PROP. XIV. XV. odsbanecer Euclid in this book has several propositions 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; though these last occur as frequently in the reason on, which is, that the last may be all demonstrated by help of the first; for if a magnitude, together with a given magnitude, bas 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 : 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 abovementioned, 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 magnitude have 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 shewn. bar PROP. XVI. XVII. In the third part of Prop. 10. in the Greek text, which is the 16th 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 de. monstrated to be given ; but without these two steps, the conclusion should have been made only by citing the 6th Proposition. And in like manner, in the first part of Prop. 11. in the Greek, which in this edition is the 17th, from the ratio of DB to DC 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. giocx odtah PROP. XXI. XXII. 3039er aidT for Sale in posto svitseridecto These now are added, as being wanting to complete the subject treated of in the four preceding propositions. ou 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 same with Prop. 14. in that text, which is here Prop. 18. T. PROP. XXIV. This, which is Prop. 13. in the Greek, is now pat into its proper place, having been disjoined from the three following it in this edition, which are of the same kind. PROP. XXVIII. Dok 1 This, which in the Greek text is Prop. 25. and se. veral of the following propositions, are there deduced from Def. 4. which is not sufficient, as has been mentioned in the note on that definition. They are therefore now shewn more explicitly. Ici PROP. XXXIV. XXXVI. Each of these has a determination, which is now added, which occasions a change in their demonstrations. PROP. X-XXVII. XXXIX. XL. XLI. The 35th and 36th Propositions in the Greek text are joined into one, which makes the 39th in this edition, because the same enunciation and demonstration serves both: and for the same reason Prop. 37. 38. in the Greek are joined into one, which is here the 40th. Prop. 37. is added to the Data, as it frequeutly occurs in the solution of problems; and Prop. 41. is added, to complete the rest. to brain toistoorinco bra. 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. 090311 mol disse to birats PROP XLV. This is Prop. 42. in the Greek, where the three straight lines made use of in the construction are said, 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 cases of it are shewn, which, though necessary, is not done in the Greek. PROP. XLVIII. There are two cases in this proposition, arising from the two cases of the third part of Prop. 47. on wbich the 48th 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. vil 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 58th is true of similar figures, though they be not given in species, and as it frequentJy occurs, it was necessary to add it. tak 1 A Stab PROP. LIX. LXI. 200019 This is the 54th in the Greek; and the 77th in the Greek, being the very same with it, is left out, and a shorter demonstration is given of Prop 61. 22319941? PROP. LXII. T This, which is most frequently useful, is not in the Greek, and is necessary to Prop. 87. 88. in this edi-; tion, as also, though 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 74th is a repetition of Prop. 56. from which it is separated in that text by many propositions; and as there is no order in these propositions, as they stand in the Greek, they are now pút into the order which seemed most convenient and natural. csir 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 observed 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 sake of distinctness; and the demonstration of the 67th 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 49th 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 49th 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 species. Mano s PROP. LXXIII. Prop. 83. in the Greek text, is neither well enunciated nor demonstrated. The 73d, which in this edition is Home 2 considering (see Dr. Gregory's edition), that A, B, C, E, in the Greek text, are four proportionals, and that the proposition is to shew that A, which has a given ratio to E, is to I, as B is to a straight line to which A has a given ratio; or, by inversion, that r is to A, as a straight line to which A has a given ratio is to B; that is, if the proportionals be placed in this order, viz. T, E, A, B, that the first r is to A, to which the second E has a given ratio, as a straight line to which the third A has a given ratio is to the fourth B; which is the enunciation of this 73d, and was thus changed that it might be made like to that of Prop. 72. in this edition, which is the 82d in the Greek text: and the demon-3 stration 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. this edition, must meet the circumference, but does not demonstrate it, which is done here at the end of Cor. S. Prop. 77. in the construction for finding a triangle similar to ABC. bioPROP. LXXVIII. The demonstration of this, which is Prop. 80. in |