7
“How Fine a Mind”: Mathematicians Past
INTEREST IN MATHEMATICIANS PAST BEGAN IN THE SIXTEENTH AND SEVEN-teenth centuries with the rediscovery, publication, and commentary on ancient mathematical texts, part of the more general Renaissance concern to rediscover and surpass ancient learning (and a process which continues to this day: “new” texts by Archimedes have been in the news in the last few years). This side of mathematical history writing is represented elsewhere in this book by the passage from Euclid in Chapter 4.
Also on display in other chapters is guessing—or, to be kinder, speculation—about the origins of mathematical techniques. We’ve seen both Joseph Fenn (Chapter 2) and Edmund Scarburgh (Chapter 4) try their hand at that in the contexts of algebra and geometry, and in this chapter we’ll see Charles Hutton do something similar for the supposed beginnings of arithmetic.
Both of these strategies receive their modern transformation in such a work as Thomas Heath’s edition of Euclid, where detailed research into ancient mathematical techniques and discoveries complements precise work on the ancient texts themselves. This textual and thematic approach to the mathematical past has been both fruitful and important and promises to continue in the future.
In popular writing it is complemented by an equally common—perhaps a more common—kind of mathematical history: biography. Several kinds of biography are showcased here: celebratory accounts of great men (Newton, in this case), eulogy, and the drier approaches of writers of historical textbooks, who until well into the twentieth century still relied very much on (sometimes dubious) anecdotes as a way to depict the characters and the activities of the past. This approach, again, blossoms into the modern school of mathematical biography—like Robert Kanigel’s—in which the individual scene may be a springboard to a really meaningful exploration of themes or technical content in the mathematics of the past.
In this chapter, too, we witness the abiding tendency to “domesticate” the mathematics of (for the historians) distant times and places by finding modern or western equivalents for it, but also the slow rise of a consciousness that the long-excluded—women, the poorly educated, the subjects of colonial rule—had a place in history as creators of mathematics.
The Labyrinth and Abyss of Infinity
Voltaire’s 1733 Letters Concerning the English Nation (published in French a year later as Lettres philosophiques) are sometimes called the first work of the Enlightenment; they commented on many aspects of English life, from science and religion to literature and theatre. Three of the twenty-four letters discussed Sir Isaac Newton at length; Voltaire was one of his greatest admirers—or propagandists—and was famously impressed by the fact that Newton was buried “like a king” at a state funeral. The Newtonian subjects covered in the Letters included optics, chronology, and anti-Cartesianism; here we see Voltaire describing the genesis of integral and differential calculus with an eye to the history of the subject and the subsequent dispute rather than its technical details. (For another popular account of matters Newtonian from the same decade, see the extract from Algarotti in Chapter 5.)
Voltaire (1694–1778), trans. John Lockman (1698–1771), Letters concerning the English nation (London, 1733), pp. 151–155 (letter 17).
The Labyrinth and Abyss of Infinity, is also a new Course Sir Isaac Newton has gone through, and we are obliged to him for the Clue by whose Assistance we are enabled to trace its various Windings.
Descartes got the Start of him also in this astonishing Invention. He advanced with mighty Steps in his Geometry, and was arrived at the very Borders of Infinity, but went no farther. Dr. Wallis, about the Middle of the last Century, was the first who reduced a Fraction by a perpetual Division to an infinite Series.
The Lord Brounker employed this Series to square° the Hyperbola.
Mercator° published a Demonstration of this Quadrature,° much about which Time, Sir Isaac Newton, being then twenty three Years of Age, had invented a general Method to perform, on all geometrical Curves, what had just before been tried on the Hyperbola.
’Tis to this Method of subjecting, everywhere, Infinity to algebraical Calculations, that the Name is given of differential Calculations or of Fluxions, and integral Calculation. ’Tis the Art of numbering and measuring exactly a Thing whose Existence cannot be conceived.
And, indeed, would you not imagine that a Man laughed at you, who should declare that there are Lines infinitely great which form an Angle infinitely little?
That a 
Line, which is a Line so long as it is finite, by changing infinitely little its Direction, becomes an infinite Curve; and that a Curve may become infinitely less than another Curve?
That there are infinite Squares, infinite Cubes; and Infinities of Infinities all greater than one another, and the last but one of which, is nothing in Comparison of the last?
All these Things which at first appear to be the utmost Excess of Frenzy, are in reality an Effort of the Subtlety and Extent of the human Mind, and the Art of finding Truths which till then had been unknown.
This so bold Edifice is even founded on simple Ideas. The Business is to measure the Diagonal of a Square, to give the Area of a Curve, to find the square Root of a Number, which has none in common Arithmetic. After all, the Imagination ought not to be startled any more at so many Orders of Infinites, than at the so well known Proposition, viz. that Curve Lines may always be made to pass between a Circle and a Tangent; or at that other, namely that Matter is divisible in infinitum. These two Truths have been demonstrated many Years, and are no less incomprehensible than the Things we have been speaking of.
For many Years the Invention of this famous Calculation was denied Sir Isaac Newton. In Germany Mr. Leibniz° was considered as the Inventor of the Differences or Moments, called Fluxions, and Mr. Bernoulli° claim’d the integral Calculation. However, Sir Isaac is now thought to have first made the Discovery, and the other two have the Glory of having once made the World doubt whether ’twas to be ascribed to him or them.
Be this as it will, ’tis by the Help of this Geometry of Infinities that Sir Isaac Newton attained to the most sublime Discoveries.
Notes
square: find the area under. William Brouncker (1620–1684) was the first president of the Royal Society.
Mercator: not the map maker but Nicolaus Mercator (1620?–1687), a Danish mathematician who spent much of his life in England after leaving Copenhagen to escape the plague.
Quadrature: area finding.
Mr. Leibniz: Gottfried Wilhelm von Leibniz (1646–1716), the German philosopher and mathematician.
Mr. Bernoulli: the Swiss mathematician Jacob Bernoulli (1654–1705).
“It Must Have Commenced with Mankind”
No selection could do justice to the magnificent Dictionary compiled by the prolific English mathematician and mathematical writer Charles Hutton, which contains some of the first really accessible mathematical history to be written in English. The section chosen here illustrates a thematic rather than a biographical approach to mathematical history and gives a hint of the range of Hutton’s learning. It may be compared with the other speculations in this volume, by Joseph Fenn and by Edmund Scarburgh, on the origins of algebra and geometry.
Charles Hutton (1737–1823), A Mathematical and Philosophical Dictionary: Containing an Explanation of the Terms, and an Account of the Several Subjects, comprised under the heads Mathematics, Astronomy, and Philosophy both natural and experimental: with an Historical Account of the Rise, Progress, and Present State of these Sciences: also Memoirs of the Lives and Writings of the Most Eminent Authors, both ancient and modern, who by their discoveries or improvements have contributed to the advancement of them (London, 1796), vol. 1, pp. 142–143.
Arithmetic: the art and science of numbers; or, that part of mathematics which considers their powers and properties, and teaches how to compute or calculate truly, and with ease and expedition. It is by some authors also defined the science of discrete quantity. Arithmetic consists chiefly in the four principal rules or operations of Addition, Subtraction, Multiplication, and Division; to which may perhaps be added involution and evolution, or raising of powers and extraction of roots. But besides these, for the facilitating and expediting of computations, mercantile, astronomical, etc., many other useful rules have been contrived, which are applications of the former, such as, the rules of proportion, progression, alligation, false position, fellowship, interest, barter, rebate, equation of payments, reduction, tare and tret, etc. Besides the doctrine of the curious and abstract properties of numbers.
Very little is known of the origin and invention of arithmetic. In fact it must have commenced with mankind, or as soon as they began to hold any sort of commerce together; and must have undergone continual improvements, as occasion was given by the extension of commerce, and by the discovery and cultivation of other sciences. It is therefore very probable that the art has been greatly indebted to the Phœnicians or Tyrians; and indeed Proclus, in his commentary on the first book of Euclid, says, that the Phœnicians, by reason of their traffic and commerce, were accounted the first inventors of Arithmetic. From Asia the art passed into Egypt, whither it was carried by Abraham, according to the opinion of Josephus. Here it was greatly cultivated and improved; insomuch that a considerable part of the Egyptian philosophy and theology seems to have turned altogether upon numbers. Hence those wonders related by them about unity, trinity, with the numbers 4, 7, 9, etc. In effect, Kircher, in his Oedipus Ægyptus shews, that the Egyptians explained every thing by numbers; Pythagoras himself affirming, that the nature of numbers pervades the whole universe; and that the knowledge of numbers is the knowledge of the deity.
