The incommunicable Doctor Pell
31 October 2001
I will begin with a confession: I neither wanted nor intended to become interested in John Pell. It is impossible, however, to venture far into seventeenth-century English mathematics without encountering him. Pell was born in 1611 and died in 1685, so he spanned much of the seventeenth century, and his reputation as a mathematician was established early and stayed with him all his life. He was well known to all the leading mathematicians and scientists of the day: at the age of seventeen he was already discussing mathematics with Henry Briggs, Gresham’s first professor of geometry. Later, during the 1650s he participated in mathematical meetings with those then most closely associated with Gresham College: Lawrence Rooke, Gresham Professor first of astronomy then of geometry, Christopher Wren, Professor of astronomy, William Brouncker, Paul Neile, Jonathan Goddard and others. Like all those mentioned, Pell became an early member of the Royal Society and was its Vice-President in 1676. From what was written and said of him there is no doubt that he was held in esteem, and especially as a mathematician. John Collins, clerk to the Royal Society and a loyal friend of Pell’s, wrote of him: ‘I take him to be a very learned man. More knowing in algebra, in some respects, than any other …’
When we try to discover, however, what exactly Pell’s reputation was based on, the picture becomes strangely cloudy. If you ask mathematicians what they know of Pell they will probably mention an equation known as Pell’s equation, but the better informed will also tell you that the name is a mistake: the equation was neither proposed nor solved by Pell. The attribution was an error on the part of Euler early in the eighteenth century. If you ask school children in Pell’s home town of Southwick in Sussex what they know of him, they will tell you that he invented the division sign. They are right, but in a century that saw the invention of algebraic geometry and the calculus, a single symbol, however useful, is hardly the stuff reputations are made on.
If we turn to Pell’s mathematical publications we are disappointed again, for they are few, and for the most part of little significance. The most important of them, and the book for which Pell is best known is an introduction to algebra, which we will look at later, but it is a book he did not originally write himself, and the extent of his contribution to it is not entirely clear. There were always hints that Pell was developing further ideas, but he could never be persuaded to share them. And there is the nub of the problem. Here is Collins again:
To incite him to publish anything seems to be as vain an endeavour, as to think of grasping the Italian Alps, in order to their removal. He hath been a man accounted incommunicable; the [Royal] Society (not to mention myself) have found him so.
If there is little to be hoped for from Pell’s published work, what about his unpublished papers? There are certainly plenty of them. Pell read and did mathematics for fifty years, from his student days in 1627 to his death in 1685. Over that entire period his papers piled up, and after his death they were acquired, along with his books, by Richard Busby, headmaster of Westminster School. Pell’s books are still at Westminster, but the papers are now in the British Library. His correspondence, with Charles Cavendish, Marin Mersenne, Samuel Hartlib, John Collins and others, fills three volumes; his mathematical papers fill no fewer than thirty-three. The papers contain tables, calculations, worked problems, and notes on the books Pell had read, a wealth of material, but they are chronologically and thematically in complete disorder. If I tell you that it takes the best part of a week to go through them even superficially, you can perhaps understand why I really did not want to become interested in John Pell.
You may also realise, however, from the fact that I have chosen him as my subject today, that I have become not only interested but intrigued by Pell. I have now visited five English counties and two Dutch cities in search of the places where he lived or worked, and I have made my way through those thirty-three volumes of papers. I have concentrated particularly on Pell’s mathematics but even there I still do not have anything like a complete picture. Trying to uncover Pell’s mathematics does feel, as Collins complained, like trying to move the Alps. It is more than one person can do, and all I have done is shift a few stones, but I think I can offer you today a fuller account than has previously been available of Pell’s mathematics, and his mathematical influence.
I am going to take you through Pell’s life in chronological order, but towards the end we will sometimes need to go sideways and backwards as well. I would have liked to begin by showing you a picture of Pell, but I am unable to do so because we do not have one. But in a way the non-picture sets the scene quite well because this lecture is going to be at least in part about things that do not exist.
Pell was born on 1 March 1611 in Southwick in Sussex. Southwick is now almost indistinguishable from the larger conurbation of Brighton but it still has its church and its village green, and a single thatched cottage a short distance from the church. People in Southwick claim that the cottage was the Pell family home. Pell’s father, a schoolmaster, died when John was just five years old. As a boy, Pell went to the newly founded Grammar School at Steyning, just a few miles inland over the downs. After Steyning, at the age of thirteen, young even for those days, Pell went to Trinity College Cambridge. John Aubrey, who knew Pell well, wrote:
At 13 yeares and a quarter old he went as good a scholar to Cambridge, to Trinity Colledge, as most Masters of Arts in the University (he understood Latin, Greek and Hebrew) so that he played not much (one must imagine) with his school fellowes, for, when they had play-dayes, or after schoole-time, he spent his time in the Library.
