590 



NATURE 



[January 30, 1913 



tial evidence bearing on the theory that the Druids 

 came from the south, while the British Celts came 

 froni the east ; that they, or their representatives, 

 were ir; Britain before the Celts — that is, in the 

 Neolithic age or the age of the megalithic monu- 

 ments ; and that the Druids of the times of Caesar 

 and Pliny were priests or professors of a Celti- 

 cised Iberic system. As evidence, the southern 

 origin of the oak-mistletoe association would agree 

 admirably with some syntactical peculiarities of 

 British Celtic speech which are distinctly non- 

 Aryan, and have been traced southward through 

 North African dialects to ancient Coptic. 



John Griffith. 



FAGNANO'S MATHEMATICAL WORKS. 

 Opere Maiematiche del Marchese Giiilio Carlo de' 

 Toschi di Fagnano. Pubblicate sotto gli 

 auspici della Societa Italiana per il Progresso 

 delle Scienze dai soci \". \'oIierra, G. Loria, 

 D. Gambioli. \^olume Primo. Pp. ix + o— q + 474. 

 Volume Secondo. Pp. xi + 471. Volume Terzo. 

 Pp. xi + 227 + 2 plates. (Milano, Roma & 

 Napoli : Albrighi, Segati e C, igii and 1912.) 

 Price 40 lire (complete in three volumes). 



GIULIO CARLO DEI TOSCHI DI 

 FAGNANO was born in Senegal on 

 September 26, 1682, and died there on May 18, 

 1766. In his boyhood he composed poetry; later 

 on he was attracted to the study of philosophy 

 and became a follower of Leibniz and Newton. 

 At the age of fourteen he went to college at Rome, 

 but his taste for mathematics only developed later, 

 and he had to devote himself to its study without 

 assistance out in Senegal. Although he became 

 absorbed in his mathematical studies sometimes 

 for days at a time, he also occupied himself with 

 administrative work, and gave expert advice to 

 the Pope Benedict XIV. regarding the safety of 

 the cupola of St. Peter's at Rome. In return, the 

 Pope promised to publish his "Mathematical Pro- 

 ductions," but for some reason the promise 

 was not fulfilled, and they were not published until 

 1750- 



The best-known original work of Fagnano is 

 that referring to the rectification of curves. 

 "Fagnano's Theorem," relating to certain 

 properties of arcs of ellipses, has frequently figured 

 in English text-books, and is now recognised as 

 the starting-point from which sprang the modern 

 theory of elliptic functions. That such was actu- 

 ally the case is confirmed in the preface to these 

 volumes, where it is stated that the work of Euler 

 and Jacobi was initiated by the examination of 

 a presentation copy of Fagnano's works, received 

 by the Berlin Academy of Sciences. 

 NO. 2257, VOL. 90] 



This and other circumstances led the Italian 

 Association for the Advancement of Science to 

 undertake the publication, not only of Fagnano's 

 "Mathematical Productions," but also of his un- 

 published writings and correspondence, the work 

 being placed in the hands of Profs. Volterra, 

 Gino Loria and Gambioli. 



The first two volumes contain the " Produzioni," 

 as originally published by Fagnano. The greater 

 part of vol. i. is taken up with a treatise on the 

 geometrical theory of proportion, which includes 

 the whole of the propositions in Euclid's fifth 

 book. In addition, we have papers on the 

 solution of the cubic and biquadratic and on the 

 mathematics of gambling, with special reference 

 to lotto — a subject of considerable importance in 

 Italy, where the Banca di lotto is a source of 

 revenue to the State. The second volume con- 

 tains a treatise on rectilinear triangles, and 

 Fagnano's contributions to the study of analysis. 

 These latter, unlike the former, consist of a series 

 of short papers which perhaps constitute the 

 most important of Fagnano's works. The third 

 volume consists of work not included in Fagnano's 

 opus magntiui, namely, articles published after the 

 appearance of the "Productions" in 1750, as 

 well as correspondence. The collection of the 

 material for this volume is due to Prof. Gambioli. 

 A biography of Count Fagnano forms a fitting 

 conclusion. 



The series of volumes forms a useful addition 

 to the archives of mathematical history. That the 

 original work of this mathematician still opens 

 up fields for investigation is shown by a recent 

 paper on elliptic trammels and Fagnano points, 

 contributed t6 The Mathematical Gazette for May 

 and Julv, 191 1, by Mr. Percy J. Harding. 



G. H. B. 



GEOLOGY IN THE SOUTHERN 

 HEMISPHERE. 

 (i) Soutli .ifrican Geology. By Prof. E. H. L. 

 Schwarz. Pp. 200. (London : Blackie and 

 Son, Ltd., igi2.) 35. 6d. net. 



(2) Geology of New Zealand. By Dr. P. Marshall. 

 Pp. viii + 218. (Wellington, N.Z. : J. Mackay, 

 Government Printer, 191 2.) 



(3) An Introduction to the Geology of Neiv South 

 IVales. By C. A. Siissmilch. Pp. xii-f-i77. 

 (Sydney, N.S.W. : W. A. GuUick, Government 

 Printer, 191 1.) 5s. 



THESE three books are written with an 

 educational purpose, and are kept within 

 j the limits of size suitable for schools. They alike 

 I bear witness to the prominent place taken by 

 I geology in the outlook of settlers in the southern 

 1 hemisphere. Where human history may be traced 



