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Example 2. Find the number whose logarithm is 2:5354291. In the Table we find that •5354207=log 3:4310.

...(i), and •5354334=log 3:4311

(ii). Let

•5354291=log (3•4310+d) Here we have three logarithms and three numbers The difference between the first and second logs is ·0000127. The difference between the first and third logs is .0000084.

The difference between the first and second numbers is *0001.

The difference between the first and third numbers is d.
By the Rule these four differences are in proportion,

.. .0000127 : 0000084=·0001 : d, or,

127 : 84=.0001 : d;

.. d=*0001 x = '0000661, etc. Therefore from (iii) •5354291=log (3.4310+ •000066)

=log 3:431066. Hence,

2.5354291=log 343:1066, or, the required number is 343.1066.

EXAMPLES. LVIII.

(1) Find log 7.65432, having given that

log 7:6543=8839055,

log 7:6544=:8839112. (2) Find log 564:123, having given that

log 5.6412=•7513715,

log 5.6413=7513792.
(3) Find log .0008736416, having given that

log 8•7364=:9413325,
log 8.7365=.9413375.

(4) Find log 6437125, having given that

log 6.4371=:8086903,

log 6.4372=8086970.
(5) Find log 3.72456, having given that

log 37245=4:5710680,
log 37246=4:5710796.

(6) Find the number whose logarithm is •5686760, having given that

•5686710=log 3.7040,

•5686827=log 3.7041. (7) Find the number whose logarithm is 4:6602987, having given that

•6602962=log 4:5740,

•6603057=log 4:5741. (8) Find the number whose logarithm is 6.3966938, having given that

•3966874=log 2-4928,

•3967049=log 2.4929. (9) Find the number whose logarithm is 7:6431150, having given that

•6431071=log 4.3965,

•6431170=log 4:3966. (10) Find the number whose logarithm is •7550480, having given that

3:7550436=log 5689.1,
2.7550512=log 568-92.

225. The same Rule of Proportional Differences is used in the case of angles and their Trigonometrical Ratios ; and therefore also in the case of angles and the logarithms of their Ratios.

Thus the (small) differences between three angles are assumed to be proportional to the corresponding differences between the sines of those three angles; also, proportional to the corresponding differences between the logarithms of the sines of those angles.

226. Sines and cosines are always less than unity, as also are the tangents of all angles between 0 and 45°.

The logarithms of these Ratios must therefore have negative characteristics.

To avoid the inconvenience of having to print these negative characteristics, the whole number 10 is added to each logarithm of the Trigonometrical Ratios, before it is set down in the Table.

The numbers thus recorded are called the tabular logarithms of the sine, cosine, etc., of an angle.

They are indicated by the letter ‘L.'

Thus L sin 31° 15', stands for the tabular logarithm of sin 31° 15', and is equal to {log (sin 31° 15') + 10).

The words logarithmic sine are used as abbreviation for tabular logarithm of the sine.

Thus in the Tables we find

I sin 31° 15'=9.7149776.
Therefore log (sin 31° 15')=9.7149776 - 10=1:7149776.
Example 1. Find sin 31° 6' 25".
The Tables give sin 31° 6'='5165333

(i), sin 31° 7'='5167824

(ii). Let sin 31° 7' 25"=5165333+d

(üi). The difference between the first two angles is 60". The difference between the first and third angle is 25".

The differences between the corresponding sines are •0002491 and d.

By the Rule these four differences are in proportion.
Therefore 60" : 25"=.0002491 : d,

id=*0002491 x 45='0001038. Hence from (iii) sin 31° 7' 25"="5165333+.0001038=-5166371.

Example 2. Find the angle whose logarithmic cosine is 9:7858083. The Table gives 9.7857611=L cos 52° 22'... ....... (i), 9:7859249=L cos 52° 21'

..(ii). The cosine diminishes as the angle increases. Hence corresponding to an increase in the angle there is a diminution of the cosine. Hence, let 9:7858083=L cos (52° 22' D)

..(iii). Subtracting the first tabular logarithm from the second the difference is .0001638.

Subtracting the first tabular logarithm from the third, the difference is '0000472.

Subtracting the first angle from the second, the difference is -60%.

Subtracting the first angle from the third, the difference is -D.

By the Rule these four differences are in proportion.
Therefore •0001638 : •0000472= –60" ; -D,

.: D=60" x Wig=17:3". Hence 9.7858083=L cos (52° 22' – 17")

=L cos 52° 21' 43"

EXAMPLES. LIX. (1) Find sin 42° 21' 30" having given that sin 42° 21'=.6736577

sin 42° 22' =6738727.

(2) Find cos 47° 38' 30" having given that cos 47° 38'=6738727

cos 47° 39'=.6736577. (3) Find cos 21° 27' 45" having given that cos 21° 27'=.9307370

cos 21° 28'=.9306306.

(4) Find the angle whose sine is •6666666 having given that .6665325=sin 41° 48'

•6667493=sin 41° 49'. (5) Find the angle whose cosine is •3333333 having given that •3332584=

=cos 700 32 3335326=cos 70° 31',

(6) Find the angle whose cosine is .25 having given that .2498167=cos 750 32

-2500984=cos 75° 31'.

(7) Find L sin 45° 16' 30" having given that L sin 45° 16'=9.8514969

L sin 45° 17'=9.8516220. (8) Find L tan 27° 13' 45" having given that L tan 27° 13'=9.7112148

L tan 27° 14'=9.7115254

(9) Find L cot 36° 18' 20" having given that L cot 36° 18'=10.1339650

L cot 36° 19'=10:1337003.

(10) Find the angle whose Logarithmic tangent is 9-8464028, having given that 9.8463018=L tan 35° 4'

9.8465705=L tan 35° 5'.

(11) Find the angle whose Logarithmic cosine is 9.9448230, having given that 9:9447862=L cos 28° 17'

9.9448541=L cos 280 16'.

(12) Find the angle whose Logarithmic cosecant is 10:4274623, having given that 10-4273638=L cosec 21° 57'

10.4276774=L cosec 21° 56'.

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