PREFACE.

EUCLID'S DATA is the first in order of the books written by the ancient geometers to facilitate and promote the method of resolution or analysis. In the general, a thing is said to be given which is either actually exhibited, or can be found out, that is, which is either known by hypothesis, or that can be demonstrated to be known; and the propositions in the book of Euclid's Data show what things can be found out or known from those that by hypothesis are already known; so that in the analysis or investigation of a problem, from the things that are laid down to be known or given, by the help of these propositions other things are demonstrated to be given, and from these, other things are again shown to be given, and so on, until that which was proposed to be found out in the problem is demonstrated to be given, and when this is done, the problem is solved, and its composition is made and derived from the compositions of the Data which were made use of in the analysis. And thus the Data of Euclid are of the most general and necessary use in the solution of problems of every kind.

Euclid is reckoned to be the author of the Book of the Data, both by the ancient and modern geometers; and there seems to be no doubt of his having written a book on this subject, but which in the course of so many ages, has been much vitiated by unskilful editors in several

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.

THE ratios of given magnitudes to one another is given.t

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 oné 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. A B C D

PROP. II.

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 A 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

B

E

F

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

+ See Notes.

D. 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.

A

B

C

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 A B 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.

PROP. IV.

Ꭰ

E F

If 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.

A

C

B

F

E

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 D 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.

Ir 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 be equal, and because

Α ́ Β

AC is equal to BD, take away the com

mon part BC; therefore the remainder AB is equal to the remainder CD.

C D

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 AE.

If 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

C

B

› 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 D conversion (E. 5.) AB is to BC, as DE to

F E

EF. Therefore the ratio of AB to BC is given, because the ra

tio of the given magnitudes DE, EF, which is the same with it, has been found.

津 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 each of the magnitudes AB, BC a given ratio.

A

B

C

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 D (18. 5.) AC is to CB, as DF to FE; and by conversion (E. 5.), AC is to AB, as

E F

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.).

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 A 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.

C

B

D

FE