PREFACE.

THE object of this edition of the ELEMENTS of EUCLID is to present an accurate translation of the twelve books, upon such a plan, and with such illustration, as may facilitate the advancement of the student.

In the accomplishment of the latter object, the Editor has inserted a number of deductions at the end of their respective propositions, by way of exercises in developing the powers of genius and inquiry; and he hopes that, as the student performs these in his progress through the work, they will serve to render the subject of Elementary Geometry more familiar to his mind. In the selection, the Editor has taken some from Cresswell, Bland, &c. and added others of his own; and he trusts it is made so as to meet the approbation of the public in general. There are also algebraic demonstrations annexed to the second and fifth books; for in these the Editor believes that analysis is generally employed as well in the Universities as in military and naval institutions; and, considering the facility which it affords, and

the simplicity of its operations, no wonder that such a system should be universally adopted.

When the Editor first thought of undertaking the work, he purposed to make his translation from the Oxford copy, edited by Dr. David Gregory in 1703; but, after consulting the edition recently published at Paris under the superintendance of Peyrard, and reading the lectiones variantes of that work, the Editor fully resolved to make his translation from it: first, because it came out under the strongest recommendations of the best mathematicians on the Continent, such as Lagrange, Legendre, &c.; and, secondly, because the learned M. Peyrard himself bestowed the greatest labour in examining and collating all the existing MSS. and oldest editions.

The Editor has bestowed the greatest care in the execution of his undertaking; he has availed himself of the assistance of several eminent mathematicians; and he trusts that the public, in reviewing his labours, will, after an impartial criticism, be enabled to bestow upon him some commendation, the only reward which he can hope to receive.

INTRODUCTION.

THE term Geometry is derived from two Greek words, which literally signify the art of measuring the earth: it does not, however, so much imply the ascertaining the measure of the whole globe as that of certain parts of its surface; and hence we are informed by historians that the finding of the dimensions of lands, and other plane figures, with some of the most simple and obvious methods of determining their contents and relative proportions, were the first uses made of this science by the ancients. It has, however, since been extended to numberless other speculations; insomuch that, together with analysis, Geometry forms the prin cipal foundation of all the mathematics.

Like many other arts and sciences, the origin of Geometry is involved in considerable obscurity, some authors fixing it at one period, and others at another. Most, however, assign Egypt for its birth-place, and that the annual inundations of the Nile first excited attention to this science among the inhabitants of that nation; for the waters bearing away the boundaries of the land, in the lower and most fertile parts of the country, and laying waste their estates, the people were obliged to devise some method for ascertaining the

property of each person after the waters had subsided, and to establish it upon principles that would serve as a guide to posterity.

Herodotus, however, the first who wrote history in prose, assigns its origin to a different cause. The following is the account he himself gives of what he learned respecting it at Thebes and Memphis: "I was told," says he, "that Sesostris divided the kingdom among all his subjects, and that he had given each an equal quantity of land, on condition of paying annually a proportionate tribute. If the portion allotted to any one were diminished by the river, he went to the king and told him of what had happened; the king then sent and ordered the land to be measured, that he might know what diminution it had undergone, and demand a tribute only in proportion to what remained. Here, I believe," adds Herodotus, "Geometry first took its rise, and that hence it was transmitted to the Greeks."

If we wished, says Bossut, to indulge in frivolous conjectures, we should carry back the origin of Geometry to the invention of the square and compasses, since it makes the greatest use of them in practice; but the same argument of their use, continues he, will lead us to suppose that these instruments were invented at the commencement of society. Indeed some such instrument must have been used in the earliest ages of the world, as the rudest operations of nature could not be effected without them. But if we fix the period when Geometry began to assume the character of