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Perspectives on Science, Volume 22, Number 1, Spring 2014, pp. 35-55
(Article)
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For additional information about this article
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35Ideal Elements in
Hilbert’s Geometry
John Stillwell
University of San Francisco
Hilbert ărst mentioned ideal elements in his 1898–99 lectures on geometry.
He described them as important, fruitful, and of frequent occurrence in math –
ematics, pointing to the examples of negative, irrational, imaginary, idealand transănite numbers. In geometry, he had in mind the examples of points,lines, and planes at inănity, whose introduction gives geometry a certain com-pleteness, by making theorems such as those of Pappus and Desargues univer-sally valid.
In this article I will discuss how Hilbert transformed our view of the
Pappus and Desargues theorems by showing that they express the underlyingalgebraic structure of projective geometry. I will compare this result with an-other of Hilbert’s great contributions, his calculus of ends. By studying theideal elements of the hyperbolic plane, Hilbert similarly extracted algebraicstructure from the axioms of hyperbolic geometry.
Hilbert’s treatments of projective and hyperbolic geometry have another
important common element: construction of real numbers. To achieve this,Hilbert has to add an axiom of continuity to the geometry axioms, but heevidently wants to show that the real numbers can be put on a geometricfoundation.
1. Ideal Elements before Hilbert
Hilbert took to using ideal elements in the 1890’s, in both algebraic num –
ber theory and geometry . His Zahlbericht of 1897 popularized the con –
cept of the ideal introduced by Dedekind in 1871 (which in turn formal –
ized the concept of “ideal number” introduced by Kummer in the 1840’s).His geometric work likewise followed a long history of ideal elements,some that originated in geometry and others that originated elsewhereand were applied to geometry . Important examples were:
Perspectives on Science 2014, vol. 22, no. 1
©2014 by The Massachusetts Institute of Technology doi:10.1162/POSC_a_00117

1. Points at inănity in projective geometry , originating in the 1430s
from the “vanishing points” in perspective drawing.
2. Imaginary points in algebraic geometry , originating from the imagi –
nary numbers used by Bombelli (1572) to solve cubic equations.
Imaginary points are needed to give the “right” number of inter –
sections between algebraic curves.
3. The point that completes the plane Cof complex numbers to a
sphere, used by Riemann (1857) to study complex functions.
4. The boundary at inănity of the hyperbolic plane, discovered by
Beltrami (1868) when he constructed the ărst models of non-
Euclidean geometry .
We now look brieșy at each of these in turn.
1.1 Points at inănity and perspective
T oday , we barely notice when correct perspective is achieved, as in theRenaissance painting Ideal City (Figure 1).
This is perhaps because of the ubiquity of computer graphics, which
has technology for perspective built in. But rules for perspective drawingwere developed only around 1430, and blatantly incorrect perspective re –
mained common for several decades after that, even in Renaissance Italy .Consider, for example, the tiled șoor in Figure 2. This drawing is an illus –
tration from Savonarola’s Art of Dying Well, published in Florence around
1490.
Points at inănity are the key to correct perspective. The method used
by Italian Renaissance artists generally used only one such point ( a “van –
ishing point”) because they drew tiles in rows parallel to the bottom of thepicture. However, one can avoid drawing parallels, and place the tiles inan arbitrary orientation, by using three points at inănity to control the po –
sitioning of lines. Figure 3 shows how. Given a single tile, with parallelsides, all other tiles fall into position one by one.36 Ideal Elements in Hilbert’s Geometry
Figure 1. Piero della Francesca’s Ideal City.

The three ideal points are where the three families of parallels—of the
sides and diagonals of the tiles—meet on the horizon (the ideal line or line
at inănity). The diagonal of the ărst tile determines the diagonal of thesecond tile, because they have the same point at inănity . The diagonal ofthe second tile in turn determines the missing side of the second tile,which determines the diagonal of the third tile, and so on.
1.2 Imaginary points
Newton (1665) stated the following result about algebraic curves, whichhe called “lines,” and their degrees, which he called their “dimensions.”
For y
enumber of points in wchtwo lines may intersect can never bee
greater ynyerectangle of yenumbers of their dimensions. And theyPerspectives on Science 37
Figure 2. Tiled șoor in the Art of Dying Well.

