ANIMAL

Effect of centrifugal

force.

8. We might now proceed immediately to the appliSTRENGTH. cation of the preceding formule, but it may not be amiss, in the first place, to say a few words relative to the centrifugal force which takes place in the motion. In order to conceive this effect, it must be observed, that the centre of gravity describes a curve, which has for its radius of curvature the leg on which we advance in walking. Let us denote this radius of curvature by r, the velocity of the centre of gravity by v, and the centrifugal force by f. We have, therefore, by Dynamics,

ƒ = = (P + 1);

75

0.1776

2.8130

Table of ve

v = √(30·196 x 2·5) 8.6884 feet.
Whence we may draw the following conclusion, viz.
that a man who runs 9 feet per second, ceases entirely
to gravitate on his feet, which is conformable to what
Mr. Lambert has observed; that is to say, that a person
running with the above velocity, remains so long in the
air, that the feet act only as they push, as it were, the
earth behind them, and have little or no effect in sup-
porting the body.*

9. Let us now consider the case of the formula

11. It is now only necessary to determine, from a Expericourse of experiments, the numerical value of n. tal M. Prony states, that he has frequently observed that sults. a man without a load can walk on a plane nearly horizontal, without fatigue, 6,000 feet in 20 minutes, viz. about 5 feet per second; we have, therefore, in this case, λ = o, v′ 5; and the equation v′ = √ (n × (A — B)) 5 = √ n x 3.8856;

when the fatigue is the least; that is, when the effort
which a man employs in walking is sensibly the same becomes
as that which he is able to continue without walking.
This condition will give k = q, and the above equa-
tion becomes

v' = {n (A — B) } 13.

We may calculate, by means of the table (art. 7), the values of (A — B) for different inclinations of the plane, agreeably to the above hypothesis, and thus form another table, which shall exhibit the value of n; for when AB is known, we shall have immediately

n = A-B'

It is to be observed, however, that the expression AB has only been considered, at present, as it applies to the case of a man ascending; when he descends, sin A becomes negative, and B changes sign. Such a table, therefore, ought to exhibit two values of n, one for each of the above cases.

TABLE II.

10. For calculating the velocities, corresponding to locities cor- different inclinations of a road, when the effort of a man responding to different upon it differs not sensibly from that necessary for him to continue to the end without walking.

ascents.

whence

And n 1.2868, a number by which we must multiply those in the table, in order to have the velocity corresponding to any proposed angle of inclination.

Lambert has observed, that he employed, without making any uncommon effort, 13 seconds in ascending a ladder of 24 steps, to the height of 13 Rhenish feet, and the angle of its slope 37°. This gives us an hypothenuse 214 French feet, which, divided by 13, gives 1.64 feet.

12. Now supposing it had been proposed to deter- App mine this velocity from the preceding table, we should the t have assumed 37° as an arithmetical mean between 35° and 40°: our tabular number would therefore have. been 1.3183, which, multiplied by the above constant value of n = 1.2868, we should have found

v' 1·3183 × 1.2868 1·696 feet; which differs as little from the preceding experimental deduction as can be expected in cases of this kind. 13. It appears, therefore, that the value n = 1·7 is very nearly conformable to experiment, and that n = 2, He deVirgil was not ignorant of this fact; he says, when speaking as assumed by Lambert, is a little in excess. rives this value of n by supposing that a man, jumping vertically, with all the force he is capable of, without a load, can raise himself 2 feet; which is perhaps rather too much for the medium strength of men.

of a certain warrior,

"Illa vel intectæ segetis per summa volaret
Gramina, nec teneras cursu læsisset aristas
Vel mare per medium, fluctu suspensa tumenti
Ferret iter, celeres nec tingeret æquore plantas."

