15836. Led by Le Brun (Théorie de l'Architecture, &c. fol. Paris, 1807.), we were many years ago induced to inquire into the doctrine of voids and solids in the Greek and Roman temples, and though we soon discovered that that author had committed manifest errors in his mode of applying his theory, there could be no doubt that if its principles were properly carried out, they would coincide with the best examples both ancient and modern. The study we have subsequently bestowed upon it has not, we regret, from various pressing occupations, received from us all the attention necessary to reduce the examples within such bounds as to make the matter subject to certain laws, though we think an approximation has been effected towards it. 1583c. It is to be lamented that, ainong the many and able writers on Gothic architecture, details, more than principles, seem to have occupied their minds. The origin of the pointed arch seems to have entirely absorbed the attention of a large proportion of them, whilst others have been mainly content with discussions on the peculiarities of style at the different periods, and watching with anxiety the periods of transition from one to another. Foliage, mouldings, and the like, have had charms for others; all, however, have neglected to bestow a thought upon the grand system of equilibrium by which such stupendous edifices were poised, and out of which system a key is to be extracted to the detail that enters into them. It is, however, to be hoped that abler bands than ours will henceforth be stimulated to the work, such being abundant in the profession whereof we place ourselves as the humblest of its members. Fig. 6102. 1583d. As on the horizontal projection or plan of a building, the ratio of the points of support have been above considered, so in the vertical projection or section of a building may the ratio of the solids to the voids be compared, as well as the ratio of the solids to the whole area. In fig. 6102, the shaded parts represent the solids, which therefore give boundaries of the voids. Worcester Cathedral is the example shown. In this mode of viewing a structure, as also in that of the points of support, there is a minimum to which art is confined, and in both cases for obvious reasons there are some dependent on the nature of the inaterials, and others on the laws of statics. Though there may be found some exceptions to the enunciation as a general rule, it may be safely assumed that in those buildings, as in the case of the points of support, wherein the ratios of the solids to the voids in section are the least, the art, not only as respects construction, but also in point of magnificence in effect, is most ad. vantageously displayed. In every edifice like a cathedral, the greater the space over which the eye can range, whether horizontally or vertically, the more imposing is its effect on the spectator, provided the solids be not so lessened as to induce a sensation of danger. 1583e. The subjoined table contains, with the exception of Notre Dame de Paris, the same buildings as those already cited. It will be seen that the ratios of the solids to the voids varies from .472 to 1•118, a little less than half to a little more than a whole. But if in their sections we compare the ratios of the solids to the whole area, there results a set of numbers varying from 321 to ·528, and that nearly following the order of the ratios of the points of support. Though the coincidence between the ratios of increase, in the points of support, does not run quite concurrently with the ratios of the solids and the areas in comparing the cathedrals of the different centuries, yet sufficient appears to show an intimate connection between them. Where the discrepancy occurs, the points of support seem inversely set out. Such, for instance, will be seen in Ely Cathedral, wherein, though the ratio of the solids to the voids in section is as high as 1 (or ratio of equality), that of the points of support is as low as 0·182, so that the space, or airiness, which is lost in the former, is compensated by the latter. Generally speaking, however, the points of support diminish as the orpament of the style increases. Thus, in Norwich Cathedral (the nave), of the early part of tbe twelfth century, the ratio of the points of support is 0.238, that of the solids to the ruids being 0*603 ; while at Salisbury (latter part of the thirteenth century) the ratio of the points of support is only 0.190, and that of the solids to the voids, 0.472. From the foregoing examination, there can scarcely exist a doubt that the first and leading lines of these fabrics were based upon a geometrical calculation of extremely simple nature, but most rigidly adhered to. Thus, taking a single bay in the nave, say, from centre to centre, and ascertaining the area, that has only to be multiplied by the ratio, to give the superficies necessary for the points of support, which, as the tables indicate, were diminished as experience taught they might be. These matters then being adjusted, and falling as they might, the system of ornamentation was applied altogether subsidiary to the great and paramount consideration of stability, 1583. A very ingenious writer and skilful architect (Mr. Sam. Ware), some years ago, took great trouble to deduce the stability of the buildings in question, from the general mass of the walls and vaulting containing within them some hidden catenarian curve. If such were the case, which can hardly be admitted, in as much as a chain for such purpose might be made to hang in all of them, it is quite certain this property was unknown to those who erected them. Dr. Hooke was the first who gave the hint that the figure of a Alexible cord, or chain, suspended from two points, was a proper form for an arch. PRESSURE OF Earth AGAINST Walls. 