Chapter XX: Book I: Of Euclid's "elements."

S 6. According to the third postulate it is possible to draw in any
plane a circle which has its centre at any given point, and its radius
equal to the distance of this point from any other point given in the
plane. This makes it possible (Prop. 1) to construct on a given line
AB an equilateral triangle, by drawing first a circle with A as centre
and AB as radius, and then a circle with B as centre and BA as radius.
The point where these circles intersect--that they intersect Euclid
quietly assumes--is the vertex of the required triangle. Euclid does
not suppose, however, that a circle may be drawn which has its radius
equal to the distance between any two points unless one of the points
be the centre. This implies also that we are not supposed to be able
to make any straight line equal to any other straight line, or to
carry a distance about in space. Euclid therefore next solves the
problem: It is required along a given straight line from a point in it
to set off a distance equal to the length of another straight line
given anywhere in the plane. This is done in two steps. It is shown in
Prop. 2 how a straight line may be drawn from a given point equal in
length to another given straight line not drawn from that point. And
then the problem itself is solved in Prop. 3, by drawing first through
the given point some straight line of the required length, and then
about the same point as centre a circle having this length as radius.
This circle will cut off from the given straight line a length equal
to the required one. Nowadays, instead of going through this long
process, we take a pair of compasses and set off the given length by
its aid. This assumes that we may move a length about without changing
it. But Euclid has not assumed it, and this proceeding would be fully
justified by his desire not to take for granted more than was
necessary, if he were not obliged at his very next step actually to
make this assumption, though without stating it.

S 7. We now come (in Prop. 4) to the first theorem. It is the
fundamental theorem of Euclid's whole system, there being only a very
few propositions (like Props. 13, 14, 15, I.), except those in the 5th
book and the first half of the 11th, which do not depend upon it. It
is stated very accurately, though somewhat clumsily, as follows:--

_If two triangles have two sides of the one equal to two sides of the
other, each to each, and have also the angles contained by those sides
equal to one another, they shall also have their bases or third sides
equal; and the two triangles shall be equal; and their other angles
shall be equal, each to each, namely, those to which the equal sides
are opposite._

That is to say, the triangles are "identically" equal, and one may be
considered as a copy of the other. The proof is very simple. The first
triangle is taken up and placed on the second, so that the parts of
the triangles which are known to be equal fall upon each other. It is
then easily seen that also the remaining parts of one coincide with
those of the other, and that they are therefore equal. This process of
applying one figure to another Euclid scarcely uses again, though many
proofs would be simplified by doing so. The process introduces motion
into geometry, and includes, as already stated, the axiom that figures
may be moved without change of shape or size.

If the last proposition be applied to an isosceles triangle, which has
two sides equal, we obtain the theorem (Prop. 5), _if two sides of a
triangle are equal, then the angles opposite these sides are equal_.

Euclid's proof is somewhat complicated, and a stumbling-block to many
schoolboys. The proof becomes much simpler if we consider the
isosceles triangle ABC (AB = AC) twice over, once as a triangle BAC,
and once as a triangle CAB; and now remember that AB, AC in the first
are equal respectively to AC, AB in the second, and the angles
included by these sides are equal. Hence the triangles are equal, and
the angles in the one are equal to those in the other, viz. those
which are opposite equal sides, i.e. angle ABC in the first equals
angle ACB in the second, as they are opposite the equal sides AC and
AB in the two triangles.

There follows the converse theorem (Prop. 6). _If two angles in a
triangle are equal, then the sides opposite them are equal_,--i.e. the
triangle is isosceles. The proof given consists in what is called a
_reductio ad absurdum_, a kind of proof often used by Euclid, and
principally in proving the converse of a previous theorem. It assumes
that the theorem to be proved is wrong, and then shows that this
assumption leads to an absurdity, i.e. to a conclusion which is in
contradiction to a proposition proved before--that therefore the
assumption made cannot be true, and hence that the theorem is true. It
is often stated that Euclid invented this kind of proof, but the
method is most likely much older.

S 8. It is next proved that _two triangles which have the three sides
of the one equal respectively to those of the other are identically
equal, hence that the angles of the one are equal respectively to
those of the other, those being equal which are opposite equal sides_.
This is Prop. 8, Prop. 7 containing only a first step towards its
proof.

These theorems allow now of the solution of a number of problems,
viz.:--

_To bisect a given angle_ (Prop. 9).

_To bisect a given finite straight line_ (Prop. 10).

_To draw a straight line perpendicularly to a given straight line
through a given point in it_ (Prop. 11), _and also through a given
point not in it_ (Prop. 12).

The solutions all depend upon properties of isosceles triangles.

S 9. The next three theorems relate to angles only, and might have
been proved before Prop. 4, or even at the very beginning. The first
(Prop. 13) says, _The angles which one straight line makes with
another straight line on one side of it either are two right angles or
are together equal to two right angles_. This theorem would have been
unnecessary if Euclid had admitted the notion of an angle such that
its two limits are in the same straight line, and had besides defined
the sum of two angles.

