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[PDF page 313] CH. XIII] TIME AND ITS MEASUREMENT. 297 style cast on the plate by the sun falls on lines engraved there which are marked with the corresponding hours. The earliest sun-dial, of which I have read, is that made by Berosus in 540 b.c. One was erected by Meton at Athens in 433b.c. The first sun-dial at Rome was constructed by Papirius Cursor in 306b.c. Portable sun-dials, with a compass fixed in the face, have been long common in the East as well as in Europe. Other portable instruments of a similar kind were in use in medieval Europe, notably the sun-rings, hereafter described, and the sun-cylinders*. I believe it is not generally known that a sun-dial can be so con- structed that the shadow will, for a short time near sunrise and sunset, move backwards on the dial†. This was discovered by Nonez. The ex- planation is as follows. Every day the sun appears to describe a circle round the pole, and the line joining the point of the style to the sun describes a right cone whose axis points to the pole. The section of this cone by the dial is the curve described by the extremity of the shadow, and is a conic. In our latitude the sun is above the horizon for only part of the twenty-four hours, and therefore the extremity of the shadow of the style describes only a part of this conic. LetQQ′ be the arc described by the extremity of the shadow of the style from sunrise at Q to sunset at Q′, and let S be the point of the style and F the foot of the style,i.e. the point where the style meets the plane of the dial. Suppose that the dial is placed so that the tangents drawn from F to the conic QQ′ are real, and that P and P ′, the points of contact of these tangents, lie on the arcQQ′. If these two conditions are fulfilled, then the shadow will regrede through the angleQF P as its extremity moves from Q to P, it will advance through the angle P F P′ as its extremity moves fromP to P ′, and it will regrede through the angle P ′F Q′ as its extremity moves fromP ′ to Q′. If the sun’s apparent diurnal path crosses the horizon—as always happens in temperate and tropical latitudes—and if the plane of the dial is horizontal, the arcQQ′ will consist of the whole of one branch of a hyperbola, and the above conditions will be satisfied ifF is within the space bounded by this branch of the hyperbola and its asymptotes. * Thus Chaucer in the Shipman’s Tale, “by my chilindre it is prime of day,” and Lydgate in theSiege of Thebes, “by my chilyndre I gan anon to see...that it drew to nine.” † Ozanam, 1803 edition, vol.iii, p. 321; 1840 edition, p. 529.

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[PDF page 314] 298 TIME AND ITS MEASUREMENT. [CH. XIII As a particular case, in a place of latitude12◦ N. on a day when the sun is in the northern tropic (of Cancer) the shadow on a dial whose face is horizontal and style vertical will move backwards for about two hours between sunrise and noon. If, in the case of a given sun-dial placed in a certain position, the conditions are not satisfied, it will be possible to satisfy them by tilting the sun-dial through an angle properly chosen. This was the rational- istic explanation, offered by the French encyclopaedists, of the miracle recorded in connection with Isaiah and Hezekiah*. Suppose, for in- stance, that the style is perpendicular to the face of the dial. Draw the celestial sphere. Suppose that the sun rises atM and culminates at N, and let L be a point between M and N on the sun’s diurnal path. Draw a great circle to touch the sun’s diurnal pathM LN at L, let this great circle cut the celestial meridian inA and A′, and of the arcs AL, A′L suppose that AL is the less and therefore is less than a quadrant. If the style is pointed to A, then, while the sun is ap- proaching L, the shadow will regrede, and after the sun passesL the shadow will advance. Thus if the dial is placed so that a style which is normal to it cuts the meridian midway between the equator and the tropic, then between sunrise and noon on the longest day the shadow will move backwards through an angle sin−1(cos ω sec 1 2 ω) − cot−1{sin ω cos(l − 1 2 ω)(cos2 l − sin2 ω)− 1 2 } , where l is the latitude of the place andω is the obliquity of the ecliptic. The above remarks refer to the sun-dials in ordinary use. In 1892 General Oliver brought out in London a dial with a solid style, the section of the style being a certain curve whose form was determined empirically by the value of the equation of time as compared with the sun’s declination†. The shadow of the style on the dial gives the local mean time, though of course in order to set the dial correctly at any place the latitude of the place must be known: the dial may be also set so as to give the mean time at any other locality whose longitude relative to the place of observation is known. The sun-ring or ring-dial is another instrument for measuring solar time‡. One of the simplest type is figured in the accompanying diagram. * 2 Kings, chap. xx, vv. 9–11. † An account of this sun-dial with a diagram was given inKnowledge, July 1, 1892, pp. 133, 134. ‡ See Ozanam, 1803 edition, vol.iii, p. 317; 1840 edition, p. 526.

