The Elements of Euclid

EUCLID'S DATA.

DEFINITIONS.

SPACES, lines, and angles, are said to be given in magnitude, 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 position, which have always the same situation, and which are either actually exhibited, or can be found.

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

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.

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 position 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.

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.

THE ratios of given magnitudes to one another are given.†

Let A, B be two given magnitudes, the ratio of A to B is given. Because A is a given magnitude, there may (1. def. Dat.) 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 (7. 5.) 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 with it, has been found.

If 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 equal to it (1. def.); let this be 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 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. 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

E F

The figures in the margin show the number of the propositions in the other editions.

† See Notes.

But A is equal to C; therefore (14. 5.) 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 upon it.

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

E F

Ir 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 (1. def.) be found; 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 is equal to the remainder FE. CB is there

fore given (1. def.), because FE which is equal to it has been

found.

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, Either AB is equal to CD, or one of

that is, BD, is given.

them is greater than the other by a given magnitude.

A B

C D

Because AC, BD are each of them given, they are either equal to one another, or not equal. First, let them 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; there

fore (4. dat.) AE the remainder is A E B C D given. And because EC is equal to

BD, by taking BC from both, the re

mainder EB is equal to the remainder CD.

And AE is given;

wherefore AB exceeds EB, that is CD, by the given magnitude

Ir 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 ratio may be found (2. def.) which is the same to it: let this be the ratio of DE a given magnitude to the given magnitude DF. And because DE, DF are given, the remainder FE is (4. dat.) given: and because AB is to AC, as DE to DF, by conversion (E. 5.) 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.

See Notes.

COR. From this it follows, that the parts AC, CB have a given ratio to one another: because as AB to BC, so is DE to EF; by division (17. 5.), AC is to CB, as DF to FE: and DF, FE are given; therefore (2. def.) the ratio of AC to CB is given.

IF two magnitudes which have a given ratio to one another, 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 (2. def.) which is the same with it; let this be the ratio of the given magnitudes DE, EF: and because DE, EF are given, the whole DF is given (3. dat.): and because as AB to BC, so is DE to EF; by composition (18. 5.), AC is to CB, as DF to FE; and by conversion (E. 5.), AC is to AB, as DF to DE: wherefore because 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 (2. def. ).

PROP. VIII.

E F

If the given magnitude be divided into two parts which have a given ratio to one another, and if a fourth proportional can be found to the sum 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 named magnitudes; AC and CB are each of them given.

Because the ratio of AC to CB is given, the ratio of AB to BC is given (7. dat.); therefore a ratio which is the

See Notes.

Ꭰ

F E

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The Elements of Euclid: Viz, the First Six Books, Together with the Eleventh ...