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# Catalogue record: 51849 — [Math. Aufzeichnung] Ad literas [33940+38279] duo quidam Grandio scriptas Martij 1714
# Incipit: Si ex puncto lucido A emittantur radii AF
# Date: 1714-03
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=51849&katnr=54727
# Work: https://leibnizlegible.com/work/DE-611-HS-973069
# Title: LBr. 324 · Grandi
# Folios: Bl. 18 (from the shelfmark LBr 324 Bl. 18): 2 pages, canvases 35–36, page ids DE-611-HS-973069:0035 to DE-611-HS-973069:0036
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=DE-611-HS-973069&page=35
# Lines: 191 recognised on 2 of 2 pages, mean confidence 0.78
# Catalogue: Leibniz-Katalog / Arbeitskatalog der Leibniz-Edition (https://leibniz-katalog.bbaw.de/), ed. Arbeitsstelle Potsdam I der Berlin-Brandenburgischen Akademie der Wissenschaften, CC BY 4.0.
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# After the first blank line, page by page: a "## Folio <label> — <page id>" line ("## Canvas <n>" where no folio label is recorded), "# " lines naming its source image and model, then its recognised lines in reading order, one per line.

## Folio 18r — DE-611-HS-973069:0035
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-973069/jpgs/default/00000035.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 82, 2026-09-09T18:17:48Z — 105 lines, status machine
A. literas duo quidum grandio scripras Martii 1714
lucido
Si ex puncto A emittantur radii AF, occurrentes curva AFF, tali angulo, ut
Sinus angulorum sint _ reciproce ut vires lam illuminationes  vires radiorum
erit illuminatio ubique aqualis.
quiaa Hinc posito vires esse in
ratione reciproca radiorum recipioca duplicata; erunt sims in ratione radiorum
directa duplicata, seu HL ad El, ut AFqu. ad A f qu.
Fl ut sinus, FL, fl, ut sinus, si FH, th sint aequales.
sunt autem
Aequivalent
autem sinus h ipsae  ipsis FF elementis curvae. Hinc perinde est ac si
coincidant puncta F et h, laideoque dlementa curvae HF, FF aequalia assumantur
AF, ramus ex puncto A, sit Y, Et  l+ erit dr  erit F5 elementum arcus
quae veceta de
constans  sit deo et FL sims eritq (dco deo tdi)  Centro A, radis aliquo constante AB,
et sinus
vocetm de
vocetur du
qui vocetur a, describatur circumferent a circuli secans AF, AF, in V, v, et Vo it 
vocetur do Eritque do: deo = a: Y seu ded = rdv: a, et vet FL erit
Vter dodo = aa drar): a set de dast ipsi sinus de: deo = rr: aa ex
hyp. et du: de = a: Y seu de = rdv: a = dcorr:aa, seu
do deo = a  a dv: r, et deo dco (= aa dudv: rr (= drdr + dede) =
drdr Frr dodo: aa seu at do dv = adtrdrdr + r dodv.
fief
aa de = ardr: Y (a - r) sit YE = at et fiet dv = _Z dt a : Y (aa - Et). Quod
est summabile et fit-acat. aaC= 1.
cum t Ergo debet esse b = - a seu frit Caa at) =a =  itaque habetur
constructio curvae, si angulis CAF e (posita CA tangente curvae in A) positis
ut L(aa-at) = a, radu AP radii AF sint ut dat.
