Geometry

GEOMETRY, the general term for the branch of. mathematics which has for its province the study of the properties of space. From experience, or possibly intuitively, we characterize existent space by certain fundamental qualities, termed axioms, which are insusceptible of proof; and these axioms, in conjunction with the mathematical entities of the point, straight line, curve, surface and solid, appropriately defined, are the premises from which the geometer draws conclusions. The geometrical axioms are merely conventions; on the one hand, the system may be based upon inductions from experience, in which case the deduced geometry may be regarded as a branch of physical science; or, on the other hand, the system may be formed by purely logical methods, in which case the geometry is a phase of pure mathematics. Obviously the geometry with which we are most familiar is that of existent spacethe three-dimensional space of experience; this geometry may be termed Euclidean, after its most famous expositor. But other geometries exist, for it is possible to frame systems of axioms which definitely characterize some other kind of space, and from these axioms to deduce a series of non-contradictory propositions; such geometries are called nori-Euclidean.

It is convenient to discuss the subject-matter of geometry under the following headings:

I. Euclidean Geometry: a discussion of the axioms of existent space and of the geometrical entities, followed by a synoptical account of Euclids Elements.

II, Projective Geometry: primarily Euclidean, but differing from I. in employing the notion of geometrical continuity points and lines at infinity.

III. Descriptive Geometry: the methods for representing upon planes figures placed in space of three dimensions.

IV. Analytical Geometry: the representation of geometrical figures and their relations by algebraic equations.

V. Line Geometry: an analytical treatment of the line regarded as the space element.

VT. Non-Euclidean Geometry: a discussion of geometries other than that of the space of experience.

VII. Axioms of Geometry: a critical analysis of the foundations of geometry.

Special subjects are treated under their own headings: e.g. Projection, Perspective; Curve, Surface; Circle, Conic section; triangle, Polygon, Polyhedron; there are also articles on special curves and figures, e.g. Ellipse, Parabola, hyperbola; Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron, Cardioid, Catenary, Cissoid, Cochleoid, Cycloid, Epicycloid, Lfnaox, Oval, Quadratrix, Spiral, &c.

History

The origin of geometry (Gr. 1i~, earth, uiTPOP, a measure) is, according to Herodotus, to be found in the etymology of the word. Its birthplace was Egypt, and it arose from thi need of surveying the lands inundated by the Nile floods. Jr its infancy it therefore consisted of a few rules, very rough and approximate, for computing the areas of triangles and quadrilaterals; and, with the Egyptians, it proceeded no further, the geometrical entitiesthe point, line, surface and solidbeing only discussed in so far as they were involved in practical affairs, The point was realized as a mark or position, a straight line as a stretched string or the tracing of a pole, a surface as an area; but these units were not abstracted; and for the Egyptians geometry was only an artan auxiliary to surveying.1 The first step towards its elevation to the rank of a science was made by Thales of Miletus, who transplanted the elementary Egyptian mensuration to Greece. Thales clearly abstracted the notions of points and lines, founding the geometry of the latter unit, and discovering per saltum many propositions concerning areas, the circle, &c. The empirical rules of the Egyptians were corrected and developed by the Ionic School which he founded, especially by Anaximander and Anaxagoras, and in the 6th century B.C. passed into the care of the Pythagoreans. From this time geometry exercised a powerful influence on Greek thought. Pythagoras, seeking the key of the universe in arithmetic and geometry, investigated logically the principles underlying the, known propositions; and this resulted in the formulation of definitions, axioms and postulates which, in addition to founding a science of geometry, permitted a crystallization, fractional, it is true, of the amorphous collection of material at hand. Pythagorean geometry was essentially a geometry of areas and solids; its goal was the regular solids the tetrahedron, cube, octahedron, dodecahedron and icosahedronwhich symbolized the five elements of Greek cosmology. The geometry of the circle,, previously studied in Egypt and much more seriously by Tbales, was somewhat neglected, although this curve was regarded as the most perfect of all plane figures and the sphere the most perfect of all solids. The circle, however, was taken up by the Sophists, who made most of their discoveries in attempts to solve the classical problems of squaring the circle, doubling the cube and trisecting an angle. These problems, besides stimulating pure geometry, i.e. the geometry of constructions made by the ruler and compasses, exercised considerable influence mother directions. The first problem led to the discovery of the method of exhaustion for determining areas. Antiphon. inscribed a square in a circle, and on each side an isosceles triangle having its vertex on the circle; on the sides of the octagon so obtained, isosceles triangles were again constructed, the process leading to inscribed polygons of 8, 16 and 32 sides; and the areas of these polygons, which are easily determined, are, successive approximations to the area of the circle. Bryson of Heraclea took an important step when he circumscribed, in addition to inscribing, polygons to a circle, but he committed an error in treating the circle as the mean of the two polygons. The method of Antiphon, in assuming that by continued division. a polygon can. be constructed coincident with the circle, demanded that magnitudes are not infinitely divisible. Much controversy ranged about this point; Aristotle supported the doctrine of infinite divisibility; Zeno attempted to show its absurdity. The mechanical tracing of loci, a principle initiated by Archytas of Tarentum to solve the last two problems, was a frequent subject for study, and several mechanical curves were thus discovered at subsequent dates (cissoid, conchoid, quadratrix). Mention may be made of Hippocrates, who, besides developing the known methods, made a study of similar figures, and, as a consequence, of proportion. This step is important as bringing into line discontinuous number and continuous magnitude.

~A fresh stimulus was given by, the succeeding Platonists, who, accepting in part the Pythagorean. cosmology, made the study of geometry preliminary to that of philosophy. The many discoveries made by this school were facilitated in no small measure by the clarification of the axioms and definitions, thc logical sequence of propositions which was adopted, and, mor especially, by the formulation of the analytic method, i,e. ol assuming the truth of a proposition and then reasoning to 1

i For Egyptian geometry see EGYPT. Science and Matherna~ics.

known truth. The main strength of the Platonist geometers lies in stereometry or the geometry of solids. The Pythagoreans had dealt with the sphere and regular solids, but the pyramid, prism, cone and cylinder were but little known until the Platonists took them in hand. Eudoxus established their mensuration, proving the pyramid and cone to have one-third the content of a prism and cylinder on the same base and of the same height, and was probably the discoverer of a proof that the volumes of spheres are as the cubes of their radii. The discussion of sections of the cone and cylinder led to the discovery of the three curves named the parabola, ellipse and hyperbola (see COMc SEcTIoN); it is difficult to over-estimate the importance of this discovery; its investigation marks the crowning achievement of Greek geometry, and led in later years to the fundamental theorems and methods of modern geometry.

The presentation of the subject-matter of geometry as a connected and logical series of propositions, prefaced by ~pof or foundations, had been attempted by many; but it is to Euclid that we owe a complete exposition. Little indeed in the Elements is probably original except the arrangement; but in this Euclid surpassed such predecessors as Hippocrates, Leon, pupil of Neocleides, and Theudius of Magnesia, devising an apt logical model, although when scrutinized in the light of modern mathematical conceptions the proofs are riddled with fallacies. According to the commentator Proclus, the Elements were written with a twofold object, first, to introduce the novice to geometry, and secondly, to lead him to the regular solids; conic sections found no place therein. What Euclid did for the line and circle, Apollonius did for the conic sections, but there we have a discoverer as well as editor. These two works, which contain the greatest contributions to ancient geometry, are treated in detail in Section I. Euclidean Geometry and the articles EUCLID; CONIC SECTION; AP0LL0NIUs. Between Euclid and Apollonius there flourished the illustrious Archimedes, whose geometrical discoveries are mainly concerned with the mensuration of the circle and conic sections, and of the sphere, cone and cylinder, and whose greatest contribution to geometrical method is the elevation of the method of exhaustion to the dignity of an instrument of research. Apollonius was followed by Nicomedes, the inventor of the conchoid; Diodes, the inventor of the cissoid; Zenodorus, the founder of the study of isoperimetrical figures; Hipparchus, the founder of trigonometry; and Heron the elder, who wrote after the manner of the Egyptians, and primarily directed attention to problems of practical surveying.

Of the many isolated discoveries made by the later Alexandrian mathematicians, those of Menelaus are of importance. He showed how to treat spherical triangles, establishing their properties and determining their congruence; his theorem on the products of the segments in which the sides of a triangle are cut by a, line was the foundation on which Carnot erected his theory of transversals. These propositions, and also those of Hipparchus, were utilized and developed by Ptolemy, the expositor of trigonometry and discoverer of many isolated propositions. Mention maybe made of the commentator Pappus, whose Mathematical Collections is valuable for its wealth of historical matter; of Theon, an editor of Euclids Elements and commentator of Ptolemys Almagest; of Proclus, a commentator of Euclid; and of Eutocius, a commentator of Apollonius and Archimedes.

The Romans, essentially practical and having no inclination to study science qua science, only had a geometry which sufficed for surveying; and even here there were abundant inaccuracies, the empirical rules employed being akin to those of the Egyptians and Heron. The Hindus, likewise, gave more attention to computation, and their geometry was either of Greek origin oi~ in the form presented in trigonometry, more particularly connected with arithmetic. It had no logical foundations; each proposition stood alone; and the results were empirical. The Arabs more closely followed the Greeks, a plan adopted as a sequel to the translation of the works of Euclid, Apollonius, Archimedes and many others into Arabic. Their chief contribution to geometry is exhibited in their solution of algebraic equations by intersecting conics, a step already taken by the Greeks in isolated cases, but only elevated into a method by Omar al Hayyami, who flourished in the IIth century. During the middle ages little was added to Greek and Arabic geometry. Leonardo of Pisa wrote a Practica geometriae (1220), wherein Euclidean methods are employed; but it was not until the 14th century that geometry, generally Euclids Elements, became an essential item in university curricula. There was, however, no sign of original development, other branches of mathematics, mainly algebra and trigonometry, exercising a greater fascination until the 16th century, when the subject again came into favor.

The extraordinary mathematical talent which came into being in the 16th and 17th centuries reacted on geometry and gave rise to all those characters which distinguish modern from ancient geometry. The first innovation of moment was the formulatkni of the principle of geometrical continuity by Kepler. The notion of infinity which it involved permitted generalizations and systematizations hitherto unthought of (see GEOMETRICAL CONTINUITY); and the method of indefinite division applied to rectification, and quadrature and cubature problems avoided the cumbrous method of exhaustion and provided more accurate results. Further progress was made by Bonaventura Cavalieri, who, in his Geometria indivisibilibus continuorutn (1620), devised a method intermediate between that of exhaustion and the infinitesimal calculus of Leibnitz and Newton. The logical basis of his system was corrected by Roberval and Pascal; and their discoveries, taken in conjunction with those of Leibnitz, Newton, and many others in the fluxional calculus, culminated in the branch of our subject known as differential geometry (see INTINITESIMAL CALCULUS; CURVE; SURFACE).

A second important advance followed the recognition that conics could be regarded as projections of a circle, a conception which led at the hands of Desargues and Pascal to modern projective geometry and perspective. A third, and perhaps the most important, advance attended the application of algebra to geometry by Descartes, who thereby founded analytical geometry. The new fields thus opened up were diligently explored, but the calculus exercised the greatest attraction and relatively little progress was made in geometry until the beginning of the 19th century, when a new era opened.

