=

P and taking the place of x and y in the XY plane. The polar co-ordinates of a point P, Fig. 12, are the radius vector p, and the vectorial angles and o, reckoned respectively from the pole O and the polar axes OE and ON. If the pole coincide with the origin, and the polar axes with OX and OZ, then x P cos 0 sin, y=p sin sin 9. z = p cos 0. P √x2 + y2+z2, tan 0=y:x, Φ COS = 2 :P, formulæ of transformation from one system to the other. If a, ẞ, y, are the direction angles (made respectively with OX, OY, OZ) of the radius vector p of any point P, Fig. 11, then r=p cos a, y=p cos B, z=p cos y, and 1=cos'a + cos + cos'y, sum of squares of direction-cosines of p. Any linear equation Ax+By+Cz+D=0 represents a plane. The x У с symmetric equation of a plane is ++. =1, b a

2

a, b, c being the axial intercepts extending from O to the plane. In normal form the equation of the plane is x cos a + y cos ẞ+z cos y=p where is the length, and cos a, cos B, cos Y are the direction-cosines, of the perpendicular from to the plane. To convert Ar + By + Cz+D=0 to the normal form it suffices to multiply it by the normalizing factor, 1: VÃ3+B2+C2, the new coefficients A: √ B: V C: V

being cos a, cos ẞ, ços y, and

D: V being. The angle between two planes Ax+ By + Cz + D=0 and A'x + B'y +C'z+D'=0 is determined by the relation cos (AA' + BB' + CC') :

V (A+B+C1) (A” + B” +C"), whence the planes are parallel when and only when A:A'B:B'= C:C', and are perpendicular when and only when AA'+ BB'+ CC′ −0. The equations of any two of the planes containing a line, together represent the line. Accordingly in space the line has two equations. Its simplest equations are those of any two of the three planes containing the line and being perpendicular respectively to the co-ordinate planes, as x = m2 + p, y=nz+q. Such a pair are unsymmetric. Symmetric equations of the line directed by a, B, Y, and going through the point (x1, y1, z1) are

(x-x1): cosa (y-y1): cosẞ=(2—21): cos Y in number three of which but (any) two are independent. The angle between two lines whose direction-cosines are proportional to L, M, N and L', M', N', respectively, is determined by the relation

cos 0=(LL'+ MM'+NN') :

√(L2+M2 + N2) (L”2 + M2 +N"), whence the lines are parallel if and only if L:L' M:M'=N:N', and are perpendicular if and only if LL' + MM' + NÑ' = 0. angle between a line of direction-cosines proportional to L, M, N and a plane

Ax+By+Cz+D=0

is given by the relation sin 0

(AL+BM+CN):

The

√ (A2+ B2 + C2) (L2 + M2 + N2), whence the line and plane are parallel if and only if AL+ BM + CN= 0 and are perpen

dicular if and only if A:LB:M-C:N. The necessary and sufficient condition that the line x = m1z + p1, y=n1z+q1, shall intersect the line x = m2 + p2, y = n2+ q2, is that (mm): (n1 — n2)=(p1 — P2) : (qı — q2).

The literature of the analytical geometry of space, herewith barely introduced, is extensive. For some account of further developments of the subject, see SURFACES, THEORY OF; CURVES OF DOUBLE CURVATURE. In the doctrine above introduced the point is employed as element. Some account of the theories that arise on choosing for element some other geometric entity, as the plane, the line, the sphere, etc., may be found in the articles, GEOMETRY, MODERN ANALYTICAL, and GEOMETRY, LINE, AND ALLIED THEORIES, in this work.

Bibliography.- College text-books of analytical geometry abound. One of the scientifically best American texts is W. B. Smith's 'Coordinate Geometry.' The most comprehensive English works are Salmon's 'Conic Sections' and Geometry of Three Dimensions' (both of which have been translated into French by O. Chemin, and supplemented and translated into German by Wilhelm Fiedler), and Frost's 'Solid Geometry.'

