# Encyclopædia Britannica, Ninth Edition/Measurement

MEASUREMENT. We propose in the first place to enter into some detail on the fundamental principles of the theory of measurement, and in doing so it will be necessary to sketch the very remarkable theory established by Riemann and other mathematicians as to the foundations of our geometrical knowledge.

Every system of geometrical measurement, as indeed the whole science of geometry itself, is founded on the possibility of transferring a fixed figure from one part of space to another with unchanged form. We are so familiar with this process that we are apt not to realize its importance until very special attention has been directed to the subject. We therefore propose to make a logical examination of the nature of the assumptions involved in the possibility of moving a figure in space so that it shall undergo no alteration. We shall find that we require to postulate certain suppositions with regard to the nature of space and to the measurement of distances.

It will facilitate the conception of this somewhat difficult subject to consider the case of hypothetical reasoning beings which Sylvester described as being infinitely attenuated bookworms confined to infinitely thin sheets of paper. We suppose such two-dimensional beings to be absolutely limited to a certain surface. They could have no conception of space except as of two dimensions. The movement of a point would for them form a line, the movement of a line would form a surface. They could conduct their measurements and form their geometrical theories. They would be able to draw the shortest lines between two points, these lines being what we would call geodesies. To these two-dimensioned geometers geodesies would possess many of the attributes of straight lines in ordinary space. If the surface to which the beings were confined were actually a plane, then the geometry would be the same as our geometry. They would find that only one straight line could be drawn between two points, that through a point only one parallel to a given line could be drawn, and that the ends of a line would never meet even though the line be prolonged to infinity.

We might also suppose that intelligent beings could exist on the surface of a sphere. Their straightest line between two points would be the arc of the great circle joining those two points. They would also find that a second geodesic could be drawn joining the two points, this being of course the remaining part of the great circle. A curious exception would, however, be presented by two points diametrically opposite. An infinite number of geodesies can be drawn between these points and all those geodesies are of equal length. The axiom that there is one shortest line between two given points would thus not hold without exception. There would be no parallel lines known to the dwellers on the sphere. It would be found by them that every two geodesies must intersect, not only in one, but even in two points. The sum of the three angles of a triangle would for them not be constant. It would always be greater than two right angles, and would increase with the area of the triangle. They would thus have no conception of similarity between two geometrical figures of different sizes. If two triangles be constructed which have their sides proportional, the angles of the larger triangle would be greater than the corresponding angles of the smaller triangle.

It is thus plain that the geometrical axioms of the sphere-dwellers must be very different from those of the plane-dwellers. The different axioms depend upon the different kinds of space which they respectively inhabit, while their logical powers are identical. In one sense, however, the dwellers on the sphere and on the plane have an axiom in common. In each case it will be possible for a figure to be moved about without alteration of its dimensions. A spherical triangle can be moved on the surface of a sphere without distortion just as a plane triangle may be moved in a plane. The sphere-dwellers and the plane-dwellers would be equally able to apply the test of congruence. It is, however, possible to suppose reasoning beings confined to a space in which the translation of a rigid figure is impossible. Take, for instance, the surface of an ellipsoid or even a spheroid such as the surface of the earth itself. A triangle drawn on the earth at the equator could not be transferred to the surface of the earth near the pole and still preserve all its sides and all its angles intact.

If a surface admits of a figure being moved about thereon so as still to retain all its sides and all its angles unaltered, then that surface must possess certain special properties. It can be shown that, if a surface is to possess this property, a certain function known as the “measure of curvature” is to be constant. The measure of curvature is the reciprocal of the product of the greatest and least radii of curvature. We do not now enter into the proof, but it is sufficiently obvious that a sphere of which the radius is the geometric mean between the greatest and least radii of curvature at each point will to a large extent osculate the surface, so that a portion of the surface in the neighbourhood of the point will, generally speaking, have the same curvature as the sphere. If the sphere thus determined be the same at all the different points of the surface, then the curvature of the different parts of the surface will on the whole resemble that of the sphere, and therefore we cannot be surprised that the surface possessing this property will admit the displacement of a rigid figure thereon without derangement of its form.

