The heliocentric path of the Moon

Introduction

The idea of considering the path of the Moon around the Sun could not even arise until there was a heliocentric conception of the planetary system. Nevertheless, Copernicus did not consider the problem. Part of the reason may be that this question, which is surprisingly complicated, has no significant application in astronomy. This is particularly the case for the works discussed in this paper, since the various authors generally stick to the first approximation of the Earth and Moon having coplanar and circular uniform motions relative to the stars and centred respectively on the Sun and Earth. In what follows, we refer to these simplifying conditions as ‘classical hypotheses’. The heliocentric lunar path is then situated in the ecliptic plane and periodically oscillates from one side to the other of the terrestrial orbit, following the rhythm of successive lunations, while remaining an unclosed curve. It is more appropriate, therefore, to speak of a path rather than an orbit of the Moon round the Sun, a lunar trajectory that authors refer to as ‘in space’, ‘real’ or ‘absolute’, to distinguish it from the lunar orbit around the Earth. However, considered under the classical hypotheses, the problem does not address the movement of our real satellite, only that of an imaginary Moon.

It seems that Kepler was the first to refer to this question. In the Astronomia nova ¹ he represents a lunar path that snakes to either side of the Earth’s orbit, with outer arcs (including the full moon) far too long compared to the inner ones (including the new moon). Except for this excessive asymmetry, this representation introduces the sinuous lunar paths that are subsequently, and for a long time, mentioned in astronomy and cosmography books.

It should be noted that the heliocentric path of the Moon is, in a sense, always sinuous in relation to the Earth’s orbit through which it periodically passes. In this paper, however, the ‘sinuous paths’ of the Moon or other satellites around the Sun always designate, more particularly, paths whose concavity is alternately turned towards the Sun or to the opposite side. Such paths are sometimes referred to as ‘wiggly’.

Few astronomers have tackled the subject of the heliocentric path of the Moon. In France, Jean-Dominique Cassini, D’Alembert, Lalande and Delambre, for example, do not seem to have considered it. Even François Arago makes no mention of it in his Astronomie populaire. However, Nicolas-Louis de Lacaille (1713–1762) and Pierre-Simon de Laplace (1749–1827) respectively note:

Only, in respect of the Sun or any other fixed point taken outside the Earth in absolute space, does the movement of the Moon follow protracted epicycloids. ²

The moon describes an almost circular orb around the earth: but seen from the sun, it seems to trace a series of epicycloids whose centres are on the circumference of the earth’s orb. ³

Laplace does not specify the shapes of the epicycloids of the Moon, and Lacaille ⁴ seems to consider his protracted epicycloids as sinuous curves, which is not always the case, as we shall see.

A ‘rather bizarre’ circumstance

In Astronomie populaire (1880), Camille Flammarion (1842–1925) describes the path of the Moon around the Sun, noting that it does not have the sinuous aspect normally expected:

By some rather bizarre and generally overlooked circumstance, this sinuous curve is so elongated that it hardly differs from the one the Earth traces annually around the Sun, and instead of being (as usually drawn in astronomy treatises) convex towards the Sun at the time of each new moon, it is always concave towards the Sun! ⁵

It may be difficult to imagine a curve that oscillates from one side to the other of the quasi-circular orbit of the Earth with its concavity always being turned towards the Sun. A simple example of such a configuration, however, is that of a circle with centre O, crossed at four points by an ellipse with the same centre O and whose major and minor axes are respectively greater and smaller than the diameter of the circle. Moreover, as we shall see, Flammarion’s statement about the shape of the Moon’s path around the Sun corresponds well to reality, at least under the classical hypotheses. Thus, he is right to denounce as incorrect the sinuous representation of this path. Furthermore, the simple example of the ellipse suggests that when the Moon is at its maximum distance relative to the Sun (full moon), the radius of curvature of the Moon’s heliocentric path will be smaller than the radius of the Earth’s orbit. The opposite will be true at each minimum distance (new moon).

However, astronomy treatises concerned with this lunar path or which seek to represent it are rare. According to Maurice Fouché (1855–1929), the author of an article published under the pseudonym Philippe Gérigny in the first issue of L’Astronomie (1882), the erroneous impression one usually has of the shape of the trajectory ‘is unfortunately confirmed by the figures given in the treatises on Astronomy to explain some phenomena, particularly the phases of the Moon, and where we are obliged to considerably alter the ratio between the distances of the Moon and the Sun to the Earth’. ⁶ It seems, however, that the use of such figures of the lunar phases for educational purposes was rare.

That said, Flammarion may have come across some erroneous texts. That is the case with the Grand dictionnaire universel du XIX^e siècle by Pierre Larousse (1817–1875) where, in the article ‘Lune’ (Moon) an erroneous figure borrowed from Charles Delaunay – see below – is supported by the text: ‘Thus, seen from the Sun, the orb of the moon is a sinuous curve, which sometimes passes within and sometimes passes outside the orb of the Earth’. ⁷ Likewise, one can read in the cosmography of Jules Pichot (b. 1820), ex-student of the École polytechnique: ‘it is a sinuous curve of which one can have a rather precise insight with our figure’. ⁸ The drawing mentioned is of the sinuous paths and is preceded by another figure that is also incorrect. Both figures come from Le Ciel by Amédée Guillemin (see below), in which Flammarion may have found the error he denounces. More curiously, he may have noticed the same error in the work of Charles-Eugène Delaunay (1816–1872). This astronomer, famous for his work on lunar theory, illustrates the lunar path around the Sun with a figure (Figure 1), of which he notes:

The different parts of these curves are, by the way, much closer to the orbit of the Earth than is shown on the figure (. . .). To make the shape of this sinuous line more noticeable, the figure has been drawn by exaggerating the distance from the Moon to the Earth relative to that from the Earth to the Sun. ⁹

Whether exaggerated or not, the undulations do exist in Delaunay’s mind.

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Figure 1.

C. Delaunay (see Note 9).

Flammarion refrained from identifying specific erroneous books on this point, which may be understandable, but neither does he quote texts where a discussion of the error of sinuous paths might be found. This last omission is embarrassing as it may lead the reader to believe that he is the author of this discovery, which is far from being the case. This paper therefore attempts to trace the story back through several more or less imperfect solutions to the author who was the first to establish the theoretical result stated by Flammarion.

A diversity of paths

Before resuming the historical discussion, the various a priori possible forms for the heliocentric path of a satellite – when the movements of the planet and its satellite are supposed to operate in conditions similar to the classical hypotheses of the Moon – should be clarified in a general way. These different forms can be easily obtained from an ‘analytical study’ to which we shall refer under this name in what follows and whose results are stated below. Without any intervention of gravitational forces, we are dealing here with pure kinematics in a plane that can be identified with that of the Ecliptic. An observer situated to the north of this plane will observe the planet and its satellite turn in the direct sense.