CASSIUS J. KEYSER, Adrain Professor of Mathematics, Columbia University.

GEOMETRY, Elementary. Geometry is the science of space. Its object is the study of the properties of forms (configurations, figures) of every conceivable kind. The subject is thus endless in two ways: in the first place, the number of configurations is infinite this is so even if we restrict ourselves to curves and surfaces; in the second place, any one type of figure has an inexhaustible variety of properties.

Elementary geometry may be roughly described as the study of the simpler or more evident properties of the simpler configurations. Specifically, the title refers to the body of geometric truths incorporated by Euclid in his famous 'Elements. The text-books on elementary geometry used throughout the civilized world for the last 20 centuries are in fact merely revisions of the 'Elements'; so that the subject itself is often referred to, especially in England, as the study of Euclid. The majority of the theorems refer to points, straight lines, and planes, with their combinations. Of curves, only the circle is considered, and of surfaces those related to the circle (spherical, cylindrical, conical).

It is not the object of this article to give a résumé of the standard theorems of elementary geometry, but rather to indicate some of the more significant general features, especially in the light of the more recent developments.

LOGICAL FOUNDATIONS.

The most prominent aspect of elementary geometry is the logical aspect: a great number of propositions, termed theorems, are deduced from a comparatively few propositions assumed at the outset and termed axioms or postulates. In the ideal treatment of the subject, all the assumptions should be enumerated explicitly, so that, if the question is asked, "Are the theorems of geometry true," the mathematician can answer correctly, "Yes, if my

postulates are true." As to whether the postulates are true, that is not a matter for the mathematician as such to consider, but rather comes within the province of the physicist, psychologist or philospher.

The ordinary course in geometry, modeled after Euclid, does not carry out this ideal. Assumptions are continually being made as they may be needed for the purpose of proof, in addition to those explicitly enunciated as axioms and postulates. For example, in the first proposition of Euclid, dealing with the construction of an equilateral triangle on a given segment AB, circles are drawn with centres A and B and common radius AB, and it is then assumed that these circles intersect. The only justification given is the diagram or the appeal to spatial intuition. Again, in dealing with the congruence of triangles, it is assumed that a triangle may be moved about without altering its sides or angles, though the stated axioms do not even mention displacement. spite of himself, Euclid's treatment is (partly) physical or intuitional, instead of purely mathematical, that is, purely logical.

In

It is only within the last few years that the ideal has been (practically) attained; that is, a set of explicit assumptions (termed axioms or postulates indifferently) drawn up, from which the propositions of ordinary geometry follow by purely logical processes. Geometry becomes then a branch of pure mathematics. As Poincaré expresses it, in this ideal treatment "We might put the axioms into a reasoning apparatus like the logical machine of Stanley Jevons, and see all geometry come out of it."

Many contributors have aided in this development, among whom may be mentioned Gauss, Lobachevsky, Pasch, Veronese, and especially Peano and his coworkers in symbolic logic, Pieri and Peano. The first elaborately worked out system is that of Hilbert (1900). We give a brief account of his axioms. Since then Veblen and Huntington in this country have developed much clearer and simpler systems. Consult Veblen, O., 'A System of Axioms for Geometry) (trans. Am. Math. Soc. 1904).

Geometry deals with three systems of objects or elements termed points, lines (used here in sense of straight lines), and planes, connected by certain relations expressed by the words lying in, between, etc. It is not necessary for the development of the subject that these words should suggest visual images; in fact the concrete nature of the elements and relations is to be eliminated from the discussion. To emphasize this abstract aspect, it is convenient to use symbols, say capital letters for points, Roman minuscules for lines, and Greek letters for planes. The axioms are arranged in five groups as follows:

I. Axioms of Association or Connection.-1. Any two different points A, B determine a line a. (Such points are then said to lie on the line.)

2. Any two different points on a line determine that line.

3. In any line there are at least two points, and in a plane there are at least three points not on a line.

4. Any three non-collinear points A, B, C determine a plane a.

5. Three non-collinear points of a plane determine that plane.

6. If two points A, B of a line a are in a plane a, then every point of a is in a. (The line is then said to lie in the plane).

