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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 13r, page id 00068196:0025
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inaequu ex
sed jam video novissimum casum sic rectius exber   laitie) x firii pt (a.: X) X:h. Ergo
 ergo et idem in eq.3 fieri pt. Ego  hoc moddo h non occurret bis. Et res ita prodibit:
(omo dy: dx = XiVia (edo) dy: dx = ( : X:) X. h et stio) y = (a: X ) X: t2 unde
deduda aq. 6 est differentialis quadraturabilis. Sed ex eg. 2 B sequitumo X: h = y:  (57
et (2mo) X. H =  X ay: aedx 6) Ergo (13ho) fit dy: dx = a Y. X (7) pro X: a fiat
(ato)Z: a erit et X x = a dq seu dx =  bu _ ex=  AaC a Z dq. Ergo fit
 yZ: B) Et talis differentoris, aequatio erit
glo dy: aq= sa: q er (qi a0  pi utabia
xhine etiam dueitZ= y. ): aa _, quadvatutabili
ee
quadraturabilis, hac
hodus curvarum positionedatarum
si yz = va, fiet qy =va
et habetimus a dev. 2): dq  (a. 2) 2 . a v: a 18) vul (1800) (Vdz - 240). 42: d2) = a.  1 a ra
Hil seo videamus an aliquid generaliis obtineri possit. Et omnia ab abintio resumum 
(Imo dy: dx =  1y: at; X: a2 do) _ assumta dy: dx = (X: a) ; h: a (1. unde (ytro)
de
 pra unde ex1 et 2 fil (9: a); (X: a) = x: adithia) V unde (4Fo) y. a = (: a); ch: a) ~ et
  d0 = dx dec. x) (r: a); B: a) (2 et (to) d() = dh  sec H) (X: a), (t: ) 2). jam
17 mo ) dy = do + (dl. Ergo d (4per 7. c. 6 fit (8vd) d asec. X) etc. + dh dsec. hete D: dx =
(X: a)(h: a)  seu (gno) dx (X: a) H: a) 1
Desideratur jam in hac dequatione dividendo
ec. X) (xa) 1) 2 = dh d bec h) X. a) dh: a) 2)
cvanescere a
sitome e
sa
(sit sit stomo) (x. axth. a( = X. h3. h; a et eodem modo
sit cmo) (x: a, h: ad 2 = X  (S) Xi h; a (∞) Sa sit xasta  x. hB
Eritus somo) X. a, h. a = X. hB, (d (ec. X) X; a; h: a (2   +( ex 1) pra, h: a. 2Y).
sit (1dmo) Xia; h: cd  = X; hia 3) in X. h (∞ et simill (mo) X: a; h: a 2) = x; h; a  in X: h6) = (12mo) y
Ergo do  d(V) seu hy = (13ti0)  X;h; a5. de d(ecX) Xih (O + X; h; aC) dh, assecit) X: h6, 1
= Xih 63 dx A (seC X) Rihia 3) + X: hE, dh d (sec. h) X; hia 5) Ergo hoc ad Ax, ut ; h; a (3) in X. h B ad 1
et (15to) dx, xihia E. d (secX) X: B5 + X. G +. (ecX) xitia 5) = Xihia 3) in xt V 
dh, xihia 57. d (sec. h)  ih 6) + X: h(6) d (sec. h) Xihi a (5) oportat haec duo aequationis
latera haber divisorem communem, per quem dividendo utrobique, evanescat litera  a
li
Ergo firt (16to) Xihia 5  (sec. X) x. h(6)+ X: h 6 a (sec X) X,h; a ) - xihia B, in t 
Xihia 5) a (sec.h.) X: H6. + x: h6) d (sec. h) X; h;a 25)
Kon
Ex aeq. 16 fiel fiymo) Xihia 3) inritia  decx) nih G + x. h ) as sec. X)a, b, a b -xih; a 5) Asec.h. ) X. h 6 x. h (2
s Ergo repertum est qualis assumenda aequatio secunda, ex data ata vel3tia
-xot 6, A sre. h) Rih; a V X. 13)
Et ita loco aeq. 2 reperta est aeq. 13ba jago ent aq 12 et (8 t ollenda est
jam literam ho tollenda est lit t litera h exprodinet aequ in quae coincidere debrult
