NUMBER ONE.
However accurate an instrument for the mensuration of time may be, it would be of little use for close observation unless we have some standard by which to test its performance. We look to Astronomy to furnish us with this desideratum, nor do we look in vain. The mean sidereal day, measured by the time elapsed between any two consecutive transits of any star at the same meridian, and the mean sidereal year—which is the time included between two consecutive returns of the sun to the same star—are immutable units with which all great periods of time are compared; the oscillations of an isochronous pendulum affording us a means of correctly dividing the intermediate space into hours and days.
We must premise that the whole theory of taking time by sidereal observations is based on angular motion, the mensuration of one of the angles of motion giving a measurement of space, so that to say space, or distance, is equivalent to saying time. From noon of one day to noon of another is the whole problem to be solved by correct division. The astronomical day begins at noon, but in civil law the day is dated from midnight. So in the year the astronomical day is dated December 31, while in common reckoning the 1st of January is the initial point. This day is divided into twenty-four hours, counted in England, America, and the most of the Continental nations of Europe, by twelve and twelve. The French astronomers, however, adopted the decimal system, for ease in the computation. Thus they divided the day into ten hours, the hour into one hundred minutes, and the minute into one hundred seconds. This plan was in conformity with the French system of decimal weights and measures. Again, in Italy, the day was divided into twenty-four hours, but counting from one to twenty-four o’clock. The French system presents some features well worthy of adoption, as it gives results so much more easy in computation—a facility unattainable in the common division; yet it did not come into general use in other countries, and although some French astronomers still hold to the system, it is gradually dying out.
At one time during the Revolution in France a clock in the gardens of the Tuileries was regulated to show time by the decimal system.
For the Horologist the mean length of the day is sufficient to show the rate of his instrument for that particular day, but the astronomical and civil division requires a much longer period of observation. This is obtained by the position of the mean annual equinoxes or solstices, and is estimated from the winter solstice, the middle of the long annual night under the North Pole; and the period between this solstice and its return is a natural cycle, peculiarly suited for a standard of measurement.
Even with such a standard as the civil year of 365d. 5h. 48m. 49.7s., the incommensurability that exists between the length of the day and the real place of the sun makes it very difficult to adjust the ratio of both in whole numbers. Were we to return to the point in the earth’s orbit in exactly 365 days, we would have precisely the same number of days in each year, and the sun would be at the same point on the ecliptic at the same second at the beginning and end of the year. There is, however, a fraction of a day, so that a solar year and civil are not of equal duration.
It is thus we have our bissextile year, from the fact that the inequality amounts to nearly a quarter of a day, so that in four years we have a whole day’s gain; but not exactly, because a fraction still remains to be accounted for. Now, if we should suppress the one day of leap-year once at the end of each three out of four centuries, the civil would be within a very small fraction equal to the solar year, as given by observation; this small fraction would be almost entirely eliminated, provided we suppressed the bissextile at the end of every four thousand years. Were this fraction neglected, the beginning of the new civil year would precede the tropical by just that much, so that in the course of 1507 years the whole day’s difference would obtain.
The Egyptian year was dated from the heliacal rising of the star Sirius; it contained only 365 days. By easy computation it can be shown that in every 1461 years a whole year was lost; this cycle was called the Sothaic period, in which the heliacal rising of Sirius passed through the whole year and took place again on the same day. The commencement of that cycle took place 1322 years before Christ. The year by the Roman calendar was dated by Julius Cæsar the 1st of January, that being the day of the new moon immediately following the winter solstice in the 707th year of Rome. Christ’s nativity is dated on the 25th of December, in Cæsar’s 45th year, and the 46th year of the Julian calendar is assumed to be the 1st year of our era. The preceding year is designated by chronologists the 1st year before Christ, the dates thence running backward the same as they run forward subsequent to that period.
Astronomically, that year is registered 0; the astronomical year begins at noon on the 31st of December, and the date of any observation expresses the number of days and hours which have actually elapsed since that time, the 31st of December—Year 0.
The year is divided into months by old and almost universal consent, but the period of seven days is by far the most permanent division of a rotation of the earth around the sun. It was the division long before the historic period. The Brahmins in India used it with the same denominations as at the present day the Jews, Arabs, Egyptians, and Assyrians. “It has survived the fall of empires, and has existed among all successive generations, a proof of their common origin.”
