In the 13. Prop. in the Greek text, which here is the 12. the words "un TS Urs de" are wrong tranflated by Claud. Hardy in his Edition of Euclid's Data printed at Paris Ann., 1625, which was the .it 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 eas in Prop. 9. of the Greek text. Lld's meaning is that the ratios mentioned in the Propofition must not be the f me; for if they were, the Propofition would not be nue. 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 fift 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 thefe words must be rendered by non autem eafdem," but not the fame ratios, as Zambertus has tranilated them in his Edition.

66

PROP. XIII.

Some very ignorant Editor has given a fecond Demo istration of this Propofition in the Greek text, which has been as ignoranty 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 to gether 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 circies, &c. and it is to be obferved that the moderns frequently take given ratios, and ratios that are always the fame for one and the fame thing, and Sir fiac Newton has failen into this mistake in the 17th Lemma of his Principia, Ed. 1713. and in other places. but this thould be carefully avoided, as it may lead into other errors.

PROP. XIV. XV.

Euclid in this book has feveral Propofitions concerning magnitudes, the ex efs 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' thefe laft occur as frequently in the folution o. Problems as the firft. the reafon of which is, that the laft 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 excels of this other above a given magnitude fhall have a given ratio to the firft, and on the contrary; as we have demonftrated in Prop. 14. and for a like reafon 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 refpect to the last kind of magnitudes above-mentioned, that the first part of Prop. 16. in this Edition is in refpect of the fift kind. this is fhewn thus; from the hypothefis, and by the firft part of Prop. 14. the excess of B above a given magnitude has unto A a given ratio; and therefore, by the firit part of Prop. 17. the excefs of B above a given magnitude has unto B and A together a given ratio; and by the fecond part of Prop. 14. A and B together with a given magnitade has unto B a given ratio; which is the thing that was to be demonftrated. in like manner the other Propofitions concerning the laft kind of magnitudes may be fhewn.

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 conclufion fhould have been made only by citing the 6. Propofition. and in like manner, in the firft part of Prop. 11. in the Greck, which in this Edition is the 17. from the ratio of DB to BC being given, the ratio of DC to DB is thewn to be given, by invertion and Compofition, inftead of citing Prep. 7. and the fanie fault occurs in the fecond part of the fame Prop. 11.

PROP. XXI. XXII.

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

This which is Prop. 20. in the Greek text, was feparated from Piop. 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 several of the following Propofitions, are there deduced from Def. 4. which is not fufficient, as has been mentioned in the Note on that Defini, tion; they are therefore now fhewn more explicitly.

PRO P. XXXIV. XXXVI.

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

PROP. XXXVII. XXXIX. XL. XLI.

The 35. and 36. Propofitions in the Greck text are joined into one, which makes the 39. in this Edition, becaufe the fame Enuntiation and Demonftration ferves both. and for the fame reason Trop. 37. 3. 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 Frop. 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 use of in the conftruction 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 feveral cafes of it are shewn, which, tho' neceflary, 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 Compofition these two cafes are explicitly given.

PRO P. LII.

The Construction and Demonftration 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.

This is not in the Greek text, but its Demonfiration 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 fpecies, and as it frequently occurs, it was neceffary 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 Demonftiation 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 alfo, 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 Greck text; the first cafe of the 74. is a repetition of Prop. 56. from which it is feparated in that text by many Propofitions; and as there is no order in thefe Propofitions, as they ftand in the Greck, they are now put into the order which feemed moft convenient and natural.

The Demonftration of the first part of Prop. 73. in the Greek is grofsly 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 [fee his figure] of Ar to PK is given, which by the Hypothefis at the beginning of the Propofition 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 these two, for the fake of diftinctnefs; and the Demonftration of the 67. is rendered fhorter than that of the first part of Prop. 70. in the Grock by means of Prop. 23. of Book 5. of the Elements.

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

Dr. Gregory in the Demonflration of Prop. 62. cites the 49. Prop. Dat. to prove that the ratio of the figure AEB to the parailelogram AH is given, whereas this was thewn a few lines before; and befides the 49. Prop. is not applicable to there two figures, because Al is not given in fpecies, bac 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 confidering [fee Dr. Gregory's Edition] that A, B, F, 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 I, as B is to the ftraight line to which A has a given ratio; or, by invertion, that F is to A, as the straight line to which A has a given ratio is to B; that is, if the proportionals be placed in this order, viz. I, E, A, B, that the firft is to a to which the fecond E has a given ratio, as the ftraight 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 ufe of Prop. 23. inftead of Prop. 22. of Book 5. of the Elements.

This is put in place of Prop. 79. in the Greek text which is not a Datum, but a Theorem premifed 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. 8o. of the Greck text, the parallel KL in the figure of Prop. 77. in this Edition must meet the circumference, but does not demonftrate it, which is doue 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 fhorter by help of Prop. 77.