Astrophysics and Space Science 16 (1972) 179-200. All Rights Reserved
Copyright © (1972) by D. Reidel Publishing Company, Dordrecht-Holland


* Paper presented at the AAAS Symposium on the Early History of the Earth and Moon in Philadelphia on 28 December 1971.


The Johns Hopkins University, Applied Physics Laboratory, Silver Spring, Md., U.S.A.

(Received 5 January, 1972)

Abstract. The most conspicuous effects of non-gravitational forces in the Earth-Moon system are the accelerations of the Earth's spin and of the Moon's mean angular velocity. Evidence indicates that the present acceleration of the Moon is between -20  and -52 s of arc per century per century and that the present average acceleration of the Earth is between -5 and -23 parts in 109 per century. Over the past 2000 yr, the average for the Moon has been about -42 s per century per century and for the Earth has been about -28 parts in 109 per century; these values are probably correct within 10%. Evidence that does not involve any assumptions about the present values shows strongly that there was a 'square wave' in the accelerations that lasted from about 700-1300, and that the accelerations were different by a factor of perhaps 5 during the time of this wave from what they were at neighboring times.

An effect that seems to be changing the obliquity of the ecliptic has been reported in recent literature, on the basis of data obtained within the past century. The effect amounts to about 1 s of arc per century if it is real. Older data are not accurate enough to give information about an effect this small.

There are no satisfactory explanations of the accelerations. Existing theories of tidal friction are quite inadequate.

1. Introduction

If one is going to calculate the early history of the Earth-Moon system by starting from present conditions, he must know quantitatively the forces acting within the system. The forces can be divided broadly into gravitational and non-gravitational ones, and we probably know the gravitational forces more accurately than the non-gravitational ones in the present state of affairs. Thus our main interest here is to review the astronomical evidence concerning non-gravitational forces. For convenience, the word 'forces' is being used to include some effects that are not forces under tile strict physical definition of the word.

The parameters most obviously affected by non-gravitational forces are the orbital angular velocity nM of the Moon and the spin angular velocity wE of the Earth, and most study of non-gravitational effects has been directed toward estimating the corresponding accelerations n·M and w·E. No definitive history of these studies has been written, so far as I know. Martin (1969) gives a detailed history through the work of Newcomb, but with little detail for the later period. Munk and MacDonald (1960. Chapter 11), Newton (1970. Chapter I), and Jeffreys (1970. Chapter VIII) give additional historical information. In the interest of brevity. I shall not go into the history of the subject in this paper.

In this paper, the acceleration n·M will always be given in units of seconds of arc per century per century, and for brevity the units will usually be omitted. Instead of giving w·E directly, most writers give the logarithmic derivative w·E/wE. There is no consensus on the units to use. For definiteness, the value of w·E in this paper will be specified by means of a quantity y defined by

always in units of reciprocal centuries. The units will usually be omitted. When the units of n·M and y are stated, they will be denoted by "/cy 2 and cy-1 respectively. The time derivaties involved in n·M and y are to be taken with respect to ephemeris time.

There is an acceleration of the Moon that results from a strictly gravitational theory of the solar system, and earlier writers often included this component in what they called the lunar acceleration. In this paper, the word 'acceleration' and the symbol n·M will be used only for the contribution arising from non-gravitational forces.

Some recent papers have reported that the obliquity of the ecliptic is decreasing more rapidly than standard theory (Explanatory Supplement [1961, Section 4B], for example) indicates. Results that have not yet been published suggest that the effect is not real. Possible non-gravitational effects in the obliquity will be discussed in Section 6 below.

Besides the angular velocities nM and wE, and the obliquity e, two parameters of the Earth-Moon system that are important in studying its history are the eccentricity e and the inclination i of the lunar orbit. Forces that tend to change nM and wE also tend to change e and i. The astronomical evidence now available not indicate any defects in the gravitational theory of e and i, but this does not rule out non-gravitational effects on them. If the non-gravitational forces are as small as we deduce from the observed changes in nM and wE we would not be able to find effects on e and i from the astronomical evidence.

The longitude W of the ascending node of the lunar orbit, and its perigee position w. process rapidly and have no direct importance for the history of the Earth-Moon system. However, non-gravitational effects on them, if known, would provide tests of non-gravitational forces beyond the tests that n·M and w·E provide, and hence they would provide useful indirect information. There have been no attempts to study the long-term behavior of w from astronomical evidence, so far as I know. Neugebauer (1929) and Newton (1970, Chapter IX) have attempted to study W, but their results do not allow us to draw any significant conclusions.

2. Tidal Friction and Other Non-Gravitational Forces

There have been many descriptions (Jeffreys [1970, Chapter VIII], for example) of the mechanism by which tidal friction affects nM and wE, and the description will not be repeated here. For our purposes, the mechanism can be represented by means of the average phase angle of the maximum tide with respect to the direction of the tide-raising body. This angle is a few degrees.

In most discussions, it has been assumed that almost all friction occurs in the ocean tides, and that friction in the bodily tide of the Earth is negligible. However, recent results (Melchior and Georis, 1968) on important components of the bodily tide show phase angles large enough to cause appreciable accelerations. Since the phase angles are positive for some parts of the Earth and negative for others, we do not yet know the world-wide average, but we cannot automatically dismiss the bodily tide as a significant source of tidal friction.

Friction in the lunar tide affects both nM and wE. Friction in the solar tide affects both wE and the orbital angular momentum of the Earth around the Sun. The orbital angular momentum of the Earth is so large, however, that changes produced in it by the solar tide are many orders of magnitude too small to concern us at present. Thus. for present purposes, we can say that the solar tide affects wE only.

The tide-raising force due to an external body is proportional to its mass divided by the cube of its distance. Although the Moon is smaller than the Sun, it is also closer, and the ratio of the lunar to the solar force is about 2.17. The ratio depends in the third figure upon whether we use the 1964 set of astronomical constants or the preceding set. If the mechanism of friction is linear in the tide-raising force, the phase angle is the same for the solar and lunar tides. The energy dissipated by friction is then proportional to the size of the tide times the tide-raising force acting on it; that is, frictional dissipation is proportional to the square of the tide-raising force for linear friction. This ratio is about 4.7. For some reason, it is customary in the literature to say that the ratio is 5.1.

