Selected quad for the lemma: book_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
book_n	county_n	line_n	residue_n	120	3	16.9006	5	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A00429	The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed; Elements. English	Euclid.; Dee, John, 1527-1608.; Candale, François de Foix, comte de, 1502-1594.; Billingsley, Henry, Sir, d. 1606.	1570 (1570)	STC 10560; ESTC S106699	1,020,889	884

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 f●_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 fb●_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 ot●●r_n 〈…〉_o flussas_n 〈…〉_o line_n ●d_a whos●_n name_v a●_z and_o ●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 a●_z which_o let_v be_v of_o and_o fb_o and_o let_v the_o li●e_z ae●_n be_v to_o the_o line_n ed●_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 f●_n and_o let_v the_o right_a line_n ●_o contain_v in_o power_n the_o superficies_n d●_n then_o i_o say_v tha●_z the_o li●e_z ●_o be_v a_o rational_a lin●_n 〈…〉_o l●ne_n construction_n which_o l●●_n b●●_n and_o upon_o the_o line_n ●●_o describe_v by_o the_o 4●_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 ●●_o and_o make_n in_o breadth_n the_o line_n dc_o wherefore_o by_o the_o ●12_n of_o this_o book_n cd_o be_v a_o residu●ll_a line●_z who_o name_n which_o let_v be_v ●●_o and_o odd_a shall_v be_v co●mensurabl●_n in_o length_n unto_o the_o name_n a●_z and_o ●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 ed●_n but_o as_o the_o line_n a●_z be_v to_o the_o line_n ●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 f●●_v wherefore_o the_o line_n county_n and_o odd_a be_v commensurable_a with_o the_o line_n a●_z and_o ●●_o by_o the_o ●●_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 a●_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 d●_n but_o the_o parallelogram_n ●●_o i●_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 ●_o wherefore_o the_o parallelogram_n ●d_a ●s_n also_o rational_o wher●fore_o the_o line_n ●_o which_o by_o supposition_n contain_v in_o power_n the_o superficies_n ●d●_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 ●_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

to_o the_o same_o and_o so_o the_o line_n bd_o be_v a_o six_o residual_a line_n and_o the_o line_n kh_o be_v a_o six_o binomial_a line_n wherefore_o kh_o be_v a_o binomial_a line_n who_o name_n kf_a and_o fh_o be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n bd_o namely_o to_o bc_o and_o cd_o and_o in_o the_o self_n same_o proportion_n and_o the_o binomial_a line_n kh_o be_v in_o the_o self_n same_o order_n of_o binomial_a line_n that_o the_o residual_a bd_o be_v of_o residual_a line_n wherefore_o the_o square_a of_o a_o rational_a line_n apply_v unto_o a_o residual_a line_n make_v the_o breadth_n or_o other_o side_n a_o binomial_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n and_o in_o the_o self_n same_o proportion_n and_o moreover_o the_o binomial_a line_n be_v in_o the_o self_n same_o order_n of_o binomial_a line_n that_o the_o residual_a line_n be_v of_o residual_a line_n which_o be_v require_v to_o be_v demonstrate_v the_o assumpt_v confirm_v now_o let_v we_o declare_v how_o as_o the_o line_n kh_o be_v to_o the_o line_n eh_o so_o to_o make_v the_o line_n hf_o to_o the_o line_n fe_o add_v unto_o the_o line_n kh_o direct_o a_o line_n equal_a to_o he_o and_o let_v the_o whole_a line_n be_v kl_o and_o by_o the_o ten_o of_o the_o six_o let_v the_o line_n he_o be_v divide_v as_o the_o whole_a line_n kl_o be_v divide_v in_o the_o point_n h_o let_v the_o line_n he_o be_v so_o divide_v in_o the_o point_n f._n wherefore_o as_o the_o line_n kh_o be_v to_o the_o line_n hl_o that_o be_v to_o the_o line_n he_o so_o be_v the_o line_n hf_o to_o the_o line_n fe_o an_o other_o demonstration_n after_o flussas_n suppose_v that_o a_o be_v a_o rational_a line_n and_o let_v bd_o be_v a_o residual_a line_n and_o upon_o the_o line_n bd_o apply_v the_o parallelogram_n dt_n equal_a to_o the_o square_n of_o the_o line_n a_o by_o the_o 45._o of_o the_o first_o make_v in_o breadth_n the_o line_n bt_n flussas_n then_o i_o say_v that_o bt_n be_v a_o binominall_a line_n such_o a_o one_o as_o be_v require_v in_o the_o proposition_n forasmuch_o as_o bd_o be_v a_o residual_a line_n let_v the_o line_n convenient_o join_v unto_o it_o be_v gd_o wherefore_o the_o line_n bg_o and_o gd_o be_v rational_a commensurable_a in_o power_n only_o construction_n upon_o the_o rational_a line_n bg_o apply_v the_o parallelogram_n by_o equal_a to_o the_o square_n of_o the_o line_n a_o and_o make_v in_o breadth_n the_o line_n be._n wherefore_o the_o line_n be_v be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n bg_o by_o the_o 20._o of_o the_o ten_o now_o forasmuch_o as_o the_o parallelogram_n by_o and_o td_v be_v equal_a for_o that_o they_o be_v each_o equal_a to_o the_o square_n of_o the_o line_n a_o demonstration_n therefore_o reciprocal_a by_o the_o 14._o of_o the_o six_o as_o the_o line_n bt_n be_v to_o the_o line_n be_v so_o be_v the_o line_n bg_o to_o the_o line_n bd._n wherefore_o by_o conversion_n of_o proportion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o as_o the_o line_n bt_n be_v to_o the_o line_n te_fw-la so_o be_v the_o line_n bg_o to_o the_o line_n gd_o as_o the_o line_n bg_o be_v to_o the_o line_n gd_o so_o let_v the_o line_n tz_n be_v to_o the_o line_n see_fw-mi by_o the_o corollary_n of_o the_o 10._o of_o the_o six_o wherefore_o by_o the_o 11._o of_o the_o five_o the_o line_n bt_n be_v to_o the_o line_n te_fw-la as_o the_o line_n tz_n be_v to_o the_o line_n ze._n for_o either_o of_o they_o be_v as_o the_o line_n bg_o be_v to_o the_o line_n gd_o wherefore_o the_o residue_n bz_o be_v to_o the_o residue_n zt_n as_o the_o whole_a bt_n be_v to_o the_o whole_a te_fw-la by_o the_o 19_o of_o the_o five_o wherefore_o by_o the_o 11._o of_o the_o five_o the_o line_n bz_o be_v to_o the_o line_n zt_n as_o the_o line_n zt_n be_v to_o the_o line_n ze._n wherefore_o the_o line_n tz_n be_v the_o mean_a proportional_a between_o the_o line_n bz_o and_o ze._n wherefore_o the_o square_a of_o the_o first_o namely_o of_o the_o line_n bz_o be_v to_o the_o square_n of_o the_o second_o namely_o of_o the_o line_n zt_n as_o the_o first_o namely_o the_o line_n bz_o be_v to_o the_o three_o namely_o to_o the_o line_n see_fw-mi by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o and_o for_o that_o as_o the_o line_n bg_o be_v to_o the_o line_n gd_o so_o be_v the_o line_n tz_n to_o the_o line_n see_fw-mi but_o as_o the_o line_n tz_n be_v to_o the_o line_n see_fw-mi so_o be_v the_o line_n bz_o to_o the_o line_n zt_n wherefore_o as_o the_o line_n bg_o be_v to_o the_o line_n gd_o so_o be_v the_o line_n bz_o to_o the_o line_n zt_n by_o the_o 11._o of_o the_o five_o wherefore_o the_o line_n bz_o