Prof. Young on the Theory of Vanishing Fractions. 515 
Euphrates between these two points and the breadth of its 
bed in the lower portion of its course, both which causes must 
produce a corresponding diminution in its speed, and on the 
other hand the contraction of the channel of the Tigris, which 
must be attended with a corresponding acceleration of the mo- 
tion of its waters. 
Col. Chesney is referred to as describing the Euphrates as 
in the present day flowing in a dull and lingering stream: 
Herodotus, also an eye-witness, in his description of Babylon 
talks of the “deep and rapid streams of the great Euphrates *.” 
No one will for a moment doubt the accuracy of Col. Chesney’s 
observation; but is not credit also due to Herodotus? and is 
he, in like manner as Arrian, to be “unceremoniously thrown 
overboard,” whilst the facts respecting the former condition of 
these rivers remain unascertained? In the passage last cited, 
the Halicarnassian traveller further expressly asserts that the 
Euphrates “discharges itseif into the Persian Gulf;” which 
assertion he confirms in his more detailed statement that that 
river, “ which before flowed in an almost straight line,” had 
its course so turned by Nitocris, that in his time, “those who 
wished ¢o go from the sea up to Babylon were compelled to touch 
at Ardericca three times on three different days+.” Surely such 
unqualified and unequivocal assertions of plain matters cf fact 
are entitled to consideration, and are not to be put aside as 
errors simply because they are not applicable to the present 
state of things, or rather, perhaps, because they do not coin- 
cide with what we have been taught by former commentators 
to receive as the truth. 
[To be continued. ] 
LXXXVIII. On the Theory of Vanishing Fractions. By 
J. R. Youne, Esg., Professor of Mathematics in Belfast 
College. 
N a letter inserted in the April number of this Journal (p. 295) 
I ventured to offer some objections to certain novel positions, 
lately advanced by an ingenious mathematician, in an Essay 
on the Fundamental Principles of the Differential and Inte- 
gral Calculus. To these objections the author of the Essay 
as furnished a reply, in the number for May (p. 393); and I 
am happy to find, from the general tone of it, that Mr. Wool- 
house has considered my scruples with the same good feeling 
in which they were avowedly offered. 
* Ea wiyas, nal Babds, nal raxds, eSler d€ odtos és Thy “Epudeqv Sarno- 
cav.—Clio, 180. 
+ Clio, 185, t Communicated by the Author, 
s8G2 