Let us designate by a = kb, where k > 1 and b > 0, the respective radii of the circular orbits of the planet and its satellite, and by ω > 0 and Ω = λω, where λ > 1, the respective angular speeds of their uniform circular motions, in relation to a fixed direction in the plane of the ecliptic. Here the heliocentric path of the satellite only depends on three parameters, a, b and λ, while its shape is linked only to the parameters k and λ. Five cases may arise and will be referred to in the historical discussion that follow. Two of these (cases 2 and 4) are limiting cases:

Conjunctions and oppositions are those of the satellite and the Sun observed from the planet and which correspond respectively to new moon and full moon in the case of the Moon. In cases 3 and 5, the path is, in particular, concave towards the Sun at opposition and convex towards the Sun at conjunction. Case 3 is the only one where the trajectory shows inflexions. The probability of observing limiting cases 2 and 4 is obviously zero.

With λ² ≈ 178.7 and k ≈ 389.2, the Moon is the only satellite in case 1 and the lunar path is relatively very close to the Earth’s orbit. Moreover, the radii of curvature of the lunar path at the points of the full and new moon, are respectively b (k + λ)² (k + λ²)⁻¹ ≈ 285.3 × b and b (k − λ)² (k − λ²)⁻¹ ≈ 671.1 × b, values situated on each side of the radius a = kb ≈ 389.2 × b of the Earth’s orbit, in accordance with what is stated above. The Galilean satellites of Jupiter accord with case 3 (Ganymede, Callisto) or case 5 (lo, Europa).

As well this classification, the analytical study provides the radius of curvature of the path at each of its points. However, the main distinctions between the different shapes of the path can be obtained more directly. Firstly, what distinguishes case 5 from the other cases is the retrograde motion of the satellite at conjunction. Now, this retrogradation is only possible if the linear orbital speed of the satellite exceeds that of the planet, that is to say if bΩ > aω, which is equivalent to k < λ. Likewise, case 3 differs from cases 1 and 2 in the convexity of the path towards the Sun at conjunction, which relates to the direction of the heliocentric acceleration of the satellite at this point. Thus, this convexity towards the Sun at conjunction requires the heliocentric acceleration of the satellite to be directed away from the Sun. So, its orbital acceleration bΩ² must exceed in intensity that of the planet, aω², and this condition bΩ² > aω² is equivalent to k < λ². Finally, it is now possible to guess the particularities of the path in the limiting cases 2 (equality of linear speeds) and 4 (equality of heliocentric accelerations).

According to the classical hypotheses, the heliocentric path of a satellite, obtained above with the combination of two uniform circular motions, is likely to accord to another generation known as an ‘epitrochoid’. Such a curve is generated within the plane by a point P attached to a circle C₁ of radius r and situated at a distance δ from the centre of the circle, when C₁ rolls at the outside and without slipping, on a fixed circle C₂ with radius R. If δ = r, the epitrochoid is an epicycloid in the strict sense (simply called ‘epicycloid’ below), with its usual cusps. In the other cases, it is an epicycloid in the broad sense of the term, called contracted or protracted depending on whether δ > r or δ < r. The latter case corresponds to the protracted epicycloids about the Moon mentioned by Lacaille, which are not all sinuous, as this category is composed of cases 1, 2 and 3. It follows that the paths described in the analytical study are epitrochoids obtained by taking r = a/λ, R = r (λ − 1) and δ = b = a/k. Conversely, taking R > 0, r > 0 and 0 < δ < R + r it is possible to find a, k and λ since a = R + r and kδ = λr = a, so that the different shapes of the path depend only on the ratios R/r and r/δ. Hence, it follows that δ < r is equivalent to k > λ, whereas the sinuous paths of case 3 (λ < k < λ²) are protracted epicycloids such as r²/(R + r) < δ < r. Authors undertaking a geometrical study of the heliocentric path of a satellite will always use this trochoidal generation, which gives the normal at any point of the path. From a kinematic viewpoint, the point of contact I between the circles C₁ and C₂ is an instant centre of rotation, so that the normal at any point P of the path is the straight line PI.

Camille Flammarion, a reader of Le Ciel

Amédée Guillemin (1826–1893) was quoted above among the authors in favour of the sinuous lunar path. Indeed, when the first edition of his work Le Ciel was published (1863 and 1864), Guillemin referred to this path ‘as a wiggly line whose development for a whole lunation is shown in figure 60, and for a whole year in figure 61’ ¹⁰ (Figures 2 and 3). This error persists in the second edition ¹¹ (1865), and it is perhaps after reading Guillemin on the Moon that Jules Verne thought it right to evoke ‘the wiggly line of its orbit’. ¹²

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Figure 2.

A. Guillemin (see Note 10, fig. 60).

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Figure 3.

A. Guillemin (see Note 10, fig. 61).

But Guillemin corrected his text in later editions. Thus, in the fifth edition (1877), he wrote: ‘the curve described by the Moon in space is always concave on the Sun side: it therefore does not show any inflexion’. ¹³ While preparing his Astronomie populaire, it seems likely Flammarion would not overlook similar work written by his contemporary and predecessor in the popularisation of astronomy, who had already published his sixth edition. Moreover, Flammarion had probably also consulted the Éléments de cosmographie, in which Guillemin specifies, concerning the path of the Moon around the Sun: ‘It has been shown that it does not have any inflexion and that it always turns its concavity towards the Sun’. ¹⁴ It is therefore likely that Flammarion owes his knowledge of the phenomenon to Guillemin, even if he may have come across this information elsewhere, as for example in the Dictionnaire universel d’histoire naturelle by Charles d’Orbigny (1806–1876), in which the heliocentric path of the Moon is viewed in space and not in the ecliptic plane: ‘the real orbit (of the Moon) is some sort of epicycloid with double curvature (skew epicycloid) whose sinuosities are such that it always presents its concavity to the Sun’. ¹⁵

However, even in the fifth edition of Le Ciel, the figure that illustrates the ‘course of the Moon in space during one of its revolutions around the Earth’ ¹⁶ is still equally erroneous, with a convex lunar path towards the Sun at new moon. That said, it is practically impossible to draw a figure ¹⁷ showing the correct shape of this path along an arc of the Earth’s orbit on the page of a book of the usual format. The ratio of the Earth-Sun and Earth-Moon distances should be about 390, whereas in the drawing by Guillemin it is close to 120. Flammarion’s figure is closer to reality: the same ratio, about 280, exceeds the limit value close to 179, thus ensuring the concavity of the lunar path towards the Sun, as shown in the analytical study.

The error of Edmond Dubois

Going further back in time, we discover that Guillemin was not the origin of the discovery. The correction of his initial error, before it appeared in the main body of Le Ciel, was made in the third edition (1866) in a footnote added to the text of the 1865 edition: ‘The calculation shows that the curve described by the Moon in space is always concave on the Sun side; it therefore does not display the inflexions as shown in our figures’. ¹⁸ It is probably not by chance that this correction was made in 1866.

Edmond Dubois (1822–1891), professor of hydrography at the École navale, published an astronomy course in 1858, a new edition of which, ‘corrected and considerably increased’, was published in 1865. Among the additions is section 243, in which the author develops a calculation under the classical hypotheses that is similar to that of the analytical study. From this he concludes that the heliocentric path of the Moon ‘has therefore no inflexion points and its concavity is always turned towards the Sun’. ¹⁹ It is likely that the footnote added by Guillemin found its root in this study by Dubois, although the latter is incorrect.

