Geometrical Fallacies

"Holmes," I cried, "this is impossible."

"Admirable!" he said. "A most illuminating remark. It is impossible as I state it, and therefore I must in some respect have stated it wrong. . . ."

—Sir Arthur Conan Doyle, The Adventure of the Priory School

It is commonly supposed that Euclid, the ancient Greek geometer, wrote only one book, his classic Elements of Geometry. Actually he wrote at least a dozen, including treatises on music and branches of physics, but only five of his works survived. One of his lost books was a collection of geometric fallacies called Pseudaria. Alas, there are no records of what it contained. It probably discussed illicit proofs that led to absurd theorems but in which the errors were not immediately apparent.

Since Euclid's time hundreds of amusing examples of geometric fallacies have been published, some of them genuine mistakes and some deliberately contrived. This month we consider five of the best. All are theorems that could have been in Euclid's Pseudaria, since none requires more than a knowledge of elementary plane geometry to follow their steps down the garden path to the false conclusion. (Q.E.D.: Quite Entertainingly Deceptive.) The reader is urged to examine each proof carefully, step by step, to see if he can discern exactly where the proof goes wrong before the errors are revealed in the Answer Section.

All triangles are isoceles ABCD is a parallelogram

Theorem 1: An obtuse angle is sometimes equal to a right angle. This was gtte of Lewis Carroll's favorites. Figure 23, left, reproduces Carroll's diagram and labeling. I know of no better way for a high School geometry teacher to convey the importance of deductrve rigor than to chalk this diagram on the blackboard and challenge a class to find where the fallacy lies. The construction and proof are described by Carroll as follows (I quote from The Lewis Carroll Picture Book, edited by Stuart Dodgson Collingwood, London, 1899; reprinted in the Dover paperback Diversions and Digressions of Lewis Carroll, 1961):

Let ABCD be a square. Bisect AB at E, and through E draw EF at right angles to AB, and cutting DC at F. Then DF = FC.

From C draw CG = CB. Join AG, and bisect it at H, and from H draw HK at right angles to AG.

Since AB, AG are not parallel, EF, HK are not parallel. Therefore they will meet, if produced. Produce EF, and let them meet at K. Join KD, KA, KG, and KC.

The triangles KAM, KGH are equal, because AH = HG, HK is common, and the angles at H are right. Therefore KA=KG.

The triangles KDF, KCF are equal, because DF = FC, FK is common, and the angles at F are right. Therefore KD = KC, and angle KDC = angle KCD.

Hence the triangles KDA, KCG have all their sides equal. Therefore the angles KDA, KCG are equal. From these equals take the equal angles KDC, KCD. Therefore the remainders are equal: i.e., the angle GCD = the angle ADC. But GCD is an obtuse angle, and ADC is a right angle.

Therefore an obtuse angle sometimes = a right angle.

Theorem 2: Every triangle is isosceles. This marvelous absurdity is also in The Lewis Carroll Picture Book. Carroll probably came on both proofs in the first (1892) edition of W. W. Rouse Ball's Mathematical Recreations and Essays, where they appeared for the first time. Carroll has explained it so well that again I give his diagram [see Figure 23, middle] and quote his wording:

Let ABC be any triangle. Bisect BC at D, and from D draw DE at right angles to BC. Bisect the angle BAC.

(1) If the bisector does not meet DE, they are parallel. Therefore the bisector is at right angles to BC. Therefore AB = AC, i.e., ABC is isosceles.

(2) If the bisector meets DE, let them meet at F. Join FB, FC, and from F draw FG, FH, at right angels to AC, AB.

Then the triangles AFC, AFH are equal, because they have the side AF common, and the angles FAG, AGF equal to the angles FAH, AHF. Therefore AH = AG, and FH = FG.

Again, the triangles BDF, CDF are equal, because BD=DC, DF is common, and the angles at D are equal. Therefore FB = FC.

