24 C Ä II L I J 11 A N N I S M A T T II E S 



Soi, ÜTIO AfTALYTrCA BECENTIORUM, flg. 1 ß. 



Ponamus AP = />., BP = y , problemati.s conditio tunc cxprimitur per p ± fj z=. a ^ 

 lecimus ut supra AB := b, alquc iiisupcr AX = c^ Siinul ac cogiiilum liabcamus spa- 

 tiuin PX, de loco puiicti P non aimplius dubium exsistcie potcsl , (piare spatium i:ilud 

 dixi X. Quae si ila siiil , duo considereiitur trinngiila PAX et PAß , ex (juibus obtiiicu- 

 fur aequationcs 



p- = JK* + C' lex . COS. X 



y" = X- + (6 -\- c)- — 2 {/> + c)x . COS. X 

 quae si conibiitentiir praebeiit : 



1/ [ x^ + (■= — 2cx . COS. X ] ± \/[x^-^{b + o)' — 2(b+ c) X . COS. X] = a 

 nudc \/[x- + c' —2rx . cos.X] = « + ^/[^^ -j- (6 + c)'- — 2(0 + c) x . cos.X] 



x^ + c- — 2cx .cos.X ip: a^ + 2a \/ [x' + (b + c)' — 2 {b + c) x. cos.X] 



+ x''+{b+cy-—2{b+c}x'.oos.X. 

 + 2rf[/[x' + {b+cY—2{b + c)x.cos.X]=x' + c^—2cx.cos.X—a''—x'~(b-{-c)-+2{b + c)x.cos.X 



=. 2bx . COS. X — b(b+ 2c) — a- 

 Aa^ [.i'4-(Ä+c)= — 2(6+c)^ . COS. X] = [2i* . cos:X — b (b +2c) — «=]"- 

 e.\. quo post reduclionem ; 



4r=[«= — 6».coÄ.=XJ —4x[2a^{b+c) cos. X — b-{b +2c) cos. X— a^b . cos. X] 



= b^(b + 2cy- + a* + 2a'b (b + 2c}—'ia'{b + c)^ 

 4x- [a' — b- . cos.= X] — 4x (b + 2c} (a=- — b') cos. X = [a» — (b + 2c) -] (a^ — b^) 

 , _ (3+ 2c) (g-- — 6') cQjf.X _ [«= — (6 4- 26-)'] («> — b'} 

 a^ — b^. cos.'X ' ^ 4~(a» — bKcos.' X) 



^_ (f>i2c)(a-—b'-)cos.X±\/l{b+2c)^a=—b'-)'cos.'-X-\-[a^—{b+2c)^](a'-b-}{a'-b\cos.''X)] 



2 («= — 6» . co4.= X ) 

 _ ( ''> 4-2c) («'— 6') co^.X ± ^[ (a= — 5'-) [ ic{b+c) COS.' X + a- — {b + 2c)' ^ 



2 (a' ~ b- .cosJX) 

 In liac aequatione nunc si substiluas a =: — b , hoc est aliis verbis si casum ponas 

 9—P = f>^ sivc BP _ AP r= AB, debet enim esse differentia , invenies ji = PX=0. 



Si sit a z= , sive AP = BP , reperies x zu , quod abunde patet ex fignra, 



2 cos. X 



si ex dimidio C rectae AB erigas perpendiculum CP' , quando eril 



PX = ^^ — i^+ c _ b+2c 

 cos. X cos.X 2co«.X' 



Si deniquc sumatur a = 6 . C0.9. X , acque ad differentiam pertinens quia /5.co«. X<4, 

 prodit r =: oo , quod item egregie cum figura conspirat. Quod ad summam attinet , 

 •,ubslitui potcst a = b + 2c, duos tunc obliaemus valores , fil enim 



jt =r 



