入 Hodgaon 7-5-39 38627 T PREFACE. HE Opinions of the Moderns concerning the Author of the Elements of Geometry which go under Euclid's Name, are very different and contrary to one another. Peter Ramus afcribes the Propositions, as well as their Demonftrations, to Theon; others think the Propositions to be Euclid's, but that the Demonftrations are Theon's; and others maintain that all the Propofitions and their Demonftrations are Euclid's own. John Buteo and Sir Henry Savile are the Authors of greatest Note who affert this last, and the greater part of Geometers have ever fince been of this Opinion, as they thought it the most probable. Sir Henry Savile, after the several Arguments he brings to prove it, makes this Conclufion (Pag. 13. Praelect.) That excepting a very few Interpolations, Explicati ons and Additions, Theon altered nothing in Euclid." But, by often confidering and comparing together the Definitions and Demonftrations as they are in the Greek Editions we now have, I found that Theon, or whoever was the Editor of the prefent Greek Text, by adding fome things, fuppreffing others, and mixing his own with Euclid's Demonftrations, had changed more things to the worse than is commonly supposed, and those not of small moment, efpecally in the Fifth and Eleventh Books of the Elements, which this Editor has greatly vitiated. for instance, by substituting a shorter, but infufficient Demonftration of the 18th Prop. of the 5th Book, in place of the legitimate one which Euclid had given ; and by taking out of this Book, befides other things, the good Definition which Eudoxus or Euclid had given of Compound Ratio, and giving an absurd one in place of it in the 5th Definition of the 6th Book, which neither Euclid, Archimedes, Apollonius, nor any Geometer before Theon's Time, ever made ufe of, and of which there is not to be found the leaft appearance in any of their Writings. and as this Definition did much embarrafs Beginners, and is quite useless, it is now thrown out of the Elements, and another which without doubt Euclid had given, is put in its proper place among the Definitions of the 5th Book, by which the Doctrine of Compound Ratios is rendered plain and easy Besides, among the Definitions of the 11th Book, there is this, which is the 10th, viz. Equal and fimilar folid figures are those which are contained by " fimilar fimilar planes of the fame number and magnitude." Now this Propofition is a Theorem, not a Definition, because the equality of figures of any kind must be demonftrated, and not affumed. and therefore, tho' this were a true Propofition, it ought to have been demonftrated. But indeed this Proposition, which makes the 10th Definition of the 11th 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 cafes, two folid figures may be contained by fimilar planes of the fame number and magnitude, and yet be unequal to one another; as shall be made evident in the Notes fubjoined to these Elements. In like manner, in the Demonftration of the 26th Prop. of the 11th 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 fame order; but neither is this universally true, except in the cafe in which the folid angles are contained by no more than three plane angles; nor of this cafe is there any Demonstration in the Elements we now have, tho' it be quite necessary there should be one. Now upon the 10th Definition of this Book depend the 25th and 28th Propofitions of it; and upon the 25th and 26th depend other eight, viz. the 27th, 31ft, 3d, 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 Prop. 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 unto one another, in the true fense of similarity received by all Geometers. and all these Propofitions have, for these reasons, been infufficiently demonftrated fince Theon's time hitherto. Besides, there are feveral 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 fince 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 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 Demonftrations of Euclid, who has ever been justly celebrated as the most accurate of Geometers, and by reftoring to him those Things which Theon or others have fuppressed, and which have these many ages been buried in Oblivion. In this second Edition Ptolomy's Propofition concerning a pròperty of quadrilateral figures in a circle is added at the end of the fixth Book. Also the Note on the 29th Prop. Book 1st is altered, and made more explicit. And a more general Demonftration is given instead of that which was in the Note on the 10th Definition of Book 11th. befides the Tranflation is much amended by the friendly affistance of a learned Gentleman. |