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Professn W. W. Bemans 3-25-1933

PREFACE.

ELEMENTS of geometry are by no means numerous in this country, a circumstance to be attributed to the almost universal preference given to Euclid; not, indeed, because the elements of Euclid is a faultless performance, but because its blemishes are so inconsiderable when compared with its extraordinary merits, that to reach higher perfection in this department of science has been generally supposed to be scarcely within the bounds of possibility, an opinion which the fruitless efforts of succeeding geometers to establish a better system have in a great measure confirmed. The The superiority of Euclid's performance consists chiefly in the rigorous and satisfactory manner in which he establishes all his assertions, preferring in every case the most elaborate reasoning rather than weaken the evidence of his conclusions by the introduction of the smallest assumption.

On the continent, however, this high opinion of Euclid does not appear to prevail, and the rigour and elegance of his demonstrations seem to be less appre

ciated. In all the modern French treatises on geometry, it is easy to discover a wide departure from that rigorous and accurate mode of reasoning so conspicuous in the writings of the ancient geometer. From this imputation even the celebrated Elémens de Géométrie of Legendre, "the first geometer in Europe," is not exempt, notwithstanding the masterly manner in which he has treated certain difficult parts of the subject. The greatest difficulty, however, in the whole compass of geometry is doubtless the doctrine of geometrical proportion. The manner in which Euclid establishes this doctrine is remarkable for the same rigour of proof that manifests itself throughout the other parts of his work, although it is universally acknowledged that from the difficulty of the subject his reasoning is so subtle and intricate, that to beginners it opposes a very serious obstacle. The grand aim, therefore, of geometers has been to deliver this part of Euclid's performance from its peculiar difficulties, without destroying the rigour and universality of his conclusions. All attempts to accomplish this important object have been unsuccessful; and those who have abandoned Euclid's method, and have treated the subject in a more concise and easy way, have greatly fallen short of that accuracy of reasoning so essential to geometrical investigations, and have arrived at conclusions that are not indisputably established, but only approximately true:- such is the doctrine of proportion as treated by geometers of the present day. It appears, therefore, that notwithstanding the recent translation of Legendre's

celebrated work into our own language-the unqualified praise which has been bestowed upon it, and its extensive circulation throughout Europe, there are still blemishes to be removed and defects to be supplied; for extraordinary as it may appear, Legendre has not, unfortunately, exercised his powerful talents upon the doctrine of proportion, but has entirely excluded the consideration of it from his elements, referring the student for requisite information " to the common treatises on arithmetic and algebra *." books on arithmetic and algebra can unfold the properties of proportion only as regards numbers, and numbers cannot extend to all classes of geometrical magnitudes, for some when compared are found to be incommensurable. The doctrine of proportion, therefore, in reference to these latter, cannot be rigorously inferred from any thing that may be established with regard to numbers or commensurable magnitudes.

Now

Having adverted to these defects it remains for me now to give some brief account of the present attempt, and to state wherein I have endeavoured to render it more particularly worthy of examination.

And first it may be remarked, in reference to the general plan of the work, that I have taken a more enlarged and comprehensive view of the elements of geometry than I believe has hitherto been done, as I have paid particular attention to the converse of every proposition throughout these elements, having demon

* Dr. Brewster's translation of Legendre, page 48.

strated the converse wherever such demonstration was possible, and in other cases shown that it necessarily failed. There can be no doubt that this comprehensive mode of proceeding, embracing as it does every thing connected with the subject, must afford the student entire satisfaction, and must also increase the accuracy, as well as the extent, of his geometrical knowledge ; since he not only learns that under certain conditions a certain property must have place, but also whether or not it is possible for the same property to exist under any change of those conditions. The first, and I believe the only work in which converse propositions are fully considered, is that of M. Garnier, entitled Réciproques de la Géométrie, and which, it appears, was intended to accompany the geometry of Legendre. In the present performance I have, in several instances, availed myself of this work of Garnier, although, in many other cases, I have found it expedient to adopt a different course. The only book on geometry with which I am acquainted, where the converse accompany the direct propositions to any extent, is the Elémens de Géométrie par M. Develey, a very comprehensive performance; but in many instances the converse propositions are not noticed, and in but very few cases is their failure shown to take place. This plan, therefore, is not systematic and uniform.

With regard to other, and more particular improvements, introduced into this work, may be noticed proposition XIII. of the first book, taken, with little alteration, from the Principes Mathématiques of M. da