PAR 
also, every triangle is half a parallelogram 
upon tlie same or an equal base, and of tlie 
same altitude, or between the same pa- 
rallels. Hence, also, a triangle is equal to 
a parallelogram, having the same base, and 
half tlie altitude, or half the base, and the 
same altitude. 
Parallelograms, therefore, are in a given 
ratio compounded of their bases and al- 
titudes. If then the altitudes be equal, 
they are as the bases, and conversely. 
In similar parallelograms and triangles, 
the altitudes are proportional to the ho- 
mologous sides, and the bases are cut pro- 
portionably thereby. Hence similar pa- 
rallelograms and triangles are in a duplicate 
ratio of their homologous sides ; as also of 
their altitudes, and the segments of their 
bases ; they are, therefore, as the squares of 
the sides, altitudes, and homologous seg- 
ments of the bases. 
In every parallelogram, the sum of the 
squares of the two diagonals is equal to the 
sum of the squares of the four sides. For if 
the parallelogram be rectangular, it follows 
that tlie two diagonals are equal ; and, con- 
sequently, the s(|uare of a diagonal, or, 
which comes to the same thing, the square 
of the hypothennse of aright angle, is equal 
to the squares of the sides. See Geome- 
try. 
Parallelogram, or Parallelism, a 
machine for the ready reduction of designs ; 
it is the same with the Pentagrapii, which 
see. 
PARAMETER, in conic sections, a con- 
stant line, otherwise called latus rectum. 
The parameter is said to be constant, be- 
cause in the parabola, the rectangle under 
it and any absciss, is always equal to the 
square of the corre.eponding semi- ordinate; 
and in the ellipsis and hyperbola, it is a 
third proportional to the conjugate and 
transverse axis. 
If t and c be the two axes in the ellipse 
and hyperbola, and x and y an absciss and 
its ordinate in the parabola : then 
t:c::c:p = ~ = the parameter in the 
former ; 
a: ; y :: y : p = ^ = the parameter in the 
last. 
The parameter is equal to the double or- 
dinate drawn through the focus of one of 
the three conic sections. 
PARAMECIUM, in natural history, a 
genus of the Vermes Infusoria class and or- 
der. Worm invisible to the naked eye, 
PAR 
simple, pellucid, flattened, oblong. There 
are seven species, of which P. aurelia is ra- 
ther a large animalculum, membranaceous, 
pellucid, and about four times longer than it 
is broad ; the fore-part obtuse, transparent, 
without intestines; the hind-part replete 
with molecules of various sizes ; the fold 
which goes from the middle to the apex is 
a.striking characteristic of the species, form- 
ing a kind of triangular aperture, and giv- 
ing it somewhat the appearance of a gim- 
blet. Its motion is rectilinear, reeling or 
staggering, and generally vehement. They 
are frequently found cohering lengthwise ; 
the lateral edges of both bodies appear 
bright. They may also be/seen sometimes 
lying on one another alternately, at others 
adhering by the middle. They will live 
many months in tlie same water without its 
being renewed. They are found in the be- 
ginning of summer, in those ditches in 
which duck-weed abounds. P. chrysalis is 
found plentifully in salt water. 
PARAPET, in fortification, an elevation 
of earth designed for covering the soldiers 
from the enemies’ cannon or small shot. 
The thickness of the parapet is from 
eighteen to twenty feet; its height is six 
feet on the inside, and four or five on the, 
outside. It is raised on the rampart, and 
has a slope above called the superior talus, 
and sometimes the glacis of the parapet. 
The exterior talus of the parapet is the slope 
facing the country : there is a banquette or 
two for the soldiers who defend the para- 
pet, to mount upon, that they may tlie bet- 
ter discover the country, fosse, and coun- 
terscarpe, and fire as they find occasion. 
Parapet of the covert-way, or corridor, is 
what covers that way from the sight of the 
enemy, which renders it the most danger- 
ous place for the besiegers, because of the 
neighbourhood of the faces, flanks, and cur- 
tins of the place. 
Parapet is also a little wall raised 
breast-high, on the banks of bridges, keys, 
or high buildings, to serve as a stay, and 
prevent people’s falling over, 
PARAPHRASE, an explanation of some 
text, in clearer and more ample terms, 
whereby is supplied what the author might 
have said or thought on the subject; such 
are esteemed Erasmus’s Paraphrase on 
the New Testament, the Chaldee Para- 
phrase on the Pentateuch, &c. 
PARASANG, an ancient Persian mea- 
sure, different at different times, and in dif- 
ferent places; being sometimes thirty, 
