ters V, 2; and before Commandine, the learned John Dee in the Book XII. Commentary he annexes to this Propofition in Henry Billingsley's Tranflation of the Elements printed at London Ann. 1570, expressly takes notice of this error, and gives a Demonstration suited to the Construction in the Greek Text, by which he shews that the perpendicular drawn from the point K to BD, muft neceffarily fall upon the point V.

Likewise it is not demonstrated that the quadrilateral figures SOPT, TPRY, and the triangle YRX do not meet the leffer sphere, as was necessary to have been done. only Clavius, as far as I know, has observed this, and demonstrated it by a Lemma, which is now premised to this Proposition, something altered and more briefly demonstrated.

In the Corollary of this Proposition it is supposed that a folid polyhedron is described in the other sphere similar to that which is described in the sphere BCDE. but as the Construction by which this may be done is not given, it was thought proper to give it, and to demonstrate that the pyramids in it are fimilar to those of the same order in the solid polyhedron described in the sphere BCDE.

From the preceding Notes it is sufficiently evident how much the Elements of Euclid, who was a most accurate Geometer, have been vitiated and mutilated by ignorant Editors. The opinion which the greatest part of learned men have entertained concerning the present Greek edition, viz. that it is very little or nothing different from the genuine work of Euclid, has, without doubt deceived them, and made them less attentive and accurate in examining that Edition; whereby several errors, some of them grofs enough, have escaped their notice from the age in which Theon lived to this time. Upon which account there is some ground to hope that the pains we have taken in correcting those errors, and freeing the Elements as far as we could from blemishes, will not be unacceptable to good Judges who can difcern when Demonftrations are legitimate, and when they are not.

The objections which, fince the first Edition, have been made against some things in the Notes, especially against the doctrine of Proportionals, have either been fully anfwered in Dr. Barrow's Lect. Mathemat. and in these Notes; or are such, except one which has been taken notice of in the Note on Prop. 1. Book 11. as shew that the person who made them has not fufficiently confidered the

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Book XII. things against which they are brought; so that it is not necessary to make any further answer to these objections and others like them against Euclid's Definition of Proportionals, of which Definition Dr. Barrow justly says in page 297 of the above-named book, that "Nisi machinis impulsa validioribus aeternum persistet " inconcuffa."

FINIS.

IN THIS EDITION

SEVERAL ERRORS ARE CORRECTED,

AND

SOME PROPOSITIONS ADDED,

BY

ROBERT SIMSON, M. D.

EMERITUS PROFESSOR OF MATHEMATICS

IN THE UNIVERSITY OF GLASGOW.

0000-00

GLASGOW:

PRINTED BY J. & M. ROBERTSON,

SALTMARKET,

1

PREFACE.

UCLID'S DATA is the first in order of the books written by the antient Geometers to facilitate and promote the method of Resolution or Analysis. In the general, a thing is said to be given which is either actually exhibited, or can be found out, that is, which is either known by Hypothesis, or that can be demonstrated to be known; and the Propositions in the Book of Euclid's Data shew what things can be found out or known from those that by Hypothesis are already known; so that in the Ana, lyfis or Investigation of a Problem, from the things that are laid down to be known or given, by the help of these Propositions other things are demonstrated to be given, and, from these, other things are again shewn to be given, and so on, until that which was proposed to be found out in the Problem is demonstrated to be given, and when this is done the Problem is solved, and its Composition is made and derived from the Compositions of the Data which were made use of in the Analysis. And thus the Data of Euclid are of the most general and necessary use in the solution of Problems of every kind.

Euclid is reckoned to be the Author of the Book of the Data both by the antient and modern Geometers; and there seems to be no doubt of his having written a Book on this subject, but which in the course of so many ages has been much vitiated by unskilful Editors in several places, both in the order of the Propositions, and in the Definitions and Demonftrations themselves. To correct the errors which are now found in it, and bring it nearer to the accuracy with which it was, no doubt, at first written by Euclid, is the design of this Edition, that so it may be rendered more useful to Geometers, at least to beginners who defire to learn the investigatory method of the antients. And for their fakes the Composition of most of the Data are subjoined to their Demonstrations, that the Compositions of Problems folved by help of the Data may be the more easily made.

Marinus the Philosopher's preface, which in the Greek Edition is prefixed to the Data, is here left out, as being of no use to understand them. at the end of it he says that Euclid has not used