Which is the Log. of the quotient required, and whofe corresponding number is 21.3.

To find the Complement of a Logarithm.

Begin at the left-hand, and fet down the difference between each figure and 9, except the first, which must be fubtracted from ten. Thus you will find the complement of the Log. of 456, viz. of 2.6589648 to be 7.3410352.

2. To perform the rule of Proportion by the Logarithms. Add the Logarithms of the fecond and third terms together, and fubtract the Log. of the first from that sum ; the remainder will be the Log. of the fourth term or number fought. Or inftead of fubtracting the Log. of the first term, add its complement, and the refult will be the fame: But obferve that in adding a complement you must always cancel 1 in the tens place of the index, for every complement fo added.

Thus, let it be required to find a fourth number in geometrical proportion to the three following, viz. 4, 9, 12.

The fame as before, except in the index, where I in the place of tens must be cancelled, as we before observed. Example 2. Suppofe 100l. in one year, or 365 days, gain 67. intereft; what will 51737. gain in 321 days? Anfwer, 272.96431. or 272%. 195. 3d.

nearly.

Comp.

In the above operation, 2 in the place of tens in the index is cancelled, because two complements are added.

3. To raise Powers by Logarithms.

Multiply the Logarithm of the given number by the index of the required power, the product will be the Logarithm of the power fought.

Example. Let it be required to find the cube of 32 by the Logarithms.

Example 2. Let it be required to find the cube of 0.009

by the Logarithms.

N.B. When the Log. has a negative index, as in the above example, whaterever is carried from the decimal part of the Log. muft be confidered as affirmative, and fubtracted from the multiplication of the negative index; for the decimal part of every Log. is affirmative, whether the whole Log, taken together be fo or not. Thus in the above example, the negative index 3 being multiplied by 3, produces 9, from which the 2 carried from the decimal part being fubtracted, gives 7 for the index of the Log. required.

Example 3. Let the 6th power of 0.0032 be required.
The Log. of 0.0032 is
Which multiplied by

The product is

6

3.50515500

Which is the Log. of

15.03093000

0.0000000000000010739

4. To

4. To extract the Roots of Powers by the Logarithms. Divide the Log. of the number by the index of the power, the quotient is the index of the root fought.

Example, Let it be required to extract the cube root of 6751269.

The Log. of 6751269 is 6.8293854, which being divided by 3, the index of the power, gives 2.2764618. the Log. of 189, the cube root fought.

Example 2. Let it be required to extract the root of the 6th power of 0.00000,00000,00000,00000,00000,00000,63 by the Logarithms.

The Log. of 0,00000,00000,00000,00000,00000,00000, 63 is 31.8061800, which being divided by 6, the index. of the power gives 5.3010300, which is the Log, of 0.00002, the root of the fixth power required.

And after the fame manner may the root of any other power be extracted.

5. To find mean proportionals between any two numbers. Subtract the Log. of the leaft term from the Log. of the greater, and divide the remainder by a number which exceeds by one the number of means fought; then add the quotient to the Log. of the leffer term, or fubtract it from the Log. of the greateft, continually, and you will have the Logarithms of all the mean proportionals required. Example. Let it be required to find three mean proportionals between 106 and 100.

The use of the tables of Logarithmic fines, tangents and fecants.

1. To find the fine, tangent and fecant of any arch to ninety degrees.

Find