Let = the money he had at first in shillings,

7. A company of foot march 1165 of their own paces ahead of a troop of horse; now. if the foot take 5 paces to every 4 of the horse, but 3 paces of a horse be equal in extent to 4 paces of the foot, how many paces will the horse have marched before they overtake the foot?

Let x be the number of steps the horse must take,

= number of footsteps the infantry march from the time

they start till overtaken.

= number of footsteps the infantry march from the time

8. A waterman finds by experience that he can, with the advantage of a common tide, row down a river from A to B, which is 18 miles, in an hour and a half; and that to return from B to A against an equal tide, though he rows back along the shore, where the stream is only three-fifths as strong as in the middle, takes him just two hours and a quarter. It is required from hence to find at what rate per hour the tide runs in the middle, where it is strongest.

Let the rate per hour the tide runs in the middle,

18

at the side.

And = 12 miles the rate per hour he can row in the 11/1/20

middle, with the advantage of the tide;

.. 12 x = the rate per hour he can row without the tide.

18

Again, = 8 miles = the rate per hour he can row

21

x = 24 miles the rate per nour

the tide runs in the middle, where it is strongest.

1. Ten years ago a boy's age was of his father's, but now it is of it; what are both their ages? Ans. 15 and 60.

2 If from three times a certain number we subtract 8, half the remainder will be equal to the number itself diminished by 2; what is the number? Ans. 4.

3 If to the numerator of a certain fraction we add one, its value will be, but if from its denominator we subtract one, its value will be ; what is the fraction? Ans.

4. £132 is to be divided between A, B, and C, so that B will receive as much as A, and C 3 as much as A and B together; find the share of each.

of

Ans. A £40, B £32, C £60. 5. A, having a certain number of sovereigns in his purse, and meeting two of his creditors, B and C, gave to B the money, and to C of the remainder; he then found that he had left exactly eleven pounds; what money had he at first? Ans. £110.

3

6. A grazier spent of his money in horses, in oxen, and 10 of the remainder in sheep; after which he found he had £98 left; how much had he at first? Ans. £240.

The mth root of a" is "a", or am.

Hence, to find any root of a quantity, we divide the index by the number indicating the root required;

Hence the square root of a quantity is either positive, or negative.

Hence the cube root of a positive quantity is positive, but the cube root of a negative quantity is negative.

Since there is no quantity, which, being multiplied by itself, will produce a, . ✔a is an impossible quantity.

a2+2ab+b2 being the square of a + b, we may, by means of this formula, investigate a rule for extracting the square root of a binomial; for a2+2ab + b2 = a2 + (2a + b)b, and the first term of the root is a, the square root of a2, we have only to find a divisor which will give the quotient + b, and this divisor is evidently 2a +b; that is, twice the first term of the root together with the second term of the root.

Hence, to find the square root of a2 + 2ab+ b2, we first take a, the square root of a2, and, placing it as we should a quotient, we subtract its square from a2 + Qab + b2; we then double the root a for a divisor, and divide the remaining portion of the quantity, namely (2a + b) b, by 2a, and annex the quotient b to the a, and also to the divisor; we then multiply 2a + b by this second term b and subtract the result.

The process may be thus exhibited,

a2 + 2ab+b2 (a + b

Let us apply this rule to the extraction of the square root of the number 1225.

1225 (305, or 35

900*

60+5, or 65) 325

325

Let it be required to extract the square root of a2 +2ab+ 2ac2bc + b2 + c2.

a2+2ab+2ac + 2bc + b2 + c2 (a + b + c

In this example, having obtained a + b, this portion of the root is doubled for a new divisor, in order to obtain + c, the remaining portion of the root.

Since the cube root of a3 + 3a2b+3ab2 + b3 is a + b, we may readily derive a rule for the extraction of the cube root. The cube root of the first term a3, is a, the first term of the root; we subtract its cube from the whole quantity; then, for a divisor we take 3 a2, and dividing the first term of the remainder by it, we obtain b, the second term of the root; then annexing 3ab + b2 to the divisor, and multiplying it by b, we obtain 3a2b+3ab2 + b3, which is equal to the remainder.

* In the ordinary arithmetical process these two ciphers would be omitted.