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# Title: Leibniz-Handschriften zur Mathematik LH 35, 2, 1
# Folio 261v, page id 00067993:0524
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va
a
Habemus ergo hes distantias DG. OH. 1X qua acantur D . BC. et BH in NB et M.: AN. erit IX AN = BG, BCtOH 1B
ca = ex a- 430C
acrc
2xaC
ve aret
ab+ cb+
R
a+ CX
6acX p 
a
qxa
4Xac
a + a-
ia
ca - X
ac + EK - 2Y
ea
db +c
eX
4
aC+CX +6acX= 4 Xa
4c 4 4d d b + CB +b4 BC,
+4 X -8Xd +
q0
u
e
ac +C
e
ve
-4X’ ac
4+6aCX - 4X a c - 4X ac
4c + 4d  d b+ 5b 4 b - bC, ac = C x 16a c - 4u a  - 4 Xa5, - 4X + Brd. d. b + C64 b bc.
Hrc aequatio redudi tun potent quando  in casu aliquo occorente ipsius b. et C. ad t  ratio ad a data ert
resolvendi in
duo vero Methodus Generalis est redmandi problema datum ad per puram geometriam, illustri admodum
exemplo imperfectionis Algebrae in Geometria. Nam rectam FE transeuntem per D. et lineas datas
ita secantem in N, ut  1X in H sit = DGCBC+ oH NB. necesse est transiae per centrum gravitatis
livearum AB. et BC. Quod ita invenemus, libratae intelligantur primum ex axe AP. deinde ex axoDC.
si librentur exe AP. momentum earum ita facile habebitur, cum una sit axi parallela altra perpendicularis,
paralletae momentum erit rectangulum ABC. = ab. perpendicularis AB. momentum eris semiquadiatum AB,
seu a. Jam ab + a dividantur per eotum summam a + b. erit ab +  distantia centri gravitatis
inearum
a + b.
linearum AB. BC abaxe AP
momentum corum ab axe PC. eodem modo initum erit: ab + et distantia tentue gravitalis lineari ab axe BC erit.
crit. ergo aentrum gravitatis linearnum normalium AB BC in intersectiore
ab + b
harum distautiarum, pe quam du quod si jam recta per istud centrum pauter et D. ducatur, ea praesta
am
jounqu
a +b
Ex his intelligi potest  problematico Analysis simplex tenori quaestionis
exacte insistens irreducibilia relinquit insoluta, an ingentiss aequatione involuta relinquit, saepe preclaris
bit desideratum.
Eils Geometriae curae Meorematis, plana reddi posse
sit jam a = b. erit distantia utrobique rquales: a + a
15
10
2a caneu
2a = 3a
4a
A ue
a-a
ob
2 = = X ego  2 == X. ergo distantia gravitatis semiparabolae ab axc ad distantiam centrigravitatis
recta gradrati seu a est ut 3 ad 4 ergo 3a est distantia centugra-
vilatis smiparatialae  ab axe. At distantium centri gravitatis Tulinci parabolici abi axe AB ita investigabim
 cum solidum huitricei crca AB revolutione genituo sit supplementum solidi purabolici circa axem quod
est dimidium olindri ad cylindium, eut etiau dimidium cylindri, ergo 2 =
Z. ergo 2 - y.
ergo A si distantia centri gravitatis colino rect Quadinti est L- Iitinci erit 3. Ergo 2
62a. unde apparet centrum gravitatis linearum A BC et Tilimei APCA anincadere in candem rectam
15
ipso AB paralle lam.
Nnc quaeramus centrum pravitatis lam sermipartolae quam Tilinei a basi BC. vel AB s
une
ses
ilidem per salida. Constat Trilimi applicalas BC. paralletis esse in duplicala raoo altiendinum. Porro circui
revolutione geniti, sunt utque quadrata radior seu ordinatorum ergo si quadratum de R9. sit 1. quadrulum de
DC. erit 16. ergo solidum hilinci erit Homogeneum hilinea quadrato quadratico id est quinta
pars cylindriasolidum ipsius semiparabolae revulatione circa AP erit 4 colindri. His possius calculus
cntrorum gravitatis ert facilis. Nam E = = Y. ergo 4-
1075
12(3 = Y ergo si distantia antu gravitalis
quadrali  ab AP t seu a sit ut 5. distantia centrigravitati rect anguli semiparabolae eritut 6. ergo 6a)3a
est distantur antu gravitatis semiparabo lae aB AP. Hinc cum patet centrum gravitatis semiparabolae non cadere
 in rectam S T. SF B P centro guvilass linearum AB+ BC. vel AP + PC. conjngentem seu line dragoam
eb quadrati, quia non est ejus diptantia 3a aequais abab AB aequalis diptar tae 3 ab alio labee AP. CD
a- 3a -
per
ert
et 
10
4~ + x + d = 2Xd,