Ancient Trigonometry & Astronomy

Transcription

1 Ancient Trigonometry & Astronomy Astronomy was hugely important in ancient culture. For some, this meant measuring the passage of time via calendars based on the seasons or the phases of the moon. For others tracking and predicting heavenly behavior was a large part of the ritual and religious practice. For example, the complex religion of the ancient Egyptian s included the belief that the region around the pole star (the region of the sky that other stars seemed to circle around) was the location of their heaven. Pyramids were built with narrow shafts pointing directly from the burial chamber of a pharaoh to this region so that the deceased could ascend to the stars. As humanity begain to travel further, astronomy came to be increasingly important for navigation. Trigonometry (lit. triangle-measure) is the essential computational tool of ancient astronomy, although the term was not used until Ancient Greek analysis of the heavens was based on several assumptions. Spheres are the most perfect solid. The earth is a sphere and the fixed stars (constellations) lie on a second celestial sphere. The gods built the universe around spheres, so all explanations of heavenly motion should rely on spheres and circles. The Earth is stationary, so the celestial sphere must rotate around the Earth once per day. The perfect design of the gods would also imply motion at a constant rate. In this way, the fixed stars (outer celestial sphere) form a background canvas for the observation of planets (known as wandering stars ). Two major contradictions were apparent when the above philosophy was tested by observation. The apparent brightness of heavenly bodies is not constant. In particular, the brightness of the planets is highly variable, though the sun also exhibits this behavior. Retrograde motion: planets mostly follow the East-West motion of the heavens, however they occasionally slow down and reverse course. If all heavenly bodies are simply moving in circles centered on the Earth at a constant speed, then how can the above be explained? The attempt to produce accurate models while saving the phenomena of spherical and circular motion, led to the development of new mathematics. Our discussion begins with the approaches of two previously-encountered Greek mathematicians. Eudoxus of Knidos (c BCE) Developed a multiple-sphere model. The outer sphere contained the fixed stars, attached to it were the poles of another sphere to which was attached a planet, or the sun. Eudoxus spheres were not considered real or made of anything. This was an early form of modelling, i.e. just for computation. The motion generated by nested spheres becomes highly complex. 1 It is capable of describing retrograde motion, but not the variable brightness of stars and planets. 1 The link directs to a rather nice flash animation of Eudoxus model.

2 Apollonius of Perga ( nd /3 rd C BCE: dates unknown) Apollonius proposed two solutions. In his eccenter model, the sun is assumed to orbit a circle (the deferent) whose center is not the Earth. This addressed the problem of variable brightness and was fairly simple to work with. The criticism is obvious: why? What is the philosophical justification for the eccenter? Eudoxus model may have been complex and essentially impossible to compute with, but it was more in line with the assumptions of spherical motion. But Apollonius wasn t done... Planet/Sun Eccenter Earth His second approach introduced epicycles: for a video animation, follow this link. An epicycle is a small circle moving around a larger one. A planet was modelled by the curve thus generated: if you ve ever seen the toy Spirograph, you ll be familiar with these curves! The observer at the center will see the apparent brightness change, as well as potentially observing retrograde motion if the motion of the epicycle is significant enough. In modern language, the motion could be parametrized by the vectorvalued function x(t) = R ( cos ωt sin ωt ) + r ( ) cos ψt sin ψt where R, r, ω, ψ are the radii and circular frequencies (rad/s) of the two circles. Combining the two models gave Apollonius the ability to describe quite complex motion. Calculation was difficult, requiring the calculation of the lengths of chords of circles from a given angle, and vice versa. It is from this requirement that the earliest notions of trigonometry arise. One might reasonably ask why the Greeks didn t make what seems to us as the obvious fix, by placing the sun at the center of the cosmos. In fact Aristarchus of Samos (c BCE) did precisely this, and opined that the fixed stars were really just other suns at exceptional distance. His writing was known to Archimedes but his heliocentric theory was largely rejected in favor of geocentric descriptions. Indeed the great thinkers of the time (Plato, Aristotle, etc.) had a strong objection to Aristarchus argument: parallax. θ distant stars nearer star Earth and Sun If the Earth moves round the sun, and the fixed stars are really independent objects, then the position of the a nearer star should appear to change throughout the year. The angle θ in the picture is the

