69 
Mr Galbraith on Vanishing Fractions. 
stood when they left it. They seemed not only puzzled but 
angry, and I was obliged to keep at a respectful distance. 
When the hive was replaced, those bees that had left it after its 
first removal, stopped for a little at the place where they had quit- 
ted it ; but in a few minutes all seemed to be quiet and regular 
as formerly. I removed another hive in the evening, when 
none of the inhabitants were abroad, and in the morning there 
was no resumption of confusion whatever. Yours very sin- 
cerely, G. S. Mackenzie. 
CouL, 9>Qth April 1820. 
Art. X . — On vanishing Fractions. By Mr William Gal- 
braith. Communicated by the Author. 
The subject of vanishing fractions, ever since the early part 
of the eighteenth century, has been frequently treated by ma- 
thematicians. The idea of these fractions first originated about 
the year 1702, in a dispute between Varignon and Bolle, rela- 
tive to the differential calculus. Rolle opposed the legitimacy 
of the conclusions derived from that calculus, because, in an 
expression for drawing a tangent to a curve, a quantity was ob- 
tained, which had both its numerator and denominator equal 
to 0, and this he regarded as a result so absurd, as to amount 
to a proof of the fallacy of the method of solution. John Ber- 
noulli soon afterwards, according to Montucla, attempted to 
clear up the difficulty ; and Saurin, upon a renewal of the dis- 
pute, farther shewed that §, in the case alluded to, had a real 
value. Still, however, the explanations hitherto given seem to 
have been rather obscure, as these fractions were also the cause 
of a violent controversy between Waring and Powell in 1760, 
when they were candidates for the Mathematical Professorship 
at Cambridge; Waring maintaining that the value of 
is equal to 4 when a? ~1, and Powell, or perhaps rather Ma- 
seres, who is understood to have conducted the controversy, that 
it is equal to 0. 
