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EUCLID'S DATA.

DEFINITIONS.

I.

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

nitude, when equals to them can be found.

II.

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 posi

tion, which have always the same situation, and which are either actually exhibited, or can be found.

A.

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 centre to the circumference is given in magnitude.

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

the centre of which is given in position, and a straight line from it to the circumference is given in magnitude.

z

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 bases are given both in posi-
tion and magnitude.

IX.

A magnitude is said 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.

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.

PROPOSITION I.

See N.

*

* be

1 Def.

Dat.

The 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
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
magnitudes C, D, which is the same A B C D
with it, has been found.

* 7.5.

The figures in the margin shew the other editions.

number of propositions in

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If a given magnitude have a given ratio to another magni- See N.

tude, 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 magni

tude 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 equal to it * ; let this be

* 1 Def. C: and because the ratio of A to B is given, a ratio which is the same with it may

be found; let this be the ratio of A B C D
the given magnitude E to the given
magnitude F: unto the magnitudes E,

EF
F, C, find a fourth proportional D,
which, by the hypothesis, can be done.
Wherefore, because A is to B, as E to
F; and as E to F, so is C to D; A is *

* 11.5. to B, as C to D. But A is equal to C: therefore * B, is equal to D. The magnitude B is therefore given * * 1 Def. because 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 same must be understood in all the propositions of the book which depend upon this second proposition, where it is not expressly mentioned. See the note

* 14. 5.

upon it.

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If 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.

Because AB is given, a magnitude equal to it may be found ; let this be DE: and because BC is given, one equal A B C to it may be found ; let this be EF: wherefore because AB is D

E equal to DE, and BC equal to

1 Def.

F

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EF; the whole AC is equal to the whole DF; AC is
therefore given, because DF has been found which is
equal to it.

PROP. IV.

4.

* 1 Def.

If a given magnitude be taken from a given magnitude ;

the remaining magnitude shall be given.
From the given magnitude AB, let the given magni-
tude AC betaken; the remaining magnitude CB is given.

Because AB is given, a magnitude equal to it may
be found ; let this be DF:
and because AC is given, A

ç B
one equal to it may be found;
let this be DE: wherefore,

D
because AB is equal to DF,

E
and AC to DE; the remain-
der CB is equal to the remainder FE. CB is therefore
given *, because FE which is equal to it has been found.

* 1 Def.

12.

PROP. V.

See N.

If 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 not equal. First, let them

A
B

С D
be equal, and because 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 given ;

B C C D
therefore * AE the remain-
der is given. And because

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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.

5.

PROP. VI. If a magnitude have a given ratio to a part of it, it shall See N.

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 ratio may be found* which is the same to it: let this be the ratio * 2 Def. of DE, a given magnitude to the given magnitude DF. And be

A

С B cause DE, DF, are given, the remainder FE is * given : and D

E because AB is to AC, as DE to DF, by conversion * AB is to BC, as DE to EF. Therefore the ratio of AB to BC is given, because the ratio of the given magnitudes DE, EF, which is the same with it, has been found.

Cor. From this it follows, that the parts AC, CB, have a given tio to one another: because as AB to BC, so is DE to EF; by division * AC is to CB, as * 17.5. DF to FE; and DF, FE, are given; therefore * the * 2 Def.ratio of AC to CB is given.

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4 Dat.

* E. 5.

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If two magnitudes which have a given ratio to one another See N.

be added together; the whole magnitude shall have to each of them a given ratio.

Let the magnitudes AB, BC, which have a given ratio to one another, be added together: the whole AC has to each of the magnitudes AB, BC, a given ratio.

Because the ratio of AB to BC is given, a ratio may be found * which is the same with it; let this be the * 2 Def. ratio of the given magnitudes DE, EF: and because DE,

B C EF, are given, the whole DF is given *: and because as

D AB to BC, so is DE to EF; by composition * AC is to CB

3 Dat.

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18, 5.

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