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...
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