Cand. Scient Thesis in Mathematics

Department of Mathematics

University of Oslo

1995

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

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

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

we never know what we are talking about,

nor whether what we are saying is true.

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.

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)

When it comes to negative numbers, Kline writes:

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

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.

The problem of convergence was very difficult. For
instance, Guido Grandi noted that
1/(1+*x*)=1-*x*+*x*²-*x*³+ ..., 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)

-And all was light.

(Pope)

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):

b.........d.......f.......h...

Figure

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:

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*.

(...) Let there be given, accordingly, any equation *x*³-*ax*²+*axy*-*y*³=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.

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

This does not seem convincingly rigorous to me. In fact, I find it difficult to understand Newton's point. The exact speed with which the body reaches its last position has to be zero - otherwise it would continue beyond this last position. I have no doubt that Newton had some idea of a limit concept here, but the difference between an idea and a fully explained and understood concept is large.

I will take a look at a
few examples where Newton computes fluxions, which were also to have an
important role in the Analyst-debate. First, Newton wants to compute
the fluxion of the product *AB*.
This is taken from *Principia*, where he
defined *moment* this way:

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

One would think that the reasonable way to do this would be to
calculate *(A+a)(B+b)- AB= Ab+aB+ab*. The problem would be to get rid of
the *ab* - many would do this by saying that *ab* is infinitely less
than *Ab+aB*. It seems clear that Newton's proof was made to avoid
this kind of infinitesimal-argument. But then it looks very arbitrary.

However, Newton's *intuition* is right. When trying to find the
rate of increase, you may calculate *(A-a)(B-b)-AB*,
*(A+ ½a)(B+ ½b)- (A- ½a)(B- ½b)* or
even
*(A+ (2/3)a)(B+ (2/3)b)-(A- a/3)(B- b/3)*,
because all you are interested in is the rate of increase in some
"small" neighbourhood of *A* and *B*. This is just as we in modern
notation may choose to calculate *lim{h->0}(f(x+h)-f(x))/h*,
*lim{h->0}(f(x+h/2)-f(x-h/2))/h* or even
*lim{h->0}(f(x+(2/3)h)-f(x-(1/3)h))/h*, when we want to
find the derivative of *f*.

Newton seems to have seen that all of the different answers are
essentially the same, and therefore chosen the one which gave the
simplest calculation. If this interpretation is correct, we have here
an extreme example of Newton's failure to write down what his
intuition told him. And without an argument saying *why* the different
results are essentially the same, the procedure still seems very mysterious.

Afterwards, Newton wants to calculate the fluxion of *x^n*:

Newton has found what we would write as *((x+o)^n -x^n)/o*, and
tries to see what happens when *o* approaches zero. Without recourse
to the limit concept, however, he seems to let the divisor
become zero, which of course must be an invalid way of reasoning. This
was the way Berkeley interpreted him. I will come back to this later.

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

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

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

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

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

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.

of that sort of man who wants

to be known for his paradoxes.

(Leibniz

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

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.

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

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

We see that Berkeley thinks that Newton's method was "illegitimate". But even if the method had been legitimate, the problem remains: there are two methods giving (seemingly) different answers to the same question. What was needed, and what Newton failed to give, was a proof that the two answers were in some sense equivalent.

Likewise, Berkeley disliked Newton's calculation of the fluxion of
*x^n* (see here)):

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

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.

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:

- Obscurity of this doctrine.
- False reasoning used in it by Sir Isaac Newton, and implicitely received by his followers.
- Artifices and fallacies used by Sir Isaac Newton, to make this false reasoning pass upon his followers.(50)

He also attacks Berkeley for misrepresenting Newton;

Of course, Newton never used these expressions. Philalethes therefore advises both Berkeley and the readers to read look at Newton's own writings.

Philalethes does not try to explain the "doctrine". In fact, he doesn't have to explain it - Berkeley has presented a parody on Newton, and Philalethes has pointed it out.

