# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00068196:0505
# Permalink to this reading: https://leibnizlegible.com/page/00068196:0505?run=74
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 255r, page id 00068196:0505
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 254v); this page is its right half.
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=505
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000505.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 86 lines, status machine
# Lines: 86 recognised, mean confidence 0.87
# Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German.
# Licence: CC BY 4.0, Leibniz Legible. Machine output: do not ingest it as verified text; keep this provenance with the text.
# Wording rule: say "the machine reads it as …", never "Leibniz wrote".
# After the first blank line: the recognised lines in reading order, one per line.

AC Ellipsit
AB X, BC, y
ab
 yy = rax - (2: b) _x. Ergo fiet
vies rabiuscilauli Esisit osa B P = a - (a.6) X.
dys: de-a  X: y. =
Ergo si X = 0 fiet BP = a
jam FA HO = AZ =  FA _e BZb FB- 2f
_ AB CB  FA = iG. B: seg FBZ es X gX et fiet
sRo cgNtr tgu = FC O - Lb + Lf = h firt hha gy=
fiet t, ttra itx 4Xx Agy  C)C. seu vel ft jam FC + OC = b
et siet ggag +bb + Lsf L bs 2bx +fx + XX + 2ax - (a: b) x x
+ + bb +  ff +  bf t bx - fx + 2ax (a. b) XX
FA = Rb -Bf ()A = Yb + Ef. si y sit hG, ta semi axis transversus
fiet x = h b et firt yy = 2 ab = a b = ab ergo KG = tab, seu semaxis
transuersis est med. prop. inter ax latera rectum et transversum. 
jam FG = AK = b Ergo ff + ab = bb. Ergo ss = bb = ab. et AF=
b - f = b - sbb - ab = AF et OA = b + Nbb - ab = b+f Hinc dato
circuli osculantes radio a, et dato joco _ F, seu distantia ejus AF, datur alter
quoque focus 3 seu datur AO sive datur Ellipsis, sive b praeter a. Hinc
patet eundem ciiculum cujus radius a, quadrare ad infinitas Ellipses. Est
oc + +TCF + LCDAet autem Egu on Ellipsis maxime accedens ipse circulus, quo A et C
coincidunt. sit a 3ped. sit  +pos f epedi b - f = E ped -  seu
2dCF+PCF = reA b) b - sbb -3b = 1. fiet bb + bb 2b - bb - 3 2 b + 4 = bb -  3b ergo b = 4
() =92 rect.
i esastosneu b et OA = 4 + seu AF = 4 = 2 et Sbb - ab = f16- 3. 4 =. 2
Bt0) LCP = PCF
2 s LocAC = 4 + 2. = 6
Ergb per2 sit res-, ) itaque dato puncto radiantes intra circa circulares sphaei speculi sphaerici
2
cud
 BCZ hCF+ ZFF= 2ra et AF =
Erguy sit rF- oet Concavi verticem p et centrumue in axeAk, dabitur focus recolligens 
A5bs  (OCT+TCF. 2. +pes = rect
sit enim radius a sit foci a  AF = 9 = b - sbb - ab. Ergo fiet
(66) = 2bg +g9. =. (b6) - a b. Ergo fit _ b = gg:, 29-a.. Ru Ergo
bC et bb gt- 29a + aa99,: 2g- a. seu g,8a-g :, 29- a = r bb- ab
Ergo g = gg - ga + gg, : 2g - a = g r et AO = bF 5bb- ab siet
ergo fiet AB = 89 + ga(=99) 4, 2g - a.
quod et sic
seu AO = 9 +25 =. (g -(g (trag  99) (dggC ag + gd99):, 29 - a
AO= _= (2gg( a9) + (29 a (-299), :, 29 - a = h
Itaque Dato speculo concavo sphaerico, cujus tentrum X radius vertex A,
radius  radius AK  si inter versicem et  poterit assumto inter
verticem At centrum in axe, puncto quocunque A, poteri tanquam foco radiante,
poterit inveniri focus recolligens , regula tali. Radius AK vocetur A A
Et AF distantia foci F intra verticem et centrum, vocetur f, distantia vero fort 0.
trans vertidem centrum, nempo recta AO vocetur h, prodibit aequatio talis,
29
m
a ag+ ah
l = m: m-n = l: n fit ln = mn = lm - ln, seu
Et in Ellipsiseilatus rectum est modium harmonicum inter distan
El n : l+n = m
tias focorum a vertice, quodidem est medium harmo diameter circuli
in vertice osculantis.
tantum quod pro b scribitur = B.
scilicet gu = pax generaliter yy = 2ax + (a: b) Xx BP = a + (a: b) X
sed tamen quoad focos investigandos methido
priore  non licet uti, quia G non cadit in Hyperbolam de qua agitur neque
 BP in vertice  fit a
 opposatam, sed in conjugatam, cum tamen in Ellipsi conjagata Ellipsis t incidat
praestat ergo foco) investigare
ex visectione anguli. Nempe, in Ellipso recta quae venit ab uno foco
in nossram, non minus quam opposita
generaliter paracentricae ad unum focum est reflexa paracentricae ad alterum
voeum  paracentricam voco, quae in datum ad punctum quae si opus producta in ipsum
incidit. possis etiam vo dicere incidentem. Et rem ita pronuntiare: incidentis smunum focum
reflexa est incidens in alterum. Incidens est vel convergens vel divergens.
in Ellipsi inte radius veniens
externus a foco uno reflectitur ad alium focum, et rudicas ten in partem convena concavam
ves potius incidentis portro est convergens vel divergens
incidens, cum veniph ab uno foco reflectitur ad alium focum; et in partem convexam incidens
cum tendat ad unum focum reflectitur ab aliu foco ut divergat ab alio foco. in Hyperbola
radius in partem concavam iicidens cum tend tendens ad unum focum
reflectitur ad alium focum
Internum; ves in partem convexam incidens tende vel veniens ab, ano foco interno reflectitur ab alio foco externo
Et in partem cqnvexa incidens vel gemiens a foco externo reflectitur ut vemens at interno
ens ad fbem interum reflectitur ad s externum
in Diyseibita equs angulus compeiosi prendinlum e
lementu
eu  in tus
Eum in Ellips angulus FCO
AC perpendi culari ad curvam. Hanc
2a9-
t