Hodgson 7-5-39 38627 TH 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 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 Demonstrations are Euclid's own. John Buteo and Sir Henry Savile are the Authors of greatcft 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 (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 fuppofed, and thofe not of fmall moment, efpecally in the Fifth and Eleventh Books of the Elements, which this Editor has greatly vitiated. for inftance, by fubflituting 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, Apollonius, nor any Geometer before Theon's Time, ever made ufe of, and of which there is not to be found the least appearance in any of their Writings. and as this Definition did much embarrafs Beginners, 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 rendered 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 those which are contained by "fimilar 66 "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 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 man- . ner, 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 plane angles of the fame number and magnitude placed in the fame order; but neither is this univerfally 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 Demonftration in the Elements we now have, tho' it be quite neceflary there fhould 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, 31st, 3 d, 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 17 th, 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 fenfe of fimilarity received by all Geometers. and all thefe Propofitions have, for thefe reafons, 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 tho' these 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 Mathematical Learning, to remove fuch blemishes, and restore 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 Sci ence ence by which the Investigation and Discovery of useful Truths, at leaft in Mathematical Learning, is promoted as far as the limited Powers of the Mind allow; and which likewife is of the greatest Ufe in the Arts both of Peace and War, to many of which Geometry is abfolutely neceffary. This I have endeavoured to do by tak ing 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 thefe many ages been buried in Oblivion. In this fecond Edition Ptolomy's Propofition concerning a pròperty of quadrilateral figures in a circle is added at the end of the fixth Book. Alfo the Note on the 29th Prop. Book 1ft 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 affiftance of a learned Gentleman. |