n CORNELL UNIVERSITY LIBRARIES )&()^ematics (Uiknary White HaH CORNELL UNIVERSITY LIBRARY 3 1924 060 184 DATE DUE 4- m THE EK4NSTON COLLOQUIUM Lectures on Mathematics DELIVERED From Aug. 28 to Sept. 9, 1893 BEFORE MEMBERS OF THE CONGRESS OF MATHEMATICS HELD IN CONNECTION WITH THE WORLD'S FAIR IN CHICAGO AT NORTHWESTERN UNIVERSITY EVANSTON, ILL. BY FELIX KLEIN REPORTED BY ALEXANDER ZIWET PUBLISHED FOR H. S. WHITE AND A. ZIWET Nefa gork MACMILLAN AND CO. AND LONDON 1894 All rights reserved •n" Copyright, 1893, By MACMILLAN AND CO. Norhiaoti ^ress ; S. Gushing & Co. — Berwick & Smith. Boston, Mass., U.S.A. PREFACE. The Congress of Mathematics held under the auspices of the World's Fair Auxiliary in Chicago, from the 2 1st to the 26th of August, 1893, was attended by Professor Felix Klein of the University of Gottingen, as one of the commissioners of the German university exhibit at the Columbian Exposition. After the adjournment of the Congress, Professor Klein kindly consented to hold a colloqidnin on mathematics with such mem- bers of the Congress as might wish to participate. The North- western University at Evanston, 111., tendered the use of rooms for this purpose and placed a collection of mathematical books from its library at the disposal of the members of the collo- quium. The following is a list of the members attending the colloquium : — W. W. Beman, A.m., professor ot mathematics, University of Michigan. E. M. Blake, Ph.D., instructor in mathematics, Columbia College. O. BoLZA, Ph.D., associate professor of mathematics. University of Chicago. H. T. Eddy, Ph.D., president of the Rose Polytechnic Institute. A. M. Ely, A.B., professor of mathematics, Vassar College. F. Franklin, Ph.D., professor of mathematics, Johns Hopkins University. T. F. HOLGATE, Ph.D., instructor in mathematics. Northwestern University. L. S. HULBURT, A.M., instructor in mathematics, Johns Hopkins University. F. H. Loud, A.B., professor of mathematics and astronomy, Colorado College. J. McMahon, A.M., assistant professor of mathematics, Cornell University. H. Maschke, Ph.D., assistant professor of mathematics. University of Chicago. E. H. Moore, Ph.D., professor of mathematics. University of Chicago. vi PREFACE. J. E. Oliver, A.M., professor of mathematics, Cornell University. A. M. Sawin, Sc.M., Evanston. W. E. Story, Ph.D., professor of mathematics, Clark University. E. Study, Ph.D., professor of mathematics. University of Marburg. H. Taber, Ph.D., assistant professor of mathematics, Clark University. H. W. Tyler, Ph.D., professor of mathematics, Massachusetts Institute of Technology. J. M. Van Vleck, A.M., LL.D., professor of mathematics and astronomy, Wesleyan University. E. B. Van Vleck, Ph.D., instructor in mathematics. University of Wis- consin. C. A. Waldo, A.M., professor of mathematics, De Pauw University. H. S. White, Ph.D., associate professor of mathematics. Northwestern Uni- versity. M. F. Winston, A.B., honorary fellow in mathematics. University of Chicago. A. ZiWET, assistant professor of mathematics. University of Michigan. The meetings lasted from August 28th till September 9th ; and in the course of these two weeks Professor Klein gave a daily lecture, besides devoting a large portion of his time to personal intercourse and conferences with those attending the meetings. The lectures were delivered freely, in the English language, substantially in the form in which they are here given to the public. The only change made consists in oblit- erating the conversational form of the frequent questions and discussions by means of which Professor Klein understands so well to enliven his discourse. My notes, after being written out each day, were carefully revised by Professor Klein him- self, both in manuscript and in the proofs. As an appendix it has been thought proper to give a transla- tion of the interesting historical sketch contributed by Professor Klein to the work Die deutschen Universitdten. The translation was prepared by Professor H. W. Tyler, of the Massachusetts Institute of Technology. It is to be hoped that the proceedings of the Chicago Con- gress of Mathematics, in which Professor Klein took a leading PREFACE. vii part, will soon be published in full. The papers presented to this Congress, and the discussions that followed their reading, form an important complement to the Evanston colloquium. Indeed, in reading the lectures here published, it should be kept in mind that they followed immediately upon the adjournment of the Chicago meeting, and were addressed to members of the Congress. This circumstance, in addition to the limited time and the informal character of the colloquium, must account for the incompleteness with which the various subjects are treated. In concluding, the editor wishes to express his thanks to Professors W. W. Beman and H. S. White for aid in preparing the manuscript and correcting the proofs. ALEXANDER ZIWET. Ann Arbor, Mich., November, 1893. CONTENTS. Lecture Page I. Clebscb . . 1 II. Sophus Lie . 9 III. Sophus Lie . 18 IV. On the Real Shape of Algebraic Curves and Surfaces 25 V. Theory of Functions and Geometry . . 33 VI. On the Mathematical Character of Space-Intuition, and the Relation of Pure Mathematics to the Applied Sciences . 41 VII. The Transcendency of the Numbers e and tt 51 VIII. Idea) Numbers . .... 58 IX. The Solution of Higher Algebraic Equations . 67 X. On Some Recent Advances in Hyperelliptic and Abelian Func- tions -75 XI. The Most Recent Researches in Non-Euclidean Geometry 85 XII. The Study of Mathematics at Gottingen . . -94 The Development of Mathematics at the German Universities . 99 LECTURES ON MATHEMATICS. o>Ho Lecture I. : CLEBSCH. (August 28, 1893.) It will be the object of our Colloquia to pass in review some of the principal phases of the most recent development of math- ematical thought in Germany. A brief sketch of the growth of mathematics in the German universities in the course of the present century has been con- tributed by me to the work Die dentschen Utiiversitdten, com- piled and edited by Professor Lexis (Berlin, Asher, 1893), for the exhibit of the German universities at the World's Fair.* The strictly objective point of view that had to be adopted for this sketch made it necessary to break off the account about the year 1870. In the present more informal lectures these restrictions both as to time and point of view are abandoned. It is just the period since 1870 that I intend to deal with, and I shall speak of it in a more subjective manner, insisting par- ticularly on those features of the development of mathematics in which I have taken part myself either by personal work or by direct observation. The first week will be devoted largely to Geometry, taking this term in its broadest sense ; and in this first lecture it will surely be appropriate to select the celebrated geometer Clebsch * A translation of this sketch will be found in the Appendix, p. 99. 2 LECTURE I. as the central figure, partly because he was one of my principal teachers, and also for the reason that his work is so well known in this country. Among mathematicians in general, three main categories may be distinguished ; and perhaps the names logicians, formalists, and intiiitionists may serve to characterize them, (i) The word logician is here used, of course, without reference to the mathe- matical logic of Boole, Peirce, etc. ; it is only intended to indi- cate that the main strength of the men belonging to this class lies in their logical and critical power, in their ability to give strict definitions, and to derive rigid deductions therefrom. The great and wholesome influence exerted in Germany by Weiei'strass in this direction is well known. (2) The formalists among the mathematicians excel mainly in the skilful formal treatment of a given question, in devising for it an "algorithm." Gordan, or let us say Cayley and Sylvester, must be ranged in this group. (3) To the intziitionists, finally, belong those who lay particular stress on geometrical intuition {Anschauimg\ not in pure geometry only, but in all branches of mathematics. What Benjamin Peirce has called " geometrizing a mathematical question " seems to express the same idea. Lord Kelvin and von Staudt may be mentioned as types of this category. Clebsch must be said to belong both to the second and third of these categories, while I should class myself with the third, and also the first. For this reason my account of Clebsch's work will be incomplete ; but this will hardly prove a serious drawback, considering that the part of his work characterized by the second of the above categories is already so fully appre- ciated here in America. In general, it is my intention here, not so much to give a complete account of any subject, as to supplement the mathematical views that I find prevalent in this country. CLEBSCH. 3 As the first achievement of Clebsch we must set down the introduction into Germany of the work done previously by Cayley and Sylvester in England. But he not only trans- planted to German soil their theory of invariants and the inter- pretation of projective geometry by means of this theory ; he also brought this theory into live and fruitful correlation with the fundamental ideas of Riemann's theory of functions. In the former respect, it may be sufficient to refer to Clebsch's Vorlesungen iiber Geometrie, edited and continued by Linde- mann ; to his Bindre algebraische Formen, and in general to what he did in co-operation with Gordan. A good historical account of his work will be found in the biography of Clebsch published in the Math. Ajtnalett, Vol. 7. Riemann's celebrated memoir of 1857* presented the new ideas on the theory of functions in a somewhat startling novel form that prevented their immediate acceptance and recogni- tion. He based the theory of the Abelian integrals and their inverse, the Abelian functions, on the idea of the surface now so well known by his name, and on the corresponding funda- mental theorems of existence {Existenztheoreme). Clebsch, by taking as his starting-point an algebraic curve defined by its equation, made the theory more accessible to the mathema- ticians of his time, and added a more concrete interest to it by the geometrical theorems that he deduced from the theory of Abelian functions. Clebsch's paper, Ueber die Aiiwendimg der Abel' schen Functioiien in der Geometrie,-\ and the work of Clebsch and Gordan on Abelian functions.