The Teaching of Geometry Part 26

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(For [triangle]_ABA'_ + [triangle]_BC'A'_ + [triangle]_A'C'C_ is the second member of both equations.)

= 1/2_A'C'_ _AP_ + 1/2_A'C'_ _PC_ = 1/2_A'C'_ _AC_ = 1/2(_AC_)^2.

[therefore] (_AB_)^2 + (_BC_)^2 = (_AC_)^2.

The Pythagorean Theorem, as it is generally called, has had other names.

It is not uncommonly called the _pons asinorum_ (see page 174) in France. The Arab writers called it the Figure of the Bride, although the reason for this name is unknown; possibly two being joined in one has something to do with it. It has also been called the Bride's Chair, and the shape of the Euclid figure is not unlike the chair that a slave carries on his back, in which the Eastern bride is sometimes transported to the wedding ceremony. Schopenhauer, the German philosopher, referring to the figure, speaks of it as "a proof walking on stilts," and as "a mouse-trap proof."

An interesting theory suggested by the proposition is that of computing the sides of right triangles so that they shall be represented by rational numbers. Pythagoras seems to have been the first to take up this theory, although such numbers were applied to the right triangle before his time, and Proclus tells us that Plato also contributed to it.

The rule of Pythagoras, put in modern symbols, was as follows:

_n_^2 + ((_n_^2 - 1)/2)^2 = ((_n_^2 + 1)/2)^2,

the sides being _n_, (_n_^2 - 1)/2, and (_n_^2 + 1)/2. If for _n_ we put 3, we have 3, 4, 5. If we take the various odd numbers, we have

_n_ = 1, 3, 5, 7, 9, ,

(_n_^2 - 1)/2 = 0, 4, 12, 24, 40, ,

(_n_^2 + 1)/2 = 1, 5, 13, 25, 41, .

Of course _n_ may be even, giving fractional values. Thus, for _n_ = 2 we have for the three sides, 2, 1 1/2, 2 1/2. Other formulas are also known. Plato's, for example, is as follows:

(2_n_)^2 + (_n_^2 - 1)^2 = (_n_^2 + 1)^2.

If 2_n_ = 2, 4, 6, 8, 10, ,

then _n_^2 - 1 = 0, 3, 8, 15, 24, ,

and _n_^2 + 1 = 2, 5, 10, 17, 26, .

This formula evidently comes from that of Pythagoras by doubling the sides of the squares.[81]

THEOREM. _In any triangle the square of the side opposite an acute angle is equal to the sum of the squares of the other two sides diminished by twice the product of one of those sides by the projection of the other upon that side._

THEOREM. _A similar statement for the obtuse triangle._

These two propositions are usually proved by the help of the Pythagorean Theorem. Some writers, however, actually construct the squares and give a proof similar to the one in that proposition. This plan goes back at least to Gregoire de St. Vincent (1647).

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It should be observed that

_a_^2 = _b_^2 + _c_^2 - 2_b'c_.

If [L]_A_ = 90, then _b'_ = 0, and this becomes

_a_^2 = _b_^2 + _c_^2.

If [L]_A_ is obtuse, then _b'_ pa.s.ses through 0 and becomes negative, and _a_^2 = _b_^2 + _c_^2 + 2_b'c_.

Thus we have three propositions in one.

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At the close of Book IV many geometries give as an exercise, and some give as a regular proposition, the celebrated problem that bears the name of Heron of Alexandria, namely, to compute the area of a triangle in terms of its sides. The result is the important formula

Area = [sqrt](_s_(_s_ - _a_)(_s_ - _b_)(_s_ - _c_)),

where _a_, _b_, and _c_ are the sides, and _s_ is the semiperimeter 1/2(_a_ + _b_ + _c_). As a practical application the cla.s.s may be able to find a triangular piece of land, as here shown, and to measure the sides. If the piece is clear, the result may be checked by measuring the alt.i.tude and applying the formula _a_ = 1/2_bh_.

