and Vulgar Arithmetic. 89 
thoroughly understood and profoundly taught, I feel persuaded that 
you will permit me, through the medium of your Magazine, to 
express my disapprobation of the present method, which preyails 
in the public seminaries throughout the united kingdom, of 
solving such questions as relate to mixing of merchandises, as 
rums, wines, &c. commonly termed Alligation, which, save the 
medial case, is in all respects very abstruse, frequently, false, 
and generally detrimental to the merchant. Nay more, indeter- 
minate equations (of which questions in Alligation are a species) 
are in works of the greatest celebrity very frequently most erro- 
neously solved, where the great power of algebra in the present 
mode of application is insufficient to demonstrate the truth; 
whilst, if rightly handled, vulgar arithmetic is abundantly sufii- 
cient to obtain all the integral answers to such questions. 
That this is not on my part a mere assertion, but a well- 
grounded fact, I shail adduce, for the satisfaction of the public, 
a most glaring instance, which may be found in that well-known 
work, Dedson’s Mathematieal Repository, where, vol. ii. ques- 
tion XII. page 39, we read as follows : 
“ Let 2x + Sy + 52+ 350u= 100003,” the number of integral 
answers (see fol. 44) are stated to be “ 160,190,378,249.” 
Now, sir, you will give me leave to state, that the actual 
number of integral equivalent answers to that equation are just 
185,090,752,407; therefore the error in Dodson’s computation 
is 24,900,374,158 answers too few. And you will also give me 
leave to state, that such erroneous computation arises not from 
accident, but from the adopted method of solution. 
Authors have uniformly avowed that vulgar arithmetic would 
not obtain all the possible answers to questions of this sort, and 
that algebraic reasoning was indispensable. I am ready to prove 
the contrary, and to show that arithmetic is not only equal, but 
infinitely superior, in all cases of alternation. 
Errors equally glaring are to be found in the algebra of the 
celebrated Bonnycastle, who, so recently as his 7th edition of that 
great art, fails in some of these equations. And, what is more 
extraordinary, the ingenious Mr. Davis’s Key to that celebrated 
work corresponds with the author's errors ;—a proof of the ne- 
cessity of adopting a different system, whereby the truth may be 
discovered. : 
To enter into sufficient detail would engross too many pages 
of your Magazine ; for 1 am well aware, that in instances where 
errors have become habitual, nothing short of an elaborate pro~ 
cess can possibly convince. I shall therefore content myself, 
for the present, with a general disapproval of the inadequacy of 
that system universally prevailing; but should any gentleman 
think 
