CRITICAL AND GEOMETRICAL;

CONTAINING

An account of thofe Things in which this Edition differs from the Greek Text; and the Reasons of the Alte rations which have been made. As alfo Obfervations on some of the Propofitions.

By ROBERT SIMSON, M. D.

Emeritus Profeffor of Mathematics in the University of Glasgow.

EDINBURG H:

Printed for J. NoURSE, London; and J. BALFOUR, Edinburgh.

M,DCC,LXXXI.

is neceffary to confider a folid, that is, a magnitude which has length, breadth, and thickness, in order to understand aright the definitions of a point, line, and fuperficies; for thefe all arife from a folid, and exift in it: The boundary, or boundaries which contain a solid are called fuperficies, or the boundary which is common to two folids which are contiguous, of which divides one folid into two contiguous parts, is called fuperficies; Thus, if BCGF be one of the boundaries which contain the folid ABCDEFGH, or which is the common boundary of this solid, and the folid BKLCFNMG, and is there, fore in the one as well as the other folid, is called a fuperficies, and has no thicknefs: For if it have any, this thickness must either be a part of the thickness of the folid AG, or the folid BM, or a part of the thicknefs of It cannot be a each of them. part of the thickness of the folid BM; because, if this folid be removed from the folid AG, the fuperficies BCGF, the boundary of the folid AG, remains ftill the fame as it was. part of the thickness of the folid AG; because, if this be removed from the folid BM, the fuperficies BCGF, the boundary of the folid BM, does nevertheless remain; therefore the superficies BCGF has no thickness, but only length and breadth.

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G
FAN

M

D

C

B

K

A
Nor can it be a

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 fuperficies, and of the fuperficies 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 fuperficies ABCD, or of the fuperficies KBCL, or part of each of them. It is not part of the breadth of the fuperficies KBCL; for, if this fuperficies be removed from the fuperficies ABCD,

the

Book 1. the line BC which is the boundary of the fuperficies ABCD remains the fame as it was: Nor can the breadth that BC is fupposed to have, be a part of the breadth of the fuperficies ABCD; becaufe, if this be removed from the.fuperficies KBCL, the line BC which is the boundary of the fuperficies KBCL does nevertheless remain: Therefore the line BC has no breadth: And because the line BC is in a fuperficies, and that a fuperficies has no thickness, as was fhewn; 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 of two lines that are contiguous : Thus, if B be the extremity of the line AB, or the common extre- E mity of the two lines AB, KB, this extremity is called a point,

H G
FAN

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and has no length: For, if it have
any, this length must either be
part of the length of the line AB,

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B

B

K

or of the line KB. It is not part A

of the length of KB; for, if the line KB be removed from AB, the point B which is the extremity of the line AB remains the fame 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 becaufe a point is in a line, and a line has neither breadth nor thicknefs, 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 diftinct one is given from a property of a plane fuperficies, which is manifeftly fuppofed 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.

DE F. VIII. B. I.

It seems that he who made this definition defigned that it fhould comprehend not only a plane angle contained by two ftraight lines, but likewife the angle which fome conceive to be made by a ftraight line and a curve, or by two curve lines, which meet one another in a plane: But, tho' the meaning of

the