Foundations for fluxions

Bjørn Smestad

Cand. Scient Thesis in Mathematics

Department of Mathematics

University of Oslo

1995

Contents

Preface to www-edition
A note of thanks
Preface
1 Introduction: The mathematical world 1650-1750
2 Newton's foundation for the fluxional calculus
3 The Analyst controversy
4 Colin MacLaurin
5 Roger Paman
6 The Analyst Controversy's effect on England's mathematical isolation
7 Conclusion
Appendix A The Newton-Leibniz controversy
Appendix B Chronology
Appendix C Some tables of contents
Appendix D Some short "biographies"
Footnotes
Bibliography

Preface to www-edition

The only changes I have made to the thesis are those necessary to transform the thesis from a printed paper to a html-document. Sadly, this means that any ambition of retaining the typographical peculiarities of Newton and others, had to go. I have also had to adopt the ascii-convention of writing x^n for x to the nth degree.

I would like to note two papers, though: After completing this thesis, I was made aware of a paper written about Roger Paman some years ago at Monash University, Australia. I have not been able to locate this paper now, but will update this preface as soon as it resurfaces. Secondly; in 1999, Jarle Bø wrote a cand. scient thesis titled "Begrepsapparatet i britisk analyse på 1700-tallet" at University of Oslo. This thesis studies Roger Paman further.

Alta 27/7-1999, Bjørn Smestad

A note of thanks

Few papers in mathematics have been written without any help. In history of mathematics such a paper is inconceivable, since a historian of mathematics at least will depend on the people who possess (or at a point of time have possessed) relevant books and manuscripts. Usually, the list of helpers includes many more than that.

All the same, when writing this foreword, I am surprised at the number of individuals and institutions who have, in one way or another, given their assistance. My thanks are due to them all:

First of all, of course, to my supervisor, Bent Birkeland, and to the Institute of Mathematics at the University of Oslo, who let me do this work under their wings. The latter also provided the necessary finances to take me to London for a week of October 1995.

Several libraries have been of assistance. The University Library here in Oslo has made lots of books and articles available to me. I must especially mention the library at the Institute of Mathematics, with Tina Mannai and Leyla Rezaye Golkar, who never gave up looking for all the 18th century books I asked for. In London, I was allowed unlimited access to the collections of the British Library, which were very helpful. The Royal Society helped me by sending me a copy of Roger Paman's Certificate of Election.

Douglas M. Jesseph of North Carolina State University and Wolfgang Breidert of the University of Karlsruhe have helped me by sending me copies of books after the libraries gave up. Jesseph has also kindly answered some questions of mine.

The organizers of The European Honours Course in History of Mathematics (in Utrecht) made it possible for me (and 23 others) to attend top-quality lectures for 3 weeks in July 1995, as well as discussing our respective subjects. I will mention Henk Bos and Klaske Blom in particular. In Utrecht, I also got the chance to discuss my paper with Niccolò Guicciardini of University of Bologna. That was very helpful. In addition, I got to know Adrian Rice, who later helped me find my way around London.

I would also like to thank the following persons who have not contributed in any way, but whom I have enjoyed being around all the same:

The people in C207; for instance Eivind, Helge, Ingvar, Kaija, Kjell, Kristine, Marianne, Morten H., Morten T., Roy, Runhild and Terje.

The people in Utrecht; for instance Anastasia, Inge, Katja, Mikhela, Nils and Per.

The people neither in C207 nor in Utrecht; for instance Olav Håkon, Phuoc and Tom.

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

(Mathematics is) the subject in which
we never know what we are talking about,
nor whether what we are saying is true.
(Bertrand Russell
(2))

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

Introduction: The mathematical world 1650-1750

A number of area and tangent problems could be solved as early as in the time of Archimedes, by double reductio ad absurdum proofs. For some reason, this number was not increased considerably in the 1800 years from 200 B.C. to 1600 A.D. But then, suddenly,
(5) a flow of new results emerged, by Kepler, Cavalieri, Fermat, Pascal, Roberval, Torricelli and others, using brave new methods - the infinitely small and the infinitely large were no longer banned from mathematics. The use of algebra to solve problems in geometry, was to be crucial. An important part of this development was Descartes' Geometria, published in 1637.

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

Pascal, Barrow and Newton said that irrational numbers could be understood only as geometrical magnitudes, they have no existence outside geometry. Others, such as John Wallis, accepted irrationals as numbers in its full sense.(7).

When it comes to negative numbers, Kline writes:

On the whole not many sixteenth- and seventeenth-century mathematicians felt at ease with or accepted negative numbers as such, let alone recognizing them as true roots of equations.(7)

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

1.2 Functions

The concept of a function, or a relation between variables, had one of its roots in the study of motion.(8). A curve was often considered as the path of a moving point.

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

Newton gave the series for sin x, cos x, arcsin x and exp (x). Others found other series, such as tan x and sec x.

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

In the years before the period we are looking at, the demand of proofs was reduced. Rigorous proofs in the way of Euclid were still the ideal, but in practice it became more and more usual to use induction from special cases, loose geometrical arguments and intuition. This was because mathematicians felt it was better to find many useful results they believed in, than one result they were absolutely certain of. When a result was found, it was more important to use it than to prove it rigorously.

1.5 Conclusion

In a way, it may have been fortunate that "the world of mathematics" was thus restricted in this period. It would have been extremely difficult to create a theory if all kinds of numbers and all the strange functions we know today should be included, the convergence problems sorted out and rigorous proofs given. But this restriction necessarily made the results less general than we would like them to be, and we would certainly not accept all the proofs. This must be remembered when studying mathematical works of this period.

Chapter 2

Newton's foundation for the fluxional calculus

God said, Let Newton be!
-And all was light.
(Pope)

2.1 Introduction

Isaac Newton is perhaps most famous for his physics, but his work on mathematics was also impressive. Among other things, he is considered the co-founder of calculus with Leibniz.
(10) One of the main virtues of their calculus is the simplicity with which the rate of growth can be found for many important expressions - and thereby tangents, maxima and minima and so on. The main rules for finding this rate of growth, given an equation of the curve in question, were given by both Newton and Leibniz (For instance, in modern notation: (x^n)'=nx^(n-1), (f+g)'=f'+g', (f(g(x)))'=f'(g(x))g'(x) and so on). These we still use today, of course. But the definitions and arguments underlying these rules caused considerable problems, and neither Newton nor Leibniz ever succeeded in giving a rigorous foundation, according to modern standards. Newton's point of view changed throughout his life. I will give an overview of this development in this chapter.

2.2 1665-1680: Fluxions

In his method of fluxions, Newton considered lines as generated by points in motion, planes as generated by lines in motion and bodies as generated by planes in motion.(11) What was in motion he called fluents, their velocity he called fluxions. To him, as a physicist, it was obvious that every moving body has an instantaneous velocity, independent on how the velocity will change at a later time.

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):

a...........c.........e......g
b.........d.......f.......h...
Figure

Lemma. If two bodys A and B move uniformly y'e one from a to c, e, g & c and y'e other from b to d, f, h & c in y'e same time. y'n are y'e lines ac & ce & eg & c and bd & df & fh & c as their velocitys p and q.

And though they move not uniformly yet are y'e infinitely little lines w'ch each moment they describe as their velocitys are w'ch they have while they describe them. As if y'e body A w'th y'e velocity p describe y'e infinitely little line o in one moment. In y't moment y'e body B w'th y'e velocity q will describe y'e line oq/p. For p:q::o:(oq/p). So y't if y'e described lines be x & y in one moment, they will bee x+o & y+(oq/p) in y'e next.

Now if y'e Equation expressing y'e relation of y'e lines x & y be rx+xx-yy=0. I may substitute x+o & y+(qo/p) into y'e place of x & y because (by y'e lemma) they as well as x & y doe signifie y'e lines described by y'e bodys A & B. By doeing so there results rx+ro+xx+2ox+oo-yy-(2qoy/p)-(qqoo/pp)=0. But rx+xx-yy=0 by supposition: there remaines therefore ro+2ox+oo-(2qoy/p)-(qqoo/pp)=0. Or divideing it by o tis r+2x+o-(2qy/p)-(oqq/pp)=0. Also those termes in w'ch o is are infinitely less y'n those in w'ch o is not therefore blotting y'm out there rests r+2x-(2qy/p)=0. Or pr+2px=2qy.(13)

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:

Hence I observe. First y't those termes ever vanish w'ch are not multiplyed by o, they being y'e propounded equation. Secondly those termes also vanish in w'ch o is of more y'n one dimension, because they are infinitely lesse y'n those in w'ch o is but of one dimension. Thirdly y'e still remaining termes, being divided by o will have y't form w'ch (...) they should have (...)(15)

Here, he uses the word "infinitely" in a very uncritical way. In June 1669(?) he mentioned

Nor am I afraid to talk of a unity in points or infinitely small lines inasmuch as geometers(16) now consider proportions in these while using indivisible methods.(17)

Thus he takes comfort in the other geometers' habits (talking of infinitely small lines), and uses that as an excuse for having these habits himself.

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.

