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The minus sign is placed over the characteristic to show that this alone is negative, while the decimal part of the logarithm is positive.

To find the Logarithm of a Vulgar Fraction.

We may reduce the vulgar fraction to a decimal, and find its logarithm by the preceding rule; or, since the value of a fraction is equal to the quotient of the numerator divided by the denominator, we may subtract the logarithm of the denominator from that of the numerator; the difference will be the logarithm of the quotient.

Required the logarithm of, or 0.1875.

From the logarithm of 3,

0.477121,

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To find the natural Number corresponding to any Logarithm. Look in the table in the column headed 0 for the first two figures of the logarithm, neglecting the characteristic; the other four figures are to be looked for in the same column, or in one of the nine following columns; and if they are exactly found, the first three figures of the corresponding number will be found opposite to them in the column headed N, and the fourth figure will be found at the top of the page. This number must be made to correspond with the characteristic of the given. logarithm by pointing off decimals or annexing ciphers. Thus

the natural number belonging to the logarithm 4.370143 is 23450 ; 1.538574 is 34.56.

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If the decimal part of the logarithm can not be exactly found in the table, look for the nearest less logarithm, and take out the four figures of the corresponding natural number as before; the additional figures may be obtained by means of the Proportional Parts at the bottom of the page.

Required the number belonging to the logarithm 4.368399.
On page 6, we find the next less logarithm.

.368287.

The four corresponding figures of the natural number are 2335. Their logarithm is less than the one proposed by 112. The tabular difference is 186; and, by referring to the bottom of page 6, we find that, with a difference of 186, the figure corresponding to the Proportional Part 112 is 6. Hence the five figures of the natural number are 23356; and, since the characteristic of the proposed logarithm is 4, these five figures are all integral.

Required the number belonging to logarithm 5.345678.
The next less logarithm in the table is
Their difference is

.345570

108.

The first four figures of the natural number are 2216.

With the tabular difference 196, the fifth figure corresponding to 108 is seen to be 5, with a remainder of 10, which furnishes a sixth figure 5 nearly. Hence the required number is 221655.

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TABLE OF NATURAL SINES AND TANGENTS, pp. 116–133. This is a table of natural sines and tangents for every degree and minute of the quadrant, carried to six places of figures. Since the radius of the circle is supposed to be unity, the sine of every arc below 90° is less than unity. These sines are expressed in decimal parts of the radius; and, although the decimal point is not written in the table, it must always be prefixed. The degrees are arranged in order at the top of the page, and the minutes in the left hand vertical column. Directly under the given number of degrees at the top of the page, and opposite to the minutes on the left, will be found the sine required. The two leading figures are repeated at intervals of ten minutes. Thus

the sine of 6° 27' is .112336;

66 "28° 53' is .483028.

The same number in the table is both the sine of an arc and the cosine of its complement. The degrees for the cosines must be sought at the bottom of the page, and the minutes on the right. Thus

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If a sine is required for an arc consisting of degrees, minutes, and seconds, it may be found by means of the line at the bottom of each page, which gives the proportional part corresponding to one second of arc Required the sine of 8° 9' 10".

The sine of 8° 9' is .141765.

By referring to the bottom of page 116, in the column under 8°, we find the correction for 1" is 4.80; hence the correction for 10" must be 48, which, added to the number above found, gives for the sine of 8° 9' 10", .141813.

In the same manner we find

the cosine of 56° 34' 28" is .550853.

It will be observed, that since the cosines decrease while the arcs in crease, the correction for the 28" is to be subtracted from the cosine of 56° 34'.

The arrangement of the table of natural tangents is similar to that of the table of sines. The tangents for arcs less than 45° are all less than radius, and consist wholly of decimals. For arcs above 45°, the tangents are all greater than radias, and contain both integral and decimal

figures. The proportional parts at the bottom of each page enable us readily to find the correction for seconds. Thus

the natural tangent of 32° 29′ 18′′ is .636784 ;

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To find the Number of Degrees, Minutes, and Seconds belonging to a given Sine or Tangent.

If the given sine or tangent is found exactly in the table, the corresponding degrees will be found at the top of the page, and the minutes on the left hand. But when the given number is not found exactly in the table, look for the sine or tangent which is next less than the proposed one, and take out the corresponding degrees and minutes. Find, also, the difference between this tabular number and the number proposed, and divide it by the proportional part for 1" found at the bottom of the page; the quotient will be the required number of seconds. Required the arc whose sine is .750000.

The next less sine in the table is .749919, the arc corresponding to which is 48° 35'. The difference between this sine and that proposed is 81, which, divided by 3.21, gives 25. Hence the required arc is 48° 35' 25".

In the same manner we find

the arc whose tangent is 2.000000, to be 63° 26′ 6′′.

TABLE OF NATURAL SECANTS, pp. 134-5.

This is a table of natural secants for every ten minutes of the quadrant carried to seven places of figures. The degrees are arranged in order in the first vertical column on the left, and the minutes at the top of the page. Thus

the secant of 21° 20' is 1.073561;

"81° 50' is 7.039622.

If a secant is required for a number of minutes not given in the table, the correction for the odd minutes may be found by means of the last vertical column on the right, which shows the proportional part for one minute.

