ETERNAL TRIANGLES Eternal Triangles Euclidean geometry is based on triangles, mainly because every polygon can be built from triangles, and most other interesting shapes, such as circles and ellipses, can be approximated by polygons. The metric properties of triangles - those that can br measured, such as the lengths of sides, the sizes of angles or the total area - are related by a variety of formulas, many of them elegant. The practical use of these formulas, which arc extremely useful in navigation and surveying, required the development of trigonometry, which basically means 'measuring triangles'. Trigonometry Trigonometry spawned a number of special functions -mathematical rules for calculating one quantity from another.Thosr functions go by names like sine, cosine and tangent.The trigonometric functions turned out to be of vital importance for the whole of I mathematics, not just for measuring triangles. Trigonometry is one of the most widely used mathematical te< I.....|ues, involved in everything from surveying to navigation to I IPS satellite systems in cars. Its use in science and technology is ■ .....imon that it usually goes unnoticed, as befits any universal tool. Historically, it was closely associated with logarithms, a clever ini'iliod for converting multiplications (which are hard) into ■dditions (which are easier). The main ideas appeared between .11 •■ mt 1400 and 1600, though with a lengthy prehistory and plenty ' 'I later embellishments, and the notation is still evolving. In this chapter we'll take a look at the basic topics: trigonometric Junctions, the exponential function and the logarithm. We also .....:;ider a few applications, old and new. Many of the older «l>|ilications are computational techniques, which have mostly .....me obsolete now that computers are widespread. Hardly anyone ■ iw uses logarithms to do multiplication, for instance. No one uses ibles at all, now that computers can rapidly calculate the values of i.....i ions to high precision. But when logaridmis were first invented, ii Was the numerical tables of them that made them useful, especially in areas like astronomy, where long and complicated numerical i .ill illations were necessary. And the compilers of the tables had to Bend years - decades - of their lives doing the sums. Humanity owes i i-11-.ii deal to these dedicated and dogged pioneers. The origins of trigonometry 1 In- liasic problem addressed by trigonometry is the calculation of properties of a triangle - lengths of sides, sizes of angles - from ■thei .such properties. It is much easier to describe the early history ■•in i gonometry if we first summarize the main features of modern Ii iy;i mometry, which is mostly a reworking in 18th century notation dl topics that go right back to the Greeks, if not earlier. This ■i111111.try provides a framework within which we can describe the lilr.i-, of the ancients, without getting tangled up in obscure and .....illy obsolete concepts. trigonometry seems to have originated in astronomy, where it i i ■ 1.11 i vely easy to measure angles, but difficult to measure the vast i H vs. The Greek astronomer Aristarchus, in a work of around Ml, 87 TAMING THE INFINITE ETERNAL ffllANOLES Trigonometry - a Primer Trigonometry relies on a number of special functions, of which the most basic are the sine, cosine and tangent. These functions apply to an angle, traditionally represented by the Greek letter 9 (theta). They can be defined in terms of a right triangle, whose three sides a, b, c are called the adjacent side, the opposite side and the hypotenuse. c (hypotenuse) b (opposite) a (adjacent) Tlieii: The sine of theta is sin 9 = b/c The cosine of theta is cos 8 - a /c The tangent of theta is tan 9 = b / a As it stands, the values of these three functions, for any given angle 9, are determined by the geometry of the triangle. (The same angle may occur in triangles of different sizes, but the geometry of similar triangles implies that the stated ratios are independent of size.) However, once these functions have been calculated and tabulated, they can be used to solve (calculate all the sides and angles of) the triangle from the value of 9. The three functions are related by a number of beautiful formulas. In particular, Pythagoras's Theorem implies that sin2 9 + cos2 9 = 1 |60 BC, On the Sizes and Distances of the Sun and Moon, deduced ili.u 11 it-Sun lies between 18 and 20 times as far from the Earth as the Moon i 11 .. (The correct figure is closer to 400, but Eudoxus and Phidias .1 argued for 10.) His reasoning was that when the Moon is hall 1)11, the angle between the directions from the observer