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Table of contents

As late as , the infinitesimals are still there: The moments of the fluent quantities that is, their indefinitely- small parts, by addition of which they increase during each in- finitely small period of time are as their speeds of flow. But further, since o is supposed to be infinitely small so that it be able to express the moments of quantities, terms which have it as a factor will be equivalent to nothing in respect to the others.

This is Whiteside's translation from Latin, where he has also replaced Newton's original n and m by x and y. It is therefore difficult for me to say anything about Newton's reasons for writing this. I, in fact, shall consider them as generated by grow- ing, arguing that they are greater, equal or less according as they grow more swiftly, equally swiftly or more slowly from their be- ginning. And this swiftness of growth I shall call the fluxion of a quantity. But earlier, he had used infinitely small quantities to find these fluxions.

Now he tried to do without them: Fluxions of quantities are in the first ratio of their nascent parts or, what is exactly the same, in the last ratio of those parts as they vanish by defluxion. Yet, because the hypothesis of indivisibles is a rather harsh one, and for this reason that method is reckoned less geometrical, I have preferred to reduce proofs of following matters to the last sums and ratios of vanishing quantities and the first ones of nascent quantities.

But what are these "last sums and ratios of vanishing quantities"? Newton saw that this could be difficult, and tried to explain: There is the objection to this—a somewhat futile one, however— that there exists no last proportion of vanishing quantities, inas- much as, before they vanish, there is no last one while, once they have vanished, there is none at all. By the same argument it can be asserted that there is no last speed of a body proceeding to a specified position: for, before the body reaches the place, there can be no last one while, once it has reached it, there is none at all.

But the answer is easy. Translation from Latin. And similarly by the last ratio of vanishing quantities you must un- derstand not the ratio of the quantities before they vanish, nor that afterwards, but that with which they vanish. There ex- ists a limit which their speed can at the end of its motion attain, but not, however, surpass. This is their last speed. In fact, I find it difficult to understand Newton's point. The exact speed with which the body reaches its last position has to be zero — otherwise it would continue beyond this last position.

I have no doubt that Newton had some idea of a limit concept here, but the difference between an idea and a fully explained and understood concept is large. I will take a look at a few examples where Newton computes fluxions, which were also to have an important role in the Analyst-debate. First, Newton wants to compute the fluxion of the product AB.

This is taken from Principia, where he defined moment this way I here consider as variable and indetermined, and increasing or decreasing as it were by perpet- ual motion or flux; and I understand their momentaneous incre- ments or decrements by the name of Moments We are to conceive them as the just nascent principles of finite magnitudes. It will be the same thing, if, instead of moments, we use ei- ther the Velocities of the increments and decrements which may also be called the motions, mutations, and fluxions of quantities or any finite quantities proportional to those velocities.

Translated from Latin. A somewhat similar quote can be found in Principia [10] p. See [40] p. It seems clear that Newton's proof was made to avoid this kind of infinitesimal-argument. But then it looks very arbitrary. However, Newton's intuition is right. Newton seems to have seen that all of the different answers are essentially the same, and therefore chosen the one which gave the simplest calculation. If this interpretation is correct, we have here an extreme example of Newton's failure to write down what his intuition told him. And without an argument saying why the different results are essentially the same, the procedure still seems very mysterious.

Afterwards, Newton wants to calculate the fluxion of xn: Let the quantity x flow uniformly and the fluxion of the quantity xn need to be found. Without recourse to the limit con- cept, however, he seems to let the divisor become zero, which of course must be an invalid way of reasoning. This was the way Berkeley interpreted him. I will come back to this later. Barrow had said clearly that "There is no absolute minimum in any species of magnitude", while Berkeley was of the opposite belief. See [25] p. Newton, as Robins conclusively proved, used at first the methods and the language of indivisibles, but after discovering the method of prime and ultimate ratios he discarded that of indivisibles, though he often used language borrowed from the older writings.

The heading of this section is taken from here. De Morgan also pointed out that the infinitely little quantity returned in , in the second edition of Principia. Fluxions are finite motions, moments are infinitely little parts We have no Ideas of infinitely little Quantities, and therefore Mr. Newton introduces Fluxions into his Method, that it might proceed by finite Quantities as much as possible. Lai29 solves this disagreement by interpreting the quotes as a rejec- tion only of the traditional infinitesimal methods, not of his own fluxional method, that was also dependent on infinitesimals.

Kitcher,30 on the other hand, has a theory that Newton considered the usual infinitesimal argu- ments, the fluxional calculus and the method of first and last ratios as three different parts of his theory, with three different goals: The theory of fluxions yielded the heuristic methods of the calcu- lus. Those methods were to be justified rigorously by the theory of ultimate ratios. Quoted in [9] p. Rather than competing for the same position, the three theories were designed for quite distinct tasks.

To investigate this would take a more thorough investigation of Newton's papers than is possible in this paper. On the other hand, it is not difficult to find other possible explanations for why Newton never completely got rid of the infinitesimals. One thing was Newton's con- servatism and unwillingness to throw away anything at all. It was more important, perhaps, that if Newton really rejected his previous methods, it would make his priority struggle with Leibniz more difficult. Moreover, the infinitesimal and fluxional calculus were and are better than the method of first and last ratios when it comes to intuitive comprehensibility.

And espe- cially if the public almost started to understand his first two explanations, it would be pedagogically unfortunate to change to yet another one. All the same, the following should be clear: Newton used several differ- ent explanations of his fluxional calculus, without making the relationship between them clear. He was not good at defining and clarifying his con- cepts, and he used intuition as a strong tool, without giving a "rigorous alternative" to the intuition.

These taken together gave room for different interpretations. The method was relatively easy to get an idea of, because of the strong connection with intuitive concepts like movement and velocity. On the other hand it was difficult to understand it completely, because of the unclear definitions. All of this paved the way for long discussions. Chapter 3 The Analyst controversy I suspect that he is one of that sort of man who wants to be known for his paradoxes.

Leibniz1 3. This book was a strong criticism of Newton's fluxional calculus, and the goal was to show that modern mathematics was accepted because people believed in it, the logic was so full of holes that it could not be said to be known that it was correct. It was therefore meaningless of mathematicians to criticize religion for being based on belief. The book was the start of a long debate,3 a part of which I will look at in this chapter. Roger Ariew and Daniel Garter. Indianapolis, Ind. Quoted in [25] p.

These are said not to be moments, but quantities generated from moments, which last are only the nascent principle of finite quantities. It is said that the minutest errors are not to be neglected in math- ematics: that the fluxions are celerities, not proportional to the finite increments, though ever so small; but only to the mo- ments or nascent increments, whereof the proportion alone, and not the magnitude, is considered.

And of the aforesaid fluxions there be other fluxions which fluxions of fluxions are called sec- ond fluxions. And the fluxions of these second fluxions are called third fluxions: and so on, fourth, fifth, sixth. Now, as our sense is strained and puzzled with the perception of objects extremely minute, even so the imagination, which fac- ulty derives from sense, is very much strained and puzzled to frame clear ideas of the least particles of time, or the least incre- ment generated therein: and much more so to comprehend the moments, or those increments of the flowing quantities in statu nascenti, in their very first origin or beginning to exist, before they become finite particles.

And it seems still more difficult to conceive the abstracted velocities of such nascent imperfect entities. However, it must be said,5 that many of these terms were never used by Newton, so Berkeley's attack cannot be said to be entirely fair. Generally, it is easy to make a theory seem incomprehensible, but difficult to prove that it is. 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 loss to conduct your selves, cannot with any decency set up for guides to other men.

Nor will it avail to say that ab is a quantity exceedingly small: since we are told that in rebus mathematicis errores quam minimi non sunt contemnendi. But even if the method had been legitimate, the problem remains: there are two methods giving seemingly different answers to the same question. What was needed, and what Newton failed to give, was a proof that the two answers were in some sense equivalent.

Likewise, Berkeley disliked Newton's calculation of the fluxion of xn see section 2. For when it is said, let the increments vanish, i. 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. However, Berkeley's interpretation of Newton will be addressed later see p. This was because there in every calculation was done two errors which cancelled each other. I will not go into Berkeley's calculations — suffice it to say that it is generally accepted that Berkeley was wrong.

But it is singular that Berkeley This point of view seems to be supported by the fact that the compensation of errors thesis is incorrect. This is made clear by Phi- lalethes, for instance in the quote on p. If Berkeley was to believed, Newton's mathematics was incomprehensible and some of his most impor- tant proofs were invalid. How could Berkeley be answered? One way was to try to show that Berkeley had misunderstood Newton. The less Newton- bound could alternatively try to show that Newton had thought something else than what he wrote. Finally, it could be tried to build a new foundation for the fluxional calculus, independent of infinitesimals, and of Newton's texts.

All of these ways were tried in the debate that followed. The real author is believed to have been Dr. James Jurin of Cambridge. Philalethes was concerned about Berkeley's attack on mathematicians. Therefore, he did not attempt to build a new foundation for fluxional calcu- lus. A new foundation, though mathematically interesting, would be irrele- vant in this respect, as it would not help Newton and other mathematicians. Instead, Philalethes had to defend what Newton had written. This was not an easy task.

Further, Philalethes' book was not aimed at mathematicians. Just like Berkeley, he wrote for the general public, his aim was to save mathematicians from Berkeley's criticism. The mathematics is kept to a minimum, and the polemic at times gives the reader a good laugh — at Berkeley's expense, of course. The same can hardly be said of Robins', MacLaurin's or Paman's contributions to the debate. I would like to stress that in my opinion, Philalethes' book is not a posi- tive contribution to this debate — his aim is to destroy Berkeley's criticism by finding errors in it and counter-attacking, not to explain and clarify the theory.

In fact, Philalethes spends the first 25 pages on non-mathematical themes, claiming that mathematicians are not infidels, that if they were, it should not be published, and if it was published, they would still not be able to make others become infidels. Making the point that he knew of no Frenchman who had given up Catholicism just because Newton was not a catholic. I will hurry on to the more mathematical discussion. Philalethes writes: Your objections against this method may, I think, all of them be reduced under these three heads.

Obscurity of this doctrine. False reasoning used in it by Sir Isaac Newton, and im- plicitely received by his followers. Artifices and fallacies used by Sir Isaac Newton, to make this false reasoning pass upon his followers. He also attacks Berkeley for misrepresenting Newton; Have you not altered his expressions in such a manner, as to mislead and confound your readers, instead of informing them? Philalethes therefore advises both Berkeley and the readers to read look at Newton's own writings. Philalethes does not try to explain the "doctrine".

In fact, he doesn't have to explain it — Berkeley has presented a parody on Newton, and Philalethes has pointed it out. However, the next theme is more difficult: 3. Given the object of his book, Philalethes was obliged to come up with an explanation of Newton's seemingly inexplicable calculation of the fluxion of AB see p. Newton offered no explanation for his proof, and I sincerely doubt that Newton's explanation would have been more convincing than Philalethes'.

