1333. When the friction was across the grain, or at right angles to the direction of the fbres, oak against oak was 376 The rat.os above given are constant quantities, and not dependent upon the velocities, excepting in the case of elm, when the pressures are very all, for then the friction is sensibly increased by the velocity. 1834. (II.) Friction is found to increase with the time of contact. It was ascertained that when wood moved upon wood in the direction of the fibres, the friction gradually increased, and reached its maximum in 8 or 10 seconds. When across the grain of the wood, it took a longer time to reach its maximum. 1935. (III.) For illustration of the friction of metals upon metals after a certain time of rest, the subjoined experiments were made with two flat rulers of iron, 4 feet long and 2 inches wide, attached to the fixed plank of the apparatus used for the investigation. Four other rulers, two of iron and two of brass, 15 inches long and 18 lines wide, were also used. The angles of each of the rulers were rounded off, and the rubbing surfaces of the rulers were 45 square inches. 1 With iron upon iron and a pressure of 53 lbs., the friction in parts of the pressure was S.5 453 lbs., 1 36 With iron upon brass and a pressure of 52 lbs., the friction in parts of the pressure was 4-2° 452 lbs., 1 1 1336. In these experiments each set gives nearly the same result, though the second pressures are nearly nine times the first; from which we learn that, in metals, friction is independent of the extent of the rubbing surfaces. Coulomb, moreover, found that the friction is independent of the velocities. The ratio of 4 to 1 between the pressure of friction, in the case of iron moving upon brass, is only to be considered accurate when the surfaces are new and very large. When they are very small the ratio varies from 4 to 1 to 6 to 1; but this last ratio is not reached unless the friction has been continued more than an hour, when the iron and brass have taken the highest polish whereof they are susceptible, free of all scratches. 1537. IV. In the friction of oak upon oak, when greased with tallow, which was renewed at every experiment, some days were required for obtaining, when the surfaces were considerable, the maximum of friction or adhesion. It was nearly similar to that without grease sometimes rather greater. For iron or copper with tallow, during rest, the increase is not 30 considerable as with oak. At first the friction was of the weight, besides a small force of a pound for every 30 square inches independent of the weight. The friction after some time changes to for. Olive oil alters the condition of the friction to }, and old soft grease to about 4. 1338. V. In the case of friction of bodies, oak upon oak for instance, in motion in the direction of its fibres, the friction was nearly constant in all degrees of velocity, though with Jarge surfaces it appeared to increase with the velocities; but when the touching surfaces were very small compared with the pressures, the friction diminished or the velocities intreased. For a pressure of 100 to 4000 pounds on a square foot, the friction is about 9-5' sides for each square foot a resistance of 13 pounds, exclusive of pressure increasing a httle with the velocity, occasioned perhaps by a down on the surface. If the surface be Very small the friction is lessened. When the narrow surface was cross-grained, the friction was invariably. In the case of oak on fir, the friction was; of fir on fir, }; of elm on but varying according to the extent of surface; for iron or copper on wood, †, which was at first doubled by increasing the velocity to a foot in a second, but on a continuance of the operation for some hours it again diminished. For iron on iron, 3.55 on copper, ; after long attrition, in all velocities. Upon the whole, in the case of most machines, of pressure may be considered a fair estimate of the friction. the 1 1 1 4:15 1339. In the experiments to ascertain the friction of axles, Coulomb used a simple pulley, here the friction of the axis and that of the rigidity of the rope produce a joint resistance. With guaiacum moving upon iron, the friction was 1 5.4 1 64 or of the weight in all velocities or, with a small weight, a little 1 11.5 exclusive of the rigidity of the rope; the mean was greater. In the cases of axles of iron on copper, or the velocity is small; the friction being always somewhat less than for plane surfaces. With grease, the friction was about With an axis of green oak or elm, and a pulley of guaiacum, the friction with tallow : without, with a pulley of elm, the quantities in question became and An s of box with a pulley of guaiacum gave and; with an elm pulley, and An as of iron and a pulley of guaiacum gave. with tallow, The velocity had but small A A effect on the rigidity of ropes, except in slightly increasing the resistance when the press was small. 