and_o zt_n be_v commensurable_a in_o power_n only_o as_o also_o be_v the_o line_n bg_o and_o gd_o which_o be_v the_o name_n of_o the_o residual_a line_n bd_o by_o the_o 10._o of_o this_o book_n wherefore_o the_o right_a line_n bz_o and_o see_fw-mi be_v commensurable_a in_o length_n for_o we_o have_v prove_v that_o they_o be_v in_o the_o same_o proportion_n that_o the_o square_n of_o the_o line_n bz_o and_o zt_n be_v and_o therefore_o by_o the_o corollary_n of_o the_o 15._o of_o this_o book_n the_o residue_n be_v which_o be_v a_o rational_a line_n be_v commensurable_a in_o length_n unto_o the_o same_o line_n bz_o wherefore_o also_o the_o line_n bg_o which_o be_v commensurable_a in_o length_n unto_o the_o line_n be_v shall_v also_o be_v commensurable_a in_o length_n unto_o the_o same_o line_n ez_o by_o the_o 12._o of_o the_o ten_o and_o it_o be_v prove_v that_o the_o line_n rz_n be_v to_o the_o line_n zt_n commensurable_a in_o power_n only_o wherefore_o the_o right_a line_n bz_o and_o zt_n be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n bt_n be_v a_o binomial_a line_n by_o the_o 36._o of_o this_o book_n and_o for_o that_o as_o the_o line_n bg_o be_v to_o the_o line_n gd_o so_o be_v the_o line_n bz_o to_o the_o line_n zt_n therefore_o alternate_o by_o the_o 16._o of_o the_o five_o the_o line_n bg_o be_v to_o the_o line_n bz_o as_o the_o line_n gd_o be_v to_o the_o line_n zt_n but_o the_o line_n bg_o be_v commensurable_a in_o length_n unto_o the_o line_n bz_o wherefore_o by_o the_o 10._o of_o this_o book_n the_o line_n gd_o be_v commensurable_a in_o length_n unto_o the_o line_n zt_n wherefore_o the_o name_n bg_o and_o gd_o of_o the_o residual_a line_n bd_o be_v commensurable_a in_o length_n unto_o the_o name_n bz_o and_o zt_v of_o the_o binomial_a line_n bt_n and_o the_o line_n bz_o be_v to_o the_o line_n zt_n in_o the_o same_o proportion_n that_o the_o line_n bg_o be_v to_o the_o line_n gd_o as_o before_o it_o be_v more_o manifest_a and_o that_o they_o be_v of_o one_o and_o the_o self_n same_o order_n be_v thus_o prove_v if_o the_o great_a or_o less_o name_n of_o the_o residual_a line_n namely_o the_o right_a line_n bg_o or_o gd_o be_v commensurable_a in_o length_n to_o any_o rational_a line_n put_v the_o great_a name_n also_o or_o less_o namely_o bz_o or_o zt_n shall_v be_v commensurable_a in_o length_n to_o the_o same_o rational_a line_n put_v by_o the_o 12._o of_o this_o book_n and_o if_o neither_o of_o the_o name_n of_o the_o residual_a line_n be_v commensurable_a in_o length_n unto_o the_o rational_a line_n put_v neither_o of_o the_o name_n of_o the_o binomial_a line_n shall_v be_v commensurable_a in_o length_n unto_o the_o same_o rational_a line_n put_v by_o the_o 13._o of_o the_o ten_o and_o if_o the_o great_a name_n bg_o be_v in_o power_n more_o than_o the_o less_o name_n by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n bg_o the_o great_a name_n also_o bz_o shall_v be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n bz_o and_o if_o the_o one_o be_v in_o power_n more_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n the_o other_o also_o shall_v be_v in_o power_n more_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n by_o the_o 14._o of_o this_o book_n the_o square_a therefore_o of_o a_o rational_a line_n etc_n etc_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 90._o theorem_a the_o 114._o proposition_n joint_o if_o a_o parallelogram_n be_v contain_v under_o a_o residual_a line_n &_o a_o binomial_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n and_o in_o the_o sel●e_a same_o proportion_n the_o line_n which_o contain_v in_o power_n

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 ●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 segment●_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 ●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 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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

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 also●_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 o●_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 note●_n note●_n you_o know_v full_a well_o that_o in_o the_o superficies_n of_o the_o sphere_n ●●●ly_o the_o circumference_n of_o the_o circle_n be_v but_o by_o th●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 ●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 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construction_n demonstration_n construction_n demonstration_n this_o in_o manner_n of_o a_o lemm●_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 i●_z it_o be_v move_v ●●out_v his_o centre_n q_o etc_n etc_n lib_fw-la 2_o prop_n 2._o de_fw-fr sphe●a_n &_o cylindr●_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 ●c_z the_o chie●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 poi●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 ●the_v converse_v of_o the_o former_a former_a as_o we_o ha●e_z note_v the_o place_n of_o the_o peculiar_a pro●e_n there_o ●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 th●_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 the●4_n ●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 constr●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 o●_n a_o pyramid_n both_o here_o &_o i●_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 f●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

e_o produce_v d_z wherefore_o a_o measure_v d_z but_o it_o also_o measure_v it_o not_o which_o be_v impossible_a wherefore_o it_o be_v impossible_a to_o find_v out_o a_o four_o number_n proportional_a with_o these_o number_n a_o b_o c_o whensoever_o a_o measure_v not_o d._n ¶_o the_o 20._o theorem_a the_o 20._o proposition_n prime_n number_n be_v give_v how_o many_o soever_o there_o may_v be_v give_v more_o prime_a number_n svppose_v that_o the_o prime_a number_n give_v be_v a_o b_o c._n proposition_n then_o i_o say_v that_o there_o be_v yet_o more_o prime_a number_n beside_o a_o b_o c._n take_v by_o the_o 38._o of_o the_o seven_o the_o lest_o number_v who_o these_o number_n a_o b_o c_o do_v measure_n and_o let_v the_o same_o be_v de._n and_o unto_o de_fw-fr add_v unity_n df._n now_o of_o be_v either_o a_o prime_a number_n or_o not_o first_o let_v it_o be_v a_o prime_a number_n case_n then_o be_v there_o find_v these_o prime_a number_n a_o b_o c_o and_o of_o more_o in_o multitude_n then_o the_o prime_a number_n ●irst_n give_v a_o b_o c._n but_o now_o suppose_v that_o of_o be_v not_o prime_a case_n wherefore_o some_o prime_a number_n measure_v it_o by_o the_o 24._o of_o the_o seven_o let_v a_o prime_a number_n measure_v it_o namely_o g._n then_o i_o say_v that_z g_z be_v none_o of_o these_o number_n a_o b_o c._n for_o if_o g_z be_v one_o and_o the_o same_o with_o any_o of_o these_o a_o b_o c._n but_o a_o b_o c_o