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that is, from 450 to 750 pounds; but a horse can draw 2000 pounds up a steep hill which is but short. The most disadvantageous mode of applying a horse's force is to make him carry or draw up hill; for if it be steep, he is not more than equal to three men, each of whom would climb up faster with a burden of 100 pounds weight than a horse that is loaded with 300 pounds. And this arises from the different construction of what may be called the two living machines.
1348. Desaguliers observes, that the best and most effectual action of a man is that exerted in rowing, in which he not only acts with more muscles at once for overcoming resistance than in any other application of his strength, but that, as he pulls backwards, his body assists by way of lever.
1349. There are cases in which the architect has to avail himself of the use of horse power; as, for instance, in pugmills for tempering mortar, and occasionally when the stones employed in a building may be more conveniently raised by such means. We therefore think it proper to observe, that, for effectually using the strength of the animal, the track or diameter of a walk for a horse should not be less than 25 to 30 feet.
1350. We close this section by observing, more for the curiosity of the thing than for the service it will be to the architect, that some horses have carried 650 or 700 lbs., and that for seven or eight miles, without resting, as their ordinary work; and, according to Desaguliers (Experiment. Philos. vol. i.), a horse at Stourbridge carried 11 cwt. of iron, or 1232 lbs., for eight miles.
1351. The construction of arches may be considered in a threefold respect. I. As respects their form. II. As respects the mode in which their parts are constructed. III. As respects the thrust they exert. 1352. The first category involves rather the mode of tracing the right lines and curves whereof their surfaces are composed, and has been partially treated of in Section VI. on Descriptive Geometry, and will be further shortly discussed in future pages of this work. The 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 not appear to have at all entered into the thoughts of the ancient architects. Experience, imitation, and a sort of mechanical intuition seem to have been their guides. They appear to have preferred positive solidity to nice balance, and the examples they have left are rather the result of art than of science. Vitruvius, who speaks of all the ingredients necessary to form a perfect architect, does not allude to the assistance which may be afforded in the construction of edifices by a knowledge of the resolution of forces, nor of the aid that may be derived from the study of such a science as Descriptive Geometry, though of the latter it seems scarcely possible the ancients could have been ignorant, seeing how much it must have been (practically, at least) employed in the construction of such vast buildings as the Coliseum, and other similarly curved structures, as respects their plan. 1354. The Gothic architects seem, and indeed must have been, guided by some rules which enabled them to counterpoise the thrusts of the main arches of their cathedrals with such extraordinary dexterity as to excite our amazement at their boldness. But they have left us no precepts nor clue to ascertain by what means they reached such heights of skill as their works exhibit. We shall hereafter offer our conjectures on the leading principle which seems as well to have guided them in their works as the ancients 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 seventeenth century, the first mathematicians who considered an arch as an assemblage of wedge-formed stones, capable of sliding down each other's surfaces, which they considered in a state of the highest polish. In this hypothesis M. de la Hire has proved, in his Treatise on Mechanics, printed in 1695, that in order that a semicircular arch, whose joints tend to the centre, may be able to stand, the weights of the voussoirs or arch stones whereof it is composed must be to each other as the differences of the tangents of the angles which form each voussoir; but as these tangents increase in a very great ratio, it follows that those which form the springings must be infinitely heavy, in order to resist the effects of the superior voussoirs. Now, according to this hypothesis, not only would the construction of a semicircular arch be an impossibility, but also all those which are greater or less than a semicircle, whose centre is level with or in a line parallel with the tops of the piers; so that those only would be practicable whose centres were formed by curves forming angles with the piers, such as the parabola, the hyperbola, and the catenary. And we may here remark, that in parabolic and hyperbolic arches, the voussoir forming the keystones should be heavier on
greater in height, and that from them the weight or size of the keystones should diminish from the keystone to the springing; the catenary being the only curve to which an horizontal 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. 1356. M. Danisy, a member of the Academy of Montpellier, liking neither of these hypotheses, endeavoured from experiments to deduce a theory. He made several models whose extradosses were equal in thickness, and divided into equal voussoirs, with piers sufficiently 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. 1857. Derand had thereupon found one which appears in his Architecture of Arches, 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 Treatise on Bridges, 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 applicate 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 Woutes en Dome, wherein he 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 formulae 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.
obseruvATIONs on FR1CTION.