7. Two planes a, ẞ which have a point A in common have at least a second point B in common.

8. There exist at least four points not in one plane.

II. Axioms of Order.- These deal with the relation expressed by the term between.

1. If A, B, C are points of a line and B is between A and C, then B is between C and A.

2. If A and C are points of a line a, then there exists on a at least one point B between A and C, and at least one point D such that C is between A and D.

3. Of any three collinear points, one and only one is between the other two.

These three axioms deal with the line, while the fourth deals with the plane.

4. If A, B, C are any three non-collinear points, and a is a line in their plane passing through a point of the segment AB, but not through A or B or C, then a contains a point of either the segment AC or the segment BC.

III. Axioms of Congruence. The first five axioms of this group relate to congruent segments and congruent angles. For example, a segment AB is congruent to itself and to the reversed segment BA; and segments congruent to the same segments are congruent to each other. Finally, the sixth is a metrical axiom concerning triangles: if two sides and the included angle of one triangle are congruent respectively to two sides and the included angle of another triangle, then the remaining angles are also congruent.

The fact that the remaining sides are congruent is not included as a part of the axiom because it may be proved. The other cases of congruent triangles are theorems. In Euclid the above statement is a theorem, but this is possible, as already observed, merely on account of unstated assumptions relating to displacement. Euclid's axiom that all right angles are congruent, in Hilbert's system becomes a theorem.

IV. The Axiom of Parallels.This contains only the so-called Euclidean axiom, in the form: Given a line a and a point A not on a, then in the plane a determined by a and a there is only one line through A which does not intersect a

V. Axioms of Continuity.- The continuity notion is analyzed into two parts of which the first (1) is stated in the axiom Archimedes. 1. On a straight line consider any two points A, B and a point A between them; construct the points A2, A3, in order, so that A1 is between A and A2, A2 between A1 and As, etc., and so that the segments AA1, A1A2, A2A3, are congruent; then among the points so constructed there exists a point An such that B is between A and An. That is, by repeatedly laying off a given segment however small any assigned point of the line will be passed after a finite number of steps.

.....

This axiom is sufficient for the development of the usual theorems of geometry. However the space to which the theorems apply would not be continuous in ordinary sense. It would in fact contain only those points of the space considered in analytical geometry whose co-ordinates are rational or expressible by radicals

of the second order. To identify with continuous space it is necessary to add a final axiom (2) relating to convergent point sets, or else the so-called axiom of completeness which states that the system of elements (points, lines, planes) cannot be enlarged by adjoining other elements in such a way that all the previous axioms are preserved.

The fact that this set of axioms is sufficient is shown by actually deducing the usual body of theorems. This is done in Hilbert's 'Grundlagen.' Diagrams are here used, it is true, but only for convenience; the proofs can be given without any reference to the diagrams. Often the deduction of those results which are evident to the intuition is long and complicated. This is the case, for example, in showing that a triangle (or any simple polygon) has the properties expressed by the terms inside and outside. It must be shown from the axioms that the given triangle brings about a division of the points in its plane into three classes, namely, points P, points I, and points O, such that any two I points or any two points may be connected by a broken line not containing any P point, while any broken line from an I point to an O point necessarily contains P points. To the intuition, of course, the P points are the points on the perimeter of the triangle, the I points are those inside, and the P points are those outside.

In the development it is important to observe that some theorems depend upon only part of the axioms. Thus from group I alone it follows that two planes having a point in common necessarily have a line in common, and that a line and a point determine a plane. The property of triangles and polygons stated above follows from group I and II. The theory of proportion may be established without employing group V. The theorem that if two triangles in one plane have their sides respectively parallel, the lines joining corresponding vertices are either parallel or concurrent (a special case of Desargues' theorem), can be proved without using axiom III 6 if and only if the spatial axioms in addition to the plane axioms are employed.

