August 8, 1895] 



NATURE 



341 



which contains the perpendicular PQ to L. l'(J and the S/ . , 

 determine a space Sx, and the proposition is that any hne 

 through r in this S^ is perpendicular to L. Through I'O con- 

 struct a plane space 2*., in S* perpendicular to I,. It must exist, 

 according to hypothesis. Si_, cuts the S^. into two parts, 

 because ever)- straight line in S/.. (as easily follows from the as- 

 sumptions) has one point in common with the S/- ,, we therefore 

 have no means of passing from one point of such straight line to 

 its other points without passing the S^.,. S/--, and 2x-^, cut 

 the Si- therefore into four dift'erent parts, which have the cut of 

 2i_, and S/._,, that is a certain S^-j, in common. Let the 

 four departments, into which the .S<. is cut, lie called A, B, C, D. 

 A straight line through P, not contained by the Sx-,, will be 

 situated (as it passes P, that is a point of the S.(-..) in two 

 dift'erent departments ; and if we change the situation of this line 

 continuously, without passing either the Sx - , or the 2*- 1, it 

 will remain in the same two departments. Tlie departments 

 are therefore arranged by two. If a straight line through P, 

 belonging to A, also belongs to B, then A and H shall be called 

 opposite to each other. Let A, B and C, I) be opposite to each 

 (■ther. We have no means wha^tever of distinguishing two 

 (_>liposite fle]-iartments, unless we assume at the very least another 

 arbitrary point, because every plane configuration through the 

 Si--.., extending into one department, equally extends into the 

 opposite one. Whatever is true for the one department must 

 therefore be true also for the opposite one. 



Now construct any line L' through P in the S*. Let L' 

 belong to A and B. If L and L' are not perpendicular, then 

 the angle I.L' contained in A must be larger or smaller than 

 the corresponding angle LL' contained in B. Let L' change its 

 position continuously ; if the angle LL' contained in A would 

 be always larger than the corresponding angle in B, this would 

 amount to a permanent property of A distinguishing it from B, 

 which it cannot possess. Therefore, whichever evolution L' 

 may ])erform from the Si.i to the 2x-i in A (and B), it must 

 have at least one intermediate situation in which L and L' are 

 jierpendicular. The aggregate of such situations form a surface 

 in .-\ and B. Let L', . . . L'/-i be /■ - i lines contained in 

 that surface ; then the plane space of /• - i manifoldness contain- 

 ing these k-\ lines, must, according to the hypothesis, be 

 jierpendicular to L. The surface must therefore contain this 

 jilane space. If now we replace one of the two Sx , or 2x-i by 

 this space, the argument will still hold. However, near the 

 two borderings plane spaces will finally approach, there will 

 always be at least one intermediate plane i>erpendicular space, 

 all of which are contained in the Si. It is therefore nothing 

 left but to concede that the Si in question has the property 

 established in the proposition. 



Through any point P only one line L «ill pass, which is 

 perpendicular to a space .S. Assume indeed two such lines, 

 H hich may have with S respectively < J an<l R in common. Then 

 I'tjR woidd form a triangle, of which i I'<^)R as well as ^ PKQ, 

 according to the foregoing, will be = a right angle. This, 

 however, is impossible, unless O and R coincide. 



A point and a plane space therefore determine a certain line, 

 the perpendicular to that space through the point, a certain point 

 — the one in which the line aiiove mentioned ciUs the space — 

 and a magnitude, the distance of the two points above mentioned. 

 Tliis is always true, unless the point belongs to the space. Let 

 the point approach the space. If the two points in question 

 coincide, then the point will belong to the space. The conditions, 

 therefore, that a point and a space are iniited, is (distance of 

 point antl space) — o. 



Let P move continuously so that its distance froma plane space 

 .S remains unaltered ; P and S may determine a space 2 ; then 

 the aggregate of such points in 2 is another plane space. Let 

 P and (J be two situations of I*. Then all points of the line 

 I'lJ have the same distance from .S, as is easily seen to rest 

 on I*'uclid"s parallel axiom by means of parallelograms. The 

 general proposition can, from this, be established by considera- 

 tions anah)gous to the proof of our first theorem, independent 

 of any new assumption. Two such spaces 2 anil S art; called 

 parallel, and determine a certain magnitude, whtjse disappear- 

 ance is the condition cif coincidence of 2 antl S. 



