In Kepler's Equation it does make sense to set Area = x + e*sin(x). This area can be constructed by compass and straightedge as an n-gon, with n -> Infinity. Kepler must have thought of this equation in terms of trig, because modern trig. function theory essentially starts with Euler. As for your question about pi-dimensions, it's difficult to interpret what you are asking for without explicit examples. For some integrals, it is easy to predict, the output pi exponent, see for example [1]. A major preventative difficulty to the dimensional analysis of integrals is well stated in K-Z monograph on periods, Conjecture 1: "If a period has two integral representations, then one can pass from one formula to another using only [additivity, change of variables, and anti- differentiation] in which all functions and domains of integration are algebraic with coefficients in Q." The situation is only worse when considering larger classes of integrable functions. Before a comprehensive dimensional system could be developed for pi integrals, I think that the transformation theory needs more work. This Conjecture 1 (and possibly more) would probably need to be settled. --Brad [1] https://mathoverflow.net/questions/328657/why-these-surprising-proportionali... On Thu, Jan 30, 2020 at 5:14 AM James Propp <jamespropp@gmail.com> wrote:
Trig provides another arena in which there are different types of scalars. Angles and ratios-of-lengths are scalars, but in applications of trig it makes no sense to add an angle to the sine of another angle; angles and ratios are different types of numbers.
Has anyone worked out a framework for this perspective? (I don’t think George treats trig in his book, though it’s possible that he did and I’m forgetting it.)
Such a framework would rewrite the standard power series formula for sin x as sin x = r x / 1! - r^3 x^3 / 3! + ... where r is a type-conversion coefficient.
What would the point of such a framework be (besides an idiosyncratic form of ideological purity)? Well, think of all those formulas in which some integral is shown to equal some rational coefficient times pi to the power of some rational exponent. Wouldn't it be nice to have a “dimensional” way to predict the exponent?
Jim Propp