The subject matter of this chapter is somewhat removed from the ideas of symmetry and their evolution through the 19th century, which form the core of the book, but is needed to understand Klein's and Lie's work.
In antiquity, mathematicians had already noticed the analogies between a segment of a straight line, a triangle in a plane, and a tetrahedron (a triangular pyramid) in space (Fig. 16(a)); pairs of points on a line, a circle in a plane, and a sphere in space (Fig. 16(b)); a closed interval on a line, a parallelogram in a plane, and a parallelepiped in space (Fig. 16(c)).174. Even before the introduc
tion of coordinates, mathematicians could express the difference between the straight line, where figures possess one dimension-their length a; the plane, where figures can be characterized by two dimensions-length a and breadth b; and space, where every solid has three dimensions-length a, breadth b, and height c (see Fig. 17). Once coordinates are introduced, this difference can be conveniently expressed as follows: a point on a (one-dimensional) line is characterized by a single number-its single coordinate (the abscissa) x; a point in a two-dimensional plane has two coordinates (its abcissa x and its ordinate y); finally, in three-dimensional space the position of a point is determined by three numbers (coordinates)-the abscissa x, the ordinate y, and the applicate z (compare Fig. 16(c)). A segment on a line can be described as a set of points M(x) with 0 � x � a, a rectangle in the plane as a set of points M(x,y) with 0 � x � a, 0 � y � b, and a parallelepiped in space as a set of points M(x, y, z) with 0 � x � a, 0 � y � b, 0 � z � c (see Fig. 16(c)).
Rudiments of the idea of "multidimensional" spaces-spaces of four or more dimensions-turn up, in implicit, and sometimes semi-mystical form, in the works of many mathematicians and philosophers of different nation
alities.175 A clear understanding of the fact that the world we live in is four-dimensional, and that, therefore, every event is characterized by three
"space" coordinates x, y, z, and time t, was expressed by the famous French mathematician and philosopher Jean le Rond d'Alembert (1717-1783).176 D'Alembert and Denis Diderot (1713-1764) were the joint editors of the 35-volume Encyclopedie or Dictionnaire raisonne des sciences, des arts and des metiers (Paris, 1751-1780). The volume which appeared in 1764 contained an
72 Felix Klein and Sophus Lie y
--��---4---. x
0 A B
0 X y
X (a)
--·----··--··--�·�_.. X
0 Q
X (b)
0
(c) FIGURE 16
z y
X X
(a) (b) (c)
FIGURE 17
Arthur Cayley
entry ("dimension") remarkable for an eighteenth-century encyclopaedia, in which d' Alembert wrote that "One could consider time as a fourth dimension, so that the product of time by volume would, in a certain sense, be the product of four dimensions; this idea is perhaps debatable, but I feel that it has certain merits . . . . "
It seems that the term "n-dimensional geometry" was first used by the outstanding English algebraist Arthur Cayley (1821-1895), repeatedly men
tioned above, in the article "Chapters on the Analytical Geometry of n Dimensions." 177 Cayley came from a well-to-do English family. Owing to business affairs, Cayley's father lived in St. Petersburg, in Russia, where young Arthur spent his childhood. Cayley studied at Cambridge, where he became senior wrangler and the first winner of Smith's prize. His first scientific papers appeared in the year of his graduation. But since he felt that mathematics did not guarantee sufficient material rewards, he began to study law. Soon he became a well-known and successful London lawyer. However, in contrast to M. Chasles, who also postponed the pursuit of an academic career because of financial considerations (see page 37), Cayley never suspended his mathe
matical work during his time as a lawyer, combining his practice with intense and extremely fruitful research activity. In 1864, considering himself sufficiently well off, Cayley left the bar for a professorship at Cambridge University, where he remained until his death.
74 Felix Klein and Sophus Lie
Paradoxically, Cayley's science combined profound creativity with caution and conservatism which often hampered his deeper understanding of the achievements of others as well as his own. Thus while he was practically the first important mathematician to comment in writing on Lobachevsky's non
Euclidean geometry (see above), he did not for a time understand its real significance. Similarly, he never acquired a deep appreciation of the geo
metric systems now known as projective metrics or Cayley geometries1 78 (see page 67). Cayley's conservatism also appears in the paper "Chapters on the Analytical Geometry of n Dimensions." In his profound analysis of certain facts of (n - I)-dimensional projective geometry177 and in his proofs of im
portant theorems in this field, Cayley is very cautious in the use of geometric terminology (except for the title) and symbolism which was obviously appro
priate but not yet fully established. Only at the end of the paper does he note that in the three-dimensional case (why only this one?) his algebraic results (algebraic only in form, not in content) are equivalent to meaningful theorems of geometry.1 79 Similarly, in his famous "Memoirs upon Quantics," particu
larly in the Sixth Memoir where the foundations of "Cayley metrics or geom
etries" are set forth, he deals exclusively with second-order curves in the projective plane and with surfaces in three-dimensional space, although the extension of all these results to the general (n-dimensional) case is obvious.