Gaspard Monge was the first important contributor, stimulating analytical and differential geometry and founding descriptive geometry in a series of papers and especially in his lectures at the cole polytechnique. Projective geometry, founded by Desargues, Pascal, Monge and L. N. M. Carnot, was crystallized by J. V. Poncelet, the creator of the modern methods. In his Trait des proprits des figures (1822) the line and circular points at infinity, imaginaries, polar reciprocation, homology, crossratio and projection are systematically employed. In Germany, A. F. Mobius, J. Plucker and J. Steiner were making far-reaching contributions. Mbius, in his Barycentrische Calcul (1827), introduced homogeneous co-ordinates, and also the powerful notion of geometrical transformation, including the special cases of collineation and duality; Plucker, in his Analytischgeometrische Eniwickelungen (1828-1831), and his System der anal ytischen Geometrie (1835), introduced the abridged notation, line and plane co-ordinates, and the conception of generalized space elements; while Steiner, besides enriching geometry in numerous directions, was the first to systematically generate figures by projective pencils. We may also notice M. Chasles, whose Aperu historique (1837) is a classic. Synthetic geometry, characterized by its fruitfulness and beauty, attracted most attention, and it so happened that its originally weak logical foundations became replaced by a more substantial set of axioms. These were found in the anharmonic ratio, a device leading to the liberation of synthetic geometry from metrical relations, and in involution, which yielded rigorous definitions of imaginaries. These innovations were made by K. J. C. von Staudt. Analytical geometry was stimulated by the algebra of invariants, a subject much developed by A. Cayley, G. Salmon, S. H. Aronhold, L. 0. Hesse, and more particularly by R. F. A. Clebsch.

The introduction of the line as a space element, initiated by H. Grassmann (1844) and Cayley (1859), yielded at the hands of Plucker a new geometry, termed line geometry, a subject developed more notably by F. Klein, Clebsch, C. T. Reye and F. 0. R. Sturm (see Section V., Line Geometry). N~n-euclidean geometries, having primarily their origin in the discussion of Euclidean parallels, and treated by Wallis, Saccheri and Lambert, have been especially developed during the f9th century. Four lines of investigation may be distinguished: the naive-synthetic, associated with Lobatschewski, Bolyai, Gauss; the metric differential, studied by Riemann, Helmholtz, Beltrami; the projective, developed by Cayley, Klein, Clifford; and the critical-synthetic, promoted chiefly by the Italian mathematicians Peano, Veronese, Burali-Forte, Levi Civitt, and the Germans Pasch and Hilbert. (C. E.*)

I. EUCLIDEAN GEOMETRY

This branch of the science of geometry is so named since its methods and arrangement are those laid down in Euclids Elements.

I. Axioms.The object of geometry is to investigate the properties 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 characterize space, and that nothing may be omitted without making the system incomplete. They must, in fact, completely define space.

2. Definitions.The axioms of Euclidean Geometry are obtained from inspection of existent space and of solids in existent space,hence 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 experience; we have only to abstract all material from it in order to gain the notion of a geometrical solid. This has 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 thickness. Their boundaries are curves or lines, and these have length only. Their boundaries, again, are points, which have no magnitude but only position. We thus come in three steps from solids to points which have no magnitude; 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 that a 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. I, I. A point is that which has no parts, or which has no magnitude.

Def. 2, I. A line is length without breadth.

Def. 5, I. A superficies is that which has only length and breadth.

Def. 1, XI. A solid is that which has length, breadth and thickness. It is to be noted that the synthetic method is adopted by Euclid; the analytical derivation of the successive ideas of surface, line, and point from the experimental realization of a solid does not find a place in his system, although possessing more advantages.

If we allow motion in geometry, 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, iii 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 manner. From this generation of the entities it follows again that the boundaries the first and last position of the moving elementof 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, I., Def. 6, I., and Def. 2, XI. He does not, however, show the connexior which these definitions have with those mentioned before When ,points and lines have been defined, a statement lik Def. 3, I., The extremities of a line are points, is a proposition which either has to be proved, and then it is a theorem, or which has to be taken for granted, in which case it is an axiom. And so with Def. 6, I., and Def. 2, XI.

3. Euclids definitions mentioned above are attempts to describe, in 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, principally those of the simplest linethe straight line, and of the simplest surfacethe flat surface or plane. These notions we possess, but to define them accurately is difficult. Euclids Definition 4, I., A straight line is that which lies evenly between its extreme points, must be meaningless to any one who has not the notion of straightness in his mind. Neither does it slate a property of the straight line which can be used in any further investigation. Such a property is given in Axiom lo, I. It is really this axiom, together with Postulates 2 and 3, which characterizes the straight line.

Whilst for the straight line the verbal definition and axiom are kept apart, Euclid mixes them up in the case of the plane. Here the Definition 7, I., includes an axiom. It defines a plane as a surface which has the property that every straight line which joins any two points in it lies altogether in the surface. But if we take a straight line and a point in such a surface, and draw all straight lines which join the latter to all points in the first line, the surface will be fully determined. Thisconstruction is therefore sufficient as a definition. That every other straight line which joins any two points in this surface lies altogether in it is a further property, and to assume it gives another axiom.

Thus a number of Euclids axioms are hidden among his first definitions. A still greater confusion exists in the present editions of Euclid between the postulates and axioms so called, but this is due to later editors and not to Euclid himself. The latter had the last three axioms put togetherwith the postulates (air*uara), so that these were meant to include all assumptions relating to space. The remaining assumptions, which relate to magnitudes in general, viz, the first eight axioms in modern editions, were called common notions (icou-ai ~vpoiai). Of the latter a few may be said to be definitions. Thus the eighth might be taken as a definition of equal, and the seventh of halves. If we wish to collect the axioms used in Euclids Elements, we have therefore to take the three postulates, the last three axioms as generally given, a few axioms hidden in the definitions, and an axiom used by Euclid in the proof of Prop. 4, I, and on a few other occasions, viz, that figures may be moved in space without change of shape or size.

4. Postulates.The assumptions actually made by Euclid may be stated as follows:

(I) Straight lines exist which have the property that any one of them may be produced both ways without limit, that through any two points in space such a line may be drawn, and that any two of them coincide throughout their indefinite extensions as soon as two points in the one coincide with two points in the other. (This gives the contents of Def. 4, part of Def. 35, the first two Postulates, and Axiom 10.)

(2) Plane surfaces or planes exist having the property laid down in Def. 7, that every straight line joining any two points in such a surface lies altogether in it.

(3) Right angles, as defined in Def. 10, are possible, and all right angles are equal; that is to say, wherever in space we take a plane, and wherever in that plane we construct a right angle, all angles thus constructed will be equal, so that any one of them may be made to coincide with any other. (Axiom 11.)

(4) The 12th Axiom of Euclid. This we shall not state now, but only introduce it when we cannot proceed any further without it.

(5) Figures maybe freely moved in space without change of shape or size. This is assumed by Euclid, but not stated as an axiom.

(6) In any plane a circle may be described, having any point in that plane as centre, and its distance from any other point in that plane as radius. (Postulate 3.)

The definitions which have not been mentioned are all nominal definitions, that is to say, they fix a name for a thing described. Many of them overdetermine a figure.

5. Euclids Elements (see EUCLID) are contained in thirteen books. Of these the first four and the sixth are devoted to plane geometry, as the investigation of figures in a plane is generally called. The 5th book contains the theory of proportion which is used in Book VI. The 7th, 8th and oth books are purely arithmetical, whilst the 10th contains a most ingenious treatment of geometrical irrational quantities. These four books will be excluded from our survey. The remaining three books relate to figures in space, or, as it is generally called, to solid geometry. The 7th, 8th, 9th, 10th, i3th and part of the 11th and 12th books are now generally omitted from the school editions of the Elements. In the first four and in the 6th book it is to be understood that all figures are drawn in a plane.

BOOK I. OF EUcLIDs ELEMENTS.

6. According to the third postulate it is possible to draw in any plane a circle which has its centre at any given point, and its radius equal to the distance of this point from any other point given in the plane. This makes it possible (Prop. 1) to construct on a given line AB an equilateral triangle, by drawing first a circle with A as centre and AB as radius, and then a circle with B as centre and BA as radius. The point where these circles intersect that they intersect Euclid quietly assumesis the vertex of the required triangle. Euclid does not suppose, however, that a circle may be drawn which has its radius equal to the distance between any two points unless one of the points be the centre. This implies also that we are not supposed to be able to make any straight line equal to any other straight line, or to carry a distance about in space. Euclid therefore next solves the problem: It is required along a given straight line from a point in it to set off a distance equal to the length of another straight line given anywhere in the plane. This is done in two steps. It is shown in Prop. 2 how a straight line may be drawn from a given point equal in length to another given straight line not drawn from that point. And then the problem itself is solved in Prop. 3, by drawing first through the given point some straight line of the required length, and then about the same point as centre a circle having this length as radius. This circle will cut off from the given straight line a length equal to the required i one. Nowadays, instead of going through this long process, we take a pair of compasses and set off the given length by its aid. This assumes that we may move a length about without changing it. But Euclid has not assumed it, and this proceeding wotild be fully justified by his desire not to take for granted more than was necessary, if he were not obliged at his very next step actually to make this assumption, though without stating it.

7. \Ve now come (in Prop. 4) to the first theorem. It is the fundamental theorem of Euclids whole system, there being only a very few propositions (like Props. 13, 14, 15, I.), except those in the 5th hook and the first half of the 11th, which do not depend upon it. It is stated very accurately, though somewhat clumsily, as follows: -

If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by those sides equal to one another, they shall also have their bases or third sides equal; and the two triangles shall be equal; and their other angles shall be equ at, each to each, namely, those to which the equal sides are opposite.

That is to say, the triangles are identically equal, and one may be considered as a copy of the other. The proof is very simple. The first triangle is taken up and placed on the second, so that the parts of the triangles which are known to be equal fall upon each other. It is then easily seen that also the remaining parts of one coincide with those of the other, and that they are therefore equal. This process of applying one figure to another Euclid scarcely uses again, though many proofs would be simplified by doing so. The process introduces motion into geometry, and includes, as already stated, the axiom that figures may be moved without change of shape or size.

If the last proposition be applied to an isosceles triangle, which has two sides equal, we obtain the theorem (Prop. 5), if two sides of a triangle are equal, then the angles opposite these sides are equal.

Euclids proof is somewhat complicated, and a stumbling-block to many schoolboys. The proof becomes much simpler if we consider the isosceles triangle ABC (AB = AC) twice over, once as a triangle BAC, and once as a triangle CAB; and now remember that AB, AC in the first are equal respectively toAC, AB in the second, and the angles included by these sides are equal. Hence the triangles are equal, and the angles in the one are equal to those in the other, viz. those which are opposite equal sides, i.e. angle ABC in the first equals angle ACB in the second, as they are opposite the equal sides AC and AB in the two triangles.

There follows the converse theorem (Prop. 6). If two angles ir~ a triangle are equal, then the sides oppbsite them are equali.e. the triangle is isosceles. The proof given consists in what is called a reductio ad absurdum, a kind of proof often used by Euclid, and principally in proving the converse of a previous theorem. It assumes that the theorem to be proved is wrong, and then shows that this assumption leads to an absurdity, i.e. to a conclusion which is in contradiction to a proposition proved beforethat therefore the assumption made cannot be true, and hence that the theorem is true. It is often stated that Euclid invented this kind of ~g,roof, hut the method is most likely much older.

