688 
as a theorem in molecular dynamics lies in the 
remarkable agreement between the results ob- 
tained by the methods described in the three 
different sections of this report, all of which 
are based on different fundamental hypotheses.’? 
Epwin H. HAL. 
CAMBRIDGE, October 28, 1899. 
Elementi di calcolo infinitesimale con numerose ap- 
plicazione geometriche. Per ERNESTO CESARO, 
professore ordinario della R. Universita di 
Napoli. Naples, Lorenzo Alvano. 1899. 8vo. 
Pp. 400. 
The absence of a text-book on the calculus 
from a too well-known series of American math- 
ematical text-books was recently remarked. 
The omission was excused by the observation 
that the author of the series knew nothing about 
the calculus. It might have been well for the 
cause of secondary and superior mathematical 
education in this country had the same modest 
confession been called into execution earlier and 
prevented the construction of the patch-work, 
fragmentary, stereotyped algebra of the same 
series. Contrast the confession of the razor- 
maker with the refusal made lately by a mathe- 
matician who declined to prepare an elementary 
treatise on the infinitesimal calculus on the 
ground that he knew too little arithmetic and 
algebra. 
Cesaro had the courage to learn and make his 
mathematics before he began to publish any of 
his courses. His treatise* on algebraical analysis 
appeared five years ago and was most favorably 
received, although published against the advice 
-of his friends. This work naturally contained an 
introduction to the infinitesimal calculus which 
gave full promise of the superb treatise which 
comes from the press this year. The former, 
which is by no means so finished a work of art 
as the latter, is a collection of sixty lectures on 
substitutions and determinants, linear forms, 
quadratic forms, irrational numbers, limits, 
series, functions, developments in series, com- 
plex numbers, quaternions, elimination, sym- 
metric functions, enumeration of roots, numeric 
and algebraic resolution of equations, differences 
and interpolation, and factorial developments. - 
*Cesairo, Corso di Analisi algebrica con introduzione 
al Calcolo infinitesimale, Turin, Bocca, 1894. 
SCIENCE. 
[N. S. Vou. X. No. 254. 
Cesiro’s course in the calculus is designed 
after the following plan the style of whose 
exposition is a most fortunate combination 
of mathematical rigor and poetic expression. 
There are three grand divisions occupied in 
order with fundamental theories, the differential 
calculus, and the integral calculus. The first 
of these consists of four chapters devoted to 
functions, derivatives, developments in series, 
and functions of several variables ; the second 
part also contains four chapters presenting the 
theory of differentiation and its applications to 
the theories of plane curves, space curves and 
surfaces ; the last division comprises five chap- 
ters on integration, applications to the evalua- 
tion of certain remarkable classes of integrals, 
applications to geometrical mensuration, differ- 
ential equations and variations. 
The reviewer has space to analyze but few 
of the chapters of this valuable work. The 
first chapter exhibits the principal properties of 
functions in all their modern refinement by the 
evolution of the following theorems: 1° If a 
function is finite throughout an interval it al- 
ways admits of an inferior limit and a superior 
limit ; 2° Ifa function is finite for all the num- 
bers of an interval it is finite throughout the 
interval; 3° The first theorem of Weierstrass, 
ifa function is finite in a finite interval, the latter 
contains at least one number for which the func- 
tion has the same limits, inferior and superior, as 
the interval itself; 4° For the existence of a finite 
limit of f(x) to the right of a it is sufficient that, 
given « positive and as small as we wish, there 
can always be found a positive number h, such 
that, for every pair of values x’ and 2/’ taken 
within the interval (a, a + h), excluding the in- 
ferior limit, the absolute value of f(x’) — f(x’) 
is less than « ; 5° If f(x) is continuous and dif- 
ferent from zero for x = a, it possesses at a the 
sign of f(a); 6° If a function is continuous in 
an interval it is also finite in the interval ; 7° A 
function continuous in an interval at the ex- 
tremities of which it takes opposite signs must 
vanish at least once in the interval ; 8° A con- 
tinuous function cannot pass from one value to 
another without passing through all the inter- 
mediate values; 9° Second theorem of Weier- 
strass, every function continuous in a finite in- 
terval takes the maximum and minimum value 
