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Second Law of Motion, F=ma, was discovered after 48 years of death of Newton
Ajay Sharma
Fundamental Physics Society. His Mercy Enclave Post Box 107 GPO Shimla 171001 HP
India
Email [email protected]
Abstract
Neither prevalent definition nor equation F=ma was given by Newton , but credited to Newton
unscientifically. Euler gave three equations of force F=ma/2 or F=ma/n in 1736, F=2ma in 1750,
(the coefficient 2 depended on the unity of measure) and F=ma in 1775 in different ways. Which
of three should be regarded as equation of force? Realistically F =ma is speculated not derived
by Euler. Further, F=ma is used to derive rest mass energy equation (E0=M0c2 ), but if F =2ma ,
F=ma/2 are used then equations for rest mass energy become E0=2M0c2 and E0=M0c2/2. These
aspects are undiscussed yet. If the magnitudes equations of force F=ma and F=2ma are equal in
magnitude then 1=2, which is not true. Or for equality of force, it has to be assumed that Euler
capriciously divided or multiplied right hand sides by 2 to obtain F =ma. There is no other way
to obtain F=ma from F=2ma and F=ma/2. The coefficients must be same in both equations of
force i.e. 2. Euler used two primary or fundamental units L (length) and F( force), hence value
of coefficient is 2. The systems of primary units L (length)-F(force) –T(Time ) and L (length)M(mass) –T(Time ) were introduced in the following century. Thus F=ma may be understood
as a postulate, and must be credited to Euler who speculated it. The Principia‘s second law of
motion i.e. F =k(v-u) is not discussed at all. In highly specialized literature it is mentioned that
Euler invented F=ma, but in standard literature or textbook level physics Newton is originator of
F =ma. This important debate concerning F =ma is wide open and will reveal new facts in
science.
Keywords Newton, Euler, F=ma
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1.0 Newton’s second law of motion in the Principia
Euler has given three equations of force i.e. F =2ma, F =ma/2 and F =ma. In the derivation of
rest mass energy (E0=M0c2 ) F =ma is used, if other equations are used then results are
E0=2M0c2 and E0=M0c2/2 [1]. So for same physical quantity different equations may be there,
but all must reduce to same equation. Thus a physical quantity must be represented by a single
equation. The Principia‘s second law of motion states [2] as
The alteration of motion is ever proportional to the motive force impressed; and is made in
the direction of the right line in which that force is impressed.
Newton explained the definition of second law of motion but did not give any mathematical
equation for it. In fact mathematical equation is F= k(v-u).
(i) Absolute Motion (motus absolutus) and Quantis motus or Quantity of motion
Newton [3] has defined motion as absolute motion in Scholium after DEF.VIII, page 10.
It is translation of body from one absolute space to other. In Physics translation is defined as
movement of body from one point to other i.e. body possesses velocity.
Newton defined ‗Quality of motion‘ in DEF. II at page 2.
‗The Quantity of Motion is the measure of the same, arising from the velocity and quantity of
matter conjunctively.‘ Now we understand the quantity of motion as momentum. Thus definition
of Quantity of motion and absolute motion are entirely different hence defined under different
headings , so ‗motion‘ cannot replace ‗Quantity of motion‘ and vice versa. If both are regarded
as having same meaning, then it is misinterpretation and contradictory to concepts laid down by
Newton in the Principia. Further the force depends upon velocity if the definitions of force and
first law of motion are interpreted as given in the Principia. Also the alternation in motion (
change in motion i.e. v=u) cannot be rate of change of velocity.
