VALUATION. Under thin haul comes the explanation of a part of the language of proportion which Is much used, and which was once very prominent in English mathematical writings. We refer to such phrases as the following varies ns B—A varies Inversely as n—tho gravitation of particles varies Inversely as the squares of their distances —the time of oscillation of a pendulum varies as the square root of its IVIien we way that one thing varies as another, we mean that there are two variable magnitndes which have this property, that If when the first changes from A to n the second change from a to b, then A to a In the same proportion as a to b. And when we say that one thing varies Inversely as another, we mean that if when the first :Lenges from • to a the second changes from a to b. then— The modes of denoting there laws of connection used to Ito, In English works— These were in fact but modes of writing the equations in a manner which should recognise their existence without obliging ua to think of the particular value of the constant c. According to the equations, if we take tho first, and suppose that A changes into B when a changes Into 6, we see obviously that A÷B is the same as a ÷ b, both being equal to e. And A et a informs us that a+ a is always the same quantity, without saying what it is.
When one quantity varies as both of two others jointly, it means that if either of the second and third mentioned remain constant, the first varies as the other. Thns the price of a quantity of goods varies jointly as the number of things and the price of each. At a given price per article, the whole price varies as the number of things; for a given number of things, the whole price varies as the price of ono. When x varies as y and a jointly, the equation x=cyz is We are rather inclined to regret the complete disappearance of the notatini of variations which has taken place within the last thirty years, though the phraseology is still in some degree of use. It is now usual either to write equations at full length, or to make an equation of the variation itself, which can always be done by a proper choice of units. Thus A cc a, or A=c a, can always be made A= a, if such choice of units be made in which to measure the magnitudea • and a as will make e = 1. This must be done by contriving that A and a shall become unity together. But this, however convenient for mere calcu lation, is likely enough to produce confusion in the mind of the learner, and actually does so In many Instances. It is obvious enough that of two different kinds of magnitude one may vary as the other : thus the height of the barometer (a length) varies as the pressure of the atmosphere on a given surface (a weight). But it is just as obvious that one magnitude cannot be equal to another, unless the two be of the same kind. When therefore a writer on mechanics, with little or no previous explanation about the units employed, states that the weight of a body is its mass multiplied by the force of gravity, or that the pressure on a mass is equal to the mass multiplied by its accele ration, he writes effectively only for a reader who knows the subject already. The weight of a body varies jointly as its mass and the
acceleration which the force of gravity would create in one second. Alter either of these alone, and the weight is altered in the same pro portion. Hence, if w, y be the numbers of units of their several kinds in the weight, the mass, and the acceleration caused by gravity, the equation le = cmy must subsist, where c is a numerical constant depending on the units employed. If the weight which is called 10 (pounds, ounces, or whatever they may be) belong to the mass called 5, when acted on by such gravity as produces an acceleration of 4 (feet, yards, or whatever the unit of length may be) in the time nailed 1 (second, minute, or other unit of time), then 10=c x S x 4, or c So long as the same units of length, time, mass, and weight are employed, the equation ta=4 mg must subsist : change the units, end the constant c must have another value, to be again determined from en instance. When the writer above mentioned says that to = In g, he means, or ought to mean, that it is an agreement between him and his reader that whatever mass may be called 1, and whatever may be meant by I of length and 1 of time, the weight which is called 1 shall be that of the mass 1 acted on by the force of gravity 1. The older writers, who used variations, needed no specifications of this kind, since the actual concretes themselves were the subjects of reasoning, and the variation asserted was true both of the concrete magnitudes and of any system of units which they might adopt. The Introduction of their units was naturally and easily made ; and when variations became equations, the rstndent could not help seeing the introduction of all conditions depending on the mode of measurement. In dropping the notation of variations, our writers passed into that want of distinct explanations of primary terms which was the characteristic of many of the French writers.
Tho beginner must carefully in mind that one quantity does not vary as another, because it varies with that other. A square and its root vary together, but the square does not vary as its root if, for instance, the root be doubled, the square is not doubled, but quad rupled.
It is however most important to remember that when two quantities change together, in any manner whatsoever, the increment of the one varies as the increment of the other very nearly, if both the increments be small, and the more nearly the smaller they are. Thus, if we know that when x has a certain value, the addition of .01 to .r gives an addi tion of .001 to its logarithm, we may be sure that the addition of *01 x It to x will give an addition of .001 x is to the logarithm, very nearly, as long as .01 x h is email.