14.

nitude 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 thefe laft occur as frequently in the folution of problems as the firft; the reafon of which is, that the laft may be all demonftrated 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 firft, and on the contrary; as we have demonstrated in prop. And for a like reafon prop. 15. has been added to the data. One example will make the thing clear; fuppofe it were to be demonftrated, 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 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 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 fecond 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 demonftrated. In like manner, the other propofitions con cerning the last kind of magnitudes may be fhewn.

PROP. XVI. XVII.

In the third part of prop. 1o. in the Greek text, which is the 16th in this edition, after the ratio of EC to CB has been fhown to be given; from this, by inverfion and converfion the ratio of BC to BE is demonftrated to be given; but without these two steps, the conclufion fhould have been made only by citing the 6th propofition. 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 BC being given, the ratio of DC to DB is fhewn to be given, by invertion and compofition, instead of citing prop. 7. and the fame fault occurs in the fecond part of the fame prop. II.

PROP. XXI. XXII.

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

PROP. XXIII.

This, which is prop. 20. in the Greek text, was feparated from prop. 14. 15. 16. in that text, after which it should have been immediately 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 thefe has a determination, which is now added, which occafions a change in their demonstrations.

PRO P. XXXVII. XXXIX. XL. XLI.

The 35th and 36th propofitions in the Greek text are joined into one, which makes the 9th in this edition, because the fame enunciation and demonftration ferves both: And for the fame reafon prop. 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 folution of problems; and prop. 41. is added to complete the reft.

PROP. XLII.

This is prop. 39. in the Greek text, where the whole conftruction of prop. 22. of book I, 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 straight lines made ufe of in the conftruction are faid, but not fhewn, to be fuch that any two of them is greater than the third, which is now done.

PROP.

PROP. XLVII.

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, though neceffary, is not done in the Greek.

There are two cafes in this propofition, arifing from the two cafes of the third part of prop. 47. on which the 48th depends; and in the compofition these two cafes are explicitly given.

PROP. LII.

The conftruction and demonstration of this, which is prop. 48. in the Greek, are made fomething fhorter than in that text.

PRO P. 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 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 58th is true of fimilar figures, though they be not given in fpecies, and, as it frequently occurs, it was neceflary to add it.

PRO P. LIX. LXI.

This is the 54th in the Greek; and the 77th in the Greek, being the very fame with it, is left out, and a fhorter demonftration is given of prop. 61.

PROP. LXII.

This, which is most frequently useful, is not in the Greek, and is necessary to prop. 87. 88. in this edition, as alfo, 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 cafe of the 74th 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 ftand in the Greek, they are now put into the order which feemed moft convenient and natural.

1

The demonftration of the first part of prop. 73. in the Greek is grofsly vitiated. Dr Gregory fays, that the fentences he has inclosed betwixt two ftars are fuperfluous, and ought to be cancelled; but he has not obferved, that what follows them is abfurd, being to prove that the ratio [See his figure] of Ar to r K is given, which by the hypothefis at the beginning of the propofition is exprefsly given; fo 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 diftinctnefs; and the demonstration of the 67th 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. LXX.

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 49th 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 49th 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 enunciated nor demonftrated. The 73d, which in this edition is put in place of it, is really the fame, as will appear by confidering [See Dr Gregory's Edition] that A, B, r, 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 F, as B is to a ftraight line to which A has a given ratio; or, by inverfion, that 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. r, E, A, B, that the firft r is to A to which the fecond E has a given ratio, as a ftraight 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

the 82d in the Greek text: And the demonftration of prop. 73. is the fame 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 premifed 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 conftruction for finding a triangle. fimilar to ABC.

PROP. LXXVIII.

The demonftration of this, which is prop. 8o. in the Greek, is rendered a good deal shorter by help of prop. 77.

PROP. LXXIX. LXXX. LXXXI.

Thefe are added to Euclid's data, as propofitions which are often useful in the folution of problems.

PROP. 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 demonftration of these two in this edition are deduced from that of prop. 82. from which they naturally follow.

PROP. LXXXVIII. XC.

Dr Gregory, in his preface to Euclid's works, which he published at Oxford in 1703, after having told that he had fupplied the defects of the Greek text of the data in innu merable places from feveral manufcripts, and corrected Cl. Hardy's tranflation by Mr Bernard's, adds, that the 86th theorem," or propofition," feemed to be remarkably vitiated, but which could not be reftored by help of the manufcripts; then he gives three different tranflations of it in Latin, according to which he thinks it may be read; the two first have no diftinct meaning, and the third, which he fays is the best, though it contains a true propofition, which is the goth in this e

Gg 4

dition,