in which it is more applicable than in mathematics. Folks do not understand books without first knowing letters and words, and yet this method is not unfrequently attempted in Geometry, in Algebra, and in Arithmetic.

It often happens, in complicated diagrams, that the same angle belongs to more than one figure, and this is also a source of annoyance to beginners. As an instance of this, we may mention the fifth proposition of the first book of Euclid's Elements, the far-famed pons asinorum, or asses' bridge, the demonstration of which is very simple, as well as beautiful; but there is a perplexity in the angles, one of which belongs to three triangles; and of two other sets of angles, at two points, one belongs to one triangle, a second part to a second triangle, the third and second to a third triangle, and the second with the third on to a fourth triangle. As the diagram is a good study for those who wish to understand such representations, we subjoin it, and append the several triangles, which the reader can easily trace. The following is the diagram as it appears in the book:

F

The object here is to prove that, if the sides AB and ac, in the triangle ABC, which comprises the upper part of the diagram, are equal, the angles at в and c must also be equal; and all that is admitted to be known about triangles is, that, if two sides, and the angle included between them in one triangle,

are equal to two sides and the included angle in another, the. two triangles are equal in every respect; that is, the third side of the one is equal to the third side of the other, and the remaining angles which are opposite the equal sides are also equal.

It must be admitted that the truth which is to be proved in this case, comes as nearly as possible to a legitimate deduction, or corollary, as it is called, from the truth by means of which we are to prove it; for if, in two equal triangles, the angles opposite to the equal sides are necessarily and in every case equal, it seems to follow that, as one triangle is in every respect equal to itself, the angles opposite to equal sides in it must be equal.

But though this would be a sound argument in ordinary reasoning, it does not come up to the rigour of geometrical demonstration, and so we must have equal triangles to compare with each other. For this purpose AB is extended to D, and Ac to E; the parts BF and CG are taken equal to each other, and BG and C F are joined, which give four additional triangles, which, taken two and two, are equal to each other.

But this, though true, is not apparent to one unacquainted with diagrams. Only three additional triangles are apparent in the diagram, and we are not in possession of the means of proving that any two of them are equal in any respect; and though we were, they could prove nothing respecting the triangle ABO, for they are all external of it, and quite unconnected with the angles ABC and ACB, the equality of which is to be proved.

But let us analyse the diagram, and see what other triangles we can get out of it, without altering the relative positions of any of the lines.

The following figures contain the real and palpable analysis, which the student is called upon to make virtually, at the same

'

THE PONS ASINORUM."

205

time that he is making his first infantine attempt to wrestle with the giant of geometry :

The middle figure will be perceived to be exactly the same as the diagram given above, the given triangle of which the angles at B and c are to be proved equal to each other. We have marked this triangle with the number 1, and strengthened the sides of it in order to distinguish it from those other triangles which are made for the purpose of the demonstration,—and as it should be in cases of teaching the first elements of geometry, where it is done by diagrams ready made, and not constructed (as they always should be when it it is possible) in the presence of the student. The two triangles marked 2, to the right and left, can be traced as answering to two equal ones which lie across each other in the central diagram, and have at their angles respectively the letters ACF and ABG. So also the two triangles farther to the left and right, marked 3, can be traced as corresponding with the two triangles in the diagram which lie partly across each other below the side BC, or base of the original triangle. The triangles 2 have each the angle at A equal to the angle at A in the original triangle, and their sides, CA and AF in that to the left, are equal to BA and AG in that to the right, and therefore they are equal in every respect, and FC and BG are equal to one another, and so are the angles at c and B, and also those at F and G.

But the trianeles 3, 3, have BF and CG made equal, and FC and GB proved equal, and also the contained angle at F equal to the contained angle at G; therefore they are equal in every respect, and the angles opposite equal sides are of course equal, that is, the angle c in the left-hand one is equal to the angle B in the right, and the angle в in the left-hand one is equal to the angle c in the right. But the angle c, of triangle 2 on the left, is equal to the whole angle ACF in the diagram; and the angle F, in triangle 3 on the left, is equal to the part BCF in the same. So also the angle в, in triangle 2 on the right, is equal the whole angle ABG in the diagram; and the angle B, in triangle 3 on the right, is equal the part сBG in the diagram. Now,

From ABC ABC + CBG, and ACF ACB + BCG
Subtract

CBG and

BCG

There remains

ABC

=

ACB;

and they are the angles opposite the equal sides, or at the base

of the given triangle.

Again, FBC, in triangle 3 on the left, has been shown equal to GBC in triangle 3 on the right, and they are respectively equal to FBC and GCB in the diagram; and these last are the angles on the other side of the base, formed by the base and the equal sides produced. Therefore, if a triangle has two equal sides, the two angles at the base, and the two angles on the opposite side the base, are equal to each other.

In geometrical language, the words "each to each" are made use of for shortness of expression, when any number of pairs of quantities have each pair equal to each other.

In the analysis of this diagram we have rather anticipated, in introducing the demonstration, but a very little attention will

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207

suffice for understanding it; and the analysis of the mere diagram would not be so well appreciated, unless the use of it were shown at the same time.

The ready understanding of diagrams, so as virtually and at a glance to dissect or analyse them into all the parts of which they are made up, is a most essential qualification in those who wish to understand easily even the simplest elements of geometry; but it is almost, or altogether, omitted in the books, and the omission is, in our opinion, chargeable with much of the perplexity and failure which so many meet with in this science. It would be too much to suppose that we have furnished, in this section, the means of wholly overcoming the difficulty; but if what we have stated is read with attention, and some practice is taken with the analysis of diagrams, either in Euclid's Elements, or in any other elementary work, the the student will be enabled to proceed to the work of investigation with much more ease and prospect of success, than if he were not so prepared. We shall examine the principles and processes of geometrical investigation in the next section.

SECTION XI.

PRINCIPLES OF GEOMETRICAL INVESTIGATION.

INVESTIGATION, taken in a general sense, means systematic and accurate inquiry, in order to determine whether something which is alleged is true or not true, or whether something proposed to be done is possible or not possible; and of course, geometrical investigation includes every instance of both of these, of which the subject can be considered as geometrical, that is,