Incidentally, Klein's famous 1871 paper "On so-called non-Euclidean geom
etry" considers in detail only plane (two-dimensional) and space (three
dimensional) geometries, and mentions the possibility of extensions to n dimensions only in a single sentence at the end of the paper.
Despite all his cautiousness, it is hard to overestimate Cayley's role in the creation of the concept of n-dimensional space and his contribution, by no means limited to the Chapters article, to the development of"multi-dimensional intuition". Cayley's research, as well as that of his followers, friends, and sometime rivals (in particular the indomitable traveller, the Anglo-American James Joseph Sylvester (1814-1897) who moved from country to country and from one activity to another, 180 and the Irish mathematician and theologian George Salmon (1819-1904)),181 was mainly devoted to the development of linear algebra (which, in its geometric aspect, reduces to the study of linear and quadratic loci planes and quadrics in multidimensional affine and pro
jective space) and to the theory of invariants, arising in the affine and projec
tive classification of quadrics in the plane and in space. And perhaps it is to this "invariant trio," as they were called by the famous French analyst Charles Hermite ( 1822-1901 ), that we owe the fact that by the end of the 1870s the notion of an n-dimensional vector space became accessible even to average students of mathematics. In connection with Cayley, we should also mention that he created the theory of matrices, i.e., tables of numbers (square at first and then rectangular when the theory developed), which provide an important tool in the theory of n-dimensional vector spaces as well as a meaningful example of an n2-dimensional space (cf. page 99; see A. Cayley, "Memoir on the Theory of Matrices" (Philos. Transactions, 1858) which also appears in Cayley's Collected Works).
Hermann Gunther Grassmann
And yet the true founder of the concept of n-dimensional space is probably the German Hermann Gunther Grassmann (1809-1877), one of the most original (and for that reason unrecognized in his time) mathematicians of the nineteenth century. Grassmann's Die lineale Ausdehnungslehre182 appeared in 1844, the same year as Cayley's Chapters, while the second, completely revised, text of the same work (Die Ausdehnungslehre) came out seventeen years later (in 1861). The vector calculus, closely related to n-dimensional geometry, was created simultaneously and independently by Grassman and by the famous Irishman William Rowan Hamilton (1805-1865). (This is yet another instance of a phenomenon that we encountered and described on a few occasions above.) Hamilton was an outstanding physicist (we recall that according to some modern views physics "belongs" to the right hemisphere of the brain) and Grassmann a leading linguist (everything having to do with linguistics is associated with the left hemisphere.
Hermann Grassmann183 was born in Stettin, a provincial German town with which practically his entire life is connected. His family had long been known for its religious and scientific interests, and both aspects of intellectual life were always very close to Grassmann. His grandfather was a pastor; his father, who influenced Hermann greatly, was also a clergyman until his scientific interests prevailed and he became professor of physics and mathematics in the Stettin gymnasium from which his son graduated and where he taught for many years.
76 Felix Klein and Sophus Lie
The senior Grassmann also wrote several works on physics, technology, and elementary mathematics. After graduating from the gymnasium, Hermann Grassmann studied for three years at Berlin University. He studied not mathematics but philosophy, psychology, philology, and theology; following family tradition, he thought seriously of becoming a clergyman, and only abandoned the idea completely while working on his Ausdehnungslehre (later he repeatedly expressed regret at not having become a pastor). After complet
ing the university course, Grassmann passed an examination granting him the right to teach, and he taught at a secondary school in Berlin for one-and-a-half years. Then, after passing two comprehensive examinations in theology (at the time he still intended to become a pastor), and then a state examination
for the right to teach in the higher forms of the gymnasium (all these exami
nations took place in Berlin), Grassmann received a document certifying that he was superbly qualified to teach mathematics, physics, mineralogy, and chemistry, as well as theology or religion. From 1836 onwards Grassmann worked exclusively in Stettin, where at first he taught at the Friedrich Wilhelm secondary school, and then, after his father's death, occupied the latter's post in the city gymnasium He moved to Stettin, where at different times he taught German, Latin, chemistry, mineralogy, physics, and mathematics. Most of the time his teaching load was very heavy-up to twenty hours a week of different subjects. It is remarkable then that Grassman found the time to do significant research in various fields-in addition to mathematics he obtained significant results in physics; in particular in the theory of electricity and colors (this work was highly valued by the famous Helmholtz). Grassmann studied music and its theory and the theory of vowels; he had a very keen ear, which in these fields served him well. For a long time he was also an editor or (with his younger brother Robert) a co-editor of, and an active contributor to the local newspaper, as well as a prominent Freemason and church figure. We will yet have occasion to discuss Grassmann's philological interests and contribu
tions, which are regarded by some as the most important part of his intense intellectual life. Klein ends his highly sympathetic biographical essay on Grassmann with the following words: "It is not surprising that in view of such a variety of activities, there was one field Grassmann failed to master: he was a very poor teacher" (compare this with what we said about Jacob Steiner on pages 42-43). Mildmannered and friendly with everyone, Grassmann was unable to maintain the needed discipline in class. He only talked with a few of the most interested pupils, while the rest "had a good time".
