Foundations for fluxions

Contents

Still more mysterious greetings to T. E. O., O . A., E. M., O. C. W.(1)

And then there's Mom and Dad, of course.

The work on this paper was done in Oslo, Utrecht and London from August 1994 to November 1995, in which period I was a student of the University of Oslo.

Blindern, November 24th, 1995

Bjørn Smestad

Preface

I will not go into the reasons for studying the history of mathematics here, just as students of algebra or logic don't have to defend their choice of study. But I will say something about my choice of subject. At the time when I had to make this decision, I didn't think I would be able to study a subject where the literature was in French or German, or even worse, Latin. Therefore, I chose British mathematics. And when my supervisor proposed the subject of post-Newtonian fluxions, I found it interesting.

It soon became painfully clear that I could not study all of the 18th. century works on fluxions, especially when I "discovered" Cajori's book,(3) where lots of them are treated. Therefore I chose the obvious ones, Philalethes', Robins' and MacLaurin's, and added Paman's work, which seemed very interesting from Jesseph's short account of it.

Studying 18th. century mathematics gives some special problems, of course. The problem of getting access to sources was partly remedied by going to the British Library in London, but the limited time I had there meant that I did not have the time to look at more than the first contributions of Philalethes and Robins. Another problem was to get an understanding of the environment of these people. The quote at the top of this preface illustrates this problem; today all of mathematics is neatly defined in mathematical terms, but is not supposed to represent physical realities. In this sense, mathematicians are only treating abstract objects, with no connection with what is true in the real world. In the 18th. century, on the other hand, mathematics was supposed to represent reality - and therefore mathematics could give answers to problems in the real world. This difference is not unimportant when studying the mathematics of the time. Therefore, I have seen my opinions change as my understanding of the time have changed (to the better, I hope).

I must mention one methodological problem: My discussion of Newton is based partly on manuscripts which he never published. It is, in theory, possible that they were not published precisely because he didn't think they were good enough for publication. I have ignored this possible objection, however, as his published papers would not be sufficient to give a clear picture of his theory, and I think that Newton's reasons for not publishing his mathematics were most often others than this.

As many of the works treated here are not easily accessible, I have felt obliged to include lots of quotes from the works. My choice of language was partly motivated by this, and partly by the fact that the number of people who understand Norwegian is quite limited.

The structure of my paper is as follows:

In chapter 1 I will try to give a very quick overview of "the mathematical world" in the period in question. In chapter 2 I will consider Newton's own foundations for his calculus, and argue that the confusion which followed was partly his fault.

In 1734, Berkeley published his criticism of the calculus, the Analyst. This, and the answers from Philalethes and Robins, are treated in chapter 3. I will try to show that some of the criticism against Philalethes in the literature have been unjust.

Colin MacLaurin's contribution is the subject of chapter 4; I have tried to understand this "incomprehensible"(4) book.

In chapter 5, I will be discussing Roger Paman. As he is virtually unknown, I have tried to assemble some information about his life. Thereafter I will try to show that Paman's work gave a very interesting foundation for the method of fluxions.

Chapter 6 is a result of a comment of Jesseph, concerning one possible effect of the Analyst controversy.

After the conclusion, I have added a few appendices, which I hope will prove useful - most notably a list of micro-biographies of most of the people mentioned in this text (excluding historians of mathematics) and the tables of contents from the books of Philalethes, Robins and MacLaurin that I have studied. A little note on typography is perhaps useful, too: I have tried to give the quotes as much as they were actually written as possible - within the bounds of what is reasonable. For instance, Newton sometimes writes "the" as "ye" with e raised (as we would write y in e'th degree). In this www-version I will write this as "y'e". In the same manner: "y'n" means "then", "y'm" means "them" and "w'ch" means "which". As usual, "(...)" means that I have omitted a part of the quote, "[comment]" is my inserted comment. Quotes are centered if not given inside quotation marks.

The full titles of the books are given in the bibliography.

Chapter 1

In this sense, the mathematical world was changing rapidly in this period. But mathematics was still supposed to explain the physical world, and physical realities were used to explain mathematics.

This connection was strengthened by Newton's theories, with mathematics explaining the motion of planets, and motion explaining his mathematics.

In this exciting world worked not only professors of mathematics and lecturers at the universities. Professionals from all areas of science found new tools to use and explore, but also lots of amateurs had the opportunity to investigate beyond the current frontiers of knowledge. The art of printing, still not more than 2-300 years old (in Europe), made the new results accessible to many, and even made it possible for amateurs to publish their results. The role of journals and of societies, such as the Royal Society, must have been considerable.

But which parts of "the world of mathematics", as we know it today, did they know?(6)

1.1 Numbers

When it comes to negative numbers, Kline writes:

Complex numbers were obviously much more difficult than negative or irrational numbers.

1.2 Functions

During the period in question, several functions were studied and better understood - such as ln x, exp(x) and sin x. The hyperbolic functions were introduced late in this period.

The function concept developed gradually through this period, and in 1748 Euler defined a function as any analytical expression formed in any manner from a variable quantity and constants, including polynomials, power series, logarithmic and trigonometric expressions. Every function considered could be expanded in power series.

It seems that violently oscillating functions, like sin (1/x), were not considered.

1.3 Infinite series

The problem of convergence was very difficult. For instance, Guido Grandi noted that 1/(1+x)=1-x+x²-x³+ ..., therefore 1/2=1-1+1-1+..., but at the same time (1-1)+(1-1)+...=0, therefore the world could be created out of nothing!(9)

1.4 Proofs

1.5 Conclusion

Chapter 2

2.2 1665-1680: Fluxions

Fluxions were introduced in the middle of 1665, perhaps inspired by Barrow's lectures on motion of the previous year.(12) The point of using motion to define fluxions probably was to give a better foundation than the one based on infinitesimals. But infinitesimals were not excluded, for instance, on November 13th, 1665, Newton wrote (see Figure):

As we see, this argument is strongly dependent on infinitesimals. Moreover, it resembles Fermat's method for finding the subtangent, and other methods of the time, which Newton had read about in Descartes' Geometria in van Schooten's second Latin edition.(14)

He was quite explicit in using and accepting infinitesimals: In October 1666 he wrote:

As late as 1671--2, the infinitesimals are still there: (in the following, I will use the notation x* instead of Newton's dotted x.

