General theory of algebraic equations

Cover......Page 1
Title Page......Page 2
Table of Contents......Page 6
Translator’s Foreword......Page 12
Dedication from the 1779 edition......Page 14
Preface to the 1779 edition......Page 16
Definitions and preliminary notions......Page 26
About the way to determine the differences of quantities......Page 28
A general and fundamental remark......Page 32
Reductions that may apply to the general rule to differentiate quantities when several differentiations must be made.......Page 33
Remarks about the differences of decreasing quantities......Page 34
About sums of quantities......Page 35
Remarks......Page 36
About sums of rational quantities with no variable divider......Page 37
About complete polynomials and complete equations......Page 40
Problem I: Compute the value of N(u . . . n)T......Page 41
Problem II......Page 42
Problem III......Page 44
Remark......Page 45
Initial considerations about computing the degree of the final equation resulting from an arbitrary number of complete equations with the same number of unknown......Page 46
Determination of the degree of the final equation resulting from an arbitrary number of complete equations containing the same number of unknowns......Page 47
Remarks......Page 49
About incomplete polynomials and first-order incomplete equations......Page 51
Problem IV......Page 53
Problem V......Page 54
Problem VII: We ask for the degree of the final equation resulting from an arbitrary number n of equations of the form (ua . . . n)t = 0 in the same number of unknowns......Page 57
Remark......Page 59
Problem VIII......Page 60
Problem X......Page 61
Problem XI......Page 62
(3) The other unknowns do not exceed a given degree (different or the same for each), but, when combined groups of two or three among themselves as well as with the first two, they reach all possible dimensions until that of the polynomial or the equation......Page 63
Problem XII......Page 64
Problem XIII......Page 65
Problem XIV......Page 66
Problem XVI......Page 67
We further assume that the degrees of the n   3 other unknowns do not exceed given values we also assume that the combination of two, three, four, etc. of these variables among themselves or with the first three reaches all possible dimensions, up to the......Page 70
Problem XVII......Page 71
Problem XVIII......Page 72
Summary and table of the different values of the number of terms sought in the preceding polynomial and in related quantities......Page 81
Problem XIX......Page 86
Problem XX......Page 87
Problem XXII......Page 88
About the largest number of terms that can be cancelled in a given polynomial by using a given number of equations,  without introducing new terms......Page 90
Determination of the symptoms indicating which value of the degree of the final equation must be chosen or rejected, among the different available expressions......Page 94
Expansion of the various values of the degree of the final equation, resulting from the general expression found in (104), and expansion of the set of conditions that justify these values......Page 95
Application of the preceding theory to equations in three unknowns......Page 96
General considerations about the degree of the final equation, when considering the other incomplete equations similar to those considered up until now......Page 110
Problem XXIII......Page 111
General method to determine the degree of the final equationfor all cases of equations of the form (ua . . . n)t = 0......Page 119
General considerations about the number of terms of other polynomials that are similar to those we have examined......Page 126
Conclusion about first-order incomplete equations......Page 137
About incomplete polynomials and second-, third-, fourth-, etc. order incomplete equations......Page 140
Problem XXIV......Page 143
About the form of the polynomial multiplier and of the polynomials whose number of terms impact the degree of the final equation resulting from a given number of incomplete equations with arbitrary order......Page 144
Useful notions for the reduction of differentials that enter in the expression of the number of terms of a polynomial with arbitrary order......Page 146
Problem XXV......Page 147
Table of all possible values of the degree of the final equations for all possible cases of incomplete, second-order equations in two unknowns......Page 152
Conclusion about incomplete equations of arbitrary order......Page 159
General observations......Page 162
A new elimination method for first-order equations with an arbitrary number of unknowns......Page 163
General rule to compute the values of the unknowns, altogether or separately, in first-order equations, whether these equations are symbolic or numerical......Page 164
A method to find functions of an arbitrary number of unknowns which are identically zero......Page 170
About the form of the polynomial multiplier, or the polynomial multipliers, leading to the final equation......Page 176
About the requirement not to use all coefficients of the polynomial multipliers toward elimination......Page 178
About the number of coefficients in each polynomial multiplier which are useful for the purpose of elimination......Page 180
About the terms that may or must be excluded in each polynomial multiplier......Page 181
About the best use that can be made of the coefficients of the terms that may be cancelled in each polynomial multiplier......Page 183
Other applications of the methods presented in this book for the General Theory of Equations......Page 185
Useful considerations to considerably shorten the computation of the coefficients useful for elimination.......Page 188
Applications of previous considerations to different examples interpretation and usage of various factors that are encountered in the computation of the coefficients in the final equation......Page 199
General remarks about the symptoms indicating the possibility of lowering the degree of the final equation, and about the way to determine these symptoms......Page 216
About means to considerably reduce the number of coefficients used for elimination. Resulting simplifications in the polynomial multipliers......Page 221
More applications, etc.......Page 230
About the care to be exercised when using simpler polynomial multipliers than their general form (231 and following), when dealing with incomplete equations......Page 234
More applications, etc.......Page 238
About equations where the number of unknowns is lower by one unit than the number of these equations. A fast process to find the final equation resulting from an arbitrary number of equations with the same number of unknowns......Page 246
About polynomial multipliers that are appropriate for elimination using this second method......Page 248
Details of the method......Page 250
First general example......Page 251
Second general example......Page 253
Third general example......Page 259
Fourth general example......Page 262
Observation......Page 266
Considerations about the factor in the final equation obtained by using the second method......Page 276
About the means to recognize which coefficients in the proposed equations can appear in the factor of the apparent final equation......Page 278
Determining the factor of the final equation: How to interpret its meaning......Page 294
About the factor that arises when going from the general final equation to final equations of lower degrees......Page 295
Determination of the factor mentioned above......Page 299
About equations where the number of unknowns is less than the number of equations by two units......Page 301
Form of the simplest polynomial multipliers used to reach the two condition equations resulting from n equations in n   2 unknowns......Page 303
About a much broader use of the arbitrary coefficients and their usefulness to reach the condition equations with lowest literal dimension......Page 326
About systems of n equations in p unknowns, where p < n......Page 332
When not all proposed equations are necessary to obtain the condition equation with lowest literal dimension......Page 339
About the way to find, given a set of equations, whether some of them necessarily follow from the others......Page 341
About equations that only partially follow from the others......Page 343
Reflexions on the successive elimination method......Page 344
About equations whose form is arbitrary, regular or irregular.Determination of the degree of the final equation in all cases......Page 345
Remark......Page 352
Follow-up on the same subject......Page 353
About equations whose number is smaller than the number of unknowns they contain. New observations about the factors of the final equation......Page 358