Nothing can be more interesting in the study of astronomy than its chronological value. La Place says: “Whole nations have been swept from the earth, with their languages, arts, and sciences, leaving but confused masses of ruins to mark the place where mighty cities stood; their history, with but the exception of a few doubtful traditions, has perished; but the perfection of their astronomical observations marks their high antiquity, fixes the periods of their existence, and proves that even at that early time they must have made considerable progress in science.”
The earth revolving around the sun in an ellipse, the position of the major axis of the orbit would indicate something in regard to eras in astronomy extending not only beyond the historical period, but so far back in the past that imagination is almost at fault. The position of the major axis of the orbit depends on the direct motion of the perigee and the precession of the equinoxes conjointly, the annual motions respectively being 11´´.8 and 50´´.1, the two combined motions being 61´´.9 annually. A tropical revolution is made in 209.84 years. This being a constant quantity, we may ascertain when the line of the major axis coincided with the line of the equinoxes. This occurrence took place about 4,000 or 4,090 years before the year 0. In the year 6,483 the major axis will again coincide with the line of the equinoxes, but then the solar perigee will coincide with the vernal equinox. So, it will be seen that the period of revolution is 20,966 years. But in the progress of this revolution there must have been a time when the major axis was perpendicular to the line of the equinoxes. A simple calculation will show that the eventful year was 1250; and so important is this event considered, that La Place, the immortal author of the Méchanique Céleste, proposed to make the vernal equinox of this year the initial day of the year 1 of our era. Again, at the solstices the sun is at the greatest distance from the equator; consequently the declination of the sun is equal to the obliquity of the ecliptic. The length of a shadow cast at noonday from the stile of an ordinary sun-dial would accurately determine the precise time on which this position occurs.
Though wanting in accuracy, such a measurement is of interest, from the fact that there are recorded observations of this kind that were taken in the city of Layang, in China, 1100 years before our present era is dated. This observation gives the zenith distance of the sun at the moment of the observation. Half the sum of the zenith distances gives the latitude, and half their difference gives the obliquity of the ecliptic at the period. Now the law of the variation of the ecliptic is well known, and modern computation has verified both the moment of taking the observation and the latitude of the place. Eclipses were the foundation of the whole of Chinese chronology, and recorded observations prove the civilization of that strange race for 4700 years.
Horology, with astronomy, was not neglected even as early as 3102 years before Christ, as the following will show.
The cycles of Jupiter and Saturn are very unequal, the latter being a period of 918 years; the mean motion of the two planets was determined by the Indians in that part of the respective orbits where Saturn’s motion was the slowest and Jupiter’s the most rapid. This observed event must have been 3102 years before, and 1491 after the year 0; but the record shows that the observation was taken before the last-named date.
Since both solar and sidereal time is estimated from the passage of the sun and the equinoctial point across the meridian of the place of observation, the time will vary in different places by as much as the passage precedes each. It being obvious that when the sun is in the meridian at any one place, it is midnight at a point on the earth’s surface diametrically opposite; so an observation taken at different places at the same moment of absolute time, will be recorded as having happened at different times. Therefore when a comparison of these different observations is to be made, it becomes necessary to reduce them by computation to what the result would have been had they been taken under the same meridian at the same moment of absolute time. Sir John Herschel proposed to employ mean equinoctial time, which is the same for all the world. It is the time elapsed from the moment the mean sun enters the mean vernal equinox, and is reckoned in mean solar days and parts of days. This difference in time is really the angular motion of the earth, and by measuring it the longitude of any place on the surface of the earth can be determined, provided we have a standard point of departure, and an instrument capable of accurately dividing the time into small quantities during its transit from the meridian on which it was rated.
As will be hereafter shown, the axis of the earth’s rotation is invariable. Were the position of the major axis of the earth’s orbit as immutable, an observation of any star on the meridian taken at any place would always be the same. Again, the form of the earth has an important effect; the equatorial diameter exceeds the polar, thus giving a large excess of matter at the equator. Now the attraction of an external body not only draws another to it in its whole mass, but, as the force of attraction is inversely as the square of the distance, it follows that the attracted body would be revolved on its own centre of gravity until its major diameter was in a straight line with the attracting body.
