part and lesser seg. together with the of lesser seg, . two rectangles under intermediate and lesser seg. with the2 of lesser are together to a rect. under the unequal segments, if .. the square of intermediate part be added to both, two rectangles under intermediate and lesser segs. with the rs of inter. and lesser (viz. * of half the given line) are to a rectangle under the unequal parts with of intermediate.

Or thus in numbers:

Suppose the given line to be 12 feet, let it be divided equally into 6 and 6, and unequally into 8 and 4 feet, the intermediate part will be 2 feet.

Then 8 X 4+22 = 36, is to 62 =

36.

Cor. 1. Hence it is evident that if a right line be bisected, the rectangle under the parts is greater than if it be cut unequally, and .. the sum of the squares of the parts is less.

2

For then the rect. is the ☐ of half the line, and is .. greater than if the line was cut unequally.

NOTE Since the lines are cut equally the rectangles are a maximum, but they with the rs of the parts are = to the 2 of the sum, (prop. 4, 2.) and if the line was cut unequally the rect. would be less than those, but they with thers of the unequal parts are also to the * of the sum, .. the sum of the *rs of the parts is less than that of the unequal parts. In numbers 6 and 6 = parts 8, and 4 unequal; then 6+62 72 and 82+42 = 80.

Cor. 2. If two equal right lines be so divided that the rectangle under the segments of one be equal to the rectangle under the segments of the other, their segments shall be equal.

If one of the lines be bisected the other also is bisected, for the rectangle under the parts is then equal to the of the half line, whence it is evident that the segments themselves are equal.

2

If they are not bisected let them be cut unequally; bisect them; then since the lines are their halves are

and.. the 2rs of their halves, but the rectangles under the unequal parts are also, take them from the O'rs of the halves and the remainders, viz. thers of the intermediate parts are,.. the intermediate parts themselves are,.. the sums of them and the halves are, and also the differences between them and the balves, i. e. the segments, are =.

Cor. 3. The rectangle under the sum and difference of two right lines is to the differences between their squares.

Because the rectangle together with the 2 of the less is to the of the greater, as is evident from the preceding proposition, for the half line is the greater, the intermediate part the less, and the lesser of the unequal segments the difference.

2

of

2 of

This might be proved, (by prop. 3, 2.) since the the less the rect. under it and dif. is to the greater the rect. under it and dif... the difference between those 'rs is to the rect. under the greater and dif. the rect. under the lesser and dif.

Schol. 1. Hence it is also evident, that in a right angled triangle, the rectangle under the sum and difference of the hypothenuse and one side, is to the square of the other side.

Schol. 2. Hence it is also evident, that if from any angle of a triangle, a perpendicular be drawn to the op posite side, the rectangle under the sum and difference of the sides is to the rectangle under the sum and difference of the segments of the side to which the dicular is drawn.

perpen

For the rectangle under the sum and difference of the sides is to the difference between the 'rs of the sides, and the rect. under the sum and difference of the segments of the base is also to the difference between the squares of the segments, and these differences are =, (cor. 4, prop. 47, 1,) ... &c. &c.

=

Cor. 4. The difference between the squares of two sides of any triangle is to double the rectangle under the remaining side and its distance from the middle point of the perpendicular which is drawn to it from the opposite side.

For the difference between the ers of the sides is = to the difference between the rs of the segments of the side upon which the perpendicular falls, (cor. 4, prop. 47, 1,) and.. when the perpendicular falls within the triangle to the rectangle under the sum of the segments and the difference between them, which difference is double the seg. between the middle point and perpendicular, .. to double the rectangle under the sum and distance from middle point to perpendicular.

But if the perpendicular falls outside the triangle, the segments are the sum of the side on which perpendicular falls and intercept between pcrpendicular and side and said intercept,.. the difference is the side opposite the angle from which the perpendicular is let fall.

Schol. Hence given in numbers the side of any triangle we can find its area; divide the difference between the squares of two sides of the triangle by the remaining side, to half this side add half the quotient and subtract the square of the sum from the square of the greater side, the remainder is the square of the perpendicular, thence the perpendicular itself is found, which multiplied into half the side on which it falls, gives the area of the triangle.

