Popular Science Monthly/Volume 58/January 1901/Geometry: Ancient and Modern


By Professor EDWIN S. CRAWLEY,


AMONGST the records of the most remote antiquity we find little to lead to the conclusion that geometry was known or studied as a branch of mathematics. The Babylonians had a remarkably well-developed number system and were expert astronomers; but, so far as we know, their knowledge of geometry did not go beyond the construction of certain more or less regular figures for necromantic purposes. The Egyptians did better than this, and Egypt is commonly acknowledged to be the birthplace of geometry. It was a poor kind of geometry, however, from our point of view, and should rather be designated as a system of mensuration. Nevertheless it served as a beginning, and probably was the means of setting the Greek mind, at work upon this subject. Our knowledge of Egyptian geometry is obtained from a papyrus in the British Museum known as the Ahmes Mathematical Papyrus. It dates from about the eighteenth century B. C, and purports to be a copy of a document some four or five centuries older. It is the counterpart of what to-day is called an engineer's hand-book. It contains arithmetical tables, examples in the solution of simple equations, and rules for determining the areas of figures and the capacity of certain solids. There is no hint of anything in the nature of demonstrational geometry, nor any evidence of how the rules were derived. In fact, they could not have been obtained as the result of demonstration, for they are generally wrong. For example, the area of an isosceles triangle is given as the product of the base and half the side, and that of a trapezoid as the product of the half-sums of the opposite sides. These rules give results which are approximately correct so long as they are applied to triangles whose altitude is large compared with the base, and to trapezoids which do not depart very far from a rectangular shape. Whether the Egyptians ever came to realize that these rules were erroneous we cannot say, but it is known that long after the Greeks had discovered the correct ones they were still in use. Thus Cajori, 'History of Mathematics,' page 12, says: "On the walls of the celebrated temple of Horus at Edfu have been found hieroglyphics written about 100 B. C, which enumerate the pieces of land owned by the priesthood and give their areas. The area of any quadrilateral, however irregular, is there found by the formula a+b2 x c+d2." [a and b for one pair of opposite sides and c and d for the others.] It is plausibly argued that a superstitious traditionalism made it an act of sacrilege to alter what had become part of the sacred writings.

When we consider the conditions of life in Egypt we can easily see why this particular kind of geometric knowledge so early gained currency. The annual inundation of the Nile was continually altering the minor features of the country along its course, and washing away landmarks between adjacent properties. Some means of re-establishing these marks and of determining the areas of fields was therefore essential. To meet this demand the surveyors devised the rules which Ahines has given us. The further necessity of ascertaining the contents of a barn of given shape and dimensions likewise gave rise to the rules for determining volumes.

We learn also that the Egyptians were acquainted with the truth of the Pythagorean theorem, that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides, for they applied this knowledge practically by means of a triangle whose sides were 3, 4 and 5 respectively, in laying down right angles. This general truth was derived in all probability by deduction from a large number of individual cases. The Egyptian rule for the area of a circle was remarkably accurate for such an early date. It consisted in squaring eight-ninths of the diameter. This gives to n the value 3.1605.

