134 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
but the term infinite division probably does not represent the same 
conception to all mathematicians. If we suppose a quantity divided 
into a number of parts, and each of these parts subdivided, and 
similar subdivisions to go on forever, each requiring finite time, we 
have a conception to which the name infinite division may be given 
with some appropriateness, but which might better be called eternal 
division. Such division never reaches a result. But if we suppose 
the time of each subdivision to be proportional to the magnitude of 
each part, the entire process is completed in finite time, although 
no limit can be given to the number of subdivisions. If a point 
be supposed to have passed with constant velocity over a given 
distance, there was a time when it had passed over half the distance ; 
afterward a time when the remaining distance was one-fourth of the 
original distance; the number of such successive halvings is cer- 
tainly unlimited; and the result is that there is no remaining dis- 
tance. This is division infinite but not eternal, and the result seems 
to be zero. 
As a point is defined to be position without magnitude, so may an 
infinitesimal be defined to be quantitative relation without magnitude, 
The terms infinitesimal, differential, nothing, and zero, are not 
synonyms. They have the same logical denotation but differ in 
connotation. Mathematicians usually speak of “the value” or 
“the true value” of a vanishing fraction, as though any quantity 
whatever were not a true value. The term serial value is proposed 
as conducive to clearness of thought. A differential coefficient is 
the serial value of a vanishing fraction; and a differential or infi- 
nitesimal may be further defined as zero in serial relation to con- 
tinuously diminishing quantity. 
The term infinitesimal is however frequently employed like other 
terms to denote the symbol of its exact signification. We speak of 
drawing and erasing lines, meaning the visible symbols of Euclidean 
lines. Even in our purely mental processes we give the name 
points to the imagined small volumes that symbolize positions with- 
out magnitude. In like manner the term infinitesimal is employed 
to denote the imagined small quantity in approximate relation that 
symbolizes a relation which becomes exact only when magnitude 
disappears. e 
A line is infinite relatively to a point, but infinitesimal, 2. e., zero, 
relatively to a surface or volume. Every quantity is finite rela- 
tively to other quantities of its own order—zero relatively to orders 
I, OE ee 
