by E. T. Bell
Gauss spent the autumn of 1798 (he was then twenty one) in Brunswick, with occasional trips to Helmstedt, putting the finishing touches to the Disquisitiones. He had hoped for early publication, but the book was held up in the press owing to a Leipzig publisher’s difficulties till September, 1801. In gratitude for all that Ferdinand had done for him, Gauss dedicated his book to the Duke—“Serenissimo Principi ac Domino Carolo Guilielmo Ferdinando.”
If ever a generous patron deserved the homage of his protégé, Ferdinand deserved that of Gauss. When the young genius was worried ill about his future after leaving Göttingen—he tried unsuccessfully to get pupils—the Duke came to his rescue, paid for the printing of his doctoral dissertation (University of Helmstedt, 1799), and granted him a modest pension which would enable him to continue his scientific work unhampered by poverty. “Your kindness,” Gauss says in his dedication, “freed me from all other responsibilities and enabled me to assume this exclusively.”
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Before describing the Disquisitiones we shall glance at the dissertation for which Gauss was awarded his doctor’s degree in absentia by the University of Helmstedt in 1799: Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus revolvi posse (A New Proof that Every Rational Integral Function of One Variable Can Be Resolved into Real Factors of the First or Second Degree).
There is only one thing wrong with this landmark in algebra. The first two words in the title would imply that Gauss had merely added a new proof to others already known. He should have omitted “nova.” His was the first proof. (This assertion will be qualified later.) Some before him had published what they supposed were proofs of this theorem—usually called the fundamental theorem of algebra—but none had attained a proof. With his uncompromising demand for logical and mathematical rigor Gauss insisted upon a proof, and gave the first. Another, equivalent, statement of the theorem says that every algebraic equation in one unknown has a root, an assertion which beginners often take for granted as being true without having the remotest conception of what it means.
If a lunatic scribbles a jumble of mathematical symbols it does not follow that the writing means anything merely because to the inexpert eye it is indistinguishable from higher mathematics. It is just as doubtful whether the assertion that every algebraic equation has a root means anything until we say what sort of a root the equation has. Vaguely, we feel that a number will “satisfy” the equation but that half a pound of butter will not.
Gauss made this feeling precise by proving that all the roots of any algebraic equation are “numbers” of the form a + bi, where a, b are real numbers (the numbers that correspond to the distances, positive, zero, or negative, measured from a fixed point O on a given straight line, as on the x-axis in Descartes’ geometry), and i is the square root of −1. The new sort of “number” a + bi is called complex.
Incidentally, Gauss was one of the first to give a coherent account of complex numbers and to interpret them as labelling the points of a plane, as is done today in elementary textbooks on algebra.
The Cartesian coordinates of P are (a, b); the point P is also labelled a + bi. Thus to every point of the plane corresponds precisely one complex number; the numbers corresponding to the points on XOX’ are “real,” those on YOY’ “pure imaginary” (they are all of the type ic, where c is a real number).
The word “imaginary” is the great algebraical calamity, but it is too well established for mathematicians to eradicate. It should never have been used. Books on elementary algebra give a simple interpretation of imaginary numbers in terms of rotations. Thus if we interpret the multiplication i × c, where c is real, as a rotation about O of the segment Oc through one right angle, Oc is rotated onto 0Y; another multiplication by z, namely i × i X c, rotates Oc through another right angle, and hence the total effect is to rotate Oc through two right angles, so that +Oc becomes —Oc. As an operation, multiplication by i × i has the same effect as multiplication by −1; multiplication by i has the same effect as a rotation through a right angle, and these interpretations (as we have just seen) are consistent. If we like we may now write i × i = −1, in operations, or i2 = −1; so that the operation of rotation through a right angle is symbolized by
All this of course proves nothing. It is not meant to prove anything. There is nothing to be proved; we assign to the symbols and operations of algebra any meanings whatever that will lead to consistency. Although the interpretation by means of rotations proves nothing, it may suggest that there is no occasion for anyone to muddle himself into a state of mystic wonderment over nothing about the grossly misnamed “imaginaries.” For further details we must refer to almost any Schoolbook on elementary algebra.
