Ancient Wisdom and Natural Philosophers

Abstract

In 1881, the Assyrian-British archaeologist Hormuzd Rassam (1826–1910) discovered a clay tablet covered with cuneiform inscriptions in the ruins of the ancient city Sippar, some 30 km southwest of Baghdad, the capital of Iraq (then part of the Ottoman Empire).

人法地.

地法天.

天法道.

道法自然.

He models himself on Earth,

Earth on Heaven,

Heaven on the Tao,

The Tao models itself.

On Nature,

On the So-of-Itself.

From Tao Te Ching, Chapter 25, by Lao Tzu (571–471 BC), an ancient Chinese philosopher.

Translated by John Minford (b. 1946).

Appendices

Appendix 1.1: The Catapult of Thales

Here we have recreated a catapult design based on the knowledge in geometry and technical craftsmanship in Thales’ time, as depicted in Fig. 1.3. The main purpose of the catapult (diagram’s right-hand side) is to throw a projectile (e.g. an iron cannonball) on the platform over a castle’s rampart of height h (diagram’s left-hand side) along a parabolic trajectory. The principal mechanism of the catapult is a lever installed on the platform.

On the left of the lever’s fulcrum, there is a rock of mass M, whose center of mass is located l from the fulcrum. To the right of the fulcrum is an iron cannonball of mass m used to bombard the wall, with its center of mass at a distance of nl from the fulcrum. When preparing for an attack, the projectile end of the lever is pulled down to a position where it forms an angle of $\beta_{1}$ with the vertical line. When an attack command is issued, a soldier will immediately let go of the tight drawstring and allow the lever to rotate quickly via the moment of force generated by the rock’s weight. The projectile at the other end of the lever will accelerate along with a circular motion of radius nl, until the rock at the other end hits the stopping plate on the platform. The projectile will then be hurled into the air upon the sudden cessation of the lever’s rotational movement, at which time the lever forms an angle of $\beta_{2}$ with the vertical line and the speed at which the iron cannonball is projected along the circumferential tangent line is v.

$$U\left( \beta \right) = mg \cdot nl\cos \beta - Mg \cdot l\cos \beta$$

(1.13)

Therefore, when the lever changes from its attack position, i.e., being at an angle of $\beta_{1}$ with the vertical line, to the instant when the iron cannonball is released, i.e., being at an angle of $\beta_{2}$ with the vertical line, the gravitational potential energy of the lever will experience the following changes:

$$U\left( {\beta_{1} } \right) - U\left( {\beta_{2} } \right) = \left( {M - nm} \right)\left( {\cos \beta_{2} - \cos \beta_{1} } \right)gl\cos \beta$$

(1.14)

From the above equations, we can see that the gravitational potential is reduced in the process. Ignoring the loss in potential energy due to the friction of the overall lever mechanism, the change in quantity of the potential energy in (1.14) is converted entirely into kinetic energy. From this energy conservation relationship, we can deduce the catapult’s attack position (location from the rampart) and other required conditions. The derivations are as follows.

$$E_{K} = \frac{1}{2}M\left( {l\omega } \right)^{2} + \frac{1}{2}m\left( {nl\omega } \right)^{2} = \frac{1}{2}\left( {M + n^{2} m} \right)l^{2} \omega^{2}$$

(1.15)

$$\omega = \sqrt {\frac{{2\left( {M - nm} \right)\left( {\cos \beta_{2} - \cos \beta_{1} } \right)g}}{{\left( {M + n^{2} m} \right)l}}}$$

(1.16)

$$v = n\sqrt {\frac{{2\left( {M - nm} \right)\left( {\cos \beta_{2} - \cos \beta_{1} } \right)gl}}{{M + n^{2} m}}}$$

(1.17)

$$\alpha = \beta_{2}$$

(1.18)

$$\Delta x = v_{x} t$$

(1.19)

$$\Delta y = v_{y} t - \frac{1}{2}gt^{2}$$

(1.20)

Substituting the relevant physical quantities into (1.19) and (1.20) and eliminating the time variable t:

$$\left( {D + nl\sin \beta_{2} } \right)\tan \beta_{2} - \frac{1}{2}g\left( {\frac{{D + nl\sin \beta_{2} }}{{v\cos \beta_{2} }}} \right)^{2} = H - L - nl\cos \beta_{2}$$

(1.21)

