of this Book depend the 25th and 28th Propofi tions of it; and upon the 25th and 26th depend other eight, viz. the 27th, 31st, 32d, 33d, 34th, 36th, 37th, and 40th of the fame Book; and the 12th of the 12th Book depends upon the 8th of the fame; and this 8th, and the Corollary of Proposition 17th and Propofition 18th 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 fimilar plane figures, which are not fimilar to one another, in the true sense of fimilarity received by all Geometers; and all these Propofitions have, for these reasons, been infufficiently demonftrated fince Theon's time hitherto. Befides, 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, though 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 fuch blemishes, and restore the principal Books of the Elements to their original accuracy, as far as I was able; especially fince these Elements are the foundation of a Science 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 likewife 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 fome 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. PRE THE PREFACE. I Nthe preceding Preface, Dr Simfon has shown how much the Elements of Euclid have fuffered from the Greek Editors; and in the Work, he has corrected many errors, and restored feveral of Euclid's Demonstrations; by which means, the Elements are in a great measure restored to their original accuracy. But there are some things of great importance overlooked by him, which need correction; and others, though corrected, are not restored to their original accuracy, because his corrections are less extenfive than the blemishes, or are not adapted to Euclid's defign. For instance, he did not observe, that the Demonstration of the 28th Propofition of the Eleventh Book was infufficient, though that Propofition be the foundation of the principal part of folid Geometry; and in correcting the 26th of the fame Book, he overlooked the defign of the Propofition, and inftead of changing the Enunciation, as he ought to have done, he attempted to accommodate the Demonftration to the Enunciation, as it is in the Greek, in which he did not fucceed: likewife, in correcting the Definition of fimilar folids, he has gone too far from the text, and changed the order of the Definitions, A tions, which he would have had no occasion to do, if he had properly attended to Euclid. Again, he very properly changed the Demonstration of the 13th Propofition of the Third Book, but he did not take notice, that the alteration he complains of there, is only one of a series of alterations made in every Proposition from the 9th to it: and in the same manner, when he corrected the 5th Propofition of the Fourth Book, he did not observe, that the want which he blames in it, is common to it, with many other Propositions throughout the Elements. In the Notes, it shall be made evident, that some corrections are necessary in all these instances. They are accordingly corrected here, together with feveral other er rors. To attempt fuch alterations as these, does not feem to need an apology; their neceffity and usefulness are fufficiently obvious; and in making them, the author walks in a beaten path. But there is another class of alterations introduced, that is, the explanation of obfcurities, which, though not less useful, are not thought to be so necessary as the former. As to these, it seems to be enough, if the expreffion be more perfpicuous than before, and no other objection lie against it than what lay against the former; and this, it is hoped, is the cafe at present. Thus, the Enunciations of the 7th Proposition of the First Book, and of the 27th, 28th, and 29th of the Sixth Book, are changed; as are also the second Definition of the Sixth Book, and the 5th and 7th Definitions of the Fifth Book, befides several others of less importance. Now, in all these places mentioned, the literal tranflation from the Greek is acknowledged to be very obfcure, so that an alteration can scarcely be objected to: and the meaning of the present 1 |