Janos Bolyai's Non-Euclidean Geometry: Trigonometric Formulae & F-Surface

9. Janos Bolyai
To make use of the F-surface and its Euclidean geometry, Bolyai compared
triangles in non-Euclidean geometry with those on the F-surface. He moved
between them by considering a non-Euclidean triangle, ABC, right-angled at B,
and the lines AM, BN, and GP, which are perpendicular to the sides of triangle
ABC and parallel to AM (see Figure 9.6).
In a move that took him even further from Euclid's way of thinking about
geometry, he introduced trigonometric formulae. He considered the surface
consisting of all lines parallel to BN and GP through points on BC, and showed
that it is perpendicular to the similar surface consisting of all lines parallel to BN
and AM through points on AB.
He then took the F-surface through C, perpendicular to GP, and supposed that
BN meets it at D and AM meets it at E. He deduced that the angle between the
L-curves CD and DE at D is a right angle. It is also clear that the angle between
the L-curves DC and EC at C is equal to the angle between the non-Euclidean
lines BC and AC at C.
Moreover, rotating the figure about the axis GP shows that the ratio of the
circumferences of circles with radii CD and CE on the F-surface is equal to the
ratio of the circumferences of circles with radii CB and CA in the non-Euclidean
plane. This means that the elementary trigonometric formulae connecting the
sides and angles of the F-triangle CDE can be transformed into results
concerning the sides and angles of the corresponding non-Euclidean triangle.
Bolyai next considered the formulae relating the sides and angles of a triangle
on a Euclidean sphere, which in his day were taught to every land surveyor and
military engineer. He showed, strikingly, that these too are absolute theorems.
This meant that he could also use spherical trigonometry in the new setting.
9.2 Janos Bolyai's Geometry
To find the appropriate trigonometric formulae in a geometry where the parallel
postulate is false, Bolyai began with the fundamental figure of such a geometry.
This consists of a straight line AB of length y, meeting a line AM at right angles,
and a line BN at an acute angle u, where the lines AM and BN are parallel. The
length y determines the angle u, and vice versa. The angle u is called the angle
of parallelism corresponding to the length y and is sometimes written as u =
Π(y). Bolyai then sought, in [19, §29], an expression for the length y as a
function of the angle u, as shown in Figure 9.7.
His argument was long and difficult. He first considered an arbitrary point C on
the line a and the L-curve passing through it, which he called L' (see Figure
9.8).
He showed that the ratio AB:CD is independent of AB and depends only on the
length AC = x. He observed that if AB is doubled, so is CD; more generally, if AE
is increased by a factor of k, then so is CD. He denoted this ratio by X and set
himself the task of evaluating it.
His answer is a formula relating u and Y, where Y is the same function of y that
X is of x, and u is the angle of parallelism corresponding to the length y. He
showed, by a simple scaling argument, that given lengths x and x', the
corresponding values X and X' satisfy Y'/Y = X'/X. From this it follows, although
Bolyai did not state it explicitly, that X = e^(kx) for some arbitrary constant k.
To determine X, he considered the curve that is everywhere equidistant from a
straight line. In Euclidean geometry, this is another straight line, but if the
parallel postulate is false, it is not.
Bolyai imagined that the curve was traced out by the tip of a line segment
moving perpendicular to the given line. In Figure 9.10, the triangle AEC is
assumed to slide rigidly along the line a, with the edge AC always remaining
perpendicular to a. The point C traces the equidistant curve c to the line a. The
segment ED represents another position of the segment AC, so the point D lies
on the curve c. The angles u and v are defined as follows: u = ∠CAD and v =
∠ADE.
Bolyai showed that the ratio of the length of the segment AE to the length of the
segment CD on the curve c is equal to sin(v)/sin(u). He established this by
dropping the perpendicular DE from D to AC and applying his trigonometric
formulae to the relevant triangles.
He then used a limiting argument to determine the ratio of the lengths AE and
CD. He observed that, since the ratio AE/CD is constant and depends only on
the height AC rather than on the width AE, it can be evaluated in the limit as AC
moves infinitely far away, when the angle u tends to a right angle and the angle
v tends to the angle β.
However, he also showed that, in Figure 9.10, where x = AC, the ratio X
introduced earlier is equal to sin(u)/sin(v). From this, he deduced the result he
sought: the angle of parallelism is given by the formula Y = cot(u/2). Using the
relation Y = e^(ky), this can be rewritten as sinh(y) = cot(u), or equivalently,
sinh(y) = cot Π(y).
This formula allowed Bolyai to derive a number of useful results rapidly. He
obtained a formula for the circumference of a circle of radius r and then
established the formulae connecting the sides and angles of triangles when the
parallel postulate is false.
These are precisely the formulae that contain an arbitrary constant and that had
concerned his father. Bolyai observed that, when the parallel postulate is false,
any three pieces of information about a triangle are sufficient to determine all
the others. He provided all the formulae for the simpler case of a right triangle;
the formulae for the general case follow with only minor additional work.
This differs from the Euclidean case, where knowing the angles of a triangle
determines its shape but not its size. Of course, one may argue that only two
pieces of information are required to determine the angles of a Euclidean
triangle, since their sum must equal two right angles.
Bolyai also showed that the new formulae reduce to their Euclidean equivalents
when the arbitrary constant tends to infinity. In particular, the non-Euclidean
analogue of Pythagoras's theorem reduces to the familiar Euclidean theorem in
this limit.