^ are well known to American mathematicians ; and in accordance with my plan, I proceed to give merely some critical remarks. * Theorie der AbeVschen Functionen, Journal fiir reine und angewandte Mathe- matik, Vol. 54 (1857), pp. 1 15-155; reprinted in Riemann's Werke, 1876, pp. 81-135. t Journal fur reine und angewandte Mathematik, Vol. 63 (1864), pp. 189-243. \ Theorie der AbeV schen Functionen, Leipzig, Teubner, 1866. 4 LECTURE I. However great the achievement of Clebsch's in making the work of Riemann more easy of access to his contempo- raries, it is my opinion that at the present time the book of Clebsch is no longer to be considered as the standard work for an introduction to the study of Abelian functions. The chief objections to Clebsch's presentation are twofold : they can be briefly characterized as a lack of mathematical rigour on the one hand, and a loss of intuitiveness, of geometrical perspicuity, on the other. A few examples will explain my meaning. (a) Clebsch bases his whole investigation on the considera- tion of what he takes to be the most general type of an algebraic curve, and this general curve he assumes as having only double points, but no other singularities. To obtain a sure foundation for the theory, it must be proved that any algebraic curve can be transformed rationally into a curve having only double points. This proof was not given by Clebsch ; it has since been supplied by his pupils and follow- ers, but the demonstration is long and involved. See the papers by Brill and Nother in the Math. Annalen, Vol. 7 (1874),* and by Nother, ib., Vol. 23 (1884).! Another defect of the same kind occurs in connection with the determinant of the periods of the Abelian integrals. This determinant never vanishes as long as the curve is irredu- cible. But Clebsch and Gordan neglect to prove this ; and however simple the proof may be, this must be regarded as an inexactness. The apparent lack of critical spirit which we find in the work of Clebsch is characteristic of the geometrical epoch in which • Ueber die algebraischen Functionen und ihre Anwendung in der Geometric, pp. 269-310. t Rationale Ausjiihrung der Operationen in der Theorie der algebraischen Func- tionen, pp. 311-358. CLEBSCH. 5 he lived, the epoch of Steiner, among others. It detracts in no- wise from the merit of his work. But the influence of the theory of functions has taught the present generation to be more exacting. (b) The second objection to adopting Clebsch's presentation lies in the fact that, from Riemann's point of view, many points of the theory become far more simple and almost self-evident, whereas in Clebsch's theory they are not brought out in all their beauty. An example of this is presented by the idea of the deficiency p. In Riemann's theory, where / represents the order of connectivity of the surface, the invariability of / under any rational transformation is self-evident, while from the point of view of Clebsch this invariability must be proved by means of a long elimination, without affording the true geometrical insight into its meaning. For these reasons it seems to me best to begin the theory of Abelian functions with Riemann's ideas, without, however, neglecting to give later the purely algebraical developments. This method is adopted in my paper on Abelian functions ; * it is also followed in the work Die elliptisclien Modulfitnctionen, Vols. I. and II., edited by Dr. Fricke. A general account of the historical development of the theory of algebraic curves in con- nection with Riemann's ideas will be found in my (lithographed) lectures on Riemann sche Fldchen, delivered in 1891-92.! If this arrangement be adopted, it is interesting to follow out the true relation that the algebraical developments bear to Riemann's theory. Thus in Brill and Nother's theory, the so-called fundamental theorem of Nother is of primary impor- * Zur Theorie der Abel'schen Functioiien, Math. Annalen, Vol. 36 (1890), pp. 1-83. t My lithographed lectures frequently give only an outline of the subject, omit- ting details and long demonstrations, which are supposed to be supplied by the student by private reading and a study of the literature of the subject. 6 LECTURE I. tance. It gives a rule for deciding under what conditions an algebraic rational integral function f of x and j can be put into the form /=A+i " diasymmetric " and " orthosymmetric " cases. If we denote as a line of symmetry any line whose points ,2 LECTURE IV. remain unchanged by the conformal transformation, the dia- symmetric cases contain respectively p, p—i,---2, i, o lines of symmetry, and the orthosymmetric cases contain p+\, p—i, ^_3, ... such lines. A surface is called diasymmetric or ortho- symmetric according as it does not or does break up into two parts by cuts carried along all the lines of symmetry. This enumeration, then, will contain a general classification of real curves, as indicated first in my pamphlet on Riemann's theory.* In the summer of 1892 I resumed the theory and developed a large number of propositions concerning the reality of the roots of those equations connected with our curves that can be treated by means of the Abelian integrals. Compare the last volume of the Math. A>malen\ and my (lithographed) lectures on Riemami scJie Fldchen, Part II. In the same manner in which we have to-day considered ordinary algebraic curves and surfaces, it would be interesting to investigate all algebraic configurations so as to arrive at a truly geometrical intuition of these objects. In concluding, I wish to insist in particular on what I regard as the principal characteristic of the geometrical methods that I have discussed to-day : these methods give us an actual mental image of the configuration under discussion, and this I consider as most essential in all true geometry. For this reason the so-called synthetic methods, as usually developed, do not appear to me very satisfactory. While giving elaborate constructions for special cases and details they fail entirely to afford a general view of the configurations as a whole. ♦ Ueher Riemann's Theorie der algehraischen Functionen und ihrer Integrate, Leipzig, Teubner, 1882. An English translation by Frances Hardcastle (London, Macmillan) has just appeared. t Ueber Kealitdtsverhdltnisse bei der einem beliebigen Geschlechte zu^ehorigen Normalcurve der (p. Vol. 42 (1893), PP- I-29. Lecture V.: THEORY OF FUNCTIONS AND GEOMETRY. (September i, 1893.) A GEOMETRICAL representation of a function of a complex variable w=f{2), where w = u + iv and s=x+ij, can be ob- tained by constructing models of the two surfaces n = (^{x,y), v = yjr(x,j/). This idea is realized in the models constructed by Dyck, which I have shown to you at the Exhibition. Another well-known method, proposed by Riemann, consists in representing each of the two complex variables in the usual way in a plane. To every point in the ^-plane will correspond one or more points in the ly-plane ; as z moves in its plane, w describes a corresponding curve in the other plane. I may refer to the work of Holzmiiller* as a good elementary intro- duction to this subject, especially on account of the large number of special cases there worked out and illustrated by drawings. In higher investigations, what is of interest is not so much the corresponding curves as corresponding areas or regions of the two planes. According to Riemann's fundamental theorem concerning conformal representation, two simply con- nected regions can always be made to correspond to each other conformally, so that either is the conformal representation * Einfiihrung in die Theorie der isogonalen Verwandtschaften und der conformcn Abbildungen, verbunden mit Anwendungen auf mathematische Physik, Leipzig, Teubner, 1882. 34 LECTURE V. {Abbildung) of the other. The three constants at our disposal in this correspondence allow us to select three arbitrary points on the boundary of one region as corresponding to three arbi- trary points on the boundary of the other region. Thus Riemann's theory affords a geometrical definition for any func- tion whatever by means of its conformal representation. This suggests the inquiry as to what conclusions can be drawn from this method concerning the nature of transcen- dental functions. Next to the elementary transcendental func- tions the elliptic functions are usually regarded as the most important. There is, however, another class for which at least equal importance must be claimed on account of their numerous applications in astronomy and mathematical physics ; these are the hypergeoinetric ftmctions, so called owing to their connection with Gauss's hypergeometric series. The hypergeometric functions can be defined as the integrals of the following linear differential equation of the second order: d-w . z — a z — b '<'->('-')]f + ' \'\"{a-b){a-c) 2 — a ^ t^<^"{b-c^{h-a) ^ v'v<'(c-a){c-b) ' z — b z— c {z—d) {z—b) (z—c) where z=a, b, c are the three singular points and X', X" ; /i', /i" ; v' , v" are the so-called exponents belonging respectively to a, b, c. If Wj be a particular solution, w,^ another, the general solution can be put in the form aw-^^^w^, where a, y8 are arbitrary con- stants ; so that az£/j + ^W2 and yzc/j -|- hw^ represent a pair of general solutions. THEORY OF FUNCTIONS AND GEOMETRY. 