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 class 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, wishing 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 circumference. (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 circumscribed 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.

There is, however, a much more worthy interest than the mere mensuration of the circle, namely, the construction of such polygons as can readily be formed by the use of compasses and straightedge alone. The pleasure of constructing such figures and of proving that the construction is correct is of itself sufficient justification for the work. As to the use of such figures in geometric design, some discussion will be offered at the close of this chapter.

The first few propositions include those that lead up to the mensuration of the circle. After they are proved it is assumed that the circle is the limit of the regular inscribed and circumscribed polygons as the number of sides increases indefinitely. This may often be proved with some approach to rigor by a few members of an elementary class, but it is the experience of teachers that the proof is too difficult for most beginners, and so the assumption is usually made in the form of an unproved theorem.

The following are some of the leading propositions of this book:

THEOREM. _Two circumferences have the same ratio as their radii._

This leads to defining the ratio of the circumference to the diameter as [pi]. Although this is a Greek letter, it was not used by the Greeks to represent this ratio. Indeed, it was not until 1706 that an English writer, William Jones, in his "Synopsis Palmariorum Matheseos," used it in this way, it being the initial letter of the Greek word for "periphery." After establishing the properties that _c_ = 2[pi]_r_, and _a_ = [pi]_r_^2, the textbooks follow the Greek custom and proceed to show how to inscribe and circumscribe various regular polygons, the purpose being to use these in computing the approximate numerical value of [pi]. Of these regular polygons two are of special interest, and these will now be considered.

PROBLEM. _To inscribe a regular hexagon in a circle._

That the side of a regular inscribed hexagon equals the radius must have been recognized very early. The common divisions of the circle in ancient art are into four, six, and eight equal parts. No draftsman could have worked with a pair of compasses without quickly learning how to effect these divisions, and that compasses were early used is attested by the specimens of these instruments often seen in museums. There is a tradition that the ancient Babylonians considered the circle of the year as made up of 360 days, whence they took the circle as composed of 360 steps or grades (degrees). This tradition is without historic foundation, however, there being no authority in the inscriptions for this assumption of the 360-division by the Babylonians, who seem rather to have preferred 8, 12, 120, 240, and 480 as their division numbers. The story of 360° in the Babylonian circle seems to start with Achilles Tatius, an Alexandrian grammarian of the second or third century A.D. It is possible, however, that the Babylonians got their favorite number 60 (as in 60 seconds make a minute, 60 minutes make an hour or degree) from the hexagon in a circle (1/6 of 360° = 60°), although the probabilities seem to be that there is no such connection.[83]

The applications of this problem to mensuration are numerous. The fact that we may use for tiles on a floor three regular polygons--the triangle, square, and hexagon--is noteworthy, a fact that Proclus tells us was recognized by Pythagoras. The measurement of the regular hexagon, given one side, may be used in computing sections of hexagonal columns, in finding areas of flower beds, and in other similar cases.

This review of the names of the polygons offers an opportunity to impress their etymology again on the mind. In this case, for example, we have "hexagon" from the Greek words for "six" and "angle."

PROBLEM. _To inscribe a regular decagon in a given circle._

Euclid states the problem thus: _To construct an isosceles triangle having each of the angles at the base double of the remaining one._ This makes each base angle 72° and the vertical angle 36°, the latter being the central angle of a regular decagon,--essentially our present method.

This proposition seems undoubtedly due to the Pythagoreans, as tradition has always asserted. Proclus tells us that Pythagoras discovered "the construction of the cosmic figures," or the five regular polyhedrons, and one of these (the dodecahedron) involves the construction of the regular pentagon.

Iamblichus (_ca._ 325 A.D.) tells us that Hippasus, a Pythagorean, was said to have been drowned for daring to claim credit for the construction of the regular dodecahedron, when by the rules of the brotherhood all credit should have been assigned to Pythagoras.

If a regular polygon of _s_ sides can be inscribed, we may bisect the central angles, and therefore inscribe one of 2_s_ sides, and then of 4_s_ sides, and then of 8_s_ sides, and in general of 2^{_n_}_s_ sides. This includes the case of _s_ = 2 and _n_ = 0, for we can inscribe a regular polygon of two sides, the angles being, by the usual formula, 2(2 - 2)/2 = 0, although, of course, we never think of two equal and coincident lines as forming what we might call a _digon_.

We therefore have the following regular polygons:

From the equilateral triangle, regular polygons of 2^_n_ · 3 sides; From the square, regular polygons of 2^_n_ sides; From the regular pentagon, regular polygons of 2^_n_ · 5 sides; From the regular pentedecagon, regular polygons of 2^_n_ · 15 sides.