The calculations developed by Dubois concerning the Moon may of course be applied to other planetary satellites, except those of Uranus, whose orbital planes are too far away – around 90° – from that of the planet for the classical hypotheses to be applicable. In 1894, the Belgium astronomer, Paul Stroobant (1868–1936) rightly pointed out this exception, ²⁰ which had not been observed by the German astronomer Georg Daniel Eduard Weyer (1818–1896) – not quoted by Stroobant – who applies the theory to the four satellites of Uranus then known. ²¹ Dubois, for his part, focused on the Galilean satellites of Jupiter and concluded: ‘Like our Moon, the first two satellites describe a curve whose concavity is always turned towards the Sun’. ²² But, this contradicts what was noted above, that is to say that Io and Europa follow case 5. The error made by Dubois was pointed out by Stroobant in 1894:

(. . .) our satellite alone has this particularity of circulating in an orbit that always presents its concavity towards the Sun; all the others have a path sometimes concave, sometimes convex towards the Sun.

This conclusion is not consistent with the one found by Mr Dubois; this is because this astronomer has lost sight of the fact that a curve can be alternately concave and convex (towards the Sun) without presenting any inflexion, which occurs for satellites when their orbit around the Sun intersects itself. ²³

We know, in fact, that the paths in case 5, as well as those in case 1, do not have an inflexion point. Stroobant is perfectly correct and Dubois’s proof is wrong in principle. Indeed, the latter succeeded in showing that a trajectory with inflexions necessarily accords with case 3, but he made a mistake in believing that the absence of inflexion necessarily put it in case 1. The mistake made by Dubois is far from insignificant as it led him to an erroneous conclusion concerning the satellites of Jupiter. However, this mistake escaped Fouché, who wrote in his study quoted above and of which Stroobant’s is a generalisation: ‘M. Edmond Dubois is the first who, as far as we know, showed that the path of the Moon is not sinuous, as it is shown here, and that this curve is without inflexion points’. ²⁴

In their papers quoted above, Weyer and Stroobant each develop an analysis of the heliocentric path of a satellite but others had done so before, including the Danish astronomer Hans Schjellerup (1827–1887) ²⁵ in 1865. Finally, apart from the analytical solution, Weyer and Stroobant also give a dynamic explanation of the direction, in relation to the Sun, of the concavity of the heliocentric path of a satellite: ‘To sum up, the path of a satellite turns its concavity towards the Sun when the attraction of the latter prevails over that of the planet’. ²⁶ The question of the lunar path was thus easily solved. However, although the explanation is correct, it ignores the fact that the results of such an approach, in which the masses of the three bodies normally interact, may strictly coincide with those of a purely kinematic analysis. Moreover, the classical hypotheses are only approximations that are theoretically incompatible with the gravitational interactions between three bodies.

An unlikely path

In 1852, before the Cours d’astronomie by Dubois, an unexpected answer to the problem of the heliocentric lunar path had already been given. It was the astronomer Hervé Faye (1814–1902) who takes credit for this in his Leçons de cosmographie. He deals in particular with the apparent movements of the planets, which describe a curve around the Earth ‘known, in Geometry, under the name of epicycloid’. Faye refers here to epicycloid in the broad sense, that is to say epitrochoid. Indeed, curves of this type are involved when, under the usual simplifying hypotheses – uniform circular and coplanar movements – the geocentric movement of planets is modelled by deferent and epicycle. More precisely, the planets describe epitrochoids with loops around the Earth, as illustrated by Faye’s figure 103. To be convinced of this, one merely has to apply the analytical study, replacing the couple of movements (planet/Sun; satellite/planet) with the couples (Sun/Earth; planet/Sun) or (Sun/planet; Earth/Sun), depending on whether the planet is inferior or superior. ²⁷ Accordingly, all the geocentric paths of the planets are indeed in case 5.

But here Faye takes the opportunity to briefly mention, in a footnote, the case of the heliocentric path of the Moon:

It is such a curve that the Moon actually follows in space, because of its double translation motion around the Earth and around the Sun. But, in the lunar epicycloid, the loops of figure 103 are replaced by simple cusp points. ²⁸

Thus, Faye claims that the heliocentric path of the Moon is an epicycloid! How did he manage to reach such a conclusion? Influenced by the case of the planets, could it be that he persuaded himself that paths of this kind were necessarily epitrochoid with loops (case 5) or, at the very least, epicycloids (limiting case 4)? The latter solution is assumed to have occurred to him then as the only one that prevented the heliocentric movement of the Moon from impossible retrogradations. But he had missed the point that his epicycloids could have no loops or inflexion points, and we know that Faye’s lunar path is incorrect, even if his epicycloid always turns its concavity towards the Sun except at its cusps. It should be remembered that a satellite would trace such a path only in the highly unlikely case where its linear speed was equal to that of the planet (k = λ), which is not achieved in the case of the Moon. Among the biggest satellites, only the moons Europa of Jupiter (k ≈ 1160, λ ≈ 1220) and Dione of Saturn (k ≈ 3785, λ ≈ 3930) approach this condition.

This oddity of a path with cusps does not seem to have excited further discussion. We may, however, suspect the mathematician Eugen Catalan (1814–1894) of having been influenced by Faye when he wrote in 1858 and subsequently: ‘It is proved that the path described by the Moon is a rather simple line similar to the ones geometricians call epicycloids’. ²⁹ But nothing is mentioned in the numerous Cours d’astronomie et de géodésie that Faye wrote for the École polytechnique, where he taught from 1873 to 1893. Yet, as General Inspector of Public Education from 1857 to 1877, ³⁰ Faye may have had something to do with a passage in the 1866 syllabus concerning special secondary education created the previous year by Victor Duruy. In the third year, the following lines can be read in the cosmography syllabus: ‘Annual revolution of the moon. – Every month it traces a circle around the centre of the earth, which itself traces another circle in twelve months around the sun. – Epicycloid’. ³¹ We can imagine that the mention of the epicycloid in this official syllabus embarrassed the authors of school cosmographies. Thus, in 1866, it is only in the conclusion of his Traité élémentaire d’astronomie that Alexis Boillot (b. 1813) complied with the requirements of the programme and he still limited himself to introducing the cycloid instead of the epicycloid. Concerning the Moon and the different ways of understanding its path in space, he then states that, ‘with respect to the terrestrial orbit, the Moon traces a wiggly curve’. ³² This is not what Faye said and Boillot repeats the usual error.