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 heights at which they act, it follows that a parallelopiped will slide whenever the force which is necessary to overturn it is greater than
that necessary to make it slide, and, reciprocally, it will be overturned when less force is necessary to produce that effect than to make it slide. 1866. II. When the parallelopiped is placed on an inclined plane, it will slide so long as the vertical QS drawn from its centre of gravity does not fall without the base A.A. Hence, to ascertain whether a parallelopiped ABCD with a rectangular base (fig. 564.) will slide down or overturn; from the point B we must raise the perpendicular BE: if it pass out of the centre of gravity, it will slide; if on the contrary, the line BE passes within, it will overturn. 1367. If the surfaces of stones were infinitely smooth, as they are supposed to be in the application of the principles of mechanics, they would begin to slide the moment the plane upon which they are placed ceases to be perfectly horizontal; but as their surfaces are full of little inequalities which catch one another in their positions, Rondelet found, by repeated experiments, that even those whose surfaces are wrought in the best manner do not begin to slide upon the best worked planes of similar stone to the solids until such planes are inclined at angles varying from 28 to 36 degrees. This difficulty of moving one stone upon another increases as the roughness of their surfaces, and, till a certain point, as their weight: for it is manifest, 1st, That the rougher their surfaces, the greater are the inequalities which catch one another. 2d. That the greater their weight, the greater is the effort necessary to disengage them; but as these inequalities are susceptible of being broken up or bruised, the maximum of force wanting to overcome the friction must be equal to that which produces this effect, whatever the weight of the stone. 3d. That this proportion is rather as the hardness than the weight of the stone. 1368. In experiments on the sliding of hard stones of different sizes which weighed from 2 to 60 lbs., our author found that the friction which was more than half the weight for the smaller was reduced to a third for the larger. He remarked that after each experiment made with the larger stones a sort of dust was disengaged by the friction. In soft stones this dust facilitated the sliding. 1369. These circumstances, which would have considerable influence on stones of a great weight, were of little importance in the experiments which will be cited, the object being to verify upon hard stones, whose mass was small, the result of operations which the theory was expected to confirm. By many experiments very carefully made upon hard freestone well wrought and squared, it was found, 1st, That they did not begin to slide upon a plane of the same material equally well wrought until it was inclined a littlemore than 30 degrees. 2d. That to drag upon such stone a parallelopiped of the same material, a little more than half its weight was required. Thus, to drag upon a level plane a parallelopiped 6 in long, 4 in. wide, and 2 in thick, weighing 4 lbs. 11 oz., (the measures and weights are French, as throughout"), it was necessary to employ a weight equal to 2 lbs. 7 oz. and 4 drs. 3d. That the size of the rubbing surface is of no consequence, since exactly the same force is necessary to move this parallelopiped upon a face of two in wide as upon one of 4. 1870. Taking then into consideration that by the principles of mechanics it is proved, that to raise a perfectly smooth body, or one which is round upon an homogeneous plane inclined at an angle of 30 degrees, a power must be employed parallel to the plane which acts with a force rather greater than half its weight, we may conclude that it requires as much force to drag a parallelopiped of freestone upon an horizontal plane of the same material as to cause the motion up an inclined plane of 30 degrees of a round or infinitely polished body. 1371. From these considerations in applying the principles of mechanics to arches composed of freestone well wrought, a plane inclined at 30 degrees might be considered as one upon which the voussoirs would be sustained, or, in other words, equivalent to an horizontal plane. 1372. We shall here submit another experiment, which tends to establish such an hypothesis. If a parallelopiped C (fig. 565.) of this stone be placed between two others, BD, RS, whose masses are each double, upon a plane of the same stone, the parallelopiped C is sustained by the friction alone of the vertical surfaces that touch it. This effect is a consequence of our hypothesis; for, the inequalities of the surfaces of bodies being stopped by one another, the parallelopiped C, before it can fall, must push aside the HAL. two others, BD, RS, by making them slide along the horizontal Fig. 565. plane of the same material, and for that purpose a force must be employed equal to double the weight sustained.
* The Paris pound=7561 Troy grains.
1373. If to this experiment the principles of mechanics be applied considering the plane of 30 degrees inclination as a horizontai plane, the vertical faces ED FR may be considered as inclined planes of 60 degrees. On this hypothesis it may be demonstrated by mechanics, that to sustain a body between two planes forming an angle of 60 degrees (fig. 566.), the resist- ance of each of these planes must be to half the weight sustained –5 as HD is to DG, as the radius is to the sine of 30 degrees, or Fig. 566. as 1 is to 2.
EQUILIBRIUM OF ARCHES.