Philalethes first asks if leaving out ab really is an error at all: Do not [mathematicians] know that in estimating any finite quan- tity how great soever, proposed to be found by the method of Fluxions, a globe, suppose, as big as that of the earth, or, if you please, of the sun, or of the whole planetary system, or even the 12[11] p. But this interpretation is wrong, as he goes on to say [I have] observed that this obmission, or error as you are pleased to call it, in rejecting the rectangle ab, is at most such an one as can cause no assignable difference, how small soever, in the conclusions drawn from the method of Fluxions This argument could have been written by anyone, with any founda- tion — infinitesimals, fluxions, first and last ratios — and perhaps even by a modern user of the Cauchy limit concept.

But it is elucidated by the following example: Suppose two Arithmeticians to be disputing whether vulgar frac- tions are to be preferred to decimal; would it be fair in him who is for expressing the third part of a farthing by the vulgar frac- tion , to affirm that his antagonist proceeded blindfold, and without knowing what he did, when he pretended to express it by 0. Might not the other reply that, if this expression was not rigorously exact, yet it could not be said he proceeded blindfold, or without clearness and science in using it, because by adding more figures he could approach as near as he pleased, and wherever he thought fit to stop, he could clearly and distinctly find and demonstrate how much he fell short of the rigorous and exact value?

Of course, Philalethes is no Cauchy, but this example shows that he had some intuition of what was going on. But the main problem remained; Newton's mysterious calculation of the fluxion of AB. This argument has no support in the definitions. And even if the definitions had supported it, we would want a proof that these three "moments" are the same. Philalethes denies that Newton tried to find the increment of AB; On the contrary, it plainly appears that what he endeavours to obtain by these suppositions, is no other than the increment of the rectangle A — a X B — 6, and you must own that he takes the direct and true method to obtain it.

You know very well that the moment of the rectangle AB is proportional to the velocity of that rectangle, with which it al- ters, either in increasing, or in diminishing. Now, I ask, in Ge- ometrical rigour what is properly the velocity of this rectangle? But methinks I hear the venerable Ghost of Sir Isaac Newton whis- per me, that the velocity I seek for, is neither the one nor the other of these, but is the velocity which the flowing rectangle has, not while it is greater or less than AB, neither before, nor after it becomes AB, but at that very instant of time that it is AB.

And this is the method Sir Isaac Newton has taken in the demonstration you except against. Similarly, I see nothing wrong with Philalethes' argument, except the usual objection; that the definitions are too unclear — in fact, it seems that both Newton and Philalethes are guided more by their intuition than by the definitions. Philalethes agrees that Newton did this to get rid of ab, but sees nothing wrong in that, as long as the demonstration was correct.

He concludes this discussion by repeating that leaving out ab is no error — this time by an argument that is clearly meaningless: 19[11] p. Gibson's discussion of this22 makes it clear, however, that it is the first of these interpretations which cover Philalethes' meaning. Philalethes goes on to consider Newton's calculation of the fluxion of xn see p. For God's sake let us examine it once more. Evanescant jam augmenta ilia, let now the incre- ments vanish, i. Hold, Sir, I doubt we are not right here. Do not the words ratio ultima stare us in the face, and plainly tell us that though there is a last proportion of evanescent increments, yet there can be no proportion of increments which are nothing, of increments which do not exist?

I believe, Sir, every thinking reader will ac- quit Sir Isaac Newton of the gross oversight you ascribe to him - 25 20 Principia vol. Philalethes points out that Berkeley has published a new proof of God's existence, and wonders if that means that Berkeley doubts all the previous proofs. He goes on: You are all in the dark, and yet are angry at his giving you so much light. Surely the fault is not in Sir Isaac Newton, but in your own eyes. He begins by making fun of his theory: Now truly, Sir, if this Paradox of yours should be well made out, I must confess it ought very much to alter the opinion the world has had of Sir Isaac Newton, and occasion our talking of him in a very different manner from what we have hitherto done.

What think you if, instead of the greatest that ever was, we should call him the most fortunate, the most lucky Mathematician that ever drew a circle? Methinks I see the good old Gentleman fast asleep and snoring in his easy-chair, while Dame Fortune is bringing him her apron full of beautiful Theorems and Problems which he never knows or thinks of. For what else but extreme good fortune could occation the conclusions arising from his method to be always true and just and accurate, when the premisses were inaccurate and erroneous and false, and only led to right conclusions by means of two errors ever compensating one another to the utmost exactness?

What luck [11] p. That when he had made one capital, fundamental, general mistake, he should happen to make a second as capital, as fundamental, as general as the first; that he should not proceed to commit three or four such mistakes, but stop at the second: That these two mistakes should chance not to lie both the same way, but on contrary sides, so that the one might help to correct the other; and lastly, that the two contrary errors, among all the infinite proportions which they might bear to one another, should happen upon that of a perfect equality; so that one might in all possible cases be exactly balanced or compensated by the other.

With a quarter of this good fortune a man might get the 1. Therefore, Philalethes goes on to see what happens if only one of the two errors is commited. The argument is essentially the same as before; he claims that the errors are nothing, but does not prove it. Therefore, I will not go into details on this. For instance Wisdom writes: " Jurin was clearly not the man to be entrusted with the task of clarifying and defending the calculus.

For some reason, Philalethes chose to defend Newton in several different ways. He first seems to say that the errors are so small that they are not important see p. Berkeley saw it like this and wrote I had observed that the great author [ In answer to this you alledge that the errour arising from the omission of such rectangle allowing it to be an errour is so small that it is insignificant. Philalethes does not say that the error is "so small that it is insignificant" — he says that it is "at most such an one as can cause no assignable difference, how small soever".

This must be the right answer to Berkeley's criticism, but it is not very helpful as long as he doesn't prove his assertion. This must be the only way to make some sort of sense of Newton's calculation, even though Newton himself says that he calculates the increment of AB, as Berkeley points out. As many have noted, this does not help defending Newton. The calculation does not fit the defini- tion, however, but it seems to me that many mathematicians at the time, including Newton, considered the definitions as little more than explana- tions, bearing the concepts and not the definitions in mind when doing mathematics.

Therefore I am not as surprised as Cajori, even though the soundness of Berkeley's criticism at this point is clear. Philalethes pointed out that Berkeley had given an incorrect translation of Newton's Latin, and explained that Newton considered the "last proportion of evanescent increments. Only if they keep to the misunderstanding, it is the time to start explaining.

We would have liked him to explain what exactly was meant by "last proportion of evanescent increments", but that was not necessary, as he had shown that Berkeley's criticism at this point was based on a faulty translation of Newton's words. He does not, however, give an explanation of the theory of fluxions. That task was left to the next writer on the subject. While Philaleth.

Robins distinguishes clearly between Newton's two methods, the method of fluxions and the method of prime and ultimate ratios. And the velocity, or degree of swiftness, with which this point moves in any part of the line AB, is called the fluxion of this line at that place. But in using the definitions, Robins is much more elaborate. He wants to show how "the proportion between the fluxions of magnitudes is assignable from the relation known between the magnitudes themselves First, he shows that the proportion of the velocity of the point at F to the velocity of the point at E is less than FH to EG.

For, while that point is advanced from E to G, the point moving on CD has passed from F to H, and has moved through that space with a velocity continually accelerated; therefore, if it had moved during the same interval 40 [43] p. For as the point in CD, in moving from K to F, proceeds with a velocity continually accelerated; with the velocity, it has acquired at F, if uniformly continued, it would describe in the same space of time a line longer than KF.

In the series, whereby CH is denoted, the line e can be taken so small, that any term proposed in the series shall exceed all the following terms together; so that the double of that term shall be greater than the whole collection of that term, and all that follow. Thus have we here made appear, that from the relation between the lines AE and CF, the proportion between the velocities, wherewith they are described, is discoverable; for we have shewn, that the proportion of na;"-1 to a"-1 is the true proportion of the velocity, wherewith CF, or augments, to the velocity, wherewith AE, or x is at the same time augmented.

He has a similar proof of the fluxion of AB. We note that through all of this, instantaneous velocity has not been defined, only used. Robins defines second fluxions etc. He argues that all orders of fluxions exist in nature. These higher orders of fluxions are then used to find the radius of curvature, for example. The main objection to Robins' method is its strong connection to phys- ical considerations. This is also a virtue, however, since it makes the theory easy to understand.

We see that Robins quickly translates the geometry into algebraic terms, and gives a solid proof. This is because The concise form, into which Sir Isaac Newton has cast his demonstration, may very possibly create a difficulty of appre- hension in the minds of some unexercised in these subjects.

But otherwise his method of demonstrating by the prime and ulti- mate ratios of varying magnitudes is not only just, and free from any defect in itself; but easily to be comprehended, at least by those who have made these subjects familiar to them by reading the ancients. The words "perpetually approach" seem to suggest monotonity. The same can be defined for ratios: Ratios also may so vary, as to be confined after the same manner to some determined limit, and such limit of any ratio is here considered as that, with which the varying ratio will ultimately coincide This terminology is perhaps unfortunate as it may suggest that this "ultimate ratio" is the ratio of the "ultimate magnitudes".

Robins therefore hastens to add that 45 [43] p. Here these quadrilaterals can never bear one to the other the proportion between AB and BE, nor have either of them any fi- nal magnitude, or even so much as a limit, but by the diminution of the distance between DF and AE they diminish continually without end: and the proportion between AB and BE is for this reason called the ultimate proportion of the two quadrilaterals, Figure 3. The quadrilaterals may be continually diminished, either by di- viding BC in any known proportion in G drawing HGI parallel to AE, by dividing again BG in like manner, and by continuing this division without end; or else the line DF may be supposed to advance towards AE with an uninterrupted motion, 'till the quadrilaterals quite disappear, or vanish.

And under this latter notion these quadrilaterals may very properly be called vanish- ing quantities, since they are now considered, as never having any stable magnitude, but decreasing by a continued motion, 'till they come to nothing. And since the ratio of the quadrilat- eral ABCD to the quadrilateral BEFC, while the quadrilaterals diminish, approaches to that of AB to BE in such manner, that this ratio of AB to BE is the nearest limit, that can be assigned to the other; it is by no means a forced conception to consider the ratio of AB to BE under the notion of the ratio, wherewith the quadrilaterals vanish; and this ratio may properly be called the ultimate ratio of two quantities.

Suppose the points B and D to move in equal spaces of time into two other positions E and F; then DF will be to BE in the ratio of the velocity, wherewith DF would be described with an uniform motion, to the velocity, wherewith BE will be described [43] p.

But if the point de- scribing the line AB moves uniformly; the velocity, wherewith the line CD is described will not be uniform. Therefore the space DF is not described with a uniform velocity; in so much that the velocity, wherewith DF would be uniformly described, is never the same with the velocity at the point D. But by diminish- ing the magnitude of DF, the uniform velocity, wherewith DF would be described, may be made to approach at pleasure to the velocity at the point D. Therefore the velocity at the point D is the ultimate magnitude of the velocity, wherewith DF would be uniformly described.