We are thus conducted to a kind of surface the geometry of which is similar to that of the plane, but in which the axiom of parallels does not hold good. In this surface the radii of curvature at every point have opposite signs, so that the measure of curvature which is zero for the plane and positive for the sphere is negative for the surface now under consideration. This surface has been called the “pseudosphere,” and its nature has been investigated by Beltrami. In the geometry of two dimensions we can thus have either a plane or a sphere or a pseudosphere which are characterized by the property that a surface may be moved about in all directions without any change either in the lengths of its lines or in the magnitudes of its angles. The axiom which assumes that there is only one geodesic connecting two points marks off the plane and the pseudosphere from the sphere. The axiom that only one parallel can be drawn through a given point to a given line marks off the plane from the pseudosphere. The geometry of Euclid is thus specially characterized among all conceivable geometries of two dimensions by the following three axioms—(1) the mobility of rigid figures, (2) the single geodesic between two points, (3) the existence of parallels.

A very interesting account of this theory will be found in Clifford's Lectures and Essays, vol. i. p. 317. We shall follow to some extent the method employed by him in order to obtain an idea of the important conception which is called the “curvature of space.” Suppose a geodesic be drawn on a surface of constant curvature. Then a piece of the surface adjoining this geodesic can be slid along the curve so as all the time to fit in close contact therewith. If the piece of surface be turned to the other side of the geodesic it will still fit along this side. A line possessing this property is called by Leibnitz a straight line. It can be easily shown that a geodesic drawn on a figure will also be a geodesic when the figure is transferred to any other position. Suppose that the figure be divided into two parts A and B by the geodesic; then the part B can be moved round so as to lie upon A, and the boundary lines of the two portions will be coincident. Now let the two parts while superposed be translated to any other position, then the piece B may be slid off and back to its original position with regard to A. It must still fit, because the whole figure might have been translated before the subdivision took place. It follows that the division between A and B having been a geodesic in its original position will continue to be a geodesic

however the figure may be translated.

In a similar way we obtain the conception of a plane. According to Leibnitz's definition a plane is a surface such that if a portion of the space contiguous thereto be slid along the surface it will continuously fit, and if the portion of space be transferred to the other side of the surface it will fit also. This definition has no meaning except we assume that the bodies may be translated in space without derangement of their dimensions. From any point we can imagine a doubly infinite number of geodesies radiating in all directions; if a plane be drawn through the point, then all the geodesies touching the plane at that point form what may be called a “geodesic surface.” It is shown that geodesic surfaces of this description can alone fulfil the conditions by which planes are to be defined. A doubly infinite number of geodesic surfaces can be drawn through every point. If a rigid body be divided into two parts by a geodesic plane, then no matter how the body be displaced the plane of section will still be geodesic. The plane of section may be made to pass through any point, and the body may then be given such an aspect as shall cause the section to coincide with any geodesic surface through the point, but this necessarily involves that the section shall fit each geodesic surface, in other words, that all the geodesic surfaces shall have a constant curvature.

The point which we have now gained is one of very great importance. In our ordinary conceptions of space the geodesic surfaces are of course our ordinary planes, and the common curvature they possess is zero, but the condition that rigid bodies shall be capable of translation with unaltered features does not require that the curvatures shall be zero, it merely requires that the curvatures shall be constant. If we add, however, the postulate of similarity, then the curvatures must be zero. The postulate of similarity requires that it shall be possible to construct a figure on any scale and anywhere similar to a given figure. This practically includes the ordinary doctrine of parallels. Lobatchewsky developed the system of geometry on the supposition that the space had a constant curvature different from zero. In this geometry the parallels can be drawn through a given point to a given line, and, to quote Clifford—

“The sum of the three angles of a triangle is less than two right angles by a quantity proportional to the area of the triangle. The whole of this geometry is worked out in the style of Euclid, and the most interesting conclusions are arrived at, particularly in the theory of solid space, in which a surface turns up which is not plane relatively to that space but which for the purpose of drawing figures upon it is identical with the Euclidean plane.”

The most comprehensive mode of viewing the whole theory is that adopted by Riemann in his celebrated memoir “Ueber die Hypothesen welche der Geometrie zu Grunde liegen,” 1854 (Abhandl. der königl. Gesellsch. zu Göttingen, vol. xiii.). The analytical treatment of this subject possesses one obvious advantage. The use of symbols only admits of deductions on purely logical principles. There is not therefore the risk of tacitly introducing other axioms in addition to those with which we started.

