In the 23. Prop. in the Greek text, which here is the 12. the words " Tès aurès dé” 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 sus as in Prop 9. of the Greek text. Euclid's meaning is that the ratios mentioned in the Propofition must not be the same; 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 consequently cannot have a given ratio to it. wherefore these words must be rendered by "non autem eafdem," but not the same ratios, as Zambertus has tranflated them in his Edition.

Some very ignorant Editor has given a second Demonftration of this Propofition 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 fhewn to be the fame to it, tho' 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 fquares 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 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 magnitude 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 occur as frequently in the solution of Problems as the firit. the reafon of which is, that the last may be

all demonftrated by help of the firft; for if a magnitude together with a given magnitude has a given ratio to another magnitude; the excefs of this other above a given magnitude fhall 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; fuppofe 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 Propofition 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 fhewn thus; from the hypothefis, 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 excefs 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 Propofitions concerning the last kind of magnitudes may be shewn.

PROP. XVI. XVII.

In the third part of Prop. 10. in the Greek text, which is the 16. in this Edition, after the ratio of EC to CB has been shown to be given; from this, by inverfion and converfion, the ratio of BC to BE is demonftrated to be given; but, without these two steps, the conclusion should have been made only by citing the 6. Propofition. and in like manner, in the first part of Prop. 11. in the Greek, which in this Edition is the 17. from the ratio of DB to BC being given, the ratio of DC to DB is fhewn to be given, by inverfion and Compofition, inftead of citing Prop. 7. and the fame fault occurs in the second part of the fame Prop. 11.

PROP. XXI. XXII.

These are now added, as being wanting to complete the subject treated of in the four preceding Propofitions.

PROP. XXIII.

This which is Prop. 20. in the Greek text, was separated from Prop. 14. 15. 16. in that text, after which it fhould have been

immediately placed, as being of the fame kind. it is now put into its proper place. but Prop. 21. in the Greek is left out, as being the fame 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 fame kind.

PROP. XXVIII.

This which in the Greek text is Prop. 25. and feveral of the following Propofitions, are there deduced from Def. 4. which is not fufficient, as has been mentioned in the Note on that Definition; they are therefore now shewn more explicitly.

PROP. XXXIV. XXXVI.

Each of these has a Determination, which is now added, which occafions a change in their Demonstrations.

PROP. XXXVII. XXXIX. XL. XLI.

The 35. and 36. Propofitions in the Greek text are joined into one, which makes the 39. in this Edition, because the fame Enuntiation and Demonftration ferves both. and for the fame reason Prop. 37. 38. in the Greek are joined into one which here is the 40.

Prop. 37. is added to the Data, as it frequently occurs in the folution of Problems. and Prop. 41. is added to complete the rest. PROP. XLII.

This is Prop. 39. in the Greek text, where the whole conftruction of Prop. 22. of Book 1. of the Elements is put without need into the Demonftration, but is now only cited.

PROP. XLV.

This is Prop. 42. in the Greek, where the three ftraight lines made ufe of in the construction are faid, but not fhewn, to be fuch that any two of them is greater than the third, which is now done.

PROP. XLVII.

This is Prop. 44. in the Greek text, but the Demonftration of it is changed into another wherein the several cafes of it are shewn, which, tho' neceffary, is not done in the Greek.

PROP. XLVIII.

There are two cafes in this Propofition, arifing from the two cafes of the 3d part of Prop. 47. on which the 48. depends. and in the Composition these two cafes are explicitly given.

4

The Conftruction and Demonstration of this which is Prop. 48. in the Greek, are made fomething shorter than in that text.

PROP. LIII.

Prop. 63. in the Greek text is omitted, being only a cafe of Prop. 49. in that text, which is Prop. 53. in this Edition.

PROP. LVIII.

This is not in the Greek text, but its Demonstration is contained in that of the first part of Prop. 54. in that text; which Propofition is concerning figures that are given in fpecies; this 58. is true of fimilar figures, tho' they be not given in species, and as it frequently occurs, it was necessary to add it.

PROP. LIX. LXI.

This is the 54. in the Greek; and the 77. in the Greek, being the very fame with it, is left out. and a fhorter Demonstration is given of Prop. 61.

PROP. LXII.

This which is most frequently useful is not in the Greek, and is neceffary to Prop. 87. 88. in this Edition, as also, tho' 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 74. is a repetition of Prop. 56. from which it is separated in that text by many Propofitions; and as there is no order in these Propofitions, as they stand in the Greek, they are now put into the order which feemed most convenient and natural.

The Demonftration of the first part of Prop. 73. in the Greek is grossly vitiated. Dr. Gregory fays that the fentences he has inclofed betwixt two ftars are fuperfluous and ought to be cancelled; but he has not obferved that what follows them is abfurd, being to prove that the ratio [see his figure] of AT to TK is given, which by the Hypothesis at the beginning of the Proposition is .exprefsly given; fo that the whole of this part was to be altered, which is done in this Prop. 64.

PROP. LXVII. LXVIII.

Prop. 70. in the Greek text is divided into thefe two, for the fake of distinctness; and the Demonstration of the 67. is rendered

, shorter than that of the first part of Prop. 70. in the Greek by means of Prop. 23. of Book 6, of the Elements.

. PROP. LXX.

This is Prop. 62. in the Greek text; Prop. 78. in that text is only a particular case of it, and is therefore omitted.

Dr. Gregory in the Demonstration of Prop. 62. cites the 49. Prop. Dat. to prove that the ratio of the figure AEB to the parallelogram AH is given, whereas this was fhewn a few lines before; and befides the 49. Prop. is not applicable to these two figures, because AH is not given in fpecies, but is, by the step for which the citation is brought, proved to be given in fpecies.

PROP. LXXIII.

Prop. 83. in the Greek text is neither well enuntiated nor demonstrated. the 73. which in this Edition is put in place of it, is really the fame, as will appear by considering [fee Dr. Gregory's Edition] that A, B, T, E in the Greek text are four proportionals, and that the Propofition is to fhew that A, which has a given ratio to E, is to г, as B is to the straight line to which A has a given ratio; or, by inverfion, that I is to A, as the ftraight line to which A has a given ratio is to B; that is, if the proportionals be placed in this order, viz. г, E, A, B, that the first r is to A to which the second E has a given ratio, as the straight line to which the third A has a given ratio is to the fourth B; which is the Enuntiation of this 73. and was thus changed that it might be made like to that of Prop. 72. in this Edition, which is the 82. in the Greek text. and the Demonftration of Prop. 73. is the fame with that of Prop. 72. only making use of Prop. 23. instead of Prop. 22. of Book 5. of the Elements.

PROP. LXXVII.

This is put in place of Prop. 79. in the Greek text which is not a Datum, but a Theorem premised as a Lemma to Prop. 80. in that text. and Prop. 79. is made Cor. 1. to Prop. 77. in this Edition. Cl. Hardy in his Edition of the Data takes notice, that, in Prop. 80. of the Greek text, the parallel KL in the figure of Prop. 77. in this Edition must meet the circumference, but does not demonftrate it, which is done here at the end of Cor. 3. of Prop. 77. in the conftruction for finding a triangle fimilar to ABC.

PROP. LXXVIII.

The Demonftration of this which is Prop. 8o. in the Greek is rendered a good deal shorter by help of Prop, 77,