UP | HOME

Halsted’s review of Wentworth’s Geometry
Aug 04, 2020

“\(M = E - \sin E\)” —J. Kepler

From the February 1, 1902 edition of the American Mathematical Monthly, vol 9 no. 2:

Some Fallacies in Wentworth’s Geometry

by Dr. George Bruce Halsted

Often both to teacher and pupil instruction seems satisfactory which to the expert is deliciously absurd.

The American-University-girl’s French in Paris is a field of rich humor. Fashionable restaurants have hired waiters for their ability to understand this strange language, American French.

I went once with a party of Americans to buy gloves at a famous place on the Avenue de la Opera. “How does it happen that you understand our French so well?” was asked the attendant.

“Why, that’s not strange,” she answered, “I am an English-woman!”

Again, a young lady from Texas, a student of French, bought a parasol at the Bon Marché, that store which has the genius to say “We will exchange what you buy, whenever it ceases to please you.” The very next day she took it back, but what she really did say in French to the attendant was “I bought this here tomorrow.” The polite attendant never smiled, but, though he spoke English fluently, answered slowly in French: “If mademoiselle bought it here tomorrow, we are glad to receive it back or exchange it.”

The French words for horses (chevaux) and for hair (cheveux) differ only in the one vowel, and Americans are vowel deaf.

This explains the apparently mysterious statement of the young coiffeur who said, “I begin to learn that American French. When an American lady drives up and asks me to wash her horses, that means she wishes her hair dressed.”

A gushing southern young lady who felt sure of the word ’fiancé’ for ’engaged’, astounded a young hack-man by asking him in French, “Are you betrothed?” and on his answering “no,” said: “Then I take you.” He was so surprised that when asked his fare, always a mistake in Paris, he told the truth, but when it came to pay, asked thrice as much; when the girl exclaimed in American French, “You are more dear to me now than when we were first engaged.”

Our books in elementary mathematics are American French, deliciously absurd to the expert.

Almost the only geometry used is that of my friend the New Hampshire banker, Wentworth, who begins with a mental confusion between a terminated piece of a straight line, which completely satisfies his newest definition for a straight line, and the straight line itself. This confusion is emphasized by his “Postulate: A straight line can be produced.”

If this is what he means, it is a definite magnitude, like an inch or a yard. But then about it his axiom is not true.

Axiom: “Two points determine a straight line.”

On the same page 8, is “§49. Axiom. A straight line is the shortest line that can be drawn from one point to another.”

But as the great Hilbert well says: “Such an axiom is senseless, if one has not defined the concept of length of a curve.”

So it need not surprise us that in attempting to attain to the length of the simplest curve, the circle, Mr. Wentworth falls into a shocking non-sequitur, which blunder is now on page 95 of his revised edition, 1899, and is being duly taught to thousands upon thousands of our young Americans, and by them duly swallowed as mathematics.

On page 9 he defines an angle as an opening [the italics are his].

On page 12, he is evidently standing on his head, since he talks upside down when he says “the magnitude of the angle depends upon the amount of rotation of the line.”

But anyone who even contemplates owning a watch should have realized that it is the angle which determines the amount of rotation.

The demonstrations of even his simplest theorems are fallacious or beg the question.

Thus to prove “In an isosceles triangle the angles opposite the equal sides are equal,” he uses a hypothetical construction on page 36 which he does not justify by any reference for the obvious reason that such reference would be to page 115 of his own book.

In what he gave as proof of the theorem: “Two triangles are equal when the three sides of the one are equal respectively to the three sides of the other,” up to 1886 he said, page 44, “place triangle \(A'B'C'\) in the position \(AB'C\), having its greatest side \(A'C'\) in coineidence with its equal \(AC\), and its vertex at \(B'\), opposite \(B\). Draw \(BB'\) intersecting \(AC\) at \(H\).“

Of course he did not prove this intersection, since it is enormously harder to prove than the whole theorem. But its standing unjustified must have been a source of trouble, since in his later editions it is simply omitted, though as before his proof depends upon it.

His treatment of tangent circles has always been consistently fallacious. Two circles are called tangent when they have only one point in common. It follows then as a theorem that the line of centers passes through the point of contact. Wentworth has this theorem, page 91, but proves it by assuming it in his definition, page 75, §221. “Two circles are tangent to each other, if both are tangent to a straight line at the same point,” which is of course only another form of the corollary: a perpendicular to the center-straight through the point of contact is a common tangent to the two circles.

The rest of the book is equally vulnerable, but, not thrice to slay the slain, we desist.

University of Texas.