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19. To find, in the table, a number answering to a given cogarithm.

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 be the logarithm sought (9). 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 next less logarithm; take also from the table the corresponding difference in the column D: then, subtract this less logarithm from the given logarithm ; divide the remainder 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. To find the number answering to the logarithm 1.532708. Here,

Next less log. is 1.532627, its number 34.09, the tab. diff. 128. The difference between the given log. 1.532708 and 1.532627 is 81 ; therefore, 128) 8100 (63 which, being decimals of a unit, in respect of the 9 in the number 34.09, must be annexed, and being so annexed, gives 34.0963 for the number answering to the log. 1.532708.

Ex. 2. Required the number answering to the logarithm 3.233568.

The given logarithm = 3.233568
The next less tabular logarithm of 1712 = 3.233504

64

Diff. =
Tab. Diff. = 253) 64.00 (25

2

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Hence the number sought is 1712.25, marking four places of integers for the characteristic 3.

TABLE OF LOGARITHMIC SINES.

20. 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 (9). 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.

21. 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 found; 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 inio the columns designated tang., cotang., sine, cosine, which correspond to the degrees indicated at the bottom of the page; the number so pointed out is the logarithm required.

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

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COS.

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 upward.

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

24. 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 log. R-log. cos. =20 -- log. cos.; that is, the logarithmi secunt is found by subtracting the logarithmic cosine from 20.

R? And cosec. = or log. cosec. =2 log. R-log. sine = 20

sine' -log. sine; that is, the logarithmic cosecant is found by subtracting the logarithmic sine from 20.

* It has been shown that R’=tang. X cotang; therefore, 2 log. R.=log. tang. +log. cotang; or, 20=log. tang. +log. cotang

25. 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 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 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, is equally applicable to either of these columns; since of the same arc, the log. tang. +log. cotang.=20 (24). Therefore, having two arcs, a and b, log. tang. b+-log. cotang. b=log. tang. a+log. cotang. a; or, log. tang. b-log. tang. a=log. cotang. b-log. cotang. a.

26. Now, if it were required to find the logarithmic sine of an arc expressed in degrees, ninutes, 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 sine 40° 26'

9.811952 Tabular difference = 247 Number of seconds =- 28

.

Product = 69.16, which being added =

69.16

Gives for the sine of 40° 26' 28" = 9.812021.16

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 hile the arcs are increased; conseqnently, the proportional numbers found for the seconds must be subtracted, not added.

Ex. To find the cosine 3° 40' 40".
Cosine 3° 40'

9.999110 Tabular difference

= 13 Number of seconds 40

Product = 5.20, which being subtracted = 5.80

Gives for the cosine of 3° 40' 40" 9.9991114.20 27. To find the degrees, minutes, and seconds answering to

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 table from the oiven logarithm, annex two ciphers, and then divide the remainder by the tabular difference: the quotient is seconds, and is to be connected with the degrees and minutes before found ; to be added for the sine and tangent, and subtracted for the cosine and cotangent. Ex. 1. To find the arc answering to the sine 9.880054

Sine 49° 20', next less in the table, 9.879963

Tab. Diff. 181)9100(50"
Hence to the given sign 9.880054 corresponds the arc 49°
20' 50".
Ex. 2. To find the arc corresponding to cotang. 10.008688.

The given cotang. 10.008688
Cotang. 44° 26', next less in the table, 10.008591

Tab. Diff. 421)9700(23" Hence, 44° 26' —23"=44° 25' 37" is the arc corresponding to the given cotangent 10.008688.

OF THE TRAVERSE TABLE.

8. A table, called a Traverse Table, is used in computing the area of a survey made with the compass. Its use will be here explained.

This table shows the difference of latitude and departure, corresponding to any bearing; and for any distance less than 100, the one hundred being regarded as links, chains, rods, or any other dimension.

In the table headed · Traverse Table,' the figures at the top and bottom show the bearings, to degrees and parts of a degree; and the columns on the left and right of each page, the distances

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