S U R 
S U R 
S U R 
73 2 
wa? reduced, parliamentary grants had be- 
come really necessary almost every year. 
It was impossible, however, for the parlia- 
ment, distrusting not only Charles’s economy, 
but his regard for the interest of his kingdoms, 
to vest considerable sums of money in such 
unsafe and improvident hands: it Was, there- 
fore, thought requisite to specity the pur- 
poses for which each sum was voted. '1 hus 
appropriating clauses came to be introduced, 
which practice has continued ever since; 
and at the commencement of each session, an 
account is presented ot the disposition of the 
grants of the preceding session, shewing how 
much has been actually paid on each branch 
of the public service, what remains unpaid 
of the sums appropriated, with the funds tor 
discharging the same*, and the surplus or de- 
ficiency of tiie ways and means. 
The supplies annually voted do not include 
the interest and charges of the national debt, 
the civil list, and some other articles which 
are provided for as permanent charges on 
the consolidated fund ; but merely the ex- 
pences of the army, navy, ordnance, and 
such miscellaneous services as are granted 
from year to year. 
SUPPORTERS. See Heraldry. 
SUPPRESSION. See Medicine. 
SUPREMACY, in our polity, the superi- 
ority or sovereignty of the king over the 
church as well as state, whereof he is establish- 
ed head. The king’s supremacy was at first 
established, or, as others say, recovered, by 
king Henry VIII. in 1534, after breaking 
with the pope. It is since confirmed by se- 
veral canons, as well as by the articles of the 
.church, and is passed into an oath which is 
required as a necessary qualification for all 
offices and employments both in church and 
state, from persons to be ordained, from the 
members of both houses of parliament, &c. 
SURA. See Anatomy. 
SURD, in arithmeticand algebra, denotes any 
number or quantity that is incommensurable to 
unity : otherwise called an irrational number or 
quantity. 
The square roots of all numbers, except L, 4, 
9 , 16, 25, 36, 49, 64, 81, 100, 121, 144, &c. 
(which are the squares of the integer numbers, 
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, &c.) are in- 
sommensurables : and after the same manner the 
cube roots of all numbers but of the cubes of 
j 2, 3, 4, 5, 6, &c. are incommensurables : and 
quantities that are to one another in the pro- 
portion of such numbers, must also have their 
square-roots, or cube-roots, incommensurable. 
The roots, therefore, of such numbers, being 
incommensurable, are expressed by placing the 
proper radical sign over them: thus ^/2,^/ 3, 
i/ 5 , */6 &c. express numbers incommensurable 
with unity. However, though these numbers 
are incommensurable themselves with unity, yet 
they are commensurable in power with it ; be- 
cause their powers are integers, that is, multi- 
ples of unity. They may also be commensur- 
able sometimes with one another, as the y/ 8 , 
and 2 because they are to one another as 2 
♦o 1 : £nd when they have a common measure^ 
as ^/2 is the common measure of both, then 
their ratio is reduced to an expression in the 
least terms, as that of commensurable quanti- 
ties, by dividing them by their greatest common 
measure. This common measure is found as in 
commensurable quantities, only the root of the 
common measure is to be made their common 
• , yi2 ,, * , V' 18 * 
divisor: thus — — ■ = y 4 = 2, and - — — 
y$ v 2 
3 y a. 
A rational quantity may be reduced to the 
form of any given surd, by raising the quantity 
to the power that is denominated by the name 
of the surd, and then setting the radical sign 
over it ; thus 
a = yv = yv = \/ a* — yv = yv , 
and 4 = yi6 = ^/64 = ^256 = ^/1024 
As surds may be considered as powers with 
fractional exponents, they are reduced to others 
of the sarre value that shall have the same radi- 
cal sign, by reducing these fractional exponents 
to fractions having the same value and a com- 
mon denominator. Thus V a — a 1 , and V a 
— a m , and — = 
; and there- 
fore, y'a and *yV reduced to the same radi- 
cal sign, become V o m and V a " . If you 
are to reduce y/ 3 andy/2 to the same denomi- 
i 
nator, consider ^/3 as equal to 3 L , and 2 
i 
as equal to 2~ 3 , whose indices reduced to a 
t j3 
common denominator, you have 3 a = 3<>,and 
I 2 
2^=26", and, consequently, \/ 3 = yV = 
^/27, and \/ 2 = %/‘2 z — %/ 4\ so that the 
proposed surds 3 and 2, are reduced to 
other equal surds \/21 and </ 4, having a com- 
mon radical sign. 
