to the next column, the first one of l-curves.  But since the curves are most advantageously added downward, it is best, when the first column is finished, simply to remember the remainder from it, and not to set down anything until the bottom is reached in the addition of the second column, when the remainders, if any, from both columns can be set down together.  In this case, starting with the eleven carried and counting the number of the l-curves, we find ourselves at the bottom with twenty-four—­twelve to carry, and nothing to set down except the degree mark from the first column.  With the twelve we go up the adjoining loop column, and the sum must be even, as this place is vacant in the answer; the r-curve column next, downward, and then another row of degree marks.  The succession must be obvious by this time.  When the last column, the one in loops to the extreme left, is added, the sum has to be reduced to unity by successive halvings.  Here we seem to have eleven; hence we enter one loop, and carry five to the next place, which, it must be remembered, is of r-curves.  Halving five we express the remainder by entering one of these curves, and carry the quotient, two, to the degree mark place.  Halving again gives one in the next place, that of l-curves; and the work is complete.

It is recommended that this work be gone over several times for practice, until the appearance and order of the characters and the details of the method become familiar; that, when the work can be done mechanically and without hesitation, the time occupied in a complete addition of the example, and the mistakes made in it, be carefully noted; that this be done several times, with an interval of some days between the trials, and the result of each trial kept separate; that the time and mistakes by the ordinary figures in the same example, in several trials, be observed for comparison.  Please pay particular attention to the difference in the kind of work required by the two methods in its bearing on two questions—­which of them would be easier to work by for hours together, supposing both equally well learned? and in which of them could a reasonable degree of skill be more readily acquired by a beginner?  The answer to these questions, if the comparison be a fair one, is as little to be doubted as is their high importance.

Example in addition by two notations

   77,823,876
   14,348,907
    8,654,912
    5,764,801
    4,635,857
    1,594,323
    6,417,728
    4,782,969
   83,886,075
   34,012,224
    2,903,111
   48,828,125
    1,724,826
    7,529,536
   43,344,817
   10,000,000
    8,334,712
    1,953,125
   11,308,417
      759,375
   21,180,840
    9,765,625
   18,643,788
    1,000,000
   44,739,243
    1,889,568
    2,517,471
   40,353,607
    4,438,414
    1,679,616