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# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 11
# Folio 10r, page id 00068056:0019
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Examinemus Calculo solationem problematis de figura in quam
c
se pondere proprio carvat funis.
S
Ca veccece
ranscementem dy=aax: Exx-aa
2
Invenimas autem Aequationem
Aequatio autem praestanda est
y Sax +du: (Nax +dus = y - X dy: dx.
seu fiet.
Fyraxrdu  (Sapidus V- (sax tau el der  dx
X dy: ax
en (y Faxadus) (Fax edis dyst y dxray) - udx +ay X dy: dx(= ssatus dus son.  dus
a
 jatum ayx per I est dy: dx = a: 1XX - aa. adeoque fiet
e
seu vax nau dy. dx = =Sranaau d ay. dx
4x
d dy: dx = axdx: _ (.2) Xx- aa adeoque per . , ex A~9. 5 peretu
co = (-1.2 Cd dc) fit EF +a a: SXK aa = = XadX. XX- aa ExK-aa seu
1.2
23: 2
sit C0 = Xx - aa. de
fiet d 1.rxa na
sex aa d
e
et a a: LXX= aa
red
ap 
 sublato communi divissre irnationali fit. ( sit Hrixa
= = (, 2 2 a X AX: XX. 04 Yer  aa
ex +que = X dx: XX- aa. Adeoque Ax idy  dx. Xx. ( nr a
seu 1 + dy idr = XX: PI xX a4
seu 1 + aa. XKG XX. 1 XX aa.
seu 1 + aa: xx aa = XX. B( ex aa
C
ergo subest error inralculo
eu XX aa +aa XX
XX aa P) XX aa
Vdx + du a: (XX -aa = - ( Sax +du a XaX: XX - aa 1XY-aa seu divisione facta
L (10)
adeoque er aequ. 5. ex n 6 et 7. fit:
eu Yax +dy - x (Sdx +du X: Ex aa. sive  Sit dy. de =- sot lyX. n a
lesen pro du: ax ponendovalorem ex aeq. 6 fit 1+ dy: du = X: rxx- aa. Ergo ex 10 et I1 fiet
(11)
1=-Fax +dg: sex aa seu Exx aa  _-Ssaxraus.  cujus differen
1
tialis est: Dx dx : Ixx -aa = - Sdx + dus seu XXdx: XX aa = dx + dy. seu
(14)
XX: XX =aa = 1 +  du: dx quod succedit test enim ipsissima aequatio 11
(16)
Itaque habemus solutionem desiteratam
si velimus a dx: exx-aa invenire rationaliter sic licebit procedere. Sumatur
XX  aa = (X ZXC DC00 et X = co:2 + aa: 2C0. fit adeoque YXX- aa = X
Ergo isiet denique
dx = deo: 2  - aa dco: 20000 sen =coco- aa . ticoco
A dx aa: YxK- aa =  cocoto coco -aa duo: 2c0co aa: aa = 4c000: 10 =- d0 : o
+aa: 2C0  W: L= aa loco: rd
Ergosetdentr
quae pendet ex quadratura hyperbolae. Itaque problema est solutum. Nam
qdordinatae hujus curvae ad tangentem verticis su basius, sunt lagarithmi ipsarum co
1 L
paxem
 seu ipsarum X = Xxaaa posito X ese abscissas seu AB