1874. The resistance of each parallelopiped represented by the prism ABDE (fig. 565.) being equal to half their weight, it follows that the weight to be sustained by the two prisms should equal one quarter of the two parallelopipeds taken together, or the half of one, which is confirmed by the experiment. This agreement between theory and practice determined Rondelet to apply the hypothesis to models of vaults composed of voussoirs and wedges disunited, made of freestone, with the utmost exactness, the joints and surfaces nicely wrought, as the parallelopipeds in the preceding example. 1375. The first model was of a semicircular arch 9 inches diameter, comprised between two concentric semi-circumferences of circles 21 lines apart. It was divided into 9 equal voussoirs. This arch was 17 lines deep, and was carried on piers 2 inches and 7 lines thick. It was found, by gradually diminishing the piers, which were at first 2 inches and 10 lines thick, that the thickness first named was the least which could be assigned to resist the thrust of the voussoirs. 1376. The model in question is represented in fig. 567., whereon we have to observe, — 1st. That the first voussoir, I, being placed on a level joint, not only sustains itself, but is able to resist by friction an effort equal to one half of its weight. 2d. That the second voussoir, M, being upon a joint inclined 20 degrees, will also, through friction, sustain itself; and that, moreover, these two voussoirs would resist, previous to giving way on the joint AB, an horizontal effort equal to one half of their weight: 3d. That the third voussoir, N, standing on a joint inclined at 40 degrees, would slide if it were not retained by a power PN acting in an opposite direction. 4th. That taking, according to our hypothesis, an inclined plane of 30 degrees, whereon the stones would remain in equilibrium as an horizontal one, the inclined point of 40 degrees may be considered as an inclined plane of Fig. 567. 10 degrees, supposing the surfaces infinitely smooth. 5th. That the effort of the horizontal power which holds this voussoir in equilibrium upon its joints will be to its weight as the sine of 10 degrees is to its cosine, as we have, in the section on Mechanics, previously shown. (1255 et seq.) 1377. The model of the vault whereon we are speaking being but 9 inches, or 108 lines in diameter, by 21 lines for the depth of the voussoirs, that is, the width between the two concentric circumferences, its entire superficies will be 4257 square lines, which, divided by 9, gives for each voussoir 473 square lines. Then, letting the weight of each voussoir be expressed by its superficies, and calling P the horizontal power, we have
The fourth voussoir, being placed upon a bed inclined at 60 degrees, will be considered as standing on a plane inclined only at 30 degrees, which gives, calling Q the horizontal power which keeps it on its joint, — Q : 473::sin. 30° : cosin. 30°. Or, Q : 473::50000: 86603=273f,
1378. The half-keystones, being placed on a joint inclined 80 degrees, are to be considered as standing on an inclined plane of 50, the area of the half key which represents its weight being 236}. If we call R the horizontal power which sustains it on its joint, we shall have the proportion
R : 2363:: sin. 50 : cosin. 50;
1379. Wishing to ascertain if the sum of these horizontal efforts, which were necessary to keep on their joints the two voussoirs N, O, and the half-keystone, was capable of thrusting away the first voussoir upon its horizontal joint AB, the half arch was laid down upon a level plane of the same stone without piers, and it was proved that to make it give way an horizontal effort of more than 16 ounces was required, whilst only 10 were necessary to sustain the half-keystone and the two voussoirs N, O, The two halves of the arches united bore a weight of 5 lbs. 2 oz. before the first voussoirs gave way. 1880. To find the effect of each of these voussoirs when the arch is raised upon its piers, let fall from the centres of gravity N, O, S of these voussoirs the perpendiculars Nn, Oo, Ss, in order to obtain the arms of the levers of the powers P, Q, R, which keep them in their places, tending at the same time to overturn upon the fulcrum T the pier which carries the half arch, and we have their effort– Px Nn + Qx Oo + R x.Ss. The height of the pier being 195 lines, we have - Nn=244.94 Oo-256-26 and Ss =260.50, whence we have The effort P × Nn= 83.4 x 244.94, which gives 20427.996 Q x Oo =273-3 x 256-26 . . . . . . . . 70035-858 R. x.Ss =281 9 x 260:50 . . . . . . . . 73434-950
Total effort in respect of the fulcrum, 163898-804
1381. The pier resists this effort, 1st, by its weight or area multiplied by the arm of the
lever determined by the distance Tu from the fulcrum T to the perpendicular let fall from
the centre of gravity Gupon the base of the pier. 2d. By the weight of the half arch
multiplied by the arm of its lever VY determined by the vertical LY let fall from the
centre of gravity L, and which becomes in respect of the common fulcrum T=Tt or
VB-BY, in order to distinguish BY, which indicates the distance of the centre of gravity of the half arch (and which is supposed known because it may be found by the rules given in 1275. et seq.) from the width VB that the pier ought to have to resist the effort of the half arch sought. In order to find it, let P, the effort of the arch above found, be 163898-804.
Let the height of the pier - d.
1382. The area of the pier which represents its weight multiplied by the arm of the ar."
lever will be air x : ="#. That of the half arch multipied by its arm of lever will be
the second power; but as ra. 1: is not a perfect square, that is to say, it wants the
square of half the known quantity * which multiplies the second term; by adding this
square, which is £, to each side of the equation, we have re-'4' * na" first member by this means having become a perfect square whose root is a +#, we shall
aa’ - 2p–2bc bö b . - - equation, *-v * +:-5, in which x being only in the first member of the equation, its value is determined from the known quantities on the other side. Substituting,
then, the values of the known quantities, we have
which becomes, by transferring # to the other side of the
1383. A proof of the truth of the hypothesis in the preceding section is to be found in the method proposed by Bossut in his Treatise on Mechanics.
Let the voussoir N (fig. 568.) standing on an inclined plane be sustained by a power Q acting horizontally. From the