PRO 
duplicate of the ratio of 2 =: A to B = 4, 
or as the square of 2 to the square of 4. 
A A 
Triplicate ratio is thus expressed, — g 
thrice ; that is, the ratio of A again == 2, to 
D =: 1 6, is triplicate of the ratio oi A = 2 
to 8 = 4, or as 8 the cube of 2, to 64 the 
cube of 4. Wherefore duplicate ratio is the 
proportion of squares, and triplicate that of 
cubes. 
And the ratio of 2 to 8 is compounded of 
the ratio of that of 2 to 4, and of 4 to 8. 
From what has been said of the nature of 
ratio and proportion, the six ways of argu- 
ing, which are often used by matliematicians, 
will evidently follow. \ 
1. Alternate proportion is the comparing 
of antecedent to antecedent, and consequent 
to consequent. As if | ^ ^ ^ ^ 
therefore alternately, or by permutation, as 
5 A : C :: B : D ) 
i 2 : 8 :: 4 : 16 S' 
2. Inverse ratio, is when the consequent 
is taken as the antecedent, and so com- 
pared to the antecedent as the consequent. 
As A : B :: C : D ; therefore inversely as 
yB : A ;; D : C ? 
i 4 ; 2 :: 16 ; 8 
■ 3. Compound ratio, is when the antece- 
dent and consequent, taken both as 000, are 
compared to the consequent itself. As 
A : B C : D ; therefore by composition, 
as A B ; B :: C -f- D : D : in numbers, as 
g 4 = 6, is to 4, so is 8 -j- 16 = 24, 
to 16. 
4. Divided ratio, is when the excess 
wherein the antecedent exceedeth the con- 
sequent is compared to the consequent. 
As A : B C : D ; therefore by division 
A B : B C — D : D in numbers, as 
16 : 8 :: 12 : 6 ; therefore 16 — 8=8, is to 
8, so is 12 — 6 =6 to 6. 
When of several quantities the difference 
or quotient of the first and second is the 
same with that of the second and third, 
they are said to be in a continued arith- 
metic or geometric proportion. 
T'hnt 5 “+2 d, «H-3d,a-}-4(i J 
\a,a — d, a — 2d, a — 3d, a — 4di 
&c. is a series of continued arithmetical pro- 
portionals, whose common difference is d. 
ra, ar, arr, arrr, arrrr, ar^'h 
And./ a ^ a a a ^ 
rrr^ rrrr' ) 
&c. is a series of continued geometric pro- 
portionals, whose common multiplier is 
— or i, or whose ratio is that of 1 to r, or 
1 r’ 
r to 1. 
PRO 
Proportion of figures. To find the 
proportion that one rectangle hath to ano- 
ther, both length and breadth must be con- 
sidered. For rectangles are to each other, 
as the products of their respective lengths 
multiplied by their breadths. Thus, if 
there be two rectangles, the former of 
which hath its length five feet, and its 
breadth three ; and the latter hath its length 
eight feet, and its breadth four. Then 
the rectangles will be to each other as 
3 X 5 (= 15), is to 4 X 8 (= 32) ; that is, 
as 15 : 32, so that all the rectangles are to 
one anotlier in a ratio compounded of that 
of their sides. 
When rectangles have their sides pro- 
, ,, AB::EH::AO: 
portionable, so that — : : — 
E F 
then is the rectangle A, to the rectan- 
gle B, in a duplicate proportion to the ra- 
tio of the sides. For the ratio of A to B, 
is compounded of the ratio of A B to E H, 
and of the ratio of A D to E F. And there- 
fore the proportion of A to B, being com- 
pounded of equal ratios, must be duplicate 
of the ratio of their sides to each other ; 
that is, duplicate of the ratio of A B : 
EH, or of AD : EF. 
Hence all triangles, paralleli^rams, 
prisms, parallelepipeds, pyramids, cones, 
and cylinders, are to one another respec- 
tively compared, in a proportion com- 
pounded of that of their heights and bases. 
All triangles, and pai allelograras, pyramids, 
prisms, and parallelepipeds ; also all cones 
and cylinders, each kind compared among 
themselves ; if they have equal altitudes, 
are in the same proportion as their bases ; 
if they have equal bases, are as their 
heights. 
For the bases, or heights, will severally 
be common efficients or multipliers ; and 
therefore must make the products be in 
the same proportion as the multiplicand 
was before. 
Thus, if the equal altitude of any two 
triangles, parallelopipeds, cones, &c, be 
called A, and their unequal bases B and D ; 
then it will be as B ; D : : A B : A D. 
Proportion, harmonic, is when three 
terms are so disposed, that as the difference 
of the first and second : the difference of 
the second and third:; first: third; and 
they are said to be harmonically propor- 
tional. Thus, 10, 15, 30, are harmonically 
proportional. For as the difference of 10 
and 15, is to the difference of 15 and 30, 
so is 10 to 30. Also 12, 6, 4, are harmo- 
