THE

ELEMENTS

EUCLI D.

VIZ.

THE FIRST SIX BOOKS,

TOGETHER WITH THE

ELEVENTH AND TWELFTH.

The Errors, by which THEON, or others, have long ago
vitiated these Books, are corrected,

And fome of EUCLID'S Demonftrations are restored."

ALSO

THE BOOK OF

EUCLID'S DATA,

İn like manner corrected.

BY

ROBERT SIMSON, M. D.

Emeritus Profeffor of Mathematics in the University of Glasgow.
To this NINTH EDITION are also annexed

ELEMENTS of PLAIN and SPHERICAL TRIGONOMETRY.

EDINBURG H:

PRINTED FOR E. WINGROVE, LONDON,

AND J. BALFOUR, EDINBURGH.

M, DCC, XCIII.

PREFACE

THE Opinions of the Moderns concerning the Author of

of Geometry, 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 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 present 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 supposed, and those not of small moment, especially in the Fifth and Eleventh Books of the Elements, which this Editor has greatly vitiated; for inftance, by fubftituting a fhorter, but infufficient 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 given an abfurd one in place of it in the 5th Definition of the 6th Book, which neither Euclid, Archimedes, Appolonius, 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 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