Dividing an Angle in an Arbitrary Ratio. 75 
cardinal number occurs in it; and, further, we shall, by 
this investigation, obtain a complete solution to the problems 
of determining : ) 
(1) The result of adding any (finite or transfinite) number 
of any cardinal numbers ; 
(2) The result of multiplying any finite number of any 
cardinal numbers. 
There will only then remain the consideration of those 
cardinal numbers of the form 
qo 
where 0 is transtinite. Some results as to this class of 
numbers, together with the detailed investigation of Cantor’s 
‘““number-classes”? in general mentioned above, I will give 
in a continuation of this paper. 
Little Close, Yateley, Hants. 
December 2nd, 1903. 



VILL. Note on Borgnet’s Method of Dividing an Angle in an 
Arbitrary Ratio. By Prof. J: D. Evrerert, F.R.S.* 
HAVE recently come across an old paper (Borgnet, 
in Rouen Acad. Travaux, 1839, pp. 113-143) containing 
a beautiful theorem which seems to have fallen into oblivion. 
The paper is devoted to what the author calls barycentrides, a 
barycentride being defined as the locus of the centroid of an 
arc (of any curve) measured from a definite initial point. 
The theorem to which I refer solves, by means of the bary- 
centride of a circle, the general problem to divide a given 
angle in the ratio of any two given straight lines. . 
In fig. 1 let P be the centroid of the circular arc AQ, and 
let the curve AP be the barycentride of the circle AQ 
described about O. Bisect the chord OP at right angles by 
MH, meeting in H the perpendicular at O to the initial 
radius OA. Let @ denote any one of the three equal angles 
AOP, OHM, MHP;; then we have 
OP=20H sin @. 
But by the rule for the centroid of a circular are 

OPaOAn=! 
= a 
These equations give 
OH .§@=4 0A, 
a constant quantity. @ therefore varies inversely as OH. 
* Communicated by the Author. 
