GEOMETRY
PART I.—PURE GEOMETRY.
GEOMETRY has been divided since the time of Euclid into an “elementary” and a “higher” part. The contents and limits of the former have been fixed by Euclid’s Elements. The latter included at the time of the Greek mathematicians principally the properties of the conic sections and of a few other curves. The methods used in both were essentially the same. These began to be replaced during the 17th century by more powerful methods, invented by Roberval, Pascal, Desargues, and others. But the impetus which higher geometry received in their works was soon arrested, in consequence of the discoveries of Descartes,——the new calculus to which these gave rise absorbing the attention of mathematicians almost ex- clusively, until lIonge., at the end of the 18th century, re-established “pure” as distinguished from Descartes's “coordinate” (or analytical) geometry. Since then the purely geometrical methods h1ve been continuously ex- tended, especially by Poncelet, Steiner, Von Staudt, and (lrcmona, and in England by Hirst aml Henry Smith, to mention only a few of the leading names.
Whilst higher geometry thus made most rapid progress, the elementary part remained almost unaltered. It has been taught up to the present day on the basis of Euclid’s Elements, the latter being either used directly as a text-book (in England), or being replaced (in most parts of the Continent) by text-books which are essentially Euclid’s Elements rewritten, with a few additions about the mensuration of the circle, cone, cylinder, and sphere. Only within a very recent period have attempts been made to change the character of the elementary part by introducing some of the modern methods.
We shall give in this article first, a survey of elementary geometry as contained in Euclid’s Elements, and then, in form of an independent treatise, an introduction to higher geometry, based on modern methods. In the former part we shall suppose that a copy of Euclid’s Elements is in the hands of the reader, so that we may dispense, as a rule, with giving proofs or drawing figures. We thus shall give only the contents of his propositions grouped together in such a way as to show their connexion, and often expressed in words which differ from the verbal translation in order to make their meaning clear. It will make little difference which of the many English editions of Euclid’s Elements the reader takes. Of these we may mention Simson’s, I’otts’s, and Todhunter’s.
Section I.—Elementary or Euclidian Geometry.
The Axioms.
§ 1. The object of geometry is to investigate the proper- ties of space. The first step must consist in establishing those fundamental properties from which all others follow by processes of deductive reasoning. They are laid down in the Axioms, and these ought to form such a system that nothing need be added to them in order fully to charac- terize space, and that nothing may be left out without 1nak- ing the system incomplete. They nmst, in fact, completely “ define ” space. Several such systems are conceivable. Eur.-li«l has given one, others have been put forward in recent times by Riemann (Abhrnull. Jar I.-o‘n[_(/l, G(’s(;Il.5'ch_
- u (:'o'lli7z_r/(72, vol. xiii.), by Ilelmholz (G'o'lz‘z'Iz_r/or ]'uch-
rfchton, June 1869), and by Grassniann (Ausrlw/mnngslehrc To/2 18-1 1). How many axioms the system ought to contain. and which system is the simplest, may be said to be for granted, in which case it is an axiom. still an open question. We shall consider only Euclid's system.
§ 2. The axioms are obtained from inspection of space and of solids in space,—hencc from experience. The same source gives us the notions of the geometrical entities to which the axioms relate, viz., solids, surfaces, lines or curves, and points. A solid is directly given by expe- rience ; we have only to abstract all material from it in order to gain the notion of a geometrical solid. This l1u.< shape, size, position, and may be moved. Its boundary or boundaries are called surfaces. They separate one part of space from another, and are said to have no tliickm-.<.<. i'1'heir boundaries are curves or lines, ar.d these have length only. Their boundaries, again, are points, which have no magnitude but only position. 'e thus come in three steps from solids to points which have no niagnitnde; in each step we lose one extension. Hence we say a solid has three dimensions, a surface two, a line one, and a point none. Space itself, of which a solid forms only a part, is also said to be of three dimensions. The same thing is intended to be expressed by saying thata solid has length, breadth, and thickness, a surface length and breadth, a line length only, and a point no extension whatsoever.
Euclid gives the essence of these statements as definitions:—
Def. 1, I. A point -is that iehich, has no parls, or zrhich has no 7n.rI;;-nitmlc.
Def. 2, I . A line is length without breadth.
Def. 5, I. A super;/(‘cies is that 1z.'hz'ch. has only length and I/)'(‘m/Ih.
Def. 1, XI. A solid is that u‘/rich. has Icnglh, breadth, and (hid.-7L€.S‘S.
If we allow motion in geometry, and it seems impos- sible to avoid it,—we may generate these entities by moving a point, a line, or a surface, thus:—
The path of a moving point is a line.
The path of a moving line is, in general, a surface.
The path of a moving surface is, in general, a solid.
And we may then assume that the lines, surfaces, and solids, as defined before, can all be generated in this man- ner. From this generation of the entities it follows again that the boundaries—the first and last position of the 1nov- ing elen1ent—-of a line are points, and so on; and thus we come back to the considerations with which we started. Euclid points this out in his definitions,——Def. 3, 1., .l)cf. 6, 1., and Def. 2, Xf. He does not, however, show the connexion which these definitions have with tlltrse mentioned before. Vhen points aml lines have been defined, a statement like Def. 3, I., “ The extremities of a line are points,” is a proposition which either has to be provcd, and then it is a theorem, or which has to be taken And so with Def. 6, 1., and Def. 2, X1.
describe, iii a few words, notions which we have obtained by inspection of and abstraction from solids. A few more notions have to be added to these, pI‘illcip'.1lly those of the simplest linc——thc straight line, and of the simplest surface —tl1e flat surface or plane. These notions we possess, but to define them accurately is diflicult. Euclid’s Definition 4, 1., “A straight line is that which lies evenly between its extreme points,” must be meaningless to any one Wlln has not the notion of straightness in his mind. Neither does it state a propert_v of the straight line which can be used in any further investigation. Such a property is
given in Axiom 10, 1. It is really this axiom, together