The moments of the fluent quantities (that is, their indefinitely small parts, by addition of which they increase during each infinitely small period of time) are as their speeds of flow.

(...) Let there be given, accordingly, any equation x³-ax²+axy-y³=0 and substitute x+no in place of x and y+mo in place of y: there will emerge (x³ +3nox² +3 n²o²x+ n³o³) - (ax² +2anox+ an²o²)+ (axy+ anoy+ amox+ anmo²) - (y³ +3moy² +3m²o²y+ m³o³)=0. Now by hypothesis x³ -ax² +axy-y³ =0, and when these terms are erased and the rest divided by o there will remain 3nx² +3n²ox+ n³o² -2anx -an²o+ any+ amx+ anmo- 3my² -3m²oy-y³o² =0. 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. I therefore cast them out and there remains 3nx²- 2anx+ any+ amx- 3my² =0 (...)(18)

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

In 1680, in his Geometria Curvilinea, Newton started to look at fluxions in a new way,(19) in an attempt to avoid infinitesimals:

Those who have taken the measure of curvilinear figures have usually viewed them as made up of infinitely many infinitely-small parts. I, in fact, shall consider them as generated by growing, arguing that they are greater, equal or less according as they grow more swiftly, equally swiftly or more slowly from their beginning. And this swiftness of growth I shall call the fluxion of a quantity.(20)

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:

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.(21)

(...) of course, proofs are rendered more compact by the method of indivisibles. 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.(22)
Newton's motive seems clear: he wants to find a ``more geometrical'' - meaning more rigorous - method. 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, inasmuch 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. By the last speed is understood that with which a body is moving, not before it attains its last position and its motion ceases, nor afterwards, but precisely when it reaches it - the exact speed, that is, with which the body reaches its last position and with which its motion ceases. And similarly by the last ratio of vanishing quantities you must understand not the ratio of the quantities before they vanish, nor that afterwards, but that with which they vanish. (...) There exists a limit which their speed can at the end of its motion attain, but not, however, surpass. This is their last speed.(23)

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:

(...)products, quotients, roots, rectangles, squares, cubes, square and cubic sides and the like (...) I here consider as variable and indetermined, and increasing or decreasing as it were by perpetual motion or flux; and I understand their momentaneous increments 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 either 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.(24)

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

Case 1. Any rectangle, as AB, augmented by a continual flux, when, as yet, there wanted of the sides A and B half their moments ½a and ½b, was A- ½a into B- ½b, or AB- ½aB- ½bA+ ¼ab; but as soon as the sides A and B are augmented by the other half-moments, the rectangle becomes A+ ½a into B+ ½b, or AB+ ½aB+ ½bA+ ¼ab. From this rectangle subtract the former rectangle, and there will remain the excess aB+bA. Therefore with the whole increments a and b of the sides, the increment aB+bA of the rectangle is generated. Q.E.D.(24)

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:

Let the quantity x flow uniformly and the fluxion of the quantity x^n need to be found. In the time that the quantity x comes in its flux to be x+o, the quantity x^n will come to be (x+o)^n, that is (when expanded) by the method of infinite series(25) x^n +nox^(n-1)+ ½(n^2 -n)o^2 x^(n-2) + ..., and so the augments o and nox^(n-1) + ½(n^2 -n)o^2 x^(n-2)+... are one to the other as 1 and nx^(n-1) + ½(n^2 -n)ox^(n-2) + .... Now let those augments come to vanish and their last ratio will be 1 to nx^(n-1); consequently the fluxion of the quantity x is to the fluxion of the quantity x^n as 1 to nx^(n-1).(26)

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"

In 1703 Newton wrote about infinitesimals in these terms:

Math. quantities I here consider not as consisting of indivisibles, either parts least possible(27) or infinitely small.(28)

(...) each time it can conveniently so be done, it is preferable to express [fluxions] by finite lines visible to the eye rather than by infinitely small ones.(29)

(...) and I wanted to show that in the method of fluxions there should be no need to introduce infinitely small figures into geometry.(30)

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:

Fluxions & moments are quantities of a different kind. Fluxions are finite motions, moments are infinitely little parts (...) [I] multiply fluxions by the letter o to make them become infinitely little (...)(33)

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

For fluxions are finite quantities but moments here are infinitely little.(34)

[The method of fluxions] is more elegant [than the Differential Method of Leibniz], because in his Calculus there is but one infinitely little Quantity represented by a symbol, the symbol o. 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.(35)

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:

The theory of fluxions yielded the heuristic methods of the calculus. Those methods were to be justified rigorously by the theory of ultimate ratios. The theory of infinitesimals was to abbreviate the rigorous proof, and Newton thought that he had shown the abbreviations to be permissible. Rather than competing for the same position, the three theories were designed for quite distinct tasks.(38)

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

This is his method at present [in 1713], this was his method when he wrote his two Letters of 1676 & five years before when he wrote the Tract mentioned in the latter of those two letters & that this was his method in the year 1669 when he communicated his Analysis to Dr. Barrow (...), appears by the Analysis itself.(39)

2.5 Conclusion

I am not sure that Newton had such a clear and intended partition of his theory as Kitcher thinks. 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 conservatism 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 especially 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 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

The Analyst controversy

(...) I suspect that he is one
of that sort of man who wants
to be known for his paradoxes.
(Leibniz
(40))

3.1 Introduction

In 1734(41) Bishop George Berkeley published The Analyst. 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,(42) a part of which I will look at in this chapter.

3.2 The Analyst - Berkeley's main points

The following were Berkeley's main points of criticism:

3.2.1 The fluxions were incomprehensible

Berkeley felt that the theory of fluxions was incomprehensible:

By moments we are not to understand finite particles. 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 mathematics: that the fluxions are celerities, not proportional to the finite increments, though ever so small; but only to the moments 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 second fluxions. And the fluxions of these second fluxions are called third fluxions: and so on, fourth, fifth, sixth. & c. ad infinitum.
Now, as our sense is strained and puzzled with the perception of objects extremely minute, even so the imagination, which faculty derives from sense, is very much strained and puzzled to frame clear ideas of the least particles of time, or the least increment 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. But the velocities of the velocities, the second, third, fourth, and fifth velocities, &c., exceeds, if I mistake not, all human understanding.(43)

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

Further, Berkeley criticized some of Newton's proofs:

(...) 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.(45)

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

(...) it is plain that the direct and true method to obtain the moment or increment of the rectangle AB, is to take the sides as increased by their whole increments, and so multiply them together, A+a by B+b, the product whereof AB+aB+bA+ab is the augmented rectangle; whence, if we subduct AB the remainder aB+bA+ab will be the true increment of the rectangle, exceeding that which was obtained by the former illegitimate and indirect method by the quantity ab. (...) 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.(45)

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)):

(...) it should seem that this reasoning is not fair or conclusive. For when it is said, let the increments vanish, i.e. let the increments be nothing, or let there be no increments, the former supposition that the increments were something, or that there were increments, is destroyed, and yet a consequence of that supposition, i.e. an expression got by virtue thereof, is retained. Which (...) is a false way of reasoning. 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.(46)

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

Berkeley also came up with a theory of why the results of the calculations were always right, even though the procedure was wrong. 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.(47)

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

If valid, Berkeley's criticism was devastating. This is made clear by Philalethes, for instance in this quote, where he says what would be the opinion of Newton if Berkeley was right. If Berkeley was to believed, Newton's mathematics was incomprehensible and some of his most important 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.

Philalethes Cantabrigiensis

The first answer to The Analyst was Geometry, no Friend to Infidelity, published under the name Philalethes Cantabrigiensis. The real author is believed to have been Dr. James Jurin of Cambridge.(49) This paper was dated April 10th, 1734, which means that Jurin had worked quite fast.

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:

Your objections against this method may, I think, all of them be reduced under these three heads.

  1. Obscurity of this doctrine.
  2. False reasoning used in it by Sir Isaac Newton, and implicitely received by his followers.
  3. Artifices and fallacies used by Sir Isaac Newton, to make this false reasoning pass upon his followers.(50)
He goes on to treat these in order:

3.3.1 Obscurity of this doctrine

Philalethes agrees that the doctrine is "not without its difficulties",(51) but declares that he and many others have understood it, and that Berkeley can, too, if he "will read it with due attention, and a desire of comprehending it, rather than an inclination to censure it."(52)

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? Where do you find Sir Isaac Newton using such expressions as the velocities of the velocities, the second, third and fourth velocities, (...)(53)

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.

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 here). 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 quantity 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 orb of the fixed stars; do not they know, I say, and are they not able clearly and invincibly to demonstrate that, in so immense a magnitude, this omission shall not cause them to deviate from the truth so much as a single pin's head, nay not the thousandth, not the millionth part of a pin's head?(54)

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

[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 (...)(55)

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:

Suppose two Arithmeticians to be disputing whether vulgar fractions 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 fraction 1/3, to affirm that his antagonist proceeded blindfold, and without knowing what he did, when he pretended to express it by 0.33333 & c. because this expression did not give the rigorous, exact value of one third of a farthing? 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? Might not he further say that as the & c. implied all the possible repetitions of the figure 3, even to infinity, therefore his expression did not differ by any the least assignable quantity from the other value, 1/3, and that as he knew and clearly conceived that it did so, he could not justly be said to be in any error, much less to act in the dark, when he used that expression?(56)

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;

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- ½a) × (B- ½b), and you must own that he takes it the direct and true method to obtain it.(57)

For what reason did Newton calculate this increment?

