308 



SCIENCE. 



[N. S. Vol. XVI. No. 399. 



On p. 5, in note to axioms of order (better 

 axioms of arrangement), W. Pascli should be 

 M. Pasch. On p. 6, first line, die Anordnung, 

 the arrangement, is rendered 'an order of 

 sequence.' 



In II., 4, the repetition of the word 'so' de- 

 stroys the statement intended. 



Could there be a more pitiful bungle than 

 that which, in the last two lines of p. 6, 

 gives 'alle iibrigen Punkte der Geraden a 

 heissen ausserhalh der Strecke AB gelegen' 

 as 'all other points are referred to the poinis 

 lying wiiJiout the segment AB.' 



The translation of the important Axiom 

 IV., 1, p. 12, is so bungled as to be worse than 

 meaningless, actually false, as will be seen 

 by comparing with the French translation: 



Si I'on designe par A, B deux points d'une 

 droite a, et par A' un point de cette meme 

 droite ou bien d'une autre droite a', I'on pourra 

 toujours, sur la droite a', d'un cote donne du 

 point A', trouver un point et un seul B', tel 

 que le segment AB soit congruent au segment 

 A'B'. 



On p. 13, 'emanating' is unfortunate. 



On p. 15, 1. 10, the angle-symbol is omitted. 



On p. 17, 1. 8 from below, 'so' should be 

 'such.' 



On p. 22, theorem 16 is mistranslated, the 

 insertion of the word 'corresponding' turning 

 it into bathos. 



But on p. 24 we have a still more ludicrous 

 misinterpretation, which shows that Professor 

 Townsend has not attempted to understand 

 the book he attempts to translate. Under the 

 heading Definitions (which should be Defini- 

 tion) he says : ' From this definition can be 

 easily deduced, with the help of the axioms 

 of groups III. and IV., all of the known 

 properties of the circle.' 



What a stupendous blunder this is we realize 

 when we recall that thus cannot even be 

 proved that a straight line which has a point 

 within a circle has a point on the circle. 



What Hilbert himself proves and what 

 Townsend translates on p. 116, demonstrates 

 that, using axioms I.-IV., we could not even 

 show that from any point without a circle 

 there is a tangent to the circle. Just so, with- 

 out an axiom of continuity we cannot demon- 



strate that a circle having a point within and 

 a point without a second circle has a point on 

 it. 



On the same p. 24 the introduction, in 1. 6 

 from below, of the word 'corresponding' is a 

 childish mistranslation. 



On p. 25 Professor Townsend puts in a lit- 

 tle from Laugel, but seems to have no better 

 luck with his French than with the German. 

 'This axiom gives us nothing directly con- 

 cerning the existence of limiting points, or of 

 the idea of convergence' is how he renders, 

 ' Get axiome ne nous dit rien sur I'existence de 

 points limites ni sur la notion de conver- 

 gence.' 



But the game would not be worth the candle 

 to go on thus through all the 132 pages. 



So I choose as a fitting climax the sentence 

 on p. 125, 'We easily see that the criterion of 

 theorem 44 is fulfilled, and, consequently, it 

 follows that every regular polygon can be con- 

 structed hy the drawing of straight lines and 

 the laying off of segments.' 



From this we should suppose that Professor 

 Townsend studied his geometry from the pop- 

 ular treatise of Mr. Wentworth between 1877 

 and 1887, which during those years contained 

 on p. 224, Proposition XIII., § 387: 'To in- 

 scribe a regular polygon of any number of 

 sides in a given circle.' 



George Bruce Halsted. 



Austin, Texas. 



SCIENTIFIC JOURNALS AND ARTICLES. 



The Journal of Physical Chemistry. March. 

 'On the Relative Velocities of the Ions in 

 Solutions of Silver Nitrate in Pyridin and 

 Acetonitril,' by Herman Schlundt. The ionic 

 velocities found are considerably lower than in 

 water, but this difference seems to decrease 

 with increasing dilution. 'On the Inversion 

 of Zinc Sulfate, II.,' by H. T. Barnes and H. 

 L. Cooke. 'Synthetic Analysis of Solid 

 Phases,' by Wilder D. Bancroft. Description 

 of a new method, applicable to alloys, efflo- 

 rescent substances, basic salts, and double salts 

 which are decomposed by the pure solvent, 

 where the solid phase cannot be conveniently 

 isolated in a pure state. 'A Derivation of the 



