by E. T. Bell
Well disciplined, the Polytechnicians observed the ban. But they were more resourceful or more courageous than the timid academicians. The King’s order covered only the funeral. The following day they marched in a body to the cemetery and laid a wreath on the grave of their master and friend, Gaspard Monge.
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I. F. J. D. Arago, 1786-1853, astronomer, physicist, and scientific biographer.
CHAPTER THIRTEEN
The Day of Glory
PONCELET
Projective geometry has opened up for us with the greatest facility new territories in our science, and has rightly been called a royal road to its own particular field of knowledge. —FELIX KLEIN
MORE THAN ONCE during the World War when the French troops were hard pressed and reinforcements nonexistent, the high command saved the day by routing some prima donna out of her boudoir, rushing her to the front, draping her from neck to heels in the tricolor, and ordering her to sing the Marseillaise to the exhausted men. Having sung her piece the lady rolled back to Paris in her limousine; the heartened troops advanced, and the following morning a cynically censored press once more unanimously assured a gullible public that “the day of glory has arrived”—with unmentioned casualties.
In 1812 the day of glory was still on its way. Prima donnas did not accompany Napoleon Bonaparte’s half-million troops on their triumphal march into Russia. The men did their own singing as the Russians retreated before the invincible Grand Army, and the endless plains rang to the stirring chant which had swept tyrants from their thrones and elevated Napoleon to their place.
All was going as gloriously as the most enthusiastic singer could have wished: six days before Napoleon crossed the Niemen his brilliant diplomatic strategy had indirectly exasperated President Madison into hurling the United States into a distracting war on England; the Russians were running harder than ever on their race back to Moscow, and the Grand Army was doing its valiant best to keep up with the reluctant enemy. At Borodino the Russians turned, fought, and, retired. Napoleon continued without opposition—except from the erratic weather—to Moscow, whence he notified the Czar of his willingness to consider an unconditional surrender of all the Russian forces. The competent inhabitants of Moscow, led by the Governor, took matters into their own hands, fired their city, burned it to the ground, and smoked Napoleon and all his men out into the void. Chagrined but still master of the situation, Napoleon disregarded this broad hint—the second or third so far vouchsafed to his military obstinacy—that “who killeth with the sword must perish by the sword,” presently ordered his driver to give the horses the lash, and dashed back post-haste over the now frozen plains to prepare for his rendezvous with Blücher at Leipzig, leaving the Grand Army to walk home or freeze as it should see fit.
With the deserted French army was a young officer of engineers, Jean-Victor Poncelet (July 1, 1788-December 23, 1867) who, as a student at the École Polytechnique in Paris, later at the military academy at Metz, had been inspired by the new descriptive geometry of Monge (1746-1818) and the Géométrie de position (published in 1803) of the elder Carnot (Lazare-Nicolas-Marguerite Carnot, May 13, 1753-August 2, 1823), whose revolutionary if somewhat reactionary program was devised “to free geometry from the hieroglyphics of analysis.”
In the preface to his classic Applications d’analyse et de géométrie (second edition 1862, of the work first published in 1822), Poncelet recounts his experiences in the disastrous retreat from Moscow. On November 18, 1812, the exhausted remnant of the French army under Marshal Ney was overwhelmed at Krasnoï. Among those left for dead on the frozen battlefield was young Poncelet. His uniform as an officer of engineers saved his life. A searching party, discovering that he still breathed, took him before the Russian staff for questioning.
As a prisoner of war the young officer was forced to march for nearly five months across the frozen plains in the tatters of his uniform, subsisting on a meagre ration of black bread. In a cold so intense that the mercury of the thermometer frequently froze, many of Poncelet’s companions in misery died in their tracks, but his ruggeder strength pulled him through, and in March, 1813 he entered his prison at Saratoff on the banks of the Volga. At first he was too exhausted to think. But when “the splendid April sun” restored his vitality, he remembered that he had received a good mathematical education, and to soften the rigors of his exile he resolved to reproduce as much as he could of what he had learned. It was thus that he created projective geometry.
Without books and with only the scantiest writing materials at first, he retraced all that he had known of mathematics from arithmetic to higher geometry and the calculus. These first labors were enlivened by Poncelet’s efforts to coach his fellow officers for the examinations they must take should they ever see France again. One legend states that at first Poncelet had only scraps of charcoal, salvaged from the meager brazier which kept him from freezing to death, for drawing his diagrams on the wall of his cell. He makes the interesting observation that practically all details and complicated developments of the mathematics he had been taught had evaporated, while the general, fundamental principles remained as clear as ever in his memory. The same was true of physics and mechanics.