35 If we now introduce the quotient ^ = 17(2) as a new variable, Ti/„ its most general value is ""'i + P^a _ <^V + P ^^^ contains there- yw^ + 6W2 717 + S fore three arbitrary constants. Hence t) satisfies a differential equation of the third order which is readily found to be (z—a){z—b){z — c) l—{a-b){_a-c)+-J—{b-c){l>-a) — a z — o + ■ ■{c-a){c-b) in which the left-hand member has the property of not being changed by a linear substitution, and is therefore called a differ- ential invariant. Cayley has named this function the Schwar- zian derivative ; it has formed the starting-point for Sylvester's investigations on reciprocants. In the right-hand member, ±X = X'-X", ±,. = /-ja", ±v = v'-v". As to the conformal representation (Fig. 6), it can be shown that the upper half of the ^■-plane, with the points a, b, c on Fig. 6. the real axis and \, /i, v assumed as real, is transformed for each branch of the 7;-function into a triangular area abc bounded by 26 LECTURE V. three circular arcs ; let us call such an area a circular triangle {Kreisbogendreieck). The angles at the vertices of this triangle are Xir, /xtt, vtt. This, then, is the geometrical representation we have to take as our basis. In order to derive from it conclusions as to the nature of the transcendental functions defined by the differential equation, it will evidently be necessary to inquire what are the forms of such circular triangles in the most general case. For it is to be noticed that there is no restric- tion laid upon the values of the constants \, /a, v, so that the angles of our triangle are not necessarily acute, nor even convex ; in other words, in the general case the vertices will be branch-points. The triangle itself is here to be regarded as something like an extensible and flexible membrane spread out between the circles forming the boundary. I have investigated this question in a paper published in the Math. Annalen, Vol. 37.* It will be convenient to project the plane containing the circular triangle stereographically on a sphere. The question then is as to the most general form of spherical triangles, taking this term in a generalized meaning as denoting any triangle on the sphere bounded by the inter- sections of three planes with the sphere, whether the planes intersect at the centre or not. This is really a question of elementary geometry ; and it is interesting to notice how often in recent times higher re- search has led back to elementary problems not previously settled. The result in the present case is that there are two, and only two, species of such generalized triangles. They are obtained from the so-called elementary triangle by two distinct operations : {a) lateral, (b) polar attachment of z. circle. • Ueber die Nullstellen de.r hypergeometrischen Reihe, pp. 573-590. THEORY OF FUNCTIONS AND GEOMETRY. 37 Let abc (Fig. 7) be the elementary spherical triangle. Then the operation of lateral attachment consists in attaching to the area abc the area enclosed by one of the sides, say be, this side being produced so as to form a complete circle. The process can, of course, be repeated any number of times and applied to each side. If one circular area be attached at be, the angles at b and e are increased each by tt ; if the whole sphere be attached, by 2ir, etc. The vertices in this way become branch-points. A triangle so obtained I call a triangle of the first species. Fig. 7. Fig. 8. A triangle of the second species is produced by the process of polar attachment of a circle, say at be; the whole area bounded by the circle be is, in this case, connected with the original triangle along a branch-cut reaching from the vertex a to some point on be. The point a becomes a branch-point, its angle being increased by 2 v. Moreover, lateral attach- ments can be made at ab and ae. The two species of triangles are now characterized as follows : the first species may have any nnmber of lateral attachments at any or all of the three sides, while the second has a polar attachment to one vertex and the opposite side, and may liave lateral attachments to the other two sides. 38 LECTURE V. Analytically the two species are distinguished by inequali- ties between the absolute values of the constants A., /*, v. .For the first species, none of the three constants is greater than the sum of the other two, i.e. for the second species, where X refers to the pole. For the application to the theory of functions, it is impor- tant to determine, in the case of the second species, the number of times the circle formed by the side opposite the vertex is passed around. I have found this number to be E \- — - — — — — — ), where E denotes the greatest positive integer contained in the argument, and is therefore always zero when this argument happens to be negative or fractional. Let us now apply these geometrical ideas to the theory of hypergeometric functions. I can here only point out one of the results obtained. Considering only the real values that 'x) = wjw^ can assume between a and b, the question presents itself as to the shape of the 97-curve between these limits. Let us consider for a moinent the curves Wj and w^. It is well known that, if w^ oscillates between a and b from one side of the axis to the other, w^ will also oscillate ; their quotient y) = wjiv^ is represented by a curve that consists of separate branches extending from —00 to -foo, somewhat like the curve y=\.z.nx. Now it appears as the result of the investigation that the number of these branches, and therefore the number of the oscillations of w-^ and w^, is given precisely by the number of circuits of the point c ; that is to say, it is Ji[ j. This is a result of importance for all THEORY OF FUNCTIONS AND GEOMETRY. 39 applications of hypergeometric functions which was derived only later (by Hurwitz) by means of Sturm's methods. I wish to call your particular attention not so much to the result itself, however interesting it may be, as to the geometrical method adopted in deriving it. More advanced researches on a similar line of thought are now being carried on at Gottingen by myself and others. When a differential equation with a larger number of singular points than three is the object of investigation, the triangles must be replaced by quadrangles and other polygons. In my lithographed lectures on Linear Differential Equations, delivered in 1890-91, I have thrown out some suggestions regarding the treatment of such cases. The difficulty arising in these generalizations is, strange to say, merely of a geometrical nature, viz. the difficulty of obtaining a general view of the possible forms of the polygons. Meanwhile, Dr. Schoenflies has published a paper on recti- linear polygons of any number of sides * while Dr. Van Vleck has considered such rectilinear polygons together with the functions they define, the polygons being defined in so general a way as to admit branch-points even in the interior. Dr. Schoenflies has also treated the case of circular quadrangles, the result being somewhat complicated. In all these investigations the singular points of the ^-plane corresponding to the vertices of the polygons are of course assumed to be real, as are also their exponents. There remains the still more general question how to represent by conformal correspondence the functions in the case when some of these elements are complex. In this direction I have to mention the name of Dr. Schilling who has treated the case of the ordinary hypergeometric function on the assumption of complex exponents. * Veber Kreisbogmpolygone, Math. Annalen, Vol. 42, pp. 377-408. 40 LECTURE V. This treatment of the functions defined by linear differential equations of the second order is of course only an example of the general discussion of complex functions by means of geometry. I hope that many more interesting results will be obtained in the future by such geometrical methods. Lecture VI.: ON THE MATHEMATICAL CHAR- ACTER OF SPACE-INTUITION AND THE RELATION OF PURE MATHEMATICS TO THE APPLIED SCIENCES. (September 2, 1893.) In the preceding lectures I have laid so much stress on geometrical methods that the inquiry naturally presents itself as to the real nature and limitations of geometrical intuition. In my address before the Congress of Mathematics at Chi- cago I referred to the distinction between what I called the naive and the refined intuition. It is the latter that we find in Euclid ; he carefully develops his system on the basis of well- formulated axioms, is fully conscious of the necessity of exact proofs, clearly distinguishes between the commensurable and incommensurable, and so forth. The naive intuition, on the other hand, was especially active during the period of the genesis of the differential and integral calculus. Thus we see that Newton assumes without hesitation the existence, in every case, of a velocity in a moving point, without troubling himself with the inquiry whether there might not be continuous functions having no derivative. At the present time we are wont to build up the infinitesi- mal calculus on a purely analytical basis, and this shows that we are living in a critical period similar to that of Euclid. It is my private conviction, although I may perhaps not be able to fully substantiate it with complete proofs, that Euclid's 41 42 LECTURE VI. period also must have been preceded by a "nafve" stage of development. Several facts that have become known only quite recently point in this direction. Thus it is now known that the books that have come down to us from the time of Euclid constitute only a very small part of what was then in existence ; moreover, much