## Answers

The errors in the fallacious geometric proofs are briefly explained as follows:

Theorem 1. An obtuse angle is sometimes equal to a right angle. The mistake lies in the location of point K. When the figure is accurately drawn, K is so far below line DC that, when G and K are joined, the line falls entirely outside the original square ABCD. This renders the proof totally inapplicable.

Theorem 2. Every triangle is isosceles. Again the error is one of construction. F is always outside the triangle and at a point such that, when perpendiculars are drawn from F to sides AB and AC, one perpendicular will intersect one side of the triangle but the other will intersect an extension of the other side. A detailed analysis of this fallacy can be found in Eugene P. Northrop's Riddles in Mathematics (1944), Chapter 6.

Theorem 3. If a quadrilateral ABCD has angle A equal to angle C, and AB equals CD, the quadrilateral is a parallelogram. The proof is correct if X and Y are each on a side of tliĀ« quadrilateral or if both X and Y are on projections of the sides. It fails if one is on a side and the other is on an extension of .a side, as shown in Figure 26. This figure meets the theorem's conditions but obviously is not a parallelogram.

Figure 26

Figure 26

Quadrilateral-theorem counterexample

Theorem 4. Pi equals 2. It is true that as the semicircles are made smaller their radii approach zero as a limit and therefore the wavy line can be made as close to the diameter of the large circle as one pleases. At no step, however, do the semicircles alter their shape. Since they always remain semicircles, no matter how small, their total length always remains pi. The fallacy is an excellent example of the fact that the elements of a converging infinite series may retain properties quite distinct from those of the limit itself.

Theorem 5. Euclid's parallel postulate can be proved by Euclid's other axioms. The proof is valid in showing that one line can be constructed through C that is parallel to AB, but it fails to prove that there is only one such parallel. There are many other methods of constructing a parallel line through C; the proof does not guarantee that all these parallels are the same line. Indeed, in hyperbolic non-Euclidean geometry an infinity of such parallels can be drawn through C, a possibility that can be excluded only by adopting Euclid's fifth postulate or one equivalent to it. Elliptic non-Euclidean geometry, in which no parallel can be drawn through C, is made possible by discarding, along with the fifth postulate, certain other Euclidean assumptions.

BIBLIOGRAPHY

Fallacies in Mathematics. E. A. Maxwell, Cambridge University Press, 1959.

Whitehead's Philosophy of Science. Robert M. Palter. University of Chicago Press, 1960. Contains a thorough discussion of Whitehead's controversy with Einstein.

Lapses in Mathematical Reasoning. V. M. Bradis, V. L. Minkovskii, and A. K. Kharcheva. Pergamon, 1963.

Mistakes in Geometric Proofs. Ya. S. Dubnov. D. C. Heath, 1963.

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