always intersect in soe many points, excepting those wchare
imaginarie onely .
This later became known as Bézout’s theorem, and a more concise statement
is as follows.
Bézout’s theorem: A curve of degree m meets a curve of degree n in exactly mn
points (if the curves have no common component).However, the theorem was not properly proved for a couple more centu-
ries, because to make it right one has to admit three kinds of ideal points.
1.Points at inănity. For example, so that parallel straight lines (which
are curves of degree 1) have one point in common.
2.Imaginary points. For example, so that the parabola y/H11005x
2/H110011 meets
the line y/H110050 in two points; namely , the points x/H11005i andx/H11005/H11002i.
3.Multiple points. For example, so that the parabola y/H11005x2(degree 2)
meets its tangent line y/H110050 (degree 1) in two points.
Imaginary points are already needed for the special case of Bézout’s the –
orem where one curve is y/H110050 and the other is y/H11005p(x), where pis a polyno –
mial of degree n.This case is essentially the fundamental theorem of alge –
bra, in which case the multiplicity of a root x/H11005a is the number of times the
factor x/H11002a occurs among the linear factors of p(x).
Thus imaginary points and multiple points have an algebraic origin.
Points at inănity have a geometric origin, but they can have implicationsfor algebra, as we will see. For the rest of this article we will be concernedwith points at inănity , and the different contexts in which they arise. Thecase of projective geometry shows that it is natural and convenient to adda whole line of points at inănity to the plane. The next case is one where itis convenient to add a single point at inănity to the plane.38 Ideal Elements in Hilbert’s Geometry
Figure 3. Using points at inănity to draw a tiled șoor in perspective.

1.3 Complex functions
Imaginary , or complex, numbers were devised to give us solutions to
equations (initially cubic equations, but also simpler equations such as
x2/H110011/H110050). But they give more than this—something that we did not ask
for—when we consider functions of a complex variable. Complex differen-
tiable functions are conformal maps of the plane (that is, maps that are an-
gle-preserving or “similar in the small”). Here is an example, the functionthat sends zto 1/z. This function maps the square grid shown in Figure 4
to the “grid” of perpendicular circles shown in Figure 5:
Thus differentiability acquires a deeper meaning when we allow the
variable to be complex. We see a property (preservation of angles) that isonly visible in two or more dimensions. This leads us to consider functionson surfaces, and it turns out that we should consider surfaces beyond justthe plane Cof complex numbers.
The Riemann Sphere Since complex functions can take the value /H11009(for ex –
ample, 1/z does at its pole z/H110050) we should think of their values lying in
the set C∪{/H11009}, where Cis the plane of complex numbers. Following
Riemann (1857), we view C∪{/H11009}a sasphere via stereographic projection
(see Figure 6, which is by Jean-Christof Benoist on Wikimedia Com –
mons). The idea of a differentiable function on the plane as one that mapsconformally can be transferred to the sphere, because stereographic projec –
tion is itself a conformal map.
The idea of “completing,” or compactifying, the plane Cto a sphere by
means of the ideal point ∞is a useful idea, not only because it includes allPerspectives on Science 39
Figure 4. The square grid. Figure 5. The grid of perpendicular
circles.

the values that a complex variable can take, but also because it makes
complex functions easier to classify . The differentiable functions on Care
hard to classify , because they include a multitude of transcendental func-tions such as e
z,cosz,and sin z.However, the latter functions are not
meaningful at /H11009, because they approach different values as zapproaches /H11009
in different directions. On the Riemann sphere, the picture is dramaticallysimpler, because we need consider only functions that are meaningfulat/H11009. With this restriction:
1. Any entire function (one that is differentiable everywhere) is con –
stant. (Liouville’s theorem).
2. Any meromorphic function (one differentiable except at poles) is
rational.
Thus the meromorphic functions on the Riemann sphere are precisely therational functions. A rational function r(z) can be written as the quotient
p(z)/q(z) of polynomials and, by the Fundamental Theorem of Algebra, the
polynomials p(z) and q(z) split into linear factors. It follows that a mero –
morphic function on C∪{/H11009} is determined, up to a constant multiple, by
its zeros a
i(the zeros of p) and poles bj(the zeros of q), and their respective
multiplicities miandnj.
Riemann’s great insight was that, by generalizing the idea of the Rie –
mann sphere to surfaces covering the sphere—Riemann surfaces—one may40 Ideal Elements in Hilbert’s Geometry
Figure 6. Stereographic projection.

