by E. T. Bell
To close this inadequate description of the kind of weapons that Poncelet used we shall mention the extremely fruitful “principle of duality.” For simplicity we consider only how the principle operates in plane geometry.
Note first that any continuous curve may be regarded in either of two ways: either as being generated by the motion of a point, or as being swept out by the turning motion of a straight line. To see the latter, imagine the tangent line drawn at each point of the curve. Thus points and lines are intimately and reciprocally associated with respect to the curve: through every point of the curve there is a line of the curve; on every line of the curve there is a point of the curve. Instead of “through” in the preceding sentence, write “on.” Then the two assertions separated by “;” after the “:” are identical except that the words “point” and “line” are interchanged.
As a matter of terminology we say that a line (straight or curved) is on a point if the line passes through the point, and we note that if a line is on a point, then the point is on the line, and conversely. To make this correspondence universal we “adjoin” to the usual plane in which Euclidean geometry (common school geometry) is valid, a so-called metric plane, “ideal elements” of the kind already described. The result of this adjunction is a projective plane: a projective plane consists of all the ordinary points and straight lines of a metric plane and, in addition, of a set of ideal points all of which are assumed to lie on one ideal line and such that one such ideal point lies on every ordinary line.II
In Euclidean language we would say that two parallel lines have the same direction; in projective phraseology this becomes “two parallel lines have the same ideal point.” Again, in the old, if two or more lines have the same direction, they are parallel; in the new, if two or more lines have the same ideal point they are parallel. Every straight line in the projective plane is conceived of as having on it one ideal point (“at infinity”); all the ideal points are thought of as making up one ideal line, “the line at infinity.”
The purpose of these conceptions is to avoid the exceptional statements of Euclidean geometry necessitated by the postulated existence of parallels. This has already been commented on in connection with Poncelet’s formulation of the principle of continuity.
With these preliminaries the principle of duality in plane geometry can now be stated: All the propositions of plane projective geometry occur in dual pairs which are such that, from either proposition of a particular pair another can be immediately inferred by interchanging the parts played by the words point and line.
In his projective geometry Poncelet exploited this principle to the limit. Opening almost any book on projective geometry at random we note pages of propositions printed in double columns, a device introduced by Poncelet. Corresponding propositions in the two columns are duals of one another; if either has been proved, a proof of the other is superfluous, as implied by the principle of duality. Thus geometry at one stroke is doubled in extent with no expenditure of extra labor. As a specimen of dual propositions we give the following pair.
Two distinct points are on one, and only one, line.
Two distinct lines are on one, and only one, point.
It may be granted that this is not very exciting. The mountain has labored and brought forth a mouse. Can it do any better?
The proposition in the left-hand column (page 217) is Pascal’s concerning his Hexagrammum Mysticum which we have already seen; that on the right is Brianchon’s theorem, which was discovered by means of the principle of duality. Brianchon (1785-1864) discovered his theorem while he was a student at the École Polytechnique; it was printed in the Journal of that school in 1806. The figures for the two propositions look nothing alike. This may indicate the power of the methods used by Poncelet.
Brianchon’s discovery was the one which put the principle of duality on the map of geometry. Far more spectacular examples of the power of the principle will be found in any textbook on projective geometry, particularly in the extension of the principle to ordinary three-dimensional space. In this extension the parts played by the words point and plane are interchangeable; straight line stays as it was.
If A, B, C, D, E, F are any points on a conic section, the points of intersection of the pairs of lines AB and DE, BC and EF, CD and FA are on a straight line; and conversely.
If A, B, C, D, E, F are tangent straight lines on a conic section, the lines joining the pairs of intersections of A with B and D with E, B with C and E with F, C with D and F with A, meet in one point; and conversely.