It is generally supposed that the Greeks had their attention drawn to geometry through intercourse with the Egyptians. It was but a step, however, for them to pass beyond the latter, and with them we find the birth of the true mathematical spirit which refuses to accept anything upon authority, but requires a logical demonstration. It is well known what an important place was held by geometry in Greek philosophy. The Pythagorean school contributed much that was important along with a great deal that was fanciful and of little value. Pythagoras himself was the first to prove the theorem referred to above, which goes by his name. The Greeks for the most part pursued the study of geometry as a purely intellectual exercise. Anything in the nature of practical applications of the subject was repugnant to them, and hence but little attention was paid to theorems of mensuration. This reminds one of the story told of a professor of mathematics in modern times who, in beginning a course of lectures, made the remark: "Gentlemen, 'to my mind the most interesting thing about this subject is that I do not see how under any circumstances it can ever be put to any practical use." Euclid in his 'Elements' does not mention the theorem that the area of a triangle is equal to half the product of its base and altitude, nor does he enter into any discussion of the ratio of the circumference to the diameter of a circle. This last, however, was a problem which as early as the time of Pythagoras had attracted much attention. 'Squaring the circle' was a stumbling block to the Greeks and has been ever since. The pursuit of the impossible seems to have an irresistible attraction for some minds. This remark applies only to the modern devotees of the subject, however. The Greeks did not know that the thing they sought was an impossibility. To square the circle, to trisect an angle and to duplicate the cube were three problems upon which the Greeks lavished more attention probably than upon any others. It was not labor wasted, because it led incidentally to many theorems, which otherwise might have remained unknown, but the principal object sought was not attained. To make matters clear it should be stated that to meet the requirements of Greek geometry the instruments used in the solution must be only the compasses and the unmarked straight edge. So that to square the circle meant to construct by these means the side of a square whose area should equal that of a given circle. The Greeks eventually succeeded in solving the last two problems by the aid of curves other than the circle, but this, of course, was unsatisfactory. As we know now they were pursuing ignes fatui. Nevertheless it is brought to the knowledge of mathematicians with painful frequency that a vast amount of energy is still wasted upon these problems, especially the first. Let me, therefore, take the space here to repeat that squaring the circle is not simply one of the unsolved problems of mathematics which is awaiting the happy inspiration of some genius, but that it has been ably demonstrated to be incapable of solution in the manner proposed.

When Euclid compiled his 'Elements' the knowledge of geometry current amongst the Greeks was about the same as that which we have to-day under the name of elementary geometry. The term Euclidean geometry has a somewhat different signification, which will be explained below.

About a century before Euclid's time the Greeks discovered the conic sections, and Apollonius of Perga, who lived about a century after Euclid, brought the geometry of these curves to a high degree of perfection. Archimedes, whose time was intermediate between that of Euclid and of Apollonius, had a more practical turn of mind and applied his mathematical knowledge to useful purposes. Amongst other things he showed that the value of π lies between 31/7 and 310/71 that is, between 3.1429 and 3,1408, a closer approximation than the Egyptian. We see, therefore, that in the few centuries during which the Greeks occupied themselves with the study of geometry the knowledge of the right line, circle and conic sections reached about as high a state of development as it was possible to attain until the invention of more powerful methods of research, and many centuries were destined to elapse before this was to occur. I do not overlook the fact that the beautiful and extensive modern geometry of the triangle and the systems of remarkable points and circles associated with it, which has been developed by Brocard, Lemoine, Emmerich, Vigarié and others, was within the reach of the Greeks; but this does not destroy the force of the remark above.

The operations of mathematics are divided fundamentally into two kinds, analytic, which employ the symbolism and methods of algebra (in its broadest sense), and geometric, which consists of the operation of pure reason upon geometric figure. The two are now only partially exclusive, however, for analysis is frequently assisted by geometry, and geometric results are frequently obtained by analytic methods.

With the Greeks, Hindoos and Arabs, the only peoples who concerned themselves to any extent with mathematics until comparatively modern times, the operations of algebra and geometry were entirely distinct. With the Hindoos and Arabs algebra received more attention than geometry and with the Greeks the reverse was true. Many of the theorems of Euclid are capable of an algebraic interpretation, and this fact was probably well known, but nevertheless the theorems themselves are expressed in geometric terms and are proved by purely geometric means; and they do not, therefore, constitute a union of analysis with geometry in the modern sense.

The seventeenth century brought the invention of analytic geometry by Descartes and that of the calculus by Newton and Leibnitz. These methods opened hitherto undreamed-of possibilities in geometric research and led to the systematic study of curves of all descriptions and to a generalization of view in connection with the geometry of the right line, circle and conics, as well as of the higher curves, which has been of the greatest value to the modern mathematician. To point out by a very simple illustration the nature of this work of generalization let us consider the case of a circle and straight line in the same plane, the line being supposed to be of indefinite extent. According to the relative position of this line and circle the Greek geometer would say that the line either meets the circle, or is tangent to the circle, or that the line does not meet the circle at all. We say now, however, that the line always meets the circle in two points, which may be real and distinct, real and coincident or imaginary. Thus a condition of things which the Greek was obliged to consider under three different cases we can deal with now as a single case. This generalized view is a direct consequence of the analytic treatment of the question.

