_o you_o for_o so_o shall_v yoâ—_n upon_o it_o 〈…〉_o the_o form_n of_o other_o body_n as_o prism_n and_o parallelipopedon_n 〈…〉_o set_v forth_o in_o these_o five_o book_n follow_v and_o see_v the_o very_a 〈◊〉_d of_o thâ—se_a body_n there_o mention_v which_o will_v make_v these_o book_n concern_v body_n as_o easy_a unto_o you_o as_o be_v the_o other_o book_n who_o figure_n you_o may_v plain_o see_v upon_o a_o plain_a superficies_n describe_v thiâ—_z figurâ—_n which_o consi_v of_o twâ—luâ—â—quilâ—â—â—â—_n and_o equiangle_n pâ—ntâ—â—â—â—â—_n upoâ—_n the_o foresay_a matter_n and_o fine_o cut_v as_o before_o be_v â—â—ught_v tâ—â—â—lâ—uâ—n_v line_n contain_v within_o thâ—_z figurâ—_n and_o bow_n and_o fold_n the_o penâ—â—gonâ—_n according_o and_o they_o will_v so_o close_o together_o thaâ—_z thâ—y_n will_v â—â—kâ—_n thâ—_z very_a form_n of_o a_o dodecahedron_n dâ—dâ—â—â—â—edron_n icosaâ—edron_n if_o you_o describe_v this_o figure_n which_o consist_v of_o twenty_o equilater_n and_o equiangle_n triangle_n upon_o the_o foresay_a matter_n and_o fine_o cut_v as_o before_o be_v show_v the_o ninâ—tâ—ne_a line_n which_o be_v contain_v within_o the_o figure_n and_o then_o bow_n and_o fold_n they_o according_o they_o will_v in_o such_o sort_n close_o together_o that_o therâ—_n will_v be_v make_v a_o perfect_v form_n of_o a_o icosahedron_n because_o in_o these_o five_o book_n there_o be_v sometime_o require_v other_o body_n beside_o the_o foresay_a five_o regular_a body_n as_o pyramid_n of_o diverse_a form_n prism_n and_o other_o i_o have_v here_o set_v forth_o three_o figure_n of_o three_o sundry_a pyramid_n one_o have_v to_o his_o base_a a_o triangle_n a_o other_o a_o quadrangle_n figure_n the_o other_o â—_o pentagonâ—_n which_o if_o you_o describe_v upon_o the_o foresay_a matter_n &_o fine_o cut_v as_o it_o be_v before_o teach_v the_o line_n contain_v within_o each_o figure_n namely_o in_o the_o first_o three_o line_n in_o the_o second_o four_o line_n and_o in_o the_o three_o five_o line_n and_o so_o bend_v and_o fold_v they_o according_o they_o will_v so_o close_o together_o at_o the_o top_n that_o they_o will_v â—ake_v pyramid_n of_o that_o form_n that_o their_o base_n be_v of_o and_o if_o you_o conceive_v well_o the_o describe_v of_o these_o you_o may_v most_o easy_o describe_v the_o body_n of_o a_o pyramid_n of_o what_o form_n so_o ever_o you_o will._n because_o these_o five_o book_n follow_v be_v somewhat_o hard_o for_o young_a beginner_n by_o reason_n they_o must_v in_o the_o figure_n describe_v in_o a_o plain_a imagine_v line_n and_o superficiece_n to_o be_v elevate_v and_o erect_v the_o one_o to_o the_o other_o and_o also_o conceive_v solid_n or_o body_n which_o for_o that_o they_o have_v not_o hitherto_o be_v acquaint_v with_o will_v at_o the_o first_o sight_n be_v somewhat_o strange_a unto_o they_o i_o have_v for_o their_o more_o â—ase_a in_o this_o eleven_o book_n at_o the_o end_n of_o the_o demonstration_n of_o every_o proposition_n either_o set_v new_a figure_n if_o they_o concern_v the_o elevate_n or_o erect_v of_o line_n or_o superficiece_n or_o else_o if_o they_o concern_v body_n i_o have_v show_v how_o they_o shall_v describe_v body_n to_o be_v compare_v with_o the_o construction_n and_o demonstration_n of_o the_o proposition_n to_o they_o belong_v and_o if_o they_o diligent_o weigh_v the_o manner_n observe_v in_o this_o eleven_o book_n touch_v the_o description_n of_o new_a figure_n agree_v with_o the_o figure_n describe_v in_o the_o plain_a it_o shall_v not_o be_v hard_o for_o they_o of_o themselves_o to_o do_v the_o like_a in_o the_o other_o book_n follow_v when_o they_o come_v to_o a_o proposition_n which_o concern_v either_o the_o elevate_n or_o erect_v of_o line_n and_o superficiece_n or_o any_o kind_n of_o body_n to_o be_v imagine_v ¶_o the_o 1._o theorem_a the_o 1._o proposition_n that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n &_o part_v elevate_v upward_o be_v impossible_a for_o if_o it_o be_v possible_a let_v part_n of_o the_o right_a line_n abc_n namely_o the_o part_n ab_fw-la be_v in_o a_o ground_n plain_a superficies_n and_o the_o other_o part_n thereof_o namely_o bc_o be_v elevate_v upward_o and_o produce_v direct_o upon_o the_o ground_n plain_a superficies_n the_o right_a line_n ab_fw-la beyond_o the_o point_n b_o unto_o the_o point_n d._n wherefore_o unto_o two_o right_a line_n give_v abc_n and_o abdella_n the_o line_n ab_fw-la be_v a_o common_a section_n or_o part_n which_o be_v impossible_a impossibility_n for_o a_o right_a line_n can_v not_o touch_v a_o right_a line_n in_o 〈◊〉_d point_n then_o one_o uâ—lesse_o those_o right_n be_v exact_o agree_v and_o lay_v the_o one_o upon_o the_o other_o wherefore_o that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n and_o part_v elevate_v upward_o be_v impossible_a which_o be_v require_v to_o be_v prove_v this_o figure_n more_o plain_o set_v forth_o the_o foresay_a demonstration_n if_o you_o elevate_v the_o superficies_n wherein_o the_o line_n bc._n an_o other_o demonstration_n after_o flâ—sâ—â—s_n if_o it_o be_v possible_a let_v there_o be_v a_o right_a line_n abg_o who_o part_n ab_fw-la let_v be_v in_o the_o ground_n plain_a superficies_n aed_n flussas_n and_o let_v the_o rest_n thereof_o bg_o be_v elevate_v on_o high_a that_o be_v without_o the_o plain_n aed_n then_o i_o say_v that_o abg_o be_v not_o one_o right_a line_n for_o forasmuch_o as_o aed_n be_v a_o plain_a superficies_n produce_v direct_o &_o equal_o upon_o the_o say_a plain_n aed_n the_o right_a line_n ab_fw-la towards_o d_z which_o by_o the_o 4._o definition_n of_o the_o first_o shall_v be_v a_o right_a line_n and_o from_o some_o one_o point_n of_o the_o right_a line_n abdella_n namely_o from_o c_o draâ—_n unto_o the_o point_n g_o a_o right_a line_n cg_o wherefore_o in_o the_o triangle_n 〈…〉_o the_o outward_a angâ—â—_n abâ—_n be_v equal_a to_o the_o two_o inward_a and_o opposite_a angle_n by_o the_o 32._o of_o the_o first_o and_o therefore_o it_o be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o same_o wherefore_o the_o line_n abg_o forasmuch_o as_o it_o make_v a_o angle_n be_v not_o â—_o right_n line_n wherefore_o that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n and_o part_v elevate_v upward_o be_v impossible_a if_o you_o mark_v well_o the_o figure_n before_o add_v for_o the_o plain_a declaration_n of_o euclides_n demonstration_n iâ—_z will_v not_o be_v hard_o for_o you_o to_o coâ—â—â—â—e_v this_o figure_n which_o ulyss_n put_v for_o his_o demonstration_n â—_o wherein_o
same_o superficies_n wherefore_o these_o right_a line_n ab_fw-la bd_o and_o dc_o be_v in_o one_o and_o the_o self_n same_o superficies_n and_o either_o of_o these_o angle_n abdella_n and_o bdc_o be_v a_o right_a angle_n by_o supposition_n wherefore_o by_o the_o 28._o of_o the_o first_o the_o line_n ab_fw-la be_v a_o parallel_n to_o the_o line_n cd_o if_o therefore_o two_o right_a line_n be_v erect_v perpendicular_o to_o one_o and_o the_o self_n same_o plain_a superficies_n those_o right_a line_n be_v parallel_n the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v here_o for_o the_o better_a understanding_n of_o this_o 6._o proposition_n i_o have_v describe_v a_o other_o figure_n as_o touch_v which_o if_o you_o erect_v the_o superficies_n abdella_n perpendicular_o to_o the_o superficies_n bde_v and_o imagine_v only_o a_o line_n to_o be_v draw_v from_o the_o point_n a_o to_o the_o point_n e_o if_o you_o will_v you_o may_v extend_v a_o thread_n from_o the_o say_a point_n a_o to_o the_o point_n e_o and_o so_o compare_v it_o with_o the_o demonstration_n it_o will_v make_v both_o the_o proposition_n and_o also_o the_o demonstration_n most_o clear_a unto_o you_o ¶_o a_o other_o demonstration_n of_o the_o six_o proposition_n by_o m._n dee_n suppose_v that_o the_o two_o right_a line_n ab_fw-la &_o cd_o be_v perpendicular_o erect_v to_o one_o &_o the_o same_o plain_a superficies_n namely_o the_o plain_a superficies_n open_v then_o i_o say_v that_o â—â—_o and_o cd_o be_v parallel_n let_v the_o end_n point_v of_o the_o right_a line_n ab_fw-la and_o cd_o which_o touch_v the_o plain_a superficies_n oâ—_n be_v the_o point_n â—_o and_o d_o fronâ—_n to_o d_o let_v a_o straight_a line_n be_v draw_v by_o the_o first_o petition_n and_o by_o the_o second_o petition_n let_v the_o straight_a line_n â—d_a be_v extend_v as_o to_o the_o point_n m_o &_o n._n now_o forasmuch_o as_o the_o right_a line_n ab_fw-la from_o the_o point_n â—_o produce_v do_v cut_v the_o line_n mn_v by_o construction_n therefore_o by_o the_o second_o proposition_n of_o this_o eleven_o book_n the_o right_a line_n ab_fw-la &_o mn_o be_v in_o one_o plainâ—_n superficies_n which_o let_v be_v qr_o cut_v the_o superficies_n open_v in_o the_o right_a line_n mn_o by_o the_o same_o mean_n may_v we_o conclude_v the_o right_a line_n cd_o to_o be_v in_o one_o plain_a superficies_n with_o the_o right_a line_n mn_o but_o the_o right_a line_n mn_o by_o supposition_n be_v in_o the_o plain_a superficies_n qr_o wherefore_o cd_o be_v in_o the_o plain_a superficies_n qr_o and_o aâ—_z the_o right_a line_n be_v prove_v to_o be_v in_o the_o same_o plain_a superficies_n qr_o therefore_o ab_fw-la and_o cd_o be_v in_o one_o plain_a superficies_n namely_o qr_o and_o forasmuch_o as_o the_o line_n aâ—_z and_o cd_o by_o supposition_n be_v perpendicular_a upon_o the_o plain_a superficies_n open_v therefore_o by_o the_o second_o definition_n of_o this_o book_n with_o all_o the_o right_a line_n draw_v in_o the_o superficies_n open_v and_o touch_v ab_fw-la and_o cd_o the_o same_o perpendiculars_n aâ—_z and_o cd_o do_v make_v right_a angle_n but_o by_o construction_n mn_v be_v draw_v in_o the_o plain_a superficies_n open_a touch_v the_o perpendicular_n ab_fw-la and_o cd_o at_o the_o point_n â—_o and_o d._n therefore_o the_o perpendicular_n aâ—_z and_o cd_o make_v with_o the_o right_a line_n mn_v two_o right_a angle_n namely_o abn_n and_o cdm_o and_o mn_v the_o right_a line_n be_v prove_v to_o be_v in_o the_o one_o and_o the_o same_o plain_a superficies_n with_o the_o right_a line_n ab_fw-la &_o cd_o namely_o in_o the_o plain_a superficies_n qr_o wherefore_o by_o the_o second_o part_n of_o the_o 28._o proposition_n of_o the_o first_o book_n the_o right_a lineâ—_z ab_fw-la and_o cd_o be_v parallel_n if_o therefore_o two_o right_a line_n be_v erect_v perpendicular_o to_o one_o and_o the_o self_n same_o plain_a superficies_n those_o right_a line_n be_v parallel_n the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o m._n dee_n hereby_o it_o be_v evident_a that_o any_o two_o right_a line_n perpendicular_o erect_v to_o one_o and_o the_o self_n same_o plain_a superficies_n be_v also_o themselves_o in_o one_o and_o the_o same_o plain_a superficies_n which_o be_v likewise_o perpendicular_o erect_v to_o the_o same_o plain_a superficies_n unto_o which_o the_o two_o right_a line_n be_v perpendicular_a the_o first_o part_n hereof_o be_v prove_v by_o the_o former_a construction_n and_o demonstration_n that_o the_o right_a line_n ab_fw-la and_o cd_o be_v in_o one_o and_o the_o same_o plain_a superficies_n qâ—_n the_o second_o part_n be_v also_o manifest_a that_o be_v that_o the_o plain_a superficies_n qr_o be_v perpendicular_o erect_v upon_o the_o plain_a superficies_n open_v for_o that_o aâ—_z and_o cd_o be_v in_o the_o plain_a superficies_n qr_o be_v by_o supposition_n perpendicular_a to_o the_o plain_a superficies_n open_v wherefore_o by_o the_o three_o definition_n of_o this_o book_n qr_o be_v perpendicular_o erect_v to_o or_o upon_o open_a which_o be_v require_v to_o be_v prove_v io._n do_v you_o his_o advice_n upon_o the_o assumpt_n of_o the_o 6._o as_o concern_v the_o make_n of_o the_o line_n de_fw-fr equal_a to_o the_o right_a line_n ab_fw-la very_o the_o second_o of_o the_o first_o without_o some_o far_a consideration_n be_v not_o proper_o enough_o allege_v and_o no_o wonder_n it_o be_v for_o that_o in_o the_o former_a bookeâ—_n whatsoeâ—â—â—â—aâ—h_v of_o line_n be_v speak_v the_o same_o have_v always_o be_v imagine_v to_o be_v in_o one_o only_a plain_a superficies_n consider_v or_o execute_v but_o here_o the_o perpendicular_a line_n ab_fw-la be_v not_o in_o the_o same_o plaint_n superficies_n that_o the_o right_a line_n db_o be_v therefore_o some_o other_o help_n must_v be_v put_v into_o the_o hand_n of_o young_a beginner_n how_o to_o bring_v this_o problem_n to_o execution_n which_o be_v this_o most_o plain_a and_o brief_a understand_v that_o bd_o the_o right_n line_n be_v the_o common_a section_n of_o the_o plain_a superficies_n wherein_o the_o perpendicular_n ab_fw-la and_o cd_o be_v &_o of_o the_o other_o plain_a superficies_n to_o which_o they_o be_v perpendicular_n the_o first_o of_o these_o in_o my_o former_a demonstration_n of_o the_o 6_o â—_o i_o note_v by_o the_o plain_a superficies_n qr_o and_o the_o other_o i_o note_v by_o the_o plain_a superficies_n open_v wherefore_o bd_o be_v a_o right_a line_n common_a to_o both_o the_o plain_a supâ—rficieces_n qr_o &_o op_n thereby_o the_o ponit_n b_o and_o d_o be_v common_a to_o the_o plain_n qr_o and_o op_n now_o from_o bd_o sufficient_o extend_v cut_v a_o right_a line_n equal_a to_o ab_fw-la which_o suppose_v to_o be_v bf_o by_o the_o three_o of_o the_o first_o and_o orderly_a to_o bf_o make_v de_fw-fr equal_a by_o the_o 3._o oâ—_n the_o first_o if_o de_fw-fr be_fw-mi great_a than_o bf_o which_o always_o you_o may_v cause_v so_o to_o be_v by_o produce_v of_o de_fw-fr sufficient_o now_o forasmuch_o as_o bf_o by_o construction_n be_v cut_v equal_a to_o ab_fw-la and_o de_fw-fr also_o by_o construction_n put_v equal_a to_o bf_o therefore_o by_o the_o 1._o common_a sentence_n de_fw-fr be_v put_v equal_a to_o ab_fw-la which_o be_v require_v to_o be_v do_v in_o like_a sort_n if_o de_fw-fr be_v a_o line_n give_v to_o who_o ab_fw-la be_v to_o be_v cut_v and_o make_v equal_a first_o out_o of_o the_o line_n db_o sufficient_o produce_v cut_v of_o dg_o equal_a to_o de_fw-fr by_o the_o three_o of_o the_o first_o and_o by_o the_o same_o 3._o cut_v from_o basilius_n sufficient_o produce_v basilius_n equal_a to_o dg_v then_o be_v it_o evident_a that_o to_o the_o right_a line_n de_fw-fr the_o perpendicular_a line_n ab_fw-la be_v put_v equal_a and_o though_o this_o be_v easy_a to_o conceive_v yet_o i_o have_v design_v the_o figure_n according_o whereby_o you_o may_v instruct_v your_o imagination_n many_o such_o help_n be_v in_o this_o book_n requisite_a as_o well_o to_o inform_v the_o young_a student_n therewith_o as_o also_o to_o master_n the_o froward_a gaynesayer_n of_o our_o conclusion_n or_o interrupter_n of_o our_o demonstration_n course_n ¶_o the_o 7._o theorem_a the_o 7._o proposition_n if_o there_o be_v two_o parallel_n right_a line_n and_o in_o either_o of_o they_o be_v take_v a_o point_n at_o all_o adventure_n a_o right_a line_n draw_v by_o the_o say_a point_n be_v in_o the_o self_n same_o superficies_n with_o the_o parallel_n right_a line_n svppose_v that_o these_o two_o right_a line_n ab_fw-la and_o cd_o be_v parallel_n and_o in_o either_o of_o they_o take_v a_o point_n at_o all_o adventure_n namely_o e_o and_o f._n then_o i_o say_v that_o a_o right_a line_n draw_v from_o the_o point_n e_o to_o the_o point_n fletcher_n be_v in_o the_o self_n same_o plain_a superficies_n that_o the_o
angle_n bac_n god_n &_o dab_n be_v equal_a the_o one_o to_o the_o other_o then_o be_v it_o manifest_v that_o two_o of_o they_o which_o two_z so_o ever_o be_v take_v be_v great_a than_o the_o three_o but_o if_o not_o let_v the_o angle_n bac_n be_v the_o great_a of_o the_o three_o angle_n and_o unto_o the_o right_a line_n ab_fw-la and_o from_o the_o point_n a_o make_v in_o the_o plain_a superficies_n bac_n unto_o the_o angle_n dab_n a_o equal_a angle_n bae_o and_o by_o the_o 2._o of_o the_o first_o make_v the_o line_n ae_n equal_a to_o the_o line_n ad._n now_o a_o right_a line_n bec_n draw_v by_o the_o point_n e_o shall_v cut_v the_o right_a line_n ab_fw-la and_o ac_fw-la in_o the_o point_n b_o and_o c_o draw_v a_o right_a line_n from_o d_z to_o b_o and_o a_o outstretch_o from_o d_z to_o c._n and_o forasmuch_o as_o the_o line_n dam_n be_v equal_a to_o the_o line_n ae_n demonstration_n and_o the_o line_n ab_fw-la be_v common_a to_o they_o both_o therefore_o these_o two_o line_n dam_n and_o ab_fw-la be_v equal_a to_o these_o two_o line_n ab_fw-la and_o ae_n and_o the_o angle_n dab_n be_v equal_a to_o the_o angle_n bae_o wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a db_o be_v equal_a to_o the_o base_a be._n and_o forasmuch_o as_o these_o two_o line_n db_o and_o dc_o be_v great_a than_o the_o line_n bc_o of_o which_o the_o line_n db_o be_v prove_v to_o be_v equal_a to_o the_o line_n be._n wherefore_o the_o residue_n namely_o the_o line_n dc_o be_v great_a than_o the_o residue_n namely_o than_o the_o line_n ec_o and_o forasmuch_o as_o the_o line_n dam_n be_v equal_a to_o the_o line_n ae_n and_o the_o line_n ac_fw-la be_v common_a to_o they_o both_o and_o the_o base_a dc_o be_v great_a than_o the_o base_a ec_o therefore_o the_o angle_n dac_o be_v great_a than_o the_o angle_n eac_n and_o it_o be_v prove_v that_o the_o angle_n dab_n be_v equal_a to_o the_o angle_n bae_o wherefore_o the_o angle_n dab_n and_o dac_o be_v great_a than_o the_o angle_n bac_n if_o therefore_o a_o solid_a angle_n be_v contain_v under_o three_o plain_a superficial_a angle_n every_o two_o of_o those_o three_o angle_n which_o then_o so_o ever_o be_v take_v be_v great_a than_o the_o three_o which_o be_v require_v to_o be_v prove_v in_o this_o figure_n you_o may_v plain_o behold_v the_o former_a demonstration_n if_o you_o elevate_v the_o three_o triangle_n abdella_n aâ—c_n and_o acd_o in_o such_o â—orâ—that_o they_o may_v all_o meet_v together_o in_o the_o point_n a._n the_o 19_o theorem_a the_o 21._o proposition_n every_o solid_a angle_n be_v comprehend_v under_o plain_a angle_n less_o than_o four_o right_a angle_n svppose_v that_o a_o be_v a_o solid_a angle_n contain_v under_o these_o superficial_a angle_n bac_n dac_o and_o dab_n then_o i_o say_v that_o the_o angle_n bac_n dac_o and_o dab_n be_v less_o than_o four_o right_a angle_n construction_n take_v in_o every_o one_o of_o these_o right_a line_n acab_n and_o ad_fw-la a_o point_n at_o all_o adventure_n and_o let_v the_o same_o be_v b_o c_o d._n and_o draw_v these_o right_a line_n bc_o cd_o and_o db._n demonstration_n and_o forasmuch_o as_o the_o angle_n b_o be_v a_o solid_a angle_n for_o it_o be_v contain_v under_o three_o superficial_a angle_n that_o be_v under_o cba_o abdella_n and_o cbd_n therefore_o by_o the_o 20._o of_o the_o eleven_o two_o of_o they_o which_o two_z so_o ever_o be_v take_v be_v great_a than_o the_o three_o wherefore_o the_o angle_n cba_o and_o abdella_n be_v great_a than_o the_o angle_n cbd_v and_o by_o the_o same_o reason_n the_o angle_n bca_o and_o acd_o be_v great_a than_o the_o angle_n bcdâ—_n and_o moreover_o the_o angle_n cda_n and_o adb_o be_v great_a than_o the_o angle_n cdb_n wherefore_o these_o six_o angle_n cba_o abdella_n bca_o acd_o cda_n and_o adb_o be_v great_a they_o these_o three_o angle_n namely_o cbd_v bcd_o &_o cdb_n but_o the_o three_o angle_n cbd_v bdc_o and_o bcd_o be_v equal_a to_o two_o right_a angle_n wherefore_o the_o six_o angle_n cba_o abdella_n bca_o acd_o cda_n and_o adb_o be_v great_a they_o two_o right_a angle_n and_o forasmuch_o as_o in_o every_o one_o of_o these_o triangle_n abc_n and_o abdella_n and_o acd_o three_o angle_n be_v equal_a two_o right_a angle_n by_o the_o 32._o of_o the_o first_o wherefore_o the_o nine_o angle_n of_o the_o three_o triangle_n that_o be_v the_o angle_n cba_o acb_o bac_n acd_o dac_o cda_n adb_o dba_n and_o bad_a be_v equal_a to_o six_o right_a angle_n of_o which_o angle_n the_o six_o angle_n abc_n bca_o acd_o cda_n adb_o and_o dba_n be_v great_a than_o two_o right_a angle_n wherefore_o the_o angle_n remain_v namely_o the_o angle_n bac_n god_n and_o dab_n which_o contain_v the_o solid_a angle_n be_v less_o than_o sour_a right_a angle_n wherefore_o every_o solid_a angle_n be_v comprehend_v under_o plain_a angle_n less_o than_o four_o right_a angle_n which_o be_v require_v to_o be_v prove_v if_o you_o will_v more_o full_o see_v this_o demonstration_n compare_v it_o with_o the_o figure_n which_o i_o put_v for_o the_o better_a sight_n of_o the_o demonstration_n of_o the_o proposition_n next_o go_v before_o only_o here_o be_v not_o require_v the_o draught_n of_o the_o line_n ae_n although_o this_o demonstration_n of_o euclid_n be_v here_o put_v for_o solid_a angle_n contain_v under_o three_o superficial_a angle_n yet_o after_o the_o like_a manner_n may_v you_o proceed_v if_o the_o solid_a angle_n be_v contain_v under_o superficial_a angle_n how_o many_o so_o ever_o as_o for_o example_n if_o it_o be_v contain_v under_o four_o superficial_a angle_n if_o you_o follow_v the_o former_a construction_n the_o base_a will_v be_v a_o quadrangled_a figure_n who_o four_o angle_n be_v equal_a to_o four_o right_a angle_n but_o the_o 8._o angle_n at_o the_o base_n of_o the_o 4._o triangle_n set_v upon_o this_o quadrangled_a figure_n may_v by_o the_o 20._o proposition_n of_o this_o book_n be_v prove_v to_o be_v great_a than_o those_o 4._o angle_n of_o the_o quadrangled_a figure_n as_o we_o see_v by_o the_o discourse_n of_o the_o former_a demonstration_n wherefore_o those_o 8._o angle_n be_v great_a than_o four_o right_a angle_n but_o the_o 12._o angle_n of_o those_o four_o triangle_n be_v equal_a to_o 8._o right_a angle_n wherefore_o the_o four_o angle_n remain_v at_o the_o top_n which_o make_v the_o solid_a angle_n be_v less_o than_o four_o right_a angle_n and_o observe_v this_o course_n you_o may_v proceed_v infinite_o ¶_o the_o 20._o theorem_a the_o 22._o proposition_n if_o there_o be_v three_o superficial_a plain_a angle_n of_o which_o two_o how_o soever_o they_o be_v take_v be_v great_a than_o the_o three_o and_o if_o the_o right_a line_n also_o which_o contain_v those_o angle_n be_v equal_a then_o of_o the_o line_n couple_v those_o equal_a right_a line_n together_o it_o be_v possible_a to_o make_v a_o triangle_n svppose_v that_o there_o be_v three_o superficial_a angle_n abc_n def_n and_o ghk_v of_o which_o let_v two_o which_o two_o soever_o be_v take_v be_v great_a than_o the_o three_o that_o be_v let_v the_o angle_n abc_n and_o def_n be_v great_a than_o the_o angle_n ghk_o and_o let_v the_o angle_n def_n and_o ghk_o be_v great_a than_o the_o angle_n abc_n and_o moreover_o let_v the_o angle_n ghk_v and_o abc_n be_v great_a than_o the_o angle_n def_n and_o let_v the_o right_a line_n ab_fw-la bc_o de_fw-fr of_o gh_o and_o hk_o be_v equal_a the_o one_o to_o the_o other_o and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n c_o and_o a_o other_o from_o the_o point_n d_o to_o the_o point_n fletcher_n and_o moreover_o a_o other_o from_o the_o point_n g_o to_o the_o point_n k._n proposition_n then_o i_o say_v that_o it_o be_v possible_a of_o three_o right_a line_n equal_a to_o the_o line_n ac_fw-la df_o and_o gk_o to_o make_v a_o triangle_n that_o be_v that_o two_o of_o the_o right_a line_n ac_fw-la df_o and_o gk_o which_o then_o soever_o be_v take_v be_v great_a than_o the_o three_o now_o if_o the_o angle_n abc_n def_n case_n and_o ghk_v be_v equal_a the_o one_o to_o the_o other_o it_o be_v manifest_a that_o these_o right_a line_n ac_fw-la df_o and_o gk_o be_v also_o by_o the_o 4._o of_o the_o first_o equal_a the_o one_o to_o the_o other_o it_o be_v possible_a of_o three_o right_a line_n equal_a to_o the_o line_n ac_fw-la df_o and_o gk_o to_o make_v a_o triangle_n case_n but_o if_o they_o be_v not_o equal_a let_v they_o be_v unequal_a and_o by_o the_o 23._o of_o the_o first_o unto_o the_o right_a line_n hk_o construction_n and_o at_o the_o point_n in_o it_o h_o make_v unto_o the_o angle_n abc_n a_o equal_a angle_n khl._n and_o by_o the_o â—_o of_o the_o first_o to_o one_o of_o the_o line_n
spherâ—_n contain_v the_o dodecahedron_n of_o this_o pentagon_n and_o the_o icosahedron_n of_o this_o triangle_n by_o the_o 4._o of_o this_o book_n â—_o and_o the_o line_n cl_n fall_v perpendicular_o upon_o the_o side_n of_o the_o icosahedron_n and_o the_o line_n ci_o upon_o the_o side_n of_o the_o dodecahedron_n that_o which_o be_v 30._o time_n contain_v under_o the_o side_n and_o the_o perpendicular_a line_n fall_v upon_o it_o be_v equal_a to_o the_o superficies_n of_o that_o solid_a upon_o who_o side_n the_o perpendicular_a fall_v if_o therefore_o in_o a_o circle_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n order_n the_o superficiece_n of_o a_o dodecahedron_n and_o of_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n be_v the_o one_o to_o the_o other_o as_o that_o which_o be_v contain_v under_o the_o side_n of_o the_o one_o and_o the_o perpendicular_a line_n draw_v unto_o it_o from_o the_o centre_n of_o his_o base_a to_o that_o which_o be_v contain_v under_o the_o side_n of_o the_o other_o and_o the_o perpendicular_a line_n draw_v to_o it_o from_o the_o centre_n of_o his_o base_a for_o aâ—_z thirtyâ—_n timâ—s_o be_v to_o thirty_o time_n so_o be_v once_o to_o once_o by_o the_o 15._o of_o thâ—_z five_o the_o 6._o proposition_n the_o superficies_n of_o a_o dodecahedron_n be_v to_o the_o superficies_n of_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n campane_n in_o that_o proportion_n that_o the_o side_n of_o the_o cube_n be_v to_o the_o side_n of_o the_o icosahedron_n contain_v in_o the_o self_n same_o sphere_n construction_n svppose_v that_o there_o be_v a_o circle_n abg_o &_o in_o it_o by_o the_o 4._o of_o this_o book_n let_v there_o be_v inscribe_v the_o sideâ—_n of_o a_o dodecahedron_n and_o of_o a_o icosahedron_n contain_v in_o onâ—_n and_o the_o self_n same_o sphere_n and_o let_v the_o side_n oâ—_n the_o dodecahedron_n be_v agnostus_n and_o the_o side_n of_o the_o icosahedron_n be_v dg_o and_o let_v the_o centre_n be_v the_o point_n e_o from_o which_o draw_v unto_o those_o sâ—des_n perpendicular_a line_n ei_o and_o ez_o and_o produce_v the_o line_n ei_o to_o the_o point_n b_o and_o draw_v the_o linâ—_n bg_o and_o let_v the_o side_n of_o the_o cube_fw-la contain_v in_o the_o self_n same_o sphere_n be_v gc_o then_o i_o say_v that_o the_o superficies_n of_o the_o dodecahedron_n iâ—_z to_o the_o superficies_n of_o the_o icosahedron_n as_o the_o line_n â—g_z iâ—_z to_o the_o liâ—â—_n gd_o for_o forasmuch_o as_o the_o line_n ei_o beinâ—_n divide_v by_o a_o extreme_a and_o mean_a proportion_n the_o great_a segment_n thereof_o shall_v be_v the_o linâ—_n ez_o by_o the_o corollary_n of_o the_o first_o of_o this_o book_n demonstration_n and_o the_o line_n cg_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n his_o great_a segment_n be_v the_o line_n agnostus_n by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o wherefore_o the_o right_a line_n ei_o and_o cg_o â—râ—_n cut_v proportional_o by_o the_o second_o of_o this_o bâ—oke_n whâ—râ—fore_o as_o the_o line_n cg_o be_v to_o the_o line_n agnostus_n so_o be_v the_o line_n ei_o to_o the_o line_n ez_fw-fr wherâ—fore_o that_o which_o it_o contain_v under_o the_o extreme_n cg_o and_o ez_o be_v squall_n to_o that_o which_o iâ—_z contaynâ—d_v under_o the_o mean_n agnostus_n and_o ei._n by_o the_o 16._o of_o the_o six_o but_o as_o that_o which_o iâ—_z contain_v under_o the_o linâ—â—_n cg_o and_o â—z_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n dg_o and_o ez_o so_o by_o the_o first_o of_o the_o six_o iâ—_z the_o linâ—_n cg_o to_o the_o line_n dg_o for_o both_o those_o parallelogram_n have_v oâ—â—_n and_o the_o self_n same_o altitude_n namely_o the_o line_n ez_fw-fr wherefore_o as_o that_o which_o be_v contain_v under_o the_o line_n ei_o and_o agnostus_n which_o iâ—_z prove_v equal_a to_o that_o which_o be_v contain_v under_o the_o lineâ—_z cg_o and_o ez_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n dg_o and_o ez_o so_o be_v the_o line_n cg_o to_o the_o liâ—â—_n dg_o but_o as_o that_o which_o be_v contain_v under_o the_o line_n ei_o and_o agnostus_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n dg_o and_o ez_o so_o by_o the_o corollary_n of_o the_o former_a proposition_n be_v the_o superficies_n of_o the_o dodecahedron_n to_o the_o superficies_n of_o the_o icosahedron_n wherefore_o as_o the_o superficies_n â—â—_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v cg_o the_o side_n of_o the_o cube_fw-la to_o gd_v the_o side_n of_o the_o icosahedron_n the_o superficies_n therefore_o of_o a_o dodecahedron_n be_v to_o the_o superficiesâ—_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v require_v to_o be_v prove_v a_o assumpt_n the_o pentagon_n of_o a_o dodecahedron_n order_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o perpendicular_a line_n which_o fall_v upon_o the_o base_a of_o the_o triangle_n of_o the_o icosahedron_n and_o five_o six_o part_n of_o the_o side_n of_o the_o cube_fw-la the_o say_v three_o solid_n be_v describe_v in_o one_o and_o the_o self_n same_o sphere_n suppose_v that_o in_o the_o circle_n abeg_n the_o pentagon_n of_o a_o dodecahedron_n be_v aâ—cig_a and_o let_v two_o side_n thereof_o ab_fw-la and_o agnostus_n be_v subtend_v of_o the_o right_a line_n bg_o and_o let_v the_o triangle_n of_o the_o icosahedron_n inscribe_v in_o the_o self_n same_o sphere_n construction_n by_o the_o 4._o of_o this_o book_n be_v afh_n and_o let_v the_o centre_n of_o the_o circle_n be_v the_o point_n d_o and_o let_v the_o diameter_n be_v ade_n cut_v fh_o the_o side_n of_o the_o triangle_n in_o the_o point_n z_o and_o cut_v the_o line_n bg_o in_o the_o point_n k._n and_o draw_v the_o right_a line_n bd._n and_o from_o the_o right_a line_n kg_v cut_v of_o a_o three_o part_n tg_n by_o the_o 9_o of_o the_o six_o now_o than_o the_o line_n bg_o subtend_v two_o side_n of_o the_o dodecahedron_n shall_v be_v the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o same_o sphere_n demonstration_n by_o the_o 17._o of_o the_o thirteen_o and_o the_o triangle_n of_o the_o icosahedron_n of_o the_o same_o sphere_n shall_v be_v aâ—h_a by_o the_o 4._o of_o this_o book_n and_o the_o line_n az_o which_o pass_v by_o the_o centre_n d_o shall_v fall_v perpendicular_o upon_o the_o side_n of_o the_o triangle_n for_o forasmuch_o as_o the_o angle_n gay_a &_o bae_o be_v equal_a by_o the_o 27._o of_o the_o thirdâ—_o for_o they_o be_v see_v upon_o equal_a circumference_n therefore_o the_o â—ases_v bk_o and_o kg_n be_v by_o the_o â—_o of_o the_o first_o equal_a wherefore_o the_o line_n bt_n contain_v 5._o six_o part_n of_o the_o line_n bg_o then_o i_o say_v that_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o bt_n be_v equal_a to_o the_o pentagon_n aâ—câ—g_n for_o forasmuch_o as_o the_o line_n â—z_n be_v sesquialter_n to_o the_o line_n ad_fw-la for_o the_o line_n dâ—_n be_v divide_v into_o two_o equal_a part_n in_o the_o point_n z_o by_o the_o corollary_n of_o the_o â—2â—_n of_o the_o thirteen_o likewise_o by_o construction_n the_o line_n kg_n be_v sesquialter_fw-la to_o the_o line_n kt_n therefore_o as_o the_o line_n az_o be_v to_o the_o line_n ad_fw-la so_o be_v the_o line_n kg_v to_o the_o 〈◊〉_d â—t_a wherefore_o that_o which_o be_v contain_v undeâ—_n the_o 〈◊〉_d az_o and_o kt_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o mean_n ad_fw-la and_o kg_n by_o the_o 16._o of_o the_o six_o but_o unto_o the_o line_n kg_n be_v the_o line_n â—k_v â—roved_v equal_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o bk_o but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o bk_o be_v by_o the_o 41._o of_o the_o first_o double_a to_o the_o triangle_n abdella_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n be_v double_a to_o the_o same_o triangle_n abdella_n and_o forasmuch_o as_o the_o pentagon_n abcig_n contaynethâ—_n 〈…〉_o equal_a â—o_o the_o triangle_n abdella_n and_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n contain_v two_o such_o triangle_n therefore_o the_o pentagon_n abcig_n be_v duple_fw-fr sesquialter_fw-la to_o the_o rectangle_n parallelogram_n contain_v under_o the_o line_n az_o and_o kt_n and_o 〈…〉_o 1._o of_o the_o six_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o bt_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n as_o the_o base_a bt_n be_v to_o the_o base_a â—â—tâ—_n therefore_o that_o which_o be_v contain_v under_o the_o line_n az_o
the_o whole_a line_n mg_o to_o the_o whole_a line_n ea_fw-la by_o the_o 18._o of_o the_o five_o wherefore_o as_o mg_o the_o side_n of_o the_o cube_fw-la be_v to_o ea_fw-la the_o semidiameter_n so_o be_v the_o line_n fghim_n to_o the_o octohedron_n abkdlc_n inscribe_v in_o one_o &_o the_o self_n same_o sphere_n if_o therefore_o a_o cube_fw-la and_o a_o octohedron_n be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n they_o shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o semidiameter_n of_o the_o sphere_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n distinct_o to_o notify_v the_o power_n of_o the_o side_n of_o the_o five_o solid_n by_o the_o power_n of_o the_o diameter_n of_o the_o sphere_n the_o side_n of_o the_o tetrahedron_n and_o of_o the_o cube_fw-la do_v cut_v the_o power_n of_o the_o diameter_n of_o the_o sphere_n into_o two_o square_n which_o be_v in_o proportion_n double_a the_o one_o to_o the_o other_o the_o octohedron_n cut_v the_o power_n of_o the_o diameter_n into_o two_o equal_a square_n the_o icosahedron_n into_o two_o square_n who_o proportion_n be_v duple_n to_o the_o proportion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n who_o less_o segment_n be_v the_o side_n of_o the_o icosahedron_n and_o the_o dodecahedron_n into_o two_o square_n who_o proportion_n be_v quadruple_a to_o the_o proportion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n who_o less_o segment_n be_v the_o side_n of_o the_o dodecahedron_n for_o ad_fw-la the_o diameter_n of_o the_o sphere_n contain_v in_o power_n ab_fw-la the_o side_n of_o the_o tetrahedron_n and_o bd_o the_o side_n of_o the_o cube_fw-la which_o bd_o be_v in_o power_n half_a of_o the_o side_n ab_fw-la the_o diameter_n also_o of_o the_o sphere_n contain_v in_o power_n ac_fw-la and_o cd_o two_o equal_a side_n of_o the_o octohedron_n but_o the_o diameter_n contain_v in_o power_n the_o whole_a line_n ae_n and_o the_o great_a segment_n thereof_o ed_z which_o be_v the_o side_n of_o the_o icosahedron_n by_o the_o 15._o of_o this_o book_n wherefore_o their_o power_n be_v in_o duple_n proportion_n of_o that_o in_o which_o the_o side_n be_v by_o the_o first_o corollary_n of_o the_o 20._o of_o the_o six_o have_v their_o proportion_n duple_n to_o the_o proportion_n of_o a_o extreme_a &_o mean_a proportion_n far_o the_o diameter_n contain_v in_o power_n the_o whole_a line_n of_o and_o his_o less_o segment_n fd_o which_o be_v the_o side_n of_o the_o dodecahedron_n by_o the_o same_o 15._o of_o this_o book_n wherefore_o the_o whole_a have_v to_o the_o less_o â—_o double_a proportion_n of_o that_o which_o the_o extreme_a have_v to_o the_o mean_a namely_o of_o the_o whole_a to_o the_o great_a segment_n by_o the_o 10._o definition_n of_o the_o five_o it_o follow_v that_o the_o proportion_n of_o the_o power_n be_v double_a to_o the_o double_a proportion_n of_o the_o side_n by_o the_o same_o first_o corollary_n of_o the_o 20._o of_o the_o six_o that_o be_v be_v quadruple_a to_o the_o proportion_n of_o the_o extreme_a and_o of_o the_o mean_a by_o the_o definition_n of_o the_o six_o a_o advertisement_n add_v by_o flussas_n by_o this_o mean_n therefore_o the_o diameter_n of_o a_o sphere_n be_v give_v there_o shall_v be_v give_v the_o side_n of_o every_o one_o of_o the_o body_n inscribe_v and_o forasmuch_o as_o three_o of_o those_o body_n have_v their_o side_n commensurable_a in_o power_n only_o and_o not_o in_o length_n unto_o the_o diameter_n give_v for_o their_o power_n be_v in_o the_o proportion_n of_o a_o square_a number_n to_o a_o number_n not_o square_a wherefore_o they_o have_v not_o the_o proportion_n of_o a_o square_a number_n to_o a_o square_a number_n by_o the_o corollary_n of_o the_o 25._o of_o the_o eight_o wherefore_o also_o their_o side_n be_v incommensurabe_n in_o length_n by_o the_o 9_o of_o the_o ten_o therefore_o it_o be_v sufficient_a to_o compare_v the_o power_n and_o not_o the_o length_n of_o those_o side_n the_o one_o to_o the_o otherâ—_n which_o power_n be_v contain_v in_o the_o power_n of_o the_o diameter_n namely_o from_o the_o power_n of_o the_o diameter_n let_v there_o ble_v take_v away_o the_o power_n of_o the_o cube_fw-la and_o there_o shall_v remain_v the_o power_n of_o the_o tetrahedron_n and_o take_v away_o the_o power_n of_o the_o tetrahedron_n there_o remain_v the_o power_n of_o the_o cube_fw-la and_o take_v away_o from_o the_o power_n of_o the_o diameter_n half_o the_o power_n thereof_o there_o shall_v be_v leave_v the_o power_n of_o the_o side_n of_o the_o octohedron_n but_o forasmuch_o as_o the_o side_n of_o the_o dodecahedron_n and_o of_o the_o icosahedron_n be_v prove_v to_o be_v irrational_a for_o the_o side_n of_o the_o icosahedron_n be_v a_o less_o line_n by_o the_o 16._o of_o the_o thirteen_o and_o the_o side_n of_o the_o dedocahedron_n be_v a_o residual_a line_n by_o the_o 17._o of_o the_o same_o therefore_o those_o side_n be_v unto_o the_o diameter_n which_o be_v a_o rational_a line_n set_v incommensurable_a both_o in_o length_n and_o in_o power_n wherefore_o their_o comparison_n can_v not_o be_v define_v or_o describe_v by_o any_o proportion_n express_v by_o number_n by_o the_o 8._o of_o the_o ten_o neither_o can_v they_o be_v compare_v the_o one_o to_o the_o other_o for_o irrational_a line_n of_o diverse_a kind_n be_v incommensurable_a the_o one_o to_o the_o other_o for_o if_o they_o shall_v be_v commensurable_a they_o shall_v be_v of_o one_o and_o the_o self_n same_o kind_n by_o the_o 103._o and_o 105._o of_o the_o ten_o which_o be_v impossible_a wherefore_o we_o seek_v to_o compare_v they_o to_o the_o power_n of_o the_o diameter_n think_v they_o can_v not_o be_v more_o apt_o express_v then_o by_o such_o proportion_n which_o cut_v that_o rational_a power_n of_o the_o diameter_n according_a to_o their_o side_n namely_o divide_v the_o power_n of_o the_o diameter_n by_o line_n which_o have_v that_o proportion_n that_o the_o great_a segment_n have_v to_o the_o less_o to_o put_v the_o less_o segment_n to_o be_v the_o side_n of_o the_o icosahedron_n &_o devide_v the_o say_a power_n of_o the_o diameter_n by_o line_n have_v the_o proportion_n of_o the_o whole_a to_o the_o less_o segment_n to_o express_v the_o side_n of_o the_o dodecahedron_n by_o the_o less_o segment_n which_o thing_n may_v well_o be_v do_v between_o magnitude_n incommensurable_a the_o end_n of_o the_o fourteen_o book_n of_o euclides_n element_n after_o flussas_n ¶_o the_o fifteen_o book_n of_o euclides_n element_n this_o finetenth_n and_o last_o book_n of_o euclid_n or_o rather_o the_o second_o book_n of_o appollonius_n or_o hypsicles_n book_n teach_v the_o inscription_n and_o circumscription_n of_o the_o five_o regular_a body_n one_o within_o and_o about_o a_o other_o a_o thing_n undouted_o pleasant_a and_o delectable_a in_o mind_n to_o contemplate_v and_o also_o profitable_a and_o necessary_a in_o act_n to_o practice_v for_o without_o practice_n in_o act_n it_o be_v very_o hard_a to_o see_v and_o conceive_v the_o construction_n and_o demonstration_n of_o the_o proposition_n of_o this_o book_n unless_o a_o man_n have_v a_o very_a deep_a sharp_a &_o fine_a imagination_n wherefore_o i_o will_v wish_v the_o diligent_a student_n in_o this_o book_n to_o make_v the_o study_n thereof_o more_o pleasant_a unto_o he_o to_o have_v present_o before_o his_o eye_n the_o body_n form_v &_o frame_v of_o past_v paper_n as_o i_o teach_v after_o the_o definition_n of_o the_o eleven_o book_n and_o then_o to_o draw_v and_o describe_v the_o line_n and_o division_n and_o superficiece_n according_a to_o the_o construction_n of_o the_o proposition_n in_o which_o description_n if_o he_o be_v wary_a and_o diligent_a he_o shall_v find_v all_o thing_n in_o these_o solid_a matter_n as_o clear_v and_o as_o manifest_v unto_o the_o eye_n as_o be_v thing_n before_o teach_v only_o in_o plain_a or_o superficial_a figure_n and_o although_o i_o have_v before_o in_o the_o twelve_o book_n admonish_v the_o reader_n hereof_o yet_o because_o in_o this_o book_n chief_o that_o thing_n be_v require_v i_o think_v it_o shall_v not_o be_v irksome_a unto_o he_o again_o to_o be_v put_v in_o mind_n thereof_o far_o this_o be_v to_o be_v note_v that_o in_o the_o greek_a exemplar_n be_v find_v in_o this_o 15._o book_n only_o 5._o proposition_n which_o 5._o be_v also_o only_o touch_v and_o set_v forth_o by_o hypsicy_n unto_o which_o campane_n add_v 8._o and_o so_o make_v up_o the_o number_n of_o 13._o campane_n undoubted_o although_o he_o be_v very_o well_o learn_v and_o that_o general_o in_o all_o kind_n of_o learning_n yet_o assure_o be_v bring_v up_o in_o a_o time_n of_o rudeness_n when_o all_o good_a letter_n be_v darken_v &_o barberousnes_n have_v
