6. What number of bottles, containing a pint and a half each, can be filled with a barrel of cider? Ans. 168.

7. How many pints, quarts, and two quarts, each an equal number, may be filled from a pipe of wine? Ans. 144.

8. LONG MEASURE.

1. In 51 miles, how many furlongs and poles?

Ans. 40 fur.. 16320 poles. 2. In 49 yards, how many feet, inches, and barley-corns? Ans. 147 ft.. 1764 inch. 5292 b. c. 3. How many inches from Boston to New-York, it being 248 miles? 4. In 4352 inches, how many yards?

Ans. 15713280 inch.

Ans. 120 yds. 2 ft. 8 in.

5. In 682 yards, how many rods?

Ans. 652×2÷ 11-124 rods.

6. In 15840 yards, how many miles and leagues?

Ans. 9 m. 3 lea.

7. How many times will a carriage wheel, 16 feet and 9 inches in circumference, turn round in going from NewYork to Philadelphia; it being 96 miles?

Ans. 30261 times, and 8 feet over. 8. How many barley-corns will reach round the globe, it being 360 degrees? Ans. 4755801600.

9. LAND OR SQUARE MEASURE,

1. In 241 acres, 3 roods, and 25 poles, how many square rods or perches?

Ans. 38705 perches. 2. In 20692 square poles, how many acres? Ans. 129 a. Ir. 12po. 3. If a piece of land contain 24 acres, and an enclosure of 17 acres, 3 roods, and 20 rods, be taken out of it, how many perches are there in the remainder?

Ans. 980 perches.

4. Three fields contain, the first 7 acres, the second 10 acres, the third 12 acres, 1 rood; how many shares can they be divided into, each share to contain 76 rods?

Ans. 61 shares and 44 rods over.

10. SOLID MEASURE.

1. In 14 tons of hewn timber, how many solid inches? Ans. 14 50 × 1728—1209600.

2. In 19 tons of round timber, how many inches?

Ans. 1313280.

3. In 21 cords of wood, how many solid feet?

Ans. 21 128=2688.

4. In 12 cords of wood, how many solid feet and inches? Ans. 1536 ft. and 2654208 inch.

5. In 4608 solid feet of wood, how many cords?

Ans. 36 cds.

1. In 41 weeks, how many days, hours, minutes, and seconds? Ans. 287 d. 6888 h. 413280 min. and 24796800 sec. 2. In 214 d. 15 h. 31 m. 25 sec. how many seconds? Ans. 18545485 sec. 3. In 24796800 seconds, how many weeks? Ans. 41 wks. 4. In 184009 minutes, how many days?

Ans. 127 d. 18 h. 49 min. 5. How many days from the birth of Christ, to Christmas, 1797, allowing the year to contain 365 days, 6 hours? Ans. 676354 d. 6 h.

6. Suppose your age to be 16 years and 20 days, how many seconds old are you, allowing 365 days and 6 hours to the year? Ans. 506649600 sec. 7. From March 24, to November 19th following, inclusive, how many days?

12. CIRCULAR MOTION.

Ans. 262.

1. In 7 signs, 15 24′ 40′′, how many degrees, minutes, and seconds? Ans. 225° 13524' and 811480".

2. Bring 1020300 seconds intó signs.

Ans. 9 signs, 13° 25′.

Questions to exercise Reduction.

1. In 1259 groats, how many farthings, pence, shillings,

and guineas, at 28s. ?

Ans. 20144grs. 5036d. 419s. 8d.

and 14 guineas, 27s. 8d.

2. Borrowed 10 English guineas at 28s. each, and 24 English crowns at 6s. and 8d. each; how many pistoles at 22s. each, will pay the debt? Ans. 20. 3. Four men brought each 177. 10s. sterling value in gold into the mint, how many guineas at 21s. each must they receive in return? Ans. 66 guin. 14s.

