the angle DFE equal to CDF, meeting AB in H: The point
H is given; as also the rectangle
CD, KH.

Let CD, FH meet one another in the point K, from which draw KL perpendicular to DF; and let DC meet the circumference again in M, and let FH meet the same in E, and join MG, GF, GH.

b

D

A

M

B

a 26.3.

K

M

K

b 8. 1.

GL

Because the angles MDF, DFE. are equal to one another, the circumferences MF, DE are equal; and adding or taking away the common part ME; the circumference DM is equal to EF; therefore the straight line DM is equal to the straight line. EF, and the angle GMD to the angle GFE; and the angles GMC, E GFH are equal to one another, because they are either the same with the angles GMD, GFE, or adjacent to them: And because the angles KDL, LKD are together equal to a right angle, that is, by the hypothesis, to the angles KDL, GCB; the angles GCB or GCH is equal to the angle (LKD, that is, to the angle) LKF or GKH: Therefore the points C, K, H, G are in the circumference of a circle: and the angle GCK is therefore equal to the angle GHF: and the angle GMC is equal to GFH, and the straight line GM to GF; therefore CG is equal to GH, and CM to HF: 26. 1. And because CG is equal to GH, the angle GCH is equal to GHC; but the angle GCH is given: Therefore GHC is given, and consequently the angle CGH is given; and CG is given in position, and the point G; therefore GH is given in position; and CB is also given in position, wherefore the 32. dat. point H is given.

A C

© 32. 1.

F

HB

e

And because HF is equal to CM, the rectangle DC, FH is f 95. or 96. equal to DC, CM: But DC, CM is given, because the point dat. C is given, therefore the rectangle DC, FH is given.

END OF THE DATA.

NOTES

ON

EUCLID'S DATA.

DEFINITION II.'

THIS is made more explicit than in the Greek text, to prevent a mistake which the author of the second demonstration of the 24th proposition in the Greek edition has fallen into, of thinking that a ratio is given to which another ratio is shown to be equal, though this other be not exhibited in given magnitudes. See the Notes on that Proposition, which is the 13th in this edition. Besides, by this definition, as it is now given, some propositions are demonstrated, which in the Greek are not so well done, by help of Prop. 2.

DEF. IV.

In the Greek text, def. 4. is thus: "Points, lines, spaces, "and angles are said to be given in position which have always "the same situation ;" but this is imperfect and useless, because there are innumerable cases in which things may be given according to this definition, and yet their position cannot be found for instance, let the triangle ABC be given in position, and let it be proposed to draw a straight line BD from the angle at B to the opposite side AC,

which shall cut off the angle DBC, which shall be the seventh part of the angle ABC; suppose this is done, therefore the straight line BD is invariable in B its position, that is, has always the same

situation; for any other straight line drawn from the point B on either side of BD cuts off an angle greater or lesser than the seventh part of the angle ABC; therefore, according to this definition, the straight line BD is given in position, as also the 22. dat. point D in which it meets the straight line AC which is given in position. But from the things here given, neither the straight line BD nor the point D can be found by the help of Euclid's Elements only, by which every thing in his data is

455

supposed may be found. This definition is therefore of no use. We have amended it by adding, "and which are either "actually exhibited, or can be found," for nothing is to be reckoned given, which cannot be found, or is not actually exhibited.

The definition of an angle given by position is taken out of the 4th, and given more distinctly by itself in the definition marked A.

DEF. XI. XII. XI. XIV. XV.

THE 11th and 12th are omitted, because they cannot be given in English so as to have any tolerable sense; and therefore, wherever the terms defined occur, the words which express their meaning are made use of in their place.

The 13th, 14th, 15th are omitted, as being of no use.

It is to be observed in general of the data in this book, that they are to be understood to be given geometrically, not always arithmetically, that is, they cannot always be exhibited in numbers; for instance, if the side of a square be given, the 44. dat. ratio of it to its diameter is given geometrically, but not in e 2. dat. numbers; and the diameter is given; but though the number example 10, the number of them in the diameter cannot be given: And the like holds in many other cases.

a 1. def.

2. def.

of any equal parts in the side be given u

PROPOSITION I.

IN this it is shown that A is to B as C to D, from this, that A is to C, as B to D, and then by permutation; but it follows directly without these two steps, from 7. to 5.:

PROP. II.

or & do datined THE limitation added at the end of this proposition between the inverted commas is quite necessary, because without it the proposition cannot always be demonstrated: For the author having said, " because A is given, a magnitude equal to it

can be founda: let this be C; and because the ratio of A to "B is given, a ratio which is the same to it can be found"," adds, let it be found, and let it be the ratio of C to A." Now, from the second definition, nothing more follows than that some ratio, suppose the ratio of E to Z, can be. found, which is the same with the ratio of A to B; and when the author supposes that the ratio of C to A, which ist

* See Dr. Gregory's edition of the Data.

B

also the same with the ratio of A to B, can be found, he necessarily supposes that to the three magnitudes E, Z, C, a fourth proportional A may be found; but this cannot always be done by the Elements of Euclid; from which it is plain Euclid must have understood the proposition under the limitation which is now added to his text. An example will make this clear: Let A be a given angle, and B another angle to which A has a given ratio; for instance, the ratio of the given straight line E to the given one Z; then, having found an angle C equal to A, how can the angle A be found to which C has the same ratio that E has to Z? Certainly no way, until it be shown how to find an angle to which a given angle has a given ratio, which

M

cannot be done by Euclid's Elements, nor probably by any Geometry known in his time, Therefore in all the propositions of this book which depend upon this second, the abovementioned limitation must be understood, though it be not explicitly mentioned.

PROP. V.

THE order of the Propositions in the Greek text between Prop. 4. and Prop. 25. is now changed into another which is more natural, by placing those which are more simple before those which are more complex; and by placing together those which are of the same kind, some of which were mixed among others of a different kind. Thus, Prop. 12. in the Greek is now made the 5th, and those which were the 2d and 23d are made the 11th and 12th, as they are more simple than the propositions concerning magnitudes, the excess of one of which above a given magnitude has a given ratioto the other, after which these two were placed; and the 24th in the Greek text is, for the same reason, made the 13th.

PROP. VI. VII.

THESE are universally true, though, in the Greek text, they are demonstrated by Prop. 2. which has a limitation; they are therefore now shown without it.