PROP. I. 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.

These propofitions, which are not in EUCLID, may, by fome, be thought unneceffary; but they must either be demonstrated, or affumed; as the firft, in particular, is wanted in almoft 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 demonstrated.

PROP. 5. Book II.

This propofition is placed in the fecond book, for the purpose of demonftrating it in a more general manner than has been done by EUCLID; and in order that fome 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 coneife 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. Воок ІІ.

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

PROP. I. 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, ufed 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 de-monftration of this propofition, that escaped his notice; which is, that the fictitious centre, or point G, may be 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 imagined 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 £, is fo 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.

1

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 test 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

here avoided. It may also be obferved that the demonftrations of these propofitions are not strictly scientific; fince, for aught that appears to the contrary, the circles may touch each other in more points than one, in which cafe the proof he has given would be nugatory. To avoid this, the fucceeding propofition fhould have been placed prior in order to the present ones, and demonftrated independently of them; which, however, cannot eafily be done. For this reason, and to avoid as much as poffible all theorems which are otherwife of little importance, the touching of the circles in one point only, has been here inferred from the definition. A fimilar objection may, likewise, be made against the 5th, 6th, and 10th theorems of EUCLID, Book III.; the laft of which should have been demonftrated firft; for, as they now stand, feveral things which require proof, as much as the propofitions themselves, are taken for granted.

PROP. IO. Book III.

In this propofition, no mention is made of the cornicular angle, or that which is fuppofed to be formed by the circumference and tangent, at the point of contact; as it is of no ufe whatever in Geometry, and ought never to have been admitted into the Elements, DR. SIMSON fufpects, with VIETA, that it is an interpolation, and on that account has properly rejected it; but there are ftill fome particulars, in his enunciation of this propofition, which appear to be equally unneceffary. The theorem is, therefore, here propofed in as fimple a manner as poffible, and reftricted to that cafe which moft frequently

pccurs.

PROP. 14. Book III.

In EUCLID's demonftration of the fecond cafe of this theorem, the following propofition has been taken for granted, viz. "If one magnitude be double of another, and a part taken from the first, be double of a part taken from the fecond, the remainder of the firft will be double the remainder of the fecond." But as this affumption, which has hitherto been tacitly acquiefced in, is not derived from the axioms, or any thing which has been previously demonstrated, it is certainly improper, and unjuftifiable. In order, therefore, to render the demonftration of this cafe more ftrict and scientific, it is here given in a different manner, which is equally eafy with the former, and not liable to the fame objection,

PROP. 15. Book III.

DR. AUSTIN in his examination of the first fix books of the Elements, is of opinion that the fecond cafe of this propofition, which has been added by DR, SIMSON, and other modern Editors, is unneceffary. "The former propofition, he obferves, is general; and, therefore, it is immaterial, whether the part of the circumference upon which the angles at the centre and circumference ftand, be greater or less than a femicircle." But this obfervation is foreign to the purpofe; for as the arc which fubtends an angle at the centre, muft always be lefs than a femicircle, no fuch angle can be introduced into the conftruction of this cafe; and therefore the demonftration of it must neceffarily be obtained in fome way different from the former.