THE

ELEMENTS

OF

EUCLID;

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 SOME of

EUCLID'S DEMONSTRATIONS ARE RESTORED.

ALSO THE BOOK OF

EUCLID'S DATA,

IN LIKE MANNER CORRECTED.

By ROBERT SIMSON, M. D.
Emeritus Professor of Mathematics in the University of Glasgow..

THE FOURTEENTH EDITION,

CAREFULLY REVISED AND IMPROVED,

TO WHICH IS ADDED,

A TREATISE ON THE CONSTRUCTION OF THE TRIGONOMETRICAL
CANON; AND A CONCISE ACCOUNT OF LOGARITHMS.

LONDON:

PRINTED FOR F. WINGRAVE; J. JOHNSON; F. C. AND J. RIVINGTON;
J. WALKER; SCATCHERD AND LETTERMAN; WILKIE AND ROBINSON;
LONGMAN, HURST, REES, AND ORME; CADELL AND DAVIES; J.
MAWMAN; C. LAW; J, RICHARDSON; AND J. M. RICHARDSON.

$861454

516.209(P)

Printed by S. Hamilton, Weybridge.

то

THE KING,

THIS EDITION

OF THE

PRINCIPAL BOOKS

OF THE

ELEMENTS OF EUCLID,

AND OF THE

BOOK OF HIS DATA,

IS MOST HUMBLY DEDICATED,

BY

HIS MAJESTY'S

MOST DUTIFUL,

AND MOST DEVOTED

SUBJECT AND SERVANT,

ROBERT SIMSON.

PREFAC E.

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 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 Propositions and their Demonstrations are Euclid's own. John Buteo and Sir Henry Savile are the authors of greatest note who assert 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 several arguments he brings to prove it, makes this conclusion (Page 13. Prælect.) "That, excepting a very few interpolations, expli"cations, and additions, Theon altered nothing in Euclid." But, by often considering 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 editor of the present Greek text, by adding some things, suppressing others, and mixing his own with Euclid's Demonstrations, has 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, Apollonius, 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 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