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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 shewn to be equal, though this other be not exhibited in given magnitudes. See the Notes on the 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.
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,
DBC, which shall be the seventh part of the angle ABC; suppose this is done, therefore the straight line BD is invariable in 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 less than the seventh part of the angle ABC; therefore, according to this definition, the straight line BD is given in position, as also * the point D in which it * 28 Dat. 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 6 found," for nothing is to be reckoned given, which cannot be found, or is not actually exhibited. Too
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. Jousboot od
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 cur, 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 not always be exhibited in numbers; for instance, if the side of a square be given, the ratio of 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 side be given, for example the number of them in the diameter cannot be and the like holds in
other cases. Retropora PROPOSITION I.
* 44 Dat.
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, fron 7.5.
The limitation added to 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 t, “because “ A is given, a magnitude equal to it can be found*; « let this be C; and because the ratio of A to B is
+ See Dr. Gregory's edition of the Data. tas nav
56 given, a ratio which is the same to it can be found *," * 2 Def.
th the ratio of A to
Β Δ angle to 8 ratio; for instance, the ratio of the given straight line E to
given ing found an angle C equal to C A, how can the angle A be found to which has the
7 ratio that E has to Z? Certainly no way, until it be? shewn how to find an angle to which a given angle has not 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 inentioned.
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 22d and 23d are made the uth and 12th, as they are more simple ethan 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 15th. the Greek text
es émotioqjor PROP. VI. VII. isi ga bilan
These are universally true, though, in the Greek text, they are demonstrated by Prop. 2. which has a limitation; they are therefore now
w shewn without it.
19tagol ook of PROP. XII. dorolt Tagso oris
In the 23d Prop. in the Greek text, which here is the 12th, the words “un tous aurous dren 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 εί και μη τους αυτούς, as im 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
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 sanie ratios, as Zambertus bas translated them in his edition.
PROP. XIII. Some very ignorant editor has given a second de monstration 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 waso given,
if another ratio could be shewn 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.7 And it is to be observed that the moderns frequently take given ratios, and ratios that are always the sames v for one and the same thing; and Sir Isaac Newton hasn 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.