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their boundaries, they may all be reduced to a mean length and breadth ;) the following is that which is applied to solid bodies, or magnitudes of three dimensions--that is, having length, breadth, and thickness. The scholar has already seen that the content of such is estimated in cubes, and is the product of the length, breadth, and thickness, all taken in the
Solid or Cubic Measure. 1000 Eng.or 1728 Fr. cubic lines, c. l., make 1 cubic inch, c.in. 1728 cubic inches......
.1 cubic foot, c. ft. 27 cubic feet.....
.1 cubic yard, c. yd. 40 feet of round, or 50 ft. of hewn timber 1 ton, t. 42 cubic feet.....
.1 ton of shipping. Firewood, 8 ft. long, 4 ft. broad, and
4 ft.high, or 128 cubic feet..........1 cord, c. 16 c. ft. or s of a cord, in the N.E. States, 1 foot of wood. How many Eng. cubic lines in a cubic mile?
Ans. Two hundred and fifty-four quadrillions, three hundred and fifty-eight trillions, sixty-one billions, and fifty-six millions.
288. The following measure is applied to dry articles, such as corn, fruit, seed, roots, salt, sand, oysters, coal, &c. The principal unit of this measure is the Winchester bushel, which is a round vessel 18.1 inches in diameter throughout, and 8 inches deep, and contains 21503 + cubic inches. The dry gallon contains, of course, 2684 cubic inches.
Dry Measure. 2 pints make...
l quart, qt. 4 quarts....
1 gallon, gal. 2 gallons, or 8 quarts ...... 1 peck, pk.
4 pecks, 8 gals., or 32 qts.... 1 busbel, bush. As 1738=}, the cubic foot, 1728 cubic inches, is to the bushel, 2150 cubic inches, very nearly as 4 to 5. But the greater the magnitude of the unit, the less must be the number to make a certain quantity, and the less the magnitude, the greater the number; wherefore, the numbers are inversely as the magnitudes; that is, the magnitude of the foot being to that of the bushel as 4 to 5, the number of feet equivalent to a given number of bushels must be as 5 to 4. Therefore, 5 feet are nearly equivalent to 4 bushels; and hence we have the
following rule, which will be found very useful in rural economy, commerce, and shipping :
To reduce bushels to feet: To the number of bushels add
of itself To reduce feet to bushels : From the number of feet subtract of itself.
If we have 1728 for the given number of bushels, we shall, by the rule, find its equivalent in cubic feet to be 2160, which is too great by 10, or zic part of itself, the true equivalent being 2150. Therefore, in finding by the rule, the equivalent, in feet, of a given number of bushels, we must, if we wish it exact, diminish it by zie of itself.
Again, if we have 2150 for the given number of cubic feet, we shall, by the rule, find, for its equivalent in bushels, 1720. But this is too little by 8 bushels, or zis part of itself, the true equivalent being 1728. Therefore, in finding by the rule the equivalent in bushels, of a given number of cubic feet, we must, if we wish it exact, increase it by zis of itself.
289. We may however observe, that for any practical purpose, such as the construction of corn-cribs, caves, bins, granaries, pits, boxes, &c., for the preservation of corn, roots, or any other articles measured by Dry Measure, or for the calculation of the capacity of such cribs, granaries, &c., the rule will be found sufficiently exact without the addition or subtraction of the fractional part. Suppose we would construct a crib to hold 1500 bushels of
As corn is put away in the ear, and as it requires, in measuring, to allow two bushels for one, and one of these well heaped, we shall allow 2} times the whole quantity of shelled bushels. We therefore say, 1500 X 2}=3300; adding to this į of itself, we have 3300 + 825 4125, the content of the crib in cubic feet. Now as this is a quantity of 3 dimensions, one of which is arbitrary—seeing that the average width of a corn-crib is generally not more than 4 or 5 feet'inside, that the air may pass freely-we find the factors of 4125, or inside dimensions of the crib thus : dividing by any convenient width, say 5 feet, we have 4125
= 825, for the area or content of the side. Then, if we wish the crib to be of a certain height, or length, we must divide the area of the side by that height or that length, and the quotient will be the remaining factor; that is the length or height accordingly. Thus, if we say 10 feet high, we have
4PA - PA
84 = 824 feet, for the length. If we say 55 feet long, we have 8 = 15 feet, for the height of the crib.
If the crib be made only 4 feet wide at the bottom, it must be 6 feet wide at the top, because the sum of these will give the average width, 5 feet.
What must be the length of a crib to hold 3600 bushels of corn,
Answer, 165 feet. What must be the height of a crib to hold 500 bushels of corn, the average width being 4 ft. 2 in., and the length 40 ft. 6 in ?
