of exhaustions, in which they varied the forms of rectilineal figures by continually increasing the number of their sides by successive bisection, the direct introduction of the idea of motion being thus avoided. This mode of proceeding might be very proper in the earlier stages of the science, but surely there can be no good reason now for excluding the other mode of demonstration from any part of the elements, where it can be introduced with advantage, by leading to more satisfactory conclusions, and obviating difficulties which admit of no other mode of solution. It might also prove very useful to students to have some acquaintance with the nature and use of variation before they advanced to that part of mathematical science where it is almost the only instrument of proof which is employed.

The Elements of Euclid have, for the space of nearly 2000 years, been considered and generally preferred as the most approved introduction to the study of the mathematical sciences. Their excellence in this respect, and the wisdom of the preference which has in consequence been given them, are fully confirmed, both by long experience, and by the failure of the greater number of the attempts which in modern times have been made to supersede them. The mode of arrangement which he has devised is very happily adapted to the purposes of instruction, as every preceding proposition prepares the way for understanding that which is to follow; and thus the mind of the learner is gradually led on from the consideration of the simplest truths to the comprehension of the most recondite theorems. In this respect, the elements may be regarded as an uninterrupted chain of reasonings rather than a regular arrangement of magnitudes, although attention is paid to this

B

also where it did not interfere with the other, which is certainly the most important. He has also been exceedingly careful to assume nothing which admitted of proof; and every proposition is established by the most rigorous demonstration, without allowing any doubtful element to enter which might in the smallest degree impair the validity of his conclusions. Almost the only exception to this is the assumption which he has made in his twelfth axiom, and which he was compelled to introduce in order to effectuate a demonstration of the properties of parallel lines. It is, however, rather a singular circumstance, that the introductory part of his first book forms the most clumsy and repulsive portion of his whole performance. Its position also, at the very commencement of the study, is peculiarly unfortunate. The fifth and seventh propositions in particular have always presented a formidable obstacle to beginners, and have tended greatly to discourage their minds and impede their progress. In the present treatise, much care and attention has been bestowed in simplifying and improving this part of the elements, and, without deviating in the least degree from Euclid's accuracy, to render the approach to the science as easy and inviting as possible.

The definitions here given are, with a few exceptions, nearly the same, in signification at least, with those generally found in other treatises of geometry. The first of these exceptions is that of a straight line, which is defined by that property which distinguishes it from all other kinds of lines whatsoever, that only one straight line can be drawn from one point to another, and is thus expressed, "When one line only of the kind can be drawn from one point to another, it is called a straight line." From this definition, all the inferences necessary for the pur

It appears

poses of demonstration can easily be deduced. also to be preferable to that given by Professor Playfair, that "If two straight lines are such that they cannot coincide in two points without coinciding altogether, each of them is called a straight line," a sentence which some geometers have considered as an axiom, and others as involving a theorem requiring demonstration. A proposition of a character so equivocal appears ill adapted to serve the purposes of a definition, which ought to contain a clear and accurate description of the thing defined, without any superfluity or any improper restriction. The definition of parallel lines here adopted is different from that of Euclid; but it has been greatly commended by many eminent geometers, and has been selected in preference to all others, because it is most suitable to the method of demonstration here employed. The third is that of the circle, which is represented as described by the revolution of a straight line about one of its extremities which remains fixed. The describing line is called the radiant or straight line, which determines the magnitude of the radii. The term is borrowed from the higher geometry, and the distinction between it and a radius is introduced for the purpose of giving a simple and easy demonstration of the first proposition. This distinction may also be sometimes found convenient in the construction of figures.

The postulates are three in number, as in Euclid's Elements, the only difference being that the third is a little more definitely expressed. It may be proper to observe, that in this third postulate the geometer requires, as a necessary preliminary of construction, that he shall be allowed to open his compasses until the distance between the legs shall be exactly equal to a given straight line, and that he may, with this distance as a

radiant, describe a circle,―concessions which must be granted before it is possible for him to construct the simplest rectilineal figure.

The axioms are here reduced to nine. Euclid's eighth axiom is transferred to the definitions, and divided into two, in order to mark out clearly in what respects the magnitudes are equal. His ninth axiom is divided into two; his tenth is comprised in the definition of a straight line; his eleventh, that all right angles are equal, is demonstrated as a theorem; and his twelfth axiom is dispensed with altogether. The axioms of geometry are self-evident truths, naturally suggested to the mind by our ideas of extension, and accommodated to our notions of that species of equality which arises from the mutual coincidence of figures. They have a close connexion with the demonstrations, and are employed as tests of the truth and validity of conclusions deduced from the suppositions and reasonings brought forward in the proof of a proposition. If the conclusions are in harmony with the axiom applicable to the particular case, they are received as veritable truths; but if they are at variance with it, they are rejected as false and absurd. Axioms should be as few as possible, for nothing ought to be admitted as self-evident which is capable of demonstration.

Euclid has employed the two first propositions of his book in pointing out how to perform the simple operation of drawing from a given point a straight line equal to a given straight line; and the solution of this problem which he has given, besides being deficient in generality, is so awkward and tedious that no geometer would ever think of putting it in practice. In the present treatise, this operation is performed in the

simplest manner, by making a distinction between the radiant and a radius, that thus a medium of comparison may be obtained between the lines there specified, by means of which they may be proved to be equal. This mode of proceeding also harmonizes theory and practice,-an object which ought always to be kept steadily in view. As a visible representative of the radiant, the student may be referred to the distance between the two points of the compasses with which the circle was described, both lines being equal to that distance are therefore equal to one another.

The ninth proposition in this treatise, that the circumferences of circles cannot cut one another in more than two points, is assumed without proof by Euclid in his first two propositions, and must be assumed unless previously established in the construction of every figure which requires the introduction of two circles which intersect. The demonstration here given is new, but it is presumed that it is quite valid. It has also this important advantage, that it furnishes the means of deducing from the next proposition a corollary equivalent to Euclid's seventh proposition, by assigning and bringing fully into view the true reason why, upon the same base and on the side of it, there cannot be two triangles which have their two sides which are terminated in one extremity of the base equal to one another, and likewise those terminated in the other extremity equal to one another. It is proper also to mention, that no more problems are introduced than such as are necessary to construct the theorems.

The method of indirect demonstration adopted in these Elements is in some respects different from that of Euclid, which modern geometers have generally followed. The indi