The greatest defect of Euclid's definition, since it applies even to the view just taken of its intent, is the want of words signifying that f Icrou refers equally to all adjoining parts of space : Euclid is thinking too much of a plane before ho has defined a plane. Suppose, for in stance, a sphere, and that lines on a sphere only are contemplated : the line which joins two points it tcrou with reference to all adjacent parts of that sphere is not a straight line, but an arc of a great circle.
Is it possible, taking such allowances as Euclid sanctions in the use of figure, to give what shall be, whether difficult or not difficult, capable of use or not capable, a just definition of a straight line f We think it is, as follows :—The Greek geometer implicitly allows (i. 4) a TRANSLATION of figure without change of form or properties : from this, by first defining the plane, a definition of the straight line may be proposed, which we bring forward, not for any value which it has, but because the stipulations of geometry are better understood by consideration of cases proposed for acceptance or rejection, than by any other method.
1. Let two points (A and a) be said to be at the same distance from a third (c), when A and c being joined by any line, the line OA can be translated, c remaining fixed, so that A shall be brought to coincide with n.
2. A plane is a surface any point of which is equally distant from two given points.
3. A straight line is the intersection of two planes.
In the debates of the normal school, which were taken down in shorthand, and published in 1800, is a discussion on this subject. Lagrange presiding, Fourier, then one of the pupils, proposed the pre ceding second and third definitions, but without assigning a definition of equidistance independently of the straight line. He also proposed as the definition of a straight line the locus of a point which is equi distant from three given points; which is faulty, inasmuch as the three given points should not be in one straight line, which cannot be sup posed until the straight line is defined. Lagrange admitted the rigor of the definition, but considered that it failed in presenting a sensible image of the thing defined. Another of the pupils however insisted
that the idea of distauce involved that of a straight line, which is true of distance as a quantity, though not necessarily so of equidistance as a relation.
General Thompson to define a straight line as one which being turned about its extreme points suffers no change of place. Lagrange, in the debate above alluded to, suggested the same notion. This definition, we think, offers the most tangible illustration of that of Euclid. Let the two extremities of the intended straight line be situated in a solid; and let them remain fixed in space while the solid takes such motion as, under that condition, it is capable of. The straight line, the line which lies it toot) with regard to the extreme points, theu remains fixed. For if any part of it moved, there would be in every position a relation to adjoining parts of space, which would be in a state of continual change. The connexion between this defini tion by rotation and that of Euclid might require more development to render it as clear as possible : but wo think the student's own reflection will lead him to make it satisfactorily. But whatever may be thought of the endeavour to exercise the discrimination of which geometry points out the possibility by training or arguiug on defini tions, we do not remember to have seen one so well calculated for the mere beginner as the following :—" A straight line is a straight line." The postulates relative to a straight line demanded by Euclid (wo do not speak of his translators) are : 1. That such a lino can be drawn from any one point to any other. 2. That when terminated, it can be lengthened indefinitely. 3. That two such lines cannot inclose (a); stpifx(4e) a space. It is also tacitly assumed that every part of a straight line is a straight line : that every straight line, infinitely pro duced, divides a plane in which it lies into two parts, and will be cut by any line drawn from a point on one side of it to a point on the other. It might also have been assumed that two straight lines which coincide in two points, coincide when produced beyond those points; but here Euclid has preferred to assume that all right angles are equal.