![xvi Preface
what might be the variety of the subgeometries involved in a partition
of a projective space?
Definition 1. Let P be a projective space. A ‘
parallelism’of P
is an equivalence relation on the set L of lines such that the Euclidean
parallel postulate holds.
Let ` be a line and denote by [`], the equivalence class containing `.
Then let P be any point of P that is not incident with `. Then there
exists a unique line `P , incident with P and ‘
parallel’to `. Hence, there
is a partition of the points of P by any equivalence class.
Definition 2. A partition of the points of a projective space P by
a set of disjoint lines is said to be a ‘
line spread’of P .
Just as we have pointed out previously:
Remark 1. Since partitions of the line set of a projective space
and equivalence relations are equivalent, we see that a partition of the
lines of a projective space by spreads is equivalent to a parallelism of a
projective space.
Why are parallelisms of projective spaces of interest to geometers?
Suppose the spreads of a parallelism produce ‘
…eld’ a¢ ne planes (or
projective planes). We shall get to the exact meaning of this in due
course, but merely think of the a¢ ne Euclidean plane over the …eld
of real numbers R, where we insert an arbitrary …eld for the …eld R.
Since …eld planes are so very easy to construct, it would seem likely
that there are many parallelisms consisting of …eld spreads. However, it
turns out that such parallelisms are quite hard to come by. For reasons
that will come later, we call such parallelisms ‘
regular’and the e¤ort
to …nd regular parallelisms does pay o¤ in the construction of new and
quite amazing a¢ ne planes.
In graduate school in 1964 at Washington State University, I was
studying algebraic geometry when my advisor thought that I should
broaden my reading and suggested that I take a course in …nite geom-
etry from T.G. Ostrom, who was charting some very new directions in
the construction of a¢ ne and projective planes. The short version of
this story is that I never went back to algebraic geometry and became
fascinated by the ideas of ‘
derivation’ that Ostrom was developing.
Derivation is a construction method by which a renaming of certain
subplanes (called ‘
Baer subplanes’
) as ‘
new’lines together with certain
of the ‘
old’lines sometimes can produce a totally di¤erent a¢ ne plane
from a given one. Projectively, a projective Baer subplane is a sub-
plane that touches all points and lines, in the sense that every point is](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-18-320.jpg)
![Combinatorics of Spreads and Parallelisms xvii
incident with some line of the subplane and every line is incident with
some point of the subplane. The derivation process is perhaps the most
important construction procedure in the construction of a¢ ne and pro-
jective planes. The reader interested in more details on derivation is
directed to the author’
s text Subplane Covered Nets, [114].
Actually, this text forms the fourth volume of a series of works on
…nite geometry and a word on where the present volume …ts is appro-
priate. In the text Subplane Covered Nets, [114], there is a complete
theory of derivation. The analysis of a derivable a¢ ne plane focuses
more properly on the associated net that contains the Baer subplanes
that are used in the process and which are rede…ned as lines in the
derived a¢ ne plane. Considered in this manner, a derivable net is a
net that is covered by subplanes in the sense that given any two a¢ ne
points P and Q of the net that are collinear, there is a unique subplane
P;Q of the net that contains P and Q and which shares the set of
parallel classes of the net as the line at in…nity of the corresponding
projective plane. The text then concerns the analogous theory of nets
that are covered by subplanes as in the previous manner. In this text,
there is the introduction of the direct product of a¢ ne planes of the
same order (or cardinality), which becomes a net containing the two
a¢ ne subplanes. When the two a¢ ne subplanes are Pappian, the con-
structed net is a derivable net, and in this case, we identify the a¢ ne
subplanes. If the Pappian subplanes correspond to quadratic …eld ex-
tensions of a given …eld K, then repeated application of these ideas
produces a set of derivable nets that share a regulus net coordinatized
by K. If L is a component of this regulus net and is a four-dimensional
K-space, then any Pappian K-spread of L induces a direct product
derivable net sharing the regulus net. Considering L as a projective
space PG(3; K), then any Pappian parallelism of PG(3; K) induces a
set of derivable nets, the union of which produces a spread in PG(7; K)
for a translation plane admitting SL(2; K).
Thus, the text Subplane Covered Nets, provides ideas for the analy-
sis of parallelisms of projective spaces. In this work, we shall repeat
parts of the theory of direct products that applies to parallelisms, so
as to keep this text as self-contained as possible.
The study of parallelisms in PG(3; q) involves, of course, a set of
q2
+ q + 1 spreads in PG(3; q) that construct translation planes. In the
work Foundations of Translation Planes, [21], a complete background
is given not of only of translation planes of "dimension two," which
arise from spreads in PG(3; q), but also of general translation planes.
The reader is directed to the foundations text for general information
on translation planes, but again, we shall include enough of the theory](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-19-320.jpg)
![xviii Preface
here so that the reader is not necessarily dependent on this previous
text.
Suppose that a parallelism in PG(3; q) admits a collineation group
that leaves one spread invariant and acts transitively on the remain-
ing q2
+ q spreads. Then we shall show that the …xed spread de…nes a
Desarguesian translation plane and the remaining spreads de…ne trans-
lation planes of order q2
with associated spreads in PG(3; q) that admit
Baer groups of order q (a group of order q that …xes a Baer subplane
pointwise). Such translation planes actually correspond to ‡
ocks of
quadratic cones and so certain of the theory of ‡
ocks of quadratic sets
shall be developed in this text.
Considering that a parallelism is de…ned as a covering of the line set
by a spread, a natural generalization would be to consider analogous
geometries as follows: If a spread is a covering of the point set by
mutually disjoint ‘
blocks,’then a covering of the block set by spreads
can be called a parallelism. When we consider parallelisms of quadric
sets in PG(3; K), for K a …eld, we would used the term ‘
‡
ock’instead
of spread so that a parallelism of a quadric set is a covering of the
conics by a set of mutually disjoint ‡
ocks. Generalizations include
parallelisms of sharply k-transitive geometries, for k 2.
In general, there is a tremendous variety of connections with the
theory of translation planes and other point-line geometries, all of
which are explicated in the text Handbook of Finite Translation Planes,
[138]. What has not been done is to give a comprehensive and "up to
date" treatment of parallelisms of projective spaces and other geome-
tries, and, of course, the intent here is to do just that. We have men-
tioned translation planes and the vector space partitions that produce
them but there are other spreads that do not produce translation planes
but may produce Sperner Spaces. One important class of geometric
structures associated with Sperner Spaces is that of a subgeometry
partition of a projective space. More generally, a subgeometry parti-
tion of a projective space need not produce or correspond to a Sperner
space or a translation plane but always gives rise to a general partition
of a vector space. Even though there are some examples of partitions
of vector spaces known, there is almost no theory that has developed
from such covers. Here we also provide a census of most of the known
examples of partitions and what can be said theoretically of associated
point-line geometries. Since there are many new subgeometry parti-
tions constructed, we then provide a complete description of all known
examples.
Therefore, it is felt that this fourth text Combinatorics of Spreads
and Parallelisms, is a strong complement to the previous three texts](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-20-320.jpg)
![xx Preface
Mauro Biliotti and Vik Jha for twenty-…ve years or so of great collab-
oration and lasting friendship (and many wonderful joint articles and
two great books Foundations of Translation Planes and Handbook of
Finite Translation Planes).
On another continent, later in 1985, I began collaborating with
Rolando Pomareda of Santiago, Chile, also many times on parallelisms
and returned to Santiago a number of times. I thank Sylvia and
Rolando for their gracious hospitality and to Rolando for his help and
insights with our joint work and for his friendship over these twenty-…ve
years.
Parts of this text also appear in certain of the previous three works
on …nite geometry, predominantly in the Subplanes text [114], and I am
grateful to Taylor and Francis for allowing the inclusion of this material
here. Certain of the material found in this text follows work previously
published and which has been suitably modi…ed for the treatment given
here. Much of the work, however, is quite new, and a set of open
problems is given at the end of the text that should be accessible to
most researchers in …nite incidence geometry.
When thinking of this time and how it is that I actually became
a mathematician interested in …nite geometry, it is also impossible for
me not to mention the in‡
uence my mother Catherine Elizabeth Lamb
(Gabriel) has had on my career in mathematics. My mother was a
talented mathematician in spirit and ability, but my grandfather was
rather old-fashioned and was not willing to support a university educa-
tion for his children. So, I feel that I was the recipient from my mother
of whatever ability I may have in mathematics.
Of course, there is another Catherine Elizabeth, with very similar
talents— my daughter, and yes, she was named after her grandmother.
Her very strong analytical abilities are now being applied in the law
as she is a disability rights attorney, graduated the University of Iowa,
with a J.D. and a Master’
s in Education and now is specializing in the
important area of defending the rights of children.
My son Garret Norman also inherited his grandmother’
s mathemat-
ical abilities and has opted rather for the …eld of engineering, holding
a Bachelor’
s and Master’
s Degree in Electrical Engineering from the
University of Florida he has become a genuine leader in his …eld.
All of these years later, I might add that the son who accompa-
nied me to Italy, Scott Hamilton, pursued a Ph.D. from the University
of Nebraska and is now a Professor of Psychology at Western Illinois
University and is a very active researcher in a variety of exciting new
directions in psychology.](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-22-320.jpg)
![4 Partitions of Vector Spaces
(1) Given two distinct points P and Q; there is a unique line `
incident with P and Q, and which we shall denote by PQ.
(2) Given two distinct lines p and q; there is a unique point L
incident with p and q, and which we shall denote by p q.
(3) There are at least four points no three of which are ‘
collinear’
(are not incident with a common line).
Remark 3. It is well known and straightforward to check that given
a projective plane +
, it is possible to construct an a¢ ne plane by
removing a line ` and the subset of the points incident with `, [`], and
where incidence of the new structure is inherited from the projective
plane. Hence, if (P; L; I) is a triple that de…nes a projective plane,
then (P [`]; L f`g; I f(P; `); P 2 [`]g) is an a¢ ne plane .
The a¢ ne plane is called the ‘
a¢ ne restriction of the projective
plane by `.’ The line ` then de…nes the set of parallel classes of the
a¢ ne plane and is sometimes called the ‘
line at in…nity.’
If is an a¢ ne plane, for each parallel class , formally adjoin a
‘
point’to each line of and let the union of the parallel classes be a
formal line `1, again called the ‘
line at in…nity’of . With the natural
extension of the point, line, and incidence sets, an associated projective
plane +
is constructed, called the ‘
projective completion of the a¢ ne
plane .’
We shall also be interested in point-line geometries with line par-
allelism with a slightly weaker set of axioms than that of an a¢ ne
plane.
Definition 5. A ‘
Sperner Space’ is a triple (P; L; I), of sets of
‘
points P,’ ‘
lines L,’ and ‘
incidence I P L,’ with the following
properties:
(1) Given two distinct points P and Q; there is a unique line `
incident with P and Q, and which we shall denote by PQ.
(2) There is an equivalence relation on the line set called ‘
paral-
lelism.’
(3) Given a line ` and a point P not incident with `, there is a
unique line m that is parallel with ` and is incident with P.
(4) For any two distinct lines ` and m; the cardinal number of the
set of points incident with `, [`]; is equal to the cardinal number of the
set of points incident with m, [m].
Remark 4. Let be an a¢ ne plane and and be distinct parallel
classes and let lines ` 2 and m 2 . Then clearly m and ` are
incident with a unique common point, for otherwise m would be parallel
to `.](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-27-320.jpg)
![Combinatorics of Spreads and Parallelisms 7
A ‘
translation focal-plane’is a translation geometry whose underly-
ing point-line geometry is a focal-plane.
A ‘
double-plane’ is a point-line geometry of type fs1; s2g and a
translation double-plane is a translation geometry whose underlying
point-line geometry is a double-plane.
In the following, we shall give constructions of focal-spreads and of
more general double spreads, where the ‘
spread’(or perhaps general-
ized spread or partition) is the underlying set of vector subspaces of a
translation geometry that corresponds to a vector space.
0.2. Beutelspacher’
s Construction. Let Vt+k be a vector space
of dimension t + k over GF(q) for t > k and let L be a subspace of
dimension t. Let V2t be a vector space of dimension 2t containing Vt+k
(t > k required here) and let St be a t-spread containing L. There are
always at least Desarguesian t-spreads with this property. Let Mt be a
component of the spread St not equal L. Then Mt Vt+k is a subspace
of Vt+k of dimension at least k. But, since Mt is disjoint from L, the
dimension is precisely k and we then obtain a focal-spread with focus
L. This construction may be found in Beutelspacher [14], Lemma 2,
page 205. The corresponding focal-spread is said to be a ‘
k-cut of a
t-spread,’ and we shall adopt this terminology and use the notation
F = StnVt+k for the focal-spread F.
The similar formal de…nition is
Definition 10. A partition of a …nite-dimensional vector space
of dimension t + k by a partial Sperner k-spread and a subspace of
dimension t 6= k shall be called a ‘
focal-spread of type (t; k).’ The
unique subspace of dimension t of the partition shall be called the ‘
focus’
of the focal-spread.
We de…ne a ‘
planar extension’ of a focal-spread as a t-spread such
that the focal-spread is of type (t; k) and arises from the t-spread as a
k-cut.
The ‘
kernel’ of a focal-spread of dimension t+k, over GF(q), with
focus of dimension t, shall be de…ned as the endomorphism ring of the
vector space that leaves the focus and each k-component invariant.
Now we turn to a similar construction of more general double-
spreads.
0.3. Double-Spreads Constructed from t-Spreads. Of course,
the idea of cutting or slicing a spread could conceivably lead to other in-
teresting partitions. For example, from a t-spread of a 2t-dimensional
vector space, we could simply select a 2t 1 dimensional subspace](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-30-320.jpg)
![Combinatorics of Spreads and Parallelisms 9
0.4. Main Theorem on Finite Translation Geometries. The
following material also is valid for arbitrary translation planes and the
reader is referred to the Foundations text [21] to see this. The proof of
the theorem stated below for …nite translation geometries is more gen-
erally valid for arbitrary translation planes, with appropriate changes.
Theorem 2. Every …nite translation geometry corresponds to a
…nite vector space over a …eld K. The …eld K may be identi…ed with
the endomorphism ring of the associated translation group that leaves
invariant each parallel class.
We shall give the proof as a series of lemmas, parts of which are left
to the reader to complete. In the following, we assume the hypothesis
of Theorem 2. We …rst point out some trivial facts.
Remark 8. (1) Each point is incident with a unique line of each
parallel class.
(2) A non-identity translation …xes a unique parallel class linewise.
(3) A non-identity translation is …xed-point-free so that the group
T of translations is …xed-point-free.
Proof. Part (1) follows immediately from the de…nitions. Let
be a translation. Then …xes all parallel classes and …xes one parallel
class linewise. Suppose that …xes another parallel class 6=
linewise. Choose any point P, so there is a line ` of and a line `
of both incident with P. Since …xes both of the indicated lines, it
follows that …xes P, so that …xes all points and hence is the identity
mapping. If …xes an a¢ ne point P, then …xes all lines incident
with P as it …xes all parallel classes. Choose any point Q distinct from
P and form the line QP. Let m be the line of incident with Q. Since
…xes QP and m, it follows that it also …xes Q. Hence, …xes all
points and so is the identity translation. If is in the translation group
it then …xes all parallel classes. If …xes an a¢ ne point P, then …xes
all lines incident with P.
Definition 12. If is a translation of a translation geometry, the
unique parallel class linewise …xed by is called the ‘
center’of . If
is a parallel class, let T( ) denote the group of translations with center
.
The …rst two parts of the following lemma are immediate.
Lemma 1. Let T denote the group of translations of a translation
geometry.
(1) Then T( ) is a normal subgroup for each parallel class .
(2) T = [ T( ), a partition of T.](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-32-320.jpg)
![10 Partitions of Vector Spaces
(3) T is Abelian.
Proof. (3): Let g and h be elements of T( ) and T( ), for 6= .
Then the commutator ghg 1
h 1
is in the intersection of both trans-
lation subgroups, which are disjoint. Hence, elements of T( ) and
elements of T( ) commute. Let g and k be elements of T( ) and
h an element of T( ). Then (gk)h = g(kh) = g(hk) = (gh)k, and
kh 2 T( ), for a parallel class distinct from or , implying that
g(kh) = (kh)g = k(hg) = (hg)k, using the same principle. Hence,
(gh)k = (hg)k, so that gh = hg. This completes the proof.
Lemma 2. Let K denote the set of endomorphisms of T that …x each
T( ). Then K is a …eld and T is a vector space over K. Identifying
the point set with T, then the translation geometry becomes a vector
space over K and the lines incident with 0 are vector subspaces over K
so that we have a partition of a vector space.
Proof. Let be a non-zero endomorphism. If is not injective,
then since T is Abelian, we may assume that (z) = 1; for some non-
identity element z of a partition element, say T( ). For u in T( ),
for 6= , then zu 2 T( ), for distinct from or . So, (zu) =
(z) (u) = (u), but (zu) 2 T( ) and (u) 2 T( ) so both are in
the intersection T( ) T( ) = h1i. Therefore, T( ) Kernel , which
also implies that [ 6= T( ) Kernel . Now repeat the argument for
a kernel element w in T( ), to obtain T( ) Kernel , so that is
the identity endomorphism. Since the translation geometry is …nite, it
follows that K is a …eld. Then considering the natural endomorphism
ring of T, T becomes a K-vector space. Now identifying the points
of the translation geometry with the elements of T and using additive
notation, choose the point 0 (the ‘
origin’
) then it follows immediately
that lines through the origin may be identi…ed with the groups T( ),
which are clearly K-vector subspaces. This completes the proof of the
theorem.
1. Collineation Groups of Translation Geometries
In this section, we show that the full collineation group of a trans-
lation geometry with underlying vector space V over the …eld K (or
skew…eld if the translation geometry is a translation plane) is a semi-
direct product of a subgroup of L(T; K) by the translation subgroup
T. The proof given is essentially the same as that in Lüneburg [167]
(1.10), determining isomorphisms of …nite translation planes.
If G is a collineation group of a translation geometry (…nite with
…eld kernel K or an arbitrary translation plane with skew…eld kernel](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-33-320.jpg)
![12 Partitions of Vector Spaces
1.1. Collineation Groups of Translation Planes. We now spe-
cialize to …nite translation planes. We shall be mostly interested in …-
nite translation planes whose underlying vector space is 4-dimensional
over a …eld K isomorphic to GF(q), where q = pr
, for p a prime and r
a positive integer. We shall refer to the parallel classes of a translation
plane as the points on the line at in…nity.
Definition 13. If V is a vector space of dimension z over a …eld L,
the ‘
projective geometry PG(z 1; L)’ is the lattice of vector subspaces
of V . A 1-dimensional subspace is said to be a ‘
point,’a 2-dimensional
vector subspace a ‘
line,’and a 3-dimensional vector space a ‘
plane.’
An ‘
a¢ ne space AG(z; L),’is the geometry of translates of the sub-
spaces of a vector space.
Definition 14. A collineation of a (an a¢ ne) translation plane
is a ‘
central collineation’provided …xes a line ` of the plane pointwise.
The collineation is an ‘
elation’if …xes all lines parallel to `, and we
say the ‘
center’ of is incident with ` (on the line at in…nity) and `
is the ‘
axis’of . The collineation is a ‘
homology’if …xes all lines of
a parallel class not containing `. If the line of incident with 0 is
denoted by 0 , we shall say that ` is the ‘
axis,’0 is the ‘
coaxis’and
is the ‘
center’of .
Definition 15. If +
is a …nite projective plane of order q2
(q2
+1
is the number of points per line), a projective subplane of order q is
said to be a ‘
Baer’subplane +
0 . Any associated a¢ ne restriction of
+
is also said to have order q2
. If the Baer subplane of +
contains
the line of restriction, then the corresponding a¢ ne subplane of order
q is a subplane of .
A ‘
Baer collineation group’of a …nite a¢ ne plane is a collineation
group that …xes a Baer subplane pointwise.
Remark 9. For …nite translation planes of order q2
, it is known
that any Baer collineation group has order dividing q(q 1). For more
details, the reader is directed to Biliotti, Jha, Johnson [21].
Notation 1. In this text, when considering translation planes of
order q2
, constructed or equipped with a 2-spread over GF(q), we may
always represent vectors in the form (x1; x2; y1; y2), for all xi, yi 2
GF(q); i = 1; 2, and further let x = (x1; x2), y = (y1; y2). Then there
is a set of q2
2 2 matrices over GF(q), including the zero matrix, so
that the components of the 2-spread are as follows:
x = 0; y = 0; y = x
g(t; u) f(t; u)
t u
; u; t 2 GF(q),](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-35-320.jpg)
![CHAPTER 2
Finite Focal-Spreads
In the previous chapter, we brie‡
y discussed focal-spreads and the
Beutelspacher’
s construction. Although there are a number of examples
of such partitions, there has been no theory developed about general
partitions. Part of our treatment in this text is modeled from the article
by Jha and Johnson [77], and the reader is directed to this article for
additional details.
Although it is certainly possible to consider focal-spreads over arbi-
trary …elds, all of the material that is presented in this text is for …nite
focal-spreads. The reader is directed to the open problem chapter 40
for more information.
In this chapter, we begin to build a theory of partitions based on
focal-spreads and their obvious connections to translation planes. For
general focal-spreads, however, nothing is known. For example, it is
not known whether collineations that …x components pointwise or ho-
mologies are elations or whether an involution that does not …x a com-
ponent pointwise becomes a Baer collineation or a kernel involution.
In the ‘
Handbook of Finite Translation Planes’[138], it is mentioned
that there are not very many known partitions of vector spaces, in
the sense that the partitions are not re…nements of Sperner t-spreads.
Furthermore, the existence of such partitions has been established by
Beutelspacher [14], who proves if the dimension of the vector space is n
and it is required to …nd a partition, where the dimensions of the sub-
spaces are ft1; t2; ::; tkg, where ti < ti+1; then if gcdft1; t2; ::; tkg = d and
n > 2t1([tk=(d k)]+t2+:::+tk), a partition may be constructed with var-
ious subspaces of dimension ti. However, these partitions are basically
constructed using focal-spreads and general spreads. We give some of
the constructions later in the section on ‘
towers of focal-spreads.’
The main point of this chapter is to use of ideas and theory of
…nite translation planes in order to develop a certain theory of focal-
spreads and, since these seem to be important building blocks of general
spreads, such work might prove useful to the general theory of parti-
tions.
15](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-37-320.jpg)
![16 Finite Focal-Spreads
Given a k-cut, of course, there is a corresponding spread for a trans-
lation plane that produces it. It is a central and potentially important
question to ask if every focal-spread is a k-cut; if it can ‘
extended’to
a spread for a translation plane.
