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.

Of Compound Interest.

AS Simple Interest is performed by a Serie•• of Musical, so is Compound Interes•• wrought by a Rank of Geometrical continua•• Proportionals. The operation whereof by th•• Canon of Logarithms, take under these four Considerations.

Prop. I.

If you shall put p = the Logarithm of a Prin∣cipal or Sum forborn, and t = the time o•• forbearance in years, quarters, months, or day, r = the Logarithm of the Rate of Interest, per cent. per annum, per mensem, or per diem, a = the Logarithm of the Amount of the said Principal for the said time, at the Rate also aforesaid: Then Q. The Amount = a?

Equation, a = rt + p.

That is, Multiply the Logarithm of the Rate by the Number of Years, Quarters, &c. to which Product add the Logarithm of the Principal, and the Aggregate is equal to the Logarithm of the Amount.

Example.

Quest. 1. If 175 l. be forborn 7 years, what will it amount to at 6 per Cent. per Annum, Compound Interest?

Log. of the Rate = 0,02530586=r 〈 math 〉〈 math 〉

Log. of the Sum = 2,24303805 = 175 = p 〈 math 〉〈 math 〉

The Answer 263 l. 2 s. 8 d. ¼ ferè.

Quest. 2. If 1000 l. be forborn for 6 months, at 6 per Cent. per Annum, Compound In∣terest, what will it amount to?

Log. of the former Rate divided by 12, the months in a year, is = 0,00210882 = r 〈 math 〉〈 math 〉

Add the Log. of 1000 viz. 3,00000000

1029.563 = Log. Amount 3,01265292 = a

The Answer 1029 l. 11 s. 3 d. ferè.

Prop. II.

A Sum of Money unknown, being forborn a cer∣tain time = t, at a given Rate of Interest = r, is amounted to a given Sum = a; Q What was p?

Equation, p = a − rt.

From the Logarithm of the Amount, sub∣duct the Logarithm of the Rate, multiplied by the time, and the Remainder is the Logarithm of the Principal.

Example.

Quest. 1. If 263 l. 2 s. 8 d. ¼ be the Amount of a Sum forborn 7 years, at 6 per Cent. per Annum, Compound Interest, what was the Principal?

Log. of the Rate = 0,02530586 = r 〈 math 〉〈 math 〉

Log. of the Amount 2,42017910

Log. of the Principal 2,24303805 = p = 175

The Answer 175 l.

Quest. 2. If 102 l. 11 s. 3 d. be the Principal and Interest of a Sum of Money forborn 6 months, at 6 per Cent. per Annum, Compound Interest, what was the Principal?

Log. of Rate for 1 m^o. 0,00210882 〈 math 〉〈 math 〉

Log. of 1029.563 = 3,01265292

Log. of the Principal 3,00000000 = 1000

The Answer 1000 l.

Prop. III.

A Sum of Money = p, being forborn for a time = t, did amount to a given Sum = a, at a Rate of Interest unknown: Q. The Rate per Cent. per Annum = r?

Equation, 〈 math 〉〈 math 〉

Divide the Logarithm of the Amount, less the Logarithm of the Principal, by the Time, and the Quote is the Logarithm of the Rate.

Example.

If 25 l. forborn 4 years, did amount to 31 l. 11 s. 2 d. ¼; at what Rate of Compound Interest did it so increase?

Logarithm of the Amount = 1,49808345

Logarithm of the Principal = 1,39794001

a − p divide by 4 ) 0,10014344

The Log. of the Rate = r = 0,02503586

Prop. IV.

A Sum of Money being forborn, at a given Rate, for a time unknown, but the Amount is known, how long was it so forborn?

Equation, 〈 math 〉〈 math 〉

Example.

If 1000 l. be increased to 10••9 l. 11 s. 3 d. at 6 per Cent. per Annum, Compound Interest, in what time was it so increased?

〈 math 〉〈 math 〉

The Answer 6 months.

It may here be expected that I should lay down the Construction of the Logarithms, having made use of them in these Calculations, but this being design'd a small Enchiridion, and there being large Volumns of that Subject in the World already, by several more learned Pens, I think it unnecessary to say any thing

further thereof, for as they are of excellent use, so are they easie to be had.

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.

There are many Tables of Compound In∣terest Printed in sundry Books for the valuation of Leases and Annuities, but they are generally made for yearly Payments, when indeed by the common and most usual Covenants in Leases the Tenant is obliged to pay quarterly; and in Leases of great value, there will be found a con∣siderable difference in the true worth, (so great, that 25 l. per Quarter is as good as 102 l. 5 s. per Annum.) I have therefore presented the Reader with a Table fitted to such Quarterly Payments, the Use of which Table I doubt not but will be very easily found by the Examples that follow.

A Table of Interest, for the Valuation of Leases or Annuities for Quarterly Payments, at 6 per Cent. per Annum, Compound Interest, for 31 Years.