to the Sun ni. I the Moon is about 87° (in modern units). Using properties of ii i.mgles that amount to trigonometric estimates, he deduced (in .....dern notation) that sin 3° lies between '/is and '/zo, leading to In. rstimate of the ratio of the distances to the Sun and the Moon. IIm' method was right, but die observation was inaccurate; the lorrect angle is 89.8°. Hie first trigonometric tables were derived by Hipparchus around i 50 bc. Instead of the modern sine function, he used a closely ii dated quantity, which from the geometric point of view was • ■ |• ully natural. Imagine a circle, with two radial lines meeting at .in angle a.The points where these lines cut the circle can be joined Chord Relation between the Sun, Moon, and Earth when the Moon is halt full Arc and chord corresponding to an angle 6 Mil 813 by a straight line, called a chord. They can also be thought of as th| end points of a curved part of the circle, called an arc. Hipparchus drew up a table relating arc and chord length for range of angles. If the circle has radius 1, then the arc length is equal to 9 when this angle is measured in units known as radians. Sonif easy geometry shows that the chord length in modern notation it 2sin eh ■ So Hipparchus s calculation is very closely related to a table of sines, even though it was not presented in that way. Astronomy Remarkably, early work in trigonometry was more complicated than most of what is taught in schools today, again because of the needi of astronomy (and, later, navigation). The natural space to work in was not the plane, but the sphere. Heavenly objects can be thouglu of as lying on an imaginary sphere, the celestial sphere. Effectively, dusky looks like the inside of a gigantic sphere surrounding the observer, and the heavenly bodies are so distant that they appear to lie on this sphere. Astronomical calculations, in consequence, refer to the geometry of a sphere, not that of a plane. The requirements are therefore no( plane geometry and trigonometry, but spherical geometry and trigonometry. One of the earliest works in this area is Menelans's Sphaerica of about ad 100. A sample theorem, one that has no analogue North Pole |p i lulidean geometry, is this: if two triangles have the same angles i« Nch other, then they are congruent - they have the same size and i . (In the Euclidean case, they are similar - same shape but ilily different sizes.) In spherical geometry, the angles of a i.■ >Ie do not add up to 180°, as they do in the plane. For ii 111 >le, a triangle whose vertices lie at the North Pole and at two lnts on the equator separated by 90° clearly has all three angles ill il lo a right angle, so the sum is 270°. Roughly speaking, the ■1'r the triangle becomes, the bigger its angle-sum becomes. In m i, i his sum, minus 180°, is proportional to the triangle's total area. II irse examples make it clear that spherical geometry has its own piracteristic and novel features, The same goes for spherical ■ tonometry, but the basic quantities are still the standard ii ii" mometric functions. Only the formulas change. Ptolemy ■ far and above the most important trigonometry text of antiquity i i he Mathematical Syntaxis of Ptolemy of Alexandria, which dates I ■ about ad150. It is better known as the Almagest, an Arabic term iiI i ng' the greatest'. It included trigonometric tables, again stated .....this of chords, together with the methods used to calculate ■ I" in, and a catalogue of star positions on the celestial sphere. An !".■,(■ utial feature of the computational method was Ptolemy's The angles of a spherical triangle do not add up to 180° Cyclic quadrilateral and its diagonals !l(l 91 TAMING THE INFINITE ť 11 H N AI IKIANlit I Tlieorem which states that if ABCD is a cyclic quadrilateral (one whose vertices lie on a circle) then AB X CD + BC x DA = AC x BD (the sum of the products of opposite pairs of sides is equal to thfl product of the diagonals). A modern interpretation of this fact is the remarkable pair formulas sin (0 + 9) = sin 6 cos 9 + cos 6 sin 9 cos (6 + 9) = cos 6 cos 9 - sin 6 sin 9 The main point about these formulas is that if you know the sines and; cosines of two angles, then you can easily work the sines and cosinci out for the sum of those angles. So, starting with (say) sin 1° and coi 1°, you can deduce sin 2 and cos 2° by taking 6 — 9- l°.Then you] can deduce sin 3° and cos 3° by taking 6 - \°,9= 2°, and so on.You j had to know how to start, but after that, all you needed was aridimetii - rather a lot of it, but nothing more complicated. Getting started was easier than it might seem, requiring! arithmetic and square roots. Using the obvious fact that 6/2 + 6/1 = 6, Ptolemy's Theorem implies that 0 / 1 - cos 9 sin — = / - 2 V 2 Starting from cos 90° = 0, you can repeatedly halve the ang!