1 THE ELEMENTS OF EUCLID, VIZ. THE FIRST SIX BOOKS, WITH THE ELEVENTH AND TWELFTH. IN WHICH THE CORRECTIONS OF DR SIMSON ARE GENERALLY ADOPTED, BUT ALSO, SOME OF EUCLID'S DEMONSTrations are RESTORED, OTHERS MADE TOGETHER WITH ELEMENTS OF PLANE AND SPHERICAL TRIGONOMETRY, AND A TREATISE ON PRACTICAL GEOMETRY. BY ALEXANDER INGRAM, PHILOMATH. EDINBURGH: PRINTED BY J. PILLANS & SONS; AND SOLD BY W. CREECH, EDINBURGH; W. COKE, LEITH; J. BURNET, R, & E, MERCIER, DUBLIN; AND W. MAGEE, BELFast. 1799. 7-6-35. HOP Bowes 8-22-33 30809 DR SIMSON's 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 Demonstrations, to Theon; others think the Propofitions to be Euclid's, but that the Demonstrations 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 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, (p. 13. Prælect.) "That, excepting a แ very few Interpolations, Explications, and Ad66 ditions, 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 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 fmall moment, efpecially in the Fifth and Eleventh Books of the Elements, which this Editor has greatly vitiated; for inftance, by fubftituting a fhorter, but in fufficient Demonftration of the 18th Propofition 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 embarrass 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 eafy. Befides, among the 1 ! the Definitions of the 11th Book, there is this, which is the 10th, viz. " Equal and fimilar folid 46 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 Propofition, it ought to have been demonftrated. But, indeed, this Propofition, which makes the roth 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 cases, 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 manin the Demonstration of the 26th Propofition 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 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, though it be quite neceffary there fhould be one. Now, upon the 10th Definition ner, of |