John Herschel, populariser of astronomy

Pursuing our research in a reverse chronological order, it is in the astronomer John Herschel (1792–1871), son of William Herschel who discovered Uranus in 1781, that we next encounter the heliocentric path of the Moon. He tackled the subject in an elementary but very successful book. Published in 1833, A Treatise on Astronomy ³³ went through numerous editions and was translated into French several times, even making up a volume of the collection of the Manuels Roret that were designed for a wide readership. ³⁴ Using a more precise framework than that of the classical hypotheses, Herschel focuses on the undulations of the lunar path on either side of the curve traced by the centre of inertia of the Earth-Moon system and specifies that:

If we trace therefore, the real curve actually described by either the moon’s or earth’s centers (. . .) it will appear to be, not an exact ellipse, but an undulated curve, like that represented in the figure to article 272 [about precession and nutation], only that the number of undulations in a whole revolution is but 13, and their actual deviation from the general ellipse, which serves them as a central line, is comparatively very much smaller – so much so, indeed that every part of the curve described by either the earth or moon is concave towards the sun. ³⁵

Herschel reiterated this property of concavity in 1849 in his Outlines of Astronomy, an extended version of the previous book which was also highly successful: ‘The real orbit of the moon is everywhere concave towards the sun’. ³⁶ But Reverend Lewis Tomlinson had probably not read the Treatise, or thought it unnecessary to follow Herschel on that point, when he wrote in his Recreations in Astronomy in 1840:

But here a singular effect results:–if the Earth were stationary, the Moon’s orbit would be found, as in the case of the planets, to be an oval with respect to the Earth; as however, the Earth revolves round the Sun and, of course, carries the Moon with it, the real path of the latter becomes a very singular and complicated curve,–it is a zig-zag circle round the Sun, with several indentations and as many protuberances. ³⁷

As well as from translations of the Treatise, the concavity of the lunar path was also relayed to France in 1836, ³⁸ and once more in 1857: ‘The real orbit of the moon has, in all its positions, its concavity turned towards the sun’. ³⁹ So, Dubois was probably influenced by one of these readings to undertake his own proof of the phenomenon.

As for John Herschel, considering that his assertions of concavity are not accompanied by any proof or reference, we might wonder whether he was the author of the discovery. But this is not the case, since concavity is mentioned in several astronomy texts of the early 19th century. These include the article ‘Moon’ in the Oxford Encyclopaedia ⁴⁰ in 1828, and a strange book published in 1819 by a playwright. ⁴¹ It also appears in reprints of texts by James Ferguson ⁴² discussed below. Moreover, in the same period, other books ⁴³ attribute a proof of this concavity to the mathematician Colin Maclaurin, so the discovery seems to date to the first half of the 18th century.

One of the authors mentioned above, Olinthus Gregory, published a proof of the property of concavity ⁴⁴ in 1802. However, while he quotes Maclaurin on this matter, he curiously omits to mention a book published half a century earlier from which he heavily borrowed his proof, often word for word. This is An Introduction to the doctrine of Fluxions (1751) by John Rowe. Rowe’s geometric proof ⁴⁵ is not always very clear but he did not make the same error as Dubois. The proof is founded on a trochoid generation (Figure 4) by implicitly supposing (following our notation) that δ < r, that is to say k > λ. This is indeed the case in the figure accompanying the text and it is satisfied in the case of the Moon. Under these conditions, Rowe established that the path presents inflexions only if r²/(R +  r ) < b < r, which is equivalent to λ < k < λ² (case 3). Now, in the case of the Moon, the numerical data show that b < r²/(R +  r ), so that the lunar path does not present any inflexions. As it is implied that this path is concave towards the Sun at opposition, and that the direction of the concavity could only reverse by passing through an inflexion point, all that remains is to conclude that: ‘consequently the Curve Mmμ generated by the Center of the Moon has not a Point of contrary Flexure, or is no where Convex towards the Sun’.

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Figure 4.

J. Rowe (see Note 45).

Mathematics professor Samuel Bamfield did not believe in the concavity of the lunar path and was opposed to ‘the unnatural imagination of Maclaurin’. It is probably Rowe to whom he refers when he writes in 1764:

The Author, or Editor, I don’t know which, of a late Publication, styled Mathematics, endeavours to demonstrate that the Moon’s path will always be concave towards the Sun; but borrowed it I suppose from Maclaurin (. . .) ⁴⁶

This is because Rowe does not quote Maclaurin, from whom he probably drew inspiration, even if their proofs differ in their details. As we shall see later, the way he proceeds is, in its principle, the same as the approach of another author not mentioned by Rowe, who refers only to two texts published in 1743 in The Gentleman’s Magazine, in which readers debated the heliocentric path of the Moon.

A lively discussion

In the first half of the 18th century the question of the shape of the heliocentric path of the Moon temporarily interested a broader audience than professional mathematicians and astronomers. The Gentleman’s Magazine, a monthly founded in London in 1731, dealt with a wide range of topics likely to interest educated readers, among them astronomy. Articles were often letters from readers. In 1742, such letters increasingly concern the heliocentric path of the Moon. This started after a first correspondent, writing under the name Philaster, noted that he was disconcerted by the double movement of the Moon around the Earth and around the Sun and expressed a wish that: ‘I should be glad to be informed, “What is the real Path that the Moon, in one of her Revolutions round the Earth, describes.”’ ⁴⁷ Philaster’s letter resulted in an often passionate debate between seven participants from May 1742 to December 1743.

Under the pseudonym Philalethes, a first respondent expresses his views concerning an epicycloid and thus appears as a distant predecessor of Hervé Faye. ⁴⁸ Then a certain J.G. suggests a sinuous path, noting that it does not close, contrary to the epicycloid of Philalethes. ⁴⁹ His representation of the path over a lunar month may have been inspired by a similar figure, provided by the Dutch author Willem Jacob ’s Gravesande (1688–1742), in a book translated into English ⁵⁰ in 1731, despite the fact that ’s Gravesande mistakenly designates the lunar path on the figure as being the Earth’s. It should be added that ’s Gravesande himself believes that this path ‘is twice inflected in each Lunation’, that is to say, sinuous. John Badder (c. 1700–1756) ⁵¹ supports the epicycloid path of Philalethes, ⁵² but the astronomer Edmund Weaver (c. 1683–1748) agrees with J.G. ⁵³ Philalethes then requests that Weaver give a proof of what he has put forward, namely that ‘the Path of the Moon is Convex to the Sun in the first and last Quarters’, that is to say between the last quarter and the next first quarter. The editor also notes that ‘J.G. and Mr Badder are of the same Sentiment’. ⁵⁴

The debate continues the following year: Weaver makes ironical remarks on J.G.’s change of opinion, ⁵⁵ whereas a certain Richard Yate pretends to draw the debate to a close with a ‘demonstration’ ⁵⁶ of the opposite of Weaver’s suggestion. To do so he develops an argument based solely on the comparison of the linear speeds of the satellite and the planet in their respective orbits, which, following the notation of the analytical study, boils down to comparing k and λ. Thus, he starts from k < λ, in which he recognises the paths of case 5. Then, increasing k/λ, he dwells on the equality k = λ (equality of speeds, case 4), which is the case of the epicycloid for which all arcs of the path that would be convex towards the Sun have disappeared. Finally, if the speed of the planet is greater than that of the satellite (k > λ), ‘it is most evident’ that no convexity towards the Sun will reappear. Yate concludes:

If what I have said be applied to the Earth and the Moon in their Orbits, the Moon’s Path will evidently appear to be according to the figure described by Mr Badder and the Convexity of any Part of the Moon’s Path to the Sun, will appear to be impossible.