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The conspicuous beauty of projective geometry and the supple elegance of its demonstrations made it a favorite study with the geometers of the nineteenth century. Able men swarmed into the new goldfield and quickly stripped it of its more accessible treasures. Today the majority of experts seem to agree that the subject is worked out so far as it is of interest to professional mathematicians. However, it is conceivable that there may yet be something in it as obvious as the principle of duality which has been overlooked. In any event it is an easy subject to acquire and one of fascinating delight to amateurs and even to professionals at some stage of their careers. Unlike some other fields of mathematics, projective geometry has been blessed with many excellent textbooks and treatises, some of them by master geometers, including Poncelet himself.
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I. In what precedes the tangents are real (visible) if the point P lies outside the circles; if P is inside, the tangents are. “imaginary.”
II. This definition, and others of a similar character given presently, is taken from Projective Geometry (Chicago, 19S0) by the late John Wesley Young. This little book is comprehensible to anyone who has had an ordinary school course in common geometry.
CHAPTER FOURTEEN
The Prince of Mathematicians
GAUSS
The further elaboration and development of systematic arithmetic, like nearly everything else which the mathematics of our [nineteenth] century has produced in the way of original scientific ideas, is knit to Gauss.—LEOPOLD KRONECKER
ARCHIMEDES, NEWTON, AND GAUSS, these three, are in a class by themselves among the great mathematicians, and it is not for ordinary mortals to attempt to range them in order of merit. All three started tidal waves in both pure and applied mathematics: Archimedes esteemed his pure mathematics more highly than its applications; Newton appears to have found the chief justification for his mathematical inventions in the scientific uses to which he put them, while Gauss declared that it was all one to him whether he worked on the pure or the applied side. Nevertheless Gauss crowned the higher arithmetic, in his day the least practical of mathematical studies, the Queen of all.
The lineage of Gauss, Prince of Mathematicians, was anything but royal. The son of poor parents, he was born in a miserable cottage at Brunswick (Braunschweig), Germany, on April 30, 1777. His paternal grandfather was a poor peasant. In 1740 this grandfather settled in Brunswick, where he drudged out a meager existence as a gardener. The second of his three sons, Gerhard Diederich, born in 1744, became the father of Gauss. Beyond that unique honor Gerhard’s life of hard labor as a gardener, canal tender, and bricklayer was without distinction of any kind.
The picture we get of Gauss’ father is that of an upright, scrupulously honest, uncouth man whose harshness to his sons sometimes bordered on brutality. His speech was rough and his hand heavy. Honesty and persistence gradually won him some measure of comfort, but his circumstances were never easy. It is not surprising that such a man did everything in his power to thwart his young son and prevent him from acquiring an education suited to his abilities. Had the father prevailed, the gifted boy would have followed one of the family trades, and it was only by a series of happy accidents that Gauss was saved from becoming a gardener or a bricklayer. As a child he was respectful and obedient, and although he never criticized his poor father in later life, he made it plain that he had never felt any real affection for him. Gerhard died in 180
6. By that time the son he had done his best to discourage had accomplished immortal work.
On his mother’s side Gauss was indeed fortunate. Dorothea Benz’s father was a stonecutter who died at the age of thirty of tuberculosis, the result of unsanitary working conditions in his trade, leaving two children, Dorothea and her younger brother Friederich.
Here the line of descent of Gauss’ genius becomes evident. Condemned by economic disabilities to the trade of weaving, Friederich was a highly intelligent, genial man whose keen and restless mind foraged for itself in fields far from his livelihood. In his trade Friederich quickly made a reputation as a weaver of the finest damasks, an art which he mastered wholly by himself. Finding a kindred mind in his sister’s child, the clever uncle Friederich sharpened his wits on those of the young genius and did what he could to rouse the boy’s quick logic by his own quizzical observations and somewhat mocking philosophy of life.
Friederich knew what he was doing; Gauss at the time probably did not. But Gauss had a photographic memory which retained the impressions of his infancy and childhood unblurred to his dying day. Looking back as a grown man on what Friederich had done for him, and remembering the prolific mind which a premature death had robbed of its chance of fruition, Gauss lamented that “a born genius was lost in him.”