Let 2 and S be [larallel. Through any ixiint .\ outside the 

 same draw two lines, which cut both 2 and S, in B, C and 

 B', C respectively, then the lines .VBB' and ACC have a 

 point \\\ connnon, the)' are therefore in the same i)lane, BC and 

 B'C must therefore either have a point in commtm, or be 

 parallel. A point in common they have not, as they are con- 



NO. 



1345, VOL. 52] 



tained in 2 and S, and these t«o have no point in common. It 

 follows thai 



AB : AB' = AC : AC. 



We now add to our assumption.s another one. n-\-\ points- 

 determine, as already slated, a plane space S„, and besides a 

 certain pyramid of 11 dimensions : of w hich we assume that it 

 shall possess magnitude. Let the 11+ I points Vje A, . . . A,i+i. 

 A., . . . A,i+i determine a certain space S„-i. Draw any line 

 through A]. It cuts S„_i in a point B. Choose A'l on this 

 line so that A,B = BA',. Then the two points A,, A'l have 

 an exactly symmetrical position to S,,.. 1. No property can be 

 valid for the one which is not valid for the other (as long as nO' 

 elements are introduced to disturb the symmetr)). We caimot 

 therefore assume that one of the two pyramids, determined re- 

 spectively by A, and the A„ . . . A„+i or A'j and A, . . . 

 A„+i, should be larger than the other. Now the locus of points 

 A'j is, according to the foregoing, a parallel 2„_i to S„-i. It 

 follows : The magnitude of the pyramid is dependent (i) on « of 

 its points (2) and the distance of the n-VV' from the plane 

 space determined by these ii points. 



What we have in mind, when we speak of the magnitude of a 

 pyramid, will come out clearer when we give a theorem of 

 addition. Let .\ be any point collinear with and intermediate 

 betw een An and Ag. Then we say : 



The pyramid determined by AjX and any other points 

 -r the pyramid determined by XA., and those other points = 

 to the pyramid formed by AjAj and the rest of the points. 



This explanation, combined with the above, shows that the 

 magnitude of a pyramid is equal to some constant multiple (say 



-) of the product of the magnitude of the pyramid A„ . . . A„-|-i^ 



and the distance of K^ from the space fixed by the other points. 

 We shall write this number (A,Aj . . . A„+i). (AjAj) is simply 

 the distance of the two points, and according to a convention 

 necessitated by considerations of continuity, we assume 



(AjA,) -V (.V.A,) = o. 



Generally, if we transpose any two letters, the magnitude desig- 

 nated changes sign. 



If A, B, C are three collinear points, and if we designate by 

 the single letters A, B, C the distances from these points of any 

 fixed point O on that line, then we have identically 



(AB)C -h (BC)A -h (CA)B = o. 



This is an algebraical identity easily established. The same 

 holds also when the single letters A, B, C are made to denote 

 the distance of these points from any space 2, which either is 

 parallel to line ABC, or has with it a point in common, as is 

 easily established by proportions. 



If between three points of a line such an equation exists, this 

 must be true also for it + i points in a S„. The proof of this 

 by induction is perfectly easy Let for instance A, B, C, D, be 

 four iwints in a plane, and let 2 be any space, that has with it 

 a line in connnon. Join CI) : it may meet AB in E. Then we 

 ha\'e some linear identity 



a.\ -<- bW -t- <E = o 

 where a, h, c denote constants independent from 2, and also 



dC -I- eV) +/V. = o. 

 Eliminating E, we obtain some linear identity between 

 A, B, C, D. 

 In order to determine the constants, let us assume the space- 

 2 (which is permitted) to be parallel to the plane ABCI> : then 

 we have if 



aA + iB + <C -^ </I) = o 

 a + /> + <■ + J = o. 



If we place 2 so that it cuts ABCD in CD, and if then we 

 make a = (BCD), < follows = (CDA). We therefore olUain 



(BCD)A + (CDA)B + (DAB)C + (ABC)n = o 

 and just so in the general case 



(BCD . . . L)A -KCD . . . LA) B -KD . . . LAB) C H- . . . = o. 

 The use of the distances of points from variable plane spaces 

 enables us to do away with fixed coordinate s)stems. The proof 

 of projective theorems becomes perfectly lucid, while at each 

 stage of the proceedings we are always able to give the 

 geometrical significance of the constants employed. To give a 