NOTE 1-In the foregoing prop. the intereept is the segment between the points of equal and unequal section and is to the difference between half the line and either of the unequal parts and is.. half the difference between the unequal parts.

2. It is evident from this proposition that the excess of the of half the sum of two lines above half their difference is to a rectangle under them, for half the given line is half the sum of the unequal parts and the intermediate part is half their difference.

Hence and from prop. 4, we might infer that the ' of the difference of any two lines is to the excess of the sum of their sqares above the double rectangle under them.

For the sum of the rs is to the of the difference, two rectangles under the lesser and dif. with two 'rs of the lesser, but two rectangles under less aud dif. with two rs of lesser is to two rectangles under the given lines, ... &c. &c.

NOTE 3. COR. 2. the greater will be will be the rectangle will be o, and the of the sum of the the least is twice the

The more unequally a line is cut the intermediate part, and.. the less under the parts, the limits of which of half the line; the greatest limit rs is the of the whole line, and of half.

2

2

2

NOTE 4. By being given the rectangle under any two lines and their difference, we can find the lines.

For if the square of half the difference be added to the given rect. the product will be to the 2 of half the sum, (for half the difference is to the intermediate part,) then add to a side of this product half the difference and you will have the greater line, and subtract the half dif. from said product and you will have the lesser line. In numbers, suppose the rect. to be 40 square feet and the dif. to be six feet. Then the of 39 +40 = 49, the square root of which is 7, which is half the sum of the given lines, then 7+3 10 the greater line and 7-3 4 the lesser side.

2

5. Given the difference between any two lines and the difference of their squares to find the lines.

to the

Describe on the difference of the lines a rect. dif. of the squares, its other side shall be to the sum of the required lines; for the difference of the 'rs is to a rectangle under their sum and difference.

Thus in numbers, suppose the dif. of the lines is 8 feet and the dif. of the rs 128 square feet; then 8 being one side of the rect. of 128 square feet, 16 must be the other side, the half of 16+48 12 the greater line, and ₫ 16 84 the lesser line.

The difference of the 2rs of any two lines is greater than the 2 of their difference, by two rectangles under the less and difference; and is greatest when the less is half the difference or one-third of the greater (the sum being suppoed given.)

of one

6. To cut a given 'right line so that the part shall be to half the of the given line. Describe on it an isosceles right angled triangle, and cut from the given line a part to a side of this triangle: the of this part is evidently to half the given line.

of the

7. To cut a right line so that the rectangle under the segments shall be equal to the square of the intermediatepart.

2

Bisect the given right line, and then cut its half (according to the foregoing note) so that the part whose is to half the 2 of the half line be the intermediate part. 2 For the of the intermediate part being to half the of the line, is to two rectangles under the immediate and lesser segments with a of lesser; but those are together to a rectangle under the unequal segments, .'. &c.

8.

2

To cut a line so that the rectangle under the parts shall be any multiple of the square of the intermediate part.

The rectangle with the 2 of the intercept is = to the 2 of half the line; then suppose we want to cut it so that the rectangle may be three times the of the intercept; let the rectangle be r, the 2 of the intercept s, and the 2 of half the line x.

Then r+s x · · · 3 s + 8 = x

= 8

i. e. The 2 of the intercept must be the fourth part of the 2 of half the given line.

9. The sum of the squares of any two lines exceeds the difference of the squares by two squares of the lesser..

2

For the difference of the rs is to a rect. under sum and dif. or to a 2 of the dif. and two rect. under lesser and difference, but those together with two 2rs of the lesser are to the sum of the rs, ... &c. &c.

10, The sum of the squares exceeds the square of the difference by two rectangles under the given lines.

For the sum of thers is to a of the difference, two'rs of the lesser and two rect. under lesser and dif. but two rs of the lesser with two rectangles under lesser and dif. are to two rect. under the lines.