( 542 1 



the foiir-dimoiisional fiunre into two aronps f J,, .1,. J„, J ,) and 

 (/>,, B.,, />:,, />',) of' iioii-adjacciil Ncrliccs, tlic sum of llic sipiai-cs of the 

 diaudiials tcriiiiiialiiiu' in llie t'<»iii- [»oiiils A is (Mjiial lo llic sum of 

 tlir s(|uar('s of the four I'emaiiiijiii' ones. lermi]ialiii,U" in llic poiuls />'. 

 And tVom this ensues, ihc eonnnon cciilrc of llic eiii'lil diagonals hciiiu- 

 in(Healed i»v (K the e(iuation 



or in woi-ds: If we (hxide ihe ci.U'hl aiiuidar poiiils of a paralhdcpiiicdon 

 iulo two ui'oups of foul- non-adjaecnl poiiils. the sum of the s(|uar('s 

 of tlie (Hstanecs from an ai'hilrai'v poiiil I) lo die poinis of each of 



die Iwo (pia(h-uph's is die same If we now suppose in die ///7// plaee 

 dial tins poini () hes w idi ihe paradeie|)i|»edon in die same diree- 

 (nmensional spare, our s|)ace I may say. wc linady liiid die followiiiLi" 

 dieoivm iKdoiiiiiuLi' to our solid Licoiuelry: 



•'If \\(' eonneel [Vvj:. .*>) an arbilrary point (^ of space with the 

 tw(» (piath-iipies of non-adjareiit \('rtices of a parailelepip<'doii. we 

 ohiain t\\o (piadniples of line-seLiuients U)V which the sum of the 

 sipiai-es has the same \;due.'" 



This siinph' llieorem whicdi up till now 1 nexcr cauK^ across in 

 aii\ lian<ll)ook is of coui-se ea>ily |)i-o\-ed : we lia\-e hut to know the 

 h)niiiila h)i- the median line in a triau.ule. \\\\\\ the Indp of this 

 formula wc lind that. disre,uar<liiiLi- ([iiantities not dependiji.Li- on the 

 place of <K the sum of O A^' and OA.^' can he rejilaced by t\vo 

 times <)(';.., the sum of '^A,' and ,1^-',' l»y two limes OC;^ and 

 l\vice the sum of ^>r';, and ^^f'',, l»y ^'^>^iy limes ''V .)/"-'; from which 

 is evident that hir the two sums named in the theorem, <lisre,Liar<lin,U' 

 Ihe stniH' (pianlities not depeiidiiiii- on <K the same \alue is foun<l, 

 namelv four limes U M \ etc, 