We see clearly that at this point infinitely small quantities played an important part in Newton's method of fluxions.

2.3 1680--1703: Fluxions founded on prime and ultimate ratios

This is not different from his previous definitions. But earlier, he had used infinitely small quantities to find these fluxions. Now he tried to do without them:

This does not seem convincingly rigorous to me. 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:

Here he has a reasoning that has not been popular with later critics:

One would think that the reasonable way to do this would be to calculate (A+a)(B+b)- AB= Ab+aB+ab. The problem would be to get rid of the ab - many would do this by saying that ab is infinitely less than Ab+aB. 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. When trying to find the rate of increase, you may calculate (A-a)(B-b)-AB, (A+ ½a)(B+ ½b)- (A- ½a)(B- ½b) or even (A+ (2/3)a)(B+ (2/3)b)-(A- a/3)(B- b/3), because all you are interested in is the rate of increase in some "small" neighbourhood of A and B. This is just as we in modern notation may choose to calculate lim{h->0}(f(x+h)-f(x))/h, lim{h->0}(f(x+h/2)-f(x-h/2))/h or even lim{h->0}(f(x+(2/3)h)-f(x-(1/3)h))/h, when we want to find the derivative of f.

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 x^n:

Newton has found what we would write as ((x+o)^n -x^n)/o, and tries to see what happens when o approaches zero. Without recourse to the limit concept, 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.

2.4 1703-: "Newton renounces and abjures infinitesimals"

Judging by these quotes, it may seem that after 1703, Newton rejects infinitesimals.(31) Augustus De Morgan claimed this in 1852.(32) However, we see that Newton is quite careful - he says "each time it can conveniently so be done" and "I wanted to show", not "always" and "I showed". De Morgan also pointed out that the infinitely little quantity returned in 1713, in the second edition of Principia. His argument for this was a letter from Newton to Keill, in which Newton wrote:

I will include two more Newton-quotes from after 1703:

These quotes do not seem to agree with the previous ones.

Lai(36) solves this disagreement by interpreting the 1703-quotes as a rejection only of the traditional infinitesimal methods, not of his own fluxional method, that was also dependent on infinitesimals. Kitcher,(37) on the other hand, has a theory that Newton considered the usual infinitesimal arguments, the fluxional calculus and the method of first and last ratios as three different parts of his theory, with three different goals:

It must be added that Newton himself denied ever having changed his method:

2.5 Conclusion

All the same, the following should be clear: Newton used several different explanations of his fluxional calculus, without making the relationship between them clear. He was not good at defining and clarifying his concepts, 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

3.1 Introduction

3.2 The Analyst - Berkeley's main points

3.2.1 The fluxions were incomprehensible

That such an intelligent man as Berkeley understood so little of the theory, is certainly a strong indication that the explanations had not been good enough. However, it must be said,(44) 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.

3.2.2 Invalid proofs

For instance, Berkeley disliked Newton's calculation of the fluxion of AB (see here):

This Latin quote is from Newton's Quadraturam Curvarum and means that "In mathematics, even the smallest errors are not to be neglected."

We see that Berkeley thinks that Newton's method was "illegitimate". 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 x^n (see here)):

If this is a correct interpretation of Newton, the criticism is valid - it is not allowed to divide by o and then let o equal 0, just as it is not allowed to divide by 0 in the first place. However, Berkeley's interpretation of Newton will be addressed later (see here).

3.2.3 The compensation of errors thesis

De Morgan writes that "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 (...) has generally been treated as a real opponent of fluxions."(48) I think (as most others) that Berkeley was pointing at problems that he didn't know how to solve himself. This point of view seems to be supported by the fact that the compensation of errors thesis is incorrect.

3.2.4 Conclusion

Philalethes Cantabrigiensis

Philalethes was concerned about Berkeley's attack on mathematicians. Therefore, he did not attempt to build a new foundation for fluxional calculus. A new foundation, though mathematically interesting, would be irrelevant 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 positive 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:

3.3.1 Obscurity of this doctrine

He also attacks Berkeley for misrepresenting Newton;

Of course, Newton never used these expressions. 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:

3.3.2 False reasoning used in the method of Fluxions by Sir Isaac Newton, and implicitely received by his followers.

Philalethes first asks if leaving out ab really is an error at all:

One possible interpretation of this could be that Philalethes just says that the error is so small that it is of no practical consequence. But this interpretation is wrong, as he goes on to say

Thus I think it is clear that what he wants to say is that there is no error - there exists no number small enough to quantify the "error" done by omitting the term ab.

This argument could have been written by anyone, with any foundation - 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:

I think that here, much more clearly than in the previous quote, a limit argument is involved. 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. First, Philalethes claimed that as aB+bA+ab is the increment of AB, and aB+bA-ab is the decrement, and the moment can be both the increment and the decrement, then the mean aB+bA must also be the moment. 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;

For what reason did Newton calculate this increment?