In September, 1814, Poncelet returned to France, carrying with him “the material of seven manuscript notebooks written at Saratoff in the prisons of Russia (1813 to 1814), together with divers other writings, old and new,” in which he, as a young man of twenty four, had given projective geometry its strongest impulse since Desargues and Pascal initiated the subject in the seventeenth century. The first edition of his classic, as already mentioned, was published in 1822. It lacked the intimate “apology for his life” which has been used above, but it started a tremendous nineteenth century surge forward in projective geometry, modern synthetic geometry generally, and the geometric interpretation of the “imaginary” numbers that present themselves in algebraic manipulations, giving to such “imaginaries” geometrical interpretations as “ideal” elements of space. It also proposed the powerful and (for a time) controversial “doctrine of continuity,” to be described presently, which greatly simplified the study of geometric configurations by unifying apparently unrelated properties of figures into uniform, self-contained complete wholes. Exceptions and awkward special cases appeared under Poncelet’s broader point of view as merely different aspects of things already familiar. The classic treatise also made full use of the creative “principle of duality” and introduced the method of “reciprocation” devised by Poncelet himself. In short, a whole arsenal of new weapons was added to geometry by the young military engineer who had been left for dead on the field of Krasnoï, and who might indeed have died before morning had not his officer’s uniform distinguished him as a likely candidate for questioning by the Russian staff.
For the next decade (1815–25) Poncelet’s duties as a military engineer left him only odd moments for his real ambition—the exploitation of his new methods in geometry. Relief was not to come for many years. His high sense of duty and his fatal efficiency made Poncelet an easy prey for short-sighted superiors. Some of the tasks he was set could have been done only by a man of his calibre, for example the creation of the school of practical mechanics at Metz and the reform of mathematical education at the Polytechnique. But the reports on fortifications, his work on the Committee of Defense, and his presidency of the mechanical sections at the international expositions of London and Paris (1851-58), to mention only a few of his numerous routine jobs, could all have been done by lesser men. His high scientific merits, however, were not unappreciated. The Academy of Sciences elected him (1831) as successor to Laplace. For political reasons Poncelet declined the honor till three years later.
Poncelet’s whole mature life was one long internal conflict between that half of him which was born to do lasting work and the other half which accepted all the odd or dirty jobs shortsighted politicians and obtuse militarists shoved in its way. Poncelet himself longed to escape, but a mis
taken sense of duty, drilled into his very bones in Napoleon’s armies, impelled him to serve the shadow and turn his back on the substance. That he did not suffer an early and permanent nervous breakdown is a remarkable testimonial to the ruggedness of his physique. And that he retained his creative abilities almost to his death at the age of seventy nine is a shining proof of his unquenchable genius. When they could think of nothing better for this splendidly endowed man to do with his time they sent him traipsing about France to inspect cotton mills, silk mills, and linen mills. They did not need a Poncelet to do that sort of thing, and he knew it. He would have been the last man in France to object had his unique talents been indispensable in such affairs, for he was anything but the sort of intellectual prude who holds that science loses her perennial virginity every time she shakes hands with industry. But he was not the only man available for the work, as possibly Pasteur was in the equally important matters of the respective diseases of beer, silkworms, and human beings.
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We now glance at one or two of the weapons either devised or remodelled by Poncelet for the conquest of projective geometry. First there is his “principle of continuity,” which refers to the permanence of geometrical properties as one figure shades, by projection or otherwise, into another. This no doubt is rather vague, but Poncelet’s own statement of the principle was never very exact and, as a matter of fact, embroiled him in endless controversies with more conservative geometers whom he politely designated as old fossils—always in the dignified diction suitable to an officer and a gentleman, of course. With the caution that the principle is of great heuristic value but does not always of itself provide proofs of the theorems which it suggests, we may see something of its spirit from a few simple examples.
Imagine two intersecting circles. Say they intersect in the points A and B. Join A and B by a straight line. The figure presents ocular evidence of two real points A, B and the common chord AB of the two circles. Now imagine the two circles pulled gradually apart. The common chord presently becomes a common tangent to the two circles at their point of contact. At any stage so far the following theorem (usually set as an exercise in school geometry) is true: if any point P be taken on the common chord, four tangent lines may be drawn from it to the two circles, and if the points in which these tangent lines touch the circles are T1 T2, T3, T4, then the segments PT1, PT2, PT3, PT4, are all equal in length. Conversely, if it is asked where do all the points P lie such that the four tangent-segments to the two circles shall all be equal, the answer is on the common chord. Stating all this briefly in the usual language, we say that the locus (which merely means place) of a point P which moves so that the lengths of the tangent-segments from it to two intersecting circles are equal, is the common chord of the two circles.I All this is familiar and straightforward; there is no element of mystery or incomprehensibility as some may say there is in the next where the “principle of continuity” enters.
Pull the circles completely apart. Their two intersections (or in the last moment their one point of contact) are no longer visible on the paper and the “common chord” is left suspended between the two circles, cutting neither visibly. But it is known that there is still a locus of equal tangent-segments, and it is easily proved that this locus is a straight line perpendicular to the line joining the centres of the two circles, just as the original locus (the common chord) was. Merely as a manner of speaking, if we object to “imaginaries,” we continue to say that the two circles intersect in two points in the infinite part of the plane, even when they have been pulled apart, and we say also that the new straight-line locus is still the common chord of the circles: the points of intersection are “imaginary” or “ideal,” but the straight line joining them (the new “common chord”) is “real”—we actually draw it on the paper.
