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# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 7
# Folio 21v, page id 00068052:0042
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GWd: WC: FG:: HC: CK: KH
NM = AK CK = CMM
CKin FG = WC in K H
sit CP, _X sit M et CN = a fiet NM. Saa -XX: :  a : X
seu NM = a raa -XX: X = AK. sed AH: X:: a: saa - X
fit AH = Xa: Vaa -xx et AH + NM = K H dabit
a( aXX + aXX: XTaa - XX - v  X quorum proindedatur
summa, ex suppo sita summa secantium ad arcum
porro KH in WE = Xq in GW
purr 
et KG = K R seu a, +R9
  est autem  CR: XR seu gK: CA = Rg et fiet RG = aa = X Y Aa : XX X
seu AQ = aa -xX a sit FP = _ fiet a a = R9 quae summa pendet ex
XX
aagu
Quadentura Hyperbola et KG = a + aedto - a= alqu + ab et summ
da-ec
aa 
dunium kq 1d Gu erit lx reducetur ad loganthmos.
sed Si ag= 1og la  et 2  log. a
X adss
ho
sa . Evu
a
. 2 at
Ca loque sario
sca ysto
0
NM. IF ix NNE CN: XM: MC:: GW: NF: FG.
a90
a-
NM NSM sim GWF NM: SM:: GF: WF et VM WoF (autadx saa- xX)
=(GF SM. GF. Sed MM(in Flementum arcu) aeqo secant( in oliam
demto C5 seu Cp (in elem arcus.
sinum CP: CF:: GW: bF
M
Sed CP in GF aequ CF in W
Ergo si sit arcus AF (ipsis X a minimo inceientitus
fie it seuentes complement in Elementa arcs dent cinta demto volumt
om
rectangulo ATFS sub finu opeen et et siun versone aequates sib secantia, complement
omnibus ad basin tangentibus complementi ad basins sipplicatis, seu omnibus tangentibus
compl.
residuorum artuum ad axem applicatis, seu su omnibus tangentibus
pesiduorum arcuum ad axem applicatis porro secantes
complimenti ad arcum applicate sic invessigantur Ek CM vel Ckin GF
= RH in WF. sed WC Est autem KH = a3: X (aa XX et WdF: GC Sanx. X
 KH in Wd F∞ =  sed (H in WF = Kg in GW. Gu sit de 
seu FPl) fiet HG in WF KCQ in S W = a deo. t purre si porro
Habuimus curnan paulo ante
ac
jam haec inventa ad nostrum scopum applicemus:
quaeritur Sadn: datnn, ipsa n, incipiente a nihilo, prima orchuata 10, 11 erit a
ultima 12.13 erit a, posito 10. 12 scilicet esse a. sit ut 10. 14 vel H 1s est N
sic 10. ns vocetur X et erit X = ~: aa + n
quaeritur (a du: Saatun ja Cit 10. 11 vel terce, n (usque ad ultimam 10 12 (agra))
sit n et res 11. 14 eri sit X = a: Saa tnn, naxima xordinata 10. 13, quando
n est minima seu 0, erit a, et minima, quan fiet _ X n + X a = a seu
_ n 14. 1b vel n = a Saa =XX : X. ergo cum n est d, fit X = a
ergo incipiendum ab X = a jam faax taa - Xx +X Ergo ad circulo si CP
sinus compl. anguli NF (ut incipiamus a CP = a seu a CN) sit X, erit
tangens NM, s  araa-xx: X = n, res ergo eo redit
ut inveniatur summa tangentium adm axem, et habebitur area
13. 17. 15, 13. jam NSM sim: GWE. Ergo NM:  GF:: SM S WF
sit SM. GF = MM. WF, jam SM = CM - C5, et CS = CP. Ergo SM. GF= CM. GF-CP. GF
sed Cp. GF = CF. GW: et  omn. CF. GW = FPC rectang. FPC. (porro CMM. GF= MV. WF
(MV = KH sppra) vil CsA. GF = jam CM.  : CM: MV = GW ergo
CM. GF = M. GW. sed Mp = M9 + 94 tq = MN: m e. MN:C
in nn: a = _ a. aa = XX: XX. Ergo Me D a3: XX a: aa- 
GW =ded WE- ax. siet  MN. WF, aded: aa   adeo
ut Et cum X crese. (per 1, 4, 9, 14, 4, 5) = n dx. et su  13. 17. 15. B
asc: aa-  = aed. sed (X dx  13. 10. 12. 15. 13(2) B. 17. 15. 13 4 10. 15. et
10. 15 = (10. 12 (seu n) in 12. 15 (seu X) ) nX siu sx dx = (n x = nx (quod et
ex caljulo patet)  jam Ergo ex 16 sit   sed x a: da + nt. ergo fit
Caedu: saetr = (a) B. Aa- Gy -ay n X log. satY add + nX. sed ay+nxo
a
quia nL a taax: X et C iaa - XX. Ergo ex (27 per 24) fit sa aa  = a log Vat
=W
eton (a dn: sa +=ordinata curvae catenanae posito n’esse catenam inde a vertice
sirum n et tmad et g esse et fit ec
iteo
est =c
posita 1parametro B numem primario, et C = 1- XX
itnn-n
si CP, X.
NM, a Taa -XX: X
t
iler
diAe
ad sin vers o
sit ene
io
alsie
n
(11)
sin est
C
seu diametres
verse
, ad scy: an -s
l0g
entn. Mtum.n
LEn Ln