1 54.3 1340. The friction and rigidity of ropes was supposed by Amontons and Desaguliers vary as the diameter as the curvature and as the tension. By Coulomb the power of diameter expressing the rigidity was found generally to be 17 or 1.8, never less than and that a constant quantity must be supposed as added to the weight. Wet ropes, if sm are more flexible than such as are dry, and tarred ones stiffer by about one sixth, and cold weather somewhat more. After rest, the stiffness of ropes increases. A rope of th strands, each having two yarns 12 lines in circumference, whose weight was 125 gra being bent upon an axis 4 inches in diameter, required a constant force of one pound (Fren and of the weight to overcome its rigidity. The same rope tarred, required one f of a pound and one fiftieth of the weight. When the strands were of fine yarns, the cumference 20 lines, and the weight 347 grains, the rigidity was equal to half a pound 1 of the weight to move it. With strands of 10 yarns, and a circumference of 28 li and a weight of 680 grains to 6 inches, the rigidity of the untarred rope was 2 lbs. of the weight, and the tarred rope of 3.3 lbs. and of the weight. Exp 13 33 ments which confirmed the above were made on a roller moving on a horizontal pla while a rope was coiled completely round it, whence an allowance must be made for friction of the roller on the plane, which varies as its weight and inversely as its diame With a roller of guaiacum or lignum vitæ, 3-6 inches in diameter, moving on oak, it was of the weight; for a roller of elm, 3 more. 23:1 1 10:34 1341. This subject has, we conceive, been pursued as far as is necessary for the archite seeing that his further investigation of it, should necessity arise, may be accomplished reference to the works of Amontons, Bulfinger, Parent, Euler, Bossut, and Coulo upon whom we have drawn for the information here given. We shall therefore clude these remarks by subjoining some of the practical results which experiments animal power afford, extracted from the celebrated Dr. Thomas Young's Natural Phil phy, vol. ii. 1342. In comparing the values of the force of moving powers, it is usual to assume unit, which is considered as the mean effect of the labour of an active man working to greatest advantage; this on a moderate calculation will be found sufficient to raise 10 to the height of 10 feet in one second for 10 hours in a day; or 100 lbs. 1 foot in a seco that is 36,000 feet in a day, or 3,600,000 lbs. 1 foot in a day. The following exhibi tabular view of the immediate force of men, without deduction for friction. Such a d work is the measuring unit in the third column of the table. A man can raise 60 French lbs. 1 French foot in A man can exert a force of 40 lbs. for a whole day For a short time, a man may exert a force of 80 lbs. with a fly when the motion is pretty quick. Desaguliers. 6.0 145 sec. A man going up stairs ascends 14 metres (35 43 feet) in 1 minute. Coulomb. 1.22 2-00 1343. Coulomb's maximum of effect is, when a man weighing 70 kilogramines 154-21 lbs. avoirdupois), carries a weight of 53 (116.76 lbs. avoirdupois,) up stairs. But this appears too great a load. 1344. Porters carry from 200 to 300 lbs., at the rate of 3 miles an hour. Chairmen walk 4 miles an hour with a load of 150 lbs. each; and in Turkey there are found porters who, it is said, by stooping forwards, carry from 700 to 900 lbs. very low on their backs. 1945. The most advantageous weight for a man of common strength to carry horizontally, is 111 pounds; or, if he return unladen, 135. With wheelbarrows, men will do half as much more work, as with hods. Coulomb. 1 The following table exhibits the performance of men by machines. A man can raise by a good common pump 1 hogshead 0.436 145 sec. 0.875 In pile driving, 55 French lbs. were raised 1 French A young man, the last-named author says, weighing 0.875 1346. In respect of the force of horses, we do not think it necessary to do more than observe that the best way of applying their force is in an horizontal direction, that in which a man acts least to advantage. For instance, a man weighing 140 lbs., and drawing a boat ang by means of a rope over his shoulders, cannot draw above 27 lbs. ; whereas a horse Laployed for the same purpose can exert seven times that force. 1847. Generally, a horse can draw no more up a steep hill than three men can carry, that is, from 450 to 750 pounds; but a horse can drav 2000 pounds up a steep hill w is but short. The most aisadvantageous mode of applying a horse's force is to m him carry or draw up hill; for if it be steep, he is not more than equal to three r each of whom would climb up faster with a burden of 100 pounds weight than a h that is loaded with 300 pounds. And this arises from the different construction of w may be called the two living machines. 