Preliminary version – accepted for publication as Chapter 1 of forthcoming MAA Notes volume

*Using Recent
History in the Teaching of Mathematics*

Common Confusions in Geometry[1]

Most people judge the size of cities simply from their
circumference. So that when one says that Megalopolis is fifty stades in
contour and Sparta forty-eight, but that Sparta is twice as large as
Megalopolis, what is said seems unbelievable to them. And when in order to
puzzle them even more, one tells them that a city or camp with the
circumference of forty stades may be twice as large as one of the circumference
of which is one hundred stades, what is said seems to them absolutely
astounding. The reason of this is that we have forgotten the lessons in
geometry we learnt as children.

-- Polybius, *The **Histories*, 9.26a, 2^{nd} century B.C.
(from the English translation by W.R. Paton, Cambridge, MA: Harvard University
Press, 1922-1927)

The great mathematicians do not allow themselves to make
their science comprehensible to us beginners. Is it because they imagine that
others have the same degree of understanding that they do and hence they
consider it superfluous to expound those statements that firm the bases of
their arguments? Or is it that they consider that those who do not have as much
sagacity as they do are incapable of understanding the mathematical sciences
and therefore they think it not worth their while to instruct them? Or is it
that they feel it is unpleasant and repulsive to explain matters, which, in
their opinion, are just too simple?

-- Georg Ludewig Spohr, *Anweisungen zur Differential- und Integral
Rechnung fuer Anfaenger*. Leipzig, 1793. Reprinted in* Drawbridge Up:
Mathematics-a Cultural Anathema*, H. M. Enzensberger (1998)

We have found that students who take our upper level geometry course (Math 451) at Cornell University usually have very little background in geometry. We have lead many week-long UFE and PREP workshops (funded by the National Science Foundation) for professors on teaching geometry and we found that even mathematicians are often confused about the history of geometry. In addition, many expository descriptions of geometry (especially non-Euclidean geometry) contain confusing and sometimes-incorrect statements – this is true even in expositions written by well-known research mathematicians. Therefore, we found it very important to give some historical perspective of the development of geometry, clearing up many common misconceptions and increasing people’s interest both in geometry and in the history of mathematics.

Questions that often have confusing or misleading (or incorrect) answers:

1. What is the first non-Euclidean geometry?

2. Does Euclid’s parallel postulate distinguish the non-Euclidean geometries from Euclidean geometry?

3. Is there a potentially infinite surface in 3-space whose intrinsic geometry is hyperbolic?

4. In what sense are the Models of Hyperbolic Geometry “models”?

5. What does “straight” mean in geometry? How can we draw a straight line?

These questions are asked by our students and by mathematicians in our workshops. In the literature we often do not find clear answers.

In order to clarify these questions for ourselves and for our students, we find that it is important to talk about the development of different ideas in mathematics. It also gives us a way to bring the history of mathematics into the geometry course. Talking with our students, we noticed that most confusions related to the above questions come from not recognizing certain strands in the history of geometry. Thus, even though the topic of this volume is the use of more recent history of mathematics in our courses, we will start (as we do with our students) with a quick look back to various historical strands that can help us understand the issues surrounding these questions. More recent history will come later.

It is important to talk with students about deep roots of
human experiences of mathematics and how it connects with modern theories. One
of our student responses demonstrates how learning is motivated by history:

I found 451(Spherical and Euclidian Geometry) to be my favorite math class so far. I feel the way we are learning math is like collecting pieces of a puzzle through our education. We need lots of those pieces to form a picture (or I should say "part of a picture" here), but if we haven't collected "enough" of them, we won't be able to put them together and thus can't see anything. I think this is why my interest in mathematics is growing as I am learning more of it, and want to take [the history of mathematics class]. I want to know how and why people started to think about "mathematics".

It was helpful to our students to talk about how the main
aspects of geometry today emerged from four strands of early human activity
that seem to have occurred in most cultures: art/patterns, building structures,
making machines, and navigation/stargazing. These strands developed more or
less independently into varying studies and practices that eventually from the
19^{th} century on were woven into what we now call *geometry*.

Art/Patterns Strand

To produce decorations for their weaving, pottery, and other objects, early artists experimented with symmetries and repeating patterns. Later the study of symmetries of patterns led to tilings, group theory, crystallography, finite geometries, and in modern times to security codes and digital picture compression. Early artists also explored various methods of representing existing objects, and living things. These explorations led to the study of perspective and then projective geometry and descriptive geometry, and (in 20th century) to computer-aided graphics, the study of computer vision in robotics, and computer-generated movies (for example, Toy Story). For more details, see [1], [3], [13], [16], [20], [22], [23], [25], [26], [38], [39].