If friction arises mostly in the oceans, it is plausible that the force is proportional to the square of the velocity. Jeffreys (1970) cosntructed the function that gives the ratio of dissipation by the tides on this assumption. He found the first two terms in the power series, and from them calculated that the ratio is 2/3 of what it is for linear friction. Newton (1968) calculated the ratio to high order on a digital computer and found that it is almost exactly 0.75. Thus the dissipation ratio is 3.5 if the square-law mechanism of friction is correct.

However, this mechanism does not seem to provide anything like the amount of dissipation required. The most recent estimate, based upon estimates of tidal heights and currents (Miller, 1966), is that the frictional power in the tides is between 1.4 and 1.7*1019 erg s-1. This is about half of what is needed to account for even the lowest estimate of n·M that anyone has made recently, as we shall see in Section 4.

The calculations of Jeffreys and Miller assume that tidal currents are the only important ones present. Since tidal currents are only of the order of a few centimeters per second, this assumption requires all currents of non-tidal origin to be extremely small. If the non-tidal currents are appreciably larger than the tidal currents at the place where friction occurs, and if their time scale of persistence is 12h (the tidal period) or longer, the situation changes considerably (Newton. 1968). In the first place, the force of friction becomes proportional to the tidal velocity times the non-tidal velocity, and this product is appreciably larger than the square of the tidal velocity. That is, we find more friction. Second, the ratio of dissipation by the lunar and solar tides reverts to the linear value 4.7 even though the detailed mechanism is non-linear.

So far as we know, friction in the lunar tide is the only appreciable source of an orbital acceleration of the Moon. Thus an estimate of n·M furnishes an estimate of tidal friction. The only independent method of estimating tidal friction that is presently available comes from an analysis of perturbations on near-Earth artificial satellites (Newton, 1968); this estimate will be discussed further in Section 4 below.

Friction in the lunar tide also contributes to the spin parameter y. Since the lunar tide is internal to the Earth-Moon system, it does not change the total angular momentum of the system. It takes angular momentum from the spin motion of the Earth and puts it into the orbital motion of the Moon. Since an increase in the orbital angular momentum of a satellite requires a decrease in its angular velocity, the result is to make n·M be negative. From a knowledge of the moments of inertia and other parameters, we can relate n·M to the contribution that tidal friction makes to y, as follows:
Contribution of lunar tidal friction to y=0.935n·M

Note that this contribution, like n·M, is negative.

There are many contributions to y besides that given in Equation (2). One is the solar tide already mentioned; according to the usual tidal theory, its contribution should be between 1/4.7 and 1/3.5 times the lunar contribution. The only measurement of friction in the solar tide has been made by means of artificial satellites (Newton, 1968 and Kozai. 1967). The result lies outside the range stated, but we cannot say yet whether this implies inadequate measurement or inadequate theory. For the present. the best course is to assume that dissipation by the solar tide is the average of the two theoretical limits. The ratio 1/4.7 implies a contribution of 0.20n·M to y and the ratio 1/3.5 implies one of 0.26n·M. If we use the mean value 0.23n·M, and if we let yf denote the total frictional contribution to y, we find

The coefficient of n·M is probably correct within about 0.03.

There is an appreciable variation of atmospheric pressure with a period of 0.5 solar days, but there is no sizeable variation with a period of 0.5 'lunar days'. Hence this 'atmospheric tide' is probably of thermal rather than gravitational origin. The peaks in the half-daily tide lie about 34° W of the Earth-Sun line. and the Sun exerts a torque directly on the atmospheric tide that contributes (Holmberg, 1952) about 2.0 to y. However, the matter is more complicated. First, the gravitational potential exerted on the Earth and oceans by the mass distribution in the atmospheric tide tends to distort them and to give an additional positive contribution to y. Second, the weight of the mass distribution loads the Earth and the resultant distortion gives a negative contribution to y. It is fairly simple to estimate that the negative contribution is roughly equal to the sum of the two positive ones. Thus. to the accuracy needed at present. we can neglect the contribution to y produced by the atmospheric tide.

Many effects besides tidal friction can conceivably contribute to y. We can lump them together to make up a quantity ynf the non-frictional component of y. Our ability to calculate ynf on the basis of fundamental mechanisms is even poorer than our ability to calculate yf. Order-of-magnitude calculations indicate that several effects may be significant. They include:

(1) Core-mantle coupling, including a possible transfer of angular momentum to the mantle through the medium of the magnetic field.

(2) An interaction of the Earth's magnetic field with interplanetary fields or plasmas.

(3) Changes in the Earth's moment of inertia because of changes of internal temperature.

(4) A time dependence of the gravitational constant.

Item (3) includes some of the effects of changing glaciation and sea level. Dicke (1966) has discussed these effects, as well as item (4). Munk and MacDonald (1960. pp. 227-46) have discussed all of the items, as well as several contributions that are certainly too small to concern us at present.

3. Osculating and Average Accelerations

Up to this point we have been writing as if the accelerations n·M and w·E were constant. However, it is highly probable that the accelerations have varied over wide ranges even within historic times, and we shall henceforth need to distinguish three meanings that can be attached to each acceleration.

First, there is the 'osculating' acceleration, by which we mean the acceleration that applies over a very short time interval. As we shall see later, the osculating acceleration can change by large amounts in times that are probably less than a year. These short-term variations do not concern us when we are studying the long-term behavior of the Earth-Moon system.

Second is the quantity that 1 shall call the 'epochal average acceleration'; it is what has been called simply 'the acceleration' in most previous writing. I deliberately use the cumbersome term 'epochal average acceleration' in order to emphasize that we must define and use this quantity carefully.