An important result which has been obtained recently is that while the areas of plane polygons may be treated without appealing to the continuity axioms, this is not possible with the volumes of polyhedrons. The difference is observed in Euclid's proofs: the proposition that triangles having the same base and altitude are equal in area is demonstrated by adding or taking away congruent parts from congruent figures, while the corresponding proposition concerning triangular pyramids is proved by the method of limits, or the equivalent method of exhaustion. That this difference in treatment is not avoidable was established by Dehn (1901), who showed that there exist polyhedrons of equal volumes which cannot be formed by the addition or subtraction of respectively congruent polyhedrons. In plane geometry this formation applies to any two polygons with the same area.

The most fundamental question concerning the set of axioms is that of consistency. In the development of geometry no contradiction has thus far presented itself; but will this always be the case? Can it be shown

that no inconsistency can ever arise? The only known method of answering this question depends upon establishing a correspondence between the geometrical elements and certain numerical entities, and showing that any inherent contradiction in geometry would involve contradictory relations among these entities. The question is thus transferred to the field of arithmetic. Are the axioms of number (commutative, associative, distributive, etc.) inconsistent? No perfectly satisfactory disproof of this has yet been devised. See however ALEGEBRA: DEFINITIONS AND FUNDAMENTAL CONCEPTS.

Another question to be considered is the independence of the axioms. If any one axiom can be deduced from the others, it may be omitted from the list and introduced as a theorem. It is therefore desirable that the axioms should express mutually independent statements. The standard method employed in proving the independence of an axiom or group of axioms consists in devising a set of objects of any kind which, when considered as elements, fulfill the relations expressed in remaining axioms, but for which the axiom or group in question is not satisfied. Thus the fact that the axiom of parallels (IV) cannot be deduced from the other axioms is shown by the non-Euclidean geometry of Lobachevsky. Similarly, the independence of axiom VI is proved by means of the non-Archimedean geometries of Veronese and Hilbert. Various apparently artificial systems have been devised in this connection, which, while not amenable to the intuition, are conceivable and mathematically true because based on assumptions which may be shown to be free from inconsistency.

The set of axioms presented above is of course not the only one which may serve as foundation for ordinary geometry. Thus the axiom of parallels may be replaced by the statement that the sum of the angles of a triangle is two right angles. In general the propositions of a given collection may be derived from various sets selected from the total collection. In the present case the possibilities are endless.

Geometry may also be founded on other primative (undefined) concepts than those introduced above. Thus in the discussions inaugurated by Helmholtz and continued by Lie and Poincaré, the principal concept is that of transformation (displacement, rigid motion) and the axioms include the group property (the resultant of two displacements is itself a displacement). The straight line is then no longer, as in Hilbert's system, a primitive concept, but receives definition: if in a displacement two points are fixed, there are an infinite number of fixed points forming, by definition, a straight line (the axis of rotation).

In the usual intuitional treatment the concept of general surface is assumed as a startingpoint and the plane is then defined as a surface such that if any two of its points are joined by a straight line, the latter lies entirely in the surface. This obviously states more than is required for the determination of the surface. To meet this objection the plane is sometimes defined as generated by drawing straight lines from a fixed point A to all the points of a straight line a. To obtain the entire plane

it is necessary to add the line through A parallel to a This definition is therefore unsatisfactory, because parallel lines require in their definition the previous definition of the plane. Peano has met the difficulty by this definition: Consider three fixed points A, B, C not in a straight line; take a fixed point D within the segment BC, and on the segment AD take a fixed point E; a plane is then generated by the lines (rays) from E to every point of the perimeter ABC. It may then be proved that a straight line connecting any two points of such a surface lies on the surface.

PROBLEMS AND CONSTRUCTIONS.