Therefore, here we give the basics of focal-spreads and use some
theory from translation planes for their study. We also show that the
existence of focal-spreads of type (k + 1; k) leads to a construction of
designs of type 2 (qk+1
; q; 1) and to other double-spreads or triple-
spreads (partitions where there are subspaces in the partition of either
two or three di¤erent dimensions, respectively).
The formal de…nition of a focal-spread has been given in De…nition
10.
1. Towers of Focal-Spreads
As we begin our discussions on t-spreads, we have brie‡
y discussed
focal-spreads and found that they are readily constructed, but we might
also just as well ask the obvious question: How hard is it to …nd general
partitions of vector spaces? That is, partitions of the non-zero vectors
by a set of subspaces of dimensions ti, for i = 1; 2; ::; k.
In Beutelspacher [16], and more generally in Heden [65], there are a
variety of results of the existence of partitions of type T = ft1; t2; ::; tkg,
in vector spaces of dimension n over GF(q), where the partition admits
exactly subspaces over GF(q) of dimension ti, for i = 1; 2; ::; n, for
t1 < t2 < :::tk. (The reader is cautioned that our previous notation of
partitions has the inequalities in the opposite order.) Many of these
results rely basically on focal-spreads of type (t; k); in the sense that a
variety of focal-spreads or t-spreads are used in the construction. We
give here one construction that provides a partition of type T in any
vector space of dimension n =
Pk
i=1 iti over GF(q), showing that
again general partitions are readily available.
First, we note the following lemma.
Lemma 3. In an st-dimensional vector space over GF(q), there is
always a t-spread over GF(q).
Proof. Let the vectors be represented by GF(qst
). Take the cosets
of GF(qt
) in GF(qst
) . Note that GF(qt
) g[f0g is a GF(q)-subspace.
Since the cosets partition GF(qst
) , representing the non-zero vectors,
the lemma is proved.
Lemma 4. Let V be a t1 +t2-dimensional vector space over GF(h),
for h a prime power. If t2 > t1, then there is focal-spread of type
(t2; t1).](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-38-320.jpg)
![Combinatorics of Spreads and Parallelisms 19
2.1. Matrix Representation of Focal-Spreads. As for planar
spreads, it is usually helpful to express these results in terms of matri-
ces. The reader might wish to work through this material carefully as
the ideas closely parallel analogous matrix coordinatization for trans-
lation planes. For addition background, the Foundations text can be
consulted [21].
Let B be a focal-spread of dimension t + k over GF(q) with focus
L of dimension t. Fix any k-component M. We may choose a basis
so that the vectors have the form (x1; x2; ::; xk; y1; y2; ::; yt). Let x =
(x1; x2; ::; xk) and y = (y1; y2; ::; yt), where the focus L has equation x =
0 = (0; 0; 0; ::; 0) (k-zeros) and the …xed k-component M has equation
y = 0 = (0; 0; 0; ::; 0) (t-zeros). We note that qt+k
qt
= qt
(qk
1), which implies that there are exactly qt
k-subspaces in the focal-
spread. We refer to this as the ‘
partial Sperner k-spread’
. Take any
k-component N distinct from y = 0. There are k basis vectors over
GF(q), which we represent as follows: y = xZk;t, where Zk;t is a k t
matrix over GF(q), whose k rows are a basis for the k-component.
It is clear that we obtain a set of qt
k-components, which we also
represent as follows: Row 1 shall be given by [u1; u2; ::; ut], as the ui
vary independently over GF(q). Then the rows 2; ::; k have entries
that are (turn out to be) linear functions of the ui. It then now follows
directly that the k t matrices in the focal-spread have rank k and the
di¤erence of any two distinct matrices associated with k-components
also has rank k.
The following is the analogous result of translation planes:
Theorem 5. Let Vt+k be a t + k-dimensional vector space over
GF(q) and let S be a set of qt
1 k t matrices of rank k such that
the di¤erence of any two distinct matrices also has rank k. Then there
is an associated focal-spread constructed as
x = 0; y = 0; y = xM; M 2 S,
where x is a k-vector and y is a t-vector over GF(q), where the focus
is x = 0.
In particular, it is possible to choose one k-space to have 1’
s in the
(i; i), position and 0’
s elsewhere in the k t matrices. We denote this
matrix by Ik t.
Conversely, any focal-spread has such a representation.
The reader is directed to the open problem chapter 40 for more
discussion on how a generalization for arbitrary partitions might be
phrased using a matrix approach.
There is, of course, a coordinate-free approach given as follows:](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-41-320.jpg)
![20 Finite Focal-Spreads
Definition 17. Let V = Vt Vk be a …nite vector space of rank t+k
expressed as a direct sum of subspaces Vt and Vk, having dimensions
t and k, respectively. Then a ‘(t;k)-spread set on V ,’based on axes
(Vk; Vt), is a collection S of linear maps from Vk to Vt, such that
(1) 0 2 S;
(2) the nonzero maps in S are injective;
(3) the di¤erence between any two members of S is injective or zero;
(4) S is transitive in the sense that for any pair of non-zero vectors
The following remarks are then left to the reader to verify.
Remark 10. Every (t; k)-spread set S as above yields a focal-spread
of type (t; k), with component set fMS : S 2 Sg [ fVtg, where MS :=
f(x; xS) : x 2 Vkg.
Every focal-spread of type (t; k) may be coordinatized by a (t; k)-
spread set, and any focal-spread of type (t; k) arises by coordinatization
by a (t; k)-spread set, which is uniquely determined by the focus and
any other component chosen as ‘
basis.’
3. k-Cuts and Inherited Groups
In this section, if a focal-spread is a k-cut of a translation plane,
we ask what sorts of collineations of the translation plane become
collineations of the focal-spread? More generally, given an abstract
focal-spread, how are central collineations de…ned? We begin with the
simplest type of collineation of a focal-spread, a ‘
homology.’
Definition 18. In a focal-spread of dimension t + k over GF(q)
and focus L of dimension t, every collineation is assumed to be an
element of L(t + k; q) that leaves invariant the focus L and permutes
the components of the partial Sperner k-space S.
A ‘
homology’h is a collineation of GL(t + k; q) with the following
properties:
(1) h leaves invariant the focus L and another k-subspace of S,
(2) h …xes one of the two …xed components pointwise and acts …xed-
point-free on another …xed component (in the case that a k-component
is …xed pointwise, we assume that the group acts …xed-point-free on the
focus).
To show why there is a potential problem with dealing with collineations
that one might wish to call a homology, consider the focal-spread of
type (2; 1) over GF(q), represented as
x = 0; y = x[t; u], for all t; u 2 GF(q),](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-42-320.jpg)
![Combinatorics of Spreads and Parallelisms 21
where the focus is x = 0, a 1-vector. Let B =
1 1
0 1
, and consider
the collineation rB : (x; y) ! (x; yB). We note that (0; 1)
1 1
0 1
=
(0; 1), so that rB is not semi-regular on the focus, and since y = x[0; 1] is
…xed by rB, the collineation is also not semi-regular on the components
other than the putative coaxis x = 0.
In terms of homologies that are not a¢ ne collineations that …x the
line at in…nity pointwise (the axis) and …x an a¢ ne point (the center),
the situation is much more manageable. This is because a …nite focal-
spread is a …nite translation geometry, and Theorem 3 shows that the
subgroup that …xes each component (k-component and focus) is the
cyclic subgroup of a …eld; the kernel is a …eld.
We now intend to show that a non-identity homology (see De…nition
18) …xes exactly two components. The pointwise …xed subspace is
called the ‘
axis’of h and the …xed subspace is the ‘
coaxis’of h. Thus,
the focus is either the axis or coaxis of any homology.
Proposition 1. A non-identity homology of a focal-spread …xes
exactly two components and permutes the remaining components semi-
regularly.
Proof. We represent the focal-spread in the form
x = 0; y = 0; y = xM; M is a k t matrix in set M
for x a k-vector and y a t-vector, where x = 0 is the focus, and y = 0
is a k-space, and where we choose Ik t 2 M. Assume that y = xM1
is …xed pointwise by a collineation h. Choose a basis by the mapping
(x; y) ! (x; xM1 + y), to change the form to
x = 0; y = 0; y = x(M1 M); M is a k t matrix in set M.
Now choose any of these matrices M1 M2, of rank k, and column-
reduce to Ik t. The other matrices M1 M reduce to rank k matri-
ces, whose di¤erences are also of rank k. Hence, if y = xM1 is …xed
pointwise by h, we may assume without loss of generality that y = 0
is …xed pointwise by h. Therefore, any collineation in GL(t + k; q)
that …xes a component pointwise may be represented in either the
form lA : (x; y) ! (xA; y), where A is a non-singular k k matrix
or rB : (x; y) ! (x; yB), where B is a non-singular t t matrix.
First, assume that rB is a homology, so that B acts …xed-point-free
on the focus. Assume that rB …xes x = 0; y = 0; and y = xM. So,
xMB = xM. Choose then any x0M = z0, for z0 a t-vector, so that
z0B = z0. However, this is contrary to the action of B. Since the](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-43-320.jpg)
![26 Finite Focal-Spreads
We shall return to these ideas in Chapter 39, when we consider
doubly transitive focal-spreads. In this setting, it is possible to produce
elation groups as well as homology groups.
4. Spread-Theoretic Dual of a Semi…eld
We interrupt our discussion of focal-spreads to prepare for the ideas
of an ‘
additive’focal-spread, which would be the natural analogue of
a semi…eld spread. Although all of the material in this chapter can
be given by various other methods, we treat the concepts here strictly
from a spread-theoretic viewpoint. This chapter is modi…ed from work
of Jha and Johnson [81].
Given an a¢ ne translation plane, complete to the projective exten-
sion and dualize the projective plane. If the translation plane is a semi-
…eld plane and if the line at in…nity becomes the parallel class (1), then
restricting to the a¢ ne plane from the dual semi…eld plane, another
semi…eld plane is constructed. Normally all of this is done using the
coordinate semi…eld, as taking the reverse or opposite multiplication,
a coordinate semi…eld for the dual ‘
a¢ ne’semi…eld plane may be con-
structed. Speci…cally, given a …nite semi…eld (S; +; ), then the ‘
dual
semi…eld’(S; +; ) may be de…ned by the multiplication a b = b a.
It is also true that for …nite translation planes, a duality of the
associated projective space provides what we call a ‘
dual spread’, which
is always a spread in the …nite case. From a matrix spread set, if
x = 0; y = x; y = xM, for M 2 SMat
is a matrix spread set for a semi…eld plane, then
x = 0; y = x; y = xMt
, for M 2 SMat,
where Mt
denotes the transpose of the matrix M, also gives a semi…eld
(see, e.g., Johnson [133]). In general, there are six semi…elds associated
with a given one, and in this chapter, we develop a notion of the dual of
a semi…eld strictly from the matrix spreadset viewpoint. This material
explicates the original results of Kantor [156] but given completely in
terms of the spreads and gives a spread description of the six semi…elds
arising from a given semi…eld.
5. The Dual Semi…eld Plane
We begin with the dual semi…eld. Of course, any …nite semi…eld
spread of order pn
, for p a prime, may be written in the form
x = 0; y = xM,](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-48-320.jpg)
![In what times the
Planets would fall to
the Sun by the
power of gravity.
The prodigious
attraction of the
Sun and Planets.
cause them to ascend again towards the higher parts of their Orbits;
during which time, the Sun’s attraction acting so contrary to the
motions of those bodies, causes them to move slower and slower,
until their projectile forces are diminished almost to nothing; and
then they are brought back again by the Sun’s attraction, as before.
157. If the projectile forces of all the
Planets and Comets were destroyed at their
mean distances from the Sun, their
gravities would bring them down so, as that
Mercury would fall to the Sun in 15 days 13 hours; Venus in 39 days
17 hours; the Earth or Moon in 64 days 10 hours; Mars in 121 days;
Jupiter in 290; and Saturn in 767. The nearest Comet in 13 thousand
days; the middlemost in 23 thousand days; and the outermost in 66
thousand days. The Moon would fall to the Earth in 4 days 20 hours;
Jupiter’s first Moon would fall to him in 7 hours, his second in 15, his
third in 30, and his fourth in 71 hours. Saturn’s first Moon would fall
to him in 8 hours; his second in 12, his third in 19, his fourth in 68
hours, and the fifth in 336. A stone would fall to the Earth’s center, if
there were an hollow passage, in 21 minutes 9 seconds. Mr. Whiston
gives the following Rule for such Computations. “[31]
It is
demonstrable, that half the Period of any Planet, when it is
diminished in the sesquialteral proportion of the number 1 to the
number 2, or nearly in the proportion of 1000 to 2828, is the time
that it would fall to the Center of it’s Orbit.” This proportion is, when
a quantity or number contains another once and a half as much
more.
158. The quick motions of the Moons of
Jupiter and Saturn round their Primaries,
demonstrate that these two Planets have
stronger attractive powers than the Earth has. For, the stronger that
one body attracts another, the greater must be the projectile force,
and consequently the quicker must be the motion of that other body,
to keep it from falling to it’s primary or central Planet. Jupiter’s
second Moon is 124 thousand miles farther from Jupiter than our](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-50-320.jpg)
![Archimedes’s
Problem for raising
the Earth.
Moon is from us; and yet this second Moon goes almost eight times
round Jupiter whilst our Moon goes only once round the Earth. What
a prodigious attractive power must the Sun then have, to draw all
the Planets and Satellites of the System towards him; and what an
amazing power must it have required to put all these Planets and
Moons into such rapid motions at first! Amazing indeed to us,
because impossible to be effected by the strength of all the living
Creatures in an unlimited number of Worlds, but no ways hard for
the Almighty, whose Planetarium takes in the whole Universe!
159. The celebrated Archimedes affirmed
he could move the Earth if he had a place
to stand on to manage his machinery[32]
.
This assertion is true in Theory, but, upon examination, will be found
absolutely impossible in fact, even though a proper place and
materials of sufficient strength could be had.
The simplest and easiest method of moving a heavy body a little
way is by a lever or crow, where a small weight or power applied to
the long arm will raise a great weight on the short one. But then,
the small weight must move as much quicker than the great weight
as the latter is heavier than the former; and the length of the long
arm of the lever to the length of the short arm must be in the same
proportion. Now, suppose a man pulls or presses the end of the long
arm with the force of 200 pound weight, and that the Earth contains
in round Numbers 4,000,000,000,000,000,000,000 or 4000 Trillions
of cubic feet, each at a mean rate weighing 100 pound; and that the
prop or center of motion of the lever is 6000 miles from the Earth’s
center: in this case, the length of the lever from the Fulcrum or
center of motion to the moving power or weight ought to be
12,000,000,000,000,000,000,000,000 or 12 Quadrillions of miles;
and so many miles must the power move, in order to raise the Earth
but one mile, whence ’tis easy to compute, that if Archimedes or the
power applied could move as swift as a cannon bullet, it would take
27,000,000,000,000 or 27 Billions of years to raise the Earth one
inch.](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-51-320.jpg)
![The Planets disturb
one another’s
motion.
The consequences
thereof.
The World not
eternal.
carry off the Planets in straight lines from those parts of their Orbits
where Gravity left them. But, the Planets being once put into motion,
there is no occasion for any new projectile force, unless they meet
with some resistance in their Orbits; nor for any mending hand,
unless they disturb one another too much by their mutual
attractions.
163. It is found that there are
disturbances among the Planets in their
motions, arising from their mutual
attractions when they are in the same
quarter of the Heavens; and that our years
are not always precisely of the same
length[33]
. Besides, there is reason to believe that the Moon is
somewhat nearer the Earth now than she was formerly; her
periodical month being shorter than it was in former ages. For, our
Astronomical Tables, which in the present Age shew the times of
Solar and Lunar Eclipses to great precision, do not answer so well
for very ancient Eclipses. Hence it appears, that the Moon does not
move in a medium void of all resistance, § 174; and therefore her
projectile force being a little weakened, whilst there is nothing to
diminish her gravity, she must be gradually approaching nearer the
Earth, describing smaller and smaller Circles round it in every
revolution, and finishing her Period sooner, although her absolute
motion with regard to space be not so quick now as it was formerly:
and therefore, she must come to the Earth at last; unless that Being,
which gave her a sufficient projectile force at the beginning, adds a
little more to it in due time. And, as all the Planets move in spaces
full of æther and light, which are material substances, they too must
meet with some resistance. And therefore, if their gravities are not
diminished, nor their projectile forces increased, they must
necessarily approach nearer and nearer the Sun, and at length fall
upon and unite with him.
164. Here we have a strong philosophical
argument against the eternity of the World.
For, had it existed from eternity, and been left by the Deity to be](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-54-320.jpg)
![The amazing
smallness of the
particles of light.
The dreadful effects
that would ensue
from their being
larger.
CHAP. VIII.
Of Light. It’s proportional quantities on the different
Planets. It’s Refractions in Water and Air. The
Atmosphere; it’s weight and properties. The
Horizontal Moon.
165. Light consists of exceeding small
particles of matter issuing from a luminous
body; as from a lighted candle such particles
of matter continually flow in all directions. Dr. Niewentyt
[34]
computes,
that in one second of time there flows
418,660,000,000,000,000,000,000,000,000,000,000,000,000,000
particles of light out of a burning candle; which number contains at
least 6,337,242,000,000 times the number of grains of sand in the
whole Earth; supposing 100 grains of sand to be equal in length to an
inch, and consequently, every cubic inch of the Earth to contain one
million of such grains.
166. These amazingly small particles, by
striking upon our eyes, excite in our minds
the idea of light: and, if they were so large
as the smallest particles of matter
discernible by our best microscopes, instead of being serviceable to
us, they would soon deprive us of sight by the force arising from their
immense velocity, which is above 164 thousand miles every
second[35]
, or 1,230,000 times swifter than the motion of a cannon
bullet. And therefore, if the particles of light were so large, that a
million of them were equal in bulk to an ordinary grain of land, we](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-56-320.jpg)
![How objects
become visible to
us.
PLATE II.
The rays of Light
naturally move in
straight lines.
A proof that they
hinder not one
another’s motions.
Fig. XI.
In what proportion
durst no more open our eyes to the light than suffer sand to be shot
point blank against them.
167. When these small particles, flowing
from the Sun or from a candle, fall upon
bodies, and are thereby reflected to our
eyes, they excite in us the idea of that body
by forming it’s picture on the retina[36]
. And
since bodies are visible on all sides, light must be reflected from them
in all directions.
168. A ray of light is a continued stream
of these particles, flowing from any visible
body in straight lines. That they move in
straight, and not in crooked lines, unless
they be refracted, is evident from bodies not
being visible if we endeavour to look at
them through the bore of a bended pipe;
and from their ceasing to be seen by the interposition of other
bodies, as the fixed Stars by the interposition of the Moon and
Planets, and the Sun wholly or in part by the interposition of the
Moon, Mercury, or Venus. And that these rays do not interfere, or
jostle one another out of their ways, in flowing from different bodies
all around, is plain from the following Experiment. Make a little hole
in a thin plate of metal, and set the plate upright on a table, facing a
row of lighted candles standing by one another; then place a sheet of
paper or pasteboard at a little distance from the other side of the
plate, and the rays of all the candles, flowing through the hole, will
form as many specks of light on the paper as there are candles
before the plate, each speck as distinct and large, as if there were
only one candle to cast one speck; which shews that the rays are no
hinderance to each other in their motions, although they all cross in
the hole.
169. Light, and therefore heat so far as it
depends on the Sun’s rays (§ 85, towards
the end) decreases in proportion to the](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-57-320.jpg)
![light and heat
decrease at any
given
distance from the
Sun.
PLATE II.
Why the Planets
appear dimmer
when viewed thro’
telescopes than by
the bare eye.
squares of the distances of the Planets from
the Sun. This is easily demonstrated by a
Figure which, together with it’s description, I
have taken from Dr. Smith’s Optics[37]
. Let the
light which flows from a point A, and passes
through a square hole B, be received upon a
plane C, parallel to the plane of the hole; or, if you please, let the
figure C be the shadow of the plane B; and when the distance C is
double of B, the length and breadth of the shadow C will be each
double of the length and breadth of the plane B; and treble when AD
is treble of AB; and so on: which may be easily examined by the light
of a candle placed at A. Therefore the surface of the shadow C, at
the distance AC double of AB, is divisible into four squares, and at a
treble distance, into nine squares, severally equal to the square B, as
represented in the Figure. The light then which falls upon the plane
B, being suffered to pass to double that distance, will be uniformly
spread over four times the space, and consequently will be four times
thinner in every part of that space, and at a treble distance it will be
nine times thinner, and at a quadruple distance sixteen times thinner,
than it was at first; and so on, according to the increase of the
square surfaces B, C, D, E, built upon the distances AB, AC, AD, AE.
Consequently, the quantities of this rarefied light received upon a
surface of any given size and shape whatever, removed successively
to these several distances, will be but one quarter, one ninth, one
sixteenth of the whole quantity received by it at the first distance AB.
Or in general words, the densities and quantities of light, received
upon any given plane, are diminished in the same proportion as the
squares of the distances of that plane, from the luminous body, are
increased: and on the contrary, are increased in the same proportion
as these squares are diminished.
170. The more a telescope magnifies the
disks of the Moon and Planets, they appear
so much dimmer than to the bare eye;
because the telescope cannot magnify the
quantity of light, as it does the surface; and,
by spreading the same quantity of light over a surface so much larger](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-58-320.jpg)
![Fig. VIII.
Refraction of the
rays of light.
than the naked eye beheld, just so much dimmer must it appear
when viewed by a telescope than by the bare eye.
171. When a ray of light passes out of one
medium[38]
into another, it is refracted, or
turned out of it’s first course, more or less,
as it falls more or less obliquely on the
refracting surface which divides the two mediums. This may be
proved by several experiments; of which we shall only give three for
example’s sake. 1. In a bason FGH put a piece of money as DB, and
then retire from it as to A, till the edge of the bason at E just hides
the money from your sight: then, keeping your head steady, let
another person fill the bason gently with water. As he fills it, you will
see more and more of the piece DB; which will be all in view when
the bason is full, and appear as if lifted up to C. For, the ray AEB,
which was straight whilst the bason was empty, is now bent at the
surface of the water in E, and turned out of it’s rectilineal course into
the direction ED. Or, in other words, the ray DEK, that proceeded in a
straight line from the edge D whilst the bason was empty, and went
above the eye at A, is now bent at E; and instead of going on in the
rectilineal direction DEK, goes in the angled direction DEA, and by
entering the eye at A renders the object DB visible. Or, 2dly, place the
bason where the Sun shines obliquely, and observe where the
shadow of the rim E falls on the bottom, as at B: then fill it with
water, and the shadow will fall at D; which proves, that the rays of
light, falling obliquely on the surface of the water, are refracted, or
bent downwards into it.