AF do = AF de (N rde = seu ra rrdv: a seu Y3dr: ( - r +)
sit re = aP siet L AF = _ c dB: dlaa - a ) = AHF, ns smmeli ta  adar
fiet Lrde
AHF irde duuadq. a a d adq la aa; Haa- an)  a: ea- aq- s
a = a 5- 498 seu 9 =  a5: a 8e- a,: 88 qu- ads - a4g. 99
Squa = a + a. 5) b, Ergo cum et 0, sit rest q est o, sit q= a, unde
a aa + aa - b = 0 fit b = 2a. Et S9des = ac +  aa (a: ) 2 a
si fuisset d2: deo = aa: Yr seu dE = deo aa: rr =  vdo: a. fiet
do = deo a: r, seu ded = (r3: a) de et deo dco = (r: a) dodo,
et fier cumque sat deodco = dede + d + de sint (N: a avdo  = d rdr L (rra: aa) v d
seu dudo = a ardr:, V - at rY seu dv = d0 = dr A: Yt (- a). seu
seu V = - seu dv = a rdr: rr 1(14- a) seu sit rr = at, et fiet
A =  Za dt: at LCaa H - a) seu dv = aa dt: 2t (H - aa)
Ergo deo
4)
ipsius Y’dr:  (aa + Y)  momentum ex axa est verticis tangente est p dr. Vaa + rY) = rE dr
supra deo = a a dr: -
di
sed idem momentum habetur per xedi
b. aa +. rr = ∞ Fap seu _ r = aPE + arrE
vel e i
et pro rr ponendo ab fiet +3 = _ 21 + EE quaertur EdZ. siat Y- tMra, et fret
aa V= tt (n 4 (3 . et fict t = a, a M(t:, aa-  e et p = a , ad- mpn 5+ + dx
it + seu fiet 5dt = t at  t. sit T., a +t = nei  de aat = a e . + tnn fit t= apmm: aa- p
et df = (aa -  2) 2m den + M n 2d , :, qu (aa = m n) = aa en de: qu (aa - n pr) Ebgo fer
EF a (A (M(M: a) Ea n L (M D:; cub (aa - (n(M) un pro  ponendo ad fiet
EA  adada
s
sic. sit  a t(t:, a + t) = H d
ipsi tt dd. Est autem at. F a dF + t55 fit + = add: aa - dd, Et
videtur res brevius investigari posse
seex
Ld td at sae quatur
 aa. d
d
H
et (, (a - d) =
= aa 8+dd, qusaa = 55
A(a+) = 5+: (a +8) qu (a - 8)
igitur summari debet P4(d: qu( aa - d5)
A(a- 8)
A: (a+5) gu (a-5) + 3t: (a - d) qu ( at S)
: Caa - dd
5: (a - 6) qu (a +8) ergo a+-
Rusus=C + D. mesus C (a +d) d: qu (a- 3)
+:  (aa-d5) = 8 +4
fiat g-5 = Ei
et fiot 2aC = 5: qu (a-S)
at - 4 a H +  aa e N - A a H3 + R: HN = 4 a - 4 a + 6aa - DaTt
2 N
  seu 8 = a - _ et firt 6 =
si a+ d = 5, + 8 = 8- a et 52 = 5 - 4a8 + 5aasp - 4 aC. + a et G(at d) = 5 - 4455 + saac 4a + (a: )
V X Ergo (2 aG = at (I: H) - 4a + 6aa H = 4a n n + X + C3 - 4a55+ 6na 8 - 4a + a+ (1 :5). jam ta O = 2 (1 2a
2IFo ro Ea 0= 28 8 +1a M-gaaK Fra r  ) + a-4a + baar qanntu te tassfare
4a+ a
am modo permatatur habetur +aD, per maliehatim s a
Est autem V arcus BV
et a( dt: + (aa - 113
et etiam arcus cujus
lana sinus ver
sinus complementi
est +. Ergo habetur
t etusque q adeor 
in
et adet
Iqe E
angulusque cum
hmpg. AHF- Zrde
eAt t t
FE.
E
L. 
5)
F AE.go 2a

## Folio 18v — DE-611-HS-973069:0036
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-973069/jpgs/default/00000036.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 82, 2026-09-09T18:17:48Z — 86 lines, status machine
quod pendet ex rr d
C0 = aa dr :  (a - , 4) -rads Assum ati v = a: Y I Ca - r) quod pendet ex rr do
VV =  a:, rY a - r seu X N ∞ + rY ~V = a sit ro = a)
fint  aa + a = a, fit ) - a: s(a r) at de =aYd  - avar,: 1Y
Ergus YY d∞ ard)  arsto nihit hinc. illu depemus Nostrium est exaegr.
