NOVEMBER 10, 1899.] 
in the same interval ; 10° Cantor’s theorem, if 
a function is continuous in a finite interval, we 
can determine for every positive number « as 
small as we wish, a number hf, such that in any 
interval of magnitude A contained within the 
given interval, the oscillation of the function 
shall be less than e. 
The second chapter deduces by the method 
of limits the rules of derivation of standard 
functions, together with the properties of de- 
rived functions, and concludes with the comple- 
ments of the theory of limits introduced in the 
first chapter. The third chapter devoted to 
series discusses in order the convergence cri- 
teria, the Taylor-Maclaurin formula, the asymp- 
totic evaluation of power-series, the technical 
discussion of functions, the interpolation form- 
ula and the decomposition of rational functions 
into sums of simple fractions. The notions of 
the first chapter are extended to functions of 
more than one variable in the fourth chapter, 
with special reference to the problems of max- 
ima and minima. The examples and exercises 
of these chapters, most of which are resolved 
in full, are especially valuable ; the collection of 
classic ones of derivativeless functions calls for 
remark ; the character of these exercises is well 
exemplified by the following which are given in 
illustration of the theory of maxima and minima : 
1° Calculate the lengths of the axes of the gen- 
eral conic ; 2° Determine the lengths of the axes 
of the section of an ellipsoid made by a given 
diametral plane ; 3° Find the minimum distance 
between two right lines; 4° Seek the minimum 
value of the sum of the squares of n variables 
connected by m<n linear equations; 5° The 
method of least squares. 
In this day of multiple algebras and multiple 
geometries it is not surprising to find Cesaro 
proposing multiplications of the differential 
ealeculus. These observations form an inter- 
esting section of the fifth chapter which gives 
the-ordinary methods for the differentiation of 
explicit and implicit functions of one or several 
variables. The differential dx of the independ- 
ant variable, arbitrary for each value of 2, 
Cesaro considers as the product of an infinites- 
imal ¢ independent of « by an arbitrary function 
of w, 7. e., dx = ay (“). Differentiating this ex- 
pression we have 
SCIENCE. 
689 
da = day = ady = ay/dx =a? yzx’, 
Gx =: ayy!” + 77/), 
day = a8 (yx + Ax2y/ x! + 84/1), o> 
The results of these successive differentiations 
become rapidly more complicated, and would 
as rapidly rob the calculus of most of its ad- 
vantages if the function vy be allowed to retain 
its arbitrary character. For convenience x (x) is 
made equal unity and we have dx = dx = -.. 
= 0, which expresses that x is equicrescent, i. e., 
that the differential of the independent variable 
is independent of the variable. However it is 
only necessary to call in the fundamental prin- 
ciple of the integral calculus to show that every 
form of calculus resulting from a change of 
form in the function x reduces to the ordinary 
calculus; the reduction is effected in precisely 
the same manner that a change of independent 
variable is made. Thus, there is always a func- 
tion t of a whose derivative is 1: x (x), then 
il 
ey ees Bl No Bf as gape ated 
dt — t/a) — rate) ay(a) =a, dt—d%t 0. 
The possibility of a calculus in which no 
variable possesses a constant differential is not 
excluded, but it is certain that the simplicity 
and homogeneity of its formule and the pre- 
cision with which the ordinary calculus assigns 
the orders of its infinitesimals will not be 
among the advantages of the new calculus. 
It may be remarked here in passing that a 
Norwegian mathematician attempted a few 
years ago to found a new calculus, in which the 
fundamental rdle taken by addition and sub- 
traction in the ordinary calculus was assigned 
to the operations of multiplication and division, 
The resulting forms yielded certain continued 
products, but were otherwise fruitless. 
The sixth and seventh chapters contain the 
geometrical applications to plane and space 
curves. These chapters must have offered a 
sore temptatation to the author to make exclu- 
sive use of his own elegant method of intrinsic 
analysis, but the reader finds no method em- 
ployed to the exclusion of all others. The ap- 
plications follow the usual order of tangents, 
normals, curvature, asymptotes, singularities, 
contacts, and envelopes. The examples are 
happily chosen, and the chapters amply illus- 
trated with well executed figures. 
