Less exclusive than Steiner was C. G. Von Staudt (1798–1867),1 who did not completely separate himself from the influence of his great master Gauss and dedicated a part of his own intellectual activity to researches on the theory of numbers. A part, but not the most elevated and decisive, for this was absorbed by the solution of the great problem to treat the geometry of position without introducing any metrical concept; the Geometrie der Lage (Nürnberg, 1847), and the Beiträge, with which he adorned it in 1856, 1857, and 1860, certify to the complete result of his assiduous and genial efforts to rid of any extraneous element that projective geometry of which a little while before Poncelet and Chasles in France, Steiner and Möbius in Germany had laid the foundations. We cannot in few words delineate the content of these writings which justify for Von Staudt the enviable epithet of the "Euclid of the nineteenth century"; we shall only mention the new definition of projectivity there given, which contains only what is necessary and sufficient, and those of the conic and of the quadric, which, including also imaginary curves and surfaces, are able to rival the analytical definitions, and like these are susceptible of being extended to curves and surfaces of any order; we shall add that the researches on the imaginary in geometry availed to put to flight the "spectre" which had pursued Steiner in the last years of his life, and to render superfluous the "principle of continuity" which had multiplied the opponents of the doctrine of Poncelet. Other minor works of

1 See the study by C. Segre prefixed to Geometria di posizione di C. G. von Staudt, trad, dal tedesco a curo del Dr. M. Pieri (Torino, 1888).

2 Comp. A. Ramorius, "Gli Elementi imaginari nelle Geometria" (Giorn, di Mat., 35, 1897, e 36, 1898).

3 Among all the investigations of Von Staudt these are undoubtedly the most abstruse; on this account their results spread with more difficulty in the mathematical world; to facilitate their comprehension, efforts were made in various directions by Lüroth (Math. Annalen, 8, 1875, and 11, 1877), August (Programm der Friedrichs-Realschule, Berlin, 1872), Stolz (Math. Ann., 4, 1871); Henry J. Stephen Smith (Ann. di Mat., II., 3, 1869–1870), H. Wiener (Rein geometrische Theorie der Darstellung binärer Formen durch Punktgruppen auf den Geraden, Darmstadt, 1885), Segre (Torino Mem., II., 38, 1886, and Journ. f. Math., 100, 1886) and Servais (Belgique Mém., 49, 1896). To the theory of imaginaries of Von Staudt is allied the "Rechnung mit Würfen," to which Lüroth (Mem, cited), Sturm (Math. Ann., 9, 1876), Schröder (ibid., 10, 1876) and G. Kohn (ibid., 46,

Von Staudt contain applications of the aforesaid doctrines, and, demonstrating how he knew duly to appreciate and in a masterly manner to treat metrical questions, make us regret that it was not given him to give to the Geometrie der Lage a sister work in the Geometrie des Masses which he had projected.1

A direction completely different from the investigations of Von Staudt have the publications of Julius Plücker (1801-1868),2 to whom analytical geometry is indebted for decisive advancements; to him in fact we owe the development of homogeneous and polyhedral coördinates, of the coördinates of the straight in the plane and of the plane in space; to him, omitting for a moment

1895) have devoted special researches. In this last work is extended the concept of Würfe to a group of n elements of a form of I species; such extension is applied in the later work of the same geometer entitled "Die homogenen Coordinaten als Wurfcoordinaten" (Wien Ber., 104, 1895). A theory of imaginary elements different from that of Von Staudt is created by A. Mouchot and expounded in the works: La réforme Cartésienne étendue aux diverses branches des mathématiques (Paris, 1876) and Les nouvelles bases de la géométrie supérieure (Paris, 1892).

It is by inspiring themselves with the conceptions of Von Staudt, or rather by unfolding them excellently, that Juel (in the dissertation "Bildrag til den imaginaere Linies og den imaginaere Planis Geometri," Kopenhagen, 1885, and in the article Ueber einige Grundgebilde der projectiven Geometrie," Acta, 14, 1889) and Segre (in the interesting group of notes on "Un nuovo campo di ricerche geometriche," inserted in Torino Atti, t. 25 and 26, 1890 and 1891) discovered new correspondences and new figures which it is necessary to consider to exhaust completely the projective geometry of the plane (that is with real and complex points). In the same direction Sforza proceeded in writing the "Contributo alla geometria complessa" (Giorn. di Mat., 30, 1892).

2 See A. Clebsch, "Zum Gedächtniss an Julius Plücker" (Götting. Abh., 15, 1872). Beltrami justly notes to this discourse that "the best eulogy which can be made of Plücker, considered as a geometer, is this, that Clebsch was not able to weave the account of his works, without recounting to a large extent the history of modern analytical geometry" (Giorn. di Mat., 11, 1873, p. 153). Besides numerous memoirs published mostly in the Journ. f. Math., we are indebted to Plücker for five great geometrical works; they are, Analytisch-geometrische Entwickelungen (Essen, 1828-1831), System der analytischen Geometrie (Berlin, 1835; see the Anzeige given of it by Plücker himself in Journ. f. Math., 10, 1833), Theorie der algebraischen Curven (Bonn, 1839), System der Geometrie des Raumes (Düsseldorf, 1846), and Neue Geometrie des Raumes (Leipzig, 1868-1869).