172. The less obliquely the rays of light fall upon the surface of any
medium, the less they are refracted; and if they fall perpendicularly
thereon, they are not refracted at all. For, in the last experiment, the
higher the Sun rises, the less will be the difference between the
places where the edge of the shadow falls, in the empty and full
bason. And, 3dly, if a stick be laid over the bason, and the Sun’s rays
be reflected perpendicularly into it from a looking-glass, the shadow](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-59-320.jpg)
![The Atmosphere.
The Air’s
compression and
rarity at different
heights.
of the stick will fall upon the same place of the bottom, whether the
bason be full or empty.
173. The denser that any medium is, the more is light refracted in
passing through it.
174. The Earth is surrounded by a thin
fluid mass of matter, called the Air, or
Atmosphere, which gravitates to the Earth,
revolves with it in it’s diurnal motion, and
goes round the Sun with it every year. This
fluid is of an elastic or springy nature, and
it’s lowermost parts being pressed by the weight of all the Air above
them, are squeezed the closer together; and are therefore densest of
all at the Earth’s surface, and gradually rarer the higher up. “It is well
known[39]
that the Air near the surface of our Earth possesses a space
about 1200 times greater than water of the same weight. And
therefore, a cylindric column of Air 1200 foot high is of equal weight
with a cylinder of water of the same breadth and but one foot high.
But a cylinder of Air reaching to the top of the Atmosphere is of equal
weight with a cylinder of water about 33 foot high[40]
; and therefore if
from the whole cylinder of Air, the lower part of 1200 foot high is
taken away, the remaining upper part will be of equal weight with a
cylinder of water 32 foot high; wherefore, at the height of 1200 feet
or two furlongs, the weight of the incumbent Air is less, and
consequently the rarity of the compressed Air is greater than near the
Earth’s surface in the ratio of 33 to 32. And having this ratio we may
compute the rarity of the Air at all heights whatsoever, supposing the
expansion thereof to be reciprocally proportional to its compression;
and this proportion has been proved by the experiments of Dr. Hooke
and others. The result of the computation I have set down in the
annexed Table, in the first column of which you have the height of
the Air in miles, whereof 4000 make a semi-diameter of the Earth; in
the second the compression of the Air or the incumbent weight; in
the third it’s rarity or expansion, supposing gravity to decrease in the
duplicate ratio of the distances from the Earth’s center. And the small](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-60-320.jpg)
![It’s weight how
found.
PLATE II.
numeral figures are here used to shew what number of cyphers must
be joined to the numbers expressed by the larger figures, as 0.17
1224
for 0.000000000000000001224, and 2695615
for
26956000000000000000.
Air’s
Height. Compression. Expansion.
0 33 1
5 17.8515 1.8486
10 9.6717 3.4151
20 2.852 11.571
40 0.2525 136.83
400 0.17
1224 2695615
4000 0.105
4465 73907102
40000 0.192
1628 26263189
400000 0.210
7895 41798207
4000000 0.212
9878 33414209
Infinite. 0.212
6041 54622209
From this Table it appears that the Air in proceeding upwards is
rarefied in such manner, that a sphere of that Air which is nearest the
Earth but of one inch diameter, if dilated to an equal rarefaction with
that of the Air at the height of ten semi-diameters of the Earth,
would fill up more space than is contained in the whole Heavens on
this side the fixed Stars, according to the preceding computation of
their distance[41]
.” And it likewise appears that the Moon does not
move in a perfectly free and un-resisting medium; although the air at
a height equal to her distance, is at least 34000190
times thinner than
at the Earth’s surface; and therefore cannot resist her motion so as to
be sensible in many ages.
175. The weight of the Air, at the Earth’s
surface, is found by experiments made with
the air-pump; and also by the quantity of
mercury that the Atmosphere balances in](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-61-320.jpg)
![A common mistake
about the weight of
the Air.
Without an
Atmosphere the
Heavens would
always appear dark,
and we should have
no twilight.
the barometer; in which, at a mean state; the mercury stands 291
⁄2
inches high. And if the tube were a square inch wide, it would at that
height contain 291
⁄2 cubic inches of mercury, which is just 15 pound
weight; and so much weight of air every square inch of the Earth’s
surface sustains; and every square foot 144 times as much, because
it contains 144 square inches. Now as the Earth’s surface contains
about 199,409,400 square miles, it must be of no less than
5,559,215,016,960,000 square feet; which, multiplied by 2016, the
number of pounds on every foot, amounts to
11,207,377,474,191,360,000; or 11 trillion 207 thousand 377 billion
474 thousand 191 million and 360 thousand pounds, for the weight
of the whole Atmosphere. At this rate, a middle sized man, whose
surface may be about 14 square feet, is pressed by 28,224 pound
weight of Air all round; for fluids press equally up and down and on
all sides. But, because this enormous weight is equal on all sides, and
counterbalanced by the spring of the internal Air in our blood vessels,
it is not felt.
176. Oftentimes the state of the Air is
such that we feel ourselves languid and dull;
which is commonly thought to be occasioned
by the Air’s being foggy and heavy about us. But that the Air is then
too light, is evident from the mercury’s sinking in the barometer, at
which time it is generally found that the Air has not sufficient
strength to bear up the vapours which compose the Clouds: for, when
it is otherwise, the Clouds mount high, the Air is more elastic and
weighty about us, by which means it balances the internal spring of
the Air within us, braces up our blood-vessels and nerves, and makes
us brisk and lively.
177. According to [42]
Dr. Keill, and other
astronomical writers, it is entirely owing to
the Atmosphere that the Heavens appear
bright in the day-time. For, without an
Atmosphere, only that part of the Heavens
would shine in which the Sun was placed:
and if an observer could live without Air, and should turn his back](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-62-320.jpg)
![It brings the Sun in
view before he
rises, and keeps
him in view after he
sets.
Fig. IX.
PLATE II.
towards the Sun, the whole Heavens would appear as dark as in the
night, and the Stars would be seen as clear as in the nocturnal sky.
In this case, we should have no twilight; but a sudden transition from
the brightest sunshine to the blackest darkness immediately after
sun-set; and from the blackest darkness to the brightest sun-shine at
sun-rising; which would be extremely inconvenient, if not blinding, to
all mortals. But, by means of the Atmosphere, we enjoy the Sun’s
light, reflected from the aerial particles, before he rises and after he
sets. For, when the Earth by its rotation has withdrawn the Sun from
our sight, the Atmosphere being still higher than we, has his light
imparted to it; which gradually decreases until he has got 18 degrees
below the Horizon; and then, all that part of the Atmosphere which is
above us is dark. From the length of twilight, the Doctor has
calculated the height of the Atmosphere (so far as it is dense enough
to reflect any light) to be about 44 miles. But it is seldom dense
enough at two miles height to bear up the Clouds.
178. The Atmosphere refracts the Sun’s
rays so, as to bring him in sight every clear
day, before he rises in the Horizon; and to
keep him in view for some minutes after he
is really set below it. For, at some times of
the year, we see the Sun ten minutes longer above the Horizon than
he would be if there were no refractions: and about six minutes every
day at a mean rate.
179. To illustrate this, let IEK be a part of
the Earth’s surface, covered with the
Atmosphere HGFC; and let HEO be the[43]
sensible Horizon of an observer at E. When the Sun is at A, really
below the Horizon, a ray of light AC proceeding from him comes
straight to C, where it falls on the surface of the Atmosphere, and
there entering a denser medium, it is turned out of its rectilineal
course ACdG, and bent down to the observer’s eye at E; who then
sees the Sun in the direction of the refracted ray edE, which lies](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-63-320.jpg)
![Our imagination
cannot judge rightly
of the distance of
inaccessible objects.
horizontal refractions are near a third part less at the Equator than at
Paris, as mentioned by Dr. Smith in the 370th remark on his Optics,
where the following account is given of an extraordinary refraction of
the sun-beams by cold. “There is a famous observation of this kind
made by some Hollanders that wintered in Nova Zembla in the year
1596, who were surprised to find, that after a continual night of three
months, the Sun began to rise seventeen days sooner than according
to computation, deduced from the Altitude of the Pole observed to be
76°: which cannot otherwise be accounted for, than by an
extraordinary quantity of refraction of the Sun’s rays, passing thro’
the cold dense air in that climate. Kepler computes that the Sun was
almost five degrees below the Horizon when he first appeared; and
consequently the refraction of his rays was about nine times greater
than it is with us.”
184. The Sun and Moon appear of an oval figure as FCGD, just
after their rising, and before their setting: the reason is, that the
refraction being greater in the Horizon than at any distance above it,
the lowermost limb G appears more elevated than the uppermost.
But although the refraction shortens the vertical Diameter FG, it has
no sensible effect on the horizontal Diameter CD, which is all equally
elevated. When the refraction is so small as to be imperceptible, the
Sun and Moon appear perfectly round, as AEBF.
185. We daily observe, that the objects
which appear most distinct are generally
those which are nearest to us; and
consequently, when we have nothing but our
imagination to assist us in estimating of distances, bright objects
seem nearer to us than those which are less bright, or than the same
objects do when they appear less bright and worse defined, even
though their distance in both cases be the same. And as in both
cases they are seen under the same angle[44]
, our imagination
naturally suggests an idea of a greater distance between us and
those objects which appear fainter and worse defined than those
which appear brighter under the same Angles; especially if they be](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-66-320.jpg)
![Nor always of those
which are
accessible.
The reason
assigned.
PLATE II.
Fig. XII.
such objects as we were never near to, and of whose real
Magnitudes we can be no judges by sight.
186. But, it is not only in judging of the
different apparent Magnitudes of the same
objects, which are better or worse defined
by their being more or less bright, that we may be deceived: for we
may make a wrong conclusion even when we view them under equal
degrees of brightness, and under equal Angles; although they be
objects whose bulks we are generally acquainted with, such as
houses or trees: for proof of which, the two following instances may
suffice.
First, When a house is seen over a very
broad river by a person standing on low
ground, who sees nothing of the river, nor
knows of it beforehand; the breadth of the
river being hid from him, because the banks seem contiguous, he
loses the idea of a distance equal to that breadth; and the house
seems small, because he refers it to a less distance than it really is
at. But, if he goes to a place from which the river and interjacent
ground can be seen, though no farther from the house, he then
perceives the house to be at a greater distance than he imagined;
and therefore fancies it to be bigger than he did at first; although in
both cases it appears under the same Angle, and consequently
makes no bigger picture on the retina of his eye in the latter case
than it did in the former. Many have been deceived, by taking a red
coat of arms, fixed upon the iron gate in Clare-Hall walks at
Cambridge, for a brick house at a much greater distance[45]
.
Secondly, In foggy weather, at first sight,
we generally imagine a small house, which is
just at hand, to be a great castle at a distance; because it appears so
dull and ill defined when seen through the Mist, that we refer it to a
much greater distance than it really is at; and therefore, under the
same Angle, we judge it to be much bigger. For, the near object FE,
seen by the eye ABD, appears under the same Angle GCH, that the
remote object GHI does: and the rays GFCN and HECM crossing one](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-67-320.jpg)
![Fig. IX.
Why the Sun and
Moon appear
biggest in the
Horizon.
another at C in the pupil of the eye, limit the size of the picture MN
on the retina; which is the picture of the object FE, and if FE were
taken away, would be the picture of the object GHI, only worse
defined; because GHI, being farther off, appears duller and fainter
than FE did. But if a Fog, as KL, comes between the eye and the
object FE, it appears dull and ill defined like GHI; which causes our
imagination to refer FE to the greater distance CH, instead of the
small distance CE which it really is at. And consequently, as mis-
judging the distance does not in the least diminish the Angle under
which the object appears, the small hay-rick FE seems to be as big as
GHI.
187. The Sun and Moon appear bigger in
the Horizon than at any considerable height
above it. These Luminaries, although at
great distances from the Earth, appear
floating, as it were, on the surface of our
Atmosphere HGFfeC, a little way beyond the
Clouds; of which, those about F, directly over our heads at E, are
nearer us than those about H or e in the Horizon HEe. Therefore,
when the Sun or Moon appear in the Horizon at e, they are not only
seen in a part of the Sky which is really farther from us than if they
were at any considerable Altitude, as about f; but they are also seen
through a greater quantity of Air and Vapours at e than at f. Here we
have two concurring appearances which deceive our imagination, and
cause us to refer the Sun and Moon to a greater distance at their
rising or setting about e, than when they are considerably high as at
f: first, their seeming to be on a part of the Atmosphere at e, which is
really farther than f from a spectator at E; and secondly, their being
seen through a grosser medium when at e than when at f; which, by
rendering them dimmer, causes us to imagine them to be at a yet
greater distance. And as, in both cases, they are seen[46]
much under
the same Angle, we naturally judge them to be biggest when they
seem farthest from us; like the above-mentioned house § 186, seen
from a higher ground, which shewed it to be farther off than it](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-68-320.jpg)
![PLATE IV.
Fig I.
The Moon’s
horizontal Parallax,
what.
The Moon’s distance
determined.
CHAP. IX.
The Method of finding the Distances of the Sun, Moon, and
Planets.
190. Those who have not learnt how to take the
[47]
Altitude of any Celestial Phenomenon by a common
Quadrant, nor know any thing of Plain Trigonometry, may pass over the first Article
of this short Chapter, and take the Astronomer’s word for it, that the distances of
the Sun and Planets are as stated in the first Chapter of this Book. But, to every
one who knows how to take the Altitude of the Sun, the Moon, or a Star, and can
solve a plain right-angled Triangle, the following method of finding the distances of
the Sun and Moon will be easily understood.
Let BAG be one half of the Earth, AC it’s semi-diameter,
S the Sun, m the Moon, and EKOL a quarter of the Circle
described by the Moon in revolving from the Meridian to the Meridian again. Let
CRS be the rational Horizon of an observer at A, extended to the Sun in the
Heavens, and HAO his sensible Horizon; extended to the Moon’s Orbit. ALC is the
Angle under which the Earth’s semi-diameter AC is seen from the Moon at L, which
is equal to the Angle OAL, because the right lines AO and CL which include both
these Angles are parallel. ASC is the Angle under which the Earth’s semi-diameter
AC is seen from the Sun at S, and is equal to the Angle OAf because the lines AO
and CRS are parallel. Now, it is found by observation, that the Angle OAL is much
greater than the Angle OAf; but OAL is equal to ALC, and OAf is equal to ASC. Now,
as ASC is much less than ALC, it proves that the Earth’s semi-diameter AC appears
much greater as seen from the Moon at L than from the Sun at S: and therefore the
Earth is much farther from the Sun than from the Moon[48]
. The Quantities of these
Angles are determined by observation in the following manner.
Let a graduated instrument as DAE, (the larger the
better) having a moveable Index and Sight-holes, be fixed
in such a manner, that it’s plane surface may be parallel to
the Plan of the Equator, and it’s edge AD in the Meridian:
so that when the Moon is in the Equinoctial, and on the
Meridian at E, she may be seen through the sight-holes
when the edge of the moveable index cuts the beginning of the divisions at o, on
the graduated limb DE; and when she is so seen, let the precise time be noted.](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-71-320.jpg)
![The Sun’s distance
cannot be yet so
exactly determined
as the
Moon’s;
How near the truth
it may soon be
determined.
Now, as the Moon revolves about the Earth from the Meridian to the Meridian again
in 24 hours 48 minutes, she will go a fourth part round it in a fourth part of that
time, viz. in 6 hours 12 minutes, as seen from C, that is, from the Earth’s center or
Pole. But as seen from A, the observer’s place on the Earth’s surface, the Moon will
seem to have gone a quarter round the Earth when she comes to the sensible
Horizon at O; for the Index through the sights of which she is then viewed will be
at d, 90 degrees from D, where it was when she was seen at E. Now, let the exact
moment when the Moon is seen at O (which will be when she is in or near the
sensible Horizon) be carefully noted[49]
, that it may be known in what time she has
gone from E to O; which time subtracted from 6 hours 12 minutes (the time of her
going from E to L) leaves the time of her going from O to L, and affords an easy
method for finding the Angle OAL (called the Moon’s horizontal Parallax, which is
equal to the Angle ALC) by the following Analogy: As the time of the Moon’s
describing the arc EO is to 90 degrees, so is 6 hours 12 minutes to the degrees of
the Arc DdE, which measures the Angle EAL; from which subtract 90 degrees, and
there remains the Angle OAL, equal to the Angle ALC, under which the Earth’s
Semi-diameter AC is seen from the Moon. Now, since all the Angles of a right-lined
Triangle are equal to 180 degrees, or to two right Angles, and the sides of a
Triangle are always proportional to the Sines of the opposite Angles, say, by the
Rule of Three, as the Sine of the Angle ALC at the Moon L is to it’s opposite side AC
the Earth’s Semi-diameter, which is known to be 3985 miles, so is Radius, viz. the
Sine of 90 degrees, or of the right Angle ACL to it’s opposite side AL, which is the
Moon’s distance at L from the observer’s place at A on the Earth’s surface; or, so is
the Sine of the Angle CAL to its opposite side CL, which is the Moon’s distance from
the Earth’s centre, and comes out at a mean rate to be 240,000 miles. The Angle
CAL is equal to what OAL wants of 90 degrees.
191. The Sun’s distance from the Earth is found the
same way, but with much greater difficulty; because his
horizontal Parallax, or the Angle OAS equal to the Angle
ASC, is so small as, to be hardly perceptible, being only 10
seconds of a minute, or the 360th part of a degree. But
the Moon’s horizontal Parallax, or Angle OAL equal to the
Angle ALC, is very discernible; being 57ʹ 49ʺ, or 3469ʺ at
it’s mean state; which is more than 340 times as great as
the Sun’s: and therefore, the distances of the heavenly bodies being inversely as
the Tangents of their horizontal Parallaxes, the Sun’s distance from the Earth is at
least 340 times as great as the Moon’s; and is rather understated at 81 millions of
miles, when the Moon’s distance is certainly known to be 240 thousand. But
because, according to some Astronomers, the Sun’s horizontal Parallax is 11
seconds, and according to others only 10, the former Parallax making the Sun’s
distance to be about 75,000,000 of miles, and the latter 82,000,000; we may take it
for granted, that the Sun’s distance is not less than as deduced from the former, nor
more than as shewn by the latter: and every one who is accustomed to make such](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-72-320.jpg)
![Why the celestial
Poles seem to keep
still in the same
points of the
Heavens,
notwithstanding the
Earth’s motion
round the Sun.
Relative mean distances from the Sun.
38710 72333 100000 152369 520096 954006
From these numbers we deduce, that if the Sun’s horizontal Parallax be 10ʺ, the
real mean distances of the Planets from the Sun in English miles are
31,742,200 59,313,060 82,000,000 124,942,580 426,478,720 782,284,920
But if the Sun’s Parallax be 11ʺ their distances are no more than
29,032,500 54,238,570 75,000,000 114,276,750 390,034,500 715,504,500
Errors in distance a rising from the mistake of 1ʺ in the Sun’s Parallax
2,709,700 5,074,490 7,000,000 10,665,830 36,444,220 66,780,420
195. These last numbers shew, that although we have the relative distances of
the Planets from the Sun to the greatest nicety, yet the best observers have not
hitherto been able to ascertain their true distances to within less than a twelfth part
of what they really are. And therefore, we must wait with patience till the 6th of
June, A. D. 1761; wishing that the Sky may then be clear to all places where there
are good Astronomers and accurate instruments for observing the Transit of Venus
over the Sun’s Disc at that time: as it will not happen again, so as to be visible in
Europe, in less than 235 years after.
196. The Earth’s Axis produced to the Stars, being
carried [50]
parallel to itself during the Earth’s annual
revolution, describes a circle in the Sphere of the fixed
Stars equal to the Orbit of the Earth. But this Orbit, though
very large in itself, if viewed from the Stars, would appear
no bigger than a point; and consequently, the circle
described in the Sphere of the Stars by the Axis of the
Earth produced, if viewed from the Earth, must appear but as a point; that is, it’s
diameter appears too little to be measured by observation: for Dr. Bradley has
assured us, that if it had amounted to a single second, or two at most, he should
have perceived it in the great number of observations he has made, especially upon
γ Dragonis; and that it seemed to him very probable that the annual Parallax of this
Star is not so great as a single second: and consequently, that it is above 400
thousand times farther from us than the Sun. Hence the celestial poles seem to
continue in the same points of the Heavens throughout the year; which by no](https://image.slidesharecdn.com/3256-250509074347-2fe523b0/85/Combinatorics-of-Spreads-and-Parallelisms-1st-Edition-Norman-Johnson-74-320.jpg)
Combinatorics of Spreads and Parallelisms 1st Edition Norman Johnson
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- 5. Combinatorics of Spreads and Parallelisms
- 6. PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes EXECUTIVE EDITORS EDITORIAL BOARD Earl J. Taft Rutgers University Piscataway, New Jersey Zuhair Nashed University of Central Florida Orlando, Florida M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology S. Kobayashi University of California, Berkeley Marvin Marcus University of California, Santa Barbara W. S. Massey Yale University Anil Nerode Cornell University Freddy van Oystaeyen University of Antwerp, Belgium Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University David L. Russell Virginia Polytechnic Institute and State University Walter Schempp Universität Siegen
- 7. MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS Recent Titles M. M. Rao, Conditional Measures and Applications, Second Edition (2005) A. B. Kharazishvili, Strange Functions in Real Analysis, Second Edition (2006) Vincenzo Ancona and Bernard Gaveau, Differential Forms on Singular Varieties: De Rham and Hodge Theory Simplified (2005) Santiago Alves Tavares, Generation of Multivariate Hermite Interpolating Polynomials (2005) Sergio Macías, Topics on Continua (2005) Mircea Sofonea, Weimin Han, and Meir Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage (2006) Marwan Moubachir and Jean-Paul Zolésio, Moving Shape Analysis and Control: Applications to Fluid Structure Interactions (2006) Alfred Geroldinger and Franz Halter-Koch, Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory (2006) Kevin J. Hastings, Introduction to the Mathematics of Operations Research with Mathematica®, Second Edition (2006) Robert Carlson, A Concrete Introduction to Real Analysis (2006) John Dauns and Yiqiang Zhou, Classes of Modules (2006) N. K. Govil, H. N. Mhaskar, Ram N. Mohapatra, Zuhair Nashed, and J. Szabados, Frontiers in Interpolation and Approximation (2006) Luca Lorenzi and Marcello Bertoldi, Analytical Methods for Markov Semigroups (2006) M. A. Al-Gwaiz and S. A. Elsanousi, Elements of Real Analysis (2006) Theodore G. Faticoni, Direct Sum Decompositions of Torsion-Free Finite Rank Groups (2007) R. Sivaramakrishnan, Certain Number-Theoretic Episodes in Algebra (2006) Aderemi Kuku, Representation Theory and Higher Algebraic K-Theory (2006) Robert Piziak and P. L. Odell, Matrix Theory: From Generalized Inverses to Jordan Form (2007) Norman L. Johnson, Vikram Jha, and Mauro Biliotti, Handbook of Finite Translation Planes (2007) Lieven Le Bruyn, Noncommutative Geometry and Cayley-smooth Orders (2008) Fritz Schwarz, Algorithmic Lie Theory for Solving Ordinary Differential Equations (2008) Jane Cronin, Ordinary Differential Equations: Introduction and Qualitative Theory, Third Edition (2008) Su Gao, Invariant Descriptive Set Theory (2009) Christopher Apelian and Steve Surace, Real and Complex Analysis (2010) Norman L. Johnson, Combinatorics of Spreads and Parallelisms (2010)
- 8. Combinatorics of Spreads and Parallelisms Norman L. Johnson University of Iowa Iowa City, U.S.A.
