251. To measure a quantity of any kind is to find times it contains another known quantity of the san Thus, to measure a line is to find how many ti tains another known line, called the linear unit. The number which expresses how many times contains the unit, joined with the name of the uni the numerical measure of that quantity; as, 5 yard 252. The magnitude of a quantity is always rel magnitude of another quantity of the same kind. N is great or small except by comparison. This relat tude is called their ratio, and is expressed by th quotient of their numerical measures when the sar measure is applied to both. The ratio of a to b is written α b' or a b. 253. Two quantities that can be expressed in terms of a common unit are said to be commensur common unit is called a common measure, and eac is called a multiple of this common measure. Thus, a common measure of 21 feet and 32 fe foot, which is contained 15 times in 21 feet, and 3 feet. Hence, 2 feet and 3 feet are multipl foot, 2 feet being obtained by taking 3 by taking of a foot 22 times. of a foot 15 254. When two quantities are incommensurab have no common unit in terms of which both quanti expressed in integers, it is impossible to find a fr will indicate the exact value of the ratio of the giv ties. It is possible, however, by taking the unit. small, to find a fraction that shall differ from the Thus, suppose a and b to denote two lines, such that α If, then, a millionth part of b be taken as the unit, the value of the ratio lies between 1118818 and 1111314, and there1414218 fore differs from either of these fractions by less than 1000000 b By carrying the decimal farther, a fraction may be found. that will differ from the true value of the ratio by less than a billionth, a trillionth, or any other assigned value whatever. Expressed generally, when a and b are incommensurable, and b is divided into any integral number (n) of equal parts, if one of these parts is contained in a more than m times, but less than m+1 times, then The error, therefore, in taking either of these values for made to decrease indefinitely, and to become less than any assigned value, however small, though it cannot be made absolutely equal to zero. Hence, the ratio of two incommensurable quantities cannot be expressed exactly by figures, but it may be expressed approximately within any assigned measure of precision. 255. The ratio of two incommensurable quantities is called an incommensurable ratio; and is a fixed value toward which 256. THEOREM. Two incommensurable ratios are when the unit of measure is indefinitely diminished, proximate values constantly remain equal. Let a:b and a': b' be two incommensurable ratios, wh m values lie between the approximate values an n when the unit of measure is indefinitely diminished 1 they cannot differ so much as n. Now the difference (if any) between the fixed val and a': b', is a fixed value. Let d denote this differen esis can be indefinitely diminished, can be made less t Therefore d cannot have any value; that is, d= there is no difference between the ratios a: b and a': b fore a ba' : b'. : THE THEORY OF LIMITS. 257. When a quantity is regarded as having a fix throughout the same discussion, it is called a consto when it is regarded, under the conditions imposed up having different successive values, it is called a variab When it can be shown that the value of a variable, m at a series of definite intervals, can by continuing th be made to differ from a given constant by less t assigned quantity, however small, but cannot be ma lutely equal to the constant, that constant is called t of the variable, and the variable is said to approac nitely to its limit. If the variable is increasing, its limit is called a limit if decreesing en inferior limit. 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 AB 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 limit, by v: Now the sum of the series 1+1+1+1, etc., is 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 difference between this variable sum and 2. CCESIUVI པཿསv IVP་་མས Vཔཔཔ་པ 10+ 180 + 1000 + 10000+ However great the number of terms of this series the sum of these terms will be less than ; but t terms we take the nearer does the sum approach. the sum of the series, as the number of terms is in approaches indefinitely the constant as a limit. 258. In the right triangle ACB, if the vertex A ap] indefinitely the base BC, the angle B diminishes, and approaches zero indefinitely; if the vertex A moves away from the base indefinitely, the angle B increases and approaches a right angle indefinitely ; but B cannot become zero or a right angle, so long as ACB is a triangle; for if B be- Ba comes zero, the triangle becomes the straight line BC B becomes a right angle, the triangle becomes two lines AC and AB perpendicular to BC. Hence the v B must lie between 0° and 90° as limits. E 259. Again, suppose a square ABCD inscribed in a and E, F, H, K the middle points of the arcs subten the sides of the square. If we draw the straight lines AE, EB, BF, etc., we shall have an inscribed polygon of double the number of sides of the square. The length of the perimeter of this polygon, represented by the dotted lines, is greater than that of the square, since two sides replace each K H side of the square and form with it a triangle, and tw of a triangle are together greater than the third side; b than the length of the circumference, for it is made |