# Tag Archives: Mathematics

# Measuring the Earth this (Southern) Christmas

In around 240 BC, Eratosthenes calculated the circumference of the Earth. The diagram above (from NOAA) shows how he did it. This Christmas, people in the Southern Hemisphere can repeat his work!

Eratosthenes knew that, at the summer solstice, the sun would be directly overhead at Syene (on the Tropic of Cancer) and would shine vertically down a well there. He also knew the distance to Syene.

On 21 December, the sun will be directly overhead on the Tropic of Capricorn at local noon. This table show the time of local noon on 21 December 2017, and the distance to the Tropic of Capricorn, for some Southern Hemisphere cities:

City | Local Noon | Distance to Tropic (km) |
---|---|---|

Adelaide | 13:14 | 1270 |

Auckland | 13:19 | 1490 |

Brisbane | 11:46 | 450 |

Buenos Aires | 12:52 | 1240 |

Darwin | 12:45 | 1220 |

Hobart | 13:09 | 2160 |

Johannesburg | 12:06 | 310 |

Melbourne | 13:18 | 1590 |

Perth | 12:15 | 940 |

Santiago | 13:41 | 1110 |

Sydney | 12:53 | 1160 |

At exactly local noon, Eratosthenes measured the length (*s*) of the shadow of a tall column in his home town of Alexandria. He knew the height (*h*) of the column. He could then calculate the angle between the column and the sun’s rays using (in modern terms) the formula θ = arctan(*s* / *h*).

You can repeat Eratosthenes’ calculation by measuring the length of the shadow of a vertical stick (or anything else you know the height of), and using the **arctan** button on a calculator. Alternatively, the table below show the angles for various shadow lengths of a 1-metre stick. You could also attach a protractor to the top of the stick, run a thread from the to of the stick to the end of the shadow, and measure the angle directly.

The angle (θ) between the stick and the sun’s rays will also be the angle at the centre of the Earth (see the diagram at top). You can then calculate the circumference of the Earth using the distance to the Tropic of Capricorn and the fact that a full circle is 360° (the circumference of the Earth will be *d* × 360 / θ, where *d* is the distance to the Tropic of Capricorn).

Height (h) |
Shadow (s) |
Angle (θ) |
---|---|---|

1 | 0.02 | 1° |

1 | 0.03 | 2° |

1 | 0.05 | 3° |

1 | 0.07 | 4° |

1 | 0.09 | 5° |

1 | 0.11 | 6° |

1 | 0.12 | 7° |

1 | 0.14 | 8° |

1 | 0.16 | 9° |

1 | 0.18 | 10° |

1 | 0.19 | 11° |

1 | 0.21 | 12° |

1 | 0.23 | 13° |

1 | 0.25 | 14° |

1 | 0.27 | 15° |

1 | 0.29 | 16° |

1 | 0.31 | 17° |

1 | 0.32 | 18° |

1 | 0.34 | 19° |

1 | 0.36 | 20° |

1 | 0.38 | 21° |

1 | 0.4 | 22° |

1 | 0.42 | 23° |

1 | 0.45 | 24° |

1 | 0.47 | 25° |

1 | 0.49 | 26° |

1 | 0.51 | 27° |

1 | 0.53 | 28° |

1 | 0.55 | 29° |

1 | 0.58 | 30° |

1 | 0.6 | 31° |

1 | 0.62 | 32° |

1 | 0.65 | 33° |

1 | 0.67 | 34° |

1 | 0.7 | 35° |

1 | 0.73 | 36° |

1 | 0.75 | 37° |

1 | 0.78 | 38° |

1 | 0.81 | 39° |

1 | 0.84 | 40° |

1 | 0.87 | 41° |

1 | 0.9 | 42° |

1 | 0.93 | 43° |

1 | 0.97 | 44° |

1 | 1 | 45° |

# The Plimpton 322 tablet re-examined

Several years back I blogged about the Plimpton 322 tablet – a Babylonian clay tablet from around 1,800 BC. It contains four columns of numbers, written in base 60 (with a small number of errors, as well as some numbers missing through damage – these are corrected below). For example, `1.59:00:15`

= 1 + 59/60 + 0/3600 + 15/216000 = 1.983402777777778.

Column B of the tablet (with a label on the tablet containing the word “width”) is one of the sides of a Pythagorean triangle, and column C (with a label on the tablet containing the word “diagonal”) is the hypotenuse, such that C^{2} − B^{2} is always a perfect square (yellow in the diagram). Column A is exactly equal to C^{2} / (C^{2} − B^{2}), the ratio of blue to yellow.

