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# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 1
# Folio 81r, page id 00068194:0161
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De Variatione constructio um per conicam
AB sit X, et BC sit y,  latus semilatus rectum a, semilatus transversum q
aequatio generalis 2ax f 4 Xx = y1, vel
UIt. calculemus tutius, sumamus casum, quo problema succedit numerice
quo ordinata et abscissa, succedunt numerice, nempe sit y = nx fiet
2 a f a: q X = hhX. seu X = 2a: hh + _ a: g. sit hh + a: g = 1. seu + a = L hh - 1
sit h = 2 fiet hh - 1 = 3 ergo + a = 3 sit a = 6 et 9 = 2 et + = + fiett a: 9 = 3 et X = 12 et y - 24
et fier a bis 6. 12 + 6: 2 12. 12 = 2. 4. 24 seu qu
Itaque sit linea conica A,  cujus verlex Aaxis ABB et abscissa AB sit 12
ordinata BC sit a 2 4 latus rectum semilatus rectur FO (seu odinata ad focum D F sit 6
et semilatus transversum seu distantia verticis a centro AC, sit 3, fiet aequatio
itaque sit linea Conica AL cujus axis ABB et A distantia verticis Aientro
et distantia verticis A, a centro C, seu semilatus transversum fit 3 AC sit 2, et distn
et oedic perpendicularis axi et semiordinata ad et ordinata ad focum F seu semi atque
rectum FO sit 6; et abscissa AB sit 12,  ordinata BL, 24. et
equatio ita stabit: bic 6. 12 +6: 2 12 =24 quae aequatio in Hyperbola
sposita + esset +) succedit in his ipsis numeris, nobis autem, nunc erit generalis.
pro 12 ponamus 9+3, et pro 24 sumamus  20 + 4 fiet 9= 12-3 et 20-244
fiet   bis 6. 9 = +bis. 6. 3 + bis 6. 9 +6, 2 9 = bis 4. 20 + 20
46. 2 32 4bis 6. 23.9
si jam velimus tollere terminum purum sufficit poni bis fiet relictis caeteris
siet 4 = +6 f b56. 3 bis 6. 3 4 6: 2 3 - 4 = 0 et aliqua est arbitraria
nempe 3 vel 2. si ambos terminos  simplicium incognitarum
tollere velimus, fiet 4 = 0 et 3 = e _ + Z si terminum
simplicis abscissae tollere velimus, fit 3 = +2 si simplicis ordinatae
fit 4 =0
ordinalis rectilineis
Specialis est aequatio ad Hyperbola Asymptoton (sit, AZ 4, cujue
potentia constans 16a sit AB, L et BL 8 fit 2. 8= 4. 4 ponamus
jam 2 - 5 3 et 5 = ere 14 - 6. fiet 8544 5b. 55. 14 - 6. 5- 3. 1
Atque hi sunt omnes modi quibus conica sectio
Sumatur jam alia recta quaevis, angulo ad arem
Quocunque EMM axem secans in M et in ea sumatur punctum F
ita ut 3 F M suit abscissae, et LM ad eam normales ordiaat aexccuir
ut numeris porro insistamusque omnesque calculi errures evitemus
ponamus LB, BN, NL latera trianguli rectanguli esse ut 3, 4, 5, seu t quia
VB et quia BL est 24 eit BN 32, et LN, 40, jam EN sib O fiet BF
et AE(= AB 12 + BE24) erit 34 et _ EM MN, NE, erunt 6, 8, 10 sit EF
fiet FB, 4, et (normalis ex FE eductus axi occurrens in P) et PE, s. FB 9. (Acrem) tin Cmmrssr
Ex his jam calculum instituamue resumta aequatione I. AB erat 1Z
BL erat 24 latus rectum 6, Transver sum 2, aequalio 1, erut 
aequatio 1 erat . bis 6. 12 +6. 2 127 = 242; jam FP EF3. FB, 4, AP. 29 et EP (5) = LFPE+ 
P
ut non su
FM, Quumeros parentresibus includo, ququi non quo
E S+ FM 9= FEden) etq. BL24: B Ne B ACG + CN10 C AB  seu AC5q in
ita enim facilivi fil
jam B N (2) = AE(+) + AE, 34 + CN (10) AB C12. et E X(10) = EM, (6). EF(). FPq da
 SN eo) = C NM (8) FM = FM9 - FC3 + CM (5) ee qusilu
AB  AB 12 = AP 29 +. EP = BE122) BE(22) = BN (32) - NE (10)
BN(32) =N L(40) Fp4: Ep() (evitando hic BL, ut in ipsius AB valore implicetur
alioqui potuissemus faere BN: BL:: Fp 4: EF3) NL (40) = _LM ( LM 32 +NM (8)
et NM  (8) = _ EM (6). F  4 EFFp 4, : EF3 ac denique EM (6) = FM 9 - EF 3
ita lan habemus BN () = A NM (8) = FMg - EF3. FP 4: EF3
et
NL = NL (40) = LM 32 + FM9 - CF3EP+: EF3 et BN = LM 3E. FP4: Ep5
et BN = LM 3 FB + FMg. EPA LM B92. FP. 4. EP: 5 + FMg - EFZ FP 4: EF3. EP5
ult Cat
deadseu M (6). EPG:
ret
denique
n  A0 in = AC ig CEd elq nte aM 3. Fpa. Eei A ay (Y4. 3F3 185 + + P X. Cp. 9 Cn 8. 1si 45()
(10
pL
yF1-3
id
it
36+27  = te cex
47+108 + 5. 81 = 560
400 4 160
29 1b + 3 16 16, 10
9 + 48 + 15- 128+ 16
rgriGxIsa (b1 2) BrI
tm
MaveaM, et MZ habentum
vares, in quam
utraque AB, et B)
ided  habeatur alteratus
sola ex FM, et ML
calitera prius est
soll
ut mii aliquantoalud