It will be seen from the illustration used above that two very important conceptions are introduced into geometry by the use of the analytic method. One of these is the conception of coincident or consecutive points of intersection, as in the case of a tangent, and the other is that of imaginary elements, as in the case of the imaginary points of intersection of a line and circle which are co-planar and non-intersecting in the ordinary sense. It is impossible to exaggerate the importance of these conceptions. Without them the beautiful fabric of modern geometry would not stand a moment. It will be seen to many readers, no doubt, that a fabric built upon such a foundation will have very much the same stability as a 'castle in Spain.' Such, however, is far from the case. The analysis by which our operations proceed is a thoroughly well founded and trustworthy instrument, and when we give to it the geometric interpretation which we are entirely justified in doing, we find frequently that it reveals to us facts which our senses unaided by its finer powers of interpretation could not have discovered. These facts require for their adequate explanation the recognition of the so-called imaginary elements of the figure. Let us take one more illustration. If from a point outside of, but in the same plane with, a circle we draw two tangents to the circle and connect the points of tangency with a straight line, the original point and the line last mentioned stand in an important relation to each other and are called respectively pole and polar with regard to the circle. Now suppose the point is inside the circle. The whole construction just described becomes then geometrically impossible, but analytically we can draw from a point within a circle two imaginary tangents to the circle, and similarly we can connect the imaginary points of tangency by a straight line, and this straight line is found to be a real line. Moreover, in its relations to the point and circle it exhibits precisely the same properties which are found in the case of the pole and polar first described. Hence this point and line are also included in the general definition of pole and polar. Such examples might be multiplied indefinitely, but they would all go to emphasize the fact of the great power of generalization which resides in the methods of analytic geometry.

While the power of the analytic method as an instrument of research is far greater than that of the older pure geometric method, yet to many minds it lacks somewhat the beauty and elegance of that method as an intellectual exercise. This is due to the fact that its operations, like all algebraic operations, are largely mechanical. Given the equations representing a certain geometric condition, we subject these equations to definite transformations and the results obtained denote certain new geometric conditions. We have been whisked from the data to the result very much as we are hurried over the country in a railroad train. We may have noted the features of the country as we passed through it or we may not; we arrive at our destination just the same. Pure geometric research, on the other hand, resembles travel on foot or horseback. We must scrutinize the landmarks and keep a careful watch on the direction in which we are traveling, lest we take •-a wrong turn and fail to reach our destination. The result is that we acquire a thorough familiarity with the country through which we pass. The analytical method, however, affords abundant opportunity for mental activity, although of a different kind from that required in the other. First, the most advantageous analytic expression for the given geometric conditions must be sought; then the proper line of analytic transformation must be determined upon; and finally the result must be interpreted geometrically. This last step requires keen insight in order to ensure the full value of the result, for it is here that we often find far more than we anticipated, or than a casual glance will reveal.

The obligation thus incurred by geometry to analysis has been largely repaid by the aid which analysis has derived from geometry. The study of pure analysis is unquestionably the most abstruse branch of mathematics, but it is now advancing with rapid strides and demands less and less the aid of geometry. The results of the analytic method in geometry, however, are too fruitful for it to be either desirable or possible for us to go back to a condition of complete separation of these two methods.

Amongst the distinctly modern developments of geometry is what is known as hyper-geometry, the geometry of space of more than three dimensions. The fact that the product of two linear dimensions is representable by an area, and the product of three linear dimensions by a volume, naturally leads us to ask what is the geometric representative of the product of four or more linear dimensions. The answer to this question leads to the ideal conception of space of four or more dimensions. Just as in space of three dimensions, the space of our every-day experience, we can draw three concurrent straight lines such that each one is perpendicular to each of the other two, so in space of four dimensions it must be possible to draw four concurrent straight lines such that each one is perpendicular to each of the other three. It is needless to say it transcends the power of the human mind to form such a conception, nevertheless it is possible to study the geometry of such a space, and much has been done in this way both analytically and by the methods of pure geometry. If our space has a fourth dimension (not to speak of any higher dimension) as some mathematicians seem disposed seriously to maintain, a body moved from any position in the direction of the fourth dimension will disappear from view. In fact, it will be annihilated so far as we are concerned. Again, a body placed in an inclosed space can be removed therefrom while the walls of the envelope remain intact; or the envelope itself can be turned inside out without rupturing the walls. For example, it would be possible to extract the meat from an egg and leave the shell unbroken. For most persons, however, the geometry of four-dimensional space is likely to remain a mathematical curiosity, serving no useful purpose except to furnish an opportunity for acute logical reasoning, for in studying the geometry of such space we have only our reasoning powers to guide us and cannot fall back upon experience, as we so often do more or less unconsciously, perhaps, in ordinary geometry.