1	.985538
2	1.956824
3	2.914064
4	3.857459
5	4.787213
6	5.836372
7	6.609520
8	7.496573
9	8.373700
10	9.238139
11	10.090079
12	10.929724
13	11.757176
14	12.572685
15	13.376402
16	14.168496
17	14.949134
18	15.718484
19	16.476707
20	17.223982
21	17.960417
22	18.686219
23	19.401524
24	20.106484
25	20.801250
26	21.485968
27	22.160789
28	22.825841
29	23.481282
30	24.127242
31	24.763844
32	25.391256
33	26.009595
34	26.618988
35	27.219206
35	27.811474
37	28.394861
38	28.969764
39	29.536352
40	30.094714

41	30.645034
42	31.187396
43	31.721914
44	32.248000
45	32.767870
46	33.279531
47	33.783794
48	34.280753
49	34.843000
50	35.253244
51	35.728999
52	36.197819
53	36.659861
54	37.115237
55	37.564028
56	38.006330
57	38.442234
58	38.871893
59	39.295222
60	39.712487
61	40.123777
62	40.529043
65	40.928469
64	41.322133
65	41.710087
66	42.092531
67	42.469245
68	42.841611
69	43.206601
70	43.567307
71	43.922768
72	44.273138
73	44.618415
74	44.958701
75	45.194062
76	45.624577
77	45.950311
78	46.271331
79	46.587715
80	46.899521
81	47.206817
82	47.509668
83	47.808141
84	48.102298
85	48.392200
86	48.677877
87	48.959492
88	49.236993
89	49.510486
90	49.780023

91	50.045601
92	50.307460
93	50.665471
94	50.819753
95	51.070356
96	51.317335
97	51.560742
98	51.841984
99	52.037044
100	52.271047
101	52.499677
102	52.725986
103	52.949021
104	53.134301
105	53.385474
106	53.598963
107	53.809375
108	54.016743
109	54.221113
110	54.422527
111	54.620966
112	54.816601
113	55.009461
114	55.199474
115	55.386751
116	55.571297
117	55.753185
118	55.932443
119	56.109107
120	56.283219
121	56.454811
122	56.623921
123	56.790588
124	56.954843

The Calculation of a Number in the precedent Table, by aid of the Canon.

The Question being, What is the present worth of 1 l. per Quarter for 21 Years?

The Logarithm of the Increase of 1l. at 6 per Cent. for three months = 0,0063264664

The Number of Quarters = 84 〈 math 〉〈 math 〉

Discompt of 1l. for 84 Qrs. 0,5314231776

Then by the Rule of Proportion:

If 1,467384617 com. arithm. 9,8334569382
have 100l. for its Principal, 2,0000000000
what shall 1l. have for its Pr. 0,0000000000

68.148601 = 1,8334569382

The Log. of the Discompt = 0,5314231776

〈 math 〉〈 math 〉

The Answer is = 48 l. 2 s. 0 d. ½ ferè.

And after this manner may a Table be Calcu∣lated, or the Value of a Lease for any Number of Years, may be found at any Rate of Inte∣rest required.

The VSE of the TABLE.

This Table sheweth the Discompt of 1 l. per Quarter at 6 per Cent. per Annum, Com∣pound Interest, and if the Tabular Number for so many Quarters as the Lease is to continue be multiplied by the Quarterly Payment, that Product is the present Value of that Lease in ready money.

Example.

A Lease of 40 l. per Annum (viz. 10 l. per quarter) for 21 years, being to be sold, what is it worth in ready money?

84 quarters per Table = 48.102221

The quarterly Rent = 10 〈 math 〉〈 math 〉

The Answer is 481 l. 0 s. 5 d. ••/4; ferè.

But if the question be, What quarterly Rent for 21 years will a given Sum purchase? Then divide the given Sum by the Tabular Number for so many quarters.

Example.

〈 math 〉〈 math 〉

A Gentleman having a Lease of certain Church Lands, worth 200 l. per Annum more than the reserved Rent, for 14 years to come, surrenders the same, upon condition the Chapter shall make him a new Lease for 31 years without a present Fine, but advan∣cing the old Rent 10 l. per quarter during the whole term of 31 years; what doth he gain by the bargain, accompting Compound Interest on both sides?

56 quarters per Table = 38.006

The quarterly Rent = 50 〈 math 〉〈 math 〉

124 quarters per Table = 56.955 〈 math 〉〈 math 〉

The Answer is 377 l. 18 s. the new Lease being so much more worth than the old one.

240 l. is demanded for the Lease of a House for 7 years, the Tenant offers 100 l. and an advance of Rent equivalent to the rest of the Fine required, what ought this Rent to be?

〈 math 〉〈 math 〉

The Advance of Rent ought to be 6 l. 2 s. 8 d. per quarter.

There is a Lease of 200 l. per Annum, viz. 50 l. per quarter, for 13 ¼ years, to be sold, what is it worth at 6 per Cent. Simple, and what at 6 per Cent. Compound Interest?

Simple.

53 quarters per Table = 38.779748 〈 math 〉〈 math 〉

Compound.

53 quarters per Table = 36.659861 〈 math 〉〈 math 〉 Which subtracted from 1938.987440 Leaves 105.99439

Whereby it appears, that it is cheaper to the Purchaser at Compound Interest than at Simple Interest by 106 l.

Six Questions performed by the aid of the Canon of Logarithms.

