PREFACE. THE 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 Propofitions, as well as their Demonftrations, to Theon; others think the Propofitions 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 laft, 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 feveral Arguments he brings to prove it, makes this Conclufion (Page 13. Praelect) "That, excepting a very few Interpolations, Explications, 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 Demonstrations, had changed more things to the worse than is commonly fuppofed, and those not of imall moment, especially in the Fifth and Eleventh Books of the Elements, which this Editor has greatly vitiated; for instance, by fubftituting a fhorter, 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 abfurd one in place of it in the 5th Definition of the 6th Book, which neither Euclid, Archimedes, Appollonius, 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 embarafs Eeginners, and is quite ufelefs, 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 ren dered plain and eafy. Befides, among the Definitions of the 11th Book, there is this, which is the 10th, 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; because the equality of Figures of any kind muft be demonftrated, and not affumed; and, therefore, though this were a true Propofition, it ought to have been demonftrated. But, indeed, this Propofition, 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 fhall be made evident in the Notes fubjoined to thefe Elements. In like manner, in the Demonftration of the 26th Prop. of the 11th Book, it is taken for granted, that thofe folid Angles are equal to one another which are contained by plain Angles of the fame Number and Magnitude, placed in the same Order; but neither is this univerfally true, except in the cafe in which the folid Angles are contained by no more than three plain Angles; nor of this Cate is there any Demonftration in the Elements we now have, though it be quite neceffary there fhould be one. Now, upon the toth Definition of this Book depend the 25th and 28th Propofitions 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 eighth of the fame, and this 8th, and the Corollary of Propofition 17th, and Prop. 18th of the 12th Bcok, depend upon the 9th Definition of the 11th Book, which is not a right Definition; because there may be Solids contained by the fame number of fimilar plane Figures, which are not fimilar to one another, in the true Senfe of Similarity received by all Geometers; and all these Propofitions have, for thefe Reatons, been infufficiently demonftrated fince 'Theon's time hitherto. Befides, there are several 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 thefe Accounts it appeared neceffary, and I hope will prove acceptable to all Lovers of accurate Reafoning, and of Ma Mathematical Learning, to remove fuch Blemishes, and reflore the principal Books of the Elements to their original Accuracy, as far as I was able; efpecially fince thefe Elements are the Foundation of a Science by which the Investigation and Discovery of ufeful Truths, at least in Mathematical Learning, is promoted as far as the limited Powers of the Mind allow; and which likewife is of the greateft Ufe 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 falfe Reafonings which unfkilful Editors have put into the place of fome of the genuine Demonstrations of Euclid, who has ever been juftly celebrated as the most accurate of Geometers, and by reftoring to him thofe Things. which Theon or others have fuppreffed, and which have these many ages been buried in Oblivion. In this Sixth Edition, Ptolemy's Propofition concerning a Property of quadrilateral Figures in a Circle is added at the End of the fixth Book. Alfo 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 10th Definition of Book 11th; besides, the Translation is much amended by the friendly Affiftance of a learned Gentleman. To which are alfo added, the elements'of Plane and Spherical Trigonometry, which are commonly taught after the Elements of Euclid. |