Navigation/Stargazing Strand

For astrological, religious, agricultural, and other
purposes, ancient humans attempted to understand the movement of heavenly
bodies (stars, planets, sun, and moon) in the apparently hemispherical sky.
Early humans used the stars and planets as they started navigating over long
distances. They used this understanding to solve problems in navigation and in
attempts to understand the shape of the Earth. Ideas of trigonometry apparently
were first developed by Babylonians in their studies of the motions of heavenly
bodies. Even Euclid wrote an astronomical work, *Phaenomena*, [17], in which he studied properties of curves on a
sphere. Navigation and large-scale surveying developed over the centuries
around the world and along with it cartography, trigonometry, spherical
geometry, differential geometry, Riemannian manifolds, and thence to many
modern spatial theories in physics and cosmology. For more details, see [2], [8], [17], [19], [21], [52], [59].

Motion in Machines Strand

Early human societies used the wheel for transportation,
making pottery, and in pulleys. In ancient Greece, Archimedes and Hero used
linkages and gears to solve geometrical problems – such as trisecting an angle,
duplicating a cube, and squaring a circle. These solutions were not accepted in
the Building Structures (Euclid) Strand, which leads to a common misconception
that these problems are unsolvable and that Greeks did not allow motion in
geometry. See [47] and [40]. We do not know much about the influence of
machines on mathematics after Ancient Greece until the advent of machines in
the Renaissance. In the 17^{th} century, Descartes used linkages and
the ideas of linkages to study curves. This study of curves led to the
development of analytic geometry. Computing machines were also developed in the
17^{th} century. As we will discuss below, there was an interaction
between mathematics and mechanics that leads to the modern mathematics of
rigidity and robotics. For more detail see [12], [14], [18], [50], [55], [62].

Building Structures Strand

As humans built shelters, altars, bridges, and other
structures, they discovered ways to make circles of various radii, and various
polygonal/polyhedral structures. In the process they devised systems of
measurement and tools for measuring. The (2000-600 BC) *Sulbasutram* [57] is written for altar builders and contains at the
beginning a geometry handbook with proofs of some theorems and a clear general
statement of the “Pythagorean” Theorem. Building upon geometric knowledge from
Babylonian, Egyptian, and early Greek builders and scholars, Euclid (325-265
BC) wrote his *Elements,* which became
the most used mathematics textbook in the world for the next 2300 years and
codified what we now call Euclidean geometry. Using the *Elements* as a basis in the period 300 BC to about 1000 AD, Greek
and Islamic mathematicians extended its results, refined its postulates, and
developed the study of conic sections and geometric algebra. Within Euclidean
geometry, analytic geometry, vector geometry (linear algebra and affine
geometry), and algebraic geometry developed later. The *Elements* also started what became known as the axiomatic method in
mathematics. For the next 2000 years mathematicians attempted to prove Euclid's
Fifth (Parallel) Postulate as a theorem (based on the other postulates); these
attempts culminated around 1825 with the discovery of hyperbolic geometry.
Further developments with axiomatic methods in geometry led to the axiomatic
theories of the real numbers and analysis, and to elliptic geometries and
axiomatic projective geometry. For more
detail see [5], [7], [8], [10],
[12], [27], [56], [57].

We will now describe how these strands can be used to clarify issues surrounding the questions identified in the introduction. We include those aspects of the recent history of mathematics that are either related to these issues or are difficult to find in the current literature.

Spherical geometry can be said to be the first non-Euclidean geometry and developed early in the Navigation/Stargazing Strand. The connections of spherical geometry with other strands will be covered in the following sections. For at least 2000 years humans have known that the earth is (almost) a sphere and that the shortest distances between two points on the earth is along great circles (the intersection of the sphere with a plane through the center of the sphere). For example:

...it will readily be seen how much space lies between the two
places themselves on the circumference of the large circle which is drawn
through them around the earth. ... [W]e grant that it has been demonstrated by
mathematics that the surface of the land and water is in its entirety a
sphere,... and that any plane which passes through the center makes at its
surface, that is, at the surface of the earth and of the sky, great circles,
and that the angles of the planes, which angles are at the center, cut the
circumferences of the circles which they intercept proportionately,...

— Claudius Ptolemy, *Geographia* (ca. 150 AD), Book One,
Chapter II

Spherical geometry is the geometry of a sphere. The great
circles are intrinsically straight on a sphere in the sense that the shortest
distances on a sphere are along arcs of great circle and because great circles
have the same symmetries on a sphere as straight lines have on the Euclidean
plane. The geometry on spheres of different radii is different; however, the
difference is only one of scale. In Aristotle we can find evidence that
spherical non-Euclidean geometry was studied even before Euclid. (See [29, p 57]
and [61].) Even Euclid in his work on astronomy, *Phaenomena* [17], discusses propositions of spherical geometry.
Menelaus, a Greek of the first century AD, published a book *Sphaerica*, which contains many theorems
about spherical triangles and compares them to triangles on the Euclidean
plane. (*Sphaerica* survives only in an
Arabic version. For a discussion see [43], page 119–120.)