- 9. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number: 978-1-4398-1946-3 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit- ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Johnson, Norman L. (Norman Lloyd), 1939- Combinatorics of spreads and parallelisms / Norman L. Johnson. p. cm. -- (Pure and applied mathematics) Includes bibliographical references and index. ISBN 978-1-4398-1946-3 (hardcover : alk. paper) 1. Vector spaces. 2. Algebraic spaces. 3. Geometry, Projective. 4. Incidence algebras. I. Title. II. Series. QA186.J64 2010 512’.52--dc22 2010018110 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
- 10. I dedicate this text to my wife Bonnie Lynn Hemenover and to my children Catherine Elizabeth Johnson Garret Norman Johnson Scott Hamilton Hemenover vii
- 11. Contents Preface xv Part 1. Partitions of Vector Spaces 1 Chapter 1. Quasi-subgeometry Partitions 3 1. Collineation Groups of Translation Geometries 10 Chapter 2. Finite Focal-Spreads 15 1. Towers of Focal-Spreads 16 2. Focal-Spreads and Coordinatization 18 3. k-Cuts and Inherited Groups 20 4. Spread-Theoretic Dual of a Semi…eld 26 5. The Dual Semi…eld Plane 26 6. The Six Associated Semi…elds 29 7. Symplectic Spreads 32 8. Additive Focal-Spreads 35 9. Designs, and Multiple-Spreads 38 10. Focal-Spreads from Designs 39 Chapter 3. Generalizing André Spreads 43 1. A Primer on André Planes 44 2. r-(sn; q)-Spreads 45 3. Multiple Extended Replacements 55 4. Large Groups on t-Spreads 57 Chapter 4. The Going Up Construction for Focal-Spreads 61 1. The ‘ Going Up’Construction 63 2. Generalization to Partitions with Many Spread Types 65 3. Going Up– Direct Sums 69 Part 2. Subgeometry Partitions 71 Chapter 5. Subgeometry and Quasi-subgeometry Partitions 73 Chapter 6. Subgeometries from Focal-Spreads 83 1. k-Cuts of Subgeometry Partitions 83 ix
- 12. x CONTENTS 2. Additive k-Cuts 85 3. Right/Left Focal-Spreads 86 4. Hyperplane Constructions 88 Chapter 7. Extended André Subgeometries 91 Chapter 8. Kantor’ s Flag-Transitive Designs 97 1. Kantor’ s Class I 97 2. Kantor’ s Class II 99 3. mth -Root Subgeometry Partitions 101 4. Subgeometries from Kantor’ s Class I 101 5. Subgeometries from Kantor’ s Class II 104 Chapter 9. Maximal Additive Partial Spreads 111 1. Direct Sums of Semi…elds 114 2. Sub…elds of Order 4 in Knuth Semi…elds 117 3. The Commutative Kantor Semi…elds 120 Part 3. Subplane Covered Nets and Baer Groups 125 Chapter 10. Partial Desarguesian t-Parallelisms 127 1. A Primer on Subplane Covered Nets 127 2. Projective Spreads and A¢ ne Nets 130 Chapter 11. Direct Products of A¢ ne Planes 141 1. Desarguesian Products 144 2. Left Spreads and Left Partial Spreads 147 3. Desarguesian Left Parallelisms– Right Spreads 150 Chapter 12. Jha-Johnson SL(2; q) C-Theorem 155 Chapter 13. Baer Groups of Nets 163 1. Left and Right in Direct Product Nets 165 2. Point-Baer Subplanes in Vector Space Nets 170 3. Translation Planes with Baer Groups 178 4. Spreads in PG(3; K) 187 Chapter 14. Ubiquity of Subgeometry Partitions 195 1. Jha-Johnson Lifting Theorem 196 2. Double-Baer Groups 201 3. Fusion of Baer Groups 203 4. Double-Homology Groups 206 5. Dempwol¤’ s Double-Baer Groups 208 6. Subgeometry Partitions from Going Up 210 7. Algebraic Lifting of Focal-Spreads 211
- 13. CONTENTS xi Part 4. Flocks and Related Geometries 215 Chapter 15. Spreads Covered by Pseudo-Reguli 217 1. Pseudo-Reguli 217 2. Conical and Ruled Spreads over Skew…elds 223 Chapter 16. Flocks 231 1. Conical Flocks 231 2. Hyperbolic Flocks 235 3. Partial Flocks of De…ciency One 240 4. Point-Baer Groups and Partial Flocks 240 5. De…ciency One Partial Conical Flocks 242 6. De…ciency One Partial Hyperbolic Flocks 246 7. The Theorem of Johnson, Payne-Thas 250 Chapter 17. Regulus-Inducing Homology Groups 253 Chapter 18. Hyperbolic Fibrations and Partial Flocks 267 Chapter 19. j-Planes and Monomial Flocks 273 1. Hyperbolic Fibrations over the Reals 277 2. Classi…cation of the Real j-Planes 281 Part 5. Derivable Geometries 285 Chapter 20. Flocks of -Cones 287 1. Maximal Partial Flokki and -Flocks 290 2. De…ciency One and Baer Groups 293 3. K-Flokki and Algebraic Lifting 294 4. Net Replacement in the Hughes-Kleinfeld Planes 295 Chapter 21. Parallelisms of Quadric Sets 301 1. The Thas, Bader-Lunardon Theorem 302 2. Parallelisms of Hyperbolic Quadrics 303 3. Bol Planes 305 4. Parallelisms of -Cones 309 Chapter 22. Sharply k-Transitive Sets 313 1. Subsets of P L(n; K) 314 2. Subsets in P L(n; q) of De…ciency 1 317 3. The Parallelisms of Bonisoli 322 Chapter 23. Transversals to Derivable Nets 327 1. Algebraic Extensions of Derivable Nets 328 2. Geometric Extension of Derivable Nets 330 3. Planar Transversal Extensions 332
- 14. xii CONTENTS 4. Semi…eld Extension-Nets 337 Chapter 24. Partially Flag-Transitive A¢ ne Planes 345 1. Derivable A¢ ne Planes with Nice Groups 350 Chapter 25. Special Topics on Parallelisms 365 1. Transversal Spreads and Dualities of PG(3; K) 365 2. Skew Parallelisms 365 Part 6. Constructions of Parallelisms 369 Chapter 26. Regular Parallelisms 371 1. In…nite Regular Parallelisms 371 2. The Set of 2-Secants 374 3. Isomorphisms of Transitive Parallelisms 377 4. Finite Regular Parallelisms 379 5. The Penttila-Williams Construction 380 Chapter 27. Beutelspacher’ s Construction of Line Parallelisms 387 1. Extension of Beutelspacher’ s Theorem 388 2. Applications of Beutelspacher’ s Construction. 391 Chapter 28. Johnson Partial Parallelisms 395 1. Isomorphisms 399 2. The Derived Parallelisms 411 Part 7. Parallelism-Inducing Groups 415 Chapter 29. Parallelism-Inducing Groups for Pappian Spreads 419 Chapter 30. Linear and Near…eld Parallelism-Inducing Groups 427 1. Parallelism-Inducing Subgroups of GL(2; q2 ) 431 2. The General Near…eld Parallelism-Inducing Group 433 3. Isomorphisms of Group-Induced Parallelisms 434 4. Near…eld Parallelism-Inducing Groups 436 5. Finite Regular Near…eld Groups 439 6. m-Parallelisms 441 Chapter 31. General Parallelism-Inducing Groups 449 1. The Isomorphisms of Kantor-Knuth Type Parallelisms 454 2. Finite Regulus-Inducing Elation Groups 457 3. Finite General Parallelism-Inducing Groups 461 4. Minimal Linear Parallelism-Inducing Groups 464 5. Relative Linear Parallelism-Inducing Groups 465 6. Existence of Minimal Groups 472
- 15. CONTENTS xiii 7. Determination of the Minimal Groups 475 8. General Isomorphisms from Minimal Groups 478 9. Isomorphic Kantor-Knuth Parallelisms 481 10. Even Order 484 11. Derived Parallelisms 486 Part 8. Coset Switching 489 Chapter 32. Finite E-Switching 491 1. E-and Desarguesian Switches 491 2. The Switches of Desarguesian Spreads 493 3. Coset Switching again with Desarguesian Spreads 496 4. An Upper Bound 499 5. Upper Bound, Even Order 502 Chapter 33. Parallelisms over Ordered Fields 507 Chapter 34. General Elation Switching 515 1. De…ciency One Transitive Groups 520 2. Switching Theorem over Ordered Fields 524 Chapter 35. Dual Parallelisms 529 1. The Isomorphisms of the Dual Parallelisms 534 Part 9. Transitivity 543 Chapter 36. p-Primitive Parallelisms 545 1. Biliotti-Jha-Johnson p-Primitive Theorem 546 2. Transitive Parallelisms 552 3. Biliotti, Jha, Johnson— Transitive Classi…cation 552 4. Johnson’ s Classi…cation of 2-Transitive Parallelisms 552 Chapter 37. Transitive t-Parallelisms 555 1. Johnson-Montinaro t-Transitive Theorem 555 2. Transitive Parallelisms of PGL(2r 1; 2) 555 3. Isomorphisms of the Transitive Parallelisms 559 Chapter 38. Transitive De…ciency One 561 1. BJJDJM-Classi…cation Theorem 561 2. De…ciency One— The Spreads Are Isomorphic 576 3. The Full Group 576 Chapter 39. Doubly Transitive Focal-Spreads 583 1. Johnson-Montinaro; Elementary Abelian Case 585
- 16. xiv CONTENTS Part 10. Appendices 591 Chapter 40. Open Problems 593 1. Non-standard Groups and Non-standard Parallelisms 594 Chapter 41. Geometry Background 609 Chapter 42. The Klein Quadric 613 1. The Thas-Walker Construction 614 Chapter 43. Major Theorems of Finite Groups 617 1. Subgroups of PSL(2; q) 617 2. The Lists of Mitchell and Hartley 618 3. Finite Doubly Transitive Groups 620 4. Primitive Subgroups of L(4; q) 620 5. Aschbacher’ s Theorem 621 6. Guralnick-Penttila-Praeger-Saxl Theorem 623 7. O’ Nan-Scott Theorem 624 8. Dye’ s Theorem 625 9. Johnson-Montinaro t-Transitive Theorem 625 Chapter 44. The Diagram 631 Bibliography 633 Index 643
- 17. Preface It may be said that a substantial part of the theory of incidence geometry is devoted to questions about ‘ covers’of the set of points or lines (or ‘ blocks’ ) by various means. It has been fashionable recently to consider ‘ ‡ ocks’of quadric sets, which are covers of elliptic, hyperbolic quadrics or quadratic cones by maximal sets of mutually disjoint conics. All of these covers are, in turn, equivalent to certain partitions of a four- dimensional vector space by sets of mutually disjoint two-dimensional vector subspaces, which is one example of a ‘ spread’of a vector space. Indeed, vector space ‘ spreads’in this context are equivalent to ‘ pro- jective spreads’of a three-dimensional projective space. In this context, projective spreads are sets of mutually skew lines that form a disjoint cover of the set of points. These objects are of considerable interest in that they construct very interesting examples of projective or a¢ ne planes. Going a bit further with the ideas of covers, a ‘ packing’or ‘ par- allelism’of the three-dimensional projective space is a disjoint cover of the set of lines by a set of spreads. In a very real sense, this is perhaps the most fundamental of the covering objects in incidence geometry and at the same time one that has, until recently, very few ‘ interesting’ or ‘ nice’examples. Many geometers are drawn to the subject of incidence geometry by the elegant beauty and simplicity of the examples. Of course, there now are a number of fascinating examples of ‘ parallelisms,’ and the point of the text is to show how to construct most if not all of the more interesting ones. It seems that spreads are also very much of interest in the more general sense, that of a set of mutually disjoint vector subspaces of the same dimension of a given …nite dimensional vector space. Not much is really known of spreads that have nice group properties, but what is known shall be essentially given in this text. More generally, arbitrary partitions of vectors by subspaces not necessarily of the same dimension are beginning to have applications. Recently, general spreads of vector spaces have been used to study and construct ‘ subgeometry partitions of projective spaces.’Here there are some very basic questions, such as xv
- 18. xvi Preface what might be the variety of the subgeometries involved in a partition of a projective space? Definition 1. Let P be a projective space. A ‘ parallelism’of P is an equivalence relation on the set L of lines such that the Euclidean parallel postulate holds. Let ` be a line and denote by [`], the equivalence class containing `. Then let P be any point of P that is not incident with `. Then there exists a unique line `P , incident with P and ‘ parallel’to `. Hence, there is a partition of the points of P by any equivalence class. Definition 2. A partition of the points of a projective space P by a set of disjoint lines is said to be a ‘ line spread’of P . Just as we have pointed out previously: Remark 1. Since partitions of the line set of a projective space and equivalence relations are equivalent, we see that a partition of the lines of a projective space by spreads is equivalent to a parallelism of a projective space. Why are parallelisms of projective spaces of interest to geometers? Suppose the spreads of a parallelism produce ‘ …eld’ a¢ ne planes (or projective planes). We shall get to the exact meaning of this in due course, but merely think of the a¢ ne Euclidean plane over the …eld of real numbers R, where we insert an arbitrary …eld for the …eld R. Since …eld planes are so very easy to construct, it would seem likely that there are many parallelisms consisting of …eld spreads. However, it turns out that such parallelisms are quite hard to come by. For reasons that will come later, we call such parallelisms ‘ regular’and the e¤ort to …nd regular parallelisms does pay o¤ in the construction of new and quite amazing a¢ ne planes. In graduate school in 1964 at Washington State University, I was studying algebraic geometry when my advisor thought that I should broaden my reading and suggested that I take a course in …nite geom- etry from T.G. Ostrom, who was charting some very new directions in the construction of a¢ ne and projective planes. The short version of this story is that I never went back to algebraic geometry and became fascinated by the ideas of ‘ derivation’ that Ostrom was developing. Derivation is a construction method by which a renaming of certain subplanes (called ‘ Baer subplanes’ ) as ‘ new’lines together with certain of the ‘ old’lines sometimes can produce a totally di¤erent a¢ ne plane from a given one. Projectively, a projective Baer subplane is a sub- plane that touches all points and lines, in the sense that every point is
- 19. Combinatorics of Spreads and Parallelisms xvii incident with some line of the subplane and every line is incident with some point of the subplane. The derivation process is perhaps the most important construction procedure in the construction of a¢ ne and pro- jective planes. The reader interested in more details on derivation is directed to the author’ s text Subplane Covered Nets, [114]. Actually, this text forms the fourth volume of a series of works on …nite geometry and a word on where the present volume …ts is appro- priate. In the text Subplane Covered Nets, [114], there is a complete theory of derivation. The analysis of a derivable a¢ ne plane focuses more properly on the associated net that contains the Baer subplanes that are used in the process and which are rede…ned as lines in the derived a¢ ne plane. Considered in this manner, a derivable net is a net that is covered by subplanes in the sense that given any two a¢ ne points P and Q of the net that are collinear, there is a unique subplane P;Q of the net that contains P and Q and which shares the set of parallel classes of the net as the line at in…nity of the corresponding projective plane. The text then concerns the analogous theory of nets that are covered by subplanes as in the previous manner. In this text, there is the introduction of the direct product of a¢ ne planes of the same order (or cardinality), which becomes a net containing the two a¢ ne subplanes. When the two a¢ ne subplanes are Pappian, the con- structed net is a derivable net, and in this case, we identify the a¢ ne subplanes. If the Pappian subplanes correspond to quadratic …eld ex- tensions of a given …eld K, then repeated application of these ideas produces a set of derivable nets that share a regulus net coordinatized by K. If L is a component of this regulus net and is a four-dimensional K-space, then any Pappian K-spread of L induces a direct product derivable net sharing the regulus net. Considering L as a projective space PG(3; K), then any Pappian parallelism of PG(3; K) induces a set of derivable nets, the union of which produces a spread in PG(7; K) for a translation plane admitting SL(2; K). Thus, the text Subplane Covered Nets, provides ideas for the analy- sis of parallelisms of projective spaces. In this work, we shall repeat parts of the theory of direct products that applies to parallelisms, so as to keep this text as self-contained as possible. The study of parallelisms in PG(3; q) involves, of course, a set of q2 + q + 1 spreads in PG(3; q) that construct translation planes. In the work Foundations of Translation Planes, [21], a complete background is given not of only of translation planes of "dimension two," which arise from spreads in PG(3; q), but also of general translation planes. The reader is directed to the foundations text for general information on translation planes, but again, we shall include enough of the theory
- 20. xviii Preface here so that the reader is not necessarily dependent on this previous text. Suppose that a parallelism in PG(3; q) admits a collineation group that leaves one spread invariant and acts transitively on the remain- ing q2 + q spreads. Then we shall show that the …xed spread de…nes a Desarguesian translation plane and the remaining spreads de…ne trans- lation planes of order q2 with associated spreads in PG(3; q) that admit Baer groups of order q (a group of order q that …xes a Baer subplane pointwise). Such translation planes actually correspond to ‡ ocks of quadratic cones and so certain of the theory of ‡ ocks of quadratic sets shall be developed in this text. Considering that a parallelism is de…ned as a covering of the line set by a spread, a natural generalization would be to consider analogous geometries as follows: If a spread is a covering of the point set by mutually disjoint ‘ blocks,’then a covering of the block set by spreads can be called a parallelism. When we consider parallelisms of quadric sets in PG(3; K), for K a …eld, we would used the term ‘ ‡ ock’instead of spread so that a parallelism of a quadric set is a covering of the conics by a set of mutually disjoint ‡ ocks. Generalizations include parallelisms of sharply k-transitive geometries, for k 2. In general, there is a tremendous variety of connections with the theory of translation planes and other point-line geometries, all of which are explicated in the text Handbook of Finite Translation Planes, [138]. What has not been done is to give a comprehensive and "up to date" treatment of parallelisms of projective spaces and other geome- tries, and, of course, the intent here is to do just that. We have men- tioned translation planes and the vector space partitions that produce them but there are other spreads that do not produce translation planes but may produce Sperner Spaces. One important class of geometric structures associated with Sperner Spaces is that of a subgeometry partition of a projective space. More generally, a subgeometry parti- tion of a projective space need not produce or correspond to a Sperner space or a translation plane but always gives rise to a general partition of a vector space. Even though there are some examples of partitions of vector spaces known, there is almost no theory that has developed from such covers. Here we also provide a census of most of the known examples of partitions and what can be said theoretically of associated point-line geometries. Since there are many new subgeometry parti- tions constructed, we then provide a complete description of all known examples. Therefore, it is felt that this fourth text Combinatorics of Spreads and Parallelisms, is a strong complement to the previous three texts
- 21. Combinatorics of Spreads and Parallelisms xix Subplane Covered Nets, Foundations of Translation Planes, and Hand- book of Finite Translation Planes. Acknowledgments. It seems that perhaps an indication of how I became interested in the general study of parallelisms might be appro- priate, and for which I feel I need to acknowledge a number of people. In 1985, my son Scott and I spent the spring semester and summer in Italy, where I met many Italian geometers and many still creat- ing wonderful mathematics. In particular, in Lecce, I am indebted to Giuseppe (Pino) Micelli and Rosanna Marinosci for their wonderful hospitality to Scott, my wife Bonnie (who joined Scott and me later in the spring semester) and of course, to me. Also, many thanks are due to Gabriella Murciano, who was instrumental in introducing Scott to a wonderful group of young people. While I was working mostly with Mauro Biliotti of Lecce, we per- suaded Vik Jha to spend a few weeks with us and about that time we began looking at the work of Mike Walker that connected regular par- allelisms with direct products of Desarguesian a¢ ne planes. I was also privileged to be invited to visit Naples and to work with Guglielmo Lu- nardon, who had independently determined the connection with regular parallelisms and direct products. And, this is where my real interest in ‘ parallelisms’was brought to light, so to speak. My wife Bonnie had joined Scott and myself for a month in Palermo while I was working with Claudio Bartolone and most of that time we persuaded Vik Jha to come to Palermo as well. Sometime during that visit and the previous period in Lecce, Vik and I proved that regular parallelisms could be realized from translation planes of order q4 that admit SL(2; q) C1+q+q2 , with the idea of applying the general the- ory of Lunardon and Walker. However, there were only two examples of regular parallelisms known at that time, something that has since dramatically changed. For that great trip, I am very much indebted to Claudio Bartolone. When Bonnie, Scott, and I …nally arrived in Naples, after a very, very long night ship from Palermo, we found Guglielmo at the dock waving wildly to us. Bonnie and Scott went o¤ to bed and Guglielmo and I went to talk— and we talked continuously for a week about ‘ par- allelisms,’and more ‘ parallelisms.’ Guglielmo and Vik really did hook me on this subject that year, and I very much appreciate both of their in‡ uences and help on my work, as well as on the general area. I might add that I returned to Italy many times, and many times, I have worked with Mauro Biliotti and Vik Jha on various aspects of parallelisms and related geometries. I gladly acknowledge my debt to
- 22. xx Preface Mauro Biliotti and Vik Jha for twenty-…ve years or so of great collab- oration and lasting friendship (and many wonderful joint articles and two great books Foundations of Translation Planes and Handbook of Finite Translation Planes). On another continent, later in 1985, I began collaborating with Rolando Pomareda of Santiago, Chile, also many times on parallelisms and returned to Santiago a number of times. I thank Sylvia and Rolando for their gracious hospitality and to Rolando for his help and insights with our joint work and for his friendship over these twenty-…ve years. Parts of this text also appear in certain of the previous three works on …nite geometry, predominantly in the Subplanes text [114], and I am grateful to Taylor and Francis for allowing the inclusion of this material here. Certain of the material found in this text follows work previously published and which has been suitably modi…ed for the treatment given here. Much of the work, however, is quite new, and a set of open problems is given at the end of the text that should be accessible to most researchers in …nite incidence geometry. When thinking of this time and how it is that I actually became a mathematician interested in …nite geometry, it is also impossible for me not to mention the in‡ uence my mother Catherine Elizabeth Lamb (Gabriel) has had on my career in mathematics. My mother was a talented mathematician in spirit and ability, but my grandfather was rather old-fashioned and was not willing to support a university educa- tion for his children. So, I feel that I was the recipient from my mother of whatever ability I may have in mathematics. Of course, there is another Catherine Elizabeth, with very similar talents— my daughter, and yes, she was named after her grandmother. Her very strong analytical abilities are now being applied in the law as she is a disability rights attorney, graduated the University of Iowa, with a J.D. and a Master’ s in Education and now is specializing in the important area of defending the rights of children. My son Garret Norman also inherited his grandmother’ s mathemat- ical abilities and has opted rather for the …eld of engineering, holding a Bachelor’ s and Master’ s Degree in Electrical Engineering from the University of Florida he has become a genuine leader in his …eld. All of these years later, I might add that the son who accompa- nied me to Italy, Scott Hamilton, pursued a Ph.D. from the University of Nebraska and is now a Professor of Psychology at Western Illinois University and is a very active researcher in a variety of exciting new directions in psychology.
