FEBRUARY 26, 1914] 
entirely supports the validity of Darwin’s theory, in 
the locality selected as a test. 
Epwarp B. Poutton. 
Oxford, February 20. 
The Accuracy of the Principal Triangulation of the 
United Kingdom. 
THERE was some discussion of this question at the 
last meeting of the British Association, and an inves- 
tigation by Capt. H. St. G. L. Winterbotham has 
been published by the Ordnance Survey (Professional 
Papers, new series, No. 2). It appears, however, that 
_ the measurement of Lossiemouth Base, valuable as it 
_ was, did not definitely decide the question, but that 
i a moderate amount of further computation would 
— do so. 
(1) It will be generally admitted that statistical 
evidence as to the precision of any kind of observation 
is of no great value unless it is based on a large 
population. For example, if we may assume the prin- 
cipal triangulation to have been all executed by similar 
_ observers under similar conditions, and no unjustifiable 
rejections to have been made, we may accept with 
' great confidence the probable error of + 1-23” computed 
from the closure of 552 triangles. But nowadays one 
_ would not think of estimating the precision of a base- 
_ line by the discrepancy between two measurements of 
it, though one might reject the measurements if they 
- did not agree as well as was to be expected from the 
_ known usual probable error of a measurement. Thus 
_ discussion of a large population of discrepancies gives 
a good estimate of the probable accidental error of a 
measurement, but a small population only fixes a 
lower limit to that probable error. Now we have only 
a population of three independent discrepancies be- 
_ tween the four bases of Lough Foyle, Salisbury Plain, 
Lossiemouth, and Paris, and even if it is brought up 
to six by the inclusion of the three bases measured 
with steel chains, yet it must be considered a very 
small population upon which to base any estimate of 
precision. 
(2) But the original question, to what extent the 
strength of the figure compensates for the large prob- 
able error of an angle, is not necessarily a question for 
experiment, but it is essentially a question for com- 
Benn: Given the probable error of an angle in a 
lock of triangulation adjusted rigorously by least 
squares, if it is required to find the probable error of 
the distance from any point of it to any other point, 
the first step is to express the unknown error in that 
length as a linear function of the unknown errors in 
a number of independent angles (i.e. two angles of 
each triangle in a chain of independent triangles 
stretching from one point to the other, and in another 
chain stretching from one of the points to the nearest 
base). It then remains only to add another column 
to the least square computation by the method de- 
scribed in Wright and Hayford’s “Adjustment of 
Observations,” paragraph 123. May I venture to sug- 
gest that if two or three such cases could be worked 
out for strong figures in the United Kingdom, and 
for comparison two or three cases for weaker figures, 
either there or elsewhere, it would not only set at rest 
the immediate question, but would also. establish 
results of great importance for surveyors in general. 
T. L.. BENNETT. 
Computation Office, Egyptian Survey Department. 
One can but agree with Mr. Bennett in insisting 
on a large ‘‘population”’ of discrepancies upon which 
to found a calculation of a probable error, whether 
it be of the measurement of a base-line or of an angle. 
His example of a base-line measurement is not, how- 
NO. 2313, VOL. 92] 
NATURE 
73 
ever, strictly comparable with the investigation in 
question. In the former a large number of independent 
measurements must be made from which to deduce 
the most probable length and the individual dis- 
crepancies from this length. In the latter the measured 
bases may be regarded as errorless compared with the 
triangulation, and the actual errors can therefore be 
deduced with surety. 
Neglecting the steel chain bases, we have four in- 
dependent measures upon any one of which the tri- 
angulation can be made to depend. These four are 
widely distributed. 
The longest line from base to base is that from the 
new base at Lossiemouth to the Paris base, and along 
this line the old steel chain bases add additional proof 
that no serious errors are inherent in the triangula- 
tion. 
I agree that, ‘if we may assume the principal 
triangulation to have been all executed by similar 
observers under similar conditions and no unjustifiable 
rejections to have been made, we may accept with 
confidence the probable error of 1-23" computed from 
the closure of 552 triangles,” but it must be remem- 
bered that this is the probable error of an observed 
angle. The date of the work makes this a matter of 
no surprise. The point at issue, however, is not the 
probable error of an observed angle, but the probable 
error of an adjusted angle, or, in other words, to find 
out how far the intricacy of the figure has compen- 
sated for the lack of precision of angular measure- 
ment, 
To say that this question is essentially one for com- 
putation is not, to my mind, correct. 
The probable error of the ratio of any two sides 
as derived from the triangulation depends upon the 
probable error of an adjusted angle. This calculation 
is possible, but from the complexity of the figure and 
the intricate system of weighting the angles, imprac- 
ticable. Moreover, the answer to it would still be the 
probable, and not the actual, error. 
It is, however, possible to pick out of the general 
figure chains of simple triangles connecting the bases. 
Supposing that these chains had been the only paths 
of calculation, and that they had shown the same 
errors of ratio between the bases as are actually found, 
we can deduce what the probable error of an observed 
angle would have been to have effected this result. 
Such an investigation has been made, and the probable 
error of an observed angle in these “equivalent” 
simple chains is 0-85", or approximately the same as 
that given by General Ferrero, in his 1892 report, as 
the mean figure for the probable error of an observed 
angle in the triangulations of those twenty countries 
represented on the International Geodetic Association. 
Although, therefore, further investigation on the 
lines advocated by Mr. Bennett would be one of the 
greatest interest, I do not think that it promises a 
result commensurate with the time and expense it 
would entail. H. S. L. Wintersoruam. 
Atomic Models 
I am indebted to Mr. Chalmers for pointing out 
(Nature, February 19, p. 687), what I had indeed 
suspected, that the magnetic moment due to an elec- 
tron moving in a circular orbit, assuming the angular 
momentum to be h/2z, is exactly five times the mag- 
netic moment of the magneton. The original value 
(15:94 x 10-*") of the latter quantity, given by Weiss, 
and quoted in my former letter, was based on the 
value of Avogadro’s constant found by Perrin. If we 
divide the magnetic moment of the atom gram, 1123-5, 
by the more recent value for Avogadro’s constant given 
by Millikan (60-62 x 1022) we obtain as the magnetic 
| moment of the magneton 18:54 x 10-2, which is exactly 