Dorothea moved to Brunswick in 1769. At the age of thirty four (in 1776) she married Gauss’ father. The following year her son was born. His full baptismal name was Johann Friederich Carl Gauss. In later life he signed his masterpieces simply Carl Friedrich Gauss. If a great genius was lost in Friederich Benz his name survives in that of his grateful nephew.
Gauss’ mother was a forthright woman of strong character, sharp intellect, and humorous good sense. Her son was her pride from the day of his birth to her own death at the age of ninety seven. When the “wonder child” of two, whose astounding intelligence impressed all who watched his phenomenal development as something not of this earth, maintained and even surpassed the promise of his infancy as he grew to boyhood, Dorothea Gauss took her boy’s part and defeated her obstinate husband in his campaign to keep his son as ignorant as himself.
Dorothea hoped and expected great things of her son. That she may sometimes have doubted whether her dreams were to be realized is shown by her hesitant questioning of those in a position to judge her son’s abilities. Thus, when Gauss was nineteen, she asked his mathematical friend Wolfgang Bolyai whether Gauss would ever amount to anything. When Bolyai exclaimed “The greatest mathematician in Europe!” she burst into tears.
The last twenty two years of her life were spent in her son’s house, and for the last four she was totally blind. Gauss himself cared little if anything for fame; his triumphs were his mother’s life.I There was always the completest understanding between them, and Gauss repaid her courageous protection of his early years by giving her a serene old age. When she went blind he would allow no one but himself to wait on her, and he nursed her in her long last illness. She died on April 19, 1839.
Of the many accidents which might have robbed Archimedes and Newton of their mathematical peer, Gauss himself recalled one from his earliest childhood. A spring freshet had filled the canal which ran by the family cottage to overflowing. Playing near the water, Gauss was swept in and nearly drowned. But for the lucky chance that a laborer happened to be about his life would have ended then and there.
In all the history of mathematics there is nothing approaching the precocity of Gauss as a child. It is not known when Archimedes first gave evidence of genius. Newton’s earliest manifestations of the highest mathematical talent may well have passed unnoticed. Although it seems incredible, Gauss showed his caliber before he was three years old.
One Saturday Gerhard Gauss was making out the weekly payroll for the laborers under his charge, unaware that his young son was following the proceedings with critical attention. Coming to the end of his long computations, Gerhard was startled to hear the little boy pipe up, “Father, the reckoning is wrong, it should be . . . .” A check of the account showed that the figure named by Gauss was correct.
Before this the boy had teased the pronunciations of the letters of the alphabet out of his parents and their friends and had taught himself to read. Nobody had shown him anything about arithmetic, although presumably he had picked up the meanings of the digits 1, 2, . . . along with the alphabet. In later life he loved to joke that he knew how to reckon before he could talk. A prodigious power for involved mental calculations remained with him all his life.
Shortly after his seventh birthday Gauss entered his first school, a squalid relic of the Middle Ages run by a virile brute, one Büttner, whose idea of teaching the hundred or so boys in his charge was to thrash them into such a state of terrified stupidity that they forgot their own names. More of the good old days for which sentimental reactionaries long. It was in this hell-hole that Gauss found his fortune.
Nothing extraordinary happened during the first two years. Then, in his tenth year, Gauss was admitted to the class in arithmetic. As it was the beginning class none of the boys had ever heard of an arithmetical progression. It was easy then for the heroic Büttner to give out a long problem in addition whose answer he could find by a formula in a few seconds. The problem was of the following sort, 81297 + 81495 + 81693 + . . . + 100899, where the step from one number to the next is the same all along (here 198), and a given number of terms (here 100) are to be added.
It was the custom of the school for the boy who first got the answer to lay his slate on the table; the next laid his slate on top of the first, and so on. Büttner had barely finished stating the problem when Gauss flung his slate on the table: “There it lies,” he said—“Ligget se’ ” in his peasant dialect. Then, for the ensuing hour, while the other boys toiled, he sat with his hands folded, favored now and then by a sarcastic glance from Büttner, who imagined the youngest pupil in the class was just another blockhead. At the end of the period Büttner looked over the slates. On Gauss’ slate there appeared but a single number. To the end of his days Gauss loved to tell how the one number he had written was the correct answer and how all the others were wrong. Gauss had not been shown the trick for doing such problems rapidly. It is very ordinary once it is known, but for a boy of ten to find it instantaneously by himself is not so ordinary.