From Egypt arithmetic was transmitted to the Greeks, by means of Pythagoras and other travellers; amongst whom it was greatly cultivated and improved, as appears by the writings of Euclid, Archimedes, and others: with these improvements it passed to the Romans, and from them it has descended to us.
The nature of the arithmetic however that is now in use, is very different from that above alluded to; this art having undergone a total alteration by the introduction of the Arabic notation, about 800 years since, into Europe: so that nothing now remains of use from the Greeks, but the theory and abstract properties of numbers, which have no dependence on the peculiar nature of any particular scale or mode of notation. That used by the Hebrews, Greeks, and Romans, was chiefly by means of the letters of their alphabets. The Greeks, particularly, had two different methods; the first of these was much the same with the Roman notation, which is sufficiently well known, being still in common use with us, to denote dates, chapters and sections of books, etc. Afterwards they had a better method, in which the first nine letters of their alphabet represented the first numbers, from one to nine, and the next nine letters represented any number of tens, from one to nine, that is, 10, 20, 30, etc., to 90. Any number of hundreds they expressed by other letters, supplying what they wanted with some other marks or characters: and in this order they went on, using the same letters again, with some different marks, to express thousands, tens of thousands, hundreds of thousands, etc: In which it is evident that they approached very near to the more perfect decuple scale of progression used by the Arabians, and who acknowledge that they had received it from the Indians. Archimedes also invented another peculiar scale and notation of his own, which he employed in his Arenarius, to compute the number of the sands. In the 2nd century of Christianity lived Claudius Ptolemy, who, it is supposed, invented the sexagesimal division of numbers, with its peculiar notation and operations: a mode of computation still used in astronomy etc, for the subdivisions of the degrees of circles. Those notations however were ill adapted to the practical operations of arithmetic: and hence it is that the art advanced but very little in this part; for, setting aside Euclid, who has given many plain and useful properties of numbers in his Elements, and Archimedes, in his Arenarius, they mostly consist in dry and tedious distinctions and divisions of numbers; as appears from the treatises of Nicomachus, supposed to be written in the 3rd century of Rome, and published at Paris in 1538; as also that of Boethius, written at Rome in the 6th century of Christ. A compendium of the ancient arithmetic, written in Greek, by Psellus, in the 9th century, was published in Latin by Xylander, in 1556. A similar work was written soon after in Greek by Jodocus Willichius; and a more ample work of the same kind was written by Jordanus, in the year 1200, and published with a comment by Faber Stapulensis in 1480.
Since the introduction of the Indian notation into Europe, about the 10th century, arithmetic has greatly changed its form, the whole algorithm, or practical operations with numbers, being quite altered, as the notation required; and the authors of arithmetic have gradually become more and more numerous. This method was brought into Spain by the Moors or Saracens; whither the learned men from all parts of Europe repaired, to learn the arts and sciences of them. This, Dr. Wallis proves, began about the year 1000; particularly that a monk, called Gilbert, afterwards pope, by the name of Sylvester II, who died in the year 1003, brought this art from Spain into France, long before the date of his death: and that it was known in Britain before the year 1150, where it was brought into common use before 1250, as appears by the treatise of arithmetic of Johannes de Sacro Bosco, or Halifax, who died about 1256. Since that time, the principal writers on this art have been, Barlaam, Lucas de Burgo, Tonstall, Aventinus, Purbach, Cardan, Scheubelius, Tartalia, Faber, Stifelius, Recorde, Ramus, Maurolycus, Hemischius, Peletarius, Stevinus, Xylander, Kersey, Snellius, Tacquet, Clavius, Metius, Gemma Frisius, Buteo, Ursinus, Romanus, Napier, Ceulen, Wingate, Kepler, Briggs, Ulacq, Oughtred, Cruger, Van Schooten, Wallis, Dee, Newton, Morland, Moore, Jeake, Ward, Hatton, Malcolm, etc, etc.
Kepler’s Astronomical Publications
Robert Small, Doctor of Divinity and Fellow of the Royal Society of Edinburgh (indeed, one of that society’s founding fellows in 1783), contributed to the Transactions of the society on mathematical subjects. His other publications were a work of political statistics and a sermon connected with the charitable support of Sunday schools; in 1791 he was Moderator of the General Assembly of the Church of Scotland. His interest in the history of mathematics was at the time a relatively uncommon way of representing mathematics to nonspecialist readers, but it was a sign of things to come.
Robert Small (1732–1808), An Account of the Astronomical Discoveries of Kepler: Including An Historical Review of the Systems which had succesively prevailed before his time (London, 1804), pp. 146–159.
The first fruits of 
application to astronomical studies appeared in his Mysterium Cosmographicum, published 
about two years after his settlement in Graz. Though he had adopted the Copernican arrangement° of the planets, the principles of connection subsisting between its parts appeared to him to be in a great degree unknown. No causes especially were assigned for the different positions of the centre of uniform motion in the different orbits, nor for the irregular distances of the planets from the sun, and no proportion was discovered between these distances and the times of the planetary revolutions. He considered the arrangement as deficient and unsatisfactory, till all those causes should be discovered; and in the work now mentioned he supposed that he had discovered them, at least he traced out several similar analogies, in the Pythagorean, or Platonic doctrines concerning numbers, in the proportions of the regular solids in geometry, and in the divisions of the musical scale; and these analogies seemed to assign the reason why the primary planets should be only six in number.
Hasty and juvenile as this production was, it displayed so many marks of genius, and such indefatigable patience in the toil of calculation, that, on presenting it to T. Brahé,° it procured him the esteem of this illustrious astronomer, and even excited his anxiety for the proper direction of talents so uncommon. Accordingly, not contented with exhorting Kepler to prefer the road of observation to the more uncertain one of theory, T. Brahé added a generous and unsolicited invitation to live with him at Uraniburg, where his whole observations would be open to Kepler’s perusal, and those advantages found for making others, which his situation at Graz denied. The opinion of other astronomers concerning this production was no less favourable, and, were there no other evidence of his just and general conceptions, the remark, which afterward led to such important consequences, that the cause of the equant,° whatever it might be, ought to operate universally, is a sufficient attestation of them, and a proof that, even in this early period of his studies, the possibility at least of deducing the equations of a planet from the relation between its different degrees of velocity and distances from the sun had presented itself to his thoughts.
Notwithstanding the ardent desire of Kepler to be admitted to the perusal of T. Brahé’s observations, it was probable that the distance of the place, and the difficulty of the journey, to one in Kepler’s situation, would have for ever prevented the gratification of it. But the persecutions now arose which drove T. Brahé from his native country, and from which he at last found a refuge in Prague, the capital of Bohemia. Thither, accordingly, Kepler repaired in the year 1600; and, that he might be under no necessity of returning to Graz, he obtained, by T. Brahé’s interest with the Emperor, his patron, the appointment of imperial mathematician.
On his arrival at Prague another circumstance occurred, equally important to his success and fame, and which he piously ascribes to the kind direction of Providence. T. Brahé, with his assistant Longomontanus,° was employed about his theory of Mars, no doubt induced by the favourable opportunity of verifying it by observations of this planet in its approaching opposition in the sign of Leo. Kepler’s attention was therefore, of course, directed to the same planet, where the greater degree of eccentricity renders its inequalities peculiarly remarkable, and leads with proportional advantage to the discovery of their laws; and it would have been directed to observations much less instructive, had those astronomers been engaged about any other planet.
The former planetary theories were in general unlike, and founded upon different principles. The ancients, as Ptolemy and his followers, considered every planet separately, and supposed that their several motions and inequalities arose from causes peculiar to every orbit, and with which the other orbits had no connection. The moderns, again, considered the orbits as connected by a common principle, and, remarking all that was similar in their motions, endeavoured to derive it from a common cause. But they disagreed about this principle, for the cause of the second inequalities° of the planets, according to Copernicus, was the annual revolution of the earth, while T. Brahé and his disciples ascribed them to the attendance of the planets upon the sun, in his annual revolution. But, notwithstanding these and other less important differences, the effects of all the theories, in representing the motions of the planets, and on the calculations of their places, were very nearly equal, and Kepler found them almost equally erroneous: for the longitudes of Mars, calculated for August 1598, and August 1608, even from the Prutenic tables of Reinholdus,° fell short of the observed longitudes, the first nearly 4°, and the last no less than nearly 5°; and errors of this magnitude were not peculiar to the Prutenic tables, for similar and even greater errors were found in all the others.
Notes
Copernican arrangement: with the sun at the centre rather than the Earth.
