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pass from 3 to 4 not only by going through 3.1, 3.2, 3.3, and if we pass from 3 to 3.1 not only by going through 3.01, 3.02, 3.03, ... and if we pass from 3 to 3.01 not only by going through 3.001, 3.002, ... and so on to any finite number of divisions, the increase is discontinuous; but if the number of divisions be infinite, and if the lesser number pass into the greater number by receiving at each successive step an infinitesimal increase, the mode of increase is continuous. For the sake of illustration let us consider the case of motion. Consider the gliding motion of a worm, and suppose it to pass uniformly over an inch in a minute; if the space through which the worm has passed be estimated at the end of each minute only, the space will apparently be discontinuously increased by an inch: but if the space be measured at the end of each infinitesimal or very short lapse of time, the increase during that instant will be very small, and if the instants be infinitesimal, the space, we say, will have increased by infinitesimal increments. The earth's motion in its orbit, the running out of water, the gradual radiation of heat, the growth of a tree, are all instances of a similar continuous increase; and it is worth observing, that it was from such cases that the Calculus had its origin. But if we count horses or men, we count discontinuously: we pass "per saltum" from one man to two men; we cannot divide a man into infinitesimal elements; each man is an unit whose personal existence does not admit of such infinitesimal subdivision. Hence it appears, that numerical continuity requires infinite numerical divisibility, and expresses the property of quantity considered under the aspect of generation by growth: thus the difference of the two modes of increase is one of degree and not of kind. Hence also we have a criterion of them; the difference between two successive numbers is finite or infinitesimal, according as the mode of increase is discontinuous or continuous.

7.] The subject-matter of arithmetic and of algebra (commonly so called) is discontinuous number. The numbers 8, 9, 10,... a, b, c,... x, y, z,... as they are commonly employed, are discontinuous; we pass from one to another "per saltus," and do not contemplate the mode of continuous increase. The distinction between the two sciences appears to be the following: In arithmetic are discussed the properties of numbers which have certain determinate values, and can have none other; in algebra we treat of symbols which are general in form, and

either have specific values, as the constants a, b, c; or admit of having one or more such values, as the variables x, y, z.

Infinitesimal Calculus, on the contrary, considers number in its aspect of continuous growth. In this lies its distinctive. character for whereas Arithmetic and Algebra treat of finite and discontinuous number, it treats of continuous, and especially of infinite and infinitesimal number.

These terms however are at present too vague to be the foundations of philosophical investigations, and therefore I proceed in this and the four following Articles to enuntiate certain axioms concerning infinitesimals and infinities, their orders and their relations, which flow from an adequate conception of continuous number, and which do not admit of proof by deduction from more general principles. Thus our method of inquiry is similar to that of pure geometry; and we shall shortly find our notions of the materies of infinitesimal calculus sufficiently definite to enable us to deduce from them results which will not be wanting in precision.

The value towards which an expression converges nearer than by any assignable difference, while the symbol on which it depends approaches to any assigned value, is called a limit or limiting value. If the assigned value of the symbol be zero, the limit is called the inferior limit; and if the value be infinity, it is called the superior limit.

1

Thus the inferior limit of of a greater than 0 the quantity is less than 1, yet the nearer x approaches to 0, the less becomes the difference between

is 1; although for every value 1 + x

1

1 + x

1

1+x

and 1; and the superior limit is 0; for as a increases, the quantity becomes less and less, and ultimately, when x is greater than any assignable quantity, the difference between and O is less than any quantity, and thus the limit is zero. So again, as the difference between a and 1 becomes less than any assignable quantity, approaches to infinity; similarly

1

1 + x

π

the inferior limit of tan x is 0, and, as a becomes the differ2'

π

ence between tan and infinity vanishes, and infinity is the

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Again: suppose that we have a series of the form

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the number of terms of which is infinite; although the terms become less and less as we proceed from left to right, yet the value of the last term does not approach to 0 but to unity, as is manifest from the form of it; unity then is the limit of the last term of the series.

In geometry a circle is the limit towards which the perimeter of an inscribed polygon converges, as the number of sides is infinitely increased, and as thereby the lengths of the sides become infinitesimally small. Some examples of finding limits will be given in the sequel. Hence absolute zero is the inferior limit of an infinitesimal, and absolute infinity is the superior limit of a quantity which is greater than any assignable quantity.