But Yate is wrong to believe that all paths are epicycloid, or curves of this appearance, when k > λ. Finally, he adds a practical procedure for drawing a cycloidal curve, so as to convince Weaver that the path of the Moon is not convex towards the Sun around new moon. At the same time, we learn that Bladder sent to the Magazine a ‘Trigonometrical Calculation’ also meant to prove that Weaver was wrong. This ‘Calculation’ was published later and only Badder’s conclusion is revealed: ‘If Mr Weaver (or even an Angel) was to assert afresh, that in the first and last Quarters of the Moon, the Curve she describes is convex towards the Sun, I declare I could not believe him’.

A new correspondent, X.Y., starts by sending a criticism to the editor: ‘I Am of Opinion that the Lucubrations (sic) of your Mathematical Friends, ought, generally speaking, to pass a stricter review before they are presented to the Publick’. ⁵⁷ He contradicts Yate’s conclusion about the Moon and correctly describes the trochoidal generation of the lunar path which, he suggests, is a ‘protracted epicycloid’, convex towards the Sun around conjunction (new moon) and which presents inflexions. In other words, it is a sinuous path. Thus, he concludes that this ‘ought to put an end to the Debate: And at the same time Mr Badder too, without any Asseveration of an Angel, may be brought to believe that Mr Weaver is in the right’. Yate responds, however, that he is not convinced by the X.Y.’s argument. ⁵⁸

However, the readers of the Magazine are beginning to tire of this never-ending controversy, and the editorial board refrains from publishing ‘some long Letters with Calculations’ received, although they eventually decide to publish the ‘Trigonometrical Calculation’ by Badder mentioned above. Before that, Thomas Sparrow (1700–c.1760) ⁵⁹ sends a letter to X.Y., in which he points out that the latter, as well as Yate and Weaver, have not demonstrated anything. He continues:

But as the Diameter of the Earth’s Orbit as well as that of the Moon are sufficiently known or at least their Proportions, I still flatter myself that we shall shortly see (. . .) a Mathematical (not a Mechanical) demonstration, not only which way the Convexity of the Lunar Orbit lieth with respect to the Sun, in the first and last Quarters of the Moon, but also the true Nature of the Convexity of that Curve.

And Sparrow adds: ‘Nor can I conceive why the Debate should cease before the Point in Dispute be determined’. ⁶⁰

In this often poorly argued and at times confused discussion, Badder’s ‘Trigonometrical Calculation’ ⁶¹ is something new. As he is in favour of the epicycloid of Philalethes, he offers to confirm the concavity of the lunar path towards the Sun outside the new moons, by means of trigonometrical and numerical calculations. Although these calculations are correct, they do not constitute the mathematical proof expected by Sparrow. This is firstly because it is not really a proof but only calculations whose results seem to point towards a concave lunar path towards the Sun around new moons. Indeed, Badder satisfies himself by showing that the positions a, n, s of the Moon (Figure 5) are such that ∠eas > ∠ean. Secondly, assuming k = 340 (in the notation of the analytical study, a value used by Badder) does not make it possible to trace a usable figure, so his calculations are based on a figure that has practically no relation to reality or to the data of these calculations (∠ASB = 1″). In other words, it is an erroneous figure in many ways, in which the line ans of the lunar path, concave towards the Sun in accordance with what the calculations show and which incidentally goes through o, is purely fictional. This figure, where k ≈ 7.7, corresponds to case 3 in which the lunar path is, on the contrary, convex towards the Sun around new moon.

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Figure 5.

J. Badder (see Note 61) (redrawn).

As for the two texts mentioned by Rowe for his own proof, they are Yate’s ‘demonstration’ and Badder’s ‘Trigonometrical Calculation’, whose publication concludes the debate in the Gentleman’s Magazine.

An anonymous text saved from oblivion

It may seem regrettable that the Gentleman’s Magazine decided to put an end to the discussion of the lunar path, but it should be acknowledged that the debate did not seem to be on the verge of achieving the correct result. Nevertheless, one of the ‘long letters with calculations’ the Magazine chose not to publish did not entirely sink into oblivion.

The Miscellaneous Correspondence, nine volumes of which were published from 1742 to 1748, consists of texts received but not published in the Gentleman’s Magazine. Volume 3 contains a study addressed to the Magazine and entitled ‘The Curve which the Moon, or any other Satellite, describes about the Sun, determined on all possible Suppositions of Distance and Velocity’. ⁶² Now, this study is of particular interest, not only because it deals with the subject of the previous discussion but also because it is signed X.Y., one of its protagonists.

Within the implicit framework of the classical hypotheses, the author begins by recalling that ‘the Moon’s Path is a protracted Epicycloid, as was asserted in my last [letter]’, ⁶³ and adds that a figure found in the work of ‘s Gravesande ⁶⁴ – a figure already highlighted in another work by the same author – had made him believe that this path was ‘partly convex towards the Sun’, as ‘s Gravesande represented it, ‘a mistake which he was probably led into by a mechanical Construction that did not nearly enough exhibit the true Distances of the Earth and Moon from the Sun’. But X.Y. has now corrected his error and states the correct property: ‘All is set to rights again, and the Moon restored to her Path every where (tho’ not equally) concave towards the Centre’, where the ‘Centre’ is the Sun.

Before getting into the details of his ‘demonstration’, X.Y. presents his result in a short paragraph that concludes: ‘then will the Epicycloid have a Point of Contrary Flexion [inflexion point], or not, as d × m is greater or less than Unity’. This is reminiscent of the way in which the analytical study distinguishes between cases 1 and 3. Unfortunately, omissions and other transcription errors make this paragraph quite absurd.

However, things become clearer when the author applies to the particular case of the Moon his result supposed to be valid for any satellite of a planet. Where the notation of the analytical study might confuse what follows, modern interpretations are given in brackets. The author adopts 12.5 for the value of number n of ‘Synodical Months in a Year’ and sets down d = n + 1. (The introduction of d is justified if by ‘year’ the author means ‘sidereal year’, because this year A and the synodical S and sidereal T revolutions of the Moon are linked by 1/S = 1/T − 1/A and, if n = A/S, thus d = A/T = Ω/ω = λ.) He then takes 327 as the value of the ratio between the Earth-Sun and Earth-Moon distances and calculates m = 13.5/327 = 0.0413. (In other words, m = d/k = λ/k.) Finally, he asserts that the criterion of the shape of the lunar path (the distinction between cases 1 and 3) is the position of dm = λ²/k with respect to 1, in accordance with the result of the analysis (if we already know that k > λ). In addition, in the limiting case dm = 1 (case 2), the author correctly notes that the path has an infinite radius of curvature at conjunction. But the text mistakenly associates this case to m = 1/12.5, when it should correspond to m = 1/(n + 1) = 1/13.5. Finally, after his proof, to which we will return, the author asserts: ‘If m = 1 (. . .) [the] two points of contrary Flexions meeting are changed into a Point of reflexion [cusp point]’ (case 4), and ‘If m is greater than Unity, [the Curve] becomes an Epicycloid contracted with Nodes’ (case 5).

