# Riemann, Bolyai, Lobatschewski and Gauss.

V. A Memoir on Abstract Geometry. Phil.
Trans. CLX., pp. 51-63 (1870).

VI. On the superlines of a quadric surface in
five dimensional space. Quar. Jour., Vol. XII.,
pp. 176-180 (1871-72).

VII. On the Non-Euclidean Geometry.
Clebsch Math. Ann. V., pp. 630-634 (1872).

Four of these pertain to Hyper-Space, and in
that Bibliography I quoted Cayley as to its
geometry as follows :

"The science presents itself in two ways — –
as a legitimate extension of the ordinary two,-
and i^ree-dimensional geometries, and as a need
in these geometries and in analysis gener-
allj'. In fact, whenever we are concerned with
quantities connected together iu any manner,
and which are or are considered as variable or
determinable, then the nature of the relation
between the quantities is frequently rendered
more intelligible by regarding them (if only two
or three in number) as the coordinates of a
point in a plane or in space : for more than
three quantities there is, from the greater com-
plexity of the case, the greater need of such a
representation ; but this can only be obtained
by means of the notion of a space of the proper
dimensionality ; and to use such a representa-
tion we require the geometry of such space.

An important instance in plane geometry has

Januaey 13, 1899.]

SCIENCE.

61

actually presented itself in the question of the
determination of the number of curves which
satisfy given conditions ; the conditions imply
relations between the coefficients in the equa-
tion of the curve ; and for the better under-
standing of these relations it was expedient to ■
consider the coeflficients as the coordinates of a
point in a space of the proper dimensionality."

For a dozen years after it was written the
Sixth Memoir on Quantics would not have been
enumerated in a Bibliography of non-Euclidean
geometry, for its author did not see that it gave
a generalization which was identifiable with
that initiated by Bolyai and Lobach^vski, though
afterwards, in his address to the British Asso-
ciation, in 1883, he attributes the fundamental
idea involved to Riemann, whose paper was
written in 1854.

Says Cayley : "In regarding the physical
space of our experience as possibly non-Euclid-
ean, Riemann's idea seems to be that of modify-
ing the notion of distance, not that of treating
it as a locus in four-dimensional space."

What the Sixth Memoir was meant to do was
to base a generalized theory of metrical geome-
try on a generalized definition of distance.

As Cayley himself says: " * -» * the
theory in effect is that the metrical properties
of a figure are not the properties of the figure
considered per se apart from everything else,
but its properties when considered in connection
with another figure, viz., the conic termed the
absolute."

The fundamental idea that a metrical property
could be looked at as a projective property of
an extended system had occurred in the French
school of geometers. Thus Laguerre (1853) so
expresses an angle. Cayley generalized this
French idea, expressing all metrical properties
as projective relations to a fundamental config-
uration.

We may illustrate by tracing how Cayley
arrives at his projective definition of distance.
Two projective primal figures of the same kind
of elements and both on the same bearer are
called conjective. When in two conjective
primal figures one particular element has the
same mate to whichever figure it be regarded
as belonging, then every element has this
property.

Two conjective figures, such that the elements
are mutually paired (coupled), form an involu-
tion. If two figures forming an involution have
self-correlated elements these are called the
double elements of the involution.

An involution has at most two double ele-
ments, for were three self-correlated all would
be self-correlated. If an involution has two
double elements these separate harmonically
any two coupled elements. An involution is
completely determined by two couples.

From all this it follows that two point-pairs A,
B and A^, i?, define an involution whose double
points D, Z>j are determined as that point-pair
which is harmonically related to the two given
point-pairs.

Let the pair A, B ha fixed and called the
Absolute. Two new points A^, B^ are said (by
definition) to be equidistant from a double point
D defined as above. D is said to be the ' center '
of the pair A^, i?,. Inversely, if A^ and D be
given, i?j is uniquely determined.

