APPENDIX TO FERGUSON'S LECTURES ON MECHANICS, &c. THE object of the following Appendix to Ferguson's Lectures is to supply in some measure their defects and omissians, and at the same time to give a full account of the discoveries and improvements in Practical Mechanics, and other branches of Natural Philosophy, in a manner suited to the popular and unpretending character of the original work. In order to do this in a somewhat systematic manner, we shall follow as much as possible the order of the Lectures, and give such a form to the Supplementary Chapters, that they may be read with advantage without any particular reference to the corresponding subjects in the first volume. CHAPTER I. ON MECHANICAL AGENTS, OR THE FIRST MOVers of MACHINERY. As VOL. II. B These powers are— 1. The force of men and animals; 2. The force of water; 3. The force of wind; and 4. The elastic force of steam and heated air. To these might be added the force of springs, and the force of gravity; but as springs require the force of man to renew their state of tension, and as weights require to be elevated by the same means, they ought rather to be considered as intermediate or secondary agents for producing small mechanical effects during the intermissions of the first mover. SECT. I.-On the Strength of Men and Animals. The force of man was no doubt the earliest mechanical agent that was employed to produce any useful effect, and he formed not only the first mover, but the machine itself. The singular adaptation of the human frame to the purposes of active life renders the force of man one of the most valuable, and, at the same time, one of the most universally applicable first movers of machinery. It is therefore a matter of great importance to ascertain the manner in which his exertions will yield the greatest quantity of useful work with the least quantity of bodily fatigue, or with a degree of fatigue compatible with the preservation of his bodily health. Excepting a few insulated, though useful observations, by Bernoulli, Smeaton, Desaguliers, Emerson, and other authors, the subject of the strength of men has been indebted almost wholly for its illustration to the labours of M. Coulomb. 1. The first object of this able philosopher was to ascertain the quantity of action which was exerted when ascending a stair either loaded or unloaded. He found that a man could ascend the stairs of a house at the rate of 16 yards in a minute, provided he did not ascend more than 20 or 30 yards. Hence he concluded, that if the average weight of a man is taken at 150 lbs. avoirdupois, the quantity of action which he thus furnishes will be 2560 lbs. avoirdupois, raised 1 yard in 1 minute; and if he is supposed capable of continuing this action 4 hours a-day, his daily action will be 614,400 lbs. avoirdupois, raised 1 yard in 1 hour. M. Coulomb likewise proved that a man unloaded ascended to the height of 164 yards by steps cut in the rock in 20 minutes, but none of the workmen would engage to continue this for 6 hours, or do it 18 times a-day, for the ordinary wages. By examining the rate at which M. Borda, with a party of sailors, ascended the Peak of Teneriffe, Coulomb concludes that a man in ascending with ease a convenient stair, will give daily a quantity of action equal to 480 lbs. avoirdupois, raised 1000 yards; or, since Coulomb considers this as decidedly too low, we may safely take the easily recollected expression of 500 lbs. avoirdupois, raised 1000 yards. In order to compare this quantity of action with that which a man can furnish loaded, Coulomb found that a man loaded with 150 lbs. of firewood could raise daily about 250 lbs. avoirdupois 1000 yards, and that a strong man could by great exertion raise about 300 lbs. avoirdupois 1000 yards. Hence, taking the first result, which is an average one, the quantity of action of a man when unloaded and loaded, is in the ratio of 500 to 250, or two to one nearly. When a man ascends a stair loaded, he raises his own weight along with the load, but the only useful effect which he produces is the elevation of the load. If the load were gradually increased, the total quantity of his daily action would diminish, and if it amounted to about 330 lbs. avoirdupois, he would scarcely be able to move it. If, on the contrary, he ascended without a load, the useful effect would be nothing, although his actual quantity of action was then a maximum. There must therefore be between these limits a certain load which will give a maximum effect. By following Coulomb's method, we shall find that the effect will be a maximum when the load is about 3-4ths, or 0.756 of the weight of the man, or about 113 lbs. avoirdupois; and that the useful effect of a man ascending a stair and properly loaded, is 137 lbs. avoirdupois raised 1000 yards. Consequently this mode of employing a man consumes nearly 3-4ths, or 0.756 of his real action; and he will cost five times