The Antikythera mechanism: Discerning its role in ancient navigation

The paper delineates an important potential role for the Mechanism in a navigational process practiced by mariners in the 18th and 19th centuries

James E. (Jim) Wyse Chief Scientific and Technology Officer, Maridia Research Associates Professor (Ret’d), Memorial University, Canada

Abstract

Since its recovery in 1901 from the seabed near the small Greek island of Antikythera northwest of Crete, a 2000- year old laptop-sized lump of severely corroded bronze known as the Antikythera Mechanism has awed the scientific and archaeological communities with its capabilities in cosmological computation. Over the past 120 years, much has been revealed about the Mechanism but much remains shrouded in mystery: the time and place of its origin, its purpose, the methods by which it was constructed, and the identity of its designer. Here we explore the second of these, the Mechanism’s ‘purpose’ mystery. Many researchers and writers proffer that the Antikythera Mechanism served no substantive navigational purpose. This paper challenges that assertion. Through an examination of the lunar distance method of position estimation in the context of the Mechanism’s structure and capabilities, the paper delineates an important potential role for the Mechanism in a navigational process practiced by mariners in the 18th and 19th centuries.

Introduction

Location: 35° 53’N, 23° 19’E Antikythera, Greece; Year: 1901: An exhausted sponge diver had just surfaced holding a nondescript, greenish blob and was sorely tempted to dismissively toss it back into the sea. The diver was part of a team searching an underwater location off the Island’s northeast coast where a ship had been storm wrecked sometime in the first century BCE. Ancient and elegant Greek artworks were being brought to the surface by divers who plumbed the dangerous depths with little more than heavy helmets and unwieldy suits. The blob was certainly less than an elegant piece of art; and, given its highly corroded state appeared even less worthy of the limited time that the diver could safely spend on the seafloor. The disappointed diver was ultimately convinced to resist the urge to toss the encrusted blob back from whence it came, a change of heart that not only spawned a century of research on a device with the capability to track the then-known cosmos with surprising accuracy but also preserved compelling evidence that our knowledge of ancient Greek technological capabilities falls far short of what the Antikythera Mechanism suggests might actually be the case [Rehm, 1906; Price, 1974; Marchant, 2009; BBC, 2012; Jones, 2017, Budiselic, 2017-2020].

Although much has been learned about the Mechanism since it narrowly escaped being re-interred in a watery Antikythera grave, many questions and much mystery still remain. Questions about its origin, methods of construction, even its designer will be left to others whilst this paper concerns itself with one of several purposes it might have served. Specifically, the paper explores the Mechanism’s potential to function as a specialized nautical almanac providing the celestial ephemerides required for positional determination by the method of lunar distance. Some in the Mechanism’s research community have suggested a direct measurement role for the Mechanism in geo-positional determination: Periklis Rediadis, who is reputed to have intervened as the frustrated diver contemplated disposing of the greenish blob, held the view that the Mechanism was a highly sophisticated astrolabe [Jones, 2017, pp. 20-23], a device designed to measure the height of an object above an observer’s celestial horizon. Others prominent in Antikythera Mechanism reconstruction such as Michael Wright assert that the Mechanism functioned as some sort of planetarium and served no role in navigation [CHM, 2015].

With respect to Rediadis’s assertion, there is no reported physical evidence of an astrolabe function or a fitting compatible with an astrolabe attachment; and, despite Wright’s firmly held ‘no-role’ assertion, the discourse on a navigational purpose for the Mechanism is, at best, equivocal. However, its potential use in lunar distance positional estimation has received no direct assessment. Doing so is the essential purpose of this report and is motivated in no small way by the Mechanism’s perceived ability to generate information useful to an obsolete method of position estimation that was once extensively practiced by both marine navigators and terrestrial surveyors. As a prelude to examining the Mechanism’s role in lunar distance navigation, let’s first address the assertion that it functioned as a planetarium since there is much related to this issue that establishes the cosmological context in which the Mechanism functioned.

Geocentrism in a heliocentric age

One of the seemingly contradictory aspects underlying modern day celestial navigation is its reliance on a geocentric model of the cosmos (depicted in Figure 2). Despite today’s wide-spread acceptance of the heliocentric cosmology proposed by Nicolaus Copernicus in the 16th century, the earth-centered model of the solar system, and not the suncentered model, persists as the basis for modern navigational practice. Celestial navigators, well aware of the conceptual clash of geocentric and heliocentric views, take mischievous delight in pointing out how they arrive at valid navigational positions on the basis of applying an invalid model of the cosmos. Of course, this apparent contradiction is satisfactorily resolved when one realizes that the geocentric celestial ephemerides are a convenient transformation of data derived from observations of celestial objects moving in heliocentric patterns. Multiple cosmological paradigms can productively coexist depending on the problem at hand: a committed heliocentrist readily embraces geocentrism when navigating an ocean passage but morphs into a zealous flat-earther when leveling the foundation for a new house.

