ALGEBRA. 
life. Afterwards algebra was applied to 
geometry, and all the other sciences in their 
turn. The application of algebra to geome- 
try, is of two kinds ; that which regards the 
plane or common geometry, and that which 
respects the higher geometry, or the nature 
of curve lines. 
The first of these, or the application of 
algebra to common geometry, is concern- 
ed in the algebraical solution of geome- 
trical problems, and finding out theorems in 
geometrical figures, by means of algebrai- 
cal investigations or demonstrations. This 
kind of application has been made from the 
time of the most early writers on algebra, 
as Diophantus, Cardan, &c. &c. down to 
the present times. Some of the best pre- 
cepts and exercises of this kind of applica- 
tion are to be met with in Sir I. Newton’s 
“ Universal Arithmetic,” and in Thomas 
Simpson’s “ Algebra and Select Exercises.” 
Geometrical Problems are commonly resolv- 
ed more directly and easily by algebra, than 
by the geometrical analysis, especially by 
young beginners ; but then the synthesis, or 
construction and demonstration, is most 
elegant as deduced from the latter method. 
Now it commonly happens that tire alge- 
braical solution succeeds best in such pro- 
blems as respect the sides and other lines in 
geometrical figures ; and, on the contrary, 
those problems in which angles are con- 
cerned, are best effected by the geometri- 
cal analysis. Sir Isaac Newton gives these, 
among many other remarks on this branch. 
Having any problem proposed, compare to- 
gether the quantities concerned in it ; and, 
making no difference between the known 
and unknown quantities, consider how they 
depend, or are related to, one another ; 
that we may perceive what quantities, if 
they are assumed, will, by proceeding syn- 
thetically, give the rest, and that in the 
simplest manner. And in this comparison, 
the geometrical figure is to be feigned and 
constructed at random, as if all the parts 
were actually known or given, and any 
other lines drawn that may appear to con- 
duce to the easier and simpler solution of 
the problem. Having considered the me- 
thod of computation, and drawn out the 
scheme, names are then to be given to the 
quantities entering into the computation, 
that is, to some few of them, both known 
and unknown, from which the rest may most 
naturally and simply be derived or express- 
ed, by means of the geometrical proper- 
ties of figures, till an equation be obtained, 
by which the value of tire unknown quan- 
tity may be derived by the ordinary me- 
thods of reduction of equations, when only 
one unknown quantity is in the notation ; or 
till as many equations are obtained as there 
are unknown letters in the notation. 
For example : suppose it were required 
to inscribe a square in a given triangle. Let 
ABC, (Plate Miscellanies, fig. t.) be the 
given triangle ; and feign DEFG to be the 
required square ; also draw the perpendicu- 
lar BP of the triangle, which will be given, 
as well as all the sides of it. Then, consider- 
ing that the triangles BAC, BEF are simi- 
lar, it will be proper to make the notation 
as follows, viz. making the base AC — b, 
the perpendicular BP — p, and the side of 
the square DE or EF = x. Hence then 
BQ = BP — ED = p — x ; consequently, 
by the proportionality of the parts of those 
two similar triangles, viz. BP : AC :: BQ ; 
EF, it is p:b::p—x:x; then, multiply 
extremes and means, &c. there arises px = 
bp 
bp — bx, oxbx-\-px = bp, and 
the side of the square sought ; that is, a 
fourth proportional to the base and perpen- 
dicular, and the sum of the two, taking this 
sum for the first term, or AC BP : BP :: 
AC : EF. 
The other branch of the application of 
algebra to geometry, was introduced by 
Descartes, in his Geometry, which is the 
new or higher geometry, and respects the 
nature and properties of curve lines. In 
this branch, the nature of the curve is ex- 
pressed or denoted by an algebraic equa- 
tion, which is thus derived : A line is con- 
ceived to be drawn, as the diameter or some 
other principal line about the curve ; and 
upon any indefinite points of this line other 
lines are erected perpendicularly, which 
are called ordinates, whilst the parts of the 
first line cut off by them are called abscis- 
ses. Then, calling any absciss x, and it* 
corresponding ordinate y, by means of the 
known nature, or relations, of the other 
lines in the curve, an equation is derived, 
involving x and y, with other given quanti- 
ties in it. Hence, as x and y are common 
to every point in the primary line, that equa- 
tion, so derived, will belong to every posi- 
tion or value of the absciss and ordinate, 
and so is properly considered as expressing 
the nature of the curve in all points of it ; 
and is commonly called the equation of the 
curve. 
In this way it is found that any curve line 
has a peculiar form of equation belonging 
to it, and which is different from that of 