This opened the door through which Gauss passed on to immortality. Büttner was so astonished at what the boy of ten had done without instruction that he promptly redeemed himself and to at least one of his pupils became a humane teacher. Out of his own pocket he paid for the best textbook on arithmetic obtainable and presented it to Gauss. The boy flashed through the book. “He is beyond me,” Büttner said; “I can teach him nothing more.”
By himself Büttner could probably not have done much for the young genius. But by a lucky chance the schoolmaster had an assistant, Johann Martin Bartels (1769-1836), a young man with a passion for mathematics, whose duty it was to help the beginners in writing and cut their quill pens for them. Between the assistant of seventeen and the pupil of ten there sprang up a warm friendship which lasted out Bartels’ life. They studied together, helping one another over difficulties and amplifying the proofs in their common textbook on algebra and the rudiments of analysis.
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Out of this early work developed one of the dominating interests of Gauss’ career. He quickly mastered the binomial theorem,
in which n is not necessarily a positive integer, but may be any number. If n is not a positive integer, the series on the right is infinite (nonterminating), and in order to state when this series is actually equal to (1 + x)n, it is mandatory to investigate what restrictions must be imposed upon x and n in order that the infinite series shall converge to a definite, finite limit. Thus, if x = −2, and n = −1, we get the absurdity that (1 −2)–1, which is ( −1)–1 or 1/(–1), or finally –1, is equal to 1 + 2 + 22 + 23 + . . . and so on ad infinitum; that is, −1 is equal to the “infinite number” 1 + 2 + 4 + 8 +
. . ., which is nonsense.
Before young Gauss asked himself whether infinite series converge and really do enable us to calculate the mathematical expressions (functions) they are used to represent, the older analysts had not seriously troubled themselves to explain the mysteries (and nonsense) arising from an uncritical use of infinite processes. Gauss’ early encounter with the binomial theorem inspired him to some of his greatest work and he became the first of the “rigorists.” A proof of the binomial theorem when n is not an integer greater than zero is even today beyond the range of an elementary textbook. Dissatisfied with what he and Bartels found in their book, Gauss made a proof. This initiated him to mathematical analysis. The very essence of analysis is the correct use of infinite processes.
The work thus well begun was to change the whole aspect of mathematics. Newton, Leibniz, Euler, Lagrange, Laplace—all great analysts for their times—had practically no conception of what is now acceptable as a proof involving infinite processes. The first to see clearly that a “proof” which may lead to absurdities like “minus 1 equals infinity” is no proof at all, was Gauss. Even if in some cases a formula gives consistent results, it has no place in mathematics until the precise conditions under which it will continue to yield consistency have been determined.
The rigor which Gauss imposed on analysis gradually overshadowed the whole of mathematics, both in his own habits and in those of his contemporaries—Abel, Cauchy—and his successors—Weierstrass, Dedekind, and mathematics after Gauss became a totally different thing from the mathematics of Newton, Euler, and Lagrange.
In the constructive sense Gauss was a revolutionist. Before his schooling was over the same critical spirit which left him dissatisfied with the binomial theorem had caused him to question the demonstrations of elementary geometry. At the age of twelve he was already looking askance at the foundations of Euclidean geometry; by sixteen he had caught his first glimpse of a geometry other than Euclid’s. A year later he had begun a searching criticism of the proofs in the theory of numbers which had satisfied his predecessors and had set himself the extraordinarily difficult task of filling up the gaps and completing what had been only half done. Arithmetic, the field of his earliest triumphs, became his favorite study and the locus of his masterpiece. To his sure feeling for what constitutes proof Gauss added a prolific mathematical inventiveness that has never been surpassed. The combination was unbeatable.