root, &c., and the natural number answering to the result will be the root required.

But if it be a compound root, or one that consists both of a root and a power, multiply the logarithm of the given number by the numerator of the index, and divide the product by the denominator, for the logarithm of the root sought.

Observing, in either case, when the index of the logarithm is negative, and cannot be divided without a remainder, to increase it by such a number as will render it exactly divisible; and then carry the units borrowed, as so many tens, to the first figure of the decimal part, and divide the whole accordingly.

EXAMPLES.

1. Find the square root of 27.465, by logarithms.

2. Find the cube root of 35 6415, by logarithms.

3. Find the 5th root of 7'0825, by logarithms.

4. Find the 365th root of 1.045, by logarithms.

5. Find the value of (001234) by logarithms.

Here, the divisor 3 being contained exactly twice in the negative index 6, the index of the quotient, to be put down, will be -2.

6. Find the value of ('024554), by logarithms.

Here 2 not being contained exactly in -5, 1 is added to it, which gives 3 for the quotient; and the 1 that is borrowed being carried to the next figure, makes 11, which, divided by 2, gives 5851834 for the decimal part of the logarithm.

7. Required the square root of 365 5674, by logarithms. Ans. 19 11981. 8. Required the cube root of 2.987635, by logarithms. Ans. 1.440265.

9. Required the 4th root of 967845, by logarithms. Ans. 9918624.

10. Required the 7th root of 098674, by logarithms.

3. Required the '07 power of 00563, by logarithms.

1. A person being asked what o'clock it was, said it is between eight and nine, and the hour and minute hands are exactly together; what was the time?

Ans. Sh. 43 min. 38 sec.

2. A certain number, consisting of two places of figures, is equal to the difference of the squares of its digits, and if 36 be added to it the digits will be inverted; what is the number? Ans. 48.

3. What two numbers are those, whose difference, sum, and product, are to each other as the numbers 2, 3, and 5, respectively? Ans. 2 and 10.

4. A person, in a party at cards, betted three shillings to two upon every deal, and after twenty deals found he had gained five shillings; how many deals did he win?

Ans. 13.

5. A person wishing to inclose a piece of ground with. palisades, found, if he set them a foot asunder, that he should have too few by 150, but if he set them a yard asunder, he should have too many by 70; how many had he? Ans. 180.

6. A cistern will be filled by two cocks, A and B, running together, in twelve hours, and by the cock A alone in twenty hours, in what time will it be filled by the cock B alone? Ans. 30 hrs.

7. A grocer bought a lot of tea at 10s. a lb., and a quantity of coffee at 2s. 6d. a lb., which cost him altogether 31. 5s. but the state of the market having changed, he sold the tea at 8s. a lb. and the coffee at 4s. 6d. a lb., and gained upon the whole 5l., how much of each did he buy? Ans. 40 lbs. of tea, and 90 lbs. of coffee. 8. What number is that, which, being severally added to 3, 19, and 51, shall make the results in geometrical progression? Ans. 13.

9. It is required to find two geometrical mean proportionals between 3 and 24; and four geometrical means between 3 and 96.

Ans. 6 and 12; and 6, 12, 24, and 48. 10. It is required to find six numbers in geometrical progression, such that their sum shall be 315, and the sum of the two extremes 165.

Ans. 5, 10, 20, 40, 80, and 160.

11. It is required to find the length and breadth of a rectangular field, consisting of two acres of ground, that shall have the same perimeter as a square field consisting of four acres. Ans. 43.1868, and 7·4097 poles.

12. After a certain number of men had been employed on a piece of work for 24 days, and had half finished it, 16 men more were set on, by which the remaining half was completed in 16 days: how many men were employed at first; and what was the whole expense, at 1s. 6d. a day per man? Ans. 32 the number of men; and the

whole expense 1157. 4s..

13. It is required to find two numbers, such that if the square of the first be added to the second, the sum shall be 62, and if the square of the second be added to the first, it shall be 176. Ans. 7 and 13.

14. The forewheel of a carriage makes six revolutions more than the hind wheel, in going 120 yards; but if the circumference of each wheel was increased by three feet, it would make only four revolutions more than the hind wheel in the same space; what is the circumference of each wheel? Ans. 12 and 15 feet.

15. A person bought as many sheep as cost him 981. 16s.; one-third of which he sold again at 40s. apiece, one-fourth at 36s., and the rest at 34s. apiece; and found his gain upon the whole to be 10l. 148.; what number of sheep had he?

Ans. 60.

16. A bankrupt owes a twice as much as he owes B, and c as much as he owes A and B together; now, out of 300%., which is to be divided amongst them, what must each receive? Ans. A 100l., в 50l., and c 1502.

17. A sum of money is to be divided equally among a certain number of persons; now if there had been 3 claimants less, each would have had 150l. more, and if there had been 6 more, each would have had 1207. less; required the number of persons, and the sum divided.

Ans. 9 persons, sum 27001.

18. From each of sixteen foreign pieces of gold, of the same denomination, a person filed a fifth of its value, and then offered them all in payment at their nominal currency; but the fraud being detected, and the pieces weighed, they were found to be worth no more than 11l. 4s.; what was the original value of each piece? Ans. 17s. 6d.

19. A composition of tin and copper, containing 100 cubic inches, was found to weigh 505 ounces; how many ounces of each did it contain, supposing the weight of a cubic inch of copper to be 5 ounces, and that of a cubic inch of tin 4 ounces.

Ans. 420 oz. of copper, and 85 oz. of tin.

20. A and B formed a joint stock in trade of 500l., and cleared by the first bargain they made 1607.; out of whicit