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 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, not as a 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 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. Vicarage-House, |