1. What will be the diameter of a steam cylinder sufficient to work a pump 16 inches diameter and 20 yards deep; the piston to be loaded with 10 pounds upon the inch?

In the first table you will find the gauge point for a 16-inch pump, to be 327.

Set unity upon B under 327 on A, and against 20 yards upon C, is 25.5 or 25 inches upon D, the diameter of the steam cylinder required. When the slide is thus set to its proper gauge point, for any diameter of a pump, the lines C and D are a table for that same diameter; for against any length, in yards, upon C, you have the diameter of the steam cylinder in inches upon D. For example: under 15 on C are 22.1 inches, the diameter, on D; under 20, 25.5; under 25, 28.5; under 30, 31.3; under 35, 33.8; and so of all the rest above or below 20 yards.

2. What will be the diameter of a cylinder to work a pump, 12 inches diameter, at 70 yards deep, and loaded with 7 lbs. on the inch?

In the second table for a 12-inch pump is 264.

Set 1 on B under 264 on A, and under 70 yards on C, is 48 inches on D, the answer.

This is likewise a table, as under 15 yards on C, are 19.8 inches, the diameter, on D; under 20, 23; under 25, 25.6; under 30, 28.1; and under 35, 30.4.

To find the size of a steam-pan or boiler, sufficient to supply with steam a cylinder of any given diameter:

Set 1.3 (the gauge point) on Cover 1 on D, and over the diameter of the cylinder or piston, found on D, will be found the number of square feet that must be contained in the surface of the water in the boiler, (whether it be round or square, or any other shape.)

When the slider is set, the C line is a table of areas, and the D line is a table of diameters. Thus, over 15 inches on D are 29.2 square feet on C; over 20, 52; over 25, 81; over 30, 117; over 35, 160; over 40, 207; and under 29.2 on C is 15 inches on D, the diameter of a piston requiring 29.2 square feet in the surface of the water in the boiler, &c.

STEAM-ENGINE.

The power of a steam-engine, other things being the same, is as the square of the diameter of the working piston, the power increasing or diminishing as the area of the end of the piston increases or diminishes.

A piston 10 inches in diameter is usually reckoned to be equal to 30 horse-power. Therefore, To find the horse-power of any engine :

Divide the square of the diameter of the piston by 100, and multiply the quotient by 30. Or, by the sliding rule:-Place 30 on Cover 10 on D, and over the diameter of the piston, found on D, will be found its horse-power on C. Or, under any horsepower found on C, will be found the diameter of the piston on D. Thus, (having set the slider,) over 12 on D will be found 43 horse-power on C; over 20 on D, 120 horse-power; over 30, 270 horse-power; over 40, 480 horse-power; and over 60

inches, 1,080 horse-power. And under 750 horse-power, found on C, will be found 50 inches on D, the diameter of a piston equivalent to a horse-power of 750; and under 2,000 found on C, will be found 81 inches; and under 3,000 horse-power will be found 100 inches, and so on.

See ¶76, and ¶ 81, problems 122 and 123.

FRICTION.

Friction, in the use of machinery, may be reduced to a minimum by polishing the surfaces of the axles, gudgeons, and bearings, and keeping them continually well greased, and wetted with water. If, however, the unguents used are olive oil, or pure soft tallow or lard, the application of water will be of no service, except it be to keep the running-gears cool. Wood should run upon a bearing of metal, cast-iron upon bell-metal, and wrought-iron upon cast-iron or bell-metal. Friction may likewise be greatly diminished by the application of friction wheels, which are substituted in place of the bearings.

T80. STRENGTH OF MATERIALS.

It is necessary, both in architecture and in the construction of machinery, that due regard should be paid to the strength of the materials employed; and that every part should be made in due proportion to the stress, or the force, or pressure to be endured.

The strength of any beam, whether of wood or metal, is directly as the square of the depth multiplied by the breadth, and inversely as the length. Hence the strengths of several pieces of timber, their lengths being the same, are to one another as their breadths multiplied by the squares of their depths: if, therefore, their breadths be the same, their relative strengths will be as the squares of their depths; and if their depth be the same, their strengths will be as their breadths. Conse

quently, if two beams are of the same length and breadth, and one be twice as deep as the other, it will be 4 times as strong; and if 3 times as deep, it will be 9 times as strong; and if 4 times as deep and twice as broad, it will be 32 times as strong.

If a beam project from a wall, and a weight be suspended from the end of it, the stress suffered by any part of the beam will be as its distance from the weight; and if the weight be augmented, the stress at any point in the beam is augmented in the same ratio; and the stress at any point is measured by the product of the weight into its distance from that point. If the weight be uniformly dispersed throughout the whole length of a beam, the stress upon any point of the beam will be only half as great as it would if the whole weight were suspended at the end of the beam. If a beam be supported at both ends, the breaking weight must be double that required to break a beam of half the length with one end firmly fixed in a wall: the length of a beam supported at both ends may be four times as great as that of the same beam supported at one end only, and having the weight suspended at its other extremity. If a beam be firmly fixed in a wall at both ends, it will sustain as much weight at its centre as it would if the ends were merely supported. When a beam, instead of being laid horizontally, is inclined, its strength is increased nearly in proportion to the angle of elevation; and consequently, it will bear the greatest stress when set in an upright, or vertical position. The figure of a beam projecting from a wall, every point in which is equally capable of sustaining a given weight suspended at the end, will be that of a direct wedge, whose upper and under sides are parallel with the horizon; or, it may be of the form of a parabola, whose vertex is at the end of the beam, the upper and under sides being curved, and the lateral sides plane surfaces. The weight which is required to break any beam, whether of wood or metal, is considerably greater than that which is required to bend or deflect it; and the quantity by which a beam is bent from its position of rest, is called the deflection. Some kinds of timber, (as, for example, ash, fir, and larch,) though capable of sustaining a great weight, are deflected by a comparatively small force or weight. The

deflection of a beam, other things being the same, is directly as the cube of the length, and inversely as the cube of depth multiplied by the breadth. The deflection of a beam fixed at one end and loaded at the other, is double that of a beam of twice the length supported at both ends, and loaded in the middle with a double weight; consequently, when the weights are the same, the deflection in the first case is to that in the second as 4: 1; and when the length and weight are both the same, the deflections (which vary as the cubes of the lengths) will be to each other as 32: 1. If the ends of a beam be firmly fixed in a wall, its deflection under a given weight will be only as great as it would if it were merely supported at the ends.

The measure of the absolute strength of any material is the greatest weight that a prism one inch square is capable of supporting, acting in the direction of its length; and the absolute strength of any beam, other things being the same, is as the area of the end of the beam. The absolute strength of the various woods varies from 6,000 pounds to 17,000 pounds per square inch, elm being the weakest, and ash the strongest of the woods; and the strength of iron, measured by the same standard, varies from 16,300 to 60,000, cast-iron being much the weakest, and malleable iron the strongest.

The resistance of beams to forces tending to crush them, and acting in the direction of the fibres, is usually much less than the force required to draw them asunder: and some of the weakest of the woods, in respect to absolute strength, are among the strongest in resisting a crushing force. The resistance of short pillars of well-seasoned wood, one inch square, to a crushing force, varies from 5,000 pounds to 10,000 pounds; and the resistance of beams, which are round or square, (that is, their power of resisting a crushing force,) is as the square of the side or diameter directly, and inversely as the square of the length consequently, the strength of beams or pillars to resist a crushing force, is measured by dividing the product of the area of the end into the number of pounds of resistance opposed by a pillar one inch square, (of the given substance,) by the square of the length. Mr. Tredgold found that short pillars