Cocker's decimal arithmetick wherein is shewed the nature and use of decimal fractions ... together with tables of interest and rebate ... : whereunto is added, his Artificial arithmetick, shewing the genesis ... of the logarithmes ... : also, his Algebraical arithmetick, containing the doctrine of composing and resolving an equation, with all other rules requisite for the understanding of that mysterious art according to the method used by Mr. John Kerley in his incomparable treatise of algebra / composed by Edward Cocker ... ; perused, corrected, and published by John Hawkins ...

CHAP. XII. Concerning Simple Interest. (Book 12)

I. VVHen Money pertaining, or belonging to one person is in the hands, pos∣session, or keeping, or is lent to another, and the Debtor payeth or alloweth to the Creditor, a certain sum in Consideration of forbearance for a certain time, such consideration for for∣bearance is called Interest, loane, or use money; and the money so lent, and forborne is called the principal.

II. Interest is either Simple, or Compound.

III. When for a sum of money lent there is loane, or interest allowed, and the same is not paid when it becomes due, and if such Interest doth not then become a part of the Principal, it is called Simple Interest.

IV, In the taking of Interest for the conti∣nuance or forbearance of Money, respect must be had to the rate limited by Act of Parliament, which Act now in force, forbiddeth, or restrain∣eth all persons whatsoever, from taking more than 6 l. for the Interest of an 100 l. for a year, and according to the same proportion for a grea∣ter or a lesser sum, not confining the lender or borrower to the space of one year, no more than

it confineth him or them to the limitation of the sum to be lent, or borrowed, but that the sum may be either more or less than 100 l, and may continue in the hands of the Debtor, either a longer, or a shorter time than one year, ac∣cording as the Lender and Borrower do agree, and oblige each other; Now for any time grea∣ter than one year, the rate or proportion of In∣terest is by Act of Parliament limited, but the Act doth not say what part of 6 l. shall be the interest of an 100 l. for half a year, a quarter of a year, a month, a day, or for any time les∣ser than one year, and in this case several Ar∣tists do differ in their opinions, some would have the true proportional interest for any time less than a year to be discovered by continual mean proportionals; as suppose it were required to know the interest of 100 l. for half a year at 6 per Cent. per Annum, they would have the In∣terest to be reckoned after the Rule of Com∣pound interest, and so 3l. is not the interest of a 100 l. for half a year, but is too much: But say they, to find out the true interest thereof, you are to find a mean proportional between 100, and 106, and that made less by 100, will give you the interest of 100 l. for half a year, and so by extracting of Roots they find out the interest for any time less than one year, but this is sufficiently laborious and painful if it be done without the help of Logarithms; but to per∣form this work to the 12 power for a Moneth, or to the 52 for a Weak, is very tedious, and to the 365 power for one Day is scarcely possi∣ble to be effected by natural Numbers; but cu∣stom and dayly practice tell us that the interest of Money for any time less than one year ought

to be computed according to the Rules of Sim∣ple Interest, and so 3 l. is the undoubted interest of 100 l. for 6 moneths, and 30 shillings is the interest of 100 l. for a quarter of a year; but here note by the way that by 6 moneths is not meant 6 times 4 weeks, or 6 times 28 dayes, but by six moneths, or half a year is to be understood the half of 365 dayes, and a quarter of a year is ¼ of 3•…•…5 dayes, and by 1 moneth is understood 1/12 of 365 dayes, so that a moneth consisteth of 305/12 dayes.

Upon the foresaid custom of computing the interest of money for time less than one year, this following Analogie seems to be assumed for a safe expo∣sition * 1.1 of the statute (and which is indeed the ground, and rea∣son it self of Simple Interest) viz. That such proportion as 365 dayes (or one year) hath to the interest of any sum for a year, such proportion hath any part of one year, or any number of dayes pro∣pounded, to the interest of the same sum, for that time propounded. And this (as was said before) is the whole ground work, and very foundation of the manner of computing of Sim∣ple Interest.