Again, the triangles FHB, FGC are right-angled. Therefore the square on FB = the squares on FH, HB; and the square on FC = the squares on FG, GC. But FB = FC, and FH = FG. Therefore the square on HB = the square on GC. Therefore HB = GC. Also, AH has been proved = to AG. Therefore AB = AC; i.e., ABC is isosceles.

Therefore the triangle ABC is always isosceles.

Theorem 3: If a quadrilateral ABCD has angle A equal to angle C, and AB equals CD, the quadrilateral is a parallelogram. P. Halsey of London contributed this subtle fallacy to The Mathematical Gazette, October, 1959, pages 204—205. On the quadrilateral shown in Figure 23 right, draw BX perpendicular to AD, and DY perpendicular to BC. Join BD. Triangles ABX and CYD are congruent, therefore BX equals DY and AX equals CY. It follows that triangles BXD and DYB are congruent, consequently XD equals YB. Since AB equals CD and AD equals BC, the quadrilateral ABCD must be a parallelogram. The proof is strongly convincing, yet the theorem is false. Can the reader provide a counterexample?

Theorem 4: Pi equals 2. Figure 24 is based on the familiar yin-yang symbol of the Orient. Let diameter AB equal 2. Since a circle's circumference is its diameter times pi, the largest semicircle, from A to B, has a length of 2tt/2 = tt. The two next-

Pi equals 2

smallest semicircles, which form the wavy line that divides the yin from the yang, are each equal to tt/2 and so their total length is pi. In similar fashion the sum of the four next-small-est semicircles (each tt/4) also is pi, and the sum of the eight next-smallest semicircles (each ir/8) also is pi. This can be continued endlessly. the semicircles grow smaller and more numerous, but they always add to pi. Clearly the wavy line approaches diameter AB as a limit. Assume that the construction is carried out an infinite number of times. The wavy line must always retain a length of pi, yet when the radii of the semicircles reach their limit of zero, they coincide with diameter AB, which has a length of 2. Consequently pi equals 2.

Theorem 5: Euclid's parallel postulate can be proved by Euclid's other axioms. First, some historical background. Among Euclid's 10 axioms, his fifth postulate states that if a line A crosses two other lines, making the sum of the interior angles on the same side of A less than 180 degrees, the two lines will intersect on that side of A. A variety of seemingly unrelated theorems can be substituted for this axiom since they require it for their proof: The theorem that the interior angles of every triangle add to 180 degrees, or that a rectangle exists, or that similar noncongruent triangles exist, or that through three points not in a straight line only one circle can be drawn, and many others.

Hundreds of attempts have been made since Euclid's time to replace his cumbersome fifth postulate with one that is simpler and more intuitively obvious. The most famous became known as "Playfair's postulate" after the Scottish mathematician and physicist John Playfair. In his popular 1795 edition of Euclid's Elements he substituted for the fifth postulate the equivalent but more succinct statement, "Through a given point can be drawn only one line parallel to a given line." Actually this form of the fifth postulate was suggested by Proclus, in a fifth-cen-tury Greek commentary on Euclid, as well as by later mathematicians who preceded Playfair, but the parallel postulate still bears Playfair's name.

Whatever form the fifth axiom was given, it always seemed less self-evident than Euclid's other axioms, and some of the greatest mathematicians labored to eliminate it entirely by proving it on the basis of the other nine. (For a good account of this history see W. B. Frankland, Theories of Parallelism, an Historical Critique, Cambridge University Press, 1910.) The 18th-century French geometer Joseph Louis Lagrange was convinced that he had produced such a proof by showing (without assuming Euclid's fifth postulate) that the angles of any triangle add to a straight angle. In the middle of the first paragraph of a lecture to the French Academy on his discovery, however, he suddenly said, "II faut que j'y songe encore" ("I shall have to think it over again"), put his papers in his pocket and abruptly left the hall.

