Nineteenth-Century Contributions: Lobachevsky to Hilbert

Nineteenth-Century Contributions:Lobachevsky to Hilbert

Geometry in every proposition speaks a language which experience never dares to utter; and indeed of which she but halfway comprehends the meaning.-WI L L IAM WHEWE L L

Attempts to Prove the Parallel PostulateThe Efforts of Proclus, Playfair,And WallisIn writing the Elements, Euclid rested his imposing structure on certain statements, grounded in physical experience, whichhe regarded as self-evident truths. Distinct among these postulates is one whose tone differs greatly from the terseness and simple comprehensibility of the others. This is the famous parallel postulate, which Euclid phrased as, If a straight line falling on two straight lines makes the interior angles on the sameside less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles. Not only did this statement lack the quality of self-evidence; it was actually open to doubt.

The ancient Greeks were aware that certain curves may approach nearer and nearer and yet not meet, as the hyperbola approaches but does not meet its asymptotes. They saw no reason why the two straight lines in the parallel postulate might not exhibit the same behavior. Evidently Euclid himself did not quite trust the postulate, for he postponed using it in his proofs until he had reached Proposition 29 of Book I. The wish to remove the unwieldyand suspectthing from the set of postulates was quite natural. Starting in Euclids own lifetime, attempts were made to change the definition of parallel lines, or to replace the troublesome postulate by a more acceptable but equivalent assertion, or to deduce it as a theorem from the other nine unquestioned axioms.

Many substitutes for Euclids parallel postulate have been suggested through the ages. These alternatives are more intuitively appealing statements, but when they are carefully examined, they turn out to be logically equivalent to Euclids axiom. The most frequentsubstitute is known as Playfairs axiom, although it was stated as early as the fifth century by Proclus. John Playfair (17481819) held a chair in mathematics at the Universityof Edinburgh from 1785 until 1805.

His Elements of Geometry was a detailed presentationof the first six books of Euclid, along with a supplement embracing the approximate calculation of and some solid geometry. Finding Euclids axiom on parallels unsatisfactory, Playfair proposed replacing it with the following statement: Through a given point,not on a given line, only one parallel can be drawn to the given line. The popularity of Playfairs Elements of Geometry, of which 10 editions were published between 1795 and 1846, doubtless led to the appearance of his version of the postulate in most modern geometry textbooks.

The parallel postulate, an assertion about intersecting lines, and Playfairs axiom, anassertion about parallel lines, are equivalent statements. This means that they mutually imply each other (the other, unquestioned axioms holding). That the parallel postulate implies its modern counterpart is evident from our earlier study of Euclidean geometry. Thus it remains for us to derive the parallel postulate, accepting as data the remaining nine axioms of Euclids geometry and Playfairs axiom.

T H E O R E MPlayfairs axiom implies the parallel postulate.Proof. Let the lines l and l be cut by a transversal t in points P and Q, forming a pair of interior angles 1 and 2 on one side of the transversal. Assume that the sum of 1 and 2 is less than two right angles.If 3 is the supplement of 2 on the side of t opposite 1 and 2, then 3 + 2 = 180 > 1 + 2,which implies that 3 > 1. Through P, construct a line l making with t an angleequal to and alternatively interior to 3. (Proposition 23 of Euclid I permits this.) Then by Proposition 27, l is parallel to l. Because 3 > 1, we have l and l as distinct lines. From Playfairs axiom, which tells us that only one line can be drawn through P parallel to l, we conclude that l and l must meet at some point R. If R were on the side of t opposite 1 and 2, then 1 would be an exterior angle of triangle PQR. The exterior angle theorem (Proposition 16) would then guarantee that 1 is greater than 3, which is impossible. Thus l and l intersect on the side of t containing 1 and 2,proving the parallel postulate.

At t emp t s t o P r o v e t h e P a r a l l e l P o s t u l a t eMathematicians from the Greeks onward have compiled a long list of statements equivalent to the parallel postulate. Among them are: A line that intersects one of two parallel lines intersects the other also. There exist lines that are everywhere equidistant from one another. The sum of the angles of a triangle is equal to two right angles. For any triangle, there exists a similar non congruent triangle. Any two parallel lines have a common perpendicular. There exists a circle passing through any three non collinear points. Two lines parallel to the same line are parallel to each other.

