The Garden of Archimedes   A Museum for Mathematics

Brief history of trigonometry

Greek trigonometry The invention of trigonometry can be associated with certainty to the studies of astronomy of the geometric school of Alexandria. The Egyptian city of Alexandria, which bears the name of ALEXANDER THE GREAT who founded it in the III century B.C. was the capital of the Hellenic kingdom of the PTOLEMY until the Romans conquered it. It had a central position in the Mediterranean world of antiquity and an enlightened cultural policy on the part of the rulers, who equipped it with a library famous for over a millennium, one of the seven beauties of the world. They made of Alexandria the centre of Greek mathematics almost until the Arab conquest, and the "bridge" that allowed classic geometry to reach modern times through the Arab tradition. One of the trends of Alexandrine mathematics, together with the studies of pure mathematics that continued vigorously for various centuries, was constant attention to scientific and technological applications, and consequently to quantitative Mathematics, through which the theoretical results of classic geometry could find their equivalent in the natural sciences. Thus a series of new disciplines developed, together with traditional mathematical ones, that today we would call "applied mathematics", ranging from optics to pneumatics, from mechanics to geodetics. This new point of view found a particularly fertile ground in astronomy, where a prevalently cosmological investigation, aiming at looking into the structure of the universe and the causes of the celestial motion, with its greatest example in the works of Aristotle, and in particular the Physics and the treatise On the Heavens, was substituted by a quantitative astronomy, capable of predicting the celestial phenomena (position of the planets, eclipses, conjunctions). This aided a series of human activities, such as the identification of time, the quantification of the year, geography, navigation and not least the compilation of horoscopes. Quantitative astronomy requires quantitative geometry, in particular a geometry of the globe, since it is on the celestial sphere that certain motions take place, the motions on which a theory needs to be built. Hence the explanation of the fact that the inventors of trigonometry are the same astronomers who applied it to the study of the sky, and to the paradox of spheric trigonometry (the study of spherical triangles, traced on the surface of the sphere and having arcs of circle as sides) historically preceding plane trigonometry, even though it is the more difficult. The founder of trigonometry probably was HYPPARCHUS of Rhodes (II century. B.C.), who mostly lived in Alexandria, even if EUDOXUS of Cnidos (c. 408-365 B. C.) and EUCLID from Alexandria (III century B.C.), better known as the author of the Elements, had briefly discussed spherical geometry before him. Fundamental contributions to spherical trigonometry are attributed to THEODOSIUS of Tripoli (I century. B. C.) e MENELAUS of Alexandria (I-II century B. C.), both authors of the volumes known under the title of Sphaerica. But the biggest part of the information about trigonometrical methods from Alexandria derive from the astronomer from antiquity: PTOLEMY CLAUDE (II century B. C.), whose work entitled Mathematical Composition, which later became known as The Great composition ( $\mu\epsilon\gamma\acute{\alpha}\lambda\eta \; \; \sigma\acute{\upsilon}\nu\tau\alpha\xi\iota\varsigma$) by his admirers, to take the final Arab title of Almagesto (a derivation from the Greek) $\mu\epsilon\gamma\acute{\iota}\sigma\tau\eta$, the greatest), laid the foundation of the astronomic theory that dominated the scientific scene until the XVII century. The fundamental difference between Greek and modern trigonometry is that Alexandrian trigonometry uses chords instead of sines. Following the Babylonian tradition, that is still in partial use today, the semicircumference was divided in $180$ equal parts, the degrees, and its diameter in $120$. the result is a kind