PRO 
PRO 
PRO 
507 
3 If a »et of continued proportionals art 1 cither 
augmented or diminished- by the same part or 
jparts of themselves, the sums or differences will 
also be proportionals. 
1 hus if a, b, c, d, &c. are propors. 
then are a± -, b, ±,~,c,± & c. also pro- 
>i n n r 
portionals, where the common ratio is 1 J. 
n 
And if any single quantity is either augment- 
ed or diminished by some part of itself, and the 
result is also increased or diminished by the 
same part ol itself, and this third quantity treated 
in the same manner, and so on; then shall all 
these quantities be continued proportionals. So, 
beginning with the quantity a, and taking al- 
ways the nth part, then shall 
. a 2 a a 1 
a > a ± > + , , &c. be propor- 
n n n c 1 
tionals, or a, a + , (« + i) 2 , ( a + &c. 
n n n 
proportionals the common ratio being 1 4- — . 
n 
4. If one set of proportionals is multiplied or 
divided by any other set of proportionals, each 
term by each, the products or quotients will 
also be proportionals. 
Thus, 
if a na 
i : 
: nb. 
and 
c \ 
\ me 
:: d ; 
’ md ; 
then is 
ac \ 
\ mnuc 
: id ; 
mnbd. 
and 
a 
na 
b ’ 
nb 
— • — 
c me 
d 
mb 
5. If therp are several continued proportion- 
als, then whatever ratio the 1st has to the 2d, 
the 1st to the 3d shall have the duplicate of the 
ratio, the 1st to the 4th the triplicate of it, and 
so on. 
be found from the definition of them, viz. that 
i a * c * * a — b \b — c\ for hence 
2 ac 
l ~ — , — the harmonical mean and 
a -j- c 
ab 
c = , the third harmon. to d and b. 
2a — l 
3. And of the four harmonicals, a , b, c, d, any 
three being given, the fourth can be found from 
the definition of them, viz. that a \ d ” a — b * 
c — <•/ ; for thence the three b, c, d, will be thus 
found, viz. 
^ 2 ad — ac 2nd — bd ab 
d a ’ 2 a — b 
4. If there are four numbers disposed in order, 
as 2. 3, 4, 6, of which one extreme and the two 
middle terms are in arithmetical proportion, 
and the other extreme and the same middle 
terms are in harmonical proportion ; then are 
the four terms in geometrical proportion: so 
here 
the three 2, 3, 4 are arithmeticals, 
and the three 3, 4, 6 are harmonicals, 
then the four 2, 3, 4, 6 are geometricals. 
5. If between any two numbers, as 2 and 6, 
there are interposed an arithmetical mean 4, and 
also a harmonical mean 3, the four will then be 
geometricals, viz. 2 ; 3 ” 4 ] 6. 
6 Between the three kinds of proportion, 
there is this remarkable difference viz. that 
from any given number there can be raised a 
continued arithmetical series increasing ad in- 
finitum, but not decreasing ; while the harmoni- 
cal can be decreased ad infinitum, but not in- 
creased ; and the geometrical admits of both. 
Proportions of the human body. See 
Drawing. 
Proportions of the antique statues. See 
Statues, and Sculpture. 
So in a, na, n 2 a, ri'a, &c. the ratio being ti ; 
then a * n a, or 1 to n 2 , the duplicate ratio, 
and a * ri'a, or 1 to « 2 , the triplicate ratio, 
and so on. 
6. In three continued proportionals, the dif- 
ference between the 1st and 2d term, is a mean 
proportional between the 1st term and the se- 
cond difference of all the terms. 
Thus, in the three propor. a , na, n 2 a ; 
Terms 
n'a 
1st difs. 
2d dif. 
then a l na — a * * na 
Or in the numbers 2, 6, 1 
13 
- 2 tut -{- a, 
n 2 a — 2 na -|~ a. 
8 the 2d difference ; 
then 2, 4, 8 are proportionals. 
7. When four quantities are in proportion, 
they are also in proportion by inversion, com- 
position, division, & c. ; thus, a, na, b, nb, being in 
proportion, viz. 
1. a 
2. Inversion na 
3. Alternation a 
4. Composition a -f- na 
5. Conversion a -j- na 
6. Division ~ 2,(2 
£ a — na 
na 
a 
b 
na 
a 
a 
na 
b l nb-, then by 
nb l b ; 
na * nb\ 
b -j- nb l nb l 
b -j- nb * b ; 
b — nb * b ; 
b — nb \ nb\ 
III. Properties of harmonical proportionals. 
1. If three or four numbers in harmonical 
proportion, are either multiplied or divided by 
any number, the products or quotients will also 
be harmonical proportions. 
Thus, 0, 3, 2, being harmon. propor. 
then 12, 6, 4, are also harmon. propor. 
and A, A, A are also harmon. propor. 
