in the place of this, by DR. SIMSON, and other Editors, fo fimple and perfpicuous as could be wifhed. Nothing is gained by the explanation of a term, if the words in which it is expreffed are equally, or more, ambiguous, than the term itfelf: for this reafon, that which is here given, has been preferred to either of thofe abovementioned; though, perhaps, it may not be equally commodious in certain cafes.

It is also to be remarked, that EUCLID never defines one thing by the intervention of another, as is the cafe in DR. SIMSON'S emendation; fo that if this method had occured to him, he would certainly have rejected it.

DEF. 7. BOOK I.

The general definition of an angle in EUCLID, has been properly objected to, by feveral of the modern Editors, as being unneceffary, and conveying no distinct meaning; and in DR. SIMSON's emendation of the ninth, there feems to be ftill a fuperfluous condition. He defines a rectilineal angle, to be "the inclination of two straight lines to one another, which meet together, but are not in the fame ftraight line." Now their not being in the fame ftraight line, is a neceffary consequence, obviously included in their having an inclination to each other; and, therefore, to make this an effential part of the definition, is certainly improper, and unscientific.

DEF. 8, 9. Book I.

EUCLID includes a right angle and a perpendicular in the fame definition, which appears to be immethodical, and contrary to his ufual cuftom. They are certainly diftinct things, though dependent upon each other, and

have as much claim to be feparately defined, as a circle and its diameter,

DEF. 13. Book I.

The definition of a circle from its generation, has been thought by DR. BARROW and others, to be preferable to EUCLID's, or the one here given; as it is fupposed to furnish its properties more readily, and to have the still farther advantage of fhewing the actual existence of such a figure, independent of any hypothefis, but that of granting the poffibility of motion. But the requifition of this poftulatum, appears to be a fufficient reafon why EUCLID rejected fuch a definition. The principles of pure Geometry, have no dependence upon motion, and it is, therefore, never used in the Elements, but in two or three places of the eleventh book, where it could not, without much obfcurity and circumlocution, have been eafily avoided. It is, befides, neither fo fimple, nor convenient to refer to, as EUCLID's; which, in these respects, is as commodious as could be wifhed.

DEF. 20. BOOK I.

DR. BARROW, and other writers of confiderable eminence, have cenfured EUCLID for defining parallel lines, from the negative property of their never meeting each other; and to this they attribute all the perplexity and confufion, which has hitherto attended this delicate fubject affirming it as an utter impoffibility, that any of the properties of thefe lines, can be derived from a definition which contains only a fimple negation. But these affertions appear to be groundless; for the definition is founded on one of the most familiar, fimple and obvious. properties

properties of parallel lines, which either reafon or science can difcover and, on this account, it is certainly preferable to any other that could have been formed from more abstruse and complicated affections of thofe lines, how ready and useful foever such a definition might have been found in its application.

The affertion, likewife, that none of the other properties of parallel lines can be derived from this. definition, has been unadvifedly made; for the 27th Prop. of the first Element, which is the fame as the 22d of the prefent performance, is fairly and elegantly demonftrated by it; and by means fomething fimilar to those made ufe of by DR. SIMSON, in his Notes upon the 29th Prop. it would not be difficult to fhew that all the other properties of thofe lines may be derived from this definition, without the affiftance of the 12th axiom, or any other of the fame kind. DR. SIMSON, indeed, in his attempt to demonftrate this axiom, has made several paralogifms which render his reafonings altogether inyalid, and nugatory. Paffing by others, of lefs confequence, it will be fufficient to obferve, that in his fifth Prop. he takes it for granted, that a line, which is perpendicular to one of two parallel lines, may be produced till it meets the other: now this is a particular cafe of the very thing he is endeavouring to prove, which is so strange an overfight, that it is remarkable how it could escape his observation.

This, however, is not the only inftance of an unfuccessful attempt to prove the truth of the 12th axiom; for CLAVIUS and others have committed fimilar miftakes, and DR. AUSTIN, who has endeavoured to demonftrate it by means of a new definition of parallel

lines, has made ufe of an affumption equally unwarrantable with that mentioned above. That the theory of parallel lines, as it is given in the Elements, is very imperfect, cannot be denied; but no one has yet been. fubftituted in its place which is not equally defective; and in fome inftances ftill more exceptionable: particularly as they are founded on a definition which is derived from an adventitious property of thofe lines, instead of one which is inherent and neceffary, as the nature of the fubject requires.

Whether EUCLID was the author of this axiom cannot perhaps, at this time, be eafily determined; but it is certainly a difgrace to the Elements. The truth of the property here affumed as a thing to be granted, is fo far from being obvious, that it requires demonstration as much as any Prop. in the Elements; and it is always obferved that learners, instead of giving that ready affent to it which an axiomatical principle requires, receive it with doubt and hefitation, and are fcarcely able to comprehend the meaning of it. The one which is here made the 4th postulate, though nearly the fame thing in effect, is much more clear and intelligible.

In the demonftration of this propofition, by EUCLID, that part which relates to the interfection of the circles is, very improperly, omitted; for in a work of this kind, nothing, however evident, ought to be taken for granted; and particularly at the firft outfet, where a ftrictness of elucidation is peculiarly neceffary. The paffing of the circles through each other's centres is, indeed, a fufficient reafon why they muft cut each other; but this fhould certainly have been mentioned in the demonstration.

PROP. 2. BOOK I.

PROCLUS, and other writers, have obferved, that this problem admits of feveral cafes, according to the fituation of the point a; but there is only one of them that can properly be called a separate cafe, which is when the point A is at either of the extremities of the given line and in this cafe, if a circle be described from the given point, at the distance CB, any of the radii of that circle will be the line required. In all other fituations of the point A, whether in the line AB, or out of it, the conftruction and demonflration will be the fame as that given in the text; which differs from EUCLID's only in the producing of the line DA, after the circle FHG is described; this being thought more conformable to the terms of the propofition.

In the construction of this problem the line AD may fall upon the line AB, and then the thing required is done. The given lines c and AB may alfo meet each other, at the point A; and then a circle defcribed from that point, with the radius c, 'will cut off from AB the part required. This cafe occurs in the conftruction of the 5th propofition following, and in feveral other parts of the Elements, and, for that reafon, ought to have been mentioned. In all other pofitions of the two given lines EuCLID's conftruction and demonftration are general.

PROP. 4. BOOK I.

The demonftration of this propofition has been frequently objected against, as being too mechanical. But this complaint is frivolous and ill founded; for the ope