Instead of subtracting one logarithm from another, add its arithmetical complement, the result will be the same, thus:

Note. The arithmetical complement of a logarithm is what it wants of 10.000000.

To find which, begin at the left hand, and subtract each figure from 9, except the last on the right hand, which must be subtracted from 10, and for every complement which is added subtract 10 from the last sum of the indices.

The arithmetical complement is commonly used in trigonometrical calculations, when radius is not the first term in the analogy.

It is further to be noted that the arithmetical complement of any sine is the same as the co-secant of the same number of degrees.

Involution is the raising of powers, from any given number, as a root; a power is a quantity produced by multiplying any given number, called the root, a certain number of times continually by itself, thus:

2 is the root of the 1st power of 2
2 x 2 is the 2nd power or square of 2
2×2×2 is the 3rd power or cube of 2
2×2×2×2=is the 4th power of 2, &c.

BY LOGARITHMS.

Take out the logarithm of the given number from the table, multiply the logarithm thus found by the index of the power proposed, find the number answering to the product, and it will be the power required.

Note.-In multiplying a logarithm with a negative index by an affirmative number, the product will be negative; but what is to be carried from the decimal part of the logarithm will always be affirmative, and therefore their difference will be the index of the product, and is always to be made of the same kind with the greater.

EVOLUTION, OR EXTRACTING ROOTS.

Evolution, or the reverse of involution, is the extracting or finding the roots of any given power.

Rule. Take the logarithm of the given number from the table, divide the logarithm thus formed by the index of the root, then the number answering to the quotient will be the

root.

Note.-When the index to the logarithm to be divided is negative, and does not exactly contain the division without some remainder, increase the index by such a number as will make it exactly divisable, by the index carrying the unity borrowed as so many times to the left hand place of the decimal, and thus divide as in whole numbers.

SINES AND TANGENTS.

The tables (both logarithmic and natural) of sines and tangents are calculated to every five minutes of a degree, by which all trigonometrical operations are performed.

The degrees descend from the top to the bottom, that is, from 0 to 90; the minutes are placed in the top column.

The co-sine, co-tangent, &c., reads from the bottom upwards. To find the logarithm of any sine, &c., to 5 minutes of a degree, look in the table for the degrees in the first column on the left hand, and the minutes on the top, opposite to which will be the logarithm required, thus:

To find the logarithm of any intermediate minute, take the difference of the logarithm next less and next greater, multiply this difference by the number of intermediate minutes, and divide the product by 5.

Add the quotient to the foregoing less logarithm, the product will be the logarithm required, nearly.

To find the degrees and minutes to any logarithm, seek in the table for the logarithms, on the top for the minutes, and on the left for the degrees as before, will give the degrees and minutes, nearly.

If the logarithm should not agree with any degree and minute in the table, find the next greatest and next less, and take their difference, thus:

* The angle exceeding 90° look for 50° in the 2nd column of co-sine.

As this difference is to 5 minutes, so is the difference between the given logarithm and that next less to a fourth number of minutes.

Add this fourth number to the minutes in the less logarithm, it will be the degree and minutes required.

Required the degrees and minutes of the sine answering to 9.540248: .

Then as 1708:5:: 1025: 3 minutes, which, added to the less logarithm 20.15, will give 20° 18' for the sine required.

DECIMAL FRACTIONS.

A decimal fraction derives its name from the Latin, decem (10), which denotes the nature of its numbers, representing the parts of an integral quantity, divided into a tenfold proportion. NUMERATION. Teaches to read and write any number proposed, either by words or characters.

In decimal fractions, the integer, or whole thing, as a gallon, a pound, a yard, an acre, &c., is supposed to be divided into ten equal parts, called tenths; those tenths into ten equal parts, called hundredths; and so on without end.

So that the denomination of a decimal being always known to consist of an unit with as many cyphers as the numerator has places, is therefore never expressed, being understood to be 10, 100, 1000, &c., according as the numerator consists of 1, 2, 3, 4, or more figures. Thus, instead of 2, 24, 211, the numbers only are written with a full point before them, thus: .2 .24 .211.

If a unit of any kind, as an acre, a gallon, &c., be divided into ten equal parts, then the decimal represents as many of

those parts as the decimal figures express; thus: .7 means seven of those parts, or seven-tenths. If the decimal consists of two figures, unity would be understood to be divided into a hundred equal parts, of which the decimal represents as many as the figures express, thus: .65 means 65 of those parts, or sixtyfive hundredths; if the decimal consisted of three figures, it would be a thousand equal parts, .625 is six hundred and twenty-five of those parts; or if the decimal .0625, unity would be divided into 10,000 equal parts. The value of the figures are more clearly described by the following table:

Thus .5 is read five-tenths, .56 is read fifty-six hundredths, .567 thousandths, and so on, as in the table.

50

1009

1000

Cyphers to the right hand of decimals cause no difference in their value, as .5 .50 .500 are decimals of the same value, being each equal to ; that is, .5, .50 = 15%, .500 = -50%. If cyphers are placed on the left hand of decimals, they diminish their value in a tenfold proportion, thus: .3 .03 .003 are three tenths, three hundredths, three thousandths, and answer to the vulgar fractions, 10, 180, 1000, respectively.

3

A whole number and decimal are thus expressed: 8.75, 85.04, &c.

When decimals terminate after a certain number of figures, they are called finite, as 125 = 10 = 1, 958=100% = 231.

125 1000

250

When one or more figures in the decimals become repeated, it is called a repeating or circulating decimal; as, .333333, &c., = 1, .666666, &c.,, .428571428571, &c., and many

others.

=

Note.-A finite decimal may be considered as infinite, by making cyphers to recur, as they do not alter the value of the decimal.

In all operations, if the result consists of several nines, reject them, and make the next superior place an unit more; thus for 26.25999, write 26.26.

In all circulating numbers dot the last figure, as 8.54666.