If we write the equations of the circles and lines algebraically in the manner of Descartes, all that we do in the algebra of solving the equations for the intersections has its unique correlate in the enlarged geometry, whereas if we do not first expand our geometry—or at least increase its vocabulary, to take account of “ideal” elements—much of the meaningful algebra is geometrically meaningless.
All this of course requires logical justification. Such justification has been given so far as is necessary, that is, up to the stage which includes the applications of the “principle of continuity” useful in geometry.
A more important instance of the principle is furnished by parallel straight lines. Before describing this we may repeat the remark a venerable and distinguished judge relieved himself of a few days ago when the matter was revealed to him. The judge had been under the weather; an amateur mathematician, thinking to cheer the old fellow up, told him something of the geometrical concept of infinity. They were strolling through the judge’s garden at the time. On being informed that “parallel lines meet at infinity,” the judge stopped dead. “Mr. Blank,” he said with great emphasis, “any man who says parallel lines meet at infinity, or anywhere else, simply hasn’t got good sense.” To obviate an argument we may say as before that it is all a way of speaking to avoid irritating exceptions and separations into exasperating distinct cases. But once the language has been agreed upon, logical consistency demands that it be followed to the end without traversing the rules of logical grammar and syntax, and this is what is done.
To see the reasonableness of the language, imagine a fixed straight line l and fixed point P not on l. Through P draw any straight line l’ intersecting l in P’9 and imagine l’ to rotate about P, so that Pf recedes along l. When does P’ stop receding? We say it stops when l, l’ become parallel or, if we prefer, when the point of intersection P’ is at infinity. For reasons already indicated this language is convenient and suggestive—not of a lunatic asylum, as the judge might think, but of interesting and sometimes highly practical things to do in geometry.
In a similar manner the visualizable finite parts of lines, planes and three-dimensional space (also of higher space) are enriched by the adjunction of “ideal” points, lines, planes, or “regions” at infinity. If the judge happens to see this he may enjoy the following shocking example of the behavior of the infinite in geometry: any two circles in a plane intersect in four points, two of which are imaginary and at infinity. If the circles are concentric, they touch one another in two points lying on the line at infinity. Further, all circles in a plane go through the same two points at infinity—they are usually denoted by I and J, and are sometimes called Isaac and Jacob by irreverent students.
In the chapter on Pascal we described what is meant by projective properties in distinction to metrical properties in geometry. At this point we may glance back at Hadamard’s remarks on Descartes’ analytic geometry. Hadamard observed among other things that modern synthetic geometry repaid the debt of geometry in general to algebra by suggesting important researches in algebra and analysis. This modern synthetic geometry was the object of Poncelet’s researches. Although all this may seem rather involved at the moment, we shall close the chain by taking a link from the 1840’s, as the matter really is important, not only for the history of pure mathematics but for that of recent mathematical physics as well.
The link from the 1840’s is the creation by Boole, Cayley, Sylvester and others, of the algebraic theory of invariance which (as will be explained in a later chapter) is of fundamental importance in current theoretical physics. The projective geometry of Poncelet and his school played a very important part in the development of the theory of invariance: the geometers had discovered a whole continent of properties of figures invariant under projection; the algebraists of the 1840’s, notably Cayley, translated the geometrical operations of projection into analytical language, applied this translation to the algebraic, Cartesian mode of expressing geometric relationships, and were thus enabled to make phenomenally rapid progress in the elaboration of the theory of algebraic invariants. If Desargues, the daring pioneer of the seventeenth century, could have foreseen what his i
ngenious method of projection was to lead to, he might well have been astonished. He knew that he had done something good, but he probably had no conception of just how good it was to prove.
Isaac Newton was a young man of twenty when Desargues died. There is no evidence that Newton ever heard the name of Desargues. If he had, he also might have been astonished could he have foreseen that the humble link forged by his elderly contemporary was to form part of the strong chain which, in the twentieth century, was to pull his law of universal gravitation from its supposedly immortal pedestal. For without the mathematical machinery of the tensor calculus which developed naturally (as we shall see) from the algebraic work of Cayley and Sylvester, it is improbable that Einstein or anyone else could ever have budged the Newtonian theory of gravitation.
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One of the useful ideas in projective geometry is that of cross-ratio or anharmonic ratio. Through a point O draw any four straight lines l, m, n, p. Across these four draw any straight line x, and label the points in which x cuts the others L, M, N, P respectively. We thus have on x the line segments LM, MN, LP, PN. From these form the ratios and Finally we take the ratio of these two ratios, and get the cross-ratio The remarkable thing about this cross-ratio is that it has the same numerical magnitude for all positions of the line x.
Later we shall refer to Felix Klein’s unification of Euclidean geometry and the common non-Euclidean geometries into one comprehensive geometry. This unification was made possible by Cayley’s revision of the usual notions of distance and angle on which metrical geometry is founded. In this revision, cross-ratio played the leading part, and through it, by the introduction of “ideal” elements of his own devising, Cayley was enabled to reduce metrical geometry to a species of projective geometry.
