S E R 
8 ER 
f reat, because it would be so if the terms con- 
tinued all equal, and therefore will be more 
so since they increa-e. But if we suppose 
the series increases infinitely, either by equal 
ratios or differences, or. by increasing differ- 
ences or ratios of each term to the preced- 
ing; then the reason of the sums being infi- 
nite will appear from the first theorem ; for 
in such a series, a term can be found greater 
than any assignable number, and much more 
therefore the sum of that and all the pre- 
ceding. 
Tiveor. IV. The sum of an infinite series 
of numbers decreasing in the same ratio is a 
finite number, equal to the quote arising from 
the division of the product of the ratio and 
first term, by the ratio less by unity ; that is, 
the sum of no assignable nujnber of terms of 
the series can ever be equal to that quote; 
and yet no number less than it is equal to ti;e 
value of the series, or to what we can actu- 
ally determine in it; so that we can carry 
the series so far, that the sum shall want of 
this quote less than any assignable difference. 
Demonstration. To whatever assigned number 
of terms the series is carried, it is so far finite ; 
and if the greatest term is /, the least A, and the 
. . rl — a 
ratio r, then the sum is S = — . See Geo- 
r — 1 
METiiicAt. Progression-. 
Now, in a decreasing series from /, the more 
terms we actually raise, the last of them, A, be- 
comes the lesser, and the lesser A be, rl — A is 
the greater, and so also is — 
but rl— A 
vl A 
being still less than rl> therefore is still 
r — 1 
rl 
less than — , that is, the sum of anv assign- 
r — 1 J 
able number of terms of the series is still less 
rl 
than the quote mentioned, which is, 
r — 1 
and 
this is the first part of the theorem. 
Again : The series may be actually continued 
so far, that — shall want of — — - less than 
> — 1 r — 1 
any assignable difference ; for, as the series goes 
on, A becomes less and less in a certain ratio, 
and so the series may be actually continued till 
A becomes less than any assignable number ; 
(by Theorem II ). Now 
r — 1 
>- — J 
— , and is less than A ; therefore, let 
r — 1 r — 1 
any number assigned lie called N, we can carry 
the series so far till the last term A is less than 
JM ; and because — wants of - — — - — , the 
. r — 1 r — 1 
dfference — - — , which is less than A, which is 
r — 1 
also less than N, therefore the second part of 
the theorem is also true, and is the 
true value of the series. 
Scholium. The sense in which 
r — I 
rl 
capacity, as if the whole were actually deter- 
mined and added together, can never he the oc- 
casion of any assignable error in any operation 
or demonstration where it is used in that sense; 
because if it is said that it exceeds that adequate 
value, yet it is demonstrated that this excess 
must he less than any assignable difference, 
which is in effect no difference, and so tlie con- 
sequent error wiil be in effect no error : for if 
any error can happen from — ~~ being great- 
er than it ought to be, to represent the com- 
plete value of the infinite series, that error de- 
rl 
pends upon the excess of over that com- 
r — 1 
plete value ; but this excess being unassignable, 
that consequent error must be so too ; because 
still the less the excess is, the less will the error 
be that depends upon it. And for this reason 
w r e may justly enough look upon — — — as ex- 
pressing the adequate value of the infinite se- 
ries. But we are farther satisfied of the reason- 
ableness of this, by finding in fact, that a finite 
quantity does actually convert into an infinite 
series, which happens in the case of infinite de- 
S E R 
cimals. For example, 
is plainly a geometrical series from 
6\[) 
.6 6 6 6, &c. which 
6 
To 
6 
in the 
continual ratio of 10 to 1 ; for it is — + 
10 
6 
100 
, 6 6 
' looo 10600’ &c ' 
And reversely ; if we take this series, and 
find its sum by the preceding theorem, it comes 
to the same f ; for l — — , r — 10, therefore 
, 60 
rl — 1Q = 6 ; and r 
1 = 9; whence 
rl 
6 2 
~ TT ~~ IT' 
_ We have added here a table of all the varie- 
ties of determined problems of infinite, decreas- 
ing, geometrical progressions, which all depend 
upon these three things, viz. the greatest term l , 
the ratio r, and the sum S ; by any two of which 
the remaining one may be found : to which we 
have added some other problems, wherein 
S — L is considered as a thing distinct by itself, 
that is, without considering S and L separately. 
Given 
rl 
r.s — l 
Sought 
s-l, 
S, l , 
r — 1 
s X r — 1 
/- — 1 
— /: 
5 s = s / x i l 
h = t - / x r — 1 5 
Solutions. 
of / — 
— h l — M 
! = — of 
l — M X 
/ h c , M/ 
S — l = of / = 
a — b l — M 
— / 
b M.? 
— of s — 
a l 
s — , • of s — / — 
/X>-1 
l — a - ^ of s — l : 
b 
~ M X S — l 
is called 
r — 1 
the sum of the series, has been sufficiently ex- 
plained ; to which, however, we shall add this ; 
that whatever consequences follow from the 
rl . 
Supposition of being the true and ade- 
r — 1 
quate value of the series taken in all its infinite 
VOL. 11. 
Theorem V. In the arithmetical progression 
1, 2, 3, 4, &c. the sum is to the product of the 
last term, by the number of terms, that is, to 
the square of the last term, in- a ratio always 
greater than 1 : 2, but approaching infinitely 
near it. But it the arithmetical series begins 
with 0, thus, 0, 1,2, 3, 4, &c then the sum is 
to the product of the last term, by the number 
of terms, exactly in every step as 1 to 2. 
Theorem VI. Take the natural progression 
beginning with 0, thus, 0, 1, 2, % &c and take 
the squares of any the like powers of the former 
series, as the squares, j, 1, 4, 9, &c. or cubes, 
0, 1, 8, 27 ; and then again take the sum of the 
series of powers to any number of terms, and 
also multiply the last of the terms summed by 
the number of terms, (reckoning always 0 for 
the first term;) the ratio of that sum io that 
product is more than 1 , (n being the index 
n X 1 s 
of the powers) that is, in the series of squares it 
is more than § ; in the cubes more than - and 
so on. But the series going on in infinitum, we 
may take in more and more terms without end 
into the sum and the more we ake, the ratio 
of the sum to the product mentioned grows less 
and less ; yet so as it never can actually be 
1 
equal to 
* X 
-p but approaches infinitely near 
4 N 
to it, or within less than any assignable dif- 
ference. 
SERlOLA, a genus of plants belonging to 
the order of polygamia axjualis, and to the 
class of syngenesis:, and in the natural system 
ranged under the 49th order, composita?. 
The receptacle is paleaceous; the calvx 
simple; and the pappus is somewhat plumose. 
There are four species : 1. r J he levigata: 2. 
EEthnensis-: 3. Creuinsis : 4. Urens. The 
first is a native of the . island of Candia, and 
flowers in July and August; the second is a 
native of Italy ; and the fourth is a native of 
the south or Europe. 
SERUTilU U, a genus of plants belonging 
to th e order of monogamia, and to the class 
of syngenesia. The calyx is imbricated; 
the corolla is monopefalous and regular, 
with one oblong seed n ider it. Phere are 
our species, natives oi the Cape of Good 
Hope. 
SERPENS. See Astronomy. 
SERPEN EES, in natural history, an or- 
der ot the amphibia das , the characteristics 
o' which are, a mourn breathing by the 
lungs only; body tapering; neck not dis- 
tinct; jaws dilatable, not articulate; no feet, 
tins, or ears : motion undulatory. r J hey are 
