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given base; and we can thus arrange and collect into groups all those which are of the same order. Primarily, the chief work of the Calculus will he the formation of subsidiary infinitesimals and infinities by means of, and from, finite quantities: but ere we enter on that work, we must plainly state the laws to which such infinitesimals and infinities are to be subject: these laws are of the utmost importance; they are indeed the conditions under which our subsidiary infinitesimals and infinities exist at all; to us, at present, it is immaterial whether they are geometrical or other representatives, or only analogues: but it is necessary that they should be of a nature which is not inconsistent with the nature of number. If any one hesitates to accept the following propositions, which I have called Theorems, but which are in fact axiomatic statements of infinitesimals and infinities, let him bear in mind, that our quantities are what they are only by virtue of these conditions. In all the following Theorems the base is assumed to be the same.

9.] Theorem I.—Infinities and infinitesimals, like finite quantities, admit of being multiplied and divided by finite numbers, and their order is not thereby changed: but multiplication or division by the base or any power of it changes the order of the infinity and of the infiuitesimal.

Thus x% and 2x2 are infinities of the same order, and i and

- are infinitesimals of the same order; i" x im = i"+m, that is, 4

by multiplication of the base raised to a power the order of the infinitesimal is changed.

Theorem II.—The product of an infinity and of an infinitesimal of the same order is a finite number.

Thus i!x4 = s; i x % = a. xi i

Theorem III.—The product of an infinity and of an infinitesimal of different orders is an infinity or an infinitesimal, according as the order of the infinity is higher or lower than that of the infinitesimal; and the order of the product depends on the difference of the orders of the component factors.

Thus x3 x - = ax2: x" x — = axn~m = ° , the former or

3C 3C W

latter form being taken according as n is greater or less than m, and therefore the result accordingly is an infinity or an infinitesimal. Similarly, i2 x -ry =

t %

Theorem IV.—The ratio to each other of two infinities or infinitesimals of the same order is finite.

Let a and b be two finite numbers, and thus let -r~ be

two infinities of the same (viz. the rath) order; then their ratio is a : b; similarly, if at", bi" be two infinitesimals of the »th order, their ratio is a : b, that is, the same as before. This is also manifest geometrically. Let there be two concentric circles, the radius of one of which is double that of the other, and in them let two regular polygons of the same number of sides be described; each side of the larger is always double each side of the smaller; and as this is true whatever is the number of the sides, it is true when the number is infinitely great; in which case each side becomes infinitesimally small: and if the number in both polygons is the same, the sides are infinitesimals of the same order, and bear to each other the finite ratio of 2 : 1.

Hence also it follows that quantities, whose symbolical form

is 5, are indeterminate by virtue of that form, and may be

either infinite, finite, or infinitesimal, and that such determinate values depend on the relation of the order of infinitesimal in the numerator to that in. the denominator; that is, if the infinitesimal in the denominator be of a higher order than that in the numerator, the determinate value is infinite; if the orders are the same, the value is finite; and if that in the numerator is higher than that in the denominator, the value is infinitesimal.

Thus = - when x = a; but dividing out (a— x)*,

a—x 0 ' 6 v ;,

the result is (fl~*''r) _ [^fjj _ K when x a. (a-x)i 0

«(1— x2) sxO a(l + x) 2a

Similarly -^—j = -—when x = l, = —- =

o(l-x) 4x0 b b

. (a—x)2_0 _t_ a—x 0 _L_

Similar results are also manifestly true of infinities and their different orders.

Theorem V.—The sum of two infinities or infinitesimals of the same order is the product of the infinity or infinitesimal by the sum of their coefficients; and the difference is an infinity or infinitesimal of the same order, except when the coefficients are equal, in which case it is absolutely zero.

Thus ai" + bin=(a + b)i", ain bin=(a b)i", ain—ai" = 0.

Theorem VI.—Since an infinitesimal is derived from a Finite number by the resolution of the finite number into an infinity of parts, the ratio of a Finite number of such infinitesimals to the original number, is that of 0 to 1; a finite number therefore of such infinitesimal parts can have no value at all when added to a finite quantity: it must be neglected.

Thus if a and b are finite numbers, and i be an infinitesimal, of such an expression as a + bi, the latter part must be neglected; bi has no value at all when added to a.

