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4 in. square is how many square inches? 5 ft. square is how many square feet? 10 miles square is how many square miles?

§ 167. Cubic or Solid Measure

Is used in measuring solids, or any extension in length, breadth and thickness.

A cubic inch is an inch long, an inch wide, and an inch thick; a cubic foot is a foot long, a foot wide, and a foot thick, and

so on.

2 in. long, 1 in. wide, and 1 in. thick would make how many cubic inches? 2 in. long, 2 in, wide, and 1 in. thick, would make how many cubic inches? 2 in. long, 2 in. wide, and 2 in. thick, would make how many cubic inches?

1728 cubic inches (cu. in.) make 1 cubic foot, (cu. ft.)

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1 cubic yard, (cu. yd.);

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1 cord.

A cord of wood is usually put up 8 ft. long, 4 ft. wide, and 4 feet high. One foot in length of such a pile is called a cord foot, and contains 16 cubic feet.

50 cubic feet of timber are allowed to weigh a ton. Of round timber such a quantity is allowed for a ton as, when hewn, will make 40 cubic feet.

A perch of stone is estimated at 1 rod or perch, which is 16 ft., in length, 1 ft. in thickness, and 1 ft. in height; and contains 24 cu. ft.

231 cu. in. is the capacity of a gallon in Wine Measure, and 282 cu. in. is the capacity of a gallon in Beer Measure.

The British Imperial gallon contains 277.274 cu. in., and the Imperial bushel, being 8 Imp. gal. contains 2218.192 cu. in. The British Winchester bushel, which is the standard bushel in the United States, contains 2150.4 cubic inches.

$168. Circular Measure

Is used in measuring any part of the circumference of a circle, in reckoning latitude and longitude, and the motions of the heavenly bodies.

60 seconds (") make 1 minute, (');

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1 degree, (°);

360 degrees, the circumference of any circle.

A degree it is evident, has no determinate linear extent; being always the 360th part of the circumference on which it is taken, it is greater or less as that circumference is greater or less.

A degree on the circumference of the earth, is about 69 miles.

One minute on the circumference of the earth, is called a geographical or nautical mile; the mile of linear measure being denominated a statute mile.

$ 169. Measure of Time.

TIME is measured in days by the revolution of the earth round its axis, and in years by the revolution of the earth around he sun.

60 seconds (sec.) make 1 minute, (min.);

60 minutes,.

24 hours,.

7 days,

365 days,

1 hour, (hr.);

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366 days,

100 years,

1 week, (wk.);

1 common year, (yr.);

1 leap year;

1 century.

A year also consists of 12 months, viz: January, February, March, April, May, June, July, August, September, October, November, and December.

The number of days in each is as follows:

Thirty days has September, April, June, and November;

February has twenty-eight alone, and all the rest have thirty-one;

But Leap Year comes one year in four, when February has one day more.

Or, the fourth, eleventh, ninth, and sixth, have 30 da. to each affixed;
And every other 31, except the second month alone,
To which we 28 assign, till Leap Year gives it 29.

Solar, Civil, and Leap Years

§ 170. The period of the earth's revolution around the sun, is 365 da. 5 hr. 48 min. 49.6 sec. This constitutes the solar year, being 5 hr. 48 min. 49.6 sec. longer than the common civil year of 365 days.

To correct the error which arises from reckoning only 365 days to a year, one day is added to February every fourth year; and this makes the Leap Year of 366 days. But one day is more than the excess (5 hr. 48 min. 49.6 sec.) of the solar above the civil year, amounts to in 4 years.

To correct this second error, so as to preserve the civil in agreement with the solar years, the following rule has been adopted; viz: if the number of the year is divisible by 4 without a remainder, it is made LEAP YEAR; but the closing year of a century, as 1700, 1800, &c., is not made Leap Year, unless the number is divisible by 400, without a remainder.

REDUCTION OF MONOMIALS AND POLYNOMIALS.

§ 171. A monomial quantity, or simply a monomial, is a quantity expressed by a single name of measuring units, (§ 154). Thus 5 dollars is a monomial; 10 shillings is a monomial. $172. A polynomial quantity, or simply a polynomial, is a quantity expressed by two or more names of measuring units. Thus 5 dollars 25 cents is a polynomial; 3 pounds, 10 shillings, and 6 pence, is a polynomial.

A polynomial is composed of two or more monomials, which may thence be called the terms of the polynomial.

Thus in the first example given, the terms are 5 dol. and 25 c.; and in the second, 3£, 10 s. 6 d.

Note.-Monomial quantities have by some been called denominate_numbers, and polynomials have usually been called compound numbers.

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§ 173. Reduction descending consists in finding the value of a given quantity in measuring units of a lower order, (§ 155). The quantity is then said to be reduced to a lower name or denomination.

RULE XXXIII.

§ 174. To reduce a Quantity to a LOWER DENOMINATION.

1. Multiply a monomial of a higher denomination, or the highest term of a polynomial, by that number of the next lower denomination which makes a unit of the higher: the product will be in the lower denomination.

2. This product may, in like manner, be reduced to a still lower denomination, and so on, observing that each lower term in a polynomial must be added to the product in the same denomination with itself.

