In order to add them, the object for which these forty numbers are here presented in two notations, it is not necessary to know just _why_ the figures on the right are equal to those on the left, or to know anything more than the order in which the different forms are to be taken, and the fact that any one has twice the value of one in the column next succeeding it on the right. The addition may be made from the printed page, first covering over the answer with a paper held fast by a weight, to have a place for the figures of the new answer as successively obtained. The fingers will be found a great assistance, especially if one of each hand be used, to point off similar marks in twos, or threes, or fours--as many together as can be certainly comprehended in a glance of the eye. Counting by fours, if it can be done safely, is preferable because most rapid. The eye can catch the marks for even powers more easily in going up and those for odd powers (the _l_ and _r_ curves) in going down the columns. Beginning at the lower right hand corner, we count the right hand column of small circles, or degree marks, upward; they are twenty-three in number. Half of twenty-three is eleven and one over; one of these marks has therefore to be entered as part of the answer, and eleven carried 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.