measure_v the_o number_n de_fw-fr wherefore_o g_z also_o measure_v de_fw-fr and_o it_o also_o measure_v the_o whole_a ef._n wherefore_o g_o be_v a_o number_n shall_v measure_v the_o residue_n df_o be_v unity_n which_o be_v impossible_a wherefore_o g_z be_v not_o one_o and_o the_o same_o with_o any_o of_o these_o prime_a number_n a_o b_o c_o and_o it_o be_v also_o suppose_a to_o be_v a_o prime_a number_n wherefore_o there_o be_v ●ound_v these_o prime_a number_n a_o b_o c_o g_o be_v more_o in_o multitude_n then_o the_o prime_a number_n give_v a_o b_o c_o which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n by_o this_o proposition_n it_o be_v manifest_a that_o the_o multitude_n of_o prime_a number_n be_v infinite_a ¶_o the_o 21._o theorem_a the_o 21._o proposition_n if_o even_o number_n how_o many_o soever_o be_v add_v together_o the_o whole_a shall_v be_v even_o svppose_v that_o these_o even_a number_n ab_fw-la bc_o cd_o and_o de_fw-fr be_v add_v together_o then_o i_o say_v that_o the_o whole_a number_n namely_o ae_n be_v a_o even_a number_n demonstration_n for_o forasmuch_o as_o every_o one_o of_o these_o number_n ab_fw-la bc_o cd_o and_o de_fw-fr be_v a_o even_a number_n therefore_o every_o one_o of_o they_o have_v a_o half_a wherefore_o the_o whole_a ae_n also_o have_v a_o half_a but_o a_o even_a number_n by_o the_o definition_n be_v that_o which_o may_v be_v divide_v into_o two_o equal_a part_n wherefore_o ae_n be_v a_o even_a number_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 22._o theorem_a the_o 22._o proposition_n if_o odd_a number_n how_o many_o soever_o be_v add_v together_o &_o if_o their_o multitude_n be_v even_o the_o whole_a also_o shall_v be_v even_o svppose_v that_o these_o odd_a number_n ab_fw-la bc_o cd_o and_o de_fw-fr be_v even_o in_o multitude_n be_v add_v together_o then_o i_o say_v that_o the_o whole_a ae_n be_v a_o even_a number_n for_o forasmuch_o as_o every_o one_o of_o these_o number_n ab_fw-la bc_o cd_o and_o de_fw-fr be_v a_o odd_a number_n be_v you_o take_v away_o unity_n from_o every_o one_o of_o they_o demonstration_n that_o which_o remain_v o●_n every_o one_o of_o they_o be_v a_o even_a number_n wherefore_o they_o all_o add_v together_o be_v by_o the_o 21._o of_o the_o nine_o a_o even_a number_n and_o the_o multitude_n of_o the_o unity_n take_v away_o be_v even_o wherefore_o the_o whole_a ae_n be_v a_o even_a number_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 23._o theorem_a the_o 23._o proposition_n if_o odd_a number_n how_o many_o soever_o be_v add_v together_o and_o if_o the_o multitude_n of_o they_o be_v odd_a the_o whole_a also_o shall_v be_v odd_a svppose_v that_o these_o odd_a number_n ab_fw-la bc_o and_o cd_o be_v odd_a in_o multitude_n be_v add_v together_o then_o i_o say_v that_o the_o whole_a ad_fw-la be_v a_o odd_a number_n take_v away_o from_o cd_o unity_n de_fw-fr wherefore_o that_o which_o remain_v ce_fw-fr be_v a_o even_a number_n but_o ac_fw-la also_o by_o the_o 22._o of_o the_o nine_o be_v a_o even_a number_n demonstration_n wherefore_o the_o whole_a ae_n be_v a_o even_a number_n but_o de_fw-fr which_o be_v unity_n be_v add_v to_o the_o even_a number_n ae_n make_v the_o whole_a ad_fw-la a●_z odd_a number_n which_o be_v require_v to_o be_v proved●_n ¶_o the_o 24._o theorem_a the_o 24._o proposition_n if_o from_o a_o even_a number_n be_v take_v away_o a_o even_a number_n that_o which_o remain_v shall_v be_v a_o even_a number_n svppose_v that_o ab_fw-la be_v a_o even_a number_n and_o from_o it_o take_v away_o a_o even_a number_n cb._n demonstration_n then_o i_o say_v that_o that_o which_o remain_v namely_o ac_fw-la be_v a_o even_a number_n for_o forasmuch_o as_o ab_fw-la be_v a_o even_o even_a number_n it_o have_v a_o half_a and_o by_o the_o same_o reason_n also_o bc_o have_v a_o half_a wherefore_o the_o residue_n ca_n have_v a_o half_a wherefore_o ac_fw-la be_v a_o even_a number_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 25._o theorem_a the_o 25._o proposition_n if_o from_o a_o even_a number_n be_v take_v away_o a_o odd_a number_n that_o which_o remain_v shall_v be_v a_o odd_a number_n svppose_v that_o ab_fw-la be_v a_o even_a number_n and_o take_v away_o from_o it_o bc_o a_o odd_a number_n then_o i_o say_v that_o the_o residue_n ca_n be_v a_o odd_a number_n demonstration_n take_v away_o from_o bc_o unity_n cd_o wherefore_o db_o be_v a_o even_a number_n and_o ab_fw-la also_o be_v a_o even_a number_n wherefore_o the_o residue_n ad_fw-la be_v a_o even_a number_n by_o the_o ●ormer_a proposition_n but_o cd_o which_o be_v unity_n be_v take_v away_o from_o the_o even_a number_n ad_fw-la make_v the_o residue_n ac_fw-la a_o odd_a number_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 26._o theorem_a the_o 26._o proposition_n if_o from_o a_o odd_a number_n be_v take_v away_o a_o odd_a number_n that_o which_o remain_v shall_v be_v a_o even_a number_n svppose_v that_o ab_fw-la be_v a_o odd_a number_n and_o from_o it_o take_v away_o a_o odd_a number_n bc._n then_o i_o say_v that_o the_o residue_n ca_n be_v a_o even_a number_n demonstration_n for_o forasmuch_o as_o ab_fw-la be_v a_o odd_a number_n take_v away_o from_o it_o unity_n bd._n wherefore_o the_o residue_n ad_fw-la be_v even_o and_o by_o the_o same_o reason_n cd_o be_v a_o even_a number_n wherefore_o the_o residue_n ca_n be_v a_o even_a number_n by_o the_o 24._o of_o this_o book_n ●_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 27._o theorem_a the_o 27._o proposition_n if_o from_o a_o odd_a number_n be_v take_v a_o way_n a_o even_a number_n the_o residue_n shall_v be_v a_o odd_a number_n svppose_v that_o ab_fw-la be_v a_o odd_a number_n and_o from_o it_o take_v away_o a_o even_a number_n bc._n demonstration_n then_o i_o say_v that_o the_o residue_n ca_n be_v a_o odd_a number_n take_v away_o from_o ab_fw-la unity_n ad._n wherefore_o the_o residue_n db_o be_v a_o even_a number_n &_o bc_o be_v by_o supposition_n even_o wherefore_o the_o residue_n cd_o be_v a_o even_a number_n wherefore_o dam_n which_o be_v unity_n be_v add_v unto_o cd_o which_o be_v a_o even_a number_n make_v the_o whole_a ac_fw-la a_o ●dde_a number_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 28._o theorem_a the_o 28._o proposition_n if_o a_o odd_a number_n multiply_v a_o even_a number_n produce_v any_o number_n the_o number_n produce_v shall_v be_v a_o even_a number_n svppose_v that_o a_o be_v a_o odd_a number_n multiply_v b_o be_v a_o even_a number_n do_v produce_v the_o number_n c._n then_o i_o say_v that_o c_z be_v a_o even_a number_n demonstration_n for_o forasmuch_o as_o a_o multiply_v b_o produce_v c_z therefore_o c_z be_v compose_v of_o so_o many_o number_n equal_a unto_o b_o as_o there_o be_v in_o unity_n in_o a._n but_o b_o be_v a_o even_a number_n wherefore_o c_o be_v compose_v of_o so_o many_o even_a number_n as_o there_o be_v unity_n in_o a._n but_o if_o even_a number_n how_o many_o soever_o be_v add_v together_o the_o whole_a by_o the_o 21._o of_o the_o nine_o be_v a_o even_a number_n wherefore_o c_o be_v a_o even_a number_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 29._o theorem_a the_o 29._o proposition_n i●_z a_o odd_a number_n multiply_v a_o

