CHAPTER XX.

TIME AND ITS MEASUREMENT.
The problems connected with time are totally different in character from those concerning space which I discussed in the last chapter. I there stated that the life of people living in space of one dimension would be uninteresting, and that probably they would find it impossible to realize life in space of higher dimensions. In questions connected with time we find ourselves in a somewhat similar position. Mentally, we can realize a past and a future — thus going backwards and forwards — actually we go only forwards. Hence time is analogous to space of one dimension. Were our time of two dimensions, the conditions of our life would be infinitely varied, but we can form no con- ception of what such a phrase means, and I do not think that any attempts have been made to work it out.
The idea of time, when we examine it carefully, involves many difficulties. For instance, we speak of an instant of time as if it were absolutely definite. If so we could represent it by a point on a line, and the idea of simultaneity would be simple, for two events could be regarded as simultaneous when their representative points were coincident. But in reality sensations have an appreciable duration, even though it be very small. This duration may be represented by an interval on a line, and it would seem reasonable to say that two events are simultaneous when their representative intervals have a common part ; hence two events which are simultaneous with the same event are not necessarily simultaneous with one another. Here, however, I exclude these quasi-metaphysical questions, and concern myself
CH. XX] TIME AND ITS MEASUREMENT 439
mainly with questions concerninc^ the measurement of time, and I shall treat these rather from a historical than from a physical point of view.
In order to measure anything we must have an unalterable unit of the same kind, and we must be able to determine how often that unit is contained in the quantity to be measured. Hence only those things can be measured which are capable of addition to things of the same kind.
Thus to measure a length we may take a foot-rule, and by applying it to the given length as often as is necessary, we shall find how many feet the length contains. But in comparing lengths we assume as the result of experience that the length of the foot-rule is constant, or rather that any alteration in it can be determined ; and, if this assumption was denied, we could not prove it, though, if numerous repetitions of the experiment under varying conditions always gave the same result, probably we should feel no doubt as to the correctness of our method.
It is evident that the measurement of time is a more difficult matter. We cannot keep a unit by us in the same way as we can keep a foot-rule ; nor can we repeat the measurement over and over again, for time once passed is gone for ever. Hence we cannot appeal directly to our sensations to justify our measurement. Thus, if we say that a certain duration is four hours, it is only by a process of reasoning that we can show that each of the hours is of the same duration.
The establishment of a scientific unit for measuring dura- tions has been a long and slow affair. The process seems to have been as follows. Originally man observed that certain natural phenomena recurred after the interval of a day, say from sunrise to sunrise. Experience — for example, the amount of work that could be done in it — showed that the length of every day was about the same, and, assuming that this was accurately so, man had a unit by which he could measure durations. The present subdivision of a day into hours, minutes, and seconds is ai'titicial, and apparently is derived £iom the Babyionians,
440 TIME AND ITS MEASUREMENT [CH. XX
Similarly a month and a year are natural units of time though it is not easy to determine precisely their beginnings and endings.
So long as men were concerned merely with durations which were exact multiples of these units or which needed only a rough estimate, this did very well ; but as soon as they tried to compare the different units or to estimate durations measured by part of a unit they found difficulties. In particular it cannot have been long before it was noticed that the duration of the same day differed in different places, and that even at the same place different days differed in duration at different times of the year, and thus that the duration of a day was not an invariable
unit.
The question then arises as to whether we can jfind a fixed unit by which a duration can be measured, and whether we have any assurance that the seconds and minutes used to-day for that purpose are all of equal duration. To answer this we must see how a mathematician would define a unit of time. Probably he would say that experience leads us to believe that, if a rigid body is set moving in a straight line without any external force acting on it, it will go on moving in that line ; and those times are taken to be equal in which it passes over equal spaces : similarly, if it is set rotating about a principal axis passing through its centre of mass, those times are taken to be equal in which it turns through equal angles. Our experiences are consistent with this statement, and that is as high an authority as a mathematician hopes to get.