obtain a similarly simple view of algebraic functions of z.Namely , an alge –
braic function on a Riemann surface is determined, up to a constant mul –
tiple, by its zeros and poles.
1.4 Ideal points in non-Euclidean geometry
In his Euclides ab Omni Naevo Vindicatus (Euclid cleared of every defect), of
1733, Girolamo Saccheri explored the non-Euclidean geometry that we
callhyperbolic, and which he called geometry with “the hypothesis of the
acute angle.” He used this term because in this geometry a quadrilateralwith three right angles has a fourth angle which is acute. Saccheri investi-gated the acute angle hypothesis in an attempt to refute it, because he be-
lieved that:
The hypothesis of the acute angle is absolutely false; because it isrepugnant to the nature of the straight line.
The consequence of the acute angle hypothesis that he found repugnantwas a pair of lines that meet at inănity and have a common perpendicular
there. However, repugnance is in the eye of the beholder, and in 1868Beltrami found models of non-Euclidean hyperbolic geometry in whichthe “lines” have precisely the behavior that Saccheri rejected. Moreover,Beltrami’s models are rather attractive. In them, a pair of lines meeting atinănity look like those in Figure 7. Each of Beltrami’s models has a natu –
ral line at inănity , and the line at inănity is a common perpendicular tolines that meet at inănity . For example, in the half-plane model
•Each “point” is a point of the upper half plane {(x,y): y/H110220}.
•Each “line” is either a vertical half line or a semicircle with cen –
ter on the x-axis.
•The “line at inănity” is the x-axis (to which all proper “lines” are
perpendicular).Perspectives on Science 41
Figure 7. Non-Euclidean lines meeting at inănity.

T wo of Beltrami’s models are particularly noteworthy because they are
conformal, or angle-preserving. This means that non-Euclidean geometry
has a natural role in complex analysis, and indeed Poincaré (1882) noticed
that some results already known in complex analysis had a non-Euclideaninterpretation. For example, the modular function, which goes back toGauss, is a periodic function on the half-plane whose periodicity is de-scribed by Figure 8.
The picture is from Klein and Fricke (1890). The values of the modular
function repeat under the transformations that send ztoz/H110011 and to /H110021/z.
These are among the transformations of the linear fractional form
f(z)/H11005(az/H11001b)/(cz/H11001d ),
for real numbers a,b,c,d with ad-bc nonzero. Poincaré was studying func –
tions invariant under linear fractional transformations around 1880 whenhe noticed that such transformations are isometries of non-Euclidean geom –
etry . They preserve the hyperbolic distance between points, which is deăned
by the distance element /H20844(dx
2/H11001dy2)/y. In particular, the curvilinear trian –
gles shown in Figure 8 are congruent in the hyperbolic sense, because any
one of them can be mapped onto any other by a map that is a composite ofthe hyperbolic isometries sending ztoz/H110011 and to /H110021/z.
Moreover, if we take zin the line at inănity , R∪ {/H11009}, rather than in the42 Ideal Elements in Hilbert’s Geometry
Figure 8. Periodicity pattern of the modular function.

upper half-plane itself, these are nothing but the projective transformations
of the real projective line. Thus, non-Euclidean plane geometry has the
same group of transformations as one-dimensional projective geometry .Also, a non-Euclidean motion of the half-plane is completely determinedby the corresponding transformation of its line at inănity , so the idealpoints control the motion of the actual points in the non-Euclidean plane.
The projective transformations of the line a R∪ {/H11009} are made visible, so
to speak, by interpreting this line as the ideal boundary of the hyperbolic
plane. Poincaré (1883) made a spectacular extension of this idea. T o un-derstand transformations of Cof the form
f(z)/H11005(az/H11001b)/(cz/H11001d ),
with complex a, b, c, d, he viewed Cas the ideal boundary of the upper half-
space, which turns out to be a model of the 3-dimensional non-Euclidean
geometry .
2. Geometry without Coordinates
Now we come back to projective geometry , taking up the story in the19th century , when points at inănity had been thoroughly assimilatedinto the subject. Indeed, it was customary at this time to use the so-calledhomogeneous coordinates, which put ordinary points and points at inănity on
the same footing, describing them both by triples of real numbers. Thisalso allowed the use of algebraic methods in projective geometry , much asDescartes had introduced into Euclidean geometry .Perspectives on Science 43
Figure 9. The Pappus conăguration.

