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The original version of the second law of thermodynamics emerged as a
CONCLUSION and a SECOND CONCLUSION in the Carnot theorem:

FALSE PREMISE: Heat is conserved (cannot be converted into work in the
heat engine).
CONCLUSION: Any two reversible heat engines working between the same
two temperatures T1 and T2 (T2>T1) have the same efficiency.
SECOND CONCLUSION: If T2 = T1, the combination of the two reversible
heat engines, but also any other device, cannot convert, cyclically,
heat into work (perpetuum mobile of the second kind is impossible).

Clausius and Kelvin did not like that fact that the CONCLUSION and the
SECOND CONCLUSION had been validly deduced from a FALSE PREMISE so in
1850 Clausius found it suitable to rededuce the CONCLUSION, this time
from a TRUE PREMISE:

TRUE PREMISE: Heat always flows, spontaneously, from hot to cold.
CONCLUSION: Any two reversible heat engines working between the same
two temperatures T1 and T2 (T2>T1) have the same efficiency.
SECOND CONCLUSION: If T2 = T1, the combination of the two reversible
heat engines, but also any other device, cannot convert, cyclically,
heat into work (perpetuum mobile of the second kind is impossible):

http://web.lemoyne.edu/~giunta/Clausius.html
"Ueber die bewegende Kraft der Warme" 1850 Clausius: "If we now
suppose that there are two substances of which the one can produce
more work than the other by the transfer of a given amount of heat,
or, what comes to the same thing, needs to transfer less heat from A
to B to produce a given quantity of work, we may use these two
substances alternately by producing work with one of them in the above
process. At the end of the operations both bodies are in their
original condition; further, the work produced will have exactly
counterbalanced the work done, and therefore, by our former principle,
the quantity of heat can have neither increased nor diminished. The
only change will occur in the distribution of the heat, since more
heat will be transferred from B to A than from A to B, and so on the
whole heat will be transferred from B to A. By repeating these two
processes alternately it would be possible, without any expenditure of
force or any other change, to transfer as much heat as we please from
a cold to a hot body, and this is not in accord with the other
relations of heat, since it always shows a tendency to equalize
temperature differences and therefore to pass from hotter to colder
bodies."

Clearly, the verification of the second law of thermodynamics could be
LOGICAL: Do the CONCLUSION and the SECOND CONCLUSION follow VALIDLY
from Clausius' TRUE PREMISE? However LOGICAL verification has not been
legitimate in science for a very long time so EXPERIMENTAL
verification is what could be done. One should only notice that, in
both Carnot's analysis and Clausius' analysis, the two reversible heat
engines do not interact, that is, as one of them does work, this does
not change the working force of the other. If they do interact, that
would make the CONCLUSION meaningless but the SECOND CONCLUSION
remains actual and can be tested.

Consider a system of two reversible engines - e.g. the system consists
of two suspended and initially stretched springs. The maximum work
(Wm) the system can do, ISOTHERMALLY, can be presented as

dWm = -F1 dL1  -  F2 dL2               /1/

where F > 0 is the force of contraction and L is the length of the
spring. L1 and L2 are independent variables which, under the
isothermal conditions, fully determine the state of the system. Hence
any state function would be a function of L1 and L2. Now we can ask:
Is Wm a state function or not? If not, this would mean that there
exists a reversible (or close to reversible) cycle undergone by the
system such that, at the end of the cycle, the net work done by the
system is positive, in violation of the second law. Also, since we do
not doubt the first law, this positive net work would have been done
at the expense of heat absorbed from the surroundings, i.e. at the end
of the cycle the net heat absorbed by the system is positive. In
accordance with the definition dS=dQrev/T, this would also mean that
the entropy is not a state function.

All these sad conclusions would be true IF Wm is not a state function,
that is, IF Wm is not a function of L1 and L2. There is a mathematical
theorem stating that, for any function f(x,y), the mixed partial
derivatives d^2f/dxdy  and d^2f/dydx  are EQUAL. If they are not, that
means that f is not a function of x and y. In our case, if Wm is a
state function, the equality of the mixed partial derivatives takes
the form

dF1 / dL2  =  dF2 / dL1                     /2/

(Note that the derivatives are partial - L1 is fixed (constant) on the
left and L2 is fixed on the right.)

For any system in which two forces F1 and F2 can do reversible work,
the partial derivatives analogous to those in /2/ have physical
meanings and are MEASURABLE. This means that thermodynamics possesses
an easy method for testing the validity of the second law and the
"statefunctionness" of the entropy. If the experimentally measured
partial derivatives prove equal, that is, if the equation /2/ is
confirmed, so much the better for the second law and the entropy. If
not (e.g. measurements show that dF1/dL2   is positive whereas dF2/dL1
is negative), the second law is incorrect and the entropy is not a
state function. For instance, this could happen if the springs are
some contractile polymers that interact through some chemical agent.

More on this in a (not well written) paper of mine:

http://www.wbabin.net/valev/valev2.htm

Pentcho Valev