Those geometers who were not content to accept Euclids postulate on parallels did not wish merely to know whether some simpler equivalent assumption could be substituted for it (although this question is not without interest). Their real aim was to learn whether Euclids form of the postulate was deducible from the other nine axioms and the first 28 propositions of the Elements, which did not depend on the parallel postulate.

One of the earliest efforts to prove the parallel postulate was made by Proclus (410485) in his Commentary on the First Book of Euclids Elements. After remarking that others before us have classed it among the theorems and demanded a proof of this which was taken as a postulate by the author of the Elements, he went on to point out a fallacy contained ina demonstration offered by the noted astronomer Ptolemy, and then to submit a proof of his own. Proclus derived what we call Playfairs axiom, hence by implication, the parallel postulate. In substance, the argument ran as follows.

Let l be a given line, with P any point not on it. Assume that Q is the foot of the perpendicular from P to l and let l be the line perpendicular to PQ at P. The lines l and l, forming equal alternate interior angles with PQ, are parallel. Thus it suffices to show that any other line l through P meets l. Pick any point R on l in the region between l and l, and take S to be the foot of the perpendicular from R to l. As the point R moves alongl, away from P, the length of the segment RS increases. Eventually, this length becomes greater than the distance between the parallels, namely, the length of the segment PQ. Then R will cross over to the other side of l, so that l cuts l. This shows that l is the only linethrough P parallel to l.

It is easy enough to pick out the flaw in Procluss reasoning. He made free with the assumption that two parallel lines are everywhere the same distance apart. (His argument would suffice if one merely granted that the perpendicular distance between them is bounded.) Yet there is no justification in Postulates 14 for this. Indeed, the supposition that parallel lines are a constant distance from one another can be shown to imply the parallel postulate, so that Proclus was tacitly assuming a property of parallel lines that is equivalent to Postulate 5. Despite his ambitious efforts, he succeeded only in begging the question.

All the numerous and varied attempts to prove the postulate on parallels failed, although for a time some mathematicians thought they had succeeded. Like Proclus, many failed because their arguments were marred by the use, open or hidden, conscious or unconscious, of some assumption equivalent to the fifth postulate. Some failed because their reasoningwas otherwise fallacious. Some of these alleged proofs, such as that presented by John Wallis (16161703), had a certain seductive appeal.

While lecturing at Oxford Universityin 1663, Wallis proposed replacing the parallel postulate by a new axiom he felt was moreplausible and should therefore be given precedence. He suggested: To each triangle, there exists a similar triangle of arbitrary magnitude. Using this assumption and the other axioms of Euclid, Wallis was able to demonstrate that Playfairs axiom holds, hence so does its equivalent, the parallel postulate.

He proceeded thus:Given a point P not on the line l, construct a line l through P parallel to l in the familiar way, that is, by dropping a perpendicular PQ to l and erecting a perpendicular l to PQ at P. The crux of the argument is to show that if l is any other line through P, then l necessarily meets l. Pursuing this aim, pick R to be any point on l in the region between l and l and drop a perpendicular RS to PQ.

Because we are taking Walliss axiom as part of our hypothesis, there exists a triangle PQT similar to triangle PSR and such that T is on the same side of PQ as R is. Then, in compliance with the definition of similar triangles, TPQ = RPS. The implication is thatsince these angles have the segments PS and PQ as a common side, PR and PT coincide. This puts the point T on the line l. Again, by similarity, PQT = PSR = 90, which makes TQ a perpendicular to PQ at Q. But l is the unique perpendicular to PQ at Q, so that T lies on l. Thus l and l meet at the point T , allowing us to conclude that the only line through P parallel to l is l.

Actually Wallis fared no better than Proclus. Although the above proof seems perfectly reasonable, it rests on the assumption of the existence of two similar but non congruent triangles. This turns out to be another of the equivalents of the parallel postulate. Like themany others who tried to demonstrate the enemy of geometers, Wallis was guilty of the circular reasoning of assuming what he set out to prove.

Saccheri QuadrilateralsThis is not the place to review the entire history of the vain efforts to establish the truthof Euclids fifth postulate as a matter of proof; it is enough to note that all inevitably failed.Although the results of these investigations were on the whole negative, some writers, such as Saccheri, Lambert, and Legendre, made important contributions to what we now call non-Euclidean geometry, though each was unaware of the true meaning at the time.

