TRANSACTIONS OP SECTION A. 531 



7. On the Need of a Non-Euclidean Bibliography. 

 By Duncan M. Y. Sommerville, M.A., D.Se. 



Thirty years ago Halsted published the first bibliography of non-euclidean 

 Geometry and Hyperspace, and one still finds it referred to as a standard work. 

 But the amount of literature published on the subject since that time is enormous, 

 and the present yearly output would almost equal Halsted's whole collection. So, 

 in spite of the excellence of this bibliography and the existence of others which 

 have been more recently compiled, the student of non-euclidean geometry is pretty 

 much in the same position as the student of any other branch of mathematics, 

 and is confined for his sources to the volumes of the 'Jahrbuch,' the 'Revue 

 Semestrielle,' and the ' International Catalogue.' A notable exception exists in 

 the subject of Quaternions. In 1904 Macfarlane published an extensive biblio- 

 graphy of this subject, and the work is being continued by the International 

 Association on a still broader basis. In many ways the two subjects are akin. 

 The one is concerned with the foundations of geometry, the examination, modifi- 

 cation, and extension of the geometrical ideas, and the investigation of all the 

 various geometries which arise therefrom; the other is concerned with exactly 

 analogous questions relating to arithmetic and algebras. As instances of the close 

 connection which exists between their lines of development, one may mention 

 finite geometries and galois fields, non-archimedean geometry and non-archi- 

 medean algebra. It is natural therefore to wish for the same facilities for the 

 one department as exist for the other, but non-euclidean geometry is handicapped 

 by having no International Association to promote its interests. 



Several years ago the present writer started to collect material for a biblio- 

 graphy on the lines of Halsted's, but it soon became evident that the growth of the 

 subject rendered such diffuse treatment practically impossible. Then in 1902 

 Bonola's catalogue appeared; but, though this is the most extensive bibliography 

 that has yet been published, it still leaves something to be desired. It will be 

 convenient to describe here the existing bibliographies and the present state of 

 the material that has been collected. We may divide the subject roughly into 

 three main branches : Theory of Parallels, Foundations of Geometry and Non- 

 Euclidean Geometry, N Dimensions. Quite a number of bibliographies relating 

 to the theory of parallels exist, dating mostly from the earlier portion of the 

 nineteenth century, but they are all included in 



1. P. Stackel and F. Engel, ' Die Theorie der Parallellinien von Euklid bis 

 auf Gauss,' Leipzig, 1895. Supplemented by a list in ' Bibliotheca Mathematical 

 1899. 



A chronological list of works on the theory of parallels, including the early 

 papers on non-euclidean geometry, from 1482 to 1837, with careful references to 

 sources. Almost complete. 



In the remaining divisions only three bibliographies need be mentioned. 



2. G. B. Halsted, ' Bibliography of Hyperspace and Non-Euclidean Geometry,' 

 'Amer. Jour. Math.,' vols. i. and ii. (1878-79). 



This is a general list, including both non-euclidean geometry and n dimensions 

 from about 1830 to 1879. The order is mainly chronological, but all the works by 

 the same author are collected together. Short notes are added to the chief works. 



3. V. Schlegel, ' Sur le Developpement et l'Etat Actuel de la Geometrie a n 

 Dimensions,' ' Enseign. Math.,' vol. ii. (1900). First published in German in 

 ' Leopoldina,' vol. xxii. (1886). 



This includes only n dimensions up to 1897. The order is alphabetical under 

 the authors. Letterpress of twenty pages contains an historical sketch corre- 

 sponding to Halsted's notes. 



4. R. Bonola, ' Index Operum ad Geometriam Absolutam Spectantium.' Pub- 

 lished in ' Ioannis Bolyai, in Memoriam,' Klausenburg, 1902. First published in 

 'Boll. Bibliogr.,' Torino, 1899-1902. 



A chronological list, not including n dimensions, from 1S39 to 1902. A classifi- 

 cation is attempted under a schedule of seven classes, with class-letters which are 

 affixed to the titles. It is accompanied by an author-index. There is no subject- 

 index, and the classification is not sufficiently detailed to be of much use. 