(...) in order to find the moment of the rectangle AB it is more consonant to strict Geometrical rigour to take the increment of the rectangle (A- ½a) × (B- ½b), than to take the increment of the rectangle AB itself (...)

You know very well that the moment of the rectangle AB is proportional to the velocity of that rectangle, with which it alters, either in increasing, or in diminishing. Now, I ask, in Geometrical rigour what is properly the velocity of this rectangle? Is it the velocity with which the rectangle from AB becomes (A+a) × (B+b); or the velocity with which from AB it becomes (A-a) × (B-b)? I find my self exactly in the case of the Ass between the two bottles of hay: I see no reason, nor possibility of a reason to determine me either one way, or the other. But methinks I hear the venerable Ghost of Sir Isaac Newton whisper 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. In like manner the moment of the rectangle is neither the increment from AB to (A+a) × (B+b); nor is it the decrement from AB to (A-a) × (B-b): It is not a moment common to AB and (A+a) × (B+b), which may be considered as the increment of the former, or as the decrement of the latter: Nor is it a moment common to AB and (A-a) × (B-b), which may be considered as the decrement of the first, or as the increment of the last: But it is the moment of the very individual rectangle AB itself, and peculiar to that only; and such as being considered indifferently either as an increment or decrement, shall be exactly and perfectly the same. And the way to obtain such a moment is not to look for one lying between AB and (A+a) × (B+b); nor to look for one lying between AB and (A-a) × (B-b): that is, not to suppose AB as lying at either extremity of the moment; but as extended to the middle of it; as having acquired the one half of the moment, and as being about to acquire the other; or as having lost one half of it, and being about to lose the other. And this is the method Sir Isaac Newton has taken in the demonstration you except against.(58)

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:

Lastly, to remove all scruple and difficulty about this affair, I must observe, that the moment of the rectangle AB, determined by Sir Isaac Newton, namely aB+bA, and the increment of the same rectangle, determined by yourself, namely aB+bA+ab, are perfectly and exactly equal, supposing a and b to be diminished ad infinitum; and this by the Lemma(59) just quoted.(60)

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).

(...) this is so great, so unaccountable, so horrid, so truly Boeotian a blunder, that I know not how to think a Great Genius, a Newton could be guilty of it. For God's sake let us examine it once more. Evanescant jam augmenta illa, let now the increments vanish, i. e. let the increments be nothing, or let there be no increments. Hold, Sir, I doubt we are not right here.(62)

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

What then must we think of your interpretation, Let the increments be nothing, let there be no increments? 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 acquit Sir Isaac Newton of the gross oversight you ascribe to him (...)(64)

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

In The Analyst, Berkeley mentions that Newton used several different ways of explaining his theory, and says that this is because Newton doubted his previous explanations. 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.(65)

3.3.4 The compensation of errors thesis

Philalethes goes on to consider Berkeley's compensation of errors thesis (see here). 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: just as the Athenians once painted her dragging towns and cities to her favourite General. 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 was here? 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 10000 l. prize in the present Lottery, with a single Ticket.(66)

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

When Boyer describes Philalethes' answer to Berkeley with the words "weak in the extreme",(67) he agrees with most historians of mathematics who have discussed Philalethes. For instance Wisdom writes: "(...) it follows that Philalethes did not understand either Berkeley, Newton or fluxions!", (68) Jesseph feels that "(...) Jurin was clearly not the man to be entrusted with the task of clarifying and defending the calculus.", (69) while Gibson claims that "It is impossible by means of extracts to convey a sufficient sense of the extreme vagueness and want of precision on the part of Philalethes when treating the crucial points of a theory of limits (...)". (70). An exception is Buffon, who calls Philalethes' defence "solid, brilliant, admirable".(71) In this section I will examine the main points of criticism that have been raised against Philalethes:

The fluxion of AB

The main parts of the criticism concerns Philalethes' defence of Newton's calculation of the fluxion of AB. 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 here). Berkeley saw it like this and wrote

I had observed that the great author [...] did not fairly get rid of the rectangle of the moments. 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.(72)

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:

That [Philalethes] should fail to see the soundness of Berkeley's criticism of Newton's proof (A+ ½a)(B+ ½b)- (A- ½a)(B- ½b) for the increment of AB is somewhat surprising, even if it must be admitted that neither Walton nor any other eighteenth-century mathematician appears to have seen and admitted the defect.(74)

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

Berkeley claimed that if o <> 0, nx^(n-1) + ½(n^2 -n)ox^(n-2) + ... will not be equal to nx^(n-1), and if o=0, the division by o is invalid. Berkeley did not seem to understand that Newton considered what happened to nx^(n-1) + ½(n^2 -n)ox^(n-2) + ... when o was positive, and was interested in what happened when o approached 0, and that he used his theory of prime and ultimate ratios to find this out. 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."

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

Apart from one major error, I would say that Philalethes gives an adequate answer to Berkeley - in the sense that his polemic piece probably countered the Analyst's effect on non-mathematicians' opinion of mathematics. He does not, however, give an explanation of the theory of fluxions. That task was left to the next writer on the subject.

3.4 Benjamin Robins

The next answer to Berkeley came from the scientist Benjamin Robins and was called A Discourse Concerning the Nature and Certainty of Sir Isaac Newton's Methods of Fluxions, and of Prime and Ultimate Ratios (1735). While Philalethes concentrates on finding faults in Berkeley's criticism, Robins is more concerned with explaining the fluxional calculus.

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

(...) though Sir Isaac Newton has very distinctly explained both these subjects, the first in his treatise on the Quadrature of curves, and the other in his Mathematical principles of natural philosophy; yet as the author's great brevity has made a more dissusive illustration not altogether unnecessary; I have here endeavored to consider more at large each of these methods; whereby, I hope, it will appear, they have all the accuracy of the strictest mathematical demonstration.(76)

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

3.4.1 Of fluxions

Robins starts by giving an explanation of the main ideas:

IN the method of fluxions geometrical magnitudes are not presented to the mind, as compleatly [sic] formed at once, but as rising gradually before the imagination by the motion of some of their extremes *. [*: Newt. Introd. ad Quad. Curv.](77)

THUS the line AB may be conceived to be traced out gradually by a point moving on from A to B, either with an equable motion, or with a velocity in a manner varied. 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.(77)

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)

FLUXIONS then in general are the velocities, with which magnitudes varying by a continued motion increase or diminish; and the magnitudes themselves are reciprocally called the fluents of those fluxions **. [** Motuum vel incrementorum velocitates nominando fluxiones, & quantitates genitas nominando fluentes. Newton. Introd. ad. Quadr. Curv.](79)

A.......I....E...G..............B
C..........K....F....H........D
Figure

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.

When the number n is greater than unite, while the line AB is described with a uniform motion, the point, wherewith CD is described, moves with a velocity continually accelerated; for if IE be equal to EG, FH will be greater than KF. Now, here, I say, that neither the proportion of FH to EG, nor the proportion of KF to IE is the proportion of the velocity, which the point moving on CD has at F, to the uniform velocity, wherewith the point moves on the line AB. 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 of time with the velocity, it has at F, uniformly continued, it would not have passed over so long a line; consequently FH bears a greater proportion to EG, than what the velocity, which the point moving on CD has at F, bears to the velocity of the point moving uniformly on AB.(81)

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:

IN like manner KF bears to IE a less proportion than that, which the velocity of the point in CD has at F, to the velocity of that in AB. 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.(81)

After these fundamental observations, the result can be found:

IN the last place I say, that no line whatever, that shall be greater or less than the line represented by the second term of the foregoing series (viz. the term (nx^(n-1)e)/(a^(n-1))) will bear to the line denoted by e the same proportion, as the velocity, wherewith the point moves at F, bears to the velocity of the point moving in the line AB; but that the velocity at F is to that at E as (nx^(n-1)e)/(a^(n-1) to e, or as nx^(n-1) to a^(n-1).(81)

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

IF possible let the velocity at F bear to the velocity at E a greater ratio than this, suppose the ratio of p to q.