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
demonstration_n demonstration_n the_o circle_n so_o make_v or_o so_o consider_v in_o the_o sphere_n be_v call_v the_o great_a circle_n all_o other_o not_o have_v the_o centre_n of_o the_o sphere_n to_o be_v their_o centre_n alsoâ—_n be_v call_v less_o circle_n note_v these_o description_n description_n an_o other_o corollary_n corollary_n an_o other_o corollary_n construction_n construction_n this_o be_v also_o prove_v in_o the_o assumpt_n before_o add_v out_o oâ—_n flussas_n note_v what_o a_o great_a or_o great_a circle_n in_o a_o spear_n be_v first_o part_n of_o the_o construction_n noteâ—_n noteâ—_n you_o know_v full_a well_o that_o in_o the_o superficies_n of_o the_o sphere_n â—â—â—ly_o the_o circumference_n of_o the_o circle_n be_v but_o by_o thâ—se_a circumference_n the_o limitation_n and_o assign_v of_o circle_n be_v use_v and_o so_o the_o circumference_n of_o a_o circle_n usual_o call_v a_o circle_n which_o in_o this_o place_n can_v not_o offend_v this_o figure_n be_v restore_v by_o m._n dee_n his_o diligence_n for_o in_o the_o greek_a and_o latin_a euclides_n the_o line_n gl_n the_o line_n agnostus_n and_o the_o line_n kz_o in_o which_o three_o line_n the_o chief_a pinch_n of_o both_o the_o demonstration_n do_v stand_v be_v untrue_o draw_v as_o by_o compare_v the_o studious_a may_v perceive_v note_n you_o must_v imagine_v 〈◊〉_d right_a line_n axe_n to_o be_v perpendicular_a upon_o the_o diameter_n bd_o and_o ce_fw-fr though_o here_o ac_fw-la the_o semidiater_n seem_v to_o be_v part_n of_o ax._n and_o so_o in_o other_o point_n in_o this_o figure_n and_o many_o other_o strengthen_v your_o imagination_n according_a to_o the_o tenor_n of_o construction_n though_o in_o the_o delineation_n in_o plain_a sense_n be_v not_o satisfy_v note_n boy_n equal_a to_o bk_o in_o respect_n of_o m._n dee_n his_o demonstration_n follow_v follow_v note_v â—his_fw-la point_n z_o that_o you_o may_v the_o better_a understand_v m._n dee_n his_o demonstration_n second_o part_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n demonstration_n which_o of_o necessity_n shall_v fall_v upon_o z_o as_o m._n dee_n prove_v it_o and_o his_o proof_n be_v set_v after_o at_o this_o mark_n ✚_o follow_v i_o dee_n dee_n but_o az_o be_v great_a them_z agnostus_n as_o in_o the_o former_a proposition_n km_o be_v evident_a to_o be_v great_a than_o kg_v so_o may_v it_o also_o be_v make_v manifest_a that_o kz_o do_v neither_o touch_n nor_o cut_v the_o circle_n fgâ—h_n a_o other_o prove_v that_o the_o line_n ay_o be_v great_a they_o the_o line_n ag._n ag._n this_o as_o a_o assumpt_n be_v present_o prove_v two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o impossibility_n second_o case_n case_n as_o it_o be_v â—asiâ—_n to_o gather_v by_o the_o assumpt_n put_v after_o the_o secoâ—â—_n of_o this_o booâ—â—_n note_v a_o general_a rule_n the_o second_o part_n of_o the_o problem_n two_o way_n execute_v a_o upright_a cone_n the_o second_o part_n of_o the_o problem_n the_o second_o â—aâ—â—_n oâ—_n the_o problem_n ☜_o ☜_o this_o may_v easy_o be_v demonstrate_v as_o in_o thâ—_z 17._o proposition_n the_o section_n of_o a_o sphere_n be_v prove_v to_o be_v a_o circle_n circle_n for_o take_v away_o all_o doubt_n this_o aâ—_z a_o lemma_n afterward_o be_v demonstrate_v a_o lemma_n as_o it_o be_v present_o demonstrate_v construction_n demonstration_n the_o second_o part_n of_o the_o problem_n *_o construction_n demonstration_n an_o other_o way_n of_o execute_v this_o problem_n the_o converse_n of_o the_o assumpt_n a_o great_a error_n common_o maintain_v between_o straight_o and_o crooked_a all_o manner_n of_o proportion_n may_v be_v give_v construction_n demonstration_n the_o definition_n of_o a_o circled_a â—â—apâ—â—d_n in_o a_o spâ—erâ—_n construction_n demonstration_n this_o be_v manifest_a if_o you_o consider_v the_o two_o triangle_n rectangle_v hom_o and_o hon_n and_o then_o with_o all_o use_v the_o 47._o of_o the_o first_o of_o euclid_n construction_n demonstration_n construction_n demonstration_n this_o in_o manner_n of_o a_o lemmâ—_n be_v present_o prove_v note_v here_o of_o axe_n base_a &_o solidity_n more_o than_o i_o need_n to_o bring_v any_o far_a proof_n for_o note_n note_n i_o say_v half_a a_o circular_a revolution_n for_o that_o suffice_v in_o the_o whole_a diameter_n n-ab_n to_o describe_v a_o circle_n by_o iâ—_z it_o be_v move_v â—â—out_v his_o centre_n q_o etc_n etc_n lib_fw-la 2_o prop_n 2._o de_fw-fr spheâ—a_n &_o cylindrâ—_n note_n note_n a_o rectangle_n parallelipipedon_n give_v equal_a to_o a_o sphere_n give_v to_o a_o sphere_n or_o to_o any_o part_n of_o a_o sphere_n assign_v as_o a_o three_o four_o five_o etc_o etc_o to_o geve_v a_o parallelipipedon_n equal_a side_v colume_n pyramid_n and_o prism_n to_o be_v give_v equal_a to_o a_o sphere_n or_o to_o any_o certain_a part_n thereof_o to_o a_o sphere_n or_o any_o segment_n or_o sector_n of_o the_o same_o to_o geve_v a_o cone_n or_o cylinder_n equal_a or_o in_o any_o proportion_n assign_v far_o use_v of_o spherical_a geometry_n the_o argument_n of_o the_o thirteen_o book_n construction_n demonstration_n demonstration_n the_o assumpt_v prove_v prove_v because_o ac_fw-la be_v suppose_v great_a than_o ad_fw-la therefore_o his_o residue_n be_v less_o than_o the_o residue_n of_o ad_fw-la by_o the_o common_a sentence_n wherefore_o by_o the_o supposition_n db_o be_v great_a than_o â—c_z the_o chieâ—e_a line_n in_o all_o euclides_n geometry_n what_o be_v mean_v here_o by_o a_o section_n in_o one_o only_a poiâ—t_n construction_n demonstration_n demonstration_n note_v how_o ce_fw-fr and_o the_o gnonom_n xop_n be_v prove_v equal_a for_o it_o serve_v in_o the_o converse_n demonstrate_v by_o m._n dee_n here_o next_o after_o this_o proposition_n â—the_v converse_v of_o the_o former_a former_a as_o we_o haâ—e_z note_v the_o place_n of_o the_o peculiar_a proâ—e_n there_o â—in_a the_o demonstration_n of_o the_o 3._o 3._o therefore_o by_o my_o second_o theorem_a add_v upon_o the_o second_o proposition_n dc_o be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n a._n and_o because_o ac_fw-la be_v big_a than_o cb_o therefore_o dam_n be_v great_a than_o ac_fw-la wherefore_o if_o a_o right_a line_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v demonstrate_v therefore_o by_o my_o second_o theorem_a add_v upon_o the_o second_o proposition_n dc_o be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n a._n and_o because_o ac_fw-la be_v big_a than_o cb_o therefore_o dam_n be_v great_a than_o ac_fw-la wherefore_o if_o a_o right_a line_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v construction_n construction_n though_o i_o say_v perpendicular_a yes_o you_o may_v perceive_v how_o infinite_a other_o position_n will_v serve_v so_o that_o diego_n and_o ad_fw-la make_v a_o angle_n for_o a_o triangle_n to_o have_v his_o side_n proportional_o cut_v etc_n etc_n demonstration_n demonstration_n i_o dee_n this_o be_v most_o evident_a of_o my_o second_o theorem_a add_v to_o the_o three_o proposition_n for_o to_o add_v to_o a_o whole_a line_n a_o line_n equal_a to_o the_o great_a segment_n &_o to_o add_v to_o the_o great_a segment_n a_o line_n equal_a to_o the_o whole_a line_n be_v all_o one_o thing_n in_o the_o line_n produce_v by_o the_o whole_a line_n i_o mean_v the_o line_n divide_v by_o extreme_a and_o mean_a proportion_n this_o be_v before_o demonstrate_v most_o evident_o and_o brief_o by_o m._n dee_n after_o the_o 3._o proposition_n note_n note_v 4._o proportional_a line_n note_v two_o middle_a proportional_n note_v 4._o way_n of_o progression_n in_o the_o proportion_n of_o a_o line_n divide_v by_o extreme_a and_o middle_a proportion_n what_o resolution_n and_o composition_n be_v have_v before_o be_v teach_v in_o the_o begin_n of_o the_o first_o book_n book_n proclus_n in_o the_o greek_a in_o the_o 58._o page_n construction_n demonstration_n two_o case_n in_o this_o proposition_n construction_n thâ—_z first_o case_n demonstration_n the_o second_o case_n construction_n demonstration_n construction_n demonstration_n this_o corollary_n be_v the_o 3._o proposition_n of_o theâ—4_n â—4_o book_n after_o campane_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n construction_n demonstration_n construstion_n demonstration_n constrâ—yction_n demonstration_n this_o corollary_n be_v the_o 11._o proposition_n of_o the_o 14._o book_n after_o campane_n this_o corollary_n be_v the_o 3._o corollary_n after_o the_o 17._o proposition_n of_o the_o 14_o book_n after_o campane_n campane_n by_o the_o name_n oâ—_n a_o pyramid_n both_o here_o &_o iâ—_z this_o book_n follow_v understand_v a_o tetrahedron_n an_o other_o construction_n and_o demonstration_n of_o the_o second_o part_n after_o fâ—ussas_n three_o part_n of_o the_o demonstration_n this_o corollary_n be_v the_o 15._o proposition_n of_o the_o 14._o book_n after_o campane_n this_o corollary_n campane_n put_v as_o a_o corollary_n after_o
the_o 17._o proposition_n of_o the_o 14._o book_n construction_n â—rist_n part_n of_o the_o demonstration_n demonstration_n for_o the_o 4._o angle_n at_o the_o point_n king_n be_v equal_a to_o four_o right_a angle_n by_o the_o corollary_n of_o tâ—e_z 15._o of_o the_o first_o and_o those_o 4._o angle_n be_v equal_a the_o one_o to_o the_o other_o by_o the_o â—_o of_o the_o â—irst_n and_o therefore_o each_o be_v a_o right_a angle_n second_o part_n of_o the_o demonstration_n demonstration_n for_o the_o square_n of_o the_o line_n ab_fw-la which_o be_v prove_v equal_a to_o the_o square_n of_o the_o line_n lm_o be_v double_a to_o the_o square_n of_o the_o line_n bd_o which_o be_v also_o equal_a to_o the_o square_n of_o the_o line_n le._n this_o corollary_n be_v the_o 16._o proposition_n of_o the_o 14._o book_n after_o campane_n first_o part_n of_o the_o demonstration_n second_o part_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n demonstration_n by_o the_o 2._o assumpt_v of_o the_o 13._o of_o this_o book_n three_o part_n of_o the_o demonstration_n second_o part_n of_o the_o demonstration_n for_o the_o line_n qw_o be_v equal_a to_o the_o line_n iz_n &_o the_o line_n zw_n be_v common_a to_o they_o both_o this_o part_n be_v again_o afterward_o demonstrate_v by_o flussas_n the_o pentagon_n vbwcr_n prove_v to_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n the_o pentagon_n vbwcz_n it_o prove_v equiangle_n equiangle_n look_v for_o a_o far_a construction_n after_o flussas_n at_o the_o end_n of_o the_o demonstration_n that_o the_o side_n of_o the_o dodecahedron_n be_v a_o residual_a line_n draw_v in_o the_o former_a figure_n these_o line_n cta_n ctl_n câ—d_v the_o side_n of_o a_o pyramid_n the_o side_n of_o a_o cube_fw-la the_o side_n of_o a_o dodecahedron_n comparison_n of_o the_o five_o side_n of_o the_o foresay_a body_n an_o other_o demommonstration_n to_o prove_v that_o the_o side_n of_o the_o icosahedron_n be_v great_a than_o the_o side_n of_o the_o dodecahedron_n that_o 3._o square_n of_o the_o line_n fb_o be_v great_a they_o 6._o square_n of_o the_o line_n nb._n that_o there_o can_v be_v no_o other_o solid_a beside_o these_o five_o contain_v under_o equilater_n and_o equiangle_n base_n that_o the_o angle_n of_o a_o equilater_n and_o equiangle_n pentagon_n be_v one_o right_a angle_n and_o a_o 〈◊〉_d part_n 〈◊〉_d which_o thing_n be_v also_o before_o prove_v in_o the_o 〈◊〉_d of_o the_o 32._o of_o the_o â—irst_n the_o side_n of_o the_o angle_n of_o the_o inclâ—â—â—tion_n of_o the_o 〈◊〉_d of_o the_o 〈◊〉_d be_v 〈◊〉_d rational_a the_o side_n of_o the_o angle_n of_o the_o inclination_n of_o the_o 〈◊〉_d â—f_n tâ—e_z 〈…〉_o that_o the_o plain_n of_o a_o octohedron_n be_v in_o liâ—e_z sort_n incline_v that_o the_o plain_n of_o a_o icosahedron_n be_v in_o like_a sort_n incline_v that_o the_o plain_n of_o a_o dâ—â—â—â—â—hedron_n be_v 〈◊〉_d like_o sort_n incline_v the_o side_n of_o the_o angle_n of_o the_o inclination_n of_o the_o supeâ—ficieces_n of_o the_o tetrahedron_n be_v prove_v rational_a the_o side_n of_o the_o angle_n of_o the_o inclination_n of_o the_o superficiece_n of_o the_o cube_fw-la prove_v rational_a the_o side_n of_o thâ—_z angle_n etc_n etc_n of_o the_o octohedron_n prove_v rational_a the_o side_n of_o the_o angle_n etc_n etc_n of_o the_o icosahedron_n prove_v irrational_a how_o to_o know_v whether_o the_o angle_n of_o the_o inclination_n be_v a_o right_a angle_n a_o acute_a angle_n or_o a_o oblique_a angle_n the_o argument_n of_o the_o fourteen_o book_n first_o proposition_n after_o flussas_n construction_n demonstration_n demonstration_n this_o be_v manifest_a by_o the_o 12._o proposition_n of_o the_o thirtenh_o book_n as_o campane_n well_o gather_v in_o a_o corollary_n of_o the_o same_o the_o 4._o pâ—posâ—tion_n after_o flussas_n flussas_n this_o be_v afterward_o prove_v in_o the_o 4._o proposition_n this_o assumpt_n be_v the_o 3._o proposition_n after_o flussas_n construction_n of_o the_o assumpt_n demonstration_n of_o the_o assumpt_n construction_n of_o the_o proposition_n demonstration_n of_o the_o proposition_n proposition_n thâ—_z 5._o proposition_n aâ—tâ—r_a 〈◊〉_d construction_n demonstration_n the_o 5._o proposition_n aâ—ter_a fâ—ussas_n demonstration_n demonstration_n this_o be_v the_o reason_n of_o the_o corollary_n follow_v a_o corollary_n which_o also_o flussas_n put_v as_o a_o corollary_n after_o the_o 5._o proposition_n in_o his_o order_n the_o 6._o pâ—â—positionâ—â—ter_a flussas_n construction_n demonstration_n demonstration_n this_o be_v not_o hard_a to_o prove_v by_o the_o 15._o 16._o and_o 19_o of_o the_o â—â—â—eth_v â—â—â—eth_v in_o the_o corollary_n of_o the_o 17._o of_o the_o tâ—irtenth_o tâ—irtenth_o 〈◊〉_d again_o be_v require_v the_o assumpt_n which_o be_v afterward_o prove_v in_o this_o 4_o proposition_n proposition_n but_o first_o the_o assumpt_n follow_v the_o construction_n whereof_o here_o begin_v be_v to_o be_v prove_v the_o assumpt_n which_o also_o flussas_n put_v as_o a_o assumpt_n aâ—ter_v the_o 6._o proposition_n demonstration_n of_o the_o assumpt_n construction_n pertain_v to_o the_o second_o demonstration_n of_o the_o 4._o proposition_n second_o demonstration_n oâ—_n the_o 4._o proposition_n the_o 7â—_n proposition_n after_o flussas_n construction_n demonstration_n demonstration_n here_o again_o be_v require_v the_o assumpt_n afterward_o prove_v in_o this_o 4._o proposition_n proposition_n as_o may_v by_o the_o assumpt_n afterward_o in_o this_o proposition_n be_v plain_o prove_v the_o 8._o prodition_n aâ—ter_a flussas_n flussas_n by_o the_o corollary_n add_v by_o flussas_n after_o have_v assumpt_v put_v after_o the_o 17._o proposition_n of_o the_o 12._o book_n corollary_n of_o the_o 8._o after_o flussas_n this_o assumpt_n be_v the_o 3._o proposition_n aâ—ter_v â—lussas_n and_o be_v it_o which_o 〈◊〉_d time_n have_v be_v take_v aâ—_z grant_v in_o this_o book_n and_o oâ—ce_n also_o in_o the_o last_o proposition_n of_o the_o 13._o book_n as_o we_o have_v beâ—ore_o note_v demonstration_n demonstration_n in_o the_o 4._o section_n â—f_o this_o proposition_n proposition_n in_o the_o 1._o and_o 3_o section_n of_o the_o same_o proposition_n proposition_n in_o the_o 5._o section_n of_o the_o same_o proposition_n a_o corollary_n the_o first_o proposition_n after_o campane_n construction_n demonstration_n the_o 2._o proposition_n after_o campane_n demonstration_n lead_v to_o a_o impossibility_n the_o 4._o proposition_n after_o campane_n construction_n demonstration_n this_o corollary_n campane_n also_o â—utteth_v after_o the_o 4._o proposition_n in_o his_o order_n the_o 5._o proposition_n after_o campane_n construction_n demonstration_n this_o be_v the_o 6._o and_o 7._o proposition_n after_o campane_n construction_n demonstration_n this_o corollary_n campane_n also_o add_v after_o the_o 7._o proposition_n iâ—_z his_o order_n the_o 5._o proposition_n aâ—ter_v campane_n construction_n demonstration_n this_o assumpt_a campane_n also_o have_v after_o the_o 8._o proposition_n in_o his_o order_n construction_n demonstration_n the_o 9_o proposition_n after_o campane_n construction_n demonstration_n this_o campane_n put_v aâ—_z a_o corollary_n in_o the_o 9_o proposition_n after_o his_o order_n this_o corollary_n be_v the_o 9_o proposition_n after_o campane_n the_o 12._o proposition_n after_o campane_n construction_n demonstration_n the_o 13._o proposition_n after_o campane_n the_o 14._o proposition_n after_o campane_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n the_o 17._o proposition_n after_o campane_n firâ—t_o part_n of_o the_o construction_n first_o part_n of_o the_o demonstration_n second_o parâ—_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n the_o 18._o proposition_n after_o campane_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n the_o corollary_n of_o the_o 8._o proposition_n after_o campane_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n the_o argument_n of_o the_o 15._o book_n book_n in_o this_o proposition_n as_o also_o in_o all_o the_o other_o follow_v by_o the_o name_n of_o a_o pyramid_n understand_v a_o tetrahedron_n as_o i_o have_v before_o admonish_v construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n that_o which_o here_o follow_v concern_v the_o inclination_n of_o the_o plain_n of_o the_o five_o solid_n be_v before_o teach_v â—hough_o not_o altogether_o after_o the_o same_o manner_n out_o of_o flussas_n in_o the_o latter_a â—nde_n of_o the_o 13_o book_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n first_o part_n of_o the_o construction_n first_o part_n of_o the_o demonstration_n second_o part_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n three_o part_n of_o the_o construction_n three_o part_n of_o the_o demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n produce_v in_o the_o figure_n the_o line_n tf_n to_o the_o point_n b._n construction_n demonstration_n this_o proposition_n campane_n have_v &_o be_v the_o last_o also_o in_o order_n of_o the_o 15._o book_n with_o he_o the_o argument_n of_o the_o 16._o book_n construction_n demonstration_n demonstration_n by_o a_o pyramid_n understand_v a_o tetrahedron_n throughout_o all_o this_o book_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n demonstration_n that_o be_v aâ—_z 18._o to_o 1._o demonstration_n demonstration_n that_o iâ—_z as_o 9_o to_o 2._o 2._o that_o be_v as_o 18._o to_o 2._o or_o 9_o to_o 1._o draw_v in_o the_o figure_n a_o line_n from_o b_o to_o h._n h._n what_o the_o duple_n of_o a_o extreme_a and_o mean_a proportion_n be_v construction_n demonstration_n demonstration_n constrution_n demonstration_n construction_n demonstration_n demonstration_n demonstration_n that_o be_v at_o 13._o 1_o â—_o be_v toâ—_n â—_z demonstration_n demonstration_n construction_n demonstration_n construction_n demonstration_n extend_v in_o the_o figure_n a_o line_n â—rom_o the_o point_n e_o to_o the_o point_n b._n extend_v in_o the_o figure_n a_o line_n from_o the_o point_n e_o to_o the_o point_n b._n demonstration_n construction_n demonstration_n second_o part_n of_o the_o demonstration_n icosidodecahedron_n exoctohedron_n that_o the_o exoctohedron_n be_v contain_v in_o a_o sphere_n that_o the_o exoctohedron_n be_v contain_v in_o the_o sphere_n give_v that_o the_o diaâ—â—â—ter_n of_o the_o sphere_n be_v doâ—ble_a to_o the_o side_n â—f_o the_o exoctohedron_n that_o the_o icosidodecahedron_n be_v contain_v in_o the_o sphere_n give_v give_v that_o be_v as_o 8._o 103â—_n â—_z fault_n escape_v â—cl_n â—ag_n line_n faultesâ—_n co_n 〈◊〉_d 〈◊〉_d  _o  _o  _o errata_fw-la lib._n 1._o  _o 1_o 2_o 41_o point_n b._n at_o campane_n point_n c_o aâ—_z campane_n 3_o 1_o 22_o aâ—l_a line_n draw_v all_o righâ—_n 〈…〉_o 3_o 1_o 28_o line_n draw_v to_o the_o superficies_n right_a line_n draw_v to_o the_o circumference_n 9_o 1_o 42_o liâ—es_n ab_fw-la and_o ac_fw-la line_n ab_fw-la and_o bc_o 15_o 1_o 35_o be_v equal_a be_v prove_v equal_a 20_o 2_o 28_o by_o the_o first_o by_o the_o four_o 21_o 1_o 39_o tâ—e_z centre_n c._n the_o centre_n e_o 2d_o 2_o â—_o iâ—â—ower_a right_n if_o two_o right_a 25_o 2_o 3_o fâ—â—â—_n petition_n five_o petition_n 49_o 2_o 7_o 14._o â—â—_o 32.64_o etc_n etc_n 4.8.16.32.64_o &_o 53_o 1_o 39_o the_o triangle_n ng_z the_o triangle_n kâ—_n 54_o 2_o 25_o by_o the_o 44_o by_o the_o 42_o 57_o 2_o 23_o and_o câ—g_v in_o the_o and_o cgb_n be_v thâ—_z  _o  _o  _o in_o stead_n of_o â—lussates_n through_o out_o 〈◊〉_d whole_a book_n read_v â—lusâ—as_v  _o  _o  _o errata_fw-la lib._n 2._o  _o 60_o 2_o 29_o gnomon_n fgeh_a gnomon_n ahkd_v  _o  _o 30_o gnomon_n ehfg_n gnomon_n â—ckd_v 69_o 1_o 18_o the_o whole_a line_n the_o whole_a â—igure_n 76_o 2_o 9_o the_o diameter_n cd_o the_o diameter_n ahf_n  _o  _o  _o errata_fw-la lib._n 3._o  _o 82_o 2_o 36_o angle_n equal_a to_o the_o angle_n 92_o 1_o last_o the_o line_n ac_fw-la be_v the_o line_n of_o 〈◊〉_d  _o  _o  _o errata_fw-la lib._n 4._o  _o 110_o 2_o 10_o cd_o touch_v the_o ed_z touch_v the_o  _o  _o 12_o side_n of_o the_o other_o angle_n of_o the_o other_o 115_o 1_o 21_o and_o hb_o and_o he_o 117_o 2_o 44_o the_o angle_n acd_o the_o angle_n acb_o 118_o 1_o 2_o into_o ten_o equal_a into_o two_o equal_a 121_o 1_o 3â—_o cd_o and_o ea_fw-la cd_o de_fw-fr &_o ea_fw-la â—â—â—_o 1_o 29_o the_o first_o the_o three_o  _o  _o  _o errata_fw-la lib._n 5._o  _o 126_o 1_o 43_o it_o make_v 12._o more_o than_o 17._o by_o 5._o it_o make_v 24._o more_o than_o 17._o by_o 7._o 129_o 1_n  _o in_o stead_n of_o the_o figure_n of_o the_o 6._o definition_n draw_v in_o the_o magâ—â—_n a_o figure_n like_o unto_o thâ—s_n 134_o 2_o 4_o as_o ab_fw-la be_v to_o a_o so_o be_v cd_o to_o c_o as_z ab_fw-la be_v to_o b_o so_o be_v cd_o to_o d_o 141_o 2_o last_v but_o if_o king_n exceed_v m_o but_o if_o h_o exceed_v m_o lief_a be_v death_n and_o death_n be_v lief_a aetatis_fw-la svae_fw-la xxxx_n at_o london_n print_v by_o john_n day_n dwell_v over_o aldersgate_n beneath_o saint_n martins_n ¶_o these_o book_n be_v to_o be_v sell_v at_o his_o shop_n under_o the_o gate_n 1570._o
the_o lengthe_n of_o day_n and_o night_n the_o hour_n and_o time_n both_o night_n and_o day_n be_v know_v with_o very_a many_o other_o pleasant_a and_o necessary_a use_n whereof_o some_o be_v know_v but_o better_a remain_v for_o such_o to_o know_v and_o use_v who_o of_o a_o spark_n of_o true_a fire_n can_v make_v a_o wonderful_a bonfire_n by_o apply_v of_o due_a matter_n due_o ☜_o of_o astrology_n here_o i_o make_v a_o arâ—e_n several_a from_o astronomieâ—_n â—_z not_o by_o new_a devise_n but_o by_o good_a reason_n and_o authority_n for_o astrology_n be_v a_o art_n mathematical_a which_o reasonable_o demonstrate_v the_o operation_n and_o effect_n of_o the_o natural_a beam_n of_o light_n and_o secret_a influence_n of_o the_o star_n and_o planet_n in_o every_o element_n and_o elemental_a body_n at_o all_o time_n in_o any_o horizon_n assign_v this_o art_n be_v furnish_v with_o many_o other_o great_a art_n and_o experience_n as_o with_o perfect_v perspective_n astronomy_n cosmography_n natural_a philosophy_n of_o the_o 4._o element_n the_o art_n of_o graduation_n and_o some_o good_a understanding_n in_o music_n and_o yet_o moreover_o with_o a_o other_o great_a art_n hereafter_o follow_v though_o i_o here_o set_v this_o before_o for_o some_o consideration_n i_o move_v sufficient_a you_o see_v be_v the_o stuff_n to_o make_v this_o rare_a and_o secret_a art_n of_o and_o hard_a enough_o to_o frame_v to_o the_o conclusion_n syllogistical_a yet_o both_o the_o manifold_a and_o continual_a travail_n of_o the_o most_o ancient_a and_o wise_a philosopher_n for_o the_o attain_n of_o this_o art_n and_o by_o example_n of_o effect_n to_o confirm_v the_o same_o have_v leave_v unto_o we_o sufficient_a proof_n and_o witness_v and_o we_o also_o daily_o may_v perceive_v that_o man_n body_n and_o all_o other_o elemental_a body_n be_v alter_v dispose_v order_a pleasure_v and_o displeasure_v by_o the_o influential_a work_n of_o the_o sun_n moon_n and_o the_o other_o star_n and_o planet_n and_o therefore_o say_v aristotle_n in_o the_o first_o of_o his_o meteorological_a book_n in_o the_o second_o chapter_n est_fw-la autem_fw-la necessariò_fw-la mundus_fw-la iste_fw-la supernis_fw-la lationibus_fw-la ferè_fw-la continuus_fw-la ut_fw-la inde_fw-la vis_fw-la eius_fw-la universa_fw-la regatur_fw-la ea_fw-la siquidem_fw-la causa_fw-la prima_fw-la putanda_fw-la omnibus_fw-la est_fw-la unde_fw-la motus_fw-la principium_fw-la existit_fw-la that_o be_v this_o elemental_a world_n be_v of_o necessity_n almost_o next_o adjoin_v to_o the_o heavenly_a motion_n that_o from_o thence_o all_o his_o virtue_n or_o force_n may_v be_v govern_v for_o that_o be_v to_o be_v think_v the_o first_o cause_n unto_o all_o from_o which_o the_o beginning_n of_o motion_n be_v and_o again_o in_o the_o ten_o chapter_n opâ—rtet_fw-la igitur_fw-la &_o horum_fw-la principia_fw-la sumamus_fw-la &_o causas_fw-la omnium_fw-la similiter_fw-la principium_fw-la igitur_fw-la vy_z movens_fw-la praecipuumque_fw-la &_o omnium_fw-la primum_fw-la circulus_fw-la ille_fw-la est_fw-la in_fw-la quo_fw-la manifest_a solis_fw-la latio_fw-la etc_n etc_n and_o so_o forth_o his_o meteorological_a book_n be_v full_a of_o argument_n and_o effectual_a demonstration_n of_o the_o virtue_n operation_n and_o power_n of_o the_o heavenly_a body_n in_o and_o upon_o the_o four_o element_n and_o other_o body_n of_o they_o either_o perfect_o or_o unperfect_o compose_v and_o in_o his_o second_o book_n de_fw-fr generatione_fw-la &_o corruption_n in_o the_o ten_o chapter_n quocirca_fw-la &_o prima_fw-la latiâ—_n orâ—us_n &_o interitus_fw-la causa_fw-la non_fw-la est_fw-la sed_fw-la obliqui_fw-la circuli_fw-la latio_fw-la ea_fw-la namque_fw-la &_o continua_fw-la est_fw-la &_o dvobus_fw-la motibus_fw-la fit_a in_o english_a thus_o wherefore_o the_o uppermost_a motion_n be_v not_o the_o cause_n of_o generation_n and_o corruption_n but_o the_o motion_n of_o the_o zodiac_n for_o that_o both_o be_v continual_a and_o be_v cause_v of_o two_o move_n and_o in_o his_o second_o book_n and_o second_o chapter_n of_o his_o physike_n homo_fw-la namque_fw-la generate_fw-la hominem_fw-la atque_fw-la sol._n for_o man_n say_v he_o and_o the_o son_n be_v cause_n of_o man_n generation_n authority_n may_v be_v bring_v very_o many_o both_o of_o 1000_o 2000_o yea_o and_o 3000._o year_n antiquity_n of_o great_a philosopher_n expert_a wise_a and_o godly_a man_n for_o that_o conclusion_n which_o daily_a and_o hourly_o we_o man_n may_v discern_v and_o perceive_v by_o sense_n and_o reason_n all_o beast_n do_v feel_v and_o simple_o show_v by_o their_o action_n and_o passion_n outward_a and_o inward_a all_o plant_n herb_n tree_n flower_n and_o fruit_n and_o final_o the_o element_n and_o all_o thing_n of_o the_o element_n compose_v do_v geve_v testimony_n as_o aristotle_n say_v that_o their_o whole_a disposition_n virtue_n and_o natural_a motion_n depend_v of_o the_o activity_n of_o the_o heavenly_a motion_n and_o influence_n whereby_o beside_o the_o specifical_a order_n and_o form_n due_a to_o every_o seed_n and_o beside_o the_o nature_n proper_a to_o the_o individual_a matrix_fw-la of_o the_o thing_n produce_v what_o shall_v be_v the_o heavenly_a impression_n the_o perfect_a and_o circumspect_a astrologien_n have_v to_o conclude_v not_o only_o by_o apotelesme_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d but_o by_o natural_a and_o mathematical_a demonstration_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d whereunto_o what_o science_n be_v requisite_a without_o exception_n i_o partly_o have_v here_o warn_v and_o in_o my_o bropâ—deâ—â—â—â—_n beside_o other_o matter_n there_o disclose_v i_o have_v mathematical_o furnish_v up_o the_o whole_a method_n to_o this_o our_o age_n not_o so_o careful_o handle_v by_o any_o that_o ever_o i_o see_v or_o hear_v of_o i_o be_v for_o lovayn_n â—1_n year_n ago_o by_o certain_a earnest_a disputation_n of_o the_o learned_a gâ—â—ardus_n mâ—rcâ—tâ—â—_n and_o 〈◊〉_d gogaâ—a_n and_o other_o thereto_o so_o provoke_v and_o by_o my_o constant_a and_o invincible_a zeal_n to_o the_o verity_n in_o observation_n of_o heavenly_a influence_n to_o the_o minâ—te_n of_o time_n than_o so_o diligent_a and_o chief_o by_o the_o supernatural_a influence_n from_o the_o star_n of_o jacob_n so_o direct_v that_o any_o modest_a and_o sober_a student_n careful_o and_o diligent_o selâ—ing_v for_o the_o truth_n will_v both_o find_v &_o confess_v therein_o to_o be_v the_o verity_n of_o these_o my_o word_n and_o also_o become_v a_o reasonable_a reformer_n of_o three_o sort_n of_o people_n about_o these_o influential_a operation_n great_o err_v from_o the_o truth_n whereof_o the_o one_o be_v light_a believer_n the_o other_o note_n light_a despiser_n and_o the_o three_o light_a practiser_n the_o first_o &_o most_o common_a sort_n think_v the_o heaven_n and_o star_n to_o be_v answerable_a to_o any_o their_o doubt_n or_o desire_n which_o be_v not_o so_o and_o in_o deed_n they_o to_o much_o over_o reach_n the_o second_o sort_n think_v no_o influential_a virtue_n from_o the_o heavenly_a body_n to_o bear_v any_o sway_n in_o generation_n and_o corruption_n in_o this_o elemental_a world_n and_o to_o the_o sun_n moon_n and_o star_n be_v so_o many_o so_o pure_a so_o bright_a so_o wonderful_a big_a so_o far_o in_o distance_n so_o manifold_a in_o their_o motion_n so_o constant_a in_o their_o periode_n etc_n etc_n they_o assign_v a_o sleight_n simple_a office_n or_o two_o and_o so_o allow_v unto_o they_o according_a to_o their_o capacity_n as_o much_o virtue_n and_o power_n influential_a as_o to_o the_o sign_n of_o the_o sun_n moon_n and_o seven_o star_n hang_v up_o for_o sign_n in_o london_n for_o distinction_n of_o house_n &_o such_o gross_a help_n in_o our_o worldly_a affair_n and_o they_o understand_v not_o or_o will_v not_o understand_v of_o the_o other_o work_n and_o virtue_n of_o the_o heavenly_a sun_n moon_n and_o star_n not_o so_o much_o as_o the_o mariner_n or_o husband_n manâ—_n no_o not_o so_o much_o as_o the_o elephant_n do_v as_o the_o cynocephalus_n as_o the_o porpentine_n doâ—h_v nor_o will_v allow_v these_o perfect_a and_o incorruptible_a mighty_a body_n so_o much_o virtual_a radiation_n &_o force_n as_o they_o see_v in_o a_o little_a piece_n of_o a_o magnes_n stone_n which_o at_o great_a distance_n show_v his_o operation_n and_o perchance_o they_o think_v the_o sea_n &_o rivers_n as_o the_o thames_n to_o be_v some_o quick_a thing_n and_o sâ—_n to_o ebb_v aâ—d_z slow_a run_v in_o and_o out_o of_o themselves_o at_o â—heiâ—_n own_o fantasy_n god_n help_v god_n help_v sure_o these_o man_n come_v to_o short_a and_o either_o be_v to_o dull_a or_o wilful_o blind_a or_o perhaps_o to_o malicious_a the_o three_o man_n be_v the_o common_a and_o vulgar_a astrologien_n or_o practiser_n who_o be_v not_o due_o artificial_o and_o perfect_o furnish_v yet_o either_o for_o vain_a glory_n or_o gain_v or_o like_o a_o simple_a dolt_n &_o blind_a bayardâ—_n both_o in_o matter_n and_o manner_n err_v to_o the_o discredit_n of_o the_o wary_a and_o modest_a astrologien_n and_o to_o the_o rob_n of_o those_o most_o noble_a corporal_a
compare_v they_o all_o with_o triangle_n &_o also_o together_o the_o one_o with_o the_o other_o in_o it_o also_o be_v teach_v how_o a_o figure_n of_o any_o form_n may_v be_v change_v into_o a_o figure_n of_o a_o other_o form_n and_o for_o that_o it_o entreat_v of_o these_o most_o common_a and_o general_a thing_n this_o book_n be_v more_o universal_a than_o be_v the_o second_o three_o or_o any_o other_o and_o therefore_o just_o occupy_v the_o first_o place_n in_o order_n as_o that_o without_o which_o the_o other_o book_n of_o eâ—clide_n which_o follow_v and_o also_o the_o work_n of_o other_o which_o have_v write_v in_o geometry_n can_v be_v perceive_v nor_o understand_v and_o forasmuch_o â—s_v all_o the_o demonstration_n and_o proof_n of_o all_o the_o proposition_n in_o this_o whole_a book_n depend_v of_o these_o ground_n and_o principle_n follow_v which_o by_o reason_n of_o their_o plainness_n need_v no_o great_a declaration_n yet_o to_o remove_v all_o be_v it_o never_o so_o little_a obscurity_n there_o be_v here_o set_v certain_a short_a and_o manife_a exposition_n of_o they_o definition_n 1._o a_o sign_n or_o point_n be_v that_o which_o have_v no_o part_n point_n the_o better_a to_o understand_v what_o manâ—r_n of_o thing_n a_o sign_n or_o point_n be_v you_o must_v note_v that_o the_o nature_n and_o property_n of_o quantity_n whereof_o geometry_n entreat_v be_v to_o be_v divide_v so_o that_o whatsoever_o may_v be_v divide_v into_o sundâ—y_a part_n be_v call_v quantity_n but_o a_o point_n although_o it_o pertain_v to_o quantity_n and_o have_v his_o be_v in_o quantity_n yet_o be_v it_o no_o quantity_n for_o that_o it_o can_v be_v divide_v because_o as_o the_o definition_n say_v it_o have_v no_o part_n into_o which_o it_o shall_v be_v divide_v so_o that_o a_o point_n be_v the_o least_o thing_n that_o by_o mind_n and_o understanding_n can_v be_v imagine_v and_o conceive_v than_o which_o there_o can_v be_v nothing_o less_o as_o the_o point_n a_o in_o the_o margin_n a_o sign_n or_o point_n be_v of_o pythagoras_n scholar_n after_o this_o manner_n define_v pythagoras_n a_o point_n be_v a_o unity_n which_o have_v position_n number_n be_v conceive_v in_o mind_n without_o any_o form_n &_o figure_n and_o therefore_o without_o matter_n whereon_o to_o 〈◊〉_d figure_n &_o consequent_o without_o place_n and_o position_n wherefore_o unity_n be_v a_o part_n of_o number_n have_v no_o position_n or_o determinate_a place_n whereby_o it_o be_v manifest_a that_o â—umbâ—â—_n iâ—_z more_o simple_a and_o pure_a then_o be_v magnitude_n and_o also_o immaterial_a and_o so_o unity_n which_o iâ—_z the_o beginning_n of_o number_n be_v less_o material_a than_o a_o â—igne_n or_o poyâ—â—_n which_o be_v the_o begin_n of_o magnitude_n for_o a_o point_n be_v material_a and_o require_v position_n and_o place_n and_o â—â—â—rby_o differ_v from_o unity_n â—_o a_o line_n be_v length_n without_o breadth_n liâ—â—_n there_o pertain_v to_o quantiâ—â—e_v three_o dimension_n length_n 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describe_v a_o equilater_n triangle_n which_o be_v the_o thing_n which_o be_v require_v to_o be_v do_v a_o triangle_n or_o any_o other_o rectiline_v figure_n be_v then_o say_v to_o be_v set_v or_o describe_v upon_o a_o line_n when_o the_o line_n be_v one_o of_o the_o side_n of_o the_o figure_n this_o first_o proposition_n be_v a_o problem_n because_o it_o require_v act_n or_o do_v namely_o to_o describe_v a_o triangle_n and_o this_o be_v to_o be_v note_v that_o every_o proposition_n whether_o it_o be_v a_o problem_n or_o a_o theorem_a common_o contain_v in_o it_o a_o thing_n give_v and_o a_o thing_n require_v to_o be_v search_v out_o although_o it_o be_v not_o always_o so_o and_o the_o thing_n give_v give_v be_v ever_o
about_o the_o diameter_n together_o with_o the_o two_o supplement_n make_v a_o gnomon_n as_o the_o parallelogram_n ebkh_n with_o the_o two_o supplement_n aegk_n and_o khfd_n make_v the_o gnomon_n fgeh_a likewise_o the_o parallelogram_n gkcf_n with_o the_o same_o two_o supplement_n make_v the_o gnomon_n ehfg_n and_o this_o definition_n of_o a_o gnomon_n extend_v itself_o and_o be_v general_a to_o all_o kind_n of_o parallelogram_n whether_o they_o be_v square_n or_o figure_n of_o one_o side_n long_o or_o rhombus_fw-la or_o romboide_n to_o be_v short_a if_o you_o take_v away_o from_o the_o whole_a parallelogram_n one_o of_o the_o partial_a parallelogram_n which_o be_v about_o the_o diameter_n whether_o you_o will_v the_o rest_n of_o the_o figure_n be_v a_o gnomon_n campane_v after_o the_o last_o proposition_n of_o the_o first_o book_n add_v this_o proposition_n book_n two_o square_n be_v give_v to_o adjoin_v to_o one_o of_o they_o a_o gnomon_n equal_a to_o the_o other_o square_n which_o for_o that_o as_o than_o it_o be_v not_o teach_v what_o a_o gnomon_n be_v i_o there_o omit_v think_v that_o it_o may_v more_o apt_o be_v place_v here_o the_o do_v and_o demonstration_n whereof_o be_v thus_o suppose_v that_o there_o be_v two_o square_n ab_fw-la and_o cd_o unto_o one_o of_o which_o namely_o unto_o ab_fw-la it_o be_v require_v to_o add_v a_o gnomon_n equal_a to_o the_o other_o square_n namely_o to_o cd_o produce_v the_o side_n bf_o of_o the_o square_a ab_fw-la direct_o to_o the_o point_n e._n and_o put_v the_o line_n fe_o equal_a to_o the_o side_n of_o the_o square_a cd_o and_o draw_v a_o line_n from_o e_o to_o a._n now_o then_o forasmuch_o as_o efa_n be_v a_o rectangle_n triangle_n therefore_o by_o the_o 47._o of_o the_o first_o the_o square_a of_o the_o line_n ea_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n of_o &_o fa._n but_o the_o square_n of_o the_o line_n of_o be_v equal_a to_o the_o square_a cd_o &_o the_o square_a of_o the_o side_n favorina_n be_v the_o square_a ab_fw-la wherefore_o the_o square_a of_o the_o line_n ae_n be_v equal_a to_o the_o two_o square_n cd_o and_o ab_fw-la but_o the_o side_n of_o and_o favorina_n be_v by_o the_o 21._o of_o the_o first_o long_o than_o the_o side_n ae_n and_o the_o side_n favorina_n be_v equal_a to_o the_o side_n fb_o wherefore_o the_o side_n of_o and_o fb_o be_v long_o they_o the_o side_n ae_n wherefore_o the_o whole_a line_n be_v be_v long_o than_o the_o line_n ae_n from_o the_o line_n be_v cut_v of_o a_o line_n equal_a to_o the_o line_n ae_n which_o let_v be_v bc._n and_o by_o the_o 46._o proposition_n upon_o the_o line_n bc_o describe_v a_o square_a which_o let_v be_v bcgh_n which_o shall_v be_v equal_a to_o the_o square_n of_o the_o line_n ae_n but_o the_o square_n of_o the_o line_n ae_n be_v equal_a to_o the_o two_o square_n ab_fw-la and_o dc_o wherefore_o the_o square_a bcgh_n be_v equal_a to_o the_o same_o square_n wherefore_o forasmuch_o as_o the_o square_a bcgh_n be_v compose_v of_o the_o square_a ab_fw-la and_o of_o the_o gnomon_n fgah_n the_o say_a gnomon_n shall_v be_v equal_a unto_o the_o square_a cd_o which_o be_v require_v to_o be_v do_v a_o other_o more_o ready_a way_n after_o pelitarius_n suppose_v that_o there_o be_v two_o square_n who_o side_n let_v be_v ab_fw-la and_o bc._n it_o be_v require_v unto_o the_o square_n of_o the_o line_n ab_fw-la to_o add_v a_o gnomon_n equal_a to_o the_o square_n of_o the_o line_n bc._n set_v the_o line_n ab_fw-la and_o bc_o in_o such_o sort_n that_o they_o make_v a_o right_a angle_n abc_n and_o draw_v a_o line_n from_o a_o to_o c._n and_o upon_o the_o line_n ab_fw-la describe_v a_o square_n which_o let_v be_v abde_n and_o produce_v the_o line_n basilius_n to_o the_o point_n fletcher_n and_o put_v the_o line_n bf_o equal_a to_o the_o line_n ac_fw-la and_o upon_o the_o line_n bf_o describe_v a_o square_n which_o let_v be_v bfgh_o which_o shall_v be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la when_o as_o the_o line_n bf_o and_o ac_fw-la be_v equal_a and_o therefore_o it_o be_v equal_a to_o the_o square_n of_o the_o two_o line_n ab_fw-la and_o bc._n now_o forasmuch_o as_o the_o square_a bfgh_o be_v make_v complete_a by_o the_o square_a abde_n and_o by_o the_o gnomon_n fegd_v the_o gnomon_n fegd_v shall_v be_v equal_a to_o the_o square_n of_o the_o line_n bc_o which_o be_v require_v to_o be_v do_v the_o 1._o theorem_a the_o 1._o proposition_n if_o there_o be_v two_o right_a line_n and_o if_o the_o one_o of_o they_o be_v divide_v into_o part_n how_o many_o soever_o the_o rectangle_n figure_n comprehend_v under_o the_o two_o right_a line_n be_v equal_a to_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n undivided_a and_o under_o every_o one_o of_o the_o part_n of_o the_o other_o line_n svppose_v that_o there_o be_v two_o right_a line_n a_o and_o bc_o and_o let_v one_o of_o they_o namely_o bc_o be_v divide_v at_o all_o adventure_n in_o the_o point_n d_o and_o e._n then_o i_o say_v that_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n a_o and_o bc_o be_v equal_a unto_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n a_o and_o bd_o &_o unto_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n a_o and_o de_fw-fr and_o also_o unto_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n a_o and_o ec_o construction_n for_o from_o the_o point_n brayse_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o right_a line_n bc_o a_o perpendicular_a line_n bf_o &_o unto_o the_o line_n a_o by_o the_o three_o of_o the_o first_o put_v the_o line_n bg_o equal_a and_o by_o the_o point_n g_z by_o the_o 31._o of_o the_o first_o draw_v a_o parallel_n line_n unto_o the_o right_a line_n bc_o and_o let_v the_o same_o be_v gm_n and_o by_o the_o self_n same_o by_o the_o poor_n d_z e_o and_o c_o draw_v unto_o the_o line_n bg_o these_o parallel_a line_n dk_o demonstration_n el_n and_o ch._n now_o then_o the_o parallelogram_n bh_o be_v equal_a to_o these_o parallelogram_n bk_o dl_o and_o eh_o but_o the_o parallelogram_n bh_o be_v equal_a unto_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o bc._n for_o it_o be_v comprehend_v under_o the_o line_n gb_o &_o bc_o and_o the_o line_n gb_o be_v equal_a unto_o the_o line_n a_o and_o the_o parallelogram_n bk_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o bd_o for_o it_o be_v comprehend_v under_o the_o line_n gb_o and_o bd_o and_o bg_o be_v equal_a unto_o a_o and_o the_o parallelogram_n dl_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o de_fw-fr for_o the_o line_n dk_o that_o be_v bg_o be_v equal_a unto_o a_o and_o moreover_o likewise_o the_o parallelogram_n eh_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n a_o &_o ec_o wherefore_o that_o which_o be_v comprehend_v under_o the_o line_n a_o &_o bc_o be_v equal_a to_o that_o which_o be_v comprehend_v under_o the_o line_n a_o &_o bd_o &_o unto_o that_o which_o be_v comprehend_v under_o the_o line_n a_o and_o de_fw-fr and_o moreover_o unto_o that_o which_o be_v comprehend_v under_o the_o line_n a_o and_o ec_o if_o therefore_o there_o be_v two_o right_a line_n and_o if_o the_o one_o of_o they_o be_v divide_v into_o part_n how_o many_o soever_o the_o rectangle_n figure_n comprehend_v under_o the_o two_o right_a line_n be_v equal_a to_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n undivided_a and_o under_o every_o one_o of_o the_o part_n of_o the_o other_o line_n which_o be_v require_v to_o be_v demonstrate_v because_o that_o all_o the_o proposition_n of_o this_o second_o book_n for_o the_o most_o part_n be_v true_a both_o in_o line_n and_o in_o number_n and_o may_v be_v declare_v by_o both_o therefore_o have_v i_o have_v add_v to_o every_o proposition_n convenient_a number_n for_o the_o manifestation_n of_o the_o same_o and_o to_o the_o end_n the_o studious_a and_o diligent_a reader_n may_v the_o more_o full_o perceive_v and_o understand_v the_o agreement_n of_o this_o art_n of_o geometry_n with_o the_o science_n of_o arithmetic_n and_o how_o never_o &_o dear_a sister_n they_o be_v together_o so_o that_o the_o one_o can_v without_o great_a blemish_n be_v without_o the_o other_o i_o have_v here_o also_o join_v a_o little_a book_n of_o arithmetic_n write_v by_o one_o barlaam_n a_o greek_a author_n a_o man_n of_o great_a knowledge_n in_o which_o book_n be_v by_o the_o author_n demonstrate_v many_o of_o the_o self_n same_o propriety_n and_o passion_n in_o number_n which_o euclid_n in_o this_o his_o second_o book_n have_v demonstrate_v in_o magnitude_n