By 1627, now 16 and still at Cambridge, Pell had taken to mathematics and started writing small booklets on mathematical subjects. There are many of these among Pell’s papers from his early years, neatly written and with carefully designed title pages. These early booklets demonstrate two things that were to be characteristic of Pell for the rest of his life. The first is a fascination with tables and table-making. One of the earliest booklets, dated 1628 is on the construction and use of multiplication tables but Pell soon moved on to more difficult things. In that same year, 1628, he wrote to Henry Briggs with questions about antilogarithms and about interpolating tables of sines. It seems that Briggs and Pell already knew each other well, because Briggs signed himself: ‘Yr very lovinge frende, Henrie Briggs’. Pell went on making tables for the rest of his life: sheet after sheet of the thirty-three volumes of his papers is filled with tables, and calculations for tables: tables of squares, of sums of squares, of primes and composites, constant difference tables, tables of logarithms, antilogarithms, trigonometrical functions, and so on.
Only one set of tables was published: the first 10,000 square numbers, which came out in 1672. There was no name on the title page though, and that brings us to the second thing that characterised Pell throughout his life: a strange desire for anonymity. Even in the early handwritten booklets of his student days, few of the title pages gave his name. There are, for example, two copies of one entitled Linea proportionata: one carries the pseudonym ‘By A Fale, Gent.’, while the other only has ‘By’, followed by a blank space, as though Pell could not decide what name to insert. This intense desire for secrecy was evident throughout Pell’s life: he was always extremely reluctant to reveal his name in print. Some have attributed this to modesty, but I think it was a stranger and more obsessional quality than that, and it is one of the things that has made Pell such a difficult subject of study for later historians.
It was probably in Cambridge in 1628 or 1629 that Pell met Samuel Hartlib, newly arrived as a protestant exile from Poland. Hartlib was to work tirelessly both for the European Protestant cause and for educational and social reform. In 1630 he set up a school in Chichester in Sussex which would both realise his educational ideals and provide employment for exiled scholars and others. Pell was employed to teach mathematics. The school lasted only a few months, but Pell remained teaching in Sussex for some years, and while there, he married Ithumaria Reginalds with whom he eventually had eight children.
Throughout this time he remained in contact with Hartlib, who did his best to find Pell a post in London but Pell was not an easy man to help. Aubrey described him as ‘naturally averse from suing or stooping much for what he was worthy of’, but the fact of the matter was probably that Pell was choosy about what he would or would not do. Theodore Haak, for instance, a close friend of Hartlib’s, introduced Pell to the Bishop of Lincoln who offered Pell a benefice, but Pell turned it down on the grounds that he would rather devote himself wholly to mathematics: ‘for the great publick need and usefullness thereof.’
In 1638 Pell composed and sent to Hartlib a tract entitled An idea of mathematics. It was published in English and in Latin that same year, but Pell’s name did not appear on it, and only later did he admit publicly to being its author. There is only one surviving copy of the original, but it also appeared in a later printed edition: in 1650 it was appended to The reformed librarie keeper written by Hartlib’s friend John Dury. It opens with an admonition:
As long as men want will, wit, means or leisure to attend those studies, it is no marvail if they make no great progress in them.
In the Idea Pell proposed three things: first, the setting up of a catalogue of mathematicians and mathematical texts; second, the establishment of a library containing those texts; and third, to save time and labour for future generations, that there should be written a concise summary of all mathematical discoveries so far. These were practical projects, which Pell began to work on but never completed. Of greater interest to us now, however, is Pell’s perception of how mathematics could advance. It should be possible, he argued, to set out the method or process of mathematical argument in such a way that it would be possible to deduce from first principles:
… not onely all that ever is to bee found in our Antecessor’s writing, and whatsoever they may seeme to have thought on, but also all the Mathematicall inventions, Theoremes, Problemes and Precepts, that it is possible for the working wits of our successors to light upon and that in one certain, unchanged order, from the first seeds of Mathematics, to their highest and noblest applications.