Nauenberg [4] while commenting on Pourciau‘s paper [5], ‗Is Newton‘s second
law really Newton‘s ?‘ has misunderstood the ‗motion‘ and ‗quantity of motion ‗ as same
hence drew incorrect conclusions. The arbitrary conclusions are not allowed. There is clear
difference between ‗motion‘ and ‗quantity of motion ‗ in the Principia. It is clear from the
Definition II and scholium. Nauenberg implied that Newton gave it a precise mathematical form
of velocity –independent central forces in Proposition I (Theorem 1) and Proposition 6 (Theorem
V) of Book I. But F=ma =m(v-u)/t is velocity dependent. Further Prop 1 and Prop 6 , in Book I
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do not lead to F=ma { depends upon velocity i.e. F = m(v-u)/t }. So these are not related to
second law of motion as given in the Principia. These propositions are discussed under the
Section: Of the Invention of Centripetal Forces. No equation is derived which relates to F=ma
=m(v-u)/t. The both LHS and RHS of an equation are separated by equality sign.
There is nothing in the Principia‘s second law about acceleration and nothing about a rate of
change as confirmed by Pourciau, article titled Is Newton’s second law really Newton’s ? [5,6].
Truesdell [7], it has been clearly stated that F=ma appears nowhere in Newton's Principia, but
was given by Euler. The extensively used equation F=ma nowhere occurs in any edition of
the Principia [8].The authors did not discuss that why credit of discovery of F=ma is given to
Newton?
2.0 In the Principia F=ma was not derived.
The equation F=ma was derived (but arbitrarily) for first time by Leonhard Euler [8,9]9,10
in 1736, 1750 and 1775. Newton started the beginning of the laying of foundations of Physics
by defining the basic terms such as mass, inertia, force, rest, motion, gravity, centripetal force
etc. in the Principia. At that time physical phenomena were expressed in terms of laws, axioms,
propositions etc. The formulation of equations and mathematical interpretations were next phase
of development of concepts. Newton has stated in Propositions (I-IX) Book III of the Principia
.These are regarded as law of gravitation. But Newton did not give any mathematical equation to
law of gravitation, like the Principia‘s second law of motion. Thus it is correct conclusion that
Newton did not derive F=ma.
According to Stanford Encyclopedia of Philosophy [8]8 — ―The
modern F=ma form of Newton's second law nowhere occurs in any edition of the Principia even
though he had seen his second law formulated in this way in print during the interval between the
second and third editions in Jacob Hermann's Phoronomia of 1716. Instead, it has the following
formulation in all three editions: A change in motion is proportional to the motive force
impressed and takes place along the straight line in which that force is impressed.‖
―The obvious question with the second law is what Newton means by ―a change in
motion.‖ If he had meant a change in what we call momentum — that is, if he had meant, in
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modern notation, Δmv — the proper phrasing would have been ―a change in the quantity of
motion.‖
―If this way of interpreting the second law seems perverse, keep in mind that
the geometric mathematics Newton used in the Principia — and others were using before him —
had no way of representing acceleration as a quantity in its own right. Newton, of course, could
have conceptualized acceleration as the second derivative of distance with respect to time within
the framework of the symbolic calculus. This indeed is the form in which Jacob Hermann
presented the second law in his Phoronomia of 1716 (and Euler in the 1740s). But the geometric
mathematics used in the Principia offered no way of representing second derivatives.‖
Thus Newton‘s second law means, F  (v-u) or
F = k(v-u)
(1)
The Principia involves geometry excessively. But the geometric mathematics and methods used
by Newton, and before him, did not at all represent acceleration and second derivative. The
dimensional analysis was started by Fourier in 1822 [11,12] and dimensions of force are based
upon F =ma and not on Principia‘s F =k(v-u). Also the unit of force dyne [13] was initially
defined in 1861 about 184 years after publication of the first edition of the Principia. But this
definition was unacceptable to Committee of the British Association for the advancement of
Science [14]. The 9th Conférence Générale des Poids et Mesures held in 1948, then adopted the
name "newton" for unit of force in resolution 7 [15]. The concept of force was initially defined
in 1687, so it took considerable time for developments of law and process is still on. In case
concepts of units, dimensions, acceleration, second derivative and related mathematical methods
were available at Newton‘s time, then interpretation would have been different.
