circumferences so that the rectangle contained by the two chords

may be equal to a given square.

31. To draw a line parallel to a given line, which shall be termi-

nated by two others given in position, so as to form with them

a triangle equal to a given rectilineal figure.

32. To bisect a triangle by a line drawn parallel to one of its

sides..

33. To divide a given triangle into any number of parts, having

a given ratio to each other, by lines drawn parallel to one of the

sides of the triangle.

34. To divide a given triangle into any number of equal parts,

by lines drawn parallel to a given line.

35. To divide a trapezium, which has two sides parallel, into

any number of equal parts, by lines drawn parallel to those sides.

36. From one of the angular points of a given square, to draw

a line meeting one of the opposite sides and the other produced, in

such a manner, that the exterior triangle formed thereby may have

a given ratio to the square.

37. From a given point in the side produced of a given rectangu-

lar parallelogram, to draw a line which shall cut the perpendicular

sides and the other side produced, so that the trapezium cut off,

which stands on the aforesaid side, may be to the triangle which

stands upon the produced part of the opposite side, in a given ratio.

38. Through a given point between two straight lines containing

a given angle, to draw a line which shall cut off a triangle equal to

a given figure.

39. Between two lines given in position, to draw a line equal to

a given line, so that the triangle thus formed may be equal to

a given rectilineal figure.

40. From two given lines to cut off two others, so that the re-

mainder of one may have to the part cut off from the other, a given

ratio; and the difference of the squares of the other remainder and

part cut off from the first may be equal to a given square.

41. From two given lines to cut off two others which shall have

a given ratio, so that the difference of the squares of the remainders

may be equal to a given square.

42. From two given lines to cut off two others, so that the

remainders may have a given ratio, and the sum of the squares of

the parts cut off may be equal to the square of a given line.

43. Two points being given in a given straight line; to determine

a third, such that the rectangles contained by its distances from each

extremity and the given point adjacent to that extremity may be

equal.

44. Through the point of intersection of two given circles, to

draw a line in such a manner, that the sum of the respective

rectangles contained by the parts thereof, which are intercepted

between the said point and their circumferences, and given lines

A and B, may be equal to a given square.

45. Through a given point, to draw an indefinite line such, that

if lines be drawn from two other given points and forming given

angles with it, the rectangle contained by the segments intercepted

between the given point and the two lines so drawn, shall be equal to

the square of a given line.

46. Through a given point between two straight lines containing

a given angle, to draw a line such that a perpendicular upon it from

the given angle may have a given ratio to a line drawn from one

extremity of it, parallel to a line given in position.

47. Through a given point between two indefinite straight lines,

not parallel to one another, to draw a line, which shall be terminated

by them, so that the rectangle contained by its segments shall be less

than the rectangle contained by the segments of any other line drawn

through the same point.

line.

SECTION VI. Page 184.

1. To describe an isosceles triangle on a given finite straight

2. To describe a square which shall be equal to the difference

of two squares, whose sides are given.

COR. Hence a mean proportional between the sum and difference

of two given lines may be determined.

3. To describe a rectangular parallelogram, which shall be equal

to a given square, and have its adjacent sides together equal to

a given line.

4. To describe a rectangular parallelogram, which shall be equal

to a given square, and have the difference of its adjacent sides equal

to a given line.

5. To describe a triangle, which shall be equal to a given equi-

lateral and equiangular pentagon, and of the same altitude.

6. To describe an equilateral triangle, equal to a given isosceles

triangle.

7. To describe a parallelogram, the area and perimeter of which

shall be respectively equal to the area and perimeter of a given

triangle.

8. To describe a parallelogram, which shall be of given altitude,

and equiangular and equal to a given parallelogram.

9. To describe a square which shall be equal to the sum of any

number of given squares.

10. Having given the difference between the diameter and side

of a square; to describe the square.

11. To divide a circle into any number of concentric equal

annuli.

COR. To divide it into annuli which shall have a given ratio.

12. In any quadrilateral figure circumscribing a circle, the oppo-

site sides are equal to half the perimeter.

13. If the opposite angles of a quadrilateral figure be equal to

two right angles, a circle may be described about it.

14. A quadrilateral figure may have a circle described about it,

if the rectangles contained by the segments of the diagonals be equal.

15. If from any point within a regular figure circumscribed about

a circle, perpendiculars be drawn to the sides; they will together be

equal to that multiple of the semi-diameter which is expressed by the

number of the sides of the figure.

16. If the radius of a circle be cut in extreme and mean ratio;

the greater segment will be equal to the side of an equilateral and

equiangular decagon inscribed in that circle.

17. Any segment of a circle being described on the base of

a triangle; to describe on the other sides segments similar to that on

the base.

18. If an equilateral triangle be inscribed in a circle; the square

described on a side thereof is equal to three times the square

described upon the radius.

19. To inscribe a square in a given right-angled isosceles triangle.

20. To inscribe a square in a given quadrant of a circle.

21. To inscribe a square in a given semicircle.

22. To inscribe a square in a given segment of a circle.

23. Having given the distance of the centres of two equal circles

which cut each other; to inscribe a square in the space included

between the two circumferences.

24. In a given segment of a circle to inscribe a rectangular

parallelogram, whose sides shall have a given ratio.

25. In a given circle to inscribe a rectangular parallelogram

equal to a given rectilineal figure.

26. In a given segment of a circle to inscribe an isosceles

triangle, such that its vertex may be in the middle of the chord, and

the base and perpendicular together equal to a given line.

27. In a given triangle, to inscribe a parallelogram similar to

a given parallelogram.

28. In a given triangle, to inscribe a triangle similar to a given

triangle.

29. In a given equilateral and equiangular pentagon, to inscribe

a square.

30. In a given triangle, to inscribe a rhombus, one of whose

angles shall be in a given point in the side of the triangle.

31. To inscribe a circle in a given quadrant.

32. To describe a circle, the circumference of which shall pass

through a given point, and touch a given straight line in a given

point.

33. To describe a circle, which shall pass through a given point,

have a given radius, and touch a given straight line.

e

34. To describe a circle, which shall pass through two given

points, and touch a given straight line.

35. To describe a circle, the circumference of which shall pass

through a given point, and touch a circle in a given point; the two

points not being in a tangent to the given circle.

36. To describe a circle, the centre of which may be in the per-

pendicular of a given right-angled triangle, and the circumference

pass through the right angle and touch the hypothenuse.

37. To describe a circle, which shall pass through the extremities

of a given line, so that if from any point in its circumference a line

be drawn making a given angle with the given line; the rectangle

contained by the segment it cuts off and the given line, may be

equal to the square of the line drawn from the same point to the

farther extremity of the given line.

38. To determine a point in the perpendicular let fall from the

vertical angle of any triangle on the base; about which as a centre

a circle may be described touching the longer side, and passing

through the opposite angular point.

39. To describe a circle which shall have a given radius, its

centre in a given straight line, and shall also touch another given

straight line inclined at a given angle to the former.

40. To describe a circle which shall touch a straight line in

a given point, and also touch a given circle.

41. To describe two circles, each having a given radius, which

shall touch each other, and the same given straight line on the same

side of it.

42. To describe a circle passing through two given points, and

touching a given circle.

43. To describe a circle which shall pass through a given point,

and touch a given circle and a given straight line.

44. To describe a circle which shall touch a straight line and

two circles given in magnitude and position.

45. To describe a circle which shall touch two given straight

lines, and pass through a given point between them.

46. To describe a circle which shall touch two given straight

lines, and also touch a given circle.