The contemporaneous existence of geocentric and heliocentric cosmology was also the case for the Antikythera Mechanism. Aristarchus of Samos (310 BCE – 230 BCE) proposed a heliocentric model of the solar system that predates by more than a century the construction circa 100 BCE of the specific unit recovered from Antikythera wreck site. Whether the Mechanism was constructed as an explicit heliocentric transformation is unknown to us; however, its use would have served ancient sailors engaged in celestial navigation in much the same way as today’s geocentric nautical almanacs serve navigators in the current heliocentric age.

Thus, we have two noteworthy, albeit highly circumstantial, notions favouring the Mechanism as a navigational device: (1) it was found at the wreck site of substantial ship as a one-of-a-kind item seemingly out of place amidst the vessel’s extensive cargo of art objects and luxury goods, and (2) the Mechanism’s design and function appear consistent with the geocentric nature of devices used in celestial navigation during a time when heliocentrism would likely only have been of interest to the astrophilosophical elites. This suggests that the geocentrically-oriented Mechanism would be of limited value to those engaged in leading edge, investigative astronomical pursuits; however, it would have been of value as an instructional device and/ or planetarium if one was a teacher or student of navigational astronomy.

In what follows, the case for a navigational role for the Mechanism is further developed from the perspective of the lunar distance method of geo-position estimation. The paper proceeds by decomposing the lunar distance method in the context of the Mechanism’s cosmological computational capabilities. This is followed by the formulation of a trigonometrically-based framework within which a vessel’s (or a landmark’s) latitude and longitude may be estimated. The formulation reveals where the Mechanism’s provision of Moon-Sun and Moon-planet angles directly addresses the method’s requirements for angular lunar distances and prime meridian times needed for positional determinations.

The mechanism and the method

Lunar Angular Distance: Depictions of the celestial sphere in Figure 2 show the same angular lunar distance, L, between the Moon and another celestial object from two perspectives: a ‘North-Up’ orientation and a ‘Zenith-Up’ earth-bound observer orientation. Two aspects of this distance (often referred to as a ‘lunar’) are noteworthy: (1) a lunar changes measurably as the Moon and the celestial object, C, move toward, or recede from, each other and (2) the same lunar measure will be observed at the same time at all earthly locations where both objects are simultaneously visible. The first implies that changes in a lunar reflect changes in time whilst the second implies that a lunar measure at some reference mark such as a prime meridian corresponds to a specific time at that reference mark.

Lunar measures were compiled into nautical almanacs in a manner that associated measures of lunar angular distances with prime meridian times. Whenever a navigator “shoots a lunar” and then “clears” the observation thereby adjusting for refraction, horizontal parallax, semi-diameter, etc. [Karl, 2011, Ch. 8], the resulting geocentric lunar, L, may be used to enter an almanac’s lunar distance table (shown in left plate of Figure 3 for August 1843) to obtain, either directly or by interpolation, the prime meridian time corresponding to L. Traditionally, tabulated lunar values were generated using . . .

L = arccos [sin D_m sin D_c + cos D_m cos D_c cos (G_c – G_m)] . . . . . (I),

an outcome of applying the Spherical Law of Cosines to ΔP_NMC in Figure 2 (a result receiving further attention in what follows). In ancient times, instead of turning almanac pages, the Mechanism-wielding navigator would turn the dial handle (seen in the right plate of Figure 1) to a date and thereby obtain, not L directly, but a subset of L’s determinants: the lunar’s longitudinal components G_m and G_c. Obtaining values for G_m and G_c without the Mechanism would have required investments in observational facilities, astronomical skills, and computational capabilities well beyond those available to day-today shipboard navigational processes. The last of these refers primarily to the computational burden of achieving a tabulation of time as a function of lunar distance, a tabulation format that is an inversion of the conventional manner in which ephemerides are observed, tabulated, and subsequently retrieved.

Almanac Inversion: When using a published almanac [e.g., NACE, 2022], daily ephemeris tables are entered using prime median date and time to obtain D_m, D_c, G_m, and G_c which are, in turn, used to determinate a lunar value via Equation (I) or its equivalent. An almanac structure of this form is an un-inverted almanac, i.e., it is ordered by time to yield the determinants of lunar distance. However, prime meridian time is not known to the navigator at the outset when applying the lunar distance method; in fact, prime meridian time is not an input but an essential output of the method. Thus, the un-inverted almanac does not directly or readily support the lunar method. What is needed is an inverted almanac; one that is ordered by lunar distance to yield prime meridian time. With inverted almanac tabulations, such as that shown in Figure 3 for August 1834, an observed lunar, L, may be used to enter the almanac to obtain a corresponding prime meridian time.

