ALGEBRA. 
282 
■was applied to the laws of chance and gaming, by R. de 
Montmort; and fince by de Moivre and Janies Bernouilli. 
The elements of the art were compiled and .publiihed 
"by Kerfey, in 1671; wherein the fpecious arithmetic, and 
the nature of equations, are largely explained, and illuf- 
trated by a variety of examples; the whole fubftance of 
Diaphantus is here delivez'ed, and many things added con¬ 
cerning mathematical compofition and resolution from 
Ghetaldus. The like has been fince done by Preflet in 
1694, and by Ozanam in 1703. But thefe authors omit 
the application of algebra to geometry; which defeat is 
Supplied by Guifnec in a French treatife exprefsly on the 
fiibjedt publiihed in 1704, and l’FIopital in his analytical 
treatife of the Conic Sections in 1707. The rules of al¬ 
gebra are alio compendioufly delivered by Sir Ifaac New¬ 
ton, in his Arithmctica Univerfalis, firft publiihed in 1707, 
which abounds in felett examples, and contains feveral 
rules and methods invented by the author. 
Algebra has fince been much improved, and applied to 
the confideration and calculus of infinities, by Dr. Brooke 
Taylor, Nicole, Sterling, Campbell, Wolfius, Clairaut, 
Fontaine, Emerfon, Landen, Euler, Malbranche, Waring, 
Hales, and others; from whence a new and extenlive 
branch of knowledge has arifen, called the DoElrine f 
Fluxions, or Analyfis oj Infinites , or the Calculus Differcntialu. 
Avery ufeful work, on the Principles of the Mathe¬ 
matics, in 4 vols. containing the elements of Algebra, 
Fluxions, Optics, &c. was lately publiihed in the univerfity 
of Cambridge, by J. Wood, B. D. and the Rev. Mr. Vince; 
on the plan of which the following treatife is given : 
PART I. 
Definitions and Explanations of Signs. 
KNOWN or determined quantities are ufually repre- 
fented by the firil letters of the alphabet, a,b.c,d, &c. 
and unknown or undetermined quantities by the lad, as 
w, x, y, &c. And the following ligns are made ufe of to 
exprefs the relations which thefe quantities bear to each 
other : 
Plus -f- fignifies that tlic quantity to which it is prefixed 
mu ft be added. Thus a-\-b fignifies that the quantity re- 
prefented by b is to be added to that reprefented by a ; if 
a reprefents 5, and b 7, a-\-b reprefents 12. If no fign be 
placed before a quantity the fign -J- is underftood. Thus 
<?, fignifies +*• Such quantities are called pefitive quan¬ 
tities. 
Minus — fignifies that the quantity to which it is prefix¬ 
ed muft be fubtracled. Thus a — b, fignifies that b mull 
be taken from a ; if a be 7, and b 5, a—b expreil’es 7 dimi- 
niflied by 3, or 2. Quantities to which the fign — is pre¬ 
fixed are called negative quantities. 
Into X fignifies that the quantities between which it 
Hands are to be multiplied together. Thus fignifies 
that the quantity reprefented by a is to be multiplied by 
the quantity reprefented by b ; for, fince quantities may in 
each particular cafe be reprefented by numbers, we may 
without impropriety fpeak of the multiplication, divifion, 
&c. of quantities by each other. This fign is frequently 
omitted, thus abc fignifies a X b X c. Or a full point is 
ufed inftead of it; thus 1X2X3 and 1-2,3 fignify the 
fame tiling. If in multiplication the faiiie quantity be 
repeated any number of times, the product is ufually ex- 
preffed by placing above the quantity the number which 
reprefents how often it is repeated, thus.a, a X a, a X a X a t 
cX‘tX a Xa> an d a', ad, a 3 , a 4 , have refpedtively the 
fame Signification. Thefe quantities are called powers, thus 
a', is called the firft power of a; a * 2 , the fecond power or 
fquare ; a 3 , the third power or cube of a, Sc c. The num¬ 
bers 1, 2, 3, &c. are called the indices of a. 
Divided by — fignifies that the former of the quantities 
between which it is placed is to be divided by the latter. 