3. In reducing a MONOMIAL FRACTION to lower denominations, the integers in the successive products may be reserved, and afterwards arranged as the terms of a polynomial.

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We multiply 5 £ by 20, because 20 s. make 1 £; the product is shillings-to which adding the 14 s., we have 114 s.

We next multiply the 114s. by 12, because 12 d. make 1 s.; the product is pence-to which adding the 9d., we have 1377 d. Thus we find 5 £ 14 s. 9 d. to be equal to 1377 d.

EXERCISES.

1. Reduce 4 lb. 7 oz. 13 dut. to dwt.

Recollect that 12 oz. make 1 lb., and 20 dwt. make 1 oz.

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2. Reduce 7 lb. 10 dwt. 2 gr. to gr.. 3. Reduce 3 T. 2 cwt. 3 qr. to qr. 4. Reduce 9 cwt. 1 qr. 13 oz. to oz.. 5. Reduce 14 3, 2 12 gr. to gr. 6. Reduce 8 ib, 3, 15 gr. to gr. 7. Reduce 15 bu. 2 pk. 7 qt. to qt.. 8. Reduce 9 bu. 5 qt. 1 pt. to pt.. 9. Reduce 3 pi. 1 hhd. 40 gal. to gal. 10. Reduce 4 tuns, 5 hhd. 3 qt. to qt.. 11. Reduce 13 m. 7 fur. 25 r. to r. 12. Reduce 10 L. 16 fur. 15 p. to p. 13. Reduce 20 yd. 3 qr. 2 na. to na. 14. Reduce 31 yd. 3 na. 2 in. to in.. 15. Reduce 14 A. 1 R. 20 P. to P.

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Ans. 43957 sq. yd.

Ans.

Ans.

16. Reduce 9 A. 13 P. 4 sq. yd. to sq. yd..
17. Reduce 10 cu. yd. 17 cu. ft. to cu. ft.
18. Reduce 4 cu. yd. 100 cu. in. to cu. in.
19. Reduce 20 wk. 5 da. 33 hr. 5 min. to min. Ans.
20. Reduce 1 yr. 100 da. 20 hr. 5 min. to min. Ans.
21. Reduce 7 T. 13 cwt. 1 qr. 4 lb. to oz. .
22. Reduce 75 th, 10 3,7 3,2,11 gr. to gr. Ans.
23. Reduce 3 hhd. 40 gal. 3 qt. 1 pt. to gills.
24. Reduce 5 L. 2 m. 4 fur. 15 r. to yards.

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Ans.

287 cu. ft. 186724 cu. in.

210785 min. 670805 min.

274688 oz.

437271 gr.

Ans.
Ans.

7356 gills.

30882 yd.

25. Reduce 11A. 2 R. 25 P. 25 sq.yd. to sq.yd. Ans. 564414 sq. yd.

Monomial Fractions Reduced to Integers.

26. Reduce £ to integers in shillings, &c.

£ multiplied by 20 produces s. 5 shillings, (§ 174). Reserving the integer 5 s., and reducing the fraction & s. to

pence,

s. multiplied by 12 produces 9 d.=84 pence. Reserving the integer 8 d., and reducing the fraction d. to farthings,

d. multiplied by 4 produces 6 qr.-2 farthings. Arranging the integers reserved as the terms of a polynomial, we find £5 s. 8 d. 23 qr.

27. Reduce .23 £ to integers in shillings, &c.

.23 £ multiplied by 20 produces 4.60 shillings, (§ 174). Reserving the integer 4 s., and reducing the fraction .60 s. to pence,

.60 s. multiplied by 12 produces 7.20 pence. Reserving the integer 7 d., and reducing the fraction .20 d. to 20 d. multiplied by 4 produces .80 farthings. Arranging the integers reserved as the terms of a polynomial, we find .23 £ 4 s. 7 d. 0.80 qr.

qr.,

The integers found in reducing are arranged with the quantity in the lowest denomination, whether that quantity be an integer or otherwise.

28. Reduce lb. to integers in oz., &c.

Ans. 5 oz. 6 dut. 16 gr.

29. Reduce .17 lb. to integers in oz., &c.

Ans. 2 oz. O dwt. 19.2 gr.

30. Reduce gr. to integers in lb., &c.

31. Reduce .19 T. to integers in cwt.,

Ans. 18 lb. 10 oz. 10 dr.
&c.

Ans. 3 cwt. 3 qr. 5.6 lb.
Ans. 2 3,0 9, 8 gr.
Ans. 2 3, 29, 8 gr.
Ans. 4 qt. 1 pt. 1 gi.

32. Reduce 3 to integers in 3, &c.
33. Reduce .35 3 to integers in 3, &c.
34. Reduce pk. to integers in qt., &c.
35. Reduce pi. to integers in hhd., &c.

Ans. 1 hhd. 12 gal. 2 qt.

36. Reduce .31 bu. to integers in pk. &c.

Ans. 1 pk. 1 qt. 1.84 pt.

37. Reduce .6 tun, to integers in pi., &c.

Ans. 1 pi. 25 gal. 1.6 pt,

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