November 27, 1902] 



NATURE 



79 



of form, harmonies of colour, and so forth. Dr. Gold- 

 5chmidt's object appears to us to be to reduce all such 

 harmonies to a common formula, and he considers that 

 the different kinds of harmony are governed by a common 

 law, the " law of complication." 



As an example of the arguments employed, a series of 

 numbers is obtained from the intervals of the musical 

 scale which coincide with numbers in another series 

 alleged to be obtainable from crystallography. But the 

 numbers in the case of the music do not represent actual 

 intervals, but are derived from them by a homographic 

 transformation, according to which the keynote and its 

 octave are represented by o and co and the major fifth 

 by 1, and the series is incomplete unless the minor 

 seventh be included in the list. And the identity of the 

 two series is by no means complete ; for there are terms 

 in the series derived from music which are absent from 

 that obtained from crystallography. 



It is easy to find connections as close as those dealt 

 with in the present work between phenomena which have 

 nothing whatever in common. For a considerable period, 

 the number of wranglers in the Cambridge mathematical 

 tripos was observed to be intimately related to the fre- 

 quency of sun-spots, and anyone who should seek to 

 establish a connection between the notes of the musical 

 scale and the courses of a tabic d'hote dinner might 

 easily make out a very strong case. What is most sur- 

 prising is that the analogy which a priori exists between 

 musical intervals and colour intervals, both of which 

 depend on ratios of vibration-frequency, appeals but 

 little to our senses, so little, in fact, that certain writers 

 have even sought to establish relations between chords 

 and colours quite independently of the known relations 

 of pitch. As for the connection which no doubt exists 

 between a love of music and a talent for mathematics, 

 its cause is not difficult to find. A mind like that of 

 Beltrami, who could discover in the purely abstract ideas 

 of geometry and algebra truths applicable to spaces 

 other than that in which we live, was necessarily well 

 trained to appreciate that beauty of form dissociated 

 from worldly matters which exists in the sonatas and sym- 

 phonies of the older composers. In order, on the other 

 hand, to make it more palatable to a mind that wants to 

 grasp something tangible, music is commonly associated 

 with such mundane ideas as love, vice, battle and murder, 

 and sudden death, the triumph of the victorious, the 

 wails of the vanquished. 



Opere Matematiche di Francesco Brioschi. Vol. ii. Opere 

 Matematiche di Eugenic* Beltrami. Vol. i. Pp. 456 

 and 437. (Milan : Ulrico Hoepli, 1902.) Price 25 lire. 



THE second volume of Brioschi's works contains thirty- 

 five papers contributed to the Annali di Matematica 

 pura ed applica/a, series 1 and series 2, vols, i.-xiv., 

 between the years 1858 and 1887. These papers have 

 all been carefully revised by Profs. Cerruti (Rome), 

 Gerbaldi (Palermo), Loria (Genoa), Pascal (Pavia), 

 Pittarelli (Rome), Reina (Rome) and Tonelli (Rome). 

 A considerable number of them deal with linear dif- 

 ferential equations, but elliptic and hyperelliptic 

 functions, curvilinear coordinates, binary forms and 

 many other subjects are treated ; and the papers also 

 include obituary notices of Borehardt and Chasles. 



After the death of Prof. Beltrami, in 1900, the Faculty 

 of Science of the University of Rome resolved to estab- 

 lish a memorial of the distinguished mathematician, 

 and it was decided that the most fitting form for the 

 memorial would be a complete edition of Beltrami's col- 

 lected works ; to quote Prof. Tonelli, monumentum aere 

 perennius. In this case, the work of preparing the 

 volumes has been carried out entirely under the direc- 

 tion of Profs. Cremona, Castelnuovo and Tonelli, as 

 representatives of the Roman Faculty of Science, 

 who have been aided by the collaboration of Profs. 



NO. 1726, VOL. 67] 



Bianchi, Burgatti, Cerruti, Dini, Pittarelli, Reina and 

 Volterra. The order of arrangement differs from that 

 adopted for Brioschi's works. Instead of being grouped 

 according to journals, Beltrami's papers are arranged in 

 strict chronological order, and this volume represents the 

 work of eight years, from 1861 to 1868. That these first 

 eight years of Beltrami's career as a mathematician were 

 productive of work of great value is shown by the list 

 of titles, which include researches on analysis applied to 

 geometry, the flexure of ruled surfaces, resolution of the 

 problem of transforming geodesies on a surface into 

 straight lines in a plane, complex variables on any sur- 

 face, fundamental theories of space of constant curva- 

 ture, and last, but not least, the " Saggio d'interpretazione " 

 of non-Euclidean geometry. The portrait of Beltrami 

 which forms the frontispiece is due to Prof. Pittarelli. 



Beltrami's works are published in uniform style with 

 those of Brioschi, and both are printed by the Mathe- 

 matical Press, of Palermo. 



Handbook of the Trees of Nezu England. By L. L. 



Dame and Henry Brooks. Pp. xv + 196. (Boston, 



U.S.A. : Ginn and Co., 1902.) 

 The interest connected with the flora of the New 

 England States lies in the fact that situated between 

 Canada and the Alleghany Mountains they furnish the 

 meeting point of a northern and a more southern flora. 

 Since the book is limited to such a relatively small part 

 of the country, it does not possess the general interest 

 which would attach to one which included, for instance, 

 the trees of all the eastern States. What it loses in 

 comparative value, perhaps it gains in definiteness ; it 

 contains useful and succinct descriptions, good illustra- 

 tions specially drawn, and states the horticultural value 

 of all the indigenous species. The Latin nomenclature 

 is satisfactory and correct, except in the case of a 

 species of Acer, and for Ouercus Muhlenbergii, which is 

 considered by some authorities to be a variety of 

 Quercus primes ; but the popular names are in utter 

 confusion, and we cannot agree with the authors that it is 

 wiser " to record what is, and not what ought to be." 

 Taking Popitlus balsamifera as an illustration , the 

 names recorded are " Balsam. Poplar. Balm of Gilead." 

 Now this tree is certainly not a balsam, and Populus 

 candicans is the real Balm of Gilead ; while the name 

 balsam-poplar would be sensible and correct. Apart 

 from this and within its limits, the book may be recom- 

 mended either to enable one to identify the trees or to 

 ascertain their characteristics. English readers will find 

 that only about half-a-dozen species are the same as 

 those indigenous to this country. 



Lake-Country Rambles. By William T. Palmer. Pp. 



viii + 334. (London : Chatto and Windus, 1902.) 



Price 6s. 

 Mr. Palmer has here collected a series of papers he 

 has from time to time contributed to vaiious magazines. 

 For many years the author has been a rambler in the 

 lake-country, and has learned to love its inhabitants and 

 to study its varied scenes. The essays are good ex- 

 amples of descriptive writing, but the aspects of nature 

 and the incidents of outdoor life are treated rather from 

 the point of view of the general observer than that of the 

 inquiring naturalist. 



Junior Arithmetic Examination Papers. Arranged by 

 W. S. Beard. Pp. vi +106. (London: Methuen and 

 Co., 1902.) Price is. 

 The ninety examination papers contained in this col- 

 lection cover all the parts of arithmetic generally studied 

 in schools. The first third of the papers gradually in- 

 crease in difficulty from paper 1, on the first four rules, 

 to paper 30, on the mensuration of rectangular solids. 

 The remaining papers are made up of mixed questions 

 and are all well graduated. The questions should be 

 useful to teachers. 



