The woodworker soon discovers that arithmetic is a very practical and necessary subject. He will meet many problems both in drawing and in actual construction which test his ability and call for some knowledge even of elementary geometry. It is important to be able to estimate from his drawing just how much lumber will be needed. He will soon discover through intercourse with dealers in lumber that there are certain standard sizes, and he should make his designs as far as possible conform at least to standard thicknesses.

Common boards are sawed 1 inch thick. When dressed on two sides the thickness is reduced to ⁷/₈ . In planning some part of a structure to be 1 inch thick it is better to make the dimension ⁷/₈ inch, else it will be necessary to have heavier material planed to 1 inch and the cost will be that of the heavier lumber, plus the expense of planing. In buying ¹/₂-inch dressed lumber, very often inch boards are dressed down to the required thickness, and the purchaser pays for 1-inch wood, in addition to the dressing. The boy is surprised to find that it costs more for ¹/₂-inch than for inch material. Standard lengths are 10, 12, 14, 16, etc., feet. Widths vary, and as wood shrinks only across the grain—with one or two exceptions—this dimension cannot be depended on, as the amount of shrinkage depends somewhat on the age after cutting. Whenever possible, it is wise to go to the lumber yard and select your own material, choosing boards that are free from knots, shakes, etc. Clear lumber—free from knots—costs more, but is worth the difference.

MEASUREMENTS

A measurement is a comparison. We measure the length of a lot by comparing it with the standard of length, the yard or foot. We measure a farm by comparing its area with the standard unit of surface measure, the acre, square rod, or square yard.

In every measurement we must first have an accepted standard unit. The history of units of measurement is a very interesting one, and its difficulty arises from the fact that no two things in nature are the same. One of the ancient units of length was the cubit, supposed to be the length of a man's forearm, from the elbow to the end of the middle finger. This, like other natural units, varied and was therefore unreliable. As civilization progressed it became necessary for the various governments to take up the question of units of measurements and to define just what they should be.

Our own standards are copied from those of Great Britain, and although congress is empowered to prescribe what shall be our units, little has been changed, so that with few exceptions we are still using English measurements.

The almost hopeless confusion and unnecessary complication of figures is shown in the following tables as compared with the metric system:

The original English definition of an inch was "three barley corns" with rounded ends. The meter is 1/10,000,000 (one ten-millionth) of a quadrant of the earth's circumference, i. e., the distance from the pole to the equator measured along one of the meridians of longitude. The length of three barley corns might be different from the next three, so here was the original difficulty again. The designers of the metric system went back to the earth itself as the only unchangeable thing—and—are we sure there is no change in the earth's circumference? The great advantage of the metric is that it is a decimal system and includes weights as well as surfaces and solids. Our weights are even more distracting than our long measure. We have in fact two kinds of weight measure—troy and avoirdupois.

In surface measurements, the same differences are seen:

In measures of volumes we are as badly off:

As if this were not enough, when we go to sea we use another system. The depth of water is measured in fathoms (6 feet = 1 fathom), the mile is 6086.07 feet long = 1.152664 land miles, and 3 sea miles = 1 league. In our cubic measure:

Isn't it about time we used the metric system? The reader will not mind one more standard unit. Lumber is measured by the board foot. Its dimensions are 12 × 12 × 1 inches; it contains 144 cubic inches and is 1¹/₁₂ of a cubic foot. A board 10 feet long, 1 foot wide and 1 inch thick contains 10 board feet. One of the same length and width but only ¹/₂ inch thick contains 5 board feet.

The contents of any piece of timber reduced to cubic inches can be found in board feet by dividing by 144, or from cubic feet by multiplying by 12. As simple examples: How many board feet in a piece of lumber containing 2,880 cubic inches? ²⁸⁸⁰/₁₄₄ = 20 board feet. How much wood in a joist 16 feet long, 12 inches wide and 6 inches thick? 16 × 1 × ¹/₂ = 8 cubic feet: 8 × 12 = 96 board feet. A simpler method may be used in most cases. How much wood in a beam 9 inches × 6 inches, 14 feet long? Imagine this timber built up of 1-inch boards. As there are nine of them, and each 14 ft. × ¹/₂ foot × 1 inch and contains 7 board feet (Fig. 235), 7 × 9 = 63 board feet. Again, how much wood in a timber 8 inches × 4 inches, 18 feet long? This is equivalent to 4 boards 1 inch thick and 8 inches or ²/₃ foot wide. Each board is 18 × ²/₃ × 1 = 12 board feet, and 12 × 4 = 48, answer. (See b, Fig. 235).To take a theoretical case: How much wood in a solid circular log of uniform diameter, 16 inches in diameter, 13 feet and 9 inches long? Find the area of a 16-inch circle in square inches, multiply by length in inches and divide by 144.

