Two Historical Puzzles
1.1 Archimedes' Puzzle
1.1.1 Archimedes' Number This is a story of two intriguing, yet familiar events in the history of mathematics. All of you know from your high school course in Geometry the symbol p for the ratio of the circumference of a circle to its diameter. The letter p is the first letter of the Greek word perijereia which means circle. This symbol p was first introduced in 1706 by the English mathematician Jones. Leonard Euler adopted this notation in 1736 instead of the symbol p he had used previously. Since then p has been used by everyone.
From the very beginning, mathematicians have searched for a value of the number p. Archimedes gave for it the approximate value 22/7*. This fact is so well known that hardly anyone will suspect that it contains a mystery. Who ever asks why Archimedes chose 7 for the denominator of his fraction? What would happen if it were approximated by a fraction with the denominator 8? These questions turn out to be most interesting.
* In his book On the Measurement of the Circle, Archimedes actually presented a different formulation. He determined for p its bounds: 310/71 < p < 31/7. He wrote: The circumference of any circle equals three times its diameter plus an excess which is less than one seventh of the diameter, but larger than 10/71 of it.
1.1.2 Approximation Mathematicians encounter frequently problems of having to replace one object (number, function, figure, etc. ) by another object of the same kind, which is in a sense sufficiently near to, but simpler than the original object. This action called an approximation. In the general case, it requires to select a set of objects and to define the phrase sufficiently close to. We will not discuss the general problem and direct our attention at real numbers.
Consider the set of all real numbers R. They may be very complicated, for example, irrational or fractions with large denominators.
Why is the complicated nature of a fraction assessed by its denominator? (Recall that a fraction is a number p/q where p and q are integers and q ¹ 0, and that Ö 3 and p/2 are not fractions.) If you are mainly interested not in the magnitude of a real number a, but in its arithmetic nature, we need to know its position between two consecutive integers n and n + 1. The addition of an integer to the number a will not change its arithmetic nature (this statement does not apply to that branch of arithmetic which deals with integers!) Fig. 1 shows two numbers a and 3 + a, located identically within the corresponding segments [0,1] and [3,4] (cf. 4.1.2 for a definition of the term segment). For example, the numbers 391/4 = 973/4 and 3/4 are located identically within the corresponding segments [97,98] and [0,1], whence there are no reasons to regard the former as being more complicated than the latter and an analysis of the nature of the numbers within the segment [0,1] would be quite sufficient, the same pattern being reproduced within each segment [n, n + 1]. This is the reason why we are only concerned with the denominator during a study of the complicated nature of fractions.
Consider now a subset of fractions with a given denominator q from the set R of all real numbers. The distance between a number a and a fraction p/q is |a-p/q|. We can now interpret the problem of the approximation of real numbers as follows: In order to approximate a real number a by a fraction with the denominator q which is closest to a among all fractions with that denominator.
If you mark all fractions with the denominator q on the number line, the number a will lie within an interval between two fractions or coincide with one of them. The latter case is trivial, so that you can write
(p - 1)/q < a < p/q.
From these two fractions, you chose the one nearest to as its approximation (Fig. 2).
It can happen that is the central point of the segment [(p - 1)/q, p/q]. In that case, and only in that case, there exist two solutions of the problem. For the sake of definiteness, we shall then adopt the left-end point of the segment as the approximation of a. Hence it is clear that a fraction with any denominator can approximate the number a, that is, the choice of the denominator q is a matter of preference.
You employ an approximation when you want to use a rational instead of an irrational number. You can also use it to replace a rational number by a less complicated one, that is, by a number with a smaller denominator. For example, the approximation to the number 2936/7043 by a fraction with the denominator 12 is
2936/7043 » 5/12,
5/12 < 2936/7043 < 6/12,
where 2936/7043 lies closer to 5/12 than to 6/12.
Approximations of real numbers by decimal fractions have been used for a long time. However, decimal fractions were unknown in Archimedes' time* and he could only employ fractions with arbitrary denominators to approximate p. Why did he prefer fractions with the denominator 7? Was it an accident?
