Example 1.

Example 2.

And so of any other rate of Interest.

Example.

To know the Amount of 140 l. for 121 days, at 6 l. per Cent. Simple In­terest.

Example.

If 142 l. 15 s. 8 d. be due at the end of 121 days, what is it worth in ready Money? [...]

[Page 7] Worth in ready Money 140 l. at 6 l. per Cent. Simple Interest, and may be done for any other rate of In­terest, working by the first Proportion and this former Rule.

Example.

Let it be required to find the A­mount of 1 l. for 50 years.

and is the Number sought, omit­ting the five superfluous places of De­cimals.

Another Example.

It is required to find the Amount of 1 l. forborn 20 years three quarters.

And the like may be done for days, and the Converse when an Amount is given, the time thereto may be found by Division, searching in the Table what Number amongst the Decimals for time agrees to the Divisors and last Quote. See Prop. 8. the First Section.

And here it is worth noting, That many Questions may be put con­cerning Compound Interest, which are of the like difficulty, as to raise the printed Logarythmetical Cannon.

For Example such a Question may be put;

One pound was put out at Com­pound Interest, and in 10 years time amounted to 10 l. in what space of time did it amount to 2 l. the answer is the Logarythm of the Number two (to wit) 3,01023 years which was not raised without much toyl, and the rate of Interest in those Logarythms is near 26 l. per Cent. to wit 25,89292.

The Uses of the said Table.

Example.

What shall 136 l. 15 s. 06 d. a­mount unto being forborn 20 years at 6 per Centum?

Reduced is 438 l. 13 s. 1 d. 4 [...]

Example first, for making the Table of Discount.

An Unit divided by 3,20713, the Quotient is 311804, the present Worth of 1 l due 20 years hence.

Example second.

If 400 l. be due 20 years hence, [Page 17] What is it worth in ready Money, aba­ting Compound Interest at 6 per Cen­tum per Annum?

Divide 400 by 3,20713, the A­mount of 1 l. forborn 20 years at Compound Interest, and the Quotient is 124 l. 722, or 124 l. 14 s. 5 d. ¼ And how to reduce sundry Payments, to an Equation of time at Compound Interest. See first Example of Prop. 8.

Example.

Let it be required to find what one pound a year Annuity forborn for 30 years at 6 per Centum shall amount to.

One pound forborn at Compound Interest so long amounts to 5, 74349 which lessened by an Unit is 4, 74349 which divided by 06, the Quotient is 79, 0581 and this is the Number [Page 19] found in the Vulgar Tables for for­bearance of Annuities.

Second Example.

Let it be required to find what 20 l. Annuity forborn for 15 years shall amount unto at 6 per Centum.

1 l. Principal forborn 15 years amounts to 2, 39655 from which subtracting an Unite it holds.

As, 06 to 1,39655 so 20 to 465, 516, that is 475 l. 10 s. 4 d.

Third Example.

A Quarterly Rent of 25 l. was re­spited 20¾ years, by the first Propor­tion the amount of 1 l. so long for­born was 3, 34978. And the Interest of one pound for a quarter is 0, 14675 Wherefore by Proportion, As, 014675 Is to 2, 34978: So is 25 to 4003, 032 that is 4003 l. 00 s. 7¾ d.

[Page 20] This Useful Proportion I thus demonstrate which the Rea­der may pass by.

Imagine the Land or Stock that yields an Annuity to be such a Prin­cipal sent out for the whole term as will bring in so much yearly Interest as the Annuity comes to, then at last the whole at Compound Interest is to be repaid, whereof so much is supposed to be repaid in the Value of the Land, as its first Principal came to, and the rest in Money; wherefore out of the whole Amount of that Principal and its Interest, the Principal must be de­ducted unless to shun it by that which Geometers call conversion. See Com­mentators on 16 Def. Quinti Euclidis &c. we say,

As the first term is to the diffe­rence of the first and second;

So the Third Term to the diffe­rence [Page 21] of the third and fourth.

