PREFACE.

Two thousand years have now rolled away since Euclid's Elements were first used in the School of Alexandria, and to this day they continue to be esteemed the best introduction to mathematical science. They have been adopted as the basis of geometrical instruction in every part of the globe to which the light of science has penetrated; and, while in every other department of human knowledge there have been almost as many manuals as schools, in this, and in this only, one work has, by common consent, been adopted as an universal standard. Euclid has been translated into the languages of England, France, Germany, Spain, Italy, Holland, Sweden, Denmark, Russia, Egypt, Turkey, Arabia, Persia, and China. This unprecedented unanimity in the adoption of one Work as the basis of instruction, has not arisen from the absence of other Treatises on the same subject. Some of the most eminent mathematicians have written, either original Treatises, or modifications and supposed improvements of the Elements; but still the "Elements" themselves have been invariably preferred. To what can a preference so universal be attributed, if not to that singular perspicuity of arrangement, and that rigorous exactitude of demonstration, in which this celebrated Treatise has never been surpassed? To this,' says Playfair, ' is added every association which can render a Work venerable. It is the production of a man distinguished among the first instructors of the human race. It was almost the first ray of light which penetrated the darkness of the middle ages; and men still view with gratitude and affection the torch which

rekindled the sacred fire, when it was nearly extinguished upon earth.'

It must not, however, be concealed, that, excellent as this Work is, many, whose opinions are entitled to respect, conceive that it needs much improvement; and some even think that it might be superseded with advantage by other Treatises. The Elements, as Dr. Robert Simson left them, are certainly inadequate to the purposes of instruction, in the present improved state of science. The demonstrations are characterised by prolixity, and are not always expressed in the most happy phraseology. The formalities and paraphernalia of rigour are so ostentatiously put forward, as almost to hide the reality. Endless and perplexing repetitions, which do not confer greater exactitude on the reasoning, render the demonstrations involved and obscure, and conceal from the view of the student the consecution of evidence. Independent of this defect, it is to be considered that the "Elements" contain only the naked leading truths of Geometry. Numerous inferences may be drawn, which, though not necessary as links of the great chain, and therefore subordinate in importance, are still useful, not only as exercises for the mind, but in many of the most striking physical applications. These, however, are wholly omitted by Simson, and not supplied by Playfair.

When I undertook to prepare an elementary geometrical text-book for students in, and preparing for, the University of London, I wished to render it useful in places of education generally. In this undertaking, an alternative was presented, either to produce an original Treatise on Geometry, or to modify Simson's Euclid, so as to supply all that was necessary, and to remove all that was superfluous; to elucidate what was obscure, and to abridge what was prolix; to retain geometrical rigour and real exactitude, but to reject the obtrusive and verbose display of them. The consciousness of inability to originate any work, which would bear even a remote comparison with that of the ancient Greek Geometer, would have

been reason sufficient to decide upon the part I should take, were there no other considerations to direct my choice. Other considerations, however, there were, and some which seemed of great weight. The question was not, whether an elementary Treatise might not be framed superior to the " Elements," as given by Simson and Playfair; but whether an original Treatise could be produced superior to what these Elements would become, when all the improvements of which they were susceptible had been made, and when all that was found deficient had been supplied. Let us for the present admit, that a new Work were written on a plan different from that of Euclid, constructed upon different principles, built upon different data, and exhibiting the leading results of geometrical science in a different order. Let us wave also the great improbability, that even an experienced instructor should execute a Work superior to that which has been stamped with the approbation of ages, and consecrated, as it were, by the collective suffrage of the whole civilized globe. Still it may be questioned whether, on the whole, any real advantage would be gained. It is certain that all would not agree in their decision on the merits of such a Work. Euclid once superseded, every teacher would esteem his own Work the best, and every school would have its own class-book. All that rigour and exactitude, which have so long excited the admiration of men of science, would be at an end. These very words would lose all definite meaning. Every school would have a different standard; matter of assumption in one, being matter of demonstration in others, until, at length, GEOMETRY, in the ancient sense of the word, would be altogether frittered away, or be only considered as a particular application of Arithmetic and Algebra.

Independently of the disadvantages which would attend the introduction of a great number of different geometrical classbooks into the schools, nearly all of which must be expected to be of a very inferior order, inconveniences of another kind

would, I conceive, be produced by allowing Euclid's Elements to fall into disuse. Hitherto Euclid has been an universal standard of geometrical science. His arrangement of principles is registered in the memory of every mathematician of the present times, and is referred to in the works of every mathematician of past ages. The books of Euclid and their propositions are as familiar to the minds of all who have been engaged in scientific pursuits, as the letters of the alphabet. The same species of inconvenience, differing only in degree, would arise from disturbing this universal arrangement of geometrical principles as would be produced by changing the names and power of the letters. It is very probable, nay it is certain, that a better classification of simple sounds and articulations could be found than the commonly received vowels and consonants; yet who would advocate a change?

In expressing my sentiments respecting Euclid's Work, as compared with others which have been proposed to supersede it, I may perhaps be censured for an undue degree of confidence in a case where some respectable opinions are opposed to mine. Were I not supported in the most unqualified degree by authorities ancient and modern, the force of which seems almost irresistible, I should feel justly obnoxious to this charge. The objections which have been from time to time brought against this Work, and which are still sometimes repeated, may be reduced to two classes; those against the arrangement, and those against the reasoning. My business is not to show that Euclid is perfect either in the one respect or the other, but to show that no other elementary writer has approached so near to perfection in both. It is important to observe, that validity of reasoning and rigour of demonstration are objects which a geometer should never lose sight of, and to which arrangement and every other consideration must be subordinate. LEIBNITZ, an authority of great weight on such a subject, and not the less so as being one of the fathers of modern analysis, has declared that the geometers who have