28._o proposition_n to_o find_v out_o medial_a right_a line_n commensurable_a in_o power_n only_o contain_v a_o medial_a parallelogram_n let_v there_o be_v put_v three_o rational_a right_a line_n commensurable_a in_o power_n only_o namely_o a_o b_o and_o c_o construction_n and_o by_o the_o 13._o of_o the_o six_o take_v the_o mean_a proportional_a between_o the_o line_n a_o and_o b_o &_o let_v th●_z same_o be_v d._n and_o as_o the_o line_n b_o be_v to_o the_o line_n c_o so_o by_o the_o 12._o of_o the_o six_o let_v the_o line_n d_o be_v to_o the_o line_n e._n and_o forasmuch_o as_o the_o line_n a_o and_o b_o be_v rational_a commensurable_a in_o power_n only_o demonstration_n therefore_o by_o the_o 21._o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o that_o be_v the_o square_a of_o the_o line_n d_o be_v medial_a wherefore_o d_z be_v a_o medial_a line_n and_o forasmuch_o as_o the_o line_n b_o and_o c_o be_v commensurable_a in_o power_n only_o and_o as_o the_o line_n b_o be_v to_o the_o line_n c_o so_o be_v the_o line_n d_o to_o the_o line_n e_o wherefore_o the_o line_n d_z and_o e_o be_v commensurable_a in_o power_n only_o by_o the_o corollary_n of_o the_o ten_o of_o this_o book_n but_o d_o be_v a_o medial_a line_n wherefore_o e_o also_o be_v a_o medial_a line_n by_o the_o 23._o of_o this_o book_n wherefore_o d_z &_o e_o be_v medial_a line_n commensurable_a in_o power_n only_o i_o say_v also_o that_o they_o contain_v a_o medial_a parallelogram_n for_o for_o that_o as_o the_o line_n b_o be_v to_o the_o line_n c_o so_o be_v the_o line_n d_o to_o the_o line_n e_o therefore_o alternate_o by_o the_o 16_o of_o the_o five_o as_o the_o line_n b_o be_v to_o the_o line_n d_o so_o be_v the_o line_n c_o to_o the_o line_n e._n but_o as_o the_o line_n b_o be_v to_o the_o line_n d_o so_o be_v the_o line_n d_o to_o the_o line_n a●_z by_o converse_n proportion_n which_o be_v prove_v by_o the_o corollary_n of_o the_o four_o of_o the_o five_o wherefore_o as_o the_o line_n d_o be_v to_o the_o line_n a_o so_o be_v the_o line_n c_o to_o the_o line_n e._n wherefore_o that_o which_o be_v contain_v under_o the_o line_n a_o &_o c_o be_v by_o the_o 16._o of_o the_o six●_o equal_a to_o that_o which_o be_v contain_v under_o the_o line_n d_o &_o e._n but_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o c_o be_v medial_a by_o the_o 21._o of_o the_o ten_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o be_v medial_a wherefore_o there_o be_v find_v out_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o medial_a superficies_n which_o be_v require_v to_o be_v do_v a_o assumpt_n to_o find_v out_o two_o square_a number_n which_o add_v together_o make_v a_o square_a number_n let_v there_o be_v put_v two_o like_o superficial_a number_n ab_fw-la and_o bc_o which_o how_o to_o find_v out_o have_v be_v teach_v after_o the_o 9_o proposition_n of_o this_o book_n and_o let_v they_o both_o be_v either_o even_a number_n or_o odd_a and_o let_v the_o great_a number_n be_v ab_fw-la and_o forasmuch_o as_o if_o from_o any_o even_a number_n be_v take_v away_o a_o even_a number_n or_o from_o a_o odd_a number_n be_v take_v away_o a_o odd_a number_n the_o residue_n shall_v be_v even_o by_o the_o 24._o and_o 26_o of_o the_o nine_o if_o therefore_o from_o ab_fw-la be_v a_o even_a number_n be_v take_v away_o bc_o a_o even_a number_n or_o from_o ab_fw-la be_v a_o odd_a number_n be_v take_v away_o bc_o be_v also_o odd_a the_o residue_n ac_fw-la shall_v be_v even_o divide_v the_o number_n ac_fw-la into_o two_o equal_a part_n in_o d_o wherefore_o the_o number_n which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o cd_o be_v by_o the_o six_o of_o the_o second_o as_o barlaam_n demonstrate_v it_o in_o number_n equal_a to_o the_o square_a number_n of_o bd._n but_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v a_o square_a number_n for_o it_o be_v prove_v by_o the_o first_o of_o the_o nine_o that_o if_o two_o like_o plain_a number_n multiply_v the_o one_o the_o other_o produce_v any_o number_n the_o number_n produce_v shall_v be_v a_o square_a number_n wherefore_o there_o be_v find_v out_o two_o square_a number_n the_o one_o be_v the_o square_a number_n which_o be_v produce_v of_o ab_fw-la into_o bc_o and_o the_o other_o the_o square_a number_n produce_v of_o cd_o which_o add_v together_o make_v a_o square_a number_n namely_o the_o square_a number_n produce_v of_o bd_o multiply_v into_o himself_o forasmuch_o as_o they_o be_v demonstrate_v equal_a to_o it_o corollary_n a_o corollary_n ●umber_n and_o hereby_o it_o be_v manifest_a that_o there_o be_v find_v out_o two_o square_a number_n namely_o the_o 〈◊〉_d the_o square_a number_n of_o bd_o and_o the_o other_o the_o square_a number_n of_o cd_o so_o that_o that_o number_n wherein_o the_o one_o exceed_v the_o other_o the_o number_n i_o say_v which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v also_o a_o square_a number_n namely_o when_o a●_z &_o bc_o be_v like_o plain_a number_n but_o when_o they_o be_v not_o like_o plain_a number_n then_o be_v there_o find_v out_o two_o square_a number_n the_o square_a number_n of_o bd_o and_o the_o square_a number_n of_o dc_o who_o excess_n that_o be_v the_o number_n whereby_o the_o great_a exceed_v the_o less_o namely_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v not_o a_o square_a number_n ¶_o a_o assumpt_n to_o find_v out_o two_o square_a number_n which_o add_v together_o make_v not_o a_o square_a number_n assumpt_n let_v ab_fw-la and_o bc_o be_v like_o plain_a number_n so_o that_o by_o the_o first_o of_o the_o nine_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v a_o square_a number_n and_o let_v ac_fw-la be_v a_o even_a number_n and_o divide_v c●_n into_o two_o equal_a par●es_n in_o d._n now_o by_o that_o which_o have_v before_o be_v say_v in_o the_o former_a assumpt_n it_o be_v manifest_a that_o the_o square_a number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o cd_o be_v equal_a to_o the_o square_a number_n of_o bd._n take_v away_o from_o cd_o unity_n de._n wherefore_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_n of_o ce_fw-fr be_v less_o than_o the_o square_a number_n of_o bd._n now_o then_o i_o say_v that_o the_o square_a num●er_n produce_v of_o ab_fw-la into_o bc_o add_v to_o the_o square_a number_n of_o ce_fw-fr make_v not_o a_o square_a number_n for_o if_o they_o do_v make_v a_o square_a number_n than_o that_o square_a number_n which_o they_o make_v be_v either_o great_a they_o the_o square_a number_n of_o be_v or_o equal_a