10. A cask containing 480 bottles of wine, has been filled with wines at 6s. and 4s. a bottle, and the whole cask is worth £108. how many bottles of each does it contain?

11. A wine merchant has wine at 4s., 5s. and 6s. a bottle, which he desires to mix in equal parts with 108 bottles of another wine at 5s. 8d., so that the mixture be worth 5s. 6d., what quantity of each sort of wine is he to mix?

12. A jeweller has gold at 15, 17, 18 and 22 carats fine, with which he wants to make a composition of 40oz. 20 carats fine, how much of each must he take?

13. How many gallons of wine at £75, £90, £1.20, and £60 are required to make 150 gallons, at £1. per gallon?

14. A spirit merchant, who has 500 gallons of spirits, worth 13s. 4d. per gallon, wishes to mix it with three other kinds, worth 12s. 6d., 14s., and 16s. 6d. per gallon, in order to sell the whole at 15s. 4d. per gallon. How much of each must he take?

15. A druggist has two the other 10s. per lb. to make a mixture of per lb. ?

sorts of bark, one worth 5s. 9d., and What portion of each must he take 24 cwt. that will be worth 8s. 6d.

16. It is required to mix British spirits at 9s., French wine at 17s., ginger wine at 3s., and water at 0 per gallon together, so that the mixture may be worth 6s. per gallon. How much of each must be taken ?

17. If I melt 8 lbs. 5 oz. of bullion, of gold 14 carats fine, with 12 lbs. 8 oz., of 18 carats fine, how many carats fine is the mixture?

INVOLUTION AND EVOLUTION OF NUMBERS.

321. A square is a figure, the four sides of which are equal, and the angles right-angled. A square surface, the side

of which is one foot or one inch is called a square foot or a square inch.

If the side were two feet, the surface would be four square feet.

one inch.

If the surface were three feet long and three feet broad the square would, consequently, be nine feet

square.

Then, to find the number of square feet contained in a square surface the dimension of one side must be known, since that dimension is to be multiplied by itself; for, if each of the sides be 8 inches, joining the points of division of the opposite sides in order, we have 64 squares, each

For this reason the name of square numbers has been given to the product of any number multiplied by itself, or to the product of two equal factors. 16 is a square number, because it is the product of 4×4. 25, 36, 49, 64, 81, 100, &c., are square

numbers.

Therefore, to find the square of any number it must be multiplied by itself,

322. The number expressing the dimension of one side of a square, or one of the two equal factors, is called the square root. Thus, 4 is the square root of 16, 7 of 49, &c.

Let it be required to find the square of. For that purpose, suppose a line divided into two equal parts, and a square described upon that line, it will contain 4 small equal squares, and

the square formed upon the half of the line is of the whole square; therefore, the square of is 1, viz., 1×1 =1. It would appear that the square of a fraction is smaller than that fraction; but we must remember that the square root of is a line or length, whilst the square is a surface, which is one-fourth part of the whole square.

The square of is 25. Let a base be divided into 7 equal parts, and a square described upon it, that square contains 49

small squares, but the square described upon of the base contains only 25 of them. Therefore, to square a fraction, multiply the numerator by itself and divide it by the denominator multiplied by itself. The square of is X, or 35. The square of 23, or of }=}×}=4, or 5. The square of 3.4=3.4 × 3.4, or 11.56.

323. The cube is a geometrical solid in the form of a die, viz., having the same dimensions in every way. A solid of this shape, measuring one foot or one inch in every way, is called a cubic foot or a cubic inch.

On a line 2 feet long let a square be described, and with a height equal to the breadth or the length, complete the scheme as in the figure, we obtain a solid composed of 8 cubes, one foot every way. The same may be exemplified by placing 8 small cubes, supposed to be one foot every way, so that four stand on a square base of four square feet, and four on the top of these, thus a solid having 2 feet every way is formed. The same may be done on other square bases, and we arrive at the conclusion that a cube or cubic number is the product of a number multiplied twice by itself, or is the product of three equal factors.

324. The number expressing the length of one side of the cube, or one of the equal factors, is called the cube root, thus: 3 is the cube root of 27, because 3 × 3 × 3=27.

The cube of 4=4×4×4=64.

The cube of 10=10×10 x10=1000.

The cube of 1×1×1. (See preceding page.)
The cube of 3=2×3×3=27.

The cube of 2=××==1839.

The cube of 4.24 4.24 × 4.24 × 4.24 =76.225024.

325. A number which is multiplied once, twice, three times, four times, &c., by itself, is said to be raised or involved to the second, third, fourth, fifth power, &c. The number itself is in the first power. The second power of a number is the same as its square, and the third power the same as its cube.

As an exercise, the pupil may construct a table similar to the following, which can easily be extended

This process is termed Involution, and the reverse process, viz., that of finding the original number, called the root, is termed Evolution.

326. If the same number be repeated or multiplied by itself, it is expressed by placing rather above the number a figure called index or exponent, denoting how often it is repeated. Thus, if 7 were to be repeated three times, or raised to the third power, it is expressed in this manner:-73 and 208 is the eighth power of 20.

327, A sign is also used to express the root of a number, ✔or V, V, V, &c, denote the square root, the third or cube root, the fourth root, &c. The same is also expressed by ,,, &c. placed a little above the number.

Thus 36 or 36+ express the square root of 36.

4

20 or 20 represent the third root of the fourth power

328. We shall consider, in the first place, the squares of the natural series of numbers, from which we shall make some observations which will be found useful for explaining the extraction of the square root.

The

squares of the numbers of one digit

The square of 10 is 100, of 99 is 9801, of 100 is 10000, of 999 is 998001, &c., &c., &c

From this we infer that the figures of a square are twice those of the root, except when the first figure of the root is 1, 2, or 3, then the figures are twice as many, minus one.

Therefore-1st, every square of one or two figures has only one figure in its root; 2nd, every square of three or four figures has only two figures in its root; 3rd, every square of five or six figures has only three figures in its root; 4th, &c.

By this law the square roots of 567 and 4236 have two figures, 53641 and 168478 have three figures. For this reason before