Book IV treats of the area of polygons, and offers a large number of

practical applications. Since the number of applications to the measuring of areas of various kinds of polygons is unlimited, while in the first three books these applications are not so obvious, less effort is made in this chapter to suggest practical problems to the teachers. The survey of the school grounds or of vacant lots in the vicinity offers all the outdoor work that is needed to make Book IV seem very important.

THEOREM. _Two rectangles having equal altitudes are to each other as their bases._

Euclid's statement (Book VI, Proposition 1) was as follows: _Triangles and parallelograms which are under the same height are to one another as their bases_. Our plan of treating the two figures separately is manifestly better from the educational standpoint.

In the modern treatment by limits the proof is divided into two parts: first, for commensurable bases; and second, for incommensurable ones. Of these the second may well be omitted, or merely be read over by the teacher and class and the reasons explained. In general, it is doubtful if the majority of an American class in geometry get much out of the incommensurable case. Of course, with a bright class a teacher may well afford to take it as it is given in the textbook, but the important thing is that the commensurable case should be proved and the incommensurable one recognized.

Euclid's treatment of proportion was so rigorous that no special treatment of the incommensurable was necessary. The French geometer, Legendre, gave a rigorous proof by _reductio ad absurdum_. In America the pupils are hardly ready for these proofs, and so our treatment by limits is less rigorous than these earlier ones.

THEOREM. _The area of a rectangle is equal to the product of its base by its altitude._

The easiest way to introduce this is to mark a rectangle, with commensurable sides, on squared paper, and count up the squares; or, what is more convenient, to draw the rectangle and mark the area off in squares.

It is interesting and valuable to a class to have its attention called to the fact that the perimeter of a rectangle is no criterion as to the area. Thus, if a rectangle has an area of 1 square foot and is only 1/440 of an inch high, the perimeter is over 2 miles. The story of how Indians were induced to sell their land by measuring the perimeter is a very old one. Proclus speaks of travelers who described the size of cities by the perimeters, and of men who cheated others by pretending to give them as much land as they themselves had, when really they made only the perimeters equal. Thucydides estimated the size of Sicily by the time it took to sail round it. Pupils will be interested to know in this connection that of polygons having the same perimeter and the same number of sides, the one having equal sides and equal angles is the greatest, and that of plane figures having the same perimeter, the circle is the greatest. These facts were known to the Greek writers, Zenodorus (_ca._ 150 B.C.) and Proclus (410-485 A.D.).

The surfaces of rectangular solids may now be found, there being an advantage in thus incidentally connecting plane and solid geometry wherever it is natural to do so.

THEOREM. _The area of a parallelogram is equal to the product of its base by its altitude._

The best way to introduce this theorem is to cut a parallelogram from paper, and then, with the class, separate it into two parts by a cut perpendicular to the base. The two parts may then be fitted together to make a rectangle. In particular, if we cut off a triangle from one end and fit it on the other, we have the basis for the proof of the textbooks. The use of squared paper for such a proposition is not wise, since it makes the measurement appear to be merely an approximation. The cutting of the paper is in every way more satisfactory.

THEOREM. _The area of a triangle is equal to half the product of its base by its altitude._

Of course, the Greeks would never have used the wording of either of these two propositions. Euclid, for example, gives this one as follows: _If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle._ As to the parallelogram, he simply says it is equal to a parallelogram of equal base and "in the same parallels," which makes it equal to a rectangle of the same base and the same altitude.

The number of applications of these two theorems is so great that the teacher will not be at a loss to find genuine ones that appeal to the class. Teachers may now introduce pyramids, requiring the areas of the triangular faces to be found.

The Ahmes papyrus (_ca._ 1700 B.C.) gives the area of an isosceles triangle as 1/2 _bs_, where _s_ is one of the equal sides, thus taking _s_ for the altitude. This shows the primitive state of geometry at that time.

THEOREM. _The area of a trapezoid is equal to half the sum of its bases multiplied by the altitude._

An interesting variation of the ordinary proof is made by placing a trapezoid _T'_, congruent to _T_, in the position here shown. The parallelogram formed equals _a_(_b_ + _b'_), and therefore

_T_ = _a_ · (_b_ + _b'_)/2.

The proposition should be discussed for the case _b_ = _b'_, when it reduces to the one about the area of a parallelogram. If _b'_= 0, the trapezoid reduces to a triangle, and _T_ = _a_ · _b_/2.

