254 Opuscdla 



£;nlorura fjiiailrati A BCD cimilo inscripti, necnon rectae 

 OX=jr, OY=:>', OZ = z, 0\] = H nonnaliler latcribus qua- 

 drali EFGH circumsciijni , cujiis latera tangnnt peripheriam 

 ad vertices angulorutn qaadrali iiisciipli, posito radio circuli 

 = r, erit 



OA^= OX"4- AX'= x^ +- (r — z)^ 

 Ul3"= OU Vl3U^- f*- 4- (r— x)2 

 0G^=: 0\- Vc\^=;-2 ^ („ _ ,.)2 



OL)- -"OZ^-I- DZ^- ^2 ^ (j _ ,.)2 

 Addantur hac aeqiialiones, eritque 



JJI-. i-0BV0GV0l)^^2x2-h2j2 -{-2;2 ^_2«2 J^/^i^^at^X-hf+Z + u) 



Sed 2x2^-2j»-2 4-2c"+2?f?-i'v.r2 § 17, x-\-j+z+u=^r 



Ergo __ ^ ^ 



0A'-^0BV0C'-f-0D"= 1 2^2 4-4r2_8/-2- 8/2-2 . 4r2 



THEOREMA XIII. 



32.Dactis e qaovis pnnclo pcripheriac circuli reclis lineis ad 

 vertices anguloram qviadrali inscripti, aggregatum quadratorum 

 harum lincarum aeqiiat diiplum quadrati diametri ipsius cir- 

 culi. 



33.Aquocumque puiicio Operipberiae circuli A BCD (fig. 8.) 

 dncantur reclae lineae OX'=ar' , OY'=r' , OZ' = z' , OU' 

 =11 perpendiculares lateribus quadrati circumsrripti ABC D. 

 Proibiratnr O Y' usque ad puncium M peripberiae ^ e- 

 rit OX'=Y'M, ideoque X'M=OY'^ ergo OX' . X' M 

 = 0X' . OY' = xy •, sed OY' . X'M = X'D . X'A = OZ' . 



0\]' = z'u'v ergo 



xy=z'u' 



Ergo 



THEOREMA XIV. 



34. Ductis e quovis puncto peripberiae circuli rectis lineis 

 nonualibusad latera quadrati inscripti, reclangula, quae fieri pos- 

 sunt normalibus opposilis, sunt aequalia inter se. 



35. Posito radio circuli =r , erit lalus quadrati inscripti 