1584. It is not our intention to pursue this branch of the practice of walling to any extent, the determination of the thickness of walls in this predicament being more useful, perhaps, to the engineer than to the architect. We shall therefore be contented with but a concise mention of it. Rondelet has (with, as we consider, great judgment) adopted the theory of Belidor, in his Science des Ingenieurs, and we shall follow him. Without the slightest disrespect to later authors, we know from our own practice that walls of revêtement may be built, with security, of much less thickness than either the theories of Belidor, or, latterly, of modern writers require. We entirely leave out of the question the rules of Dr. Hutton in his Mathematics, as absurd and incomprehensible. The fact is, that in carrying up walls to sustain a bank of earth, nobody, in the present day, would dream of constructing them without carefully ramming down the earth, layer by layer, as the wall is carried up, so as to prevent the weight of the earth, in a triangular section, pressing upon the wall, which is the foundation of all the theory on the subject. With this qualification, therefore, we shall proceed ; premising, that if the caution whereof we speak be taken, the thickness resulting from the following investigations will be much more than the outside of enough. 1585. Earth left to itself takes a slope proportionate to its consistence; but for our purpose it will sufficiently exhibit the nature of the investigation, to consider the substance pressing against the wall as dry sand or pounded freestone, which will arrange itself in a slope of about 55° with the vertical plane, and therefore of 34, with an horizontal plane, as Rondelet found to be the case when experimenting on the above materials in a box, one of whose sides was removable. Ordinarily, 45° is taken as the mean slope into which earths recently thrown up will arrange themselves. 1586. Belidor, in order to form an estimate for the thrust or pressure into which we are inquiring, divides the triangle EDF (fig. 611.) representing the mass of earth which creates the thrust, by parallels to its base ED, forming slices or sections of equal thickness and similar form; whence it follows, that, taking the first triangle a Fb as unity, the second slice will be 3, the third 5, the fourth 7, and so on in a progression whose difference is 2. 1587. Each of these sections eing supposed to slide upon an inclined plane parallel to ED, so as to act upon the face FD, if we multiply them by the mean height at which they collectively act, the sum of the products will give the total effort tending to overturn the wall ; but as this sum is equal to the product of the whole triangle by the height determined by a line drawn from its centre of gravity parallel to the base, this last will be the method followed, as much less complicated than that which Belidor adopts, independent of some of that author's suppositions not being rigorously correct. 1588. The box in which the experiment was tried by Rondelet was 16) in. (French) long, 12 in. wide, and 174 in. high in the clear. As the slope which the pounded freestone took when unsupported in front formed an angle with the horizon of 34, the height AE is 11, so that the part acting against the front, or that side of the box where would be the wall, is represented by the triangle EDF. 1589. To find by calculation the value of the force, and the thickness which should be given to the opposed side, we must first find the area of the triangle EDF= 16x1 1} =93! ; but as the specific gravity (or equal mass) of the pounded stone is only if of that of the stone or other species of wall which is to resist the effort, it will be reduced to 73} =81. This mass being supposed to slide upon the plane ED, its effort to its weight will be as AE is to ED::111: 20, or 81 x ug =45.9, which must be considered as the oblique power qr passing through the centre of gravity of the mass, and acting at the extremity of the lever ik. To ascertain the length of the lever, upon whose length depends the thickness of the side which is unknown, we have the similar triangles qsr, qho, and kio, whose sides are proportional: whence qs : sr::qh : ho; and as ko=hk - hn, we have gr: qs::hk ho : ik. Whence, ik = (nk-ho)xqe, The three sides of the triangle qsr are known from the position of the angle q at the centre of gravity of the great triangle EFD, whence each of the sides of the small triangle is equal to one third of those of the larger one, to which it is correspondent. Fig. 611. 20 Thus, making the side gr=a, qs=b, The unknown side r8=C, sh=x, hk=f, nd ad ad ad a = ad pc n = Ву The pressure 45.9 found =P The height DF =d, We have b: c::b+r: bc+cx = ho, and hk – ho will be f-bctct. To obtain ik, we have the proportion a : 6::$_cact: ik. Whence ik= bf-bc-cr ; so that the pressure p x ik is represented by p (Of= -C*), to which the resistance expressed by dro must equilibrate. =p( 2pcr _2p(bf-bc). or r2+ For easier solution, make 2pbf-2pbc 2pc = 2m, and = 2n, and we have x2 + 2nx = 2m, an equation of the second degree, which makes x= V2m + 123 n, which is a general formula for problems of this sort. Returning to the values of the known quantities, in which f=73 b=51 p=45% c=3 d=113 75 - 31 m=pb x becomes m=45.9 x 55 x = 12.70 and 2m = 25.4 ; 6} + 114 - 2.28 and = 5.20. 