unto_o it_o or_o less_o than_o it_o first_o great_a it_n can_v be_v for_o it_o be_v already_o prove_v that_o the_o square_a number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v less_o than_o the_o square_a number_n of_o bd._n but_o between_o the_o square_a number_n of_o bd_o and_o the_o square_a number_n of_o be_v there_o be_v no_o mean_a square_a number_n for_o the_o number_n bd_o exceed_v the_o number_n be_v only_o by_o unity_n which_o unity_n can_v by_o no_o mean_n be_v divide_v into_o number_n or_o if_o the_o number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_n of_o the_o number_n ce_fw-fr shall_v be_v great_a than_o the_o square_a of_o the_o number_n be_v then_o shall_v the_o self_n same_o number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_n of_o the_o number_n ce_fw-fr be_v equal_a to_o the_o square_n of_o the_o number_n bd_o the_o contrary_a whereof_o be_v already_o prove_v wherefore_o if_o it_o be_v possible_a let_v that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o the_o number_n ce_fw-fr be_v equal_a to_o the_o square_a number_n of_o be._n and_o let_v ga_o be_v double_a to_o unity_n de_fw-fr that_o be_v let_v it_o be_v the_o number_n two_o now_o forasmuch_o as_o the_o whole_a number_n ac_fw-la be_v by_o supposition_n double_a to_o the_o whole_a number_n cd_o of_o which_o the_o number_n agnostus_n be_v double_a to_o unity_n de_fw-fr therefore_o by_o the_o 7._o of_o the_o seven_o the_o residue_n namely_o the_o number_n gc_o be_v double_a to_o the_o residue_n namely_o to_o the_o number_n ec_o wherefore_o the_o number_n gc_o be_v divide_v into_o two_o equal_a part_n in_o e._n wherefore_o that_o which_o be_v produce_v of_o gb_o into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v equal_a to_o the_o square_a number_n

of_o these_o triangle_n ekf_a flg_n gmh_o and_o hne_n be_v great_a than_o the_o half_a of_o the_o segment_n of_o the_o circle_n which_o be_v describe_v about_o it_o now_o then_o devide_v the_o circumference_n remain_v into_o two_o equal_a part_n and_o draw_v right_a line_n from_o the_o point_n where_o those_o division_n be_v make_v &_o so_o continual_o do_v this_o we_o shall_v at_o the_o length_n by_o the_o 1._o of_o the_o ten_o leave_v certain_a segment_n of_o the_o circle_n which_o shall_v be_v less_o than_o the_o excess_n whereby_o the_o circle_n efgh_o exceed_v the_o superficies_n s._n for_o it_o have_v be_v prove_v in_o the_o first_o proposition_n of_o the_o ten_o book_n that_o two_o unequal_a magnitude_n be_v give_v if_o from_o the_o great_a be_v take_v away_o more_o than_o the_o half_a and_o likewise_o again_o from_o the_o residue_n more_o than_o the_o hal●e_n and_o so_o continual_o there_o shall_v at_o the_o length_n be_v leave_v a_o certain_a magnitude_n which_o shall_v be_v less_o than_o the_o less_o magnitude_n give_v let_v there_o be_v such_o segment_n leave_v &_o let_v the_o segment_n of_o the_o circle_n efgh_o namely_o which_o be_v make_v by_o the_o line_n eke_o kf_n fl_fw-mi lg_n gm_n mh_o hn_o and_o ne_fw-fr be_v less_o than_o the_o excess_n whereby_o the_o circle_n efgh_o exceed_v the_o superficies_n s._n wherefore_o the_o residue_n namely_o the_o poligonon_n figure_n ekflgmhn_n be_v great_a than_o the_o superficies_n s._n inscribe_v in_o the_o circle_n abcd_o a_o poligonon_n figure_n like_o to_o the_o poligonon_n figure_n ekflgmhn_n and_o let_v the_o same_o be_v axbocpdr_n wherefore_o by_o the_o proposition_n next_o go_v before_o as_o the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o so_o be_v the_o poligonon_n figure_n axbocpdr_n to_o the_o poligonon_n figure_n ekflgmhn_n but_o as_o the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fg_o so_o be_v the_o circle_n abcd_o suppose_v to_o be_v to_o the_o superficies_n s._n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o circle_n abcd_o be_v to_o the_o superficies_n s_o so_o be_v the_o poligonon_n figure_n axbocpdr_n to_o the_o poligonon_n figure_n ekflgmhn_n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o circle_n abcd_o be_v to_o the_o poligonon_n figure_n describe_v in_o it_o so_o be_v the_o superficies_n s_o to_o the_o poligonon_n figure_n ekflgmhn_n but_o the_o circle_n abcd_o be_v great_a than_o the_o poligonon_n figure_n describe_v in_o it_o wherefore_o also_o the_o superficies_n saint_n be_v great_a than_o the_o poligonon_n figure_n ekflghmn_a but_o it_o be_v also_o less_o which_o be_v impossible_a wherefore_o as_o the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o so_o be_v not_o the_o circle_n abcd_o to_o any_o superficies_n less_o than_o the_o circle_n efgh_o case_n in_o like_a sort_n also_o may_v wprove_v that_o as_o the_o square_n of_o the_o line_n fh_o be_v to_o the_o square_n of_o the_o line_n bd_o so_o be_v not_o the_o circle_n efgh_o to_o any_o superficies_n less_o than_o the_o circle_n abcd._n i_o say_v namely_o that_o as_o the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o so_o be_v not_o the_o circle_n abcd_o to_o any_o superficies_n great_a they_o than_o the_o circle_n efgh_o for_o if_o it_o be_v possible_a let_v it_o be_v to_o a_o great_a namely_o to_o the_o superficies_n s._n wherefore_o by_o conversion_n as_o the_o square_n of_o the_o line_n fh_o be_v to_o the_o square_n of_o the_o line_n bd_o so_o be_v the_o superficies_n s_o to_o the_o circle_n abcd._n prove_v but_o as_o the_o superficies_n saint_n be_v to_o the_o circle_n abcd_o so_o be_v the_o circle_n efgh_o to_o some_o superficies_n l●sse_v they_o the_o circle_n abcd._n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o square_n of_o the_o line_n fh_o be_v to_o the_o square_n of_o the_o line_n bd_o so_o be_v the_o circle_n efgh_o to_o some_o superficies_n less_o than_o the_o circle_n abcd_o which_o be_v in_o the_o first_o case_n prove_v to_o be_v impossible_a wherefore_o as_o the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o so_o be_v not_o the_o circle_n abcd_o to_o any_o superficies_n great_a than_o the_o circle_n efgh_o and_o it_o be_v also_o prove_v that_o it_o be_v not_o to_o any_o less_o wherefore_o as_o the_o square_n of_o the_o l●ne_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o so_o be_v the_o circle_n abcd_o to_o the_o circle_n efgh_o wherefore_o circle_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o the_o square_n of_o their_o diameter_n be_v which_o be_v require_v to_o be_v prove_v ¶_o a_o assumpt_n i_o say_v now_o that_o the_o superficies_n saint_n be_v great_a than_o the_o circle_n efgh_o as_o the_o superficies_n saint_n be_v to_o the_o circle_n abcd_o so_o be_v the_o circle_n efgh_o to_o some_o superficies_n less_o than_o the_o circle_n abcd._n for_o as_o the_o superficies_n saint_n be_v to_o the_o circle_n abcd_o so_o let_v the_o circle_n efgh_o be_v to_o the_o superficies_n t._n now_o i_o say_v that_o the_o superficies_n t_n be_v less_o than_o the_o circle_n abcd._n for_o