CASE II.

When both, or either, fractions are lefs than unity?

NOTE. Whatever index you make reprefent unity, omit it in the fum of the indices, and borrow it in the fubtraction of indices, the fum or remainder will be the true index required.

To EXTRACT THE ROOTS IN LOGARITHMS.

As the multiplying the logarithm of any number by the index. of its power produces the logarithm of that power; fo the division of any logarithm by its propofed index, the quotient will be the logarithm of the root required.

What is the fquare root of 324 ? 324 Its logarithm is 2)2.51054

What is the cube root of 10618? 10648 Its log. is

18 Log. of the root is 1.25527 22 leg. of the root is

3)4.02627

1.34209

thus :

:

To find any propofed root of any decimal fraction, you must first prepare the index for the divifion of the propofed power, For the fquare you must add 10 to the index before you divide it; for the cube you must add 20 to its in ex before you divide it; and fo on for the root of any power propofed.

The APPLICATION of LOGARITHMS in measuring Boards, Timber, Glass, Stone, and all kinds of Packages, ufually taken on board Ships*.

Required the content of a board or | Required the content of a piece plank 9 feet long and 14 foot of glass 2.9 foot long. and 1.75

broad?

broad?

In like manner may any dimenfions be squared, and the content be found.

If the folid content be required of any box, bale, &c. add the logarithms of the length, breadth, and depth together, the fum will be the log. of the folid content.

EXAMPLE.What is the folid content of a box whose depth is 2.7, breadth 2. 3, and length 4. 5

feet.

2. 7 Its log. is

0.43136

1.44630amber 27.95 or 28

Sum equal the log. of the content feet nearly.

The diameter of a cafk at the head and bung, and also its length, being given, to find its content in beer and in wine measure?

Iit. Multiply the difference of the head and bung diameter by 0,7, and add the product to the head diameter for a mean diameter.

RULE FOR WINE MEASURE.

Place down the log. of the mean diameter twice the log. of the length, and under these two the conftant log. 7.53148, the fum of thefe four logarithms will be the log. of the content, abating 10 in the fum of the indices,

RULE FOR BEER MEASURE.

Put this conftant log. under the two former logs. always 7.44484 the fum of the four logs. will be the content for beer gallons, abating 10 in the index.

The AUTHOR has lately published an improved GUNTER'S SCALE, on which the foot is divided into ten equal parts. and these parts fubdivided into ten equal parts, for the purpose of taking dimensions, and calculating by logarithms or decimal fractions.

EXAM

EXAMPLE. What is the content of a cafk whofe head diameter is 20, the bung diameter 28, and length 40 inches ?

Bung diameter
Head diameter

The way these two conftant multiplying logarithms were found is thus:

ift. The area of a circle, whofe diameter is unity, is 7854 decimal parts of the fquare thereof; fo that if the fquare of the diameter of any circle be multiplied by ,7854, the product will be the area of the given circle: hence,7854 is always a constant quantity whofe logarithm is 9.89509.

2d. If the area of a circle be divided by 231, the number of cubic inches there are in a wine gallon, the quotient will be the number of gallons that circular area contains, at I inch deep: hence 231 is a conftant divifor. Its logarithm is 2.36361, the arithmetical complement of which is 7.63639, which I add to the former conftant Pogarithm 9.89509

The fum 7.53148 abating 10 in the indices, is the constant logarithm to be added, as per rule, for wine measure.

For beer measure the divifor is always 282, its log. is 2.45025, whofe arithmetical complement is 7.54975

Add the conftant log. 9.89509

Sum 7.44484, the conftant loga

rithm for beer meafure, as per rule, omitting 10 in the index, or fubtract 2.45025 from 9.89509

Take 2.45025

Remains 7.44484, the fame as above.
D

The

The common Way of finding a Ship's Tonnage at London.