The spaces and the angles can be measured, and thus dura- tions can be compared. Now the earth may be taken roughly as a rigid body rotating about a principal axis passing through its centre of mass, and subject to no external forces affecting its rotation : hence the time it takes to turn through four right angles, i.e. through 360°, is always the same; this is called a sidereal day: the time to turn through one twenty-fourth part of 360°, i.e. through 15°, is an hour: the time to turn through one-sixtieth part of 15°, i.e. through 15', is a minute : and so oJi.
CH. XX] TIME AND ITS MEASUREMENT 441
If, by the progress of astronomical research, we find that there are external forces affecting the rotation of the earth, mathematics would have to be invoked to find what the time of rotation would be if those forces ceased to act, and this would give us a correction to be applied to the unit chosen. In the same way we may say that although an increase of temperature affects the length of a foot-rule, yet its change of length can be determined, and thus applied as a correction to the foot-rule when it is used as the unit of length. As a matter of fact there is reason to think that the earth takes about one sixty-sixth of a second longer to turn through four right angles now than it did 2500 years ago, and thus the duration of a second is just a trifle longer to-day than was the case when the Romans were laying the foundations of the power of their city.
The sidereal day can be determined only by refined astro- nomical observations and is not a unit suitable for ordinary purposes. The relations of civil life depend mainly on the sun, and he is our natural time-keeper. The true solar day is the time occupied by the earth in making one revolution on its axis relative to the sun ; it is true noon when the sun is on the meridian. Owing to the motion of the sun relative to the earth, the true solar day is about four minutes longer than a sidereal day.
The true solar day is not however always of the same duration. This is inconvenient if we measure time by clocks (as now for nearly two centuries has been usual in Western Europe) and not by sun-dials, and therefore we take the average duration of the true solar day as the measure of a day : this is called the mean solar day. Moreover to define the noon of a mean solar day we suppose a point to move uniformly round the ecliptic coinciding with the sun at each apse, and further we suppose a fictitious sun, called the mean sun, to move in the celestial equator so that its distance from the first point of Aries ii) the same as that of this point : it is mean noon when this mean sun . is on the meridian. The mean solar day is divided into hours, minutes, and seconds ; and these are the usual units of time in civil life.
44)2 TIME AND ITS MEASUREMENT [CH. XX
The time indicated by our clocks and watches is mean solar time; that marked on ordinary sun-dials is true solar time. The difference between them is the equation of time : this may amount at some periods of the year to a little more than a quarter of an hour. In England we take the Greenwich meridian as our origin for longitudes, and instead of local mean solar time we take Greenwich mean solar time as the civil standard.
Of course mean time is a comparatively recent invention. The French were the last civilized nation to abandon the use of true time : this was in 1816.
Formerly there was no common agreement as to when the day began. In parts of ancient Greece and in Japan the interval from sunrise to sunset was divided into twelve hours, and that from sunset to sunrise into twelve hours. The Jews, Chinese, Athenians, and, for a long time, the Italians, divided their day into twenty-four hours, beginning at the hour of sunset, which of course varies every day : this method is said to have been used as late as the latter half of the nineteenth century in certain villages near Naples, except that the day began half-an-hour after sunset — the clocks being re-set once a week. Similarly the Babylonians, Assyrians, Persians, and until recently the modern Greeks and the inhabitants of the Balearic Islands counted the twenty-four hours of the day from sunrise. Until 1750, the inhabitants of Basle reckoned the twenty-four hours from our 11.0 p.m. The ancient Egyptians and Ptolemy counted the twenty-four hours from noon: this is the practice of modern astronomers. In Western Europe the day is taken to begin at midnight — as was first suggested by Hipparchus — and is divided into two equal periods of twelve hours each.
The week of seven days is an artificial unit of time. It had its origin in the East, and was introduced into the West probably during the second century by the Poman emperors, and, except during the French Pevolution, has been subsequently in general use among civilized races. The names of the days are derived from the seven astrological planets, arranged, as was customary, in the order of their apparent times of rotation round tlie earth,
CH. XX] TIME AND ITS MEASUREMENT 443
namely, Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon. The twenty-four hours of the day were dedicated successively to these planets : and the day was consecrated to the planet of the first hour.