for_o that_o as_o the_o superficies_n saint_n be_v to_o the_o circle_n abcd_o so_o be_v the_o circle_n efgh_o to_o the_o superficies_n t_o therefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o superficies_n saint_n be_v to_o the_o circle_n efgh_o so_o be_v the_o circle_n abcd_o to_o the_o superficies_n t._n but_o the_o superficies_n saint_n be_v great_a than_o the_o circle_n efgh_o by_o supposition_n wherefore_o also_o the_o circle_n abcd_o be_v great_a than_o the_o superficies_n t_n by_o the_o 14._o of_o the_o five_o wherefore_o as_o the_o superficies_n saint_n be_v to_o the_o circle_n abcd_o so_o be_v the_o circle_n efgh_o to_o some_o superficies_n less_o than_o the_o circle_n abcd_o which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o corollary_n add_v by_o flussas_n circle_n have_v the_o one_o to_o the_o other_o that_o proportion_n that_o like_o poligonon_n figure_n and_o in_o like_a sort_n describe_v in_o they_o have_v for_o it_o be_v by_o the_o first_o proposition_n prove_v that_o the_o poligonon_n figure_n have_v that_o proportion_n the_o one_o to_o the_o other_o that_o the_o square_n of_o the_o diameter_n have_v which_o proportion_n likewise_o by_o this_o proposition_n the_o circle_n have_v ¶_o very_o needful_a problem_n and_o corollarye_n by_o master_n john_n dee_n invent_v who_o wonderful_a use_n also_o be_v partly_o declare_v a_o problem_n 1._o two_o circle_n be_v give_v to_o find_v two_o right_a line_n which_o have_v the_o same_o proportion_n one_o to_o the_o other_o that_o the_o give_v circle_n have_v o●e_v to_o the_o other●_n suppose_v a_o and_o b_o to_o be_v the_o diameter_n of_o two_o circle_n give_v i_o say_v that_o two_o right_a line_n be_v to_o be_v find_v have_v that_o proportion_n that_o the_o circle_n of_o a_o have_v to_o the_o circle_n of_o b._n let_v to_o a_o &_o b_o by_o the_o 11_o of_o the_o six_o a_o three_o proportional_a line_n be_v find_v which_o suppose_v to_o be_v c._n construction_n i_o say_v now_o that_o a_o have_v to_o c_o that_o proportion_n which_o the_o circle_n of_o a_o have_v to_o the_o circle_n of_o b._n for_o forasmuch_o as_o a_o b_o and_o c_o be_v by_o construction_n three_o proportional_a line_n demonstration_n the_o square_a of_o a_o be_v to_o the_o square_n of_o b_o as_z a_o be_v to_o c_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o ●_o but_o as_o the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o so_o be_v the_o circle_n who_o diameter_n be_v the_o line_n a_o to_o the_o circle_n who_o diameter_n be_v the_o line_n b_o by_o this_o second_o of_o the_o eleven_o wherefore_o the_o circle_n of_o the_o line●_z a_o and_o b_o be_v in_o the_o proportion_n of_o the_o right_a line_n a_o and_o c._n therefore_o two_o circle_n be_v give_v we_o have_v find_v two_o right_a line_n have_v the_o same_o proportion_n between_o they_o that_o the_o circle_n give_v have_v one_o to_o the_o other_o which_o ought_v to_o be_v do_v a_o problem_n 2._o two_o circle_n be_v give_v and_o a_o right_a line_n to_o find_v a_o other_o right_a line_n to_o which_o the_o line_n give_v shall_v have_v that_o proportion_n which_o the_o one_o circle_n have_v to_o the_o other_o suppose_v two_o circle_n give_v which_o let_v be_v a_o &_o b_o &_o a_o right_a line_n give_v which_o let_v be_v c_o i_o say_v that_o a_o other_o right_a line_n be_v to_o be_v ●ounde_v to_o which_o the_o line_n c_o shall_v have_v that_o proportion_n that_o

the_o residue_n or_o of_o this_o excess_n but_o a_o pyramid_n be_v to_o the_o same_o cube_fw-la inscribe_v in_o it_o nonecuple_n by_o the_o 30._o of_o this_o book_n wherefore_o the_o dodecahedron_n inscribe_v in_o the_o pyramid_n and_o contain_v the_o same_o cube_fw-la twice_o take_v away_o the_o self_n same_o three_o of_o the_o less_o segment_n and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o residue_n shall_v contain_v two_o nine_o part_n of_o the_o solid_a of_o the_o pyramid_n of_o which_o nine_o part_v each_o be_v equal_a unto_o the_o cube_fw-la take_v away_o this_o self_n same_o excess_n the_o solid_a therefore_o of_o a_o dodecahedron_n contain_v of_o a_o pyramid_n circumscribe_v about_o it_o two_o nine_o part_n take_v away_o a_o three_o part_n of_o one_o nine_o part_n of_o the_o less_o segment_n of_o a_o line_n divide_v by_o a_o extmere_n and_o mean_a proportion_n and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o residue_n ¶_o the_o 36._o proposition_n a_o octohedron_n exceed_v a_o icosahedron_n inscribe_v in_o it_o by_o a_o parallelipipedon_n set_v upon_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n and_o have_v to_o his_o altitude_n the_o line_n which_o be_v the_o great_a segment_n of_o half_a the_o semidiameter_n of_o the_o octohedron_n svppose_v that_o there_o be_v a_o octohedron_n abcfpl_n construction_n in_o which_o let_v there_o be_v inscribe_v a_o icosahedron_n hkegmxnudsqt●_n by_o the_o ●6_o of_o the_o fifteen_o and_o draw_v the_o diameter_n azrcbroif_n and_o the_o perpendicular_a ko_o parallel_n to_o the_o line_n azr_n then_o i_o say_v that_o the_o octohedron_n abcfpl_n be_v great_a th●n_n the_o icosahedron_n inscribe_v in_o it_o by_o a_o parallelipipedon_n set_v upon_o the_o square_n of_o the_o side_n hk_o or_o ge_z and_o have_v to_o his_o altitude_n the_o line_n ko_o or_o rz_n which_o be_v the_o great_a segment_n of_o the_o semidiameter_n ar._n forasmuch_o as_o in_o the_o same_o 16._o it_o have_v be_v prove_v that_o the_o triangle_n kdg_n and_o keq_n be_v describe_v in_o the_o base_n apf_n and_o alf_n of_o the_o octohedron_n demonstration_n therefore_o about_o the_o solid_a angle_n there_o remain_v upon_o the_o base_a feg_fw-mi three_o triangle_n keg_n kfe_n and_o kfg_n which_o contain_v a_o pyramid_n kefg_n unto_o which_o pyramid_n shall_v be_v equal_a and_o like_v the_o opposite_a pyramid_n mefg_n set_v upon_o the_o same_o base_a feg_n by_o the_o 8._o definition_n of_o the_o eleven_o and_o by_o the_o ●ame_n reason_n shall_v there_o at_o every_o solid_a angle_n of_o the_o octohedron_n remain_v two_o pyramid_n equal_a and_o like_a namely_o two_o upon_o the_o base_a ahk_n two_o upon_o the_o base_a bnv_n two_o upon_o the_o base_a dp_n and_o moreover_o two_o upon_o the_o base_a qlt._n now_o they_o there_o shall_v be_v make_v twelve_o pyramid_n set_v upon_o a_o base_a contain_v of_o the_o side_n of_o the_o icosahedron_n and_o under_o two_o le●●e_a segment_n of_o the_o side_n of_o the_o octohedron_n contain_v a_o right_a angle_n as_o for_o example_n the_o base_a gef_n and_o forasmuch_o as_o the_o side_n ge_z subtend_v a_o right_a angle_n be_v by_o the_o 47._o of_o the_o ●irst_n in_o power_n duple_n to_o either_o of_o the_o line_n of_o and_o fg_o and_o so_o the_o ●●de●_n kh_o be_v in_o power_n duple_n to_o either_o of_o the_o side_n ah_o and_o ak_o and_o either_o of_o the_o line_n ah_o ak_o or_o of_o fg_o be_v in_o power_n duple_n to_o either_o of_o the_o line_n