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[PDF page 315] CH. XIII] TIME AND ITS MEASUREMENT. 299 The sun-ring consists of a thin brass band, about a quarter of an inch wide, bent into the shape of a circle, which slides between two fixed circular rims—the radii of the circles being about one inch. At one point of the band there is a hole; and when the ring is suspended from a fixed point attached to the rims so that it hangs in a vertical plane containing the sun, the light from the sun shines through this hole and makes a bright speck on the opposite inner or concave surface of the ring. On this surface the hours are marked, and, if the ring is properly adjusted, the spot of light will fall on the hour which indicates the solar time. The adjustment for the time of year is made as follows. The rims between which the band can slide are marked on their outer or convex side with the names of the months, and the band containing the hole must be moved between the rims until the hole is opposite to that month for which the ring is being used. For determining times near noon the instrument is reliable, but for other hours in the day it is accurate only if the time of year is properly chosen, usually near one of the equinoxes. This defect may be corrected by marking the hours on a curved brass band affixed to the concave surface of the rims. I possess two specimens of rings of this kind. These

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[PDF page 316] 300 TIME AND ITS MEASUREMENT. [CH. XIII rings were distributed widely. Of my two specimens, one was bought in the Austrian Tyrol and the other in London. Astrolabes and sea-rings can be used as sun-rings. Clepsydras or water-clocks, andhour-glasses or sand-clocks, afford other means of measuring time. The time occupied by a given amount of some liquid or sand in running through a given orifice under the same conditions is always the same, and by noting the level of the liquid which has run through the orifice, or which remains to run through it, a measure of time can be obtained. The burning of graduated candles gives another way of measuring time, and we have accounts of those used by Alfred the Great for the purpose. Incense sticks were used by the Chinese in a similar way. Modern clocks and watches * comprise a train of wheels turned by a weight, spring, or other motive power, and regulated by a pendulum, balance, fly-wheel, or other moving body whose motion is periodic and time of vibration constant. The direction of rotation of the hands of a clock was selected originally so as to make the hands move in the same direction as the shadow on a sun-dial whose face is horizontal—the dial being situated in our hemisphere. The invention of clocks with wheels is attributed by tradition to Pacificus of Verona, circ. 850, and also to Gerbert, who is said to have made one at Magdeburg in 996: but there is reason to believe that these were sun-clocks. The earliest wheel-clock of which we have historical evidence was one sent by the Sultan of Egypt in 1232 to the Emperor Frederick II, though there seems to be no doubt that they had been made in Italy at least fifty years earlier. The oldest clock in England of which we know anything was one erected in 1288 in or near Westminster Hall out of a fine imposed on a corrupt Lord Chief Justice. The bells, and possibly the clock, were staked by Henry VIII on a throw of dice and lost, but the site was marked by a sun-dial, destroyed about sixty years ago, and bearing the inscription Discite justiciam moniti. In 1292 a clock was erected in Canterbury Cathedral at a cost of £30. One erected at Glastonbury Abbey in 1325 is at present in the Kensington Museum and is in regular action. Another made in 1326 for St Alban’s Abbey showed the astro- nomical phenomena, and seems to have been one of the earliest clocks that did so. One put up at Dover in 1348 is still in good working order. * See Clock and Watch Makingby Lord Grimthorpe, 7th edition, London, 1883.

