PER AQUATIONES INFINITAS*

^nde, pro uequatione s/aa + xx =_y, nova producitur, viz.

&c. Et (per Reg. 2 .) area qiiaefita7 &c» Et haec efi: quadra-

’ Cl -f-

X '

A * A'

^ i6«’

Q-

128 a 7x 5 x 1

-3 + -i

40<r 112 ar

1 l Z2a

^Lvc erit - ax + ~tllra hyperbolae (").

Eodem modo, fi lit v 7 aa-xx =y, ejus ra- c

. ..2 _4. ..s - fi *

erit a - -

xr

sT 3

A

l6a s

il-

1 zSa'

8cc. Adeoque

^ rea qu relita abdc erit aequalis ax — x -r —

X

ba 40a 3

s* 9

IIc 2 a‘

Culi (e)/

&x. Et haec eft Quadratura Gir-

Ari

, 7- Vel fi ponas Vx-xx =y, erit radii aequalis infinitae feriei

— ^x* —-^X 7 * - T~X* &c. Et

L v£ --- _i

arp B ' 6 ^ 128 . 3

, s a quae fi ta abd aequalis erit jX i% —

l Xi ~~ ~r%x l - —x* - Lee. five x* in

y* " 4 a 2 - JL_ v3 _ _l v 4 - _i_ r 5 fec EtW i%X llX 7°4^

c e ft Areae Circuli Quadratura (*).

Llli

8. si a/i +** 1

VT^Tx*

A B

=_y, (Cujus Quadratura dat Longitudinem curvae

Micae (g), extrahendo radicem utramque prodit

1 +

( r ) Sit hyperbola ftpc qux centrum habeat f., afymptotas adperpendiculum compofitas ef, esi. In afymptota alterutra, putaEli, datum intelligatur punftum a» Sitque bd cum altera afymp-tota parallela. Jam e natura curva; reftangulum eb x bd dabi-tur. Si igitur reftangulum illud datum symbolum aa defignet zredtam datam ea, litera b ; Indefinitas ab, bd, literx x,y; erit

by -f xy — aa i et

b- |-a*

-7'

Et feties Nevrtoniana infinite

. produ&a arex hyperbolicx abdc ultimd xqualis erit.

5Xis hyperbola xquilatera cd, cujus centrum Aj iemiaxis tranfverfus ac — a, abfcifTa abec 'tndi — a, ordinata ed — y. (Vid. fig. pag. 264») Erit aced area illa hyperboliea

" ^ er 'es ax + — _ ^° a + ultimo xqualis erit.

(J o- * S

' '"' lrc uU centrum fit a, femidiameter ae=«. Quadrans aeDc. Abfciffa femidiametri stPart e ce • ... x 3 x s

. etl tri, AB—A; ordinata bd —y. Brit abdc area circularis, cui feries ax- ---» -—~

'Mir.h' ba 4 c a

(J\ J^oduOa ultimo xqUalis erit»

Sit AE circuli diameter —i ; ade femicircultls ; ab abfciffa eea ; bd ordirlata =j>; feries

— vJj-a' 3 infinite produfta arex adb ultimd xqualis erit»

^!- r j ■y

Y 1 eni pe hoc modo» Sit ellipfis cdbe cujus centrum a, axes cab, DAE» Abfciffa axis mi-0i -1» M m Eis

L6Z

Qj'A''R ATII»» A CoMPO-SIIAttM.