# 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.^{[1]} 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

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.).^{[2]} 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.

In the operations of analytical geometry we are accustomed to specify the position of a point by the relation which it bears to certain fixed axes. By means of certain quantities, either altogether linear or partly linear and partly angular, we are enabled to specify the position of the point with absolute definiteness. These quantities are called the coordinates of the point. In a similar though more extended sense we may use the word " coordinates " to express the group of eight magnitudes which we have found to be adequate to the complete specification of the field. By the measurement of the field in the most complete sense of the term we mean the measurement of its eight coordinates. Suppose that an object is completely specified by n coordi nates, then every different group of n coordinates will specify a different object. The entire group of such objects will form what is called a continuously extended manifold- ness. The singly extended manifoldness may be most conveniently illustrated by the conception of time, the various epochs of which are the elements in the manifold- ness. Space is a triply extended manifoldness whereof the elements are points. All conceivable spheres form a quadrnply extended manifoldness. All conceivable triangles in space form a manifoldness of nine dimensions. The number of coordinates required to specify the position of an element in a manifoldness is thus equal to the order of the manifoldness itself. It is important to observe that the elements of the manifoldness may be themselves objects of no little complexity. Thus, for instance, the conies forming a confocal group constitute the elements of a singly ex tended manifoldness. The essential feature of a singly extended manifoldness is that a continuous progress of an element can take place only in two directions, either forwards or backwards. But a singly extended manifoldness may be regarded as itself an element in a manifoldness of a higher order. Thus the points on a circle form a singly extended manifold- ness, while the circle itself is one element of the manifold- ness which consists of a series of concentric circles. The system of concentric circles may in like manner be regarded as an element in the manifoldness which embraces all systems of concentric circles whose centres lie along a given line. We are thus led to conceive of a multiply extended manifoldness as made up by the successive com position of singly extended manifoldnesses. It follows from the conception of a manifoldness that in the case of a singly extended manifoldness the position of every element must be capable of being completely specified by a single quantity. It becomes natural to associate with each element of the manifoldness a special numerical magni tude. These .magnitudes may vary from -co to + oo ; to each magnitude will correspond one element of the manifold- ness, and conversely each, element of the manifoldness is completely specified whenever the appropriate number has been assigned. It is quite possible to have this association of numerical magnitude with the actual position of an element independent of any ordinary metrical relations of the system; it will, however, most usually be found that the numerical magnitudes chosen are such as admit of direct interpretations for the particular manifoldness under con sideration. Thus, for instance, in the case of the system of concentric circles it will be natural to associate with each circle its radius, and the position of each circle in the manifoldness will thus be completely defined by the radius. So also in the case of that singly extended manifoldness whicli consists of colours, it will be natural to employ as the number which specifies each particular colour the wave-length to which that particular colour corre sponds. If the elements of such a manifoldness can receive a simultaneous displacement, then it is plain that to each element in the original position will correspond an element in the second position. Let x and y bo the numerical magnitudes correlated to these two elements. Then, since the relation must be of the one-to-one type, it is necessary that the magnitudes x and y must be connected by an equation of the type axy + Ix + cy + d = 0. It follows from this that there are a pair of elements which are common to both systems, for if x=y we have the equation The original equation may be written in the form axy + (b- ta)x + (c + w)y + d*= , and whatever value o> may have this equation will lead to the same quadratic for the two common elements. We thus have a singly infinite number of displacements which are compatible with the condition that the two fundamental elements shall remain unaltered, and it is displacements of this kind which express the movements of a rigid system. The position of a point is to be defined by three coordinates. In our ordinary conception of coordinates the