1348. Desaguliers observes, that the best and most effectual action of a man is exerted in rowing, in which he not only acts with more muscles at once for overcom resistance than in any other application of his strength, but that, as he pulls backwa his body assists by way of lever. 1349. There are cases in which the architect has to avail himself of the use of ho power; as, for instance, in pugmills for tempering mortar, and occasionally when stones employed in a building may be more conveniently raised by such means. effectually using the strength of the animal, the track or diameter of a walk for a ho should not be less than 25 to 30 feet. A steam horse-power is reckoned as equal to th actual horses' power, and a living horse is equal to seven men. 1350. We close this section by observing that some horses have carried 650 or 700 1 and that for seven or eight miles, without resting, as their ordinary work; and, accord to Desaguliers (Experiment. Philos. vol i ), a horse at Stourbridge carried 11 cwt. of it or 1232 lbs., for eight miles. SECT. VIII. PIERS AND VAULTS. Authors on equilibrium of arches. 1351. The construction of arches may be considered in a threefold respect. I. respects their form. II. As respects the mode in which their parts are construct III. As respects the thrust they exert. 1352. In the first category is involved the mode of tracing the right lines and cur whereof their surfaces are composed, which has been partially treated in Section V. on 1 scriptive Geometry, and will be further discussed in future pages of this work. 1 other two points will form the subject of the present section. 1353. The investigation of the equilibrium of arches by the laws of statics does : appear to have at all entered into the thoughts of the ancient architects. Experien imitation, and a sort of mechanical intuition seem to have been their guides. They app to have preferred positive solidity to nice balance, and the examples they have left rather the result of art than of science. Vitruvius, who speaks of all the ingredie necessary to form a perfect architect, does not allude to the assistance which may afforded in the construction of edifices by a knowledge of the resolution of forces, nor the aid that may be derived from the study of such a science as Descriptive Geomet though of the latter it seems scarcely possible the ancients could have been ignorant, seei how much it must have been (practically, at least) employed in the construction of su vast buildings as the Coliseum, and other similarly curved structures, as respects their pl 1354. The Gothic architects seem, and indeed must have been, guided by some ru which enabled them to counterpoise the thrusts of the main arches of their cathedr with such extraordinary dexterity as to excite our amazement at their boldness. they have left us no precepts nor clue to ascertain by what means they reached si heights of skill as their works exhibit. We shall hereafter offer our conjectures on leading principle which seems as well to have guided them in their works as the ancie in their earliest, and perhaps latest, specimens of columnar architecture. 1355. Parent and De la Hire seem to have been, at the latter end of the seventee century, the first mathematicians who considered an arch as an assemblage of wedge-form stones, capable of sliding down each other's surfaces, which they considered in a state of highest polish. In this hypothesis M. de la Hire has proved, in his Treatise on Mechan printed in 1695, that in order that a semicircular arch, whose joints tend to the centre, n be able to stand, the weights of the voussoirs or arch stones whereof it is composed m be to each other as the differences of the tangents of the angles which form each vousso but as these tangents increase in a very great ratio, it follows that those which form springings must be infinitely heavy, in order to resist the effects of the superior vousso Now, according to this hypothesis, not only would the construction of a semicircular a be an impossibility, but also all those which are greater or less than a semicircle, wh centre is level with or in a line parallel with the tops of the piers; so that those only wo be practicable whose centres were formed by curves forming angles with the piers, such the parabola, the hyperbola, and the catenary. And we may here remark, that in p bolic and hyperbolic arches, the voussoir forming the keystones should be heavier greater in height, and that from it the weight or size of the voussoirs should diminish from the keystone to the springing; the catenary being the only curve to which an horirontal extrados, or upper side, can be properly horizontal. In the Memoirs of the Academy of Sciences, 1729, M. Couplet published a memoir on the thrusts of arches, wherein he adopts the hypothesis of polished voussoirs; but, finding the theory would not be applicable to the materials whereof arches are usually composed, he printed a second memoir in 1730, wherein the materials are so grained that they cannot slide. But in this last he was as far from the truth as in his first. 