on the line of the equator, the circumference being 24899 miles?

23. Two men purchase in equal shares, a stick of hewn timber, 40 feet long, 2 feet square at the larger end, and 1 foot square at the smaller end; how far from the larger end shall they cut it in two, so that each may have exactly one half?

24. A surveyor, in laying out a lot of land, first runs a line due North, to a certain tree; from the tree he runs between South and West till he comes to a point due West from the place he started from; the whole of these two lines is 212 rods, but those who measured it neglected to note how far the tree was from the starting point; on measuring a third line, connecting the extremities of the two first lines, they find it 98 rods; how many acres does the triangle contain ?

Specific Gravity.

The specific gravity of a body, is its weight compared with the weight of an equal bulk of water. To find the specific gravity of a body heavier than water.

Weigh the body in water, and out of water, and find the difference in the weight; then, as the difference in the weight is to the weight out of water, so is 1 to the specific gravity. The weight of a cubic foot of water is 62 lbs. av. specific gravity of the most important of the metals is as follows:

The

From the above table, we may find the weight of any mass of one of the above metals the magnitude of which is known. 25. What is the weight of a cubic foot of iron?

26. What is the weight of an iron ball six inches in diameter?

27. What is the diameter of a 24 lb. cannon ball? 28. What is the diameter of a 48 lb. cannon ball?

29. What is the weight of a cannon ball one foot in diameter?

30. If the column of mercury in the barometer be 29

inches high, what would be the weight of a column of mercury of that height, one inch square?

31. As the weight of mercury in the barometer equals the weight of the atmosphere on the same base, what is the pressure of the atmosphere on a square foot, when the mercury in the barometer is 29 inches high?

The height to which water will rise in a suction pump, and the height of the mercury in the barometer, are in inverse proportion to the specific weight of those two bodies; that is, the water is as much higher than the mercury, as mercury is heavier than water.

32. How high will water rise in a suction pump, when the mercury in the barometer is 29 inches high?

33. What is the weight of a copper prism, its base being an equilateral triangle, 3 inches on a side, and its height 15 inches?

Mechanical Powers.

The object to be gained by the application of mechanical powers, is to overcome a large weight or resistance, by means of a comparatively small power.

In doing this, however, the power must move through a space as much larger than the space which the weight moves through, as the weight is heavier than the power.

Or, the distance Xweight, of the power distance weight, of the weight, or mass to be moved.

This is the great law of mechanical powers, and applies to them all, without exception. In the practical application of them, a certain allowance must be made on account of the friction in the machine. The amount of friction differs in different powers. No account of this will be taken in the examples which follow, unless it is particularly mentioned; nor will any difference be made between the power when in motion, and when in equilibrium, or at rest.

The Lever.

The lever is a straight bar used to support or raise heavy weights. It is supported by a prop or fulcrum, placed near the weight, and the power is applied at the other end of the lever. The distance from the fulcrum to the weight, is called

the shorter arm; the distance from the fulcrum to the power, the longer arm of the lever. If the lever were to turn over the fulcrum as a centre, the longer arm would describe a larger circle, and the shorter arm would describe a smaller circle. The circumferences of these two circles, or an arc of the same number of degrees in both, would be the distances passed through by the power and the weight respectively. But we may take the arms themselves as representing these distances, for they are the radii of the two circles; and the radii of different circles have the same ratio to each other as the circumferences.

We have therefore this proportion:

The longer arm is to the shorter arm, as the weight to the power. Or, let 1. a. stand for the longer arm; s. a., for the shorter; W. for the weight, and p. for the power.

1. a. s. a. w. p.; and any change admissible in the terms of a proportion, may be made in these terms.

34. If a lever 10 feet long have its fulcrum one foot from the weight, how great must the power be, to raise a weight of 1640 lbs. ?

35. If a lever 10 feet long have its fulcrum 18 inches from the weight, how great a weight will be raised by a power of 160 lbs.?

36. A lever 18 feet long rests on a fulcrum 2 feet from the end; how large a weight can two men raise, one weighing 164 lbs., the other 172 lbs.,- by applying their weight at the longer arm?

