Mathematicians Deceived

15. No just Conclusion can be directly drawn from 2 inconsistent Suppositions.

You cannot suppose any thing that destroys what you first supposed. If you do, you must begin de novo.

If therefore you suppose that the Augments vanish, i. e. that there are no Augments, you are to begin again, and see what follows from such Supposition.

But nothing will follow to your purpose. You cannot by that means ever arrive at your Conclusion, or succeed in, what is called by the celebrated Author, the Investigation of the first or last Proportions of nascent and evanescent Quantities, by instituting the Analysis in finite ones. I repeat it again: You are at liberty to make any possible Supposition:

You may destroy one Supposition by another: But then you may not retain the Consequences, or any part of the Consequences of your first Supposition so destroyed. I admit that Signs may be made to denote either any thing or nothing: And consequently that in the original Notation x + o, o might have signified either an Increment or nothing.

But then which of these soever you make it signify, you must argue consistently with such its Signification, and not proceed upon a double Meaning: Which to do were a manifest Sophism.

Whether you argue in Symbols or in Words, the Rules of right Reason are still the same. Nor can it be supposed, you will plead a Privilege in Mathematics to be exempt from them.

16. If you assume at first a Quantity increased by nothing, and in the Expression x + o, stands for nothing, upon this Supposition as there is no Increment of the Root, so there will be no Increment of the Power; and consequently there will be none except the first, of all those Members of the Series constituting the Power of the Binomial;

You will therefore never come at your Expression of a Fluxion legitimately by such Method.

Hence you are driven into the fallacious way of proceeding to a certain Point on the Supposition of an Increment, and then at once shifting your Supposition to that of no Increment. There may seem great Skill in doing this at a certain Point or Period.

Since if this second Supposition had been made before the common Division by o, all had vanished at once, and you must have got nothing by your Supposition.

Whereas by this Artifice of first dividing, and then changing your Supposition, you retain 1 and .., notwithstanding all this address to cover it, the fallacy is still the same.

For whether it be done sooner or later, when once the second Supposition or Assumption is made, in the same instant the former Assumption and all that you got by it is destroyed, and goes out together. And this is universally true, be the Subjectt what it will, throughout all the Branches of humane Knowledge; in any other of which, I believe. Men would hardly admit such a reasoning as this, which in Mathematics is accepted for Demonstration.

17. It may not be amiss to observe, that the Method for finding the Fluxion of a Rectangle of two flowing Quantities, as it is set forth in the Treatise of Quadratures, differs from the abovementioned taken from the second Book of the Principles, and is in effect the same with that used in the calculus differentialis[5].

For the supposing a Quantity infinitely diminishied and therefore rejecting it, is in effect the rejecting an Infinitesimal; and indeed it requires a marvellous sharpness of Discernment, to be able to distinguish between evanescent Increments and infinitesimal Differences. It may perhaps be said that the Quantity being infinitely diminished becomes nothing, and so nothing is rejected.

But according to the received Principles it is evident, that no Geometrical Quantity, can by any division or subdivision whatsoever be exhausted, or reduced to nothing.

Considering the various Arts and Devices used by the great Author of the Fluxionary Method: in how many Lights he placeth his Fluxions: and in what different ways he attempts to demonstrate the same Point: one would be inclined to think, he was himself suspicious of the justness of his own demonstrations; and that he was not enough pleased with any one notion steadily to adhere to it.

Thus much at least is plain, that he owned himself satisfied concerning certain Points, which nevertheless he could not undertake to demonstrate to others[6].

Whether this satisfaction arose from tentative Methods or Inductions; which have often been admitted by Mathematicians, (for inftance by Dr. Wallis in his Arithmetic of Infinites) is what I shall not pretend to determine. But, whatever the Case might have been with respect to the Author, it appears that his Followers have shewn themselves more eager in applying his Method, than accurate in examining his Principles.

18. It is curious to observe, what subtilty and skill this great Genius employs to struggle with an insuperable Difficulty; and through what Labyrinths he endeavours to escape the Doctrine of Infinitesimals; which as it intrudes upon him whether he will or no, so it is admitted and embraced by others without the least repugnance.

Leibnitz and his Followers in their calculus differentialis making no manner of scruple, first to suppose, and secondly to reject Quantities infinitely small: with what clearness in the Apprehension and justness in the reasoning, any thinking Man, who is not prejudiced in favour of those things, may easily discern.

The Notion or Idea of an infinitesimal Quantity, as it is an Object simply apprehended by the Mind, hath been already considered[7]. I shall now only observe as to the method of getting rid of such Quantities, that it is done without the least Ceremony.

As in Fluxions the Point of first importance, and which paves the way to the rest, is to find the Fluxion of a Product of two indeterminate Quantities, so in the calculus differentialis (which Method is supposed to have been borrowed from the former with some small Alterations) the main Point is to obtain the difference of such Product.

Now the Rule for this is got by rejecting the Product or Rectangle of the Differences. And in general it is supposed, that no Quantity is bigger or lesser for the Addition or Subduction of its Infinitesimal: and that consequently no error can arise from such rejection of Infinitesimals.

19. And whatever errors are admitted in the Premises, proportional errors ought to be apprehended in the Conclusion, be they finite or infinitesimal: and that therefore the ἀχρίβεια of Geometry requires nothing should be neglected or rejected. In answer to this you will perhaps say, that the Conclusions are accurately true, and that therefore the Principles and Methods from whence they are derived must be so too.