Geometry of three-dimensional space is often studied by projecting the solid in question upon two or more planes and working with these plane projections instead of with the solid itself. This is done exclusively in descriptive geometry, the geometry of the engineer and builder with their plan and elevation, so called. The geometry of four-dimensional figures has been studied in an analogous way. A four-dimensional figure, it should be remarked, is a figure whose bounding parts are three dimensional figures, just as a three-dimensional figure is one whose bounding parts are surfaces or two-dimensional figures. A four-dimensional figure can be projected on a three-dimensional space and its properties studied from such projections made from different points of view, corresponding to the plan and elevation of ordinary geometry. The mathematical department of the University of Pennsylvania has in its possession wire models of solid projections of all the possible regular four-dimensional hyper-solids, the number of which is limited in the same way as is the number of regular three-dimensional solids. These models were constructed, after a careful study of the question, by Dr. Paul E. Heyl, a recent student and graduate of the University.

Amongst the subjects of most profound interest to mathematicians of recent years has been an investigation into the foundations of geometry and analysis. It was found, as the growth of the science proceeded, that much of fundamental importance, which hitherto had been accepted without question, would not bear searching scrutiny, and it began to be feared that the foundation might collapse in places altogether. We are concerned here with this only so far as it relates to geometry. Whatever may be said of geometry as a science which proceeds by pure reason from certain axioms, postulates and definitions, it is undoubtedly true that for at least the most fundamental conceptions we are thrown back upon experience; and that in the matter of axioms or postulates there is some latitude as to what we shall accept. Amongst the axioms or postulates given by Euclid is one known as the parallel-postulate, which states that if two coplanar straight lines are intersected by a third straight line (transversal) and if the interior angles on one side of the transversal are together less than two right angles, the two straight lines, if produced far enough, will meet on the same side of the transversal on which the sum of the interior angles is less than two right angles. This is, in fact, a theorem, and it is hardly possible to suppose that Euclid did not adopt it as a postulate only after finding that he could neither prove it nor do without it. It belongs to a set of theorems which are so connected that if the truth of any one of them be assumed the others are readily proved. The theorem that the sum of the three angles of a triangle is equal to two right angles belong to this set. Ptolemy (Claudius Ptolemæus, second century A. D.) seems to have been the first to publish an attempted proof of this postulate of Euclid. Almost all mathematicians down to the beginning of the nineteenth century have given more or less attention to this question, and the account of their efforts to prove the postulate forms one of the most interesting chapters in the history of mathematics. Cajori, in his 'History of Elementary Mathematics' says, page 270: "They all fail, either because an equivalent assumption is implicitly or explicitly made, or because the reasoning is otherwise fallacious. On this slippery ground good and bad mathematicians alike have fallen. We are told that the great Lagrange, noticing that the formulas of spherical trigonometry are not dependent upon the parallel-postulate, hoped to frame a proof on this fact. Toward the close of his life he wrote a paper on parallel lines and began to read it before the Academy, but suddenly stopped and said: 'Il faut que j'y songe encore' (I must think it over again); he put the paper in his pocket and never afterwards publicly recurred to it."

About the time to which I have referred, the end of the eighteenth and beginning of the nineteenth century, the idea began to force itself upon mathematicians that perhaps there was more in the question than appeared on the surface. It was one of the many instances which have occurred in all branches of human knowledge where some truth of fundamental importance has begun to force itself simultaneously on a number of minds. We leave the significance of this aspect of the question to the psychologists. Another curious fact to be noted in connection with the writings which have finally shown us the true meaning, of the parallel-postulate is that either they attracted little or no general attention when they first appeared, or else they remained unpublished. The names of Lobatchewsky and the Bolyais have been made immortal by their writings on this subject, but it was not until long after they were published that their vast importance was recognized. The inimitable Gauss wrote on the same subject, but left his work unpublished, and Cajori (ibid., p. 274) mentions two writers of much earlier date who anticipated in part the theories of Lobatchewsky and the Bolyais. These are Geronimo Saccheri (1667-1733), a Jesuit father of Milan, and Johann Heinrich Lambert (1728-1777), of Mühlhausen, Alsace.

