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

SP

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 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 position.

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 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 bases, are given in magnitude. VIII.

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

I.

See N.

a 1. def. dat.

7.5.

See N.

1. def.

b Ix. 5.

X.

A magnitude is faid to be less than another by a given magn 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

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 ⚫ be found one equal to it; let this be C: And becaufe 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, becaufe the ratio of the given magnitudes 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, as 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.

Becaufe A is given, a magnitude may be found equal to it; let this be C: And becaufe the ratio of A to B is given, a ratio which is the fame with it may be found, let

EF

this be the ratio of the given magnitude E AB CD
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.

с

D. But A is equal to C; therefore B is equal to D. The c 14. 5. magnitude B is therefore given a, 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. MI.

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.

A

3.

Because AB is given, a magnitude equal to it may be found; a 1. def. let this be DE: And because BC is given, one equal to it may be found; let this be EF: Wherefore, because AB is equal to DE, and BC equal to EF; the whole AC is equal to the whole DF:

B

C

EF

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

PROP. IV.

I

F a given magnitude be taken from a given magnitude; the remaining magnitude thall be given.

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

a

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

found; let this be DE: And because

AC is given, one equal to it may be A

found; let this be DF: Wherefore, be

caufe AB is equal to DE, and AC to D DF; the remainder CB is equal to the

C B

FE

remainder FE. CB is therefore given, because FE which is

equal to it has been found.

PROP. V.

IF of three magnitudes, the first together with the fecond be given, and alfo the fecond 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 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 e-
qual to one another, or not equal. A B
B
First, let them be equal, and becaufe

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

ven; therefore * AE the remainder AE B

is given. And becaufe EC is equal

to BD, by taking BC from both,

B C

D

And AE is

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

5.

See N.

a 2. def.

IF

PROP. VI.

a magnitude has a given ratio to a part of it, it shall alfo 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 fame to it: Let this be the ratio of DE a given magnitude to the given mag- A nitude DF. And because DE,

b 4. dat. given, the remainder FE is

c E. 5.

DF are

given :

DE to D

And because AB is to AC, as
DF, by converfion AB is to BC, as

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.

COR. From this it follows, that the parts AC, CB have a given ratio to one another: Becaufe as AB to BC, fo is DE to

EF; by divifion, AC is to CB, as DF to FE; and DF, FE d 17. 5. are given; therefore the ratio of AC to CB is given.

a

PROP. VII.

a 2. def.

6.

IF two magnitudes which have a given ratio to one an- See N.

other, 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 fame with it; let this be the ratio of the a 2. def.

given magnitudes DE, EF: And be-A caufe DE, EF are given, the whole'

DF is given: And becaufe as AB to

BC, fo is DE to EF; by compofition

AC is to CB, as DF to FE; and by

B

C

D

E F

b 3. dat.

c 18. 5.

converfion, AC is to AB, as DF to DE: Wherefore becaufe a E. 5. AC is to each of the magnitudes AB, BC, as DF to each of the others DE, EF; the ratio of AC to each of the magnitudes AB, BC is given.

F

IF

a given magnitude be divided into two parts which See N. have a given ratio to one another, and if a fourth proportional can be found to the fun of the two magnitudes by which the given ratio is exhibited, one of them, and the given magnitude; each of the parts is given.

Let the given magnitude AB be divided into the parts AC, CB which have a given ratio to one another; if a fourth proportional can be found to the above A

named magnitudes; AC and CB are

each of them given.

Because the ratio of AC to CB isD.

C B

FE

given, the ratio of AB to BC is given; therefore a ratio; a 7. dat.

A a 2

which