For example: With only ten iterations, a "standard" angle is obtained that is less than one-thousandth of a circle, which would permit the measurement of any other angle to within 1/3 of one degree. Expressed another way, with only ten iterations, 1,024 angles are exactly defined with the difference between them being less than 1/3 of one degree. If that were not enough, twenty iterations would produce almost a million exactly defined angles, with a difference between them so small that it would be "impossible" for practical purposes to discern the difference between neighboring angles.

Yet, angles that are not members of that family are known that are not exactly measurable by even the smallest standard of that family. Whether by accident, or by magic, the radius of a circle divides a circle into six equal parts, each equal to the angle of an equilateral triangle, which cannot be constructed from that first family of angles. Thus, no matter how far such a system were developed, it would not be complete, since it would "skip over"such angles as a 33 1/3 % angle, or one third of a circular angle, or an angle of 120 degrees, and angles obtained by bisections thereof.

It might be interesting to note in passing that the set of "standard angles" obtained by bisections of a circle as indicated above represent a family of angles that can be trisected easily by the succesive members of a second set of angles obtained by bisections of the first trisection of a circle. In other words: Given the first division of a circle into three equal parts, then each successive bisection of one third the circle is equal to one third of each successive bisection of the full circle, as indicated by the table below.

First Family . . Second Family . . Relation

(trisection) . . . 360/3 = 120 . .120=1/3 of 360
 

(then bisections)
360/2 = 180 . . 120/2 = 60 . . . 60 =1/3 of 180
180/2 = 90 . . . 60/2 = 30 . .. . 30 = 1/3 of 90
.90/2 = 45 . . . .30/2 = 15 . .. . 15 = 1/3 of 45
et cetera.

Once the second family of angles is established, there really is no more need for the first one, since those angles can be obtained easily from the second family. In fact, they are part of the second family. Another thing that might be noted is that the tenth iteration (bisection) of the second family permits the measurement of angles to within approximately one-tenth of a degree.

Although a system of direct comparison of angles to a family of angular standards is simple, as so far developed it must be theoretically incomplete, and it still needs a more precise means of assuring the equality of angles.

The more traditional approach to the measurement of angles utilizes linear measures, both; to assure the equality of angles, and to define the relative size of angles as in the trigonometric functions. It might also be noticed that a singular linear measure is sufficient for the first purpose, but at least two linear measures are required for ths second. The use of linear measures appears to be simpler and more precise than curvilinear ones, yet similar problems arise when lengths such as the diagonal and the side of a square are encountered.

Starting with a pair of intersecting lines and the primary reference of their intersection being already established, simply measuring off equal lengths from the vertex of the angles just produces secondary reference points with no indication at all about the relative sizes of any of the angles. Knowing, or measuring, the distance between two of the secondary reference points not on the same line defines or determines or measures a very specific set of angles.

Thus, at least three points are required to measure a single angle, but those same three points also define a triangle with at least three angles, if only the internal angles are considered. In addition to defining a specific triangle, three points can define or determine specific circles such as the circumscribed circle of the triangle, or in the special case in which two sides are equal, three points can determine the reference circle of measure that contains the angle to be measured either as a central angle, or as the special inscribed angle that was previously described as a regular V. Three points could also define other special angles, such as the one that has one side equal to the diameter of the circle, which was previously described as a lopsided V.

It should be noted that specifying the relative measures of just two lines, whether directly, or as a relationship or ratio between them, is insufficient to define a specific angle. Either three lines must be specified, or two lines and their included angle must be specified. For example: If I specify that I measured an angle by measuring a length of one unit from the vertex and, from that point, I measured one unit to the opposite side of the angle, what angle was measured?

If the included angle between those two unit lengths was 60 degrees, then that procedure would specify a measure of a 60 degree angle and the third length would also be equal to one unit. However, if the included angle were 90 degrees, then the angle measured would be 45 degrees, and the third length could be specified as the square root of two. If any other included angle were used, the result would be an isosceles triangle, with the included angle at the vertex and the measured angle being one of its base angles. Without knowing either the included angle or the third length, the base angle could not be determined.

It would seem reasonable to conclude from this that any system of angle measure that is not purely intuitive requires the inclusion of a reference angle of measure, either explicitly or implicitly, as well as a reference unit of measure. The reference angle of measure in traditional trigonometry can be considered to be a right angle, with the reference unit of length being the diameter of a circle.

