Chapter XXVI: Book II: It will also be seen that if we write the ratios in Book V

as quotients, or rather as fractions, then most of the theorems state
properties of quotients or of fractions.

S 64. Prop. 17. _If three straight lines are proportional the
rectangle contained by the extremes is equal to the square on the
mean;_ and conversely, is only a special case of 16. After the
problem, Prop. 18, _On a given straight line to describe a rectilineal
figure similar and similarly situated to a given rectilineal figure_,
there follows another fundamental theorem:

Prop. 19. _Similar triangles are to one another in the duplicate ratio
of their homologous sides._ In other words, the areas of similar
triangles are to one another as the squares on homologous sides. This
is generalized in:

Prop. 20. _Similar polygons may be divided into the same number of
similar triangles, having the same ratio to one another that the
polygons have; and the polygons are to one another in the duplicate
ratio of their homologous sides._

S 65. Prop. 21. _Rectilineal figures which are similar to the same
rectilineal figure are also similar to each other_, is an immediate
consequence of the definition of similar figures. As similar figures
may be said to be equal in "shape" but not in "size," we may state it
also thus:

"Figures which are equal in shape to a third are equal in shape to
each other."

Prop. 22. _If four straight lines be proportionals, the similar
rectilineal figures similarly described on them shall also be
proportionals; and if the similar rectilineal figures similarly
described on four straight lines be proportionals, those straight
lines shall be proportionals._

This is essentially the same as the following:--

_If_ a : b = c : d,
_then_ a^2 : b^2 = c^2 : d^2.

S 66. Now follows a proposition which has been much discussed with
regard to Euclid's exact meaning in saying that a ratio is
_compounded_ of two other ratios, viz.:

Prop. 23. _Parallelograms which are equiangular to one another, have
to one another the ratio which is compounded of the ratios of their
sides._

The proof of the proposition makes its meaning clear. In symbols the
ratio a : c is compounded of the two ratios a : b and b : c, and if a
: b = a' : b', b : c = b" : c", then a : c is compounded of a' : b'
and b" : c".

If we consider the ratios as numbers, we may say that the one ratio is
the product of those of which it is compounded, or in symbols,

a a b a' b" a a' b b"
-- = -- . -- = -- . --, if -- = -- and -- = --.
c b c b' c" b b' c c"

The theorem in Prop. 23 is the foundation of all mensuration of areas.
From it we see at once that two rectangles have the ratio of their
areas compounded of the ratios of their sides.

If A is the area of a rectangle contained by a and b, and B that of a
rectangle contained by c and d, so that A = ab, B = cd, then A : B =
ab : cd, and this is, the theorem says, compounded of the ratios a : c
and b : d. In forms of quotients,

a b ab
-- . -- = --.
c d cd

This shows how to multiply quotients in our geometrical calculus.

Further, _Two triangles have the ratios of their areas compounded of
the ratios of their bases and their altitude._ For a triangle is equal
in area to half a parallelogram which has the same base and the same
altitude.

S 67. To bring these theorems to the form in which they are usually
given, we assume a straight line u as our unit of length (generally an
inch, a foot, a mile, &c.), and determine the number [alpha] which
expresses how often u is contained in a line a, so that [alpha]
denotes the ratio a : u whether commensurable or not, and that a =
[alpha]u. We call this number [alpha] the numerical value of a. If in
the same manner [beta] be the numerical value of a line b we have

a : b = [alpha] : [beta];

in words: _The ratio of two lines (and of two like quantities in
general) is equal to that of their numerical values._

This is easily proved by observing that a = [alpha]u, b = [beta]u,
therefore a : b = [alpha]u : [beta]u, and this may without difficulty
be shown to equal [alpha] : [beta].

