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Hist. of sci. - Spec.

Thin

9-17-37

34691 PREFACE

THE 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 Propofitions, 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 Propositions and their Demonstrations 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 since 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 present Greek Text, by adding some things, fuppreffing others, and mixing his own with Euclid's Demonstrations, had changed more things to the worse than is commonly supposed, and those not of small 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 Demonstration 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, besides 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, 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 embarafs 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 gth 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 fimilar folid Figures are those 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 must be demonftrated, and not affumed; and, therefore, though this were a true Proposition, it ought to have been demonstrated. But indeed this Proposition, which makes the roth Definition of the 11th Book, is not true universally, except in the case 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 subjoined to these Elements. In like manner, in the Demonstration of the 26th Prop. of the 11th Book, it is taken for granted, that those solid Angles are equal to one another which are contained by plain 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 solid Angles are contained by no more than three plain Angles; nor of this Cafe is there any Demonftration in the Elements we now have, though it be quite necessary there should be one. Now, upon the roth 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 Book, 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 similar to one another, in the true Senfe of Similarity received by all Geometers; and all these Propofitions have, for these Reasons, been infufficiently demonstrated 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, though these are not fo gross as the others now mentioned, they ought by no means to remain un

corrected.

Upon these Accounts it appeared neceffary, and I hope will prove acceptable to all Lovers of accurate Reasoning, and of Ma

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 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 absolutely neceffary. This I have endeavoured to do, by taking away the inaccurate and false Reasonings which unskilful Editors have put into the place of some of the genuine Demonstrations of Euclid, who has ever been justly celebrated as the most accurate of Geometers, and by restoring to him those Things which Theon or others have suppressed, and which have these many Ages been buried in Oblivion.

In this fifth Edition, Ptolemy's Propofition concerning a Property of quadrilateral Figures in a Circle is added at the End of the xth Book. Also the Note on the 29th Prop. Book ist, 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; befides, the Translation is much amended by the friendly Assistance 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.