< obtaining sines and cosines of angles as small as you please, (Ptolemy used V4°.) Then you can work back up through all integer j multiples of that small angle. In short, starting with a few general! trigonometric formulas, suitably applied, and a few simple valnc\ for specific angles you can work out values for pretty much an/ J angle you want. It was an extraordinary tour de force, and it put] astronomers in business for well over a thousand years. A final noteworthy feature of the Almagest is how it handled ilic "dnis of the planets. Anyone who watches the night sky regularly • 11■ i■ kly discovers that the planets wander against the background of prd stars, and that the paths they follow seem rather complicated, •'inn•Limes moving backwards, or travelling in elongated loops, i udoxus, responding to a request from Plato, had found a way represent these complex motions in terms of revolving spheres .........led on other spheres. This idea was simplified by Apollonins ■ iid I lipparchus, to use epicycles — circles whose centres move along ■iln i circles, and so on. Ptolemy refined the system of epicycles, so 11i.ii ii provided a very accurate model of the planetary motions. Marly trigonometry I trigonometric concepts appear in the writings of Hindu I! i' maticians and astronomers: Varahamihira's Pancha Siddhanta of DO, Brahmagupta's Brahma Sputa Siddhanta of 628 and the more ilt i.ulcd Siddhanta Siromani of Bhaskaracharya in 1150. July 1 June I February 1 January 1 Mars s Motion of Mars as viewed from the Earth 'I." 93 TAMING THE INFINITE Indian mathematicians generally used the half-chord, or jya-ardh which is in effect the modern sine. Varahamihira calculated thi: function for 24 integer-multiples of 3°45', up to 90°. Aroun 600, in the Maha Bhaskariya, Bhaskara gave a useful approximate formula for the sine of an acute angle, which he credited ti Aryabhata. These authors derived a number of basic trigonometri formulas. The Arabian mathematician Nasir-Eddin's Treatise on thi Quadrilateral combined plane and spherical geometry into a single unified development, and gave several basic formulas for spherical triangles. He treated the topic mathematically, rather than as a paifl of astronomy. But his work went unnoticed in the West until aboni 1450. Because of the link with astronomy, almost all trigonometry wall spherical until 1450. In particular, surveying - today a major useffl of trigonometry - was carried out using empirical methods, codified by the Romans. But in the mid-15th century, planfl trigonometry began to come into its own, initially in the northl German Hanseatic League. The League was in control of most trade, and consequently was rich and influential. And it needed improved navigational methods, along with improved timekeeping and! practical uses of astronomical observations. A key figure was Johannes Müller, usually known all Regiomontanus. He was a pupil of George Peuerbach, who beg.ui working on a new corrected version of the Almagest. In 1471, (inauced by his patron Bernard Walther, he computed a new table II of sines and a table of tangents. Other prominent mathematicians of the 15th and 16th centuriej computed their own trigonometric tables, often to extreme accuracy, (ieorge Joachim Rhaeticus calculated sines for a circle of radius 10" - effectively, tables accurate to 15 decimal places, but multiplying all .....ill it • is by I015 to get integers —for all multiples of one second of .im. I !<■ si.iicd the law of sines for spherical triangles ETERNAL I IIIANCil.l-S Ml sin a sin A sin b sin B sin c sin C y< a is to perfect and publish it. It seems likely diat he started with geometric progressions, sequences of numbers in which each term i ■ < il it.lined from the previous one by multiplying by a fixed number in h as the powers of 2 Plane Trigonometry Nowadays trigonometry is lirst developed in the plane, where the geometry is simpler and the basic principles are easier to grasp. (It is curious how often new mathematical ideas are first developed in a complicated context, ;uid the underlying simplicities emerge much later.) There is a law of sines, and a law of cosines, for plane triangles, and it is worth a quick digression to explain these. Consider a plane triangle with angles a, d, c and sides a, b, c. Now the law of sines takes the form a b c sin A sin B sin C .md the law of cosines is a1 = b1 + c1 - 2bc cos A, with companion formulas involving the other angles We can use the law of cosines