Girolamo Saccheri

The work of Girolamo Saccheri (16671733), a professor at the University of Pavia, deserves attention, because he seems to have been the first to study the logical consequences of an actual denial of the famous postulate. Saccheri was a Jesuit priest who taught in a succession of colleges of his order in Italy. He was considered a brilliant teacher and a man of such remarkable memory, we are told, that he could play three games of chess at a time without seeing any of the boards. Before going to Pavia in 1697 to occupy the chair in mathematics, Saccheri taught philosophy for three years in Turin.

The result of this Experience was the publication of a work on logic, Logica Demonstrativa (1697), concerningthe compatibility of definitions and postulates. As a logician, Saccheri became impressed with the deductive power of reductio ad absurdum, or the indirectmethod of reasoning, used early in the Elements. According to this method, one shows that if the desired conclusionwere not true, then a contradiction (absurdity) would follow. The question of Euclids fifth postulate interested Saccheri throughout his lifetime and it is natural that in searching for material to which his logical principles could be applied, he would turn to the problem of parallels.

In the year of his death, Saccheri published a little treatise of some 101 pages, bearing the intriguing title Euclides ab omni naevo vindicatus, commonly translated Euclid Vindicated of Every Blemish. This was an attempt to clear Euclid of the criticisms arising from the doubtful status of the parallel postulate; and the title was perhaps intended in response to Sir Henry Savile, who in his Praelectiones on Euclid called the theory of parallels a blemish (naevus) on geometry.

Saccheris novel procedure for vindicating Euclid was to assume the parallel axiom to be false and to develop the resulting consequences, hoping to reach a contradiction; this would entitle him to affirm the parallel postulate by reductio ad absurdum. Denying the parallel postulate gave Saccheri a new premise for his reasonings and led him to expect more success than his predecessors had found.

The fundamental figure of Saccheris investigations was a quadrilateral ABCD in which the sides AD and BC were equal and perpendicular to the base AB. These quadrilaterals have subsequently become known as Saccheri quadrilaterals. Without using the parallelpostulate or any of its consequences, Saccheri was able to prove that the angles at C and D, the summit angles of the quadrilateral, were equal. For triangles BAD and ABC are congruent by the side-angle-side proposition (Proposition 4 of Euclid I), hence BD = AC. But this in its turn makes triangles ADC and BCD congruent by the side-side-side proposition (Proposition 8 of Euclid I)

As corresponding parts of congruent triangles, it follows that C = D. Now there are three possible alternatives, giving rise to three hypotheses:

C = D > 90 (hypothesis of the obtuse angle).

C = D < 90 (hypothesis of the acute angle).

3. C = D = 90 (hypothesis of the right angle).

Euclids parallel postulate is equivalent to the third of these possibilities, so that to assumethat the summit angles are either obtuse or acute is an implicit denial of the parallel axiom.This is exactly what Saccheris proof by reductio ad absurdum demanded, to show that thehypothesis of the obtuse angle and the hypothesis of the acute angle both led to contradictions. The hypothesis of the right angle, and in consequence the parallel postulate, should then hold.

Saccheri proved that if any one of the three hypotheses were true for one of his quadrilaterals, then it would be true for every such quadrilateral. He went on to show that according to the hypothesis of the obtuse angle, the hypothesis of the acute angle, or the hypothesis of the right angle held, the sum of the angles of a triangle would be, respectively, greater than, less than, or equal to two right angles.

The reasoning was as follows: Given a triangle ABC, let D and E be the midpoints of sides AC and BC, respectively. Drop perpendiculars AF,BG, and CH from the vertices A, B, and C to the line through D and E. The triangles AFD and CHD are congruent; and also the triangles BGE and CHE are congruent by the side angle- angle proposition (Proposition 26 of Euclid I). This means that AF = CH = BG, making ABGF a Saccheri quadrilateral with right angles at F and G. Then FAB = FAD + DAB = HCD + DABAnd GBA = GBE + EBA = HCE + EBA.