Thus, starting from two points P and Pj, we
determine P.^ such that Pj is the center of P and
Pj, then determine P3 so that P^ is the center of
Pi and P3, etc. ; also in the opposite direction
we determine an analogous series of points
P — 1, P — 2, …. We have, therefore, a series
of points

….,P— , P-i, P, P„P„P3

at ' equal intervals of distance.' Taking the
points P, Pi to be indefinitely near to each other,
the entire line will be divided into a series of
equal infinitesimal elements.

In determining an analytic expression for the
distance of two points Cayley introduced the
inverse cosine of a certain function of the coor-
dinates, but in the Note which he added in the
Collected Papers he recognizes the improve-
ment gained by adopting Klein's assumed defi-
nition for the distance of any two points P, Q :

, AP.BQ
dist. (P® = clog^^^^^,

where A, B are the two fixed points giving the
Absolute.

This definition preserves the fundamental re-
lation

dist. (P® + dist. {QR) = dist. (PP).

In this note (Col. Math. Papers, Vol. 2, p.

62

SCIENCE.

[N. S. Vol. IX. No. 211.

604) Cayley discusses the question whether the
new definitions of distance depend upon that of
distance in the ordinary sense, since it is obvi-
ously unsatisfactory to use one conception of
distance in defining a more general conception
of distance.

His earlier view was to regard coordinates
' not as distances or ratios of distances, but as
an assumed fundamental notion, not requiring
or admitting of explanation.' Later he re-
garded them as ' mere numerical values, at-
tached arbitrarily to the point, in such wise
that for any given point the ratio x : y has a de"
terminate numerical value,' and inversely.

But in 1871 Klein had explicitly recognized
this difficulty and indicated its solution. He
says : " The cross ratios (the sole fixed ele-
ments of projective geometry) naturally must
not here be defined, as ordinarily happens, as
ratios of sects, since this would assume the
knowledge of a measurement. In von Staudt's
Beitriigen zur Geometric der Lage (§ 27. n.
393), however, the necessary materials are given
for defining a cross ratio as a pure number.
Then from cross ratios we may pass to homo-
geneous point- and plane-coordinates, which, in-
deed, are nothing else than the relative values
of certain cross ratios, as von Staudt has like-
wise shown (Beitraege, § 29. n. 411)."

This solution was not satisfactory to Cayley,
who did not think the difficulty removed by the
observations of von Staudt that the cross ratio
{A, B, P, Q) has "independently of any notion
of distance the fundamental properties of a
numerical magnitude, viz. : any two such ratios
have a sum and also a product, such sum and
product being each of them a like ratio of four
points determinable by purely descriptive con-
structions."

Consider, for example, the product of the
ratios (A, S, P, Q) and {A/ B/ P/ Q'). We
can construct a point B such that (A/ B/ P/
Q') = {A,B, Q, B). The product of (^, B, P, Q)
and {A, B, Q, B) is said to be {A, B, P, B).
This last definition of a product of two cross
ratios, Cayley remarks, is in effect equivalent
to the assumption of the relation dist. (PQ)
+ dist. {QB) = dist. (PB).

The original importance of this memoir to
Cayley lay entirely in its exhibiting metric as a

branch of descriptive geometry. That this gen-
eralization of distance gave pangeometry was
first pointed out by Klein in 1871.

Klein showed that if Cayley's Absolute be
real we get Bolyai's system ; if it be imaginary
we get either spheric or a new system called by
Klein single elliptic ; if the Absolute be an im-
aginary point pair we get parabolic geometry ;
and if, in particular, the point pair be the cir-
cular points we get ordinary Euclid.

It is maintained by B. A. W. Russell, in his
powerful essay on the Foundations of Geometry
(Cambridge, 1897), "that the reduction of met-
rical to projective properties, even when, as in
hyperbolic geometry, the Absolute is real, is
only apparent, and has merely a technical
validity. ' '

Cayley first gave evidence of acquaintance
with non-Euclidean geometry in 1865 in the
paper in the Philosophical Magazine, above-men-
tioned.