to find the angles n[ a triangle from its sides. Sides and angles of a triangle 1 2 4 8 16 32 ... IK >wers of 10 1 10 100 1000 10,000 100,000 •ii, '1/ TAMING THE INFINITE ETERNAL I HIANIil.ES Here it had long been noticed that adding the exponents equivalent to multiplying the powers. This was fine if you wante to multiply two integer powers of 2, say, or two integer powers 10. But there were big gaps between these numbers, and powers i 2 or 10 seemed not to help much when it came to problems likr 57.681 x 29.443, say. Napierian logarithms While the good Baron was trying to somehow fill in the gaps in geometric progressions, the physician to King James VI of Scotland James Craig, told Napier about a discovery that was in widesprea use in Denmark, with the ungainly name prosthapheiresis.This referrc to any process that converted products into sums. The main methc in practical use was based on a formula discovered by Vieta: x + y x — y sin x + sin y sin-- cos-u = -- 2 2 2 If you had tables of sines and cosines, you could use this formi: to convert a product into a sum. It was messy, but it was still quicker than multiplying the numbers directly. Napier seized on the idea, and found a major improvement. I It formed a geometric series with a common ratio very close to l.Tlul is, in place of the powers of 2 or powers of 10, you should usf j powers of, say, 1.0000000001. Successive powers of such a numl>pf j are very closely spaced, which gets rid of those annoying gaps. I'<>f I some reason Napier chose a ratio slightly less than 1, namely 0.9999999. So his geometric sequence ran backwards from a largfl .....uber lo successively smaller ones. In fact, he started witfl 10,000,000 and then multiplied this by successive powers of 0.4999999. If we write Naplog x for Napier's logarithm of x, it I »-* the curious feature that Naplog 10,000,000 = 0 Naplog 9,999,999 = 1 ■ I I so on.The Napierian logarithm, Naplog x, satisfies the equation Naplog (107xy) = Naplog (x) +Naplog(y) .....can use this for calculation, because it is easy to multiply or I i• le by a power of 10, but it lacks elegance. It is, however, much hit thanVieta's trigonometric formula. Inst: ten logarithms I In next improvement came when Henry Briggs, the first Savilian ■ofessor of geometry at the University of Oxford, visited Napier. :. suggested replacing Napier's concept by a simpler one: the 11mm' len) logarithm, L = log,0 x, which satisfies the condition x = 10l. log,„ xy - log,„ x + logl0 y mnl everything is easy.To findxy, add the logarithms of x and y and iln n lind the antilogarithm of the result. Before these ideas could be disseminated, Napier died; the " was 1617, and his description of his calculating rods, hdologia, had just been published. His original method for • .1' nl.uing logarithms, the Mirifici Logarithmorum Cunonis Constructio, ,ued two years later. Briggs took up the task of computing I ile of Briggsian (base 10, or common) logarithms. He did this i in nig from logn,10 = 1 and taking successive square roots. | i '• I 7 he published Logarithmorum Chilias Prima, the logarithms '.I iln integers from 1 to 1000, stated to 14 decimal places. His ! Arithmetic Logarithmica tabulated common logarithms of l»rs from 1 to 20,000 and from 90,000 to 100,000, also Id I I places. •IH 99 ETEHNAL IMIANCLES What trigonometry did for them_ Ptolemy's Almagest formed the basis of all studies of planetary motion prior to Johannes Kepler's discovery that orbits are elliptical. The observed movements of a planet are complicated by the relative motion of the Earth, which was not recognized in Ptolemy's time. Even if planets moved at uniform speed in circles, the Earth's motion round the Sun would effectively require a combination of two different circular motions, and an accurate model has to be distinctly more complicated than Ptolemy's model. Ptolemy's scheme of epicycles combines circular motions by making the centre of one circle revolve around another circle. This circle can itself revolve round a third circle, and so on. The geometry of uniform circular motion naturally involves trigonometric functions, and later astronomers used these for calculations of orbits. An epicycle. Planet P revolves uniformly around point D, which in turn revolves uniformly around point C. I___J| The idea snowballed. John Speidell worked out logarithms of trigonometric functions (such as log sin x) published as No* 11iiiuiilimits in 1619.The Swiss clockmaker Jobst Biirgi published Ins own work on logarithms in 1620, and may well have possessed '11«■ basic idea in 1588, well before Napier. But the historical development of mathematics depends on