The month mentioned here is the lunar month, but a common year consists of 12 calendar months, of 30 days nearly.

The true solar year is 365 days 5 hrs. 48 min. 48 sec. (which is 11 min. 12 sec. less than the Julian year), but as it was most convenient to begin the year on the commencement of a day, it was agreed, in the time of Julius Cæsar, that the year should, during 3 successive times, consist of 365 days, called a common year, and of a year of 366 days called a leap year; and it is so managed that whenever the number of years is divisible by 4 the corresponding year is a leap year, the month of February having 29 days instead of 28.

But as the true solar year is 365.242264, and not 365.25, the correction is too much by 0.007736 day; by finding how many times this is contained in 1 day, we shall know in how many years the error amounts to 1 day, which is every 129.2657 years nearly; or in every 400 years the error is very nearly 3.0944 days. So it was ordained that whenever the number expressing the centuries is not divisible by 4, the corresponding year shall not be a leap year; thus 1600 was a leap year, because 16 is divisibe by 4; but 1700 and 1800 are not, since 17 and 18 are not divisible by 4. This correction is too great, but the error only amounts to 28 hours in 5000 years.

The calendar, thus rectified, is called the Gregorian calendar, and is used throughout Europe, with the exception of Russia and Greece, where the Julian calendar is still employed. The first was promulgated by Pope Gregory, in 1582; and England adopted it on the 2nd of September, 1752. The error then amounted to 11 days, which were omitted, and the 3rd of September was called the 14th. This calendar is called the new style, the Julian being the old style.

REMARKABLE CHRONOLOGICAL ERAS.

The account of time from any particular date or epoch is called an era.

The Christian, vulgar, or Dionysian era dates from January 1st, 4004 years after the creation of the world.

The era of the deluge dates from February 18th, 2358 years before Christ.

The era of Abraham dates from October 1st, 1996 years before Christ.

The era of the Olympiads dates from July 1st, 776 years before Christ.

The era of the building of Rome dates from April 22nd, 753 years before Christ.

The era of Nabonassar dates from 747 years before Christ.

The era of Constantinople, used by the Greeks, dates from the creation of the world, and the year 5509 corresponds to the 1st of September, before the Christian era.

The era of the Seleucidæ dates from October 1st, 312 years before Christ.

The Spanish era dates from January 1st, 45 years before Christ.

The era of Diocletian dates from August 29th, 284 years after Christ.

The era of the Hejira dates from July 15th, 622 years after Christ.

The last day of the old style (catholic nations), October 4th, 1582.

The last day of the old style in England, September 2nd, 1752.

New style in catholic nations, October 15th, 1582.
New style in England, September 14th, 1752.

230. TABLE OF UNITS, &c.

REDUCTION.

232. A quantity expressed in different units, such as £3. 11s. 6d. or 15 cwt. 3 qr. 16 lbs., is called a compound quantity; and by the tables, it is evident that any compound quantity can be expressed in several ways. For instance, £4. 9s. 6d. is the same as 89s. 6d., or 1074 pence, or 4296 farthings.

When a quantity is expressed in one or more denominations, reduction shows the method of converting it into one or more others. Since quantities may be either reduced from a high denomination to a lower one, or raised from a lower one to a higher, then there are two kinds of reduction, reduction descending and reduction ascending.

233. First, reduce £24. 8s 74d. into farthings.

Since £1.20s., £24.=24 × 20s. =480s. ; therefore, £24. 8s =480+8=488s. Since 1s. = 12d., 488s.=488 × 12d.=5856d.; therefore, £24. 8s. 7d. 5856d.+7d.=5863d. And since 1d. =4 farthings, 5863 pence=5863 × 4f.=23452f.; and, there fore, £24. 8s. 73d.=23452f.+3f.=23455 farthings.

234. Secondly, reduce 37849 farthings to pounds, shillings,

and pence

Since 4f. 1d., 1f.d.; therefore, 37849f.=

37849 × 1

4

d. = 94621d.; also, since 12d. 1s., 1d.s.; therefore, 9462d. 9462 × 1 =788s. 6d.; and also, since 20s. £1, 1s. = £20,

12

=

therefore, 788s. = 788 x £= £39. 8s.