of goniometer, the round part of which is used to measure arcs, and the flat one to measure the relative chords. For example the chord of an arc of $180$ degrees (a straight angle) is $120$, the measure of the diameter on a flat scale; that of an angle of $60^{\circ}$ (the angle of the regular hexagon) è $60$.1.1 In general, in an angle $AOB$, the arc $AB$ is measured in degrees (that is with a unit so that the circumference measures $360$) and the chord $AB$ is measured in units such that the radius $OA$ is $60$. We now observe that since a diameter $120$ long, corresponds to a semi-circumference $60 \pi$ long , the units of measure of the arcs and of the chords are different. They would be equal if $\pi$ was equal to $3$; given the antiquity of the system of measure, it is not impossible that its origin is due to an approximation $\pi =3$ which is found in quite primitive times. It is not difficult to find the relation between the chord and the sine of an angle. In fact if we divide the angle $\alpha$ in two we have $$\begin{eqnarray*} OA &=& 60, \\ \stackrel{\frown}{AB} &=& \alpha \; \; \mbox{... ... &=& 120 \times \mathop{\rm sin}\nolimits \, \frac{\alpha}{2} , \end{eqnarray*}$$ and then, if we indicate with $c(\alpha)$ the measure of the chord $AB$, $$c(\alpha) = 120 \mathop{\rm sin}\nolimits \, \frac{\alpha}{2}$$ or $$\mathop{\rm sin}\nolimits \, \alpha = \mbox{$\frac{1}{120}$} c( 2\alpha).$$ In the first book of the Almagesto by PTOLEMY we find, among other things, a table of the chords, which precedes in half degrees from $1^{\circ}$ to $180^{\circ}$. They can be used to solve triangles. We see how to operate for a right-angled triangle $ABC$. If two sides are given, the third can be found through the Theorem of PYTHAGORAS (VI century B. C.). In order to find the angles, note that a right angled triangle can be inscribed to a circumference with a diameter equal to the hypotenuse. If we now choose a unit of measure so that the hypotenuse $AC$ measures $120$, the side $BC$ will be the side corresponding to the angle at the center $2\alpha$; from the table of the chords $2\alpha$ can be obtained, and so $\alpha$. Naturally, one could avoid to bring back to the hypotenuse with a length of $120$; it will be enough to find the arc $2\alpha$ corresponding to the chord $BC \times \frac{120}{AC}$. The procedure is the same if one side and one angle are known; for example the hypotenuse $AC$ and the angle $\alpha$. Having chosen a unit of measure so that $AC=120$, from the table of the chords one can have the chord $BC$ corresponding to the angle $2\alpha$, and therefore the side $AB$ with the Theorem of PYTHAGORAS. Once again, avoid going back to the hypotenuse $120$; and in fact we have $$BC = \frac{AC}{120} c(2\alpha).$$ In this manner all right angled triangles can be solved. General triangles were then brought back to right angle triangles. More complex is the construction of the table of the chords. These are, in fact, known only for few angles (that of $60^{\circ}$, as we saw , or that of $36^{\circ}$, the chord of which is the side of the regular decagon), in too small a number to be sufficient to compile tables precise enough. These values, known exactly or with extreme precision, are used as starting points. To find the others, formulae that are analogous to the ones of addition and subtraction and to those of bisection, are necessary. The key to obtain such results is a theorem, which was found for the first time in the Almagest, and which is known as the theorem of Ptolemy. It says that in a quadrilateral inscribed in a circle, the product of the multiplication of the diagonals is equal to the sum of the products of the two opposite sides. In formulae, given the quadrilateral $ABCD$ in the figure, we have $$\overline{AC} \times \overline{BD} = \overline{AB} \times \overline{CD} + \overline{AD} \times \overline{BC}.$$ Tolomeo uses this theorem in the case where $AD$ is a diameter. Supposing $\alpha = \stackrel{\frown}{AB}$ and $\beta = \stackrel{\frown}{AC}$, we have $\stackrel{\frown}{BC}= \beta- \alpha$, $\stackrel{\frown}{BD}=180-\alpha$ e $\stackrel{\frown}{CD}=180-\beta$. From the theorem of Ptolemy it then follows that $$c(\beta) c(180-\alpha) = c(\alpha) c(180-\beta)+120 \, c(\beta-\alpha).