Mathematicians Deceived

7. All these Points are believed by Men who believe only what they see.

They hesitate to admit that there might be obscure Points.

They believe that Mens Faculties are made alike. They use this as basis for their debates.

So what is impossible and repugnant to one person is presumed to be impossible and repugnant to another person.

But with what appearance of Reason shall any Man presume to say, that Mysteries may not be Objects of Faith, at the same time that he himself admits such obscure Mysteries to be the Object of Science?

8. Modern Mathematicians do not consider these Points as Mysteries, but as clearly conceived and mastered by their comprehensive Minds.

They scruple not to say, that by the help of these new Analytics they can penetrate into Infinity it self: That they can even extend their Views beyond Infinity: that their Art comprehends not only Infinite, but Infinite of Infinite (as they express it) or an Infinity of Infinites.

Mathematicians are wonderfully deceived and deluded by their own peculiar Signs, Symbols, or Species just as Men in other Inquiries are often deceived by Words and Terms.

It is easy to devise Expressions or Notations for Fluxions and Infinitesimals of the first, second, third, fourth and subsequent Orders, proceeding in the same regular form without end or limit &c. or dx. ddx. dddx. ddddx &c.

These Expressions are clear and distinct. The Mind finds no difficulty in conceiving them to be continued beyond any assignable Bounds.

But if we remove the Veil and look underneath and see what they represent, we see much Emptiness, Darkness, and Confusion, direct Impossibilities and Contradictions.

9. What are the Principles of this new Analysis by Momentums, Fluxions, or Infinitesimals?

Their capital Points, upon which the rest are supposed to depend, include Error and false Reasoning.

This means that they cannot guide others.

The main Point in the method of Fluxions is to obtain the Fluxion or Momentum of the Rectangle or Product of 2 indeterminate Quantities.

Inasmuch as from thence are derived Rules for obtaining the Fluxions of all other Products and Powers; be the Coefficients or the Indexes what they will, integers or fractions, rational or surd.

Now this fundamental Point one would think should be very clearly made out, considering how much is built upon it, and that its Influence extends throughout the whole Analysis.

This is given for Demonstration.[2]

Suppose the Product or Rectangle AB increased by continual Motion.

• The momentaneous Increments of the Sides A and B are a and b.

When the Sides A and B were deficient, or lesser by one half of their Moments, the Rectangle was

…

As soon as the Sides A and B are increased by the other two halves of their Moments, the Rectangle becomes

…

or

…

From the latter Rectangle subduct the former.

The remaining difference will be aB + bA.

Therefore the Increment of the Rectangle generated by the intire Increments a and b is aB + bA. Q.E.D.

But the direct and true Method to obtain the Moment or Increment of the Rectangle AB is to take the Sides as increased by their whole Increments

• A + a should be multiplied by B + b
• Its Product AB + aB + bA + ab is the augmented Rectangle whence if we subduct AB, the Remainder aB + bA + ab will be the true Increment of the Rectangle, exceeding that which was obtained by the former illegitimate and indirect Method by the Quantity ab.

This holds universally be the Quantities a and b what they will, big or little, Finite or Infinitesimal, Increments, Moments, or Velocities.

Nor will it avail to say that ab is a Quantity exceeding small: Since we are told that in rebus mathematicis errores quàm minimi non sunt contemnendi.[3]

Such reasoning as this, for Demonstration, nothing but the obscurity of the Subject could have encouraged or induced the great Author of the Fluxionary Method to put upon his Followers, and nothing but an implicit deference to Authority could move them to admit.

The Case indeed is difficult.

There can be nothing done till you have got rid of the Quantity ab.

In order to this the Notion of Fluxions is shifted: It is placed in various Lights: Points which should be clear as first Principles are puzzled’ and Terms which should be steadily used are ambiguous.

But despite this, getting rid of ab cannot be obtained by legitimate reasoning.

A man might establish certain Rules as Truths which he demostrates subtley.

