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# Title: Leibniz-Handschriften zur Mathematik LH 35, 10, 1
# Folio 12r, page id 00068154:0023
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2x-4- 
=4
Elementis radiorum
Calculus de crentis adioum Ellipseos ex umbilico eductorum
x radies velocitatem circulationis inveniendis, et speciatim de casu quo circulationes
rectangulum
Ellipseos ALGV iaxis major AV. foci OF centrum C triangulum characteristicum GDL
e descendendi impesus, versus
Goccurra
irmportionale
in quo  GL velocitas in curva, Gd circulatio, D L  descensus
focum O, _  seu  decrementum radii LD. Cducta ex Fbisecanti e angulum FGC perpendiculanter
occurrat in , et ipsi  OG in M, patet triangula GMH, et GDL esse similia. quaeritur
ratio GM: MH seu GD: DL. ex OG. Habebitur autem FM si drea trianguli OFG dividatur per OG.
habetur et MN ex FM et GF (seu AV-(,OG) quia sinus anguli dimi MN sinus anguli dimidi MGN
habetur ex simu toto GF, et FM siu anguli integri MGF. Area autem Trianguli habetur ex tatvibus
quod hic peculiaria involvit toremata compendia sit (0g, , AV, 4. OF, e. OG.r FG r. q
summus laterum +C
differentia prima je
differentia 2da AC
differentia 3tia -e
+4 - Y
= +e+ q
+Y - 9+r = + e-9+21
-Y +q. Y = +C + 9 - 27
+r +g. Y = -e+ 9
Ergo area trianguli 49 - ee, in eC - 2r - 9: 4 = OFG -
Ergo erit FM- V . LuI
jam ob angulum MGF ab ipsa GN bisectum erit MN ME: MG: MGGF
et GM= P GOFG FM seu qui 4F.Y agrra
Ergo fiet SpN. A 2X = jam quin MN: Et quia MN= MF. MG jam Et GM. M N_ seu GD: DL est:: MG+GF: MF
ergo GD: DL:: FG _ = FM + FG,: FM. seu Gd: DL:: r-g - 4Y: Y + q. Y
jam in eo casu quo circulationes sunt radii reciproce AG posito temporis incremento , ducto in aestarie
sut  est GD = aD: Y ergo fiet D L = 2Y. aD:  . r9- 4r. r +9- Y, seu
postremo DL = aba 14 ee in ∞ -2r. g+AY (4r-9 r - 49- ee. ec- 2r. q + 29 Y r.
nR: Y = 0
Ergo 0: D.. M: Y
Eandem quantitatem ad huc aliter quaeramus alia via. Angulum OGF bisecans axi occurratim 
et ex F et O eductis ad ipsam normaibus in Q et H. sunt autem Friangula GQF GHO,GDe, similo
et triangula RQF, RHO etiam similia, et ex natura anguli bissecti FR: FO: FG: AV.
et OR: FO:: OG: AV. quaeritur GD: DL seu GH: HO. Et primum GE: GF: GH: HDGC
QR: RH(:: FR: BR (per 4): re BGF FG: OC (per
1092 Et 3
5et 6) et similiter per 4, 5,6, F. Q: OH: FG: 0
Ex  his quaerendae GH, et HO
denique FL + Gq= FG2 et GH + HC
Ex9 et 10 fit FG. DH: OGFG Gq et cessat aequasio ex duabus difficilior, 10, substituta in
ejus locum 13. in qua tollatur GQ per per fiet FG CH: OG- FGaFG. GH: OG
in qua tollatur  OH per 11. fiet FE. OG- GH OG GG2 et habetum GE. K
FG = FG- FG. GH: 05
G2 G FG. GH: OG et per 10 F
Rursus H  per 7 fit
et ob i fit HO 05 - OG GH: et RQ = FN - F 92 seu ob 7 RQ = FO. FG. 5 
=FG + FG. GH: OG, et ob 4, 5, 6, RH = FB ∞G 0G-G + EBG. GH: 9G
RH2  F02. OG2: AV - 0G + GH:. Ergo peraequ. 12. cum 13. 16. et 17 con-
punctam, fit: FG. GH: _OG + FG. VFO: AV - 1 + GH: 65
+OG. V FO: AV - i + GH: OG = GH. unde fet.
