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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 antients. And for their fakes, the compofitions of moft of the Data are fubjoined to their demonftrations, 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 says, that Euclid has not used the synthetical, but the analytical method in delivering them; in which he is quite miftaken; 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 analysis of the problem.

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

EUCLID's

EUCLID's DATA.

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DEFINITION S.

I.

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

II.

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

III.

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

IV.

Points, lines, and fpaces, 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 straight 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 straight 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 bases are given both in position and magnitude.

IX.

A magnitude is faid to be greater than another by a given magnitude, when this given magnitude being taken from it, the remainder is equal to the other magnitude.

X.

X.

Sce N.

a 1. def.

dat.

67.5.

3.

Sec N.

A magnitude is faid to be less than another by a given magni-
tude, when this given magnitude being added to it, the
whole is equal to the other magnitude.

PROPOSITION I.

HE ratios of given magnitudes to one another is

TH
T 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 fince A is equal to C,
and B to D; therefore A is to B, as C to
D; and confequently the ratio of A to B is
given, because the ratio of the given magni-
tudes C, D which is the fame with it, has been A B C D

found.

PROP. II.

F a given magnitude has a given ratio to another

I magnitude, and if unto the two magnitudes by

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which the given ratio is exhibited, and the given
magnitude, a fourth proportional can be found;" the
other magnitude is given.

Let the given magnitude A have a given ratio to the mag-
nitude 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

1. def. found equal to it; let this be C; And be-
cause the ratio of A to B is given, a ratio
which is the fame with it may be found, let

b II. 5.

EF

this be the ratio of the given magnitude E ABCD
to the given magnitude F: Unto the magni-
tudes E, F, C find a fourth proportional
D, which, by the hypothefis, can be done.!
Wherefore, because A is to B, as E to F; and
as E to F, fo is C to D; A is to B, as C to

D.

The figures in the margin fhow the number of the propofitions in the other

editions.

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D. But A is equal to C; therefore B is equal to D. The c 14. 5. magnitude B is therefore given, because a magnitude D equal a 1. def. to it has been found.

The limitation within the inverted commas is not in the Greek text, but is now neceffarily added; and the fame must be understood in all the propofitions of the book which depend upon this fecond propofition, where it is not exprefsly mentioned. See the note upon it.

I

PROP. II.

any given magnitudes be added together, their fum fhall be given.

Let any given magnitudes AB, BC be added together, their

fum AC is given..

3.

Because AB is given, a magnitude equal to it may be found; a 1. def.

let this be DE: And becaufe BC is gi

ven, one equal to it may be found: fetA

this be EF: Wherefore, because AB is equal to DE, and BC equal to EF; the whole AC is equal to the whole DF:

D

B

C

EF

AC is therefore given, because DF has been found which is e

qual to it.

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PROP. IV.

a given magnitude be taken from a given magnitude; the remaining magnitude shall be given.

From the given magnitude AB, let the given magnitude AC be taken; the remaining magnitude CB is given.

Because AB is given, a magnitude equal to it may a be a I. def.

found; let this be DE: And because

AC is given, one equal to it may be A

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C B

FE

DF; the remainder CB is equal to the

remainder FE. CB is therefore given ", becaufe FE which is

equal to it has been found.

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12

See Ne

2 4. dat.

5.

See N.

a 2. def.

IF

PROP. V.

of three magnitudes, the first together with the fe. cond be given, and also the second together with the third; either the firft is equal to the third, or one of them greater than the other by a given magnitude.

is

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Let AB, BC, CD be three magnitudes, of which AB together with BC, that is AC, is given; and alfo BC together with CD, that is BD, is given. Either AB is equal to CD, or one of them is greater than the other by a given magnitude.

Because AC, BD are each of them given, they are either equal to one another, or not equal. AB

First, let them be equal, and because

CD

AC is equal to BD, take away the common part BC; therefore the remainder AB is equal to the remainder CD.

But if they be unequal, let AC be greater than BD, and make CE equal to BD. Therefore CE is given, because BD is given. And the whole AC is gi

a

ven; therefore AE the remainder AE

is given. And becaufe EC is equal

to BD, by taking BC from both,

B C D

the remainder EB is equal to the remainder CD. And AE is given; wherefore AB exceeds EB, that is CD, by the given magnitude AE.

IF

PROP. VI.

a magnitude has a given ratio to a part of it, it shall also have a given ratio to the remaining part of it.

Let the magnitude AB have a given ratio to AC a part of it; it has also a given ratio to the remainder BC.

Because the ratio of AB to AC is given, a found which is the fame to it: Let this be the a given magnitude to the given mag- A nitude DF. And because DE, DF are b 4. dat. given, the remainder FE is given: And becaufe AB is to AC, as DE to D

c E. 5.

DF, by converfion AB is to BC, as

ratio may be

ratio of DE

C B

FE

DE to EF. Therefore the ratio of AB to BC is given, because the ratio of the given magnitudes DE, EF, which is the fame with it, has been found.

COR.

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