Through the ostensibly infallible process of logical deduction, Euclid of Alexandria (ca. 300 b.c.) derived a colossal body of geometric facts from a bare minimum of genetic material: fivepostulates—five simple geometric assumptions that he listed at the beginning of his masterpiece, the Elements. That Euclid could produce hundreds of unintuitive theorems from five patently obvious assumptions about space, and, still more impressively, that he could do so in a manner that precluded doubt, sufficed to establish theElementsas mankind’s greatest monument to the power of rational organized thought. As a logically impeccable, tightly wrought description of space...

In 1794, when Lobachevski was an infant, Adrien Marie Legendre published his famousÉléments de Géométrie, a textbook that attempted to improve Euclid’s presentation of geometry by simplifying the proofs in Euclid’sElements, and reordering its propositions. In subsequent editions and translations, Legendre’s text became a 19^th-century educational staple. Its admirers were legion; they taught and learned from it in locations throughout Europe, the antebellum United States, and even in Lobachevski’s remote Russian city of Kazan. Legendre died in 1833, but his textbook remained immensely popular for the duration of the 19th century. In the epigraph for this section, Herr...

Mathematical terms cannot be definedex nihilo. The words that one uses in any given definition require further definitions of their own; these secondary definitions necessitate tertiary definitions; these in turn require still others. To escape infinite regress, geometers must leave a handful of so-called primitive terms undefined. These primitive terms represent the basic building blocks from which the first defined terms may be constructed. From there, one may build upward indefinitely; all subsequent development will be grounded upon the primitive terms, and circular definitions will be avoided.

Only in the late 19th-century was such clarity achieved in the foundations...

Lobachevski commences hisTheory of Parallelsby redefining parallelism. This is no mere preliminary matter, but a bold decision to alter a definition that had stood largely unquestioned since ancient times. For a first-time reader, accustomed to the simplicity of Euclid’s definition of parallels (coplanar non-intersecting lines), Lobachevski’s replacement will no doubt seem mysterious, if not presumptuous. What exactly does it mean? Is it permissible to redefine a familiar term? What is wrong with the classical definition? Why does Lobachevski not simply contrive a new name for his “boundary-line” relation instead of appropriating the term “parallelism”?

We shall answer all...

In TP 17, Lobachevski proves that his new sense of parallelism¹ is well defined.

Recall thatAB‖CDonly ifABadmits no wiggle room aboutA. (i.e.ABmust exhibit the “mark of parallelism” atA.) SinceAhas no particular significance among the infinitely many points on lineAB, its conspicuous presence in the definition of parallelism is disconcerting. To set our minds at ease, Lobachevski demonstrates thatA’s ostensibly special role is an illusion: he proves that if the line exhibits the mark of parallelism (lack of wiggle room) at any one of its points, then it will...

Lobachevski is going to show that parallelism is asymmetricrelation: givenAB‖CD, he will prove thatCD‖AB. To do so, he must verify that every rayCEentering⊾DC AintersectsAB. Clever though his proof is, Lobachevski’s obscures his geometric artistry under murky exposition. Accordingly, I shall follow his proof with an alternate explanation of my own.

LetACbe perpendicular toCD, a line to whichABis parallel. FromC, draw any lineCEmaking an acute angle⊾ECDwithCD. FromA, drop the perpendicularAFtoCE. This produces a...

In proposition I.32 of theElements, Euclid demonstrates that the angle sum of every triangle isπ. It was well-known in Lobachevski’s day that this theorem islogically equivalentto the parallel postulate. In fact, the equivalence of the two statements was such common knowledge that Lobachevski apparently felt no need to prove it, or even to mention it explicitly, inThe Theory of Parallels. It is not a hard equivalence to establish.

Claim 1. Given Euclid’s first four postulates, the parallel postulate holds if and only if the sum of the angles in every triangle isπ.

Proof. ⇒)...

This proposition, like the previous one, played an important role in Legendre’s purported proofs of the parallel postulate. It is sometimes calledLegendre’s second theorem, but a pleasantly literary alternative,The Three Musketeers Theorem,¹ has gained favor in recent years. Whatever its name, Legendre was not the first to prove it. Saccheri preceded him once again. Nonetheless, Legendre’s proof, which Lobachevski largely follows, is particularly elegant. The enormous popularity of Legendre’s Éléments was due, in no small measure, to his artful proofs, which seem even to invite names: we have seen his “siphon construction” in TP 19, and we will...

