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I

N the definitions a few changes have been made, of which Book I.

it is necessary to give some account. One of these changes respects the first definition, that of a point, which -Euclid has said to be, " That which has no parts, or which • has no magnitude. Now, it has been objected to this definition, that it contains only a negative, and that it is not convertible, as every good definition qught certainly to be. That it is not convertible is evident, for though every point is unextended, or without magnitude, yet every thing unextended, or without magnitude, is not a point. To this it is impossible to reply, and therefore it becomes necessary to change the definition altogether, which is accordingly done here, a point being defined to be, that which has postion but not magnitude. Here the affirmative part includes all that is essential

to

Book I. to a point, and the negative part excludes every thing that

is not essential to it. I am indebted for this definition to a friend, by whose judicious and learned remarks I have often profited.

II.

After the second definition, Euclid has introduced the following, “ the extremities of a line are points.”

Now this is certainly not a definition, but an inference from the definitions of a point and of a line. That which terminates a line can have no breadth, as the line in which it is has none, and it can have no length, as it would not then be a termination, but a part of that which it is supposed to ter. minate. The termination of a line can therefore have no magnitude, and having necessarily position, it is a point. But as it is plain, that in all this we are drawing a consequence from two definitions already laid down, and not giving a new definition, I have taken the liberty of putting it down as a co. rollary to the second definition, and have added, that the interSections of one line with another are points, as this affords a good illustration of the nature of a point, and is an inference exactly of the same kind with the preceding. The same thing nearly has been done with the fourth definition, where that which Euclid gave as a separate definition, is made a corollary to the fourth, because it is in fact an inference deduced from comparing the definitions of a superficies and a line.

As it is imposible to explain the relation of a superficies, a line and a point to one another, and to the folid in which they all originate, better than Dr Simson has done, I shall here add, with very little change, the illustration given by that excellent Geometer,

“ It is necessary to consider a solid, that is, a magnitude which has length, breadth and thickness, in order to understand aright the definitions of a point, line and superficies; for thefe all arise from a folid, 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

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Book I.

contain the folid ABCDEFGH, or which is the common boundary of this solid, and the solid BKLCFNMG, and is therefore in the one as well as the other solid, it is called a su. perficies, and has no thickness: 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 thickness of each of them. It cannot be a part of the thickness of the solid BM; because, if this solid be removed from the solid AG, the superficies BCGF, the boundary of the solid AG, remains still the same as it was. Nor can it be a part of the thickness of the folid 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 thick. ness, 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 it is that which divides one superficies 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

HL
G

M
breadth of the superficies
ABCD, or of the super-
ficies KBCL, or part of each
of them. It is not part of

F

N
the breadth of the super-
ficies KBCL ; for, if this su.
perficies be removed from D
the superficies ABCD, the
line BC which is the boun.
dary of the superficies ABCD
remains the same as it was.

А
B

K к
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
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 superficies has no thickness, as was

Сс

shewn;

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Book I. shewn; therefore a linę hąs 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 conti-
guous: 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

1

M
AB, 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

F

IN
B, which is the extremity
of the line AB, remains the
same as it was: Nor is it D)
part of the length of the
line AB; for, if AB be re-
moved from the line KB,

A

B
the point B, which is the

K
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.”

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III.

Euclid has defined a straight line to be a line which (as we translate it), “ lies evenly between its extreme points.” This definition is obviously faulty, the word evenly standing as much in need of an explanation as the word straight, which it is intended to define. In the original, however, it must be confelsed, that this inaccuracy is at least less striking than in our translation ; for the word which we render evenly is ekos equally, and is accordingly translated ex æquo, and equaliter by Commandinę and Gregory. The definition, therefore, is

, that a straight line is one which lies equally between its extreme points; and if by this we understand a line that lies

between

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