invenore TY ex N fiet vel at ex a N fint -at v + at va a = a, seu
3 aat = -a: N~, sit d: V = HD FB, fiet It - aa = a: (, t BB) = + (PA
aa t + at BB = qa seu aa Ht + at BB + Lβt = a + 4 . seu at + L BB - (  )
Lde V = a: +B = a( H(a r aB  a:  -  β + β ) (a + 1 34)
et t = (a + 3) = LBB,: a unde haberi potest Ede, ingredrente  (a  =3)
ubi notandum, ab.  (a - r2) transiri ad (a Y Ba
Complacto res Generaliter resolvamus 1: Im 4 sit l = Y 10k + na et m = 20X + 21a
fiet L sit 1: lm = O fiet, Ol = 1: m Om = 1: l fiet 20( 10MO = (11. 20 - 10. 21) a C
seu 1. tm = a. 20l se (20 + 10) : (t 20 1 21) = 1.. Et - 20 + 10, :, 11.20+ 10. 21 = 
PErgo =
m
e
mn
F10. 2 = H. 2
nn In
bn LI V. 305
20, f. 20. 31 7 (110. 21)) +30p +0. 2) + si20. 317) (+10. 214
- 1. 305- n. 208  mlL v1. 305]- 11205
t
36.(-
1620
R
L 10. 31 - 11. 30 10. 21- 11. 20
20. 3] -2. 30 n 10. 2- H1. 20
10. 31 - 11. 30, 20.31-21
0
10. 10
10.21 - 11. 20; 10.
Imnp 10 - 11. 20, 10. 31 - 11. 30, 10H - 1. 40,L
1
-2 - 11. 30)
21. 36)). - 11.20)
230 10.
1. 31- 11.30 10. 2 - 11. 20
2
N 1. 31 -11. 30 104-420 R V3. 430
10. 31 - 11. 30, 20. 31 = 21.30, 10-21 -H. 20
8.30
n
30. 30
2. 2L∞, 2031- 21. 30, 0 M- 240, 1
30
hot- g 1o. Xo rt 3. 2 t. M3 FuM)
40. 11 - H. v, 1. 4- H 20, 1o. p - H. 3, P
+ 
10 tax So ax +104) = 50 lbx + sra fiet S+ 60 CX. + 51 a, et firi 10 ax + uaa = 5obx +5I
606x
50rbx +x + 5Ia + Zta 50lbX + s1a + 70sGX + 71 a
olx + 16o ex +60aa)
30
20. 11 - 21. 10, 20. 31-21. 30, M 30. 11 - 31. 10, 30. 21 - 31. 20)
20. 2
10X sOx+ 1ta4) _ Sos t sta, sit se sonX L. 2. sa sraX  staam 1oa X + uaa
Io ax + 1 aa)=
L
20 A, 30 errgo (10. 20. 31. f 10. 2. 30 (F 10. 20. 31 11. 20. 30
10.50 1
240
Ergo tandem  -
lmn
Leg 1 = 
26
1: (a - r) = N(a - r): a - r. Resolvamus jam  1: a - Y = C O(atr) = +, atr a. 1d
Cd Ca - 7)  a Y. 1 : (a- Y) (a a 4 rY) = ut  Het 8+ Q = 2 aC. Bursus
m
sit n = 30X + 3la fiet  L =  + 20 - 30 . 20. 31 - 21. 30. Ergo tandem fiet
1
et eodem modo f = 10 - 38,i, 10. 31 - 11. 30, Ergo landem L = 10 (H. 3)
10. 31 - 11. 30
Et continuatas
M
Los r.  3.f10. 21
lnn
10
60+ cX + 610
+94. 50 bX + AaSISIaa +2.50. 5i aa bx +
aa 50 bx + aa. 51. staa + 2. 50. 51 aa Vbx +  qu (70 1 fX + 71a) (60 ex+ 61 a) = 2(650 xr suidlsrrna o lerhia
elace
60. 70 fx)fC
10 aX +11 aa, 6060 CX + 60. 61 a tex +61 aa
comparandi termini XX, X (x, X, ex, a