The Analyst: a Discourse addressed to an Infidel Mathematician

24. For the fuller illustration of this Point, I shall consider it in another light, and proceeding in finite Quantities to the Conclusion, I shall only then make use

of one Infinitesimal. Suppose the straight Line MQ cuts the Curve AT in the Points R and S. Suppose LR a Tangent at the Point R, AN the Abscisse, NR and OS Ordinates. Let AN be produced to O, and RP be drawn parallel to NO. Suppofe AN = x, NR = y, NO = v, PS = z, the subsecant MN = S, Let the Equation y = xx express the nature of the Curve: and supposing y and x increased by their finite Increments, we get y + z = xx + 2xv + vv: whence the former Equation being subducted there remains z = 2xv + vv. And by reason of similar Triangles PS : PR :: NR : NM, i.e. z : v :: y : s =

wherein if for y and z we substitute their values, we get

Supposing NO to be infinitely diminished, the subsecant NM will in that case coincide with the subtangent NL, and v as an Infinitesimal may be rejected, whence it follows that

which is the true value of the Subtangent. And since this was obtained by one only error, i. e. by once rejecting one only Infinitesimal, it should seem, contrary to what hath been said, that an infinitesimal Quantity or Difference may be neglected or thrown away, and the Conclusion nevertheless be accurately true, although there was no double mistake or rectifying of one error by another, as in the first Case.

But if this Point be throughly considered, we shall find there is even here a double mistake, and that one compensates or rectifies the other.

For in the first place, it was supposed, that when NO is infinitely diminished or becomes an Infinitesimal, then the Subsecant NM becomes equal to the Subtangent NL.

But this is a plain mistake, for it is evident, that as a Secant cannot be a Tangent, so a Subsecant cannot be a Subtangent. Be the Difference ever so small, yet still there is a Difference.

And if NO be infinitely small, there will even then be an infinitely small Difference between NM and NL.

Therefore NM or S was too little for your supposition, (when you supposed it equal to NL) and this error was compensated by a second error in throwing out v, which last error made s bigger than its true value, and in lieu thereof gave the value of the Subtangent.

This is the true State of the Case, however it may be disguised. And to this in reality it amounts, and is at bottom the same thing, if we should pretend to find the Subtangent by having first found, from the Equation of the Curve and similar Triangles, a general Expression for all Subsecants, and then reducing the Subtangent under this general Rule, by considering it as the Subsecant when v vanishes or becomes nothing.

25. Upon the whole I observe the following:

26. v can never be nothing so long as there is a secant.

27. The same Line cannot be both tangent and secant.

28. When v or NO[8] vanisheth, PS and SR do also vanish, and with them the proportionality of the similar Triangles.

Consequently the whole Expression, which was obtained by means thereof and grounded thereupon, vanisheth when v vanisheth.

4. The Method for finding Secants or the Expression of Secants, be it ever so general, cannot in common sense extend any further than to all Secants whatsoever.

As it necessarily supposeth similar Triangles, it cannot be supposed to take place where there are not similar Triangles.

5. The Subsecant will always be less than the Subtangent, and can never coincide with it; which Coincidence to suppose would be absurd; for it would be supposing, the same Line at the same time to cut and not to cut another given Line, which is a manifest Contradiction, such as subverts the Hypothesis and gives a Demonstration of its Falshood.

6. If this be not admitted, I demand a Reason why any other apagogical Demonstration, or Demonstration ad absurdum should be admitted in Geometry rather than this: Or that some real Difference be assigned between this and others as such.

7. It is sophistical to suppose NO or RP, PS, and SR to be finite real Lines in order to form the Triangle RPS, in order to obtain Proportions by similar Triangles; and afterwards to suppose there are no such Lines, nor consequently similar Triangles, and nevertheless to retain the Consequence of the first Supposition, after such Supposition hath been destroyed by a contrary one.

8. In the present case, by inconsistent Suppositions Truth may be obtained, yet that such Truth is not demonstrated: That such Method is not conformable to the Rules of Logic and right Reason: That, however useful it may be, it must be considered only as a Presumption, as a Knack, an Art or rather an Artifice, but not a scientific Demonstration.

9. The Doctrine premised may be farther illustrated by the following simple and easy Case, wherein I shall proceed by evanescent Increments. Suppofe AB = x, BC = y, BD = o, and that xx is equal to the Area ABC:

It is proposed to find the Ordinate y or BC. When x by flowing becomes x + o, then xx becomes xx + 2xo + oo: And the Area ABC becomes ADH, and the Increment of xx will be equal to BDHC the Increment of the Area, i. e. to BCFD + CFH.

If we suppose the curvilinear Space CFH to be qoo, then 2xo + oo = yo + qoo which divided by o gives 2x + o = y + qo. And, supposing o to vanish, 2x = y, in which Case ACH will be a ‘straight Line, and the Areas ABC, CFH, Triangles.

