When the earth of the bank or terrace is liable to be much saturated with water, the proportional thickness of wall must be at least doubled.*

8. For walls with an interior slope, or a slope towards the bank, let the base of the slope be of the height, and let s and

1

n

s, as before, be the specific gravities of the wall and of the earth; then

=

where m 0424, for vegetable or clayey earth mixed with large gravel; m = 0464, if the earth be mixed with small gravel; m = 1528, for sand; and m = 166, for semi-fluid earths.

Example. Suppose the height of a wall to be 20 feet, and of the height for the base of the talus or slope; suppose, also, the specific gravities of the wall and of the bank to be 2600, and 1400, and the earth semi-fluid: what, then, must be the thickness of the wall at the crown?

1. Let there be any number of lines, or bars, or beams, A B, B C, C D, DE, &c. all in the same vertical plane, connected together, and freely moveable about the joints or angles, A, B, C, D, E, &c. and kept in equilibrio by weights laid on the angles : It is required to assign the proportion of those weights as also the force or push in the direction of the said lines; and the horizontal thrust at every angle.

20
11

* When weights of French cubic feet are given in kilogrammes, of them will be the corresponding weight of an English cubic foot in pounds avoirdupois.

Through any point, as D, draw a vertical line a DHg, &c. ; to which, from any point, as c, draw lines in the direction of, or parallel to, the given lines or beams, viz. c a pa

rallel to A B, cb parallel to в C, ce to D E, c fto E F, cg to F G, &c.; also c H parallel to the horizon, or perpendicular to the vertical line a d g, in which also all these parallels terminate.

Then will all those lines be exactly proportional to the forces acting or exerted in the directions to which they are parallel, and of all the three kinds, viz. vertical, horizontal, and oblique. That is, the oblique forces or thrusts in direction of the bars

A B, B C, C D, D E, E F, F G,

are proportional to their parallels ca, c b, c d, ce, cf, cg; and the vertical weights on the angles B, C,

D, E, F, &c. are as the parts of the vertical. . . . a b, a b, b D, De, ef, fg, and the weight of the whole frame A B C D E F G, is proportional to the sum of all the verticals, or to a g; also the horizontal thrust at every angle, is everywhere the same constant quantity, and is expressed by the constant horizontal line C H.

Corol. 1. It is worthy of remark that the lengths of the bars A B, B C, &c. do not affect or alter the proportions of any of these loads or thrusts; since all the lines c a, c b, a b, &c. remain the same, whatever be the lengths of A B, B C, &c. The positions of the bars, and the weights on the angles depending mutually on each other, as well as the horizontal and oblique thrusts. Thus, if there be given the position of D c, and the weights or loads laid on the angles D, C, B ; set these on the vertical, DH, D b, b a, then c b, c a, give the directions or positions of c B, BA, as well as the quantity or proportion c H of the constant horizontal thrust.

Corol. 2. If c н be made radius ; then it is evident that нa is the tangent, and c a the secant of the elevation of c a or A B above the horizon; also н b is the tangent and c b the secant of the elevation of cb or c в; also H D and CD the tangent and secant of the elevation of c D; also He and сe the tangent and secant of the elevation of c e or DE; also н ƒ and of the tangent and secant of the elevation of E F ; and so on; also the parts of the vertical a b, b D, ef, fg, denoting the

weights laid on the several angles, are the differences of the said tangents of elevations. Hence then in general,

1st. The oblique thrusts, in the directions of the bars, are to one another, directly in proportion as the secants of their angles of elevation above the horizontal directions; or, which is the same thing, reciprocally proportional to the cosines of the same elevations, or reciprocally proportional to the sines of the ver tical angles, a, b, D, e, f, g, &c. made by the vertical line with the several directions of the bars; because the secants of any angles are always reciprocally in proportion to their cosines.

2. The weight or load laid on each angle is directly proportional to the difference between the tangents of the elevations above the horizon, of the two lines which form the angle.

3. The horizontal thrust at every angle is the same constant quantity, and has the same proportion to the weight on the top of the uppermost bar, as radius has to the tangent of the elevation of that bar. Or, as the whole vertical a g, is to the line CH, so is the weight of the whole assemblage of bars, to the horizontal thrust.

