infinite quantities

pl. Math.: [INFINITY.] infinite series, s. Math.: A series, the terms of which go on increas. ing or decreasing without coming to an end. Arithmetic of infinites : A method invented by For the difference between infinite and bound le , see BOUNDLESS. Dr. Wallis for the summation of infinite series. In-fin-ite-ly, adv . [Eng. infinite ; -ly .] 1. To an infinite degree or extent; without limits or bounds. "Whatever is le than infinite, is still infinitely distant from infinity." -Brooke: Universal Beauty , bk. iv. (Note.) 2. To a very great degree or extent; immensely; exceedingly. "He knows their value when he finds them, and is inf in itely pleased." -Dryden: Parallel of Poetry and Painting. In-fin-ite-ne , s . [Eng. infinite; -ne .] The quality or state of being infinite; infinity; boundle ne ; immensity; exceeding greatne or extent. Sermons, vol. i., ser. 11. "The infinitene of the divine con descent i on."- Sharp: In-fin-1-tes-I-mal, a. & s. [Formed from Lat. inf in it us , on analogy of Eng. centesimal ; Fr. infini tésimal ; Ital.infinites i male ; Sp. infinitesimal , from Fr. infinitésime , Sp. infinitisimo , Ital. infiniti imo = exceedingly small.] A. As adj .: Infinitely, immeasurably, or indefinitely small; smaller or le than any a ignable quantity. "Vibrations of the small, and, as one may say, infinites imal medullary particles." -Hartley: Observations of Man , pt. i., prop. 4. B. As substantive: 1. Ord. Lang .: An infinitely or indefinitely small quantity. "Infinites are composed of finites, in no other sense, than as finites are composed of infinitesimals." Dr. Clarke: Fourth Reply to Leibnitz. 2. Math .: A quantity le or smaller than any a ignable quantity; a quantity so small as not to be comparable with any finite quantity. Infinitesimals are of different orders. No quantity is great or small except in comparison with some other quantity. An infinitely small quantity of the first order is one that is infinitely small with respect to a finite quantity, that is, so small that it may be contained in it an infinite number of times. An infinitely small quantity of the second order is one that is infinitely small with respect to an infinitely small quantity of the first order. In general, an infinitely small quantity of nth order is one which is infinitely small with respect to an infinitely small quantity of the (n-1)th order. When several quantities, either finite or infinitesimal, are connected by the signs plus or minus, all except those of the lowest order may be neglected without affecting the value of the expre ion. In-fin-1-tès-i-mal-lý, adv . [Eng. infinitesimal : -ly .) In an infinitesimal manner or degree; to an infinitesimal degree or extent; in an infinitely small quantity, degree, or extent.