260 



ARITHMETIC AUK. 



follow any law, however complicated, may he formed. 

 to a greater or lcsse\i.ni. solely l>y ilir proper ar- 

 r..HiM'inenl of the successive addition and slikr.lCtion 

 of numbers befitting each Uililf. is a general principle 

 \vliicli can be dflmOtttmted to those only who are 

 well acquainted with mathemat'cs ; but tin- mind, 

 even of the reader who is but very slightly acquainted 

 with that science, will readily conceive that it is not 

 impossible, by attending to the following example. 

 l^el us consider the subjoined table. This table is 

 the beginning of one in very extensive n-e, which 

 en printed and re-printed very frequently in 

 many countries, and is called a table of square num- 



l.rrs. 



Any number in the table, column A, may be ob- 

 tained by multiplying the number which expresse 

 the distance of tliat term from the commencement ol 

 the table by itself; thus 25 is the fifth term from the 

 beginning of the table, and 5 multiplied by itself, or 

 by 5, is equal to 25. Let us now subtract each term o: 

 this table from the next succeeding term, and place the 

 result in another column (B), which may be callec 

 first-difference column. If we again subtract each 

 term of this first-difference from the succeeding term 

 we find the result is always the number 2 (column C) 

 .and that the same number will always recur in tha 

 column, which may be called the second-difference 

 will appear to any person who takes the trouble to 

 carry on the table a few terms further. Now, when 

 once this is admitted as a known fact, it is quite clea 

 that, provided the first term (I) of the table, the firs 

 term (3) of the first-differences, and the first term (2 

 of the second or constant difference are originally 

 given, we can continue the table to any extent, merel; 

 by simple addition : for the series of first-difference 

 may be formed by repeatedly adding the constant dif 

 ference 2 to (3) the first number in column B, an< 

 we then necessarily have the series of odd numbers 

 3, 5, 7, &c. ; and again, by successively adding eacl 

 of these to the first number (1) of the table, we pro- 

 duce the square numbers." Having thus thrown som 

 light on the theoretical part or the question, M 

 liabbage proceeds to show that the mechanical exe 

 cution of such an engine as would produce this serie 

 of numbers, is not so far removed from that of ordi 

 nary machinery as. might be conceived. He imagine 

 three clocks to be placed on a table, side by side 

 each having only one hand, and a thousand division 

 instead of twelve hours marked on the face ; an 

 every time a string is pulled, each strikes on a be! 

 the numbers of the divisions to which the hand points 

 Let it lie supposed that two of the clocks, for th 

 sake of distinction called B and C, have some mecha 

 nism by which the clock C advances the hand of th 

 clock B one division for each stroke it makes on i 

 own bell ; and let the clock B by a similar contr 

 vance advance the hand of the clock A one divisio 



r each stroke it makes on it< own I)' II. Having 

 t tin- hand of the clock A. to the di\ison |. that 

 fBtollI, and that of ( to II, pull the string of 

 ock A, which will strike one ; pull tlmtof clock B, 

 Inch will strike three, and al the s;mic time, in coiv 

 quence of the mechanism we have referred to above, 

 ill advance the hand of A three di\ isituis. I'ull the 

 tring of C, which will strike i\\o and advance the 

 and of 15 I wo divisions, or to division V. Let this 

 peration be repeated ; A will then strike four ; J* 

 ill strike five, and In BO doing will advance the hand 



f A five divisions; andC will aain strike two, al 

 lie same time advancing the hand of B twodi\ isions. 

 Again pull A, and it will strike nine ; B will strike 

 even, and C two. If now thus-' disisions struck, or 

 lointed at by the clock A be attend* d to and written 

 own, it will be found that they j-.roduee a sen- s of 

 he squares of the natural numbers; and this will 

 ie the more evident, if the operation be continued 

 urther than we have carried it. Such a series could 

 if course be extended by this mechanism only so far 

 as the three first figures ; but this may be sufficient to 

 jive some idea of the construction, and was in frcl, 



Babbage states, the point to which th' 

 nodel of his calculating engine was dinned. In 

 >rder to convey some 'idea of the power of thi- 

 stupendous machine, we may mention the effects pro- 

 duced by a small trial engine constructed by the in- 

 ventor, and by which he computed the following 

 table from the formula ^ + a- -f 41. The figi i 

 hey were calculated by the machine, were not exhi- 

 bited to the eye as in sliding-rulesand similar instru- 

 ments, but were actually presented to it on two 

 opposite sides of the machine, the number IJX.'J. for 

 example, appearing in figures before the person em- 

 ployed in copying. The following table was calcu- 

 lated by the engine referred to : 



11 



131 



383 



791 



While the machine was occupied in calculating this 

 table, a friend of the inventor undertook to write 

 down the numbers as they appeared. In consequence 

 of the copyist writing quickly, he rather more than 

 kept pace with the engine at first, but as soon as 

 five figures appeared, the machine was at least equal 

 in speed to the writer. At another trial, thirty-two 

 numbers of the same table were calculated in the 

 space of two minutes and thirty seconds, and as these 

 contained eighty-two figures, the engine produced 

 thirty-three figures every minute, or more than one 

 figure in every two seconds. On a subsequent oc- 

 casion, it produced forty-four figures per minute ; 

 and this rate of computation could be maintained for 

 any length of time. See Mathematics. 



ARIUS. See Ariuns. 



ARK ; the name applied, in our translation of tlte 

 Bible, to the boat or floating edifice in which Noah 

 resided during the flood or deluge ; derived, undoubt- 

 edly, from the Latin area, a chest, or vessel. (See 

 Deluge.) In the synagogue of the Jews, the chest, 

 in which the tables of the law were preserved, bore 

 the name of the ark of the covenant. This was a 

 small chest or coffer, three feet nine inches in length, 

 two feet three inches in breadth, and the same in 

 height, in which were contained the various sacred 

 articles mentioned in the quotations. It was mt.de 