Gauss thought the theorem that every algebraic equation has a root in the sense just explained so important that he gave four distinct proofs, the last when he was seventy years old. Today some would transfer the theorem from algebra (which restricts itself to processes that can be carried through in a finite number of steps) to analysis. Even Gauss assumed that the graph of a polynomial is a continuous curve and that if the polynomial is of odd degree the graph must cross the axis at least once. To any beginner in algebra this is obvious. But today it is not obvious without proof, and attempts to prove it again lead to the difficulties connected with continuity and the infinite. The roots of so simple an equation as x2 −2 = 0 cannot be computed exactly in any finite number of steps. More will be said about this when we come to Kronecker. We proceed now to the Disquisitiones Arithmeticae.
The Disquisitiones was the first of Gauss’ masterpieces and by some considered his greatest. It was his farewell to pure mathematics as an exclusive interest. After its publication in 1801 (Gauss was then twenty four), he broadened his activity to include astronomy, geodesy, and electromagnetism in both their mathematical and practical aspects. But arithmetic was his first love, and he regretted in later life that he had never found the time to write the second volume he had planned as a young man. The book is in seven “sections.” There was to have been an eighth, but this was omitted to keep down the cost of printing.
The opening sentence of the preface describes the general scope of the book. “The researches contained in this work appertain to that part of mathematics which is concerned with integral numbers, also fractions, surds [irrationals] being always excluded.”
The first three sections treat the theory of congruences and give in particular an exhaustive discussion of the binomial congruence xn m A (mod p), where the given integers n, A are arbitrary and p is prime; the unknown integer is x. This beautiful arithmetical theory has many resemblances to the corresponding algebraic theory of the binomial equation xn = A, but in its peculiarly arithmetical parts is incomparably richer and more difficult than the algebra which offers no analogies to the arithmetic.
In the fourth section Gauss develops the theory of quadratic residues. Here is found the first published proof of the law of quadratic reciprocity. The proof is by an amazing application of mathematical induction and is as tough a specimen of that ingenious logic as will be found anywhere.
With the fifth section the theory of binary quadratic forms from the arithmetical point of view enters, to be accompanied presently by a discussion of ternary quadratic forms which are found to be necessary for the completion of the binary theory. The law of quadratic reciprocity plays a fundamental part in these difficult enterprises. For the first forms named the general problem is to discuss the solution in integers x, y of the indeterminate equation
ax2 + 2bxy + cy2 = m,
where a, b, c, m are any given integers; for the second, the integer solutions x, y, z of
ax2 + 2bxy + cy2 + 2dxz + 2eyz +fz2 = m,
where a, 6, c, d, e, f, m, are any given integers, are the subject of investigation. An easy-looking but hard question in this field is to impose necessary and sufficient restrictions upon a, c, f, m which will ensure the existence of
a solution in integers x, y, z of the indeterminate equation
ax2 + cy2 + fz2 = m.
The sixth section applies the preceding theory to various special cases, for example the integer solutions x, y of mx2 + ny2 = A, where m, n, A are any given integers.
In the seventh and last section, which many consider the crown of the work, Gauss applies the preceding developments, particularly the theory of binomial congruences, to a wonderful discussion of the algebraic equation xn = 1, where n is any given integer, weaving together arithmetic, algebra, and geometry into one perfect pattern. The equation xn = 1 is the algebraic formulation of the geometric problem to construct a regular polygon of n sides, or to divide the circumference of a circle into n equal parts (consult any secondary text book on algebra or trigonometry); the arithmetical congruence xM ≡ 1 (mod p), where m, p are given integers, and p is prime, is the thread which runs through the algebra and the geometry and gives the pattern its simple meaning. This flawless work of art is accessible to any student who has had the usual algebra offered in school, but the Disquisitiones is not recommended for beginners (Gauss’ concise presentation has been reworked by later writers into a more readily assimilable form).