To define the epochal average acceleration of the Moon, suppose that the observed mean longititude L of the Moon at some epoch t exceeds that calculated from the gravitational theory by an amount DL. The epochal average acceleration of the Moon, to be written <n·M> between the epoch t and some reference epoch t0 will mean

We may call this the 'displacement' average, as opposed to the 'time' average DnM/(t-t0). The distinction between the two kinds of average is crucial to much of the work.

We must similarly define an epochal average acceleration <w·E> of the Earth's spin. The Earth's angular position in space is the source of the time measure called universal time (UT). The measure of time used in the gravitational theory of the solar system is called ephemeris time (ET). If we have correctly matched UT and ET and their rates at some point, the effect of non-gravitational forces on the Earth's spin is represented by the difference

If w·E were positive, we would find DT<0. Hence we define <w·E> by


in some units. If DT is in seconds of time and if t is in centuries, <w·E> is in seconds of time per century per century. If we divide by the number of seconds in a century and multiply by 109, we get the 'epochal average y' given by
<y>=-2"*109DT/(t-t0)2/3.156*109= -0.6338 DT/(t-t0)2

In this paper, the epochal average accelerations will always be calculated between some epoch t and the epoch 1900.

If the epochal average tended to approach some limit as we go back in time, this limit would be a convenient description of the acceleration. However, as we shall see, the epochal average does not behave this way within the historical period. Hence, for the third meaning of acceleration, we adopt a 'running' average. The running average will mean the average over an interval long enough that its value does not change 'suddenly' when we change the length of the averaging interval. I shall not try to develop a rigorous definition of the running average. In practice, so far as we know now, we get averages that behave satisfactorily for our purpose if we take running averages over an interval of two or three centuries.

We shall have to deal with the osculating accelerations only in the next section, and I shall not use a separate notation for them. The notations n·M, w·E, and y will henceforth mean running averages, except in the next section where it will be explicitly pointed out that they mean osculating values.

4. Modern Data that Concern the Accelerations

Since modern astronomical data are so accurate, one might think that it would be a simple matter to calculate the accelerations from them. Surprisingly, however, estimates of the accelerations from ancient data may be more accurate than estimates from modern data.

Part of the problem is that there are large fluctuations of short duration in the Earth's rotation. Markowitz (1970) has studied the fluctuations since the introduction of the cesium clock in 1955, and his results confirm what had been suspected for some time before then. He finds that there are sudden changes in wE (or y) that occur within a few months; the data are not yet able to reveal the details within a period of change. (Here wE and y denote osculating values.) Once a new value of y is established, it stays substantially constant for some time. The average interval between changes is about 4 yr. The changes in y look random, but the type of statistics has not been determined. The standard deviation of the changes in y is about 350.

This standard deviation is an order of magnitude greater than the epochal average y. Thus, no matter how accurate the data are, we cannot estimate an epochal average unless the time span of the data is long enough to remove the effect of the fluctuations. The time span needed can be estimated from the work of Martin (1969).

Martin has analyzed about two thousand records of occultations of stars by the Moon between 1627 and 1860; the data had been collected and used by Newcomb in earlier work that Martin describes. Briefly, Martin calculates the value of ET when the occultation would have occurred, using the present orbital theory of the Moon. He then finds DT by comparing the recorded time with ET. The results are shown in Figure 1. Data in Figure 1 for years after 1860 are taken from page 90 of the Explanatory Supplement (1961).

Whith the imprecise early data it is necessary to use 5-yr averages of DT, and even this averaging is probably not long enough for the first few points. For consistency, I have used 5-yr averages even for the later data that are precise. The averaging removes much detail, but the sporadic nature of the changes is still evident.

In order to study the effect of the averaging interval, I fitted four parabolas in turn to increasingly long segments of the data. The four curves obtained are plotted in Figure 1, and the corresponding accelerations, which are approximately epochal averages, are listed in Table I.

It is clear from Figure 1 and Table I that the epochal average acceleration depends drastically upon the time interval unless the interval is considerably greater than a century. The average for the twentieth century even has a different sign from the longer averages. <y> does not become reasonably constant unless the averaging interval is about 3 centuries. If we had used the 'time averaging' definition of <y>, the averaging period would have had to be measured in millennia.

Fig. 1. DT, the difference between ephemeris and universal time, as derived by Martin (1969) from lunar occulations. It should be emphasized that the primary quantity determined by the data is the parameter called <n'M> in the text, and that additional assumptions are needed in order to get DT. The four curves are arcs of parabolas that have been fitted to different spans of the data.

From the statistical properties of the fluctuations, and from the definition given in Equation (6), we can calculate that the expected contribution of the fluctuations to |<y>| is about 1 when |t-t0| is 3 cy. This agrees excellently with Table I.

Effect of the averaging interval upon
the recent epochal average spin
acceleration of the Earth

Interval <y>, cy-1
1900-1960 +63.2
1810-1960 -27.8
1710-1960 - 6.4
1627-1960 - 5.7

The lunar occultations used by Martin do not actually yield estimates of DT or of <y> directly. Instead, they yield a parameter <n'M> which will be defined later. First we must discuss some estimates of <n·M>, and then we can return to Martin's work.

Hopefully, we do not need as long a time interval to estimate <n·M> reliably as we need to estimate <y>, but the matter is not certain, van Flandern (1970) has used observations of the Moon since the introduction of the cesium clock to estimate the osculating n·M. His estimate, for the interval 1955 to 1969, is
n·M=-52±16 "/cy2

This differs widely from the value accepted as standard in the literature, which comes from the work of Spencer Jones (1939).

Spencer Jones used telescopic observations of the Moon, the Sun, Mercury, and Venus for the period 1677 to 1935. He combined these data with analyses of ancient Greek and Babylonian data that had been made by others, and estimated a parameter that he called 'the acceleration of the Sun'; this parameter is directly proportional to <w·E>. However, it was soon recognized that his work has a quite different significance. In effect, his analysis of the solar and planetary data furnishes the scale of ephemeris time back to 1677, and his study of the Moon, when combined with ephemeris time, gives an estimate of <n·M> since about 1677. Perhaps most importantly, Spencer Jones's assumptions about the ancient data drop out of the estimate of <n·M>, so that the estimate of <n·M> depends only upon the data from 1677 to 1935. His work furnishes an example of something that is often seen, namely that the first method adopted for the solution of a problem is seldom the simplest one. The value obtained for <n·M>, which is usually called Spencer Jones's value, is
<n·M>= -22.44±s "/cy2

The value of s based only upon internal consistency of the data is about 1 "/cy2, but this neglects possible systematic errors in the analysis, van Flandern says that systematic errors in Spencer Jones's estimate could be large enough to alter the derived acceleration by 100%.' Thus we should perhaps take s in Equation (8) to be about 20.

