 | Mathematics - 1836 - 488 pages
...How — the ratio of lesser inequality ? If, in several couplets the ratios are equal, what ratio has the sum of all the antecedents to the sum of all the consequents ? How ¡я a ratio of greater inequality affected by adding the same quantity to both terms ? How —... | |
 | James Bryce - Algebra - 1837 - 322 pages
...ôrta : : d : d±c, and b±a:b::d±c:d. 178. V. When any number of quantities are proportionals, as one antecedent is to its consequent, so is the sum of...antecedents to the sum of all the consequents. Let there be any number of proportionals, a:b::c:d::e:f; then ad=bc, and afz=be; also ab = ba; therefore,... | |
 | John Hind - Algebra - 1837 - 584 pages
...+ d +/+ &c.) = b(a + c + e + &c.): , , aa + c + e + &c. whence, - = : b 6 + d+/+&c. that is, as one antecedent is to its consequent, so is the sum of...the antecedents to the sum of all the consequents. Similarly, if a : b = b : c = c : d = &c., we shall have aa + b + c + &c. 6 b + c + d + &c. ' (J f*... | |
 | Andrew Bell - Euclid's Elements - 1837 - 290 pages
...10). PROPOSITION XII. THEOREM. If any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so is the sum of all the antecedents to that of the consequents. If A:B::C:D,andC:D::E:F;A:B:':A + C + E: B + D + F. nB + wD + «F ; and if... | |
 | Euclides - 1838 - 264 pages
...than proportional. PROP. V. THEOR. If any number of homogeneous magnitudes be proportional, as one antecedent is to its consequent, so is the sum of...the antecedents to the sum of all the consequents. First, let there be four magnitudes, A being one antecedent, B its consequent ; C the other antecedent,... | |
 | John Radford Young - 1839 - 332 pages
...&c. ' aei, &c. : bfk, &c. : : cgl, &c. : dhm, &c. THEOREM 7. In any number of equal ratios, as one antecedent is to its consequent, so is the sum of...the antecedents to the sum of all the consequents. Т Ï /• • Let the ratios be - = - = - = &c. : Put - = a ; ace a then b — aq, d = cq, f= eq,... | |
 | Bewick Bridge - Algebra - 1839 - 280 pages
...proportional quantities, a : b :: c : d :: e :f:: g : h &c., then will the first be to the second as the sum of all the antecedents to the sum of all the consequents." For, since ab=ba, and (by THS. 1 and 5)ad=bc,af=be,ah= bg, &c-, we have ab+ad+af+ah+&!.c.—ba+bc+be+bg+f&c.,... | |
 | Euclides - 1840 - 192 pages
...ABC : FGH : : ACD : FHK : : ADE : FKL ; and as one of the antecedents is to one of the consequents, so is the sum of all the antecedents to the sum of all the consequents, that is to say, the similar triangles have to one another the same ratio as the polygons (v. Prop.... | |
 | John D. Williams - Algebra - 1840 - 216 pages
...proportional quantities, a : b : '• c : d : : e : f: : g : h, &c., then will the first be to the second as the sum of all the antecedents to the sum of all the consequents. For, since ab=ba, and (by theorems 1 and 5,) ad = 4c, af=be, ah=bg, &c., we have a4+iM?+a/+aA+&c.=4a+4c+4e... | |
 | John D. Williams - Algebra - 1840 - 634 pages
...proportional quantities, a : b : : с : d : : e : f : : g : h, Sax,., then will the first be to the second as the sum of all the antecedents to the sum of all the consequents. For, since a6=ba, and (by theorems 1 and 6,) ad = be. а с then -j-^ о а , m eg • hence —=-jj,OTae:bf::cg:t... | |
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When any number of quantities are proportionals, as one antecedent is to its consequent, so is the sum of all the antecedents to the sum of all the consequents. 
