PG26839.B00303
[PDF page 302] 286 HYPER-SPACE. [CH. XII that the selection of a geometry is really a matter of convenience, and that that geometry is the best which enables us to state the physical laws in the simplest form. This opinion has been strongly controverted, but at any rate it expresses one view of the question. The above refers only to hyper-space of two dimensions. Natu- rally there arises the question whether there are different kinds of Non- Euclidean space of three or more dimensions. Riemann showed that there are three kinds of Non-Euclidean space of three dimensions hav- ing properties analogous to the three kinds of Non-Euclidean space of two dimensions already discussed. These are differentiated by the test whether at every point no geodetical surface, or one geodetical surface, or a fasciculus of geodetical surfaces can be drawn parallel to a given surface: a geodetical surface being defined as such that every geodetic line joining two points on it lies wholly on the surface. It may be added that each of the three systems of geometry of two dimensions described above may be deduced as properties of a surface in each of these three kinds of Non-Euclidean space of three dimensions. It is evident that the properties of Non-Euclidean space of three dimensions are deducible only by the aid of mathematics, and cannot be illustrated materially, for in order to realize or construct surfaces in Non-Euclidean space of two dimensions we think of or use models in space of three dimensions; similarly the only way in which we could construct models illustrating Non-Euclidean space of three dimensions would be by utilizing space of four dimensions. We may proceed yet further and conceive of Non-Euclidean ge- ometries of more than three dimensions, but this remains, as yet, an unworked field. Returning to the former question of Non-Euclidean geometries, I wish again to emphasize the fact that, if the axioms enunciated by Euclid are replaced by others, it is possible to construct other con- sistent systems of geometry. Some of these are interesting, but those which have been mentioned above have a special importance, from the somewhat sensational fact that they lead to no results necessarily in- consistent with the properties, as far as we can observe them, of the space in which we live; we are not at present acquainted with any other systems which are consistent with our experience.

PG26839.B00304
[PDF page 303] CHAPTER XIII. TIME AND ITS MEASUREMENT. Theproblemsconnectedwithtimearetotallydifferentincharacter 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 torealizelifeinspaceofhigherdimensions. Inquestionsconnectedwith 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 conception of what such a phrase means, and I do not think that any attempts have been made to work it out. I shall concern myself here mainly with questions concerning the measurement of time, and shall treat them rather from a historical than from a philosophical 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 resultofexperiencethatthelengthofthefoot-ruleisconstant, orrather that any alteration in it can be determined; and, if this assumption was denied, we could not prove it, though, if numerous repetitions of 287

PG26839.B00305
[PDF page 304] 288 TIME AND ITS MEASUREMENT. [CH. XIII the experiment under varying conditions always gave the same result, probably we should feel no doubt as to the correctness of our method. Itisevidentthatthemeasurementoftimeisamoredifficultmatter. 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 durations 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 wasaccuratelyso, manhadaunitbywhichhecouldmeasuredurations. The present subdivision of a day into hours, minutes, and seconds is artificial, and apparently is derived from the Babylonians. 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 find 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

PG26839.B00306
[PDF page 305] CH. XIII] TIME AND ITS MEASUREMENT. 289 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 durations 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 of360◦, i.e.through 15◦, is an hour: the time to turn throughone-sixtiethpartof 15◦,i.e.through 15′, isaminute: andsoon. 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 astronomical 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

PG26839.B00307
[PDF page 306] 290 TIME AND ITS MEASUREMENT. [CH. XIII 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 is 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. 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 be- gan. In parts of ancient Greece and in Japan the interval from sunrise to sunset was divided into 12 hours, and that from sunset to sunrise into 12 hours. The Jews, Chinese, Athenians, and, for a long time, the Italians, divided their day into 24 hours, beginning at the hour of sun- set, which of course varies every day: this method is said to be still used in certain villages near Naples, except that the day begins half-an-hour after sunset—the clocks being re-set once a week. Similarly the Baby- lonians, 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 the middle of last century, 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 by the Roman emper- ors, and, except during the French Revolution, has been subsequently in general use among civilized races. The names of the days are de- rived from the seven astrological planets, arranged, as was customary,

PG26839.B00308
[PDF page 307] CH. XIII] TIME AND ITS MEASUREMENT. 291 in the order of their apparent times of rotation round the earth, namely, Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon. The twenty-four hours of the day were dedicated successively to these plan- ets: 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 about29 1 2 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, 365d. 5h. 48m. 47s..4624. The Egyptians knew that it 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 46b.c. Julius Caesar decreed that thenceforth the year should contain 365 days, except that in every fourth or leap year one additional day should be introduced. He ordered this rule to come into force on January 1, 45b.c. The change was made on the advice of Sosigenes of Alexandria. It must be remembered that the year 1a.d. follows immediately 1 b.c., that is, there is no year 0, and thus 45b.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 * Herschel, Astronomy, London, 11th ed. 1871, arts. 916–919.

PG26839.B00309
[PDF page 308] 292 TIME AND ITS MEASUREMENT. [CH. XIII first 36 years after 45b.c., intercalated the extra day every third year, thus producing 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 yearsa.u.c. 746 to 757 inclusive, but that the intercalation should be again made in the yeara.u.c. 761 (that is, 8 a.d.) and every succeeding fourth year. The Julian calendar made the year, on an average, contain 365.25 days. The actual value is, very approximately, 365.242216 days. Hence the Julian year is too long by about11 1 4 minutes: this produces an error of nearly one day in 128 years. If the extra day in every thirty- secondleapyearhadbeenomitted—aswassuggestedbysomeunknown 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 min- utes 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 year 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 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 Stöffler, made in 1518, to omit one day in every 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 Riccioli† tells us that, as those * Romani Calendarii a Greg. XIII, restituti Explicatio, Rome, 1603. † Chronologia Reformata, Bonn, 1669, vol.ii, p. 206.

