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1. Leonhard Euler and the Mechanics of Rigid Bodies
JE Marquina1
, ML Marquina1
, V Marquina1
and JJ Hernández-Gómez2,a
1 Facultad de Ciencias, Universidad Nacional Autónoma de México
Circuito Exterior, Ciudad Universitaria, Coyoacán, Ciudad de México, 04510, México.
2 Centro de Desarrollo Aeroespacial, Instituto Politécnico Nacional
Belisario Dominguez 22, Centro, Ciudad de México, 06610, México.
E-mail: ajjhernandezgo@ipn.mx
Abstract. In this work, we present the original ideas as well as the construction of the rigid bodies theory realised by
Leonhard Euler, between 1738 and 1775. The amount of treatises done by Euler in this subject is enormous, being the most
notorious Scientia Navalis [Euler 1749], Decouverte d’un noveau principe de mecanique [Euler 1752], Du mouvement de rotation
des corps solides autour d’un axe variable [Euler 1765a], Theoria motus corporum solidorum seu rigidorum [Euler 1765b] and Nova
methodus motu corporum rigidorum determinandi [Euler 1776], in which he developed the ideas of instantaneous rotation
axis, the so called Euler’s equations and angles, the components of what is now known as the inertia tensor, the principal
axes of inertia, and finally, the generalisation of the translation and rotation movement equations for any system.
PACS numbers: 01.65.+g, 45.20.Dd, 45.40.-f
Keywords: History of physics; classical mechanics; dynamics of rigid bodies.
Submitted to: Eur. J. Phys.

2. Euler and Rigid Bodies 2
EULER, THE MAN WHO “PUT MOST OF MECHANICS INTO ITS MODERN FORM”
[TRUESDELL 1968a, P. 106].
1. Introduction
Last year we commemorated the 250th anniversary of the publication of Theoria motus corporum solidorum seu
rigidorum (Theory of the motion of solid or rigid bodies) [Euler 1765b] †, published by Leonhard Euler in
1765.
When the reader hear this title, he tends to think that this text features all the very important contributions made
by Euler to the rigid bodies dynamics, but a careful study of this work reveals, as it is pointed out by Truesdell
[Truesdell 1968b, p. 603], that it is disperse and full of examples which deviate the reader’s attention from the
theory corpus. Actually, the relevant results for the treatment of rigid bodies, are found in previous manuscripts.
Euler somehow tried in this book, to make a compendium of all his previous results, but he clearly could not
accomplish this goal. In this work we present the lenghty path traveled by Euler to build his rigid bodies theory,
from his Scientia Navalis (Naval science) [Euler 1749] ‡ probably finished in 1738, until his Nova methodus motum
corporum rigidorum determinandi (A new method for generating the motion of a rigid body) [Euler 1776] §, written
in 1775. We ought to state that we have tried to maintain, as much as it is possible, the original notation which
Euler used at each step.
2. Discussion
It could be said that Euler began to develop the theory of rigid bodies, due to his great interest in the motion of
ships, so in 1735 or 1736 he began to write his second treatise on mechanicsk, Scientia Navalis [Euler 1736]. Is in
this work where Euler proposes that the movement of a ship could be described by a translation plus a rotation
about an axis that passes through the ship’s centre of gravity. This is achieved by thinking in a force opposite to the
resultant of all the forces acting on the ship’s centre of gravity, so this point would be at rest. It is possible then, to
think in the separation of the progressive movement (translational) from the rotational one about an axis passing
through the gravity centre, because the force he introduces shall not affect the rotation (i.e., it will not generate a
torque). This way, two independent principles are required to describe the motion of the ship (and of a rigid body
in general). The translation was already described by Newton in his second law, so an equation to describe the
rotation movement is now necessary.