position of the point is defined by certain measure ments, and thus it would seem that the very mention of coordinates had already presupposed the idea of distance. This, however, need not be the case. We can assume a point in space to be completely defined by three purely numerical quantities. It will be supposed that to each group of three coordinates corresponds one point, and that conversely to one point will correspond three coordinates and no ambiguity is to be present. This latter considera tion will exclude from our present view such cases as those where the position of a point is defined by a line and two angles, because angles are subject to a well-known ambiguity amounting to any even multiple of TT. In this case it would not be true that to one point corresponds one set of coordinates, although the converse may be correct. It is necessary to understand clearly the nature of the suppositions which are made with regard to space by this assumption. Let x, y, z and x, y , z be the coordinates of two points a and a . Now x, y, z can change continu ously by any law into x , y , z. Each intermediate stage will give the coordinates of a point. It must thus be possible to pass continuously in an infinite number of ways from the point a to the point a. We thus assume that space is continuous when we have assumed that its points are represented by coordinates. It must be observed that we predicate nothing as to space which is not involved in the fact that to each point corresponds one group of three coordinates. To some extent the considerations now before us will apply to any other continuous manifoldness which requires three coordinates for the complete specification of its elements. Take, for instance, a musical note. It can be specified accurately by its pitch, intensity, and timbre. These three quantities may be regarded as the three coordi nates which will discriminate one sound from the rest. The manifoldness comprising all musical notes is, however, very different from the manifoldness which embraces all the points of space. Each of these manifoldnesses is no doubt continuous, and each of them is of three dimensions, but the conception of distance can have no place in the musical manifoldness. This is due to the absence from the musical manifoldness of anything parallel with the conception of rigidity in the space manifoldness. These remarks will show that the conception of " distance " is something of a special type even in a three-dimensioned continuous manifoldness. There are also other three-dimensioned and continuous manifoldnesses from which the conception of distance is also absent. Take, for instance, the manifoldness which embraces all the circles that can lie in a given plane. The points of such a manifoldness are the circles. It is three- dimensioned, for two coordinates will be required for the centre of each circle and one for its radius. It is obviously a continuous manifoldness, for one circle may by infinitely graduated modifications pass into any other. Yet from this manifoldness also the conception of distance is absent. There is no intelligible relation of one circle to another which is analogous to the distance which we require to determine. We shall now give the investigation of Helmholtz, by which the analytical form of the function expressing the distance is to be ascertained (Gottingen Nachrichten, 1868, pp. 193 sq.). It must be remembered that our definition of a point will be purely analytical. Suppose three different scales of pure quantity each extending from -co to + GO . Each of these scales is perfectly continuous, so that, no matter how close any two elements in the scale may be, it is always possible to conceive the insertion of an infinite number of intermediate elements. A point is to be defined for our present purpose as a group of three numerical magnitudes taken one from each of the three scales. This conception may be stated more generally. We can conceive n different numerical scales. Then a grottp of n numbers, one from each scale, will define an element of a continuous re-fold manifoldness. It will be obvious that unless the theory of distance possess a special character it will not be possible for a rigid body to exist. Take, for instance, five points in a rigid body ABODE. There are ten different pairs of points and ten corresponding distances ; all these ten distances must remain unchanged when the body is displaced. We may assume the position of A arbitrarily. Then after the dislpacement B must be placed at the right distance from A, but will only be limited by this condition to a certain surface, C must be placed at the right distance from A and from B, thus C will be limited to a certain curve, D must be placed at the proper distances from A, from B, and from C. These conditions will be sufficient to define D with complete definiteness. In the same way E will be completely defined by its distances from A, B, and C, but as D and E are thus fully defined we have no guarantee that the distance DE shall retain, after the translation, the same value which it had before. This then indicates that the