Difference between revisions of "Astronomical Prediction Based on Patterns"
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==Galactic Dynamics== | ==Galactic Dynamics== | ||
| − | Lindblad’s epicycles – valid method or bad science? | + | '''Lindblad’s epicycles – valid method or bad science?''' |
https://arxiv.org/ftp/arxiv/papers/0911/0911.1594.pdf | https://arxiv.org/ftp/arxiv/papers/0911/0911.1594.pdf | ||
Revision as of 18:28, 1 October 2019
This page will demonstrate that prediction in astronomy is based solely on patterns in the sky. Celestial events come in patterns and trends. By analyzing the patterns of past behaviors from historic tables it is possible to create an equation that will predict a future event. This is how prediction in astronomy has been performed for thousands of years, and how it is still performed today.
Ancient Astronomy
Ancient Babylonians
Astronomy for Physical Science - Cal State Long Beach (Archive)
“ The Babylonians accumulated records of astronomical observations for many centuries. The records enabled them to see repeated patterns in the motions of the celestial objects. They used the patterns to predict the positions of the Moon and planets. ”
Mathematical Thought from Ancient to Modern Times: Volume One
“ Babylonians calculated the first and second differences of successive data, observed the consistency of the first or second differences, and extrapolated or interpolated data. Their procedure was equivalent to using the fact that the data can be fit by polynomial functions and enabled them to predict the daily positions of the planets. They knew the periods of the planets with some accuracy, and also used eclipses as a basis for calculation. There was, however, no geometrical scheme of planetary or lunar motion in Babylonian astronomy. ”
Modern Astronomy
Perturbations
Description and Function
Gravitation Vs. Relativity
Charles Lane Poor, Ph.D. (bio)
Professor Emeritus of Celestial Mechanics,
Columbia University
“ The deviations from the “ideal” in the elements of a planet’s orbit are called “perturbations” or “variations”.... In calculating the perturbations, the mathematician is forced to adopt the old device of Hipparchus, the discredited and discarded epicycle. It is true that the name, epicycle, is no longer used, and that one may hunt in vain through astronomical text-books for the slightest hint of the present day use of this device, which in the popular mind is connected with absurd and fantastic theories. The physicist and the mathematician now speak of harmonic motion, of Fourier’s series, of the development of a function into a series of sines and cosines. The name has been changed, but the essentials of the device remain. And the essential, the fundamental point of the device, under whatever name it may be concealed, is the representation of an irregular motion as the combination of a number of simple, uniform circular motions. ”
“ The Tide Predicting Machine of the Coast and Geodetic Survey at Washington is a note-worthy example of the application of the mechanical method [of prediction via epicycles]. The rise and fall of the tide at any port is a periodic phenomenon, and it may, therefore, be analyzed, or separated into a number of simple harmonic, or circular components. Each component tide will be simple, will have a definite period and a constant amplitude; and each such component may be represented mechanically by the arm of a crank, the length of which represents the amplitude; each crank arm being, in fact, the radius of one of the circles in our diagram.
Such a machine was invented by Sir William Thomson and was put in operation many years ago. The machine at present in use at Washington was designed by William Ferrell. It provides for nineteen components and directly gives the times and heights of high and low waters. In order to predict the tides for a given place and year, it is necessary to adjust the lengths of the crank arms, so that each shall be the same proportion of the known height of the corresponding partial tide, and to adjust the periods of their revolutions proportionally to the actual periods. Each arm must also be set at the proper angle to represent the phase of the component at the beginning of the year. When all these adjustments have been made, the machine is started and it takes only a few hours to run off the tides for a year, or for several years. This machine probably represents the highest possible development of the graphical or mechanical method. It is a concrete, definite mechanical adaptation of the epicyclic theory of Hipparchus.
But, because the Coast Survey represents and predicts the movements of tidal waters by a complicated mass of revolving cranks and moving chains, does anyone imagine for a moment that the actual waters are made up of such a system of cranks? No more did Hipparchus believe that the bodies of the solar system were actually attached to the radial arms of his epicycles; his was a mere mathematical, or graphical device for representing irregular, complicated motions.