Let it be required to find the secant of 30° 33'.

The secant of 30° 30' is 1.160592.

The correction for 1' is 198.9, which, multiplied by 3, gives 597 Adding this to the number before found, we obtain 1.161189.

For a cosecant, the degrees must be sought in the right-hand vertical column, and the minutes at the bottom of the page. Thus

the cosecant of 47° 40' is 1.352742.

TABLE OF LOGARITHMIC SINES AND TANGENTS, pp. 21-115.

This is a table of the logarithms of the sines and tangents for every ten seconds of the quadrant, carried to six places of decimals The de

grees and seconds are placed at the top of the page, and the minutes in the left vertical column. After the first two degrees, the three leading figures in the table of sines are only given in the column headed 0", and are to be prefixed to the numbers in the other columns, as in the table of logarithms of numbers. Also, where the leading figures change, this change is indicated by dots, as in the former table. The correction for any number of seconds less than 10 is given at the bottom of the page.

To find the Logarithmic Sine or Tangent of a given Arc.

Look for the degrees at the top of the page, the minutes on the left hand, and the next less tenth second at the top; then, under the seconds, and opposite to the minutes, will be found four figures, to which the three leading figures are to be prefixed from the column headed 0"; to this add the proportional part for the odd seconds from the bottom of the page.

Required the logarithmic sine of 24° 27' 34".

The logarithmic sine of 24° 27' 30" is 9.617033.
Proportional part for 4′′ is

Logarithmic sine of 24° 27′ 34′′ is

18.

9.617051.

This is the logarithm of .414049 found in the table of natural sines on page 120. The natural sine being less than unity, the characteristic of its logarithm is negative. Tc obviate this inconvenience, the char acteristics in the table nave all been increased by 10; or the logarithmic sines may be regarded as the logarithms of natural sines computed for a radius of 10,000,000,000.

Required the logarithmic tangent of 73° 35′ 43′′.

The logarithmic tangent of 73° 35′ 40′′ is 10.531031.
Proportional part for 3" is

Logarithmic tangent of 73° 35′ 43′′

23.

10.531054.

When a cosine is required, the degrees and seconds must be sought at the bottom of the page, and the minutes on the right, and the correction for the odd seconds must be subtracted from the number in the table. Required the logarithmic cosine of 59° 33' 47".

The logarithmic cosine of 59° 33' 40" is 9.704682.
Proportional part for 7" is

Logarithmic cosine of 59° 33' 47" is

25.

9.704657.

So, also, the logarithmic cotangent of 37° 27′ 14′′ is found to be 10.115744. The proportional parts given at the bottom of each page correspond to the degrees at the top of the page increased by 30', and are not strictly applicable to any other number of minutes; nevertheless, the differences of the sines change so slowly, except near the commencement of the quadrant, that the error resulting from using these numbers. for every part of the page will se dom exceed a unit in the sixth decimal place. For the first two degrees, the differences change so rapidly

that the proportional part for 1" is given for each minute in the righthand column of the page. The correction for any number of seconds less than ten will be found by multiplying the proportional part for 1" by the given number of seconds.

Required the logarithmic sine of 1° 17′ 33′′.

The logarithmic sine of 1° 17′ 30′′ is 8.352991

The correction for 3" is found by multiplying 93.4 by 3, which gives 280. Adding this to the above tabular number, we obtain

the sine of 1° 17' 33", 8.353271.

A similar method may be employed for several of the first degrees of the quadrant, if the proportional parts at the bottom of the page are not thought sufficiently precise. This correction may, however, be obtained pretty nearly by inspection from comparing the proportional parts for two successive degrees. Thus, on page 26, the correction for 1", corresponding to the sine of 2° 30', is 48; the correction for 1", corresponding to the sine of 3° 30', is 34. Hence the correction for 1", corresponding to the sine of 3° 0', must be about 41; and in the same manner we may proceed for any other part of the table.

Near the close of the quadrant, the tangents vary so rapidly, that the same arrangement of the table is adopted as for the commencement of the quadrant. For the last as well as the first two degrees of the quadrant, the proportional part to 1" is given for each minute separately. These proportional parts are computed for the minutes placed opposite to them, increased by 30', and are not strictly applicable to any other number of seconds; nevertheless, the differences for the most part change so slowly, that the error resulting from using these numbers for every part of the same horizontal line is quite small. When great accuracy is required, the table on page 114 may be employed for arcs near the limits of the quadrant. This table furnishes the differences between the logarithmic sines and the logarithms of the arcs expressed in seconds. Thus

the logarithmic sine of 0° 5' is 7.162696;
the logarithm of 300" (=5') is 2.477121;

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This is the number found on page 114, under the heading log. sine A-log. A", opposite 5 min.; and in a similar manner the other numbers in the same column are obtained. These numbers vary quite slowly for two degrees; and hence, to find the logarithmic sine of an arc less. than two degrees, we have but to add the logarithm of the arc expressed in seconds to the appropriate number found in this table.

Required the logarithmic sine of 0° 7' 22".

Tabular number from page 114, 4.685575.
The logarithm of 442′′ is

2.645422.

Logarithmic sine of 0° 7′ 22′′ is 7.330997

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