Its converse (Prop. 14) is of great use, inasmuch as it enables us in
many cases to prove that two straight lines drawn from the same point
are one the continuation of the other. So also is

Prop. 15. _If two straight lines cut one another, the vertical or
opposite angles shall be equal._

S 10. Euclid returns now to properties of triangles. Of great
importance for the next steps (though afterwards superseded by a more
complete theorem) is

Prop. 16. _If one side of a triangle be produced, the exterior angle
shall be greater than either of the interior opposite angles._

Prop. 17. _Any two angles of a triangle are together less than two
right angles, is an immediate consequence of it._ By the aid of these
two, the following fundamental properties of triangles are easily
proved:--

Prop. 18. _The greater side of every triangle has the greater angle
opposite to it_;

Its converse, Prop. 19. _The greater angle of every triangle is
subtended by the greater side, or has the greater side opposite to
it_;

Prop. 20. _Any two sides of a triangle are together greater than the
third side_;

And also Prop. 21. _If from the ends of the side of a triangle there
be drawn two straight lines to a point within the triangle, these
shall be less than the other two sides of the triangle, but shall
contain a greater angle._

S 11. Having solved two problems (Props. 22, 23), he returns to two
triangles which have two sides of the one equal respectively to two
sides of the other. It is known (Prop. 4) that if the included angles
are equal then the third sides are equal; and conversely (Prop. 8), if
the third sides are equal, then the angles included by the first sides
are equal. From this it follows that if the included angles are not
equal, the third sides are not equal; and conversely, that if the
third sides are not equal, the included angles are not equal. Euclid
now completes this knowledge by proving, that "_if the included angles
are not equal, then the third side in that triangle is the greater
which contains the greater angle_"; and conversely, that "_if the
third sides are unequal, that triangle contains the greater angle
which contains the greater side_." These are Prop. 24 and Prop. 25.

S 12. The next theorem (Prop. 26) says that _if two triangles have one
side and two angles of the one equal respectively to one side and two
angles of the other, viz. in both triangles either the angles adjacent
to the equal side, or one angle adjacent and one angle opposite it,
then the two triangles are identically equal_.

This theorem belongs to a group with Prop. 4 and Prop. 8. Its first
case might have been given immediately after Prop. 4, but the second
case requires Prop. 16 for its proof.

S 13. We come now to the investigation of parallel straight lines,
i.e. of straight lines which lie in the same plane, and cannot be made
to meet however far they be produced either way. The investigation
which starts from Prop. 16, will become clearer if a few names be
explained which are not all used by Euclid. If two straight lines be
cut by a third, the latter is now generally called a "transversal" of
the figure. It forms at the two points where it cuts the given lines
four angles with each. Those of the angles which lie between the given
lines are called interior angles, and of these, again, any two which
lie on opposite sides of the transversal but one at each of the two
points are called "alternate angles."

We may now state Prop. 16 thus:--_If two straight lines which meet are
cut by a transversal, their alternate angles are unequal_. For the
lines will form a triangle, and one of the alternate angles will be an
exterior angle to the triangle, the other interior and opposite to it.

From this follows at once the theorem contained in Prop. 27. _If two
straight lines which are cut by a transversal make alternate angles
equal, the lines cannot meet, however far they be produced, hence they
are parallel._ This proves the existence of parallel lines.

Prop. 28 states the same fact in different forms. _If a straight line,
falling on two other straight lines, make the exterior angle equal to
the interior and opposite angle on the same side of the line, or make_
_the interior angles on the same side together equal to two right
angles, the two straight lines shall be parallel to one another_.

Hence we know that, "if two straight lines which are cut by a
transversal meet, their alternate angles are not equal"; and hence
that, "if alternate angles are equal, then the lines are parallel."

The question now arises, Are the propositions converse to these true
or not? That is to say, "If alternate angles are unequal, do the lines
meet?" And "if the lines are parallel, are alternate angles
necessarily equal?"

The answer to either of these two questions implies the answer to the
other. But it has been found impossible to prove that the negation or
the affirmation of either is true.

The difficulty which thus arises is overcome by Euclid assuming that
the first question has to be answered in the affirmative. This gives
his last axiom (12), which we quote in his own words.

Axiom 12.--_If a straight line meet two straight lines, so as to make
the two interior angles on the same side of it taken together less
than two right angles, these straight lines, being continually
produced, shall at length meet on that side on which are the angles
which are less than two right angles._

The answer to the second of the above questions follows from this, and
gives the theorem Prop. 29:--_If a straight line fall on two parallel
straight lines, it makes the alternate angles equal to one another,
and the exterior angle equal to the interior and opposite angle on the
same side, and also the two interior angles on the same side together
equal to two right angles_.

S 14. With this a new part of elementary geometry begins. The earlier
propositions are independent of this axiom, and would be true even if
a wrong assumption had been made in it. They all relate to figures in
a plane. But a plane is only one among an infinite number of
conceivable surfaces. We may draw figures on any one of them and study
their properties. We may, for instance, take a sphere instead of the
plane, and obtain "spherical" in the place of "plane" geometry. If on
one of these surfaces lines and figures could be drawn, answering to
all the definitions of our plane figures, and if the axioms with the
exception of the last all hold, then all propositions up to the 28th
will be true for these figures. This is the case in spherical geometry
if we substitute "shortest line" or "great circle" for "straight
line," "small circle" for "circle," and if, besides, we limit all
figures to a part of the sphere which is less than a hemisphere, so
that two points on it cannot be opposite ends of a diameter, and
therefore determine always one and only one great circle.