The remainder of the letter consists mainly of the short ‘demonstration’. Like Rowe a century later and without saying it explicitly, the author limits himself to the case of a heliocentric path that would be a protracted epicycloid and investigates the conditions under which this path would have inflexion points. Thanks to this limitation, the author does not commit Dubois’s mistake any more than Rowe does. His proof uses the trochoidal generation (Figure 6) – which he had already presented in his previous letter – and this property of the normal to the curve, whose slope goes through an extremum at the inflexion point. But the exploitation of this property is seriously altered by typographical or transcription errors, improper manipulation of fluxions and false equalities. However, the correction of these errors, as well as the reconstitution of an unexplained ‘easy Reduction’, makes it possible to restore a proper formulation of this proof. Thus, we obtain that the existence of an inflexion point requires the double inequality 0 < (d²m² − 1)/(d² − 1) < 1, as this ratio is the square of a sine. It is necessary to exclude the values 0 and 1, which would only occur at opposition and conjunction, where no inflexion is possible. Without recalling that d > 1, the author concludes with the inequality dm > 1 but forgets that m < 1 is also necessary. Finally, with modern notations and after correction, the author’s proof states that the double inequality λ < k < λ² is necessary for inflexions to exist. The reasoning is no more detailed than Rowe’s, but we can see that the result will make it possible to treat the case of the Moon, of which the path will necessarily always be concave towards the Sun.

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Figure 6.

X.Y. (see Note 62).

In spite of its defects, the text shows considerable progress since the author’s previous submission to the Gentleman’s Magazine, dated 13 October 1743, just 3 months earlier. It might even pretend to offer the mathematical proof Sparrow wanted. It should be borne in mind, then, that the five cases of the analytical study have now been studied and more or less characterised, which is impressive and unexpected for the time, most of all by an anonymous author. It is also possible that this study, rather than Maclaurin’s, was a source of inspiration for Rowe. However, at least concerning the particular case of the Moon, X.Y. had been preceded by Maclaurin, whose proof, published only in 1748, he did not know about. This will be discussed after focusing on a completely different approach to the lunar path.

James Ferguson: The Moon’s path delineated by a mechanical device

At the time of the discussions in the Gentleman’s Magazine, the Moon’s path around the Sun not only gave rise to mathematical speculations, but was also the subject of practical approaches with the simple aim of drawing the path by mechanical means.

The extraordinary Scot James Ferguson (1710–1776) was a largely self-made man, author of several books, particularly in astronomy and mechanics, and distinguished himself through the combination of the two. His career from shepherd in the north of Scotland to Fellow of the Royal Society in 1763, has been described in several publications including those by Ferguson himself and, in particular, by Ebenezer Henderson (1809–1879). ⁶⁵ Although little known in France, he is often quoted in the Bibliographie astronomique by Lalande who writes that his popular astronomy book for children ⁶⁶ was greatly appreciated by Madame de Genlis. Delambre devotes a few pages to him in his Histoire de l’astronomie au dix-huitième siècle, but is often critical of Ferguson’s astronomy, concluding: ‘In general, Ferguson’s System is to speak mainly to the eyes, either with large illustrations or with machines: this is where his particular merit lies’. ⁶⁷

In 1739, Ferguson began an astronomical device, his astronomical rotula. Composed of four concentric and abundantly graduated moving plates, the device indicates, on the ecliptic, the positions of the Sun, the Moon and the nodes of its orbit, and gives the dates of solar and lunar eclipses, as well as other information. In 1740, Ferguson contacted the mathematician Colin Maclaurin, who suggested some improvements, and an engraving of the rotula was published in 1742. Ferguson met Maclaurin in Edinburgh in 1741 and, at his request, Maclaurin showed him an orrery demonstrating the motions of the Earth and Moon around the Sun, although he could not view the mechanism. A short while later, Ferguson built a similar orrery in wood, which he showed to Maclaurin in spring 1742. Impressed, Maclaurin asked Ferguson to give lectures on the orrery to his students.

Ferguson settled in London in 1743 and, according to Henderson, it was in late 1744 that he became interested in the Moon’s heliocentric path. He then undertook to build a device to model the paths of the Earth and the Moon around the Sun :

Soon afterward it appeared to me, that, although the moon goes round the earth and that the sun is far on the outside of the moon’s orbit, yet, the moon’s motion must be in a line that is always concave toward the sun: and upon making a delineation representing her absolute path in the heavens, I found it to be really so. I then made a simple machine for delineating both her path and the earth’s on a long paper laid on the floor. ⁶⁸

In 1745, Ferguson showed his apparatus (Figure 7), which he named trajectorium lunare, as well as the delineations, to Martin Folkes, President of the Royal Society, where he was immediately invited to show his achievements. An engraving (Figure 8) dedicated to Folkes was published by Ferguson that same year, with the format of ‘2 feet 10 3 8 inches in length, by 6 1 8 in breath (about 87 × 16 cm)’. It represents the heliocentric paths of the Moon and the Earth over slightly more than a lunar month, accompanied by a rather long text which begins:

The Line delineated which y^e Moon describes in y^e Heavens, during y^e time of somewhat more than a Month; showing that her real Path is constantly curved or concave towards y^e Sun.

Following this demonstration at the Royal Society, the clockmaker John Ellicott (1706–1772), a Fellow of the Society, told Ferguson that he had had the same idea 20 years before. In addition, he showed him a drawing and part of the device by which he had obtained it. Ferguson acknowledged that, ‘it must have been done many years before I saw it’. ⁶⁹ So, highlighting the correct shape of the heliocentric path of the Moon by a mechanical process can be attributed to Ellicott. As we shall see, this clockmaker, who published nothing on the subject, is likely to have been the first to understand, as early as 1724, that this path is everywhere concave towards the Sun.

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Figure 7.

J. Ferguson, Trajectorium lunare (ref. 73).

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Figure 8.

J. Ferguson, J. Bickham (engraver). Inv. 13552. © History of Science Museum, University of Oxford.

Ferguson adds that, ‘from that time, till his death, Mr Ellicott was one of my best friends’. ⁷⁰ By contrast, his relations with Thomas Hawkes, ⁷¹ another tracer of the lunar path, was not as cordial. In December 1752, this mathematical instrument maker from Norwich published in the Gentleman’s Magazine a description of an orrery fitted with a device to trace the lunar path. It is similar to Ferguson’s apparatus, which he does not mention at all. ⁷² Ferguson had certainly been told of this publication because, as early as January 1753, he published a description of his own device in the same journal. ⁷³ Introducing himself as a ‘limner’, that is to say as a portrait painter in Chinese ink (as he also had artistic talents), Ferguson explained that:

Last summer, I gave a draught of the wheel work of my orrery (. . .) together with a scheme of the moon’s concave path, and a black lead sketch of the instrument above described, for drawing her path, to Mr Hawkes of Norwich, tho’ in his account published in the Gent.’ Mag. for December last, he has not thought proper to mention it.

Moreover, a defect in the mechanism devised by Hawkes did not escape Ferguson:

Besides, it seems plain to me that he never tried it before he published his account; for if it should be done according to the figure he has given, the moon must go the wrong way round the earth in his orrery, to make the pencil which describes her path (. . .) go the right way round the pencil which describes the annual path of the earth.

