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 compa.s.ses 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 a.s.sumed that the circle is the limit of the regular inscribed and circ.u.mscribed 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 cla.s.s, but it is the experience of teachers that the proof is too difficult for most beginners, and so the a.s.sumption is usually made in the form of an unproved theorem.
The following are some of the leading propositions of this book:
THEOREM. _Two circ.u.mferences have the same ratio as their radii._
This leads to defining the ratio of the circ.u.mference 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 circ.u.mscribe 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 compa.s.ses without quickly learning how to effect these divisions, and that compa.s.ses 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 a.s.sumption 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 a.s.serted. 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 a.s.signed 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.
[Ill.u.s.tration: ROMAN MOSAIC FOUND AT POMPEII]
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 compa.s.ses 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 ill.u.s.tration 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.
[Ill.u.s.tration: MOSAIC FROM DAMASCUS]
[Ill.u.s.tration: MOSAIC FROM AN ANCIENT BYZANTINE CHURCH]
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._
[Ill.u.s.tration: ARABIC PATTERN]
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 circ.u.mscribed 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 circ.u.mscribing as well as inscribing a regular polygon.
PROBLEM. _To find the numerical value of the ratio of the circ.u.mference 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 circ.u.mference exactly in terms of the radius, since c = 2[pi]_r_. If we could find the circ.u.mference 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 circ.u.mference; in other words, we could "rectify the circ.u.mference." 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 compa.s.ses, 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 compa.s.ses 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 circ.u.mference of the earth to the fraction of an inch, and thirty decimal places would give the circ.u.mference of the whole visible universe to a quant.i.ty 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 circ.u.mference 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 Heng (78-139 A.D.). A little later in China, w.a.n.g 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) + ...).