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No T E s, CRITICAL AND GEOMETRICAL;

CoNTAINING

AN A CCOUNT OF THOSE TH INGS IN W H ICH THIS EID it ION
Di FFERS FROM THE GREEK TExt; AND
THE REASONS OF THE ALTERATIONS
WHICH HAVE BEEN MADE. -

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E.M. Ehritus PROFESSOR OF MATHEMATICs in Tile UNIVERSITY OF GLASGOW.

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IT is necessary to consider a solid, that is, a magnitude Boo [.
which has length, breadth, and thickness, in order to un- w
derstand aright the definitions of a point, line, and superfi-
cies; for these all arise from a solid, and exist in it: The
boundary, or boundaries, which contain a solid are called su-
perficies, 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

must either be a #. of the thick- H G M
ness of the solid AG, or of the so- F EZ | N /
lid BM, or a part of the thickness jo
of each of them. It cannot be a
part of the thickness of the solid I) C I
BM; because if this solid be re- ---
moved from the solid AG, the . .
superficies BCGF, the boundary A. B
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 contigu-
ous, 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 con-
tiguous 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: o, if this superficies be

removed from the superficies ABCD, the line BC, which 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 su

perficies has nothickness, as was shown, therefore a line has

The boundary of a line is called a point, or a point is the Thus, if B be the extremity of F. this extremity is called a point, D C L. part of the length of the line AB, A. K.

neither breadth nor thickness, but only length. common boundary or extremity . of two lines that are contiguous: H G M

- 21 E/TN/ the line AB, or the common ex- tremity of the two lines AB, KB, and has go length. For if it have *- Z any, this length must either be or of the line KB. It is 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 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 curve lines which meet one another in a plane: But, though,

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