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 23,708,715 11,890,625 945,754 823,543 15,308,805 60,466,176 30,685,377 10,077,696 19,416,381 43,046,721 =========== 740,685,681 [Illustration]