Also, the fact that the result is in radians makes sense with my description of radians, since sine describes the ratio of the the opposite side of a right triangle to the hypoteneuse. This is an untransformed ratio of two quantities having the same dimension, which is, I believe uncoincidentally, exactly how I described radians. They are merely untransformed ratios which we only most commonly explicitly use when refering to angles. I believe the abstract concept of "radians" is always there for ratios of quantities having the same classification, it's just that we only write them down on paper when working with angles since it's used for disambiguation with common transformations of ratios such as degrees. I know it's strange, but comparing radians to decibels I think is another good way to attempt to explain the standpoint. Decibels and bels and nepers are all dimensionless types which represent the ratio of two values in logarithmic scales. My system can represent them perfectly fine and in a way that I believe makes logical sense. They are related to radians and degrees since all of them are just ways of representing ratios, either transformed or untransformed, just like other quantities having an empty classification, and a conversion exists between them. Since they are just transformations of ratios, expressions such as my_decibel_quantity = length / length should work fine, since the result of the right-hand operation is just a ratio, and decibels are merely transformations of ratios -- exactly like the relationship of two units of another classification such as time, energy, etc. Raw ratios are convertible to decibels since they both have units of empty classification and a defined relationship exists between them. I believe that stating that ratios exist in the same classification as quantities with an empty classification, only with no units attached is a mistake. My stance is that the unit type is not "unitless", but that the ratio has untransformed units in that derived classification. This unit type happens to be analogous to radians only perhaps with a broader definition than most people use. I wonder if the name is all that we are arguing about, so call it what you want -- "untransformed ratio" or something else, as the name is unimportant, but I believe the abstraction definately exists which is why we have different unit types which describe ratios, such as decibels. Using raw values without units associated loses that abstraction. That natural form isn't unitless, it's just that we don't normally attach a name to it. Following with geometry, radians are untransformed ratios (note here I'm purposely stating that radians are untransformed ratios not that untransformed ratios are radians, since I apparently have hit a nerve with some of you). If you don't want to call all ratios radians, a conversion still definately exists between them -- that conversion happens to be an identity conversion, where you leave the value the same, which is why the length of an arc divided by the radius can be converted to radians without changing the value. If you want, then look at the relationship similar to that of kelvin vectors and celsius vectors -- conversion between them is to simply leave the values the same and is implicit. Both concepts have units and can be looked at logically differently, but they are represented exactly the same, and they can be converted between. -- -Matt Calabrese