As I noted when treating Newton's proof (here), I think this is something like what Newton thought, but did not write down. We would certainly agree that, in order to find the derivative of f(x), we could, instead of calculating lim {h->0} (f(x+h)-f(x))/h, calculate lim{h->0} (f(x+½h)-f(x-½h))/h. 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:

It is of course true that lim{a,b->0} aB+bA equals lim{a,b->0} aB+bA+ab, since both of them are 0. But it is also true that the ratio of aB+bA to aB+bA+ab tends to equality as a and b are diminished. Gibson's discussion of this(61) makes it clear, however, that it is the first of these interpretations which cover Philalethes' meaning. But then this argument is meaningless - we are not interested in the quantities aB+bA and aB+bA+ab when they are zero!

Philalethes goes on to consider Newton's calculation of the fluxion of x^n (see here), and Berkeley's criticism of it (see here).

Philalethes' translation is "Let the augments now become evanescent, let them be upon the point of evanescence".(63)

If Philalethes' translation is the correct one, Berkeley has again misrepresented Newton, and Philalethes does not have to go into mathematical detail to defend him.

3.3.3 Arts and fallacies used by Sir Isaac Newton to make his false reasoning pass upon his followers

3.3.4 The compensation of errors thesis

While this parody of Berkeley's theory was probably well received by Newton's followers, and is still funny, it is of course not an adequate refutation of Berkeley's thesis. 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.

3.3.5 Criticism of Philalethes

The fluxion of AB

As noted earlier , I do not share this interpretation. 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.

Philalethes' second explanation of why Newton is correct, is that Newton calculates the moment of AB by calculating the increment of (A- ½a)(B- ½b), and that this is a correct method of doing it. But Newton himself says that he calculates the increment of AB, as Berkeley points out.(73) But Newton's calculations does not fit the definition he had given of moment, even if it fitted his intuition.

His last explanation amounts to saying that aB+bA+ab=aB+bA, "supposing a and b to be diminished ad infinitum". As many have noted, this does not help defending Newton.

Cajori writes:

This may be because Newton's proof is perfectly understandable from the intuitive view-point, given Philalethes' explanation, which was soon to be repeated by Robins (see here). The calculation does not fit the definition, however, but it seems to me that many mathematicians at the time, including Newton, considered the definitions as little more than explanations, 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.

The fluxion of x^n

Jesseph says that Philalethes' response at this point "hardly comes up to the mark".(75) I think Philalethes' answer is adequate - the first thing to do when someone has misunderstood a sentence is to point out where the misunderstanding is. 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.

3.3.6 Conclusion

3.4 Benjamin Robins

Robins distinguishes clearly between Newton's two methods, the method of fluxions and the method of prime and ultimate ratios.

The book is divided into two main parts, and I will follow this division.

3.4.1 Of fluxions

He goes on to explain how a space can be described by motion, and how the fluxion of a space is defined as the fluxion of a line that "augment in the same proportion with the space (...)"(78)

These definitions are very similar to Newton's own. 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 (...)".(80) The example he chooses is the one where AE=x, CF= (x^n)/(a^(n-1)) (see Figure). He shows that if EG is denoted by e, FH will be denoted by (nx^(n-1)e)/(a^(n-1))+ (n×(n-1)x^(n-1)ee)/(2a^(n-1))+ & c.; and KF will be denoted by (nx^(n-1)e)/(a^(n-1))- (n×(n-1)x^(n-1)ee)/(2a^(n-1)+ & c.

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.

Similarly, the proportion of the velocity of the point at F to the velocity of the point at E is greater than KF to IE:

After these fundamental observations, the result can be found:

Robins proves this by reductio ad absurdum index reductio ad absurdum , over two pages. I will include one half of this:

After showing the opposite case, Robins says that the demonstrations are the same if n is less than 1.

This seems to be a correct proof, although helplessly long. 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. in the usual way. 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 physical 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.

3.4.2 Of prime and ultimate ratios

The principal definitions are as follows:

This fixed quantity is called the ultimate magnitude of the varying quantity.p> The words "perpetually approach" seem to suggest monotonity.

The same can be defined for ratios:

This limit is called the ultimate ratio of the ratios.

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

He gives a couple of examples, of which this is the most interesting (see Figure):

This is an illuminating example, in that it clearly shows that there may exist ultimate ratios between vanishing quantities, in the precise sense of the words given by Robins.

3.4.3 Of Sir Isaac Newton's method of demonstrating his rules for finding fluxions

The proof is shorter than the previous one, but Robins still uses the undefined term velocity in his argument. Therefore this way of doing it is not much better or worse than his previous one.

3.4.4 Explanation of the term momentum

It is difficult to see that this definition is the same as Newton's own.

Then Robins comes with the long-sought-for argument for why the "moments" Ab+aB+ab and Ab+aB are essentially the same:

This is surely a correct argument, but we note that the definition of momentum used is Robins' own - thus this cannot be seen as a valid answer to Berkeley's criticism of Newton's argument.

Robins has the following comment on Newton's calculation of the same fluxion:

I do not see a great difference between this argument and Jurin's argument.

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 theme was what Newton's view had been, and did not help the science of mathematics much. I therefore refer to Cajori(94) on this subject.(95)

3.5 Comparison Philalethes-Robins

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 unnecessary), gives illuminating examples and proves his propositions. Except the term "momentum", he also keeps close to Newton's definitions. His uncritical use of the notion of instantaneous velocity would probably not have satisfied Berkeley,(96) but for the less philosophically inclined, I think Robins' book gave explanation enough.

3.6 Philalethes-Robins in the literature

During the last century, however, Robins has been much more highly regarded than Philalethes. Gibson is one of the writers most critical of Philalethes, and he writes:

The main reason why Gibson was "at a loss" to understand this is probably, in my view, that he treats Philalethes as if he, too, in his first book tried to explain and clarify the theory. 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 mathematics teacher.

3.7 Other answers

3.8 Conclusion

Paman noted this, and used it as an excuse to publish yet another one...(100)

What happened to English mathematics after the Analyst Controversy will be treated briefly in chapter 6. Suffice it here to say that English mathematics 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 Berkeley had not published his work. As these are extremely difficult questions, I will only say that Berkeley's work was very important for British mathematics. 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.

