Archimedes

Archimedes

Archimedes worked in Syracuse in the third century BCE. His conventional dates are about 287–212 BCE, but the evidence for his mathematical works is stronger than the evidence for a detailed personal biography. Later narratives about his death, a bath, a crown, and siege machines must not be treated as eyewitness accounts merely because they are memorable. Correspondence with mathematicians associated with Alexandria establishes intellectual contact; it does not by itself establish a documented course of study there. MacTutor discusses these different sources.

His works offer several distinct lessons: how to bound a quantity, how to name very large finite numbers, and how a discovery argument differs from a proof. None requires an invented straight line from ancient mechanics to modern computer architecture.

Bounds make an approximation meaningful

In Measurement of a Circle, Archimedes obtained the inequalities now written 223/71 < π < 22/7, using inscribed and circumscribed polygons. The two sides matter together: a plausible decimal value alone does not certify its distance from the true value. The historical account and discussion of his surviving works are in MacTutor.

The following calculation is an original application of that result, not a reconstruction of his geometric proof. Suppose a circular disk has exact radius 10 length units. Its area is 100π, so:

22300/71 < area < 2200/7

The width of this interval is exactly 100/497, about 0.20121 square units. Its midpoint is an approximation with absolute error smaller than 50/497, about 0.10061. The guarantee follows from interval containment; it does not depend on how many decimal digits a calculator displays.

If the measured radius is instead only known to lie between 9.9 and 10.1, the previous interval is unjustified. Because radius and π are positive, valid bounds become (223/71)*9.9² and (22/7)*10.1². Uncertainty in the input must be propagated as well as uncertainty in the constant. A more accurate value of π cannot compensate for an inaccurately measured radius.

Worked mechanism: preserving a certified interval

Let an unknown nonnegative number x satisfy L <= x <= U. The midpoint m = (L+U)/2 has error at most (U-L)/2. This statement follows by considering x <= m and x >= m; in either case the farther endpoint is half the interval width away.

Suppose we are locating sqrt(2). Start with [1,2] because 1² <= 2 <= 2². Squaring is increasing on nonnegative inputs, so testing the midpoint gives a valid update:

Midpoint	Square	New interval
3/2	9/4 > 2	[1,3/2]
5/4	25/16 < 2	[5/4,3/2]
11/8	121/64 < 2	[11/8,3/2]
23/16	529/256 > 2	[11/8,23/16]

After four steps the width is 1/16, so its midpoint has error at most 1/32. Each update preserves containment and halves width. After k steps, the bound is 1/2^(k+1).

This is modern bisection pedagogy inspired by the importance of bounds. It is not the polygon algorithm of Archimedes. Keeping those mechanisms distinct lets us transfer an idea without rewriting history. An implementation using rounded arithmetic needs additional care: a rounded comparison that puts the true value outside the interval destroys the certificate.

Naming large numbers is a separate problem

The Sand Reckoner addresses objections that the number of sand grains is infinite or cannot be named. Archimedes sets assumptions about sizes and constructs orders and periods of numbers. In Heath's presentation, a myriad myriads is B = 10^8; successive orders use powers of this unit. The hierarchy reaches extraordinarily large but finite values. It is not a floating-point format with a fixed mantissa, rounding rules, and a hardware exponent field. Read the primary text in Heath's translation and mathematical paraphrase.

An original miniature illustrates the distinction. With base 100, the expression 7*100² + 3*100 + 9 denotes exactly 70309. Representing it by the coefficients (7,3,9) is exact. Rounding it to 7.03*10^4 gives 70300, losing 9. Large scale and approximate representation are independent choices.

The surviving text also reports Aristarchus's hypotheses. Reporting another astronomer's model is not evidence that Archimedes adopted every premise as his own cosmology.

Reading recovered texts responsibly

The Archimedes Palimpsest preserves material including The Method. Conservation and imaging recovered writing obscured by later reuse. The institutional project report distinguishes imaged material from text successfully recovered; “fully read” is too strong. The SLAC project account describes both discoveries and limitations. A recovery project supplies evidence, not permission to fill gaps with attractive stories.

Exercises

1. Using the π interval above and an exact radius of 3, calculate an area interval, its width, and a midpoint error guarantee.
2. How many bisection steps from [1,2] suffice to make midpoint error at most 1/1000? Prove the count from the width bound rather than trusting printed digits.
3. Represent 120304 in base 100. Contrast the exact representation with rounding to three significant decimal digits, and explain why neither operation by itself demonstrates a historical floating-point system.

Compare your reasoning with the worked solutions

1. Scaling a bound

The area is between 2007/71 and 198/7. Their difference is 9/497, because 22/7 - 223/71 = 1/497. The midpoint therefore has error below 9/994. Multiplying an interval by a positive constant preserves endpoint order; that assumption is why the transformation is valid.

2. Counting refinement steps

After k steps the midpoint error is at most 2^(-k-1). We need 2^(k+1) >= 1000. Since 2^9 = 512 and 2^10 = 1024, k = 9 suffices, while eight steps do not supply this guarantee. A particular midpoint could happen to be better, but the general certificate is what the question asks for.

3. Representation and rounding

120304 = 12*100² + 3*100 + 4, so the base-100 digits are (12,3,4), each between 0 and 99. Three significant decimal digits give 120000, with error 304. An exact positional expansion and an approximate significant-digit encoding are mathematically different. Historical attribution would require evidence of the actual representation, its permitted operations, and its rounding behavior.

Sources

• Archimedes biography and works, University of St Andrews: historical context and circle bounds; its biographical conjectures remain conjectures.
• The Sand Reckoner, T. L. Heath, 1897, public-domain transcription: the opening is translated more literally and later mathematics more freely presented, as the editor explicitly states.
• Archimedes Palimpsest Project, SLAC and project researchers: conservation, imaging, and recovery limitations. The numerical examples and bisection proof here are original modern teaching constructions.