The Mechanism provides a rudimentary almanac inversion via two of its outputs: (1) the front cover dials (seen in Figure 3) for G_m along with the solar and various planetary G_c’s whose differences, │G_m – G_c│, represent equatorial projections for a specific lunar value, and (2) the upper back dial (seen on the extreme right of Figure 1’s right panel) which tracks the Metonic Cycle indicating the date of the Moon’s equatorial position [Wright, 2005]. The Metonic Cycle (attributed to Meton of Athens, 5th century BCE) is the 235-month (approximately 19-year) period taken by the Moon to return to the same stellar proximity with the same lunar phase. The Mechanism’s Metonic dial and its front cover dials jointly operate to yield an almanac inversion wherein a lunar’s equatorial projection │G_m – G_c│ may be associated with a specific prime meridian date and time.

Nonius Interpolation: Budiselic at al. [2020] concluded from an analysis of the Mechanism’s Fragment C that the divisional spacing on the outer dial was consistent with a 354-day lunar calendar, and not the 365-day solar calendar proffered in the preponderance of the Mechanism’s research literature. The inner zodiacal scale, the partial remains of which may be seen in Figure 4, is believed to consist of 360 divisions corresponding to the common notion of a 360-degree circle. Findings of the Budiselic et al. study raise the possibility of a relationship between the zodiacal and calendrical scales with implications for the determination of prime meridian time. The ratio (354:360), which may be reduced to 59:60, places the calendrical scale in a nonius relationship with the zodiacal scale. A “nonius” [Porta Editora, 2022], latterly known as a Vernier scale, facilitates the mechanical interpolation of the wider (calendrical) divisions by the narrower (zodiacal) divisions. The nonius relationship in this case will result in a zodiacal degree mark being located between adjacent calendrical marks at a point corresponding to a specific prime meridian time.

The nonius ratio of 59:60 implies that marked times will differ by one 60th of a day for each of the 59 days covered by each 60-degree span in the 360-degree zodiacal scale. Thus, with zero points on the two scales aligned, the 1° mark for Day 1 will indicate a prime meridian time (PMT) of 2336 corresponding to 59 sixtieths of a day, whilst the 2° mark indicates a PMT on Day 2 of 2312 corresponding to 58 sixtieths of a day, with the 3° mark indicating a PMT on Day 3 of 2248 (corresponding to 57 sixtieths of a day), and so on. Although we may find working in ‘60ths’ unduly awkward in today’s decimal-based world, Van Brummelen [2009] reminds us that ancient mathematicians and navigators would have been quite comfortable performing computations in the sexagesimal-based world of the Mechanism’s epoch.

Those with experience reading Vernier scales will readily see how this interpolative process uses the inter-scale nonius relationship to yield a specific PMT. Readers may find the content and commentary in Figure 5 of some assistance in following the seemingly cryptic explanation provided above. The Appendix presents a pro forma inverted almanac table (i.e., angle → time) that could have assisted the Mechanism-using navigator to translate zodiacal angle into calendrical time of day. The table enables the navigator to identify up to 360 PMTs throughout the lunar year with the potential to permit an observable lunar distance to be reduced to a geographical position. Also, the table permits an interpolation to be achieved without requiring the visual acuity to distinguish gradational alignments between the Mechanism’s zodiacal and calendrical scales.

Whenever a potential lunar observation is indicated by the proximity of G_m and G_c, one member (say G_m) of a {G_m, G_c} pair defining a lunar is set to the nearest zodiacal marker. This action, specifically and exactly, sets its value to one of PMTs listed in the Appendix. The remaining member of the pair (G_c in this case) will not generally correspond to a marker and its value will need to be estimated. Had an additional nonius scale been available to ‘slide’ to the pointer for G_c (in a manner similar to a current-day Vernier scale) then its value and, consequently, the value of │G_m – G_c│ could be determined with greater precision. However, there is no reported evidence, circumstantial or otherwise, that a nonius of this type was fitted to the Mechanism. In their virtual reconstruction of the front dial seen in Figure 6, Freeth et al. [2021] show dials and scales in addition to those for the zodiac and calendar. Although none of the additional scales appear to form a sliding nonius, there is clearly ample space for one. However, this report chooses to avoid assuming its existence but will assert that (1) its presence was certainly a possibility, and (2) its contrivance appears to be well within the capability of the Mechanism’s designer(s). In any case, the fixed nonius relationship between the zodiacal and calendrical scales is sufficient to identify prime meridian times and thereby retrieve and apply appropriate declination values (Figure 2’s D_m and D_c) to construct a lunar value against which an observed lunar may be compared.

