Second. The last figure of the prepared divisor being 3, this is the extraneous multiplier.  Multiplying the truncated divisor, 7, by the extraneous multiplier, 3, and adding 1, we have 22 for the current multiplier.

Third. The dividend, 1883, has now to be multiplied by the product of 3, the extraneous multiplier, and 2, the multiplier used in preparing the divisor.  The product, 11298, is the prepared dividend.

Fourth. From the prepared dividend, 11298, we cut off the last figure 8, and multiply this by the current multiplier, 22.  The product, 176, is added to the truncated dividend, 1129, and gives 1305 for the first modified divisor.  The whole operation is shown thus: 

1 8 8 3
6
-------
1 1 2 9|8
1 7 6 —
-----
1 3 0|5
1 1 0 —
-----
2|4 0
8 8 —–­
—–­
|9 0
-----
1 9|8
1 7 6 —
-----
1 9|5
1 1 0 —
-----
1 2|9
1 9 8 —
-----
2|1 0
2 2 —–­
2 4

We stop at this point because 24 was a previous modified dividend, written under the form 240 above.  Our two infinite numbers (which need not in practice be written down) are, with their difference: 

. .
10,958,904,058 . .
10,958,904,109.5890410958904
----------------------------
. .
51.5890410958904
. . 
Hence the quotient sought is 5.158904109.

Example 2. Find the reciprocal of 333667.

The whole work is here given: 

3 3 3 6 6|7 |7
2 3 3 5 6 7 — 1 6 3 4 9 6|9
2 1 0 2 1 0 3 —
-------------
2 2 6 5 5 9|9
2 1 0 2 1 0 3 —
-------------
2 3 2 8 6 6|2
4 6 7 1 3 4 —
-----------
7 0 0 0 0 0

. .
Answer, 0.000002997.

Example 3. Find the reciprocal of 41.

Solution.—­

4|1                 |9
-----              -----
3 7|9              3 3|3
—            1 1 1 —
-----
1 4|4
1 4 8 —
-----
1 6|2
7 4 —
—–­
9 0
.   .
Answer, 0.02439.

C.S.  PEIRCE.

* * * * *

[SCIENCE.]

EXPERIMENTS IN BINARY ARITHMETIC.

Those who can perform in that most necessary of all mathematical operations, simple addition, any great number of successive examples or any single extensive example without consciousness of a severe mental strain, followed by corresponding mental fatigue, are exceptions to a general rule.  These troubles are due to the quantity and complexity of the matter with which the mind has to be occupied at the same time that the figures are recognized.  The sums of pairs of numbers from zero up to nine form fifty-five distinct propositions that must be borne in memory, and the “carrying” is a further complication.  The strain and consequent weariness are not only felt, but seen, in the mistakes in addition that they cause.  They are, in great part, the tax exacted of us by our decimal system of arithmetic.  Were only quantities of the same value, in any one column, to be added, our memory would be burdened with nothing more than the succession of numbers in simple counting, or that of multiples of two, three, or four, if the counting is by groups.