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. Again, by diminishing e, the ratio of the second term in this series to twice the third, that is, of (nx^(n-1)e)/(a^(n-1)) to (n×(n-1)x^(n-2)ee)/(a^(n-1)) or the ratio of x to (n-1)×e, shall be greater than any, that shall be proposed, consequently the line e may be taken so small, that twice the third term, that is (n×(n-1)x^(n-2)ee)/(a^(n-1)) shall be greater than all the terms following the second, and also, that the ratio of (nx^(n-1)e)/(a^(n-1)) + (n×(n-1)x^(n-2)ee)/(a^(n-1)) to e shall less exceed the ratio of (nx^(n-1))/(a^(n-1)) to e, than any other ratio, that can be proposed. Therefore let the ratio of (nx^(n-1)e)/(a^(n-1)) + (n×(n-1)x^(n-2)ee)/(a^(n-1)) to e be less than the ratio of p to q; then, if (n×(n-1)x^(n-2)ee)/(a^(n-1)) be also greater than the third and all the following terms of the series, the ratio of the series (nx^(n-1)e)/(a^(n-1)) + (n×(n-1)x^(n-2)ee)/(2a^(n-1)) + & c. to e, that is, the ratio of FH to EG shall be less than the ratio of p to q, or of the velocity at F to the velocity at E, which is absurd; for it has above been shewn, that the first of these ratios is greater than the last. Therefore the velocity at F cannot bear to the velocity at E any greater proportion than that of (nx^(n-1)e)/(a^(n-1)) to e.(82)

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

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 nx^(n-1) to a^(n-1) is the true proportion of the velocity, wherewith CF, or (x^n)/(a^(n-1)) augments, to the velocity, wherewith AE, or x is at the same time augmented.(83)

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

As an introduction to the method of prime and ultimate ratios, Robins explains the method of exhaustions . This is because

THE concise form, into which Sir Isaac Newton has cast his demonstration, may very possibly create a difficulty of apprehension in the minds of some unexercised in these subjects. But otherwise his method of demonstrating by the prime and ultimate 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.(84)

The principal definitions are as follows:

IN this method any fix'd quantity, which some varying quantity, by a continual augmentation or diminuition, shall prepetually [sic] approach, but never pass, is considered as the quantity, to which the varying quantity will at last or ultimately become equal; provided the varying quantity can be made in its approach to the other to differ from it by less than by any quantity how minute soever, that can be assigned * [* Princ. Philos. Lib. I. Lem. I.](85)

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:

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 (...)(86)

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

FROM any ratio's having such a limit, it does not follow, that the variable quantities exhibiting the ratio have any final magnitude, or even limit, which they cannot pass.(87)

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

THE quadrilateral ABCD bears to the quadrilateral EBCF the proportion of AB+CD to BE+CF, provided the two lines AE and DF are parallel. Now if the line DF be drawn nearer to AE, this proportion of AB+DC to BE+CF will not remain the same, unless the lines DA, CB, FE produced will meet in the same point; and this proportion, by diminishing the distance between DF and AE may at last be brought nearer to the proportion of AB to BE, than to any other whatever. Therefore the proportion of AB to BE is to be considered as the ultimate proportion of AB+DC to BE+CF, or as the ultimate proportion of the quadrilateral ABCD to the quadrilateral EBCF.

HERE these quadrilaterals can never bear one to the other the proportion between AB and BE, nor have either of them any final 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, because it is the proportion, which those quadrilaterals can never actually have to each other, but the limit of that proportion.

THE quadrilaterals may be continually diminished, either by dividing 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 vanishing 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 quadrilateral 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.(88)

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

Using the method of prime and ultimate ratios, Robins is now able to give a shorter proof concerning the fluxion of x^n (see the figure)

A..........B.......E....................
C..............D......F.................
Figure

FOR determining the fluxion of a simple power suppose the line AB to be denoted by x, and another line CD to be denoted by (x^n)/(a^(n-1)), or by considering a as unite, CD will be denoted by x^n. 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 in the same time with an uniform motion. But if the point describing 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 diminishing 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. Therefore the ratio of the velocity at D to the velocity at B, that is, of the fluxion of CD to the fluxion of AB, is the same with the ultimate ratio of DF to BE.

IF now the augment BE be denoted by o, the augment DF will be denoted by nx^(n-1)o+ (n×(n-1))/2 × x^(n-2)o^2+ (n×(n-1)×(n-2))/6× x^(n-3)o^3+ & c. 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. Consequently the proportion of the first term nx^(n-1)o to the whole augment may be made to approach within any degree whatever of the proportion of equality; and therefore the ultimate proportion of nx^(n-1)o+ (n×(n-1))/2 × x^(n-2)o^2+ (n×(n-1)×(n-2))/6× x^(n-3)o^3+ & c. to o, or of DF to BE, is that of nx^(n-1)o only to o, or the proportion of nx^(n-1) to 1.

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 x^n to the fluxion of x, as nx^(n-1) to 1.(89)

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

Robins chose to avoid the term momentum for the first 74 pages - and with a good reason: this is possibly the most obscure of all of Newton's terms. 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 Newton's definition of momenta, That they are the momentaneous increments or decrements of varying quantities, may possibly be thought obscure.(90)

IN determining the ultimate ratios between contemporaneous differences of quantities, it is often previously required to consider each of these differences apart, in order to discover, how much of those differences is necessary for expressing that ultimate ratio. 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.(91)

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:

(...) if A and B denote varying quantities, and their contemporaneous increments be represented by a and b; the rectangle under any given line M and a is the contemporaneous increment of the rectangle under M and A, and A × b + B × a + a × b is the like increment of the rectangle under A,B. And here the whole increment M × a represents the momentum of the rectangle under M,A; but A × b + B × a only, and not the whole increment A × b + B × a + a × b, is called the momentum of the rectangle under A,B; because so much only of this latter increment is required for determining the ultimate ratio of the increment of M × A to the increment of A × B, this ratio being the same with the ultimate ratio of M × a to A × b + B × a; for the ultimate ratio of A × b + B × a to A × b + B × a + a × b is the ratio of equality. Consequently the ultimate ratio of M × a to A × b + B × a differs not from the ultimate ratio of M × a to A × b + B × a + a × b.(92)

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:

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 momentum of any quantity may be determined, either by considering 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.(93)

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

In my opinion, the most important difference between the Philalethes' Geometry and Robins' Discourse, is their aim. 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 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

Gibson writes that at their own time

It is usually Jurin who obtains the credit for refuting Berkeley, and when Robins is mentioned at all, his criticism is put alongside that of Jurin.(97)

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:

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.(98)

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

Philalethes and Robins were not the only ones to write an answer to Berkeley. 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).

3.8 Conclusion

It is noteworthy that Berkeley withdrew from the discussion at an early stage, and left the mathematicians quarrelling - and thereby proving his point. He commented briefly on the dispute in a footnote in Siris (1744), saying that

(...) witness their doctrine of fluxions, about which, within these ten years, I have seen published about twenty tracts and dissertations, whose authors being utterly at variance with each other, instruct bystanders what to think of their pretensions to evidence.(99)

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

Colin MacLaurin

The Analyst has met with
universal contempt, I am glad
you have undertaken him.
(Conduitt
(101))

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

Besides an answer to The Analyst that appeared very early under the name of Philalethes Cantabrigiensis (...), a second, by the same hand, in Defence of the first, a Discourse by Mr. Robins, a Treatise of Sir Isaac Newton's, with a Commentary by Mr. Colson, and several other Pieces, were published on this Subject.(104)

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.

In explaining the Notion of a Fluxion, I have followed Sir Isaac Newton in the first Book, imagining that there can be no difficulty in conceiving Velocity wherever there is Motion; nor do I think that I have departed from his Sense in the second Book (...)(105)

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

In the beginning of Book I, MacLaurin discusses the ancient geometry, giving several of Archimedes' proofs. Then he goes on to explain the theory:

(...) we conceive the quantities to be increased and diminished, or to be wholly generated by motion, or by a continual flux analogous to it. The quantity that is thus generated is said to flow, and called a Fluent.(106)

The velocity with which a quantity flows, at any term of the time while it is supposed to be generated, is called its Fluxion, which is therefore always measured by the increment or decrement that would be generated in a given time by this motion, if it was continued uniformly from that term without any acceleration or retardation (...)(107)

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):

The sides AD, AE (...), of the triangle ADE being given in position, and the angle ADE being also given; in the same time that the motion with which the base AD flows, continued uniformly, would generate any right line DG, the motion with which the triangle ADE flows, continued uniformly would generate the parallelogram EG. Or, the fluxion of the base AD being represented by DG, the fluxion of the triangle ADE is accurately measured by the parallelogram EG.(109)
The proof of this theorem occupies more than pages. I will not quote it - but the main idea is to consider an invariable line, moving at such speed that the rectangle generated by it has the same area as the triangle generated by the motion of DE. 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 quantity increases or decreases, is supposed to be infinitely small, and is generally expressed by two or more terms, some of which are infinitely less than the rest, which being neglected as of no importance, 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.(110)

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:

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!]), and may obviate some objections that have been made to it.(112)

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

In Book II, MacLaurin discusses the "algebraic part"(113) of the method of fluxions, after explaining the use of negative and imaginary numbers.

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:

(...) it does not seem necessary to have always recourse to such suppositions [as quantities being generated by motions index motion etc.](114)

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

By the fluxions index fluxions of quantities we shall therefore now understand, any measures of their respective rates of increase or decrease, while they vary (or flow) together.(115)

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.

