Infinite.

Infinite. In philosophy, infinite is that which is without any limitation, and, like absolute and unconditioned, is used especially of the Infinite, of God. As to our knowledge of the infinite, some (as Hamilton and Mansel) hold that the idea is purely negative; Descartes affirmed that the idea of the infinite was not merely the idea of an objective reality, but is implied as a necessary condition of every other. See ABSOLUTE, CONDITION, RELATIVITY OF KNOWLEDGE; Hamilton's Discussions, Mansel's Limits, Calderwood's Philosophy of the Infinite, Spencer's First Principles.

In mathematics, the term infinity and the phrases infinitely great and infinitely small are of constant occurrence; and the symbol $\backslash infty$ is usually said to denote a magnitude infinitely great, the symbol 0 a magnitude infinitely small. Are these magnitudes infinitely great and infinitely small to be reasoned about in the same way as ordinary finite magnitudes? Are these symbols $\backslash infty$ and 0 to be treated in the same way as ordinary algebraic symbols, $a$, $b$, $x$, $y$, &c.? With respect to the symbol 0 there seems at first sight to be little difficulty, for we are accustomed to regard it as denoting the absence of all quantity, or as the result obtained by subtracting any finite quantity from a quantity equal to it. It is found convenient however, though it would be impossible to explain in short compass the grounds of the convenience, to give another meaning to the symbol 0. The new meaning will perhaps be understood from the following illustration. Take the algebraical expression $\backslash frac\{1\}\{x\}$, and suppose $x$ capable of increasing so that it may become greater than any assignable quantity; then the value of $\backslash frac\{1\}\{x\}$ will diminish and become less than any assignable quantity, and the limit towards which it tends, that is to say, the value from which it may be made to differ as little as we please, is symbolised by 0. The same expression will enable us to give a meaning to the symbol $\backslash infty$. Suppose $x$ capable of diminishing so that it may become less than any assignable quantity; then the value of $\backslash frac\{1\}\{x\}$ will increase and become greater than any assignable quantity, and the limit towards which it tends, that is to say, the value from which it may be made to differ as little as we please, is symbolised by $\backslash infty$. The symbols 0 and $\backslash infty$ therefore, denoting the limits towards which certain variable quantities tend when particular suppositions are made, cannot be used absolutely like the symbols denoting finite quantities: because $a\; \backslash div\; a\; =\; 1$, it would be erroneous to conclude that $0\; \backslash div\; 0\; =\; 1$ or $\backslash infty\; \backslash div\; \backslash infty\; =\; 1$. Expressions such as $0\; \backslash div\; 0$, $\backslash infty\; \backslash div\; \backslash infty$, $\backslash infty\; -\; \backslash infty$, $0\; \backslash times\; \backslash infty$, $\backslash infty^\backslash infty$, and some others are called indeterminate forms: for methods of evaluating them, see Chrystal's Algebra, chap. xxv., or De Morgan's Differential and Integral Calculus, chap. x.

Infinitesimals is the name applied to the method adopted by Leibnitz as the foundation of his Differ- ential Calculus. Leibnitz considered magnitudes as composed of infinitely small elements or infinitesimals. Those elements which are infinitely small compared to any finite magnitude are infinitesimals of the first degree; those which are infinitely small compared to infinitesimals of the first degree are infinitesimals of the second degree; and so on. The principle of the method briefly stated is that two finite magnitudes are equal if they differ only by an infinitely small magnitude. Though the results obtained by the application of infinitesimals are seen to be always in accord with the results obtained by other methods, and a method which always leads to correct conclusions must be logically sound, yet the fundamental principle does not at first sight seem rigorously exact, and the method looks as if it were merely one of approximation. In consequence it has now come to be usual to found the calculus on the doctrine of limits.