420 OpnSCULA. 



_ ,. m'-t-m n'-\-n P^-'r-P , •z'-i-'/ , 



Generaliter aggrcgato — ^^ "*" — o" "* 2 *"• o etc. ad-» 



daiur(w-H » ) C""^ O + C'^^ O ()"+ O + C^'-^O ('/^ 0-f- 

 oic. 4- (n -HI )(;>■+. 1 )-t-(n4- 1 )(r/-t- 1 )+ etc. 4-(p+ 1) 

 (^_j_ 1 )-t-elc. scilicet omnia producta ex radicibus inajori- 

 bus biais suinplis, crit sutnma 



m^+ m -j- n-->r n ~>r p'' -\- p ■+- q''-'r q +etc. -^ (/»■+■ 1 ) (,14-1 ) _f.(r?j4-1 ) 



2 

 (;'-l-1)+(m-f.1)(7-Hl)4-etc. +(n-4-1)(/; + 1 ) + (» + !) 



(V + M+eic+Cr-T-l ) (9-1-1 ) + etc = 



(m+l-Hn-H'l +/)+1+y-t-.1 etc. — 1 ) ( /n-J-l -f-n+l 4- ;»-4-1 -t-^+l -t-etc.) 



"2 ~~ 



qiji numerus est triangularis 5 ergo 



THEOREM A VI. 



Si plurlum numerorum Iriangularlum aggregate addantur om- 

 nia producta ex radicibus majoribus binis sumptis, suiama 

 C[uae eniergit erit numerus triangularis. 



YW aggregate — ^ r— — detrahatur m\n.^\) vel 



n(OT-f-i) productum scilicet radicis minoris unius per ra- 



dicem majorem alterius, erit differentia vel - — h — - — 



-m(n + 1) = i i_^ ^^vel — 2 — ^"2 "("'-M) 



_(m— «)=4-(m— n) . 



-„ . Utrocjue casu erit diilerentia numerus 



triangularis j ergo 



THEOREMA VII. 



Si ex aggregate duorum numerorum Iriangularium detra- 

 hatur productum radicis minoris unius per radicem majorem 

 alterius, residuum crit numerus triangularis. 



Series numerorum triangularium iu qualuor, quae sequua- 

 lur, decomponatur. 



0, 10, 36, 78, 136 etc. 



1 , 15 , 45 , 91 , 153 etc. 