This strong sense of an inherent logical structure in mathematics sets Pell apart, I think, from most other seventeenth-century mathematicians, who generally took a much more pragmatic approach to their subject, and it brings Pell in some ways closer to the twentieth century, when it was again believed by some that mathematics could be developed in a purely logical manner. There are other respects too in which I think Pell would have been at home in the twentieth century, as we shall see later.
By 1639 Pell had made his way into other circles of acquaintance than those surrounding Hartlib. He had become known also to Charles Cavendish, interested in all things mathematical, and to the men around Cavendish, including Thomas Aylesbury and Walter Warner. Aylesbury and Warner had both been active in editing and publishing some of the posthumous mathematical papers of Thomas Harriot, who had died some twenty years earlier in 1621. Harriot had made important contributions to algebraic notation and to understanding the structure and solution of equations, and Pell, through his friendship with Aylesbury and Warner, came to know Harriot’s work well. Both his skill and his reputation in algebra probably rested in great part on this intimate knowledge of Harriot’s work.
With Warner, Pell embarked on an ambitious project, the construction of tables of antilogarithms, something he had already discussed with Briggs some ten years earlier. Aubrey, incidentally, knew about these tables, which he described as being ‘like a Dictionary of the Latine before the English’, but was puzzled as to the use of them: ‘Quaere Dr Pell’, he wrote, ‘what is the use of those Inverted Logarithmes? For W. Warner would not doe such a thing in vaine. ’ Unfortunately Warner died in 1643 before the calculations were finished, and Pell later argued that the work was too incomplete for him to continue, and the tables were abandoned.
In any case, by the end of 1643, the efforts of Hartlib and others to establish Pell in a paid post had finally met with success: Pell was offered a teaching post at the Gymnasium in Amsterdam and left England in December 1643. He was to remain in Amsterdam until 1646. During this time Pell put a great deal of time and energy into refuting a quadrature of the circle by the Danish mathematician Longomontanus. Pell’s refutation was published almost immediately, but it was hardly a major piece of work. At the same time he was working on two rather more important projects: an edition of Books V to VII of the Conics of Apollonius, which at that time existed only in manuscript, and an edition of the Arithmetic of Diophantus. In 1646, however, Pell received a visit from Descartes, and he reported afterwards that he was disinclined to continue with these classical texts. Whether Descartes’ discouragement was real or perceived, we cannot know, but Pell’s editions were never published and no draft of either has ever been found. As I said earlier, the story of Pell’s life is often the story of things that do not exist.
In 1646 William, Prince of Orange, founded the Illustre School in Breda, and personally invited Pell to be its professor of both mathematics and philosophy. Pell accepted but later, rather to his relief, was required to teach only mathematics. We have Pell’s inaugural lecture, and we have a poster advertising the first term’s lectures with Pell listed as the professor for mathematics. (The original is not in the archives at Breda, where one might expect it to be, but in England, in the library of Lincoln Cathedral. It was deposited there in 1660 by Michael Honeywood, Dean of Lincoln, who like many other Anglican clergymen had lived in exile in the Netherlands during the interregnum.)
We know a little about some of Pell’s acquaintances and pupils in Breda. The exiled English king, Charles II, spent some weeks there in 1649 and there is reason to believe that Pell became friends and discussed mathematics with one of his young courtiers, Silas Titus. This mathematical friendship between Pell and Titus was to continue for many years: we will see more of it later. Another of Pell’s pupils was William Brereton, then aged seventeen, who had been forced to flee his home, Brereton Hall in Cheshire, in the early stages of the Civil War and who was sent to Breda for his education. He too became a lifelong friend and supporter of Pell, an example of the sort of loyalty Pell could inspire. The same cannot be said, however, of another pupil, the young Christian Huygens, who studied in Breda while Pell was there and visited Pell on several occasions but complained that Pell refused to discuss certain books with him and taught him nothing.
Pell returned to England in 1652 and became involved in the scientific meetings held at Gresham College that were to lead eventually to the founding of the Royal Society. His stay in London was short-lived, however. In 1654 he was offered a post as Cromwell’s envoy to the Protestant Cantons of Switzerland, and spent the next four years in Zurich. This part of Pell’s life is relatively well known and documented from the many letters that passed between him and John Thurloe, head of Cromwell’s intelligence service.
