To place under the radical sign a factor that is without it Remarks on peculiar cases which occur in the Calculus of Radi- Of the decomposition of an equation into simple factors An equation composed of simple factors Formation of its coefficients - ib. Of Elimination among Equations exceeding the First Degree 186 By substituting the value of one of the unknown quantities Rule for making the radical sign to disappear General formulas for equations having two unknown quantities - 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 other Singular cases, in which the proposed equations are contradic tory, or leave the question indeterminate To eliminate one unknown quantity in any two equations Euler's method of solving the above problem 190 192 193 Inconvenience of the successive elimination of the unknown Of Commensurable Roots, and the equal Roots of Numerical 198 199 Every equation, the coefficients of which are entire numbers, that of the first term being 1, can only have for roots entire numbers or incommensurable numbers ib. Method of clearing an equation of fractions ib. 202 Investigation of commensurable divisions of the first degree How to obtain the equation, the roots of which are the differences between one of the roots of the proposed equation and each of the others 205 Of equal roots 206 To form a general equation, which shall give all the differences between the several roots combined two and two 209 210 Method of clearing an equation of any term whatever Of the Resolution of Numerical Equations by Approximation Note on the changes of the value of polynomials To assign a number which shall render the first term greater Determination of the limits of roots, example Application to this example of Newton's method for approximat ing the roots of an equation How to determine the degree of the approximation obtained 219 ib. 220 221 222 To prove the existence of real and unequal roots Use of division of roots for facilitating the resolution of an equa- ib. 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 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, The greater added to the less forms the number to be divided. Substituting in this last proposition, instead of the words, the greater part, the equivalent expression given above, namely, the less part added to the given excess, we find that The less part, added to the given excess, added moreover to the less part, forms the number to be divided. But the language may be abridged, thus, Twice the less part, added to the given excess, forms the number to be divided; whence we infer, that, Twice the less part is equal to the number to be divided diminished by the given excess ; and that, Once the less part is equal to half the difference between the number to be divided and the given excess. Or, which is the same thing, The less part is equal to half the number to be divided, diminished by half the given excess. The proposed question then is resolved, since to obtain the parts sought it is sufficient to perform operations purely arithmetical upon the given numbers. divided were 9, and the 5, the less part would be, less, or 4, or 2; and If, for example, the number to be excess of the greater above the less according to the above rule, equal to the greater, being composed of the less plus the excess 5, would be equal to 7. 2. The reasoning, which is so simple in the above problem, but which becomes very complicated in others, consists in general of a certain number of expressions, such as added to, diminished by, is equal to, &c. often repeated. These expressions relate to the operations by which the magnitudes, that enter into the enunciation of the question, are connected among themselves, and it is evident, that the expressions might be abridged by representing each of them by a sign. This is done in the following manner. To denote addition we use the sign +, which signifies plus. For subtraction we use sign —, which signifies minus. For multiplication we use the sign X, which signifies multiplied by. To denote that two quantities are to be divided one by the |