Euclidean Geometry

MATHEMATICS is the science of measurement of physical quantity. Quantity is that property of matter by which it is divisible into parts. All division of matter is however not quantitative division; the division must be into parts of the same kind. Thus the division of water into oxygen and hydrogen is not a division of quantity, but division into elements. Quantity is of two kinds, multitude and magnitude, or discrete quantity and continuous quantity. Multitude or discrete quantity is a natural plurality or multiplicity, that is, it is quantity that is divisible into really separate or non-continuous parts. Magnitude or continuous quantity is divisible into continuous parts, that is, there is no natural separation or division between them.

Both magnitude and multitude can therefore be considered under the aspect of a numerable multitude, or a multitude that can be counted. This counting is measurement. Whenever the parts of quantity are equal, or can be considered as equal, the quantity is measured by enumerating the parts. The natural parts of discrete quantity are the parts to be numbered. Since there is no natural division in continuous quantity, that is, since it is a quantity not composed of separate individual parts, no distinction can be found except by referring it to some standard outside of itself. Such a standard, which must be a standard of magnitude, may be arbitrarily chosen, and then the whole quantity can be enumerated in terms of this unit. Measuring a quantity means therefore counting or otherwise determining a multitude whether that multitude be a natural one or an arbitrary one.

In modern physics another division of quantity has been introduced. This division is into scalar quantity and vector quantity, to which correspond scalar and vector measurement. The terms scalar and vector are quite modern importations into the science of mathematics. They were introduced by Sir William Hamilton to denote magnitude with and without direction. Scalar quantity and scalar measurement correspond to both multitude and magnitude and their measurement. Scalar quantity is quantity considered merely as a multiplicity, or a quantity consisting of the repetition of a certain unit, whether this be a natural or an arbitrary unit; scalar measurement is therefore the measurement of a quantity by means of some accepted standard unit. The total quantity is measured by this unit and is expressed in terms of the unit. Vector quantity is quantity that has both magnitude and direction. A vector is a mathematical line having magnitude and direction, and is used to represent a mechanical quantity. The magnitude is measured as a scalar quantity and the direction is a separate element.

Of the former or scalar type of measurement we have those branches of pure mathematics which are arithmetic, algebra, and again the analytical mathematics, such as analytical geometry and the calculus. Examples of vector mathematics are to be seen in the sciences of geometry and trigonometry. When these are reduced to scalar mathematics, we have analytical geometry and some branches of the calculus. The former is the application of scalar algebra to vector geometry, and the latter, by means of reducing the scalar units progressively till they approach a limit, attempts to solve the problems of the vector continuum directly.

Geometry is the science of possible determinations of space. Geometric determinations may then contain three elements, distance, direction, figure. Geometry therefore studies the properties of space. With regard to space we have two elementary notions with which the science of geometry starts; these are distance and direction. Geometry studies the relations of space with regard to these two fundamental concepts, and establishes the relations of equality and inequality, of ratio and proportion, and of direction and form among the various figures and parts of figures into which general space may be broken up and given form. Geometry is not so much a study of space itself, which is a question of metaphysics, as of the measurable quantity units for which our idea of space furnishes the background. Space itself does not enter further into this science than to supply the material frame into which these formal figures fit. The science of geometry is therefore restricted to determining the properties and relations of limited figures located in space.

Geometry is the simplest of sciences, and one of the purest in the sense that it is built on the fewest and best controlled assumptions of the human intellect and of experience, and from these assumptions, which are its axioms and postulates, it proceeds by a pure a priori method of deductive reasoning to all the various conclusions of the science. For this reason geometry made great progress in ancient times. The clearness and certainty of its method, the few facts of observation required for its development, the ease with which its conclusions might be controlled by experience, caused it to be a science peculiarly adapted to the clarity of the Greek intellect and to the scientific conditions of the age.

The primitive notions on which it is founded were easy of acquirement, being among the first perceptions of sensible intuition; moreover, measurements of limited portions of space and the determination of the accompanying space relationships were not difficult, and did not require a more intimate knowledge of the universe or a more complicated physical apparatus of observation and measuring than could be expected of so early a time. This enabled the Greek geometers to apply the acuteness of their minds to the development of a science out of the fundamental ideas of practical measurement, and do it in such a masterful way, that there has been very little improvement in the particular field they developed since their day.