But this inverted way of demonstrating your Principles by your Conclusions, as it would be peculiar to you Gentlemen, so it is contrary to the Rules of Logic.

The truth of the Conclusion will not prove either the Form or the Matter of a Syllogism to be true: inasmuch as the Illation might have been wrong or the Premises false, and the Conclusion nevertheless true, though not in virtue of such Illation or of such Premises. I say that in every other Science Men prove their Conclusions by their Principles, and not their Principles by the Conclusions.

But if in yours you should allow your selves this unnatural way of proceeding, the Consequence would be that you must take up with the Induction, and bid adieu to Demonstration. And if you submit to this, your Authority will no longer lead the way in Points of Reason and Science.

20. I have no Controversy about your Conclusions, but only about your Logic and Method. How you demonstrate? What Objects you are conversant with, and whether you conceive them clearly?

What Principles you proceed upon; how sound they may be; and how you apply them? It must be remembred that I am not concerned about the truth of your Theorems, but only about the way of coming at them; whether it be legitimate or illegitimate, clear or obscure, scientific or tentative.

To prevent all possibility of your mistaking me, I beg leave to repeat and insist, that I consider the Geometrical Analyst as a Logician, i. e. so far forth as he reasons and argues; and his Mathematical Conclusions, not in themselves, but in their Premifes; not as true or false, useful or insignificant, but as derived from such Principles, and by such Inferences.

And forasmuch as it may perhaps seem an unaccountable Paradox, that Mathematicians should deduce true Propositions from false Principles, be right in the Conclusion, and yet err in the Premises; I shall endeavour particularly to explain why this may come to pass, and shew how Error may bring forth Truth, though it cannot bring forth Science.

21. In order therefore to clear up this Point, we will suppose for instance that a Tangent is to be drawn to a Parabola, and examine the progress of this Affair, as it is performed by infinitesimal Differences.

Let AB be a Curve, the Abscisse AP = x, the ordinate PB = y, the Difference of the Abscisse PM = dx the Difference of the Ordinate RN = dy. Now by supposing the Curve to be a Polygon, and consequently BN, the Increment or Difference of the Curve, to be a straight Line coincident with the Tangent, and the differential Triangle BRN to be similiar to the triangle TPB the Subtangent PT is found a fourth Proportional to RN: RB:PB: that is to dy: dx:y. Hence the Subtangent will be …

But herein there is an error arising from the forementioned false supposition, whence, the value of PT comes out greater than the Truth: for in reality it is not the Triangle RNB but RLB, which is similar to PBT, and therefore (instead of RN) RL should have been the first term of the Proportion, i. e. RN + NL, i. e. dy + z: whence the true expression for the Subtangent should have been …

There was therefore an error of defect in making dy the divisor: which error was equal to z, i. e. NL the Line comprehended between the Curve and the Tangent. Now by the nature of the Curve yy = px, supposing p to be the Parameter, whence by the rule of Differences 2ydy = pdx and …

But if you multiply y + dy by it self, and retain the whole Product without rejecting the Square of the Difference, it will then come out, by substituting the augmented Quantities in the Equation of the Curve, that .. truly. There was therefore an error of excess in making , which followed from the erroneous Rule of Differences. And the measure of this second error is …

Therefore the two errors being equal and contrary destroy each other; the first error of defect being corrected by a second error of excess.

22. If you had committed only one error, you would not have come at a true Solution of the Problem. But by virtue of a twofold mistake you arrive, though not at Science, yet at Truth. For Science It cannot be called, when you proceed blindfold, and arrive at the Truth not knowing how or by what means. To demonstrate that z is equal to .., let BR or dx be m and RN or dy be n. By the thirty third Proposition of the first Book of the Conics of Apollonius, and from similar Triangles, as 2x to y so is m to …

Likewise from the Nature of the Parabola yy + 2yn + nn = xp + mp, and 2yn + nn = mp: wherefore:

.. and because yy = px,

.. will be equal to x, Therefore substituting these values instead of m and x we shall have

i. e.

which being reduced gives

23. In the first place, the Conclusion comes out right, not because the rejected Square of dy was infinitely small; but because this error was compensated by another contrary and equal error.

I observe in the second place, that whatever is rejected, be it ever so small, if it be real and consequently makes a real error in the Premises, it will produce a proportional real error in the Conclusion. Your Theorems therefore cannot be accurately true, nor your Problems accurately solved, in virtue of Premises, which themselves are not accurate, it being a rule in Logic that Conclusio sequitur partem debiliorem.

Therefore I observe in the third place, that when the Conclusion is evident and the Premises obscure, or the Conclusion accurate and the Premises inaccurate, we may safely pronounce that such Conclusion is neither evident nor accurate, in virtue of those obscure inaccurate Premises or Principles;

But in virtue of some other Principles which perhaps the Demonstrator himself never knew or thought of. I observe in the last place, that in case the Differences are supposed finite Quantities ever so great, the Conclusion will nevertheless come out the same: inasmuch as the rejected Quantities are legitimately thrown out, not for their smallness, but for another reason, to wit, because of contrary errors, which destroying each other do upon the whole cause that nothing is really, though something is apparently thrown out.

This Reason holds equally, with respect to Quantities finite as well as infinitesimal, great as well as small, a Foot or a Yard long as well as the minutest Increment.