7. To find the interest of 355/ 15s for 4 years, at 4 per cent.

per annum.

Ex. 8. To find the interest of 321 58 8d,

per cent. per annum.

per annum.

Ans. 56/ 188 43d. for 7 years, at 4 Ans. 97 128 1d.

9. To find the interest of 1701, for 1 year, at 5 per cent. Ans. 12/ 158. 10. To find the insurance on 205/ 15s, for of a year, at 4 per cent. per annum. Ans. 21 18 13d. 11. To find the interest of 3191 6d, for 5 years, at 33 per cent. per annum. Ans. 68/ 158 94d. 12. To find the insurance on 1077, for 117 days, at 43 per cent. per annum. Ans. 1/ 128 7d. 13. To find the interest of 171 58, for 117 days, at 43 per cent. per annum. Ans. 58 3d. 14. To find the insurance on 7121 6s, for 8 months, at 7 per cent. per annum. Ans. 351 128 31d. Note. The Rules for Simple Interest, serve also to calculate Insurances, or the Purchase of Stocks, or any thing else that is rated at so much per cent.

See also more on the subject of Interest, with the algebraical expression and investigation of the rules at the end of the Algebra, next following.

COMPOUND INTEREST, called also Interest upon Interest, is that which arises from the principal and interest, taken together, as it becomes due, at the end of each stated time of payment. Though it be not lawful to lend money at Compound Interest, yet in purchasing annuities, pensions, or leases in reversion, it is usual to allow Compound Interest to the purchaser for his ready money.

RULES.-1. Find the amount of the given principal, for the time of the first payment, by Simple Interest. Then consider this amount as a new principal for the second payment, whose amount calculate as before. And so on through all the payments to the last, always accounting the last amount as a new principal for the next payment. The reason of which is evident from the definition of Compound Interest. Or else,

2. Find the amount of 1 pound for the time of the first payment, and raise or involve it to the power whose index is denoted by the number of payments. Then that power multiplied by the given principal, will produce the whole

amount,

108

Numb. Square.

901 811801

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902 813604

733870808

30-0333148

9.662040

736314327

30.0499584

9.665609

738763264

30-0665928

9.669176

741217625

30.0832179

9.672740

906

820836 743677416

30.0998339

9.676301

907

822649

746142643

30-1164407

9.679860

748613312

30.1330383

9.683416

751089429

30.1496269

9.686970

910 828100

753571000

30'1632063

9.690521

756058031

30*1827765

9.694069

912 831744

758550528

30 1993377

9.697615

761048497

30.2158899

9.701158

763551944

30-2324329

9.704698

766060875

30.2489669

9.708236

768575296

30.2654919

9.711772

771095213

30.2820079

9.715305

773620632

30.2985148

9.718835

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30.3150128

9.722363

778688000

30.3315018

9.725888

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9-729410

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30.3644529

9-732930

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30.3809151

9.736448

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30.3973683

9.739963

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30.4138127

9.743475

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30.4302481

9.746985

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30.4466747

9.750493

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30.4630924

9.753998

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9.757500

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30.4959014

9.761000

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30.5122926

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30.5286750

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9.771484

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9 827025

9.830475

ALLIGATION.

ALLIGATION teaches how to compound or mix together several simples of different qualities, so that the composition may be of some intermediate quality or rate. It is commonly distinguished into two cases, Alligation Medial, and Alligation Alternate.

ALLIGATION MEDIAL.

ALLIGATION MEDIAL is the method of finding the rate or quality of the composition, from having the quantities and rates or qualities of the several simples given. And it is thus performed:

* MULTIPLY the quantity of each ingredient by its rate or quality; then add all the products together, and add also all

* Demonstration. The rule is thus proved by Algebra.

Let a, b, c be the quantities of the ingredients,

and m, n, p their rates, or qualities, or prices;

then am, bn, cp are their several values,

and ambn + cp the sum of their values,

also a+b+c is the sum of the quantities,

and if r denote the rate of the whole composition,

then a+b+cxr will be the value of the whole,

conseq. a + b + c xr = am + bn + cp,

and r = am + bn + cp ÷ a+b+c, which is the Rule.