One could intuitively think that the equation for rotation should be similar to the translation one, i.e. that if the
second law (of motion) is written as:
F = ma
then in the rotation equation should appear an equivalent to the mass and the angular acceleration. In the work
called Dissertation sur la meilleure construction du cabestan (Dissertation on the best construction of a winch) [Euler
1745] - in the same age of Scientia Navalis - Euler gives the explicit relation between the equation describing the
translation and the relative one to the rotation when he points out that: “Thus [the moment] of the acting forces
divided by the moment of the matter [the moment of inertia], gives the force of rotation [the angular acceleration],
exactly in the same way as in the rectilinar motion the [...] force divided by the [...] mass of the body gives the
† This work is in The Euler Archive under the code E-289. This code refers to the Eneström index, made by the mathematician Gustav Eneström
to classify the very wide work done by Euler. This index enumerates 866 different manuscripts of Euler.
‡ Scientia Navalis was much probably written in 1738, but it was not published until 1749. A severe problem to analyse the genesis of the
concepts in the oeuvre of Euler, is the fact that given its wideness, publishing policies, as well as Euler’s own attitude, there are manuscripts
written with anticipation with respect to others, that were published subsequently. This fact remarks the necessity to track and distinguish the
dates of writing, presentation and publication, so to be able to build a consistent picture of the development of the eulerian corpus.
§ This work was written and presented in 1765, and it was published until 1776.
k The first was Mechanica sive motus scientia analitice (Mechanics or analytical science of motion), in 1736, oeuvre in two volumes (E-15 y E-16),
treatise in which Euler formulated the newtonian mechanics using mathematical analysis. A detailed analysis of the mechanics of Euler, is
found in [Suisky 2008].

3. Euler and Rigid Bodies 3
acceleration. This remarkable analogy well deserves to be underlined” [Maltese 2000, p. 327]. This way and for the
first time, Euler sets that for the movement of rotation of a ship about a rotation axis passing thought its gravity
centre and perpendicular to the movement plane, that:
Z
r2
dm
dω = Mdt (1)
being M the moment of all the forces producing rotation (i.e. the torque),
R
r2
dm
= I the inertia moment and ω
the angular velocity, so that:
M = Fr = I
dω
dt
(2)
where F is the magnitude of the resultant of the forces producing the torque, and r is the lever arm.
The problem concerning ships was important for Euler, so he spent much time researching about it, but there was
another more fundamental one in that time: the precession of the equinoxes. Newton in his Principia [Newton
1999, p. 751-757] ¶ was the first one to give an explanation of this phenomenon. Afterwards and parallely to Euler,
D’Alembert approached the problem, improving the newtonian approach, but it was Euler who gave a turn by
approaching it in a general way, giving birth to the rigid body mechanics.
In 1750 Euler presents, in the Academy of Berlin, a manuscript entitled Decouverte d’un noveau principe de mecanique
(Discovery of a new principle in mechanics) [Euler 1752] +
, in which he poses the first version of the equations that
feature his name, the “Euler equations”. For their deduction, he begins with a “determination of the movement
in general of which a solid body is capable, while its center of gravity remains at rest” [Wilson 1987, p. 259]. To
achieve this, Euler introduces the idea of an instantaneous rotation axis. In fact, the concept was already set by
D’Alembert in his Recherches sur la precession de equinoxes (Researches on the precession of equinoxes) [D’Alambert
1749] ∗ for the case of Earth, but it is Euler who demonstrates the existence of such an axis in a general way. In Euler
words: “Assuming, then, the gravity centre of whichever solid body at rest ... I shall demonstrate in what follows
that independently of the movement of such body, it shall occur that not only the gravity centre shall remain at
rest, but that there would also exist an infinity of points situated along a straight line passing thought the gravity
centre which would likewise find itselves without movement. This is, that independently of the movement of the
body, it shall exist in each instant, a rotation movement about an axis passing thought the gravity centre” [Euler
1752, p. 188].