function which is to express the distance must have a very special form. Any arbitrary function of the six coordinates of the two points would in general not fulfil the condition that the distance DE after the transforma tion will retain the same value as it had before. If a greater number of points than five be taken, the conditions which a rigid system must fulfil become still more nume rous. Let x, y, z be the coordinates of a point in a rigid body free to rotate around a point. We shall assume that x, y, z is in the vicinity of the fixed point, and that the displacement of the body is such that a second point remains unaltered. Tn other words, the displacement is to be a rotation around a line joining the two points, and ve shall also assume that when this rotation has been completed every point will be restored to its original position. Let i be the angle of rotation around the axis, then x, y, z will all be functions of vj, and we may assume that the following equations will hold dr] In the first place it is plain that these differential coefficients must be functions of x, y,z, and, these functions being expanded in ascend ing powers, we may omit all powers above the first. It will also be obvious that the absolute terms must be zero as the origin is by hypothesis to be a fixed point. As the displacement is a rotation, it follows that the differential coefficients must be zero for certain values of x, y, z different from zero, but this involves the condition = 0. We now proceed to solve the three linear differential equations by the well-known process. If wo multiply the three equations by I, m, n respectively, and if we determine I, m, n so that Ih = Ia + ma-^ + na. + nc. 2 , where h is another constant determined by the equation a h , KI , # a = , b , b c , c- the differential equations then give - (Ix + my + nz) = h(lx + my + nz) , whence Ix + my + nz = Cc hr i . We have already seen that one of the values of h must be zero, whence if the other values be h l and h. 2 we have the three equations l. 2 x + m> 2 y + n, 2 z = Cc 7 ^ . It is plain that h^ and A 2 cannot be real quantities, for then the quantities l^x + 111$ + n-^z and l. 2 x + m<$ + n z z could attain any values from- oo to + co according to the variations in rj. If h l and h. 2 are imaginary then will also the corresponding values of I, m, n be imaginary. We therefore write 7^ = +<mi h. 2 = - coi L == A0 T A^ i so that A a; + ^y + v z = Ae 6r i cos (co?/ + c) tX + ^y + v^ = Ae 6r > sin (<ay + c) ; in which case we have But it is plain that unless 9 be zero the left-hand side of tliis equation will be susceptible of indefinite increase, which is contrary to our hypothesis. We are therefore entitled to assume that = 0. The two roots of the cubic for h must, therefore, be pure imaginaries, and thus we have the condition ffl + &! + c 2 = . Finally we have for the determination of x, y, z the following threo equations : I x + m y + n z = const. z = A cos (wy + c) = A sin (cay + c) . It will simplify the subsequent calculations if we now make such a transformation of the coordinates as will enable us to write We shall then have from the results just obtained = dZ -j- = +o> . dt The movement corresponding to t is such as leaves unaltered all points of which the Y and the Z are equal to zero. 1 Let us now suppose another displacement to be given to the system by a rotation 77 about another axis so chosen that all the points for which X = and Z = shall remain unaltered. For this condition to be fulfilled we must have the equation for tlien each side of these equations will be equal to zero for points which make X and Z zero. The condition that the roots of h shall be purely imaginary gives us If the body receives both the rotation 77 and the rotation 77 then the joint effect of these two rotations must be equal to that of a single rotation <, so that dX dX

^Y = ^Y <2Y d<f> dr, dr, dZ dZ dZ d<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 > . - This condition reduces to coa ( (/! &>) - 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> 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> 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 > 0o > 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 ^

dX drj" = _ dty dr) dri" dZ dZ dZ dty dr) dr)" we shall then have by substitution But, if this is to represent a rotation, As this is always to be true, even suppose g , /,, and g< 2 for instance were multiplied by a common factor, it is plain that we must have and BS^O"! = ^ The first condition is to be satisfied by making g 2 zero ; for neither of the other possible solutions is admissible if the coordi nates are to be presented from indefinite increase. In a similar way the second condition requires that v l must be zero. Resuming now the three groups of equations, they are as follows: ^7V ^7V .JV CtA. _ W-.A Cl-A. ._ df) dY 7 -r- = - UlL dr, dZ v -7- = +oiY dr, drf fr dZ v J- =2 X Finally let us suppose that the body receives all three motions simultaneously. The resulting motion must still be a rotation, and thus we have the condition , ff > v o /x , , -a. a., . + u , = 