738 
Proceedings of the Royal Society 
plane CBB ; and swing it round till B^ conies into the line BB. 
This being done, the proportionality wanted, 
that AjAg : B^B2 : : A^A^ : B^B^ 
will generally not have been instituted. But now, while maintain- 
ing all the conditions already attained to, move AA varying its 
distance from C, till that proportionality takes place as indicated by 
the mechanism of parallel rulers, &c., already used, but with some 
obviously necessary additions sufficiently suggested by the explana- 
tions already given. Now the whole system becomes clamped, and 
the problem is solved, or the rest is easy as in Method I., the two 
straight lines AA and BB, being lines fixed in the sought-for frame, 
and it being possible to find them for any prophesied future con- 
figuration of the set of three points A, B, and C, as has been ex- 
plained at the close of the explanations for Method I. 
It becomes now convenient and desirable to examine into some 
questions as to how many distinct elements of data from measure- 
ment, or how many ascertained conditions from measurement, are 
required for the solution of the problem ; and as to whether there 
are more essentially distinct solutions than one in various cases of 
the number of points used and the number of conditions from 
measurement ascertained. 
It is to be recollected, as was pointed out near the end of the 
explanation of Method I. that the lengths Aj^Ag, AgAg, AgA^, &c., 
and BjBg, B^Bg, BgB^, &c., found in the model as representatives of 
simultaneous travels of the real points A and B relative to the 
desired frame, must be mutually proportional. But the conditions 
of this mutual proportionality have been left unused in the solution, 
and so we may see that in Method I. we have used redundant data. 
It is to be understood that the introduction of any one complete 
templet brings in just two not three new conditions. Though for it 
there are three sides measured, yet the only conditions thereby 
ascertained are that when some one side has a certain stated length, 
a second side has another ascertained length, which is one condition, 
and that also at the same instant the third side has another ascer- 
tained length which is another condition ; and this makes with the 
previous one only two in all. 
In Method I. after the stage up to which the procedure is 