• His Disciples might confound a Rule as a Truth in order to save themselves the trouble of thinking, especially if they are used to computing instead of thinking.

11. The Points or meer Limits of nascent Lines are undoubtedly equal, as having no more magnitude one than another, a Limit as such being no Quantity.

If by a Momentum you mean more than the very initial Limit, it must be either a finite Quantity or an Infinitesimal.

But all finite Quantities are expresly excluded from the Notion of a Momentum.

Therefore the Momentum must be an Infinite.

Much Artifice has been employed to avoid admitting infinitely small quantities, yet it seems ineffedtual.

For ought I see, you can admit no Quantity as a Medium between a finite Quantity and nothing, without admitting Infinitesimals.

An Increment generated in a finite Particle of Time, is it self a finite Particle; and cannot therefore be a Momentum.

You must therefore take an Infinitesimal Part of Time wherein to generate your Momentum.

It is said, the Magnitude of Moments is not considered: And yet these same Moments are supposed to be divided into Parts.

This is not easy to conceive, no more than it is why we should take Quantities less than A and B in order to obtain the Increment of AB, of which proceeding it must be owned the final Cause or Motive is very obvious; but it is not so obvious or easy to explain a just and legitimate Reason for it, or shew it to be Geometrical.

12. From the foregoing Principle so demonstrated, the general Rule for finding the Fluxion of any Power of a flowing Quantity is derived[4].

But I have shown its defect in the foregoing Demonstration.

The finding the Fluxion of a given Power is a Point of primary Importance.

That is why I will dissect it too.

The Lemma states:

“Use a Point to come up with other Points in order to demonstrate any Proposition.

If this Point is itself later destroyed or rejected by a contrary Supposition, then all those other Points must also be destroyed and rejected.

This is so plain as to need no Proof.

13. Here is the other Method of obtaining a Rule to find the Fluxion of any Power:

Let the Quantity x flow uniformly. Find the Fluxion of xn.

By flowing, x becomes x + o. At the same timme, the Power xn becomes .., i. e. by the Method of infinite Series &c. and the Increments o and &c. are one to another as 1 to .. &c.

Let the Increments vanish, and their last Proportion will be 1 to ..

But this reasoning is not fair nor conclusive.

Letting the Increments vanish also destroys the former Supposition that the Increments were something.

Which, by the foregoing Lemma, is a false way of reasoning. Certainly when we suppose the Increments to vanish, we must suppose their Proportions, their Expressions, and every thing else derived from the Supposition of their Existence to vanish with them.

14. To make this Point plainer, I shall unfold the reasoning, and propose it in a fuller light to your View. It amounts therefore to this, or may in other Words be thus expressed. I suppose that the Quantity x flows, and by flowing is increased, and its Increment I call o, so that by flowing it becomes x + o.

And as x increaseth, it follows that every Power of x is likewise increased in a due Proportion. Therefore as x becomes x + o, xn will become ..: that is, according to the Method of infinite Series, .. &c.

And if from the two augmented Quantities we subduct the Root and the Power respectively, we shall have remaining the two Increments, to wit, o and .. &c. which Increments, being both divided by the common Divisor o, yield the Quotients 1 and .. &c. which are therefore Exponents of the Ratio of the Increments.

Hitherto I have supposed that x flows, that x hath a real Increment, that o is something. And I have proceeded all along on that Supposition, without which I should not have been able to have made so much as one single Step.

From that Supposition it is that I get at the Increment of x^n, that I am able to compare it with the Increment of x, and that I find the Proportion between the two Increments.

I now beg leave to make a new Supposition contrary to the first, i. e. I will suppose that there is no Increment of x, or that o is nothing; which second Supposition destroys my first, and is inconsistent with it, and therefore with every thing that supposeth it.

I do nevertheless beg leave to retain .., which is an Expression obtained in virtue of my first Supposition, which necessarily presupposeth such Supposition, and which could not be obtained without it: All which seems a most inconsistent way of arguing, and such as would not be allowed of in Divinity.