Though this is six years after the Sixth Me-
moir, and though another six was to elapse
before the two were connected, yet this is, I
think, the very first appreciation of Lobachev-
sky in any mathematical journal.

Baltzer has received deserved honor for in
1866 calling the attention of Holiel to Lobachev-
sky's ' GeometrischeUntersuchungen,' and from
the spring thus opened actually flowed the flood
of ever-broadening °non-Euclidean research.

But whether or not Cayley's path to these
gold-fields was ever followed by any one else,
still he had therein marked out a claim for
himself a whole year before the others.

In 1872, after the connection with the Sixth
Memoir had been set up, Cayley takes up the
matter in his paper, in the Mathemaiische An-
nalen,' On the Non-Euclidean Geometry,' which
begins as follows: "The theory of the non-
Euclidean geometry, as developed in Dr. Klein's
paper ' Ueber die Nicht-Euclidische Geometrie,'
may be illustrated by showing how in such a
system we actually measure a distance and an
angle, and by establishing the trigonometry of
such a system."

I confine myself to the ' hyperbolic ' case of
plane geometry : viz., the Absolute is here a real
conic, which for simplicity I take to be a circle ;
and I attend to the points within the circle.

January 13, 1899.]

SCIENCE.

63

I use the simple letters, a, A, . . to denote
(linear or angular) distances measured in the
ordinary manner ; and the same letters with a
superscript stroke (i, ^, . . to denote the same
distances measured according to the theory.
The radius of the Absolute is for convenience
taken to be = 1 ; the distance of any point from
the center can, therefore, be represented as the
sine of an angle.

The distance BC, or say a, of any two points
B,C is by definition as follows :

BICJ
BJ^I

:*]

(where I, J" are the intersections of the line BC
with the circle).

As for the trigonometry " the formulas are, in
fact, similar to those of spherical trigonometry
with only cosh a, sinh a, etc., instead of cos a,
sin a, etc."

Cayley returned again to this matter in his
celebrated Presidential Address to the British
Association (1883), saying there: "It is well
known that Euclid's twelfth axiom, even in
Play fair's form of it, has been considered as
needing demonstration ; and that Lobatsch^v-
sky constructed a perfectly consistent theory,
whei'ein this axiom was assumed not to hold
good, or say a system of non-Euclidean plane
geometry. There is a like system of non-
Euclidean solid geometry."

"But suppose the physical space of our ex-
perience to be thus only approximately Euclid-
ean space, what is the consequence which fol-
lows?"

The very next year this ever-interesting sub-
ject recurs in the paper (May 27, 1884) ' On the
Non-Euclidean Plane Geometry.' "Thus the
geometry of the pseudo-sphere, using the ex-
pression straight line to denote a geodesic of
the surface, is the Lobatschevskian geometry;
or, rather, I would say this in regard to the
metrical geometry, or trigonometry, of the sur-
face; for in regard to the descriptive geometry
the statement requires some qualification * * *
this is not identical with the Lobatschevskian
geometry, but corresponds to it in a manner such
as that in which the geometry of the surface of
the circular cylinder corresponds to that of the
plane. I would remark that this realization of.

the Lobatschevskian geometry sustains the
opinion that Euclid's twelfth axiom is un-
demoustrable."

But here this necessarily brief notice must
abruptly stop. Cayley, in addition to his won-
drous originality, was assuredly the most learned
and erudite of mathematicians. Of him in his
science it might be said he knew everything,
and he was the very last man who ever will
know everything. His was a very gentle,
sweet character. Sylvester told me he never
saw him angry but once, and that was (both
were practicing law !) when a messenger broke
in on one of their interviews with a mass of
legal documents — new business for Cayley. In
an access of disgust, Cayley dashed the docu-
ments upon the floor.

George Bruce Halsted.

Austin, Texas.

Euclid has had critics and detractors in all ages. All have been battling against the prevailing doctrine of the meaning of Euclid, the applicability of Euclid. Some have demonstrated theoretical or logical arrogance in the face of a religio-cultural establishment attempting to defend the status quo.

