after the equated time, from that time to the times at which they respectively become due.' But the argument by which he attempts to prove the truth of the rule is, according to Mr. Malcolm, very erro

neous.

3. Mr. Hatton supposes the equated time to be true, 'When the interest of the sum of the debts or bills, from the time of the question to the equated time, is equal to the sum of the interests of the several debts or bills from the time of the question to the several terms of payment ;' and then, by an example, shews that the rule agrees with this supposition.

4. Mr. R. Burrow, in his Diary for the year 1777, reduces the subject, To find in what time the whole sum of the single payments will produce the same amount as that which arises from the sum of all the single payments, together with the interest of each payment from the time of its becoming due to the time of the last payment;' and then gives an algebraical demonstration, which shews that the rule is true according to this supposition.

5. That the rule is universally true, according to any of these suppositions, or that, if it be true according to one of them, it must necessarily be true according to the whole, may easily be demonstrated.

6. The following is KERSEY'S RULE.-Find the present worth of each debt or bill, discounting from the time at which it is payable, (by the rule of Discount,) then find (by Prop. 6. of Simple Interest) in what time the sum of these present worths will amount to the sum of the debts or bills, and that is the time sought. There are other rules given by different authors, as Sir Samuel Moreland's, Ward's, &c.; but, upon a close attention to their principles, they will be found exactly the same as one or other of the rules already given: indeed, the foundation of Burrow's demonstration seems to have been taken from Moreland's rule. Malcolm's rule will be given in the second part of this treatise: its requiring an extraction of the square-root makes it inadmissible in this place.

Examples.

(1.) A owes B 1107. whereof 50l. is to be paid at two years' end, 407. at 3 years' end, and 207. at 4 years' end; at what time may B receive the whole at once, without prejudice to either party?

110 sum of the payments. 330 sum of the products. Then, 330 divided by 110 gives 3 years, the answer.

ILLUSTRATION.

Suppose the interest of inoney to be at 5 per cent. and that 3 years is the true equated time as found above. It is evident that A gains the interest of 501. for one year, which is 2l. 10s., by extending the term of payment to 3 years instead of 2; and that he loses the interest of 401. for half a year, and the interest of 201. for 1 year, by paying 401. half a year before it becomes due, and 201. 11⁄2 year before it becomes due; which interests, added together, make 21. 10s., so that his gain and his loss, on this consideration, appear to be equal. But, we must recollect, that B is not intitled to the interest of 401. for half a year, and of 201. for 1 year, but to the discount of each of these sums for those times; so that the rule cannot be precisely accurate, though it be near enough to the truth for any practical purpose to which it can be applied.

(2.) I am to pay 500l. at three diffèrent payments, viz. 1001. at 2 months, 2007. at 4 months, and the rest at 6 months; but the person who is to receive the money has agreed to take a single note for the payment of the whole at once, for what length of time must the note be given?

(3.) A debt of 7001. is to be discharged thus: 150l. present, 300l. at 6 months, 2001. at 9 months, and the rest at 12 months; what is the equated time for the payment of the whole ?

(4.) A merchant buys goods to the amount of 7501. 3501. of which is to be paid at 3 months, and the rest at 9 months; to prevent farther trouble, it is agreed to pay the whole at once, and to prolong the time of the first payment in proportion to the shortening the time of the second; at what time must the whole be discharged without prejudice to either?

(5.) A debt of 5007. 15s. is payable as follows: 1507. at two months, 1477. 17s. at 74 days, 1377. 18s. at 95 days, and the rest at 5 months. It is to be discharged at one payment; what is the equated time, reckoning 30 days to a month?

CLASS II.

(6.) A traveller received 12007. in 4 bills, all payable at Newcastle-upon-Tyne; viz. 6007. due at 4 months, 8007. at 5 months, 2001. at 7 months, and 100%. at 10

months: he agreed to pay the banker there, a reasonable commission, and the expense of the stamps, provided he would give him a single bill on London for the payment of the whole at once; for what length of time after date ought this bill to be drawn?

