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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 3 (shelfmark LH 13, 3)
# Folio 227v, page id 00068200:0456
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Ut jam ex hac progressione rationem Aggregalorum ex applicatis, seu
assignata
II sequitur
si ex data quadam altitudine X, applicata sit y. erit ex altitudine 2 X, appli.
calam fore R4c8, RCy3. et ex altitudine 38 applicatam fore Rger7 DYe
Applicatae Parabolae ita procedunt Paraboloeidis ita procedunt:
Si ex assigna
ta quadam altitudine X. (eadem quae ant) applicata sit Z. ex altitudine 2X.
applicatam fore RgCB RCZa.. Ex altitudine 3X. fore R9C 27 _Rc 7a.
futurae sunt ergo applicatae Parabolae ad applicatas parabo loeidis, distantiis
a vertice proportionalibus ut RC
Esto figura ABC semiparabola Alticudo abscissa
a vertice AD = X. applicata DE = y.
duplicata abscissa ut fiat AF = ZX. Quadratum
applicatae DE =  y. duplicetur  producti
Radix quadrata R9 y erit = FG. applicatae quaesita.
applicatae ergo aequalium  axis portionum
crescent ut Radices quadratae numerorum naturalium
deinceps ab unitate; Rg. 1. Rg. 2. Rg 3. )Rg4
Esto Figura AB C Sarabolocis Heuratiana, abscissa
a vertice X AD =X. applicata DEE = Z.  Duplum
abscissae AF = 2X. cubus ipsius AD est X. Cubus
ipsius AF est 8X. quadratum applicatae D. EE est Z2.
Ergo Quadralum applicatae F- GG erit 87. Ergo.
applicata FGG erit R48Z. applicatae erga aequalium
axis portionum in larabolocide Heuratiana erunt ut
Radices quadratae numerorum cubicorum deinceps ab
unitate. Rq. 1. eg8 (R9 27
Restat figura, quae.
Nunc quemad modum reperta a Geometris ratio est, ope
Arithmeticae infinitorum, quadrandi parabolam, seu colligendi
harum applicatarum summam, ita tentandum est, an colligi
quoque summa paraboloeidis possit, quod ita tentabimus.
Applicatae parabolae ita crescunt, ut dictum est:
seu
Rg. 1. R92. R43. Rg. +.
10R91. 22892. 3.R43. 4084 4.
Applicatae parabolocidis ita crescunt:
RgI. 148. sq27. R964
seu A41(1. 292,2 A43, 3. R44,4
Ryt
12 16
3
a
Si utrobique multiplicentur per Radices Numerorum naturalium, fient Aexplicapplicatis parabolae
Ex applicatis paraboloeidis
4 et applicatae parabole
erunt
1
a
Agt. RG2 43 R44.
16 et applicatae pcrabolyi. 1
s
2
a
16
aei ago Ex et
9b
4. 16 36 64
A91. R92 A93. 194
X RqBL B R, 128
ers
y64 Aqu R, 2S6