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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 wellknown 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, those, which constitute the first Part, have been disposed according to Euclid's arrangement. And not only have the propositions contained in 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 be found, also, 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

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Euclid's Elements, before he enters upon the perusal of the first Part 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 theorems and problems which 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 propositions, 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 simplest elements 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, bare 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 the first Part 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 Second Part, which has been added to this edition, the arrangement has not been made to follow that of Euclid. This part may, therefore, be considered as a separate collection of geometrical propositions, promiscuous in its composition, but yet admitting of a certain degree of classification, which the reader will find adopted in it. Its last two Books relate to subjects, which are almost wholly omitted in the greater number of treatises on Geometry, but which are not deficient in interest.

A general Index to the whole work has, also, been annexed to this second edition, exhibiting

, the enunciations of all its Propositions, apart from their proofs ; in order that the student may use it, notas mere table of contents only, but as manual problems; not having recourse to the printed demonstrations, until he has exercised his own ingenuity in discovering solutions.


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

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 beforehand 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 enunciation 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 general plan of the work meet with the approbation of those who are competent to decide upon it.




Shening the Order in which the Propositions of the first part of

the following Supplement may be read along with the Pro-
positions contained in Euclid's Elements.



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