P R E F A C E. 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 ascribes the Propositions, as well as their Demonstrations, to Theon; others think the Propositions to be Euclid's, but that the Demonstrations are Theon's; and others maintain that all the PropoGtions and their Demonstrations are Euclid's own. John Buteo and Sir Henry Savile are the Authors of greatest Note who affert this latt, and the greater part of Geometers have ever since 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 Con. clusion (Page 13. Praelect ) " That, excepting a very few " Interpolations, Explications, and Additions, Theon altered “ nothing in Euclid.” But, by often conûdering and comparing together the Definitions and Demonstrations as they are in the Greek Editions we now have, I found that Theon, or whoever was the Eaitor of the present Greek, Text, by adding some things, suppressing others, and mixing his own with Euclid's Demonstrations, had changed more things to the worse than is commonly supposed, 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 substituting a shorter, but insufficient 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, belides 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 oth Book, which neither Euclid, Archimedes, Appollonius, nor any Geometer before Theon's time, ever made use of, and of which there is not to be found the least appearance in any of their Writings; and, as this Definition did much embarass Eeginners, 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 sch 5th Book, by which the Doctrine of Compound Ratios is ren. dered plain and easy. Besides, among the Definitions of the 11th Book, there is this, which is the 10th, viz.“ Equal “ and similar folid Figures are those which are contained by « Gmilar Planes of the fame Number and Magnitude.” Now, this Proposition is a Theorem, not a Definition ; because the equality of Figures of any kind must be demonstrated, and not assumed; and, therefore, though this were a true Proposition, it ought to have been demonstrated. But, indeed, this Propofition, which makes the Toth Definition of the uth Book, is not true universally, except in the case in which each of the solid angles of the Figures is contained by no more than three plane Angles; for, in other Cases, two folid Figures may be contained by similar Planes of the same Number and Mag. nitude, and yet be unequal to one another; as shall be made evident in the Notes, subjoined to these Elements. In like manner, in the Demonftration of the 26th Prop. of the isth Book, it is taken for granted, that those solid Angles are equal to one another which are contained by plain Angles of the same Number and Magnitude, placed in the same Order ; but neither is this universally true, except in the cate in which the folid Angles are contained by no more than three plain Angles; nor of this Cate is there any Demonstration in the Elements we now have, though it be quite neceffary there should be one. Now, upon the oth Definition of this Book depend the 25th and 28th Propositions of it; and, upon the 25th and 26th depend other eight, viz. the 27th, 3iit, 320, 33d, 34th, 36th, 37th, and 40th of the same Book; and the 12th of the 12th Book depends upon the eigbth of the same, and this 8th, and the Corollary of Propofition 17th, and Prop. 18th of the 12th Bcok, depend upon the oth 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 similar to one another, in the true Sense of si. milarity received by all Geometers; and all these Propofitions have, ior these Reaions, been insufficiently demonstrated tince Theon's time hitherto. Befides, there are several other things, which have nothing of Euclid's accuracy, and which plainly thew, 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 tbefe Accounts it appeared necesary, and I hope will prove acceptable to all Lovers of accurate Realoning, and of Ma. Mathematical Learning, to remove such Blemishes, and reflore the principal Books of the Elements to their original Accuracy, as far as I was able; especially since 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 likewise is of the greatest Use in the Arts both of Peace and War, to many of which Geometry is abso. Jutely neceffary. This I have endeavoured to do, by taking a. way the inaccurate and false Reasonings which unskiltul Edi. tors have put into the place of some of the genuine Demonstrations of Euclid, who has ever been juftly celebrated as the most accurate of Geometers, and by restoring to him those Things which Theon or others have luppressed, and which have these many ages been buried in Oblivion. In this Sixth Edition, Ptolemy's Proposition concerning a Pro. perty of quadrilateral Figures in a Circle is added at the End of the fixth Book. Also the Note on the 29th Prop. Book ist, is altered, and made more explicit, and a more general Demonftration is given, instead of that which was in the Nore on the Toth Definition of Book rith; besides, the Translation is much amended by the friendly Aslistance of a learned Gentleman. To which are also added, the elements of Plane and Spherical Trigonometry, which are commonly taught after the Elements of Euclid. |