INDETERMINED
, or Indeterminate, in Geometry, is understood of a quantity, which has no certain or definite bounds.
Indeterminate Problem, is that which admits of innumerable different solutions, and sometimes perhaps only of a great many different answers; otherwise called an unlimited problem.
In problems of this kind the number of unknown quantities concerned, is greater than the number of the conditions or equations by which they are to be found; from which it happens that generally some other conditions or quantities are assumed, to supply the defect, which being taken at pleasure, give the same number of answers as varieties in those assumptions.
As, if it were required to find two square numbers whose difference shall be a given quantity d. Here, if x2 and y2 denote the two squares, then will , by the question, which is only one equation, for finding two quantities. Now by assuming a third quantity z so that the sum of the two roots; then is , and , which are the two roots having the difference of their squares equal to the given quantity d, and are expressed by means of an assumed quantity z; so that there will be as many answers to the question, as there can be taken values of the Indeterminate quantity z, that is, innumerable.
Diophantus was the first writer on Indeterminate problems, viz, in his Arithmetic or Algebra, which was first published in 1575 by Xilander, and afterwards in 1621 by Bachet, with a large commentary, and many additions to it. His book is wholly upon this subject; whence it has happened, that such kind of questions have been called by the name of Diophantine problems. Fermat, Des Cartes, Frenicle, in France, and Wallis and others in England, particularly cultivated this branch of Algebra, on which they held a correspondence, proposing difficult questions to each other; an instance of which are those two curious ones, proposed by M. Fermat, as a challenge to all the mathematicians of Europe, viz 1st, To find a cube number which added to all its aliquot parts shall make a square number; and 2d, To find a square number which added to all its aliquot parts shall make a cubic number; which problems were answered after several ways by Dr. Wallis, as well as some others of a different nature. See the Letters that passed between Dr. Wallis, the lord Brounker, Sir Kenelm Digby, &c, in the Doctor's Works; and the Works of Fermat, which were collected and published by his son. Most authors on Algebra have also treated more or less on this part of it, but more especially Kersey, Prestet, Ozanam, Kirkby, &c. But afterwards, mathematicians seemed to have forgot such questions, if they did not even despise them as useless, when Euler drew their attention by some excellent compositions, demonstrating some general theorems, which had only been known by induction. M. la Grange has also taken up the subject, having resolved very difficult problems in a general way, and discovered more direct methods than heretofore. The 2d volume of the French translation of Euler's Algebra contains an elementary treatise on this branch, and, with la Grange's additions, an excellent theory of it; treating very generally of Indeterminate problems, of the sirst and second degree, of solutions in whole numbers, of the method of Indeterminate coefficients, &c.
Finally, Mr. John Leslie has given, in the 2d volume of the Edinburgh Philos. Transactions, an ingenious paper on the resolution of Indeterminate problems, resolving them by a new and general principle. “The doctrine of Indeterminate equations,” says Mr. Leslie, “has been seldom treated in a form equally systematic with the other parts of Algebra. The solutions commonly given are devoid of uniformity, and often require a variety of assumptions. The object of this paper is to resolve the complicated expressions which we obtain in the solution of Indeterminate problems, into simple equations, and to do so, without framing a number of assumptions, by help of a single principle, which though extremely simple, admits of a very extensive application.”
“Let A × B be any compound quantity equal to another, C × D, and let m be any rational number assumed at pleasure; it is manifest that, taking equimultiples, . If, therefore, we suppose that A = mD, it must follow that mB = C, or B = C/m. Thus two equations of a lower dimension are obtained. If these be capable of farther decomposition, we may assume the multiples n and p, and form four equations still more simple. By the repeated application of this principle, an higher equation admitting of divisors, will be resolved into those of the first order, the number of which will be one greater than that of the multiples assumed.”
For example, resuming the problem at first given, viz, to find two rational numbers, the difference of the squares of which shall be a given number. Let the given number be the product of a and b; then by hypothesis, ; but these compound quantities admit of an easy resolution, for . If therefore we suppose , we shall obtain ; where m is arbitrary, and if rational, x and y must also be rational. Hence the resolution of these two equations gives the values of x and y, the numbers sought, in terms of m; viz, , and .
