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PROP. XII. In the 23. Prop. in the Greek text, which here is the 12. the words“ ja tos aires des are wrong translated 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 translating them by the words “et G non eafdem," as if the Greek had been outis aires as in Prop 9. of the Greek text, Euclid's meaning is that the ratios mentioned in the Proposition must not be the same; for if they were, the Proposition would not be true. whatever satio 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 çannot 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.
PROP. XIII. Some very ignorant Editor has given a second Demonstration of this Proposition in the Greek text, which has been as ignorantly kept in it 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 shewn to be the same to it, tho' 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 two 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 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 fisit. 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 firft, 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 shewn 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 magpitude has unto B a 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 fhewn.
PROP. XVI. XVII. In the third part of Prop. 19. in the Greek text, which is the 16. in this Edition, after the ratio of EC to CB has been shewn 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 6. Proposition. 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 shewn 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 are now 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 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 sufficient, as has been mentioned in the Note on that Defini. tion; they are therefore now shewn 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 35. and 36. Propofitions in the Greek text are joined into one, which makes the 39. in this Edition, because the same Enuntiation and Demonstration serves both. and for the same reason Prop. 37. 38. in the Greek are joined into one which here is
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 construction of Prop. 22. of Book 1. 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 ufe 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.
PROP. XLVII. This is Prop. 44. in the Greek text, but the Demonstration of it is changed into another wherein the several cafes of it are shewn, which, tho' necessary, is not done in the Greek.
PROP. XLVIII. There are two cases in this Propofition, arising from the two cases of the 3d part of Prop. 47. on which the 48. depends. and in the Composition these two cases are explicitly given.
PROP. LII. The Construction and Demonstration of this which is Prop. 48. in the Greek, are made something shorter than in that text.
PROP. LIII. 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.
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 Proposition 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.
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 shorter 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 seemed most convenient and natural.
The Demonstration of the first part of Prop. 73. in the Greek 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 obferved that what follows them is absurd, being to prove that the ratio [see his figure] of Ar to TK 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 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 shewn a few lines before; and besides the 49. Prop. is not applicable to these two figures, because AH is not given in species, 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 same, as will appear by considering [fee Dr. Gregory's Edition) that A, B, F, E in the Greek text are four proportionals, and that the Proposition is to fhew that A, which has a given ratio to E, is to I, as B is to the straight line to which A has a given ratio; or, by inversion, that I 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 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 same 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 demonstrate 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 shorter by help of Prop. 77,