and h and k be diminished indefinitely, hk may be neglected, when compared with xk+yh.

47. The principle of limiting ratios is in itself free from all difficulty. The first and last ratios of the magnitudes are as precise and definite, and as well adapted in every respect to be the subject of calculation, as any of the intermediate ratios. This principle is frequently represented by foreign writers to be different from that upon which Newton founded his system of Fluxions, and they are fond of considering it as an improvement due to D'Alembert. But that the theories are the same, is sufficiently evident from the second Lemma of the second volume of the Principia, without appealing to the treatise on fluxions, which is a posthumous work. In this Lemma the theory is fully developed; the fluxions are called Momenta, because the increments are supposed to be generated by material particles. He repeats the former caution, "Cave tamen intellexeris particulas finitas ;" and then adds, with respect to the magnitudes of the momenta generated, "Neque enim spectatur in hoc lemmate magnitudo momentorum; sed prima nascentium proportio:" and he then proceeds to show that the momentum of Am. B = maam−1‍ + nbв2-1 where a and b are the momenta of A and B. Newton saw that the ratio of the velocities of the fluents is the same as the limiting ratio of their increments, and adopted the former phraseology as being less liable to metaphysical objections.

It has been objected to Newton's mode of considering the genesis of quantity, that it includes the idea of motion and time, which, it is urged, belong to physical science, and are foreign to the spirit of pure analysis. This would be an unanswerable objection, if in the principle of fluxions, or in its application, we either assumed or even referred to the Laws of Motion, which are founded upon observation and experiment. The science of Analytical Calculation, though it renders great assistance to Natural Philosophy, should accept of none in return *. But the introduction of the simple idea of motion is so far from being an objection, that

There are elementary writers who have not sufficiently attended to this distinction. I have seen a demonstration, in which it is assumed that the motion of the generating point of a curve is compounded of two motions in the directions of its Co-ordinates.

it enters into every system of calculation which has hitherto been proposed. In Arbogast's "Calcul des Derivations" and Lagrange's "Théorie des Fonctions Analytiques," while certain parts of the function are constant, it is supposed that the remaining parts change their state or value, and all change implies motion; so that, with respect to this objection at least, the principle of limiting ratios is as purely algebraical as the theories of these eminent analysts.

The doctrine of Infinitesimals, which was admitted both by Newton and Leibnitz, is not so easily vindicated. The supposition that a quantity may become infinitely smaller than another which has been already assumed to be infinitely small, is obviously absurd or unintelligible. To avoid this inconsistency, we must be careful to consider an infinitesimal, not as representing an absolute quantity, however small, but as representing a ratio which is either infinitely great or infinitely small in the limit, according as we compare it with an infinitesimal of a lower or of a higher order than itself.

Influenced partly by objections such as these, and partly, I fear, by a spirit of jealousy, many writers of the last century looked out for some other principle of calculation than that upon which Sir I. Newton has founded his doctrine of Fluxions. As my object is not only to teach the elements of the science, but also to induce the reader to study the writings of those who have unquestionably greatly enlarged its boundaries, I shall give, in the third Chapter, the principle upon which Lagrange rests his Theory, which the reader will perceive is independent of the doctrine of limits. The great difficulty, however, is to establish the truth of the proposition contained in (44. Cor. 2.) which follows so immediately from the doctrine of infinitesimals. We shall insert Lagrange's demonstration of this proposition in his own words: the student will then be enabled to form an opinion of the relative merit of the two systems.

CHAPTER II.

On Integration.

1. HAVING established in the preceding chapter rules for the differentiation of quantity both algebraick and transcendental, we shall now consider the inverse problem; or the method of finding the fluent of a given fluxion.

If it were required to give a definition of the fluent of a proposed fluxion, we must suppose that two fluxions are under consideration, and their fluents may be defined to be two magnitudes, the limiting ratio of whose cotemporary increments is the ratio of the proposed fluxions.