Lobatchewsky (Nicholaus Ivanovitch Lobatchewsky, 1793-1856) conceived the brilliant idea of cutting loose from the parallel-postulate altogether and succeeded in building up a system of geometry without its aid. The result is startling to one who has been taught to look upon Ihe facts of geometry (that is, of the Euclidean geometry) as incontrovertible. The denial of the parallel-postulate leaves Lobatchewsky to face the fact that under the conditions given in the postulate the two lines, if continually produced, may never meet on that side of the transversal on which the sum of the interior angles is less than two right angles. In other words, through a given point we may draw in a plane any number of distinct lines which will never meet a given line in the same plane. A result of this is that the sum of the angles of a triangle is variable (depending on the size of the triangle), but is always less than two right angles. Notwithstanding the shock to our preconceived notions which such a statement gives, the geometry of Lobatchewsky is thoroughly logical and consistent. What, then, does it mean? Simply this: We must seek the true explanation of the parallel-postulate in the characteristics of the space with which we are dealing. The Euclidean geometry remains just as true as it ever was, but it is seen to be limited to a particular kind of space, space of zero-curvature the mathematicians call it; that is, for two dimensions, space which conforms to our common notion of a plane. Lobatchewsky's geometry, on the other hand, is the geometry of a surface of uniform negative curvature, while ordinary spherical geometry is geometry of a surface of uniform positive curvature. The Lobatchewskian geometry is sometimes spoken of as geometry on the pseudo-sphere.

The 'absolute geometry' of the Bolyais (Wolfgang Bolyai de Bolya, 1775-1856, and his son, Johann Bolyai, 1802-1860) is similar to that of Lobatchewsky. 'The Science Absolute of Space,' by the younger Bolyai, published as an appendix to the first volume of his father's work, has immortalized his name.

The work of Lobatchewsky and the Bolyais has been rendered accessible to English readers by the translations and contributions of Prof. George Bruce Halsted, of the University of Texas.

If we proceed beyond the domain of two-dimensional geometry we merge the ideas of non-Euclidean and hyper-space. The ordinary triply-extended space of our experience is purely Euclidean; and if we approach the conception of curvature in such a space it must be curvature in a fourth dimension, and here the mind refuses to follow, although by pure reasoning we can show what must take place in such a space.

H. Grassman, Blemann and Beltrami have written profoundly on these questions, and it is to the last that is due the discovery that the theorems of the non-Euclidean or Lobatchewskian geometry find their realization in a space of constant negative curvature.

We naturally ask the question: Is there any reason to suppose that the space which we inhabit is other than Euclidean? To this a negative reply must be returned. We may have suspicions, but we have no evidence. If we could discover a triangle the sum of whose angles by actual measurement departs from two right angles, the fact of the non-Euclidean character of our space would be established at once. But no such triangle has been discovered. Even the largest, which are concerned in the measurement of stellar parallax, do not help us, and it does not seem possible to get larger ones. Nevertheless Clifford and others have shown that some physical phenomena, which require the conception of elaborate and complex machinery for their explanation, are capable of very simple explanation upon the hypothesis of a fourth dimension. Then, too, in the domain of pure mathematics several phenomena find a ready explanation upon the basis of such an assumption. In the theory of curves we constantly make use of the assumption that a curve may return into itself after passing through infinity, which is only another aspect of the same hypothesis. In fact, without this aid our processes of generalization, so important to the development of modern geometry, would be sadly hampered. Professor Newcomb has carried this matter to its logical conclusion and has deduced the actual dimensions of the visible universe in terms of the measurement of curvature in the fourth dimension. In such a space it becomes actually possible for a curve with infinite branches to pass through infinity (so-called) and return into itself. Upon this hypothesis our universe is unbounded in the sense that however far we travel we can never reach its limits, for it has none, but it is not infinite. Just as we can travel forever on the surface of the earth without reaching any limits, but that surface is not infinite. But even supposing that all this is true, the question still presses home: What is beyond?