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[PDF page 317] CH. XIII] TIME AND ITS MEASUREMENT. 301 The clocks at Peterborough and Exeter were of about the same date, and portions of them remainin situ. Most of these early clocks were regulated by horizontal balances: pendulums being then unknown. Of the elaborate clocks of a later date, that at Strasburg made by Dasy- podius in 1571, and that at Lyons constructed by Lippeus in 1598, are especially famous: the former was restored in 1842, though in a manner which destroyed most of the ancient works. In 1370, Vick constructed a clock for Charles V with a weight as motive power and a vibrating escapement—a great improvement on the rough time-keepers of an earlier date. The earliest clock regulated by a pendulum seems to have been made in 1621 by a clockmaker named Harris, of Covent Garden, Lon- don, but the theory of such clocks is due to Huygens*. Galileo had dis- covered previously the isochronism of a pendulum, but did not apply it to the regulation of the motion of clocks. Hooke made such clocks, and possibly discovered independently this use of the pendulum: he invented or re-invented the anchor pallet. A watch may be defined as a clock which will go in any position. Watches, though of a somewhat clumsy design, were made at Nurem- berg by P. Hele early in the sixteenth century—the motive power being a ribbon of steel, wound round a spindle, and connected at one end with a train of wheels which it turned as it unwound—and possibly a few similar time-pieces had been made in the previous century. By the end of the sixteenth century they were not uncommon. At this time they were usually made in the form of fanciful ornaments such as skulls, or as large pendants, but about 1620 the flattened oval form was introduced, rendering them more convenient to carry in a pocket or about the person. In the seventeenth century their construction was greatly improved, notably by the introduction of the spring balance by Huygens in 1674, and independently by Hooke in 1675—both mathe- maticians having discovered that small vibrations of a coiled spring, of which one end is fixed, are practically isochronous. The fusee had been used by R. Zech of Prague in 1525, but was re-invented by Hooke. Clocks and watches are usually moved and regulated in the manner indicated above. Other motive powers and other devices for regulating the motion may be met with occasionally. Of these I may mention a clock in the form of a cylinder, usually attached to another weight as * Horologium Oscillatorium, Paris, 1673.

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[PDF page 318] 302 TIME AND ITS MEASUREMENT. [CH. XIII in Atwood’s machine, which rolls down an inclined plane so slowly that it takes twelve hours to roll down, and the highest point of the face always marks the proper hour*. A water-clock made on a somewhat similar plan is described by Ozanam† as one of the sights of Paris at the beginning of the last century. It was formed of a hollow cylinder divided into various com- partments each containing some mercury, so arranged that the cylinder descended with uniform velocity between two vertical pillars on which the hours were marked at equidistant intervals. Other ingenious ways of concealing the motive power have been described in the columns ofLa Nature‡. Of such mysterious timepieces the following are not uncommon examples, and probably are known to most readers of this book. One kind of clock consists of a glass dial suspended by two thin wires; the hands however are of metal, and the works are concealed in them or in the pivot. Another kind is made of two sheets of glass in a frame containing a spring which gives to the hinder sheet a very slight oscillatory motion–imperceptible except on the closest scrutiny–and each oscillation moves the hands through the requisite angles. Some so-called perpetual motion timepieces were described above on pages 77–78. Lastly, I have seen in France a clock the hands of which were concealed at the back of the dial, and carried small magnets; pieces of steel in the shape of insects were placed on the dial, and, following the magnets, served to indicate the time. The position of the sun relative to the points of the compass de- termines the solar time. Conversely, if we take the time given by a watch as being the solar time—and it will differ from it by only a few minutes at the most—and we observe the position of the sun, we can find the points of the compass§. To do this it is sufficient to point the hour-hand to the sun, and then the direction which bisects the angle between the hour and the figurexii will point due south. For instance, if it is four o’clock in the afternoon, it is sufficient to point the hour- hand (which is then at the figureiiii) to the sun, and the figureii on the watch will indicate the direction of south. Again, if it is eight * Ozanam, 1803 edition, vol. ii, p. 39; 1840 edition, p. 212; orLa Nature, Jan. 23, 1892, pp. 123, 124. † Ozanam, 1803 edition, vol.ii, p. 68; 1840 edition, p. 225. ‡ See especially the volumes issued in 1874, 1877, and 1878. § The rule is given by W.H. Richards,Military Topography, London, 1883, p. 31, though it is not stated quite correctly. I do not know who first enunciated it.