This proposition is the basis of the theory of land surveying, a piece of land being, for purposes of measurement, divided into trapezoids and triangles, the latter being, as we have seen, a kind of special trapezoid.

The proposition is not in Euclid, but is given by Proclus in the fifth century.

The term "isosceles trapezoid" is used to mean a trapezoid with two opposite sides equal, but not parallel. The area of such a figure was incorrectly given by the Ahmes papyrus as 1/2(_b_ + _b'_)_s_, where _s_ is one of the equal sides. This amounts to taking _s_ = _a_.

The proposition is particularly important in the surveying of an irregular field such as is found in hilly districts. It is customary to consider the field as a polygon, and to draw a meridian line, letting fall perpendiculars upon it from the vertices, thus forming triangles and trapezoids that can easily be measured. An older plan, but one better suited to the use of pupils who may be working only with the tape, is given on page 99.

THEOREM. _The areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles._

This proposition may be omitted as far as its use in plane geometry is concerned, for we can prove the next proposition here given without using it. In solid geometry it is used only in a proposition relating to the volumes of two triangular pyramids having a common trihedral angle, and this is usually omitted. But the theorem is so simple that it takes but little time, and it adds greatly to the student's appreciation of similar triangles. It not only simplifies the next one here given, but teachers can at once deduce the latter from it as a special case by asking to what it reduces if a second angle of one triangle is also equal to a second angle of the other triangle.

It is helpful to give numerical values to the sides of a few triangles having such equal angles, and to find the numerical ratio of the areas.

THEOREM. _The areas of two similar triangles are to each other as the squares on any two corresponding sides._

This may be proved independently of the preceding proposition by drawing the altitudes _p_ and _p'_. Then

[triangle]_ABC_/[triangle]_A'B'C'_ = _cp_/_c'p'_.

But _c_/_c'_ = _p_/_p'_,

by similar triangles.

[therefore] [triangle]_ABC_/[triangle]_A'B'C'_ = _c_^2/_c'_^2,

and so for other sides.

This proof is unnecessarily long, however, because of the introduction of the altitudes.

In this and several other propositions in Book IV occurs the expression "the square _on_ a line." We have, in our departure from Euclid, treated a line either as a geometric figure or as a number (the length of the line), as was the more convenient. Of course if we are speaking of a line, the preferable expression is "square _on_ the line," whereas if we speak of a number, we say "square _of_ the number." In the case of a rectangle of two lines we have come to speak of the "product of the lines," meaning the product of their numerical values. We are therefore not as accurate in our phraseology as Euclid, and we do not pretend to be, for reasons already given. But when it comes to "square _on_ a line" or "square _of_ a line," the former is the one demanding no explanation or apology, and it is even better understood than the latter.

THEOREM. _The areas of two similar polygons are to each other as the squares on any two corresponding sides._

This is a proposition of great importance, and in due time the pupil sees that it applies to circles, with the necessary change of the word "sides" to "lines." It is well to ask a few questions like the following: If one square is twice as high as another, how do the areas compare? If the side of one equilateral triangle is three times as long as that of another, how do the perimeters compare? how do the areas compare? If the area of one square is twenty-five times the area of another square, the side of the first is how many times as long as the side of the second? If a photograph is enlarged so that a tree is four times as high as it was before, what is the ratio of corresponding dimensions? The area of the enlarged photograph is how many times as great as the area of the original?

THEOREM. _The square on the hypotenuse of a right triangle is equivalent to the sum of the squares on the other two sides._

Of all the propositions of geometry this is the most famous and perhaps the most valuable. Trigonometry is based chiefly upon two facts of plane geometry: (1) in similar triangles the corresponding sides are proportional, and (2) this proposition. In mensuration, in general, this proposition enters more often than any others, except those on the measuring of the rectangle and triangle. It is proposed, therefore, to devote considerable space to speaking of the history of the theorem, and to certain proofs that may profitably be suggested from time to time to different classes for the purpose of adding interest to the work.

Proclus, the old Greek commentator on Euclid, has this to say of the history: "If we listen to those who wish to recount ancient history, we may find some of them referring this theorem to Pythagoras and saying that he sacrificed an ox in honor of his discovery. But for my part, while I admire those who first observed the truth of this theorem, I marvel more at the writer of the 'Elements' (Euclid), not only because he made it fast by a most lucid demonstration, but because he compelled assent to the still more general theorem by the irrefragable arguments of science in Book VI. For in that book he proves, generally, that in right triangles the figure on the side subtending the right angle is equal to the similar and similarly placed figures described on the sides about the right angle." Now it appears from this that Proclus, in the fifth century A.D., thought that Pythagoras discovered the proposition in the sixth century B.C., that the usual proof, as given in most of our American textbooks, was due to Euclid, and that the generalized form was also due to the latter. For it should be made known to students that the proposition is true not only for squares, but for any similar figures, such as equilateral triangles, parallelograms, semicircles, and irregular figures, provided they are similarly placed on the three sides of the right triangle.

