Otherwise. To find ANY ROOT by approximation. RULE. Let g denote the given number or power; n, the index of the power; a, an assumed power nearly equal to g; r, its root, and R, the required root.

2

: a cs g : : r: Rúr;*

which difference or correctional number, being added or subtracted (as required) will give R: and by repeating the process, any degree of accuracy may be obtained.

(1) What is the square root of 141376? + (2) What is the cube root of 53157376? Ans. 376. (3) What is the fourth root of 19987173376? Ans. 376. Required the fifth root of 2508-474615614240625.

Ans. 4.785.

(5) Required the sixth root of 3·1416. Ans. 1.210201+

SINGLE POSITION.

Is the method of using one supposed number, and working with it as the true one, to find the real number required.‡

RULE. As the result from the supposition, is to the true result; so is the supposed number, to the true one required. PROOF. Add the several parts together, according to the conditions of the question.

(1) A schoolmaster being asked how many scholars he had, said, "If I had as many, half as many, and one quarter as many more, I should have 88." How many had he?§

• This for tl'e Cube Root will be, As 2a +g: a ∞g::r: Ror. + 141376(376 the root.

9

6)51 dividend.

= 1369 subtrahend.

74)447 dividend.

141376 subtrahend.

Questions belonging to this Rule have the results proportional to their suppositions: the conditions requiring the number sought to be increased by the addition of itself, or of some known multiple or part thereof; or to be diminished by the subtraction of such part. § Suppose he had 40.-Then 40+40+20+10=110.

And, as 110: 88: : 40:

88×4/0 352

11/0 11

-32 Ans.

ر

(2) A person who had a certain number of antique coins, said, If the third, fourth, and sixth parts of the number were added together, they would make 54. How many had he? Ans. 72.

(3) A chaise, a horse, and harness, cost £60; the horse being double the price of the harness, and the chaise double the price of the horse and harness. What was given for each? Ans. Horse £13..6..8. harness £6.,13..4. chaise £40.

(4) What sum of money will amount to £300. in ten years, at £6. per cent. per annum, simple interest? Ans. £187..10. (5) A, B, and C, dividing a quantity of goods, which cost £120. mutually agreed that B should have a third part more than A, and C a fourth part more than B. What must

each man pay?

Ans. A £30. B £40. C £50. (6) A gentleman bought a house, with a garden, and a horse in the stable for £500. He paid four times the price of the horse for the garden, and 5 times the price of the garden for the house. What were their respective prices?

Ans. horse £20. garden £80. house £400.

DOUBLE POSITION

REQUIRES the use of two supposed numbers to find the true one required.*

RULE Work with the two supposed numbers, and mark the errors in the results with or, according as they exceed or fall short of the true result: then place the errors against their respective positions, and multiply them cross

wise.

If the errors be of like kinds, i. e. both greater, or both less than the given number, take their difference for a divisor, and the difference of the products for a dividend. But if unlike, take their sum for a divisor, and the sum of their products for a dividend: the quotient will be the answer.t

Questions belong to this Rule which require the addition or subtraction of a number, &c., which is not any known part of the number required. The results are, therefore, not proportional to their suppositions.

The following Rule will, in some cases, be found more eligible : Multiply the difference of the supposed numbers by the less error; and divide the product by the difference of the errors when they are of

(1) A, B, and C, would divide £200. among them, so that B may have £6. more than A, and C £8. more than B. How much must each have?*

(2) A man had two silver cups of unequal weight, having one cover to both of 5 ounces. Now if the cover is put on the less cup, it will double the weight of the greater; and put on the greater cup, it will be thrice as heavy as the less. What is the weight of each?

Ans. 3 ounces the less, and 4 the greater. (3) Three persons conversing about their ages; says K, My age is equal to that of H, and of L's;" and L says, "I am as old as both of you together." Required the ages of K and L; H's being 30. Ans. K 50, and L 80.

(4) D, E, and F, playing at cards, staked 324 crowns; but disputing about the tricks, each man seized as many as he could: E got 15 more than D; and F got a fifth part of both their sums added together. How many did each person get? Ans. D 127, E 1424, and F 54.

(5) A gentleman meeting with some ladies, said to them, "Good morning to you, ten fair maids." "Sir, you mistake,”

like kinds, or by their sum, when unlike: the quotient will be a correctional number; which being added to the nearest supposition when defective, or subtracted from it, when excessive, will give the number required.

1

answered one of them, "we are not ten: but if we were three times as many as we are, we should be as many above ten as we are now under." How many were they?

ARITHMETICAL PROGRESSION.

Ans. 5.

AN Arithmetical Progression, is a series of numbers increasing or decreasing uniformly by a continued equal difference. Thus, 1, 2, 3, 4, 5, &c. 2, 5, 8, 11, 14, &c. 9, 8, 7, 6, 5, &c. 16, 12, 8, 4,

are increasing Arithmetical Series.

0, &c. are decreasing Arithmetical Series.

Observe, that the terms of the first series are formed by adding successively the common difference 1, and the second by the common difference 3. The terms of the third and the fourth diminish continually by the subtraction of 1 and 4 respectively.

In an odd number of terms, the double of the mean (or middle term) is equal to the sum of the extremes, or of any two terms equidistant from the mean. Thus, in 1, 2, 3, 4, 5,

the double of 3=1+5=2+4= 6.

In an even number of terms, the sum of the two means is equal to the sum of the extremes, or of any two equidistant terms. Thus, in 2, 4, 6, 8, 10, 12; 6 + 8 = 2 + 12 = 4 +10=14.

To give Theorems or Rules for the solution of the various cases, the terms are represented by symbols, or letters. Thus, let a denote the less extreme, or least term,

the greater extreme, or greatest term,
the common difference,

the number of terms, and

the sum of all the terms.

Any three being given, the others may be found. NOTE. The twenty cases in this Rule may be resolved by the following Theorems.

Moreover, when the least term a=nothing, the Theorems become, z=d(n-1), and s={nz.

Case 1. The two extremes, and the number of terms being given, to find the sum.

RULE. Multiply the sum of the extremes by the number of terms, and half the product will be the answer.*

*

(1) How many strokes does the hammer of a clock strike in 12 hours?t

(2) A man bought 17 yards of cloth, and gave for the first yard 2s. and for the last 10s. What was the price of the 17 yards? Ans. £5..2.

(3) If 100 eggs be placed in a right line, exactly a yard from each other, and the first a yard from a basket, how far must a person travel to gather them all up singly, and return with every egg to put it into the basket?

Ans. 5 miles, 1300 yards.

Case 2. The same three terms given, to find the common

difference.

RULE. Divide the difference of the extremes by the number of terms less 1; and the quotient will be the answer.

(4) A man had eight sons, whose ages were in arithmetical progression; the youngest being 4 years old, and the eldest 32. What was the common difference of their ages?

(5) A man travelling from London to a certain place, went 3 miles the first day, and increased every day by an equal excess, making the twelfth day's journey 58 miles.

The learner should find each of these cases among the preceding Theorems. Thus, the present Rule will be found designated by s = } n (a + z), &c.

+32

+12+1 x 6 13 X 6 = 78. Anɛ.

48 -1=28÷ 7 = 4 years. Ans.