Thus if the first hour was dedicated to Saturn, the second would be dedicated to Jupiter, and so on ; but the day would be Saturn's day. The twenty-fourth hour of Saturn's day would be dedicated to Mars, thus the first hour of the next day would belong to the Sun ; and the day would be Sun's day. Similarly the next day would be Moon's day ; the next, Mars's day ; the next, Mercury's day; the next, Jupiter's day; and the next, Venus's day.
The astronomical month is a natural unit of time depending on the motion of the moon, and containing about 29J- days. The months of the calendar have been evolved gradually as convenient divisions of time, and their history is given in numerous astronomies. In the original Julian arrangement the months in a leap year contained alternately 31 and 30 days, while in other years February had 29 days. This was altered by Augustus in order that his month should not be inferior to one named after his uncle.
The solar tropical year is another natural unit of time. According to a recent determination, it contains 365*242216 days, that is, 365^- 5^- 48°^ 47'--4624. Civilized races usually number the passing years consecutively from some fixed date. The Romans reckoned from the traditional date of the foundation of their city. In the sixth century of our era it was suggested that the birth of Christ was a more fitting epoch from which to reckon dates, but it was not until the ninth century that this suggestion was generally adopted.
The Egj^ptians knew that the year contained between 365 and 366 days, but the Romans did not profit by this information, for Numa is said to have reckoned 355 days as constituting a year — extra months being occasionally intercalated, so that the seasons might recur at about the same period of the year.
In 46 B.C. Julius Caesar decreed that thenceforth the year should contain 365 days, except that in every fourth or leap
444> TIME AND ITS MEASUREMENT [CH. XX
year one additional day should be introduced. He ordered this rule to come into force on January 1, 45 B.C. The change was made on the advice of Sosigenes of Alexandria.
It must be remembered that the year 1 A.D. follows imme- diately 1 B.C., that is, there is no year 0, and thus 45 B.C. would be a leap year. All historical dates are given now as if the Julian calendar was reckoned backwards as well as forwards from that year*. As a matter of fact, owing to a mistake in the original decree, the Romans, during the first 36 years after 45 B.C., intercalated the extra day every third year, thus pro- ducing an error of 3 days. This was remedied by Augustus, who directed that no intercalation of an extra day should be made in any of the twelve years A.u.c. 746 to 757 inclusive, but that the intercalation should be again made in the year A.U.C. 761 (that is, 8 A.D.) and every succeeding fourth year.
The Julian calendar made the year, on an average, con- tain 365*25 days. The actual value is, very approximately, 365"242216 days. Hence the Julian year is too long by about 11 J minutes: this produces an error of nearly one day in 128 years. If the extra day in every thirty-second leap year had been omitted — as was suggested by some unknown Persian astronomer — the error would have been less than one day in 100,000 years. It may be added that Sosigenes was aware that his rule made the year slightly too long.
The error in the Julian calendar of rather more than eleven minutes a year gradually accumulated, until in the sixteenth century the seasons arrived some ten days earlier than they should have done. In 1582 Gregory XIII corrected this by omitting ten days from that year, which therefore contained only 355 days. At the same time he decreed that thenceforth every year which was a multiple of a century should be or not be a leap j^ear according as the multiple was or was not divisible by four.
The fundamental idea of the reform was due to Lilius, who died before it was carried into effect. The work of framing the new calendar was entrusted to Clavius, who explained the * Herscbel, Astronomy, London, lltli ed. 1871, arts. 916 — 919.
CH. XX] TIME AND ITS MEASUREMENT 445
principles and necessary rules in a prolix but accurate work* of over 700 folio pages. The plan adopted was due to a suggestion of Pitatus made in 1552 or perhaps 1537 : the alternative and more accurate proposal of Stoffler, made in 1518, to omit one day in eveiy 134 years, being rejected by Lilius and Clavius for reasons which are not known.
