remainder, divided by the latter will be the tabular number; for quarterly payments, use the 4th root, and take one quarter of the quotient.

Table 1st is calculated thus: take the first year's amount, which is D1, multiply it by 1.06+1=2.06, second year's amount, which also multiply by 1.06+1=3.1836, third year's amount, &c., at 6 per cent., and in this manner calculate the other tables.

RULE III.

Multiply the tabular number under the rate, and opposite to the time, by the annuity, and the product will be the amount. 3. What will an annuity of D60 per annum amount to in 20 years, allowing 6 per cent., compound interest?

=

Thus 36.785592 × D60 ann. D2207.13.5520. Ans. 4. What will an annuity of D60 per annum, payable yearly, amount to, in 4 years, at 6 per cent.? Thus 1+1.06+1.062 +1.063=4.374616, sum, ×D60, ann. =D262.47.696. Ans. (Rule 2.)

5. What will a pension of D75 per annum, payable yearly, amount to in 9 years, at 5 per cent., compound interest? Ans. D826.99.27%.

The annuity, time, and rate given, to find the present worth.

RULE IV.

Divide the annuity by the amount of D1 for 1 year, and the quotient will be the present worth of one year's annuity.

2. Divide the annuity by the square of the ratio, and the quotient will be the present worth for 2 years.

3. In like manner, find the present worth of each year by itself, and the sum of all these will be the present value of the annuity.

6. What ready money will purchase an annuity of D60, to continue 4 years, at 6 per cent., compound interest?

1.06)60.00000(56.603=present worth 1 year. 1.1236)60.00000(53.399

1.191016)60.00000(50.377

66

2 years.

ratio 4= 1.26247696)60.00000(47.525

D207.904 Ans.

(See table, compound interest.)

OR, BY TABLE 2.

Multiply the number under the rate, and opposite the time, in

the table, by the annuity, the product will be the present worth for yearly payments.

When the payments are to be made quarterly, or half-yearly, the present worth, found as above, must be multiplied by the proper number in table 3d.

:

Thus question 6, tabular No. 3, 46510 × D60=207.906. Ans. 7. What is the present worth of an annuity of D60 per an num, for 20 years, at 6 per cent. ? Ans. D688.19.50. 8. What is the present worth of D75 per annum, for 7 years, at 5 per cent. ? Thus: 5.78637 × 75=D433.97.775. Ans. Table 2d is thus made: Divide D1 by 1.06=.94339 the present worth of the first year, which, divided by 1.06 is equal to .88999, which added to the first year's present worth is 1.83339, the second year's present worth; then .88999 divided by 1.06, and the quotient added to 1.83339 gives 2.67301 for the 3d year's present worth, &c.

Annuities in reversion at compound interest.

RULE V.

=

Take two numbers under the given rate in table 2d, that stand opposite the sum of the two given times, and the number opposite the time when the annui.y is to commence, or time of reversion, and multiply their difference by the annuity for the present worth.

9. The reversion of a freehold estate of D60 per annum, for 4 years, to commence 2 years hence; what is the present worth, allowing 4 per cent. for present payment? Thus tab. no.

5.24214-1.88609=3.35605 × D60=D201.36.300. Ans.

10. What is the present worth of a reversion of a lease for D120 per annum, to continue 9 years, but not to commence till the end of 4 years, at 4 per cent., to the purchaser ? Thus

9.98565-3.62989-6.35576 × 120D762.69.1. Ans.

PERPETUITIES AT COMPOUND INTEREST.

Perpetuities are such annuities as continue for ever.
The annuity and rate given, to find the present worth.

RULE VI.

Multiply the amount of D1 for one year, at the given rate per cent., involved to the time of reversion, by the ratio, for a divisor, by which divide the yearly payments; the quotient will be the answer.

11. Suppose a freehold estate of D140 per annum, to com mence three years hence, is to be sold; what is it worth, allow ing the purchaser 7 per cent.? Thus, 1.07×1.07 × 1.07 ×.07= .08575301; then D140÷08575301=1632.59.5D. Ans.

12. What is an estate of D260 per annum, to continue for ever, worth in present money, allowing 6 per cent. to the pur chaser ? D4333.33.3. Ans.

T find the present worth of a freehold estate, or an annuity to continue for ever at compound interest.

RULE VII.

