AuecustT 17, 1899] 
NATURE 
363 
triously and ably worked out, and, on the whole, well 
written. At the same time, it should be pointed out 
that such a work was especially in need of a good and 
exhaustive index, and that it is a pity the author did not 
compile one himself. 
CALCULATION BY ABACUS. 
Traité de Nomographie. Par Maurice d’Ocagne. 
xiv + 480. (Paris: Gauthier-Villars, 1899.) 
HIS is a book which ought to make even the 
ordinary reader appreciate the perennial fresh- 
ness of mathematics. The method of ‘“‘ Nomography ” 
(X3 of the international catalogue), recent as it is in its 
more important developments, is based upon a very 
simple idea which has long been familiar—that of the 
indexed scale. The ever-recurring problem of applied 
mathematics is to calculate an unknown numerical 
quantity from its relation to other quantities that are 
known. The simplest case is when two quantities x, y 
are connected by a relation /(x, y)=o or y=(x). For 
practical purposes it is convenient to have a permanent 
record of a large number of corresponding values of + 
and y so that for any given value of + the approximate 
value of y may be at once found or obtained by simple 
interpolation. Three methods are available: the first 
is that of a numerical table, such asa table of logarithms ; 
the second that of the graph, for instance the curve 
f(x, y)=0 or y=$(x) referred to rectangular coordinates ; 
the third is that of the indexed scale, that is to saya 
straight line or curve at different points of which the 
corresponding values of + and y are shown infigures. A 
familiar example is given by a thermometer with Centi- 
grade and Fahrenheit readings, or by a measuring tape 
with centimetres marked along one edge and inches 
along the other. 
In this very simple case the advantage of the indexed 
scale is not very obvious; even here, however, the 
method combines much of the vividness of the graph 
with a considerable saving of space. It is when three or 
more variables are connected by a relation that the great 
value of the scale method becomes apparent. Suppose, 
for instance, we have a relation 
Fig(x), x(¥), ¥(2), a(Z)}=0 
where +, y, 2, f are the variables and F, 9, y, W, are 
known functions. The essence of the nomographic 
function consists in first plotting off in a suitable way in- 
dexed scales of (x), x(y), ¥(z), (2), and then employing 
a linkage or similar mechanism to associate four corre- 
sponding values, x’ y’, 2’, ¢. In the case of two variables 
x, y the “linkage” consists merely in the juxtaposition 
of the scales; when a proportion sum is done with a 
slide-rule, the scales are moved relatively to each other ; 
in most of M. d’Ocagne’s illustrations, involving several 
variables, the scales are either superposed in a two- 
dimensional grating or a movable linkage is used con- 
sisting of a transparent sheet with lines of reference 
ruled upon it, or a combination of both devices is 
employed. 
Of course a method so elastic leaves ample room for 
ingenuity in constructing an “abacus,” as M. d’Ocagne 
calls it, suited to any particular problem. The author 
NO. 1555, VOL. 60] 
Pp. 
gives an abundant variety of illustrations, many of great 
practical importance to the physicist and engineer : it is 
by studying these, and actually taking readings for him- 
self, that the reader will succeed in appreciating the value 
of the method. For of this, as of other graphical methods, 
it may be said that merely reading it up, or understand- 
ing its principles in ‘a general way, is of little use as 
compared with a thorough working knowledge of its 
application. 
At the same time, M. d’Ocagne has done really good 
service in devoting his final chapter to the general 
theory. This has, in its way, the same kind of special 
value as Reuleaux’s “Kinematics of Machinery” in 
relation to the ordinary treatises on mechanism. For 
in this chapter we have a clear conspectus of the general 
principles which underlie the construction of ay abacus ; 
and, what is still more remarkable, all possible varieties 
of abacus are classified into perfectly definite types which 
can be expressed by a simple abstract notation. Oddly 
enough, the enumeration of the different types leads to a 
difficult problem in the partition of numbers, happily 
solved by Major MacMahon. 
It is not impossible that the human race may ulti- 
mately set off against the ravages of warfare the indirect 
stimulus which it has given to mathematics; nomo- 
graphy, at any rate, has been developed in great measure 
to meet the demands of civil and military engineering. 
M. d’Ocagne’s numerous bibliographical notes will enable 
his readers to follow in detail, if they wish, the history of 
the subject. Pure and applied mathematicians alike 
will be grateful to him for a work so full of novelty and 
interest ; while its subject-matter, as well as its clearness 
and simplicity, ought to make it eminently acceptable to 
the engineer. Ga Be Mi. 
OUR BOOK SHELF. 
Die Spiele der Menschen. By Karl Groos. 
(Jena: G. Fischer, 1899.) 
Pror. Groos will add by the present volume to the 
reputation he has already earned by his well-known 
work on the “‘Games of Animals.” <A really compre- 
hensive account, at once sympathetic and intelligent, of 
the games of both children and adults has long been a 
desideratum with the psychologist as well as with the 
anthropologist, and Prof. Groos’s new work goes very far 
indeed towards permanently supplying the want. As is 
only right and proper, by far the larger part of the book 
is given up to an exhaustive description of the facts as 
far as they are known; the “ Theory of Play” enunciated 
in the second part of the treatise can thus be judged by 
the reader upon a sufficiently wide basis of empirical 
information. The range and the accuracy of Prof. Groos’s 
knowledge are alike surprising ; not only is he a mine of 
information about the amusements of his own country, 
but he appears, for instance, as much at home in the 
English nursery and playground as though he had been 
brought up amongst us. Almost the only signs of im- 
perfect knowledge of English games to be detected in 
the whole book are the author’s ascription of ‘“‘ Hare and 
Hounds,” in its familiar form, exclusively to America, 
and his apparent ignorance of the continued vitality of 
“Hunt the Slipper.” As a psychologist Prof. Groos is 
distinguished by a singular subtlety of discrimination ; 
his account, for instance, of the various elements which 
enter into the gambler’s enjoyment of high play, or, again, 
Pp. vi+538. 