This gives us, for successive values of _n_, the following regular polygons of less than 100 sides:

From 2^_n_ · 3, 3, 6, 12, 24, 48, 96; From 2^_n_, 2, 4, 8, 16, 32, 64; From 2^_n_ · 5, 5, 10, 20, 40, 80; From 2^_n_ · 15, 15, 30, 60.

Gauss (1777-1855), a celebrated German mathematician, proved (in 1796) that it is possible also to inscribe a regular polygon of 17 sides, and hence polygons of 2^_n_ · 17 sides, or 17, 34, 68, ..., sides, and also 3 · 17 = 51 and 5 · 17 = 85 sides, by the use of the compasses and straightedge, but the proof is not adapted to elementary geometry. In connection with the study of the regular polygons some interest attaches to the reference to various forms of decorative design. The mosaic floor, parquetry, Gothic windows, and patterns of various kinds often involve the regular figures. If the teacher uses such material, care should be taken to exemplify good art. For example, the equilateral triangle and its relation to the regular hexagon is shown in the picture of an ancient Roman mosaic floor on page 274.[84] In the next illustration some characteristic Moorish mosaic work appears, in which it will be seen that the basal figure is the square, although at first sight this would not seem to be the case.[85] This is followed by a beautiful Byzantine mosaic, the original of which was in five colors of marble. Here it will be seen that the equilateral triangle and the regular hexagon are the basal figures, and a few of the properties of these polygons might be derived from the study of such a design. In the Arabic pattern on page 276 the dodecagon appears as the basis, and the remarkable powers of the Arab designer are shown in the use of symmetry without employing regular figures.

PROBLEM. _Given the side and the radius of a regular inscribed polygon, to find the side of the regular inscribed polygon of double the number of sides._

The object of this proposition is, of course, to prepare the way for finding the perimeter of a polygon of 2_n_ sides, knowing that of _n_ sides. The Greek plan was generally to use both an inscribed and a circumscribed polygon, thus approaching the circle as a limit both from without and within. This is more conclusive from the ultrascientific point of view, but it is, if anything, less conclusive to a beginner, because he does not so readily follow the proof. The plan of using the two polygons was carried out by Archimedes of Syracuse (287-212 B.C.) in his famous method of approximating the value of [pi], although before him Antiphon (fifth century B.C.) had inscribed a square (or equilateral triangle) as a basis for the work, and Bryson (his contemporary) had attacked the problem by circumscribing as well as inscribing a regular polygon.

PROBLEM. _To find the numerical value of the ratio of the circumference of a circle to its diameter._

As already stated, the usual plan of the textbooks is in part the method followed by Archimedes. It is possible to start with any regular polygon of which the side can conveniently be found in terms of the radius. In particular we might begin with an inscribed square instead of a regular hexagon. In this case we should have

_Length of Side_ _Perimeter_

_s__{4} = 1.414... = 1.41 5.66 _s__{8} = [sqrt](2 - [sqrt](4 - 1.414^2)) = 0.72 5.76

and so on.

It is a little easier to start with the hexagon, however, for we are already nearer the circle, and the side and perimeter are both commensurable with the radius. It is not, of course, intended that pupils should make the long numerical calculations. They may be required to compute _s__{12} and possibly _s__{24}, but aside from this they are expected merely to know the process.

If it were possible to find the value of [pi] exactly, we could find the circumference exactly in terms of the radius, since c = 2[pi]_r_. If we could find the circumference exactly, we could find the area exactly, since _a_ = [pi]_r_^2. If we could find the area exactly in this form, [pi] times a square, we should have a rectangle, and it is easy to construct a square equivalent to any rectangle. Therefore, if we could find the value of [pi] exactly, we could construct a square with area equivalent to the area of the circle; in other words, we could "square the circle." We could also, as already stated, construct a straight line equivalent to the circumference; in other words, we could "rectify the circumference." These two problems have attracted the attention of the world for over two thousand years, but on account of their interrelation they are usually spoken of as a single problem, "to square the circle."

Since we can construct [sqrt]_a_ by means of the straightedge and compasses, it would be possible for us to square the circle if we could express [pi] by a finite number of square roots. Conversely, every geometric construction reduces to the intersection of two straight lines, of a straight line and a circle, or of two circles, and is therefore equivalent to a rational operation or to the extracting of a square root. Hence a geometric construction cannot be effected by the straightedge and compasses unless it is equivalent to a series of rational operations or to the extracting of a finite number of square roots. It was proved by a German professor, Lindemann, in 1882, that [pi] cannot be expressed as an algebraic number, that is, as the root of an equation with rational coefficients, and hence it cannot be found by the above operations, and, furthermore, that the solution of this famous problem is impossible by elementary geometry.[86]

It should also be pointed out to the student that for many practical purposes one of the limits of [pi] stated by Archimedes, namely, 3 1/7, is sufficient. For more accurate work 3.1416 is usually a satisfactory approximation. Indeed, the late Professor Newcomb stated that "ten decimal places are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimal places would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope."