Magnitudes which have only one dimension present the theory of measurement in its simplest form. The length of a straight line may be taken as an illustration of a one-dimensioned magnitude. The velocity of a moving particle, the temperature of a heated body, the electric resistance of a metal, all these and many others are instances of one-dimensioned magnitude, the measure of which is to be expressed by a single quantity. But there may be magnitudes which require more than a single measurement for their complete specification. Take, for instance, a four-sided field which has been duly surveyed. Of what is the measurement of this field to consist? If the number of acres in the field be all that is required then the area is expressed by a simple reference to a number of standard acres. If, however, the entire circumstances of the field are to be brought into view, then a simple statement of the area is not sufficient. It can be easily shown that the surveyor must ascertain five independent quantities before the details of the shape of the field can be adequately defined. Four of these quantities- may naturally be the lengths of the four sides of the field, the fifth may be one of the angles, or the area, or the length of one of the diagonals. Speaking generally, we may say that five distinct measurements will be necessary to define the field adequately. The actual choice of the particular measurements to be made is to a great extent arbitrary. The only condition absolutely necessary is that they shall be all independent and free from ambiguity. Once these five quantities are ascertained then all the other features of the figure are absolutely determined. For instance, the four sides and the diagonal being ascertained by measurement, then the other diagonal, the four angles, and the area can all be computed by calculation. The five quantities would determine everything about the field except its actual position on the surface of the earth. If we further desired to have the field exactly localized certain other quantities must be added. The latitude and the longitude of one specified corner of the field would completely indicate that corner, while the azimuth of one side from that corner would complete the definition of its position. We are thus led to see that for the complete delineation of every circumstance relating to the shape of the field and its locality eight different measurements have been required. Two sets of eight measurements differing in any particular can never indicate the same field. It is very important to notice that the number of quantities required is quite independent of the particular nature of the measurements adopted. We might for instance have simply measured the latitude and the longitude of each of the four corners of the field. Once these quantities are known, then the shape of the field, its area, its angles, and its diagonals have all been implicitly determined. Here again we see that as two quantities are required to localize each of the four corners, so eight quantities will be required to fully determine the whole field.

^Y = ^Y &lt;2Y d&lt;f&gt; dr, dr, dZ dZ dZ d&lt;p dr, dr, or uniting the two sets of equations we have dX Y-Lrt 7 -r- = a X + +f Z As this movement must also be a rotation, the three right-hand mem bers must be capable of being rendered zero for certain values of X, Y, Z, and therefore we have (remembering that o "o = 0. 2 &gt; . - This condition reduces to coa ( (/! &&gt;) - or a () co 2 +eo(a 1 i - a v 1 ) = 0. This equation must be satisfied for every value of u ; for, whatever be the amplitudes of the two rotations, they must when com pounded be equal to a single rotation. We therefore have the conditions "i^e = . To satisfy the latter condition either o^ or v g must be equal to zero. We must examine which of these two conditions is required by the problem. Since o 8 is equal to zero we have dX dY dZ If t&gt; were zero then the first equation would show X to be con stant ;-and the result would be that Y = 1^X77 + ^o-j^Xr; 2 -f const. ; or, in other words, Y would be susceptible of indefinite increase with the increase of 77. The supposition j/ = is therefore precluded, and we are forced to admit that o 1 = 0. The three equations then reduce to = Z =vZ dr, l dZ _ ,, If the body receives a rotation 77" about an axis which leaves X and Y unaltered, we then have The condition that the two roots of h shall be purely imaginary gives us /o + fc-0. Let this rotation and the first rotation be communicated together. The resulting rotation could have been produced by a rotation x&gt; and thus we have ^X = ^X + ^X dx dr) drj" ^Y = ^Y + dY dx dr, dr," dZdZ dZ Substituting, we obtain as before and as before the condition must be fulfilled fo &gt; 0o &gt; A , Si or, expanding, This can only be satisfied for all values of if / = and if To determine whether g Q can be zero, we have the equations dX dY dr, It can be shown that if g were zero then we should have Z capable of indefinite increase ; and hence we see that/ 2 must be zero, so that the three equations have the form Let us now see whether these equations will fulfil the necessary condition for a rotation . If ^

remain unaltered notwithstanding the rotation of the plane which

contains them around their intersection. The two lines here referred to are of course those which are drawn through the two circular points at infinity. This paradox is therefore only a degraded form of the property of the tangents to the fundamental conic.

