1

CRITICAL AND GEOMETRICAL;

CONTAINING,

An Account of thofe Things in which this Edition differs from the Greek Text; and the Reafens of the Alterations which have been made. As alfo Obfervations on fome of the Propofitions.

BY

ROBERT SIMSON, M. D.

Emeritus Professor of Mathematics in the Univerfity of Glasgow.

GLASGOW:

PRINTED BY ROBERT AND ANDREW FOULIS

PRINTERS TO THE UNIVERSITY

M.DCC.LXII.

NOTES &c.

DEFINITION I. ВО ОК

a

I.

T 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 these all arife from a folid, and exist in it. the boundary, or boundaries which contain a folid are called fuperficies, or the boundary which is common to two folids which are contiguous, or which divides one folid into two contiguous parts, is called a fuperficies. 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 folid, is called a fuperficies, and has no thickness. for if it have any, this thickness muft either be a part of the thickness of the folid AG, or of the folid BM, or a part of the thicknefs of each of them. It cannot be a E part of the thicknefs of the folid BM, because if this folid be removed from the folid AG, the fuperficies BCGF, the boundary of the folid AG, remains still the fame as it was. Nor can it be a 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 fuperficies BCGF has no thickness; but only length and breadth.

A

H

G

M

F

N

C

D

B

K

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 line BC which is the boundary of the fuperficies ABCD

Book I. remains the fame as it was. nor can the breadth that BC is fuppofed to have be a part of the breadth of the fuperficies ABCD, because 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 thicknefs, 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 extremity of the two E
line AB, KB, this extremity is cal-
led a point, and has no length. for
if it have any, this length muft ei-
ther be part of the length of the
line AB, or of the line KB. It is A

H

G

M

F

N

C

D

L

B

K

not part
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 extre-
mity 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 nei-
ther 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 underflood.

DE F. VII. B. I.

Inflead of this Definition as it is in the Greek copies, a more diflinct one is given from a property of a plane fuperficies, which is manifeftly supposed in the Elements, viz. that a ftraight 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 should comprehend not only a plane angle contained by two straight lines, but likewife the angle which fome 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 in' leias; that is, in

a ftraight