Finally, Ferguson defends himself from the plagiarism he was accused of in relation to his tracing of the lunar path. To do so, he uses the words from a letter he wrote to the historian Thomas Birch (1705–1766) on 12 December 1752:

I know it has been insinuated that I pirated this machine from a Magazine published some years before my scheme of the moon’s path was printed. But I am ready and willing to declare, in the most solemn manner, that I never saw that Magazine, nor heard of any such thing being in it until last summer, when it was handed about by a certain person, with an intent to do me all the prejudice that lay in his power. But the figure of the moon’s path in the Magazine, instead of being always concave to the sun, turns off from it in a sharp angle at every new moon. ⁷⁴

Hawkes’s accusation is of course totally unfounded. We recognise that Ferguson here alludes to the epicycloidal path suggested by Philalethes and then by Badder in the Gentelman’s Magazine in May and August 1742. Let us conclude by pointing out that the clockmaker John Neale (before 1725–after 1758), a maker of orreries, ⁷⁵ also made, probably after Ferguson, a device intended to draw the Moon’s path, ⁷⁶ which was described in 1764:

By the help of this machine, an idea of the motion of the moon, and the curve she describes, may be obtained in an easy and entertaining manner. It will also appear evident, that her path is always concave towards the sun, notwithstanding her motion round the earth, an idea which, to beginners, has always been attended with difficulty. ⁷⁷

Henderson pays tribute to the concavity towards the Sun of the lunar path as ‘the first of Ferguson’s important discoveries in Astronomy (and which added much to his reputation)’. Ferguson was, indeed, the first to publish this result, but he had been anticipated not only by Ellicott but also probably by Maclaurin.

Colin Maclaurin: The first mathematical proof

The mechanical delineation of the heliocentric path of the Moon achieved by Ferguson, and before him by Ellicott, gave a practical answer to the problem of the shape of this curve by revealing that it was clearly concave everywhere towards the Sun. But the satisfaction of the mind still required that this property be confirmed by a mathematical proof, which could only be produced by a skilled mathematician. In fact, when Ferguson completed his delineation, the proof probably already existed in the work of Colin Maclaurin (1698–1746). The name of this famous Scottish mathematician ⁷⁸ is associated in particular in analysis with the ‘Maclaurin series’ and the ‘Euler-Maclaurin summation’, and in geometry with the curve called the ‘Maclaurin trisector’, as well as with ‘Maclaurin ellipsoids’ in mechanics.

After being communicated by Maclaurin in a letter mentioned below, the proof was published for the first time in 1748 in a posthumous book entitled An account of Sir Isaac Newton’s philosophical discoveries, ⁷⁹ which was translated into French the following year. ⁸⁰ It was reprinted again in 1764 ⁸¹ and also in 1777 in an astronomical lesson written in Latin by the Austrian Jesuit and mathematician Karl Sherffer (1716–1783), ⁸² who refers to the original text dated 1748. As early as 1731, Maclaurin had begun to write a book on Newton’s work, but it was only after his death that the mathematician Patrick Murdoch (deceased in 1774) was able to publish the book thanks to a subscription.

Chapter V of book IV is entitled ‘On the path of a secondary planet upon an immoveable plane; with an illustration of Sir Isaac Newton’s account of the motions of the satellites from the theory of gravity’. ⁸³ The first part of the title corresponds well to the lunar motion problem: to study the heliocentric path of a satellite (‘secondary planet’) in the orbital plane of its planet (‘primary planet’) correlated to fixed directions in relation to the stars. In fact, chapter V had not been planned by Maclaurin, as Murdoch added a note to the title: ‘The following chapter, as belonging properly to this place, is inferred from a letter of the author, to his learned friend Dr. Benjamin Hoadly, physician to his Majesty’s household’. ⁸⁴ Murdoch does not indicate the date of this letter to Hoadly (1706–1757), but since Maclaurin does not mention Ferguson’s delineation in it, we can assume that it was written before 1745.

This dating may be refined if we speculate as to why Maclaurin became interested in the question of the lunar path that was never mentioned by Newton. The letter notes that:

Because the gravity of the moon towards the sun is found to be greater, at the conjunction, than her gravity towards the earth, so that the point of equal attraction, where these two powers would sustain each other, falls between the moon and the earth, some have apprehended that either the parallax of the sun is very different from that which is assigned by astronomers, or that the moon ought necessarily to abandon the earth. ⁸⁵

On this subject, Murdoch refers in a note to a book by the Scottish philosopher Andrew Baxter (1686–1750), published in 1738, in which this thesis is indeed presented, ⁸⁶ while a reaction to Baxter’s remarks appears in a short anonymous article published in October the same year. ⁸⁷ Was it written or at least inspired by Maclaurin? Whatever the case, the objection raised by this article to Baxter’s thesis is the same as the one Maclaurin adds to the above quotation, namely an implicit reference to a Newtonian statement on relative motions. ⁸⁸ It seems that Maclaurin’s refutation of Baxter’s thesis is the real subject of what Murdoch describes in the letter to Hoadly, since Maclaurin uses his study of the heliocentric path of a satellite to analyse the gravitational forces at work in its motion, and concludes:

Thus we arrrive at the same conclusion which Sir Isaac Newton, more briefly, derived from an analysis of the motions of the satellite; that while the satellite gravitates towards the primary, if, at the same time, it be acted on by the same solar force as the primary, and with a parallel direction, it will remove about the primary, in the same manner as if the last was at rest, and there was no solar action. ⁸⁹

The theme of the letter to Hoadly therefore seems to show that it might have been written in the late 1730s, or at the very beginning of the next decade, in the context of discussions generated by Baxter’s comments. It would thus be significantly earlier than the delineation by Ferguson. Incidentally, it should be noted that Maclaurin’s argument was taken up by Ferguson in 1756, though without refering to the mathematician. ⁹⁰ Finally, we should understand that Maclaurin, may have become interested in the movement of the Moon around the Sun if he had been familiar with the mechanical tracing made by Ellicot about 15 years earlier.

That being said, what matters most in this letter is the way in which Maclaurin dealt with the question of the Moon’s heliocentric path. The anonymous author of an article published in the Athenæum in 1868 on the occasion of the publication of the Life of James Ferguson, ⁹¹ correctly writes that, ‘Maclaurin’s letter to Dr. Hoadly is probably the first publication – in the scientific sense – on the subject’. ⁹² Maclaurin begins with general conclusions founded on the comparison of gravitational forces:

The force that bends the course of the satellite into a curve, when the motion is referred to an immoveable plane, is, at the conjunction, the difference of its gravity towards the sun, and of its gravity towards the primary. When the former prevails over the latter, the force that bends the course of the satellite tends towards the sun; consequently the concavity of the path is towards the sun: and this is the case of the Moon, as will apppear afterwards. When the gravity towards the primary exceeds the gravity towards the sun, at the conjunction, then the force that bends the course of the satellite tends towards the primary, and therefore towards the opposition of the sun; consequently the path is there convex towards the sun: and this the case of the satellites of Jupiter. When these two forces are equal, the path has, at the conjunction, what mathematicians call a point of rectitude, in which case, however, the path is concave towards the sun throughout. ⁹³

Thus the question is rapidly settled as the data at the time were indeed already sufficient to classify the two ‘gravities’ in the order indicated by Maclaurin, both in the case of the Moon and for Jupiter’s satellites. It is this gravitational argument that was taken up by Weyer and Stroobant, then by Fouché in the case of the Moon. ⁹⁴ There is no doubt that Dubois would not have commited his error concerning the satellites of Jupiter if he had known this text.