At the same time it was noticed that the projective plane could also be
given an axiomatic description: the points and lines of a projective plane
satisfy the following axioms.
1. Any two points determine a unique line.2. Any two lines meet in a unique point.3. There exist at least four points, no three of which are in a line.
These axioms do not sufăce to prove all the theorems one can prove with
the help of homogeneous coordinates, so it remained to determine whataxioms should be added to the “obvious” axioms above. Hopefully , onecould ănd axioms from which the existence of coordinates could be de –
duced. The ărst such investigation was in the book Geometrie der Lage of
von Staudt (1847).
Following von Staudt (1847) and Wiener (1891), Hilbert in the 1890’s
analyzed the axiomatic approach to projective geometry , and how it maybe used to introduce coordinates “from inside.” The keys to this projectare the so-called “theorems” of Pappus andDesargues. Pappus discovered
his theorem around 350 CE, at a time when it was still part of Euclideangeometry; Desargues discovered his theorem around 1640, along withother pioneering results in projective geometry , which he was the ărst toview as a new branch of geometry .
Theorem of Pappus. For any hexagon with vertices alternately on two lines,
the intersections of opposite sides lie on a line (at the top of the picture in the example
shown).
(In Figure 9 we have deliberately drawn the line where the pairs of oppo-
site sides meet so that it looks like the horizon, in which case the pairs ofopposite sides would be pairs of parallels. The freedom to interpret anylineLas the horizon, and hence to interpret lines that meet on Las paral –
lels, is something we will exploit below, when we deăne “addition ofpoints” in projective geometry .)
This theorem states a projective property of points and lines; that is,
one involving only points, lines, and their intersections. But its proof re –
quires non-projective concepts, such as lengths, and the congruence axioms
that govern them. The theorem of Desargues is a second theorem whosestatement involves only projective concepts, illustrated in Figure 10.
Theorem of Desargues. For any two triangles in perspective, the intersections
of corresponding sides lie on a line.(Again, the picture is arranged so that the line where three intersections
occur looks like the horizon.)44 Ideal Elements in Hilbert’s Geometry

This result really isa projective theorem, because it can be proved using
only projective concepts, but only if we work in projective space (see Fig-
ure 11). As one can see from Figure 11, corresponding sides of the trian-
gles in perspective must meet in a line; namely , the line of intersection ofthe two planes in which the triangles lie.
However, the proof breaks down if the two triangles lie in the same
plane. Like the Pappus theorem, the Desargues theorem in the plane is notprovable without the help of congruence axioms.
In fact, Hilbert showed that the Desargues theorem takes the place of
spatial axioms, and the Pappus theorem takes the place of congruence axi –
oms. More precisely , using points at inănity , one can deăne addition and
multiplication of points on a line. With the additional assumption of
Desargues or Pappus, this system of ‘coordinates’ has additional algebraicstructure, described by the following ăeld axioms.
a/H11001b/H11005b/H11001a, ab /H11005ba (commutativity)
a/H11001(b/H11001c)/H11005(a/H11001b)/H11001c a(bc) /H11005(ab)c (associativity)
a/H110010/H11005aa 1 /H11005a (identity
a/H11001(/H11002a)/H110050a a
-1/H110051 when a /HS110050 (inverse)
a(b/H11001c)/H11005ab/H11001ac (distributivity)Perspectives on Science 45
Figure 10. The Desargues conăguration.

•Assuming Desargues, the coordinate system is a skew ăeld (that
is, it satisăes all ăeld axioms except possibly commutative
multiplication). Assuming the projective space axioms, we getexactly the same result, so Desargues replaces the space axioms.
•Assuming Pappus and Desargues, the coordinate system is a ăeld.
Assuming congruence axioms, we get exactly the same result, soPappus replaces the congruence axioms.46 Ideal Elements in Hilbert’s Geometry
Figure 11. Why the Desargues theorem holds in projective space.
Figure 12. Projective addition of atob.