While the graphical, or mechanical method is limited to a few terms, the trigonometrical, or analytical method is unlimited. It is possible to pile epicycle upon epicycle, the number being limited only by the patience of the mathematician and computer. ”
Use in Astronomical Almanacs
From the Wikipedia section on Special Perturbations (Archive):
“ In methods of special perturbations, numerical datasets, representing values for the positions, velocities and accelerative forces on the bodies of interest, are made the basis of numerical integration of the differential equations of motion.[6] In effect, the positions and velocities are perturbed directly, and no attempt is made to calculate the curves of the orbits or the orbital elements.[2] Special perturbations can be applied to any problem in celestial mechanics, as it is not limited to cases where the perturbing forces are small.[4] Once applied only to comets and minor planets, special perturbation methods are now the basis of the most accurate machine-generated planetary ephemerides of the great astronomical almanacs.[2][7] ”
General Application
Perturbation methods are, in fact, prevalent in many areas of science. From Perturbations in Complex Molecular Systems (Archive) we read the following:
“ In general perturbation methods starts with a known exact solution of a problem and add “small” variation terms in order to approach to a solution for a related problem without known exact solution. Perturbation theory has been widely used in almost all areas of science. Bohr's quantum model, Heisenberg's matrix mechanics, Feynman diagrams, and Poincare's chaos model or “butterfly effect” in complex systems are examples of perturbation theories. ”
The Wikipedia article on Perturbation Theory (Archive) echoes the same:
“ This general procedure is a widely used mathematical tool in advanced sciences and engineering: start with a simplified problem and gradually add corrections that make the formula that the corrected problem becomes a closer and closer match to the original formula. ”
The book Approximate Analytical Methods for Solving Ordinary Differential Equations states on p.65 (Archive):
“ The perturbation theory had its roots in early studies of celestial mechanics, for which the theory of epicycles was used to make small corrections to the prediction of the path of planets. Later, Charles Eugene used it to study the solar system, in particular the earth-sun- moon system. Now, it finds applications in many fields, such as fluid dynamics, quantum mechanics, quantum chemistry, quantum field theory, and so on.
The idea behind the perturbation method is that we start with a simplified form of the original problem (which may be extremely difficult to handle) and then gradually add corrections or perturbations, so that the solution matches that of the original problem. The simplified form is obtained by letting the perturbation parameter take the zero value. ”
History
The Wikipedia article on Perturbation Theory also provides a history:
“ Perturbation theory was first devised to solve otherwise intractable problems [Links us to the Three Body Problem Wikipedia page] in the calculation of the motions of planets in the solar system.
...Since astronomic data came to be known with much greater accuracy, it became necessary to consider how the motion of a planet around the Sun is affected by other planets. This was the origin of the three-body problem; thus, in studying the system Moon–Earth–Sun the mass ratio between the Moon and the Earth was chosen as the small parameter. Lagrange and Laplace were the first to advance the view that the constants which describe the motion of a planet around the Sun are "perturbed", as it were, by the motion of other planets and vary as a function of time; hence the name "perturbation theory". ”
University of Toronto Quarterly
At the end of the 19th century, long after Copernicus, Kepler and Newton's contributions to astronomy, we read about the state of astronomy in University of Toronto Quarterly (1895):
“ Of the modern employment of the Ptolemaic epicycles, De Morgan, secretary of the Royal Astronomical Society of London, wrote in 1844: “ The common notion is that the theory of epicycles was a cumbrous and useless apparatus, thrown away by the moderns and originating in the Ptolemaic or rather Platonic notion that all celestial motions must either be circular and uniform motions or compounded of them. But, on the contrary, it was an elegant and most efficient mathematical instrument which enabled Hipparchus and Ptolemy to represent and predict much better than their predecessors had done; and it was probably at least as good a theory as their instruments and capabilities of observation required or deserved. And many readers will be surprised to hear that the modern astronomer to this day resolves the same motions into epicyelic ones. When the latter expresses a result by series of sines and cosines (especially when the angle is a mean motion or a multiple of it) he uses epicycles; and for one which Ptolemy scribbled on the, heavens, to use Milton’s phrase, he scribbles twenty. The difference is that the ancient believed in the necessity of these instruments, the modern only in their convenience; the former used those which do not sufficiently represent actual phenomena, the latter knows how to choose better; the former taking the instruments to be the actual contrivances of nature was obliged to make one set explain everything, the latter will adapt one set to latitude, another to longitude, another to distance. Difference enough no doubt, but not the sort of difference which the common notion supposes.”
