1890-91.] Dr Sang on the Extension of Brounckers Method. 341 
On the Extension of Brouncker’s Method to the Com- 
parison of several Magnitudes. By Edward Sang, 
LL.D. 
(Read December 15, 1890.) 
In the paper on this subject, printed in the twenty-sixth volume 
of the Society’s Transactions , the general principles only of this 
extension were explained; since then the subject has lain aside. 
An accident has drawn attention to the use of this method for 
determining cube-roots, and particularly in regard to the duplication 
of the cube. The consideration of this matter has led to the 
observation of certain relations which deserve to be recorded. 
Here we have to compare three quantities which are in continued 
geometrical progression. From the greatest of these we deduct 
multiples of the others to obtain a fourth magnitude or remainder 
less than the least of the preceding ; leaving off now the greatest, 
we treat the remaining three quantities in the same way, and so 
proceed until the remainder become insignificant or, in the case of 
commensurables, become zero. 
In the former explanation the three quantities were arranged in 
the order of decreasing magnitude, and the second was taken as 
often as possible from the first before the subtraction of the third 
was attempted, and the three quantities in hand remained arranged in 
the order of decreasing magnitude. 
For the sake of illustration we may take the case of the cube- 
root of 2, in which case we have the three quantities ^4, ^/2 and 
1, to be compared. Expressing them numerically, we have 
1.5874011 =A, 
1.2599210 = B, 
1.0000000 = 0. 
The second of these deducted once from the first leaves a remainder 
less thai| the third, therefore we write A=1.B + 0.C + D, leaving 
D = . 3274801. Treating B, Cj D in the same way we find 
VOL. XVIII. 13/1/92 2 B 