For the demonstration of the existence of the instantaneous rotation axis [Euler 1752, p. 197-204], Euler introduces
a cartesian axes system fixed in the absolute space, which origin is the gravity centre of a rigid body. Afterwards
he considers a point Z of the body that moves with a velocity (P, Q, R) with coordinates (x, y, z), and a point
Z0 ] with coordinates (x + dx, y + dy, z + dz), that moves with a velocity (P + dP, Q + dQ, R + dr) † separated by
a distance d, given by
d2
(Z, Z0) = dx2
+ dy2
+ dz2
(3)
After a time interval dt, the coordinates of the two points shall be
Z = (x + Pdt, y + Qdt, z + Rdt)
and
Z0 = (x + dx + (P + dP)dt, y + dy + (Q + dQ)dt, z + dz + (R + dR)dt)
¶ Newton tackled the problem of the precession of the equinoxes in Proposition XXXIX of Book III in the Principia, but he was not able to give
an answer with the rigour that he used to, because he overestimated by a factor of two, the influence of the Moon on tides, relative to the sun
and to the ellipticity of Earth. Nevertheless, in Lemmas I, II and III preceding Proposition XXXIX, he prefigures the concepts of inertia moment
and moment of moments, in the context of the analysis of a rigid body rotating about a fixed axis passing throught the mass centre.
+ Although he presented it in 1750, the publication of the Memories of the Academy of Berlin is in 1752
∗ In this manuscript, D’Alambert was the first one whom deduced correctly precession and nutation.
] Euler denotes the second point as z, but for the sake of clarity, we shall call it Z0
† Euler do not denotes the coordinates nor the velocity in vector form. Here we do so to ease the following of the argument.

4. Euler and Rigid Bodies 4
Due to the fact that it is a rigid body, the distance between the two points ought to be the same, so after the time
interval dt is elapsed, we have:
d2
(Z, Z0) = [x + Pdt − (x + dx + (P + dP)dt)]
2
+
h
y + Qdt − (y + dy + (Q + dQ))dt)
2
i
+
h
z + Rdt − (z + dz + (R + dR))dt)
2
i
=
h
(−dx − dPdt)
2
+ (−dy − dQdt)
2
+ (−dz − dRdt)
2
i
= dx2
+ dy2
+ dz2
+ 2(dxdP + dydQ + dzdR)dt +
+ dP2
+ dQ2
+ dR2
dt2
(4)
Equating (3) and (4), and neglecting the terms dP2
+ dQ2
+ dR2
dt2
, we have that:
(dxdP + dydQ + dzdR)dt = 0
so
dxdP + dydQ + dzdR = 0 (5)
Afterwards, Euler assumes the case in which dx = dy = 0, that implies dR = 0, so we have that R do not depend
on z. Repeating the same argument, Euler finds that P do not depend on x, and that Q do not depend on y, so
that:
P = Ay + Bz
Q = Cz − Ax (6)
R = − Bx − Cy
with A, B, C constants.
Inasmuch as the instantaneous rotation axis ought to be instantaneously at rest, Euler determines by a variable
change, that the points for which the speed (P, Q, R) is equal to zero in the dt interval are: x = Cu, y = −Bu, z =
Au, where u is the new variable. This points determine a rect line passing throught the origin, the instantaneous
rotation axis. Afterwards, Euler identifies the angular velocity of the body as‡:
A2
+ B2
+ C2
1/2
(7)
Once demonstrated the existence of the instantaneous rotation axis, Euler is interested in calculating the
acceleration of a mass element dM, so to later be able to apply the second law of Newton. With this goal [Euler 1752,
p. 205-210] he assumes a system of three fixed axes in absolute space, mutually perpendicular. From equations (6),
the displacement of a point with coordinates x, y, z during the interval dt, could be expressed as:
dx = (λy − µz)dt
dy = (νz − λx)dt (8)
dz = (µx − νy)dt
where λ = A, µ = −B and ν = C, so now the angular velocity is
ν2
+ µ2
+ λ2
1/2
(9)
and differentiating equations (8) it is obtained that:
ddx = (ydλ − zdµ)dt + (λνz + µνy − (λλ + µµ)x)dt2
ddy = (zdν − xdλ)dt + (µνx + λµz − (νν + λλ)y)dt2
(10)
ddz = (xdµ − ydν)dt + (λµy + λνx − (µµ + νν)z)dt2
‡ In modern mathematical language, it is obtained by setting
v = ω × r = (ωyz − ωzy)î + (ωzx − ωxz)ĵ + (ωxy − ωyx)k̂
and by a simple inspection of equations (6), it is clear that A = ωy, B = −ωz, C = ωx and the magnitude of the angular velocity is
A2 + B2 + C2
1/2
=
ωx
2 + ωy
2 + ωz
2
1/2
, which is equation (7).

