We may state it more generally, thus:Ifa : 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.:—a : b : c : d : e : ... = 1/a′ : 1/b′ : 1/c′ : 1/d′ : 1/e′ ...,thena : 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, thena + 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.§ 57. 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 asquotientsof 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 theElementsshows 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. 1.Triangles and parallelograms of the same altitude are to one another as their bases.The proof has already been considered in § 49.From this follows easily the important theoremProp. 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 straight 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 divides the opposite side internally in a ratio equal to the ratio of the two sides including that angle;and conversely,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.Simson’s 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 Euclid’s, 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. 11.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 quantity—viz. if we call the given straight lines a, b, c, and the required line x, we have to find a line x so thatProp. 11.a : b = b : 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 reciprocally proportional; and triangles 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.The latter proposition is really the same as the former, for if, as in the accompanying diagram, in the figure belonging to the former the two equal parallelograms AB and BC be bisected by the lines DF and EG, and if EF be drawn, we get the figure belonging to the latter.It is worth noticing that the lines FE and DG are parallel. We may state therefore the theorem—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 isProp. 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 proportionals.In symbols, if a, b, c, d are the four lines, andifa : b = c : d,thenad = bc;and conversely, ifad = bc,thena : 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 a/b of two lines as thenumberwhich multiplied into b gives a, so thatab = a,bwe see that from the equality of two quotientsa=cbdfollows, if we multiply both sides by bd,ab·d =cd·b,bdad = cb.But from this it follows, according to the last theorem, thata : b = c : d.Hence we conclude that the quotient a/b and the ratio a : b are different forms of the same magnitude, only with this important difference that the quotient a/b 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 fifth book which enables us to extend the geometrical calculus given before in connexion 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 rectilineal figure, 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 rectilineal 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 proportionals; and if the similar rectilineal figures similarly described on four straight lines be proportionals, those straight lines shall be proportionals.This is essentially the same as the following:—Ifa : b = c : d,thena² : b² = c² : d².§ 66. Now follows a proposition which has been much discussed with regard to Euclid’s exact meaning in saying that a ratio iscompoundedof 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 their sides.The proof of the proposition makes its meaning clear. In symbols the ratio a : c is compounded of the two ratios a : b and b : c, and if a : b = a′ : b′, b : c = b″ : c″, then a : c is compounded of a′ : b′ and b″ : c″.If we consider the ratios as numbers, we may say that the one ratio is the product of those of which it is compounded, or in symbols,a=a·b=a′·b″, ifa=a′andb=b″.cbcb′c″bb′cc″The theorem in Prop. 23 is the foundation of all mensuration of areas. From it we see at once that two rectangles have the ratio 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 that of a rectangle contained by c and d, so that A = ab, B = cd, then A : B = ab : cd, and this is, the theorem says, compounded of the ratios a : c and b : d. In forms of quotients,a·b=ab.cdcdThis shows how to multiply quotients in our geometrical calculus.Further,Two triangles have the ratios of their areas compounded of the ratios of their bases and their altitude.For a triangle is equal in area to half a parallelogram which has the same base and the same altitude.§ 67. To bring these theorems to the form in which they are usually given, we assume a straight line u as our unit of length (generally an inch, a foot, a mile, &c.), and determine the number α which expresses how often u is contained in a line a, so that α denotes the ratio a : u whether commensurable or not, and that a = αu. We call this number α the numerical value of a. If in the same manner β be the numerical value of a line b we havea : b = α : β;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 = αu, b = βu, therefore a : b = αu : βu, and this may without difficulty be shown to equal α : β.If now a, b be base and altitude of one, a′, b′ those of another parallelogram, α, β and α′, β′ their numerical values respectively, and A, A′ their areas, thenA=a·b=α·β=αβ.A′a′b′α′β′α′β′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 α′ = β′ = 1, A′ = u², and we haveA= αβ or A = αβ·u².A′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 greater 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 rectangles 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(a − x)y = k²,x=p,yqk² 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 the required parallelograms.If we substitute the value of y, we get(a − x)x =pk²,qor,ax − x² = b²,where a and b are known quantities, taking b² = pk²/q.The second case (Prop. 29) gives rise, in the same manner, to the quadraticax + x² = b².The next problem—Prop. 30.To cut a given straight line in extreme and mean ratio, leads to the equationax + 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. 11.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 the sides containing the right angle,—is a pretty generalization of the theorem of Pythagoras (I. 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 angles by arcs.With this closes that part of theElementswhich 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 afterwards 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 line 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 Euclid’s Prop. 1, viz.:—Prop. 1.One part of a straight line cannot be in a plane and another part without 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 points which do not lie in a straight line; for if two of these points be joined by a straight line we have case 1;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 Euclid’sProp. 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 Euclid’sProp. 7.If two straight 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 followsProp. 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 beAll 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 line.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 shall 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 perpendicular to the same plane are parallel to each other;and converselyProp. 