PRO 
nically proportional ; for 12—6:6 — 4:: 
12 : 4. So -f 3 A n 4- 2 n% A’ + 2 A 71, 
h? h n, are harmonically proportional. 
For A 7t 4- 2 7t^ A 7t : : A^ -jl 3 A 7t 4- 2 71^ : 
A^ 4” ^ Whence if the two first terms of 
an harmonic proportion be given, tho third 
is readily found. 
For if A, B, C, be harmonically propor- 
tional. Then A — B: B — C: : A: C,and 
AC — BC=AB — AC. Therefore AB 
= 2A — B X C, andBC = 2C — B 
X A. Consequently C = — — , and 
^ ^ 2 A — B’ 
A F C , . , 
A =: Again, when four terms 
are so disposed, that as the difference of the 
1st and 2d : the difference of the 3d and 
4th: : 1st: 4th, they are also harmonically 
proportional. As 10, 16, 24, 60; for as 
10 — 16 : 24 — 60 10 : 60. Whence if 
the three first terms of such an harmonic 
proportional be given, the 4th is easily 
found. 
For if a, h, c, d, be harmonic proportion- 
als, then a — 6;c — d: : a: d; andad — 
b d — a c — ad, therefore d = 
and a = 
bd 
ad — c 
a c 
aa — b’ 
If the terms of an harmonic propor- 
tion be continued, then it is called an har- 
monic progression. Thus, supposing 
< A, to be the 2d term, > 
I d, the difference of the 1st and 2d \ 
and that the ist exceeds the 2d. The pro- 
gression will be 
L I j I. b d A^.4“ b d -i- h d 
/ 74 - 4 0 ’ 
^74.^,/ 
^ Whence, if out of a rank 
of harmonic proportionals, there be taken 
any series of equidistant terms, that series 
will be harmonically proportional. And this 
kind of proportion has several other pro- 
perties common with arithmetic and geo- 
metric proportions. 
When three terms are so disposed, that 
the difference of the 1st and 2d ; difference 
of the ?d and 3d : : 3d : Ist, they are said 
to be m a contra-harmonic proportion, 
lints, 6, 5, 3, and 12, 10, 4, are, contra- 
harmonics. For 6 — 5: 5 — 3::3;6- 
and 12 — 10 : 10 — 4 : : 4 : 12. Or, sup- 
jtosing A greater than n, if the 2d term be 
greater than the 1st : 
Then A Tt -f- n% V 4- A^ -f- A n, are 
coiitra hannonics, fnr A n — A^ : hn 
; ; A^ 4~ b n ; A n n^. 
But if the 1st term exceeds the 2d, then 
PRO 
A^ 4- A 77, A’ 4- 71% A 71 -f 71% are contra, 
harmonics. For A — 71^ : A^ — A 71 " 
A 71 4- 7i^ ; A^ 4- A 71. 
PROPOSITION, in logic, part of an 
argginent wherein some quality, either ne- 
ptive or positive, is attributed to a sub- 
ject, as “ God is just.” While the comparing 
of our ideas is considered merely as the act 
of the mind, assembling them together, and 
joining or disjoining them according to the 
result of its perceptions, this operation is 
called judgment. But when these judg- 
ments are expressed in words, they then 
bear the name of propositions. Hence a 
proposition is a sentence expressing some 
judgment of the mind, whereby two or 
more ideas are affirmed to agree or disa- 
gree : and as our judgments include at least 
two ideas, one of which is affirmed or de- 
nied of the other; so a proposition must 
have terms corresponding to these ideas. 
The idea of which we affirm or deny, and of 
course the term expressing that idea, js 
called the subject of the proposition ; and 
the idea affirmed or denied, as also the 
term answering to it, is called its preilicate ; 
thus in the proposition, God is omnipotent, 
God is the subject, it being of him that we 
affirm omnipotence ; and omnipotent is the 
predicate, because we affirm the idea ex- 
pressed by that word to belong to God. 
Proposition, in mathematics, is either 
some trutli advanced and shown to be such 
by demonstration, or some operation pro- 
posed and its solution shown. If the pro- 
position be deduced from several tlieo- 
retical definitions compared tof'ether, it is 
called a theorem ; if from a praxis, or se-, 
ries of operations, it is called a problem. 
PROPosiTtON, in poetry, the first part 
of a poem wherein the author proposes 
briefly, and in general, what he is to say in 
the body of his work. It should compre- 
hend only the matter of the poem, that k 
the action and the persons that act. Ho;- 
race prescribes modesty and simplicity in 
the proposition, and would not have tlie 
poet promise too much, nor raise in tlie 
reader too great ideas of what he is going to 
relate. 
PROSERBINACA, in botany, a geinis. 
of the Trian(iria 'rriirynia ohiss and order. 
^Natural order of Inimdatvi?, Hydroclia- 
rides, Jnssieti. Essential ciiaracter : calyx 
three-parted, superior ; corolla none ; drupe 
with a three- celled nut. There is but one 
species, viz. P, pahistris, a uatiye ot Vir- 
ginia in marshes. 
PROSOD Y, that part of grammar which 
