 | Mathematics - 1860 - 294 pages
...cy-j-az ax-\-by-\-cz a — bb — cc — aa -f- 5 -I- e t ions = — . I Since these ratios are equal, any antecedent is to its consequent as the sum of...the antecedents is to the sum of all the consequents ; therefore either fraction equals the sum of all the numerators divided by the mm of all the denominators,... | |
 | Horatio Nelson Robinson - Geometry - 1860 - 470 pages
...magnitudes are proportional, any one of the antecedents will be to its consequent as the sum of all tht antecedents is to the sum of all the consequents. Let A, B, C, D, JB, etc., represent the several inagm tudes whi )h give the proportions To which we may annex the identical... | |
 | Charles Hutton - Mathematics - 1860 - 1020 pages
...THEOREM I.XXIl. If any number of quantities be proportional, then any one of tne antecedent* "¡/I be to its consequent, as the sum of all the antecedents, is to the aim of ¡ли the consfqnents. Let А:В::тА:тпВ::пЛ:пВ, &С.; then will Л : JÎ : : Л -f... | |
 | Elias Loomis - Conic sections - 1861 - 244 pages
...are proportional, any one ante e&dent is to its consequent's the sum of all the antecedents* i& t& the sum of all the consequents. Let A : B : : C : D : : E : F, &c. ; then will A : B : : A+C+E : B+D+F For, since A : B : : C : D, we have AxD=BxC. And, since A :... | |
 | Benjamin Greenleaf - Geometry - 1862 - 532 pages
...Therefore, by alternation (Prop. VI.), if two proportions have the two antecedents or the two consequents the same in both, the remaining terms will be in proportion....+ E:B + D + F. For, from the given proportion, we hate AXD = BXC, and AXF = BX E. By adding AXB to the sum of the corresponding sides of these equations,... | |
 | Benjamin Greenleaf - Geometry - 1861 - 638 pages
...antecedent and its consequent the same in both, the remaining terms will be in proportion. scquents the same in both, the remaining terms will be in proportion....consequents. Let A : B : : C : D : : E : F ; then will A:B::A + C + E:B + D + F. For, from the given proportion, we hare AXD = BXC, and " AXF = BX E. By adding... | |
 | Benjamin Greenleaf - Geometry - 1863 - 504 pages
...proportions have an antecedent and its consequent the same in both, the remaining terms will be iu proportion. sequents the same in both, the remaining...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 = BX E. By adding... | |
 | Benjamin Greenleaf - Geometry - 1863 - 502 pages
...antecedent and its consequent the same in both, the remaining terms will be in proportion. sequerits the same in both, the remaining terms will be in proportion....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 = BX E.... | |
 | 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,... | |
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If any number of magnitudes 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; then will A : B : : A + C + E : B + D + F. 