The result of adding these equalities is that FAB + GBA = DAB + ( HCD + HCE) + EBA= BAC + ACB + ABC, or in words, the sum of the summit angles of the Saccheri quadrilateral ABGF equals the sum of the angles of the triangle ABC. Knowing this, one can easily see that if Saccherishypothesis of the obtuse angle holds (so that in particular, each of FAB and GBA is obtuse), then the sum of the angles of any triangle is greater than two right angles. If the hypothesis of the acute holds, the same sum is less than two right angles. If the hypothesis of the right angle holds, this sum equals two right angles.

Saccheri had no trouble in eliminating the hypothesis of the obtuse angle. After a carefully reasoned chain of 13 propositions, he found that the hypothesis of the obtuse angle implied the parallel postulate, which in its turn implied that the sum of the angles ofa triangle would have to equal two right angles. But this second implication contradicts the just-demonstrated result that on the supposition of the hypothesis of the obtuse angle, the sum of the angles of a triangle is greater than two right angles.

In Saccheris own colorful words from Euclides Vindicatus: The Hypothesis of the Obtuse Angle is absolutely false because it destroys itself.We might remark that the unconscious assumption that a straight line is infinite played a part in reaching this conclusion, since Euclids Proposition 16 (the exterior angle theorem) was used.

The task of disposing of the hostile Acute Angle Hypothesis, as Saccheri called it, was much more elusive. Though he proved one theorem after another, he never found the sought-after contradiction. Without realizing it, Saccheri was on the threshold of discovering the first non-Euclidean geometry. Although the theorems he was obtaining seemed at Variance with experience, they formed a geometry that in internal coherence was not inferior to Euclids. But Saccheri was so convinced by what he was trying to do that he permitted his faith in Euclidean geometry to interfere with his logical perseverance.

When no contradiction presented itself, the devoted geometer forced his mathematical development into an unsatisfactory ending by treating a point at infinity (a limit point of the plane) as if it were a point of the plane. Lamely, he concluded that two distinct lines that meet at an infinitely distant point can both be perpendicular at that point to the same straight line. Saccheri viewed this as a contradiction to Proposition 12 of Euclid I, according to which there is a unique perpendicular to a line at each point of the line.

Triumphantly he announced, The Hypothesis of the Acute Angle is absolutely false, being repugnant to the nature of a straight line. Saccheri was satisfied that he had finally removed the uncertainty surrounding Euclids axiom system, declaring at the close of his work, The foregoing considerations seem to me sufficient to clear Euclid of the faults with which he has been reproached. But a violation of intuitive ideas does not make a logical contradiction, and the problem of the necessity of the parallel postulate remained.

Had Saccheri unflinchingly admitted that his reasoning led to no contradiction but merely results that seemed paradoxical in that they clashed sharply with the familiar geometry,he would have anticipated the discovery of non-Euclidean geometry by at least 100 years. As it was, his treatise was not read widely enough to have much influence and was virtually forgotten until the Italian mathematician Eugenio Beltrami rescued it from oblivion in 1889.

Beltrami hailed Saccheri as the Italian precursor of the Hungarian Bolyai and the Russian Lobachevsky, the two individuals who are usually credited with the concept of a logically consistent geometry based on a denial of the parallel postulate. But the final discovery of non-Euclidean geometry did not depend on the pioneering work of Saccheri, because neither Bolyai nor Lobachevsky seemed to have heard of the book or its author.

Indeed, had Saccheri actually accomplished his purpose and proved the parallel postulate from the remaining axioms of Euclidean geometry, he would not have vindicated Euclid. Quite to the contrary, he would have dealt a terrible blow to Euclid. Euclid was vindicated by the discovery of non-Euclidean geometry, for its existence demonstrates that the parallel postulate is independent of Euclids other axioms, so that it truly widens the axiomatic base on which Euclids geometry stands . We must admire the Great Geometer allthe more; the introduction of the fifth postulate, so decidedly un axiomatic in appearance, yet an independent postulate, was a stroke of pure genius.

There is a notable resemblance between Saccheris Euclides Vindicatus and the Theorie der Parallellinien (written in 1766) of the German mathematician Johann Heinrich Lambert (17281777), a colleague of Euler and Lagrange at the Berlin Academy of Sciences. For the fundamental figure of his investigations, Lambert adopted a quadrilateral having three rightangles and then examined the hypotheses in which the remaining angle was in turn obtuse, acute, and right. Lambert disposed of the obtuse-angle hypothesis in much the manner of Saccheri by establishing the contradictory result that the parallel postulate could be proved as a theorem from it.