5. Euler and Rigid Bodies 5
At this point, Euler is able to apply the second law for a mass element dM. In this way, the force in the direction of
the three axes shall be [Euler 1716] §:
2dMddx
dt2
,
2dMddy
dt2
,
2dMddz
dt2
(11)
i.e.:
2
dM
dt
(ydλ − zdµ) + 2dM(λνz + µνy − (λλ + µµ)x)
2
dM
dt
(zdν − xdλ) + 2dM(µνx + λµz − (νν + λλ)y) (12)
2
dM
dt
(xdµ − ydν)dt + 2dM(λµy + λνx − (µµ + νν)z) .
From this point, Euler states that “... in order to know exactly the status of this forces, it is necessary to take into
account only their moments with respect to our three axes ...” [Euler 1752, p. 208]. Each one of the three components
of the force that act on the mass dM, shall have a moment (the torque components, in modern language), about
two of the three axes.
Thereby, Euler determines the moments on the three axes [Euler 1752, pp. 208-209]k in such a way, that the moment
on z shall be:
2
dM
dt
(yydλ + xxdλ − yzdµ − xzdν) + 2dM(λνyz − λµxz + µνyy − µνxx − (µµ − νν)xy) (13)
the y component,
2
dM
dt
(xxdµ + zzdµ − xydν − yzdλ) + 2dM(λµxy − µνyz + λνxx − λνzz − (νν − λλ)xz) (14)
and lastly, on x
2
dM
dt
(zzdν + yydν − xzdλ − xydµ) + 2dM(µνxz − λνxy + λµzz − λµyy − (λλ − µµ)yz) (15)
Integrating these equations on a mass element, and defining:
Z
dM(xx + yy) = Mff
Z
xydM = Mll
Z
dM(xx + zz) = Mgg
Z
xzdM = Mmm
Z
dM(yy + zz) = Mhh
Z
yzdM = Mnn
where Euler emphasises that Mff, Mgg y Mhh are the inertia moments of the body along the three fixed axes, it
is obtained [Euler 1752, p. 209-210]:
2M
ff
dλ
dt
− nn
dµ
dt
− mm
dν
dt
+ λνnn − λµmm − (µµ − νν) ll + µν (hh − gg)
2M
gg
dµ
dt
− ll
dν
dt
− nn
dλ
dt
+ λµll − µνnn − (νν − λλ) mm + λν (ff − hh)
2M
hh
dν
dt
− mm
dλ
dt
− ll
dµ
dt
+ µνmm − λνll − (λλ − µµ) nn + λµ (gg − ff)
.
§ In this work, written and presented in 1747, and published in 1749, is where for the first time, he expresses the second law of Newton in
terms of the components of the force in the three cartesian axes. The factor of 2 appearing in equations (11), is due to the units used by Euler.
First, what Euler denominates as M, is the weight of the body, i.e. M = mg, and he chooses the units of position x and time t in a way that the
velocity acquired by a body falling from a height h, is (dx/dt)2
= h. From this, it is clear that the units used by Euler, are such that g = 1/2, so
the second law (of Newton) is expressed as F = mr̈ = M
g
r̈ = 2Mr̈.
k In modern language, this equations are obtained in the following way. The torque N is equal to r × F, where F is the force and r the body’s
position vector, so that
N = (yFz − zFy)ı̂ + (zFx − xFz)̂ + (xFy − yFx)k̂.
Substituting the components of the force given by equations (13), we obtain expressions (13), (14) and (15).