703. The successive values of the root A being represented by A-a, : A, : A+a, : & c. which increase by any constant difference a, let the corresponding values of any quantity produced from A by any algebraic operation (or that has a dependance upon it so as to vary with it) be B-b, : B, : B+ þ, : & c.(116) Then if the successive differences b, þ, & c. of the latter quantity always increase, how small soever a may be, then B cannot be said to increase at so great a rate as a quantity that increases uniformly by equal successive differences greater than þ, or at so small a rate as any quantity that increases uniformly by equal successive differences less than b. In like manner, if the relation of the quantities is such, that the successive differences, b, : þ, : & c. continually decrease; then B cannot be said to increase at the same rate as a quantity that increases uniformly by equal successive differences greater than b, or less than þ.

704. Therefore the fluxion of A being supposed equal to the increment a, the fluxion of B cannot be greater than þ or less than b, when the successive differences b, : þ, : & c. continually increase; and cannot be greater than b, or less than þ, when these successive differences always decrease.(117)

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

Boyer writes that "This work, however, was as little read as it was widely praised (...)",(121) while Kline thinks that MacLaurin's work was "no doubt profound, but incomprehensible".(122) That the book is incomprehensible is an exaggeration. And the fact that it was little read must not be held against MacLaurin either. As MacLaurin himself writes (in defence of Archimedes):

(...) the number of steps is not the greatest fault a demonstration may have; nor is this number to be always computed from those that may be proposed in it, but from those that are necessary to make it full and conclusive(123)

- 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

[Colin MacLaurin] attempted to establish the rigor of the calculus. It was a commendable effort but incorrect.(124)

without giving any reasons for his claim.

Turnbull, on the other hand, writes that

'The Treatise of Fluxions' (...) is a masterpiece of reasoning, in which MacLaurin gave a systematic account of Newton's theory, set out in both geometrical and analytical form, with a wealth of applications and many discoveries. (...) In point of rigor it is a worthy link between the ancient method of exhaustions and the subsequent work of Cauchy and of Weierstrass.(125)

Paman writes about MacLaurin that his

Performance, as it is the last, so it is, without Doubt, the clearest, best guarded, and most elegant, of any general Treatise of Fluxions.(126)

and that

(...) MacLaurin had published his Treatise of Fluxions, and (...) it was the general Opinion of Mathematicians, he had fully confuted the Analyst, and rendered any farther Notice thereof unnecessary (...)(127)

4.4 Conclusion

MacLaurin's foundation in Book I is based on the intuitive concepts of motion and instantaneous velocity, but the proofs are given by means of geometry. 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. 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

Roger Paman

Paman's work was crippled
by his extensive use of
new terminology (...)
(Sageng
(128))

5.1 The life of Roger Paman

In 1745 there appeared a book which did not get very much attention, neither when it was published nor later. This was Roger Paman's The Harmony of the Ancient and Modern Geometry Asserted. 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. The most well-known one is Henry Paman, Professor of physics at Gresham College. He was born at his father's estate at Chevington, Suffolk in 1629.(129) In the village Chevington, the Paman family was an important one,(130) but no Roger Paman seems to have been born here in the period in question.(131)

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:

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.(136)

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:

The scheme which Commodore Anson was sent to execute, was certainly well laid; and if the two ships that repassed the Streights 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 (...)(137)

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:

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 recommend him as personally known to us and likely to become a usefull Member thereof(139)

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":

The harmony of the antient and modern geometry asserted; in answer to the Analyst, & c. pr. 7s. 6d. sew'd. Nourse(140)

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

(...) the height of the Mercury in the Thermometer every Day at Noon, during the Months of February, March, April and May, between the Latitudes of 40° and 60° South.

An accurate Account of the Variations of the Needle, at different Distances, on the same Parallels from the Coasts of Brazil, Patagonia, and Terra del Fuego.

With such Curious Particulars relating to different Parts of South America, as the Author had an Opportunity of remarking himself, or procuring from Persons of Credit and Distinction, during Seven Months that he lived in Brazil.

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 1748.(141)

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

In 1745 there appeared an anonymous(142) publication on fluxions which we have not had the opportunity to examine; it was entitled, The Harmony of the Ancient and Modern Geometry asserted. In A. C. Fraser's edition of Berkeley's Works, vol. iii, Oxford, 1871, p. 301, it is referred to as follows: 'This last and forgotten tract consists of papers given in to the Royal Society in 1742, and treats fluxions as a particular branch of an alleged more general reasoning, called the doctrine of maximinority and minimajority'.(143)

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

Paman's ambitious aim in writing this book is clear from its very first words:

HAVING undertaken to cultivate the Discoveries of the Moderns upon the Principles of the Ancients, without any Considerations of Velocity, Time or Motion of Indivisibility or Infinity, in such a Manner that, whilst I omitted those Considerations, I might not neglect the Design (...) of introducing them first into Geometry, and that whilst I aimed at the Rigour of the Ancients, I might avoid the Tædium and Perplexity of their Demonstrations ad absurdum.(144)

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:

(...) by the first State of x, I do not mean the nascent or evanescent State of Sir Isaac Newton, nor, by any of the Values of x in its first State, the Minimum Magnum of Dr. Barrow, or the Infiniment Petit of the Marquis de l'Hospital; but all the finite Values of x less than a particular Value, which particular Value is assignable from the Quantities compared: And in the last State of x 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 x greater than a particular Value, the Assignability whereof depends upon the Quantities compared.(147)

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

Another important concept is the (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:

However harshly the Names of Maximinus or Minimajus may sound, their Existence is evident every where (...)

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

After defining one last term, Paman will be ready to define fluxion:

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:

Thus it appears, that the radical Quantity has no second, third, & c. 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.(172)

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

When writing history of mathematics, there is always an option to modernize the notation and concepts to make it more understandable to our time, with the risk of writing something completely different from what the original author intended. 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 "variable", even though Paman implies that an expression can be composed of this variable only by taking powers of it, and by multiplicating by scalars, which means that Paman is thinking of polynomials or power series.

The "first State of x", we would write x :0 < x < c for some c in R , or simply "a neighbourhood of 0 in R+". Similarly, the "last State of x", we would write x: c < x < oo for some c in R, or simply "a neighbourhood of oo in R"

The notion of a "x Quantity" is a bit unusual, but it is clear that Paman means that ax^m is a x^m quantity if and only if a is a (real, nonzero) constant.

5.3.1 Maximinus and Minimajus

Breidert
(173) writes: "Was Paman mit 'Maximinus' bezeichnet ist die Größte untere Schranke (Infimum), das Minimajus ist die kleinste obere Schranke (Supremum)."(174)

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

(...) all that is understood by a Maximinus, is such a Quantity as being less than another, cannot be augmented by any Quantity of the same Kind, that is by any Part of itself, without becoming greater; thus A is the Maximinus of B, when it is the greatest of all those Quantities of the same Kind that are less than B.

And all that is understood by a Minimajus is such a Quantity, as being greater than another, cannot be diminished by any Quantity of the same Kind without becoming less.(175)

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:

(...) 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, or rather the same Quantity (...)

Breidert goes on:

Paman definiert die Fluxion als das Supremum bzw. Infimum des Differenzenquotienten, d. h. z. B. für(177) y=x^m
y* = Sup (x^m - (x- Delta x)^m)/(Delta x) bzw. Inf ((x+ Delta x)^m - x^m)/(Delta x)

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

The fluxion of y is the x* Quantity which is the first Maximinus or Minimajus of y(x+ x*)-y(x) (where x* is the radical Quantity). If y can be written as a power series sum{k=n to oo} a_k x^k, then

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

It is perhaps time to finish the arguments about x^m and AB. Paman has arguments too, of course:

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

There are some interesting points to note about Paman's approach:

5.5.1 The non-generative approach

Newton uses motion and velocity to define fluxions. These are powerful concepts, and Philalethes, Robins and MacLaurin (in Book I) copy Newton. Paman, on the other hand, does not use these concepts, in fact he makes a point of not using them. 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.

5.5.2 The fluxion is a number, not a ratio

Boyer writes: "(...) Newton never calculated a single fluxion, but always a ratio (...)".
(184) Here, too, Philalethes, Robins and MacLaurin follow suit. Paman, on the other hand, calculates the fluxion as a number. The difference is perhaps not very great; when Newton says that the fluxion of x is to the fluxion of x^n as 1 is to nx^(n-1),(185) Paman says that the fluxion of y^m is my^(m-1) q x*, if the fluxion of y is q x*, that is: The fluxion of y^m is my^(m-1) y*.(186) However, Paman's way of doing it is less cumbersome.

