Elements of Algebra

CHAP. VII. OF ALGEBRA. 359 supposed $a = 83$, we should have $e = 9$, and $83n^2 + 1 = m^2$, where $n = 9$, and $m = 82$ *. * Our author might have added here another very obvious case, which is when $a$ is of the form $e^2 \pm \frac{2}{c}e$; for then by mak- ing $n = c$, our formula $an^2 + 1$, becomes $e^2c^2 \pm 2ce + 1 =$ $(ec \pm 1)^2$. I was led to the consideration of the above form, from having observed that the square roots of all numbers in- cluded in this formula are readily obtained by the method of continued fractions, the quotient figures, from which the fractions are derived, following a certain determined law, of two terms, readily observed, and that whenever this is the case, the method given above is also applied with great facility. And as a great many numbers are included in the above form, I have been in- duced to place it here, as a means of abridging the operations in those particular cases. The reader is indebted to Mr. P. Barlow of the Royal Aca- demy, Woolwich, for the above note; and also for a few more in this Second Part, which are distinguished by the signature, B.