$$ (1.1) On the other hand, the triangles $ACD$ and $ABD$ are right angled, being inscribed in a circumference; for the theorem of Pythagoras we then have $$\begin{eqnarray*} \overline{AC}^{2}&+&\overline{CD}^{2}=\overline{AD}^{2}=120^{... ...overline{AB}^{2}&+&\overline{BD}^{2}=\overline{AD}^{2}=120^{2} \end{eqnarray*}$$ and therefore $$\begin{eqnarray*} c^{2}(180-\beta)&=&120^{2}-c^{2}(\beta)\\ c^{2}(180-\alpha)&=&120^{2}-c^{2}(\alpha). \end{eqnarray*}$$ Consequently, in the (1.1) all the terms are known, and the chord of $\beta-\alpha$ can be obtained: $$c(\beta-\alpha)= \frac{c(\beta)c(180-\alpha)-c(\alpha)c(180-\beta)}{120}.$$ (1.2) The (1.2) is equivalent to the theorem of subtraction of the sines; in fact if we remember that $c(\alpha) = 120 \mathop{\rm sin}\nolimits \, \frac{\alpha}{2}$, we could write it in the form $$120 \mathop{\rm sin}\nolimits \, \frac{\beta-\alpha}{2} = \fr... ...s 120 \mathop{\rm sin}\nolimits \, \frac{180-\beta}{2}}{120}$$ from which, to make things simple $$\mathop{\rm sin}\nolimits \, \frac{\beta-\alpha}{2} = \mathop... ...{\alpha}{2} \mathop{\rm sin}\nolimits \, (90-\frac{\beta}{2}),$$ or rather, given that $\mathop{\rm sin}\nolimits \, (90-\frac{\alpha}{2})=\cos \, \frac{\alpha}{2}$, $$\mathop{\rm sin}\nolimits \, \frac{\beta-\alpha}{2} = \mathop... ...\rm sin}\nolimits \, \frac{\alpha}{2} \cos \, \frac{\beta}{2}.$$ With a similar method, the formula of the bisection can be found If in fact we take $\beta = 2\alpha$, we have from the (1.1) $$\begin{eqnarray*} c(2\alpha) c(180-\alpha) &=& c(\alpha) c(180-2\alpha)+120 \, c(\alpha) = \\ &=& c(\alpha) (120+c(180-2\alpha)). \end{eqnarray*}$$ Raising both terms to the second power , and considering that $c^{2}(180- \alpha)=120^{2}-c^{2}(\alpha)$, we find $$c^{2}(2\alpha) (120^{2}-c^{2}(\alpha)) = c^{2}(\alpha)(120+c(180-2\alpha))^{2}$$ and solving $$c^{2}(\alpha) = \frac{120^{2} c^{2}(2\alpha)}{[120+c(180- 2\alpha)]^{2}+c^{2}(2\alpha)}.$$ This last one is already a formula of bisection, since from the chord of the arc $2\alpha$ can be obtained the one of the arc $\alpha$. But we could obtain a better formula evolving the denominator : $$\begin{eqnarray*} (120+c(180-2\alpha))^{2}+c^{2}(2\alpha)&=&120^{2}+c^{2}(180-2... ...lpha) = \\ & & \hspace{-5cm}= 2 \times 120 (120+c(180-2\alpha)) \end{eqnarray*}$$ and therefore in conclusion $$c^{2}(\alpha) = \frac{60 c^{2}(2\alpha)}{120+c(180-2\alpha)} .$$ (1.3) Through this equation, PTOLEMY can calculate the chords corresponding to smaller and smaller angles. In particular, knowing the angles of $60^{\circ}$ and of $72^{\circ}$ he obtains the chord of $12^{\circ}$, and then with the following division, the ones of $6^{\circ}$, of $3^{\circ}$, of $1^{\circ} \; 30^{\prime}$, and of $45^{\prime}$, that he used to make the table of the chords. What is missing is only the calculus of the chord of $1^{\circ}$, that PTOLEMY obtains approximately, using a result, which in different forms is already found in ARISTARCHUS of Samo (III century B. C.) and ARCHIMEDES from Syracuse (287-212 B. C.), and that says that for two arcs $\alpha$ e $\beta$, con $\alpha>\beta$, results $$\frac{c(\alpha)}{c(\beta)}<\frac{\alpha}{\beta}.$$ Applying this inequality to angles $\alpha=1^{\circ} \; 30^{\prime}$ and $\beta=1^{\circ}$ one finds $$\frac{c(1^{\circ} \; 30^{\prime})}{c(1^{\circ})} <\frac{3}{2},$$ while if one takes $\alpha=1^{\circ}$ and $\beta=45^{\prime}$, is obtained $$\frac{c(1^{\circ})}{c(45^{\prime})} < \frac{4}{3}$$ and then $$\mbox{$\frac{2}{3}$} c(1^{\circ} \; 30^{\prime}) <c(1^{\circ} )<\mbox{$\frac{4}{3}$} c(45^{\prime}),$$ a formula that allows to find the chord of $1^{\circ}$ with an approximation of less that 0,1 percent. Really the tables of PTOLEMY $\frac{1}{2}^{\circ}$ starting from the one of $1^{\circ}$.

Exercise.

1.1   Get the formulas of addition and duplication of the chords.

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