Up into the 19^{th} century, spherical geometry
occurred almost entirely in the Navigation/Stargazing Strand and was used by
Brahe and Kepler in studying the motion of stars and planets and by navigators
and surveyors. The popular book *Spherical
Trigonometry: For the use of colleges and schools *(1886) by Todhunter [59] contains several discussions of the use
of spherical geometry in surveying and was used in British schools before
hyperbolic geometry was widely known in the British Isles. On the continent,
C.F. Gauss (1777-1855) was using spherical geometry in various large-scale
surveying projects before the advent of hyperbolic geometry. Currently,
3-dimensional spherical geometry is being considered as one of the possible
shapes for our physical universe. (See [61] and [31].)

After developing for some 2000 years, spherical geometry
was related to the Euclidean postulates of the Building Structure Strand in the
19^{th} and 20^{th} centuries. There are many popular accounts
(for example, Katz [40, p 782] and Davis & Hersh [11, pp 219-221]) that
attempt to distinguish between Euclidean and spherical geometries on the basis
of the parallel postulate. In Euclid’s *Elements*, the “parallel
postulate” is the Fifth Postulate:

(**EFP**) If a straight line intersecting two straight lines
makes the interior angles on the same side less than two right angles, then the
two lines (if extended indefinitely) will meet on that side on which the angles
are less than two right angles.

The interested reader can check that **EFP** is not
needed to assert the *existence *of parallel lines. In fact Euclid in I.31
proves (in modern wording):

Given a line *l* and a point *P* not on *l *there
exists a line through *P* parallel to *l*.

In his proof, Euclid constructs
first a line *n* through *P* perpendicular to *l* and then
another line *m* through *P* and perpendicular to *n*. Then
Euclid concludes (I.27) that *m* is parallel to *l* because the
alternate interior angles are equal. In none of these constructions or
conclusions does Euclid use his Fifth Postulate. Now, in this situation, the
reader can easily see that **EFP** immediately implies that any line through
*P*, other than *m*, will intersect *l* and thus *m* is the
*unique* parallel line through *P* parallel to *l*.

Note that **EFP** is true in spherical geometry
(because all lines intersect) and is even provable in the strong sense that the
two lines meet closest on that side on which the angles are less than two right
angles (see, for example, [31, Chapter 10]). However, the *existence *of
parallel lines is false in spherical geometry and thus it must be one of the
other postulates that fails on the sphere. Thus it is not true that **EFP**
distinguishes Euclidean from non-Euclidean geometries.

Over the years, **EFP** has been replaced by many other
postulates (see for example, [28, p 220], [30, p 203], [11, p 219]. The most
common of these postulates used in the modern high school textbooks and popular
accounts is:

(**HSP**) Given a line and a point not on the line there is *one
and only one* line through the point that is parallel to the given line.

We will call this the High School
Postulate (**HSP**). Note that this postulate posits both the *existence*
and *uniqueness*. In the context of all the other postulates of Euclidean
geometry **HSP** is equivalent to **EFP**. However, unlike the Fifth
Postulate, the High School Postulate is not true in spherical geometry. Often, **HSP**
is called by many authors (including us in [31, p 124]!) “Playfair’s Parallel
Postulate”. However, Playfair actually stated ([53, p 11], 1839, and in the
editions Philadelphia, 1806; Boston,
1814; Edinburgh, 1819; and New York, 1835):

(**PP**) Two straight lines which intersect one another,
cannot be both parallel to the same straight line.

Many other authors (including David Hilbert in his axiomation of Euclidean geometry [33]) state the Playfair postulate in is logically equivalent form:

(**PP**) Given a
point *P* and a line *l*, there exists at most one line through *P*
and parallel to *l*.

This postulate only posits *uniqueness*
and thus it is true in both spherical and Euclidean geometries.

Nevertheless, many writers are confused, for example: Katz
in the excellent history text [40, p 782] says that one of the “two possible
negation of Euclid’s parallel postulate” leads to spherical geometry. Davis and
Hersh [11, pp 219-221] state **HSP**, but call it Playfair’s postulate, and
then state that the sphere satisfies the first four postulates of Euclid but
not **HSP**. Heiede in [30, p 203] states (correctly) that **PP** is true
on a sphere, but continues “so [**PP**] can only be equivalent to [**EFP**]**
**under some additional condition …”

All of these confusions seem to come from replacing **EFP**
with **HSP**. We do not know in detail how this happened but there are two
hints: Heath’s translation of Euclid has been for the past century the de facto
standard and in his commentary he states [28, p 220] “Playfair’s Axiom” as:

*Through a given point only one parallel can be drawn to a
given straight line* or, *Two straight lines which intersect one another
cannot both be parallel to one and the same straight line*.

