admitted fact and this is his apology for introducing the statements on page 11. They are not advanced as supplementary general principles necessary to the rigid support of Euclid's system of Geometry but rather as answers to the questions which intelligent students are sure to ask with regard to Euclid's unsupported statements.

(4) All indirect demonstrations are concluded in the form,

If A is B,

then C is D, which is impossible;

therefore A is not B.

The objectionable phrase "which is absurd" is thereby avoided and the logical inference made more manifest.

It is impossible that a student can clearly understand indirect demonstrations without knowing something of logic. The author has therefore called attention to the few logical facts which are necessary to the support and explanation of Euclid's system, and added in italics, a few other such facts, the consideration of which will afford useful mental exercise.

With the hope of encouraging the student to apply, when practicable, superposition in the demonstration of theorems, some examples of the application of this method are given.

With a view to guarding the student against some of the principal errors which beginners are liable to make when attempting to demonstrate theorems, examples of erroneous demonstrations are given, the student being required to detect the errors.

Marked attention has also been called to Euclid's principal errors, and in this, and other ways, the author has attempted to encourage the most exact and rigid reasoning.

The propositions are so arranged in the book, that the whole text and diagram of each can be seen without turning over the page; and opposite each of the first 26 propositions, the various definitions, postulates, axioms and former propositions, necessary for the construction and demonstration are stated. The blank spaces left may be found useful for notes.

The only abbreviations used are:

A, for 'the point A.'

AB, for 'the straight line AB.'

Join AB, for 'draw from the point A to the point B a straight 'line AB.'

Experience having shewn the author that in the vast majority of cases the notes on the propositions are not observed, he has given more prominence to his notes than is usual, and, by advising the student when to read certain propositions, and when to work sets of examples, proposes a regular course of geometrical study, which he hopes will be found useful to his readers.

The examples have been selected chiefly with a view to teach the properties of the magnitudes treated of by Euclid in his first book; but, considering it necessary to commence with the very simplest ideas, the author has been obliged to set questions of a different nature, some of which are very much easier than those generally given in text books of geometry. He has also given exercises in the use of instruments, and endeavoured to encourage the habit of demonstrating theorems without using diagrams.

With reference to the progressive nature of the exercises, the author has taken the opinions of a great many young beginners as to their relative difficulty, and arranged them accordingly. The first 163 examples are of such a nature that at least 50 per cent. of his pupils have been able to originate the solutions.

At the end of the book are given particular enunciations of the propositions with diagrams different from those used in the text.

FIRST BOOK.

CHAPTER I.

GEOMETRY, which originally meant the art of measuring portions of the earth's surface or distances on it, now denotes the science which treats of the measurement, properties, and relations of space magnitudes generally.

It is the most important of all mathematical sciences; Astronomy, Navigation, Surveying, Perspective, Mechanics etc., all depend upon it, and before the student can proceed to the study of such sciences he must know geometry.

It is however chiefly as mental discipline and intellectual education that the study of Euclid's geometry is so valuable. The facts which are proved are very useful and indeed essential stepping stones to the study of other mathematical sciences, it is not however only what is proved, but the rigid and exact system of reasoning employed which makes the study so useful.

Euclid wrote nearly 2200 years ago and his book is the text-book of Geometry in many English schools at the present day, although other systems have, from time to time, been compiled in different countries.

Every assertion Euclid makes is proved at the time, has been proved before, or is assumed to be true.

Euclid's statement of what he assumes consists of the definitions, postulates, and axioms; but, as will be pointed out, these do not include all the truisms that he asserts as self-evident.

Let the student before proceeding read carefully the definitions, postulates and axioms on page 63.

It must not be imagined that Euclid was allowed to take for granted as true anything he liked;-far from it. He was engaged in a controversy in which he only assumed what he knew would be accepted by his adversaries, and they did not admit anything which could possibly be denied.

In the definitions he explains what he wishes certain words to be

P. G.

1

understood to express, and clearly his adversaries could not object to such abbreviations.

In the postulates he assumes that he can effect three processes, but in imagination only; for, as will be pointed out later on, no one could really draw a line, produce a line, or describe a circle, but it is perfectly reasonable, for the sake of argument, to suppose such things can be done.

In the axioms he assumes as true certain statements which his adversaries must have more or less readily admitted.

Every body, or particle of matter, however small, has length, breadth and thickness, and occupies a portion of the boundless space around us. Length, breadth, and thickness are called dimensions of space. Space can be of either one, two, or three dimensions.

The distance of one star from another is a space of one dimension, viz. length.

A shadow on a wall occupies a space of two dimensions, viz. length and breadth.

The wall itself occupies a space of three dimensions, viz. length, breadth and thickness.

That which in imagination separates one part of a space having length, breadth, and thickness, from another part, without being in itself part of either of them, and having only length and breadth, is a superficies. The surface of this paper is a superficies, but the superficies is not part of the paper, nor of the adjacent air. That which in imagination separates one part of a superficies from another, being part of neither, and having length only, without breadth, is a line. A point is that which in imagination divides one part of a line from another, without being part of either, and it has neither length, breadth, nor thickness. A point has only position, a line has only position and length, and a superficies has only position, length, and breadth. Neither a point, a line, nor a superficies can be either felt or seen; it is only in imagination that a line can be drawn, or a point placed in position. The so-called lines supposed to be drawn on paper are really accumulations of particles of some kind, each of which is a solid substance having length, breadth and thickness. Of such lines Euclid's arguments treat of their length only, independently of any idea of their breadth and thickness. A point in motion passes through a space called a line, and a line in motion, not in its own direction, passes through a space called a superficies.

A carpenter applies in practice the test which Euclid applies in theory, to determine whether a superficies is plane, for, after he has