T. Brahé: the Danish astronomer Tycho Brahe (1546–1601) on whose observational data Kepler eventually relied.
equant: a point from which a particular planet appears to move at constant angular speed, or a circle about this point.
Longomontanus: Christen Sørensen Longomontanus (1562–1647), Danish astronomer, important for promoting Tycho’s theory of planetary motion.
second inequalities: nonuniformities in planetary motion; specifically, those not explained at the first stage of constructing a model of planetary motion.
the Prutenic tables of Reinholdus: the German astronomer and mathematician Erasmus Reinhold (1511–1553) published his astronomical tables, based on Copernicus’s theory of planetary motion, in 1551.
Isaac Newton, a Good and Great Man
The nineteenth century saw the considerable development of the “Newton myth,” the publication of more of Newton’s papers and the appearance of the first biography based on them. This charming (though not exactly accurate) portrait gives a sense of how Newton was seen at a more popular level; see Figure 7.1 for the accompanying illustration, complete with the famous apple. (George Bernard Shaw, in the following extract, will show us in some ways a fairly similar Newton.) He is, aside from Christopher Wren, the only mathematician among the fifty “greats” in the book.
The Children’s Picture-Book of Good and Great Men (London, 1860), pp. 150–156.
Sir Isaac Newton, the greatest philosopher and astronomer of modern days, was born on Christmas day, 1642, at Woolsthorpe, in Lincolnshire.
He was such a very little creature that his mother declared he might have been put into a quart-mug, and so feeble that no one thought he could live. But that poor little weak baby was to be one of our greatest instructors, and to find out more of God’s wonderful power and wisdom, in the creation of the earth, and those heavenly bodies that shine out to us in the sky, than anyone had ever before done.
Isaac’s mother, who was a widow, carefully nursed him up till he was three years old, when his grandmother took charge of the fatherless child, and after sending him for a while to a day-school in the neighbourhood, placed him, that he might have better teaching, at the Grammar-school of Grantham.
Figure 7.1. Newton meditating on the fall of the apple. (Picture-book, p. 155. © The Bodleian Libraries, University of Oxford. Nuneham 2526 e.4.)
He was here chiefly noticed as an ingenious lad, being fond, as boys generally are, of using carpenter’s tools, only that he managed them better than boys usually do. Among the various articles that his nimble fingers and thoughtful little head contrived was a water-clock. A box filled with water, which was permitted to escape drop by drop from the lower part, formed the body of this clock, while a piece of wood floating on the surface of the water was so contrived as, by its gradual sinking, to point out, one after the other, the hours of the day, which were marked on another portion of his ingenious contrivance.
At the age of fourteen, he was taken from school to be made a farmer of. But it was soon plain that Isaac was no farmer, nor man of business either. Each market-day that he was sent to Grantham, instead of busying himself to make good bargains for his corn or hay, he slunk off to an attic in the house where he had lodged while at school, and there sat poring over some old mathematical books, till the servant, who had been sent with him, called to take him home. Sometimes he never reached Grantham at all, but would stop by the road-side, quite taken up with a water-mill, or some such machine, till the returning waggon picked him up again. This would never do, and his mother, knowing that his inattention to business proceeded neither from idleness nor perverseness, but from his whole soul being taken up with study, instead of forcing him to be a farmer, sent him to college, where he might have his fill of learning. Trinity College, Cambridge, has the honour of having given young Newton (he was only eighteen) his first training in science.
He was in his right place there, and speedily showed how fine a mind he possessed. Within six years, that is, by the time he had reached his twenty-fifth year, he had made some of his greatest discoveries, including the one for which he is most celebrated—that of finding out how it is that the moon and stars keep their places, circling in wide space around the sun. This had long puzzled learned men. Newton, sitting in his garden one day, saw an apple fall from the tree, and, strangely enough, as he sat there thinking why it should come to the ground, he found out that the very same thing that made it do so was that which worked all the wonders of the regular movements of the heavenly bodies. To understand this would require more learning than you children have got. But people with sufficient learning see very well how it is, and that it is perfectly true.
Newton made many important discoveries in other sciences. But his vast knowledge did not make him think highly of himself. Much as he had discovered of God’s works, he knew there was so much more still unknown, that, at the close of his long life he said that to himself he appeared like a child picking up a few shells on the sea-shore, while the “great ocean of truth” lay all undiscovered before him.
In the year 1705 he received the honour of knighthood from Queen Anne.
This eminent philosopher died on the 20th of March, 1727, in the eighty-fifth year of his age, and was buried in Westminster Abbey; his pall was borne by the lord chancellor, two dukes, and three earls.
He was a man of great amiability and goodness. He was a sincere Christian, and spent much of his time in the study of the Holy Scriptures; nor could anything cause him greater grief than to hear the subject of religion spoken of in a light and irreverent manner.
Pythagoras and His Theorem
Still the definitive English version of Euclid a century after publication, Thomas Heath’s magnificent edition is made with rather different intentions from many earlier versions, and indeed it comes with a wealth and depth of commentary which seem intended to place it outside the category of “popular mathematics.” The commentary on Pythagoras’s theorem continues for another seventeen pages after the extract given here, which is presented with some simplifications. But Heath made clear that he nonetheless considered the Elements above all an elementary textbook, indeed the supreme example of its kind, “the greatest . . . that the world is privileged to possess.”
Sir Thomas L. Heath (1861–1940), The Thirteen Books of Euclid’s Elements, Translated from the text of Heiberg with introduction and commentary by Sir Thomas L. Heath (Cambridge, 1908, 1925), vol. 1, pp. 349–352.
“If we listen,” says Proclus,° “to those who wish to recount ancient history, we may find some of them referring this theorem to Pythagoras and saying that he sacrificed an ox in honour of his discovery. But for my part, while I admire those who first observed the truth of this theorem, I marvel more at the writer of the Elements, not only because he made it fast by a most lucid demonstration, but because he compelled assent to the still more general theorem by the irrefragable arguments of science in the sixth Book. For in that Book he proves generally that, in right-angled triangles, the figure on the side subtending the right angle is equal to the similar and similarly situated figures described on the sides about the right angle.”
In addition, Plutarch, Diogenes Laertius and Athenaeus° agree in attributing this proposition to Pythagoras. It is easy to point out, as does G. Junge, that these are late witnesses, and that the Greek literature which we possess belonging to the first five centuries after Pythagoras contains no statement specifying this or any other particular great geometrical discovery as due to him. Yet the distich of Apollodorus the “calculator,” whose date (though it cannot be fixed) is at least earlier than that of Plutarch and presumably of Cicero,° is quite definite as to the existence of one “famous proposition” discovered by Pythagoras, whatever it was. Nor does Cicero, in commenting apparently on the verses, seem to dispute the fact of the geometrical discovery, but only the story of the sacrifice. Junge naturally emphasises the apparent uncertainty in the statements of Plutarch and Proclus. But, as I read the passages of Plutarch, I see nothing in them inconsistent with the supposition that Plutarch unhesitatingly accepted as discoveries of Pythagoras both the theorem of the square of the hypotenuse and the problem of the application of an area, and the only doubt he felt was as to which of the two discoveries was the more appropriate occasion for the supposed sacrifice.
There is also other evidence not without bearing on the question. The theorem is closely connected with the whole of the matter of Euclid Book II, in which one of the most prominent features is the use of the gnomon.° Now the gnomon was a well-understood term with the Pythagoreans. Aristotle also clearly attributes to the Pythagoreans the placing of odd numbers as gnomons round successive squares beginning with 1, thereby forming new squares, while in another place the word gnomon occurs in the same (obviously familiar) sense: “e.g. a square, when a gnomon is placed round it, is increased in size but is not altered in form.” The inference must therefore be that practically the whole doctrine of Book II is Pythagorean.
Again Heron (?3rd cent. A.D.), like Proclus, credits Pythagoras with a general rule for forming right-angled triangles with rational whole numbers for sides. Lastly, the “summary” of Proclus appears to credit Pythagoras with the discover of the theory, or study, of irrationals. But it is now more or less agreed that the reading here should be 
“of proportionals,” and that the author intended to attribute to Pythagoras a theory of proportion, i.e. the (arithmetical) theory of proportion applicable only to commensurable magnitudes, as distinct from the theory of Euclid Book V, which was due to Eudoxus.° It is not however disputed that the Pythagoreans discovered the irrational.
