We have grown up in an era in which the idea of higher dimensions is a widely accepted, if little understood notion. It is useful to think beyond everyday acceptance and realize that people did not 'just know' to think about four-dimensional geometry- -centuries of thought and work went into creating what we now know about higher dimensional mathematics. Likewise, the field is always widening as mathematicians continue to contribute their time and life's work to advance our understanding of what the f ourth dimension is and how the geometry within it works.

The earliest references to the number of dimensions of space date back to Aristotle (384-322 B.C.), who stated in the first book of *Heaven*, "The line has magnitude in one way, the plane in two ways, and the solid in three ways, and beyond these
there is no other magnitude because the three are all," and "There is no transfer into another kind, like the transfer from length to area and from area to a solid." (Manning, p. 1) Ptolemy apparently took this thought a step further, for he gave a proof
that there are not more than three distances in his book *On Distance*. According to Simplicius (sixth century A.D.), Ptolemy wrote in the middle of the second century A.D. that this is so "because of the necessity that distances should be defined,
and that the distances defined should be taken along perpendicular lines, and because it is possible to take only three lines that are mutually perpendicular, two by which the plane is defined and a third measuring depth; so that if there were any other
distance after the third it would be entirely without measure and without definition." (Ibid.)

Early algebraists ran into powers higher than three, which caused some confusion. In the third century, Diophantus used the terms *square-square, square-cube, and cube-cube* to denote a square times a square, a square times a cube and a cube time
s a cube, respectively. Soon these terms were being used to mean a square squared, a square cubed and a cube cubed. New terms were clearly necessary to denote powers of prime orders, most immediately, the fifth power, which came to be called a *sursol
id*. (The seventh power was referred to as a *Bsursolid*, and so on.) The ancients were accustomed to being able to match their algebra to geometric concepts, so these higher powers were viewed as unreal and were avoided whenever possible.

By the late fifteenth century, higher equations were considered, allowing only for a purely numerical conception of the nature of algebraic quantities. Higher dimensions were seen in a most unnatural light by the mathematicians of the day. Stifel (14 86?-1567) speaks of "going beyond the cube just as if there were more than three dimensions," "which is," he adds, "against nature." (Ibid., p. 3) John Wallis (1616-1703) describe s one of the objects of a power higher than three as a "Monster in Nature, less possible than a Chimaera or Centaure." He says: "Length, Breadth and Thickness, take up the whole of Space. Nor can Fansie imagine how there should be a Fourth Local Dimens ion beyond these Three." (Ibid.) Ozanam (1640-1717) claims that the product of any number of letter will be a magnitude of "as many dimensions as there are letters, but it will only be imaginary because in nature we do not know of any quantity which has more than three dimensions." (Ibid.)

Writers in the eighteenth century began to discuss the possibility of representing mechanics as a geometry of four dimensions, considering time to be the fourth dimension. Lagra
nge (1736-1813) published this idea in 1797 in his *Theorie des fonctions analytiques*, but it was previously expressed by d'Alembert (1717-1783) in an article on "
Dimension", published in 1754 in the *Encyclopedie* edited by Diderot and himself. (Ibid., p. 4)

In the early part of the nineteenth century, the history of higher dimensional geometry becomes a bit blurred by the increase in the study both of higher synthetic geometry and of analysis. Synthetic geometry requires that one build up one's knowledge
dimension by dimension, one at a time, while analysis is developed for an arbitrary dimension, so that once something is known for any dimension, one is ready to consider *n* dimensional objects. Analysis has therefore been of some use to scholars
of four dimensional geometry, for some problems, once understood for n dimensions can be seen more clearly in four dimensions, but this was not directly recognized.