Thus a~b fignifies that the quantity a is to be divided by 
b. The divifion of one quantity by another is frequently 
reprefented by placing the dividend over the divifor with 
a line between them, in which cafe the expreflion is call¬ 
ed a fraftion. Thus, j fignifies a divided by b, and a is 
0 
alfo the numerator and b the denominator of the fraction; 
fife fignifies that a, b, and c, added together, are to 
f 4:/+0 
be divided by e,f, and g, added together. A quantity in 
the denominator of a fraction is alfo expreffed by placing 
it in the numerator, and prefixing the negative fign to its 
index; thus, a a 2 , a 3 , a n , fignify -L, A, _L 
j a a a 3 
refpe&ively; thefe are called negative powers of a. 
The fign ** between two quantities fignifies their diffe¬ 
rence ■, a^x, is a—x or x — a, according as a or x is the 
greater. 
A line drawn over feveral quantities fignifies that they 
are to be taken collectively, and it is called a vinculum. 
Thus a —^ —j—r X d—e fignifies that the quantity repre¬ 
fented by a — b-\-c is to be multiplied by the quantity re¬ 
prefented by d — c. Let a Hand for 6 ; b, 5; c, 4 ; <4 3; 
and e, r; then <2—^-(-cis6—5-(-4, 01-5; and d—e is 3—1, 
or 2; therefore a — b-\~cX d. — e is 3X2, or 10; ab—cdX 
ab—cd or ab — cdY fignifies that the quantity reprefented 
by ad — cd, is to be multiplied by itfelf. 
Equal to j== fignifies that the quantities between which 
it is placed are equal to each other ; tints ax—by — rdf-qd 
fignifies that the quantity ax — by is equal to the quantity 
-cd-\-ad. 
The fquare root of any propofed quantity is that quantity 
whofe fquare or fecond power gives the propofed quan¬ 
tity. The cube root is the quantity whofe cube gives the 
2 3 4 
propofed quantity, Sec. ff or ff, ff, ff, See. are ufed to 
exprefs the fquare, cube, biquadrate, See. roots of the quan- 
2 •_ 3 _ 
tities before which they are placed : as, ff a*—a, f a 3 —a, 
4 — 
a?—a. See. If thefe roots cannot be exaftly determin¬ 
ed, the quantities are called irrational or furds. 
Points are made ufe of to denote proportion, thus a\b 
:: c: d, fignifies that a bears the fame proportion to b that 
c bears to d. 
The number prefixed to any quantity, and which fhews 
how often it is to be taken, is called its coefficient. Thus, 
7, 6, and 3, are called the coefficients of 7 ax, 6by, and 7 dz, 
ref’peCIively. When no number is prefixed, the quantity 
is to be taken once, or the coefficient 1 is underflood. 
Thefe numbers are fometimes reprefented by letters, which 
are alfo called coefficients. 
Similar or like algebraical quantities are fuch as differ on¬ 
ly in their coefficients ; 4a, Gab, 9 ar, q,adbc, are respectively 
fimilar to 15a, 3 ab, izad, ie,adbc,Sec. Unlike quantities are 
different combinations of letters ; thus, ab, adb, ab 2 , abc. 
Sec. are unlike. 
A quantity is faid to be a multiple of another, when it 
contains it a certain number of times exaftly; thus i6ais 
a multiple of 4 a, as it contains it exactly four times. A 
quantity is called a mcafure of another when it is contained 
in it a certain number of times exadtly; thus 4 a is a mea- 
fure of \Ga. 
A Jimp/e algebraical quantity is one which confifts of a 
fingle term, as a 2 bc. A binomial is a quantity confiding 
of two terms, as a -J-/5, or ia —3^. A trinomial is a quan¬ 
tity confiding of three terms, as 2 a-\~bd —3c. 
The following examples will ferve to illudrate the mc» 
thod of reprefenting quantities algebraically: 
Let a=. 3, b— 7, c—6 , d— 5, and e— 1 ; then 
3<z—2^+44—-e=24—14+24—1=33. 
3 C6 