Consequently the ratio of the velocity at D to the velocity at B is the ultimate ratio of the velocity, wherewith DF would be uniformly described, to the velocity, wherewith BE is uniformly described. But DF being to BE as the velocity, wherewith BE is uniformly described, the ultimate ratio of DF to BE is also the ultimate ratio of the first of these velocities to the last; because all the ultimate ratios of the same varying ratio are the same with each other.

And here it is obvious, that all the terms after the first taken together may be made less than any assignable part of the first. And we have already proved, that the proportion of the velocity at D to the velocity at B is the same with the ultimate proportion of DF to BE; therefore the velocity at D is to the velocity at B, or the fluxion of xn to the fluxion of x, as nx to l.

Therefore this way of doing it is not much better or worse than his previous one. But at the very end of his discourse, Robins tries to explain it: And in this I shall be the more particular, because Sir Isaac New- ton's definition of momenta, That they are the momentaneous increments or decrements of varying quantities, may possibly be thought obscure. In this case Sir Isaac Newton distinguishes, by the name of momentum, so much of any difference, as constitutes the term used in expressing this ultimate ratio.

Robins has the following comment on Newton's calculation of the same fluxion: These momenta equally relate to the decrements of quantities, as to their increments, and the ultimate ratio of increments, and of decrements at the same place is the same; therefore the mo- mentum of any quantity may be determined, either by consider- ing the increment, or the decrement of that quantity, or even by considering both together.

And in determining the momentum of the rectangle under A and B Sir Isaac Newton has taken the last of these methods; because by this means the superfluous rectangle is sooner disengaged from the demonstration. Robins' Discourse was unfortunately only the beginning of a long and wordy debate between Philalethes and Robins, later with Henry Pemberton in Robins' place.

I have not had the opportunity to study the contributions in this debate, but the secondary literature suggests that the debate's main [43] p. I therefore refer to Cajori55 on this subject. Philalethes clearly does not try to give a comprehensive account of the theory of fluxions. Instead he faces Berkeley's objections one by one, avoiding technicalities whenever possible, perhaps because the book is not aimed only at mathematicians.

Robins, on the other hand, wants to explain the theory. He gives the definitions, and examples with long reductio ad absurdum proofs. He doesn't mention the Analyst or Berkeley. Does Philalethes succeed in refuting Berkeley's criticism? In my opinion, he partly does succeed: He shows that Berkeley has misrepresented Newton and he gives an explanation of Newton's AB calculation that makes sense.

However, in some points he is too unclear to succeed fully, especially when he argues that the errors for instance ab are nothing. Here we would want proofs, not just claims. Does Robins succeed in explaining the theory? Yes, certainly. He defines his terms except "velocity" — probably considering a definition of it un- necessary , gives illuminating examples and proves his propositions. Except the term "momentum", he also keeps close to Newton's definitions.

His un- critical use of the notion of instantaneous velocity would probably not have satisfied Berkeley,57 but for the less philosophically inclined, I think Robins' book gave explanation enough. It is not in my purpose to enter into the details of this quarrel which, strangely enough, has been given great importance in Cajori" [21] note 4 p. Gibson is one of the writers most critical of Philalethes, and he writes: There can be no question, that there is a profound difference of conception in the views of Philalethes and Robins, and I confess myself at a loss quite to understand the favour shown to the work of Philalethes, and the comparative neglect of the brilliant essays of Robins.

The reason for the "favour shown" to Philalethes may have been that he wrote a book that could easily be read by anyone at least large parts of it , which was at times funny, and which at several points refuted Berkeley. Moreover, it is not surprising that the writings of a Cambridge scholar should be taken more seriously at first than the writings of a simple math- ematics teacher. Two of the other answers, those of MacLaurin and Paman, have been given their own chapters.

The rest will not be treated here at all. Some of them were not very bad, while some were very confused. The number of works is large, and many of them are at least mentioned in Cajori [9]. Suffice it here to say that English math- ematics stayed more geometrical than the mathematics on the Continent, and that most of the interesting developments happened elsewhere than in England. Therefore a much discussed question in the literature has been: Was Berkeley's work good or bad for British mathematics?

To answer this question it is necessary to have an idea of what "good or bad" means in this context, and of what would have happened to British mathematics if Berke- ley had not published his work. As these are extremely difficult questions, I will only say that Berkeley's work was very important for British mathemat- ics. This is clearly shown from the number of answers he received, and the amount of time great mathematicians as MacLaurin spent to write them. It should be clear from Philalethes' and Robins' work that the answers were of varying scope and quality, and that the method of fluxions did not have a clear foundation at the time of The Analyst.

Robins provided one, however, and in the next two chapters we will see two others. MacLaurin originally planned to write a shorter answer to Berkeley,2 but was encouraged to do more out of it. Col- son, and several other Pieces, were published on this Subject. But MacLaurin's Treatise became much more than an answer to Berkeley; it included a mathematical treatment of centres of gravity and oscillation, lines of swiftest descent, the figure of the planets, the tides, wind-mills, vibration of chords and so on.

But it also gave a 1John Conduitt to MacLaurin in a letter dated August 24th, , see [37] letter 40, p. The book is relatively unreadable, but gives the method of fluxion a foundation independent of infinitesimals. Like Robins, MacLaurin divides his treatise in two main parts. MacLaurin's book, however, is even more geometrical than Robins'. The second book is much more algebraic and avoids using velocities. The proofs' are given by double reductio ad absurdum, however. Then he goes on to explain the theory The quantity that is thus generated is said to flow, and called a Fluent.

These can hardly be unproblematic concepts on which to found a mathematical method, but, as mentioned before, they seem to have been accepted at the time. The rest of Book I consists of lots of propositions, with long, geometrical proofs which seem unreadable to the modern reader. For instance, he proves the following proposition see Figure 4. I will not quote it — but the main idea is to consider an invariable line, moving at such speed 'MacLaurin says that "we shall always measure this velocity by the space that would be described by it continued uniformly for some given finite time.

What velocity is "continued uniformly"? The instantaneous velocity, of course. Guicciardini objects to this definition [21] p. Jesseph does not agree [25] p. Thereafter it takes some pages of double reductio ad absurdum proofs in four different cases to prove the proposition. In a brilliant passage, he explains the connection between his geometrical method of Book I and the method of infinitesimals: In the method of infinitesimals, the element, by which any quan- tity increases or decreases, is supposed to be infinitely small, and is generally expressed by two or more terms, some of which are irifinitely less than the rest, which being neglected as of no im- portance, the remaining terms form what is called the difference of the proposed quantity.

The terms that are neglected in this manner, as infinitely less than the other terms of the element, are the very same which arise in consequence of the acceleration, of retardation, of the generating motion, during the infinitely small time in which the element is generated; so that the remaining terms express the element that would have been produced in that time, if the generating motion had continued uniform.

Therefore those differences are accurately in the same ratio to each other as the generating motions or fluxions. He has a similar argument, by way of an example, concerning Newton's method of first and last ratios. To quote MacLaurin: The method of demonstration, which was invented by the author of fluxions, is accurate and elegant; but we propose to begin with one that is somewhat different; which, being less removed from that of the antients, may make the transition to his method more easy to beginners for whom chiefly this treatise is intended [sic!

There is one important difference between MacLaurin's way of doing things in Book II and the ways of Newton and Robins that we have seen earlier; MacLaurin does not use the intuitive concept of velocity here I will give a little example to show what I think is MacLaurin's meaning: Example 4. This is what MacLaurin calls that the "successive differences Therefore it is clear that B is not growing uniformly, it is accelerating, so the fluxion must be less than 3 and greater than b. Now MacLaurin is ready to compute the fluxion of A2, by reductio ad absurdum: Proposition 4.

Suppose now that u is any increment of A less than o; and because a is to u 17[36] vol. But it was shown, from art. And these being contradictory, it follows that the fluxion of A being equal to a, the fluxion of AA cannot be greater than 2Aa. How does he do it? The crucial point is that the differences increase. Since they always increase, the "limit" has to be between the two differences, for any choice of u, and it can easily be shown what it is.

MacLaurin then goes on to consider the inverse method of fluxions what we call integration. And the fact that it was little read must not be held against MacLaurin either. As MacLaurin himself writes in defence of Archimedes Kline also wrote that [Colin MacLaurin] attempted to establish the rigor of the calcu- lus. It was a commendable effort but incorrect. Turnbull, on the other hand, in felt that 'The Treatise of Fluxions' In point of rigor it is a worthy link between the ancient method of exhaustions and the subsequent work of Cauchy and of Weierstrass.

MacLaurin had published his Treatise of Fluxions, and They are painstakingly long and not much of an improvement on Robins' earlier proofs. Book II, on the other hand, takes a more promising approach, by being less geometrical and more algebraic. His wordy proofs seem to be extendible to a large class of functions — all functions that are concave or convex in a neighbourhood of 0 — and they are not dependent on the intuitive concept of instantaneous velocity. The foundations for the method of fluxions were only a small part of the Treatise — the books were filled with applications of the method.

This was obviously part of the explanation of why his work was treated as the author- itative answer to Berkeley. Perhaps his foundation of fluxions was important mostly because everyone believed that the theory of fluxions were given a geometric, rigorous foundation without actually examining the foundation in detail. Chapter 5 Roger Paman Paman's work was crippled by his extensive use of new terminology Sageng1 5. We do not know anything about where and when Roger Paman was born. Unfamiliar as the name may seem, however, he is not the only Paman we know of.

He was born at his father's estate at Chevington, Suffolk in He was not himself registered as a student in Cambridge,5 but in the preface to his book he mentions Mr. Prank, who belonged to St. John's College, Cambridge, and who was the one 'In [45] p. Only three male Paman's were baptized in the period ; John, William and Henry.

BSee Venn's Alumni Cantabrigienses [52]. Paman wrote a paper on this, which was communicated to several members of The Royal Society, and which kept circulating until Helen's for a journey round the world. Of the eight ships that set out, only one ship, the Centurion, managed to get around the world and return to England, reaching Spithead on June 15th Five of the ships, the Gloucester, the Wager, the Tryal, the Anna and the Industry, were destroyed during the journey.

Paman must therefore have been on one of the remaining ships: the Severn and the Pearl. Severn and Pearl left England together with the other ships, and an- chored upon the coast of Patagonia Southern Argentina February 18th, March 7th, they passed the Straits of Le Maire,7 still together. But on April 10th, they lost sight of the other ships,7and on April 25th they even lost sight of each other,8 but were rejoined May 21st.

The weather had been terrible and most of the men were ill, and both ships had to wait before going on. July 4th, , the ships arrived in Rio de Janeiro, and Captain Legge of the Severn wrote: And I arrived by the great mercy of Almighty God safe in this port the 6th of June, not having above thirty men in the ship, myself, lieutenants, officers and servants besides three men I had at sea from the Pearl that were able to assist to the working of the ship; and all of us so weak and so much reduced that we could hardly walk along the deck.