Since vieta, postulate 5 has been hotly debated in europe, unaware that the spherical geometers in Persia and Arabia had already considered it and come up with modifications, and some new relationships between the angles and trig ratios.

A new geometry is often declared, but this is a subjective description of a few axiom/ postulate changes. The main difference is straight and curved lines are combined in forms i call shunyasutras.

The new tool was the differential calculus, and with the prevailing opinion being that euclid was static it was easy to declare adversely against Euclid.Pythaoras was split off and new notions gathered round it.

The calculus approach is not that hew, but the notation was. Newtom invented the binomial series, and discovered infinitesimals,vanishingly small numbers.Differential geonetry was what Descartes started, but Leibniz and Newton finished. In doing so they constructed an appropriate geometry from Euclid.What was constructed is the infinitessimal Euclidean geometry. The idea is to reduce the size to a scale where everything can be well approximated by Euclidean forms, and then to aggregate all the aproximations to get a proper value of magnitude.

Certain theorems associated with plane figures came to signify Euclid, Whereas Euclid is more than the theorems. The theorems arise out of he euclidean oraxis applied to the forms in a plane. Different theorems arise out of different formms, different spatial circumstances. The praxis is the same.

Euclid himself drew on the praxis of many skilled artisans, of which Hipparchus and Pythagoras are noteable. The pythagorean reserarches produced the remarkableplane and solid theorems and predicted forms based on the assumptions. But every artisan knew that "Euclid's" assumption was "ideal". Every engineer knew that the ideal figures forms and relationships had to be adapted to the real pragmatic expeerience, and this was done through scale change.The change of scales was what made Greek geometry and all the former geometries of Babylon,Egypt,China and The Dravidian cultures of India so dynamic, Calculus arises naturally out of euclidean geometry applied to pragmatic situations. Trigonometry is the calculating heart of Euclidean geometry and from that ALL geometers have derived all praxis using the ideal relationships . Euclid's efficacy is in collating the ideal forms that assist in developing a praxis, and showing that a consistent set of deductions can be arrived at ,that a set of inductions can also be arrived at through consistency constraints and that a" thinking". a "manthematos", a "logical", that is a "kairos" and a "logos", proportion and ratio, can guide the pragmatic application of ideal or subjective forms to derive solutions to empirical forms.

The problem has never been Euclid or Euclidean Geometry, but rather interpreters and commentators on the geometry. When the higher echelons of the european churches decided that Euclidean geometry was of divine origin and immutable, they went against the very empirical core of geometry. They took the Logos of pythagoras and applied it to a divine ommutable circumstance. They then took the kairos and from that flowed the whole measurement of a fixed and immutable eternal divine order.

To be fair, pythaforas and many mystics were of the opinion that such a divine order existed in the realms of the gods while we humans had to make do with bodging and fudging things! The rise of the Newtonian Paradigm, especially the power and success of the differential and integral calculus, along with the sheer ingenuity and experimental, that is empirical investigation of reality lead to an explosion of technological success both in engineering but also in alchemy/chemistry. Culturally the renaissance meant that the power of the hierarchy was being resisted and challenged at every level, and those artisans who empirically knew what was according to nature were being vindicated against the arrogant rationalists who thought reason alone provided the "true" desription, the "ideal" solution.

It is easy to understand why the baby got thrown out with the bathwater, but time to state that opinions and misconceptions of the past do not need to be continued into the future. Euclidean geometry has always been empirical, dynamic and adaptable. Modern geometers should be proud to update the basic ideal forms of Euclidean geometry, to uodate the calculus of euclidean geometry by ever more applicable trigonometries because that is all they are doing.

When Pythagoras proved a relationship between 3 gnomons, he merely showed how a "manthematic", a thinking approach can order real forms and divine invariant relationships. These invariant relationships he sought for all his life, and he advised that they be looked for in "arithmoi".

