ALGEBRA. 
the word species. Thus, they use the names 
Titius, Sempronius, Caius, and the like, to 
represent indefinitely any person in such 
circumstances; and cases so propounded, 
they call species. Vieta, accustomed to 
the language of the civil law, gave, as Wallis 
supposes, the name of species to the letters 
A, B, C, &c. which he used to represent 
indefinitely any nuihber or quantity, so cir- 
cumstanced as the occasion required. This 
mode of expression frees the memory and 
imagination from that stress or effort, which 
is required to keep several matters, neces- 
sary tor the discovery of the truth investi- 
gated, present to the mind ; for which rea- 
son this art may be properly denominated 
metaphysical geometry. Specious algebra 
is not like the numeral, confined to certain 
kinds of problems; but serves universally 
for the investigation or invention of theo- 
rems, as well as the solution and demonstra- 
tion of ail kinds of problems, both arithme- 
tical and geometrical. The letters used in 
algebra, do each of them, separately, repre- 
sent either lines or numbers, as the prdblem 
is either arithmetical of geometrical; and 
together, they represent planes, solids, and 
powers, more or less high, as the letters are 
in a greater or less number. For instance, 
if there be two letters, a A, they represent 
a rectangle, whose two sides are expi essed, 
one by the letter a, and the other by b; so 
that by their mutual multiplication they 
produce the plane a b. Where the same 
letter is repeated twice, as a a, they denote 
a square. Three letters abc, represent a 
solid or a rectangular parallelepiped, whose 
three dimensions are expressed by the three 
letters abc; the length by o, the breadth 
by b, and the depth by c ; so that by then- 
mutual multiplication, they produce the 
solid abc. As the multiplication of dimen- 
sions is expressed by the multiplication of 
letters, and as the number of these may be 
so great as to become incommodious, the 
method is only to write down tiie root, arid 
on the right hand to write the index of the 
power, that is, the number of letters of 
which the quantity to be expressed consists ; 
as a\ a\ a\ &c. the last of which signifies as 
much as a multiplied four times into itself ; 
and so of the rest. But as it is necessary, 
before any progress can be made in the 
science of algebra, to understand the me- 
thod of notation, we shall here give a general 
view of it. In algebra, as we have already 
stated, every quantity whether it be known 
or given, or unknown or required, is usually 
represented by some letter of the alphabet ; 
and the given quantities are commonly de- 
noted by the initial letters, v,h, c,d, &c. 
and the unknown ones by the final letters, 
u, w, x, y, z. These quantities are connected 
together by certain signs or symbols, which 
serve to shew their mutual relation, and at 
the same time to simplify the science, and 
to reduce its operations into a less compass. 
Accordingly the sign plus, or more, signi- 
fies that the quantity to which it is prefixed 
is to be added, and it is called a positive or 
affirmative quantity. Thus, a -j- A expresses 
the sum of the two quantities a and b, so that 
if a were 5, and A, 3, a -{- A would be 5 -J- 3, or 
8. If a quantity have no sign, -j-> plus, is 
understood, and the quantity is affirmative 
or positive. The sign — , minus, or less, 
denotes that the quantity which it precedes 
is to be subtracted, and it is called a nega- 
tive quantity. Thus a — b expresses the 
difference of a and b ; so that a being 5, 
and b, S, a — b or 5 — 3 would be equal to 
2. If more quantities than two were con- 
nected by these signs, the sum of those 
with the sign — must be subtracted from 
the sum of those with the sign Thus, 
a -j- A — c — d represents the quantity which 
would remain, when c and d are taken from 
a and b. So that if a were 7, b, 6, c, a, and 
d, 3, a -f- A — c — d, or 7 -j- 6 — 5 — 3, or 
13 — 8, would be equal to 5. If two quan- 
tities are connected by the sign *, as a co ft, 
this mode of expression represents the dif- 
ference of a and b, when it is not known 
which of them is the greatest The sign x 
signifies that the quantities between which 
it stands are to be multiplied together, 
or it represents their product. Thus, a X b 
expresses the product of a and b ; a x A x e 
denotes the product of «, b, and c ; a -J- b 
X c denotes the product of the compound 
quantity a -j- h by the simple quantity c ; 
anda + A + c X a-b-\-c x « + A repre- 
sents tiie product of tiie three compound 
quantities, multiplied continually into one 
another ; so that if a wer e o, h, 4, and c, 3, 
then would a -)- ft -j- c X a — A -}- c x u-\-c 
be 1-2 x 4 X 8, or 384. The line connect- 
ing the simple quantities and forming a 
compound one, placed over them, is called 
a vinculum. Quantities that are joined to- 
gether without any intermediate sign form 
a product ; thus a A is the same with a x A, 
and abc the same with a X A X c. When 
a quantity is multiplied into itself, or raised 
to any power, the usual mode of expression 
is to draw a line over the quantity, and to 
place the number denoting the power at the 