5.5.3 It is clear what is the variable

Boyer writes: "Newton, Leibniz and D'Alembert had not distinguished clearly between independent and dependent infinitesimals (...)".(187) In this respect, Paman is very clear. 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.(188)

5.5.4 The terminology

Paman's terminology - States of x, x^m-Quantities, first and last Maximinus' and Minimajus' - will of course seem strange at first glance. 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 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

Paman claims that this book shows how

(...) the Practice of Fluxions may be derived from the Principles of the Antients, whose Method of approximating by Exhaustions (...) cannot, I think, be taken in a different Sense from what I have done [in] the Doctrine of Maximinority and Minimajority (...).(189)

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

He goes on to say that

My Principles and Axioms are fetched from the Writings of Euclid, Apollonius and Archimedes, in pure Geometry; not from Infinites, or Non-Entities; not from any abstract Considerations of Velocity, Time, and Motion, those great Objects of metaphysical Enquiries.(190)

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

What, then, made MacLaurin the hero, while Paman fell into oblivion? 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 see(191) at least five different persons writing to MacLaurin about the Treatise before it was published.(192) MacLaurin was "an important key figure in the Scottish Enlightenment"(193) and a professor, while Paman was neither. 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.(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

One of the first to discuss Paman, was Sageng (1989).(195) I will quote the main part of this discussion:

Paman made a very early attempt in which he divided all values of the variable x into "the first state of x," which make some expression in terms of x greater than a given value p; and "the last state of x," which make the expression less than p. 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 x. Besides the problems we would recognize in assigning a first or last member of these sets, Paman's work was crippled by his extensive use of new terminology such as this.(196)
This presentation of Paman's definitions has surprisingly little connection with the actual work of Paman. One thing is that "maximinus" is not confined to the last state of x, in fact, only the first maximinus is involved in the definition of fluxion. But much more importantly, Paman nowhere says that the (last) Maximinus is "the last value" in the last state of x, just as present mathematicians do not define lim{x -> oo} f(x) as "the last value" of f(x). Moreover, Paman's work perhaps included more new terminology than usual at the time, but at least it was well defined and motivated. 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, 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

Paman's theory gives a good foundation for the theory of fluxions, without use of intuitive concepts like velocity. 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 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

The Analyst Controversy's effect on England's mathematical isolation

In a 1971 article, Elaine Koppelman wrote the following about the mathematical development in the eighteenth century:

During the eighteenth century, England remained in intellectual isolation from the Continent. The work of great Continental analysts - the Bernoullis, Euler, Lagrange and Laplace - was not assimilated. This had several causes. One external factor, 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 admiration of Continental philosophers for the English political system and her achievements in industry and commerce. Furthermore, the Newton-Leibniz priority battle(199) left those in academic circles with the belief that it was a dishonor to Newton to abandon his notation or methods.(200)

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:

Many British presentations of the calculus in the 1730s and 1740s were concerned with answering Berkeley's charge in The Analyst (1734) 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 infinitesimal methods in favor on the Continent. Whether such an explanation can hold may be a matter for separate investigation (...).(202)

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

In order to undrestand Newton's different foundations for his theory of fluxions, it is important to see that they are different from Leibniz' foundation. In the years before the Newton-Leibniz controversy, however, there seems to have been a bit of confusion. For instance, Abraham de Moivre uses "fluxions" in the meaning of "infinitely small" (1695), as does Newton's once very close friend Fatio de Duillier (1699), Roger Cotes (1701) and John Harris (1702),(204) who writes

These Infinitely small Increments or Decrements, our incomparable Mr. Isaac Newton calls very properly by this name of Fluxions.(205)

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

1710 can be seen as the starting year of the Newton-Leibniz controversy,(207) and we would perhaps suppose people to become more aware of the difference between the two. 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 1711 John Keill wrote

If in place of the letter o, which represents an infinitely small quantity in James Gregory's GeometriÆ pars universalis (1667), 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.(208)

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

(...) the Differential Method is one and the same with the Method of Fluxions, excepting the name and the notation; Mr. 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.(210)

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

In 1736, George Berkeley published The Analyst. 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.(212)

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

The Differential Method teaches us to consider Magnitudes as made up of an infinite Number of very small constituent Parts put together; whereas the Fluxionary Method teaches us to consider Magnitudes as generated by Motion (...); so that to call a Differential a Fluxion, or a Fluxion a Differential is an Abuse of Terms.(213)

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

1742 was the publication year of MacLaurin's A Treatise of Fluxions, where he gave his foundation for the fluxional calculus. From 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.(215)

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:

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.(216)

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

That the principles of Fluxions stand in need of demonstration, especially since the publication of MacLaurin's works, is certainly a mere pretence, made only to cover the ignorance of the objector (...)(217)

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):

[I have] long been of the opinion that, in point of intellectual conviction and certainty, the fluxional calculus is decidedly superior to the differential and integral calculus.(218)

and Florian Cajori agreed (in 1919):

From the standpoint of rigour, the British treatment of the calculus was far in advance of the Continental. It is certainly remarkable 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 (...)(219)

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

These quotes suggest the following pattern: until 1710, many considered fluxions as infinitely small quantities. From 1710 to 1736, 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 1736 to 1741, people were more careful to avoid infinitesimals, while after 1742, there existed a solid foundation for the calculus, and many recognized this.

6.6 Conclusion

It is not possible to draw absolute conclusions from this material. But it seems that the recognition that Newton's and Leibniz' theories were fundamentally different grew from 1730/6 onwards. 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.

6.7 The rest of the story

Whatever the reasons for this isolation, the result was that the centre of mathematics moved away from England. England's contributions in the following years were meagre in comparison to those of the Continent. This is the reason why Berkeley's Analyst has 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 satisfactory 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 method 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 chenged 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 basing 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 succedded 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 definitons and rigorous proofs, while still depending on intuitive concepts. MacLaurin managed to give a foundation using neither infinitesimals nor motion or velocities. 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 neighbourhood concept and lim inf and lim sup. 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 foundational questions, may well have contributed to English mathematicians' preference of their own notation and foundation.

Appendix A

The Newton-Leibniz controversy

Newton and Leibniz developed their theories independent of each other. 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 1699 Fatio de Duillier publicly called Leibniz a "second discoverer", but Leibniz' only reaction was to write a private complaint to Wallis.

In 1705 it was Leibniz' turn - he wrote (anonymously) about Newton that

in place of Leibnizian differences Mr. Newton employs fluxions, and has ever employed them.

- perhaps implying that Newton had copied Leibniz' method, only changing the notation. The storm did not break loose yet, however - it seems that Newton didn't read these lines until later.

In 1708, John Keill wrote a letter to Edmond Halley which was published in the Royal Society's Philosophical Transactions;

(...) arithmetic of fluxion; whose first inventor was beyond all doubt Mr. Newton (...) the same arithmetic was, however, afterwards published with changes in names and notation by Mr. Leibniz in the Acta Eruditorum.

This insinuation is even clearer than Leibniz', and Whiteside writes that

from then on it would have taken an angel such as none of the participants were in order not to be stung into direct response.(220)

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:

(...) each of these independently framed variant modes of analysis of the infinitely small and the instantaneously moving drew so heavily on the insights of so many who had gone before, the priority in time of creation of his fluxional method which Newton indubitably has must seem of minimal significance. The rest, as they say, is 'history'.(222)

Appendix B

Chronology

1642: Newton born
1669: Newton's De Analysi
ca. 1680:Newton's The Geometry of Curved Lines
1687: Newton's Principia
1704: Newton's De Quadratura Curvarum
1727: Newton died
1734: Berkeley's The Analyst
1734: Philalethes' Geometry, no Friend to Infidelity
1735: Berkeley's A Defence of Free-thinking in Mathematics
1735: Philalethes' The Minute Mathematician
1735: Robins' A Discourse...
1742: MacLaurin's A Treatise of Fluxions
1745: Paman's The Harmony of the Ancient and Modern Geometry Asserted
1746: MacLaurin died
1748: Paman died
1750: Philalethes died
1751: Robins died
1753: Berkeley died

Appendix C

Some tables of contents

The table of contents often give a good impression of a work's scope and composition. Some of the works treated in this paper are not easily accessible, therefore I give their tables of contents here.

