UNIFORM. Though this word mean nothing more than "of one form," it has a signification in mathematics which might be better rendered by "of one value" or one degree," when we speak to the mathematical proficient. But it is a convenience, though only an accidental one, that the word does not imply the idea of value abso lutely; a circumstance which may serve us to elucidate a point of great importance in the differential calculus. The commencement is made in the present article the continuation will follow in VELOCITY.
In order to understand any application of mathematics, whether to space or matter, It is necessary that a perfect conception should be formed of the quality of apace or matter which is to come under consideration. By a perfect mathematical conception, we mean that it must be distinctly seen, first, that the object under considera tion is of the nature of magnitude ; secondly, that it is of a measurable kind, that is, is capable of being measured, and can actually have a mode of measuring it assigned. Why do so many persona talk and write vaguely about force, velocity, density, acceleration, &c. Simply because they are only conversant with the first consideration, and have no precision in their Ideas of the second : they feel that they are speaking of magnitudes, of things which they know may be snore or less, but they have not that familiarity with the precise way of ascer taining the how much more or the how much less, without which deduc tion cannot be made intelligible.
Now we say that in every instance in which measurement is shown to be attainable, there is a notion of uniformity which precedes or ought to precede that of mensurability ; and that emphatic mention of this circumstance, and full development of its truth and meaning, ought to be the preliminary step to actual measurement. Moreover, we say that, inasmuch as this idea of uniformity is to be gained pre viously to that of measurement, we must forego the notion of " uni form and of one value " being convertible terms, and illustrate the word by considerations independent of value ; for this last term implies measurement, as is easily seen.
If we were to take velocity as our instance, most readers would be able to appeal to ideas of measurement and value established in their minds, whether vaguely or precisely : we therefore prefer to choose curvature, a term which will be quite new as meaning a measurable magnitude to all except those who have more than an elementary knowledge of mathematics. Curvature is, as the name imports, the bending, the gradual bending, which distinguishes a curve from a straight line. It is a magnitude, that is, it allows of the application of the idea of more and less : one curve may bend more than another, or more in one place than in another. So much every one can be aura of at the first announcement : the next step would be to imagine it passible, that one curve might, say at and about a point A, bend exactly twice as much as another at and about a point B. But here
the ordinary reader can only imagine a poasibility : no distinct criterion will at once present itself for determining what proportion the bendings or curvatures of two curves are to be stated as having to one another at two given points. If two tangents be drawn at the two given points, it is obvious that, according as the curve bends more or less, there will be more or less deflection from the tangent. Thus the curve A I., at the point A, has as much curvature as A @, or more ; certainly not less. Now as in other cases, if we measure curvature, it must be by curva ture, as length by length, weight by weight, &o. ; and as a preliminary, it will be desirable to have that curve which has everywhere the same curvature. This curve is obviously a circle, which is throughout its circumference bent in exactly the same manner. Those who cannot imagine how curvatures are to be measured can always ace this much, that a true mode of measurement will give the same result to what ever point of a given circle it may be applied. A method of deter mining value must be false which gives at one point of the same circle a greater curvature than at another. Here we say that any one may see that a notion of uniformity has a useful existence previously to that of any mode of comparing the values of different cases of this uniformity. The circle A may have a radius twice as large as that of : are we then to say that the curvature of B is double that of a ? That the smaller circle bends most is certain ; whence it is equally certain that curvature or bending is a magnitude : it has its more and less. Again, it is obvious that the circle a has the same curvature in all its parts, and that the circle a has the same ; though the parts of A have a curvature which is not the same as that of the parts of n. Hence it is certain that uniformity of curvature is perfectly conceivable. Now what we have to enforce is, that all this takes place in the mind, before any mode can be given of answering the question how much the curvature of B exceeds that of A. The greater the radius the less the curvature, and A has twice as great a radius as B. If it be proper to say (VARIATION] that the curvature varies inversely as the radius, then n is twice as much curved as A; but if it be proper to say that the curvature varies inversely as the square of the radius, then that of B is four times as great as that of A. Here the object of this article ends, and we have referred to VELOCITY the manner of making the next step. At the risk of undue repetition, we state again, that a perfect idea of a magnitude, as a magnitude, and of ita uniformity, or total absence of change of value, may exist in cases in which the accurate comparison of values, or measurement, is not attained, and may even exist in a mind which has not the means of conceiving the possibility of such comparison or measurement being accurately made,