a right angle and the angle contained by the straight line given in position, and the straight line which joins the given point and the centre of the circle; and from the point in which the second line meets the circumference again, a third straight line be drawn, making with the second, an angle equal to that which the first makes with the second: the point in which this third line meets the straight line given in position is given; as also the rectangle contained by the first straight line and the segment of the third betwixt the circumference and the straight line given in position, is given.

Let the straight line CD be drawn from the given point C in the straight line AB given in position, to the circumference of the circle DEF given in position, of which G is the centre; join CG, and from the point D let DF be drawn, making the angle CDF equal to the difference of a right angle and the angle BCG, and from the point F let FE be drawn, making the angle DFE equal to CDF, meeting AB in H: the point H is given; as also the rectangle CD, FH.

Let CD, FH meet one another in the

point K, from which draw KL perpendi- D cular to DF; and let DC meet the circumference again in M, and let FH meet the same in E, and join MG, GF, GH.

K D

F

B

E

Because the angles MDF, DFE are A equal to one another, the circumferences MF, DE are equal (26. 3.); 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 (8. 1.) GFE; and the angles GMC, GFH, are equal to one another, because they are either the same with the E angles GMD, GFE or adjacent to them: and because the angles KDL, LKD are together equal (32. 1.) to a right angle, that is, by the hypothesis, to the angles KDL, GCB; the angle GCB, or GCH, is equal to the angle (LKD, that is, to the angle) LKF or GHK: 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 straight line GM to GF; therefore (26. 1.) CG is equal to GH, and CM to HF and because CG is equal to GH, the angle GCH is equal to GHC; but the angle GCH is given: therefore GHC is

47

M

K

AC

Fˋ

H B equal to GFH, and the

given, and consequently the angle CGH is given; and CG is given in position, and the point G; therefore (32. dat.) GH is given in position; and CB is also given in position, wherefore the point H is given.

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

FINIS.

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

A

straight line BD is invariable in its position, that is, has always the same situa

B

D

C

tion; 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 (28. dat.) the 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 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. XIII. 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 ratio of it to its diameter is given (44. dat.) geometrically, but not in numbers; and the diameter is given (2. dat.); but though the number of any equal parts in the side be given, for example, 10, the number of them in the diameter cannot be given: and the like holds in many other cases.

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, 5.

PROP. II.

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 found (1. def.); 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 (2. def.);" 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 is 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 ▲ 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 ▲ 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 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 above mentioned limitation must be understood, though it be not explicitly mentioned.

PROP. V.

A

C

E

Z

BA

The order of the propositions in the Greek text, between prop. 4, and prop. 25, is now changed into another which is more natu

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

ral, 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 22d 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 ratio to 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 Grcek text they are demonstrated by prop. 2, which has a limitation; they are therefore now shown without it.

66

PROP. XII.

In the 23d prop. in the Greek text, which here is the 12th, the words, 'un 785 AUT85 SE," are wrong translated by Claud. Hardy, in his edition of Euclid's Data, printed at Paris, anno 1625, which was the first edition of the Greek text: and Dr. Gregory follows him in translating them by the words, "etsi, non easdem," as if the Greek had been ɛ xaι μn 785 aur85, as in prop. 9. of the Greek text. Euclid's meaning is, that the ratios mentioned in the proposition must not be the same; for, if they were, the proposition would not be true. Whatever ratio the whole has to the whole, if the ratios of the parts of the first to the parts of the other be the same with this ratio, one part of the first may be double, triple, &c. of the other part of it, or have any other ratio to it, and consequently cannot have a given ratio to it; wherefore, these words must be rendered by non autem easdem," but not the same ratios, as Zambertus has translated them in his edition.

PROP. XIII.

66

Some very ignorant editor has given a second demonstration of this proposition in the Greek text, which has been as ignorantly kept in by Claud. Hardy and Dr. Gregory, and has been retained in the translations of Zambertus and others; Carolus Renaldinus gives it only: the author of it has thought that a ratio was given, if another ratio could be shown to be the same to it, though this last ratio be not found: but this is altogether absurd, because from it would be deduced, that the ratio of the sides of any two squares is given, and the ratio of the diameters of any two circles, &c. And it is to be observed, that the moderns frequently take given ratios, and ratios that are always the same, for one and the same thing; and Sir Isaac Newton has fallen into this mistake in the 17th lemma of his Principia, edit. 1713, and in other places: but this should be carefully avoided, as it may lead into other errors.

PROP. XIV. XV.

Euclid, in this book, has several propositions concerning magnitudes, the excess of one of which above a given magnitude has