Of two infinitely small quantities, one may be infinitely smaller than the other. For this read—When two quantities diminish without limit, it is also possible that their ratio may diminish without limit.
Hitherto we have been dealing with considerations which may appear purely verbal, but which are nevertheless connected with the trans lation of two very different modes of conception each into the other. But were they declared to be merely verbal, they would not be unim portant. It is of great consequence that the fundamental notions of mathematics should be expressed in those terms which have always represented the rude and unrigorous form in which they are expressed in common life : and also, when the form just alluded to has given birth to several different modes of expression, it is necessary to point out the connection of these with each other, and to assimilate their defined meanings. But, so far as demonstration is concerned, wo have made no step by using one form of words instead of another, or even by substituting the notion of a limit unattainable for that of the same magnitude attained by the supposition of absolute infinity. The theorem by which rigorous results are obtained is the following : If two variable magnitudes, A and B, be always equal, and if they have limits, namely, r the limit of A, and Q of B ; then P and Q must be equal. This proposition may seem almost self-evident ; it is not however a perfect axiom, and the method of exhaustions [GEOMETRY] was employed by Archimedes to prove it, or rather, to prove the pro position that if two variable magnitudes be always in a given ratio, their limits are in that ratio. The latter form of the proposition is requisite in Geometry [PROPORTION]; the former is sufficient in Algebra; and the proof is as follows : Supposing A and B for instance to be varying lines, always equal, let their limits, if possible, be the unequal lines KL and MN.
Since A and 13 are equal, and since the first can be made 'as near as we please to x L, and the second to m N, it follows that the latter pair are as nearly equal as we please. But this is not true, since the limits are fixed and invariable magnitudes, differing (if they differ at all) by a fixed and invariable quantity. Consequently the limits cannot be other than equal. The proof of the proposition of Archimedes is given in GEOMETRY.
This proposition, being once understood, is more fruitful in applica tions than almost any other. We shall give one instance from geometry and one from algebra.
Circles are to one another as the squares on their diameters. For this proposition is evidently true of the regular polygons inscribed in the two circles with the same number of sides ; and the polygons may be made as nearly equal as we please to the circles. The limits of the polygons then (or the circles themselves) are in that ratio which the polygons always preserve.
As an instance from algebra, apply the BINomIAL THEOREM to the development of (I + nx)* which gives, by an easy transformation, I —n 1—n 1-211 „ + (A), 1+ x + + a series which (by the method in CONVERGENT) is always convergent when nx is less than unity. Apply the same method to the develop ment of (14-nx).•=s; which gives in the same manner y—n y-2:t1+ yx + y 2 + (n).
Now 13 is evidently A ; and if when n diminishes without limit, and A approach the limits P and Q, then B and A s (equal quantities) will approach the limits Q and r V, which are therefore equal. But the limit of A,when n diminishes without limit, is xs 1+x+ + 2.3 F =r. That of B, on the same supposition, is 1+ xy + + + Hence the second of these series is the yth power of the first ; a theorem which the algebraical student will recognise as one of the most important in that science.
The method of limits generally means the Differential Calculus exhibited upon the principles explained in the article DIFFERENTIAL COEFFICIENT. It is admitted, by a large majority of those who are capable of forming a judgment, that the method by which this theory should be established is either the method of limits, or that of La grange [FUNCTIONS, THEORY OF], or a mixture of the two. The number of those who contend for the second has very much diminished of late years; and the controversy (if such a thing can be said to exist) lies between the first and third. The reader will find in the eighth number of the Treatise on the subject, published by the Society for the Diffusion of Useful Knowledge, some additional reasons for considering the use of assumed expansions as fallacious. See also SERIES.
It has been 'customary in elementary mathematical works to en deavour to postpone the theory of limits as late as possible. Such an attempt can never be very successful ; a clear understanding of the notion of a limit may easily be, and often is, deferred sine die, but the necessity for such an understanding enters with the sixth book of Euclid. We shall even undertake to show [PROPORTION] that the fifth book cannot be properly understood without it.
Wherever we have occasion to speak of a supposition as true, /tow ever small a quantity may be, we are really involving the notion of a limit. We cannot, for example, even define uniform velocity without It. be the bomber of feet described by a point in the number of Item& I, there is uniform velocity when si-t is the name fur all values of 1, Aesiserer assail, begin to reckon t at what part or the motion we may. This is saying that, begin where we may, the limit of s÷1 is always the same.
One of the best studies In the theory of limits is the first section of Newton's Principle. In the article PRIME AND ULTIMATE 11..artoa we &hail present one or two of the leading propositions.