because the index of the given log. is 5, its number will be 125605.7.

From what has been said on this head, the following problems may be easily solved by logarithms, viz.

The number answering to which sum, viz. 3430, is nearly the product of 134 by 25.6 and is the answer.

Again, multiply 234 by 36.

Again, what is the quotient of 30550 by 47?

Three numbers in direct proportion given, to find a fourth.

From the sum of the logarithms of the second and third numbers; deduct the logarithm of the first, the remainder will be the logarithm of the fourth required.

EXAMPLE I.

Let the three proposed numbers be 36, 48, 66, to find a fourth proportional.

Again, let three numbers be 240, 1440, 1230, to find the fourth proportional.

Multiply the given number's logarithm by 2, and the product is the logarithm of its square.

PROB. V.

To extract the square root of any given number.

Take half of the logarithm of the number, and that is the logarithm of its square root.

square root of the number required.

By the manner of projecting the lines of chords, sines, tangents, and secants (being prob. 20 of geometry) it is evident, that if the radius be supposed any number of equal parts (as 1000 or 10000, &c.) the sine, tangents, &c. of every arc, must consist of some number of those equal parts: and by computing them in parts of the radius, we have tables of sines, tangents, &c. to every arc of the quadrant, called natural sines, tangents, &c. and the logarithms of these give us tables of logarithmick sines, tangents, &c. and such are usually bound up with logarithms of numbers.

In which you may observe, that each page is divided into 8 columns, the first and last of which are minutes, and the intermediate ones contain the sines, tangents, and secants, the upper and lower columns contain degrees, the column of the minutes on the left

hand of each page, answers to the degrees in the top column; and the sines, tangents, and secants belonging to those degrees and minutes, are in the columns marked at the top with the words sine, tangent, and secant; the column of minutes on the right hand of each page, answers to the degrees in the bottom of the page; and the sines, tangents, and secants, answering to those degrees and minutes, are in the columns, marked at the bottom with the words sine, tangent, secant; the degrees in the top column beginning at 0, proceed to 44, where they end; and those at the bottom of the page begin at 89, and proceed to 45 in a decreasing series; the degrees in the different columns being the complement of each other. From what has been said, we may easily find the sine, tangent, or secant of any arc, from the tables, by looking for the given number of degrees at the head or foot of the page, according as they are less or greater than 45, and in the proper side column for the odd minutes, if there be any; then below or above the word sine, tangent, or secant, and on the same line with the minutes, we shall have that which was required.

EXAMPLE I.

Required, the sine of 36 degrees 40 minutes.

Look at the head of the page for 36 degrees, and in the side column on the left hand, for 40 minutes; then below the word sine, on the same line with 40,