exponent ò, the number is from 1 to 10; if the exponent is 2, the number is from 10 to 100; if 3, from 100 to 1000, &c. But, as the following table confifts only of 4 places, or from I to 10,000, which is for the moft part fufficient; but, if not, by the following eafy proportion, it may be extended to any num ber of places thought neceffary. If of 5 places, find the first four places, three of which are in the column below N, and their log. in the column next, on the right hand, in the column below o; and the others below the fourth figure, which will be found along the top of the column, except fuch figures as are in common with thofe in the first columns; but, if the fourth figure to the right hand is a cypher, its log. is the fame as the three firft figures, but the exponent 3; the fame, if all the places to the right hand of the fignificant figures are cyphers, only putting their exponent according the number of places, as before directed; but, if the figures to the right hand are fome fignificant figures, the first four being found, find the log, of the next greater; then, as that difference is to 1, and cyphers to the number of places, fo is the figure given to the log. of these figures.

Example 1. What is the log. of 7675? The log. of 47670 is is 4.678452, the next greater is 47 80-4-6-83362; then, as 10, the diff. of the numbers, is to 910, the diff. of their log. fo is, the number wanted, to 455, the log. to be added to the log. 4 6782452 = 4.6782007.

Ex. 2. What is the log. of 47675178? The log. of 47670000 is 7.6782452, next greater, viz. 47680000, is 7.6783362; then, as 10000, the diff. of the number, is to 910, the diff. of the log. fo is 5478 to 498, the log. to be added to 7.6782452=7.6782950.

Any Logarithm being given, to find the number anfwering to it.

Ex. What is the number anfwering to the log. of 7.5571689? which number, from the exponent, muft confift of 8 places. The nearest log. in the table is .5571461, their difference is 228; the difference between the log. found and next greater, is 1204. Then, as 1204 is to 10000, the diff. in places from the number found in the table, and that wanted; so is 228 to 1893.6; which, placed to the right hand of the number already found, is the number anfwering to the given log. thus 36071893.6; and fo of any other.

If the log. of a fraction is wanted, fubtract the log. of the denominator from the log. of the numerator; the remainder is the log fought; but its exponent negative the fame of a decimal fraction.

Y

1

If the log. of a mixed number is wanted, find the log. of the whole number, and the fraction, as above, whose log. add to the log. of the whole number, or reduce the mixed number to an improper fraction, and find its log. as above. If the log. of a whole number, and decimal fraction is wanted, find the log. as if all were whole numbers; but prefix the exponents only for the whole numbers. If, at any time, the hyperbolic, or Neper's log. is wanted, the modulus betwixt which and Brigg's log. is the decimal .434294481903, &c. Suppole the hyperbolic log. 10 is wanted, divide Brigg's log, of 10=1,0000000 by the above modulus, gives 2,302585092994; the fame of any other. If the hyperbolic log. is given, being multiplied by the modulus, gives Brigg's log.

Of SINES, TANGENTS, &c.

THE radius of the circle being fuppofed divided into any number of parts, the fines of fuch a number of these parts as the arches, of which they are the fines, are of a quadrant; and, as the tangent of 45° is equal radius, the parts of the tangent above 45° will be proportionally greater than radius: The fame of fecants, as they are always greater than radius; but the fines being given, the tangents and fecants can be found from the proportion given page 150; the verfed fine may be found thus, because the cofine of any arch + the verfed fine, is equal to rad. Therefore, from rad. fubtract the cofine, gives the versed fine required. The above are natural fines, tangents, &c.

The log. fines, tangents, &c. are only the log. of the natural fines, tangents, &c. fo that from the log. tables of numbers, find the log. anfwering the number of the log. fine, tangent, fecants, verfed fines, gives the log. fine, tangent, &c. of the arch required.

Ex. If the log. fine of 50° is required, its natural fine is .7660,444, of fuch parts as the rad. is 1.000,000, the log. of which fine is 9.8842540; and as many places as the natural fine wants of 1. fo many units will the log. fine want of index 9; and as many places as the natural tangent or fecant exceeds units, fo many units will the log. tangent, or fecant, exceed index ro; but the log. fine, tangent, &c. may be found without the log. tables, by an infinite feries, the rad. of the log. being 10.0000000; the fame of any other fine. If the tangent of any a fchol. 1. arch, as of 50° is wanted, then,becaufe the tangent of any arch is pl. trig. a fourth proportional to the cof. fine, and rad. *, therefore adding the log fine to rad. and, fubtracting its cofine, gives the tangent of that arch.

Ex. Log. S. of 50° + rad. 19.8842540-log. of 40°9.8080675 10.0761865 tang. 50°; and, because the fecant is a third proportional to the cofine and rad. therefore, from twice rad. fubtract the cofine of 50°, gives 10.1919325, the fecant of 50°. If the verfed fine is wanted, fuppofe of 50°, fubtract its cofine from rad. gives 3572.124, its natural fine; its log. fine, found, as before directed, is 9.5529265: The fame thing of any other. But, becaufe the verfed fine of an arch above 90° is frequently wanted, the excefs of the arch above rad. + rad. = the verfed fine above 90°, Suppose the verfed fine of 130° is wanted, its excefs above rad. is 40°, the natural fine of which is 6427.876+ R. 16427.876, its log. is 10.2155814; or, without the natural fines, thus, the log. of 2 30103000 + twice the log. fine of half the arch, 20.2155814 rad. 10.2155814= log. verfed fine of 130°.

In the table of fines, tangents, &c. begin, as usual, in all tables of the like nature with one minute, increafing to 45°, the degrees below 45° are placed on the left fide of the page; the minutes anfwering to them increase downwards; thofe above 45° are placed on the right fide of the page; the minutes anfwering to them increase from the bottom of the page upwards, as ufual.

To find the nat. or log. fine, tangent, &c. of fecond and third Minutes, for which a Table is calculated, for every fecond; and each 6 to 5 places, at the end of Table of fines, tangents, &c.

TO find which, find the degrees and minutes in the table of fines, tangents, &c. and take the difference betwixt that found and next greater; then, as is to that difference, fo is fecond and third minutes, required in the denomination of a minute, which is done in decimals, in the table, at the end of the tables of fines, &c. to a fourth proportional; which add to the fine or tangent found in the table of fines, &c.

Ex. If the nat. fine of 58°, 19, 22", 36", is wanted, find the fine of 58°.19', the difference betwixt which and 58°.20', the next greater is 1,528; which, x the decimal of 22", 36", as found in the table, gives 576.056+8509, 6348510,215, the decimal .056 being rejected. If the log. fine of 580 19, 22", 36", is wanted, find log. fine 58° 19', the difference betwixt which and 58° 20' is 779 × 3-7, found, as before, is 293.683 + log. finé of 58°199.9299112=9.9299405, the decimal 683 being rejected, as before: The fame of any other fine, tangent, fecant, whether above or below 45°, or of any verfed fine, whether above or below 90o.

If