namely_o the_o first_o ten_o proposition_n as_o they_o follow_v in_o order_n which_o be_v undoubted_o great_a pleasure_n to_o consider_v also_o great_a increase_n &_o furniture_n of_o knowledge_n who_o proposition_n be_v set_v orderly_a after_o the_o proposition_n of_o euclid_n every_o one_o of_o barlaam_n correspondent_a to_o the_o same_o of_o euclid_n and_o doubtless_o it_o be_v wonderful_a to_o see_v how_o these_o two_o contrary_a kind_n of_o quantity_n quantity_n discrete_a or_o number_n and_o quantity_n continual_a or_o magnitude_n which_o be_v the_o subject_n or_o matterâ—_n of_o arithmitique_a and_o geometry_n shall_v have_v in_o they_o one_o and_o the_o same_o propriety_n common_a to_o they_o both_o in_o very_a many_o point_n and_o affection_n although_o not_o in_o all_o for_o a_o line_n may_v in_o such_o sort_n be_v divide_v that_o what_o proportion_n the_o whole_a have_v to_o the_o great_a part_n the_o same_o shall_v the_o great_a part_n have_v to_o the_o less_o but_o that_o can_v not_o be_v in_o number_n for_o a_o number_n can_v not_o so_o be_v divide_v that_o the_o whole_a number_n to_o the_o great_a part_n thereof_o shall_v have_v that_o proportion_n which_o the_o great_a part_n have_v to_o the_o less_o as_o jordane_n very_o plain_o prove_v in_o his_o book_n of_o arithmetic_n which_o thing_n campane_n also_o as_o we_o shall_v afterward_o in_o the_o 9_o book_n after_o the_o 15._o proposition_n see_v prove_v and_o as_o touch_v these_o ten_o first_o proposition_n of_o the_o second_o book_n of_o euclid_n demonstrate_v by_o barlaam_n in_o number_n they_o be_v also_o demonstrate_v of_o campane_n after_o the_o 15._o proposition_n of_o the_o 9_o book_n who_o demonstration_n i_o mind_n by_o god_n help_n to_o set_v forth_o when_o i_o shall_v come_v to_o the_o place_n they_o be_v also_o demonstrate_v of_o jordane_n that_o excellet_fw-la learned_a author_n in_o the_o first_o book_n of_o his_o arithmetic_n in_o the_o mean_a time_n i_o think_v it_o not_o amiss_o here_o to_o set_v forth_o the_o demonstration_n of_o barlaam_n for_o that_o they_o geve_v great_a light_n to_o the_o second_o book_n of_o euclid_n beside_o the_o inestimable_a pleasure_n which_o they_o bring_v to_o the_o studious_a considerer_n and_o now_o to_o declare_v the_o first_o proposition_n by_o number_n i_o have_v put_v this_o example_n follow_v take_v two_o number_n the_o one_o undivided_a as_o 74._o the_o other_o divide_v into_o what_o part_n and_o how_o many_o you_o listen_v as_o 37._o divide_v into_o 20._o 10._o 5._o and_o 2d_o which_o altogether_o make_v the_o whole_a 37._o then_o if_o you_o multiply_v the_o number_n undivided_a namely_o 74_o into_o all_o the_o part_n of_o the_o number_n divide_v as_o into_o 20._o 10._o 5._o and_o 2._o you_o shall_v produce_v 1480._o 740._o 370._o 148._o which_o add_v together_o make_v 2738_o which_o self_n number_n be_v also_o produce_v if_o you_o multiply_v the_o two_o number_n first_o give_v the_o one_o into_o the_o other_o as_o you_o see_v in_o the_o example_n on_o the_o other_o side_n set_v so_o by_o the_o aid_n of_o this_o proposition_n be_v get_v a_o compendious_a way_n of_o multiplication_n by_o break_v of_o one_o of_o the_o number_n into_o his_o part_n which_o oftentimes_o serve_v to_o great_a use_n in_o working_n chiâ—â—ly_o in_o the_o rule_n of_o proportion_n the_o demonstration_n of_o which_o proposition_n follow_v in_o barlaam_n but_o â—irst_n be_v put_v of_o the_o author_n these_o principle_n follow_v barlaam_n ¶_o principles_n 1._o a_o number_n be_v sâ—yd_v to_o multiply_v a_o other_o number_n when_o the_o number_n multiply_v be_v so_o oftentimes_o add_v to_o itself_o as_o there_o be_v unity_n in_o the_o number_n which_o multiply_v whereby_o be_v produce_v a_o certain_a number_n which_o the_o number_n multiply_v measure_v by_o the_o unity_n which_o be_v in_o the_o number_n which_o multipli_v 2._o and_o the_o number_n produce_v of_o that_o a_o multiplication_n be_v call_v a_o plain_a or_o superficial_a number_n 3._o a_o square_a number_n be_v that_o which_o be_v produce_v of_o the_o multiplicatian_n of_o any_o number_n into_o itself_o 4._o every_o less_o number_n compare_v to_o a_o great_a be_v say_v to_o be_v a_o part_n of_o the_o great_a whether_o the_o less_o measure_n the_o great_a or_o measure_v it_o not_o 5._o number_n who_o one_o and_o the_o self_n same_o number_n measure_v equal_o that_o be_v by_o one_o and_o the_o self_n same_o number_n be_v equal_a the_o one_o to_o the_o other_o 6._o number_n that_o be_v equemultiplâ—ces_n to_o one_o and_o the_o self_n same_o number_n that_o be_v which_o contain_v one_o and_o the_o same_o number_n equal_o and_o alike_o be_v equal_a the_o one_o to_o the_o other_o the_o first_o proposition_n two_o number_n be_v give_v if_o the_o one_o of_o they_o be_v divide_v into_o any_o number_n how_o many_o soever_o barlaam_n the_o plain_n or_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o two_o number_n first_o give_v the_o one_o into_o the_o other_o shall_v be_v equal_a to_o the_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n not_o divide_v into_o every_o part_n of_o the_o number_n divide_v suppose_v that_o there_o be_v two_o number_n ab_fw-la and_o c._n and_o divide_v the_o number_n ab_fw-la into_o certain_a other_o number_n how_o many_o soever_o as_o into_o ad_fw-la de_fw-fr and_o ebb_v then_o i_o say_v that_o the_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n c_o into_o the_o number_n ab_fw-la be_v equal_a to_o the_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n c_o into_o the_o number_n ad_fw-la and_o of_o c_o into_o de_fw-fr and_o of_o c_o into_o ebb_n for_o let_v fletcher_n be_v the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n c_o into_o the_o number_n ab_fw-la and_o let_v gh_o be_v the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o c_o into_o ad_fw-la and_o let_v high_a be_v produce_v of_o the_o multiplication_n of_o c_o into_o de_fw-fr aâ—d_v final_o of_o the_o multiplication_n of_o c_o into_o ebb_n let_v there_o be_v produce_v the_o number_n ik_fw-mi now_o forasmuch_o as_o ab_fw-la multiply_v the_o number_n c_o produce_v the_o number_n f_o therefore_o the_o number_n c_o measure_v the_o number_n f_o by_o the_o unity_n which_o be_v in_o the_o number_n ab_fw-la and_o by_o the_o same_o reason_n may_v be_v prove_v that_o the_o number_n c_o do_v also_o measure_v the_o number_n gh_o by_o the_o unity_n which_o be_v in_o the_o number_n ad_fw-la and_o that_o it_o do_v measure_v the_o number_n high_a by_o the_o unity_n which_o be_v in_o the_o number_n df_o and_o final_o that_o it_o measure_v the_o number_n ik_fw-mi by_o the_o unity_n which_o be_v in_o the_o number_n ebb_n wherefore_o the_o number_n c_o measure_v the_o whole_a number_n gk_o by_o the_o unity_n which_o be_v in_o the_o number_n ab_fw-la but_o it_o before_o measure_v the_o number_n f_o by_o the_o unity_n which_o be_v in_o the_o number_n ab_fw-la wherefore_o either_o of_o these_o number_n fletcher_n and_o gk_o be_v equemultiplex_n to_o the_o number_n c._n but_o number_n which_o be_v equemultiplices_fw-la to_o one_o &_o the_o self_n same_o number_n be_v equal_a the_o one_o to_o the_o other_o by_o the_o 6._o definition_n wherefore_o the_o number_n fletcher_n be_v equal_a to_o the_o number_n gk_o but_o the_o number_n fletcher_n be_v the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n c_o into_o the_o number_n ab_fw-la and_o the_o number_n gk_o be_v compose_v of_o the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n c_o not_o divide_v into_o every_o one_o of_o the_o number_n ad_fw-la de_fw-fr and_o ebb_v if_o therefore_o there_o be_v two_o number_n give_v and_o the_o one_o of_o they_o be_v divide_v etc_n etc_n which_o be_v require_v to_o be_v prove_v the_o 2._o theorem_a the_o 2._o proposition_n if_o a_o right_a line_n be_v divide_v by_o chance_n the_o rectangle_v figure_n which_o be_v comprehend_v under_o the_o whole_a and_o every_o one_o of_o the_o part_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o whole_a svppose_v that_o the_o right_a line_n ab_fw-la be_v by_o chance_n deny_v in_o the_o point_n c._n then_o i_o say_v that_o the_o rectangle_n figure_n comprehend_v under_o ab_fw-la and_o bc_o together_o with_o the_o rectangle_n comprehend_v under_o ab_fw-la and_o ac_fw-la be_v equal_a unto_o the_o square_n make_v of_o ab_fw-la construction_n describe_v by_o the_o 46._o of_o the_o first_o upon_o ab_fw-la a_o square_a adeb_n and_o by_o the_o 31_o of_o the_o first_o by_o the_o point_n c_o draw_v a_o line_n parallel_n unto_o either_o of_o these_o line_n ad_fw-la aâ—d_fw-la be_v and_o let_v the_o same_o
other_o number_n de_fw-fr be_v of_o a_o other_o number_n fletcher_n and_o let_v ab_fw-la be_v less_o than_o de._n then_o i_o say_v that_o alternate_o also_o what_o part_n or_o part_n ab_fw-la be_v of_o de_fw-fr the_o self_n same_o part_n or_o part_n be_v c_o of_o f._n forasmuch_o as_o what_o part_v ab_fw-la be_v of_o c_o the_o self_n same_o part_n be_v de_fw-fr of_o fletcher_n construction_n therefore_o how_o many_o part_n of_o c_o there_o be_v in_o ab_fw-la so_o many_o part_n of_o fletcher_n also_o be_v there_o in_o de._n divide_v ab_fw-la into_o the_o part_n of_o c_o that_o be_v into_o ag_z and_o gb_o and_o likewise_o de_fw-fr into_o the_o part_n of_o fletcher_n that_o be_v dh_o and_o he._n now_o then_o the_o multitude_n of_o these_o agnostus_n and_o gb_o be_v equal_a unto_o the_o multitude_n of_o these_o dh_o and_o he._n demonstration_n and_o forasmuch_o as_o what_o part_n agnostus_n be_v of_o c_o the_o self_n same_o part_n be_v dh_o of_o fletcher_n therefore_o alternate_o also_o by_o the_o former_a what_o part_n or_o part_n agnostus_n be_v of_o dh_o the_o self_n same_o part_n or_o part_n be_v c_o of_o f._n and_o by_o the_o same_o reason_n also_o what_o part_n or_o part_n gb_o be_v of_o he_o the_o same_o part_n or_o part_n be_v c_o of_o f._n wherefore_o what_o part_n or_o part_n agnostus_n be_v of_o dh_o the_o self_n same_o part_n or_o part_n be_v ab_fw-la of_o de_fw-fr by_o the_o 6._o of_o the_o seven_o but_o what_o part_n or_o part_n agnostus_n be_v of_o dh_o the_o self_n same_o part_n or_o part_n be_v it_o prove_v that_o c_z be_v of_o f._n wherefore_o what_o part_n or_o part_n ab_fw-la be_v of_o d_o e_o the_o self_n same_o part_n or_o part_n be_v c_o of_o fletcher_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 9_o theorem_a the_o 11._o proposition_n if_o the_o whole_a be_v to_o the_o whole_a as_o a_o part_n take_v away_o be_v to_o a_o part_n take_v away_o then_o shall_v the_o residue_n be_v unto_o the_o residue_n as_o the_o whole_a be_v to_o the_o whole_a quantity_n svppose_v that_o the_o whole_a number_n ab_fw-la be_v unto_o the_o whole_a number_n cd_o as_o the_o part_n take_v away_o ae_n be_v to_o the_o part_n take_v away_o cf._n then_o i_o say_v that_o the_o residue_n ebb_n be_v to_o the_o residue_n fd_o as_o the_o whole_a ab_fw-la be_v to_o the_o whole_a cd_o for_o forasmuch_o as_o ab_fw-la be_v to_o cd_o as_o ae_n be_v to_o cf_o therefore_o what_o part_n or_o part_n ab_fw-la be_v of_o cd_o the_o self_n same_o part_n or_o part_n be_v ae_n of_o cf._n wherefore_o also_o the_o residue_n ebb_n be_v of_o the_o residue_n fd_o by_o the_o 8._o of_o the_o seven_o the_o self_n same_o part_n oâ—_n part_n that_o ab_fw-la be_v of_o cd_o demonstration_n wherefore_o also_o by_o the_o 21._o definition_n of_o this_o book_n as_o ebb_n be_v to_o fd_o so_o be_v ab_fw-la to_o cd_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 10._o theorem_a the_o 12._o proposition_n if_o there_o be_v a_o multitude_n of_o number_n how_o many_o soever_o proportional_a as_o one_o of_o the_o antecedentes_fw-la be_v to_o one_o of_o the_o consequentes_fw-la so_o be_v all_o the_o antecedentes_fw-la to_o all_o the_o consequentes_fw-la svppose_v that_o there_o be_v a_o multitude_n of_o number_n how_o many_o soever_o proportional_a namely_o a_o b_o c_o d_o so_fw-mi that_o as_o a_o be_v to_o b_o so_o let_v c_o be_v to_o d._n quantity_n then_o i_o say_v that_o as_o one_o of_o the_o antecedentes_fw-la namely_o a_o be_v to_o one_o of_o the_o consequentes_fw-la namely_o to_o b_o or_o as_z c_z be_v to_o d_z so_o be_v all_o the_o antecedentes_fw-la namely_o a_o and_o c_o to_o all_o the_o consequentes_fw-la namely_o to_o b_o and_o d._n for_o forasmuch_o as_o by_o supposition_n as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z therefore_o what_o part_n or_o part_n a_o be_v of_o b_o the_o self_n same_o part_n or_o part_n be_v c_z of_o d_z by_o the_o 21._o definition_n of_o this_o book_n wherefore_o alternate_o what_o part_n or_o part_n a_o be_v of_o c_o the_o self_n same_o part_n or_o part_n be_v b_o of_o d_z by_o the_o nine_o and_o ten_o of_o the_o seven_o wherefore_o both_o these_o number_n add_v together_o demonstration_n a_o and_o c_o be_v of_o both_o these_o numbers_z b_o and_o d_z add_v together_o the_o self_n same_o part_n or_o part_n that_o a_o be_v of_o b_o by_o the_o 5._o and_o 6._o of_o the_o seven_o wherefore_o by_o the_o 21._o definition_n of_o the_o seven_o as_o one_o of_o the_o antecedent_n namely_o a_o be_v to_o one_o of_o the_o consequentes_fw-la namely_o to_o b_o so_o be_v all_o the_o antecedentes_fw-la a_fw-la and_o c_o to_o all_o the_o consequentes_fw-la b_o &_o d._n which_o be_v require_v to_o be_v prove_v ¶_o the_o 11._o theorem_a the_o 13._o proposition_n if_o there_o be_v four_o number_n proportional_a then_o alternate_o also_o they_o shall_v be_v proportional_a svppose_v that_o there_o be_v four_o number_n proportional_a quantity_n a_o b_o c_o d_o so_fw-mi that_o as_o a_o be_v to_o b_o so_o let_v c_o be_v to_o d._n then_o i_o say_v that_o alternate_o also_o they_o shall_v be_v proportional_a that_o be_v as_o a_o be_v to_o c_o so_o be_v b_o to_o d._n for_o forasmuch_o as_o by_o supposition_n as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z therefore_o by_o the_o 21._o definition_n of_o this_o book_n what_o part_n or_o part_n a_o be_v of_o b_o the_o self_n same_o part_n or_o part_n be_v c_o of_o d._n therefore_o alternate_o what_o part_n or_o part_n a_o be_v of_o c_o the_o self_n same_o part_n or_o part_n be_v b_o of_o d_z by_o the_o 9_o of_o the_o seven_o &_o also_o by_o the_o 10._o of_o the_o same_o wherefore_o as_o a_o be_v to_o c_o so_o be_v b_o to_o d_z by_o the_o 21._o definition_n of_o this_o book_n which_o be_v require_v to_o be_v prove_v here_o be_v to_o be_v note_v that_o although_o in_o the_o foresay_a example_n and_o demonstration_n the_o number_n a_o be_v suppose_v to_o be_v less_o than_o the_o number_n b_o and_o so_o the_o number_n c_o be_v less_o than_o the_o number_n d_o note_n yet_o will_v the_o same_o serve_v also_o though_o a_o be_v suppose_v to_o be_v great_a than_o b_o whereby_o also_o c_o shall_v be_v great_a than_o d_z as_z in_o thâ—s_n example_n here_o put_v for_o for_o that_o by_o supposition_n as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z and_o a_o be_v suppose_v to_o be_v great_a than_o b_o and_o c_z great_a than_o d_z therefore_o by_o the_o 21._o definition_n of_o this_o book_n how_o multiplex_n a_o be_v to_o b_o so_o multiplex_n be_v c_o to_o d_z and_o therefore_o what_o part_n or_o part_n b_o be_v of_o a_o the_o self_n same_o part_n or_o part_n be_v d_z of_o c._n wherefore_o alternate_o what_o part_n or_o part_n b_o be_v of_o d_o the_o self_n same_o part_n or_o part_n be_v a_o of_o c_o and_o therefore_o by_o the_o same_o definition_n b_o be_v to_o d_z as_z a_o be_v to_o c._n and_o so_o must_v you_o understand_v of_o the_o former_a proposition_n next_o go_v before_o ¶_o the_o 12._o theorem_a the_o 14._o proposition_n if_o there_o be_v a_o multitude_n of_o number_n how_o many_o soever_o and_o also_o other_o number_n equal_a unto_o they_o in_o multitude_n which_o be_v compare_v two_o and_o two_o be_v in_o one_o and_o the_o same_o proportion_n they_o shall_v also_o of_o equality_n be_v in_o one_o and_o the_o same_o proportion_n quantity_n svppose_v that_o there_o be_v a_o multitude_n of_o number_n how_o many_o soever_o namely_o a_o b_o c_o and_o let_v the_o other_o number_n equal_a unto_o they_o in_o multitude_n be_v d_o e_o f_o which_o be_v compare_v two_o and_o two_o let_v be_v in_o one_o and_o the_o same_o proportion_n that_o be_v as_z a_o to_o b_o so_o let_v d_o be_v to_o e_o and_o as_o b_o be_v to_o c_o so_o let_v e_o be_v to_o f._n then_o i_o say_v that_o of_o equality_n as_o a_o be_v to_o c_o so_o be_v d_z to_o f._n for_o forasmuch_o as_o by_o supposition_n as_o a_o be_v to_o b_o so_o be_v d_z to_o e_o therefore_o alternate_o also_o by_o the_o 13_o of_o the_o seven_o as_o a_o be_v to_o d_o so_o be_v b_o to_o e._n again_o for_o that_o as_z b_o be_v to_o c_o so_o be_v e_o to_o fletcher_n demonstration_n therefore_o alternate_o also_o by_o the_o self_n same_o as_z b_o be_v to_o e_o so_o be_v c_o to_o f._n but_o as_o b_o be_v to_o e_o so_o be_v a_o to_o d._n wherefore_o by_o the_o seven_o common_a sentence_n of_o the_o seven_o as_o a_o be_v to_o d_o so_o be_v c_o to_o f._n wherefore_o alternate_o by_o the_o 13._o of_o the_o seven_o as_o a_o be_v to_o c_o so_o be_v d_z to_o fletcher_n which_o
the_o double_a of_o b_o be_v prime_a unto_o the_o double_a of_o b_o then_o the_o two_o number_n whereof_o the_o number_n be_v compose_v namely_o the_o number_n compose_v of_o a_o and_o c_o and_o the_o double_a of_o b_o shall_v be_v prime_a the_o one_o to_o the_o other_o by_o the_o 30_o of_o the_o seven_o and_o therefore_o the_o number_n compose_v of_o a_o and_o c_o shall_v be_v prime_a to_o b_o take_v once_o for_o if_o any_o number_n shall_v measure_v the_o two_o number_n namely_o the_o number_n compose_v of_o a_o and_o c_o and_o the_o number_n b_o it_o shall_v also_o measure_v the_o number_n compose_v of_o a_o and_o c_o and_o the_o double_a of_o b_o by_o the_o 5._o common_a sentence_n of_o the_o seven_o which_o be_v not_o possible_a for_o that_o they_o be_v prove_v to_o be_v prime_a number_n here_o have_v i_o add_v a_o other_o demonstration_n of_o the_o former_a proposition_n after_o campane_n which_o prove_v that_o in_o number_n how_o many_o soever_o which_o be_v there_o prove_v only_o touch_v three_o number_n and_o the_o demonstration_n seement_n somewhat_o more_o perspicuous_a than_o theon_n demonstration_n and_o thus_o he_o put_v the_o proposition_n if_o number_n how_o many_o soever_o be_v in_o continual_a proportion_n be_v the_o least_o that_o have_v one_o &_o the_o same_o proportion_n with_o they_o every_o one_o of_o they_o shall_v be_v to_o the_o number_n compose_v of_o the_o rest_n prime_a second_o i_o say_v that_o this_o be_v so_o in_o every_o one_o of_o they_o namely_o that_o c_z be_v a_o prime_a number_n to_o the_o number_n compose_v of_o a_o b_o d._n for_o if_o not_o then_o as_o before_o let_v e_o measure_v c_o and_o the_o number_n compose_v of_o a_o b_o d_o which_o e_o shall_v be_v a_o number_n not_o prime_a either_o to_o fletcher_n or_o to_o g_z by_o the_o former_a proposition_n add_v by_o campane_n wherefore_o let_v h_o measure_v they_o and_o forasmuch_o as_o h_o measure_v e_o it_o shall_v also_o measure_v the_o whole_a a_o b_o c_o d_o who_o e_o measure_v and_o forasmuch_o as_o h_o measure_v one_o of_o these_o number_n fletcher_n or_o g_o it_o shall_v measure_v one_o of_o the_o extreme_n a_o or_o d_o which_o be_v produce_v of_o fletcher_n or_o g_o by_o the_o second_o of_o the_o eight_o if_o they_o be_v multiply_v into_o the_o mean_n l_o or_o k._n and_o moreover_o the_o same_o h_o shall_v measure_v the_o meame_n bc_o by_o the_o 5._o common_a sentence_n of_o the_o seven_o when_o as_o by_o supposition_n it_o measure_v either_o fletcher_n or_o g._n which_o measure_n b_o c_o by_o the_o second_o of_o the_o eight_o but_o the_o same_o h_o measure_v the_o whole_a a_o b_o c_o d_o as_o we_o have_v prove_v for_o that_o it_o measure_v e._n wherefore_o it_o shall_v also_o measure_v the_o residue_n namely_o the_o number_n compose_v of_o the_o extreme_n a_o and_o d_z by_o the_o 4._o common_a sentence_n of_o the_o seven_o and_o it_o measure_v one_o of_o these_o a_o or_o d_z for_o it_o measure_v one_o of_o these_o fletcher_n or_o g_o which_o produce_v a_o and_o d_z wherefore_o the_o same_o h_o shall_v measure_v one_o of_o these_o a_o or_o d_o and_o also_o the_o other_o of_o they_o by_o the_o former_a common_a sentence_n which_o number_n a_o and_o d_o be_v by_o the_o 3._o of_o the_o eight_o prime_n the_o one_o to_o the_o other_o which_o be_v absurd_a this_o may_v also_o be_v prove_v in_o every_o one_o of_o these_o number_n a_o b_o c_o d._n wherefore_o no_o number_n shall_v measure_v one_o of_o these_o number_n a_o b_o c_o d_o and_o the_o number_n compose_v of_o the_o rest_n wherefore_o they_o be_v prime_a the_o one_o to_o the_o other_o if_o therefore_o number_n how_o many_o soever_o etc_o etc_o which_o be_v require_v to_o be_v prove_v here_o as_o i_o promise_v i_o have_v add_v campane_n demonstration_n of_o those_o proposition_n in_o number_n which_o euclâ—de_n in_o the_o second_o book_n demonstrate_v in_o line_n and_o that_o in_o this_o place_n so_o much_o the_o rather_o for_o that_o theon_n as_o we_o see_v in_o the_o demonstration_n of_o the_o 15._o proposition_n seem_v to_o allege_v the_o 3._o &_o 4._o proposition_n of_o the_o second_o book_n which_o although_o they_o concern_v line_n only_o yet_o as_o we_o there_o declare_v and_o prove_v be_v they_o true_a also_o in_o number_n ¶_o the_o first_o proposition_n add_v by_o campane_n that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o one_o number_n into_o number_n how_o many_o soever_o be_v equal_a to_o that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o same_o number_n into_o the_o number_n compose_v of_o they_o this_o prove_v that_o in_o number_n which_o the_o first_o of_o the_o second_o prove_v touch_v line_n suppoâ—â—_n that_o the_o number_n a_o be_v multiply_v into_o the_o number_n b_o and_o into_o the_o number_n c_o demonstratiou_n and_o into_o the_o number_n d_o do_v produce_v the_o number_n e_o f_o and_o g._n then_o i_o say_v that_o the_o number_n produce_v of_o a_o multiply_a into_o the_o number_n compose_v of_o b_o c_o and_o d_z be_v equal_a to_o the_o number_n compose_v of_o e_o f_o and_o g._n for_o by_o the_o converse_n of_o the_o definition_n of_o a_o number_n multiply_v what_o part_n unity_n be_v of_o a_o the_o self_n same_o part_n be_v b_o of_o e_o and_o c_o of_o fletcher_n and_o also_o d_z of_o g._n wherefore_o by_o the_o 5._o of_o the_o seven_o what_o part_n unity_n be_v of_o a_o the_o self_n same_o part_n be_v the_o number_n compose_v of_o b_o c_o and_o d_o of_o the_o number_n compose_v of_o e_o f_o and_o g._n wherefore_o by_o the_o definition_n that_o which_o be_v produce_v of_o a_o into_o the_o number_n compose_v of_o b_o c_o d_o be_v equal_a to_o the_o number_n compose_v of_o e_o f_o g_o which_o be_v require_v to_o be_v prove_v the_o second_o proposition_n that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o number_n how_o many_o soever_o into_o one_o number_n be_v equal_a to_o that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n compose_v of_o they_o into_o the_o same_o number_n this_o be_v the_o converse_n of_o the_o former_a as_o if_o the_o â—â—â—bers_n â—_o and_o g_z and_o d_o multiply_v into_o the_o number_n a_o do_v produce_v the_o number_n e_o and_o f_o and_o g._n former_a then_o the_o number_n compose_v of_o b_o c_o d._n multiply_v into_o the_o number_n a_o shall_v produce_v the_o number_n compose_v of_o the_o number_n e_o f_o g._n which_o thing_n be_v easy_o prove_v by_o the_o 16._o of_o the_o seven_o and_o by_o the_o former_a proposition_n ¶_o the_o three_o proposition_n that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o number_n how_o many_o soever_o into_o other_o number_n how_o many_o soever_o be_v equal_a to_o that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n compose_v of_o those_o first_o number_n into_o the_o number_n compose_v of_o these_o latter_a number_n as_o if_o the_o number_n a_o b_o c_o do_v multiply_v the_o number_n d_o e_o f_o each_o one_o each_o other_o and_o if_o the_o number_n produce_v be_v add_v together_o then_o i_o say_v that_o the_o number_n compose_v of_o the_o number_n produce_v be_v equal_a to_o the_o number_n produce_v of_o the_o number_n compose_v of_o the_o number_n a_o b_o demonstration_n c_o into_o the_o number_n compose_v of_o the_o number_n d_o e_o f._n for_o by_o the_o former_a proposition_n that_o which_o be_v produce_v of_o the_o number_n compose_v of_o a_o b_o c_o into_o d_z be_v equal_a to_o that_o which_o be_v produce_v of_o every_o one_o of_o the_o say_a number_n into_o d_o and_o by_o the_o same_o reason_n that_o which_o be_v produce_v of_o the_o number_n compose_v of_o a_o b_o c_o into_o e_o be_v equal_a to_o that_o which_o be_v produce_v of_o every_o one_o of_o the_o say_a number_n into_o e_o and_o so_o likewise_o that_o which_o be_v produce_v of_o the_o number_n compose_v of_o a_o b_o c_o into_o fletcher_n be_v equal_a to_o that_o which_o be_v produce_v of_o every_o one_o of_o the_o say_a number_n into_o f._n but_o by_o the_o first_o of_o these_o proposition_n thâ—â—_n which_o be_v produce_v of_o the_o number_n compose_v of_o these_o number_n a_o b_o c_o into_o every_o one_o of_o these_o number_n d_o e_o fletcher_n be_v equal_a to_o that_o which_o be_v produce_v of_o the_o number_n compose_v into_o the_o number_n compose_v wherefore_o that_o be_v manifest_v which_o be_v require_v to_o be_v prove_v ¶_o the_o four_o proposition_n if_o a_o number_n be_v deuâ—â—â—d_v into_o part_n how_o many_o soever_o that_o number_n which_o be_v produce_v of_o the_o whole_a into_o himself_o be_v equal_a to_o that_o number_n which_o be_v produce_v of_o
number_n be_v make_v of_o unity_n and_o therefore_o can_v a_o point_n be_v a_o common_a part_n of_o all_o line_n and_o measure_v they_o as_o unity_n be_v a_o common_a part_n of_o all_o number_n and_o measure_v they_o unity_n take_v certain_a time_n make_v any_o number_n for_o there_o be_v not_o in_o any_o number_n infinite_a unity_n but_o a_o point_n take_v certain_a time_n yea_o as_o often_o as_o you_o listen_v never_o make_v any_o line_n for_o that_o in_o every_o line_n there_o be_v infinite_a point_n wherefore_o line_n figure_n and_o body_n in_o geometry_n be_v oftentimes_o incommensurable_a and_o irrational_a now_o which_o be_v rational_a and_o which_o irrational_a which_o commensurable_a and_o which_o incommensurable_a how_o many_o and_o how_o sundry_a sort_n and_o kind_n there_o be_v of_o they_o what_o be_v their_o nature_n passion_n and_o property_n do_v euclid_n most_o manifest_o show_v in_o this_o book_n and_o demonstrate_v they_o most_o exact_o this_o ten_o book_n have_v 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his_o book_n be_v definition_n definition_n 1_o magnitude_n commensurable_a be_v suchwhich_v one_o and_o the_o self_n same_o measure_n do_v measure_n first_o he_o show_v what_o magnitude_n be_v commensurable_a one_o to_o a_o other_o to_o the_o better_a and_o more_o clear_a understanding_n of_o this_o definition_n note_v that_o that_o measure_v whereby_o any_o magnitude_n be_v measure_v be_v less_o than_o the_o magnitude_n which_o it_o measure_v or_o at_o least_o equal_a unto_o it_o for_o the_o great_a can_v by_o no_o mean_n measure_v the_o less_o far_o it_o behove_v that_o that_o measure_n if_o it_o be_v equal_a to_o that_o which_o be_v measure_v take_v once_o make_v the_o magnitude_n which_o be_v measure_v if_o it_o be_v less_o than_o oftentimes_o take_v and_o repeat_v it_o must_v precise_o render_v and_o make_v the_o magnitude_n which_o it_o measure_v which_o thing_n in_o number_n be_v easy_o see_v for_o that_o as_o be_v before_o say_v all_o number_n be_v commensurable_a one_o to_o a_o other_o and_o although_o euclid_n in_o this_o definition_n comprehend_v purpose_o only_o magnitude_n which_o be_v continual_a quantity_n as_o be_v line_n superficiece_n and_o body_n yet_o undoubted_o the_o explication_n of_o this_o and_o such_o like_a place_n be_v apt_o to_o be_v seek_v of_o number_n as_o well_o rational_a as_o irrational_a for_o that_o all_o quantity_n commensurable_a have_v that_o proportion_n the_o one_o to_o the_o other_o which_o number_n have_v to_o number_n in_o number_n therefore_o 9_o and_o 12_o be_v commensurable_a because_o there_o be_v one_o common_a measure_n which_o measure_v they_o both_o namely_o the_o number_n 3_n first_o it_o measure_v 12_o for_o it_o be_v less_o than_o 12._o and_o be_v take_v certain_a time_n namely_o 4_o time_n it_o make_v exact_o 12_o 3_o time_n 4_o be_v 12_o it_o also_o measure_v 9_o for_o it_o be_v less_o than_o 9_o and_o also_o take_v certain_a time_n namely_o 3_o time_n it_o make_v precise_o 9_o 3_o time_n 3_o be_v 9_o 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thereof_o be_v 〈◊〉_d this_o iâ—_z also_o to_o be_v note_v that_o of_o line_n some_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o and_o some_o be_v commensurable_a the_o one_o to_o the_o other_o in_o power_n of_o line_n commensurable_a in_o length_n the_o one_o to_o the_o other_o be_v give_v a_o example_n in_o the_o declaration_n of_o the_o first_o definition_n namely_o the_o line_n a_o and_o b_o which_o be_v commensurable_a in_o length_n one_o and_o the_o self_n measure_n namely_o the_o line_n c_o measure_v the_o length_n of_o either_o of_o they_o of_o the_o other_o kind_n be_v give_v this_o definition_n here_o set_v for_o the_o open_n of_o which_o take_v this_o example_n let_v there_o be_v a_o certain_a line_n namely_o the_o line_n bc_o and_o let_v the_o square_n of_o that_o line_n be_v the_o square_n bcde_v suppose_v also_o a_o other_o line_n namely_o the_o line_n fh_o &_o let_v the_o square_v thereof_o be_v the_o square_a fhik_n and_o let_v a_o certain_a 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common_o of_o the_o most_o part_n of_o writer_n line_n but_o some_o there_o be_v which_o mislike_n that_o it_o shall_v be_v call_v a_o rational_a line_n &_o that_o not_o without_o just_a cause_n in_o the_o greek_a copy_n it_o be_v call_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d rete_fw-la which_o signify_v a_o thing_n that_o may_v be_v speak_v &_o express_v by_o word_n a_o thing_n certain_a grant_v and_o appoint_v wherefore_o flussates_n a_o man_n which_o bestow_v great_a travel_n and_o diligence_n in_o restore_v of_o these_o element_n of_o euclid_n leave_v this_o word_n rational_a call_v this_o line_n suppose_v and_o first_o set_v a_o line_n certain_a because_o the_o part_n thereof_o into_o which_o it_o be_v divide_v be_v certain_a certain_a and_o know_v and_o may_v be_v express_v by_o voice_n and_o also_o be_v count_v by_o number_n other_o line_n be_v to_o this_o line_n incommensurable_a who_o part_n be_v not_o distinct_o know_v but_o be_v uncertain_a nor_o can_v be_v express_v by_o name_n nor_o assign_v by_o number_n which_o be_v of_o other_o man_n call_v irrational_a he_o call_v uncertain_a and_o surd_a line_n petrus_n montaureus_n although_o he_o do_v not_o very_o well_o like_a of_o the_o name_n yet_o he_o alter_v it_o not_o but_o use_v it_o in_o all_o his_o book_n likewise_o will_v we_o do_v here_o for_o that_o the_o word_n have_v be_v and_o be_v so_o universal_o receive_v and_o therefore_o will_v we_o use_v the_o same_o name_n and_o call_v it_o a_o rational_a line_n for_o it_o be_v not_o so_o great_a a_o matter_n what_o name_n we_o geve_v to_o thing_n so_o that_o we_o full_o understand_v the_o thing_n which_o the_o name_n signify_v this_o rational_a line_n thus_o here_o define_v be_v the_o ground_n and_o foundation_n of_o all_o the_o proposition_n almost_o of_o this_o whole_a ten_o book_n book_n and_o chief_o from_o the_o ten_o proposition_n forward_o so_o that_o unless_o you_o first_o place_n this_o rational_a line_n and_o have_v a_o special_a and_o continual_a regard_n unto_o it_o before_o you_o begin_v any_o demonstration_n you_o shall_v not_o easy_o understand_v it_o for_o it_o be_v as_o it_o be_v the_o touch_n and_o trial_n of_o all_o other_o line_n by_o which_o it_o be_v know_v whether_o any_o of_o they_o be_v rational_a or_o not_o note_n and_o this_o may_v be_v call_v the_o first_o rational_a line_n purpose_n the_o line_n rational_a of_o purpose_n or_o a_o rational_a line_n set_v in_o the_o first_o place_n and_o so_o make_v distinct_a and_o sever_v from_o other_o rational_a line_n of_o which_o shall_v be_v speak_v afterward_o and_o this_o must_v you_o well_o commit_v to_o memory_n definition_n 6_o line_n which_o be_v commensurable_a to_o this_o line_n whether_o in_o length_n and_o power_n or_o in_o power_n only_o be_v also_o call_v rational_a this_o definition_n need_v no_o declaration_n at_o all_o but_o be_v easy_o perceive_v if_o the_o first_o definition_n be_v remember_v which_o show_v what_o magnitude_n be_v commensurable_a and_o the_o three_o which_o show_v what_o line_n be_v commensurable_a in_o power_n here_o notâ—_n how_o apt_o &_o natural_o euclid_n in_o this_o place_n use_v these_o word_n commensurable_a either_o in_o length_n and_o power_n or_o in_o power_n only_o because_o that_o all_o line_n which_o be_v commensurable_a in_o length_n be_v also_o commensurable_a in_o pour_v when_o he_o speak_v of_o line_n commensurable_a in_o length_n he_o ever_o add_v and_o in_o power_n but_o when_o he_o speak_v of_o line_n commensurable_a in_o power_n he_o add_v this_o word_n only_o and_o add_v not_o this_o word_n in_o length_n as_o he_o in_o the_o other_o add_v this_o word_n in_o power_n for_o not_o all_o line_n which_o be_v commensurable_a in_o power_n be_v straight_o way_n commensurable_a also_o in_o length_n of_o this_o definition_n take_v this_o example_n let_v the_o first_o line_n rational_a of_o purpose_n which_o be_v suppose_v and_o lay_v forth_o who_o part_n be_v certain_a &_o know_v and_o may_v be_v express_v name_v and_o number_v be_v ab_fw-la the_o quadrate_n whereof_o let_v be_v abcd_o then_o suppose_v again_o a_o other_o line_n namely_o the_o line_n of_o which_o let_v be_v commensurable_a both_o in_o length_n and_o in_o power_n to_o the_o first_o rational_a line_n that_o be_v as_o before_o be_v teach_v let_v one_o line_n measure_v the_o length_n of_o each_o line_n and_o also_o lâ—t_v one_o superficies_n measure_v the_o two_o square_n of_o the_o say_v two_o line_n as_o here_o in_o the_o example_n be_v suppose_v and_o also_o appear_v to_o the_o eye_n then_o be_v the_o line_n e_o fletcher_n also_o a_o rational_a line_n moreover_o if_o the_o line_n of_o be_v commensurable_a in_o power_n only_o to_o the_o rational_a line_n ab_fw-la first_o set_v and_o suppose_v so_o that_o no_o one_o line_n do_v measure_v the_o two_o line_n ab_fw-la and_o of_o as_o in_o example_n yâ—_n see_v to_o be_v for_o
