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In words.-From half the fum fubtract half the difference, the remainder will be the less.

To half the sum add half the difference, the sum will be the greater.

Ex. 2. The powers or forces of three different agents being given, to find a general theorem for determining the time in which they would, all three together, produce a given effect.

Three day-labourers, A, B, and C, have undertaken a piece of work, which A could perform in a days, B in b days, and C in odays. In what time will they perform it, if all the three work together?

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The rule obtained may be translated thus. Divide the product of the three given times by the sum of the products of each two taken separately.

3. Required a theorem for determining two numbers, whose sum (s) and sum of their squares (9) are given.

Anf

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Trans. From half the sum of their squares subtract the square of their sum, and to the square root or the remainder add or subtract the half sum for the numbers required.

In like manner any other theorem may be tranflated.

4. Required a theorem for determining two numbers x and y, whose product, p, and proportion a to b, are given.

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5. Required a theorem for determining two numbers, x and y', whose sum, e, and proportion a to b, are given.

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6. Required a theorem for determining two numbers, x and y, whose sum, s, and product, P, are given.

Ans. aste

7. Required a theorem for determining two numbers, x andy, whose proportion, a to b, and the sum of their squares, cubes, &c. are given.

Let

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8. Required a theorem for determining the time and place in which two bodies, moving towards each other, will meet, their velocity, their distance, and the difference of the time of their first motion, being given. Put a = the velocity of the one;

b = the velocity of the other,
d = their distance, and
t = the time the one moves before the other.

dt
Anf. = the time of their meeting.

a+b And the product of the velocity of either body, multiplied by the time of its motion, will give the space passed over, and confequently the place of meeting.

GEOMETRICAL PROBLEMS.

When a geometrical problem or question is proposed you are first to construct a figure representing the true one; prepare the figure (if necessary) by drawing more lines as you see cause, according to the method of solution you have chosen, that so, by the help of these lines and mediums, you may deduce a connection between known and unknown quantities or lines. Then proceed to the operation as before directed, which, with a competent knowledge of Euclid's Elements, will be your guide : but the exact manner of proceeding can scarlely be re

duced

duced to general rules, but must be collected by a careful examination of each step, assisted by practice.

PROBLEM I.

Given the hypothenuse of a right-angled triangle, and the sum

of the sides, to find each of the sides.

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Given the hypothenuse and the difference of the legs, to find

the legs.

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PROBLEM III. Fig. 1.

Given the hypothenuse and the product of the two legs, to

find each of the legs.

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Given the hypothenuse, and the proportion of the two legs, to

find each of them.

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Given one of the legs, and the sum of the hypothenuse and the

other leg, to find the hypothenuse and that leg.

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