6. Euler and Rigid Bodies 6
(16)
It could be said that this expressions are the first version of Euler equations, which would be expressed in a modern
fashion as:
Nz = Izzω̇z − Iyzω̇y − Ixzω̇x + Iyzωzωx − Ixzωzωy − Ixy(ω2
y − ω2
x) + (Ixx − Iyy)ωyωx
Ny = Iyyω̇y − Ixyω̇x − Iyzω̇z + Ixyωyωz − Iyzωxωy − Ixz(ω2
x − ω2
z) + (Izz − Ixx)ωzωx
Nx = Ixxω̇x − Ixzω̇z − Ixyω̇y + Ixzωyωx − Ixyωzωx − Iyz(ω2
z − ω2
y) + (Iyy − Izz)ωzωy .
(17)
where N = (Nx, Ny, Nz) is the torque, ω = (ωx, ωy, ωz) the angular velocity, Ixx, Iyy, Izz the inertia moments and
Ixy, Iyz, Izx the inertia products. The components are represented by a two order tensor, expressed by a 3 × 3
symmetric matrix, where the diagonal elements are the inertia moments and the leftover are the inertia products.
The I tensor receives the inertia tensor name.
This equations are a big step in the understanding of a rigid body, but it is clear that their integration is a complex
problem and aditionally, as it is pointed out by the same Euler, “... from these formulae it shall be known for each
instant, the change in the position of the rotation axis and of the angular velocity. It is then necessary to change
each instant the position of the three axes ... This coerces to calculate for each instant the values ll, mm, nn, ff,
gg, hh, insomuch as the change in the position of the body with respect to the three axes shall cause continuous
variations” [Euler 1752, p. 214].
In 1751 Euler in Du mouvement de rotation des corps solides autour d’un axe variable (On the movement of rotation of
solid bodies around a variable axis) [Euler 1765a] ¶, considering the problem that represents the calculation with
time of the inertia tensor components, he changes the scope introducing axes fixed to the body, choosing them as
the principal of inertia, i.e. the axes in which the inertia tensor is diagonal [Wilson 1987, p. 264-270] +
.
Euler chooses a system of axes fixed in the absolute space IA, IB, IC and determines the position of the rotation
axis IO through the angles AIO = α, AIB = β and AIC = γ. Afterwards, he considers a point Z of the rigid
body, with coordinates (x, y, z) with respect to axes IA, IB, IC, and he chooses three mutually perpendicular axes
Za, Zb and Zc, parallel to the axes in absolute space, in such a way that they pass through Z and are principal
of inertia. If the body is rotating along the IO axis with an angular velocity ω, the components on Za, Zb and Zc
could be expressed by means of the direction cosines
ω = (ω cos α, ω cos β, ω cos γ)
with cos2
α + cos2
β + cos2
γ = 1.
The velocity (u, v, w) of the point Z, with respect to the principal axes, shall be:
u =
dx
dt
= ω (z cos β − ycosγ)
v =
dy
dt
= ω (x cos γ − z cos α) (18)
w =
dz
dt
= ω (y cos α − x cos β)
differentiating the component u with respect to time, it is obtained:
du
dt
= z
d
dt
(ω cos β) − y
d
dt
(ω cos γ) + ω cos β
dz
dt
− ω cos γ
dy
dt
substituting dy = vdt and dz = wdt, and using the fact that the sume of the square of the direction cosines is 1, it
is arrived to:
du = zd(ω cos β) − yd(ω cos γ) + ω2
dt y cos α cos β + z cos α cos γ − x sin2
α
(19)
¶ Although this work is from 1751, it was presented in 1758, but was published until 1765.
+ The existence of the principal inertia axes was published by the first time by Johannes Andreas Segner in his Specimen theoriae turbinum
in 1755 [Segner 1765]. Segner was member of the Academy of Berlin and he knew Euler and his oeuvre very well, and although surely this
knowledge was of great utility for him, the work of Segner is independent. For this reason, years later Laplace shall assure that the discovery
of principal axes is of Segner [Wilson 1987, pp. 264-270].

