points of intersection to the extremities of the diameter, cutting each

other, may have a given ratio.

22. From the circumference of a given circle, to draw to a straight

line given in position, a line which shall be equal and parallel to a

given straight line.

23. The bases of two given circular segments being in the same

straight line; to determine a point in it such, that a line being drawn

through it making a given angle, the part intercepted between the

circumferences of the circles may be equal to a given line.

24. If two chords of a given circle intersect each other, the

angle of their inclination is equal to half the angle at the centre

which stands on an arc equal to the sum or difference of the arcs

intercepted between them, according as they meet within or without

the circle.

25. If from a point without two circles which do not meet each

other, two lines be drawn to their centres, which have the same

ratio that their radii have; the angle contained by tangents drawn

from that point towards the same parts will be equal to the angle

contained by lines drawn to the centres.

26. To determine the Arithmetic, Geometric and Harmonic

means between two given straight lines.

27. If on each side of any point in a circle any number of equal

arcs be taken, and the extremities of each pair joined: the sum of

the chords so drawn will be equal to the last chord produced to

meet a line drawn from the given point through the extremity of the

first arc.

28. If the circumference of a semicircle be divided into an odd

number of equal parts, and through the points which are equally

distant from the diameter lines be drawn; the segments of these

lines intercepted between radii drawn to the extremities of the most

remote, will together be equal to a radius of the circle.

29. If from the extremities and the point of bisection of
any arc

of a circle, lines be drawn to any point in the opposite circumference;

the sum of those drawn from the extremities will have to that from

the point of bisection, the same ratio that the line joining the extre-

mities has to that joining one of them and the point of bisection.

30. If two equal circles cut each other, and from either point

of intersection a circle be described cutting them; the points where

this circle cuts them, and the other point of intersection of the equal

circles are in the same straight line.

31. If two equal circles cut each other, and from either point of

intersection a line be drawn meeting the circumferences; the part of

it intercepted between the circumferences will be bisected by the

circle whose diameter is the common chord of the equal circles.

32. If two circles touch each other externally or internally; any

straight line drawn through the point of contact will cut off similar

segments.

33. If two circles touch each other externally or internally; two

straight lines drawn through the point of contact will intercept arcs,

the chords of which are parallel.

34. If two circles touch each other externally or internally; any

two straight lines drawn through the point of contact, and terminated

both ways by the circumference, will be cut proportionally by the

circumference.

35. If two circles touch each other externally, and parallel

diameters be drawn; the straight line joining the extremities of these

diameters will pass through the point of contact.

36. If two circles touch each other and also touch a straight

line; the part of the line between the points of contact is a mean

proportional between the diameters of the circles.

37. If two circles touch each other externally, and the line join-

ing their centres be produced to the circumferences; and from its

middle point as a centre with any radius whatever a circle be de-

scribed, and any line placed in it passing through the point of contact;

the parts of the line intercepted between the circumference of this

circle and each of the others will be equal.

38. If from the point of contact of two circles which touch each

other internally, any number of lines be drawn; and through the

points, where these intersect the circumferences, lines be drawn from

any other point in each circumference, and produced to meet; the

angles formed by these lines will be equal.

39. If two circles touch each other internally, and any two per-

pendiculars to their common diameter be produced to cut the cir-

cumferences; the lines joining the points of intersection and the

point of contact are proportional.

40. If three circles, whose diameters are in continued proportion,

touch each other internally, and from the extremity of the least

diameter passing through the point of contact a perpendicular be

drawn, meeting the circumferences of the other two circles; this

diameter and the lines joining the points of intersection and contact

are in continued proportion.

41. If a common tangent be drawn to any number of circles

which touch each other internally, and from any point in this tangent

as a centre, a circle be described cutting the others, and from this

centre lines be drawn through the intersections of the circles respec-

tively; the segments of them within each circle will be equal.

42. If from any point in the diameter of a circle produced, a

tangent be drawn; a perpendicular from the point of contact to the

diameter will divide it into segments which have the same ratio that

the distances of the point without the circle from each extremity of

the diameter, have to each other.

