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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 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 an- А
B A gle 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 a be found to which C has the same ratio that E has to Z? с certainly no way, until it be shown how
Eto find an angle to which a given angle has a given ratio which cannot be done
Zby Euclid's Elements, nor probably by any Geometry known in his time. Therefore, in all the propositions of this book which depend upe on this second, the abovementioned limitation must be under stood, though it be not explicitly mentioned.
The order of the propositions in the Greek text, betwech 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 Greck, 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 24ch 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 art therefore now shown without it.
In the 23d prop. in the Greek text, which here is the 12th, the words, "Men 786 euros de," 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 worris, “etsi non easdem," as if the Greek had been anal than 785 &s, as in prop. 9, of the Greek text.
Euclid's meaning is, thit 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.
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, ihough 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 iwo 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 bas 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 magninitude 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 à 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 being the same with prop. 14, in that text, which is here
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 defini. tion: 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 I. 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.
This is prop. 44, in the Greek text; but the demonstration of it is changed into another, wherein the several cases of it are shown, which, though necessary, is not done in the Greek.
There are two cases in this proposition, arising from the two cases of the third part of prop. 47, on which the 48th depends ; and in the composition these two cases are explicitly given..
The construction and demonstration of this, which is prop. 48, in the Greek, are made something shorter than in that text.
Prop. 63, in the Greek text is omitted, being only a case of prop. 49, in that text, which is prop. 53, in this edition.
This is not in the Greek text, but its denionstration is contained in that of the first part of prop. 54, in that text; whichi proposition is concerning figures that are given in species: this 58th is true of similar figures, though they be not given in species, and as it frequently occurs, it was necessary to add it..
PROP. LIX, LXI. This is the 54th in the Greek; and the 77th in the Greek, being the very same with it, is left out, and a shorter demon, stration is given of prop. 61.
PROP. LXII. This, which is most frequently useful, is not in the Greek, and is necessary to prop. 87, 88, in this edition, as also, though not mentioned, to prop. 86, 87, in the former editions. Prop. 66, in the Greek text, is made a corollary to it.
PROP. LXIV. This contains both prop. 74, and 73, in the Greek text; the first case of the 74th is a repetition of prop. 56, from which it is separated in that text by many propositions; and as there is no order in these propositions, as they stand in the Greek, they are now put into the order which seemed most convenient and natural.