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This equation, supposing W and v variable, is an equation to a parabola, the construction of which will serve to represent this law more clearly to the imagination.
A formula for expressing the law of animal action was first proposed by EULER, in a Dissertation on the Force of Oars, Mém. Acad. de Berlin, 1747. That which he employed was W = P(1), different from both those we have mentioned, but a function of the first, and one that becomes 0 when v = c. EULER, however, changed this to another, Mém. Acad. de Berlin, 1752, and Nov. Com. Petrop, viii. p. 244, the same that we have adopted. He appears to have done so merely on account of the analogy thus preserved between the action of animals and of fluids. The physical fact mentioned above, is a better reason for the preference.
'178. The effect of animal force then, or the quantity of work done in a given time, will be proportional to W v, or to P v (1 — 2), and
will be a maximum when v= and W = that is, when the ani
3' mal moves with one third of the speed with which it is able only to move itself, and is loaded with four-ninths of the greatest load it is able to put in motion.
The quantities P and c can only be determined by experience, and as they must differ for different individuals, an average estimation of them is all that can be obtained. Even that average is but imperfectly known; EULER supposes, that for the work of men P may be taken = 60lb. and c = 6 feet per second, or a little more than four miles an hour.
A man, according to this estimate, when working to the greatest advantage, should carry a load of 271b. and walk at the rate of two feet in a second, or a mile and one-third an hour.
A horse, according to Desaguliers, drawing a weight out of a well, over a pulley, can raise 200lb. for eight hours together, at the rate of two miles and a quarter an hour. Supposing the horse in this case, to 200 × 9 = 450, and c = 2.25 × 4
work to the greatest advantage, P =
363 miles per hour.
This estimate seems to give too high a value to P. It will suit better with general experience to make P = 420, and c = 7.
'When a horse's work is estimated by the load he draws in a cart or waggon, a great reduction must be made, in order to compare the force he exerts with that which is necessary for raising a weight, by drawing it over a pulley. Though accurate experiments on the friction of wheel carriages are wanting, we probably shall not err much in supposing the friction on a road, and with a carriage of the ordinary construction, to amount to a twelfth part of the load. If then, a horse draws a ton in a cart, which a strong horse will continue to do for several hours together, we must suppose his action the same as if he raised up the twelfth part of a ton, (2240lb.), or 1861b. perpendicularly against the force of gravity. To raise a weight of 186lb. therefore, at
the rate of two miles, or two miles and an half an hour, (that is, 2.9 or 3.6 feet per second) may be taken as the average work of a strong draught horse in good condition.
A different view of the manner of estimating animal force, has been taken by COULOMB, Mém. de l'Instit. Nat. tom. 11. p. 380, &c.
179. It appears to be a certain fact that when a man carries only his own weight, the quantity of his action, that is, the height he is able to ascend in a given time, multiplied into his weight is greater than when he carries any additional load; and COULOMB thinks it probable that this diminution of action, is in proportion to the additional load carried. Now it appeared from his experiments, that when a man carried a load equal to his own weight, his action was reduced nearly one half; and, therefore supposing the reduction always proportional to the load, if w be the weight of the man's body, I an additional load which he is made to carry, H the height to which he ascends in a given time, when walking freely, and h the height to which he ascends in the same time, with the load 7; then his action in the latter case or (w + 1) ι
w H (1
h, is reduced to w H (1 — ~); and therefore also ✯ = w + l Suppose that a man is loaded with one-fourth of his own weight; w H (1) H (1 품) then h =H (.699).
w (1 + 4) 1 + 1
The value of H is deduced from the ascent of the Peak of Teneriffe. BORDA, accompanied by eight men on foot, ascended in the first day (7h 45m), to the height of 2923 metres, or 9593 feet. This was at the rate of 1225 feet in an hour. Had each of the men carried a load equal to the fourth part of his weight, they would only have ascended at the rate of 857 feet an hour.
= 0, or 1 = 2w, the height h= 0. With a load
• When 1 equal to twice a man's weight, he could not ascend.
180. The strength of a man being supposed to follow the law now laid down, its greatest effect in raising a weight will be when the weight of the man is to that of his load, as 1 to 1+ √ 3, or nearly as 4 to 3.
H w (1 "Because h = ; now I h, or the w+1 wt l weight multiplied into the height into which it is to be raised, is the measure of the effect, or of the work done, which, therefore, will be a maximum when the last formula is so, that is, when = w(− + √ 3.)
If in the equation h ==
we suppose h and I to be variable, the other quantities being constant, the locus of the equation is a hyperbola, which may be easily constructed.
The theorems just given only differ from COULOMB'S, by being somewhat simpler, and free from all reference to any particular mea
sure of length or of weight. On this subject, however, many more experiments are wanting.'
The preceding is an interesting quotation, to which we have only one objection to offer. It might be imagined from the language of the learned professor, that the formula WP (1 – 3) for the estimation of animal energy, had little besides its simplicity to recommend it; and that scarcely any thing was known from experience as to the safety with which "we may suppose the strength of animals to follow the law expressed by that formula." But the truth is, that it is extremely easy to ascertain by experiment the correctness of any assumed formula, and that the requisite experiments have long ago been made. M. Schulze has recorded in the Memoirs of the Berlin Academy of Sciences for 1783, a tolerably extensive series of experiments; from which he has shown, that the above formula is a sufficiently correct expression for human mechanical energy under the supposed circumstances, and that Euler's other theorem, WP(1 leads to extreme anoca), malies in many of its applications. Indeed, the results offered by the two theorems for the case of a maximum effect, are enough to determine the point: according to the first formula, we should have c, according to the latter vc, when the maximum occurs; and this last result is well known to be completely at variance with experiment. With respect to the action of horses, Mr. Playfair's predecessor, Professor Robison, made many experiments, and found that when drawing a lighter in a canal, and loaded so as not to be able to trot, that action varied nearly as (1 · — 2)1.7 or as (cv); which corresponds much better with the first than with the second of the theorems just given. So that there can be no occasion for the doubtful language employed by Professor Playfair on this subject.