For spherical triangles, therefore, all the important propositions 4,
8, 26; 5 and 6; and 18, 19 and 20 will hold good.

This remark will be sufficient to show the impossibility of proving
Euclid's last axiom, which would mean proving that this axiom is a
consequence of the others, and hence that the theory of parallels
would hold on a spherical surface, where the other axioms do hold,
whilst parallels do not even exist.

It follows that the axiom in question states an inherent difference
between the plane and other surfaces, and that the plane is only fully
characterized when this axiom is added to the other assumptions.

S 15. The introduction of the new axiom and of parallel lines leads to
a new class of propositions.

After proving (Prop. 30) that "_two lines which are each parallel to a
third are parallel to each other_," we obtain the new properties of
triangles contained in Prop. 32. Of these the second part is the most
important, viz. the theorem, _The three interior angles of every
triangle are together equal to two right angles_.

As easy deductions not given by Euclid but added by Simson follow the
propositions about the angles in polygons, they are given in English
editions as corollaries to Prop. 32.

These theorems do not hold for spherical figures. The sum of the
interior angles of a spherical triangle is always greater than two
right angles, and increases with the area.

S 16. The theory of parallels as such may be said to be finished with
Props. 33 and 34, which state properties of the parallelogram, i.e. of
a quadrilateral formed by two pairs of parallels. They are--

Prop. 33. _The straight lines which join the extremities of two equal
and parallel straight lines towards the same parts are themselves
equal and parallel_; and

Prop. 34. _The opposite sides and angles of a parallelogram are equal
to one another, and the diameter (diagonal) bisects the parallelogram,
that is, divides it into two equal parts._

S 17. The rest of the first book relates to areas of figures.

The theory is made to depend upon the theorems--

Prop. 35. _Parallelograms on the same base and between the same
parallels are equal to one another_; and

Prop. 36. _Parallelograms on equal bases and between the same
parallels are equal to one another_.

As each parallelogram is bisected by a diagonal, the last theorems
hold also if the word parallelogram be replaced by "triangle," as is
done in Props. 37 and 38.

It is to be remarked that Euclid proves these propositions only in the
case when the parallelograms or triangles have their bases in the same
straight line.

The theorems converse to the last form the contents of the next three
propositions, viz.: Props, 40 and 41.--_Equal triangles, on the same
or on equal bases, in the same straight line, and on the same side of
it, are between the same parallels_.

That the two cases here stated are given by Euclid in two separate
propositions proved separately is characteristic of his method.

S 18. To compare areas of other figures, Euclid shows first, in Prop.
42, how _to draw a parallelogram which is equal in area to a given
triangle, and has one of its angles equal to a given angle_. If the
given angle is right, then the problem is solved _to draw a
"rectangle" equal in area to a given triangle_.

Next this parallelogram is transformed into another parallelogram,
_which has one of its sides equal to a given straight line_, whilst
its angles remain unaltered. This may be done by aid of the theorem in

Prop. 43. _The complements of the parallelograms which are about the
diameter of any parallelogram are equal to one another._

Thus the problem (Prop. 44) is solved to _construct a parallelogram on
a given line, which is equal in area to a given triangle, and which
has one angle equal to a given angle_ (generally a right angle).

As every polygon can be divided into a number of triangles, we can now
construct a parallelogram having a given angle, say a right angle, and
being equal in area to a given polygon. For each of the triangles into
which the polygon has been divided, a parallelogram may be
constructed, having one side equal to a given straight line and one
angle equal to a given angle. If these parallelograms be placed side
by side, they may be added together to form a single parallelogram,
having still one side of the given length. This is done in Prop. 45.

Herewith a means is found to compare areas of different polygons. We
need only construct two rectangles equal in area to the given
polygons, and having each one side of given length. By comparing the
unequal sides we are enabled to judge whether the areas are equal, or
which is the greater. Euclid does not state this consequence, but the
problem is taken up again at the end of the second book, where it is
shown how to construct a square equal in area to a given polygon.

Prop. 46 is: _To describe a square on a given straight line_.

S 19. The first book concludes with one of the most important theorems
in the whole of geometry, and one which has been celebrated since the
earliest times. It is stated, but on doubtful authority, that
Pythagoras discovered it, and it has been called by his name. If we
call that side in a right-angled triangle which is opposite the right
angle the hypotenuse, we may state it as follows:--

Theorem of Pythagoras (Prop. 47).--_In every right-angled triangle the
square on the hypotenuse is equal to the sum of the squares of the
other sides._

And conversely--

Prop. 48. _If the square described on one of the sides of a triangle
be equal to the squares described on the other sides, then the angle
contained by these two sides is a right angle._

On this theorem (Prop. 47) almost all geometrical measurement depends,
which cannot be directly obtained.