▁rig orous ness , ▁and ▁then ▁arg ues ▁that ▁there ▁can ▁be ▁no ▁meaning ful ▁ratio ▁between ▁two ▁infin ities , ▁and ▁therefore ▁it ▁is ▁meaning less ▁to ▁compare ▁one ▁to ▁another . ▁▁ ▁Caval ieri ' s ▁Ex erc itation es ▁geomet rica e ▁sex , ▁or ▁Six ▁Ge ometric ▁Ex er cis es ▁( 1 6 4 7 ) ▁was ▁written ▁in ▁direct ▁response ▁to ▁G uld in ' s ▁criticism . ▁It ▁was ▁initially ▁draft ed ▁as ▁a ▁dialog ue ▁in ▁the ▁manner ▁of ▁Gal ile o , ▁but ▁correspond ents ▁ad vised ▁against ▁the ▁format ▁as ▁being ▁unnecess arily ▁infl am mat ory . ▁The ▁charges ▁of ▁pl ag iar ism ▁were ▁without ▁subst ance , ▁but ▁much ▁of ▁the ▁Ex erc itation es ▁de alt ▁with ▁the ▁mathematical ▁subst ance ▁of ▁G uld in ' s ▁arguments . ▁He ▁argued , ▁dis ingen ously , ▁that ▁his ▁work ▁regarded ▁' all ▁the ▁lines ' ▁as ▁a ▁separate ▁entity ▁from ▁the ▁area ▁of ▁a ▁figure , ▁and ▁then ▁argued ▁that ▁' all ▁the ▁lines ' ▁and ▁' all ▁the ▁plan es ' ▁de alt ▁not ▁with ▁absolute ▁but ▁with ▁relative ▁infinity , ▁and ▁therefore ▁could ▁be ▁compared . ▁These ▁arguments ▁were ▁not ▁convin cing ▁to ▁contempor aries . ▁The ▁Ex erc itation es ▁non eth eless ▁represented ▁a ▁significant ▁improvement ▁to ▁the ▁method ▁of ▁ind iv is ibles . ▁By ▁applying ▁transformations ▁to ▁his ▁variables , ▁he ▁general ised ▁his ▁previous ▁integral ▁result , ▁showing ▁that ▁for ▁n =