506 
V R O 
PRO 
the ground, supported only by a piece of 
wood in the middle, of about five or six 
inches thick, to raise the muzzle a little, 
and then the piece is fired against a solid 
butt of earth. 
Proof of mortars and hozvitzers, is made 
to ascertain their being well cast, and of 
strength to resist the effort of their charge. 
For this purpose the mortar or howitzer is 
placed upon the ground, with some part of 
the trunnions or breech sunk below the 
surface, and resting on wooden billets at an 
elevation of about seventy degrees. The 
mirror is generally the only instrument to 
discover the defects in mortars and howitzers. 
In order to use it, the sun must shine; the 
breech must be placed towards the sun, and 
the glass over-against the mouth of the piece 
which illuminates the bore and chamber suf- 
ficiently to discover the flaws in it. 
Proof of foreign brass artillery. 1st, 
The Prussians. Their battering-train and 
garrison artillery are proved with a quantity 
of powder equal to half the weight of the 
shot, and fired seventy-five rounds as fast as 
in real service ; that is, two or three rounds 
in a minute. Their light field-train, from a 
12-pounder upwards, are proved with a quan- 
tity of powder = 1 -3d of the weight of the 
shot, and fired 150 rounds, at three or four 
rounds in a minute. From a 12-pounder 
downwards, are proved with a quantity 
= l-5tb of the shot’s weight, and fired 300 
rounds, at five or six rounds each minute, 
properly spunged and loaded. Their mor- 
tars are proved with the chambers full of 
powder, and the shells loaded. Three rounds 
are fired as quick as possible. 2d, The 
Dutch prove all their artillery by firing each 
piece five times : the two first rounds with a 
quantity of powder = 2-3ds of the weight 
of the shot; and the three last rounds with 
a quantity of powder = half the weight of 
the shot. 3d, The French the same as the 
Dutch. 
Proof, in brandy and other spirituous 
liquors, is a little white froth which appears 
on the top of the liquor when poured into a 
glass. This froth, as it diminishes, forms 
itself into a circle called by the French the 
chapelet, and by the English the bead or 
bubble. 
PROPOLIS. See Resins. 
PROPORTION, in arithmetic, &c. See Al- 
gebra, p. 54. 
Proportion is often confounded with ratio ; 
but they are quite different things. For, ratio 
is properly the relation of two magnitudes or 
quantities" of one and the same kind; as the ratio 
of 4 to 8, or of 15 to SO, or of 1 to 2 and so 
implifes or respects only two terms or things. 
But proportion respects four terms or things, 
or two ratios which have each two terms ; 
though the middle term may be common to 
both ratios, and then the proportion is expressed 
by three terms only, as 4, 8, 64, where 4 is to 8 
as 8 to 64. 
Proportion is also sometimes confounded with 
progression. In fact, the two often coincide ; 
the difference beiween them only consisting in 
this, that progression is a particular species of 
proportion, being indeed a continued propor- 
tion, or such as has all the terms in the same 
-ratio, viz the 1st to the 2d, the 2d to the 3d, 
the 3d to the 4th, &c ; as the terms 2, 4, 8, 16, 
&c.' so that progression is a series or continu- 
ation of proportions. See Progression. 
PRO 
Proportion is either continual, or discrete, or 
interrupted. 
The proportion is continual when every two 
adjacent terms have the same ratio, or when the 
consequent of each ratio is the antecedent of the 
next following ratio, and so all the terms form a 
progression ; as 2, 4, 8, 16, &c.; where 2 is to 4 
as 4 to 8, and as 8 to 16, & c. 
Discrete or interrupted proportion, is when 
the consequent of the first ratio is different from 
the antecedent of the 2d, &c. ; as 2, 4, and 3, 6. 
Proportion is also either direct or inverse. 
Direct proportion is when more requires 
more, or less requires less; as it will require 
more men to perform more work, or fewer men 
for less work, in the same time. 
Inverse or reciprocal proportion, is when 
more requires less, or less requires more. As it 
will require more men to perform the same 
work in less time, or fewer men in more time. 
Ex. If 6 men can perform a piece of work in 15 
days, how many men can do the same in 10 
days ? Then, 
reciprocally as to fo so is 6 [ 9? the an- 
or inversely as 10 to 15 so is 6 * 95 swer. 
Proportion, again, is distinguished into arith- 
metical, geometrical, and harmonical. 
Arithmetical proportion is the equality of 
two arithmetical ratios, or differences : as in 
the numbers 12, 9, 6 ; where the difference 
between 12 and 9, is the same as the difference 
between 9 and 6, viz. 3. 
And here the sum of the extreme terms is 
equal to the sum of the means, or to double the 
single mean when there is but one. As 12 6 
= 9 -f 9= 18. 
Geometrical proportion is the equality be- 
tween two geometrical ratios, or between the 
quotients of the terms. As in the three 9, 6, 4, 
where 9 is to 6 as 6 is to 4, thus denoted 9 * 6 
‘ * 6 * 4 ; for 2. — F, being each equal A or 
And in this proportion, the rectangle or pro- 
duct of the extreme terms, is equal to that of 
the two means, or the square of the single mean 
when there is but one. For 9 X 1= 6x6 = 
36. 
