The Arab-Indian contribution

The Roman conquest didn't contribute in any way to the development of the mathematical sciences but neither did it hinder its continuation, especially around the school of Alexandria that continued on well beyond the Roman conquest of Egypt in the first century B.C.

After the fall of the Western Roman Empire, and cultural retreat of the Eastern one, the natural successors of the Greek geometers - at least from the IX century - were the Arabs.

Placed at the crossroads of a mathematical tradition in which the inheritance of the Egyptian and Babylonian cultures merged with the texts of classic Greek geometry and the innovations of Indian mathematicians, the Arabs quickly assimilated most of these different traditions. This they incorporated into an original method, that a few centuries later, they bequeathed to the scholars of an emerging Europe.

Some fundamental discoveries, both technological and on paper, reached the West through Arab influence, and were to be crucial in the diffusion of culture and the development of science. These are both scientific, like the use of the numeric characters commonly called Arab (which would more accurately be called Indian), and the positional notation. ^1.2

The first innovation related to Alexandrine trigonometry came from India: the use of the sine instead of the chord. The first work containing the table of the sines, which dates from the IV or V century of our era, is known by the name of Surya Siddhanta. There we find the calculus of sines of the angles' multiple of $3^{\circ} \; 45^{\prime}$ , until $90^{\circ}$ . Indian astronomers added the cosine to the sine and also the sine "towards" (a circular function equal to $1-\cos \, \alpha$ , today no longer in use) and probably the tangent and the cotangent.

Indian astronomic texts, containing the tables of the sines, had already been translated into Arab during the eighth century A.D. at the court of the sultan of Baghdad, AL-MANS¯UR. The Arab astronomers systematically studied circular functions, and introduced important changes and improvements.

On the other hand, besides astronomy reasons, trigonometry - and spheric trigonometry in particular - was especially important for religious reasons. It is well known that Muslims recite their prayers in the direction of the Mecca, the native town of Mohammed. In the Arab world, the direction of the Mecca, the Qibla, is indicated by a niche, the mihrab. This is traced on all public sundials, and its direction is determined by the solution of the spheric triangle with Mecca and the North Pole as vertexes, starting from the knowledge of the latitude and longitude of the specific location and of Mecca.

Let's return to trigonometric functions. The cosine was simply defined as the sine of the complementary arc: $\cos \, \alpha = \mathop{\rm sen}\nolimits \, (90^{\circ} -\alpha)$ ; in general there were no tables of the cosine, since it could be found directly in the tables of the sines.

The tangent and the cotangent were instead related to gnomonics, the science of sundials. The tangent, in particular, is the shade that a gnomon (a pole fixed perpendicularly on a vertical wall) with length projects a shadow onto a wall for a given height of the sun.
In correspondence, the cotangent is the shade of the gnomon vertically fixed on a horizontal plane. In both cases, the angle $\vartheta$ is the height of the sun on the horizon, which could thus way be determined by measuring the shadows.

Similarly, the secant and the cosecant represent the hypotenuse of the triangles having the gnomon and its shade as cathets.

In this manner the tangent and the cotangent (and the secant and the cosecant) are originally connected to the construction of, respectively, horizontal and vertical sundials. Among these only the tangent is tabulated, since, as for the cosine, it is soon recognized that the cotangent is the tangent of the complement.

It is worth remembering that the original terms to identify these two functions were zill and zill màkus, translated in Latin as umbra recta and umbra versa. The term tangent was only introduced in by T. FINK (1561-1656), the one of the cotagent in by E. GUNTER (1581-1626).

Once these functions were introduced, it was necessary to prepare their tables, and to improve the existing ones. Many Arab mathematicians and, later, Europeans worked very hard on this matter. Among them, we'll only discuss AL- KASH¯I (XV century), because his method is connected to the third degree equation.

As we already mentioned , PTOLEMY finds an approximation of the chord corresponding to $1^{\circ}$ . An improvement in precision could not provene from the method PTOLEMY'S, method without replacing the angles of $1^{\circ} \; 30^{\prime}$ and $45^{\prime}$ with the other angles nearer to $1^{\circ}$ , with which it is possible to calculate exactly the chords (or sines). This can be done by continuing the subdivision of the angle of $45^{\prime}$ , up to angles as small as necessary, and then use the formulas of addition to calculate the sine of the multiples of this last quantity. $1^{\circ}$ . This is though a very long operation, and mainly useless, since one calculates the sines of numerous small angles, that do not finally appear in the tables. AL-KASH¯I'S idea is, instead, based on the formula that gives the sine of $3\vartheta$ in terms of the one of $\vartheta$ , analogous to that for the cosine:

$\begin{displaymath}\mathop{\rm sen}\nolimits \, 3\vartheta = 3 \mathop{\rm sen}\nolimits \, \vartheta -4\mathop{\rm sen}\nolimits ^{3}\vartheta. \end{displaymath}$

Now, taking $\vartheta=1^{\circ}$ , and assuming for the sake of simplicity $x=\mathop{\rm sen}\nolimits \, 1^{\circ}$ , the previous relation can be written in the form

$\begin{displaymath}x = \frac{4x^{3}+\mathop{\rm sen}\nolimits \, 3^{\circ}}{3}. \end{displaymath}$

On the other hand, $a=\mathop{\rm sen}\nolimits \, 3^{\circ}$ is known with arbitrary precision, since it can be obtained with the bisection and therefore is reduced to the calculus of the roots. It is then a matter of solving the equation

$\begin{displaymath}x = \frac{4x^{3}+a}{3}. \end{displaymath}$

At first sight, it seems that a dead end is reached, since there wasn't, at the time, any method to solve the third degree equation. But AL-KASH¯I'S had a brilliant idea . The figure $x=\mathop{\rm sen}\nolimits \, 1^{\circ}$ is rather small, and therefore it is even smaller when elevated to the third power. We can then find a first approximation for

simply ignoring the term $x^{3}$ :

$\begin{displaymath}x_{1}=\frac{a}{3}. \end{displaymath}$

At this point we can obtain a second approximation

$\begin{displaymath}x_{2}= \frac{4x_{1}^{3}+a}{3}, \end{displaymath}$

then a third with the value of $x_{2}$ , and so on. Thus way it is possible to find $\mathop{\rm sen}\nolimits \, 1^{\circ}$ with the necessary approximation, and above all, to improve the approximation it is sufficient to take value that was first obtained and start the process again from that point.

Index: brief history of trigonometry

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