The logarithms.

A decisive impulse to the developments of trigonometric techniques comes from the invention of logarithms by JOHN NAPIER (1550-1617), which greatly simplified computations. Multiplications and divisions were reduced to additions and subtractions, the powers and the extractions of roots were reduced to products and quotients by an integer. It was soon noticed that even trigonometrical calculus could be significantly simplified by a combined use of the tables of circular functions and of the logarithmic ones. For example, the solution of a triangle, with a given side and angles, for which the theorem of the sines is used:

$\begin{displaymath}b = \frac{a \mathop{\rm sen}\nolimits \, \beta}{\mathop{\rm sen}\nolimits \, \alpha} \end{displaymath}$

could be simplified taking the logarithms:

$\begin{displaymath}\log b = \log a + \log \mathop{\rm sen}\nolimits \, \beta - \log \mathop{\rm sen}\nolimits \, \alpha, \end{displaymath}$

since instead of a multiplication and a division, one only need perform an addition and a subtraction. From here the development of trigonometry moved in two directions: on one side tables were needed which reported directly the logarithm of the sine and of the other trigonometric functions, so that it was no longer necessary to first find the sine in the trigonometric tables and then its logarithm in the logarithmic ones. In other words, to the usual tables of sines and other circular functions were added those of their logarithms, thus creating logarithmic-trigonometric tables. On the other side, for these tables to be used with maximum profit, it was necessary to employ trigonometrical formulas containing only additions and subtractions (in which case there was no need for logarithms), or only multiplications and divisions, that with the use of logarithms were reduced to the first ones. On the contrary, mixed formulae, in which appeared both sums and products, were less practical because they could not be reduced to additions and subtractions. Thus for each problem one needs to search for the form of the solving formulas that would allow the use of logarithms. For example, in the solution of the triangle with given sides, the formula of Carnot

$\begin{displaymath}\cos \, \alpha= \frac{b^{2}+c^{2}-a^{2}}{2bc}\end{displaymath}$

cannot be computed by means of logarithms, and therefore it is preferable to use the formula of Briggs

$\begin{displaymath}\cos \, \frac{\alpha}{2} = \sqrt{\frac{p(p-a)}{bc}} \end{displaymath}$

that taking the logarithms becomes

$\begin{displaymath}\log \cos \, \frac{\alpha}{2} = \mbox{$\frac{1}{2}$} \log p +... ...- \mbox{$\frac{1}{2}$} \log b - \mbox{$\frac{1}{2}$} \log c. \end{displaymath}$

Needless to say, the appearance of calculating machines made all these precautions totally useless, since with them to multiply is as easy as to add.

Index: brief history of trigonometry

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