...: 3 is to the second : : 2 is to the third. Now, we have seen (186) that the sum of the antecedent? is to the sum of the consequents, as any one antecedent is to its consequent : we can then say here, that the sum 9 of the three parts proportional to the parts that we seek, is...

...= dq, e =fq, g = hq. Adding these equations q, or Dividing by 6 + d+f+h a-\-c-4-e-\-gac - ' In any series of equal ratios, the sum of the antecedents is to the sum of the consequents, as any one of the antecedents is to its consequent. 16. If a : b = c : d, and e :f= g : h, that is, — = —...

...antecedents, is to the sum or difference of the consequents, as either antecedent is to its consequent ; the sum of the antecedents is to the sum of the consequents, as the difference of the antecedents is to the difference of the consequents; also, the sum of the, antecedents...

...A4 : A4' : : be : be' : : cd : cd', &c. PLANE SAILING. therefore, since by the theory of proportion the sum of the antecedents is to the sum of the consequents as any one antecedent is to its consequent, A* : A6' : : A4 + be + cd + &c., : A4' + be' + cd' + >fec. But AJ + be + cd + Ac., is the whole distance...

...-f- d : d. Comparing this with the proposed, we derive the following principle : In every proportion, th~e sum of the antecedents is to the sum of the consequents, as any one anetcedent is to its consequent. In like manner it may be shown, that the difference of the antecedents...

...-\-d : d. Comparing this with the proposed, we derive the following principle : In every proportion, the sum of the antecedents is to the sum of the consequents as any one antecedent is to its consequent. Let there be next a series of quantities, a, b, c, d, e, f, g h... forming, two and two, a series of...

...§1O«5. If any number of like magnitudes be proportionals, one antecedent will be to its consequent as the sum of the antecedents is to the sum of the consequents. Let a : Ъ : : с : d : : e :f, then a:b::a-\-c-{-e:b-{-d -)-/• For since a : b : : с : d, and a...

...the proposed, we derive the following principle : In every proportion, the sum of the antecedents u to the sum of the consequents as any one antecedent is to its consequent. Let there be next a series of quantities, a, b, c, d, e, f, g h . . . forming, two and two, a series...

...there be a series of equal ratios represented by f we shall have - « - * - V - ««• T • - Tj Therefore, in a series of equal ratios, the sum of...its consequent. If there be two proportions, as 30 : 1 5 : : 6 : 3, and 2 : 3 : : 4 : 6, then multiplying them term by term we shall have 30 x 2 : 15...

...shall have, A±PA : B±*-B :: C ±2,0 : 2>±^D; PEOPOSITION XI. THEOEEM. In any continued proportion, the sum of the antecedents is to the sum of the consequents, as any antecedent to its corresponding consequent. From the definition of a continued proportion (D. 3), A...