(7.) A debt is to be discharged thus,

present, at

25 days, at 3 months, and the rest at 4m. 17d. what time may the whole be paid at once?

(8.) Three legacies are left by a gentleman, in his will, payable by his executors, to one person, or his heirs. The first legacy of 500l. 18s. is payable in a year, the 2d of 9007. 178. 6d. is payable in 1 year 114 days, and the 3d of 17007. 18s. 4 d. is payable in 22 years. The legatee and executors have agreed, that the payment of these sums shall be made at once; at what time must that be, that neither party may be injured, allowing simple interest?

COMPOUND INTEREST.

Definition. Compound Interest is that which is produced not only from the sum of money lent as the principal, but also from the interest, which, (when unpaid,) as it becomes due, is added to the principal.

Proposition. To find the interest of any sum of money, unpaid, for any equal number of payments at any rate per cent.

Rule I. Find the amount of the given principal for the time of the first payment by Simple Interest; then consider this amount, as the principal for the second payment, and find its amount as before. Proceed thus through all the payments, always considering the last amount ast he principal of the next payment; then, if the given principal, or money lent, be deducted from the last amount, the remainder will be the interest required.

Or, Rule II.

Reduce the given sum into farthings, which multiply by the rate per cent. and cut off two figures from the right hand of each successive product, (or place each successive product two figures farther towards the righthand,) and the last result will be farthings.

Note. The above rules will be true, whether the payments are made yearly, half-yearly, quarterly, monthly, or by any other aliquot part of a year: thus, for half-yearly payments, take half the rate per cent., and twice the number of years;-for quarterly payments, take of the rate per cent. and four times the number of years, &c. But the given time must be complete years, half-years, or quarters; thus, you cannot find the interest of a given sum payable yearly, for 44 years, 4 years, &c. by the above rules, as directed by several authors. The truth of this remark will easily appear to those who are acquainted with logarithmical arithmetic, and the involution of numbers to fractional powers.-For other rules, see Compound Interest by Decimals.

Examples.

(1.) What is the compound interest of 3577. 10s. for 3 years, at 5 per cent, per annum ?

Answer, 561. 7s. Od. whole interest, which is 2.14..6 -more than the simple interest of the same sum. See Example 1, Simple Interest.

(2.) What is the compound interest of 7007. 18s. for 4 years, at 5 per cent. per annum ?

(3.) What is the compound interest of 10577. 178. 6d. for 6 years, at 4 per cent. per annum?

(4.) Required the amount of 500l. 17s. for 5 years, at 44 per cent. compound interest?

(5.) What will 7001. amount to in 7 years, at 4 per cent. per annum, compound interest?

CLASS II.

(6.) Find the several amounts of 5007. payable yearly, half-yearly, and quarterly, for 4 years, at 5 per cent, per annum. Answ. 6071. 158. 02d. for yearly, 6091, 4s. 04d. for half-yearly, and 6097. 18s. 101d. for quarterly pay

ments.

(7.) What is the amount of 7157. for 6 years, the interest payable half-yearly, at 44 per cent. per annum? (8.) What is the compound interest of 7407. 188. for 9 years, by quarterly payments, at 4 per cent. per annum ?

FELLOWSHIP, OR PARTNERSHIP.

Definition. Fellowship, or Partnership, is a general rule by which the accounts of merchants, &c. trading in company, with a joint stock, are adjusted; so that every partner may have his due share of the gain, or sustain a proportional part of the loss, according to the money he has advanced in the stock, and the time of its continuance therein.

SINGLE FELLOWSHIP, or PARTNERSHIP for ANY EQUAL TIME.

for

Definition. Single Fellowship, or Partnership for any equal time, is when different stocks are employed any certain equal time. The effects of bankrupts are by this rule properly divided among their creditors, legacies adjusted in deficiencies of assets, &c.-It likewise teaches us to divide any given number into unequal parts, proportional to certain other given numbers.

Proposition. Having each man's particular stock and the whole gain or loss given, to find each man's part of the gain or loss.