A question in which one of the unknown quantities has the The combination, with respect to their signs, of insulated simple How to find the true enunciation of a question involving negative 69 ib. Note on the use of the term identical Of changing the signs of quantities to comprehend several ques- Solution of the preceding questions by employing only one un- Of the resolution of any given number of equations of the first degree, containing an equal number of unknown quantities General formulas for the resolution of equations of the first General process for exterminating, in two equations, an unknown Equations of the second degree, having only one unknown quantity Examples of equations of the second degree, which contain only 99 ib. 100 105 ib. 106 Of numbers which are not perfect squares Method of approximating roots Method of abridging, by division, the extraction of roots ib. 107 To approximate a root indefinitely, by means of vulgar fractions ib. Most simple method of obtaining the approximate root of a fraction, the terms of which are not squares Resolution of equations, involving only the second power of the unknown quantity The square root of a quantity may have the sign + or The square root of a negative quantity is imaginary Complete equations of the second degree General formula for resolving equations of the second degree, having only one unknown quantity General rule for the above process Examples showing the properties of negative solutions In what cases problems of the second degree become absurd Expressions called imaginary An equation of the second degree has always two roots To divide any number into two parts, the squares of which shall be in a given ratio Extraction of the square root of algebraic quantities. Transformation for simplifying radical quantities Extraction of the square root of polynomials The formation of powers of simple quantities, and the extraction of their roots 136 · 137 ib. Table of the first seven powers of numbers from 1 to 9 139 Manner of denoting these powers ib, Form of the product of any number whatever of factors of the first degree 145 Method of deducing from this product the development of any power of a binomial 145 Theory of permutations and combinations 146 Rule for the development of any power whatever of a binomial 149 Method of approximating the cube root of numbers which are To extract the roots of literal quantities Of equations with two terms am by x-a The division of xm tinction between arithmetical and algebraical determinations 166 Of equations which may be resolved in the same manner as those of the second degree To determine their several roots ib. 167 160 162 163 Process for performing on radicals of the same degree the four fundamental operations To raise a radical quantity to any power whatever To extract the root of any degree 168 To reduce to the same degree, any number of radical quanti- To place under the radical sign a factor that is without it To determine the product of a X √ 175 ib. ib. 174 ib. 176 177 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 tional exponents General theory of equations The form which equations assume Of the root of an equation Fundamental proposition of the theory Of the decomposition of an equation into simple factors The number of divisors of the first degree, which any equation How far an equation may have factors of any given degree 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 proposed A common divisor of two equations leads to the elimination of one of the unknown quantities 189 ib. 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- To eliminate one unknown quantity in any two equations Inconvenience of the successive elimination of the unknown Of commensurable roots, and the equal roots of numerical equa tions 190 192 193 194 198 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 Method of clearing an equation of fractions 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 To form a general equation, which shall give all the differences ib. ib. 202 205 206 209 210 212 ib. To resolve equations into factors of the second and higher degrees Of the resolution of numerical equations by approximation Principle on which the method of finding roots by approximation, depends Note on the changes of the value of polynomials To assign a number which shall render the first term greater than the sum of all the others 213 214 216 Every equation denoted by an odd number has necessarily a real root, with a sign contrary to that of its real term Every equation of an even degree, the last term of which is negative, has at least two real roots, the one positive and the other negative Determination of the limits of roots, example Application to this example of Newton's method for approximat, How to determine the degree of the approximation obtained 219 ib. • 220 221 222 |