Chapter 4

In 1742 appeared Colin MacLaurin's attempt at explaining the method of fluxions, A Treatise of Fluxions. MacLaurin originally planned to write a shorter answer to Berkeley,(102) but was encouraged to do more out of it.(103) While he wrote his Treatise, he obviously followed the controversy with interest, and mentions that

I find it probable that MacLaurin had read all or most of these before publishing his own answer. 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 rigorous foundation for the method of fluxions, with long, double reductio ad absurdum proofs which Eudoxus might have appreciated.

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.

The first book explains the theory of fluxions in much the same way as Robins did in the first part of his book - considering quantities as generated by motion and using velocities as a basic, undefined tool. 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.

4.1 Book I

Here, MacLaurin has already (like Newton and Robins) used the intuitive concept of velocity without further explanation, looked at the velocity in a point and considered the effect if that velocity is held constant(108) . 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):

In a brilliant passage, he explains the connection between his geometrical method of Book I and the method of infinitesimals:

Therefore, the method of infinitesimals gives the same, correct results as the method of fluxions. He has a similar argument, by way of an example, concerning Newton's method of first and last ratios.(111)

In short, Book I gives long, geometrical proofs of geometrical propositions. To quote MacLaurin:

I pity the beginners who started studying fluxions by reading the 575 pages of Book I. In my view, Book II is far more interesting and important.

4.2 Book II

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:

Therefore, the definition of fluxion in Book II is slightly different from the one in Book I:

The following arguments play an important part in finding the fluxion of x^n, which of course is one of the key results of the whole theory.

This is not altogether clear, especially the part "b, : þ , : & c. (...) always increase, how small soever a may be (...)" would profit from a little clearing up. I will give a little example to show what I think is MacLaurin's meaning:

Example 4.1 If the successive values of A is represented by A-a,A,A+a, & c., where a is the fluxion of A and B=A^2, then the corresponding values of B are (A-a)^2 , : A^ , : (A+a)^2 , : & c. = A^2 -2Aa+a^2 , : A^2, : A^2+2Aa+a^2 , : & c.= B-b, : B, : B+ þ , : & c., where b=2Aa-a^2, : þ =2Aa+a^2. Here we see that þ > b whatever a is. This is what MacLaurin calls that the "successive differences (...) always increase, how small soever a may be (...)", and which we would call that the sequence b, þ, & c. is increasing. Therefore it is clear that B is not growing uniformly, it is accelerating, so the fluxion must be less than þ and greater than b.

Now MacLaurin is ready to compute the fluxion of A^2, by reductio ad absurdum:

Proposition 4.2 "The fluxion of the root A being supposed equal to a, the fluxion of the square AA will be equal to 2A × a".(118)

Proof "Let the successive values of the root be A-u, A, A+u, and the corresponding values of the square will be AA-2Au+uu,AA,AA+2Au+uu, which increase by the differences 2Au-uu, : 2Au+uu : & c. and because those differences increase, it follows from art. 704, that if the fluxion of A be represented by u, the fluxion of AA cannot be represented by a quantity that is greater than 2Au+uu, or less than 2Au-uu. This being premised, suppose, as in the proposition, that the fluxion of A is equal to a; and if the fluxion of AA be not equal to 2Aa, let it first be greater than 2Aa in any ratio, as that of 2A+o to 2A, and consequently equal to 2Aa+oa. Suppose now that u is any increment of A less than o; and because a is to u as 2Aa+oa to 2Au+ou, it follows (art. 706(119)) that if the fluxion of A should be represented by u, the fluxion of AA would be represented by 2Au+ou, which is greater than 2Au+uu. But it was shown, from art. 704, that if the fluxion of A be represented by u, the fluxion of AA cannot be represented by a quantity greater than 2Au+uu. And these being contradictory, it follows that the fluxion of A being equal to a, the fluxion of AA cannot be greater than 2Aa."
Likewise he shows that the fluxion of AA is not less than 2Aa.(120) QED.

We see that he proves the proposition by showing that the fluxion of AA is not unequal to 2A × a, without using infinitesimals or velocities. 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. The same argument would be possible for any expression with the same characteristic, that is (to use modern notation): f(x+ delta x)-f(x) < (or >) f(x)-f(x+ delta x) for all delta x in a neighbourhood of 0 (in R+), which is equivalent to that f'(x) is monotonously increasing (or decreasing) in a neighbourhood of x, that is; f is convex or concave in a neighbourhood of x. However, this argument can obviously not be used when the differences are not increasing or decreasing, that is if f is not convex or concave in a neighbourhood of x.

MacLaurin uses the same argument to prove that the fluxion of A^n equals naA^(n-1) (for integer n). Thereafter he proves that the fluxion of A^(m/n) equals (ma)/n A^(m/n-1) and that the fluxion of ABCDE... equals aBCDE ... + AbCDE ... + .... MacLaurin then goes on to consider the inverse method of fluxions (what we call integration).

4.3 Views on MacLaurin

- meaning that it is not a virtue to use a small number of steps if these steps are not sufficient to make the proof complete.

Kline also wrote that

without giving any reasons for his claim.

Turnbull, on the other hand, writes that

Paman writes about MacLaurin that his

and that

4.4 Conclusion

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. Therefore, Book II gives a solid foundation compared to Newton and Robins.

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 authoritative 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

5.1 The life of Roger Paman

We do not know how Paman was educated. He was not himself registered as a student in Cambridge,(132) but in the preface to his book he mentions Mr. Frank, who belonged to St. John's College, Cambridge, and who was the one to give Paman the Analyst to consider. Paman wrote a paper on this, which was communicated to several members of The Royal Society, and which kept circulating until 1739.