0, which when expanded gives It is therefore necessary that /i"o- a 2</o = - But, as this is homogeneous in the two component rotations involved, it does not follow that the separate terms of this equation must necessarily be zero. We satisfy this condition by writing kfi--ffo and ka. 2 =i . Let the body next receive any displacement, 8?;, 877 , and 877", then we have in general , v rfX . dX. , dX , SA= -, STJ+ T~? J ? + TT/OTJ . drj di di)" with similar equations for 8Y and 8Z. By substitution these equations become SX = - .j/tZ8ij - SY= - SZ= + If we multiply the first of these equations by X, the second by Y, and the third by kZ, and add, we find const. Here we have attained the fundamental property which the coordinates must satisfy. If k be equal to unity then we have the well-known condition of ordinary space and ordinary rectangulat coordinates, but it will be seen that there is nothing in the preced ing investigation to make it necessary that k should be unity. There are therefore a singly infinite variety of spaces in which it is possible for a rigid body to be displaced. The different values of k thus correspond to the different "curvatures" which a space might have while it still retained the fundamental property which is necessary for measurement by congruence. It will now be proper to study the special charac teristics of the space with which we are familiar. It has been called flat space or faomaloidal space to distinguish it from other spaces in which the curvature is not zero. It is manifest that the characteristic features of our space are not necessarily implied in the general notion of an extended quantity of three dimensions and of the mobility of rigid figures therein. The characteristic features of our space are not necessities of thought, and the truth of Euclid s axioms, in so far as they specially differentiate our space from other conceivable spaces, must be established by experience and by experience only. The special feature of our space, by which it is distin guished from spherical space on the one hand and pseudo- spherical space on the other hand, depends upon what Riemann calls the measure of curvature. If the sum of the three angles of a triangle is to be two right angles, and if the geometrical similarity of large figures and small figures is to be possible, then the measure of curvature must be zero. Now all measurements that can b.e made seem to confirm the axiom of parallels and seem to show the measure of curvature of our space to be indistinguish able from zero. It can be proved that the amount by which the three angles of a triangle would differ from two right angles in curved space depends upon the area of the triangle. The greater the area of the triangle the greater is the difference. To test the famous proposition, Euclid i. 32, it will there fore not be sufficient to measure small triangles. It might be contended that in small triangles the difference between the sum of the three angles and two right angles was so small as to be inextricably mixed up with the unavoidable errors of measurement. Seeing therefore that small triangles obey the law, it is necessary to measure large triangles, and the largest triangles to which we have access are, of course, the triangles which the astronomers have found means of measuring. The largest available triangles are those which have the diameter of the earth s orbit as a base and a fixed star at the vertex. It is a very curious circumstance that the investigations of annual parallax of the stars are precisely the investigations which would be necessary to test whether one of these mighty triangles had the sum of its three angles equal to two right angles. It must be admitted that the parallax-seeking astronomers have never yet found any reason to think that there is any measurable difference. If there were such a difference it would probably be inextricably mixed up with the annual parallax itself. Were space really pseudospherical, then stars would exhibit a real parallax even if they were infinitely distant. Astronomers have sometimes been puzzled by obtaining a negative parallax as the result of their labours. No doubt this has generally or indeed al ways arisen from the errors which are inevitable in inquiries of this nature, but if space were really curved then a nega tive parallax might result from observations which possessed mathematical perfection. It must, however, be remembered that even the triangles of the parallax investigations are utterly insignificant when compared with the dimensions of space itself. Even the whole visible universe must be only an uncon- ceivably small atom when viewed in its true relation to infinite space. It may well be that even with the parallax triangles our opportunities of testing the proposition are utterly inadequate to pronounce on the presence or absence of curvature in space. It must remain an open question whether if we had large enough triangles the sum of the three angles would still be two right angles. Helmholtz illustrates the subject by considering the representation of space which is obtained in a spherical mirror. A