S 58. The sixth book contains the theory of similar figures. After a
few definitions explaining terms, the first proposition gives the
first application of the theory of proportion.
Prop. 1. _Triangles and parallelograms of the same altitude are to one
another as their bases._
The proof has already been considered in S 49.
From this follows easily the important theorem
Prop. 2. _If a straight line be drawn parallel to one of the sides of
a triangle it shall cut the other sides, or those sides produced,
proportionally; and if the sides or the sides produced be cut
proportionally, the straight line which joins the points of section
shall be parallel to the remaining side of the triangle._
S 59. The next proposition, together with one added by Simson as Prop.
A, may be expressed more conveniently if we introduce a modern
phraseology, viz. if in a line AB we assume a point C between A and B,
we shall say that C divides AB internally in the ratio AC : CB; but if
C be taken in the line AB produced, we shall say that AB is divided
externally in the ratio AC : CB.
The two propositions then come to this:
Prop. 3. _The bisector of an angle in a triangle divides the opposite
side internally in a ratio equal to the ratio of the two sides
including that angle;_ and conversely, _if a line through the vertex
of a triangle divide the base internally in the ratio of the two other
sides, then that line bisects the angle at the vertex_.
Simson's Prop. A. _The line which bisects an exterior angle of a
triangle divides the opposite side externally in the ratio of the
other sides;_ and conversely, _if a line through the vertex of a
triangle divide the base externally in the ratio of the sides, then it
bisects an exterior angle at the vertex of the triangle_.
If we combine both we have--
_The two lines which bisect the interior and exterior angles at one
vertex of a triangle divide the opposite side internally and
externally in the same ratio, viz. in the ratio of the other two
sides._
S 60. The next four propositions contain the theory of similar
triangles, of which four cases are considered. They may be stated
together.
_Two triangles are similar_,--
1. (Prop. 4). _If the triangles are equiangular:_
2. (Prop. 5). _If the sides of the one are proportional to those of
the other_;
3. (Prop. 6). _If two sides in one are proportional to two sides in
the other, and if the angles contained by these sides are equal_;
4. (Prop. 7). _If two sides in one are proportional to two sides in
the other, if the angles opposite homologous sides are equal, and if
the angles opposite the other homologous sides are both acute, both
right or both obtuse; homologous sides being in each case those which
are opposite equal angles_.
An important application of these theorems is at once made to a
right-angled triangle, viz.:--
Prop. 8. _In a right-angled triangle, if a perpendicular be drawn from
the right angle to the base, the triangles on each side of it are
similar to the whole triangle, and to one another_.
_Corollary._--From this it is manifest that the perpendicular drawn
from the right angle of a right-angled triangle to the base is a mean
proportional between the segments of the base, and also that each of
the sides is a mean proportional between the base and the segment of
the base adjacent to that side.
S 61. There follow four propositions containing problems, in language
slightly different from Euclid's, viz.:--
Prop. 9. _To divide a straight line into a given number of equal
parts_.
Prop. 10. _To divide a straight line in a given ratio_.
Prop. 11. _To find a third proportional to two given straight lines_.
Prop. 12. _To find a fourth proportional to three given straight
lines_.
Prop. 13. _To find a mean proportional between two given straight
lines_.
The last three may be written as equations with one unknown
quantity--viz. if we call the given straight lines a, b, c, and the
required line x, we have to find a line x so that
Prop. 11. a : b = b : x;
Prop. 12. a : b = c : x;
Prop. 13. a : x = x : b.
We shall see presently how these may be written without the signs of
ratios.
S 62. Euclid considers next proportions connected with parallelograms
and triangles which are equal in area.
Prop. 14. _Equal parallelograms which have one angle of the one equal
to one angle of the other have their sides about the equal angles
reciprocally proportional; and parallelograms which have one angle of
the one equal to one angle of the other, and their sides about the
equal angles reciprocally proportional, are equal to one another_.
Prop. 15. _Equal triangles which have one angle of the one equal to
one angle of the other, have their sides about the equal angles
reciprocally proportional; and triangles which have one angle of the
one equal to one angle of the other, and their sides about the equal
angles reciprocally proportional, are equal to one another_.
[Illustration]
The latter proposition is really the same as the former, for if, as in
the accompanying diagram, in the figure belonging to the former the
two equal parallelograms AB and BC be bisected by the lines DF and EG,
and if EF be drawn, we get the figure belonging to the latter.
It is worth noticing that the lines FE and DG are parallel. We may
state therefore the theorem--
_If two triangles are equal in area, and have one angle in the one
vertically opposite to one angle in the other, then the two straight
lines which join the remaining two vertices of the one to those of the
other triangle are parallel_.
S 63. A most important theorem is
_Prop. 16. If four straight lines be proportionals, the rectangle
contained by the extremes is equal to the rectangle contained by the
means; and if the rectangle contained by the extremes be equal to the
rectangle contained by the means, the four straight lines are
proportionals_.
In symbols, if a, b, c, d are the four lines, and
if a : b = c : d,
then ad = bc;
and conversely, if ad = bc,
then a : b = c : d,
where ad and bc denote (as in S 20), the areas of the rectangles
contained by a and d and by b and c respectively.
This allows us to transform every proportion between four lines into
an equation between two products.
It shows further that the operation of forming a product of two lines,
and the operation of forming their ratio are each the inverse of the
other.
If we now define a quotient a/b of two lines as the _number_ which
multiplied into b gives a, so that
a
-- b = a,
b
we see that from the equality of two quotients
a c
-- = --
b d
follows, if we multiply both sides by bd,
a c
-- b.d = -- d.b,
b d
ad = cb.
But from this it follows, according to the last theorem, that
a : b = c : d.
Hence we conclude that the quotient a/b and the ratio a : b are
different forms of the same magnitude, only with this important
difference that the quotient a/b would have a meaning only if a and b
have a common measure, until we introduce incommensurable numbers,
while the ratio a : b has always a meaning, and thus gives rise to the
introduction of incommensurable numbers.
Thus it is really the theory of ratios in the fifth book which enables
us to extend the geometrical calculus given before in connexion with
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.