Declination: There is much to suggest that during the Mechanism’s epoch the notion of declination as the height of a celestial object above or below the celestial equator was not only well understood but also creatively used and systematically recorded. Eratosthenes of Cyrene (276 BCE – 194 BCE) estimated the earth’s circumference based on, among other things, his understanding of the Sun’s maximum northerly zenith at summer solstice. Posidonius of Apameia (135 BCE – 51 BCE) also estimated the circumference of the earth but in this case based on the angular height of the star Canopus at two different locations: Rhodes and Alexandria. Whilst Eratosthenes and Posidonius took very different approaches [Nicastro, 2008], both yielded surprisingly accurate estimates of the size of the earth. Eratosthenes’ method recognized the variable nature of the declination of some celestial objects (the Sun in this case) whilst Posidonius’ method recognized the fixed nature of the declination of other celestial objects (stars generally, and the star Canopus in particular).

Hipparchus of Nicaea (190 BCE – 120 BCE), the reputed founder of spherical trigonometry and who, along with Archimedes of Syracuse (287 BCE – 212 BCE) and Posidonius, is held to be one of the leading contenders as the Mechanism’s chief designer, created a catalogue documenting the positions of over 850 stars [Stewart, 2016]. One of the components of each star’s position is a measure of declination. Any observer of stellar movement readily sees that stellar declinations are essentially constant. However, the declination measures of other celestial objects are not. The 235-month Metonic cycle, reflected in the movements of the larger, upper pointer on the Mechanism’s back cover (seen on the extreme right in Figure 1) and its smaller companion, the callippic dial, along with the 223-month Saros cycle of solar and lunar eclipses, reflected in the movement of the lower pointer (also seen on Figure 1’s extreme right) and its smaller companion, the exeliG_mos dial, together imply that lunar and solar declinations were available to the Mechanism’s designer(s). The Metonic and Saros cycles hold that both the Sun and the Moon return to the same celestial location on a periodic basis. These ‘return-to’ locations are not generally points (G_m and G_c) lying along the celestial equator but locations displaced north and south of the equator by the extent of their respective declinations. Thus, the Mechanism’s tracking of the Metonic and Saros cycles implies that declination values were known during the Mechanism’s epoch. In addition to the direct observations on which the Metonic and Saros cycles were based, the star positions in Hipparchus’ catalogue would have provided an indirect, comparative basis for declination determination with respect to the Sun, Moon, and the five known plants through comparisons with proximal star positions extracted from the catalogue. The Sun’s declination would have been essentially constant when employing ecliptical coordinates but would have varied in a sinusoidal pattern with the equinoxes and solstices as primary nodes, when using equatorial coordinates. The latter, presumed by this paper, is essentially that which several scholars of antiquity suggest Hipparchus favoured [Delambre, 1817; Duke, 2002], possibly because it simplified the construction and presentation of his star catalogue.

A recent report suggests that the Mechanism itself provided direct declination data. Freeth et al. [2021] propose the existence of a rotating indicator on the Mechanism’s front dial referred to as the “Dragon Hand” (seen in Figure 6 as the only doubleheaded dial pointer). The Dragon Hand indicates the celestial sphere locations for the lunar nodes (points where the ecliptic and celestial equator intersect) and describe the Hand’s interaction with the Moon’s pointer (seen in Figure 6 extending from the Moon’s phase ball) indicating when the Moon’s declination is south or north of the celestial equator. With only about a third of the Mechanism surviving its 2000-year aquatic internment it would not be surprising to learn that it could have provided a full range of declination measures; however, to date there is no evidence that the missing two thirds performed such a function. In summary, several sources and approaches yielding declination values were available to determine the ephemerides constituting a nautical almanac in which declination measures were associated with temporal measures. Whatever the means by which the measures of declination were determined, their accessibility and applicability requires, then as now, a globally unique longitudinal reference mark: the prime meridian.

Prime Meridian Identification: With considerable (but not complete) agreement, the current reference mark for nautical ephemerides and their associated measures of time is the meridian passing through the Royal Observatory at Greenwich, England. This has not always been the case nor does it need to be; other reference marks may be, and throughout history have been, used. Eratosthenes proposed a charting system wherein the prime meridian passed through Egypt’s Alexandria and his parallels of latitude corresponded to horizontal lines passing through important city states of the ancient Greek world. Eratosthenes’ latitudinal schema resembled that used by Viking navigators wherein (for example) Bergen in modern day Norway was the latitudinal departure point for a voyage to Greenland whilst Trondheim’s location served the same purpose for a voyage to Iceland, and so on [Karlsen, 2003]. Viking navigators intending to reach Iceland would coastal cruise northward or southward from their respective homeports to Trondheim and then sail westward toward Iceland whilst maintaining Trondheim’s latitude. Although this worked reasonably well for the northsouth oriented coastlines of Norway, Eratosthenes’ ‘city-state’ latitude scheme was ill-suited to sailing the archipelagostrewn, irregular coastline of the Aegean, Ionian, and other waters within, and adjacent to, the Mediterranean Sea.

