ALG 
every other curve, either as to the number 
of the terms, the powers of the unknown let- 
ters .r and y, or the signs or co-efficients of 
the terms of the equation. Thus, if the 
curve line HK, (fig. 2.) be a circle, of which 
HI is part of the diameter, and IK a per- 
pendicular ordinate : then put HI = x, IK 
— y, and p = the diameter of the circle, 
the equation of the circle will be p x — x 1 = 
y 2 . But if HK be an ellipse, an hyperbola, 
or parabola, the equation of the curve will 
be different, and for all the four curves, will 
be respectively as follows : viz. 
For the circle 
II 
1 
For the ellipse 
P 2 2 
For the hyperbola ... 
i V 2 2 
For the parabola ... 
% 
II 
1 
where t is the transverse axis, and p its pa- 
rameter. And in like manner for other 
curves. 
This way of expressing the nature of 
curve lines, by algebraic equations, has 
given occasion to the greatest improvement 
and extension of the geometry of curve 
lines ; for thus, all the properties of alge- 
braic equations, and their roots, are trans- 
ferred and added to the curve lines, whose 
abscisses and ordinates have similar proper- 
ties. Indeed the benefit of this sort of 
application is mutual and reciprocal, the 
known properties of equations being trans- 
ferred to the curves they represent; and, 
on the contrary, the known properties of 
curves transferred to their representative 
equations. 
Besides the use and application of the 
higher geometry, namely of curve lines, to 
detecting the nature and roots of equations, 
and to the finding the values of those roots 
by the geometrical construction of curve 
lines, even common geometry may be made 
subservient to the purposes of algebra. 
Thus, to take a very plain and simple in- 
stance, if it were required to square the 
binomial a -j- b ; (fig. 3.) by forming a 
square, as in the figure, whose side is 
equal to « -(- b, or the two lines or parts 
added together denoted by the letters a and 
b ; and then drawing two lines parallel to 
the sides, from the points where the two 
parts join, it will be immediately evident 
that the whole square of the compound 
quantity a -f- 6 is equal to the squares of 
both the parts, together with two rectan- 
gles under the two parts, or a 2 and b 2 and 
A LI 
$ a b, that is, the square of a -f- b is equal to 
a 2 b 2 L 1 ab, as derived from a geome- 
trical figure or construction. And in this 
very manner it was, that the Arabians, and 
the early European writers on algebra, de- 
rived and demonstrated the common rule 
for resolving compound quadratic equations. 
And thus also, in a similar way, it was, that 
Tartalea and Cardan derived and demon- 
strated all the rules for the resolution of cu- 
bic equations, using cubes and parallelopipe- 
dons instead of squares and rectangles. 
Many other instances might be given 
of the use and application of geometry in 
algebra. 
ALGOL, the name of a fixed star of the 
third magnitude in the constellation Per- 
seus, otherwise called Medusa’s head. This 
star has been subject to singular variations, 
appearing at different times of different 
magnitudes, from the fourth to the second, 
which is its usual appearance. These vari- 
ations have been noticed with great ac- 
curacy and the period of their return is 
determined to be 2 d 20 h 48' 56". The 
cause of this variation, Mr. Goodricke, who 
has attended closely to the subject, conjec- 
tures, may be either owing to the in- 
terposition of a large body revolving round 
Algol, or to some motion of its own, in 
consequence of which, part of its body, 
covered with spots or some such like mat- 
ter, is periodically turned towards the 
earth. 
ALGORITHM, gn Arabic term, not 
unfrequently used to denote the practical 
rules of algebra, ahd sometimes for the 
practice of common arithmetic ; in which 
last sense it coincides with logislica nume- 
ralis, or the art of numbering truly and 
readily. 
ALIEN, in law, a person born in a 
strange country, not within the king’s alle- 
giance, in contradistinction from a denizen 
or natural subject. 
An alien is incapable of inheriting land* 
in England, till naturalized by an of 
parliament. No alien is entitled v °te in 
the choice of members of parliament, has 
a right to enjoy offices., or can be returned 
on any jury, railed where an alien is party 
in a cause • !rtfa then the inquest of jurors 
shall h® one dhJf denizens and the other 
a ums. 
Every alien neglecting the king’s procla- 
mation directing him to depart from the 
realm within a limited time, shall, on con- 
viction, for the first offence be imprisoned 
for any time not exceeding one month, and 