8.If of two parallel straight lines one is perpendicular to a plane, the other is so also.Prop. 7.If two straight lines are parallel, the straight line which joins any point in one to any point in the 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 in the same plane with it,” may be omitted, for they exclude the case of three parallels in a plane, which has been proved before; andProp. 10.If two angles in different 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 later on in Prop. 15.This theorem is not necessarily true, for the angles in question may be supplementary; but then the one angle will be equal to 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 the 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 from a given point which lies1.Not in the plane(Prop. 11).2.In the plane(Prop. 12).The second case is easily reduced to the first—viz. 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 given 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 interest 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 perpendicular BC be drawn to any straight line in the plane, then the straight 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 shownthat 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(Prop. 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); andTwo 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 of the problem: “Through a given point to draw a plane parallel 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, every 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 atrihedralangle, is formed, and similarly by more planes a solid angle of more faces, or apolyhedralangle. 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 angles—has no analogous theorem in the plane.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 the planes containing the solid angle are called itsfaces. We get, for instance, from I. 4, thetheorem,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 equal faces.The solid angles themselves are not necessarily equal, for they may be only symmetrical like the right hand and the left.The connexion indicated between triangles and trihedral angles will also be recognized inProp. 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 them being greater than the third.It is, of course, analogous to the problem of constructing a triangle having its sides of given length.Two other theorems of this kind are added by Simson in his edition of Euclid’sElements.§ 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 worddistance. 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-intersecting 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 asbase, and its distance from the opposite face asaltitude.Prop. 25.If a solid parallelepiped 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. 1) that parallelograms between the same parallels are 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 toProp. 28.If a solid parallelepiped be cut by a plane passing through the 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.Prop. 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 the proof of the general theorem.As consequences of this fundamental theorem we getProp. 32.Solid parallelepipeds, which have the same altitude, are to one another as their bases;andProp. 33.Similar 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.Parallelepipeds which are not similar but equal are compared by aid of the theoremProp. 34.The bases and altitudes of equal solid parallelepipedsarereciprocally proportional; and if the bases and altitudes be reciprocally proportional, the solid parallelepipeds are equal.§ 83. Of the following propositions the 37th and 40th are of special interest.Prop. 37.If four straight lines be proportionals, the similar solid parallelepipeds, similarly described from them, shall also be proportionals; and if the similar parallelepipeds similarly described from four straight lines be proportionals, the straight lines shall be proportionals.In symbols it says—If a : b = c : d, then a³ : b³ = c³: d³.Prop. 40 teaches how to compare the volumes of triangular prisms with those of parallelepipeds, by provingthat a triangular prism is equal in volume to a parallelepiped, which has its altitude and its base equal to the altitude and the base of the triangular prism.§ 84. From these propositions follow all results relating to the mensuration of volumes. We shall state these as we did in the case of areas. The starting-point is the “rectangular” parallelepiped, which has every edge perpendicular to the planes it meets, and which takes the place of the rectangle in the plane. If 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 parallelepiped 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, say 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 their 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 V1and V2. We then haveV : V1= a : a′; V1: V2= b : b′, V2: V′ = c : c′.Compounding these, we haveV : V′ = (a : a′) (b : b′) (c : c′),orV=a·b·c.V′a′b′c′Hence, as a special case, making V′ equal to the unit cube U on u we getV=a·b·c= α·β·γ,Uuuuwhere α, β, γ are the numerical values of a, b, c; that is,The number of unit cubes in a rectangular parallelepipedis 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 altitude, which in this case is the same.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 done by theTheorem.—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 Euclid’sProp. 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 triangular pyramids have equal 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 divided into 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 smaller than any assignable quantity, however small. It follows, so we should conclude at present, that the two volumes must be equal, for they cannot differ by any assignable quantity.To Greek mathematicians this conclusion offers far greaterdifficulties. They prove elaborately, by areductio ad absurdum, that the volumes cannot be unequal. This proof must be read in theElements.We must, however, state that we have in the above not proved Euclid’s 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 haveVolume of prism= Bh,Volume of pyramid=1⁄3Bh,statements which have to be taken in the sense that B means the number of square units in the base, h the number of units of length in the altitude, or that B and h denote the numerical values of base and altitude.§ 87. 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. 11.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 the diameters of their bases have.Prop. 13.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 the 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=1⁄3Bh.§ 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 Euclid’sElementsis devoted to the regular solids (seePolyhedron). It is shown that there are five of them, viz.:—1. The regulartetrahedron, with 4 triangular faces and 4 vertices;2. Thecube, with 8 vertices and 6 square faces;3. Theoctahedron, with 6 vertices and 8 triangular faces;4. Thedodecahedron, with 12 pentagonal faces, 3 at each of the 20 vertices;5. Theicosahedron, 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 theElements, 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 form 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 equals6⁄5of a right angle. Hence three of them equal18⁄5(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.
