ISO 
CHRONOus, is applied to such vibrations of 
a pendulum as are performed in the same 
space of time, as all tlie vibrations or swings 
of tlie same pendulum are, whether the 
arches it describes be longer or shorter ; 
for when it describes a shorter arch it moves 
so much the slower, and when a long one 
proportionably faster. 
Isochronal line, that in which a heavy 
body is supposed to descend without any 
acceleration. 
M. Leibnitz shows, that an heavy body, 
with a degree of velocity acquired by the 
descent from any height, may descend from 
the same point by an infinite number of 
isochronal curves, all which are of the same 
species, differing from one another only in 
the magnitude of their parameters ; such 
are all the (juadrato-cubical paroboloids, 
and consequently .similar to one another. 
He shows also there, how to find a line in 
which a heavy body descending shall recede 
uniformly from a given point, or approach 
uniforndy to it. 
ISOETES, in botany, a genus of the 
Cryptogamia Filices class and order. Na- 
tural order of Filices, or Ferns. Essentia! 
character : male, anther within the base of 
the frond ; female, capsule two-celled, with- 
in the base of the frond. There are two 
species, viz. I. laciistris, common quillwort, 
and I. coromandelina, Coromandel quill- 
wort, both natives of mountain lakes, and 
in wet places that are inundated in the 
rainy season. 
ISOPERIMETRICAL figures, in geo- 
metry, are such as have equal perimeters, 
or circumferences. 
Isoperimetrical lines and figures have 
greatly engaged the attention of mathema- 
ticians at all times. The fifth hook of Pap- 
pus’s Collections is cliiefly upon this sub- 
ject ; where a great variety of curious and 
important properties are demonstrated, 
both of planes and solids, some of which 
were then old in his time, and many new 
ones of his own. Indeed, it seems, lie has 
here brought together into this book all the 
properties relating to isoperimetrical figures 
then known, and their ditferent degrees of 
capacity. The analysis of tlie general pro- 
blem concerning figures, that, among all 
those of the same perimeter, produce maxi- 
ma and minima, was given by Mr. James 
Bernoulli, from computations that involve 
tlie second and third fluxions. And several 
enquiries of this nature have lieen since pro- 
secuted in like manner, but not always 
with equal success. Mr. Maclanrin, to vin- 
ISO 
dicate the doctrine of fluxions from the im- 
putation of uncertainty or obscurity, has 
illustrated this subject, which is considered 
as one of the most abstruse parts of this doc- 
trine, by giving the resolution and composi- 
tion of these problems by first fluxions 
only ; and in a manner that suggests a syn- 
thetic demonstration, serving to verify the 
solution. See Maclaurin’s Fluxions. Mr. 
Crane also, in the Berlin Memoirs for 17S2, 
has given a paper in which he proposes to 
demonstrate, in general, what can be de- 
monstrated only of regular figures in the 
elements of geometry, viz. that the circle 
is the greatest of all isoperimetrical figures, 
regular or irregular. AVe shall now mention 
a few of tlie properties of isoperimetrical 
figures. 
1. Of isoperimetrical figures, that is the 
greatest that contains the greatest number 
of sides, or the most angles, and conse- 
quently a circle is the greatest of all figures 
that have the same ambit as it has. 
2. Of two isoperimetrical triangles, hav- 
ing the same base, whereof two sides of 
one are equal, and of the other unequal, 
that is the greater whose two sides are 
equal. 
3. Of isoperimetrical figures, whose sides 
are equal in number, that is the greatest 
which is equilateral and equiangular. From 
hence follows that common problem of 
making the hedging or walling that will 
wall in one acre, or even any determinate 
number of acres, a ; fence or wall in any 
greater number of acres whatever, h. In 
order to the solution of this problem, let 
the greater number, Ji, be supposed a 
square ; let x be one side of an oblong, 
whose area is a ; then will ~ be the other 
X 
side ; and 2 - 4- 2 .v will be the ambit of the 
oblong, which must be equal to four times 
the square root of b ; that is, 2 ~ -1- 2 x = 4 
s/ b. Whence the value of x may be easily 
had, and you may make infinite numbers of 
squares and oblongs that have the same 
ambit, and yet shall have different given 
areas. 
Let .y b = d 
2 a -4- 4 a; a: ^ , 
Then ‘ = 4 d 
X 
a-{- 2xx — 2dx 
^xx — Sdx = — a 
\ 