Pell came back to England for good in 1658, just three weeks before Cromwell’s death, and was immediately invited to rejoin the Gresham meetings. We have a letter written to Pell only days after his return, which reads as follows:
Mr. Pell – There is this day a meeting to be in the Moor-fields of some mathematical friends (as you know our custom hath been). There will be Mr. Rook and Mr. Wren, my Lord Brouncker, Sir Paul Neale, Dr Goddard, Dr Scarborough etc. I had notice last night of your being in town, from some of the gentlemen now named, and of their desire to enjoy your company, …
It is clear from this that Pell was both well known and respected, but for him there was still no paid post in mathematics, and it is not clear how he made his living over the next two or three years. In 1661 he did what he had refused to do thirteen years earlier: he entered the church. His first living was at Fobbing in Essex, a church that was in the direct gift of Charles II. This seems a generous reward to someone who had served Cromwell so faithfully, and I wonder if Pell’s friend Titus, by now a Member of the King’s Bedchamber, brought some influence to bear here? Two years later, Pell received a second living at Laindon, four miles from Fobbing, from Gilbert Sheldon, Archbishop of Canterbury. Both Fobbing and Laindon, in the low-lying Essex marshes, were notoriously unhealthy, and according to Aubrey, Pell once complained to the Archbishop. ‘I doe not intend that you shall live there’, said Sheldon. ‘No, sayd Pell, but your Grace does intend that I shall die there.’
In 1663 Pell was made a Doctor of Divinity at Lambeth, and by now was also a Fellow of the Royal Society, and looked set to enjoy a comfortable middle age. It was not to be: he became increasingly impecunious and from the mid 1660s was rescued over and over again by the kindness of his friends. He lodged for a time with the ever patient John Collins, and then moved with part of his family to Brereton Hall, the home of William Brereton who had been his pupil in Breda.
During his time in Cheshire, Pell was engaged on the mathematical work for which he is now best known, An introduction to algebra. Some of this, as the title page suggests, was translated by Thomas Brancker from an earlier German text, the Teutsche algebra of Johann Rahn, which had been published in Zurich in 1659. We know that Rahn had been Pell’s pupil in Zurich in the 1650s, and there is ample evidence that Pell made considerable input into the original German edition; indeed, Rahn claimed in his foreword that his methods were learned from a high and very learned person who, however, would not allow his name to be published. This can only have been Pell. When Brancker translated the book into English as An introduction to algebra during the 1660s, Pell made extensive additions and corrections to the original.
We have some of the correspondence that passed between Pell and Brancker over this matter, and we can only admire the tact and patience that Brancker displayed in his dealings with Pell. It will not surprise you by now to learn that a major problem for Pell was whether to allow his name to be added to the book or not. A letter drafted but never sent describes his dilemma exactly; he was reluctant to put his own name to the book but equally reluctant to let others claim credit that was rightly his:
I know not how my mind may alter but for the present, I think it best not to name mee at all in the title or preface: and yet you may be more ingenuous than Rahn was and not vent all for you owne devices. You may say, that the alterations and additions etc. were made by the advice of one of good reputation in those studies.
When the book finally appeared, the compromise was this: ‘Much Altered and Augmented by D.P.’ A somewhat feeble compromise, it has to be said, because it was well enough known that D.P. was Doctor Pell. In the British Library we have the original draft of the title page in Pell’s own hand, and there ‘D.P.’ is crossed out and then written in again as though even at that stage Pell was not at all sure how much to give away.
The book is, as its title indicates, an elementary introduction to algebra, but there are one or two features that distinguish it from any other text of the time. The first, obvious on every page, is Pell’s layout. He set out all his examples in three columns: on the left in each line he wrote an instruction and on the right the result of carrying out that instruction; down the middle he entered line numbers, very much as in a modern computer programme. The first few lines of work were a little different: here Pell set out on the left a list of unknown quantities and on the right a list of given or known conditions. If the solution can be uniquely determined, then the number of unknowns is exactly matched by the number of conditions.
Pell took a further step, however, and in doing so differed from other contemporary writers, by showing also how to handle indeterminate problems, those in which there are more unknowns than conditions. His Problem 24 (page 100), for example, was: ‘Find two numbers, either of which being subtracted from the square of their sum, will leave a square number’. Pell called these remaining square numbers dd and ee, so he had five unknown quantities: a, b, c, d and e, but only three conditions. The two non-existent conditions were marked by asterisks, then at a later stage Pell replaced each asterisk by an arbitrary condition to aid the solution; he replaced the first asterisk, for instance, by the condition e = 2d.