4. It may hence be deduced also, that the weight or pressure laid on any angle, is directly proportional to the continual product of the sine of that angle and of the secants of the elevations of the bars or lines which form it.

Scholium. This proposition is very fruitful in its practical consequences, and contains the whole theory of centerings, and indeed of arches, which may be deduced from the premises by supposing the constituting bars to become very short, like arch stones, so as to form the curve of an arch. It appears too, that the horizontal thrust, which is constant or uniformly the same throughout, is a proper measuring unit, by means of which to estimate the other thrusts and pressures, as they are all determinable from it and the given positions; and the value of it, as appears above, may be easily computed from the uppermost or vertical part alone, or from the whole assemblage together, or from any part of the whole, counted from the top downwards.

In all the useful cases, a model of the structure may be made, and the relations of the pressures at any angle, whether horizontal, vertical, or in the directions of the beams, may be determined by a spring steel-yard applied successively in the several directions, after the manner described in Art. 4. Sect. 1. Statics.

2. If the whole figure in the preceding problem be inverted, or turned round the horizontal line A G as an axis, till it be

completely reversed, or in the same vertical plane below the first position, each angle D, d, &c. being in the same plumb line; and if weights i, k, l, m, n, which are respectively equal to the weights laid on the angles B, C, D, E, F, of the first figure, be now suspended by threads from the corresponding angles h, c, d, e, f, of the lower figure; those weights keep this figure in exact equilibrio, the same as the former, and all the tensions or forces in the latter case, whether vertical or horizontal or oblique, will be exactly equal to the corresponding forces of weight or pressure or thrust in the like directions of the first figure.

This, again, is a proposition most fertile in its consequences, especially to the practical mechanic, saving the labour of tedious calculations, but making the results of experiment equally accurate. It may thus be applied to the practical determination of arches for bridges, with a proposed road way; and to that of the position of the rafters in a curb or mansard roof.

3. Thus, suppose it were required to make such a roof, with a given width A E, and of four proposed rafters A B, B C, C D, D E. Here, take four pieces that are equal or in the same given proportions as those proposed, and connect them closely together at the joints A, B, C, D, E, by pins or strings, so as to be freely moveable about them; then suspend this from two pins, A E, fixed in a horizontal line, and the chain of the piece will arrange itself in such a festoon or form, a b c d E, that all its parts will come to rest in equili

A

E

brio. Then, by inverting the figure, it will exhibit the form and

frame of a curb roof A B C D E, which will also be in equilibrio, the thrusts of the pieces now balancing each other, in the same manner as was done by the mutual pulls or tensions of the hanging festoon A b c d E.

4. If the mansard be constituted of four equal rafters; then, if angle c A E = m, angle c A B = x; it is demonstrable that 2 sin 2 x = sin 2 m. So that if the span A E, and height м C, be given, it will be easy to compute the lengths A B, B C, &c. Example. Suppose A E = 24 feet, м c 12.

Then

MC

M A

=

1 = tan 45° angle c A M = m.

.. sin 2 m = sin 90° = 1, and sin 2 x = = }
30°, and x = 15° CAB

.. 2 x =

and A M B 180° - (75° +60°)

90° -15° = 75°

= 45°

Lastly, sin 75°: sin 45 :: A M = 12 : A B = 8.7846 feet.

Note. In this example, since A M = M C, as well as a B = B C, it is evident that M B bisects the right angle A м c; yet it seemed preferable to trace the steps of a general solution.

Stability of Arches.

1. If the effect of the force of gravity upon the ponderating matter of an arch and pier, be considered apart from the operation of the cements which unite the stones, &c. the investigation is difficult to practical men, and it furnishes results that require much skill and care in their application. But, in an arch whose component parts are united with a very powerful cement, those parts do not give way in vertical columns, but by the separation of the entire mass, including arches and piers, into three, or, at most, into four parts. In this case, too, the conditions of equilibrium are easily expressed and easily applied.

Let f F, f' F', be the joints of rupture, or places at which the arch would most naturally separate, whether it yield in two pieces or in one.

Let G be the centre of gravity of the semi-arch fF K k, and G' that of the pier A B F ƒ. Let FI be drawn parallel to the horizon, and & н be demitted perpendicularly