Eratosthenes’ coordinate scheme was eventually superceded by that of Hipparchus, who created the earliest known method of identifying a location by latitude and longitude in a manner similar to that in current use today. His scheme, developed during the Mechanism’s epoch, differed from both that of Eratosthenes and current day position plotting conventions in its placement of a prime meridian – it passed through neither Greenwich nor Alexandria but Rhodes. The choice of prime meridian is arbitrary; however, its function as a reference for longitudinal measures and its role in tying an object’s ephemerides (G_x ’s and D_x ’s) to a measure of time is essential to position determination generally and to the method of lunar distance in particular. It should be noted that informal discussions about the Mechanism’s Olympiad dial, which reports the locations of the Panhellenic games through their 4-year cycle, raise the possibility of a movable prime meridian. This is certainly a possibility albeit one that would complicate positional calculations and Mechanism calibration; however we are not aware of anything that supports this. Furthermore, a movable prime meridian would neither support nor negate a navigational role for Mechanism.

Position Determination by Navigational Triangle: The Navigational Triangle joins the Celestial Sphere as one of the fundamental concepts in celestial navigation [Bowditch, 2017]. Seen in Figure 7 as ΔP_NZX, it is formed from the arcs of circles whose planes intersect the center of the Celestial Sphere, thereby permitting the Spherical Law of Cosines to be applied to obtain the coordinates (φ, λ) of the observer’s zenith, Z, which directly correspond to the observer’s latitude and longitude. Table 1 demonstrates the Navigational Triangle’s application to three celestial objects visible to an observer located at a selected position in the Ionian Sea on the date and time indicated. Conventional “Line of Position (LOP)” site reduction methods, attributed to 19th century mariners T. Sumner and M. Saint-Hilaire [Vanvaerenbergh and Ifland, 2003] will yield positional estimates (‘est. φ’ and ‘est. λ’ in Table 1) with respect to each of the three celestial objects. The lines of position (LOPs) for the objects taken pairwise, or all together, will yield a positional fix for each of four combinations. LOPbased methods require an initial dead reckoning (DR) position and an accurate measure of local time. The first requires a navigator to assume a specific position for the vessel whilst the second presumes the availability of a chronological device. The method of lunar distances requires neither.

The lunar distance method

Although the lunar method is now only familiar to a relatively small group of navigational enthusiasts, jokingly referred to as celestial navigation’s lunatic fringe, the method was common practice amongst 18^th and 19^thcentury mariners and achieved considerable prominence in position determination both on land and at sea. Solitary circumnavigator Joshua Slocum’s June 16^th 1896 lunar, the only one of his entire voyage [Werf, 1997, Slocum 1899], may well be the most prominent celestial measurement in lunar method history; the Lewis and Clarke expedition across the continental United States in the early 1800s collected lunar distance measures for post-expedition mapping and journey reconstruction [Bergantino and Mussulman, n. d.]; Captain James Cook used the method for ocean navigation [Keir, 2010] as well as land surveying; notable among his surveying applications of the method is his surprisingly accurate 18^th century map of the entire Island of Newfoundland [Snowdon, 1984]; and, Astronomer Royal Nevil Maskelyne’s use of the method to determine longitude on a voyage to Barbados featured prominently in the saga of John Harrison’s chronometers [Sobel, 1995; Baker, n.d.]. The method’s complexity was undoubtedly an important factor in its losing out to the simplicity of determining longitude with chronometers, however, the method’s approach to position determination, its use of inverted almanacs, and its focus on lunar movements relative to other celestial objects, suggests there is much that is methodologically compatible and complementary between the method and the Mechanism.

The Mechanism predicts the occurrence of astronomical events with the potential to simplify a positional determination. Whenever the zodiacal dial shows G_M = G_C the term, cos (G_C – G_M) equals 1(one) thereby simplifying the determination of lunar distance, L. Should the G_M = G_C event also correspond to a geocentric occultation, many further simplifications are induced: D_C = D_M, H_C = H_M, the polar triangle P_NMC vanishes as does the zenith triangle ZMC, and the two navigational triangles shown in Figure 8 collapse into a single navigational triangle similar to that depicted in Figure 7. In view of what is known about computational capabilities during the Mechanism’s epoch, any computational simplifications would likely have been appreciated.

The trigonometric formulation presented above is specific to the circumstances depicted in Figure 8 and, although valid for a wide range of lunar phenomena, would need to be extended to remain applicable to situations such as the zenith lying south of the lunar, and/ or the term (θ – G_M) falling outside the interval (-180, 180), and/or the referent celestial object (the Moon in this case) positioned east of the zenith, etc. Despite these limitations, which are essentially simplifications for discussion purposes here, the above formulation when applied to Table 1’s ephemerides for a MoonVenus lunar yields latitude, φ = 35° 42.1’N and longitude, λ = 18° 36.0’E, a result aligning with the actual position and the various positional determinations appearing in the Table. Note that no DR position was needed and no explicit measure of local time was required.

