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PRO P. XII.

In the 23. Prop. in the Greek text, which here is the 12. the words" Tosaurs dé" are wrong tranflated by Claud. Hardy in μὴ τὸς αὐτὸς δέ his Edition of Euclid's Data printed at Paris Ann. 1625, which was the fift 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 en 8 as as in Prop. 9. of the Greek text. Euclid's meaning is that the ratios mentioned in the Propofition muft not be the fame; 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 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 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 laft ratio be not found. but this is altogether abfurd, becaufe 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 obferved 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 fhould be carefully avoided, as it may lead into other errors.

PRO P. 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' thefe laft occur as frequently in the folution of Problems as the firft, the reafon of which is, that the last may

be

all

all demonstrated 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 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. 1 5. has been added to the Data. one example will make the thing clear; fuppofe 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 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 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 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 demonstrated. 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 fhewn to be given; from this, by inversion and converfion, the ratio of BC to BE is demonftrated to be given; but, without these two fteps, the conclufion fhould have been made only by citing the 6. Propofition. and in like manner, in the first part of Prop. 1 1. 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 Composition, instead 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 preceeding Propofitions.

PROP. XXIII.

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

mediately

mediately 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 fhewn more explicitly.

PROP. XXXIV. XXXVI.

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

PRO P. 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 Demonstration serves both. and for the fame reafon Propp. 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 fotion 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 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.

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

PRO P. XLVIII.

There are two cases in this Propofition, arising 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.

PROP.

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The Conftruction and Demonftration of this which is Prop. 48. in the Greek, are made fomething fhorter 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 Demonftration 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 necellary to add it.

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tho' not men

This which is most frequently ufeful is not in the Greek, and is neceffary to Propp. 87. 88. in this Edition, as alfo, tioned, to Propp. 86. 87. in the former Editions. the Greek text is made a Corollary to it.

PROP. LXIV.

Prop. 66. in

This contains both Prop. 74. and 73. in the Greek text; the firft 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 stand in the Greek, they are now put into the order which seemed most convenient and natural.

The Demonstration of the firft part of Prop. 73. in the Greek is grofsly vitiated. Dr. Gregory fays that the sentences he has inclosed betwixt two stars are fuperfluous and ought to be cancelled; but he has not observed that what follows them is abfurd, being to prove that the ratio [fee his figure] of Ar to гK is given, which by the Hypothefis at the beginning of the Propofition is exprefly given; fo that the whole of this part was to be altered, which is done in this Prop. 64.

PRO P. 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 Greek by means of Prop. 23. of Book 6. of the Elements.

PROP.

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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 Demonftration 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 demonftrated. 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, I, 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 inverfion, that F 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. I, E, A, B, that the firft r 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.

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 Construction for finding a triangle similar to ABC.

PROP. LXXVIII.

The Demonstration of this which is Prop. 80. in the Greek is rendered a good deal fhorter by help of Prop. 77.

PROP.

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