After what we have said of the general merits of the work before us; we trust the author will not impute it to any unfriendly motive, if we devote the remainder of this article to the less grateful, but more useful task of suggesting alterations and improvements. And first, we will point to a few places where the professor may be inclined to supply omissions. We cannot but express some surprize that at page 189, there is no mention of Dr. Abram Robertson's theorems respecting rotatory motion, in the Philosophical Transactions for 1807; that at p. 160, Mr. Playfair should have forgotten Girard's valuable work on the resistance of solids; that at p. 270, he should neglect to mention Dr. T. Young's interesting inquiries respecting the motion of musical
cords, as well as omit the curious subject of temperament entirely; and that at p. 150, he should make no reference to Berard's Treatise on the Theory of Arches and Domes, though it is doubtless the best which has been published out of England. This is the more remarkable, as the professor has referred to Bossut, whose Essay on Arches exhibits three or four very gross blunders; and as he has noticed in terms of high commendation, Mr. Atwood's Dissertations on Arches, though they are well known to have been written after the mind of that excellent mathematician had been greatly impaired by long sickness, and to be so tedious, prolix, inelegant, and sometimes erroneous, that the friends to his reputation sincerely wish he had never published them. If it were not that an attention to the sublimer sciences is generally acknowledged to check the indulgence of prejudice and partiality, we should really be apt to suspect that Professor Playfair had more than once felt their influence.
But we proceed to a few omissions of another kind. Thus, at p. 9, the author has neglected to distinguish between adhesion and cohesion, though such distinction is perfectly conformable to the precision of modern philosophical writers. So again, p. 43, the student is not told where it is that a heavy body falls from quiescence 16 feet and 1 inch, in the first second of time; though it would be very wrong for him to conclude that the space would be the same in all places. At p. 61, where Mr. Playfair treats of the motion of the centre of gravity, he forgets to introduce the leading theorems of the centrobaric method,' as it is technically called; although it is one of the most useful as well as curious applications of the centre of gravity, and although the relation between that centre and the figure generated by the revolution of a line or plane, which is the foundation of the method, is distinctly stated by Pappus in his Mathematical Collections, a work to which our author has referred at the page just specified. At p. 125, art. 203, it should, we think, have been shown that when ar, or the cone is right angled, the centre of oscillation is in the centre of the base; and that in oblique angled ones the centre of oscillation falls entirely below the solid; this would, at least, have led Mr. Playfair to correct the definition given in the preceding page, where he says of a compound pendulum, that the centre of oscillation is a point in it'; which very frequently is not the case. Again, at p. 127, where it is affirmed that the vibrations of a cycloidal pendulum whether great or small are isochronal; it ought to have been added, that this is merely true on the supposition that the whole mass of the pendulum is concentrated into a point; and that cycloids, when used to regulate the motions of pendulums, produce errors of another kind much greater than those which they
are intended to obviate. This, we think, has been remarked both by Atwood and Gregory. So again, at p. 148, when treating on the theory of arches, the author acknowledges that what he has advanced, rests on a defective hypothesis; it is therefore extraordinary, that he did not introduce at least one other hypothesis, and exhibit the equation of equilibrium between the arch and the piers, supposing the latter not susceptible of a motion of rotation, but one of translation; that is, not likely to turn, but to slide. Once more, when treating of the resistance of fluids, pp. 201, 202, Mr. Playfair speaks both of the experiments of Bossut and those of Mr. Vince; yet he does not seem to have instituted any comparison of their results, as M. Lacroix has done in the Bulletin de la Société Philomathique. Such a comparison developes some singular irregularities in those results, which Dr. T. Young has endeavoured to explain, (Journal Royal Institution, vol. ii. and more fully Nat. Phil. ii. 229.), and which should not have escaped the professor's notice in this place.
We are aware that the various particulars which we regard as omissions, may be brought forward in their proper order and connection in the lecture room; and therefore, that the specifying them here may be represented as a kind of hyper-criticism, But we wish it to be recollected that the book will be seen by many who never belonged to Professor Playfair's class, that to such the information, of which we here regret the omission, is in most cases essential, and that the Synopses of Atwood, Vince, and Young, though not half the size of Mr. Playfair's, are not chargeable with a fourth of the number of similar defects.
We shall next glance at a few points which we consider as at least doubtful. Such, for example, is our author's definition of a hypothesis. A fact (says he) assumed in order to explain appearances, and having no other evidence of its reality, but the explanation it is supposed to afford, is called a hypothesis.' The professor has probably a right notion of what he here intends to define; but we suspect that those who know no more of an hypothesis than can be learnt from this definition, will be far to seek. It may serve to designate the Ptolemaic or Tychonic hypothesis, but will not be very appropriate if applied to the Newtonian hypothesis in astronomy.
Bodies differ in their capacity for receiving and maintaining different figures.
Some receive new figures with difficulty, but maintain them easily. Such are the bodies usually called solid.
'Others receive any figure easily, but cannot maintain it without the assistance of other bodies. Fluid bodies are of this kind.'
These are more like enigmatical than philosophical representa