by E. T. Bell
In this Address he outlined what he hoped to do (he did it) in his lectures and researches.
“There are things called Algebraical Forms. Professor Cayley calls them Quantics. [Examples: ax2 + 2bxy + cy2, ax3 + 3bx2y + 3cxy2 + dy3; the numerical coefficients 1, 2, 1 in the first, 1, 3, 3, 1 in the second, are binomial coefficients, as in the third and fourth lines of Pascal’s triangle (Chapter 5); the next in order would be x4 + 4x3y + 6x2y2 + 4xy3 + y4]. They are not, properly speaking, Geometrical Forms, although capable, to some extent, of being embodied in them, but rather schemes of process, or of operations for forming, for calling into existence, as it were, Algebraic quantities.
“To every such Quantic is associated an infinite variety of other forms that may be regarded as engendered from and floating, like an atmosphere, around it—but infinite as were these derived existences, these emanations from the parent form, it is found that they admit of being obtained by composition, by mixture, so to say, of a certain limited number of fundamental forms, standard rays, as they might be termed in the Algebraic Spectrum of the Quantic to which they belong. And, as it is a leading pursuit of the Physicists of the present day [1877, and even today] to ascertain the fixed lines in the spectrum of every chemical substance, so it is the aim and object of a great school of mathematicians to make out the fundamental derived forms, the Covariants [that kind of ’invariant’ expression, already described, which involves both the variables and the coefficients of the form or quantic] and Invariants, as they are called, of these Quantics.”
To mathematical readers it will be evident that Sylvester is here giving a very beautiful analogy for the fundamental system and the syzygies for a given form; the nonmathematical reader may be recommended to reread the passage to catch the spirit of the algebra Sylvester is talking about, as the analogy is really a close one and as fine an example of “popularized” mathematics as one is likely to find in a year’s marching.
In a footnote Sylvester presently remarks “I have at present a class of from eight to ten students attending my lectures on the Modern Higher Algebra. One of them, a young engineer, engaged from eight in the morning to six at night in the duties of his office, with an interval of an hour and a half for his dinner or lectures, has furnished me with the best proof, and the best expressed, I have ever seen of what I call [a certain theorem]. . . .” Sylvester’s enthusiasm—he was past sixty—was that of a prophet inspiring others to see the promised land which he had discovered or was about to discover. Here was teaching at its best, at the only level, in fact, which justifies advanced teaching at all.
He had complimentary things to say (in footnotes) about the country of his adoption: “. . . I believe there is no nation in the world where ability with character counts for so much, and the mere possession of wealth (in spite of all that we hear about the Almighty dollar), for so little as in America. . . .”
He also tells how his dormant mathematical instincts were again aroused to full creative power. “But for the persistence of a student of this University [Johns Hopkins] in urging upon me his desire to study with me the modern Algebra, I should never have been led into this investigation. . . . He stuck with perfect respectfulness, but with invincible pertinacity, to his point. He would have the New Algebra (Heaven knows where he had heard about it, for it is almost unknown on this continent), that or nothing. I was obliged to yield, and what was the consequence? In trying to throw light on an obscure explanation in our text-book, my brain took fire, I plunged with re-quickened zeal into a subject which I had for years abandoned, and found food for thoughts which have engaged my attention for a considerable time past, and will probably occupy all my powers of contemplation advantageously for several months to come.”
Almost any public speech or longer paper of Sylvester’s contains much that is quotable about mathematics in addition to technicalities. A refreshing anthology for beginners and even for seasoned mathematicians could be gathered from the pages of his collected works. Probably no other mathematician has so transparently revealed his personality through his writings as has Sylvester. He liked meeting people and infecting them with his own contagious enthusiasm for mathematics. Thus he says, truly in his own case, “So long as a man remains a gregarious and sociable being, he cannot cut himself off from the gratification of the instinct of imparting what he is learning, of propagating through others the ideas and impressions seething in his own brain, without stunting and atrophying his moral nature and drying up the surest sources of his future intellectual replenishment.”
As a pendant to Cayley’s description of the extent of modern mathematics, we may hang Sylvester’s beside it. “I should be sorry to suppose that I was to be left for long in sole possession of so vast a field as is occupied by modern mathematics. Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined: it is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer’s gaze; it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness, the life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud and cell, and is forever ready to burst forth into new forms of vegetable and animal existence.”
In 1878 the American Journal of Mathematics was founded by Sylvester and placed under his editorship by Johns Hopkins University.
The Journal gave mathematics in the United States a tremendous urge in the right direction—research. Today it is still flourishing mathematically but hard pressed financially.
Two years later occurred one of the classic incidents in Sylvester’s career. We tell it in the words of Dr. Fabian Franklin, Sylvester’s successor in the chair of mathematics at Johns Hopkins for a few years and later editor of the Baltimore American, who was an eye (and ear) witness.