Now everything goes to show that this discovery of the irrational was made with reference to 
, the ratio of the diagonal of a square to its side. It is clear that this presupposes the knowledge that I.47 is true of an isosceles right-angled tringle; and the fact that some triangles of which it had been discovered to be true were rational right-angled triangles was doubtless what suggested the inquiry whether the ratio between the lengths of the diagonal and the side of a square could also be expressed in whole numbers. On the whole, therefore, I see no sufficient reason to question the tradition that, so far as Greek geometry is concerned (the possibly priority of the discovery of the same proposition in India will be considered later), Pythagoras was the first to introduce the theorem of I.47 and to give a general proof of it.
Notes
Proclus: Proclus Diadochus (411–485), a Greek philosopher important for his commentaries on various mathematicians.
Plutarch, Diogenes Laertius and Athenaeus: Greek writers on philosophy and (Athenaeus) rhetoric and grammar of, respectively, the 1st–2nd, 3rd, and 2nd–3rd centuries.
Cicero: the great Roman orator and statesman (106–43 BC).
gnomon: “the part of a parallelogram which remains after a similar parallelogram is taken away from one of its corners.” (OED).
Eudoxus of Cnidus: Greek mathematician and astronomer (408–355 BC), to whom (part of) the theory of ratios in Euclid’s Elements is often attributed.
Seki Kōwa
David Eugene Smith was a professor of mathematics at Teachers College, Columbia University, involved with the American Mathematical Society, the International Commission on the Teaching of Mathematics, the Mathematical Association of America, and the History of Science Society. He took a keen interest in the history of mathematics, publishing a sourcebook and a textbook on the subject. Yoshio Mikami published a history of mathematics in China and Japan.
In our extract the authors give an outline of the life and achievements of Seki Kōwa, one of the great Japanese mathematicians of the seventeenth century, with an example of his work. Although the approach is basically biographical, there is much more mathematical detail here than in earlier works like Small’s discussion of Kepler; indeed, the presentation perhaps has more in common with Thomas Heath’s annotated translation of Euclid. The footnotes in this extract are by the original authors.
David Eugene Smith (1860–1944) and Yoshio Mikami (1875–1950), A History of Japanese Mathematics (Chicago, 1914), pp. 91–100.
In the third month according to the lunar calendar, in the year 1642 of our era, a son was born to Uchiyama Shichibei, a member of the samurai class living at Fujioka in the province of Kōzuke. While still in his infancy this child, a younger son of his parents, was adopted into another noble family, that of Seki Gorozayemon, and hence there was given to him the name of Seki by which he is commonly known to the world. Seki Shinsuke Kōwa was born in the same year in which Galileo died, and at a time of great activity in the mathematical world both of the East and the West. And just as Newton, in considering the labors of such of his immediate predecessors as Kepler, Cavalieri, Descartes, Fermat, and Barrow, was able to say that he had stood upon the shoulders of giants, so Seki came at an auspicious time for a great mathematical advance in Japan, with the labors of Yoshida, Imamura, Isomura, Muramatsu, and Sawaguchi° upon which to build. The coincidence of birth seems all the more significant because of the possible similarity of achievement, Newton having invented the calculus of fluxions in the West, while Seki possibly invented the yenri or “circle principle” in the East, each designed to accomplish much the same purpose, and each destined to material improvement in later generations. The yenri is not any too well known and it is somewhat difficult to judge of its comparative value, Japanese scholars themselves being undecided as to the relative merits of this form of the calculus and that given to the world by Newton and Leibniz.
Seki’s great abilities showed themselves at an early age. The story goes that when he was only five he pointed out the errors of his elders in certain calculations which were being discussed in his presence, and that the people so marveled at his attainments that they gave him the title of divine child.
Another story relates that when he was but nine years of age, Seki one time saw a servant studying the Jinkō-ki of Yoshida. And when the servant was perplexed over a certain problem, Seki volunteered to help him, and easily showed him the proper solution. This second story varies with the narrator, Kamizawa Teikan telling us that the servant first interested the youthful Seki in the arithmetic of the Jinkō-ki, and then taught him his first mathematics. Others say that Seki learned mathematics from the great teacher Takahara Kisshu, who 
had sat at the feet of Mōri as one of his san-shi, although this belief is not generally held. Most writers agree that he was self-made and self-educated, his works showing no apparent influence of other teachers, but on the contrary displaying an originality that may well have led him to instruct himself from his youth up. Whatever may have been his early training Seki must have progressed very rapidly, for he early acquired a library of the standard Japanese and Chinese works on mathematics, and learned, apparently from the Suan-hsiao Chi-mêng, the method of solving the numerical higher equation. And with this progress in learning came a popular appreciation that soon surrounded him with pupils and that gave to him the title of The Arithmetical Sage. In due time he, as a descendent of the samurai class, served in public capacity, his office being that of examiner of accounts to the Lord of Kōshū, just as Newton became master of the Mint under Queen Anne. When his lord became heir to the Shōgun, Seki became a Shogunate samurai, and in 1704 was given a position of honor as master of ceremonies in the Shōgun’s household. He died on the 24th day of the 10th month in the year 1708, at the age of sixty-six, leaving no descendents of his own blood. He was buried in a Buddhist cemetery, the Jorinji, at Ushigome in Yedo (Tōkyō), where eighty years later his tomb was rebuilt, as the inscription tells us, by mathematicians of his school.
Several stories are told of Seki, some of which throw interesting side lights upon his character. One of these relates that he one time journeyed from Yedo to Kōfu, a city in Kōshū, or the Province of Kai, on a mission from his lord. Traveling in a palanquin he amused himself by noting the directions and distances, the objects along the way, the elevations and depressions, and all that characterized the topography of the region, jotting down the results upon paper as he went. From these notes he prepared a map of the region so minutely and carefully drawn that on his return to Yedo his master was greatly impressed with the powers of description of one who traveled like a samurai but observed like a geographer.
Another story relates how the Shōgun, who had been the Lord of Kōshū, once upon a time decided to distribute equal portions of a large piece of precious incense wood among the members of his family. But when the official who was to cut the wood attempted the division he found no way of meeting his lord’s demand that the shares should be equal. He therefore appealed to his brother officials who, with one accord, advised him that no one could determine the method of cutting the precious wood save only Seki. Much relieved, the official appealed to “The Arithmetical Sage,” and not in vain.
It is also told of Seki that a wonderful clock was sent from the Emperor of China as a present to the Shōgun, so arranged that the figure of a man would strike the hours. And after some years a delicate spring became deranged, so that the figure would no longer strike the bell. Then were called in the most skilful artisans of the land, but none was able to repair the clock, until Seki heard of his master’s trouble. Asking that he might take the clock to his own home, he soon restored it to the Shōgun successfully repaired and again correctly striking the hours.
Such anecdotes have some value in showing the acumen and versatility of the man, and they explain why he should have been sought for a post of such responsibility as that of examiner of accounts.
The name of Seki has long been associated with the yenri, a form of the calculus that was possibly invented by him . It is with greater certainty that he is known for his tenzan method, an algebraic system that improved upon the method of the “Celestial element” inherited from the Chinese, for the Yendan jutsu, a scheme by which the treatments of equations and other branches of algebra is simpler than by the methods inherited from China and improved by such Japanese writers as Isomura and Sawaguchi, and for his work in determinants that antedated what has heretofore been considered the first discovery, namely the investigations of Leibniz.
As to his works, it is said that he left hundreds of unpublished manuscripts, but if this be true most of them are lost. He also published the Hatsubi Sampō in 1674. In this he solved the fifteen problems given in Sawaguchi’s Kokon Sampoō-ki of 1670, only the final equations being given.
As to Seki’s real power, and as to the justice of ranking him with his great contemporaries of the West, there is much doubt. He certainly improved the methods used in algebra, but we are not at all sure that his name is properly connected with the yenri.
Figure 7.2. In a circle three other circles are inscribed. . . .
For this reason, and because of his fame, it has been thought best to enter more fully into his work than into that of any of his predecessors, so that the reader may have before him the material for independent judgment.
First it is proposed to set forth a few of the problems that were set by Sawaguchi, with Seki’s equations and with one of Takebe’s° solutions.
Sawaguchi’s first problem is as follows: “In a circle three other circles are inscribed as here shown (see Figure 7.2), the remaining area being 120 square units. The common diameter of the two smallest circles is 5 units less than the diameter of the one that is next in size. Required: to compute the diameters of the various circles.”
Seki solves the problem as follows: “Arrange the ‘celestial element,’° taking it as the diameter of the smallest circles. Add to this the given quantity and the result is the diameter of the middle circle. Square this and call the result A.
“Take twice the square of the diameter of the smallest circles and add this to A, multiplying the sum by the moment of the circumference.1 Call this product B.