It can also be readily shown that, if a plane receive two small rotations round two points, then the total rotation produced could have been produced by a single rotation about a certain point on the line joining the two points.

Let A, B be the two points and P the pole of the line AB, then a rotation round A will displace B along the line PB to an adjacent point B′. The rotation around B will displace A to A′ along the line PA; but, if A′B′ intersects AB in O, then a single rotation about O would have effected the required displacement of A and B, and therefore of the whole line. For, as the point O in the line AB could only move by displacement into the line A′B′, while it can also only move in the direction OP, it must obviously remain unaltered.

We are now in a position to inquire how the magnitude of an angle is to be expressed in the present system of measurement. Our definition of the magnitude of an angle must be made consistent with the supposition that when the angle is carried round by rotation about the vertex the magnitude shall remain unaltered. As anharmonic ratios are unaltered by the rotation, it follows that the anharmonic ratio of the pencil formed by the two legs of the angle and the two tangents to the fundamental conic must remain unaltered. Remembering that the tangents do not move by the rotation, it is natural to choose a function of this anharmonic ratio as the appropriate measure of an angle. The question still remains as to what function should be chosen. The student of ordinary geometry is doubtless aware that the angle between two lines multiplied into 2i is equal to the logarithm of the anharmonic ratio of the pencil formed by joining the intersection of the two lines to the two imaginary circular points at infinity. This consideration suggests that the angle between the straight lines in the generalized sense may be appropriately measured by the logarithm of the anharmonic ratio of the pencil formed by the two legs of the angle and the two tangents drawn from their point of intersection to the fundamental conic. There is also a convenience in assuming the angle to be actually equal to c times the logarithm of the anharmonic ratio, where c is the same constant as is employed in the expression of the distance. In this case the angle between two lines is by a well-known theorem equal to the distance between their poles. There is here an analogy to a well-known theorem in spherical geometry.

It will now be obvious that, however the angle be situated, its magnitude is unchanged by any displacement of the plane; for, as we have already seen that the displacement does not alter the distance between the poles of the two lines forming the angle, it follows that the magnitude of the angle itself is unaltered.

Just as in the measurement of distance we find a pair of fundamental points on each straight line, so in the measurement of angles we find a pair of fundamental rays in each plane pencil. These rays are the two tangents from the vertex of the pencil to the fundamental conic. In ordinary geometry the two fundamental points on each straight line coalesce into the single point at infinity; but it is exceedingly interesting to observe that even in ordinary geometry the two fundamental rays on each pencil do not coincide. It should also be observed that in the degraded circumstances of ordinary geometry it would be impracticable to employ the same constant c for the purpose of both linear and angular measurement.

It is easy to see that the definition of a right angle in the generalized sense is embodied in the statement that “if two corresponding legs of an harmonic pencil touch the fundamental conic then the two other legs are at right angles.” We also see that all the perpendiculars to a given line pass through a point, i.e., the pole of the given line; and from a given point a perpendicular can be drawn to a given line by joining the point to the pole of the line. The common perpendicular to two lines is obtained by joining their poles.

The student of modern geometry is already accustomed to think of parallel lines as lines which intersect at infinity, or as lines whose inclination is zero. In speaking of the generalized geometry in a plane, we may define that two straight lines which intersect upon the fundamental conic are parallel. It thus follows that through any point two distinct parallels can be drawn to a given straight line. The only exception will arise in the case where the given line touches the fundamental conic. This is precisely the case in which the generalized system of measurement degrades to the ordinary system. It will follow that in the present theory of measurement the three angles of a triangle are together not equal to two right angles. In fact, to take an extreme case, we may suppose the three vertices of the triangle to lie upon the fundamental conic. In this case each of the three angles, and therefore their sum, is equal to zero.

A sphere in the generalized system of measurement is the locus of a point which moves at a constant distance from a fixed point. It can therefore be easily shown that a sphere is a quadric which touches the fundamental quadric along its intersection with the polar plane of the centre of the sphere.