However, on February 5th, , they arrived in Barbados on their way home. For instance, John Campbell wrote: The scheme which Commodore Anson was sent to execute, was certainly well laid; and if the two ships that repassed the Streights "See [64] p. ROGER PAMAN 60 of Le Maire, and thereby exposed themselves to greater dangers, than they could have met with by continuing their voyage, had either proceeded with the Commodore, or had followed him to the island of Juan Fernandez, he would have had men enough to have undertaken something of consequence either in Chile or Peru Before leaving England, Paman had given his paper to his friend Dr.

Hartley, and when he returned, in February , Paman sent it to the Royal Society. He was recommended by Abraham de Moivre, R. Barker and G. Scott February 10th, , with the following description: Mr. Roger Paman of London A Gentleman Extremely well versed in all the Parts of the higher Mathematicks desiring to be a member of this Society we rec- ommend him as personally known to us and likely to become a usefull Member thereof13 He was elected May 12, The preface was dated August 1st, , the Postscript of the preface August 24th, Nourse14 This book, which is the main subject of this chapter, also included an advertisement call for subscriptions for another book of Paman's, giving nJohn Campbell ed.

I, pp. Jesseph writes that Paman returned in [25] p. This must be a misunderstanding. This seems to fit the description of the movements of the Severn and the Pearl given above. No trace of this book has been found, and it is probable that it was not published, due to too few subscribers. We do not know more about Paman, except that he died in Fraser's edition of Berkeley's Works, vol.

To avoid breaking up Paman's exposition too much, I will give my interpretation and comments in sections 5. Definition 5. Barrow, or the Infiniment Petit of the Marquis de l'Hospital; but all the finite Values of a; less than a particular Value, which particular Value is assignable from the Quantities compared: And in the last State of a; I do not consider any of its Values as infinitely great, or as the Maximum Magnum of Dr. Barrow; but I mean thereby all the Values of a; greater than a particular Value, the Assignability whereof depends upon the Quantities compared.

I will skip the proofs here. Proposition 5. Paman neglects the problems of convergence, as usual at the time. Ill, p. In a footnote,29 Paman writes: However harshly the Names of Maximinus or Minimajus may sound, their Existence is evident every where I have put together this proof from several proofs from Paman, to avoid having to give all of the propositions and corollaries in full: Proposition 5.

IV, p. A similar definition is given for last Maximinus and Minimajus, but this is not used in the definition of fluxions. State of x State of a;, any xm Quantity, as pxm, will be greater than the Sum of, or Difference between A and B, in the same State of x. It must have seemed probable that all of these series converge in the first State of a;. Nobody seems to have considered this problem in the 18th. In the following propositions, Paman proves the uniqueness of Maximi- nus and Minimajus: Proposition 5.

Rule 5. See the comment after the proof. The other instances should be similar. I will not discuss this, as it is not necessary for the definition of fluxion. Fluxions; and it's first Fluxion is the same as its Increment or Decrement, and is that Fluxion, to which all the rest are referred, and may be called the radical Fluxion, or fluxionary Unit; and it may be observed, that the making any Quantity the radical Quantity of the rest answers to making one of the Fluents to flow uniformly, in the Method of Fluxions.

Now it is time to study the definitions a bit closer. Therefore, this extra option is necessary in the definition, if the second fluxion of for instance x2 is to be defined. But the same is true of the definition of first fluxion, where Paman has forgotten this, see p. In this section, I will take this risk, as I will be looking at Paman's mathematics from our point of view, testing it on functions he never considered. The following can therefore not be anything else than my interpretation of Paman. A "radical Quantity" is about the same thing as what we call a "vari- able" , even though Paman implies that an expression can be composed of this variable only by taking powers of it, and by multiplying by scalars, which means that Paman is thinking of polynomials or power series.

And all that is understood by a Minimajus is such a Quantity, as being greater than another, cannot be diminished by any Quan- tity of the same Kind without becoming less. Another difference is that Paman says that "a Maximinus, is such a Quantity as being less than another Taken literally, Paman's explanation means that the constant function 1 has no Maximinus.

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It would be nice to say that this is only an oversight of Paman, or a modern misinterpretation of the words "less than", but we see from his definition of second fluxion p. This is also seen another place,49 where he writes Breidert goes on: Paman definiert die Fluxion als das Supremum bzw. Infimum des Differenzenquotienten, d. Thus the concepts of first Maximinus and Minimajus become very simple when dealing with power series.

Paman did not think of this kind of functions. Using y x — y x — A gives the same result. Today we would expect a mathematician using the expression xiA in a definition to prove that the two possibilities give the same result. Paman, however, leaves this unsaid. There is one minor error in Paman's definition of fluxion, however.

With the current definition, y — 2x has no fluxion, because the difference will be 2A, which has no first Maximinus or Minimajus. Therefore it is necessary to change into as Paman has done in the definition of second Fluxion : "And I call that A Quantity, which is either equal to, or the first Maximinus or Minimajus His functions could not oscillate, and "his variable was always less than or greater than his limit", according to Boyer see [6] p. Paman has arguments too, of course: Proposition 5. Sect, iv. II, p. For instance, he defines tangent using Maximinus' and Minimajus'.

I will not go into the details of Paman's geometrical propositions. Paman, on the other hand, does not use these concepts, in fact he makes a point of not using them.

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This is, in a way, a positive development, since the intuitive concepts hide the underlying limit arguments, while Paman has to argue without these "short cuts", and thereby make the limit arguments clearer. Newton never calculated a single fluxion, but always a ratio Paman, on the other hand, calculates the fluxion as a number.

He always points out what is the radical quantity as he calls it , and that the fluxion of x is 1 if x is the radical quantity.

First Name Basis: Kai Mana Henry

But as all of these terms cover concepts developed by Paman, we can not blame him for using new names. But it must be considered whether these concepts are really useful. In a limited sense, they certainly are — Paman managed to give a foun- dation using these concepts, and he needed all of them. But we would not be able to define the derivative using these terms, as we have to consider more complicated functions than Paman did. However, the states of a; are closely related to the very important concept of neighbourhoods the latter of course being used far more generally than just on R , and the first Maximums' and Minimajus' are cousins of Hminf and Mm sup although more powerful.

It must therefore be said that far from introducing concepts for the sake of introducing them, Paman introduced interesting new concepts that were 67[6] p. But at the same time he wanted to avoid "the Taedium and Perplexity" of their ad absurdum proofs see p. Paman managed to keep to the rigorous proofs and breaking with geometry at the same time; in much of his book geometry plays little role. The answer, I think, is that the question is wrong. MacLaurin already was a "hero", while Paman, as far as we know, was unknown.

It is interesting to see64 at least five different persons writing to MacLaurin about the Treatise 62[42] preface p. It must also be said that MacLaurin's work included much more than Paman's in the way of interesting mathematical theorems and methods. It would have been nice if mathematicians of the time had read and understood Paman's book. It would certainly have been a more suitable starting point for getting where we are today, than MacLaurin's geometry- oriented treatise. But then, that is not what history is about.

It must be mentioned here that Paman himself points out that his work is independent of MacLaurin's A Treatise of Fluxions, and at the same time mentions that the two books at times agree strongly with each other. Then the "maximinus" of the expression is the last value in the last state of x, and the "minimajus" was the first value in the first state of a:. George Mark "All are impatient for your Treatise of Fluxions Patrick Murdock "All the world is impatient for the Publication of it, and none more than the Parisians I do not doubt how much pleasure reading a book by you will give me, announcing subjects of great interest.

David Hartley. I mention this, because at my Return I found Mr. MacLaurin had published his Treatise of Fluxions, in which I have the Pleasure to see my Agreement with that ingenious Gentleman, in several Conclusions concerning Flexures and the greatest and least Ordinate. One thing is that "maximums" is not confined to the last state of x, in fact, only the first maximinus is involved in the definition of fluxion. Moreover, Paman's work perhaps included more new terminology than usual at the time, but at least it was well defined. If 18th century mathematicians understood Descartes and Leibniz, they would certainly have understood Paman as well, given time to read it.

In the same year, , Breidert too included a discussion on Paman. Douglas Jesseph in included a somewhat more lengthy account,71 which is both clear and correct, and includes a lot more mathematical detail than Breidert. His use of the concepts "first State of x" and "Maximinus and Minimajus" make his theory remarkably modern — and Paman's proofs are short compared with the more geometrical proofs of his predecessors. In my view, Paman's work is therefore superior to Robins' and MacLau- rin's concerning the foundation of the method of fluxions — in addition to introducing important concepts which could have been used in other con- nections.

Chapter 6 The Analyst Controversy's effect on England's mathematical isolation In a article, Elaine Koppelman wrote the following about the mathe- matical development in the eighteenth century: During the eighteenth century, England remained in intellectual isolation from the Continent. The work of the great Continen- tal analysts — the Bernoullis, Euler, Lagrange and Laplace — was not assimilated. This had several causes. One external fac- tor, undoubtedly, was the fact that England and France were at war much of the time. Also, there may have been a feeling of arrogance among English intellectuals raised by the admira- tion of Continental philosophers for the English political system and her achievements in industry and commerce.

Furthermore, the Newton-Leibniz priority battle1 left those in academic circles with the belief that it was a dishonor to Newton to abandon his notation or methods. This idea is present in Jesseph, where he says: Many British presentations of the calculus in the s and s were concerned with answering Berkeley's charges in The Ana- lyst to the effect that the calculus of fluxions was obscure and unrigorous.

This may account for the British preference for the Newtonian formulation of the calculus, since the method of fluxions was frequently touted as a rigorous alternative to the in- finitesimal methods in favor on the Continent. Whether such an explanation can hold may be a matter for separate investigation - -4 I will try to face this question in this chapter. To do this, it is natural to look at how the views on the relationship between Newton's and Leibniz' foundations changed in time — looking at the situation both before and after the publication of The Analyst.

As I do not have unlimited access to primary sources,5 I will not be able to do a thorough investigation, but I hope I will find some points of interest. In the years before the Newton-Leibniz controversy, however, there seems to have been a bit of confusion. For instance, Abraham de Moivre uses "flux- ions" in the meaning of "infinitely small" , as does Newton's once very close friend Fatio de Duillier , Roger Cotes and John Harris ,6 who writes ern Geometry Asserted.

I have tried to include all the information I have found which bear on this subject in these three books, instead of choosing some and discarding some. But of course the conclusion of this chapter will be dependent on the selection process done by Cajori, as well as whatever unintentional omissions I have made. Isaac Newton calls very properly by this name of Fluxions. But as Newton's supporters wanted to show that Leibniz had plagiarized Newton, and Leibniz' supporters accused Newton of the same, the outcome was more confusion — both sides focusing on the similarities and not on the differences between the methods.

In John Keill wrote: If in place of the letter o, which represents an infinitely small quantity in James Gregory's Geometric pars universalis , or in place of the letters a or e which Barrow employs for the same thing, we take the x or y of Newton or the dx or dy of Leibniz, we arrive at the formulas of fluxions or of the differential calculus. Leibniz calling those Quantities Differences, which Mr. Newton calls Moments or Fluxions; and marking them with the letter d, a mark not used by Newton.