As the rate per cent. is to D100, so is the yearly rent to the value required.

13. What is the worth of a freehold estate of D40 per annum, allowing 5 per cent. to the purchaser?

Thus, D5: 100 :: 40: 800. Ans. 14. What is the amount of an annuity of D180 for 9 years at 5 per cent. ?* Ans. D1984.781520. 15. What will an annuity of D200 amount to in 5 years, to be paid by half-yearly payments at 6 per cent.? Thus, 5.637093 ×200=1127.4186 x 1.014781=1144.08.2D. Ans.

16. Required the amount of an annuity of D150 for 10 years at 5 per cent.? Ans. 1886.68.3. 17. If a salary of D100 per annum, to be paid yearly, be forborne 5 yrs. at 6 per cent., what is the amount? Ans. D563.70.9. 18. What sum of ready money will buy an annuity of D300, to continue 10 years, at 6 per cent.

10 years 7.36008 × 300=2208.024D. Ans. 19. What salary, to continue 10 years, will D2208.024 buy? This example is the reverse of the one above, consequently D2208.0247.36008=300. Ans.

20. If the annual rent of a house, which is D150, remain in arrears for 3 years at 6 per cent., what will be the amount due for that time? Ans. D477.54.

Note. From the nature of an annuity, as explained in the proof of the rule in annuities at simple interest, there is due one year's interest less than the number of years the annuity has been continued. By the rule and examples in compound interest, the amount of D1 at the given rate is equal to that power of the amount for one year, which is indicated by the number of years. This amount is obtained for one less than the numbe of years by forming the geometrical series as directed in the rule, or beginning with unity; thus, in question 1 (annuities) the series * See proceding rules.

is 1, 1.06, 1.062, 1.063, and the last term is the amount of D1 for 1 less than 4, the number of years. The sum of this series is the amount at compound interest, of an annuity of D1 for 4 years. The amount of any other annuity, for the same time and rate, will be as much greater or less as the annuity is greater or less than D1; that is, the amount of the annuity of D1 must be multiplied by the annuity to obtain its amount, &c.

REVIEW.

What

What is an annuity? How will you find the amount of an annuity at simple interest? How will you find the amount of an annuity at compound interest? What is the rule? will you do when the payments are half-yearly, or quarterly? How is table 1st calculated? How will you find the present worth of an annuity? What is the rule? How is table 2d calculated? When quarterly payments are required, what is to be done? How is table 3 constructed? What can you say of annuities in reversion? What is the rule? What name is given to annuities which continue for ever? What is the rule for perpetuities? What more can you say of annuities?

DUODECIMALS, OR CROSS MULTIPLICATION.

DUODECIMALS, that is, numbering by 12, are fractions of a foot, or of an inch, or parts of an inch, having 12 for their denominator. It is a rule in general use with artificers in casting up the contents of their work, and an excellent and useful rule for that purpose. Considering a foot as the measuring unit, a prime is the 12th part of a foot, a second the 12th part of a prime, &c. It is to be observed, that in measures of length, inches are primes, but in superficial measure they are seconds. In both, a prime is of a foot; but of a square foot is a parallelogram, a foot long and an inch broad. The 12th part of this is a square inch, which is T of a square foot. This method of multiplying crosswise is not confined to twelves, but may be greatly, extended, for any number, whether its inferior denominations decrease from the integer in the same ratio or not, may be multiplied crosswise. Thus, pounds multiplied by pounds are pounds, pounds multiplied by shillings are shillings, &c.; shillings multiplied by shillings are twentieths of a shilling; shillings multiplied by pence are twentieths of a penny, pence multiplied by pence are 240ths of a penny, &c. The word

duodecimal is derived from the Latin word duodecim, signifying The denominations are

twelve.

ADD as in Compound Addition, carrying 1 for each 12 to the next denomination.

3. Five floors in a certain building contain each 1295 feet, 9 inches, 8′′; how many feet in all?

Ans. 6479ft., Oin., 4". 4. Several boards measure as follows, namely: 27ft., 3in.; 25ft., 11in.; 23ft., 10in.; 20ft., 9in.; 20ft., 6in.; and 18ft., 5 in.; what number of feet in all ?

Ans. 136ft., 8in.

SUBTRACTION OF DUODECIMALS.

RULE.

WORK as in Compound Subtraction, borrowing 12, &c.