Such was the state of affairs fifty years ago; today epicycles may be said to possess the heavens above and the earth beneath and the waters and the air between, nor has the all-pervading ether escaped them. In analytic guise they dominate the mathematics of hydrokinetics and sound, of heat, light and electricity; in fact, wherever there is either periodic or irregular motion, there the mathematician “ scribbles ” his epicycles, and not content like Ptolemy to wheel them on simple circles he rolls epicycle on epicycle to the third, the fourth or the fifth degree. Nor does their influence end here. Machines have been made to record for a sufficient length of time any motions for the character of which a working theory has to be found; other machines analyse the records into epicyclic movements, smoothing out or rejecting accidental irregularities, and still other machines recombine the epicycles to predict the motions as they will occur at a future time or under given changes of condition. Thus we have mechanical tide-predictors, harmonic analyzers of meteorological phenomena, epicyclic tracers of deviation curves for the compasses in iron ships, and a fast increasing array of other such machines. ”
VSOP
VSOP (French: Variations Séculaires des Orbites Planétaires) is a popular software package used to generate planetary ephemeris, which are the positions of the planetary objects in the sky. It is used in astronomy software such as Stellarium and Celestia. It has been alleged that VSOP uses a geometric RET model to make its predictions, and so VSOP and the astronomy software which uses it is therefore a validation of the theory. On assessment we find, however, that VSOP is based on the ancient pattern methods of epicycles and perturbations:
Comparing VSOP to the Ptolemaic System
The following is left by an editor on VSOP's Wikipedia Talk Page (Archive):
“ Modelling VSOP on a ubiquitous PC computer program, starting with only one element for each of the three parameters (L, B R) and then slowly incrementing the number of elements, gives a sense of irony that it is in fact nothing more than a more complex development of the ancient deferent / epicycle system used by Ptolemy. A system that despite being totally dismissed out of hand for being intellectually "wrong", was able to provide a prediction service accurate enough to match the observational resolution available (naked eye, with no reliable mechanical timekeeping). A system that, astoundingly to this author, was able to detect and measure, accurately, the lunar evection, one of the still-used perturbations of the Earth-Moon system. Summing powers of sines and cosines is certainly tantamount to circles upon (or perhaps within) circles; recursing, or perhaps simply nesting, almost endlessly. Whilst of course this is totally irrelevant to the mathematics, it perhaps behoves Wikipedia's wider terms of reference to include this as a philosophical point. ”
Comments from Celestia Developers
Celestia Developers comment on the large number of planet-specific terms used in computing positions:
https://celestia.space/forum/viewtopic.php?f=2&t=8285 (Archive)
“ VSOP87 is a set of polynomials describing the orbits of the major planets. There are over 1000 terms in each series. ”
https://celestia.space/forum/viewtopic.php?f=3&t=2592 (Archive)
“ I could add more terms to the VSOP-87 series, but there are already over 1000 per major planet ”
Replacing the Foundations of Astronomy
In Replacing the Foundations of Astronomy (Archive) its author Dr. Gopi Krishna Vijaya (bio) gives us a historic overview of the foundations of Astronomy. He describes that perturbations are epicycles, and that they are used in astronomy to 'make the observations fit' the theory:
Epicycles Once More
“ Following the Newtonian era, in the 18th century there were a series of mathematicians – Bernoulli, Clairaut, Euler, D’Alembert, Lagrange, Laplace, Leverrier – who basically picked up where Newton left off and ran with it. There were no descendants to the wholistic viewpoints of Tycho and Kepler, but only those who made several improvements of a mathematical nature to Newtonian theory. Calculus became a powerful tool in calculating the effects of gravitation of all the planets upon each other, due to their assumed masses. The motion of the nearest neighbor – the Moon – was a surprisingly hard nut to crack even for Newton, and several new mathematical techniques had to be invented just to tackle that.