It should be noticed that van Flandern's estimate of the error in his own determination (Equation (7)) includes a generous allowance for systematic errors. The error estimate based solely upon internal consistency is 2 rather than 16.

Newton (1968), using perturbations on near-Earth satellites, estimated the potential due to the lunar tide and calculated that its contribution to y is between  -16 and -22. depending upon assumptions made in the analysis. With the aid of Equation (2), we can calculate the corresponding values of n·M (which will be osculating values), and we can represent the mean of the results by
n·M = -20 ±3 "/cy2

The standard deviation quoted reflects mostly the uncertainty introduced by assumptions made in the analysis. There is also the possibility of a large systematic error, because the available data were poorly distributed with respect to an important parameter.

To sum up the situation about the present n·M, there are three estimates, given by Equations (7), (8), and (9). The ratio of the largest to the smallest estimate is 2.6. The situation should improve quickly as additional data become available, and we can hope that the estimates will become consistent in perhaps a decade from now (1971). However, we cannot yet rule out the possibility that the differences reflect real changes in the osculating acceleration n·M.

It is often said that Spencer Jones's data show that tidal friction has been constant for the past two centuries, but Spencer Jones did not say this and his data do not support the statement. Put into our terms, what Spencer Jones said was that the acceleration -22.44 is the constant that best fits the data from 1677 to 1935, but this does not say that the acceleration was constant. After all, 4 is the constant that best fits the set of numbers 2, 3, and 7, but this does not say that the numbers are constant. In the case of Spencer Jones's data, let T denote time in centuries from 1900. Then a fit using n·M= -17.2+6.0T does not change any residual in the Moon's longitude by as much as half a second of arc from the fit using n·M=-22.44, in the presence of residuals as large as 10". However, the value of 17.2+6.0T changes by a factor of 2 between 1677 and 1935.

It is also said often that theoretical considerations indicate that tidal friction cannot have changed by more than a small amount, say a few per cent, within historic times. This statement also does not stand up to analysis. Spencer Jones's value of n·M requires a power dissipation of about 2.9*1019 erg s-1 and van Flandern's value requires about 6.7 x 1019 erg s-1. Miller's estimate (Miller. 1966) that was mentioned in Section 2 above is 1.4 to
1.7*1019 and he seems to favor the lower value. Thus we have no knowledge at all about the mechanism that produces half or more of the dissipation. There can be no theoretical considerations about a mechanism of which we know nothing.

Let us now return to the question of the parameter that is furnished by the lunar occultations. Martin estimated the values of DT by taking <n·M>= -22.44. A different value of <n·M> would lead to different values of DT, which would lead to different values of <y> in Table I. When we go through the algebra of finding how much <y> changes when we change <n·M>, we find that the parameter <n'M> = <n·M> - 1.7373<y> is independent of the value used for <n·M>. In other words, <n'M> is the parameter determined by the occultation data. Physically, <n'M> is the epochal average acceleration of the Moon with respect to solar time, as contrasted with ephemeris time.

From Table I, we can say that <y> = -6.0±0.4 represents the situation fairly well when we use <n·M> = -22.44. Thus the lunar occultations yield
<n'M> = <n·M> - 1.7373<y> = -12.1 ± 0.7 

for the epochal average from 1627 to the present.

From the geophysical viewpoint, the independent parameters are n·M, which we assume is wholly of tidal origin, and ynf, the part of y that does not have a tidal origin. It is interesting to calculate the values of ynfthat result from combining the estimate of n'M with the three estimates relating to n·M in Equations (7), (8), and (9). The results are shown in Table II. Both osculating and epochal average values appear in Table II and the notation used there does not try to distinguish them.

Some important parameters that correspond
to different estimates of the present lunar
acceleration n·E as constrained by
recent lunar occultations

-20  - 4.6  -23.3  18.7
-22.44  6.0  -26.2 20.2
-52  -23.0  -60.7 37.7

5. Ancient and Medieval Data that Concern the Accelerations

Strictly speaking, it is necessary to distinguish between universal time and solar time in analyzing ancient and medieval data. Luckily, we shall not make much error in the accelerations if we ignore the difference, and I shall use the two terms as equivalents in the rest of this paper. The difference between the two kinds of time is probably of the order of a minute in the past 2000 yr.

In two recent studies (Newton, 1970 and 1972), I have analyzed about 600 observations, with dates ranging from June 15, 763 BCE to April 2, 1288 CE, for the purpose of studying the accelerations of the Earth and Moon. This is many times the number of observations that had been used before for this purpose. Further, except for one oversight, I believe that I have analyzed every ancient astronomical record that has been used by any earlier worker for the purpose of studying the accelerations. For these reasons, the results that I shall present in this section will be taken only from my own work. It is desirable, however, to mention two differences in viewpoint between my work and that of some earlier workers in the field.

(1) There are a number of ambiguous references to solar eclipses in Greek literature, and many workers have leaned heavily on them, for some reason, instead of using straightforward accounts. For example, Fotheringham (1920) finally used only three 'eclipses', which are often called the eponym canon eclipse, the eclipse of Plutarch. and the eclipse of Hipparchus. (See the discussion by Newton [1970, Chapter IV].) The first is a record of the eclipse of June 15, 763 BCE from the official annals of Nineveh, and it seems to pose no problems. The second is a description of a solar eclipse from Plutarch's book about the Moon (Plutarch, ca. 90). I see no reason to suppose that it is a record of a specific eclipse at all; it seems to me that Plutarch merely put it in his book because he needed a description of an eclipse. If it is a specific record, there is no way to know either the time or place of the observation, although various people have 'proved' various times and places. The third probably refers to a real eclipse, but the source directly tells us only that the eclipse was total near (not at) the Hellespont, and it tells us indirectly that the eclipse occurred after the founding of Alexandria (in 332 BCE) and before the death of the astronomer Hipparchus (in about 125 BCE). I have shown that at least four eclipses, with dates from August 15, 310 BCE to November 20. 129 BCE, satisfy all the conditions of the record.

