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# Title: Leibniz-Handschriften zur Mathematik LH 35, 8, 3
# Folio 4r, page id 00068098:0007
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 Juspicet hic figniam Y. ducatur AF. paralleta MC, et FG parallela AAMM. Fantem est in langenta
 summa munum AF ad AM. applicatarum aequatur segiento AKE duplcato, ut alibi demonstratum 
igitur summae omnium, GC. aequatur Trrangulo An. duplicato Haec con alibi a me inventa ac
apmonsnata, convertere hoc loco tentenus in rem nostram Quaeritur fgma cuus producta Ma, sit
semper dupla AM seu 2X. omanitestum est differ applicata posita , minimam applcalar
ejus differentiam ab applicata sequenti esse 3
C+X
2X
Quare minima applicata posita y, erit peneminium
y + Y, et posita abscissa X, minima =1, erit pene minima applicata: y + y. Jam posita MC.
applicatarum minimay. et AM = 1. eit AF = Y. et GC = y atqui  C+  seu GC A MM-
libia me
M porro demonstratum est MG  summam aequar segmento AXCA ergo posito NG- AF
summa omnium Ne aequabitur Triangulo AMC. ergo ha loco GC.  + NG= 2. seu A
ductum in AM = 1. aqui seu 37  X = AMC - 
Griangulo
7
xin eNc = a
si sit polysgonum quodcum laterum AB, BC. esc. suntaque diiectrix ad q
usi DEp
referantur DE. duatur tangens CBF quaredam D A productum  secet in G. recta
DG applocata ad HL, aequatur triangulo DBC dupbeato, ut alibi demonstratum est,
HI
et ideo Hi dimidiata in M ducta in LH, seu rectangulum L H M Tri angulo
DBC aequatur. Eodem modo LBV = DAB.
data jam LD, et DP = AQ et, PS, quaeritur DA, = Lq, et EB
Esto Ld - x. DP= p DA = y. eret qB = YX. sed ad ipsim y. cognoscendum opus
foret alia adhiec aequatione, sed cujus ego pinciium invenio nullum verum est
quidem summam omnium GB RB posita R9- 2tq, semper aequari Triangulo sed non video
quid inde duci queat, tentandum tamen.
x aequatur esto quaedam recta sumta pro applicata = b. positaque minima
X = v. fiet by = b- y. posito y esse applicutam sequentem quae quaeritur.
erit b-
13-y. P posieqe 3 1. et p- 2X fiet, b - b = y. Jano fiat dG. g.
fia   y VX patet esse (DG.3
LB- b
bt hoc loco = =. et L  
fiet g-
DF.X CFp- 2x 
entue 2B- 1, et R1. Zb et erit 2 summma omuilim RB = bx.
sic; sumta sit non b, sed prox eiiei  mina nempe y du B = B est pru X, poptuchi Xui e
pro P. ponendum p. 2X 2 fcetu et  Y pro DX fiet b-
2( - 1 -  - 2Y
at
2X- 2
cur subtiahatur ab y= b-  restat b- , cui addatur paoris dimidium  x
fiet: 3b - eb -D Ecderatur B X - B, Fed a 3b X- Ecb 3  eb= X
aex
de
At Ergo summa omnium AB, denta A9 ultima RB, seu demta 3. b, aequatur y X-1
b- Ex x1 seu dessia dnter
mermxam, CH, + me tux in sive
27 -
Cx - b - Ex, 2 x - 1 = =b
C
1= Ex, X 1, =2 ita evanesct b. et cum eo calculus
b2x
b X -
e
3b
sit quod supra r y = b - h
X - 1, = 2 b, vel bCx -
2b =  y X- 1
y. Ergo = - 2  Y Unde nihil novi sed idim
x - 1
Esto in figs. 2. portio Hyperbolae vel Hypeboloendis cujusdam Mchi. maxama
assumta
applicata ma
= m minima assumta 3I = h. differentiae A (vel etios
HN, =EFX
AH) X G et semper Een = M.eritque differentia inter duas quaslibet
appheatas proximas
Cedentem proginias
C+X, erit diffeienta inter duas quaslibet applicatas arte
~, et quia 7 =
C+)
efiet diffeentia inter duas applicatas
Under patet diffcentias duphieme inao modum exprimi posse
vel per midum loci vel terminis semper in se reflexis, ut si nesatiemus esse 
locus prima y nota est nempel) m fietmm
(4)
in4 EX, vel potus erunt differentiae. C +X
in
+X
in unm enunter ec.
proginie major ()m 
cix 
vel
nc +nx + M
c+X
ve
C+ X
e+2XC
Eac ergo resulum progressum, si relinquas 
semper M, ducta in spolesatem
aliquam ipsius c+X et ponver 
aucta eadum intancta in potesta-
tem propime inferiorem ipsus
C+x dsque per pobestatem
seu  posito C+X = g
tred. m
m
mg 4m
Mo+ mgam
ng3+ m
via
+X