the construction or the proof is attempted. Every teacher knows how carelessly these references, generally inserted in text-books in the margin, or in the body of the proposition, are slurred over, or repeated mechanically, without any reference to their import, by boys in class, and how confused and imperfect is their conception of the reasoning in consequence.

2. In describing the figures, the parts which are given in the enunciation are represented by dark lines, and those which are added in the course of the demonstration by dotted lines. The process of the construction is thus exhibited to the eye, and the data and the quæsita of the problem can always be distinguished at a glance.

3. In the demonstrations, the several steps of the proof are arranged in a logical form, by giving the premisses and the conclusion always in separate lines, and in a different type; and, as a further aid to the learner, the enunciations are broken into paragraphs, and the demonstrations into corresponding divisions, wherever the proposition consists of more than one case.

In this way the constituent parts of a proposition are presented separately, part by part, and the learner, knowing exactly where one begins and the other ends, is enabled to make himself master of the one before he proceeds to the other.

The symbolical editions of Hill, Blakelock, and Williams, were among the earliest to show the advantages of printing separately the parts of a proposition and its demonstration; and they have been followed with great success by Mr. Potts and other recent writers. The plan adopted by them of printing every sentence, or part of a sentence, which contains a new step in the reasoning, in a separate line, has been followed in the present edition, but to a greater extent, and with increased distinctness. To avoid the confused appearance produced by the lines being scattered irregularly over the page, as in previous texts of the Elements' on this plan, the lines have been printed so as to commence uniformly from the side of the page. Every conclusion is what printers technically term indented,' and the applicable part of it- that is, the part

made use of or referred to in the subsequent reasoning is, to mark its importance, printed in italics.

The conclusions thus distinguished, or the most important of them, if entered in the Copy-books' prepared for this purpose, and published in connexion with this work, will supply a kind of Analysis of Euclid, to those who have gone through the subject, but who wish at any time, as on the approach of an examination, to refresh their memory by a cursory re-perusal. To such, the reading of the several steps, and an inspection of the figure, will, in most cases, be sufficient to recall the complete proof to the mind, without the trouble of going over the entire proposition a second time; and this will be a pleasing and most improving exercise, and tend strongly to impress not only the proof itself, but also the principle of the proof, on the memory.

Although the text of Dr. Simson has been, in the main, adhered to in the present edition, alterations have been made wherever there appeared to be any obscurity in the language which could be removed by the introduction of a step, or the variation or transposition of a sentence.

As examples of such alteration may be mentioned the introduction of an additional figure in prop. 27, book i., and the use of a definite form of expression to mark the distinction in indirect demonstrations between a conclusion true in itself and one correctly deduced, but from an incorrect hypothesis. In the former case, the conclusion is expressed in the ordinary way; in the latter, the word 'assumed' is employed in the premiss, and the word 'must' in the conclusion, to indicate that the reasoning proceeds on a false assumption, although the reasoning itself is correct. This distinction, it is believed, will be found of considerable practical importance in teaching students.

Euclid to young

STATIONERS' SCHOOL:
August 1, 1862.

A. K. ISBISTER.

THE SCHOOL EUCLID.

BOOK I.

DEFINITIONS.

I.

A POINT is that which hath no parts, or which hath no magnitude.

A line is length without breadth.

II.

III.

The extremities of a line are points.

IV.

A straight line is that which lies evenly between its extreme points.

V.

A superficies is that which hath only length and breadth.

VI.

The extremities of a superficies are lines.

VII.

A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.

VIII.

A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction.

N.B. When several angles are at one point B, any one of them is expressed by three letters, of which the letter that is at the vertex of the angle, that is, at the point in which the straight lines containing the angle meet one another, is put between the other two letters, and one of these two is somewhere upon one of those straight lines, and the other upon the other line: Thus the angle which is contained by the straight lines, AB, CB, is named the angle ABC, or CBA; that which is contained by AB, DB, is named the angle ABD, or DBA; and that which is contained by DB, CB, is called the angle DBC, or CBD; but, if there be only one angle at a point, it may be expressed by a letter placed at that point: as the angle at E.

X.

When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.

XI.

An obtuse angle is that which is greater than a right angle.

XII.

An acute angle is that which is less than a right angle.

XIII.

A term or boundary is the extremity of any thing.

XIV.

A figure is that which is enclosed by one or more boundaries.

XV.

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

XVI.

And this point is called the centre of the circle.

Ө

XVII.

A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.