mirror of this kind represents the objects in front of it in apparently fixed positions behind the mirror. The images of the sun and of other distant objects will lie behind the mirror at a distance equal to its focal length, or, to quote the description of Helmholtz " The image of a man measuring with a rule a straight line from the mirror would contract more and more the farther he went, but with his shrunken rule the man in the image would count out exactly the same number of centimetres as the real man. And in general all geometrical measurements of lines or angles made with regularly varying images of real instruments would yield exactly the same results as in the outer world. All congruent bodies would coincide on being applied to one another in the mirror as in the outer world. All lines of sight in the outer world would be repre sented by straight lines of sight in the mirror. In short I do not see how men in the mirror are to discover that their bodies are not rigid solids and their experiences good examples of the correctness of Euclid s axioms. But if they could look out upon our world as we can look into theirs, without overstepping the boundary, they must declare it to be a picture in a spherical mirror, and would speak of us just as we speak of them ; and if two inhabitants of the different worlds could communicate with one another, neither so far as I can see would be able to convince the other that he had the true, the other the distorted relations. Indeed I cannot see that such a question would have any meaning at all so long as mechani cal considerations are not mixed up with it." A very important contribution to this subject has been made by Professor Simon Newcomb, entitled " Elementary Theorems Relating to the Geometry of a Space of Three Dimensions and of Uniform Positive Curvature in the Fourth Dimension," see Jour. f. d. reine und angeivandte Math ., vol. Ixxxiii., Berlin, 1877. He commences by assuming the three following postulates : 1 " That space is triply extended, unbounded, without properties dependent either upon position or direction, and possessing such planeness ^in its smallest parts that both the postulates of the Euclidean geometry and our common conceptions of the relations of the parts of space are true for every indefinitely small region in space." 2. " That this space is affected with such curvature that a right line shall always return into itself at the end of a finite and real distance 2D without losing in any part of its course that symmetry with respect to space on all sides of it which constitutes the funda mental property of our conception of it." 3. That if two right lines emanate from the same point making the indefinitely small angle a with each other, their distance apart at the distance Y from the point of intersection will be given by the equation 2aD . 7-7T Newcomb also assumes that two straight lines intersect only in a single point. lie defines a " complete right line " as one returning into itself, as supposed in postulate 2. Any small portion of it is to be conceived as a Euclidean right line. The locus of all complete right lines passing through the same point and touching a Euclidean plane through that point will be called a " complete plane." A " region " will mean any indefinitely small portion of space iu which we are to conceive of the Euclidean geometry as holding true. Newcomb then proceeds to demonstrate the following proposi tions. I. All complete right lines are of the same length 2D. Hence D is the greatest possible distance at which the points can be situated, it being supposed that the distance is measured on the line of least absolute length. II. The complete plane is a Euclidean plane in every region of its extent. III. Every system of right lines passing through a common point A and making an indefinitely small angle with each other are parallel to each other in the region A at distance 1). IV. If a system of right lines pass in the same plane through A the locus of their most distant points will be a complete right line. Y. The locus of all the points at distance D from a fixed point A is a complete plane, and indeed a double plane if we consider as distinct the coincident surfaces in which the two opposite lines meet. VI. Conversely, all right lines perpendicular to the same com plete plane meet in a point at the distance D on each side of the plane. VII. For every complete right line there is a conjugate com plete right line such that every point of the one is at distance D from every point of the other. VIII. Any two planes in space have as a common perpendicular the right line joining their poles, and intersect each other in the conjugate to that right line. IX. If a system of right lines pass through a point, their con jugates will be in the polar plane of that point. If they also be in the same plane the conjugates will all pass through the pole of that plane. X. The relation between the sides a, &, c of a plane triangle in curved space and their opposite angles A, B, C will be the same as in a Euclidean spherical triangle of which the corresponding sides are ^ 2D 20 2D