Substituting (1.16) into (1.21), we can derive the relationship between $\beta_{1}$ and $\beta_{2}$:

$$\cos \beta_{1} = \cos \beta_{2} - \frac{1}{{4n^{2} {\text{co}}s^{2} \beta_{2} }}\frac{{M + n^{2} m}}{M - nm}\frac{{\left( {D + nl\sin \beta_{2} } \right)^{2} }}{{l\left[ {\left( {D + nl\sin \beta_{2} } \right)\tan \beta_{2} - H + L + nl\cos \beta_{2} } \right]}}$$

(1.22)

Now, let us consider a siege that would have likely occurred at Thales’ time. The relevant parameters are as follows: $M = 10$ metric tons, $m = 100\,{\text{kg}}$, $L = 10\,{\text{m}}$, $l = 2\, {\text{m}}$, $\beta_{1} = 180^{{^\circ }}$ (the projectile end of the lever lowered to the fullest extent possible; M at the highest level), $\beta_{2} = 45^{{^\circ }}$, $n = 6$. We substitute these numbers into (1.22) to obtain the relationship between D and H, which specifies how far the catapult should be positioned to be able to throw projectiles over a wall of a specific height. Simplifying (1.23) further we have

$$\begin{aligned} D & = \xi \tan \beta_{2} + \sqrt {\xi^{2} \tan^{2} \beta_{2} + 2\left( { - H + L + nl\cos \beta_{2} } \right)\xi } - nl\sin \beta_{2} \\ & = 76.46 + \sqrt {10357 - 169.9H} \\ \end{aligned}$$

(1.23)

The units of the quantities D and H above are in meter (m), and the parameter $\xi$ satisfies the following:

$$\xi = 2n^{2} {\text{co}}s^{2} \beta_{2} \left( {\cos \beta_{2} - \cos \beta_{1} } \right)\frac{M - nm}{{M + n^{2} m}}l \approx 84.95{\text{m}}$$

(1.24)

If the height of the rampart is $H = 20\;{\text{m}}$, then the catapult should be placed at a distance of $D = 160\,{\text{m}}$. Since the expression inside the square root sign of (1.24) is a non-negative real number, we have $10357 - 169.9H \ge 0$. This means that the maximum height of the rampart the catapult is capable of besieging cannot exceed $\frac{10357}{169.9} \approx 61\,{\text{m}}$ and the corresponding distance of projection is 76 m.

Note that we set $\beta_{1} = 180{{^\circ }}$ above to make (1.24) more tractable, although this would create issues in real-world designs. The interested reader may assume an arbitrary value for $\beta_{1}$ and repeat the derivation process. The equations also reveal a design limitation: the catapult’s position from the rampart cannot exceed 160 m.

Appendix 1.2: Proofs of the Pythagorean Theorem

There are a number of different ways to prove the Pythagorean theorem. Apart from the proof provided by Euclid in his Elements, the 12th U.S. President James A. Garfield also published a proof when he was a U.S. Representative (1876). There is also one that uses the principle of similar right-angled triangles and the ratio of their sides. Both are outlined below.

• 1. President Garfield’s proof

  As depicted in Fig. 1.12a of the diagram below, the trapezoid is composed of two congruent right-angled triangles and an isosceles triangle. The area of the trapezoid is easily calculated as

  Fig. 1.12

  

  The proofs of Pythagorean theorem

  $$\frac{1}{2}\left( {a + b} \right)\left( {a + b} \right) = \frac{1}{2}ab + \frac{1}{2}ab + \frac{1}{2}c^{2}$$

  The left-hand side is simply (based + height)/2, and the right-hand side is the sum of the areas of three triangles. After simplifying the expression, we obtain the Pythagorean theorem $a^{2} + b^{2} = c^{2}$.

• 2. A proof based on similar right triangles

  As shown in Fig. 1.12b. From the similarity of the triangles $\Delta ABC$ and $\Delta DBA$, we have $\overline{AB}^{2} = \overline{BD} \times \overline{BC}$;. Likewise, the similarity of $\Delta ABC$ and $\Delta DAC$ implies $\overline{AC}^{2} = \overline{CD} \times \overline{CB}$. Adding these two equations together, we have $\overline{AB}^{2} + \overline{AC}^{2} = \left( {\overline{BD} + \overline{DC} } \right) \times \overline{BC} = \overline{BC}^{2}$, which is the Pythagorean theorem.