: (FG+ 6G seu) q = FO: AV - 1 + GH. OG
1-2FG+F
seu GH = B S - FS: AV : Y1: 0G, - _ 1 : 1 - 2FG + FG = 51)
03 dist
BH = OG TAV = FO :AVY 1 = 0G: 0G -. OGFG+ FG et substitutis literis
GH = Y sgq- ee:  1 - rr: r - g. 1 = r. 2r- 999- ee: 4 ar= a9r + 99a rv
c jam OH = (OG =GH et Ergo OH: GH seu A DL: DG:: (OG - GH: GH. jam
DG = ab: Y. et fiet: DL = a SC GH: YSH ad Sr= GH: r.GH
seu DL jam si compendii causa Erq vocemus p, et 19q ee vocemus h, fiet
GH = rph: q S pp- rr. ergo  r = GH =  gqrr pp. rr-  hhrrpp: 49. pp. YV
seu =  ee rrpp - 99r: 4qpp - rr et fiet DC = aD eexrpp - 99rt: hxrp
seu DL = aB seepp - Ggrr: hrp
In numeris sit g5 e = 3  erit h = 4
Eautem quantitatem ad huc aliter quaeramus flagulum OGFbisecans axi sccurrat in A, et
h =, 12
4- t3. ex 5
ex F et O eductis ad ipsam normalibus, t occurrat in Q et H, quaeritur GH: HO:: GD: DL.  sit AV, 9
et FB, e et DG, Y et FG n = q- Y. et GH, sit X, et HO y. ob angulum bissectum cunt 
OR: FR:: OG: FG ergo OR+FR, seu C,: FR:: OG+FG seu q: FGsO n. ergo FR = en: 9, et similiter
OR = er9 ob AC Ob RQF sim. RHO est FQ. FR:: OH: OR dver seu FQ- en 9 y; errque = ny: r
Q: OH seu y:: FR: OR:: n: r. Ergo F9 = ny: Y et G9 YFG - Fq2 = p TrKyy∞: et QR = FR -F =
49 + nn -9 + 29u -nn) Seenn: 9q - nnyy: rr = n Seerr - ggyy: gr = eR et RH = SOR - H~ - cerr: 9q - yy = Seerr - 99yy:9
4+ p= (2Y
49 + G- 29r + rr
q- p = n
un-rr-
Ergo . G jam GH = GQ + eR + RH. ErgolX = n Frr-yy: Y + Seerr- qqyy n: r F 1: 9seu) R: Y
sed rursus X2+  = rY, quin GH + HO= OG. ergo fit r rryy=rrnn-nnyy+ eerr-qquytrry. er. q
H rrnnt eerr vocemus AEY, et  99 + nn =rY vocemus B, et fiet Byy = AEr + 2S.. seu
B1t = 2 A B ry + r A+ = a r- gy. eerr -qqyy = 4 40r4- 4ggrryy -4 eerr yy+ 497,
jam nar=q et r-n = p siet B- 9 q+ P et q+p nn -rr = pG et B- 9 - 9. 9p. sed 4 - 2r.
Ergo B 2n sed q. p = n. firt B =  gn. Aursus    Et A2 - ee -pq et fiet: . (+qqui
(99nn y= 2 gurryy ee + Egurryy pus + rt  - 2r eepq. + rp 9= 4e rn ggrryy-Anneerris
Ergo denique yy = 4Cren  etr+  cerq reerpq. rpq: aggrrunt aeerrun ycern
et dit i. vi dtubus
F- F
Y = q = 2Y-
 Pr
4n qrqtq
Ser 3r 4 p 4)
nam si