Then arch B D + GH = 2E F, if A is within the circle; or G H = 2 E F, if A is without.

arch B D

18. If from a point without, two lines touch a circle: the angle made by them is equal to the angle at the centre, standing on half the difference, of these two parts of the circumference.

19. The angle A = ▲ BH D + HDG, when A is within; or A B HD HD G, when a is without the circle.

H

G

20. In a circle, the angle made at the point of contact between the tangent and any chord, is equal to the angle in the alternate segment; E CF E B C, and E CA

=EG C.

B

B

H

D

A

G

21. A tangent to the middle point of an arch, is parallel to the chord of it.

22. If from any point в in a semicircle, a perpendicular в D be let fall upon the diameter, it will be a mean proportional between the segments of the diameter;

ADD BD BD C.

A

B

D

B

C

23. The chord is a mean proportional between the adjoining segment and the diameter, from the similarity of the triangles: that is, A D A B A B AC; and C D C B :: CB: CA. 24. In a circle if the diameter A D be drawn, and from the ends of the cords a B, a c, perpendiculars be drawn upon the diameter; the squares of the chords will be as the segments of the diameter; AE AF :: A B3: A c2.

A

25. If two circles touch one another in P, and the line P DE be drawn through their centres; and any line P A B is drawn through that point to cut the circles, that line will P be divided in proportion to the diameters

PA: PB: PD: PE

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. FDA F. F B also GF.

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.

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

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

F

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

P

D

D P

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

D

d

31. In a circle E D F whose centre is c, and radius c E, if the points в 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 D o drawn from D, then A B, A C, A o, will be continually proportional.

E

PBC

F

P

D

BA

1

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: AC; and A B.AC=B E. E C+A Eo.

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

A

D

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.

A

E

F

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.

E

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+PC+ P D2= diam."

47. If the diameter P Q be divided into two parts at any point R, and if R s be drawn perpendicular to PQ; also R T applied equal to the radius, and TR produced to the circumference at v then, between the two segments PR,

R Q,

R T is the arithmetical mean,
Rs is the geometrical mean,

R v is the harmonical mean.

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

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. P A X PCX PE &C. be="x" when P is within the circle, or when P is without the circle; and the product of the rest of the lines, viz. P B X PD X PF, &c. a where r = A o the radius, and x = o P the distance of P from the centre.

= xn

=

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 QR 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 R V, RT, RS; So shall the parts, 17, 3 5, 5 3,7 1, 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 perpendi cular 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.