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Logarithmic Sines, Tangents, and Secants.
This table contains the logarithmic, or, as they are sometimes called, the artificial sines, tangents, and secants, to each degree and minute of the quadrant, with their complements or co-sines, co-tangents, and co-secants, to six places of figures besides the index.
To find the Logarithmic Sine, Co-sine, &c. of any JWumber of Degrees and JMinutes.
If the given degrees be under 45, they are to be taken from the top, and the minutes from the left side column, opposite to which in that column with the name of the legarithm at the top, will be found the required logarithm. But if the degrees be more than 45, they will be found at the bottom of the page, and the minutes in the right side column ; likewise the name of the logarithm is to be taken from the bottom of the page.
When the given degrees exceed 90, they are to be subtracted from 180 degrees, and the logarithm of the remainder taken out as before. Or the logarithmic, sine tangent, &c. of degrees more than 90, is the logarithmic co-sine, co-tangent, &c. of their excess above 90 degrees.
sine of 108 36
oo." logarithm, Required the log, sine of 36 82 - 9.774729 - - co-sine of 61 18 - 9.681443
- tangent of 54 17 - 10,143263
- co-tang. of 42 50 - 10.032877
- secant of 19 27 • * 10,025519
- - 10,025519
To find the Degrees and JMinutes nearest corresponding to a given Logarithmic Sine,
Look in the eolumn marked at the top or bottom with the name of the given logarithm, and when the nearest to it is found, the corresponding degrees and minutes will be those required, observing that when the name is at the top of the column, the degrees are to be taken from the top and the minutes from the left side column, but if the name is at the bottom, the corresponding degrees will be there likewise, and the minutes in the right side column.
The logarithmic sines, &c. taken out to degrees and minutes only are in general sufficiently accurate, but in some of the more rigid astronomical calculations, it is frequently necessary to take them out to the nearest second ; when this is the case they are to be found in the following manner:
To find the sine, tangent, &c. of an arch expressed in degrees, minutes and seconds.
Find the sine, tangent, &c. answering to the given degree and minute, and also that answering to the next greater minute; multiply the difference between them by the given number of seconds, and divide the product by 60; then the quotient added to the sine, tangent, &c, of the given degree and minute, or subtracted from the co-sine, co-tangent, &c. will give the quantity required, nearly.
If the arch be less than three degrees it will be necessary to use the following rule :
To the arithmetical compliment of the given degrees and minutes reduced to seconds, add the logarithm of the given degrees, minutes, and seconds, reduced to seconds, and the log: sine, tangent, &c. of the given degrees, and minutes, the sum, rejecting 10 from the index, will be the log sine, tangent, &c, of the proposed number of degrees, minutes, and seconds.
The arithmetical compliment of any log, is found as in the common log. but when the index is 10 or greater than 10, the left hand figure of it must be rejected.
Required the arithmetical complement of 9,265390. -
For the first figure 9, write 0; for 2, 7; for 6, 3; for 5, 4; for 3, 6; for 9, 1 ; and for 0, 9; thus the arithmetical complement is 0.234610,
In the same manner, the arithmetical complement of 9.528461 is 0.471539, the arithmetical complement of 9.701560 is 0.298440, and the arith. complement offio.254130 is 9.745870.
To find the degrees, minutes, and seconds, answering to a given logarithmic sine, tangent,&c. RULE.
Find the degrees and minutes answering to the next less logarithmic sine, tangent, &c. which subtract from that given; multiply the remainder by 60, and divide the product by the difference between the next less and next greater logarithms, and the quotient will be the seconds to be annexed to the degrees and minutes before found; or, which is the same, as the difference between the next less and the next greater logarithms is to the difference between the next less and the given logarithms, so is 60 seconds to a number of seconds to be annexed to the number of degrees and minutes, answering to the less logarithm found.
1. Find the degrees, minutes, and seconds (less than 90) answering to the log sine 9,828846 next less log. 42° 23' 9.828716 given log. 9.828840 greater 42 24, 9,828855 next less 9.828716
2. Required the degrees, minutes, and seconds answering to the log tangent 9.975120.
next less log. 43° 21' 9.974973 given log. 9.975120
As 243: 147 : : 60’’: 32” which annexed to 43° 21' gives 43° 21' 32” the degrees, minutes and seconds required.
If the given logarithm is that of the sine or tangent of a small arch—then, to the arithmetical complement of the next lesslogarithm in the tables, add the given logarithm and the logarithm of the degrees and minutes, in seconds, answering to the next less logarithm, the sum, rejecting radius, will be the logarithm of the number of seconds in the required arch.
* This difference is to be added when the log, of the degrees and minutes next below is less than the log. next above the given sine, tangent, &c.; but when greater it is to be subtracted.
i In this last example the index is 10, therefore I reject the left hand figure, and the remaining figures, 9,254130, are to be subtracted from 10.000000 according to the Rule.