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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 20r, page id 00068196:0039
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Exe
Investigatio Analytica Tangentialium
Tangenetiales voco aequationes adrs differentiales
primi gradus. Assumo aequationem ordinariam quoad unam
indete ordinatam, cujus coefficientes dentur utcunque ex abscissa, et hujus
hanc differentiactore comparo datae. Vel quod eodem redit et syntheticum
 est, sed si omnia percurras analysin absoboit or dine assamo aequationes
ordinarias ad crdinatam, et video quae differentiates datae per quamque solvi
9+ m = 0 hinc dy + dn = 0 quod hinc ducetur, palet per se
possint.
hytam = hdy + ydh + adm =0 sit data aadus + (By dx + a) dx = o
quae multiplicetur per E: aa fiet prodit hdy + βh ydx: aa + h) dx: a = 0 Et
et fit aadm _ aadh = Bt dx se aa logh = (Bdx et aam = (h) dx
adeoque aequatio quaesita est hy+ sh) dx: a =0 posito logh = (βdx: aa
a) hdyt ydhiel + by + m -0 fit hydy eah dy + aydh + aa dl = 0
ral dy + hydm + at dm
+ab dys
hy+ al + ab
+ ab dm
Sit data aequalio agdys + a Bdys+ Vydx 1ad ex ∞ e, fiet h- a
Sit data squatio aydys ta βdy + Vydx. + aBdx = 0 quae multiplicetur per h: a
fiet hydy + h(Bdy + hVydx: a + hbdx =c. Et comparanda
fiet ex dy fiet h∞ al+ ab , βua al = h B ee βh - an - ab
et denique ex ex fiet a aBdh + ah d(3 - aa dh H(Bh - ah - ab) V dx: a - adh: h)
et ex ydx fiet dm= m = (Vdx: a sadh: h
= hBdx sit b = 0 et fiet aBdh: h + ad (3- aa dh: h + (3 - a)) dx:a  (3 a) adhi. B
-Badh: h + a a dh: h = hAdx. et evanescit h, nihilque agitur
Ergo maneat b, et fet Ca(B- a) dh: h) + a d β + B- a)) dx:a  a (B3. a) dh : h)
Lb) dx: A + aabah: hih = Adx quae aequatio solvi potest
ex Canone proxime praecedente nam bdh: hisi est differentia a - b: h
a Adx = ac dβ + (β- a))) dx - ad ax et sit Lsc Z = su Adx: a
1
20
Si AdX posito logto = s) axiaa
t
tietque
sit b: h = Z. a et aequatio  erit talis
et excanor
posito logW = (Vdx: au seu b =
aleue (hyraltab) + g + m = c. et t = βh. a h b et m= sidxia = dogh
et posito A = aa dβ + (3 - a)) - ab,: a
Data ergo aequatione solvenda aydus + aBdy + Vydx + aD dx = 0. fiet
aloge (hy + al tab) + y +m =0 posito esse - Stonax logro= fdx: aa
U
A = a aadβ + β a) dx abdX,: a dx  l+bBh: a - h -b et
henique M - Sydx. a = alogh, et b manet arbitraria toussans
seu f aloge (hy+ Bh - ah) + y + sy dx: a - alogh = 0
posito
posito  = 1 fuit et W log W = f) dxea posito h = bW: sondx
et legu = (V dx: aa et A = aa d β + (3- a Vdx- abdx: adx
i
repetamus calculum: ashdys +ydh +a dt +94 m 20 Ergo ta hdy F aydhta d +hidy =
tigdm 
im ydx sit
superest x et dabit
t data aequatio Hydus + Baus fVgaX + abi dx nd quae multiplicasa per Ai a dabit hyputhedy+ s ydxia a sd tu) = v. Ergo
(4s p atdB anat  (a Bah at a β-aadhe ((3- a) H) dx: a ((β- a) dh = ABaX. Ergo heva nescit
xtaddx =o
Adtro m= Paxia  log.t ex du fit an + al + ab = hB sit a l+ ab = hB at Bh- ah.
t hypusdy-
kgys dk
Alter si nsNaq Ry  P aaya  + RX y ay + altay +
dk raatam Tal ycus quae multiplicerur per rhy +P fictque
indecoparendo ex udus fit hN+ 2aC - n B seu Al = hB . HH. + xphyam ahgus dus + Ehise ge + 2bst ex taBodus + 2hv rylxia + 2kB yex + apB ax = 
x
K=Fit
et ex dysit lK Tan = C sO ABK - HKK + apt = a (β3Y et ex yudx sit aK = 2hE dX e seu
et exlude it et ex ax ter aa dl + aldm= sp B ux  seu at al aBF a Bah EA ex- = aF d + Bam  PK an = siK. HKK tuas 
supere s ex yax
sit unde ex se pvicem habentum habetur af (B-  K) (t ah +  dM) adeoqe habetur asahih +Mper Quadratura
equai aantin n aleat in ente tacia- aptax  de  atk r uua scy t jusu habesur Sahisi ta e prvinde
caleulus nion succedit
alnil
hunc hac
pagina
aβ est homo-
genea uniteti
ut sax seu
ut loguhuu
ut logX