Note, If an ounce or any other quantity of pure gold be reduced in. to 24 equal parts, these parts are called Caracts; but gold is often mixed with some base metal, which is called the Alloy, and the mixture is said to be of so many caracts fine, according to the proportion of pure gold contained in it; thus, if 22 caracts of pure gold, and 2 of alloy be mixed together, it is said to be 22 caracts fine.

If any one of the simples be of little or no value with respect to the rest, its rate is supposed to be nothing; as water mixed with wine, and alloy with gold and silver.

VOL, I.

S

the

OF RATIOS, PROPORTIONS, AND PROGRESSIONS.

NUMBERS are compared to each other in two different ways the one comparison considers the difference of the two numbers, and is named Arithmetical Relation; and the differénce sometimes the Arithmetical Ratio: the other considers their quotient, which is called Geometrical Relation; and the quotient is the Geometrical Ratio. So, of these two numbers 6 and 3, the difference, or arithmetical ratio, is 63 or 3, but the geometrical ratio is § or 2.

There must be two numbers to form a comparison: the number which is compared, being placed first, is called the Antecedent; and that to which it is compared, the Consequent. So, in the two numbers above, 6 is the antecedent, and 3 the consequent.

If two or more couplets of numbers have equal ratios, or equal differences, the equality is named Proportion, and the terms of the ratios Proportionals. So, the two couplets, 4, 2 and 8, 6, are arithmetical proportionals, because 4 - 2 2; and the two couplets 4, 2 and 6, 3, are geometrical proportionals, because 433 2, the same ratio.

6

11

8

To denote numbers as being geometrically proportional, a colon is set between the terms of cach couplet, to denote their ratio; and a double colon, or else a mark of equality, between the couplets or ratios. So, the four proportionals, 4, 2, 6, 3 are set thus, 4:2:: 6:3, which means, that 4 is to 2 as 6 is to 3; or thus, 4: 2 6 3, or thus, both which mean, that the ratio of 4 to 2, is equal to the ratio of 6 to 3.

Proportion is distinguished into Continued and Discontinued. When the difference or ratio of the consequent of one couplet, and the antecedent of the next couplet, is not the same as the common difference or ratio of the couplets, the proportion is discontinued. So, 4, 2, 8, 6 are in discontinued arithmetical proportion, because 4 2=8 62, whereas 8-2 = 6 and 4, 2, 6, 3 are in discontinued geometrical proportion, because == 2, but = 3, which is not the same.

But when the difference or ratio of every two succeeding terms is the same quantity, the proportion is said to be Continued, and the numbers themselves make a series of Continued Proportionals,

ALLIGATION ALTERNATE.

ALLIGATION ALTERNATE is the method of finding what quantity of any number of simples, whose rates are given, will compose a mixture of a given rate. So that it is the reverse of Alligation Medial, and may be proved by it.

RULE 1*.

1. SET the rates of the simples in a column under each other.-2. Connect, or link with a continuéd line, the rate of each simple, which is less than that of the compound, with one, or any number, of those that are greater than the compound; and each greater rate with one or any number of the less.-3. Write the difference between the mixture rate, and that of each of the simples, opposite the rate with which they are linked.-4. Then if only one difference stand against any rate, it will be the quantity belonging to that rate; but if there be several, their sum will be the quantity.

The examples may be proved by the rule for Alligation. Medial.

Demonst. By connecting the less rate to the greater, and placing the difference between them and the rate alternately, the quantities resulting are such, that there is precisely as much gained by one quantity as is lost by the other, and therefore the gain and loss upon the whole is equal, and is exactly the proposed rate: and the same will be true of any other two simples managed according to the Rule. In like manner, whatever the number of simples may be, and with how many soever every one is linked, since it is always a less with a greater than the mean price, there will be an equal balance of loss and gain between every two, and consequently an equal balance on the whole. QE. D.

It is obvious, from this Rule, that questions of this sort admit of a great variety of answers; for, having found one answer, we may find as many more as we please, by only multiplying or dividing each of the quantities found, by 2, or 3, or 4, &c: the reason of which is evident; for, if two quantities, of two simples, make a balance of loss and gain, with respect to the mean price, so must also the double or treble, the or part, or any other ratio of these quantities, and so on ad infinitum.

These kinds of questions are called by algebraists indeterminate or unlimited problems; and by an analytical process, theorems may be raised that will give all the possible answers.

EXAMPLES.