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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 3 (shelfmark LH 13, 3)
# Folio 198r, page id 00068200:0397
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figurae
perutile videtur consiverare quaenam curvae possint quadrari
e
(27 81
per meras curvas paraboloeidum sit co = Z fit deo = e. Z. dZ. et
curvae longitudo erit , due S1+ ec Z = W.
Sit Sydx = v et tiat WV + 2= Y
posito p esse quantitatem ordinariam. quaeritur v, id est quaeritur relatio ipsorum
L et aY ad ipsam X, posito Y datam esse ex X. Ex 5
15s
+ = fydex. seu dz Sitec  + d p = ydx
si dy esset irrationalis simplem
Ex ∞g. 5 per 3 et 4, fit (dZ S1t ee Ze
en d3 + ec ZiC.
quodprica cujus quadralum sit rationale; et irrationalitas ejus contincatur et in V adeoque et in dis
sub latu foret irrationalitas in aeq. 8. sit y = m Y et p ner fiet d
et p nifien sr, fiet di = dn tr + ur: r seu dip er, rdnt Lnar, : Y. Ergo ex 6per 11
(10
fet d7 + ee2 dz = Gmmrax-Emran-mndr +rdn + ndran + nar: Y.
2
A
dantur m et Y. Hinc cum terminus mnydx sit datus; patet quousque assurgere possint
termini reliqui Q et nE ut possibile sit desideratum. Quanquam si Z vel n. sint
rationaliter expressae per X ut possibile sit
fractiones altius assurgi, possit
BGeneralius
seu yax ∞ dz rdco. sit
Etvius sed ut Sydx conginat cum S Vao
yu rationalis integra  quoad X et sit y rationalis, itemqu integra, itemque co
et altissimus ipsius Z terminus sit  X et ipsius cd sit X   seu Z= X etc. et Co = X et
ets
vi (0)
st
m-1
fiet dq =  M. X. dx et dcd =n. X dxinfiet  Dentque sit y = X  etc.
2n -2
()
siet X et mm. X. dx at nnX. x etc. itaque apparet quam alte assurga-
2m- 2 f2
tur inquaesitistur, non ultra scilicet, quam ut sit m = r- 1 et  n = r-,
imi addamus V. fiet (ydx +a = sor 12 Z + dcodio et y = m Ir exa i
fiet di Tr ven + ndr ergo ( v mr + Vdn + ndr, tr = Y fasd + dco ded
seu um fiet di= (A du Tr + ndr. Er =) irdu + nar,: Etr
et seu dip = Erdntndr, Er : EY. Ergo ejam ex 1 fl ydx + d ep= a dx +dcodeo
Ergo per 6 et v ex 6 per 2 et 5 fiet, 2mdr 2md
Erm de + 2r dn tandr  Yr = ar SaZ AZ + dio deo
Unde quadrando
arrmmaxt Barrmdxdn + 4Mrmdr + gaarrandu + 4aarn dndr + aunn drdr = aardydu t aardiolles
ubi m et r sunt datae, ex X, at n, et Z, et cd, quaesitae, quae proarbitrio assumi possunt
ng
et comparatio institui veq verb. g. si m - a et exa  raa fi et p= a. X, a
4∞ 4 ax AX dxx + Sa dxd ex dxdn t gas et dr= - 3X dx: a et rr- a- 2a X + X: a
quibus substitutis in aeq. 8. licebit deinde sic explicare n et q et eo rationaliter per X, u onia
ut aequatio  8t praestetur. Nempe si n assurgat ad X oporet videtur n non essuns si nolumus assurgi
ultra X_ debet n non assurgere ultra X, et q non ntera dzdque non uetra X ergo dque non ultra px, ergo
Ergo Z non ultra X; idemque est des qu de re co quod de Z.  sed ita non siabemus sefficientia