Mathematics. The method of exhaustions is a way of comparing or estimating fixed magnitudes by means of variable magnitudes, which may be brought to approach or exhaust them more nearly than any assigned interval or remainder.

Thus, a regular polygon may be inscribed in a circle, and by repeatedly doubling the multitude of sides, or increasing it by other given laws, this variable figure may be made so to exhaust the circle as to leave a remainder less than any given area. And because similar polygons inscribed in different circles have always the duplicate ratio of the diameters, it has been perceived that the circles themselves must be as these exhaustive polygons, which Euclid proves strictly by the method of exhaustions.

A rectilineal figure may be inscribed in a segment of a parabola, so as to leave less of the parabola than any area assigned, and also to be nearer four thirds of the greatest inscribed triangle than any given difference. Hence Archimedes proves, by the method of exhaustions, that the segment of the parabola exceeds the triangle by one third of the same triangle.

The first lemma of Newton's Principia brings the method of exhaustions to general application. It is in these words:

"Quantitates, ut et quantitatum rations, qua ad aquatilitatem tempore quovis finito constanter tendunt, et ante finem temporis illius propius ad invicem accedunt quam pro data quavis differentia, flunt ultimo aequales."

"Si negas; fiant ultimo inaequales, et sit eorum ultima differentia D. Ergo nequeant propius ad aquatilitatem accedere quam pro data differentia D: contra hypothesin."

Many authors have not rightly understood this lemma, as Horsley in his note upon it; Hutton on Prime and Ultimate Ratios, in his Dictionary; and Woodhouse, Principles of Analytical Calculation, paragraph 105, where it is said to be "merely a definition," and, again, that "it gave a new signification to the term equality."

If our opinion be just, it gives no signification to the term equality which was not given by Euclid and Archimedes. The novelty is in giving to the word ultimate, or the adverb ultimately, an extension beyond what it ordinarily includes; as, in the technical language of the Principia, things are often said to be ultimately what, by their definition, they are incapable of becoming.

In this language, as explained by its author, an ultimate quantity, or that which a quantity ultimately becomes (fit ultimo), is what is otherwise called the limit of the quantity; that to which, repeating the words of this lemma, tempore quovis finito constanter tendit, et ante finem temporis illius proprius accedit quam pro data quavis differentia. Thus, in the Scholium, after the eleventh lemma, our author says, ad ultimas quantitatum evanescunt summam, id est, ad limites summorum.

Few as are the words of the demonstration of this celebrated lemma, it must be concluded from them that it only concerns quantities of which it could be questioned whether they have a fixed difference. Applied to variables, it can therefore only compare them in some simultaneous magnitudes. Among simultaneous magnitudes, Newton, for the sake of brevity, includes limits; and, in fact, only these limits are here compared. Perhaps a slight change in the language may show more clearly the intention.

"Quantities, as also the ratios of quantities, which in any finite time soever constantly tend to equality, and before the end of that time approach nearer than any difference, have equal limits."

"If not, let their limits be unequal, and differ by D. Then the variables, in their approach, cannot come nearer than D, contra hypothesein."

This may be too concise, but is fully made out in the following manner:

Let X and Y be two quantities, continually approaching, and capable of being brought nearer than any difference within a finite time. Let A be the limit of X, and B the limit of Y; that is, let A and B be fixed quantities, to which X and Y, during their approach, may be brought nearer than any difference assignable. Then shall A and B be equal.

For, if not, let A = B + D. And at any time during the approach of X and Y, let m be the difference of X from its limit, and n the contemporary difference of Y from its limit. And,

I. Suppose the variables to approach by the decrease of the greater and the increase of the less towards their respective limits (like a circumscribed and inscribed polygon); or X = A + m, Y = B - n. Then X - Y = A - B + m + n = D + m + n, so that X and Y necessarily differ by more than D, contra hypothesein.

II. Suppose both variables to decrease towards their limits; or X = A + m and Y = B + n. Then X - Y = A - B + m - n = D + m - n.