“Multiply 4 times the remaining area by the moment of diameter.2
“This being added to B the result is the product of the square of the diameter of the largest circle multiplied by the moment of circumference. This is called C.3
“Take the diameter of the smallest circle and multiply it by A and by the moment of the circumference. Call the result D.4
“From four times the diameter of the middle circle take the diameter of the smallest circle, and from C times this product take D. The square of the remainder is the product of the square of the sum of four times the diameter of the middle circle and twice the diameter of the smallest circle, the square of the diameter of the middle circle, the square of the moment of circumference, and the square of the diameter of the largest circle. Call this X.5
“The sum of four times the diameter of the middle circle and twice the diameter of the smallest circle being squared, multiply it by A and by C and by the moment of circumference.6 This quantity being canceled with X we get an equation of the 6th degree.7 Finding the root of this equation according to the reversed order we have the diameter of the smallest circle.
“Reasoning from this value the diameters of the other circles are obtained.”
It may add to an appreciation or an understanding of the mathematics of this period if we add Takebe’s analysis.
Let x be the diameter of the largest circle, y that of the middle circle, and z that of the smallest circles.
Then let AC = a, AD = b, AB = c, and BC = d, these being auxiliary unknowns at the present time.
Then
2a = −z + x,
and
4a2 = z2 − 2zx + x2
or
4a2 − z2 = −2zx + x2.
Therefore
If we take y from x we have −y + x, which is 2c.
Squaring,
To y add z and we have
2d = y + z.
Squaring,
4d2 = y2 + 2yz + z2.
Subtracting z2,8 we have
4(b + c)2 = y2 + 2yz.
Subtract from this (1) and (2) and we have
b × 8c = 2yz + (2z + 2y)x − 2x2.
Dividing by 2,
b × 4c = yz + (z + y)x − x2.
Squaring,
Multiplying (7.1) by (7.2) we also have
b2 × 16c2 = −2y2zx + (y2 + 4yz)x2 − (2y + 2z)x3 + x4,
which being canceled with the expression in (7.3) gives
y2z2 + (4y2z + 2yz2)x + (−4yz + z2)x = 0,
from which, by canceling z,
y2z + (4y2 + 2yz)x + (−4y + z)x2 = 0.
This may be written in the form
y2z + (x2z − 4x2y) + (4y2 + 2yz)x = 0.
Takebe has now eliminated his auxiliary unknowns, and he directs that the quantity in the first parenthesis be squared and canceled with the square of the rest of the expression,9 and that the rest of the steps be followed as in Seki’s solution. In this he expresses y and z in terms of x and given quantities and thus finds an equation of the sixth degree in x. Without attempting to carry out his suggestions, enough has been given to show his ingenuity in elimination.
Notes
Yoshida, Imamura, Isomura, Muramatsu, and Sawaguchi: Japanese mathematicians active in roughly the first half of the seventeenth century.
Takebe: Takebe Hikojirō Kenkō was active in the early part of the eighteenth century.
“celestial element”: This may be understood as what we would call the unknown quantity.
“Her Absolute, Incomparable Uniqueness”
Emmy Noether (1882–1935) worked in several areas of mathematics, including most notably abstract algebra. Hermann Weyl, in his memorial address, described her as “a great mathematician, the greatest, I firmly believe, that her sex has ever produced.” The passages reproduced here are taken from a translation of her 1935 obituary by Bartel Leendert van der Waerden, a Dutch mathematician and historian of mathematics.
B. L. van der Waerden (1903–1996), trans. Christina M. Mynhardt, “Obituary of Emmy Noether,” in J. W. Brewer and Martha K. Smith (eds.), Emmy Noether: A Tribute to Her Life and Work (New York, 1981), pp. 93–94, 97–98. Reprinted by permission.
Fate has tragically taken from our science a very important, entirely unique personality. Our faithful journal co-worker Emmy Noether died on April 14, 1935 as a result of an operation. She was born in Erlangen on March 23, 1882, the daughter of the well-known mathematician Max Noether.
Her absolute, incomparable uniqueness cannot be explained by her outward appearance only, however characteristic this undoubtedly was. Her individuality is also by no means exclusively a consequence of the fact that she was an extremely talented mathematician, but lies in the whole structure of her creative personality, in the style of her thoughts, and the goal of her will. For as these thoughts were primarily mathematical thoughts and the will primarily intent on scientific recognition, so must we first analyze her mathematical accomplishments if we want to understand her personality at all.
One could formulate the maxim by which Emmy Noether always let herself be guided as follows: All relations between numbers, functions, and operations become clear, generalizable, and truly fruitful only when they are separated from their particular objects and reduced to general concepts. For her this guiding principle was by no means a result of her experience with the importance of scientific methods, but an a priori fundamental principle of her thoughts. She could conceive and assimilate no theorem or proof before it had been abstracted and thus made clear in her mind. She could think only in concepts, not in formulas, and this is exactly where her strength lay. In this way she was forced by her own nature to discover those concepts that were suitable to serve as bases of mathematical theories.
Indefatigable and in spite of all external difficulties, she proceeded along the way indicated by the concepts she formed. Also, when she lost her teaching rights in Göttingen in 1933 and was appointed to the women’s college in Bryn Mawr (Pennsylvania), she soon gathered a school around her there and in nearby Princeton. Her research, which covered commutative algebra, commutative arithmetic, and noncommutative algebra, now turned to noncommutative arithmetic, but was suddenly terminated by her death.
As characteristic features we have found: An exceptionally energetic and consistent pursuit of abstract elucidation of the material to complete methodological clarity; a stubborn clinging to methods and concepts once they had been acknowledged as being correct, even when they still appeared to her contemporaries as abstract and futile; an aspiration to classify all special relationships under specific general abstract models.
Indeed, her thoughts deviated in some respects from those of most other mathematicians. We are all so dependent on figures and formulas. For her these resources were useless, rather annoying. Her sole concern was with concepts, not with intuition or calculations. The German letters which she scribbled down hurriedly on the blackboard or on paper in typical simplified form were for her representations of concepts, not objects of a more or less mechanical calculation.
This totally unintuitive and unanalytical attitude was undoubtedly also one of the main causes of the complexity of her lectures. She had no didactical gifts, and the great pains she took to explain her remarks by quickly spoken interjections even before she had finished speaking were more likely to have the opposite effect. And still how exceptionally great was the impact of her talks, everything notwithstanding! The small, faithful audience, mostly consisting of a few advanced students and often just as many lecturers and foreign guests, had to exert themselves to the utmost to keep up. When that was done, however, one had learned far more than from the most excellent lecture. Completed theories were almost never presented, but usually those that were still in the making. Each of her lecture series was a paper. And nobody was happier than she herself when such a paper was completed by her students. Completely unegotistical and free of vanity, she never claimed anything for herself, but promoted the works of her students above all. For all of us she always wrote the introductions in which the main ideas of our work, which we initially never could understand and express in such clarity on our own, were explained. She was a faithful friend to us and at the same time a strict and unprejudiced judge. As such she was also invaluable to Mathematische Annalen.°
As mentioned earlier, her abstract nonintuitive concepts initially found little acknowledgement. As the success of her methods also became clear to those of a different mind, this situation changed accordingly, and during the last eight years, prominent mathematicians from home and abroad went to Göttingen to ask her advice and listen to her lectures. In 1932 she shared the Ackermann-Teubner commemorative prize for arithmetic and algebra with E. Artin.° And throughout the world today the triumphant progress of modern algebra which developed from her ideas seems to be unending.
Notes
Mathematische Annalen: founded in 1868, a journal with wide coverage in modern mathematics.
E. Artin: Emil Artin (1898–1962), Austrian mathematician whose major work was in the branch of abstract algebra dealing with the mathematical structures called rings.
“One of Your Calculating Fits”
Isaac Newton, that “good and great man” (see earlier extract), has been the subject of a number of plays, among them than this one, relatively little known, by George Bernard Shaw. It features an unlikely procession of historical figures through Newton’s drawing room and a good deal of dialogue attempting to educate us about their political problems. By contrast with more recent dramatic depictions of Newton, which have tended toward the psychologically sensational, Good King Charles represents him as something close to an old-fashioned dotty professor, an image possibly influenced by the image of Albert Einstein, a very prominent contemporary at the time this was written.
George Bernard Shaw (1856–1950), In Good King Charles’s Golden Days: A True History That Never Happened (London, 1939; collected edition, 1946), pp. 163–166. Reproduced with the permission of the Society of Authors, on behalf of the estate of Bernard Shaw.
Newton, aged 38, comes in from the garden, hatless, deep in calculation, his fists clenched, tapping his knuckles together to tick off the stages of the equation. He stumbles over the mat.
