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space. But that it must vary upon the least change of space." Now admitting thus much to be demonstrated, yet I am still at a loss to conceive how Mr. Walton's conclusion will follow, to wit, " that I am greatly mistaken in imagining there can be no motion, no velocity, in a point of space." (P. 20.) Pray, Sir, consider his reasoning. The same velocity cannot be in two points of space; therefore velocity can be in a point of space. Would it not be just as good reasoning to say, the same man cannot be in two nutshells; therefore a man can be in a nutshell? Again, velocity must vary upon the least change of space; therefore there may be velocity without space. Make sense of this if you can. What have these consequences to do with their premises? Who but Mr. Walton could have inferred them? Or how could even he have inferred them had it not been in jest?
V. Suppose the centre of a falling body to describe a line, divide the time of its fall into equal parts, for instance, into minutes. The spaces described in those equal parts of time will be unequal. That is, from whatsoever points of the described line you measure a minute's descent, you will still find it a different space. This is true. But how or why from this plain truth a man should infer, that motion can be conceived in a point, is to me as obscure as any the most obscure mysteries that occur in this profound author. Let the reader make the best of it. For my part, I can as easily conceive Mr. Walton should walk without stirring, as I can his idea of motion without space. After all, the question was not whether motion could be proved to exist in a point, but only whether it could be conceived in a point. For, as to the proof of things impossible, some men have a way of proving that may equally prove any thing. But I much question whether any reader of common sense will undertake to conceive what this pleasant man at inference undertakes to prove.
VI. If Mr. Walton really meant to defend the au
thor of the fluxionary method, would he not have done it in a way consistent with this illustrious author's own principles? Let us now see what may be Sir Isaac's notion about this matter. He distinguisheth two sorts of notion, absolute and relative. The former he defineth to be a translation from absolute place to absolute place, the latter from one relative place to another.* Mr. Walton's is plainly neither of these sorts of motion. But some third kind, which, what it is, I am at a loss to comprehend. But I can clearly comprehend that, if we admit motion without space, then Sir Isaac Newton's account of it must be wrong: for place by which he defines motion is, according to him, a part of space. And if so, then this notable defender hath cut out new work for himself to defend and explain. But about this, if I mistake not, he will be very easy. For, as I said before, he seems at bottom a back friend to that great man; which opinion you will see farther confirmed in the sequel.
VII. I shall no more ask Mr. Walton to explain any thing: for I can honestly say, the more he explains, the more I am puzzled. But I will ask his readers to explain, by what art a man may conceive motion without space. And supposing this to be done, in the second place to explain, how it consists with Sir Isaac Newton's account of motion. Is it not evident, that Mr. Walton hath deserted from his old master, and been at some pains to expose him, while he defends one part of his principles by overturning another? Let any reader tell me, what Mr. Walton means by motion, or, if he can guess, what this third kind is, which is neither absolute nor relative, which exists in a point, which may be conceived without space. This learned professer saith, "I have no clear conception of the principles of motion." (P. 24.) And in another place (p. 7) he saith, "I might have conceived velocity in a point,
* See Schol. def. viii. Philos. Nat. Princip. Math.
if I had understood and considered the nature of motion." I believe I am not alone in not understanding his principles. For myself, I freely confess the case to be desperate. I neither understand them, nor have any hopes of ever being able to understand them.
VIII. Being now satisfied, that Mr. Walton's aim is not to clear up or defend Sir Isaac's principles, but rather to contradict and expose them, you will not, I suppose, think it strange, if instead of putting questions to this intrepid answerer, who is never at a loss, how often soever his readers may, I entreat you, or any other man of plain sense, to read the following passage cited from the thirty-first section of the Analyst, and then try to apply Mr. Walton's answer to it: whereby you will clearly perceive what a vein of raillery that gentleman is master of. 66 Velocity necessarily implies both time and space, and cannot be conceived without them. And if the velocities of nascent or evanescent quantities, i. e. abstracted from time and space, may not be comprehended, how can we comprehend and demonstrate their proportions? Or consider their rationes primæ et ultimæ ? For to consider the proportion or ratio of things, implieth that such things have magnitude: that such their magnitudes may be measured, and their relations to each other known. But, as there is no measure of velocity except time and space, the proportion of velocities being only compounded of the direct proportion of the spaces and the reciprocal proportion of the times; doth it not follow, that to talk of investigating, obtaining, and considering the proportions of velocities, exclusively of time and space, is to talk unintelligibly?" Apply now, as I said, Mr. Walton's full answer, and you will soon find how fully you are enlightened about the nature of fluxions.