Now if n were greater than m, we might decrease Y by a part of n, which we will call y, greater than m itself, and still greater than its part (x), the contemporary decrement of X; and putting X' and Y' for the variables thus decreased, we should have X' = X - x, Y' = Y - y, and thence X - Y' = X - Y + (y - x); so that the difference of the variables would be augmented by bringing them nearer to their limits. Therefore n cannot be greater than m; consequently D + m - n cannot be less than D, or the difference of X and Y cannot become less than D, contra hypothesein.

III. It only remains to examine the case where both increase towards their limits, or X = A - m, Y = B - n, and X - Y = A - B + n - m = D + n - m.

Here if m exceeded n, we might increase X by a quantity x greater than n, and still more exceeding y, the contemporary increment of Y, and putting X' and Y' to signify the variables thus increased, X' - Y' = X + x - (Y + y), or X' - Y' = X - Y + (x - y), so that the variables would again diverge. Therefore again D + n - m cannot be less than D, or X and Y cannot differ by less than D, contra hypothesein.

If X, Y, A, B, D, m, n, x, y, represent measures of ratios, the proof includes the quantumitum rations.

There results no absurdity from supposing the limits of two quantities which continually approach to have a difference D; but then the approaching quantities must always differ still more, contrary to one of the conditions.

By making n nothing in Case II, we make Y constant, and coincident in magnitude with B. The proposition obtained by the reasoning is, for this case, that whatever quantity A (suppose it a parabolic segment) may be an acknowledged limit of a variable X, if X can at the same time be proved to approach indefinitely to Y or B (suppose it a triangle equal to four thirds of the greatest triangle in the parabolic segment), A is equal to B. This is to prove that limits of the same variable are equal, which is neither giving a new signification to equality, nor concluding an identical proposition, nor indeed one which an ancient geometer would have admitted without formal proof. We come to a similar conclusion by making n nothing in Case III.

It is true that, in the Principia, when a fixed quantity is proved to be a limit to a variable, and the variable then pronounced ultimately equal to the fixed quantity, the author or editors refer to the first lemma, when reference might as well be made to the definition of ultimate quantities there implied; only (what is the main thing) the demonstration of the lemma teaches how, by admitting the limit as a particular value of the variable, we cannot produce an error measured by any fixed quantity, while if an error were admitted at all, it must be of fixed amount.

Thus in Lemma IV, where the same multitude of rectangles is supposed to be inscribed in the two figures AceE, PprT, and when this multitude is indefinitely increased, and consequently the breadth of the rectangles and the remainder of the areas indefinitely diminished, each rectangle in the one figure has, by hypothesis, the same ratio to a corresponding rectangle in the other; it is concluded that this is also the ratio of the whole figures. For the ratio of the area AceE to PprT may be viewed as compounded of AceE to the rectangles within itself, of these to the contemporary rectangles within PprT, and of these latter rectangles to PprT. But the first and last of the component ratios being ultimately of equality, the compound ratio is the constant ratio of the contemporary rectangles. For let L be the measure of the ratio of the inscribed rectangles, and let it be asserted that the compound ratio differs from L by a ratio whose measure is D. The first and last of the component ratios being ultimately of equality, their contemporary values may be made such that the measures (m and n) of the ratios by which the former exceeds equality and the latter falls below it, may be each less than D, and, a fortiori, their difference less than D. But L + m - n is the true measure of the compound ratio; therefore L does not differ from the true measure by D. Thus, by admitting ultimate values, no error is admitted. The method of exhaustions, which Newton's first lemma reduces to an enunciable proposition, only requires it to be allowed that fixed quantities either have a fixed difference or are equal. What is often called the principle of limits is to comprehend limits at once in the conclusion of a demonstration which does not apply to them. Newton's lemma serves to admit them through the method of exhaustions. According to the foregoing exposition, it is neither an identical proposition nor an arbitrary extension of the meaning of a term, though these inconsistent charges have both been brought against it.

A and B are proved to be equal in the old signification of equality; and whilst it is said that X and Y are ultimately equal, X and Y are also ultimately A and B. (o.o.o.)