Implicit Measures of Local Time: The altitudes of the two objects, H_M and H_C (seen in Figures 2, 8, and 9 as well as in Figure 7 as H_X) implicitly capture local time. Their measured values change as the two celestial objects rise from the east and set into the west. Their heights above the celestial horizon vary directly with local time. Apparent altitude values (i.e., values unadjusted for atmospheric refraction, semi-diameter, parallax, etc.) could have been ascertained using a marine astrolabe, an instrument known to be available to ancient navigators and whose creation is often attributed to Hipparchus [Marchant, 2008]. Avoiding the direct measurement of local time would have been advantageous given the poor accuracy and reliability of time-keeping instrumentation available during the late BCE/early CE epoch. Thus, capturing local time indirectly through altitude measurements reduces the navigator’s burden in two respects: (1) time-keeping efforts were reduced to merely keeping track of the date and (2) implicit measurements of local time (H_M and H_C) were readily available at the moment observations are made.

Graphical Methods for Position Determination: There is much that is uncertain about the capability of ancient shipboard navigators vis-à-vis their ability to apply the spherical Law of Cosines as outlined above. Various scholars (Van Brummelen [2009] among others) equivocally suggest that doing so was possibly within the knowledge and capability of leading mathematicians and astronomers of the Mechanism’s epoch; however, much uncertainty pervades the literature about the general availability of the required skills in this respect. Even for the reputed founder of spherical trigonometry, Hipparchus, it is not at all clear that the spherical Law of Cosines was known to him and, even if it was, the computational processes required by its application would likely have been unfeasibly burdensome.

A similar situation exists for modern navigators. Not that the Law of Cosines remains unknown nor is there any lack of computational capability, it is more the case now that applying the Law of Cosines in particular and spherical trigonometry in general remain impenetrable for many navigational practitioners. This problem is not only well recognized but also widely and effectively accommodated by graphical methods that greatly minimize computational effort. Comprehensive texts on navigational methods are replete with charting templates of various formats that yield usable answers to a wide range of practical navigational questions based on graphical methods.

It is not known whether graphical methods were in use during the Mechanism’s epoch; however, Figure 9 depicts a proposed graphical method functionally equivalent to the algebraic representation derived above to yield a geo-position without the computational burden of applying the Law of Cosines. It assumes that plotting would take place on a sphere, a task that would not have been out of the ordinary for ancient navigators and spherical drawing surfaces were seemingly more common in the Mechanism’s epoch than they are today. Four concepts underlie the graphical method illustrated in Figure 9: (1) celestial altitudes (H_M and H_C) are always perpendicular to an observer’s celestial horizon, (2) altitude-horizon perpendicularity occurs on an circular arc of radius H_X centered on celestial object X at the point of altitude-horizon tangency, (3) all points of altitude-horizon tangency with respect to a specific zenith lie along that zenith’s celestial horizon, and (4) a zenith is located 90° from a point of tangency along the great circle that includes the point of tangency and the location of its respective celestial body. From what we know about the capabilities of, and the instrumentation available to, ancient spherical geometers. the graphical method would not have been unduly challenging.

Lunar Measurement Practicalities: 18^th and 19^th century mariners used the term “in distance” to refer to a celestial circumstance where the Moon and another celestial object were conveniently juxtaposed to take the measurements required by the method of lunar distance. The Mechanism identified potential indistance lunar opportunities through its G_M and G_C dial positions. The zodiacal-calendrical nonius provided a prime meridian time at which the in-distance lunar may be anticipated. The navigator would know the date but not the time of the anticipated indistance event. Instrumentation would be preset: the lunar angle measuring instrument, perhaps a dioptra (also often attributed to Hipparchus), would be preset to L (adjusted to include semi-diameter, atmospheric refraction, horizontal parallax and so on) while the marine astrolabes would be preset to approximate measures for H_M and H_C (also appropriately adjusted). Presetting sight-taking instrumentation is common practice even in modern times since it greatly eases the burden of taking sights on the heaving deck of a ship.

At the moment that L is registered by the dioptra, final altitude astrolabe angles are taken, and the navigator has the measurements needed for position determination at the prime meridian time corresponding to the lunar distance, L.

It may be useful to note at this point that positional errors arise from three sources: (1) the sight-taking instrumentation (astrolabes, dioptrae, etc.) used to measure H_M, H_C, and L, (2) the Mechanism’s readings for G_M and G_C, and (3) an appropriate source (as discussed above) that provides the values of declination, D_M and D_C. The interplay of quantities from these three sources in positional determination is seen above in two respects: computationally when applying the Law of Cosines and graphically when using the method illustrated in Figure 9. Each source carries its own level of accuracy with the result that positional error is a function of the accuracy associated with each source. Whilst instrumentation and declination errors are important concerns, it is the Mechanism that is the primary focus of this report and the issues surrounding its accuracy or lack thereof are of primary concern here.

