To extract the root of any power whatever of a simple quantity 138 Of negative exponents Of the Formation of Powers of Compound Quantities Manner of denoting these powers Form of the product of any number whatever of factors of the General law for the number of roots of an equation, and the dis- Of Equations which may be resolved in the same manner as To place under the radical sign a factor that is without it Remarks on peculiar cases which occur in the Calculus of Radi- How to deduce the rules given for the calculus of radical quantities ib. Examples of the utility of signs, shown by the calculus of frac- An equation composed of simple factors ib. Formation of its coefficients ib. Of Elimination among Equations exceeding the First Degree General formulas for equations having two unknown quantities ib. and how they may be reduced to equations having only one 188 Formula of elimination in two equations of the second degree To determine whether the value of any one of the unknown quantities satisfies, at the same time, the two equations pro- How to proceed after obtaining the value of one of the unknown quantities in the final equation in order to find that of the Singular cases, in which the proposed equations are contradic- tory, or leave the question indeterminate Inconvenience of the successive elimination of the unknown 198 ib. ib. Every equation, the coefficients of which are entire numbers, 205 Method of clearing an equation of any term whatever To resolve equations into factors of the second and higher de- Of the Resolution of Numerical Equations by Approximation Principle on which the method of finding roots by approxima- To assign a number which shall render the first term greater than the sum of all the others Every equation denoted by an odd number has necessarily a real Every equation of an even degree, the last term of which is ne- gative, has at least two real roots, the one positive and the Determination of the limits of roots, example Application to this example of Newton's method for approximat- Note-respecting equal roots. To prove the existence of real and unequal roots Use of division of roots for facilitating the resolution of an equa- tion when the coefficients are large Method of approximation according to Lagrange Fundamental principles of proportion and equidifference Of the changes which a proportion may undergo To determine any term whatever of this progression 235 Progressions by quotients, the sum of which has a determinate Manner of reducing all the terms of a progression by quotients In what cases the quotient of this operation is converging and may be taken for the approximate value of the fraction Theory of Exponential Quantities and of Logarithms Remarkable fact, that all numbers may be produced by means of What is meant by the term logarithm, and the base of logarithms 246 ELEMENTS OF ALGEBRA. Preliminary Remarks upon the Transition from Arithmetic to Algebra-Explanation and Use of Algebraic Signs. 1. It must have been remarked in the Elementary Treatise of Arithmetic, that there are many questions, the solution of which is composed of two parts; the one having for its object to find to which of the four fundamental rules the determination of the unknown number by means of the numbers given belongs, and the other the application of these rules. The first part, independent of the manner of writing numbers, or of the system of notation, consists entirely in the development of the consequences which result directly or indirectly from the enunciation, or from the manner in which that which is enunciated connects the numbers given with the numbers required, that is to say, from the relations which it establishes between these numbers. If these relations are not complicated, we can for the most part find by simple reasoning the value of the unknown numbers. In order to this it is necessary to analyze the conditions, which are involved in the relations enunciated, by reducing them to a course of equivalent expressions, of which the last ought to be one of the following; the unknown quantity equal to the sum or the difference, or the product, or the quotient, of such and such magnitudes. This will be rendered plainer by an example. To divide a given number into two such parts, that the first shall exceed the second by a given difference. In order to this we would observe 1, that, The greater part is equal to the less added to the given excess, and that by consequence, if the less be known, by adding to it this excess we have the greater; 2, that, |