C.1 Philalethes Cantabrigiensis' Geometry no friend to infidelity

MAthematicians accused of Infidelity, of perverting other persons to Infidelity, and of error in their own science. p. 6
Title page to the Analyst gives hopes of a Mathematical Demonstration of the Christian Religion. 7
This not attempted. No more certainty in the modern Analysis, than in the Christian Religion. No honour to Christianity from this comparison. 8
Design to lessen the reputation of Sir Isaac Newton and his followers and their science. Mathematics a useful science. 9
Not too much studied. Ought not to be depretiated. Reason for this design. 10
If Mathematicians are Infidels, it ought not in prudence to be published. Objection against our Saviour. 11
No room for this objection in our days. The reputation of adversaries to be ruined. 12
Odium Theologicum. Not the practice of our Saviour and his Apostles. 13
The allowed wisdom and reason to Infidels. The Church of Christ in no danger. 14
The proper method of opposing Mathematicians. An inscription for pulpits. mbox 15
Zeal of the Clergy. 16
Example set them by the Author of the Analyst 17
Solemn hymn proposed to be sung by them to his honour. 18
Mathematicians mistaken in the method of Fluxions. May be good reasoners notwithstanding. 19
A dangerous undertaking. Unnecessary. The best reasoners, the best Christians. Doubt whether zeal for Chrisianity were the motive to writing the Analyst. 20
Reason for that doubt. That Author's former behaviour to Mathematicians mbox 21
True motive to this undertaking. 22
The treatment he has given to some of the greatest men. 23
His presumption and vanity. 24
His proof of Infidelity against Mathematicians. 25
A proposal to hang or burn all the Mathematicians in Great Britain. 26
Wickedness and flooy of their Accusers. Extreme credulity of the Author of the Analyst. 27
Unlikelyhood of Infidelity in the clearest reasoners. 28
Reputation for Mathematicks gives no authority in Divinity, Law, or Physick. Proved from the example of Dr. Barrow and Sir Isaac Newton. 29
Objections against the method of Fluxions. 30
Fluxions obscure to what readers. Clear to others. 31
Disingenuity of the Author of the Analyst. 32
False reasoning in Fluxions. First instance of error. 33
Great triumph upon this. No great occasion for it. 34
Case proposed for unmathematical readers 36
A French Marquis accused of using too little ceremony. 38
Injustice in this accusation 38,39
Sir Isaac Newton charged with using tricks and artifices. 40
Blindness of the accuser. 44
A difficult case. 45
Two ways of ending a Mathematical dispute, Sir Isaac in the right. 46
Final cause of his proceeding. 47
A just reason for his proceeding. Velocity of a rectangle what? Ass between two bottles of hay. Whisper from a Ghost. 48
Moment of a rectangle what? 49
Sir Isaac's proceeding more geometrical than that proposed by his censurer. The censurer's want of caution. 50
Advice to him. Sir Isaac's foresight, humanity, prudence and caution. Danger of those who unadvisedly attack him. 51
An objection prevented. 52
A scruple removed. 53
Second instance of error in the method of Fluxions. Two inconsistent suppositions. 54
No danger to Religion from such reasoners 54
Horrible blunder charged upon Sir Isaac Newton. Inquired into. 56
Proved to belong to the Author of the Analyst 57
Arts and fallacies imputed to Sir Isaac Newton. Not wanted, nor used by him. 58
Sir Isaac Newton supposed not to be satisfied with his own notions. Injustice of such a supposition with regard to him. 59
Or to preachers. Or to the Author of the Minute Philosopher. 60
Sir Isaac's words misrepresented on purpose to draw a false inference from them. 61
Truth supposed to arise from the contrast of two errors. 62
A ghost exorcised with the Principia Mathematica. 63
Sir Isaac Newton proceeds blindfold. 64
Fast asleep. Monstrously lucky. 65
The two errors examined into. 66
Are at most infinitely small. No errors at all. A motto unluckily chosen. A beam less than a mote. 68
Excellency of the method of Fluxions owing to these pretended errors. 69
Mathematicians when they commit them, know what they are doing. 70
Sir Isaac Newton was aware of this objection, and provided against it. 71
Mr. Locke charged with contradicting himself. 72
In two instances. 73
General Ideas necessary to science. Distinction between abstract and general Ideas. Abstract Ideas how acquired. General Ideas how acquired. 74
Example of the method of acquiring abstract and general Ideas, taken from Botany. 75
Another example taken from Geometry. General Idea of a Triangle. 76
Easily acquired by a learner. 77
How the Author of the Analyst may acquire it. Not more difficult to conceive than the Idea of any particular species of Triangles. Or than the Idea of an Angle. 78
Mr. Locke grossly misrepresented. 79
First instance of contradiction examined. 80
Second instance. 81
One of Mr. Locke's traps for Cavillers. 82
Conclusion. 84

C.2 Benjamin Robins' Discourse Concerning...

INTRODUCTION: of the rise of these methods. Page 1
Fluxions described, and when they are velocities in a literal sense, when in a figurative, explained. p. 3.
General definition of fluxions and fluents. p. 6.
Wherein the doctrine of fluxions consists. Ibid.
The fluxions of simple powers demonstrated by exhaustions. p. 7.
The fluxion of of a rectangle demonstrated by the same method. p. 13.
The general method of finding all functions observed to depend on these two. p. 20.
The application of fluxions to the drawing tangents to curve lines. Ibid.
The application to the mensuration of curvilinear spaces. p. 23.
The superior orders of fluxions described. p. 29.
Proved to exist in nature. p. 31.
The method of assigning them. p. 32.
The relation of the orders of fluxions to the first demonstrated. p. 34.
Second fluxions applied to the comparing the curvature of curves. p. 38.
That fluxions do not imply any motion in their fluents, are the velocities only, wherewith the fluents vary in magnitude, and appertain to all subjects capable of such variation. mbox p. 42.
Transition to the doctrine of prime and ultimate ratios. p. 43.
A short account of exhaustions. p. 44.
The analogy betwixt the method of exhaustions, and the doctrine of prime and ultimate ratios. p. 47.
When magnitudes are considered as ultimately equal. p. 48.
When ratios are supposed to become ultimately the same. Ibid.
The ultimate proportions of two quantities assignable, though the quantities themselves have no final magnitude. p. 49.
What is to be understood by the ultimate ratios of vanishing quantities, and by the prime ratios of quantities at their origine. p. 50.
The doctrine treated under a more diffusive form of expression. p. 53.
Ultimate magnitudes defined. Ibid.
General proposition concerning them. p. 54.
Ultimate ratios defined. p. 57.
General proposition concerning ultimate ratios. Ibid.
How much of this method was known before Sir Isaac Newton. p. 58.
This doctrine applied to the mensuration of curvilinear spaces. p. 59.
And to the tangents of curves. p. 64.
And to the curvature of curves. p. 65.
That this method is perfectly geometrical and scientific. p. 68
Sir Isaac Newton's demonstration of his general rule for finding fluxions illustrated. mbox p. 71.
Conclusion, wherein is explained the meaning of the word momentum, and the perfection shewn of Sir Isaac Newton's demonstration of the momentum of a rectangle; also the essential difference between the doctrine of prime and ultimate ratios, and that of indivisibles set forth. p. 75.

C.3 MacLaurin's A Treatise of Fluxions

VOLUME I.
INTRODUCTION
The design of this Treatise Page 1
Of the method of exhaustions, from the 12th book of the Elements 4
Elliptic and circular areas compared by this method 11
A general theorem concerning figures described about a conic section, or inscribed in it, 8
Propositions from Archimedes concerning spheres, spheroids, & c. 9
A general property of the solid that is generated by a conic section revolving about its axis 26
The quadrature of the parabola after Archimedes 27
Of the spiral of Archimedes 30
The quadrature of a spiral by Pappus 31
Remarks on the method of the Antients 33
On the methods of indivisibles and infinitesimals 37
BOOK I
CHAPTER I. Of the grounds of the method of fluxions
Definitions and illustrations, Article 1
The axioms, 15
Theorems concerning uniform motions from Archimedes, 16
Theorems concerning variable motions, 18
Of comparing the fluxions of quantities by determining the limit of the ratio of their increments or decrements, 66
Of second fluxions, 70
CHAP. II. Of the fluxions of plane rectilineal figures.
Of the fluxion of a parallelogram of an invariable altitude, Art. 78
Of the fluxion of a triangle, 81
The increment of the triangle resolved into two parts, --- that which measures the generating motion, and that which measures its acceleration, 93
The theory of motions that are accelerated or retarded uniformly, 94
Of the fluxion of a rectangle, 98
CHAP. III. Of the fluxions of plane curvilineal figures.
Of the fluxion of an area, the ordinates being supposed parallel, Art. 105
General corollaries relating to the theory of motion, 114
Of the fluxion of the area generated by a ray revolving about a given centre, 116
Of similar curvilineal figures, 128
CHAP. IV. Of the fluxions of solids, Art. 124
Illustrations of second and third fluxions, 128
CHAP. V. Of the fluxions of quantities that are in a continued geometrical progression, the first term of which is incariable, Art. 140
CHAP. VI. Of logarithms, and the fluxions of logarithmic quantities. An account of logarithms from Napier the inventor, Art. 151
Of the fluxions of quantities that increase or decrease proportionally, 158
Of the fluxions of quantities, when their logarithms are in an invariable ratio, 165
Of the fluxions of quantities that are represented by powers with irrational or variable exponents, 168
Of the second, third, and higher fluxions of a quantity that increases proportionally, 169
Theorems for approximating to the value of logarithms, 171
Of the different logarithmic systems and the ratio modularis, 174
Of the logarithmic curve, 176
Of hyperbolic areas, 177
Of the analogy betwixt circular arks and logarithms, 178
CHAP. VII. Of tangents.
Definitions, Art. 180
Of the fluxion of the base, ordinate, and curve, 184
Of the fluxion of the ark, sine, tangent, secant, & c. 192
Of the fluxions of the curve, the ray drawn to the curve from a given point, and the circular ark descirbed from that point as centre, 199
Of the fluxions of angles, 203
Theorems concerning tangents, 211
CHAP. VIII. Of the fluxions of curve surfaces.
Lemmas concerning conical surfaces, Art. 216
Of the fluxion of a curve surface, 228
Of the surfaces generated by a circular arch about any chord, 231
Of the surfaces generated by any arks, the centre of gravity of an arch, and the theorem Guldinus, 233
CHAP. IX. Of the usual rule for determining the greatest and least ordinates, Art. 238
Of the analogy betwixt the inverse method of tangents, and the quadrature of figures, 247
A more accurate rule for finding the greatest and least ordinates, 261
A similar rule for finding the points of contrary flexure, 263
Of cuspids of various kinds, 268
Of the greatest and least rays that can be drawn from a given point to a curve, 277
Other rules for finding the points of contrary flexure and cuspids, 279
CHAP. X. Of the asymptotes of curve lines, & c.
Definition of asymptotes, with examples, Art. 286
Of the parts of geometrical magnitude, 290
Of asymptotes, and the areas bounded by them and the curves, 292
Of the solid generated by this area, 307
Examples of constructions for determining the tangents and asymptotes of curves that are described by the revolution of lines or angles, 318
Theorems for discovering whether a figure hath an asymptote, and the area bounded by it and the curve hath an assignable limit which it cannot exceed, 326
Of the surface generated by the curve about the asymptote, 339
Of spiral lines and their areas, 340
Of the limits to which the sums of progressions approach, with examples and theorems for approximating to those limits, 350
CHAP. XI. Of the curvature of lines, & c.
Definitions, Art. 363
Theorems for finding the curvature and its variation in geometrical figures, and for comparing the different degrees of contact of the curve and circle of curvature, 365
Examples in the conic sections, 371
Of the curvature that is less than that of any circle, 377
Of the curvature that is greater than in any circle, 378
Other theorems concerning the curvature and its variation, 381
A general property of the lines of the third order, when two tangents can be drawn to the line from a point in it, 401
Of the evolution of lines, 402
Of the proporties of the cycloid, and the descent of a heavy body along it.
Of the caustics by reflexion, 409
Caustics by refraction, 413
Of the rays that define the first and second rainbow, 415
Of centripetal forces, 416
The ratio of the velocity in a curve to the velocity in a circle at the same distance from the centre in a void or medium, 424
The construction of the trajectory, when the velocity is such as would be acquired by an infinite descent, 436
Of motions in a conic section, 445
The cases distinguished wherein a body may revolve betwixt the higher and lower apsides, and when it continually approaches to the centre or recedes from it, 447
Of the resistance and density of the medium in which a given trajectory is described, 452
Of gravitation towards several centres, 462
Of the motion in the nodes of the moon, 480
Of the variation of the inclination of the plane of the lunar orbit, 487
Of the acceleration of the area described by the moon about the earth, 490
Of fluids that gravitate towards several centres, 491
Of the figure of a fluid that gravitates towards a centre and revolves about an axis, 492
Of the intersection of the curve and circle of curvature, 493
Of Remarks on the preceding Part, 494