Besides Proclus, Plutarch testifies to the fact that Pythagoras was the discoverer, saying that "Pythagoras sacrificed an ox on the strength of his proposition as Apollodotus says," but saying that there were two possible propositions to which this refers. This Apollodotus was probably Apollodorus, surnamed Logisticus (the Calculator), whose date is quite uncertain, and who speaks in some verses of a "famous proposition" discovered by Pythagoras, and all tradition makes this the one. Cicero, who comments upon these verses, does not question the discovery, but doubts the story of the sacrifice of the ox. Of other early writers, Diogenes Laertius, whose date is entirely uncertain (perhaps the second century A.D.), and Athenæus (third century A.D.) may be mentioned as attributing the theorem to Pythagoras, while Heron (first century A.D.) says that he gave a rule for forming right triangles with rational integers for the sides, like 3, 4, 5, where 3^2 + 4^2 = 5^2. It should be said, however, that the Pythagorean origin has been doubted, notably in an article by H. Vogt, published in the _Bibliotheca Mathematica_ in 1908 (Vol. IX (3), p. 15), entitled "Die Geometrie des Pythagoras," and by G. Junge, in his work entitled "Wann haben die Griechen das Irrationale entdeckt?" (Halle, 1907). These writers claim that all the authorities attributing the proposition to Pythagoras are centuries later than his time, and are open to grave suspicion. Nevertheless it is hardly possible that such a general tradition, and one so universally accepted, should have arisen without good foundation. The evidence has been carefully studied by Heath in his "Euclid," who concludes with these words: "On the whole, therefore, I see no sufficient reason to question the tradition that, so far as Greek geometry is concerned ..., Pythagoras was the first to introduce the theorem ... and to give a general proof of it." That the fact was known earlier, probably without the general proof, is recognized by all modern writers.

Pythagoras had studied in Egypt and possibly in the East before he established his school at Crotona, in southern Italy. In Egypt, at any rate, he could easily have found that a triangle with the sides 3, 4, 5, is a right triangle, and Vitruvius (first century B.C.) tells us that he taught this fact. The Egyptian _harpedonaptae_ (rope stretchers) stretched ropes about pegs so as to make such a triangle for the purpose of laying out a right angle in their surveying, just as our surveyors do to-day. The great pyramids have an angle of slope such as is given by this triangle. Indeed, a papyrus of the twelfth dynasty, lately discovered at Kahun, in Egypt, refers to four of these triangles, such as 1^2 + (3/4)^2 = (1 1/4)^2. This property seems to have been a matter of common knowledge long before Pythagoras, even as far east as China. He was, therefore, naturally led to attempt to prove the general property which had already been recognized for special cases, and in particular for the isosceles right triangle.

How Pythagoras proved the proposition is not known. It has been thought that he used a proof by proportion, because Proclus says that Euclid gave a new style of proof, and Euclid does not use proportion for this purpose, while the subject, in incomplete form, was highly esteemed by the Pythagoreans. Heath suggests that this is among the possibilities:

[triangles]_ABC_ and _APC_ are similar.

[therefore] _AB_ × _AP_ = (_AC_)^2.

Similarly, _AB_ × _PB_ = (_BC_)^2.

[therefore] _AB_(_AP_ + _PB_) = (_AC_)^2 + (_BC_)^2,

or (_AB_)^2 = (_AC_)^2 + (_BC_)^2.

Others have thought that Pythagoras derived his proof from dissecting a square and showing that the square on the hypotenuse must equal the sum of the squares on the other two sides, in some such manner as this:

Here Fig. 1 is evidently _h_^2 + 4 [triangles].

Fig. 2 is evidently _a_^2 + _b_^2 + 4 [triangles].

[therefore] _h_^2 + 4 [triangles] = _a_^2 + _b_^2 + 4 [triangles], the [triangles] all being congruent.

[therefore] _h_^2 = _a_^2 + _b_^2.