The second statement here clearly
posits *uniqueness* while the first statement could be mistaken for the **HSP**.
And then Kline, in [43, p 865] which was for many years the de facto standard
history of mathematics, repeats the first statement of Heath’s and then adds:

This is the axiom used in modern books (which for simplicity usually say there is “one and only one line …”).

Because of the 2000-year long history of investigations
into parallel postulates within the Building Structures Strand, many books
misleadingly call hyperbolic geometry “*the*
non-Euclidean geometry” or “the first non-Euclidean geometry”. Spherical
geometry has been studied since ancient times but it did not fit easily into
this axiomatic approach and thus was (and still is) left out of many
discussions of non-Euclidean geometry.

In the discussions that do include spherical geometry it is called by various names which causes more confusions: Riemannian, projective, elliptic, double elliptic, and spherical. These different labels for spherical geometry usually imply different settings and contexts as summarized in the following:

·
Bernhard
Riemann (1826–1866) pioneered the intrinsic (and analytic) view for surfaces
and space; and, in particular, he introduced an intrinsic analytic view of the
sphere that became known as the ** Riemann Sphere**. The Riemann Sphere
is usually studied in a course on complex analysis. Some writers use the term
“Riemannian geometry” to describe spherical geometry (usually in parallel with
“Lobatchevskian geometry”),
but this practice has led to confusion among students and mathematicians
because the term “

·
** Double-elliptic geometry** usually indicates an axiomatic formalization of
spherical geometry in which antipodal points are considered as separate points.
This axiom system for spherical geometry is in the spirit of Euclid’s Axioms
(as embellished by Hilbert [33]) for Euclidean geometry (see for example, [6]),
in this context spherical geometry is usually called

·
** Elliptic geometry** usually indicates an axiomatic formalization of
spherical geometry in which each pair of antipodal points is identified as one
point.

·
** Projective geometry** originally developed within the Art/Patterns Strand
in the study of perspective drawings. If you project space (or a plane) onto a
sphere from the center of the sphere (called a

·
** Spherical geometry** often indicates the geometry of the sphere sitting
extrinsically in Euclidean 3-space.

Starting soon after the *Elements* were written and continuing for the next 2000 years
mathematicians attempted to either prove Euclid's Fifth Postulate as a theorem
(based on the other postulates) or to modify it in various ways. These attempts
culminated around 1825, when Nicolai Lobatchevsky (1792-1850) and János Bolyai
(1802-60) independently discovered a geometry that satisfies all of Euclid's
Postulates and Common Notions except that the Fifth Postulate does not hold. It
is this geometry that is called *hyperbolic
geometry*.

One of the first open questions about hyperbolic geometry
was whether it is the geometry of any surface in Euclidean space, in the same
sense that spherical geometry is the geometry of the sphere in Euclidean
3-space. In the mid-19th century beginnings of differential geometry it was
shown that hyperbolic surfaces would be precisely surfaces with constant negative
curvature. This aspect of hyperbolic geometry belongs in the Navigation/Star
Gazing strand of geometry, in the sense that differential geometry of surfaces
(and higher-dimensional manifolds) uses calculus to study the geometric
properties that are *intrinsic* – properties of the surface that a bug
crawling on the surface could detect. Intrinsic properties include geodesics
(intrinsically straight lines), length of paths, shortest paths (which are
almost always segments of geodesics), angles, surface area, and so forth.

Students (and mathematicians) desire to touch and feel a hyperbolic surface in order to experience its intrinsic properties. Many people have trouble with standard “models” and pictures of hyperbolic geometry in textbooks, because the intrinsic meanings of geodesics, lengths, angles, and areas cannot be directly seen. However, it is a common misconception that you cannot have a surface in 3-space whose intrinsic geometry is hyperbolic geometry.

Mathematicians looked for surfaces whose intrinsic
geometry is complete hyperbolic geometry in the same sense that the intrinsic
geometry of a sphere has the complete spherical geometry. In 1868, Eugenio
Beltrami (1835-1900) described a surface, called the *pseudosphere*, whose local intrinsic geometry is hyperbolic geometry
but is not complete in the sense that some geodesics (intrinsically straight
lines) cannot be continued indefinitely. (See [24, p 218] for photos of the
surfaces that Beltrami constructed and further discussion of the pseudosphere.)
In 1901, David Hilbert (1862-1943) [34] proved that it is impossible to define
by (real analytic) equations a complete hyperbolic surface. In those days
"surface" normally meant one defined by real analytic equations and
so the search for a complete hyperbolic surface was abandoned and still today
many works state that a complete hyperbolic surface is impossible. For
popularly written examples, see [51, p 158] and [35, p 243]. In 1964, N. V.
Efimov [15] extended Hilbert’s result by proving that there is no isometric
embedding defined by functions whose first and second derivatives are
continuous. However, in 1955, Nicolas
Kuiper [46] proved, without giving an explicit construction, the existence of
complete hyperbolic surfaces defined by continuously differentiable functions.
Then in some workshops in the 1970's William Thurston described the
construction of complete hyperbolic surfaces (that can be made out of paper),
see [58, p 49 and p 50].