Answer, 8 ft. 12 in. 290. If, of several factors, any one be diminished in a certain ratio, that is, by a certain part of itself, the whole product of those factors will be diminished in the same ratio; that is, by the same part of itself.
Let P represent the product of the undiminished factors, and A the factor we intend to diminish. Then PA is the total product. Now, if we diminish A by 1 of itself; thus, A-Å, the remainder 34, multiplied by P, will be 3pl But PA
=s Wherefore, the total product of PA is diminished in the same ratio as the factor A.
Having a granary 43 feet long, 10 ft. wide, and 10 ft. high, we wish to know how many bushels of grain it will hold,
Now, as by the rule (228) we must, from the content of the granary in cubic feet, which is 43 X 10 X 10, subtract of itself; we subtract from 10, which is of the factors, of itself, which gives 43 X 10 X 8 3440 for the content in bushels. But this is too little by zig of itself. We therefore say,
16, and 3440 + 16 3456, the number of bushels required.
43 x 10 x 10 x 1728 43 x 1728 Proof.
2 x 1728 = 2150
21,5 3456 bush., as before.
Having 6912 bushels of grain, we wish to know to what depth it will cover a floor that is 43 ft. long and 25 ft. wide.
1728 one-fourth added.
which we subtract.
40 is dit?
43 x 25
1075 area, or surface of floor. 8600 -- 1075=8 feet, the required depth. Proof. 43 X 20 X 8=6880, too little by 15 of itself.
= 32; then, 6880 + 32 6912 bushels. 291. In the heaped bushel, used in measuring fruit, roots, &c., it is found that, according to pretty general usage, the heaped part is only about } of the capacity of the vessel, which, as we shall shortly see, is by no means a large allowance. But custom is law, at least to the calculator; wherefore, applying to this our approximate rule, (228,) as the internal capacity of the bushel (called the strick bushel) compared with the cubic foot, is as 5 to 4, and the heaped part 1 of 5, or 1, we have, for the ratio of the heaped bushel to the foot 6 to 4, or 3 to 2; hence the following rule :
To reduce heaped bushels to cubic feet: To the number of heaped bushels, add its half. To reduce cubic feet to heaped bushels: From the number of cubic feet, subtract its third part.
To what length must a pit, 4 ft. wide and 3 ft. deep, be dug, in order to contain 3000 bushels of potatoes ?
Ans. 375 feet. How many
bushels of coal can be stored in a cellar which is 16 ft. long, 10 ft. wide, and 9 ft. high ? Ans. 960.
292. In England, a bushel of potatoes is used consisting of 80lbs. weight Avoirdupois. This was probably based upon the following experiment: Let a tight half-bushel measure be filled with good-sized potatoes, by packing them with the band; then, if water be poured into it amongst the potatoes, it will contain about 8 quarts before the water begins to run over, which proves that, when a half-bushel is thus filled, only one half its solid capacity is occupied by the potatoes; much less when, as usual, it is filled with the shovel.
If we mieasure 80 lbs. of potatoes, shovelling them on the halfbushel as long as they will conveniently lie, the measure will be thus filled 3 times. A bushel, therefore, as they are frequently measured, will only weigh about 53 lbs. Now, as a bushel of good wheat, which weighs from 62 to 64 lbs., is often lower in price than a bushel of potatoes, the student will see that it is important to estimate as well as calculate. In Maine the bushel of potatoes is 60 lbs , Avoirdupois.
293. By the following measure are measured brandy, spirits, cider, vinegar, molasses, oil, &c. 'Honey is sold by the pound Avoirdupois.
The gallon contains 231 cubic inches. Ten gallons make one anker.
Wine Measure. 4 gills, gi., make...
1 pint, pt. 2 pints
.1 quart, qt. 4 quarts.
1 gallon, gal 314 gallons..
.1 barrel, übl. or bur. 42 gallons..
.1 tierce, tier. 63 gallons...
1 hogshead, hhd.
.1 puncheon, pun.
2 pipes, or 252 gallons...... 1 tun, T. The following measure is used for malt liquor only. The gallon contains 282 cubic inches.
1 gallon, gal.
.1 firkin, fir. 2 firkins
1 kilderkin, kil.
.1 barrel, bar.
.1 butt, b.
1 tun, T.
OF TIME, AND THE REGULATION OF CLOCKS-ASTRONO.
AND GEOGRAPHICAL MEASURE.
294. No subject that we can contemplate is of deeper interest, or more serious consequence to mankind, than time. The wise in all ages have testified that on the proper improvement of
depends our present as well as future happiness; and, though to some it may appear irrelevant to introduce in this place subjects of graver import than mere calculation, it may not be amiss to remind the student, in the words of Don José de Cadalso, (in a little satirical work entitled Los Eruditos á la