Sloane of , quoted in [9] p. Newton wrote it, and the paper claimed, not surprisingly, that Newton was the sole inventor of the calculus. See Appendix A for more on this. Much later, in , Edmund Stone published his translation of l'Hopital's Analyse des Infinements Petits, where every occurrence of the word "differ- ence" was translated with "fluxion", and dx was replaced by i.

This year could therefore well mark the beginning of a new awareness of the problems of rigour. At least, it seems that people had started to study Newton's explanations, and seen that they were different from the ones on the Continent. Cajori writes about this period: Excepting only in Benjamin Martin, the definition of a fluxion as a 'differential' nowhere appears. Therein we see a step in advance. Prom this year on, the fluxional calculus could be treated as any other part of mathematics, many people thought that the "foundational crisis" was over.

But it was also a common view that Newton's own method was solid enough. MacLaurin himself, for instance, wrote that Sir Isaac Newton accomplished what Cavalerius wished for, by inventing the method of fluxions, and proposing it in a way that admits of strict demonstration, which requires the supposition of no quantities but such as are finite, and easily conceived. Thomas Simpson, for instance, in , wrote: And it appears clear to me, that, it is by a diligent cultivation of the Modern Analysis, that Foreign Mathematicians have, of late, been able to push their Researches farther, in many particulars, than Sir Isaac Newton and his Followers here, have done: tho' it must be allowed, on the other hand, that the same Neatness, and Accuracy of Demonstration, is not every-where to be found in those Authors, owing in some measure, perhaps, to too great a disregard for the Geometry of the Ancients.

This information is taken from Cajori [9] p. Olynthos Gregory wrote in : [I have] long been of the opinion that, in point of intellectual con- viction and certainty, the fluxional calculus is decidedly superior to the differential and integral calculus. It is certainly remark- able that in Great Britain there was achieved in the eighteenth century, in the geometrical treatment of fluxions, that which was not achieved in the algebraical treatment until the nineteenth century From to , people were more careful about which notation to use, but not so much about the foundation.

This might be because in the Newton-Leibniz controversy both sides claimed that the methods were the same, but with different notations. From to , people were more careful to avoid infinitesimals, while after , there existed a solid foundation for the calculus, and many recognized this. But it seems that the recognition that Newton's and Leibniz' theories were fun- 20Olynthos Gregory in the eleventh edition of Hutton's Course of Mathematics vol. Quoted in [21] p. Please note that Cajori never read Paman's work. His assertion about geometrical vs. It also appears that the English mathematicians saw that their way of viewing the calculus could be given a rigorous foundation MacLaurin , while not believing that the same was possible for the Continental way.

This must necessarily be connected with the Analyst controversy, especially when we see MacLaurin being used as proof that the principles of fluxions are valid. It is not unreasonable to believe that this feeling of superiority when it comes to foundations might work against changing into the Continental notation and foundation. This, together with the state of war and the respect for Newton, might have been enough to stop this change. England's contributions in the following years were meagre in comparison to those of the Continent.

This is the reason why Berkeley's Analysthas been treated as a disaster for British mathematics by some writers. Chapter 7 Conclusion Isaac Newton, the inventor of the method of fluxions, never found a satis- factory foundation for it. Instead, he used two different ways of explaining it — the method of fluxions and the method of prime and ultimate ratios.

The metod of fluxions relied heavily on intuitive concepts like motion and instantaneous velocity, and did not get rid of infinitesimals. The method of prime and ultimate ratios, on the other hand, relied on some sort of limit idea that he never managed to explain well. In addition, he pretended that he never had changed his mind at all. George Berkeley saw that this was the perfect area in which to attack mathematicians. How could mathematicians attack faith in religion, and at the same time base mathematics on it? His critique was just — although he was guilty of misrepresenting the theory to make it seem worse than it was.

Some of the answers to him were unconvincing. But more important, the mathematicians began to quarrel about what was Newton's true meaning. Thereby, Berkeley was proven right. The first to answer Berkeley, was Philalethes Cantabrigiensis. He did not try to give an explanation of the theory, and therefore he is mostly interesting for being the first to answer Berkeley, and for being so much criticized by historians of mathematics. He succeeded in finding errors in Berkeley's criticism, and thereby injured the credibility of The Analyst. Some of the answers managed to give a firm foundation for the method of fluxions.

Robins, MacLaurin and Paman did this, in different manners. Robins gave an explanation of Newton's theories, with clearer definitions and rigorous proofs, while still depending on intuitive concepts. Given his position in the learned world, it was only natural that his answer was the one to be remembered by most, especially when his work also included much interesting mathematics. However, it is interesting to see Paman's remarkably modern work — with concepts resembling our neigh- bourhood concept and liminf and limsup.

Sadly, his work was apparently not studied at the time, and did not influence the later developments. The growing awareness that there existed good answers to the foun- dational questions, may well have contributed to English mathematicians' preference of their own notation and foundation. Newton was the first inventor, Leibniz was the first one to publish. This situation was the perfect opportunity for a controversy. In a way it is strange that the controversy did not start sooner — already in Fatio de Duillier publicly called Leibniz a "second discoverer", but Leibniz' only reaction was to write a private complaint to Wallis.

In it was Leibniz' turn — he wrote anonymously about Newton that in place of Leibnizian differences Mr. Newton employs fluxions, and has ever employed them. Doubtless this use of "moment" with reference to time suggested the more extended and general use of the term "momentum" or "momenta" as found in the Principia' 1 and later publications. It indicates the almost complete exclusion 1 See our In this connection De Morgan's remarks are of interest: 5 "In , Newton in the Quadratura Curvarum renounced and abjured the infinitely small quantity ; but he did it in a manner which would lead any one to suppose that he had never held it.

And yet, there is something like a recognition of some one having used infinitely small quantities in Fluxions, contained in the following words : volui ostendere quod in Methodo Fluxionum non opus sit figuras infinite parvas in Geometriam introducere : nothing is wanted except an avowal that the some one was Newton himself.

The want of this avowal was afterwards a rock of offence. Berkeley, in the Analyst, could not or would not see that Newton of an d Newton of were of two different modes of thought. NEWTON 35 fluxions cannot be given on the basis of infini- tesimals or that infinitely small quantities are impossible; for he says, 1 "the analysis may be performed in any kind of figures whether finite or infinitely small, which are imagined similar to the evanescent figures.

In , fluxions are "in the first ratio of the nascent augments," or "in the ultimate ratio of the evanescent parts. At any rate, the history of fluxions shows that these expressions did not meet the demands for clearness and freedom from mysticism. Newton himself knew full well the logical difficulty involved in the words "prime and ultimate ratios"; for in he said, 3 "it is objected, that there is no ultimate proportion of evanescent quantities ; because the proportion, before the quantities have vanished, is not ultimate; and, when they have vanished, is none.

He does so simply by stating the difficulty in another 1 See our 33, But the answer is easy : for by the ultimate velocity is meant that It might be argued that such a return was necessary in the second edition of the Principia, , unless the work were largely re- written. Newton's Analysis per cequationes numero terminorum infinitas was first printed in , and might have been rewritten so as to exclude infini- tesimals as fully as was done in the Quadrature of Curves of But the infinitely little is per- mitted to remain.

THE earliest printed publication in Great Britain on the new calculus was from the pen of John Craig, a Scotsman by birth, who settled in Cambridge and became a friend of Newton. Later he was rector of Gillingham in Dorsetshire. He was "an inoffensive, virtuous man," fond of mathe- matics. In ne published at London a book entitled, Methodus figurarum.

At that time nothing could be known about fluxions except through private com- munication. Craig used in the calculus of Leibniz and also the notation of Leibniz. The meanings of dp, dy, dx, etc. In Craig published another book in which the notation of Leibniz is used. In preparing the book of he had received from Newton the binomial theorem which he used before it had appeared in print, but he had no communication about fluxions. In the issue No. Craig submitted to Newton one of his early manuscripts probably the one printed in With regard to this event De Morgan wrote to Hamilton, the inventor of 'quater- nions : Few of us know that Leibniz was perfectly well known in England before the dispute, and that Newton's first provocative to an imperfect publica- tion was ds and infinitely small quantities paraded under his own eyes by an English writer Craig , who lent him his MSS.

The book does not discuss fundamentals, and no explanation of x is given. Graves, vol. At any rate, in there appeared the account of fluxions in Wallis's Algebra. Abraham De Moivre, a French mathemati- cian who in , after the revocation of the Edict of Nantes, came to London, contributed in to No. In the same number of the Transac- tions, the astronomer Edmund Halley has an article on logarithms in which he uses infinitely small ratiunculce and differentiolce, but neither the nota- tion of Leibniz nor that of Newton.

In , David Gregory used in No. This publi- cation is noted as containing a statement which started the Newton-Leibniz controversy on the invention of the Calculus. Other writings that do not define their terms are the Fluxionum methodus inversa, , by the London physician, George Cheyne, and De Moivre's Animadversions in D. Georgii Cheynai Tractatum, London, He was at one time Secretary of the Royal Society.

In he published at London A New Short Treatise of Algebra, which devotes the last 22 pages, out of a total of pages, to fluxions. It is the first book in the English language in which this subject is treated. The doctrine of fluxions is the " Arithmetick of the Infinitely small Increments or Decrements of Indeterminate or Variable Quan- tities, or as some call them the Moments or Infin- itely small Differences of such Variable Quantities.

These Infinitely small Increments or Decrements, our incomparable Mr.

Isaac Newton calls very pro- perly by this name of Fluxions " p. A few lines further on it says that Newton "calls the celerity or Velocity of the Augmentation o Diminu- tion of these Flowing Quantities, by the name of Fluxions. As authors on fluxions, Harris in 1 J. John Harris also published a Lexicon Tech- nicum, of which the second volume, London, , contains an article, "Fluxions.

Robartes, Esq. If a Quantity gradually increases or decreases, its immediate Increment or Decre- ment is called its Fluxion, Or the Fluxion of a Quantity is its Increase or Decrease indefinitely small. Since xx Authors' Names who have written of Fluxions: D. Part' Hospital, Fr.

Carre, Paris, ; Mr. See P kilos. Humphry Dittoris Institution of Fluxions. In the above list of writers are Charles Hayes and Humphry Ditton, authors of English texts now demanding our attention. Hayes starts his elucidation of fundamentals p. Now those infinitely little Parts being extended, are again infinitely Divisible ; and these infinitely little Parts of an infinitelylittle FlG Part of a given Quantity, are by Geometers call'd Infinite - siuice Infinitesimarum or Fluxions of Fluxions.

Again, one of those infinitely little Parts may be conceiv'd to be Divided into an infinite Number of Parts which are call'd Third Fluxions, etc. He rejects xzy and xzy ' ' as being incomparably less " than xzy. The same year in which Hayes wrote this first English book on fluxions which could make any claim to attention, saw the appearance of Newton's Quadratura Curvarum. The contrast in the defini- ' tion of "fluxion" was sharp. Hayes called it "an infinitely small increment " ; Newton called it a "velocity," a finite quantity.