8. It is next proved that two triangles which have the three sides of the one equal respectively to those of the other are identically equal, hence that the angles of the one are equal respectively to those of the other, those being equal which are opposite equal sides. This is Prop. 8, Prop. 7 containing only a first step towards its proof.

These theorems allow now of the solution of a number of problems, viz.: To bisect a given angle (Prop. 9).

To bisect a given finite straight line (Prop. 10).

To draw a straight line perpendicularly to a given straight line through a given point in it (PrOp. If), and also through a given point not in it (Prop. 12).

The solutions all depend upon properties of isosceles triangles, 9. The next three theorems relate to angles only, and might have been proved before Prop. 4, or even at the very beginning. The first (Prop. 13) says, The angles which one straight line makes with another straight line on one side of it either are two right angles or are together equal to two right angles. This theorem would have been unncce~sary if Euclid had admitted the notion of an angle such that its two limits are in the same straight line, and had besides defined the sum of two angles.

Its converse (Prop. 14) is of great use, inasmuch as it enables us in many cases to prove that two straight lines drawn from the same point are one the continuation of the othei. So also is Prop. 15. If two straight lines cut one another, the vertical or opposite angles shall be equal. -

10. Euclid returns now to properties of triangles. Of great importance for the next steps (though afterwards superseded by a more complete theorem) is Prop. 16. If one side of a triangle be produced, the exterior angle shall be greater than either of the interior opposite angles.

Prop. 17. Any two angles of a triangle are together less than two right angles, is an immediate consequence of it. By the aid of these two, the following fundamental properties of triangles are easily proved: Prop. 18. The greater side of every triangle has the greater angle opposite to it; Its converse, Prop. 19. The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it; Prop. 20. Any two sides of a triangle are together greater than the third side; And also Prop. 21. If from the ends of the side of a triangle there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.

11. Having solved two problems (Props. 22, 23), he returns to two triangles which have two sides of the one equal respectively to two sides of the other. It is ,known (Prop. 4) that if the included angles are equal then the third sides are equal; and conversely (Prop. 8), if the third sides are equal, then the angles included by the first sides are equal. From this it follows that if the included angles are not equal, the third sides are not equal; and conversely, that if the third sides are not equal, the included angles are not equal. Euclid now completes this knowledge by proving, that if the included angles are not equal, then the third side in that triangle is the greater which contains the greater angle; and conversely, that if the third sides are unequal, that triangle contains the greater angle which contains the greater side. These are Prop. 24 and Prop. 25.

12. The next theorem (Prop. 26) says that if two triangles have one side and two angles of the one equal respectively to one side and two angles of the other, viz, in both triangles either the angles adjacent to the equal side, or one angle adjacent and one angle opposite it, then the two triangles are identically equal.

This theorem belongs to a group with Prop. 4 and Prop. 8. Its first case might have been given immediately after Prop. 4, but the second case requires Prop. 16 for its proof.

13. We come now to the investigation of parallel straight lines, i.e. of straight lines which lie in the same plane, and cannot be made to meet however far they be produced either way. The investigation which starts from Prop. 16, will become clearer if a few names be explained which are not all used by Euclid. If two straight lines be cut by a third, the latter is now generally called a transversal of the figure. It forms at the two points where it cuts the given lines four angles with each. Those of the angles which lie between the given lines are called interior angles, and of these, again, any two which lie on opposite sides of the transversal but one at each of the two points are called alternate angles.

We may now state Prop. 16 thus:If two straight lines which meet are cut by a transversal, their alternate angles are unequal. For the lines will form a triangle, and one of the alternate angles will be an exterior angle to the triangle, the other interior and opposite to it.

From this follows at once the theorem contained in Prop. 27. If two straight lines which are cut by a transversal make alternate angles equal, the lines cannot meet, however far they be produced, hence they are parallel. This proves the existence of parallel lines.

Prop. 28 states the same fact in different forms. If a straight line, falling on two other straight lines, make the exterior angle equal to the interior and opposite angle on the same side of the line, or make the interior angles on the same side together equal to two right angles, the two straight tines ihall be parallel to one another.

Hence we know that, if two straight lines which are cut by a transversal meet, their alternate angles are not equal and hence that, if alternate angles are equal, then the lines are parallel.

The question now arises, Are the propositions converse to these true or not ? That is to say,, If alternate angles are unequal, do the lines meet ? And if the lines are parallel, are alternate angles necessarily equal ?

The answer to either of the~e two questions implies the answer to the other. But it has been found impossible to prove that the negation or the affirmation of either is true.

The difficulty which thus arises is overcome by Euclid assuming that the first question has to be answered in the affirmative. This gives his last axiom (12), which we quote in his own words.

Axiom Ia-If a straight line meet two straight lines, so as ,to make the Iwo interior angles ,on the same side of it taken together less than two right angles, these straight lines, being continually produced, shall at length meet on that side oil which are the angles which areless than two right angles.

The answer to the second of the above questions follows from this, and gives the theorem Prop. 29~If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior andopposite angle on the same side, and also the two interior angles on the same side together equal to two right angles.

4- \Vith this a new part of elementary geometry begins. The earlier propositions are independent of this axiom, and would be true even if a wrong assumption had been made in it. They all relate to figures in a plane. But a, plane is,only oneamong an infinite number of conceivable surfaces. We may draw figures on any one of them and study their prope.rties. We may, for instance, take a sphere instead.of the plane, and obtain spherical in the place of plane geometry. If on one of these surfaces lines and figures could be drawn, answering to all the definitions of our plane figures, and if the axioms with the exception of the last ~ll hold, then all propositions up to the 28th will be true for these figures. ,This is the case in spherical geometry if. we substitute shortest line or great circle for straight line, small circle for circle, and if, besides, we limit all figures to a part of the sphere which is less than a hemisphere, so that two, points on it cannot be opposite ends of a diameter, and therefore determine always one and only one great circle.

For spherical triangles, therefore, all the important propositions 4, 8, 26; 5 and 6; and 18, 19 and 20 will hold good.

This remark will be sufficientto show the impossibility of proving Euclids last axiom, which would mean proving that this axiom is a consequence of the others, and hence that the theory of parallels would hold on a spherical surtace, where the other axioms do hold, whilst parallels do not even exist.

It follows that the axjom ip question states an inherent difference between the plane and other, surfaces, and that the plane is only fully characterized wheti this axiom is added to the other assumptiop~.

15. The introduction of, the new axiom and of parallel lines leads to a new class of propositions.

After proving (Prop. 30) that two lines which are each parallel to a third are parallel to eec/i other, we obtain the new properties of triangles contained in Prop. 32.01 these the second part is, the most important, viz, the theorem, The three interior angles of every triangle are together equal to two right angles.

As easy deductions not given by Euclid but added by Simson follow the propositions about the angles in polygons; they are given in English editions as corollaries to Prop. 32.

These theorems do not hold for spherical figures. The sum of the interior angles of a spherical triangle is always greater than two right angles, and increases with the area.

16. The theory of parallels as such may be said to be finished with Props. 33 and 34, which state properties of the parallelogram, i.e. of a quadrilateral, formed by two pairs of parallels. They arc Prop. 33. The straight lines which join the extremities of two equal and parallel straight lines towards life same parts are themselves equal and parallel; and Prop. 34. The opposite sides and angles of a parallelogram are equal to, one another, and the diameter (diagonal) bisects the parallelogram, that is, divides it into two equal parts.

17. The rest of, the first boo~c relates to areas of figures. The theory is made to depend upon the theorems Prop. 35. Parallelograms on the same base and between the same parallels areequal to one another; and Prop. 36. Parallelograms on equal bases and between the same parallels are equal to one another.

As each parallelogram is bisected by a diagonal, the last theorems hold also if the word parallelogram be replaced by triangle, as is done in Props. 37 and 38.

It is .to be remarked that Euclid proves these propositions only in the case when the parallelograms or triangles have their bases in the sanie straight line.

The theorems converse to the last form the contents of the next three prooositions. viz.: Props. 40 and 41.Equal triangles, on the same or on equal bases, in the same straight line, and on the same side of it, are between the same parallels.

That the two cases here stated are given by Euclid in two separate propositions proved separately is characteristic of his method.

18. To compare areas of other figures, Euclid shows first, in Prop. 42, how to draw a parallelogram which is equal in area to a given triangle, and has one of its angles equal to a given angle. If the given angle is right, then the problem is solved to draw a rectangle equal in area to a given triangle.

Next this parallelogram is transformed into another parallelogram, which has one of its sn/es equal to a given straight line, whilst its angles remain unaltered. This may be done by aid of the theorem in Prop. 43. The complements of the parallelograms which are about the diameter of any parallelogram are equal to one another.

Thus the problem (Prop. 44) iS solved to construct a parallelogram on a given line, which is equal in area to a given triangle, and which has one angle equal to a given angle (generally a right angle).

As every polygon can be divided into a number of triangles, we can now construct a parallelogram having a given angle, say a right angle, and being equal in area to a given polygon. For each of the triangles into which the polygon has been divided,, a parallelogram may be constructed, having one side equal to a given straight line and one angle equal to a given angle. If these parallelograms be placed side by side, they may be added together to form a single parallelogram, having still one side of the given length. This is done in Prop. 45.

Herewith a means is found to compare areas of different polygons. We need only construct two rectangles equal in area to the given polygons, and having each one side of given length. By comparing the unequal sides we are enabled to judge whether the areas are equal, or which is the greater. Euclid does not state thisconsequence, but the problem is taken up again at the end of the second book, where it is shown how to construct a square equal in area to a given polygon.

Prop. 46 is: To describe a square on a given straight line.

19. The first book concludes with one of the most important theorems in the whole of geometry, and one which has been celebrated since the earliest times. It is stated, but on doubtful authority, that Pythagoras discovered it, and it has been called by his name. If we call that side in a right-angled triangle which is opposite the right angle the hypotenuse, we may state it as follows: Theorem of Pythagoras (Prop. 47).In every right-angled triangle the square on the hypotenuse is equal to the sum of the squares of the other sides.

And conversely Prop. 48. If the square described on one of the sides of a triangle be equal to the squares described on the other sides, then the angle contained by these two sides is a right angle.

On this theorem (Prop. 47) almost all geometrical measurement depends, which cannot be directly obtained.

Booa II.

20. The propositions in the second book are very different in character from those in the first; they all relate to areas of rectangles and squares. Their true significance is best seen by stating them in an algebraic form. This is often done by expressing the lengths of lines by aid of numbers, which tell how many times a chosen unit is contained in the lines. If there is a unit to be found which is contained an exact number of times in each side of a rectangle, it is easily seen, and generally shown in the teaching of arithmetic, that the rectangle contains a number of unit squares equal to the product of the numbers which measure the sides, a unit square being. the square on the unit line. If, however, no such unit can be found, this process requires that connexiun between lines and numbers which is only established by aid of ratios of lines, and which is therefore at this stage altogether inadmissible. But there exists another way of connecting these propositions with algebra, based on modern notions which seem destined greatly to change and to simplify mathematics. We shall introduce here as much of it as is required for our present purpose.

At tije beginning of the second book we find a definition according to which a rectangle is said to be contained by the two sides which contain one of its right angles; in the text this phraseology is extended by speaking of rectangles contained by any two straight lines, meaning the rectangle which has two adjacent sides equal to the two straight lines.