XL Space is finite, and its total volume admits of being definitely expressed by a number of Euclidean solid units which is a function of D. 8D : * XII. The total volume of space is - . IT XIII. The two sides of a complete plane are not distinct, as in a Euclidean surface. XIV. If moving along a right line we erect an indefinite series of perpendiculars each in the same Euclidean plane with the one which precedes it, then on completing the line and returning to our starting point, the perpendiculars will be found pointing in a direction the opposite of that with which we started. Newcomb concludes thus : " It may be also remarked that there is nothing within our experience which will justify a denial of the possibility that the space in which we find ourselves may be curved in the manner here supposed. It might be claimed that the dis tance of the farthest visible star is but a small fraction of the greatest distance D, but nothing more. The subjective impossibility of conceiving of the relation of the most distant points in such a space does not render its existence incredible. In fact our difficulty is not unlike that which must have been felt by the first man to whom the idea of the sphericity of the earth was suggested in con ceiving how by travelling in a constant direction he could return to the point from which he started without during his journey feel ing any sensible change in the direction of gravity." A sketch of the non-Euclidean geometry is given by Professor O. Chrystal in the Proc. Roy. Soc. Edin., vol. x., session 187_9-80. The study of this paper is recommended to all who desire to study the elements of what has been called "pangeo- metry." A more extensive work, which contains the theories of Riemann and Helmholtz, is J. Frischauf s Elementc dcr absolutcn Geometric, Leipsic, 1876. A fundamental step in the abstract theory of measure ment was taken by Professor Cayley in his " Sixth Memoir upon Quantics," Philosophical Transactions, vol. cxlix. (1859). The theory thus originated by Cayley has been 665 more fully developed by Klein in his memoir " Ueber dio nicht-Euclidische geometrie," Mathematische Annalen, vol. iv. p. 573. We shall here enter into this theory in some detail, for in it lies the true foundation of geometrical measurement. A sketch of the theory was given by the author of the present article in Hermathena, No. vi. pp. 500-541, Dublin, 1879. This theory may be regarded merely as a more general ized method of measuring distances and angles in ordinary space, but the results to which it leads are in many respects identical with those to which we are conducted by the theory just discussed. For instance, Newcomb s principle as to the length of the shortest distance between two points never exceeding a certain magnitude is common to his theory and to Cayley s. The theory of Cayley has, however, claims on our attention of a special kind. We here deal with the space with which Euclid has made us familiar, only observing that it is the measurements in that space which are to be conducted on a more general principle. We commence by assuming the existence of a certain surface called the "fundamental quadric," often called " the absolute." By the aid of this quadric and an arbitrary constant c we determine the generalized distance between the points in accordance with the following definition : The distance between two points is equal to c times the logarithm of the anharmonic ratio in which the line joining the two points is divided by the fundamental quadric. Let us first test this theory by a very obvious principle which any theory of distance ought to fulfil. It is plain that, if P, Q, R be three collinear points, then in ordinary measurement we ought to have but it is easy to see that this condition is fulfilled in the generalized measurement. Let the line PQ cut the fundamental quadric in the two points X, Y, then we have PQ = clog(PX-f-PY)-clog(QX-=-QY) QR = clog(QX-f-QY)-clog(KX-^RY) PR = clog (PX^PY) - clog(RX^RY) ; whence, as in the ordinary measures, PQ+QR=PR. It is also obvious that ir the generalized as in the ordinary measures (PQ)--(QP), and that the distance between the coincident points is equal to zero. From an obvious property of logarithms we also learn the im portant fact that the generalized distance between the points is indeterminate to the extent of any integral number of the periods 2ciir. The distance from any point to its harmonic conjugate witli respect to the two fundamental points is equal to civ. We thus see that the distance between any two harmonic conjugates is con stant. It is usual to make the arbitrary constant c equal to - i-^-2, in which case we see that the distance between the two harmonic conjugates is equal to ir-^-2. It can also be shown that, if the two absolute points on a right line coalesce, then the generalized s} T stem of measurement degrades to the ordinary system. The two abso lute points are at an infinite distance from every other point, so that in the generalized system of measurement every right line has two points at infinity, and in general all the points in space which