Figure 1. Crocheted
Hyperbolic Planes of Radius 4 cm and 8 cm

**crocheted by Daina
Taimina, photo by David Henderson, first published in [31] – used here by
permission**

Directions for constructing Thurston’s surface out of paper can be found in [31] or [32]. See Figure 1 for crocheted hyperbolic planes with different radii. These references also contain a description of the method (invented by the first author of this article) for crocheting these surfaces. In addition, there is in these references a description of an easily constructible polyhedral hyperbolic surface, called the "hyperbolic soccer ball", that consists of heptagons (7-sided regular polygons) each surrounded by 7 hexagons (the usual spherical soccer ball consist of pentagons each surrounded by 5 hexagons) – this construction was discovered recently by the second author’s son, Keith Henderson. The intrinsic straight lines ("geodesics") on a hyperbolic surface can be found by folding the surface (in the same way that folding a sheet of paper will produce a straight line on the paper). This folding also determines a reflection about the intrinsic straight line.

To study the geometry of the sphere analytically we
develop coordinate systems and projections (maps) of the sphere onto the plane.
(Note that the usual spherical coordinates give a projection of the sphere onto
a rectangle in the plane.) For the same reason, coordinate systems and
projections of the hyperbolic plane are useful for systematic analytic study.
At the same time as he discovered the pseudosphere, Beltrami described a
projection of the hyperbolic plane onto a disk in the plane in 1868, this
projection was more fully developed in 1871 by Felix Klein (1849-1925). This
projection, which can also be thought of as describing a coordinate system, is
now called in the literature *projective
disk model*, *Beltrami-Klein model*,
or *Klein model*.

The Beltrami-Klein and other model are perhaps called “model” instead of a “coordinate system” or “projection” because of the absence of a surface whose intrinsic geometry was complete hyperbolic geometry. But after Thurston’s construction we can now describe this model (and the ones described below) as projections or coordinate systems of the complete hyperbolic surface – for details, see [31, Chapter 16].

The Beltrami-Klein model was based on projective geometry and projective transformations, which had their origins in the Art/Pattern Strand. The Beltrami-Klein model (thought of as a transformation) takes straight lines in the hyperbolic plane to straight lines in the Euclidean plane in the same way that gnomic projection takes straight lines (great circles) on the sphere to straight lines on the Euclidean plane. As with gnomic projection, the measure of angles is not preserved in the Beltrami-Klein model. (See Chapters 15 and 16 of [31].)

Other well known models of the hyperbolic plane (the *upper half-plane model* and *Poincaré
disk model*) were developed in 1882 by Henri Poincaré and were based on
circles and inversions in circles. Both Poincaré models preserve the measure of
angles but take straight lines in the hyperbolic plane to semicircles in the
upper half-plane or disk. Reflection through a straight line on the hyperbolic
plane corresponds in the Poincaré models to inversion in the semicircles. See
[27, Section 39], and [31, Chapter 16].

Inversions in a circle are a necessary component of the
analytic study of hyperbolic geometry. As we will see below they are also
useful to understanding linkages that convert straight-line motion into
circular motion. Our students want to know the origins of inversions. We have
found that it is not easy to find a history of inversions. We describe
inversions as emerging from the Art/Pattern Strand and trace the ideas in the
theory of inversions back to Apollonius of Perga (225 B.C. - 190 B.C.), who
investigated one particular family of circles and straight lines. We know about
this from a commentary of Pappus of Alexandria (290-350). Apollonius defined
the curve *c _{k}*(

Figure 2. *B*
is the inversion of *A *with respect to *C* (and vice versa)

A systematic study of inversions in circles started only in the 19th century. Jakob Steiner (1796-1863) was among the first to use extensively the technique of inversions in circles. Steiner had no early schooling and did not learn to read or write until he was age 14. Against the wishes of his parents, at age 18 he went to the Pestalozzi School at Yverdon, Switzerland, where his extraordinary geometric intuition was discovered. By age 28 he was making many geometric discoveries using inversions. At age 38 he occupied the chair of geometry established for him at the University of Berlin, a post he held until his death. We find these facts inspiring to students. For more details, see [45]. A complete recent history of inversions can be found in the forthcoming book [48].

Inversions were also used in the 19^{th} century
to solve a long-standing engineering problem (from the Motion in Machines
Strand) we describe in the next section.

When using a compass to draw a circle, we are not starting with a model of a circle; instead we are using a fundamental property of circles that the points on a circle are a fixed distance from a center. Or we can say we use Euclid's definition of a circle. So, now what about drawing a straight line: Is there a tool (serving the role of a compass) that will draw a straight line? One could say: We can use a straightedge for constructing a straight line. Well, how do you know that your straightedge is straight? How can you check that something is straight? What does "straight" mean? Think about it!

