P ft o 
they ara dispersed over the 
whole Atlantic ocean. 
3. The pdagica, <*>r stormy petrel, is about 
t-lie bulk of the hous«.-swaUo\v ; the length 
six inches - r the extent of wings thirteen. 
The whole bird is black, except the coverts 
of the tail and vent-feathers, which are white; 
the bill is hooked at the end; the nostrils 
tubular; the lugs slender and long. It has 
tire same faculty of spouting oil from its bill 
as the other species ; and Mr. Brunuich tells 
« that the inhabitants ot the Eerroe islands 
this bird serve the purposes, of a 
«-* ii 
Biake — ; • , * 
candle, by drawing a wick through the 
mouth and rump, which being lighted, the 
flame is fed by the fat and oil of the body. 
Except in breeding-time, it is always at sea, 
and is seen all over the vast Atlantic ocean, 
at the greatest distance from land ; ollen 
following the vessels in great iiocks, to pick 
op any thing that foils from on board : for 
trial sake, chopped straw has been Hung over, 
which tlrey would stand on with expanded 
wings, but were never observed to settle or 
swim in the water; it presages bad weather, 
and cautions the seamen of the approach of a 
tempest, by collecting under the stern of the 
ships; it braves the utmost fury of the storm, 
sometimes skimming with incredible velo- 
city along the hollows of the waves, some- 
times on the summits. 
PROCESS, is the manner of proceeding 
in every cause, being the writs and precepts 
that proceed or go forth upon the original 
in every action, being either original or judi- 
cial.. Britton, 138. Process is only meant 
to bring the defendant into court, in order 
to contest the suit, and abide the determina- 
tion of the law. See Impey’s Practice. 
PROCKIA, a genus of the polyandria 
monogynia class and order. I lie cal. is 
three-leaved ; cor. none ; berry five-corner- 
ed, many-seeded. There is one species, a 
shrub of Santa Cruz. 
PROCLAMATION, a public notice 
given of any thing of which the king thinks 
proper to advertise his subjects. Procla- 
mations are a branch of the king s preroga- 
tive, and no person can make them without 
the king’s authority, except mayors of towns, 
&c. by custom or privilege. Proclamations 
which require the people to do, or not to 
do, certain things, have the force of laws ; 
but then they are supposed to be consistent 
with the laws already in being, otherwise they 
are superseded. 
PROCREATION- See Physiology. 
PROCTOR, a person commissioned to 
manage another person’s cause in any court of 
the civil or ecclesiastical law. T1 he proctors 
of the clergy, are the representatives chosen 
by the clergy to sit in the lower house of 
convocation ; of these there are two for each 
diocese, and one for each collegiate chuich. 
PROCURATOR, a person who has a 
charge committed to him to act tor another. 
Thus the proxies of the lords in parliament 
are, in our law-books, called procurators ; 
the’ bishops are sometimes called procura- 
tores ecclesiarum ; and the representatives 
sent by the clergy to convocation, procura- 
tors clench r I he word is also used tor a 
vicar or lieutenant ; and vye read of a pro- 
curator regni, who was an untie nt magistrate. 
Those who manage causes ia Doctors’ Com- 
PRO 
mons, are also railed procurators or proctor*. 
In our statutes, he who gathers the iruit* 
of a benefice for another is jfortieulai ly cak- 
ed a procurator, and the instrument in. pow- 
ering him to receive them is termed a pro- 
curacy. 
PROCYON, in astronomy, a fixed star of 
the second magnitude in the constellation 
called (ranis minor. See Cams. 
PRODUCING, in geometry, signifies the 
drawing out a line farther till it lias any as- 
signed length. 
PRODUCT, in arithmetic and geometry, 
the factum of two or more numbers, or lines, 
&c\. into one another : thus 3 X 4 = 20 the 
product required. 
In lines it is always (and in numbers some- 
times) called the rectangle between the two 
lines, or numbers, multiplied by one ano- 
ther. 
PROFILE, the draught of a building, for- 
tification, &c. See Architecture. 
Profile also denotes the outline of a 
figure, building, member of architecture, 
&c. 
Profile, in sculpture and painting, de- 
notes a head, portrait, &c. when represent- 
ed sideways, or in a side view. On almost 
all medals, faces are represented in profile. 
