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THE following pages contain a collection of Problems, which are for the most part an easy application of the Elements of Euclid. They are arranged in what seemed to be the most natural order: The 1st section comprises such as contain the properties of straight lines and angles; the 2nd straight lines and circles: the 3rd straight lines and triangles; and the 4th parallelograms, squares and polygons. The 5th section contains those which require lines to be drawn in certain directions, but which involve properties of rectangles or squares, or such others as were excluded from the three first. The 6th comprises those by which figures are described, and also inscribed in or circumscribed about each other. The 7th comprehends such as contain the properties of triangles described in or about circles; the 8th those which contain the squares or rectangles of lines connected with circles; and the 9th the construction of triangles. To these is added an Appendix, intended to contain so much of the Elements of Plane Trigonometry, as is necessary for understanding those parts of Natural Philosophy which are the common subjects of Lectures in the Uni

versity. The Reader who wishes for farther information,
is referred to Mr. Woodhouse's treatise, or that of
Cagnoli, to the latter of which are appended extensive
Tables of trigonometrical formulæ.

From this performance the only credit expected is
that of having endeavoured to place principles in a clear
light, and to render a service to the younger students by
setting before them a series of Problems, on the solution
of which they are recommended to exercise their own
ingenuity; for which purpose a table of Contents has
been prefixed.

To the Syndics of the University Press, who from
the funds which are placed at their disposal, with great
readiness agreed to bear a considerable part of the ex-
pence of printing, most sincere thanks are due; and this
opportunity is taken of gratefully acknowledging the
obligation.

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.

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

fortning 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

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