In discussing the general case of the displacement of a rigid system it will simplify matters to suppose that the fundamental quadric has real rectilinear generators. It must, however, be understood that the results are not on that account less general. A displacement must not alter the quadric, and must not deform a straight line. Hence it follows that the only effect of a displacement upon a generator of the fundamental quadric will be to convey it to a position previously occupied by a different generator. We shall further suppose that the displacement is such that the two generators to which we have referred belong to the same system. Let A, B, C, D be four generators of the first system which by displacement are brought to coincide with four other generators A′, B′, C′, D′. Let X be one generator of the second system which the displacement brings to X′. Since the anharmonic ratio of the four points in which four fixed generators of the one system are cut by any generator of the other system is constant, we must have, using an obvious notation for anharmonic ratio,

X(ABCD) = X′(ABCD);

but, since anharmonic ratios are unaltered by displacement, we have

X(ABCD) = X′(A′B′C′D′),

whence

X′(ABCD) = X′(A′B′C′D′).

It therefore follows that the anharmonic ratio in which four generators cut a fixed generator X′ is equal to the anharmonic ratio in which the four generators after displacement cut the same generator X′.

If P be a generator which passes through one of the double points on X′ determined by the two systems of points in which X′ is cut by the four generators before and after displacement, we must have

X(A, B, G, P) = X′(A′, B′, C′, P′);

hence we see that the generator P will be unaltered by displacement. Similar reasoning applies to the generator which passes through the other double point, and of course to a pair of generators of the second system, and hence we have the following remarkable theorem:—

In the most general displacement of a rigid system two generators of each of the systems on the fundamental quadric remain unaltered.

These four fixed generators are the edges of a tetrahedron. Denoting the four faces of this tetrahedron by the equations

x = 0, y = 0, z = 0, w = 0,

the equation of the fundamental quadric is

xz + h²yw = 0.

If the quadric be unaltered by the transformation

x′ = αx, y′ = βy, z′ = γz, w′ = δw,

then we must have

αγ = βδ

When this condition is satisfied, then, whatever h may be, every quadric of the family

xz + hyw = 0

will remain unaltered.

The family of quadrics here indicated are analogous to the right circular cylinders which have for a common axis the screw along which any displacement of a rigid body in ordinary space may bo effected.

The two lines

x = 0, z = 0

and

y = 0, w = 0

are conjugate polars with respect to the fundamental quadric, and both these lines are unaltered by the displacement. Hence we see that in any displacement of a rigid system there are two right lines which remain unaltered, and these lines are conjugate polars with respect to the fundamental quadric.

Since the pole of a plane through one of these lines lies on the other line, it appears that a rotation of a rigid system about a straight line is identical with a translation of the system along its conjugate polar.

Clifford has pointed out the real nature of the lines which are to be called parallel in the generalized system of measurement. We have explained that in the plane two parallel lines intersect upon the fundamental conic; in a certain sense also we may consider two lines in space of three dimensions to be parallel whenever they intersect upon the fundamental quadric. This is the view of parallel lines to which we are conducted by simply generalizing the property that two parallel lines intersect at infinity. But we can take a different definition of two parallel lines. Let us, for example, call two lines parallel when they admit of an indefinitely large number of common perpendiculars. It is exceedingly interesting

to observe that when this condition is fulfilled in the generalized system of measurement the parallel lines so obtained enjoy many of the properties of ordinary parallel lines. The perpendicular distance between such a pair of parallels is constant, and the angles which they make with any common transversal are equal.

It will be shown in a moment that any pair of straight lines which intersect the same two generators of the same system on the fundamental quadric are parallel in this new sense. The fact is that in the degraded circumstances of ordinary geometry two quite different conceptions have become confused, A pair of lines which intersect on the fundamental quadric and a pair of lines which intersect the same pair of generators of the same kind on the fundamental quadric are quite different conceptions; but when the fundamental quadric degrades to the ordinary infinity then the conceptions coalesce, and each of them is merely a pair of parallel lines in the ordinary sense of the word. The ordinary properties of parallel lines have all their analogues in the generalized geometry, but these analogues are distributed between the two original sources of parallels. Clifford proposes to retain the word “parallel” in non-Euclidean space for that conception which exhibits the more remarkable properties of ordinary space, and defines as follows:—

Straight lines which intersect the same two generators of the same system on the fundamental quadric are parallel.

