by E. T. Bell
The total curvature of any part of a surface bounded by an unlooped closed curve C is defined as follows. The normal to a surface at a given point is that straight line passing through the point which is perpendicular to the plane which touches the surface at the given point. At each point of C there is a normal to the surface. Imagine all these normals drawn. Now, from the center of a sphere (which may be anywhere with reference to the surface being considered), whose radius is equal to the unit length, imagine all the radii drawn which are parallel to the normals to C. These radii will cut out a curve, say C, on the sphere of unit radius. The area of that part of the spherical surface which is enclosed by C′ is defined to be the total curvature of the part of the given surface which is enclosed by C. A little visualization will show that this definition accords with common notions as required.
Another fundamental idea exploited by Gauss in his study of surfaces was that of parametric representation.
It requires two coordinates to specify a particular point on a plane. Likewise on the surface of a sphere, or on a spheroid like the Earth: the coordinates in this case may be thought of as latitude and longitude. This illustrates what is meant by a two-dimensional manifold. Generally: if precisely n numbers are both necessary and sufficient to specify (individualize) each particular member of a class of things (points, sounds, colors, lines, etc.,) the class is said to be an n-dimensional manifold. In such specifications it is agreed that only certain characteristics of the members of the class shall be assigned numbers. Thus if we consider only the pitch of sounds, we have a one-dimensional manifold, because one number, the frequency of the vibration corresponding to the sound, suffices to determine the pitch; if we add loudness—measured on some convenient scale—sounds are now a two-dimensional manifold, and so on. If now we regard a surface as being made up of points, we see that it is a two-dimensional manifold (of points). Using the language of geometry we find it convenient to speak of any two-dimensional manifold as a “surface,” and to apply to the manifold the reasoning of geometry—in the hope of finding something interesting.
The foregoing considerations lead to the parametric representation of surfaces. In Descartes’ geometry one equation between three coordinates represents a surface. Say the coordinates (Cartesian) are x, y, z. Instead of using a single equation connecting x, y, z to represent the surface, we now seek three:
x = f(u, v), y = g(u, v), z = h(u, v),
where f(u, v), g(u, v), h(u, v) are such functions (expressions) of the new variables u, v that when these variables are eliminated (got rid of—“put over the threshold,” literally) there results between x, y, z the equation of the surface. The elimination is possible, because two of the equations can be used to solve for the two unknowns u, v; the results can then be substituted in the third. For example, if
x = u + v, y = u—v, z = uv,
we get u = ½(x + y), v = ½(x—y) from the first two, and hence 4z = x2—y2 from the third. Now as the variables u, v independently run through any prescribed set of numbers, the functions f, g, h will take on numerical values and x, y, z will move on the surface whose equations are the three written above. The variables u, v are called the parameters for the surface, and the three equations x = f(u, v), y = g(u, v), z = h(u, v) their parametric equations. This method of representing surfaces has great advantages over the Cartesian when applied to the study of curvature and other properties of surfaces which vary rapidly from point to point.
Notice that the parametric representation is intrinsic; it refers to the surface itself for its coordinates, and not to an extrinsic, or extraneous, set of axes, not connected with the surface, as is the case in Descartes’ method. Observe also that the two parameters u, v immediately show up the two-dimensionality of the surface. Latitude and longitude on the earth are instances of these intrinsic, “natural” coordinates; it would be most awkward to have to do all our navigation with reference to three mutually perpendicular axes drawn through the center of the Earth, as would be required for Cartesian sailing.
Another advantage of the method is its easy generalization to a space of any number of dimensions. It suffices to increase the number of parameters and proceed as before. When we come to Riemann we shall see how these simple ideas led naturally to a generalization of the metric geometry of Pythagoras and Euclid. The foundations of this generalization were laid down by Gauss, but their importance for mathematics and physical science was not fully appreciated till our own century.