We shall denote a finite straight line by a single small letter, a, b, c, - - - x, and the area of the rectangle contained by-two lines a and b by ab, and this we shall call the product of the two lines a and b. It will be understood that this definition has nothing to do with the definition of a product of numbers.

We define as follows: The sum of two straight lines a and b means a straight line c,which may be divided in two parts equal respectively to a and b. This sum is denoted by a+b.

The differehce of two lines a and b (in symbols, ab) means a line c which when added to b gives a; that is, ab=c if b+c=a.

The prod~cl of two lines a and b (in symbols, ab) means the area of the rectangle contained by the lines a and b. For aa, which means the square on the line a, we write a2.

21. The first ten of the fourteen propositions of the second book may then be written in the form of formulae as follows: Prop. I. a(b+c+d+.. .)ab+ac+ad+

2. ab+ac=a2ifb+c=a.

3. a(a+b) =a2+ab.

4. (a+b)1=a2-4-2ab-l-b2.

5. (a+b) (ab)-l-b2=a2.

6. (a+b) (ab)+bi=at.

7. a2+(ab)2=2a(ab)+b2.

8.4(a+h)a+b2= (2a+b)2.

9. (a-~-b)i-{-(a b)2=2a2+2b2.

10. (a+b)2+(a b)22a2+2b2.

It will be seen that 5 and 6, and also 9 and To, ire id,~ntical. In Euclids statement they do not look the same, the figures being arranged differently.

If the letters a, b, c,. .. denoted numbers, it follows from algebra that each of these formulae is true. But this does not prove them in our case, where the letters denote lines, and their products areas without any reference to numbers. To prove them we have to discover the laws which rule the operations introduced, viz, addition and multiplication of segments. This we shall do now; and we shall find that these laws are the same with those which hold in algebraical addition and multiplication.

22. In a sum of numbers we may change the order in which the numbers are added, and we may also add the numbers together in groups and then add these groups. But this also holds for the sum of segments and for the sum of rectangles, a~ a little consideration shows. That the sum of rectangles has always a meaning follows from the Props. 43-45 in the first book. These laws about addition are reducible to the two a+b=b+a. .. (I),

a+(b+c)sa+b+c. .. (2);

or, when expressed for rectangles, ab-fed=ed+ab. .. (3),

ab+(cd+ef)=ab+cd+ef. .. (4).

The brackets mean that the terms in the bracket have been added together before they are added to another term. The more general cases for more terms may be deduced from the above.

For the product of two numbers we have the law that it remains unaltered if the factors be interchanged. This also holds for our geometrical product. For if ab denotes the area of the rectangle which has a as base and b as altitude, then ba will denote the area of the rectangle which has b as base and a as altitude. But in a rectangle we may take either of the two lines which contain it as base, and then the other will be the altitude. This gives ab=ba. ... (5).

In order further to multiply a sum by a number, we have in algebra the rule :Multiply each term of the sum, and add the products thus obtained. That this holds for our geometrical products is shown by Euclid in his first proposition of the second book, where he proves that the area of a rectangle whose base is the sum of a number of segments is equal to the sum of rectangles which have these segments separately as bases. In symbols this gives, in the simplest case, a(b+c)=ab-l-ac 6

and (b+c)a=ba+ca .

To these laws, which have been investigated by Sir William Hamilton and by Hermann Grassmann, the former has given special names. He calls the laws expressed in (I)and (~) the commutative law for addition; (5) ,, ,, multiplication; (2) and (4) the associative laws for addition; (6) the distributive law.

23. Having proved that these six laws hold, we can at once prove every one of the above propositions in their algebraical form.

The first is proved geometrically, it being one of the fundamental laws. The next two propositions are only special cases pf the first. Of the others we shall prove one, viz. the fourth: (a-+b)=(a+b) (a+b)=(a+b)a+(a+b)b by (6).

But (a-l-b)aaa+ba by (6),

=aa+ab by (5);

and (a+b)b=ab+bb by (6).

Therefore (a+b)2 = a-a +ab+ (ab+bb))

aa+(ab+ab)+bb f by (4);-

=aa+2ab+bb)

This gives the theorem in question.

In the same manner every one of the first ten propositions is proved.

It will be seen that the operations performed arc exactly the same as if the letters denoted numbers.

Props. 5 and 6 may also be written thus (a+b) (ab)=aZb2.

Prop. 7, which is an easy consequence of Prop. 4, may be transformed. If we denote by c the line a+b, so that c=a+b, acb, we get c2+(c b)2=2c(c b)+b2

=2ci 2bC+bi.

Subtracting c2 from both sides, and writing a for c, we get (a b)2 =a2 2ab+b2.

In Euclids Elements this formS of the theorem does not appear, all propositions being so stated that the notion of subtraction does not enter into them.

24. The remaining two theorems (Props. 12 and 13) connect the square on one side of a triangle with the sum of the squares on the other sides, in case that the angle between the latter is acute or obtuse. They are important theorems in trigonometry, where it is possible to include them in a single theorem.

25. There are in the second book two problems, Props. II and 14.

If written in the above symbolic language, the former requires to find a line x such that a(ax) =xf. Prop. II contains, therefore, the solution of a quadratic equation, which we may write x2+ax =a2. The solution is required later on in the construction of a regular decagon.

More important is the problem in the last proposition (Prop. 14). It requires the construction of a square equal in area to a given rectangle, hence a solution of the equation x1=ab.

In Book 1., 42-45, it has been shown howa rectangle may be constructed equal in area to a given figure bounded by straight lines. By aid of the new proposition we may therefore now determine a line such that the square on that line is equal in area to any given rectilinear figure, or we can square any such figure.

As of two squares that is the greater which has the greater side, it follows that now the comparison of two areas has been reduced to the comparison of two lines.

The problemof reducing other areas to squares is frequently met with among Greek mathematicians. We need only mention the problem of squaring the circle (see CIRCLE).

In the present day the comparison of areas is performed in a simpler way by reducing all areas to rectangles having a common base. Their altitudes give then a measure of their areas.

The construction of a rectangle having the base u, and being equal in area to a given rectangle, depends upon Prop. 43, 1. This therefore gives a solution of the equation a-b =ux, where x denotes the unknown altitude.

BooK III.

26. The third book of the Elements relates exclusively to properties of the circle. A circle and its circumference have been defined in Book I., Def. 15. We restate it here in slightly different words: Definition.The circumference of a circle is a plane curve such that all points in it have the same distance from a fixed point in the plane. This point is called the centre of the circle.

Of the new definitions, of which eleven are given at the beginning of the third book, a few only require special mention. The first,, which says that circles with equal radii are equal,is in part a theorem, but easily proved by applying the one circle to the other. Or it may be considered proved by aid of Prop. 24, equal circles not being used till after this theorem.

In the second definition is explained what is meant by a line which touches a circle. Such a line is now generally called a tangent to the circle. The introduction of this name allows us to state many of Euclids propositions in a much shorter form.

For the same reason we shall call a straight line joining two points on the circumference of a circle a chord.

Definitions 4 and 5 may be replaced with a slight generalization by the following:- Definition.By the distance of a point from a line is meant the length of the perpendicular drawn from the point to the line.

27. From the definition of a circle it follows that every circle has a centre. Prop. 1 require~s to find it when the circle is given, i.e. when its circumference is drawn. -

To solve this problem a chord is drawn (that is, any two points in the circumference are joined), and through the point where this is bisected a perpendicular to it is erected. Euclid then proves, first, that no point off this perpendicular can be the centre, hence that the centre must lie in this line; and, secondly, that of the points on the perpendicular one only can be the centre, viz, the one which bisects the parts of the perpendicular bounded by the circle. In the second part Euclid silently assumes that the perpendicular there used does cut the circumference in two, and only in two points. The proof therefore is incomplete. The proof of the first part, however, rs exact. By drawing two non-parallel chords, and the perpendiculars which bisect them, the centre will be found as the point where these perpendiculars intersect.

28. In Prop. 2 it is proved that a chord of. circle lies altogether within the circle.

What we have called the first part of Euclids solution of Prop. I may be stated as a theorem: Every straight line which bisects a chord, and is at right angles to it, passes through the centre of the circle.

The converse to this gives Prop. 3, which may be stated thus: If a straight line through the centre of a circle bisect a chord, then it is perpendicular to the chord, and if it be perpendicular to the chord it bisects it.

An easy consequence of this is the following theorem, which is essentially the same as Prop. 4: Two chords of a circle, of which neither passes through the centre, cannot bisect each other.

These last three theorems are fundamental for the theory of the circle. It is to be remarked that Euclid never proves that a straight line cannot have more than two points in common with a circumference.

29. The next two propositions (5 and 6) might be replaced by a single and a simpler theorem, viz: Two circles which have a common centre, and whose circumferences have one point in common, coincide.

Or, more in agreement with Euclids form: Two different circles, whose circumferences have a point in common, cannot have the same centre.

That Euclid treats of two cases is characteristic of Greek mathematics.

The next two propositions (7 and 8) again belong together. They may be combined thus: If from a point in a plane of a circle, which is not the centre, straight lines be drawn to the different points of the circumference, then of all these lines one is the shortest, and one the longest, and these lie both in that straight line which joins the given point to the centre, Of all the remaining lines each is equal to one and only one other, and these equal lines lie on opposite sides of the shortest or longest, and make equal angles with them.

Euclid distinguishes the two cases where the given point lies within or without the circle, omitting the case where it lies in the circumference.

From the last proposition it follows that if from a point more than two equal straight lines can be drawn to the circumference, this point must be the centre. This is Prop. 9.

As a consequence of this we get If the circumferences of the two circles have three points in common they coincide.

For in this case the two circles havefl a common centre, because from the centre of the one three equal Lines can be drawn to points on the circumference of the other. But two circles which have a common centre, and whose circumferences have a point in common, coincide. (Compare above statement of Props. 5 and 6.)

This theorem may also be stated thus: Through three points only one circumference may be drawn; or, Three points determine a circle.

Euclid does not give the theorem in this form. He proves, however, that the two circles cannot cut another in more than two points (Prop. 10), and that two circles cannot touch one another in more points than one (ProD. 13).

30. Propositions II and 12 assert that if two circles touch, then the point of contact lies on the line joining their centres. This gives two propositions, because the circles may touch either internally or externally.

31. Propositions 14 and 15 relate to the length of chords. The first says that equal chords are equidistant from the centre, and that chords which are equidistant from the centre are equal; Whilst Prop. 15 compares unequal chords, viz. Of all chords the diameter is the greatest, and of other chords that is the greater which is nearer to the centre; and conversely, the greater chord is nearer to the centre.

32. In Prop. 16 the tangent to a circle is for the first time introduced. The proposition is meant to show that the straight line at the end point of the diameter and at right angles to it is a tangent.

The proposition itself does not state this. It runs thus: Prop. 16. The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and no straight line cau be drawn from the extremity, between that straight line and the circumference, so as not to cut the circle.

Cbrollary.The straight line at right angles to a diameter drawn through the end point of it touches the circle.

The statement of the proposition and its whole treatment show the difficulties which the tangents presented to Euclid.

Prop. 17 solves the problem through a given point, either in the circumference or without it, to draw a tangent to a given circle.