A | B (“width”) | C (“diagonal”) | D |
---|---|---|---|

1.59:00:15 = 1.983402777777778 | 1:59 = 119 | 2:49 = 169 | #1 |

1.56:56:58:14:50:06:15 = 1.949158552088692 | 56:07 = 3367 | 1:20:25 = 4825 | #2 |

1.55:07:41:15:33:45 = 1.918802126736111 | 1:16:41 = 4601 | 1:50:49 = 6649 | #3 |

1.53:10:29:32:52:16 = 1.886247906721536 | 3:31:49 = 12709 | 5:09:01 = 18541 | #4 |

1.48:54:01:40 = 1.815007716049383 | 1:05 = 65 | 1:37 = 97 | #5 |

1.47:06:41:40 = 1.785192901234568 | 5:19 = 319 | 8:01 = 481 | #6 |

1.43:11:56:28:26:40 = 1.719983676268861 | 38:11 = 2291 | 59:01 = 3541 | #7 |

1.41:33:45:14:03:45 = 1.692709418402778 | 13:19 = 799 | 20:49 = 1249 | #8 |

1.38:33:36:36 = 1.642669444444444 | 8:01 = 481 | 12:49 = 769 | #9 |

1.35:10:02:28:27:24:26:40 = 1.586122566110349 | 1:22:41 = 4961 | 2:16:01 = 8161 | #10 |

1.33:45 = 1.5625 | 45 | 1:15 = 75 | #11 |

1.29:21:54:02:15 = 1.489416840277778 | 27:59 = 1679 | 48:49 = 2929 | #12 |

1.27:00:03:45 = 1.450017361111111 | 2:41 = 161 | 4:49 = 289 | #13 |

1.25:48:51:35:06:40 = 1.430238820301783 | 29:31 = 1771 | 53:49 = 3229 | #14 |

1.23:13:46:40 = 1.38716049382716 | 28 | 53 | #15 |

What is this table all about? A good discussion is by Eleanor Robson [“Words and pictures: new light on Plimpton 322,” *American Mathematical Monthly*, **109** (2): 105–120]. Robson sees Plimpton 322 as fitting into standard Babylonian mathematics, and interprets it as a teacher’s effort to produce a list of class problems.

Specifically, Robson believes that the table was generated by taking values of *x* (in descending order of *x*) from standard Babylonian reciprocal tables (specifically the values 2:24, 2:22:13:20, 2:20:37:30, 2:18:53:20, 2:15, 2:13:20, 2:09:36, 2:08, 2:05, 2:01:30, 2, 1:55:12, 1:52:30, 1:51:06:40, and 1:48) and then using the relationship (*x* − 1 / *x*)^{2} + 2^{2} = (*x* + 1 / *x*)^{2} to generate Pythagorean triples. If we let *y* = (*x* − 1 / *x*) / 2 and *z* = (*x* + 1 / *x*) / 2, then B and C are multiples of *y* and *z*, and A = *z*^{2} / (*z*^{2} − *y*^{2}).

Just recently, Daniel F. Mansfield and N. J. Wildberger [“Plimpton 322 is Babylonian exact sexagesimal trigonometry,” *Historia Mathematica*, online 24 August 2017] interpret the table as proto-trigonometry. I find their explanation of the first column (“a related squared ratio which can be used as an index”) unconvincing, though. Why such a complex index? Robson calls such trigonometric interpretations “conceptually anachronistic,” and points out that there is no other evidence of the Babylonians doing trigonometry.

Mansfield and Wildberger also suggest that “the numbers on P322 are just too big to allow students to reasonably obtain the square roots of the quantities required.” However, I don’t think that’s true. The Babylonians loved to calculate. Using the standard square-root algorithm, even simplistic starting guesses for the square roots of the numbers in column A give convergence in 2 or 3 steps every time. For example, to get the square root of `1.59:00:15`

(1.983402777777778), I start with `1.30:00:00`

(1.5) as a guess. That gives `1.24:40:05`

as the next iteration, then `1.24:30:01`

, and then `1.24:30:00`

(1.408333333333333), which is the exact answer. That said, however, calculating those square roots was not actually necessary for the class problems envisaged by Robson.

Sadly, I do not think that Mansfield and Wildberger have made their case. Robson is, I believe, still correct on the meaning of this tablet.

# A History of Science in 12 Books

Here are twelve influential books covering the history of science and mathematics. All of them have changed the world in some way:

**1**: Euclid’s

*Elements*(c. 300 BC). Possibly the most influential mathematics book ever written, and used as a textbook for more than 2,000 years.

**2**:

*De rerum natura*by Lucretius (c. 50 BC). An Epicurean, atomistic view of the universe, expressed as a lengthy poem.

