The product is formed in precisely the same manner in the usual mode of multiplication, as may be seen, if the products are written down as they are formed, without carrying.

27

27

49

140

140

400

729

Here we observe, 7 times 7 is 49, 7 times 20 is 140, 20 times 7 is 140, and lastly 20 times 20 is 400. These added together make 729, which is the second power of 27.

We observe,

1st. When the root or first power consists of two figures, the second power consists of the second power of the tens, plus the product of twice the tens by the units, plus the second power of the units.

2d. The second power of 9, the largest number consisting of one figure, is 81; and the second power of 10, the smallest number consisting of two places, is 100; and the second power of 100, the smallest number consisting of three places, is 10000. Hence, when the root consists of one figure, the second power cannot exceed two figures; and when the root consists o two figures, the second power consists of not less than three figures, nor more than four figures.

From these remarks it appears, that we must first endeavour to find the second power of the tens, and that it will be found among the hundreds and thousands.

Let it be required to find the root of 729. This number contains hundreds, therefore the root will contain tens. The second power of the tens is contained in the 700. 20 x 20 is 400, and 30 x 30 is 900. 400 is the greatest second power of tens contained in 700. The root of 400 is 20. Subtract 400 from 729, and the remainder is 329. This must contain 2 a b + b2, that is, the product of twice the tens by the units, plus the second power of the units. If it contained exactly the product 2 a b of twice the tens by the units, the units of the root would be found by dividing 329 by twice 20, or 40; for 2 ab divided by 2 a gives b. As it is, if we divide by twice 20 or 40, we shall obtain a quotient either exact, or too large by 1 or 2. 40 is contained in 329, 8 times. Write 8 in the root and raise the whole to the second power. 28 x 28 784, which is larger than 729. Next try 7 in the place of 8. 27 × 27 = 729. Therefore 7 is right, and 27 is the

root required.

The operation may stand as follows.

729 (20 + 7 = 27 root

400

329 (40 divisor

27 × 27729.

What is the root of 1849 ?

18,49 (40 + 3 = 43 root.

16,00

249 (80 divisor.

43 × 43 1849.

In this example, the second power of the tens will be found in the 1800. 30 x 30 900; 40 40 1600;

X

50 x 50 2500. The greatest second power in 1800 is 1600, the root of which is 40. Write 40 in the place of a quotient. Subtract 1600 from 1829. The remainder is 249, which divided by twice 40, or 80, gives 3. Add 3 to the root, and raise the whole to the second power. 43 × 43 = 1849. Therefore 43 is the

root required.

It is evident that the result will not be affected, if instead of writing 40 in the root at first, we omit the zero, and then subtract the second power of 4, viz. 16 from the 18, omitting the two zeros which come under the other period. Then to fo m the divisor, the 4 may be doubled, and the divisor will be 8 instead of 80, and the dividend must be 24, the right hand figure being rejected.

6. What is the root of 8281?

The second power of a+b+c, or (a+b+c)(a+b+c)is a2 + 2 a b + b2 + 2 a c + 2 b c + C$ a2 + 2 a b + b2 + 2 (a + b) c + c2.

To find the second power of 726

726

726

4356

1452

5082

527076

The first three terms of the formula, viz.

a2 + 2 a b + b2,

are the second power of a + b or of the hundreds and tens, viz. 720. The second power of 720 can have no significant figure below hundreds, and the significant figures of the second power of 720 and of 72 are the same; the former is 518400, the latter 5184. If from the whole number 527076 the two right hand figures be rejected, the number is 5270. This contains the second power of 72 and something more, viz. a part of the product 2 (700 + 20) × 6 = 2 (a + b) c.

The method of procedure then, is to find the largest root contained in 5720. The first three terms of the above formula, viz. a2+2ab+b2, show, that this is to be found by the method given above for finding a root consisting of two figures.