2. In the three harmonical proportionals, a, b, 
c, when any two of these are given, the third can 
PROPOSITION, in logic, part of an ar- 
gument wherein some quality, either nemi- 
tive or positive, is attributed to a subject ; or 
according to Chauvinus, it is a complete con- 
sistent sentence, indicating or expressing 
something either true or false, without ambf- 
guit v ; as, God is just. 
Proposition, in mathematics, is either 
some truth advanced and shewn to be such 
by demonstration, or some operation pro- 
posed and its solution shewn. If the propo- 
sition is deduced from several theoretical 
definitions compared together, it is called a 
theorem ; if from a praxis, or series of opera- 
tions, it is called a problem. 
PROSERPINACA, a genus of the trigynia 
order, in the triandria class of plants ; and in 
the natural method ranking under the 15th 
order, inundatar. The calyx is tripartite su- 
perior ; there is no corolla ; there is one tri- 
locular seed. I here is one species, a marsh 
plant of Virginia. 
PROSODA , that part of grammar which 
treats of the quantities and accents of sylla- 
bles, and the manner of making verses. 
PROSOPIS, a genus of the monogynia 
order, in the decandria class of plants. The 
calyx is hemispherical and quadridentate; 
the stigma is simple, the legumen inflated 
and monospermous. There is one species, a 
tree of the East Indies. 
PROSOPOPOEIA, a figure in rhetoric, 
whereby we raise qualities, or things inani- 
mate, into persons. See Rhetoric. 
PROSTATE. See Anatomy. 
PROSTYL E, in antient architecture, a 
range of columns in front of a temple. 
PROTEST, when one openly affirms, that 
lie does either not at all, or but conditionally, 
3 S2 
yield his consent to any act, or unto the 
proceeding of a judge in court wherein lhs 
jurisdiction is doubtful, or to answer upon 
his oath any farther than by law he is bound. 
Protest, is also that act by which the 
holder of a foreign bill of exchange declares 
that such bill is dishonoured. 
Protest, is also that act of a master, on 
his arrival with his ship from parts beyond 
the seas, to save him and his owners harmless 
and indemnified from any damage sustained 
in the goods of her lading, on account of 
storms. See Bills of Exchange, and In- 
surance. 
PROTESTANT, a name first given in 
Germany to those who adhered to the doc- 
trine of Luther; because in 1529, they pro- 
tested against a decree of the emperor 
Charles V. and the diet of Spires; declaring 
that they appealed to a general council. 
The same name also has been given to those 
of the sentiments of Calvin, and is now be- 
come a common denomination for all those 
of the reformed churches. 
PROTEA, the silver-tree, a genus of the 
monogynia order, in the tetrandria class of 
plants ; and in the natural method ranking 
under the 47th order, stellatfe. There is 
one quadrifid petal surrounding the germ ; 
there is no proper calyx ; the receptacle is 
paleaceous. There are sixty-four species, 
chiefly natives of the Cape of Good Hope; 
ot which the most remarkable are, 1 . The 
conifera, with linear, spear-shaped, entire 
leaves, .grows to the height of ten or twelve 
feet, with a straight regular stem. The 
branches naturally form a large regular head. 
I he leaves are long and narrow, of a shining 
silver-colour, and, as they remain the whole 
year, make a line appearance in the green- 
house. 2. The argeutea, commonly called 
silver-tree, has a strong upright stem co- 
vered with purplish bark, dividing into sever- 
al branches which grow erect, with broad, 
shining, silvery, leaves, which make aline ap- 
pearance when intermixed with other exotics. 
Through the whole year it exhibits its glossy 
white or silvery leaves. It has at first a very- 
uncommon and beautiful appearance; and 
sometimes in the course of twelve or fifteen 
years, reaches the height of twenty feet, 
which it never exceeds. In a rich' soil it 
grows twice as quick, and is by far the lar- 
gest of the protea kind. They are generally 
planted near some farms, and very seldom 
grow wild ; Mr. Sparrman thinks it was pro- 
bably brought to the Cape of Good PI ope 
from Anamaqua ; for he had travelled over 
the whole north-east side of Hottentots’ Hol- 
land, without finding it either in its wild 
state or planted. 3. The nitida, or wage- 
boom, greatly resembles the second sort ; 
the leaves are very silky and white, with 
erect purple branches. 
Ail these plants being tender exotics, re- 
quire to be continually kept in the green- 
house during winter. The first may be pro- 
pagated by cuttings, which should be cutoff 
in April, just before the plants begin to 
■shoot; the second and third sorts may be 
propagated by seeds. 
PROTHONOTARY, a term which pro- 
perly signifies first notary, and which was 
antiently the title of the principal notaries of 
the emperors of Constantinople. 
Prothonotary with us is used for an officer, 
•in the courts of king’s bench and common 