==

Hence, 37849f.= 94624d.=788s. 64d. = £39. 8s. 64d., which process is thus

written :

4)37849

12)9462. If.

20)788. 6d.

£39. 8s. 6d.

As a proof of the correctness of these results, operate in a converse way.

235. From what has been said in (§ 233 and 234), the operations in reduction are evident; it is only required to refer to the tables.

236. EXERCISES.

1. Reduce £164. 19s. 74d. to farthings, and prove the operation. 2. In £764. 11s. 114d. how many farthings? Verify the

result.

3. What number of pounds, &c., are contained in 3769473 farthings? and prove the correctness of the result.

4. Change 48493 pence to pounds, &c., and conversely.

5. Find the number of farthings in £100. 10s. 10d., and prove it.

6. Required, the number of pence in 564 guineas, and verify the operation.

7. Convert 112546 grains into pounds Troy, and prove the result.

8. Bring 54 lbs. 10 oz. 16 dwts. 19 grs. to grains.

9. In 24 cwt. 2 qrs. 14 lbs., how many pounds?

10. How many drams are there in 3 tons 17 cwt. 2 qrs. 24 lbs. 15 oz. 3 drs. ?

11. Bring 24895 lbs. to cwts., &c.

12. Change 24 lasts 1 wey 4 tods 1 stone 0 cl. 5 lbs. to pounds.

13. Find the number of miles, &c., in 463972 inches.

14. Bring 5 mi. 4 fur. 3 pl. 4 yds. 1 ft. 7 in. to inches.

15. Reduce 14 yds. 3 qrs. 2 nl. 2 in. to inches.

16. How many inches are there in 54 E. e. 4 qrs. 3 nl. 1 in. ?

17. Change 714 inches to Flemish ells.

18. Convert 1476 French ells to inches.

19. Reduce 3 ac. 3 rd. 33 pl. 27 yds. 7 ft. 15 in. to inches

20. In 964893 yds., how many acres?

21. Bring 174 cubic yds. 24 ft. 362 in. to cubic inches.

22. What number of cubic yards, &c., are equivalent to 7945673 cubic inches?

23. Convert 64596 pints to lasts, &c.

24. Reduce 6 lasts 0 wey 3 qrs. 6 bu. 3 pk. 1 gal. 7 pts. to pints.

25. In 37496 pecks how many chaldrons ?

26. In 7 scores 20 ch. 35 bu. 3 pk. how many pecks?

27. Bring 434 hhds. 48 gal. 1 qt. of ale to pints.

28. 76465 pints are equivalent to how many butts, &c.

29. How many tuns, &c., are there in 264054 pints of wine?

30. How many pints are there in 9 hhds. 55 gals. 3 qts. 1 pt. of wine?

31. Convert 365 days 5 hrs. 48 min. 48 sec. to seconds.

32. Convert 4673360 seconds to years, &c.

33. Reduce 24 reams 16 quires 20 sheets of paper to sheets.

34. Convert 1649750 sheets of paper to reams, &c.

35. Find the number of seconds in 164° 45′ 34′′.

36. Find the number of degrees, &c., in 6849".

37. How many barleycorns will reach round the earth, which is 25000 miles in circumference?

38. The wheel of a railway carriage makes 369600 revolutions in 700 miles. What is the circumference of the wheel?

How

39. The distance between Dover and Calais is 21 miles. many arches, each of 75 feet span, would a bridge between the two places have?

40. The hind wheel of a carriage is 15 ft. 8 in. in circumference, and the fore wheel 12 ft. 3 in. How many revolutions will the latter make more than the former between London and Edinburgh, the distance being 389 mi. 6 fur. 20 p. ? 41. An equal number of moidores, guineas, pounds, shillings, and pence make £389. 8s. 4d. How many are there of each? 42. A sack of flour weighs 22 stones 12 lbs. of 4 lbs., 3 lbs., 3 lbs. 2 lbs., and 1 from it?

How many loaves lb. can be made