It is surprising that the nine ăeld axioms are implied by just ăve projec-
tive plane axioms (the three projective plane axioms plus Pappus andDesargues). Indeed it is surprising that Pappus and Desargues have any al-gebraic content at all. In the next section we will say more about why thisis so.
2.1 Projective addition and multiplication
T o illustrate how the Pappus theorem leads to a coordinate system whoseelements form a ăeld, we ărst show how to add points on a line, and ex –
plain why the Pappus theorem ensures that addition is commutative. T oshow the construction as clearly as possible, we draw certain pairs of linesas parallels—exploiting the freedom to call any line the “horizon,” and tocall lines “parallel” when they meet on the “horizon.” Figure 12 showshow to form the sum a/H11001b of points aandb,using parallel lines to simu –
late the process of translating a ăgure along a line.
Intuitively , the line sloping upwards out of O and the line sloping
downwards to atogether form a ‘pair of dividers’ spanning the interval
from 0 to a.When we “translate” the dividers parallel to themselves so
that the left end moves to b,the right end moves to what we call a/H11001b.
(The faint lines mark the initial position of the “dividers,” the dark linesPerspectives on Science 47
Figure 13. Projective addition of btoa.
Figure 14. Comparing a/H11001b with b/H11001a.

indicate the ănal position.) This is a simple and natural construction of
a/H11001b, but notice that the construction of b/H11001a is different. Indeed, b/H11001a is
constructed as shown in Figure 13.
Thus it is not immediate that a/H11001b/H11005b/H11001a. Luckily , the Pappus theorem
comes to the rescue. If we superimpose the two constructions, we get thefollowing Figure 14.
The ăgure contains a Pappus conăguration (in which the opposite sides
of the hexagon are parallel), which ensures that lines ending at a/H11001b and
b/H11001a end at the same point. Thus, a/H11001b/H11005b/H11001a.
There is an equally easy construction of the product of points, using
parallel lines to simulate a process of magniăcation. One begins with aline with points marked 0 and 1, and arbitrary points aandb,and
‘magniăes’ abybusing the construction shown in Figure 15. The key
is the second line out of 0, along which we slide the joint in the “divid -48 Ideal Elements in Hilbert’s Geometry
Figure 15. Multiplying abyb.
Figure 16. Multiplying bbya.

ers.” The “dividers” initially have their left end on 1 and the right end
ona.Then we move the two lines in the “dividers” parallel to themselves
until the left end is on b.
This magniăes the interval from 1 to abyb,so that the right end of the
“dividers” lands on ab.
Again, it is not clear that the process is commutative, because the con-
struction of bais different from the construction of ab; namely , bais con-
structed as shown in Figure 16.
But again the two constructions lead to the same point because of the
Pappus theorem, as the Pappus conăguration in Figure 17 makes clear.
These proofs show that a/H11001b/H11005b/H11001a and ab/H11005ba are very natural conse-
quences of the Pappus theorem. In fact, with some ingenuity , it is possibleto prove all the ăeld axioms. The Desargues theorem is particularly help –
ful in proving the associative laws.
However, this is as much as we can prove with purely projective axi –
oms, since there are ănite projective planes, whose coordinates come from
ănite ăelds. T o extract the real numbers from geometry , Hilbert needed
extra geometric axioms of “betweenness” and “continuity .” With theseaxioms, one recovers the intuitive projective plane used by artists. Thebetweenness axioms guarantee that points on a projective line (minus itspoint at inănity) have a left-to-right order, and that they are densely or –
dered; that is, between any two of them there is a third. Continuity guar –
antees that the line has no gaps, which makes it isomorphic to the realnumber line.
Hilbert’s work on extracting algebra from geometry with the help of
ideal elements was the culmination of the investigations that began withvon Staudt (1847) and Wiener (1891). The latter authors attempted toPerspectives on Science 49
Figure 17. Why ab/H11005ba.

