LEAP YEARS.

548. An astronomical or solar year

takes the earth to revolve round the

that is, the time it sun-is 365 days, A civil year

5 hours, 48 minutes, 50 seconds, nearly.

closes with an even number of days, at 12 P. M., Dec. 31.

Reckoning 365 days a year, we lose

5 h. 48 m. 50 sec.

each year, and in 4 years we lose

4 × (5 h. 48 m. 50 sec.)

= 23 h. 15 m. 20 sec.

If, then, in the fourth year we add a day, or 24 hours, giving the year 366 days, we shall add too much by

24 h. (23 h. 15 m. 20 sec.) = 44 m. 40 sec.

If we continue for 100 years, with 366 days every fourth year, and 365 in the other years, we shall have added too much by

If now we omit the leap day in the hundredth year we shall omit too much by

24 h. (18 h. 36 m. 40 sec.) ·

= 5 h. 23 m. 20 sec.

If we continue in this way (having a leap year every 4 years except the hundredth years) we shall in 4 centuries omit too much by

4 × (5 h. 23 m. 20 sec.) = 21 h. 33 m. 20 sec.

Adding a day in the 400th year, we shall add too much by

24 h. (21 h. 33 m. 20 sec.) = 2 h. 26 m. 40 sec. Continuing in this way for 40 centuries, or 4000 years, we shall have added too much by

10 × (2 h. 26 m. 40 sec.) = 24 h. 26 m. 40 sec. Therefore the 4000th year ought not to be a leap year. By the present method of having leap years (Art. 254), at the end of 4000 years the civil and solar years will coincide within about a day, and if the 4000th year is not made a leap year the civil and solar years will coincide within half an hour.

The calendar, as arranged by Julius Cæsar, added a leap day every 4 years. This method of reckoning, in 1582, had caused an error of

10 days. What ought to have been, for example, called March 20, was called March 10. The spring months were moving forward into the summer, the summer into the fall, and so, in 1582, Pope Gregory XIII. ordered that 10 days should be dropped from the calendar, and that the leap years should occur as stated in Art. 254.

In 1752 the change was adopted in England by act of Parliament directing what otherwise would be Sept. 3 to be Sept. 14, the error having at that time increased to 11 days.

INTEREST.

549. Legal Rates of Interest in the Several States.

When no rate is specified in a note, or document, the rate in the left-hand column is the legal rate. Any rate up to that in the righthand column is legal if specified in writing.

1. When payments are made one year or more from the time from which the interest is reckoned, or when any payment is less than the interest then due, the work of computing

the interest is performed according to THE UNITED STATES RULE.

2. When payments exceeding the interest then due, are made within a year from the time from which the interest is reckoned, the amount of the principal must be found for a full year, and the amount of the payment from the time of payment to the end of such full year, and this, deducted from the amount of the principal previously obtained, will form the new principal.

3. If the year extends beyond the time of settlement, the last computation is to the time of settlement.

$875.

(58.)

HARTFORD, CONN., Dec. 18, 1876. On demand, I promise to pay Samuel Hopkins, or order, eight hundred seventy-five dollars, with interest. Value received. RICHARD JAMESON.

INDORSEMENTS: Feb. 6, 1877, $ 250; July 12, 1877, $ 150; Nov. 9, 1878, $20; Jan. 21, 1879, $375; June 12, 1879, $25. What was due Sept. 5, 1879?

OPERATION.

Am't. of $875 Dec. 18, 1876, to Dec. 18, 1877

$927.50

(Payment $20 Nov. 9, 1878, less than int. due.)

Int. of $ 510.60 Dec. 18, 1877, to Jan. 21, 1879
Amount

66

$ 510.60

33.44

66

66

$544.04

Int. of $149.04 Jan. 21, 1879, to Sept. 5, 1879

5.59

$154.63

25.35

Ans. $129.28

(59.)

$1675.

SAYBROOK, CONN., July 7, 1876.

On demand, I promise to pay George A. Howe, or order, sixteen hundred seventy-five dollars, with interest. Value received. JOSEPH A. Davis.

INDORSEMENTS: Dec. 8, 1876, $300; Oct. 19, 1877, $60; Aug. 15, 1878, $500. What was due Sept. 1, 1879 ?

ANNUAL INTEREST.

551. In some States when notes are written "with interest annually," the interest, if it is not paid at the end of the year, when it is due, draws simple interest until paid.

NOTE 1. In Massachusetts and Maine interest is not allowed on annual interest, unless after demand for its payment or by special agreement.

(60.)

$350.

MANCHESTER, N. H., May 17, 1875. On demand, I promise to pay John Cogswell, or order, three hundred fifty dollars, with interest annually. Value received. FRANCIS WHITNEY.

What is due on this note May 17, 1879 ?

OPERATION.

Int. due on principal annually.......$350 x 0.06 = $21.00

Principal...................

Total annual interest

Int. for 3 yr. on the $21 due May 17, 1876.........

×

350.00

.$21 × 4 = 84.00 X

3.78

NOTE 2. The simple interest on the several interests due annually can be computed at once by computing the interest on the interest due annually for a number of years equal to the sum of the number of years the several annual

= the in

interests remain unpaid. Thus, in Ex. 60, the interest on the $21 for 3 yr. + the interest on the $21 for 2 yr. + the interest on the $21 for 1 yr. terest on $21 for 6 yr.

552. Hence, to find the amount due on a note "with interest annually," when the interest remains unpaid,

Rule.

Add to the principal the simple interest on the principal for the given time, and also the simple interest on one year's interest of the principal for a number of years equal to the sum of the number of years the several annual interests remain unpaid.

61. What would be due on a note of $642 "with interest annually" in 7 yr. from its date?

62. What is the excess of annual over simple interest of $520 for 4 yr. 6 m.?

63. What is the amount of $ 465 at annual interest for 8 yr. 6 m. ?

64. What is the amount of $562 at annual interest for 3 yr. 3 m. 18 d.?

553. When partial payments are made on a note that is written "with interest annually" the following is the

common

Rule.

Find the amount of the principal by the annual interest method, to the end of the first year in which a payment, or payments, plus the interest on the payment, or payments, not less than the interest due at the end of the year has been made. From this amount subtract the payment, or payments, plus the simple interest on the payment, or payments, to the end of the year. The remainder will be a new principal, with which proceed as before.