Appendix 1.3: Pythagoras’ Vibrating String: Length and Harmony

In a block print dating back to the Renaissance period, Pythagoras is vividly portrayed to be next to a five-stringed instrument, tuning it with two wooden sticks in his hands. The protagonist is striking a string with one of the sticks and using the other to adjust the length of the vibrating portion of the string. One end of each of the five strings is supported by the instrument’s bridge, and the other end is attached to a stone used as a counterweight to set the string’s tension. The artist’s inspiration must have come from the desire to appreciate the music theory legacy of Pythagoras, which contributed to the development of Western music for over a thousand years.

In modern terms, Pythagoras would be described as a mathematician with absolute pitch (often called “perfect pitch”), since he used clear logical reasoning to define the musical scales audible to the human ear. According to legend, Pythagoras formulated the scales by dividing the sounds produced by two strings with a 2:1 ratio in length into seven notes (frequencies), thus forming an octave. He also divided the sounds produced by two strings with a 3:2 ratio in length into four notes to produce a “fifth”. Finally, he put these two scales together by ordering the notes to get 12 semitones.

However, to produce the beauty of harmony, someone later made gradual adjustments to the octave to produce the so-called “just intonation” on the basis of these 12 semitones. The purpose of tuning was then to make the harmonies of musical chords more natural and pleasing. However, the frequency difference between each pair of notes is not the same, which means that the chords of musical pieces of different keys are not interchangeable, causing many problems in composing and performing music. A good example is that when a musical piece changes keys midway, the performer may make a mistake without realizing it.²⁰ To solve this problem, the German musicians known for their insistence on uniformity, introduced the 12-tone equal temperament toward the end of the 17th century. The person who advocated this system of tuning was Andreas Wreckmeister (1645–1706). He divided the octave into 12 parts (semitones), all of which are equal on a logarithmic scale. See the keyboard shown in Fig. 1.13.

Fig. 1.13

The notes of the modern keyboard

Given a note, the note one octave above it has a frequency twice that of the first note, so the frequency of the note one semitone higher would be $\sqrt[{12}]{2} = 1.059463094$ times that of the first note. If the frequency of the C note is taken to be 1, then the frequency of C# (one semitone above C) is the product of 1 and $\sqrt[{12}]{2}$, the frequency of D is ($\sqrt[{12}]{2})^{2}$, and D# is ($\sqrt[{12}]{2})^{3}$, and so on. The keyboard of the modern piano, depicted in Fig. 1.13, is constructed using the 12-tone equal temperament scale.

This common multiple of $\sqrt[{12}]{2}$, between two adjacent semitones, can be explained by the following mathematical relationship. Modern wave theory, familiar to many, points out that the frequency f of the sound emitted by a vibrating string is related to the length of the string L, the density ρ of the string, and the string’s tension T. Therefore, the velocity of propagation of a wave in the string v can be expressed as $v = \sqrt {\frac{T}{\rho }}$. When the string is vibrating at the fundamental frequency or first harmonic $\left( {n = 1} \right)$, the following relationship holds:

$$f = \frac{1}{2L}\sqrt {\frac{T}{\rho }}$$

(1.25)

This equation tells us that if we reduce the string’s density, or shorten the string’s length, or increase the string’s tension, we will produce a higher vibrating frequency. Now suppose a string is fixed at both ends. Provided that the density and force of tension are both constant, plucking the string will produce a standing (or stationary) wave of wavelength $\lambda /2 = L/n$ and frequency $f = nv/2L$, where n is a positive integer and v is the velocity of wave propagation on the string. The equation $\lambda /2 = L/n$ can then be interpreted as follows: when the string’s length is an integer multiple of half of the wavelength, the frequency of vibration is inversely proportional to the string’s length.

Now suppose we have two strings with a length ratio of 1:2. This implies the ratio of their frequencies is 2:1. In other words, the notes they produce are one octave, or 12 semitones, apart. This can be expressed mathematically as ${ \log }_{{\sqrt[{12}]{2}}} \left( {\frac{2}{1}} \right) = 12$. Since a note of a semitone higher has a frequency being $\sqrt[{12}]{2}$ times of the original note, if we want to double the frequency, we need to multiply the original frequency by 12th power of $\sqrt[{12}]{2}$. Similarly, if two strings have a length ratio of 2:3, their frequency ratio will be 3:2, and the difference in pitch will be ${ \log }_{{\sqrt[{12}]{2}}} \left( {\frac{3}{2}} \right) = 7.02 \approx 7$ semitones, so it is necessary to multiply the original frequency by the 7th power of $\sqrt[{12}]{2}$, about five scale degrees apart. If the two strings have a length ratio of is 3:4, their frequency ratio is 4:3, and the difference in pitch will be ${ \log }_{{\sqrt[{12}]{2}}} (\left. {\frac{4}{3}} \right) = 4.98 \approx 5$ semitones, so it is necessary to multiply the original frequency by the 5th power of $\sqrt[{12}]{2}$, about four scale degrees apart. These notes with integer multiples of frequencies are important components of musical chords. If the frequencies of the notes do not have integer ratios, the notes produced will not sound harmonious at all and will even be harsh to the ear. Examples are alarms, sirens, noisy machines, screams, and the fluttering sounds made by insects.

