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# Title: Leibniz-Handschriften zur Mathematik LH 35, 2, 1
# Folio 235r, page id 00067993:0471
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2. D  Extractio Radiies ex
quadcat
6 quadquad quod ex Za.
pqualquad qusu
noinde Extractio reotu
EH = BF 1 - 50._
LM = 2 - Se -  Y+
sequer
progressio non estaco
n
i e sfintum
Item ope Logarithmoeum inveniri
potest, atque cujulibet polugon subsectionibus arcus cujusdam
dati; sine ullo calculo
Ac proinde appropinquatio quan-
tupnoe excuta. Tantum enim ex prima
is quolmetrica du
st
Di, etra–
se porens terminiprogres
sola oefiore
ili
vavis alia
trica omnium Radicum
nug
cuti
 ad
itur eadem radixfit
lita sit, et quadiulo quadratuta
jE = 1B = a = 1. AD- b BD - N. AD = Pa X =x. AB - S2aX HB = ax  IH = a - ax. EH = a - sa  av
Et EH = a+ a x -  A rata  EG EB = Ca = 2a sae - ax
ABC = ABC = BD AD = TraX - X seu X Srax- 
1 - si-
AD spax
AEB = EBF =  EH HB = ax  2 a + a - ax - 2a sa ax, an a X - eX - Xa a
ABC. XHS~ -X = ABC
AEB = +X, Si = X - 5 1 - Z Ergo ratio ahriusque
e
s
axY
-4
Si X sit to fiet
- aX y + +Xy
12- te
11. - 64 X + X 5,
2662 2)feset eu (, poncat B
et
b a Xa b na b-X a b + n
usa )n
ta +X3a
+y
intra quos terminos cudat vabi
X N2-X
S1 - X - 1 X
4X45 etc.
1673
ABC = B vel posit EH - y. erit = X  2XY
la
AEB
A potus 1e
F-X
aVy.
X S2 - X = CS1 - X - s1 - Z. Ergo 2X - X  = B = x - β,1 - Z
Egx β - βX - 2X + x = B51 - = vel.
v
6- C5 - 4RX + 2X3 + BX + Cx - Y + ax - 4X X -
(3.- . β aiui 
Ut (3 inveniatur velut numerus, eopitandumet
X esse numerum
∞ SXC SS+
14 X + X
+2a
-a +xC
= β - βX
+Xa = O. Inveniatur per dioristrcen
+X 4 42 4
intra quos terminos cadat
valor ipsius B. inprimis si
Ponatur maximus vator possibilis ipsius X quo
utemu qui est vator lipsius BD, posito radio = 1.
Juocntus Bdplaetur, e smmma valor ipsius (3 magor in extrer po limiles ipsius
3
 oint ratio ne mnajor minorve vera
nianguli
ABC ad AEB + BFC
et brminorum progipsionis Germetrice d ratome cen tentonum prossim
tium in infinilum quaeratur summa
circuli intra hosduos terminos cadet
quam exporoptes numen
e
major minorve vera. Area potr tonas
mdugit
Cii hoc modo daretur extractio quarum libet radicum, dalettur inventro quotcunus mediarum proportionatium ba
per sectine anguloruti mnersalem Et huius sips o angulorum sectioni opte, etiam extractio radicum affecttarum
omnium, et construitio geometrisa omnium problematum huginpudit. quod duplicem habetusum. tum ut per
numeros inventamo statim latus quocurgcunque polysoni collinua arcuum dati subdiorsione genidi, ac proinde
appropinquationem quantumlibot exactam Jam vicissim, ut  Eusmodi extacthones si opus sit ope
angulorum copendro geometriceefficiamus
3’exponen radicis
Dati M statia dater AM, nempe SLLM = LM e Aea ALE+ ENB = LM 52Lm - LM = S2 LM- 1pi
LM sLM Sa - Zm seu L  1 - ()) + -=
1 1(V) 1 X C) LM
1+ 1() 12 2 -. 2 d) 1 = 2,
1 - V)- =
p - b
Se 1D -2 -
um
a +b = a- b
S. - )1 -X = AIE , CNC
AVI  ) 1 - Z + 2 S ) cab. de 1 - Z
1 = 2 1 (Y) 1- 2 + 2, , ) lnb e 1 -X - s(V), - 2 = ALE + ENB.
sunt nuneri progressioris Gomretricae dupla continne crescentes. Unde patet terminos ipsos non esse
progressionis Geometrice, sed ipsolum Eaponentest. Uti fit quoque in radicibus pr humeros rationales extractis.
ga
ad
Hinc E a
Sectione haberi potest
ad Hugen. d loc. Magui.
fig. 1. Hugen. de
CucMagna
aucta C
st2 sba