In the process, a new form of theory became popular: Perturbation theory. In this approach, a small approximate deviation from Newton's law is assumed, based on empirical data, and then a rigorous calculation of differential equation is used to nail down the actual value of the deviation. It does not take much to recognize that this was simply the approach taken before Kepler by Copernicus and others for over a thousand years – adding epicycles to make the observations fit. It is the same concept, but now dressed up in gravitational disguise: ”
| Copernican Theory | Newtonian Theory |
|---|---|
| Circular motion is primary | Inverse square law is primary |
| Variations: epicycles | Variations: perturbations |
| 30 - 50 circle variables | 100’s of perturbation variables |
| Geometric techniques | Calculus techniques |
| Variables fit to observation | Variables fit to observation |
“ In other words, the entire thought process took several steps backwards, to redo the same process as the Ptolemaic - Copernican epicycle theory, only with different variables. The more logical way of approach would have been to redirect the focus of the improved mathematical techniques to the assumptions in Newton’s theory, but instead the same equations were re-derived with calculus, without examining the assumptions. Hence any modern day textbook gives the same derivation for circular and elliptical motion that Newton first derived in his Principia. The equivalence of the epicycle theory and gravitational theory has not been realized, and any new discovery that fits in with the mathematical framework of Newtonian gravity is lauded as a “triumph of the theory of gravitation.” In reality, it is simply the triumph of fitting curves to the data or minor linear extrapolations – something that had already been done at least since 2nd century AD. Yet the situation is conceptually identical.
~
The Dead End
In the late 19th century, one of the French mathematicians – Henri Poincaré – had already discovered that many of the terms being used in the “perturbation” series by mathematicians like Laplace and Lagrange were becoming infinite for long periods of time, making the system unstable. In simple words, the solutions ‘blow up’ fairly quickly. He also showed that the general problem of 3 mutually gravitating bodies was insoluble through any mathematical analysis! Many physicists and mathematicians built up modern “Chaos theory” based on these ideas, to show simply that one cannot calculate the movements of the planets accurately. Thus began the field of non-linear dynamics.
In the middle of the 20th century, with computers entering the field, the mathematicians pretty much gave up on calculating the orbits by themselves and programmed the computer to do it, even though it was mathematically shown that these orbits were incalculable. They had to be satisfied with approximations or numerical methods (or “brute force” methods.) The result of it all was that after 300 years, Newtonian/Einsteinian thought lands in the same spot that Kepler ended: the orbits point to a living or chaotic system. Only now, there is the additional baggage of all the wrong concepts introduced with regard to “inverse-square law”, “gravitational attraction”, “gravitational mass” and “curved space-time” along with uncountable number of minor assumptions. In this process, an enormous amount of human effort was put to derive thousands of terms in equations over centuries. The entire enterprise has been a wild goose chase ”
Perturbation Search
A search for 'perturbation' on nasa.gov shows that perturbations are used widely for many applications, including heavy use in celestial mechanics.
https://www.google.com/search?&q=perturbation+site:nasa.gov
No. of Results: 56,000
Motion of the Planets
A work on by Dr. Hannes Alfvén (bio) titled Evolution of the Solar System (1976) shows that epicycles are still in use in celestial mechanics, more than 365 years after Kepler's discovery of elliptical orbits. Dr. Alfvén uses a combination of Kepler and epicycles:
https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19770006016.pdf
https://history.nasa.gov/SP-345/contents.htm
The Motion of Planets and Satellites
3.1 The guiding-center approximation of celestial mechanics
3.2 Circular orbits
3.3 Oscillations modifying the circular orbit
p.40
"FIGURE 3.3.1.—The guiding-center method of approximating the Kepler motion. The guiding center moves with constant velocity along the dashed circle of radius r0 in the center of which the gravitating mass Mc is situated. The body M moves in an “epicycle” around the guiding center. The epicycle is an ellipse with the axis ratio 2/1 and semiminor axis of er0. The epicycle motion is retrograde. The resulting motion of M is an ellipse which almost coincides with the undashed circle which has its center at O. The distance from O to Mc is er0. The position of the pericenter is given by Φp. The difference between the undashed circle and the exact Kepler ellipse is really less than the thickness of the line."
Galactic Dynamics
Lindblad’s epicycles – valid method or bad science?
