2. A grocer having four sorts of tea worth 5s, 6s, 8s and 9s per lb. wishes a mixture of 871b. worth 7s per lb.: how much must be taken of each sort?

291b.

Ans. {144lb. at 8s; and 2916. at 9s. at 5s; 14176. at 6s;

3. A vintner has four sorts of wine, viz., white wine at 4s per gallon, Flemish at 6s per gallon, Malaga at 8s per gallon, and Canary at 10s per gallon: he would make a mixture of 60 gallons to be worth 5s per gallon: what quantity must be taken of each?

Ans. S 45gal. of white wine; 5gal. of Flemish;

5gal. of Malaga; and 5gal. of Canary.

4. A silver-smith has four sorts of gold, viz.; of 24 carats fine, of 22 carats fine, of 20 carats fine, and of 15 carats fine he would make a mixture of 42oz. of 17 carats fine: how much must be taken of each sort?

Q. How do you find the proportional parts when the price only is given? What is the rule when a given quantity of one of the simples is to be taken? What is the rule when the quantity of the compound, as well as the price, is given?

INVOLUTION.

§ 177. If a number be multiplied by itself, the product is called the second power, or square of that number. Thus 4x4 16: the number 16 is the 2nd power or square of 4.

If a number be multiplied by itself, and the product arising be again multiplied by the number, the second product is called the 3rd power, or cube of the number. Thus 3x3x3=27: the number 27 is the 3rd power, or cube of 3.

The term power designates the product arising from multiplying a number by itself a certain number of times, and the number multiplied is called the root.

Thus, in the first example above, 4 is the root, and 16 the square or 2nd power of 4.

In the 2nd example, 3 is the root, and 27 the 3rd power or cube of 3. The first power of a number is the number itself.

Q. If a number be multiplied by itself once, what is the product called? If it be multiplied by itself twice, what is the product called? What does the term power mean? What is the root?

§ 178. Involution teaches the method of finding 'the powers of numbers.

The number which designates the power to which the root is to be raised, is called the index or exponent of the power. It is generally written on the right, and a little above the root. Thus 42 expresses the second power of 4, or that 4 is to be multiplied by itself once: hence, 42=4x4=16.

For the same reason 33 denotes that 3 is to be raised to the 3rd power, or cubed: hence

33=3×3×3=27: we may therefore write,

4=4

42=4x4=16

4=4X4X4=64

the 1st power of 4.

the 2nd power of 4.

the 3rd power of 4.

4=4x4x4x4-256 the 4th power of 4. 45=4x4x4x4x4=1024 the 5th power of 4.

Q. What is Involution? What is the number called which designates the power? Where is it written?

Hence, to raise a number to any power, we have the following

RULE.

Multiply the number continually by itself as many times less 1 as there are units in the exponent: the last product will be the power sought.

9. What is the cube of ? 10. What is the square of,01? 11. What is the square of 2,04? 12. What is the 5th power of 10? 13. What is the cube of 21?

Q. How do you raise a number to any power?

EVOLUTION.

§ 179. We have seen (§ 178,) that Involution teaches how to find the power when the root is given. Evolution is the reverse of Involution: it teaches how to find the root when the power is known. The root is that number which being multiplied by itself a certain number of times will produce the given power.

The square root of a number is that number which being multiplied by itself once will produce the given

number.

The cube root of a number is that number which being multiplied by itself twice will produce the given number. For example, 6 is the square root of 36; because 6×8 =36; and 3 is the cube root of 27, because 3×3×3= 27. The sign✔ placed before a number denotes that its square root is to be extracted. Thus, √36-6. The sign is called the sign of the square root.

When we wish to express that the cube root is to be extracted, we place the figure 3 over the sign of the square root: thus, 8-2 and 27-3.

Q. What is Evolution? What does it teach? What is the square root of a number? What is the cube root of a number? Make the sign denoting the square root? How do you denote the cube root?

EXTRACTION OF THE SQUARE ROOT.

§ 180. To extract the square root of a number, is to find a number which being multiplied by itself once, will produce the given number. Thus

√4=2; for 2x2=4;

Also ✓93; for 3x3=9.

Before proceeding to explain the rule for extracting the square root, let us first see how the squares of numbers are formed.

The first ten numbers are

1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Roots. 1 4 9 16 25 36 49 64 81 100 Squares. the numbers in the second line are the squares of those in the first: and the numbers in the first line are the square roots of the corresponding numbers of the second.

Now, it is evident that, the square of a number expressed by a single figure will not contain any figure of a higher order than tens. And also, that if a number contains three figures its root must contain tens and units.

The numbers 1, 4, 9, &c. of the second line, are called perfect squares, because they have exact roots.

Let us now see how the square of any number may be formed: say the number 36. This number is made up of 3 tens or 30, and 6 units.

Let the line AB represent the 3 tens or 30, and BC the six units.

Let AD be a square on H AC, and AE a square on the tens line AB.

Then ED will be a square on the unit line 6, and the rectangle EF will be the product of HE which is equal to the tens line, by IE which is equal to the unit line.

F

Also, A

the rectangle BK will be the product of EB which is equal to the tens line, by the unit line BC. But the whole square on AC is made up of the square AE, the two rectangles FE and EC, and the square ED: Hence

The square of two figures is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units.

Let it now be required to extract the square root of 1296.

Since the number contains more than two places, its root will contain tens and units. But as the square of one ten is one hundred, it follows that the ten's place quired root must be found in the figures on the Hence, we point off the number into periods of two figures each.

We next find the greatest square con

tained in 12, which is 3 tens or 30.

of the releft of 96. 12 96(36

9

We

66)396

396

then square 3 tens which gives 9 hundred,

and then place 9 under the hundred's place, and subtract.

This takes away the

square AE and leaves the two rectangles FE and BK, together with the square ED on the unit line.

Now, since tens multiplied by units will give at least tens in the product, it follows that the area of the two rectangles FE and EC must be expressed by the figures at the left of the unit's place A

6, which figures may also express a part of the square ED. If, then, we divide the figures 39, at the left of 6, by twice the tens, that is, by twice AB or BE, the quotient will be BC or EK, the unit place of the root.

Then, placing BC or 6, in the root, and also in the divisor, and then multiplying the whole divisor 66 by 6, we obtain for a product the two rectangles, FE and EC together with the square ED.