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.:—
a : b : c : d : e : ... = 1/a′ : 1/b′ : 1/c′ : 1/d′ : 1/e′ ...,
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.
§ 57. 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 asquotientsof 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 theElementsshows 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. 1.Triangles and parallelograms of the same altitude are to one 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 straight 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 divides the opposite side internally in a ratio equal to the ratio of the two sides including that angle;and conversely,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.
Simson’s 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 Euclid’s, 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. 11.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 quantity—viz. 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. 11.
a : b = b : 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 reciprocally proportional; and triangles 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.
The latter proposition is really the same as the former, for if, as in the accompanying diagram, in the figure belonging to the former the two equal parallelograms AB and BC be bisected by the lines DF and EG, and if EF be drawn, we get the figure belonging to the latter.
It is worth noticing that the lines FE and DG are parallel. We may state therefore the theorem—
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 proportionals.
In symbols, if a, b, c, d are the four lines, and
if
a : b = c : d,
then
ad = bc;
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 a/b of two lines as thenumberwhich multiplied into b gives a, so that
we see that from the equality of two quotients
follows, if we multiply both sides by bd,
ad = cb.
But from this it follows, according to the last theorem, that
a : b = c : d.
Hence we conclude that the quotient a/b and the ratio a : b are different forms of the same magnitude, only with this important difference that the quotient a/b 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 fifth book which enables us to extend the geometrical calculus given before in connexion 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 rectilineal figure, 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 rectilineal 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 proportionals; and if the similar rectilineal figures similarly described on four straight lines be proportionals, those straight lines shall be proportionals.
This is essentially the same as the following:—
If
a : b = c : d,
then
a² : b² = c² : d².
§ 66. Now follows a proposition which has been much discussed with regard to Euclid’s exact meaning in saying that a ratio iscompoundedof 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 their sides.
The proof of the proposition makes its meaning clear. In symbols the ratio a : c is compounded of the two ratios a : b and b : c, and if a : b = a′ : b′, b : c = b″ : c″, then a : c is compounded of a′ : b′ and b″ : c″.
If we consider the ratios as numbers, we may say that the one ratio is the product of those of which it is compounded, or in symbols,
The theorem in Prop. 23 is the foundation of all mensuration of areas. From it we see at once that two rectangles have the ratio 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 that of a rectangle contained by c and d, so that A = ab, B = cd, then A : B = ab : cd, and this is, the theorem says, compounded of the ratios a : c and b : d. In forms of quotients,
This shows how to multiply quotients in our geometrical calculus.
Further,Two triangles have the ratios of their areas compounded of the ratios of their bases and their altitude.For a triangle is equal in area to half a parallelogram which has the same base and the same altitude.
§ 67. To bring these theorems to the form in which they are usually given, we assume a straight line u as our unit of length (generally an inch, a foot, a mile, &c.), and determine the number α which expresses how often u is contained in a line a, so that α denotes the ratio a : u whether commensurable or not, and that a = αu. We call this number α the numerical value of a. If in the same manner β be the numerical value of a line b we have
a : b = α : β;
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 = αu, b = βu, therefore a : b = αu : βu, and this may without difficulty be shown to equal α : β.
If now a, b be base and altitude of one, a′, b′ those of another parallelogram, α, β and α′, β′ their numerical values respectively, and A, A′ their areas, then
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 α′ = β′ = 1, A′ = u², and we have
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 greater 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 rectangles 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
(a − x)y = k²,
k² 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 the required parallelograms.
If we substitute the value of y, we get
or,
ax − x² = b²,
where a and b are known quantities, taking b² = pk²/q.
The second case (Prop. 29) gives rise, in the same manner, to the quadratic
ax + x² = b².
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. 11.
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 the sides containing the right angle,—is a pretty generalization of the theorem of Pythagoras (I. 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 angles by arcs.
With this closes that part of theElementswhich 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 afterwards 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 line 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 Euclid’s Prop. 1, viz.:—
Prop. 1.One part of a straight line cannot be in a plane and another part without 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 points which do not lie in a straight line; for if two of these points be joined by a straight line we have case 1;
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 Euclid’s
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 Euclid’s
Prop. 7.If two straight 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 line.