- 23. Combinatorics of Spreads and Parallelisms xxi My three children have been wonderful sources of support and en- couragement for my work, for which I am most thankful. Considering support, I am happy to acknowledge support from the University of Iowa who awarded me a Career Development Award in 2009 in order to complete this text. I am indebted to Doug Slauson for help with the technical aspects of the preparation and with the electronic version of the diagram for the cover of the book, which, by the way, is encrypted in a more or less obvious manner. At the end of the text, all will be revealed. I am glad to thank Andre Barnett for her copious editing of this text. Finally, I gratefully acknowledge my wife Bonnie Lynn Hemenover, to whom I owe much, if not everything, and without whom none of the various books on geometry would have been written. Norman L. Johnson University of Iowa, Iowa City, IA. 2010
- 24. Part 1 Partitions of Vector Spaces
- 25. In this part, we consider a variety of partitions of vector spaces, which are more general than the spreads that construct translation planes. This material is very general, and there are many new areas of research presented. We also focus on the so-called focal-spreads, which are partitions of a vector space of dimension t + k by subspaces of two di¤erent dimensions, t and k, where there is a unique t-subspace. It will turn out that such focal-spreads can become the building blocks for general partitions. 2
- 26. CHAPTER 1 Quasi-subgeometry Partitions 0.1. Partitions of Vector Spaces. A ‘ t-spread’ of a …nite di- mensional vector space over a skew…eld K is a partition of the non-zero vectors by vector subspaces of the same dimension t. Apart from their intrinsic interest, these partitions are important geometrically, which we shall see in due course. We shall assume, for the most part, that the reader is familiar with basic aspects of a¢ ne and projective planes and their collineation groups. In our previous texts on …nite geometry, the main emphasis has been on the type of projective and a¢ ne planes called ‘ translation planes.’ Of course, the main reason for this is that all known …nite projective planes are intrinsically related to …nite vector spaces and for this reason are connected somehow to translation planes. In the following, we note the connection between translation planes and vector spaces. We start at the beginning with the fundamental de…nitions. Definition 3. An ‘ a¢ ne plane’ is a triple (P; L; I), of sets of ‘ points P,’ ‘ lines L,’ and ‘ incidence I P L,’ with the following properties: (1) Given two distinct points P and Q; there is a unique line ` incident with P and Q, and which we shall denote by PQ. (2) Given a line ` and a point P not incident with `, there is a unique line m that is not incident with any points (is ‘ disjoint from’ ) with ` and is incident with P. The line m is said to be ‘ parallel’to `. (3) There are at least four points no three of which are ‘ collinear’ (are not incident with a common line). Remark 2. It is easy to show that ‘ parallelism’is an equivalence relation on the set of lines of an a¢ ne plane. The equivalence classes are called ‘ parallel classes.’ So, given two lines from distinct parallel classes, there is a unique point incident with the lines, and each point is incident with a unique line of each parallel class. Definition 4. A ‘ projective plane’is a triple (P; L; I), of sets of ‘ points P,’‘ lines L,’and ‘ incidence I,’with the following properties: 3
- 27. 4 Partitions of Vector Spaces (1) Given two distinct points P and Q; there is a unique line ` incident with P and Q, and which we shall denote by PQ. (2) Given two distinct lines p and q; there is a unique point L incident with p and q, and which we shall denote by p q. (3) There are at least four points no three of which are ‘ collinear’ (are not incident with a common line). Remark 3. It is well known and straightforward to check that given a projective plane + , it is possible to construct an a¢ ne plane by removing a line ` and the subset of the points incident with `, [`], and where incidence of the new structure is inherited from the projective plane. Hence, if (P; L; I) is a triple that de…nes a projective plane, then (P [`]; L f`g; I f(P; `); P 2 [`]g) is an a¢ ne plane . The a¢ ne plane is called the ‘ a¢ ne restriction of the projective plane by `.’ The line ` then de…nes the set of parallel classes of the a¢ ne plane and is sometimes called the ‘ line at in…nity.’ If is an a¢ ne plane, for each parallel class , formally adjoin a ‘ point’to each line of and let the union of the parallel classes be a formal line `1, again called the ‘ line at in…nity’of . With the natural extension of the point, line, and incidence sets, an associated projective plane + is constructed, called the ‘ projective completion of the a¢ ne plane .’ We shall also be interested in point-line geometries with line par- allelism with a slightly weaker set of axioms than that of an a¢ ne plane. Definition 5. A ‘ Sperner Space’ is a triple (P; L; I), of sets of ‘ points P,’ ‘ lines L,’ and ‘ incidence I P L,’ with the following properties: (1) Given two distinct points P and Q; there is a unique line ` incident with P and Q, and which we shall denote by PQ. (2) There is an equivalence relation on the line set called ‘ paral- lelism.’ (3) Given a line ` and a point P not incident with `, there is a unique line m that is parallel with ` and is incident with P. (4) For any two distinct lines ` and m; the cardinal number of the set of points incident with `, [`]; is equal to the cardinal number of the set of points incident with m, [m]. Remark 4. Let be an a¢ ne plane and and be distinct parallel classes and let lines ` 2 and m 2 . Then clearly m and ` are incident with a unique common point, for otherwise m would be parallel to `.
- 28. Combinatorics of Spreads and Parallelisms 5 More generally, if S is a Sperner space the lines from distinct par- allel classes need not share a common incident point. For example, an a¢ ne space is a Sperner space. Definition 6. A ‘ translation’of an a¢ ne plane (respectively, of a Sperner Space) is a collineation that leaves invariant all parallel classes and …xes each line of some parallel class. The group generated by the set of translations is called the ‘ translation group.’ Definition 7. A ‘ translation plane’ (respectively, a ‘ translation Sperner Space’ ) is an a¢ ne plane (respectively, Sperner Space) that admits a translation group that acts transitively on the points (i.e., ‘ a¢ ne points’ ). It turns out that a translation plane or a translation Sperner Space has an underlying vector space whose vectors are the points of the structure. We consider a construction that turns out to be canonical. Theorem 1. Let V be a vector space of dimension kt over a skew- …eld K, whose non-zero vectors are partitioned by a set of mutually disjoint t-dimensional subspaces. We de…ne such a partition to be a ‘ t-spread.’ De…ne ‘ points’ to be vectors, de…ne a ‘ t-component’ to be any t-subspace of the t-spread, and de…ne ‘ lines’ to be vector translates of t-components, with incidence inherited from the vector space. (1) Then the point-line geometry constructed is a translation Sperner Space. (2) If the dimension of the vector space is 2t, then the point-line geometry is a translation plane. Proof. Let P and Q be distinct points. Translate Q to the zero vector 0 by a translation Q. Then QP is incident with a unique t- component `t. P and Q are incident with a unique line 1 Q `t. Two lines are said to be parallel if and only if one is an image of the other by a translation. The translations act on the points and hence induce mappings on the corresponding points. Given any component `t, there is a translation subgroup that …xes `t and acts transitively on the points incident with `t. The parallel classes of lines are, then the set of orbits of the components under the vector space translation group (which, of course, is transitive on the points). Note that every vector space translation is a ‘ translation’in the sense of De…nition 6, since every group element 1 Q that maps 0 to Q will …x the unique t-space 0Q that contains 0 and Q and hence must …x each element of the orbit of 0Q. It now follows directly that a translation Sperner Space is obtained. Now assume that the dimension of the vector space is 2t. Let i and j
- 29. 6 Partitions of Vector Spaces be distinct lines and let `t and mt be the two distinct components such that i is in the translation orbit of `t and j is in the translation orbit of mt. Then `t mt = V , so that if T`t and Tmt are the translation subgroups that …x `t and mt, respectively, then the full translation group T = T`t Tmt . Note that there exists an element mt of Tmt that maps `t to i and an element `t that maps mt to j. Then = `t mt maps (`t; mt) to (i; j), as the group is Abelian. Hence, i j = 0. Therefore, the point-line geometry is a translation plane. Remark 5. The construction in the preceding can be made much more general. For example, we illustrate this in the …nite case. De- …ne a ‘ partition’of a …nite vector space over a …eld K isomorphic to GF(q), for q = pr , a prime power, to be a set of vector subspaces whose union is a exact cover of the non-zero vectors of V . Then a point-line geometry P may be de…ned by taking the ‘ points’as vectors and ‘ lines’ as translates of the partition subspaces (‘ components’of the partition), and where two lines are said to be ‘ parallel’if and only if they belong to the same orbit under the translation group of the vector space. In this setting, lines do not have to be incident with the same number of points. Definition 8. Let P be a point-line geometry such that any two distinct points are incident with a unique line, and which admits an equivalence relation on the line set satisfying the Euclidean parallel ax- iom. A translation is de…ned as a collineation that …xes all parallel classes and …xes one parallel class linewise. In the …nite case, if there are at least three parallel classes and if there is a set fs1; ::; skg such that each line of a parallel class is incident with exactly si points and each sj is the number of points of some line, we shall say that the point-line geometry is of type (s1; ::; sk), for s1 > s2 > ::: > sk. The translation group is the group of translations (each element is a translation) and if the translation group is transitive on the points of P then we shall say that we have a ‘ translation geometry.’ In the partition case, we then say that we have a ‘ translation geom- etry of type (s1; ::; sk).’ We shall have the occasion to study translation geometries of var- ious types. Later in this text, we shall be considering the theory of ‘ focal-planes’and, more generally, ‘ double-planes.’ Definition 9. A ‘ focal-plane’is a point-line geometry of type fs1; s2g, there is exactly one parallel class each of whose lines are incident with s1 points and all remaining lines are incident with s2 points.
- 30. Combinatorics of Spreads and Parallelisms 7 A ‘ translation focal-plane’is a translation geometry whose underly- ing point-line geometry is a focal-plane. A ‘ double-plane’ is a point-line geometry of type fs1; s2g and a translation double-plane is a translation geometry whose underlying point-line geometry is a double-plane. In the following, we shall give constructions of focal-spreads and of more general double spreads, where the ‘ spread’(or perhaps general- ized spread or partition) is the underlying set of vector subspaces of a translation geometry that corresponds to a vector space. 0.2. Beutelspacher’ s Construction. Let Vt+k be a vector space of dimension t + k over GF(q) for t > k and let L be a subspace of dimension t. Let V2t be a vector space of dimension 2t containing Vt+k (t > k required here) and let St be a t-spread containing L. There are always at least Desarguesian t-spreads with this property. Let Mt be a component of the spread St not equal L. Then Mt Vt+k is a subspace of Vt+k of dimension at least k. But, since Mt is disjoint from L, the dimension is precisely k and we then obtain a focal-spread with focus L. This construction may be found in Beutelspacher [14], Lemma 2, page 205. The corresponding focal-spread is said to be a ‘ k-cut of a t-spread,’ and we shall adopt this terminology and use the notation F = StnVt+k for the focal-spread F. The similar formal de…nition is Definition 10. A partition of a …nite-dimensional vector space of dimension t + k by a partial Sperner k-spread and a subspace of dimension t 6= k shall be called a ‘ focal-spread of type (t; k).’ The unique subspace of dimension t of the partition shall be called the ‘ focus’ of the focal-spread. We de…ne a ‘ planar extension’ of a focal-spread as a t-spread such that the focal-spread is of type (t; k) and arises from the t-spread as a k-cut. The ‘ kernel’ of a focal-spread of dimension t+k, over GF(q), with focus of dimension t, shall be de…ned as the endomorphism ring of the vector space that leaves the focus and each k-component invariant. Now we turn to a similar construction of more general double- spreads. 0.3. Double-Spreads Constructed from t-Spreads. Of course, the idea of cutting or slicing a spread could conceivably lead to other in- teresting partitions. For example, from a t-spread of a 2t-dimensional vector space, we could simply select a 2t 1 dimensional subspace
- 31. 8 Partitions of Vector Spaces and form the possible intersections to construct a partition, leading ultimately to a focal-spread of type (t; t 1). In general, it is very di¢ cult to achieve any sort of uniformity with such slicing with explic- itly determined subspaces. However, one type of cut that does produce interesting partitions arises as follows: Let Pt be a t-spread of a st- dimensional vector space over GF(q), where k > 1. Let U denote any vector subspace of dimension st 1. Consider any component of Pt, it follows easily that Pt U has dimension either t or t 1. Let a denote the number of components of intersection of dimension t 1 so that (qst 1)=(qt 1) a components then have dimension t. Therefore, by intersections, we clearly have a cover of U so that qst 1 1 = a(qt 1 1) + ((qst 1)=(qt 1) a)(qt 1). This implies that aqt 1 (q 1) = qst 1 (q 1); so that a = qt(s 1) . Therefore, we have constructed a double-spread (a partition of a vector space with exactly two cardinalities of components) of type (t; t 1) with (q(s 1)t 1)=(qt 1) subspaces of dimension t and qt(s 1) subspaces of dimension t 1. Definition 11. The notation becomes also problematic, since we use only the dimensions and say that the double-spread is of type (t; t 1). However, in this case, there are qt(s 1) 1 qt 1 subspaces of dimension t and qt(s 1) subspaces of dimension t 1. The double-spread of this type is said to be a ‘ (t 1)-cut’of a t-spread for an st-dimensional subspace. Remark 6. Note that t + k = st 1 if and only if t(s 1) = k + 1 and for k < t. So, t(s 1) t 1 + 1, implying that s 2. So, when s = 2, we see that we have a focal-spread of type (t; t 1). Hence, the previous construction provides a focal-spread of type (t; k), exactly when s = 2 and otherwise, a double-spread that is not a focal-spread is constructed. Remark 7. Of course, given any double-spread of type (t; t 1), in a vector space of dimension st 1, the natural question is whether it arises from a t-spread of a st-dimensional vector space, as a (t 1)-cut. We shall develop the theory of focal-spreads more completely in a later chapter and further give several other constructions of double- spreads and triple-spreads.
- 32. Combinatorics of Spreads and Parallelisms 9 0.4. Main Theorem on Finite Translation Geometries. The following material also is valid for arbitrary translation planes and the reader is referred to the Foundations text [21] to see this. The proof of the theorem stated below for …nite translation geometries is more gen- erally valid for arbitrary translation planes, with appropriate changes. Theorem 2. Every …nite translation geometry corresponds to a …nite vector space over a …eld K. The …eld K may be identi…ed with the endomorphism ring of the associated translation group that leaves invariant each parallel class. We shall give the proof as a series of lemmas, parts of which are left to the reader to complete. In the following, we assume the hypothesis of Theorem 2. We …rst point out some trivial facts. Remark 8. (1) Each point is incident with a unique line of each parallel class. (2) A non-identity translation …xes a unique parallel class linewise. (3) A non-identity translation is …xed-point-free so that the group T of translations is …xed-point-free. Proof. Part (1) follows immediately from the de…nitions. Let be a translation. Then …xes all parallel classes and …xes one parallel class linewise. Suppose that …xes another parallel class 6= linewise. Choose any point P, so there is a line ` of and a line ` of both incident with P. Since …xes both of the indicated lines, it follows that …xes P, so that …xes all points and hence is the identity mapping. If …xes an a¢ ne point P, then …xes all lines incident with P as it …xes all parallel classes. Choose any point Q distinct from P and form the line QP. Let m be the line of incident with Q. Since …xes QP and m, it follows that it also …xes Q. Hence, …xes all points and so is the identity translation. If is in the translation group it then …xes all parallel classes. If …xes an a¢ ne point P, then …xes all lines incident with P. Definition 12. If is a translation of a translation geometry, the unique parallel class linewise …xed by is called the ‘ center’of . If is a parallel class, let T( ) denote the group of translations with center . The …rst two parts of the following lemma are immediate. Lemma 1. Let T denote the group of translations of a translation geometry. (1) Then T( ) is a normal subgroup for each parallel class . (2) T = [ T( ), a partition of T.
- 33. 10 Partitions of Vector Spaces (3) T is Abelian. Proof. (3): Let g and h be elements of T( ) and T( ), for 6= . Then the commutator ghg 1 h 1 is in the intersection of both trans- lation subgroups, which are disjoint. Hence, elements of T( ) and elements of T( ) commute. Let g and k be elements of T( ) and h an element of T( ). Then (gk)h = g(kh) = g(hk) = (gh)k, and kh 2 T( ), for a parallel class distinct from or , implying that g(kh) = (kh)g = k(hg) = (hg)k, using the same principle. Hence, (gh)k = (hg)k, so that gh = hg. This completes the proof. Lemma 2. Let K denote the set of endomorphisms of T that …x each T( ). Then K is a …eld and T is a vector space over K. Identifying the point set with T, then the translation geometry becomes a vector space over K and the lines incident with 0 are vector subspaces over K so that we have a partition of a vector space. Proof. Let be a non-zero endomorphism. If is not injective, then since T is Abelian, we may assume that (z) = 1; for some non- identity element z of a partition element, say T( ). For u in T( ), for 6= , then zu 2 T( ), for distinct from or . So, (zu) = (z) (u) = (u), but (zu) 2 T( ) and (u) 2 T( ) so both are in the intersection T( ) T( ) = h1i. Therefore, T( ) Kernel , which also implies that [ 6= T( ) Kernel . Now repeat the argument for a kernel element w in T( ), to obtain T( ) Kernel , so that is the identity endomorphism. Since the translation geometry is …nite, it follows that K is a …eld. Then considering the natural endomorphism ring of T, T becomes a K-vector space. Now identifying the points of the translation geometry with the elements of T and using additive notation, choose the point 0 (the ‘ origin’ ) then it follows immediately that lines through the origin may be identi…ed with the groups T( ), which are clearly K-vector subspaces. This completes the proof of the theorem. 1. Collineation Groups of Translation Geometries In this section, we show that the full collineation group of a trans- lation geometry with underlying vector space V over the …eld K (or skew…eld if the translation geometry is a translation plane) is a semi- direct product of a subgroup of L(T; K) by the translation subgroup T. The proof given is essentially the same as that in Lüneburg [167] (1.10), determining isomorphisms of …nite translation planes. If G is a collineation group of a translation geometry (…nite with …eld kernel K or an arbitrary translation plane with skew…eld kernel
- 34. Combinatorics of Spreads and Parallelisms 11 K), let T denote the translation group and let G0 denote the subgroup of G that …xes the zero vector 0. Clearly, T is a normal subgroup so that for g 2 G, then if g0 6= 0, there is a translation so that g0 = 0, so that g 2 G0, which implies that G G0T. Now if h is a collineation that …xes 0, h will permute the lines incident with 0 (components), which means that h permutes the parallel classes. We wish to prove that h 2 L(T; K), that is, h is a semi-linear group element, where T is a K-vector space. For P 2 T, where P (0) = P, we, of course, know that h P h 1 is a translation Q, where Q(0) = Q, for points P and Q of . Since h leaves 0 invariant, we note that Q = hPh 1 = h P h 1 = hP . Now consider h(P+Q)h 1 = h(P+Q) : h(P+Q) = h(P+Q)h 1 = h P+Qh 1 = h P Qh 1 = h P h 1 h Qh 1 = hPh 1 hQh 1 = hP hQ = hPh 1+hQh 1 . Therefore, h(P + Q) = hP + hQ. It remains to show that if k is any element in the kernel K of T then h(kP) = k h(P), for all points P, where is an automorphism of K. It is clear that K may be considered a normal subgroup of the collineation group that …xes 0. Let hkh 1 = k . First, h(kP) = hkh 1 (hP) = k h(P), and for k; k0 2 K, we have that h(k + k0 )P = (k + k0 ) hP = h(kP + k0 P) = h(kP) + h(k0 P) = k h(P) + k0 h(P) = (k + k0 )(hP), implying that (k + k0 ) = (k + k0 ). Similarly, h(kk0 P) = (kk0 ) h(P) = h(k(k0 P) = k h(k0 P) = k k0 h(P), implying that (kk0 ) = k k0 . Theorem 3. In any translation geometry (or translation plane) with ambient vector space V over a …eld (or skew…eld K), the full collineation group is a semi-direct product of the subgroup of L(V; K) by the translation subgroup T. The subcollineation group that …xes the zero vector shall be called the ‘ translation complement.’ The subgroup of the translation complement of GL(V; K) is called the ‘ linear translation complement.’
- 35. 12 Partitions of Vector Spaces 1.1. Collineation Groups of Translation Planes. We now spe- cialize to …nite translation planes. We shall be mostly interested in …- nite translation planes whose underlying vector space is 4-dimensional over a …eld K isomorphic to GF(q), where q = pr , for p a prime and r a positive integer. We shall refer to the parallel classes of a translation plane as the points on the line at in…nity. Definition 13. If V is a vector space of dimension z over a …eld L, the ‘ projective geometry PG(z 1; L)’ is the lattice of vector subspaces of V . A 1-dimensional subspace is said to be a ‘ point,’a 2-dimensional vector subspace a ‘ line,’and a 3-dimensional vector space a ‘ plane.’ An ‘ a¢ ne space AG(z; L),’is the geometry of translates of the sub- spaces of a vector space. Definition 14. A collineation of a (an a¢ ne) translation plane is a ‘ central collineation’provided …xes a line ` of the plane pointwise. The collineation is an ‘ elation’if …xes all lines parallel to `, and we say the ‘ center’ of is incident with ` (on the line at in…nity) and ` is the ‘ axis’of . The collineation is a ‘ homology’if …xes all lines of a parallel class not containing `. If the line of incident with 0 is denoted by 0 , we shall say that ` is the ‘ axis,’0 is the ‘ coaxis’and is the ‘ center’of . Definition 15. If + is a …nite projective plane of order q2 (q2 +1 is the number of points per line), a projective subplane of order q is said to be a ‘ Baer’subplane + 0 . Any associated a¢ ne restriction of + is also said to have order q2 . If the Baer subplane of + contains the line of restriction, then the corresponding a¢ ne subplane of order q is a subplane of . A ‘ Baer collineation group’of a …nite a¢ ne plane is a collineation group that …xes a Baer subplane pointwise. Remark 9. For …nite translation planes of order q2 , it is known that any Baer collineation group has order dividing q(q 1). For more details, the reader is directed to Biliotti, Jha, Johnson [21]. Notation 1. In this text, when considering translation planes of order q2 , constructed or equipped with a 2-spread over GF(q), we may always represent vectors in the form (x1; x2; y1; y2), for all xi, yi 2 GF(q); i = 1; 2, and further let x = (x1; x2), y = (y1; y2). Then there is a set of q2 2 2 matrices over GF(q), including the zero matrix, so that the components of the 2-spread are as follows: x = 0; y = 0; y = x g(t; u) f(t; u) t u ; u; t 2 GF(q),
- 36. Combinatorics of Spreads and Parallelisms 13 and g and f are functions from GF(q) GF(q) such that the matrices and their di¤erences are either non-singular or the zero matrix. Conversely, any set of q2 2 2 matrices, including the zero matrix with the above properties, determines a translation plane of order q2 . This representation shall become extremely important when we con- sider ‡ocks of quadratic cones. The translation complement of any translation plane of this type is a subgroup of L(4; q).