lie at infinity are situated on the fundamental quadric. In ordinary geometry we define a circle to be the locus of a point which is at a constant distance from a given point. In the more generalized geometry we retain the same definition of the circle, only observing that the distance to be constant must be expressed in the generalized manner. The plane of course cuts the absolute in a conic section, so that the determination of the circle whose centre is C is the following problem in conic sections : Through a fixed point a straight line OP is drawn which cuts a given conic in the points X, Y ; determine the locus of P so that the anhar monic ratio (0, P, X, Y) shall remain constant. This problem is most readily solved by projecting the conic into a circle the centre of which is the projection of O. The problem then assumes the very simple form. On the diameter of a fixed circle a point P is taken so that the anharmonic ratio of the four points consisting of P , the centre O , and the two points in which the line O P cuts the circle remains constant. It is required to find the locus of P . The solution is obvious, and hence we learn that & conic which has double contact with the fundamental conic is a circle in the generalized sense, and the centre of that circle is the pole of the chord of contact. A system of conies which have double contact with the fundamen tal conic in the same two points form a system of concentric circles, and the centre of the system is the pole of the chord of contact. We are accustomed in ordinary geometry to admit that every circle passes through the two imaginary circular points at infinity. This is the specialized form of the general theorem which asserts that every circle has double contact with the fundamental conic. The two theorems indeed coincide if the fundamental conic degrades to the infinity of ordinary measurement. A critical case is presented when the chord through coincides with either of the two tangents which may be drawn from to the fundamental conic. The two fundamental points then coincide, and hence the distance between any two points on a tangent to the fundamental conic is equal to zero. We have thus the curious result that in every system of concentric circles, including even the fundamental conic itself, the two points common to the system of circles are at the distance zero from the centre of the system. In fact the pair of tangents from the centre may be regarded as a conic having double contact with the fundamental conic, and therefore forming one of the circles of the concentric system of which the radius is zero. The reader will at once perceive the analogy to a well-known phenomenon in ordinary geometry. The equation in rectangular coordinates a; 2 + 7/ 2 =0 denotes either a circle of which the radius is zero or the pair of lines in the latter case we are obliged to admit that the distance of any point on either of these lines from their intersection is equal to zero. We have now to consider the displacement of a rigid figure, and we shall for the present speak only of a plane movement. We shall first show that it is possible for a plane figure to receive such a displacement that the distance between every two points in the figure after the displacement is equal to what it was before. Let x, y, z be the trilinear coordinates of a point in a plane, and suppose that x , y , z are the coordinates of the position to which this point is transferred in accordance with the linear transfor mation x =ax +by +cz y = a x + b y + c z z = a"x + b"y + c"z . There are in general three points in the plane which are not altered by this transformation ; for, if we assume x = px, y = P y, z =oz, we have for p the cubic equation a p b b -p b" c c"- P = 0. The three values of p which satisfy this equation determine the coordinates of the three points. It is natural to take the sides of the triangle formed by these three points as the three lines of refer ence, in which case, if o, & 7 be constants, the system of equations assume the simple form x = ax, y = frj, z = yz. It is easily shown that four collinear points before the transforma tion are collinear after the transformation, and that their anhar- monic ratio is unaltered. This general form of linear transformation must be specialized in order to represent the movement. As no finite movement can either bring a point to infinity or from infinity, it is obvious that the displacement must be such as to leave the fundamental conic unaltered. It is easily seen that the specification of the transforma tion in its general form requires eight constants ; viz., the ratios of the nine quantities a, b, c, a , b , c , a", b", c". We may imagine five_of these constants to be disposed of by the provision that the conic shall remain unaltered. There will still remain three dis posable constants to give variety to the possible displacements. Although the fundamental conic will coincide with itself after the transformation, yet it generally happens that each point thereon will slide along the conic during the transformation. It is, how ever, very important to observe that there