[77] These two and several which follow are from Stark, loc. cit.
[78] The author has a beautiful ivory specimen of the Sixteenth century.
CHAPTER XVII
THE LEADING PROPOSITIONS OF BOOK IV
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 alt.i.tudes 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 cla.s.s and the reasons explained. In general, it is doubtful if the majority of an American cla.s.s in geometry get much out of the incommensurable case. Of course, with a bright cla.s.s 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 alt.i.tude._
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 cla.s.s 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 alt.i.tude._
The best way to introduce this theorem is to cut a parallelogram from paper, and then, with the cla.s.s, 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 alt.i.tude._
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 alt.i.tude.
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 cla.s.s. 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 alt.i.tude. 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 alt.i.tude._
[Ill.u.s.tration]
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._
[Ill.u.s.tration]
This may be proved independently of the preceding proposition by drawing the alt.i.tudes _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 alt.i.tudes.
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 s.p.a.ce to speaking of the history of the theorem, and to certain proofs that may profitably be suggested from time to time to different cla.s.ses 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 a.s.sent 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.