470 



SCIENCE. 



[N. S. Vol. XVI. No. 403. 



of ordinary algebra, of determinants, of sub- 

 stittition and group theory, of the theory of 

 numbers, and of the theory of probability. 

 Netto's book is of value as a reference book, 

 ■especially as no text of importance on com- 

 binatoric has been published for sixty-five 

 years. In arrangement and selection of mate- 

 rial it resembles somewhat Netto's brief article 

 'Kombinatorik' in the ' Encyklopadie der 

 Mathematischen Wissenschaften.' The book 

 takes notice of researches of recent date, in- 

 cluding several papers by A m erican authors. 

 Starting _out with the fundamental defini- 

 tions the author treats of combinations, per- 

 mutations, and variations under different 

 limiting conditions, leading up to various 

 problems, as, for instance, Tait's problem of 

 knots. Combinations and variations are con- 

 sidered under the restriction of a definite sum 

 or a definite product of the elements. The 

 partition of numbers and Durfee's graphs are 

 taken up. In the course of further combina- 

 torial operations the author studies systems 

 of triads arising in connection with Kirk- 

 mann's and Steiner's problems. Steiner's 

 queries have not yet been fully answered. 

 Kirkmann's is the 'Fifteen School Girl Prob- 

 lem': 'To walk out fifteen girls by threes, 

 daily for a week, without ever having the 

 same two together.' In the discussion of this 

 it is to be regretted that Netto overlooked E. 

 W. Davis's pretty 'geometric picture,' given 

 in the Annals of Mathematics, Vol. XL, 1897, 

 where a one-to-one correspondence is estab- 

 lished between the fifteen girls and fifteen 

 points on a cube; eight points at the corners, 

 six at the mid-points of the faces, one at the 

 cube-center; the thirty-five triads are then 

 easily found. 



Netto's book is substantial food for the 

 average reader. Yet some topics in combina- 

 toric were originally suggested by questions 

 propounded for amusement. The 'problem of 

 the eight queens' is of this nature. Eight 

 queens are to be placed upon a chessboard so 

 that none of them can capture any other. It 

 was first propounded in Berlin in 1848 and 

 has 92 solutions. J. Bernoulli, in his 'Ars 

 Conjectandi' (1Y13), gives certain hexameter 

 lines in which the words were to be changed 



about in every possible way, yet so that every 

 new arrangement still conformed to the laws 

 of verse. Thus the hexameter. 



Tot tibi sunt dotes, Virgo, quot sidera coelo, 

 studied by several pious mathematicians, ad- 

 mits, according to Bernoulli, of 3,312 such 

 arrangements. An interesting recent book, 

 taking up combinatorial and other mathe- 

 matical topics for the purpose of recreation, 

 is W. Ahrens' ' Mathematische Unterhaltungen 

 und Spiele' (Teubner, 1901). 



Florian Cajori. 



Colorado College, 

 Colorado Springs. 



DISCUSSION AND CORRESPONDENCE. 



THE OPPORTUNITY FOE FURTHER STUDY OF VOL- 

 CANIC PHENOMENA. 



To THE Editor of Science : It has been just 

 four months to-day since the terrible calamity 

 at St. Pierre occurred. Much has been writ- 

 ten and said concerning it. Many able scien- 

 tific men — French, British and American — • 

 have examined the locality and published 

 thereon, but so far as I am aware their obser- 

 vations and conclusions only point to one 

 deduction — that the terrible secret of Pelee's 

 destructive clouds is still unsolved, and that 

 the volcano still exhibits a deadly unex- 

 plained force, as attested by two thousand ad- 

 ditional victims last week. 



I think I may speak correctly, when I say 

 that all the visiting geologists agree upon 

 the major geological facts and only diverge 

 seriously when they reach the field of specu- 

 lation concerning the nature and behavior of 

 the mysterious gases and clouds of lapilli, 

 which descend instead of arising, which de- 

 veloped marvelous electric effects, after pass- 

 ing away from the crater, and which create 

 powerful destructive forces. 



So far as I am aware there was not a single 

 member of the American scientific corps who 

 did not leave the scene with a knowledge of 

 the incompleteness of his studies and the 

 lack of facilities for study during the brief 

 time he was there. One of these. Professor 

 Heilprin, has returned to the scene at his own 

 expense, but, alas, even if he has survived 