7. Euler and Rigid Bodies 7
and analogously, differentating the other two components and substituting dx = udt, dy = vdt and dz = wdt, it is
obtained respectively:
dv = xd(ω cos γ) − zd(ω cos α) + ω2
dt x cos α cos β + z cos β cos γ − y sin2
β
(20)
dw = yd(ω cos α) − xd(ω cos β) + ω2
dt x cos α cos β + y cos β cos γ − z sin2
γ
(21)
At this point, Euler is in conditions to apply the second law of Newton, so the components of the force on a mass
element dM (lets remember that in Euler, M is the weight) shall be: dM
2g
du
dt , dM
2g
dv
dt , dM
2g
dw
dt
∗.
Substituting equations (19), (20) and (21), we have:
dM
2g
du
dt
=
dM
2g
z
d
dt
(ω cos β) − y
d
dt
(ω cos γ) + ω2
y cos α cos β + z cos α cos γ − x sin2
α
dM
2g
dv
dt
=
dM
2g
x
d
dt
(ω cos γ) − z
d
dt
(ω cos α) + ω2
x cos α cos β + z cos β cos γ − y sin2
β
dM
2g
dw
dt
=
dM
2g
y
d
dt
(ω cos α) − x
d
dt
(ω cos β) + ω2
x cos α cos β + y cos β cos γ − z sin2
γ
.
(22)
From here, it is possible to calculate the moments of the forces in each axis, obtaining, for the component along the
Za axis
dM
2g
y2 d
dt
(ω cos α) − xy
d
dt
(ω cos β) + ω2
xy cos α cos β + y2
cos β cos γ − yz sin2
γ
−
dM
2g
xz
d
dt
(ω cos γ) − z2 d
dt
(ω cos α) + ω2
xz cos α cos β + z2
cos β cos γ − yz sin2
β
from where, integrating on the mass element dM, it is arrived to
1
2g
d
dt
(ω cos α)
Z
(y2
+ z2
)dM −
d
dt
(ω cos β)
Z
(xy − xz)dM
+
1
2g
ω2
cos α cos β
Z
(xy + xz)dM+ω2
cos β cos γ
Z
(y2
−z2
)dM
which can be rewritten as:
1
2g
d
dt
(ω cos α)
Z
(y2
+ z2
)dM −
d
dt
(ω cos β)
Z
(xy − xz)dM
+
1
2g
ω2
cos α cos β
Z
(xy + xz)dM+ω2
cos β cos γ
Z
(x2
+ y2
)dM−
Z
(x2
+z2
)dM
Now, inasmuch as the axes are principal of inertia, the inertia products are zero, i.e.
Z
xydM =
Z
xzdM =
Z
yzdM = 0
and defining the moments of inertia as:
Z
dM(x2
+ y2
) = Maa
Z
dM(x2
+ z2
) = Mbb
Z
dM(y2
+ z2
) = Mcc
doing the same calculation for the other two components, and defining P, Q and R as the moments of the forces
along the principal axes, Euler arrives to:
P = Maa
dω cos α
2gdt
+ M(cc − bb)
ω2
cos β cos γ
2g
∗ Here, Euler changed again the units of space and time, and he define them in terms of the distance g travelled by a body that falls from rest,
during one second, i.e. g = 1
2
gt2 = g/2. So the second law (of Newton) can be written as F = ma = M
g
a = Ma
2g
.

8. Euler and Rigid Bodies 8
Q = Mbb
dω cos β
2gdt
+ M(aa − cc)
ω2
cos α cos γ
2g
(23)
R = Mcc
dω cos γ
2gdt
+ M(bb − aa)
ω2
cos α cos β
2g
which are the known “Euler equations” for a rigid body, referred to principal inertia axes, and with the angular
velocity components in terms of the angles α, β, γ, which are the angles subtended by the rotation axes with the
principal ones fixed in the body. It could be said that this are the Euler angles, although actually they are usually
defined by applying the rotation operator to the axes fixed on the body, for the sake of that each angle to be related
with the angular velocities of rotation known as precession, nutation and spin.
In modern language, this equations are presented as:
N1 = I11ω̇1 + (I33 − I22)ω2ω3
N2 = I22ω̇2 + (I11 − I33)ω1ω3 (24)
N3 = I33ω̇3 + (I22 − I11)ω1ω2
where 1, 2, 3 are the principal axes of inertia fixed to the body; the components of angular velocity in this system
are ω = (ω1, ω2, ω3), the torque is N = (N1, N2, N3) and the diagonal elements of the inertia tensor are I11, I22 e
I33.