MRS BASHAM. Oh, do look where youre° going, Mr Newton. Someday youll walk into the river and drown yourself. I thought you were out at the university.
NEWTON. Now dont scold, Mrs Basham, dont scold. I forgot to go out. I thought of a way of making a calculation that has been puzzling me.
MRS BASHAM. And you have been sitting out there forgetting everything else since breakfast. However, since you have one of your calculating fits on I wonder would you mind doing a little sum for me to check the washing bill. How much is three times seven?
NEWTON. Three times seven? Oh, that is quite easy.
MRS BASHAM. I suppose it is to you, sir; but it beats me. At school I got as far as addition and subtraction; but I never could do multiplication or division.
NEWTON. Why, neither could I: I was too lazy. But they are quite unnecessary: addition and subtraction are quite sufficient. You add the logarithms of the numbers; and the antilogarithm of the sum of the two is the answer. Let me see: three times seven? The logarithm of three must be decimal four seven seven or thereabouts. The logarithm of seven is, say, decmial eight four five. That makes one decimal three two two, doesnt it? What’s the antilogarithm of one decimal three two two? Well, it must be less than twentytwo and more than twenty. You will be safe if you put it down as—
SALLY. Please, maam, Jack says it’s twentyone.
NEWTON. Extraordinary! Here was I blundering over this simple problem for a whole minute; and this uneducated fish hawker solves it in a flash! He is a better mathematician than I.
MRS BASHAM. This is our new maid from Woolsthorp, Mr Newton. You havent seen her before.
NEWTON. Havent I? I didnt notice it. [To Sally] Youre from Woolsthorp, are you? So am I. How old are you?
SALLY. Twentyfour, sir.
NEWTON. Twentyfour years. Eight thousand seven hundred and sixty days. Two hundred and ten thousand two hundred and forty hours. Twelve million six hundred and fourteen thousand, four hundred minutes. Seven hundred and fiftysix million eight hundred and sixtyfour thousand seconds. A long long life.
MRS BASHAM. Come now, Mr. Newton: you will turn the child’s head with your figures. What can one do in a second?
NEWTON. You can do, quite deliberately and intentionally, seven distinct actions in a second. How do you count seconds? Hackertybackertyone, Hackertybackertytwo, Hackertybackertythree and so on. You pronounce seven syllables in every second. Think of it! This young woman has had time to perform more than five thousand millions of considered and intentional actions in her lifetime. How many of them can you remember, Sally?
SALLY. Oh, sir, the only one I can remember was on my sixth birthday. My father gave me sixpence: a penny for every year.
NEWTON. Six from twentyfour is eighteen. He owes you one and sixpence. Remind me to give you one and sevenpence on your next birthday if you are a good girl. Now be off.
SALLY. Oh, thank you, sir. [She goes out].
NEWTON. My father, who died before I was born, was a wild, extravagant, weak man: so they tell me. I inherit his wildness, his extravagance, his weakness, in the shape of a craze for figures of which I am most heartily ashamed. There are so many more important things to be worked at: the transmutations of matter, the elixir of life, the magic of light and color, above all, the secret meaning of the Scriptures. And when I should be concentrating my mind on these I find myself wandering off into idle games of speculation about numbers in infinite series, and dividing curves into indivisibly short triangle bases. How silly! What a waste of time, priceless time!
MRS BASHAM. There is a Mr Rowley going to call on you at half past eleven.
NEWTON. Can I never be left alone? Who is Mr Rowley? What is Mr Rowley?
MRS BASHAM. Dressed like a nobleman. Very tall. Very dark. Keeps a lackey. Has a pack of dogs with him.
NEWTON. Oho! So that is who he is! They told me he wanted to see my telescope. Well, Mrs Basham, he is a person whose visit will be counted a great honor to us. But I must warn you that just as I have my terrible weakness for figures Mr Rowley has a very similar weakness for women; so you must keep Sally out of his way.
MRS BASHAM. Indeed! If he tries any of his tricks on Sally I shall see that he marries her.
NEWTON. He is married already. [He sits at the table.]
MRS BASHAM. Oh! That sort of man! The beast!
NEWTON. Shshsh! Not a word against him, on your life. He is privileged.
MRS BASHAM. He is a beast all the same!
NEWTON [opening the Bible] One of the beasts in the Book of Revelation, perhaps. But not a common beast.
MRS BASHAM. Fox the Quaker,° in his leather breeches, had the impudence to call.
NEWTON. [interested] George Fox? If he calls again I will see him. Those two men ought to meet.
MRS BASHAM. Those two men indeed! The honor of meeting you ought to be enough for them, I should think.
NEWTON. The honor of meeting me! Dont talk nonsense. They are great men in their very different ranks. I am nobody.
MRS BASHAM. You are the greatest man alive, sir. Mr Halley° told me so.
NEWTON. It was very wrong of Mr Halley to tell you anything of the sort. You must not mind what he says. He is always pestering me to publish my methods of calculation and to abandon my serious studies. Numbers! Numbers! Numbers! Sines, cosines, hypotenuses, fluxions, curves small enough to count as straight lines, distances between two points that are in the same place! Are these philosophy? Can they make a man great?
He is interrupted by Sally, who throws open the door and announces visitors.
SALLY. Mr Rowley and Mr Fox.
King Charles the Second, aged 50, appears at the door, but makes way for George Fox the Quaker, a big man with bright eyes and a powerful voice in reserve, aged 56. He is decently dressed but his garments are made of leather.
CHARLES. After you, Mr Fox. The spiritual powers before the temporal.
FOX. You are very civil, sir; and you speak very justly. I thank you [he passes in].
Sally, intensely impressed by Mr Rowley, goes out.
FOX. Am I addressing the philosopher Isaac Newton?
NEWTON. You are, sir.
Notes
youre: Shaw’s practice was not to mark contractions with apostrophes.
Fox the Quaker: George Fox (1624–1691), one of the founders of the Religious Society of Friends (Quakers).
Mr Halley: Edmond Halley (1656–1742), an early supporter of Newton.
Analysis Incarnate
During the twentieth century the discipline of the history of mathematics burgeoned, and several textbooks were produced which remain well-loved classics. One is this, Carl Boyer’s 1968 A History of Mathematics, which the author intended as a textbook in which the history of mathematics was presented “with fidelity, not only to mathematical structure and exactitude, but also to historical perspective and detail.” This passage about Leonhard Euler exemplifies the attractive blend which results.
Carl B. Boyer (1906–1976), A History of Mathematics (New York, 1968), pp. 481–488. Copyright © 1968 by John Wiley & Sons, Inc. Reproduced by permission of John Wiley & Sons, Inc.
The history of mathematics during the modern period is unlike that of antiquity or the medieval world in at least one respect: no national group remained the leader for any prolonged period. In ancient times Greece stood head and shoulders over all other peoples in mathematical achievement; during much of the Middle Ages the level of mathematics in the Arabic world was higher than elsewhere. From the Renaissance to the eighteenth century the center of mathematical activity had shifted repeatedly—from Germany to Italy to France to Holland to England. Had religious persecution not driven the Bernoulli family from Antwerp, Belgium might have had its turn; but the family emigrated to Basel, and as a result Switzerland was the birthplace of many of the leading figures in the mathematics of the early eighteenth century. We have already mentioned the work of four of the mathematicians of the Bernoulli clan, as well as that of Hermann, one of their Swiss protégés. But the most significant mathematician to come from Switzerland during that time—or any time—was Leonhard Euler (1707–1783), who was born at Basel.
Euler’s father was a clergyman who, like Jacques Bernoulli’s father, hoped that his son would enter the ministry. However, the young man studied under Jean Bernoulli and associated with his sons, Nicolaus and Daniel, and through them discovered his vocation. The elder Euler also was adept in mathematics, having been a pupil under Jacques Bernoulli, and helped to instruct the son in the elements of the subject, despite his hope that Leonhard would pursue a theological career. At all events, the young man was broadly trained, for to the study of mathematics he added theology, medicine, astronomy, physics, and oriental languages. This breadth stood him in good stead when in 1727 he heard from Russia that there was an opening in medicine in the St. Petersburg Academy, where the young Bernoullis had gone as professors of mathematics. This important institution had been established only a few years earlier by Catherine I along lines laid down by her late husband, Peter the Great, with the advice of Leibniz. On the recommendation of the Bernoullis, two of the brightest luminaries in the early days of the academy, Euler was called to be a member of the section on medicine and physiology; but on the very day that he arrived in Russia, Catherine died. The fledgling Academy very nearly succumbed with her, because the new rulers showed less sympathy for learned foreigners than had Peter and Catherine.
