of half the given line, (prop. 7, b. 2.) Therefore the 2 of the whole produced line together with the 2 of the produced part is to double the 2 of half the given line and of the line made up of half and produced part.

Cor. The sum of the squares of two lines is equal to double the square of half their sum and double the square of half their difference. For the line made up of half and produced part is = to half the sum and half the given line to half their difference.

NOTE-The geometrical construction of this and the foregoing proposition being rather prolix and of little or no use, they are not necessary here; but in Dr. Elrington's Euclid they are very elegantly demonstrated. Thus in numbers: suppose the given line is 8 feet and the produced part 2, the compound line will be 10 feet. Then 10+2 = 104 is to 42 x 2 + 62 x 2 = 104.

PROP. 11, PROB.

To divide a given finite right line so that the rectangle under the whole line and one segment, shall be equal to the square of the other segment.

From one extremity of the given line erect a perpendicular and make it to the given line, bisect the perpendicular and connect the point of bisection with the other extremity of the given line, produce the perpendicular below the given line until the part compounded of the half and produced part is to the connecting line, and cut from the given line a part to the produced part, the of this shall be to the rectangle under the given line and the other segment.

Complete the on the given line, and draw through the point of section a line parallel to the perpendicular, draw through the extremity of the produced part to meet this parallel a line par to given line.

Then because the perpendicular is bisected and produced, the rectangle under the whole produced line and produced part together with the 2 of the half line, is

to the of the line made up of the half and produced part, (prop. 6, 2,) or to the of the connecting line, which is to the rs of given line and half per

pendicular, (prop. 47, 1,) take away the of the half line, then the rect. under the whole produced line and produced part is to the off the given line, if .. the rectangle under the perpendicular and produced part be taken away from both, (it being common to them,) the of the produced or cut off segment shall be to a rectangle under given line and the other segment.

Cor. The rectangle under the greater segment and difference between the segments is to the square of the less segment, as is evident by taking the rectangle under the segments from the of the greater segment and the rectangle under given line and lesser.

NOTE 1. Given the greater segment (AB) of a line, cut in extreme and mean ratio, to find the lesser.

Cut the given segment in extreme and mean ratio in C, and its greater segment AC shall be the segment required.

Fig. 1, plate 2.

Produce AB 'till BD is to AC.

Then the rectangle under BA and AC or under AB BD is with the of AC or BD to the 2 of AB, but the rectangle under AB, BD with the of BD is to the rectangle ADB, .. AC is the line required to be found.

2

2. Given the lesser segment of a line cut in extreme and mean ratio to find the greater.

Fig. 2.

Cut the given segment AB in extreme and mean ratio, in C; produce AB 'till BD is to AC.

Then since AB is cut in extreme and mean ratio, the rectangle ABC or the of AC (or of BD) with the rectangle BAC (or ABD) is to the ' of AB, AD .. is cut in extreme and mean ratio and is evidently the line required to be fouud.

3. When a right line is cut in extreme and mean ratio, the square of the whole line is equal to the rect angle under the whole line and greater segment with the square of that segment.

For it is to the rectangles under the whole and greater, and whole and less; but the rectangle under the whole and less is to the of the greater,.. &c. &c,

2

P

4. The square of the whole line is also equal to two squares of the greater with a rectangle under the seg

ments.

2

For one of the greater with a rectangle under the segments is to a rectangle under the whole and greater, ... &e.

5. The square of the whole line is also equal to three rectangles under the segments with two squares of the less.

For it is to two rectangles under the segments with their 'rs; but the of the greater is to a rect→ angle under the segments with a of the less, .. &c. &c.

2

6. The rectangle under the segments is equal to twice the rectangle under the whole line and intermediate part.

2

For the rectangle under the segments is to a2 of the less a rect. under the less and difference, and the of the less is to a rect. under the greater and difference, .. the rect, under the segments is to a rect. under the sum and difference, or to two rectangles under the sum and less.

