Chapter XXV: Book VI

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_.

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