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Page 1: Non-Euclidean Geometry: a mathematical revolution · Non-Euclidean Geometry: a mathematical revolution during the long 19th century Lobachevskii Properties of the hyperbolic plane,

Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Non-Euclidean Geometry:a mathematical revolution

during the long 19th century

Waseda University, SILS,History of Mathematics

Page 2: Non-Euclidean Geometry: a mathematical revolution · Non-Euclidean Geometry: a mathematical revolution during the long 19th century Lobachevskii Properties of the hyperbolic plane,

Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Outline

Introduction

Saccheri

Lobachevskii

Poincare

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Introduction

What is geometry?

Until the 19th century, geometry was the study of themathematical properties of figures. All figures were regarded asembedded in Euclidean space. (Ex. Spherical geometry.)

During the 19th century, geometry became the study of physicaland mathematical space itself. It was an investigation of theproperties of the types of spaces in which figures could befound. Hyperbolic space was put forward as a viablealternative to Euclidean space.

By the end of the century, geometry had come to be seen as theabstract investigation of the consequences of some set ofaxioms. Space was understood as the set of points determinedby these axioms. Geometers realized that elliptical (spherical)spaces, and many more, were also possible.

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Introduction

Was there a revolution in the study of geometry?

Since antiquity, mathematicians had sought to demonstrate theparallel postulate, but by the beginning of the 19th century,there was a growing anxiety that it could not be done. (Crisis.)

At the beginning of the 19th century, three men, independently,arrived at a geometry of hyperbolic space (Gauss, Bolyai,Lobachevsky). In the middle of the century, Riemann showedthat there are an infinite number of non-Euclidean geometries,and pointed out that elliptical spaces are possible.

By the end of the century, mathematicians were no longerasking, “what is the correct geometry?” but, rather, trying todetermine what sets of axioms would produce what kinds ofspaces. . .

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Introduction

Topics in the non-euclidean revolution

We will look at:

I An extended attempt to prove that euclidean space is theonly “correct” space. (We can now see this as logicallyflawed.)

I An abstract definition of parallel lines, which leads to a newkind of space – hyperbolic space.

I A model for showing that a hyperbolic plane is logicallyconsistent with, or mathematically mappable to, aeuclidean plane.

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Saccheri

Giovanni Girolamo Saccheri (1667–1733)

I Saccheri was an Italian Jesuit, who worked as professor ofphilosophy, theology and mathematics at Turin and Pavia.

I In 1733, he published Euclides ab omni naevo vindicatus(Euclid freed from all flaws), the first part of which was anattempt to demonstrate Euclid’s 5th postulate. (The other“flaws” concern fourth proportionals and compoundratios.)

I Many of Saccheri ideas were predated, and perhapsinfluenced, by the work of Omar Khayyam and Ibnal-Haytham.

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Saccheri

The Saccheri quadralateral

A B

C D

Hypothesis of the Right Angle (HRA) : ∠C = ∠D = R = 90◦

Hypothesis of the Obtuse Angle (HOA) : ∠C = ∠D > R = 90◦

Hypothesis of the Acute Angle (HAA) : ∠C = ∠D < R = 90◦

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Saccheri

The structure of the “proof”

The entire first part of Euclides vindicatus is structured like anextended indirect proof:

I A): He uses HRA to demonstrate the parallel postulate.I B): He uses HOA to show that the geometry so created is

inconsistent with Elem. I 16 & 17.1 (These contain theArchimedean postulate.)2

I C): He uses HAA to argue that it leads to consequencesthat are “repugnant to the nature of the straight line.”

1Elem. I 16: an exterior ∠ of a4 is greater than the sum of the 2 interior∠s. Elem. I 17: the sum of 2∠s of a4 are less than 2R.

2A line, multiplied by itself, can be made to exceed any given length.

