If great accuracy be required, cut the plan into 4 portions, called 1, 2, 3, 4. First, weigh 1 and 2 together, 3 and 4 together, and take their sum. Then weigh 1 and 3 together, 2 and 4 together, and take their sum. Lastly, weigh 1 and 4 together, 2 and 3 together, and take their sum. The mean of the four aggregate weights thus obtained, compared with the weight of the standard square, will give the ratio of their surfaces very nearly.

** I have employed in this operation a balance which turns with the 100th part of a grain. The results are proportionally

accurate.

9. Prism.-L= length, B=breadth, D=depth, all in inches: then T L B D = content in yards, nearly.

If L, B, D, be in feet, as suppose the dimensions of a corn bin; then '8 L B D = the content in Winchester bushels. This is about one bushel in 200 in defect.

Ex. Suppose L= 125 inches, B == = 25, D = 24.

Then 125 × 25-12500-3125;

and 3125 x 24=6250 × 12=75000

For wrought iron square bars allow 100 inches in length of an inch square bar to a quarter of a cut. in weight; and so in proportion. This is easily remembered, because the word hundred occurs twice.

An inch square cast iron bar would require 9 feet in length for a quarter of a cwt.

Or, take of the product of the breadth and thickness, each in eighths of an inch, the result is the weight of one foot in length, in avoirdupois pounds.

Or, one foot in length of an inch square bar weighs 3 pounds.

Bricks of the usual size require 384 to a cubic yard. A rod of brick work, brick and a half thick, requires 4356 bricks.

10. Cylinder. One tenth diameter squared, (d", d being taken in inches,) gives the content in ale-gallons of a yard in length.

This rule gives a result defective only by the 376th part. I d2 x 00283257 = imperial gallons in a cylinder, dimensions in inches.

11. Timber measuring.-Let L denote the length of a tree in feet and decimals, and & the mean girth, taken in inches: then the following rules given by Mr. Andrews may be employed.

40 × 602 2304

EXAMPLE by Rule 1.-No allowance for bark.

A tree 40 feet long, and 60 inches whole girth or circumference.

= 62 cubic ft. customary, and Ex. by Rule 2.-A tree feet 50

50 X 492

3009

40 cubic ft. customary, and

40 X 602

79 cubic feet, true content.

1807

long, and 49 inches circumference.
50 × 492
2360

=

50 cubic feet, true content.

If G as well as L be in feet, then '08 L G2 content, nearly.

When the difference between the girths at the two ends is considerable, it is best to find the content of the tree as though it were a conic frustrum, and make the usual allowances afterwards.

12. Sphere of the cube of the diameter = capacity. Use the component factors, 3 and 7, in dividing by 21.

This rule gives a result true to its 2600th part.

Or, multiply the cube of the circumference by 0169, for the capacity.

13. To find the capacity, or solid content of an irregular body.-Procure a prismatic or cylindric vessel that will hold it. Put in the body, and then pour in water to cover it, marking the height to which the water reaches. Then take out the body, and observe accurately how much the water has descended in consequence. The capacity of the prism or cylinder thus left dry by the water will be evidently equal to that of the body.

If the vessel will not hold water, sand may be employed, though not with quite so much accuracy.

In this manner, too, a portion of a body may be measured without detaching it from the rest, by simply immersing that portion.

Even an irregular vessel may be employed for this purpose. In which case it should be placed in a larger vessel, and then filled with water. Then submerge the body whose magnitude you wish to determine, the quantity of water that has run over, and is caught in the exterior vessel, will be the measure. may be weighed, and its cubic measure estimated by allowing 1000 avoirdupois ounces to the cubic foot. (See farther, Hydrostatics and Specific Gravity.)

B

D

F

нкмо

It

14. To find the contents of surfaces and solids not reducible to any known figure, by the equidistant ordinate method. The general rule is included in this proposition: viz.-If any right line A N be divided into any even number of equal parts, a C, C E, E G, &c. and at the points of division be erected perpendicular ordinates A B, C D, E F, &c. terminated by any curve в HO: then, if A be put for the sum of the first and last ordinates, A B, N 0, B for the sum of the even ordinates, C D, G H, L M, &c. viz. the second, fourth, sixth, &c. and c for the sum of all the rest, E F, I K, &c. viz. the third, fifth, &c. ordinates, excepting the first and last then, the common distance A C, C E, &c. of the ordinates being multiplied into the sum arising from the addition of A, four times B, and twice c, one third of the product will be the area A B O N, very nearly.

A or the odd

CEGILN

D= area, D being =A C=c E, &c.

The same theorem will equally serve for the contents of all solids, by using the sections perpendicular to the axis instead of the ordinates. The proposition is quite accurate, for all parabolic and right lined areas, as well as for all solids generated by the revolutions of conic sections or right lines about axes, and for pyramids and their frustrums. For other areas and solidities it is an excellent approximation.

The greater the number of ordinates, or of sections, that are taken, the more accurately will the area or the capacity be determined. But in a great majority of cases five equidistant ordinates, or sections, will lead to a very accurate result.

In cask-guaging, indeed, three sections will be usually sufficient. Thus, taking the bung and head diameters, and

a diameter mid-way between them; the sum of the squares of the bung and head diameters, and of the square of double the middle diameter, multiplied into the length of the cask, and then into 785398, will give six times the content of the cask, very nearly. Or, if H, B, and м, represent the head, bung and middle diameters respectively, and L the length, all in inches; then (H3 + 4 M2 + B3) X L X 1309=content of the cask in inches.

A similar method may be advantageously adopted in all cases of ullaging either standing or lying casks, by taking the areas at the top, bottom, and middle, of the liquor. See Hutton's Mensuration, part iv.

EXAMPLE. The bung diameter of a cask is 32 inches, the head diameter 24; the middle 30-2, the length 40. Required the content in ale gallons of 282 cubic inches, and in wine gallons of 231.

The former multiplier, divided by 282 and 231 respectively, gives 00047 and 0005, for the proper multipliers.

Hence (32+24+4. 30.2") x 40 x 00047-97-44 ale gallons :

And (322+24+4. 30-2) x 40 x 00053-118.95 wine gallons:

Or of 97.44 (the ale gallons) gives 119, the wine gallons, very nearly.

Also of 118.95 (the wine gallons)=99-125 imperial gallons. See the table of factors, p. 19.

MECHANICS.

1. Mechanics is the science of equilibrium and of motion.

2. Every cause which tends to move a body, or to stop it when in motion, or to change the direction of its motion, is called a force or power.

3. When the forces that act upon a body, destroy or annihilate each other's operation, so that the body remains quiescent, there is said to be an equilibrium.

4. Statics has for its object the equilibrium of forces applied to solid bodies.

5. Dynamics relates to the circumstances of the motion of solid bodies.

6. Hydrostatics is devoted to the equilibrium of fluids.

7. Hydrodynamics relates to their motion, and connected circumstances.

8. The properties and operation of elastic fluids are often treated distinctly, under the head of Pneumatics.

9. A single force, which would give to a physical point, or to a body, the same motion both in velocity and direction as several forces acting simultaneously, is called the resultant of those forces, while they are called the constituents or the composants of the single resulting force.

10. The action of a force is the same in which ever point of its direction it is applied; unless the manner of its action be changed.

11. Vis inertiæ, or power of inactivity, is defined by Newton to be a power implanted in all matter, by which it resists any change attempted to be made in its state, that is, by which it requires force to alter its state, either of rest or motion.