- io3 



ex parte sinista prodibit (a -h ]) (j3 -f- i) (y -j- i) , at 

 A^ero ex parte dextra aP-f-ay + Py -f- 2 (a-i-p-l-'y) 4-3, 

 quae formula manifestb resolvitur in bas partes : ' . 



(a-f-i)(p-}-i)-f-(a+ i)(y-hi) + (p-Hi)(y-4- 1). 

 Hac igitur forma substituta , dividatur utrinque pec pro- 

 ductum (a4- 1) (|3 H- 1) (y + 1) , ac prodibit 



1 == ^ir + pi- H- ^- • a . E . D. 

 §. i3. Hinc qiioque derivari potest ista memorabilis 

 proprietas: ^^^r^ + 0::?— H- ^^^7 =: 2. Si enira huic adda- 

 tur praecedens aequatio , orietur ista aeqiiatio identica^: 

 1 + i -h 1 = 3. 



Demonstratio slmpliclssima. 

 Elementis vulgaribiis innixa. 



§. 14- Per ptinctum O singulis trianguli lateribus TaS. I. 

 parallelae ducantar /^ ipsi BC, gyf ipsi A C et h& ipsi S- ^• 

 AB, et statim evidens est fore xi ~^ ri ~^ ^Â3 -— ^ ' Nimc 

 vero ob triangula ABa et A/O similia erit 



Bf : BA — Oa : Aa, 

 sicque prima fractio evadit ^ n: g{. Dcinde y quia 

 ABAboo AB>|0, erit A>) : AB =: Ob : Bb, unde ergo fit 

 ^^ =: 3 ■ . Denique A/0 >, cv A B C A , hiinc /-v) : B A =/0 : BC, 

 unde ergo fit j^ = 4c- ^^^ "^^'*^ /0=:BA, hincque, quia 

 triangulura BCccv>AeCO, erit H — ^l, unde fit^ — ^^, 