V. Rebate, or Discount, is, when there is an allowance of so much per Cent. for money paid before it be due, and * 1.2 as the increase of money at interest is found out by continual proportio∣nals Arithmetical or Geometrical increasing, so is the Rebate or discount of Money found out by continual proportionals decreasing Arithmetical∣ly or Geometrically, that is according as the al∣lowance

is, either after Simple or Compound In∣terest; Now the nature of Rebate or discount is thus; When there is a sum of Money, (suppose 100 l.) to become due at the end of a certain time to come, (viz. at the end of 12 Moneths;) and it is agreed upon by the Debtor and Credi∣tor that there shall be made present payment of the whole Debt, and it is likewise agreed that in consideration of this present payment, that the Creditor shall allow the Debtor after the rate of 6 per cent. per annum: Now upon this agree∣ment the Creditor ought to receive so much mo∣ney as being put out at interest for the same time it was paid before 'twas due, and at the same rate of interest, that the discount was reckoned at, then would it amount or be increased to the sum that was first due.

The manner of working Questions in Rebate at Simple Interest shall be shewn in the ninth Rule of this Chapter, and of working Questions in Rebate at Compound Interest shall be shewen in the Fourth Rule of the next Chapter.

VI. When the interest of a 100 l. for a year is known, the interest of any other sum, for the same time, is also found out, by one single rule of direct proportion, viz. The Interest of a 100 l. for a year by the statute is 6l, I demand what is the interest of 75 l. for the same time, and at the same Rate of Interest? The proportion is as fol∣loweth.

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Or if you would have the Answer to produce both principal and interest, then make the se∣cond

number to be the sum of the given princi∣pal and interest, and the fourth proportional will answer your desire. Thus,

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VII. When the Interest of 100 for a year is given, and the interest of any other sum of pounds, shillings, and pence is required for a year, the answer may be easily found after the practical method delivered in the following Ex∣ample.

Let it be required to find the interest of 148l.−13s. 04. for one year after the rate of 6 per cent. per annum, simple interest?

First, I place the given Numbers according to the Direction given for the Rule of 3, which will then stand thus, viz.

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Now it is evident that if I multiply 148l.−13s−04d. (which is the third num∣ber) by 6 (which is the second number) and di∣vide the product by 100 (which is the first num∣ber) the Quotient will be the answer; Therefore I proceed thus, viz first I multiply the pence by 6, which makes 24 pence, or two shillings, therefore I set down o under the pence, and carry 2 to the next, then I go to the 13s. say∣ing 6 times 13 is 78, and 2 that I carryed is 80 s. which is 4 l. therefore I set down o under the shillings, and carry 4 to the pounds, then I proceed, saying 6 times 8 is 48, and 4 that I

carry is 52, then I set down 2, and carry 5, &c. proceeding thus till the work be finished, and then will the product be 890l.−00s.−00d, which product should be divided by 100 (the first number) but it being an Unite with two Cyphers, I cut off two figures from the right hand of the pounds, with a dash of the pen, and the figures on the left hand of the said dash, are so many pounds, and those on the right hand of it, are the Decimal parts of a pound, whose value may be found out by the 3 R•…•…le of the 2 Chap. But remember, that if there be any shillings or pence, in the product you are to add them to their respective products in your Reduction.

The work of the foregoing Example is as followeth.

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So that by the work I find the interest of 148 l.−13s.−4d. for one year after the rate of 6 per Cent. per An. to be 8l.−18s.−04d−3.2qu.

Another Example may be this, viz. I demand the Interest of 368l. 15s.−3d. for one year, at 6 per Cent. per An. Answer 22l.−02s.−6d. as by the work following.

