The Moon And The Differential

The Moon and the Diﬀerential
October 2009 – A Guest Column by Rob Bradley
Euler’s output was split fairly evenly between pure and applied mathemat-
ics, the latter including many topics that we would today classify as physics.
Most of his papers fall decisively into one category or the other, but it wasn’t
at all rare for one of his works of applied mathematics to include new tech-
niques or results in analysis. This frequently happened in the Paris Prize
competition, for example, where the questions were generally of a practical
nature. This month, we’ll look at an astronomical paper [E401] that proposes
numerical techniques for approximating a body’s velocity and acceleration.
Remarkably, one of the results in E401 was probably the ﬁrst step in the
development of the calculus of operations and seems to have inﬂuenced La-
grange’s foundational program for the calculus.
Euler read E401, “A New Method for Comparing Observations of the
Moon to the Theory,” to the Berlin Academy on February 6, 1766, just a few
months before his return to St. Petersburg. Because he quoted astronomical
data from the summer of 1765, it’s nearly certain that his results date that
year. Nevertheless, the paper appeared in the Berlin Academy’s volume for
1763, which was only published in 1770. Euler probably had it included in
this volume, despite its later composition date, because it’s a follow-up to
E398, “A New Method for Determining the Perturbations in the Motion of
Heavenly Bodies Caused by their Mutual Attraction,” which he read to the
Academy on July 8, 1762. Because of the long publication delay in the 1763
volume, Euler was able to arrange matters so that E398 was immediately fol-
lowed by three papers that build upon it results: E399, read on December 18,
1763, which applies the methods E398 to the moon, E400, read on December
4, 1765, which considers the general three body problem, and E401.
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Euler begins E401 by summarizing what he had done in E398. Supposing
“. . . ﬁrst that both the position of the body in question, as
well as its motion, i.e. its speed and direction, are exactly known
for a given epoch; and further, knowing at the same time the ac-
celerations produced by the forces exerted on the body, I showed
how we may assign the position and motion of the body from
this, not just an instant later, but for a considerable enough time
after the ﬁrst instant.”
Euler observes that it would be quite useful to apply this method to the
determination of the motion of the moon and the construction of lunar tables.
However, he says that he initially despaired of ever being able to measure
the moon’s velocity with suﬃcient accuracy, having “neither the fortitude
nor the patience to undertake a task of this kind; but. . . I found a method,
by using various observations of the moon, made on several consecutive days,
to ascertain for each one the true speed and direction of the moon.” That is,
he had ﬁgured out a numerical method for determining the moon’s velocity
(and for that matter its acceleration) from a sequence of observations of the
moon’s position. He describes this in the following proposition.
“Lemma. For the abscissas ζ = 0, ζ = 1, ζ = 2, ζ = 3, ζ = 4, etc.,
knowing their ordinates p, q, r, s, t, etc. on a curve, it is required to ﬁnd
the diﬀerential values both of the ﬁrst degree dp , dq , dr , ds , etc., as well
dζ
dζ
dζ
dζ
as the second degree ddp , ddq , ddr , dds , etc., taking the diﬀerential dζ to be
dζ2
dζ2
dζ2
dζ2
constant.”
Euler’s notation may seem a little strange to modern readers. The letters
p, q, r, etc., don’t represent diﬀerent functions, only diﬀerent values of a
given function, which we might call z = f (ζ). Therefore we would write
something more like
dz
dz
dz
,
,
,
. . .
dζ
dζ
dζ
ζ=0
ζ=1
ζ=2
where Euler has written dp , dq , dr , etc. It’s even clearer in Lagrange’s deriva-
dζ
dζ
dζ
tive notation, not yet invented in 1765: Euler is simply trying to ﬁnd f (0),
f (1), f (2), f (3), . . . as well as f (0), f (1), f (2), f (3), . . ..
Euler’s solution to the problem posed in this lemma begins with some
notation. He deﬁnes z as we have done, then he sets q − p = ∆p, r − 2q + p =
2

∆2p, s − 3r + 3q − p = ∆3p, etc. If we let zn = f (n), then these are the
forward diﬀerences ∆z0, ∆2z0, ∆3z0, . . .. “Given this,” says Euler, “we know
that we have:
ζ
ζ(ζ − 1)
ζ(ζ − 1)(ζ − 2)
z = p + ∆p.
+ ∆2p.
+ ∆3p.
etc., or
(1)
1
1.2
1.2.3
1
1
= p + ∆p ζ +
∆2p(ζζ − ζ) +
∆3p(ζ3 − 3ζ2 + 2ζ)
2
6
1
+
∆4p(ζ4 − 6ζ3 + 11ζ2 − 6ζ)
24
1
+
∆5p(ζ5 − 10ζ4 + 35ζ3 − 50ζ2 + 24ζ)
120
etc.”
Equation (1) is sometimes called Newton’s Forward Diﬀerence Formula.
Many of us have come across it in a numerical methods course, see e.g.
