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well ss the drawings) were oanted for 
Lord Berwick, beam with an attractive 
fimplicity, that faicinates the eve, and 
awakens in the heart the moft pathetic 
fenfations. His drawings of Sappho, 
chanting the Hymna of Love ; Hefiod in- 
ftructing the Greeks in the Arts of Peace ; 
and a Storm in Harveft, are admirably 
thought, ana exquifitely coloured... “Whe 
different effets of the thunder ftorm on 
the old and the young, are delineated in 
amanner that evinces the tafte, truth, 
and genius of the mafter, and diiplays 
nature as in a mirror. 
Among the novelties of the exhibition, 
are two Artifts, who have trod in the 
path of William Hogarth. IMr. Smirke, 
in his little piéture of The Conqueft, has 
given us a reprefentation fo irrefiftibly 
fudicrous, fo wellimagined, fo well drawn, 
and fo admirably pencilled and coloured, 
as places him at the head of that clafs, 
and entitles him to the place of lineal 
fucceffor to that great mafter, whofe 
works have been properly confidered 
rather as comedies, written with a pen- 
cil, than as piétures; for when we con- 
fider the clear mannerin which the ftory 
is told, the nice difcrimination cf cha- 
rafter, and variety of incident, heightened 
by broad and genuine humour, they are 
fairly entitled to the appellation. To 
this may be added, what 1s a ftill higher 
see that his ftories. have almoft inva- 
riably a moral aim, and tend to the. pro- 
motion of virtue. That Mr. Northcote 
had the fame view in the ten pictures, 
which he has entitled, Diligence and Du1/- 
Sipation, we can readily fuppofe ; but to 
give, what he calls, she Progrefs of a Mo- 
defi Girl and a Wanton, after Hogarth’s 
Harlot’s Progrefs, and Two Apprentices, re- 
quired a knowledge of life, an intimate 
acquaintance with the manners cf the 
times, a power of marking characters 
appropriate to’ fituations, and ai num- 
ber of other. ef ceferas, whiclvit is nogreat 
reproach to Mr. Northcote to fay he does 
not poffefs ; for few men, of any age or 
country, have poffefledthem. The ae is 
not very fkilfuily told, neither is the draw- 
ing, difpofition of the figures, or colouring, 
equal to many of Mr. Northcote’s other 
produétions; but the four prints which 
are engraved, are ina good flile, and the 
whole is intended to have a good tenden- 
cy. We with him fuccefs in the prints, 
tnough we think, that in painting fuch 
a’ feries of: pidures; he has miftaken’ his 
powers; they are not calculated for this 
branch:of the arts, nor will he add to 
his reputation by quitting hiftory, in 
Mathematical Correfpondence. 
[May 
which he is fo defervedly diftinguithed, 
for morality or humour. 
May 15, 1796. Your’s, &c, 

MATHEMATICAL CORRESPONDENCE, 
For the Monthly Magazine. 
A NEW AND GENERAL METHOD FOR 
EXTRACTING THE Roots oF Num- 
BERS. 
HE common method for extraéting the 
roots of numbers, deduced from the 
involution of. the binomial, is, in moft 
cafes, too laborious to be employed i in prac- 
tice. Authors have, therefore, had reccurfe 
to the method of approximation, and, by 
“means of the great improvements in the 
modern algebra, have difcovered many 
fimple and ufeful rules for this purpofe. 
The celebrated Dr. Halley, in the Phi- 
Jofophical Tranfaétions for 1694, publifh- 
ed a very eafy and general method for 
finding the reots of equations in numbers, 
The Gea rems which he there determines, 
are fallin bigh efteem among mathemati- 
Cians, and the method of invettigation is 
followed in moft refearches of a familar 
nature. He firft affumes the required 
root, as nearly true as poffible;. and then, 
by fubftiruting this value, increafed_or di- 
minifhed. by an unknown quantity, in- 
ftead of the true root, a new equation 
emerges, affected only with that difference 
and known quantitics. The root of this 
new-found cquation, on account of its 
{mallnefs, is eafily determined to any de- 
gree of exadtnefs, and from thence the 
root of the-given equation. Itis manifeft, 
that the extraction of roots is the fame 
with finding the roots of pure equations ; 
and, therefore, from the general formule 
of Dr. Halley’s accurate aud ufeful rules 
may be deduced for this purpofe. Many 
other mathematicians, proceeding in the 
fame track, have confiderably improved: 
and fimplified thefe rules: the names of 
Taylor, Simpfon, and Emerfon, are among 
the number. 
Dr. Hutton, the prefent profeffor of 
mathematics, in the Royal Academy at’ 
Woolwich, has likewife paid attention to 
this fabject: in his Traéts,. Mathematical 
and Philofophical, a general rule is given 
for the extraétion of roots, which, though 
not effentially new, is much fimpler than 
any other in its form, and better adapted 
for common ufe. 
But all thefe rules, however elegant and 
ufeful, are deficient; for when very great 
accuracy is required, it is neceflary to re- 
peat the operation with the new-found 
roet; which additional procefs more than 
doubles 