- 37. CHAPTER 2 Finite Focal-Spreads In the previous chapter, we brie‡ y discussed focal-spreads and the Beutelspacher’ s construction. Although there are a number of examples of such partitions, there has been no theory developed about general partitions. Part of our treatment in this text is modeled from the article by Jha and Johnson [77], and the reader is directed to this article for additional details. Although it is certainly possible to consider focal-spreads over arbi- trary …elds, all of the material that is presented in this text is for …nite focal-spreads. The reader is directed to the open problem chapter 40 for more information. In this chapter, we begin to build a theory of partitions based on focal-spreads and their obvious connections to translation planes. For general focal-spreads, however, nothing is known. For example, it is not known whether collineations that …x components pointwise or ho- mologies are elations or whether an involution that does not …x a com- ponent pointwise becomes a Baer collineation or a kernel involution. In the ‘ Handbook of Finite Translation Planes’[138], it is mentioned that there are not very many known partitions of vector spaces, in the sense that the partitions are not re…nements of Sperner t-spreads. Furthermore, the existence of such partitions has been established by Beutelspacher [14], who proves if the dimension of the vector space is n and it is required to …nd a partition, where the dimensions of the sub- spaces are ft1; t2; ::; tkg, where ti < ti+1; then if gcdft1; t2; ::; tkg = d and n > 2t1([tk=(d k)]+t2+:::+tk), a partition may be constructed with var- ious subspaces of dimension ti. However, these partitions are basically constructed using focal-spreads and general spreads. We give some of the constructions later in the section on ‘ towers of focal-spreads.’ The main point of this chapter is to use of ideas and theory of …nite translation planes in order to develop a certain theory of focal- spreads and, since these seem to be important building blocks of general spreads, such work might prove useful to the general theory of parti- tions. 15
- 38. 16 Finite Focal-Spreads Given a k-cut, of course, there is a corresponding spread for a trans- lation plane that produces it. It is a central and potentially important question to ask if every focal-spread is a k-cut; if it can ‘ extended’to a spread for a translation plane. Therefore, here we give the basics of focal-spreads and use some theory from translation planes for their study. We also show that the existence of focal-spreads of type (k + 1; k) leads to a construction of designs of type 2 (qk+1 ; q; 1) and to other double-spreads or triple- spreads (partitions where there are subspaces in the partition of either two or three di¤erent dimensions, respectively). The formal de…nition of a focal-spread has been given in De…nition 10. 1. Towers of Focal-Spreads As we begin our discussions on t-spreads, we have brie‡ y discussed focal-spreads and found that they are readily constructed, but we might also just as well ask the obvious question: How hard is it to …nd general partitions of vector spaces? That is, partitions of the non-zero vectors by a set of subspaces of dimensions ti, for i = 1; 2; ::; k. In Beutelspacher [16], and more generally in Heden [65], there are a variety of results of the existence of partitions of type T = ft1; t2; ::; tkg, in vector spaces of dimension n over GF(q), where the partition admits exactly subspaces over GF(q) of dimension ti, for i = 1; 2; ::; n, for t1 < t2 < :::tk. (The reader is cautioned that our previous notation of partitions has the inequalities in the opposite order.) Many of these results rely basically on focal-spreads of type (t; k); in the sense that a variety of focal-spreads or t-spreads are used in the construction. We give here one construction that provides a partition of type T in any vector space of dimension n = Pk i=1 iti over GF(q), showing that again general partitions are readily available. First, we note the following lemma. Lemma 3. In an st-dimensional vector space over GF(q), there is always a t-spread over GF(q). Proof. Let the vectors be represented by GF(qst ). Take the cosets of GF(qt ) in GF(qst ) . Note that GF(qt ) g[f0g is a GF(q)-subspace. Since the cosets partition GF(qst ) , representing the non-zero vectors, the lemma is proved. Lemma 4. Let V be a t1 +t2-dimensional vector space over GF(h), for h a prime power. If t2 > t1, then there is focal-spread of type (t2; t1).
- 39. Combinatorics of Spreads and Parallelisms 17 Proof. Form a 2t2-dimensional vector space over GF(h) and con- struct a t2-spread. Since t2 > t1, there is a proper focal-spread of type (t2; t1) constructed as a t1-cut. Theorem 4. A partition of type ft1; t2; ::; tkg exists in any vector space of dimension Pk i=1 iti over GF(q). Proof. First assume that i = 1, for all i = 1; 2; ::; k. Construct a (t2; t1)-focal-spread in a t1 + t2-dimensional vector space over GF(q). Now there are three cases (1) t1+t2 > t3, (2) t1+t2 = t3, (3) t1+t2 < t3. In case (1), in a t1 + t2 + t3-dimensional vector space, construct a (t1 + t2; t3)-focal-spread. We consider the direct sum of a vector space of dimension t1 + t2 with a vector space of dimension t3 and make the obvious identi…cations. On the unique focus, we identify our original (t2; t1)-focal-spread. We then would have subspaces of dimensions ex- actly t1; t2; t3. In case (2), if t1 +t2 = t3, in a 2t3-dimensional subspace, we always have a t3-spread. In any proper subset of the components, we may consider forming (t2; t1)-focal-spreads, again providing the re- quired subspaces of dimensions t1; t2; t3. Finally, in case (3), we con- struct a (t3; t2 + t1)-focal-spread. Then on all (t2 + t1)-components, we construct (t2; t1)-focal-spreads. Clearly, this sort of argument may be continued and completes the proof of the theorem, in the case when all i = 1. Now more generally, we claim that in a vector space of dimension 1t1 + 2t2, there is a partition of type ft1; t2g. Since t2 > t1, form a focal-spread of type ( 1t2 + 2t2; t1). Then on the unique focus of dimension 1t1 + 2t2 construct another focal-spread of type (( 1 t)t1 + 2t2; t1). If we continue with the same line of argument, we end up with a focal-spread with focus of dimension 2t2, which always contains a t2-spread by Lemma 3. Now consider a vector space of dimension 1t1 + 2t2 + 3t3 = z + 3t3 and form a sequence of (e; t3)- focal spreads until we obtain a focus (or vector space) of dimension 1t1 + 2t2 + t3. We consider the cases (1) 1t1 + 2t2 > t3, (2) 1t1 + 2t2 = t3, and (3) 1t1 + 2t2 < t3. In case (1), we may form a ( 1t1 + 2t2; t3)-focal-spread. Then, on the focus of dimension 1t1 + 2t2, by the previous argument, we may …nd a ft1; t2g-partition. In case (2), we have a 2t3-dimensional vector space, which allows a t3-partition. Take any component of dimension t3 = 1t1 + 2t2 and again …nd a ft1; t2g-partition. In case (3), we form a series of (r; ti)-focal-spreads on the corresponding foci, for i = 1; 2, possible since t3 > ti, for i = 1; 2. This leaves us with a focus of dimension t3. Clearly, this argument is inductive, thus completing the proof of the theorem. Alternatively, in this last case (3), form a
- 40. 18 Finite Focal-Spreads (t3; 1t1 + 2t2)-focal-spread and on each subspace of dimension 1t1 + 2t2 form ft1; t2g-partitions. Definition 16. A partition of a vector space V of dimension n over GF(q) shall be called a ‘ tower of general focal-spreads’ if and only if there is a sequence of vector subspaces F1 F2 F3; ::; Fk = V , such that Fj is a vector space of dimension Pj i=1 iti admitting a partition by a set of focal-spreads of type (e; tz), for z = 1; 2; ::; j, or by tz-spreads, for i a positive integer, i = 1; 2; ::; j. We note that built into the construction of towers of general focal- spreads, there is a certain variation. For example, consider the con- struction of a f4; 5; 9g-partition in a 18-dimensional vector space over GF(q). We begin with the construction of a (5; 4)-focal spread in a 9-dimensional vector space and form the direct sum of a 9-dimensional and a second 9-dimensional vector space. In this 18-dimensional vec- tor space, we take any 9-spread. Now take any proper subset of 9- dimensional components and form (5; 4)-focal-spreads. Since we may do this construction for any proper subset, we always end up with a partition of type f4; 5; 9g that arises from a tower of focal-spreads. This complexity makes the problem listed above potentially quite challeng- ing. So, we are interested in what can be said about the general the- ory of focal-spreads, as focal-spreads and t-spreads seem to form basic building blocks of arbitrary partitions, about which we know essentially nothing beyond existence. We shall see in the next chapters that there are a number of ways of constructing focal-spreads that do not appear to arise from the Beutelspacher cut-method. 2. Focal-Spreads and Coordinatization It is tempting to de…ne a focal-spread simply as a partition by sub- spaces of two di¤erent dimensions, of which there is a unique subspace of one of the dimensions. For example, if the dimension of the ambient vector space is n and the subspaces have dimensions t and k, we call such partitions of ‘ type (n; (t; k))’and if n = t + k, then of type (t; k), over a …nite …eld GF(q). In fact, it is possible to construct such par- titions as we have previously seen and which, we shall also do in due course. In the beginning, we insist that the dimension is t+k, for focal- spreads of type (t; k). We …rst consider how to represent focal-spreads using matrices, in a manner analogous to that of the representation of translation planes.
- 41. Combinatorics of Spreads and Parallelisms 19 2.1. Matrix Representation of Focal-Spreads. As for planar spreads, it is usually helpful to express these results in terms of matri- ces. The reader might wish to work through this material carefully as the ideas closely parallel analogous matrix coordinatization for trans- lation planes. For addition background, the Foundations text can be consulted [21]. Let B be a focal-spread of dimension t + k over GF(q) with focus L of dimension t. Fix any k-component M. We may choose a basis so that the vectors have the form (x1; x2; ::; xk; y1; y2; ::; yt). Let x = (x1; x2; ::; xk) and y = (y1; y2; ::; yt), where the focus L has equation x = 0 = (0; 0; 0; ::; 0) (k-zeros) and the …xed k-component M has equation y = 0 = (0; 0; 0; ::; 0) (t-zeros). We note that qt+k qt = qt (qk 1), which implies that there are exactly qt k-subspaces in the focal- spread. We refer to this as the ‘ partial Sperner k-spread’ . Take any k-component N distinct from y = 0. There are k basis vectors over GF(q), which we represent as follows: y = xZk;t, where Zk;t is a k t matrix over GF(q), whose k rows are a basis for the k-component. It is clear that we obtain a set of qt k-components, which we also represent as follows: Row 1 shall be given by [u1; u2; ::; ut], as the ui vary independently over GF(q). Then the rows 2; ::; k have entries that are (turn out to be) linear functions of the ui. It then now follows directly that the k t matrices in the focal-spread have rank k and the di¤erence of any two distinct matrices associated with k-components also has rank k. The following is the analogous result of translation planes: Theorem 5. Let Vt+k be a t + k-dimensional vector space over GF(q) and let S be a set of qt 1 k t matrices of rank k such that the di¤erence of any two distinct matrices also has rank k. Then there is an associated focal-spread constructed as x = 0; y = 0; y = xM; M 2 S, where x is a k-vector and y is a t-vector over GF(q), where the focus is x = 0. In particular, it is possible to choose one k-space to have 1’ s in the (i; i), position and 0’ s elsewhere in the k t matrices. We denote this matrix by Ik t. Conversely, any focal-spread has such a representation. The reader is directed to the open problem chapter 40 for more discussion on how a generalization for arbitrary partitions might be phrased using a matrix approach. There is, of course, a coordinate-free approach given as follows:
- 42. 20 Finite Focal-Spreads Definition 17. Let V = Vt Vk be a …nite vector space of rank t+k expressed as a direct sum of subspaces Vt and Vk, having dimensions t and k, respectively. Then a ‘(t;k)-spread set on V ,’based on axes (Vk; Vt), is a collection S of linear maps from Vk to Vt, such that (1) 0 2 S; (2) the nonzero maps in S are injective; (3) the di¤erence between any two members of S is injective or zero; (4) S is transitive in the sense that for any pair of non-zero vectors The following remarks are then left to the reader to verify. Remark 10. Every (t; k)-spread set S as above yields a focal-spread of type (t; k), with component set fMS : S 2 Sg [ fVtg, where MS := f(x; xS) : x 2 Vkg. Every focal-spread of type (t; k) may be coordinatized by a (t; k)- spread set, and any focal-spread of type (t; k) arises by coordinatization by a (t; k)-spread set, which is uniquely determined by the focus and any other component chosen as ‘ basis.’ 3. k-Cuts and Inherited Groups In this section, if a focal-spread is a k-cut of a translation plane, we ask what sorts of collineations of the translation plane become collineations of the focal-spread? More generally, given an abstract focal-spread, how are central collineations de…ned? We begin with the simplest type of collineation of a focal-spread, a ‘ homology.’ Definition 18. In a focal-spread of dimension t + k over GF(q) and focus L of dimension t, every collineation is assumed to be an element of L(t + k; q) that leaves invariant the focus L and permutes the components of the partial Sperner k-space S. A ‘ homology’h is a collineation of GL(t + k; q) with the following properties: (1) h leaves invariant the focus L and another k-subspace of S, (2) h …xes one of the two …xed components pointwise and acts …xed- point-free on another …xed component (in the case that a k-component is …xed pointwise, we assume that the group acts …xed-point-free on the focus). To show why there is a potential problem with dealing with collineations that one might wish to call a homology, consider the focal-spread of type (2; 1) over GF(q), represented as x = 0; y = x[t; u], for all t; u 2 GF(q),
- 43. Combinatorics of Spreads and Parallelisms 21 where the focus is x = 0, a 1-vector. Let B = 1 1 0 1 , and consider the collineation rB : (x; y) ! (x; yB). We note that (0; 1) 1 1 0 1 = (0; 1), so that rB is not semi-regular on the focus, and since y = x[0; 1] is …xed by rB, the collineation is also not semi-regular on the components other than the putative coaxis x = 0. In terms of homologies that are not a¢ ne collineations that …x the line at in…nity pointwise (the axis) and …x an a¢ ne point (the center), the situation is much more manageable. This is because a …nite focal- spread is a …nite translation geometry, and Theorem 3 shows that the subgroup that …xes each component (k-component and focus) is the cyclic subgroup of a …eld; the kernel is a …eld. We now intend to show that a non-identity homology (see De…nition 18) …xes exactly two components. The pointwise …xed subspace is called the ‘ axis’of h and the …xed subspace is the ‘ coaxis’of h. Thus, the focus is either the axis or coaxis of any homology. Proposition 1. A non-identity homology of a focal-spread …xes exactly two components and permutes the remaining components semi- regularly. Proof. We represent the focal-spread in the form x = 0; y = 0; y = xM; M is a k t matrix in set M for x a k-vector and y a t-vector, where x = 0 is the focus, and y = 0 is a k-space, and where we choose Ik t 2 M. Assume that y = xM1 is …xed pointwise by a collineation h. Choose a basis by the mapping (x; y) ! (x; xM1 + y), to change the form to x = 0; y = 0; y = x(M1 M); M is a k t matrix in set M. Now choose any of these matrices M1 M2, of rank k, and column- reduce to Ik t. The other matrices M1 M reduce to rank k matri- ces, whose di¤erences are also of rank k. Hence, if y = xM1 is …xed pointwise by h, we may assume without loss of generality that y = 0 is …xed pointwise by h. Therefore, any collineation in GL(t + k; q) that …xes a component pointwise may be represented in either the form lA : (x; y) ! (xA; y), where A is a non-singular k k matrix or rB : (x; y) ! (x; yB), where B is a non-singular t t matrix. First, assume that rB is a homology, so that B acts …xed-point-free on the focus. Assume that rB …xes x = 0; y = 0; and y = xM. So, xMB = xM. Choose then any x0M = z0, for z0 a t-vector, so that z0B = z0. However, this is contrary to the action of B. Since the
- 44. 22 Finite Focal-Spreads previous argument is also valid for any power Bj 6= It, then hrBi …xes exactly two components and acts semi-regularly on the remaining com- ponents. Now assume that lA is a non-identity collineation that …xes the focus pointwise. Assume that lA …xes x = 0; y = 0 and y = xN, so that A 1 N = N. Column reduce to A 1 Ik t = Ik t, and then realize that this forces A 1 = Ik, a contradiction. This proves (1) and (2). Now we prove that, contrary to when a k-component is …xed point- wise, if a collineation …xes the focus pointwise, it must be a homology, provided it …xes another component. Proposition 2. If a collineation h …xes the focus pointwise and …xes another component then h is a homology. Proof. Assume that h …xes a k-component y = xM and …x the focus pointwise. Change representation so that the …xed k-component is y = 0. Then represent h as lA : (x; y) ! (xA; y), where A is a non-singular k k matrix. The proof of the previous result shows that if A is not Ik, then lA is semi-regular on the remaining components. Assume that x0A = x0. Then (x0; x0Ik t) is on both y = xIk t and y = xA 1 Ik t, a contradiction. Hence, h is a homology. We now de…ne the concept of an ‘ elation’of a focal-spread. Definition 19. An ‘ elation’e of a focal-spread of type (t; k) over GF(q) is a collineation of GL(t+k; q) with the following two properties: (1) e …xes the focus L pointwise, and (2) if V is the associated vector space of dimension t + k over GF(q), e …xes V=L pointwise. Note that a collineation e in GL(t + k; q) that …xes the focus x = 0 pointwise may be represented in the form eA;C : (x; y) ! (xA; xC + y), where A is a non-singular k k matrix, where C is a k t matrix of rank k. Without additional assumptions, there is no reason to assume that A is necessarily I. But, under the assumption of De…nition 19, condition (2), a collineation eA;C is an elation if and only if A = I, since on V=(x = 0), eA;C acts on (z; w)+(x = 0) as (zA; zC +w)+(x = 0) = (z; w)+(x = 0) if and only if (zA z; zC) 2 (x = 0), so A = I. We note then that when A = I, the collineation is additive and semi-regular or order p, because ej I;C : (x; y) ! (x; jxC + y) (for q = pr , p a prime) on M [ f0g. Furthermore, any group E of elations is an elementary Abelian p-group that acts semi-regularly on the partial Sperner spread. In terms of inherited groups, the following is immediate: Lemma 5. Let V = V (2t; q) and let S be a t-spread on V and assume that G is a collineation group of S. Let W be a subspace of
- 45. Combinatorics of Spreads and Parallelisms 23 dimension t+k containing a component L and assume that both W and L are G invariant. Then G acts faithfully as a group of collineations of the k-cut F = S n W. So, obviously, any kernel homology group of the translation plane of order qt that leaves k-subspaces invariant inherits as a collineation group of any k-cut. Therefore, we obtain the following corollary in- volving the inheritance of central collineations. Corollary 1. (1) Any a¢ ne homology group with axis y = 0, and coaxis x = 0, of the t-spread inherits as a collineation group of a k-cut focal-spread with focus x = 0. (2) Any a¢ ne elation group with axis x = 0 of the t-spread inherits as a collineation group of any k-cut focal-spread with focus x = 0. Proof. Consider the axis y = 0 of an a¢ ne homology group. Take the subspace Vt+k generated by any k-subspace of y = 0 and x = 0. Then the a¢ ne homology group will leave Vt+k invariant. This proves (1). Let E be an elation group with axis x = 0. An elation group acts trivially on the quotient space V2t=(x = 0) and hence will leave Vt+k invariant. This proves (2). We note that planar extensions of focal-spreads do not have to be unique, as one extension might admit a particular collineation group that the other does not. We need, therefore, a de…nition of what it might be for a planar extension to admit a given collineation group. Definition 20. Let F be a focal-spread of type (t; k) with collineation group G in L(t+k; q) and let V be a 2t-dimensional vector space over GF(q). Let W be a t + k-dimensional subspace of V containing the focus L of F. Then we shall say that F has a ‘ planar extension with group G’if and only if there is a t-spread S of V and a group G0 in L(2t; q), such that there is a GF(q)-linear monomorphism : F ! V and a group isomorphism f : G ! G0 such that (L) lies in a unique spread component L0 of S, where (L ) L0f( ) , for 2 G. When G = h1i, we simply say that F has a ‘ planar extension.’ Now we show at least one situation where a focal-spread necessarily is a k-cut. As a guide to the argument that we give, we formulate a matrix-based method to construct k-cuts from t-spreads (corresponding to translation planes of order qt with kernel containing GF(q)). The situation that we then consider will show almost immediately that the focal-spread in question can arise as a k-cut. 3.1. Matrix Representation of k-Cuts.
- 46. 24 Finite Focal-Spreads Theorem 6. Let be a translation plane of order qt , and ker- nel containing GF(q). Represent points of the 2t-dimensional GF(q)- vector space as (x1; ::; xt; y1; ::; yt); where xi; yi 2 GF(q); for i = 1; 2; ::; t: Assume that we have a matrix t-spread set x = 0; y = 0; y = xM; M 2 M, where M is a set of non-singular t t matrices, where the di¤erences of distinct matrices are also non-singular, and where x = (x1; ::; xt); y = (y1; ::; yt): Let Vt+k be the t + k-dimensional subspace of vectors (x1; ::; xk; 0; ::; 0; y1; y2; ::; yt) for all xj; yi 2 GF(q); for j = 1; 2; ::; k; i = 1; 2; ::; t: If we form the k-cut, Vt+k Z, where Z is one of the components of the matrix t-spread, we have the focal-spread: Vt+k (x = 0) = (x = 0); (note the two uses of x = 0) Vt+k (y = 0) = f(x1; ::; xk; 0; 0; ::; 0); xi 2 GF(q); i = 1; 2; ::; kg; Vt+k (y = xM) = f(x1; ::; xk; 0; ::; 0); (x1; ::; xk; 0; ::; 0)Mg, 8M 2 M. Now if Ik t is the k t matrix with 1 in the (i; i) positions, for i = 1; 2; ::; k and zeros in all other positions, then (x1; ::; xk; 0; :::0)M = (x1; ::; xk)Ik tM: Now suppress the t = k zeros in x and now use x to represent (x1; ::; xk). Then the focal spread of type (t; k) over GF(q) is x = 0; y = 0; y = xIk tM, for M 2 M, where x is a k-vector and y is a t-vector. Two major questions regarding focal-spreads and homology or ela- tion groups arise. If a focal-spread admits a collineation group …xing a line pointwise and acting regularly on the remaining non-…xed compo- nents, is the focal-spread a k-cut of a spread of a translation plane? If the group is a putative homology group, then we may prove that this is always the case. We begin with a fundamental proposition. Proposition 3. A collineation h of a focal-spread that …xes a k- component pointwise and is semi-regular on the remaining components is a homology.
