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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 101r, page id 00068199:0201
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Ex puncto proposito
ducere normalem
ad curvam est ducere aximam velillinis
Hoce aoplcees Hinc culculus
Maximo. Aiulimarum synboliZat.
differentialium,  est cum calculo tque
Ducamus hnc expressionem
parametrilogarithmicae, unde pendet, dimensio areae Hyscerbi les 
AH sit a, cujus lig. ponitur esse 0. ipsius HeB seele ABC seu l log. sit P 
AX sit X, et XY sit y. sit AX = 3b, Et fiet
Y = _ da (2b. a
AB sit k
seu quia 3b = X. hoc loco
fiet V =  BX: a: a: a
21, quae est aequatio generalis
a
Quaenitur jam modus ex puncto, dato P in axe AB, dato
ducendi minimam ad lineam rogarithmicam
quia punctum p est datum ideoc data est Ap, et quia punctum
Y est quaesitum, quaeritur AN. Et  porro p Y Minima
vocetur M, fiet p - X + Gy= mm. Et
=X: a (4)
Pro y ponendo valorem, fiet: pp- 2pX+XX + aa b bb. aa. = mm
AC proinde indenire X talem, ut sit A- 2pX +XX F bb
seu - 2p X +X X + aa bb: aa   ap posito nim - pp - a a = a M
a
nam et haec maximum, ac minimum non mutant. Unde
res eo redit, pro invenienda quadratura Hyperbolae, ut sit
0
invenire X talem, ut sit
XX - 2p X + aa bb: aa
omniam possibilium maxima
seu. (MX XX - 2pX + a a bb: aq   a=
X
L.a: p= X= C
quanquam
constanti quod
aliunde scimus in casu X ad maximam. Nempe est subtangentralis
sufficiunt aeqqus 4 vel 5. da bb: aa
Lmum (Xa (S()
constans. sit Fit ergo (p(p) - 2p () + (1) + aa bb: aa. = (mi siet
1+ (sLia ) - (X) = C
ob: a
ader  ob aeq. 7 potest inveneri tolli X.
e
no
sit
XX - 2p X + pp = nm seu X = Am ( 4 pp  4C p (+ 4pP (4 pCH CC 2p+C, : 2
fit 1. ab
nam fbt: ad
= CX + cp 1 seu X = Amm +CC = p + LC, nisi calculus fallit sed
fiet
haec mittamus.
si aeg. 4 differentiemus
a
(14
 fit X: a log b: a = log. y xea
(16)
jam si AB sit h fiat  loq. b: a = h: 4 et fiet hX: aa  log. y. jam
47
2p au
prius differentiari debet bra b: a:y
log. y = ee sdy: y: aa Ergo hX _ = ee (ay: 9 Ergo hdx = CC, dy .t9
E sed y = a b: a x. a  Ergo fit  dy 72)a h b: a dx: ee. Ergo
a 4)
e
3
26
o
unde sed ex
Ergo Ret
terentiare jam possumus aeq. 4 et fit- p + (2x + Daah bb: aa
eq es L unde per aeg. 7. tollendo fit - p+ X + p- YC h: eC L0.
Hyperboliaae est media pro
C= ee: R. seu parameter latus potestatis
portionalis inter paracpetrum Logarithmicae cujus arcae sunt ut logari
jem Ergo ex 19 fit dx = C du: y. Ergo si datae he Hyperbolae si
Linea
est c
sit dx = cdg. y erit  esunt X e y covodinata potentiae logarit hmilae, cujus parameter
et du
it
Etgo d. aa lbiaa
pahba
eydy
raotbli an dx: et