of the teaching was done by oral tradition. Not many of the books had that artistic finish that we admire in Euclid's " Elements " ; the majority were in the form of improvised lectures, written out for the use of the students. The investigations of Zeuthen * and Allman f have done much to clear up these historical conditions. If we now ask how we can account for this distinction between the naive and refined intuition, I must say that, in my opinion, the root of the matter lies in the fact that the naive mtiiition is not exact, while the refined intuition is not properly intuition at all, but arises through the logical develop- ment from axioms considered as perfectly exact. To explain the meaning of the first half of this statement it is my opinion that, in our naive intuition, when thinking of a point we do not picture to our mind an abstract mathemati- cal point, but substitute something concrete for it. In imagin- ing a line, we do not picture to ourselves "length without breadth," but a strip of a certain width. Now such a strip has of course always ^ a tangent (Fig. 9) ; i.e. we can always imagine a straight strip having a small portion (element) in common with the curved strip ; similarly with respect to the osculating circle. The definitions in this case are regarded as holding only approximately, or as far as may be necessary. * Die Lehre von den KegekchniUen im Altertum, ubersetzt von R. v. Fischer- Benzon, Kopenhagen, Host, 1886. t Greek geometry from Thales to Euclid, Dublin, Hodges, 1889. MATHEMATICAL CHARACTER OF SPACE-INTUITION. 43 The " exact " mathematicians will of course say that such definitions are not definitions at all. But I maintain that in ordinary life we actually operate with such inexact definitions. Thus we speak without hesitancy of the direction and curvature of a river or a road, although the " line" in this case has certainly considerable width. As regards the second half of my proposition, there actually are many cases where the conclusions derived by purely logical reasoning from exact definitions can no more be verified by intuition. To show this, I select examples from the theory of automorphic functions, because in more common geometrical illustrations our judgment is warped by the familiarity of the ideas. Let any number of non-intersecting circles i, 2, 3, 4, •••, be given (Fig. 10), and let every circle be reflected {i.e. transformed Fig. la by inversion, or reciprocal radii vectores) upon every other circle ; then repeat this operation again and again, ad infinitiiin. The question is, what will be the configuration formed by the totality 44 LECTURE VI. of all the circles, and in particular what will be the position of the limiting points. There is no difficulty in answering these questions by purely logical reasoning; but the imagination seems to fail utterly when we try to form a mental image of the result. Again, let a series of circles be given, each circle touching the following, while the last touches the first (Fig. 1 1). Every circle is now reflected upon every other just as in the preceding exam- ple, and the process is repeated indefinitely. The special case when the original points of contact happen to he on a circle Fig. II. being excluded, it can be shown analytically that the continuous curve which is the locus of all the points of contact is not an analytic curve. The points of contact form a manifoldness that is everywhere dense on the curve (in the sense of G. Cantor), although there are intermediate points between them. At each of the former points there is a determinate tangent, while there is none at the intermediate points. Second deriv- atives do not exist at all. It is easy enough to imagine a strip covering all these points ; but when the width of the strip is reduced beyond a certain limit, we find undulations, and it seems impossible to clearly picture to the mind the final outcome. It is to be noticed that we have here an example of a curve MATHEMATICAL CHARACTER OF SPACE-INTUITION. 45 with indeterminate derivatives arising out of purely geometrical considerations, while it might be supposed from the usual treatment of such curves that they can only be defined by artificial analytical series. Unfortunately, I am not in a position to give a full account of the opinions of philosophers on this subject. As regards the more recent mathematical literature, I have presented my views as developed above in a paper published in 1873, and since reprinted in the Math. Annalen* Ideas agreeing in general with mine have been expressed by Pasch, of Giessen, in two works, one on the foundations of geometry,f the other on the principles of the infinitesimal calculus.^ Another author, Kopcke, of Hamburg, has advanced the idea that our space-intuition is exact as far as it goes, but so limited as to make it impossible for us to picture to ourselves curves with- out tangents.§ On one point Pasch does not agree with me, and that is as to the exact value of the axioms. He believes — and this is the traditional view — that it is possible finally to discard intuition entirely, basing the whole science on the axioms alone. I am of the opinion that, certainly, for the purposes of research it is always necessary to combine the intuition with the axioms. I do not believe, for instance, that it would have been possible to derive the results discussed in my former lectures, the splendid researches of Lie, the continuity of the shape of algebraic curves and surfaces, or the most general forms of triangles, without the constant use of geometrical intuition. * Ueber den allgemeinen Functionsbegriff und desscn Darstellung durch eine willkiirliche Curve, Math. Annalen, Vol. 22 (1883), pp. 249-259. t Vorlesungen iiber neuere Geometrie, Leijjrig, Teubner, 1882. J Einleitung in die Differential- und Integralrechnung, Leipzig, Teubner, 1882. § Ueber Differentiirbarkeit und Amchaulichkeit der sietigen Functionen, Math. Annalen, Vol. 29 (1887), pp. 123-140. 46 LECTURE VI. Pasch's idea of building up the science purely on the basis of the axioms has since been carried still farther by Peano, in his logical calculus. Finally, it must be said that the degree of exactness of the intuition of space may be different in different individuals, per- haps even in different races. It would seem as if a strong naive space-intuition were an attribute pre-eminently of the Teutonic race, while the critical, purely logical sense is more fully developed in the Latin and Hebrew races. A full investi- gation of this subject, somewhat on the lines suggested by Francis Galton in his researches on heredity, might be inter- esting. What has been said above with regard to geometry ranges this science among the applied sciences. A few general remarks on these sciences and their relation to pure mathe- matics will here not be out of place. From the point of view of pure mathematical science I should lay particular stress on the heuristic value of the applied sciences as an aid to discov- ering new truths in mathematics. Thus I have shown (in my little book on Riemann's theories) that the Abelian integrals can best be understood and illustrated by considering electric currents on closed surfaces. In an analogous way, theorems concerning differential equations can be derived from the con- sideration of sound-vibrations ; and so on. But just at present I desire to speak of more practical mat- ters, corresponding as it were to what I have said before about the inexactness of geometrical intuition. I believe that the more or less close relation of any applied science to mathematics might be characterized by the degree of exactness attained, or attainable, in its numerical results. Indeed, a rough classifi- cation of these sciences could be based simply on the number of significant figures averaged in each. Astronomy (and some branches of physics) would here take the first rank ; the num- MATHEMATICS AND THE APPLIED SCIENCES. 47 ber of significant figures attained may here be placed as high as seven, and functions higher than the elementary transcendental functions can be used to advantage. Chemistry would probably be found at the other end of the scale, since in this science rarely more than two or three significant figures can be relied upon. Geometrical drawing, with perhaps 3 to 4 figures, would rank between these extremes ; and so we might go on. The ordinary mathematical treatment of any applied science substitutes exact axioms for the approximate results of experi- ence, and deduces from these axioms the rigid mathematical conclusions. In applying this method it must not be forgotten that mathematical developments transcending the limit of exact- ness of the science are of no practical value. It follows that a large portion of abstract mathematics remains without finding any practical application, the amount of mathematics that can be usefully employed in any science being in proportion to the degree of accuracy attained in the science. Thus, while the astronomer can put to good use a wide range of mathemati- cal theory, the chemist is only just beginning to apply the first derivative, i.e. the rate of change at which certain processes are going on ; for second derivatives he does not seem to have found any use as yet. As examples of extensive mathematical theories that do not exist for applied science, I may mention the distinction between the commensurable and incommensurable, the investigations on the convergency of Fourier's series, the theory of non-analytical functions, etc. It seems to me, therefore, that Kirchhoff makes a mistake when he says in his Spectral-Analyse that absorption takes place only when there is exact coincidence between the wave-lengths. I side with Stokes, who says that absorption takes place in the vicinity