The Plain Proportion grounded up­on the former Considerations runs thus,

As 1 l. Principal. Is to its Amount for the time forborn;

So the Principal that shall bring in any Annuity proposed. To the Sum of the said Principal and of the Arrea­rages of the Annuity.

Then it will hold by conversion of Reason. As 1 l. forborn at Compound Interest is to its Amount less by an Unit for the time forborn;

So is the Principal of an Annuity, forborn the like time,

To the Arrearages of the Annui­ty.

And instead of the third term of this Proportion, we may take in a fraction equivalent thereto, the Nu­merator whereof is the Annuity or yearly payment of Rent, and the De­nominator the Interest of 1 l. for a year; for to find the Principal of an An­nuity say,

[Page 22] As 6 is to 100. Or rather, 06: 1.

So is the Annuity to its Princi­pal.

And both these latter Proportions compounded into one will be the pro­portion first delivered, the Units in each being expunged as insignificant either in Multiplication or Division.

Example. First for making the Tables.

To find the present worth of an Annuity of 1 l. per Annum, to conti­nue 25 years at 6 per Centum com­pound Interest.

The Amount of 1 l. for that time is 4,29187 which Multiplyed by ,06 the fact is ,257512, whereby dividing 3,29187 the Quote is 12 l. ,78335 the present worth sought.

Example. Secondly for half-year­ly Payments.

An Annuity of 40 l. payable, 20 l. each half year is to befold for 12 years at 6 per Centum.

Multiply these two together, and that added together makes the fact of both;

Which is—,058487 It therefore holds,

As ,058487 is to ,1,012196.

So is 20 to 346, 166 that is 346 l. 3 s. 4 d. the present worth thereof.

Example the first.

To find what Annuity 1 l. shall purchase to continue 30 years, it holds;

As 4,74349 to ,06. So 5,74349 to 0,07264.

Example Second.

Let it be required to find what rent payable yearly 8 l. shall purchase at 6 per Centum to continue 21 years.

As 2,39956 the Amount less an U­nit of 1 l. for 21 years is to ,20397 the fact of ,06 and the amount,

So is 8 to ,68 or 13 s. 7¼ d. the An­nuity sought.

Memorandum, That by the Fact is meant that you should multiply the foregoing Figures by ,06. Viz. 2,39956 by ,06,

Which makes 2,2039736.

[Page 28] Now whereas the Lease of a house of 1 l. per Annum to continue 21 years is commonly sold for 8 l. or 8 years purchase, and your mony will purchase a certainty but of 13 s. 7 d. [...] per Annum, you see by this supposition you are abated 6 s. 5 d. 3/ [...]nt of Taxes Reparations and Casualties; and verygood Reason there is for great abatements, for a Tenant taking a Lease on a Tunber house, if it be burnt down by a Fire beginning at his Neighbours as leases commonly Run, is bound to build it up again and hath no relief ei­ther in Law or Equity against his Landlord, as I am informed by able Council, only he hath the benefit of a Benevolence, his Action against them where the fire began (who perchance are ruined.)

[Page 29]

Hence it appears that the Value of Leases of Houses cannot be estimated near the Truth by the Common Ta­bles for Annuities at the currant rate of Interest, and that if any one would use them to this purpose it were much nearer the truth first to abridge the Rent as aforesaid.

Example:

Let the Fee Simple or Copy-hold Lands be worth 16 years 8 months Purchase, then dividing 100 by 16⅔ the Quotient is 6, whereof 6 pound in the 100 is an equitable Rate of Interest whereby to compute the present worth of a Lease of any number of [Page 31] years therein, and so è contra if mony were at 8 per Centum, the Laws of Arithmetick allow the worth of the Inheritance of the best Land that is, to be but 12½ years Purchase, which some would confirm, from this reason, because otherwise their money would yield a better income at Simple or Compound Interest, but the most pro­per Reason is derived from the Na­ture of a Geometrical Progression de­creasing ad Infinitum; for instance, ad­mit you have a Tenant in the Te­nure or Possession of 1 l. per Annum, and you say to him, pay the rent now that will be due at the end of

and you will rebate him after the rate of Compound Interest. I say the Total of all those Payments shall never exceed 12 l. 10 s. 00 d.

