Figs. 3 and 4.-To develop the surface of a cylinder intersected by another cylinder, as in the formation of a Ɩ pipe.

The construction is similar to the preceding, and as the same letters and figures are preserved relatively, the demonstration will be easily understood from the foregoing.

The development of the surface of a right cone (figs. 5 and 6). From C' (fig. 6) as a centre, with a radius C' A' equal to the inclined side A C of the cone (fig. 5), describe an arc of a circle A'B' A′′; on this arc lay off the distance A'B' A" equal to the circumference of the base of the cone; connect A'C' and C' A", and A' B' A" C' is the developed surface required.

To develop the surface of a frustrum of a cone, D A B E (fig. 5).

On fig. 6 develop the cut-off cone C D E as in preceding construction, and we have A' B' A" D" E' D as the developed surface of the right frustrum.

To develop the surface of a frustrum of a cone, when the cutting plane a b (fig. 5) is inclined to the base.

On A B the base describe the semicircle A 3' B; divide the semicircle into any number of equal parts, six for instance; from each point of division 1', 2′,3′,4′, 5′, let fall perpendiculars to the base; at 1, 2, 3, 4, 5, connect each of these last points with the apex C. Divide now the arc A'B' (fig. 6) into six equal parts, or the arc A' B' A" into twelve; each of these parts by the construction is equal to the arc A 1', 1' 2′ (fig. 5); connect these points of division with the point C'; on C'A' (fig. 6) take C'a' equal to C a of fig. 5, a being the point at which the plane cuts the inclined side of the cone; in the same way on C' B', lay off C' b' equal to C b.

It is evident that all the lines connecting the apex C with the base, included within the two inclined sides, are represented as less than their actual length in fig. 5, and must be projected on the inclined sides to determine their absolute dimensions; project, therefore, the points 1", 2", 3", 4", 5′′, at which the cutting plane intersects the lines C 1, C 2, C 3, C 4, C 5, by drawing parallels to the base through these points to the inclined side C B'. On fig. 6 lay off C' 1"", C′ 2′′", &c., equal to C 1"", C 2," !!! &c. (fig. 5); connect the points a', 1"", 2"", a", and we have the developed surface a' A' B′ A′′ a′′ b' required.

- b',

To develop the surface of a sphere or ball (figs. 189, 190).

It is evident that the surface cannot be accurately represented on a plane surface. It is done approximately by a number of gores. Let CAB (fig. 189) be the eighth of a hemisphere; on C D describe the quarter circle D A c; divide the arc into any number of equal parts, six for in

stance; from the points of division 1, 2, 3,... let fall perpendiculars on CD, and from the intersections with this line describe arcs 1'1′′, 2′ 2′′, 3′ 3′′, cutting the line C B at 1", 2", 3′′, ....; on the straight line C' D' (fig. 190), lay off C' D' equal to the arc D A c, with as many equal divi

..

sions; then from either side of this line lay off 1""' 1"""', 2"" 2""" . . . . D'B' equal to the arcs 1'1", 2' 2", .... DB (fig. 189). Connect the points C', 1′′", 2′′",. . . . and C'A' B' is the developed surface.

It is to be remarked, that in the preceding demonstrations, the forms are described to cover the surface only; in construction, allowance is to be made for lap by the addition of margins on each side as necessary. It is found difficult in the formation of hemispherical ends of boilers, to bring all the gores together at the apex; it is usual, therefore, to make them, as shown (fig. 191), by cutting short the gores, and surmounting the centre with cap piece.

Fig. 191.

MECHANICS.

THE profession of an architectural or mechanical draughtsman should embrace not merely the mere copying of examples which may be furnished him, but also the designing of new edifices and machines, in which he may draw from the results of his own experience; from good models, by collating suitable parts from divers designs; or by the rules of mechanics, proportioning the parts according to the magnitude and direction of the strains to which they are to be subject, and the materials of which they are to be composed; introducing as much of ornament as the subject may require.

