INFINITE
, is applied to quantities which are either greater or less than any assignable ones. In which sense it differs but little from the terms Indefinite and Indeterminate. Thus, an
Infinite, or Infinitely great line, denotes only an indefinite or indeterminate line; or a line to which no certain bounds or limits are prescribed.
Infinite Quantities. Though the idea of magnitude infinitely great, or such as exceeds any assignable quantity, does include a negation of limits, yet all such magnitudes are not equal among themselves; but besides Infinite length, and Infinite area, there are no less than three several sorts of Infinite solidity; all of which are quantities sui generis; and those of each species are in given proportions.
Infinite length, or a line infinitely long, may be considered, either as beginning at a point, and so infinitely extended one way; or else both ways from the same point.
As to Infinite surface or area, any right line infinitely extended both ways on a plane infinitely extended every way, divides that plane into two equal parts, one on each side of the line. But if from any point in such a plane, two right lines be infinitely extended, making an angle between them; the Infinite area, intercepted between these Infinite right lines, is to the whole Infinite plane, as that angle is to 4 right angles. And if two Infinite and parallel lines be drawn at a given distance on such an Infinite plane, the area intercepted between them will be likewise Infinite; but yet it will be infinitely less than the whole plane; and even infinitely less than the angular or sectoral space, intercepted between two Insinite lines, that are inclined, though at never so small an angle; because in the one case, the given finite distance of the parallel lines diminishes the Infinity in one of the dimensions; whereas in a sector, there is Infinity in both dimensions. And thus there are two species of Infinity in surfaces, the one infinitely greater than the other.
In like manner there are species of Infinites in solids, according as only one, or two, or as all their three dimensions, are Infinite; which, though they be all infinitely greater than a finite solid, yet are they in succession infinitely greater than each other.
Some farther properties of Infinite quantities are as follow:
The ratio between a finite and an Infinite quantity, is an Infinite ratio.
If a finite quantity be multiplied by an infinitely small one, the product will be an infinitely small one; but if the former be divided by the latter, the quotient will be infinitely great.
On the contrary, a finite quantity being multiplied by an infinitely great one, the product is infinitely great; but the former divided by the latter, the quotient will be infinitely little.
The product or quotient of an infinitely great or an infinitely little quantity, by a finite one, is respectively infinitely great, or infinitely little.
An infinitely great multiplied by an infinitely little, is a finite quantity; but the former divided by the latter, the quotient is infinitely Infinite.
The mean proportional between infinitely great, and infinitely little, is finite.
Arithmetic of Infinites. See Arithmetic. Also Wallis's treatise of this subject; and another by Emerson, at the beginning of his Conic Sections; also Bulliald's treatise Arithmetica Infinitorum.
Infinite Decimals, such as do not terminate, but go on without end; as .333 &c = 1/3, or .1.42857. &c = 1/7. See Repetend.
Infinitely Infinite Fractions, or all the powers of the fractions whose numerator is 1; which are all together equal to unity, as is demonstrated by Dr. Wood, in Hook's Philos. Coll. N<*> 3, p. 45; where some curious properties are deduced from the same.
Infinite Series, a series considered as infinitely continued as to the number of its terms. See Series.