If we suffered from a paucity of straightforward records, it might be desirable to see what information we could rescue from such ambiguous records. However, we have a large body of clear and useful records, and it seems to me that we should be conservative, in choosing the ones that we use. It is much safer to run the risk of omitting some valid data than to run the risk of using erroneous data.

(2) Aside from observations of large or total solar eclipses, the observations that are useful in finding the accelerations involve the measurement of time. There is a strong tendency to use eclipses and to ignore the other data. Apparently this tendency comes from the fact that totality lasts only a few minutes at a particular place, which leads to the belief that an ancient eclipse observation is more precise than an ancient measurement of time. The data provide no basis for this belief. If only a place and not a time of totality is recorded, the question is not how long the eclipse was total there. It is rather: Through what angle could the Earth turn and still let the place lie within the zone of totality? This angle averages about 8°, the equivalent of 32 min of time. However, an analysis of measured times of lunar eclipses (Newton, 1970, Section X.3) indicates that the standard error in an ancient time measurement is about 20 min. Thus ancient observations of totality and ancient measurements of time (as made by professional astronomers) have comparable precision.

In the first study, I used observations of these kinds: conjunctions or occultations involving the Moon or a planet, magnitudes and times of solar and lunar eclipses. times of equinoxes or solstices, and places where solar eclipses were total or nearly so. Most of the 200 or so observations were made by professional astronomers.

Most of the observations (including the eclipses) involve the position of the Moon. The observations that involve both the Moon and the Earth, even if they involve the Earth only to the extent of a time measurement, strongly determine a parameter <D"> defined by

Some observations yield exactly this parameter. Others, like the lunar occultations discussed in the preceding section, yield a slightly different parameter such as <n'M> (Equation (10)), from which we can calculate <D"> with little uncertainty. For example, <D"> - <n'M> = 0.1300<y>. If <y> = -6, we get <D"> = -12.8 ± 0.7 from Equation (10), and <y> = -23.0, which is the largest value in Table II, we get <D"> = -15.0±0.7. Physically, <D"> means the epochal average value of the second derivative of the angle from the Sun to the Moon. The angle involved is called the lunar elongation, and the second derivative is taken with respect to solar rather than to ephemeris time.

Before we look at the accelerations themselves, it is instructive to look at estimates of <D"> obtained from the ancient data. Newton (1970. Table XIV.4) gives fourteen estimates of -0.622 <D"> at different epochs from 700 BCE to 1050 CE, as derived from six types of observation. The values of <D"> that are calculated from his table are plotted in Figure 2. The figure suggests that <D"> has been far from constant. All the values before 700 are positive and all the values after 700 are negative. <D"> apparently increased steadily from the time of the earliest data until about 700. and then abruptly began to decrease. The values fit two straight lines quite well, and I have indicated this by the broken line in Figure 2. Since it is physically impossible for the slope to, be discontinuous, I have joined two straight segments by a short curve.

Fig.2. Fourteen estimates of D", the epochal average of the second derivative of the lunar elongation with respect to solar time, at different epochs. The broken line has been drawn by eye to fit the data approximately, but it has no theoritical basis and it has not been found by any formal procedure. A negative epoch means an epoch BCE.

It should be emphasized that there is no phenomenological basis for the function indicated by the broken line in Figure 2. It has merely been drawn by eye to fit the points plotted.

The hypothesis that <D"> varied in the manner suggested, instead of being constant, has very high statistical significance. However, since there is no theoretical basis either for or against the variation, and since sudden changes sometimes merely mean bad data, it is unsafe to accept the hypothesis without more evidence. Therefore I decided to test it further. The critical period is clearly not the period of classical antiquity but rather the period from, say, 400 to 1200 CE, within which the hypothesized change took place.

During the most critical part of this period, around 700. there was little professional astronomical activity anywhere in the world, so far as we know. Therefore 1 decided to perform the test by using records of solar eclipses preserved in medieval European annals and chronicles. Anyone, and not just a trained astronomer, can note reliably the occurrence of a large solar eclipse. An individual record may not be a precise observation, but this difficulty is overcome by the large number of observations that have been preserved. I have found about 375 records that seem to be independent and reliable, which we can date accurately, and for which we know the place of observation with useful accuracy (Newton. 1972).

I divided the medieval records chronologically into twelve sets and formed an estimate of <D"> for each set. The results are shown in Figure 3, which is reproduced from the reference. The straight lines in the figure are those from Figure 2, without the refinement of the curved section. The twelve points and error bars are the estimates of <D"> formed from the twelve sets of data. The figure provides almost overwhelming confirmation of the hypothesis that <D"> is far from constant and that its behavior changed suddenly near the year 700, if the analysis has been done correctly. It is desirable that someone perform an independent analysis of the data. In the remaining discussion. I shall assume that the analysis is correct.

Fig. 3. Twelve estimates of D" formed from about 375 reports of solar eclipses found in medieval European chronicles. The two straight lines are the same as those in Figure 2, without the refinement of the short curve joining them. The dashed horizontal line is the best-fitting constant. This figure is taken from Newton (1972) by permission of the Johns Hopkins Press.

<D"> is a combination of <n·M> and <w·E>. We cannot tell from its behavior whether <n·M> alone has changed, whether <w·E> alone has changed, or whether both have changed simultaneously. However, since <D"> has changed, we can be sure that at least one of <n·M> and <w·E> has changed also.