43. If from the extremity of the diameter of a given semicircle

a straight line be drawn in it, equal to the radius, and from the centre

a perpendicular let fall upon it and produced to the circumference;

it will be a mean proportional between the lines drawn from the point

of intersection with the circumference to the extremities of the

diameter.

44. If from the extremity of the diameter of a circle, two lines

be drawn, one of which cuts a perpendicular to the diameter, and

the other is drawn to the point where the perpendicular meets the

circumference; the latter of these lines is a mean proportional between

the cutting line, and that part of it which is intercepted between the

perpendicular and the extremity of the diameter.

45. In the diameter of a circle produced, to determine a point,

from which a tangent drawn to the circumference, shall be equal to

the diameter.

46. To determine a point in the perpendicular at the extremity

of the diameter of a semicircle, from which if a line be drawn to the

b

other extremity of the diameter, the part without the circle may be

equal to a given straight line.

47. Through a given point without a given circle, to draw a

straight line to cut the circle, so that the two perpendiculars drawu

from the points of intersection to that diameter which passes through

the given point, may together be equal to a given line, not greater

than the diameter of the circle.

48. If from each extremity of any number of equal adjacent

arcs in the circumference of a circle, lines be drawn through two

given points in the opposite circumference, and produced till they

meet; the angles formed by these lines will be equal.

49. To determine a point in the circumference of a circle, from

which lines drawn to two other given points, shall have a given ratio.

50. If any point be taken in the diameter of a circle, which is

not the centre; of all the chords which can be drawn through that

point, that is the least which is at right angles to the diameter.

51. If from any point without a circle lines be drawn touching

it; the angle contained by the tangents is double the angle contained

by the line joining the points of contact and the diameter drawn

through one of them.

52. If from the extremities of the diameter of a circle tangents

be drawn, and produced to intersect a tangent to any point of the

circumference; the straight lines joining the points of intersection

and the centre of the circle form a right angle.

53. If from the extremities of the diameter of a circle tangents

be drawn; any other tangent to the circle, terminated by them, is

so divided at the point of contact, that the radius of the circle is a

mean proportional between its segments.

54. Two circles being given in magnitude and position; to find

a point in the circumference of one of them, to which if a tangent be

drawn cutting the circumference of the other, the part of it inter-

cepted between the two circumferences may be equal to a given line.

55. To draw a straight line cutting two concentric circles, so

that the part of it which is intercepted by the circumference of the

greater may be double the part intercepted by the circumference of

the less.

36. If two circles intersect each other, the centre of the one

being in the circumference of the other, and any line be drawn from

that centre; the parts of it which are cut off by the common chord

and the two circumferences will be in continued proportion.

57. If a semicircle be described on the side of a quadrant, and

from any point in the quadrantal arc a radius be drawn, the part of

this radius intercepted between the quadrant and semicircle, is equal

to the perpendicular let fall from the same point on their common

tangent.

COR. Any chord of the semicircle drawn from the centre of the

quadrant is equal to the perpendicular drawn to the other side,

from the point in which the chord produced meets the quadrantal arc.

58. If a semicircle be described on the side of a quadrant, and

a line be drawn from the centre of the quadrant to a common tan-

gent; this line, the parts of it cut off by the circumferences of the

quadrant, and of the semicircle, and the segment of the diameter of

the semicircle made by a perpendicular from the point where the

line meets its circumference, are in continued proportion.

59. If the chord of a quadrant be made the diameter of a semi-

circle, and from its extremities two straight lines be drawn to any

point in the circumference of the semicircle; the segment of the

greater line intercepted between the two circumferences shall be

equal to the less of the two lines.

60. If two circles cut each other, so that the circumference of

one passes through the centre of the other, and from either point of

intersection a straight line be drawn cutting both circumferences; the

part intercepted between the two circumferences will be equal to the

chord drawn from the other point of intersection to the point where

it meets the inner circumference.

61. If from each extremity of the diameter of a circle lines be

drawn to any two points in the circumference; the sums of the lines

so drawn to each point will have to one another the same ratio that

the lines have, which join those points and the opposite extremity

of a diameter perpendicular to the former.

62. If from any two points in the circumference of a circle there

be drawn two straight lines to a point in a tangent to that circle;

they will make the greatest angle when drawn to the point of contact.