Socius mercatoris: or The merchant's companion: in three parts. The first, being a plain and easie introduction to arithmetick, vulgur and decimal, the extraction of the square and cube roots, with a table of 200 square roots, and their use in the resolution of square equations. The second, a treatise of simple and compound interest and rebate, with two tables for the calculation of the value of leases or annuities, payable quarterly, the one for simple, the other compound interest, at 6 per cent. per annum, with rules for making the like for any other rate. The third, a new and exact way of measuring solids in the form of a prismoid and cylindroid, with the frustums of pyramids and of a cone: whereunto is added, some practical rules and examples for cask-gauging. By John Mayne, philo-accomptant.

COmpound Interest Infinite, may be so called as it relates to divers equal Payments at equal times, but the number of those equal times are infinite, (i. e.) when an Estate in Fee-Simple shall be sold for ever. Now there being usually an interval of time, between the Purchasers Payment and the reception of his first Rent, be it yearly, half yearly, or quar∣terly;

Any Question of this Nature may be wrought by the following Analogism:

Putting V = the Rent (yearly or quarterly)and S = the Price paid for the Land, also R = the Common Factor of the Rate of Interest, per Cent. per Annum.

Hence then may arise these three Propositions.

Prop. I.

There is a Fee Simple to be sold, what is it worth in ready money, so that the Purchaser may have 6 per Cent. per Annum, Compound Interest, allowed for his money.

Quest. 1. There is a Manour to be sold of the clear yearly value of 969 l. 18 s. what Sum of ready money is this Estate worth, 6 per Cent. per Annum Compound Interest being allowed the Purchaser for his money?

Equation, 〈 math 〉〈 math 〉

The Annual (or Quarterly) Payment, divided by the Ratio, less Unity, exhibits the Sum in the Quotient.

〈 math 〉〈 math 〉

The Answer is 16165.

Quest. 2. There is an Estate of 969 l. 18 s. per Annum▪ payable Quarterly, what is it worth in ready money, allowing the Pur∣chaser 6 per Cent. per Annum Compound Interest?

〈 math 〉〈 math 〉

The Answer is 16524 l. 2 s. 6 d. ferè.

The difference between Yearly and Quarterly Payments in this Purchase raiseth the value 359 l. 2 s. 6 d.

☞ Having the increase of 1 l for a Year, at any Rate of Interest, the Biquadrate Root of that Increase, is the Increase of 1 l. for a Quarter at Compound Interest.

Prop. II.

A Sum of money lying ready for a Purchase, and it be desired to know what Free-hold Estate such a Sum will purchase, if laid out at a given Rate per C. per Ann. Compound Interest.

Theorem, V = S × R − 1.

Or, in other terms, the Sum of Money mul∣tiplied by the Rate, less Unity, the Product shall be equal to the Annual half quarterly or quar∣terly Payment.

Quest. A Gentleman upon Marriage of his Daughter promiseth to lay out 1600 l. for a Free-hold Estate, to be settled upon her and her Heirs, provided he meet with such a Pennyworth as shall bring 8 per Cent. per Annum, Compound Interest for his money: Q. What Annual Rent must it be?

〈 math 〉〈 math 〉

The Answer 128 l. per Annum.

Prop. III.

An Estate being offered for a certain Sum of money, the annual Rent is also known: Q What

Rate of Interest upon Interest shall the Pur∣chaser have for his money?

Equation, V ÷ S = R − 1.

The annual Rent being divided by the Sum demanded, quotes the Rate less Unity.

Example.

Quest. 1. There is a Free-hold Estate to be sold for 1600 l. the yearly Rent being 128 l. what Rate of Interest shall the Purchaser have for his money?

〈 math 〉〈 math 〉

Quest. 2. Admit there be a small Farm to be sold of the Value of 35 l. per Annum for 500 l. what Rate of Compound Interest shall the Purchaser have for his money at that price?

〈 math 〉〈 math 〉

Furthermore, if it be inquired how many years Purchase any Annuity is worth, putting R = the Ratio as before, and Y the number of Years, the Rule is: 〈 math 〉〈 math 〉

That is, Divide Unity by the Ratio less 1, and the Quote informs the Number of Years.

Example.

There is a Free-hold Estate to be sold, Q. How many Years Purchase is it worth at 5 per Cent. per Annum?

〈 math 〉〈 math 〉

The Answer is 20 Years Purchase.

What is it worth at 6 per Cent. p••r Annum?

〈 math 〉〈 math 〉

The Answer is 16 Years, and ⅔ of a Year.

Again, if an Estate be offered at any num∣ber of Years Purchase, and it be demanded what Rate of Interest the Purchaser shall have for his Money, the Rule is: 〈 math 〉〈 math 〉

That is, Divide Unity by the number of Years propos'd, and the Quote exhibits the Ratio, less Unity.

Example.

An Estate is offered at 20 Years Purchase, what Rate of Interest shall the Purchaser then have?

〈 math 〉〈 math 〉

The Answer is 5 per Cent. per Annum.