The Academy somehow managed to survive, and Euler, in 1730, found himself in the chair of natural philosophy rather than in the medical section. His friend Nicolaus Bernoulli had died, by drowning, in St. Petersburg the year before Euler arrived, and in 1733 Daniel Bernoulli left Russia to occupy the chair in mathematics at Basel. Thereupon Euler at the age of twenty-six became the Academy’s chief mathematician. He married and settled down to pursue in earnest mathematical research and raise a family that ultimately included thirteen children. The St. Petersburg Academy had established a research journal, the Comentarii Academiae Scientiarum Imperialis Petropolitanae, and almost from the start Euler contributed a spate of mathematical articles. The editor did not have to worry about a shortage of material as long as the pen of Euler was busy. It was said by the French academician François Arago that Euler could calculate without any apparent effort, “just as men breathe, as eagles sustain themselves in the air.” As a result, Euler composed mathematical memoirs while playing with his children. In 1735 he had lost the sight of his right eye—through overwork, it is said—but this misfortune in no way diminished the rate of output of his research. He is supposed to have said that his pencil seemed to surpass him in intelligence, so easily did memoirs flow, and he published more than 500 books and papers during his lifetime. For almost half a century after his death, works by Euler continued to appear in the publications of the St. Petersburg Academy. A bibliographical list of Euler’s works, including posthumous items, contains 886 entries; and it is estimated that his collected works, now being published under Swiss auspices, will run close to seventy-five substantial volumes. His mathematical research during his lifetime averaged about 800 pages a year; no mathematician has ever exceeded the output of this man whom Arago characterizes as “Analysis Incarnate.”
Euler early acquired an international reputation; even before leaving Basel he had received an honorable mention from the Parisian Académie des Sciences for an essay on the masting of ships. In later years he frequently entered essays in the contests set by the Académie, and twelve times he won the coveted biennial prize. The topics ranged widely, and on one occasion, in 1724, Euler shared with Maclaurin and Daniel Bernoulli a prize for an essay on the tides. (The Paris prize was won twice by Jean Bernoulli and ten times by Daniel Bernoulli.) Euler was never guilty of false pride, and he wrote works on all levels, including textbook material for use in the Russian schools. He generally wrote in Latin, and sometimes in French, although German was his native tongue. Euler had an unusual language facility, as one should expect of a person with a Swiss background. This was fortunate, for one of the distinguishing marks of eighteenth-century mathematics was the readiness with which scholars moved from one country to another, and here Euler encountered no language problems. In 1741 Euler was invited by Frederick the Great to join the Berlin Academy, and the invitation was accepted. (Jean and Daniel Bernoulli also were invited from Switzerland, but they declined.) Euler spent twenty-five years at Frederick’s court, but during this period he continued to receive a pension from Russia, and he submitted numerous papers to the St. Petersburg Academy, as well as to the Prussian Academy.
Euler’s stay at Berlin was not entirely happy, for Frederick preferred a scholar who scintillated, as did Voltaire. The monarch, who valued philosophers above geometers, referred to the unsophisticated Euler as a “mathematical cyclops,” and relationships at the court became intolerable for Euler. Catherine the Great was only too eager to have the prolific mathematician resume his place in the St. Petersburg Academy, and in 1788 Euler returned to Russia. During this year Euler learned that he was losing by cataract the sight of his remaining eye, and he prepared for ultimate blindness by practicing writing with chalk on a large slate and by dictating to his children. An operation was performed in 1771, and for a few days Euler saw once more; but success was short-lived and Euler spent almost all of the last seventeen years of his life in total darkness. Even this tragedy failed to stem the flood of his research and publication, which continued unabated until in 1783, at the age of seventy-six, he suddenly died while sipping tea and enjoying the company of one of his grandchildren.
From 1727 to 1783 the pen of Euler had been busy adding to knowledge in virtually every branch of pure and applied mathematics, from the most elementary to the most advanced. Moreover, in most respects Euler wrote in the language and notations we use today, for no other individual was so largely responsible for the form of college-level mathematics today as was Euler, the most successful notation-builder of all times. The definitive use of the Greek letter π for the ratio of circumference to diameter in a circle also is largely due to Euler, although a prior occurrence is found in 1706, the year before Euler was born—in the Synopsis Palmariorum Matheseos, or A New Introduction to the Mathematics, by William Jones (1675–1749). It was Euler’s adoption of the symbol π in 1737, and later in his many popular textbooks, that made it widely known and used.
It is not only in connection with designations for important numbers that today we use notations introduced by Euler. In geometry, algebra, trigonometry, and analysis we find ubiquitous use of Eulerian symbols, terminology, and ideas. The use of the small letters a, b, c for the sides of a triangle and of the corresponding capitals A, B, C for the opposite angles stems from Euler, as does the application of the letters r, R, and s for the radius of the inscribed and circumscribed circles and the semiperimeter of the triangle respectively. The beautiful formula 4rRs = abc relating the six lengths also is one of the many elementary results attributed to him, although equivalents of this result are implied by ancient geometry. The designation l x for logarithm of x, the use of the now-familiar ∑ to indicate a summation, and, perhaps most important of all, the notation f (x) for a function of x (used in the Petersburg Commentaries for 1734–1735) are other Eulerian notations related to ours. Our notations today are what they are more on account of Euler than of any other mathematician in history.
The first volume of the Introductio° is concerned from start to finish with infinite processes—infinite products and infinite continued fractions, as well as innumerable infinite series. In this respect the work is the natural generalization of the views of Newton, Leibniz, and the Bernoullis, all of whom were fond of infinite series. However, Euler was very incautious in his use of such series. Although upon occasion he warned against the risk in working with divergent series,° he himself used the binomial series 1/(1 − x) = 1 + x + x2 + x3 + . . . for values of x ≥ 1. In fact, by combining the two series x/(1 − x) = x + x2 + x3 + . . . and x/(x − 1) = 1 + 1/x + 1.x2 + . . . Euler concluded that . . . 1/x2 + 1/x + 1 + x + x2 + x3 + . . . = 0.
Despite his hardihood, through manipulations of infinite series Euler achieved results that had baffled his predecessors. Among these was the summation of the reciprocals of the perfect squares—1/12 + 1/22 + 1/32 + 1/42 + . . . . Oldenburg,° in a letter to Leibniz in 1673, had asked for the sum of this series, but Leibniz failed to answer; in 1689 Jacques Bernoulli had admitted his own inability to find the sum. Euler began with the familiar series sin z = z − z3/3! + z5/5! − z7/7! + . . . . Then sin z = 0 can be thought of as the infinite polynomial equation 0 = 1 − z2/3! + z4/5! − z6/7! + . . . (obtained by dividing through by z), or, if z2 is replaced by w, as the equation 0 = 1 − w/3! + w2/5! − w3/7! +. . . . From the theory of algebraic equations it is known that the sum of the reciprocals of the roots is the negative of the coefficient of the linear term—in this case 1/3!. Moreover, the roots of the equation in z are known to be π, 2π, 3π, and so on; hence the roots of the equation in w are π2, (2π)2, (3π)2, and so on. Therefore
Through this carefree application to polynomials of infinite degree of algebraic rules valid for the finite case Euler had achieved a result that had baffled the older Bernoulli brothers; Euler in later years repeatedly made discoveries in similar fashion. When Jean Bernoulli learned of Euler’s triumph, he wrote:
And so is satisfied the burning desire of my brother who, realizing that the investigation of the sum was more difficult than anyone would have thought, openly confessed that all his zeal had been mocked. If only my brother were alive now.
Euler’s summation of the reciprocals of the squares of the integers seems to date from about 1736, and it is likely that it was to Daniel Bernoulli that he promptly communicated the result. His interest in such series always was strong, and in later years he published the sums of the reciprocals of other powers of the integers. Using the cosine series instead of the sine series, Euler similarly found the result
Many of these results appeared also in the Introductio of 1748, including the sums of reciprocals of even powers from n = 2 through n = 26. The series of reciprocals of odd powers are so intractable that it still is not known whether or not the sum of the reciprocals of the cubes of the positive integers is a rational multiple of π3, whereas Euler knew that for the 26th power the sum of the reciprocals is
Notes
Introductio: Euler’s Introductio in analysin infinitorum (Lausanne, 1748), “Introduction to the analysis of infinities.”
divergent series: a series whose sum becomes larger without limit as the number of terms increases.
Oldenburg: Henry Oldenburg (c. 1619–1677) was secretary to the Royal Society from its inception until his death, developing a very wide scientific correspondence.
