OCTOBEE 25, 1907] 



SCIENCE 



551 



lineata Say, and colored figures of the three 

 larval stages are given on plate 17, facing 

 page 146, of more than ten of the species. 

 Elsewhere (pp. 164, 219) general statements 

 to this effect occur. Three larval instars are 

 therefore implied for all, or almost all, of the 

 species of the genus. And it is to this state- 

 ment, in so far as it concerns the species 

 decemlineata, that I desire to call attention. 

 Rearings of this species, both in nature and 

 the laboratory, carried on in Georgia in 1906 

 and in Ohio in 1907, showed in both places 

 four larval instars, all of which were distinct, 

 and which have been described." These rear- 

 ings involved a total of not more than seventy 

 specimens, and while this is very small in 

 comparison with the large total reared by 

 Tower, I can not think otherwise than that 

 they represent the average for the species, and 

 were not exceptions. All of the lots were 

 small and under normal conditions, and the 

 rearings were made especially with the view 

 of determining the duration and number of 

 the larval instaj-s, so that errors in observa- 

 tion were eliminated. As Dr. Tower had 

 . other objects in view, I believe his observa- 

 tions in this respect were faulty, at least one 

 ecdysis being overlooked in the larval devel- 

 opment of decemlineata; and if in that spe- 

 cies then as well perhaps in the others, though 

 I am not concerned with them here. 



A. Arsene Gieault 



Washington, D. C, 

 September 16, 1907 



EVEN PERFECT MEASURESTG IMPOTENT 



The attention of geometers should be 

 directed to a remarkable article by Dr. R. L. 

 Moore, of Princeton, whose extraordinarily 

 elegant proof of the redundancy of Hilbert'a 

 axioms first appeared in The American Mathe- 

 matical Monthly. 



The new article, in the Transactions of the 

 American Mathematical Society^ Vol. 8, No. 

 3, pp. 369-378, July, 1907, is also a perfect- 

 ing of the work of the Hilbert school, but 



"Girault and Rosenfeld, Psyche, XIV., 1907, 

 pp. 47-52. 



reaches new results so unexpected, so profound 

 as to be nothing less than epoch making. 



We knew that the so-called laboratory 

 method for mathematics, the " measuring " 

 method, was rotten at the core, since mathe- 

 matics is not an experimental science, since 

 no theorem of arithmetic, algebra or geometry 

 can be proved by measurement. 



Our argument was suificiently cogent: that 

 the theorems of mathematics are absolutely 

 exact, while no human measurement ever can 

 be exact. 



But Dr. Moore shows that even granting the 

 impossible, granting the super-human power of 

 precise measurement, we could not thereby 

 ever prove our space Euclidean, ever prove it 

 the space taught in all our text -books. 



The title of his article is : " Geometry in 

 which the Sum of the Angles of Every Tri-. 

 angle is Two Right Angles." But, omitting 

 the Archimedes assumption, if this postulate 

 be substituted for Euclid's, there results a 

 geometry not necessarily Euclidean. Never- 

 theless, no human being confined therein could 

 ever distinguish it from a Euclidean space 

 even though he were supplied with instru- 

 ments which could decide for him whether any 

 two sects were exactly equal. 



The Euclidean space would contain other 

 points, points ideal or ultra as regards this 

 " angle-sum " space. 



But, most extraordinarily, no ultra point is 

 ever between two ordinary points. 



Geoege Bruce Halsted 



Gbeelet, Colo. 



SPECIAL ARTICLES 



PLANKTON FISHING OFF THE ISLE OF MAN* 



During recent years a good deal of atten- 

 tion has been paid by naturalists in various 

 parts of the world to the quantitative dis- 

 tribution of organisms in the sea. It is 

 obvious that exact information in regard to 

 such a matter may be of enormous importance 

 in connection with the fishing industries. 

 Notable methods of work, and instruments for 



'Read before Section D (Zoology) of the Brit- 

 ish Association meeting at Leicester on August 

 6, 19.07. 