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Saccheri

The hypothesis of the obtuse angle

I EV Props. 11 & 12 (HRA, HOA): If a line falls on two givenlines such that it is perpendicular to one and makes anacute angle with the other, then the two given lines willmeet in the direction of the said acute angle. (If ∠LPD isright and ∠PAD is acute, then AD will intersect PL.)

I EV Props. 13 (HRA, HOA): If a line falls on two given linessuch that it makes the two interior angles on the same sideless than two right angles, then the two lines will meet inthe direction of the said interior angles. (If∠AXL + ∠XAD < 2R, then AD will meet XL.)

I Contradiction with Elem. I.17.

I EV Props. 14 (HOA): “The hypothesis of the oblique angleis absolutely false, because it destroys itself.”

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Saccheri

The hypothesis of the acute angle

I EV Prop. 32 (HAA): If HAA is true, there there will exist aline with the following properties:

I AX is a limit to the set of lines passing through point A thatmeet BX and also a limit to the set of lines passing throughpoint A that have two distinct perpendiculars with line BX– one in each direction.

I AX meets BX at one point, infinitely distant.I AX is asymptotic to BX.3I AX is a straight line.

I EV Prop. 33 (HAA): “The hypothesis of the acute angle isabsolutely false, because it is repugnant to the nature of astraight line.”

3That is the distance between the two lines will become less than anygiven value.

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Lobachevskii

Nikolai Ivanovich Lobachevsky (1792–1856)

I Born to a provincial,middle class family. Hisfather died when he was 8.

I Graduated from KazanUniversity with a degree inphysics and mathematics.

I Worked at KazanUniversity his whole life.

I He developed his ideas onnon-Euclidean geometryfrom 1826 to 1855.

I Died in poverty.

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Lobachevskii

An abstract theory of parallels

I Lobachevsky begins by defining a line, `p, through a givenpoint P, as parallel to a given line, `0, when it divides all thelines that pass through point P into two mutually exclusivesets, those which are intersecting, `i, and those which arenon-intersecting, `n. (`p ‖ `0.) That is, where the angle at Pis written Π(α):

I lines, `i, where Π(α`i) < Π(α`p), intersect `0, andI lines, `n, where Π(α`n) > Π(α`p), do not intersect `0.

I Where Π(α`p) = 1/2π = 90◦, then there is a unique parallel,and it is parallel in both directions.

I Where Π(α`p) < 1/2π = 90◦, then there is another line onthe opposite side at the same angle that is parallel in theopposite direction.

I With respect to `0, all lines through P can be classified as 1)intersecting, 2) parallel, or 3) non-intersecting.

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Lobachevskii

Properties of the hyperbolic plane, H 2

I The sum of the angles of a triangle are less than 2R = π.I There is no distinction between similarity and congruency.

(There are four congruency theorems SAS, SSS, AAS, andAAA.)

I As triangles get larger and smaller, the angles also change.

I There are no straight lines everywhere equidistant fromone another.

I Lines parallel to the same line need not be parallel to oneanother. (Parallelism is non-transitive, because ofdirectionality.)

I Two lines which intersect one another may both be parallelto the same line. (Again, because of directionality.)

I and so on. . .

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Lobachevskii

A model of the Euclidean plane in hyperbolic spaceLobachevskii then constructed a three dimensional model ofEuclidean space within his hyperbolic space. (Just as we canconstruct a three dimensional model of spherical space inEuclidian space.)

I He shows that the planes containing three non-coplanarparallel lines contain solid angles that sum to 2R. (Prismtheorem.)

I He develops a horocycle as a curve perpendicular to a set ofcoplanar parallel lines, and a horosphere as the solid ofrotation of a horocycle.

I He shows that the geometry of the horosphere is Euclideanby mapping it to the Euclidean plane. In this way, heshows that a Euclidean plane can be embedded inhyperbolic space – just as a spherical plane can beembedded in Euclidean space.

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Poincare

Jules Henri Poincare (1854–1912)

I Born to an influential, bourgeoisfamily.4

I Graduated from the EcolePolytechnique and the Ecole desMines.