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 shall 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 perpendicular to the same plane are parallel to each other;and conversely
Prop. 8.If of two parallel straight lines one is perpendicular to a plane, the other is so also.
Prop. 7.If two straight lines are parallel, the straight line which joins any point in one to any point in the 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 in 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 different 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 later on in Prop. 15.
This theorem is not necessarily true, for the angles in question may be supplementary; but then the one angle will be equal to 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 the 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 from 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 first—viz. 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 given 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 interest 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 perpendicular BC be drawn to any straight line in the plane, then the straight 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 shownthat 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(Prop. 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); andTwo 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 of the problem: “Through a given point to draw a plane parallel 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, every 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 atrihedralangle, is formed, and similarly by more planes a solid angle of more faces, or apolyhedralangle. 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 angles—has no analogous theorem in the plane.
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 the planes containing the solid angle are called itsfaces. We get, for instance, from I. 4, thetheorem,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 equal faces.The solid angles themselves are not necessarily equal, for they may be only symmetrical like the right hand and the left.
The connexion 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 them being greater than the third.It is, of course, analogous to the problem of constructing a triangle having its sides of given length.
Two other theorems of this kind are added by Simson in his edition of Euclid’sElements.
§ 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 worddistance. 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-intersecting 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 asbase, and its distance from the opposite face asaltitude.
Prop. 25.If a solid parallelepiped 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. 1) that parallelograms between the same parallels are 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 parallelepiped be cut by a plane passing through the 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.
Prop. 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 the proof of the general theorem.
As consequences of this fundamental theorem we get
Prop. 32.Solid parallelepipeds, which have the same altitude, are to one another as their bases;and
Prop. 33.Similar 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.
Parallelepipeds which are not similar but equal are compared by aid of the theorem
Prop. 34.The bases and altitudes of equal solid parallelepipedsarereciprocally proportional; and if the bases and altitudes be reciprocally proportional, the solid parallelepipeds are equal.
§ 83. Of the following propositions the 37th and 40th are of special interest.
Prop. 37.If four straight lines be proportionals, the similar solid parallelepipeds, similarly described from them, shall also be proportionals; and if the similar parallelepipeds similarly described from four straight lines be proportionals, the straight lines shall be proportionals.
In symbols it says—
If a : b = c : d, then a³ : b³ = c³: d³.
Prop. 40 teaches how to compare the volumes of triangular prisms with those of parallelepipeds, by provingthat a triangular prism is equal in volume to a parallelepiped, which has its altitude and its base equal to the altitude and the base of the triangular prism.
§ 84. From these propositions follow all results relating to the mensuration of volumes. We shall state these as we did in the case of areas. The starting-point is the “rectangular” parallelepiped, which has every edge perpendicular to the planes it meets, and which takes the place of the rectangle in the plane. If 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 parallelepiped 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, say 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 their 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 V1and V2. We then have
V : V1= a : a′; V1: V2= b : b′, V2: V′ = c : c′.
Compounding these, we have
V : V′ = (a : a′) (b : b′) (c : c′),
or
Hence, as a special case, making V′ equal to the unit cube U on u we get
where α, β, γ are the numerical values of a, b, c; that is,The number of unit cubes in a rectangular parallelepipedis 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 altitude, which in this case is the same.
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 done 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 Euclid’s
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 triangular pyramids have equal 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 divided into 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 smaller than any assignable quantity, however small. It follows, so we should conclude at present, that the two volumes must be equal, for they cannot differ by any assignable quantity.
To Greek mathematicians this conclusion offers far greaterdifficulties. They prove elaborately, by areductio ad absurdum, that the volumes cannot be unequal. This proof must be read in theElements.We must, however, state that we have in the above not proved Euclid’s 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
statements which have to be taken in the sense that B means the number of square units in the base, h the number of units of length in the altitude, or that B and h denote the numerical values of base and altitude.
§ 87. 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. 11.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 the diameters of their bases have.
Prop. 13.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 the 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,
§ 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 Euclid’sElementsis devoted to the regular solids (seePolyhedron). It is shown that there are five of them, viz.:—
1. The regulartetrahedron, with 4 triangular faces and 4 vertices;
2. Thecube, with 8 vertices and 6 square faces;
3. Theoctahedron, with 6 vertices and 8 triangular faces;
4. Thedodecahedron, with 12 pentagonal faces, 3 at each of the 20 vertices;
5. Theicosahedron, 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 theElements, 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 form 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 equals6⁄5of a right angle. Hence three of them equal18⁄5(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.