WESSEL, Caspar.

Om Directionens analytiske Betegning, et forsøg, anvendt fornemmelig til plane og sphæriske Polygoners Opløsning [Read 10 March 1797], pp. 469-518 and 3 folded engraved plates in: Nye Samling af det Kongelige Danske Videnskabernes Selskabs Skrifter, Vol. V.

Copenhagen: Johan Rudolph Thiele, 1799.

First edition, journal issue, of the first work to propose a geometric representation of complex numbers. “Wessel’s fame as a mathematician is based entirely on one paper, written in Danish and published in the Mémoires of the Royal Danish Academy, that established his priority in publication of the geometric representation of complex numbers. John Wallis had given a geometric representation of the complex roots of quadratic equations in 1685; Gauss had had the idea as early as 1799 but did not explicitly publish it until 1831. Robert Argand’s independent publication in 1806 must be credited as the source of this concept in modern mathematics because Wessel’s work remained essentially unknown until 1895, when its significance was pointed out by Christian Juel. Despite its lack of influence upon the development of mathematics, Wessel’s publication was remarkable in many ways: he was not a professional mathematician; Norway and Denmark were not mathematically productive or stimulating at that time; he was not a member of the Royal Danish Academy (he had been helped and encouraged by J. N. Tetens, councillor of state and president of the science section of the Academy); and yet his exposition was, in some respects, superior to and more modern in spirit than Argand’s. The title of Wessel’s treatise calls it an ‘attempt’ to give an analytic representation of both distance and direction that could be used to solve plane and spherical polygons. The connection of this goal with Wessel’s work as a surveyor and cartographer is obvious. The statement of the problem also suggests that Wessel should be credited with an early formulation of vector addition. In fact, Michael J. Crowe, in A History of Vector Analysis (University of Notre Dame Press, 1967), defines the first period in that history as that of a search for hypercomplex numbers to be used in space analysis and dates it from the time of Wessel, whom he calls the first to add vectors in three-dimensional space … Wessel’s initial formulation was remarkably clear, direct, concise, and modern. It is regrettable that it was not appreciated for nearly a century and hence did not have the influence it merited” (DSB). We have never seen an offprint of this paper, and it is probable that none exists. Not on RBH.

“To have a short title for Wessel’s On the Analytical Representation of Direction. An Attempt Applied Chiefly to Solving Plane and Spherical Polygons, I call it his Attempt. He opened this by giving the impression that it was born of a wish to find expressions for line segments from which their lengths and directions can be determined and to find a way of calculating with such expressions. This wish seems very natural for a surveyor who had solved an endless number of problems concerning triangles. Wessel added, however, another motivation: he had been seeking a method whereby he could avoid ‘the impossible operations’. He also stated that this had paradoxically shown that the possible sometimes has to be sought by ‘impossible means’. He did not explain what he meant by ‘impossible operations’, but it is likely that he thought of extracting the square root of -1, and generally operations with impossible, i.e., complex numbers. If this is the case, it appears that complex numbers were also a source of inspiration for Wessel.

“Irrespective of whether Wessel originally started with complex numbers or with line segments, he structured his Attempt as a pursuit of algebraic operations for directed line segments. This idea might be seen as an extension of what Descartes did in La géométrie, namely to show how the operations +, -, •, :, and √ can be interpreted for traditional line segments. Wessel did not refer to Descartes, but he spent more than a page arguing that it is allowed to conceive of the algebraic operations in a broader sense so that they can be applied to an extended field of quantities …

“Wessel's line segments were defined by their directions and their lengths. He himself just called them lines, whereas I use the expression directed line segments and sometimes only line segments. His conception involves that he considered all parallel line segments which have the same length and the same orientation to be equivalent though he did not use the term. When operating with line segments Wessel in fact chose convenient representants from the equivalence classes. In the following I assume this convention to be understood when I write the directed line segment.

“Wessel started by defining addition and multiplication geometrically. His definition of addition is in accordance with the tradition of adding velocities by the parallelogram rule and corresponds to addition of vectors … In a geometrical language he expressed that his addition is commutative and associative. He furthermore emphasized a result concerning polygons which turned out to be essential for his trigo nometric applications of his new algebra. For a quadrangle with vertices a, b, c, and d his theorem states that

ab+bc+cd+da = 0.

Wessel did not spend many words on subtraction, but noticed that -ba = ab.

“The real novelty in Wessel's Attempt is his geometric definition of a product of two directed line segments, and here he took quite an abstract step. First, he introduced a unit, 1, as Descartes had done for his geometric product of two (non-directed) line segments … While Wessel’s definition of the sum of two directed line segments applies in three-dimensional space, he only defined multiplication for line segments that are situated in a plane containing the unit. Let oa and ob be two such segments and let oc be the, not yet defined, product. Wessel required that oc shall be situated in the plane determined by oa and ob, and moreover

that the product of two straight lines should in every respect be formed from the one factor, in the same way as the other factor is formed from the … unit.

If we conceive of oa to be formed from oe [the unit on a chosen, fixed, and oriented line of reference] by letting the latter be rotated a certain angle and by letting its length be multiplied by a certain number, then a geometrical interpretation of Wessel’s requirement leads to the conclusion that the triangles oea and obc must be similar and similarly oriented. This implies, when | | signifies length,

(*)

and

|oc| : |ob| = |oa| : |oe| (**)

or

|oc| = |oa| . |ob|.

Reckoning angles with signs Wessel used precisely the relations (*) and (**) to define the product oc. It should be noticed, as Wessel himself did, that when oa and ob have the same direction as the unity, then the product oc corresponds to a usual product of two line segments.

