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We may therefore find the square of 257.4 in the following way: 257266 049

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In many cases only one more figure will be required in the square than in the given number. The square can then be interpolated with all required accuracy by the differences, the last two figures of which are found in the last column of the table, while the remaining figures are found by taking the difference between two consecutive numbers in the principal column.

To return to the last example, we find the difference between 257 and 258' to be 515, the first figure being the difference between 660 and 665, and the last two, 15, in the last column.

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Then

It will be remarked that the two methods are substantially the same when only five figures are sought in the result. The substantial identity rests upon the general theorem that

The difference of the squares of two consecutive numbers is equal

to the sum of the numbers.

We prove this theorem thus:

(n + 1)2 — n2 = 2n + 1 = n + (n + 1).

When the tabular difference is taken in the way already described, it will often happen that the difference between the numbers in the columns of hundreds is to be diminished by unity.

41734160 13, the difference between 645' and

=

Thus, although 646' is not 1391, but 1291. These cases are noted by the asterisk after the number in the last column.

The squares of numbers of more than four figures may be found in the same way, but in such cases it will generally be easier to use logarithms than the table of squares.

TABLE VIII.

TO CONVERT HOURS, MINUTES, AND SECONDS INTO DECIMALS OF A DAY, AND VICE VERSA.

22. The familiar method of solving this problem is to convert the seconds into decimals of a minute, and the minutes into decimals of an hour, by dividing by 60, and then the hours into decimals of a day by dividing by 24. The reverse problem is solved by multiply

ing by 24, 60, and 60.

Table VIII. enables us to perform these operations without division. Column D gives each hundredth of a day, but its numbers may also be regarded as ten thousandths or millionths of a day, according to which of the following three columns is used. In column H.M.S. are found the hours, minutes, and seconds corresponding to these hundredths. In the next column is one hundredth of column H. M.S., or the minutes and seconds in the number of ten thousandths of a H.M.S. day in column D. Finally, column 100❜

shows the number of

seconds in the number of millionths of a day found in column D. Example. To convert 0.532 946 into hours, minutes, and seconds. = 12h 43m 128

04.53

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It will be seen that we divide the figures of the given decimal of a day into pairs, and enter the three columns of time with these three pairs in succession.

If seven decimals are given, we may interpolate the last number, as in taking out a logarithm.

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In practice the computer will perform the interpolation mentally, adding.7 × .08.06 to the number 5.36 of the table in his head, and writing down 5.42 as the last quantity to be added.

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To use the table for the reverse operation, we proceed as in the following example:

It is required to convert 17h 29m 30.93 into decimals of a day. Looking in the table, we find that the required decimal is between Hence the first two figures are 0.72, the equivalent Subtracting the lat

0.72 and 0.73.

17h 29m 30.93 =17h 16m 48s

of 17h 16m 48.

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12m 42.93

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.0088 or 12m 40.32, we have left 2.61, which we are to seek in

column

H.M.S. 1002 be 302. Hence

We find the corresponding number of column D to

17h 29m 30.93 = 0.728 830 2.

In solving this problem the computer should be able, after a little practice, to perform the subtractions and carry the remainders mentally, thus saving himself the trouble of writing down the numbers.

EXERCISES.

Take the answers obtained from the four preceding exercises, subtract each result from 24h 0m 0s, change the remainder to decimals of a day, and see if when added to the decimals of the preceding exercises the sum is 1a.000 000 0, as it should be.

TABLE IX.

TO CONVERT TIME INTO ARC, AND VICE VERSA.

23. In astronomy the right ascensions of the heavenly bodies are commonly given in hours, minutes, and seconds, the circumference being divided into 24 hours, each hour into 60 minutes, and each minute into 60 seconds.

Since we have

360° one circumference,

1h = 15°;

1m = 15';

h

8

1° = 15′′;

the signs ",", and indicating hours, minutes, and seconds of time. Hence we may change time into arc by multiplying by 15, and arc into time by dividing by 15, the denominations being changed in each case. Table IX. enables us to do this by simple addition and subtraction by a process similar to that employed in changing hours, minutes, and seconds into decimals of a day.

To turn time into arc, we find in the table the whole number of degrees contained in the time denomination next smaller than the given one, and subtract the former time denomination from the latter.

Next we find the minutes of arc corresponding to the given time next smaller than the remainder, and again subtract.

Lastly we interpolate the seconds corresponding to the second remainder.

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The computer should be able to go through this operation without writing down anything but the result.

The operation of changing arc into time is too simple to require description, but it is more necessary to write down the work.

EXERCISES.

Change the following times to arc, and then check the results by changing the arcs into time and seeing whether the original times are reproduced:

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