Hardy and Littlewood Rummage
Srinivasa Ramanujan (1887–1920), one of the great mathematicians of the twentieth century, came to prominence when, as an unknown young man, he wrote to the famous British pure mathematician G. H. Hardy (1877–1947) at Cambridge University. The papers he sent included profound new results in number theory and infinite series. Kanigel’s biography of Ramanujan, as well as giving a flavor of a modern tradition of mathematical biography which does not shy away from technical details, provides a vivid picture of one of the great dramatic set pieces in the history of mathematics: Hardy and his colleague Littlewood reading over the “Indian clerk’s” mathematics for the first time.
Robert Kanigel (dates unknown), The Man Who Knew Infinity (London, 1991), pp. 165–169. Reprinted by permission of the author.
About nine o’clock, they met, probably in Littlewood’s rooms, and soon the manuscript lay stretched out before them. Some of the formulas were familiar while others, Hardy would write, “seemed scarcely possible to believe.” Twenty years later, in a talk at Harvard University, he would invite his audience into the day that had so enriched his life. “I should like you to begin,” he said, “by trying to reconstruct the immediate reactions of an ordinary professional mathematician who receives a letter like this from an unknown Hindu clerk.” It was a mathematical audience, so Hardy introduced them to some of Ramanujan’s theorems. Like this one, on the bottom of page three:
The elongated S-like symbol appearing on the left-hand side of this equation, and in many other equations all through the letter, was an integral sign, a notation originating with Newton’s competitor Leibniz. An integral—the idea goes back to the Greeks—is essentially an addition, a sum, but one of a peculiar, precise, and, at first glance, infuriating kind.
Imagine cutting a hot dog into disclike slices. You could wind up with ten sections half an inch thick or a thousand paper-thin slices. But however thin you sliced it, you could, presumably, reassemble the pieces back into a hot dog. Integral calculus, as this branch of mathematics is called, adopts the strategy of taking an infinite number of infinitesimally thin slices and generating mathematical expressions for putting them back together again—for making them whole, or “integral.” This powerful additive process can be used to determine the drag force buffeting a wing as it slices through the air, or the gravitational effects of the earth on a man-made satellite, or indeed to solve any problem where the object is to piece together the contributions of many small influences.
You don’t need integral calculus to determine the area of a neat rectangular plot of farmland; you just multiply length times width. But you could use it. And you could use the same additive methods applicable to wings and satellites to calculate the area of an irregularly shaped plot where length-times-width won’t work. Furnish the function that mathematically defines its shape, and in principle you can get its area by “integrating” it—that is, by performing the additive process in a particular, precisely defined way.
Calculus books come littered with hundreds of ways to integrate functions. And yet, pick a function at random and chances are it can’t be integrated—at least not straightforwardly. With “definite integrals” like those Ramanujan offered in his letter to Hardy, however, you’re offered a back-door route to a solution.
A definite integral is “definite” in that you seek to integrate the function over a definite numerical range; the little numbers at top and bottom of the elongated S—the ∞ and 0 in Ramanujan’s equation—tell us what it is. (In other words, you mark off a piece of the farm plot whose area you want reckoned.) When you evaluate a definite integral, you don’t wind up with a general algebraic formula (as you do with indefinite integrals) but, in principle, an actual number. And sometimes, by applying the right mathematical tools, you can determine this number without integrating the function first—indeed, without being able to integrate it at all.
Broadly, this was what Ramanujan was doing in the theorem on page 3 of his letter to Hardy and all through the section labeled “IV. Theorems on Integrals.”
This particular integral, he was saying, could be represented in terms of gamma functions. (The gamma function is like the more familiar “factorial”—4!, read “four factorial,” = 4 × 3 × 2 × 1—except that it extends the idea to numbers other than integers.) Hardy figured he could prove this theorem. Later he tried, and succeeded, though it proved harder than he thought. None of Ramanujan’s other integrals were trifling exercises, either, and all would wind up, years later, the object of papers devoted to them. Still, Hardy judged, these were among the least impressive of Ramanujan’s results.
More so were the infinite series, two of which were:
The first wasn’t new to Hardy, who recognized it as going back to a mathematician named Bauer. The second seemed little different. To a layman, in fact, it and kindred ones in Ramanujan’s letter might seem scarcely intimidating at all; save for pi and the gamma function they were nothing but ordinary numbers. But Hardy and others would show how these series were derived from a class of functions called hypergeometric series first explored by Leonhard Euler and Carl Friedrich Gauss and as algebraically formidable as anybody could want.
Sometime before 1910, Hardy learned later, Ramanujan had come up with a general formula, later to be known as the Dougall–Ramanujan Identity, which under the right conditions could be made to fairly spew out infinite series. Just as an ordinary beer can is made in a huge factory, the ordinary numbers in Ramanujan’s series were the deceptively simple end product of complex mathematical machinery. Of course, on the day he got Ramanujan’s letter, Hardy knew nothing of this. He knew only that these series formulas weren’t what they seemed. Compared to the integrals, they struck him as “much more intriguing, and it soon became obvious that Ramanujan must possess much more general theorems and was keeping a great deal up his sleeve.”
Some theorems in Ramanujan’s letter, of course, did look comfortably familiar. For example,
If αβ = π2, then
Hardy had proved theorems like it, had even offered a similar one as a mathematical question in the Education Times fourteen years before. Some of Ramanujan’s formulas actually went back to the days of Laplace and Jacobi a century before. Of course, it was quite something that this Indian had rediscovered them.
But now, then, what was Hardy to make of this one, which he found on the last page of Ramanujan’s letter?
then
This was a relationship between continued fractions, in which the compressed notation for, say, the function u actually means this:
The publication of this result some years hence would set off a flurry of work by English mathematicians. Rogers would furnish one ten-page proof for it in 1921. Darling would explore it, too. In 1929, Watson would approach it from a different angle, trying to steer clear of the tricky mathematical terrain of theta functions. But in 1913, Hardy could make nothing of it, classing it among a group of Ramanujan’s theorems which, he would write, “defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written down by a mathematician of the highest class.” And then, in a classic Hardy flourish, he added: “They must be true because, if they were not true, no one would have the imagination to invent them.”
As Hardy and Littlewood probed the theorems before them, trying to make out what they said, where they fit into the mathematical canon, and how they might be proved or disproved, they began to reach a judgment. That the Indian’s mathematics was strange and individual had been evident from the start. But now they were coming to see his work as something more. It was not “individual” in the way a rebellious teenager tries to be, camouflaging his ordinariness behind bizarre dress or hair. It was much more. “There is always more in one of Ramanujan’s formulae than meets the eye, as anyone who sets to work to verify those which look the easiest will soon discover,” Hardy would write later. “In some the interest lies very deep, in others comparatively near the surface; but there is not one which is not curious and entertaining.”
The more they looked, the more dazzled they became. “Of the theorems sent without demonstration, by this clerk of whom we had never heard,” one of their Trinity colleauges, E.H. Neville, would later write, “not one could have been set in the most advanced mathematical examination in the world.” Hardy would rank Ramanujan’s letter as “certainly the most remarkable I have ever received,” its author “a mathematician of the highest quality, a man of altogether exceptional originality and power.”
And so, before midnight, Hardy and Littlewood began to appreciate that for the past three hours they had been rummaging through the papers of a mathematical genius.
1 By the “moment of the circumference” is meant the numerator of the fractional value of π. This is 22 in case π is taken as 
.
2 “Moment of diameter” means the denominator of the fractional value of π. In the case of , this is 7. That is, we have 7 × 120.
3 Thus far the solution is as follows: Let x = the diameter of the smallest circle, and y = the diameter of the largest circle. Then x + 5 is the diameter of the so-called “middle circle.”
Then
That the formula for C is correct is seen by substituting for 120 the difference in the areas as stated. We then have
or 22(y2 − x2 − 10x − 25 − 2x2 + 3x2 + 10x + 25) = C, or 22y2 = C, which is, as stated in the rule, “the product of the square of the diameter of the largest circle multiplied by the moment of circumference.”
4 I.e., 22x(x2 + 10x + 25) = D.
5 I.e., {C[4(x + 5) − x] − d}2 = X.
6 I.e., 22 · 22y2(x + 5)2[2(x + 5) + 2x]2. This is merely the second part of the preceding paragraph stated differently.
7 I.e., X = 222(3xy2 + 5y2 − x2)2, and this quantity equals 22y2(x + 5)2(6x + 20)2. Their difference is a sextic.
8 And noting that d2 − (
)z2 = (b + c)2.
9 This amounts to equating x2z − 4x2y to −[y2z + (4y2 + 2yz)x], and then squaring and canceling out like terms.