https://arxiv.org/ftp/arxiv/papers/0911/0911.1594.pdf
“ In popular culture, epicycles have become almost synonymous with bad science; “adding epicycles” refers to a process of introducing fudges to make a theory fit data, when actually the theory needs to be replaced in its entirety. It is generally believed that epicycles were banished from science when Newton solved his equations of motion and showed that it follows from the inverse square law of gravity that planetary orbits are ellipses. So, it comes as something of a surprise to those unfamiliar with galactic dynamics that the galactic orbits of stars are treated in textbooks using a theory of epicycles revitalized by Bertil Lindblad in the 1920s, and used to introduce density wave theory, which, as reinforced by Lin & Shu (1964), by Lin, Yuan and Shu (1969) and by Kalnajs (1973), has been the leading model of spiral structure for nearly 40 years. ”
Britannica
Encyclopedia Britannica says:
https://www.britannica.com/science/celestial-mechanics-physics
“ Nicolaus Copernicus assumed that Earth was just another planet that orbited the Sun along with the other planets. He showed that this heliocentric (centred on the Sun) model was consistent with all observations and that it was far simpler than Ptolemy’s scheme. His belief that planetary motion had to be a combination of uniform circular motions forced him to include a series of epicycles to match the motions in the noncircular orbits. The epicycles were like terms in the Fourier series that are used to represent planetary motions today. (A Fourier series is an infinite sum of periodic terms that oscillate between positive and negative values in a smooth way, where the frequency of oscillation changes from term to term. They represent better and better approximations to other functions as more and more terms are kept.) ”
Quotes
R. J. Morrison, F.A.S.L., R.N., in his "New Principia," says:
“ Eclipses, occultations, the positions of the planets, the motions of the fixed stars, the whole of practical navigation, the grand phenomena of the course of the sun, and the return of the comets, may all and every one of them be as accurately, nay, more accurately, known without the farrago of mystery the mathematicians have adopted to throw dust in the eyes of the people, and to claim honors to which they have no just title.
The public generally believe that the longitudes of the heavenly bodies are calculated on the principles of Newton's laws. Nothing could be more false. ”
Sir Richard Phillips in his Million Facts (Archive) reports:
“ The precision of Astronomy arises not from theories, but from prolonged observations, and the regularity of the mean motions, and the ascertained uniformity of their irregularities.
...Nothing therefore can be more impertinent than the assertion of modern writers that the accuracy of astronomical predictions arises from any modern theory. Astronomy is strictly a science of observation, and far more indebted to the false theory of Astrology, than to the equally false and fanciful theory of any modern.
We find that four or five thousand years ago, the mean motion of the Sun, Moon and Planets were known to a second, just as at present, and the moon's nodes, the latitudes of the planets, &c., were all adopted by Astrologers in preparing horoscopes for any time past or present. Ephemerides of the planet's places, of eclipses, &c., have been published for above 600 years, and were at first nearly as precise as at present.' ”
The Eclipses
In Chapter 11 of Earth Not a Globe (Archive) its author gives us an overview of the eclipse calculations:
“ Those who are unacquainted with the methods of calculating eclipses and other phenomena, are prone to look upon the correctness of such calculations as powerful arguments in favour of the doctrine of the earth's rotundity and the Newtonian philosophy, generally. One of the most pitiful manifestations of ignorance of the true nature of theoretical astronomy is the ardent inquiry so often made, "How is it possible for that system to be false, which enables its professors to calculate to a second of time both solar and lunar eclipses for hundreds of years to come?" The supposition that such calculations are an essential part of the Newtonian or any other theory is entirely gratuitous, and exceedingly fallacious and misleading. Whatever theory is adopted, or if all theories are discarded, the same calculations can be made. The tables of the moon's relative positions for any fraction of time are purely practical--the result of long-continued observations, and may or may not be connected with hypothesis. The necessary data being tabulated, may be mixed up with any, even the most opposite doctrines, or kept distinct from every theory or system, just as the operator may determine.
...The simplest method of ascertaining any future eclipse is to take the tables which have been formed during hundreds of years of careful observation; or each observer may form his own tables by collecting a number of old almanacks one for each of the last forty years: separate the times of the eclipses in each year, and arrange them in a tabular form. On looking over the various items he will soon discover parallel cases, or "cycles" of eclipses; that is, taking the eclipses in the first year of his table, and examining those of each succeeding year, he will notice peculiarities in each year's phenomena; but on arriving to the items of the nineteenth and twentieth years, he will perceive that some of the eclipses in the earlier part of the table will have been now repeated--that is to say, the times and characters will be alike. If the time which has elapsed between these two parallel or similar eclipses be carefully noted, and called a "cycle," it will then be a very simple and easy matter to predict any future similar eclipse, because, at the end of the "cycle," such similar eclipse will be certain to occur; or, at least, because such repetitions of similar phenomena have occurred in every cycle of between eighteen and nineteen years during the last several thousand years, it may be reasonably expected that if the natural world continues to have the same general structure and character, such repetitions may be predicted for all future time. The whole process is neither more nor less--except a little more complicated--than that because an express train had been observed for many years to pass a given point at a given second--say of every eighteenth day, so at a similar moment of every cycle or eighteenth day, for a hundred or more years to come, the same might be predicted and expected. To tell the actual day and second, it is only necessary to ascertain on what day of the week the eighteenth or "cycle day" falls.
