one is π, what is the area of a circle whose radius is r? What, when r = 4? 11. The surfaces of spheres are to each other as the squares of their diameters. If the surface of a sphere whose radius is one is 4, what is the surface of a sphere whose radius is r? What, when r=5? 12. The volumes of spheres are to each other as the cubes of their radii. If the volume of a sphere whose radius is one is 3 π, what is the volume of a sphere whose radius is? What, when r = 6 ? Surfaces and volumes that have the same shape are similar. To have the same shape, they must have their corresponding angles equal and their corresponding dimensions proportional. 13. Similar surfaces are to each other as the squares of their like dimensions. If a field a rods long contains m acres, what will a similar field c rods long contain? 14. Similar volumes are to each other as the cubes of their like dimensions. If a keg whose bung diameter is c inches holds n gallons, what will a similar keg d inches in bung diameter hold? 15. The quantities of water that flow through circular pipes are to each other as the squares of the diameters of the pipes. If c gallons flow through a pipe m inches in diameter in one minute, how many gallons will flow through a pipe n inches in diameter in the same time? Limiting Ratios. Definitions and Principles. 309. A quantity that retains the same value throughout an operation or discussion is a constant. 310. A quantity that continuously changes its value— that is, passes from one value to another by successively assuming all values lying between them-is a variable. Illustration.-A line a foot long is a constant. A line traced by a point moving according to some well-defined law is a variable. 311. A finite unit is a unit of comprehensible size or value. 312. A quantity that can be expressed in finite units is a finite quantity. 313. A quantity too small to be expressed in finite units is said to be infinitely small. An infinitely small variable is called an infinitesimal, and may be expressed by the character, read an infinitesimal or zeroid. 314. A quantity too large to be expressed in finite units is said to be infinitely large. An infinitely large variable is called an infinite, and may be expressed by the character x, read an infinite. 315. The entire absence of quantity is called zero, and is expressed by the character 0, read zero. 316. The unlimited whole of quantity, or rather unlimited quantity, is called infinity, and is expressed by the character, read infinity. х 317. If, in the fraction x decreases by a constant α ratio until it becomes an infinitesimal and a remains a finite constant, the value of the fraction decreases in the same ratio [P. 55], and becomes an infinitesimal. Therefore, Prin. 123. a An infinitesimal divided by a finite constant is an infinitesimal. Prin. 124. Xa. An infinitesimal multiplied by a finite constant is an infinitesimal. 319. Since Xa, it follows that, Prin. 125. ==a. An infinitesimal divided by an infinitesimal may be any finite constant. α 320. If, in the fraction, a increases by a constant ratio until it becomes an infinite and a remains a finite. constant, the value of the fraction increases in the same ratio [P. 54], and becomes an infinite. Therefore, finite constant is an infinite. 322. Since x X a = x, it follows that, Prin. 128. α - α = a. An infinite divided by an infinite may be any finite constant. 323. If, in the fraction, a decreases by a constant х ratio until it becomes an infinitesimal and a remains a finite constant, the value of the fraction increases in the same ratio [P. 54], and becomes an infinite. Therefore, Prin. 129. a Ξα, A finite constant divided by an infinitesimal is an infinite. α 324. Since = x, it follows that, Prin. 130. • × α = a. The product of an infinitesimal and an infinite may be any finite constant. 325. Since X ∞ = a, it follows that, 326. Since and X are each satisfied by any finite constant, they are symbols of indetermination. 327. The limit of a variable is a value which the variable continually approaches but which it can never reach, but may be made to differ from it by less than any assignable quantity. Illustration. If a point starts at A in the direction of B, and goes 1/2 the distance the first second, 1⁄2 the remaining distance the next, 1/2 the remaining distance the third, and so on, the distance passed A B over constantly approaches the distance from A to B, and will eventually differ from this distance by an infinitesimal, but it can never equal this distance. From A to B is therefore the limit of the distance the point can go. 328. The limit of a variable that decreases by a constant ratio is zero. Illustration. If 1⁄2 a line be cut off, then 11⁄2 the remainder, and so on indefinitely, the part retained continually approaches zero, from which it will eventually differ by less than any assignable quantity. Therefore, zero is the limit of the remainder. 329. A variable quantity that increases by a constant ratio has no limit. This fact is sometimes expressed by saying that its limit is infinity. 330. A function of a variable quantity is any expression that contains the variable. Thus, a b is a function of x. 331. A function of a variable is generally a variable also. It is then called the dependent variable, and the variable upon which it depends the independent variable. 332. The limit of a function, when the independent variable approaches its limit, may be zero, infinity, or a finite quantity. Illustration.-1. If x approaches a as a limit, the func 2. If x approaches a as a limit, the function. α approaches, or∞ as a limit. 3. If x approaches a as a limit, the function 333. Sometimes a factor whose limit is zero is common to both terms of a fraction. The limit of the fraction will then assume the irreducible form 0 The true limit is then found by removing the common factor before passing to the limit. Illustration.—If x approaches a as a limit, the func a3 tion x2 a2 0 approaches as a limit. This form results, because the common factor a moving this factor, we a has 0 for its limit. Rex2 + a x + a2 have which has for |