Surds of the same rational quantity are multi- 
plied by adding their exponents, and divided by 
subtracting them; thus, ^/ a X ^/a —a^x a ~ 3 
3 i~ s — V “ a? 
= * =« 6 =^; and = 
i 5 - 3 
5 — a 1 5 
m -}- n m \/ * 
mtt ’ n y<7 
V 2 _ 
aT 5 = t \/a 2 - y \/a X V a 
V 2 X V 2 = 
V flS = V 2 - 
If the surds are of different rational quanti- 
ties, as y' a 2 and V b ■> and have the same sign, 
multiply these rational quantities into one an- 
other,- or divide them by one another, and set 
the common radical sign, over their product or 
quotient. Thus, yV X V ^ = j \/ 2 X 
2 /,_2/ ]n . V* __ m / ^ _ m f a} . 
V V ’ 
VY _ 3 / T _ 3 / F _ x , . 
^/24 7 24 S' 8 2 , \ • 
If surds have not the same radical sign, re- 
duce them to such as shall have the same radi- 
cal sign, and proceed as before ; ^/ a X \/l>~ 
’•y—; iV = ”7‘- ; V 2 x V4 = 
' VT 
i I 3 2 6. 
2 1 X 4 J = 26 X 46 =y2 X = 
8 /8Xl6= *4=^ = ^ = 
%/2 2 * 26 
6 / $ /l6 
yj 7 >t = \/ jf = V 2 - If thc surds have 
any rational co-efficients, their product or quo- 
tient must be prefixed ; thus, 2\/ 3 X 5 V 6 = 
10 y/ 18. The powers of surds are found as the 
powers of their quantities, by multiplying their 
exponents by the index of the power required ; 
thus the square of V 2 is 2 :> * = 2^ — 
the cube of \/ 5 = 5 a ^ = 5 2 = %J 125. Or 
you need only, in involving surds, raise the 
quantity under the radical sign to the power 
required, continuing the same radical sign ; un- 
less the index of that power is equal to tha 
name of the surd, or a multiple of it, and in 
that case the power of the surd becomes ra- 
tional. Evolution is performed by dividing the 
fraction, v/hich is the exponent of the surd, by 
the name of the root required. Thus the square 
root of iy/V is y/a-^or y /a 4 . 
The surd *y/ a m x — a ’ij x ; and, in like man- 
ner, if a power of any quantity of the same 
name with the surd divides the quantity under 
the radical sign without a remainder, as here 
a m divides a m x, and 25 t.he square of 5 divides 
75, the quantity under the sign in V" 75, without 
a remainder ; then place the root of that power 
rationally before the sign, and the quotient un- 
der the sign, and thus the surd will be reduced 
to a more simple expression. Thus y/7 5 — $ 
y/3; ^48 = v / 3"x’l6 = 4 y/3; ^ 8I = 
\/ 27~X^ — 3 V3. 
When surds are reduced to their least ex- 
pressions. if they have the same irrational part, 
they are added or subtracted, by adding or sub- 
tracting their rational co-efficients, and prefix- 
ing the sum or difference to the common irra- 
tional part. Thus, 
V 75 V 48 = 5y , 3-|-4,y/3:=:9y3; 
y/81 — {- i/24 — 3 V 3 -f- 2 V 3 — 5 y/ 3 ; 
V 150 — 2 \Z54 = 5 ye — 3 ye = 2 y6 ; 
V a2x + = a V x + b V x = a b X 
V X. 
Compound surds are such as consist of two or 
more joined together; the simple surds are com- 
mensurable in power, and by being multiplied 
into themselves, give at length rational quanti- 
ties ; yet compound surds multiplied into them- 
selves, commonly give still irrational products. 
But, when any compound surd is proposed, 
there is another compound surd which, multi- 
plied into it, gives a rational product. Thus, if 
y a -[- yi were proposed, multiplying it by 
y « — y//4, the product will be a — b. 
The investigation of that surd, which, multi- 
plied into the proposed surd, gives a rational 
product, is made easy by three theorems, deli- 
vered by Mr. Maclaurin, in his Algebra, p. 109, 
seq. to which we refer the curious. 
This operation is of use in reducing surd ex- 
pressions to more simple forms. Thus, sup- 
pose a binominal surd divided by another, as 
2 2 2 2 
y/ 20 -j- yi2, by y5 — y.3, the quotient 
. , , , , y 20 -fyi 2 
might be expressed by £ ut this 
might be expressed in a more simple form, by 
multiplying both numerator and denominator 
by that surd which, multiplied into the deno- 
minator, gives a rational product ; thus, 