In fact it is the Greeks who have given us nearly all that we possess of pure geometry. Although they may have taken much from elsewhere, especially, if we may credit Herodotus, from the Egyptians, who certainly made many geometric discoveries, yet the Greeks have the credit of being the first to draw it up into the compact body of connected propositions we know. The arrangement of the science into a severely symmetrical and orderly body of doctrine, having nothing superfluous, but just what would be required in a book of Elements for the development of the art, has come down to us in the thirteen books of the Elements of Euclid (about 300 B.C.). There were other books of Elements before his time, and doubtless he made sufficient use of them in his work. He was rather at the end of a line of scientific discoverers than at the beginning; for geometry was always highly regarded in Greek education. We all know Plato’s inscription over the porch of the Academy: “Let no one unversed in geometry enter my doors.”

Nearly all we know of Greek geometry is contained in these Elements of Euclid, which is a logical digest of the geometric science of his time, with perhaps a great deal of his own added. This digest both in doctrine and method endures till today. For twenty centuries the geometry of Euclid has been the geometry of the world, with little or no improvement from other sources. It has always been considered the perfect science, and it has remained for a few of the more modern mathematicians to doubt the complete usefulness of his method, and the reliability of the assumptions on which it is based. These mathematicians have given us a new Geometry, which has thence acquired the generic name of non-Euclidean Geometry. We have as our purpose to investigate the reasons for this departure, and to see on what foundations of fact they rest. If any exception is to be taken to the validity of the science of Euclidean geometry, it must be in answer to one of the three following questions:—

1. —Are the assumptions from which the logical method of Euclid proceeds themselves correct, and proper postulates of the science?
2. —Is the method of deduction used in the science strictly applied, so as to leave no loophole for the entrance of logical and real error?
3. —Can the ideas of space postulated and deduced in Euclidean geometry be predicated of reality in the physical world?

If Euclidean geometry is not scientifically and therefore absolutely true, it must be impugned on one or more of these three topics, and thus under these headings we shall treat the question, with this proviso, however, that we shall only take up the disputed points. We do not intend at this time to study the whole basis of geometry.

The second question may be disposed of immediately. All without exception admit the rigor of Euclidean deduction, and acknowledge that the geometry of Euclid is a compact, perfectly logical body of doctrine based on well defined principles, and that once these principles are admitted, the rest of the science follows with strict and unassailable method engendering complete certitude. If we start with Euclid’s assumptions, all his propositions and constructions follow with rigorous logical necessity. The dictum of Cicero, placed at the head of this chapter, “In geometry, if we grant the first principles, all must be granted,” has never been gainsaid. We can, therefore, pass over this one point as universally accepted, and hence the consistency of Euclidean geometry remains assured. For what is not attacked need not be defended. The position of geometry can be held to be impregnable on this front.

The case for or against Euclidean geometry reverts to one or other of the two remaining questions. The first one is, do all Euclid’s assumptions contain their own proper evidence, or are certain non-evident assumptions made? In this latter case, can these be proved or established from his evident assumptions and deductions? This question we shall treat first. The other question whether the geometry of Euclid is such that it can be applied to actual physics, and to the real relations of space, or whether it is so notional and out of touch with reality that it cannot be universally used, so that its use is limited to practical human measurements, we shall leave for later consideration.

There is one great weakness that has always been apparent in Euclidean geometry that subsequent mathematicians have never been able to overcome, and that is in what is called the Parallel Postulate. Non-Euclidean Geometers were willing, at least originally, to admit the validity of all Euclid’s assumptions except what are sometimes known as the fifth and sixth postulates with regard to parallel lines. These are as follows:—

1. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
2. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

These two statements are not always numbered and classified in the same way. The first book of Euclid especially shows differences of arrangement in different editions. The second of the two statements is evidently a definition, and is usually so classed. It is the twenty-third definition in the editions of Heiberg and of Heath. The first statement is also differently classified. It is really a postulate, and is usually so given. It is the fifth in the editions of Heiberg and of Heath. It has, however, also been termed an axiom; it was called axiom 13 in the edition of Clavius, axiom 12 in that of Simpson, and John Bolyai called it axiom 11. We shall classify it as the fifth postulate, where it more properly belongs. It is not of the nature of an axiom. The impossibility of establishing this parallel postulate is the fundamental assumption of non-Euclidean Geometry. The whole question then depends on this one point. There are few problems of mathematics about which so much has been written from Euclid’s time to our day. A brief sketch of the history of these writings may not be amiss.