Grassmann's "theory of extension" was certainly not his only achievement in pure mathematics, 184 but here it is appropriate to examine only his two basic books in the 1 844 and 1861 editions. In effect both of these books present the theory of n-dimensional space (nonmetric "linear" or "affine" space in the 1 844 book; n-dimensional Euclidean space in the 1 861 book) and employ, respectively, the two basic methods adopted today. In Die lineale Ausdehnungslehre, the author, unfortunately, keeps his promise, given to the readers in the introduction, to set forth his work "proceeding from general philosophical notions without the help of any formulas." These notions correspond to a modern theory of linear (vector) spaces; more specifically, to their axiomatic, i.e., descriptive, development. Nowadays, all mathematics students (and even some secondary school pupils) know that a vector space is a set of (undefined) objects a, b, c, . . . , called vectors (Hamilton's term;
Grassmann talks about "extensive magnitudes") closed under two opera
tions, namely addition and scalar multiplication, and satisfying the following axioms:
a + b = b + a (commutativity of addition);
A.(Jw) = (A.Jl)a;
78 Felix Klein and Sophus Lie
(a + b) + c = a + (b + c) (associativity of addition);
la = a; a + 0 = a; a + ( - a) = 0
(A + JJ.)a = Aa + Jla, A(a + b) = Aa + Ab (distributive laws);
here 0 is called the zero vector, and ( - a) the additive inverse of the vector a.
(Dieudonne's Linear Algebra and Elementary Geometry, intended for second
ary school teachers, opens with the axioms for a vector space; it seems that Dieudonne believes that the school geometry course should begin in that way).185 However, in Grassmann's time, the reader was totally unprepared for such a manner of presentation and for such an approach to the essence of mathematics (regardless of the fact that it goes back to Pythagoras and Plato, or, at the very least, to Leibniz). Neither could Grassmann's general philo
sophical standpoint, 186 completely shared by Hankel (see Note 189) and George Boole (1815-1864), 187 be generally absorbed at that time (and perhaps not even today). Readers were unprepared for Grassmann's approach and for his idiosyncratic style, in particular, his use of many strange terms of his own invention; if the lawyer Cayley was excessively cautious in using new terms, the linguist Grassmann obviously enjoyed inventing them! Hence Grassmann's first book was ignored by mathematicians. There was not a single reference to it, nor one review; and, 20 years after its appearance, about 600 copies of the book (of the 900 which seem to have been printed) were disposed of as waste (the other unsold copies were distributed free to anyone who wanted one). Neither was the book supported by Gauss, to whom Grassmann sent a copy. Gauss responded as usual, thanking the author in a short letter; instead of appraising the book, he said: "The tendencies in your book partly intersect the roads along which I wandered for almost half a century." Gauss failed to respond to Grassmann's calculus (or algebra), the most important thing in Die Ausdehnungslehre. We deal with it below.
Grassmann was not at all so philosophical as to be indifferent to his failure.
In the introduction to the book's second version (1861) he stressed that it "was completely revised and presented in the rigorous language of mathematical formulas." Indeed, the underlying approach here is the "arithmetic" (construc
tive) approach, i.e., all the constructions are based on "arithmetic" or "co
ordinate" space-a set of"points" defined by their coordinates (x1 , x2, •••, xn), or, in modern terms, the set of n-tuples of (real) numbers (x1, x2, •••, xn).
The respective operations of addition of points x = (x1 , x2, •••, Xn), and y =(y1 , y2, •••, yn) and of multiplication of a point x by a number A, are defined as follows:
X + Y = (x1 + Y1 , X2 + Yz, . . . , Xn + Yn) Ax = (Ax1 , Ax2, ••., Axn).
Linear subspaces of the space are defined as the solution sets of one (or several) linear homogeneous equations relating the coordinates of points:
A 1x1 + A2x2 + · · · + AnXn = 0.