Harmonical proportion, is when the first term 
is to the third, as the difference between the 1st 
and 2d is to the difference between the 2d and 
3d ; or in four terms, when the 1st is to the 4th, 
as the difference between the 1st and 2d is to the 
difference between the 3d and 4th ; or the re- 
ciprocals of an arithmetical proportion are in 
harmonical proportion. As 6, 4, 3 , because 
6* 3**6 — 4 = 2*4 — 3=1 or because 
4 -> 4 >X> are in arithmetical proportion, making 
A -|- i- == 5 4 — a- Also the four 24, 16, 
12, 9, are in harmonical proportion, because 24 * 
9 : : s * 3 . 
PROPORTIONAL compasses, are compasses 
with two pair of opposite legs, like a St. An- 
drew’s cross, by which any space is enlarged or 
diminished in any proportion. 
Proportional scales , called also logarithmic 
scales, are the logarithms, or artificial numbers, 
placed on lines, for the ease and advantage of 
multiplying and dividing, &c. by means of com- 
passes, or of sliding rulers. These are in effect 
so many lines ot numbers, as they are called by 
Gunter, but made single, double, triple, or qua- 
druple; beyond which they seldom go. 
Pff IPORTIONaLS, are the terms of a pro- 
portion: consisting of two extremes, which are 
the first and last terms of the set, and the means, 
which are the rest of the terms. These propor- 
tionals may be either arithmeticals, geometricals, 
or harinonicals, and in any number above two, 
and also either continued or discontinued. 
Pappus gives this beautiful and simple com- 
parison of the three kinds of proportionals, 
arithmetical, geometrical, and harmonical, viz. 
a, b , c, being the first, second, a r.d third terms in 
any such proportion, then 
In the arithmeticals, a * a~) 
in the geometricals, — b \ b — c. I 
in the harmonicals, a \ c j 
Continued proportionals form what is called 
a progression. See Progression. 
I. Properties of arithmetical proportionals. 
For what respects progressions and mean 
proportionals of all sorts, see Mean and Pro- 
gression. 
1. Four arithmetical proportions, as 2, 3, 4, 5, 
are still proportionals when inversely 5, 4, 3,2; 
or alternately, thus, - 2, 4, 3, 5; 
or inversely and alternately, thus, 5, 3, 4, 2. 
2. If two arithmeticals are added to the like 
terms of other two arithmeticals, of the same 
difference or arithmetical ratio, the sums will 
have double the same difference or arithmetical 
ratio. 
So, to 3 and 5, whose difference is 2, 
add 7 and 9, whose difference is also 2, 
the sums 10 and 14 have a double diff. viz. 4. 
And if to these sums are added two other num- 
bers also in the same difference, the next sums 
-will have a triple ratio or difference ; and so on. 
Also, whatever are the ratios of the terms that 
are added, whether the same or different, the 
sums of the terms will have such arithmetical 
ratio as is composed of the sums of the others 
that are added. 
So 3 , 5, whose diff. is 2 
and 7 , 10, whose diff. is 3 
and 12 , 16, whose diff. is 4 
make 22 , 31, whose diff. is 9. 
On the contrary, if from two arithmeticals are 
subtracted others, the difference will have such ^ 
arithmetical ratio as is equal to the differences 
of those. 
So from 12 and 16, whose dif. is 4 
take 7 and 10, whose dif. is 3 
leaves 5 and 6, whose dif. is 1 
Also from 7 and 9, whose dif. is 2 
take 3 and 5, whose dif. is 2 
leaves 4 and 4, whose dif. is O 
Hence, if arithmetical proportionals are multi- 
plied or divided by the same number, their diffe- 
rence, or arithmetical ratio, is also multiplied or 
divided by the same number. 
II. Properties of geometrical proportionals. 
The properties relating to mean propor- 
tionals are given under the term Mean Pro- 
portional; some are also given under the ar- 
ticle Propo ktion ; and some additional ones 
are as below : 
1. To find a 3d proportional to two given num- 
bers, or a 4th proportional to three in the 
former case, multiply the 2d term by itself, and 
divide the product by the 1 st ; and in the latter 
case, multiply the 2d term by the 3d, and divide 
the product l>y the 1st. 
So 2 ’ 6 ** 6 * 18, the 3d prop, to 2 and 6: 
and 2 * 6 ** 5 * 15, the 4th prop, to 2, 6, and 5. 
2. If the terms of any geometrical ratio are 
augmented or diminished by any others in the 
same ratio, or proportion, the sums or differences 
will still be in the same ratio or proportion. 
So if a * b * * c * d, 
then is a ] b * * a -j— c * h d * * c * d. 
And if the terms of a ratio, or proportion, are 
multiplied or divided by any one and the same 
number, the products and quotients will still 
be in the same ratio, or proportion. 
Thus, a J b ** na ’ nb 7* — I — » 
n u 