Pell invented a number of special symbols which enabled him to keep each instruction on the left on a single line: 8 @ 2 meant ‘take the square of Line 8’, while 11 vvv 2 meant ‘take the square root of Line 11’. The only one of these symbols to have remained in use is the division sign, and for exactly the reason that Pell invented it, because it allows division to be written without resorting to two-line fractions. There are several examples on page 59, where Pell also explained the use of the inequality signs first devised by Thomas Harriot. Pell used the same layout for geometrical problems: on page 93, for example, is an algebraic proof of a geometrical proposition. What is strikingly clear in all of this is Pell’s very strong sense of order and logic in mathematics. A book by Pell would hardly be complete without tables. There were tables relating to particular problems (and from the headings you can imagine how much Pell would have enjoyed spreadsheets), and a more general table of primes, or incomposites, up to 10,000.
There are no other books by Pell to tell you about. The Introduction to algebra has long been regarded as the only significant publication of Pell’s mathematics. What I have to tell you next rather changes that view. I believe that more of Pell’s mathematics did appear in print, but in a rather curious way. To untangle the story we now have to go sideways a little, and look at another mathematician, Pell’s contemporary, John Wallis.
Wallis was born five years after Pell, in 1616. Like Pell he was educated at Cambridge, but unlike Pell did not embark on his mathematical career until he was over thirty, by which time Pell was already in the Netherlands. Wallis became Savilian Professor of Geometry at Oxford in 1649, and the two may have met when Pell returned to England in 1652; they certainly knew each other soon after Pell’s final return in 1658, because some of Wallis’s mathematical notes from that period use Pell’s three-column layout, and since that had not yet been published, Wallis presumably learned it from Pell himself.
Then in 1662, we find from Pell’s manuscripts that he and Wallis were discussing this problem: find numbers a, b and c such that aa + bc = 16, bb + ac = 17, cc + ab = 18. Why this problem should have been of such great interest to either of them I do not understand, for it eventually leads to an unpleasant quartic equation that has to be solved by brute force:
e8 – 80e6 + 1998e4 – 14937e2 + 5000 = 0
Pell devoted pages and pages to this. He wrote that he had first come across the problem in Breda in 1649:
Mr. William Brereton of Breda anno 1649 brought me an example of this question aa + bc = 16, bb + ac = 17, cc + ab = 22 [which] as triall of logisticall skill I transformed to:
aa + bc = 16
bb + ac = 17
cc + ab = 18
To which I gave this answer …………………………………………….
but the manner of investigation I did not shew him. Neither do I now at all remember what course I tooke … but I will heere endeavour to show a way …
No answer was given. Reconstructing the solution was evidently not as easy as Pell supposed, and at some stage Wallis joined in. Another page of working carries this note:
Ergo conveniunt DIW & MIP in omnibus coefficientibus aequationis
(Therefore Doctor John Wallis and Mr. John Pell agree in all the coefficients of the equation)
It seems that Silas Titus was also interested in the solution, for on yet another sheet we have a complete solution, with a note in Pell’s hand that reads:
Ex Iohannis Wallisii autographo exscripsi, Aprilis 14. 1663. quod reddidi Cap. Tito Novemb. 14. 1663.
(I wrote this from a copy in Wallis’s hand, April 14 1663, which I delivered to Captain Titus, November 14 1663)
We have already seen that Titus was in Breda in 1649 and he may well have discussed the problem with Pell at that time, but in 1663 it was Pell and Wallis who came up with the solution.
Now I am going to jump ahead twenty-two years to 1685. In that year Wallis published A treatise of algebra both historical and practical. Of the one hundred chapters of this large volume, seven are devoted to the algebra of John Pell. What did Wallis have to say about it? First he drew attention to Pell’s three-column layout and quoted a long example of it. Then he noted Pell’s particular interest in indeterminate equations. Thus Wallis correctly and astutely pointed out what I think are indeed the most important features of Pell’s algebra.
Then Wallis suddenly changed tack and began to discuss the problem we just looked at: aa + bc = 16 etc , and he presented the solution in Pell’s three-column style. In fact his solution is an exact copy of the one he worked on with Pell in 1663 and so it is not too surprising to find it at this point in his book under the running head of ‘Dr Pell’s algebra’. What was strange, however, was Wallis’s way of introducing it, because he managed to do so without actually attributing anything to Pell. First his chapter heading was rather vague: it refers not to ‘An Example of Pell’s’, but merely to an ‘Example in Imitation of his’. Then Wallis wrote this:
I here subjoin another Question, proposed to my self, long since, by Colonel Silas Titus (then of his Majesties Bed-Chamber;) a very Ingenious Person, and well skilled in affairs Civil and Military, and very well accomplished in Mathematical and other learning.
