1.9. Marks on the straightedge 29 Figure 1.9.2 begins to rise. Continue this until the second mark lies on the x-axis. Draw this line and let C be the point where meets the circle and D the point where meets the x-axis. This produces a picture such as in Figure 1.9.2. Note that from the construction |CD| = |OC| = 1 so that ΔODC is isosceles hence ∠ODC = ∠COD = α radians. Similarly |OA| = |OC| = 1 so that ΔOAC is isosceles and ∠OCA = ∠CAO = β radians. Let γ be the number of radians in ∠AOC. This yields the equations π = α + γ + θ, π = γ + 2β, π = 2α + (π − β). The last equation gives β = 2α and so π = α + (π − 2β) + θ, or 0 = α−2β +θ = α−4α+θ, from which we see that α = θ 3 and we have trisected the angle with θ radians. So what is wrong with the preceding argument? It is not the act of putting marks on the straightedge, even though to mathematicians the idea of physically marking something seems strange. If you prefer, we could have been presented with a standard ruler with inches (or centimeters) marked on it as well as marks for various fractions of length since all rational numbers are constructible. The diﬃculty lies in obtaining the line = CD — there is no guarantee that it is a constructible line and hence the points C and D may not be constructible. Thus even if the angle above with θ radians is constructible, ∠ODC may not be constructible. I must confess here that I have never researched the original documents from the Greeks that detail the trisection problem, but I doubt they ever discussed the concepts of constructible points, lines, and circles. I suspect, however, these notions were intuitive for them. They certainly never dis- cussed the algebra we have presented here in fact, algebra was not so highly developed then. So it is fair to say they did not allow points on their straight- edge. On the other hand, would they have considered the construction above

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