The 8th Book of Tan by Sam Loyd

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Trisecting the angle.

The tri-section of an angle is the most interesting of the unsolved problems of mathematics, because of the extreme simplicity of the proposition: "How can you divide any given angle into three equal parts?"

Here is the way to bisect an angle so as to divide it into two parts: Take any angle formed by the junction of two straight lines, and with the leg of the compasses placed at A mark off the arc BB. Then place the compasses on B and B, describe the arcs C and C, and a straight line drawn from the intersection of these two arcs to A bisects the angle.

Now discover how to trisect the angle by rule without recourse to experimental measurements.

In mechanical practice we should describe the semicircle BB, as shown in Fig. 2, and by trial measurements find approximately correct points for C and D. At the first glance it would seem as if by laying off the three arcs respectively one, two, and three inches from A as shown, that B is exactly three times as large as E. Such actually is the case, but, as the curves of the two circumferences are different, we can find no way to measure that of the smaller upon the greater, as shown by F.

F is the same as the arc E, and if it could be represented by a strip of flexible paper which accommodates itself to the form of the arc BB, we would readily locate the point.

The problem is a fascinating one, which has interested all mathematicians from the time of Archimedes, Pythagoras, Euclid, and Euler to the present day, although a simple one, which can be solved readily by rule and compass alone, and might be guessed by any bright little boy or girl.

The writer is prepared to show that the answer was known to the ancients many thousand years ago, and a simple demonstration of the method of getting a correct answer has been passed upon as correct and absolutely satisfactory by the highest mathematical authority of the present day.

The interest which clings to the problem of trisecting the angle turns upon the fact of such a simple proposition, demolishing the great axiom that a good rule should work both ways.

Any measurement repeated three times furnishes an angle correctly trisected, but if we start from the larger angle first no rule has yet been discovered which will work backwards so as to trisect it.

Take three pieces of Tangrams and place them together, and we find they show the angle of 135 degrees properly trisected; but how is it possible to work the other way so as to divide any given angle into three parts?

The accompanying Tangram illustration shows the four lines converging to one point:

Sam Loyd's Illustration #1

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