construct projective geometry without numbers, but it was Hilbert who
clearly identiăed the role of the Desargues theorem (implying the struc –
ture of a skew ăeld, that is, a structure satisfying all the ăeld axioms ex –
cept commutative multiplication) and the Pappus theorem (implyingcommutative multiplication). Thus if both the Desargues and Pappustheorems hold, we have the structure of a ăeld. In a surprising late devel –
opment (more than 250 years after the discovery of the Desargues theo –
rem!), Hessenberg (1905) discovered that Pappus implies Desargues, sothe Pappus theorem alone implies all the ăeld properties.
3. Extracting the Real Numbers from Non-Euclidean Geometry
One sees in retrospect that Hilbert’s approach to geometry has the aim ofextracting the ăeld of real numbers from geometry . If he had wanted onlyto derive the theorems of Euclid, or of projective geometry , he need nothave included axioms of continuity . For Euclid’s geometry one needs onlya ăeld that includes the rationals and is closed under the square root (ofpositive numbers), since these are the numbers that arise from ruler andcompass constructions.
In 1902–3, Hilbert carried out a similar program of extracting the real
numbers from axioms for the non-Euclidean hyperbolic plane. Again thepoints at inănity are crucial; they serve as real numbers, and Hilbert addsand multiplies them in what is called his calculus of ends. A point /H9251at
inănity is called an end because it is the common “end” of a family of as-
ymptotic or “parallel” lines (Figure 18). Hilbert then uses the geometry of
asymptotic lines to deăne the sum and product of ends, and these turn outto have the same behavior as the sum and product of real numbers.
The deănitions are based on certain propositions that Hilbert proves
axiomatically , but we will look at them in the half-plane, where they areeasily seen to be true. Once it is seen why they are true, it is fairly easy toconstruct axiomatic proofs. An example is the proposition that if the per –
pendicular bisectors of two sides of a triangle have the same end, so does the thirdside. T o see why this is true, let the end of two of the perpendicular bisec –
tors be /H11009in the half-plane model. Then all the perpendicular bisectors are
vertical lines, as Figure 19 shows, so they have the same end.
Given ends /H9251and/H9252, view them as points on the x-axis, and draw the
vertical lines from 0, /H9251, and /H9252to/H11009. Then the construction shown in Fig –
ure 20, which obviously produces /H9251/H11001/H9252, is expressible in the language of
hyperbolic geometry .
The construction is the following:
1. Choose a point Xon the line from 0 to /H11009.
2. Find the reșection X
/H9251ofXin the line from /H9251to/H11009.50 Ideal Elements in Hilbert’s Geometry

3. Find the reșection X/H9252ofXin the line from /H9252to/H11009.
4. Construct the line through X/H9251andX/H9252. The ends of its perpendicu-
lar bisector are /H9251/H11001/H9252 and/H11009.
3.1 Algebraic properties of sum and product
It is clear from the half-plane model that the sum construction is inde-
pendent of the point X,and no surprise that this fact can be proved from
the axioms of hyperbolic geometry . It is also clear that /H9251/H11001/H9252/H11005/H9252/H11001/H9251,b e –
cause we have the same triangle in both cases. Thus /H9251/H11001/H9252/H11005/H9252/H11001/H9251 is atheo-
rem of hyperbolic geometry, which can be proved from the axioms. The same
applies to properties of the product of ends, which we deăne shortly . We
ănd that sum and product satisfy all the ăeld properties, and that theseproperties are theorems of hyperbolic geometry . (However, unlike thederivation of ăeld properties from the Pappus and Desargues theorems, weneed to use axioms about length and angle.)
The product of ends can be expected to exist, because of the role of
multiplication in the half-plane model. Since the element of length in thehalf-plane model is /H20844(dx
2/H11001dy2)/y, sending (x,y)t o( /H9252x,/H9252y) preserves
hyperbolic length for any /H9252/H11022 0. This allows “multiplication by magni –
ăcation,” not unlike the projective construction, except that insteadof magnifying by parallel displacement we magnify by displacementthrough equal hyperbolic distances. Thus, in the situation shown in Fig –
ure 21, the line segments iAandBC have equal hyperbolic length, so we
can multiply the end /H9251by/H9252by displacing Bupwards to Cthrough a hy –
perbolic distance equal to iA.Perspectives on Science 51
Figure 18. The end of a family of asymptotic lines.

In more detail, to construct /H9251/H9252 from ends /H9251and/H9252:
1. Draw the line from 1 to its reșection /H110021 in the line 0/H11009, meeting
0/H11009 ati.
2. Draw the line from /H9251to its reșection /H11002/H9251 in 0/H11009, meeting 0/H11009 atA.
3. Draw the line from /H9252to its reșection /H11002/H9252 in 0/H11009, meeting 0/H11009 atB.
4. Make BC on 0/H11009 with the same (hyperbolic) length as iA.52 Ideal Elements in Hilbert’s Geometry
Figure 19. Perpendicular bisectors with a common end.
Figure 20. Constructing the sum of two ends.

