463

ON

EUCLID's DATA.

TH

DEFINITION II.

HIS is made more explicit than in the Greek text, to prevent a mistake which the author of the fecond demonftration of the 24th propofition in the Greek edition has fallen into, of thinking that a ratio is given to which another ratio is fhown to be equal, though this other be not exhibited in given magnitudes. See the notes on that propofition, which is the 13th in this edition. Befides, by this definition, as it is now given, fome propofitions are demonftrated, which in the Greek are not fo well done by help of prop. 2.

DEF. IV.

In the Greek text, def. 4. is thus: "Points, lines, spaces, " and angles are faid to be given in pofition which have always "the fame fituation ;" but this is imperfect and ufelefs, becaufe there are innumerable cafes in which things may be given according to this definition, and yet their polition cannot be found; for inftance, let the triangle ABC be given in pofition, and let it be proposed to draw a ftraight line BD from the angle at B to the oppofite fide AC which shall cut off the angle DBC which fhall be the feventh part of the angle ABC; suppose this is done, therefore the ftraight line BD is invariable in its pofition, that is, B has always the fame fituation; for any

A

other ftraight line drawn from the point B on either fide of BD cuts off an angle greater or leffer than the feventh part of the angle ABC; therefore, according to this definition, the ftraight line BD is given in pofition, as alfo the point D in a 28. dat. which it meets the flraight line AC which is given in pofition. But from the things here given, neither the ftraight 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 fuppofed may

be

be found.
We have a
mended 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.

This definition is therefore of no ufe.

The definition of an angle given by pofition is taken out of the 4th, and given more diftinctly by itfelf in the definition marked A.

DE F. XI. XII. XIII. XIV. XV.

The 11th and 12th are omitted, because they cannot be given in English fo as to have any tolerable fenfe; and, therefore, wherever the terms defined occur, the words which exprefs their meaning are made ufe of in their place.

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

It is to be obferved 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 inftance, if the fide of a fquare be given, the ratio of b 44. dat. it to its diameter is given geometrically, but not in numbers; and the diameter is given ; but though the number of any equal parts in the fide be given, for example 10. the number of them in the diameter cannot be given: And the like holds in many other cafes.

C 2. dat.

a I. def. b 2. def.

PROPOSITION I.

In this it is fhown 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 thefe two fteps, from 7. 5.

PROP. II.

The limitation added at the end of this propofition between the inverted commas is quite neceflary, because without it the propofition cannot always be demonftrated: For the author having faid "becaufe A is given, a magnitude equal to it "can be found, let this be C; and because the ratio of A "to B is given, a ratio which is the fame to it can be found"," adds, "let it be found, and let it be the ratio of C to A.” Now, from the fecond definition nothing more follows than that fome ratio, fuppofe the ratio of E to Z, can be found, which is the fame with the ratio of A to B; and when the author fuppofes that the ratio of C to A, which is allo

alfo the fame with the ratio of A to B, can be found, he neceffarily fuppofes 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 muft have underftood the Propofition 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 inftance, the ratio of the given ftraight 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 fame ratio that E has to Z? certainly no way, until it be fhewn how to find an angle to which a given angle has a given ratio, which cannot be done

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C

E

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

PROP. V.

The order of the Propofitions in the Greek text between prop. 4. and prop. 25. is now changed into another which is more natural, by placing thofe which are more fimple before those which are more complex; and by placing together thofe which are of the fame kind, fome 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 fimple than the propofitions concerning magnitudes, the excefs of one of which above a given magnitude has a given ratio to the other, after which thefe two were placed; and the 24th in the Greek text is, for the fame reason, made the 13th.

PROP. VI. VII.

These are univerfally true, tho' in the Greek text they are demonftrated by prop. 2. which has a limitation; they are therefore now fhewn without it.

PROP. XII.

In the 23d prop. in the Greek text, which here is the 12th, the words, 66 μen THE MUTHS de," are wrong translated by Claud. Hardy, in his edition of Euclid's Data, printed at Paris, ann. 1625, which was the first edition of the Greek text; and Dr Gregory follows him in tranflating them by the words, "eth non eafdem," as if the Greek had been exμan Tus aures, as in prop. 9. of the Greek text. Euclid's meaning is, that the ratios mentioned in the propofition must not be the fame; for, if they were, the propofition 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 fame 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 confequently cannot have a given ratio to it; wherefore, these words muft be rendered by non autem eafdem," but not the fame ratios, as Zambertus has tranflated them in his edition.

PROP. XIII.

Some very ignorant editor has given a fecond demonftration of this propofition in the Greek text, which has been as ignorantly kept in by Claud. Hardy and Dr Gregory, and has been retained in the tranflations 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 fhown to be the fame to it, though this last ratio be not found: But this is altogether abfurd, becaufe from it would be deduced that the ratio of the fides of any two fquares is given, and the ratio of the diameters of any two circles, &c. And it is to be obferved, that the moderns frequently take given ratios, and ratios that are always the fame, for one and the fame thing; and Sir Ifaac Newton has fallen into this mistake in the 17th Lemma of his Principia, ed. -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 feveral propofitions concerning magnitudes, the excess of one of which above a given mag

nitude