Closely connected with Prop. 16 are Props. 18 and 19, which state (Prop. i 8), that the line joining the centre of a circle to the point of contact of a tangent is perpendicular to the tangent; and conversely (Prop. 19), that the straight line through the point of contact of, and perpendicular to, a tangent to a circle passes through the centre of the circle.

33. The rest of the book relates to angles connected with a circle, viz, angles which have the vertex either at the centre or on the circumference, and which are called respectively angles at the centre and angles at the circumference. Between these two kinds of angles exists the important relation expressed as follows Prop. 20. The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc.

This is of great importance for its consequences, of which the two following are the principal: Prop. 21. The angles in the same segment of a circle are equal to one another; Prop. 22. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.

Further consequences are: Prop. 23. On the same straight line, and on the same side of it, there cannot be two similar segments of circles, not coinciding with one another; Prop. 24. Similar segments of circles on equal straight lines are equal to one another.

The problem Prop. 25. A segment of a circle being given to describe the circle of which it is a segment, may be solved much more easily by aid of the construction described in relation to Prop. 1, III., in 27.

34. There follow four theorems connecting the angles at the centre, the arcs into which they divide the circumference, and the chords subtending these arcs. They are expressed for angles, arcs and chords in equal circles, but they hold also for angles, arcs and chords in the same circle.

The theorems are: Prop. 26. In equal circles equal angles stand on equal arcs, whether they be at the centres or circumferences; Prop. 27. (converse to Prop. 26). In equal circles the angles which stand on equal arcs are equal to one another, whether they be at the centres or the circumferences; Prop. 28. In equal circles equal straight lines (equal chords) cut off equal arcs, the greater equal to the greater, and the less equal to the less; Prop. 29 (converse to Prop. 28). In equal circles equal arcs are subtended by equal straight lines.

i5. Other important consequences of Props. 20-22 are: Prop. 31. In a circle the angle in a semicircle is a right anile; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle; Prop. 32. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which this line makes with the line touching the circle shall be equal to the angles which are in the alternate segments of the circle.

~6. Propositions 30, 33, 34, contain problems which are solved by aid of the propositions preceding them: Prop. 30. To bisect a given arc, that is, to divide it into two equal parts; Prop. 33. On a given straight line to describe a segment of a circle containing an angle equal to a given rectilineal angle; Prop. 34. From a given circle to cut off a segment containing an angle equal to a given rectilineal angle.

i~. If we draw chords through a point A within a circle, they will each be divided by A into two segments. Between these segments the law holds that the rectangle contained by them has the same area on whatever chord through A the segments are taken. The value of this rectangle changes, of course, with the position of A.

A similar theorem holds if the point A be taken without the circle. On every straight line through A, which cuts the circle in two points B and C, we have two segments AB and AC, and the rectangles contained by them are again equal to one another, and equal to the square on a tangent drawn from A to the circle.

The first of these theorems gives Prop. 35, and the second Prop. 36, with its corollary, whilst Prop. 37, the last of Book III., gives the converse to Prop. 36. The first two theorems may be combined in one: If through a point A in the plane of a circle a straight line be drawn cutting the circle in B and C, then the rectangle AB.A C has a constant value so long as the point A be fixed; and if from A a tangent AD can be drawn to the circle, touching at D, then the above rectangle equals the square on AD.

Prop. 37 may be stated thus: If from a point A without a circle a line be drawn cutting the circle in B and C, and another line to a point D on the circle, and AB.A C= AD2, then the line AD touches the circle at D.

It is not difficult to prove also the converse to the general proposition as above stated. This proposition and its converse may be expressed as follows: If four points A BCD be taken on the circumference of a circle, and if the lines AB, CD, produced if necessary, meet at E, then EA.EB =EC.ED;

and conversely, if this relation holds then the four points lie on a circle, that is, the circle drawn through three of them passes through the fourth.

That a circle may always be drawn through three points, provided that they do not lie in a straight line, is proved only later on in Book IV.

BooK IV.

38. The fourth book contains only problems, all relating to the construction of triangles and polygons inscribed in and circumscribed about circles, and of circles inscribed in or circumscribed about triangles and polygons. They are nearly all given for their own sake, and not for future use in the construction of figures, as are most of those in the former books. In seven definitions at the beginning of the book it is explained what is understood by figures inscribed in or described about other figures, with special reference to the case where one figure is a circle. Instead, however, of saying that one figure is described about another, it is now generally said that the one figure is circumscribed about the other. We may then state the definitions 3 or 4 thus: Definition.A polygon is said to be inscribed in a circle, and the circle is said to he circumscribed about the polygon, if the vertices of the polygon lie in the circumference of the ircle.

And definitions 5 and 6 thus: Definition.A polygon is said to be circumscribed about a circle, and a circle is said to be inscribed in a polygon, if the sides of the. polygon are tangents to the circle.

,p~. The first problem is merely constructive. It requires to draw in a given circle a chord equal to a given straight tine, which is not greater than the diameter of the circle. The problem is not a determinate one, inasmuch as the chord may be drawn from any point in the circumference. This may be said of almost all problems in this book, especially of the next two. They are: Prop. 2. In a given circle to inscribe a triangle equiangular to a given triangle; -

Prop. 3. About a given circle to circumscribe a triangle equsangular to a given triangle.

40. Of somewhat greater interest are the next problems, where the triangles are given and the circles to be found.

Prop. 4. To inscribe a circle in a given triangle.

The result is that the problem has always a solution, viz, the centre of the circle is the point where the bisectors of two of the interior angles of the triangle, meet. The solution shows, though Euclid does not state this, that the problem has but one solution; and also, The three bisectors of the interior angles of any triangle meet in a point, and this is the centre of the circle inscribed in the triangle.

The solutions of most of the other problems contain also theorems. Of these we shall state those which are of special interest; Euclid does not state any one of them.

41. Prop. 5. To circumscribe a circle about a given triangle. The one solution which always exists contains the following: The three straight lines which bisect the sides of a triangle at right angles meet in a point, and this point is the centre of the circle circumscribed about the triangle.

Euclid adds in a corollary the following property: The centre of the circle circumscribed about a triangle lies within, on a side of, or without the triangle, according as the triangle is acute-angled, right-angled or obtuse-angled.

42. Whilst it is always possible to draw a circle which is inscribed in or circumscribed about a given triangle, this is not the case with quadrilaterals or polygons of more sides. Of those for which this is possible the regular polygons, i.e. polygons which have all their sides and angles equal, are the most interesting. In each of them a circle may be inscribed, and another may be circumscribed about it.

Euclid does not use the word regular, but he describes the polygons in question as equiangular and equilateral. We shall use the name regular polygon. The regular triangle is equilateral, the regular quadrilateral is the square.

Euclid considers the regular polygons of 4, 5, 6 and 15 sides. For each of the first three he solves the problems(1) to inscribe such a polygon in a given circle; (2) to circumscribe it about a given circle; (3) to inscribe a circle iii, and (4) to circumscribe a circle about, such a polygon.

For the regular triangle the problems are not repeated, because more general problems have been solved.

Props. 6, ~, 8 and 9 solve these problems for the square.

The general problem of inscribing in a given circle a regular polygon of n sides depends upon the problem of dividing the circumference of a circle into n equal parts, or what comes to the same thing, of drawing from the centre of the circle n radii such that the angles between consecutive radii are equal, that is, to divide the space about the centre into n equal angles. Thus, if it is required to inscribe a square in a circle, we have to draw four lines from the centre, making the four angles equal. This is done by drawing two diameters at right angles to one another. The ends of these diameters are the vertices of the required square. If, on the other hand, tangents be drawn at these ends, we obtain a square circum~cribed about the circle.

43. To construct a regular pentagon, we find it convenient first to construct a regular decagon. This requires to divide the space about the centre into ten equal angles. Each will be fieth of a right angle, or lth of two right angles. If we suppose the decagon constructed, and if we join the centre to the end of one side, we get an isosceles triangle, where the angle at the centre equals 1/2th of two right angles; her,ce each of the angles at the base will be iths of two right angles, as all three anglea together equal two right -angles. Thus we have to construct an isosceles triangle, having the angle at the vertex equal to half an angle at the base. This is solved in Prop. 10, by aid of the problem in Prop. II of the second book. If we make the sides of this triangle equal to the radius of the given circle, then the base will be the side, of the regular decagon inscribed in tile circle. This side being known the decagon can be constructed, and if the vertices are joined alternately, leaving out half their number, we obtain the regular pentagon. (Prop. If.)

Euclid does not proceed thus. He wants the pentagon before the decagon. This, however, does not change the real flature of his solution, nor does his solution become simpler by not mentioning the decagon.

Once the regular pentagon is inscribed, it is easy to circumscribe another by drawing tangents at the vertices of the inscribed pentagon. This is shown in Prop. 12.

Props. 13 and 14 teach how a circle may be inscribed in or circumscribed about any given regular pentagon.

44. The regular hexagon is more easily constructed, as shown in Prop. 15. The result is that the side of the regular hexagon inscribed in a circle is equal to the radius, of the circle.

For this polygon the other three problems mentioned are not solved.

45. The book closes with Prop. 16. To inscribe a regular quindecagon in a given circle. If we inscribe a regular pentagon and a regular hexagon in the circle, having one vertex in common, then the arc from the common vertex to the next vertex of the pentagon is 1/2th of the circumference, and to the next vertex of the hexagon is lth of the circumference. The difference between these arcs is, therefore, 1/2, 11-sth of the circumference. The latter may, therefore, be divided into thirty, and hence also in fifteen equal parts, and the regular quindecagoa be described.

46. We conclude with a few theorems about regular polygons which are not given by Euclid.

The straight lines perpendicular to and bisecting the sides of any regular polygon meet in a point. The straight lines bisecting, the angle,s in the regular polygon meet in the sense point. This point is the centre of the circles circumscribed about and inscribed in the regular polygon.

Wecan bisect any given arc (Prop. 30, III.). Hence we can divide a circumference into 2n equal parts as soon as it has been divided into n equal parts, or as soon as a regular polygon of n sides has been constructed. Hence If a regular polygon of n sides has been constructed, then a regular polygon of 2n sides, of 40, of 8n sides, &c., may also be constructed. Euclid shows how to construct regular polygons of 3, 4, 5 and 15 sides. It follows that we can construct regular polygon.s of 3, 6, 12, 24.. - sides 4, 8, 16, 32...,,

5, 10, 20, 40...,,

15, 30, 60, 120

The construction of anynew regular polygon not included in one of these series will give rise to a new series. Till the beginning of the 19th century nothing was added to the knowledge ofregular polygons as given by Euclid. Then Gauss, in his celebrated Arithmetic, proved thatevery regular polygon of 2+I sides may be cOnstructed if this number 2+f be prime, and that no others except those with 2(2+1) sides can be constructed by elementary methods. This shows that regular polygons of 7, 9, 13 sides cannot thus be constructed, but that a regular polygon of 1~ sides ispossible; for 17=2i-i-I. The next polygon isone of 257 sides. The construction becomes already rather complicated for I 7 sides.