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[PDF page 319] CH. XIII] TIME AND ITS MEASUREMENT. 303 o’clock in the morning, we must point the hour-hand (which is then at the figureviii) to the sun, and the figurex on the watch gives the south point of the compass. Between the hours of six in the morning and six in the evening the angle between the hour andxii which must be bisected is less than 180◦, but at other times the angle to be bisected is greater than180◦; or perhaps it is simpler to say that at other times the rule gives the north point and not the south point. The reason is as follows. At noon the sun is due south, and it makes one complete circuit round the points of the compass in24 hours. The hour-hand of a watch also makes one complete circuit in 12 hours. Hence, if the watch is held in the plane of the ecliptic with its face upwards, and the figurexii on the dial is pointed to the south, both the hour-hand and the sun will be in that direction at noon. Both move round in the same direction, but the angular velocity of the hour-hand is twice as great as that of the sun. Hence the rule. The greatest error due to the neglect of the equation of time is less than2◦. Of course in practice most people, instead of holding the face of the watch in the ecliptic, would hold it horizontal, and in our latitude no serious error would be thus introduced. In the southern hemisphere where at noon the sun is due north the rule requires modification. In such places the hour-hand of a watch (held face upwards in the plane of the ecliptic) and the sun move in opposite directions. Hence, if the watch is held so that the figurexii points to the sun, then the direction which bisects the angle between the hour of the day and the figurexii will point due north.

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[PDF page 320] CHAPTER XIV. MATTER AND ETHER THEORIES. Matter, like space and time, cannot be defined, but either the statement that matter is whatever occupies space or the statement that it is anything which can be seen, touched, or weighed, suggests its more important characteristics to anyone already familiar with it. The means of measuring matter and some of its properties are treated in most text-books on mechanics, and I do not propose to dis- cussthem. Iconfinethechaptertoanaccountofsomeofthehypotheses by physicists as to the ultimate constitution of matter, but I exclude metaphysical conjectures, which from their nature are mere assertions incapable of proof and are not subject to mathematical analysis. The question is intimately associated with the explanation of the phenom- ena of attraction, light, chemistry, electricity, and other branches of physics. I commence with a list of some of the more plausible of the hy- potheses formerly proposed which accounted for the obvious properties of matter, and shall then discuss how far they explain or are consis- tent with other facts†. The interest of the list is largely historical, for within the last few years new views as to the constitution of matter † I have based my account mainly onRecent Advances in Physical Science, by P.G.Tait, Edinburgh, 1876(chaps,xii, xiii); andonthearticle Atom byJ.Clerk Maxwell in the Encyclopaedia Britannica or hisCollected Works, vol.ii, pp. 445– 484: see also W.M. Hicks’s address, Report of the British Association (Ipswich meeting), 1895, vol. lxv, pp. 595–606. For the more recent speculations see J.J. Thomson,Electricity and Matter, Westminster, 1904, and J. Larmor,Aether and Matter, Cambridge, 1900. 304

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[PDF page 321] CH. XIV] MATTER AND ETHER THEORIES. 305 have been propounded, but the details of these more recent hypotheses are so complicated and technical that only professional mathematicians can understand them. Accordingly I allude to them only briefly. I. Hypothesis of Continuous Matter. It may be sup- posed that matter is homogeneous and continuous, in which case there is no limit to the infinite divisibility of bodies. This view was held by Descartes*. This conjecture is consistent with the facts deducible by untrained observation, but there are many other phenomena for which it does not account; moreover there seems to be no way of reconciling such a structure of matter either with the facts of chemical changes or with the results of spectrum analysis. At any rate the theory must be regarded as extremely improbable. II. Atomic Theories. If matter is not continuous we must suppose that every body is composed of aggregates of molecules. If so, it seems probable that each such molecule is built up by the association of two or more atoms, that the number of kinds of atoms is finite, and that the atoms of any particular kind are alike. As to the nature of the atoms the following hypotheses have been made. (i) Popular Atomic Hypothesis. The popular view is that every atom of any particular kind is a minute indivisible article possessing definite qualities, everlasting in its form and properties, and infinitely hard. This statement is plausible, but the difficulties to which it leads appeartobeinsuperable. Infactwehavereasontothinkthattheatoms which form a molecule are composite systems in incessant vibration at a rate characteristic of the molecule, and it is most probable that they are elastic. Newton seems to have hazarded a conjecture of this kind when he suggested† that the difficulties, connected with the fact that the velocity of sound was one-ninth greater than that required by theory, might be overcome if the particles of air were little rigid spheres whose distance from one another under normal conditions was nine times the * Descartes, Principia, vol. ii, pp. 18, 23. † Newton, Principia, bk. ii, prop. 50.

