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# Title: LBr. 437 · Huygens
# Folio 41r, page id DE-611-HS-3595205:0081
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2XXy - a ax: ha - 2XY cl: y:: dx: ay
2XXy - a aX dy = 3a a - 2xy y dx. Seu 2Xxyy = aa Xy dy = 3a a = 2Xis yy dx
L et propus scribendolr fit X + ydx = ldv seu ydx = bd v Liv: y dy
Ergo denique Lovaq - ap bv ly = 3aq - va ybav - vb dus
faciamus 2aa acb - 3aab + 2 a b dv: dy = 2Wv a  = afbv: 3a f - 2V + vb
duo mihi problemata proposuit Hugenius, quae redeunt ad Methodum
Tangentium inversam, in literis datis 24 Aug. 1690.
quaeritur scilicet curva in qua  abscissa AX AB existente X, es
ordinata CBv et faugente CD an eju Axi Asdoccurrente in D
ipsa sBd (inter vallum ordinatae et langentis in axo) sit  gy. 2X = 2X
ve vel quaeritur curva in qua iisdem posidis Dd sit rary- aur: dua 
Quoad primum problema, in quo Bd = jus: 2x = 2X se  fit
Labad ex mei calculi legibus: dx: dy: : y: 2X - 2X : y
faciamus 58 y: x = v: a
fiet du = Xde + vdx: a.  et dy: dx = xdvrcdx + v. a 
Et quia ex, L
ex it est x: dy B= v: 2a = 2a: v fiet  _2av: Wv -4aa = (dy idx D feuper 3) = xdv a 
seu (2aaV) - v - Daav : pv - 4 44 do = X: pax vel - (Lx : x = vy - 4aa dv: v + 2aa
seu log X = (vav: vrt raa - 4 aa flv: v - 2aav quoramus ergo
has quadraturas  Et primum v quaeramus S2 Radv: 2aa + vv
vyzet) est momentum curvae meae quadratricis cujus aequatio Z = 2a. taa + v, seu
vo u y = (3. bb + vV. ejus momentum ex EK est (vb2 bb +VV. nempe momentum figurae
HCFGH. sed idem momentum aeqo (VvaZ2 + BZ. 2 et suq = b (aZ. Z - bbZ
Ergo fitavdo. v0+b  Ergossosito b = a st) fit B a rE svar: ve + caa = a 2 sch. 2, Z
Jam veniamus ad (dv: v- bbv estea quae  quantitas constat, duabus partibus
nempe (do. V et C  sed haec ve est loga de V= bbV sed haec summa etiam
 reduci potest ad logarithmos
vib.
vb
fiet p= n et m = - 2B Ergo R -.
jam quaeritur summa de = 2 bb _ v2 - bbp. Ergo - mbb = - 2bb. fit M = 2 et n = - 1 = p
3= bbv
M + P, +  b
v+b
v
ergo bl (dv: p -bbo = 2(dr: v - sav: v+b = Sdv. v- b seu = duplo log. V - log. B + b. - log. v - b
seu def. g. = 2log. V = log.  diff.  vvo bb. Ergo andem fit logex
log. X = log. Z = Z: a s2 + veZ: 4a 2 - 2log. V + log. vv - bb quam aquationem
trantformando in Transcendentem potentialen fiet:
seu log XVZ: vv- bb  =  voz- 4442,: 4a s2. seu ssiansformando aequationem
loganituicam in potentialem, fiet: 2XVZ: vv- bbd  ) vZ. 4a a2: 4 a 52). posito log a = 1
jam XV = ya et Z=  a 2 (2: 2a a + vD fiet vVZ - 44aZ: 4 a s2 = vv- +a4: 2 2ac + vY
et Z: vr - 2aa = 2 a F2 : v - 4a.
Et siet tandem.
 4a y s2: v - 4 a = _) ivv- 2aa : vB + 2 aa) sumb C quocunque
Log Hyperbolitus ipsius C
bg a . log. n tat  fI AXI ax. dy - Gy. 2X + 2X: y
dx sit gB  a + bx axidus = + yy +4XX,: 2Xy
seu 2Xy dx = +yy dy 4 4Xx du da X Y + by X
log ab= c
L ab = v
Exso tog. ab = c log. m
sid a exy (+ df x9) = rexy ex + ex dy +25 8 xy dy + fy de 99id 7 x- 4 ut aquaretur
A...
= -2xy dx 4 +X dy
ponamus h2 et + = + et + = - et
tiuti
T hyydus
summadem
2e = 2h. Ergo C = h, restat ergo +3hX dus quod impoddit
sit y = a+ 10 + LX + 12 x + 13, X + 19P + 15N + y = 10 + 2. 1B.11X F2. 10. 2
11. 2. 13 X 13, 19X 4 f. 14X1515X et
F12. 12
yX+ B sit ( sit X-10+1g19+9y et
Teguy ectu ec = 20+ 2 + 22 y + 23 et fil, 10. 20 X102 y + 10. 22. 7 + 10 37 11024) 1
dy=
eim 24= 1: 10 et 21 = - 11. . 15 et 22 = 11. 11 : 10 - 12: 10 fit . = +4 2 + 11. 2 +11, 2F +11. 27 er
Cxax:d = y - 2Xx :  sit 1X = 10 +_ ∞ + 12y + 13 y) ea y e,
 +1L. 22 et
F13. 2C
y: 2X =2x: y
X = 10 ∞210. 11]4 2, 10. 12 y = 2. 10. 13) + 2. 10. 19 g et Xdx = BCX n d X = 10. 11+ E. 10. 14+, 1012 y 4. 11 4Y eri
1
1t f.t 10 114 2 10. 12 96+ 3. 10. 1315+
 aseudinius) reiisi haua curys A gy lg. 10-0 et 1- 10 allii 0
p
12. IL
14 y3
S
lit
3
se esles yy 4a a =0 fieret
alone wey
1X.  2a
fil  a fieret y infinit
. sit yv-aaa: er +2a
constanti seg tant itim