(81) = dy
cum q3 cum aequ Hec intthodus esset (actuoen s  et promtior si Xhia S) idopanstet quid pade do
Aliavia st) e quemus duos valores ipsius dy: dx ex aeq. 1. et ex equ 18, et fipt 9. a, Xia  aihi; a V H sc. X) et, ut in aeq. 8
ubi pra y substitu enda valirem ex aeq. 12;  quod eo facilius quia subosti butuo tantum facienda in datis; superent
aequatio 3I, in qua solum X, h, a quae debet esse identica. compendii cansa faliamus X hic 5) LI) 
et X: h6, = f2d0) D) a hic sigto) giasria= e d (c.X) D + V dsse cx) a -a ec. s) v-D d sect) ae
it pihi = bino, (d.  seu Esto yla  d(scc x) Da -D d (sec. h) v_ = X: a; BaeL
cuEsto) d (SecX) va = co d (sec. h) Ba = _ L; Ba , vel (sto) Aster x) vN - X: a; v_: aa l,: A ecti) vae
= X: h(2 seu co, vel (27mo) d (dec. X) X: h6). x;hial = X: a; X: h (6, Xihic Bdat: d (ec ) N. R6. X. 1, a D = x. 7)
vel si malimus hypothesi aeq. u. carere sufferert:  28 v) Affer. _) Xih; a 3) - X: a; X; hia 2L: A (sect) 2, h, a 2) = X: L 12)
Retumamus ab initio. data  Emo) ay. d = g X: L; y: a (2do) ese alsumenda () dy: ax = X;tia es unde (91) Xa; yc a  =
Xihia V. seu ato (4H) y= Xihia 3 assumenda  jam (5t0) du = d0 + de) et L  = (6to) eld (ec. X) val. g eY
(ymo) do = d ssech)  val. y. Et fit dy: dx (8v0) =di d Bec. X) X; hia 2) ) dh a bec ) Xi hia 2: xn fay. de- Irtiab
compendii carsa X,  X; hia  vel copendii causa  ex  ec. X) val y + dh. d (sec Li) val. y = bomo ) val. dy: dx intelliquane erloe
per X, h, a -  Ergo _. fit dh d sec. h) val. y = (mo) dX, ( val (dy: dx) - + (sec. X) val y). oportet iam
seu db = (12mo) val (di. dx) - d (seC. X) val. Y  quod o portet ese = (13tr0) X. F 2). seu oportela divide
tem et nominatorem per
numera
munem sivisorem evanesere
et fiet (14to) dh: dx = x: h2Z quae est quadraturabilis. jam ex aeq. 13. fit (15to) val  d ec) cly x. h B. A sec h. vol y  +d cxd
a ec. k) val. 9
val. dg: dx) seu = X; hia L habemus ergo qualis ex assumta caq. 4, fit assumenda aequ 2.. AC proinde loco a 12.
ex aeg. 15 scribemus, () 16to) dis. X. h  d ec. h) val y + dbec X) val. y = dy: dx. sed exer unde per aeg. 1.
fit (ermo) xh ∞ dsec. h) val y + d (sec. X) val. y = X: a; y. a, vel _ in parte dextra hujus aequationis pro y substituenda
dxJa; val. y seu fiet xhes X. a, val.L
X: 14 X,a; val. y.  (sec. X) val. y,: d. (Nec E) val. y. Ergo valor y in aeg. 4. seu X ihi d ) sic assimi
peratorem et oominatorem per communem
Xih; a 2 esse = (20mo) X. h16,  X: h, aS) vel per comp
ad = (21 mo) V de, posilo v - x. h16, et
val. y = v abec. h)ae + a d (sec. h) V
jam balorem y e
ergo per 2, 23 se ex +g. c9 fiet X: h(B = X; a; vaV = V dsec. X)a = a d (sec X) v, L: V 2bec. h) ae + a + ec. h) V
multiplicatae itanta m per a vel ejus potentiam. in g. M19 etiam
consdorandum, quia a (sec. h) valor. y non immutat X ideo posse facilique conferri cum xia; val. 
is
an
Ex aeq. 19 patet non licere fucere val. y = formulae per X et p,
u
ae e ta
i  me ne