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P R E F A C E.

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 Ihew what things can be found out or known from those that by Hypothesis are already known; so 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 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 Demonstrations themselves. To correct the · errors which are now found in it, and bring it nearer to the acc??racy 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 iQ Geometers, at least to beginners who desire 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 solved 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 underftand them. at the end of it he says that Euclid has not used 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 demonstrated is assumed in the Analysis; but in the Demonstrations of the Data, the thing to be demonstrated, which is that something or other is given, is never once assumed in the Demonstration, from which it is manifest that every one of them is demonstrated synthetically; tho' indeed if a Proposition 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 Proposition becomes the Analysis of the Problem.

Wherein this Edition differs from the Greek, and the reasons of the alterations from it will be thewn in the Notes at the end of the Data.

EUCLID'S

EU CLI D'S DA TA.

D E F INI TI ON S.

I.

SPACES, lines and angles are said to be given in magnitude,

when equals to them can be found.

A ratio is said to be given, when a ratio of a given magnitude to a given magnitude which is the same ratio with it can be found.

III.
Rectilineal figures are said to be given in species, which have each
of their angles given, and the ratios of their sides given.

IV.
Points, lines and spaces are said to be given in position, which have

always the same situation, and which are either actually exhibi-
ted, or can be found.

A. An angle is said to be given in position, which is contained by straight lines given in position.

V.
A circle is said to be given in magnitude, when a Itraight line froin
its center to the circumference is given in magnitude.

VI.
A circle is said to be given in position and magnitude, the center of

which is given in position, and a straight line from it to the cir-
cumference is given in magnitude.

VII.
Segments of circles are said to be given in magnitude, when the
angles in them, and their bases are given in magnitude.

VIII.
Segments of circles are said to be given in position and magnitude,

when the angles in them are given in magnitude, and their basés
are given both in position and magnitude.

IX.
A magnitude is said to be greater than another by a given magni-

tude, when this given magnitude being taken from it, the re-
mainder is equal to the other magnitude.
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X. A

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X.
A magnitude is said to be less than another by a given magnitude,

when this given magnitude being added to it, the whole is equal
to the other magnitude.

* 1.

See N.

TH

a 1. Def.

Dat.

PROPOSITION I.
HE ratios of given nagnitudes to one another is

given.
Let A, B be two given magnitudes, the ratio of A to B is given.

Because A is a given magnitude, there may a be found one equal to it; let this be C. and because B is given, one equal to it may be found; let it be D. and since A is equal to C, and B to D; therefore 6 A is to B, as C to D; and consequently the ratio of A to B is given, because the ratio of the given magnitudes C, D which is the fame with it has been found.

A B C D

b. 7. 5.

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2.

See N.

PRO P. II.
IF F a given magnitude has a given ratio to another mag.

nitude, “ and if unto the two magnitudes by which “ the given ratio is exhibited, and the given inagnitude, a fourth proportional can be found;" the other inagnitude is given.

Let the given magnitude A have a given ratio to the magnitude B; if a fourth proportional can be found to the three magnitudes above named, B is given in magnitude.

Because A is given, a magnitude may be found 2.1. Def. equal to it? ; let this be C. and because the ratio

of A to B is given, a ratio which is the fame with ABCD
it may be found; let this be the ratio of the given
magnitude E to the given magnitude F. unto the
magnitudes E, F, C find a fourth proportional D,
which, by the Hypothesis, can be done. where-
fore because A is to B, as E to F; and as E to F,

• The figures in the margia hew the number of the Propositions in the other Editions.

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