However, the next theme is more difficult:

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

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

This argument could have been written by anyone, with any foundation - infinitesimals, fluxions, first and last ratios - and perhaps even by a modern user of the Cauchy limit concept. But it is elucidated by the following example:

I think that here, much more clearly than in the previous quote, a limit
argument is involved. Of course, Philalethes is no Cauchy, but
this example shows that he had some intuition of what was going on.
But the main problem remained; Newton's mysterious calculation of the
fluxion of *AB*.
First, Philalethes claimed that as *aB+bA+ab* is the increment of
*AB*, and *aB+bA-ab* is the decrement, and the moment can be both the
increment and the decrement, then the mean *aB+bA* must also be the
moment. This argument has no support in the definitions. And even if
the definitions had supported it, we would want a proof that these
three "moments" are the same.

Philalethes denies that Newton tried to find the increment of *AB*;

For what reason did Newton calculate this increment?

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:

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

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

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

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

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.

As noted earlier , I do not share this interpretation. Philalethes does not say that the error is "so small that it is insignificant" - he says that it is "at most such an one as can cause no assignable difference, how small soever". This must be the right answer to Berkeley's criticism, but it is not very helpful as long as he doesn't prove his assertion.

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

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

Cajori writes:

This may be because Newton's proof is perfectly understandable from the intuitive view-point, given Philalethes' explanation, which was soon to be repeated by Robins (see here). The calculation does not fit the definition, however, but it seems to me that many mathematicians at the time, including Newton, considered the definitions as little more than explanations, bearing the concepts and not the definitions in mind when doing mathematics. Therefore I am not as surprised as Cajori, even though the soundness of Berkeley's criticism at this point is clear.

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.

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

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

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)

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*.

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

After these fundamental observations, the result can be found:

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

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.

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.

The principal definitions are as follows:

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

The same can be defined for ratios:

This limit is called the *ultimate ratio* of the ratios.

This terminology is perhaps unfortunate as it may suggest that this "ultimate ratio" is the ratio of the "ultimate magnitudes". Robins therefore hastens to add that

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

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.

C..............D......F.................

Figure

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.

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

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

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

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

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

Robins' *Discourse* was unfortunately only the beginning of a long and
wordy debate between Philalethes and Robins, later with Henry
Pemberton in
Robins' place. I have not had the opportunity to study the
contributions in this debate, but the secondary literature suggests
that the debate's main theme was what Newton's view had been,
and did not help the science of mathematics much. I therefore refer to
Cajori(94) on this
subject.(95)

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.

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

The main reason why Gibson was "at a loss" to understand this is probably, in my view, that he treats Philalethes as if he, too, in his first book tried to explain and clarify the theory. The reason for the "favour shown" to Philalethes may have been that he wrote a book that could easily be read by anyone (at least large parts of it), which was at times funny, and which at several points refuted Berkeley.

Moreover, it is not surprising that the writings of a Cambridge scholar should be taken more seriously at first than the writings of a simple mathematics teacher.

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.

universal contempt, I am glad

you have undertaken him.

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

I find it probable that MacLaurin had read all or most of these before
publishing his own answer. But MacLaurin's *Treatise* became much
more than an answer to Berkeley; it
included a mathematical treatment of centres of gravity and oscillation, lines
of swiftest descent, the figure of the planets, the tides, wind-mills,
vibration of chords and so on. But it also gave a rigorous foundation for
the method of fluxions, with long, double reductio ad
absurdum proofs which Eudoxus might have appreciated.

The book is relatively unreadable, but gives the method of fluxion a foundation independent of infinitesimals.

Like Robins, MacLaurin divides his treatise in two main parts.

The first book explains the theory of fluxions in much the same way as Robins did in the first part of his book - considering quantities as generated by motion and using velocities as a basic, undefined tool. MacLaurin's book, however, is even more geometrical than Robins'.

The second book is much more algebraic and avoids using velocities. The proofs are given by double reductio ad absurdum, however.

Here, MacLaurin has already (like Newton and Robins) used the intuitive concept of velocity without further explanation, looked at the velocity in a point and considered the effect if that velocity is held constant(108) . These can hardly be unproblematic concepts on which to found a mathematical method, but, as mentioned before, they seem to have been accepted at the time.

The rest of Book I consists of lots of propositions, with long, geometrical proofs which seem unreadable to the modern reader. For instance, he proves the following proposition (see figure):

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

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

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

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

There is one important difference between MacLaurin's way of doing things in Book II and the ways of Newton and Robins that we have seen earlier; MacLaurin does not use the intuitive concept of velocity here:

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

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

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).