Sept. 18th. 1740, Roger Paman was on board, he claims, when George Anson's ships set out from St. 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 1744.(133) Paman, however, was back in England long before this. 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 anchored upon the coast of Patagonia (Southern Argentina) February 18th, 1741. March 7th, they passed the Straits of Le Maire,(134). still together. But on April 10th, they lost sight of the other ships,(134) and on April 25th they even lost sight of each other,(135). 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, 1741, the ships arrived in Rio de Janeiro, and Captain Legge of the Severn wrote:

They stayed in Rio for a long time, trying to get their ships fixed and their men well, while quarrelling over what to do. However, on February 5th, 1742, they arrived in Barbados on their way home.(136i)

It seems that much of the blame for making Anson's voyage relatively unsuccessful, was given to these two ships. For instance, John Campbell wrote:

It is therefore no reason to think that the men from the Severn and the Pearl were heroes when they came back to England.

Before leaving England, Paman had given his paper to his friend Dr. Hartley, and when he returned, in February 1742, Paman sent it to the Royal Society.(138) This must probably be the main reason why he was elected Fellow of the Royal Society. He was recommended by Abraham de Moivre, R. Barker and G. Scott February 10th, 1743, with the following description:

He was elected May 12, 1743.

In 1745 he published the paper as a book, The Harmony of the Ancient and Modern Geometry asserted. The preface was dated August 1st, 1745, the Postscript of the preface August 24th, 1745. It probably appeared in October, as The Gentleman's Magazine includes this book in the list of "Books and Pamphlets published this Month":

This book, which is the main subject of this chapter, also included an advertisement (call for subscriptions) for another book of Paman's, giving

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 1748.(141)

In 1919, Florian Cajori mentioned Paman's work in a footnote:

The first treatments of Paman's work in works on history of mathematics, seem to be Breidert (7) and Sageng (45), both from 1989.

5.2 The definitions

I will now give Paman's way of defining fluxions. To avoid breaking up Paman's exposition too much, I will give my interpretation and comments in sections 5.3 and 5.5, where I will also argue that he succeeded in his task.

Definition 5.1 "I call one Expression the radical Quantity of another; when the latter is compos'd of any Power or Powers of the former, their Parts or Multiples."(145)

Definition 5.2 "By the first State of x, I mean all the Values of x, between some certain assignable Value and Nothing."(146)

Definition 5.3 "By the last State of x, I mean all the Values of x, greater than, or above some certain assignable Value."(146)

Paman hastens to give an explanation:

Definition 5.4 "Quantities are distinguish'd in the following Pages, by the Powers of x, which they involve, thus I call ax an x Quantity; bx^2 an x^2 Quantity; and in general (a+b+c)x^m an x^m Quantity."(148)

Then Paman proves a proposition which shows some of the strength of these definitions:

Proposition 5.5 "Any determinate Quantity p is greater than any x^m Quantity, as ax^m in the first; and than any x^(-m) Quantity, as ax^(-m) in the last State of x."(149)

Proof "For p is greater than ax^m, in all the Values of x, between (p^(1/m))/a^(1/m) and Nothing; and than ax^(-m) or a/(x^m) in all the Values of x, greater than, or above (a^(1/m))/p^(1/m) therefore p must be greater than ax^m in the first, and than ax^(-m) in the last State of x." QED.

He goes on to prove the following propositions, among others. I will skip the proofs here.

Proposition 5.6 "(...) Any x^m Quantity, as px^m, is greater than any Series of higher Powers, as ax^(m+n), bx^(m+n+o), cx^(m+n+o+r), & c. in the first State of x; and than any Series of lower Powers, as ax^(m+n), bx^(m+n+o), cx^(m+n+o+r), & c. in the last State of x; the Converse is also true."(150)

This must be a misprint. The latter series should have been ax^(m-n), bx^(m-n-o), cx^(m-n-o-r), & c. instead of ax^(m+n), bx^(m+n+o), cx^(m+n+o+r), & c.

A comment on notation is necessary here: The notation ax^(m+n), bx^(m+n+o), cx^(m+n+o+r) means ax^(m+n) + bx^(m+n+o) + cx^(m+n+o+r), but it is unclear what the "Series" ax^(m+n), bx^(m+n+o), cx^(m+n+o+r), & c. is - is it a (possibly long) polynomial, or an infinite series? Given the important part infinite series have in Newton's theory, and seeing that Philalethes uses the " & c." in the meaning "all the possible repetitions (...), even to infinity." (here), I find it likely that infinite series are covered by this notation. Paman neglects the problems of convergence, as usual at the time.

Proposition 5.7 "If any x^m Quantity be greater than A, any x^m Quantity, as px^m, will be greater than any Quantity, which is to A in a given ratio; or, any x^m Quantity will be greater than any Part or Multiple of A (as d × A) in the same State of x."(151)

5.2.1 (First and last) Maximinus and Minimajus

Definition 5.8 If one Expression be less (greater) than another, in the first State of their radical Quantity, and yet no Quantity of the same Kind can be added to (subtracted from) the former, without making the Sum (Remainder) greater (less) than the latter in the first State of their radical Quantity; then I call the former the first Maximinus (Minimajus) of the latter.(152)

Example 5.9 "(...) ax is the first Maximinus of ax+bx² for ax is less than ax+bx², in the first State of x; yet, if any x Quantity, as px, be added to ax, the Sum (a+p) × x will be greater than ax+bx², in the first State of x; because it will be greater in all Values of x between p/b and Nothing."(153)

Paman lets a dotted = denote Maximinority or Minimajority in the first State.(154). In this version of the paper, I will have to use =* for this (and x* for a dotted x etc.)

Clearly, this is not exactly the same as a limit, as lim{x->0} (x+x²+x³) =0, while the x Quantity that is the first Maximinus of x+x²+x³ is x.