Many parts of all this had been done otherwise before—by Fermat, Euler, Lagrange, Legendre and others; but Gauss treated the whole from his individual point of view, added much of his own, and deduced the isolated results of his predecessors from his general formulations and solutions of the relevant problems. For example, Fermat’s beautiful result that every prime of the form 4n + 1 is a sum of two squares, and is such a sum in only one way, which Fermat proved by his difficult method of “infinite descent,” falls out naturally from Gauss’ general discussion of binary quadratic forms.
“The Disquisitiones Arithmeticae have passed into history,” Gauss said in his old age, and he was right. A new direction was given to the higher arithmetic with the publication of the Disquisitiones, and the theory of numbers, which in the seventeenth and eighteenth centuries had been a miscellaneous aggregation of disconnected special results, assumed coherence and rose to the dignity of a mathematical science on a par with algebra, analysis, and geometry.
The work itself has been called a “book of seven seals.” It is hard reading, even for experts, but the treasures it contains and (partly conceals) in its concise, synthetic demonstrations are now available to all who wish to share them, largely the result of the labors of Gauss’ friend and disciple Peter Gustav Lejeune Dirichlet (18051859), who first broke the seven seals.
Competent judges recognized the masterpiece for what it was immediately. LegendreIII at first may have been inclined to think that Gauss had done him but scant justice. But in the preface to the second edition of his own treatise on the theory of numbers (1808), which in large part was superseded by the Disquisitiones, he is enthusiastic. Lagrange also praised unstintedly. Writing to Gauss on May 31, 1804 he says “Your Disquisitiones have raised you at once to the rank of the first mathematicians, and I regard the last section as containing the most beautiful analytical discovery that has been made for a long time. . . . Believe, sir, that no one applauds your success more sincerely than I.”
Hampered by the classic perfection of its style the Disquisitiones was somewhat slow of assimilation, and when finally gifted young men began studying the work deeply they were unable to purchase copies, owing to the bankruptcy of a bookseller. Even Eisenstein, Gauss’ favorite disciple, never owned a copy. Dirichlet was more fortunate. His copy accompanied him on all his travels, and he slept with it under his pillow. Before going to bed he would struggle with some tough paragraph in the hope—frequently fulfilled—that he would wake up in the night to find that a re-reading made everything clear. To Dirichlet is due the marvellous theorem, mentioned in connection with Fermat, that every arithmetical progression
a, a + b, a + 2b, a + 3b, a + 4b, . . . ,
in which a, b are integers with no common divisor greater than 1, contains an infinity of primes. This was proved by analysis, in itself a miracle, for the theorem concerns integers, whereas analysis deals with the continuous, the non-integral.
Dirichlet did much more in mathematics than his amplification of the Disquisitiones, but we shall not have space to discuss his life. Neither shall we have space (unfortunately) for Eisenstein, one of the brilliant young men of the early nineteenth century who died before their time and, what is incomprehensible to most mathematicians, as the man of whom Gauss is reported to have said, “There have been but three epoch-making mathematicians, Archimedes, Newton, and Eisenstein.” If Gauss ever did say this (it is impossible to check) it deserves attention merely because he said it, and he was a man who did not speak hastily.
Before leaving this field of Gauss’ activities we may ask why he never tackled Fermat’s Last Theorem. He gives the answer himself. The Paris Academy in 1816 proposed the proof (or disproof) of the theorem as its prize problem for the period 1816-18. Writing from Bremen on March 7, 1816, Olbers tries to entice Gauss into competing: “It seems right to me, dear Gauss, that you should get busy about this.”
But “dear Gauss” resisted the tempter. Replying two weeks later he states his opinion of Fermat’s Last Theorem. “I am very much obliged for your news concerning the Paris prize. But I confess that Fermat’s Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.”
