such that A: B :: B: C, B is said to be the mean proportional between A and C, and C the third proportional to A and B.

DEF. 7. If there are two ratios A: B, P: Q, and C be taken such that B: C: P: Q, then A is said to have to C a ratio compounded of the ratios A: B, P: Q. Thus if there are three magnitudes A, B, C, then A has to C the ratio compounded of the ratios A: B, B: C. DEF. 8. A ratio compounded of two equal ratios is called the duplicate of either of these ratios.

GENERAL PROPOSITIONS OF PROPORTION.

(1.) Ratios that are equal to the same ratio are equal to one another.

(2.) Equal magnitudes have the same ratio to the same. or to equal magnitudes.

(3.) Magnitudes that have the same ratio to the same or equal magnitudes are equal.

(4.) The ratio of two magnitudes is equal to that of their halves or doubles.

(5.) If A : B :: P: Q, then B: A :: Q: P.

(invertendo)

(6.) If A: B: C: D, all the four being of the same kind,

THEOR. I. If two straight lines are cut by three parallel straight

lines, the intercepts on the one are to one another in the rame ratio as the corresponding intercepts on the other.

COR. I. If the sides of a triangle are cut by a straight line parallel to the base, the segments of one side are to one another in the same ratio as the segments of the other side.

COR. 2. If two straight lines are cut by four parallel straight lines the intercepts on the one are to one another in

the same ratio as the corresponding intercepts on the other.

THEOR. 2. A given finite straight line can be divided internally into segments having any given ratio, and also externally into segments having any given ratio except the ratio of equality: and in each case there is only one such point of division.

THEOR. 3. A straight line which divides the sides of a triangle proportionally is parallel to the base of the triangle. THEOR. 4. Rectangles of equal altitude are to one another in the same ratio as their bases.

COR. Parallelograms or triangles of the same altitude are to one another as their bases.

THEOR. 5. In the same circle or in equal circles angles at the centre and sectors are to one another as the arcs on

which they stand.

SECTION I.

SIMILAR FIGures.

DEF. 1. Similar rectilineal figures are those which have their

angles equal, and the sides about the equal angles

proportional.

DEF. 2. Similar figures are said to be similarly described upon given straight lines, when those straight lines are homologous sides of the figures.

THEOR. I. Rectilineal figures that are similar to the same rectilineal figures are similar to one another.

THEOR. 2. If two triangles have their angles respectively equal, they are similar, and those sides which are opposite

to the equal angles are homologous.

THEOR. 3. If two triangles have one angle of the one equal to one angle of the other and the sides about these angles proportional, they are similar, and those angles which are opposite to the homologous sides are equal.

THEOR. 4. If two triangles have the sides taken in order about each of their angles proportional, they are similar,

and those angles which are opposite to the homologous sides are equal.

THEOR. 5. If two triangles have one angle of the one equal to one angle of the other, and the sides about one other angle in each proportional, so that the sides opposite the equal angles are homologous, the triangles have their third angles either equal or supplementary, and in the former case the triangles are similar.

COR. Two such triangles are similar

(1.) If the two angles given equal are right angles or obtuse angles.

(2.) If the angles opposite to the other two homologous sides are both acute or both obtuse, or if one of them is a right angle.

(3.) If the side opposite the given angle in each triangle is not less than the other given side.

THEOR. 6. If two similar rectilineal figures are placed so as to have their corresponding sides parallel, all the straight lines joining the angular points of the one to the corresponding angular points of the other are parallel or meet in a point; and the distances from that point along any straight line to the points where it meets corresponding sides of the figures are in the ratio of the corresponding sides of the figures.

COR. Similar rectilineal figures may be divided into the same number of similar triangles.

DEF. 3. The point determined as in Theor. 6 is called a centre of similarity of the two rectilineal figures.

THEOR. 7. In a right-angled triangle if a perpendicular is drawn from the right angle to the hypotenuse it divides the triangle into two other triangles which are similar to the whole and to one another.

COR. Each side of the triangle is a mean proportional between the hypotenuse and the adjacent segment of the hypotenuse; and the perpendicular is a mean proportional between the segments of the hypotenuse. THEOR. 8. If from any angle of a triangle a straight line is drawn perpendicular to the base, the diameter of the circle. circumscribing the triangle is a fourth proportional to the perpendicular and the sides of the triangle which contain that angle.

THEOR. 9. If the interior or exterior vertical angle of a triangle is bisected by a straight line which also cuts the base, the base is divided internally or externally in the ratio of the sides of the triangle. And, conversely, if the base is divided internally or externally in the ratio of the sides of the triangle, the straight line

drawn from the point of division to the vertex bisects

the interior or exterior vertical angle.

SECTION 2.

AREAS.

THEOR. IO. If four straight lines are proportional the rectangle

contained by the extremes is equal to the. rectangle contained by the means; and, conversely, if the rectangle contained by the extremes is equal to the rectangle contained by the means the four straight lines are proportional.

COR. If three straight lines are proportional the rectangle contained by the extremes is equal to the square on the mean; and, conversely, if the rectangle contained by the extremes of three straight lines is equal to the square on the mean the lines are proportional.

THEOR. II. If two chords of a circle intersect either within or without a circle the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.

OBS. This theorem has been proved in Book III, to which reference may be made for the corollaries. THEOR. 12. The rectangle contained by the diagonals of a quadrilateral is less than the sum of the rectangles contained by opposite sides unless a circle can be circumscribed about the quadrilateral, in which case it is equal to that sum.

THEOR. 13. If two triangles or parallelograms have one angle of the one equal to one angle of the other, their areas have to one another the ratio compounded of the