az_o or_o zk_v which_o contain_v a_o right_a angle_n make_v in_o the_o triangle_n or_o base_a ahk_n by_o the_o perpendicular_a az_o wherefore_o it_o follow_v that_o the_o side_n ge_z or_o hk_o be_v in_o power_n quadruple_a to_o the_o triangle_n efg_o or_o ahk_n but_o the_o pyramid_n kefg_n have_v his_o base_a efg_o in_o the_o plain_a flbp_n of_o the_o octohedron_n shall_v have_v to_o his_o altitude_n the_o perpendicular_a ko_o by_o the_o 4._o definition_n of_o the_o six_o which_o be_v the_o great_a segment_n of_o the_o semidiameter_n of_o the_o octohedron_n by_o the_o 16._o of_o the_o fifteen_o wherefore_o three_o pyramid_n set_v under_o the_o same_o altitude_n and_o upon_o equal_a base_n shall_v be_v equal_a to_o one_o prism_n set_v upon_o the_o same_o base_a and_o under_o the_o same_o altitude_n by_o the_o 1._o corollary_n of_o the_o 7._o of_o the_o twelve_o wherefore_o 4._o prism_n set_v upon_o the_o base_a gef_n quadruple_v which_o be_v equal_a to_o the_o square_n of_o the_o side_n ge_z and_o under_o the_o altitude_n ko_o or_o rz_n the_o great_a segment_n which_o be_v equal_a to_o ko_o shall_v contain_v a_o solid_a equal_a to_o the_o twelve_o pyramid_n which_o twelve_o pyramid_n make_v the_o excess_n of_o the_o octohedron_n above_o the_o icosahedron_n inscribe_v in_o it_o a_o octohedron_n therefore_o exceed_v a_o icosahedron_n inscribe_v in_o it_o by_o a_o parallelipipedon_n set_v upon_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n and_o have_v to_o his_o altitude_n the_o line_n which_o be_v the_o great_a segment_n of_o half_a the_o semidiameter_n of_o the_o octohedron_n ¶_o a_o corollary_n a_o pyramid_n exceed_v the_o double_a of_o a_o icosahedron_n inscribe_v in_o it_o by_o a_o solid_a set_v upon_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o and_o have_v to_o his_o altitude_n that_o whole_a line_n of_o which_o the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n for_o it_o be_v manifest_a by_o the_o 19_o of_o the_o fivetenth_n that_o a_o octohedron_n &_o a_o icosahedron_n inscribe_v in_o it_o be_v inscribe_v in_o one_o &_o the_o self_n same_o pyramid_n it_o have_v moreover_o be_v prove_v in_o the_o 26._o of_o this_o book_n that_o a_o pyramid_n be_v double_a to_o a_o octohedron_n inscribe_v in_o it_o wherefore_o the_o two_o excess_n of_o the_o two_o octohedron_n unto_o which_o the_o pyramid_n be_v equal_a above_o the_o two_o icosahedrons_n inscribe_v in_o the_o say_v two_o octohedron_n be_v bring_v into_o a_o solid_a the_o say_v solid_a shall_v be_v set_v upon_o the_o self_n same_o square_n of_o the_o side_n of_o the_o icosahedron_n and_o shall_v have_v to_o his_o altitude_n the_o perpendicular_a ko_o double_a who_o double_a couple_v the_o opposite_a side_n hk_o and_o xm_o make_v the_o great_a segment_n the_o same_o side_n of_o the_o icosahedron_n by_o the_o first_o and_o second_o corollary_n of_o the_o 14._o of_o the_o fiu●●en●h_n the_o 37._o proposition_n if_o in_o a_o triangle_n have_v to_o his_o base_a a_o rational_a line_n set_v the_o side_n be_v commensurable_a in_o power_n to_o the_o base_a and_o from_o the_o top_n be_v draw_v to_o the_o base_a a_o perpendicular_a line_n cut_v the_o base_a the_o section_n of_o the_o base_a shall_v be_v commensurable_a in_o length_n to_o the_o whole_a base_a and_o the_o perpendicular_a shall_v be_v commensurable_a in_o power_n to_o the_o say_v whole_a base_a and_o now_o that_o the_o perpendicular_a ap_n be_v commensurable_a in_o power_n to_o the_o base_a bg_o demonstration_n i●_z thus_o prove_v forasmuch_o as_o the_o square_n of_o ab_fw-la be_v by_o supposition_n commensurable_a to_o the_o square_n of_o bg_o and_o unto_o the_o rational_a square_n of_o ab_fw-la be_v commensurable_a the_o rational_a square_n of_o bp_o by_o the_o 12._o of_o the_o eleven_o wherefore_o the_o residue_n namely_o the_o square_a of_o pa_o be_v commensurable_a to_o the_o same_o square_n of_o bp_o by_o the_o 2._o part_n of_o the_o 15._o of_o the_o eleven_o wherefore_o by_o the_o 12._o of_o the_o ten_o the_o square_a of_o pa_o be_v commensurable_a to_o the_o whole_a square_n of_o bg_o wherefore_o the_o perpendicular_a ap_n be_v commensurable_a in_o power_n to_o the_o base_a bg_o by_o the_o 3._o definition_n of_o the_o ten_o which_o be_v require_v to_o be_v prove_v in_o demonstrate_v of_o this_o we_o make_v no_o mention_n at_o all_o of_o the_o length_n of_o the_o side_n ab_fw-la and_o ag_z but_o only_o of_o the_o length_n of_o the_o base_a bg_o for_o that_o the_o line_n bg_o be_v the_o rational_a line_n first_o set_v and_o the_o other_o line_n ab_fw-la and_o ag_z be_v suppose_v to_o be_v commensurable_a in_o power_n only_o to_o the_o line_n bg_o wherefore_o if_o that_o be_v plain_o demonstrate_v when_o the_o side_n be_v commensurable_a in_o power_n only_o to_o the_o base_a much_o more_o easy_o will_v it_o follow_v if_o the_o same_o side_n be_v suppose_v to_o be_v commensurable_a both_o in_o length_n and_o in_o power_n to_o the_o base_a that_o be_v if_o their_o lengthe_n be_v express_v by_o the_o root_n of_o square_a number_n ¶_o a_o corollary_n 1._o by_o the_o former_a thing_n demonstrate_v it_o be_v manifest_a that_o if_o from_o the_o power_n of_o the_o base_a and_o of_o one_o of_o the_o side_n be_v take_v away_o the_o

in_o the_o point_n e._n and_o unto_o the_o line_n cg_o put_v the_o line_n cl_n equal_a now_o forasmuch_o as_o the_o line_n agnostus_n and_o gc_o be_v the_o great_a seg●●●tes_n of_o half_a the_o line_n ab_fw-la for_o ●che_fw-mi of_o they_o be_v the_o half_a of_o the_o great_a segment_n of_o the_o whole_a line_n ab_fw-la the_o line_n ebb_v and_o ec_o shall_v be_v the_o less_o segment_n of_o half_a the_o line_n ab_fw-la wherefore_o the_o whole_a line_n c●_n be_v the_o great_a segment_n and_o the_o line_n ce_fw-fr be_v the_o less_o segment_n but_o as_o the_o line_n cl_n be_v to_o the_o line_n ce_fw-fr so_o be_v the_o line_n ce_fw-fr to_o the_o residue_n el._n wherefore_o the_o line_n el_n be_v the_o great_a segment_n of_o the_o line_n ce_fw-fr or_o of_o the_o line_n ebb_v which_o be_v equal_a unto_o it_o wherefore_o the_o residue_n lb_n be_v the_o less_o segment_n of_o the_o same_o ebb_n which_o be_v the_o les●e_n segment_n of_o halfa_n the_o side_n of_o the_o cube_fw-la but_o the_o line_n agnostus_n gc_o and_o cl_n be_v three_o great_a segment_n of_o the_o half_a of_o the_o whole_a line_n ab_fw-la which_o three_o great_a segment_n make_v the_o altitude_n of_o the_o foresay_a solid_a wherefore_o the_o altitude_n of_o the_o say_v solid_a want_v of_o ab_fw-la the_o side_n of_o the_o cube_fw-la by_o the_o line_n lb_n which_o be_v the_o less_o segment_n of_o the_o line_n be._n which_o line_n be_v again_o be_v the_o less_o segment_n of_o half_a the_o side_n ab_fw-la of_o the_o cube_fw-la wherefore_o the_o foresay_a solid_a consist_v of_o the_o six_o solid_n whereby_o the_o dodecahedron_n exceed_v the_o cube_fw-la