In other words, Wallis attributed the problem to Titus who, I believe, was only a minor character in the story, and not to its real originator, and the person with whom he had worked most closely on it, John Pell. Given what we already know of Pell’s desire for anonymity, I think that what we have here is an example of Wallis deliberately deflecting attention from Pell to Titus. I think so because this was not the only occasion when Wallis did this, as I shall now show.
In Chapter 11 of A treatise of algebra Wallis looked for fractional equivalents for the number we now know as π, the ratio between the perimeter and diameter of a circle. He began with the well known 22 to 7 but ended four pages later with ratios of 18- or 19-figure numbers. There was a phenomenal amount of calculation behind all this, but Wallis never suggested that anyone but he worked on it. I was somewhat astonished, therefore, when amongst Pell’s papers I found this:
July 2 1636
Of the proportion of ye periphery to ye diameter of a circle
Archimedes [about 210 years before Christ] determined it as 22+ to 7.
Ludolph van Ceulen [A0 Christi 1599] determined it to be as 314159 26535 89793 to 1 at 15 cyphers.
Lansberg [A0 Christi 16..] determined it to be as 314159 26535 89793 23846 26433 833 to 1 at 28 cyphers.
Mr. Henry Briggs [A0 Christi 16..] determined it, to a radius of 1, at 40 cyphers.
So that to seeke a greater proportion is meerely needlesse, it is better to seeke to bring it to some smaller one for common use.
1. Some doe by cutting off as many cyphers as they thinke Goode from ye end, so Mr Oughtred saise 3|1416 and Mr Gunter 3142
2. Some seeke ye ratio in some other Diameter, hoping by yt means to find it rationall, so …………………... determined it to be as 355 to 113.
(The missing name here was that of Adriaen Anthoniusz Metius, a Dutch engineer who discovered the ratio 355 to 113 in 1584.) Pell continued:
Concerning which we will enquire:
1. How farre this proportion of 355 to 113 will hold?
2. How he found this proportion in small numbers?
To reduce this fraction to an equall one of its least termes, this he might doe by a perpetuall dividing ye greater by ye lessser, and ye lesser by ye relique …
Now compare the way Wallis introduced the problem:
The proportion of the Diameter to the Perimeter of a Circle, is by Archimedes shewed to be (very near) as 7 to 22, in small numbers;
Then Wallis went on:
Amongst others Metius hath pursued the same Inquiry, and gives us the proportion of 113 to 355; which is nearer than that of Archimedes, but in Greater numbers; … I find some have been wondering by what means Metius came to light upon those Numbers …
That was just what Pell wondered, in 1636. And Wallis’s method of tackling the problem was exactly the method described then by Pell, essentially an application of the Euclidean algorithm, except that where Pell used only 7 figures Wallis used 35.
Wallis’s introduction and method were so similar to Pell’s that it leaves me in no doubt that Wallis had the idea for this piece of work from Pell. Did he say anything about Pell? He did not. His opening words were these:
[The work] was occasion’d by a Problem sent to me (as I remember) about the Year 1663, or 1664, by Dr Lamplugh the present Bishop of Exeter, from (his Wives Father) Dr Davenant, then one of the Prebends Residentiaries of the Church of Salisbury, a very worthy Person, of great Learning and Modesty, as I since understand from persons well acquainted with him, and by divers Writings of his which I have seen, though I never had the opportunity of being personally acquainted with him ….
This was what we might call ‘flannel’: Dr John Davenant, one time Bishop of Salisbury, was a friend of Samuel Hartlib’s, and therefore possibly also of Pell’s, and he may at some time have discussed this problem with Pell, but I think he was introduced here, as Silas Titus was introduced later, simply to divert attention from the real originator of the problem, who was none other than Pell himself.
Let me give you another example of Wallis hiding Pell. In Chapter 56 of A treatise of algebra Wallis presented Descartes’ rule for solving a biquadratic equation as a product of two quadratics. But, Wallis complained, Descartes did not explain how he arrived at his rule, and Wallis proceeded to demonstrate how the method worked. It was not difficult for him to do so because Pell had written it all out years before. There is a neatly written sheet of solutions in Pell’s handwriting, this time not among Pell’s own papers but among those of Charles Cavendish. We know that Pell and Cavendish were discussing Descartes’ equations in 1646 and Pell would have written out his solutions at about that time, forty years before they went into Wallis’s Treatise of algebra.
