CASE III.

When the number exceeds 10,000, or consists of five or more places of figures.

Consider all the figures after the fourth from the left hand, as ciphers. Find, from the table, the logarithm of the first four places, and prefix a characteristic which shall be one less than the number of places including the ciphers. Take from the last column on the right of the page, marked D, the number on the same horizontal line with the logarithm, and multiply this number by the numbers that have been considered as ciphers: then, cut off from the right hand as many places for decimals as there are figures in the multiplier, and add the product, so obtained, to the first logarithm: this sum will be the logarithm sought.

Let it be required to find the logarithm of 672887. The log. of 672800 is found, on the 11th page of the table, to be 5.827886, after prefixing the characteristic 5. The corresponding number in the column D is 65, which being multiplied by 87, the figures regarded as ciphers, gives 5655; then, pointing off two places for decimals, the number to be added is 56.55. This number being added to 5.827886, gives 5.827942 for the logarithm of 672887; the decimal part .55, being omitted.

This method of finding the logarithms of numbers, from the table, supposes that the logarithms are proportional to their respective numbers, which is not rigorously true. In the example, the logarithm of 672800 is 5.827886; the logarithm of 672900, a number greater by 100, 5.827951: the difference of the logarithms is 65. Now, as 100, the difference of the numbers, is to 65, the difference of their logarithms, so is 87, the difference between the given number and the least of the numbers used, to the difference of their logarithms, which is 56.55: this difference being added to 5.827886, the logarithm of the less number, gives 5.827942 for the logarithm of 672887. The use of the column of differences is therefore manifest.

When, however, the decimal part which is to be omitted exceeds .5, we come nearer to the true result by increasing the next figure to the left by 1; and this will be done in all the calculations which follow. Thus, the difference to be added, was nearer 57 than 56; hence it would have been more exact to have added the former number.

The logarithm of a vulgar fraction is equal to the logarithm of the numerator, minus the logarithm of the denom

mator. The logarithm of a decimal fraction is found, by considering it as a whole number, and then prefixing to the decimal part of its logarithm a negative characteristic, greater by unity than the number of ciphers between the decimal point and the first significant place of figures. Thus, the logarithm of .0412. is

2.614897.

PROBLEM.

To find from the table, a number answering to a given logarithm.

XXXII Search, in the column of logarithms, for the decimal part of the given logarithm, and if it be exactly found, set down the corresponding number. Then, if the characteristic of the given logarithm be positive, point off, from the left of the number found, one place more for whole numbers than there are units in the characteristic of the given logarithm, and treat the other places as decimals; this will give the number sought.

If the characteristic of the given logarithm be 0, there will be one place of whole numbers; if it be -1, the number will be entirely decimal; if it be -2, there will be one cipher between the decimal point and the first significant figure; if it be -3, there will be two, &c. The number whose logarithm is 1.492481 is found in page 5, and is 31.08.

But if the decimal part of the logarithm cannot be exactly found in the table, take the number answering to the nearest less logarithm; take also from the table the corresponding difference in the column D: then, subtract this less logarithm from the given logarithm; and having annexed a sufficient number of ciphers to the remainder, divide it by the difference taken from the column D, and annex the quotient to the number answering to the less logarithm: this gives the required number, nearly. This rule, like the one for finding the logarithm of a number when the places exceed four, supposes the numbers to be proportional to their corresponding logarithms.

Ex. 1. Find the number answering to the logarithm 1.532708. Here,

The given logarithm, is

Next less logarithm of 34,09, is

Their difference is

1.532708

1.532627

81

And the tabular difference is 128: hence

128) 81.00 (63

which being annexed to 34,09, gives 34.0963 for the number

answering to the logarithm 1.532708.

Ex. 2. Required the number answering to the logarithm

3.233568.

The given logarithm is

3.233568

The next less tabular logarithm of 1712, is 3.233504

Tab. Diff.=253) 64.00 (25 Hence the number sought is 1712.25, marking four places of integers for the characteristic 3.

TABLE OF LOGARITHMIC SINES.

XXXIII. In this table are arranged the logarithms of the numerical values of the sines, cosines, tangents, and cotangents, of all the arcs or angles of the quadrant, divided to minutes, and calculated for a radius of 10,000,000,000. The logarithm of this radius is 10. In the first and last horizontal line, of each page, are written the degrees whose logarithmic sines, &c. are expressed on the page. The vertical columns on the left and right, are columns of minutes.

CASE I.

To find, in the table, the logarithmic sine, cosine, tangent, or cotangent of any given arc or angle.