In a footnote,(155) Paman writes:

and he also proves the existence for power series with a lowest power. 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 5.10 "In any Series ax^m , : bx^(m+n) , : cx^(m+n+o) , : & c., the lowest Term, as ax^m, is the first Maximinus or Minimajus of the Series".(156)

Proof "(...) p is greater than ax^m, in all the Values of x, between p^(1/m)/a^(1/m) and Nothing.(157)

"In like manner any x^m Quantity, as px^m, is greater than any higher Power of x, as ax^(m+n) in the first (...) State of x (...)"(158)

"If any x^m Quantity be greater than A, and if any x^m Quantity also be greater than B, in the first (...) State of x, any x^m Quantity, as px^m, will be greater than the Sum of, or Difference between A and B, in the same State of x. For, if any x^m Quantity be greater than A or B, ²px^m will be greater than A, and ²px^m will be greater than B, consequently px^m will be greater than A+B, in the same State of x. (...)"(159)

"Hence it appears, that any x^m Quantity, as px^m, is greater than any Series of higher Powers, as ax^(m+n), bx^(m+n+o), cx^(m+n+o+r), & c. in the first State of x (...)"(160)

"(...) if any x^m Quantity be greater than the Difference between ax^m and A, ax^m will be either the Maximinus or Minimajus of A, in the same State of x, unless A represents any single Power of x; for then ax^m and A will be equal (...)"(161)

Therefore ax^m will be the Maximinus or Minimajus of the series. QED.

Paman does not mention convergence in this connection. It must have seemed probable that all of these series converge in the first State of x. But this is wrong, for instance sum n! x^n {n=1 to infinity} converges only for x=0. Nobody seems to have considered this problem in the 18th. century.

In the following propositions, Paman proves the uniqueness of Maximinus and Minimajus:

Proposition 5.11 "If ax^m be the Maximinus of A, no other x^m Quantity can be the Minimajus of A, in the same State of x; and, if ax^m be the Minimajus of A, no other x^m Quantity, as dx^m, can be the Maximinus of it in the same State of x."(162)

Paman goes on to give some rules, after the following definition:

Definition 5.12 "Maximinus's and Minimajus's are said to be similar, when they are referred to the same State, and involve equal Powers of the same radical Quantity."(163)

Rule 5.13 "The Sum of, or Difference between two similar Maximinus's or Minimajus's, or a similar Maximinus and Minimajus, will constitute the Maximinus or Minimajus [or be equal to(164)] of the Sum of, or Difference between, the two Expressions they belong to."(163)

Paman does not give a proof of this or the other rules, although he could probably have done so; for instance:

(My) Proof If ax^m is the first Maximinus of A, and bx^m is the first Maximinus of B, then (a+b)x^m is less than A+B, but (a+b+p)x^m is greater than A+B, in the first State of x, since (a+p/2)x^m > A and (b+p/2) x^m > B, in the first State of x.
If ax^m is the first Maximinus of A and bx^m is the first Minimajus of B, then let C=A-ax^m, D=bx^m-B. Then (a+b)x^m-(A+B)=D-C. If D > C in the first State of x, then (a+b)x^m > A+B, but (a+b-p)x^m < A+B, since ax^m < A and (b-p)x^m < B in the first State of x, so (a+b)x^m is first Maximinus of A+B. If D < C, (a+b)x^m is first Minimajus of A+B by a similar argument. QED.

Comment If C=D, which happens for instance if A=2x+2x^2, B=2x-2x^2, we have an exception to the rule, since we get (a+b)x^m=A+B, and therefore (a+b)x^m is not the Maximinus or Minimajus of A+B).

Rule 5.14 "If either a Maximinus or a Minimajus, and the Expression it belongs to, be multiplied into, or divided by, the same Quantity, the former Product or Quotient will be the Maximinus or Minimajus of the latter, in the same State of x."(165)

Example 5.15 "If ax^m =* by then dax^m =* dby and ax^m/d =* by/d."

Rule 5.16 "The Product or Quotient of two first Maximinus's, or Minimajus's, or a first Maximinus or Minimajus, will constitute the first, and the Product or Quotient of two last will constitute the last Maximinus or Minimajus of the Product or Quotient of the two Expressions they belong to."(166)

(My) Proof For instance: If ax^m is the first Maximinus of A and bx^n is the first Maximinus of B (a,b>0), then abx^(m+n) will be less than AB, but (ab+p)x^(m+n) will be greater than AB, since (ab+p)x^(m+n) > (a+p/3b)x^m(b+p/3a)x^n > AB (whenever p < 3ab) so abx^(m+n) will be first Maximinus of AB.

Another example: If ax^m is the first Maximinus of A and bx^n is the first Minimajus of B (a,b>0) and abx^(m+n) > AB, then (ab-p)x^(m+n) < ax^m(b-p/a)x^n < AB in the first state of x, which means that abx^(m+n) is the first Minimajus of AB.

The other instances should be similar. QED.

Rule 5.17 "Any Power or Root of a first Maximinus of Minimajus will constitute the first, and any Power or Root of a last will constitute the last Maximinus or Minimajus of the same Power or Root of the corresponding Expression."(167)

Example 5.18 "If ax^m =* by then a^n x^(nm) =* b^n y^n and a^(1/n) x^(m/n) =* b^(1/n) y^(1/n)."

Paman also includes a discussion on what he calls "Approximating Series", that is infinite power series, and he explains how to find these series for a fraction and for y given an equation in x and y - the binomial series is an example of this when y=(a+x)^n. I will not discuss this, as it is not necessary for the definition of fluxion.

