...bd In a similar manner we may prove that : a : c = b : d, b : d = a : c, с : d = a : b, etc. 297. In any proportion the terms are in proportion by Alternation;...is to the fourth. Let a : b = с : d. Then, by Art. 293, ad=bc. Whence, by Art. 296, a : с = b : d. 298. In any proportion the terms are in proportion...

...In a similar manner we may prove that : a : с = b : d, b : d = a : с, с : -d = a : b, etc. 297. In any proportion the terms are in proportion by Alternation;...as the second term is to the fourth. Let a : b = с :ß. Then, by Art. 293, ad = be. Whence, by Art. 296, a : с = b : d. 298. In any proportion the terms...

...manner we may prove that : a : с = b : d, b : d = a : c, c: d = a: b, etc. 297. In any proportion t)ie terms are in proportion by Alternation; that is, the...is to the fourth. Let a : b = с : d. Then, by Art. 293, ad= 6e. Whence, by Art. 296, a : с = b : d. 298. In any proportion the terms are in proportion...

...gJÏ-i-7 In a similar manner we may prove that : a : с = b : cZ, b : d = a : c, с : d = a : b, etc. 297. In any proportion the terms are in proportion by Alternation;...third, as the second term is to the fourth. Let a : Ь = с : d. Then, by Art. 293, ad=bc. Whence, by Art. 296, a : с = b : d. 298. In any proportion...

...a:b = c:d. In a similar manner it may be proved that a : c = b : d, PROPOSITION III. THEOREM. 245. In any proportion the terms are in proportion by ALTERNATION...the third as the second term is to the fourth. Let a : 6 = c : d. Then by §242, ad = 6c. Whence by § 244, a : c = 6 : d. PROPOSITION IV. THEOREM. 246....

...&e. In a similar manner we may prove that : a : с = b : d, b : d = a : c, с : d = a : b, etc. 313. In any proportion the terms are in proportion by Alternation...the third, as the second term is to the fourth. Let a:b = c:d. Then by Art. 309, ad = be. Whence by Art. 312, a:r. = b:d. 314. In any proportion the terms...

...= - , ac . • . I : a = d : c. QED Proposition 4. 286. If four quantities are in proportion, they are in proportion by alternation; that is, the first...to the third as the second term is to the fourth. Hyp. Let a : b = c : d. To prove a : c — b : d. Proof. Since a : b = c : d, . • . ad = be. (281)...

...ae or, - = - . bd That is, a:b = c:d. In like manner, a : c = b : d, PROPOSITION III. THEOREM. 234. In any proportion, the terms are in proportion by...the third as the second term is to the fourth. Let the proportion be a : b — c : d. (1) To prove a : c = b : d. We have from (1), ad = be. (§231.)...

...That is, a:b = c:d. In like manner, a : c = b : d, b : a = d: c, etc. PROPOSITION III. THEOREM. 234. In any proportion, the terms are in proportion by...ALTERNATION ; that is, the first term is to the third as the se.-ond term is to the fourth. Let the proportion be a: b = c:d. (1) To prove a : c = b : d. We have...

...§ 301, a : 6 = с : d. In like manner, we may prove that a : с = b : d, с : d = a : b, etc. 310. In any proportion, the terms are in proportion by...the third as the second term is to the fourth. Let the proportion be a : b = с : d. Then, ad = bc. (§ 306) Whence, a : с = b : d. (§ 309) 311. In...