BOOK V~

4~. The fifth book of the Elements is not exclusively geometrical. It contains the theory of ratios and proportion of quantities, in general. The treatment, as here given, is admirable, and in every respect superior to the algebraical method by which Euclids theory is now generally replaced. We shall treat the subject in1order to show why the usual algebraical treatment of proportion is not really sound. We begin by quoting those definitions at the beginning of Book V. -which are most important. These definitions have given rise to much discussion, The only definitions which are essential for the fifth book are Defs. I, 2, 4, 5, 6 and 7. Of the remainder 3, 8 and 9 are more than useless, and probably not Euclids, but additions of later editors, of whom Theon of Alexandria was the most prominent. Defs. 10 and II belong rather to the sixth book, whilst all the others are merely nominal. The really important ones are 4,5, 6 and 7.

48. To define a magnitude is not attempted by Euclid. The first two definitions state what is meant by a part, that is, a submultiple or measure, and by a ~ multiple. Of .a given magnitude. The meaning of Def. 4 is that two given quantities can have a ratio to one another only in case that they are comparable as to their magnitude, that is, if they are of the same kind.

Def. 3, which is probably due to Theon, professes to define a ratio, but is as meaningless as it is uncalled for, for all that is wanted is given in Defs. 5 and 7.

In Def. 5 it is explained what is meant by saying that two magnitudes have the same ratio to one another as two other magnitudes, and it a>c, then d>f, but if a=c, then d=f, and if a

By aid of these two propositions the following two are proved.

55. Prop. 22. If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in order, the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last.

We may state it more generally, thus:

If a:b:c:d:e:...~a:b:c:d:e:...,

then not only have two consecutive, but any two magnitudes on the first side, the same ratio as the corresponding magnitudes on the other. For instance a: c=a: c; b: e=b: e, &c.

Prop. 23 we state only in symbols, viz.:

.1.. ..i _I,I I I I

a.v.c ~

then a: c=c: a, b: e=e: b, and so on.

Prop. 24 comes to this: If a: b=c: d and e: b=f: d, then a+e: b=c+f: d.

Some of the proportions which are considered in the above propositions have special names. These we have omitted, as being of no use, since algebra has enabled us to bring the different operations contained in the propositions under a common point of view.

56. The last proposition in the fifth book is of a different character.

Prop. 25. If four magnitudes of the same kind be proportional, the greatest and least of them together shall be greater than the other two together. In symbols If a, b, c, d be magnitudes of the same kind, and if a: b=c: d, and if a is the greatest, hence d the least, then a+d> b+c.

5~. We return once again to the question, What is a ratio? We have seen that we may treat ratios as magnitudes, and that all ratios are magnitudes of the same kind, for we may compare any two as to their magnitude. It will presently be shown that ratios of lines may be considered as quotients of lines, so that a ratio appears as answer to the question, How often is one line contained in another? But the answer to this question is given by a number, at least in some cases, and in all cases if we admit incommensurable numbers. Considered from this point of view, we may say the fifth book of the Elements shows that some of the simpler algebraical operations hold for incommensurable numbers. In the ordinary algebraical treatment of numbers this proof is altogether omitted, or given by a process of limits which does not seem to be natural to the subject.

BooK VI.

58. The sixth book contains the theory of similar figures. After a few definitions explaining terms, the first proposition gives the first application of the theory of proportion.

Prop. I. Triangles and parallelograms of the same altitude are to cne another as their bases.

The proof has already been considered in 49.

From this follows easily the important theorem Prop. 2. If a straight line be drawn parallel to one of the sides of a triangle it shall cut the other sides, or those sides produced, proportionally; and if the sides or the sides produced be cut proportionally, the sirai~ht Line which joins the points of section shall be parallel to the remaining side of the triangle.

59. The next proposition, together with one added by Simson as Prop. A, may be expressed more conveniently if we introduce a modern phraseology, viz, if in a line AB we assume a point C between A and B, we shall say that C divides AB internally in the ratio AC: CB; but if C be taken in the line AB produced, we shall say that AB is divided externally in the ratio AC: CB.

The two propositions then come to this:

Prop. 3. The bisector of an angle in a triangle div-ides the opposite side internally in a ratio equal to the ratio of the two sides including that angle; and convetsely, if a line through the vertex of a triangle divide the base internally in the ratio of the two other sides, then that line bisects the angle at the vertex.

Simsons Prop. A. The line which bisects an exterior angle of a triangle divides the opposite side externally in the ratio of the other sides; and conversely, if a line through the vertex of a triangle divide the base externally in the ratio of the sides, then it bisects an exterior angle at the vertex of the triangle. -

If we combine both we have The two lines which bisect the interior and exterior angles at one vertex of a triangle divide the opposite side internally and externally in the same ratio, viz, in the ratio of the other two sides.

60. The next four propositions Contain the theory of similar triangles, of which four cases are considered. They may be stated together.

Two triangles are similar, 1. (Prop. 4). If the triangles are equiangular:

2. (Prop. 5). If the sides of the one are proportional to those of the other; 3. (Prop. 6). If two sides in one are proportional to two sides in the other, and if the angles contained by these sides are equal; 4. (Prop. 7). If two sides in one are proportional to two sides in the other, if the angles opposite homologous sides are equal, and if the angles opposite the other homologous sides are both acute, both right or both obtuse, homologous sides being in each case those which are opposite equal angles.

An important application of these theorems is at once made to a right-angled triangle, viz.: Prop. 8. In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.

Corollary.From this it is manifest that the perpendicular drawn from the right angle of a right-angled triangle to the base is a mean proportional between the segments of the base, and also that each of the sides is a mean proportional between the base and the segment of the base adjacent to that side.

61. There follow four propositions containing problems, in language slightly different from Euclids, viz.: Prop. 9. To divide a straight line into a given number of equal parts.

Prop. 10. To divide a straight line in a given ratio.

Prop. II. To find a third proportional to two given straight lines.

Prop. 12. To find a fourth proportional to three given straight lines.

Prop. 13. To find a mean proportional between two given straight lines.

The last three may be written as equations with one unknown quantityviz, if we call the given straight lines a, b, c, and the required line x, we have to find a line x so that Prop. 1I a: bb,: x; Prop. 12. a: b=c: x; Prop.13. a:x=x:b.

We shall see presently how these may be written without the signs of ratios.

62. Euclid considers next proportions connected with parallelograms and triangles which are equal in area.

Prop. 14. Equal parallelograms which have one angle of the one equal to one angle of the other have their sIdes about the equal angles reciprocally proportional; and parallelograms which have one - angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.

Prop. 15. Equal triangles which have one angle of the one equal to one angle of the other, have their sides about the equal angles reci procally proportional; and triangles which have one angle of the one equal to one angle of the other, and their s-ides about the equal angles reciprocally pro portional, are equal to one another.

The latter p?oposition is really the same as the former, for if, as in the accompanying diagram, in the figure belonging to the A former the two equal parallelograms AB and BC be bisected by the lines DF and EG, and if EF be drawn, we get the D -.. B figure belonging to the latter.

It is worth noticing that the lines FE and DG are parallel. We may state there- fore the theorem G

If two triangles are equal in area, and have one angle in the one vertically opposite to one angle in the other, then the two straight lines which join the remaining two vertices of the one to those of the other triangle are parallel.

63. A most important theorem is Prop. 16. If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means~ and if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines, are pro portionals.

In symbols, if a, b, c, d are the tour lines, and if a:b=c:d, then adbc; and conversely, if ad = bc, then a:b=c:d, where ad and bc denote (as in 20), the areas of the rectangles contained by a and d and by b and c respectively.

This allows us to transform every proportion between four lines into an equation between two products.

It shows further that the operation of forming a product of two lines, and the operation of forming their ratio are each the inverse of the other.

If we now define a quotient ~ of two lines as the number which multiplie4 into b gives a, so that we see that from the equality of two quotients follows, if we multiply both sides by lid, ~b.d=~d.b, ad = cb.

But from this it follows, according to the last theorem, that a: b~c: d.

Hence we conclude that the quotient ~ and the ratio a: b are different forms of the same magnitude, only with this important difference that the quotient ~ would have a meaning only if a and b have a common measure, until we introduce incommensurable numbers, while the ratio a: b has always a meaning, and thus gives rise to the introduction of incommensurable numbers.

Thus it is really the theory of ratios in the filth book which enables us to extend the geometrical calculus given before in connection with Book II. It will also be seen that if we write the ratios in Book V. as quotients, or rather as fractions, then most of the theorems state properties of quotients or of fractions.

64. Prop. 17. If three straight lines are proportional the rectangle contained by the extremes is equal to the square on the mean; and conversely, is only a special case of 16. After the problem, Prop. 18, On a given straight line to describe a rectilineal figure similar and similarly situated to a given rectilinealfigure, there follows another fundamental theorem:

Prop. 19. Similar triangles are to one another in the duplicate ratio of their homologous sides. In other words, the areas of similar triangles are to one another as the squares on homologous sides. This is generalized in:

Prop. 20. Similar polygons may be divided into the same number of similar triangles, having the same ratio to one another that the polygons have; and the polygons are to one another in the duplicate ratio of their homologous sides.

• 65. Prop. 21. Rectilineal figures which are similar to the same rectilinea,l figure are also similar to each other, is an immediate consequence of the definition of similar figures. As similar figures may be said to be equal in shape but not in size, we may state it also thus:

Figures which are equal in shape to a third are equal in shape to each other.

Prop. 22. If four straight lines be proportionals, the similar rectilineal figures similarly described on them shall also be pro portionals; and if the similar rectilineal figures similarly described on four straight lines be pro portionals, those straight lines shall be proportionals.

This is essentially the same as the following:

if a :b~c :d, then a2: b2c2: d2.

66. Now follows a proposition which has been much discussed with regard to Euclids exact meaning in saying that a ratio is corn pounded of two other ratios, viz.:

Prop. 23. Parallelograms which are equiangular to one another, have to one another the ratio which is compounded of the ratios of thel, sides.

The proof of the proposition makes its meaning clear. In symbols the ratio a: c is com,pounded of the two ratios a: b and b: c, and ii a: b=a: b, b: cb: c~, then a: c is compounded of a: b anc If we consider the ratios as numbers, we may say that the om ratio is the product of those of which it is compounded, or in symbols a ab ab.a a b b ~==~i~w, if ~ and The theorem in Prop. 23 is the foundation of all mensuration 0 areas. From it we see at once that two rectangles have the rath of their areas compounded of the ratios of their sides.

If A is the area of a rectangle contained by a and b, and B tha~ of a rectangle contained by c and d, so that A=ab, B=cd, thet A: B =ab: cd, and this is, the theorem says, compounded of th~ ratios a: c and b: a. In forms of quotients, a b ab, cdc~

This shows how to multiply quotients in our geometrical calculus Further, Two triangles have the ratios of their areas com~oundei of the ratios of their bases and their altitude. For a triangle is equa in area to half a parallelogram which has the same base and th same altitude.

67. To bring these theorems to the form in which they are usuall given, we assume a straight line u as our unit of length (generallan inch, a foot, a mile, &c.), and determine the number a whici expresses how often u is contained in a line a, so that a denotes th ratio a: u whether commensurable or not, and that a=au. W

call this number a the numerical value of a. If in the same manner $ be the numerical value of a line b we have a: b=a :

in words: The ratio of two lines (and of two like quantities in general) is equal to that of their numerical values.

This is easily proved by observing that a = au, b = $u, therefore a: b=au: $u, and this may without difficulty be shown to equal a:$.