William Jones, in his Synopsis Palmariorum Matties eos, London, , devotes a few pages to fluxions and fluents, using the Newtonian notation. Like so many other English writers on fluxions during the eighteenth century, he had not been at either of the great universities. He states in his preface that he has also consulted and drawn from the writings of John Bernoulli and some other Continental writers.

The reader of Ditton's book is impressed by the fact that he labours strenuously to make every- thing plain. He takes the reader fully into his confidence. This is evident in the extracts which follow pp. I conceive we may say without Scruple, that the Fluxions are the velocities of those Increments, con- sider'd not as actually generated, but quatenus Nascentia, as arising and beginning to be generated.

As there is a vast difference between the Increments considered as Finite, or really and actually generated ; and the same considered only as Nascentia or in the first Moment of their Generation : So there is as great a difference also between the Velocities of the Increments, consider'd in this two fold respect. By Humphry Ditton, London, Because there is speaking strictly and accurately an Infinity of Velocities to be consider'd, in the Generation and Production of a Real Increment ; So that if we conceiv'd the Fluxion, to be the Velocity of the Increment, as actually Generated ; we must conceive it to be an Infinite Variety or Series of Velocities.

Whereas the Velocity, with which any sort of In- crement arises, or begins to be generated ; is a thing that one may form a very clear and distinct Idea of, and leaves the Mind in no Ambiguity or Confusion at all. However, if we take those Particles of time exceeding small indeed, and Neglect the Acceleration of the Velocity as inconsiderable, we may say the Fluxions are proportional to those In- crements ; remembering at the same time, that they are but nearly, and not accurately so.

If in the Differential Calculus, some Terms are rejected and thrown out of an Equation, because they are nothing Comparatively, or with respect to other Terms in the same Equation ; that is, because they are infinitely small in proportion to those other Terms, and so may be neglected upon that Score: On the other hand, in the Method of Fluxions, those same Terms go out of the Equation, because they are multiplied into a Quantity, which. Neiwentiit chuse to term them, I cannot but take notice of a notion, which that Excellent and In- genious Person advances in.

I confess I cannot discover the truth of this. Now Mr. Neiwentiit will hardly allow his Infinitesimal to be nothing ; and yet Next he takes up algebraical expressions. To find the fluxion of x n , he lets x flow uniformly and re- presents the augment of x in a given particle of time by the symbol o. Taking o as a very small quantity, he lets the ex- pressions ox, oy represent the moments, or increments of the flowing quantity z, y generated in a very small part of time.

Ditton considers the in- crements as finite p. Newton had before expresly told us ; that the Increments generated in a very small Particle of time were very nearly, as the Fluxions. Ditton's first edition appeared at a time when the Newton-Leibniz controversy was under way. Leibniz had appealed to the Royal Society for justice. From this book the early use of infinitely small quantities on the part of Newton is conspicuously evident.

The book makes it clear also that some of Newton's warmest supporters were guilty of gross inaccuracy in the use of the word ' ' fluxion. Newton's Analysis per cequationes numero terminorum infinitas, which was sent on July 31, , through Barrow to Collins, and which was first published at London in , was reprinted in the Commercium Epistolicum. In this Analysis in- finitely small quantities are used repeatedly, but the word "fluxion" and the fluxional notation do not occur. In a letter to H. Sloane, who was then Secretary of the Royal Society of London, written in answer to a letter of Leibniz dated March 4, 17 1 1, John Keill, professor of astronomy at Oxford, re- counts the achievements of Isaac Barrow and James Gregory, and says: "If in place of the letter o, which represents an infinitely small quantity in James Gregory's Geometric pars universalis , or in place of the letters a or e which Barrow em- ploys for the same thing, we take the x or y of Newton or the dx or dy of Leibniz, we arrive at the formulas of fluxions or of the differential calculus.

He comes so near to this as to be guilty of lack of caution, if not of inaccuracy. More serious is a statement further on. The en- 1 "Nam si pro Litera o, quae in Jacobi Gregorii Parte Matheseos Uni- versali quantitatem infinite parvam reprsesentat ; aut pro Literis a vel e quas ad eandem designandam adhibet Barrovius ; ponamus x vel y Newtoni, vel dx seu dy Leibnitii, in Formulas Fluxionum vel Calculi Differ en tialis incidemus " p.

Leibniz calling those Quantities Differences, which Mr. Newton calls Moments or Fluxions ; and marking them with the letter d, a mark not used by Newton. Joseph Raphson, in his History of Fluxions which appeared as a posthumous work at London, in , printed in English, and in the same year also in Latin, the Latin edition containing new corre- spondence bearing on the Newton-Leibniz contro- versy , says on p.

Brook Taylor brought out at London in his Methodus incrementorum directa et inversa, in which he looks upon fluxions strictly from the stand- point of the Newtonian exposition in the Quadrature of Curves, James Stirling uses x and y as infinitesimals in his Linece tertii or dints, Oxford, He draws the infinitely small right triangle at the contact of a curve with its asymptote, the horizontal side being "quam minima" and equal to x, the vertical side being y. In the appendix to this booklet of , x and y are again infinitely small.

In his Methodus dijferentialis , London, , there is no direct attempt to explain fundamentals, any more than there was in , but on p. For twenty-four years after Ditton no new text appeared. This Analysis may ever be said to go beyond the Bounds of Infinity itself; as not being confined to infinitely small Differences or Parts, but discovering the Ratio's of Differences of Differ- ences, or of infinitely small Parts of infinitely small Parts, and even the Ratio's of infinitely small Parts of these again, without End.

So that it not only contains the Doctrine of Infinites, but that of an Infinity of Infinites. It is an Analysis of this kind that can alone lead us to the Knowledge of the true Nature and Principles of Curves : For Curves being no other than Polygons, having an Infinite Number of Sides, and their Differences arising altogether from the different Angles which their infinitely small Sides make with each other, it is the Doctrine of Infinites alone that must enable us to determine the Position of these Sides, in order to get the 1 The Method of Fluxions, both Direct and Inverse.

Stone, F. It was the Discovery of the Analysis of Infinites that first pointed out the vast Extent and Fecundity of this Principle. Yet this itself is not so simple as Dr. Barrow afterwards made it, from a close Consideration of the Nature of Polygons, which naturally represent to the Mind a little Triangle consisting of a Particle of a Curve contained between two infinitely near Ordinates , the Differ- ence of the correspondent Absciss's ; and this Triangle is similar to that formed by the Ordinate, Tangent, and Subtangent.

The Defect of this Method was supplied by that of Mr. Barrow and others left off: His Calculus has carried him into Countries hitherto unknown. But the Method of Mr. Leibnitz'z is much more easy and expeditious, on account of the Notation he uses. That of our Author is much easier, tho less Geo- metrical, who calls a Differential or Fluxion the infinitely small Part of a Magnitude.

In De PHospitaPs treatise, as translated by Stone, we read : "The infinitely small Part whereby a variable Quantity is continually increased or decreas'd, is called the Fluxion of that Quantity. De 1'HospitaPs wording is "La portion infiniment petite dont une quantite variable augmente ou diminue continuelle- ment, en est appellee la Difference. Then follow two postulates : "Grant that two Quantities, whose Difference is an infinitely small Quantity, may be taken or used indifferently for each other : or which is the same thing that a Quantity, which is increased or decreas'd only by an infinitely small Quantity, may be consider'd as remaining the same.

Further on in Stone's translation p. The earliest treatment of the new analysis which became current in England was that of Leibniz. The Scotsman Craig used it for over a quarter of a century before rejecting it in favour of fluxions. Harris, Hayes, and Stone drew their inspiration from French writers who followed Leibniz.

He also uses the infinitesimal triangle. Hayes and Stone have no hesitation in speaking of " fluxions of fluxions," and ''infinitely little parts of an in- finitely little part. The dropping of such quantities from an equation was usually permitted without scruple. What an opportunity did this medley of untenable philosophical doctrine present to a close reasoner and skilful debater like Berkeley! The arguments in the Analyst were so many bombs thrown into the mathematical camp. The views expressed in the Analyst are fore- shadowed in Berkeley's Principles of Human Know- ledge , published nearly a quarter of a century earlier.

The "Infidel mathematician," it is generally supposed, was Dr. Mathe- maticians complain of the incomprehensibility of religion, argues Berkeley, but they do so unreason- ably, since their own science is incomprehensible. Wherein it is examined whether the Object, Principles, and Inferences of the Modern Analysis are more distinctly conceived, or more evidently deduced, than religious Mysteries and Points of Faith. And it seems still more difficult to conceive the abstracted velocities of such nascent imperfect entities.

But the velocities of the velocities the second, third, fourth, and fifth velocities, etc. That is, they consider quantities infinitely less than the least discernible quantity ; and others infinitely less than those infinitely small ones ; and still others infinitely less than the preceding infinitesimals, and so on without end or limit " 6.

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 are deficient, or lesser by one-half of their moments, the rectangle was A-itfxB-i b, i.

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The point of getting rid of ab cannot be obtained by legitimate reasoning. 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. Therefore the momentum must be an infinitesimal. For aught I see, you can admit no quantity as a medium between a finite quantity and nothing, without admitting infinitesimals" n. Berkeley argues : "But it should seem that this reasoning is not fair or conclusive.

For when it is said, let the increments vanish, i. 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 expres- sions, and everything else derived from the supposi- tion of their existence, to vanish with them All which seems a most inconsistent way of arguing, and such as would not be allowed of in Divinity Nothing is plainer than that no just conclusion can be directly drawn from two inconsistent suppositions It may perhaps be said that [in the calculus differentials] the quantity being infinitely diminished becomes nothing, and so nothing is rejected.

But, accord- ing to the received principles, it is evident that no geometrical quantity can by any division or sub- division 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 demon- strations, and that he was not enough pleased with any notion steadily to adhere to it" In answer to this you will perhaps sayj, that the conclusions are accurately true, and thajt therefore the principles and methods from whence they are derived must be so too.

Berkeley proceeds to show that correct results are derived from false principles by a compensation of errors, a view advanced again later by others, particularly by the French critic L. That is, if dy is the true increment, then in ydx I dy there is an "error of defect. There is here an ' ' error of excess.

But how can we conceive a velocity by help of such limits? It necessarily implies both time and space, and cannot be conceived without them. And if the velocities of nascent and evan- escent quantities, i. I answer that if, in order to arrive at these finite lines proportional t to the fluxions, there be certain steps made use of which are obscure and inconceivable, be those finite lines themselves ever so clearly conceived, it must nevertheless be acknowledged that your pro- ceeding is not clear nor your method scientific " Berkeley discusses this matter with reference to a geometric figure, and argues that "a point there- fore is considered as a triangle, or a triangle is supposed to be formed in a point.

