EUCLID, in his first book, treats of the construction and properties of triangles and parallelograms; and does not shew the properties of any other figures; the circle he defines and then uses simply as an instrument. The corollaries to Proposition 32 are not Euclid's.

Before proceeding further, let the student read carefully the first twelve propositions. He will notice that every assertion made is supported either by reference to one of Euclid's formerly assumed or established facts, or, in cases where this cannot be done, by letters which refer to the statements mentioned on page 11.

Euclid's first three problems establish his right to a freer use of the imaginary instruments allowed him by the postulates, by showing that he may use compasses which will remain open and a ruler with which he can add or subtract straight lines; and, when he has established the properties of triangles enunciated in the fourth and eighth propositions, he is able, with such instruments, to bisect straight lines and angles, and to erect perpendiculars.

I. 1. Euclid, in his first proposition, assumes that the circumferences of the two circles will cut one another. This assumption is a weak point in the argument, and his authority for it cannot be quoted. If a beginner will admit without question that the circumferences will cut, he will probably admit also that they will cut at two points, and at not more than two; now Euclid, whilst assuming the one fact, thinks it necessary to give, in his third book, a proof of the other. If it be admitted that the circumferences cut at two points, then two equilateral triangles can be described, one on each side of the line, and it is to be noticed that Euclid, in his ninth proposition, directs us to describe an equilateral triangle on one particular side of a line. In the first proposition he also assumes that two straight lines cannot have a

common segment; if such a thing were possible, the diagram might be

thus:

I. 2. In the second proposition it is assumed that the circumferences of the circles will cut DF and DE in G and L; that is to say, Euclid takes for granted that any straight line drawn through the centre of a circle will, if produced far enough, cut the circumference. Every other inference he draws is directly supported by a definition, postulate, axiom, or former proposition. The student must however closely consider whether the proof holds good in every relative position of the given point and line.

Let him test the accuracy of the construction by the careful use of instruments in the following cases, observing rigidly the directions and arguments of the proposition. His instruments should show him that practically the construction is correct, and the demonstration should convince him of its mathematical truth.

In each case there will be four diagrams to be drawn, for the point may be joined to either end of the line, and the equilateral triangle may be described on either side of the line so joined.

(75) Let the given point be in the given straight line, but not at an extremity of it.

(76) Let the given point be in the direction of the given line produced.

(77) Let the given point be on the circumference of the first circle.

(78) Let the given point be outside the first circle.

In some cases the problem could be solved by producing the sides of the equilateral triangle through the vertex, but this would not be a general solution.

To produce DA to E means to produce it through A, not through D.

In the above diagram, let A and BC be respectively the given point and line, and let the construction be as directed in the proposition, DA and DB being produced through D. If the text be followed with reference to this particular diagram it will be noticed that the circumference GCH does not cut DF.

If DF is held to mean FD produced, then the position of L is ambiguous; if L is between D and E the proof fails.

When the point is at an extremity of the given line, there are, by Euclid's method, only two solutions; although, by simply describing a circle, any number of lines could be drawn equal to the given line.

In any other case the construction might be worded thus: Draw a straight line from the given point to either extremity of the given line, and, on either side of the line so drawn, describe an equilateral triangle; with that extremity of the given line to which the point is joined as a centre, and the given line as a radius, describe a circle; produce the side of the equilateral triangle which is terminated at an extremity of the given line, away from the vertex to meet the circumference of the circle; with the vertex of the equilateral triangle as a centre, and the line last produced as a radius, describe another circle; produce the other side of the equilateral triangle away from the vertex to meet the circumference of the last circle. The produced part is the line required.

The following are given to show a few of the forms which the diagram of I. 2. might assume with the construction as given on page 25.

I. 3. In the third proposition Euclid assumes without authority that the circumference of the circle will cut AB. There are always two solutions to this proposition, for the part can be cut off from either end of the greater line.

I. 4. Geometrical figures exist in imagination only, and can as easily be pictured to the mind in one position as in another. Euclid, without claiming it as a postulate, directs one figure to be removed from its original position, and placed on another figure. His object in so doing is to conclude the equality of the two figures by showing that their boundaries can be made to coincide with each other. This method of testing the equality or inequality of space magnitudes is known as superposition. Euclid seems to have had some objection to the method, and did not make use of it when it could be avoided; in modern systems of geometry it is much more generally applied.

When the triangle ABC is placed on the triangle DEF, as in Prop. 4, so that A is on D, and AB on DE, then if there were no restrictions as to the relative magnitude of the sides and angles, the triangles might fall thus or otherwise, without coinciding.

DA

First, it is because AB is equal to DE, that B and E coincide; but if the angle BAC were not equal to the angle DEF, the triangles might fall thus, or otherwise;

Secondly, even when the angle BAC is equal to the angle EDF, if