75.55 From the above, then, the formula x= V 2m + n? - n becomes x= V25.4 + 5.20 – 2.28 = 3.22, a result which was confirmed by the experiment, inasmuch as a facing of the thickness of 34 inches was found necessary to resist the pressure of pounded freestone. Belidor's method, the thickness comes out 4 inches; but it has been observed that its application is not strictly correct. In the foregoing experiment, the triangular part only of the material in the box was filled with the pounded stone, the lower part being supposed of material which could not communicate pressure. But if the whole of the box had been filled with the same material, the requisite thickness would have been found to be 54 inches to bear the pressure. 1590. In applying the preceding formula to this case, we must first find the area of the trapezium BEDF (fig. 612.), which will be found 1951; multiplying this by is, to reduce the retaining wall and the material to the same specific gravity, we have 1694. This mass being supposed to slide upon the inclined plane ED, its effort parallel to that plane will be 195} * 95-76=p. Having found in the last formula that qs is represented by b=6-93, er by c=4.76, qr by a=8.40, f= 11:3, d:=17:5; the thickness of the retaining wall becomes =sh-x; m= m=pb x will become, substituting the values 11:3–4 76 95.76 x 6.93 pc 3.1, and n? 8.40 x 17:50 9:61. Substituting these values in the formula x= 12m +12 - n, we have x = N 59.04 + 9.61 -3.1=5.2, a result very confirmatory of the theory. 1591. In an experiment made on common dry earth, reduced to a powder, which took a slope of 46° 50', its specific gravity being only of that of the retaining side, it was found that the thickness necessary was 3 inches o 1592. It is common, in practice, to strengthen walls for the retention of earth with piers at certain intervals. which are called counterforts, by which the wall acquires additional Fig. 612. ad 95.76 x 76 ad strength ; but after what we have said in the beginning of this article, on the dependence that is to be placed rather on well ramming down each layer of earth at the back of the wall, supposing it to be of ordinary thickness, we do not think it necessary to enter upon any calculation relative to their employment. It is clear their use tends to diminish the requisite thickness of the wall, and we would rather recommend the student to apply him. self to the knowledge of what has been done, than to trust to calculation for stability, though we think the theory ought to be known by him. PRESSURE OR FORCE OF WIND AGAINST WALLS, &c. 1592a. Air rushes into a void with the velocity a heavy body would acquire by falling in a homogeneous atmosphere. Air is 840 times lighter than water. The atmosphere supports water at 33 ft. ; homogeneous atmosphere, therefore, is 33 x 840=27,720 it. A heavy body falling one foot acquires a velocity of eight feet per second. Velocities are as the square roots of their heights. Therefore to find the velocity corresponding to any given height, expressed in feet per second, multiply the square root of the height in feet by 8. For air we have V= 27,720 - 166,493 x 8 = 1332 feet per second : this, therefore, is the velocity with which common air would rush into a void: or 79,920 feet per minute; some say 80,880 feet. (Telford's Memorandum Book). Some authors say that the weight or pressure of the atmosphere is equal to the weight of a volume of water 34 ft. in height; or 14.7 lbs. per square inch at a mean temperature; for air and all (?) kinds of gases are rendered lighter by the application of heat, because the particles of the mass are repelled from each other, or rarefied, and occupy a greater space. 15926. The force with which air strikes against a moring surface, or with which the wind strikes against a quiescent surface, is nearly as the square of the velocity. If B be the angle of incidence; 8 the surface struck in square feet; and u the velocity of the wind in feet per second ; then if f equals the force in pounds avoirdupois, either of the two following approximations may be used, viz.,f= ve 3 sino ß; or, f= .002288v+8sin B. The first is the easiest in operation, requiring only two lines of short division, viz., by 40 and by 11. If the incidence be perpendicular, sino B= 1, and these become f=440 ·002288v 8. (Gregory). The force or pressure per square foot in lbs., is as the square of the velocity multiplied by 002288. 1592c. IMPULSE OF THE WIND ON A SQUARE Four. 1592d. The resistance of a sphere is stated not to exceed one-fourth of that of its greatest circle. Tredgold, Carpentry, and Iron, has minutely examined the effect of the above forces, and the principle of forming the necessary resistance to them in the construction of walls and roofs. See HURRICANES. Where the roofs of buildings, as in the country, are exposed to rude gusts and storms, it is necessary to increase the weight of the ridges, hips and Aasbings. 1592e. The utmost power of the wind in England is said to be 90 miles per hour, or 40 lbs. per square foot. Tredgold takes the force at 572 lbs. per square foot. Dr. Nichol, of the Glasgow Observatory, records 55 lbs. per square foot, or +382 lbs. per square inch, as the greatest pressure of wind ever observed in Britain (Rankine, Civil Eng. 538). During the extremely heavy gale of January 16, 1866, the pressure in London was recorded as 33 lbs. per square foot; at Liverpool it was 30-4 lbs. The velocity of the wind on the south coast of England, during January 11, when it uprooted old elm trees, averaged 65 miles an hour; later in the day it was 90 miles; the latter impetus is equal to the 40 lbs. per square foot, above mentioned. 1592f. Wind exercises a tendency to overthrow a building upon the external edge op. posite to the line of its advance, equivalent to the surface of the face receiving the impulsion multiplied by the force of the wind, and by a lever which on the average may be taken to be equal to half the height of the building. To secure the stability of the latter, its |