IN THIS EDITION

SEVERAL ERRORS ARE CORRECTED,

AND

SOME PROPOSITIONS ADDED.

BY

ROBERT SIMSON, M. D.

Emeritus Profeffor of Mathematics in the University of Glasgow.

EDINBURGH:

Printed for J. NOURSE, London; and J. BALFOUR, Edinburgh, M,DCC,LXXV.

E

PREFACE.

UCLID's DATA is the firft in order of the books written by the ancient geometers to facilitate and promote the method of refolution or analyfis. In the general, a thing is faid to be given which is either actually exhibited, or can be found out, that is, which is either known by hypothefis, or that can be demonftrated to be known; and the propofitions in the book of Euclid's Data fhew what things can be found out or known from thofe that by hypothefis are already known; fo that in the analysis or investigation of a problem, from the things that are laid down to be known or given, by the help of these propofitions other things are demonftrated to be given, and from these, other things are again fhewn to be given, and fo on, until that which was propofed to be found out in the problem is demonstrated to be given, and when this is done, the problem is folved, and its compofition is made and derived from the compofitions of the Data which were made use of in the analyfis. And thus the Data of Euclid are of the moft general and neceffary 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 ancient and modern geometers; and there feems to be no doubt of his having written a book on this fubject, but which, in the course of fo many ages, has been much vitiated by unskilful editors in feveral places, both in the order of the propofitions, and in the definitions and demonftrations themfelves, To cor

rect the errors which are now found in it, and bring it nearer to the accuracy with which it was, no doubt, at firft written by Euclid, is the defign of this edition, that fo it may be rendered more useful to geometers, at least to beginners who defire to learn the investigatory method of the ancients. And for their fakes, the compofitions of most of the Data are fubjoined to their demonstrations, that the compofitions of problems folved by help of the Data may be the more easily made.

Marinus the philofopher's preface, which, in the Greek edition, is prefixed to the data, is here left out, as being of no ufe to understand them. At the end of it, he fays, that Euclid has not ufed the fynthetical, but the analytical method in delivering them; in which he is quite mistaken; for in the analysis of a theorem, the thing to be demonftrated is affumed in the analysis; but in the demonftrations of the Data, the thing to be demonstrated, which is, that fomething or other is given, is never once affumed in the demonftration, from which it is manifeft, that every one of them is demonftrated fynthetically; though indeed, if a propofition of the Data be turned into a problem, for example the 84th or 85th in the former editions, which here are the 85th and 86th, the demonstration of the propofition becomes the analyfis of the problem.

Wherein this edition differs from the Greek, and the reafons of the alterations from it, will be fhewn in the notes at the end of the Data.

EUCLID's

EUCLID's

DATA.

DEFINITION S.

I.

PACES, lines, and angles, are faid to be given in magnitude, when equals to them can be found.

SPACE

II.

A ratio is faid to be given, when a ratio of a given magnitude to a give magnitude which is the fame ratio with it can be 'found.

III.

Rectilineal figures are faid to be given in fpecies, which have cach of their angles given, and the ratios of their fides given.

IV.

Points, lines, and spaces, are faid to be given in pofition, which have always the fame fituation, and which are either actually exhibited, or can be found.

A.

An angle is faid to be given in pofition, which is contained by ftraight lines given in pofition.

V.

A circle is faid to be given in magnitude, when a ftraight line from its centre to the circumference is given in magnitude.

VI.

A circle is faid to be given in pofition and magnitude, the centre of which is given in pofition, and a ftraight line from it to the circumference is given in magnitude.

VII.

Segments of circles are faid to be given in magnitude, when the angles in them, and their bafes, are given in magnitude.

VIII.

Segments of circles are faid to be given in pofition and magnitude, when the angles in them are given in magnitude, and their bafes are given both in pofition and magnitude.

IX