quadrates_n but_o all_o other_o kind_n of_o rectiline_a figure_n plain_a plait_n &_o superficiese_n what_o so_o ever_o so_o that_o if_o any_o such_o figure_n be_v commensurable_a unto_o that_o rational_a squareâ—_n it_o be_v also_o rational_a as_o suppose_v that_o the_o square_a of_o the_o rational_a line_n which_o be_v also_o rational_a be_v abcd_o suppose_v 〈◊〉_d so_o some_o other_o square_a as_o the_o square_a efgh_o to_o be_v commensurable_a to_o the_o same_o they_o be_v the_o square_a efgh_o also_o rational_a so_o also_o if_o the_o rectiline_a figure_n klmn_n which_o be_v a_o figure_n on_o the_o one_o side_n long_o be_v commensurable_a unto_o the_o say_a square_a as_o be_v suppose_v in_o this_o exampleâ—_n it_o be_v also_o a_o rational_a superficies_n and_o so_o of_o all_o other_o superficiese_n 10_o such_o which_o be_v incommensurable_a unto_o it_o be_v irrational_a definition_n where_o it_o be_v say_v in_o this_o definition_n such_o which_o be_v incommensurable_a it_o be_v general_o to_o be_v take_v as_o be_v this_o word_n commensurable_a in_o the_o definition_n before_o for_o all_o superficiese_n whether_o they_o be_v square_n or_o figure_n on_o the_o one_o side_n long_o or_o otherwise_o what_o manner_n of_o right_n line_v figure_n so_o ever_o it_o be_v if_o they_o be_v incommensurable_a unto_o the_o rational_a square_n suppose_v they_o be_v they_o irrational_a as_o let_v thâ—_z square_v abcd_o be_v the_o square_n of_o the_o suppose_a rational_a line_n which_o square_n therefore_o be_v also_o rational_a suppose_v also_o also_o a_o other_o square_a namely_o the_o square_a e_o suppose_v also_o any_o other_o figure_n as_o for_o example_n sake_n a_o figure_n of_o one_o side_n long_o which_o let_v be_v f_o now_o if_o the_o square_a e_o and_o the_o figure_n fletcher_n be_v both_o incommensurable_a to_o the_o rational_a square_n abcd_o then_o be_v 〈◊〉_d of_o these_o figure_n e_o &_o fletcher_n irrational_a and_o so_o of_o other_o 11_o and_o these_o line_n who_o powere_n they_o be_v be_v irrational_a if_o they_o be_v square_n then_o be_v their_o side_n irrational_a if_o they_o be_v not_o square_n definition_n but_o some_o other_o rectiline_a figure_n then_o shall_v the_o line_n who_o square_n be_v equal_a to_o these_o rectiline_a figure_n be_v irrational_a suppose_v that_o the_o rational_a square_n be_v abcd._n suppose_v also_o a_o other_o square_a namely_o the_o square_a e_o which_o let_v be_v incommensurable_a to_o the_o rational_a square_n &_o therefore_o be_v it_o irrational_a and_o let_v the_o side_n or_o line_n which_o produce_v this_o square_n be_v the_o line_n fg_o then_o shall_v the_o line_n fg_o by_o this_o definition_n be_v a_o irrational_a line_n because_o it_o be_v the_o side_n of_o a_o irrational_a square_n let_v also_o the_o figure_n h_o be_v a_o figure_n on_o the_o one_o side_n long_o which_o may_v be_v any_o other_o rectiline_a figure_n rectangle_v or_o not_o rectangle_v triangle_n pentagone_fw-mi trapezite_fw-mi or_o what_o so_o ever_o else_o be_v incommensurable_a to_o the_o rational_a square_n abcd_o then_o because_o the_o figure_n h_o be_v not_o a_o square_a it_o have_v no_o side_n or_o root_n to_o produce_v it_o yet_o may_v there_o be_v a_o square_n make_v equal_a unto_o it_o for_o that_o all_o such_o figure_n may_v be_v reduce_v into_o triangle_n and_o so_o into_o square_n by_o the_o 14._o of_o the_o second_o suppose_v that_o the_o square_a queen_n be_v equal_a to_o the_o irrational_a figure_n h._n the_o side_n of_o which_o figure_n q_n let_v be_v the_o line_n kl_o then_o shall_v the_o line_n kl_o be_v also_o a_o irrational_a line_n because_o the_o power_n or_o square_v thereof_o be_v equal_a to_o the_o irrational_a figure_n h_o and_o thus_o conceive_v of_o other_o the_o like_a these_o irrational_a line_n and_o figure_n be_v the_o chief_a matter_n and_o subject_n which_o be_v entreat_v of_o in_o all_o this_o ten_o book_n the_o knowledge_n of_o which_o be_v deep_a and_o secret_a and_o pertain_v to_o the_o high_a and_o most_o worthy_a part_n of_o geometry_n wherein_o stand_v the_o pith_n and_o marry_v of_o the_o hole_n science_n the_o knowledge_n hereof_o bring_v light_n to_o all_o the_o book_n follow_v with_o out_o which_o they_o be_v hard_a and_o can_v be_v at_o all_o understand_v and_o for_o the_o more_o plainness_n you_o shall_v note_v that_o of_o irrational_a line_n there_o be_v diâ—ers_n sort_n and_o kind_n but_o they_o who_o name_n be_v set_v in_o a_o table_n here_o follow_v and_o be_v in_o number_n 13._o be_v the_o chief_a and_o in_o this_o ten_o book_n sufficient_o for_o euclides_n principal_a purpose_n discourse_v on_o a_o medial_a line_n a_o binomial_a line_n a_o first_o bimedial_a line_n a_o second_o bimedial_a line_n a_o great_a line_n a_o line_n contain_v in_o power_n a_o rational_a superficies_n and_o a_o medial_a superficies_n a_o line_n contain_v in_o power_n two_o medial_a superficiece_n a_o residual_a line_n a_o first_o medial_a residual_a line_n a_o second_o medial_a residual_a line_n a_o less_o line_n a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a of_o all_o which_o kind_n the_o definition_n together_o with_o there_o declaration_n shall_v be_v set_v here_o after_o in_o their_o due_a place_n ¶_o the_o 1._o theorem_a the_o 1._o proposition_n two_o unequal_a magnitude_n be_v give_v if_o from_o the_o great_a be_v take_v away_o more_o than_o the_o half_a and_o from_o the_o residue_n be_v again_o take_v away_o more_o than_o the_o half_a and_o so_o be_v do_v still_o continual_o there_o shall_v at_o length_n be_v leave_v a_o certain_a magnitude_n lesser_a than_o the_o less_o of_o the_o magnitude_n first_o give_v svppose_v that_o there_o be_v two_o unequal_a magnitude_n ab_fw-la and_o c_o of_o which_o let_v ab_fw-la be_v the_o great_a then_o i_o say_v that_o if_o from_o ab_fw-la be_v take_v away_o more_o than_o the_o half_a construction_n and_o from_o the_o residue_n be_v take_v again_o more_o than_o the_o half_a and_o so_o still_o continual_o there_o shall_v at_o the_o length_n be_v leave_v a_o certain_a magnitude_n lesser_a than_o the_o less_o magnitude_n give_v namely_o then_o c._n for_o forasmuch_o as_o c_z be_v the_o less_o magnitude_n therefore_o c_z may_v be_v so_o multiply_v that_o at_o the_o length_n it_o will_v be_v great_a than_o the_o magnitude_n ab_fw-la by_o the_o 5._o definition_n of_o the_o five_o book_n demonstration_n let_v it_o be_v so_o multiply_v and_o let_v the_o multiplex_n of_o c_z great_a than_o ab_fw-la be_v de._n and_o divide_v de_fw-fr into_o the_o part_n equal_a unto_o c_o which_o let_v be_v df_o fg_o and_o ge._n and_o from_o the_o magnitude_n ab_fw-la take_v away_o more_o than_o the_o half_a which_o let_v be_v bh_o and_o again_o from_o ah_o take_v away_o more_o than_o the_o half_a which_o let_v be_v hk_a and_o so_o do_v continual_o until_o the_o division_n which_o be_v in_o the_o magnitude_n ab_fw-la be_v equal_a in_o multitude_n unto_o the_o division_n which_o be_v in_o the_o magnitude_n de._n so_o that_o let_v the_o division_n ak_o kh_o and_o hb_o be_v equal_a in_o multitude_n unto_o the_o division_n df_o fg_o and_o ge._n and_o forasmuch_o as_o the_o magnitude_n de_fw-fr be_v great_a than_o the_o magnitude_n ab_fw-la and_o from_o de_fw-fr be_v take_v away_o less_o than_o the_o half_a that_o be_v eglantine_n which_o detraction_n or_o take_v away_o be_v understand_v to_o be_v do_v by_o the_o former_a division_n of_o the_o magnitude_n de_fw-fr into_o the_o part_n equal_a unto_o c_o for_o as_o a_o magnitude_n be_v by_o multiplication_n increase_v so_o be_v it_o by_o division_n diminish_v and_o from_o ab_fw-la be_v take_v away_o more_o than_o the_o half_a that_o be_v bh_o therefore_o the_o residue_n gd_o be_v great_a than_o the_o residue_n ha_o which_o thing_n be_v most_o true_a and_o most_o easy_a to_o conceive_v if_o we_o remember_v this_o principle_n that_o the_o residue_n of_o a_o great_a magnitude_n after_o the_o take_v away_o of_o the_o half_a or_o less_o than_o the_o half_a be_v ever_o great_a than_o the_o residue_n of_o a_o less_o magnitude_n after_o the_o take_v away_o of_o more_o than_o the_o half_a and_o forasmuch_o as_o the_o magnitude_n gd_o be_v great_a than_o the_o magnitude_n ha_o and_o from_o gd_o be_v take_v away_o the_o half_a that_o be_v gf_o and_o from_o ah_o be_v take_v away_o more_o than_o the_o half_a that_o be_v hk_o therefore_o the_o residue_n df_o be_v great_a than_o the_o residue_n ak_o by_o the_o foresay_a principle_n but_o the_o magnitude_n df_o be_v equal_a unto_o the_o magnitude_n c_o by_o supposition_n wherefore_o also_o the_o magnitude_n c_o be_v great_a than_o the_o magnitude_n ak_o wherefore_o the_o magnitude_n ak_o be_v less_o than_o the_o magnitude_n c._n wherefore_o of_o the_o magnitude_n
under_o the_o line_n a_o and_o b_o be_v to_o the_o square_n of_o the_o line_n b_o so_o be_v the_o number_n f_o to_o the_o number_n g._n but_o as_o the_o square_n of_o the_o line_n a_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o so_o be_v the_o number_n e_o to_o the_o number_n f._n wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o so_o be_v the_o number_n e_o to_o the_o number_n g._n but_o either_o of_o these_o numbers_z e_o and_o g_z be_v a_o square_a number_n for_o e_o be_v produce_v of_o the_o number_n c_o multiply_v into_o himself_o and_o g_o be_v produce_v of_o the_o number_n d_o multiply_v into_o himself_o wherefore_o the_o square_a of_o the_o line_n a_o have_v unto_o the_o square_n of_o the_o line_n b_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n which_o be_v require_v to_o be_v demonstrate_v an_o other_o demonstration_n of_o the_o same_o first_o part_n after_o montaureus_n suppose_v that_o there_o be_v two_o line_n commensurable_a in_o length_n a_o and_o b._n then_o i_o say_v that_o the_o square_n describe_v of_o those_o linâ—s_n shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n montaureus_n for_o forasmuch_o as_o the_o line_n a_o and_o b_o be_v commensurable_a in_o length_n they_o shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_v by_o the_o 5._o of_o this_o book_n let_v a_o be_v to_o b_o in_o duple_n proportion_n which_o be_v in_o such_o proportion_n as_o number_n be_v to_o number_n namely_o as_o 4._o be_v to_o 2_o and_o 6._o to_o 3_o and_o so_o of_o many_o other_o and_o by_o the_o sâ—cond_o oâ—_n the_o eight_o take_v three_o lest_o number_n in_o continual_a proportion_n and_o in_o duple_n proportion_n &_o let_v the_o same_o be_v the_o number_n 4.2.1_o wherefore_o by_o the_o corollary_n of_o the_o second_o of_o the_o eight_o the_o number_n 4._o and_o 1._o shall_v be_v square_v number_n for_o as_o 4._o be_v a_o square_a number_n produce_v of_o 2._o multiply_v into_o himself_o so_o be_v 1._o also_o a_o square_a number_n for_o it_o be_v produce_v of_o unity_n multiply_v into_o himself_o i_o say_v moreover_o that_o those_o be_v the_o square_a number_n who_o proportion_n the_o square_n of_o the_o line_n a_o and_o b_o have_v the_o one_o to_o the_o other_o for_o as_o the_o number_n 4._o be_v to_o the_o number_n 2._o so_o be_v the_o line_n a_o to_o the_o line_n b_o â—or_a either_o proportion_n be_v double_a by_o supposition_n but_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o square_a of_o the_o line_n a_o to_o the_o parallelogram_n contain_v under_o the_o line_n a_o and_o b_o by_o the_o first_o of_o the_o six_o wherefore_o as_o the_o number_n 4._o be_v to_o the_o number_n 2_o so_o shall_v the_o square_a of_o the_o line_n a_o be_v to_o the_o parallelogram_n contain_v under_o the_o line_n a_o and_o b._n likewise_o as_o the_o number_n 2._o be_v to_o the_o number_n 1._o so_o be_v the_o line_n a_o to_o the_o line_n b_o for_o either_o proportion_n be_v duple_n by_o supposition_n but_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n a_o and_o b_o to_o the_o square_n of_o the_o line_n b_o by_o the_o self_n same_o first_o of_o the_o six_o wherefore_o as_o the_o number_n 2._o be_v to_o 1._o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n a_o and_o b_o to_o the_o square_n of_o the_o line_n b_o wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o so_o be_v the_o number_n 4._o to_o 1._o which_o be_v prove_v to_o be_v square_a number_n former_a but_o now_o suppose_v that_o the_o square_a of_o the_o line_n a_o be_v unto_o the_o square_n of_o the_o line_n b_o as_o the_o square_a number_n produce_v of_o the_o number_n c_o be_v to_o the_o square_a number_n produce_v of_o the_o number_n d._n then_o i_o say_v that_o the_o line_n a_o &_o b_o be_v commensurable_a in_o length_n for_o for_o that_o as_o the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o so_o be_v the_o square_a number_n produce_v of_o the_o number_n c_o to_o the_o square_a number_n produce_v of_o the_o number_n d_o but_o the_o proportion_n of_o the_o square_n of_o the_o line_n a_o be_v unto_o the_o square_n of_o the_o line_n b_o double_v to_o that_o proportion_n which_o the_o line_n a_o have_v unto_o the_o line_n b_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o and_o the_o proportion_n of_o the_o square_a number_n which_o be_v produce_v of_o the_o number_n c_o to_o the_o square_a number_n produce_v of_o the_o number_n d_o be_v by_o the_o 11._o of_o the_o eight_o double_a to_o that_o proportion_n which_o the_o number_n c_o have_v unto_o the_o number_n d._n wherefore_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o number_n g_o to_o the_o number_n d._n wherefore_o the_o line_n a_o have_v unto_o the_o line_n b_o the_o same_o proportion_n that_o the_o number_n c_o have_v to_o the_o number_n d._n wherefore_o by_o the_o 6._o of_o this_o book_n the_o line_n a_o and_o b_o be_v commensurable_a in_o length_n which_o be_v require_v to_o be_v prove_v an_o other_o demonstration_n to_o prove_v the_o same_o â—vppose_v again_o that_o the_o square_a of_o the_o line_n a_o have_v unto_o the_o square_n of_o the_o line_n b_o part_n the_o same_o proportion_n that_o the_o square_a number_n e_o have_v to_o the_o square_a number_n g._n then_o i_o say_v that_o the_o line_n a_o and_o b_o be_v commensurable_a in_o length_n for_o suppose_v that_o the_o side_n of_o the_o square_a number_n e_o be_v the_o number_n c_o &_o let_v the_o side_n of_o the_o square_a number_n g_o be_v the_o number_n d_o and_o let_v the_o number_n c_o multiply_v the_o number_n d_o produce_v the_o number_n f._n wherefore_o these_o number_n e_o f_o g_o be_v in_o continual_a proportion_n and_o in_o the_o same_o proportion_n that_o the_o number_n c_o be_v to_o the_o number_n d_o by_o the_o 17._o and_o 18._o of_o the_o seven_o demonstration_n and_o forasmuch_o as_o the_o mean_a proportional_a between_o the_o square_n of_o the_o line_n a_o and_o b_o be_v that_o which_o be_v contain_v under_o the_o line_n a_o and_o b._n which_o though_o it_o may_v brief_o be_v prove_v yet_o we_o take_v it_o as_o now_o and_o likewise_o the_o mean_a proportional_a between_o the_o number_n e_o and_o g_o be_v the_o number_n fletcher_n by_o the_o 20._o of_o the_o seven_o therefore_o as_o the_o square_n of_o the_o line_n a_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o so_o be_v the_o number_n e_o to_o the_o number_n fletcher_n and_o as_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o be_v to_o the_o square_n of_o the_o line_n b_o so_o be_v the_o number_n f_o to_o the_o number_n g._n but_o as_o the_o square_n of_o the_o line_n a_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o so_o be_v the_o line_n a_o to_o the_o line_n b_o by_o the_o first_o of_o the_o six_o wherefore_o the_o line_n a_o and_o b_o be_v in_o the_o same_o proportion_n that_o the_o number_n e_o be_v to_o the_o number_n fletcher_n that_o be_v that_o the_o number_n c_o be_v to_o the_o number_n d._n wherefore_o the_o line_n a_o and_o b_o be_v commensurable_a in_o length_n by_o the_o 6._o of_o this_o ten_o which_o be_v require_v to_o be_v prove_v but_o now_o suppose_v that_o the_o line_n a_o and_o b_o be_v incommensurable_a in_o length_n then_o i_o say_v that_o the_o square_a of_o the_o line_n a_o have_v not_o unto_o the_o square_n of_o the_o line_n b_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n part_n for_o if_o the_o square_a of_o the_o line_n a_o have_v unto_o the_o square_n of_o the_o line_n b_o the_o same_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n they_o shall_v the_o line_n a_o and_o b_o be_v commensurable_a in_o length_n by_o the_o second_o part_n of_o this_o proportion_n but_o by_o supposition_n they_o be_v not_o wherefore_o the_o square_a of_o the_o line_n a_o have_v not_o unto_o the_o square_n of_o the_o line_n b_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n which_o be_v require_v to_o be_v prove_v â—_o again_o suppose_v that_o
parallelogram_n shall_v afterward_o be_v teach_v in_o the_o 27._o and_o 28._o proposition_n of_o this_o book_n ¶_o a_o corollary_n hereby_o it_o be_v manifest_a that_o a_o rectangle_n parallelogram_n contain_v under_o two_o right_a line_n be_v the_o mean_a proportional_a between_o the_o square_n of_o the_o say_v line_n corollary_n as_o it_o be_v manifest_a by_o the_o first_o of_o the_o six_o that_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v the_o mean_a proportional_a between_o the_o square_n ad_fw-la and_o cx_o this_o corollary_n be_v put_v after_o the_o 53._o proposition_n of_o this_o book_n as_o a_o assumpt_n and_o there_o demonstrate_v which_o there_o in_o his_o place_n you_o shall_v find_v but_o because_o it_o follow_v of_o this_o proposition_n so_o evident_o and_o brief_o without_o far_a demonstration_n i_o think_v it_o not_o amiss_o here_o by_o the_o way_n to_o note_v it_o ¶_o the_o 23._o theorem_a the_o 26._o proposition_n a_o medial_a superficies_n exceed_v not_o a_o medial_a superficies_n by_o a_o rational_a superficies_n for_o if_o it_o be_v possible_a let_v ab_fw-la be_v a_o medial_a superficies_n exceed_v ac_fw-la be_v also_o a_o medial_a superficies_n by_o db_o be_v a_o rational_a superficies_n and_o let_v there_o be_v put_v a_o rational_a right_a line_n ef._n construction_n and_o upon_o the_o line_n of_o apply_v a_o rectangle_n parallelogram_n fh_o equal_a unto_o the_o medial_a superficies_n ab_fw-la who_o other_o side_n let_v be_v eh_o and_o from_o the_o parallelogram_n fh_o take_v away_o the_o parallelogram_n fg_o equal_a unto_o the_o medial_a superficies_n ac_fw-la absurdity_n wherefore_o by_o the_o three_o common_a sentence_n the_o residue_n bd_o be_v equal_a to_o the_o residue_n kh_o but_o by_o supposition_n the_o superficies_n db_o be_v rational_a wherefore_o the_o superficies_n kh_o be_v also_o rational_a and_o forasmuch_o as_o either_o of_o these_o superficiece_n ab_fw-la and_o ac_fw-la be_v medial_a and_o ab_fw-la be_v equal_a unto_o fh_o &_o ac_fw-la unto_o fg_o therefore_o either_o of_o these_o superficiece_n fh_a and_o fg_o be_v medial_a and_o they_o be_v apply_v upon_o the_o rational_a line_n ef._n wherefore_o by_o the_o 22._o of_o the_o ten_o either_o of_o these_o line_n he_o and_o eglantine_n be_v rational_a &_o incommensurable_a in_o length_n unto_o the_o line_n ef._n and_o forasmuch_o as_o the_o superficies_n db_o be_v rational_a and_o the_o superficies_n kh_o be_v equal_a unto_o it_o therefore_o kh_o be_v also_o rational_a and_o it_o be_v apply_v upon_o the_o rational_a line_n of_o for_o it_o be_v apply_v upon_o the_o line_n gk_o which_o be_v equal_a to_o the_o line_n of_o wherefore_o by_o the_o 20._o of_o the_o ten_o the_o line_n gh_o be_v rational_a and_o commensurable_a in_o length_n unto_o the_o line_n gk_o but_o the_o line_n gk_o be_v equal_a to_o the_o line_n ef._n wherefore_o the_o line_n gh_o be_v rational_a and_o commensurable_a in_o length_n unto_o the_o line_n ef._n but_o the_o line_n eglantine_n be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ef._n wherefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n eglantine_n be_v incommensurable_a in_o length_n unto_o the_o line_n gh_o and_o as_o the_o line_n eglantine_n be_v to_o the_o line_n gh_o so_o be_v the_o square_a of_o the_o line_n eglantine_n to_o the_o parallelogram_n contain_v under_o the_o line_n eglantine_n and_o gh_o by_o the_o assumpt_n put_v before_o the_o 21._o of_o the_o ten_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a of_o the_o line_n eglantine_n be_v incommensurable_a unto_o the_o parallelogram_n contain_v under_o the_o line_n eglantine_n and_o gh_o but_o unto_o the_o square_n of_o the_o line_n eglantine_n be_v commensurable_a the_o square_n of_o the_o line_n eglantine_n and_o gh_o for_o either_o of_o they_o be_v rational_a as_o have_v before_o be_v prove_v wherefore_o the_o square_n of_o the_o line_n eglantine_n and_o gh_o be_v incommensurable_a unto_o the_o parallelogram_n contain_v under_o the_o line_n eglantine_n and_o gh_o but_o unto_o the_o parallelogram_n contain_v under_o the_o line_n eglantine_n and_o gh_o be_v commensurable_a that_o which_o be_v contain_v under_o the_o line_n fg_o and_o gh_o twice_o for_o they_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n namely_o as_o unity_n be_v to_o the_o number_n 2_o or_o as_o 2._o be_v to_o 4_o and_o therefore_o by_o the_o 6._o of_o this_o book_n they_o be_v commensurable_a wherefore_o by_o the_o 13._o of_o the_o ten_o the_o square_n of_o the_o line_n eglantine_n and_o gh_o be_v incommensurable_a unto_o that_o which_o be_v contain_v under_o the_o line_n eglantine_n and_o gh_o twice_o this_o be_v more_o brieâ—ly_o conclude_v by_o the_o corollary_n of_o the_o 13._o of_o the_o ten_o but_o the_o square_n of_o the_o line_n eglantine_n and_o gh_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n eglantine_n and_o gh_o twice_o be_v equal_a to_o the_o square_n of_o the_o line_n eh_o by_o the_o 4._o of_o the_o second_o wherefore_o the_o square_a of_o the_o line_n eh_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n eglantine_n and_o gh_o by_o the_o 16._o of_o the_o ten_o but_o the_o square_n of_o the_o line_n fg_o &_o gh_o be_v rational_a wherefore_o the_o square_a of_o the_o line_n eh_o be_v irrational_a wherefore_o the_o line_n also_o eh_o be_v irrational_a but_o it_o have_v before_o be_v prove_v to_o be_v rational_a which_o be_v impossible_a wherefore_o a_o medial_a superficies_n exceed_v not_o a_o medial_a superficies_n by_o a_o rational_a superficies_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 4._o problem_n the_o 27._o proposition_n to_o find_v out_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o rational_a parallelogram_n let_v there_o be_v put_v two_o rational_a line_n commensurable_a in_o power_n only_o namely_o a_o and_o b._n construction_n and_o by_o the_o 13._o of_o the_o six_o take_v the_o mean_a proportional_a between_o the_o line_n a_o and_o b_o and_o let_v the_o same_o line_n be_v c._n and_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o by_o the_o 12._o of_o the_o six_o let_v the_o line_n c_o be_v to_o the_o line_n d._n demonstration_n and_o forasmuch_o as_o a_o and_o b_o be_v rational_a line_n commensurable_a in_o power_n only_o therefore_o by_o the_o 21._o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o that_o be_v the_o square_a of_o the_o line_n c._n for_o the_o square_n of_o the_o line_n c_o be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n a_o anâ—_z b_o by_o the_o 17._o of_o the_o six_o be_v medial_a â—herfore_o c_z also_o be_v a_o medial_a line_n and_o for_o that_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o line_n c_o to_o the_o line_n d_o therefore_o as_o the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o so_o be_v the_o square_a of_o the_o line_n c_o to_o the_o square_n of_o the_o line_n d_o by_o the_o 22._o of_o the_o six_o but_o the_o square_n of_o the_o line_n a_o and_o b_o be_v commensurable_a for_o the_o liâ—â—s_v a_o and_o b_o aâ—e_fw-fr suppose_a to_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o also_o the_o square_n of_o the_o line_n c_o and_o d_o be_v commensurable_a by_o the_o 10._o of_o the_o ten_o wherefore_o the_o line_n c_o and_o d_o be_v commensurable_a in_o power_n only_o and_o c_o be_v a_o medial_a line_n wherefore_o by_o the_o 23._o of_o the_o ten_o d_z also_o be_v a_o medial_a line_n wherefore_o c_z and_o d_o be_v medial_a line_n commensurable_a in_o power_n only_o now_o also_o i_o say_v that_o they_o contain_v a_o rational_a parallelogram_n for_o for_o that_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o line_n c_o to_o the_o line_n d_o therefore_o alternate_o also_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n a_o be_v to_o the_o line_n c_o so_o be_v the_o line_n b_o to_o the_o line_n d._n but_o as_o the_o line_n a_o be_v to_o the_o line_n c_o so_o be_v the_o line_n c_o to_o the_o line_n b_o wherefore_o as_o the_o line_n c_o be_v to_o the_o line_n b_o so_o be_v the_o line_n b_o to_o the_o line_n d._n wherefore_o the_o parallelogram_n contain_v under_o the_o line_n c_o and_o d_o be_v equal_a to_o the_o square_n of_o the_o line_n b._n but_o the_o square_n of_o the_o line_n b_o be_v rational_a wherefore_o the_o parallelogram_n which_o be_v contain_v under_o the_o line_n c_o and_o d_o be_v also_o rational_a wherefore_o there_o be_v find_v out_o medial_a line_n commensurablâ—_n in_o powâ—r_n only_o contain_v a_o rational_a parallelogrammeâ—_n which_o 〈◊〉_d require_v to_o be_v do_v the_o 5._o problem_n the_o
line_n be_v when_o the_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o the_o less_o part_n by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o it_o and_o neither_o part_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v as_o suppose_v the_o line_n ce_fw-fr to_o be_v a_o binomial_a line_n who_o part_n be_v join_v together_o in_o the_o point_n d_o and_o let_v the_o square_n of_o the_o line_n cd_o the_o great_a part_n exceed_v the_o square_n of_o the_o less_o part_n de_fw-fr by_o the_o square_n of_o the_o line_n fg_o and_o let_v the_o line_n fg_o be_v commensurable_a in_o length_n to_o the_o line_n cd_o the_o great_a part_n of_o the_o binomial_a moreover_o let_v neither_o the_o great_a part_v cd_o nor_o the_o less_o part_n de_fw-fr be_v commensurable_a in_o length_n to_o the_o rational_a line_n ab_fw-la then_o be_v the_o line_n ce_fw-fr by_o this_o definition_n a_o three_o binomial_a line_n a_o four_o binomial_a line_n be_v definition_n when_o the_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o the_o less_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o great_a part_n and_o the_o great_a be_v also_o commensurable_a in_o length_n to_o the_o rational_a line_n as_o let_v the_o line_n ce_fw-fr be_v a_o binomial_a line_n who_o part_n let_v be_v cd_o &_o de_fw-fr &_o let_v the_o square_n of_o the_o line_n cd_o the_o great_a part_n exceed_v the_o square_n of_o the_o line_n de_fw-fr the_o less_o by_o the_o square_n of_o the_o line_n fg._n and_o let_v the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n cd_o the_o great_a let_v also_o the_o line_n cd_o the_o great_a part_n be_v commensurable_a in_o length_n unto_o the_o rational_a line_n ab_fw-la then_o by_o this_o definition_n the_o line_n ce_fw-fr be_v a_o four_o binomial_a line_n a_o five_o binomial_a line_n be_v definition_n when_o the_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o the_o less_o part_n by_o the_o square_n of_o a_o line_n incommensurable_a unto_o it_o in_o length_n and_o the_o less_o part_n also_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v as_o suppose_v that_o ce_fw-fr be_v a_o binomial_a line_n who_o great_a part_n let_v be_v cd_o and_o let_v the_o less_o part_n be_v de._n and_o let_v the_o square_n of_o the_o line_n cd_o exceed_v the_o square_n of_o the_o line_n de_fw-fr by_o the_o square_n of_o the_o line_n fg_o which_o let_v be_v incommensurable_a in_o length_n unto_o the_o line_n cd_o the_o great_a part_n of_o the_o binomial_a line_n and_o let_v the_o line_n de_fw-fr the_o second_o part_n of_o the_o binomial_a line_n be_v commensurable_a in_o length_n unto_o the_o rational_a line_n ab_fw-la so_o be_v the_o line_n ce_fw-fr by_o this_o definition_n a_o five_o binomial_a line_n a_o six_o binomial_a line_n be_v definition_n when_o the_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o the_o less_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o it_o and_o neither_o part_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v as_o let_v the_o line_n ce_fw-fr be_v a_o binomial_a line_n divide_v into_o his_o name_n in_o the_o point_n d._n the_o square_n of_o who_o great_a part_n cd_o let_v exceed_v the_o square_n of_o the_o less_o part_n de_fw-fr by_o the_o square_n of_o the_o line_n fg_o and_o let_v the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n cd_o the_o great_a part_n of_o the_o binomial_a line_n let_v also_o neither_o cd_o the_o great_a part_n nor_o de_fw-fr the_o less_o part_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n ab_fw-la and_o so_o by_o this_o definition_n the_o line_n ce_fw-fr be_v a_o six_o binomial_a line_n so_o you_o see_v that_o by_o these_o definition_n &_o their_o example_n and_o declaration_n all_o the_o kind_n of_o binomial_a line_n be_v make_v very_o plain_a this_o be_v to_o be_v note_v that_o here_o be_v nothing_o speak_v of_o those_o line_n both_o who_o portion_n aâ—e_v commensurable_a in_o length_n unto_o the_o rational_a line_n first_o set_v for_o that_o such_o line_n can_v be_v binomial_a line_n â—or_a binomial_a line_n be_v compose_v of_o two_o rational_a line_n commensurable_a in_o power_n only_o by_o the_o 36._o of_o this_o book_n but_o line_n both_o who_o portion_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n first_o set_v be_v not_o binomial_a line_n for_o that_o the_o part_n of_o such_o line_n shall_v by_o the_o 12._o of_o this_o book_n be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o and_o so_o shall_v they_o not_o be_v such_o line_n as_o be_v require_v to_o the_o composition_n of_o a_o binomial_a line_n moreover_o such_o line_n shall_v not_o be_v irrational_a but_o rational_a for_o that_o they_o be_v commensurable_a tâ—â—ch_o of_o the_o part_n whereof_o they_o be_v compose_v by_o the_o 15._o oâ—_n this_o book_n and_o therefore_o they_o shall_v be_v rational_a for_o that_o the_o line_n which_o composâ—_n they_o be_v rational_a ¶_o the_o 13._o problem_n the_o 48._o proposition_n to_o find_v out_o a_o first_o binomial_a line_n composition_n take_v two_o number_n ac_fw-la and_o cb_o &_o let_v they_o be_v such_o that_o the_o number_n which_o be_v make_v of_o they_o both_o add_v together_o namely_o ab_fw-la have_v unto_o one_o of_o they_o 〈◊〉_d unto_o bc_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n â—ut_z unto_o the_o other_o namely_o unto_o ca_n let_v it_o not_o have_v that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n such_o as_o be_v every_o square_a number_n which_o may_v be_v divide_v into_o a_o square_a number_n and_o into_o a_o number_n not_o square_a construction_n take_v also_o a_o certain_a rational_a line_n and_o let_v the_o same_o be_v d._n and_o unto_o the_o line_n d_o let_v the_o line_n of_o be_v commensurable_a in_o length_n wherefore_o the_o line_n of_o be_v rational_a and_o as_o the_o number_n ab_fw-la be_v to_o the_o number_n ac_fw-la so_o let_v the_o square_n of_o the_o line_n of_o be_v to_o the_o square_n of_o a_o other_o â—iâ—e_n namely_o of_o fg_o by_o the_o corollary_n of_o the_o six_o of_o the_o ten_o wherefore_o the_o square_a of_o the_o line_n of_o have_v to_o the_o square_n of_o the_o line_n fg_o that_o proportion_n that_o number_n have_v to_o number_v wherefore_o the_o square_a of_o the_o line_n of_o be_v commensurable_a to_o the_o square_n of_o the_o line_n fg_o by_o the_o 6._o of_o this_o book_n and_o the_o line_n of_o be_v rational_a demonstration_n wherefore_o the_o line_n fg_o also_o be_v rational_a and_o forasmuch_o as_o the_o number_n ab_fw-la have_v not_o to_o the_o number_n ac_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n neither_o shall_v the_o square_a of_o the_o line_n of_o have_v to_o the_o square_n of_o the_o line_n fg_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n of_o be_v incommensurable_a in_o length_n to_o the_o line_n fg_o by_o the_o 9_o of_o this_o book_n wherefore_o the_o line_n of_o and_o fg_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n eglantine_n be_v a_o binomial_a line_n by_o the_o 36._o of_o the_o ten_o i_o say_v also_o that_o it_o be_v a_o â—irst_n binomial_a line_n for_o for_o that_o as_o the_o 〈◊〉_d basilius_n be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n â—g_z but_o the_o number_n basilius_n be_v great_a than_o the_o number_n ac_fw-la wherefore_o the_o square_a of_o the_o line_n â—f_n be_v also_o great_a than_o the_o square_a oâ—_n the_o line_n fg._n unto_o the_o square_n of_o the_o line_n of_o let_v the_o square_n of_o the_o line_n fg_o and_o h_o be_v equal_a which_o how_o to_o find_v out_o be_v teach_v in_o the_o assumpt_n put_v â—ftâ—r_v the_o 13._o of_o the_o tâ—nth_o and_o fâ—r_v that_o as_o thâ—_z number_n basilius_n be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n fg_o therefore_o by_o coâ—uersion_n or_o eâ—ersion_n of_o proportion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o as_o the_o number_n ab_fw-la be_v to_o the_o number_n bc_o so_o be_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n h._n but_o the_o number_n ab_fw-la have_v to_o the_o number_n bc_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o also_o the_o square_a of_o the_o line_n of_o have_v to_o the_o square_n of_o the_o line_n h_o
therefore_o be_v divide_v into_o two_o unequal_a part_n the_o square_n which_o be_v make_v of_o the_o unequal_a part_n be_v great_a they_o the_o rectangle_n parallelogram_n contain_v under_o the_o unequal_a part_n twice_o which_o be_v require_v to_o be_v demonstrate_v in_o number_n i_o need_v not_o to_o have_v so_o allege_v for_o the_o 17._o of_o the_o seven_o have_v confirm_v the_o double_n to_o be_v one_o to_o the_o other_o as_o their_o single_n be_v but_o in_o our_o magnitude_n it_o likewise_o be_v true_a and_o evident_a by_o alternate_a proportion_n thus_o as_o the_o parallelogram_n of_o the_o line_n ac_fw-la and_o cb_o be_v to_o his_o double_a so_o be_v the_o square_a of_o the_o line_n ad_fw-la to_o his_o double_a each_o be_v half_a wherefore_o alternate_o as_o the_o parallelogram_n be_v to_o the_o square_n so_o be_v the_o parallelogram_n his_o double_a to_o the_o double_a of_o the_o square_n but_o the_o parallelogram_n be_v prove_v less_o than_o thâ—_z square_a wherefore_o his_o double_a be_v less_o than_o the_o square_a his_o double_a by_o the_o 14._o of_o the_o five_o this_o assumpt_n be_v in_o some_o book_n not_o read_v for_o that_o in_o manner_n it_o seem_v to_o be_v all_o one_o with_o that_o which_o be_v put_v after_o the_o 39_o of_o this_o book_n but_o for_o the_o diverse_a manner_n of_o demonstrate_v it_o be_v necessary_a follow_v for_o the_o fear_n of_o invention_n be_v thereby_o further_v and_o though_o zambert_n do_v in_o the_o demonstration_n hereof_o omit_v that_o which_o p._n montaureus_n can_v not_o supplyâ—_n but_o plain_o doubt_v of_o the_o sufficiency_n of_o this_o proof_n yet_o m._n do_v you_o by_o only_a allegation_n of_o the_o due_a place_n of_o credit_n who_o pith_n &_o forceâ—_n theon_n his_o word_n do_v contain_v have_v restore_v to_o the_o demonstration_n sufficient_o both_o light_n and_o authority_n as_o you_o may_v perceive_v and_o chief_o such_o may_v judge_v who_o can_v compare_v this_o demonstration_n here_o thus_o furnish_v with_o the_o greek_a of_o theon_n or_o latin_a translation_n of_o zambert_n ¶_o the_o 42._o theorem_a the_o 60._o proposition_n composition_n the_o square_a of_o a_o binomial_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o first_o binomial_a line_n svppose_v that_o the_o line_n ab_fw-la be_v a_o binomial_a line_n and_o let_v it_o be_v suppose_v to_o be_v divide_v into_o his_o name_n in_o the_o point_n c_o so_fw-mi that_o let_v ac_fw-la be_v the_o great_a name_n construction_n and_o take_v a_o rational_a line_n deâ—_n â—_z and_o by_o the_o â—4_o of_o the_o first_o unto_o the_o line_n de_fw-fr apply_v a_o rectangle_n parallelogram_n defg_n equal_a to_o the_o square_n of_o theâ—line_n ab_fw-la and_o make_n in_o breadth_n the_o line_n dg_o then_o i_o say_v that_o the_o line_n dg_o be_v a_o first_o binomial_a line_n unto_o the_o line_n de_fw-fr apply_v the_o parallelogram_n dh_fw-mi equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o unto_o the_o line_n kh_o which_o be_v equal_a to_o the_o line_n de_fw-fr apply_v the_o parallelogram_n kl_o equal_a to_o the_o square_n of_o the_o line_n bc._n wherefore_o the_o residue_n namely_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la &_o cb_o twice_o be_v equal_a to_o the_o residue_n namely_o to_o the_o parallelogram_n mf_n by_o 〈◊〉_d 4â—_o of_o the_o second_o divide_v by_o the_o 1â—_o of_o the_o first_o the_o line_n mg_o into_o two_o equal_a part_n in_o the_o point_n n._n demonstration_n and_o by_o the_o 31._o of_o the_o first_o draw_v the_o line_n nx_o parallel_v to_o either_o of_o these_o line_n ml_o and_o gf_o wherefore_o either_o of_o these_o parallelogram_n mx_o and_o nf_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o once_o by_o the_o 15._o of_o the_o five_o and_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o binomial_a line_n and_o be_v divide_v into_o his_o name_n in_o the_o point_n c_o therefore_o the_o â—ines_n ac_fw-la and_o cb_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v rational_a and_o therefore_o commensurable_a the_o one_o to_o the_o other_o wherefore_o by_o the_o 15._o of_o the_o ten_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o be_v commensurable_a to_o either_o of_o the_o square_n of_o the_o line_n ac_fw-la or_o cb_o wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o lineâ—_z ac_fw-la and_o cb_o add_v together_o be_v rational_a and_o it_o be_v equal_a to_o the_o parallelogram_n dl_o by_o construction_n wherefore_o the_o parallelogram_n dl_o be_v rational_a and_o it_o be_v apply_v unto_o the_o rational_a line_n de_fw-fr wherefore_o by_o the_o 20._o of_o the_o ten_o the_o line_n dm_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n de._n again_o forasmuch_o as_o the_o line_n ac_fw-la and_o cb_o be_v rational_a commensurable_a in_o power_n only_o therefore_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o that_o be_v the_o parallelogram_n mf_n be_v medial_a by_o the_o 21_o of_o the_o ten_o and_o it_o be_v apply_v unto_o the_o rational_a line_n ml_o wherefore_o the_o line_n mg_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ml_o by_o the_o 22._o of_o this_o ten_o that_o be_v to_o the_o line_n de._n but_o the_o line_n md_o be_v prove_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n de._n wherefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n dm_o be_v incommensurable_a in_o length_n to_o the_o line_n mg_o wherefore_o the_o line_n dm_o &_o mâ—_n be_v rational_a commensurable_a in_o power_n only_o line_n wherefore_o by_o the_o 36._o of_o the_o ten_o the_o whole_a line_n dg_o be_v a_o binomial_a line_n now_o rest_v to_o prove_v that_o it_o be_v a_o first_o binomial_a line_n forasmuch_o as_o by_o the_o the_o assumpt_n go_v before_o the_o 54._o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o be_v the_o mean_a proportional_a between_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o therefore_o the_o parallelogram_n mx_o be_v the_o mean_a proportional_a between_o the_o parallelogram_n dh_o and_o kl_o wherefore_o as_o the_o parallelogame_n dhâ—_n â—_z be_v to_o the_o parallelogram_n mx_o so_o be_v the_o parallelogram_n mx_o to_o the_o parallelogram_n kl_o that_o be_v as_o the_o line_n dk_o be_v to_o the_o line_n mn_o so_o be_v the_o line_n mn_v to_o the_o line_n mk_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n dk_o and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n mn_v and_o forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la be_v commensurable_a to_o the_o square_n of_o the_o line_n cb_o the_o parallelogram_n dh_o be_v commensurable_a to_o the_o prarallelograme_n kl_o wherefore_o by_o the_o 1._o of_o the_o six_o and_o 10._o of_o the_o ten_o the_o line_n de_fw-fr be_v commensurable_a in_o length_n to_o the_o line_n km_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v great_a than_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o the_o assumpt_n go_v before_o this_o proposition_n or_o by_o the_o assumpt_n after_o this_o 39_o of_o the_o ten_o therefore_o the_o parallelogram_n dl_o be_v great_a than_o the_o parallelogram_n mf_n wherefore_o by_o the_o first_o of_o the_o six_o the_o line_n dm_o be_v great_a than_o the_o line_n mg_o and_o that_o which_o be_v contain_v under_o the_o line_n dk_o and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n mn_o that_o be_v to_o the_o four_o part_n of_o the_o square_n of_o the_o line_n mg_o but_o by_o the_o 17._o of_o the_o ten_o if_o there_o be_v two_o unequal_a right_a line_n and_o if_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a to_o the_o four_o part_n of_o the_o square_n make_v of_o the_o less_o line_n and_o want_v in_o figure_n by_o a_o square_a if_o also_o the_o parallelogram_n thus_o apply_v deâ—ide_v the_o line_n whereupon_o it_o be_v apply_v into_o part_n commensurable_a in_o length_n then_o shall_v the_o great_a line_n be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o great_a wherefore_o the_o line_n dm_o be_v in_o power_n more_o than_o the_o line_n mg_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n dm_o and_o the_o line_n dm_o and_o mg_o be_v prove_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n dm_o be_v prove_v the_o great_a name_n and_o commensurable_a in_o length_n to_o the_o rational_a line_n give_v de._n wherefore_o by_o
