5. Since all sines and cosines are, generally, less than 1, their logarithms to base 10 are negative. In order to avoid the inconvenience of printing negative characteristics, the logarithms to base of 10 of all goniometrical quantities have been increased by 10, and the resulting numbers being registered are called "Tables of logarithmic sines, cosines, &c."

Hence if any one of these tabular logarithmic quantities be given, by subtracting 10 from it we get the real logarithm of the goniometric quantity.

These tabular logarithmic quantities we shall indicate by the letter L; thus the tabular logarithmic sine of 4, or 10 + 110 sin A, will be written L sin A.

6. The common Logarithmic Tables of goniometric functions are calculated for angles which differ from one another by one minute. If, besides degrees and minutes, the angle contain some seconds, its tabular logarithmic function may, with certain exceptions, be found on the principle proved in the next Article.

7. The increments of tabular logarithmic functions of angles vary, except in certain cases, as the increment of the angle.

Let the angle A receive the increments 84", and 60", successively.

Then Sin (A + SA′′) = sin A . {

1 +

sin (A + dA′′) – sin A - 4

sin A

.. 110 sin (A + ♪ A′′) = 120sin A + 1,0(1 + cot A. sin d'A′′) ;

.. {10+]1sin (A+S4′′) } − { 10+] 10 sin A} = 110(1+cot A . sin ♪♫′′) ;

.. L sin (A + SA′′) - L sin A

1,10 {cot 4. sind 4′′ – 1 (cot 4)2. (sin d'4′′)*+.....} App. 1. (ii.)

neglecting the higher powers of cot A. sin dA", which may be done unless A = 2n. 90° nearly.

And writing 60" for 84" in this equation, it becomes.

And in a manner exactly similar it may be shewn that for the Cosine, Tangent, &c., of an angle, the increment of the tabular logarithm varies as the increment of the angle, except in those cases mentioned in the Corollaries to Arts. 60, 61, 62, 63.

8. To explain the meaning and use of the columns of differences for one second (Diff. 1"), which are placed after the columns of logarithmic functions.

If SA" become 1", the equation of Art. 7. becomes,

L sin (4+1′′) - L sin A = {L (sin A + 60') - L sin 4}.

1

A} 60

which is the difference of L sin A corresponding to one second.

Now, if this quantity be computed and registered, when SA is given, L (sin A + d A′′) − L sin A may be determined by

merely multiplying this registered difference by 84,—and when L sin (4 + 84′′) is given, d▲ may be found by dividing L sin (A + 84") – L sin A by this difference.

– =

For

– 4} .

L sin (4 + d 4′′) − L sin A = {L sin (4 + 60′′) − L sin ▲} .

SA

60

.84,

and 8A =

4130

=

68.833, which is the quantity put down in the

60

Now

Tables as the difference for one second to the sines of angles between 17° and 17o,1′.

The significant part of the difference is considered as a whole number, or the real difference is multiplied by 107, to avoid the necessity of printing the three or four cyphers which nearly in every case precede the significant part of the difference*.

The column of differences for the natural sines &c. of angles are computed after a manner similar to this, and the differences themselves are all multiplied by 1000, for the sake of leaving out the cyphers immediately following the decimal point.

Ex. 2. If L sin A = 9.4685537, required A.

1468

Now Diff. for 1" is in this case 68.400, and

21.46;

68.400

.... A = 17, 6, 2146.

Ex. 3.

If L cos A= 9.9784328, required A.

In this case, because the increase of the angle is attended by the decrease of the L cosine, Art. 61, Cor. 2, we subtract the given L cosine from that in the tables which is next greater than it.

Ex. 4. Required the L cosine of 72o, 5′, 8′′.

By the Tables, p. 173.

The difference for the seconds being in this case subtracted from L cos 72°, 5′. Art. 60.

NOTE. It may here be observed, that the difference for additional seconds must be added for L sines, L tangents, and L secants, Arts. 60, 62, 63; and subtracted for L cosines, Art. 60, L cotangents and L cosecants.

10. To shew that the same columns of " Differences for 1"" serve for L sin A and L cosec A, for L cos A and L sec A, and for L tan A and L cot A.

L sin A = 10 + 110 sin 4 = 20 - (10+1, cosec A)=20-L cosec A.

Similarly, L sin (A + 1′′) = 20 − L cosec (A + 1′′)

‚·. Į sin (A + 1′′) – Į sin A

=

{L cosec (A + 1′′) – L cosec A}.

Hence a column of "differences for 1"" is printed between the columns of logarithmic sines and cosecants; serving to the former as a column of increments for 1", and to the latter as a column of decrements for 1".

In like manner it may be shewn that

L cos (4+1′′) - L cos A = - {L sec (A+1') - L sec A}

L tan (4 + 1′′) − L tan A

==

{L cot (4+1")- L cot A}.

Wherefore the columns of cosines and secants have the same differences for 1", as also have the tangents and cotangents: and it is to be observed that these differences serve respectively as increments to the secants, (Art. 62.), and to the tangents (Art, 63.), and as decrements to the cosines (Art. 61.), and to the cotangents.

11.

Before the increment of a tabular logarithmic function of an angle can be determined from the small given increment of the angle, or conversely, the two following conditions must be fulfilled.

I. The logarithmic increment of the function must, in that particular case, vary as the increment of the angle.

II. The increment of the logarithmic function must not be an exceedingly small quantity.