\ 



COMMENT.4TrO AD QUA.ESTIONEM MA.THEMÄ.TIGAM. 45 



Est .^ = abc X ^^" + ^ -'- ''^ ^' + ; ~"i+^ ' J ^" + ''^ - '^^' + '^' (B") 



j ^. [{a + b + c)ia + c — b-) + ab1(ib+c) — a{a + c)'- 

 sive : z- = abc X ;; — ; ., ,, , -5 — ^^ 



Nobis restat , ut hos Talores ad raflios circulorum referamus. Reductione appa- 

 ret , si numerantem fractionis form. B indicemus litera iS" , esse : 



S' = a^b — ab^ + a'c — ac^ + Z.=c + bc^ + «3 _ J3 _ ^3 + Zabc ,. 

 ein si deniseris : 



/'=:8/(-ß+/) = «'Ä+ßi°+a'c+n!c=+i'c+öc»— a3_i3_c3 (Theor. XVII. Dem. form./3) 

 residuiim erit : S' — 8/(ß +r)= — 2ab^ — 2ac'^ +2a3 ->r3abc , 

 cui si addideiis: 4//i = abc 



summa efficielur: S'-^irI{R+2r)=2ala^ -■ {b — cf] 



o _ ra 

 = 2a(a + c - i) (« 4- 5 — c) = 5^ (Theor. III.) 



unde habebis : S' = 4/7 R + 2r -{- '— \ • quo Substitute in aequatione .... (B) 



invemes: a;" = 7 — — TTT; — V ^ X 4/< Ä -i- 2r + — - ^ 



16/= r „, „ 2a/Ä 7 



= ( . 4- br ,a ^ c, X [^' + =^^ + -ß^\ 



«* J^'- •• - = ia + b''a + c)ßy ^ { K^^ + ^^'•^^^^ + '"^^^'^'' 1 • • • (C^ 



Gaeterum erit : y = -. j^-y-, r — V > C(Ä^ + 2Är)«7 + 2bIJl}ar 1 • • • (<^') 



(« + o) (0 + c;«?' »- J 



{a -}- c) (ü + c);;/3 C J 



SOLUTIO Part» II. 



HF"' : HD"' = AF'" : AIT" 



at est: AF'" : AD'" =r -^ : ■^- = a + i : a-\-c 

 a -^ c a-\- U 



est ergo etiam : HF'" : HD"' =.a + b: a-\-c 



indeque compositione : HF'" + HD"' : {a + b) +{a-\-c) = HF'" -.« + & = HD'" -.a + c 



idest:F"'D'" : 2a + i + c zzrHF*«' : a + i = HD " : o: + c 



unde sequilur : 



HF'" = _^±^ X F"'D'" = , ,Jl, ^ ,v X V [ [{R-+2Rr)ßy+2aIR-[ßr ] 



2a+6+o (2a-f-6+c) (a+c)/3y c J 



et HD'"= -^±^ XF"'D'"=, j-r-^r^^-7v5-><l/f [(-S"+2-?^>-)/3?'+2«'''^-]/5?^l 



F 5. j THE-- 