In all these formulae, non-Euclidean lengths always appear divided by the
arbitrary constant k. Therefore, an approximation that holds when x/k is very
small may be interpreted as holding either when k is very large or when x is
very small.
Consequently, to a very high degree of approximation, the Euclidean version of
Pythagoras's theorem remains valid for small triangles in non-Euclidean
geometry, demonstrating that small regions of non-Euclidean space are
approximately Euclidean.
Bolyai went on to show that, in the new geometry, the area of a triangle in the
plane—not on the special surface F—is equal to its angular defect. He also
demonstrated that, if the parallel postulate is false, it is possible to construct a
square equal in area to a given circle.
Thus, Bolyai squared the circle.
Squaring the circle had long been synonymous with achieving the impossible
and had been regarded as such since at least 450 BC, when Aristophanes
referred to it in The Birds. Bolyai must therefore have been excited to realise
that, within his new geometry, the construction was indeed possible.
9.3 Janos Bolyai's Section 32
In Section 32 of the Appendix [19], Bolyai made his strongest claim that
anything achievable in Euclidean geometry could also be achieved in the new
geometry. In doing so, he effectively challenged anyone to find a flaw in it.
He presented a method for solving problems in the new geometry, stating that,
"having accomplished this through the more obvious examples, it will finally be
candidly shown what this theory demonstrates." Most commentators appear to
have underestimated the significance of this remark.
Bolyai and his readers were familiar with maps in an atlas. When the Earth's
surface is represented on a flat Euclidean plane, formulae are needed to
determine the area of a region on the sphere in terms of latitude and longitude,
as well as the length of curves on the sphere.
A glance at Mercator's projection reveals that these formulae cannot be as
simple as those of ordinary plane geometry. Equal increments of y on the map
clearly correspond, as one moves northward, to progressively smaller distances
on the Earth's surface. Although the formulae differ, the underlying idea remains
the same.
Bolyai seems to have assumed that his readers would recognise these
arguments and appreciate their application within the altered framework of his
new geometry.
He drew a figure of a curve ABG in the familiar Cartesian plane with x- and yaxes (see Figure 9.11) and explained how it should be interpreted as a
representation of non-Euclidean geometry within a Euclidean plane.
The x-axis represents a non-Euclidean straight line. Perpendiculars to the x-axis
through points C and F intersect the curve at points B and G, respectively.
Through point B, he drew the ray BH parallel to the x-axis, forming equal angles
FCB and HBC.
What Bolyai did not explicitly state is that the perpendiculars CB and FG must
be L-lines. In that case, he should also have explained how they can be
represented as Euclidean straight lines.
Bolyai then considered what happens to the triangle BGH as the distance CF
becomes very small. The arc length BG along the curve becomes
correspondingly small and is increasingly well approximated by the nonEuclidean length of the line segment BG.
Knowledge of infinitesimal lengths would then allow the determination of
arbitrary finite lengths along the curve by means of integration.
Bolyai was confident that the trigonometric formulae relating the sides of the
very small triangle BGH could be derived from the formulae for an arbitrary finite
non-Euclidean triangle, which he had established in the preceding section.
Although this claim is correct, it is far from elementary. Bolyai avoided the
pedagogical difficulties by simply stating, correctly, that "it can be demonstrated"
that the required approximation is:
BG² = dy² + BH²
He then used this formula to derive expressions for the areas and volumes of
certain non-Euclidean figures.
Here, Bolyai missed an important opportunity to begin again from first
principles. With a modest amount of additional work, he could have provided a
far more convincing account of non-Euclidean geometry.
He might have shown that the entire picture of two-dimensional non-Euclidean
geometry can be represented in the right half-plane, defined by x > 0; that
straight lines appear as curves of a particular form; that two lines are parallel
when, in a typical case, they meet on the y-axis; and that the trigonometric
formulae for triangles are precisely those he had derived earlier.
In such representations, as in conventional cartography, very small distances in
the ordinary (x, y)-plane are measured according to the new metric, while
arbitrary finite distances are obtained by integration.
As a consequence, equal non-Euclidean distances appear to contract as one
approaches the y-axis—the exact opposite of what occurs in Mercator's
projection when moving away from the equator.
Most likely, however, the mathematical community would still not have been
convinced. This interpretation effectively anticipates Riemann's later ideas.
Nevertheless, the fact that Bolyai came so close to formulating the foundations
of his new geometry in terms of calculus, thereby making it possible to address
problems of differential geometry within the new setting just as effectively as in
the familiar Euclidean one, testifies to his remarkable insight.
This achievement seems not to have been fully appreciated either in his own
time or since.
10. Lobachevskii
10.1 Lobachevskii and Kazan
The new mathematical world of a geometry distinct from Euclid's was
discovered independently by Nikolai Ivanovich Lobachevskii, born in Kazan,
Russia, in 1792, and by his Hungarian contemporary János Bolyai (1802–
1860).
Although their work was little understood during their lifetimes, it eventually
helped overturn almost every established belief about the mathematics of space
and opened the way to the modern geometries of Hilbert and Einstein.
After his death, Lobachevskii became known as the "Copernicus of Geometry."
On the occasion of his centenary, he was celebrated by Malevich and the
Russian Futurists, and although his bicentenary passed more quietly, it
remained an event worth commemorating.
