May 6, 1915] 
NATURE 
255 


his predecessors, we may now be fairly confident 
that the primes in the first nine millions have 
been correctly determined ; the introduction to the 
present table describes the checks used for the 
tenth million, and contains other very interesting 
matter. First we have an account of Kulik’s re- 
markable work, which, although not accurate, 
has been found to be of great value as a check, 
and actually goes beyond 100,000,000; unfortu- 
nately the second of the eight MS. volumes is 
missing. Then we have an admirable summary 
of the work done in the theory of the distribution 
of primes, ending with Gram’s series, which is a 
transformation of Riemann’s celebrated formula. 
Finally we have a table, at steps of 50,000, giving 
comparisons of the actual count with the values 
_ found from the formule of Legendre, Tchébicheff, 
and Riemann respectively. 
The superiority of the last-named becomes more 
and more evident the further we go; the errors 
fluctuate in sign, and their ratios to the true 
value diminish in a most remarkable manner. The 
error is actually zero for a table going from 1 
to 9,050,000; and for 10,000,000 it is only +87, 
the number of primes, as counted, being 664,580. 
The errors in the other two calculated values are 
always in excess and Legendre’s value is less 
accurate than the other; but the comparative 
smallness of the errors is noteworthy, being only 
560 and 338 respectively for a ten-million table. 
Perhaps there may be a simple modification of 
Legendre’s formula which would bring it into 
closer agreement with Gram’s. 
The liberality of the Carnegie Institution of 
Washington has made it possible to publish this 
table at a price which is remarkably low, con- 
sidering the labour involved. We hope that 
English universities and colleges will provide them- 
selves with copies, and make an announcement 
that they have done so; this would be a great 
benefit to scattered arithmeticians, who now and 
then wish to know whether a particular number is 
prime or not, and may not be able to afford evena 
sovereign for the luxury of possessing this work. 
The same thing may be said, with greater 
emphasis, about the factor-table. 
(2) Prof. Pierpont’s treatise on the complex 
variable is very good, and a judicious mean be- 
tween elaborate works addressed to the expert and 
specious outlines which ignore all difficult points, 
and tempt the reader to draw all sorts of false 
conclusions. The method is practically that of 
Cauchy, as developed by Briot and Bouquet, and 
Hermite, no use being made of Riemann surfaces. 
On the whole, we think this is the preferable 
course, because, although in simple cases the 
Riemann surface provides a visual image of great 
NO. 2375, VOL. 95] 


simplicity, and is invaluable for purposes of 
research, we cannot construct it until we have 
worked out the analytical theory of the algebraic 
functions we are considering, and this comes to 
discussing a system of Cauchy loops. 
Readers of Dedekind and Weber’s memoir on 
algebraic. functions that the 
authors laid great stress on the precise meaning 
of “a point on a Riemann surface”’; this is the 
main crux of the whole theory, and another way 
of putting it is to determine the complete charac- 
teristics of a singular point on a given algebraic 
curve. This last way was that of H. J. S. Smith 
and Halphen. The present writer is no doubt 
prejudiced; but he ventures to say that in his 
opinion Dedekind’s notion of algebraic divisors, 
as expounded, for instance, in Hensel and Lands- 
berg’s treatise on algebraic functions, is the best 
way to express the analytical facts in a concise 
symbolical form. For one thing, it brings into 
prominence the idea of a compound modulus, 
which is bound to lead eventually to a great sim- 
plification of the theory of algebraic functions. 
A good feature of the treatise is that special 
functions such as those of Legendre and Bessel, 
and the hypergeometric function and_ elliptic 
functions, are treated in the light of the general 
theory. 
There are two points about which the author 
might have written differently with advantage. 
It is not correct to say that the argument (ampli- 
tude) of x+y is tan-ly/x; otherwise we should 
have arg (x+yi)=arg (—x—y!). The author 
does not say this, but (p. 11) he invites this false 
" The correct statement may be 
will remember 
conclusion. 
written— 
arg (x+yi)=(sin, cos)“\(y/r, x/r), 
where r= + /(x2+ 2), and (sin, cos)~1(a, b) means 
an angle of which a is the sine and b is the cosine. 
Thus arg (x+yi) is determinate up to multiples 
of 27; and this is the only way to avoid error 
arising from the definition. 
The second point is this. 
infinite series— 
S=Q,+4,+ Ag+ 
s!=b,+bo+b3+ 
of such a nature that a;=bj;, where i is uniquely 
determined by 7 and conversely, we can define 
s! as a permutation of s, on the ground that every 
term of s/ occurs in s and conversely. But it 
does not follow that the sum of s/ is equal to the 
sum of s, even though both are convergent. We 
are, however, able to say that if s is absolutely 
convergent, and s’ is such a permutation of s that 
the relation 4;=a; makes j finite whenever i is 
finite, and conversely, then s/ is absolutely con- 
Suppose we have two 