five_o binomial_a line_n but_o if_o neither_o of_o the_o line_n ae_n nor_o ebb_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v neither_o also_o of_o the_o line_n cf_o nor_o fd_o shall_v be_v commensurable_a in_o length_n to_o the_o same_o and_o so_o either_o of_o the_o line_n ab_fw-la and_o cd_o shall_v be_v a_o six_o binomial_a line_n a_o line_n therefore_o commensurable_a in_o length_n to_o a_o binomial_a line_n be_v also_o a_o binomial_a line_n of_o the_o self_n same_o order_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 49._o theorem_a the_o 67._o proposition_n a_o line_n commensurable_a in_o length_n to_o a_o bimedial_a line_n be_v also_o a_o bimedial_a line_n and_o of_o the_o self_n same_o order_n svppose_v that_o the_o line_n ab_fw-la be_v a_o bimedial_a line_n and_o unto_o the_o line_n ab_fw-la let_v the_o line_n cd_o be_v commensurable_a in_o length_n then_o i_o say_v that_o the_o line_n cd_o be_v a_o bimedial_a line_n and_o of_o the_o self_n order_n that_o the_o line_n ab_fw-la be_v divide_v the_o line_n ab_fw-la into_o his_o part_n in_o the_o point_n e._n construction_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o bimedial_a line_n and_o be_v divide_v into_o his_o part_n in_o the_o point_n e_o therefore_o by_o the_o 37._o and_o 38._o of_o the_o ten_o the_o line_n ae_n and_o ebb_n be_v medial_n commensurable_a in_o power_n only_o and_o by_o the_o 12._o of_o the_o six_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o let_v the_o line_n ae_n be_v to_o the_o line_n cf._n wherefore_o by_o the_o 19_o of_o the_o five_o the_o residue_n demonstration_n namely_o the_o line_n ebb_n be_v to_o the_o residue_n namely_o to_o the_o line_n fd_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o but_o the_o line_n ab_fw-la be_v commensurable_a in_o length_n to_o the_o line_n cd_o wherefore_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o line_n cf_o and_o the_o line_n ebb_n to_o the_o line_n fd._n now_o the_o line_n ae_n and_o ebb_n be_v medial_a wherefore_o by_o the_o 23._o of_o the_o ten_o the_o line_n cf_o and_o fd_o be_v also_o medial_a and_o for_o that_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd._n but_o the_o line_n ae_n and_o ebb_n be_v commensurable_a in_o power_n only_o wherefore_o the_o line_n cf_o and_o fd_o be_v also_o commensurable_a in_o power_n only_o and_o it_o be_v prove_v that_o they_o be_v medial_a wherefore_o the_o line_n cd_o be_v a_o bimedial_a line_n i_o say_v also_o that_o it_o be_v of_o the_o self_n same_o order_n that_o the_o line_n ab_fw-la be_v for_o for_o that_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd_o but_o as_o the_o line_n cf_o be_v to_o fd_o so_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd_o by_o the_o first_o of_o the_o six_o therefore_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o by_o the_o 11._o of_o the_o five_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd_o but_o as_o ae_n be_v to_o ebb_v so_o by_o the_o 1._o of_o the_o six_o be_v the_o square_a of_o the_o line_n ae_n to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n therefore_o by_o the_o 11._o of_o the_o five_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n so_o be_v the_o square_a of_o the_o line_n cf_o to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o square_n of_o the_o line_n cf_o so_o be_v that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n to_o that_o which_o be_v contain_v under_o the_o line_n cf_o &_o fd._n but_o the_o square_n of_o the_o line_n ae_n be_v commensurable_a to_o the_o square_n of_o the_o line_n cf_o because_o ae_n and_o cf_o be_v commensurable_a in_o length_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n in_o commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd._n if_o therefore_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v rational_a that_o be_v if_o the_o line_n ab_fw-la be_v a_o first_o bimedial_a line_n that_o also_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o be_v rational_a wherefore_o also_o the_o line_n cd_o be_v a_o first_o bimedial_a line_n but_o if_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v medial_a that_o be_v if_o the_o line_n ab_fw-la be_v a_o second_o bimedial_a line_n that_o also_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o be_v medial_a wherefore_o also_o the_o line_n cd_o be_v a_o second_o bimedial_a line_n wherefore_o the_o line_n ab_fw-la and_o cd_o be_v both_o of_o one_o and_o the_o self_n same_o order_n which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o flussates_n but_o first_o note_v by_o p._n montaâ—reus_n a_o line_n commensurable_a in_o power_n only_o to_o a_o bimedial_a line_n be_v also_o a_o bimedial_a line_n and_o of_o the_o self_n same_o order_n suppose_v that_o ab_fw-la be_v a_o bimedial_a line_n either_o a_o first_o or_o a_o second_o whereunto_o let_v the_o line_n gd_v be_v commensurable_a in_o power_n only_o flussetes_n take_v also_o a_o rational_a line_n ez_o upon_o which_o by_o the_o 45._o of_o the_o first_o apply_v a_o rectangle_n parallelogram_n equal_a to_o the_o square_n of_o the_o line_n ab_fw-la which_o let_v be_v ezfc_a and_o let_v the_o rectangle_n parallelogram_n cfih_n be_v equal_a to_o the_o square_n of_o the_o line_n gd_o and_o forasmuch_o as_o upon_o the_o rational_a line_n ez_o be_v apply_v a_o rectangle_n parallelogram_n of_o equal_a to_o the_o square_n of_o a_o first_o bimedial_a line_n therefore_o the_o other_o side_n thereof_o namely_o ec_o be_v a_o second_o binomial_a line_n by_o the_o 61._o of_o this_o book_n and_o forasmuch_o as_o by_o supposition_n the_o square_n of_o the_o line_n ab_fw-la &_o gd_o be_v commensurable_a therefore_o the_o parallelogram_n of_o and_o ci_o which_o be_v equal_a unto_o they_o be_v also_o commensurable_a and_o therefore_o by_o the_o 1._o of_o the_o six_o the_o line_n ec_o and_o change_z be_v commensurable_a in_o length_n but_o the_o line_n ec_o be_v a_o second_o binomial_a line_n wherefore_o the_o line_n change_z be_v also_o a_o second_o binomial_a line_n by_o the_o 66._o of_o this_o book_n and_o forasmuch_o as_o the_o superficies_n ci_o be_v contain_v under_o a_o rational_a line_n ez_o or_o cf_o and_o a_o second_o binomial_a line_n change_z therefore_o the_o line_n which_o contain_v it_o in_o power_n namely_o the_o line_n gd_o be_v a_o first_o bimedial_a line_n by_o the_o 55._o of_o this_o book_n and_o so_o be_v the_o line_n gd_o in_o the_o self_n same_o order_n of_o bimedial_a line_n that_o the_o line_n ab_fw-la be_v the_o like_a demonstration_n also_o will_v serve_v if_o the_o line_n ab_fw-la be_v suppose_v to_o bâ—_z a_o second_o bimedial_a line_n for_o so_o shall_v it_o make_v the_o breadth_n ec_o a_o three_o binomial_a line_n whereunto_o the_o line_n change_z shall_v be_v commensurable_a in_o length_n and_o therefore_o ch_n also_o shall_v be_v a_o three_o binomial_a line_n by_o mean_n whereof_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ci_o namely_o the_o line_n gd_o shall_v also_o be_v a_o second_o bimedial_a line_n wherefore_o a_o line_n commensurable_a either_o in_o length_n or_o in_o power_n only_o to_o a_o bimedial_a line_n be_v also_o a_o bimedial_a line_n of_o the_o self_n same_o order_n but_o so_o be_v it_o not_o of_o necessity_n in_o binomial_a line_n for_o if_o their_o power_n only_o be_v commensurable_a it_o follow_v not_o of_o necessity_n that_o they_o be_v binomiall_n of_o one_o and_o the_o self_n same_o order_n note_n but_o they_o be_v each_o binomialls_n either_o of_o the_o three_o first_o kind_n or_o of_o the_o three_o last_o as_o for_o example_n suppose_v that_o ab_fw-la be_v a_o first_o binomial_a line_n who_o great_a name_n let_v be_v agnostus_n and_o unto_o ab_fw-la let_v the_o dz_o be_v commensurable_a in_o power_n only_o then_o i_o say_v that_o the_o line_n dz_o be_v not_o of_o the_o self_n same_o order_n that_o the_o line_n ab_fw-la be_v for_o if_o it_o be_v possible_a let_v the_o line_n dz_o be_v of_o the_o self_n same_o order_n that_o the_o line_n ab_fw-la be_v wherefore_o the_o line_n dz_o may_v
in_o like_a sort_n be_v divide_v as_o the_o line_n ab_fw-la be_v by_o that_o which_o have_v be_v demonstrate_v in_o the_o 66._o proposition_n of_o this_o bookeâ—_n let_v it_o be_v so_o divide_v in_o the_o point_n e._n wherefore_o it_o can_v not_o be_v so_o divide_v in_o any_o other_o point_n by_o the_o 42â—_n of_o this_o book_n and_o for_o that_o the_o line_n ab_fw-la â—â—_o to_o the_o line_n dz_o as_o the_o line_n agnostus_n be_v to_o the_o line_n de_fw-fr but_o the_o line_n agnostus_n &_o de_fw-fr namely_o the_o great_a name_n be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o by_o the_o 10._o of_o this_o book_n for_o that_o they_o be_v commensurable_a in_o length_n to_o 〈◊〉_d and_o the_o self_n same_o rational_a line_n by_o the_o first_o definition_n of_o binomial_a line_n wherefore_o the_o line_n ab_fw-la and_o dz_o be_v commensurable_a in_o length_n by_o the_o 13._o of_o this_o book_n but_o by_o supposition_n they_o be_v commensurable_a in_o power_n only_o which_o be_v impossible_a the_o self_n same_o demonstration_n also_o will_v serve_v if_o we_o suppose_v the_o line_n ab_fw-la to_o be_v a_o second_o binomial_a line_n for_o the_o less_o name_n gb_o and_o ez_o be_v commensurable_a in_o length_n to_o one_o and_o the_o self_n same_o rational_a line_n shall_v also_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o and_o therefore_o the_o line_n ab_fw-la and_o dz_o which_o be_v in_o the_o self_n same_o proportion_n with_o they_o shall_v also_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o which_o be_v contrary_a to_o the_o supposition_n far_o if_o the_o square_n of_o the_o line_n ab_fw-la and_o dz_o be_v apply_v unto_o the_o rational_a line_n cf_o namely_o the_o parallelogram_n et_fw-la and_o hl_o they_o shall_v make_v the_o breadthes_fw-mi change_z and_o hk_o first_o binomial_a line_n of_o what_o order_n soever_o the_o line_n ab_fw-la &_o dz_o who_o square_n be_v apply_v unto_o the_o rational_a line_n be_v by_o the_o 60._o of_o this_o book_n wherefore_o it_o be_v manifest_a that_o under_o a_o rational_a line_n and_o a_o first_o binomial_a line_n be_v confuse_o contain_v all_o the_o power_n of_o binomial_a line_n by_o the_o 54._o of_o this_o book_n wherefore_o the_o only_a commensuration_n of_o the_o power_n do_v not_o of_o necessity_n bring_v forth_o one_o and_o the_o self_n same_o order_n of_o binomial_a line_n the_o self_n same_o thing_n also_o may_v be_v prove_v if_o the_o line_n ab_fw-la and_o dz_o be_v suppose_v to_o be_v a_o four_o or_o five_o binomial_a line_n who_o power_n only_o be_v conmmensurable_a namely_o that_o they_o shall_v as_o the_o first_o bring_v forth_o binomial_a line_n of_o diverse_a order_n now_o forasmuch_o as_o the_o power_n of_o the_o line_n agnostus_n and_o gb_o and_o de_fw-fr and_o ez_o be_v commensurable_a &_o proportional_a it_o be_v manifest_a that_o if_o the_o line_n agnostus_n be_v in_o power_n more_o than_o the_o line_n gb_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o agnostus_n the_o line_n de_fw-fr also_o shall_v be_v in_o power_n more_o than_o the_o line_n ez_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n de_fw-fr by_o the_o 16._o of_o this_o book_n and_o so_o shall_v the_o two_o line_n ab_fw-la and_o dz_o be_v each_o of_o the_o three_o first_o binomial_a line_n but_o if_o the_o line_n agnostus_n be_v in_o power_n more_o than_o the_o line_n gb_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n agnostus_n the_o line_n de_fw-fr shall_v also_o be_v in_o powâ—r_a 〈◊〉_d than_o the_o line_n ez_o by_o the_o square_n of_o a_o line_n incomensurable_a in_o length_n unto_o the_o line_n de_fw-fr by_o the_o self_n same_o proposition_n and_o so_o shall_v eke_v of_o the_o line_n ab_fw-la and_o dz_o be_v of_o the_o three_o last_o binomial_a line_n but_o why_o it_o be_v not_o so_o in_o the_o three_o and_o six_o binomial_a line_n the_o reason_n be_v for_o that_o in_o they_o neither_o of_o the_o nameâ—_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v fc_n ¶_o the_o 50._o theorem_a the_o 68_o proposition_n a_o line_n commensurable_a to_o a_o great_a line_n be_v also_o a_o great_a line_n svppose_v that_o the_o line_n ab_fw-la be_v a_o great_a line_n and_o unto_o the_o line_n ab_fw-la let_v the_o line_n cd_o be_v commensurable_a construction_n then_o i_o say_v that_o the_o line_n cd_o also_o be_v a_o great_a line_n divide_v the_o line_n ab_fw-la into_o his_o part_n in_o the_o point_n e._n wherefore_o by_o the_o 39_o of_o the_o ten_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a and_o let_v the_o rest_n of_o the_o construction_n be_v in_o this_o as_o it_o be_v in_o the_o former_a and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o demonstration_n so_o be_v the_o line_n ae_n to_o the_o line_n cf_o &_o thâ—_z line_n ebb_n to_o the_o line_n fd_o but_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n cd_o by_o supposition_n wherefore_o the_o line_n ae_n be_v commensurable_a to_o the_o line_n cf_o and_o the_o line_n ebb_n to_o the_o line_n fd._n and_o for_o that_o as_o the_o line_n ae_n be_v to_o the_o line_n cf_o so_o be_v the_o line_n ebb_n to_o the_o line_n fd._n therefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd._n wherefore_o by_o composition_n also_o by_o the_o 18._o of_o the_o five_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cd_o to_o the_o line_n fd._n wherefore_o by_o the_o 22._o of_o the_o six_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v to_o the_o square_n of_o the_o line_n ebb_n so_o be_v the_o square_a of_o the_o line_n cd_o to_o the_o square_n of_o the_o line_n fd._n and_o in_o like_a sort_n may_v we_o prove_v that_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v to_o the_o square_n of_o the_o line_n ae_n so_o be_v the_o square_a of_o the_o line_n cd_o to_o the_o square_n of_o the_o line_n cf._n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v to_o the_o square_n of_o the_o line_n ae_n and_o ebb_n so_o be_v the_o square_a of_o the_o line_n cd_o to_o the_o square_n of_o the_o line_n cf_o and_o fd._n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v to_o the_o square_n of_o the_o line_n cd_o so_o be_v the_o square_n of_o the_o line_n ae_n and_o ebb_n to_o the_o square_n of_o the_o line_n cf_o and_o fd._n but_o the_o square_n of_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o square_n of_o the_o line_n cd_o for_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n cd_o by_o supposition_n wherefore_o also_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v commensurable_a to_o the_o square_n of_o the_o line_n cf_o and_o fd._n but_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a and_o be_v add_v together_o be_v rational_a wherefore_o the_o square_n of_o the_o line_n cf_o and_o fd_o be_v incommensurable_a &_o be_v add_v together_o be_v also_o rational_a and_o in_o like_a sort_n may_v we_o prove_v that_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n twice_o be_v commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o twice_o but_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n twice_o be_v medial_a wherefore_o also_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o twice_o be_v medial_a wherefore_o the_o line_n cf_o and_o fd_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a wherefore_o by_o the_o 39_o of_o the_o ten_o the_o whole_a line_n cd_o be_v irrational_a &_o be_v call_v a_o great_a line_n a_o line_n therefore_o commensurable_a to_o a_o great_a line_n be_v also_o a_o great_a line_n an_o other_o demonstration_n of_o peter_n montaureus_n to_o prove_v the_o same_o suppose_v that_o the_o line_n ab_fw-la be_v a_o great_a line_n and_o unto_o it_o let_v the_o line_n cd_o be_v commensurable_a any_o way_n that_o be_v either_o both_o in_o length_n and_o in_o power_n or_o else_o in_o power_n only_o montaureus_n then_o i_o say_v that_o the_o line_n cd_o also_o be_v a_o great_a
power_n a_o rational_a and_o a_o medial_a which_o be_v require_v to_o be_v demonstrate_v an_o other_o demonstration_n of_o the_o same_o after_o campane_n suppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a campane_n whereunto_o let_v the_o line_n gd_v be_v commensurable_a either_o in_o length_n and_o power_n or_o in_o power_n only_o then_o i_o say_v that_o the_o line_n gd_o be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a take_v a_o rational_a line_n ez_o upon_o which_o by_o the_o 45._o of_o the_o first_o apply_v a_o rectangle_n parallelogram_n ezfc_n equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o upon_o the_o line_n cf_o which_o be_v equal_a to_o the_o line_n ez_o apply_v the_o parallelogram_n fchi_n equal_a to_o the_o square_n of_o the_o line_n gdâ—_n and_o let_v the_o breadth_n of_o the_o say_a parallelogram_n be_v the_o line_n eglantine_n and_o ch._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n gd_o at_o the_o least_o in_o power_n only_o therefore_o the_o parallelogram_n of_o and_o fh_o which_o be_v equal_a to_o their_o square_n shall_v be_v commensurable_a wherefore_o by_o the_o 1._o of_o the_o six_o the_o right_a line_n ec_o and_o change_z be_v commensurable_a in_o length_n and_o forasmuch_o as_o the_o parallelogram_n of_o which_o be_v equal_a to_o the_o square_n of_o the_o line_n aâ—_z which_o contain_v in_o power_n â—_o rational_a and_o a_o medial_a be_v apply_v upon_o the_o rational_a ez_o make_v in_o breadth_n the_o line_n ec_o therefore_o the_o line_n ec_o be_v a_o five_o binomial_a line_n by_o the_o 64._o of_o this_o book_n unto_o which_o line_n ec_o the_o line_n change_z be_v commensurable_a in_o length_n wherefore_o by_o the_o 66._o of_o this_o book_n the_o line_n change_z be_v also_o a_o five_o binomial_a line_n and_o forasmuch_o as_o the_o superficies_n ci_o be_v contain_v under_o the_o rational_a line_n ez_o that_o be_v cf_o and_o a_o five_o binomall_a line_n change_z therefore_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ci_o which_o by_o supposition_n be_v the_o line_n gd_o be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a by_o the_o 58._o of_o this_o book_n a_o line_n therefore_o commensurable_a to_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a etc_n etc_n ¶_o the_o 52._o theorem_a the_o 70._o proposition_n a_o line_n commensurable_a to_o a_o line_n contain_v in_o power_n two_o medial_n be_v also_o a_o line_n contain_v in_o power_n two_o medial_n svppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n and_o unto_o the_o line_n ab_fw-la let_v the_o line_n cd_o be_v commensurable_a whether_o in_o length_n &_o power_n or_o in_o power_n only_o then_o i_o say_v that_o the_o line_n cd_o be_v a_o line_n contain_v in_o power_n two_o medial_n forasmuch_o as_o the_o line_n ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n construction_n let_v it_o be_v divide_v into_o his_o part_n in_o the_o point_n e._n wherefore_o by_o the_o 41._o of_o the_o ten_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o that_o also_o which_o be_v contain_v under_o they_o medial_a and_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n &_o ebb_n be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n let_v the_o self_n same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a demonstration_n and_o in_o like_a sort_n may_v we_o prove_v that_o the_o line_n cf_o &_o fd_o be_v incommensurable_a in_o power_n and_o that_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_v add_v together_o be_v commensurable_a to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o and_o that_o that_o also_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o be_v medial_a by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o and_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o be_v medial_a by_o the_o same_o corollary_n â—_o and_o moreover_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o &_o fd_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o the_o line_n cd_o be_v a_o line_n contain_v in_o power_n two_o medial_n which_o be_v require_v to_o be_v prove_v ¶_o a_o assumpt_n add_v by_o montaureus_n assumpt_n that_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o be_v thus_o prove_v for_o because_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_v add_v together_o be_v to_o the_o square_n of_o the_o line_n ae_n so_o be_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o to_o the_o square_n of_o the_o line_n cf_o as_o it_o be_v prove_v in_o the_o proposition_n go_v before_o therefore_o alternate_o as_o that_o which_o be_v make_v of_o the_o square_n of_o ae_n and_o ebb_v add_v together_o be_v to_o that_o which_o be_v make_v of_o the_o square_n of_o cf_o and_o fd_o add_v together_o so_o be_v the_o square_a of_o the_o line_n ae_n to_o the_o square_n of_o the_o line_n cf._n but_o before_o namely_o in_o the_o 68_o proposition_n it_o be_v prove_v that_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o square_n of_o the_o line_n cf_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o alternate_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n so_o be_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd._n but_o by_o supposition_n that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n &_o ebb_n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o be_v incommensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd_o which_o be_v require_v to_o be_v prove_v an_o other_o demonstration_n after_o campane_n suppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n whereunto_o let_v the_o line_n gd_v be_v commensurable_a either_o in_o length_n and_o in_o power_n or_o in_o power_n only_o campanâ—_n then_o i_o say_v that_o the_o line_n gd_o be_v a_o line_n contain_v in_o power_n two_o medial_n let_v the_o same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a and_o forasmuch_o as_o the_o parallelogram_n of_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o be_v apply_v upon_o a_o rational_a line_n ez_o it_o make_v the_o breadth_n ec_o a_o six_o binomial_a line_n by_o the_o 65._o of_o this_o book_n and_o forasmuch_o as_o the_o parallelogram_n of_o &_o ci_o which_o be_v equal_a unto_o the_o square_n of_o the_o line_n ab_fw-la and_o gd_o which_o be_v suppose_v to_o be_v commensurable_a be_v commensurable_a therefore_o the_o line_n ec_o and_o change_z be_v commensurable_a in_o length_n by_o the_o first_o of_o the_o six_o but_o ec_o be_v a_o six_o binomial_a line_n wherefore_o change_z also_o be_v a_o six_o binomial_a line_n by_o the_o 66._o of_o this_o book_n and_o forasmuch_o as_o the_o superficies_n ci_o be_v contain_v under_o the_o rational_a line_n cf_o and_o a_o six_o binomial_a line_n change_z therefore_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ci_o namely_o the_o line_n gd_o be_v a_o line_n contain_v in_o power_n two_o medial_n by_o the_o 59_o of_o
measure_v the_o square_a number_n produce_v of_o ef._n wherefore_o also_o by_o the_o 14._o of_o the_o eight_o the_o number_n g_z measure_v the_o number_n of_o and_o the_o number_n g_o also_o measure_v itself_o wherefore_o the_o number_n g_z measure_v these_o number_n of_o and_o g_z when_o yet_o they_o be_v prime_a the_o one_o to_o the_o other_o which_o be_v impossible_a wherefore_o the_o diameter_n a_o be_v not_o commensurable_a in_o length_n to_o the_o side_n b._n wherefore_o it_o be_v incommensurable_a which_o be_v require_v to_o be_v demonstrate_v an_o other_o demonstration_n after_o flussas_n suppose_v that_o upon_o the_o line_n ab_fw-la be_v describe_v a_o square_n who_o diameter_n let_v be_v the_o line_n ac_fw-la then_o i_o say_v that_o the_o side_n ab_fw-la be_v incommensurable_a in_o length_n unto_o the_o diameter_n ac_fw-la forasmuch_o as_o the_o line_n ab_fw-la and_o bc_o be_v equal_a therefore_o the_o square_a of_o the_o line_n ac_fw-la be_v double_a to_o the_o square_n of_o the_o line_n ab_fw-la by_o the_o 47._o of_o the_o first_o take_v by_o the_o 2._o of_o the_o eight_o number_n how_o many_o soever_o in_o continual_a proportion_n from_o unity_n and_o in_o the_o proportion_n of_o the_o square_n of_o the_o line_n ab_fw-la and_o ac_fw-la which_o let_v be_v the_o number_n d_o e_o f_o g._n and_o forasmuch_o as_o the_o first_o from_o unity_n namely_o e_o be_v no_o square_a number_n for_o that_o it_o be_v a_o prime_a number_n neither_o be_v also_o any_o other_o of_o the_o say_a number_n a_o square_a number_n except_o the_o three_o from_o unity_n and_o so_o all_o the_o rest_n leving_fw-mi one_o between_o by_o the_o 10._o of_o the_o nine_o wherefore_o d_z be_v to_o e_o or_o e_o to_o fletcher_n or_o fletcher_n to_o g_z in_o that_o proportion_n that_o a_o square_a number_n be_v to_o a_o number_n not_o square_a wherefore_o by_o the_o corollary_n of_o the_o 25._o of_o the_o eight_o they_o be_v not_o in_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n be_v to_o a_o square_a number_n wherefore_o neither_o also_o have_v the_o square_n of_o the_o line_n ab_fw-la and_o ac_fw-la which_o be_v in_o the_o same_o proportion_n that_o porportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o by_o the_o 9_o of_o this_o book_n their_o side_n namely_o the_o side_n ab_fw-la and_o the_o diameter_n ac_fw-la be_v incommensurable_a in_o length_n the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v this_o demonstration_n i_o think_v good_a to_o add_v for_o that_o the_o former_a demonstration_n seem_v not_o so_o full_a and_o they_o be_v think_v of_o some_o to_o be_v none_o of_o theon_n as_o also_o the_o proposition_n to_o be_v none_o of_o euclides_n here_o follow_v a_o instruction_n by_o some_o studious_a and_o skilful_a grecian_a perchance_o theon_n which_o teach_v we_o of_o far_a use_n and_o fruit_n of_o these_o irrational_a line_n see_v that_o there_o be_v find_v out_o right_a line_n incommensurable_a in_o length_n the_o one_o to_o the_o â—ther_n as_o the_o line_n a_o and_o b_o there_o may_v also_o be_v find_v out_o many_o other_o magnitude_n have_v length_n and_o breadth_n such_o as_o be_v plain_a superficiece_n which_o shall_v be_v incommensurable_a the_o one_o to_o the_o other_o for_o if_o by_o the_o 13._o of_o the_o six_o between_o the_o line_n a_o and_o b_o there_o be_v take_v the_o mean_a proportional_a line_n namely_o c_z then_o by_o the_o second_o corollary_n of_o the_o 20._o of_o the_o six_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o figure_n describe_v upon_o the_o line_n a_o to_o the_o figure_n describe_v upon_o the_o line_n c_o be_v both_o like_a and_o in_o like_a sort_n describe_v that_o be_v whether_o they_o be_v square_n which_o be_v always_o like_o the_o one_o to_o the_o other_o or_o whether_o they_o be_v any_o other_o like_o rectiline_a figure_n or_o whether_o they_o be_v circle_n about_o the_o diameter_n a_o and_o c._n for_o circle_n have_v that_o proportion_n the_o one_o to_o the_o other_o that_o the_o square_n of_o their_o diameter_n have_v by_o the_o 2._o of_o the_o twelve_o wherefore_o by_o the_o second_o part_n of_o the_o 10._o of_o the_o ten_o the_o â—_z describe_v upon_o the_o line_n a_o and_o c_o being_n like_o and_o in_o like_a sort_n describe_v be_v incommensurable_a the_o one_o to_o the_o other_o wherefore_o by_o this_o mean_n there_o be_v find_v out_o superficieces_o incommensurable_a the_o one_o to_o the_o other_o in_o like_a sort_n there_o may_v be_v find_v out_o figure_n commensurable_a the_o one_o to_o the_o other_o if_o you_o put_v the_o line_n a_o and_o b_o to_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o and_o see_v that_o it_o be_v so_o now_o let_v we_o also_o prove_v that_o even_o in_o soliâ—es_n also_o or_o body_n there_o be_v some_o commensurable_a the_o one_o to_o the_o other_o and_o other_o some_o incommensurable_a the_o one_o to_o the_o other_o for_o if_o from_o each_o of_o the_o square_n of_o the_o line_n a_o and_o b_o or_o from_o any_o other_o rectiline_a figure_n equal_a to_o these_o square_n be_v erect_v solid_n of_o equal_a altitude_n whether_o those_o solid_n be_v compose_v of_o equidistant_a supersiciece_n or_o whether_o they_o be_v pyramid_n or_o prism_n thosâ—_n solid_n sâ—_n erected_a shall_v be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v by_o the_o 32._o oâ—_n the_o eleven_o and_o 5._o and_o 6._o of_o the_o twelve_o howbeit_o there_o be_v no_o such_o proposition_n concern_v prism_n and_o so_o if_o the_o base_n of_o the_o solid_n bâ—_z commensurable_a the_o one_o to_o the_o other_o the_o solid_n also_o shall_v be_v commensurable_a the_o one_o to_o the_o other_o and_o if_o the_o base_n be_v incommensurable_a the_o one_o to_o the_o other_o the_o solid_n also_o shall_v be_v incommensurable_a the_o one_o to_o the_o other_o by_o the_o 10._o of_o the_o ten_o and_o if_o there_o be_v two_o circle_n a_o and_o b_o and_o upon_o each_o of_o the_o circle_n be_v erect_v cones_n or_o cilinder_n of_o equal_a altitude_n those_o cones_n &_o cilinder_n sâ—all_v be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o the_o circle_n be_v which_o be_v their_o base_n by_o the_o 11._o of_o the_o twelve_o and_o so_o if_o the_o circle_n be_v commensurable_a the_o one_o to_o the_o other_o the_o cones_n and_o cilinder_n also_o shall_v be_v commensurable_a the_o one_o to_o the_o other_o but_o if_o the_o circle_n be_v incommensurable_a the_o one_o to_o the_o other_o the_o cones_n also_o and_o cilinder_n shall_v be_v incommensurable_a the_o one_o to_o the_o other_o by_o the_o 10._o of_o the_o ten_o wherefore_o it_o be_v manifest_a that_o not_o only_o in_o line_n and_o superâ—icieces_n but_o also_o in_o solid_n or_o body_n be_v find_v commensurabilitie_n or_o incommensurability_n a_o advertisement_n by_o john_n dee_n although_o this_o proposition_n be_v by_o euclid_n to_o this_o book_n allot_v as_o by_o the_o ancient_a graecian_n publish_v under_o the_o name_n of_o aristoteles_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d it_o will_v seem_v to_o be_v and_o also_o the_o property_n of_o the_o same_o agreeable_a to_o the_o matter_n of_o this_o book_n and_o the_o proposition_n itself_o so_o famous_a in_o philosophy_n and_o logic_n as_o it_o be_v will_v in_o manner_n crave_v his_o elemental_a place_n in_o this_o ten_o book_n yet_o the_o dignity_n &_o perfectionâ—_n of_o mathematical_a method_n can_v not_o allow_v it_o here_o as_o in_o due_a order_n follow_v but_o most_o apt_o after_o the_o 9_o proposition_n of_o this_o book_n as_o a_o corollary_n of_o the_o last_o part_n thereof_o and_o undoubted_o the_o proposition_n have_v for_o this_o 2000_o year_n be_v notable_o regard_v among_o the_o greek_n philosopher_n and_o before_o aristotle_n time_n be_v conclude_v with_o the_o very_a same_o inconvenience_n to_o the_o gaynesayer_n that_o the_o first_o demonstration_n here_o induce_v namely_o odd_a number_n to_o be_v equal_a to_o even_o as_o may_v appearâ—_n in_o aristotle_n work_n name_v analitica_fw-la prima_fw-la the_o first_o book_n and_o 40._o chapter_n but_o else_o in_o very_a many_o place_n of_o his_o work_n he_o make_v mention_n of_o the_o proposition_n evident_a also_o it_o be_v that_o euclid_n be_v about_o aristotle_n time_n and_o in_o that_o age_n the_o most_o excellent_a geometrician_n among_o the_o greek_n wherefore_o see_v it_o be_v so_o public_a in_o his_o time_n so_o famous_a and_o so_o appertain_v to_o the_o property_n of_o this_o book_n it_o be_v most_o likely_a both_o to_o be_v know_v to_o euclid_n and_o also_o to_o have_v be_v by_o he_o in_o apt_a order_n place_v but_o of_o the_o disorder_v of_o it_o can_v remain_v no_o doubt_n if_o you_o consider_v in_o zamberts_n translation_n
at_o all_o adventure_n namely_o d_o five_o g_o s_o and_o a_o right_a line_n be_v draw_v from_o the_o point_n d_o to_o the_o point_n g_o and_o a_o other_o from_o the_o point_n five_o to_o the_o point_n s._n wherefore_o by_o the_o 7._o of_o the_o eleven_o the_o line_n dg_o and_o we_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n and_o forasmuch_o as_o the_o line_n de_fw-fr be_v a_o parallel_n to_o the_o line_n bg_o therefore_o by_o the_o 24._o of_o the_o first_o the_o angle_n edt_n be_v equal_a to_o the_o angle_n bgt_fw-mi for_o they_o be_v alternate_a angle_n and_o likewise_o the_o angle_n dtv_n be_v equal_a to_o the_o angle_n gts_fw-fr now_o then_o there_o be_v two_o triangle_n that_o be_v dtv_n and_o gts_fw-fr have_v two_o angle_n of_o the_o one_o equal_a to_o two_o angle_n of_o the_o other_o and_o one_o side_n of_o the_o one_o equal_a to_o one_o side_n of_o the_o other_o namely_o the_o side_n which_o subtend_v the_o equal_a angle_n that_o be_v the_o side_n dv_o to_o the_o side_n g_n for_o they_o be_v the_o half_n of_o the_o line_n de_fw-fr and_o bg_o wherefore_o the_o side_n remain_v be_v equal_a to_o the_o side_n remain_v wherefore_o the_o line_n dt_n be_v equal_a to_o the_o line_n tg_n and_o the_o line_n vt_fw-la to_o the_o line_n it_o be_v if_o therefore_o the_o opposite_a side_n of_o a_o parallelipipedon_n be_v divide_v into_o two_o equal_a part_n and_o by_o their_o section_n be_v extend_v plain_a superficiece_n the_o common_a section_n of_o those_o plain_a superficiece_n and_o the_o diameter_n of_o the_o parallelipipedon_n do_v divide_v the_o one_o the_o other_o into_o two_o equal_a part_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o flussas_n every_o plain_a superficies_n extend_v by_o the_o centre_n of_o a_o parallelipipedon_n divide_v that_o solid_a into_o two_o equal_a part_n and_o so_o do_v not_o any_o other_o plain_a superficies_n not_o extend_v by_o the_o centre_n for_o every_o plain_n extend_v by_o the_o centre_n cut_v the_o diameter_n of_o the_o parallelipipedon_n in_o the_o centre_n into_o two_o equal_a part_n for_o it_o be_v prove_v that_o plain_a superficiece_n which_o cut_v the_o solid_a into_o two_o equal_a part_n do_v cut_v the_o dimetient_fw-la into_o two_o equal_a part_n in_o the_o centre_n wherefore_o all_o the_o line_n draw_v by_o the_o centre_n in_o that_o plain_a superficies_n shall_v make_v angle_n with_o the_o dimetient_fw-la and_o forasmuch_o as_o the_o diameter_n fall_v upon_o the_o parallel_n right_a line_n of_o the_o solid_a which_o describe_v the_o opposite_a side_n of_o the_o say_v solid_a or_o upon_o the_o parallel_n plain_a superficiece_n of_o the_o solid_a which_o make_v angel_n at_o the_o end_n of_o the_o diameter_n the_o triangle_n contain_v under_o the_o diameter_n and_o the_o right_a line_n extend_v in_o that_o plain_n by_o the_o centre_n and_o the_o right_a line_n which_o be_v draw_v in_o the_o opposite_a superficiece_n of_o the_o solid_a join_v together_o the_o end_n of_o the_o foresay_a right_a line_n namely_o the_o end_n of_o the_o diameter_n and_o the_o end_n of_o the_o line_n draw_v by_o the_o centre_n in_o the_o superficies_n extend_v by_o the_o centre_n shall_v always_o be_v equal_a and_o equiangle_n by_o the_o 26._o of_o the_o first_o for_o the_o opposite_a right_a line_n draw_v by_o the_o opposite_a plain_a superficiece_n of_o the_o solid_a do_v make_v equal_a angle_n with_o the_o diameter_n forasmuch_o as_o they_o be_v parallel_a line_n by_o the_o 16._o of_o this_o book_n but_o the_o angle_n at_o the_o centre_n be_v equal_a by_o the_o 15._o of_o the_o first_o for_o they_o be_v head_n angle_n &_o one_o side_n be_v equal_a to_o one_o side_n namely_o half_o the_o dimetient_fw-la wherefore_o the_o triangle_n contain_v under_o every_o right_a line_n draw_v by_o the_o centre_n of_o the_o parallelipipedon_n in_o the_o superficies_n which_o be_v extend_v also_o by_o the_o say_a centre_n and_o the_o diameter_n thereof_o who_o end_n be_v the_o angle_n of_o the_o solid_a be_v equal_a equilater_n &_o equiangle_n by_o the_o 26._o of_o the_o first_o wherefore_o it_o follow_v that_o the_o plain_a superficies_n which_o cut_v the_o parallelipipedon_n do_v make_v the_o part_n of_o the_o base_n on_o the_o opposite_a side_n equal_a and_o equiangle_n and_o therefore_o like_a and_o equal_a both_o in_o multitude_n and_o in_o magnitude_n wherefore_o the_o two_o solid_a section_n of_o that_o solid_a shall_v be_v equal_a and_o like_a by_o the_o 8._o definition_n of_o this_o book_n and_o now_o that_o no_o other_o plain_a superficies_n beside_o that_o which_o be_v extend_v by_o the_o centre_n divide_v the_o parallelipipedon_n into_o two_o equal_a part_n it_o be_v manifest_a if_o unto_o the_o plain_a superficies_n which_o be_v not_o extend_v by_o the_o centre_n we_o extend_v by_o the_o centre_n a_o parallel_n plain_a superficies_n by_o the_o corollary_n of_o the_o 15._o of_o this_o book_n for_o forasmuch_o as_o that_o superficies_n which_o be_v extend_v by_o the_o centre_n do_v divide_v the_o parallelipipedom_n into_o two_o equal_a parâ—â—_n it_o be_v manifest_a that_o the_o other_o plain_a superficies_n which_o be_v parallel_n to_o the_o superficies_n which_o divide_v the_o solid_a into_o two_o equal_a part_n be_v in_o one_o of_o the_o equal_a part_n of_o the_o solid_a wherefore_o see_v that_o the_o whole_a be_v ever_o great_a than_o his_o part_n it_o must_v needs_o be_v that_o one_o of_o these_o section_n be_v less_o than_o the_o half_a of_o the_o solid_a and_o therefore_o the_o other_o be_v great_a for_o the_o better_a understanding_n of_o this_o former_a proposition_n &_o also_o of_o this_o corollary_n add_v by_o flussas_n it_o shall_v be_v very_o needful_a for_o you_o to_o describe_v of_o past_v paper_n or_o such_o like_a matter_n a_o parallelipipedom_n or_o a_o cube_n and_o to_o divide_v all_o the_o parallelogram_n thereof_o into_o two_o equal_a part_n by_o draw_v by_o the_o centre_n of_o the_o say_a parallelogram_n which_o centre_n be_v the_o point_n make_v by_o the_o cut_n of_o diagonal_a line_n draw_v from_o thâ—_z opposite_a angle_n of_o the_o say_a parallelogram_n line_n parallel_n to_o the_o side_n of_o the_o parallelogram_n as_o in_o the_o former_a figure_n describe_v in_o a_o plain_a you_o may_v see_v be_v the_o six_o parallelogram_n de_fw-fr eh_o ha_o ad_fw-la dh_o and_o cg_o who_o these_o parallel_a line_n draw_v by_o the_o centre_n of_o the_o say_a parallelogram_n namely_o xo_o or_o pr._n and_o px_n do_v divide_v into_o two_o equal_a part_n by_o which_o four_o line_n you_o must_v imagine_v a_o plain_a superficies_n to_o be_v extend_v also_o these_o parallel_a line_n kl_o ln_o nm_o and_o mâ—_n by_o which_o four_o line_n likewise_o yâ—_n must_v imagine_v a_o plain_a superficies_n to_o be_v extend_v you_o may_v if_o you_o will_v put_v within_o your_o body_n make_v thus_o of_o past_v paper_n two_o superficiece_n make_v also_o of_o the_o say_a paper_n have_v to_o their_o limit_n line_n equal_a to_o the_o foresay_a parallel_n line_n which_o superficiece_n must_v also_o be_v divide_v into_o two_o equal_a part_n by_o parallel_a line_n draw_v by_o their_o centre_n and_o must_v cut_v the_o one_o the_o other_o by_o these_o parallel_a line_n and_o for_o the_o diameter_n of_o this_o body_n extend_v a_o thread_n from_o one_o angle_n in_o the_o base_a of_o the_o solid_a to_o his_o opposite_a angle_n which_o shall_v pass_v by_o the_o centre_n of_o the_o parallelipipedon_n as_o do_v the_o line_n dg_o in_o the_o figure_n before_o describe_v in_o the_o plain_n and_o draw_v in_o the_o base_a and_o the_o opposite_a superficies_n unto_o it_o diagonal_a line_n from_o the_o angle_n from_o which_o be_v extend_v the_o diameter_n of_o the_o solid_a as_o in_o the_o former_a description_n be_v the_o line_n bg_o and_o de._n and_o when_o you_o have_v thus_o describe_v this_o body_n compare_v it_o with_o the_o former_a demonstration_n and_o it_o will_v make_v it_o very_o plain_a unto_o you_o so_o your_o letter_n agree_v with_o the_o letter_n of_o the_o figure_n describe_v in_o the_o book_n and_o this_o description_n will_v plain_o set_v forth_o unto_o you_o the_o corollary_n follow_v that_o proposition_n for_o where_o as_o to_o the_o understanding_n of_o the_o demonstration_n of_o the_o proposition_n the_o superficiece_n put_v within_o the_o body_n be_v extend_v by_o parallel_a line_n draw_v by_o the_o centre_n of_o the_o base_n of_o the_o parallelipipedon_n to_o the_o understanding_n of_o the_o say_a corollary_n you_o may_v extend_v a_o superficies_n by_o any_o other_o line_n draw_v in_o the_o say_a base_n so_o that_o yet_o it_o pass_v through_o the_o midst_n of_o the_o thread_n which_o be_v suppose_v to_o be_v the_o centre_n of_o the_o parallelipipedon_n ¶_o the_o 35._o theorem_a the_o 40._o proposition_n if_o there_o be_v
