( 541 ) 



lliic-sct/iiicni. 



parallcUximin, 



F1-. J. 



(iriiUcli'/ii/xuloii 



cwvy tiiiu' ;i iicw diiiKMisioii is ;h1(1(m1, w liilsl IIic iiiiiiilx'r of coiist.iiils 

 (ItMcnniiii]!,!»- llic li^^iirc, llioiiuli al lii'sl laffer lliaii llic miiiihcr of 

 (lia.uoiials, increases less sli-oiiuiv lliaii \\\o laller; lliis is illiisli-ale(l In 

 llie ioUow iii.U' linie laUle, \vli(M-e under each oüwv llie c()rres|)()ii(li]i,u- 

 \aliies of (Ik' iimiiher // of the (liiiieiisioiis. the number d of Ihe dia- 

 gonals and the nnnd)ei- ;/ of the deterniiniiii:' constants are indicated, 

 whilst llie nieaniuü,- of h is cx[)hii]ied further on. 



A II I I 1 I 5 |16 42i 99 | 219 ! 466 ... 



From this is evident in the scrnrul iilncc that when c(uistruclin,i:- 

 parallelogram and parallele|)i|)edon all dia.uonals can be used as 

 determininji' lines, but that this is not possible for the pai-allelotope 

 l\ with five and for the followin,2: parallelotopes /^„, l\ . . . with 

 still more dimensions; and from this ensues in lli<' flin-<l iihicc, what 

 becomes the ])rijici|>al Ihini;- here, that between the 16 dia'iouals of 

 /-*, at least one relation ninst exist and that this iniud)er ofi-elalions 

 for /*., I*.... must increase ccmsecutively to 32 21 oi' 11. (54 — 2'S 

 or 3(), ... If ill the fniiiili place we wish to trace those r(>latioiis 

 and trv to do so under the condition thai the length of all the edges 

 miisl liniire amon,<j,sl the determininu' <lata, then we find lliat the 

 sum of the s(|uares (»f all the diagonals always ecpial to the sum of 

 the scpiares of all the edues — is know u a! the same time, and thai 

 the otiier relations, between the diauonals only, always present lliein- 

 selves in the form of homogeneous e(|uatioiis, the number Aof w hicli 

 is iuilicaled aboxc. This includes that already for the parallelotope 

 /*, we come across a relation between the diagonals. This r-impic 

 relation can be expressed as follows: If we di\ide ( I'^ig. 2ilhe eight 

 \ei"lices of one of the eiglil paral leh'pipeda forniiiig llie boundary of 