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VIII. The Interest of 100l. being known for a year, or 365 dayes, the interest of any other sum may be known for any other time, or num∣ber of dayes, more or less than a year, by two single Rules of 3 Direct, viz. First, find out what is the interest of the given sum, for one year, or 365 dayes, according to the last Rule, then having found out that, you may (by ano∣ther single Rule of 3 Direct) find out its inte∣rest for any other time more or less.

Example.

What is the interest of 322l. for 6 years af∣ter the rate of 6 per Cent. per Ann•…•…m •…•…imple In∣terest?

First, I find what is the Interest of 322l. for a year by the following proportion,

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Thus having found the Interest of 322l. for a year to be 19.32l. at 6 per Cent. by the following proportion, I find out its interest for 6 years, to be 115l.−18s.−04¾d. and that added to the principal, makes 437l.−18s.−04¾d. for the sum due to the Creditor at the end of the said time.

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And here take notice that the second number in this last proportion, must always be only the interest of the sum proposed, and not the sum

of the principal and interest, as in the second proportion under the sixth Rule.

After the same manner is the interest of 1l. (at the rate of 6 per Cent. per Annum, or any other rate of interest,) discovered for a day, by the help of which the interest of any sum whatsoever may be discovered for any number of dayes as shall be shown by and by.

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So that by the foregoing proportions I have found that the interest of 1l. at 6 per Cent. per Annum for a day is .0001643835l.

Now if you would know the interest of any other sum for any number of dayes more or less than 365, you may do it by help of the said number after this manner, viz.

Multiply the sum whose interest is required by the said number, and that product will give you the interest of the said sum for one day, then multiply that product by the number of dayes given, and the last product will give you the in∣terest of the said sum for the number of dayes in the Question. Take the following Question for an example, viz.

What is the interest of 568l. for 213 dayes after the Rate of 6 per Cent. per Annum.

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Having finished the work as you see, I find the answer to be 19.8877 &c. which upon sight I discover to be 19l.−17s−09d. by the brief way of valluing a Decimal Fraction of Coyne laid down in the 4 Rule of the 2 Chapter be∣fore-going.

But when the interest of any sum of Money is required for any number of dayes as aforesaid, at any other rate of interest then at 6 per Cent. per Annum, the foresaid number will not then serve for the work, but you are to find out par∣ticular multiplyars for the several Rates of Inte∣rest as is before directed. All which I have ex∣pressed from 4 to 10 per Cent. in the following Table.

When you would find the Interest of any sum for any number of dayes at the rate of	4	per. Cent per An. the Multiplyar is	.0001095890
5	.0001369863
6	.0001643835
7	.0001917808
8	.0002191780
9	.0002465753
10	.0002739726

So that when you would find out the interest of any sum of Money for any number of dayes according to the direction before given, at any Rate from 4 to 10 per Cent. per Annum, Simple Interest, you may performe the work by the mul∣tiplyar in the foregoing Table which is placed against each respective Rate of interest.

IX. When the present worth of a sum of mo∣ney due at the end of any time to come is requi∣red, Rebate being allowed at any rate of Simple Interest, it may be found out by the following method; viz. First, Find out the Interest of 100l. for the time that the Rebate is to be al∣lowed for, and at the same rate of interest pro∣pounded, then make the sum of an 100 pound, and its interest for the proposed time, to be the first number in the Rule of 3, and 100l. the se∣cond number, and the given sum whose present worth is required, let be the third number, and the fourth number in a direct proportion shall answer the question, as in the following Example, viz.

What present Money will satisfie a debt of 100l. that is due at the end of a year yet to come, discount, or Rebate being allowed at the Rate of 6 per Cent. per Annum.

According to the foregoing Directions, I state the numbers as followeth, and the fourth pro∣portional

number or answer to the question is 94.33962 l.=94 l.−06s.−09 ½d fere.

