of this Book depend the 25th and 28th Propofi→ tions of it; and upon the 25th and 26th depend other eight, viz. the 27th, 31ft, 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 Propofition 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 fense 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 feveral other things which have nothing of Euclid's accuracy, and which plainly fhew, that his Elements have been much corrupted by unskilful Geometers; and, though thefe are not fo grofs as the others now mentioned, they ought by no means to remain uncorrected. Upon these accounts, it appeared neceffary, and I hope will prove acceptable to all lovers of accurate reasoning, and of Mathematical learning, to remove fuch blemishes, and reftore 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 abfolutely neceffary. This I have endeavoured to do, by taking away the inaccurate and false reasonings which unfkilful Editors have put into the place of fome of the genuine Demonftrations of Euclid, who has ever been justly celebrated as the most accurate of Geometers, and by reftoring to him thofe things which Theon or others have fuppreffed, and which have thefe 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 reftored feveral of Euclid's Demonstrations; by which means, the Elements are in a great measure restored to their original accuracy. But there are fome things of great importance overlooked by him, which need correction; and others, though corrected, are not reftored to their original accuracy, becaufe his corrections are lefs extenfive than the blemishes, or are not adapted to Euclid's defign. For inftance, he did not observe, that the Demonftration 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 DefiniA tions, tions, which he would have had no occafion to do, if he had properly attended to Euclid. Again, he very properly changed the Demonftration 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 Propofition from the 9th to it; and in the fame manner, when he corrected the 5th Propofition of the Fourth Book, he did not obferve, that the want which he blames in it, is common to it, with many other Propofitions throughout the Elements. In the Notes, it fhall be made evident, that fome corrections are necessary in all these inftances. They are accordingly corrected here, together with feveral other er rors. To attempt fuch alterations as thefe, 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 clafs of alterations introduced, that is, the explanation of obfcurities, which, though not lefs ufeful, are not thought to be fo neceffary as the former. As to thefe, it feems to be enough, if the expreffion be more perfpicuous than before, and no other objection lie againft it than what lay against the former; and this, it is hoped, is the cafe at prefent. Thus, the Enunciations of the 7th Propofition of the First Book, and of the 27th, 28th, and 29th of the Sixth Book, are changed; as are alfo the fecond Definition of the Sixth Book, and the 5th and 7th Definitions of the Fifth Book, befides feveral others of lefs importance. Now, in all thefe places mentioned, the literal tranflation from the Greek is acknowledged to be very obfcure, fo that an alteration can fcarcely be objected to: and the meaning of the prefent |