Mobius may have been the first contributor to the synthetic geometry of four dimensions, for he points out that symmetrical figures could be made to coincide if there were a s pace of four dimensions. Cayley was beginning to use geometry of four dimensions to study configurations of points in 1846. In 1851, Sylvester discussed tangent and polar forms in n-dimensional geometry. He went on in 1859 to make an application of hyperspace; and in 1863 he used mostly synthetic conceptions with some analytic methods to prove his t heorems for four and n dimensions. Meanwhile, Clifford was applying higher geometry to probability.

Independently from this synthetic development, work was being done on higher geometry in the field of analysis, and steps were eventually made to recognize the geometrical significance of analytic methods. Green made a move towards this synthesis in 1
833 when he used analysis to solve the problem of the attraction of ellipsoids so he could apply the idea to any number of variables, saying, "It is no longer confined as it were to the three dimensions of space." (Ibid., p. 6) Cauchy made it clear, however, that geometry and analysis could share the same language, forms and processes when he announced in 1847, "We shall call a set of n variables an analytical point, an equatio
n or system of equations an analytical locus," etc. (Ibid.) The most
noteworthy work during this period, however, was Riemann's "On the Hypotheses which Lie at the Foundations
of Geometry.", presented in 1854 and first published in 1866. This paper contains work on multiply-extended manifolds and their measure-relations. In it, Riemann discusses the line element *ds* when the manifold is expressed by means of n variables
. When *ds* is equal to the square root of the sum of the squares of the quantities *dx* (as in the ordinary plane), the manifold is *flat*. This lead to the concept of *deviation from flatness*, or *curvature* and the study of
mathematical objects with constant curvature. Riemann had introduced the concept of unbounded yet finite space--and, in fact that space cannot be infinite if it has a constant positive curvature differing at all from zero--hence Elliptic Non-Euclidean ge
ometry is attributed to Riemann. This breakthrough is doubly relevant to four dimensional geometry. Firstly, his manifold of n dimensions is a space of n dimensions, so the geometrical concepts and implications are never far from the mind. Also, the id
ea that three-space may be curved implies that it is lying in some other dimension--a fourth dimension.

Towards the end of the century, the number of books, memoirs, and papers on geometry of four or more dimensions increased dramatically. Cayley and Clifford both wrote further on the subject, and contributions were made by others, such as Nother in 187 0 and Jordan in 1875. Veronese published a memoir in 1882 which took these geometries up in from an entirely synthetic point of study. By 1911, a bibliography of Sommerville contained 1832 references on n dimensions, written in Italian, German, French, English and Dutch. (Ibid., p. 9) Clearly an area of great interest had been uncovered, although popularity had been slow in coming to the topic.

Skeptics of higher dimensional geometry argue that geometry is intended for the study of concrete objects, and since we live in a three-dimensional world, we will most probably never come across hyperobjects. Higher geometry is intimately related to m
uch of the scientific world, however, and is directly applicable to mathematical physics. D'Alembert's notion that time is a fourth dimension has obtained such a firm place in our thinking that it is at times difficult to get people to understand that ti
me is not *the* fourth dimension. Four dimensional geometry remains very important to the study of relativity. Complex numbers and quaternions are both vital to modern physics, and both can be much more clearly understood because of higher geometry
. The use of higher geometry helps us understand two- and three-dimensional geometries, as well, for it forces us to use geometric reasoning rather than intuition. We may never be able to picture what four-dimensional objects would 'look' like (that is,
if we were visually equipped to see in 4-D), but if we reason about them and come to understand more about their nature by purely intellectual means, we will have exercised a part of our minds that will then have increased capability of understanding tho
se objects of which we can see representations in the 'real' world.

It was for this reason, along with a general love and respect for Euclidean geometry, that we chose to describe an version of high school three-dimensional geometry, extended to include the fourth dimension, using commonly accepted postulates and theor ems and a synthetic method.

Sources:

Manning, Henry Parker, **Geometry of Four Dimensions**, Dover Publications, Inc., New York, NY,1956.

Smith, David Eugene, **History of Mathematics, Volume I**, Dover Publications, Inc., New York, NY, 1951.