PRO P. XII. In the a 3. Prop. in the Greek text, which here is the 12. the words us täis antes deare wrong translated by Claud. Hardy in his Edition of Euclid's Data printed at Paris Ann., 1625, which was the Edition of the Greek text; and Dr. Gregory follows him in translating them by the words “ etfi non eafdein," as if the Greek hau been ei ny uj tos 70's as in Prop. q.of the Greek text. Lulu's meaning is that the ratios mentioned in the Proposition mul not be the fi me; for if they were, the Proposition would not be rue. 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 wich 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 confequentiy cannot have a given ratio to it. wherefore these words mut be rendered by non autem eafuem,” but not the saine ratios, as Zunscrtus has traniluted thein in his Edition.

PROP. XIII. Some very ignorant Editor has given a fecond Demontration of this Propolition in the Greek text, which has been as ignoranty kept in it by Claud. Hady and Dr. Gregory, and has been retaiud in the translations of Zambertus and others; Carolus Renaitinus gives it oniy. the author or it has thought thit a ratio was giren if another ratio could be sewn to be the fa'ne to it, tho’this lady ratio be not found. but this to rether absurd, becauic from it would be deduced that the ratio of the lives of any iwo squares is given, and the ratio of the diameters of any two circies, &c. ani it is to be obierved that the moderns frequently take given ratios, and nation that are always the fame for one and the same thing, and Sir Ifiac Newton has failen into this mistake in the 17th Lem. ma of hi» Principia, Ed. 1713. and in other places. but this thould be carefully avoided, as it may lead into other errors.

PROP. XIV. XV. Luclid in this book has several Propositions concerning magnitudes, the exress of one of which above a g.ven magnitude has a given ratio to the other; but he has given none concerning magnituges whereof one together with a given magnitude has a given ratio to the other; tho' these last occur as frequently in the folution o: Problems as the first. the icafon of which is, that the last may be

all deinonstrated by help of the first; for if a magoitude 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 ; Tuppote it were to be demonturated, 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 name Proposition with respect to the last kind of magnitudes abore-mentionců, ihat the first part of Prop. 16. in this Elition is in respect of the first kind. this is Thewn 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 Band A to getier a given ratio ; and by the fecond part of Prop. 14. A and B together with a given magnitade h..s ucto B a girca ratio ; which is the thing that was to be dumontruted. in like manner the other Proportions concerning tie dat kind of magnitudes may be rewn.

PRO P. XVI. XVII. In the third part of Prop. 10. in the Greek text, which is the 16. in this Edition, afcer 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 demonstratad to be given; but, without these two steps, the conclusion thould have been made only by citing the 6. Prom position, and in like manner, in the first 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 hewn to be given, by inversion and Composition, instead of citing Prep. 7. and the fania fault occurs in the second part of the fame Prop. 11.

PRO P. XXI. XXII. These are now added, as being wanting to complete the subjest treated of in the four preceding Propolitions.

PROP. This which is Prop. 20. in the Greek text, was separated from Piop. 14. 15. 16. in that text, after which it should have buca

immediately placed, as being of the fame kind. it is now pot 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. 18.

PROP. XXIV. This, which is Prop. 13. in the Greek, is now put into its proper place, having been disjoined from the three foilowing it in this Edition, which are of the same kind.

PRO P. XXVIII. This which in the Greek ţext is Prop. 25. and several of the following Propositions, 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 shewn more explicitly.

PRO P. XXXIV. XXXVI. Each of these has a Determination, which is now added, which occasions a change in their Demonstrations.

PRO P. XXXVII. XXXIX. XL. XLI. The 35. and 36. Proposnions in the Greek text are joined into one, which makes the 39. in this Edition, because the fame Enuntiation and Demonftration serves both. and for the same reason Prop. 37. 30. in the Greck 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 reit,

PRO P. 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 geed into the Demonstration, but is now only cited.

PRO P. XLV. This is Prop. 42. in the Greek, where the three straight lines made use of in the construction are faid, but not shewn, to be such that any two of them is greater than the third, which is now done.

P R O P. XLVII. This is Prop. 44. in the Greek text, but the Demonstration of it is changed into another wherein the several cases of it are hewn, which, tho' necesary, is not done in the Greek.

PRO P. XLVIII. There are two cases in this Proposition, arising from the two cases of the 3d part of Prop. 47. on which the 49. depends, and in the Composition these two cases are explicitly given.

PROP. LII. The Construction and Denonitration 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 case of Prop. 49. in thai 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 Propolition is concerning figures that are given in species; this 58. is true of similar figures, tho' they be not given in species, and as it frequently occurs, it was necessary to add it.

PRO P. LIX. LXI. This is the 54. in the Greek; and the 77. in the Greek, being the

very same with it, is left out, and a shorter Demonitiation is given of Prop. 61.

PROP. LXIT. This which is most frequently useful is not in the Greck, and is neceffary to Prop. 87. 88. in this Edition, as also, tho' not mentioned, to Prop. 86. 87. in the former Editivas. Prop. 66. in the Greck text is made a Corollary to it.

PROP. LXIV. . This contains both Prop. 74, and 73. in the Gieck text; the first case 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 these Propositions, as they stand in the Greck, they are now put in:o the order which seemned most convenient and natural.

The Demonitration of the first part of Prop. 73. in the Greck is grossly vitiated. Dr. Gregory says that the sentences he has inclosed betwixt two stars are superfluous and ought to be cancelled; but he has not observed that what follows them is absurd, being to prove that the ratio (see his figure] of ar to CK is given, which by the Hypothesis at the beginning of the Proposition is expressly given; so 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 distinctress; and the Demonstration of the 67. is rendered shorter than that of the first part of Prop. 70. in the Grick 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.

Di. Gregory in the Demoniliation of Trop. 6 2. cites the 4). Prop. Dat. to prove that the ratio of the figure AEB to the faraj. lelogram AH is given, whereas this was thewn a few lines before; and belides the 49. Prop. is not applicable to theie two figures, because All is not given in species, bui is, by the step for wich the citation is brought, proved to be given in species.

PROP. LXXIII. Prop. S 3. in the Greek text is neither well enuntiated nor demonitrated. the 73. which in this Edition is put in place of it, is really the same, as will appear by confidering [lee Dr. Grego.y's Edition) that A, B, C, E in the Greek text are four proportives, and that the Propotition is to thew that a, which has a given raiso to L, is to l', as B is to the straight line to which A has a grea ratio ; or, by invertion, that ris to 1, as the ftraight line to waita A has a given ratio is to B; that is, it the proportionals be placed ja tuis wider, viz. C', t, 1, B, that the fifti' is to s to which tre icconi E has a given ratio, as the straight line to which the third

has a given ratio is io the fourth B; which is the Inuntiation uis 73. anů 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 Demonstration of Prop. 73. is the same with that of Prop. 72. only making use of Prop. 23. in:tead of Prop. 22. of Book 5. of the Elements,

PRO P. LXXVII. This is put in place of Prop. 79. in the Greek text which is act a Datuin, but a Thcorem premised as a Lemma to Prop. 80. in that text. and Prop. 79. is made Cor. 1. to Prop. 77. in this Editiva. Cl. Hardy in his Edition of the Data takes notice, that, in Prop. So. of the Greck text, the parallel KL in the figure of Pros; in this Llition must meet the circum'erence, but does not densestrate it, which is doue here at the end of Cor. 3. of Prop. 77. in the constrution for finding a triangle similar to ABC.

PRO P. LXXVIII. The Demonstration of this which is Prop. 83. in the Greek is rendered a good deal fhorter by help of Prop. 77.

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