# How did Roger Cotes come up with logarithm form of Euler formula?

I have been trying to get my head around how Roger Cotes first discovered Euler Formula.

I knew how Euler did it, but I wanted a new perspective, especially from someone who discovered it earlier.

Cotes used geometry to explain his discovery, but it was very terse (or at least to my preference). I tried to draw the diagram after his explanation, but I still can't understand how he suddenly conclude that $$\ln{\frac{EX+XC\sqrt{-1}}{CE}}=arc length*\sqrt{-1}$$

He used many old keywords such as "measurement of ratio" to represent natural log. How is natural log a "measurement of ratio", and why does taking "measurement of ratio" of $$\cos{\theta}+isin{\theta}$$ equate to arc length of circle intuitively?

• For equidistant points on a logarithmic scale, neighboring points are in identical ratio to each other. – njuffa May 2 at 23:14
• You may find the my answer to HSM question 13004 "Why did John Napier use logos 'ratio', to coin 'logarithm'?" helpful. – Somos May 7 at 0:24