This fact complicates attempts to estimate <n·M> and <w·E> (or <y>) individually. Most of the old observations that have been analyzed involve the Moon. The same fact that led us to study <D"> makes it difficult to determine both <n·M> and <y> from such observations. In order to determine <n·M> and <w·M> strongly, we need a large body of data, and this, in the present state of affairs, means data covering a long time span. But, if one or both accelerations change, it is not valid to use data from a long time span for the purpose of estimating them; when we must determine two or more parameters, we do not necessarily get valid estimates of their mean value under these circumstances.

In the first study (Newton, 1970, Chapter XIV), I used data from 763 BCE to 500 CE for one estimate of the accelerations. Figure 2 indicates that <D"> was reasonably constant over this time span. Thus it is a plausible assumption that both <n·M> and <y> were constant, and this estimate is probably valid. It is
<n·M>= -41.6±4.3              <y>= -27.7±3.4

These estimates can be taken as applying at the approximate epoch 200 BCE.

I used data from 500 to 1241 to form a second estimate. The change in <D"> after 500 suggests that this procedure may not be valid, but I had no clear alternative procedure at the time. Now there is another possibility. Three sets of observations after 500 do not involve the Moon; hence they involve only <y> and not <n·M> and we may validly take their average.

Estimates of the spin acceleration
of the Earth from Islamic data

Type of data  Approximate epoch  <y>, cy-1 s<y>, cy-1
Ten equinoxes or solstices  840  -17.0  6.0 
Occupations or conjunc- 
tions involving Venus 
930  +20.0  24.0 
Mean longitude of the 
Sun at the epoch 
from solar tables 
1000  -25.1  5.6 

The sets of data in question are summarized in Table III. First are ten measurements of the times of equinoxes or solstices, made by Islamic astronomers between 830 and 882. Second are twenty measurements of the time when Venus occulted or was in conjunction with some other celestial object, made by Islamic astronomers between 858 and 1003. Third is the 'mean longitude of the Sun at the epoch' from the solar tables prepared by Ibn Yunis (see Newton [1970, pp. 31-3]); it is not a single observation but instead represents the average of a number of observations made near the year 1000. I suspect that there is a systematic error in the analysis of the Venus observations, but I have not been able to find it. The estimate of <y> formed from Table III is
<y>=- 20.2 ±4.0

This applies at the approximate epoch 920.

Instead of using other data directly in order to estimate <n·M>, let us first estimate <D"> at the epoch 920 by fitting a straight line to the data in Figures 2 and 3 near that epoch. This estimate is -4.5, and a plausible estimate of its standard error is 1.0. The combination of this estimate with Equation (13) yields
<n·M> = - 37.0 ± 6.4

at the approximate epoch 920. Equations (13) and (14) should be compared with the estimates found by using all data after 500 in a single least-squares fit. That estimate (Newton, 1970, p. 272) is <n·M>= -42.3 ±6.1, <y>=-22.5 ±3.6.

It is hard to decide whether we should prefer the latter estimates or those in Equations (13) and (14). Those in Equations (13) and (14) are based upon a sounder principle of averaging time-dependent quantities. They are also based upon more data. On the other hand, data involving the Moon do give some strength toward the determination of the individual accelerations, and this strength is ignored in obtaining Equations (13) and (14). On the whole, I shall prefer the estimates in Equations (13) and (14). but the preference is not strong. Actually, it does not matter greatly which set of estimates is used, because the differences are only at the 'one sigma' level.

6. Non-Gravitational Changes in the Obliquity

The gravitational theory of the solar system yields the following result (Explanatory Supplement, 1961, p. 98, for example) for the mean obliquity e of the ecliptic as a function of time:
e = 23°.452294 - 0°.0130125T - 0°.00000164T 2 + 0°.000000503T 3

In this, T is time in centuries from the beginning of the year 1900. Within historic times, the linear term in Equation (15) dominates, and we see that measurements of the obliquity should increase steadily as we go back in time (T<0) within the historic period.

Needham (1959) discusses a number of old measurements of e. They were mostly assembled, and corrected for refraction and parallax, by Laplace, Gaubil, and others; Needham gives the original citations. He points out that all measurements made about the year 1 or earlier are considerably larger than the value calculated from Equation (15), and he suggests that Equation (15) may need substantial revision.

It seems to me that the probable explanation is an improvement in observing technique. The values ofe in the first few centuries BCE cluster around 23°50' in Needham's plot. Within three or four centuries, the values fall to about 23°40', which agrees well with Equation (15). The value of de/dT derived from the measurements after the time of Ptolemy agrees well with that from Equation (15).

Needham emphasizes the agreement between Ptolemy's value and an almost contemporaneous Chinese measurement. I am not qualified to comment on the Chinese measurement, but measurements made by Ptolemy must, unfortunately, be regarded with suspicion.

For example, it is almost certain (Newton. 1970, pp. 17-22) that Ptolemy's solar data, which he said had been observed with great care (Ptolemy, ca. 152. Chapter III.2), were in fact fudged. The errors in the data are about ten times the size of a plausible observing error for Ptolemy's time, and they are all in the same direction. On the other hand, straightforward calculations based upon results of Hipparchus and earlier astronomers duplicate Ptolemy's 'measurements' exactly. It is also probable that the coordinates in Ptolemy's star table were calculated from an older table by the use of an erroneous rate of precession.

With regard to the obliquity, Ptolemy describes (Ptolemy, ca. 152. Chapter 1.12) two instruments for measuring the meridian altitude of the Sun. He then says that he observed the solstices for several years, presumably with one or both of these instruments, and that he always found the angle between them to be between 47] and 47^ degrees. This, he says, gives the obliquity that Eratosthenes (fl. 250 BCE) had found and that Hipparchus had used. He writes Eratosthenes' value in the odd form that the angle between the solstices is 11/83 of a circle. Later (Chapter 1.14), in preparation for giving some tables, Ptolemy says that twice the obliquity, following 'the value that agrees with ours', is 47°42'40"; he writes this in the sexagesimal form given, but of course with Greek numerals.

The interesting thing is that Ptolemy's measurements would lead to the value 47°42'30" rather than to the value that he in fact adopted. However, 11/83 of a circle is slightly more than 47°42'39". Ptolemy, in the last statement cited, probably rounded 39" up to 40" in order that he could take half of it conveniently.