- 47. Combinatorics of Spreads and Parallelisms 25 Proof. Represent the focal-spread in the form x = 0; y = 0; y = xM; M is a k t matrix in set M for x a k-vector and y a t-vector, where x = 0 is the focus, and y = 0 is a k-space. Assume furthermore that Ik t 2 M. By Proposition 2, we may choose a representation so that the collineation is of the form B : (x; y) ! (x; yB), where B is a non-singular t t matrix. Assume that some B does not act …xed-point-free on the focus x = 0. Let y0B = y0, where y0 is a non-zero t-vector (that is, (0; y0) is …xed by B). There exists a non-zero k-vector x0 and a matrix M of M so that x0M = y0. Hence, (x0; x0MB) = (x0; y0B) = (x0; y0) = (x0; x0M), so that (x0; y0) is a vector on y = xMB and y = xM, so this means that B leaves y = xM, invariant, a contradiction to our assumptions. Hence, h is a homology group and as such acts sharply transitive on the non-zero vectors of the focus x = 0. 3.2. Near…eld Focal-Extension Theorem. Theorem 7. Let F be a focal-spread of type (t; k) over GF(q). Assume that there is an a¢ ne group G of order qt 1 …xing the focus, …xing a k-component pointwise and acting transitively on the remaining k-spaces of the partial Sperner k-spread. (1) Then G is a homology group (every non-identity element of G is an a¢ ne homology with the same axis and coaxis). (2) There is a near…eld plane of order qt so that F is a k-cut of . Hence, there is a planar extension with group G. Proof. By the previous proposition, since G is semi-regular, it follows that G is an a¢ ne homology group. In the context of the matrix cut procedure 3.1, recalling that we may always assume that y = xIk t is a k-component, we begin by …rst considering x and y t-vectors and form the associated 2t-dimensional vector space over GF(q) with vectors (x1; ::; xt; y1; ::; yt) = (x; y). Let C denote the group fB; B = Ik 0 0 B 2 Gg. Form the putative t-spread: x = 0; y = 0; y = xB; B 2 C. Now we claim that this is a t-spread. To see this, we note that if y = xB and y = xD, for B; D 2 C, share a vector (x0; x0B) = (x0; x0D), for x0 6= 0, then x0BD 1 = x0. However, C is …xed-point-free, as noted above. Hence, we obtain a t-spread and by the matrix cut procedure 3.1, it follows immediately that the focal-spread is a k-cut. This com- pletes the proof of the theorem.
- 48. 26 Finite Focal-Spreads We shall return to these ideas in Chapter 39, when we consider doubly transitive focal-spreads. In this setting, it is possible to produce elation groups as well as homology groups. 4. Spread-Theoretic Dual of a Semi…eld We interrupt our discussion of focal-spreads to prepare for the ideas of an ‘ additive’focal-spread, which would be the natural analogue of a semi…eld spread. Although all of the material in this chapter can be given by various other methods, we treat the concepts here strictly from a spread-theoretic viewpoint. This chapter is modi…ed from work of Jha and Johnson [81]. Given an a¢ ne translation plane, complete to the projective exten- sion and dualize the projective plane. If the translation plane is a semi- …eld plane and if the line at in…nity becomes the parallel class (1), then restricting to the a¢ ne plane from the dual semi…eld plane, another semi…eld plane is constructed. Normally all of this is done using the coordinate semi…eld, as taking the reverse or opposite multiplication, a coordinate semi…eld for the dual ‘ a¢ ne’semi…eld plane may be con- structed. Speci…cally, given a …nite semi…eld (S; +; ), then the ‘ dual semi…eld’(S; +; ) may be de…ned by the multiplication a b = b a. It is also true that for …nite translation planes, a duality of the associated projective space provides what we call a ‘ dual spread’, which is always a spread in the …nite case. From a matrix spread set, if x = 0; y = x; y = xM, for M 2 SMat is a matrix spread set for a semi…eld plane, then x = 0; y = x; y = xMt , for M 2 SMat, where Mt denotes the transpose of the matrix M, also gives a semi…eld (see, e.g., Johnson [133]). In general, there are six semi…elds associated with a given one, and in this chapter, we develop a notion of the dual of a semi…eld strictly from the matrix spreadset viewpoint. This material explicates the original results of Kantor [156] but given completely in terms of the spreads and gives a spread description of the six semi…elds arising from a given semi…eld. 5. The Dual Semi…eld Plane We begin with the dual semi…eld. Of course, any …nite semi…eld spread of order pn , for p a prime, may be written in the form x = 0; y = xM,
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- 50. In what times the Planets would fall to the Sun by the power of gravity. The prodigious attraction of the Sun and Planets. cause them to ascend again towards the higher parts of their Orbits; during which time, the Sun’s attraction acting so contrary to the motions of those bodies, causes them to move slower and slower, until their projectile forces are diminished almost to nothing; and then they are brought back again by the Sun’s attraction, as before. 157. If the projectile forces of all the Planets and Comets were destroyed at their mean distances from the Sun, their gravities would bring them down so, as that Mercury would fall to the Sun in 15 days 13 hours; Venus in 39 days 17 hours; the Earth or Moon in 64 days 10 hours; Mars in 121 days; Jupiter in 290; and Saturn in 767. The nearest Comet in 13 thousand days; the middlemost in 23 thousand days; and the outermost in 66 thousand days. The Moon would fall to the Earth in 4 days 20 hours; Jupiter’s first Moon would fall to him in 7 hours, his second in 15, his third in 30, and his fourth in 71 hours. Saturn’s first Moon would fall to him in 8 hours; his second in 12, his third in 19, his fourth in 68 hours, and the fifth in 336. A stone would fall to the Earth’s center, if there were an hollow passage, in 21 minutes 9 seconds. Mr. Whiston gives the following Rule for such Computations. “[31] It is demonstrable, that half the Period of any Planet, when it is diminished in the sesquialteral proportion of the number 1 to the number 2, or nearly in the proportion of 1000 to 2828, is the time that it would fall to the Center of it’s Orbit.” This proportion is, when a quantity or number contains another once and a half as much more. 158. The quick motions of the Moons of Jupiter and Saturn round their Primaries, demonstrate that these two Planets have stronger attractive powers than the Earth has. For, the stronger that one body attracts another, the greater must be the projectile force, and consequently the quicker must be the motion of that other body, to keep it from falling to it’s primary or central Planet. Jupiter’s second Moon is 124 thousand miles farther from Jupiter than our
- 51. Archimedes’s Problem for raising the Earth. Moon is from us; and yet this second Moon goes almost eight times round Jupiter whilst our Moon goes only once round the Earth. What a prodigious attractive power must the Sun then have, to draw all the Planets and Satellites of the System towards him; and what an amazing power must it have required to put all these Planets and Moons into such rapid motions at first! Amazing indeed to us, because impossible to be effected by the strength of all the living Creatures in an unlimited number of Worlds, but no ways hard for the Almighty, whose Planetarium takes in the whole Universe! 159. The celebrated Archimedes affirmed he could move the Earth if he had a place to stand on to manage his machinery[32] . This assertion is true in Theory, but, upon examination, will be found absolutely impossible in fact, even though a proper place and materials of sufficient strength could be had. The simplest and easiest method of moving a heavy body a little way is by a lever or crow, where a small weight or power applied to the long arm will raise a great weight on the short one. But then, the small weight must move as much quicker than the great weight as the latter is heavier than the former; and the length of the long arm of the lever to the length of the short arm must be in the same proportion. Now, suppose a man pulls or presses the end of the long arm with the force of 200 pound weight, and that the Earth contains in round Numbers 4,000,000,000,000,000,000,000 or 4000 Trillions of cubic feet, each at a mean rate weighing 100 pound; and that the prop or center of motion of the lever is 6000 miles from the Earth’s center: in this case, the length of the lever from the Fulcrum or center of motion to the moving power or weight ought to be 12,000,000,000,000,000,000,000,000 or 12 Quadrillions of miles; and so many miles must the power move, in order to raise the Earth but one mile, whence ’tis easy to compute, that if Archimedes or the power applied could move as swift as a cannon bullet, it would take 27,000,000,000,000 or 27 Billions of years to raise the Earth one inch.
- 52. Hard to determine what Gravity is. If any other machine, such as a combination of wheels and screws, was proposed to move the Earth, the time it would require, and the space gone through by the hand that turned the machine, would be the same as before. Hence we may learn, that however boundless our Imagination and Theory may be, the actual operations of man are confined within narrow bounds; and more suited to our real wants than to our desires. 160. The Sun and Planets mutually attract each other: the power by which they do so we call Gravity. But whether this power be mechanical or no, is very much disputed. We are certain that the Planets disturb one another’s motions by it, and that it decreases according to the squares of the distances of the Sun and Planets; as light, which is known to be material, likewise does. Hence Gravity should seem to arise from the agency of some subtile matter pressing towards the Sun and Planets, and acting, like all mechanical causes, by contact. But on the other hand, when we consider that the degree or force of Gravity is exactly in proportion to the quantities of matter in those bodies, without any regard to their bulks or quantity of surface, acting as freely on their internal as external parts, it seems to surpass the power of mechanism; and to be either the immediate agency of the Deity, or effected by a law originally established and imprest on all matter by him. But some affirm that matter, being altogether inert, cannot be impressed with any Law, even by almighty Power: and that the Deity must therefore be constantly impelling the Planets toward the Sun, and moving them with the same irregularities and disturbances which Gravity would cause, if it could be supposed to exist. But, if a man may venture to publish his own thoughts, (and why should not one as well as another?) it seems to me no greater absurdity, to suppose the Deity capable of superadding a Law, or what Laws he pleases, to matter, than to suppose him capable of giving it existence at first. The manner of both is equally inconceivable to us; but neither of them imply a contradiction in our ideas: and what implies no contradiction is
- 53. within the power of Omnipotence. Do we not see that a human creature can prepare a bar of steel so as to make it attract needles and filings of iron; and that he can put a stop to that power or virtue, and again call it forth again as often as he pleases? To say that the workman infuses any new power into the bar, is saying too much; since the needle and filings, to which he has done nothing, re-attract the bar. And from this it appears that the power was originally imprest on the matter of which the bar, needle, and filings are composed; but does not seem to act until the bar be properly prepared by the artificer: somewhat like a rope coiled up in a ship, which will never draw a boat or any other thing towards the ship, unless one end be tied to it, and the other end to that which is to be hauled up; and then it is no matter which end of the rope the sailors pull at, for the rope will be equally stretched throughout, and the ship and boat will move towards one another. To say that the Almighty has infused no such virtue or power into the materials which compose the bar, but that he waits till the operator be pleased to prepare it by due position and friction, and then, when the needle or filings are brought pretty near the bar, the Deity presses them towards it, and withdraws his hand whenever the workman either for use, curiosity or whim, does what appears to him to destroy the action of the bar, seems quite ridiculous and trifling; as it supposes God not only to be subservient to our inconstant wills, but also to do what would be below the dignity of any rational man to be employed about. 161. That the projectile force was at first given by the Deity is evident. For, since matter can never put itself into motion, and all bodies may be moved in any direction whatsoever; and yet all the Planets both primary and secondary move from west to east, in planes nearly coincident; whilst the Comets move in all directions, and in planes so different from one another; these motions can be owing to no mechanical cause of necessity, but to the free choice and power of an intelligent Being. 162. Whatever Gravity be, ’tis plain that it acts every moment of time: for should it’s action cease, the projectile force would instantly
- 54. The Planets disturb one another’s motion. The consequences thereof. The World not eternal. carry off the Planets in straight lines from those parts of their Orbits where Gravity left them. But, the Planets being once put into motion, there is no occasion for any new projectile force, unless they meet with some resistance in their Orbits; nor for any mending hand, unless they disturb one another too much by their mutual attractions. 163. It is found that there are disturbances among the Planets in their motions, arising from their mutual attractions when they are in the same quarter of the Heavens; and that our years are not always precisely of the same length[33] . Besides, there is reason to believe that the Moon is somewhat nearer the Earth now than she was formerly; her periodical month being shorter than it was in former ages. For, our Astronomical Tables, which in the present Age shew the times of Solar and Lunar Eclipses to great precision, do not answer so well for very ancient Eclipses. Hence it appears, that the Moon does not move in a medium void of all resistance, § 174; and therefore her projectile force being a little weakened, whilst there is nothing to diminish her gravity, she must be gradually approaching nearer the Earth, describing smaller and smaller Circles round it in every revolution, and finishing her Period sooner, although her absolute motion with regard to space be not so quick now as it was formerly: and therefore, she must come to the Earth at last; unless that Being, which gave her a sufficient projectile force at the beginning, adds a little more to it in due time. And, as all the Planets move in spaces full of æther and light, which are material substances, they too must meet with some resistance. And therefore, if their gravities are not diminished, nor their projectile forces increased, they must necessarily approach nearer and nearer the Sun, and at length fall upon and unite with him. 164. Here we have a strong philosophical argument against the eternity of the World. For, had it existed from eternity, and been left by the Deity to be
- 55. governed by the combined actions of the above forces or powers, generally called Laws, it had been at an end long ago. And if it be left to them it must come to an end. But we may be certain that it will last as long as was intended by it’s Author, who ought no more to be found fault with for framing so perishable a work, than for making man mortal.
- 56. The amazing smallness of the particles of light. The dreadful effects that would ensue from their being larger. CHAP. VIII. Of Light. It’s proportional quantities on the different Planets. It’s Refractions in Water and Air. The Atmosphere; it’s weight and properties. The Horizontal Moon. 165. Light consists of exceeding small particles of matter issuing from a luminous body; as from a lighted candle such particles of matter continually flow in all directions. Dr. Niewentyt [34] computes, that in one second of time there flows 418,660,000,000,000,000,000,000,000,000,000,000,000,000,000 particles of light out of a burning candle; which number contains at least 6,337,242,000,000 times the number of grains of sand in the whole Earth; supposing 100 grains of sand to be equal in length to an inch, and consequently, every cubic inch of the Earth to contain one million of such grains. 166. These amazingly small particles, by striking upon our eyes, excite in our minds the idea of light: and, if they were so large as the smallest particles of matter discernible by our best microscopes, instead of being serviceable to us, they would soon deprive us of sight by the force arising from their immense velocity, which is above 164 thousand miles every second[35] , or 1,230,000 times swifter than the motion of a cannon bullet. And therefore, if the particles of light were so large, that a million of them were equal in bulk to an ordinary grain of land, we
- 57. How objects become visible to us. PLATE II. The rays of Light naturally move in straight lines. A proof that they hinder not one another’s motions. Fig. XI. In what proportion durst no more open our eyes to the light than suffer sand to be shot point blank against them. 167. When these small particles, flowing from the Sun or from a candle, fall upon bodies, and are thereby reflected to our eyes, they excite in us the idea of that body by forming it’s picture on the retina[36] . And since bodies are visible on all sides, light must be reflected from them in all directions. 168. A ray of light is a continued stream of these particles, flowing from any visible body in straight lines. That they move in straight, and not in crooked lines, unless they be refracted, is evident from bodies not being visible if we endeavour to look at them through the bore of a bended pipe; and from their ceasing to be seen by the interposition of other bodies, as the fixed Stars by the interposition of the Moon and Planets, and the Sun wholly or in part by the interposition of the Moon, Mercury, or Venus. And that these rays do not interfere, or jostle one another out of their ways, in flowing from different bodies all around, is plain from the following Experiment. Make a little hole in a thin plate of metal, and set the plate upright on a table, facing a row of lighted candles standing by one another; then place a sheet of paper or pasteboard at a little distance from the other side of the plate, and the rays of all the candles, flowing through the hole, will form as many specks of light on the paper as there are candles before the plate, each speck as distinct and large, as if there were only one candle to cast one speck; which shews that the rays are no hinderance to each other in their motions, although they all cross in the hole. 169. Light, and therefore heat so far as it depends on the Sun’s rays (§ 85, towards the end) decreases in proportion to the
- 58. light and heat decrease at any given distance from the Sun. PLATE II. Why the Planets appear dimmer when viewed thro’ telescopes than by the bare eye. squares of the distances of the Planets from the Sun. This is easily demonstrated by a Figure which, together with it’s description, I have taken from Dr. Smith’s Optics[37] . Let the light which flows from a point A, and passes through a square hole B, be received upon a plane C, parallel to the plane of the hole; or, if you please, let the figure C be the shadow of the plane B; and when the distance C is double of B, the length and breadth of the shadow C will be each double of the length and breadth of the plane B; and treble when AD is treble of AB; and so on: which may be easily examined by the light of a candle placed at A. Therefore the surface of the shadow C, at the distance AC double of AB, is divisible into four squares, and at a treble distance, into nine squares, severally equal to the square B, as represented in the Figure. The light then which falls upon the plane B, being suffered to pass to double that distance, will be uniformly spread over four times the space, and consequently will be four times thinner in every part of that space, and at a treble distance it will be nine times thinner, and at a quadruple distance sixteen times thinner, than it was at first; and so on, according to the increase of the square surfaces B, C, D, E, built upon the distances AB, AC, AD, AE. Consequently, the quantities of this rarefied light received upon a surface of any given size and shape whatever, removed successively to these several distances, will be but one quarter, one ninth, one sixteenth of the whole quantity received by it at the first distance AB. Or in general words, the densities and quantities of light, received upon any given plane, are diminished in the same proportion as the squares of the distances of that plane, from the luminous body, are increased: and on the contrary, are increased in the same proportion as these squares are diminished. 170. The more a telescope magnifies the disks of the Moon and Planets, they appear so much dimmer than to the bare eye; because the telescope cannot magnify the quantity of light, as it does the surface; and, by spreading the same quantity of light over a surface so much larger
- 59. Fig. VIII. Refraction of the rays of light. than the naked eye beheld, just so much dimmer must it appear when viewed by a telescope than by the bare eye. 171. When a ray of light passes out of one medium[38] into another, it is refracted, or turned out of it’s first course, more or less, as it falls more or less obliquely on the refracting surface which divides the two mediums. This may be proved by several experiments; of which we shall only give three for example’s sake. 1. In a bason FGH put a piece of money as DB, and then retire from it as to A, till the edge of the bason at E just hides the money from your sight: then, keeping your head steady, let another person fill the bason gently with water. As he fills it, you will see more and more of the piece DB; which will be all in view when the bason is full, and appear as if lifted up to C. For, the ray AEB, which was straight whilst the bason was empty, is now bent at the surface of the water in E, and turned out of it’s rectilineal course into the direction ED. Or, in other words, the ray DEK, that proceeded in a straight line from the edge D whilst the bason was empty, and went above the eye at A, is now bent at E; and instead of going on in the rectilineal direction DEK, goes in the angled direction DEA, and by entering the eye at A renders the object DB visible. Or, 2dly, place the bason where the Sun shines obliquely, and observe where the shadow of the rim E falls on the bottom, as at B: then fill it with water, and the shadow will fall at D; which proves, that the rays of light, falling obliquely on the surface of the water, are refracted, or bent downwards into it. 172. The less obliquely the rays of light fall upon the surface of any medium, the less they are refracted; and if they fall perpendicularly thereon, they are not refracted at all. For, in the last experiment, the higher the Sun rises, the less will be the difference between the places where the edge of the shadow falls, in the empty and full bason. And, 3dly, if a stick be laid over the bason, and the Sun’s rays be reflected perpendicularly into it from a looking-glass, the shadow
- 60. The Atmosphere. The Air’s compression and rarity at different heights. of the stick will fall upon the same place of the bottom, whether the bason be full or empty. 173. The denser that any medium is, the more is light refracted in passing through it. 174. The Earth is surrounded by a thin fluid mass of matter, called the Air, or Atmosphere, which gravitates to the Earth, revolves with it in it’s diurnal motion, and goes round the Sun with it every year. This fluid is of an elastic or springy nature, and it’s lowermost parts being pressed by the weight of all the Air above them, are squeezed the closer together; and are therefore densest of all at the Earth’s surface, and gradually rarer the higher up. “It is well known[39] that the Air near the surface of our Earth possesses a space about 1200 times greater than water of the same weight. And therefore, a cylindric column of Air 1200 foot high is of equal weight with a cylinder of water of the same breadth and but one foot high. But a cylinder of Air reaching to the top of the Atmosphere is of equal weight with a cylinder of water about 33 foot high[40] ; and therefore if from the whole cylinder of Air, the lower part of 1200 foot high is taken away, the remaining upper part will be of equal weight with a cylinder of water 32 foot high; wherefore, at the height of 1200 feet or two furlongs, the weight of the incumbent Air is less, and consequently the rarity of the compressed Air is greater than near the Earth’s surface in the ratio of 33 to 32. And having this ratio we may compute the rarity of the Air at all heights whatsoever, supposing the expansion thereof to be reciprocally proportional to its compression; and this proportion has been proved by the experiments of Dr. Hooke and others. The result of the computation I have set down in the annexed Table, in the first column of which you have the height of the Air in miles, whereof 4000 make a semi-diameter of the Earth; in the second the compression of the Air or the incumbent weight; in the third it’s rarity or expansion, supposing gravity to decrease in the duplicate ratio of the distances from the Earth’s center. And the small
