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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.

form of Lord Napier, the inventor of logarithms. The manner in which Napier's logarithm of 10 is found, may be seen in most bcoks of Algebra, but it is here omitted, because students of Surveying are too generally unacquainted with the principles of that science, and the subject is too extensive for the present treatise. Those, however, who have not an opportunity for entering thoroughly into this subject, may with more propriety grant the truth of one number, and thereby be enabled to try the correctness of any logarithm in the tables, than receive those tables, as truly computed, without any means of examining their accuracy.

EXAMPLE I.

Required the Logarithm of the number 2.

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

3)0.868588964

9)0.2895296541=0.289529654

9)0.032169962- 3=0.010723321 9)0.003574440+ 5=0.000714888

9)0.000397160= 7=0.000056737

90.000044129: 9=0.000004903

90.000004903-11=0.000000446

9)0.000000545=13=0.000000042

0.00000006115=0.000000004

To this Logarithm 0.301029995 add theLogarithmof1=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 A”; 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.

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EXAMPLE 2.
Required the Logarithm of the number 3.

Here the next less number is 2; and 3+2 5=A, and A'=25.

5)0.868588964

25)0.173717793= 1=0.173717793 25)0.006948712= 3=0.002316237 25)0.000277948- 5=0.000055599 25)0.000011118= 7=0.000001588 25)0.0000004457 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=2x2, 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=3x2, therefore to the Logarithm of

3=0.477121254 add the Logarithm of 2=0.301029995

their sun=Log. of 6=0.778151249 Example 6. Required the Logarithm of 8. 8=29, therefore multiply the Logarithm of

2=0.301029995 by

3

The product=Log. of 8=0.903089985 Example 7. Required the Logarithm of 9. 9=3, therefore the Logarithm of

3=0.477121254 being multiplied by

2

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 A2=169.

13)0.868588964

169)0.066814536+1=0.066814536 169)0.000395352=-3=0.000131784

169)0.000002339-5=0.000000468

0.0000000147=0.000000002

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

The Log. of 16

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

of 3 and 4. of 14

of 7 and 2. of 15 is equal to the sum of 3 and 5. of the Logs.

of 4 and 4. of 18

of 3 and 6. of 20

of 4 and 5. 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 numberexceeds 400, the first quotient, being added to the Logarithm of its next lesser number, will give the Logarithm sought, true 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 corres

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