[Page 32] The Proportion for casting up the sum of a sinite Geometrical Progression runs thus,

As the difference of an assumed extreme and its next inward mean is to the next inward mean;

So is the difference of the remote extremes to the sum of the Progression, except the assumed extreme.

The reason wherof is, That if a rank of Numbers be in Geometrical Pro­gression their sums and differences are likewise in the same Proportion. See 35 of 9 Book of Euclid, or Briggs his Arithmetica Logarithmica.

Example.

To the Sum of all the Consequents is 1089.

Wherefore the sum of the whole progression is—1092.

And supposing this Progression to decrease infinitely, then will the first term be o, and the sum of all the Differences 729, and it holds. As [...]

Wherefore the sum of this infinite Pro­gression is 1093½, and can never exceed it, and the said progression continued but in part towards the left hand, would stand thus, &c. 1/729 1/243 1/81 1/27 1/9 ⅓ I.

2. But admit the present worth of a Lease for a certain number of years be given, some third term must be further [Page 34] given, let that be the yearly rent, and then you cannot assign the rate; (and the contrary) in this Case to find the rate is one of the most difficult Questi­ons that commonly happens about Annuities, because the Proposition in the 5, 6, (also 4th.) Prop. will not hold conversly, there are but two terms in the Proposition given, which contain but a bare ratio, &c. therefore though out of Tables of Forbearance of Money at compound Interest, you can make those for Annuities, yet the con­verse will not hold.

In this Case you must either by help of the 5 Prop. and common Lo­garithms, or of Tables of the present worth of Annuities, calculated to the best rate that shall suit the Inheritance, find the present worth of the Num­ber of years proposed according to two rates assumed as near the truth as you can possible, and then if you have not lighted upon the given [Page 35] worth of the years assigned, use the help of this Approximation.

As the difference of the present worths found, is to the difference of the assumed rates of Interest;

So the difference between the gi­ven worth and the truest of those Tryal worths;

To the difference between the rate of Interest of the tryal worth and that sought.

And when the rate of Interest is truly found, compute accordingly the present worth of the years sought.

But this were to send away the Rea­der, as if we could in this Case give no answer to the question, by help of the ta­ble here used; whereto I answer, That if the worth of the Inheritance be assigned, repair to the following Proposition.

But if not, let the Casualty as in the 6th. Proposition be reduced to a cer­tainty; viz. if it concern the Lease of a house which is a Casualty, abridge [Page 36] the Annual Rent, and then you may by the 5th. Prop. cast up the Value of any Number of Years therein.

But herein I would not be misun­derstood, as if when a Lease of a House of 1 l. yearly for 21 Years is sold for 8 l. 10 s. the which will pur­chase an Annuity or Certainty of 14 s. 5 ½ per Annum, and any Number of years in this Certainty shall be equivo­lent to as many in that Casualty, that therefore Tables made to both Rates, and a Computation to both the Year­ly Rents must needs agree, because all Tables of Annuities are made for Certainties not Casualties.

Or lastly, repair to the first and last Prop. and you will there find how to cast up the Amount of 1 l. Princi­pal for any time, and at any Rate, where the true manner of such Equa­tions is shewed.

In this second Case is couched two usual Questions, most commoly pro­pounded [Page 37] without sufficient Limits: As,

1. When a Lease is sunk by a Fine to a certain Yearly Rent, for a cer­tain term of time, What the whole Lease is worth: Or,

2. What any number of years to be added, after the term in Lease is ex­pired, is worth.

In Order to the Resolution of ei­ther of these Questions it must be a­greed how much the sunk Rent was, or at least as much given as before was required, and then as before you have a foundation whereon to raise a Rate of Interest, for there is now given the yearly Rent sunk, its pre­sent worth, and the time, and the Rate being found, you may then, ac­cording as is done in the 5th. Prop. resolve both these Questions.