The only instruments whose use is implied in the postulates of elementary geometry are the ruler (straight-edge), for drawing straight lines, and the compass, for drawing circles. Only those problems are considered as coming within the domain of elementary geometry which can be solved by a finite number of operations with these instruments. Such constructions are termed Euclidean, or sometimes simply geometric. An example is the construction for bisecting an angle. With the vertex V as centre and any radius describe a circle cutting the sides of the angle in points A and B; with these points as centres and any (sufficiently large) radius describe circles intersecting in points C and D; the line joining C and D necessarily passes through and bisects the given angle.

For

However, many problems arise which cannot be solved in this way. A well-known example is the problem of trisecting an angle. centuries the Greek geometers and their followers sought for a solution; only within the present century has it been shown that such attempts must necessarily fail. The statement that the problem is impossible does not deny that lines trisecting the given angle exist, but means simply that such lines cannot be obtained by a construction employing a finite number of straight lines and circles.

No one has yet succeeded in demonstrating this impossibility by purely geometric means. The question arises naturally in elementary geometry, but apparently cannot be answered by elementary methods. We give now an outline of the algebraic method for deciding whether a given problem comes within the class of possible or the class of impossible problems.

Any line segment may be represented by a segment, namely, the ratio of the given segment to an assumed unit segment. Conversely, any number then represents a segment. Consider now the elementary operations of arithmetic or algebra in relation to geometric constructions.

If a and b denote given segments, or the corresponding numbers, the sum a+b is constructed by transferring the segment b, by means of the compass, so that it is collinear and adjacent to a. The difference a — b is also readily constructible.

The product rab may be defined by the proportion 1:ab:x. The proper construction is then suggested by the theorem that a line parallel to the base of a triangle divides the sides proportionally. Draw any triangle with 1 and a as two of the sides; along the first VOL. 12-29

side prolonged if necessary lay off segment b; from the terminal point draw a line parallel to the base of the triangle; this cuts off on the second side a segment equal to the required r. The quotient y=a/b is obtained similarly from the proportion b:1a:y. Hence all rational expressions, that is, expressions formed by a finite number of additions, subtractions, multiplications, and divisions are constructible.

Furthermore, extraction of square roots is possible. For = Va may be defined by 1:2=s:a. Hence if on 1+a as diameter a semicircle is described, the perpendicular at the end of the unit segment is the required 2. Therefore,

Theorem I-Any expression involving only rational operations and the extraction of square roots can be constructed with ruler and compass.

Expressions which cannot be reduced to this form cannot be constructed. This we now prove in the form of the converse:

Theorem II- Any segment which can be constructed with ruler and compass is expressible algebraically by rational operations and the extraction of square roots.

For any such construction consists in drawing a finite number of straight lines and circles and finding their intersections. Employing Cartesian co-ordinates (see GEOMETRY, CARTESIAN), the equation of a straight line is of the form ar+by+c=0, and that of a circle is of the form x2 + y2+ ax + by +c=0. The intersection of two straight lines leads to the solution of two equations of the first degree, which requires only rational operations. The intersections of a straight line and circle, or of two circles, depends on the solution of quadratic equations and leads to radicals of the second degree.

We proceed to apply these theorems to several examples.

Consider first the problem of bisecting an angle. The given angle and the required 0

angle may be determined by their cosines.

2

the following general theorem taken from the theory of equations:

Theorem III.— An irreducible equation whose degree is not a power of two cannot have a root expressible by radicals of the second degree. (The term irreducible equation is here employed to describe an equation f(x)=0 with rational coefficients whose left member cannot be factored rationally).

In general the algebraic questions which arise in this connection require for their complete discussion the powerful Galois Theory of Equations. See EQUATIONS, GALOIS' THEORY OF.

A second of the so-called famous problems of elementary geometry is the Delian problem, of the duplication of the cube. Given a cube with side a, to construct a cube with side r having double the volume. The equation of the problem is x3-2a3. Theorem III and then Theorem II apply. The corresponding problem concerning the square, leading to the equation 2a, is easily solved: the side of the required square is simply the diagonal of the given square.

