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NOTE S

ON THE

FIRST BOOK OF THE ELEMENTS,

I

DEFINITIONS.

I.

N the definitions a few changes have been made, of which Book I. it is neceffary to give fome account. One of these changes refpects the first definition, that of a point, which -Euclid has faid 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 impoffible to reply, and therefore it becomes neceffary to change the definition altogether, which is accordingly done here, a point being defined to be, that which has pofition but not magnitude. Here the affirmative part includes all that is effential

to

to a point, and the negative part excludes every thing that is not effential to it. I am indebted for this definition to a friend, by whofe judicious and learned remarks I have often profited.

II.

After the fecond definition, Euclid has introduced the fol lowing," 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 fuppofed to ter minate. The termination of a line can therefore have no magnitude, and having neceffarily pofition, it is a point. But as it is plain, that in all this we are drawing a confequence from two definitions already laid down, and not giving a new definition, I have taken the liberty of putting it down as a corollary to the fecond definition, and have added, that the interfections of one line with another are points, as this affords a good illuftration of the nature of a point, and is an inference exactly of the fame kind with the preceding. The fame 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 fuperficies and a line.

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

"It is neceffary to confider a folid, that is, a magnitude which has length, breadth and thickness, in order to underftand 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 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

contain the folid ABCDEFGH, or which is the common boundary of this folid, and the folid BKLCFNMG, and is therefore in the one as well as the other folid, it is called a fuperficies, and has no thickness: For if it have any, this thicknefs 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 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 thicknefs, but only length and breadth.

H

G

M

"The boundary of a fuperficies is called a line; or a line is the common boundary of two fuperficies that are contiguous, or it is that 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 remains the fame as it was. Nor can the breadth that

E

D

A

B

K

N

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 thickness, as was

Cc

fhewn ;

Book I.

Book I. fhewn; therefore a line has neither breadth nor thickness, but

only length.

H

G

"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 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 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,

F

A

D

F

IN

K

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 underflood."

III.

Euclid has defined a ftraight line to be a line which (as we tranflate 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 confef fed, that this inaccuracy is at leaft lefs ftriking than in our translation; for the word which we render evenly is s equally, and is accordingly tranflated ex æquo, and equaliter by Commandine and Gregory. The definition, therefore, is, that a ftraight 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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