11. The square of the sum exceeds the difference of the squares by two rectangles under the sum and lesser.

12. The squares of the sum exceeds the square of the difference, by four retangles under the given lines.

13. The difference of the squares exceeds the square of the difference, by two rectangles under the lesser and difference.

14. The square of the greater of two lines exceeds the rectangle under the greater and less, by a rectangle under greater and difference. Hence it is evident that the rectangle under the lines exceeds the square of the less by the rectangle under the lesser and difference.

15. If a right line be cut twice unequally, the greater rectangle exceeds the less by a square of the greater intermediate part minus a square of the lesser intermediate, or by two rectangles under lesser inter. and dif. of inter. together with a square of said dif,

16. The rect. under the segments of the hypothenuse (of a right angled triangle) made by a perpendicular Is to a square of the perpendicular.

For the sum of the 'rs of the sides is to the 'rs of the segments with two rs of the perpendicular, and also to the 'rs of the segments with two rectangles under the segments,.. those rect. must be the two 2rs of the perpendicular, .. &c. &c.

2

to

Hence the of one segment + the rectangle under the segments, is to the 2 of the conterminous side.

PROP. 6, THEOR.

If a right line be divided into any two parts, the squares of the whole line and either segment are together equal to double the rectangle under the whole line and that segment, together with the square of the other segment.

Describe on the line made up of the half and part pre. duced a 2, from the common extremity of the given line and produced part draw a line par. to a side of the

2, from the other extremity of the produced part draw a diagonal and through the point in which it cuts the parallel line draw a line and parallel to the whole produced line and connect their other extremities.

Then because the rectangles under the halves and produced parts are and also the complemental rectangles, the gnomon formed by taking away the 2 of half the given line is to the rectangle under the whole produced line and produced part; therefore if you add to both the 2 of half the given line, the rectangle under the whole produced line and produced part together with the 2 of half the given line, is to the of the line made up of the half and produced part.

Otherwise thus:

The rectangle under the whole produced line and produced part is to the sum of two rectangles under half the given line (which may be considered an intermediate part) and produced part, together with the 2 of the prcduced part, if.. you add the 2 of half (or intermediate) to both the rectangle under the whole produced line and produced part together with the 2 of half the given line, are to two rectangles under the half and produced part together with the 2rs of half and produced part, but these latter together make up the 2 of the line made up of half and produced part, •.• &c. &c.

In numbers, suppose the given line 8 and the produced part 4 feet, then the whole line is 12 feet: then 12 x 4 4264 is to 82 (the line made up of 'and produced part.)

Cor. If a right line be drawn from the vertex of an isosceles triangle to the base, the rectangle under the segments of the base, is to the difference between the square of this line and the square of either side.

Bisect the base and connect the point of bisection with the vertical angle.

2

to the

of

Then the rect, under the former segments is difference between the of half the base and the the intercept between the point of bisection and other point of section, (prop. 5 and 6, b. 2,) but the dif. between thers of half and intercept is to the dif. between the ers of the first drawn line and a side of triangle, ... &c.

But if the line drawn to the base be perpendicular to it, it bisects the base, and the rectangle under the segments of the base is the of the half, and .. together with the of the line drawn to base is to the square of the

2

side.

PROP. 7, THEOR.

If a right line be divided into any two parts, the squares of the whole line and either segment are equal to double the rectangle under the whole line and that segment, together with the square of the other segment.

Describe on the given line a ', draw a diagonal, through the point of section draw a right line parallel to one side, and through the point in which it meets the diagonal draw a right line par. to given line.

2

Then the of given line is to a rect. under the whole line and one segment, a complemental rect. and a of the other segment; add to both the of the other segment: then the 'rs of given line and one seg. are together to a rect. under the whole line and one seg. a complemental rect. and the 'rs of the segments; but the complemental rect. and added is to a rect. under the woole line and that segment, .. the 'rs of the whole line, &c. &c.

2

Otherwise thus:

The of the whole line is to the 'rs of the segments and two rectangles under the segments; add to both the of one segment; then the rs of the whole line and one segment are together to two rectangles under the segments, a square of one seg. and two'rs of the other segment, but two rectangles under the segments with two rs of the segments are to two rectangles under the whole line and that segment,.• &c. &c.

Thus in numbers: suppose the given line to be 12 feet, and that it is divided into 8 and 4 feet, then 12' 4' is = to twice 12 X 4 +82.

For 12144+4 160, and 12 X 4 = 48 + 12 X496 +82 = 160.

PROP. 8, THEOR,

If a right line be divided into any two parts, the square of the sum of the whole line and either segment, is equal to four times the rectangle under the whole line and that segment together with the square of the other segment.

Produce the given line 'till the produced part is to one segment, on the whole compound line describe a *, through the point of section, and common extremity of given line and produced part, draw lines parallel to a side of the 2, through the other extremity of produced part draw a diagonal, and through the points in which it meets the drawn lines draw two other lines par. to the given line.

Then the of the whole produced line is

to the

'rs

of one seg. of given line, four 'rs of the other and four rectangles under the segments; but four of one seg. with four rectangles under the segments, are together to four rectangles under the given line and that seg. (of which the rs are).. &c. &c.

Otherwise thus:

Produce the given line 'till the produced part is to the adjacent segment: then the 2 of the whole produced line is to a 2 of the given line, a 2 of the produced part (which is to a 2 of one seg.) and two rectangles under the given line and produced part; and the of the given line is to the 'rs of the segments and two rectangles under the segments; but one of thosers, the of produced part, and the rectangles under the segments are together to two. rectangles under the given line and produced part. .. the of the whole produced line is to four rectangles under the given line and one segment, together the of the other segment.

Thus in numbers; suppose that the given line is 8 feet and the segments 6 and 2 feet, then the produced part will be 2 feet, the whole produced line will be 10 feet. Then 102 100 is to 8 x 2+8x2+8x2 + 8 x 2 + 6 100. =

PROP. 9, THEOR.

If a right line be cut into equal and also into unequal parts, the sum of the squares of the unequal parts is equal to double the sum of the squares of the half and of the intermediate part.

The of the greater segment is to a 2 of half a 2 of intermediate part and two rectangles under half and intermediate, add to both the 2 of the lesser segment; then the rs of the unequal segments are together = to the rs of half of intermediate and of lesser seg. with two rectangles under half and intermediate, but two rectangles under half and intermediate with the 2 of lesser segment are to a 2 of half and a 2 of intermediate, (prop. 7, 2,) ... the sum of the 21s of the unequal parts are to two grs of half and two of intermediate part.

=

PROP. 10, THEOR.

2rs

a right line be bisected and produced to any point, the square of the whole line thus produced together with the square of the produced part, is equal to double the square of the line made up of the half and produced part together with double the square of half the given line.

er

The 2 of the whole produced line is to the of half the given line and of the line made up of the half and produced part, together with twice the rectangle under the half and line made up of half and produced part. But double the rectangle under the half and line made up of half and produced part, together the 2 of the produced part is to the 2 of the line made up of half and produced part together with the