Page:The Elements of Euclid, viz. the first sixs books, together with eleventh and twelfh.djvu/8

 PREFACE.

and as this Definition did much embarrafs Beginners, and is quite ufc- lefs, it is now thrown out of the Elements, and another which with- out doubt Euclid had given, is put in its proper place among the De- finitions of the j lh Book, by which the Doctrine of Compound Ratios is rendered plain and eafy. Bcfides, among the Definitions of the 11 th Book, there is this, which is the 10 th, viz. “ Equal and fimilar folid “ figures are thofe which are contained by fimilar planes of the fame “ number and magnitude.” Now this Propofition is a Theorem, not a Definition, becaufe the equality of figures of any kind mull be dc- monllrated, and not aflumed. and therefor, tho’ this were a true Pro- pofition, it ought ,c * ^, * vc b^vn demonftrated. Rut thk Propo-

fltion, which makes the 1 o lh Definition of the 1 1 ^ Book, is not true univerfally, except in the cafe in which each of the folid angles of the figures is contained by no more than three plane angles; for, in other _ odes, two folid figures may be contained by fimilar planes of the fame number and magnitude, and yet be unequal to one another; as fhall be made evident in the Notes fubjoined to thefe Elements. In the like manner, in the Dcmonftration of the 2 6 1,1 Prop, of the 11 th Book, it is taken for granted, that thofe folid angles are equal to one another which are contained by plane angles of the fame number and magnitude placed in the fame order; but neither is this univerfally true, except in the cafe in which the folid angles arc contained by no more than three plane angles; nor of this cafe is there any Demonftration in the Elements we now have, tho’ it be quite neccflary there fhould be one. Now upon the 1 o * Definition of this Book depend the 2 j ,h and 2 8 lh Propofitions of it; and upon the 25 th and 26 th de- pend other eight, viz. the 27 th, 31®, 32^ 33 d , 34 th , 36* 37* and 4 o <h of, die fame Book, and the 1 2 * of the t2 dl Book de- pends upon the 8 th of the fame, and this 8 th, and the Corollary of

Propofidon