LOGARITHMS AND TRIGONOMETRICAL RATIOS 83

cesses, yet it can be calculated merely by dividing the number’s
logarithm by five and then treating the result as the anti-
logarithm of the root, which may easily be found from the
tables.

The invention of logarithms by Napier was undoubtedly
influenced, if not directly encouraged, by the discovery of
certain relationships between the numerical values of trigo-
nometrical ratios and is, itself, an example of the ways in
which the purely ‘number’ branch of mathematics is related
to the constructional branches such as geometry and trigono-
metry. We have already mentioned Pythagorean numbers
in Chapter 2; these were found to be related to the geo-
metric construction of right-angled triangles, and it is
from the trigonometrical ratios of these same triangles that
was first discovered a way of dealing with mathematical
processes akin to that upon which the theory of logarithms
is based.

A ratio is the result of comparing two quantities of the
same kind by dividing one by another and expressing the
result as a fraction. In trigonometrical ratios, the two quan-
tities compared are the lengths of the sides of right-angled
triangles and the results obtained are expressed as relatives
of the angles contained between the two sides compared in
any one ratio. These ratios are called sines, cosines and tan-
gents, and for any triangle ABC, such that the angle C'is a
right-angle, the respective ratios are calculated as follows:

B
o
‘ e
g Q
0' SN
tz*‘ X
W 3
9 &
3N
A Base ¢

BB AG LR RSN
WIS

B

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