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[PDF page 322] 306 MATTER AND ETHER THEORIES. [CH. XIV diameter of any one of them. This was ingenious, but obviously the view is untenable, because, if such a structure of air existed, the air couldnotbecompressedbeyondacertainlimit, namely, about 1/1021st part of its original volume, which has been often exceeded. The true explanation of the difficulty noticed by Newton was given by Laplace. (ii) Boscovich’s Hypothesis.In 1759 Boscovich suggested* that the facts might be explained by supposing that an atom was an in- finitely small indivisible mass which was a centre of force—the law of force being attractive for sensible distances, alternately attractive and repulsive for minute distances, and repulsive for infinitely small dis- tances. In this theory all action between bodies is action at a distance. He explained the apparent extension of bodies by saying that two parts are consecutive (or similarly that two bodies are in contact) when the nearest pair of atoms in them are so close to one another that the repulsion at any point between them is sufficiently great to prevent any other atom coming between them. It is essential to the theory that the atom shall have a mass but shall not have dimensions. This hypothesis is not inconsistent with any known facts, but it has been described, perhaps not unjustly, as a mere mathematical fiction, and certainly it is opposed to the apparent indications of our senses. At any rate it is artificial, though it may be a prejudice to regard that as an argument against its adoption. To some extent this view was accepted by Faraday. Lord Kelvin, better known as Sir William Thomson, has shown† that, if we assume the existence of gravitation, then each of the above hypotheses will account for cohesion. (iii) Hypothesis of an Elastic Solid Ether.Some physicists have tried to explain the known phenomena by properties of the medium through which our impressions are derived. By postulating that all space is filled with a medium possessed of many of the characteristics of an elastic solid, it has been shown by Fresnel, Green, Cauchy, Neu- mann, MacCullagh, and others that a large number of the properties of light and electricity may be explained. In spite of the difficulties to which this hypothesis necessarily leads, and of its inherent improbabil- * Philosophiae naturalis Theoria redacta ad unicam Legem Virium, Vienna, 1759. † Proceedings of the Royal Society of Edinburgh, April 21, 1862, vol.iv, pp. 604– 606.

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[PDF page 323] CH. XIV] MATTER AND ETHER THEORIES. 307 ity, it has been discussed by Stokes, Lamé, Boussinesq, Sarrau, Lorenz, Lord Rayleigh, and Kirchhoff. This hypothesis has been modified and rendered somewhat more plausible by von Helmholtz, Lommel, Ketteler*, and Voigt, who based their researches on the assumption of a mutual reaction between the ether and the material molecules located in it: on this view the prob- lems connected with refraction and dispersion have been simplified. Fi- nally, Sir William Thomson, in his Baltimore Lectures, 1885, suggested a mechanical analogue to represent the relations between matter and this ether, by which a possible constitution of the ether can be realized. He also suggested later a form oflabile ether, from whose properties most of the more familiar physical phenomena can be deduced, pro- vided the arrangement can be considered stable; a labile ether is an elastic solid, and its properties in two dimensions may be compared with those of a soap-bubble film, in three dimensions. It is, however, difficult to criticise any of these hypotheses as a theory of the constitution of matter until the arrangement of the atoms or their nature is more definitely expressed. III. Dynamical Theories. In recent years the suggestion has been made that the so-called atoms may be forms of motion (ex. gr. permanent eddies) in one elementary material known as the ether; on this view all the atoms are constituted of the same matter, but the physical conditions are different for the different kinds of atoms. It has been said that there is an initial difficulty in any such hypothesis, since the all-pervading elementary fluid must possess inertia, so that to explain matter we assume the existence of a fluid possessing one of the chief characteristics of matter. This is true as far as it goes, but it is not more unreasonable than to attribute all the fundamental properties of matter to the atoms themselves, as is done by many writers. The next paragraph contains a statement of one of the earliest attempts to formulate a dynamical atomic hypothesis. (i) The Vortex Ring Hypothesis. This hypothesis assumes that each atom is a vortex ring in an incompressible frictionless homoge- neous fluid. * Theoretische Optik, Braunschweig, 1885.
