Noticesrespecting New Books. 477 



both in detail and in theory, to another, which we will explain by an 

 example taken from p. 207: 



7276*68024=3 { 3, 3, 3, 0, 0, 0, 0, 747541336 



= { 0, 0, 0, 0, 0, 0, 0, 889287691 



= { 8 88928769lV 

 Similarly, 



>| 0, 0, 5 is -jf 3 T\ 

 and 



J, 0,0, 0,317 is i 4 3lT . 



In practice, this index, which may be always taken ad libitum, is 

 a fixed number. Mr. Byrne chooses for this the number 8 and sup- 

 presses the index. The result is that the expression X=P or ^ 

 means nothing more than that x is the logarithm of X to the base 

 1*00000001, or 1 + 10 ~ 8 , and that its use as such is the key to the 

 whole work done. 



We confess to considerable disappointment at discovering this to 

 be the "resultant" of the method. It is quite certain that arith- 

 metic is not to be revolutionized or even materially simplified by 

 such means as this ; and we are driven to the conclusion that there 

 is more merit in the attempt than value in the result. After a 

 perusal made with sufficient care to discover a few typographical 

 errors in a carefully printed book, we find ourselves obliged to state 

 that all Mr. Byrne's arithmetical skill and ingenuity do not enable 

 him to solve his problems with as little work as a good selection of 

 ordinary methods would require. We rise from our perusal far more 

 convinced of the power of the author than of his method. 



Mr. Byrne concludes his introduction as follows : — 



" In my works on Algebra and the Calculus, which are being 

 prepared for publication, the whole subject and its different applica- 

 tions will be treated in a general and exhaustive manner. My work 

 on the Calculus, to be termed the ' Calculus of Form,' unfolds a new 

 science, and establishes modern analysis on a purely mathematical 

 basis, rejecting the reasoning of the differential and other methods." 



We do not quite understand what reasoning it is that Mr. Byrne 

 proposes to reject, or why. But we gather that he entertains some 

 objection to the existing systems, probably thinking them illogical, 

 complex, and prolix. Now we are not bigoted to them. We are 

 painfully aware of their deficiencies in the last two respects, and we 

 are not without some misgivings on the first head. But we may be 

 forgiven for reminding Mr. Byrne that a proposal of change is as 

 disagreeable as a dose of physic, and requires even more tact in its 

 administration. If he expects to gain the ear of mathematicians, 

 he must do what he has not done in his ' Dual Arithmetic ' — give a 

 clear and succinct statement of his plan, his principle, and his results, 

 so that the educated reader may see the value of the prize which is 



Phil. Mag. S. 4. Vol. 25. No. 170. June 1863. 2 K 



