The School Edition. Euclid's Elements of Geometry, the First Six Books, by R. Potts. Corrected and Enlarged (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1850 Excerpt: ... 21 is to A," as a to rf; therefore A is to /, as m top. (v. 11.) Q.e.d. The propositions G and K are usually, for the sake of brevity, expressed in the same terms with propositions F and H: and therefore it was proper to shew the true meaning of them when they are so expressed; especially since they are very frequently made use of by geometers. NOTES TO BOOK V. In the first four Books of the Elements are considered, only the absolute equality and inequality of Geometrical magnitudes. The Fifth Book contains an exposition of the principles whereby a more definite comparison may be instituted of the relation of magnitudes, besides their simple equality or inequality. Def. i, Ii. In the first four Books the word part is used in the same sense as we find it in the ninth axiom, "The whole is greater than its part: " where the word part means any portion whatever of any whole magnitude: but in the Fifth Book, the word part is restricted to mean that portion of magnitude which is contained an exact number of times in the whole. For instance, if any straight line be taken two, three, four, or any number of times another straight line, by Euc. i. 3; the less line is called a part, or rather a submultiple of the greater line; and the greater, a multiple of the less line. The multiple is composed of a repetition of the same magnitude, and these definitions suppose that the multiple may be divided into its parts, any one of which is a measure of the multiple. And it is also obvious that when there are two magnitudes, one of which is a multiple of the other, the two magnitudes must be of the same kind, that is, they must be two lines, two angles, two surfaces, or two solids: thus, a triangle is doubled, trebled, &c., by doubling, trebling, &c. the...