More than a century ago it was established that it is as impossible to prove the fifth postulate as it is to trisect the angle, square the circle or duplicate the cube, yet even in this century "proofs" of the parallel axiom continue to be published. A splendid example is the heart of a 310-page book, Euclid or Einstein, privately printed in 1931 by Very Rev. Jeremiah Joseph Callahan, then president of Duquesne University. Since the general theory of relativity assumes the consistency of a non-Euclidean geometry, a simple way to demolish Einstein is to show that non-Euclidean geometry is contradictory. This Father Callahan proceeds to do by a lengthy, ingenious proof of the parallel postulate. It is a pleasant exercise to retrace Father Callahan's reasoning in an effort to find exactly where it goes astray. (For those who give up, the error is exposed by D. R. Ward's "A New Attempt to Prove the Parallel Postulate" in The Mathematical Gazette, Vol. 17, pages 101-104, May, 1933.)

A simple proof of the parallel postulate uses the diagram shown in Figure 25. AB is the given line and C the outside point. From C drop a perpendicular to AB. It can be shown, without invoking the parallel postulate, that only one such perpendicular can be drawn. Through C draw EF perpendicular to CD. Again, the parallel postulate is not needed to prove that this too is a unique line. Lines EF and AB are parallel. Once more, the theorem that two lines, each perpendicular to the same line, are parallel is a theorem that can be established

A proof of the parallel postulate without the parallel postulate, although the proof does require other Euclidean assumptions (such as the one that straight lines are infinite in length) that do not hold in elliptic non-Eu-clidean geometry. Elliptic geometry does not contain parallel lines, but given Euclid's other assumptions one can assume that parallel lines do exist.

We have apparently now proved the parallel postulate. Or have we?

This and hundreds of other false proofs of Euclid's fifth axiom, or axioms equivalent to it, show how easily intuition can be deceived. It helps one to understand why it took so long for geometers to realize that the parallel postulate was independent of the others, that one may assume either that no parallel line can be drawn through the outside point, or that at least two can. (It turns out that if two can, an infinite number can.) In each case a consistent non-Euclidean geometry is constructible.

Even after non-Euclidean geometries were found to be as free of logical contradiction as Euclidean geometry, many eminent mathematicians and scientists could not believe that non-Euclidean geometry would ever have a useful application to the actual space of the universe. It is well known that Henri Poincare argued in 1903 that if physicists ever found empirical evidence suggesting that space was non-Euclidean, it would be better to keep Euclidean geometry and change the physical laws. "Euclidean geometry, therefore," he concluded, "has nothing to fear from fresh experiments." Not so well known is the fact that Bertrand Russell and Alfred North Whitehead once voiced the same view. In 1910, in the famous 11th edition of The Encyclopaedia Britannica, the article on "Geometry, Non-Euclidean" is by Russell and Whitehead. If scientific observation were ever to conflict with Euclidean geometry, they assert, the simplicity of Euclidean geometry is so overwhelming that it would be preferable "to ascribe this anomaly, not to the falsity of Euclidean geometry (as applied to space), but to the falsity of the laws in question. This applies especially to astronomy."

Six years later Einstein's general theory of relativity made this statement, along with Poincare's, hopelessly naive. Not only does non-Euclidean geometry provide a simpler description of the space-time of general relativity; it is even possible that space may close on itself (as it does in Einstein's early model of the universe) to introduce topological properties that are in principle capable of being tested, and that could make the choice of non-Euclidean geometry as the best description of space more than a trivial matter of convention.

Russell was quick to alter the opinion expressed in the Bri-tannica article but Whitehead was slow to get the point. In 1922 he wrote an embarrassing book, The Principle of Relativity, that attacked Einstein's use of a generalized non-Euclidean geometry (in which curvature varies from spot to spot) by arguing that simplicity demands that the geometry applied to space must be either Euclidean (Whitehead's preference) or, if the evidence warrants it, a non-Euclidean geometry in which the curvature is everywhere constant.

What is the moral of all this? Intuition is a powerful tool in mathematics and science but it cannot always be trusted. The structure of the universe, like pure mathematics itself, has a way of being much stranger than even the greatest mathematicians and physicists suspect.

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