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 wished; the lying evenly between its extreme points, is too vague and indefinite a term to be used in a science fo much celebrated for its ftrictness and fimplicity as Geometry.' ARCHIMEDES defines it to be the fhorteft diftance 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, fo common and fimple, that it feems 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 itfelf; but thefe 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 fubftituted 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 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 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 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 ftill a fuperfluous condition. He defines a rectilineal angle, to be "the inclination of two ftraight 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 confequence, 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 PROP. XX. THEOREM. Every cone is equal to a pyramid of an equal bafe and altitude. Let DABC be a cone, and KEFGH a pyramid, flanding upon equal bafes ABC, EFGH, and having equal altitudes DP, KS; then will DABC be equal to KEFGH. For parallel to the bafes, and at equal distances Do, kr from the vertices, draw the planes nmp and vw. Then by the last Prop. and Prop. 13, the fquare of Do is to the fquare of DP as ump is to ABC; and the square of Kr to the fquare of Ks as vw to EG. And fince the fquares of Do, DP are equal to the fquares of Kr, Ks (Conft. and II. 2.), nmp is to ABC as vw is to EG (V. 11.) But ABC is equal to EG, by hypothefis; wherefore nmp is, alfo, equal to vw (V. 10.) And, in the fame manner, it may be fhewn, that any other fections, at equal diftances from the vertices, are equal to each other. Since, therefore, every fection in the cone is equal to its correfponding fection in the pyramid, the folids DABC, KEFGH of which they are compofed, must be equal. Q. E. D. PROP. XXI. THEOREM. Every cone is the third part of a cylinder of the fame bafe and altitude. Let EAB be a cone, and DABC a cylinder, of the fame bafe and altitude; then will EAB be a third of DABC. For let, KFG, KFGH be a pyramid and prifm, having an equal base and altitude with the cone and cylinder. Then fince cylinders and prisms of equal bases and altitudes are equal to each other (VIII. 18.) the cylinder DABC will be equal to the prism KFGH. And, because cones and pyramids of equal bases and altitudes are equal to each other (VIII. 20.), the cone EAB will be equal to the pyramid KFG. But the pyramid KFG is a third part of the prism KFGH (VIII. 16.), wherefore the cone EAB is, also, a third part of the cylinder DABC. Q.E.D. SCHOLIUM 1. Whatever has been demonftrated of the proportionality of pyramids, prifms, or cylinders, holds equally true of cones, thefe being a third of the latter. 2. It is alfo to be observed, that fimilar cones and cylinders are to each other as the cubes of their altitudes, or the diamaters of their bases; the term like fides being here inapplicable. PROP. XXII. THEOREM. If a sphere be cut by a plane the fection will be a circle. Let the fphere EBD be cut by the plane BSD; then will BSD be a circle. For let the planes ABC, ASC pass through the axis of the sphere EC, and be perpendicular to the plane BSD. Alfo draw the line BD; and join the points A, D and Then fince each of these planes are perpendicular to the plane BSD, their common section ar will also be perpendicular to that plane (VII. 14.) And, because the fides AB, Ar, of the triangle ABr are equal to the fides As, Ar of the triangle Asr, and the angles Arb, Ars are right angles, the fide ri will be equal to the fide rs (I. 4.) In like manner, the fides AD, Ar, of the triangle ADr, being equal to the fides As, Ar, of the triangle Asr, and the angles ArD, Ars right angles, the fide rD will also be equal to the fide rs (I. 4.) The lines rB, 7rD and rs, are, therefore, all equal; and the fame may be fhewn of any other lines, drawn from the |