Which to con-i ceive seems quite impossible " The Veloci- ties of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor 1 yet nothing. May we not call them the ghosts of departed quantities? Then follow sixty-seven queries, of which the sixteenth is a good specimen : " Qu. Whether certain maxims do not pass current among analysts which are shocking to good sense? And whether the common assumption, that a finite quantity divided by nothing is infinite, be not of this number?

A reply to Berkeley's Analyst was made by the noted physician, James Jurin, at one time a student in Trinity College, Cambridge, who had imbibed Newtonian teachings from Newton himself. Jurin wrote under the pseudonym of " Philalethes Cantabrigiensis. Philalethes says that the charge in the Analyst "consists of three principal points : i Of Infidelity with regard to the Christian Religion.

Wherein it is examined, How far the Conduct of such Divines as intermix the Interest of Religion with their private Disputes and Passions, and allow neither Learning nor Reason to those they differ from, is of Honour or Service to Christianity, or agreeable to the Example of our Blessed Saviour and his Apostles, By Philalethes Cantabrigiensis. Ne Deus intersit, nisi dignus vindice nodus Inciderit.

London : Printed for T. Cooper at the Globe in Ivy-Lane. Price is. The early part of Jurin's reply is given to a discussion of the religious side. If there is no more certainty in modern analysis, argues Jurin, than in the Christian religion, this comparison brings no honour to Christianity ; it is not true that mathematicians are infidels, leading others to infidelity.

If it were true, this fact ought not in prudence to be published. Even if it be shown that the method of fluxions is built upon false principles, will it follow that all other parts of mathematics rest on inaccurate and false reasoning? Your attack, I surmise, is really, not so much in the interest of Christianity, as to demonstrate your superiority as a reasoner, by showing Newton and Barrow, two of the greatest mathematicians, less clear and just than you are.

But because a mathe- matician "is thought to reason well in Geometry," his " decisions against the Christian Religion " will not "pass even upon weak and vulgar minds. You say that the Marquis de 1'Hospital, in his Analyse des infiniment petits, Prop. Which of those two will you call the moment of AB? Let me advise you hereafter to "first examine and weigh every word he [Newton] uses.

As to your second instance of false reason- ing, in Newton's book on Quadratures, apparently that is "so truly Boeotian a blunder" that I know not how "a Newton could be guilty of it. As to the third head of your objections, since Newton did not reason falsely, ' ' he had no occasion to make use of arts and fallacies to impose upon his followers. I should sally out and attack you in your own. Not as much as the thousand-millionth part of an inch. Jurin ends with a discussion of Lock on abstract ideas. Walton's First Reply. Little is known about John Walton.

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He was Professor of Mathematics in Dublin, and partici- pated in this controversy. Otherwise, practically nothing about him has been handed down. His reply to Berkeley was published in at Dublin. Walton begins by stating that inasmuch as the credulous may " become infected" by Berkeley's attack on fluxions, it seems necessary to give a short account of the nature of fluxions.

Siquid novisti rectius istis, candidus imperti : bi non, his utere mecum. In the fulness of his Sufficiency he shall be in Straits : Every Hand of the Wicked shall come upon him. Dublin, Printed ; and reprinted at London, and sold by J. There are certain determinate Limits to which all such Proportions perpetually tend, and approach nearer than by any assignable Difference, but never attain before the Quantities themselves are infinitely diminish'd ; or 'till the Instant they evanesce and become nothing.

Walton next quotes a Latin passage from the Quadratures Cutvarum. He says that Berkeley seems "to have been deceived by an Opinion that there can be no first or last Ratios of mathematical Quantities," but Walton insists that if quantities are generated together, or if they vanish together, they will do so "under certain Ratios, which are their first or last Ratios. Walton's Vindication follows Newton's ex- position closely ; Berkeley's claim that Walton followed in Jurin's track and borrowed from him, is, I believe, incorrect. Take the vital question of rejecting infinitesimals : Jurin claims that, being so very small, they do not appreciably affect the result ; Walton takes the stand that there is no rejection whatever of infinitesimals.

The main criticism to be passed on Walton's first essay con- sists, in our judgment, in a failure to meet Berkeley's objections squarely and convincingly. Berkeley's Reply to Jurin and Walton In Answer to a Pamphlet of Philalethes Cantabrigiensis. Also an Appendix concerning Mr. Waltorfs Vindication. I say that an infidel, who believes the doctrine of fluxions, acts a very inconsistent part in pretending to reject the Christian religion because he cannot believe what he doth not comprehend" 7.

In a criticism A Revieiv of the Fiery Eruption, etc. In the application of this method it became neces- sary to consider these quantities, sometimes in a nascent, and at other times in an evanescent state, by which ingenious contrivance they could be made either continually to tend to and at last absolutely to become nothing, or vice versa, according to the intention and occasions of the Artist. Now by extending this noble invention to the two religions, it evidently appeared, that, from the time of the first coming of Christ, Judaism entered into its evanescent state, as on the other hand Christianity did into a nascent state, by which means both being put into a proper flux, one was seen continually decaying, and the other continually improving, till at last by the destruction of the Temple Judaism actually vanished and became nothing, and the Christian religion then bursted out a perfectly generated Entity.

As the great author of the mathematical method of fluxions had for very good reasons studiously avoided giving any definition of the precise magni- tude of those moments, by whose help he discovers the exact magnitude of the generated quantities, so our Author [Warburton] by the same rule of application, and under the influence of the same authority, was fairly excused from defining that precise degree of perfection and imperfection in which the two religions subsisted, during the respective evanescent and nascent state of each, by the help of which he discovered the precise time when Judaism was perfectly abolished, and Christianity perfectly established.

But we may well suppose, that the most alluring charm in this extraordinary piece of ingenuity, was the creating of a new character by it : For questionless he may now be justly stiled the great founder and inventor of the ftuxionary method of theology. This fancy of a necessary connexion between the Temple-edifice, and the being of Christianity,. What have you to say in answer to this? Do you attempt to clear up the notion of a fluxion or a difference? Nothing like it" Berkeley quotes from Newton's Principia and Quadrature of Curves, and then asks, "Is it not plain that if a fluxion be a velocity, then the fluxion of a fluxion may, agreeably thereunto, be called the velocity of a velocity?

In like manner, if by a fluxion is meant a nascent augment, will it not then follow that the fluxion of a fluxion or second fluxion is the nascent augment of a nascent augment? In answer to this you allege that the error arising from the omission If you mean to defend the reasonableness and use of approxi- mations I have nothing to say.

That the method of fluxions is supposed accurate in geometrical rigour is manifest to whoever considers what the great author writes about it In rebus mathernaticis errores quam minimi non sunt contemnendi" 25 ; our Is it a finite quantity, or an infinitesimal, or a mere limit, or nothing at all?

If you take it in either of the two former senses, you con- tradict Sir Isaac Newton. And, if you take it in either of the latter, you contradict common sense ; it being plain that what hath no magnitude, or is no quantity, cannot be divided " You observe that the moment of the rectangle determined by Sir Isaac Newton, and the increment of the rectangle determined by me are perfectly and exactly equal, supposing a and b to be diminished ad infinitum : and, for proof of this, you refer to the first lemma of the first section of the first book of Sir Isaac's Principles.

I answer that if a and b are real quantities, then ab is some- thing, and consequently makes a real difference : but if they are nothing, then the rectangles whereof they are coefficients become nothing like- wise : and consequently the momentum or incre- mentum, whether Sir Isaac's or mine, are in that case nothing at all. As regards Newton's evane scant jam augmenta ilia our 32 , Berkeley argues that it means either "let the increments vanish," or else "let them become infinitely small," but the latter "is not Sir Isaac's sense," since on the very same page in the Introduction to the Quadrature of Curves he says that there is no need of considering infinitely small figures.

Taking advantage of the fact that the Newton of the Principia differed from the Newton of the Quadratura Curvarum , Berke- ley broke out into the following philippic: "You Sir, with the bright eyes, be pleased to tell me, whether Sir Isaac's momentum be a finite quantity, or an infinitesimal, or a mere limit? If you say a finite quantity ; be pleased to reconcile this with what he saith in the scholium of the second lemma of the first section of the first book of his Principles our 12 : Cave intelligas quantitates magniiudine determinatas , sed cogita semper diminuendos sine limite.

If you should say, it is a mere limit ; be pleased to reconcile this with what we find in the first case of the second lemma in the second book of his Principles our 17 : Ubi de lateribus A et B deerant momentorum dimidia, etc. I should be very glad a person of such a luminous intellect would be so good as to explain whether by fluxions we are to understand the nascent or evanescent quantities themselves, or their motions, or their velocities, or simply their proportions.

In an appendix to the Defence of Free-Think- ing in Mathematics, Berkeley replies to Walton, stating that the issues raised by him had been previously raised by "the other," that he delivered a technical discourse without elucidating anything, that his scholars have a right to be informed as to the meaning of fluxions and should therefore ask him "the following questions. Let him then be asked, what his momentums are good for, when they are thus brought to nothing? I wish he were asked to explain the differ- ence between a magnitude infinitely small and a magnitude infinitely diminished.

Let him be farther asked, how he dares to explain the method of Fluxions, by the Ratio of magnitudes infinitely diminished, when Sir Issac Newton hath expressly excluded all consideration of quantities infinitely small? If this able vindicator should say that quantities infinitely diminished are nothing at all, and consequently that, according to him, the first and last Ratio's are proportions between nothings, let him be desired to make sense of this.

If he should say the ultimate proportions are the Ratio's of mere limits, then let him be asked how the limits of lines can be proportioned or divided? In a second reply 1 to Berkeley, Walton states that in the Appendix to the Defence, Berkeley "has composed a Catechism which he recommends to my Scholars " and which Walton quotes. Printed at Dublin.

Reprinted at London, and sold by J. Roberts, It is a pamphlet of 30 pages.

1911 Encyclopædia Britannica/Infinitesimal Calculus/History 2

If by vanishing he means that they vanish and become nothing as Areas, I grant they do ; but absolutely deny, upon such an Evan- escence of the Gnomon and Sum of the two Rectangles by the moving back of the Sides of the Gnomon till they come to coincide with those of the Rectangle, that nothing remains. For there still remain the moving Sides, which are now become the Sides of the Rectangle,. If a point moves forward to generate a Line, and afterwards the same Point moves back again to destroy the Line with the very same Degrees of Velocity, in all Parts of the Line 1 If a parallelogram is extended in length and breadth and if the original parallelogram be removed, the remaining figure is called the gnomon.

And the Case is the very same with respect to the Rectangle increas- ing by the Motion of its Sides. After some illustra- tions, Walton exclaims: "This is a full and clear Answer to this part of the catechism, and shows that its Author has been greatly mistaken in supposing that I explained the Doctrine of Fluxions by the Ratio of Magnitudes infinitely diminished, or by Proportions between nothings. I do not wonder that this Author should have no clear Ideas or Conceptions of second, third or fourth Fluxions, when he has no clear Conceptions of the common Principles of Motion, nor of the first and last Ratios of the isochronal Increments of Quantities generated and destroyed by Motion.

