Hodgson 7-5:39 138627 THE Opinions of the Moderns concerning the Author of the Elements of Geometry which go under Euclid's Name, are , “ similar “ similar planes of the same number and magnitude.” Now this Proposition is a Theorem, not a Definition, because the equality of figures of any kind must be demonstrated, and not assumed and therefore, tho' this were a true Propofition, it ought to have been demonstrated. But indeed this Propofition, which makes the roth Definition of the i Ith Book, is not true universally, except in the case in which each of the folid angles of the figures is contained by no more than three plane angles; for, in other cases, two folid figures may be contained by similar planes of the fame number and magnitude, and yet be unequal to one another; as fhall be made evident in the Notes fubjoined to these Elements. In like manner, in the Demonstration of the 20th Prop. of the Ith Book, it is taken for granted, that those folid angles are equal to one another which are contained by plane angles of the fame number and magnitude placed in the same order; but neither is this universally true, except in the case in which the folid angles are contained by no more than three plane angles; nor of this case is there any Demonstration in the Elements we now have, tho' it be quite neceílary there should be one. Now upon the roth Definition of this Book depend the 25th and 28th Propositions of it; and upon the 25th and 26th depend other eight, viz. the 27th, 31st, 3 d, 33d, 3 34th, 36th, 37th, and 40th of the fame Books and the 12th of the 12th Book depends upon the 8th of the same, and this 8th, and the Corollary of Proposition 17th, and Prop. 1 8th of the 12th Book depend upon the 9th Definition of the 11th Book, which is not a right Definition, because there may be folids contained by the fame number of similar plane figures, which are not similar unto one another, in the true sense of similarity received by all Geometers. and all these Propositions have, for these reasons, been infufficiently demonstrated since Theon's time hitherto. Besides, there are several other things, which have nothing of Euclid's accuracy, and which plainly shew that his Elements have been much corrupted by unskilful Geometers. and tho' these are not so gross as the others now mentioned, they ought by no means to remain uncorrected. Upon these Accounts it appeared necessary, and I hope will prove acceptable to all Lovers of Accurate Reasoning and of Mathematical Learning, to remove such blemishes, and restore the principal Books of the Elements to their original Accuracy, as far as I was able; especially since these Elements are the foundation of a Sci ence ence by which the Investigation and Discovery of useful Truths, at least in Mathematical Learning, is promoted as far as the limited Powers of the Mind allow; and which likewise is of the greatest Use in the Arts both of Peace and War, to many of which Geometry is absolutely necessary. This I have endeavoured to do by taking away the inaccurate and false Reasonings which unskilful Editors have put into the place of some of the genuine Demonstrations of Euclid, who has ever been justly celebrated as the most accurate of Geometers, and by restoring to him those Things which Theon or others have suppressed, and which have these many ages been buried in Oblivion. In this second Edition Ptolomy's Proposition concerning a property of quadrilateral figures in a circle is added at the end of the sixth Book. Also the Note on the 29th Prop. Book ift is altered, and made more explicit. And a more general Demonstration is given instead of that which was in the Note on the roth Definition of Book 11th. besides the Translation is much amended by the friendly assistance of a learned Gentleman. |