Geodetic researches also suggested to Gauss the development of another powerful method in geometry, that of conformal mapping. Before a map can be drawn, say of Greenland, it is necessary to determine what is to be preserved. Are distances to be distorted, as they are on Mercator’s projection, till Greenland assumes an exaggerated importance in comparison with North America? Or are distances to be preserved, so that one inch on the map, measured anywhere along the reference lines (say those for latitude and longitude) shall always correspond to the same distance measured on the surface of the earth? If so, one kind of mapping is demanded, and this kind will not preserve some other feature that we may wish to preserve; for example, if two roads on the earth intersect at a certain angle, the lines representing these roads on the map will intersect at a different angle. That kind of mapping which preserves angles is called conformal. In such mapping the theory of analytic functions of a complex variable, described earlier, is the most useful tool.
The whole subject of conformal mapping is of constant use in mathematical physics and its applications, for example in electrostatics, hydrodynamics and its offspring aerodynamics, in the last of which it plays a part in the theory of the airfoil.
Another field of geometry which Gauss cultivated with his usual thoroughness and success was that of the applicability of surfaces, in which it is required to determine what surfaces can be bent onto a given surface without stretching or tearing. Here again the methods Gauss invented were general and of wide utility.
To other departments of science Gauss contributed fundamental researches, for example in the mathematical theories of electromagnetism, including terrestrial magnetism, capillarity, the attraction of ellipsoids (the planets are special kinds of ellipsoids) where the law of attraction is the Newtonian, and dioptrics, especially concerning systems of lenses. The last gave him an opportunity to apply some of the purely abstract technique (continued fractions) he had developed as a young man to satisfy his curiosity in the theory of numbers.
Gauss not only mathematicized sublimely about all these things; he used his hands and his eyes, and was an extremely accurate observer. Many of the specific theorems he discovered, particularly in his researches on electromagnetism and the theory of attraction, have become part of the indispensable stock in trade of all who work seriously in physical science. For many years Gauss, aided by his friend Weber, sought a satisfying theory for all electromagnetic phenomena. Failing to find one that he considered satisfactory he abandoned his attempt. Had he found Clerk Maxwell’s (1831–1879) equations of the electromagnetic field he might have been satisfied.
To conclude this long but still far from complete list of the great things that earned Gauss the undisputed title of Prince of Mathematicians we must allude to a subject on which he published nothing beyond a passing mention in his dissertation of 1799, but which he predicted would become one of the chief concerns of mathematics—analysis situs. A technical definition of what this means is impossible here (it requires the notion of a continuous group), but some hint of the type of problem with which the subject deals can be gathered from a simple instance. Any sort of a knot is tied in a string, and the ends of the string are then tied together. A “simple” knot is easily distinguishable by eye from a “complicated” one, but how are we to give an exact, mathematical specification of the difference between the two? And how are we to classify knots mathematically? Although he published nothing on this, Gauss had made a beginning, as was discovered in his posthumous papers. Another type of probl
em in this subject is to determine the least number of cuts on a given surface which will enable us to flatten the surface out on a plane. For a conical surface one cut suffices; for an anchor ring, two; for a sphere, no finite number of cuts suffices if no stretching is permitted.
These examples may suggest that the whole subject is trivial. But if it had been, Gauss would not have attached the extraordinary importance to it that he did. His prediction of its fundamental character has been fulfilled in our own generation. Today a vigorous school (including many Americans—J. W. Alexander, S. Lefschetz, O. Veblen, among others) is finding that analysis situs, or the “geometry of position” as it used sometimes to be called, has far-reaching ramifications in both geometry and analysis. What a pity it seems to us now that Gauss could not have stolen a year or two from Ceres to organize his thoughts on this vast theory which was to become the dream of his old age and a reality of our own young age.
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His last years were full of honor, but he was not as happy as he had earned the right to be. As powerful of mind and as prolifically inventive as he had ever been, Gauss was not eager for rest when the first symptoms of his last illness appeared some months before his death.