**3**: The Vienna

*Dioscurides*(c. 510 AD). Based on earlier Greek works, this illustrated guide to botany continued to have an influence for centuries after it was written.

**5**: Galileo’s

*Dialogue Concerning the Two Chief World Systems*(1632). The brilliant sales pitch for the idea that the Earth goes around the Sun.

**8**: Beilstein’s

*Handbook of Organic Chemistry*(1881). Still (revised, in digital form) the definitive reference work in organic chemistry.

**9**:

*Relativity: The Special and the General Theory*by Albert Einstein (1916). An explanation of relativity by the man himself.

**10**:

*Éléments de mathématique*by “Nicolas Bourbaki” (1935 onwards). A reworking of mathematics which gave us words like “injective.”

**11**:

*Algorithms + Data Structures = Programs*by Niklaus Wirth (1976). One of the early influential books on structured programming.

**12**:

*Introduction to VLSI Systems*by Carver Mead and Lynn Conway (1980). The book which revolutionised silicon chip design.

That’s four books of biology, four of other science, two of mathematics, and two of modern IT. I welcome any suggestions for other books I should have included.

# 350 years ago on Friday

On 26 May 1667, Abraham de Moivre was born. This French mathematician gave us, *inter alia*, the formula named after him:

De Moivre was born to French Protestant parents. When the Edict of Nantes was revoked, he was imprisoned for his beliefs for several years, after which he was allowed to leave for England. De Moivre made important contributions to probability theory, and was a pioneer of analytic geometry. Sadly, he was unable to get a university position in England, and he died in poverty.

# Mathematics of the Harp

After my second harp post, I thought I’d keep going with some mathematics. In particular I want to answer the question: why does a harp have that shape?

The physics of vibrating strings gives us Mersenne’s laws, which tell us that the frequency of a string of length *L* is (1 / 2*L*) √ *T* / *μ* , where *T* is the tension force on the string (in newtons), and *μ* is the density per unit length (in kg per metre).

For a string of diameter *d* and density *ρ*, we can calculate *μ* = *A* *ρ*, where *A* = *π* (*d* / 2)^{2} is the cross-sectional area. Nylon has a density *ρ* of 1150 kg/m^{3}. For a nylon string of 1 mm diameter, we get *μ* = 0.0009 kg/m. Putting 0.448 m (17.6 inches) of that string under a tension of 140 newtons (31.5 pounds force), we get a frequency of (1 / 0.896) √ 140 / 0.0009 = 440 Hz. That is, the string plays the note A.

The important thing here is that the frequency is inversely proportional to the length. Over an octave the frequency doubles, which means that the string length halves. A 36-string harp covers 5 octaves, therefore if all the strings were made of the same material under the same tension, the longest string would be 32 times the length of the shortest. Doing some calculations in R for strings of diameter 0.8 mm under a tension of 140 newtons (31.5 pounds), we would get the following harp, which has strings ranging from 7 to 224 cm in length (note that the strings run from A to A, and the C strings are red):

You can see quite clearly that, starting at the treble end, there is an exponential growth in string length. That makes for a terribly unwieldy instrument, and creates all sorts of problems in playing. In practice, we make bass strings thicker (and usually of heavier material) and we vary the tension as well – although, to make life easier for the harpist, we want the properties of the strings to vary reasonably smoothly. If we re-do our calculations with string diameters varying linearly from 3 mm to 0.6 mm, and tension varying linearly from 210 newtons at the bass end to 60 newtons at the treble end, we get a much more realistic-looking harp:

We can flatten out the curve at the top a little by changing the way we vary the strings (after all, a guitar manages to span 2 octaves with all the strings being the same length). However, we cannot eliminate that curve completely – it is the inevitable result of spanning so many octaves, combined with the mathematics of exponential growth.

**Left**: an exponential curve (red) and a similar polynomial (dashed).**Right**: the quotient of the two functions (green) compared to a straight line (grey). Click images to zoom.Mathematically speaking, the modifications to the strings have the effect of dividing an exponential function by some kind of polynomial (as shown above). Over a short range of *x* values, we can find a polynomial that fits the exponential well, and gives us strings of the same length. Over a wide range of *x* values, however, the exponential wins out. Furthermore, exponential growth is initially slow (sub-linear), so that (starting at the right of the harp), growth in string length is slower than the linear shift provided by the sloping base, which means that the top of the harp curves down. After a few octaves, growth in string length speeds up, and so the top of the harp curves up again.

A similar situation arises with the strings of a piano, although these are usually hidden from view:

And to finish, here is one of my favourite classical harpists in action: