This is just non workable mechanics you're suggesting with a point traveling on squared path. It is though true that it travels 4 x 2R in both cases, one being physically impossible to execute / follow in reality. Another thing that is an important difference in cycloid vs square path the point travels : in square path's case the point needs to go 360deg to complete the distance, while in cycloid path point travels 180deg . So it is impossible to compare the two mechanically even though the result is the same value.

I was trying to understand even further the implication of cycloid in movement of an object. And while trying to reason my claims, it turned out that I should take a look at the definition of the meter in order to get to the essence. It shows that we as humans adopted the meter as standard in late 18th century firstly by French republic and later by others internationally, and it was then defined as a

one ten millionth of an arc connecting north Pole and equator. (details here:

https://en.wikipedia.org/wiki/History_of_the_metre#M.C3.A8tre_des_Archives ) . Just note that consequently, the meter as standard exists as portion of the Earth's circumference and it is based on straightening the curve to become the line. Also note that the first measurement of just a portion of this arc was done with then French units "lignes" (

https://en.wikipedia.org/wiki/Ligne , still in use today in watchmaking business).

However, what is really interesting is that the Englishman John Wilkins some 100-150 years earlier had an alternate idea how to achieve an universal measure. I quote :

Wilkins' idea was to choose the length of a "seconds pendulum" (a pendulum with a half-period of one second) as the unit of length: such pendulums had recently been demonstrated by Christiaan Huygens, and their length is quite close to one modern meter (as well as to some other length units which were then in use, such as the yard). However, it was soon discovered that the length of a seconds pendulum varies from place to place: French astronomer Jean Richer had measured the 0.3% difference in length between Cayenne (in French Guiana) and Paris.[6]

.

link:

https://en.wikipedia.org/wiki/Christiaan_Huygens#PendulumsWhat follows is a true gem :

In 1673 Huygens published Horologium Oscillatorium sive de motu pendulorum, his major work on pendulums and horology. It had been observed by Mersenne and others that pendulums are not quite isochronous: their period depends on their width of swing, with wide swings taking slightly longer than narrow swings.[99][100]

Huygens analyzed this problem by finding the curve down which a mass will slide under the influence of gravity in the same amount of time, regardless of its starting point; the so-called tautochrone problem. By geometrical methods which were an early use of calculus, he showed it to be a cycloid, rather than the circular arc of a pendulum's bob, and therefore that pendulums are not isochronous.

. (link:

https://en.wikipedia.org/wiki/Christiaan_Huygens#PendulumsSo, historically we were choosing among 2 different methods to acquire an universal measurement, yet

they are both involving circular paths and curves in definitions to establish their fundamental units. In the prevailing standardization we chose a part of the length of Earth's circumference to be the meter.

Here it would be very brave to make a statement, that it is all connected. I somehow feel like I might be on to something more relevant than I can comprehend at this moment.

Anyway, I turned to Mathis , sending him an email on surveryor's wheel and the measurement of straight path with it....hopefully, he'll answer me and allow it to be published here.