Grassmann introduces, for the first time, the crucial notion of linear dependence of points a1, a2, •••, ak (we would now say vectors; Grassmann talks of "extensive magnitudes" instead). The points in question are linearly dependent if there is a relation
A1a1 + A.2a2 +
· · ·
+ A.kak = 0,where 0 = (0, 0, . . . , 0) is the zero point (vector) and not all the numbers A-1 , A-2, •••, A.k are equal to zero. This notion enables him to define the dimension of the entire space (or of the associated linear space) as the largest possible number oflinearly independent points. One consequence of that definition, is the simple Grassmann formula
dim U + dim V = dim(U
·
V) + dim(U + V).The notation is modern; dim means dimension, U and V are two linear subspaces, U · V = U n V is their intersection, while U + V is their (vector) sum.188 Another consequence is what is now called a Grassmann algebra (see below), described in a clear "formula form." In the second version of Die lineale Ausdehnungslehre, Grassmann introduces a metric defined by the expression xi + x� +
· · ·
+ x; into the manifold under consideration and thus transforms it into an (n-dimensional!) Euclidean space. (Riemann's famous lecture "LJber die Hypothesen," where n-dimensional manifolds with much more general metrics were considered, had already been delivered at the time; however, Grassmann had no way of knowing about it, because he lived far away from scientific centers in total isolation from mathematical circles; not even the best-known research journals reached him.) Although the book's second version could be more easily understood by a persistent and favorably-inclined reader it too was presented in severely abstract form and abounded in new terms; it seems that it attracted even less attention than the 1844 edition. In his last years Grassmann prepared a new edition of the first version of his book; this came out in 1878, a year after the author's death.It should be noted that the awkward language, more philosophical than mathematical, of both versions of his book and the abundance of new terms, unfamiliar to the reader, served as an obstacle to Grassmann's being invited to teach at the university. Twice he applied for a university post and twice he was rejected, and once a negative review of his works (stressing the above
mentioned shortcomings) was written by the well-known mathematician Ernst Eduard Kummer (1810-1893).
It seems that the first mathematician who really appreciated Grassmann's achievements was W.R. Hamilton. In 1853 Hamilton wrote a number of letters to the Cambridge algebraist and logician Augustus de Morgan in which he explained and praised Grassmann's Ausdehnungslehre. Hamilton's long Lectures on Quaternions appeared in the same year. The author attached special importance to these lectures and in the introduction gave credit to his German colleague's accomplishments. Unfortunately, Grassmann never learned of the impression his works made on Hamilton. Still, the established
80 Felix Klein and Sophus Lie
German mathematician Hermann Hankel ( 1839-1873), who learned of the Ausdehnungslehre from Hamilton's lectures, sent Grassmann an enthusiastic letter189 in 1866. In 1871 Grassmann was elected a corresponding member of the Gottingen scientific society, but this belated (and far from adequate) recognition came at a time when Grassmann had already largely forsaken mathematics for the purely philological studies to which he had always been inclined. As early as 1843, Grassmann and his brother Robert,l90 put out an elementary textbook of the German language with exercises ("mit zahlreichen Obungen").191 Its fourth edition appeared in 1876. Grassmann's extensive work on German names of plants was much more substantial, 191 as was his study of German folklore and, in particular, his collections of German folk songs. In his last years his scientific interests focussed on his study (now regarded as a classic) of the famous literary and religious work of ancient India known as the Rig-Veda. In 1873, Brockhaus, a Leipzig publishing house, issued Grassmann's extensive dictionary for the Rig-Veda (written in Sanskrit) and in 1876-1877 (Grassmann died in 1877) the same publisher issued his two
volume translation of this outstanding work into German. These were the only scientific works for which Grassmann was awarded a degree: the remark
able scholar was made doctor of philosophy honoris causa by Tiibingen University. Recognition of Grassmann's mathematical achievements came only after his death, and was due to the high praise bestowed on his work by Felix Klein and Sophus Lie. In particular, much space was devoted to Grassmann in Klein's history of nineteenth-century mathematics.192 In 1894-191 1, largely on Lie's initiative, B.G. Teubner publishers of Leipzig (with whom Lie was closely associated) issued Gesammelte mathematische und physikalische Werke von Hermann Grassmann in three volumes (six substantial books). An active part in the publication was taken by Lie's closest pupils Georg Scheffers and, especially, Friedrich Engel, who was the editor of the whole work. The last of the six books contained a biography of Grassmann, written by Engel (see Note 183).
Surprisingly, in view of the few contemporary comments on Grassmann's n-dimensional geometry, the relevant ideas very quickly became common knowledge. In 1851, the Swiss Ludwig SchHifli (1814-1895), a professor at Bern University, presented a large work, Theorie der vielfachen Kontinuitiit,
Surprisingly, in view of the few contemporary comments on Grassmann's n-dimensional geometry, the relevant ideas very quickly became common knowledge. In 1851, the Swiss Ludwig SchHifli (1814-1895), a professor at Bern University, presented a large work, Theorie der vielfachen Kontinuitiit,