But Maclaurin did not stop there. He went on to develop a kinematic proof of the concavity property by locating the centre of the curvature at any point along the path, since the direction of the concavity at that point is clearly indicated by the position of that centre. To do so, Maclaurin explicitly follows the classical hypotheses: ‘we supposed the orbits of the primary about the sun, and of the satellite about the primary, to be both circular, and the motions in these orbits to be uniform in the same plane’. ⁹⁵ Then, the examination of the properties of the figure representing two neighbouring positions of the satellite, followed by a passage to the limit when one of them tends towards the other, allows the location of the ‘centre of the curvature’. But with positions of the Earth that are sometimes considered as distinct and sometimes as practically confused, Maclaurin’s proof is not always easy to follow. However, a detailed study is not within the scope of this article. Like Rowe’s, it is based on a trochoidal generation (Figure 9) of the path of the satellite and (using the notations of the analytical study) is limited to the case where δ < r (cases 1, 2 and 3). Maclaurin did not use algebraic notation, however, so his proof is purely geometric (even more so than Rowe’s) and more difficult to achieve. He distinguishes between cases 1, 2 (with points ‘of rectitude’) and 3 (with points ‘of contrary flexure’), while cases 4 and 5, which are outside the scope of his study, were added by Murdoch as a footnote:

If AC = AE, these points meet again, and form a cusp: and if AC is greater than AE, the path has a nodus: which last is the case of the innermost of the satellites of Jupiter and Saturn. ⁹⁶

But what Murdoch says is ambiguous in that it may suggest that the paths of the other satellites of Jupiter and Saturn known at the time do not have double points, which is not the case. For each of these two planets, it is the heliocentric paths of the two innermost satellites – Io and Europa for Jupiter, Tethy and Dione for Saturn – that have loops. Finally, Maclaurin clearly places the Moon in case 1: ‘the path of the moon is concave towards the sun throughout’. ⁹⁷

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Figure 9.

C. Maclaurin (see Note 79, fig. 70).

Conclusion

The unexpected theoretical form of the heliocentric path of the Moon, everywhere concave towards the Sun, was first established by Colin Maclaurin, probably in the late 1730s. This result, which remained anecdotal in the mathematician’s scientific output, long preceded its publication in London in 1748, 2 years after the author’s death, then in Paris the following year. It confirmed the two mechanical delineations of this path made firstly by John Ellicot as early as 1724 but still little-known, secondly and more famously by James Ferguson, published in 1745. Noting that the philosopher Andrew Baxter seems to be involved in the origin of the proof by Maclaurin, the resolution of this lunar astronomy question appears as a product of the so-called ‘Scottish enlightenment’ of the 18th century. ⁹⁸ But the authorship of this mathematical result has long remained unknown, even in Britain where John Herschel makes no mention of Maclaurin in this regard. After Maclaurin’s, other geometrical proofs were proposed by the anonymous X.Y. (1744), then by Rowe (1751), and in the following century by the astronomers Schjellerup (1865), Weyer (1890) and Stroobant (1894), who produced analytical proofs that were much easier to implement.

After Lalande, neither Delambre nor Laplace mentioned this geometrical curiosity, of which they may have been ignorant, and which anyway has no particular application in astronomical science. With regard to the Moon, astronomers of the 19th century were mainly concerned to take into account the many inequalities of its geocentric movement in order to produce increasingly precise tables. The question of the shape of the heliocentric path of the Moon, most of all considered under the classical hypotheses, thus appears more as a purely mathematical problem than as a question of astronomy, since these hypotheses differed considerably from reality at a time when celestial mechanics was seeking greater precision, as shown for example, with the complex Théorie du mouvement de la Lune by Delaunay (1860, 1867). ⁹⁹ The fascination generated by Earth’s satellite, associated with the simplicity of the classical hypotheses and the counter-intuitive character of the result, may explain why, under these hypotheses, so many authors developed an interest in the shape of this fictitious path.

In his Histoire de l’astronomie au dix-huitième siècle Delambre does not quote Maclaurin and, regarding Ferguson and the movement of satellites, simply says: ‘He traces the path of a satellite in space, by the same means that he used for Mercury and Venus’. ¹⁰⁰ One consequence of this lack of interest on the part of French astronomers on this issue is that in the mid-19th century, by which time Maclaurin’s discovery must have been known in France for a century and John Herschel’s treatise on astronomy had been translated into French for some 15 years, such eminent astronomers as Hervé Faye and Charles Delaunay showed their ignorance of the British result. It would take another half century for Maclaurin to be mentioned by a French astronomer on this matter, by Jean Mascart (1872–1935) ¹⁰¹ in 1899, who had read Weyer. ¹⁰²

In France, whether before or after Camille Flammarion, the heliocentric path of the Moon is rarely mentioned by the authors of astronomy treatises. The case of Henri Bouasse (1866–1953) seems to be a rare exception. ¹⁰³ Likewise, school cosmographies most often avoid this subject, maybe considered too difficult in view of its limted interest and thus included in curricula for only a short time. In an effort to arouse the curiosity of their readers by pointing out strange facts, it is the books intended for a wide readership that are the most motivated to address this issue. Following those by Guilllemin and Flammarion already quoted, we find it mentioned more often in the 20th century, particularly in the Encyclopédie Larousse méthodique ¹⁰⁴ and in a revised edition of Flammarion’s Astronomie populaire. ¹⁰⁵ But if these books teach the result found by Maclaurin, this is not the case in Le Ciel by Alphonse Berget (1860–1933) and Lucien Rudaux (1874–1947), ¹⁰⁶ or in Astronomie by Lucien Rudaux and Gérard de Vaucouleurs (1918–1995), ¹⁰⁷ or again in the Encyclopédie générale Larousse, ¹⁰⁸ where the Moon still describes a ‘sinuous curve’ or a ‘sinuous orbit’ around the Sun.

Other than the case of the heliocentric path of the Moon, there do not seem to be other examples of efforts made to understand a problem in astronomy whose solution is of no practical use for this science. Unnecessary research may, however, be found in other fields. For example, following the publication of the Nouvelle mappemonde (1753) by the engineer Nicolas-Antoine Boulanger (1722–1759), who parted the world into two hemispheres, one of which contained a maximum of land, geographers and oceanographers tried to locate the pole of this hemisphere or continental pole. Among them was Alphonse Berget ¹⁰⁹ who located this pole in France, on Dumet island (approximately 47°24′N, 2°37′W), while a modern location puts it in the Mediterranean off Tarragona. ¹¹⁰ If the location of this pole feeds curiosity, however, it is certainly of no use.