3 parallax of the nearer star. Unfortunately for Aristarchus, the Greeks were incapable of observing any parallax. Astronomical co-ordinates The Greeks were hardly the only ancient culture attempting to calculate the motions of the heavens. The Egyptians, Babylonians and the Chinese certainly did so and, to some extent, had observed the same difficulties as the Greeks. Indeed modern astronomical coordinates come to us courtesy of the Babylonians. Their astrological system has its roots around 000 BCE, but by BCE they had settled on the structure on which the most commonly used modern astrological systems are based. 3 Here is their rough structure. The Sun s path through the fixed stars forms a great circle now known as the ecliptic. The year is roughly 360 days (a round number that divides nicely): 1 degree of measure along the ecliptic corresponds to approximately the motion of the sun each day. Celestial longitude was measured from zero to 360 around the ecliptic with zero fixed at the vernal (spring) equinox. The ecliptic was divided into 1 equal segments each of 30. These described the zodiac: in modern terms Aries corresponds to 0 30, Taurus 30 60, etc. Celestial latitude was measured in degrees north or south of the ecliptic, so that the Sun always had latitude zero. The Greeks, coming into possession of Babylonian knowledge through trade and Alexander s conquests, co-opted this system. Indeed it became standard to perform all astronomical, and later trigonometric, calculations using the base 60 degrees-minutes-seconds system inherited from Babylon. Hipparchus of Nicaea (c BCE) Hipparchus was the first of the two pre-emminent Greek astronomers (we shall encounter Ptolemy later). He made numerous observations, in particular of the moon, working with both the eccenter and epicycle models of Apollonius to compute the mean distance to the moon. Recognizing that the ecliptic is angled with respect to the Earth s equator, Hipparchus also introduced equatorial coordinates. 4 as it is typically easier to make measurements this way. As part of his work, he needed to be able to compute chords of circles with some accuracy: his chord tables are acknowledged as the earliest tables of trigonometric values. You might have heard of the astronomical distance unit the parsec. This is the distance of star exhibiting one arc-second (= ) of parallax, roughly 3.3 light-years or km, an unimaginable distance to anyone before the scientific revolution. The closest star to the Sun is Proxima Centauri at 4. light years, with a parallax of 0.77 arcseconds: is it any wonder the Greeks rejected the hypothesis?! 3 Thankfully modern astrology has got past divination using livers from sacrifices. 4 The modern terms are right ascension measured in hours, minutes, seconds East of the initial point of Aries, and declination measured in degrees north of the Earth s equator. 3

4 In a modern imitation of Hipparchus approach, we define a function chord to return the length of the chord in a given circle subtended by a given angle. In the pictured circle we have crd α = r sin α crd(180 α) α r crd α Hipparchus chose a circle with circumference 360 (in fact he used = 1600 arcminutes): the result is that r = 1600 π 57; 18. This is precisely the number of degrees per radian, which should be no surprise 5 His chord table was constructed starting with the obvious: crd 60 = r = 57; 18 crd 90 = r = 81; Pythagoras Theorem was used to obtain chords for angles 180 α. In modern language crd(180 α) = (r) (crd α) = r 1 sin (α/) = r cos α Finally, Hipparchus used Pythagoras again to halve and double angles in an approach analogous to Archimedes quadrature of the circle. We rewrite the argument in this language. In the picture, DB = crd(180 α). Since BDA = 90 and M is the midpoint of AD, we see that OMD is rightangled. Thus D C OM = 1 BD = 1 crd(180 α) Now apply Pythagoras to CMD: B O α α M A (crd α) = ( ) 1 crd α + (r 1 ) crd(180 α) = 1 4 (crd α) + r r crd(180 α) + 1 crd(180 α) 4 = 1 4 (crd α) + r r crd(180 α) (4r (crd α) ) = r r crd(180 α) In modern notation this reads 4r sin α = r r cos α cos α = 1 sin α simply one of the double-angle trigonometric identities! 5 One radian is defined as the angle subtended by an arc equal in length to the radius of a circle. Hipparchus essentially does this in reverse: if the circumference is fixed so that degree now measures both subtended angle and circumferential distance, then the radius of the circle is fixed. 4

5 Example To calculate crd 30, we start with crd(60 ) = r. Then crd(10 ) = 4r r = 3r = (crd 30 ) = r r crd(180 60) = r 3r = ( 3)r = crd 30 = 3r In modern language we have an exact value for sin 15 : crd 30 = r sin 15 = sin 15 = 1 3 Continuing this process, we obtain crd 150 = r + 3, whence (crd 15 ) = r r crd 150 = ( + 3)r = crd 15 = r + 3 Again: sin 7.5 = Using this approach, Hipparchus computed the chord of each of the angles 7.5, 15,..., 17.5, in steps of 7.5. Of course everything was an estimate since he had to rely on approximations for squareroots. All Hipparchus original work is now lost. We primarily know of his work by reference. In particular, the above method is probably due to Hipparchus, although we see it first in the work of... Claudius Ptolemy (c CE) Produced the Mathematica Syntaxis around 150 CE, better known as the Almagest: the latter term is derived from the Arabic al-mageisti, meaning great work and reflects the fact that renaissance Europe inherited the text from the Islamic world. The Almagest is essentially a textbook on geocentric cosmology. It shows how to compute the motions of the moon, sun and planets, describing lunar parallax, eclipses, the constellations, etc. It contains our best evidence as to the accomplisments of Hipparchus and describes his calculations. The text formed the basis of Western/Islamic astronomical theory through to the 1600 s: it is essentially the Elements of Astronomy. It is important to note that, following the Pythagoreans and Plato, Ptolemy did not find it necessary to seek physical causes for the motion of the planets. The mathematics and the description of what one would see was enough. From a trigonometric point of view, the most important feature of the Almagest was its dramatic improvement on the chord tables of Hipparchus, and the inclusion of elementary spherical trigonometry, probably courtesy of Menelaus (c.100 CE). Ptolemy lived in Alexandria: while the name Ptolemy is Greek, his other name Claudius is Roman, reflecting the changing cultural situation. 5