IX. In the following article of Mr. Walton's full answer, he saith divers curious things, which being derived from this same principle, that motion may be conceived in a point, are altogether as incomprehensible as
the origin from whence they flow. It is obvious and natural to suppose Ab and B a* to be rectangles produced from finite lines multiplied by increments. Mr. Walton indeed supposeth that when the increments vanish or become nothing, the velocities remain, which being multiplied by finite lines produce those rectangles (p. 13). But admitting the velocities to remain, yet how can any one conceive a rectangular surface to be produced from a line multiplied by velocity, otherwise than by supposing such line multiplied by a line or increment, which shall be exponent of or proportional to such veloctiy? You may try to conceive it otherwise. I must own I cannot. Is not the increment of a rectangle itself a rectangle? must not then Ab and B a be rectangles? and must not the coefficients or sides of rectangles be lines? Consequently are not b and a lines or (which is the same thing) increments of lines? These increments may indeed be considered as proportional to and exponents of velocity. But exclusive of such exponents to talk of rectangles under lines and velocities is, I conceive, to talk unintelligibly. And yet this is what Mr. Walton doth, when he maketh b and a in the rectangles Ab and B a to denote mere velocities.
X. As to the question, whether nothing be not the product of nothing multiplied by something, Mr. Walton is pleased to answer in the affirmative. And nevertheless when a b is nothing, that is, when a and b are nothing, he denies that Ab+ Ba is nothing. This is one of those many inconsistencies which I leave the reader to reconcile. But, saith Mr. Walton, the sides of the given rectangle still remain, which two sides according to him must form the increment of the flowing rectangle. But in this he directly contradicts Sir Isaac Newton, who asserts that Ab+ B a and not A + B is the increment of the rectangle A B. And, indeed, how is it possíble á line should be the increment of a surface? "Late* See Nat. Phil. Princip. Math. 1. ii. lem. 2.
rum incrementis totis a et b generatur rectanguli incrementum Ab+ Ba," are the words of Sir Isaac,* which words seem utterly inconsistent with Mr. Walton's doctrine. But no wonder that gentleman should not agree with Sir Isaac, since he cannot agree even with himself; but contradicts what he saith elsewhere, as the reader may see, even before he gets to the end of that same section, wherein he hath told us that "the gnomen and the sum of the two rectangles are turned into those two sides by a retroverted motion." (P. 11 and 12.) Which proposition, if you or any other person shall try to make sense of, you may possibly be convinced, that this profound author is as much at variance with common sense, as he is with himself and Sir Isaac Newton.
XI. Mr. Walton in the ninth page of his Vindication, in order to explain the nature of fluxions, saith, that "to obtain the last ratio of synchronal increments, the magnitude of those increments must be infinitely diminished." Notwithstanding which, in the twentythird page of his full answer, he chargeth me as greatly mistaken, in supposing that he explained the doctrine of fluxions by the ratio of magnitudes infinitely diminished. It is an easy matter for any author to write so as to betray his readers into mistakes about his meaning. But then it is not easy to conceive what right he hath to upbraid them with such their mistakes. If I have mistaken his sense, let any one judge if he did not fairly lead me into the mistake. When a man puzzleth his reader, saith an unsaith, useth ambiguous terms and obscure terms, and putteth them together in so preserve a manner, that it is odds you can make out no sense at all, or if any, wrong sense; pray who is in fault but the writer himself? Let any one consider Mr. Walton's own words, and then say whether I am not justified in making this remark."
XII. In the twentieth page of his full answer Mr.