26 



SCIENCE 



[N. S. Vol. XXXIV. No. 862 



peated by a large number of prominent 

 mathematicians.^ That the graph on page 

 214 is the graphic representation of the func- 

 tion in question is open to serious doubts, 

 which should not have been passed over in a 

 work on logical foundations.- The statement 

 on page 101 that " Diophantus of Alexandria, 

 who lived 300 a.d., seems to have been the first 

 actually to have made use of rational num- 

 bers " is apt to mislead the reader even if a 

 footnote helps to ascertain the author's mean- 

 ing. Taken by itself this statement seems 

 absurd. 



These are, however, matters of secondary 

 importance and the book under review seems 

 to be remarkably free from errors if we con- 

 sider its wide scope. In particular, the proof 

 seems to have been read with unusual care 

 and one can only wish that the book will be 

 very widely read, especially by those who are 

 preparing to teach secondary mathematics. 

 Its style is attractive and many of the ques- 

 tions which it treats are so far reaching that 

 one may reasonably expect that it will find a 

 considerable number of readers outside of the 

 circle of professional mathematicians. 



G. A. MiLLEE 



University op Illinois 



SPECIAL ARTICLES 



COLOR DISPERSION IN THE ASTIGMATIC EYE 



When an astigmatic eye views a bright 

 point of light in which only the rays near the 

 ends of the visible spectrum are present, the 

 image of the source is blurred by fringes or 

 wings of red and blue. If the eye has a well- 

 defined axis of astigmatism but is otherwise 

 fairly emmetropic, the appearance of such a 

 source is so curious as to compel attention. 

 The purplish image is then crossed by a pro- 

 nounced red band parallel to that meridian of 

 the eye in which the curvature is least, and by 

 a blue band at right angles to it. In the case 

 of astigmatism with the rule, the red band is 

 approximately horizontal. The experiment is 

 easily tried in a darkened room by allowing 



' Cf . ' ' Encyclopedia des Sciences Mathemat- 

 iques," Tome 2, Vol. 1, p. 3; Cantor's "Ge- 

 schichte," Vol. 3, 1901, pp. 215, 456-7. 



the light from a flame or electric lamp to pass 

 through a hole a few millimeters in diameter 

 in a screen, or better by placing the lamp in a 

 box having a small hole in one side. One or 

 two thicknesses of common " pot blue " glass 

 are placed over the opening, which is then 

 viewed from a distance of two meters or more. 

 The blue glass, as is well known, is fairly 

 transparent to red light. Distant blue lights 

 seen at night, such as the " dwarf signals " in 

 railroad yards, show the effect well.' Indeed, 

 the appearance can be seen by viewing any 

 bright light through a blue glass held in front 

 of the eye. A person free from astigmatism 

 can see the effect by holding a cylindrical lens 

 in front of the eye. 



The explanation is simple, and has very 

 likely occurred to many who have noticed the 

 effect. However, the writer has been unable 

 to find any reference to it, either in the 

 classical memoirs of Helmholtz and his prede- 

 cessors, or in such later writings as he has 

 access to. Astigmatic vision seems to have 

 been considered only on the tacit assumption 

 that dispersion could be neglected — an as- 

 sumption that is sufficient with ordinary white 

 light, in which the yellow and green rays pre- 

 dominate in determining our visual sensa- 

 tions. It is only when these intermediate rays 

 are excluded that the effects of dispersion be- 

 come noticeable.' 



Taking the type of astigmatism most com- 

 monly found, let us assume that the radius of 

 curvature of the cornea is less for the vertical 

 than for the horizontal meridian. If the eye 

 observes a distant point-source giving only 



' Among those whom the writer asked whether 

 they had noticed the crossed red and blue bands 

 was a certain raUroad employe, who not only 

 observed the appearance to a marked degree, but 

 also volunteered the explanation that the dwarf 

 signal ' ' had a dirty glass. ' ' Eemarks of this 

 sort show how unconscious we are of our own 

 defects of vision. 



-For example, Helmholtz describes some inter- 

 esting experiments on the effects of chromatic 

 dispersion in the eye; these are also recounted 

 by Lummer in Miiller-Pouillet 's "Lehrbuch der 

 Physik. " In all these cases the eye is assumed 

 to be free from astigmatic defects. 