a_o triangle_n and_o if_o the_o parallelogram_n be_v double_a to_o the_o triangle_n those_o prism_n be_v by_o the_o 40._o of_o the_o eleven_o equal_v the_o one_o to_o the_o other_o therefore_o the_o prism_n contain_v under_o the_o two_o triangle_n bkf_a and_o ehg_n and_o under_o the_o three_o parallelogram_n ebfg_n ebkh_n and_o khfg_n part_n be_v equal_a to_o the_o prism_n contain_v under_o the_o two_o triangle_n gfc_a and_o hkl_n and_o under_o the_o three_o parallelogram_n kfcl_n lcgh_n and_o hkfg_n and_o it_o be_v manifest_a that_o both_o these_o prism_n of_o which_o the_o base_a of_o one_o be_v the_o parallelogram_n ebfg_n pyramid_n and_o the_o opposite_a unto_o it_o the_o line_n kh_o and_o the_o base_a of_o the_o other_o be_v the_o triangle_n gfc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n klh_n be_v great_a than_o both_o these_o pyramid_n who_o base_n be_v the_o triangle_n age_n and_o hkl_n and_o top_n the_o point_n h_o &_o d._n for_o if_o we_o draw_v these_o right_a line_n of_o and_o eke_o the_o prism_n who_o base_a be_v the_o parallelogram_n ebfg_n and_o the_o opposite_a unto_o it_o the_o right_a line_n hk_o be_v great_a than_o the_o pyramid_n who_o base_a be_v the_o triangle_n ebf_n &_o top_n the_o point_n k._n but_o the_o pyramid_n who_o base_a be_v the_o triangle_n ebf_n and_o top_n the_o point_n king_n be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h_o for_o they_o be_v contain_v under_o equal_a and_o like_a plain_a superficiece_n wherefore_o also_o the_o prism_n who_o base_a be_v the_o parallelogram_n ebfg_n and_o the_o opposite_a unto_o it_o the_o right_a line_n hk_o be_v great_a than_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h._n but_o the_o prism_n who_o base_a be_v the_o parallelogram_n ebfg_n and_o the_o opposite_a unto_o it_o the_o right_a line_n hk_o be_v equal_a to_o the_o prism_n who_o base_a be_v the_o triangle_n gfc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n hkl_n and_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h_o be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n hkl_n and_o top_n the_o point_n d._n part_n wherefore_o the_o foresay_a two_o prism_n be_v great_a than_o the_o foresay_a two_o pyramid_n who_o base_n be_v the_o triangle_n aeg_n hkl_n and_o top_n the_o point_n h_o and_o d._n wherefore_o the_o whole_a pyramid_n who_o base_a be_v the_o triangle_n abc_n proposition_n and_o top_n the_o point_n d_o be_v divide_v into_o two_o pyramid_n equal_a and_o like_v the_o one_o to_o the_o other_o and_o like_v also_o unto_o the_o whole_a pyramid_n have_v also_o triangle_n to_o their_o base_n and_o into_o two_o equal_a prism_n and_o the_o two_o prism_n be_v great_a than_o half_a of_o the_o whole_a pyramid_n which_o be_v require_v to_o be_v demonstrate_v if_o you_o will_v with_o diligence_n read_v these_o four_o book_n follow_v of_o euclid_n which_o concern_v body_n and_o clear_o see_v the_o demonstration_n in_o they_o contain_v it_o shall_v be_v requisite_a for_o you_o when_o you_o come_v to_o any_o proposition_n which_o concern_v a_o body_n or_o body_n whether_o they_o be_v regular_a or_o not_o first_o to_o describe_v of_o pâ—sâ—ed_a paper_n according_a as_o i_o teach_v you_o in_o the_o end_n of_o the_o definition_n of_o the_o eleven_o book_n such_o a_o body_n or_o body_n as_o be_v there_o require_v and_o have_v your_o body_n or_o body_n thus_o describe_v when_o you_o have_v note_v it_o with_o letter_n according_a to_o the_o figure_n set_v forth_o upon_o a_o plain_a in_o the_o proposition_n follow_v the_o construction_n require_v in_o the_o proposition_n as_o for_o example_n in_o this_o three_o proposition_n it_o be_v say_v that_o every_o pyramid_n have_v a_o triangle_n to_o â—is_n base_a may_v be_v divide_v into_o two_o pyramid_n etc_n etc_n here_o first_o describe_v a_o pyramid_n of_o past_v paper_n haâ—ing_v his_o base_a triangle_v it_o skill_v not_o whether_o it_o be_v equilater_n or_o equiangle_v or_o not_o only_o in_o this_o proposition_n be_v require_v that_o the_o base_a be_v a_o triangle_n then_o for_o that_o the_o proposition_n suppose_v the_o base_a of_o the_o pyramid_n to_o be_v the_o triangle_n abc_n note_v the_o base_a of_o your_o pyramid_n which_o you_o have_v describe_v with_o the_o letter_n abc_n and_o the_o top_n of_o your_o pyramid_n with_o the_o letter_n d_o for_o so_o be_v require_v in_o the_o proposition_n and_o thus_o have_v you_o your_o body_n order_v ready_a to_o the_o construction_n now_o in_o the_o construction_n it_o be_v require_v that_o you_o divide_v the_o line_n ab_fw-la bc_o ca._n etc_o etc_o namely_o the_o six_o line_n which_o be_v the_o side_n of_o the_o four_o triangle_n contain_v the_o pyramid_n into_o two_o equal_a part_n in_o the_o poyntet_fw-la â—_o f_o g_o etc_n etc_n that_o be_v you_o must_v divide_v the_o line_n ab_fw-la of_o your_o pyramid_n into_o two_o equal_a part_n and_o note_v the_o point_n of_o the_o division_n with_o the_o letter_n e_o and_o so_o the_o line_n bc_o be_v divide_v into_o two_o equal_a part_n note_v the_o point_n of_o the_o division_n with_o the_o letter_n f._n and_o so_o the_o rest_n and_o this_o order_n follow_v you_o as_o touch_v the_o rest_n of_o the_o construction_n there_o put_v and_o when_o you_o have_v finish_v the_o construction_n compare_v your_o body_n thus_o describe_v with_o the_o demonstration_n and_o it_o will_v make_v it_o very_o plain_a and_o easy_a to_o be_v understand_v whereas_o without_o such_o a_o body_n describe_v of_o matter_n it_o be_v hard_o for_o young_a beginner_n unless_o they_o have_v a_o very_a deep_a imagination_n full_o to_o conceive_v the_o demonstration_n by_o the_o sigâ—e_n as_o it_o be_v describe_v in_o a_o plain_a here_o for_o the_o better_a declaration_n of_o that_o which_o i_o have_v say_v have_v i_o set_v a_o figure_n who_o form_n if_o you_o describe_v upon_o past_v paper_n note_v with_o the_o like_a letter_n and_o cut_v the_o line_n â—a_o dam_n dc_o and_o fold_v it_o according_o it_o will_v make_v a_o pyramid_n describe_v according_a to_o the_o construction_n require_v in_o the_o proposition_n and_o this_o order_n follow_v you_o as_o touch_v all_o other_o proposition_n which_o concern_v body_n ¶_o a_o other_o demonstration_n after_o campane_n of_o the_o 3._o proposition_n suppose_v that_o there_o be_v a_o pyramid_n abcd_o have_v to_o his_o base_a the_o triangle_n bcd_o and_o let_v his_o top_n be_v the_o solid_a angle_n a_o from_o which_o let_v there_o be_v draw_v three_o subtended_a line_n ab_fw-la ac_fw-la and_o ad_fw-la to_o the_o three_o angle_n of_o the_o base_a and_o divide_v all_o the_o side_n of_o the_o base_a into_o two_o equal_a part_n in_o the_o three_o point_n e_o f_o g_o divide_v also_o the_o three_o subtended_a line_n ab_fw-la ac_fw-la and_o ad_fw-la in_o two_o equal_a part_n in_o the_o three_o point_n h_o edward_n l._n and_o draw_v in_o the_o base_a these_o two_o line_n of_o and_o eglantine_n so_o shall_v the_o base_a of_o the_o pyramid_n be_v divide_v into_o three_o superficiece_n whereof_o two_o be_v the_o two_o triangle_n bef_n and_o egg_v which_o be_v like_o both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o whole_a base_a by_o the_o 2_o part_n of_o the_o second_o of_o the_o six_o &_o by_o the_o definition_n of_o like_a superâ—iciecâ—s_n &_o they_o be_v equal_v the_o one_o to_o the_o other_o by_o the_o 8._o of_o the_o first_o the_o three_o superficies_n be_v a_o quadrangled_a parallelogram_n namely_o efgc_n which_o be_v double_a to_o the_o triangle_n egg_v by_o the_o 40._o and_o 41._o of_o the_o first_o now_o then_o again_o from_o the_o point_n h_o draw_v unto_o the_o point_n e_o and_o f_o these_o two_o subtendent_fw-la line_n he_o and_o hf_o draw_v also_o a_o subtended_a line_n from_o the_o point_n king_n to_o the_o point_n g._n and_o draw_v these_o line_n hk_o kl_o and_o lh_o wherefore_o the_o whole_a pyramid_n abcd_o be_v divide_v into_o two_o pyramid_n which_o be_v hbef_n and_o ahkl_n and_o into_o two_o prism_n of_o which_o the_o one_o be_v ehfgkc_n and_o be_v set_v upon_o the_o quadrangled_a base_a cfge_n the_o other_o be_v egdhkl_n and_o have_v to_o his_o base_a the_o triangle_n egg_v now_o as_o touch_v the_o two_o pyramid_n hbef_a and_o ahkl_n that_o they_o be_v equal_v the_o one_o to_o the_o other_o and_o also_o that_o they_o be_v like_o both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o whole_a it_o be_v manifest_a by_o the_o definition_n of_o equal_a and_o like_a body_n and_o by_o the_o 10._o of_o the_o eleven_o and_o by_o 2._o part_n of_o the_o second_o of_o the_o six_o and_o that_o the_o two_o prism_n be_v equal_a it_o
it_o comprehend_v wherefore_o the_o pyramid_n who_o base_a be_v the_o square_n abcd_o and_o altitude_n the_o self_n same_o that_o the_o cone_fw-mi have_v be_v great_a than_o the_o half_a of_o the_o cone_fw-it divide_v by_o the_o 30._o of_o the_o three_o every_o one_o of_o the_o circumference_n ab_fw-la bc_o cd_o and_o dam_n into_o two_o equal_a part_n in_o the_o point_n e_o f_o g_o and_o h_o and_o draw_v these_o right_a line_n ae_n ebb_n bf_o fc_o cg_o gd_o dh_o and_o ha._n wherefore_o every_o one_o of_o these_o triangle_n aeb_fw-mi bfc_n cgd_v and_o dha_n be_v great_a than_o the_o half_a part_n of_o the_o segment_n of_o the_o circle_n describe_v about_o it_o upon_o every_o one_o of_o these_o triangle_n aeb_fw-mi bfc_n cgd_v and_o dha_n describe_v a_o pyramid_n of_o equal_a altitude_n with_o the_o cone_n and_o after_o the_o same_o manner_n every_o one_o of_o those_o pyramid_n so_o describe_v be_v great_a than_o the_o half_a part_n of_o the_o segment_n of_o the_o cone_fw-it set_v upon_o the_o segment_n of_o the_o circle_n now_o therefore_o divide_v by_o the_o 30_o of_o the_o three_o the_o circumference_n remain_v into_o two_o equal_a part_n &_o draw_v right_a line_n &_o raise_v up_o upon_o every_o one_o of_o those_o triangle_n a_o pyramid_n of_o equal_a altitude_n with_o the_o cone_n and_o do_v this_o continual_o we_o shall_v at_o the_o length_n by_o the_o first_o of_o the_o ten_o leave_v certain_a segment_n of_o the_o cone_n which_o shall_v be_v less_o then_o the_o excess_n whereby_o the_o cone_n exceed_v the_o three_o part_n of_o the_o cylinder_n let_v those_o segment_n be_v ae_n ebb_n bf_o fc_o cg_o gd_o dh_o and_o ha._n wherefore_o the_o pyramid_n remain_v who_o base_a be_v the_o poligonon_n figure_n aebfcgdh_n and_o altitude_n the_o self_n same_o with_o the_o cone_n be_v great_a than_o the_o three_o part_n of_o the_o cylinder_n but_o the_o pyramid_n who_o base_a be_v the_o polygonon_n figure_n aebfcgdh_n and_o altitude_n the_o self_n same_o with_o the_o cone_n be_v the_o three_o part_n of_o the_o prism_n who_o base_a be_v the_o poligonon_n figure_n aebfcgdh_n and_o altitude_n the_o self_n same_o with_o the_o cylinder_n wherefore_o you_o the_o prism_n who_o base_a be_v the_o polygonon_n figure_n aebfcgdh_n and_o altitude_n the_o self_n same_o with_o the_o cylinder_n be_v great_a than_o the_o cylinder_n who_o base_a be_v the_o circle_n abcd._n but_o it_o be_v also_o less_o for_o it_o be_v contain_v of_o it_o which_o be_v impossible_a wherefore_o the_o cylinder_n be_v not_o in_o less_o proportion_n to_o the_o cone_n then_o in_o treble_a proportion_n and_o it_o be_v prove_v that_o it_o be_v not_o in_o great_a proportion_n to_o the_o cone_n then_o in_o treble_a proportion_n wherefore_o the_o cone_n be_v the_o three_o part_n of_o the_o cylinder_n wherefore_o every_o cone_n be_v the_o three_o part_n of_o a_o cylinder_n have_v one_o &_o the_o self_n same_o base_a and_o one_o and_o the_o self_n same_o altitude_n with_o it_o which_o be_v require_v to_o be_v demonstrate_v ¶_o add_v by_o m._n john_n dee_n ¶_o a_o theorem_a 1._o the_o superficies_n of_o every_o upright_o cylinder_n except_o his_o base_n be_v equal_a to_o that_o circle_n who_o semidiameter_n be_v middle_a proportional_a between_o the_o side_n of_o the_o cylinder_n and_o the_o diameter_n of_o his_o base_a ¶_o a_o theorem_a 2._o the_o superficies_n of_o every_o upright_o or_o isosceles_a cone_n except_o the_o base_a be_v equal_a to_o that_o circle_n who_o semidiameter_n be_v middle_a proportional_a between_o the_o side_n of_o that_o cone_n and_o the_o semidiameter_n of_o the_o circle_n which_o be_v the_o base_a of_o the_o cone_n my_o intent_n in_o addition_n be_v not_o to_o amend_v euclideâ—_n method_n which_o need_v little_a add_v or_o none_o at_o all_o but_o my_o desire_n be_v somewhat_o to_o furnish_v you_o towards_o a_o more_o general_a art_n mathematical_a them_z euclides_n element_n addition_n remain_v in_o the_o term_n in_o which_o they_o be_v write_v can_v sufficient_o help_v you_o unto_o and_o though_o euclides_n element_n with_o my_o addition_n run_v not_o in_o one_o methodical_a race_n towards_o my_o mark_n yet_o in_o the_o mean_a space_n my_o addition_n either_o geve_v light_a where_o they_o be_v annex_v to_o euclides_n matter_n or_o geve_v some_o ready_a aid_n and_o show_v the_o way_n to_o dilate_v your_o discourse_n mathematical_a or_o to_o invent_v and_o practice_v thing_n mechanical_o and_o in_o deed_n if_o more_o leysor_n have_v happen_v many_o more_o strange_a matter_n mathematical_a have_v according_a to_o my_o purpose_n general_a be_v present_o publish_v to_o your_o knowledge_n but_o want_v of_o due_a leasour_n cause_v you_o to_o want_v that_o which_o my_o good_a will_n towards_o you_o most_o hearty_o do_v wish_v you_o as_o concern_v the_o two_o theorem_n here_o annex_v their_o verity_n be_v by_o archimedes_n in_o his_o book_n of_o the_o sphere_n and_o cylinder_n manifest_o demonstrate_v and_o at_o large_a you_o may_v therefore_o bold_o trust_v to_o they_o and_o use_v they_o as_o supposition_n in_o any_o your_o purpose_n till_o you_o have_v also_o their_o demonstration_n but_o if_o you_o well_o remember_v my_o instruction_n upon_o the_o first_o proposition_n of_o this_o book_n and_o my_o other_o addition_n upon_o the_o second_o with_o the_o supposition_n how_o a_o cylinder_n and_o a_o cone_n be_v mathematical_o produce_v you_o will_v not_o need_v archimedes_n demonstration_n nor_o yet_o be_v utter_o ignorant_a of_o the_o solid_a quantity_n of_o this_o cylinder_n and_o cone_n here_o compare_v the_o diameter_n of_o their_o base_a and_o heith_n be_v know_v in_o any_o measure_n neither_o can_v their_o crooked_a superficies_n remain_v unmeasured_a whereof_o undoubted_o great_a pleasure_n and_o commodity_n may_v grow_v to_o the_o sincere_a student_n and_o precise_a practiser_n ¶_o the_o 11._o theorem_a the_o 11._o proposition_n cones_n and_o cylinder_n be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o that_o proportion_n the_o one_o other_o that_o their_o base_n be_v in_o like_a sort_n also_o may_v we_o proveâ—_v that_o as_o the_o circle_n efgh_o be_v to_o the_o circle_n abcd_o so_o be_v not_o the_o cone_n en_fw-fr to_o any_o solid_a less_o than_o the_o cone_n al._n now_o i_o say_v that_o as_o the_o circle_n abcd_o be_v to_o the_o circle_n efgh_o so_o be_v not_o the_o cone_n all_o to_o any_o solid_a great_a than_o the_o cone_fw-mi en_fw-fr for_o if_o it_o be_v possible_a let_v it_o be_v unto_o a_o great_a namely_o to_o the_o solid_a x._o wherefore_o by_o conversion_n as_o the_o circle_n efgh_o be_v to_o the_o circle_n abcd_o so_o be_v the_o solid_a x_o to_o the_o cone_n all_o but_o as_o the_o solid_a x_o be_v to_o the_o cone_n all_o so_o be_v the_o cone_n en_fw-fr to_o some_o solid_a less_o than_o the_o cone_n all_o as_o we_o may_v see_v by_o the_o assumpt_n put_v after_o thâ—_z second_o of_o this_o book_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o circle_n efgh_o be_v to_o the_o circle_n abcgâ—_n so_o be_v the_o cone_n en_fw-fr to_o some_o solid_a less_o than_o the_o cone_n all_o which_o we_o have_v prove_v to_o be_v impossible_a wherefore_o as_o the_o circle_n abcd_o be_v to_o the_o circle_n efgh_o so_o be_v not_o the_o cone_n all_o to_o any_o solid_a great_a than_o the_o cone_fw-mi en_fw-fr and_o it_o be_v also_o prove_v that_o it_o be_v not_o to_o any_o less_o wherefore_o as_o the_o circle_n abcd_o be_v to_o the_o circle_n efgh_o so_o be_v the_o cone_n all_o to_o the_o cone_n en_fw-fr but_o as_o the_o cone_n be_v to_o the_o cone_n so_o be_v the_o cylinder_n to_o the_o cylinder_n by_o the_o 15._o of_o the_o five_o for_o the_o one_o be_v in_o treble_a proportion_n to_o the_o other_o cylinder_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o circle_n abcd_o be_v to_o the_o circle_n efgh_o so_o be_v the_o cylinder_n which_o be_v set_v upon_o they_o the_o one_o to_o the_o other_o the_o say_a cylinder_n being_n under_o equal_a altitude_n with_o the_o cones_fw-la cones_n therefore_o and_o cylinder_n be_v under_o one_o &_o the_o self_n same_o altitude_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 12._o theorem_a the_o 12._o proposition_n like_a cones_n and_o cylinder_n be_v in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o their_o base_n be_v now_o also_o i_o say_v that_o the_o cone_fw-mi abcdl_n be_v not_o to_o any_o solid_a great_a than_o the_o cone_n efghn_n case_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bd_o be_v to_o the_o diameter_n fh_o for_o if_o it_o be_v possible_a let_v it_o be_v to_o a_o great_a namely_o to_o the_o solid_a x._o wherefore_o by_o conversion_n by_o the_o
square_a of_o the_o line_n by._n and_o forasmuch_o as_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_n all_o be_v equal_a to_o the_o line_n ab_fw-la therefore_o the_o square_a of_o the_o line_n all_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la but_o unto_o the_o square_n of_o the_o line_n all_o be_v equal_a the_o square_n of_o the_o line_n agnostus_n and_o gl_n and_o to_o the_o square_n of_o the_o line_n ab_fw-la be_v equal_a the_o square_n of_o the_o line_n by_o and_o ya_fw-mi wherefore_o the_o square_n of_o the_o line_n agnostus_n and_o gl_n be_v equal_a to_o the_o square_n of_o the_o line_n by_o and_o ya_fw-mi of_o which_o the_o square_a of_o the_o line_n by_o be_v less_o than_o the_o square_a of_o the_o line_n gl_n wherefore_o the_o residue_n namely_o the_o square_a of_o the_o line_n ya_fw-mi be_v great_a they_o the_o square_n of_o the_o line_n ag._n wherefore_o also_o the_o line_n ay_o be_v great_a they_o the_o line_n ag._n wherefore_o two_o sphere_n consist_v both_o about_o one_o and_o the_o self_n same_o centre_n be_v give_v there_o be_v inscribe_v in_o the_o great_a sphere_n a_o polihedron_n or_o solid_a of_o many_o side_n which_o touch_v not_o the_o superficies_n of_o the_o less_o sphereâ—_n which_o be_v requinred_n to_o be_v do_v ¶_o corollary_n and_o if_o in_o the_o other_o sphere_n namely_o in_o the_o less_o sphere_n be_v inscribe_v a_o polihedron_n or_o solid_a of_o many_o side_n like_v to_o the_o polihedron_n inscribe_v in_o the_o sphere_n bcde_v than_o the_o polihedron_n inscribe_v in_o the_o sphere_n bcde_v be_v to_o the_o polihedron_n inscribe_v in_o the_o other_o sphere_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o the_o sphere_n bcde_v be_v to_o the_o diameter_n of_o the_o other_o sphere_n for_o those_o solid_n be_v divide_v into_o pyramid_n equal_a in_o number_n and_o equal_v in_o order_n the_o pyramid_n shall_v be_v like_a but_o like_o pyramid_n be_v by_o the_o 8._o of_o the_o twelve_o the_o one_o to_o the_o other_o in_o treble_a proportion_n of_o that_o in_o which_o side_n of_o like_a proportion_n be_v to_o side_n of_o like_a proportion_n wherefore_o the_o pyramid_n who_o base_a be_v the_o four_o side_v figure_n kbos_n and_o top_n the_o point_n a_o be_v to_o that_o pyramis_n which_o be_v of_o like_a order_n in_o the_o other_o sphere_n in_o treble_a proportion_n of_o that_o in_o which_o side_n of_o like_a proportion_n be_v to_o side_n of_o like_a proportion_n that_o be_v of_o that_o in_o which_o the_o side_n ab_fw-la which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n which_o be_v about_o the_o centre_n a_o be_v to_o the_o side_n which_o be_v draw_v from_o the_o centre_n of_o the_o other_o sphere_n and_o in_o like_a sort_n also_o every_o one_o of_o the_o pyramid_n which_o be_v in_o the_o sphere_n which_o be_v about_o the_o centre_n a_o be_v to_o every_o one_o of_o the_o pyramid_n of_o the_o self_n same_o order_n in_o the_o other_o sphere_n in_o treble_a proportion_n of_o that_o in_o which_o the_o side_n ab_fw-la be_v to_o the_o side_n which_o be_v draw_v from_o the_o centre_n of_o the_o other_o sphere_n but_o as_o one_o of_o the_o antecedentes_fw-la be_v to_o one_o of_o the_o consequence_n so_o be_v all_o the_o antecedent_n to_o all_o the_o consequentes_fw-la by_o the_o 12._o of_o the_o five_o wherefore_o the_o whole_a polihedron_n solid_a of_o many_o side_n which_o be_v in_o the_o sphere_n which_o be_v about_o the_o centre_n a_o be_v to_o the_o whole_a polihedron_n or_o solid_a of_o many_o side_n which_o be_v in_o the_o other_o sphere_n in_o treble_a proportion_n of_o that_o in_o which_o the_o side_n ab_fw-la be_v to_o the_o side_n which_o be_v draw_v from_o the_o centre_n of_o the_o other_o sphere_n that_o be_v of_o that_o which_o the_o diameter_n bd_o be_v to_o the_o diameter_n of_o the_o other_o sphere_n by_o the_o 15._o of_o the_o five_o which_o be_v require_v to_o be_v demonstrate_v m._n do_v you_o his_o devise_n to_o help_v the_o imagination_n to_o young_a student_n in_o geometryâ—_n and_o to_o make_v his_o demonstration_n more_o evident_a as_o concern_v the_o error_n by_o he_o correct_v in_o euclides_n figure_n by_o the_o ignorant_a misline_v i_o dâ—e_n this_o figure_n be_v answerable_a to_o the_o first_o plain_n which_o cut_v the_o two_o sphere_n by_o their_o common_a centre_n a_o make_v two_o concentrical_a circle_n have_v the_o same_o centre_n with_o the_o two_o sphere_n namely_o bcde_v and_o fgh_a upon_o which_o â—_o you_o apt_o rear_v perpendicular_o the_o second_o figure_n contain_v two_o concentrical_a circle_n to_o the_o first_o equal_a and_o make_v the_o point_n note_v with_o like_a letter_n to_o agree_v and_o afterward_o upon_o the_o second_o figure_n set_v on_o the_o three_o figure_n be_v here_o for_o the_o better_a handle_v make_v a_o semicircle_n which_o upon_o the_o first_o figure_n must_v also_o be_v erect_v perpendicular_o and_o last_o if_o you_o take_v the_o little_a quadrangled_a figure_n book_n and_o make_v every_o point_n to_o touch_v his_o like_a &_o then_o read_v the_o construction_n &_o weigh_v the_o demonstration_n twice_o oâ—_n thrice_o be_v read_v over_o shall_v you_o in_o this_o delineation_n in_o apt_a pastborde_n or_o like_o maâ—ter_n frame_v find_v all_o thing_n in_o this_o problem_n very_o evident_a i_o need_v not_o warn_v you_o that_o the_o line_n ay_o may_v easy_o be_v imagine_v or_o with_o a_o fine_a thread_n supply_v or_o of_o the_o right_a line_n imaginable_a between_o pea_n and_o t_o and_o between_o r_o and_o v_o i_o need_v say_v nothing_o trust_v that_o the_o great_a exercise_n past_a by_o that_o time_n you_o be_v orderly_a come_v to_o this_o place_n will_v have_v make_v you_o sufficient_a perfect_a to_o supply_v any_o far_a thing_n herein_o to_o be_v consider_v the_o little_a fowercornerd_a piece_n remain_v to_o the_o semicircle_n be_v to_o be_v let_v through_o the_o first_o ground_n plain_n thereby_o to_o stay_v this_o semicircle_n the_o better_a in_o his_o apt_a place_n and_o situation_n which_o it_o will_v the_o more_o apt_o do_v if_o you_o do_v 〈…〉_o the_o contrary_a arrase_n of_o the_o slit_v of_o it_o and_o of_o the_o slit_v of_o the_o second_o figure_n into_o which_o it_o be_v to_o be_v let_v abate_v they_o alike_o much_o a_o little_a will_v serve_v experience_n by_o advice_n will_v teach_v sufficient_o ¶_o master_n dee_fw-mi his_o advice_n and_o demonstration_n reform_v a_o great_a error_n in_o the_o designation_n of_o the_o former_a figure_n of_o euclides_n second_o problem_n with_o two_o corollary_n by_o he_o infer_v upon_o his_o say_a demonstration_n a_o theorem_a ✚_o that_o a_o right_a line_n drowen_v from_o the_o point_n king_n perpendicular_o upon_o the_o line_n bz_o do_v fall_v upon_o the_o point_n z_o we_o will_v thus_o make_v evident_a by_o the_o premise_n it_o be_v manifest_a that_o the_o point_n z_o be_v that_o point_v where_o a_o right_a line_n from_o the_o point_n oh_o be_v perpendicular_o let_v â—all_n to_o the_o circle_n bcde_v do_v touch_v the_o same_o circle_n which_o point_n z_o also_o be_v prove_v to_o be_v in_o the_o right_a line_n bad_a the_o common_a section_n of_o two_o circle_n cut_v each_o other_o be_v one_o to_o the_o other_o perpendicular_o erect_v these_o thing_n with_o other_o before_o demonstrate_v i_o here_o make_v my_o supposition_n consider_v now_o the_o two_o triangle_n rectangle_v ozb_n and_o kzb_n of_o which_o the_o angle_n ozb_n iâ—_z equal_a to_o the_o angle_n kzb_n for_o by_o construction_n they_o be_v both_o right_a angle_n prove_v and_o the_o angle_n zâ—o_n be_v equal_a to_o the_o angle_n zbk_n for_o if_o from_o d_z to_o king_n you_o imagine_v a_o right_a line_n and_o the_o like_a from_o d_o to_o oh_o you_o have_v two_o triangle_n in_o equal_a semicircle_n rectangle_v namely_o dkb_n and_o dob_n which_o have_v the_o diameter_n bd_o common_a and_o bk_o the_o chord_n equal_a to_o boy_n the_o chord_n by_o construction_n wherefore_o by_o the_o 47._o of_o the_o â—irst_n the_o three_o side_n namely_o dk_o be_v equal_a to_o the_o three_o namely_o do_v wherefore_o by_o the_o 5._o of_o the_o six_o the_o angle_n zbo_n which_o be_v dbo_n be_v equal_a to_o zbk_n which_o be_v dbk._n for_o the_o line_n zb_n by_o construction_n be_v part_n of_o db_o and_o see_v two_o angle_n of_o ozb_n be_v prove_v equal_a to_o two_o angle_n of_o kzb_n of_o necessity_n the_o three_o namely_o zob_n be_v equal_a to_o the_o three_o namely_o zkb_n by_o the_o 32._o of_o the_o first_o wherefore_o the_o two_o triangle_n rectangle_v ozb_n and_o kzb_n be_v prove_v equiangle_v by_o the_o four_o therefore_o of_o the_o six_o their_o side_n be_v proportional_a therefore_o by_o the_o premise_n prove_v as_o boy_n be_v to_o bk_o so_o be_v oz_n to_o kz_o and_o the_o three_o
line_n which_o subtend_v the_o angle_n zob_n to_o the_o three_o line_n which_o subtend_v the_o angle_n zkb_n but_o by_o construction_n boy_n be_v equal_a to_o bk_o therefore_o oz_n be_v equal_a to_o kz_o and_o the_o three_o alâ—o_o be_v equal_a to_o the_o three_o wherefore_o the_o point_n z_o in_o respect_n of_o the_o two_o triangle_n rectangle_v ozb_n and_o kzb_n determine_v one_o and_o the_o same_o magnitude_n iâ—_z the_o line_n bz_o which_o can_v not_o be_v if_o any_o other_o point_n in_o the_o line_n bz_o be_v assign_v near_o or_o far_o of_o from_o the_o point_n b._n one_o only_a point_n therefore_o be_v that_o at_o which_o the_o two_o perpendicular_n kz_o and_o oz_n fall_v but_o by_o construction_n oz_n fall_v at_o z_o the_o point_n and_o therefore_o at_o the_o same_o z_o do_v the_o perpendicular_a draw_v from_o king_n fall_v likewise_o which_o be_v require_v to_o be_v demonstrate_v although_o a_o brief_a monition_n may_v herein_o have_v serve_v for_o the_o pregnant_a or_o the_o humble_a learner_n yet_o for_o they_o that_o be_v well_o please_v to_o have_v thing_n make_v plain_a with_o many_o word_n and_o for_o the_o stiffnecked_a busy_a body_n it_o be_v necessary_a with_o my_o controlment_n of_o other_o to_o annex_v the_o cause_n &_o reason_n thereof_o both_o invincible_a and_o also_o evident_a a_o corollary_n 1._o hereby_o it_o be_v manifest_a that_o two_o equal_a circle_n cut_v one_o the_o other_o by_o the_o whole_a diameter_n if_o from_o one_o and_o the_o same_o end_n of_o their_o common_a diameter_n equal_a portion_n of_o their_o circumference_n be_v take_v and_o from_o the_o point_n end_v those_o equal_a portion_n two_o perpendicular_n be_v let_v down_o to_o their_o common_a diameter_n those_o perpendicular_n shall_v fall_v upon_o one_o and_o the_o same_o point_n of_o their_o common_a diameter_n 2._o second_o it_o follow_v that_o those_o perpendicular_n be_v equal_a ¶_o note_n from_o circle_n in_o our_o first_o supposition_n each_o to_o other_o perpendicular_o erect_v we_o proceed_v and_o infer_v now_o these_o corollary_n whether_o they_o be_v perpendicular_o erect_v or_o no_o by_o reasou_v the_o demonstration_n have_v a_o like_a force_n upon_o our_o supposition_n here_o use_v ¶_o the_o 16._o theorem_a the_o 18._o proposition_n sphere_n be_v in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o their_o diameter_n be_v svppose_v that_o there_o be_v two_o sphere_n abc_n and_o def_n and_o let_v their_o diameter_n be_v bc_o and_o ef._n then_o i_o say_v that_o the_o sphere_n abc_n be_v to_o the_o sphere_n def_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n ef._n proposition_n for_o if_o not_o than_o the_o sphere_n abc_n be_v in_o treble_a proportion_n of_o that_o in_o which_o bc_o be_v to_o of_o either_o to_o some_o sphere_n less_o than_o the_o sphere_n def_n or_o to_o some_o sphere_n great_a first_o let_v it_o be_v unto_o a_o less_o namely_o to_o ghk_v case_n and_o imagine_v that_o the_o sphere_n def_n and_o ghk_o be_v both_o about_o one_o and_o the_o self_n same_o centre_n impossibility_n and_o by_o the_o proposition_n next_o go_v before_o describe_v in_o the_o great_a sphere_n def_n a_o polihedron_n or_o a_o solid_a of_o many_o side_n not_o touch_v the_o superficies_n of_o the_o less_o sphere_n ghk_v and_o suppose_v also_o that_o in_o the_o sphere_n abc_n be_v inscribe_v a_o polihedron_n like_a to_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n wherefore_o by_o the_o corollary_n of_o the_o same_o the_o polihedron_n which_o be_v in_o the_o sphere_n abc_n be_v to_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n ef._n but_o by_o supposition_n the_o sphere_n abc_n be_v to_o the_o sphere_n ghk_v in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n ef._n wherefore_o as_o the_o sphere_n abc_n be_v to_o the_o sphere_n ghk_o so_o be_v the_o polihedron_n which_o be_v describe_v in_o the_o sphere_n abc_n to_o the_o polihedron_n which_o be_v describe_v in_o the_o sphere_n def_n by_o the_o 11._o of_o the_o five_o wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o sphere_n abc_n be_v to_o the_o polihedron_n which_o be_v describe_v in_o it_o so_o be_v the_o sphere_n ghk_v to_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n but_o the_o sphere_n abc_n be_v great_a than_o the_o polihedron_n which_o be_v describe_v in_o it_o wherefore_o also_o the_o sphere_n ghk_o be_v great_a than_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n by_o the_o 14._o of_o the_o five_o but_o it_o be_v also_o less_o for_o it_o be_v contain_v in_o it_o which_o impossible_a wherefore_o the_o sphere_n abc_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n of_o to_o any_o sphere_n less_o than_o the_o sphere_n def_n in_o like_a sort_n also_o may_v we_o prove_v that_o the_o sphere_n def_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o be_v to_o the_o diameter_n bc_o to_o any_o sphere_n less_o than_o the_o sphere_n abc_n case_n now_o i_o say_v that_o the_o sphere_n abc_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n of_o to_o any_o sphere_n great_a they_o the_o sphere_n def_n for_o if_o it_o be_v possible_a let_v it_o be_v to_o a_o great_a namely_o to_o lmn_o wherefore_o by_o conversion_n the_o sphere_n lmn_o be_v to_o the_o sphere_n abc_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o be_v to_o the_o diameter_n bc._n but_o as_o the_o sphere_n lmn_o be_v to_o the_o sphere_n abc_n so_o be_v the_o sphere_n def_n to_o some_o sphere_n less_o they_o the_o sphere_n abc_n booâ—â—_n as_o it_o have_v before_o be_v prove_v for_o the_o sphere_n lmn_o be_v great_a than_o the_o sphere_n def_n wherefore_o the_o sphere_n def_n be_v in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o be_v to_o the_o diameter_n bc_o to_o some_o sphere_n less_o they_o the_o sphere_n abc_n which_o be_v prove_v to_o be_v impossible_a wherefore_o the_o sphere_n abc_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o be_v be_v to_o of_o to_o any_o sphere_n great_a they_o the_o sphere_n def_n and_o it_o be_v also_o prove_v that_o it_o be_v not_o to_o any_o less_o wherefore_o the_o sphere_n abc_n be_v to_o the_o sphere_n def_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n of_o which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o flussas_n hereby_o it_o be_v manifest_a that_o sphere_n be_v the_o one_o to_o the_o other_o as_o like_o polihedrons_n and_o in_o like_a sort_n describe_v in_o they_o be_v namely_o each_o be_v in_o triple_a proportion_n of_o that_o in_o which_o the_o diameter_n a_o corollary_n add_v by_o mâ—_n dee_n it_o be_v then_o evident_a how_o to_o geve_v two_o right_a line_n have_v that_o proportion_n between_o they_o which_o any_o two_o sphere_n give_v have_v the_o one_o to_o the_o other_o for_o if_o to_o their_o diameter_n as_o to_o the_o first_o and_o second_o line_n of_o four_o in_o continual_a proportion_n you_o adjoin_v a_o three_o and_o a_o four_o line_n in_o continual_a proportion_n as_o i_o have_v teach_v before_o the_o first_o and_o four_o line_n shall_v answer_v the_o problem_n rule_n how_o general_a this_o rule_n be_v in_o any_o two_o like_a solid_n with_o their_o correspondent_a or_o omologall_n line_n i_o need_v not_o with_o more_o word_n declare_v ¶_o certain_a theorem_n and_o problem_n who_o use_n be_v manifold_a in_o sphere_n cones_n cylinder_n and_o other_o solid_n add_v by_o joh._n dee_n a_o theorem_a 1._o the_o whole_a superficies_n of_o any_o sphere_n be_v quadrupla_fw-la to_o the_o great_a circle_n in_o the_o same_o sphere_n contain_v it_o be_v needless_a to_o bring_v archimedes_n demonstration_n hereof_o into_o this_o place_n see_v his_o book_n of_o the_o sphere_n and_o cylinder_n with_o other_o his_o woâ—kes_n be_v every_o where_o to_o be_v have_v and_o the_o demonstration_n thereof_o easy_a a_o theorem_a 2._o every_o sphere_n be_v quadruplâ—_n to_o that_o cone_n who_o base_a be_v the_o great_a circle_n &_o height_n the_o semidiameter_n of_o the_o same_o sphere_n this_o be_v the_o 32._o proposition_n of_o archimedes_n fiâ—st_n book_n of_o the_o sphere_n and_o cylinder_n a_o problem_n 1._o a_o sphere_n be_v give_v to_o make_v a_o upright_a cone_n equal_a to_o the_o same_o or_o in_o any_o other_o proportion_n give_v between_o two_o right_a line_n and_o as_o concern_v the_o other_o part_n of_o
contain_v in_o a_o the_o sphere_n be_v the_o circle_n bcd_o construction_n and_o by_o the_o problem_n of_o my_o addition_n upon_o the_o second_o proposition_n of_o this_o book_n as_o x_o be_v to_o you_o so_o let_v the_o circle_n bcd_v be_v to_o a_o other_o circle_n find_v let_v that_o other_o circle_n be_v efg_o and_o his_o diameter_n eglantine_n i_o say_v that_o the_o spherical_a superficies_n of_o the_o sphere_n a_o have_v to_o the_o spherical_a superficies_n of_o the_o sphere_n who_o great_a circle_n be_v efg_o or_o his_o equal_a that_o proportion_n which_o x_o have_v to_o y._n demonstration_n for_o by_o construction_n bcd_o be_v to_o efg_o as_o x_o be_v to_o you_o and_o by_o the_o theorem_a next_o beforeâ—_n as_o bcd_o be_v to_o â—fg_n so_o be_v the_o spherical_a superficies_n of_o a_o who_o great_a circle_n be_v bcd_o by_o supposition_n to_o the_o spherical_a superficies_n of_o the_o sphere_n who_o great_a circle_n be_v efg_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o x_o be_v to_o you_o so_o be_v the_o spherical_a superficies_n of_o a_o to_o the_o spherical_a superficies_n of_o the_o sphere_n who_o great_a circle_n be_v efg_o wherefore_o the_o sphere_n who_o diameter_n be_v eglantine_n the_o diameter_n also_o of_o efg_o be_v the_o sphere_n to_o who_o spherical_a superficies_n the_o spherical_a superficies_n of_o the_o sphere_n a_o have_v that_o proportion_n which_o x_o have_v to_o y._n a_o sphere_n be_v give_v therefore_o we_o have_v give_v a_o other_o sphere_n to_o who_o spherical_a superficies_n the_o superficies_n spherical_a of_o the_o sphere_n geum_z have_v any_o proportion_n give_v between_o two_o right_a line_n which_o ought_v to_o be_v do_v a_o problem_n 10._o a_o sphere_n be_v give_v and_o a_o circle_n less_o than_o the_o great_a circle_n in_o the_o same_o sphere_n contain_v to_o cope_v in_o the_o sphere_n give_v a_o circle_n equal_a to_o the_o circle_n give_v suppose_v a_o to_o be_v the_o sphere_n give_v and_o the_o circle_n give_v less_o than_o the_o great_a circle_n in_o a_o contain_v to_o be_v fkg_v spâ—erâ—_n i_o say_v that_o in_o the_o sphere_n a_o a_o circle_n equal_a to_o the_o circle_n fkg_v be_v to_o be_v coapt_v first_o understand_v what_o we_o mean_v here_o by_o coapt_a of_o a_o circle_n in_o a_o sphere_n we_o say_v that_o circle_n to_o be_v coapt_v in_o a_o sphere_n who_o whole_a circumference_n be_v in_o the_o superficies_n of_o the_o same_o sphere_n let_v the_o great_a circle_n in_o the_o sphere_n a_o contain_v be_v the_o circle_n bcd_o who_o diameter_n suppose_v to_o be_v bd_o and_o of_o the_o circle_n fkg_v let_v fg_o be_v the_o diameter_n construction_n by_o the_o 1._o of_o the_o four_o let_v a_o line_n equal_a to_o fg_o be_v coapt_v in_o the_o circle_n bcd_o which_o line_n coapt_v let_v be_v be._n and_o by_o the_o line_n be_v suppose_v a_o plain_a to_o pass_v cut_v the_o sphere_n a_o and_o to_o be_v perpendicular_o erect_v to_o the_o superficies_n of_o bcd_o demonstration_n see_v that_o the_o portion_n of_o the_o plain_n remain_v in_o the_o sphere_n be_v call_v their_o common_a section_n the_o say_a section_n shall_v be_v a_o circle_n as_o before_o be_v prove_v and_o the_o common_a section_n of_o the_o say_a plain_n and_o the_o great_a circle_n bcd_o which_o be_v be_v by_o supposition_n shall_v be_v the_o diameter_n of_o the_o same_o circle_n as_o we_o will_v prove_v for_o let_v that_o circle_n be_v blem_n let_v the_o centre_n of_o the_o sphere_n a_o be_v the_o point_n h_o which_o h_o be_v also_o the_o centre_n of_o the_o circle_n bcd_o because_o bcd_o be_v the_o great_a circle_n in_o a_o contain_v from_o h_o the_o centre_n of_o the_o sphere_n a_o let_v a_o line_n perpendicular_o be_v let_v fall_n to_o the_o circle_n blem_n let_v that_o line_n be_v ho_o and_o it_o be_v evident_a that_o ho_o shall_v fall_v upon_o the_o common_a section_n be_v by_o the_o 38._o of_o the_o eleven_o and_o it_o divide_v be_v into_o two_o equal_a part_n by_o the_o second_o part_n of_o the_o three_o proposition_n of_o the_o three_o book_n by_o which_o point_n oh_o all_o other_o line_n draw_v in_o the_o circle_n blem_n be_v at_o the_o same_o point_n oh_o divide_v into_o two_o equal_a part_n as_o if_o from_o the_o point_n m_o by_o the_o point_n oh_o a_o right_a line_n be_v draw_v one_o the_o other_o side_n come_v to_o the_o circumference_n at_o the_o point_n n_o it_o be_v manifest_a that_o nom_fw-la be_v divide_v into_o two_o equal_a part_n at_o the_o point_n oh_o euclid_n by_o reason_n if_o from_o the_o centre_n h_o to_o the_o point_n n_o and_o m_o right_a line_n be_v draw_v hn_o and_o he_o the_o square_n of_o he_o and_o hn_o be_v equal_a for_o that_o all_o the_o semidiameter_n of_o the_o sphere_n be_v equal_a and_o therefore_o their_o square_n be_v equal_a one_o to_o the_o other_o and_o the_o square_a of_o the_o perpendicular_a ho_o be_v common_a wherefore_o the_o square_a of_o the_o three_o line_n more_n be_v equal_a to_o the_o square_n of_o the_o three_o line_n no_o and_o therefore_o the_o line_n more_n to_o the_o line_n no_o so_o therefore_o be_v nm_o equal_o divide_v at_o the_o point_n o._n and_o so_o may_v be_v prove_v of_o all_o other_o right_a line_n draw_v in_o the_o circle_n blem_n pass_v by_o the_o point_n oh_o to_o the_o circumference_n one_o both_o side_n wherefore_o o_n be_v the_o centre_n of_o the_o circle_n blem_n and_o therefore_o be_v pass_v by_o the_o point_n oh_o be_v the_o diameter_n of_o the_o circle_n blem_n which_o circle_n i_o say_v be_v equal_a to_o fkg_v for_o by_o construction_n be_v be_v equal_a to_o fg_o and_o be_v be_v prove_v the_o diameter_n of_o blem_n and_o fg_o be_v by_o supposition_n the_o diameter_n of_o the_o circle_n fkg_v wherefore_o blem_n be_v equal_a to_o fkg_v the_o circle_n give_v and_o blem_n be_v in_o a_o the_o sphere_n give_v wherefore_o we_o have_v in_o a_o sphere_n give_v coapt_v a_o circle_n equal_a to_o a_o circle_n give_v which_o be_v to_o be_v do_v a_o corollary_n beside_o our_o principal_a purpose_n in_o this_o problem_n evident_o demonstrate_v this_o be_v also_o make_v manifest_a that_o if_o the_o great_a circle_n in_o a_o sphere_n be_v cut_v by_o a_o other_o circle_n erect_v upon_o he_o at_o right_a angle_n that_o the_o other_o circle_n be_v cut_v by_o the_o centre_n and_o that_o their_o common_a section_n be_v the_o diameter_n of_o that_o other_o circle_n and_o therefore_o that_o other_o circle_n divide_v be_v into_o two_o equal_a part_n a_o problem_n 11._o a_o sphere_n be_v give_v and_o a_o circle_n less_o than_o double_a the_o great_a circle_n in_o the_o same_o sphere_n contain_v to_o cut_v of_o a_o segment_n of_o the_o same_o sphere_n who_o spherical_a superficies_n shall_v be_v equal_a to_o the_o circle_n give_v suppose_v king_n to_o be_v a_o sphere_n give_v who_o great_a circle_n let_v be_v abc_n and_o the_o circle_n give_v suppose_v to_o be_v def_n construction_n i_o say_v that_o a_o segment_n of_o the_o sphere_n king_n be_v to_o be_v cut_v of_o so_o great_a that_o his_o spherical_a superficies_n shall_v be_v equal_a to_o the_o circle_n def_n let_v the_o diameter_n of_o the_o circle_n abc_n be_v the_o line_n ab_fw-la at_o the_o point_n a_o in_o the_o circle_n abc_n cope_v a_o right_a line_n equal_a to_o the_o semidiameter_n of_o the_o circle_n def_n by_o the_o first_o of_o the_o four_o which_o line_n suppose_v to_o be_v ah_o from_o the_o point_n h_o to_o the_o diameter_n ab_fw-la let_v a_o perpendicular_a line_n be_v draw_v which_o suppose_v to_o be_v high_a produce_v high_a to_o the_o other_o side_n of_o the_o circumference_n and_o let_v it_o come_v to_o the_o circumference_n at_o the_o point_n l._n by_o the_o right_a line_n hil_n perpendicular_a to_o ab_fw-la suppose_v a_o plain_a superficies_n to_o pass_v perpendicular_o erect_v upon_o the_o circle_n abc_n and_o by_o this_o plain_a superficies_n the_o sphere_n to_o be_v cut_v into_o two_o segment_n one_o less_o than_o the_o half_a sphere_n namely_o hali_n and_o the_o other_o great_a than_o the_o half_a sphere_n namely_o hbli_n i_o say_v that_o the_o spherical_a superficies_n of_o the_o segment_n of_o the_o sphere_n king_n in_o which_o the_o segment_n of_o the_o great_a circle_n hali_n be_v contain_v who_o base_a be_v the_o circle_n pass_v by_o hil_n and_o top_n the_o point_n a_o be_v equal_a to_o the_o circle_n def_n demonstration_n for_o the_o circle_n who_o semidiameter_n be_v equal_a to_o the_o line_n ah_o be_v equal_a to_o the_o spherical_a superficies_n of_o the_o segment_n hal_n by_o the_o 4._o theorem_a here_o add_v and_o by_o construction_n ah_o be_v equal_a to_o the_o semidiameter_n of_o the_o circle_n def_n therefore_o the_o spherical_a superficies_n of_o the_o segment_n of_o the_o sphere_n king_n cut_v of_o by_o the_o