Gauss goes on to say that the question has induced him to recall some of his old ideas for a great extension of the higher arithmetic. This doubtless refers to the theory of algebraic numbers (described in later chapters) which Kummer, Dedekind, and Kronecker were to develop independently. But the theory Gauss has in mind is one of those things, he declares, where it is impossible to foresee what progress shall be made toward a distant goal that is only dimly seen through the darkness. For success in such a difficult search one’s lucky star must be in the ascendency, and Gauss’ circumstances are now such that, what with his numerous distracting occupations, he is unable to give himself up to such meditations, as he did “in the fortunate years 1796-1798 when I shaped the main points of the Disquisitiones Arithmeticae. Still I am convinced that if I am as lucky as I dare hope, and if I succeed in taking some of the principal steps in that theory, then Fermat’s Theorem will appear as only one of the least interesting corollaries.”
Probably all mathematicians today regret that Gauss was deflected from his march through the darkness by “a couple of clods of dirt which we call planets”—his own words—which shone out unexpectedly in the night sky and led him astray. Lesser mathematicians than Gauss—Laplace for instance—might have done all that Gauss did in computing the orbits of Ceres and Pallas, even if the problem was of a sort which Newton said belonged to the most difficult in mathematical astronomy. But the brilliant success of Gauss in these matters brought him instant recognition as the first mathematician in Europe and thereby won him a comfortable position where he could work in comparative peace; so perhaps those wretched lumps of dirt were after all his lucky stars.
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The second great stage in Gauss’ career began on the first day of the nineteenth century, also a red-letter day in the histories of philosophy and astronomy. Since 1781 when Sir William Herschel (17381822) discovered the planet Uranus, thus bringing the number of planets then known up to the philosophically satisfying seven, astronomers had been diligently searching the heavens for further members of the Sun’s family, whose existence was to be expected, according to Bode’s law, between the orbits of Mars and Jupiter. The search was fruitless till Giuseppe Piazzi (1746-1826) of Palermo, on the opening day of the nineteenth century, observed what he at first mistook for a small comet approaching the Sun, but which was presently recognized as a new planet—later named Ceres, the first of the swarm of minor planets known today.
By one of the most ironic verdicts ever delivered in the agelong litigation of fact
versus speculation, the discovery of Ceres coincided with the publication by the famous philosopher Georg Wilhelm Friedrich Hegel (1770–1831) of a sarcastic attack on astronomers for presuming to search for an eighth planet. Would they but pay some attention to philosophy, Hegel asserted, they must see immediately that there can be precisely seven planets, no more, no less. Their search therefore was a stupid waste of time. Doubtless this slight lapse on Hegel’s part has been satisfactorily explained by his disciples, but they have not yet talked away the hundreds of minor planets which mock his Jovian ban.
It will be of interest here to quote what Gauss thought of philosophers who busy themselves with scientific matters they have not understood. This holds in particular for philosophers who peck at the foundations of mathematics without having first sharpened their dull beaks on some hard mathematics. Conversely, it suggests why Bertrand A. W. Russell (1872- ), Alfred North Whitehead (1861) and David Hilbert (1862- ) in our own times have made outstanding contributions to the philosophy of mathematics: these men are mathematicians.
Writing to his friend Schumacher on November 1, 1844, Gauss says “You see the same sort of thing [mathematical incompetence] in the contemporary philosophers Schelling, Hegel, Nees von Essenbeck, and their followers; don’t they make your hair stand on end with their definitions? Read in the history of ancient philosophy what the big men of that day—Plato and others (I except Aristotle)—gave in the way of explanations. But even with Kant himself it is often not much better; in my opinion his distinction between analytic and synthetic propositions is one of those things that either run out in a triviality or are false.” When he wrote this (1844) Gauss had long been in full possession of non-Euclidean geometry, itself a sufficient refutation of some of the things Kant said about “space” and geometry, and he may have been unduly scornful.