It may be stated to the cla.s.s that Heron's formula is only a special case of the more general one developed about 640 A.D., by a famous Hindu mathematician, Brahmagupta. This formula gives the area of an inscribed quadrilateral as [sqrt]((_s_ - _a_)(_s_ - _b_)(_s_ - _c_)(_s_ - _d_)), where _a_, _b_, _c_, and _d_ are the sides and _s_ is the semiperimeter. If _d_ = 0, the quadrilateral becomes a triangle and we have Heron's formula.[82]

At the close of Book IV, also, the geometric equivalents of the algebraic formulas for (_a_ + _b_)^2, (_a_ - _b_)^2, and (_a_ + _b_)(_a_ - _b_) are given. The cla.s.s may like to know that Euclid had no algebra and was compelled to prove such relations as these by geometry, while we do it now much more easily by algebraic multiplication.

FOOTNOTES:

[79] See, for example, G. B. Kaye, "The Source of Hindu Mathematics," in the _Journal of the Royal Asiatic Society_, July, 1910.

[80] An interesting j.a.panese proof of this general character may be seen in Y. Mikami, "Mathematical Papers from the Far East," p. 127, Leipzig, 1910.

[81] Special recognition of indebtedness to H. A. Naber's "Das Theorem des Pythagoras" (Haarlem, 1908), Heath's "Euclid," Gow's "History of Greek Mathematics," and Cantor's "Geschichte" is due in connection with the Pythagorean Theorem.

[82] The rule was so ill understood that Bhaskara (twelfth century) said that Brahmagupta was a "blundering devil" for giving it ("Lilavati," -- 172).

CHAPTER XVIII

THE LEADING PROPOSITIONS OF BOOK V

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Book V treats of regular polygons and circles, and includes the computation of the approximate value of [pi]. It opens with a definition of a regular polygon as one that is both equilateral and equiangular.

While in elementary geometry the only regular polygons studied are convex, it is interesting to a cla.s.s to see that there are also regular cross polygons. Indeed, the regular cross pentagon was the badge of the Pythagoreans, as Lucian (_ca._ 100 B.C.) and an unknown commentator on Aristophanes (_ca._ 400 B.C.) tell us. At the vertices of this polygon the Pythagoreans placed the Greek letters signifying "health."

Euclid was not interested in the measure of the circle, and there is nothing in his "Elements" on the value of [pi]. Indeed, he expressly avoided numerical measures of all kinds in his geometry, wis.h.i.+ng the science to be kept distinct from that form of arithmetic known to the Greeks as logistic, or calculation. His Book IV is devoted to the construction of certain regular polygons, and his propositions on this subject are now embodied in Book V as it is usually taught in America.

If we consider Book V as a whole, we are struck by three features. Of these the first is the pure geometry involved, and this is the essential feature to be emphasized. The second is the mensuration of the circle, a relatively unimportant piece of theory in view of the fact that the pupil is not ready for incommensurables, and a feature that imparts no information that the pupil did not find in arithmetic. The third is the somewhat interesting but mathematically unimportant application of the regular polygons to geometric design.

As to the mensuration of the circle it is well for us to take a broad view before coming down to details. There are only four leading propositions necessary for the mensuration of the circle and the determination of the value of [pi]. These are as follows: (1) The inscribing of a regular hexagon, or any other regular polygon of which the side is easily computed in terms of the radius. We may start with a square, for example, but this is not so good as the hexagon because its side is incommensurable with the radius, and its perimeter is not as near the circ.u.mference. (2) The perimeters of similar regular polygons are proportional to their radii, and their areas to the squares of the radii. It is now necessary to state, in the form of a postulate if desired, that the circle is the limit of regular inscribed and circ.u.mscribed polygons as the number of sides increases indefinitely, and hence that (2) holds for circles. (3) The proposition relating to the area of a regular polygon, and the resulting proposition relating to the circle. (4) Given the side of a regular inscribed polygon, to find the side of a regular inscribed polygon of double the number of sides.

It will thus be seen that if we were merely desirous of approximating the value of [pi], and of finding the two formulas _c_ = 2[pi]_r_ and _a_ = [pi]_r_^2, we should need only four propositions in this book upon which to base our work. It is also apparent that even if the incommensurable cases are generally omitted, the notion of _limit_ is needed at this time, and that it must briefly be reviewed before proceeding further.