[Burden 2001, p. 127], where the name is usually applied to an interpolating
polynomial, rather than an inﬁnite series. If {xn} is a sequence with constant
diﬀerences h = ∆xi, then
n
t
pn(t) =
∆kf (x0)
(2)
k
k=0
is the unique polynomial of degree ≤ n with the property that pn(k) = f (xk)
for k = 0, 1, 2, . . . , n. When x ∈ [x0, xn] and x = x0 + th, then f (x) ≈ pn(t).
In Euler’s application, x0 = 0 and h = 1, so that t in equation (2) is just
his variable ζ. Furthermore, if f (x) is a well-behaved function, then under
certain conditions we will have pn(t) → f (x) as n → ∞, which more or less
justiﬁes Euler’s claim in equation (1).
Next, Euler diﬀerentiates equation (1) to get
dz
1
1
= ∆p +
∆2p (2ζ − 1) +
∆3p (3ζζ − 6ζ + 2)
dζ
2
6
1
+
∆4p (4ζ3 − 18ζ2 + 22ζ − 6)
24
1
+
∆5p (5ζ4 − 40ζ3 + 105ζ2 − 100ζ + 24)
120
etc.,
an expression that is sometimes called Markoﬀ’s Formula [MathWorld]. Euler
then diﬀerentiates a second time to ﬁnd a similar expression for ddz .
dζ2
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Finally, Euler substitutes ζ = 0, ζ = 1, ζ = 2, etc., to ﬁnd the ﬁrst and
second order diﬀerential quantities he set out in the statement of the lemma.
The ﬁrst of these is
dp
1
1
1
1
= ∆p −
∆2p +
∆3p −
∆4p +
∆5p − etc.
(3)
dζ
2
3
4
5
For good measure, he also derives formulas for the third and fourth deriva-
tives.
Now let’s skip ahead about half a century, to 1814. In that year, Fran¸cois-
Joseph Servois (1768-1847) published a paper called “Essay on a new method
of exposition for the diﬀerential calculus”[Servois 1814a] in Annales de math´
e-
matiques pures et appliqu´
es. Often called “Gergonne’s Annales” after its ed-
itor, this was the ﬁrst journal ever to be devoted entirely to mathematics.
Servois’ paper contained the following remarkable deﬁnition:
1
1
∆z −
∆2z +
∆3z − . . . = dz,
(4)
2
3
for an arbitrary function z. “This is the complete deﬁnition of a new function
of z,” says Servois, “polynomial or even inﬁnitinomial,1 in general, which I
call the diﬀerential.”
In this paper, Servois was grappling with the foundational problem of the
calculus. At the beginning of the 19th century, there were three competing
foundational schools on the European Continent: those who thought that
diﬀerentials were acceptable or at least could be made suitably rigorous,
those who wanted to base calculus on the limit – still an informal notion at
that time – and a third group who, following Lagrange (1736-1813), deﬁned
derivatives not via limits, but rather through the coeﬃcients of a function’s
power series expansion. Servois was a disciple of Lagrange and his paper was
full of formal series manipulations, including a derivation of the expansion
(1). Although he was reasonably sympathetic to the limit approach, as he
demonstrated in a philosophical essay that followed immediately in the pages
of Gergonne’s Annales [Servois 1814b], he wanted to banish the inﬁnitely
small from mathematics. However, he recognized that the use of diﬀerentials
was deeply ingrained in mathematical practice, so he sought to explain them
here through formal operations rather than through an appeal to some vague
notion of inﬁnitely small quantities.
1Servois coined this term (inﬁnitinˆome) here. Although this word never caught on, he
also introduced the mathematical terms “distributive” and “commutative” in this paper.
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Servois’ formula (4) gives dz in terms of a constant increment in the
independent variable, which he denoted by a Greek letter such as α.
If
we call it dζ instead and formally divide it through both sides of (4), we
get Euler’s formula (3).
It’s extremely unlikely that Servois gleaned his
deﬁnition of the diﬀerential directly from E401, although his publication
record make it quite clear that he was very familiar with Euler’s works.
Rather, the line from formula (3) to deﬁnition (4) passes through the works of
Lagrange, Arbogast (1759-1803) and Jacques Fran¸cais (1775-1833). Arbogast
and Fran¸cais established the calculus of operations, in which operators were
abstracted from the functions to which they were applied, so that a formula
like (3) could be re-written as
1
1
1
1
D = ∆ −
∆2 +
∆3 −
∆4 +
∆5 − . . .
2
3
4
5
where D is the derivative operator. Because the right hand side has the form
of the power series for the natural log, Fran¸cais wrote Euler’s formula as
D = ln(1 + ∆)
(5)
and Servois said that the diﬀerential is the logarithm of what he called the
“varied state,” i.e. the forward increment operator that maps z to z + ∆z.
Ivor Grattan-Guinness traces this evolution in [Grattan-Guinness 1990,
pp. 161-163, 211-219]. The next step after E401 was taken by Lagrange,
who succeeded Euler at the Berlin Academy. In [Lagrange 1774], he not only
generalized Euler’s formula (3) to the multivariable case, but he produced
its dual, by showing that
∆z = f (ζ + h) − f (ζ) = eh dz
dζ − 1
where h, the increment for the diﬀerence operator, was taken by Euler to be
1 in E401. Fran¸cais could derive the corresponding ∆ = eD − 1 by formally
solving relation (5) for ∆, but Lagrange derived his result from the Taylor
series expansion of the function z.