Clavius believed the year to contain 365*2425432 days, but he framed his calendar so that a year, on the average, contained 365*2425 days, which he thought to be wrong by one day in 3323 years : in reality it is a trifle more accurate than this, the error amounting to one day in about 3600 years.
The change was unpopular, but Ricciolif tells us that, as those miracles which take place on fixed dates — ex. gr. the liquefaction of the blood of S. Januarius — occurred according to the new calendar, the papal decree was presumed to have a di\dne sanction — Deo ipso huic correctioni Gregorianae sub- scribente — and was accepted as a necessary evil.
In England a bill to carry out the same reform was intro- duced in 1584, but was withdrawn after being read a second time; and the change was not finally eifected till 1752, when eleven days were omitted from that year. In Roman Catholic countries the new style was adopted in 1582. In the German Lutheran States it was made in 1700. In England, as I have said above, it was introduced in 1752; and in Ireland it was made in 1782. It is well known that the Greek Church still adheres to the Julian calendar.
The Mohammedan year contains 12 lunar months, or 354J days, and thus has no connection with the seasons.
The Gregorian change in the calendar was introduced in order to keep Easter at the right time of year. The date of Easter depends on that of the vernal equinox, and as the Julian calendar made the year of an average length of 365*25 days instead of 365*242216 days, the vernal equinox came earlier and earlier in the year, and in 1582 had regreded to within about ten days of February.
* Romani Galendarii a Greg. XIII Eestituti Explication Kome, 1603. t Chronologia ReJ'ormata, Bonn, 1669, vol. n, p. 206.
446 TIME AND ITS MEASUREMENT [CH. XX
The rale for determining Easter is as follows*. In 325 the Nicene Council decreed that the Roman practice should be followed ; and after 463 (or perhaps, 530) the Roman practice required that Easter-day should be the first Sunday after the full moon which occurs on or next following the vernal equinox — full moon being assumed to occur on the fourteenth day from the day of the preceding new moon (though as a matter of fact it occurs on an average after an interval of rather more than 14J days), and the vernal equinox being assumed to fall on March 21 (though as a matter of fact it sometimes falls on March 22).
This rule and these assumptions were retained by Gregory on the ground that it was inexpedient to alter a rule with which so many traditions were associated ; but, in order to save disputes as to the exact instant of the occurrence of the new moon, a mean sun and a mean moon defined by Clavius were used in applying the rule. One consequence of using this mean sun and mean moon and giving an artificial definition of full moon is that it may happen, as it did in 1818 and 1845, that the actual full moon occurs on Easter Sunday. In the British Act, 24 Geo. II. cap. 23, the explanatory clause which defines full moon is omitted, but practically full moon has been interpreted to mean the Roman ecclesiastical full moon; hence the Anglican and Roman rules are the same. Until 1774 the German Lutheran States employed the actual sun and moon. Had full moon been taken to mean the fifteenth day of the moon, as is the case in the civil calendar, then the rule might be given in the form that Easter-day is the Sunday on or next after the calendar full moon which occurs next after March 21.
Assuming that the Gregorian calendar and tradition are used, there still remains one point in this definition of Easter which might lead to different nations keeping the feast at different times. This arises from the fact that local time is introduced. For instance the difference of local time between
* De Morgan, Com;panion to the Almanac, London, 1815, pp. 1 — 36 ; ibid., 1846, pp. 1—10.
CH. XX] TIME AND ITS MEASUREMENT 447
Rome and London is about 50 minutes. Thus the instant of the first full moon next after the vernal equinox might occur in Rome on a Sunday morning, say at 12.30 a.m., while in England it would still be Saturday evening, 11.40 p.m., in which case our Easter would be one week earlier than at Rome. Clavius foresaw the difficulty, and the Roman Communion all over the world keep Easter on that day of the month which is determined by the use of the rule at Rome. But presumably the British Parliament intended time to be determined by the Greenwich meridian, and if so the Anglican and Roman dates for Easter might differ by a week ; whether such a case has ever arisen or been discussed I do not know, and I leave to ecclesiastics to say how it should be settled.