6 Ptolemy s Calculations Ptolemy computed more chords and with greater accuracy than Hipparchus. To assist with this, he made one critical change and a number of improvements to the method. Ptolemy started with r = 60 rather than r = 57; 18. This made for easier calculations, in particular, crd 60 = 60. Ptolemy then uses what was probably Hipparchus method to halve/double angles and compute chords of supplementary angles: crd α = r r crd(180 α) = 60 ( 10 crd(180 α) ) crd(180 α) = (r) crd α = 10 crd α For example, ( crd 30 = 60(10 crd 10 ) = ) 3 = 60 ( 3) = crd 30 = 60 3 Ptolemy had more initial data that Hipparchus: beyond the equilateral and 1 : 1 : triangles, he also knew the side-length of an inscribed regular decagon and pentagon crd 36 = 30( 5 1) crd 7 = 60 + crd 36 = Most usefully, Ptolemy could also effectively add and subtract angles using what is now known as Ptolemy s Theorem (see below). He computed crd 1 = crd(7 60 ), then halved this for angles of 6, 3, 1.5, and Chords for all integer multiples of 1.5 could then be computed using addition formula. For finer detail, Ptolemy used interpolation to find approximations. His essential observation was that α < β = crd β crd α < β α In modern language sin ˆβ sin ˆα β < ˆα so that sin x x increases to 1 as x 0 +. This allowed him to compute the chord for every integer multiple of 1 to the incredible accuracy of 1 part in 60 (his tables were written in a modification of the Babylonian sexagesimal system using Greek characters). Ptolemy s table contains even more than this. For approximating between the exact values for half-degrees, he supplied a second table of values indicating how much should be added for each arcminute ( 1 60 ). For example, the second line of Ptolemy s table reads 1 1;, 50 ; 1,, 50 meaning first that crd 1 = 1;, 50 to two sexagesimal places. 6 example that crd(1 5 ) 1;, (; 1,, 50) = 1; 8, 4, 10 The second entry says, for It is believed that Ptolemy computed his half-angle chords to an accuracy of five sexagesimal places in order to obtain his arcminute approximations. 6 This is = = 10 sin , an already phenomenal level of accuracy. 6

7 Ptolemy s Theorem. Suppose a quadrilateral is inscribed in a circle. Then the product of the diagonals equals the sum of the products of the opposite sides. Proof. Choose a point E on AC such that ABE = DBC. Then ABD = EBC. We obtain two pairs of similar triangles: C B C B E A E A D D ABE DBC ABD EBC The proof follows immediately: compute AE CD = AB CE and BD AD = BC BD, whence AC = AE + CE = which gives the result. AB CD + AD BC BD A special case of Ptolemy s Theorem allowed him to find chords of sums and differences of angles: these are precisely the multiple-angle formulæ from modern trigonometry. Corollary (Multiple Angle Formula). If α > β, then 10 crd(α β) = crd α crd(180 β) crd β crd(180 α) In modern language, divide out by 10 to obtain sin α β = sin α cos β sin β cos α Proof. Suppose that AD = 10 is a diameter of the drawn circle. Ptolemy s Theorem says C B AC BD = AB CD + AD BC which is simply crd α crd(180 β) = crd β crd(180 α) + 10 crd(α β) D α O β A from which the result follows. Similar expressions for crd(α + β) and crd(180 (α ± β)) were also obtained. 7

8 Examples Here is how Ptolemy might have calculated crd 4. Let α = 7 and β = 30, then 10 crd 4 = crd 7 crd 150 crd 30 crd 108 Since crd 7 = is the side-length of a regular pentagon, and crd 108 = crd(180 7 ) = 10 crd 7 = we see that crd 4 = 1 ( ) 5 10 ( = ) 5 43; 0, Note all the square-root terms which had to be approximated: the construction of the chord-table was truly a gargantuan task, a task for which Ptolemy almost certainly had assistance. The Almagest contained a huge number of examples to aid the reader in their own calculations. Here is one such. At a particular point at noon, a stick of length 1 is placed in the ground. The angle of elevation of the sun is 7. What is the length of its shadow? To solve the problem, Ptolemy instructs the reader to draw a picture and use ratios. The lower isosceles triangle has base angles 7, as required. We use modern notation: clearly 1 1 : l = crd 144 : crd 36 crd 36 = l = crd 144 = 30( 5 1) Note that this is precisely cot 7, though Ptolemy had no such notion. Further calculations and examples were far more complex! 36 l It is worth noting that Ptolemy s system was resolutely geocentric. Similarly to how the fame of Euclid s Elements shaped 000 years of mathematical principle, the massive impact of Ptolemy s Almagest and the reverence felt for ancient knowledge meant that it became increasingly difficult in medieval Europe to challenge geocentrism. By the time Galileo was advancing the Copernican theory in the 1600 s, heliocentrism was essentially heresy. 8