We can try to use Euclid's definition: "A straight line is a curve that lies symmetrically with the points on itself.” (See [31, Appendix A] for a justification of this translation; and Chapter 1 for a discussion of “What is straight?”) This leads to knowing that folding a flat piece of paper will produce a straight line. This use of symmetry, stretching, and folding can also be extended to spheres and hyperbolic planes (See [31, Chapters 2 and 5].) We can also use the usual high school definition, “A straight line is the shortest distance between two points.” This leads to producing a straight line by stretching a string. Students are confused when reading in the literature that “straight line” is an undefined term, or that straight lines on the sphere are “defined to be arcs of great circles”. We find that putting it in the context of the four strands helps clarify this: Symmetry comes from the Art/Pattern Strand, “undefined terms” come from the Building Structures Strand, and “shortest distance” from the Navigation/Star Gazing Strand.

But there is still left unanswered the question of whether there is a mechanisms analogous to a compass that will draw an accurate straight line. Discussions about straightness in class had a different perspective after we learned ourselves and then presented to our students the following not widely known history of the problem. The students responded that it was important to learn this connection of a pure geometrical problem with a mechanical problem. It also shows that you can find history of mathematical problems in history of other sciences, in this case mechanics, which leads us into the Motion in Machines Strand.

Figure 3. Up-and-down sawmill of the 13^{th}
century

After Robert Willis, trans. and ed., *Facsimile of the Sketch-Book of Wilars
de Honecort* (London, 1859, pl.43)

Turning circular motion into straight line motion has been
a practical engineering problem since at least the 13^{th} century. As
we can see in some 13^{th} century drawings of a sawmill (see Figure 3)
four bar linkage (rigid bars constrained to be near a plane and joined at their
ends by rivets) was in use and probably was originated much earlier. In 1588,
Agostino Ramelli published his book [55] on machines where linkages were widely
used. In the late 18^{th} century people started turning to steam
engines for power. James Watt (1736-1819), a highly gifted designer of
machines, worked on improving the efficiency and power of steam engines. In a
steam engine the steam pressure pushes a piston down a straight cylinder.
Watt’s problem was how to turn this linear motion into the circular motion of a
wheel (such as on steam locomotives).

It took Watt several years to design the straight-line
linkage that would change straight-line motion to circular one. Years later Watt told his son:

Though I am not over anxious after fame, yet I am more proud of the parallel motion than of any other mechanical invention I have ever made. (quoted in [18, pp 197-198])

"Parallel motion" is a name Watt used for his
linkage, which was included in an extensive patent of 1784. Watt's linkage was
a good solution to the practical engineering problem. See Figure 4 where the
linkage is the parallelogram *murt* connecting the piston *P* to the
beam *B*.

Figure 4. A steam engine with Watt’s “parallel motion” linkage

But Watt’s solution did not satisfied mathematicians who knew that it can draw only an approximate straight line. In 1853, Pierre-Frederic Sarrus (1798-1861), a French professor of mathematics at Strassbourg, devised an accordion-like spatial linkage that traced exact straight lines (Figure 5).

Figure 5. Sarrus
spatial linkage for drawing a straight line

(photo by Francis C. Moon, courtesy of the Cambridge University Engineering
Department)

Mathematics continued to look for a planar straight-line linkage. An exact straight-line linkage in a plane was not known until 1864-1871 when a French army officer, Charles Nicolas Peaucellier (1832-1913), and a Russian graduate student, Lipmann I. Lipkin (1851-1875), independently developed a linkage that draws an exact straight line. See Figure 6. (There is not much known about Lipkin. Some sources mentioned that he was born in Lithuania and was Chebyshev's student but died before completing his doctoral dissertation.)

Figure 6.
Peaucellier-Lipkin linkage from Cornell University F. Reuleaux kinematic
model collection

(photo by Francis C. Moon)

The drawing in Figure 7 depicts the working parts of the
Peaucellier-Lipkin linkage. The linkage works because the point *P* is inverted to the point *Q* through a circle with center at *C* and a radius squared equal to *s*^{2}
- *d*^{2}. The point *P* is constrained to move on a circle
that has center at *D* and that passes
through *C *and thus *Q* must move along a straight line. If
the distance between *C* and *D* is changed to *g* (not equal to *f*), then *Q* instead of moving along a straight
line will move along an arc of a circle of radius (*s*^{2} - *d*^{2})*f* */
*(*g*^{2} - *f *^{2}). This allows one to draw
an arc of a large circle without using its center. Note that the definition of
a straight line used here is: “A
straight line is a circle of infinite radius.” For a learning module on these
topics, see [44].

