EMERITUS PROFESSOR OF MATHEMATICS IN TIIE UNIVERSITY OF GLASGOW, LONDON: PRINTED FOR WINGRAVE AND COLLINGWOOD), And the rest of the Proprietors. 1816. NOTES, &c. a DEFINITION I. BOOK I. It is necessary to consider a solid, that is, a magnitude Book I. which has length, breadth, and thickness, in order to understand aright the definitions of a point, line, and superficies; for these all arise from a solid, and exist in it: The boundary, or boundaries, which contain a solid are called superficies, or the boundary which is common to two solids which are contiguous, or which divides one solid into two contiguous parts, is called a superficies: Thus, if BCGF be one of the boundaries which contain the solid ABCDEFGH, or which is the common boundary of this solid and the solid BKLCFNMG, and is therefore in the one as well as in the other solid, is called a superficies, and has no thickness : For, if it have any, this thickness G. M must either be a part of the thickness of the solid AG, or of the so- & E N lid BM, or a part of the thickness of each of them. It cannot be a part of the thickness of the solid DI L BM; because if this solid be removed from the solid AG, the superficies BCGF, the boundary A B K of the solid AG, remains still the same as it was. Nor can it be a part of the thickness of the solid AG; because if this be removed from the solid BM, the superficies BCGF, the boundary of the solid BM, does nevertheless remain; therefore the superficies BCGF has no thickness, but only length and breadth. The boundary of a superficies is called a line, or a line is the common boundary of two superficies that are contiguous, or which divides one superficies into two contiguous parts: Thus if BC be one of the boundaries which contain the superficies ABCD, or which is the common boundary of this superficies and of the superficies KBCL which is contiguous to it, this boundary BC is called a line, and has no breadth: For if it have any, this must be part either of the breadth of the superficies ABCD, or of the superficies KBCL, or part of each of them. It is not part of the breadth of the superficies KBCL: for, if this superficies be U a H Book I. removed from the superficies ABCD, the line BC, which w is the boundary of the superficies ABCD, remains the same as it was: Nor can the breadth that BC is supposed to have, be a part of the breadth of the superficies ABCD; because, if this be removed from the superficies KBCL, the line BC, which is the boundary of the superficies KBCL, does nevertheless remain: Therefore the line BC has no breadth: And because the line BC is in a superficies, and that a superficies, has nothickness, as was shown, therefore a line has neither breadth nor thickness, but only length, The boundary of a line is called a point, or a point is the common boundary or extremity G M of two lines that are contiguous: Thus, if B be the extremity of F N the line AB, or the coinmon extremity of the two lines AB, KB, this extremity is called a point, D L B К. ; if the line KB be removed from AB, the point B which is the extremity of the line AB remains the same as it was: Nor is it part of the length of the line AB; for, if AB be removed from the line KB, the point B, which is the extremity of the line KB, does nevertheless, remain : Therefore the point B has no length; And because a point is in a line, and a line has neither breadth, nor thickness, therefore a point has no length, breadth, nor thickness. And in this manner the definitions of a point, line, and superficies, are to be understood. DEF. VII. B. I. INSTEAD of this definition as it is in the Greek copies, a more distinct one is given from a property of a plane superficies, which is manifestly supposed in the Elements, viz. that a straight line drawn from any point in a plane to any other in it, is wholly in that plane. DEF. VIII. B. I. It seems that he who made this definition designed that it should comprehend not only a plane angle contained by two straight lines, but likewise the angle which some conceive to be made by a straight line and a curve, or by two eurve lines which meet one another in a plane: But, though a |