The sun and moon are both attracting bodies for the earth; the plane of the equator is at an angle to the plane of the ecliptic of 23° 27´ 34´´.69, and the plane of the moon’s orbit is inclined to it 5° 8´ 47´´.9 Now from the oblate form of the earth, the sun and moon, acting obliquely and unequally, urge the plane of the equator from its own position from east to west, thus changing the equinoctial points to the extent of 50´´.41 annually.
This action, were it not compensated by another force, would in time alter the angle of the ecliptic until the equatorial plane and the ecliptic coincided. There are few but have seen the philosophical toy called the Gyrascope. This toy, on a miniature scale, gives a fine illustration of the force brought in to correct the combined action of the sun and moon on the obliquity of the equator. The rotation of the earth is held in its own plane by its own revolution, the same as the gyrascope seems to overcome the laws of gravitation by its force of revolution.
But not only do the sun and moon disturb the plane of the ecliptic, but the action of other planets on the earth and sun is to be taken into account. A very slow variation in the position of the plane of the ecliptic, in relation to the plane of the equator, is observed from these influences. It must be remembered that a very slight deviation in the angle can and would be detected by observation with modern instruments. We do find that this attraction affects the inclination of the ecliptic to the equator of 0´´.31 annually.
This motion is entirely independent of the form of the earth. Now, if we assume that the sun and moon give the equinoctial points a retrograde motion on the ecliptic, we must deduct the influence of the planets. We may then calculate the mean disturbance by subtracting the latter from the former—the difference is settled by both theory and observation to be 50´´.1 annually. This motion of the equinoxes is called the precession of the equinoxes. Its consideration forms a very important element in the estimation of time, as the position of the various fixed stars, though so very distant, are all affected in longitude by this quantity of 50´´.1—being an increase of longitude. Therefore, if we were to calculate the position of any given star in order to get a transit for mean time, or true time, we must take this quantity into consideration. The increase is so great that the earliest astronomers, even with their imperfect modes of observation, detected it. Hipparchus, 128 years before Christ, compared his own observations with those of Timocharis, 153 years before. He found the solution of the problem the same as Diophantus found the solution of the squares and cubes, by analysis. In the time of Hipparchus, the sun was at a point 30° in advance of its present position, for it then entered into the constellation of Aries near the vernal equinox.
At the present time the position of the equinoctial points shows a recession of the whole, 30° 1´ 40´´.2. At this rate of motion the constellations called the Signs of the Zodiac are some distance from the divisions of the ecliptic that bear their names. At the rate of 50´´.1 the whole revolution of the equinoctial points will be accomplished in 25,868 years; but this is again modified because the precession must vary in different centuries for the following reasons: the sun’s motion is direct, the precession retrograde; therefore, the sun arrives at the equator sooner than he does at the same star of observation. Now, the tropical year is 365d. 5h. 48´ 49´´.7; and as the precession is exactly 50´´.1, we must suppose it takes some time for the sun to move through that arc. By direct observation it is found that the time required for such translation is 20´ 19´´.6. By adding this amount to the tropical year we have the sidereal year of 365d. 6h. 9´ 9´´.6 in mean solar days. This amount of precession has been on the increase since the days of its first recorder, Hipparchus, as the augmentation amounts to no less than 0.´´455. By adding that to the known precession we find that the civil year is shorter now by 4´´.21 than in his time; but, as a great division of time, the year can be changed by this cause not more than 43.´´
The action of the moon on the accumulation of matter at the earth’s equator is a source of disturbance that in very accurate observations for time should be eliminated. Thus the moon, with the conjoint action of the sun, depending on relative position, causes the pole of the equator to describe a small ellipse in the heavens with axes of 18´´.5 for the major, and 13´´.674 for the minor; the longer axis being directed to the pole of the ecliptic. This inequality has a period of 19 years,—it being equal to the revolution of the nodes of the lunar orbit. The combination of these disturbances changes, by a small quantity, the position of the polar axis of the earth in regard to the stars, but not in regard to its own surface. With so many disturbing causes, we must add that of Jupiter, whose attraction is diminishing the obliquity of the ecliptic by 0´´.457 according to M. Bessel.
The results of all these forces must affect the position of all the stars and planets as seen from our earth. Their longitudes being reckoned from the equinoxes, the precession of these points would increase the longitude; but as it affects all the stars and planets alike, it would make no real or apparent change in their relative positions. Nutation, however, affects the celestial latitudes and longitudes, as the real motion of the earth’s polar axis changes the relative positions. So great is the change that our present pole star has changed from 12° to 1° 24; in regard to the celestial pole, the gradual approximation will continue until it is with 0° 30´, after which it will leave the pole indefinitely until in 12,934 years α Lyræ will be the pole star.
