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The ratio of the whole perimeter to the diameter is defined as Pi. This was also given by Euler. Finally the ratio of the altitude of a point to the radius Was defined as the sine, but the ratio of the change in arc length to that radius was 1 : 1 – 1/x where x was very large. This was the basis of the Napieran logarithms. These were logarithms of the then extant and detailed sine tables, this ratio being ver close to sin (Pi/2 – I/x).
Both Cotes and Ruler recognised in Napiers method the ratio or Logos 1: 1 + 1/x where x was very large, was a significant value for the circle. This tended to the limit that Euler denoted by e. both of these were expressed in terms of the binomial theorem to obtain these results, and thus the basis of ny logarithm is the limit to which such a ratio tends as x grows very large under a binomial expansion of the form
( 1 – 1/ x)^x or (1 + 1/x)^x.
It is therefore simple to see that
Pi = 2*i
What is obscure is why i is defined as _/ -1
Euler defined it as the Sqrt of -1 because no one else before him had bothered to define the negative square area. Although Bombelli had utilised the definition, no one understood that a line or rectilineal form above the diameter had to be given a different sign to one below it . Bombelli did, but did not apparently, to my current knowledge specifically identify squares or quadrature in this way. However, he gave the exact rules for using these types of magnitudes, which he apparently claims was a daring gamble on his intuition. I believe Bombelli came to the same conclusion I did after meditation, that if you are going to define negative magnitudes for accounting then you must also define negative magnitudes for geometrical accounting.
Euler specifically defines the quadrature of the unit circle for positive and negative magnitudes of squares. What his contemporaries failed to pick up on was that the Sqrt of -1 is -1 AND +1 while the sqrtnof +1 is -1 OR +1.
