I have long ago thought that the first principles and rules of the method of Fluxions stood in need of a more full and distinct explanation and proof, that what they had received either from their first incomparable author, or any of his followers; and therefore was not at all displeased to find the method itself opposed with so much warmth by the ingenious author of the 'Analyst'; and had it been his only design to bring this point to a fair issue, whether a demonstration by the method of Fluxions be truly scientific or not, I should have heartily applauded his conduct, and have thought he deserved the thanks even of the Mathematicians themselves. But the invidious light in which he has put this debate, by representing it as of consequence to the interests of religion, is, I think, truly unjustifiable, as well as highly imprudent. Among all wise and fair inquirers, 'tis beyond all contradiction plain, that religion can be no ways affected by the truth or falsehood of the doctrine of Fluxions. And tho' prejudiced minds may be variously affected by it, yet it is not easy to be conceived what advantage this debate is likely to give to the cause of religion and virtue in general even among them. Whereas it is easy to guess of what disservice our author's representation of a controversy in which religion has no manner of concern, may be towards raising and inflaming the passions of weak men on both of the question: And I wish he had been pleased coolly to consider beforehand of what consequence the result of this dispute is likely to be to the cause of religion, among those for whose conviction his 'Analyst' is chiefly designed. If he should not be able to make out his point, will not the blind followers of the Infidel Mathematicians be more confirmed in their errors than they were before? Will they not be more prejudiced against religion, and established in their esteem and veneration of their masters by a weal and fruitless attempt to depress their characters; and by finding that a zeal for it has occasioned so strong an attempt to wound the reputation of Sir Isaac Newton as a cautious and fair reasoner? And on the other hand, if our author should carry his point, and his proofs should be allowed, that the doctrine of Fluxions is an incomprehensible mystery, and that the most accurate Mathematicians have, one after another, imposed upon themselves in the most egregious manner, by false and inconclusive reasonings, what consequences can we suppose that such persons will draw from these premises? Our Author indeed would have them only from hence make this one conclusion, That their masters, the Mathematicians, are not to be depended upon when they speak against religion. But I believe it can't in reason be expected that they should stop here.

If such men as Dr Barrow, Dr Clarke, etc. and the incomparable Sir Isaac Newton, were capable of imagining that they saw with the greatest clearness and perspicuity, where they had nothing but absolute and incomprehensible darkness before them, what conclusion will persons, used to take their opinions from authority, be likely to make from these premises, but that all pretences to knowledge in religion, and every thing else, are only confidence and presumption.

If they are taught that it is inconsistent for a person to reject the mysteries of religion, and yet believe the mystery of Fluxions, will they not know how to draw the opposite conclusion themselves, that it is inconsistent to reject the doctrine of Fluxions because mysterious, and yet receive the mysteries of religion? And when they are taught to think that a person may be justly said to have faith, because they give into what they can neither demonstrate nor conceive; if this give them a mean opinion of the Mathematicians, 'tis odds if it don't give them a mean opinion of faith itself. I am sure 'tis a very strange account of that which may justly be called faith: For without clear notions no man can believe any more than demonstrate.

Considering these things, I can't help thinking it was highly wrong to bring religion at all into this controversy, which may inflame the dispute, but can hardly do any real service: Of which, to me, it is a very strong presumption, that every thing urged by the author of the 'Analyst' against infidels in general, would have sounded full as well in the mouth of a Papist, if urged against those Mathematicians that don't believe the doctrine of transubstantiation, as it would have been peculiarly in character for such a one to have made his chief attack upon a great enemy of all superstition and tyranny, and an hearty friend to the reasonable religion of Christians and Protestants. But enough of this. I shall now consider my subject as striped of all relation it has to religion, and endeavour to show that the method of Fluxions is founded upon clear and substantial principles.

It cannot be doubted that Sir Isaac Newton well understood the doctrine of which he was the original inventor; and his proofs of it are very far from being fallacious and deceitful, or their force hard to be understood by those that are used to these kinds of subjects.

But it is also very plain, that the question, which is the main dispute between our author and his adversaries, whether Mathematicians take the notion and certainty of the method of Fluxions implicitly from him or not, does not depend upon our being able to defend the exact accuracy of the demonstrations he has made use of, and the propriety of every phrase by which he has explained his notions upon this head. He always seems to have studied conciseness of expression, and to depend on the good sense and judgment of his reader. And on this account some of his demonstrations are not the most full and complete that might be given, and must remain obscure to those who have no genius for the mathematical science, and can't find out those steps in the demonstration, which a writer often omits in confidence of the sagacity of his reader. In my opinion this is in some measure the case with respect to his proofs of the first principles of Fluxions, and therefore I don't wonder persons differ in their sentiments about them. But it is truly provoking to find that the greatest genius that ever appeared in the philosophical world, and one whom the lovers of knowledge must always think of with respect and gratitude, should be represented contrary to his known character, as craftily imposing on the world, in confidence of his own authority, and obscurity of his subject. And therefore I would hope that the author of the 'Analyst' did not design that severe reflection his words seem to carry with them, when he says, "Such reasoning as this, nothing but the obscurity of the subject could have encouraged or induced the great author of the fluxionary method to have put upon his followers; and nothing but an implicit deference to his authority could have moved them to admit." To suspect Sir Isaac Newton of the mean design of seeking reputation among the ignorant, by venting unintelligible notions, and defending them by artful and cunning sophistry, is when I think no man is capable of doing. And therefore if the author of the 'Analyst' does not think fit, for his own reputation, to revoke or explain the sentence just mentioned, it needs not a particular confutation. Nor do I propose particularly to follow him in all the objections he has made against Sir Isaac's notions and demonstrations, being of opinion that the best way of answering him is to assist persons in understanding the subject itself; for if any one can do this, he will easily see there is little weight in what he has said against it. However, as I go along, I shall endeavour to obviate anything that I think may create a difficulty; but my main view is to settle the first principles on which the doctrine of Fluxions depends, and then to show that, by just reasoning from them, the rules for finding the Fluxions of equations, as delivered by Sir Isaac, do truly follow.