The salient feature of this little result is its neutrality; it is valid in both Euclidean and imaginary geometry. Twice, in the notes following TP 19, I committed the venial sin of using this result without having proved it first, referring the reader to the present proposition for its demonstration. I am not, however, guilty of the mortal sin of circular argument; Lobachevski’s proof of TP 21 does not require the intervening proposition, TP 20, and thus it could have been given directly after TP 19, prior to the proofs in which I used it.

From the given pointA,...

A theorem of neutral geometry (I.28) guarantees that two lines with a common perpendicular (i.e. two lines perpendicular to a third) will never meet one another. Thus, in Euclidean geometry, lines with a common perpendicular are parallel to one another.

In imaginary geometry, however, non-intersection does not suffice to establish parallelism (see TP 16). In fact, the present proposition demonstrates that in imaginary geometry, lines with a common perpendicular arenotparallel to one another. (Were they parallel, all triangles would have angle sumπ, contradicting the result proved in Claim 2 of the notes for TP 20.)

Accordingly, when...

Beginning with the initial segmentAA’, the repeated construction consists of doubling the length of the segment and erecting a perpendicular at its new endpoint. In Euclidean geometry, each perpendicular would meetABat an angle of (π/2 – α), producing an endless sequence of similar right triangles. In imaginary geometry, this cannot happen, as there are no similar triangles. After Lambert’s results on area and angle defect (discussed in the notes to TP 20) we should not be surprised to learn that each successive perpendicular meetsABat a smaller angle than does its predecessor. However, Lobachevski asserts something....

To prove this proposition, Lobachevski shows that ifCG‖AB, thenGis closer toABthanCis. This is easy to miss on a first reading, since an auxiliary construction dominates the proof; Lobachevski does not even mention the parallelCGuntil the penultimate sentence of his proof.

Upon the lineAB, erect two perpendicularsAC=BD, and join their endpointsCandDwith a straight line. The resulting quadrilateralCABDwill have right angles atAandB, but acute angles atCandD(TP 22¹). These acute angles are equal to one another; one...

Whereas the classical definition of parallelism is transitive only in the presence of the parallel postulate (see “A Deeper Definition” in the notes to TP 16), Lobachevski proves here that his new notion of parallelism is transitive in imaginary geometry as well.

His proof falls into two parts: first, he establishes transitivity in the plane; then, he does the same in space. The latter part, his first foray into three-dimensional imaginary geometry, initiates a sequence of results in solid geometry (culminating in TP 28). His desire to place these three-dimensional results together is responsible for this relatively late proof of...

As Lobachevski notes in TP 12, a plane passed through a sphere produces a circle of intersection on the sphere’s surface. The closer the plane comes to the sphere’s center, the larger the circle of intersection will be; if it passes through the center itself, the resulting intersection is a great circle. On a globe, for example, lines of longitude and the equator are all examples of great circles, while the Tropic of Capricorn is not.

Diametrically opposed (antipodal) points on a sphere can be joined by many great circles; the North and South poles of the globe, for example,...

A solid angle isdihedralif it is bounded by two planes meeting at a line; trihedral if it is bounded by three planes at a point. Thus, a tetrahedron contains four trihedral angles, one at each vertex, and six dihedral angles, one at each edge. TP 27 relates the measure of a trihedral angle to the measures of the three dihedral angles at its edges. We know how to measure dihedral angles (see the TP 26 notes), but how does one measure a trihedral angle? We must answer this question before we can understand the statement of TP 27,...

This portentous result, which Jeremy Gray has namedthe prism theorem, says that if the edges of an infinitely long triangular prism are parallel to one another, then the three dihedral angles at those edges will add up toπ. What makes this theorem remarkable is its neutrality. Its independence from the parallel postulate is surprising when one considers its resemblance to another theorem, Euclid I.32 (the sum of the angles in a triangle add up toπ), which is actually equivalent to the postulate.

The prism theorem occupies a distinguished place in the structure of theTheory of Parallels....

In Euclidean geometry, every triangle’s perpendicular bisectors are concurrent. In fact, they meet at the triangle’s circumcenter (ElementsIV.5). In contrast, it is easy to construct a triangle in imaginary geometry with non-concurrent perpendicular bisectors.