But it is not legitimate or logical to suppose o to vanish, i. e. to be nothing, i. e. that there is no Increment, unless we reject at the same time with the Increment it self every Consequence of such Increment, i. e. whatsoever could not be obtained but by supposing such Increment.

It must nevertheless be acknowledged, that the Problem is rightly solved, and the Conclusion true, to which we are led by this Method. It will therefore be asked, how comes it to pass that the throwing out o is attended with no Error in the Conclusion? I answer, the true reason hereof is plainly this:

Because q being Unite, qo is equal to o: And therefore 2x + o - qo = y = 2x, the equal Quantities qo and o being destroyed by contrary Signs.

27. As on the one hand it were absurd to get rid of o by saying, let me contradict my self: Let me subvert my own Hypothesis: Let me take it for granted that there is no Increment, at the same time that I retain a Quantity, which I could never have got at but by assuming an Increment: So on the other hand it would be equally wrong to imagine, that in a geometrical Demonstration we may be allowed to admit any Error, though ever so small, or that it is possible, in the nature of Things, an accurate Conclusion should be derived from inaccurate Principles.

Therefore o cannot be thrown out as an Infinitesimal, or upon the Principle that Infinitesimals may be safely neglected. But only because it is destroyed by an equal Quantity with a negative Sign, whence o - qo is equal to nothing.

And as it is illegitimate to reduce an Equation, by subducting from one Side a Quantity when it is not to be destroyed, or when an equal Quantity is not subducted from the other Side of the Equation: So it must be allowed a very logical and just Method of arguing, to conclude that if from Equals either nothing or equal Quantities are subducted, they shall still remain equal.

This is a true Reason why no Error is at last produced by the rejecting of o. Which therefore must not be ascribed to the Doctrine of Differences, or Infinitesimals, or evanescent Quantities, or Momentums, or Fluxions.

28. Suppose the Case to be general, and that xn is equal to the Area ABC whence by the Method of Fluxions the Ordinate is found

which we admit for true, and shall inquire how it is arrived at. Now if we are content to come at the Conclusion in a summary way, by supposing that the Ratio of the Fluxions of x and xn are found [10] to be 1 and

nx^{n-1}}, and that the Ordinate of the Area is considered as its Fluxion; we shall not so clearly see our way, or perceive how the truth comes out, that Method as we have shewed before being obscure and illogical.

But if we fairly delineate the Area and its Increment, and divide the latter into two Parts BCFD and CFH[11] and proceed regularly by Equations between the algebraical and geometrical Quantities, the reason of the thing will plainly appear. For as as xn is equal to the Area ABC so is the Increment of xn equal to the Increment of the Area, i. e. to BDHC; that is, to say,

nox^{n-1}+{\frac {nn-n}{2}}oox^{n-2}+&c.=BDFC+CFH} . And only the first Members, on each Side of the Equation being retained,

nox^{n-1}=BDFC}: And dividing both Sides by o or BD, we shall get

nx^{n-1}=BC}. Admitting, therefore, that the curvilinear Space CFH is equal to the rejectaneous Quantity

and that when this is rejected on one Side, that is rejected on the other, the Reasoning becomes just and the Conclusion true.

And it is all one whatever Magnitude you allow to BD, whether that of an infinitesimal Difference or a finite Increment ever so great. It is therefore plain, that the supposing the rejectaneous algebraical Quantity to be an infinitely small or evanescent Quantity, and therefore to be neglected, must have produced an Error, had it not been for the curvilinear Spaces being equal thereto, and at the same time subducted from the other Part or Side of the Equation agreeably to the Axiom, If from Equals you subduct Equals, the Remainders will be equal.

For those Quantities which by the Analysts are said to be neglected, or made to vanish, are in reality subducted. If therefore the Conclusion be true, it is absolutely necessary that the finite Space CFH be equal to the Remainder of the Increment expressed by

equal I say to the finite Remainder of a finite Increment.

29. Therefore, be the Power what you please, there will arise on one Side an algebraical Expression, on the other a geometrical Quantity, each of which naturally divides it self into three Members:

The algebraical or fluxionary Expression, into one which includes neither the Expression of the Increment of the Absciss nor of any Power thereof, another which includes the Expression of the Increment it self, and a third including the Expression of the Powers of the Increment.

The geometrical Quantity also or whole increased Area consists of three Parts or Members, the first of which is the given Area, the second a Rectangle under the Ordinate and the Increment of the Absciss, and the third a curvilinear Space, And, comparing the homologous or correspondent Members on both Sides, we find that as the first Member of the Expression is the Expression of the given Area, so the second Member of the Expression will express the Rectangle or second Member of the geometrical Quantity; and the third, containing the Powers of the Increment, will express the curvilinear Space, or third Member of the geometrical Quantity.

This hint may, perhaps, be further extended and applied to good purpose, by those who have leisure and curiosity for such Matters. The use I make of it is to shew, that the Analysis cannot obtain in Augments or Differences, but it must also obtain in finite Quantities, be they ever so great, as was before observed.