1485. f, representing the developement of one eighth part of the circular wall, will be 421, whence=42. d, the difference between the arm of the lever and the height of the pier, being 41, we shall have pd=738974.

1486. To obtain the value of me we must first find that of m, which represents the vertical effort of the upper part of the vault, and is equal to the cube of this part multiplied by KM and divided by the arc KG. This cube is equal to the difference of the cube of the sector of a sphere in which it is comprised with that which forms its interior capacity. We will merely recall here to the reader's recollection from a previous page, that the cube of the sector of a sphere is equal to the product of the superficies of the sphere whereof it forms a part by one third of the radius, and that this superficies is equal to the product of the circumference of a great circle by the line which measures its depth. Thus the area of the great sector ORCr (fig. 588. No. 1.) is equal to the product of the great circle, whereof Aa is the diameter =126, by CS=18, which is 7308, and its cube 7308 × 21 = 153468.

1487. The area of the small sector ONDn will be equal to the product of the great circle, whereof Bb is the diameter 108 by VD=15, which gives 536934, and its cube by 53697 189664874. Deducting this last cube from that of the great sector already found 153468, the remainder 56819 will be the cube of the upper part of the vault forming the cap, whose eighth part 71023 will be the cube sought, which multiplied by KM=17 and divided by the arc KG=46, gives 26463 for the value of m in the formula; c, which represents iK, being 12, we have

mc=334614; pd-me will be 738974 – 334613=404364 ;
and for pd-
af

-mc

404361 120 × 42

=7.92.

1488. In the preceding application to the model of the coved vault, the walls being straight, the distance of their centre of gravity from the point of support was equal to half their thickness; in this, the wall being circular, its centre of gravity is so much more distant from the point of support as it takes in more or less a greater part of the circumference of the circle. By taking it only the eighth part, the centre of gravity falls without the thickness of the walls, by a quantity which we shall call e, so that the arm of the lever, instead of being, will be e+a, which changes the preceding formula to

b expresses the vertical effort of an eighth part of the vault equal to its cube, multiplied by the vertical Bf, and divided by the mean circumference TKG. This cube is equal to an eighth of the sphere, whereof Aa is the diameter, less that of the eighth part of a sphere whose diameter is Bb.

1489. The diameter Aa being 126, the eighth of the circumference of a great circle will be 49, which, multiplied by the vertical axis, which in this case is equal to the radius or 63, gives for the area of one eighth part of the sphere 3118, and for its cube 3118 × 21 =656881

1490. "The diameter Bb being 108, an eighth part of the circumference of the great circle will be 424, which, multiplied by the radius 54, gives for the area 22914, and for its cube 22914 × 18=412404; taking the smaller of these cubes from the greater, the difference 24447 will be that of this eighth part of the vault, which must be multiplied by Bf= 58, and the product 14302033, divided by the mean arc TKG-919; the quotient 15558 expresses the vertical effort of the eighth part of the vault, represented by b in the formula, whence =3.05.

b 15558
5100

af

e being 2.51, we shall have for the value of h 2-78 and h2=7·72.

Substituting the values thus found in the formula

we have = √42 + 7·92 +7·72 −2·78 = √57·64-2·78=4·72.

By using the method of the centres of gravity, Rondelet found the result rather less than that just found.

1491. The result of all these calculations induces the following facts: :

- I. That for a

semicircular cylindrical vault, whose length is equal to its diameter, the area of the two parallel walls is 4698. II. That that of the four square piers supporting a groined arch is 7056. III. That of the four walls of the coved vault, the area should be 34253. IV. That that of the spherical vault is 1238.

1492. In respect of the opening of these vaults, which is the same for all the examples, taking the area of the circular wall for the spherical vault at 1,

That of the walls of the coved vault will be a little less than 3.
That of the cylindrical vault

That of the groined arch

less than 4.

less than 6.

But if we look to the space that each of these vaults occupies in respect of walls and points of support, we shall find that in equal areas the walls of the cylindrical vault will be of such space.

Those of the coved vaulting less than

The piers of the groined arch a little more than

The circular wall for spherical vault a little more than

of such space.

So that, if we suppose the space occupied by each of these vaults to be 400,

The walls of the cylindrical vaulting will be 115
Those for the coved vault

91

Which figures therefore show the relative proportions of the points of support necessary in each case.

