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# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folio 100r, page id 00068195:0199
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Apollonias sectionem nationis quam oveat exposint 
peculiari libello, quem nuper restituit Hallerus. Nostra
problema huc redit
x
Ex punctis duobus datis, eductae sint rectae
hatidem posatio infinitae positione datae AB, LM
Et ex punctur aliudatum F educatur recta, quae ex
er
rectis inde a punctis datis abscindat partes AC,
LN, rationis inter se datae.
Hoc problema nostra hodiernae Analyseos more ita attenta
retur. jungantur puncta datap
ex puncto F agatur vcta FG(in ipsam AL, et C AL, CA
rmaltis
Cddo concurrant in 19 cui ipsa CN occurrat in G. Ex ipsis
F, C, N in AL, agantur normales FH, cd, xp
dantur AL, FH, et angulus CAD, et anus NLp
quaeratur GH
sit GH, x FH, H AL, l. ratio AD ad DC sit b: A, et ratio Z
et ratio AC ad LN
ad pN sit iH
sit GH, X i  hac ergo Assumta habetur recta FGseu CN:po
punctum C est in duabus rectis AC ABet eN Ob rectam AB erit CD ad
AD  ut sa ad b seu CD = HD - AH, CD h: b Aursus ob rectam CN, est Cd: GX dG
= f: X DG autem est GH + HD ergo fit CD= GH + Hd, f: X = Hd - AH, h: b.
Et fiet HB H GH. bf + HD fb = Hd h AHhX. Ergo HD = (B) + AH. H) X :, hX - b f
et Ad = HD = AH. Ergo Ad = bf,  AH. hX = AH. L x) + AH. bf, : hx - b f Ergo Ad = bf, X + AN; h
Eodem modo procedemus cum NP, ut cum cd. Nempe punctum N est in
duobus rectis  M, et CN. ob rectam LM est Np: PL = n: m, seu est
N p = HP = LH, h: m. Rirsus ob rectam (N, est NP: pG = f: X est autem PG = GH = HP
Ergo tit Np = GH - HP, f: x. Ergo sit Y sed rp  et n sed supra Np = Hp - LH h: m
Ergo fiet GH. mf =HPnf = HPhX = LH. hX seu HB = G mst LH. hx, chx+ nf
sed LP = Ht p LH. Ergo fit in fx(+ LH = LH h∞ = LH nf,: hx + mf = L P, seu
(P= af, X LH; tix um mfE X - LH: tx + inf = LP.
AD: AC = b: s(bb +hh).
jam Lp: L N =
Ergo fiet jam
AC: LN =, Et AC: LN = Y si Y dato rationi Ergo fiet
et
m: Amm+hh)
itaque AC = (Y (bb +hh) C: D) Of, XF AH, :, lx - bf
LN = C Semmelh) (: M) rf, X - LH, :, hx + nf
, aex, hxamf
=2 , UX, hx- bf
Et eodem modo
jam AC: LN = g. r rationi datae
et compendi  Abb +hh) =C et Noumthh) = n
4
et AH, a et LH, L
ng: CY = 5: t
aD,  a + X.:, hx - bf
seu fiet (a +X, X + m.  )5): (l - X, X - (b. h) 5) = s. +. Unde tandem
SEm. h) rat -st (b.h -ls X + as A(m :h) + (5 5 (b. h)
5+ +
Unde jam more Cartesiano facile deriva
5+ t
n(B X. : hx+ inf
Ergosiet
FXX +Am: h) GX +m .t A5 = C sia XX =
eeta
+5 XX  Sib: h) fX + 5(b. 4) (f
porrosed assumta constructiore Aptolloniana poterimus
a3aX
pitur constructio
ad hoc problema reducere ea omnia, ubi rectangulam duo sunt aequalia
quorum quaelibet latera sunt ut e  iX + K
si non ratio ipsem rectarum AC, LN data sit sed rectanentum videtur
eodem pene res redit, si heerc nostram Analus sin sequare, musatis fautum
mutandis. Daulo difficilior res videtur, q si summa aut differentia sit data
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