of such coincidence. Similarly, when the astronomer says that the periods of two planets must be exactly commensurable to admit the possibility of a collision, 48 LECTURE VI. this holds only abstractly, for their mathematical centres ; and it must be remembered that such things as the period, the mass, etc., of a planet cannot be exactly defined, and are changing all the time. Indeed, we have no way of ascertaining whether two astronomical magnitudes are incommensurable or not ; we can only inquire whether their ratio can be expressed approxi- mately by two small integers. The statement sometimes made that there exist only analytic functions in nature is in my opinion absurd. All we can say is that we restrict ourselves to analytic, and even only to simple analytic, functions because they afford a sufficient degree of approximation. Indeed, we have the theorem (of Weierstrass) that any continuous function can be approximated to, with any required degree of accuracy, by an analytic function. Thus if <^{x) be our continuous func- tion, and h a. small quantity representing the given limit of exactness (the width of the strip that we substitute for the curve), it is always possible to determine an analytic function f(x) such that ^(*)=/(^)-t-€, where | c | < | S |, within the given limits. All this suggests the question whether it would not be pos- sible to create a, let us say, abridged system of mathematics adapted to the needs of the applied sciences, without passing through the whole realm of abstract mathematics. Such a system would have to include, for example, the researches of Gauss on the accuracy of astronomical calculations, or the more recent and highly interesting investigations of Tchebycheff on interpolation. The problem, while perhaps not impossible, seems difficult of solution, mainly on account of the somewhat vague and indefinite character of the questions arising. I hope that what I have here said concerning the use of mathematics in the applied sciences will not be interpreted MATHEMATICS AND THE APPLIED SCIENCES. 49 as in any way prejudicial to the cultivation of abstract mathe- matics as a pure science. Apart from the fact that pure mathematics cannot be supplanted by anything else as a means for developing the purely logical powers of the mind, there must be considered here as elsewhere the necessity of the presence of a few individuals in each country developed in a far higher degree than the rest, for the purpose of keeping up and gradually raising the ge7ieral standard. Even a slight raising of the general level can be accomplished only when some few minds have progressed far ahead of the average. Moreover, the "abridged" system of mathematics referred to above is not yet in existence, and we must for the present deal with the material at hand and try to make the best of it. Now, just here a practical difficulty presents itself in the teaching of mathematics, let us say of the elements of the differential and integral calculus. The teacher is confronted with the problem of harmonizing two opposite and almost con- tradictory requirements. On the one hand, he has to consider the limited and as yet undeveloped intellectual grasp of his students and the fact that most of them study mathematics mainly with a view to the practical applications ; on the other, his conscientiousness as a teacher and man of science would seem to compel him to detract in nowise from perfect mathe- matical rigour and therefore to introduce from the beginning all the refinements and niceties of modern abstract mathe- matics. In recent years the university instruction, at least in Europe, has been tending more and more in the latter direc- tion ; and the same tendencies will necessarily manifest them- selves in this country in the course of time. The second edition of the Cours d'analyse of Camille Jordan may be regarded as an example of this extreme refinement in laying the foundations of the infinitesimal calculus. To place a work of this character in the hands of a beginner must necessarily CO LECTURE VI. have the effect that at the beginning a large part of the sub- ject will remain unintelligible, and that, at a later stage, the student will not have gained the power of making use of the principles in the simple cases occurring in the applied sciences. It is my opinion that in teaching it is not only admissible, but absolutely necessary, to be less abstract at the start, to have constant regard to the applications, and to refer to the refinements only gradually as the student becomes able to understand them. This is, of course, nothing but a universal pedagogical principle to be observed in all mathematical instruction. Among recent German works I may recommend for the use of beginners, for instance, Kiepert's new and revised edition of Stegemann's text-book ; * this work seems to combine sim- plicity and clearness with sufficient mathematical rigour. On the other hand, it is a matter of course that for more advanced students, especially for professional mathematicians, the study of works like that of Jordan is quite indispensable. I am led to these remarks by the consciousness of a growing danger in the higher educational system of Germany, — the danger of a separation between abstract mathematical science and its scientific and technical applications. Such separation could only be deplored ; for it would necessarily be followed by shallowness on the side of the applied sciences, and by isolation on the part of pure mathematics. * Grundriss der Differential- und Integral- Rechnung, 6te Auflage, herausgegeben von Kiepert, Hannover, Helwing, 1892. Lecture VII. : THE TRANSCENDENCY OF THE NUMBERS e AND tt. (September 4, 1893.) Last Saturday we discussed inexact mathematics ; to-day we shall speak of the most exact branch of mathematical science. It has been shown by G. Cantor that there are two kinds of infinite manifoldnesses : (a) countable {abzdJdbare) manifold- nesses, whose quantities can be numbered or enumerated so that to each quantity a definite place can be assigned in the system ; and {b) non-countable manifoldnesses, for which this is not possi- ble. To the former group belong not only the rational numbers, but also the so-called algebraic numbers, i.e. all numbers defined by an algebraic equation, a + a^x ■{■ a.^ 4- •• • + a„x"- = o with integral coefficients (« being of course a positive integer). As an example of a non-countable manifoldness I may mention the totality of all numbers contained in a continttmn, such as that formed by the points of the segment of a straight line. Such a continuum contains not only the rational and algebraic numbers, but also the so-called transcendental numbers. The actual existence of transcendental numbers which thus naturally follows from Cantor's theory of manifoldnesses had been proved before, from considerations of a different order, by Liouville. With this, however, is not yet given any means for deciding whether any particular number is transcendental or not. But 5' 52 LECTURE VII. during the last twenty years it has been established that the two fundamental numbers e and it are really transcendental. It is my object to-day to give you a clear idea of the very simple proof recently given by Hilbert for the transcendency of these two numbers. The history of this problem is short. Twenty years ago, Hermite * first established the transcendency of e ; i.e. he showed, by somewhat complicated methods, that the number e cannot be the root of an algebraic equation with integral coefficients. Nine years later, Lindemann,f taking the develop- ments of Hermite as his point of departure, succeeded in proving the transcendency of tt. Lindemann's work was verified soon after by Weierstrass. The proof that tt is a transcendental number will forever mark an epoch in mathematical science. It gives the final answer to the problem of squaring the circle and settles this vexed question once for all. This problem requires to derive the number tt by a finite number of elementary geometrical processes, i.e. with the use of the ruler and compasses alone. As a straight line and a circle, or two circles, have only two intersections, these processes, or any finite combination of them, can be expressed algebraically in a comparatively simple form, so that a solution of the problem of squaring the circle would mean that tt can be expressed as the root of an algebraic equation of a comparatively simple kind, viz. one that is solvable by square roots. Lindemann's proof shows that tt is not the root of any algebraic equation. The proof of the transcendency of tt will hardly diminish the number of circle-squarers, however ; for this class of people has always shown an absolute distrust of mathematicians and a * Comptes rendus, Vol. 77 (1873), p. 18, etc. t Math. Annalen, Vol. 20 (1882), p. 213. TRANSCENDENCY OF THE NUMBERS e AND tt. S3 contempt for mathematics that cannot be overcome by any amount of demonstration. But Hilbert's simple proof will surely be appreciated by all those who take interest in the establishment of mathematical truths of fundamental impor- tance. This demonstration, which includes the case of the number e as well as that of tt, was published quite recently in the Gottinger Nachricliten* Immediately after f Hurwitz published a proof for the transcendency of e based on still more elementary principles ; and finally, Gordan \ gave a fur- ther simplification. All three of these papers will be reprinted in the next Heft of the Math. Annalen.