Once Euler has established the moment of moment ideas, of an instantaneous rotation axis, Euler equations and
principal axes of inertia, it could be said that his contribution to the mechanics of rigid body is complete, and
it would seem that he also thinks it himself, because he devotes himself to prepare a work that, judging by his
title, represents a compendium of his work in mechanics of the rigid bodies, Theoria motus corporum rigidorum seu
solidorum, appeared in 1765. Nevertheless, this voluminous work do not result to be an adequate compendium,
insomuch as Euler disaggregates himself in excessive details without clarifying what he posed before, nor stating
anyelse].
A real contribution to mechanics of rigid bodies is found in his Nova methodus motu corporum rigidorum determinandi
(A new method for generating the motion of a rigid body), in which for the first time, Euler poses, explicitly, one
beside the other, the fundamental laws of mechanics [Euler 1776, p. 224]:
I.
Z
dM
ddx
dt2
= iP IV.
Z
zdM
ddy
dt2
−
Z
ydM
ddz
dt2
= iS
II.
Z
dM
ddy
dt2
= iQ V.
Z
xdM
ddz
dt2
−
Z
zdM
ddx
dt2
= iT
III.
Z
dM
ddx
dt2
= iR VI.
Z
ydM
ddx
dt2
−
Z
zdM
ddy
dt2
= iU
where dM is the infinitesimal mass element of the body; x, y, z, the position of the body in cartesian coordinates;
P, Q, R the resultant of the external forces in each of the axes directions; i is a constant chosen depending on the
used units; and S, T, U are the moments of the forces in the three directions x, y, z. In this equations, Euler do
not provide the vector nature to S, T, U, i.e. do not consider them as the components of unique physical entity
[Borrelli 2011] and, although it could be alleged that in this equations it is already implicit the rotation equation
of movement, i.e. N = dL
dt (with N the torque and L the angular momentum), Euler do not consider the time
evolution of this equations.
The great contribution of Euler in his Nova methodus is to set these equations as general laws, independently of the
specific problem, i.e. that they are valid for all bodies and kinds of movement.
As Maltese thinks, in this manuscript, Euler is able to separate the description of the inertial properties from the
application of the mechanics principles, inaugurating “... what we now call the Newtonian tradition in mechanics
in its modern form” [Maltese 2000, p. 340].
] In other aspects not strictly related to the ridig bodies, Theoria motus corporum rigidorum seu solidorum is interesting, being that it provides a
detailed examination of the dynamical bases of his theory, beside fathom in conceptions relative to absolute space and movement.

9. REFERENCES 9
3. Conclusions
The mechanics in general, and the rigid bodies mechanics in particular, was one of the topics that most relevantly
interested to Euler during his life. This interest allowed him to build, in a non-linear way - and in many cases in
an apparently disordered one - the fundamental concepts of the mechanics of rigid bodies, which are necessary
to be tracked in different works through 35 years, trying to save the problems relative to the writing, presentation
and publication dates.
Leaving this creative labyrinth, finally we find absolute clarity about the eulerian thought that illuminated the
XVIII century culture, with a difficultly comparable brightness.
Acknowledgments
JJHG acknowledges partial support projects 20160105 and 20160576, as well as EDI grant, all provided by
SIP/IPN.
References
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Newtonian Paris.
Euler L 1716 Recherches sur le mouvement des corps célestes en général The Euler Archive, E-112.
URL: http://eulerarchive.maa.org
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Segner J A 1765 Specimen theoriae turbinum Halle.
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Truesdell C 1968a Essays in the History of Mechanics Springer Verlag Berlin.

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Truesdell C 1968b in C Truesdell, ed., ‘Essays in the History of Mechanics’ Springer Verlag Berlin chapter 5, pp. 239–
271.
Wilson C A 1987 Arch. Hist. Exact Sci. 37(3), 233–273.