Lobachevskii came from a poor background, but his mother managed to secure
him a scholarship to the local Gymnasium in 1800. In 1805, the Gymnasium
became the nucleus of the newly founded University of Kazan, and in 1807
Lobachevskii began his studies there, following in the footsteps of his brother
Alexei, who later became a successful chemist.
Lobachevskii was fortunate because the university, unable to appoint a Russian
mathematician of sufficient distinction, had recruited Martin Bartels as Professor
of Mathematics.
Bartels had taught Gauss during the latter's youth and therefore knew
exceptional talent when he encountered it. By the time Bartels arrived in Kazan
in 1807, Gauss had already become both the leading pure mathematician and
the foremost astronomer of his age.
As contemporary humour suggested, if Gauss was the greatest mathematician
in the world, then Bartels was the greatest mathematician in Germany. By
extension, he was certainly the leading mathematician in Russia.
Bartels not only provided Lobachevskii with an excellent education but also
protected him from the consequences of his youthful indiscretions, as the young
student frequently found himself in trouble with the authorities.
On several occasions, he lost holiday privileges and was even suspected of
atheism. When he was due to graduate, the university acknowledged that his
academic work merited a degree, but considered his conduct too poor to
overlook and initially refused to award it.
Bartels lobbied the other professors on his behalf, and three days later
Lobachevskii received not the ordinary degree but a Master's qualification. His
career as a professional mathematician could then begin.
However, the university was poorly administered, and the governing body soon
split into competing factions centred around the director. Each group began
leaking information about the misconduct of its rivals to the Ministry of
Education in St Petersburg until the scandal became impossible to ignore.
The man sent to restore order was Magnitsky. He conducted a rapid inspection
of the university and the local schools, supported by professors who saw him
either as the much-needed reformer or as an opportunity to advance their own
interests.
Within a month, he had gathered all the information he considered necessary
and reported to the Minister that, although the professors were generally
capable, the university was underfunded—particularly the medical faculty—and
education throughout the region was in a deplorable state.
The province, he argued, lagged almost half a century behind the rest of
Europe, which, he added, was certainly not what the Emperor desired. The
Ministry acted accordingly and granted Magnitsky the authority he sought. He
became director of the university, with responsibility for education throughout
the Kazan region.
Once in office, he ruled with a single-minded sense of purpose that his earlier
reports had concealed. The principal aim of education, he insisted, was to instil
knowledge of the true faith, together with humility and obedience.
Scientific prizes were abolished and replaced by awards exclusively for
theology. Students who violated the rules were labelled sinners and confined in
isolation, where they were expected to contemplate images of the Last
Judgement.
Every university appointment and every school teacher was scrutinised not only
for professional competence but also for moral character. Non-Russians,
particularly Germans, were regarded as dangerous free thinkers and were
either dismissed or chose to leave.
Past records were examined, and reports were sent to St Petersburg on
everyone. Lobachevskii, undoubtedly aware of what his own records might
reveal, participated in this process, signing reports for both the Education
Committee on which he served and the University Council.
He championed the reform of the university library, which was in urgent need of
improvement, but encountered continual resistance from the librarian and
eventually resigned from the position.
At the same time, he carried a heavy teaching load, not only in mathematics but
also in theoretical physics and astronomy. As required, he submitted monthly
reports on the intellectual and moral development of his students.
He managed to avoid involvement in the effort to demonstrate how mathematics
contributed to revealing religious truths. However, when he prepared his
geometry lectures for publication, Magnitsky sent the manuscript to the
academician Fuss in St Petersburg for review.
Fuss decided that it should not be published. He objected to the use of the
metre as a unit of measurement and to the division of the quarter circle into one
hundred degrees—a system, he argued, associated with the French Revolution,
when "the desire to destroy extended even to the calendars and the division of
the circle."
Lobachevskii requested the return of his manuscript. It was rediscovered in
1898 and eventually published in 1909.
In this work, he observed that Euclidean geometry had not yet been rigorously
established, but he did not suggest the existence of an alternative geometry.
Magnitsky's rule lasted seven years, ending in 1826. Financial corruption was
once again damaging the university's reputation, but his downfall was ultimately
caused by an imprudent letter to Emperor Alexander, in which he accused the
Grand Dukes, including Nicholas, of liberal tendencies.
Alexander's unexpected death brought Nicholas to the throne. Although
Nicholas was no supporter of educational freedom, he considered the
accusation unforgivable.
Within days, Magnitsky was exiled and his appointed rector dismissed.
Three days later, Lobachevskii delivered a public lecture entitled "On the
Principles of Geometry, with a Rigorous Demonstration of the Theory of
Parallels."
The manuscript of this lecture has been lost, but later references suggest that it
marks the beginning of Lobachevskii's awareness of non-Euclidean geometry.
The new Director of Education for the Kazan region restored elections for the
university rectorship, and Lobachevskii was elected rector.
Under his leadership, the neglected library was finally reorganised properly.
New professors were appointed, and new buildings were constructed. During
his tenure, both an observatory and an anatomical theatre were established.
In 1830, a cholera epidemic swept across the country. When it reached Kazan,
Lobachevskii sealed off the university, allowing only medical personnel to enter
and leave.
Strict measures—though of uncertain effectiveness—were introduced to
prevent the disease from entering the university through food and drink
supplies. All incoming mail was fumigated, washed in a chlorine solution, and
stored until the epidemic had passed.
Consistent with the prevailing miasmatic theory of disease transmission, the
university walls were surrounded by vessels of boiling vinegar.
