The X, Y, and V relations that have already been presented to show a relationship of angles to their double and triple angles can be used not only to summarize many geometric relations of intersecting lines, but they can also summarize some relationships of right triangles that can be used as a summary of some interesting trigonometric relations. I have wondered at times whether or not the "right" angle was the "best" angle to use as a standard reference in trigonometry. Perhaps, a trigonometry based on the more "perfect" angle of an equilateral triangle would be better, but how could such a trigonometry be constructed?
I shall attempt to show that the trigonometric relationships of a right triangle are based upon a "shifting" reference when viewed from a geometric perspective. Then, from that geometric perspective, I will show by analogy that trigonometric tables could be constructed using other angles as reference. In addition, I will show that the standard trigonometric unit of linear measure is the diameter of a circle that produces values for some of the trigonometric functions, which are half of what they would be if they were based on the more "normal" standard of a unit circle, that is, on its radius.
Then - again by analogy - it can be shown that other linear measures inherent in a unit circle (such as the side of the inscribed equilateral triangle) could also be used as a linear standard, either by using a different set of trigonometric tables, or by using formulas to convert the values expressed in one linear standard to express them in terms of another - in the same way that shifting between a standard diameter and a standard radius has the effect of doubling, or halving, some of the values.
The net result of the exercise indicated in the preceding paragraphs is to raise the question: What does the relative linear dimensions of lines have to do with the relative angular dimensions of angles? That question can be put into a sharper focus by asking another question. Can a system of angle measure be developed that is independent of linear dimensions? I shall address the second question first by presenting a system of direct comparison of angles.
Since one of the most universal notions of an angle is that it is composed of two intersecting lines, I shall begin with that notion. To view an angle, it is necessary to find a point of reference from which both lines can be viewed at the same time and in such a way that there is some degree of certainty that those two lines do actually intersect. For example: A pair of parrallel lines such as might be represented by railroad tracks appear to intersect from any point of reference from which they are viewed, unless the "view" that is permitted is severely limited to only a short distance. Thus the point of intersection seems to be the best point to consider as a primary reference, since at that point there is the greatest certainty that the lines do in fact intersect.
It may, or may not, be obvious that given a pair of intersecting lines, those intersecting lines establish a "plane" that is most naturally viewed from outside the plane - in the manner of viewing a representation of intersecting lines drawn on the surface of a piece of paper from a point above the plane represented by the paper. It is also possible to represent intersecting lines in a way that the observer is limited to observations that can be made from within the plane, e.g., a traveler on intersecting roads.
Another interesting possibility is to consider that an angle is formed by "bending" a single line. A curve or circle could then be considered as an angle that has a "smooth" bend rather than a "sharp" one. In either case, it can be observed that it is always possible to consider more than one angle. Conversely , it is impossible to construct a single angle; at least one other angle, i.e., its reflex angle, is always present.
In the case of intersecting lines, four associated angles are formed, which can be grouped in different ways. Grouping three of them together would produce the reflex angle of the fourth angle. Grouping them by twos could produce two pairs of adjacent angles that are supplementary, or two pairs of equal vertical angles that alternate with each other.
Actually, there is another pair of equal angles that are present but are being overlooked, or ignored; the two intersecting lines each represent a straight angle. The fact that two intersecting lines can be considered to represent many angles at the same time does not appear to be a particularly useful observation, but straight angles are always present and implied by the notion of supplementary angles.
It seems that I have a knack for making very simple ideas very complicated. What I am trying to do is to get at some of the mystery and the beauty of the thinking processes by pointing out that the focus of attention implicitly includes - and at the same time ignores - many relationships as part of the process of focusing attention on the particular relations relevant to a particular purpose. Such processes that are a "natural" part of the thinking process most frequently go unnoticed, yet they may produce a subtle bias or limitation which also goes unnoticed.
For example: Speaking of a straight line as representing a straight angle implies a measure of angles that has not yet been defined in the present context. Notice that considering a straight line as a "standard" angle such as a straight angle implies several things, one of them being that other angles are measured by how much they deviate from it. Another interesting "bias" is that to conceive of a straight angle, attention is usually focused on only one side of the line while there is another straight angle on the opposite side of the line that is not considered. Thus a straight line can represent not one, but two straight angles.
There are other "problems", too. A straight line does not fit the notions or definitions of angles so far presented, either as an intersection of two lines, or as a bent line. Rather, it takes the position of a standard of comparison against which other angles can be "measured" by how much they deviate from that standard. As such a "standard of comparison" it also represents a "limit" on the size of angles considered; if angles on one side of a line are "x" degrees smaller than a straight angle, then angles on the other side are "x" degrees larger than the standard where "x" represents a supplementary angle that cannot exceed the "limit" of a straight angle.
The idea that a reference can represent a limit of particular items that can be considered in terms of their deviation from intuitive "norms" is interesting in and of itself. Such limits can always be expanded or contracted without having any effect on the fundamental relations. Different limits do not necessarily interfere with each other.
For example: If an angle is considered to be the result of rotating one line about a fixed point in relation to another, then the angular limit could easily be expanded to 360 degrees and beyond - even to an "infinite angle", which is as unlimited as time, or a clock whose hands will move for all eternity. Even then, the "angle of deviation" from a straight line could not exceed a straight angle, though the angle of deviation from the starting point can be considered to do so. This same view of angles as rotating lines that intersect can also establish a smaller limit of a right angle, since that is the maximum angle by which they can be separated.
The idea of a "right" angle can also be established intuitively in several ways that are relatively independent of the notion of a straight angle. In fact, intuitive notions of equality, symmetry, balance, et cetera, are probably parts of the idea of just what an angle is in terms of a deviation from some "norms" that are so basic that they approach the level of instincts.
Out of curiosity, I once performed an "experiment" that consisted merely of hanging a picture crooked enough so that almost everyone noticed it. Some felt obligated not only to comment about it, but also to demand a reason why I refused to correct it. Not being satisfied with my assertion that I liked it that way, a few felt compelled to correct it themselves - even over my objections. That "experiment" probably has more to do with psychology than it does with mathematics, but it does seem to suggest that the sense of "right" angles. of perpendicularity, of "square" corners, of a "balanced" environment, et cetera, are basic intuitive notions from which much mathematical thought has been derived.
Given the characterization, or "definitions" of angles so far developed, there is no definite method of "measuring" angles other than an intuitive sense of equality. However, assuming that a "standard angle" is the rotation of a line about an endpoint until it goes "full circle" to its starting point, and further assuming that a straight line represents one half that rotation, then a "straight angle" is exactly one half the "standard angle", or a 50% angle. If the standard angle is taken to be 360 degrees, then a straight angle is 50% of that standard, or 180 degrees, provided that it can be assumed or "proved" that the "angle" on one side of a straight line is equal to the angle on the other side of that line.
Similarly, a "right" angle would be 25% of a "full circle" angular standard, or 50% of a straight angle standard, provided only that the angles on either side of the perpendicular can be demonstrated, or assumed, to be equal. Such a system of angle measure could be continued indefinitely on the same principles to find successively smaller "standard" angles", since the idea of equal angles is equivalent to finding a balance, or midpoint, which in turn is equivalent to a process of bisections.