Force is that which tends to cause or to destroy motion. The direction of a force is that in which this tendency is exerted; thus, gravity tends to draw bodies to the earth, and its direction is therefore vertical. Any number of forces which, being applied to a body, destroy one another's tendency to communicate motion to it, and thus hold it at rest, are said to be in equilibrium. Two forces cannot hold a body at rest unless they are equal to each other, and in opposite directions, and in the same straight line. To compare different forces, they are referred to some known standard, as for instance to pounds weight. For constructive purposes it is very convenient to represent forces by lines; in this way they can be represented both in magnitude and direction. Thus, if two forces act, for instance, on the point P in the direction P A and PB (fig. 192), by laying off from a scale of equal parts a distance corresponding in number to the pounds weight exerted by each force; thus, if the force exerted in the direction of P A would be represented numerically by 4 lbs., 4 cwt., or 4 tons, and that exerted in the direction P B be represented by 2 lbs., 2 B

A

cwt., or 2 tons, lay off from any convenient scale

4 equal parts on PA from P, and from the same scale 2 parts on P B, and

B b

Fig. 193.

P

a

we have the two forces represented not only in direction but in magnitude. It is evident, that in this particular case the body cannot be at rest, as the two forces do not act in opposite directions, neither are they in the same straight line. A third force is necessary for equilibrium. We proceed to show how the magnitude and direction of this force may be determined. Through the extremities a and b (fig. 193), the distances laid off to represent the forces, draw two other lines, a c parallel to P b, and b c parallel to Pa. The four will then form a parallelogram. Join the two opposite angles, P, c; then this line will represent, in magnitude and direction, the other, which will hold the other two at rest. Or stating this in general terms :—If three forces acting on a point are in equilibrium, and lines be measured from this point in the direction of the forces, representing severally in length the relative amount of force exerted by them, then these lines will form the adjacent sides and diagonal of a parallelogram. This is called the Law of the Parallelogram of Forces, which governs the equilibrium of any three forces, and the understanding of which is important for every one having charge of constructions either in machinery or architecture. The following simple and easy experiment will illustrate the principle. Let a small cord be passed over the pulleys B and C (fig. 194), and weights be attached to each end, and also one suspended from the centre at A, then the threads will assume a certain position dependent on the relation of the weights, and in this one position only will they balance each other; and if deranged by pulling, they will return to the same position when left at liberty. If this position be laid off on paper, and from a scale, lay off A P equal to the weight W, and the line PE drawn parallel to A C, and B A continued to E, then PE on the same scale would represent the weight suspended at C, and A E the weight at B. Thus, if a body be kept at rest by three forces, and any two of them be represented in magnitude and direction by two sides of a triangle, the third side will represent in magnitude and direction the other force. To extend this illustration still farther to any number of forces in the same plane: suspend a number of weights (fig. 195) over pulleys, and after they have assumed the position of rest, lay off the position of the

B

P

W

Fig. 194.

E

C

lines on paper, making A a equal to the weight M; a b equal to N, and parallel to its direction A n; b c equal to O, and parallel to A o; c d equal to P, and parallel to Ap; de equal to Q, and parallel to a g; then the last force R will be in the direction and equal to the line A e, which completes the polygon. Hence, if we represent in direction and magnitude any number of forces by the sides of a polygon, the force necessary to keep the others at rest will be represented by the line which completes the polygon. This line is called the resultant.

As a practical illustration of the application of the Parallelogram of Forces, take the raising of a weight by a crane or derrick (fig. 196). Represent by A B the weight W; draw C B parallel to the direction of the rope A D; C B will represent the strain on the rope A D, and A C the pressure in the direction of the line of the boom. The strain in the direction A D is supported by the guy D F, and the mast D E; represent by Da the strain on D A; then by constructing the parallelogram we have Dƒ as the strain on the guy, and D d as the downward pressure or weight exerted on the top of the mast. By a similar construction, the pressure A C lengthways of the boom may be decomposed into a vertical pressure downwards on the step of the

mast, and a horizontal thrust

tending to displace it.

In the common roof truss (fig. 197), suspend a weight from the ridge representing it by

a line, and construct the paral

B

W

W

Fig. 197.

lelogram by lines parallel to the direction of the rafters, and draw dia