three_o proposition_n we_o will_v use_v the_o same_o supposition_n and_o construction_n there_o specify_v so_o far_o as_o they_o shall_v serve_v our_o purpose_n begin_v therefore_o at_o the_o conclusion_n we_o must_v infer_v the_o part_n of_o the_o proposition_n before_o grant_v it_o be_v conclude_v that_o the_o square_n of_o the_o line_n db_o be_v quintuple_a to_o the_o square_n of_o the_o line_n dc_o his_o own_o segment_n therefore_o dn_o the_o square_n of_o db_o be_v quintuple_a to_o gf_o the_o square_n of_o dc_o but_o the_o squaâ—e_n of_o ac_fw-la the_o double_a of_o dc_o which_o be_v r_n be_v quadruple_a to_o gf_o by_o the_o second_o corollary_n of_o the_o 20._o of_o the_o six_o and_o therefore_o r_n with_o gf_o be_v quintuple_a to_o gf_o and_o so_o it_o be_v evident_a that_o the_o square_a dn_o be_v equal_a to_o the_o square_a r_n together_o with_o the_o square_a gf_o wherefore_o from_o those_o two_o equal_n take_v the_o square_a gf_o common_a to_o they_o both_o remain_v the_o square_a r_n equal_a to_o the_o gnomon_n xop_n but_o to_o the_o gnomon_n xop_n the_o parallelogram_n 3._o ce_fw-fr be_v equal_a wherefore_o the_o square_a of_o the_o line_n ac_fw-la which_o be_v r_n be_v equal_a to_o the_o parallelogram_n câ—_n which_o parallelogamme_n be_v contain_v under_o be_v equal_a to_o ab_fw-la and_o cb_o the_o part_n remain_v of_o the_o first_o line_n gruen_o which_o be_v db._n and_o the_o line_n ab_fw-la be_v make_v of_o the_o double_a of_o the_o segment_n dc_o and_o of_o cbâ—_n the_o other_o part_n of_o the_o line_n db_o first_o goven_v wherefore_o the_o double_a of_o the_o segment_n dc_o with_o cb_o the_o part_n remain_v which_o altogether_o be_v the_o whole_a line_n ab_fw-la be_v to_o ac_fw-la the_o double_a of_o the_o segment_n dc_o as_o that_o same_o ac_fw-la be_v to_o cb_o by_o the_o second_o part_n of_o the_o 16._o of_o the_o six_o therefore_o by_o the_o 3._o definition_n of_o the_o six_o book_n the_o whole_a line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n &_o ac_fw-la the_o double_a of_o the_o segment_n dc_o be_v middle_a proportional_a be_v the_o great_a part_n thereof_o wherefore_o if_o a_o right_a line_n be_v quintuple_a in_o power_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v or_o thus_o it_o may_v be_v demonstrate_v forasmuch_o as_o the_o square_n dn_o be_v quintuple_a to_o the_o square_a gf_o i_o mean_v the_o square_n of_o db_o the_o line_n give_v too_o the_o square_a oâ—_n dc_o the_o segment_n and_o the_o same_o square_a dn_o be_v equal_a to_o the_o parallelogram_n under_o ab_fw-la cb_o with_o the_o square_n make_v of_o the_o line_n dc_o by_o the_o six_o of_o the_o second_o for_o unto_o the_o line_n ac_fw-la equal_o divide_v the_o line_n cb_o be_v as_o it_o be_v adjoin_v wherefore_o the_o parallelogram_n under_o ab_fw-la cb_o together_o with_o the_o square_n of_o dc_o which_o be_v gf_o be_v quintuple_a to_o the_o square_a gf_o make_v oâ—_n thâ—_z line_n dc_o take_v then_o that_o square_a gf_o â—rom_o the_o parallelogram_n under_o ab_fw-la cb_o that_o parallelogram_n under_o ab_fw-la cb_o remain_v alone_o be_v but_o quadruple_a to_o the_o say_v square_a of_o the_o line_n dc_o but_o by_o the_o 4._o of_o the_o second_o or_o the_o second_o corollary_n of_o the_o 20._o of_o the_o six_o r_n â—he_z square_a of_o the_o line_n ac_fw-la be_v quadrupla_fw-la to_o the_o same_o square_a gfâ—_n wherefore_o by_o the_o 7._o of_o the_o five_o the_o square_a of_o the_o line_n ac_fw-la be_v equal_a to_o the_o parallelogram_n under_o ab_fw-la cb_o and_o so_o by_o the_o second_o part_n of_o the_o 16._o of_o the_o six_o ab_fw-la ac_fw-la and_o cb_o be_v three_o line_n in_o continual_a proportion_n and_o see_v ab_fw-la be_v great_a they_o ac_fw-la the_o same_o ac_fw-la the_o double_a of_o the_o line_n dc_o shall_v be_v great_a than_o the_o part_n bc_o remain_v wherefore_o by_o the_o 3._o definition_n of_o the_o six_o ab_fw-la compose_v or_o make_v of_o the_o double_a of_o dc_o and_o the_o other_o part_n of_o db_o remain_v be_v divide_v by_o a_o extreme_a and_o middle_n proportion_n and_o also_o his_o great_a segment_n be_v ac_fw-la the_o double_a of_o the_o segment_n dc_o wherefore_o if_o a_o right_a line_n be_v quintuple_a in_o power_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v a_o theorem_a 2._o if_o a_o right_a line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n be_v give_v and_o to_o the_o great_a segment_n â—herof_o he_o direct_o adjoin_v a_o line_n equal_a to_o the_o whole_a line_n give_v that_o adjoin_v line_n and_o the_o say_v great_a segment_n do_v make_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n who_o great_a segment_n be_v the_o line_n adjoin_v suppose_v the_o line_n give_v divide_v by_o extreme_a and_o mean_a proportion_n to_o be_v ab_fw-la divide_v in_o the_o point_n c_o and_o his_o great_a segment_n let_v be_v ac_fw-la unto_fw-la ac_fw-la direct_o adjoin_v a_o line_n equal_a to_o ab_fw-la let_v that_o be_v ad_fw-la i_o say_v that_o ad_fw-la together_o with_o ac_fw-la that_o be_v dc_o be_v a_o divide_v by_o extreme_a and_o middle_n proportion_n who_o great_a segment_n be_v ad_fw-la the_o line_n adjoin_v divide_v ad_fw-la equal_o in_o the_o point_n e._n now_o forasmuch_o as_o ae_n be_v the_o half_a of_o ad_fw-la by_o construction_n it_o be_v also_o the_o half_a of_o ab_fw-la equal_a to_o ad_fw-la by_o construction_n wherefore_o by_o the_o 1._o of_o the_o thirteen_o the_o square_a of_o the_o line_n compose_v of_o ac_fw-la and_o ae_n which_o â—ne_n be_v ec_o be_v quintuple_a to_o the_o square_n of_o the_o line_n ae_n wherefore_o the_o double_a of_o ae_n and_o the_o line_n ac_fw-la compose_v as_o in_o one_o right_a line_n be_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n by_o the_o converse_n of_o this_o three_o by_o i_o demonstrate_v and_o the_o double_a of_o ae_n be_v the_o great_a segment_n but_o dc_o be_v the_o line_n compose_v of_o the_o double_a of_o ae_n &_o the_o line_n ac_fw-la and_o with_o all_o ad_fw-la be_v the_o double_a of_o ae_n wherefore_o dc_o be_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n and_o ad_fw-la iâ—_z hiâ—_z great_a segment_n if_o a_o right_a line_n therefore_o divide_v by_o extreme_a and_o mean_a proportion_n be_v give_v and_o to_o the_o great_a segment_n thereof_o be_v direct_o adjoin_v a_o line_n equal_a to_o the_o whole_a line_n give_v that_o adjoin_v line_n and_o the_o say_v great_a segment_n do_v make_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n who_o great_a segment_n be_v the_o line_n adjoin_v which_o be_v require_v to_o be_v demonstrate_v two_o other_o brief_a demonstration_n of_o the_o same_o forasmuch_o as_o ad_fw-la be_v to_o ac_fw-la as_o ab_fw-la be_v to_o ac_fw-la because_o ad_fw-la be_v equal_a to_o ab_fw-la by_o construction_n but_o as_o ab_fw-la be_v to_o ac_fw-la so_o be_v ac_fw-la to_o cb_o by_o supposition_n therefore_o by_o the_o 11._o of_o the_o five_o as_z ac_fw-la be_v to_o cb_o so_o be_v ad_fw-la to_o ac_fw-la demonstrate_v wherefore_o as_o ac_fw-la and_o cb_o which_o be_v ab_fw-la be_v to_o cb_o so_o be_v ad_fw-la and_o ac_fw-la which_o be_v dc_o to_o ac_fw-la therefore_o everse_o as_o ab_fw-la be_v to_o ac_fw-la so_o be_v dc_o to_o ad._n and_o it_o be_v prove_v ad_fw-la to_o be_v to_o ac_fw-la as_o ac_fw-la be_v to_o cb._n wherefore_o as_o ab_fw-la be_v to_o ac_fw-la and_o ac_fw-la to_o cb_o so_o be_v dc_o to_o ad_fw-la and_o ad_fw-la to_o ac_fw-la but_o ab_fw-la ac_fw-la and_o cb_o be_v in_o continual_a proportion_n by_o supposition_n wherefore_o dc_o ad_fw-la and_o ac_fw-la be_v in_o continual_a proportion_n wherefore_o by_o the_o 3._o definition_n of_o the_o six_o book_n dc_o be_v divide_v by_o extreme_a and_o middle_a proportion_n and_o his_o great_a segment_n be_v ad._n which_o be_v to_o be_v demonstrate_v note_v from_o the_o mark_n demonstrate_v how_o this_o have_v two_o demonstration_n one_o i_o have_v set_v in_o the_o margin_n by_o ¶_o a_o corollary_n 1._o upon_o euclides_n three_o proposition_n demonstrate_v it_o be_v make_v evident_a that_o of_o a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n if_o you_o produce_v the_o less_o segment_n equal_o to_o the_o length_n of_o the_o great_a the_o line_n thereby_o adjoin_v together_o with_o the_o say_v less_o segment_n make_v a_o new_a line_n divide_v by_o extreme_a and_o middle_a proportion_n who_o less_o segment_n be_v the_o line_n adjoin_v for_o if_o ab_fw-la be_v divide_v by_o extreme_a and_o middle_a proportion_n in_o the_o point_n c_o ac_fw-la be_v the_o great_a segment_n and_o cb_o be_v produce_v from_o the_o point_n b_o make_v a_o line_n with_o cb_o equal_a to_o ac_fw-la which_o let_v be_v cq_n and_o the_o
double_a to_o the_o side_n of_o the_o octohedron_n &_o the_o side_n be_v in_o power_n sequitertia_fw-la to_o the_o perpendiclar_a line_n by_o the_o 12._o of_o this_o book_n wherefore_o the_o diameter_n thereof_o be_v in_o power_n duple_fw-fr superbipartiens_fw-la tertias_fw-la to_o the_o perpendicular_a line_n wherefore_o also_o the_o diameter_n and_o the_o perpendicular_a line_n be_v rational_a and_o commensurable_a by_o the_o 6._o of_o the_o ten_o as_o touch_v a_o icosahedron_n it_o be_v prove_v in_o the_o 16._o of_o this_o book_n that_o the_o side_n thereof_o be_v a_o less_o line_n when_o the_o diameter_n of_o the_o sphere_n be_v rational_a and_o forasmuch_o as_o the_o angle_n of_o the_o inclination_n of_o the_o base_n thereof_o be_v contain_v of_o the_o perpendicular_a line_n of_o the_o triangle_n and_o subtend_v of_o the_o right_a line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n which_o contain_v five_o side_n of_o the_o icosahedron_n and_o unto_o the_o perpendicular_a line_n the_o side_n be_v commensurable_a namely_o be_v in_o power_n sesquitertia_fw-la unto_o they_o by_o the_o corollary_n of_o the_o 12._o of_o this_o book_n therefore_o the_o perpendicular_a line_n which_o contain_v the_o angle_n be_v irrational_a line_n namely_o less_o line_n by_o the_o 105._o of_o the_o ten_o book_n and_o forasmuch_o as_o the_o diameter_n contain_v in_o power_n both_o the_o side_n of_o the_o icosahedron_n and_o the_o line_n which_o subtend_v the_o foresay_a angle_n if_o from_o the_o power_n of_o the_o diameter_n which_o be_v rational_a be_v take_v away_o the_o power_n of_o the_o side_n of_o the_o icosahedron_n which_o be_v irrational_a it_o be_v manifest_a that_o the_o residue_n which_o be_v the_o power_n of_o the_o subtend_a line_n shall_v be_v irrational_a for_o if_o it_o shall_v be_v rational_a the_o number_n which_o measure_v the_o whole_a power_n of_o the_o diameter_n and_o the_o part_n take_v away_o of_o the_o subtend_a line_n shall_v also_o by_o the_o 4._o common_a sentence_n of_o the_o seven_o measure_n the_o residue_n namely_o the_o power_n of_o the_o side_n irrational_a which_o be_v irrational_a for_o that_o it_o be_v a_o less_o line_n which_o be_v absurd_a wherefore_o it_o be_v manifest_a that_o the_o right_a line_n which_o compose_v the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o the_o icosahedron_n be_v irrational_a line_n for_o the_o subtend_a line_n have_v to_o the_o line_n contain_v a_o great_a proportion_n than_o the_o whole_a have_v to_o the_o great_a segment_n the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o a_o dodecahedron_n be_v contain_v under_o two_o perpendicular_n of_o the_o base_n of_o the_o dodecahedron_n and_o be_v subtend_v of_o that_o right_a line_n who_o great_a segment_n be_v the_o side_n of_o a_o cube_n inscribe_v in_o the_o dodecahedron_n which_o right_a line_n be_v equal_a to_o the_o line_n which_o couple_v the_o section_n into_o two_o equal_a part_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n and_o this_o couple_a line_n we_o say_v be_v a_o irrational_a line_n for_o that_o the_o diameter_n of_o the_o sphere_n contain_v in_o power_n both_o the_o couple_a line_n and_o the_o side_n of_o the_o dodecahedron_n but_o the_o side_n of_o the_o dodecahedron_n be_v a_o irrational_a line_n namely_o a_o residual_a line_n by_o the_o 17._o of_o this_o book_n wherefore_o the_o residue_n namely_o the_o couple_a line_n be_v a_o irrational_a line_n as_o it_o be_v â—asy_a to_o prove_v by_o the_o 4._o common_a sentence_n of_o the_o seven_o and_o that_o the_o perpendicular_a line_n which_o contain_v the_o angle_n of_o the_o inclination_n be_v irrational_a be_v thus_o prove_v by_o the_o proportion_n of_o the_o subtend_a line_n of_o the_o foresay_a angle_n of_o inclination_n to_o the_o line_n which_o contain_v the_o angle_n be_v find_v out_o the_o obliquity_n of_o the_o angle_n angle_n for_o if_o the_o subtend_a line_n be_v in_o power_n double_a to_o the_o line_n which_o contain_v the_o angle_n then_o be_v the_o angle_n a_o right_a angle_n by_o the_o 48._o of_o the_o first_o but_o if_o it_o be_v in_o power_n less_o than_o the_o double_a it_o be_v a_o acute_a angle_n by_o the_o 23._o of_o the_o second_o but_o if_o it_o be_v in_o power_n more_o than_o the_o double_a or_o have_v a_o great_a proportion_n than_o the_o whole_a have_v to_o the_o great_a segmentâ—_n the_o angle_n shall_v be_v a_o obtuse_a angle_n by_o the_o 12._o of_o the_o second_o and_o 4._o of_o the_o thirteen_o by_o which_o may_v be_v prove_v that_o the_o square_a of_o the_o whole_a be_v great_a than_o the_o double_a of_o the_o square_n of_o the_o great_a segment_n this_o be_v to_o be_v note_v that_o that_o which_o flussas_n have_v here_o teach_v touch_v the_o inclination_n of_o the_o base_n of_o the_o â—ive_a regular_a body_n hypsicles_n teach_v after_o the_o 5_o proposition_n of_o the_o 15._o book_n where_o he_o confess_v that_o he_o receive_v it_o of_o one_o isidorus_n and_o seek_v to_o make_v the_o matter_n more_o clear_a he_o endeavour_v himself_o to_o declare_v that_o the_o angle_n of_o the_o inclination_n of_o the_o solid_n be_v give_v and_o that_o they_o be_v either_o acute_a or_o obtuse_a according_a to_o the_o nature_n of_o the_o solid_a although_o euclid_v in_o all_o his_o 15._o book_n have_v not_o yet_o show_v what_o a_o thing_n give_v be_v wherefore_o flussas_n frame_v his_o demonstration_n upon_o a_o other_o ground_n proceed_v after_o a_o other_o manner_n which_o seem_v more_o plain_a and_o more_o apt_o hereto_o be_v place_v then_o there_o albeit_o the_o reader_n in_o that_o place_n shall_v not_o be_v frustrate_a of_o his_o also_o the_o end_n of_o the_o thirteen_o book_n of_o euclides_n element_n ¶_o the_o fourteen_o book_n of_o euclides_n element_n in_o this_o book_n which_o be_v common_o account_v the_o 14._o book_n of_o euclid_n be_v more_o at_o large_a entreat_v of_o our_o principal_a purpose_n book_n namely_o of_o the_o comparison_n and_o proportion_n of_o the_o five_o regular_a body_n customable_o call_v the_o 5._o figure_n or_o form_n of_o pythagoras_n the_o one_o to_o the_o other_o and_o also_o of_o their_o side_n together_o each_o to_o other_o which_o thing_n be_v of_o most_o secret_a use_n and_o inestimable_a pleasure_n and_o commodity_n to_o such_o as_o diligent_o search_v for_o they_o and_o attain_v unto_o they_o which_o thing_n also_o undoubted_o for_o the_o worthiness_n and_o hardness_n thereof_o for_o thing_n of_o most_o price_n be_v most_o hard_a be_v first_o search_v and_o find_v out_o of_o philosopher_n not_o of_o the_o inferior_a or_o mean_a sort_n but_o of_o the_o deep_a and_o most_o ground_a philosopher_n and_o best_a exercise_v in_o geometry_n and_o albeit_o this_o book_n with_o the_o book_n follow_v namely_o the_o 15._o book_n have_v be_v hitherto_o of_o all_o man_n for_o the_o most_o part_n and_o be_v also_o at_o this_o day_n number_v and_o account_v among_o euclides_n book_n and_o suppose_v to_o be_v two_o of_o he_o namely_o the_o 14._o and_o 15._o in_o order_n as_o all_o exemplar_n not_o only_o new_a and_o late_o set_v abroad_o but_o also_o old_a monument_n write_v by_o hand_n do_v manifest_o witness_v yet_o it_o be_v think_v by_o the_o best_a learned_a in_o these_o day_n that_o these_o two_o book_n be_v none_o of_o euclides_n but_o of_o some_o other_o author_n no_o less_o worthy_a nor_o of_o less_o estimation_n and_o authority_n notwithstanding_o than_o euclid_n apollonius_n a_o man_n of_o deep_a knowledge_n a_o great_a philosopher_n and_o in_o geometry_n marvellous_a who_o wonderful_a book_n write_v of_o the_o section_n of_o cones_fw-la which_o exercise_n &_o occupy_v thewitte_v of_o the_o wise_a and_o best_a learned_a be_v yet_o remain_v be_v think_v and_o that_o not_o without_o just_a cause_n to_o be_v the_o author_n of_o they_o or_o as_o some_o think_v hypsicles_n himself_o for_o what_o can_v be_v more_o plain_o then_o that_o which_o he_o himself_o witness_v in_o the_o preface_n of_o this_o book_n basilides_n of_o tire_n say_v hypsicles_n and_o my_o father_n together_o scan_v and_o peyse_v a_o writing_n or_o book_n of_o apollonius_n which_o be_v of_o the_o comparison_n of_o a_o dodecahedron_n to_o a_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n and_o what_o proportion_n these_o figure_n have_v the_o one_o to_o the_o other_o find_v that_o apollonius_n have_v fail_v in_o this_o matter_n but_o afterward_o say_v he_o i_o find_v a_o other_o copy_n or_o book_n of_o apollonius_n wherein_o the_o demonstration_n of_o that_o matter_n be_v full_a and_o perfect_a and_o show_v it_o unto_o they_o whereat_o they_o much_o rejoice_v by_o which_o word_n it_o seem_v to_o be_v manifest_a that_o apollonius_n be_v the_o first_o author_n of_o this_o book_n which_o be_v afterward_o set_v forth_o by_o hypsicles_n for_o so_o his_o own_o word_n after_o in_o
the_o same_o preface_n seem_v to_o import_v the_o preface_n of_o hypsicles_n before_o the_o fourteen_o book_n friend_n protarchus_n when_o that_o basilides_n of_o tire_n come_v into_o alexandria_n have_v familiar_a friendship_n with_o my_o father_n by_o reason_n of_o his_o knowledge_n in_o the_o mathematical_a science_n he_o remain_v with_o he_o a_o long_a time_n yea_o even_o all_o the_o time_n of_o the_o pestilence_n and_o sometime_o reason_v between_o themselves_o of_o that_o which_o apollonius_n have_v write_v touch_v the_o comparison_n of_o a_o dodecahedron_n and_o of_o a_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n what_o proportion_n such_o body_n have_v the_o one_o to_o the_o other_o they_o judge_v that_o apollonius_n have_v somewhat_o err_v therein_o wherefore_o they_o as_o my_o father_n declare_v unto_o i_o diligent_o weigh_v it_o write_v it_o perfect_o howbeit_o afterward_o i_o happen_v to_o find_v a_o other_o book_n write_v of_o apollonius_n which_o contain_v in_o it_o the_o right_a demonstration_n of_o that_o which_o they_o seek_v for_o which_o when_o they_o see_v they_o much_o rejoice_v as_o for_o that_o which_o apollonius_n write_v may_v be_v see_v of_o all_o man_n for_o that_o it_o be_v in_o â—uery_n man_n hand_n and_o that_o which_o be_v of_o we_o more_o diligent_o afterward_o write_v again_o i_o think_v good_a to_o send_v and_o dedicate_v unto_o you_o as_z to_o one_o who_o i_o think_v worthy_a commendation_n both_o for_o that_o deep_a knowledge_n which_o i_o know_v you_o have_v in_o all_o kind_n of_o learning_n and_o chief_o in_o geometry_n so_o that_o you_o be_v able_a ready_o to_o judge_v of_o those_o thing_n which_o be_v speak_v and_o also_o for_o the_o great_a love_n and_o good_a will_n which_o you_o bear_v towards_o my_o father_n and_o i_o wherefore_o vouchsafe_v gentle_o to_o accept_v this_o which_o i_o send_v unto_o you_o but_o now_o be_v it_o time_n to_o end_v our_o preface_n and_o to_o begin_v the_o matter_n ¶_o the_o 1._o theorem_a the_o 1._o proposition_n flussas_n a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o a_o circle_n to_o the_o side_n of_o a_o pentagon_n describe_v in_o the_o same_o circle_n be_v the_o half_a of_o these_o two_o line_n namely_o of_o the_o side_n of_o a_o hexagon_n figure_n and_o of_o the_o side_n of_o a_o decagon_n figure_n be_v both_o describe_v in_o the_o self_n same_o circle_n svppose_v that_o the_o circle_n be_v abc_n construction_n and_o let_v the_o side_n of_o a_o equilater_n pentagon_n describe_v in_o the_o circle_n abc_n be_v bc._n and_o by_o the_o 1._o of_o the_o three_o take_v the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v d._n and_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n d_o draw_v unto_o the_o line_n bc_o a_o perpendicular_a line_n de._n and_o extend_v the_o right_a line_n de_fw-fr direct_o to_o the_o point_n f._n then_o i_o say_v that_o the_o line_n de_fw-fr which_o be_v draw_v from_o the_o centre_n to_o bc_o the_o side_n of_o the_o pentagon_n be_v the_o half_a of_o the_o side_n of_o a_o hexagon_n and_o of_o a_o decagon_n take_v together_o and_o describe_v in_o the_o same_o circle_n draw_v these_o right_a line_n dc_o and_o cf._n and_o unto_o the_o line_n of_o put_v a_o equal_a line_n ge._n and_o draw_v a_o right_a line_n from_o the_o point_n g_o to_o the_o point_n c._n demonstration_n now_o forasmuch_o as_o the_o circumference_n of_o the_o whole_a circle_n be_v quintuple_a to_o the_o circumference_n bfc_n which_o be_v subtend_v of_o the_o side_n of_o the_o pentagon_n and_o the_o circumference_n acf_n be_v the_o half_a of_o the_o circumference_n of_o the_o whole_a circle_n and_o the_o circumference_n cf_o which_o be_v subtend_v of_o the_o side_n of_o the_o decagon_n be_v the_o half_a of_o the_o circumference_n bcf_n therefore_o the_o circumference_n acf_n be_v quintuple_a to_o the_o circumference_n cf_o by_o the_o 15._o of_o the_o â—iâ—t_n wherefore_o the_o circumference_n ac_fw-la be_v qradruple_a to_o the_o circumference_n fc_o but_o as_o the_o circumference_n ac_fw-la be_v to_o the_o circumference_n fc_o so_o be_v the_o angle_n adc_o to_o the_o angle_n fdc_n by_o the_o last_o of_o the_o six_o wherefore_o the_o angle_n adc_o be_v quadruple_a to_o the_o angle_n fdc_n but_o the_o angle_n adc_o be_v double_a to_o the_o angle_n efc_n by_o the_o 20._o of_o the_o three_o wherefore_o the_o angle_n efc_n be_v double_a to_o the_o angle_n gdc_n but_o the_o angle_n efc_n be_v equal_a to_o the_o angle_n egc_n by_o the_o 4._o of_o the_o first_o wherefore_o the_o angle_n egc_n be_v double_a to_o the_o angle_n edc_n wherefore_o the_o line_n dg_o be_v equal_a to_o the_o line_n gc_o by_o the_o 32._o and_o 6._o of_o the_o first_o but_o the_o line_n gc_o be_v equal_a to_o the_o line_n cf_o by_o the_o 4._o of_o the_o first_o wherefore_o the_o line_n dg_o be_v equal_a to_o the_o line_n cf._n and_o the_o line_n ge_z be_v equal_a to_o the_o line_n of_o by_o construction_n wherefore_o the_o line_n de_fw-fr be_v equal_a to_o the_o line_n of_o and_o fc_n add_v together_o unto_o the_o line_n of_o and_o fc_n add_v the_o line_n de._n wherefore_o the_o line_n df_o and_o fc_o add_v together_o be_v double_a to_o the_o line_n de._n but_o the_o line_n df_o be_v equal_a to_o the_o side_n of_o the_o hexagon_n and_o fc_a to_o the_o side_n of_o the_o decagon_n wherefore_o the_o line_n de_fw-fr be_v the_o half_a of_o the_o side_n of_o the_o hexagon_n and_o of_o the_o side_n of_o the_o decagon_n be_v both_o add_v together_o and_o describe_v in_o one_o and_o the_o self_n same_o circle_n it_o be_v manifest_a same_o by_o the_o proposition_n of_o the_o thirteen_o book_n that_o a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o a_o circle_n to_o the_o side_n of_o a_o equilater_n triangle_n describe_v in_o the_o same_o circle_n be_v half_a of_o the_o semidiameter_n of_o the_o circle_n wherefore_o by_o this_o proposition_n a_o perpendicular_a draw_v from_o the_o câ—ntre_n of_o a_o circle_n to_o the_o side_n of_o a_o pentagon_n be_v equal_a to_o the_o perpendicular_a draw_v from_o the_o centre_n to_o the_o side_n of_o the_o triangle_n â—nd_v to_o half_a of_o the_o side_n of_o the_o decagon_n describe_v in_o the_o same_o circle_n ¶_o the_o 2._o theorem_a the_o 2._o proposition_n one_o and_o the_o self_n same_o circle_n comprehend_v both_o the_o pentagon_n of_o a_o dodecahedron_n and_o the_o triangle_n of_o a_o icosahedron_n flussas_n describe_v in_o one_o and_o the_o self_n same_o sphere_n this_o theorem_a be_v describe_v of_o aristeus_n in_o that_o book_n who_o title_n be_v the_o comparison_n of_o the_o five_o figure_n and_o be_v describe_v of_o apollonius_n in_o his_o second_o edition_n of_o the_o comparison_n of_o a_o dodecahedron_n to_o a_o icosahedron_n which_o be_v proposition_n that_o as_o the_o superficies_n of_o a_o dodecahedron_n be_v to_o the_o superficies_n of_o a_o icosahedron_n so_o be_v the_o dodecahedron_n to_o a_o icosahedron_n for_o that_o a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o a_o sphere_n to_o the_o pentagon_n of_o a_o dodecahedron_n and_o to_o the_o triangle_n of_o a_o icosahedron_n be_v one_o and_o the_o self_n same_o now_o must_v we_o also_o prove_v that_o one_o and_o the_o self_n same_o circle_n comprehend_v both_o the_o pentagon_n of_o a_o dodecahedron_n and_o also_o the_o triangle_n of_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n first_o this_o be_v prove_v flussas_n if_o in_o a_o circle_n be_v describe_v a_o equilater_n pentagon_n the_o square_n which_o be_v make_v of_o the_o side_n of_o the_o pentagon_n and_o of_o that_o right_a line_n which_o be_v subtend_v under_o two_o side_n of_o the_o pentagon_n be_v quintuple_a to_o the_o square_n of_o the_o semidiameter_n oâ—_n the_o circle_n suppose_v that_o abc_n be_v a_o circle_n assumpt_n and_o let_v the_o side_n of_o a_o pentagon_n in_o the_o circle_n abc_n be_v ac_fw-la 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 and_o let_v the_o same_o be_v d._n and_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n d_o draw_v unto_o the_o line_n ac_fw-la a_o perpendicular_a line_n df._n and_o extend_v the_o line_n df_o on_o either_o side_n to_o the_o point_n b_o and_o e._n and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n b._n now_o i_o say_v that_o the_o square_n of_o the_o line_n basilius_n and_o ac_fw-la be_v quintuple_a to_o the_o square_n of_o the_o line_n de._n draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n e._n wherefore_o the_o line_n ae_n be_v the_o side_n of_o a_o decagon_n figure_n and_o forasmuch_o as_o the_o line_n be_v be_v double_a to_o thâ—_z line_n de_fw-fr assumpt_n therefore_o the_o square_n
triangle_n wherefore_o six_o such_o triangle_n as_o dbc_n be_v 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 bc_o thrice_o but_o six_o sâ—ch_a triangle_n as_o dbc_n be_v be_v equal_a to_o two_o such_o triangle_n as_o abc_n be_v wherefore_o that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o bc_o thrice_o be_v equal_a to_o two_o such_o triangle_n as_o abc_n be_v but_o two_o of_o those_o triangle_n take_v ten_o time_n contain_v the_o whole_a icosahedron_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n de_fw-fr &_o bc_o thirty_o time_n be_v equal_a to_o twenty_o such_o triangle_n as_o the_o triangle_n abc_n be_v that_o be_v to_o the_o whole_a superficies_n of_o the_o icosahedron_n follow_v wherefore_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v that_o which_o be_v contain_v under_o the_o line_n cd_o and_o fg_o to_o that_o which_o be_v contain_v under_o the_o line_n bc_o and_o de._n ¶_o corollary_n by_o this_o it_o be_v manifest_a that_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n order_n so_o be_v that_o which_o be_v contain_v under_o the_o side_n of_o the_o pentagon_n and_o the_o perpendicular_a line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n describe_v about_o the_o pentagon_n to_o the_o same_o side_n to_o that_o which_o be_v contain_v under_o the_o side_n of_o the_o icosahedron_n and_o the_o perpendicular_a line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n describe_v about_o the_o triangle_n to_o the_o same_o side_n so_o that_o the_o icosahedron_n and_o dodecahedron_n be_v both_o describe_v in_o one_o and_o the_o self_n same_o sphere_n ¶_o the_o 4._o theorem_a the_o 4._o proposition_n flussas_n this_o be_v do_v now_o be_v to_o be_v prove_v that_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n construction_n take_v by_o the_o 2._o theorem_a of_o this_o book_n a_o circle_n contain_v both_o the_o pentagon_n of_o a_o dodecahedron_n and_o the_o triangle_n of_o a_o icosahedron_n be_v both_o describe_v in_o one_o and_o the_o self_n same_o sphere_n and_o let_v the_o same_o circle_n be_v dbc_n and_o in_o the_o circle_n dbc_n describe_v the_o side_n of_o a_o equilater_n triangle_n namely_o cd_o and_o the_o side_n of_o a_o equilater_n pentagon_n namely_o ac_fw-la 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 and_o let_v the_o same_o be_v e._n and_o from_o the_o point_n e_o draw_v unto_o the_o line_n dc_o and_o ac_fw-la perpendicular_a line_n of_o and_o eglantine_n and_o extend_v the_o line_n eglantine_n direct_o to_o the_o point_n b._n and_o draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n c._n and_o let_v the_o side_n of_o the_o cube_fw-la be_v the_o line_n h._n now_o i_o say_v that_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v the_o line_n h_o to_o the_o line_n cd_o demonstration_n forasmuch_o as_o the_o line_n make_v of_o the_o line_n ebb_v and_o bc_o add_v together_o namely_o of_o the_o side_n of_o the_o hexagon_n and_o of_o the_o side_n of_o a_o decagon_n be_v by_o the_o 9_o of_o the_o thirteen_o divide_a by_o a_o extreme_a and_o mean_a proportion_n and_o his_o great_a segment_n be_v the_o line_n be_v and_o the_o line_n eglantine_n be_v also_o by_o the_o 1._o of_o the_o foâ—retenth_n the_o half_a of_o the_o same_o line_n and_o the_o line_n of_o be_v the_o half_a of_o the_o line_n be_v by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o the_o line_n eglantine_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n â—â—â—eth_v his_o great_a segment_n shall_v be_v the_o line_n ef._n and_o the_o line_n h_o also_o be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n his_o great_a segment_n be_v the_o line_n ca_n as_o it_o be_v prove_v tâ—irtenth_o in_o the_o dodecahedron_n proposition_n wherefore_o as_o the_o line_n h_o be_v to_o the_o line_n ca_n so_o be_v the_o line_n eglantine_n to_o the_o line_n ef._n wherefore_o by_o the_o 16._o of_o the_o six_o that_o which_o be_v contain_v under_o the_o line_n h_o and_o of_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ca_n and_o eglantine_n and_o for_o that_o as_o the_o line_n h_o be_v to_o the_o line_n cd_o so_o be_v that_o which_o be_v contain_v under_o the_o line_n h_o and_o of_o to_o that_o which_o be_v contain_v under_o the_o line_n cd_o and_o of_o by_o the_o 1._o of_o the_o six_o but_o unto_o that_o which_o be_v contain_v under_o the_o line_n h_o and_o of_o be_v equal_a that_o which_o be_v contain_v under_o the_o line_n ca_n and_o eglantine_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o line_n h_o be_v to_o the_o line_n cd_o so_o be_v that_o which_o be_v contain_v under_o the_o line_n ca_n and_o eglantine_n to_o that_o which_o be_v contain_v under_o the_o line_n cd_o and_o of_o that_o be_v by_o the_o corollary_n next_o go_v before_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v the_o line_n h_o to_o the_o line_n cd_o an_o other_o demonstration_n to_o prove_v that_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n prove_v let_v there_o be_v a_o circle_n abc_n and_o in_o it_o describe_v two_o side_n of_o a_o equilater_n pentagon_n by_o the_o 11._o of_o the_o five_o namely_o ab_fw-la and_o ac_fw-la and_o draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n c._n and_o by_o the_o 1._o of_o the_o three_o take_v the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v d._n and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n d_o and_o extend_v it_o direct_o to_o the_o point_n e_o and_o let_v it_o cut_v the_o line_n bc_o in_o the_o point_n g._n and_o let_v the_o line_n df_o be_v half_a to_o the_o line_n dam_fw-ge and_o let_v the_o line_n gc_o be_v treble_a to_o the_o line_n hc_n by_o the_o 9_o of_o the_o six_o proposition_n now_o i_o say_v that_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o bh_o be_v equal_a to_o the_o pentagon_n inscribe_v in_o the_o circle_n abc_n draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n d._n now_o forasmuch_o as_o the_o line_n ad_fw-la be_v double_a to_o the_o line_n df_o therefore_o the_o line_n of_o be_v sesquialter_fw-la to_o the_o line_n ad._n again_o assumpt_n forasmuch_o as_o the_o line_n gc_o be_v treble_a to_o the_o line_n change_z therefore_o the_o line_n gh_o be_v double_a to_o the_o line_n ch._n wherefore_o the_o line_n gc_o be_v sesquialter_fw-la to_o the_o line_n hg_o wherefore_o as_o the_o line_n favorina_n be_v to_o the_o line_n ad_fw-la so_o be_v the_o line_n gc_o to_o the_o line_n gh_o wherefore_o by_o the_o 16._o of_o the_o six_o that_o which_o be_v contain_v under_o the_o line_n of_o &_o hg_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n dam_n and_o gc_o but_o the_o line_n gc_o be_v equal_a to_o the_o line_n bg_o by_o the_o 3._o of_o the_o three_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o bg_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o gh_o but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o bg_o be_v equal_a to_o two_o such_o triangle_n as_o the_o triangle_n abdella_n be_v by_o the_o 41._o of_o the_o first_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o gh_o be_v equal_a to_o two_o such_o triangle_n as_o the_o triangle_n abdella_n be_v wherefore_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o gh_o â—ive_a time_n be_v equal_a to_o ten_o triangle_n but_o ten_o triangle_n be_v two_o pentagon_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o gh_o five_o time_n be_v equal_a to_o two_o pentagon_n and_o forasmuch_o as_o the_o line_n gh_o be_v double_a to_o the_o line_n hc_n therefore_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o gh_o be_v double_a to_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o hc_n by_o the_o 1._o of_o the_o six_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o change_z twice_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n
and_o bt_n be_v duple_fw-fr sesquialter_fw-la to_o that_o which_o be_v contain_v unâ—â—r_o the_o line_n az_o &_o kt_n but_o unto_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n the_o pentagon_n abcig_n be_v prove_v duple_n sesquialter_fw-la wherefore_o the_o pentagon_n abcig_n of_o the_o dodecahedron_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o perpendicular_a line_n az_o and_o under_o the_o line_n bt_v which_o be_v five_o six_o part_n of_o the_o line_n bg_o ¶_o the_o 7._o proposition_n campane_n a_o right_a line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n what_o proportion_n the_o line_n contain_v in_o power_n the_o whole_a line_n and_o the_o great_a segment_n have_v to_o the_o line_n contain_v in_o power_n the_o whole_a and_o the_o less_o segment_n the_o same_o have_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n contain_v in_o one_o and_o the_o same_o sphere_n construction_n take_v a_o circle_n abe_n and_o in_o it_o by_o the_o 11._o of_o the_o four_o inscribe_v a_o equilater_n pentagon_n bzech_n and_o by_o the_o second_o of_o the_o same_o a_o equilater_n triangle_n abi_n and_o let_v the_o centre_n thereof_o be_v the_o point_n g._n and_o draw_v a_o line_n from_o g_z to_o b._n and_o divide_v the_o line_n gb_o by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n d_o by_o the_o 30._o of_o the_o six_o and_o let_v the_o line_n ml_o contain_v in_o power_n both_o the_o whole_a line_n gb_o and_o his_o less_o segment_n bd_o by_o the_o corollary_n of_o the_o 13._o of_o the_o ten_o and_o draw_v the_o right_a line_n bâ—_n subtend_v the_o angle_n of_o the_o pentagon_n which_o shall_v be_v the_o side_n of_o the_o cube_fw-la by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o â—_o and_o the_o line_n by_o shall_v be_v the_o side_n of_o the_o icosahedron_n and_o the_o line_n â—z_n the_o side_n of_o the_o dodecahedron_n by_o the_o 4._o of_o this_o book_n then_o i_o say_v that_o be_v the_o side_n of_o the_o cube_fw-la be_v to_o by_o the_o side_n of_o the_o icosahedron_n as_o the_o line_n contain_v in_o power_n the_o line_n bg_o &_o gd_o be_v to_o the_o line_n contain_v in_o power_n the_o line_n gb_o and_o bd._n demonstration_n for_o forasmuch_o as_o by_o the_o 12._o of_o the_o thirteen_o the_o line_n by_o be_v in_o power_n triple_a to_o the_o line_n bg_o and_o by_o the_o 4._o of_o the_o same_o the_o square_n of_o the_o line_n gb_o &_o bd_o be_v triple_a to_o the_o square_n of_o the_o line_n gd_o wherefore_o by_o the_o 15._o of_o the_o five_o the_o square_a of_o the_o line_n by_o be_v to_o the_o square_n of_o the_o line_n gb_o &_o bd_o namely_o triple_a to_o treble_v as_o the_o square_n of_o the_o line_n bâ—_n be_v to_o the_o square_n of_o the_o line_n gd_v namely_o as_o one_o be_v to_o one_o but_o as_o the_o square_n of_o the_o line_n bg_o be_v to_o the_o square_n of_o the_o gd_o so_o be_v the_o square_a of_o the_o line_n be_v to_o the_o square_n of_o the_o line_n bz_o for_o the_o line_n bg_o gd_o and_o be_v bz_o be_v in_o one_o and_o the_o same_o proportion_n by_o the_o second_o of_o this_o book_n for_o bz_o be_v the_o great_a segment_n of_o the_o line_n be_v by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o wherefore_o the_o square_a of_o the_o line_n be_v be_v to_o the_o square_n of_o the_o line_n bz_o as_o the_o square_n of_o the_o line_n by_o be_v to_o the_o square_n of_o the_o line_n bg_o and_o bd._n wherefore_o alternate_o the_o square_a of_o the_o line_n be_v be_v to_o the_o square_n of_o the_o line_n by_o as_o the_o square_n of_o the_o line_n bz_o be_v to_o the_o square_n of_o the_o line_n gb_o and_o bd._n but_o the_o square_n of_o the_o line_n bz_o be_v equal_a to_o the_o square_n of_o the_o line_n bg_o and_o gd_o by_o the_o 10._o of_o the_o thirteen_o for_o the_o line_n bg_o be_v equal_a to_o the_o side_n of_o the_o hexagon_n and_o the_o line_n gd_o to_o the_o side_n of_o the_o decagon_n by_o the_o corollary_n of_o the_o 9_o of_o the_o same_o wherefore_o the_o square_n of_o the_o line_n bg_o and_o gd_o be_v to_o the_o square_n of_o the_o line_n gâ—_n and_o bd_o as_o the_o square_n of_o the_o line_n be_v be_v to_o the_o square_n of_o the_o line_n by_o but_o the_o line_n zb_n contain_v in_o power_n the_o line_n bg_o and_o gd_o and_o the_o line_n ml_o contain_v in_o power_n the_o line_n gb_o and_o bd_o by_o construction_n wherefore_o as_o the_o line_n zb_n which_o contain_v in_o power_n the_o whole_a line_n bg_o and_o the_o great_a segment_n gd_o be_v to_o the_o line_n ml_o which_o contain_v in_o in_o power_n the_o whole_a line_n gb_o and_o the_o less_o segment_n bd_o so_o be_v be_v the_o side_n of_o the_o cube_fw-la to_o by_o the_o side_n of_o the_o icosahedron_n by_o the_o 22._o of_o the_o six_o wherefore_o a_o right_a line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n what_o proportion_n the_o line_n contain_v in_o power_n the_o whole_a line_n and_o the_o great_a segment_n have_v to_o the_o line_n contain_v in_o power_n the_o whole_a line_n and_o the_o less_o segment_n the_o same_o have_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n contain_v in_o one_o and_o the_o same_o sphere_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 8._o proposition_n order_n the_o solid_a of_o a_o dodecahedron_n be_v to_o the_o solid_a of_o a_o icosahedron_n as_o the_o side_n of_o a_o cube_n be_v to_o the_o side_n of_o a_o icosahedron_n all_o those_o solid_n be_v describe_v in_o one_o and_o the_o self_n same_o sphere_n forasmuch_o as_o in_o the_o 4._o of_o this_o book_n it_o have_v be_v prove_v that_o one_o and_o the_o self_n same_o circle_n contain_v both_o the_o triangle_n of_o a_o icosahedron_n and_o the_o pentagon_n of_o a_o dodecahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n wherefore_o the_o circle_n which_o contain_v those_o base_n be_v equal_a the_o perpendicular_n also_o which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o those_o circle_n shall_v be_v equal_a by_o the_o corollary_n of_o the_o assumpt_n of_o the_o 16_o of_o the_o twelve_o and_o therefore_o the_o pyramid_n set_v upon_o the_o base_n of_o those_o solid_n have_v one_o and_o the_o self_n same_o altitude_n for_o the_o altitude_n of_o those_o pyramid_n concurrâ—_n in_o the_o centre_n wherefore_o they_o be_v in_o proportion_n as_o their_o base_n be_v by_o the_o 5._o and_o 6._o of_o the_o twelve_o and_o therefore_o the_o pyramid_n which_o compose_v the_o dodecahedron_n arâ—_n to_o the_o pyramid_n which_o compose_v the_o icosahedron_n as_o the_o base_n be_v which_o base_n be_v the_o superficiece_n of_o those_o solid_n wherefore_o their_o solid_n be_v the_o one_o to_o the_o other_o as_o their_o superficiece_n be_v but_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n by_o the_o 6._o of_o this_o book_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o solid_a of_o the_o dodecahedron_n be_v to_o the_o solid_a of_o the_o icosahedron_n so_o be_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n all_o the_o say_v solid_n be_v inscribe_v in_o one_o and_o the_o self_n same_o sphere_n wherefore_o the_o solid_a of_o a_o dodecahedron_n be_v to_o the_o solid_a of_o a_o icosahedron_n as_o the_o side_n of_o a_o cube_fw-la be_v to_o the_o side_n of_o a_o icosahedron_n all_o those_o solid_n be_v describe_v in_o one_o and_o the_o self_n same_o sphere_n which_o be_v require_v to_o be_v prove_v a_o corollary_n the_o solid_a of_o a_o dodecahedron_n be_v to_o the_o solid_a of_o a_o icosahedron_n campane_n as_o the_o superficiece_n of_o the_o one_o be_v to_o the_o superficiece_n of_o the_o other_o be_v describe_v in_o one_o and_o the_o self_n same_o sphere_n namely_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n as_o be_v before_o manifest_v for_o they_o be_v resolve_v into_o pyramid_n of_o one_o and_o the_o self_n same_o altitude_n ¶_o the_o 9_o proposition_n if_o the_o side_n of_o a_o equilater_n triangle_n be_v rational_a the_o superficies_n shall_v be_v irrational_a campane_n of_o that_o kind_n which_o be_v call_v medial_a svppose_v that_o abg_o be_v a_o equilater_n triangle_n and_o from_o the_o point_n a_o draw_v unto_o the_o side_n bg_o a_o perpendicular_a line_n ad_fw-la and_o let_v the_o line_n ab_fw-la be_v rational_a then_o i_o say_v that_o the_o superficies_n abg_o be_v medial_a construction_n forasmuch_o as_o the_o line_n ab_fw-la be_v in_o power_n
