...difference of the last two terms is to the fourth term. 335. In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. 338. Like powers of the terms of a proportion are in proportion. 342. If a line is drawn through two...

...found sufficient to substitute ra for b and re for d. 245. In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to ifs consequent. we may put r fur each of these ratios. ,,., « xeg I hen, . . = r, — = r, = r, -...

...Division. а — с : с : : Ь — d: d. 295. In a series of equal ratios, the sum of the antecedente is to the sum of the consequents as any antecedent is to us consequent. H-;-f r may be pat for each of these ratios. i---S--7"-i--- .-. a =- br, с=- dr, e...

...each other. Example : BOOK III PROPOSITION X. THEOREM 443. In a continued proportion the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. Let To Prove Proof bdfh b+d+f+hf lj (2i H ^3> a) From (1). 7=7 ^4) H (5) a/= be (6) ] cf=de (7) «/...

...as will he seen in the solution of Ex. 4, Art. 408. 408. In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. •Ei •£ « I' e for, if - = = = •••, b 'I- j we may put each of these equal ratios equal...

...AB + BC + etc. : A'B' + B'C' + etc. = AB : A'B', § 335 (in a series of equal ratios the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent). That is, ' = AB:A'B'. QED Ex. 252. If the line joining the middle points of the bases of a trapezoid...

...equals the product of the extremes, and conversely. 2. In any number of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. 3. If two proportions be multiplied together, term by term, the resulting products will be in proportion....

...the proportion 9: 18=18: 36, we have 18=1/8 x 36. 205. In a, series of equal ratios the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. ampx Suppose that _=_=£.=_== onqy Ti a Let Ъ=г' mpx . . Then — =r. -=r. -—r, etc. Axiom 7. n...

...nth power, whether n is integral or fractional, 6"~d"' XI. In a series of equal ratios the sum of the antecedents is to the sum of the consequents as any antecedent is to its own consequent. If a_c._m_x si*. , . a с mx /г,-. let - = r, - = r, - = r, -=r, (2) bdny clearing...

...ratio a : b ; with the other ratios. PRINCIPLE 13. — In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. Thus, when a:6 = C:d = g: ft and so on, a + с + e + g + - : Ь + d +f+ Л + ••• : : a : Ь....