Understanding the Carnot cycle

[Omar Nagib]

This P-V diagram summarizes the Carnot cycle. Given a working fluid whose state $(P,V,T)$ is $(P_1,V_1,T_1)$ (at point 1) where $PV=nRT$, the working fluid undergoes a cycle of four stages and again retrieve its original state of $(P_1,V_1,T_1)$, that is it gets back again to point 1 again.

What puzzles me though is Carnot cycle as was originally formulated by Carnot himself.
This figure is from Carnot's original work Reflections on the motive power of fire, where he introduces Carnot cycle and Carnot engine for the first time(page 63).

So the cycle proceeds according to him like this:

1)The working fluid initially is confined in the region $abcd$, with temperature $T_1$, the same as that of the hot reservoir A. The hot reservoir comes into contact with the fluid, and the fluid undergoes isothermal expansion, therefore the piston moves from $cd$ to $ef$.

2)Then the reservoir is removed. And the fluid is left to undergo adiabatic expansion from $ef$ to $gh$, where its temperature drops to $T_2$.

3)The cold reservoir(body B) has temperature $T_2$. At this stage the fluid is put into contact with the cold reservoir, and made to compress isothermally. So the piston moves from $gh$ to $cd$ at constant temperature $T_2$.

4)The cold reservoir is then removed, and the gas is made to compress adiabatically from $cd$ to $ik$, and its temperature increases from $T_2$ to $T_1$.

5)Then the fluid is made to contact the hot reservoir and expands from $ik$ to $ef$ at constant temperature $T_1$

6) repeat steps 3,4,5,6,3,4,5,6 and so on.

He says such a cycle produces the maximum work.

My question:

The Carnot cycle in the P-V diagram proceeds like this :

original state$(P_1,V_1,T_1)$-->isothermal expansion-->adiabatic expansion-->isothermal compression-->adiabatic compression-->original state $(P_1,V_1,T_1)$ again.

What Carnot stated is this:

original state $(P_1,V_1,T_1)$-->isothermal expansion-->adiabatic expansion-->isothermal compression-->adiabatic compression--> isothermal expansion again(from $ik$)--> original state $(P_1,V_1,T_1)$ again.

I don't understand why he added another isothermal expansion. The one that starts from $ik$ to $ef$?

According to my understanding of the Carnot cycle, to match the diagram, he should've done the follwoing:

At the start of step 3, the fluid has temperature $T_2$ and is at $gh$, it should be isothermally compressed and before reaching $cd$ it should be made to adiabatically compress from $T_2$ to $T_1$ from some position that is between $gh$ and $cd$ to position $cd$, so that it gets back to its original state.

 

[Simon Bridge]

Looks like Carnot was contemplating an experimental setup ... he needed another step to regain the initial state.
The PV-diagram version (you should draw the PV diagram for Carnot's example) is the ideal situation.
Just because a process is named for someone does not mean the modern version agrees with what they thought.