TRANSACTIONS OF THE SECTIONS. • 39 



the integrating wheel ; and then to give to the connecting piece of the two parallel 

 rods a direction upwards instead of downwards, so that the second bar may pass to 

 the superior surface of the upper plate ; if this bar be then precisely of the length of 

 the first, the point of contact will be made to describe a path on the lower plate 

 precisely similar to the guiding point of the rail on the upper. There are, however, 

 some important applications of the instrument, in which the curve admitting of an 

 easy mechanical description, the guidance of a rail may be entirely dispensed with, 

 the point of contact being made to vary its position according to the required law, 

 by some more convenient or more accurate mechanical expedient ; of this class are 

 the eUipse and the logarithmic spiral. 



On a Neio Calculating Machine. By Mr. Fowler. 



The machine i.tself is on the principle of the old abacus, or calculating rods. At 

 the first glance it has somewhat the appearance of a pianoforte, or organ, with all 

 its keys laid bare. It would be impossible by mere description to make clear the de- 

 tails of the instrument, or the method of working it. The property of numbers on 

 ■which Mr. Fowler bases the notation of his machine is, that any number whatever 

 may be expressed by a proper combination of the powers of the number 3. The 

 powers of 3 are in succession thus : the power is 1 ; 1st, 3 ; 2nd, 9 ; 3rd, 11 ; 4th, 

 81 ; 5th, 243 ; &c. &c. Thus the number 14 may be expressed by subtracting 

 from 27, or the 3rd power, the sum of 9, 3, and 1, the 2nd, 1st, and powers, and 

 similarly for all other numbers ; the combination for some being more simple, for 

 others more compUcated. Instead of using the nine common characters with 0, 

 nought, or zero, as in our common mode of numbering, Mr. Fowler only uses three 

 marks, + when the power of 3 is to be added, — when it is to be subtracted ; and 

 the power itself is expressed by the place in which the mark stands : thus the num- 

 ber 14 would be H , where the — most to the right means that the power 



of zero, or 1, is to be subtracted; the next — to the left means, that the 1st power 

 of 3 is also to be subtracted ; the next, that the 2nd power of 3 is to be subtracted, 

 and the -\- in the 4th place of figures means, that the 4th power of 3 is to be added. 

 When any power belonging to any rank or place is not required in expressing any 

 particular number, occupies the place of either -|- or — in that place of the com- 

 bination of characters for the number : thus +0 — h would express 243 + 1, 



diminished by 3 and 27, or 214, to express which, the 3rd power of three, or 9, is 

 not required ; the then in the third place expresses this, and yet gives the proper 

 value to the — , 0, and +, in the 4th, 5th and 6th places. Arithmetical operations 

 are performed by the aid of these simple marks with all the rapidity and security of 

 the simplest algebraic processes, and pretty much in accordance with the well-known 

 algebraic rules : thus to add 214 and 14 the process would be thus : 



214 or +0 — h 



14 or H 



sum 228 or +0 j- + 



Here the + and — in the place of the power of 3 destroy one another ; if the two 

 — 's in the next place had a third with them, they would go on as one — to the 3rd 

 place ; that — is then supposed to be introduced, but to balance it a + is also intro- 

 duced and set down below : we then have two — 's in the third place, which simi- 

 larly give + below, and one — goes on to the fourth place, where the ■\- and — al- 

 ready existing balance one another, or mutually destroy the — ; that which has been 

 brought on appears below, and so do the of the 5th place, and the + of the 6th, 

 there being nothing to alter them. One other simple example must suffice : multi- 

 plication of 214 by 14 will stand thus : 



+ 0- - + 



-0 + + - 

 -0 + + — 

 — + + — 

 +0-0-+ 



+ +0+ — 

 or 2187 + 81 — 1 = 2996. 



