PROP. 28. Book I.

In most editions of EUCLID, two corollaries are affixed to this propofition; which are equally or more intricate than the propofition itself. DR. AUSTIN has endeavoured to prove that thefe, and moft of the other corollaries, to be found in the Elements, were not introduced by EuCLID, but by fome of his commentators, or interpreters; and there are many reafons for believing that this opinion, is not ill founded. It is generally allowed, that EUCLID wrote a book entitled Corollaries, which were a collection, of confequences deducible from his Elements; and, therefore, it is not to be imagined that they were originally inferted in that work; as in that cafe it would have been quite unneceffary to have published them in a separate performance. Befides this, the chain of reafoning is. complete without them, as is evident from their being feldom referred to in any propofition. In all cafes, however, where a ufeful truth of this kind can be readily de. duced from a preceding demonftration, there appears to be no impropriety in making it a corollary.

PROP. 31. Book I.

This propofition, as it ftands in most of the early editions, has three diftinct cafes, which all require to be feparately demonftrated. DR. SIMSON, by changing the mode of demonftration, has reduced it to two: but by an obvious alteration in the enunciation of the I. 26. EuC. which is the fame as our 21ft, the propofition, both for parallelograms and triangles, might have been demonfrated generally, in one cafe only; which, when it can

be done, is always to be preferred. In this part of the work, alfo, feveral other alterations have been made, the reafon for which will be given in the notes to the second book.

PROP. 33. Book I.

This propofition is fubftituted in the place of Prop. 42, 44, and 45 of Euc. B. I. as being lefs intricate, and equally useful in its application. One of the principal designs of these propofitions, is to fhew, that a parallelogram, under certain conditions, can be formed; and as this can be more readily effected by other methods, the preference has been given to that which appears the most fimple. It may also be observed, that the 44th proposition of EUCLID is not legally demonftrated; for the parallelogram BF, which makes a part of the conftruction, cannot be formed from Prop. 42, as is directed, being entirely a different cafe: and as the 45th is derived from the 44th, it must also be liable to the fame objection.

PROP. 34, 35. Book I.

These propofitions are delivered by EUCLID in a different form, and not given till the 6th book; but as they are extremely easy, and of frequent use in their application to other propofitions, in the preceding books, they have been here introduced as early as poffible, and demonftrated independently of the doctrine of proportion; which, it is imagined, beginners will confider as an advantage, as they feldom arrive to fuch a proficiency, in a knowledge of the Elements, as to obtain clear and fatisfactory ideas of that intricate subject.

PROP. 1. BOOK II.

In EUCLID's demonftration of this problem, it ought to have been proved that the lines which are directed to be drawn parallel to AB, AD, will meet each other; or otherwise it is not certain that the fquare required can be formed. On this account, a mode of conftruction has been here obferved, which is not liable to that objection.

PROP. 2, 3, 4. Book II.

Thefe propofitions, which are not in EUCLID, may by fome, be thought unneceffary; but they muft either be demonftrated, or affumed; as the firft, in particular, is wanted in almost every propofition of the second book; and the others are frequently required in feveral parts of the Elements. Why they were omitted by EUCLID does not appear; they are certainly not axiomatical, nor more evident in themselves than many others which he has fcrupulously demonftrated.

PROP. 5. BOOK II.

This propofition is placed in the second book, for the purpose of demonftrating it in a more general manner than has been done by EUCLID; and in order that some others, of little importance, might be more easily omitted. The demonftration depends principally upon the first propofition, mentioned above; and this, among other inftances, is fufficient to fhew the utility of that theorem, and the neceffity of its being introduced into the Elements.

PROP 7. BOOK II.

This theorem, which is not in EUCLID, is given chiefly on account of its application to fome of the following propofitions, the demonftrations of which are, by this means, rendered more concife and elegant.

PROP. 13. Book II.

All the theorems in EUCLID's fecond book, which relate to the divifion of a line into more than two parts, are here omitted, as they are commonly found tedious and embarraffing to beginners, and are not of any very extenfive use. The prefent propofition, which is not in EUCLID, is much more generally applicable; and this, together with the preceding ones, will be found fufficient for moft geometrical purposes.

PROP. 16, 19, 20 and 21. Book II.

Thefe propofitions, though not in EUCLID, are frequently wanted, particularly the 1st, 2d, and 3d, which are, alfo, equally remarkable, both for their elegance and utility.

PROP. 1. Book III.

It is properly obferved by DR. SIMSON, in his notes upon this book, that the objections which have been ufually made against the indirect method of proof, used in this and feveral other propofitions in the Elements, are injudicious and ill founded; as it is obvious. to every one, who has duly confidered the fubject, that there are many things which cannot be proved in any

other way. There is, however, a real defect in the demonftration of this propofition, that escaped his notice; which is, that the fictitious centre, or point G, may bè taken in the line EC'; and in this cafe the demonftration given by EUCLID will not hold.

PROP. 4. BOOK III.

This propofition is the fame as the 9th of EUCLID; Book III. but demonstrated in a manner which it is ima gined will appear fomething more clear and fatisfactory, at least to beginners According to his method the propofition admits of feveral cafes; and in that which he has chofen as a general one, the fictitious centre, or point È, is so taken, that the proof would be exactly the fame for two equal right lines as for three, which is a manifeft imperfection,

PROP. 5. BOOK III.

EUCLID has given this theorem in his 3d Book, in the form of a definition; which is the more remarkable, as he appears, in feveral parts of the Elements, to be well aware, that the equality of no two figures can be admitted but from the teft which he has laid down in the 8th axiom,

PROP. 6, 7. BOOK III.

These theorems are, in fubftance, the fame as EuCLID's, but differently enunciated, in order to accommodate beginners, who are generally embarraffed with the aukwardnefs of the figures, and the two fictitious centres in the laft propofition; the latter of which are