82 
MR. DAVIES’S GEOMETRICAL INVESTIGATIONS 
The demonstration is obvious from the analysis. 
The problem admits of several other cases, but the same analysis and construc- 
tion, mutatis mutandis, serves for them all. 
Lemma VII. Theorem. — If a ratio be one of greater inequality, the triplicate of that 
ratio is greater than the ratio itself ; but if it be a ratio of less inequality, the triplicate 
ratio is less than the ratio itself. Also conversely, the subtriplicate of a ratio of greater 
inequality is less than the ratio itself ; but of a ratio of less inequality, the subtriplicate 
is greater than the ratio itself. 
This is too obvious to need a formal proof here. 
Lemma VIII. Problem. — If a line T U be divided in any undetermined ratio, viz. 
r, : r u in C and D, and O be the middle of CD; and, if the same line be divided in Q in 
the triplicate ratio of r t : r n ; it is required to find whether Q and O can ever coincide for 
any value of the ratio r t : r n . (Plate XI. fig. 6.) 
Suppose they can ; and let us first investigate the values of T O and T Q generally. 
Then if T U = 2 a, we have 
2 a r , 1 
D T = I 
Hence 
and 
DT + TC = DC= Aar i r n 
r 2 « 2 
'll 'l 
OC = vl) C = — ,r » 
and therefore 
Again, 
TO = CO — CT = 
T Q = prf 
T l\ ~ T l 
C Z a rf 
7 n ’ i 
And since we have admitted the hypothesis of the equality of TO, TQ, we have 
2 arf 2 a rf 
which reduces to 
And this again to the two equations r, 2 = 0 and rf — 0 ; which indicates that it 
takes place at T and U, or when the ratios are infinitely great and infinitely 
small. 