The usual method of calculating the date on which Easter- day falls in any particular year is involved, and possibly the following simple rule* may be unknown to some of my readers.
Let m and n be numbers as defined below, (i) Divide the number of the year by 4, 7, 19; and let the remainders be a, 6, c respectively, (ii) Divide 19c + m by 30, and let d be the remainder, (iii) Divide 2ci + 46 + 6c^ + n by 7, and let e be the remainder, (iv) Then the Easter full moon occurs d days after March 21; and Easter-day is the (22 + c? + e)th of March or the (cZ + e — 9)th day of April, except that if the calculation gives (i = 29 and e= 6 (as happens in 1981) then Easter-day is on April 19 and not on April 26, and if the calculation gives d = 28, e = 6, and also c > 10 (as happens in 1954) then Easter- day is on April 18 and not on April 25, that is, in these two cases Easter falls one week earlier than the date given by the rule. These two exceptional cases cannot occur in the Julian calendar, and in the Gregorian calendar they occur only very rarely. It remains to state the values of m and n for the par- ticular period. In the Julian calendar we have m = 15, n = 6. In the Gregorian calendar we have, from 1582 to 1699 in- clusive, m = 22, 71 = 2; from 1700 to 1799, m=23, n = 3; from 1800 to 1899, m = 23, n==4>; from 1900 to 2099, m = 24,
*■ It is due to Gauss ; his proof is given in Zacb's Monatliche Coirespondeiiz, August, 1800, vol. u, pp. 221—230.
448 TIME AND ITS MEASUREMENT [CH. XX
w = 5; from 2100 to 2199, m = 24, n = 6; from 2200 to 2299, m = 25, n = 0; from 2300 to 2399, m=26, 7^ = 1; and from 2400 to 2499, m=25, n=l. Thus for the year 1908 we have m = 24, n = 5 ; hence a = 0, 6 = 4, c=8; d = 26; and e = 2: therefore Easter Sunday was on April 19. After the year 4200 the form of the rule will have to be slightly modified.
The dominical letter and the golden number of the ecclesi- astical calendar can be at once determined from the values of h and c. The epact, that is, the moon's age at the beginning of the year, can be also easily calculated from the above data in any particular case ; the general formula was given by Delambre, but its value is required so rarely by any but professional astronomers and almanac-makers that it is un- necessary to quote it here.
We can evade the necessity of having to recollect the values of m and n by noticing that, if iV is the given year, and if {N/oo} denotes the integral part of the quotient when N is divided by x, then m is the remainder when 15 + ^ is divided by 30, and n is the remainder when 6 + 77 is divided by 7 : where, in the Julian calendar, f = 0, and 77 = 0 ; and, in the Gregorian calendar, ^= {i\^/100} - {i\^/400} - {i\^/300}, and 97={iV/100}-{iV/400}-2.
If we use these values of m and n, and if we put for a, b, c their values, namely, a = iV^-4 (i\^/4}, h = N-'7 {N/7}, c = iV^— 19 {i\^/19}, the rule given on the last page takes the following form. Divide 19iY- {N/19} + 15 + f by 30, and let the remainder be d. Next divide 6(N+d + l)- {N/4i} +7) by 7, and let the remainder be e. Then Easter full moon is on the dth day after March 21, and Easter-day is on the (22 -\- d + e)th. of March or the {d + e-9)th of April as the case may be ; except that if the calculation gives d = 29, and e = 6, or if it gives d = 2S, e = 6, and c> 10, then Easter-day is on the (d + e — 16)th of April.
Thus, if A^= 1920, we divide
19 (1920) - 101 -1- 15 -t- (19 - 4 - 6) by 30,
CH. XX] TIME AND ITS MEASUREMENT 449
which gives d=13, and then we proceed to divide
6(1920 + 13 + l)-480 + (19-4-2) by 7, which gives e = 0 : therefore Easter-day will be on April 4.