Theorem VII.—For a similar reason a finite quantity can have no value when added to an infinity, and must therefore be neglected.

Thus of ax + b, if a is a finite number and x is an infinity, the finite quantity b must be neglected, and the expression is equal to ax.

Similarly, in expressions involving the sum or difference of infinitesimals of different orders which have finite coefficients, all the higher infinitesimals must be neglected, and the lower ones alone retained. Thus let a and b be two finite quantities, and i" and i"+r two infinitesimals; then

ai* + bin+r = in{a + bir},

the latter part of which is equal to a by Theorem VI; and there

fore at" + bi"+r = ai".

Similarly, a4 bi + ci2+ ... + ki" a.

And similarly, if an expression involves the algebraical sum of infinities of various orders, whose coefficients are finite, the expression is equal to the infinity of the highest order, and all the others, and the finite quantities, can have no value, when added to it, and must be neglected.

10.] To enable the student to appreciate the importance of the above theorems, some examples are subjoined:

Ex. 1. To find the inferior and superior limits of

ax3 + bxs + cx + e mx3 + nx2 + px + q'

By Theorem VI the inferior limits of the numerator and denominator are severally e and q, and by Theorem VII the superior limits are severally ax3 and mx3; hence the superior and


inferior limits of the fraction are severally — and -.

m q

Ex. 2. To find the inferior and superior limits of

b + az
a + b*'

If x = 0. the inferior limit becomes —^4 5 and if x = oo, the

a +1

^) , which is oo or 0, according as a is greater or less

than b.

11.] If any one idea or conception is pregnant with the whole Calculus, it is that contained in Theorems VI and VII; they enuntiate the essential properties of infinitesimals and their reciprocal infinities: such as flow immediately from, inasmuch as they are involved in, any adequate notion of such a mode of resolution as the Calculus contemplates; were not the properties of infinitesimals such as the theorems import, the Calculus would not be what it is: from them it takes its rise, and whatever its genius be, such have they imparted to it.

On inspecting the scales of infinities and infinitesimals which are given above, it will be observed that the finite quantity is represented by the symbol which has 0 for its exponent: the reason of which by the common law of indices is manifest from the examples given in illustration of Theorem IV of Article 9; and on the correctness of thus representing it more will be said hereafter. And it will also be observed, that all the symbols on one side of it represent infinities, and all on the other infinitesimals; but it is quite arbitrary which grade shall be considered finite, or the one intermediate to the infinite and the infinitesimal. Borrowing an analogy from the senses, as explained above, we make them the test of finiteness, but such is not necessary; and doubtless, were our senses much more delicate than they are, we should start from some order lower than we do, and call that finite which we now call infinitesimal; and if we were living amongst bodies and distances which were comparable with the distances of the fixed stars from the sun, they would doubtless be our finite quantities, and what are now finite would become infinitesimal.

A pertinent illustration of the preceding Theorems is found in the modern treatises on algebraical geometry. The general equation of the first degree is assumed to be of the form

AX + By + c = 0,

where A, B and c are finite constants; and in the course of the discussion of this equation we meet with the paradoxical result c = 0, where c is not and does not admit of being zero. Thus, for instance, we may in the course of an investigation arrive at the equation 5 = 0; and this equation is, whether for satisfactory reasons or otherwise I will not now inquire, said to represent a straight line at an infinite distance from the origin. Now we have here the reason for this seeming impossibility: if c = 0, it is implied that c is one term of an equation, the other terms of which are infinite; and that in addition or subtraction with them, the term c must be omitted; in other words, the other terms of the equation satisfy the equation, and the constant term c must be omitted: in order however that the nature of the equation may not be forgotten or overlooked, it is left in the seemingly paradoxical form

c = 0.

Thus if the preceding equation takes the form c = 0, where c is not zero, it is implied that a; and y are infinite, and have different signs, and that the equation is satisfied by them; and thus the equation represents a line at an infinite distance. For many cases of this curious result I may refer the reader to Salmon's Conic Sections, Dublin, 1855, 3rd edit. Art. 64.

12.] Thus far we have spoken of single symbols, and of their properties; it is of continuous variables that we shall treat, and we shall not introduce discontinuous ones without special statement. Now it is plain that two or more such variables may be combined with constants in an equation, and may be such that a change of value of one may involve a corresponding change of value of one or more of the others; when this is the case, such


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