He too derived many non-Euclidean facts from the acute angle hypothesis, but unlike Saccheri, he was well aware that he had reached no contradiction. Lambert showed what no one had noticed beforethat in this new geometry the angle sum of a triangle increases when the area decreases. (By the angle sum of a triangle, we mean the sum of its three interior angles.) He did this by proving that the area of a triangle is proportional to its angular defect, that is, the amount by which its angle sum falls short oftwo right angles. In modern symbols, for a triangle with angles a, b, and c, he demonstrated that the area A = k2[180 (a + b + c)], where k is a constant of proportionality.

Although the range of Lamberts research interests was enormous, he is remembered today mainly for having given the first rigorous proof that the number is irrational. In a paper presented to the Berlin Academy in 1768, he showed that if x is a nonzero rational number, then neither ex nor tan x can be rational (earlier, in 1737, Euler had established that e and e2 are irrational). Because tan /4 = 1, a rational number, it can be inferred that /4, hence itself, is irrational. Lambert also suggested that e and are transcendental numbers; but proof of this would have to wait a hundred years.

The Accomplishments of Legendre

Adrien-Marie Legendre

Adrien-Marie Legendre (1752 1833) came from a well-to-do family in the south of France but spent the greater part of his life in Paris. He was educated there at the College Mazarin and from 1775 until 1780 was aprofessor at the Ecole Militaire, where Laplace was also teaching. Two years after resigning his position to reserve more time for research, Legendre won a prize offered by the Berlin Academy with an essay on the path of a projectile in a resisting medium (Recherches sur la trajectoire des projectiles dans les milieux resistants). This brought him to the attention ofthe scientific community and secured his appointment to the Academie des Sciences, first as an adjunct member in 1783 and then as an associate member in 1785.

Following the closing of the Academie in 1793, Legendre was not invited to cooperate in the reorganization of public education. This may have been because he was not in favor with the revolutionarygovernment or because of some other reason. His name does not figure among those of the professors at either the Ecole Normale or the Ecole Polytechnique. Nor was he on the list of 48 scholars whom the government selected (1795) to form the nucleus of the new Institut National des Sciences et des Arts as Lagrange and Laplace were. His colleagues redressed this oversight by electing him a resident member of the mathematics section.

During this period of turmoil in France, Legendre was, however, made a member of several public commissions. He served, in particular, as one of the three commissioners who were to oversee the triangulations necessary for determining the standard meter. (This new unit of measure was originally intended to represent the 10-millionth part of the distance from the North Pole to the equator, calculated from the measured length of the meridian arc between Dunkirk and Barcelona.) He later was appointed to succeed Laplace as examiner in mathematics of the graduates of the Ecole Polytechnique destined for the artillery. It is said that Legendre was not so well regarded by his countrymen as Lagrange and Laplace were, for, according to Lao Simons, He sought recognition which apparently came unsought to both Lagrange and Laplace.

Whatever the faults of his personality, Legendre was a man of integrity who always spoke his mind, even when his own interests were adversely affected. Refusing to vote in favor of the governments candidate for the Institut National, he was deprived (1824) of his pension and died in poverty.

Legendres achievements in mathematics, although they were considerable, did not approach the achievements of his two compeers, Lagrange and Laplace. That his work was nonetheless held in high regard is indicated by a comment of the secretary of the French Academy, Jean-Baptiste Elie de Beaumont, who wrote, Laplace . . . has earned a right to bestyled the Newton of France; Legendre, more profound than popular, was our Euler. Like Euler, Legendre had interests covering the breadth of mathematics. He seems to have been most pleased with his research in number theory, celestial mechanics, and the theory ofelliptic functions, because although he took up other problems in the course of his lifetime, he always returned to these.

In 1785, Legendre read a memoir to the Academie with the title Recherches danalyse indetermine. This contained the celebrated quadratic reciprocity law, which is concerned with the solvability of the pair of quadratic congruences x2 q (mod p) and x2 p(mod q), where p and q are distinct odd primes. The following holds:Either both congruences x2 q (mod p) and x2 p (mod q) are solvable or both unsolvable; except in the case in which p and q are each of the form 4n + 3, in which event one of the congruences is solvable whereas the other is not.

