TERMINAL. 1Ve cannot say that this term is used in mathematics to the extent to which we shall carry it ; but the very great conveni ence which would arise from an extension of its use is sufficient justi fication for coining a few new meanings. Term is a word of geometry very little used, and signifying boundary or extremity ; the words ter minal value and terminal form are sometimes used to signify the last and most complete value or festal. When a finite expression, added to a certain number of terms of a series, makes up the equivalent of the expression from which the series is deduced, or stands for all the sub sequent terms of the series, this finite expression might be called the terminal expression. Thus in Taylor's Theorem we have one terminal expression in D'Alembert's form, another in that of Lagrange.
There is also another use of the word, which would convey a dis tinction much wanting words to express it : wo allude to what might be called terminal language. All the use of the words infinitely small and infinitely great [Issisers ; Liner] is entitled to this name ; as follows : —When we say, for example, that a circle is a regular polygon with on Infinitely great number of Infinitely small sides, the language used is that of an end arrived at, a transformation made ; the circle is described as actually consisting of straight lines; and the language is terminal (expressive of a boundary actually attained). But
the meaning of this language is, or is generally held to be, false: no polygon is a circle, how great Reeves the number, or how small Keever the magnitude, of the siduf. The proposition which is really true, that is, over which all shake hands, whatever their notion of infinity may be, is that the terminal proposition, true or false, is one to which an interminable and unlimited degree of approximation may be made. An inscribed regular polygon may, with sides enough, be made to coincide with the circle within any degree of nearness we please to assign : or the following proposition—" the area of the inscribed polygon may be made to differ from that of the circle by less than the nth part of the latter"—may be made true for every value of n that can be named, however great. Terminal langnage, properly employed, may be made the means of abbreviation of all those truths whose announcement contains interminable approximation : the development of this sentence is the object of the article barren&