by A, and reserve the quotient. Divide the reserved quotient by the square of A, and reserve this quotient. Divide the last reserved quotient by the square of A, reserving the quotient still; and thus proceed as long as division can be made. Write the reserved quotients orderly under one another, the first being uppermost. Divide these quotients respectively by the odd numbers 1, 3, 5, 7, 9, 11, &c.; that is, divide the first reserved quotient by 1, the second by 3, the third by 5, the fourth by 7, &c. and let these quotients be written orderly under one another; add them together, and their sum will be a logarithm. To this logarithm add the logarithm of the next less number, and the sum will be the logarithm of the number proposed.

EXAMPLE 1.

Required the Logarithm of the number 2.

Here the next less number is 1, and 2+1=3= A, and A3, or 33=9; then

3)0.868588964

9)0.289529654÷ 1-0.289529654

9)0.032169962÷ 3=0.010723321

9)0003574440÷ 5=0.000714888

9)0.000397160 7=0.000056737

9)0.000044129÷÷ 9=0.000004903

9)0.000004903÷11=0.000000446

9)0.000000545+13=0.000000042

0.000000061-15-0.000000004

To this Logarithm 0.301029995 add the Logarithm of 1=0.000000000

Their Sum-0.301029995 Log. of 2.

The manner in which the division is here carried on, may be readily perceived by dividing, in the first place, the given decimal by A, and the succeeding quotients by A2; then letting these quotients remain in their situation, as seen in the example, divide them respectively by the odd numbers, and place the new quotients in a column by themselves. By employing this process, the operation is considerably abbreviated.

EXAMPLE 2.

Required the Logarithm of the number 3. Here the next less number is 2; and 3+2=5=A, and A2=25.

5)0.868588964

25)0.173717793+1=0.173717793

25)0.006948712÷ 3=0.002316237

25)0.000277948÷ 5=0.000055590

25)0.000011118÷ 7=0.000001588

25)0.000000445÷ 9=0.000000049

0.000000018-11 0.000000002

To this Logarithm 0.176091259 add the Logarithm of 2=0.301029995

Their Sum=0.477121254-Log, of 3.

Then, because the sum of the logarithms of

numbers, gives the logarithm of their product; and the difference of the logarithms, gives the logarithm of the quotient of the numbers: from the two preceding logarithms, and the logarithm of 10, which is 1, a great many logarithms can be easily made, as in the following examples.

Example 3. Required the Logarithm of 4.

Since 4=2×2, then to the Logarithm of

2=0.301029995

add the Logarithm of 2=0.301029995

The sum Logarithm of 4-0.602059990 Example 4. Required the Logarithm of 5. 10÷2 being=5, therefore from the Log. of 10 1.000000000 subtract the Log, of 2=0.301029995

the remainder is the Log. of 5=0.698970005 Example 5. Required the Logarithm of 6 6=3×2, therefore to the Logarithm of

3=0.477121254 add the Logarithm of 2=0.301029995

their sum=Log. of 6-0.778151249

Example 6. Required the Logarithm of 8. 8 23, therefore multiply the Logarithm of

The product-Log. of 8-0.903089985

Example 7. Required the Logarithm of 9.

9=32, therefore the Logarithm of

3=0.477121254

2

being multiplied by

the product-Log. of 9-0.954242508

Example 8. Required the Logarithm of 7 Here the next less number is 6, and 7+6=13: A, and A2169.

13)0.868588964

169)0.066814536÷1=0.066814536

169)0.000395352÷3=0.000131784

169)0.000002339-5-0.000000468

0.000000014÷70.000000002

To this Logarithm=0.066946790
add the Log. of 6=0.778151249

Their sum=0.845098039-Log. of 7. (of 12

of 14

of 3 and 4.

of 7 and 2.

The Logarithms of the prime numbers 11, 13, 17, 19, &c. being computed by the foregoing general Rule, the Logarithms of the intermediate numbers are easily found by composition and division. It may however be observed, that the operation is shorter in the larger prime numbers; for when any given number exceeds 400, the first quotient, being added to the Logarithm of its next lesser number, will give the Logarithm sought, frue to 8, or 9

places; and therefore it will be very easy to examine any suspected Logarithm in the Tables.

For the arrangement of Logarithms in a Table, the method of finding the Logarithm of any natural number, and of finding the natural number corresponding to any given Logarithm, therein: likewise for particular rules concerning the Indices, the reader will consult Table 1, with its explanation, at the end of this Treatise.

MULTIPLICATION.

Two or more numbers being given, to find their product by Logarithms.

RULE.

Having found the Logarithms of the given numbers in the table, add them together, and their sum is the Logarithm of the product; which Logarithm, being found in the Table, will give a natural number, that is, the product required.

Whatever is carried from the decimal part of the Logarithm is to be added to the affirmative indices; but subtracted from the negative. Likewise the indices must be added together, when they are all of the same kind, that is, when they are all affirmative, or all negative; but when they are of different kinds, the difference must be found, which will be of the same denomination with the greater.

Example 1. Required the product of 86.25 multiplied by 6.48.

Log. of 86.25-1.935759

Log. of 6.48 0.811575

Product=558.9-2.747334*

For the method of finding the natural number, answering to the sum of the Logarithms, the reader will consult Table 1, at the end of this Treatise.