Regular Polygons.- The construction of a regular polygon of n sides is equivalent to the division of a given circumference into n equal arcs. The only cases treated by Greek geometers and the ordinary text-books are, for prime numbers, n= =3 and n=5; from these constructions of the regular triangle and pentagon, combined with the construction for bisecting an angle, the constructions for the cases 2*, 3-2, 5-2, 3-5·2*, where is any integer, are easily found.

No advance was made, that is, no new constructible polygons were discovered, until Gauss, about a century ago, applied the algebraic method. The equation of the problem may be put into the form

...

tan 2+ + x + 1 = 0, which is then termed the cyclotomic equation. When n is a prime number the equation is irreducible. Hence by Theorem III the construction is possible only when n-1 is a power of 2. That is, n must be of the form 24 +1. Prime numbers of this type are necessarily of the form 22+1, and are known as Fermat primes. The values v= =0 and v=1 give the familiar cases n= =3 and n=5; the first new case, arising from v= =2, is n=17. The construction for the regular polygon of 17 sides is complicated, but the steps are indicated definitely by the algebraic solution of the cyclotomic equation, which is in fact solvable by square roots.

The general result on regular polygons is as follows: The regular polygon of n sides can be constructed with ruler and compass if, and only if, the prime factors of n are 2 repeated any number of times and distinct Fermat primes.

The first impossible cases are n 7 and n = 9.

Quadrature of the Circle.- This most famous problem of geometry requires the construction of a square having the same area as a given circle. That this is impossible (that is, that the construction cannot be effected with the ruler and compass) was not definitely shown until 1882, although the failure of innumerable attempts had led many to suspect the true result. The rectification of the circle, that is the

construction of a straight line having the same length as a given circumference, is an equivalent problem, and hence also impossible. This is so on account of the theorem that the area of a circle equals one-half of the product of the radius into the circumference.

The ratio of the circumference to the diameter is the same for all circles: the constant thus arising has been generally denoted by the symbol since the time of Euler. It was proved quite simply by Legendre that is not rational (i.e., cannot be represented exactly by the ratio of any two integers, and hence, in particular, cannot be represented by a terminating decimal). The difficulty consists in showing that is transcendental, that is, is not the root of any algebratic equation

Aoxn+a1xn−1+...+an=0,

where n is a positive integer, and the coefficients are any integers. This was finally proved by Lindemann in 1882, after Hermite in 1873 had shown that e, the base of the Napierian system of logarithms, is transcendental. The two numbers are connected by the remarkable relation ei=1, where i is the imaginary unit number V-1. Since cannot satisfy any algebraic equation, it certainly cannot be expressed by square roots. Hence Theorem II proves the impossibility.

π

Approximate Constructions. The problems considered cannot be solved exactly by ruler and compass, but they can be solved to any required degree of approximation. Thus a simple approximate solution of the rectification problem is the following: Let O be the centre and AB any diameter of the given circle. At the middle point E of AO construct a perpendicular cutting the circumference in C and D. On AB prolonged lay off EF-CD. Draw FD, and on this line lay off FH-AB. Then the segment HD is approximately onefourth the circumference. The error is less than one part in 5,000.

Other Instruments.- The problems considered may be solved exactly if other instruments in addition to ruler and compass are allowed. Thus the trisection and duplication problems (like all problems depending on cubic and biquadratic equations) can be solved by the instruments for drawing parabolas or other conics, or by appropriate linkages. The quadrature of the circle, being a transcendental problem, cannot be effected by any instrument which draws algebraic curves. It can be solved by various transcendental curves (quadratrix, sinusoid, cycloid); or by the integraph (an instrument which draws the curve

= ƒ ƒ (x) dx, where y==ƒ (x) is a given curve).

y=

f

We consider now various restrictions which may be imposed on Euclidean constructions.

(1) Ruler Constructions. Here only the straight-edge is allowed. For the possibility of such a construction it is necessary but not sufficient that the corresponding algebraic expression should be rational. If two parallel lines are given, then through a given point a line may be drawn parallel to given lines by a ruler construction. But this is not the case when a line is to be drawn through a given point parallel to a given line. The impossibility