If any number of quantities are proportional, any antecedent is to its consequent as the sum of all the antecedents is to the sum of all the consequents. Let a : b = c : d = e :f Now ab = ab (1) and by Theorem I. An Elementary Geometry - Page 30by William Frothingham Bradbury - 1872 - 110 pagesFull view - About this book
| Benjamin Greenleaf - Geometry - 1862 - 532 pages
...remaining terms will be in proportion. PROPOSITION XI. — THEOREM. 147. If any number of magnitudes **are proportional, any antecedent is to its consequent...all the consequents. Let A : B : : C : D : : E : F** ; then will A:B::A+C + E:B + D + F. For, from the given proportion, we hate AXD = BXC, and AXF = BX... | |
| Benjamin Greenleaf - Geometry - 1863 - 504 pages
...remaining terms will be in proportion. PROPOSITION XI. — THEOREM. 147. If any number of magnitudes **are proportional, any antecedent is to its consequent...all the consequents. Let A : B : : C : D : : E : F** ; then will A:B::A + C + E:B + D + F. For, from the given proportion, we have AXD = BXC, and AXF =... | |
| Benjamin Greenleaf - Geometry - 1863 - 502 pages
...remaining terms will be in proportion. PROPOSITION XI. — THEOREM. 147. If any number of magnitudes **are proportional, any antecedent is to its consequent...all the consequents. Let A : B : : C : D : : E : F;** then will A : B : : A + C + E : B + D + F. For, from the given proportion, we have AXD = BXC, and AXF... | |
| Benjamin Greenleaf - 1863 - 338 pages
...: : с : d. THEOREM X. 324 1 If any number of quantities are proportional, any antecedent is to ils **consequent as the sum of all the antecedents is to the sum of all the consequents. Let a : b** : : с : d : : e : f; then a : b : : a -|- с -f- e : b -f- d -J- f. For, by Theo. I., , ad = bc, and... | |
| Horatio Nelson Robinson - Algebra - 1863 - 432 pages
...If there be a proportion, consisting of three or more equal ratios, then either antecedent will be **to its consequent, as the sum of all the antecedents is to the sum of all the consequents.** Suppose a : Ь = с : d — e : f= g : h =, etc. Then by comparing the ratio, a : b, first with itself,... | |
| Evan Wilhelm Evans - Geometry - 1862 - 116 pages
...: B : : .£ : R. 2 2 7. By composition, implies that if any number of magnitudes are proportionals, **the sum of all the antecedents is to the sum of all the consequents** as . any one antecedent is to its consequent. Thus, If A : B : : C : D : : E : F, Then A+C+E : B+D+F... | |
| Benjamin Greenleaf - Algebra - 1864 - 336 pages
...ratios, ^ = e-, , ce and -d=-f. Therefore, by Art. 38, Ax. 7, | — ^ or, a : b : : c : d. THEOREM X. 324 **1 If any number of quantities are proportional, any...all the consequents. Let a : b : : c : d : : e : f;** then a : b : : a-\-c-\-e : b -\-d-\- f. For, by Theo. I., od = bc, and af= be ; also, ab = ba. Adding,... | |
| Benjamin Greenleaf - Algebra - 1864 - 420 pages
...d. _ae , ce .. ac For, - = -; , and - = — ; therefore r = -7 ; whence, a : i : : c : e?. 319. -Jf **any number of quantities are proportional, any antecedent...antecedents is to the sum of all the consequents.** If a : b : : c : d : : e :/, then a : b : : a-\-c-\-e : b-\-d-\-f. For, by Art. 311, ad = be, and af... | |
| Horatio Nelson Robinson - Conic sections - 1865 - 474 pages
...: Q. THEOREM VII. If any number of magnitudes are proportional, any one of the antecedents will be **to its consequent as the sum of all the antecedents...the sum of all the consequents. Let A, B, C, D, E,** etc., represent the several magnitudes which give the proportions A : B :: C : D A : B :: E : F A :... | |
| Paul Allen Towne - Algebra - 1865 - 314 pages
...mq = np; whence am X dy = bn X cp, or am : bn :: cp : dq. (14) PROP. IX. In a continued proportion, **the sum of all the antecedents is to the sum of all the consequents** as any one antecedent is to its consequent. (Vide § SS16, def. ,7.) For, since a : b : : c : d, we... | |
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