more than a man who after having ascended the stair unloaded, raises a weight by allowing himself to descend by gravity through the height to which he ascended. If we suppose the man is so loaded as to perform no work, then the formule would give for the action 260 lbs. avoirdupois, which is not far from the weight which an ordinary man can lift or just carry.1 The next object of Coulomb's consideration is the quantity of action which a man can furnish when he walks on a horizontal road, either loaded or unloaded. When a man travels for several successive days unloaded, he can easily walk about 54,680 yards, or about 31 miles a-day, which gives for his quantity of action 7700 lbs. avoirdupois, carried 1094 yards. We likewise find that the average quantity of action furnished by porters carrying furniture or loads of any kind amounting to 130 lbs. is 4400 lbs. avoirdupois, carried 1094 yards. Hence the quantity of action lost is 3300 lbs. avoirdupois, carried 1094 yards. And if we suppose the losses of action to be proportional to the load, we shall have Hence the real quantity of action will be 7700 - 25.37 L; and making this equal to nothing, we obtain L = 304 lbs. as the greatest load which a man can carry, a result coinciding very nearly with that given in the preceding note. 1 The following very simple formulæ I have deduced from those given by Coulomb, by supposing that the number of kilogrammes which a man can raise through one kilometre daily, when ascending a stair unloaded, is thrice his own weight, and that the quantity of action lost by being loaded is 14 times the load. Putting L for the load, h the height to which the man ascends with that load daily, and W the (3 W — 13 L) L 3W-1L man's weight, then L h = and h which becomes a W+L maximum when L = 3 W-1 L W (3-1) = 0.732 W. W + L When L 2 W we have 0, which shews that when the load is equal to twice the weight of the man, or about 300 lbs. he loses the power of ascending. O and h = 2 By considering that the weight which a man can carry daily through the distance of 1 kilometre, or 1094 yards, is 50 times his own weight, supposing what is very nearly the case, that the loss of action when he is loaded with 130 lbs. is 25 times the load raised through the height above mentioned, I have deduced the following simple formula from those of Coulomb's, d being the distance through which he carries the load L. Ld= == (50 W25 L) L and d = 50 W25 L = which becomes a maximum when L = W (√ 3—1) 0.732 W. When The next case considered by Coulomb is that in which porters return unloaded to carry away a new load. His general result is, that a maximum effect will be produced when the porter carries a load equal to nine tenths of his own weight, and that his daily quantity of action will be 1533 lbs. carried 1094 yards. The load usually carried by porters under the circumstances of this case, is very nearly 135 lbs. or 0.9 W. 3 From the above results it follows, that the quantity of action furnished by a man walking unloaded, and a man walking under the circumstances of the present case, is as 505 to 100, or nearly as 5 to 1. Although the quantities of action of a man ascending a stair is not of the same kind with that of a man walking freely on a horizontal road, yet it is interesting to compare them together, as Coulomb has done. The quantities of action in these two cases are as 205 to 3500, or as 1 to 17. Now, if we take the height of a step at 6 inches, we shall have 17 × 6 = 102 inches, the length of horizontal road which a man can travel with the same degree of fatigue which he experiences in ascending a step of 6 inches, and if the pace of a man is 30 inches, it follows, that a man experiences the same degree of fatigue in ascending one step of 6 inches as in advancing about three paces and a half. L = W, then 50 W - 25 P = 0, and d = 0, which shews that when the load is equal to twice the weight of the man, or 300 lbs. he can no longer carry it. It is a singular circumstance that the ratio between L and W, when the effect is a maximum, should be so exactly the same in the two cases of a man carrying a burden on a level road and up a pair of stairs, and that this should take place by such a slight change on the numbers of Coulomb. In ascending a height, In walking along a horizontal plain, Mean, His results are L= 0.754 W L= 0.72 W 'L = 0.737 W, a mean which is nearly the same as the result L = 0.732 W, which we have just found for both cases. = 50 3 I have also simplified Coulomb's formulæ for this case by considering that Ꮃ Ꭰ = = 50 W; that D, the distance a man can travel unloaded, kilometres, and substituting 25. in place of 25.37, as before. Coulomb's for mula is L x = L (W D2 —b DL), the portion of action which is equal to the 2 WD † L (D — b)' useful effect which a man can furnish in a day. By the above substitutions we have L (2500 W -25 D2 L) which will be a maximum when Lx= L D (100 W + L) |