PG26839.B00310
[PDF page 309] CH. XIII] TIME AND ITS MEASUREMENT. 293 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 divine sanction—Deo ipso huic correctioni Gregorianae subscribente—and was accepted as a nec- essary evil. In England a bill to carry out the same reform was introduced in 1584, butwaswithdrawnafterbeingreadasecondtime; andthechange was not finally effected till 1752, when eleven days were omitted from that year. In Roman Catholic countries the new style was adopted in 1582. In Scotland the change was made in 1600. 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, or354 1 3 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. The rule 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 14 3 4 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 * De Morgan, Companion to the Almanac, London, 1845, pp. 1–36;Ibid., 1846, pp. 1–10.

PG26839.B00311
[PDF page 310] 294 TIME AND ITS MEASUREMENT. [CH. XIII 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 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 by4, 7, 19; and let the remainders bea, b, c, respectively. (ii) Divide 19c + m by 30, and let d be the remainder. (iii) Divide 2a + 4b + 6d + n by 7, and let e be the remainder. (iv) Then the Easter full moon occursd days after March 21; and Easter-day is the (22 + d + e)th of March or the(d + e − 9)th day of April, except that if the calculation gives d = 29 and e = 6 (as happens in 1981) then * It is due to Gauss, and his proof is given in Zach’sMonatliche Correspondenz, August, 1800, vol.ii, pp. 221–230.

PG26839.B00312
[PDF page 311] CH. XIII] TIME AND ITS MEASUREMENT. 295 Easter-day is on April 19 and not on April 26, and if the calculation gives d = 28, e = 6, and alsoc > 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 particular period. In the Julian calendar we have m = 15, n = 6. In the Gregorian calendar we have, from 1582 to 1699 inclusive,m = 22, n = 2; from 1700 to 1799,m = 23, n = 3; from 1800 to 1899,m = 23, n = 4; from 1900 to 2099,m = 24, n = 5; from 2100 to 2199,m = 24, n = 6; from 2200 to 2299,m = 25, n = 0; from 2300 to 2399,m = 26, n = 1; and from 2400 to 2499,m = 25, n = 1. Thus for the year 1908 we havem = 24, n = 5; hence a = 0, b = 4, c = 8; d = 26; and e = 2: therefore Easter Sunday will be on the 19th of April. After the year 4200 the form of the rule will have to be slightly modified. The dominical letter and the golden number of the ecclesiastical calendar can be at once determined from the values ofb 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 almanack-makers that it is unnecessary to quote it here. We can evade the necessity of having to recollect the values ofm and n by noticing that, ifN is the given year, and if{N/x} denotes the integral part of the quotient whenN is divided byx, then m is the remainder when 15 + ξ is divided by 30, andn is the remainder when 6 + η is divided by7: where, in the Julian calendar,ξ = 0, and η = 0; and, in the Gregorian calendar,ξ = {N/100} − { N/400} − { N/300}, and η = {N/100} − { N/400} − 2. If we use these values ofm and n, and if we put fora, b, c, their values, namely,a = N − 4{N/4}, b = N − 7{N/7}, c = N − 19{N/19}, the rule given on the preceding page takes the following form. “Divide 19N − {N/19} + 15 +ξ by 30, and let the remainder bed. Next divide 6(N + d + 1) − {N/4} + η by 7, and let the remainder be e. Then Easter full moon is on thedth 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 givesd = 29, and e = 6,

PG26839.B00313
[PDF page 312] 296 TIME AND ITS MEASUREMENT. [CH. XIII or if it gives d = 28, e = 6, and c > 10, then Easter-day is on the (d + e − 16)th of April.” Thus, ifN = 1899, we divide19(1899)−99+15+(18 −4−6) by 30, which givesd = 5, and then we proceed to divide6(1899 + 5 + 1)− 474 +(18 −4−2) by 7, which givese = 6: therefore Easter-day is on April 2. 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 Zeller, for determining the day of the week corresponding to any given date. Suppose that the pth day of theqth month of the yearN anno domini is the rth day of the week, reckoned from the preceding Saturday. Thenr is the remainder when p + 2q + {3(q + 1)/5} + N + {N/4} − η is divided by7; provided January and February are reckoned respectively as the13th and 14th months of the preceding year. For instance, Columbus first landed in the New World on Oct. 12, 1492. Here p = 12, q = 10, N = 1492, η = 0. If we divide12 + 20 + 6 + 1492 + 373 by 7 we get r = 6; hence it was on a Friday. Again, Charles I was executed on Jan. 30, 1649. Here p = 30 , q = 13 , N = 1648, η = 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 p = 18, q = 6, N = 1815, η = 12, and we find r = 1; hence it took place on a Sunday. 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 * Mostoftheabove-mentionedfactsaboutthecalendararetakenfromDelambre’s Astronomie, Paris, 1814, vol.iii, chap. xxxviii; and hisHistoire de l’astronomie moderne, Paris, 1821, vol. i, chap. i: see also A. De Morgan, The Book of Almanacs, London, 1851; S. Butcher,The Ecclesiastical Calendar, Dublin, 1877; and C. Zeller,Acta Mathematica, Stockholm, 1887, vol.ix, pp. 131–136: on the chronological details see J.L. Ideler,Lehrbuch der Chronologie, Berlin, 1831.
