In the volume now offered to the public, the author has aimed to bring the science of Elementary Geometry within limits which should adapt it to the convenience of a greater number than can afford time to acquire a competent knowledge of the subject from the treatises in common use; while, so far as this branch of elementary preparation is concerned, he should omit nothing essentially introductory to the higher geometry, or to the physico-mathematical sciences. He has endeavoured also to state the first principles of the science in a manner better suited to the apprehension of the young student, than that in which they are usually presented in elementary treatises.

Clairaut, one of the first mathematical geniuses of the eighteenth century, has said, "Though Geometry be in itself abstract, it must nevertheless be acknowledged, that the difficulties which the student encounters in the commencement of the subject, most frequently arise from the manner in which it is taught in common Elementary Treatises. These begin with a great number of definitions, postulates, axioms, and preliminary principles, which scem to promise little to interest the reader. The propositions which follow, do not fix the mind upon engaging objects; and as they are moreover difficult to be understood, it commonly happens that beginners are fatigued and dis

gusted before they have any distinct idea of what is designed to be taught them."

In the present treatise, the author has endeavoured, as far as he was able, to remove the difficulties here complained of; by avoiding the abstract phraseology and technical forms of the books in common use; by presenting to the mind of the learner, in a manner as simple as he could, the elementary truths of the science; and developing by a process of plain and, at the same time, exact examination, those which are more involved; by following in a great degree the natural order of discovery; and aiming so to conduct the investigation of any particular fact or general principle, as that other collateral truths should unfold themselves incidentally, and thus, by the very circumstance of their being unexpected, afford pleasure to the learner, and excite in his mind a curiosity to know what will next discover itself on either side of the path he is pursuing.

This desire to know with what interesting and useful truths the further pursuit of the subject will bring him acquainted, will enable the student to perform with pleasure what is commonly (and frequently with too much truth) called his task, and really to do more with less fatigue. It is, moreover, a state of the mind in which the truths actually presented to it make the greatest impression; and are most likely to become a permanent part of the student's knowledge.

If the result is kept back till the learner has gone through the process of induction by which any truth is to be established, and by which also it must have been at first discovered, his mind will be prepared to perceive the precise meaning of the proposition in which this truth is enunciated; whereas the slightest misapprehension of the nature or the extent of the truth stated beforehand in the proposition, may seriously embarrass him in applying the process of proof, to say nothing of the difficulty of ridding

himself entirely of the first impression made by a proposition endowed with the oracular name of theorem.

Another important advantage results from thus withholding the proposition till its truth is apparent. The student will early begin to anticipate, before he arrives at the statement of the result, the truths as they unfold themselves; and finding that he has discovered them from the relations presented, he is gratified by this evidence of his own power, and is encouraged to continue his exertions. He will be constantly on the watch for new discoveries; and his studies will be a salutary discipline, not only to his understanding, but to the inventive faculty of his mind.

In preparing these Elements, the author has consulted various editions of Euclid, and several modern treatises, among which may be mentioned those of Legendre, Bézout, Reynaud, Lacroix, and Clairaut; besides the meta physical writings of Lacroix, Carnot, and Laplace. He has endeavoured to combine, especially in the earlier part of the work, the simplicity of Clairaut with the certainty of what are usually called the "severer methods." In the latter part, the general arrangement of the subject is more nearly that of Lacroix than of any other. But most of what may be considered as peculiar, whether in the arrangement or the manner of conducting the inquiries by which the truths of the science should unfold themselves to the student, has been suggested by a careful observation of the difficulties which (in the course of no inconsiderable experience as an instructer in elementary mathematics) the author saw his pupils continually encountering; and by the results of constant efforts to investigate the causes of these difficulties, and of multiplied endeavours to state the elementary principles of the subject in so simple a manner as to be apprehended by every mind. Though he may not always have succeeded, yet the results were such as to con

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vince him that the mistakes and misgivings which the young student experiences in the early part of his geometrical studies, are very uncertain indications of any peculiar want of adaptation of mind to this branch of science. So far from it, these very mistakes should be expected. The subject is new; the leading ideas are many of them expressed in terms almost exclusively appropriated to this science, and must therefore be new to the beginner; though these terms seem very definite to the metaphysical geometer, there is much uncertainty whether they will convey to the mind of the student the precise idea which the writer intended; and it will frequently happen, that the ideas which many, even of the better scholars, receive from some of the principal definitions and statements in our elementary books, require to be considerably modified or entirely changed, to adapt them to the use which is to be made of them in the subsequent part of their studies.

How much perplexity and discouragement might have been spared the learner in such cases, if the mistake could have been discovered at first. In consequence of a slight misapprehension of some leading principle in the beginning, to which every mind is liable, the learner tasks himself in vain to reconcile the subsequent reasoning and results with the first notions which he received, and which, coming with all the freshness of novelty, fixed themselves in his mind. Such mistakes might be, in a great measure, prevented, if the instructer would explain and illustrate every sentence of one or two of the first lessons before they are read by his pupils; and if, wherever any new element is introduced, he would be sure that they understand it before they proceed to apply it.

There is always danger in giving the beginner two or three pages, at first, of elementary matter in the form of distinct principles and definitions, attended by little or no explanation. Such an array of detached and unconnected truths tends only to fatigue and distract him.

To prevent, in some degree, this liability to misconception and embarrassment, the definitions and principles, in the present treatise, are given only as they are to be used in the course of the investigation, either as principal or auxiliary truths.

For some of the elements new definitions have been adopted, which, it is believed, convey to the mind of the learner more accurate notions of the things defined, and very much simplify the processes into which they enter.

"A good definition should contain an enumeration of certain simple characteristic attributes of the thing defined, by which it may be clearly distinguished from all other things of a like kind. In this respect the definitions of mathematics are, in general, peculiarly happy. They usually contain some one simple but characteristic property of the thing in question, from which all its other properties may be readily and legitimately deduced."*

In conformity with the spirit of the remark here quoted, the three following definitions are adopted.

A straight line is one which has the same direction throughout its whole extent.

A plane angle is the inclination of two straight lines to each other.

Parallel lines are straight lines which have the same direction in space.

The simple characteristic property of a straight line, that is, its straightness, by which it is distinguished from all other lines, is its identity of direction in every part. The definition given in several elementary books, namely, that a straight line is the shortest way from one point to another, is a proposition which carries to the mind the fullest conviction of its truth; and whether a simple principle or a

* Prof. Elliot of Aberdeen.