- 61. It’s weight how found. PLATE II. numeral figures are here used to shew what number of cyphers must be joined to the numbers expressed by the larger figures, as 0.17 1224 for 0.000000000000000001224, and 2695615 for 26956000000000000000. Air’s Height. Compression. Expansion. 0 33 1 5 17.8515 1.8486 10 9.6717 3.4151 20 2.852 11.571 40 0.2525 136.83 400 0.17 1224 2695615 4000 0.105 4465 73907102 40000 0.192 1628 26263189 400000 0.210 7895 41798207 4000000 0.212 9878 33414209 Infinite. 0.212 6041 54622209 From this Table it appears that the Air in proceeding upwards is rarefied in such manner, that a sphere of that Air which is nearest the Earth but of one inch diameter, if dilated to an equal rarefaction with that of the Air at the height of ten semi-diameters of the Earth, would fill up more space than is contained in the whole Heavens on this side the fixed Stars, according to the preceding computation of their distance[41] .” And it likewise appears that the Moon does not move in a perfectly free and un-resisting medium; although the air at a height equal to her distance, is at least 34000190 times thinner than at the Earth’s surface; and therefore cannot resist her motion so as to be sensible in many ages. 175. The weight of the Air, at the Earth’s surface, is found by experiments made with the air-pump; and also by the quantity of mercury that the Atmosphere balances in
- 62. A common mistake about the weight of the Air. Without an Atmosphere the Heavens would always appear dark, and we should have no twilight. the barometer; in which, at a mean state; the mercury stands 291 ⁄2 inches high. And if the tube were a square inch wide, it would at that height contain 291 ⁄2 cubic inches of mercury, which is just 15 pound weight; and so much weight of air every square inch of the Earth’s surface sustains; and every square foot 144 times as much, because it contains 144 square inches. Now as the Earth’s surface contains about 199,409,400 square miles, it must be of no less than 5,559,215,016,960,000 square feet; which, multiplied by 2016, the number of pounds on every foot, amounts to 11,207,377,474,191,360,000; or 11 trillion 207 thousand 377 billion 474 thousand 191 million and 360 thousand pounds, for the weight of the whole Atmosphere. At this rate, a middle sized man, whose surface may be about 14 square feet, is pressed by 28,224 pound weight of Air all round; for fluids press equally up and down and on all sides. But, because this enormous weight is equal on all sides, and counterbalanced by the spring of the internal Air in our blood vessels, it is not felt. 176. Oftentimes the state of the Air is such that we feel ourselves languid and dull; which is commonly thought to be occasioned by the Air’s being foggy and heavy about us. But that the Air is then too light, is evident from the mercury’s sinking in the barometer, at which time it is generally found that the Air has not sufficient strength to bear up the vapours which compose the Clouds: for, when it is otherwise, the Clouds mount high, the Air is more elastic and weighty about us, by which means it balances the internal spring of the Air within us, braces up our blood-vessels and nerves, and makes us brisk and lively. 177. According to [42] Dr. Keill, and other astronomical writers, it is entirely owing to the Atmosphere that the Heavens appear bright in the day-time. For, without an Atmosphere, only that part of the Heavens would shine in which the Sun was placed: and if an observer could live without Air, and should turn his back
- 63. It brings the Sun in view before he rises, and keeps him in view after he sets. Fig. IX. PLATE II. towards the Sun, the whole Heavens would appear as dark as in the night, and the Stars would be seen as clear as in the nocturnal sky. In this case, we should have no twilight; but a sudden transition from the brightest sunshine to the blackest darkness immediately after sun-set; and from the blackest darkness to the brightest sun-shine at sun-rising; which would be extremely inconvenient, if not blinding, to all mortals. But, by means of the Atmosphere, we enjoy the Sun’s light, reflected from the aerial particles, before he rises and after he sets. For, when the Earth by its rotation has withdrawn the Sun from our sight, the Atmosphere being still higher than we, has his light imparted to it; which gradually decreases until he has got 18 degrees below the Horizon; and then, all that part of the Atmosphere which is above us is dark. From the length of twilight, the Doctor has calculated the height of the Atmosphere (so far as it is dense enough to reflect any light) to be about 44 miles. But it is seldom dense enough at two miles height to bear up the Clouds. 178. The Atmosphere refracts the Sun’s rays so, as to bring him in sight every clear day, before he rises in the Horizon; and to keep him in view for some minutes after he is really set below it. For, at some times of the year, we see the Sun ten minutes longer above the Horizon than he would be if there were no refractions: and about six minutes every day at a mean rate. 179. To illustrate this, let IEK be a part of the Earth’s surface, covered with the Atmosphere HGFC; and let HEO be the[43] sensible Horizon of an observer at E. When the Sun is at A, really below the Horizon, a ray of light AC proceeding from him comes straight to C, where it falls on the surface of the Atmosphere, and there entering a denser medium, it is turned out of its rectilineal course ACdG, and bent down to the observer’s eye at E; who then sees the Sun in the direction of the refracted ray edE, which lies
- 64. Fig. IX. The quantity of refraction. above the Horizon, and being extended out to the Heavens, shews the Sun at B § 171. 180. The higher the Sun rises, the less his rays are refracted, because they fall less obliquely on the surface of the Atmosphere § 172. Thus, when the Sun is in the direction of the line EfL continued, he is so nearly perpendicular to the surface of the Earth at E, that his rays are but very little bent from a rectilineal course. 181. The Sun is about 321 ⁄4 min. of a deg. in breadth, when at his mean distance from the Earth; and the horizontal refraction of his rays is 333 ⁄4 min. which being more than his whole diameter, brings all his Disc in view, when his uppermost edge rises in the Horizon. At ten deg. height the refraction is not quite 5 min. at 20 deg. only 2 min. 26 sec.; at 30 deg. but 1 min. 32 sec.; between which and the Zenith, it is scarce sensible: the quantity throughout, is shewn by the annexed table, calculated by Sir Isaac Newton. 182. A Table shewing the Refractions of the Sun, Moon, and Stars; adapted to their apparent Altitudes. Appar. Alt. Refraction. Ap. Alt. Refraction. Ap. Alt. Refraction. D. M. M. S. D. M. S. D. M. S. 0 0 33 45 21 2 18 56 0 36 0 15 30 24 22 2 11 57 0 35 0 30 27 35 23 2 5 58 0 34 0 45 25 11 24 1 59 59 0 32 1 0 23 7 25 1 54 60 0 31 1 15 21 20 26 1 49 61 0 30 1 30 19 46 27 1 44 62 0 28 1 45 18 22 28 1 40 63 0 27 2 0 17 8 29 1 36 64 0 26 2 30 15 2 30 1 32 65 0 25 3 0 13 20 31 1 28 66 0 24
- 65. PLATE II. The inconstancy of Refractions. A very remarkable case concerning refraction. 3 30 11 57 32 1 25 67 0 23 4 0 10 48 33 1 22 68 0 22 4 30 9 50 34 1 19 69 0 21 5 0 9 2 35 1 16 70 0 20 5 30 8 21 36 1 13 71 0 19 6 0 7 45 37 1 11 72 0 18 6 30 7 14 38 1 8 73 0 17 7 0 6 47 39 1 6 74 0 16 7 30 6 22 40 1 4 75 0 15 8 0 6 0 41 1 2 76 0 14 8 30 5 40 42 1 0 77 0 13 9 0 5 22 43 0 58 78 0 12 9 30 5 6 44 0 56 79 0 11 10 0 4 52 45 0 54 80 0 10 11 0 4 27 46 0 52 81 0 9 12 0 4 5 47 0 50 82 0 8 13 0 3 47 48 0 48 83 0 7 14 0 3 31 49 0 47 84 0 6 15 0 3 17 50 0 45 85 0 5 16 0 3 4 51 0 44 86 0 4 17 0 2 53 52 0 42 87 0 3 18 0 2 43 53 0 40 88 0 2 19 0 2 34 54 0 39 89 1 1 20 0 2 26 55 0 38 90 0 0 183. In all observations, to have the true altitude of the Sun, Moon, or Stars, the refraction must be subtracted from the observed altitude. But the quantity of refraction is not always the same at the same altitude; because heat diminishes the air’s refractive power and density, and cold increases both; and therefore no one table can serve precisely for the same place at all seasons, nor even at all times of the same day; much less for different climates: it having been observed that the
- 66. Our imagination cannot judge rightly of the distance of inaccessible objects. horizontal refractions are near a third part less at the Equator than at Paris, as mentioned by Dr. Smith in the 370th remark on his Optics, where the following account is given of an extraordinary refraction of the sun-beams by cold. “There is a famous observation of this kind made by some Hollanders that wintered in Nova Zembla in the year 1596, who were surprised to find, that after a continual night of three months, the Sun began to rise seventeen days sooner than according to computation, deduced from the Altitude of the Pole observed to be 76°: which cannot otherwise be accounted for, than by an extraordinary quantity of refraction of the Sun’s rays, passing thro’ the cold dense air in that climate. Kepler computes that the Sun was almost five degrees below the Horizon when he first appeared; and consequently the refraction of his rays was about nine times greater than it is with us.” 184. The Sun and Moon appear of an oval figure as FCGD, just after their rising, and before their setting: the reason is, that the refraction being greater in the Horizon than at any distance above it, the lowermost limb G appears more elevated than the uppermost. But although the refraction shortens the vertical Diameter FG, it has no sensible effect on the horizontal Diameter CD, which is all equally elevated. When the refraction is so small as to be imperceptible, the Sun and Moon appear perfectly round, as AEBF. 185. We daily observe, that the objects which appear most distinct are generally those which are nearest to us; and consequently, when we have nothing but our imagination to assist us in estimating of distances, bright objects seem nearer to us than those which are less bright, or than the same objects do when they appear less bright and worse defined, even though their distance in both cases be the same. And as in both cases they are seen under the same angle[44] , our imagination naturally suggests an idea of a greater distance between us and those objects which appear fainter and worse defined than those which appear brighter under the same Angles; especially if they be
- 67. Nor always of those which are accessible. The reason assigned. PLATE II. Fig. XII. such objects as we were never near to, and of whose real Magnitudes we can be no judges by sight. 186. But, it is not only in judging of the different apparent Magnitudes of the same objects, which are better or worse defined by their being more or less bright, that we may be deceived: for we may make a wrong conclusion even when we view them under equal degrees of brightness, and under equal Angles; although they be objects whose bulks we are generally acquainted with, such as houses or trees: for proof of which, the two following instances may suffice. First, When a house is seen over a very broad river by a person standing on low ground, who sees nothing of the river, nor knows of it beforehand; the breadth of the river being hid from him, because the banks seem contiguous, he loses the idea of a distance equal to that breadth; and the house seems small, because he refers it to a less distance than it really is at. But, if he goes to a place from which the river and interjacent ground can be seen, though no farther from the house, he then perceives the house to be at a greater distance than he imagined; and therefore fancies it to be bigger than he did at first; although in both cases it appears under the same Angle, and consequently makes no bigger picture on the retina of his eye in the latter case than it did in the former. Many have been deceived, by taking a red coat of arms, fixed upon the iron gate in Clare-Hall walks at Cambridge, for a brick house at a much greater distance[45] . Secondly, In foggy weather, at first sight, we generally imagine a small house, which is just at hand, to be a great castle at a distance; because it appears so dull and ill defined when seen through the Mist, that we refer it to a much greater distance than it really is at; and therefore, under the same Angle, we judge it to be much bigger. For, the near object FE, seen by the eye ABD, appears under the same Angle GCH, that the remote object GHI does: and the rays GFCN and HECM crossing one
- 68. Fig. IX. Why the Sun and Moon appear biggest in the Horizon. another at C in the pupil of the eye, limit the size of the picture MN on the retina; which is the picture of the object FE, and if FE were taken away, would be the picture of the object GHI, only worse defined; because GHI, being farther off, appears duller and fainter than FE did. But if a Fog, as KL, comes between the eye and the object FE, it appears dull and ill defined like GHI; which causes our imagination to refer FE to the greater distance CH, instead of the small distance CE which it really is at. And consequently, as mis- judging the distance does not in the least diminish the Angle under which the object appears, the small hay-rick FE seems to be as big as GHI. 187. The Sun and Moon appear bigger in the Horizon than at any considerable height above it. These Luminaries, although at great distances from the Earth, appear floating, as it were, on the surface of our Atmosphere HGFfeC, a little way beyond the Clouds; of which, those about F, directly over our heads at E, are nearer us than those about H or e in the Horizon HEe. Therefore, when the Sun or Moon appear in the Horizon at e, they are not only seen in a part of the Sky which is really farther from us than if they were at any considerable Altitude, as about f; but they are also seen through a greater quantity of Air and Vapours at e than at f. Here we have two concurring appearances which deceive our imagination, and cause us to refer the Sun and Moon to a greater distance at their rising or setting about e, than when they are considerably high as at f: first, their seeming to be on a part of the Atmosphere at e, which is really farther than f from a spectator at E; and secondly, their being seen through a grosser medium when at e than when at f; which, by rendering them dimmer, causes us to imagine them to be at a yet greater distance. And as, in both cases, they are seen[46] much under the same Angle, we naturally judge them to be biggest when they seem farthest from us; like the above-mentioned house § 186, seen from a higher ground, which shewed it to be farther off than it
- 69. Their Diameters are not less on the Meridian than in the Horizon. appeared from low ground; or the hay-rick, which appeared at a greater distance by means of an interposing Fog. 188. Any one may satisfy himself that the Moon appears under no greater Angle in the Horizon than on the Meridian, by taking a large sheet of paper, and rolling it up in the form of a Tube, of such a width, that observing the Moon through it when she rises, she may, as it were, just fill the Tube; then tie a thread round it to keep it of that size; and when the Moon comes to the Meridian, and appears much less to the eye, look at her again through the same Tube, and she will fill it just as much, if not more, than she did at her rising. 189. When the full Moon is in perigeo, or at her least distance from the Earth, she is seen under a larger Angle, and must therefore appear bigger than when she is Full at other times: and if that part of the Atmosphere where she rises be more replete with vapours than usual, she appears so much the dimmer; and therefore we fancy her to be still the bigger, by referring her to an unusually great distance; knowing that no objects which are very far distant can appear big unless they be really so. Plate IIII.
- 70. J. Ferguson delin. J. Mynde Sculp.
- 71. PLATE IV. Fig I. The Moon’s horizontal Parallax, what. The Moon’s distance determined. CHAP. IX. The Method of finding the Distances of the Sun, Moon, and Planets. 190. Those who have not learnt how to take the [47] Altitude of any Celestial Phenomenon by a common Quadrant, nor know any thing of Plain Trigonometry, may pass over the first Article of this short Chapter, and take the Astronomer’s word for it, that the distances of the Sun and Planets are as stated in the first Chapter of this Book. But, to every one who knows how to take the Altitude of the Sun, the Moon, or a Star, and can solve a plain right-angled Triangle, the following method of finding the distances of the Sun and Moon will be easily understood. Let BAG be one half of the Earth, AC it’s semi-diameter, S the Sun, m the Moon, and EKOL a quarter of the Circle described by the Moon in revolving from the Meridian to the Meridian again. Let CRS be the rational Horizon of an observer at A, extended to the Sun in the Heavens, and HAO his sensible Horizon; extended to the Moon’s Orbit. ALC is the Angle under which the Earth’s semi-diameter AC is seen from the Moon at L, which is equal to the Angle OAL, because the right lines AO and CL which include both these Angles are parallel. ASC is the Angle under which the Earth’s semi-diameter AC is seen from the Sun at S, and is equal to the Angle OAf because the lines AO and CRS are parallel. Now, it is found by observation, that the Angle OAL is much greater than the Angle OAf; but OAL is equal to ALC, and OAf is equal to ASC. Now, as ASC is much less than ALC, it proves that the Earth’s semi-diameter AC appears much greater as seen from the Moon at L than from the Sun at S: and therefore the Earth is much farther from the Sun than from the Moon[48] . The Quantities of these Angles are determined by observation in the following manner. Let a graduated instrument as DAE, (the larger the better) having a moveable Index and Sight-holes, be fixed in such a manner, that it’s plane surface may be parallel to the Plan of the Equator, and it’s edge AD in the Meridian: so that when the Moon is in the Equinoctial, and on the Meridian at E, she may be seen through the sight-holes when the edge of the moveable index cuts the beginning of the divisions at o, on the graduated limb DE; and when she is so seen, let the precise time be noted.
- 72. The Sun’s distance cannot be yet so exactly determined as the Moon’s; How near the truth it may soon be determined. Now, as the Moon revolves about the Earth from the Meridian to the Meridian again in 24 hours 48 minutes, she will go a fourth part round it in a fourth part of that time, viz. in 6 hours 12 minutes, as seen from C, that is, from the Earth’s center or Pole. But as seen from A, the observer’s place on the Earth’s surface, the Moon will seem to have gone a quarter round the Earth when she comes to the sensible Horizon at O; for the Index through the sights of which she is then viewed will be at d, 90 degrees from D, where it was when she was seen at E. Now, let the exact moment when the Moon is seen at O (which will be when she is in or near the sensible Horizon) be carefully noted[49] , that it may be known in what time she has gone from E to O; which time subtracted from 6 hours 12 minutes (the time of her going from E to L) leaves the time of her going from O to L, and affords an easy method for finding the Angle OAL (called the Moon’s horizontal Parallax, which is equal to the Angle ALC) by the following Analogy: As the time of the Moon’s describing the arc EO is to 90 degrees, so is 6 hours 12 minutes to the degrees of the Arc DdE, which measures the Angle EAL; from which subtract 90 degrees, and there remains the Angle OAL, equal to the Angle ALC, under which the Earth’s Semi-diameter AC is seen from the Moon. Now, since all the Angles of a right-lined Triangle are equal to 180 degrees, or to two right Angles, and the sides of a Triangle are always proportional to the Sines of the opposite Angles, say, by the Rule of Three, as the Sine of the Angle ALC at the Moon L is to it’s opposite side AC the Earth’s Semi-diameter, which is known to be 3985 miles, so is Radius, viz. the Sine of 90 degrees, or of the right Angle ACL to it’s opposite side AL, which is the Moon’s distance at L from the observer’s place at A on the Earth’s surface; or, so is the Sine of the Angle CAL to its opposite side CL, which is the Moon’s distance from the Earth’s centre, and comes out at a mean rate to be 240,000 miles. The Angle CAL is equal to what OAL wants of 90 degrees. 191. The Sun’s distance from the Earth is found the same way, but with much greater difficulty; because his horizontal Parallax, or the Angle OAS equal to the Angle ASC, is so small as, to be hardly perceptible, being only 10 seconds of a minute, or the 360th part of a degree. But the Moon’s horizontal Parallax, or Angle OAL equal to the Angle ALC, is very discernible; being 57ʹ 49ʺ, or 3469ʺ at it’s mean state; which is more than 340 times as great as the Sun’s: and therefore, the distances of the heavenly bodies being inversely as the Tangents of their horizontal Parallaxes, the Sun’s distance from the Earth is at least 340 times as great as the Moon’s; and is rather understated at 81 millions of miles, when the Moon’s distance is certainly known to be 240 thousand. But because, according to some Astronomers, the Sun’s horizontal Parallax is 11 seconds, and according to others only 10, the former Parallax making the Sun’s distance to be about 75,000,000 of miles, and the latter 82,000,000; we may take it for granted, that the Sun’s distance is not less than as deduced from the former, nor more than as shewn by the latter: and every one who is accustomed to make such
- 73. The Sun proved to be much bigger than the Moon. The relative distances of the Planets from the Sun are known to great precision, though their real distances are not well known. observations, knows how hard it is, if not impossible, to avoid an error of a second; especially on account of the inconstancy of horizontal Refractions. And here, the error of one second, in so small an Angle, will make an error of 7 millions of miles in so great a distance as that of the Sun’s; and much more in the distances of the superiour Planets. But Dr. Halley has shewn us how the Sun’s distance from the Earth, and consequently the distances of all the Planets from the Sun, may be known to within a 500th part of the whole, by a Transit of Venus over the Sun’s Disc, which will happen on the 6th of June, in the year 1761; till which time we must content ourselves with allowing the Sun’s distance to be about 81 millions of miles, as commonly stated by Astronomers. 192. The Sun and Moon appear much about the same bulk: And every one who understands Geometry knows how their true bulks may be deduced from the apparent, when their real distances are known. Spheres are to one another as the Cubes of their Diameters; whence, if the Sun be 81 millions of miles from the Earth, to appear as big as the Moon, whose distance does not exceed 240 thousand miles, he must, in solid bulk, be 42 millions 875 thousand times as big as the Moon. 193. The horizontal Parallaxes are best observed at the Equator; 1. Because the heat is so nearly equal every day, that the Refractions are almost constantly the same. 2. Because the parallactic Angle is greater there as at A (the distance from thence to the Earth’s Axis being greater,) than upon any parallel of Latitude, as a or b. 194. The Earth’s distance from the Sun being determined, the distances of all the other Planets from him are easily found by the following analogy, their periods round him being ascertained by observation. As the square of the Earth’s period round the Sun is to the cube of it’s distance from him, so is the square of the period of any other Planet to the cube of it’s distance, in such parts or measures as the Earth’s distance was taken; see § 111. This proportion gives us the relative mean distances of the Planets from the Sun to the greatest degree of exactness; and they are as follows, having been deduced from their periodical times, according to the law just mentioned, which was discovered by Kepler and demonstrated by Sir Isaac Newton. Periodical Revolution to the same fixed Star in days and decimal parts of a day. Of Mercury Venus The Earth Mars Jupiter Saturn 87.9692 224.6176 365.2564 686.9785 4332.514 10759.275
- 74. Why the celestial Poles seem to keep still in the same points of the Heavens, notwithstanding the Earth’s motion round the Sun. Relative mean distances from the Sun. 38710 72333 100000 152369 520096 954006 From these numbers we deduce, that if the Sun’s horizontal Parallax be 10ʺ, the real mean distances of the Planets from the Sun in English miles are 31,742,200 59,313,060 82,000,000 124,942,580 426,478,720 782,284,920 But if the Sun’s Parallax be 11ʺ their distances are no more than 29,032,500 54,238,570 75,000,000 114,276,750 390,034,500 715,504,500 Errors in distance a rising from the mistake of 1ʺ in the Sun’s Parallax 2,709,700 5,074,490 7,000,000 10,665,830 36,444,220 66,780,420 195. These last numbers shew, that although we have the relative distances of the Planets from the Sun to the greatest nicety, yet the best observers have not hitherto been able to ascertain their true distances to within less than a twelfth part of what they really are. And therefore, we must wait with patience till the 6th of June, A. D. 1761; wishing that the Sky may then be clear to all places where there are good Astronomers and accurate instruments for observing the Transit of Venus over the Sun’s Disc at that time: as it will not happen again, so as to be visible in Europe, in less than 235 years after. 196. The Earth’s Axis produced to the Stars, being carried [50] parallel to itself during the Earth’s annual revolution, describes a circle in the Sphere of the fixed Stars equal to the Orbit of the Earth. But this Orbit, though very large in itself, if viewed from the Stars, would appear no bigger than a point; and consequently, the circle described in the Sphere of the Stars by the Axis of the Earth produced, if viewed from the Earth, must appear but as a point; that is, it’s diameter appears too little to be measured by observation: for Dr. Bradley has assured us, that if it had amounted to a single second, or two at most, he should have perceived it in the great number of observations he has made, especially upon γ Dragonis; and that it seemed to him very probable that the annual Parallax of this Star is not so great as a single second: and consequently, that it is above 400 thousand times farther from us than the Sun. Hence the celestial poles seem to continue in the same points of the Heavens throughout the year; which by no
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