Quest. 1. A Gentleman pays 350 l. for a Lease in Reversion, to commence at the end of 13 years and a quarter, and to continue for 21 years and 3 quarters, what quarterly Rent may he lett the Premises for, after he comes to be in possession thereof, so as to gain 8 per Cent. Compound Interest for his money?

The Logarithm of 350l. = 2,544008

Worth of 1l. forborn 53 quarters = 0,442865

Log. of the Increase of 350l. = 2,986873

Worth of 1l. for 87 quarters = 1,621420

Log. of 23.198 = 1,365453

The Answer = 23 l. 3 s. 11 d. ½ ferè.

Quest. 2. A Citizen having taken a Lease of a House and Shop for 21 years, at 370 l. Fine, and 100 l. per Annum, viz. 25 l. per quarter, Rent, at the end of two years is willing to leave it for 300 l. and the old Rent, or to have such an increase of Rent, during the whole term yet to come, as may reim∣burse him his Fine paid, with Compound Interest at 6 per Cent. per Annum: What

ought he to receive in advance of Rent, and what doth he offer to lose of his Fine paid in taking 300 l.

2,568202 = 370l.

1,682165 = The worth of 1l. per quarter for 84 quarters.〈 math 〉〈 math 〉

0,886037 The Advance of Rent ought to be 7l. 13s. 10d. ¼ ferè.

1,652198 The worth of 1l. per quarter for 76 quarters.〈 math 〉〈 math 〉

2,545235 The present Fine ought to be 350l. 18s. 10d.

Whereby it appears, there is 50 l. 18 s. 10 d. offered to be lost in putting off the House and Shop aforementioned.

Quest. 3. A sells a House to B for 800 l. to be paid with Interest upon Interest by 100 l. per Annum, viz. 25 l. per quarter, how many quarters Rent ought B to pay before A is satisfied for his 800 l. with Compound Interest at 6 per Cent. per Annum, and what ought the last Payment be?

〈 math 〉〈 math 〉

The last Payment 13 l. 3 s. 4 d. ferè.

Quest. 4. A lends unto B a certain Sum of ready money, and accepts a Rent Charge of 40 l. quarterly for 7 years in satisfaction, finding it paid him his Principal with Interest upon Interest at 8 per Cent. within 13 l. 4 s. 6 d. what was the Money lent?

1,602060 The Logarithm of 40l.

1,565196 The Logarithm of the worth of 1l. quarterly for 48 quarters.〈 math 〉〈 math 〉

3,167256 The Logarithm of the worth of 40l. per quarter for 48 quarters.〈 math 〉〈 math 〉

0,177140 The Logarithm of the increase of 1l. forborn 28 quarters.

2,986188 The Logarithm of 968.7.

The Money lent was 968 l. 14 s.

Quest. 5. A Testator leaving one Son and two Daughters, bequeaths out of his Estate (be∣ing 600 l. per Annum for 11 years) to his eldest Daughter 500 l. per Annum for 4 years next coming, at the end whereof, to his younger Daughter 300 l. per Annum for 7 years, and to his Son the Remainder of the

Estate for the whole time: Q. Which had the greatest Portion, and by how much, calculating their several Annuities at 6 per Cent. Compound Interest?

0,539716 Logarithm of the worth of 1l. per annum for 4 years.

2,698970 The Logarithm of 500l.〈 math 〉〈 math 〉

3,238686 = 1732.55.〈 math 〉〈 math 〉

0,746820 The Logarithm of the worth of 1l. per annum for 7 years.

2,477121 The Logarithm of 300l. 〈 math 〉〈 math 〉

0,101232 The Logarithm of the worth of 1l. forborn 4 years.〈 math 〉〈 math 〉

3,122709 = 1326.47.〈 math 〉〈 math 〉

0,896905 The Log. of the present worth of 1l. per annum for11 years. 〈 math 〉〈 math 〉

2,896905 = 788.68

0,746820 The Log. of the present worth of 1l. per annum for 7 years.

2,301029 The Logarithm of 200l.〈 math 〉〈 math 〉

0,101232 The Logarithm of the worth of 1l. forborn 4 years. 〈 math 〉〈 math 〉

The Sons Portion —1673:00:05

The eldest Daughters Portion 1732:11:00

The youngest Daughters — 1326:09:04¾

〈 math 〉〈 math 〉

Proof.

0,896905 The Logarithm of the worth of 1l. per annum for 11 years.

2,778151 The Logarithm of 600l.〈 math 〉〈 math 〉

3,675056 The Logarithm of 4732l.

Quest. 6. A Merchant sold 16 Kintals of Cy∣prus Cottons for 320 l. to be paid at two six-months; the Buyer having Money by him, offers to pay the Money presently,

provided the Merchant allow him Discompt at 6 per cent. Compound Interest. Q. What ought the Merchant to receive?

2,505149 The Logarithm of 320l.

0,018979 The Logarithm of 1l. forborn 9 months, the equated time, acor∣ding to the Table of Mean Time, Pag. 110.〈 math 〉〈 math 〉

2,486170 The Logarithm of 306.316.

The Answer 306 l. 6 s. 4 d. ferè.