ON MEASUREMENT. 190. DEF. To measure a quantity of any kind is to find how many times it contains another known quantity of the same kind. Thus, to measure a line is to find how many times it contains another known line, called the linear unit. 191. DEF. The number which expresses how many times a quantity contains the unit, prefixed to the name of the unit, is called the numerical measure of that quantity; as 5 yards, etc. 192. DEF. Two quantities are commensurable if there be some third quantity of the same kind which is contained an exact number of times in each. This third quantity is called the common measure of these quantities, and each of the given quantities is called a multiple of this common measure. 193. DEF. Two quantities are incommensurable if they have no common measure. 194. DEF. The magnitude of a quantity is always relative to the magnitude of another quantity of the same kind. No quantity is great or small except by comparison. This relative magnitude is called their Ratio, and this ratio is always an abstract number. When two quantities of the same kind are measured by the same unit, their ratio is the ratio of their numerical measures. 195. The ratio of a to b is written is meant : How many times b is contained in a; or, what part a is of b. a or ab, and by this b. I. If b be contained an exact number of times in a their ratio is a whole number. If b be not contained an exact number of times in a, but if there be a common measure which is contained m times in a II. If a and b be incommensurable, their ratio cannot be exactly expressed in figures. But if b be divided into n equal parts, and one of these parts be contained m times in a with a remainder less than part of b, then is an approximate. 1 n m Again, if each of these equal parts of b be divided into n equal parts; that is, if b be divided into n2 equal parts, and if one of these parts be contained m' times in a with a remainder less than 1 222 α m' part of b, then is a nearer approximate value of the ratio correct within " b 222 1 n2 By continuing this process, a series of variable values, m' m" n n2 28, etc., will be obtained, which will differ less and α less from the exact value of We may thus find a fraction which shall differ from this exact value by as little as we please, that is, by less than any assigned quantity. Hence, an incommensurable ratio is the limit toward which its successive approximate values are constantly tending. ON THE THEORY OF LIMITS. 196. DEF. When a quantity is regarded as having only one or few definite values, it is called a Constant; but, when it is regarded, under the conditions imposed upon it, as having an indefinite number of different values, it is called a Variable. 197. DEF. When it can be shown that the value of a variable, measured at a series of definite intervals, can by indefinite continuation of the series be brought to differ from a given constant by less than any assigned quantity, without ever exceeding the constant, that constant is called the Limit of the variable, and the variable is said to approach indefinitely to its limit. If the variable be increasing, its limit is called a superior limit; if decreasing, an inferior limit. Α 198. Suppose a point 4 M M M B to move from A toward B, under the conditions that the first second it shall move one-half the distance from A to B, that is, to M; the next second, one-half the remaining distance, that is, to M'; the next second, one-half the remaining distance, that is, to M", and so on indefinitely. Then it is evident that the moving point may approach as near to B as we please, but will never arrive at B. For, however near it may be to B at any instant, the next second it will pass over one-half the interval still remaining; it must, therefore, approach nearer to B, since half the interval still remaining is some distance, but will not reach B, since half the interval still remaining is not the whole distance. Hence, the distance from A to the moving point is an increasing variable, which indefinitely approaches the constant A B as its limit; and the distance from the moving point to B is a decreasing variable, which indefinitely approaches the constant zero as its limit. If the length of AB be two inches, and the variable be denoted by x, and the difference between the variable and its variable etc., is evidently less than 2; but by taking a great number of terms, the sum can be made to differ from 2 by as little as we please. Hence 2 is the limit of the sum of the series, when the number of the terms is increased indefinitely; and 0 is the limit of the variable difference between this variable sum and 2. lim. will be used as an abbreviation for limit. 199. [1] The difference between a variable and its limit is a variable whose limit is zero. [2] If two or more variables, v, v', v', etc., have zero for limit, their sum, v + v' + v'', etc., will have zero for a limit. [3] If the limit of a variable, v, be zero, the limit of a ± will be the constant a, and the limit of a × v will be zero. [4] The product of a constant and a variable is also a variable, and the limit of the product of a constant and a variable is the product of the constant and the limit of the variable. [5] The sum or product of two variables, both of which are either increasing or decreasing, is also a variable. PROPOSITION I. [6] If two variables be always equal, their limits are equal. Let the two variables AM and AN be always equal, and let A C and A B be their respective limits. A B Since A M approaches indefinitely to c A C, we may suppose that it has reached a value A P greater than A C'. Let AQ be the corresponding value of A N. But both of these equations cannot be true, for A P> A C', and A Q<A B. .. A C cannot be greater than A B. Again, suppose ACA B. Then we may diminish A B to some value A B' such that A CA B'. Since AN approaches indefinitely to AB we may suppose that it has reached a value A Q greater than A B'. Let A P be the corresponding value of A M. AP AQ. But both of these equations cannot be true, for A P< A C, and AQA B'. .. A C cannot be less than A B. Since AC cannot be greater or less than AB, it must be equal to A B. [7] COROLLARY 1. If two their limits are in the same ratio. Q. E. D. variables be in a constant ratio, For, let x and y be two variables having the constant ratio r, then r, or, x=ry, therefore y lim. (x) = lim. (r y) = r × lim. (y), therefore [8] COR. 2. Since an incommensurable ratio is the limit of its successive approximate values, two incommensurable ratios α a' and are equal if they always have the same approximate values b' PROPOSITION II. [9] The limit of the algebraic sum of two or more variables is the algebraic sum of their limits. a Let x, y, z, be variables, a, b, and c, their respective limits, and v, v', and v', the variable differences between x, y, z, b and a, b, c, respectively. We are to prove lim. (x + y + z)=a+b+c. с - Then, x + y + z = a − v + b − v + c —v". .. lim. (x+y+2)=lim. (a−v+b-v'+c-v'). But, lim. (a- v + b − v + c − v") = a+b+c. .. lim. (x + y + 2) = a + b + c. PROPOSITION III. [10] The limit of the product of two or more variables is the product of their limits. Let x, y, z, be variables, a, b, c, their respective limits, and v, v', v', the variable differences between x, y, z, and a, b, c, respectively. We are to prove lim. (x y z) = a b c. Now, x=α- v, y=b-v', z = c — v''. Multiply these equations together. Then, xyz=abc terms which contain one or more of the factors v, v', v', and hence have zero for a limit. .. lim. (x y z) But lim. (a b c = lim. (a b c ‡ terms whose limits are zero). For decreasing variables the proofs are similar. Q. E. D. [3] NOTE. In the application of the principles of limits, reference to this section (§ 199) will always include the fundamental truth of limits contained in Proposition I.; and it will be left as an exercise for the student to determine in each case what other truths of this section, if any, are included in the reference. |