423

Anf. 2s+ 19452

Tranf. From half the fum of their fquares fubtract the fquare of their fum, and to the fquare root or the remainder add or fubtract the half fum 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.

5. Required a theorem for determining two numbers, x and y, whofe fum, e, and-proportion a to b, are given.

6. Required a theorem for determining two numbers, x and y, whofe fum, s, and product, p, are given.

7. Required a theorem for determining two numbers, and y, whofe proportion, a to b, and the fum of their fquares, cubes, &c. are given.

Let

8. Required a theorem for determining the time and place in which two bodies, moving towards each other, will meet, their velocity, their diftance, 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.

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

GEOMETRICAL PROBLEMS.

When a geometrical problem or queftion is proposed you are first to construct a figure representing the true one; prepare the figure (if necessary) by drawing more lines as you fee cause, according to the method of folution you have chosen, that fo, by the help of thefe 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 fearlely be re

duced

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

PROBLEM I.

Given the hypothenuse of a right-angled triangle, and the sum of the fides, to find each of the fides.

Given the hypothenufe and the difference of the legs, to find the legs.

PROBLEM III. Fig. 1.

Given the hypothenufe and the product of the two legs, to find each of the legs.

Given the hypothenufe, and the proportion of the two legs, to find each of them.

Given one of the legs, and the fum of the hypothenuse and the other leg, to find the hypothenufe and that leg.

PROBLEM VI. Fig. 2.

To divide any given line, AB, into two fuch parts that the rectangle, contained by the whole line and one of the parts, fhall be equal to the fquare of the other part.

Given the difference between the diagonal of a fquare and one of its fides, to find the diagonal and the side.

Let d be the difference and x the fide.

Anf. x=d+2d2

PROBLEM VIII. Fig. 1.

Given the perimeter, and the area of a right-angled triangle, to find the hypothenufe.