3.0 Euler derived F=
d 2s
1
dp
ma, F =2 m 2 = 2
=2ma, F =ma.
n
dt
dt
(i) In 14th century French philosopher Jean Buridan (1300 – 1358) propagated Impetus Theory
[16] and defined equation similar to momentum.
Impetus = weight (mass)  velocity
(2)
Also Leibniz has coined term vis viva or living force =mv2
(3)
(ii) In 1716, Jacob Hermann published a Latin text called ―Phoronomia‖, meaning the science of
motion [17]. He stated a equation dc =pdt, where c stands for ―celeritas‖ meaning speed,
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and p stands for ―potentia‖, meaning force.
or
p=
dc
= Force
dt
(4)
In today‘s notation it is acceleration. However Herman called it ‗potentia‘ or force at that time.
(iii) Further momentum was defined by J Jenning's Miscellanea [18] in Latin in 1721 as
p=mv
(5).
In the next definition the velocity or speed is defined by dividing distance with time i.e. S/t.
These developments took place after second edition (1713) and before third edition (1726) of
the Principia. But Newton preferred not to mention them at all and kept original second law of
motion in third edition, as in first edition (1687) of the Principia [8]. So Newton did not write
any equation for Principia‘s second law of motion. The topic of the comparative study of
developments of equations of motion of Newton and Euler has been recently studied with details
by various authors [19-23], but some issues further need to be discussed.
Basically Euler has given at different times, three equations of force F=ma/n
(1736), F =2ma (1750) and F=ma (1775). One physical quantity may be represented by various
equations but all should yield the same equation. Now the biggest question which is unaddressed
is that how Euler obtained F =ma from F=ma/n (1736), F =2ma (1750) ? Or how coefficients ½
and 2 are neglected. In the relativistic physics E0 =moc2 is derived by using F=ma , but it is not
obtained if F=2ma or F =ma/2 are used. So a physical quantity may be represented by many
equations but all must reduce to same equation. It must be thoroughly discussed and clearly
understood that how the most significant equation of physics is derived? Euler has given rule or
method to measure coefficient or constant in equation that it depends upon unity of measure. The
same is applied here.
(a) In 1736, Euler wrote [24, 9-10, 25-27] equation of potentia (p) meaning force which has
resemblance with eq.(4) i.e. dc=pdt,
dc =
npdt
m
or F =
1
ma
n
(6)
where m is mass, c is velocity, F is force , t is time and where n is constant and depends upon
unity of measure [24]. By unity of measure we mean, unit of measurement. Euler [25] used two
primary or fundamental units L (length) and F(force), thus coefficient/constant of proportionality
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is 2. Now co-efficients are determined experimentally. The systems of primary units L (length)F(force) –T(Time ) and L (length)-M(mass) –T(Time ) were introduced in the following
century. From eq.(6) Euler was able to derive all differential equations necessary to describe the
motion of a point-mass.
(b) In Mechanica, however, Euler used an intrinsic coordinates system. He
decomposed speeds and forces according to directions that depended upon the intrinsic nature of
the problem. In these papers, Euler used an extrinsic references frames (a system of three
orthogonal Cartesian axes) and formulated the second law of motion in this way [9-10]:
2Mddx=Pdt2, 2Mddy=Qdt2, 2Mddz=Rdt2,
or P = 2M
(7)
d 2x
d2y
d 2z
,
Q
=
,
R
=
2
M
2
M
dt 2
dt 2
dt 2
(7)
where M is the mass and P, Q, and R the components of the force on the axis (the coefficient 2
depended on the unity of measure , and Euler has chosen two primary units ).
F = 2m
d 2s
dv
dp
= 2m
=2
= 2ma
2
dt
dt
dt
(8)
In view of it (dependence upon unity of measure ) coefficient, n is 2 in eq.(6).