7. The square of the line compounded of half the greater segment and the lesser segment is to five times the square of half the greater segment.

It is evident that the of half the greater segment is the fourth part of the rect. under the whole line and lesser seg. but the rectangle under the whole line and lesser seg. together with the square of half the greater seg. is

2

to the square of the line made up of the half and lesser seg. (prop. 6, b. 2,).. the of the half is the fifth part of the of this compound line.

8.

The squares of the whole line and lesser segment are together equal to three times the square of the greater or to three times the rectangle under the whole and lesser.

2

For the of the whole line is to a 2 of the lesser and two rectangles under the lines; if .. you add a of the less to both, thers of the sum and less will be to a of the greater and two rectangles under the sum and less or two 'rs of the greater, .'. &c. &c.`

2

2

9. Therefore it is evident that the of the sum of the whole line and lesser segment is to five rectangles under them or to fivers of the greater segment.

PROP. 12. THEOR.

In an obtuse angled triangle, the square of the side subtending the obtuse angle, exceeds the sum of the squares of the sides which contain the obtuse angle, by double the rectangle under either of these sides and the external segment between the obtuse angle and the perpendicular let fall from the opposite angle,

2

The of the side subtending the obtuse angle is = to thers of the perpendicular and line made up of the side (on which per. falls) and external segment; but the

2

2

of this line is to a ☐ of the side, a of the seg. and two rectangles under them; and the of the other side is to the 'rs of the per. and seg... the 2 of the side subtending the obtuse angle is to the sum of the

rs of the other two sides and two rectangles under the side on which the perpendicular falls and external seg. ment.

NOTE-If the triangle be isosceles, the square of the side subtending the obtuse angle is equal to two rect angles under the whole produced line and the side.

=

For it is to the sum of the 'rs of the sides and two rect. under the side and seg. but those are together to two rectangles under the whole produced line and side.

In

PROP. 13. THEOR.

any triangle the square of the side subtending an acute angle, is less than the sum of the squares of the sides containing that angle, by twice the rectangle under either of them and the segment between the acute angle and the perpendicular let fall from the opposite angle.

2

The of the side on which the perpendicular falls with the of the intercept is to two rectangles under them with the of the other segment, if the of the perpendicular be added to both, the 'rs of the side, intercept and perpendicular (or the 'rs of the sides containing the acute angle) are to the double rectangle with the of the other segment and of the

2

2

perpendicular or the of the side opposite to the acute angle,.. this with double the rectangle under the side on which the perpendicular falls, and intercept, is to the sum of the rs of the other two sides.

Schol. 1. If the triangle be right angled the perpendicular coincides with the side subtending the acute angle, and the rectangle under the base and intercept is the of the base, but it is evident that in this case the 2 of the side subtending the acute angle is less than the 'rs of the sides containing it, by twice the of the base.

Schol. 2. Hence given in numbers the sides of any triangle, we can find its area: fór subtract the 'of one side which is not the greater from the sum of the 'rs of the other sides and divide half the remainder, (that is the rectangle under the side on which the perpendicular falls and the intercept,) by either of these sides, and subtract the of the quotient from the of the remaining side, the square root of the remainder multiplied into half the former divisor gives the area of the triangle.

2

Cor. If from any angle of a triangle, a right line be drawn bisecting the opposite side, the square of the sides containing the angle is double the squares of the bisecting line and of the side subtending the angle.

If the bisecting line be perpendicular to the side it is evident from prop. 47, 1.

2

If not, let fall a perpendicular; then the of one of the sides containing the angle, will be greater than the O'rs of the perpendicular and half side by a double rectangle, by which the of the other side is less than the same rs, ... &c.

2

NOTE 1. If the sides BA, AC containing the acute angle be equal, the square of the side BC subtending it is equal to two rectangles under either side A and the intercept DC.

Fig. 3.

For BC is to the sum of the 'rs of BA,ACtwo rectangles under CA and AD, but those two rectangles two rectangles under AC,CD are to two ers of either sides AC or AB, . &c. &c.