A narrow escape from a violent death had made him more reserved than ever, and he could not bring himself to speak of the sudden passing of a friend. For the first time in more than twenty years he had left Göttingen on June 16, 1854, to see the railway under construction between his town and Cassel. Gauss had always taken a keen interest in the construction and operation of railroads; now he would see one being built. The horses bolted; he was thrown from his carriage, unhurt, but badly shocked. He recovered, and had the pleasure of witnessing the opening ceremonies when the railway reached Göttingen on July 31, 1854. It was his last day of comfort.
With the opening of the new year he began to suffer greatly from an enlarged heart and shortness of breath, and symptoms of dropsy appeared. Nevertheless he worked when he could, although his hand cramped and his beautifully clear writing broke at last. The last letter he wrote was to Sir David Brewster on the discovery of the electric telegraph.
Fully conscious almost to the end he died peacefully, after a severe struggle to live, early on the morning of February 23, 1855, in his seventy eighth year. He lives everywhere in mathematics.
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I. The legend of Gauss’ relations to his parents has still to be authenticated. Although, as will be seen later, the mother stood by her son, the father opposed him; and, as was customary then (usually, also, now), in a German household, the father had the last word.—I allude later to legends from living persons who had known members of the Gauss family, particularly in respect to Gauss’ treatment of his sons. These allusions refer to first-hand evidence; but I do not vouch for them, as the people were very old.
II. Shakespeare’s King Lear, Act I, Scene II, 1-2, with the essential change of “laws” for “law.”
III. Adrien-Marie Legendre (1752-1833). Considerations of space preclude an account of his life; much of his best work was absorbed or circumvented by younger mathematicians.
IV. “Came from” is right. When the sagacious Nazis expelled Fräulein Noether from Germany because she was a Jewess, Bryn Mawr College, Pennsylvania, took her in. She was the most creative abstract algebraist in the world. In less than a week of the new German enlightenment, Göttingen lost the liberality which Gauss cherished and which he strove all his life to maintain.
CHAPTER FIFTEEN
Mathematics and Windmills
CAUCHY
A man may say even his pater noster out of turn.—SPANISH PROVERB
IN THE FIRST THREE DECADES of the nineteenth century mathematics quite suddenly became something noticeably different from what it had been in the heroic post-Newtonian age of the eighteenth. The change was in the direction of greater rigor in demonstration following an unprecedented generality and freedom of inventiveness. Something of the same sort is plainly visible again today, and he would be a rash prophet who would venture to forecast what mathematics will be like three-quarters of a century hence.
At the beginning of the nineteenth century only Gauss had any inkling of what was so soon to come, but his Newtonian reserve held him back from telling Lagrange, Laplace, and Legendre what he foresaw. Although the great French mathematicians lived well into the first third of the nineteenth century much of their work now appears to have been preparatory. Lagrange in the theory of equations prepared the way for Abel and Galois; Laplace, with his work on the differential equations of Newtonian astronomy—including the theory of gravitation—hinted at the phenomenal development of mathematical physics in the nineteenth century; while Legendre’s researches in the integral calculus suggested to Abel and Jacobi one of the most fertile fields of investigation analysis has ever acquired. Lagrange’s analytical mechanics is still modern; but even it was to receive magnificent additions at the hands of Hamilton and Jacobi and, later, Poincaré. Lagrange’s work in the calculus of variations was also to remain classic and useful, but again the work of Weierstrass gave it a new direction under the rigorous, inventive spirit of the latter half of the nineteenth century, and this in its turn has been amplified and renovated in our own times (American and Italian mathematicians taking a leading part in the development).
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Augustin-Louis Cauchy, the first of the great French mathematcians whose thought belongs definitely to the modern age, was born in Paris on August 21, 1789—a little less than six weeks after the fall of the Bastille. A child of the Revolution, he paid his tax to liberty and equality by growing up with an undernourished body. It was only by the diplomacy and good sense of his father that Cauchy survived at all in the midst of semi-starvation. Having outlived the Terror, he graduated from the Polytechnique into the service of Napoleon. After the downfall of the Napoleonic order Cauchy got his full share of deprivations from revolutions and counter-revolutions, and in a measure his work was affected by the social unrest of his times. If revolutions and the like do affect a scientific man’s work, Cauchy should be the prize laboratory specimen for proving the fact. He had an extraordinary fertility in mathematical inventiveness and a fecundity that has been surpassed only twice—by Euler and Cayley. His work, like his times, was revolutionary.
