463 ON EUCLI D's DATA. DEFINITION II. T HIS is made more explicit than in the Greek text, to pre. vent a mistake which the author of the fecond demonstration of the 24th propofition 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 that propofition, which is the 13th in this edition. Befides, by this definition, as it is now given, some propositions are demonftrated, 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 faid to be given in position which have always "the same situation;" but this is imperfect and useless, because there are innumerable cafes in which things may be given according to this definition, and yet their pofition 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 A C to the opposite fide AC which shall cut off the angle DBC which shall be the feventh part of the angle ABC, suppose this is done, therefore the straight line BD is invariable in its position, that is, B has always the same situation; for any other straight line drawn from the point B on either fide 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 pofition, as also a the point Dina 28. dat. 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 464 be found. This definition is therefore of no use. We have 1mended 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, where ever 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 b 44. dat. it to its diameter is given b geometrically, but not in numbers; • 2. dat. 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. a 1. def. b 2. def. In this it is shewn 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 neceffary, because without it the propofition cannot always be demonftrated: For the author having faid * " because A is given, a magnitude equal to it can be found a; 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 b," adds, "let it be found, and let it be the ratio of C to Now, from the second definition nothing more follows than that fome ratio, suppose the ratio of E to Z. can be found, which is the fame with the ratio of A to B; and when the author supposes that the ratio of C to A, which is • See Dr Gregory's edition of the data. alfo alfo the fame with the ratio of A to B, can be found, he neceffarily supposes that to the three magnitudes E, Z, Ca 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 B 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 fame ratio that E has to Z? certainly no way, until it be shewn 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 men- tioned. 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 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 proроfitions concerning magnitudes, the excess of one of which above a given magnitude has a given ratio to the other, after whicli these two were placed; and the 24th in the Greek text is, for the fame reafon, made the 13th. PROP. VI. VII. These are universally true, tho' in the Greek text they are demonstrated by prop. 2. which has a limitation; they are therefore now shewn without it. PROP. XII. In the 23d prop. in the Greek text, which here is the 12th, the words, “ μὴ τὸς αὐτὸς δὲ," are wrong tranflated 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, " etfi non eafdem," as if the Greek had been εἰ και μὴ Τες, αυτός, as in prop. 9. of the Greek text. Euclid's meaning is, that the ratios mentioned in the proposition must not be the fame; 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 confequently cannot have a given ratio to it; wherefore, these words must 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 demonstration of this proposition in the Greek text, which has been as ignorantly kept in it 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 shewn to be the fame to it, though this last ratio be not found: But this is altogether abfurd, because from it would be deduced that the ratio of the fides 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 fame, for one and the same 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 several propositions concerning magnitudes, the excess of one of which above a given magnitude hitude 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; tho' these last oсcur as frequently in the solution of problems as the first; the reafon 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 demonftrated 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 demonftrated, 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 fame 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 shewn 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 ravio; 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 demonftrated. In like manner, the other propofitions concerning the last kind of magnitudes may be thewn. 1 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 shewn to be given; from this, by inverfion and converfion the ratio of BC to BE is demonstrated to be given; but, without these two steps, the conclufion should have been made only by citing the 6th propofition. 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 fhewn to be given, by inverfion and compofition, instead of citing prop. 7. and the fame fault occurs in the second part of the fame prop. 11. : Gg2 PROP |