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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2b
# Folio 136r, page id 00068198:0271
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Differentiae seu elementa fguram  ubi et de locis ad superficiem.
a = a, Ct vel axL a β = a x  vel a differenta applicalarum Hyperbolae
Jam 1 +  facit X4 a, vel  Iderque lalus infinite parvum potuoni.
X + XB
X X+ B
Polygoni infinitanguli Hyperbolici SXa  8, quaeritur summa omnium V.
ea enim data, datur et quedtiora dimensio Curvse Hyperbolicae fiet aX +a yX lacu
Cueae tut aquatio figurae quae curvae Hypervolicae homogenea est, unde fit.
a X + a , YX,  divivis omnibus per X habebinus a + 4 11 3, vel _ Sy- a  aa
a prio porro datur summa omnum .
Erg. X H  Ha
summa autem omnium x facit a S quae quantum
fujus figurae ut quaeratur. Quadratura nostiganda clangen, nempe a +=- a- a
gudico nondum datur. vi 1+ = AY = XV
4xa  A2 a. Reducta L) X P
aX 1a nfier
iuter quam et productam est media proportionalis y,
ita ul sit
a 11
Cx aX + X  LX
24
reductauel sumi alia methodo investigetur
redibit s.
a x Aliter quaeramus
quaescomus differentias ordinatarum Hyperbolae
ad Asymat. quaerauus et ad axem
e
Sa + X + a +X + 2βX + B, sie 2BX = Z
vel 2X, differentia applocula Hcoperbole ad ax tirc
a
X Mk
d 80216
2
agn addantur quadrata a, et 1aX
fiest  a fier fiet 2a x + a  +2 BX +  _ - a X a X +2 + HZ
jam
274
etfa +gaX + 8aβX + A N + B PX + aBX  247- 2xZ 2 2 βx1 Z p arXEa  + 2BX
etxaX2ex
fiet 3a + bax + 3x - 2aZ - 2XZ 1 0. lnde a +X. 5 2 Heu ax 
Sae+X.
vel six rI   a 4 XX
a + deLX
quae est Hyperbolacusa
a  F  A digtu ba
expertole
sa p2 sive ZS a X () a + X hac autem figma est Hypritat  bria hade
sI
a et datur ejus quadratura.
atx ax
aAxeax
a =X
Nutabile
 a Y a ex Cat  s, a. X g X atK aFX
EXAC- a ta
ae+rea- a
a -X(7 a C - a],2a- C
ponendo e +2a V. binomium tolletur ex divisore, faciliorque
fier reductio.
a+x
a + 2 X + X
a - X,
a+XN C, fiet e Ca. Daet e
XN C- a
a
N
a-
eum un sh grebtit siReca
13 quem admod. i Locis Linearibus investigandis quaeritur semper y ita in locis
ad superficiem investigandis sufficit inveniri y. Loca ad superficiem, in quibus
duae tantum sunt incognitae sunt. Ungulae tantum cylindrici vel conici, planatum locorim
Ao Cum tres sunt intignitae aliquando fieri et ut una tolli, possit
tunc locus reducetur ad linearem. Investrgandum quodis naturae sunt loca ad superficiem
linearium.
simplicissimia et an memorabiles quasdam hubcant proprtetates.
Loca linearia sa superficiatibus in eo im agnopere differut quod loca ordinatae in in area linearium
componitur ex lineis, omnes autem lineae aeque latae, hunc longitudinis tantum iieunda computatio, at
i prea locorum ad superficiem invenienda. Etiam latitudo oidinatarum, id est plandrum consi
deranda est. Quod ut fiat lineae invesligandae, quae sut ipsis solido homogeneae, sed eae lineae
non eruut geometricae, si plana illa non suit lngula quadrabilia.
a
ry - X a
cup y2 p at  a 2
atle
X+ 2βX + (
X +2B∞+P
X
7 +(N bBX 45  
int r
dinala a
aa Aack
e7rea
C 4a +a ete
C ta
e
fiet Z13