* Decimal fractions became known in Europe at the end of the 16th Century, while they had been used in the Orient since the end of the 15th Century. They were reinvented by the Flemish scientist Simon Stevin. The English writer Jerome K.Jerome commented on them as follows: "From Gent we went to Bruges (where I had the satisfaction of throwing a stone at the statue of Simon Stevin, who added to the miseries of my school days by inventing decimals), and from Bruges we came on here." (Diary of a Pilgrimage, entry for Monday, June 9.)
1.1.3 Error of approximation A real number a is approximated by a fraction p/q with the error
D = a - /q,
where /q denotes that end point of the segment [(p - 1)/q, p/q] which is closest to a.
Thus, the error is the exact value of a minus its approximation, an error is positive if /q = p/q and negative if /q= (p - 1)/q.
The absolute value |D| of the error is called an absolute error.
It is clear now that an absolute error does not exceed 1/2q (cf. Fig. 2):
|D | £ 1/2q.
The number 1/2q is the upper bound of the absolute error. It depends on the choice of the approximation. For example, if it is agreed to approximate the number a by the left-end point of the segment [(p - 1)/q, p/q], then the upper bound would be 1/q.
1.1.4 Quality of approximation
The absolute error approaches its upper bound if a is the central point of the segment [(p - 1)/q, p/q]. This is the most unfavourable case. However, if a is very close to one of the end points, the absolute error may be considerably smaller than the upper bound.
This observation suggests that an evaluation of the quality of an approximation is required. Obviously, the approximation of a number a by a fraction with a small denominator is appropriate, if the error is small; or to be more precise, if the absolute error is substantially less than the upper bound of the error (Fig. 3).
In order to evaluate the quality of an approximation, you estimate the ratio of the actual absolute error to its upper bound:
|absolute error|||a - p/q||
|upper bound of absolute error||1/2q|
It is convenient to employ one half of this ratio, denoted by h and called the normalized error,
h = |qa -p|. (1.1)
Thus, the normalized error h is one half of the ratio of the actual absolute error and the maximum possible error. Obviously,
0 < h £ 1/2.
The quality of an approximation is the higher, the smaller is h.
You call the quantity
l = 1/2h = 1/2|qa -p| (2)
the quality factor. It has a simple and clear meaning: The quality factor l of an approximation is the factor by which the actual absolute error is less than the maximum possible error. Obviously,
1 £ l < ¥,
and the larger is l, the better is the approximation.
It would be wrong to expect fractions with larger denominators to be more useful. It could happen that an approximation of the number a by a fraction with the denominator 8 is less accurate than that by a fraction with the denominator 7. Take a look at the number p, approximated by fractions with denominators 1 to 10 (Table 1). Check these results with your calculator!
This table demonstrates that the approximation of p by a fraction with the denominator 7 is more accurate that those by the other fractions. The actual error is less than its upper bound by a factor of 56.5.
Fig. 4 shows the location of p on the number line. Accidentally and indeed so, p happens to be quite close to 31/7. If you are to approximate p with an absolute error less than or equal to 0.0013, how would you proceed? You eould write down the condition
1/2q £ 0.0013,
whence q ³ 385. Archimedes achieved the same accuracy by means of a much smaller denominator. It is worthwhile to mention here that fractions with the denominator 385 allow to approximate any real number with an error of less than 0.0013, while fractions with the denominator 7 are to be preferred for the approximation of p. Hence Archimedes' proposal cannot have been accidental. However, how did he make his choice?
In 1585, many centuries later, the Dutch scientist Adriaen Antoniszoon (also known as Adriaen Antonisz) from Metz found for p the approximate value 355/113.
His result was published after his death by his son Adriaen Metius, so that the value 355/113 is traditionally referred to as Metius' number. This number has the same striking property as Archimedes' number: The actual error is less than could be expected for the denominator 113. Examine Metius' number in the same way as we have examined Archimedes' number!
