162 
Proceedings of the Royal Society 
the Neanderthal skull, although below the European mean in its 
internal capacity, yet exceeded the dimensions of some normal 
modern European crania which had been carefully measured — its 
large transverse parietal diameter compensating for the brain space 
lost by the retreating forehead and flattened occiput- 
As the history and geological age of the Neanderthal skull were 
both unknown, and as many of its most striking anatomical 
characters were closely paralleled in some modern European 
crania, the Author considered that great caution ought to be 
exercised in coming to any conclusion, either as to the pithecoid 
affinities or psychical endowments of the man to whom it originally 
appertained. 
4. Notice of a Simple Method of Approximating to the 
Boots of any Algebraic Equation. By Edward Sang, Esq. 
M. Lagrange, on applying the method of continued fractions to 
the resolution of numerical equations, discovered that, for those of 
the second degree, the quotients recur periodically. Erom this, 
combined with the previously well known fact that all periodic 
chain fractions belong to quadratics, he inferred that periodicity is 
exclusively confined to equations of this order. 
In January 1858, I showed to the Eoyal Society that the series 
of approximating fractions obtained by M. Lagrange can be con- 
tinued in the opposite direction, and that the convergence then is 
to the other root ; and enunciated the general theorem, that if any 
two fractions be assumed, and if a progression be formed from 
them by combining fixed multiples of their members, this pro- 
gression, which I called duserr or two-headed, may be continued 
in either way, and gives on the one hand the one, on the other 
hand the other root of a quadratic. 
This would seem to confirm Lagrange’s view of the limited 
application of periodicity. 
However, we may observe that our attention has been restricted 
to one kind of periodicity; there may be recurrences of higher 
orders which may belong to equations of higher degrees. Thus if, 
instead of beginning with two fractions, we had assumed three, and 
formed the progression by combining specified multiples of the last, 
