Book VI.

1

PROP. G. THE OR.

F from the extremity of the diameter of a circle a ftraight line be drawn in the circle, and if either within the circle or produced without it, it meet a line perpendicular to the fame diameter, the rectangle contained by the ftraight line drawn in the circle, and the fegment of it intercepted between the extremity of the diameter and the perpendicular, is equal to the rectangle contained by the diameter and the fegment of it cut off by the perpendicular.

Let ABC be a circle, of which AC is a diameter, let DE be perpendicular to the diameter AC, and let AB meet DE in F the rectangle BA, AF is equal to the rectangle CA, AD. Join BC, and because ABC is an angle in a femicircle,

a 3. 3. it is a right angle a: Now, the angle ADF is also a right b Hyp. angle b; and the angle BAC is either the fame with DAF,

or vertical to it; therefore the triangles ABC, ADF are c4. 6. equiangular, and BA: AC::AD: AFC; therefore alfo the

rectangle BA, AF, contained by the extremes, is equal to 16. 6. the rectangle AC, AD contained by the means d, If therefore, &c. Q. E. D.

PROP.

THE

PROP. H. THEO R.

HE perpendiculars drawn from the three angles of any triangle to the oppofite fides interfect one another in the fame point.

A

Let ABC be a triangle, BD and CE two perpendiculars interfecting one another in F; let AF be joined, and, produced if neceffary, let it meet BC in G, AG is perpendicular to BC. Join DE, and about the triangle AEF let a circle be described, AEF; then, because AEF is a right angle, the circle defcribed about the triangle AEF will have AF for its diametera. In the fame manner, the circle described about the triangle ADF has AF for its diameter; therefore the points A, E, F and D are in the circumference of the fame circle. But because the angle EFB is equal to the angle DFC b, and also the angle BEF to the angle

E

Book VL

a 31. 3.

b 15. 1.

CDF, being both right B

angles, the triangles BEF

and CDF are equiangular, and therefore BF: EF:: CF:FD,c c 4. 6. or alternately d, BF: FC:: EF: FD. Since, then, the fides d 16. 5. about the equal angles BFC, EFD are proportionals, the triangles BFC, EFD are alfo equiangulare; wherefore the angle e 6. 6. FCB is equal to the angle EDF. But EDF is equal to

EAF, because they are angles in the fame fegment f; there- f 21. 3. fore the angle EAF is equal to the angle FCG: Now, the angles AEF, CFG are alfo equal, because they are vertical angles; therefore the remaining angles AEF, FGC are also equals: But AEF is a right angle, therefore FGC is a right g 32. 1, angle, and AG is perpendicular to BC. QE. D.

ELE

A

ftraight line is perpendicular, or at right angles to a Book VII. plane, when it makes right angles with every ftraight line meeting it in that plane.

II.

A plane is perpendicular to a plane, when the ftraight lines drawn in one of the planes perpendicularly to the common fection of the two planes, are perpendicular to the other plane.

III.

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

point

Book VII. point in which a perpendicular to the plane drawn from any point of the first line above the plane, meets the fame plane.

IV.

The inclination of a plane to a plane is the acute angle contained by two straight lines drawn from any the fame point of their common fection at right angles to it, one upon one plane, and the other upon the other plane.

Two planes are said to have the fame, or a like inclination to one another, which two other planes have, when the faid angles of inclination are equal to one another.

VI.

Parallel planes are fuch as do not meet one another though produced.

A Solid is that which has length, breadth, and thickness.

VIII. 1

See N. A folid angle is that which is made by the meeting of more than two plane angles, which are not in the fame plane, in one point.

IX.

See N. Similar folid figures are fuch as are contained by the fame number of fimilar planes fimilarly fituated, and having like inclinations to one another.

X..

A pyramid is a folid figure contained by planes that are con stituted betwixt one plane and a point above it in which they meet.

ΧΙ