The act of deriving the fluent from its fluxion is called Integration*.

In this chapter we chall only investigate the general rules and methods of integration, and apply them to such examples as occur the most frequently; reserving the canonical forms and the remainder of the subject for the second volume.

2. Whatever be the form of the fluent, its fluxion may be found; but the inverse problem cannot be solved in all cases, as the proposed fluxion may be of such a form as cannot arise from the differentiation of any fluent.

3. The symbol adopted to denote the operation of taking the fluent is .

*I use the terms Integration and Differentiation, which properly belong to the Calculus of Differences, because they are adopted into the language, and I know no other that will answer my purpose. No confusion can arise from this phraseology, as the two sciences of Fluxions and of Increments possess each a peculiar notation and algorithm. Some of our writers, in imitation of the French, even call the science of Fluxions the Differential Calculus; but, surely, the differential calculus, if the analogy of language is to be observed, means the calculus of differences.

It was shown (Ch. I. 33.) that the same function admits of successive differentiations; hence a fluxion which is the result of n differentiations may be integrated n times. Symbols for these double, triple fluents areƒƒ, SSS, &c.

Since the two symbols d and denote reverse operations, whenever they occur together they destroy each other, thus, d.frdx = Pdx.

4. The fluent may sometimes be determined at once from inspection.

The fluxion of x3 being 3x2dx, the fluent of 3x dx must be a3. Also, since d.x+1=m+1. xmdr, therefore f.m+1 xmdx = x2+i. Upon the same principle Л(ydx+xdy)=xy, rydx - xdy

=

y

and ЛAa*dx = a*.

It is evident that, in general, if the form of the proposed fluxion is derivable from its fluent by any of the preceding rules, we have only to invert that rule in order to find the fluent. We shall begin with Art. 11. Rule 1.

5. In Integration we may have to add or to subtract a constant quantity.

For (Art. 11.) d(x± a) dx, therefore the fluent of dx may be either x or xa, where a is some constant quantity. This quantity a, called the correction, or the constant, is to be determined from the conditions of the question.

6. If any two corresponding values of the variables be known, we may either eliminate the constant or determine its value.

1

Ex. Let du = m + 1.xmdx .. u = x2+1+c and when u=r let x=b.. we have r = bm+1 + c

7. By this correction we can find the value of the fluent included between two known values of its principal variable. For let it be required to find famda between the values of

x = b and x = a; we have u =

1

m+1

xm+1+c, and u=o

8. A Definite Integral is one that has been corrected, and its symbol is . The function may be enclosed in brackets, if the fluent has been corrected, and a particular value assigned to the variable.

9. The constant multiplier or divisor may be removed from under the symbol f

For (Art. 12.) d.ax=adx, therefore fadx = afdx.

1

m + 1

·Sm+1. xmdx =

xm+1 m+1

+ c.

We shall, in general, omit the correction; it being understood that it is to be taken into consideration when the form is applied to particular examples.

10. The fluent of the sum or difference of any number of fluxions is the sum or difference of their fluents.

=

For (Art. 13.) d(u+v-w+ &c.) du+dv-dw+ &c. therefore (du + dv - dw+ &c.) =fdu +fdv -fdw ± &c. +c, where c is the result of the corrections of all the fluents.

11. It appears (Art. 14.) that mom-1dx=x"; and from Art. 14. Cor. 3. that fn.(am+m)n-1 m.xm-1dx=(am+xm)": hence, to integrate a compound quantity of the above form, where the part without the vinculum is of the form of the fluxion of the part within, adopt the following:

RULE 1. Increase the index by unity, divide by the index so increased, and also by the fluxion of the root.

2. f(a+x)3 dx = ±(a + x)1.

8. J (a + x y xdz = f(a +

4. J (a + x) xdx = } (a+c) =.

Or thus, (a2 + x2)2 xdx=a*xdx+2a2x3dx+x3dx .•. (2.10.)