5.2.2 Fluxions

Definition 5.19 "If any Expression be augmented, or diminished, by the Augmentation or Diminuition of it's radical Quantity, I call the Increment, or Decrement of the Expression, the Difference of that Expression."(168)

First, he defines the fluxion of "the radical Quantity":

Definition 5.20 "If x be the radical Quantity of any Expression represented by y; and if x be augmented, or diminished, by any indeterminate Quantity z, I call z the Increment, or Decrement of x, the Fluxion of x , and denote it by x*."(169)

Then, finally, he defines the fluxion of a function y of x:

Definition 5.21 "And I call that x* Quantity, which is the first Maximinus or Minimajus of the Difference of y (arising from the Substitution of x ± x* for x in the Value of y) the first Fluxion of y , and denote it by y*."(170)

Example 5.22 If y=x^m, then y*=mx^(m-1)x*, because mx^(m-1)x* is the x*-quantity which is the first Maximinus or Minimajus of the difference between x^m and (x ± x*)^m.(170)

The second fluxion is defined similarly:

Definition 5.23 "And that x*² Quantity, which is either equal to,(171) or the first Maximinus or Minimajus of the Difference of y*, arising from the Substitution, of x ± x* for x in the Value of y*; I call the second Fluxion of y, and denote it by x**, thus, if y=x^m, y* = mx^(m-1) x*, y** =m × (m-1) x^(m-1) × x*^² for m × (m-1)x^(m-1) × x*² =* m x* × (x+ x*)^(m-1) ± mx^(m-1) x*".(170)

Paman then relates his theory to Newton's method of fluxions:

How the definitions are used is shown in section 5.4. Now it is time to study the definitions a bit closer.

5.3 Interpretation of Paman

Paman in fact has a definition of Maximinus and Minimajus, in the preface:

Paman's explanation looks like our present definition of Infimum and Supremum, but whereas our Infimum and Supremum given a set (of real numbers) gives a real number, the Maximinus and Minimajus of a quantity is a quantity of the same kind - I will come back to this shortly. Another difference is that Paman says that "a Maximinus, is such a Quantity as being less than another (...)", while today we have " < = ". Taken literally, Paman's explanation means that the constant function 1 has no Maximinus. 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 , that he is aware that equality is not covered by the concept of Maximinus and Minimajus. This is also seen another place,(176) where he writes:

Breidert goes on:

Here, Breidert seems to miss a major point: Paman's central concepts are not the Maximinus and the Minimajus, but the first Maximinus and Minimajus. The definition of first Maximinus can be "translated" into: "If there exists d > 0 such that ax^m < y(x) whenever 0 < x < d, and at the same time there exists no pair p > 0, D > 0 such that (a+p)x^m < y(x) whenever 0 < x < D, then ax^m is the first Maximinus of y."

But even the concepts of first Maximinus and Minimajus are not the same as Infimum and Supremum - or lim inf or lim sup - for instance sup{sin x : x > 0}=1 and inf{sin x : x > 0} = -1, lim inf {sin x} = lim sup {sin x} = 0 (x ->0), but the first Maximinus of sin x is x and no first Minimajus exists. Proposition 5.10 says clearly that the lowest term of a power series is the first Maximinus or Minimajus of the series. Thus the concepts of first Maximinus and Minimajus become very simple when dealing with power series.

For power series with arbitrarily large negative powers, for instance x sin(1/x), neither first Maximinus or Minimajus exist. Paman did not think of this kind of functions.(178)

5.3.2 Fluxions

y(x + x*) - y(x) = sum{k=n to oo} a_k (x+ x*)^k - sum{k=n to oo} a_k x^k =
x* sum{k=n to oo} a_k kx^(k-1) + x*² sum{k=n to oo} a_k k(k-1)x^(k-2) + ...

thus the first Maximinus or Minimajus of this Difference is x* sum{k=n to oo} a_k kx^(k-1) which is of course equal to the derivative found by modern methods.

Here I have used y(x+ x*)-y(x). Using y(x)-y(x- x*) gives the same result. Today we would expect a mathematician using the expression x ± x* 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 2 x*, 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 x* Quantity, which is either equal to, or the first Maximinus or Minimajus (...)

5.4 The use of the definitions

Proposition 5.24 "If x be the radical Quantity of y, and q x* be the first Fluxion of y, my^(m-1) × q x* will be the Fluxion of y^m".(179)

Proof "Let v represent the Difference of y, and (by Prop. Sect. iv.) my^(m-1) × v+m × (m-1) y^(m-1) × v^2 + & c. will be the real, or first approximating Value of the Difference of y^m; but, by Supposition, q x* =* v, which, substituted for v, will give my^(m-1) × q x* for that Quantity, which involves the lowest Power of x*; therefore my^(m-1) × q x* will be that x* Quantity, which is the first Maximinus or Minimajus of the Difference, and consequently the Fluxion of y^m, and equal to my^(m-1) y* (...)" QED.

To find the fluxion of AB, Paman first has to prove this Proposition:

Proposition 5.25 "If the first Maximinus or Minimajus of every particular Term of any Series, A, B, C, D, be substituted for the Terms themselves, then the Term or Quantity involving the lowest Power of x, arising from the Substitution, will be the first Maximinus or Minimajus of the Series. Thus if ax^m =* A, bx^(m+n) =* B, cx^(m+n+o) =* C, & c., then ax^m =* A,B,C,D, & c."(180)

Proof "Let the Difference between ax^m and A be called V, and put B, C, D, & c. = Q; and let the Difference between ax^m, and the Series A, B, C, D, & c. will be equal, either to the Sum of, or Difference between V and Q; but any x^m Quantity is, by Supposition, greater than V; and any x^m Quantity is (...) greater than Q, in the first State of x: Therefore, any x^m Quantity will be greater than the Difference between ax^m, and the Series, A, B, C, D, & c. in the first State of x; and ax^m will be the first Maximinus or Minimajus of the Series, A, B, C, D, & c. (...)"(181)

