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Besides the alterations in the enunciation of the propositions, others of considerable importance have also been made in the present edition. The proposition in Book V., which proves that a polygon and circle may be made to coincide so nearly, as to differ from each other by less than any assignable quantity, has been taken from the Edinburgh Encyclopedia. It is proved in the corollaries that a polygon of an infinite number of sides becomes a circle, and this principle is made the basis of several important demonstrations in Book VIII.

Book II.,on Ratios and Proportions, has been partly adopted from the Encyclopedia Metropolitana, and will, it is believed, supply a deficiency in the original work.

Very considerable alterations have also been made in the manner of treating the subjects of Plane and Spherical Trigonometry. It has also been thought best to publish with the present edition a table of logarithms and logarithmic sines, and to apply the principles of geometry to the mensuration of surfaces and solids. Military Academy,

West Point, March, 1834.

AN INDEX

SHOWING THE PROPOSITIONS OF LEGENDRE WHICH CORRESPOND TO THE PRINCIPAL PROPOSITIONS OF THE FIRST SIX BOOKS OF EUCLID.

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