1499. It is a remarkable circumstance that by the formula the coved and spherical vaults give to the walls a less thickness than that of the arch. But although experiment

has verified the formula, we cannot be supposed to recommend that they should be made of less thickness in practice; but we see that, if of the same thickness, considerable openings may be used in them. Irregular as well as regular compound vaults being only an assemblage of the parts of more simple ones, if what has already been said be well understood, and the examples given have been worked out by the student, he will not be much at a loss in determining the efforts of all sorts of vaults.

On the adhesive Power of Mortar and Plaster upon Stones and Bricks.

1494. The power of mortar and plaster will of course be in proportion to the surface of the joints, compared with the masses of stone, brick, or rubble. Thus a voussoir of wrought stone, one foot cube, may be connected with the adjoining voussoirs by four joints, each of 1 foot area, in all 4 feet. But if instead of this voussoir three pieces of

rough stone or rubble be substituted instead of 4 feet area of joints, we shall have 8. Lastly, if bricks be employed instead of rubble, we shall want 27 to form the same mass, which gives for the developement of the joints 13 feet. Thus, representing the force which connects the voussoirs in wrought stone by 4, that representing the joints of the rough stones will be 8, and that for bricks 13: whence we may infer that arches built with rough stones will have less thrust than those in wrought stone, and those in bricks more than three times less. From experiments made by Rondelet, he found that at the end of six months some species of mortar showed a capability of uniting bricks with sufficient force to overcome the efforts of thrust in a vault segmental to of a semicircle, 15 feet diameter and 4 inches thick, the extrados being 4 inches concentrically above the intrados. Plaster united a vaulted arch of 18 feet opening, of the same form and thickness. This force is, moreover, greater in arches whose voussoirs increase from the keystone to the springing, and that in proportion to the thickness at the haunches, where fracture takes place; so that whatever the diameter and form of the arch, the strength of good mortar at the end of six months, if the arches are well constructed, is capable of suppressing the thrust as long as the thickness, taken at the middle of the haunches, is stronger than the tenth part of those laid in mortar, and one twelfth of those laid in plaster. Here we have to observe, that arches laid in plaster, as long as they are kept dry and sheltered from the changes of the season, preserve their strength, but, on the contrary, they lose all their stability in seven or eight years, whilst those cemented in mortar endure for ages.

1495. The small quantity of mortar or of plaster used in vaults constructed of wrought stone, in which the joints are often little more than run, ought to make an architect cautious of depending merely on the cementing medium for uniting the voussoirs. There are other means which he may employ in cases of doubt, such as dowels and cramps, means which were much employed by the Romans in their construction; and these are far better than the chains and ties of iron introduced by the moderns.

1496. The thrust of an arch is, in practice, the constant dread of an architect; but it depends entirely on the method employed in the construction. It is only dangerous where the precautions indicated in the foregoing examples are altogether lost sight of. It has

been seen that the least fracture in too thin an arch of equally deep voussoirs may cause its ruin; and we shall here add, that this defect is more dangerous in arches wherein the number of joints is many, such as those constructed in brick; for when they are laid in mortar they are rather heaped together than well fitted to each other.

1497. Whatever materials are used in the construction of vaults, the great object is to prevent separation, which, if it occur, must be immediately met by measures for making the resistance of the lower parts capable

of counterbalancing the effort of the upper parts. Those fractures which occur in cylindrical arches are the most dangerous, because they take place in straight lines which run along parallel to the walls bearing them. To avoid the consequences of such failures, it is well to fill up the haunches to the height where the fracture is usually to be found, as in K, K', K", K"" (fig. 590.) and diminish the thickness towards the key.

1498. Rondelet found, and so indeed did Couplet before him, that the least thickness which an arch of equal voussoirs ought to have, to be capable of standing, was one fiftieth part of the radius. But as the bricks and stone employed in the construction of arches are never so perfectly formed as the theory supposes, the least thickness which can be used for cylindrical arches from 9 to 15 feet radius is 4 inches at the vertex if the lower course be laid with a course of brick on edge or two courses flatwise, and 5 inches when the material used is not a very hard stone, increasing the thickness from the keystone to the point where the extrados leaves the walls or piers. But if the haunches are filled up to the point N (fig. 590.), it will be found that for the pointed arch in the figure the thickness need not be more than the of the radius, and for the semicircular arch,. For arches whose height is less than their opening or that are segmental the thickness should be part of the versed sine ; a practice also applicable to Gothic vaults and semicircular cylindrical arches, to which for vaults cemented with plaster one line should be added for each foot in length, or part of the chord subtended by the extrados. With vaults executed in mortar may be added, the thickness of the arch increasing till it reaches the point N, where the arch becomes detached from the haunches, and where it should be once and a half the thickness of the key. It was in this way the arches throughout the Pantheon at Paris were regulated, and a very similar sort of expedient is practised in the dome of the Pantheon at Rome. A like diminution at the keystone may be used in groined, coved, and spherical vaults.