are two exceptions to this statement. Let 0, A, B, C be four points upon the fundamental conic which by transformation move into the positions , A , B , C . If OX be one of the double rays of the systems OA, OB, 00 and OA , OB , OC , and if we use the ordinary notation for anharrnonic pencils, then we have 0(A, B, C, X) = 0(A , B , C , X). But the anharrnonic ratio subtended by four points on a conic at any fifth point is constant, whence 0(A , B , C , X) = (A , B .C , X), and therefore 0(A, B, C, X) = (A , B , C , X). Suppose the transformation moved X to X , Jlien since the an- hannonic ratio of a pencil is unaltered by transformation we have 0(A, B, C, X) = (A , B , C , X ); whence (A , B , C , X) = (A B C X ); but this can only be true if the rays O X and O X are coincident, in which case X and X are coincident, whence it follows that the point X has remained unaltered notwithstanding the transforma tion. Similar reasoning applies to the point Y defined by the other double ray, and hence we have the following theorem : In that linear transformation of the points in a plane which con stitutes a generalized movement, there are two points upon the fundamental conic which remain, unchanged. It also follows that the tangents to the fundamental conic at the points X and Y, as well as the chord of contact, must remain unaltered. These two tangents and their chord of contact must therefore form the triangle of reference to which we were previously conducted by the general theory of this transformation. It will now easily appear how a transformation of this kind is really a displacement of a rigid plane. The distance between each pair of points is expressed by an anharmonic ratio; such ratios are unchanged by the transformation, and the two points which lay on the absolute originally are also there after the transformation. It therefore appears that the distance in the generalized sense between every pair of points is unchanged by the transformation. In other words, a rigid system will admit of a displacement of the kind now under consideration. If we denote the two tangents at the unaltered points on the conic by =0, y=0, and the chord of contact by z = 0, then the equation to the absolute is xy-k 2 z 2 =*Q. Transforming this equation by the substitution x = ax, y = Py, s = 7 2 . we see that the condition <xj8-=y 2 must be fulfilled. It is very remarkable that the fundamental conic is only one of a family of conies, each of which remains unaltered by the transforma tion. In fact every generalized circle of which the intersection of the two tangents is the centre has for its equation xy - h z 2 ; and, whatever h may be, this circle remains unaltered by the transfor mation. Hence we have the following remarkable theorem: When a plane rigid system is displaced upon itself there is one point of the system which remains unaltered, and all the circles which have as their centre remain unaltered also. It is quite natural to speak of this motion as a "rotation," and thus we may assert the truth in generalized measurement of the well-known theorem in ordinary geometry that Every displacement of a plane upon itself could have been prod need by a rotation of the plane around a certain point in the plane. Notwithstanding the rotation of the plane round 0, the two tangents from to the fundamental conic and also their chord of intersection, or the polar of 0, remain unaltered; each point on the polar of is displaced along the polar, and we would in ordinary geometry call this motion a translation parallel to the polar. It thus appears that, in the sense now attributed to the words rota tion and translation, a rotation round a point is identical with a translation along the polar of the point. Another point on which the present theory throws light on tho ordinary geometry must be here alluded to. We have seen that the two tangents from to the fundamental conic remain unchanged during the rotation of the plane round 0. It certainly does seem paradoxical to assert that a plane, and all it contains are rotated around a point, and that nevertheless this operation does not alter the position of a certain pair of lines in the plane which pass through the point. But have we not precisely the same difficulty in ordinary geometry ? Let us suppose that a plane pencil of rays is rotated through an angle 9 about the origin. Then a line through the origin whose equation before the rotation was becomes after the rotation xcos The lines thus represented are of course in general different, but the" will be the same if It follows that even in ordinary geometry the two lines xHy = Q

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 2*i* 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* + *h*²*yw* = 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.

*Units of Measurement*.—A most excellent account of the
units employed in scientific measurements will be found in
Professor J. D. Everett's *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.

(r. s. b.)

- ↑
*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. - ↑ 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.