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The reason of the said Analogy will appear if you consider, that there ought to be so much ready money paid, that if it were put out to inte∣rest at the same rate of Int. that Rebate was allow∣ed for, and for the same time, the same would then be augmented to the sum that was at first due, as in the last question, there is given 100 l. which is due at the end of 12 moneths, now I say, that there ought to be so much money paid down to satisfie this debt, as being put out to interest at 6 perCent. for 12 moneths, would then be increa∣sed to 100 l. which is the sum first due, and again it is as evident that if there were 106 l. due at the end of 12 moneths, or a year, and present payment is agreed upon, allowing Rebate at 6 per Cent. per Annum, that then there ought to be paid the sum of 100 l, in full discharge of the said debt of 106 l. for if when I have received the said sum of 100 l, I put it out to interest for one year at the rate of 6 per. Cent. it will then be increased to •…•…06 l.

Therefore to solve the said question, the pro∣portion here used is no more than if I should say, If 106 l. be decreased to 100 l. what will 100 l. be decreased to? The answer is, to 94l.−6s.−9d.•…•…, and for proof, if yon will seek what that sum will be increased to at the end of 12 moneths, at the rate of 6 per Cent. you will find it to be 100l.

Example 2.

How much present money will satisfie a debt

of 82l.−15s. due at the end of 126 dayes, yet to come allowing Rebate after the rate of 6 per Cent. per Annum?

First, I find the interest of 100l. at the same rate of interest for 126 dayes, by the following proportion.

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Then do I add 2.0712l. (the interest of 100l.) to 100 l. and the sum is 102.0712 which I make the first number in the Rule of 3, and 100l. the second, and 82.75l. (the sum given to be Reba∣ted) the third number, and the fourth number in a direct proportion is the answer to the que∣stion, see the work as followeth.

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So that by the work it appears that 82l.−15s. due at the end of 126 dayes yet to come, will be satisfied with the present payment of 81l.−01s.−04¾d. Rebate be allowed after the rate of 6 per Cent. per An.

The proof of the Rule.

Find out (by the eighth Rule foregoing) how much the present money that is paid upon Re∣bate, will amount to being put out to interest for the same time, and at the same Rate of inte∣rest that Rebate was allowed for, and if the a∣mount be equal to the sum that was due at the end

of that time, then you may conclude the work to be rightly performed, otherwise not.

As for Example.

In the foregoing Question it was found that 81.0708 l. being paid presently would satisfie a debt of 82.75 due at the end of 126 dayes to come, and to prove it, let us see whether 81.0708 being put out to interest for 126 dayes at the rate of 6 per Cent. per Annum, will be in∣creased to 82.75 l. (the sum which was said to be due at the end of 126 dayes to come) which I do by these two proportions following according to the Eighth Rule.

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So you see that I have found the interest of 81.0708 for 126 dayes to be 1.6791 &c. which added to the principal 81.0708 the sum is 82.7499 which by the brief way of valuing the Decimal of a pound sterling is 82 l.−15s. and indeed it doth not want 1/10 part a farthing of the exact sum, which is occasioned by the defective Decimal wherefore I conclude the work to be rightly performed.

Upon the foregoing ninth Rule is grounded the manner of calculating the ensuing Table of Multiplyars, which sheweth in Decimal parts of a pound, the present worth of a pound sterling due at the end of any number of years to come,

not exceeding 30, Simple interest being Compu∣ted at 6 per Cent. per annum.

The first number in the Table being found out by this following proportion, viz.

As 106l. is to 100l, so is 1 l. to .943396, and the second number in the Table being the present worth of 1 l due at the end of two years to come, is thus found out, viz. First I consider that 12 l. is the simple interest of 100l, for 2 years, which added to 100 l. makes 112 l. where∣fore I say, as 112 l. is to 100 l. so is 1l. to .892857 l. which is the present worth of 1l. due at the end of 2 years to come.

The several proportions and operations for the whole Calculation being as followeth, viz.

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And after the same manner are all the numbers in the following Table Calculated; which being well understood, the way of calculating most of the ensuing Tables will easily be obtained; and its use you will find immediately after the Table it self.

