A P P 
APR 
A a U 
131 
for the roots of pure powers, of which there 
are many., the best is that which was dis- 
covered by Dr. Hutton, and given in the first 
volume of his Mathematical Tracts, in point of 
ease, both of execution and for remembering it. 
The method is this : if N denote any number, 
out of which is to be extracted the root whose 
index is denoted by r, and if n be the nearest root 
_ , . , ,,r-]-l.N-|-r — l.nr 
first taken ; then shall — v » 
r — l.N-fr-f 1. nr 
be the required root of N very nearly; or as 
r — 1 times the given number added to r -|- 1 
times the given number added to r — 1 times 
the nearest power, so is the assumed root », to 
the required root, very nearly. Then this last 
value of the root, so found, if one still nearer 
is wanted, is to be used for n in the same theo- 
rem, to repeat the operation with it. And so 
on, repeating the operation as often as necessary. 
Which theorem includes all the rational formulae 
of Halley and De Lagny. 
For example, suppose it were required to 
double the cube, or to find the cube root of the 
number 2. Here r = 3 ; consequently r -f- 1 =4, 
and r — 1=2; and therefore the general theo- 
4n 4- 2 n l 2n -f- n 3 
l-em becomes — — r - , X « or 2 x n 
2n -f- 4tr n -j~ 2n 3 
for the cube root of N ; or as N -j- 2 n } ’ 2n -|- »' 
• * n * the root sought nearly. Now, in this 
case, N = 2, and therefore the nearest root « is 
1, and its cube h ! = 1 also : hence N -}- 2 « 3 — 
2 -j- 2 = 4, and -2N — j— « 3 =r 4 — (— 1 = 5 ; there- 
fore, as 4 ; 5 ” 1 ; s or lj = 1.25 the first ap- 
proximation. Again, taking r = and r % = 
125 . . . „ . 250 378 , , 
■ — ; hence k 4- 2 =24 = — , and 
64 T r 64 64 ’ ‘ 
:2n -f" n 3 = 4 -j- 
125 
64 
381, or as 126 l 127 ; 
therefore, as 378 
635 
= 1.259921, 
381 
' 64 
5 
4~ * 504 
which is the cube root of 2, true in all the 
635 
figures. And by taking — for a new value of 
n, and repeating the process, more figures may 
be found. 
Approximation, of the roots of equa- 
tion, bjj. Sir Isaac Newton’s method for 
approximating roots is this : first take a value 
of the root as near as may b«, by trials, either 
greater or less ; then assuming another letter 
to denote the unknown difference between 
this and the true value, substitute into the 
equation the sum or difference of the ap- 
proximate root and this assumed letter, in- 
stead of the unknown letter or root of the 
equation, which will produce a new equation 
having only the assumed small difference for 
its root or unknown letter ; and, by any 
means, find, from this equation, a near value 
of this small assumed quantity. Assume 
then another letter for the small difference 
between this last value and' the true one, and 
substitute the sum or difference of them into 
the last equation, by which will arise a third 
equation, involving the second assumed 
quantity ; whose near value is found as be- 
fore. Proceeding thus as fir as we please, 
all the near values, connected together by 
their proper signs, will form a series ap- 
proaching still nearer and nearer to the true 
value of the root of the first or proposed 
equation. The approximate values of the 
several small assumed differences, may lie 
found in different ways : Sir I. Newton’s me- 
thod is this ; as the quantity sought is small, its 
higher powers decrease more and more, and 
therefore neglecting them will not lead to 
any material error. lie therefore neglects 
all the terms having in them the second and 
higher powers, leaving only the (irst power 
and the absolute known term ; from which 
simple equation he always finds the value of 
the assumed unknown letter nearly, in a very 
neat and easy manner. 
For example, let it be required to find the root 
of the equation a 2 — 5x = 31 , or x 2 — 5x — 31 
= 0 : Here the root x, it is evident, is nearly 
= 8 ; for x therefore take 8 + *> and substitute 
8 + z for x in the given equation, and the 
terms will be thus ; 
a- 2 = 64 -j- 16z -p a 2 
— 5x = — 40 — 5z 
— 31 = — 31 
the sum is — 7 -|- 1 1 z -j- z 2 = 0. 
Then, rejecting z 2 , it is 1 lz — 7=0, and z = 
— 7 — = .6363, &c., or = .6 nearly. 
Assume now z = .6 + y : then 
11s 
= .36 -j- 1.2 'y -{“J 2 
= 6 . 6 -}- 1 by 
-/ = 0 , 
the sum — .04 1 2.2_y • 
, .04 
where y = = .003278 nearly. 
Assume it y = .003278 — * : then 
y 1 = .000010745284 — .006556* -{- * 2 
12.2 y =.0399916 — 12.2--* 
— .04 — — 04 
the sum .000002345284 — 1 2.206556* -j- * 2 = 0, 
where * = 
.000002345284 
.000000192133. 
12.206556 
Hence, then, collecting all the assumed differ- 
ences, with their signs, it is found that x = 8 -f- 
z -f-jy — * = 8 -}- .6 -f- .003278 — .000000192133 
= 8.6003277807867 the root of the equation re- 
quired, by Newton’s method. 
