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Again, let three numbers be 240, 1440, 1230, to find the fourth proportional.
To the log. of 1449 3.15836
From the product
6.24827 Take the log. of 240 2.38021
3.86806 its number 17380
the 4th required.
To find the square of any given number.
Multiply the given number's logarithm by 2, and the product is the logarithm of its square.
Required, the square of 36.
1.55630 Multiply by
3.11260 its number 1296
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 itft
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.
Required, the sine of 36 degrees 10 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,
Required, the tangent of 54 degrees 30 minutes.
Look at the foot of the page (because the proposed degrees are more than 45.) for 54 degrees, and in the right hand column for 30 minutes ; then in the column marked tangent at its bottom, and on the same line with the 30 minutes, in the side column, we find 10.14673, which is the log-tangent required.
The reverse of this, viz. The logarithm of a sine, tangent, or secant, being given, to find the arc belonging to it, is performed by only looking in the proper column, for the nearest logarithm to that proposed, and the degrees and minutes answering thereto, are those required.
We will now shew how any sine, tangent, or secant may be had, though the figures in the tables were defaced, mis-printed, or obliterated.
To find the tangent which is defaced, by the sine and co-sine.
The co-sine taken from the sine added to 90, or radius, which is 10.00000, the remainder is the tangent, (by part 1. theo. 24.)
4. Suppose the tangent of 41°. 20' was defaced, but the sine and co-sine of it visible.
From the sine of 41° 20'+10.00000, or radius,
Take the co-sine of 41° 20'
The rem. is the tan. of 41°. 20° req. viz. 9 94426
2. To find a sine which is mis-printed, by help of the co-sine and tangent.
From the sum of the tangent and co-sine, take 10.00000, or radius, or (which is the same thing) cut off the first figure in the index, the remainder is the sine required (by part 2. theo. 24.)
Suppose the sine of 46o. 50'. was defaced, but the tangent and co-sine visible.
To the tangent of 46°. 50
10.02781 9.835 13