Tables of the places of the sun and moon, of eclipses, and of kindred phenomena, have existed for thousands of years, and were formed independently of each other, by the Chaldean, Babylonian, Egyptian, Hindoo, Chinese, and other ancient astronomers. Modern science has had nothing to do with these; farther than rendering them a little more exact, by averaging and reducing the fractional errors which a longer period of observation has detected. ” — Samuel Birley Rowbotham
Rowbotham provides pattern-based equations for finding the time, magnitude, and duration of the Lunar Eclipse at the end of Chapter 11.
Royal Astronomer Robert Ball
The Royal Astronomer Sir Robert Ball, in his work The Story of the Heavens (Archive), on page 58, told us:
“ If we observe all the eclipses in a period of eighteen years, or nineteen years, then we can predict, with at least an approximation to the truth, all the future eclipses for many years. It is only necessary to recollect that in 6585 ⅓ days after one eclipse a nearly similar eclipse follows. For instance, a beautiful eclipse of the moon occurred on the 5th of December, 1881. If we count back 6585 days from that date, that is, 18 years and 2 days, we come to November 24th, 1863, and a similar eclipse of the moon took place then. …It was this rule which enabled the ancient astronomers to predict the occurrence of eclipses at a time when the motions of the moon were not understood nearly so well as we now know them. ”
Sommerville
Somerville in Physical Sciences pg. 46 states:
“ No particular theory is required to calculate Eclipses, and the calculations may be made with equal accuracy, independent of every theory. ”
Earth Review
T.G. Ferguson in the Earth Review for September 1894, says:
“ No Doubt some will say, 'Well, how do the astronomers foretell the eclipses so accurately.' This is done by cycles. The Chinese for thousands of years have been able to predict the various solar and lunar eclipses, and do so now in spite of their disbelief in the theories of Newton and Copernicus. Keith says 'The cycle of the moon is said to have been discovered by Meton, an Athenian in B.C. 433,' then, of course, the globular theory was not dreamt of. ”
NASA Eclipse Website
Website URL: https://eclipse.gsfc.nasa.gov
If one visits NASA's eclipse website they will find that NASA explains eclipse prediction through the ancient Saros Cycle, rather than the Three Body Problem of astronomy.
From Resources -> Eclipses and the Saros (Archive) we read a description of the Saros Cycle:
“ The periodicity and recurrence of eclipses is governed by the Saros cycle, a period of approximately 6,585.3 days (18 years 11 days 8 hours). It was known to the Chaldeans as a period when lunar eclipses seem to repeat themselves, but the cycle is applicable to solar eclipses as well. ”
The reader is encouraged to visit NASA's eclipse website and count how many times the Saros Cycle is mentioned, and then count how many times the Three Body Problem is mentioned.
Google Search Term: "saros" site:https://eclipse.gsfc.nasa.gov
No. of Results: 13,700
Google Search Term: "three body" site:https://eclipse.gsfc.nasa.gov
No. of Results: 2 (duplicate text)
From the result: “ The distance of apogee does not vary by much month to month although the value of perigee can change quite a bit. Minimum vs. maximum apogee is a 0.6% spread and minimum vs. maximum perigee is a 3.9% spread. If Newton couldn't solve the three-body problem I certainly can't ”
The Three Body Problem refers to the greatest problem in the history of astronomy. It is the inability of science to simulate or recreate a model of the Sun-Earth-Moon system. It is for this reason that pattern-based methods must be used for prediction in astronomy.
A Text-Book of Astronomy
Read any astronomy textbook and the same will be seen:
A TEXT-BOOK OF ASTRONOMY
by George C. Comstock
Director of the Washburn Observatory and
Professor of Astronomy in the
University of Wisconson
p.116
One should notice that there is nothing in the book about the three body problem, the geometry of the sun-earth-moon system, or Newton's equations as being the basis for the eclipse predictions.