So far we have given only the causes that affect the meridian, and consequently our standard for time; but that point being established for the yearly and diurnal revolutions, it becomes necessary to find some means to divide the day into minute fractional parts, such as seconds and parts of seconds. This, it has been stated, is effected by means of an isochronous pendulum. On this instrument no comment is required but of the causes that disturb its accuracy much is needed. In 1672, at Cayenne, the astronomer Richter, while taking transits of fixed stars, found his clock lost 2´ 28´´ per day. This was an error that arrested his attention, and he immediately attributed it to some variation in the length of the pendulum—due to other causes than atmospheric changes and expansion. He determined the length of a pendulum beating seconds in that latitude, which was 5° N. in South America. He found that that pendulum was shorter than one beating seconds in Paris, by 0833+ of an inch. Now, if the earth was a sphere, the attraction of gravitation at all places on its surface would be equal, and the oscillations of a pendulum would also be equal, + or - the disturbing effect of centrifugal force—an amount that can be easily determined. The real reason of the variation is found in the configuration of the earth.
The amount of the attraction of gravitation at any point of the earth’s surface is found by the distance traversed by any body during the first second of its fall. The pendulum is a falling body, and may be by the same analysis reasoned on that pertains to the laws of gravitation; the centrifugal force is measured by any deflection from a tangent to the earth’s surface in a second.
It follows that the centrifugal force at the poles, where there is the least motion, would not be equal to the force of gravitation, and at the equator must be exactly equal; but the deflection of a circle from a tangent measures the intensity of the earth’s attraction, and is equal to the versed sine of the arc described during that time, the velocity of the earth’s rotation being known, the value of the arc is deducible. The centrifugal force at the equator is equal to ¹⁄₂₈₉th part of the attraction of gravitation. Again, the uniformity of the earth’s mass becomes an object of consideration. Assuming that the figure of the earth is an ellipsoid of rotation, we will show the relation that form bears to the equal oscillation of a pendulum.
Taking the earth as a homogeneous mass, analysis gives us the certainty that if the intensity of gravitation at the equator be taken as unity, the increase of gravity to the poles eliminating the differences of the centrifugal force must be = to 2.5, the ratio of the centrifugal force to that of gravitation at the equator. Now, taking the 2.5 of .346 = 1/115.2, this then must be the total increase of gravitation. Did we know the exact amount of increase at every point, from the equator to the poles, a perfect map of the form of the earth could be produced from calculation; experiment being from physical causes totally impracticable. The following analysis, quoted from an eminent physicist, gives a very lucid idea of the reasoning:
“If the earth were a homogeneous sphere without rotation, its attraction on bodies on its surface would be everywhere equal. If it be elliptical and of variable density, the force of gravity ought to increase in intensity from the equator to the pole as unity plus a constant quantity multiplied into the square of the sine of the latitude. But for a spheroid in rotation the centrifugal varies by the law of mechanics, as the square of the sine of the latitude from the equator, where it is greatest, to the poles, where it is least. And as it tends to make bodies fly off the surface, it diminishes the force of gravity by a small quantity. Hence, by gravitation, which is the difference of these two forces, the fall of bodies ought to be accelerated from the equator to the poles proportionably to the square of the sine of the latitude, and the weight of the body ought to increase in that ratio.”
Assuming the above reasoning to be correct, it follows, that the rate of descent of falling bodies will be accelerated in the transition from the equator to the poles. Now, it has been before stated that the pendulum is a falling body; therefore, with the same length of pendulum, the oscillations at the pole should be faster than at the equator. Theory, in this case, is verified; for it has been proved by experiments, repeated again and again, that a pendulum oscillating 86,400 times in a mean day at the equator, will give the same number of oscillations at any other point, provided its length is made longer in the exact ratio as the square of the sine of the latitude.