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

Kline also wrote that

without giving any reasons for his claim.

Turnbull, on the other hand, writes that

Paman writes about MacLaurin that his

and that

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).

by his extensive use of

new terminology (...)

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

Sept. 18th. 1740, Roger Paman was on board, he claims, when George
Anson's
ships set out from St. Helen's for a journey round the world. Of the eight
ships that set out, only one ship, the *Centurion*, managed to get around
the world and return to England, reaching Spithead on June 15th
1744.(133) Paman, however, was back in England long
before this. Five of the ships, the *Gloucester*, the *Wager*,
the *Tryal*, the *Anna* and the *Industry*, were destroyed during
the journey. Paman must therefore have been on one of the remaining ships: the
*Severn* and the *Pearl*.

*Severn* and *Pearl* left England together with the other ships, and
anchored upon the coast of Patagonia (Southern Argentina) February 18th,
1741. March 7th, they passed the Straits of Le Maire,(134).
still together. But on April 10th, they lost sight of the other
ships,(134) and on April 25th they even lost sight of
each other,(135). but were rejoined May 21st.
The weather had been terrible and most of the men were ill, and both ships
had to wait before going on. July 4th, 1741, the ships arrived in Rio de
Janeiro, and Captain Legge of the *Severn* wrote:

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

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

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

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

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":

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

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.

No trace of this book has been found, and it is probable that it was not published, due to too few subscribers.

We do not know more about Paman, except that he died in 1748.(141)

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

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

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

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

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

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

Paman hastens to give an explanation:

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

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

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

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

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

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

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

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

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

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

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

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

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

In a footnote,(155) Paman writes:

and he also proves the existence for power series with a lowest power. I have put together this proof from several proofs from Paman, to avoid having to give all of the propositions and corollaries in full:

**Proposition 5.10**
"In any Series *ax^m , : bx^(m+n) , : cx^(m+n+o) , : & c.*, the lowest
Term, as *ax^m*, is the first Maximinus or Minimajus of the
Series".(156)

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

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

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

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

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

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

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

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

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

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

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

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

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

**(My) Proof**
If *ax^m* is the first Maximinus of *A*, and *bx^m* is the first
Maximinus of *B*, then
*(a+b)x^m* is less than *A+B*, but *(a+b+p)x^m* is greater than *A+B*, in the
first State of *x*, since *(a+p/2)x^m > A* and *(b+p/2)
x^m > B*, in the first State of *x*.

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

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

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

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

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

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

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

The other instances should be similar. QED.

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

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

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

**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

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

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

The second fluxion is defined similarly:

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

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

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

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.

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

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:

Breidert goes on:

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

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

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

*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 (...)

**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.

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.

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

He goes on to say that

Thus we see that Paman wanted to be in the tradition of the old Greek mathematicians. But at the same time he wanted to avoid "the Tædium and Perplexity" of their ad absurdum proofs index reductio ad absurdum (see here). Paman managed to keep to the rigorous proofs and breaking with geometry at the same time; in much of his book geometry plays little role.

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.

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.

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.

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

I will try to face this question in this chapter. To do this, it is natural to look at how
the views on the relationship between Newton's and Leibniz' foundations changed with time -
looking at the situation both before and after the publication of *The Analyst*.
As I do not have unlimited access to primary sources,(203) I
will not be able to do a thorough investigation, but I hope I will find some points of interest.

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)

In 1711 John Keill wrote

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

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

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

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

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

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

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

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

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

Olynthos Gregory wrote (in 1836-7):

and Florian Cajori agreed (in 1919):

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

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.

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.

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

- 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*;

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

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:

1669: Newton's

ca. 1680:Newton's

1687: Newton's

1704: Newton's

1727: Newton died

1734: Berkeley's

1734: Philalethes'

1735: Berkeley's

1735: Philalethes'

1735: Robins'

1742: MacLaurin's

1745: Paman's

1746: MacLaurin died

1748: Paman died

1750: Philalethes died

1751: Robins died

1753: Berkeley died

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

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.

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

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