Figure 7. Working parts of the Peaucellier-Lipkin linkage

In the late 19^{th} century Franz Reuleaux
(1829-1905), an engineer with a strong mathematical background, collected
Watt’s and 38 other linkages that had been constructed to turn circular motion
into straight-line motion. These linkages were a part of a larger collection of
about 800 models of mechanisms that he considered to be a basis of a theory of
machines. (For Reuleaux, “mechanisms” are the simple component parts that are
put together to form a machine that does useful work.) The first president of
Cornell University A. D. White purchased in 1882 a large number of these models
and there are today more than 220 remaining. A picture of this new collection
was on a cover of *Scientific American*
in 1885. See Figure 8. This Reuleaux model collection at Cornell University is
now being digitized and put on the web as part of the National Digital Science
Library complete with viewer controlled motion, simulations, learning modules,
and amazingly the possibility for printing out 3-D working models of many of
the mechanisms. See [50]. It is viewable at [44]. Among mathematicians the most
popular of these mechanisms is the Peaucellier-Lipkin linkage. A detailed
history of the Peaucellier-Lipkin linkage can be found at [44].

Fugure 8. Reuleaux
kinematic model collection in Cornell University Sibley School of Mechanical
Engineering

(reproduced from *Scientific American*, 1885 cover, Cornell University
Library)

We add here a few remarks that excited out students. In January 1874, James Joseph Sylvester (1814-1897) delivered a lecture "Recent Discoveries in Mechanical Conversion of Motion." Sylvester's aim was to bring the Peaucellier-Lipkin linkage to the notice of the English-speaking world. Sylvester learned about this problem from Chebyshev - during a visit of the Russian to England. He observed:

The perfect parallel motion of Peaucellier looks so simple, and moves so easily that people who see it at work almost universally express astonishment that it waited so long to be discovered. (quoted in [18, p 206])

Later Mr. Prim, "engineer to the Houses" (the Houses of Parliament in London) was pleased to show his adaptation of Peaucellier linkage in his new "blowing engines" for the ventilation and filtration of the Houses. Those engines proved to be exceptionally quiet in their operation. See [42]. Sylvester recalled his experience with a little mechanical model of the Peaucellier linkage at a dinner meeting of the Philosophical Club of the Royal Society. The Peaucellier model had been greeted by the members with lively expressions of admiration,

when it was brought in with the dessert, to be seen by them after dinner, as is the laudable custom among members of that eminent body in making known to each other the latest scientific novelties. (quoted in [18, p 207])

And Sylvester would never forget the reaction of his brilliant friend Sir William Thomson (later Lord Kelvin) upon being handed the same model in the Athenaeum Club. After Sir William had operated it for a time, Sylvester reached for the model, but he was rebuffed by the exclamation:

No! I have not had nearly enough of it -- it is the most beautiful thing I have ever seen in my life. (quoted in [18, p 207])

In summer of 1876 Alfred Bray Kempe (1849-1922), a barrister who pursued mathematics as a hobby, delivered at London's South Kensington Museum (now Science Museum) a lecture with the provocative title "How to Draw a Straight Line" which in the next year was published in a small book [42]. In this book, which is viewable on the web as part of Cornell University library online book collection, you can find pictures of the linkages we have mentioned here. Kempe essentially knew that linkages are capable of drawing any algebraic curve. Other authors provided more complete proofs of this fact during the period 1877-1902. The Peaucellier-Lipkin linkage is also used in computer science to prove theorems about workspace topology in robotics [36].

Using history in a mathematics class shows the human side of mathematics and explores connections within mathematics and with other sciences. There are questions, which students and mathematicians have, that can only be clarified by reference to an appropriate history.

[1]
Keith Albarn, Jenny Mial Smith, Stanford Steele, and Dinah Walker, *The
Language of Pattern*, Harper & Row, New York, 1974.

[2]
L. Bagrow, *A History of Carthography*, Harward University Press,
Cambridge, 1964.

[3]
George Bain, *Celtic Arts: The Methods of Construction*, Constable,
London, 1977.

[4]
David E. Blair, *Inversion Theory and Conformal Mapping*, Amerian
Mathematics Society, Providence, RI, 2000.

[5]
William Blackwell, *Geometry in Architecture*, John Wiley &
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[6] Karol Borsuk and Wanda Szmielew, *Foundations of geometry: Euclidean and Bolyai-Lobachevskian geometry.
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[7]
Abraham Bosse, *Traite des
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[8]
Walter Burkert, *Lore and Science in Ancient Pythagoreanism*,
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[9] J.L. Coolidge, *A treatise
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[10]
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[11]
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[12]
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[13]
Keith Devlin, *Mathematics: The Science of Patterns*, Scientific
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George B. Dyson, *Darwin among the Machines: The Evolution of Global
Intelligence*, Perseus Books, Reading, MA, 1997.

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[16]
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Rutgers University Press, New Brunswick, 1999.

[17] Euclid, *Phaenomena*,
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[18] Eugene S. Fergusson, “Kinematics of
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**Smithsonian Institute, Washington D.C., 1962, pp. 185-230.