It is not likely that a boy would often need to figure such an example, but if the approximate weight of such a timber were desired, this method could be used, reducing the answer to cubic feet and multiplying by the weight per cubic foot.

A knowledge of square root is often of great value to the woodworker for estimating diagonals or squaring foundations. The latter is usually based on the known relation of an hypothenuse to its base and altitude. It is the carpenters' 3-4-5 rule. The square of the base added to the square of the altitude = square of the hypothenuse. 3² = 9, 4² = 16; 9 + 16 = 25. The square root of 25 is 5. (See Fig. 235). To square the corner of his foundation the carpenter measures 6 feet one way and 8 the other. If his 10-foot pole just touches the two marks, the corner is square. 6² = 36, 8² = 64; 36 + 64 = 100. v100 = 10. This method was used in laying out the tennis court, the figures being 36, 48, 60—3, 4, and 5 multiplied by 12.

To take a more practical case, suppose we are called upon to estimate exactly, without any allowance for waste, the amount of lumber in a packing case built of one-inch stock, whose outside dimensions are 4 feet 8 inches × 3 feet 2 inches × 2 feet 8 inches. Referring to the drawing (Fig. 235, d), we draw up the following bill of material:

The top and bottom, extending full length and width, are the full dimensions of the box, while the sides, although full length, are not the full height, on account of the thickness of the top and bottom pieces—hence the dimensions, 2 feet 6 inches. From the ends must be deducted two inches from each dimension, for the same reason. In multiplying, simplify as much as possible. There are four pieces 2 feet 6 inches wide; as their combined length is 15 feet 4 inches, we have 15¹/₃ feet × 2¹/₂ feet = 38¹/₃ square feet. The combined length of top and bottom is 9 feet 4 inches = 9¹/₃ × 3¹/₆ = 29⁵/₅ , and 38¹/₃ + 29⁵/₅ = 67⁸/₅ or 68 board feet, ignoring such a small amount as ¹/₅ of a foot. This is close figuring, too close for practical work, but it is better to figure the exact amount, and then make allowances for waste, than to depend on loose methods of figuring, such as dropping fractions, to take care of the waste.

As a good example of estimation, take the hexagonal tabourette shown in Fig. 178; the five pieces, aside from the hexagon under and supporting the top, which may be made from scrap lumber, are shown laid out in Fig. 236. The board must be at least twelve inches wide in order to get out of it the large hexagon. The legs may be laid out as shown with space left between for sawing, yet even by this method considerable waste will result, and it should be kept constantly in mind that as far as possible waste is to be reduced to a minimum. "Wood butcher" is the common shop name for the workman who spoils more material than he uses.

The great advantage of making out a bill of material before starting is that it not only makes you study your drawing, but causes you to consider the best method of laying out the blank pieces.

It is often necessary to find the areas of figures other than the square or parallelogram. Assume that we are to floor a room in an octagonal tower or summer house. If the distance across the flat sides of the octagon is sixteen feet, leaving out the item of waste, how many square feet will be required?

The octagon may be drawn in a square and its area will be that of the square, less the four triangles in the corners. (Fig. 237). So the problem resolves itself into finding the area of one of these triangles. If we knew the length of one of the sides of the octagon, the solution would be simple, but we only know that the eight sides are equal. The following method may be worked out: Find the diagonal of the sixteen foot square. It is 22.6+. Deduct the distance across the flats, 16, leaving 6.6 feet equally divided between a and b; a = 3.3 and it may be proved that c = a = d. So in each corner we have a triangle whose base is 6.6 × 3.3. The area of a triangle equals half its base by the altitude. Therefore the area of each triangle is 3.3 × 3.3 and 3.3 × 3.3 × 4 equals 43.56 square feet, the combined area of the four corners. This deducted from the area of the square leaves the area of the octagon, or 256-43.56 = 212.44 square feet.

Assume that our problem is to find the narrowest board we can use to cut out a hexagon whose diameter is fourteen inches. As shown in Chapter IV, the hexagon is drawn in a circle. One of the sides is equal to the radius or half the diameter. This gives us the arrangement shown in Fig. 238, in which our problem is confined to the right-angled triangle whose base is seven and hypothenuse fourteen. From our knowledge of triangles, we deduct the square of seven (49) from the square of 14 (196), leaving 196-49 = 147, which is the square of the altitude. Then v147 = 12.12, which is the narrowest board from which we can obtain a hexagon 14 inches in diameter.

These examples are given to show the close connection between woodwork and arithmetic.