452 



NATURE 



[March 13, 1902 



are required who will, without doubt, introduce new ideas and 

 obtain results far transcending those we are in possession of 

 now. The latest work has shown that the method is applicable 

 to boards of triangular and trapezoidal shapes, and also to solid 

 boards in three dimensions, so that the remote ground occupied 

 by magic and Nasil; cubes will soon be invaded. 



In conclu.sion, I bring before you an interesting example of 

 magic arrangement that I founc^ whilst engaged in rummaging 

 amongst the books and documents of the old Mathematical 

 Society of Spital fields (1717-1845) for the purpose of extracting 

 something which might interest or amuse, if it might not in- 

 struct, the audience I addressed in Section A of the British 

 Association for the Advancement of Science at (Glasgow last 

 autumn. It is an arrangement of the first eighteen numbers on 

 five connected triangles ; the magical property consists in the 

 circumstance that the numbers 19, 38 and 57 appear as sums in 

 a variety of ways. The number 19 appears nine times, 38 

 twelve times and 57 fourteen times (Fig. 15). 



0= 7+12=14+ 5= 4-l-lr, 



= 6+13=17+ 2= 9+10 



=1«+ 3= 1 + 18= 8+11 

 S8z= 7+11 + 14+ 0=11+ 2 + 15+10=15+11!+ 0+ 1 



=11 + 10+ 3 + 14=10+15+ 1+12= 1+ 15+14+17 



=14+ 3+ 4 + 17= 3+10+12+13=12+ 1+17+ S 



= 3+13+18+ 4=13+12+ 8+ 5= 8+17+ 4+ !• 

 57= 7+14+ 4+ 5+12+16= 8+17+ S1+13+10+ 2=18+ 1+ 8+18+ 3+11 



= 7+11+ 2 + 15+16+ 0=11+10+15+ 1+ 6+14=14+ 3+10+12+ 1 + 17 



= 3+13 + 12+ 8+17+ 4= 4+18+13+ 5+ 8+ 9 



= 9+ 4+ 3+10+15+16=18+13+12+ 1+ 0+ 7= 5+ 8+17+14+11+ 2 



= 9+ 8+12+10+11+ 7=18+ 4+17+ 1+15+ 2= 5+13+ 3+14+ 6+10 



I should say that I feel conscious that I have not been able 

 to introduce the subject of my lecture without occasional and, 

 perhaps, in the circumstances, unavoidable obscurity. Vox 

 the rest, I have felt somewhat doubtful as to the interest I might 

 arouse in these problems, but the managers honoured me 

 by inviting me to display to you some of the chips from a 

 pure mathematician's workshop, and I felt no hesitation in 

 accepting. 



FORTHCOMING BOOKS OF SCIENCE. 



Mr. Felix Alcan (Paris) promises : — " Les Bases scientifiques 

 de I'Kducation physique," by Demeny ; "Les Grands I'he- 

 nomenesgtiologiques," by Prof. S. Meunier ; " Manuel d'iilectro- 

 theropie," by A. Weill ; " Traite d'Intubation du Larynx," by 

 Bonam ; " Manuel d'Histologie pathologique," tome ii., by 

 MM. Durante, Dominici, &c. 



Mr. Edward Arnold gives notice of: — "Elementary Princi- 



ples in Statistical Mechanics, by Dr. J. W. Gibbs, and " The 

 Elements of Experimental Phonetics," by Dr. E. W. Scripture. 



Messrs. G. Pell and Sons announce: — " Comparative Anatomy 

 of Animals, an Introduction to the Study of," by Dr. G. C. 

 Bourne, vol. ii. : — The Coelomata, illustrated; "Elementary 

 General Science," by I). E. Jones and Dr. D. S. Macnair ; " In- 

 jurious and Useful Insects," by Prof. L. C. Miall, F.K.S., 

 illustrated ; " Physiography," by H. N. Dickson ; " Electricity 

 and Magnetism," by Dr. Oliver J. Lodge, F.K.S.; "Light," 

 by A. E. Tutton, F.R.S. 



Messrs. A. and C. Black promise: — "Problems in Astro- 

 physics," by Agnes M. Gierke, and a new edition of the same 

 writer's " History of Astronomy during the Nineteenth Century." 