Probably the earliest approximation of the value of [pi] was 3. This appears very commonly in antiquity, as in I Kings vii, 23, and 2 Chronicles iv, 2. In the Ahmes papyrus (_ca._ 1700 B.C.) there is a rule for finding the area of the circle, expressed in modern symbols as (8/9)^2_d_^2, which makes [pi] = 256/81 or 3.1604....

Archimedes, using a plan somewhat similar to ours, found that [pi] lay between 3 1/7 and 3 10/71. Ptolemy, the great Greek astronomer, expressed the value as 3 17/120, or 3.14166.... The fact that Ptolemy divided his diameter into 120 units and his circumference into 360 units probably shows, however, the influence of the ancient value 3.

In India an approximate value appears in a certain poem written before the Christian era, but the date is uncertain. About 500 A.D. Aryabhatta (or possibly a later writer of the same name) gave the value 62832/20000, or 3.1416. Brahmagupta, another Hindu (born 598 A.D.), gave [sqrt](10), and this also appears in the writings of the Chinese mathematician Chang Hêng (78-139 A.D.). A little later in China, Wang Fan (229-267) gave 142 ÷ 45, or 3.1555...; and one of his contemporaries, Lui Hui, gave 157 ÷ 50, or 3.14. In the fifth century Ch'ung-chih gave as the limits of [pi], 3.1415927 and 3.1415926, from which he inferred that 22/7 and 355/113 were good approximations, although he does not state how he came to this conclusion.

In the Middle Ages the greatest mathematician of Italy, Leonardo Fibonacci, or Leonardo of Pisa (about 1200 A.D.), found as limits 3.1427... and 3.1410.... About 1600 the Chinese value 355/113 was rediscovered by Adriaen Anthonisz (1527-1607), being published by his son, who is known as Metius (1571-1635), in the year 1625. About the same period the French mathematician Vieta (1540-1603) found the value of [pi] to 9 decimal places, and Adriaen van Rooman (1561-1615) carried it to 17 decimal places, and Ludolph van Ceulen (1540-1610) to 35 decimal places. It was carried to 140 decimal places by Georg Vega (died in 1793), to 200 by Zacharias Dase (died in 1844), to 500 by Richter (died in 1854), and more recently by Shanks to 707 decimal places.

There have been many interesting formulas for [pi], among them being the following:

[pi]/2 = 2/1 · 2/3 · 4/3 · 4/5 · 6/5 · 6/7 · 8/7 · 8/9 · .... (Wallis, 1616-1703)

4/[pi] = 1 + 1/2 + 9/2 + 25/2 + 49/2 + .... (Brouncker, 1620-1684)

[pi]/4 = 1 - 1/3 + 1/5 - 1/7 + .... (Gregory, 1638-1675)

[pi]/6 = [sqrt](1/3) · (1 - 1/(3 · 3) + 1/(3^2 · 5) - 1/(3^3 · 7) + ...).

[pi]/2 = (log _i_) / _i_. (Bernoulli)

[pi]/(2[sqrt](3)) = 1 - 1/5 + 1/7 - 1/11 + 1/13 - 1/17 + 1/19..., thus connecting the primes.

[pi]^2/16 = 1 - 1/2^2 - 1/3^2 + 1/4^2 - 1/5^2 + 1/6^2 - 1/7^2 - 1/8^2 + 1/9^2 + ....

[pi]/2 = _x_/2 + sin _x_ + (sin^2 _x_) / 2 + (sin^3 _x_) / 3 + .... (0 < _x_ < 2[pi])

[pi]/4 = 3/4 + 1/(2 · 3 · 4) - 1/(4 · 5 · 6) + 1/(6 · 7 · 8) - ....

2[pi]^2/3 = 7 - (1/(1 · 3) + 1/(3 · 6) + 1/(6 · 10) + ...).

[pi] = 2^_n_[sqrt](2 - [sqrt](2 + [sqrt](2 + [sqrt](2 + [sqrt](2...))))).

Students of elementary geometry are not prepared to appreciate it, but teachers will be interested in the remarkable formula discovered by Euler (1707-1783), the great Swiss mathematician, namely, 1 + _e_^{_i_[pi]} = 0. In this relation are included the five most interesting quantities in mathematics,--zero, the unit, the base of the so-called Napierian logarithms, _i_ = [sqrt](-1), and [pi]. It was by means of this relation that the transcendence of _e_ was proved by the French mathematician Hermite, and the transcendence of [pi] by the German Lindemann.