4. A silversmith received three ingots of silver, each weighing 27 ounces, with directions to make them into spoons of 2 oz., cups of 5 oz., salts of 1 oz., and snuff-boxes of 2 oz., and deliver an equal number of each; what was the number? Ans. 8 of each, and 1 oz. over.

5. Admit a ship's cargo from Bordeaux to be 250 pipes, 130 hhds. and 150 quarter casks, [ hhds.] how many gallons in all; allowing-every pint to be a pound, what burden was the ship of? Ans. 44415 gals. and the ship's burden was 158 tons, 12 cwt. 2 qrs.

6. In 15 pieces of cloth, each piece. 20 yards, how many French Ells? Ans. 200. 7. In 10 bales of cloth, each bale 12 pieces, and each piece 25 Flemish Ells, how many yards? Ans. 2250.

8. The forward wheels of a wagon are 14 feet in circumference, and the hind wheels 15 feet and 9 inches; how many more times will the forward wheels turn round than the hind wheels, in running from Boston to New-York, it being 248 miles? Ans. 7167.

9. How many times will a ship 97 feet 6 inches long, sail her length in the distance of 12800 leagues and ten yards? Ans. 2079508.

10. The sun is 95,000,000 of miles from the earth, and a cannon ball at its first discharge flies about a mile in 71 seconds; how long would a cannon ball be, at that rate in flying from here to the sun? Ans. 22 yr. 216 d. 12 h. 40 m.

11. The sun travels through 6 signs of the zodiac in half a year; how many degrees, minutes, and seconds? Ans. 180 deg. 10800 min. 648000 sec. 12. How many strokes does a regular clock strike in 365 days, or a year? Ans. 56940. 13. How long will it take to count a million, at the rate of 50 a minute? Ans. 333 h. 20 m. or 13 d. 21 h. 20 m.

14. The national debt of England amounts to about 279 millions of pounds sterling; how long would it take to count this debt in dollars (4s. 6d. sterling) reckoning without intermission twelve hours a day at the rate of 50 dols. a minute, and 365 days to the year?

Ans. 94 years, 134 days, 5 hours, 20 min.

FRACTIONS.

FRACTIONS, or broken numbers, are expressions for any assignable part of a unit or whole number, and (in general) are of two kinds, viz.

VULGAR AND DECIMAL.

A Vulgar Fraction, is represented by two numbers placed one above another, with a line drawn between them, thus, ,, &c. signifies three fourths, five eighths, &c.

3 5

The figure above the line, is called the numerator, and that below it, the denominator;

The denominator (which is, the divisor in division) shows how many parts the integer is divided into; and the numerator (which is the remainder after division) shows how many of those parts are meant by the fraction.

A fraction is said to be in its least or lowest terms, when it is expressed by the least numbers possible, as when reduced to its lowest terms will be, and is equal to 3, &c.

PROBLEM I.

To abbreviate or reduce fractions to their lowest terms. RULE.-Divide the terms of the given fraction by any number which will divide them without a remainder, and the quotients again in the same manner; and so on, till it appears that there is no number greater than 1, which will divide them, and the fraction will be in its least terms.

To find the value of a fraction in the known parts of the integer, as to coin, weight, measure, &c.

RULE. Multiply the numerator by the common parts of the integer, and divide by the denominator, &c.

EXAMPLES.

1. What is the value of of a pound sterling?

3. Reduce of a shilling to its proper quantity. Ans. 41d. 4. What is the value of 3 of a shilling?

Ans. 44d.

5. What is the value of 12 of a pound troy? Ans. 9oz. 6. How much is of a hundred weight?

Ans. 3 qrs. 7 lb. 10 oz.

7. What is the value of 1⁄2 of a mile?

Ans. 6 fur. 26 po. 11 ft. 8. How much is of a cwt.? Ans. 3 qrs. 3 lb. 1 oz. 124 dr. 9. Reduce of an Ell English to its proper quantity.

10. How much is of a hhd. of wine?

Ans. 2 qrs. 33 na.
Ans, 54 gal.