ural ▁ment ality ', ▁which ▁can ▁be ▁ide ational ▁( re ality ▁is ▁spiritual ), ▁sens ate ▁( re ality ▁is ▁material ), ▁or ▁ideal istic ▁( a ▁synth esis ▁of ▁the ▁two ). ▁He ▁interpreted ▁the ▁contemporary ▁West ▁as ▁a ▁sens ate ▁civil ization ▁dedicated ▁to ▁techn ological ▁progress ▁and ▁proph es ied ▁its ▁fall ▁into ▁dec ad ence ▁and ▁the ▁emer gence ▁of ▁a ▁new ▁ide ational ▁or ▁ideal istic ▁era . ▁ ▁Alexandre ▁De ul of eu ▁( 1 9 0 3 – 1 9 7 8 ) ▁developed ▁a ▁mathematical ▁model ▁of ▁social ▁cycles ▁that ▁he ▁claimed ▁fit ▁historical ▁facts . ▁He ▁argued ▁that ▁civil izations ▁and ▁emp ires ▁go ▁through ▁cycles ▁in ▁his ▁book ▁ ▁Mathemat ics ▁of ▁History ▁( in ▁Catal an , ▁published ▁in ▁ 1 9 5 1 ). ▁He ▁claims ▁that ▁each ▁civil ization ▁passes ▁through ▁a ▁minimum ▁of ▁three ▁ 1 7 0 0 - year ▁cycles . ▁As ▁part ▁of ▁civil izations , ▁emp ires ▁have ▁an ▁average ▁lif es pan ▁of ▁ 5 5 0 ▁years . ▁He ▁also ▁stated ▁that ▁by ▁knowing ▁the ▁nature ▁of ▁these ▁cycles , ▁it ▁could ▁be ▁possible ▁to ▁modify ▁the ▁cycles ▁in ▁such ▁a ▁way ▁that ▁change ▁could ▁be ▁peace ful ▁instead ▁of ▁leading ▁to ▁war . ▁De ul of eu ▁believed ▁he ▁had ▁found ▁the ▁origin ▁of ▁Roman es que ▁art , ▁during ▁the ▁ 9 th ▁century , ▁in ▁an ▁area ▁between ▁Emp ord à ▁and ▁R ouss illon , ▁which ▁he ▁argued