pointer to the circumference of the section; whence BSD is a circle, as was to be fhewn.

COR. The centre of every fection of a fphere is always in a diameter of the sphere.

PROP. XXIII. THEOREM.

Every fphere is two thirds of its circumfcribing cylinder.

Let rESM be a fphere, and DABC its circumfcribing cylinder; then will rESM be two thirds of DABC.

For let AC be a fection of the fphere through its centre F; and parallel to DC, or AB, the base of the cylinder, draw the plane LH, cutting the former in n and m; and join FE, Fn, FD and Fr.

Then, if the fquare Er be conceived to revolve round. the fixed axis Fr, it will generate the cylinder EC; the quadrant FEr will alfo generate the hemifphere EMTE; and the triangle F Dr the cone FDC.

And fince FHn is a right angled triangle, and FH is equal to Hm, the fquares of FH, Hn, or of нm, Hn, are equal to the fquare of Fn.

But Fn is also equal to FE or HL; whence the fquares of нm, нn are equal to the fquare of HL: or the circular

when it requires a number of prolix arguments to establish its truth and propriety.

DEF. 5. Book I..

EUCLID's definition of a right line is not expreffed in fo accurate and scientific a manner as could be wifhed; the lying evenly between its extreme points, is too vague and indefinite a term to be used in a fcience fo much celebrated for its ftrictness and fimplicity as Geometry.' ARCHIMEDES defines it to be the fhortest distance between any two points; but this is equally exceptionable, on account of the uncertain fignification of the word distance, which, in common language, admits of various meanings. That which is here given, is, perhaps, not much preferable to either of these. The term right, or straight line, is, indeed, so common and simple, that it seems to convey its own meaning, in a more clear and fatisfactory manner than any explanation which can be given of it. DR. AUSTIN, in his Examination of the firft fix books of the Elements, proposes a fingular emendation of this definition, which includes the confideration of right lines, instead of a right line, as the cafe manifeftly requires.

DEF. 6. Book I.

Some call a plane fuperficies that which is the leaft of all thofe having the fame bounds: and others, that which is generated by the motion of a right line, not moving in the direction of itself; but these definitions are too complex and obfcure to answer the purpose required. EUCLID defines it to be that which lies evenly between its lines; which is liable to the fame exception as that given of a right line: nor is the one which has been substituted

in the place of this, by DR. SIMSON, and other Editors, fo fimple and perfpicuous as could be wished. 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 itself: for this reason, 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 alfo 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 occurred 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 still 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 necessary 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 diftinét 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 fuppofed to furnish its properties more readily, and to have the still farther advantage of fhewing the actual exiftence of fuch 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 reason 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 fubjet affirming it as an utter impoffibility, that any of the properties of these lines, can be derived from a definition which contains only a fimple negation. But these affertions appear to be groundlefs; for the definition is founded on one of the most familiar, simple and obvious

properties

properties of parallel lines, which either reason or fcience can discover: and, on this account, it is certainly preferable to any other that could have been formed from more abstruse and complicated affections of those 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 unadvisedly made; for the 27th Prop. of the firft 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 thofe made use of by DR. SIMSON, in his Notes upon the 29th Prop. it would not be difficult to fhew that all the other properties of those 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 feveral paralogisms which render his teasonings altogether invalid, 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 case of the very thing he is endeavouring to prove, which is fo ftrange an overfight, that it is remarkable how it could escape his obfervation.

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