Suppose thatAB‖CD. FromG, an arbitrary point that lies between the parallels, drop perpendicularsGEandGH, as shown in the figure. Double the lengths of these segments, extending them toFandI, respectively. Notice thatF,G, andIcannot be collinear: if they were, then the line upon which they lie would be a common perpendicular for the parallels, which is...

This proposition continues the story of its predecessor. In imaginary geometry, certain triangles lack circumcircles, since their perpendicular bisectors fail to meet. In TP 29, we saw that if two of the perpendicular bisectors of a triangle’s sides intersect one another, then all three bisectors must be concurrent. But what happens if no two bisectors meet? In TP 30, Lobachevski gives a partial answer: if two bisectors are not only non-intersecting, but also parallel to one another, then the third bisector will beparallelto them as well.

His proof falls into two cases: one in which the two given...

It is often convenient to think of a horocycle as a “circle of infinite radius”, or “a circle whose center is at infinity”, but this will not suffice as a formal definition. Although one typically defines a circle as the locus of points at a fixed distance from a given point, Lobachevski used an alternate, equivalent definition of a circle as the basis for his definition of the horocycle. Namely, a circle is “a closed curve in the plane with the property that the perpendicular bisectors of its chords are all concurrent,” and its center is the point of concurrence....

When Lobachevski introduced the horocycle in TP 31 with the words, “Grenzlinie (Oricycle) nennen wir…” (We shall define a boundary-line (horocycle) … ), he offered two names for it: the now familiar “horocycle”, with its suggestions of circle-like properties, andGrenzlinie, meaning “boundary-line” or “limiting curve”. In his final exposition of the subject,Pangeometrie(1855), he combined the circle and limit imagery into a single French name,cercle limite.¹

“Horocycle” has become the standard term, and I have taken the translator’s liberty of making exclusive use of it, despite the fact that Lobachevski favorsGrenzliniein the German original....

Recall that a horocycle is determined by two data: a point upon it, and its center (intuitively, the center is a point at infinity; formally, it is a pencil of parallels—the set of the horocycle’s axes). Naturally,concentric horocyclesare defined to be horocycles sharing the same center. In other words, two horocycles are concentric if and only if their sets of axes are identical. We define thedistance between two concentric horocyclesto be the length of any axis cut off between them; this length does not depend upon the particular axis that we choose to measure, as...

Just as we can produce a sphere by revolving a circle about one of its diameters¹, we produce ahorosphereby revolving a horocycle about one of its axes. We shall call this axis the horosphere’saxis of rotation.

As the generating horocycle revolves about the axis of rotation, each of its axes traces out a trumpet-like cylinder in space. We define all the rays that lie upon these cylinders as thehorosphere’s axes(i.e. its “other” axes, besides its axis of revolution). Clearly, every axis of the horosphere lies on some line in space, parallel to the horosphere’s axis...

Lobachevski now begins to develop imaginary trigonometry. As promised at the conclusion of TP 22, he will derive the formulae of imaginary trigonometry both in the plane and on the sphere. This project, the culmination ofThe Theory of Parallels, stretches out over three lengthy propositions. TP 35, the first of the three, is the most difficult proposition in the entire work and offers the most virtuosic display of Lobachevski’s genius. Within the pages of this proposition, Lobachevski establishes links between triangles in the plane, on the sphere, and on the horosphere. He completely elucidates the structure of imaginary spherical...

Having revealed the structure of spherical trigonometry, Lobachevski returns to the plane to settle some old business: twenty propositions after introducing it, he finally derives an explicit formula for the П-function.

Lobachevski begins by deriving a pair of relations among ∆ABC’s parts.

Lobachevski draws rayCBand two rays parallel to it: one emanating fromA, the other perpendicular to the hypotenuse. Each is uniquely determined.

The derivation of the relation in this passage is self-explanatory, but it does depend upon the fact that П(b) > П(ɑ). Lobachevski does not bother to prove this, since it is so easy to...

In this final proposition, Lobachevski develops the trigonometric formulae of the imaginary plane. Although we can now replace the П-functions in any trigonometric equation with hyperbolic functions, Lobachevski chooses not to make these helpful translations. This lack, coupled with some awkward derivations and peculiar notation, make his work in this section appear particularly opaque. This is unfortunate, since the conclusions of this section are actually quite simple and admit easy proofs. To emphasize this fact, I shall derive Lobachevski’s results in П-free notation and deviate from his unnecessarily convoluted proofs of the two laws of cosines.

We can interpret the...