—P 
Fune 30, 1870] 
NATURE 
159 
The experiments give us eight distinct equations by 
which to determine the two unknown quantities / and g ; 
and by the method of least squares we determine their 
most probable values to be— 
B= TIS 7 
g—> 39 
The formula thus becomes— 
5 Sa RB ew 
w+ 39 
And calculating thence the distances for the severai 
weights, they are :— 
Weights ... 56 28 
Calcu- 
lated dis- 
tances 
Differ- 
ences 
from ex- 
periment 
Ewes ay Pe eice i) -d 
T'°93 3°63 6'46 10°61 14°65 tg°61 23°61 26°30 
+09 —"07—"40 4°05 +'04 +'°96 +°56 —°85 
The correspondence is so close as to show that the 
formula is in all probability the true one, and the quantity 
3°9 does not differ much from half the weight of the arm, 
which might be expected to enter into the question. The 
fact is that the correspondence is embarrassingly close, 
and I am inclined to attribute it partly to chance. The 
experiments could hardly have been expected to give re- 
sults accurate to an inch or two in some cases, and though 
the formula must be considered true on the ground of ex- 
periment, I do not quite see how to explain it on mechani- 
cal principles. 
If we regard the useful effect as the moving of the 
greatest amount of matter, it is + X w, and, theoretically 
speaking, increases continually with w. For the different 
weights and calculated distances, it is as follows :— 
Weight... 56 Be. FA, 7 4 
Useful 108'I I01°6 904 74°3 586 392 23°6 13° 
2 I 
4 
= 
effect 
But in reality it was not possible to raise the larger 
weights without exerting additional force unconsidered 
in the formula, so that the practical maximum of effici- 
ency is probably about 28lb., in the case of my own 
right arm. With different people it would, of course, 
vary somewhat. 
The above experiments completely confirmed Mr. 
Babbage’s remarks, but did not seem to lead to any 
further results. I proceeded, therefore, to other experi- 
ments upon the rate of exhaustion of muscular fibre. One 
mode of trial was to raise and lower various weights by a 
pulley and cord through the convenient range of the arm, 
continuing the motion with unrelaxed rapidity until the 
power of the muscles was entirely exhausted. The 
results of more than fifty experiments were as follows :— 
Weight lifted... 56 42 28 21 14 
Average 
number of 
times 
Useful effect ... 319 500 644 790 1554 
These numbers show that the total greatest amount of 
labour can be done with small rather than large weights 
in this case ; but they fail to give any regular law, owing 
probably to the weight of the body being brought into 
use with the larger weights. 
57 11g 23'0 37°6 III‘ 
weights in the hand at the full stretch of the arm, and to 
observe the times during which they could be supported. 
No two experiments were made with the same arm, with- 
out allowing, at least, one hour to elapse, so that the 
vigour of the muscles might be restored. With the 
smaller weights there was naturally some uncertainty as 
to the time, but in the case of the large ones the time was 
very definite. Altogether 238 experiments were made, an 
equal number with each arm. Uniting all the experi- 
ments for the same weight, the results are :— 
Weight TO) om lA, | eto 7 4 2 ri 
Times i 
sevonds |! 14°8 32°5 603 874 1479 2189 3212 
These results are pretty satisfactory averages ; thus the 
| probable error, for two cases indifferently chosen, was, for 
18lbs, in the left hand about *5, and for 4lbs. in the right 
2°7, and the error of the combined results would be less. 
With the exception of the results for rolbs. in the left 
arm, which appear to be somewhat in excess, these 
numbers are very regular, and point to a systematic law 
governing the rate of fatigue. The useful effect, or the 
product of the weight and time, shows a decided 
maximum, about 7lbs., as follows :— 
Weight 18 I4 10 7 4 2 I 
Useful effect 266 455 603 612 592 438 321 
If the weight held be very small, much power is lost in 
merely sustaining the arm ; if the weight is large, there is 
comparatively little loss on that account, but the power 
of the muscles is soon run out, and no sufficient oppor- 
tunity for restoration is allowed, The weights chosen for 
dumb-bells and other gymnastic exercises appear to be 
about those which give the maximum efficiency. 
I have made several attempts to explain these numbers by 
reasonable suppositions as to the conditions of exhaustion 
and restoration of muscular power. It seemed reasonable 
to suppose that the supply of new matter from the blood 
would increase in some proportion to the vacancy or want 
of it, but all such conditions led to integrals of a logarith- 
mic form, which could not be easily compared with experi- 
mental results. No formula that I obtained could be made 
to agree properly with the figures, and all that can be said 
is that the curve representing the results has a certain 
appearance of a logarithmic character, so far agreeing with 
the formulas obtained. Those who are acquainted with 
the physiology of the subject might succeed better ; I am 
not sure, for instance, how far the failure of strength is 
due to the exhaustion of the original substance of the 
muscle, how far to the inadequacy of the current supply 
of blood. It is a question again how far in any case of 
muscular action the supply is promoted by the increased 
action of the heart, or checked by the possible constriction 
of the arteries. If these questions have not been or cannot 
be otherwise decided, they might, perhaps, be indirectly 
solved by experiments of the kind described. 
My own object, however, was not to intrude into the 
domain of physiology, but to show that definiteness might 
possibly be given by degrees to some of the principles and 
laws which form the basis of the science of political 
economy. In some speculations upon the mathematical 
theory which must underlie that science (read at the 
The mode ultimately adopted was to hold out various | British Association in 1862, and published in the Journal 
