or annuity by the rate per cent, and the quotient will be the annuity.

EXAMPLES.

1. What is the worth of a freehold estate of which the yearly rent is $50, allowing to the purchaser 5 per cent? 5 100 50 $1000 Ans.

or, ,05)50,00(1000 Answer as before.

2. What is the worth of $125 annuity to continue forever, allowing 6 per cent to the purchaser ?

Ans. $2083,333. 3. What is the value of an estate which brings in yearly $96, allowing 6 per cent to the purchaser ? Ans. $1600.

CASE V.

To find the present worth of perpetual annuities or freehold estates in reversion at Compound Interest.

RULE.

Find the present worth of the estate by the Rule, Case 4th, which will give the value if entered on immediately. Then divide that value by the power of the ratio denoted by the time of reversion, and the quotient will be the present worth of the annuity in reversion.

EXAMPLES:

1. Suppose a freehold estate of $50 per annum to commence 2 years hence, be put on sale, what is its value, allowing the purchaser 5 per cent ?

5 100 50 $1000 present worth if entered on immediately.

the ratio1,05 2d power=1,1025)1000,0000(907,02,9+Ans. By Table 2, find the present worth of the annuity or rent for the time of reversion, which subtract from the value of the annuity found by Case IV.

By Tab. 2d, the present worth of $1 for 2yrs. 1,859410

Present worth for the time of reversion,
Value of immediate possession,

1000,0000

X 50

92,970500

2. Suppose an estate of $100 per annum to commence 6 years hence, were to be sold, allowing the purchaser 5 per cent what is its value? Ans. $1492,43c.0m. 3. What is the value of the reversion of an estate of $120 per annum, to commence 15 years hence, at 6 per cent? Ans. $834,53+

Questions.

1. How do you find the amount of annuity at simple interest, by Arithmetical Progression?

2. How do you find the amount of annuities or pensions in arrears, at compound interest?

3. How do you find the present worth of annuities at compound interest?

4. How do you find the present worth

of annuities, leases, &c. taken in reversion, at compound interest?

5. How do you find the present worth of freehold estates or annuities forever?

6. How do you find the present worth of perpetual annuities, or freehold estates in reversion at compound interest ?

PERMUTATION.

Permutation is a method of finding how many different ways any given number of things may be changed.

To find the number of different changes or permutations that can be made of any given number of things, different. from each other.

RULE.

Multiply all the terms of the natural series continually together, from one up to the given number, and the last product will be the answer.

EXAMPLES.

1. How many changes can be made of the first three letters of the alphabet ?

If there were but 2 letters, we could only change them 1x2=2 ways, thus, a, b, and b, a. But three letters can be changed 1×2×3=6 different ways, as follows:

2. How many changes can be made with the nine digits?

Ans. 362880.

3. Eight gentlemen agreed to dine together so long as they could sit every day in a different position; now admitting they had fulfilled their agreement, how long must they have tarried together? Ans. 40320 days,=1103 yrs. 4. How many changes may be rung on 9 bells?—and how long will it take to ring them, allowing 20 seconds to every change? Ans. to the last, 84 days. 5. Of what number of variations will the twenty-six letters of the alphabet admit?

Ans. 403291461126605635584000000.

1. What is Permutation?

Questions.

2. How do you find the number of different changes or permutations that can be made of any given number of things?

POSITION

Is a Rule which, by the use of false, or supposed numbers, discovers the true ones required.

It is divided into two parts, Single and Double,

SINGLE POSITION.

Single Position teaches to solve those questions whose results are proportional to their suppositions.

RULE.

1. Take any number, and perform the same operations with it as are described to be performed in the question. 2. Then as the result of the operation is to the given sum, so is the supposed number to the true one required.

EXAMPLES.

1. A schoolmaster being asked how many scholars he had, replied: If I had as many more as I now have, half as many, one-third as many, and one-fourth as many, I should then have 222. How many scholars had he?

Diff. of the prod. 4500000; this divided by 2000, the difference of the errors, gives $2250, the share of the first; to which 3000=5250, the share of the second; and 2250+5250=7500, the share of the third.

2. B, C and D built a house which cost $1000. C paid $100 more than B, and D paid as much as B and C both; how much did each man pay?

Ans. B paid $200, C $300, and D $500. 3. What number is that which being increased by its 1, its, and 16 more, will be doubled ?

Ans. 64. 4. D, E and F, playing at cards staked 324 crowns, but disputing about tricks, each one took as many as he could get. D got a certain number, E got as many as D and 15 and F got part of both their sums added together; how many did each get? Ans. D got 1274, E 142, and F 54.

more,

5. A man has 100 acres of land in 3 lots. The second lot contains twice as much as the first, lacking 8 acres, and the third contains three times as much as the first, lacking 15 acres; how many acres does each lot contain ?

Ans. the 1st contains 20 acres, the 2d 33, the 3d 461a. 6. A and B laid out equal sums of money in trade; A gained $150, and B lost $225; then A's money was double that of B's. What sum did each lay out? 1st, Supp. A $300 B $300 2d Supp.

+150 -225

A $400 B $400 +150

-225

7. A man agreed to work 70 days on this condition, that for every day he worked he should receive $,80, and for every day he was idle he should pay $,30; and at the expiration of the time he received $31,80. How many days did he work and how many was he idle?

Ans. he worked 48 days, and was idle 22. 8. A farmer having driven his cattle to market received for them all $546, being paid for every ox $30, for every cow $20, and for every calf 3. There were twice as many cows as oxen, and three times as many calves as cows; how many had he of each?

Ans. 6 oxen 12 cows 36 calves. 9. A man gave his estate to his three sons in the following manner, viz. to B he gave half, lacking $130, to C one-third, and to D the rest, which was $75 less than the share of C; what was the amount of the whole estate, and how much was each one's portion ?

Ans. The whole amount was $1230, and

B had $485, C $410, and D $335. 10. Three men are to share a certain sum of money, as follows, viz.: the first is to have twice as much as the third, and the second two-thirds as much as the first, and the shares of the second and third, added together, are $1435; what is the share of each?

Ans. The first $1230, the second $820, the third $615.

Duodecimals are fractions of a foot. They are so called from the Latin word duodecim, which signifies twelve. The foot being divided into 12 equal parts, called inches, or primes, and each of these parts again divided into 12 other equal parts, called seconds, and each second again