311

"Ueber ein neues Coordinatensystem " (Journ. f. Math., 5, 1829).

"Ueber eine neue Art, in der analytischen Geometrie Punkte und Curven durch Gleichungen darzustellen" (ibid., 6, 1829).

511

"Note sur une théorie générale et nouvelle des surfaces courbes " (ibid., 9,

the geometry of the straight in space of which we shall treat ex professo later, we owe finally varied and most important applications of the "method of abridged notation" of which he is one of the creators, and of that of the "enumeration of constants" which, often but not always, he knew how to employ fitly. If we here dispense with enumerating at this moment the new results for which our science is indebted to him, it is that we consider it much more convenient to do it in describing the successive evolution of the individual theories which constitute modern geometry, and to which it is now time for us to turn, having finished the sketch of the intellectual movement which prepared the present epoch. We thus shall see how the great men of whom we just now learned have been followed by a numerous and brilliant cohort of disciples, who, gleaning in the fields plowed by the masters, proved the fecundity of the seed which these had sowed.1

G. LORIA.

UNIVERSITY of Genoa.

1 "Ueber ein neues Princip der Geometrie und den Gebrauch unbestimmter Symbole und Coefficienten" (ibid., 5, 1829); "Analytisch-geometrische Aphorismen" (ibid., 10 and 11, 1831); "Ueber Curven dritter Ordnung und analytische Beweisführung" (ibid., 34, 1847).

2 Of this procedure (invented also by Bobillier) Plücker indicates the most valuable qualities by the following words: "Meine Gleichungsformen sind vollständige Darstellungen graphischer Constructionen, in denen nichts Fremdartiges sich findet; es sind ideale, mit analytischen Symbolen hingezeichnete Figuren." (Journ. f. Math., 34, 1847, P 332.)

3 No one ignores how dangerous is this artifice, otherwise very fertile (see for this the recent memoir published by Küpper in the Math. Ann., 32, 1888); Plücker, who knew and boasted of its qualities, knew moreover its drawbacks and often succeeded in avoiding them in the manner described by Clebsch in the afore-mentioned Commemoration.

The foregoing Sketch may be followed by other extracts from Loria's work, giving the history of modern geometry as above indicated. But the editors are at present unable to promise how much will be offered to their readers.—Ed. Monist.

RELIGION IN FRANCE.

INTRODUCTORY.

HAT forms does the religious sentiment take in the various

WH

provinces of France and in the various classes of the French nation? How is this sentiment satisfied by existing positive religions? These are the questions which I propose to answer in the present paper. I make no pretence to settling the matter, and the inquiry of which I shall here give the results will doubtless throw but a pale light on the religious soul of France. To the set answers obtained from my questionnaires I shall add the thousand and one little facts of my daily observation and reading, hoping from these sources to arrive at a tolerably exact view of the situation.

If it should be asked in what manner a religion can satisfy the religious sentiment,-that particular state of the soul which involves fear, a sense of the beautiful, phantasy, and the need of knowledge, and which is apparently compounded of all forms of desire, we shall at once discover that dogmas and beliefs are not alone sufficient. A religion acts, and acts powerfully, through the personal quality of its ministers, through the forms of its observances, through the architecture of its churches, and the character of its ceremonies.

It would be useles to insist on the importance of the monuments and the ceremonies of the Catholic religion; the same flight of faith that gave birth to the cathedrals still inspires respect for it among us. Not one of the least causes of the disfavor with which the Reformation was received in our country was the indiscreet zeal of the Calvinists in demolishing statues and mutilating the portals of the sacred edifices; the genius of the nation felt itself

wounded in its dearest living creations. The absence of all pomp in the reformed religion and the bareness of its places of worship were also a cause of estrangement; the popular sou did not find in the new faith the spectacles which charmed it, the joyfulness of its old festivals, and the wealth of art which had become a common possession even of the suffering and unfortunate multitudes. There is a body of sentiment so deeply rooted in the Latin race that even its most indifferent representative, when contemplating the magnificent monuments of the past, cannot but feel some vestige of the ancient emotions, and his eyes still cling to their solemn grandeurs even when their religious significance would never affect him.

I believe, therefore, that I am justified in saying that the same frame of mind which led our people to oppose the reformation of Calvin still continues to estrange us from it. It must also be acknowledged that Protestantism, even though it offers some satisfaction to the reason and gives wider scope to criticism, is still in many respects inferior to Catholicism. Its artistic infertility alone is sufficient to justify this statement. It was never able to renew in France the miracles of the preceding centuries. Even in the countries where it triumphed, it could only bestow upon the edifices which it occupied that appearance of "devastated churches" which so forcibly impressed French, Italian, and Spanish travellers and reawakened in them, if only for a moment, the Catholic sentiment, as they stood in the presence of that "bare and sombre religion which saddens both the eyes and the heart."2 Finally, coming as late as it did, it could make no pretension to satisfying the intellect of the cultivated classes. Human reason, already exacting in its demands, clamored for more than Protestantism offered, and strong minds could not find in the new belief the complete enfranchisement for which they longed on leaving the old.

The reformed religion came, therefore, not as an alleviation,

1 The only exception perhaps is music, and that principally among the followers of Luther. Luther possessed a warmth and feeling for art which Calvin utterly lacked.

2 Emilio Castelar.