[19]
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[20]
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[21]
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African Mathematical Union and Higher Pedagogical Institute's Faculty of
Science, Mozambique, 1991.

[23]
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New York, 1977.

[24]
Livia Giacardi, “Scientific Research and Teaching Problems in Beltrami’s
Letters to Houel” in Victor Katz (editor), *Using History to Teach
Mathematics: An International Perspective*, MAA Notes #51, MAA, Washington,
DC, 2000.

[25]
Ernst Gombrich, *The Sense of Order: A study in the Psychology of
Decorative Art*, Cornell University Press, Ithaca, NY, 1978.

[26]
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Shelter Publications, Bolinas, 1994.

[27] Robin Hartshorne, *Geometry: Euclid and Beyond*, Springer, New York, 2000.

[28] T.L. Heath, *Euclid: The Thirteen Books of
The Elements*, Vol. 1, Dover Publications, New York, 1956.

[29] T.L. Heath, *Mathematics in Aristotle*, Clarendon Press, Oxford, 1949.

[30] Torkil Heiede, “The History of Non-Euclidean
Geometry” in Victor Katz (editor), *Using
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MAA, Washington, DC, 2000.

[31] David W. Henderson (with Daina Taimina), *Experiencing Geometry in Euclidean,
Spherical, and Hyperbolic Spaces*,** **Prentice Hall

[32] David W. Henderson and Daina Taimina,
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[33] David Hilbert, *Foundations of geometry* (*Grundlagen
der Geometrie*), translated by Leo Unger, Open Court, La Salle, IL, 1971.

[34] David Hilbert, “Über Flächen von konstanter
gausscher Krümmung,”* Transactions of the
A.M.S*., 1901, pp. 87-99.

[35] Hilbert, David, and S. Cohn-Vossen, *Geometry
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[36] J. Hopkroft, D. Joseph, and S. Whitesides,
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[37] A.J. Hoffman, “On the foundations of
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[38] William Ivins, Jr., *Art & Geometry: A
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[39]
Jay Kappraf, *Connections: The Geometric Bridge between Art and
Science*, McGraw-Hill, New York, 1991.

[40]
Victor J. Katz, *A History of Mathematics: An Introduction*, 2nd
Edition, Addison-Wesley Longman, Reading, MA, 1998.

[41]
Victor J. Katz (editor), *Using History to Teach Mathematics: An
International Perspective*, MAA Notes # 51, Mathematical Association of
America, Washington, DC, 2000.

[42] A. B. Kempe, *How to Draw a Straight Line, *Macmillan and Co., London, 1877.

[43] Morris Kline, *Mathematical Thought from Ancient to Modern Times*, Oxford
University Press, Oxford, 1972.

[44] <http://kmoddl.library.cornell.edu>, *Reuleaux
Kinematic Model Collection*, part of National Digital Science Library
<www.nsdl.org>.

[45] Louis Kollros, *Jacob Steiner*,
Elemente der Mathematik, Beihefte Nr. 2, Verlag Birkhauser, Basel, Dezember 1947.

[46] Nicolas Kuiper, “On C^{1}-isometric
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[47] George E. Martin, *Geometric Constructions*,
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[48] Elena Anne Marchisotto and James T. Smith, *The Legacy of Mario Pieri in Arithmetic and
Geometry*, Birkhauser,
Boston, 2003.

[49] John McCleary, *Geometry from a Differential Viewpoint*, Cambridge University Press,
Cambridge, UK, 1994.

[50] Francis C. Moon, "Franz Reuleaux:
Contributions to 19th Century Kinematics and the Theory of Machines", *Applied Mechanics Reviews*, vol 56, no.
2, March 2003, pp.1-25.

[51] Robert Osserman, *Poetry of the Universe:
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[52]
John Pennethorne, *The Geometry and Optics of Ancient Architecture*,
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[53]
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[54] V.V. Prasolov and V.M. Tikhomirov, *Geometry*, Translations of Mathematical
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[56]
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[57]
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[58] William Thurston, *Three-Dimensional Geometry and Topology*, Vol. 1, Princeton
University Press, Princeton, NJ, 1997.

[59] Isaac Todhunter, *Spherical Trigonometry*, Macmillan, London, 1886.

{Available online from the Cornell University Mathematics Library at:
http://cdl.library.cornell.edu/Hunter/hunter.pl?handle=cornell.library.math/00640001&id=5}

[60] Imre Toth, “Non-Euclidean Geometry before
Euclid”, *Scientific American*, 1969.

[61] Jeffrey Weeks, *The Shape of Space*,
Second Edition, Marcel Dekker, New York, 2002.

[62] Trevor Williams, *A History of Invention:
From Stone Axes to Silicon Chips*, Revised Edition, Checkmark Books, New
York, 2000.

[1] Partially supported by National Science Foundation's National Science, Technology, Engineering, and Mathematics Education Digital Library (NSDL) Program under grant DUE-0226238.