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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
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.
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 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
Book I. shewn; therefore a linę hąs neither breadth nor thickness, but
• The boundary of a line is called a point, or a point is the
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