(c) Later, in 1765, Euler introduced the concept of moment of inertia of a rigid body and
decomposed the motion into the rectilinear motion of the centre of mass and the rotational
motion about the centre of mass [26]; in 1775, he completed the construction of general
equations of dynamics by formulating a system of six equations determining the motion of any
body, which (except for an additional coefficient) he wrote in this way [27].
P=  dM
 zdM
d2y
d 2x
,
Q=
dM
 dt 2 , R=
dt 2
 dM
d 2z
dt 2
d2y
d 2z
d 2z
d2y
=S
,
=T,
ydM
xdM
zdM

dt 2 
dt 2
dt 2 
dt 2
Or in general, F =
 dM
 ydM
d 2x
d2y
=U
xdM
dt 2 
dt 2
(12)
d 2s
dt 2
4.0 How F=ma is obtained from Euler’s equations of force F =
1
ma and F =2ma ?
n
It is crystal clear from the existing literature that Euler gave eq.(6) and eq.(8), but with different
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coefficients/constants of proportionality. In eq.(6) it is 1/2 and in eq.(8) it is 2. Euler determined
that the value of coefficient of proportionality depends upon unity of measure. It is clearly
mentioned that coefficient 2 depended upon unity of measure, as Euler has used 2 fundamental
units [25] i.e. L(length) and F (force). Euler used L(length) and F(force) as primary
(fundamental) units , so that masses are measured in units of force, acceleration is dimensionless
and it is actually measured by the square root of length. The velocity was measured as square
root of height defined by condition that body falling from the rest [25]. At time of Euler, it was
just developmental state. The systems of primary units L (length)-F(force) –T(Time ) and L
(length)-M(mass) –T(Time ) were introduced in the following century. Such notion is
inconsistent in view of current perception or understanding of units [28]. The dimensional
analysis was stated by Fourier in 1822 [11-12] 11-12 and dimensions of force are based upon F
=ma, not on F=k(v-u) i.e. Principia‘s second law of motion. In terms of current perceptions the
value of k should have dimensions mass per unit time in eq.(1).
The other way to understand the problem is that Euler arbitrarily divided the RHS of
eq.(8) by 2 to obtain F=ma.
F=m
d 2s
dv
dp
= m
= = ma
2
dt
dt
dt
(9)
Also to obtain F =ma , from F =ma/2 , arbitrarily Euler multiplied RHS of eq.(10) by 2.
Under this condition Euler applied eq.(9) in various phenomena in mechanics.
Further, in Euler‘s eq.(6) i.e. F = ma/n, here n is coefficient which depends upon unity of
measure. The value of n here is 2 , as it depends upon unity of measure. Further Euler has taken
two primary units i.e. L (length) and F (force). Hence in this case equation of force i.e. eq.(6)
becomes
F =ma/2
(10)
So for same physical quantity different equations may be there, but all must reduce
to same equation. F=ma is used to derive rest mass energy equation (E0=M0c2 ), but if F =2ma ,
F=ma/2 then equations for rest mass energy become E0=M0c2 and E0=M0c2/2 . If a physical
quantity is represented by more than one equations, then they must reduce to single equation
logically. Thus coefficient 2 from eq.(8) and ½ from eq.(6) is eliminated in this case, thus
equation F=ma came into existence. But it is pure speculation only, especially in the sense Euler
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has given two other equations of force i.e. F =2ma and F =ma/2. These equations are forced to
reduce to F=ma. Had Euler given just one equation of force then this discussion would have not
arisen. The equations F =2ma and F=ma cannot be same in magnitude (i.e. 1  2).
If we try to obtain value of F as in previous case then RHS of eq.(10) has to be multiplied by 2.
Thus arbitrary multiplication or division as done by Euler to obtain F =ma from eq.(6) and eq.(8)
is not allowed in science or mathematics. Then coefficient cannot be neglected arbitrarily. Like
this identity of equation vanishes. Thus it is true that the method which Euler obtained is not
allowed. There is no other way that eqs.(6,8) reduce to F=ma, a physical quantity is represented
by one equation only. In the relativistic physics E0 =moc2 is derived by using F=ma , but it is not
obtained if F=2ma or F =ma/2 are used.