Example first.

As 74108, Speidells difference of the Logarithm of 13 and 14.

Is to 32184 d. Brigs his difference of those Logarithms;

So is 16000, Speidells difference of the Logarithms of 62 and 63.

To 69487, the difference of those Logarithms in Mr. Brigs, or the Common Tables.

Moreover Van Schooten in his Mis­cellanies gives you an Account of all Numbers under 1000, that are prime or incomposite, to wit, 1226 in Num­ber, viz. the which no other Number will divide, to the which if the dif­ferences be first found by Proportion, which in this Case having the two fixt Terms fixed, may be converted into a Multiplication or Division, and that Multiplier or Divisor being Multi­plied [Page 40] by all the Digits into an Addi­tion or Substraction, the Loga­rithms of all the Composite Numbers will easily be made out of the rest, by the continual Addition of the Loga­rithm of 2, or otherwise.

In the Table here used the time is the Logarithm, and the Amount the Number thereto belonging, and a Proportion accordingly may be ap­plied to any kind of Logarithms, to find the Excess of time above a year, in which a 100 l. at 6 per Centum did amount to 108 l. But it may be more easily thus done.

As, 02530586, the Logarithm of the Amount 1, 06.

Is to 1, viz. One year the time that 1 l. Principal was forborn;

So is, 03342375 the Logarithm of the Amount 1, 08.

To 1, 32079, the time required, and that is 1 Year, 3 Months, and about 26 Days, and thus the nearest way of [Page 41] resolving such a Proposition, having the Common Logarithms in Store, is by a Division of the Logarithms: But supposing no such Tables, it may be supplied by two Divisions by help of this Table, which I shall explain in two Cases.

Sect. 1. The Amount of 1 l. being pro­posed, to find what time it must be forborn, at 6 per Centum to amount unto as much.

Divide the given Amount by some Amount in the Table, next lesser, and that Quotient, again by the next les­ser Amount, reserving the Quoti­ent.

If the time in the Tables belong­ing to the two first Divisors, and last Quote be added together, it is the time sought.

Example.

1 l. in a Year at 8 per Centum did amount to 1 s., 08, in what time at 6 per Centum, shall it amount to so much.

[Page 43] But to save the Reader this trou­ble we have added the Equated time for these Rates.

And by the second Proposition the present worth of sundry payments due hereafter being computed, after the manner of this Example, a [...]ue time may be found when the total of all those Payments may equitably be paid at once.

[Page 44] Sect. 2. The Rate of Compound Inte­rest, and the time being given to find what 1 l. Principal did amount to in that time.

Or rather let it be thus proposed:

How long shall one pound at 6 per Cent. be forborn to amount to as much as 1 l. forborn any space of time at any other Rate of Interest doth amount unto, and what is the said Amount?

By the time Proposed multiply the Equated time, next before found (in the first Case) that agrees to the Rate proposed, and you have the time sought, and what it shall amount, is found by the first Proposition.

For instance, if 1 l. be forborn 18 years at 8 per Centum, what shall it amount to?

Example, 100 l. did amount to 105 l. in, 83732 Years.

[Page 83] Admit it were required to find what 1 l. amounted to in 20 Years at 8 per cent. multiply 1,32079 by 20, the Fact or Product is 26,14158, and by Prop. 1. 1 l. at 6 per cent. in that time did amount to 4 6609.

Now if the Amount of 1 l. be given, Annuity Problems are salved thereby. And for the advantage of this Proposition the Decimals of time were added.

Example.

50 l. a Year at 8 per Cent. is worth 490 l. 18 s. 2½, or 490,91, the time of continuance is 20 years. [...]

[Page 92] An Amount is proposed for 20 years to be 4,6609, what is the Rate of Interest?

1. The time in which 1 l. came to so much at 6 per Cent. is 26,4158, found by the second Proposition.

2. Divide 26,4158 by 20, the time proposed, the Quote is 132079 years.

3. 1 l. at 6 per Cent. in that time amounted to 1,08, the Ratio sought.