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PREFACE.

A MODERATE experience in tuition will show the truth of the assertion, that the Elements of Euclid is the subject that most frequently proves an insurmountable obstacle to such pupils as have not acquired habits of close application, or have no natural taste for mathematical study. For this, two principal reasons may be assigned ;

the first is the nature of the subject itself, which can only be pursued with success by a vigorous exercise of the reasoning powers; whereas many algebraical operations are purely mechanical, and are worked by rules which the student may easily apply, though profoundly ignorant of the reasoning on which they depend. But another, and perhaps the chief, cause of this difficulty, is the great want of those helps and illustrations which are so plentifully supplied in the study of every other subject. The truths of Algebra are familiarly explained in numerous popular treatises, and are thus brought down to the level of very moderate capacities; whilst Geometry, to most persons a far more difficult subject, is presented to the learner, with scarce a note or a hint to assist him.

The present little work, it is hoped, may in some degree supply this lack of assistance. Those who find Euclid an easy subject, may, perhaps, object that they find little that is new in this book, and that the system it contains is such as any private tutor might have recommended his pupils to adopt ; but we beg of such persons to reflect that many

who find great difficulties in Euclid have no private tutor : and that only those who have a taste for Geometry would, (even if advised to do so take the trouble to form such a system for themselves, as is here made ready to their hands.

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COMPANION TO EUCLID.

PRELIMINARY OBSERVATIONS AND DIRECTIONS.

The chief design of this work is to afford the beginner in Euclid an insight into the plan and connexion of argument in the demonstrations he is required to learn, in order to remove that appearance of confusion and useless repetition which many complain of on first attempting a geometrical proposition.

It should be observed that most propositions consist of four distinct parts.

1. The Enunciation, or General Statement of the Theorem or Problem.

2. The Enunciation repeated with reference to the figure.

3. The construction of those additions to the figure*, which are necessary to the proof of a Theorem or the solution of a Problem.

4. The Demonstration.

The learner should carefully observe the distinction between the three last parts. The demonstrations, however, can alone present any difficulty, and are therefore the only object of the present work.

Some propositions do not require this part. In others, this part includes a demonstration.

The learner should keep his Companion open before him while studying a proposition, and carefully observe the steps of the proof as he proceeds; and, in long propositions, he had better draw a line between them in his Euclid. Many demonstrations which at first appeared very intricate will thus become clear and intelligible; for though the proof of a long proposition may

be difficult to a learner on account of the number of the steps in it, yet the proofs of these steps themselves are always easy and always short: so that the student will find (after a little practice) that he can take up his Companion, and find out for himself, without referring to Euclid, the proofs of the several steps of a new proposition, provided he has learned perfectly those which precede; and this will be a pleasing and a most improving exercise, and tend strongly to impress not only the proof itself, but also the principle of the proof, on the understanding.

Such is our first object; the second is to 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 will bring the complete proof into the mind without reference to Euclid, and will thus ensure a great saving of their time.

A well-drawn figure is by no means a matter of minor importance. The eye is so great an assistant to the memory, that the sight of the figure alone will often recall a demonstration to the mind. A set of figures are therefore added, which, it is hoped, will prove much more intelligible than those in the com

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