756 



SCIENCi:. 



[Vol. II., No. 45. 



example slightly below the middle, is 2'° (repre- 

 sented b}- a degree-mark in the flfth row of these 

 marks, counting from the right) phis 2"-|-2° 

 (two Z-curves in the fifth and third places of 

 Z-curves) plus 2^^-|-2"-|-2'' (three loops) plus 

 2^° (the 7*-curve at the extreme left) ; wliile 

 the absence of 2^, 2^, and 2\ is shown by the 

 vertical stroke at the right. This equivalent 

 expression ma.j be verified, if desired, cither 



Example in addition by two notations. 



77,82.3,876 



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 



1,724,820 



7,529,536 



43,344,817 



10,000,000 



8,334,712 



1,953,125 



11,.30S,417 



7.59,375 



21,180,840 



9,765,625 



18,613,788 



1,000,000 



44,739,243 



1,SS9,.568 



2,517,471 



40,353,607 



4,438,414 



1,679,616 



23,708,715 



11,.390,625 



045,754 



823.543 



15,.3OS,S0'5 



60,466.176 



30,685,377 



10,077,696 



19,416,381 



43,016,721 



•^ 6 1° I d I „ 



t'tmr (Ti) I " 

 i tr^m> "D° i*> 



(T \'nr-h''(i i„ 



■^ "n "^ T^ I 

 trnr r~^°(m „ 



-^r^ -^ -5 °^ \° 



-^ "o" I rnm " 

 -iT'ti r fl 1 



r r (mrnrd if 

 mm n T" 



•H) " a m ~b I "• 

 ■-> n "H) <r I I 



° m c 0" m)" 



740,685,681 .£--l> C t ■^T' (P (f I " 



by adding the de.signated powers of two, from 

 524.288 down to G4, or bj' successive multipli- 

 cations bj' two, adding one when necessary. 

 The form of characters here exhibited was 

 thought to be the best of nearly three hundred 

 that were devised and considered, and in about 

 sixty cases tested for economic value by actual 

 additions. 



In Older to add them, the olijcct for which 

 these forty numbers are here presented in two 

 notations, it is not necessary to know just why 



the figures on tlie right arc equal to those on the 

 left, or to know any thing more than the order 

 in which the different forms are to be taken, 

 and the iact that any one has twice the value 

 of one in the column next succeeding it on 

 the right. The addition maj^ 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 successivelj' 

 obtained. The fingers will be found a great 

 assistance, especiallj' if one of each hand be 

 used, to point ofl:" similar marks in twos, or 

 threes, or fours, — -as manj' together as can be 

 certainly comprehended in a glance of the eye. 

 Counting by fours, if it can be done safeh', is 

 prelerable, because most rapid. The eje can 

 catch tlie marks for even powers more easilj' in 

 going up, and tho.se for odd powers (the I 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, upwards : the}- are twenty-three in num- 

 ber. Half of twenty-three is eleven, and one 

 over : one of these marks has therefore to be 

 entered as part of tlie answer, and eleven carried 

 to the next column, the first one of Z-curves. 

 But since the curves are most advantageously' 

 added downward, it is best, when the first col- 

 umn is finislied, siraplj- to remember the re- 

 mainder from it, and not to set down any thing 

 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 Z-curves, we 

 find ourselves at the bottom with twentj'-four, 

 — twelve to carry, and nothing to set down ex- 

 cept the degree-mark from the first column. 

 With the twelve we go up the adjoining loop-col- 

 umn, and the sum must be even, as this place 

 is vacant in the answer; the r-cnrve column 

 next, downward, and then another row of de- 

 gree-marks. The succession must be obvious 

 by this time. When the last column, the one in 

 loops to tlie 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 ;nust be remembered, is of j'-curves. 

 Halving five, we express the remainder by 

 entering one of these curves, and carry the 

 quotient, two, to the degree-mark place. Halv- 

 ing again gives one in the next place, that of 

 Z-curves ; and the work is complete. 



It is reeouimended that this work be gone 

 over several times for practice, until the ap- 

 pearance and order of the characters, and the 

 details of the method, become familiar ; that, 