A different perspective might be helpful here. In geometry there is a theorem that states that an inscribed angle is measured by one-half the intercepted arc of the circle. Since a diameter of a circle divides a circle in half, there are 180 degrees of arc on either side of it. Thus any angle inscribed in a semicircle is equal to 90 degrees.

Furthermore, since the trigonometric functions are based on right triangles, trigonometry can be viewed as a particular application of this geometric theorem in which the reference angle of measure of 90 degrees is "shifting" around the circumference of the circle. The trigonometric functions, then, are a measure of that shift; i.e., the measure of the relative position of the vertex of the reference angle in relation to two fixed points. Those fixed points could represent a straight angle, and/or a specific unit of linear measure that is represented by the hypotenuse, or the diameter, depending on whether attention is focused on the triangle or the circle.

Can I assume that the idea of a "specific unit of linear measure" is sufficiently established, and understood, that it is not necessary to show that it carries with it an implication that there is an elusive "common unit of measure" between any two lengths? I think not. Such an assumption directly contradicts the mathematical "proofs" that some lengths are incommensurable, such as the side and diagonal of a square.
(See Curious Construction 1).

Without getting involved in a deep philosophic discussion at this time, perhaps I can appeal to a common procedure that is used to establish a common "measure" or "base" of comparison that is taught in elementary school.

The idea of "percentage" is a system for expressing two quantities in terms of a standard quantity of 100. The number 100 is chosen as a standard because it is an easy number to manipulate in a base 10 number system.

Given any fraction and expressing that fraction as a decimal is a procedure peculiar to a base ten number system. The formula:
 

can be understood as follows:
Given a part, or portion (the PERCENTAGE) of a whole quantity (the BASE), both of which are measured in terms of the same unit, then the decimal equivalent (the RATE), if multiplied by 100 gives the number of "parts" that would be required if that base were divided into 100 parts. If multiplied by 1000, it would give the number of parts required if the base were divided into a thousand parts.

If a decimal is not multiplied at all, it would give the "number of parts" or "portion" of the "whole" unit, which, however, - if undivided - has no parts. Thus the idea that there exists a part or "portion" of a whole quantity requires that the whole quantity be divisible, even if that whole quantity is represented by the number "1". (unity?) If a whole quantity can be represented by "one", then there must be units less than "1", i.e., fractions, or there can be no parts or "portions" of that "whole".

In other words, simply assigning a number to a quantity can have no effect on its divisibility. Simple fractions like 1/2, 1/3, 2/3, etc., are statements that there is a whole unit "1" that is divisible into the number of parts represented by the denominator, the number of those parts of the whole represented by the fraction is given by the numerator. Since any quantity can be represented either as a whole unit, or as an aggregate of parts, the assignment of the number "1" to that quantity must be understood as merely the establishment of that quantity as the "base" measure in a particular number system as understood in the concept of percentages.

Perhaps the most important idea in this discussion, is the idea that the expression of any fraction requires that the numerator and denominator be expressed in terms of the same unit of measure. If there is no common measure between the numerator and denominator, either the fraction is illegitimate, or it is being used in a very peculiar way.

Given the idea that the numerator represents the part, and the denominator represents the whole, would it be proper to represent 2 feet (the part) of a 4 yard wire (the whole) as 2/4, or 1/2, or 50 percent? That's ridiculous, because the part and the whole are not expressed in the same units, although they could be. Thus 2feet/4yards is an illegitimate fraction, unless one of those dimensions is converted to the units of the other.

One apple divided by three bananas is perhaps a better example of an illegitimate fraction, because there is no known way to divide bananas into apple parts. How about the alchemists' quest of somehow dividing lead into its atomic, or subatomic, parts to produce gold?

What about fractions like 1 divided by the square root of 2 ? It would seem that either there must be some common measure or the fraction must be illegitimate. The fact that such a fraction cannot be evaluated exactly as a decimal seems to be irrelevant, just as much as the fact that 1/3 also results in an unending decimal. The difference between a "repeating" and "nonrepeating" decimal also appears to be irrelevant, since it may be the result of the peculiarities of a base 10 number system, not necessarily an essential part of the entities being represented by that system.

(See Curious Construction 1)