Let X and Y be two rectilinear generators of the fundamental quadric belonging to the same system, and let A and B be two straight lines which intersect both X and Y. Since AX and AY are tangent planes their poles must lie on X and Y respectively, and therefore A′ (and B′), the polar of A (and B), must intersect both X and Y. The anharmonic ratio of the four points in which X intersects AB, A′B′ respectively is equal to that of the tangent planes drawn at the points where X intersects A′B′, AB respectively ; and, as all these tangent planes contain X, their anharmonic ratio must be equal to that in which they are cut by the line Y. It hence follows that the lines X and Y are divided equianharmonically by the four rays A, B, A′, B′, and therefore the four rays A, B, A′, B′ must be all generators of the same system on an hyperboloid. An infinite number of transversals can therefore be drawn to intersect these four rays, that is to say, an infinite number of common perpendiculars can be drawn to the two rays A and B, and it is easy to show that the lengths of all these perpendiculars are equal.

Clifford has proved the very remarkable theorem that rotations of equal amplitude about two conjugate polars have simply the effect of translating every point operated on through equal distances along parallel lines. This property leads to most important consequences, but it would lead us too far to enter into the subject at present.

A memoir by the present writer on the extension of the theory of screws to space of this description will be found in the Transactions of the Royal Irish Academy, vol. xxvii. pp. 157–184.

A most excellent account of the units employed in scientific measurements will be found in Professor Units and Physical Constants, London, 1879. We shall here only give a very brief out line of this branch of the general theory of measurement, referring inquirers to Everett's volume for further details.

Most of the quantities for which measurements are needed can be ultimately expressed by means of (1) a definite length, (2) a definite mass, or (3) a definite interval of time.

It is very important that the units thus referred to should be chosen judiciously, and it must be admitted that the units ordinarily used do not fulfil the conditions which a well-chosen system of units should fulfil. The most scientific system is beyond doubt that which has been suggested by the units committee of the British Association. In this system the unit of length is the centimetre, the unit of mass is the gramme, and the unit of time is the second, and the system is therefore often referred to for brevity as the C.G.S. system. The unit of force is termed the dyne, and it is defined to be the force which, acting upon a gramme of matter for a second, generates a velocity of a centimetre per second. The unit of work is the work done by this force working through a centimetre, and this unit is termed the erg. The unit of power is the power of doing work at the rate of one erg per second, and the power of an engine can be specified in ergs per second. By the prefixes deca, hecto, kilo, mega, we can express a magnitude equal to the unit multiplied by 10, 100, 1000, or 1,000,000 respectively. On the other hand the prefixes deci, centi, milli, micro, signify the units divided by 10, 100, 1000, or 1,000,000 respectively.

For comparison with the ordinary units the following statements will be useful. The weight of a gramme at any part of the earth's surface is about 980 dynes, or rather less than a kilodyne. The weight of a kilogramme is rather less than a megadyne, being about 980,000 dynes.

The application of these units to electrical and many other measurements will be found in Professor Everett's book already referred to. On the general principles of appliances for measurement, see a paper by Clifford in the Handbook to the Special Loan Collection of Scientific Apparatus, 1876, pp. 55–59, reprinted in Clifford's Mathematical Papers, pp. 419–23.

1. Saggio di Interpretazione della Geometria non-Euclidea, Naples, 1868; “Teoria fondamentale degli sparii di curvatura constante,” Annali di Matematica, ser. ii. tom. ii. pp. 232–55. Both papers have been translated into French by J. Houël (Annales Scientifiques de l'École Normale, tom. v., 1869). An exceedingly interesting account of the whole subject will be found in Helmholtz, Popular Lectures on Scientific Subjects, translated by Atkinson, second series, London, 1881, pp. 27–71.
2. A translation of this paper was published by Clifford in Nature (vol. viii. Nos. 183, 184, pp. 14–17, 36, 37), and has been reprinted in the collected edition of Clifford's Works, 1882, pp. 55–69. For a bibliography of higher-space and non-Euclidean geometry, see articles by George Bruce Halsted in the American Journal of Mathematics Pure and Applied, i. 261-276, 384, 385 ; ii. 65-70.