The great Hindu mathematician, Bhaskara (born 1114 A.D.), proceeds in a somewhat similar manner. He draws this figure, but gives no proof. It is evident that he had in mind this relation:

_h_^2 = 4 · _ab_/2 + (_b_ - _a_)^2 = _a_^2 + _b_^2.

A somewhat similar proof can be based upon the following figure:

If the four triangles, 1 + 2 + 3 + 4, are taken away, there remains the square on the hypotenuse. But if we take away the two shaded rectangles, which equal the four triangles, there remain the squares on the two sides. Therefore the square on the hypotenuse must equal the sum of these two squares.

It has long been thought that the truth of the proposition was first observed by seeing the tiles on the floors of ancient temples. If they were arranged as here shown, the proposition would be evident for the special case of an isosceles right triangle.

The Hindus knew the proposition long before Bhaskara, however, and possibly before Pythagoras. It is referred to in the old religious poems of the Brahmans, the "Sulvasutras," but the date of these poems is so uncertain that it is impossible to state that they preceded the sixth century B.C.,[79] in which Pythagoras lived. The "Sulvasutra" of Apastamba has a collection of rules, without proofs, for constructing various figures. Among these is one for constructing right angles by stretching cords of the following lengths: 3, 4, 5; 12, 16, 20; 15, 20, 25 (the two latter being multiples of the first); 5, 12, 13; 15, 36, 39; 8, 15, 17; 12, 35, 37. Whatever the date of these "Sulvasutras," there is no evidence that the Indians had a definite proof of the theorem, even though they, like the early Egyptians, recognized the general fact.

It is always interesting to a class to see more than one proof of a famous theorem, and many teachers find it profitable to ask their pupils to work out proofs that are (to them) original, often suggesting the figure. Two of the best known historic proofs are here given.

The first makes the Pythagorean Theorem a special case of a proposition due to Pappus (fourth century A.D.), relating to any kind of a triangle.

Somewhat simplified, this proposition asserts that if _ABC_ is _any_ kind of triangle, and _MC_, _NC_ are parallelograms on _AC_, _BC_, the opposite sides being produced to meet at _P_; and if _PC_ is produced making _QR_ = _PC_; and if the parallelogram _AT_ is constructed, then _AT_ = _MC_ + _NC_.

For _MC_ = _AP_ = _AR_, having equal bases and equal altitudes.

Similarly, _NC_ = _QT_.

Adding, _MC_ + _NC_ = _AT_.

If, now, _ABC_ is a right triangle, and if _MC_ and _NC_ are squares, it is easy to show that _AT_ is a square, and the proposition reduces to the Pythagorean Theorem.

The Arab writer, Al-Nair[=i]z[=i] (died about 922 A.D.), attributes to Th[=a]bit ben Qurra (826-901 A.D.) a proof substantially as follows:

The four triangles _T_ can be proved congruent. Then if we take from the whole figure _T_ and _T'_, we have left the squares on the two sides of the right angle. If we take away the other two triangles instead, we have left the square on the hypotenuse. Therefore the former is equivalent to the latter.

A proof attributed to the great artist, Leonardo da Vinci (1452-1519), is as follows:

The construction of the following figure is evident. It is easily shown that the four quadrilaterals _ABMX_, _XNCA_, _SBCP_, and _SRQP_ are congruent.

[therefore] _ABMXNCA_ equals _SBCPQRS_ but is not congruent to it, the congruent quadrilaterals being differently arranged.

Subtract the congruent triangles _MXN_, _ABC_, _RAQ_, and the proposition is proved.[80]

The following is an interesting proof of the proposition:

Let _ABC_ be the original triangle, with _AB_ < _BC_. Turn the triangle about _B_, through 90°, until it comes into the position _A'BC'_. Then because it has been turned through 90°, _C'A'P_ will be perpendicular to _AC_. Then

1/2(_AB_)^2 = [triangle]_ABA'_,

and 1/2(_BC'_)^2 = [triangle]_BC'C_,

because _BC_ = _BC'_.

[therefore] 1/2((_AB_)^2 + (_BC_)^2) = [triangle]_ABA'_ + [triangle]_BC'C_.

[therefore] 1/2((_AB_)^2 + (_BC_)^2) = [triangle]_AC'A'_ + [triangle]_A'C'C_

(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).

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'_ passes through 0 and becomes negative, and _a_^2 = _b_^2 + _c_^2 + 2_b'c_.

Thus we have three propositions in one.

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 class 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 altitude and applying the formula _a_ = 1/2_bh_.

It may be stated to the class 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 class 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 Japanese 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).