Elements of Algebra

CHAP. I. ADDITIONS. 465 determinate equations of the first and second degree; for these I give new and general methods, both for the case in which the numbers are only required to be rational, and for that in which the numbers sought are required to be integer; and I consider some other important matters relating to the same subject. The last Chapter contains researches on the functions *, which have this property, that the product of two or more similar functions is always a similar function. I give a general method for finding such functions, and shew their use in the resolution of different indeterminate problems, to which the usual methods could not be applied. Such are the principal objects of these Additions, which might have been made much more extensive, had it not been for exceeding proper bounds; I hope, however, that the subjects here treated will merit the attention of mathematicians, and revive a taste for this branch of algebra, which appears to me very worthy of exercising their skill *** CHAPTER I. ON CONTINUED FRACTIONS. 1. As the subject of Continued Fractions is not found in the common books of arithmetic and algebra, and for this reason is but little known to mathematicians, it will be proper to begin these Additions by a short explanation of their theory, which we shall have frequent opportunities to apply in what follows. In general, we call every expression of this form, a *continued fraction*, $a + \frac{b}{\beta + \frac{c}{\gamma + \frac{d}{\delta + , \&c.}}}$ * A term used in algebra for any expression containing a certain letter, denoting an unknown quantity, however mixed and compounded with other known quantities or numbers. Thus, $ax + yx$; $2x - a \sqrt{(\frac{a^2 - x^2}{3})}$; $3xy^3 + \sqrt{(\frac{bc + yx}{2})}$, are all functions of $x$. H H