Modern mathematics is indebted to Cauchy for two of its major interests, each of which marks a sharp break with the mathematics of the eighteenth century. The first was the introduction of rigor into mathematical analysis. It is difficult to find an adequate simile for the magnitude of this advance; perhaps the following will do. Suppose that for centuries an entire people has been worshipping false gods and that suddenly their error is revealed to them. Before the introduction of rigor mathematical analysis was a whole pantheon of false gods. In this Cauchy was one of the great pioneers with Gauss and Abel. Gauss might have taken the lead long before Cauchy entered the field, but did not, and it was Cauchy’s habit of rapid publication and his gift for effective teaching which really got rigor in mathematical analysis accepted.
The second thing of fundamental importance which Cauchy added to mathematics was on the opposite side—the combinatorial. Seizing on the heart of Lagrange’s method in the theory of equations, Cauchy made it abstract and began the systematic creation of the theory of groups. The nature of this will be described later; for the moment we note only the modernity of Cauchy’s outlook.
Without enquiring whether the thing he invented had any application or not, even to other branches of mathematics, Cauchy developed it on its own merits as an abstract system. His predecessors, with the exception of the universal Euler who was as willing to write a memoir on a puzzle in numbers as on hydraulics or the “system of the world.” had found their inspiration growing out of the applications of mathematics. This statement of course has numerous exceptions, notably in arithmetic; but before the time of Cauchy few if any sought profitable discoveries in the mere manipulations of algebra. Cauchy looked deeper, saw the
operations and their laws of combination beneath the symmetries of algebraic formulas, isolated them, and was led to the theory of groups. Today this elementary yet intricate theory is of fundamental importance in many fields of pure and applied mathematics, from the theory of algebraic equations to geometry and the theory of atomic structure. It is at the bottom of the geometry of crystals, to mention but one of its applications. Its later developments (on the analytical side) extend far into higher mechanics and the modern theory of differential equations.
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Cauchy’s life and character affect us like poor Don Quixote’s—we sometimes do not know whether to laugh or to cry, and compromise by swearing. His father, Louis-François, was a paragon of virtue and piety, both excellent things in their way, but easily overdone. Heaven only knows how Cauchy senior escaped the guillotine; for he was a parliamentary lawyer, a cultured gentleman, an accomplished classical and biblical scholar, a bigoted Catholic, and a lieutenant of police in Paris when the Bastille fell. Two years before the Revolution broke he had married Marie-Madeleine Desestre, an excellent, not very intelligent woman who, like himself, was also a bigoted Catholic.
Augustin was the eldest of six children (two sons, four daughters). From his parents Cauchy inherited and acquired all the estimable qualities which make their lives read like one of those charming love stories, insipid as stewed cucumbers, concocted for French schoolgirls under sixteen, in which the hero and heroine are as pure and sexless as God’s holy angels. With parents such as his it was perhaps natural that Cauchy should have grown up to be the obstinate Quixote of French Catholicism in the 18S0’s and 1840’s when the Church was on the defensive. He suffered for his religion, and for that he deserves respect (possibly even if he was the smug hypocrite his colleagues accused him of being), but he also richly deserved to suffer on more than one occasion. His everlasting preaching about the beauty of holiness put peoples’ backs up and engendered an opposition to his pious schemes which they did not always deserve. Abel, himself the son of a minister and a decent enough Christian, expressed the general disgust which some of Cauchy’s antics inspired when he wrote home, “Cauchy is a bigoted Catholic—a strange thing for a man of science.” The emphasis of course is on “bigoted,” not on the word it qualifies. Two of the finest characters and greatest mathematicians we shall meet, Weierstrass and Hermite, were Catholics. They were devout but not bigoted.