There is just one other seventeenth-century algebra textbook that discusses this method of solving biquadratics, though Wallis did not know it. It is the Algebra ofte Stel-Konst of the Dutch writer Gerard Kinckhuysen, published in the Netherlands in 1661. Kinckhuysen’s method of setting out the work is very close to Pell’s handwritten version, which raises the question of whether they ever discussed it together. It is not impossible: Kinckhuysen lived only a few miles from Amsterdam, and though he was only eighteen years old when Pell arrived there, he had already published a book on the use of quadrants (a subject on which Pell had also written in his own student days). There is another remarkable similarity between Pell’s algebra and Kinckhuysen’s: we have seen the way Pell handled indeterminate equations by introducing arbitrary conditions; Kinckhuysen too explained that in indeterminate problems, arbitrary values could be introduced at will, leading to innumerable solutions. We also know that by 1648 Kinckhuysen was familiar with Book V of the Conics of Apollonius, which then existed only in manuscript, and that Pell was one of the few people who had access to such a manuscript. Thus, though the evidence is all circumstantial, I now think it almost certain that Kinckhuysen was Pell’s student for a time while Pell was in Amsterdam.
This possible connection between Pell and Kinckhuysen has not previously been noticed but it is of some significance because the Stel-Konst later came to be of considerable importance in English mathematics: in 1669 it was proposed to publish a Latin translation of the text, with annotations by Isaac Newton. The publication never came about, but Nicolaus Mercator’s handwritten Latin translation, interleaved into the Dutch text, and with Newton’s annotations, is preserved in the Bodleian Library in Oxford. What is less well known is that other Kinckhuysen texts were also translated, one of them into English by Thomas Brancker, the translator of Rahn’s algebra. This information comes from a note from Pell’s papers, written for Henry Oldenburg in 1672. The note gives you a sense of Pell’s idiosyncratic style; it reads:
Introductio Kinckhuyseni translated into Latin and enlarged by Mr. Newton’s notes, to serve as an introduction to his general method of Analyticall quadratures.
Kinckhuysen’s last book, of Geometricall problems, was transcribed into Latin by a German gunner blown up in trying experiments of fire-work, but the translation in the hands of M. Bernard is fitted for the press.
And Kinckhuysen’s Apol …….nicks are translated into English, and put into Dr Pell’s method by Brancker the publisher of Rhonius his Algebra.
There is a hole in the paper in the middle of the final sentence, as though something has been scratched out, but I take it to read ‘Apollonius his Conicks’. A few days after this was written, Oldenburg turned the information into better English and passed it on to Huygens, but he omitted the fate of the German gunner and he also omitted any mention of the English translation of Apollonius. We never hear of it again. Is this the final trace of work that Pell and Kinckhuysen did together thirty years earlier? Little is known about Kinckhuysen, but I do not think it is merely coincidence that Pell was so closely involved in English translations of not one but two foreign texts, one by a Swiss-German whom we know to have been his pupil, the other by a Netherlander whom I now strongly suspect to have been.
Let us return to Wallis’s Treatise of algebra because we still have not exhausted what it possibly contains of Pell. In Chapters 66 to 69 Wallis put forward a number of geometrical representations of complex numbers. He did not come up with what is now known as the Argand diagram but he did display considerable ingenuity and understanding in the ten different methods he did devise. There is nothing that overtly links any of this with Pell, but in his final summary Wallis explained that the value of these constructions was to indicate just how far complex or ‘impossible’ solutions deviated from the real or ‘possible’:
We find therefore, that in all Equations which in the strict sense, and first Prospect, appear Impossible; some mitigation is to be allowed to make them Possible; …
[The constructions] while declaring the case in Rigor to be impossible, shew the measure of the impossibility; which if removed, the case will become possible.
Compare these words with a paragraph written at about the same time by John Collins:
These impossible roots, saith Dr. Pell, ought as well to be given in number as the negative and affirmative roots, their use being to shew how much the data must be mended to make the roots possible …
It therefore seems that both Wallis and Pell were interested in what Wallis calls ‘the measure of the impossibility’, and in whether one could somehow adjust an ‘impossible’ equation to make it ‘possible’.