If now a, b be base and altitude of one, a, b those of another parallelogram, a, $ and a, $ their numerical values respectively, and A, A their areas, then A a b a$ afl~

~=~ni~=~i7 ~

In words: The areas of two parallelograms are to each other as the products of the numerical values of their bases and altitudes.

If especially the second parallelogram is the unit square, i.e. a square on the unit of length, then a=$=I, A~u2, and we have or A~a$.u2.

This gives the theorem: The number of unit squares contained in a parallelogram equals the product of the numerical values of base and altitude, and similarly the number of unit squares contained in a triangle equals half the product of the numerical values of base and altitude.

This is often stated by saying that the area of a parallelogram is equal to the product of the base and the altitude, meaning by this product the product of the numerical values, and not the product as defined above in 20.

68. Propositions 24 and 26 relate to parallelograms about diagonals, such as are considered in Book I., 43. They are Prop. 24. Parallelograms about the diameter of any parallelogram are similar to the whole parallelogram and to one another; and its converse (Prop. 26), If two similar parallelograms have a common angle, and be similarly situated, they are about the same diameter.

Between these is inserted a problem.

Prop. 25. To describe a rectilineal figure which shall be similar to one given rectilinear figure, and equal to another given rectilineal figure.

69. Prop. 27 contains a theorem relating to the theory of maxima and minima. We may state it thus:

Prop. 27. If a parallelogram be divided into two by a straight line cutting the base, and if on half the base another parallelogram be constructed similar to one of those parts, then this third parallelogram is greate? than the other part.

Of far greater interest than this general theorem is a special case of it, where the parallelograms are changed into rectangles, and where one of the parts into which the parallelogram is divided is made a square; for then the theorem changes into one which is easily recognized to be identical with the following: Of all ectangles which have the same perimeter the square has the greatest area.

This may also be stated thus: Of all rectangles which have the same area the square has the least perImeter.

70. The next three propositions contain problems which may be said to be solutions of quadratic equations. The first two are, like the last, involved in somewhat obscure language. We transcribe them as follows:

Problem.To describe on a given base a parallelogram, and to divide it either internally (Prop. 28) or externally (Prop. 29) from a point on the base into two parallelograms, of which the one has a given size (is equal in area to a given figure), whilst the other has a given shape (is similar to a given parallelogram).

If we express this again in symbols, calling the given base a, the one part x, and the altitude y, we have to determine x and y in the first case from the equations (ax)y =

,~1 being the given size of the first, and p and q the base and altitude of the parallelogram which determine the shape of the second of th~

required parallelograms.

If we substitute the value of y, we get or, (ax)x=~

axx2

where a and b are known quantities, taking b2 = s...

The second case (Prop. 29) gives rise, in the same manner, to th~ quadratic ax-{-x2=b1.

The next problem Prop. 30. To cut a given straight line in extreme and mean ratio - leads to the equation ax+x2=a2.

This is, therefore, only a special case of the last, and is, besides, an old acquaintance, being essentially the same problem as that proposed in II. ii.

Prop. 30 may therefore be solved in two ways, either by aid of Prop. 29 or by aid of II. 11. Euclid gives both solutions.

71. Prop. 31 (Theorem). In any right-angled triangle, any rectilineal figure described on the side subtending the right angle is equal to the similar and similarly-described figures on tile sides containing the right angleis a pretty generalization of the theorem of Pythagoras (1.47).

Leaving out the next proposition, which is of little interest, we come to the last in this book.

Prop. 33. In equal circles angles, whether at the centres or the circumferences, have the same ratio which the arcs on which they stand have to one another; so also have the sectors.

Of this, the part relating to angles at the centre is of special importance; it enables us to measure apgles by arcs.

With tlus closes that part of, the Elements which is devoted to the study of figures in a plane.

BOOK XI.

72. In this book figures -are considered which are not confined to a plane, viz, first relations between lines and planes in space, and af terwards properties of solids.

Of new definitions we mention those which relate to the perpendicularity and the inclination of lines and planes.

Def. 3. A straight line is perpendicular, or at right angles, to a plane when it makes right angles with every straight tine meeting it in that plane.

The definition of perpendicular planes (Def. 4) offers no difficulty. Euclid defines the inclination of lines to planes and of planes to planes (Defs. 5 and 6) by aid of plane angles, included by straight lines, with which we have been made familiar in the first books.

The other important definitions are those of parallel planes, which never meet (Def. 8), and of solid angles formed by three or more planes meeting in a point (Def. 9).

To these we add the definition of a line parallel to a plane as a line which does not meet the plane.

73. Before we investigate the contents of Book XI., it will be well to recapitulate shortly what we know of planes and lines from the definitions and axioms of the first book. There a plane has been defined as a surface which has the property that every straight line which joins two points in it lies altogether in it. This is equivalent to saying that a straight line which has two points in a plane has all points in the plane. Hence, a straight line which does not lie in the plane cannot have more than one point in common with the plane. This is virtually the same as Euclids Prop. I, viz.: Prop. I. One part of a straight line cannot be in a. plane and another part lfrithout it.

It also follows, as was pointed out in 3, in discussing the definitions of Book I., that a plane is determined already by one straight line and a point without it, viz, if all lines be drawn through the point, and cutting the line, they will form a plane.

This may be stated thus: A plane is determined 1st, By a straight line and a point which does not lie on it; 2nd, By three pointl which do not lie in a straight line; for if two of these points be joined by a straight line we have case I;

3rd, By two intersecting straight lines; for the point of intersection and two other points, one in each line, give case 2;

4th, By two parallel lines (Def. 35, I.).

The third case of this theorem is Euclids Prop. 2. Two straight lines which cut one another are in one plane, and three straight lines which meet one another are in one plane.

And the fourth is Euclids Prop. 7. If two straig lit lines be parallel, the straight line drawn from any point in one to any point in the other is in the same plane with the parallels. From the definition of a plane further follows Prop. 3. If two planes cut one another, their common section is a straight line.

74. Whilst these propositions are virtually contained in the definition of a plane, the next gives us a new and fundamental property of space, showing at the same time that it is possible to have a straight line perpendicular to a plane, according to Def. 3. It states -

Prop. 4. If a straight line is perpendicular to two straight lines in a plane which it meets, then it is perpendicular to all lines in the plane which it meets, and hence it is perpendicular to the plane.

Def. 3 may be stated thus: If a straight line is perpendicular to a plane, then it is perpendicular to every line in the plane which it meets. The converse to this would be -

All straight lines which meet a given straight line in the same point, and are perpendicular to it, lie in a plane which is perpendicular to that tine.

This Euclid states thus: - -

Prop. 5. If three straight lines meet all at one point, and a straight line stands at right angles to each of them at that point, the three straight lines shalt be in one and the same plane.

75. There follow theorems relating to the theory of parallel lines in space, viz.:

Prop. 6. Any two lines which are perpen1i~cular to the same plane are parallel to each other; and Conver~ely Prop. 8. If of two parallel straight lines one is perpendicular to a plane, the other is so also. -

Irop. 7. If two straight lines are parallel, the straight line whieh joins any point in one to any point in tile other is in the same plane as the parallels. (See above, 73.)

Prop. 9. Two straight lines which are each of them parallel to the same straight line, and not in the same plane with it, are parallel to one another; where the words, and not ifi the same plane with it, may be omitted, for they exclude the case of three parallels in a plane, which has been proved before; and Prop. 10. If two angles in diffeicent planes have the two limits of the one parallel to those of the other, then the angles are equal. That their planes are parallel is shown iatet oiiinPt-op. 15. n Tins theorem is not necessarily true, for the angles in question may be supplementary; but thed the one angle will be equal td that which is adjacent and supplementary to the other, and this, latter angle will also have its limits parallel to, those of the first.

From this theorem it follows that if we take any two straight lines in space which do not meet, and if we draw through any point P in space two lines~parallel to them, then the angle included by these lines will always be the same, whatever the position of th& point P may be. This angle has in modern times been called the angle between the given lines: By the angles between two not intersecting lines we understand the angles which two intersecting lines include that are parallel respectively to the two given lines. ..

76. It is now possible to solve the following two problems: To draw a straight line perpendicular to ~a given plane frirn a given point which lies 1. Not in the plane (Prop. 11).

2. In the plane (Prop. 12).

The second case is easily reduced to the firstviz, if by aid of the first we have drawn any perpendicular to the plane from some point without it, we need only draw through the ~1ven point in the plane a line parallel to it, in order to have the required perpendicular given. The solution of the first part is of ipterest in itself. It depends upon a construction which may be expressed as a theorem.

If from a point A without a plane a perpendicular AB be drawn to the plane, and if from the foot B of this perpendicular another, perpendici~1ar BC be drawn to any straight line in the plane, then the st~aight,line. joining A to the foot C of this second perpendicular will also, be perpendicular to the line In the plane. -

The theory of perpendiculars to a plane is, concluded by the theorem Prop. 13. Through any point in space,. whether in. or without ,a plane, only one straight line can be drawn perpendicular to the plane.,

77. The next four propositions treat of parallel~planes. It, is shown that planes which have a common perpendicular are parallel (Prop. 14); that two planes are parallel, if, two intersecting straight lines in the one are parallel respectively to two straight lines in the other plane (Prop. 15); that parallel planes are cut by any plane in parallel straight lines (Prop. 16); and lastly, that any two straight lines are cut proportionally by a series of parallel planes (Piop. 17).

This theory is made more complete by adding the following theorems, which are easy deductions from the last: Two parallel planes have common perpendiculars (converse to 14); and Two planes which are parallel to a third plane are parallel to each other.

It will be noted that Prop. 15 at once allows of the solution pf the problem: Through a given point to draw a planeparalll to a given plane. And it is also easily proved that,this problem allows always of one, and only of one, solution 78. We come now to planes which are perpendicular to one, another. Two theorems relate to them.

Prop. 18. If a straight line be at right angles to a plane, eqery plane which passes through it shall be at right angles to that plane.

Prop. 19. If two planes which cut one another be each of them perpendicular to a third plane, their common section shall be perpendicular to the same plane.

79. If three planes pass through a common point, and if they bound each other a solid angle of three faces or a trihedral angie, is formed and similarly by more plmnes a solid angle of more faces or a polyhedral angle. These have many properties which are quite analogous to those of triangles and polygons in, a plane. Euclid states some, viz. :

Prop. 20. If a solid angle be contained by three plane angles, any two of them are together greater than the third.

But the next Prop. 21. Every solid angle is contained by plane angles, which are together less than four right angleshas no analogous theorem in the plane. 1~ ,

We may mention, however, that the theorems about triangles contained in the propositions of Book I., which do not depend upon the theory of parallels (that is all up to Prop. 27), have their corresponding theorems about trihedral angles. The latter are formed, if for side of a triangle we write plane - angle or face of trihedral angle, and for angle of triangle we substitute angle between two faces where theplanes ~ontaining the solid angle are called its faces. We get, for instance, from I. 4, the theorem, If two trihedral angles have the angles of two faces in the one equal to the angles of two faces in the other, and have likewise the angles included by these faces equal, then the angles in the remaining faces are equal, and the angles between the other faces are equal each to each, viz. those which are opposite equalfaces. The solid angles themselves are not necessarily equal, for they may be only symmetrical like the right hand and the left.

The connection indicated between triangles and trihedral angles will also be recognized in Prop. 22. If every two of three plane angles be greater than the third, and if the straight lines which contain them be all equal, a triangle may be made of the straight lines that join the extremities of those equal straight lines.