RULE. Multiply the length of the keel by the breadth of the beam, and that product by half the breadth of the beam, and divide .the laft product by 94, and the quotient arifing is the tonnage. EXAMPLE. Suppofe a fhip 72 feet by the keel, and 24 feet by the beam, what is the tonnage?

Length

72

log. is 1.55733

do.

1.07918

Arith. complement of log. of 94, do.

8.02687

Tonnage 220.6

2.34359 Answer.

To find the Logarithm of the Sines, Tangents, and Secants, belonging to any Number of Degrees and Minutes required.

If the required degrees be lefs than 45, feek the degrees on the top, and the minutes in the left-hand column, marked M, against which, in the column figned at the top with the propofed name, ftands the fine, tangent, and fecant required; but when the degrees given are more than 45, feek the degrees at the bottom, and the minutes in the right-hand column, marked M at the bottom, and the proposed name at the bottom. Here it may be obferved, that the degrees at the top, and minutes at the left-hand column, added to the degrees at the bottom and minutes in the right-hand column, always make 90; hence, if a fine be looked for, the co-fine or complement will be found in the adjoining column, the fame may be obferved of tangents and fecants.

EXAMPLE I. Required the log.

fine of 28° 37'?

Find 28 at the top of the page, and, in the left-hand column, marked M at the top, find 37; against which, in the column marked with the word Sine, ftands 9.68029, the logarithm of the fine of 28° 37' required. The fame may be observed of tangents and fecants.

EXAMPLE II. Required the log.

tangent of 67° 45'?

Find 67° at the bottom of the page, and 45' at the right-hand column marked M at the bottom; against this, in the column marked Tangent at the bottom, ftands 10 38816, which is the logarithm required,

Having the fine, tangent, and fecant, the co-fine, co-tangent, co-fecant, are always found in the adjoining columns.

The logarithm to any number of degrees above 90°, is found by fubtracting the given degrees from 180°, and taking the logarithm of the remainder; or, if 90° be fubtracted from the given fine, and the log. co-fine of the remainder be taken, it will give the fame.

To find the Degrees, Minutes, and Seconds, correfponding to any given Logarithm.

If the degrees, minutes, and feconds, be wanted to a given logarithmic fine, or co-fine thus found, and the next greater, and the next less than the given logarithm, and the difference between the given logarithm and the next lefs if a fine, and the next greater if a co-fine; then fay, as the difference between the next greater and next less is to 60", fo is the difference between the next lefs, if a fine, and the next greater if a co-fine, to the number of feconds to be annexed to the degrees and minutes found before.

EXAMPLE I-Find the degrees, minutes, and feconds, correfponding to the log. fine 9.61405? Next lefs log. 9.61382 Next greater

9.61411

29

Next lefs log. 9.61382

Given log

9.61405

23

Here the given log is found ftanding between 24° 16', and 24° 17; then, as 29 is to 60, fo is 23 to 48, which, annexed to 24° 16, gives 24° 16' 48", anfwering to log. 9.61405.

EXAMPLE II. Find the degrees, minutes, and seconds, correfponding to the log. co-fine 9.43297.?

The nearest found between 74° 16', and 74° 17'.

74° 16' Next greater log. 9.43323 Next greater log. 9.43323 74° 17' Next lefs 9.43278 Given log.

Diff.

45

9.43297

Diff. 26

Now, as 45 is to 60, fo is 26 to 34", which, annexed to 74° 16 gives 74° 16' 34", the degrees, minutes, and feconds required.

To find the Logarithm of the Sine or Co-fine, for Degrees, Minutes, and Seconds.

RULE. Find the logarithm to the degrees and minutes as before; take the difference between the logarithm and the next greater in the fine; but, if a co-fine, the next lefs; multiply this difference by the odd feconds, and divide the product by 60'; add the quotient to the right hand of the log. of the degrees and minutes, if a fine, but fubtract it if a co-fine, the fum or difference will be the logarithm, fine, or co-fine required.