The isolation measures proved successful. While the epidemic caused
widespread devastation outside the university, only forty cases and sixteen
deaths occurred within its walls.
Even the Emperor wrote to express his appreciation for Lobachevskii's efforts.
The 1830s represented the high point of Lobachevskii's life. Despite his
demanding teaching responsibilities and the heavy administrative burden of
serving as rector, he found the time and energy to publish his original work for
the first time.
What he had discovered—perhaps as early as 1826 and certainly by 1829,
when he published an article in the Kazan Messenger—was an alternative to
Euclidean geometry.
In this new geometry, the theorems differ from, and often contradict, those of
Euclidean geometry, while the underlying definitions remain equally plausible.
As a result, empirical evidence becomes necessary to determine which
geometry more accurately describes physical space.
It requires little imagination to understand how such investigations would have
been received under Magnitsky's administration. Unfortunately, no surviving
documents reveal how far Lobachevskii had progressed before the liberal
reforms of 1826.
10.2 Lobachevskii's New Geometry
How can another geometry exist, and how did Lobachevskii come to discover
it?
Ironically, during his years as rector, Lobachevskii was deeply involved in the
university's ambitious construction programme, supervising even the smallest
details.
Yet he firmly believed that the foundations of Euclidean geometry were flawed.
In his view, concepts such as "line", "surface", and "position" were highly
abstract and obscure. What we truly know, he argued on several occasions, are
physical bodies and their motions.
It is through reflection on these more concrete ideas that the concepts of
geometry should be constructed.
In this respect, Lobachevskii echoed the ideas of d'Alembert as expressed in
the Encyclopédie, whose philosophy of geometry was particularly well suited to
the application of geometry and calculus to Newtonian mechanics.
According to Lobachevskii, our understanding of straight lines arises from the
study of physical bodies, and careful analysis shows that such lines need not
possess the properties attributed to them by Euclid.
Lobachevskii offered his readers the example of geometry on the surface of a
sphere. In this setting, the shortest path between two points is the arc of a great
circle—that is, the intersection of the sphere with a plane passing through its
centre.
This geometry was already familiar to navigators, and Lobachevskii proposed
placing it on an equal footing with Euclidean geometry before introducing a third
alternative: his own discovery. Together, these three geometries formed what he
regarded as an elegant trinity.
In Euclidean geometry, it is a fundamental theorem that the angles of any
triangle sum to two right angles. As noted earlier, this result follows from the
parallel postulate, which states, in one of its equivalent forms, that given any
line and any point P not lying on that line, there exists exactly one line through P
that does not intersect the given line.
On the sphere, the parallel postulate is false because any two great circles
intersect. Consequently, the sum of the angles of a spherical triangle is greater
than two right angles.
In Lobachevskii's geometry, the parallel postulate is also false, but in a different
way.
Given a line and a point P not lying on it, there are infinitely many lines through
P that do not intersect the given line. However, once a direction is specified,
there is a unique limiting line in each direction.
Lobachevskii could have defined all non-intersecting lines as parallel, but this
would have created an undesirable consequence: if line a were parallel to line
b, and line b parallel to line c, it would not necessarily follow that line a was
parallel to line c.
For this reason, he distinguished three classes of lines with respect to a given
line: those that intersect it, those that never meet it, and the two limiting lines—
one in each direction—which he designated as parallels.
Unfortunately for Lobachevskii, his abilities as a lecturer did not translate
equally well into his writing.
His papers often failed to persuade readers. In 1837, he attempted to reach a
broader European audience by publishing in French in Crelle's Journal (Journal
für die reine und angewandte Mathematik).
However, his article was heavily dominated by formulae, and much of their
geometric significance remained obscure because he assumed that readers
were already familiar with his earlier publications in Russian.
When he presented his ideas in a self-contained booklet published in Berlin in
1840, his argument appeared to take the following form: assume that lines
behave in this way, and these are the theorems that result.
Critics could still object that the resulting theorems might eventually prove selfcontradictory and that the new geometry could therefore be illusory.
Indeed, this had long been the traditional defence of Euclidean geometry.
Several prominent geometers had even claimed to derive contradictions from
assumptions similar to those proposed by Lobachevskii.
Lobachevskii himself began his most comprehensive defence of his discoveries
by refuting some of the most famous of these flawed arguments.
What gave him confidence that his geometry was free from contradiction was a
combination of two factors.
The first was his philosophical starting point: the analysis of the motion of
physical bodies, an idea that remained largely hidden from his European
readers.
The second was his mathematical conclusion: he expressed his results through
trigonometry and calculus.
Lobachevskii deliberately sought formulae because he believed that geometry
was fundamentally concerned with measurement, and that measurements—
being numerical quantities—must be related by mathematical expressions.
The validity of these formulae depended on algebra, regardless of their
geometric interpretation.
However, he confined these reflections largely to his Russian-language
publications.
Consequently, his reasoning contained a logical weakness: he inferred that his
new geometry was meaningful because it yielded coherent trigonometric
formulae.
Furthermore, these formulae approached those of Euclidean geometry when
the parameter they contained became arbitrarily large. In that limit, the two
geometries became indistinguishable.
It can also be shown that if this parameter is taken to be purely imaginary, the
formulae of spherical trigonometry emerge.
In many respects, Lobachevskii's work closely resembles that of János Bolyai.
Both developed geometries in three dimensions. Both expressed their central
results through new trigonometric formulae.