% The problem has thus been reduced to such simple terms that the proofs for the transcendency of e and tt should henceforth be introduced into university teaching everywhere. Hilbert's demonstration is based on two propositions. One of these simply asserts the transcendency of e, i.e. the impos- sibility of an equation of the form a + aie + a^-\ \-a„e" = o, (i) where a, a^ a^, ■■■a„ are integral numbers. This is the original proposition of Hermite. To prove the transcendency of tt, another proposition (originally due to Lindemann) is required, which asserts the impossibility of an equation of the form a4--f i)!. It follows that /"i is an integer, viz. />i = ±a(«!)'^i [mod. (p 4-1)]. If, therefore, p be selected so as to make the right-hand mem- ber of this congruence not divisible by p+i, the whole expres- sion Pj is different from zero. As regards the condition that P^ should be made as small as we please, it can evidently be fulfilled by selecting a suffi- ciently large value for p ; this is of course consistent with the condition of making y not divisible by p+i. For by the theorem of mean values {Mittelwertsats) the integrals can be replaced by powers of constant quantities with p in the expo- -6 LECTURE VII. nent ; and the rate of increase of a power is, for sufficiently large values of p, always smaller than that of the factorial which occurs in the denominator. The proof of the impossibility of equation (2) proceeds on precisely analogous lines. Instead of the integral / we have now to use the integral the /S's being the roots of the algebraic equation This integral is decomposed as follows : r= r+ r, Jo Jo J^ where of course the path of integration must be properly determined for complex values of /3. For the details I must refer you to Hilbert's paper. Assuming the impossibility of equation (2), the transcendency of TT follows easily from the following considerations, originally given by Lindemann. We notice first, as a consequence of our the- orem, that, with the exception of the point x=o, jy=i, the exponen- tial curve y=e^ has no algebraic point, i.e. no point both of whose co-ordinates are algebraic num- bers. In other words, however densely the plane may be covered with algebraic points, the exponential curve (Fig. 12) manages to pass along the plane without meeting them, the single point (o, i) excepted. This curious result can be deduced as follows from the impossibility of equation (2). Let j/ be any algebraic Fig. 12. TRANSCENDENCY OF THE NUMBERS e AND -k. 57 quantity, i.e. a root of any algebraic equation, and let y-^, y^,-- be the other roots of the same equation ; let a similar notation be used for x. Then, if the exponential curve have any alge- braic point (x, y), (besides ;ir=o, 7=1), the equation )■ =0 J must evidently be fulfilled. But this equation, when multiplied out, has the form of equation (2), which has been shown to be impossible. As second step we have only to apply the well-known identity which is a special case of y = e' Since in this identity ^=1 is algebraic, x=itr must be transcendental. Lecture VIII. : IDEAL NUMBERS. (September 5, 1893.) The theory of numbers is commonly regarded as something exceedingly difficult and abstruse, and as having hardly any connection with the other branches of mathematical science. This view is no doubt due largely to the method of treatment adopted in such works as those of Kummer, Kronecker, Dede- kind, and others who have, in the past, most contributed to the advancement of this science. Thus Kummer is reported as having spoken of the theory of numbers as the only pure branch of mathematics not yet sullied by contact with the applications. Recent investigations, however, have made it clear that there exists a very intimate correlation between the theory of num- bers and other departments of mathematics, not excluding geometry. As an example I may mention the theory of the reduction of binary quadratic forms as treated in the Elliptische Modul- fiinctionen. An extension of this method to higher dimensions is possible without serious difficulties. Another example you will remember from the paper by Minkowski, Ueber Eigen- schaften von ganzen Zahlen, die durch rdumliche Anschauiing erschlossen sind, which I had the pleasure of presenting to you in abstract at the Congress of Mathematics. Here geom- etry is used directly for the development of new arithmetical ideas. 58 IDEAL NUMBERS. eg To-day I wish to speak on the composition of binary algebraic forms, a subject first discussed by Gauss in his Disquisitiones arithmeticcz * and of Kummer's corresponding theory of ideal numbers. Both these subjects have always been considered as very abstruse, although Dirichlet has somewhat simplified the treatment of Gauss. I trust you will find that the geometrical considerations by means of which I shall treat these questions introduce so high a degree of simplicity and clearness that for those not familiar with the older treatment it must be diiificult to realize why the subject should ever have been regarded as so very intricate. These considerations were indicated by myself in the Gottinger Nachrichten for January, 1893 ; and at the beginning of the summer semester of the present year I treated them in more extended form in a course of lectures. I have since learned that similar ideas were proposed by Poincare in 1881 ; but I have not yet had sufficient leisure to make a comparison of his work with my own. I write a binary quadratic form as follows : f=ay?-\- bxy -f cf, i.e. without the factor 2 in the second term ; some advantages of this notation were recently pointed out by H. Weber, in the Gottinger Nachrichten, 1892-93. The quantities a, b, c, x, y are here of course all assumed to be integers. It is to be noticed that in the theory of numbers a common factor of the coefficients a, b, c cannot be introduced or omitted arbitrarily, as in projective geometry ; in other words, we are concerned with the form, not with an equation. Hence we maike the supposition that the coefficients a, b, c have no common factor ; a form of this character is called a frimitive form. * In the 5th section ; see Gauss's Werke, Vol. I, p. 239. 6o LECTURE VIII. As regards the discriminant D = lr /^ac. we shall assume that it has no quadratic divisor (and hence cannot be itself a square), and that it is different from zero. Thus D is either = o or = i (mod. 4). Of the two cases, Do, which have to be considered separately, I select the former as being more simple. Both cases were treated in my lectures referred to before. The following elementary geometrical interpretation of the binary quadratic form was given by Gauss, who was much inclined to using geometrical considerations in all branches of mathematics. Construct a parallelogram (Fig. 13) with two Fig. 13. adjacent sides equal to Va, Vt, respectively, and the included angle <^ such that cos ^ = — —, As P-4ac) of the lattice will represent simply the complex number x + V-^iB-y; such numbers we shall call principal numbers. In any system of numbers the laws of multiplication are of prime importance. For our principal numbers it is easy to prove that the product of any two of them always gives a principal number; i.e. the system of principal numbers is, for multiplication, complete in itself. We proceed next to the consideration of lattices of discrimi- nant D that do not belong to the principal class ; let us call them secondary lattices {Nebetigitter). Before investigating the laws of multiplication of the corresponding numbers, I must call attention to the fact that there is one feature of arbitrari- ness in our representation that has not yet been taken into account ; this is the orientation of the lattice, which may be regarded as given by the angles, -^ and ■)(, made by the sides 64 LECTURE VIII. Vrt, Vc, respectively, with some fixed initial line (Fig. i6). For the angle <}> of the parallelogram we have evidently ^ = ;^; — i/r. The point {x, y) of the lattice will thus give the complex number ^j, 2Va -I which we call a secondary number. The definition of a secondary number is therefore indeterminate as long as i^ or ;^ is not fixed. Now, by determining i^ properly for every secondary point- lattice, it is always possible to bring about the important result Fig. 16. that the product of any two complex numbers of all our lattices taken together will again be a complex number of the system, so that the totality of these complex numbers forms, likewise, for multiplication, a complete system. Moreover, the multiplication combines the lattices in a definite way ; thus, if any number belonging to the lattice Zj be multiplied into any number of the lattice L^, we always obtain a number belonging to a definite lattice L^. These properties will be seen to correspond exactly to the characteristic properties of Gauss's composition of algebraic forms. For Gauss's law merely asserts that the product of IDEAL NUMBERS. gc two ordinary numbers that can be represented by two primitive forms /i, /j of discriminant D is always representable by a definite primitive form /g of discriminant D. This law is included in the theorem just stated, inasmuch as the values of ^fv ^fv y/3 represent the distances of the points in the lattices from the origin. At the same time we notice that Gauss's law is not exactly equivalent to our theorem, since in the multiplication of our complex numbers, not only the distances are multiplied, but the angles ^ are added. It is not impossible that Gauss himself made use of similar considerations in deducing his law, which, taken apart from this geometrical illustration, bears such an abstruse character. It now remains to explain what relation these investigations have to the ideal numbers of Kummer. This involves the question as to the division of our complex numbers and their resolution into primes. In the ordinary theory of real numbers, every number can be resolved into primes in only one way. Does this fundamental law hold for our complex numbers .' In answering this question we must distinguish between the system formed by the totality of all our complex numbers and the system of principal numbers alone. For the former system the answer is : yes, every com- plex number can be decomposed into complex primes in only one way. We shall not stop to consider the proof which is directly contained in the ordinary theory of binary quadratic forms. But if we proceed to the consideration of the system of principal numbers alone, the matter is different. There are cases when a principal number can be decomposed in more than one way into prime factors, i.e. principal numbers not decomposable into principal factors. Thus it may happen that we have m-^m^ = n^n^; m^, m^, «i, «2 being principal primes. The reason is, that these principal numbers are no longer primes 66 LECTURE VIII. if we adjoin the secondary numbers, but are decomposable as follows : OTi = « • y3, W2 = y • 8, «i = « • y, »2 = ^ • S, ") /3. 