I He eventually became aprofessor at the Sorbonne.

I He was an influential figure inFrench mathematical sciencesand also wrote many popularworks.

4His sister married the philosopher Emile Broutroux and his cousinRaymond Poincare became president of France.

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Poincare

Some axioms (Hilbert, Grundlagen der Geometrie, 1899)

I Incidence.1: For every two points A, B there exists a line athat contains each of the points A, B.

I Incidence.2: For every two points A, B there exists no morethan one line that contains each of the points A, B.

I Congruence.2: If a segment A′B′ and a segment A′′B′′ arecongruent to the same segment AB, then segments A′B′

and A′′B′′ are congruent to each other.I Congruence.5: If for two triangles ABC and A′B′C′ the

congruences AB ∼= A′B′, AC ∼= A′C′ and ∠BAC ∼= ∠B′A′C′

are valid, then the congruence4ABC ∼= 4A′B′C′ is alsosatisfied. (SAS congruence: Elements I 4.)

I and so on. . .

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Poincare

Preliminary, polar inverse of a point by a circle

The polar inverse of a point5 with respect to a circle, or a pair oforthogonal circles is found as follows. Where A is an internalpoint, join OA and erect ⊥ AP. Join OP and draw tangent PB tomeet OA extended to B. Any circle through A and B is C′ ⊥ C.

The same construction can be used when an external point, B,is given. Find the tangent BP and drop ⊥ PA.

5Polar inversion is defined by the relation AO · BO = r2.

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Poincare

The Poincare disk

The hyperbolic plane, H 2, isdefined as the set of points of thedisk inside circle Γ, but notincluding the circumference itself.

A P-point is any point inside thecircle.

A P-line is any circle orthogonal toΓ, or a line drawn through thecenter of the circle.

The Euclidean points, lines andcircle are simply referred to assuch.

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Poincare

Consistency with the axioms of Euclidean geometry

I We can use the model to demonstrate all of the Euclidianaxioms. For example, we show that two P-points alwaysdetermine a unique P-line (Incidence 1 & 2).

I Suppose we have two P-points, A and B.I a) If line AB goes through center O, then ABO is the only

line through A and B, and it is also a P-line.I b) If not, we find the polar inverse of A as A′ and draw

circle γ through A′, A and B. Then γ is the only circlethrough A′, A and B and it is orthogonal to Γ.

I Therefore there is always a unique P-line through any twoP-points.

I We can do similar proofs for all of the Euclidean axioms,except the parallel axiom.

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Poincare

Parallels in the Poincare disk

Hilbert’s parallel axiom: “Let a be any line and A a point not onit. Then there is at most one line in the plane that contains a andA that passes through A and does not intersect a.”

I Two P-lines that are P-parallel intersect circle Γ at the samepoint. (Hence, they are tangent circles.)

I There are many P-lines through a given P-point that do notintersect a given P-line. Proof by construction. (We let γ bea given P-line and A a given P-point. We find the inverseof A, as A′. . . )

I The two angles of parallelism, Π(α), associated with agiven P-point and P-line are equal. Proof by contradiction.(Let AMB be a P-line with two parallels, AD and DBthrough P-point D. Assume, to the contrary that,∠ADM > ∠BDM. . . )

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Poincare

A Saccheri quadrilateral in the Poincare disk

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Non-Euclidean Geometry: a mathematical revolution during the long 19th century

Poincare

Overview

In the 18th century, it became increasingly clear that it wouldnot be possible to prove that Euclidean geometry was the onlyvalid geometry. We looked at Saccheri’s failed attempt to provethis.

In the 19th century, there were a number of attempts to developnon-Euclidean geometries and to show that these were valid.Mathematicians became increasingly concerned with validityover truth, and with modeling one type of geometry in another.

Around the turn of the 20th century, there was newfoundational work on Euclidean geometry. This lead, in turn,to new work on showing the logical consistency betweenvarious types of geometries. The mechanism for this was amap, or model.


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