Here is a situation analogous to that of the solar data. Ptolemy claims that he made some measurements, but the errors in them are unreasonably large. The value that he finally uses was actually obtained by calculation from an older value. It is doubtful, at least to me, that Ptolemy made any observations of the obliquity.

It is interesting that the astronomer Abu Sahl al-Kuhi, working in Baghdad around 990, said that, after a 'thorough investigation' based upon measurements of solar altitudes, he found the obliquity to be 'identical with what Ptolemy had found', namely 23°51'20". The great contemporaneous astronomer al-Biruni [1025, Section 3], from whose writing the quotations above are taken, doubts that al-Kuhi made the claimed observations, but nonetheless finds a charitable interpretation of al-Kuhi's actions. Since we do not know the conditions under which Ptolemy worked, we should extend him a similar charity; perhaps he was the victim and not the instigator of the apparent hoaxes in his work.

In summary, all measurements of the obliquity made since Ptolemy agree well with Equation (15). Measurements made in the few centuries before Ptolemy are higher than the values calculated from Equation (15) by about 10' of arc. This has been interpreted as possibly requiring a revision of the equation. If so. we need a mechanism that would change the obliquity by about 10' in about four centuries. It seems more plausible that the change is a reflection of improved measuring techniques. Someone should investigate the early methods of measurement to see if they have a natural bias toward large values of e.

Some recent papers have indicated a non-gravitational effect upon e, but at a rate too small to be revealed by the ancient data that have been analyzed. Duncombe and Clemence(1958) summarize the situation. Observations of the Sun, Moon, and planers all indicate that e is decreasing more rapidly than Equation (15) says, by an amount that is between 0".2 and 0".35 (0.00005 and 0.00010) per century; this effect is far too small to be shown by the old values of e.

Duncombe and Clemence studied the possibility that this effect could come from an overlooked term in the perturbation theory that leads to Equation (15). Recent work summarized by Aoki (1969) tends to indicate that the theory does not contain any omissions as large as the indicated effect. Aoki thereupon searched for the origin of the effect in friction between the core and the mantle.

We know little about the parameters that govern such friction and hence there is little trouble in constructing a model that leads to the observed effect in the obliquity. The difficulty lies in finding a model that is also consistent with the observed spin acceleration. Aoki (1969) noted that his model leads to a contribution of about -670 to the quantity called ynf here. He noted that this is much too large in magnitude but he apparently did not notice that it seems to have the wrong sign: according to all indications, ynf must be around +20. More recently, Kakuta and Aoki (1971) have worked out a model that accounts for e and that makes a negligible contribution to ynf. It is still not clear that this situation is satisfactory. We need a positive value of ynf, not merely a small negative one.

Luckily, we may soon be able to forget this apparent effect in e. Duncombe (private communication) says that recent study shows a systematic error in the position of the equator used in star catalogues of the 19th century. The discrepancy between the observed behavior of e and that given by Equation (15) apparently vanishes when the observations are corrected for this error. The study described by Duncombe is being prepared for publication.

We should note that the forces which produce the spin acceleration may also tend to change e. A rate of change ofe that is readily consistent with the observed spin acceleration is of the order of 0".01 or less. This is too small to be shown even by modern data.

7. Possible Time Histories of the Accelerations

The pair of straight lines in Figure 3 gives the variation of the epochal average <D"> from -700 to +1300 with an error that is probably no more than about 2"/cy2. To this we may add the estimate, from the discussion following Equation (11), that the present value of <D"> is about -14. Thus it is doubtful that the value of <D"> between 1300 and the present ever fell much below the smallest value appearing in Figure 3. If we were to complete Figure 3 by drawing a straight line from the point at 1300 to the value 14 at the present, we should probably not be far from the truth.

In addition, we have estimates of the individual parameters <n·M> and <y> at the epochs 200 BCE and 920 CE in Equations (12), (13), and (14). In order to obtain more detailed time histories of <n·M> and <y> we shall need to make some assumptions.

Let us estimate the parameter <ynf> at the same epochs, using Equation (3) for <yf> and using the fact that <y>=<yf>+<ynf>. The results are
<ynf> =
20.8 at 200 BCE,
23.0 at 920 CE.

For values at the present epoch, we have two choices. If the value of Spencer Jones (1939) for <n·M> is correct, as confirmed by Newton (1968), we see from Table II that <ynf> =-20.1 near the present time. This, in conjunction with the values in Equation (16), suggest that ynf has been nearly constant during the historical period at about the value 21.3. On the other hand, if van Flandern (1970) is correct that n·M= -52 ± 16 at present, this in conjunction with Equations (12) and (14) suggests that n·M has been nearly constant at about -45.3 during the historic period. We can combine these two assumptions with the detailed time history of <D"> to give two alternatives for the individual accelerations. Before we do this, it is desirable to take up another question.

Figure 3 gives the epochal average <D">. For the study of the basic mechanisms, we are more interested in the running average D", which we can obtain from <D"> in the following way. The accumulated change DD in D at any epoch T (measured in centuries from 1900) is DD=1/2<D">T 2. The running average of the second derivative, that is, the average over a relatively short interval, is D"=1/2 d2(<D">T 2)/dT2. We see from Figure 3 that we can approximate <D"> by
<D"> =
-14+T, -6<T<0,
-44-4T, -12.5<T<-6,
9.125+T/4, -26<T<-12.5.

This functional dependence leads to the time variation of D" that is shown by the solid line in Figure 4. The dashed line shows the function in Equation (17). The circles show the estimates plotted in Figure 2, the +'s show the additional results plotted in Figure 3, and the square is approximately the value obtained from Martin (1969).

The running average D"  from Figure 4, when combined with the assumption ynf = 21.3, leads to the running averages of n·M and y plotted in (A) of Figure 5. It is particularly notable that the dynamics of the situation requires n·M<0 and that the function in Figure 5A does not violate this condition even though the condition was not used in deriving it.