“He [Sylvester] made some excellent translations from Horace and from German poets, besides writing a number of pieces of original verse. The tours de force in the way of rhyming, which he performed while in Baltimore, were designed to illustrate the theories of versification of which he gives illustrations in his little book called ’The Laws of Verse.’ The reading of the Rosalind poem at the Peabody Institute was the occasion of an amusing exhibition of absence of mind. The poem consisted of no less than four hundred lines, all rhyming with the name Rosalind (the long and short sound of the i both being allowed). The audience quite filled the hall, and expected to find much interest or amusement in listening to this unique experiment in verse. But Professor Sylvester had found it necessary to write a large number of explanatory footnotes, and he announced that in order not to interrupt the poem he would read the footnotes in a body first. Nearly every footnote suggested some additional extempore remark, and the reader was so interested in each one that he was not in the least aware of the flight of time, or of the amusement of the audience. When he had dispatched the last of the notes, he looked up at the clock, and was horrified to find that he had kept the audience an hour and a half before beginning to read the poem they had come to hear. The astonishment on his face was answered by a burst of good-humored laughter from the audience; and then, after begging all his hearers to feel at perfect liberty to leave if they had engagements, he read the Rosalind poem.”
Doctor Franklin’s estimate of his teacher sums the man up admirably: “Sylvester was quick-tempered and impatient, but generous, charitable and tender-hearted. He was always extremely appreciative of the work of others and gave the warmest recognition to any talent or ability displayed by his pupils. He was capable of flying into a passion on sligh
t provocation, but he did not harbor resentment, and was always glad to forget the cause of quarrel at the earliest opportunity.”
Before taking up the thread of Cayley’s life where it crossed Sylvester’s again, we shall let the author of Rosalind describe how he made one of his most beautiful discoveries, that of what are called “canonical forms.” [This means merely the reduction of a given “quantic” to a “standard” form. For example ax2 + 2bxy + cy2 can be expressed as the sum of two squares, say X2 + Y2; ax5 + 5bx4y + 10cx3y2 + 10dx2y3 + 5exy4 + fy5 can be expressed as a sum of three fifth powers, X5 + Y5 + Z5.]
“I discovered and developed the whole theory of canonical binary forms for odd degrees, and, so far as yet made out, for even degreesI too, at one sitting, with a decanter of port wine to sustain nature’s flagging energies, in a back office in Lincoln’s Inn Fields. The work was done, and well done, but at the usual cost of racking thought—a brain on fire, and feet feeling, or feelingless, as if plunged in an ice-pail. That night we slept no more.” Experts agree that the symptoms are unmistakable. But it must have been ripe port, to judge by what Sylvester got out of the decanter.
* * *
Cayley and Sylvester came together again professionally when Cayley accepted an invitation to lecture at Johns Hopkins for half a year in 1881-82. He chose Abelian functions, in which he was researching at the time, as his topic, and the 67-year-old Sylvester faithfully attended every lecture of his famous friend. Sylvester had still several prolific years ahead of him, Cayley not quite so many.
We shall now briefly describe three of Cayley’s outstanding contributions to mathematics in addition to his work on the theory of algebraic invariants. It has already been mentioned that he invented the theory of matrices, the geometry of space of n dimensions, and that one of his ideas in geometry threw a new light (in Klein’s hands) on non-Euclidean geometry. We shall begin with the last because it is the hardest.
Desargues, Pascal, Poncelet, and others had created projective geometry (see chapters 5, 13) in which the object is to discover those properties of figures which are invariant under projection. Measurements—sizes of angles, lengths of lines—and theorems which depend upon measurement, as for example the Pythagorean proposition that the square on the longest side of a right triangle is equal to the sum of the squares on the other two sides, are not projective but metrical, and are not handled by ordinary projective geometry. It was one of Cayley’s greatest achievements in geometry to transcend the barrier which, before he leapt it, had separated projective from metrical properties of figures. From his higher point of view metrical geometry also became projective, and the great power and flexibility of projective methods were shown to be applicable, by the introduction of “imaginary” elements (for instance points whose coordinates involve ) to metrical properties. Anyone who has done any analytic geometry will recall that two circles intersect in four points, two of which are always “imaginary.” (There are cases of apparent exception, for example concentric circles, but this is close enough for our purpose.) The fundamental notions in metrical geometry are the distance between two points and the angle between two lines. Replacing the concept of distance by another, also involving “imaginary” elements, Cayley provided the means for unifying Euclidean geometry and the common non-Euclidean geometries into one comprehensive theory. Without the use of some algebra it is not feasible to give an intelligible account of how this may be done; it is sufficient for our purpose to have noted Cayley’s main advance of uniting projective and metrical geometry with its cognate unification of the other geometries just mentioned.