In order to prevent my being Catechised any more by this Author," Walton makes a confession "of some Part of my Faith in Religion. Jurin brought out a second publication, 1 of pages, which was in reply to Berkeley's Defence of Free-Thinking. Passing by unimportant pre- liminaries, we come to Jurin's definitions of " flow- 1 The Minute Mathematician : or. The Free- Thinker no Just- Thinker. By Philalethes Cantabrigiensis.

He then endeavours to prove the proposition : ' ' The Fluxions, or Velocities of flowing quantities. Jurin succeeds, we think, in establishing the contention that there is no greater difficulty in explaining the second or third fluxion, than there is in explaining the first. Paul's Church by Fluxions, he would be out about three quarters of a hair's breadth : But yonder is one Philalethes at Cambridge, who pretends that Sir Isaac would not be out above the tenth part of hair's breadth.

Hearing this, and that two books had been written in this controversy, the honest gentle- man flew into a great passion, and after muttering something to himself about some body's being over- paid, he went on making reflections, which I don't care to repeat, as not being much for your honour or mine.

Jurin thereupon takes up the rectangle AB. The terms "moment" and "increment" are involved in the discussion of it. Consequently you were mistaken in supposing that the moment of the rectangle AB was the increment of the same rectangle AB. Hence the rectangle ab "is by his Own confession equal to nothing. Jurin does not accept "any one of those senses. Now to me there appears no more difficulty in conceiving this, than in appre- hending how any finite quantity is divided or dis- tinguished into halves.

For nascent quantities may bear all imaginable proportions to one another, as well as finite quantities. Near the close Jurin enters upon the dis- cussion of Berkeley's Lemma, given in the Analyst : "If one supposition be made, and be afterwards destroy'd by a contrary supposition, then everything that followed from the first supposition, is destroyed with it.

Walton as well, inferred that you were charging Newton with committing double errors. The rest of Jurin's ill-arranged article is given either to a renewed and fuller elucidation of his previous contentions or to poetical outbursts. Sure of the soundness of his exposition, he exclaims, " I meet with nothing in my way but the Ghosts of departed difficulties and objections. Walton's Catechism. Walton's Full Answer, This last reply has been called "a combination of reasoning and sarcasm," in which " he affects to treat his opponent as a disguised convert. I can as easily conceive Mr.

Walton should walk without stirring, as I can his idea of motion without space. Walton's is plainly neither of these sorts of motion " ; hence, he argues against Newton. This is one of the inconsistencies which I leave the reader to reconcile. Berkeley translation and velocity, as when he says, " Walton, Catechism. Newton, Principia, Definitions, Scholium, def.

Walton begins 1 by explaining what Newton means by Velocity. And that there is no Measure of Velocity except Time and Space. The Second Edition. With an Appendix, in 'Answer to the Reasons for not replying to Mr. Walton's Fidl Answer. Dublin : Printed by S. Berkeley thinks that "from the generated Velocity not being the same in any two different Points of the described Space it will not follow that Velocity can exist in a Point of Space.

But in this he is mistaken. For the continual Action of a Moving Force necessarily preserves a continual Velocity ; and if the generated Velocity be not the same in any two different Points of the described Space, a Velocity must of Consequence exist in every Point of that Space " p. On the question of first and last ratios it cannot be said that Walton here throws new light. He insists that he explained fluxions not "by the Ratio of Magnitudes infinitely diminish'd, but by the first and last Ratios of Increments generated or destroyed in equal times : that is, by the Ratios of the Velocities with which those Increments begin or cease to exist" p.

I tell him, that Sir Isaac's Momentum is a finite quantity ; it is a Product contained under the ; moving Quantity and its Velocity, or under the moving Quantity and first Ratio of that Space described by it in a given Particle of Time. Remarks Berkeley's Analyst must be acknowledged to be a very able production, which marks a turning- point in the history of mathematical thought in Great Britain.

His contention that no geometrical quantity can be exhausted by division x is in consonance with 1 See our Nevertheless, a reader of Berkeley feels that he spoke in the Analyst with perfect sincerity. But the Analyst was intentionally a publication involving the principle of Dr. Whately's argument against the existence of Buonaparte ; and Berkeley was strictly to take what he found. The Analyst is a tract which could not have been written except by a person who knew how to answer it.

But it is singular that Berkeley, though he makes his fictitious character nearly as clear as afterwards did Whately, has generally been treated as a real opponent of fluxions. Let us hope that the arch Archbishop will fare better than the arch Bishop. Sir William Rowan Hamilton once wrote De Morgan : ' ' On the whole, I think that Berkeley persuaded himself that he was in earnest against Fluxions, especially of orders higher than the first, as well as against matter.

De Morgan, Philosophical Magazine, 4 S. One is not so easily convinced of the ability and sincerity of Jurin. That at first he should argue that quantities may be dropped because small, and afterwards admit that this argument was in- tended for popular consumption, is not reassuring. In this connection we quote from a letter which Hamilton wrote De Morgan in when Hamilton was seeing his Elements of Quaternions through the press : 2 "When your letter arrived this morning, I was deep in Berkeley's ' Defence of Freethinking in Mathematics ';!

His mode of getting rid of ab appeared to me long ago I must confess it to involve so much of artifice, as to 1 See our 97, , But by what right, or what reason other than to give an unreal air of simplicity to the calculation, does he prepare the products thus? Might it not be argued similarly that the difference, was the moment of A 3 ; and is it not a sufficient indication that the mode of procedure adopted is not the fit one for the subject, that it quite masks the notion of a limit ; or rather has the appearance of treating that notion as foreign and irrelevant, not- withstanding all that had been said so well before, I in the First Section of the First Book?

Walton's two or three articles do not seem to have been read much. They are seldom mentioned. The pamphlets are now rare. Pro- 1 H. Gibson had not seen them when he wrote on the Analyst controversy. It is worthy of notice that Walton 2 ex- pressed himself on the nature of limits, by claiming that the limit was reached.

As to the nature of " variable velocity," it is interesting to see that Berkeley realised the difficulty of the concept, and probably saw that there was no variable velocity as a physical fact, while Walton was compelled to take refuge in less primitive mechanical concepts in order to uphold his side of the argument. Berkeley's Lemma 4 was rejected by Jurin and Walton. We shall see that it found no recog- nition from mathematicians in England during the eighteenth century, but was openly and repeatedly accepted as valid in its application to limits, by Woodhouse at the beginning of the nineteenth century.

Eighteenth-century mathematicians did not attach due importance to this point. The existence of infinitesimals infinitely small quantities was denied by Berkeley, but, it would seem, not denied by Jurin and Walton. All three finally abjured the philosophy which permits their being dropped because so small. It is well known that many mathematicians of prominence have believed in the reality of such quantities. From Leibniz to Lagrange all Continental writers of note used them. Lagrange headed a small school that was opposed to them, when he pub- lished his Fonctions analytiques.

There followed a reaction against Lagrange. De Morgan once remarked: "Duhamel, Navier, Cournot, are pure infinitesimalists. Some of them say an infinitely small quantity is one which may be made as small as you please. This is an evasion ; but they do not mean that dx is finite. By-the-way, Poisson was a believer in the reality of infinitely small quantities as I am.

For myself, I am now fixed in the faith of the subjective reality of infinitesimal quantity. But what an infinitely small quantity is, I know no more than I know what a straight line is ; but I know it is ; and there I stop short. But I do not believe in objectively realised infinitesimals. Society, vol. We must not neglect to express our appre- ciation of the fact that Berkeley withdrew from the controversy after he had said all that he had to say on his subject. Some of the debates that came later were almost interminable, because the par- ticipants continued writing even after they had nothing more to say.

Benjamin Robins was a native of Bath and a self - educated mathematician of considerable reputation. Robins was a man of mathematical power ; his exposition is regarded by Professor G. Gibson as very able, and far superior to that of Jurin. The whole foundation of the doctrine of fluxions is 1 This paper is republished, along with subsequent articles on the same subject, in the Mathematical Tracts of the late Benjamin Robins, Esq. London, , vol. Gibson, loc. Definition : ". The gain from the standpoint of debating is very great ; a regular inscribed polygon whose sides are steadily doubling at set intervals of time, say, every half second, presents a picture to the imagination which is comparatively simple.

The hopeless attempt of imagining the limit as reached need not be made. But this great gain is made at the expense of generality. Robins descends to a very special type of variation which is not the variation encountered in ordinary mechanics ; it is an exceedingly artificial variation. According to Robins's definition, Achilles never caught the tortoise. It would not be difficult to assume a time rate in the doubling of the sides of a polygon inscribed in a circle, so that the cir- cumference is reached.

The process here tran- scends our power of imagination, but lies within the limits of reason. We are dwelling upon this point because of its extreme importance in the history of the theory of limits. Robins constructs upon his first definition the theorem, "that, when varying magnitudes keep constantly the same proportion to each other, their ultimate magnitudes are in the same proportion.

Definition: "If there be two quantities, that are one or both continually varying, either by being continually augmented, or continually diminished ; though the proportion, they bear to each other, should by this means perpetually vary, but in such a manner, that it constantly approaches nearer and nearer to some determined proportion, and can also be brought at length in its approach nearer to this determined proportion than to any other, that can be assigned, but can never pass it : this determined proportion is then called the ulti- mate proportion, or the ultimate ratio of those varying quantities.

Robins remarks thereupon that attempts at the exposition of this method, so far as it depends upon his first definition, were made by Lucas Valerius in a treatise on the centre of gravity, and by Andrew Tacquet in a treatise on the cylindrical and annular solids ; but the development involving his second definition was first made by Newton. There are a number of writers, not mentioned by Robins, who might be cited as forerunners in the theory of limits ; such, for instance, as Gregory St.

Vincent and Stevin. Newton's definition of momenta as the momentane- ous increments or decrements of varying quantities, " may possibly be thought obscure. Of this Discourse, a long account of twenty- six pages, written by Robins himself, although his name does not appear, 1 was given in The Present State of the Republick of Letters, London, October, , in which it is staged that Robins wrote his Discourse with the view of removing the doubts which had lately arisen concerning fluxions and 1 This account is republished in the Mathematical Tracts of the late Benjamin Robins, edited by James Wilscn, London, , vol.

After an historical excursion viewing the works of the ancients, of Cavalieri and Wallis, the introduction by Newton of the concept of motion is taken up. The "method of prime and ultimate ratios proceeds entirely upon the consideration of the increments produced. Newton's descrip- tion is capable of an interpretation too much resembling the language of indivisibles in fact, he sometimes did use indivisibles at first ; Robins has freed the doctrine from this imputation in a manner that "shall agree to t.

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In the November, , number of the Republick of Letters, Philalethes Cantabrigiensis Jurin appears with an article, Considerations upon some passages contained in two Letters to the Author of the Analyst. The two letters in question are the two replies Jurin himself had made to Berkeley. The article is really a reply to Robins, though Robins's name is not mentioned. Jurin claims to have adhered strictly to Newton's language ; some other defenders of Newton, says he, are guilty of departing from it. Their objections to his own defence are threefold : " I.