TABLE I. Which sheweth in De∣cimal parts of a pound the present worth of 1l. due at the end of any number of years to come under 31, at the rate of 6 per Cent. per Ann. Sim∣ple Interest.

years
1	.943396
2	.892857
3	.847457
4	.806451
5	.769230
6	.735294
7	.704225
8	.675675
9	.649350
10	.625000
11	.602409
12	.581395
13	.561797
14	.543478
15	.526315
16	.510204
17	.495049
18	.480769
19	.467289
20	.454545
21	.4•…•…2477
22	.431034
23	.420168
24	.409836
25	.400000
26	.390625
27	.381679
28	.373134
29	.364963
30	.357143

After the same method might this Table be continued to any number of years at pleasure, I might also have calculated for other rates of in∣terest, as those are in the next Chapter concern∣ing Compound Interest, but Simple Interest be∣ing not so generally in practice, I shall there∣fore forbear.

The use of the preceding TABLE.

It is evident (by the ninth Rule foregoing)

that if any sum be paid with an allowance of Rebate, you are to make 100 l. with its interest (for the same time you Rebate for) both in one sum, to be the first number in the Rule of 3, 100 the second, and the sum to be rebated the third, then will the fourth proportional be the answer; and the same may be wrought by any other number and its interest, as well as by 100 l. and its interest mutatis mutandis; Now in the Table before-going there is expressed in Decimal parts of a pound, the present worth of 1 l. due at the end of any number of years to come under 31, &c. that is to say, if you take the money signified by those Decimals, and put it out to interest at 6 per Cent. per Annum, Sim∣ple Interest for so many years as are expressed in the Collum of years against the said Decimal, then will that sum at the end of the said Term be augmented to 1 l, wherefore if you have any sum whatsoever to be Rebated for any number of years within the limits of the Table, make 1 l. the first number in the Rule of 3, and the Decimal in the Table against the number of years to be Rebated for, make that the second, and the sum whose present worth is required the third number, so will the fourth proportional be the Answer. But (because the first number (being Unity) neither multiplyeth nor divideth) if you take the number in the Table, correspon∣dent to the number of years for which you would reckon Rebate, and thereby multiply the sum whose present worth is required, the product will give you the Answer.

Example.

There is a sum of money, viz. 560 l. due at the end of 8 years to come, but the Debtor and Creditor agree that present payment shall be made, and the Debtor to be allowed Rebate after the rate of 6 per Cent. per Annum, Simple Interest. Now I demand how much present mo∣ney will satisfie the said Debt? Answer, 378.378l.=378l.−07s.−06¾d. see the following work.

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First (the Rebate being to be reckoned for 8 years) I look for 8 in the Collum of years, and just against it on the Right hand, I fiud .675675 which I multiply by 560 (the sum whose pre∣sent worth is required,) and the product is 378.378, which (by the brief way of valuing the fraction of a pound sterling) I find at first sight to be 378 l.=07s.−06•…•…d.

This Question, if it had been wrought by the foregoing ninth Rule, would have produced the same answer, for, The Int. of 100l. for 8 Mon. is 48l. and 100 + 48=148 wherefore by the Rule of 3 I say

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X. When an Annuity or yearly income is in ar∣rears for any number of years, and you would know the increase, or amount of it, allowing Simple Interest at a certain rate per Cent. per annum, for each yearly payment from the time it first be∣came due, the operation will be somewhat more tedious than to find the amount of one single sum, according to the eighth Rule of this Chap. which will clearly appear by solving the follow∣ing question, viz.

There is an Annuity, or an income of 100 l. per annum forborne to the end of 6 years, I de∣mand how much is due at the end of the said Term, allowing interest at the rate of 6 per Cent. per Annum Simple Interest? Answer 690 l.