There is no doubt that Metius' number was not discovered accidentally. In fact, it was known a long time before Adrian Antoniszoon found it (cf. Struik D.J. A Concise History of Mathematics, 3rd edition, Dover, New York, 1967).
1.2 The Puzzle of Pope Gregory XIII
1.2.1 The Mathematical Problem of the Calendar Pope Gregory was not a mathematician, but his name is associated with an important mathematical problem - the calendar.
Nature has supplied two obvious time units: The year and the day (solar day). You can find in an old text on cosmography:"Unfortunately, the year does not comprise an integral number of days." We can only agree with this complaint, because this fact brings many inconveniences. However, it also generates an interesting mathematical problem:
1 year = 365 days 5 hours 48 minutes 46 seconds = 365.242 199 days*.
* Neither the astronomical aspects of the calendar (such as variations in the length of the year) nor its history will be studied here in detail; we will only concentrate on one mathematical problem linked to the calendar.
It would be very difficult to enact and implement in civil life this duration of the year. In fact, what would happen if the civil year is declared to be exactly 365 days long? Fig. 5 shows Earth's orbit. At midnight, on 1st January, 1985, Earth was at Point A, on 1. January, 1986 , it will be at Point B, and on the next 1. January at Point C, etc. Consequently, if you mark on the orbit the positions of Earth corresponding to a fixed date, this position will not be the same each year, but it will move backwards by nearly 6 hours. In four years, this lag will build up to almost one day, so that the fixed date will gradually recede among the seasons, that is, the first of January will move from Winter to Autumn to Summer to Spring. This would turn out to be most inconvenient, because periodic events like crop sowing or the start of the school year could not be tied to fixed calendar dates.
We know how to remedy the situation. Some years must be given 365 days, and some 366 days, in order to have the average duration of the year as close to the true duration as possible. This approach can approximate the true duration with any prescribed precision, but the required rule of alternation of shorter (ordinary) and longer (leap) years may become undesirably complicated. A compromise must be found: A relatively simple pattern of alternation of ordinary and leap years, which takes the year's average length sufficiently close to its true value.
1.2.2 The Julian and Gregorian Calendars This problem was solved first by Julius Caesar. Or rather by the Alexandrian astronomer Sosigenes, who was called for this purpose to Rome and given the task. The following system was introduced by Caesar: Three successive shorter (ordinary) years followed by one longer (leap) year. Much later, when the Christian chronology had been introduced, it was decided to have leap years when the number of the year was an integral multiple of 4.
This calendar is the Julian one. The average length of the year in it is 365¼ days = 365 days 6 hours, that is, it is 11 minutes 14 seconds longer than the year's true length.
The Julian calendar was improved by Pope Gregory XIII. In fact, calendar reforms had been proposed and worked out long before, but they were never implemented. The Pope enacted his calendar reform in 1582. The alternation of ordinary and leap years was retained with an additional rule: If the number of the year ends with two zeroes, but the number of hundreds is not an integer multiple of 4, the year is treated as ordinary year. For example, this rule classifies the year 1700 as an ordinary year and the year 1600 as a leap year. Moreover, assuming that the error which had been accumulated since the year 1 A.D. was 10 days, the Pope ordered an addition of 10 days to the current date, namely, to consider the day following Thursday, 4th October, 1582 to be Friday, 15th October. More days have been accumulated since then (in 1700, 1800 and 1900). Consequently, at this time (1986), the discrepancy between the Julian and the Gregorian calendars is 13 days.
What is the average length of the Gregorian year? Of 400 years of the Julian calendar, 100 are leap years, while four Gregorian centuries contain only 97 leap years, whence, the average Gregorian year has 36597/400 = 365.242 500 days = 365 days 5 hours 49 minutes 12 seconds, that is, it is 26 seconds longer than the true year's length.
You see that very high precision has been reached by quite simple means. How could this result be achieved? This question is answered in Chapter VI.