The division or multiplication by 2 to RHS violates the inherent definition of
equation. If both sides of equation are divided by same number then it is permissible. If the
magnitude of one side of equation is randomly changed then that does not remain equation. Thus
equation F =ma is obtained from eqs.(6,8) in arbitrary an speculative way only. The equations F
=ma/2, F=2ma and F=ma cannot be equal. Thus it is evident that Euler‘s publications were un
reviewed at that time.. Even Einstein‘s annus mirabilis or miracle year (1905) papers were
published un reviewed in the Annalen der Physik.[29] The peer reviewed standard developed
later on. Further Euler‘s work has been critically analysed by other author‘s as well [30].
5.0 Equation F =ma should be credited to Euler who speculated it and not to Newton
(who did not contribute to it at all)
Newton in the Principia at page 19, gave second law of motion as
‗alteration of motion is proportional to impressed force‘ or F=k(v-u),which is completely
neglected. It is never taught as Newton‘s law in any way in old and modern texts, but F=ma is
regarded as second law of motion. The Principia‘s, genuine second law F=k(v-u) is not even
mentioned in physics texts at all. Is it invalid? If so then it must be debated. Scientists are
unanimous (in highly specialized literature) that Newton did not discover F=ma, even then
credited to Newton which was speculated by Euler. Who credited Euler‘s speculated equation F
=ma to Newton and what were the reasons? Why Euler was deprived of credit of discovery of
speculation of F=ma in standard textbooks, and Newton is given credit (F=ma was discovered
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after death of Newton) ? It is the rarest or rarest cases. These are very sensitive but unanswered
questions. These issues should be investigated scientifically; some more new and exciting results
are expected in research. All the aspects of law must be discussed together before reaching at
conclusions. Thus on such fundamental topic scientific discussion must continue taking all
aspects (hidden or vivid) in account, as science has beginning but no end. The contents of highly
specialized literature and standard textbooks must be the same, but it not so in case of F=ma.
Acknowledgements
Author is highly indebted to anonymous reviewer for useful comments and Dr Steve Crothers for
critical discussions and encouragements at various stages of the work.
References
[1] Sharma A , to be presented in 2014 The Euler Society Conference, 21-23 July 2014, St. Edward‘s
University, Austin USA
[2 ] Sharma A , Infinite Energy July/August 116 2014
[3] Newton , Isaac Mathematical Principles of Natural Philosophy pp.19-20 , London, 1727 ,
translated by Andrew Motte from the Latin.
http://books.google.co.in/books?id=Tm0FAAAAQAAJ&pg=PA1&redir_esc=y#v=onepage&q&
f=false
[4] Nauenberg , Michael Am. J. Phys. 80(10) 931-936 (2012)
[5] B Pourciau , ‗ Is Newton‘s second law really Newton‘s ? American Journal of Physics , 79
1015, 2011
[6] B. Pourciau, ―Newton‘s interpretation of Newton‘s second law of motion. ‖ Arch. Hist.
Exact Sci. 60, 157–207 (2006).
[7] C. Truesdell, Arch. History. Exact. Sci 1, 3 (1960),
[8]
http://plato.stanford.edu/entries/newton-principia/#ThrEdiPri
[9] Euler, Leonhard Recherches sur le mouvement des corps célestes en général, Mémoires de
l’académie des sciences de Berlin 3, 93-143 or Opera, ser. 2, vol. 25, 1 – 44 (1747).
[10] Euler, Leonhard (1750) ―Découverte d‘un nouveau principe de Mécanique‖, Mém., 6, 185217 or Opera, ser. 2, vol. 5, 81 – 108, (1750).
[101] J. Fourier. The analytical theory of heat. Cambridge University Press, 1822.
