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Spaces of constant curvature, i.e. Euclidean space, the sphere, and
Loba chevskij space, occupy a special place in geometry. They are
most accessible to our geometric intuition, making it possible to
develop elementary geometry in a way very similar to that used to
create the geometry we learned at school. However, since its basic
notions can be interpreted in different ways, this geometry can be
applied to objects other than the conventional physical space, the
original source of our geometric intuition. Euclidean geometry has
for a long time been deeply rooted in the human mind. The same is
true of spherical geometry, since a sphere can naturally be
embedded into a Euclidean space. Lobachevskij geometry, which in
the first fifty years after its discovery had been regarded only as
a logically feasible by-product appearing in the investigation of
the foundations of geometry, has even now, despite the fact that it
has found its use in numerous applications, preserved a kind of
exotic and even romantic element. This may probably be explained by
the permanent cultural and historical impact which the proof of the
independence of the Fifth Postulate had on human thought."
This book deals with various systems of "numbers" that can be
constructed by adding "imaginary units" to the real numbers. The
complex numbers are a classical example of such a system. One of
the most important properties of the complex numbers is given by
the identity (1) Izz'l = Izl.Iz'I. It says, roughly, that the
absolute value of a product is equal to the product of the absolute
values of the factors. If we put z = al + a2i, z' = b+ bi, 1 2 then
we can rewrite (1) as The last identity states that "the product of
a sum of two squares by a sum of two squares is a sum of two
squares. " It is natural to ask if there are similar identities
with more than two squares, and how all of them can be described.
Already Euler had given an example of an identity with four
squares. Later an identity with eight squares was found. But a
complete solution of the problem was obtained only at the end of
the 19th century. It is substantially true that every identity with
n squares is linked to formula (1), except that z and z' no longer
denote complex numbers but more general "numbers" where i, j, . .
., I are imaginary units. One of the main themes of this book is
the establishing of the connection between identities with n
squares and formula (1)."
Spaces of constant curvature, i.e. Euclidean space, the sphere, and
Loba chevskij space, occupy a special place in geometry. They are
most accessible to our geometric intuition, making it possible to
develop elementary geometry in a way very similar to that used to
create the geometry we learned at school. However, since its basic
notions can be interpreted in different ways, this geometry can be
applied to objects other than the conventional physical space, the
original source of our geometric intuition. Euclidean geometry has
for a long time been deeply rooted in the human mind. The same is
true of spherical geometry, since a sphere can naturally be
embedded into a Euclidean space. Lobachevskij geometry, which in
the first fifty years after its discovery had been regarded only as
a logically feasible by-product appearing in the investigation of
the foundations of geometry, has even now, despite the fact that it
has found its use in numerous applications, preserved a kind of
exotic and even romantic element. This may probably be explained by
the permanent cultural and historical impact which the proof of the
independence of the Fifth Postulate had on human thought."
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