PROB. VI. The sum of any two numbers, and their proJucts being given, to find each number.

RULE. From the square of their sum, subtract 4 times their product, and extract the square root of the remainder, which will be the difference of the two numbers; then half the said difference added to half the sum, gives the greater of the two numbers, and the said half difference subtracted from the half sum, gives the lesser number.

EXAMPLES.

The sum of two numbers is 43, and their product is 442; what are those two numbers?

The sum of the numb. 43 × 43-1849 square of do. The product of do. 442 × 4=1768 4 times the pro.

Then to the sum of 21,5

[numb.

√81-9 diff. of the

EXTRACTION OF THE CUBE ROOT.

A cube is any number multiplied by its square.

To extract the cube root, is to find à number, which, being multiplied into its square, shall produce the given number.

RULE.

1. Separate the given number into periods of three figures each, by putting a point over the unit figure, and every third figure from the place of units to the left, and if there be decimals, to the right.

2. Find the greatest cube in the left hand period, and place its root in the quotient.

3. Subtract the cube thus found, from the said period, and to the remainder bring down the next period, calling this the dividend.

4. Multiply the square of the quotien. by 300, calling the divisor.

5. Seck how often the divisor may be had in the divi dend, and place the result in the quotient; then multiply the divisor by this last quotient figure, placing the product under the dividend.

6, Multiply the former quotient figure, or figures, by the square of the last quotient figure, and that product by 301, and place the product under the last; then under these two products place the cube of the last quotient figure, and add them together, calling their sum the subtrahend.

7. Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend; with which proceed in the same manner, till the whole br finished,

NOTE. If the subtrahend (found by the foregoing rule) happens to be greater than the dividend, and con 'equently cannot be subtracted therefrom, you must make the las quotient figure one less; with which find a new subtrahend (by the rule foregoing,) and so on until you can subtrac the subtrahend from the dividend.

EXAMPLES.

1. Required the cube root of 18399,744.

18399,744(26,4 Root Ans.

8

2×2=4×300=1200)10399 first dividend

7200

6×6=36×2=72×30=2160

6×6×6 216

9576 1st subtrahend.

811200

26x26=676 × 300-202800)823744 2d dividend.

4×4=16×26=416 × 30=
= 12480

4X4×4

64

823744 2d subtranend

NOTE.-The foregoing example gives a perfect root; And if, when all the periods are exhausted, there happens to he a remainder, you may annex periods of ciphers, and cortinue the operation as far as you think it necessary.

RULE.-1. Find by trial, a cube near to the given number, and call it the supposed cube.

2. Then, as twice the supposed cube, added to the given number, is to twice the given number added to the supposed cube, so is the root of the supposed cube, to the true root, or an approximation to it.

3. By taking the cube of the root thus found, for the supposed cube, and repeating the operation, the root will be had to a greater degree of exactness.

EXAMPLES.

1. Let it be required to extract the cube root of 2. Assume 1,3 as the root of the nearest cube; then—1,3× 1,3× 1,3=2,197 supposed cube.

: : 1,3 : 1,2599 roof,

which is true to the last place of deci¬nas; but might by peating the operation be brought to greater exictuless. 2. What is the cube root of 584,§

135796744(51,4 the ro
125 1st subtrahend.

5)107 dividend.

132651 2d subtrahend. 7803) 31457-2d dividend.

135796744-3d subtrahend.

5x5x3=75 first divisor.

51×51 × 51=132651 second subtrahend. 51 × 51x3=7803 second divisor. 514×514×514=135796744 3d subtrahen 1

2. Required the sursolid or 5th root of 6436343.

6436343(23 root.

32

2×2×2×2×5=80)323 dividend.

23×23×23 × 23 × 23=6436343 subtrahend.

NOTE. The roots of most powers may be found by the square and cube roots only; therefore, when any ever power is given, the easiest method will be (especially in t very high power) to extract the square root of it, which re duces it to half the given power, then the square root o that power reduces it to half the same power; and so on till you come to a square or a cube.

For example: suppose a 12th power be given; the square root of that reduces it to a 6th power: and the square roo of a 6th power to a cube.

EXAMPLES.

3. What is the biquadrate, or 4th root of 199871733761

Ans. 376.

4. Extract the square, cubed, or 6th root of 12230590 464. Ans. 48.

5. Extract the square, biquadrate, or 8th root of 72138 95769338336. Ans 96.

ALLIGATION,

18 the method of mixing several simples of different quauties, so that the composition may be of a mean or middle quality: It consists of two kinds, viz. Alligation Medial, and Alligation Alternate.

ALLIGATION MEDIAL,

Is when the quantities aud prices of several things are giyen, to find the mean price of the mixture composed of those materials.

RULE.

As the whole composition: is to the whole value:: so is any part of the composition: to its mean price.

EXAMPLES.

1. A farmer mixed 15 bushels of rye, at 64 cents a bushe, 18 bushels of Indian corn, at 55 cts. a bushel, and 21 bushels of oats, at 28 cts. a bushel; I demand what a bushel of this mixture is worth?

bu. cts. $cts. bu. $cts. bu.

15 at 64-9,60 As 54: 25,38::1

2. If 20 bushels of wheat at 1 dol. 35 cts. per bushel be mixed with 10 bushels of rye at 90 cents per bushel, what will a bushel of this mixture be worth?

Ans. $1,20 cts.

3. A tobacconist mixed 36 lb. of tobacco, at 1s. 6d. per lb. 12 lb. at 2s. a pound, with 12 lb. at Is. 10d. per lb.; what is the price of a pound of this mixture?

Ans. 1s. 8d.

4. A grocer mixed 2 C. of sugar at 56s. per C. and 1 C. at 43s. per C. and 2 C. at 50s. per C. together; I demand the price of 3 cwt. of this mixture? Ans. £7 13s.

5. A wine merchant mixes 15 gallons of wine at 4s. 2d. per gallon, with 24 gallons at 6s. 8d. and 20 gallons 6s. 3d; what is a gallon of this composition worth? Ans. 5s. 10d. 24f grs.