Example 2. Again, taking the cubic equation 
y 1 — 2 \y — 5 = 0; Newton proceeds thus : 
y is nearly = 2 ; take it therefore y = 2 -f- / ; 
then y 2 " 8 — j— 12^ — Gp 2 — j— f 
— 2y- = — 4 — 2 p 
the sum — 1 + 10 / -T C 'P 1 +/> 3 = 0 ; 
hence p = _1_ = .1 nearly. 
Assume it/> = .1 -f- q ; 
then />’= 0.001 4* 0.03 q -j-0.3 / 4- / 
4- Op 2 = 0.06 4-1.2 -j- 6 
-f 10/ = 1 -f 10 
the sum 0.06 1 — }- 1 1 .23 q 6.3/ -J- q ' = 0 ; 
hence q = — 0.0054 nearly, 
j Assume it q = — 0.0054 -j- r ; 
then / = —0.0000001 57464 -f- 0.00008748/-, &c. 
! 4- 6.3 ? 2 = 4-0.000183708 — 0.06804/-, &c. 
-j- 1 1 .23? = — 0.060642 -f 1 1 .23r 
- j - 0.061 = 4 - 0.061 
the sum 4*0.000541550536-)- 1 1.16204748/ ; 
hence /- = — 0.000048517, & c. 
| Hence, y ~ 2 -j- p -J- q -j- r 
— 2 4- 0.1 — 0.0054 — 0.000048517 
= 2.094551483, 
the root of the equation yd — 2y = 5. And in 
the same manner Sir I. Newton performs the ap- 
proximation for the roots of literal equations. 
APPIT, in the manege, the sense of the 
action of the bridle in the horseman’s hand. 
APPULSE, in astronomy, the approach 
of a planet towards a conjunction with the 
sun, or any of the fixed stars. 
APPURTENANCES, in common law, 
signify tilings corporeal and incorporeal that 
appertain to another tiling as principal; as 
hamlets to a manor, and common of pasture 
and fishery. Things 'must agree in nature 
and quality to be appurtenant, as a turbary, 
or a seat in a church, to a house. 
A PRIORI, a kind of demonstration. 
112 
APRON, in gunnery, the piece of lead 
which covers the touch-hole of a cannon. . 
APSIS, in astronomy, a term used indif- 
ferently for either of the two points of a pla- 
net’s orbit, where it is at the greatest or least 
distance from the sun or earth. Hence the 
line connecting these points is called the line 
of the apsides. 
APTENODYTES, in ornithology, a ge- 
nus of birds that seems to hold the same 
place in the southern parts of the world as 
the awks do in the northern. According to 
Latham, this is the genus penguin. It is seen 
only in the temperate and frigid zones on 
that side of the equator which it frequents. 
The same is observed of the awk in opposite 
latitudes. Sec Alca. 
APTERA, in the Linnccan system of zoo- 
logy, the seventh and last order of insects, 
the distinguishing characteristic of which is, 
that the insects comprehended in it have no 
wings : such are the louse, the flea, the no- 
dura, the monoculus, the acarus, the spicier, 
the scorpion, and the crab. 
API 01 E, among grammarians, an inde- 
clinable noun, or one which lias no variation 
of cases. 
APUS, avis indica, a constellation of the 
southern hemisphere, near the pole, which, 
according to Payer’s catalogue, contains 12 
stars; the largest is of the fifth magnitude. 
AQUA, water, a term frequently met with 
in the writings of physicians, chemists, &c. 
for certain medicines or menstruums, in a 
liquid form, distingui -he'd from cadi other bv 
peculiar epithets, as aqua alexiteria, aqua 
aluminosa, aqua fortis, See. Though for- 
merly thought an element, pure or distilled 
water is now found to be a compound body 
of three parts hydrogen and one of oxygen. 
See Air, &c. 
Aqua fortis, nitric acid. SeeCiiEMisTRv, 
Aqua marina, or Aqua marine, a name 
by which the jewellers call the beryl, on ac- 
count of its sea-green colour. 
Aqua regia, a combination of nitric and 
muriatic acids. In the new nomenclature it 
is called nitre-muriatic acid; it is called aqua 
regia, as the only acid formerly known to 
dissolve gold. 
Aqua secunda, aquafortis diluted with 
wider, and employed in the arts. 
Aqua vita:, answers to the eau de vie of 
the I rencli, usquebaugh of the Irish; whisky 
ot the Scotch, and is a name familiarly ap- 
plied to native distilled spirits. 
AQU/LDUCT, in hydraulics and archi- 
tecture, a conveyance made for carrying 
water from one place to another. T hose of 
the antient Romans were surprisingly mag- 
nificent. 1 hat which Lewis XIV. built near 
Maintenon, for carrying the Pucq to Versail- 
les, is perhaps the greatest now in the world : 
it is seven thousand fathoms long, with two 
thousand live hundred and sixty fathoms of 
elevation, and contains two hundred and 
forty-two arcades. 
AQUARIANS, in church history, an an- 
tient sect of Christians, who, under pretence 
of abstinence, made use of water instead of 
wine in the cut harist. 
AQUARIUS, in astronomy, a constel- 
lation, which makes the eleventh sign in the 
zodiac, marked thus, m. It consists of forty- 
live stars in Plolemy’s catalogue, of forty in 
T ycho’s, and in the Britannic catalogue of 
108 . • u 
l • ■ - . - . 