The above rules cover all the cases worked out with so much labour by Clavius and others*.
I may add here a rule, quoted by Zcller, for determining the day of the week corresponding to any given date. Suppose that the pih day of the ^th month of the year N anno domini is the 7'th day of the week, reckoned from the preceding Saturday. Then r is the remainder when
;? + 2^+ {3 (5 + \)ib]-\-N[Nm-'n
is divided by 7 ; provided January and February are reckoned respectively as the 13th and 14th months of the preceding year.
For instance, Columbus first landed in the New World on October 12, 1492. Here p = 12, ^ = 10, iV=1492, 17 = 0. If we divide 12 + 20 + 6 + 1492 + 373 by 7 we get r = 6 ; hence it was on a Friday. Again, Charles I was executed on January 30, 1G49 N.S. Here ^ = 30, g=13, xV=1648, 77 = 0, and we find r = 3 ; hence it was on a Tuesday. As another example, the battle of Waterloo was fought on June 18, 1815. Here ^ = 18, q = Q, N = 1815, 7} = 12, and we find r = 1 ; hence it took place on a Sunday.
Various rules have been given for obtaining these results with less arithmetical calculation, but they depend on the construction of tables which must be consulted in all cases. One rule of this kind is given in Whitakers Almanac. Lightning calculators use such rules and commit the tables to memory. The same results can be also got by mechanical contrivances. The best instrument of this kind with which
* Most of the above-mentioned facts about the calendar are taken from Delambre's Astrononiie, Paris, 1814, vol. iii, chap, xxxviii ; and his Histoire de Vustronornie modcrnc, Paris, 1821, vol. i, chap, i: see also A. De Moi^'an, The Book of Almanacs, London, 1851 ; S. Butcher, The Ecclesiastical Calendar, Dublin, 1877; and C. Zeller, Acta Mathematica, Stockholm, 1887, vol. ix, jip. 131 — 136: on the chronological details see J. L. Ideler, Lehrbuch der Chronologie, Berlin, 1831.
B. R. 29
450 TIME AND ITS MEASUREMENT [CH. XX
I am acquainted is one called the World's Calendar, invented by J. P. Wiles, and issued in London in 1906.
I proceed now to give a short account of some of the means of measuring time which were formerly in use.
Of devices for measuring time, the earliest of which we have any positive knowledge are the styles or gnomons erected in Egypt and Asia Minor. These were sticks placed vertically in a horizontal piece of ground, and surrounded by three concentric circles, such that every two hours the end of the shadow of the stick passed from one circle to another. Some of these have been found at Pompeii and Tusculum.
The sun-dial is not very different in principle. It consists of a rod or style fixed on a plate or dial; usually, but not necessarily, the style is placed so as to be parallel to the axis of the earth. The shadow of the 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 1 have read, is that made by Berosus in 540 B.C. One was erected by Meton at Athens in 433 B.C. The first sun-dial at Rome was constructed by Papirius Cursor in 306 B.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 constructed that the shadow will, for a short time near sunrise and sunset, move backwards on the dialf. This was discovered by Nonez. The explanation 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
'■' Thus Chaucer in iheShipman's Tale, "by my chilindre it is prime of day," and Lydgate in the Siege of Thebes, "by my chilyndre I gan anon to see. ..that it drew to nine." • t Ozanam, 1803 edition, vol. ni, p. 321 ; 1840 edition, p. 529.
Cfl. XX] TIME AND ITS MEASUREMENT 451
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. Let QQ' 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 arc QQ\ If these two conditions are fulfilled, then the shadow will regrede through the angle QFF as its extremity moves from Q to P, it will advance through the angle FFF' as its extremity moves from P to P\ and it mil regrede through the angle P'FQ' as its extremity moves from P' 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 arc QQ' will consist of the whole of one branch of a hyperbola, and the above con- ditions will be satisfied if F is within the space bounded by this branch of the hyperbola and its as3niiptotes. As a particular case, in a place of latitude 12° 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 rationalistic explanation, offered by the French encyclopaedists, of the miracle recorded in connection with Isaiah and Hezekiah*. Suppose, for instance, that the style is perpendicular to the face of the dial. Draw the celestial sphere. Suppose that the sun rises at M and culminates at i\', and let Z be a point between M and N on the sun's diurnal path. Draw a great circle to touch the sun's diurnal path MLN at Z, let this great circle cut the celestial meridian in A
* 2 Kii)gs, chap, xx, vv. 9—11.