In order to the solution of this Question, I consider, First, that

It is evident that for the last year, viz. the sixth years payment, there must be no interest at all Reckoned, because it becomes not due till the end of the sixth year; Secondly, there must be reckoned the interest of 100 l. for one year, viz that which is due at the end of the fifth year; Thirdly, there must be reckoned the in∣terest of 100 l. for two years, viz. that which is due at the end of the fourth year. Fourthly, There must be reckoned the interest of 100 l. for three years, viz. that which is due at the end of the third year, Fifthly, the interest of a 100 l. for 4 years, viz. that which is due at the end of the second year: And Sixthly, The interest of 100 l. for 5 years, viz. that which is due at the end

of the first year, and is forborne the second, third, fourth, fifth, and sixth years; all which interests being added together, and their sum added to the sum of each years income, the sum will exhibit the total sum, due at the end of the said six years, which you may perceive by the following work to be 690 l. which is the an∣swer to the foregoing Question.

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The Construction of Table II.

Upon the foregoing reason is grounded the Calculation of the following Table, which shew∣eth the amount of 1l. annuity, being forborne to the end of any number of years under 31, Inte∣rest being allowed for each yearly payment af∣ter the rate of 6 per Cent. per Annum, Simple In∣terest.

The first number in the Table being 1 l. which is that due at the end of the first year, no in∣terest being due for that; the second number in

the Table is 2.06, which is the first and second years payment, and the interest of 1l. for one year, being that which was due at the end of the first year; The third number in the Table is 3.18 l. being the increase of 1 l. for 2 years ad∣ded to the second number in that Table which is 2.06, for the amount of 1 l. at the end of 3 years is 1.12 which added to 2.06 the second number it makes 3.18 for the third number; The fourth number is the amount of 1 l. for 3 years which is 1.18 added to the number before it, viz. the third number, proceeding in the same method, till you have composed the Table at your pleasure, each number in the Table being 1 l. and the amount of 1 l. (for so many years as it standeth against in the Table made less by one.) added to the number immediately pre∣ceding it.

TABLE II. Which sheweth in pounds and Decimal parts of a pound the amount of 1 l, annuity be∣ing forborne to the end of a∣ny number of years under 31, Simple Interest being compu∣ted after the Rate of 6 per Cent. per Annum.

Years
1	1.00
2	2.06
3	3.18
4	4.36
5	5.60
6	6.90
7	8.26
8	9.68
9	11.16
10	1•…•….70
11	14.30
12	15.96
13	17.68
14	19.46
15	21.30
16	23.20
17	25.16
18	•…•…7.18
19	29.26
20	31.40
21	33.60
22	35.86
23	38.18
24	40.56
25	43.00
26	45.50
27	48.06
28	50.68
29	53.36
30	56.10

The Use of Table II.

In the preceding Table in the Collum under the word Years, are set down every year suc∣cessively from 1 to 30, and the number in the Table placed against each year, is the amount of 1 l annuity, in pounds and Decimal parts of a pound, being forborne so many years as it is

placed against. The use of it will plainly ap∣pear by the solving of one, or two Questions, viz.

There is an Annuity of 134 l.−10s.−6d. all forborne to the end of 4 years; I demand how much is due to the Creditor at the end of the said Term, Simple Interest being allowed after the rate of 6 per Cent. per Annum?

Facit 586l.−10s.−07d.

To answer this Question, First, I look for 4 years, in the Collum of years, and the number against it is 4.36 which is the amount of 1 l. An∣nuity for 4 years; therefore having turned the 1 s.−6d. (in the given annuity) into a Decimal (which is .525) I say by the Rule of 3 thus.

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Thus by the work I find the answer to be 58.529 l. the value of which Decimal by the brief way of valuing a Decimal laid down in the 4th Rule of the 2d Chapter, I find to be 580l. 10s. 7d.

And it is plain that in solving Questions by this Table, that (the first number in the Rule of 3 being unite) if you multiply the given Annuity

by the proper Tabular Number, that then the product will be the answer.