Fig. 4. The epochal average <D"> and the running average D". The circles are the estimates of <D"> from Figure 2, which were mostly obtained from observations made by professional astronomers. The +'s are the estimates from Figure 3, which were derived from lay notices of solar eclipses found in medieval European chronicles; they naturally show more scatter. The square is obtained from the data of Martin (1969). The function shown by the broken line segments is a simple approximation to the observed values of <D"> and the function shown by the solid lines is the corresponding variation of the running average D".

Fig. 5. Simplified, and probably somewhat exaggerated, functions describing various running averages as functions of time. In (A), the running average D" from Figure 4 is combined with the assumption that ynf has been constant at 21.3. In (B), the same D" is combined with the assumption that n·M has been constant at -45.3. The choice between (A) and (B) requires resolving the uncertainty in the present value of n·M.

If we combine D" from Figure 4 with the assumption n·M= -45.3, we get the running averages of y and ynf plotted in (B) of Figure 5. It is interesting that y always remains negative here, although there is no dynamical requirement that it do so.

It is of course not possible for the accelerations to be discontinuous as they appear in Figure 5. The discontinuities are a consequence of the fact that we approximated <D"> by segments of straight lines in Figure 4. The extreme values of the running averages are probably also a consequence of that simplification, and hence the function in Figure 5 should be regarded as somewhat exaggerated. However, even if we use the smoothest function that we can in good conscience fit to the values of <D">, we obtain swings in the running averages that are more than half of those shown in Figure 5.

The differences between (A) and (B) in Figure 5 depend almost entirely upon the uncertainty in the present value of n·M. It is highly important that the ambiguity in n·M be removed as soon as possible. For the moment, we can say only that the evidence favors (A) somewhat.

The reader may ask why I am willing to accept sudden changes in the accelerations when I was not willing to accept them in the obliquity. There are two basic reasons.

(1) The changes in <D"> rest upon many kinds of observation, ranging from measurements made by professional astronomers who had attained a high level of observing technique to simple statements made by laymen that they had seen an eclipse of the Sun. In contrast, the suggested change in the obliquity rests only upon the earliest surviving observations, and the measurements began to agree with the modern calculations as observing techniques improved.

(2) There is also the magnitude of the effect. The accelerations involve changes in the Earth's spin that are probably less than 1 part in 107 per century. On the other hand, the change in e that Needham (1959) suggested means a change in the Earth's angular momentum vector at the rate of nearly 1 part in 103 per century, four orders of magnitude greater.

8. Summary

The acceleration of the Moon due to non-gravitational forces has averaged about -42 "/cy2 over the past 2000 yr, and the acceleration of the Earth's spin has averaged about -28 parts in 109 per century over the same interval. The errors in these estimates probably do not exceed about 10% or so. In contrast, the present values of the accelerations are uncertain by a factor of more than 2. There is also a reported decrease in the obliquity of the ecliptic by about 1/4 s of arc per century that probably results from systematic errors in observations made in the 19th century. Non-gravitational effects upon e of the size that we expect are too small to be shown by either ancient or modern data.

The parameter <D">, which is a linear combination of the accelerations of the Earth and Moon, can be followed as a function of time with high confidence from about 700 BCE to the present. From its behavior, we are apparently forced to conclude thai there was something like a 'square wave' in the non-gravitational forces that began about 700 CE and that lasted until about 1300 CE. During the time of this square wave, the accelerations apparently changed by factors of around 5.

We are seriously lacking in mechanisms to explain the non-gravitational forces. The only mechanism of tidal friction (the 'shallow seas' model) that has been evaluated quantitatively provides only about one-fourth of the necessary amount of friction, and it does not provide for much change with time within a period as short as historic times. Forces of non-tidal origin, which are of the same order as the tidal forces, may be due largely to core-mantle interactions. There are no quantitative theories of thess interactions; there are only models whose parameters are uncertain within many orders of magnitude.

The reason for including this paper in this symposium is to answer this question: Can we infer the time scale of evolution of the Earth-Moon system from astronomical evidence? At present, we cannot do so with confidence, because the forces involved seem to have changed greatly within historic times, and we know of no mechanisms that can explain either the magnitude or the time behavior of the forces. Thus any attempt to extrapolate the forces from historic time into geologic time is fraught with uncertainty.

This is not a conclusion of despair. Our knowledge of the situation has increased rapidly during the past few years and should continue to increase rapidly. The answer to the question may be quite different a few years from now.

Since we need to extend research into new areas in order to understand the situation, a little speculation is in order. It is probable that we need mechanisms of tidal friction that provide more friction than the current -shallow seas' mechanism and that provide for large changes within a period as short as 1000 yr. There is no reason to limit ourselves to only one important mechanism. Jeffreys (1968) suggests the energy dissipation in breaking waves, and their interaction with the tides, as a likely mechanism. He works out some of the necessary theory, but makes no quantitative estimates of the total dissipation. It does not seem likely that this contribution would change appreciably within historic times.

Earlier (Newton, 1970) I suggested shelf ice around Antarctica as a possible source of friction. Such ice may be an efficient absorber of tidal energy, it is closely coupled to all the major oceans, and it could vary rapidly with time since it is probably a sensitive function of climate, which has changed appreciably within historic times.

There is another possibility. It was suggested in Section 2 that the interaction between tidal currents and non-tidal ones might provide more friction than tidal currents alone. Further, there is the possibility that non-tidal currents either reflect or cause changes in climate. Hence, not only is there the possibility that this interaction may provide more friction, but there is also the possibility that it may have changed rapidly with time even within historic times.

However, it is possible that tidal friction has remained constant during the past 2000 yr. If we conclude that this is the case, the data force us to conclude that the forces of non-tidai origin have changed greatly during that time. It seems harder to explain rapid changes in the non-tidal forces than in the tidal ones.

Acknow ledgements

I thank V. L. Pisacane and R. E. Jenkins, of the Applied Physics Laboratory of Johns Hopkins University, for their comments and for many helpful discussions. This work was supported by the Department of the Navy under Contract N00017-72-C-4401 with Johns Hopkins University.


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