[12] R. Martins, The origin of dimensional analysis. J. Franklin Institute, 331, 331 (1981)
[13] Rossiter, William (1879). Dictionary of Scientific Terms. London and Glasgow: William
10
Collins, Sons, and Coy. p. 109.
[14] Everett, ed. "First Report of the Committee for the Selection and Nomenclature of
Dynamical and Electrical Units". Forty-third Meeting of the British Association for the
Advancement of Science. Bradford: Johna Murray. p. 223. Retrieved 8 April 2012
[15] International Bureau of Weights and Measures (1977), The international system of
units (330–331) (3rd ed.), U.S. Dept. of Commerce, National Bureau of Standards, p. 17
[16 ] Pedersen, Olaf (1993-03-26). Early physics and astronomy: a historical introduction. CUP
Archive. p. 210 (1993).
[17] Herman Jacob , Phoronomia, sive De viribus et motibus corporum solidorum et fluidorum
libri duo , Amstelædami, apud R. & G. Wetstenios, h. ff., 1716
[18] Jennings, John (1721). Miscellanea in Usum Juventutis Academicae. Northampton: R.
Aikes & G. Dicey. p. 67.
[19] Sébastien Maronne1,, Marco Panza , Advances in Historical Studies Vol.3, No.1, 12-21
2014
[20 ] Blay, M. La naissance de la mtcanique analytique:La science movement au tourant des
xviiie siecles , Paris, PUF, 1992
[21 ] Blay M Reasoning with the Infinite From the Closed World to the Mathematical Universe,
Chicago, University of Chicago Press, 1998
[ 22] MARONNE, Sébastien and PANZA, Marco (2010) ―Newton and Euler‖, in Mandelbrot,
Scott and PULTE, Helmut (eds.) [2010] The reception of Isaac Newton in the European
Enlightenment, (2 vols.), London, Continuum, 2010
[ 23] G Maltese (2002) ―On the changing Fortunes of the Newtonian Tradition in Mechanics‖ in Kim
William (ed) Two cultural essays in Honour of David Speiser, Birkhauser 97-113
[24] Giovanni Ferraro Euler‘s analytical program, Quaderns d'història de l'enginyeria,11
(2010), 185-208.
[25] Grattan Guinness, Companion Encyclopedia of History of Philosophy of the
Mathematical Sciences , Volume 2 The John Hopkins University Press , Baltimore pp.1011,
1994
[26] EULER, Leonhard (1765) Theoria motus corporum solidorum seu rigidorum ex primis
nostrae cognitionis principiis stabilita et ad omnes motus, qui in hujusmodi corpora cadere
11
possunt, accommodata, Rostochii et Gryphiswaldiae litteris et impensis A. F. Röse.
[27] EULER, Leonhard (1775) ―Nova methodus motum corporum solidorum rigidorum
determinandi‖, Novi Commentarii academiae scientiarum Petropolitanae (afterwards: Novi
Comm.) 20, 208-238 or Opera, ser. 2; vol. 9, 99 – 125.
[28] http://en.wikipedia.org/wiki/Units_of_measurement
[29] Sharma, A Beyond Einstein and E=mc2 , in press
[30] The use of the differential calculus in mechanics was criticized in England. For instance,
Benjamin Robins wrote: ―I have no design to charge this author [Euler] with haste or negligence
on account of these errors; but I consider them solely, as the effect of that inaccuracy in
conception, to which the differential calculus is disposed to betray its admirers ... In the
beginning of the third chapter, which treats of right-lined motion, Mister Euler has given
Galileo‘s theory of falling bodies, in its own nature no difficult subject; but it is here so
compounded with differential computations, that this subject may be much better learned from
what has been writ in a more simple manner by others‖ ( ROBINS, Benjamin, ―Remarks on Mr.
Euler‘s Treatise of Motion‖, in Mathematical Tracts of the late Robins, Benjamin … , edited by
J. Wilson, London, J. Nourse: 1761,, 197-221, on 203 and 205.)