GEOM with 1‘ostul-ates 2 and 3, which characterizes the straight line. Whilst for the straight line the verbal definition and axiom are kept apart, Euclid mixes them up in the case of the plane. Here the Definition 7, I., includes an axiom. lt defines a plane as a surface which has the property that every straight line which joins any two points in it lies altogether in the surface. But if we take a straight line and a point in such a surface, and draw all straight lines which join the latter to all points in the first line, the surface will be fully determined. This construction is therefore sutticient as a definition. That every other straight line which joins any two points in this surface lies altogether in it is a further property, and to assume it gives another axiom. Tlnis a number of Euclid’s axioms are hidden among his first definitions. A still greater confusion exists in the present editions of Euclid between the postulates and axioms so—called, but this is due to later editors and not in Euclid himself. The latter had the last three axioms put together with the postulates (a.Zr7§,ua.-ra), so that these were meant to include all assumptions relating to space. The remaining assumptions which relate to magnitudes in general, viz., the first eight “axioms” in modern editions, were called “common notions” (Kowal Evvoiat). Of the latter a few may be said to be definitions. Thus the eighth might be taken as a definition of “ equal,” and the seventh of halves. If we wish to collect the axioms used in ‘luclid's Elements, we have therefore to take the three postiilates, the last three axioms as generally given, a few axioms hidden in the definitions, and an axiom used by Euclid in the proof of Prop. 4 and on a few other occa- sioiis, viz., that figures may be moved in space without clrnige of shape or size. “'0 shall not enter into the investigation how far the assumptions which would be included in such a list are sullicient, and how far they are necessary. It may be sutlicient here to state that from the beginning of a gerunetrical science to the present century attempts without end have been made to prove the last of Euclid’s axioms, that only at the beginning of the present century the futility of this attempt was shown, and that only within the last twenty years the true nature of the connexion between the. axioms has become known through the researches of Riemann and Ilelmholz, although Grassmann had pub- lished already, in 1844, his classical but long—neglected A nsrlelz nungsleh re. § 4. The assumptions actually made by Euclid may be stated as follows :-— 1. Straight lines exist which have the property that any one of them may be produced both ways without limit, that through any two ponits in space such a line may be drawn, and that any two of them coincide throughout their indefinite extensions as soon as two points in the one coincide with two points in the other. (This gives the contents of Def. 4, part of Def. 35, the first two Postulates, and Axiom 10.) 2. Plano surfaces or planes exist having the property laid do'i1 in Def. 7, that every straight line joining any two points in such a surface lies altoget.her in it. 3. llight angles, as defined in Def. 10, are possible, and all right angles are equal; that is to say, wherever in space we take a plane, and wherever in that plane we construct a right angle, all angles thus constructed will be equal, so that any one of them may be made to coincide with any other. (Axiom 11.) 4. The 12th Axiom of Euclid. This we shall not state now, but only introduce it when we cannot proceed any further without it. 5. Figures may be freely moved in space without change of shape or size. This is assumed by Euclid, but not stated as an axiom. 6. In any plane a circle may be described_. having any point in that plane as centre, and its distance from any other point in that plane as radius. (Postulate 3.) The definitions which have not been mentioned are all “nominal definitions,” that is to say, they fix a name for a EUcLmi.-xN.] E T R Y 377 thing described. Many of them overdeterinine a figure. (Compare notes to definitions in Simson’s or Todhunter’s edition.) § 5. Euclid’s Elements are contained in thirteen books. Of these the first four and the sixth are devoted to “plane geometry,” as the investigation of figures in a plane is generally called. The 5th book contains the theory of proportion which is used in Book VI. The 7th, 8th, and 9th books are purely aritlniietical, whilst the 10tl1 contains a most ingenious treatment of geometrical irrational quantities. These four books will be excluded from our survey. The remaining three books relate to figures in space, or, as it is generally called, to “ solid geometry.” The 7th, 8th, 9th, 10th, 13th, and part of the lltli and 12th books are now generally omitted from the school editions of the Elements. In the first four and in the 6th book it is to be understood that all figures are drawn in a plane. ' lloox I. or Eu-i.in’s “ l".i.i-:.ir..'rs." § 6. According to the third postulate it is possible to draw in any plane a circle which has its centre at any given point, aml its radius cqual to the distance of this point from any other point given in the plane. This makes it possible (Prop. 1) to construct on a given line AB an equilateral triangle, by drawing first a circle with A as centre and All as radius, and then a circle with ll as centre and BA as radius. The point where these circles iiiter- sect—tliat they intersect Euclid quietly assumes—is the vertex of the required triangle. Euclid does not suppose, however, that a circle may be drawn which has its radius equal to the distance be- tween any two points unless one of the points be the centre. This implies also that we are not supposed to be able to make any straight line equal to any other straight line, or to carry a distance about in space. Euclid therefore next solves the problem: It is required along a given straight line from a point in it to set oil‘ a distance equal to the length of another straight line given anywhere in the plane. This is done in two steps. It is shown in Prop. 2 l1ov a straight line may be drawn fi'oni a given point equal in length to another given straight line not drawn from that point. And then the problem itself is solved in Prop. 3, by drawing first through the given'poiiit some straight line of the requii'cd length, and then about the same point as centre a circle having this length as radius. This circle will cut off from the given straight line a length equal to the required one. Now-a-days, instead of going through this long process, we take a pair of compasses and set oil’ the given length byits aid. This assumes that we may move a length about without changing it. lhit Euclid has not assumed it, and this proceeding would be fully justified by his desire not to take for granted more than was necessary, if he were not obliged at his very next step actually to inake this assumption, though without stating it. § 7. Ve now come (in Prop. 4) to the first theorem. It is the fundamental theorem of Euclid's whole system, there being only a very few propositions (like Props. 13, 14, 15, l. ), except those in the 5th book and the first half of the 11th, which do not depend upon it. It is stated Very accurately, though somewhat clumsily, as follows :— If two triangles have two siclcs of the one equal to tzro sirlcs of the other, each. to each, and hate also the (mglcs containcrl by those sides equal to one another, they shall also have their bases or third side.s~ equal ; and the two tr2'an_r/lcs shall be equal ,- and their other angles shall be equal, each to each, namely, those to zchich the equal sides are ipositc. That is to say, the triangles are “identically” equal, and one may be considered as a copy of the other. The proof is very simple. The first triangle is taken up and placed on the second, so that the parts of the triangles which are known to be equal fall upon each other. It is then easily seen that also the remaining parts of one eoincide with those of the other, and that they are therefore equal. This process of applying one figure to another Euclid scarcely uses again, though many proofs would be simplified by (loing so. The. process introduces motion into geometry, and inclii(lcs, as already stated, the axiom that figures may be moved without change of sha we or size. 1 the last proposition be applied to an isosceles triangle, which has two sides equal, we obtain the theorem (Prop. 5), if two sioles of a triangle are equal, than the angles opposite these sides are equal. Ei1clid’s proof is somewhat complicated, and a stiimbling-block to many schoolboys. The proof becomes nuicli simpler if we con- sider the isosceles triangle ABC (Al’=AC) twice over, once as a triangle BAG, and once as a triangle CAD; and now remember that
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