nition 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, and 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, à 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 4, which is also the same with the ratio of A to B, See Dr. Gregory's edition of the Data. 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 an A B gle, 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 an с gle a be found to which C has the E same ratio that E has to Z? Certainly no way, until it be shown how to find Zan 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. AM 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 22d and 23d are made the ilth 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 Greek text they are demonstrated by prop. 2., which has a limitation; they are therefore now shown without it. PROP. XII. In the 23d prop. in the Greek text, which here is the 12th, the words, “ un tos avres ds,” 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 ev xai un 70s avtos, 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. 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 magni tude has a given ratio to the other ; but he has given none concerning magnitudes whereof one together with a given magnitude has a given ratio to the other; though these last occur as frequently in the solution of problems as the first ; the reason of which is, that the last may be all demonstrated by help of the first ; for, if a magnitude, together with a given magnitude, has a given ratio to another magnitude, the excess of this other above a given magnitude shall have a given ratio to the first, and on the contrary ; as we have demonstrated in prop. 14. And for a like reason prop. 15 has been added to the Data. One example will make the thing clear: suppose it were to be demonstrated, that if a magnitude A together with a given magnitude has a given ratio to another magnitude B, that the two magnitudes A and B, together with a given magnitude, have a given ratio to that other magnitude B ; which is the same proposition with respect to the last kind of magnitudes above mentioned, that the first part of prop. 16, in this edition, is in respect of the first kind : this is shown thus; from the hypothesis, and by the first part of prop. 14, the excess of B above a given magnitude has unto A a given ratio ; and, therefore, by the first part of prop. 17, the excess of B above a given magnitude has unto B and A together a given ratio ; and by the second part of prop. 14, A and B together with a given magnitude has unto B a given ratio ; which is the thing that was to be demonstrated. In like manner, the other propositions concerning the last kind of magnitudes may be shown. PROP. XVI. XVII. In the third part of prop. 10, in the Greek text, which is the 16th in this edition, after the ratio of EC to CB has been shown to be given ; from this, by inversion and conversion, the ratio of BC to BE is demonstrated to be given ; but without these two steps, the conclusion should have been made only by citing the 6th proposition. And in like manner, in the first part of prop. 11, in the Greek, which in this edition is the 17th, from the ratio of DB to BC being given, the ratio of DC to DB is shown to be given by inversion and composition, instead of citing prop. 7, and the same fault occurs in the second part of the same prop. 11. PROP. XXI. XXII. These now are added, as being wanting to complete the subject treated of in the four preceding propositions. PROP. XXIII. This, which is prop. 20, in the Greek text, was separated from prop. 14, 15, 16, in that text, after which it should have been immediately placed, as being of the same kind : it is now put into its proper place; but prop. 21 in the Greek is left out, as beiog the same with prop. 14, in that text, which is here prop. 18. PROP. XXIV. This, which is prop. 13, in the Greek, is now put into its proper place, having been disjoined from the three following it in this edition, which are of the same kind. PROP. XXVIII. This, which in the Greek text is prop. 25, and several of the following propositions, are there deduced from def. 4, which is not sufficient, as has been mentioned in the note on that definition : they are therefore now shown more explicitly. PROP. XXXIV. XXXVI. Each of these has a determination, which is now added, which occasions a change in their demonstrations. PROP. XXXVII. XXXIX. XL. XLI. The 35th and 36th propositions in the Greek text are joined into one, which makes the 39th in this edition, because the same enunciation and demonstration serves both : and for the same reason prop. 37, 38, in the Greek, are joined into one, which here is the 40th. Prop. 37 is added to the Data, as it frequently occurs in the solution of problems ; and prop. 41 is added to complete the rest. PROP. XLII. This is prop. 39, in the Greek text, where the whole construction of prop. 22, of book 1. of the Elements is put, without need, into the demonstration, but is now only cited. PROP. XLV. This is prop. 42, in the Greek, where the three straight lines made use of in the construction are said, but not shown, to be such that any two of them is greater than the third, which is now done. |