The announcements of the Cambridge University Press in- 

 clude : — "Catalogue of Scientific Papers," compiled by the 

 Royal Society, vol. xii., supplementary volume; "Scientific 

 Papers," by John William Strutt, Baron Rayleigh, F.R.S. , vol. 

 iv. ; "Theory of Differential Equations," by Prof. A. R. For- 

 syth, F.R.S., part iii. : — Ordinary Linear Equations ; "Mathe- 

 matical Analysis," by E. T. Whittaker ; " The Algebra of 

 Invariants," by J. H. Grace and A. Young ; " Electric Waves, 

 being an Adams Prize Essay in the University of Cambridge," 

 by H. M. Macdonald, F.R.S. : " A Treatise on Determinants, " 

 by R. F. Scott, a new edition by G. B. Mathews, F.R.S. ; 

 "The Electrical Properties of Gases," by Prof. J. J. Thomson, 

 F.R.S. ; "A Treatise on Spherical Astronomy," by Sir 

 Robert S. Ball, F.R.S. ; " Fossil Plants, a Manual for Students 

 of Botany and Geology," by A. C. Seward, F.R.S. , vol. ii. ; "A 

 Primer of Botany," by F. F. Blacknian ; " A Primer of 

 Geology," by J. E. Marr, F.R.S.; " Immunity in Infectious 

 Diseases," by Prof. E. MetchnikofiT, authorised English trans- 

 lation by F. G. Binnie, illustrated ; " Index Nominum Ani- 

 malium," compiled by C. D. Sherborn under the supervision of 

 a Committee appointed by the British Association and with the 

 support of the British Association, the Royal Society and the 

 Zoological Society, vol. i. (175S-1800); "Zoological Results 

 based on Material from New Uritain, New (Juinea, Loyalty 

 Islands and elsewhere, collected during the years 1S95, 1896 

 and 1S97," by Dr. A. Willey, part vii. Conclusion : "Reports 

 of the Anthropological Expedition to Torres Straits by the 

 members of the Expedition," edited by Dr. A. C. Haddon, 

 F.R.S. (it is expected that the work will be completed in five 

 volumes); "Fauna Hawaiiensis, or the Zoology of the Sand- 

 wich Islands : being results of the Exploratii'ns instituted by 

 the Joint Committee appointed by the Royal Society of London 

 for promoting Natural Knowledge and the British Association 

 for the .Vdvancement of Science, and carried on with the as- 

 sistance of those bodies and of the Trustees of the Bernice 

 Pauahi Bishop Museum," edited by Dr. D. Sharp, F.R.S. ; 

 " The t'auna and Geography of the Maldive and Laccadive 

 Archipelagoes : being the Account of the Work carried on and 

 of the Collections made by an Expedition during the years 

 1899 and 1900 under the Leadership of J. S. Gardiner," vol. 

 ii. part ii. ; " The Geographical Distribution of Diseases," by 

 Dr. F. G.Clemow ; " An Introduction to Logic," by VV. E. 

 John.son ; " Euclid, Books i.-iii., with Simple Exercises," by 

 R. T. Wright ; "An Introduction to Physiography," by W. N. 

 Shaw, F.R.S.; "A Brief History of Geographical Discovery 

 since 1400," by Dr. F. II. H. Guillemard ; and a new 

 edition of " Solution and Electrolysis," by W. C. D. Whetham. 



Messrs. Cassell and Co., Ltd., give notice of: — " The .\scent 

 of Aconcagua," by Sir W. M. Conway, illustrated; Cassell's 

 "Cyclopaedia of Mechanics," edited by 1'. N. Hasluck, second 

 series, illustrated ; " The Automobile : its Construction and 

 Management," translated from Gerard I.avergne's " Manuel 

 Theoretique et Pratique de I'Aulomobile sur Route," revised 

 and edited by P. N. Hasluck, illustrated. 



Messrs. W. and R. Chambers, Ltd., call attention to : — "The 

 Nineteenth Century Series," containing " Medicine, Surgery, 

 and Hygiene in the Century," by Dr. E. H. Stafford ; " Progress 

 of India, Japan, and China in the Century, ' by the Right Hon. 

 Sir R. Temple, Bart., F.R.S.; " Progress of the United States 

 of America in the Century," by Prof. W. I' Trent ; " Progress 

 of British Empire in the Century," by J. S Little; "Progress 

 of Canada in the Century," by J. C. Hopkins ; " Progress of 

 Australasia in the Century," by T. A. Coghlan and T. T. 

 Ewing ; "Progress of New Zealand in the Century " ; "Dis- 

 coveries and Explorations of ihe Century," by Prof. C. G. D. 

 Roberts : " Economic and Industrial Progress of Ihe Century," 

 by Dr. II. de B. Gibbins ; " Invmtions of the Century," by 



NO. 1689, VOL. 65] 