8.] The symbols by which we shall represent an infinity and an infinitesimal are and 0: the relation of which is, that if a represent a finite quantity, ∞ = , and 0=. We shall at

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a

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tach a more definite meaning to these relations, if we consider a dividend to be the product of the divisor and quotient; thus there is no finite quantity which, when multiplied into an infinitesimal, will produce a finite product: nothing short of an infinity can do it; and from the illustrations of Article 5 it appears that it must be an infinity of a particular kind. It will be observed, that Ỏ does not represent absolute zero, and that does not express absolute infinity.

If then any finite numerical quantity be divided into any number of equal or unequal parts, as the case may be, the larger the number of parts is, the smaller is each part; and if the number of parts be infinitely great, each part is an infinitesimal: and the less the difference is between the number of parts and absolute infinity, the less is also the difference between each part and absolute zero. Suppose a to be a finite determinate quantity, and to be divided into a equal parts; then each part ==; and, if a is infinitely great, is an infini

a

a

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tesimal, and being thus symbols of an infinity and an infi

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nitesimal, which mutually imply and are reciprocal to each other.

1

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a

to be divided into a equal parts, then each

part is equal to, and a2 is the number of parts into which a

α

has been divided: thus 2 and severally represent an infinity

x2

and an infinitesimal, which are reciprocal to and mutually imply each other. By similar and subsequent divisions we may find

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tesimals which are relative to each other.

Or again, suppose i to be an infinitesimal element of a, so

a

α

that a is divided into equal parts, then is an infinity, and i

is relative to the infinitesimal i. And again, suppose a to be

a

a

resolved into elements each of which is equal to i2, then 12 is the number of equal parts, and is an infinity which is relative to the infinitesimal i2; similarly by subsequent resolu

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tions may other infinitesimals and infinities i3, be formed, which mutually involve each other.

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Now although generally infinities and infinitesimals are symbolized respectively by ∞ and 0, yet it is manifest that all of each kind are not equal; not only do infinitesimals differ from absolute zero, but they may also differ from each other: and so may infinities differ from each other, and from a quantity which transcends every assignable quantity, that is, from absolute infinity. Hence the need of classifying such quantities.

Assuming then the order to depend on the exponent, it is plain that such orders must exist relatively to a certain determinate quantity, which is the subject of the exponent, and which we call the base. Taking therefore x to be the base of infinities, let 3, x3, ...... x" be infinities of the second, third,

nth orders; and taking i to be the base of infinitesimals, let i2, i3, ...... i" be infinitesimals of the second, third, ...... nth orders respectively. Similarly are infinitesimals

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......

a a x'x2'

a xn

nth orders, if x be the infinity-base;

are infinities of the first, second, ...... nth

orders, if i be the infinitesimal-base: such properties evidently

involve each other, if i and x are so related that x=

1 i

1

or i==.

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These, it is to be observed, are the definitions of the orders of infinities and infinitesimals. Similarly, if x be the infinity-base, x, x3, x2 ...... would be infinities of the second, third, fourth orders; and if it be the infinitesimal-base, i, i, it...... would be infinitesimals of the second, third, fourth orders respectively. Order then, it is to be remembered, is relative to a certain base.

Hence then it appears, that there will be a scale of infinities and of infinitesimals in regular sequence: such that an infinity of the nth order must be infinitely subdivided to produce an infinity of the (n−1)th order, and infinitely quantupled to produce one of the (n+1)th order: infinitesimals also bear such relations to those, on either side of them in the scale, that they are infinitesimal parts of the one, and the aggregate of an infinity of the other. Thus, if a be the symbol of infinity as above, will be the symbol of the finite quantity, and the scale will be

x", ... x2, x1, xo, x-1, x-2, ... x-n;

and if i be the symbol of an infinitesimal, io will represent the finite quantity, and the scale will be

i-", ... i-2, i-1, io, i1, i2, ... i",

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the order in each scale being a descending one.

Hence also,

using the general symbols of such infinitesimals and infinities, viz. O and ∞, the scales become

0-",... 0-2, 0-1, 0o, 01,

∞", ... ∞2, ∞∞1, ∞, ∞0-1,

02,...

02, ... 0",
∞-2,... ∞-".

Thus then, although the mind is incapable of forming adequate notions of infinities and infinitesimals as they were described in rough outline in Article 5, yet they may be brought within its grasp when they are symbolized as above*. It is true that they do not always present themselves under the simple forms herein investigated; but in a subsequent chapter methods will be discussed for determining the orders of the more complex forms and it will then appear that we can always determine the order of an infinity or of an infinitesimal relatively to a

* See Poisson, Traité de Mécanique, Tome I, pp. 14, 16, 2de ed. Paris, 1833.

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