If now a, b be base and altitude of one, a', b' those of another
parallelogram, [alpha], [beta] and [alpha]', [beta]' their numerical
values respectively, and A, A' their areas, then

A a b [alpha] [beta] [alpha][beta]
-- = -- . -- = -------- . ------ = ---------------.
A' a' b' [alpha]' [beta]' [alpha]'[beta]'

In words: _The areas of two parallelograms are to each other as the
products of the numerical values of their bases and altitudes._

If especially the second parallelogram is the unit square, i.e. a
square on the unit of length, then [alpha]' = [beta]' = 1, A' = u^2,
and we have

A
-- = [alpha][beta] or A = [alpha][beta] . u^2.
A'

This gives the theorem: The number of unit squares contained in a
parallelogram equals the product of the numerical values of base and
altitude, and similarly the number of unit squares contained in a
triangle equals half the product of the numerical values of base and
altitude.

This is often stated by saying that the area of a parallelogram is
equal to the product of the base and the altitude, meaning by this
product the product of the numerical values, and not the product as
defined above in S 20.

S 68. Propositions 24 and 26 relate to parallelograms about diagonals,
such as are considered in Book I., 43. They are--

Prop. 24. _Parallelograms about the diameter of any parallelogram are
similar to the whole parallelogram and to one another_; and its
converse (Prop. 26), _If two similar parallelograms have a common
angle, and be similarly situated, they are about the same diameter._

Between these is inserted a problem.

Prop. 25. _To describe a rectilineal figure which shall be similar to
one given rectilinear figure, and equal to another given rectilineal
figure_.

S 69. Prop. 27 contains a theorem relating to the theory of maxima and
minima. We may state it thus:

Prop. 27. _If a parallelogram be divided into two by a straight line
cutting the base, and if on half the base another parallelogram be
constructed similar to one of those parts, then this third
parallelogram is greater than the other part._

Of far greater interest than this general theorem is a special case of
it, where the parallelograms are changed into rectangles, and where
one of the parts into which the parallelogram is divided is made a
square; for then the theorem changes into one which is easily
recognized to be identical with the following:--

_Of all rectangles which have the same perimeter the square has the
greatest area._

This may also be stated thus:--

_Of all rectangles which have the same area the square has the least
perimeter._

S 70. The next three propositions contain problems which may be said
to be solutions of quadratic equations. The first two are, like the
last, involved in somewhat obscure language. We transcribe them as
follows:

_Problem_.--To describe on a given base a parallelogram, and to divide
it either internally (Prop. 28) or externally (Prop. 29) from a point
on the base into two parallelograms, of which the one has a given size
(is equal in area to a given figure), whilst the other has a given
shape (is similar to a given parallelogram).

If we express this again in symbols, calling the given base a, the one
part x, and the altitude y, we have to determine x and y in the first
case from the equations

(a - x)y = k^2,

x p
-- = --,
y q

k^2 being the given size of the first, and p and q the base and
altitude of the parallelogram which determine the shape of the second
of the required parallelograms.

If we substitute the value of y, we get

pk^2
(a - x)x = ----,
q

or,

ax - x^2 = b^2,

where a and b are known quantities, taking b^2 = pk^2/q.

The second case (Prop. 29) gives rise, in the same manner, to the
quadratic

ax + x^2 = b^2.

The next problem--

Prop. 30. _To cut a given straight line in extreme and mean ratio_,
leads to the equation

ax + x^2 = a^2.

This is, therefore, only a special case of the last, and is, besides,
an old acquaintance, being essentially the same problem as that
proposed in II. 11.

Prop. 30 may therefore be solved in two ways, either by aid of Prop.
29 or by aid of II. 11. Euclid gives both solutions.

S 71. Prop. 31 (Theorem). _In any right-angled triangle, any
rectilineal figure described on the side subtending the right angle is
equal to the similar and similarly-described figures on the sides
containing the right angle_,--is a pretty generalization of the
theorem of Pythagoras (I. 47).

Leaving out the next proposition, which is of little interest, we come
to the last in this book.

Prop. 33. _In equal circles angles, whether at the centres or the
circumferences, have the same ratio which the arcs on which they stand
have to one another; so also have the sectors_.

Of this, the part relating to angles at the centre is of special
importance; it enables us to measure angles by arcs.

With this closes that part of the _Elements_ which is devoted to the
study of figures in a plane.