Lobachevskii was clearer in certain respects and more concerned with
discovering theorems unique to the new geometry than with identifying results
shared by both Euclidean and non-Euclidean geometries.
He also adopted the sensible practice of naming important features of the new
geometry—which he referred to as "imaginary geometry"—whereas Bolyai
generally relied on letters, a habit that Gauss criticised.
Lobachevskii called the curve perpendicular to a family of parallel lines a
horocycle and the corresponding surface a horosphere.
Like Bolyai, he demonstrated that the geometry on a horosphere is Euclidean.
10.2.1 Astronomical Evidence
Lobachevskii showed that, according to his new definition of parallels, the sum
of the angles of a triangle is always less than two right angles.
Moreover, the angle sum decreases as the triangle becomes larger.
This observation suggested to him that it might be possible to determine
whether physical space is non-Euclidean through astronomical measurements.
His idea was to study stellar parallax.
However, stellar parallax is extremely small, and in 1829 Lobachevskii could
conclude only that Euclidean geometry was sufficiently accurate for all practical
purposes.
In his final work, Pangeometrie, he proposed a different argument based on two
measurements of an arbitrarily distant star, from which the arbitrary constant
appearing in the formula for the angle of parallelism could, in principle, be
determined.
This argument relies implicitly on the fact that, in Euclidean geometry, parallax
decreases as distance increases and may become arbitrarily small for
sufficiently distant stars.
In non-Euclidean geometry, however, parallax cannot decrease below a certain
minimum value determined by the diameter of the Earth's orbit.
Therefore, if observations consistently revealed a non-zero lower limit for stellar
parallax, this would constitute evidence in favour of non-Euclidean geometry.
10.3 From Lobachevskii's Geometrische Untersuchungen (1840)
10.3.1 Opening Remarks
"In geometry, I find certain imperfections that I consider responsible for the fact
that this science, apart from its transition into analysis, has made no significant
progress beyond the state in which it came down to us from Euclid.
Among these imperfections, I include the obscurity surrounding the fundamental
concepts of geometrical magnitudes, the methods used to measure them, and,
above all, the significant gap in the theory of parallels—a gap that the efforts of
mathematicians have so far failed to fill.
Legendre's attempts contributed nothing to this theory because he abandoned
the only rigorous path, turning instead to a side route and relying on auxiliary
propositions that he sought, illogically, to elevate to the status of necessary
axioms.
I published my first essay on the foundations of geometry in the Kazan
Messenger in 1829.
Believing that I had satisfied all essential requirements, I subsequently
undertook a comprehensive treatment of the subject and published it in
separate parts in the Scientific Proceedings of the University of Kazan between
1836 and 1838 under the title New Elements of Geometry, with a Complete
Theory of Parallels.
The length of this work may have discouraged my countrymen from following a
subject that, since Legendre, had lost much of its appeal.
Nevertheless, I remain convinced that the Theory of Parallels deserves the
attention of geometers, and I therefore present here the substance of my
investigations.
I note in advance that, contrary to Legendre's opinion, all other imperfections—
for example, the definition of a straight line—prove irrelevant and exert no real
influence on the theory of parallels."
All straight lines that originate from a point in a plane can, with respect to a
given straight line in the same plane, be divided into two classes: those that
intersect the given line and those that do not.
The boundary lines separating these two classes are called parallels to the
given line.
Let AD be the perpendicular dropped from a point A to the line BC, and let AE
be another line drawn through A perpendicular to AD.
Within the right angle EAD, either every line passing through A intersects the
line BC, as does AF, or some lines, such as AE, do not intersect BC.
Since it is uncertain whether AE is the only line that fails to intersect BC, we
shall assume that other such lines may exist—for example, AG—which do not
meet BC, no matter how far they are extended.
As we pass continuously from intersecting lines, such as AF, to non-intersecting
lines, such as AG, we encounter a limiting line AH, which is parallel to BC.
This boundary line separates the two classes: on one side lie all lines that do
not intersect BC, while on the other side lie all lines that do.
The angle HAD, formed by the parallel AH and the perpendicular AD, is called
the angle of parallelism and is denoted by Π(p), where AD = p.
If Π(p) is a right angle, then the extension AE' of the perpendicular AE is
likewise parallel to the extension DB of the line BC.
Furthermore, considering the four right angles formed at A by the lines AE, AD,
and their extensions AE' and AD', every line passing through A—or at least its
extension—lies within one of the two right angles facing BC.
Consequently, except for the line EE', all such lines, if extended sufficiently in
both directions, must intersect BC.
If Π(p) is less than π/2, then on the opposite side of AD there exists another line
AK, making the same angle DAK = Π(p), which is parallel to the extension DB of
BC.
Under this assumption, it becomes necessary to distinguish between the two
directions of parallelism.
All remaining lines, or their extensions, within the two right angles facing BC
belong to the class of intersecting lines if they lie within the angle HAK = 2Π(p)
formed by the two parallels.
Conversely, they belong to the class of non-intersecting lines if they lie outside
the parallels AH and AK, within the angles EAH = π/2 − Π(p) and E'AK = π/2 −
Π(p), formed by the parallels and the perpendicular EE' to AD.
On the opposite side of the perpendicular EE', the extensions AH' and AK' of the
parallels AH and AK are likewise parallel to BC.
The remaining lines belong to the class of intersecting lines if they lie within the
angle K'AH', and to the class of non-intersecting lines if they lie within the
angles K'AE and H'AE'.
