places, which proves the proposition.

Similarly, cos (0 + h) — cos 0 = − h sin 0, approximately.

147. To prove the Rule for a Table of Natural Tangents.

If h is the circular measure of a very small angle,

tan h = h nearly.

h sec2 0

.. tan (0+ h) — tan 0

=

1-h tan 0

=h sec20+ h2 sin 0 sec3 0.

..tan (0+ h) tan 0= h sec2 0, approximately, unless sin sec3 is large, which proves the proposition.

Similarly, cot (0 − h) — cot 0 = h cosec2 0, approximately.

Sch. 1. If h is the circular measure of an angle not > 1', then h is not > .0003. Hence the greatest value of h2 sin sec3 is not>.00000009 sin 0 sec3 0. Therefore, when >

π

4'

we are liable to an error in the seventh place of decimals. Hence the rule is not true for tables of tangents calculated for every minute, when the angle is between 45° and 90°.

Sch. 2. Since the cotangent of an angle is equal to the tangent of its complement, it follows immediately that the rule must not be used for a table of cotangents, calculated for every minute, when the angle lies between 0° and 45°.

148. To prove the Rule for a Table of Logarithmic Sines.

If h is the circular measure of an angle not > 10", then h is not >.00005, and therefore, unless cot is small or cosec2 large, we have

logsin(0+h)— logsin 0 = ph cot 0,

as far as seven decimal places, which proves the rule to be generally true.

Sch. 1. When is small, cosec is large. If the log sines are calculated to every 10", then h is not >.00005, and μ is not >.5.

In order that this error may not affect the seventh decimal place, 6 cosec2 must not be > 103, that is, ◊ must not be less than about 5°.

When is small, cot is large. Hence, when the angles

are small, the differences of consecutive log sines are irregular, and they are not insensible. Therefore the rule does not apply to the log sine when the angle is less than 5°.

Sch. 2. When

is nearly a right angle, cot is small,

and cosec approaches unity. 0

Hence, when the angles are nearly right angles, the differences of consecutive log sines are irregular and nearly insensible.

149. To prove the Rule for a Table of Logarithmic Cosines.

In this case the differences will be irregular and large when is nearly a right angle, and irregular and insensible when is nearly zero. This is also clear because the sine of an angle is the cosine of its complement.

150. To prove the Rule for a Table of Logarithmic Tangents.

tan (0 + h) — tan 0 = h sec2 0 + h2 sin 0 sec3 0. (Art. 147)

151. Cases where the Principle of Proportional Parts is Inapplicable.

It appears from the last six Articles that if h is small enough, the differences are proportional to h, for values of ✪ which are neither very small nor nearly equal to a right angle.

The following exceptional cases arise :

(1) The difference sin (0+ h) — sin 0 is insensible when O is nearly 90°, for in that case h cos is very small; it is then also irregular, for h2sin◊ may become comparable with h cos 0.

(2) The difference cos (0+ h) — cos is both insensible and irregular when 0 is small.

(3) The difference tan (0+ h) — tan is irregular when 0 is nearly 90°, for h2 sin 0 sec3 may then become comparable with h sec20; it is never insensible, since sec◊ is not <1.

(4) The difference log sin (0+ h) — log sin 0 is irregular when is small, and both irregular and insensible when ✪ is nearly 90°.

(5) The difference log cos (0+ h) - log cos 0 is insensible and irregular when is small, and irregular when is nearly 90°.

(6) The difference log tan (+ h) — log tan is irregular when is either small or nearly 90°.

A difference which is insensible is also irregular; but the converse does not hold.

When the differences for a function are insensible to the number of decimal places of the tables, the tables will give the functions when the angle is known, but we cannot use the tables to find any intermediate angle by means of this function; thus, we cannot determine from the value log cos 0, for small angles, or from the value log sin 0, for angles nearly 90°.

When the differences for a function are irregular without being insensible, the approximate method of proportional parts is not sufficient for the determination of the angle by means of the function, nor the function by means of the angle; thus, the approximation is inadmissible for log sin 0, when is small, for log cos 0, when is nearly 90°, and for log tan in either case. (Compare Art. 81.)

In these cases of irregularity without insensibility, the following three means may be used to effect the purpose of finding the angle corresponding to a given value of the function, or of the function corresponding to a given angle.*

152. Three Methods to replace the Rule of Proportional Parts.

(1) The simplest plan is to have tables of log sines and log tangents, for each second, for the first few degrees of the quadrant, and of log cosines and log cotangents, for each second, for the few degrees near 90°. Such tables are generally given in trigonometric tables of seven places; we can then use the principle of proportional parts for all angles which are not extremely near 0° or 90°.

(2) Delambre's Method. In this method a table is constructed which gives the value of log +log sin 1" for

sin 0

0

every second for the first few degrees of the quadrant.

*This article has been taken substantially from Hobson's Trigonometry.