{

v = {n (A + B)}

2ng

+

or

2ng sin A 2(1+3 sin A) (1 + 3 sin2 A)

into fluxions, and equate it to zero, which will give
7- √45
sin' of 12° 44';
6

sin2 =

whence the angle answering to the greatest velocity is
12° 44', and the corresponding value of is found
to be 6 feet per second.
15. The second column of Table II. which appertains
to the velocity of ascent, presents neither a maximum
nor a minimum; we may, however, still arrive at either
by having regard to the time. Suppose we go from
one point to another in a right line, the second being
higher than the first by a given quantity H; and if
denote the angle of the inclination of the road, the
length of it will be

H

sin X

STRENGTH.

ascending a plane, of which the inclination was equal ANIMAL
to the angle formed by a tangent at that point of the
circumference where he is placed. It follows, therefore,
from what is stated above, that the most advantageous
position of that tangent will be, when it forms an angle
of 24° 6' with the horizon, in which case, the effective
weight employed in turning the wheel, will be
P sin 24° 6' 0.40833 P,

P denoting, as before, the weight of the man.

Let P+Q be the greatest effort of which a man is capable; the effort Q may be augmented by habit and exercise, and it is liable to diminution by inaction; it will also undergo some modification when either great loads are to be lifted, or great swiftness is required; but in all cases the application of the effort P+Q can be only instantaneous; that is, being the greatest the whether it be employed in lifting a great weight, or in man is capable of, it can be exerted only for a moment, running with a great velocity.

17. A man who is not loaded with any weight, whe- Time in ther he merely stand upright on his feet, or walk with- which the out using any effort, employs, at first, only the power P, strength is and in the first instants the force Q will remain to him exhausted. entire; but in either of the above cases this power Q will, after a time, be weakened, and will ultimately be extinguished; let the time from the beginning to the latter event taking place be called T. Let us suppose, now, that instead of the power P, the man from the first instant employed an effort P+K; there will then, in this case, remain to him a quantity of force Q-K,

Let t denote the time, and the velocity; then we which will also be extinguished after a certain time t,

shall have to find

or since H is constant,

and it is an important question to decide the ratio of
the times T and t, when the powers Q and K are given.
Lambert conceives that it is not far from the truth to
assume that these times are proportional to the residual.
forces, viz. by assuming that

'sinλ a maximum.

Substituting for its general value given in (art. 16), equation (3), we shall have

which gives t=

This time t, being that which has passed from the moment the man began to walk till he can, from fatigue,

mar. Which being put into fluxions, and reduced, gives walk no longer, it is evident, that at this instant, he

sin2 = This result shows that the inclination which corresponds to the minimum of time necessary to ascend a given quantity, varies with the relation between the force that we employ and the load we have to carry. Again, since sin A is necessarily less than 1, we shall have (P+ k) < 9 (P+ q) 3 (P + k)2, or 4 (Pk) 9 (P + q)2, or 2 (P+ k) <3 (P + q).

ANIMAL space s, passed over before the forces are all spent, a STRENGTH. maximum. We may hence also determine the mean resulting velocity of this effort, and the time which passes, before fatigue prevents the man from walking

Particular

any longer.

Call

the effort necessary for holding the arm straight.
Let us assume Eif: we have then f to represent
the angle AC g, and the weight Cg of the

man P; and we shall have

dg=Cf=Px tan o,

18. It will be observed, that the quantity n does not value of Q. enter into the formulæ of the preceding article, but, in order to apply them, it is necessary that we determine either by hypothesis, or experiment, the quantity Q, or which, substituted in the preceding equation, gives the greatest effort that a man can make beyond that necessary for supporting his own weight. M. Lambert supposes PQ, which changes the above formulæ into the following: P+k=

Velocity into the

2 B

3 A

·(P + q),

(a)

By this means, there is now remaining only the quantity T to be determined experimentally, and it is, of course, subject to certain variations according to the age and activity of a man, his natural strength, and the habit acquired by practice. We may, however, without any remarkable violation of probability, assume that, a man without a load can continue walking for twelve or fourteen hours in a day; from which assumption, the value of T becomes determined.

19. Every particular load q requires a particular velocity, in order that the man may pass over the greatest maximum. space possible before his strength is exhausted; for this we must have

mass a

Putting the second side of this equation into fluxions, regarding q as variable, and equating it to zero, the value of 9 will be determined; but those values of q which exceed Q, it is obvious must be rejected.