1956. M. Danisy, a member of the Academy of Montpellier, liking neither of these Lypotheses, endeavoured from experiments to deduce a theory. He made several models whose extradosses were equal in thickness, and divided into equal voussoirs, with piers sutficiently thick to resist the thrusts. To ascertain the places at which the failure would take place where the piers were too weak, he loaded them with different weights From many experiments, in 1732, he found a practical rule for the walls or piers of a cylindrical arch so as to resist the thrust. 1357. Derand had found one which appears in his L'Architecture des Voutes, 1643, but it seems to have been empirical. It was nevertheless adopted by Blondel and Deschalles, and afterwards by M. de la Rue. Gautier, in his Dissertation sur l'épaisseur ds Culées des Ponts, &c. 1727, adopts one which seems to have had no better foundation in science than Derand's. 1358. At the end of a theoretical and practical treatise on stereotomy by M. Frezier that author subjoined an appendix on the thrust of arches, which was an extract of what had theretofore been published by MM. de la Hire, Couplet, Bernouilli, and Danisy, with the applications of the rules to all sorts of arches. He seems to have been the first who considerably extended the view of the subject. 1359. Coulomb and Bossut occupied themselves on the subject. The first, in 1773, presented to the French Academy of Sciences a memoir on several architectural problems, amongst which is one on the equilibrium of arches. The last-mentioned author printed, in the Memoirs (1774 and 1776) of the same academy, two memoirs on the theory of cylindrical arches and of domed vaulting, wherein are some matters relating to the cupola of the Pantheon at Paris, whose stability was then a matter of doubt. 1360. In Italy, Lorgna of Verona considers the subject in his Saggi di Statica Mecanica applicati alle Arti; and in 1785, Mascheroni of Bergamo published, in relation to this branch of architecture, a work entitled Nuove Ricerche delle Volte, wherein he treats of cupolas on circular, polygonal, and elliptical bases. 1361. We ought, perhaps, not to omit a memoir by Bouguer in the Transactions of the French Academy of 1734, Sur les Lignes Courbes propres a former les Voutes en Dome, wherein be adduces an analogy between cylindrical and dome vaulting; the one being supposed to be formed by the movement of a catenarian curve parallel to itself, and the other by the revolution of the same curve about its axis. 1362. In this country, the equilibration of the arch, as given by Belidor and others on the Continent, seems to have prevailed, though little was done or known on the subject. Emerson seems to have been the earliest attracted to the subject, and in his Treatise on Mechanics, 1743, appears to have been the first who thought, after the Doctors Hooke and Gregory, of investigating the form of the extrados from the nature of the curve, in which he was followed by Hutton, who added nothing to the stock of knowledge; an accusation which the writer of this has no hesitation of laying at his own door, as having been the author of a Treatise on the Equilibrium of Arches, which has passed through two editions; but who, after much reflection, is now convinced, that, for the practical architect, no theory wherein the extrados is merely made to depend on the form of the intrados can ever be satisfactory or useful. It is on this account that in the following pages he has been induced to follow the doctrines of Rondelet, as much more satisfactory than any others with which he is acquainted. 1363. The formulæ of Rondelet were all verified by models, and the whole reasoning is conducted upon knowledge which is to be obtained by acquaintance with the mathematical and mechanical portions of the preceding pages. It moreover requires no deep acquaintance with the more abstruse learning requisite for following the subject as treated by later authors. OBSERVATIONS ON FRICTION. 1364. I. In order that the stone parallelopiped ABCD (fig. 563.) may be made to slide upon the horizontal plane FG, the power which draws or pushes it parallel to this plane, must not be higher than the ⚫ length of its base AB; for if it acts from a higher point, such as C, the parallelopiped will be overturned instead of sliding along it. 1365. As the effects of the powers P and M are in the inverse ratio of the neights at which they act, it follows that a parallelopiped will slide whenever the force which is necessary to overturn it is greater than |