It also appears that the least positive value of the ex pression x + is + 2, and its greatest negative value 2, reckoning, as before, that negative quantity to be greatest, which, independently of the sign, is expressed by the smallest number.
Hence, we learn that no real value of x can be found that shall make the expression (x + 1 equal to a pro per fraction either positive or negative, but that this expression may represent any positive or negative tity whatever that is not between the limits of + 1 and. 1.
From this example, it appears that the theory of impos sible quantities may sometimes be applied with great ad vantage to a very elegant and interesting class of problems, namely, such as require the determination of the greatest and least values of a variable quantity. The general me thod of proceeding is to suppose, that the quantity to be a maximum or minimum is equal to a given quantity ; and then to inquire what is the greatest or least values which this quantity can have, without introducing the imaginary sym bol 1 into the resulting formulx.
The second class of imaginary expressions, or those which indicate real quantities, involve the symbol ?-1 in such a manner, that, by suitable transformations, it may ' at last be made to disappear. The two following expres sions arc of this kind, viz.
-Fb?-1)-F 1) ; -4/ 1 By taking the square, and then again the square root of each, the former is transformed to ? ; and the latter to ./ , which are both real quantities.
The general expression for the roots of a cubic equa tion has the form 3 -f-bv I when its roots are all real ; but, unlike the two former, it cannot, by any means be transformed into a real algebraic expression, consisting of a finite number of terms ; but its value may be found by an infinite series, or a table of sines. (ALGEBRA, § 227§ 230.) It has been proved, in the ARITHMETIC OF SINES, 19, that, 7n being any whole number or fraction, (cos. x + sin. x cos. m x + sin. m This formula was first given by De Moivre, (Phil. Trans. 1707, and Miscel. Analytica, lib. 2.) Lagrange calls it " a formula as remarkable for its simplicity and elegance as its generality and fertility in consequences," (Calcul.des Fonc
tfons, p. 116) ; and Laplace considers its invention as of equal importance with the binomial theorem, (Leions des F.coles N'ormales.) As the sign of the square root of a quantity may be either or 1, we may put in the formu la instead of + i/-1; it then becomes (cos. x sin. x =COS. m x sin. m x From this, and the former expression, we find, by addition and subtraction, that the imaginary expression (cos. x ± sin. x v-1)/n+ (cos. x sin. x V-1),n is equivalent to 2 cos. m x, or real quantity ; also, that ima ginary expression 1 5 z (cos. x 4- sin. x 1)m--(cos. xsin. x is equivalent to 2 sin. m x, another real quantity. These expressions, although the representatives of real geome trical quantities, viz. 2 cos. and 2 sin.rn x, considered by themselves, are utterly without any, geometrical signi fication. It is impossible to translate the analytic formula into the language of strict geometry, because the symbols sin. x 1 1 and correspond to nothing that ad v mits of geometrical definition. Notwithstanding this in congruity, these very expressions have led to the discovery of some of the most beautiful and general theorems in ge ometry, and have enabled analysts to resolve questions which, without their aid, ivould have been altogether un tractable. We have given examples of their application to the theory of angular sections, and the investigation of that elegant property of the circle called the Cotesian the orem, in the ARITHMETIC OF SINES, §19---§ 25.
If in the formula (Cos. x sin.x V-1)m= cos. in x + sin. nix 1 we write for rn, it becomes 71 (COS. x + sin. .r,,/-1)= cos. -4- sin.
11 71 Suppose now n to be indefinitely great, then, x being supposed a finite arc, the arc be indefinitely small; in this case its cosine will be equal to the radius, and its sine equal to the arc itself, hence, n being indefinitely great, (Cos. x + sin. x V--1 = 1 + I or putting cos. n + sin. x v-1 = v.