20. Let us examine now the case where a man pushes,
or draws; and suppose, at first, that the path he has to
go over is horizontal, as also the direction in which
his exertion is made. (Fig. 2, Plate V, Miscella-
neous) may represent the attitude of this man
for the moment that he supports himself on the

leg CDB. In this attitude there are two points of
support, the one at A, the other at K: the arm KE
being supposed extended horizontally. The efforts
made by a man in this case, are those necessary to
keep his arm straight, his body erect, and that due to
his motion: but the actual force which enters into our
consideration, is gravity, and particularly the weight of
his body.

Let the vertical Cg represent the weight of the man,
and draw the horizontal lines Cf, dg, and complete the
parallelogram Cfdg: then the weight C g may be re-
presented by the components Cd, Cf. Now the effort
which exercised horizontally in the direction EK, and
which we shall represent by Ei, and as we may suppose
.t applied at the point E of the lever EA, of which of the
axis of rotation is in A, we have the proportion
Ei Cf:: AC; AE;

AC
AE

f= Px tan p.

The component Cd representing the action on the point A, which is subject to no diminution, ought to be estimated by its constant value fort made by the man to hold himself erect on his feet; we have, therefore,

that is to say, it will be the same here as would be excited by a man carrying a load q.

We might now proceed to reduce these equations to the best practicable form for solution, as we have done in the preceding part of this article, and to consider the effort of men employed in drawing loads, &c. but we are fearful it would carry us beyond the limits that can be assigned to this article; we must, therefore, refer the reader who is desirous of pursuing the subject to a greater length to Prony's Architecture Hydraulique, where he will find many important analytical results.

ANIMA

4T BENG

Experimental researches respecting animal strength. 22. We have already had occasion to remark, that Exper purely analytical investigations are of little use in such mea cases as those we have just been examining, inde-i pendent of experimental results; we propose, there- streng fore, before we conclude this article, to give a detail of a few of the best conducted experiments that have been made with reference to this subject.

Desaguliers asserts, that a man can raise water, or any other weight, about 550 lbs. (or one hogshead, the weight of the vessel included), ten feet high in a minute; but this statement, although he says it will hold good for six hours, appears, from his own facts, to be too high, and is certainly such as could not be continued one day after another. Mr. Smeaton considers this work as the effort of haste or distress; and reports, that six good English labourers will be required to raise 21141 solid feet of sea water to the height of four feet in four hours; in which case, the men would

EL

MAL raise

ton's

ing on a horizontal path, with or without a load, the ANIMAL
same author concludes that the greatest quantity of STRENGTH.
action takes place when the men are unloaded; and it
is to that of men loaded with 190lbs. nearly as seven
to four. The weight which a man ought to carry to
produce the greatest useful effect, or that effect in
which the quantity of action relative to the carrying
his own weight is deducted from the total effect, is
165lbs.

There is a particular case, which obtains with re-
spect to burdens carried in towns, where the men,
after having carried their load, return home unloaded;
the weight they ought to carry in this case, according
to M. Coulumb, is about 200 lbs. Here the quantity
of useful action, compared with that of a man who
walks freely, and without a load, is nearly as one to
five, or, in other words, he employs to pure loss four-
fifths of his power. By causing a man to mount a
flight of steps freely and without a burden, his quantity
of action is at least double of what he affords in any
other way of employing his strength.