The sequence to be derived from all the foregoing considerations is, that the whole decrease of gravitation from the equator to the poles is 0.005.1449, which subtracted from the 1/155.2 gives the amount of compression of the earth to be nearly 1/285.26. But this form of the earth would give the excess of the equatorial axis over the polar about 26¹⁄₂ miles. The measurement is confirmed by Mr. Ivory in his investigations on the five principal measurements of arcs of the meridian in Peru, India, France, England, and Lapland. He found that the law required an ellipsoid of revolution whose equatorial radius should be 3,962.824 miles, and the polar 3,949.585 miles; the difference is 13.239 miles; this quantity multiplied by two gives 26.478 as the excess of one diameter over the other. Thus, by two different processes the figure of the earth has been determined; but another remains that is the result of pure analysis, derived from the nutation and precession of the equinoxes—for, as explained before, these effects are caused by the excess of matter at the earth’s equator. The calculation does not lead us to certainty, but it does show the compression to be comprised between the two fractions ¹⁄₂₇₀ and ¹⁄₅₇₃. There is this advantage in the lunar theory, that it takes the earth as a whole, disregarding any irregularities of surface, or the local attractions that influence the pendulum—the difficulties of measuring an arc of the meridian being an obstacle to perfect accuracy.
The form of the earth has, however, a value confined not alone to those interested in horology—it furnishes us with a standard of weights and measures. In England and the United States, the pendulum is the unit of mensuration, or at least the common standard from which measurement is derived. It has been shown that, deducting the effects of nutation, the axis of the earth’s rotation is always in the same plane. Now, the mass being the same constant quantity, a pendulum oscillating seconds at the Greenwich Observatory, has been adopted by the English Government as its standard of length. Oscillating in vacuo at the level of the sea, at 62° Fahr., Captain Kater found its approximate length to be 39.1393 inches; as this must be invariable under the same circumstances, it becomes a standard for all time. The French deduced their standard from the measurement of the ten-millionth part of a quadrant of the meridian passing through Formentera and Greenwich. They have also adopted the decimal system; yet it seems to prove that nothing under the sun is new, for over forty centuries ago the Chinese used the decimal system in the division of degrees, weights, and measures.
The antiquity of the pendulum is also shown by the fact that the Arabs were in the habit of dividing the time in observations, by its oscillations, when Ibn Junis, in the year one thousand, was making his astronomical researches. Before we lose sight of the influence of the form of the earth on the pendulum, it may be well to state another source of disturbance, arising from the combined influence of the earth’s rotation and the fact that a body moving in its own plane seeks to maintain that plane. It will be seen from the very beautiful experiment showing the rotation of the earth, that if a body like a pendulum be suspended so as to be free in every direction, and not be influenced by the motion of the earth when set in oscillation in any plane, that that plane will preserve its line of motion, while the earth in its motion beneath the body can be seen to slowly move, as though the minute hand of a watch were made stationary while the dial revolved. The same principle is the one that maintains the spinning-top in a parallel position to the horizon, or the gyrascope in its apparently anomalous defiance of all the laws of gravitation. In the pendulum this tendency to preserve the same plane of motion becomes a cause of error—slight, it is true, but can be very easily remedied by so placing it that the plane of oscillation shall be parallel to the equator. It will be readily seen that this precaution will become more important as we recede from the equator; for if we were to suspend a pendulum at the pole in a true line with the axis of rotation, and if the plane of vibration remained constant, the earth would turn once around that plane in the diurnal period. During this time there would be a continuous torsion on the point of suspension, that would in time materially affect the accuracy of the instrument. The reasoning holds good for every latitude—degree of influence being the only difference.
Having given the action of the earth’s form, mass, and rotation on the pendulum, there remain the disturbances due to expansion and contraction, owing to changes of temperature and those of atmospheric causes. The astronomical points to be observed are somewhat too fully laid down, but it must be remembered that an exact science requires the premises to be fully established before a sequence can be drawn.
As the standard of time depends on the passage of a star or the sun, or any known celestial object, at a certain time across the meridian of the place where the observation is taken, it was absolutely necessary to give the modes of calculation, together with the disturbing causes. Moreover, a full appreciation of the indebtedness of horology to astronomy could not be obtained without a general knowledge of the change of the position of the major axis of the orbit described by the earth around the sun. Also, the difference between mean and apparent solar time was required to illustrate the use of the tables of equated time, the necessity of which will become patent when the use of the transit instrument for the establishment of time, or a fixed standard, is introduced. Also, the disturbing effects of the sun and moon collectively and relatively as to position, could not be passed, as they produce the precession of the equinoxes and the nutation of the pole—essential elements in the computation of time.