And Prop. 23 solves the problem, To construct a trihedral angle having the angles of its faces equal to three given plane angles, any two of 1/tern being greater than the third. It is, of course, analogous to the problem of constructing a triangle having its sides of giyen length.

Two other theorems of this kind are added by Simson in his edition of Euclids Elements.

80. These are the principal properties of lines and planes in space, but before we go on to their applications it will be well to define the word distance. In geometry distance means always shortest distance; viz, the distance of a point from a straight line, or from a plane, is the length~ of the perpendicular from the point to the line or plane. The distance between two non-interfecting lines is the length of their common perpendicular, there being but one. The distance between two parallel lines or between two parallel planes is the length of the common perpendicular between the lines or the planes.

81. Parallelepipeds.The rest of the book is devoted to the study of the parallelepiped. In Prop. 24 the possibility of such a solid is proved, viz.: Prop. 24. If a solid be contained by six planes two and two of which are parallel, the opposite planes are similar and equal parallelograms.

Euclid calls this solid henceforth a parallelepiped, though he never defines the word. Either face of it may be taken as base, and its distance from the opposite face as altitude.

Prop. 25. If a solid parallele piped be cut by a plane parallel to two of its opposite planes, it divides the whole into two solids, the base of one of which shall be to the base of the other as the one solid is to the other.

This theorem corresponds to the theorem (VI. I) that parallelograms between the silme parallelsare to one another as their bases. A similar analogy is to be observed among a number of the remaining propositions.

82. After solving a few problems we come to Prop. 28. If a solid parallele piped be cut by a plane passing through tile diagonals of two of the opposite planes, it shall be cut in two equal parts.

In the proof of this, as of several other propositions, Euclid neglects the difference between solids which are symmetrical like the right hand and the left.

Irop. 31. Solid parallelepipeds, which are upon equal bases, and of the same altitude, are equal to one another.

Props. 29 and 30 contain special cases of this theorem leading up to theproof of the general theorem.

As consequences of this fundamental theorem we get Prop. 32. Solid pci-allele pipeds, which have the same altitude, are to one aeother as their bases; and Prop. 33. Sirn7iar solid parallelepipeds are to one another in the triplicate ratio of their homologous sides.

If we consider, as in ~ 67, the ratios of lines as numbers, we may also say The ratio of the volumes of similar parallelepipeds is equal to the ratio of the third powers of homologous sides.

Parallelepipuds which are not similar btit equal are compared by aid of the theorem Prop. 34. The bases and altitudes of equal solid parallelepipeds and reciprocally proportional; and if the bases and altitudes be re~ ciprocally proportional, the solid parailelepipeds are equal.

83. Of the following propositions the 37th and 40th are of special interest, Prop. 37. If four straight lines be pro portionals, the similar solie parallelepipeds. similarly described from them, shall also be pro portio eels; and if the similar parallelepipeds similarly describec from four straight lines be pro portionals, the straight lines shall b~ pro portionals. -

In symbols it says If a: b =c: d, then a1: b=c: d.

Prop. 40 teaches how to compare the volumes of triangulw pri~ms with those of parailelepipeds, by proving that a trianguia~ prism is equal in volume to a parallelepiped, which has its altitud and its base equal to the altitude and the base of the triangula~ prism.

84. From these propositions follow all results relating to tb mensuration of volumes. We shall state these as we did in the cas of areas. The starting-point is the rectangular parallelepiped which has every edge perpendicular to the planes it meets, an which takes the place of the rectangle in the plane. It this has all its edges equal we obtain the cube If we take a certain line u as unit length, then the square on u is the unit of area, and the cube on u the unit of volume, that is to say, if we wish to measure a volume we have to determine how many unit cubes it contains.

A rectangular paralielepiped has, as a rule, the three edges unequal, which meet at a point. Every other edge is equal to one of them. If a, b, c be the three edges meeting at a point, then we may take the rectangle contained by two of them, lay by b and c, as base and the third as altitude. Let V be its volume, V that of another rectangular parallelepiped which has the edges a, b, c, hence the same base as the first. It follows then easily, from Prop. 25 or 32, that V:V=a:a; or in words, Rectangular parallelepipeds on equal bases are proportional to thei, altitudes.

If we have two rectangular parallelepipeds, of which the first has the volume V and the edges a, b, c, and the second, the volume V and the edges a, b, c, we may compare them by aid of two new ones which have respectively the edges a, b, c and a, b, c, and the volumes Vi and Vi. We then have V: Vi = a: a; Vi: Vf = be b, Vi: V = c: c. Compounding these, we have V: V=(a: a) (be b) (C: c), or V abc Va b c Hence, as a special case, making V equal to the unit cube U on u we get Vab c where a, ~9, y are the numerical values of a, b, C; that is, The number of unit cubes in a rectangular parallele piped is equal to the product of the numerical values of its three edges. This is generally expressed by saying the volume of a rectangular parallelepiped is measured by the product of its sides, or by the product of its base into its altittide, which in this case is the same. I

Prop. 31 allows us to extend this to any parallelepipeds, and Props. 28 or 40,to triangular prisms.

The volume of any parallelepiped, or of any triangular prism, is measured by the product of base and altitude.

The consideration that any polygonal prism may be divided into a number of triangular prisms, which have the same altitude and the sum of their bases equal to the base of the polygonal prism, shows further that the same holds for any prism whatever.

BOOK XII.

85. In the last part of Book XI. we have learnt how to compare the volumes of parallelepipeds and of prisms. In order to determine the volume of any solid bounded by plane faces we must determine the volume of pyramids, for every such solid may be decomposed into a number of pyramids.

As every pyramid may again be decomposed into triangular pyramids, it becomes only necessary to determine their volume. This is doae by the Theorem.Every triangular pyramid is equal in volume to one third of a triangular prism having the same base and the same altitude as the pyramid.

This is an immediate consequence of Euclids Prop. 7. Every prism having a triangular base may be divided into three pyramids that have triangular bases, and are equal to one another.

The proof of this theorem is difficult, because the three triangular pyramids into which the prism is divided are by no means equal in shape, and cannot be made to coincide. It has first to be proved that two tiiangular pyramids have eqtial volumes, if they have equal bases and equal altitudes. This Euclid does in the following manner. He first shows (Prop. 3) that a triangular pyramid may be dividedinto four parts, of which two are equal triangular pyramids similar to the whole pyramid, whilst the other two are equal triangular prisms, and further, that these two prisms together are greater than the two pyramids, hence more than half the given pyramid. He next shows (Prop. 4) that if two triangular pyramids are given, having equal bases and equal altitudes, and if each be divided as above, then the two triangular prisms, in the one are equal to those in the other, and each of the remaining pyramids in the one has its base and altitude equal to the base and altitude of the remaining pyramids in the other. Hence to these pyramids the same process is again applicable. We are thus enabled to cut out - of the two given pyramids equal parts, each greater than half the - original pyramid. Of the remainder we can again cut out equal - parts greater than half these remainders, and so on as far as we like. This process may be continued till the last remainder is smallei than any assignable quantity, however small, It follows, so wi should conclude at present, that the two volumes m~st be equal, foi they cannot differ by any assignable quantity.

To Greek mathematicians this conclusion offers far greate~

difficulties. They prove elaborately, by a reductio ad absurduin, that the volumes cannot be unequal. This proof must be read in the Elements. We must, however, state that we have in the above not proved Euclids Prop. 5, but only a special case of it. Euclid does not suppose ,that the bases of the two pyramids to be compared are equal, and hence he proves that the volumes are as the bases. The reasoning of the proof becomes clearer in the special case, from which the general one may be easily deduced.

86. Prop. 6 extends the result to pyramids with polygonal bases. From these results follow again the rules at present given for the mensuration of solids, viz, a pyramid is the third part of a triangular prism having the same base and the same altitude. But a triangular prism is equal in volume to a parallelepiped which has the same base and altitude. Hence if B is the base and h the altitude, we have Volume of prism - = Bh, Volume of pyramid = IBh, statements which have to be taken in the sense that B means the number of square units in the base, /1 the number of units 01 length in the altitude, or that B and h denote the numerical values of base and altitude.

8~. A method similar to that used in proving Prop. 5 leads to the following results relating to solids bounded by simple curved surfaces: Prop. 10. Every cone is the third part of a cylinder which has the same base, and is of an equal altitude with it.

Prop. I I. Cones or cylinders of the same altitude are to one another as their bases. -

Prop. 12. Similar cones or cylinders have to one another the triplicate ratio of that which She diameters of their bases have.

Prop. 3. If a cylinder be cut by a plane parallel to its opposite planes or bases, it divides the cylinder into two cylinders, one of which is to the other as the axis of She first to the axis of the other; which may also be stated thus: Cylinders on the same base are proportional to their altitudes. Prop. 14. Cones or cylinders upon equal bases are to one another as their altitudes.

Prop. 15. The bases and altitudes of equal cones or cylinders are reciprocally proportional, and if the bases and altitudes be reciprocally proportional, the cones or cylinders are equal to one another.

These theorems again lead to formulae in mensuration, if we compare a cylinder with a prism having its base and altitude equal to the base and altitude of the cylinder. This may be done by the method of exhaustion. We get, then, the result that their bases are equal, and have, if B denotes the numerical value of the base, and h that of the altitude, Volume of cylinder = Bh, Volume of cone jBh.

88. The remaining propositions relate to circles and spheres. Of the sphere only one property is proved, viz.: Prop. 18. Spheres have to one another the triplicate ratio of that which their diameters have. The mensuration of the sphere, like that of the circle, the cylinder and the cone, had not been settled in the time of Euclid. It was done by Archimedes.

BooK XIII.

89. The 13th and last book of Euclids Elements is devoted to the regular solids (see POLYHEDRON). It is shown that there are five of them, viz. :

i. The regular tetrahedron, with 4 triangular faces and 4 vertices; 2. The cube, with 8 vertices and 6 square faces; 3. The ockihedron, with 6 vertices and 8 triangular faces; 4. The dodecahedron, with 12 pentagonal faces, 3 at each of the 20 vertices; 5. The icosahedron, with 20 triangular faces, 5 at each of the 12 vertices.

It is shown how to inscribe these solids in a given sphere, and how to determine the lengths of their edges.

90. The 13th book, and therefore the Elements, conclude with the scholium, that no other regular solid exists besides the five ones enumerated.

The proof is very simple. Each face is a regular polygon, hence the angles of the faces at any vertex must be angles in equal regular polygons, must be together less than four right angles (XI. 21), and must be three or more in number. Each angle in a regular triangle equals two-thirds of one right angle. Hence it is possible to forth a solid angle with three, four or five regular triangles or faces. These give the solid angles of the tetrahedron, the octahedron and the icosahedron. The angle in a square (the regular quadrilateral) equals one right angle. Hence three will form a solid angle, that of the cube, and four will not. The angle in the regular pentagon equals ~ of a right angle. Hence three of them equal ljt (i.e. less than 4) right angles, and form the solid angle of the dodecahedron. Three regular polygons of six or more sides cannot form a solid angle. - Therefore no other regular solids are possible. (0. H.)

U. PROJECTIVE GEOMETRY

It is difficult, at the outset, to characterize projective geometry as compared with Euclidean. But a few examples will at least indicate the pra