Example 2.

What is the amount of an Annuity of 150 l. 10 s. being forborne to the end of 7 years, al∣lowing Simple Interest after the Rate of 6 per Cent, per Annum? Answer, 1243 l.−02 s.−07 ¼ d. fere.

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XI. When an Annuity or yearly Income, for a certain number of years to come, is to be sold for ready Money, and the seller is to allow the Buyer Rebate at * 1.3 Simple Interest for his present payment, then in this case the buyer ought to pay so much present money for each yearly payment, as being put out at Simple Interest for so many years as it is Rebated for, it would then amount to one yearly payment, and the sum of all those pre∣sent worths will be the present worth of the Annuity required, the Rule will appear very plain by the following Example.

There is an Annuity or Lease of 100 l. per Annum to continue 6 years yet to come to be sold for ready Money, the Seller being to allow the Buyer Rebate at 6 per Cent. per Annum, Simple Interest, now I desire to know how much present Money will buy out the said Lease?

Facit 499 l.−09 s.−04 ¼ d. fere.

It is evident that if we find out the present worth of 100 l. due at the end of the first year, and also the present worth of a 100 l. due at the end of the second year, and the present worth of 100 l. due at the end of the third year, and likewise the present worth of 100 l. due at the end of the fourth, fifth, and sixth years, and add all these present worths together, their sum will be the present worth of the given Annuity; which several present worths are found out accor∣ding to the ninth Rule, by the several propor∣tions following, viz.

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So that you see by the foregoing proportions, the present worth of 100 l. per Annum to conti∣nue six years, allowing Rebate at 6 per Cent. per

Annum, Simple Interest, is 499.468754 l.=499 l. 9 s. 04 ¼d.

Upon the foregoing eleventh Rule is grounded the construction and calculation of the following Table which * 1.4 sheweth the present worth of 1 pound annuity to continue a∣ny number of years under 31 Simple Interest be∣ing computed after the rate of 6 per Cent. per Annum; The first number in the Table is .943396 which is the present worth of 1 pound due at the end of a year to come. The second number in the Table is 1.836253, which is the sum of the present worths of 1 l. due at the end of two years to come, and of 1 l. due at the end of one year to come added together; And the third number in the Table is 2.683710 which is the sum of the present worths of 1 l. due at the end of 3, 2, and 1 years to come. And after the same method is the whole Table calcu∣lated.

But the numbers in the said Table may more easily be found out thus, viz. Look in the first Table, and let the first number of that be the first number of this third Table, and let the sum of the first number in this, and the second number in that be the second number in this Ta∣ble, and for the third number in this Table take the sum of the second in this, and the third in that Table, and in this manner you may proceed till you have composed the whole Table.

TABLE III. Which sheweth the present worth of 1 l. annuity to continue any num∣ber of years under 31, Simple Interest being computed at 6 per Cent. per An.

Years
1	.943396
2	1.836253
3	2.683710
4	3.490161
5	4.259391
6	4.994685
7	5.698900
8	6.374575
9	7.023925
10	7.648925
11	8.251334
12	8.832729
13	9.394526
14	9.938004
15	10.464319
16	10.974523
17	11.469572
18	11.950341
19	12.437630
20	12.892175
21	13.334652
22	13.765686
23	14.175524
24	14.585360
25	14.985360
26	15.375985
27	15.757664
28	16.120798
29	16.485761
30	16.842904

The Use of the foregoing Table. III.

In the foregoing third Table, in the left hand Collum under the Title of years, are expressed all the integral numbers, from 1 to 30, which signifie so many years, and the numbers in the Right hand Collum which are placed against the number of years are pounds, and decimal

parts of a pound sterling, and every one of them are the present worth of 1 pound Annuity to continue so many years to come as are placed against them in the Collum of years, Rebate be∣ing allowed at Simple Interest 6 per Cent. per An.

