The mystery of space

by Robert T. Browne

[CHAPTER II]

Historical Sketch of the Hyperspace Movement

Egypt the Birthplace of Geometry—Precursors: Nasir-Eddin, Christoph Clavius, Saccheri, Lambert, La Grange, Kant—Influence of the Mecanique Analytique—The Parallel-Postulate the Root and Substance of the Non-Euclidean Geometry—The Three Great Periods: The Formative, Determinative and Elaborative—Riemann and the Properties of Analytic Spaces.

The evolution of the idea of a fourth dimension of space covers a long period of years. The earliest known record of the beginnings of the study of space is found in a hieratic papyrus which forms a part of the Rhind Collection in the British Museum and which has been deciphered by Eisenlohr. It is believed to be a copy of an older manuscript of date 3400 B. C., and is entitled "Directions for Knowing All Dark Things" The copy is said to have been made by Ahmes, an Egyptian priest between 1700 and 1100 B. C. It begins by giving the dimensions of barns; then follows the consideration of various rectilineal figures, circles, pyramids, and the value of pi ([Greek: p]). Although many of the solutions given in the manuscript have been found to be incorrect in minor particulars, the fact remains that Egypt is really the birth-place of geometry. And this fact is buttressed by the knowledge that Thales, long before he founded the Ionian School which was the beginning of Greek influence in the study of mathematics, is found studying geometry and astronomy in Egypt.

The concept of hyperspace began to germinate in the latter part of the first century, B. C. For it was at this date that Geminos of Rhodes (B. C. 70) began to think seriously of the mathematical labyrinth into which Euclid's parallel-postulate most certainly would lead if an attempt at demonstrating its certitude were made. He recognized the difficulties which would engage the attention of those who might venture to delve into the mysterious possibilities of the problem. There is no doubt, too, but that Euclid himself was aware, in some measure at least, of these difficulties; for his own attitude towards this postulate seems to have been one of noncommittance. It is, therefore, not strange that the astronomer, Ptolemy (A. D. 87-165), should be found seeking to prove the postulate by a consideration of the possibilities of interstellar triangles. His researches, however, brought him no relief from the general dissatisfaction which he felt with respect to the validity of the problem itself.

For nearly one thousand years after the attempts at solving the postulate by Geminos and Ptolemy, the field of mathematics lay undisturbed. For it was at this time that there arose a strange phenomenon, more commonly known as the "Dark Ages," which put an effectual check to further research or independent investigations. Mathematicians throughout this long lapse of time were content to accept Euclid as the one incontrovertible, unimpeachable authority, and even such investigations as were made did not have a rebellious tendence, but were mainly endeavors to substantiate his claims.

Accordingly, it was not until about the first half of the thirteenth century that any real advance was made. At this time there appeared an Arab, Nasir-Eddin (1201-1274) who attempted to make an improvement on the problem of parallelism. His work on Euclid was printed in Rome in 1594 A. D., about three hundred and twenty years after his demise and was communicated in 1651 by John Wallis (1616-1703) to the mathematicians of Oxford University. Although his calculations and conclusions were respectfully received by the Oxford authorities no definite results were regarded as accomplished by what he had done. It is believed, however, that his work reopened speculation upon the problem and served as a basis, however slight, for the greater work that was to be done by those who followed him during the next succeeding eight hundred years.

About twenty years before the printing of the work of Nasir-Eddin, Christoph Clavius (1574) deduced the axiom of parallels from the assumption that a line whose points are all equidistant from a straight line is itself straight. In his consideration of the parallel-postulate he is said to have regarded it as Euclid's XIIIth axiom. Later Bolyai spoke of it as the XIth and later still, Todhunter treated it as the XIIth. Hence, there does not seem to have been any general unanimity of opinion as to the exact status of the parallel-postulate, and especially is this true in view of the uncertainty now known to have existed in Euclid's mind concerning it.

Girolamo Saccheri (1667-1733), a learned Jesuit, born at San Remo, came next upon the stage. And so important was his work that it will perpetuate the memory of his name in the history of mathematics. He was a teacher of grammar in the Jesuit Collegio di Brera where Tommaso Ceva, a brother of Giovanni, the well-known mathematician, was teacher of mathematics. His association with the Ceva brothers was especially beneficial to him. He made use of Ceva's very ingenious methods in his first published book, 1693, entitled Solutions of Six Geometrical Problems Proposed by Count Roger Ventimiglia.