CRITICAL AND GEOMETRICAL; CONTAINING, An Account of those Things in which this Edition differs from the Greek Text; and the Reasons of the Alterations which have been made. As also Observations on some of the Propositions. BY ROBERT SIMSON, M. D. Emeritus Professor of Mathematics in the University of Glasgow. GLASGOW : PRINTED AND SOLD BY ANDREW FOULIS; SOLD ALSO BY ROBERT CROSS, NEAR THE COLLEGE. M.DCC.LXXXI. T is necessary to consider a solid, that is a magnitude which hat I the Definitions of a point, line and superficies; for these al! arise from a solid, and exist in it. the boundary, or boundaries which contain a solid are called fuperficies, or the boundary which is common to two solids which are contiguous, or which divides one folid into two contiguous parts, is called a superficies. thus if BCGF be one of the boundaries which contain the folid ABCDEFGH, or which is the common boundary of this folid, and the folid BKLCENMG and is therefore in the one as well as the other solid, is called a superficies, and has no thickness. for if it have any, this thickness must either be a part of the F G M thickness of the folid AG, or of the folid BM, or a part of the thickness FIN of each of them. It cannot be a part E of the thickness of the folid PM, because if this folid be removed from D L the folid AG, the fuperficies BCGF, the boundary of the folid AG, remain's still the same as it was. Nor A B can' it be a part of the thickness of the solid AG, becaufe if this be removed from the folid BM, the fuperficies BCGF, the borindary of the folid BM, does nevertheless remain. therefore the superficies BCGF has no thickness , but only length and breadth. The boundary of a fuperficies is called a line, or a line is the common boundary of two fuperficies that are contiguous, or which divides one fuperficies into two contiguous parts. thus if BC be one of the boundaries which contain the fuperficies 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, of each of them. It is not part of the breadth of the fuperficies KBCL, for if this superficies be removed from the superficies ABCD, the line BC which is the boundary of the superficies ABCD T or part Book I remains the same as it was. nor can the breadth t'at BC is sup. posed 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 no thick. ness, as was shewn; therefore a line has neither breadth nor thicknefs, but only length. The boundary of a line is called a point, or a point is the common boundary or extremity of two lines that are contiguous. thus if B be the extremity of the line AB, or the common extremity of the two lines AB, KB, this extremity is called a point, and has no length. for if it have any, this length must either be part of the length of the line AB, or of the line H G M TO IN C L length of the line AB; for if AB be removed from the line KB, the point 3 which is the extremity of the line A B K 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 fuperficies are to be understood. DE F. VII. B. I. Inftead 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 inanifestly supposed in the Elements, viz. that a straight line drawa from any point in a plane to any other in it, is wholly in that plane. 1 DE F. 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 curve lines, which meet one another in a plane. but tho' the meaning of the words ét' tu'cías; that is, |