out 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.

PROPOSITIONS XXI. XXII.

These now are added, as being wanting to complete the subject treated of in the four preceding propositions.

PROPOSITION 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.

PROPOSITION XXIV.

This, which is Prop. 13. in the Greek, is now put into its proper place, having been disjointed from the three following it in this edition, which are of the same kind.

PROPOSITION XXVIII.

This, which in the Greek text is Prop. 25. and several 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.

PROPOSITIONS XXXIV. XXXVI.

Each of these has a determination, which is now added, which occasions a change in their demonstrations.

PROPOSITIONS XXXVII. 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.

Proposition 37. is added to the Data, as it frequently oc curs in the solution of problems; and Prop. 41. is added, to complete the rest.

PROPOSITION 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.

PROPOSITION 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 are greater than the third, which is now done.

PROPOSITION 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.

PROPOSITION XLVIII.

There are two cases in this Proposition, arising from the two cases of the third part of Prop. 47. on which the 48th depends and in the composition these two cases are explicitly given.

PROPOSITION LII.

The construction, and demonstration of this, which is Prop. 48. in the Greek, are made something shorter than in that

text.

PROPOSITION 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.

PROPOSITION 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 58th is true of similar figures, though they be not given in species, and as it frequently occurs, it was necessary to add it.

PROPOSITIONS LIX. LXI.

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.

PROPOSITION LXII.

This, which is most frequently useful, is not in the Greek, and is necessary to Prop. 87. 88. in this edition, 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.

PROPOSITION 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 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 observed that what follows them is absurd, being to prove that the ratio [see his figure] of Ar to гK 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.

PROPOSITIONS 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 Ele

ments..

E E

PROPOSITION 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.

PROPOSITION LXXIII.

Prop. 83. in the Greek text is neither well enunciated nor demonstrated. The 73d, which in this edition is put in place of it, is really the same, as will appear by considering [see Dr. Gregory's edition], that A, B, г, 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 г, as B is to a straight line to which A has a given ratio; or, by inversion, that I 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. г, 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 Greck text: and the demonstration 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.

PROPOSITION 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. Prop. 77. in the construction for finding a triangle similar to ABC.

PROPOSITION LXXVIII.

The demonstration of this, which is Prop. 80. in the Greek, is rendered a good deal shorter by help of Prop. 77.

PROPOSITIONS LXXIX. LXXX. LXXXI.

These are added to Euclid's Data, as propositions which are often useful in the solution of Problems.

PROPOSITION LXXXII.

This, which is Prop. 60. in the Greek text, is placed before the 83d and 84th, which in the Greek are the 58th and 59th, because the demonstrations of these two in this edition are deduced from that of Prop. 82. from which they naturally follow.

PROPOSITIONS LXXXVIII. XC.

Dr. Gregory, in his preface to Euclid's works, which he published at Oxford in 1703, after having told that he had supplied the defects of the Greek text of the Data in innumerable places from several manuscripts, and corrected Cl. Hardy's translation by Mr. Bernard's, adds, that the 86th theorem, "or proposition", seemed to be remarkably vitiated, but which could not be restored by help of the manuscripts; then he gives three different translations of it in Latin, according to which he thinks it may be read; the two first have no distinct meaning, and the third, which he says is the best, though it contains a true proposition, which is the 90th in this edition, has no connexion in the least with the Greek text. And it is strange that Dr. Gregory did not observe, that, if Prop. 86. were changed into this, the demonstration of the 86th must be cancelled, and another put into its place: but the truth is, both the enunciation and the demonstration of Prop. 86. are quite entire and right, only Prop. 87. which is more simple, ought to have been placed before it; and the deficiency which the Doctor justly observes to be in this part of Euclid's Data, and which, no doubt, is owing to the carelessness and ignorance of the Greek editors, should have been supplied, not by changing Prop. 86. which is both entire and necessary, but by adding the two propositions, which are the 88th and 90th in this edition.