**PLÜCKER, JULIUS** (1801-1868), mathematician and physicist, was born at Elberfeld on the 16th June 1801. After being educated at the gymnasium at Dusseldorf and studying at the universities of Bonn, Heidelberg, and Berlin, he went in 1823 for a short time to Paris, where he came under the influence of the great school of French geometers, whose founder, Monge, was only recently dead ; and there can be no doubt that his intercourse with the mathematicians of that school, more particularly with Poncelet and Gergonne, greatly helped to determine the earlier part at least of his career. In 1825 he was received as " privat-docent " at Bonn, and after three years he was made professor extraordinary. The title of his " habilitationsschrift," Generalem analyseos applicationem ad ea quse geometrise altioris et mechanics basis et funda-menta sunt e serie Taylor ia deducit Julius Plücker (Bonn, 1824), indicated the course of his future researches. The mathematical influence of Monge had two sides represented respectively by his two great works, the Géométrie Descrip-tive and the Application de I'Analyse á la Géométrie. Although fully master of those general ideas of modern geometry whose development began with the publication of the former of these works, Plücker's actual labours lay more in the direction of the latter. It was his aim to furnish modern geometry with suitable analytical methods and thus to give it an independent analytical development. In this effort he was as successful as were his great con-temporaries Poncelet and Steiner in cultivating geometry in its purely synthetical form. From his lectures and researches at Bonn sprang his first great work Analytisch-Geometrische Entwickelungen (vol. i. 1828, vol. ii. 1831).

In the first volume of this treatise Plücker introduced for the first time the method of abridged notation which has become one of the characteristic features of modern analytical geometry. The peculiarity of this method consists in this, that the letters used in the equations of curves and surfaces represent, not coordinates of a point with respect to arbitrary chosen axes, but straight lines, or it may be curves or surfaces, intrinsically related to the figure under discussion. For example, if it be wished to investigate the properties of a conic section with respect to a pair of tangents and their chord of contact, we write its equation uv + yp=0, where u=0, IJ = 0, w=0 represent the two tangents and the chord of contact respectively. This procedure has two great advantages. It enables us to greatly abridge the necessary analytical equations, to arrive at them more easily, and thus to lighten or altogether avoid the cumbersome algebraical calculations which had broken the back of the old-fashioned Cartesian geometry and arrested its progress altogether; and it greatly facilitates the geometrical interpretation of analytical results whether intermediate or final. In the first volume of the Entwickelungen, Pliicker applied the method of abridged notation to the straight line, circle, and conic sections, and he subsequently used it with great effect in many of his researches, notably in his theory of cubic curves.

In the second volume of the Entwickelungen, Plücker clearly established on a firm and independent basis the great principle of duality. This principle had originally been established by Poncelet as a corollary on the theory of the pole and polar of a conic section. Gergonne maintained the independent and fundamental nature of the principle, and hence arose a violent discussion between him and Poncelet into which Pliicker was drawn. He settled the matter in Gergonne's favour by introducing the notion of the coordinates of a line and of a plane, and showing that in plane geometry, for example, we could with equal readiness represent a point either by means of coordinates or by means of an equation, and that the same was true of a line. Hence it appeared that the point or the line in plane geometry, and the point or the plane in solid geometry, could with equal readiness and with equal reason be taken as elements. It was thus made evident that any system of equations proving a theorem regarding points and lines or regarding points and planes could at once be read as proving another in which the words point and line or the words point andptoe were everywhere interchanged.