VOLUME II.
BOOK I
CHAPTER XII. of the methods of infinitesimals, of the limits of ratios, and of the general theorems which are derived from this doctrine, for the resolution of geometrical and philosophical problems
Of the harmony betwixt the method of fluxions and of infinitesimals 495
Some objections against the method of fluxions and of infinitesimals 498
The true reason why parts of the element are to be neglected in the method of infinitesimals 501
Of Sir Isaac Newton's method by the limits of ratios 502
Propositions of the preceding chapters demonstrated briefly by this method 506
Theorems concerning the centre of gracity and its motion, and their use shown in resolving several problems concerning the collisions of bodies 510
Of the descent of bodies that act upon one another, of the descent and ascent of their centre of gravity, and the preservation of the vis ascendens, or vis viva 521
Of the centre of oscillation 534
Of the motion of water issuing from a cylindric vessel 537
Of the motion of water issuing from any vessel 550
Of the Catenaria, when gravity acts in parallel lines 551
General theorems concerning the trajectories, lines of swiftest descent, the Catenaria, & nolinebreak c. mbox 563
CHAP. XIII. The analysis of the problem concerning the lines of swiftest descent, when an uniform or variable gravity acts in parallel lines 572
The synthetic demonstration 576
The same, when gravity tends to a given centre 578
Another synthetic demonstration 584
Of the lines of swiftest descent amongst those of the same perimeter in any hypothesis of gravity 588
The first general isoperimetrical problem resolved by first fluxions, and the resolution demonstrated synthetically 592
The problem extended further by the same method 597
The second general isoperimetrical problem resolved in the same manner 601
The property of the solid of least resistance demonstrated in this manner 606
CHAP. XIV. Of the ellipse considered as the section of a cylinder 609
General properties of the conic sections transferred briefly from the circle 622
Of gravitation towards spheres and spheroids 628
Supposing the density of the planets uniform, their figure is accurately that of the oblate spheroid, which is generated by the conic ellipse about its second axis 636
Of the figure of the planets and variation of gravity towards them 641
The gravitation at the pole and equator, or any point on the surface of a spheroid, measured accurately by circular arks or logarithms 642
The gravitation in the axis or plane of the equator produced, measured accurately by the same 648
Of the figure of the earth in particular, supposing its density uniform 655
Of the gravity towards a spheroid, supposing the density variable 660
Of the figure of Jupiter, and the effects of his spheroidical form upon the motions of the satellites 682
Of the tides 686
Of other laws of attraction 696

BOOK II
Of the Computations in the Method of Fluxions.
CHAP. I. Of the fluxions of quantities considered abstractly as represented by general characters in Algebra.
Of the import of some algebraic symbols Art. 699
The principles of this method adapted to algebra 700
Of the fluxions of powers of all kinds 707
Of the fluxions of products and quotients 715
Of the fluxions of logarithms 717
Of second and higher fluxions 720
CHAP. II. Of the notation of fluxions Art. 723
The rules of the direct method 724
The fundamental rules of the inverse method 735
Of infinite series 745
An investigation of the binomial and multinomial theorems 748
Other theorems 751
Examples of their use 753
CHAP. III. Of the analogy betwixt elliptic and hyperbolic sectors Art. 758
Of resolving trinomials into quadratic divisors 765
Of reducing fluents to circular arks and logarithms when the fluxion is expressed by rational quantities 770
Of reducing fluents to the same measures when the fluxion involves an irrational binomial or trinomial 789
Of reducing fluents to hyperbolic and elliptic arks 798
Of reducing fluents of a higher kind to others of a more simple form 810
CHAP. IV. Of the area when the ordinate is expressed by a fluent Art. 813
Of the area when the ordinate and base are both expressed by fluents 819
Instances wherein the total area, or fluent, is measured by circular arks or logarithms, when it does not appear that the same fluent can be generally reduced to those measures 822
Theorems derived from the method of flusions for approximating to the sums of progressions by areas, and conversely 828
Theorems for finding the sum of any powers, positive or negative, of the terms in an arithmetical progression, and for finding the sums of their logarithms 833
Of the ratio of the sum of all the unci ae of a binomial of a very high power to the uncia of the middle term 844
Of computing the area from a few equidistant ordinates 848
Theorems derived from the method of fluxions for interpolating the intermediate terms of a series 850
CHAP. V. Of the general rules for the resolution of problems by computations, with examples
Of the rules for determining the tangents Art. 857
The greatest and least ordinates 858
The points of contrary flexure and cuspids 866
The centre of curvature 870
The caustics by reflexion and refraction 872
the centripetal forces 874
The construction of the trajectory that is described by a force which is inversely as the fifth power of the distance, by logarithms in certain cases 878
In these cases a body may recede from the centre continually, so as never to rise to a certain altitude, or may approach to it for ever, and never descend to a certain distance 879
The construction in other cases 881
The rules for computing the time of descent along a given curve 884
The time in a finite circular arch measured by the arks of conic sections 886
The same by infinite series 887
Rules concerning the computation of motions in a medium 888
Rules for determining the figure of the catenaria, and the lines of swiftest descent 889
Rules for the computation of areas, solids, curvilineal arks and surfaces 890
The meridional parts in a spheroid computed by circular arks or logarithms 895
The gravitation towards a spheroid at the pole and equator, measured by circular arks and logaritmes, when the force towards any particle is inversely as any power of the distance from it 900
Of the centres of gravity and oscillation 906
Of the proportion of the power to the weight, that a machine may have the greatest effect 907
Of the same when the friction is considered 908
The most advantageous position of a plane, which moves parallel to itself with a given direction, that a stream may impel it with the greatest force, when the velocities of the stream and plane are given 910
The wind ought to strike the sails of a wind-mill a greater angle than 54’ 44' 914
The most advantageous position of the sails that the wind may impel a ship with the greatest force in a given direction, the velocities of the wind and ship being given 916
How an ark is to be divided into any number of parts, that the product of any powers of the sines of the several parts may be a maximum 921
The most advantageous direction of the motion of a ship, and best position of the sail, that the ship may recede from a given line or coast with the greatest velocity 922
Of reducing equations from second to first fluxions, with examples 924
The construction of the elastic curve, and of other figures, by the rectification of the conic sections 927
Of the vibrations of musical chords 929
Problems concerning the maxima and minima that are proposed with limitations concerning the perimeter of the figure, its area, the solid generated by this area, & c. resolved by first fluxions 931
Examples of this kind relating to the solid of least resistance 934
An example of the method of computing from the general principles in art. 563 935
An instance of the theorems by which the value of the ordinate may be determined from the value of the area, by common algebra 936
It is relative not absolute space and motion that are supposed in the method of fluxions 937