The principles behind musical chords were applied by the clergymen in monasteries in the 12th century to design Gothic-style cathedrals and churches. For example, the ratio of the lateral width of a church’s main hall to the distance between lengthwise stone pillars is 2:1; the ratio of the main hall’s length to the altar’s width is 8:5; and the ratio of main hall’s height and width is 3:2. A church built with musical chord frequency ratios in mind not only exhibits visual beauty but also a serene auditory sensation. The reason is that the choral harmonies and musical chords performed in the church are amplified, while discordant music is suppressed. With both visual and auditory harmony, the church attendants naturally enjoy peace of mind.

Appendix 1.4: Computing Archimedes’ π Using Integration

Archimedes’ achievements in geometry are even more brilliant. He was the first to employ a purely geometric method to estimate π to be $3\frac{1}{7}$, which we already introduced in Sect. 1.4. Here we will take advantage of the techniques in modern calculus to derive the upper and lower bounds of π. We first consider the integral²¹

$$\begin{aligned} I & = \int_{0}^{1} {\frac{{x^{4} (1 - x)^{4} }}{{1 + x^{2} }}} dx = \int_{0}^{1} {\frac{{x^{4} - 4x^{5} + 6x^{6} - 4x^{7} + x^{8} }}{{1 + x^{2} }}} dx \\ & = \int_{0}^{1} {\left( {x^{6} - 4x^{5} + 5x^{4} - 4x^{2} + 4 - \frac{4}{{1 + x^{2} }}} \right)} dx \\ & = \left. {\left[ {\frac{{x^{7} }}{7} - \frac{{2x^{6} }}{3} + x^{5} - \frac{{4x^{3} }}{3} + 4x - 4\tan^{ - 1} x} \right]} \right|_{0}^{1} = \frac{22}{7} - \pi \\ \end{aligned}$$

(1.26)

From the result of this integral, we can estimate its upper and lower bounds. First, we substitute $x = 1$ into the right-hand side of the first equality in (1.26) above to get $I = \mathop \smallint \limits_{0}^{1} \frac{{x^{4} \left( {1 - x} \right)^{4} }}{2} \ge \frac{1}{1260}$. Next, we substitute $x = 0$ to get $I = \mathop \smallint \limits_{0}^{1} \frac{{x^{4} \left( {1 - x} \right)^{4} }}{1} \le \frac{1}{630}$. Combining these two results we have $\frac{1}{1260} < \frac{22}{7} - \pi \le \frac{1}{630}$, or $\frac{22}{7} - \frac{1}{630} \le \pi \le \frac{22}{7} - \frac{1}{1260}$, which gives the lower and upper bounds for π.

In arriving at the same result achieved by Archimedes over two millennia ago, both of the approaches above utilize modern trigonometry and the machinery of integral calculus, which shows that ancient natural philosophers already possessed an impressive grounding in geometric reasoning. Today almost everybody knows that quantities related to the circle (e.g., its area and circumference) and the sphere (e.g., its volume and surface area) are all associated with the number π. The circle and the sphere are the most abundant shapes in the Universe, as well as on Earth’s surface. Even in physical or mathematical equations that are ostensibly unrelated to the circle, π can be found everywhere, making this irrational number an important universal constant. Examples include: the period of a simple pendulum, $T = 2\pi \sqrt {\frac{L}{g}}$; the Stefan–Boltzmann constant in the calculation of black-body radiation, $\sigma = \frac{{2\pi^{5} k^{4} }}{{15c^{2} h^{3} }}$; and the probability density function of the normal (or Gaussian) distribution, $f\left( x \right) = \frac{1}{{\sqrt {2\pi } }}e^{{ - \frac{{x^{2} }}{2}}}$, used in computational statistics. If we look at the decimal representation of π, we can feel its profound nature: it is irrational, with an infinite, non-repeating number of digits after the decimal point. However, any short sequences of digits anyone considers important, such as birthdays, license plates and phone numbers, can all be found somewhere in this infinite sequence, and we only need a precision of only 39 decimal places to calculate the circumference of the Universe and the size of an atom.