PROPOSITION III. If four quantities are proportional, they are proportional: (39) 10. By inversion of couplets; 20. By inversion of terms; 30. By inversion of both couplets and terms. By additions and subtractions we have, If four quantities are proportional: 10. The sum or difference of the antecedents is to the sum or difference of the consequents, as either antecedent to its consequent ; 2°. The sum or difference of the terms of the first couplet is to the sum or difference of those of the second, as antecedent to antecedent or as consequent to consequent ; (40) 3o. The sums or differences, or both sums and differences of the terms, taken in the same order, whether homologous or analogous, are proportional. The principle in (40), 1°, may be extended thus: if a:c:: a'c': : a" : c' : : &c., i. e., a : c : : a' : c' and a+a+a+...: c + c' + c " + ... :: a:c:: a': c:: a": c" :: &c., PROPOSITION V. If any number of couplets have the same ratio: The sum of all the antecedents is to the sum of all the consequents, as any one antecedent to its consequent. We should also have (41) ±a±a±a"±...: ±c±c'±c" ±... :: a : c :: a' : c' :: &c., If four quantities are proportional the product of the (42) extremes is equal to the product of the means. Cor. To change an equation into a proportion, make any (43) two factors into which one member may be resolved the extremes and the factors of the other member of the equation the means of the proportion; or, to read an equation as a proportion, begin and end the reading in the same member. where m may be any quantity whatever, whole or fractional; .. PROPOSITION VII. If quantities are proportional, their like powers, or roots, (44) or powers of roots, are proportional. Again } =r' • ma' { ma = r • mc, } and { ma =1 (10) and (20) na' =r• nc' (5°) nc=r' • nc'; where m and n may be any whatever, whole or fractional; .. PROPOSITION VIII. A proportion is not destroyed by taking equal multiples (45) or submultiples of homologous or analogous terms. Thus, since we have and From we have or 4:12 8: 24, 16: 24:: 32: 48, [eq. mult. of anal, terms], : 1 3 1:3, [eq. submult. homol. terms]. a = rc, a (1m)=r⚫ (1±m) c, PROPOSITION IX. If any two quantities be increased or diminished by (46) equal multiples or submultiples of those quantities, the sums or differences thus formed will be proportional to the quantities themselves. Def. 4. Four quantities are said to be reciprocally proportional when the ratios of the couplets are the reciprocals of each other. Thus if a, c; a', c', are reciprocally proportional. and a = rc a' = +c'; then { But from the same equations, we have a = rc, Def. 5. Four quantities are inversely proportional when the first is to the second as the fourth to the third. Def. 6. If a b::b:c::cd::, &c., then a, b, c, d, ..., are said to be in continued proportion, or to constitute a geometrical progression; c is said to be a third proportional to a and b, and b is called a mean proportional between a and c. Since from ab :: b: c, &c., we have multiplying together the 1st and 2d, the 1st, 2d and 3d, &c., there results r2 a = c, ... ... ... 9 pr-1a = 1, 7 being the nth term ; PROPOSITION X. In a continued proportion we have the 1st term to the (47) 3d as the square of the 1st to the square of the 2d, or as the square of the second to the square of the 3d, &c.; and the 1st to the nth as the (n − 1)th power of the first to the (n − 1)th power of the 2d, &c. Cor. The last or nth term is equal to the first multiplied (48) by the ratio raised to a power one less than the number of the term [n-1]. It will be observed that the series is here supposed to be ascending, or that a <b< c... ; if it be descending, or a >b> c..., then r will be less than unity-and in all cases By addition we infer that r(a+b+c+...+ k) = b+c+d+...+l, we may write every proportion as a fractional equation—whereby the many and all the changes that can be rung on proportions, will be reduced to very simple operations on equations. VARIATION. Def. 7. We shall often have occasion to consider quantities, not as proportional simply, but as passing by inappreciable degrees through all magnitudes compatible with certain conditions. Such quantities are denominated VARIABLE—and are represented by the last letters of the alphabet, as x, y, z, while the first letters are used to indicate quantities regarded as constant, or such as are independent of the variables. Thus, that y varies as x, a being their constant ratio, is expressed by y = ax. In this equation it is to be understood that x, in passing from any one given value to another, is regarded as passing through all intermediate values while a remains unchanged; and that consequently y changes, taking new values depending upon the value of x. On this account r is called the independent, and y the dependent variable. Thus, if the rate of interest, r, be constant, and the principal, p, be also constant, we shall have a given sum pr, as the interest for one year on the given principal; then if y be the interest for the time x in years, there will result the variation y = prx. This manner of looking upon quantities, not so much as known and unknown, as constant and variable, is as important as it is simple. |