There should be introduced at this time, if it has not already been done, the proposition of the lunes of Hippocrates (_ca._ 470 B.C.), who proved a theorem that asserts, in somewhat more general form, that if three semicircles be described on the sides of a right triangle as diameters, as shown, the lunes _L_ + _L'_ are together equivalent to the triangle _T_.

In the use of the circle in design one of the simplest forms suggested by Book V is the trefoil (three-leaf), as here shown, with the necessary construction lines. This is a very common ornament in architecture, both with rounded ends and with the ends slightly pointed.

The trefoil is closely connected with hexagonal designs, since the regular hexagon is formed from the inscribed equilateral triangle by doubling the number of sides. The following are designs that are easily made:

It is not very profitable, because it is manifestly unreal, to measure the parts of such figures, but it offers plenty of practice in numerical work.

In the illustrations of the Gothic windows given in Chapter XV only the square and circle were generally involved. Teachers who feel it necessary or advisable to go outside the regular work of geometry for the purpose of increasing the pupil's interest or of training his hand in the drawing of figures will find plenty of designs given in any pictures of Gothic cathedrals. For example, this picture of the noble window in the choir of Lincoln Cathedral shows the use of the square, hexagon, and pentagon. In the porch of the same cathedral, shown in the next illustration, the architect has made use of the triangle, square, and pentagon in planning his ornamental stonework. It is possible to add to the work in pure geometry some work in the mensuration of the curvilinear figures shown in these designs. This form of mensuration is not of much value, however, since it places before the pupil a problem that he sees at once is fictitious, and that has no human interest.

The designs given on page 283 involve chiefly the square as a basis, but it will be seen from one of the figures that the equilateral triangle and the hexagon also enter. The possibilities of endless variation of a single design are shown in the illustration on page 284, the basis in this case being the square. The variations in the use of the triangle and hexagon have been the object of study of many designers of Gothic windows, and some examples of these forms are shown on page 285. In more simple form this ringing of the changes on elementary figures is shown on page 286. Some teachers have used color work with such designs for the purpose of increasing the interest of their pupils, but the danger of thus using the time with no serious end in view will be apparent.

In the matter of the mensuration of the circle the annexed design has some interest. The figure is not uncommon in decoration, and it is interesting to show, as a matter of pure geometry, that the area of the circle is divided into three equal portions by means of the four interior semicircles.

An important application of the formula _a_ = [pi]_r_^2 is seen in the area of the annulus, or ring, the formula being _a_ = [pi]_r_^2 - [pi]_r'_^2 = [pi](_r_^2 - _r'_^2) = [pi](_r_ + _r'_)(_r_ - _r'_). It is used in finding the area of the cross section of pipes, and this is needed when we wish to compute the volume of the iron used.

Another excellent application is that of finding the area of the surface of a cylinder, there being no reason why such simple cases from solid geometry should not furnish working material for plane geometry, particularly as they have already been met by the pupils in arithmetic.

A little problem that always has some interest for pupils is one that Napoleon is said to have suggested to his staff on his voyage to Egypt: To divide a circle into four equal parts by the use of circles alone.

Here the circles _B_ are tangent to the circle _A_ at the points of division. Furthermore, considering areas, and taking _r_ as the radius of _A_, we have _A_ = [pi]_r_^2, and _B_ = [pi](_r_/2)^2. Hence _B_ = 1/4_A_, or the sum of the areas of the four circles _B_ equals the area of _A_. Hence the four _D'_s must equal the four _C'_s, and _D_ = _C_. The rest of the argument is evident. The problem has some interest to pupils aside from the original question suggested by Napoleon.

At the close of plane geometry teachers may find it helpful to have the class make a list of the propositions that are actually used in proving other propositions, and to have it appear what ones are proved by them. This forms a kind of genealogical tree that serves to fix the parent propositions in mind. Such a work may also be carried on at the close of each book, if desired. It should be understood, however, that certain propositions are used in the exercises, even though they are not referred to in subsequent propositions, so that their omission must not be construed to mean that they are not important.

An exercise of distinctly less value is the classification of the definitions. For example, the classification of polygons or of quadrilaterals, once so popular in textbook making, has generally been abandoned as tending to create or perpetuate unnecessary terms. Such work is therefore not recommended.

FOOTNOTES:

[83] Bosanquet and Sayre, "The Babylonian Astronomy," _Monthly Notices of the Royal Asiatic Society_, Vol. XL, p. 108.

[84] This and the three illustrations following are from Kolb, loc. cit.

[85] This was in five colors of marble.

[86] The proof is too involved to be given here. The writer has set it forth in a chapter on the transcendency of [pi] in a work soon to be published by Professor Young of The University of Chicago.