The matter of n-dimensional geometry when Cayley first put it out was much more mysterious than it seems to us today, accustomed as we are to the special case of four dimensions (space-time) in relativity. It is still sometimes said that a four-dimensional geometry is inconceivable to human beings. This is a superstition which was exploded long ago by Plücker; it is easy to put four-dimensional figures on a flat sheet of paper, and so far as geometry is concerned the whole of a four-dimensional “space” can be easily imagined. Consider first a rather unconventional three-dimensional space: all the circles that may be drawn in a plane. This “all” is a three-dimensional “space” for the simple reason that it takes precisely three numbers, or three coordinates, to individualize any one of the swarm of circles, namely two to fix the position of the center with reference to any arbitrarily given pair of axes, and one to give the length of the radius.
If the reader now wishes to visualize a four-dimensional space he may think of straight lines, instead of points, as the elements out of which our common “solid” space is built. Instead of our familiar solid space looking like an agglomeration of infinitely fine birdshot it now resembles a cosmic haystack of infinitely thin, infinitely long straight straws. That it is indeed four-dimensional in straight lines can be seen easily if we convince ourselves (as we may do) that precisely four numbers are necessary and sufficient to individualize a particular straw in our haystack. The “dimensionality” of a “space” can be anything we choose to make it, provided we suitably select the elements (points, circles, lines, etc.) out of which we build it. Of course if we take points as the elements out of which our space is to be constructed, nobody outside of a lunatic asylum has yet succeeded in visualizing a space of more than three dimensions.
Modern physics is fast teaching some to shed their belief in a mysterious “absolute space” over and above the mathematical “spaces”—like Euclid’s, for example—that were constructed by geometers to correlate their physical experiences. Geometry today is largely a matter of analysis, but the old terminology of “points,” “lines,” “distances,” and so on, is helpful in suggesting interesting things to do with our sets of coordinates. But it does not follow that these particular things are the most useful that might be done in analysis; it may turn out some day that all of them are comparative trivialities by more significant things which we, hidebound in outworn traditions, continue to do merely because we lack imagination.
If there is any mysterious virtue in talking about situations which arise in analysis as if we were back with Archimedes drawing diagrams in the dust, it has yet to be revealed. Pictures after all may be suitable only for very young children; Lagrange dispensed entirely with such infantile aids when he composed his analytical mechanics. Our propensity to “geometrize” our analysis may only be evidence that we have not yet grown up. Newton himself, it is known, first got his marvellous results analytically and re-clothed them in the demonstrations of an Apollonius partly because he knew that the multitude—mathematicians less gifted than himself—would believe a theorem true only if it were accompanied by a pretty picture and a stilted Euclidean demonstration, partly because he himself still lingered by preference in the pre-Cartesian twilight of geometry.
The last of Cayley’s great inventions which we have selected for mention is that of matrices and their algebra in its broad outline. The subject originated in a memoir of 1858 and grew directly out of simple observations on the way in which the transformations (linear) of the theory of algebraic invariants are combined. Glancing back at what was said on discriminants and their invariance we note the transformation (the arrow is here read “is replaced by”) Suppose we have two such transformations,
the second of which is to be applied to the x in the first. We get
Attending only to the coefficients in the three transformations we write them in square arrays, thus
and see that the result of performing the first two transformations successively could have been written down by the following rule of “multiplication,”
where the rows of the array on the right are obtained, in an obvious way, by applying the rows of the first array on the left onto the columns of the second. Such arrays (of any number of rows and columns) are called matrices. Their algebra follows from a few simple postulates, of which we need cite only the following. The matrices and are equal (by definition) when, and only when, a = A, b =
B, c = C, d = D. The sum of the two matrices just written is the matrix The result of multiplying by m (any number) is the matrix The rule for “multiplying,” X, (or “compounding”) matrices is as exemplified for above.
A distinctive feature of these rules is that multiplication is not commutative, except for special kinds of matrices. For example, by the rule we get
and the matrix on the right is not equal to that which arises from the multiplication
All this detail, particularly the last, has been given to illustrate a phenomenon of frequent occurrence in the history of mathematics: the necessary mathematical tools for scientific applications have often been invented decades before the science to which the mathematics is the key was imagined. The bizarre rule of “multiplication” for matrices, by which we get different results according to the order in which we do the multiplication (unlike common algebra where x × y is always equal to y × x), seems about as far from anything of scientific or practical use as anything could possibly be. Yet sixty seven years after Cayley invented it, Heisenberg in 1925 recognized in the algebra of matrices exactly the tool which he needed for his revolutionary work in quantum mechanics.
Cayley continued in creative activity up to the week of his death, which occurred after a long and painful illness, borne with resignation and unflinching courage, on January 26, 1895. To quote the closing sentences of Forsyth’s biography: “But he was more than a mathematician. With a singleness of aim, which Wordsworth would have chosen for his ’Happy Warrior,’ he persevered to the last in his nobly lived ideal. His life had a significant influence on those who knew him [Forsyth was a pupil of Cayley and became his successor at Cambridge]: they admired his character as much as they respected his genius: and they felt that, at his death, a great man had passed from the world.”