We should note carefully that none of these later theoretical consequences
of E401 were foreshadowed in any way by Euler himself. He must have
observed the elegance of the expression (3) of the derivative in terms of
diﬀerences and noticed the analogy with the logarithm series. However, in
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this paper it was just a means to a practical end: the construction of accurate
lunar tables and, by extension, progress on the Longitude Problem.
To illustrate the use of his newly discovered tool, Euler gathered astro-
nomical data from the lunar tables of J´
erˆ
ome LaLande (1732-1807) for Paris
on six consecutive days, July 31 through August 5 of 1765. His coordinate
system takes the center of the earth as the origin, with the plane of the eclip-
tic as the xy-plane and the positive x-axis pointing in the direction of the
vernal equinox. With units chosen so that the mean distance from the earth
to the sun is 100,000, the position (x, y, z) of the moon at noon, ζ days after
August 1, 1765, is approximately given by the quadratic formulas
x =
166.970 + 36.090ζ
−
5.104ζζ
y = −184.039 + 40.316ζ
+
5.618ζζ
z =
−9.7545 + 5.1982ζ + 0.3020ζζ
As another consequence of his methods, Euler calculates the ratio of the
sun’s mass to that of the earth to be 309,108, assuming solar parallax to be
9 of arc. This compares reasonably well with the currently accepted ﬁgure of
332,830 and would have been much closer had he used the currently accepted
value of 8.794 seconds of arc for solar parallax.
References
[E398] Euler, Leonhard, Nouvelle m´
ethode de d´
eterminer les d´
erangemens
dans le mouvement des corps c´
elestes, caus´
es par leur action mutuelle,
Memoires de l’academie des sciences de Berlin, 19 (1763) 1770,
pp. 141-179. Reprinted in Opera Omnia II.26. Available online at
EulerArchive.org
[E399] Euler, Leonhard, R´
eﬂexions sur les diverses manieres dont on peut
reprsenter le mouvement de la Lune, Memoires de l’academie des
sciences de Berlin, 19 (1763) 1770, pp. 180-193. Reprinted in Opera
Omnia II.24, pp. 75-87. Available online at EulerArchive.org
[E400] Euler, Leonhard, Considerations sur le probleme des trois corps,
Memoires de l’academie des sciences de Berlin, 19 (1763) 1770,
pp. 194-220. Reprinted in Opera Omnia II.26. Available online at
EulerArchive.org
6

[E401] Euler, Leonhard, Nouvelle mani`
ere de comparer les observations de la
Lune avec la th´
eorie, Memoires de l’academie des sciences de Berlin,
19 (1763) 1770, pp. 221-234. Reprinted in Opera Omnia II.24, pp. 90-
100. Available online at EulerArchive.org
[Burden 2001] Burden, R. L., Faires, J. D., Numerical Analysis, 7th ed.,
Brooks Cole, Paciﬁc Grove, 2001.
[Grattan-Guinness 1990] Grattan-Guinness, Ivor, Convolutions in French
Mathematics, 1800-1840, 3 vols., Birkh¨
auser, Basel, 1990.
[Lagrange 1774] Lagrange, Joseph-Louis, Sure une nouvelle esp`
ece de cal-
cul . . . , Nouveaux m´
emoires de l’acad´
emie des sciences de Berlin, 3
(1772) 1774, pp. 185-221. Reprinted in Oeuvres de Lagrange vol. 3,
pp. 439-479.
[Servois 1814a] Servois,
Fran¸cois-Joseph,
Essai sur un nouveau mode
d’exposition des principes du calcul diﬀ´
erentiel,
Annales de
math´
ematiques pures et appliqu´
es 5 (1814-1815), pp. 93-140.
[Servois 1814b] Servois, Fran¸cois-Joseph, R´
eﬂexions sur les divers syst`
emes
d’exposition des principes du calcul diﬀ´
erentiel, et, en particulier, sur
la doctrine des inﬁniment petits, Annales de math´
ematiques pures et
appliqu´
es 5 (1814-1815), pp. 141-170.
[MathWorld] Weisstein, E. W., Markoﬀ’s Formulas. From MathWorld – A
Wolfram Web Resource.
mathworld.wolfram.com/MarkoﬀsFormulas.html, retrieved 9/12/09.
Ed Sandifer (SandiferE@wcsu.edu) is Professor of Mathematics at Western
Connecticut State University in Danbury, CT. He is an avid runner, with
37 Boston Marathons on his shoes, and he is secretary of the Euler Society
(www.EulerSociety.org). His ﬁrst book, The Early Mathematics of Leonhard
Euler, was published by the MAA in December 2006, as part of the cele-
bration of Euler’s tercentennial in 2007. The MAA published a collection of
forty How Euler Did It columns in June 2007.
How Euler Did It is updated each month.
Copyright c 2009 Rob Bradley and Ed Sandifer
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