26. If through any point F in the diameter of a circle, any chord c F D be drawn, the rectangle of the segments of the chord is equal to the rectangle of the segments of the diameter; c F. F D = A F. F B = also G F .

FE.

27. If through any point F. out of the circle in the diameter в A produced, any line F C D be drawn through the circle: the rectangle of the whole line and the external part is equal to the rectangle of the whole line passing through the centre, and the external part;

DF. F C AF. F B.

B

G

28. Let H F be a tangent at н; then the rectangle C F. FD= square of the tangent F H.

29. If from the same point F, two tangents be drawn to the circle, they will be equal; F H = F I.

F

C

E

K

B

B

F

E

30. If a line P F c be drawn perpendicular to the diameter A D of a circle, and any line drawn from A to cut the circle and the perpendicular; then the rectangle of the distances of A the sections from a, will be equal to the rectangle of the diameter and

A

P D

D P

the distance of the perpendicular from A; ABXAC = APX AD. Also, A B X A C = A K2.

D

d

31. In a circle E D F whose centre is c, and radius c E, if the points B A, be so placed in the diameter produced, that c B, C E, CA, be in continual proportion, then two lines B D, A D drawn from these points to any point in the circumference of the circle will always be in the given ratio of B E, to A E.

A

32. In a circle, if a perpendicular D B be let fall from any point D, upon the diameter c 1, and the tangent Do drawn from D, then A B, A C, A o, will be continually proportional.

33. If a triangle B DF be inscribed in a circle, and a perpendicular D P let fall from D on the opposite side B F, and the diameter DA drawn; then, as the perpendicular is to one side including the angle D, so is the other side to the diameter of the circle;

DP: DB:: D F D A.

34. The rectangle of the sides of an inscribed triangle is equal to the rectangle of the diameter, and the perpendicular on the third side; B D.D FA D. D P.

35. If a triangle B A C be inscribed in a circle, and the angle A bisected by the right line A E D, then as one side to the segment of the bisecting line, within the triangle, so the whole bisecting line to the other side; A B; A E:: AD: A C; and A B.A C B E. E C+A E3.

36. If a quadrilateral A B C D be inscribed in a circle, the sum of two opposite angles is equal to two right angles; ADC+ABC two right angles.

B

E B

D

E

C

A

37. If a quadrangle be inscribed in a circle, the rectangle of the diagonals is equal to the sum of the rectangles of the oppo

site sides.

38. A circle is equal to a triangle whose base is the circumference of the circle; and height, its radius.

39. The area of a circle is equal to the rectangle of half the circumference and half the diameter.

40. Circles (that is, their areas) are to one another as the squares of their diameters, or as the squares of the radii, or as the squares of the circumferences.

41. Similar polygons inscribed in circles, are to one another as the circles wherein they are inscribed.

42. A circle is to any circumscribed rectilineal figure, as the

circle's periphery to the periphery of the figure.

43. If an equilateral triangle A B C be inscribed in a circle; the square of the side thereof is equal to three times the square of the radius: A B2-3 A D3.

17

A

E

44. A square inscribed in a circle, is equal to twice the square of the radius.

45. The side of a regular hexagon inscribed in a circle, is equal to the radius of the circle;

BE B C.

B

46. If two chords in a circle mutually intersect at right angles, the sum of the squares of the segments of the chords is equal to the square of the diameter of the circle. A p2 + PB2 + P C2 + P D2= diam.o

47. If the diameter P Q be divided into two
parts at any point R, and if R S be drawn perpen-
dicular to PQ; also R T applied equal to the
radius, and т в produced to the circumference P
at v then, between the two segments P R,
R Q,-

RT is the arithmetical mean,
RS is the geometrical mean,"
RV is the harmonical mean.

48. If the arcs P Q, Q R, RS, &c. be equal, and there be drawn the chords P Q, PR, P S, &c. then it will be PQ: PR:: PR PQ PS: PS: PR+PT:: PT: PS + P V, &c.

B

A

P

R

49. The centre of a circle being o, and Pa point in the radius, or in the radius produced; if the circumference be divided into as many equal parts A B, B C, C D, &c. as there are units in 2 n, and lines be drawn from P to all the points of division; then shall the continual product of all the alternate lines, viz. PAX PCX PE &C. be"-" when P is within the

circle, or
= xn - when P is
without the circle; and the product
of the rest of the lines, viz. PBX PD
X PF, &c. = p2 + xn : where r =
A o the radius, and x = 0 P. the dis-
tance of P from the centre.

50. A circle may thus be divided into any number of parts that shall be equal to one another both in area and perimeter. Divide the diameter Q R into the same number of equal parts at the points s, T, v, &c. ; then on one side of the diameter describe semicircles on the diameters Q s, QT, Q v, and on the other side of it describe semicircles on RV, RT, RS; so shall the parts, 17, 3 5, 5 3, 71, be all equal, both in area and peri

meter.

SECTION V.-Of Planes and Solids.

Definitions.

1. The common section of two planes, is the line in which they meet, or cut each other.

2. A line is perpendicular to a plane, when it is perpendicular to every line in that plane which meets it.

3. One plane is perpendicular to another, when every line of the one, which is perpendicular to the line of their common section, is perpendicular to the other.

4. The inclination of one plane to another, or the angle they form between them, is the angle contained by two lines, drawn from any point in the common section, and at right angles to the same, one of these lines in each plane.

5. Parallel planes are such as being produced ever so far both ways, will never meet, or which are everywhere at an equal perpendicular distance.

6. A solid angle is that which is made by three or more plane angles, meeting each other in the same point.

7. Similar solids, contained by plane figures, are such as have all their solid angles equal, each to each, and are bounded by the same number of similar planes, alike placed.

8. A prism is a solid whose ends are parallel, equal, and like plane figures, and its sides, connecting those ends, are parallelograms.

9. A prism takes particular names according to the figure of its base or ends, whether triangular, square, rectangular, pentagonal, hexagonal, &c.

10. A right or upright prism, is that which has the planes of the sides perpendicular to the planes of the ends or base.

11. A parallelopiped, or a parallelopipedon, is a prism bounded by six parallelograms, every opposite two of which are equal, alike, and parallel.

12. A rectangular parallelopipedon is that whose bounding planes are all rectangles, which are perpendicular to each

other

13. A cube is a square prism, being bounded by six equal square sides or faces, which are perpendicular to each other.

14. A cylinder is a round prism having circles for its ends; and is conceived to be formed by the rotation of a right line about the circumferences of two equal and parallel circles, always parallel to the axis.

15. The axis of a cylinder is the right line joining the centres of the two parallel circles about which the figure is described.