7. S being primes in the enlarged system. /« investigating the laws of division it is therefore not convenient to consider the principal system by itself ; it is best to introduce the secondary systems. Kummer, in studying these questions, had originally at his disposal only the principal system ; and noticing the imperfection of the resulting laws of division, he introduced by definition his «ir where e = es. These 60 quantities, then, form a group of 60 linear substitutions. Fig. 17. 2"=-0O (2) Let us next illustrate geometrically the dependence of 77 on z by establishing the conformal representation of the ^--plane on the 17-plane, or rather (by stereographic projection) on a sphere (Fig. 17). The triangles corre- sponding to the upper (shaded) half of the r-plane are the alternate (shaded) triangles on the sphere determined by inscribing a regular icosahedron and dividing each of the 20 triangles so obtained into six equal and symmetrical triangles by drawing the altitudes (Fig. 18). This conformal representation on the sphere assigns to every root a definite region, and is therefore equivalent to a Fig. 18. 70 LECTURE IX. perfect separation of the 60 roots. On the other hand, it cor- responds in its regular shape to the 60 linear substitutions indicated above. (3) If, by putting v^yi/^v ^^ make the 60 expressions of the roots homogeneous, the different values of the quan- tities y will all be of the form » and therefore satisfy a linear differential equation of the second order p and q being definite rational functions of z. It is, of course, always possible to express every root of an equation by means of a power series. In our case we reduce the calculation of T) to that of jj and j/j, and try to find series for these quanti- ties. Since these series must satisfy our differential equation of the second order, the law of the series is comparatively simple, any term being expressible by means of the two preceding terms. (4) Finally, as mentioned before, the calculation of the roots may be abbreviated by the use of elliptic functions, provided tables of such elliptic functions be computed before- hand. Let us now see what corresponds to each of these four points in the case of the pure equation 7f = z. The results are well known : (i) All the n roots can be expressed as linear functions of any one of them, 17 : € being a primitive «th root of unity. SOLUTION OF HIGHER ALGEBRAIC EQUATIONS. 71 (2) The conformal representation (Fig. 19) gives the division of the sphere into 2 n equal lunes whose great circles all pass through the same two points. Fig. 19. (3) There is a differential equation of the first order in t), viz., nz ■ -q' — r) = o, from which simple series can be derived for the purposes of actual calculation of the roots. (4) If these series should be inconvenient, logarithms can be used for computation. The analogy, you will perceive, is complete. The principal difference between the two cases lies in the fact that, for the pure equation, the linear substitutions involve but one quantity, while for the quintic equation we have a group of binary linear substitutions. The same distinction finds expression in the differential equations, the one for the pure equation being of the first order, while that for the quintic is of the second order. Some remarks may be added concerning the reduction of the general equation of the fifth degree, M^) = °, to the icosahedron equation. This reduction is possible because the Galois group of our quintic equation (the square root of the discriminant having been adjoined) is isomorphic with the group 72 LECTURE IX. of the 60 linear substitutions of the icosahedron equation. This possibility of the reduction does not, of course, imply an answer to the question, what operations are needed to effect the reduc- tion. The second part of my Vorlesuugen iiber das Ikosaeder is devoted to the latter question. It is found that the reduction cannot be performed rationally, but requires the introduction of a square root. The irrationality thus introduced is, however, an irrationality of a particular kind (a so-called accessory irration- ality) ; for it must be such as not to reduce the Galois group of the equation. I proceed now to consider the general problem of an analo- gous treatment of higher equations as first given by me in the Math. Annalen, Vol. 15 (1879).* ^ must remark, first of all, that for an accurate exposition it would be necessary to dis- tinguish throughout between the homogeneous and projective formulations (in the latter case, only the ratios of the homoge- neous variables are considered). Here it may be allowed to disregard this distinction. Let us consider the very general problem : a finite group of homogeneous linear substitutions of n variables being given, to calculate the "values of the n variables from the invariants of the group. This problem evidently contains the problem of solving an algebraic equation of any Galois group. For in this case all rational functions of the roots are known that remain unchanged by certain permutations of the roots, and permutation is, of course, a simple case of homogeneous linear transformation. Now I propose a general formulation for the treatment of these different problems as follows : among the problems having isomorphic groups ive consider as the simplest the one that has the * Ueber die .Aufl'osung gewisser Gleichungen vom siebenten and achten Grade, pp. 251-282. SOLUTION OF HIGHER ALGEBRAIC EQUATIONS. 73 least number of variables, and call this the normal problem. This problem must be considered as solvable by series of any kijid. The question is to reduce the other isotnorphic problems to the normal problem. This formulation, then, contains what I propose as a gen- eral solution of algebraic equations, i.e. a reduction of the equa- tions to the isomorphic problem with a minimum number of variables. The reduction of the equation of the fifth degree to the icosahedron problem is evidently contained in this as a special case, the minimum number of variables being two. In conclusion I add a brief account showing how far the gen- eral problem has been treated for equations of higher degrees. In the first place, I must here refer to the discussion by myself* and Gordanf of those equations of the seventh degree that have a Galois group of 168 substitutions. The minimum number of variables is here equal to three, the ternary group being the same group of 168 linear substitutions that has since been discussed with full details in Vol. I. of the Elliptischc Modulfunctionen. While I have confined myself to an expo- sition of the general idea, Gordan has actually performed the reduction of the equation of the seventh degree to the ternary problem. This is no doubt a splendid piece of work ; it is only to be deplored that Gordan here, as elsewhere, has dis- dained to give his leading ideas apart from the complicated array of formulae. Next, I must mention a paper published in Vol. 28 (1887) of the Math. Annalen,X where I have shown that for the general * Math. Annalen, Vol. 15 (1879), pp. 251-282. t Ueber Gleichungen siebenten Grades mit einer Gruppe von 1 68 Suhstitutionen, Math. Annalen, Vol. 20 (1882), pp. 515-530, and Vol. 25 (1885), pp. 459-521. X Zur Theorie der allgemeinen Gleichungen sechsten utid siebenten Grades, pp. 499-532. 74 LECTURE IX. equations of the sixth and seventh degrees the minimum num- ber of the normal problem is four, and how the reduction can be effected. Finally, in a letter addressed to Camille Jordan* I pointed out the possibility of reducing the equation of the 27th degree, which occurs in the theory of cubic surfaces, to a normal prob- lem containing likewise four variables. This reduction has ultimately been performed in a very simple way by Burkhardtf while all quaternary groups here mentioned have been con- sidered more closely by Maschke.if This is the whole account of what has been accomplished ; but it is clear that further progress can be made on the same lines without serious difficulty. A first problem I wish to propose is as follows. In recent years many groups of permutations of 6, 7, 8, 9, . . . letters have been made known. The problem would be to determine in each case the minimum number of variables with which isomor- phic groups of linear substitutions can be formed. Secondly, I want to call your particular attention to the case of the general equation of the eighth degree. I have not been able in this case to find a material simplification, so that it would seem as if the equation of the eighth degree were its own normal problem. It would no doubt be interesting to obtain certainty on this point. * Journal de mathematiques, annee 1888, p. 169. t Untersuchtingen aus dem Gebiete der hyperelliptischen Modulfunctionen. Dritter Theil, Math. Annalen, Vol. 41 (1893), PP- 313-343. X Ueber die quaterndre, endliche, lineare Substituiiomgruppe