Arithmoi are a special set of geometrical forms based on the articulated or "joined" relationship. The forms are not solely rectilinear, and the more general form i have called shunyasutras. They are however jointed or joined forms because the notion of a relationship requires at least two forms to compare, two regions to contrast, two distinguishable entities from which to draw the distinctions, comparisons and contrasts. By joining them the effect of one on the other can be surmised by geometrical relationships. Without joining it is fair to assume that there is no relationship.

However, one day Pythagoras found out that music related even forms that were not connected! He found out what we now call resonance, but what was more stunning to him was that it related arithmoi to disconnected arithmoi!

We now study this under the guise of fields and field effects, but it is still the same thing: a geometry in one place has a resonant effect on a geometry in another place through special geometrical forms called ARITHMOI.

What a poor description we have accepted in the use of the term "number"!

Although mystical to the core we study it in the same way Pythagoras did by empirical experiment, by science! And what we have found leads to the electronic and nuclear and quantum technologies we have today.

Within Euclid is the powerful praxis of science as devised by ancient greek philosophers, of whom Pythagoras appears to be the founder. The empirical approach is Euclidean, the testing of postulates is Euclidean, the setting out of Postulates is Euclidean. Thus to describe the new geometries as nonEuclidean is a bit too much hyperbole.

Riemann in his Book commissioned it seems by Gauss as a draft thesis may be the originator of the term non Euclidean, but every inventor wants to distinguish themselves in some way. Riemann apparently was not full aware of how deep the arabian philosophers and geometers had gone in there investigations of spherical trigonometry.

Spherical Geometry and trigonometry is a forgotten study nowadays, because it is apparently subsumed in the new riemannian geometries which exposit the subject in terms of the differential geometry of today. As a student one would not know any better, so one can easily pick up the dismissive attitude of those who for their own personal reasons wish to give the primacy to themselves, or their academy or nation. Nevertheless, the old methods are sometimes better than the new, and sometimes worse than the new. This should still be laid out before a student to avoid stultification of thinking.

The notion of a metric is a geometrical notion of a projective relationship. Projective geometry is where the real heart of new postulates for Euclidean geometry lie. Transformation of surfaces, dynamic surfaces and projection onto different surfaces so necesaary in map making was the major driver for the study of the logarithmic series after f course napier had invented the trigonometric extension called logarithms ,

|These were times of immense calculation. The methods of calculation themselves gave rise to the difference equations which were founded on geometrical differences , and these were formalised into the differential calculus.

There is no real surprise that several individuals would arrive at form of calculus, it is after all a result of deriving a method to do immense calculation schemes systematically. What Newton and Leibniz fell out over is the geometical theorems that were established by its use. The suspicion was that if the same theorem was established with the same wording and the same proof then plagiarism had taken place! It is to be noted that tis was an argument between geometers who applied a Cartesian like praxis which was heavily modelled on the Euclidean forms. However Euclid was only recently made available in latin translation to Them. Whether Descartes had Prior access i do not know, but he certainly was taught by a Euclidean tutor.

Newton in his notes was already exploring applications of Euclidean forms to irregular shapes, and applying differential techniques to derive formulae and theorem. With his help, Mercator was able to find a solution to his projection problem, based on the Logarithmic series, Cotes was able to devise the exponential formula for imaginary numbers, and together they calculated e the value of a constant needed in compound interest formulae, but also applicable to a geometric form called a loxodrome, a rhumb line. This was especially important for commerce, since navigation on a globe required an invariant solution.

Everyone of these successes is an application of Euclidean praxis and geometry in a pragmatic way. From Ptolemy to the present day The scheme that Euclid drew together from the insight of artisans and engineers and thinkers has found fundamental applicability and advancement into the subject areas of physics and engineering, and mathematics, but the praxis has spread to all the sciences, and the philosophical rationale has guided thinking until now.

The Euclid you know is not the Euclid that is discoverable within his elements.http://aleph0.clarku.edu/~djoyce/java/elements/bookI/defI8.html