inscribe_v in_o it_o be_v set_v upon_o a_o base_a which_o want_v of_o the_o base_a of_o the_o cube_fw-la by_o a_o three_o part_n of_o the_o less_o segment_n and_o be_v under_o a_o altitude_n want_v of_o the_o side_n of_o the_o cube_fw-la by_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o side_n of_o the_o cube_fw-la the_o solid_a therefore_o of_o a_o dodecahedron_n exceed_v the_o solid_a of_o a_o cube_fw-la inscribe_v in_o it_o by_o a_o parallelipipedon_n who_o base_a want_v of_o the_o base_a of_o the_o cube_fw-la by_o a_o three_o part_n of_o the_o less_o segment_n and_o who_o altitude_n want_v of_o the_o altitude_n of_o the_o cube_fw-la by_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o side_n of_o the_o cube_fw-la ¶_o a_o corollary_n a_o dodecahedron_n be_v double_a to_o a_o cube_n inscribe_v in_o it_o take_v away_o the_o three_o part_n of_o the_o less_o segment_n of_o the_o cube_fw-la and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a of_o that_o excess_n for_o if_o there_o be_v give_v a_o cube_fw-la from_o which_o be_v cut_v of_o a_o solid_a set_v upon_o a_o three_o part_n of_o the_o less_o segment_n of_o the_o base_a and_o under_o one_o and_o the_o same_o altitude_n with_o the_o cube_fw-la that_o solid_a take_v away_o have_v to_o the_o whole_a solid_a the_o proportion_n of_o the_o section_n of_o the_o base_a to_o the_o base_a by_o the_o 32._o of_o the_o eleven_o wherefore_o from_o the_o cube_fw-la be_v take_v away_o a_o three_o ●art_n of_o the_o less_o segment_n far_o forasmuch_o as_o the_o residue_n want_v of_o the_o altitude_n of_o the_o cube_fw-la by_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o altitude_n or_o side_n and_o that_o residue_n be_v a_o parallelipipedon_n if_o it_o be_v cut_v by_o a_o plain_a superficies_n parallel_v to_o the_o opposite_a plain_a superficiece_n cut_v the_o altitude_n of_o the_o cube_fw-la by_o a_o point_n it_o shall_v take_v away_o from_o that_o parallelipipedon_n a_o solid_a have_v to_o the_o whole_a the_o proportion_n of_o the_o section_n to_o the_o altitude_n by_o the_o 3._o corollary_n of_o the_o 25._o of_o the_o eleven_o wherefore_o the_o excess_n want_v of_o the_o same_o cube_fw-la by_o the_o thi●d_o part_n of_o the_o less_o segment_n and_o moreover_o by_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a of_o that_o excess_n ¶_o the_o 34._o proposition_n the_o proportion_n of_o the_o solid_a of_o a_o dodecahedron_n to_o the_o solid_a of_o a_o icosahedron_n inscribe_v in_o it_o consist_v of_o the_o proportion_n triple_a of_o the_o diameter_n to_o that_o line_n which_o couple_v the_o opposite_a base_n of_o the_o dodecahedron_n and_o of_o the_o proportion_n of_o the_o side_n of_o the_o cube_n to_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n svppose_v that_o ahbck_n be_v a_o dodecahedron●_n who_o diameter_n let_v be_v ab_fw-la and_o let_v the_o line_n which_o couple_v the_o centre_n of_o the_o opposite_a base_n be_v kh●_n and_o let_v the_o icosahedron_n inscribe_v in_o the_o dodecahedron_n abc_n be_v do_v you_o who_o diameter_n let_v be_v de._n now_o forasmuch_o a●_z o●e_n and_o the_o self_n same_o circle_n contain_v the_o pentagon_n of_o a_o dodecahedron_n &_o the_o triangle_n of_o a_o icosahedro●_n describe_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o 14._o of_o the_o fourteen_o let_v that_o circle_n be_v igo._n wherefore_o io_o be_v the_o side_n of_o the_o cube_fw-la and_o ig_v the_o side_n of_o the_o icosahedron_n by_o the_o same_o then_o i_o say_v that_o the_o proportion_n of_o the_o dodecahedron_n ahbck_n to_o the_o icosahedron_n def_n inscribe_v in_o it_o consist_v of_o the_o proportion_n triple_a of_o the_o line_n ab_fw-la to_o the_o line_n kh_o and_o of_o the_o proportion_n of_o the_o line_n io_o to_o the_o line_n ig_n for_o forasmuch_o as_o the_o icosahedron_n def_n be_v inscribe_v in_o the_o dodecahedron_n abc_n demonstration_n by_o supposition_n the_o diameter_n de_fw-fr shall_v be_v equal_a to_o the_o line_n kh_o by_o the_o 7._o of_o the_o fifteen_o wherefore_o the_o dodecahedron_n set_v upon_o the_o diameter_n kh_o shall_v be_v inscribe_v in_o the_o same_o sphere_n wherein_o the_o icosahedron_n def_n be_v inscribe_v but_o the_o dodecahedron_n ahbck_n be_v to_o the_o dodecahedron_n upon_o the_o diameter_n kh_o in_o triple_a proportion_n of_o that_o in_o which_o the_o diameter_n ab_fw-la be_v to_o the_o diameter_n kh_o by_o the_o corollary_n of_o the_o 17._o of_o the_o twelve_o and_o the_o same_o dodecahedron_n which_o be_v set_v upon_o the_o diameter_n kh_o have_v to_o the_o icosahedron_n def_n which_o be_v set_v upon_o the_o same_o diameter_n or_o upon_o a_o diameter_n equal_a unto_o it_o namely_o de_fw-fr that_o proportion_n which_o io_o the_o side_n of_o the_o cube_fw-la have_v to●_n ig_n the_o side_n of_o the_o icosahedron_n inscribe_v in_o one_o &_o the_o self_n same_o sphere_n by_o the_o 8_o of_o the_o fourteen_o wherefore_o the_o proportion_n of_o the_o dodecahedron_n ahbck_n to_o the_o icosahedron_n def_n inscribe_v in_o it_o consist_v of_o the_o proportion_n triple_a of_o the_o diameter_n ab_fw-la to_o the_o line_n kh_o which_o couple_v the_o centre_n of_o the_o opposite_a base_n of_o the_o dodecahedron_n which_o proportion_n be_v that_o which_o the_o dodecahedron_n ahbck_n have_v to_o the_o dodecahedron_n set_v upon_o the_o diameter_n kh_o and_o of_o the_o proportion_n of_o io_o the_o side_n of_o the_o cube_fw-la to_o ig_o the_o side_n of_o the_o icosahedron_n which_o be_v the_o proportion_n of_o the_o dodecahedron_n set_v upon_o the_o diameter_n kh_o to_o the_o icosahedron_n def_n describe_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o 5._o definition_n of_o the_o six_o the_o proportion_n therefore_o of_o the_o solid_a of_o a_o dodecahedron_n to_o the_o solid_a of_o a_o icosahedron_n inscribe_v in_o it_o consi_v of_o the_o proportion_n triple_a of_o the_o diameter_n to_o that_o line_n which_o couple_v the_o opposite_a base_n of_o the_o dodecahedron_n and_o of_o the_o proportion_n of_o the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n the_o 35._o proposition_n the_o solid_a of_o a_o dodecahedron_n contain_v of_o a_o pyramid_n circumscribe_v about_o it_o two_o nine_o part_n take_v away_o a_o three_o part_n of_o one_o nine_o part_n of_o the_o less_o segment_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o residue_n it_o have_v be_v prove_v that_o the_o dodecahedron_n together_o with_o the_o cube_fw-la inscribe_v in_o it_o be_v contain_v in_o one_o and_o the_o self_n same_o pyramid_n by_o the_o corollary_n of_o the_o first_o of_o this_o book_n and_o by_o the_o corollary_n of_o the_o 33._o of_o this_o book_n it_o be_v manifest_a that_o the_o dodecahedron_n be_v double_a to_o the_o same_o cube_fw-la take_v away_o the_o three_o part_n of_o the_o less_o segment_n and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a