def_n who_o side_n let_v be_v de_fw-fr and_o let_v the_o right_a line_n subtend_v the_o angle_n of_o the_o pentagon_n make_v of_o the_o side_n of_o the_o icosahedron_n be_v the_o line_n ef._n then_o i_o say_v that_o the_o side_n ed_z be_v in_o power_n double_a to_o the_o line_n h_o the_o less_o of_o those_o segment_n forasmuch_o as_o by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n demonstration_n it_o be_v manifest_a that_o ed_z the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n of_o the_o line_n efâ—_n and_o that_o the_o diameter_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o of_o namely_o the_o whole_a and_o the_o great_a segment_n but_o by_o supposition_n the_o side_n ab_fw-la contain_v in_o power_n the_o two_o line_n c_o &_o h_o join_v together_o in_o the_o self_n same_o proportion_n wherefore_o the_o line_n of_o be_v to_o the_o line_n ed_z as_z the_o line_n c_o be_v to_o the_o line_n h_o by_o the_o â—_o oâ—_n this_o bokeâ—_n and_o alternally_n by_o the_o 16._o of_o the_o five_o the_o line_n of_o be_v to_o the_o line_n c_o as_o the_o line_n ed_z be_v to_o the_o line_n h._n and_o forasmuch_o as_o the_o line_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o of_o and_o the_o line_n ab_fw-la contain_v in_o power_n the_o two_o line_n c_o and_o h_o therefore_o the_o square_n of_o the_o line_n of_o and_o ed_z be_v to_o the_o square_n of_o the_o line_n df_o as_o the_o square_n of_o the_o line_n c_o and_o h_o to_o the_o square_a ab_fw-la and_o alternate_o the_o square_n of_o the_o line_n of_o and_o â—d_a be_v to_o the_o square_n of_o the_o line_n c_o and_o h_o as_o the_o square_n of_o the_o line_n df_o be_v to_o the_o square_n of_o the_o line_n abâ—_n but_o df_o the_o diameter_n be_v by_o the_o 14._o of_o the_o thirtenâ—h_o iâ—_z power_n double_a to_o ab_fw-la the_o side_n of_o the_o octohedron_n inscribe_v by_o supposition_n in_o the_o same_o sphere_n wherefore_o the_o square_n of_o the_o line_n of_o and_o ed_z be_v double_a to_o the_o square_n of_o the_o line_n c_o and_o h._n and_o therefore_o one_o square_a of_o the_o line_n ed_z be_v double_a to_o one_o square_a of_o the_o line_n h_o by_o the_o 12._o of_o the_o five_o wherefore_o ed_z the_o side_n of_o the_o icosahedron_n be_v in_o power_n duple_n to_o the_o line_n h_o which_o be_v the_o less_o segment_n if_o therefore_o the_o poweâ—_n of_o the_o side_n of_o a_o octohedron_n be_v express_v by_o two_o right_a line_n join_v together_o by_o a_o extreme_a and_o mean_a proportion_n the_o side_n of_o the_o icosahedron_n contain_v in_o the_o same_o sphere_n shall_v be_v duple_v to_o the_o less_o segment_n the_o 17._o proposition_n if_o the_o side_n of_o a_o dodecahedron_n and_o the_o right_a line_n of_o who_o the_o say_a side_n be_v the_o less_o segment_n be_v so_o set_v that_o they_o make_v a_o right_a angle_n the_o right_a line_n which_o contain_v in_o power_n half_o the_o line_n subtend_v the_o angle_n be_v the_o side_n of_o a_o octohedron_n contain_v in_o the_o self_n same_o sphere_n svppose_v that_o ab_fw-la be_v the_o side_n of_o a_o dodecahedron_n and_o let_v the_o right_a line_n of_o which_o that_o side_n be_v the_o less_o segment_n be_v agnostus_n namely_o which_o couple_v the_o opposite_a side_n of_o the_o dodecahedron_n by_o the_o 4._o corollary_n of_o the_o 17._o of_o the_o thirteen_o construction_n and_o let_v those_o line_n be_v so_o set_v that_o they_o make_v a_o right_a angle_n at_o the_o point_n a._n and_o draw_v the_o right_a line_n bg_o and_o let_v the_o line_n d_o contain_n in_o power_n half_o the_o line_n bg_o by_o the_o first_o proposition_n add_v by_o flussas_n after_o the_o last_o of_o the_o six_o then_o i_o say_v that_o the_o line_n d_o be_v the_o side_n of_o a_o octohedron_n contain_v in_o the_o same_o sphere_n forasmuch_o as_o the_o line_n agnostus_n make_v the_o great_a segment_n gc_o the_o side_n of_o the_o cube_fw-la contain_v in_o the_o same_o sphere_n by_o the_o same_o 4._o corollary_n of_o the_o 17._o of_o the_o thirteen_o demonstration_n and_o the_o square_n of_o the_o whole_a line_n ag._n and_o of_o the_o less_o segment_n ab_fw-la be_v triple_a to_o the_o square_n of_o the_o great_a segment_n gc_o by_o the_o 4._o of_o the_o thirteen_o moreover_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n triple_a to_o the_o same_o line_n gc_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o the_o thirteen_o wherefore_o the_o line_n bg_o be_v equal_a to_o the_o 〈◊〉_d for_o it_o contain_v in_o power_n the_o two_o line_n ab_fw-la and_o ag_z by_o the_o 47._o of_o the_o first_o and_o therefore_o it_o contain_v in_o power_n the_o triple_a of_o the_o line_n gc_o but_o the_o side_n of_o the_o octohedron_n contain_v in_o the_o same_o sphere_n be_v in_o power_n triple_a to_o half_a the_o diameter_n of_o the_o sphere_n by_o the_o 14._o of_o the_o thirteen_o and_o by_o supposition_n the_o line_n d_o contaiâ—â—â—â—_n in_o powâ—â—_n the_o half_a of_o the_o line_n bg_o wherefore_o the_o line_n d_o contain_v in_o power_n the_o half_a of_o the_o same_o diameter_n be_v the_o side_n of_o a_o octohedron_n if_o therefore_o the_o side_n of_o a_o dodecahedron_n and_o the_o right_a line_n of_o who_o the_o say_a side_n be_v the_o less_o segment_n be_v so_o set_v that_o they_o make_v a_o right_a angle_n the_o right_a line_n which_o contain_v in_o power_n half_o the_o line_n subtend_v the_o angle_n be_v the_o side_n of_o a_o ocâ—â—â—edron_n contain_v in_o the_o self_n same_o sphere_n which_o be_v require_v to_o be_v prove_v a_o corollary_n unto_o what_o right_a line_n the_o side_n of_o the_o octoâ—edron_n be_v in_o power_n sesquialter_fw-la unto_o the_o same_o line_n the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o sphere_n be_v the_o great_a segment_n for_o the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o the_o segment_n cg_o unto_o which_o d_z the_o side_n of_o the_o octohedron_n be_v in_o power_n sesquialter_n that_o be_v be_v half_a of_o the_o power_n of_o the_o line_n bg_o which_o be_v triple_a unto_o the_o line_n cg_o ¶_o the_o 18._o proposition_n if_o the_o side_n of_o a_o tetrahedron_n contain_v in_o power_n two_o right_a line_n join_v together_o by_o a_o extreme_a and_o mean_a proportion_n the_o side_n of_o a_o icosahedron_n describe_v in_o the_o self_n same_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o less_o right_a line_n svppose_v that_o abc_n be_v a_o tetrahedron_n and_o let_v his_o side_n be_v ab_fw-la construction_n who_o power_n let_v be_v divide_v into_o the_o line_n agnostus_n and_o gb_o join_v together_o by_o a_o extreme_a and_o mean_a proportion_n namely_o let_v it_o be_v divide_v into_o agnostus_n the_o whole_a line_n and_o gb_o the_o great_a seâ—ment_n by_o the_o corollary_n of_o the_o first_o proposition_n add_v by_o flussas_n after_o the_o last_o of_o the_o six_o and_o let_v ed_z be_v the_o side_n of_o the_o icosahedron_n edf_o contain_v in_o the_o self_n same_o sphere_n and_o let_v the_o line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n describe_v of_o the_o side_n of_o the_o icosahedron_n be_v ef._n then_o i_o say_v that_z ed_z the_o side_n of_o the_o icosahedron_n be_v in_o power_n sesquialter_fw-la to_o the_o less_o line_n gb_o demonstration_n forasmuch_o as_o by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n the_o side_n ed_z be_v the_o greâ—ter_a segment_n of_o the_o line_n of_o which_o subtend_v the_o angle_n of_o the_o pentagon_n but_o as_o the_o whole_a line_n of_o be_v to_o the_o great_a segment_n ed_z so_o be_v the_o same_o grâ—â—ter_a segment_n to_o the_o less_o by_o the_o 30._o of_o the_o six_o and_o by_o supposition_n agnostus_n be_v the_o whole_a line_n and_o gâ—_n the_o great_a segment_n wherefore_o as_o of_o be_v to_o ed_z so_o be_v agnostus_n to_o gâ—_n by_o the_o second_o of_o the_o fourteen_o and_o alternate_o the_o line_n of_o be_v to_o the_o line_n agnostus_n as_o the_o line_n ed_z be_v to_o the_o line_n gb_o and_o forasmuch_o as_o by_o supposition_n the_o line_n ab_fw-la contain_v in_o power_n the_o two_o line_n agnostus_n and_o gb_o therefore_o by_o the_o 4â—_o of_o the_o first_o the_o angle_n agb_n be_v a_o right_a angle_n but_o the_o angle_n def_n be_v a_o right_a angle_n by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n wherefore_o the_o triangle_n agâ—_n and_o feed_v be_v equiangle_n by_o the_o â—_o of_o the_o six_o wherefore_o their_o side_n be_v proportional_a namely_o as_o the_o line_n ed_z be_v to_o the_o line_n gb_o so_o be_v the_o line_n fd_o to_o the_o line_n ab_fw-la by_o the_o 4._o of_o the_o six_o but_o by_o that_o which_o have_v before_o
by_o the_o 4._o and_o eight_o of_o the_o first_o and_o forasmuch_o as_o upon_o every_o one_o of_o the_o line_n cut_v of_o the_o cube_fw-la be_v set_v two_o triangle_n as_o the_o triangle_n alm_n and_o blmâ—_n there_o shall_v be_v make_v 12._o triangle_n and_o forasmuch_o as_o under_o every_o one_o of_o the_o â—_o angle_n of_o the_o cube_fw-la be_v subtend_v the_o other_o 8._o triangle_n as_o the_o triangle_n amp._n etc_n etc_n of_o 1â—_o and_o 8._o triangle_n shall_v be_v produce_v 20._o triangle_n equal_a and_o equilater_n contain_v the_o solid_a of_o a_o icosahedron_n by_o the_o 25._o definition_n of_o the_o eleven_o which_o shall_v be_v inscribe_v in_o the_o cube_fw-la give_v abc_n by_o the_o first_o definition_n of_o this_o book_n the_o invention_n of_o the_o demonstration_n of_o this_o depend_v of_o the_o ground_n of_o the_o former_a wherefore_o in_o a_o cube_fw-la give_v we_o have_v describe_v a_o icosahedron_n which_o be_v require_v to_o be_v do_v first_o corollary_n the_o diameter_n of_o a_o sphere_n which_o contain_v a_o icosahedron_n contain_v two_o side_n namely_o the_o side_n of_o the_o icosahedron_n and_o the_o side_n of_o the_o cube_fw-la which_o contain_v the_o icosahedron_n for_o if_o we_o draw_v the_o line_n ab_fw-la it_o shall_v make_v the_o angle_n at_o the_o point_n a_o right_a angle_n for_o that_o it_o be_v a_o parallel_n to_o the_o side_n of_o the_o cube_fw-la wherefore_o the_o linâ—_n which_o couple_v the_o opposite_a angleâ—_n of_o the_o icosahedron_n at_o the_o point_v fletcher_n and_o b_o contain_v in_o power_n the_o line_n ab_fw-la the_o sidâ—_n of_o thâ—_z cube_fw-la and_o the_o line_n of_o the_o side_n of_o the_o icosahedron_n by_o the_o 47._o of_o the_o first_o which_o line_n fb_o be_v equal_a to_o the_o â—iameter_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n by_o the_o demonstration_n of_o the_o â—â—_o of_o the_o thirteen_o second_o corollary_n the_o six_o opposite_a side_n of_o the_o icosahedron_n divide_v into_o two_o equal_a part_n their_o section_n be_v couple_v by_o three_o equal_a right_a line_n cut_v the_o one_o the_o other_o into_o two_o equal_a part_n and_o perpendicular_o in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n for_o those_o three_o line_n be_v the_o three_o line_n which_o couple_v the_o centre_n of_o the_o base_n of_o the_o cube_fw-la which_o do_v in_o such_o sort_n in_o the_o centre_n of_o the_o cube_fw-la cut_v the_o one_o the_o other_o by_o the_o corollary_n of_o the_o three_o of_o this_o book_n and_o therefore_o be_v equal_a to_o the_o side_n of_o the_o cube_fw-la but_o right_a line_n draw_v from_o the_o centre_n of_o the_o cube_fw-la to_o the_o angle_n of_o the_o icosahedron_n every_o one_o of_o they_o shall_v subtend_v the_o half_a side_n of_o the_o cube_fw-la and_o the_o half_a side_n of_o the_o icosahedron_n which_o half_a side_n contain_v a_o right_a angle_n wherefore_o those_o line_n be_v equal_a whereby_o it_o be_v manifest_a that_o the_o foresay_a centre_n be_v the_o centre_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n three_o corollary_n the_o side_n of_o a_o cube_fw-la divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n the_o side_n of_o a_o icosahedron_n describe_v in_o it_o for_o the_o half_a side_n of_o the_o cube_fw-la make_v the_o half_a of_o the_o side_n of_o the_o icosahedron_n the_o great_a segment_n wherefore_o also_o the_o whole_a side_n of_o the_o cube_fw-la make_v the_o whole_a side_n of_o the_o icosahedron_n the_o great_a segment_n by_o the_o 15._o of_o the_o fifthe_o for_o the_o section_n be_v like_a by_o the_o â—_o of_o the_o fourteen_o ¶_o four_o corollary_n the_o side_n and_o base_n of_o the_o icosahedron_n which_o be_v opposite_a the_o one_o to_o the_o other_o be_v parallel_n forasmuch_o as_o every_o one_o of_o the_o opposite_a side_n of_o the_o icosahedron_n may_v be_v in_o the_o parallel_a line_n of_o the_o cube_fw-la namely_o in_o those_o parallel_n which_o be_v opposite_a in_o the_o cube_fw-la and_o the_o triangle_n which_o be_v make_v of_o parallel_a line_n be_v parallel_n by_o the_o 15._o of_o the_o eleven_o therefore_o the_o opposite_a triâ—ngleâ—_n of_o the_o icosahedron_n as_o also_o the_o side_n be_v parallel_n the_o one_o to_o the_o other_o ¶_o the_o 15._o problem_n the_o 15._o proposition_n in_o a_o icosahedron_n give_v to_o inscribe_v a_o octohedron_n svppose_v that_o the_o icosahedron_n give_v be_v acdf_n and_o by_o the_o former_a second_o corollary_n construction_n let_v there_o be_v take_v the_o three_o right_a line_n which_o cut_v the_o one_o the_o other_o into_o two_o equal_a part_n perpendicular_o and_o which_o couple_n the_o section_n into_o two_o equal_a part_n of_o the_o side_n of_o the_o icosahedron_n which_o let_v be_v be_v gh_o and_o kl_o cut_v the_o one_o the_o other_o in_o the_o point_n i_o and_o drawâ—_n these_o riâ—ht_a line_n â—g_z ge_z eh_o and_o hb_o and_o forasmuch_o as_o the_o angle_n at_o the_o point_n i_o be_v by_o construction_n right_a angle_n demonstration_n â—nd_v be_v conâ—â—ined_v under_o equal_a linesâ—_n the_o 〈◊〉_d gâ—_n and_o â—â—_o shall_v 〈…〉_o squâ—re_o by_o the_o â—_o of_o the_o â—irst_n likewise_o â—nto_o thoâ—e_a 〈◊〉_d shall_v be_v squall_v the_o line_n drâ—wâ—n_v from_o 〈◊〉_d point_n king_n and_o â—_o to_o every_o one_o of_o the_o poinâ—â—s_n â—_o g._n â—_o h_o and_o therefore_o the_o triangle_n which_o 〈◊〉_d the_o â—â—ramis_n dgenk_n shall_v be_v equal_a 〈…〉_o and_o by_o 〈…〉_o ¶_o the_o 16._o problem_n the_o 16._o proposition_n in_o a_o octohedron_n give_v to_o inscribe_v a_o icosahedron_n let_v there_o be_v take_v a_o octohedron_n who_o 6._o angle_n let_v be_v a_o b_o c_o f_o p_o l._n and_o draw_v the_o line_n ac_fw-la bf_o pl_z construction_n cut_v the_o one_o the_o other_o perpendicular_o in_o the_o point_n r_o by_o the_o 2._o corollary_n of_o the_o 14._o of_o the_o thirteen_o and_o let_v every_o one_o of_o the_o 12._o side_n of_o the_o octohedron_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n h_o x_o m_o king_n d_o s_o n_o g_o five_o e_o q_o t._n and_o let_v the_o great_a segment_n be_v the_o line_n bh_o bx_n fm_o fk_o ad_fw-la aq_n cs_n et_fw-la pn_n pg_n lv_o le_fw-fr and_o draw_v these_o line_n hk_o xm_o ge_z nv_n d_n qt_n now_o forasmuch_o as_o in_o the_o triangle_n abf_n the_o side_n be_v cut_v proportional_o namely_o as_o the_o line_n bh_o be_v to_o the_o line_n ha_o so_o be_v the_o line_n fk_o to_o the_o line_n ka_o by_o the_o 2._o of_o the_o fourteen_o demonstration_n therefore_o the_o line_n hk_o shall_v be_v a_o parallel_n to_o the_o line_n bf_o by_o the_o 2._o of_o the_o six_o and_o forasmuch_o as_o the_o line_n ac_fw-la cut_v the_o line_n hk_o in_o the_o point_n z_o and_o the_o line_n zk_n be_v a_o parallel_n unto_o the_o line_n rf_n the_o line_n ramires_n shall_v be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n z_o by_o the_o 2._o of_o the_o six_o namely_o shall_v be_v cut_v like_o unto_o the_o line_n favorina_n and_o the_o great_a segment_n thereof_o shall_v be_v the_o line_n zr_n unto_o the_o line_n zk_v put_v the_o line_n ro_n equal_a by_o the_o 3._o of_o the_o first_o and_o draw_v the_o line_n ko_o now_o then_o the_o line_n ko_o shall_v be_v equal_a to_o the_o line_n zr_n by_o the_o 33._o of_o the_o â—irst_n draw_v the_o line_n kg_v ke_o and_o ki_v and_o forasmuch_o as_o the_o triangle_n arf_n and_o azk_v be_v equiangle_n by_o the_o 6._o of_o the_o six_o the_o side_n az_o and_o zk_v shall_v be_v equal_a the_o one_o to_o the_o other_o by_o the_o 4._o of_o the_o six_o for_o the_o side_n be_v and_o rf_n be_v equal_a wherefore_o the_o line_n zk_n shall_v be_v the_o less_o segment_n of_o the_o line_n ra._n but_o if_o the_o great_a segment_n rz_n be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n the_o great_a segment_n thereof_o shall_v be_v the_o line_n zk_n which_o be_v the_o less_o segment_n of_o the_o whole_a line_n ramires_n by_o the_o 5._o of_o the_o thirâ—enth_o and_o forasmuch_o as_o the_o two_o line_n fe_o and_o fg_o be_v equal_a to_o the_o two_o line_n ah_o and_o ak_o namely_o each_o be_v less_o segment_n of_o equal_a side_n of_o the_o octohedron_n and_o the_o angle_n hak_n and_o efg_o be_v equal_a namely_o be_v right_a angle_n by_o the_o 14._o of_o the_o thirteen_o the_o base_n hk_o and_o gf_o shall_v be_v equal_a by_o the_o 4._o of_o the_o first_o and_o by_o the_o same_o reason_n unto_o they_o may_v be_v prove_v equal_a the_o line_n xm_o nv_n d_n and_o qt_o and_o forasmuch_o as_o the_o line_n ac_fw-la bf_o and_o pl_z do_v cut_v the_o one_o the_o other_o into_o two_o equal_a part_n and_o perpendicular_o by_o construction_n the_o line_n hk_o and_o ge_z which_o
subtend_v angle_n of_o triangle_n like_v unto_o the_o triangle_n who_o angle_n the_o line_n ac_fw-la bf_o and_o pl_z subtend_n be_v cut_v into_o two_o equal_a part_n in_o the_o point_n z_o an_o i_o by_o the_o 4._o of_o the_o six_o so_o also_o be_v the_o other_o line_n nv_n xm_o d_n qt_fw-la which_o be_v equal_a unto_o the_o line_n hk_o &_o ge_z cut_v in_o like_a sort_n and_o they_o shall_v cut_v the_o line_n ac_fw-la bf_o and_o pl_z like_z wherefore_o the_o line_n ko_o which_o be_v equal_a to_o rz_n shall_v make_v the_o great_a segment_n the_o line_n ro_n which_o be_v equal_a to_o the_o line_n zk_n for_o the_o great_a segment_n of_o the_o rz_n be_v the_o line_n zk_n and_o therefore_o the_o line_n oi_o shall_v be_v the_o less_o segment_n when_o as_o the_o whole_a line_n ri_n be_v equal_a to_o the_o whole_a line_n rz_n wherefore_o the_o square_n of_o the_o whole_a line_n ko_o and_o of_o the_o less_o segment_n oi_o be_v triple_a to_o the_o square_n of_o the_o great_a segment_n ro_n by_o the_o 4._o of_o the_o thirteen_o wherefore_o the_o line_n ki_v which_o contain_v in_o power_n the_o two_o line_n ko_o and_o oi_o be_v in_o power_n triple_a to_o the_o line_n ro_n by_o the_o 47._o of_o the_o first_o for_o the_o angle_n koi_n be_v a_o right_a angle_n and_o forasmuch_o as_o the_o line_n fe_o and_o fg_o which_o be_v the_o less_o segment_n of_o the_o side_n of_o the_o octohedron_n be_v equal_a and_o the_o line_n fk_o be_v common_a to_o they_o both_o and_o the_o angle_n kfg_n and_o kfe_n of_o the_o triangle_n of_o the_o octohedron_n be_v equal_a the_o base_n kg_v and_o ke_o shall_v by_o the_o 4_o of_o the_o first_o be_v equal_a and_o therefore_o the_o angle_n kie_fw-mi and_o kig_n which_o they_o subtend_v be_v equal_a by_o the_o 8._o of_o the_o first_o wherefore_o they_o be_v right_a angle_n by_o the_o 13._o of_o the_o first_o wherefore_o the_o right_a line_n ke_o which_o contain_v in_o power_n the_o two_o line_n ki_v and_o â—e_n by_o the_o 47._o of_o the_o first_o be_v in_o power_n quadruple_a to_o the_o line_n ro_n or_o je_n for_o the_o line_n ri_n be_v prove_v to_o be_v in_o power_n triple_a to_o the_o same_o line_n ro_n but_o the_o line_n ge_z be_v double_a to_o the_o line_n je_n wherefore_o the_o line_n ge_z be_v also_o in_o power_n 〈…〉_o pf_n and_o by_o the_o same_o reason_n may_v be_v prove_v that_o the_o â—est_n of_o the_o eleven_o solid_a angle_n of_o the_o 〈◊〉_d be_v 〈…〉_o the_o section_n of_o every_o one_o of_o the_o side_n of_o the_o octohedron_n namely_o in_o the_o point_n e_o n_o five_o h_o â—_o m_o â—_o d_o s_o q_o t._n wherefore_o there_o be_v 12._o angle_n of_o the_o icosahedron_n moreover_o forasmuch_o as_o every_o one_o of_o the_o base_n of_o the_o octohedron_n do_v each_o contain_v triangle_n of_o the_o icosahedron_n 〈…〉_o pyramid_n abcâ—fp_n which_o be_v the_o half_a of_o the_o octohedron_n the_o triangle_n fcp_n receive_v in_o thâ—_z section_n of_o his_o side_n the_o â—_o triangle_n gm_n and_o the_o triangle_n cpb_n contain_v the_o triangle_n nxs_n and_o thâ—_z triangle_n â—ap_n contain_v the_o triangle_n hnd_n and_o moreover_o the_o triangle_n apf_n contain_v the_o triangle_n edge_n and_o the_o same_o may_v be_v prove_v in_o the_o opposite_a pyramid_n abcfl_fw-mi wherefore_o there_o shall_v be_v eight_o triangleâ—_n and_o forasmuch_o as_o beside_o these_o triangle_n to_o every_o one_o of_o the_o solid_a angle_n of_o the_o octohedron_n 〈◊〉_d subtend_v two_o triangle_n as_o the_o triangle_n keg_n and_o meg_n to_o the_o angle_n f_o and_o the_o triangle_n hnv_n and_o xnv_n to_o the_o angle_n b_o also_o the_o triangle_n nd_n and_o â—ds_n to_o the_o angle_n p_o likewise_o the_o triangleâ—_n dhk_n and_o qhk_v to_o the_o angle_n a_o moreover_o the_o triangle_n eqt_a and_o vqt_a to_o the_o angle_n l_o and_o final_o the_o triangle_n sxm_n and_o txm_n to_o the_o angle_n c_o these_o 12._o triangle_n be_v add_v to_o thâ—â—_n for_o 〈◊〉_d triangle_n shall_v produce_v _o triangle_n equal_a and_o equilater_v couple_v together_o which_o shall_v male_a a_o icosahedron_n by_o the_o 25._o definition_n of_o the_o eleven_o and_o it_o shall_v be_v inscribe_v in_o the_o octohedron_n give_v abcâ—â—l_o by_o the_o first_o definition_n of_o this_o book_n for_o the_o 1â—_o angle_n thereof_o be_v set_v in_o 1â—_o like_a section_n of_o the_o side_n of_o the_o octohedron_n wherefore_o in_o a_o octohedron_n give_v be_v inscribe_v a_o icosahedron_n ¶_o first_n corollary_n the_o side_n of_o a_o equilater_n triangle_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n a_o right_a line_n subtend_v within_o the_o triangle_n the_o angle_n which_o be_v contain_v under_o the_o great_a segment_n and_o the_o less_o be_v in_o power_n duple_n to_o the_o less_o segment_n of_o the_o same_o side_n for_o the_o line_n ke_o which_o subtend_v the_o angle_n kfe_n of_o the_o triangle_n afl_fw-mi which_o angle_n kfe_n be_v contain_v under_o the_o two_o segment_n kf_a &_o fe_o be_v prove_v equal_a 〈◊〉_d the_o line_n hk_o which_o contain_v in_o power_n the_o two_o less_o segment_n ha_o and_o ak_o by_o the_o 47._o of_o the_o â—â—rst_a foâ—_n 〈◊〉_d angle_n hak_n be_v 〈…〉_o second_o corollary_n the_o base_n of_o the_o icosahedron_n be_v concentrical_a that_o be_v have_v one_o and_o the_o self_n same_o centre_n with_o the_o base_n of_o the_o octohedron_n which_o contain_v it_o for_o suppose_v that_o 〈…〉_o octohedron_n 〈◊〉_d ecd_v the_o base_a of_o a_o icosahedron_n and_o let_v the_o centre_n of_o the_o base_a abg_o be_v the_o point_n f._n and_o draw_v these_o right_a line_n favorina_n fb_o fc_o and_o fe_o now_o than_o the_o 〈…〉_o to_o the_o two_o line_n fb_o and_o bc_o for_o they_o be_v line_n draw_v from_o the_o centre_n and_o be_v also_o less_o segment_n and_o they_o contain_v the_o 〈…〉_o ¶_o the_o 17._o problem_n the_o 17._o proposition_n in_o a_o octohedron_n give_v to_o inscribe_v a_o dodecahedron_n construction_n svppose_v that_o the_o octohedron_n give_v be_v abgdec_n who_o 12._o â—ides_n let_v be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n as_o in_o the_o former_a proposition_n it_o be_v manifest_a that_o of_o the_o right_a line_n which_o couple_v thâ—se_a section_n be_v make_v 20._o triangle_n of_o which_o 8._o be_v concentrical_a with_o the_o base_n of_o the_o octohedron_n by_o the_o second_o corollary_n of_o the_o former_a proposition_n if_o therefore_o in_o every_o one_o of_o the_o centre_n of_o the_o 20._o triangle_n be_v inscribe_v by_o the_o 1._o of_o this_o book_n every_o one_o of_o the_o â—â—_o â—â—gles_n of_o the_o dodecahedron_n demonstration_n we_o shall_v find_v that_o â—_o angle_n of_o the_o dodecahedron_n be_v set_v in_o the_o 8._o centre_n of_o the_o base_n of_o the_o octohedron_n namely_o these_o angle_n i_o u._fw-mi ct_n oh_o m_o a_o p_o and_o x_o and_o of_o the_o other_o 12._o solid_a angle_n there_o be_v two_o in_o the_o centre_n of_o the_o two_o triangle_n which_o have_v one_o side_n common_a under_o every_o one_o of_o the_o solid_a angle_n of_o the_o octohedron_n namely_o under_o the_o solid_a angle_n a_o the_o two_o solid_a angle_n king_n z_o under_o the_o solid_a angle_n b_o the_o two_o solid_a angle_n h_o t_o under_o the_o solid_a angle_n g_o the_o two_o solid_a angle_n in_o v_z under_o the_o solid_a angle_n d_o the_o two_o solid_a angle_n f_o l_o under_o the_o solid_a angle_n e_o the_o two_o solid_a angle_n s_o n_o under_o the_o solid_a angle_n c_o the_o two_o solid_a angle_n queen_n r_o and_o forasmuch_o as_o in_o the_o octohedron_n be_v six_o solid_a angle_n under_o they_o shall_v be_v subtend_v 12._o solid_a angle_n of_o the_o dodâ—cahedron_n and_o so_o be_v mâ—de_v 20._o solid_a angle_n compose_v of_o 12._o equal_a and_o equilater_v superficial_a pentagon_n as_o it_o be_v 〈◊〉_d by_o the_o 5._o of_o this_o book_n which_o therefore_o contain_v a_o dodecahedron_n by_o the_o 24._o definition_n of_o the_o eleven_o and_o it_o be_v inscribe_v in_o the_o octohedron_n by_o the_o 1._o definition_n of_o this_o book_n for_o that_o every_o one_o of_o the_o base_n of_o the_o octohedron_n do_v receive_v angle_n thereof_o wherefore_o in_o a_o octohedron_n give_v be_v inscribe_v a_o dodecahedron_n ¶_o the_o 18._o problem_n the_o 18._o proposition_n in_o a_o trilater_n and_o equilater_n pyramid_n to_o inscribe_v a_o cube_n construction_n svppose_v that_o there_o be_v a_o trilater_n equilater_n pyramid_n who_o base_a let_v be_v abc_n and_o â—oppe_v the_o point_n d._n and_o let_v it_o be_v comprehend_v in_o a_o sphereâ—_n by_o the_o 13._o of_o the_o 〈◊〉_d and_o lâ—â—_n the_o centre_n of_o that_o sphere_n be_v the_o point_n e._n and_o from_o the_o solid_a angle_n a_o b_o c_o d_o draw_v right_a line_n pass_v by_o the_o centre_n e_o unto_o the_o opposite_a base_n of_o
two_o line_n hif_n and_o tio_n cut_v the_o one_o the_o other_o be_v in_o one_o and_o the_o self_n same_o '_o plain_n by_o the_o 2._o of_o the_o eleven_o and_o therefore_o the_o point_n h_o t_o f_o oh_o be_v in_o one_o &_o the_o self_n same_o plain_a wherforeâ—_n the_o rectangle_n figure_n hoff_z be_v quadrilater_n and_o equilater_n and_o in_o one_o and_o the_o self_n same_o plain_n be_v a_o square_a by_o the_o definition_n of_o a_o square_a and_o by_o the_o same_o reason_n may_v the_o rest_n of_o the_o base_n of_o the_o solid_a be_v prove_v to_o be_v square_n equal_a and_o plain_a or_o superficial_a now_o than_o the_o solid_a be_v comprehend_v of_o 6._o equal_a square_n which_o be_v contain_v of_o 12._o equal_a side_n which_o square_n make_v 8._o solid_a angle_n of_o which_o four_o be_v in_o the_o centre_n of_o the_o base_n oâ—_n the_o pyramid_n and_o the_o other_o 4._o be_v in_o the_o middle_a section_n of_o the_o four_o perdendiculars_n wherefore_o the_o solid_a hoftpgrn_n be_v a_o cube_fw-la by_o the_o 21._o definition_n of_o the_o eleven_o and_o be_v inscribe_v in_o the_o pyramid_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o trilater_n equilater_n pyramid_n give_v be_v inscribe_v a_o cube_fw-la ¶_o a_o corollary_n the_o line_n which_o cut_v into_o two_o equal_a part_n the_o opposite_a side_n of_o the_o pyramid_n be_v triple_a to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n and_o pass_v by_o the_o centre_n of_o the_o cube_fw-la for_o the_o line_n sev_n who_o three_o part_n the_o line_n si_fw-it be_v cut_v the_o opposite_a side_n cd_o and_o ab_fw-la into_o two_o equll_a part_n but_o the_o line_n ei_o which_o be_v draw_v from_o the_o centre_n of_o the_o cube_fw-la to_o the_o base_a be_v prove_v to_o be_v a_o three_o part_n of_o the_o line_n es_fw-ge wherefore_o the_o side_n of_o the_o cube_fw-la which_o be_v double_a to_o the_o line_n ei_o shall_v be_v a_o three_o part_n of_o the_o whole_a line_n we_o which_o be_v as_o have_v be_v prove_v double_a to_o the_o line_n es._n the_o 19_o problem_n the_o 19_o proposition_n in_o a_o trilater_n equilater_n pyramid_n give_v to_o inscribe_v a_o icosahedron_n svppose_v that_o the_o pyramid_n be_v give_v 〈◊〉_d abâ—dâ—_n every_o one_o of_o who_o sâ—des_n 〈◊〉_d be_v divide_v into_o two_o equal_a part_n in_o the_o poyâ—â—â—â—â—_n m_o king_n l_o p_o n._n construction_n and_o iâ—_z every_o one_o of_o the_o base_n of_o that_o pyramid_n descry_v the_o trianglâ—â—_n lâ—â—_n pmn_n nkl_n and_o 〈…〉_o which_o triangle_n shall_v be_v equilater_n by_o the_o 4._o of_o the_o firâ—t_o â—or_a the_o side_n subtend_v equal_a angle_n of_o the_o pyramid_n contain_v under_o the_o half_n of_o the_o side_n of_o the_o same_o pyramisâ—_n wherefore_o the_o side_n of_o the_o say_a triangle_n be_v equal_a let_v those_o side_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n by_o the_o 30._o of_o the_o six_o in_o the_o point_n c_o e_o q_o r_o s_o t_o h_o i_o oh_o five_o a'n_z x._o now_o then_o those_o side_n be_v cut_v into_o the_o self_n same_o proportion_n by_o the_o 2._o of_o the_o fourtenth_n and_o therefore_o they_o make_v the_o liâ—e_z section_n equal_a by_o the_o â—_o part_n of_o the_o nine_o of_o the_o five_o now_o i_o say_v that_o the_o foresay_a point_n doâ—_n receive_v the_o angle_n of_o the_o icosahedron_n inscribe_v in_o the_o pyramid_n abâ—d_n in_o the_o foresay_a triangle_n let_v there_o again_o be_v make_v other_o triangle_n by_o couple_v the_o section_n and_o let_v those_o triangle_n be_v tr_n joh_n ceq_n and_o vxy_n which_o shall_v be_v equilater_n for_o every_o one_o of_o their_o side_n do_v subâ—â—â—d_v equal_a angle_n of_o equilater_n triangle_n and_o those_o say_v equal_a angle_n be_v contain_v under_o equal_a sideâ—_n namely_o under_o the_o great_a segment_n and_o the_o less_o â—_o and_o therefore_o the_o side_n which_o subtend_v those_o angle_n be_v equal_a by_o the_o 4._o of_o the_o first_o now_o let_v we_o prove_v that_o at_o each_o of_o the_o foresay_a point_n as_o for_o example_n at_o t_o be_v set_v the_o solid_a angle_n of_o a_o icosahâ—dronâ—_n demonstration_n forasmuch_o as_o the_o triangle_n trs_n and_o tqo_n be_v equilater_n and_o equal_a the_o 4._o right_a line_n tr_z it_o be_v tq_n and_o to_o shall_v be_v equal_a and_o forasmuch_o as_o â—pnk_v be_v a_o square_n cut_v the_o pyramid_n abâ—d_v into_o two_o equal_a paâ—â—â—â—_n by_o the_o corollay_n of_o the_o second_o of_o this_o bookeâ—_n the_o line_n th._n shall_v be_v in_o power_n duple_n to_o the_o line_n tn_n or_o nh_v by_o the_o 47._o of_o the_o first_o for_o the_o line_n tn_n or_o nh_v be_v equal_a for_o that_o by_o construction_n they_o be_v each_o less_o segment_n and_o the_o line_n rt_n or_o it_o be_v be_v in_o power_n duple_n to_o the_o same_o line_n tn_n or_o nh_v by_o the_o corollary_n of_o the_o 16._o of_o this_o book_n for_o it_o subtend_v the_o angle_n of_o the_o triangle_n contain_v under_o the_o two_o segment_n wherefore_o the_o line_n th._n it_o be_v tr_z tq_n and_o to_o be_v equal_a and_o so_o also_o be_v the_o line_n his_o sir_n rq_n qo_n and_o oh_o which_o subtend_v the_o angle_n at_o the_o point_n t_o equal_a for_o the_o line_n qr_o contain_v in_o power_n the_o two_o line_n pq_n and_o pr._n the_o less_o segment_n which_o two_o line_n the_o line_n th._n also_o contain_v in_o power_n and_o the_o rest_n of_o the_o line_n do_v subtend_v angle_n of_o equilater_n triangle_n contain_v under_o the_o great_a segment_n and_o the_o less_o wherefore_o the_o five_o triangle_n trs_n tsh_o tho_o toq_fw-la tqr_n be_v equilater_n and_o equal_a make_v the_o solid_a angle_n of_o a_o icosahedron_n at_o the_o point_n t_o by_o the_o 16._o of_o the_o thirteen_o in_o the_o side_n pn_n of_o the_o triangle_n p_o nm_o and_o by_o the_o same_o reason_n in_o the_o other_o side_n of_o the_o 4._o triangle_n pnm_n nkl_n fmk_n &_o lfp_n which_o be_v inscribe_v in_o the_o base_n of_o the_o pyramid_n which_o side_n be_v 12â—_n in_o number_n shall_v be_v set_v 12._o angle_n of_o the_o icosahedron_n contain_v under_o 20._o equal_a &_o equilater_n triangle_n of_o which_o fowere_n be_v set_v in_o the_o 4._o base_n of_o the_o pyramid_n namely_o these_o four_o triangle_n trs_n hoi_n ceq_n vxy_n 4._o triangle_n be_v under_o 4._o angle_n of_o the_o pyramid_n that_o be_v the_o four_o triangle_n cix_o ysh_a erv_n tqo_n and_o under_o every_o one_o of_o the_o six_o side_n of_o the_o pyramid_n be_v set_v two_o triangle_n namely_o under_o the_o side_n of_o the_o triangle_n this_o and_o thoâ—_n under_o the_o side_n db_o the_o triangle_n rqe_n and_o rqt_n under_o the_o side_n da_z the_o triangle_n coq_fw-la and_o coi_fw-fr under_o the_o side_n ab_fw-la the_o triangle_n exc_a and_o exuâ—_n under_o the_o side_n bg_o the_o triangle_n sur_n and_o svy_n and_o under_o the_o side_n agnostus_n the_o triangle_n jyh_n and_o jyx._n wherefore_o the_o solid_a be_v contain_v under_o 20._o equilater_n and_o equal_a triangle_n shall_v be_v a_o icosahedron_n by_o the_o 23._o definition_n of_o the_o eleven_o and_o shall_v be_v inscribe_v in_o the_o pyramid_n abâ—d_v by_o the_o first_o definition_n of_o this_o book_n for_o all_o his_o angle_n do_v at_o one_o time_n touch_v the_o base_n of_o the_o pyramid_n wherefore_o in_o a_o trilater_n equilater_n pyramid_n give_v we_o have_v inscribe_v a_o icosahedron_n ¶_o the_o 20._o proposition_n the_o 20._o problem_n in_o a_o trilater_n equilater_n pyramid_n give_v to_o inscribe_v a_o dodecahedron_n svppose_v that_o the_o pyramid_n give_v be_v abgd_v â—che_fw-mi of_o who_o side_n let_v be_v cut_v into_o two_o equal_a part_n and_o draw_v the_o line_n which_o couple_v the_o section_n which_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o right_a line_n be_v draw_v by_o the_o section_n shall_v receive_v 20._o triangle_n make_v a_o icosahedron_n as_o in_o the_o former_a proposition_n it_o be_v manifest_a now_o than_o if_o we_o take_v the_o centre_n of_o those_o triangle_n we_o shall_v there_o find_v the_o 20._o angle_n of_o the_o dodecahedron_n inscribe_v in_o it_o by_o the_o 5._o of_o this_o book_n and_o forasmuch_o as_o 4._o base_n of_o the_o foresay_a icosahedron_n be_v concentricall_a with_o the_o base_n of_o the_o pyramid_n as_o it_o be_v prove_v in_o the_o 2._o corollary_n of_o the_o 6._o of_o this_o book_n there_o shall_v be_v place_v 4â—_o angle_n of_o the_o dodecahedron_n namely_o the_o 4._o angle_n e_o f_o h_o d_o in_o the_o 4._o centre_n of_o the_o base_n and_o of_o the_o other_o 16._o angle_n under_o every_o one_o of_o the_o 6._o side_n of_o the_o pyramid_n be_v subtend_v two_o namely_o under_o the_o side_n ad_fw-la the_o angle_n ck_o under_o the_o side_n bd_o the_o angle_n li_n under_o the_o
second_o part_n of_o the_o first_o case_n the_o second_o case_n first_o part_n of_o the_o second_o case_n second_o part_n of_o the_o second_o case_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o first_o part_n of_o the_o first_o case_n 〈◊〉_d second_o 〈◊〉_d of_o the_o 〈◊〉_d case_n the_o second_o case_n a_o corollary_n the_o first_o senary_a by_o substraction_n demonstration_n an_o other_o demonstration_n after_o campane_n definition_n of_o the_o eight_o irrational_a line_n definition_n of_o 〈◊〉_d â—inth_o irrational_a line_n an_o other_o demonstration_n after_o campane_n construction_n demonstration_n definition_n of_o the_o ten_o irrational_a line_n definition_n of_o the_o eleven_o irrational_a line_n â—â—â—â—iâ—ition_n of_o the_o twelve_o irraâ—ionall_a line_n definition_n of_o the_o thirteen_o and_o last_o irrational_a line_n a_o assumpt_n of_o campane_n i_o dee_n though_o campane_n lemma_n be_v true_a yeâ—_z the_o manner_n of_o demonstrate_v it_o narrow_o consider_v be_v not_o artificial_a second_o senary_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o absurdity_n construction_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n 〈◊〉_d a_o abjurdâ—tâ—â—â—_n six_o kind_n of_o reâ—iduall_a line_n first_o definition_n second_o definition_n three_o definition_n four_o definition_n five_o definition_n six_o definition_n three_o senary_n construction_n demonstration_n construction_n demoâ—stratiâ—â—_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o other_o more_o ready_a way_n to_o find_v out_o the_o six_o residual_a line_n four_o senary_n the_o â—irst_n parâ—_n of_o the_o construction_n the_o first_o part_n of_o the_o demonstration_n note_n a_z and_o fk_o conclude_v rational_a parallelogram_n note_n dh_o and_o fk_o parallelogram_n medial_a second_o part_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n ln_o be_v the_o only_a liâ—e_z â—hat_n we_o seek_v &_o consider_v first_o part_n of_o the_o construction_n the_o first_o part_n of_o thâ—_z demonstration_n a_z and_o fk_o conclude_v parallelogram_n medial_a dh_o &_o eke_o rational_a the_o second_o part_n of_o the_o construction_n the_o second_o part_n of_o the_o demonstration_n demonstration_n analytical_o the_o proâ—e_n hereof_o follow_v amongâ—_n other_o thing_n the_o line_n ln_o find_v which_o be_v the_o principal_a drift_n of_o all_o the_o former_a discourse_n the_o first_o part_n of_o the_o construction_n the_o fiâ—st_a part_n of_o the_o demonstration_n note_n a_z and_o fk_o medial_a note_n dh_o and_o eke_o medial_a note_n ai_fw-fr incommensurable_a to_o eke_o second_o part_n of_o the_o construction_n the_o principal_a line_n ln_o find_v find_v because_o the_o line_n of_o and_o â—g_z be_v prove_v commensurable_a in_o length_n length_n by_o the_o first_o oâ—_n the_o six_o and_o ten_o of_o the_o ten_o the_o first_o part_n of_o the_o construction_n the_o first_o part_n of_o the_o demonstration_n note_n ak_o rational_a note_n dk_o medial_a a_z and_o fk_o incommensurable_a the_o second_o part_n of_o the_o construction_n the_o second_o part_n of_o the_o demonstration_n ln_o the_o chief_a line_n of_o this_o theorem_a find_v demonstration_n the_o line_n ln_o demonstration_n the_o five_o senary_a these_o six_o proposition_n follow_v be_v the_o conuâ—rses_n of_o the_o six_o former_a proposition_n construction_n demonstration_n demonstration_n by_o the_o 20._o of_o the_o ten_o ten_o by_o the_o 21._o of_o the_o ten_o ten_o by_o the_o 22._o of_o the_o ten_o â—f_o conclude_v a_o residual_a line_n construction_n demonstration_n cf_o conclude_v a_o residual_a line_n construction_n demonstration_n cf_o conclude_v a_o residual_a line_n construction_n demonstration_n cf_o prove_v a_o residual_a line_n cf_o prove_v a_o residual_a line_n construction_n demonstration_n cf_o â—roved_v â—_o residual_a the_o six_o senary_n construction_n demonstration_n cd_o conclude_v a_o residual_a line_n note_n construction_n demonstration_n cd_o prove_v a_o medial_a construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n seven_o senary_n constraction_n demonstration_n construction_n demonstration_n demonstration_n construction_n demonstration_n on_o lead_v to_o a_o impossibility_n a_o corollary_n the_o determination_n have_v sundry_a part_n orderly_a to_o be_v prove_v construction_n demonstration_n this_o be_v a_o assumpt_n problematical_a artificial_o use_v and_o demonstrate_v demonstrate_v therefore_o those_o three_o line_n be_v in_o continual_a proportion_n fe_o conclude_v a_o residual_a liâ—â—_n which_o be_v sumwhat_o prepâ—â—icroâ—sly_o in_o respect_n oâ—_n the_o â—â—der_n propound_v both_o in_o the_o proposition_n and_o also_o in_o the_o determination_n construction_n demonstration_n construction_n demonstration_n here_o be_v the_o â—ower_a part_n of_o the_o propositiâ—_n more_o orderly_a huddle_v theâ—_z in_o the_o former_a demöstration_n construction_n demonstration_n a_o assumpt_n an_o other_o demonstration_n after_o flussas_n construction_n demonstration_n this_o be_v in_o a_o manner_n the_o converse_n of_o both_o the_o former_a proposition_n joint_o construction_n demonstration_n construction_n demonstration_n demonstration_n an_o other_o demonstration_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n lead_v to_o a_o impossibility_n the_o argument_n of_o the_o eleven_o book_n a_o point_n the_o beginning_n of_o all_o quantity_n continual_a the_o method_n use_v by_o euclid_n in_o the_o ten_o â—â—â—mer_a boot_n â—irst_n boâ—â—e_n second_o â—â—oâ—e_n three_o booâ—e_n forth_o bâ—oâ—e_n â—iveth_v boâ—â—e_n six_o booâ—e_n seven_o bookâ—_n â—ight_a booâ—â—_n nine_o book_n ten_o booâ—e_n what_o be_v entreaâ—ea_n of_o in_o the_o fiâ—e_a boot_n follow_v 〈◊〉_d â—â—â—ular_a bodiesâ—_n the_o â—â—all_a end_n 〈…〉_o oâ—_n i_o uâ—â—â—â—es_v â—eomeâ—â—â—all_a element_n comparison_n â—â—_o the_o 〈◊〉_d â—â—oâ—e_n and_o 〈◊〉_d book_n 〈◊〉_d first_o definition_n a_o solid_a the_o most_o perfect_a quantity_n no_o science_n of_o thing_n infinite_a second_o definition_n three_o definition_n two_o difâ—initions_n include_v in_o this_o definition_n declaration_n of_o the_o first_o part_n declaration_n of_o the_o second_o part_n four_o definition_n five_o definition_n six_o definition_n seven_o defâ—inition_n eight_o definition_n nine_o dissilition_n ten_o definition_n eleven_o definition_n an_o other_o definition_n of_o a_o prism_n which_o be_v a_o special_a definition_n of_o a_o prism_n as_o it_o be_v common_o call_v and_o use_v this_o body_n call_v figura_fw-la serratilis_fw-la psellus_n twelve_o definition_n what_o be_v to_o be_v taâ—â—n_v heed_n of_o in_o the_o definition_n of_o a_o sphere_n give_v by_o johannes_n de_fw-fr sacro_fw-la busco_n theodosiuâ—_n definition_n of_o a_o sphere_n the_o circumference_n of_o a_o sphere_n galens_n definition_n 〈◊〉_d a_o sphâ—râ—_n the_o dignity_n of_o a_o sphere_n a_o sphere_n call_v a_o globe_n thirteen_o definition_n theodosius_n definition_n of_o the_o axe_n of_o a_o sphere_n fourtenth_n definition_n theodosius_n definition_n of_o the_o centre_n of_o a_o sphere_n flussas_n definition_n of_o the_o centre_n of_o a_o sphere_n fivetenth_fw-mi definition_n difference_n between_o the_o diameter_n &_o axe_n of_o a_o sphere_n sevententh_n definition_n first_o kind_n of_o cones_n a_o cone_n call_v of_o campane_n a_o roâ—â—de_a pyramid_n sevententh_n definition_n a_o conical_a superficies_n eightenth_n definition_n ninetenth_fw-mi definition_n a_o cillindricall_a superficies_n corollary_n a_o roundâ—_n column_n or_o sphere_n a_o corollary_n add_v by_o campane_n twenty_o 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