“To be able to work with his line segments algebraically Wessel took a step which today seems very natural, but which was quite remarkable at his time. He introduced a second unit which he let be defined by having the length 1 and the direction 90° and which he denoted e. Applying definition (*) Wessel found that the direction of e is 180°, that is the direction of -1, and from (**) he concluded that e.e has the length 1, that is the same length as -1, so that

e.e = -1 …

and then laconically remarked ‘from this it follows that e becomes = ,/-1 ‘. This observation shows quite clearly that he was aware that he had given a geometric interpretation of ,/-1, but he did not make any point out of this achievement.

“From multiplying the units Wessel proceeded towards obtaining general algebraic expressions for directed line segments and an algebraic multiplication rule. He did this in two steps; first he dealt with directed line segments of length 1 and secondly with general line segments …

“Starting with geometric definitions of addition and multiplication Wessel deduced that algebraically his multiplication follows the same rule as the product of complex numbers (when e is treated as ,/-1). Moreover, he had implicitly shown how complex numbers as well as a sum and a product of complex numbers can be interpreted geometrically. This is quite an accomplishment. Wessel was too modest to express this in strong words, but he did remark that he had shown ‘that the direction of all lines in the same plane maybe expressed as analytically as their length, and this without burdening the memory by new symbols or rules’. Although satisfied with this result he did not hesitate to mention its limitations, namely that the two factors and 1 have to be in the same plane. Wessel continued with division and root extraction and confirmed the usual results ...

“To demonstrate that his directed line segments could be of use Wessel sophisticatedly showed how the theorem of Cotes follows from his calculation rules. His main concern, however, was to solve trigonometric problems. For this purpose he derived a universal formula from which unknown elements in a polygon can be determined …

“Wessel was also interested in solving spherical triangles so he was naturally led to extend his calculations with directed line segments to three dimensions. For this purpose he considered a sphere with radius r and introduced so to speak two complex planes in this. He let the planes be perpendicular to each other sharing the real axis and termed the second imaginary unit h, i.e., h²= -1 … Quite easily Wessel found that any directed radius in the sphere can be written in the form

x+hy+ez. (***)

His definition of the sum of directed line segments covered the three-dimensional situation and resulted in an obvious rule for how two of these expressions should be added . Multiplication, however, was a problem. In introducing the product of two line segments Wessel stressed the fact that he had not been able to introduce an algebraic product in three dimensions. Undoubtedly, he had searched for a meaning of e.h and had found none.

“Wessel did not let himself stop by the lack of a general product, but used the product which he had introduced in a plane and linked this to rotations. This step actually turned out to be sufficient for solving spherical triangles …

“He ended the Attempt by indicating how his theory can be applied to a rectilinear polygon having sides in different planes. Altogether, his Attempt is a remarkable piece of mathematics, although it was not appreciated in its own time

“From the end of the nineteenth century every historical exposition touching upon com-plex numbers has mentioned Wessel’s ideas, but they were not, as told earlier, noticed at the time of their publication. In fact, Wessel’s Attempt only became known after Sophus Andreas Christensen had mentioned it in his thesis (1895). Christensen did not pay much attention to the contents, but his remarks inspired the Danish mathematician Christian Juel to study Wessel’s work and publish a paper on it (1895). Sophus Lie also got interested and had Wessel’s Attempt reprinted in 1896. It was moreover translated into French the following year, and part of it into English three decades later.

“The belated fame of Wessel’s achievement has been explained by the fact that it was presented in Danish – a language read by few. In my opinion the reason was rather that around 1800 the problem of representing the complex numbers geometrically was not considered important in the leading circle of mathematicians, and I shall give three arguments to support this view. First, if the problem really had been considered an essential issue, then some mathematicians able to read Danish would have been familiar with it and would have called attention to Wessel’s solution. En passant it can be remarked that Niels Henrik Abel belonged to the group who could have made Wessel’s result known. Viggo Brun has documented that in August 1822 Abel borrowed the issue of the Kongelige Danske Videnskabernes Selskabs Skrifter which contains Wessel’s Attempt. However, as Brun has pointed out, Abel’s interest was presumably not caught by Wessel’s paper but by one on the theory of equations written by Carl Ferdinand Degen.

“Secondly, it is essential to observe that at the beginning of the nineteenth century none of the influential mathematicians published on the geometrical representation of complex numbers. This was done by a number of other persons who in general did not work on mainstream mathematics and who did not write any other really remarkable piece of mathematics. Moreover, at first their publications were not noticed by the first-rank mathematicians.

“The third argument is slightly more complicated; it involves Gauss, and his motives are often difficult to make out. A drawing in one of his notebooks indicates that by 1805 he worked with the complex plane, and in 1811 he explicitly referred to this in a letter to Friederich Wilhelm Bessel. He did not, however, publish anything about his understanding of complex numbers until 1831. This took place in a paper which contains the claim that already when he composed his first proof of the fundamental theorem of algebra (published in 1799) he had an understanding of the complex plane. Irrespective of whether Gauss remembered correctly in 1831 he could, as just shown, have published much earlier on his interpretation of complex numbers. So the question becomes why he did not. He is known for having kept some of his results unpublished for a long time; in the case of non-Euclidean geometry he later maintained that he feared the philosophical reactions to his insights. I do not think that he saw similar troubles in introducing the complex plane and believe that if he really had found his understanding important he would have published it earlier. That he later would indicate that he had a kind of priority to the result was not atypical of his behaviour” (Andersen, pp. 71-82).

4to, pp. xii, 670, with 15 engraved plates (some light dampstains). Contemporary calf, spine lettered and decorated in gilt in compartments, sides with gilt border (a little rubbed, joints starting).

Item #6397

Price: $3,500.00