...oblique cone with a circular base. 5 66 The same proportion is true for every other element ; and since the sum of the antecedents is to the sum of the consequents as any one antecedent is to its consequent, we have I. Ad' : I. dc :: AD : DC; in which the symbol 2* is used to denote the sum of all the terms,...

...their difference as the sum of the third and fourth is to their difference. COR. 3. — In any number of equal ratios, the sum of the antecedents is to...consequents as any one antecedent is to its consequent. riH a + e + e + &c. a „ . . , , 5 -2 — - — = r = r = &c. (Art. 200.) 6 + Й+/+&С. 6 369. These...

...circle as the conjugate semi-axis is to the transverse semi-axis. VIII. HIGHER ALGEBRA. 1. Prove that in a series of equal ratios, the sum of the antecedents is to the sums of the consequents as any one antecedent is to its consequent. 2. Insert three arithmetical, three...

...the equation. Then thatis, bd a~bc~d bd or, a — b : b : : с — d : d. QED PROPOSITION VIII. 266. 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. Let a : b = с : d = e :/=<7 : h. We are to prove a + c+e + g: b +...

...obtained by: VI. Composition. a + c: c : : b -\- d : d. VII. Division. a — c:c::b — d:d. ' 350. 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. r may be put for each of these ratios. Then " = r, l=r, l=r, ?-=r,...

...£ _ i = 1 _ i b ' ' d l> that is, -, bd or, a — b : b : : c — d : d. QED PROPOSITION VIIL 266. In a series of equal ratios, the sum of the antecedents is to the sum of tlie consequents as any antecedent is to its consequent. Let a : b = c : d = e :f — g : h. We are...

...on \ Then or a ± £- a : b ± £- b :: a : b. QED THEOREM XIII. 168. In any continued proportion, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. Let a : b :: c : d :: e : f :: g : h. To prove that a -\- c -\- e...

...QED PROPOSITION VIII. 266. In a series of equal ratios, of which all the terms are of the same kind, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. Let a :6 = c : d — e : f = g : h. We are to prove -a + c+e + g:...

...-\- nc : d + nd. 4. State ' 2' and " 3' in general terms. r THEOREM XII. In a continued proportion, the sum of the antecedents is to the sum of the consequents as any antecedent is to its corresponding consequent. Let a : b :: c : d :: e : f :: g : h ; then will a +...

...G.— 8. Then (Theo. IlI), (a + c + e + g):b+d a : b ; that is, in a set of continued proportionals, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. Cor. — If any number of fractions are equal each to each, the sum...