Proposition 5.26 "If x be the radical Quantity of the two Expressions represented by A and B, and d x* and q x* be their respective Fluxions, d x* ± q x* will be the Fluxion of A ± B, and A × q x* + B × d x* will be the Fluxion of the Product A × B."(182)

Proof "Call the Differences of A and B, arising from the Substitution of x ± x* for x, a and b, and, by Supposition, d x* =* a, and q x* =* b; therefore, (...) (d ± q) × x* =* A ± B, and by [the definition] the Fluxion of A ± B. Also Ab+Ba+,ab, is the Difference of A × B, for b and a substitute their first Maximinus's or Minimajus's q x* and d x*, and you will have A × q x* +B × d x* +dq x*²; therefore (Aq+Bd) × x* being that Quantity, which involves the lowest Power of x* will (by [Proposition 5.25]) be the first Maximinus or Minimajus of the Difference of A × B, and consequently [by the definition] the Fluxion of A × B."(183) QED.

Paman goes on to prove that

Proposition 5.27 "In any Equation the Fluxions of the Quantities on one Side, will be Equal to the Fluxions of the Quantities on the other Side of the same Order."

In section VI Paman considers geometry. For instance, he defines tangent using Maximinus' and Minimajus'. I will not go into the details of Paman's geometrical propositions.

5.5 Comments on Paman

5.5.3 It is clear what is the variable

5.5.4 The terminology

In a limited sense, they certainly are - Paman managed to give a foundation 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 x 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 Maximinus' and Minimajus' are cousins of liminf and limsup (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 useful to him and that would have been useful to mathematics if other mathematicians had noticed them.

5.5.5 Ancient and modern geometry

He does not, however, give a sufficiently clear explanation of this.

He goes on to say that

Thus we see that Paman wanted to be in the tradition of the old Greek mathematicians. But at the same time he wanted to avoid "the Tædium and Perplexity" of their ad absurdum proofs index reductio ad absurdum (see here). 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.

5.6 Paman - MacLaurin

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.(194) I don't think we have any way of finding out whether Paman did change his manuscript considerably after reading MacLaurin's book.

5.7 Paman in the literature

In the same year, 1989, Breidert too included a discussion on Paman.(197) He does not include much mathematical detail, and the little there is, is not totally convincing (see here).

Douglas Jesseph in 1993 included a somewhat more lengthy account,(198) which is both clear and correct, and includes a lot more mathematical detail than Breidert.

5.8 Conclusion

In my view, Paman's work is therefore superior to Robins' and MacLaurin's concerning the foundation of the method of fluxions - in addition to introducing important concepts which could have been used in other connections.

Chapter 6

English mathematicians may have had another reason for keeping to Newton's/MacLaurin's formulation: Perhaps the Analyst controversy(201) made them feel that their way of doing mathematics was more rigorous than the Continental one, and they therefore stuck to the Newtonian foundation and notation? This idea is present in Jesseph, where he says:

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 with time - looking at the situation both before and after the publication of The Analyst. As I do not have unlimited access to primary sources,(203) I will not be able to do a thorough investigation, but I hope I will find some points of interest.

6.1 Before 1710

Charles Hayes (1704) and William Jones (1706) are also guilty of this, while Humphry Ditton (1706) is a lot more careful in this respect.(206)

6.2 1710-1736

In 1711 John Keill wrote

and in 1712 this strange sentence appeared in the Commercium Epistolicum:(209)

In both of these quotes, the authors "forget" to say that the letters o, a, e, x* and dx denote very different things.

Much later, in 1730, Edmund Stone published his translation of l'Hôpital's Analyse des Infinements Petits, where every occurence of the word "différence" was translated with "fluxion", and dx was replaced by x*.(211)

6.3 1736-1741

Some of the writers were very clear, for instance James Hodgson, who in 1736 wrote

He also writes that in the method of fluxions, "Quantities are rejected, because they really vanish", in the differential method they are rejected "because they are infinitely small."(214)

Obviously, it would be difficult to make sense of what Newton said about fluxions and at the same time look at them from Leibniz' point of view.

6.4 After 1742

The important point, however, is that from the publication of MacLaurin's Treatise onwards, many English mathematicians felt that a sound foundation existed. Thomas Simpson, for instance, in 1757, wrote:

In the same year, a non-specialist's criticism of fluxions was met by one single sentence:

These two quotes suggest that MacLaurin's way of connecting the theory of fluxions with geometrical proofs were supposed to have given the theory a sound foundation - superior to the one on the Continent.

Olynthos Gregory wrote (in 1836-7):

and Florian Cajori agreed (in 1919):

We see that from 1742 to our own century, many English mathematicians have regarded the English (Newton/MacLaurin) foundation as better than Leibniz' foundation.

6.5 Pattern

6.6 Conclusion

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.

6.7 The rest of the story

Chapter 7

This volume of the Philosophical Transactions did not reach Leibniz until January 1711, but from then on things happened quickly: Leibniz demanded of the Royal Society (of which he was also a member) that Keill withdrew his charge - the President of the Royal Society was Newton himself. Keill drew Newton's attention to Leibniz' 1705 assertions, and got permission to write an answer to Leibniz. This answer, written in May 1711, was anything else than an apology. When Leibniz read it in December, he wrote a new letter to the Royal Society, calling upon both the President (Newton) and the Secretary (Sloane) to intervene.

The Royal Society appointed a committee which was to investigate the case - the report of this committee, however, was drafted by Newton himself. He was also the editor of the printed version of the report, the Commercium Epistolicum, which also included lots of previously unpublished evidence.

This was of course not enough to stop the controversy, nor was Leibniz' death in 1716 - Johann Bernoulli stepped in to defend Leibniz' honour.

Whiteside, in his article(221) on which this appendix is based, concludes:

Appendix B