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. But, notwithstanding all these assertions and pretensions, it may be justly questioned whether, as other men in other inquiries are often deceived by words or terms, so they likewise are not wonderfully deceived and deluded by their own peculiar signs, symbols or species. Nothing is easier than 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, indeed, are clear and distinct, and the mind finds no difficulty in conceiving them to be continued beyond any assignable bounds. But if we remove the veil and look underneath, if, laying aside the expressions, we set ourselves attentively to consider the things themselves which are supposed to be expressed or marked thereby, we shall discover much emptiness, darkness, and confusion; nay, if I mistake not, direct impossibilities and contradictions. Whether this be the case or no, every thinking reader is entreated to examine and judge for himself.
9. Having considered the object, I proceed to consider the principles of this new analysis by momentums, fluxions, or infinitesimals;wherein if it shall appear that your capital points, upon which the rest are supposed to depend, include error and false reasoning; it will then follow that you, who are at a loss to conduct yourselves, cannot with any decency set up for guides to other men. The main point in the method of fluxions is to obtain the fluxion or momentum of the rectangle or product of two 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. But let the reader judge. This is given for demonstration. [`Naturalis Philosophiae Principia Mathematica,' lib.
ii., lem. 2.] Suppose the product or rectangle AB increased by continual motion: and that 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 , i.e. , . And 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, and the remaining difference will be aB + bA . Therefore the increment of the rectangle generated by the entire increments a and b is aB + bA . Q.E.D. But it is plain that 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, and so multiply them together, A + a by B + b , the product whereof 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 .
And 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 quam minimi non sunt contemnendi .
10. Such [`Introd. ad Quadraturam Curvarum.'] 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 as clear as first principles are puzzled; and terms which should be steadily used are ambiguous. But, notwithstanding all this address and skill, the point of getting rid of ab cannot be obtained by legitimate reasoning. If a man, by methods not geometrical or demonstrative, shall have satisfied himself of the usefulness of certain rules; which he afterwards shall propose to his disciples for undoubted truths; which he undertakes to demonstrate in a subtile manner, and by the help of nice and intricate notions; it is not hard to conceive that such his disciples may, to save themselves the trouble of thinking, be inclined to confound the usefulness of a rule with the certainty of a truth, and accept the one for the other; especially if they are men accustomed rather to compute than to think; earnest rather to go on fast and far, than solicitous to set out warily and see their way distinctly.
11. The points or mere 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 expressly excluded from the notion of a momentum.
