I am looking at time dilation.

It occurs to me that this Lorentzian transformation has more to do with the rotational attributes of the circle than The limit of light speed. The ratio appears to be the relation between the tangent and the circle, the tangent ratio relative to the circle arc.

Originally posted by author:

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You are looking at Roger Cotes Harmonium Mensuraram. For Millenia scientists have measured the arc after the Babylonians. After Robert Cotes scientists measured the arc after π, but it was Cotes who set the unit which later became known as the Radian.

Cotes found out , under the noses of Newton, Wallis and De Moivre that this one standard unified all measurement.

Of course like Newton he stood on the shoulders of giants to see a little further. When Cotes died Newton expressed genuine and heartfelt regret at what he may have brought to science and astronomy.

This i believe is what he saw.

This i think is something like how he thought

Anyone care to post info about how trigonometric ratios encouraged the development of Fractions? Spherical trig nad Regiomontanus had something to do with it.

In the meantime a ratio called the versine was more significant in navigation and surveying than the the sine, as was the haversine.

The versine was ≡ to 1-cosø and the coversine1+cosø.

In the unit circle these are magnitudes the diameter is sectioned into, therefore the geometric mean of the products is the sine as the product (1-cosø)*(1+cosø) is sin2ø.

This enabled a geometrical tool and a calculation for finding the square roots of values whose factors formed the diameter of a circle.

In addition the surprising omplexity of the trig idenits is revealed especially in the regime of directed numbers. Simply changing the signs and applying the bombelli operatoe lead to varying but relate results depending on the internal relations!

Thus was revealed by Wallis et al the strange an complex relations in measurement.

Today we begin to realise that these complex relations are not artifacts but "revealings" of a richer and more dynamic relationship in "what" we measure, and how we measure.

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