Pell repeatedly asserted that he had his own method for finding numerical solutions for polynomial equations, and claimed that Viète’s method, by comparison, was ‘work unfit for a Christian.’ It will not surprise you that Pell’s method involved the use of tables, but it will also not surprise you that beyond that it remains a mystery because Pell could never be persuaded to explain it. Collins despaired of him: ‘We have been fed with vain hopes from Dr Pell about twenty or thirty years,’ he wrote, ‘Dr Pell communicates nothing.’ The originals of the crucial correspondence on this matter have been in private hands and inaccessible to scholars for many years, but earlier this year they were acquired by Cambridge University Library, and my hope is that when the collection is opened up for research it might at last provide some firmer clues as to Pell’s mysterious method.
There is one further section of A treatise of algebra where Pell’s unseen hand was at work and where I believe his secrecy had harmful and long-lasting consequences. It is Wallis’s account of the mathematics of Thomas Harriot. Wallis devoted twenty-six chapters, a quarter of his book, to describing Harriot’s algebra, and his account is marked by what seem like grossly exaggerated claims for Harriot, interspersed with accusations that Descartes used Harriot’s work without acknowledgement.
Wallis knew a great deal about Harriot’s mathematics, more than he could have learned from Harriot’s published work, and the source of his information was undoubtedly Pell who had studied Harriot’s papers long ago in the 1630s. I believe that Wallis’s antagonism toward Descartes was also encouraged by Pell, whose personal dislike of Descartes was well known. Wallis, however, as we have seen him do so often, kept Pell’s name out of it. To disparage Descartes openly, though, was a foolish thing to do. French readers, especially, reacted with scorn and Wallis was forced to defend himself. Seven years after A treatise of algebra was first published, and seven years after Pell himself had died, Wallis at last revealed the extent of Pell’s influence. In a piece entitled: De Harrioto addenda (‘What must be added concerning Harriot’) he wrote this important paragraph:
Pell urged me all the more often in this; and from his words I have written everything I have said; and afterward I showed him what I wrote (to be examined, altered, corrected as he decided, or preferred) before it went to press, and everything that was published was said with Pell’s assent and approval.
If Wallis had said this earlier, his account of Harriot might have been taken more seriously. Instead, it seemed to most of his readers that his claims for Harriot were simply unfounded. As a result, Wallis’s reputation as a historian, and Harriot’s as a mathematician, have suffered ever since, just the opposite of what he and Pell intended.
If we take all the chapters of A treatise of algebra where Pell’s influence can be detected, they account for more than forty percent of the book. Yet Wallis gave Pell no direct acknowledgement; indeed he seems at times to have gone to some lengths to divert attention away from Pell. I believe he did so out of regard for Pell’s obsessional desire for anonymity, and he succeeded so well that Pell’s part in A treatise of algebra has passed unnoticed for 350 years. Even Pell could hardly have hoped for more than that.
There are still many unanswered questions about Wallis and Pell. How, for instance, did they communicate? Since Wallis lived in Oxford and Pell was based in London, you might imagine that letters would have passed between them, but if so they have vanished. We have dozens of letters written by Pell and hundreds by Wallis, but not one that links them directly: something else that does not exist. Pell and Wallis; Pell and Kinckhuysen: two relationships that have remained unexplored until now because the evidence for them is written not in English or Latin or Dutch, but in a language sometimes a little harder to penetrate, mathematics.
I will bring the story to an end. By 1680 Pell’s financial difficulties were dire and for a while he was actually imprisoned for debt. For the remaining five years of his life he lived on the charity of friend and relatives, until he died in December 1685. Aubrey tells us poetically but without justification, that he ‘died of a broaken heart’.
John Pell devoted most of his life to mathematics. He held no post of long term significance, wrote no great work, made no important discovery. Yet he knew mathematics better than most. He knew its history, and he knew its practitioners, both in England and on the continent. Above all, he had a sense, unparalleled in England at the time, of mathematics as a profoundly logical subject.
Unusual, difficult, enigmatic, witty, erudite, incommunicable: Pell remains an elusive character. He does not quite fit into any category, can not quite be pinned down. Trying to follow Pell’s line of thought or sift his work from his volumes of papers are not easy things to do; trying to understand why he behaved as he did and what inspired or motivated him is even harder. But as Pell evades us for the first or the hundredth time, we can at least smile and remind ourselves that that is exactly what he intended to do.
The Queen’s College, Oxford