what people publish In (lie original sense of make public - and ideas that remain private ■ i\ e no influence on anyone else. So credit, probably rightly, has In go to those people who put their ideas into print, or at least into Idely circulated letters. (The exception is people who put the Heas of others into print without due credit. This is generally l" snnd. the pale.) The number e \v.ciciated with Napier's version of logarithms is one of the most I m 11< irtant numbers in mathematics, now denoted by the letter e. Its > llue is roughly 2.7128. It arises if we try to form logarithms by i iiMug from a geometric series whose common ratio is very lightly larger than l.This leads to the expression (1 + l/n)n, v\ I ii're n is a very large integer, and the larger n becomes, the closer xpression is to one special number, which we denote by e. This formula suggests that there is a natural base for logarithms, 'n.l ii is neither 10 nor 2, but c. The natural logarithm of x is whichever lumber y satisfies the condition x = &. In today's mathematics the '* Mural logarithm is written y - log x. Sometimes the base e is made i In it, as y = logc x, but this notation is mainly restricted to school .....hematics, because in advanced mathematics and science the only 11 ii 11 un of importance is the natural logarithm. Base ten logarithms in best for calculations in decimal notation, but natural logarithms .......ire fundamental mathematically. I he expression cY is called the exponential of x, and it is one of the i important concepts in the whole of mathematics. The number .......of those strange special numbers that appear in mathematics, ■d have major significance. Another such number is Jt. These two .....libers are the tip of an iceberg — there are many others. They are ■I V.i i ably the most important of the special numbers because they imp up all over the mathematical landscape. TAMING THE INFINITE ETĽHNAL lUIANC.tFS What trigonometry does for us Trigonometry is fundamental to surveying anything from building sites to continents. It is relatively easy to measure angles to high accuracy, but harder to measure distances, especially on difficult terrain. Surveyors therefore begin by making a careful measurement of one length, the baseline, that is, the distance between two specific locations. They then form a network of triangles, and use the measured angles, plus trigonometry, to calculate the sides of these triangles. In this manner, an accurate map of the entire area concerned can be constructed. This process is known as triangulation. To check its accuracy, a second distance measurement can be made once the triangulation is complete. The figure here shows an early example, a famous survey carried out in South Africa in 1751 by the great astronomer Abbé Nicolas Louis de Lacaille. His main aim was to catalogue the stars of the southern skies, but to do this accurately he first had to measure the arc of a suitable line of longitude. To do this, he developed a triangulation to the north of Cape Town. His result suggested that the curvature of the earth is less in southern latitudes than in northern ones, a surprising deduction which was verified by later measurements. The Earth is slightly pear-shaped. His cataloguing activities were so successful that he named 15 of the 88 constellations now recognized, ^ ';. having observed more than 10,000 stars using a small refracting telescope. Laciiille's triangulation of South Africa Where would we be without them? ■ would be difficult to underestimate the debt that we owe to tin >sc in .i ring individuals who invented logarithms and trigonometry. I pent years calculating the first numerical tables. Their efforts I the way to a quantitative scientific understanding of the hi. 11 world, and enabled worldwide travel and commerce by , I ■ iving navigation and map-making. The basic techniques of • ■inrying rely on trigonometric calculations. Even today, when tin wytng equipment uses lasers and the calculations are done on ii .i<>m-built electronic chip, the concepts that the laser and the I embody are direct descendants of the trigonometry that Inn i^ued the mathematicians of ancient India and Arabia. I. igarithms made it possible for scientists to do multiplication It kly and accurately. Twenty years of effort on a book of tables, me mathematician, saved tens of thousands of man-years of ill later on. It then became possible to carry out scientific I, ,es using pen and paper that would otherwise have been loo ..... I ons timing. Science could never have advanced without some mi. Ii method. The benefits of such a simple idea have been .....I Tillable. 103