As, suppose there were a Lease of 20 shillings per annum to continue 6 years, to be sold for pre∣sent money, allowing the buyer Rebate at 6 per Cent. per annum Simple Interest. I desire to know how much is its present worth? To answer this, I look in the Collume of years for 6, and in the next Collume on the right hand just against 6 you have 4.99468 5 l.=4 l.−19 s.−10 ¾ d. which is the answer to the Question. And by the help of this Table may the present worth of any Annuity to continue any number of years under 31 be found out, allowing Rebate at 6 per Cent. per Annum, Simple Interest, by one single Rule of 3 Direct, according to the manner of solving the follow∣ing question, viz.

Quest. 1.

There is a Lease of 18 years yet to come, of the yearly value of 130 l. to be sold for ready Money, and the purchaser is to be allowed Re∣bate after the rate of 6 per Cent. per Annum, Sim∣ple Interest, now I demand how much is the present worth of this Lease?

Facit 1553 l.−10 s.−10 ¾ d.

First, I look in the Table for 18 years, and over against it on the right hand I find •…•…1.950341 which is the present worth of 1 pound annuity to continue 18 years, &c. Therefore by the Rule of 3 Direct, I say

〈 math 〉〈 math 〉

So that by the work you find the answer to be 1553.544 l. &c. or 1553 l.−10 s.−10 ¾ d. ve∣ry near, which said answer is nothing else but the product of the Tabular number, (11.950341l.) multiplyed by the given annuity (130 l.) For it is evident, that if the present worth of 1 pound annuity to continue 18 years be 11.950341 l. then the present worth of 130 l. per annum, to continue the same number of years (and Rebate being allowed at the same Rate per Cent. per an. for the one as for the other) must be 130 times as much. But when Rebate is to be allowed af∣ter any other rate then 6 per Cent. per annum, then the foregoing Table will not at all be useful, but you must have recourse to a Table calculated for the same rate of interest, which you may easily perform at leisure by the foregoing Rules.

Quest. 2.

What Annuity to continue 18 years will 1553.5443 30 purchase, allowing the Buyer Sim∣ple Interest at 6 per Cent. per annum?

Facit 130 l.

This Question is but the converse of the for∣mer, and may be thus Resolved, viz. Táke the Tabular number corresponding to 18 years, which is 11.950341 by which divide the given purchase Money, and the Quotient will give you the annuity that it will purchase, viz.

〈 math 〉〈 math 〉

So that by the work I find it will purchase an Annuity of 130l. to continue 18 years.

The reason of the work is plain, for if the Tabular number correspondent to 18 years be the present worth of 1 l. Annuity to continue 18 years to come, then it is certain that so much money as is expressed by that Tabular number, will purchase an Annuity of 1 l. to continue 18 years: And consequently we may find by help of the said Table what annuity any other sum of money will purchase, to continue any number of years not exceeding 30, by a sin∣gle Rule of 3 Direct, as in the last Question, the proportion is as followeth, viz.

〈 math 〉〈 math 〉

And it is no more in effect than a sum in Divi∣sion, for the second number (being 1) neither multiplyeth, nor divideth, &c.

By what hath been said concerning the use of the foregoing Table, you may perceive that the p•…•…esent worth of an Annuity is found out by mul∣tiplication, and to know what annuity any

sum will purchase is performed by Division.

I might have made Tables for other Rates of Interest, but Simple Interest being seldom allowed in the purchasing or valuing of Leases and Annuities, (they being generally purchased at Compound Interest, or Interest upon Interest) makes me forbear, and indeed at Simple Interest a Lease is over-valued.

Notes

• * 1.1

  Vide Sect. 6 of the 5 chap of Mr. Ker∣sies Appendix to Wing. Arith.
• * 1.2

  Of Rebate, what it is.
• * 1.3

  The Rebate of Annuities at Simple Interest
• * 1.4

  The construction of the 3 Table.