" When piers indeed are to be exceedingly high, as in the columns which are sometimes employed in supporting a lofty aqueduct, the best way is to make them hollow, and give them stability, by enlarging the base. They will, in that case, press less on the foundations, be less expensive, and they may be gi-eatly stiffened by hooping.
" Indeed, it is not usual to make piers solid all the way up to the road ; the spandrel-walls are carried back so far as to unite with those of the neighbouring arch, are locked together by a cross wall just over the middle of the pier, b havina also walls longitudinally, and the whole arched or flagged from spandrel to spandrel just under the roadway.
"Nevertheless, as the case of will enable us to assign a limit to the breadth of piers, which it may be proper to be acquainted with, we shall proceed in that investigation.
"The weight of the pier in that case will lie as the rect angle under its height and thickness, expressing the weight of arch and pier by the cubic feet of stone. The pier indeed will be somewhat more ; for the sterlings, or breakwaters, at each end, will add something to its stability ; and this will be still farther increased in proportion to the horizontal push, if the whole bridge be wider at the foundation than at top, as is very common. Excluding these collateral advantages, we shall consider the whole as rectangular, and then the stability may be found in the longitudinal section. We h have already b = p and in the case of a parallelo gram 1p = z b (h + c), c being the height from springing to the roadway. By substitution, there arises 1- c) and by resolving this quadratic equation, 3a we have b 2 it t b, + ( h + c) ) 4 (h+c) or thus, y2 (it + c)h as a formula h+ c for the thickness of solid piers to support equilibrated arches; and it must be observed, that if the arch be understood to act otherwise than at three-fourths the thickness of the pier, this co-efficient may be altered accordingly.
" When the arch is a segment less than a semicircle, a greater thickness of pier becomes necessary. For, the span continuing the same, we must either make the arch a part of a circle of greater radius, which would increase the horizontal thrust, or we must, in order to obviate that, diminish the thickness at the crown. In either case the weight of the arch
is diminished, and with it the assistance which it gives to the stability of the pier.
"There is an interesting subject of inquiry, which might not be 'inappropriately noticed here ; we mean the lowest versed sine that can be used for arches in proportion to the span. We conceive this, however, as in a great measure a practical question. We have already given some idea of the greatest possible arch of stone, or brick ; a segment of that circle may, of course, be employed in any situation, but the piers (if the arch be of considerable span and height to the springing) must be made very great. Indeed, the investigation depends intimately on the thickness of piers. We ought to know the dimensions of the largest pier that can be trusted, and this, we conceive, depends chiefly on the care of the mason ; for stone, and especially cement, is a com pressible substance ; and when an arch is very flat, a very small yielding at the springing produces an enormous depres sion at the crown, insomuch that there may be reason to dread, lest the arch pass down Mow the horizontal line, and fall to pieces before the stability of the abutments can be acted upon. A compression in the joints is equivalent to a yielding at the abutments, and appears equally difficult of remedy.
"In great horizontal thrusts, where the segment is flat, the immersion of the pier in water conies to have an impor tant effect. On the weight of the pier, in those cases, the stability chiefly depends, and a deduction from that of two fifths must be compensated by enlarging the thickness.
" But, indeed, the immersion of the pier, if it be very tall, that is, if the depth of water be great in proportion to the span, will demand attention, although the arch should not be very flat. In such a case, the stability arising from the pier is often as great as that which is derived from the weight of the arch. It can seldom be greater, and consequently can seldom require, addition of more than one-fifth of that breadth, which would be sufficient were there no immersion.