very little more than six cubic feet of fresh water NGTH. each to the height of ten feet in a minute. Now, the hogshead containing about 84 cubic feet, Smeaton's allowance of work proves less than Desaguliers' in about the ratio of 6 to 8. And his good English labourers, who can work at this rate, are estimated by him to be equal to a double set of common workmen; it appears, therefore, that with the probabilities of voluntary interruptions, and other incidents, a man's work, for several successive days, ought not to be valued at more than half a hogshead raised ten feet high in a minute. Smeaton likewise states that two ordinary horses will do the above work in 34 hours, which is at the rate of little more than two and a half hogsheads ten feet high in a minute; so that, if these statements can be depended upon, one horse will do the work of five men. According to Emerson's statement, a man of ordinary strength, turning a roller by the handle, can act for a whole day against a resistance equal to 30 lbs. weight; and if he works ten hours per day, he will raise a weight of 30lbs. through 3 feet in a second of time; or, if the weight be greater, he will raise it to a proportional less height; so that, under all circumstances, 30 x 3 = 105, the momentum of his effort. If two men work at a windlass, or roller, they can more easily raise up 70lbs. than one man can 30 lbs., provided the elbow of one of the handles be at right angles to that of the other. Men accustomed to bear loads, such as porters, will carry from 150 lbs. to 200 lbs. or 250 lbs., according to their strength. A man cannot well draw. more than 70lbs. or 80lbs. horizontally; and he cannot thrust with a greater force acting horizontally at the height of his shoulders than 27 lbs. or 30 lbs. But one of the most advantageous ways in which a man can exert his force is to set and pull towards him, as in rowing.

Coulumb, so well known as an accurate experimental philosopher, in a memoir communicated to the French Institute, states that the quantity of action which a man can produce, when during a day he is employed in mounting a flight of steps without a burden, is double that which the same man could produce tf loaded with a weight of 223 lbs., continuing his exertions, in both cases, through the day. Hence it appears how much, with equal fatigue and time, the total or absolute effect may obtain different values, by varying the combination of effort and velocity. This fact is immediately applicable to the formulæ investigated in the preceding part of this article.

It will of course be observed by the reader that the term effect here denotes the total quantity of labour necessary to raise not only the burden but the man himself; the useful effect is very different, and it is this, as M. Coulumb observes, which it is most important to determine. For instance, we have seen that the total effect is the greatest when without a burden, but the useful effect is then nothing; it is also nothing when the man is so loaded as not to be able to move; and it is between these limits that the useful effect is a maximum; this we have already determined analytically in the foregoing part of this article, but the above results of Coulumb will be found to change somewhat the ultimate value; the principle, however, remains, and other experiments are perhaps still necessary to arrive at a satisfactory conclusion.

23. From an examination of the work of men walk

This seems to be understood by our coal merchants, who thus employ manual labour in emptying the coal vessels of their loads in the river Thames, where we frequently see four or five men perpetually ascending a step-ladder and jumping down, so as by their weight to bring up the coals from the hold by means of a rope passing over a pulley. Here the useless action is in ascending, and the useful in descending.

When labour is applied to cultivating the ground, the whole quantity afforded by one man, during a day, amounts to about the same as 328 lbs. raised 1094 yards; and M. Coulumb comparing this work with that of men employed to carry burdens up an ascent of steps, or at a pile-engine, finds a loss of about 'th part only of the quantity of action, which may be neglected in researches of this kind.

It may not be improper to observe, that in estimating mean results, we should not determine from experiments of short duration, nor should we make any deductions from the exertions of men of more than ordinary strength. The mean results have also a relation to climate. M. Coulumb observes, that he has directed extensive works at Martinico, where Fahrenheit's thermometer is seldom less than 77°, and similar works in France; and he affirms that not more than half the work can be done in similar cases in the one climate to what can be effected in the other.

Feats of strength, either natural or artificial.

24. We have already observed, that unusual strength Feats of is not to be considered in forming any mechanical de- strength. ductions relative to the employment of animal exertion as a first mover of machinery, but still any extraordinary power, whether natural or artificial, cannot but be considered as an interesting subject for philosophical reflection, and we must not, therefore, pass over certain surprising facts of this kind; but we shall confine our remarks principally to those recorded by Desaguliers, of Thomas Topham, a man, at the time he exhibited before the author, thirty-one vears of age, but who had practised the same feats for five or six years preceding that time. The exploits of this man, which Desaguliers witnessed, were as follow:

"1. By the strength of the fingers (only rubbed in