Another subject of importance which Plücker took up in the Entwickelungen was the curious paradox noticed by Euler and Cramer, that, when a certain number of the intersections of two algebraical curves are given, the rest are thereby determined. Gergonne had shown that when a number of the intersections of two curves of the (p + q)Üi degree lie on a curve of the pth. degree the rest lie on a curve of the qth. degree. Pliicker finally (Gergonne Ann., 1828-29) showed how many points must be taken on a curve of any degree so that curves of the same degree (infinite in number) may be drawn through them, and proved that all the points, beyond the given ones, in which these curves intersect the given one are fixed by the original choice. Later, simultaneously with Jacobi, he extended these results to curves and surfaces of unequal order. Allied to the matter just mentioned was Plticker's discovery of the six equations connecting the numbers of singularities in algebraical curves. It will be best described in the words of Clebsch :—" Cramer was the first to give a more exact discussion of the singularities of alge-braical curves. The consideration of singularities in the modern geometrical sense originated with Poncelet. He showed that the class k of a curve of the ?ith order, which Gergonne by an extra-ordinary mistake had considered to be identical with its order, is in general n(n — 1) ; and hence arose a paradox whose explanation became possible only through the theory of the simple singularities. By the principle of duality the order n of a curve should be derived in the same way from the class k as k is from n. But if we derive n in this way from k we return not to » but to a much greater number. Hence there must be causes which effect a reduction during this operation. ' Poncelet had already recognized that a double point reduces the class by 2, a cusp at least by 3, and a multiple point of the pth order, all of whose tangents are distinct, by ip(p-l). Here it was that Pliicker took up the question. By first directly determining the number of the points of inflexion, considering the influence of double points and cusps, and finally applying the principle of duality to the result obtained, he was led to the famous formulae for the singularities of curves which bear his name, and which completely resolve the paradox of Poncelet — formulae which already in the year 1854 Steiner could cite as the 'well known,' without, however, in any way mentioning Pliicker's name in con-nexion with them. Pliicker communicated his formula? in the first place to Grelles Journal, vol xii. (1834), and gave a further exten-sion and complete account of his theory in his Theorie der Algebraischen Curven, 1839."

In 1833 Plücker left Bonn for Berlin, where he occupied for a short time a post in the Friedrich Wilhelm's Gymnasium. He was then called in 1834 as ordinary professor of mathematics to Halle. While there he published his System der Analytischen, Geometrie, auf neue Betrachtungsweisen gegrün-det, und insbesondere eine Ausführliche Theorie der Curven dritter Ordnung enthaltend, Berlin, 1835. In this work he introduced the use of linear functions in place of the ordinary coordinates, and thereby increased the generality and elegance of his equations; he also made the fullest use of the principles of collineation and reciprocity. In fact he develops and applies to plane curves, mainly of the third degree, the methods which he had indicated in the Entwickelungen and in various memoirs published in the interim. His discussion of curves of the third order turned mainly on the nature of their asymptotes, and depended on the fact that the equation to every such curve can be put into the form pqr + pis =0. He gives a com-plete enumeration of them, including two hundred and nineteen species. In 1836 Pliicker returned to Bonn as-ordinary professor of mathematics. Here he published his Theorie der Algebraischen Curven which formed a con-tinuation of the System der Analytischen Geometric The work falls into two parts, which treat of the asymptotes and singularities of algebraical curves respectively; and extensive use is made of the method of counting constants which plays so large a part in modern geometrical researches. Among the results given we may mention the enumeration of curves of the fourth order according to the nature of their asymptotes, and according to the nature of their singularities, and the determination for the first time of the number of double tangents of a curve of the fourth order devoid of singular points.

From this time Pliicker's geometrical researches practically ceased, only to be resumed towards the end of his life. It is true that he published in 1846 his System der Geometrie des Baumes in neuer Analytischer Behandlungs-weise, but this contains merely a more systematic and polished rendering of his earlier results. It has been said that this cessation from pure mathematical work was due to the inappreciative reception accorded by his country-men to his labours, and to their jealousy of his fame in other lands; it seems likely, however, that it was due in some degree to the fact that he was called upon to under-take the work of the physical chair at Bonn in addition to his proper duties. In 1847 he was made actual professor of physics, and from that time his wondrous scientific activity took a new and astonishing turn.

Plücker now devoted himself to experimental physics in the strictest sense as exclusively as he had formerly done to pure mathematics, and with equally brilliant results. His first physical memoir, published in Poggendorf's Annalen, vol. lxxii., 1847, contains his great discovery of magnecrystallic action. Then followed a long series of researches, mostly published in the same journal, on the properties of magnetic and diamagnetic bodies, establishing results which are now part and parcel of our magnetic knowledge. It is unnecessary here to analyse these re-searches, of which an account has been given in the article MAGNETISM (vol. xv. p. 262 sq.); it will be sufficient to say that in this work Pliicker was the worthy collaborateur, and, had it not been that their fast friendship and mutual admiration renders the word inappropriate, we might have said rival, of Faraday.

In 1858 (Pogg. Ann., vol. ciii.) he published the first of his classical researches on the action of the magnet on the electric discharge in rarefied gases (see ELECTRICITY, vol. viii. p. 74). It is needless now to dilate upon the-great beauty and importance of these researches, which remain the leading lights in one of the darkest channels of magnetic science. All the best work that has recently been done on this important subject is simply development of what Pliicker did, and in some instances (notably in many of the researches of Crookes) merely reproduction on a larger scale of his results.

