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tions to these are not only simple, but also practical. This is important, as the constructions generally given in theoretical works on Geometry are not employed in practice, whilst those in use are selected from treatises on Practical Geometry, and have no proofs attached to them. The Problems are mainly those of the first four Books of Euclid. The demonstrations of these, as also of the Theorems, assume no more than Euclid does in the corresponding Propositions, and may accordingly be substituted for them in examinations. This is therefore not merely a course preparatory to a more extended study of Geometry, but, in the event of its being followed by Euclid as a text-book, a great deal of the ground of his first four Books will then have been already covered by it, and in such a manner as to render the completion a matter of no great labour.


October 1875.


THE following axioms will be assumed :—

Things which are equal to the same thing are equal to one another.

The whole is equal to the sum of its parts.

If equals be added to equals the wholes are equal.

If equals be taken from equals the remainders are equal. Things which are double of equal things are equal to one another.

Things which are halves of equal things are equal to one another.

After the first few Propositions the following symbols will be introduced :






equal to.






A straight line is one which lies evenly between its extremities.

EUCLID'S AXIOM. Two straight lines cannot enclose a space.

If two straight lines be drawn from a point, they will form an angle. Those two lines are called the arms, and that point is called the vertex of the angle.

The size of an angle does not depend on the length of its arms, but only on their direction.

If one angle can be placed on another so that the vertex of one may coincide with the vertex of the other, and the arms of one lie along the arms of the other, the angles are said to be equal to one another.



A triangle is a plane figure bounded by three straight lines, called the sides of the triangle.

The three angles formed by the sides are called the angles of the triangle.

If one triangle can be placed on another so that the boundary of one may coincide with the boundary of the other, then the triangles are equal to one another, the sides of the one are equal to the sides of the other, and the angles of the one to the angles of the other. The triangles are then said to be equal in all respects.

A circle is a plane figure bounded by one line, called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal.

That point is called the centre, and those straight lines are called radii.

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