Accuracy: Throughout navigational history the issue of positional accuracy has not just been a concern but a longstanding angst amongst ocean-going mariners. Positional ignorance and navigational error have taken numerous lives. England’s institution of the Board of Longitude in the years following the destruction in 1707 of four naval ships killing over 2000 sailors in a single incident was a response to the poor level of accuracy with which a vessel’s position could be determined (Sobel, 1995). The Board offered substantial financial awards and set specific standards of accuracy: ₤10,000 for a method and/or device that could reliably determine longitude within 1°; ₤20,000 if within 0.5°. The latter would roughly correspond to about half a kilometer in the latitudes of the Ionian and Aegean Seas.

Although the Board’s financial terms were substantial (a top 3 percent yearly family income at the time was ₤200 [Hume, 2015]), its accuracy standards seemed quite low, and not just by today’s submeter accuracy standard. However, the specification was thought to be consistent with (1) a level of navigational safety that adequately responded to the losses that had lead to the Board’s creation and (2) the level of accuracy thought to be achievable by whatever method or device might result. It was during this time that Nevil Maskelyne was formalizing the method of lunar distance, instituting the publication of the (inverted) nautical almanac needed for the method’s use, and conducting sea trials to assess its practicality in harsh marine conditions. Although Maskelyne was not an award recipient, his work resulted in the use of the method for well over a hundred years. Much in the preceding sections suggests that the Mechanism is amenable to the method but a standard of accuracy would be an important aspect of assessing the value of its use. Could it have achieved the modest standard of accuracy set by the Board of Longitude?

Jones deals extensively with the issue of accuracy throughout his 2017 book on the Mechanism. In a section on “Imperfections and inaccuracies” (p. 225) he identifies two categories of inaccuracy: random and systematic. The first refers to issues internal to the Mechanism such as gearing “backlash”, an issue that plagues users of marine sextants even today; the second refers to issues external to the Mechanism such as the validity of astronomical theories that guided the Mechanism’s design and, more subtly, various ways and means by which the Mechanism might be calibrated and corrected. Beyond keeping an instrument in good repair and properly maintained, random errors may be accommodated with multiple measurements, a tactic that avails of the central tendency of random measures to converge on a true value. With respect to systematic inaccuracies, tactics may take the form of (1) identifying the error amounts with respect to known, perhaps formally catalogued, location coordinates and applying them as positional corrections (discussed next) and/or (2) incorporating procedural ‘tricks’ during celestial sight reduction such as using the “navigators right angle”, a term that referred to an angular value of 89° 48’ used by navigators during the Age of Sail (18th and 19th centuries) to accommodate corrective factors (atmospheric refraction, etc.) when clearing lunar observations (Reed, 2022).

The mapping methods devised by Hipparchus used measures of latitude and longitude for prominent locations, a development suggesting that an informational infrastructure was available during the Mechanism’s epoch to determine and apply corrections in response to systematic errors. Position differentials could have been ascertained and applied in a manner conceptually similar to how American Differential GPS and Russian Differential GLONASS have been used in recent times to correct satellite-based positional determinations [Beser at al., 1995]. The historical record does not reveal how random and systemic errors were taken into account; however, careful and clever navigators in the Mechanism’s epoch would likely have mastered the Mechanism-based methods along with the use of any available correctional information to the extent that longitudinal inaccuracies would be reliably within the one degree standard of accuracy set almost two millennia later by the Board of Longitude.

Concluding remarks

Luna-centric Functionality. Although the Mechanism’s cosmological paradigm and physical structure are clearly geocentric, its functionality is primarily luna-centric. The Moon is the single central subject of the Metronic and Callippic scales. The Saros and Exeligmos scales combine to yield a 669-lunar month cycle for eclipse prediction that brings the Sun into Mechanism’s cosmos in terms of its positional relationship with the Moon. The Mechanism’s gearing captures elliptical retrograde planetary motion, a capability enhancing the validity of lunar-planetary angles and permitting the zodiacal scale to provide critical ephemerides required to determine lunar angular distances. The (hypothesized) nonius inter-scale relationship ties zodiacal angles and prime meridian times to a 354-day lunar year and not the 365-day solar year. The Mechanism’s offset axial pin-and-slot gearing tracks the Moon’s anomalous motion, the Dragon Hand indicates the Moon’s nodes and their precession, and contrate gearing drives a ½-black-½-white ball mimicking the monthly progression of the Moon’s phases.

