1890 - 91 .] Dr Sang on the Extension of Broimcker s Method. 347 
In the same way we may proceed to compute four geometrical 
means between a line and its double thus : — 
1 5 
35 
235 
1580 
10626 71460 etc. 
1 6 
40 
270 
1815 
12206 82086 etc. 
1 7 
46 
310 
2085 
14021 94292 etc. 
1 8 
53 
356 
2395 
16106 108313 etc. 
1 9 
61 
409 
2751 
18501 124419 etc. 
and we have the six numbers 71460, 82086, 94292, 108313, 124419, 
142920 in 
. continued proportion to within one-tenth part of unit 
in any of them. 
The same process may be 
used for the roots of the number 3. 
Thus, if we write r for the rt 
th root of 3, and work out the successive 
powers of 
+ r *~2 
+ . . . 
-hr 1 -!-!) we shall find that the 
coefficients 
may 
be computed in a 
manner quite analogous to the 
preceding. 
Thus for 
n = 5. 
, that 
is, for ^3, the scheme is as 
under : — 
1 
5 
45 
365 
2965 
24141 196485 etc. 
1 
7 
55 
455 
3695 
30071 244767 etc. 
1 
9 
69 
565 
4605 
37461 304909 etc. 
1 
11 
87 
703 
5735 
46671 379831 etc. 
1 
13 
109 
877 
7141 
58141 473173 etc. 
Here the number placed at the top of a column is the sum of the 
numbers in the preceding column, and the following terms are 
got by adding thereto the double of the preceding number, thus 
365 + 2.45 = 455 ; 455 + 2.55 = 565, and so on, as is seen whenever 
we proceed to collect the surd elements of the successive powers of 
^/3 4 + ^3 3 + + ^3 + 1 ; and it is manifest that the subsequent 
or sixth line would be just the triple of the first one. The approxi- 
mation is, in this case, much slower than in the preceding. 
The same method is applicable to the roots of rational fractions. 
Thus we may take the case of or as we may write it ^(1 +§). 
The arrangement is thus : — 
15 
285 
5295 
98445 
etc. 
17 
315 
5865 
109035 
etc. 
19 
349 
6495 
120765 
etc. 
21 
387 
7193 
133755 
etc. 
23 
429 
7967 
148141 
etc. 
