THE PRINCIPAL THEOREMS IN BOOK VI.

Triangles, and also parallelograms, of the same altitude are to one another as their bases.

If a straight line be drawn parallel to one of the sides of a triangle it will cut the other two sides proportionally.

If any angle of a triangle be bisected by a straight line which cuts the base or opposite side, the segments of the base will have the same ratio which the other two sides have to each other.

The sides about the equal angles of equiangular triangles are proportional.

If the sides of two triangles about each of their angles be proportional, the triangles will be equiangular.

If two triangles have one angle of one triangle equal to one angle of the other, and the sides about the equal angles proportional, the triangles will be equiangular.

Equal parallelograms, and also equal triangles, which have one angle of one equal to one angle of the other, have their sides about the equal angles reciprocally proportional; and parallelograms, and also triangles, which have one angle of one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.

In a right angled triangle if a perpendicular be drawn from the right angle to the hypothenuse, it will divide the triangle into two triangles which are similar to the whole triangle, and also to each other.

In a right angled triangle if a perpendicular be drawn from the right angle to the hypothenuse, it will

be a mean proportional between the segments of the hypothenuse; and each of the sides about the right angle will be a mean proportional between the hypothenuse and the segment adjacent to that side.

If a perpendicular be drawn from any point in the circumference of a sircle to the diameter, it will be a mean proportional between the segments of the dia

meter.

Similar triangles, and all similar figures, are to one another as the squares of their corresponding sides.

Equiangular parallelograms, and also equiangular triangles, are to one another as the the rectangles under the sides about the equal angles.

If three straight lines be proportional, the first is to the third as any rectilineal figure described on the first is to a similar figure similarly described on the second.

In a right angled triangle if similar rectilineal figures be similarly described on the three sides, the figure on the hypothenuse will be equal to both the figures on the other two sides.

In equal circles angles either at the centres or at the circumferences have the same ratio to one another as the arches on which they stand have to one another.

An angle at the centre of a circle is to four right angles as the arch on which it stands is to the circumference of the circle.

In unequal circles arches which subtend equal angles at the centres are to one another as the circumferences of the circles.

The rectangle under the diagonals of a quadrilateral figure inscribed in a circle is equal to both the rectangles under its two opposite sides.

The homologous sides, and also the perimeters of similar polygons inscribed in circles, are to one an. other as the diameters of the circles.

Equilateral polygons of the same number of sides,

U

inscribed in circles, are similar, and are to one another as the squares of the diameters of the circles.

The diameter of a circle is to the circumference as the square of the radius is to the surface of the circle.

The surface of any circle is equal to the rectangle contained by the radius and a straight line equal to half the circumference; or, it is equal to a triangle whose base is equal to the circumference and altitude equal to the radius of the circle.

Of all plane figures having equal perimeters the circle is the most capacious.

The circumferences of circles are to one another as the diameters; and the surfaces of circles are to one another as the squares of the diameters, or circumferences.

END OF BOOK VI.

ELEMENTS OF GEOMETRY.

BOOK XI.

OF THE INTERSECTION OF PLANES.

DEFINITIONS.-See Notes.

1. A straight line is perpendicular to a plane, when it is perpendicular to every straight line meeting it in that plane.

2. A plane is perpendicular to a plane, when all the straight lines drawn in one of the planes perpendicular to the common section of the two planes are perpendicular to the other plane.

3. The inclination of a straight line to a plane is the acute angle contained by that line and another straight line drawn from the point in which the first line meets the plane to the point in which a perpendicular drawn from any point in the first line to the plane meets the plane.

4. The inclination of two planes is the angle contained by two straight lines drawn from any point in the line of their common section perpendicular to that line, one line in one plane, and the other line in the other plane.

5. Two planes are said to have the same, or a like inclination to each other, which two other planes have, when the angles of inclination above defined are equal to each other.

6. A straight line is said to be parallel to a plane, when it has no inclination to the plane, or is equidistant from it.

7. Planes are said to be parallel to one another which are equidistant, or do not meet, though produced ever so far.

8. A solid angle is an angle made by the meeting of more than two plane angles in one point, which are not in the same plane.

PROPOSITION I. THEOREM.

One part of a straight line cannot be in a plane, and another part above it.

For if one part of a line were in a plane and another part above it, all its parts would not lie in the same direction, and therefore it could not be a straight line (3 Def. 1). Therefore, one part &c. Q. E. D.

PROPOSITION II. THEOREM.

ED.

Any three straight lines which meet one another, but not in the same point, are in one plane.

Let the three straight lines AB, CD, CB meet one another in the points B, C, E; they are in one plane.

E

Let any plane pass through the line EB, and let the plane revolve round EB as an axis until it pass through the point C; then, because the points E, C are in this plane, the line EC is in it (Def. 8. 1). For the same reason BC is in the same plane; and, by the hypothesis, EB is in it. Therefore the three lines EC, CR, BE are in one plane. But the whole lines DC, AB, BC, produced, are in the same plane with the parts of them EC, EB, BC (1.11). Therefore AB, CD, CB are all in one plane. Wherefore, any three &c. Q. E. D.

B

COR. It is manifest that any two straight lines which eut each other are in one plane; and that any three points whatever are in one plane.

PROPOSITION III. THEOREM.

If two planes cut each other, their common section is a straight line.