38 OBSERVATIONES DE SlNGVLARl 



arithmetice progredientiiim , conflantur; veniam me 

 facilc impetraturum cxilUmaui , fi ea quac de hcc 

 argumento meditata habni euulgauerim , licet cum 

 inu.ntis tantorum virorum vix comparari mereantur. 

 2. Theorema i. Propofita pro^effione finuum ^ 

 pro arcuhus arithmetkam progrejfionem cwfiituentibus 



eius fumma erit 



•^ afin.ii; "~ fin.iv ' 



Demonftratio. 



Ponatur fumma progrcflionis propofitae — S 

 et multiphcando vtrinque per a cof. v prodibit : 



ftScof.'yz:2fin.2;cor'y+2fin.(s+'yjcof.i'-i-2fin (xr+a-y^cofc^... 



+2fin {z-\nv)zo{\v^ 



— ^1^.(2-1;)+^^ 5!+2fin (5;+i;)4-2fin (c-f2i.') 



+ 2 fin.(5;+(M- jj-y^+fin.^js+wyj-i- fin.(24 («+ 1 ^-y), 



€X quo deducitur 



ftS(i-cofv)-fin.2-fin.(2;-v)+fin.(s-lwv)-fin (z+Cw+O-y), 

 hincque 



cof^ g-i^y^-cof.fg+^w+s XO^ fin.fg+^ n a;)(i n. J («+i)u 

 ^ 2f1n.ii; "" im.lv 



Idem vero et hoc modo demonflrari poteft , .multi- 

 cetur prcgreflSo noftra per 2 fin. v , vt prodcat 



ftSfin.i;— afin.jzfin.v+afin.f^+v^fin.v+afin.fjg-i-ci;)^!! 1; 



. . . +-2fin.(jr + «a>)fin.v 



= COf (« - V) + COf. 5!-C0f.(«+«4;)-C0f.(2+(«+l) V), 



Hinc 