Legendre tried a long and imperfect demonstration of this, but slipped in assuming an obvious theorem that was as difficult to prove as the law itself. Undaunted, he tried another proof in his Essai sur la theorie des nombres (1798). This too contained a gap, becauseLegendre took for granted that there were an infinite number of primes in certain arithmetic progressions. At 18, Gauss (in 1795) had independently discovered this reciprocity law, and after a years unremitting labor, he obtained the first complete proof. It tortured me, saidGauss, for awhole year and eludedmymost strenuous efforts before, finally, I got the proof explained in the fourth section of the Disquisitiones Arithmeticae.

In the Disquisitiones Arithmeticae (published in 1801, though finished in 1798) Gauss attributed the quadraticreciprocity law to himself. Two pages after its proof, he vaguely alluded to Legendres contribution, saying, Legendre in his excellent tract in Mem. Acad. des Sci. 1785 arrived at a theorem which is basically the same as the fundamental theorem. If Legendre was nettledby this passing reference, It would not be the last time that he would feel that Gauss had given him insufficient recognition. His Essai sur la theorie des nombres, which represented the first modern treatise devoted exclusively to number theory, passed through two editions, one appearing in 1798 and the other in 1808

In the second edition, Legendre adopted theproof of the quadratic reciprocity law given by the young Gauss. Legendres Essai was later expanded into his Theorie des nombres. The two-volume third edition (1830), together with Gausss Disquisitiones Arithmeticae, became the dominant work on the subject for the remainder of the century.

Although the primes occur among the positive integers in a most irregular manner, there are ways in which their overall distribution seems quite regular. By inspecting tables of prime numbers, both Legendre and Gaussthe latter still in his early teenshoped to find a simple function whose values approximated those of (x), the function which gives the number of primes less than or equal to x.

Legendre, in his Essai sur la theorie desnombres, stated that the function x log x 1.08366agrees very well with (x) as long as x is not greater than 1 million. Although Gaussdiscovered that the integral

dt log tproduces a better approximation than Legendres guess, he never published the resultsof his investigation.

The first major progress concerning (x) is attributed to the Russianmathematician P. L. Chebyshev (18211894). Around 1850, he showed that the inequalities(0.92)xlog x< (x) < (1.11)xlog xare valid for all x sufficiently large; and if the limitlimx(x)x/ log xexists, then its value must be 1

However, the existence of the limit turned out to be very difficult to prove. It was finally established in 1896, a hundred years after Legendres and Gausss conjectures, by means of new and powerful methods primarily from analysis. This celebrated achievement has since become known as the prime number theorem.

The usual procedure in astronomy is for different observers, using different instruments, to make numerous observations at a variety of locations. Because the results of these observations are subject to errors arising from the reactions of the observers or the precision of their instruments, determining the most probable value of an observed quantity becomes a question. The method of least squares is so called because the sum of the squares of the differences between the observed values and the true value of the observed quantity must be minimized. This method was discovered independently, and almost simultaneously, by Legendre and Gauss.

Gauss had been using the basic idea since 1794 or 1795, when he was a student at Caroline College in Brunswick preparing for his university studies. But the first explicit account of the method was published by Legendre in 1805 in his paper Nouvelles methodes pour la determination des orbites des com`etes. He gave it the name methode des moindres quarres, showed that the rule of the arithmetic mean is a particular case of the general principle, and presented examples of its application to the determination of the orbit of a comet.

He offered no probabilistic justification, however, that the results thus obtained were the best or the most plausible results the observations were capable of affording. It remained for Gauss in his famous astronomical text Theoria Motus Corporum Coelestium (Theory of the Motion of the Heavenly Bodies) of 1809 to derive the law of probability oferror that is the basis for the method of least squares. This was the work in which Gauss explained his solution of one of the most interesting and difficult problems in astronomy, finding the orbit of the lost planet Ceres.

At one point he stated, Our principle, which wehave use of since the year 1795, has lately been published by Legendre. Legendre was disturbed by Gausss use of the phrase our principle and wrote to him in censure:There is no discovery that one cannot claim for himself by saying that one had found the same thing some years previously; but if one does not supply the evidence by citing the place where one has published it, this assertion becomes pointless and serves only to do a disservice to the true author of the discovery.