Fig. 590.

1499. For vaultings of large openings, Rondelet (and we fully concur with him) thinks wrought stone preferable to brick or rubble stone. They have the advantage of being liable to less settlement and to stand more independent of any cementitious medium employed. It is indeed true that this cannot connect wrought stone so powerfully as it does rubble; but in the former we can employ cramps and dowells at the joints, which are useful in doubtful cases to prevent derangement of the parts. In many Roman ruins the surfaces of the voussoirs were embossed and hollowed at the joints, for the purpose of preventing their sliding upon each other; and expedients of the same nature are frequently found in Gothic ruins.

SECT. X.

WALLS.

1500. The thickness which is to be assigned to walls and points of support, that their stability may be insured, depends on the weight they have to sustain, and on their formation with proper materials; still more on the proportion which their bases bear to their heights. The crushing of stone and brick, by mere superimposed weight, is of such extremely rare occurrence in practice, even with soft stone and with bad bricks, that we think it sufficient to give the result of the some few experiments that have been made in that respect, to give the reader some notion of the resistance of our bricks and stones to a

1501. The above experiments lose much of their practical value from our knowledge that the interior particles in granulated substances are protected from yielding by the lateral resistance of the exterior ones; but to what extent it is impossible to estimate, because so much depends on the internal structure of the body. We are, however, thus far informed, that, taking into account the weight with which a point of support is loaded, its thickness ought to be regulated in an inverse ratio to the crushing weight of the material employed. In Gothic structures we often see, for instance, in chapter houses with a central column, a prodigious weight superimposed. It is needless to say, that, in such instances, the strongest material was necessary, and we always find it so employed. So, in the columns, or rather pillars, of the naves in such edifices, the greatest care was taken to select the hardest stone.

1502. Generally speaking, the thickness of walls and piers should be proportioned rather to their height than to the weight they are to bear; hence often the employment of a better material, though more costly, is in truth the most economical.

Of the Stability of Walls.

1503. In the construction of edifices there are three degrees of stability assignable to walls. I. One of undoubted stability; II. A mean between the last; and the III. The least thickness which they ought to possess.

1504. The first case is that in which from many examples we find the thickness equal to one eighth part of the height: a mean stability is obtained when the thickness is one tenth part of the height; and the minimum of stability when one twelfth of its height. We are, however, to recollect that in most buildings one wall becomes connected with another, so that stability may be obtained by considering them otherwise than as independent walls. 1505. That some idea may be formed of the difference between a wall entirely isolated and one connected with one or two others at right angles, we here give figs. 591, 592, and 593. It is obvious that in the first case (fig. 591.), a wall acted upon by the horizontal force MN, will have no resistance but from the breadth of its base; that in the second

case (fig. 592.) the wall GF is opposed to the force MN, so that only the triangle of it HIF can be detached; lastly, in fig. 593. the force MN would only be effective against

the triangle CGH, which would, of course, be greater in proportion to the increased distance of the walls CD, HI.

1506. In the first case, the unequal settlement of the soil or of the construction may produce the effect of the force MN. The wall will fall on the occurrence of an horizontal

disunion between the parts.

1507. In the second case the disunion must take place obliquely, which will require a greater effort of the power MN.

1508. In the third case, in order to overturn the wall, there must be three fractures through the effort of MN, requiring a much more considerable force than in the second case. 1509. We may easily conceive that the resistance of a wall standing between two others will be greater or less as the walls CD, HI are more or less distant; so that, in an extreme approximation to one another, the fracture would be impossible, and, in the opposite case, the intermediate wall approaches the case of an isolated wall.

1510. Walls enclosing a space are in the preceding predicament, because they mutually tend to sustain one another at their extremities; hence their thickness should increase as their length increases.

1511. The result of a vast number of experiments by Rondelet, whose work we are still using, will be detailed in the following observations and calculations.

B

1512. Let ABCD (fig. 594.) be the face of one of the walls for enclosing a rectangular