Aprit 28, 1923] 
The reduction has been repeated ab initio, the data 
being entered from the original observing books. The 
clockandazimuth errors and equator-points were derived 
by the use of Auwers’s positions of fundamental stars. 
Pulkovo refractions were used. The probable error of a 
re-reduced catalogue place (depending on 1-8 observa- 
_ tions, the average number) is 0-083* secant decl. in R.A., 
and to” indecl. The difference of magnitude equation 
_ for 3:5 mag. and 8 mag. is about 0:08*. This has not been 
applied. The star places were compared individually 
with those of the Astronomische Gesellschaft Catalogues 
and the differences are given beside the star places, 
_ though the interval in years is not given. 
_ Many errata in Riimker’s reductions were detected 
_and corrected in the course of this comparison. These 
are mentioned in footnotes. There were some stars 
_ for which Riimker did not read the full number of 
microscopes, but in all cases there is ample material 
_to determine the necessary correction. Finally a list 
_ is given of the proper motions that have been published 
for Riimker stars, some 6000 in number. It should now 
_ be possible to increase this list with the aid of the newly 
_ published positions. 
_ To save expense the catalogue was not set up in type, 
but written by hand and multiplied by a mechanical 
process. It is, however, quite clear and legible. 
ASC. DYE 







Mathematik und Physik: Eine erkenntnistheoretische 
Untersuchung. Von E. Study. (Sammlung Vieweg, 
Heft 65.) Pp. 31. (Braunschweig: F. Vieweg und 
Sohn, 1923.) 675 marks. 
In this tract Prof. Study’s chief aim is to discuss the 
question: What is to be regarded as mathematical 
and what as specifically physical in theoretical physics ? 
How comes it that parts of mathematics and of physics 
can be combined so as to form a higher unity? For 
the purposes of his discussion he defines mathematics 
as the limit towards which present-day mathematics 
seems to him to be tending, in which it will include 
calculation by means of natural numbers (positive 
integers) with all that is based thereon, and nothing 
besides. When, for example, projective geometry is 
“arithmetised ” by identifying a point, or a straight 
line, with the set of homogeneous co-ordinates repre- 
senting it, the word point, or straight line, as the case 
may be, becomes merely a symbol bearing no logical 
_ relation either to the material world or to our concept 
of space. 
Thus all branches of geometry, Euclidean or other, 
are logically independent of experience. Similarly 
“arithmetical physics,” arising from the arithmetisation 
of the mathematical portions of physics, is based 
logically on calculation by means of numbers alone, 
developed in one particular direction, chosen from 
many possible alternatives on the basis of a judgment 
of value, not of cause, in so far as it is desired to make 
only investigations closely related to experience. Thus 
the relation of theoretical physics to the content of 
experience appears to be not logical, but only psycho- 
logical and historical. The content of theoretical 
physics is threefold: (1) a purely mathematical part, 
characterised by the method of deduction; (2) an 
experimental part, characterised by the method of 
(incomplete) induction ; and (3) an intermediate part, 
NO. 2791, VOL. 111] 
NATURE 

565 
characterised by an independent method, that of 
“idealisation.” By idealisation Prof. Study means 
the process whereby we substitute the simple abstract 
reality of mathematics for the infinitely complex and 
barely comprehensible reality of physics. 
This tract can be recommended as a very stimulating 
introduction to the philosophical aspects of mathe- 
matics and physics by a writer who is eminently 
fitted for the task by the wide range of his knowledge 
as well as the importance of his own contributions to 
science. 
Musical Acoustics based on the Pure Third System. 
By Thorvald Kornerup. Translated by Phyllis A. 
Petersen. Pp. 56. (Copenhagen and Leipzig : Wil- 
helm Hansen, 1922.) 2s. 6d. 
In this little book the author discusses very fully the 
relations of the pitches of the notes in the various scales 
in just intonation and in a variety of temperaments. 
Instead of Ellis’s logarithmic cents, the millioctave is 
here used, which (as its name implies) is one-thousandth 
of an octave instead of Ellis’s one-twelve-hundredth 
of the octave. It is pointed out early in the work that 
for a pure intonation of the minor triad, D F A, the 
D must be only a small tone above C and a large 
tone below the just E. The fact that the major chord, 
G B D, equally needs a D which is a small tone below 
the just E and a large tone above C, does not seem to 
receive equal emphasis. 
The book contains very many diagrams and tables. 
One of the most striking diagrams is the author’s 
tonal circle in which the circumference contains a 
single octave, equal angles corresponding to equal 
differences of frequencies. Thus, putting one C at the 
starting-point on the circumference, the other notes 
occur at the following angles, the D being what is 
called in England grave D and denoted by D’. The 
ordinary D would be at 45°. 
Angles . 0° 40° go° 120° 180° 240° 315° 360° 
Notes... C: Dare Gowan BC 
Quite a number of scales and temperaments are treated 
at length, special attention being directed to the nine- 
teen steps to the octave, which is considered to be the 
consequence of the third system and the practical ideal. 
Other temperaments considered are as follows, and 
illustrate the fulness of the treatment : 
No. Steps No. of Steps in Steps in 
ieee toe, ween) oe ee et 
5 x 2 + 2 x 1 =12 Aristoxenos. C. 350 B.C. 
5 xX 3 +2 x 2=19 Elsasz. c. A.D.1590. 
pe 5 4 eX FS Vicentino. c. 1546. 
he ‘ a 4+ 2 x 4= 41 Paulv. Janko. 1882-1901. 
t: % 4 + 2 x 5 = 53 Nicholas Mercator. c. 1675. 
The work is in some respects rather fanciful but will 
repay cateful study. E. H. B. 
Production économique de la vapeur. Par Dr. O. 
Manville. Pp.vii+4o7. (Paris: Gaston Doin, 1923.) 
25 francs. 
M. MAnvitte’s work is timely. While French industries 
in pre-War days consumed 64 million tons of coal, the 
addition of Alsace-Lorraine has increased the potential 
demand to 80 million tons. To meet this there exists 
