NA TURE 



97 



THURSDAY, NOVEMBER 30, 1905. 



LAGUERRE'S MATHEMATICAL PAPERS. 

 CEuvres de Laguerre. Tome ii. Pp. 715. (Paris : 

 Gauthier-Villars, 1905.) Price 22 francs. 



THE publication of the mathematical papers of 

 Laguerre, undertaken after his death in 1886 

 by MM. Hermite, Poincare - , and Rouche under the 

 auspices of the Academy of Sciences, has at length 

 been completed. A first volume, 10 which M. 

 Poincare contributed as a preface an admirable 

 appreciation of the author, appeared in 1898; and 

 now some eighty papers which treat of geometrical 

 subjects have been collected from the various scientific 

 journals and reprinted in a second and final volume 

 of more than 700 pages. Most of these papers are 

 (if but four or five pages in length, for it was 

 Laguerre's habit, when a mathematical investigation 

 had aroused his interest, to return to it again and 

 again as new ideas occurred to him ; and so it comes 

 to pass that the majority of his writings on geometry 

 may be classified as dealing with one or other of 

 >ome half-dozen wide but distinct subjects. 



The discovery with which Laguerre made his entry 

 into the ranks of original investigators is of such 

 moment in the history of modern mathematics that 

 we will pause in order to realise the condition of 

 geometrical knowledge at the time, and the circum- 

 stances in which it was made. The first three of the 

 papers now under review bear the dates 1852-3, and 

 were written when the author, a student eighteen 

 years of age at the Institution Barbet, was still only 

 a candidate for admission to the Ecole Polytechnique. 

 It was a time when a great change in geometrical 

 thought had been initiated. Poncelet and Chasles 

 had begun to build up the theory of projective geo- 

 metry, and through it mathematicians had been made 

 to recognise that theorems previously regarded as 

 wholly without connection might in reality be but 

 different presentments of the same more fundamental 

 fact. But the structure was as vet very far from 

 complete ; many of the chief features of the theory 

 as we now know it were still obscure and needed 

 explanation. It fell to Laguerre to provide the most 

 important and prolific discovery on which modern 

 geometrical theory has been founded, the inner mean- 

 ing of angular magnitude. In the second of his 

 three early papers was enunciated for the first time 

 the proposition that the sides of an angle form with 

 the two isotropic lines through the vertex a pencil 

 the anharmonic ratio of which depends only upon the 

 magnitude of the angle. 



This was no chance discovery, lighted on by a 

 stroke of undeserved good fortune, for the rest of the 

 paper shows how true a grasp of the new principles 

 Laguerre had already obtained. He goes on to point 

 out that the proposition furnishes the solution of the 

 problem of homographic transformation of angular 

 relations — a problem which had baffled the founders 

 of projective geometry — and gives many further 

 developments and results, a special case of one of 

 which may be cited as illustrative. The well known 

 NO. 1883, VOL - 73] 



theorem of Menelaus concerning the ratios of the 

 segments into which the sides of a triangle are divided 

 by any straight line is identical with the theorem 

 that the angles of a triangle make up two right 

 angles ; either theorem can be deduced from the other. 

 It is regrettable that English treatises upon analytical 

 geometry so rarely attribute theorems to their authors, 

 for it is with this discovery, rather than any achieved 

 later in more advanced subjects, that we should wish 

 the name of Laguerre to be always associated. 

 Certainly no discovery has had so far-reaching an 

 influence upon geometrical research during the past 

 half-century. 



In the twelve years which followed Laguerre pub- 

 lished nothing. His military duties as an officer of 

 artillery at first absorbed him; then, having been 

 transferred to the manufactory of arms at Mutzig, he 

 found leisure to take up once more his favourite 

 study; it was not, however, until 1S65, after his 

 recall to Paris to the Ecole Polytechnique, that he 

 published the first of the series of original papers 

 which he continued without intermission until his 

 death. 



A large number of these are concerned with 

 analytical geometry, and through an interesting 

 section of them runs an idea allied to his earlier 

 work, the use of imaginaries in geometry. Thus 

 in one of the first papers we find a full exposition of 

 the theory of foci as extended from conies to plane 

 curves of any class, and the distinction is drawn 

 between ordinary foci and singular foci of a curve 

 which passes through the circular points. Another 

 such matter with which Laguerre frequently occupied 

 himself, and which seems never to have received the 

 attention it deserves, is the means of representing 

 in a concrete manner the points of a plane or in 

 space the coordinates of which are complex quantities. 

 Imaginary values of the coordinates may satisfy the 

 equation of a curve, and are then often spoken of as 

 the coordinates of an imaginary point on the curve ; 

 cannot geometry suggest some mode of representing 

 these points similar to that of Argand, which plays 

 so important a part in the theory of equations in- 

 volving a single variable? No solution could be 

 satisfactory which varied with the coordinate-s)'stem 

 employed. Laguerre's method was to represent a 

 pair of points the coordinates of which are conjugate 

 complex quantities by a real segment of a line, dis- 

 tinguishing the two imaginary points when necessary 

 by the sense in which the segment is measured ; the 

 ends of the segment are the intersections of the lines 

 which join the imaginary points to the circular points. 

 Thus the line joining two real foci of a conic repre- 

 sents the two imaginary foci. It now becomes 

 feasible to express the conditions of collinearity, &c, 

 of imaginary points by properties of their represent- 

 ative segments. For points in space a similar pro- 

 cedure is adopted ; a point the coordinates of which 

 are complex is represented by a real circle, which 

 must be described in a definite sense. 



Having extended the notion of a focus to curves 

 other than conies, it was natural that Laguerre should 

 study curves which possess focal properties ; accord 



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