Accordingly, if Π(p) = π/2, lines can only be either intersecting or parallel.
However, if Π(p) < π/2, there exist two parallels—one on each side—and all
remaining lines must be distinguished as either intersecting or non-intersecting.
In both cases, the defining property of a parallel is that an arbitrarily small
deviation towards the side containing the parallel transforms the line into an
intersecting one.
Thus, if AH is parallel to BC, every line AF intersects BC, no matter how small
the angle HAF may be.
10.3.2 Concluding Remarks
The four equations expressing the relationships among the sides a, b, c and the
opposite angles A, B, C of a rectilinear triangle are as follows:
sin A · tan Π(a) = sin B · tan Π(b)
cos A = cos Π(b) · cos Π(c) + sin Π(b) · sin Π(c) · cos C
cot A · sin C · sin Π(b) + cos C = cos Π(c) / cos Π(b)
cos A + cos B · cos C = sin B · sin C · cot Π(a)
If the sides a, b, and c of the triangle are very small, we may use the following
approximations:
cot Π(a) = a
sin Π(a) = 1 − ½a²
cos Π(a) = a
and similarly for the sides b and c.
Under these approximations, the preceding equations reduce to:
b sin A = a sin B
a² = b² + c² − 2bc cos A
a sin(A + C) = b sin A
cos A + cos(B + C) = 0
From the first two equations, which correspond to the familiar laws of ordinary
geometry, the remaining two imply:
A+B+C=π
Thus, imaginary geometry reduces to ordinary geometry whenever the sides of
a rectilinear triangle are sufficiently small.
In the scientific publications of the University of Kazan, I have presented
investigations concerning the measurement of curved lines, plane figures,
surfaces, and the volumes of solids, as well as applications of imaginary
geometry to analysis.
The four equations given above provide sufficient grounds for regarding the
assumptions of imaginary geometry as possible.
Consequently, astronomical observation remains the only means of judging the
degree of accuracy associated with the calculations of ordinary geometry.
As I have shown elsewhere, this accuracy extends remarkably far. For example,
for triangles whose sides are accessible to measurement, the sum of the three
angles differs from two right angles by less than one hundredth of a second of
arc.
It is also worth noting that the four equations of plane geometry transform into
the equations of spherical trigonometry if we replace the sides a, b, and c by a/i,
b/i, and c/i, respectively.
Under this substitution, we must also write:
sin Π(a) = cos(a/i)
cos Π(a) = sin(a/i)
tan Π(a) = cot(a/i)
and similarly for the sides b and c.
In this way, the equations of imaginary geometry pass naturally into those of
spherical geometry.
11. Publication and Initial Reception up to 1855
11.1 Minding's Surface
In the 1830s, H. F. Minding discussed an interesting surface in a series of
papers published in Crelle's Journal. Although these papers were largely
forgotten for the next thirty years, they described a surface of constant negative
curvature with the property that geodesics between two points are not
necessarily unique.
Minding's surface is generated by rotating a tractrix about its vertical axis.
The tractrix—named by Huygens—is sometimes known as the curve of the
obstinate dog. A point Q moving along a straight line is connected to a point P
by a segment of fixed length. As Q, representing the owner, moves along the
line, P, representing the dog, is dragged behind it. The path traced by P is called
the tractrix.
Conventionally, the motion begins with the segment PQ perpendicular to the
line of motion.
Minding's surface, also known as the pseudosphere, is formed by rotating the
tractrix about the line of motion. In this process, the point P generates a circle of
singular points.
At every point, the surface locally resembles a saddle. Its curvature is negative
everywhere and, moreover, constant throughout.
This follows from the properties of the tractrix: as one moves upward along the
surface, the principal vertical curvature decreases while the principal horizontal
curvature increases in such a way that their product remains constant.
The geometry of Minding's surface is described by figures composed of
geodesic segments—that is, curves constrained to lie entirely on the surface
and representing the shortest paths between points.
The geometry of Minding's surface is described by figures composed of
geodesic segments—that is, curves constrained to lie entirely on the surface
and representing the shortest paths between points.
Minding was interested in surfaces that are, at least locally, metrically
equivalent, such as the plane and the cylinder.
He observed a property of surfaces with constant curvature that would later
prove fundamental: any figure drawn on such a surface—whether a plane, a
sphere, or a pseudosphere—can be moved freely across the surface without
changing its shape.
To understand this, imagine drawing a figure on a curved surface and
attempting to slide it elsewhere. Variable curvature distorts the figure as it
moves, whereas constant curvature preserves its dimensions and angles.
Thus, the only surfaces that admit the unrestricted movement of congruent
figures are those with constant curvature.
The tractrix and its axis closely resemble the image of asymptotic parallels
described by Saccheri and Gauss, and this resemblance is no coincidence.
Gauss himself was familiar with the surface. In an unpublished note written
between 1823 and 1827, he referred to it as the surface of revolution that is the
opposite of the sphere.
However, he does not appear to have connected it with non-Euclidean
geometry.
Minding also demonstrated that the pseudosphere is not the only surface of
revolution with constant negative curvature. He provided additional examples,
although the tractrix generates the simplest one.
11.2 The Bolyais Read Lobachevskii
The Bolyais first learned of Lobachevskii's work in 1844 through the Hungarian
mathematician and physicist Franz Mentovich, who had met Gauss in
Göttingen.
Unable to obtain copies of Lobachevskii's publications, Wolfgang Bolyai wrote
to Gauss in 1848 seeking advice.