It was not that Gauss treated casually what others did, only that he considered a result to be his if he gave the first rigorous demonstration of it. Legendre, on the other hand, felt that publication established priority, even if an argument that was merely plausible took the place of a complete proof. All discussion of priority rights between the two was futile. Because each clung to the correctness of his position, neither took heed of the other. Gausscould only express regret about the competition between them.Writing to Heinrich Olbers, his associate and a noted astronomer, Gauss was led to remark:

It seems to be my fate to compete with Legendre in almost all my theoretical works. So it is in the higher arithmetic, in the researches on transcendental functions connected with the rectification of the ellipse, in the fundamentals of geometry, and now here again.

A large portion of Legendres research was devoted to elliptic functions. (The name elliptic is apt because these functions exist in the expression for finding the length of an arc of an ellipse.) A systematic account of the theory of elliptic functions is contained in Legendres three-volume Exercises du calcul integral (1811, 1817, 1819), a great and rich textbook on the integral calculus that rivaled Eulers Institutiones Calculi Integralis in its comprehensiveness. The third volume includes long tables of elliptic integrals calculated by Legendre himself at immense labor. This material was further developed in another three volume treatise, Traite des fonctions elliptiques (1825, 1826, 1830), the most significant of Legendres works in higher mathematics.

The third volume, which although composedin 1830 did not appear until a few weeks before his death, presents an account of Abels contemporaneous research and also Jacobis in the same area . Notwithstanding the disparityin their ages, Legendre had developed an active correspondence with Carl Gustav Jacobi (18041851) about elliptic functions.

On hearing that many of Jacobis results had been obtained independently by Gauss but never published, Legendre wrote (1827) to the young man to share his indignation:How can M. Gauss have dared to tell you that the greater part of your theorems on elliptic functions were known to him and that he discovered them as early as 1808? . . . This extreme impertinence is incredible on the part of a man who has sufficient personal merit to have no need of appropriating the discoveries of others. . . .But this is the same man who, in 1801,wished to attribute to himself the discovery of the law of quadratic reciprocity published in 1785 and who wanted to appropriate in 1809 the method of least squares published in 1805.

Legendres Elements de geometrie

The work of Legendre that mainly concerns us is his Elements de geometrie (1795). As the teaching of geometry began to be stressed in the eighteenth-century universities, many geometry textbooks were published, some introducing novel ideas. Legendres Elements de geometrie presents an interesting twofold contrast when compared with contemporaryEnglish and French geometries. In the universities of Great Britain, Euclid met little competition, being practically the only geometry textbook used. To English authors of versions of the Elements, the order of the propositions in Euclid was absolutely essential to rigorous demonstration. Rather than depart from the original, they spent their efforts purging the text of faults that had crept in over time.

The French, on the other hand, maintained a critical attitude toward Euclid as a textbook for beginners. Many elementary geometries appeared in France in the 1700s in which rigor and formalism were sacrificed for the sake of a new sequence of propositions better fitted to the needs of mathematical novices. These endeavors tomake geometry palatable were frequently criticized for lightening work for the examiner as well as for the student. In a move away from the loose, intuitive presentations, Legendres Elements de geometrie undertook to revive a taste for rigorous demonstration in France.

Legendre simplified and rearranged many of the proofs of Euclids Elements, but on the whole approached the severity of the ancient treatment more closely than his predecessors did. The clarity of Legendres exposition and the attractiveness of his style made the Elements de geometrie one of the most successful textbooks ever written. Twentyeditions of this book, comprising some 100,000 copies, appeared in France alone before the authors death in 1833. Perhaps the only complaint was that Legendre, prepossessed by the methods of Euclid, relied excessively on reductio ad absurdum, a method that convinces but does not satisfy the mind.

When the American Academy of Arts and Sciences was formed in Boston in 1780 at the suggestion of John Adams, its founders publicly stated that it was their intention to give it the air of France rather than that of England and to follow the Royal Academy rather than the Royal Society. Little wonder that in mathematics, as in the physical sciences, translations of French textbooks came to dominate teaching in the colleges of the United States. Legendres name became known to a great army of students, young people who had never heard of Lagrange or Laplace, when the Elements de geometrie completely supplanted Euclid in American schools.