The perpendicular to BC atCthen has ends /H9251/H9252 and/H11002/H9251/H9252. With this con-
struction it is not hard to prove /H9251/H9252/H11005/H9252/H9251, and in fact all the ăeld axioms
are provable as theorems of hyperbolic geometry .
Thus, Hilbert has again succeeded in extracting the real numbers from
geometry , with the help of ideal elements. Indeed, in hyperbolic geometry ,
the real numbers arethe ideal elements.
4. Summary of Hilbert’s Theory of Ends
1. The theory is probably motivated by the half-plane model,
which Hilbert used in all his early courses on the foundations ofgeometry .
2. Ends themselves are motivated by the ideal points on the boundary
of the half-plane, that is, by real numbers.
3. By replacing ideal points by sets of objects inside the abstract
hyperbolic plane (families of asymptotic lines), Hilbert was able tobuild the arithmetic of real numbers by pure geometry .
4. Thus numbers, and hence the half-plane model of hyperbolic geom –
etry , is implicit in the geometry itself.
5. Therefore, what holds in hyperbolic geometry is what holds in the half-
plane model. In particular, the Hilbert axioms of hyperbolic geome –
try are complete, because what follows from them is what is true in
the half-plane model.Perspectives on Science 53
Figure 21. Multiplication and hyperbolic length.

Hilbert’s experience with ideal elements in geometry clearly had an im –
portant inșuence on his later thought. In On the Inănite (1925), he re –
șected as follows on ideal elements, revisiting many of the ideas we have
touched on above, and envisaging a continuing role for ideal elements inall ăelds of mathematics that involve the concept of inănity:
[. . .] as is well known, the introduction of ideal elements, namely ,points at inănity and a line at inănity , renders the proposition thattwo straight lines always intersect in a unique point universallyvalid [. . .]
The ordinary complex magnitudes of algebra likewise are [. . .]
ideal elements; they serve to simplify the theorems on the existenceand number of roots of an equation.
Just as in geometry inănitely many straight lines, namely , a fam –
ily of parallels, are used to deăne an ideal point, so in higher arith –
metic certain systems of inănitely many numbers deăne a number
ideal, and indeed probably no use of ideal elements is a greater
stroke of genius [. . .]
Now we come to analysis [. . .] in a sense mathematical analysis
is but a single symphony of the inănite.
5. Conclusion
As Hilbert intimated in the above remarks from On the Inănite, most of
the ideal elements in mathematics are related to inănity , and indeed toinănite sets. The points at inănity that occur in projective and non-Euclidean geometry are admittedly motivated by the idea of inănite dis-tance rather than inănite sets. But, as Hilbert realized, each point atinănity corresponds to an inănite set of lines (a set of parallels in the pro –
jective case, a set of asymptotic lines in the non-Euclidean case). Thus, asin the case of ideals in algebraic number theory , one does not have to go“outside” the domain of actual elements—an ideal element is simply acertain inănite set of actual elements.
Hilbert seems to be fascinated by the construction of the real numbers
in geometric systems. In part, this may be due to the ancient roots of realnumbers in Euclid’s Elements and earlier; from the discovery of irrational
quantities in Pythagorean times to the development of the “theory of pro –
portions” by Eudoxus and its exposition in Euclid’s Book V . The Greeksbelieved that geometric quantities were more general than numericalquantities, since they did not consider /H208442 to be a number. Hilbert of
course had no such qualms, but he evidently sympathized with the ideathat geometry is logically prior to the theory of real numbers. At any rate,54 Ideal Elements in Hilbert’s Geometry

he wanted to show how the real numbers could be constructed from geo –
metric foundations.
In the long run, however, the real numbers are too important to be
taken as a special feature of geometric systems, and inănity is too impor –
tant to be left as some unexplained ideal element. As Hilbert well knew,
the “symphony of the inănite” that is analysis leads to questions aboutinănity deeper than those posed by the existence of ideal elements. Never –
theless, ideal elements provide a model for the use of inănity in mathe –
matics, and they make a convincing case for its indispensability .
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