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# Catalogue record: 15586 — [Mat. Problem]
# Incipit: Ex puncto dato in plano perpendicularem ducere ad rectam datam extra p
# Date: 1695
# Akademie-Ausgabe: AA VII (assigned, not yet published)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=15586&katnr=39619
# Work: https://leibnizlegible.com/work/00068195
# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folios: Bl. 82 (from the shelfmark LH 35, 12, 2 Bl. 82): 2 pages, canvases 163–164, page ids 00068195:0163 to 00068195:0164
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068195&page=163
# Lines: 119 recognised on 2 of 2 pages, mean confidence 0.85
# 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 82r — 00068195:0163
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000163.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 52 lines, status machine
EX puncto dato in plano perpendicularem ducere ad rectam datam extra planum
sit planum BDF, ex cujus puncto F ducenda ad rectam datam AD pta pl planum perfodientem
in D, rect perpendicularis FL. quia data Ad positione, dabitur D. ergo et FD, et DB in plano verticaliter subjecta
ipsi AD, seu in quam incidunt omnes rectae ad planum perpendiculares ex punctis ipsius Ad demissae, unic norma
B) normatis ipsi D. jungatur
AP. Eritque ob angulos ABP, BPD rectos, angulus quoque APD rectus rg ex puncto l demiss
liter occurrat FB et ex B erigsatur
id planum normalis ipsi DA occurrens in A
Analysis linearis. quia Et Ponata data quaesita ponatur inventa, ergo habita FC
intelligatur lh. perp. ad D)
(ob ang. FLd rectum, et Fd datam) habebitur lD. Ergo ob triangs. FID, lhD similia
habebsitur et lh. Rursus qquia lh ad AP ut B ID ad DA, habebitur ad huc semet
unde habetur aequatio, et proinde FL squaesita
Calculus: A Fd AP aeq. AH + BP BPC DB. FB: FD.
Ergo AP aeq. AB. FD + DB. FB: FD. jam DL aeq. FD - FL
1h
et In aequ  LD. FI. FD aequ. SFD - FL, FL: FD. Rusus Ih aego. SAB. FD + B∞r. eB : FD, SfDF;
quos duos valores aequando, fiet: FL aeg.  TAB. FD + DB. Fp. AD. seu seu fier
FL aequ. F D AP FD. AP: AD. idque potuissemus praexidere et calculum contrahere
nam in tiangs. DAF utique AP in FD aeg. AD in FL. sed ut solis rectis in plano datis
rectas quas per alias possumus explicem
FL aequ. AB. DB+ AB. FB + DB. FB: Ad, sed si brevitatem expressionis quaerammus,
e
quaeratur recta quae sit aq. eB seciamus FD aequ F BB.  B  YFB: AB, et fiet
utamur
FL aeg. FB frrA DB: AD Aliter Fd aeq. DB. FB: q, erit q. aeqBP. et fier.
FL aeg. DB. FB (AB + BP2 seu FLaego FD. + AB + BP: AD quod etiam
jam constabat BP. Anam habebamus FL aeqo FD. AP: AD. Co
Constructio per B ducatur paralleta ipsi. fd, et in ea sumatur BQ aergo. AP. junga–
tur BZ. D recta ex _ per F et alia ex F per  produtatur utounque, et in recta
ep producta si opus, sumaturR
sit solidum coniforne AFF generatum recta Pait+A
est puncto immoso A circa planum FC finmum planum 
 plano sit GG (.
datur punctum B, et magnitudo rectae AB, parpeni
es
dicularis ad planum FBF secans
E. ad quod plasso
planu
similiter datur puncta
in plano GCG et recta AC perp. ad hoc planuis
recansplunum F BF in D. jam
datu ex natura curvae FF, rela–
tione inter BF, BF. Opoaeritur
relatio inter cG, EG. seue natura
curve GG.
CE
CE tG normales ad AC, et BD, BF pormates ad AB.
IL parallita CG, est BM parillela FB
datio FL, et AL per leu praecedens problema datur
et AC, ergo datur B CG. similiter quiadatur
Am et BM dabitur CE  entu

## Folio 82v — 00068195:0164
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000164.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 67 lines, status machine
sD
Pigranis AB duti (idiatio, et pum a, et   perp. ad ad, Bpad
atque in plono BDF ducenda FLinormalis
ix busa pucatur b porgilis ad FB inaguntur AF, AB. qua t
IF
go
B. Ergo)
aa dis
ponut
et ad ur.
Non videtur
simul pp..
Fraeque rd
ost eus, eneme
ornalis B)
ex puncto dato in plano
perpendicularem ducere
t adt rectatu extra planum
positione datam
datur Fd, datur AB
perpendicularis ad planu
cui paralleta In
fie
In: nd: dl:: AB. Bd: Ad
ncatur
perp. ad D
erg Aerdi dn: ur:: gd: db
eta prpo ad Bd si Y quperr ad FD. erit 
aequin In +rn
rarsus tr itf In: nd:: AB: Bd. ncl: nX: DB: Bq
Ergo ln: nN:: AB. DB: DD. B8. Ergo
lr:m:: 1 pe e  sae
Ergo br t. lr:nr:: SAB+Z BqB
at nr: Yd:: B9: 4D. Ergo Ir. rd:: AB : 2
seu lraego Yd (AB, Bqr: 2D s
Ex aego ra AB+B2: QD. et quia l ide
prop. inter Ererre. eut
a
ra F Fngo 
BAQ, AB+BB,: gDEm 
E r AB aqu. FB
Ergo poaequ. DGB (EDL AC+B2)eFAVoC
sed r. Dt: PL: FB
AB+ BF aequ A
Est est aulem  constans perpendi.
cularis ad planum BdF ergo
datur AB datur et BF, DF
ex natura curvae 
simililer est AC constans per
pendicularis ad planum CEG
quaeritur relatio inter CE
et CG. Dantur autem
rest DRtem
rectae AB, AC, AD, AE,
Laequ Ird. F
n BL aeg. FD. 2D
seu DLaego FD. 8D
Nota duto
datur ergo et AF
et ut perpendiutar
duximus
ep
Ad extraplanum.
a
s auu
(AD +2DQ
et 60, CE