9. From a given point between two indefinite straight lines given

in position, to draw a line which shall be terminated by the given

lines, and bisected in the given point.

10. From a given point without two indefinite straight lines

given in position, to draw a line such that the parts intercepted by

the point and the lines may have a given ratio.

11. From a given point to draw a straight line which shall cut

off from lines containing a given angle, segments that shall have a

given ratio.

12. If from a given point any number of straight lines be drawn

to a straight line given in position; to determine the locus of the

points of section, which divide them in a given ratio.

13. A straight line being drawn parallel to one of the lines con-

taining a given angle, and produced to meet the other; through a

given point within the angle to draw a line cutting the other three,

so that the part intercepted between the two parallel lines may have

a given ratio to the part intercepted between the given point and

the other line.

14. Two parallel lines being given in position; to draw a third

such, that if from any point in it lines be drawn at given angles to

the parallel lines, the intercepted parts may have a given ratio.

15. If three straight lines drawn from the same point and in the

same direction be in continued proportion, and from that point also

a line equal to the mean proportional be inclined at any angle; the

lines joining the extremity of this line and of the proportionals will

contain equal angles.

16. To trisect a right angle.

17. To trisect a given finite straight line.

18. To divide a given straight line into any number of equal

parts.

COR. To divide a straight line into any number of parts having

a given ratio.

19. To divide a given finite straight line harmonically.

20. If a given finite straight line be harmonically divided, and

from its extremities and the points of division lines be drawn to meet

in any point, so that those from the extremities of the second pro-

portional may be perpendicular to each other; the line drawn from

the extremity of this proportional will bisect the angle formed by the

lines drawn from the extremities of the other two.

21. If a straight line be drawn through any point in the line

bisecting a given angle, and produced to cut the sides containing

that angle, as also a line drawn from the angle perpendicular to the

bisecting line; it will be harmonically divided.

22. If from a given point there be drawn three straight lines

forming angles less than right angles, and from another given point

without them a line be drawn intersecting the others, so as to be

harmonically divided; then will all lines drawn from that point

meeting the three lines be harmonically divided.

23. If a straight line be divided into two equal and also into

two unequal parts, and be produced, so that the part produced may

have to the whole line so produced, the same ratio that the unequal

segments of the line have to each other; then shall the distances of

the point of unequal section from one extremity of the given line,

from its middle point, from the extremity of the part produced, and

from the other extremity of the given line, be proportionals.

24. Three points being given; to determine another, through

which if any straight line be drawn, perpendiculars upon it from two

of the former shall together be equal to the perpendicular from the

third.

25. From a given point in one of two straight lines given in

position, to draw a line to cut the other, so that if from the point

of intersection a perpendicular be let fall upon the former, the seg-

ment intercepted between it and the given point together with the

first drawn line may be equal to a given line.

26. One of the lines which contain a given angle is also given.

To determine a point in it such that if from thence to the indefinite

line there be drawn a line having a given ratio to that segment of it

which is adjacent to the given angle; the line so drawn and the

other segment of the given line may together be equal to another

given line.

27. Two straight lines and a point in each are given in position;

to determine the position of another point in each, so that the straight

line joining these latter points may be equal to a given line, and their

respective distances from the former points in a given ratio.

28. If a straight line be divided into any two parts and pro-

duced, so that the segments may have the same ratio that the whole

line produced has to the part produced, and from the extremities of

the given line perpendiculars be erected; then any line drawn through

the point of section, meeting these perpendiculars, will be divided at

that point into parts which have the same ratio, that those lines have

which are drawn from the extremity of the produced line to the

points of intersection with the perpendicular.

29. From two given points to draw two straight lines which

shall contain a given angle, and meet two lines given in position, so

that the parts intercepted between those points and the lines may

have a given ratio.

30. The length of one of two lines which contain a given angle

being given; to draw, from a given point without them, a straight

line which shall cut the given line produced, so that the part pro-

duced may be in a given ratio to the part cut off from the indefinite

line.

31. From two given straight lines to cut off two parts which

may have a given ratio; so that the ratio of the remaining parts may

also be equal to the ratio of two other given lines.

32. Three lines being given in position; to determine a point in

one of them, from which if two lines be drawn at given angles to the

other two, the two lines so drawn may together be equal to a given

line.

33. If from a given point two straight lines be drawn including

a given angle and having a given ratio, and one of them be always

terminated by a straight line given in position; to determine the

locus of the extremity of the other.

34. If from two given points straight lines be drawn containing

a given angle, and from each of them segments be cut off having

a given ratio; and the extremities of the segments of the lines drawn

from one of the points be in a straight line given in position; to deter-

mine the locus of the extremities of the segments of lines drawn from

the other.

SECTION II. Page 24.

1. IF

a straight line be drawn to touch a circle, and be parallel

to a chord; the point of contact will be the middle point of the arc

cut off by that chord.

COR. 1. Parallel lines placed in a circle cut off equal parts of the

circumference.

COR. 2. The two straight lines in a circle which join the ex-

tremities of two parallel chords are equal to each other.

2. If from a point without a circle two straight lines be drawn

to the concave part of the circumference, making equal angles with

the line joining the same point and the centre, the parts of these lines

which are intercepted within the circle are equal.

3. Of all straight lines which can be drawn from two given points

to meet on the convex circumference of a given circle; the sum of

those two will be the least, which make equal angles with the tangent

at the point of concourse.

4. If a circle be described on the radius of another circle; any

straight line drawn from the point where they meet to the outer cir-

cumference, is bisected by the interior one.

5. If two circles cut each other, and from either point of inter-

section diameters be drawn; the extremities of these diameters and

the other point of intersection shall be in the same straight line.

6. If two circles cut each other, the straight line joining their

two points of intersection is bisected at right angles by the straight

line joining their centres.

7. To draw a straight line which shall touch two given circles.

8. If a line touching two circles cut another line joining their

centres, the segments of the latter will be to each other, as the dia-

meters of the circles.

9. If a straight line touch the interior of two concentric circles,

and be placed in the outer; it will be bisected at the point of

contact.

10. If any number of equal straight lines be placed in a circle;

to determine the locus of their points of bisection.

11. If from a point in the circumference of a circle any number

of chords be drawn; the locus of their points of bisection will be a

circle.

12. If on the radius of a given semicircle, another semicircle be

described, and from the extremity of the diameters any lines be

drawn cutting the circumferences, and produced, so that the part

produced may always have a given ratio to the part intercepted

between the two circumferences; to determine the locus of the ex-

tremities of these lines.

13. If from a given point without a given circle straight lines be

drawn and terminated by the circumference; to determine the locus

of the points which divide them in a given ratio.

14. Having given the radius of a circle; to determine its centre

when the circle touches two given lines which are not parallel.

15. Through three given points which are not in the same

straight line, a circle may be described; but no other circle can pass

through the same points.

16. From two given points on the same side of a line given

in position, to draw two straight lines which shall contain a given

angle, and be terminated in that line.

17. If from the extremities of any chord in a circle perpendi-

culars be drawn, meeting a diameter; the points of intersection are

equally distant from the centre.

18. If from the extremities of the diameter of a semicircle per-

pendiculars be let fall on any line cutting the semicircle; the parts

intercepted between those perpendiculars and the circumference are

equal.

19. In a given circle to place a straight line parallel to a given

straight line, and having a given ratio to it.

20. Through a given point, either within or without a given

circle, to draw a straight line, the part of which intercepted by the

circle shall be equal to a given line, not greater than the diameter of

the circle.

21. From a given point in the diameter of a semicircle produced,

to draw a line cutting the semicircle, so that lines drawn from the