Page:Halliwell Collection of Letters.djvu/17

Rh $$\gamma = 17$$, in which case the problem remains in the same position as before with regard to a solution; but it is singular that Pell's ingenuity should not have suggested another method of solution in the case which he gives where $$\alpha = 7$$, $$\beta = 7$$, $$\gamma = 11$$. In this case we have



\begin {align} a^2 + bc = b^2 & +ac \\ a^2 -b^2 = ac & -bc = c (a-b) \\ \text{or, } & a  +  b = c \end{align} $$

It is unnecessary to pursue this any further, for by substituting this value of $$c$$ in $$(3)$$ and $$(2)$$, and adding the two equations together, we obtain $$2 (a+b)^{2} = 18$$, or $$c = 3$$. The values of $$a$$ and $$b$$ are $$1$$ and $$2$$ respectively, and this is, perhaps, the simplest case which could be selected.

To return to the contents of our volume. The notes of inventions of Ralph Rabbards at p. 7, may be noticed as somewhat similar to the far-famed "Century of Inventions" of the Marquis of Worcester. The number of such proposals is great, and several seem to include discoveries generally considered as belonging to a more modern period. The letter of Tycho Brahe, at p. 32, may be mentioned as a curious notice of the intercourse between the mathematicians of this and foreign countries. The letters of Thomas Lydyat are more valuable for