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THE propositions contained in the following compilation are either obvious deductions from those of Euclid, or such as exhibit some remarkable properties of lines, angles, or figures, which are not to be found in Euclid's work; or, lastly, they are the geometrical solutions of many well-known problems in the different branches of Natural Philosophy. But although the propositions, which have here been collected for the use of the academical student, are of these three kinds, it has not been thought advisable to class them according to that threefold division. Designed as a supplement to the Elements of Euclid, they have been disposed according to Euclid's arrangement. And

not only have those which constitute the first book been made to depend upon the first book of the Elements, and so on; but the propositions in each separate book will also be found to follow the order of the propositions of the corresponding book of Euclid. There is no necessity, therefore, for the student to wait until he has gone through Euclid's Elements, before he enters upon the perusal of this Supplement. It will, perhaps, be more to his advantage to read the original work and this, which is principally intended to supply its deficiencies, together; especially if he has the assistance of a tutor, who will point out to him those propositions which may be considered as best deserving his attention. Some regard has, indeed, been paid to the probability of such a plan being thought worthy of adoption, in the distribution of the matter of this present publication. An endeavour has been made to offer something to the notice of the

reader, after almost every one of the most important propositions, in each of the books of Euclid's Elements: so that, supposing him not to advance beyond the first book, or beyond the first four books, of Euclid, a field, more or less contracted, is still open to his research, for the exploring of which he will find himself already sufficiently furnished with previous knowledge. With this view, especially, many of the following theorems and problems, which might undoubtedly have been demonstrated more concisely, if they had been put after Euclid's fifth book, have had a place assigned to them nearer to the beginning. For thus is the learner shewn how extensive an application may be made of some of the easiest propositions of Geometry; and thus is a scope afforded to the study of those, who cannot at first encounter, without reluctance, the somewhat abstruse reasonings, upon which the ancients, with so

much acuteness and solidity of judgment, have founded the doctrine of proportionality.

In order to facilitate the execution of the plan here recommended, an index has been constructed, by means of which the Geometry of this Supplement may be incorporated, as it were, with that of Euclid, and the reading of both the treatises may be made to go on together.

In the demonstrations of the propositions recourse has been had to symbols. But these symbols are merely the representatives of certain words and phrases, which may be substituted for them at pleasure, so as to render the language employed strictly conformable to that of ancient Geometry. The consequent diminution of the bulk of the whole book is the least advantage which results from this use of symbols. For the demonstrations themselves are sooner read and more easily comprehended by means

of these useful abbreviations, which will, in a short time, become familiar to the reader, if he is not already perfectly well acquainted with them.

It appeared to be unnecessary to print the formal and logical conclusion which belongs to every geometrical demonstration, and which consists in repeating the enuntiation of the proposition which was to be proved, and in asserting that it has been proved. This last step, is, therefore, left for the reader in all cases mentally to supply. And if some omissions of a weightier kind, and some errors, be discoverable in the following pages, it is hoped that they will be found neither too great, nor too many to be forgiven, if the genéral plan of the work meet with the approbation of those who are competent to decide upon it.

Trinity College, April 27th, 1819.

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