Gauss replied that they would have greater success searching for the German
booklet Lobachevskii had published in 1840.
The Bolyais eventually obtained a copy on 17 October 1848.
After studying it, they identified numerous details they considered open to
criticism.
In some instances, their objections appear justified, particularly regarding the
proof of the exponential relationship between lengths and the angle of
parallelism.
Nevertheless, they recognised the fundamental importance of Lobachevskii's
achievement.
The similarities between the two approaches were striking. Both Lobachevskii
and János Bolyai had independently discovered a geometry in which Euclid's
parallel postulate was false, and both had developed trigonometric formulae to
describe it.
Yet there were significant differences in style and emphasis.
Bolyai devoted considerable effort to demonstrating that the new geometry
could reproduce the results of ordinary geometry wherever appropriate.
Lobachevskii, by contrast, focused more directly on the novel properties of the
geometry itself.
Although neither mathematician succeeded in convincing the wider
mathematical community during his lifetime, their work established the
foundations of non-Euclidean geometry and profoundly transformed the
understanding of space.
11.3 Why Non-Euclidean Geometry Was Not Immediately Accepted
The failure of mathematicians to embrace non-Euclidean geometry during the
first half of the nineteenth century is not difficult to understand.
For more than two thousand years, Euclidean geometry had been regarded not
merely as a successful mathematical theory but as a necessary description of
space itself.
The proposition that an entirely different geometry could be equally valid
seemed, to many, absurd.
Moreover, neither Bolyai nor Lobachevskii provided what later mathematicians
would regard as a definitive proof of consistency.
Their arguments relied heavily on analogy, trigonometric formulae, and the
absence of contradictions within their calculations.
While persuasive to some, these considerations fell short of establishing that
the new geometry was logically sound.
Another obstacle was the style in which their work was presented.
Lobachevskii's writings were often difficult to follow, and many appeared in
Russian journals that were inaccessible to most European mathematicians.
Bolyai's Appendix, although concise and elegant, was published as an appendix
to his father's textbook and therefore attracted limited attention.
Even Gauss, who privately expressed admiration for both men and
acknowledged the correctness of their ideas, refused to endorse them publicly.
His silence deprived the new geometry of the authority that only he could have
provided.
As a result, non-Euclidean geometry remained a marginal subject for many
years.
Its status changed only when mathematicians began constructing explicit
models within ordinary geometry that demonstrated the consistency of the new
theory, provided Euclidean geometry itself was consistent.
These developments, which emerged after 1855, transformed non-Euclidean
geometry from a speculative possibility into an accepted branch of
mathematics.
11.4 The Search for Models
The decisive breakthrough in the acceptance of non-Euclidean geometry came
when mathematicians shifted their attention from proving the parallel postulate
to constructing models of geometries in which it was false.
A model demonstrates that a set of axioms is free from contradiction by
interpreting its undefined terms in such a way that all the axioms become true
statements about familiar mathematical objects.
If a consistent model of non-Euclidean geometry could be constructed within
Euclidean geometry, then any contradiction in the new geometry would
necessarily imply a contradiction in Euclidean geometry itself.
This approach transformed the problem completely.
Instead of asking whether non-Euclidean geometry described physical space,
mathematicians could investigate whether it was logically possible.
The distinction between mathematical truth and physical truth, which had often
remained implicit, now became explicit.
Questions concerning the geometry of physical space could be answered only
through observation and experiment.
Questions concerning the consistency of geometry belonged to mathematics.
This change in perspective marked one of the most profound conceptual shifts
in the history of mathematics.
11.5 Riemann's New Conception of Space
In 1854, Bernhard Riemann delivered his celebrated habilitation lecture, "On the
Hypotheses Which Lie at the Foundations of Geometry."
Rather than asking whether Euclid's parallel postulate could be proved,
Riemann considered a more general question: what properties must a space
possess in order to support geometry?
He introduced the idea that space need not be limited to three dimensions and
that geometry should be understood as the study of abstract manifolds.
According to Riemann, the distance between neighbouring points in a space
could be determined by a metric, expressed through a quadratic differential
form.
Different choices of metric give rise to different geometries.
Euclidean geometry appears as only one possibility among infinitely many.
Riemann also generalised Gauss's concept of curvature.
Whereas Gauss had studied the curvature of two-dimensional surfaces
embedded in ordinary three-dimensional space, Riemann developed an intrinsic
theory of curvature applicable to spaces of any dimension.
Curvature, in this new framework, is a property of the space itself and does not
require the existence of a surrounding higher-dimensional space.
This insight fundamentally altered the nature of geometry.
Geometry was no longer the study of a unique physical space but the
investigation of all possible spaces satisfying specified axioms.
Riemann's ideas would eventually become essential to twentieth-century
physics, particularly in Einstein's general theory of relativity.
At the time, however, his lecture attracted little immediate attention.
Its significance became clear only decades later.
11.6 Beltrami and the Validation of Non-Euclidean Geometry
The decisive step towards the acceptance of non-Euclidean geometry was
taken by Eugenio Beltrami in 1868.
Beltrami showed that the geometry developed by Lobachevskii could be
realised on surfaces of constant negative curvature.
His work established that non-Euclidean geometry was at least as consistent as
Euclidean geometry.
Beltrami began by examining the pseudosphere, Minding's surface of revolution
generated by a tractrix.
He demonstrated that geodesics on the pseudosphere satisfy many of the
properties predicted by Lobachevskii's geometry.
