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the synthetical, 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 affumed 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 shewn in the Notes at the end of the Data.

S

EUCLID'S DATA.

DEFINITIONS.

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 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 faid to be given in pofition, which have always the same situation, and which are either actually exhibited, 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 straight line from 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 circumference is given in magnitude.

VII.

Segments of circles are faid 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 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.

See N.

* I.

a. 1. Def. Dat.

b. 7.5.

See N.

2.

Χ.

A magnitude is faid 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.

T

PROPOSITION I.

HE ratio of given magnitudes 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 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 A is to

B, as C to D; and consequently the ratio of A

to B is given, because the ratio of the given mag

nitudes C, D which is the fame with it has been ABCD

found.

I

PROP. II.

F a given magnitude has a given ratio to another magnitude, " and if unto the two magnitudes by "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 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 a. 1. Def. equal to it a; let this be C. and because the ratio

EF

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 Hypothefis, can be done. where-
fore because A is to B, as E to F; and as E to F,

* The figures in the margin shew the number of the Propositions in the other Editions.

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so is C to D; A is to B, as C to D. but A is equal to C, there- b. 11. 5. fore B is equal to D. the magnitude B is therefore given a, be- c. 14. 5. cause a magnitude D equal to it has been found.

The limitation within the inverted commas is not in the Greek text, but is now necessarily added; and the fame must be underftood in all the Propositions of the Book which depend upon this second Propofition, where it is not expressly mentioned. See the Note upon it,

I

PROP. III.

F any given magnitudes be added together, their sum shall be given.

Let any given magnitudes AB, BC be added together, their sum AC is given.

3.

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

let this be DE. and because BC is given,

one equal to it may be found; let this be

A

B

C

fore given, because DF has been found, which is equal to it.

I

۱

PROP. IV.

F 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.

4.

Because AB is given, a magnitude equal to it may a be found; a. r. Def.

let this be DE. and because AC is given,

one equal to it may be found; let this be

DF. wherefore because AB is equal to

DE, and AC to DF, the remainder CB D

F

E

is equal to the remainder FE. CB is there

fore given, because FE which is equal to it has been found.

See N.

12.

PROP. V.

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

Let AB, BC, CD be three magnitudes, of which AB together with BC, that is AC, is given; and also 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 notequal. first, let them

be equal, and because AC is equal to A

BD, take away the common part BC;

B

C

D

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 given, therefore

a. 4. Dat. a AE the remainder is given. and A E

5.

BC

D

because EC is equal to BD, by
taking BC from both, 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.

PROP. VI.

See N.

F

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.

a. 2. Def.

Because the ratio of AB to AC is given, a ratio may be found a which is the fame to it. let this be the ratio of DE a given magni

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.