37. If a lever 7 feet long rest on a fulcrum 15 inches from the end, how heavy must the power be to support a ton, gross weight?

38. If the weight be 3600 lbs., and the power 140 lbs., how far from the weight must the fulcrum be placed under a lever 12 feet long, so as to have the weight and power balance?

39. If the weight be 6480 lbs., the power 312, and the lever 16 feet long, how far from the weight must the fulcrum be, to have the weight and power balance?

40. In a certain machine, it is necessary to adjust a lever 3 feet long, so that a power of 1 lbs. shall balance 13 lbs. ; how far from the weight must the fulcrum be placed?

The Wheel and Axle.

In this case the power is applied at the circumference of the wheel, and the weight is drawn up by a rope passing round the axle, which is a smaller wheel. The principle, therefore, is the same as in the lever; the semi-diameter of the wheel is the longer arm; the semi-diameter of the axle, the shorter arm.

41. In a grocery store the wheel and axle used in raising heavy articles, are of the following dimensions, viz. : the wheel 5 feet in diameter the axle 7 inches in diameter; what power must be applied to the rope passing over the wheel, to balance a barrel of flour weighing 205 lbs., suspended by a rope passing over the axle?

42. With the same wheel and axle, what power will raise a box of sugar weighing 431 lbs., adding to the power, to overcome the friction?

43. In digging a well, the wheel employed in raising stones and earth, is 6 feet in diameter; the axle 6 in diameter; what power will raise a rock weighing 640 lbs., adding to the power, to overcome the friction?

44. If a wheel is 14 feet in diameter, what must be the diameter of the axle, in order that a power of 140 lbs. may balance 5760 lbs. ?

45. If an axle is 16 inches in diameter, what must be the diameter of the wheel in order that a power of 56 lbs., may balance a weight of 1344 lbs.?

The Screw.

In this case, the distance passed through by the power in one revolution, is equal to the circumference of the circle described by the lever which turns the screw; the distance passed by the weight, is the distance between two threads of the screw, measured in the direction of its axis.

In the practical application of this power, a large allowance must be made to compensate for the friction.

46. If the lever of a screw is 11 feet in length, and the distance of the threads 14 inches, what power will raise a weight of 6431 lbs., making no allowance for friction?

47. With the same conditions as in the last example, what weight will be raised by a power of 124 lbs.?

48. What must be the length of the lever of a screw the threads of which are 1 inch asunder, in order that a power of 3 lbs. may balance a weight of 1640 lbs., making no allowance for friction?

49. How far asunder must the threads of a screw be, so that, with a lever of 8 feet in length, 26 lbs. will balance 6590 lbs.?

Strength of Beams to resist Fracture.

[See Section XX. Part I.]

In addition to the principles that have already been stated in estimating the strength of timbers, the following are among the most important. It is understood in all cases, when timbers are compared, that they are of the same wood, and equally good in quality.

When the depth of two beams is the same, and the thickness the same, the strength is inversely as the length.

50. There are two beams of the same depth and thickness; one 18 feet in length, the other 13; the longer beam will sustain a weight of 68 cwt.; what weight will the shorter beam sustain ?

51. Two beams of the same size, measure in length 22 and 17 feet; the shorter beam will sustain 76 cwt.; how much will the longer beam sustain?

52. Two beams of equal thickness have a depth of 14 and 16 inches respectively; the deeper beam is 20 feet long, and will sustain 84 cwt.; the other is 17 feet in length; what weight will it sustain?

First take into view the length; then, in a second proportion, the depth.

53. If a beam 25 feet in length and 9 in. in depth, will sustain a weight of 12 cwt., what weight will be sustained by a beam of the same thickness 18 feet long, and 10 in. in depth? When beams are of the same length and depth, the strength varies directly as the width.

54. There are two beams of equal length and depth; one 9 inches in width, the other 7 inches; the wider beam will sustain 47 cwt.; what weight will the narrower beam sustain?