Although the pseudosphere represents only a portion of the entire nonEuclidean plane, it provided the first concrete realisation of the new geometry.
Beltrami later introduced more general models that represented the whole of
non-Euclidean geometry within Euclidean space.
These models transformed the debate.
No longer could critics dismiss non-Euclidean geometry as an exercise in formal
manipulation or as a system destined to collapse into contradiction.
If Euclidean geometry was consistent, then non-Euclidean geometry had to be
consistent as well.
This conclusion established the principle of relative consistency, one of the
central ideas of modern mathematical logic.
Beltrami's work also clarified the relationship between geometry and physical
space.
The existence of a mathematical model does not imply that physical space
possesses the corresponding structure.
Instead, it shows only that the geometry is logically possible.
Determining which geometry best describes the physical universe remains a
matter for empirical investigation rather than pure reason.
With Beltrami's results, the long controversy surrounding Euclid's parallel
postulate finally came to an end.
The problem that had occupied mathematicians for more than two millennia was
resolved not by proving the postulate but by demonstrating that alternatives to it
are possible.
11.7 The Legacy of Non-Euclidean Geometry
The discovery of non-Euclidean geometry transformed mathematics in ways
that extended far beyond geometry itself.
It challenged the traditional belief that mathematical truths are necessarily selfevident descriptions of reality and encouraged mathematicians to distinguish
more clearly between axioms and experience.
Axioms came to be understood not as obvious truths but as assumptions whose
consequences could be explored systematically.
This new viewpoint contributed directly to the development of modern
axiomatics in the work of David Hilbert and others.
It also influenced philosophy by raising fundamental questions about the nature
of knowledge and the relationship between mathematics and the physical world.
For centuries, philosophers had regarded Euclidean geometry as an example of
certain knowledge obtained independently of experience.
The existence of logically consistent alternatives called this assumption into
question.
At the same time, non-Euclidean geometry provided essential tools for later
developments in mathematics and physics.
Riemannian geometry became central to differential geometry, topology, and the
study of manifolds.
In the twentieth century, Einstein used Riemann's ideas to formulate general
relativity, according to which gravitation arises from the curvature of spacetime.
The discovery of non-Euclidean geometry therefore represents far more than
the solution to a technical problem concerning parallels.
It marks a fundamental turning point in the history of human thought,
demonstrating that even the most deeply rooted assumptions about space may
be reconsidered and replaced by richer and more powerful ideas.
The Bolyais eventually learned of Lobachevskii's work in 1844 from the
Hungarian mathematician and physicist Franz Mentovich, who had met Gauss
in Göttingen.
Unable to obtain copies of Lobachevskii's publications, Wolfgang Bolyai wrote
to Gauss in 1848 seeking advice. Gauss replied that they might have better luck
locating the German booklet that Lobachevskii had published in 1840.
They obtained a copy on 17 October 1848 and examined it carefully. Although
they identified numerous points open to criticism, they recognised that
Lobachevskii had reached conclusions remarkably similar to their own.
Some of their objections concerned details of presentation and proof. In
particular, they believed that their own demonstration of the exponential
relationship between lengths and the angle of parallelism was more satisfactory.
Nevertheless, the similarities between the two systems were undeniable. Both
Lobachevskii and János Bolyai had independently discovered a geometry in
which Euclid's parallel postulate does not hold, and both had developed
trigonometric formulae to describe it.
The differences between their approaches lay primarily in style and emphasis.
Bolyai sought to demonstrate that the new geometry could reproduce the
familiar results of Euclidean geometry whenever appropriate and devoted
considerable effort to establishing analogies between the two systems.
Lobachevskii, by contrast, focused more directly on the distinctive properties of
the new geometry itself. He introduced a richer terminology, naming important
concepts such as the horocycle and the horosphere, and emphasised the role
of measurement and analysis.
Despite their differences, the work of both mathematicians shared a common
fate. Neither succeeded in convincing the mathematical community during his
lifetime.
The decisive acceptance of non-Euclidean geometry would require a new idea:
the construction of explicit models showing that these geometries are logically
consistent if Euclidean geometry itself is consistent.
11.3 The Significance of Minding's Surface
Minding's investigation of surfaces with constant negative curvature would later
prove essential to the development of non-Euclidean geometry.
The pseudosphere provided a concrete example of a surface whose local
geometry exhibits many of the properties described by Lobachevskii.
In particular, geodesics on the pseudosphere behave analogously to straight
lines in the non-Euclidean plane.
The existence of such a surface suggested that non-Euclidean geometry might
not merely be a formal system of equations but could correspond to an actual
geometric structure.
Although neither Minding nor Gauss explicitly made this connection, their work
laid the foundations for later developments.
The crucial step would eventually be taken by Beltrami, who showed that the
geometry of the pseudosphere provides a model for a portion of Lobachevskii's
geometry.
This discovery transformed the debate surrounding the parallel postulate.
For centuries, mathematicians had sought either to prove Euclid's postulate or
to derive a contradiction from its negation.
Beltrami's work demonstrated that a different approach was possible: instead of
proving the postulate, one could investigate the consequences of replacing it.
If these consequences could be realised within ordinary geometry, then nonEuclidean geometry would be as logically consistent as Euclidean geometry
itself.
The problem of parallels, which had dominated geometry since antiquity, was
therefore resolved not by a proof but by a change in perspective.
This shift marked the beginning of modern geometry.