29-2
452 TIME AND ITS MEASUREMENT [CH. XX
and A\ and of the arcs AL, A'L suppose that AL i's> the less and therefore is less than a quadrant. If the style is pointed to Ay then, while the sun is approaching L, the shadow will regrede, and after the sun passes L 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~^ (cos (o sec \(o) — cot~^ {sin © cos (Z — Jo) (cos^ I — sin^ to) ~ ^),
where I is the latitude of the place and w 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 ov ring-dial is another instrument for measuring solar time-|*. One of the simplest type is figured in the diagram below. 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 con- taining 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
'* An account of this sun-dial with a diagram was given in Knowledge^ London, July 1, 1892, pp. 133, 134.
t See Ozanam, 1803 edition, vol. iii, p. 317 ; 1810 edition, p. 526.
CH. XX]
TIME AND ITS MEASUREMENT
4.33
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 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, and hour-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.
454 TIME AND ITS MEASUREMENT [CH. XX
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 regu- lated 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 early in the nineteenth century, and bearing the inscription Discite justiciam moniti. In 1292 a clock was erected in Canter- bury Cathedral at a cost of £30. One erected at Glastonbury Abbey in 1325 is at present in the Kensington Museum and is in excellent condition. Another made in 1326 for St Albans Abbey showed the astronomical 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. The clocks at Peterborough and Exeter were of about the same date, and portions of them remain in situ. Most of these early clocks
* See Clock and Watch Maldng by Lord Griinthorpe, 7th edition, London, 1883.
CH. XX] TIME AND ITS MEASUKEMENT 455
were regulated by horizontal balances: pendulums being then unknown. Of the elaborate clocks of a later date, that at Strasburg made by Dasypodius in 1571, and that at Lyons constructed by Lippcus in 1598, are especially famous: the former was restored in 1842, though in a manner which de- stroyed 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 Co vent Garden, London, but the theory of such clocks is due to Huygens*. Galileo had discovered 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 dis- covered 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 Nuremberg by P. Hole early in the sixteenth century — the motive power being a ribbon of steel, wound round a spindle, and comiected at one end with a train of wheels which it turned as it unwound. Possibly a few similar timepieces had been made in the previous century; by the end of the sixteenth century they were not uncommon. At first 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 mathematicians 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.
* Jlorologiuiu Oscillatoiium, Paris, 1G73.
456 TIME AND ITS MEASUREMENT [CH. XX
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 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 Ozanamf as one of the sights of Paris at the beginning of the last century. It was formed of a hollow cylinder divided into various compartments 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 of La KatureX- 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 — imper- ceptible 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 page 96. 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
* Ozanam, 1803 edition, vol. n, p. 39 ; 1840 edition, p. 212 ; or La Nature^ Jan. 23, 1892, pp. 123, 124.
t Ozanam, 1803 edition, vol. ii, p. 68 ; 1840 edition, p. 225. X See especially the volumes issued in 1874, 1877, and 1878.
CH. XX] TIME AND ITS MEASUREMENT 457
determines 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 figure xii 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 figure iiii) to the sun, and the figure ii on the watch will indicate the direction of south. Again, if it is eight o'clock in the morning, we must point the hour-hand (which is then at the figure viii) to the sun, and the figure X 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 and xil which must be bisected is less than 180°, but at other times the angle to be bisected is greater than 180°; 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 in 24 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 figure xii 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 than 2°. 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.
* 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.
458 TIME AND ITS MEASUREMENT [CH. XX
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 figure xii points to the sun, then the direction which bisects the angle between the hour of the day and the figure XII will point due north.
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