Pliicker, first by himself and afterwards in conjunction with Hittorf, made many important discoveries in the spectroscopy of gases. He was the first to use the vacuum tube with the capillary part now called a Geissler's tube,.

by means of which the luminous intensity of feeble electric discharges was raised sufficiently to allow of spectroscopic investigation. He anticipated Bunsen and Kirchhoff in announcing that the lines of the spectrum were character-istic of the chemical substance which emitted them, and in indicating the value of this discovery in chemical analysis. According to Hittorf he was the first who saw the three lines of the hydrogen spectrum, which a few months after his death were recognized in the spectrum of the solar protuberances, and thus solved one of the mysteries of modern astronomy. For a fuller account of the important discoveries regarding the influence of tempera-ture and pressure on the nature of gaseous spectra made in conjunction with Hittorf see SPECTRUM ANALYSIS.

Hittorf, who had good means of knowing, tells us that Pliicker never attained great manual dexterity as an experimenter. He had always, however, very clear con-ceptions as to what was wanted, and possessed in a high degree the power of putting others in possession of his ideas and rendering them enthusiastic in carrying them into practice. Thus he was able to procure from the Sayner Hiitte in 1846 the great electromagnet which he turned to such noble use in his magnetic researches; thus he attached to his service his former pupil the skilful mechanic Fessel; and thus he discovered and fully availed himself of the ability of the great glass-blower Geissler, in conjunction with whom he devised many of those physical instruments whose use all over the civilized world has rendered the name of the artificer of Bonn immortal. It was thus also that, when he felt his own want of chemical knowledge and manipulative skill, he sought and obtained the assistance of Hittorf, one of the ablest of German experimenters.

Induced by the encouragement of his mathematical friends in England, Pliicker in 1865 returned once more to the field in which he first became known to fame, and adorned it by one more great achievement—the invention of what is now called Line Geometry. A remark containing the fundamentally new idea of this new geometry had, as Clebsch remarks, already been embodied in the System der Geometric des Raumes :—" A straight line depends on four linear constants. The four magnitudes which we consider as variables receive for any given line constant values, which may be easily constructed and are the four coordin-ates of the straight line. A single equation between these four coordinates does not determine a locus for the straight line, but merely a law according to which infinite space is made up of straight lines." Here we have the new idea of the straight line considered as an element of space, and of the "complex," as Pliicker afterwards called it, made up of a threefold infinity of straight lines subject to a onefold relation. Space thus becomes as it were four-dimensioned, and we have, instead of the three degrees of freedom of space considered as an aggregate of points, four degrees of freedom according as the linear element is (1) absolutely unconditioned, (2) subject to a onefold, (3) subject to a twofold, or (4) subject to a threefold relation. In the first case we have the complex of straight lines, in the second the congruency of lines, in the third the regulus or ruled surface. The last of these geometri-cal figures had been considered long before, and even the congruency had been discussed before or independently of Pliicker, notably by Hamilton and Kummer. The general conception of the linear complex seems to be entirely due to Pliicker. At all events he developed the notion to such an extent that he is entitled to be called the founder of Line Geometry, in which the theory of the complex holds a fundamental position. His first memoir on the subject was published in the Philosophical Transactions of the Royal Society of London. It attracted much attention, and almost at once became the source of a large literature in which the new science was developed. Pliicker himself worked out the theory of complexes of the first and second order, introducing in his investigation of the latter the famous complex surfaces of which he caused those models to be constructed which are now so well known to the student of the higher mathematics. He was engaged in bringing out a large work embodying the results of his researches in Line Geometry when he died on the 2 2d May 1868. The work was so far advanced that his pupil and assistant Klein was able to complete and publish it, there-by erecting the worthiest monument to the genius of his master, whose wonderful scientific activity endured to the very last. Of the very numerous honours bestowed on Pliicker by the various scientific societies of Europe it may suffice to mention here the Copley medal, awarded to him by the Royal Society two years before his death.

Most of the facts in the above notice have been taken from Clebsch's obituary notice of Plücker (Abh. d. Kim. Ges. d. Wiss. z. Gottingen, xvi., 1871), to which is appended an appreciation of Plücker's physical researches by Bittoff, and a list of Plücker's

works by F. Klein. See also Gerhardt, Geschichte der Mathematik in Deutschland, p. 282; and Pliicker's life by Dronke (Bonn, 1871). (G. CH.)

**Footnote**

The independent development of a similar idea by the brilliant young French geometer Bobillier (1797-1832) was cut short by his premature death.