der Borchardfschen Moduln, Math. Annalen, Vol. 30 (1887), pp. 496-515; Aufslellung des vollen For- mensystems einer quaternaren Gruppe von 51840 linearen Suhstitutionen, ib., Vol. 33 ('889), pp. 317-344; Ueber eine merkwiirdige Configuration gerader Linien im Raume, ib., Vol. 36 (1890), pp. 190-215. Lecture X.: ON SOME RECENT ADVANCES IN HYPERELLIPTIC AND ABELIAN FUNCTIONS. (September 7, 1893.) The subject of hyperelliptic and Abelian functions is of such vast dimensions that it would be impossible to embrace it in its whole extent in one lecture. I wish to speak only of the mutual correlation that has been established between this subject on the one hand, and the theory of invariants, projective geometry, and the theory of groups, on the other. Thus in particular I must omit all mention of the recent attempts to bring arithmetic to bear on these questions. As regards the theory of invariants and projective geometry, their introduction in this domain must be considered as a realization and farther extension of the programme of Clebsch. But the additional idea of groups was necessary for achieving this extension. What I mean by establishing a mutual correlation between these various branches will be best understood if I explain it on the more familiar example of the elliptic functions. To begin with the older method, we have the fundamental elliptic functions in the Jacobian form sin am (v, ^\, cos am [v, —\ A am fv, — j, as depending on two arguments. These are treated in many works, sometimes more from the geometrical point of view of Riemann, sometimes more from the analytical standpoint of 75 76 LECTURE X. Weierstrass. I may here mention the first edition of the work of Briot and Bouquet, and of German works those by Konigs- berger and by Thomae. The impulse for a new treatment is due to Weierstrass. He introduced, as is well known, three homogeneous arguments, u, (Ml, ojg, instead of the two Jacobian arguments. This was a necessary preliminary to establishing the connection with the theory of linear substitutions. Let us consider the dis- continuous ternary group of linear substitutions, i' = au>i + j8i, our first task is to construct all its sub-groups. Among these the simplest and most useful are those that I have called congruence sub-groups ; they are obtained by putting mi = o, z«2 — °> 1 a=i, /3 = o, \ (yaodi. n) . I y = o, 8= I, J The second problem is to construct the invariants of all these groups and the relations between them. Leaving out of consideration all sub-groups except these congruence sub- groups, we have still attained a very considerable enlargement of the theory of elliptic functions. According to the value assigned to the number n, I distinguish different stages {Stufen) of the problem. It will be noticed that Weierstrass's theory corresponds to the first stage («=i), while Jacobi's answers, generally speaking, to the second (« = 2) ; the higher stages have not been considered before in a systematic way. Thirdly, for the purpose of geometrical illustration, I apply Clebsch's idea of the algebraic curve. I begin by introducing 78 LECTURE X. the ordinary square root of the binary form which requires the axis of X to be covered twice ; i.e. we have to use a C^ in an ^j. I next proceed to the general cubic curve of the plane (C3 in an S^, to the quartic curve in space of three dimensions (Q in an S^, and generally to the elliptic curve C„+i in an 5„. These are what I call the normal elliptic curves; they serve best to illustrate any algebraic relations between elliptic functions. I may notice, by the way, that the treatment here proposed is strictly followed in the Elliptische Modulfimctionen, except that there the quantity ic is of course assumed to be zero, since this is precisely what characterizes the modular functions. I hope some time to be able to treat the whole theory of elliptic functions {i.e. with u different from zero) according to this programme. The successful extension of this programme to the theory of hyperelliptic and Abelian functions is the best proof of its being a real step in advance. I have therefore devoted my efforts for many years to this extension ; and in laying before you an account of what has been accomplished in this rather special field, I hope to attract your attention to various lines of research along which new work can be spent to advantage. As regards the hyperelliptic functions, we may premise as a general definition that they are functions of two variables 2ii/r5 = <^3-o. It is here, of course, essen- tial to adopt the system of von Staudt and not that of Steiner, since the latter defines the anharinonic ratio by means of distances of points, and not by pure projective constructions. (3) Finally, we have the point of view of Riemann and Helm- holtz. Riemann starts with the idea of the element of distance ds, which he assumes to be expressible in the form ds = ^"Siai^XidXf Helmholtz, in trying to find a reason for this assumption, con- siders the motions of a rigid body in space, and derives from these the necessity of giving to ds the form indicated. On the other hand, Riemann introduces the fundamental notion of the measure of curvature of space. The idea of a measure of curvature for the case of two variables, i.e. for a surface in a three-dimensional space, is due to Gauss, who showed that this is an intrinsic characteristic of the surface quite independent of the higher space in which the surface happens to be situated. This point has given rise to a misunderstanding on the part of many non-Euclidean writers. When Riemann attributes to his space of three dimensions a measure of curvature k, he only wants to say that there exists an invariant of the "form" ^.a^dXidxt; he does not mean to imply that the three-dimensional space necessarily exists as a curved space in a space of four dimensions. Similarly, the illustration of a space of constant positive measure of curvature by the familiar example of the sphere is somewhat misleading. Owing to the fact that on the sphere the geodesic lines (great circles) issuing from any point all meet again in another definite RESEARCHES IN NON-EUCLIDEAN GEOMETRY. 87 point, antipodal, so to speak, to the original point, the existence of such an antipodal point has sometimes been regarded as a necessary consequence of the assumption of a constant positive curvature. The projective theory of non-Euclidean space shows immediately that the existence of an antipodal point, though compatible vi^ith the nature of an elliptic space, is not necessary, but that two geodesic lines in such a space may intersect in one point if at all.* I call attention to these details in order to show that there is some advantage in adopting the second of the three points of view characterized above, although the third is at least equally important. Indeed, our ideas of space come to us through the senses of vision and motion, the "optical properties" of space forming one source, while the "mechanical properties" form another ; the former corresponds in a general way to the pro- jective properties, the latter to those discussed by Helmholtz. As mentioned before, from the point of view of projective geometry, von Staudt's system should be adopted as the basis. It might be argued that von Staudt practically assumes the axiom of parallels (in postulating a one-to-one correspondence between a pencil of lines and a row of points). But I have shown in the Math. Annalen^ how this apparent difficulty can be overcome by restricting all constructions of von Staudt to a limited portion of space. I now proceed to give an account of the most recent re- searches in non-Euclidean geometry made by Lie and myself. Lie published a brief paper on the subject in the Berichte of the Saxon Academy (1886), and a more extensive exposition of his views in the same Berichte for 1890 and 1891. These * This theory has also been developed by Newcomb, in the jfournal fur reine und angewandte Mathematik, Vol. 83 (1877), pp. 293-299. t Utber die sogenannte Nicht-Euklidische Geometrie, Math. Annalen, Vol. 6 («873). PP- II2-I4S- 88 LECTURE XI. papers contain an application of Lie's theory of continuous groups to the problem formulated by Helmholtz. I have the more pleasure in placing before you the results of Lie's investi- gations as they are not taken into due account in my paper on the foundations of projective geometry in Vol. 37 of the Math. Amialen (1890),* nor in my (lithographed) lectures on non-Euclidean geometry delivered at Gottingen in 1889-90; the last two papers of Lie appeared too late to be considered, while the first had somehow escaped my memory. I must begin by stating the problem of Helmholtz in modern terminology. The motions of three-dimensional space are so^, and form a group, say Cg. This group is known to have an invariant for any two points p, p' , viz. the distance fi {p, p') of these points. But the form of this invariant (and generally the form of the group) in terms of the co-ordinates x-^, x^, x^ Ji> ^2' J's °f the points is not known a priori. The question arises whether the group of motions is fully characterized by these two properties so that none but the Euclidean and the two non-Euclidean systems of geometry are possible. For illustration Helmholtz made use of the analogous case in two dimensions. Here we have a group of oo'' motions ; the distance is again an invariant ; and yet it is possible to construct a group not belonging to any one of our three systems, as follows. Let ^^ be a complex variable ; the substitution characterizing the group of Euclidean geometry can be written in the well- known form z' = e*H + m + in= (cos + i sm)z+ m + in. Now modifying this expression by introducing a complex number in the exponent, z' = f'«+"*z -\-m + in = tf«*(cos + i sin