1. If the angle be less than 45°, look in the first horizontal line of the different pages, until the number of degrees be found; then descend along the column of minutes, on the left of the page, till you reach the number showing the minutes; then pass along the horizontal line till you come into the column designated, sine, cosine, tangent, or cotangent, as the case may be: the number so indicated, is the logarithm sought. Thus, the sine, cosine, tangent, and cotangent of 19° 55', are found on page 37, opposite 55, and are, respectively, 9.532312, 9.973215, 9.559097, 10.440903.

2. If the angle be greater than 45°, search along the bottom line of the different pages, till the number of degrees are fourd; then ascend along the column of minutes, on the right hand side of the page, till you reach the number expressing the minutes; then pass along the horizontal line into the columns designated tang., cotang., sine, cosine, as the case may be the number so pointed out is the logarithm required.

It will be seen, that the column designated sine at the top of the page, is designated cosine at the bottom; the one designated tang., by cotang., and the one designated cotang., by tang.

The angle found by taking the degrees at the top of the page, and the minutes from the first vertical column on the left, is the complement of the angle, found by taking the corresponding degrees at the bottom of the page, and the minutes traced up in the right hand column to the same horizontal line. This being apparent, the reason is manifest, why the columns designated sine, cosine, tang., and cotang., when the degrees are pointed out at the top of the page, and the minutes counted downwards, ought to be changed, respectively, into cosine, sine, cotang., and tang., when the degrees are shown at the bottom of the page, and the minutes counted upwards.

If the angle be greater than 90°, we have only to subtract it from 180°, and take the sine, cosine, tangent, or cotangent of the remainder.

The secants and cosecants are omitted in the table, being easily found from the cosines and sines.

R2

For, sec.=. ; or, taking the logarithms, log. sec.-2

COS.

log. R-log. cos. 20-log. cos. ; that is, the logarithmic secant is found by substracting the logarithmic cosine from 20. And R2 or log. cosec.=2 log. R-log. sine-20—log. sine; that is, the logarithmic cosecant is found by subtracting the logarithmic sine from 20.

cosec.

sine

It has been shown that R-tang. x cotang.; therefore, 2 log. R=log. tang.+log. cotang.; or 20=log. tang. +log. cotang.

The column of the table, next to the column of sines, and on the right of it, is designated by the letter D. This column is calculated in the following manner. Opening the table at any page, as 42, the sine of 24° is found to be 9.609313; of 24 1', 9.609597: their difference is 284; this being divided by 60, the number of seconds in a minute, gives 4.73, which is entered in the column D, omitting the decimal point. Now, supposing the increase of the logarithmic sine to be proportional to the increase of the arc, and it is nearly so for 60", it follows, that 473 (the last two places being regarded as decimals) is the increase of the sine for 1". Similarly, if the arc be 24° 20', the increase of the sine for 1", is 465, the last two places being decimals. The same remarks are equally applicable in respect of the column D, after the column cosine, and of the column D, between the tangents and cotangents. The column D between the tangents and cotangents, answers

to either of these columns; since of the same arc, the log. tang.+log. cotang=20. Therefore, having two arcs, a and b, log. tang blog. cotang blog. tang a+log. cotang a; or, log. tang blog. tang a=log. cotang a-log. cotang b.

Now, if it were required to find the logarithmic sine of an arc expressed in degrees, minutes, and seconds, we have only to find the degrees and minutes as before; then multiply the corresponding tabular number by the seconds, cut off two places to the right hand for decimals, and then add the product to the number first found, for the sine of the given arc. Thus, if we wish the sine of 40° 26′ 28′′.

The tangent of an arc, in which there are seconds, is found in a manner entirely similar. In regard to the cosine and cotangent, it must be remembered, that they increase while the arcs decrease, and decrease while the arcs are increased, consequently, the proportional numbers found for the seconds must be subtracted, not added.

Ex. To find the cosine 3° 40′ 40′′.

Cosine 3° 40'

Tabular difference = 13

9.999110

Number of seconds = 40

Product 5.20, which being subtracted = 5.20

Gives for the cosine of 3° 40′ 40' 9.999104.80

CASE II.

To find the degrees, minutes, and seconds answering to any given logarithmic sine, cosine, tangent, or cotangent.

Search in the table, and in the proper column, until the number be found; the degrees are shown either at the top or bottom of the page, and the minutes in the side columns, either at the left or right. But if the number cannot be exactly found in the table, take the degrees and minutes answering to the nearest less logarithm, the logarithm itself, and also the corresponding tabular difference. Subtract the logarithm taken, from the