EXAMPLES.

The degrees and minutes corresponding to the

log. sine 9.265390 are 10o 37'

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.

RULE.

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 complement 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.

To find the degrees, minutes, and seconds, answering to a given logarithmic sine, tangent, &c.

RULE.

Find the degrees minutes and seconds answering to the next less loga rithmic 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.

If the given logarithm is that of the sine or tangent of a small archthen, to the arithmetical complement of the next less logarithm 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.

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