PROGRESSION, an orderly advancing or 
proceeding in the same manner, course, tenor, 
proportion, &c. 
Progression is either arithmetical, -or geome- 
trical. 
v ii o 
Thai, 1, \ 5, 7, f), 11, ii, 
IS, II, 9, 7, 5, S, I, 
14 
fiOi 
Arithmetical Progression, is a series of quan- 
tities proceeding by continued equal differences, 
either increasing or decreasing. 1 hus, 
increasing 1, 3, 5, 7, 9, &c. or 
decreasing 21, 18, 15, 12, 9, Ac; 
where the former progression increases continu- 
ally by the common difference 2, and the latter 
scries "or progression decreases continually by 
the common difference 3. 
1 . And hence, to construct an arithmetical 
progression, from any given first term, and with 
a given common difference ; add the common 
difference to the first term, to give the 2d ; to 
the 2d, to give the 3d ; to the 3d, to give the 
4th ; and so on ; when the series is ascending or 
increasing * hut subtract the common difference 
continually, when the series is a descending one. 
2. The chief property of an arithmetical pro- 
gression, and which arises immediately from the 
nature of its construction, is this ; that the sum 
of its extremes, or first and last terms, is equal 
to the sum of every pair of intermediate terms 
that are equidistant front the extremes, or to the 
double of the middle term when there is an un- 
even number of the terms. 
Sum? 14 14 14 14 14 14 14, 
where the sunt of every pa'f of terms is the sam* 1 
number, 14. 
Also, a. <*-)- d, a -j- 2 J, A-j- 3d, 
ci — 4</j a j** * 3f/j ct -j 2</j a -j- d, a 
slims 2a -f- Ad 2a -j- Ad 2a -j- Ad 2<z — j — 4^/ 2 ./-j-4/. 
3. And hence it follows, that double the sunt 
of all the terms in the series, is equal to the siinu 
of the two extremes multiplied by the number 
of the terms • and consequently, that the single 
sum of all the terms of the series, is equal to 
half the said product. So the sum of the 7 terms • 
1, 3, 5, 7, 9, 11, 13, 
1 + 13 X t = V X 7 = 49. 
And the sum of the five terms 
a , a -J- d , a -j- 2d, a -1— 3./, a -f- Ad, • 
is a -{- Ad X 
4. Hence also, if the first term of the progres-' 
sion is 0, the sum of the series will he equal to-' 
half the product of the last term multiplied by' 
the number of terms : /. e. the sum of 
0 + d 2d A. 3 J -L Ad ■ 
l.d 
1 ,d, 
where n is the number of terms, supposing 0 to 
lie one of them. That is, in other words, t'ic 
sum of an arithmetical progression, whether fi- 
nite or infinite, whose first term is 0, is to the 
sum of as many times the greatest term, in the 
ratio of 1 to 2. 
5. In like manner, the sum of the squares of 
the terms of such a series, beginning at 0, is to 
the sum of as many terms each equal to the 
greatest, in the ratio of 1 to 3. And, 
6. The sum of the cubes of the terms of such 
a series, is to the sum of as many times the 
greatest term, in the ratio of 1 to 4. 
7. And universally, if every term of such a 
progression is raised to the mth power, then the 
sum of all those powers will be to the sum of as 
many terms equal to the greatest, in the ratio of 
m -f- 1 to 1. That is, 
the sum 0 2d 3d l , 
is to l n -f- l m -J- l m -f l' n - r, 
in the ratio of 1 to m 1. 
8. A synopsis of all the theorems, or relatio ns, 
in an arithmetical progression, between the (ex- 
tremes or first and last term, the sum of the (se- 
ries, the number of terms, and the common dlif* 
ference, is as follows : viz. if 
a denotes the least term, 
z- the greatest term, 
d the common difference, 
n the number of terms, 
s the sum of the series ; 
then will each of these five quantities (ex- 
pressed in terms of the others, as below ; - 
a — a — » — 1 • d — * = — ~ V / l^+ — 2 dt -f \d. 
n n 2 
\d — a -f- \f\d — aV 2ds -\--z — y' \d ■ 
#-f-z * — » Af d ia -J- a — 1 • 4 ^ __ 2z 71 
1 . J 