In all of the above and other respects the Mechanism provides a functional environment within which navigational processes based on the Moon-based method of lunar distance could be feasibly utilized. The method directly avails of (1) the Mechanism’s capability to provide the ephemerides derived from almanac inversion, (2) its noniusbased determination of the prime meridian times for Moon-Sun and Moon-planet angular differences, and (3) the amenability of spherical-based methods of positional determinations.

Case ‘Not’ Dismissed. Evidence that the Mechanism served a navigational purpose has been deepened but remains inconclusive; however, three assertions are proffered: (1) a navigational role cannot be dismissed and should now stand as a strengthened hypothesis; (2) the Mechanism functioned, at least partially, as the inverted almanac required for position determination by the method of lunar distance; and (3) several areas hold considerable potential for productive investigation: the utilization of nonius scales during the Mechanism’s epoch; the availability of declination ephemerides either externally on a medium separate from the Mechanism or internally by Mechanism functionality, possibly by geared components in Mechanism’s missing two-thirds; the use of graphical methods and ancillary instruments for positional determination on spherical surfaces; and, the existence of catalogues or compendia of coordinates for differential corrections and Mechanism calibrations. Investigative pursuits in the above respects will undoubtedly shed light on the Mechanism’s other mysteries. The work undertaken for this report encountered Hipparchus along many pathways, notably those related to astronomical concepts, trigonometrical developments, and instrument creation. As with the Mechanism’s purported navigational role, the evidence for Hipparchus as the Mechanism’s designer also remains circumstantial; but as the work behind this report progressed an intellectual nexus was coming into view that placed Hipparchus at its focal point.

Acknowledgement

The spherical illustrations shown in Figures 2, 7, 8, and 9 were constructed using Geogebra, an open software facility available for use and/or download at https://www.geogebra.org/.

References

Antikythera Mechanism Research Project, [2008]. Fragment C-1, Image File AK34a. http://www.antikytheramechanism.gr/data/ptm/full-resolutionptm. Accessed February 7, 2022.

Baker, A. [n. d.]. Log Book of Voyage to Barbados. Royal Greenwich Observatory Archives, Papers of Nevil Maskelyne. Cambridge University Digital Library. http://cudl.lib.cam. ac.uk/view/MS-RGO-00004-00321/1. Accessed February 7, 2022.

BBC. [2012]. Antikythera Mechanism: The 2000 Year Old Computer. https://www. youtube.com/watch?v=3T1n7RjCMfQ. British Broadcasting Corporation, May 10: 58min41sec. Accessed February 7, 2022.

Beser, J., A. Balendra, E. Erpelding, and S. Kim. [1995]. Differential GLONASS, Differential GPS, and Integrated Differential GLONASS/GPS. Proceedings of the 8th International Technical Meeting of the Satellite Division of the Institute of Navigation. Palm Springs, CA, September 1995, pp. 507-515.

Bergantino, R. and J. Mussulman. [n. d.]. Deciphering Lewis’s Data. Lewis’s Celestial Navigation. Lewis & Clark Trail Heritage Foundation. http:// www.lewis-clark.org/article/3357. Accessed February 7, 2022.

Bowditch, N. [2017]. The Navigational Triangle. American Practical Navigator: An Epitome of Navigation, 2017 Edition. United States Government, National Geospatial-Intelligence Agency, Section 1331, pp. 235-237.

Budiselic, C. [2017-2020]. Reconstructing the Antikythera Mechanism. (10 episodes). Clickspring. http://www.clickspringprojects. com/the-antikythera-mechanism. html. Accessed February 7, 2022.

Budiselic, C., A. Thoeni, M. Dubno, and A. Ramsey. [2020]. The Antikythera Mechanism: Evidence of a Lunar Calendar Parts – 1&2. Horological Journal, The British Horological Institute. pp. 1-14.

CH_M. [2015]. Secrets of the Antikythera Mechanism: Session 1. https://www. youtube.com/watch?v=cSh551cdIEY, Computer History Museum, May 22: 22m44sec. Accessed February 7, 2022.

J. B. J. Delambre, Histoire de l’Astronomie Ancienne, (1817, reprinted New York, 1965), v. 1, p. 117, 172, 184.

Delambre, J. [1817]. Histoire da l’Astronomie Ancienne. Ve Courier, Imprimeur-Libraire Pour les Sciences, Paris.

Duke, D. [2002]. Hipparchus’ Coordinate System. https:// www.researchgate.net/ publication/226337656_ Hipparchus’_Coordinate_System. Accessed February 7, 2022.

Freeth, T., D. Higgon, A. Dacanalis. et al. [2021]. A Model of the Cosmos in the Ancient Greek Antikythera Mechanism. Sci Rep 11, 5821 (2021).

HMNAO. [1843]. The Nautical Almanac and Astronomical Ephemeris. HM Nautical Almanac Office, page 168.

Hume, R. [2015]. The Value of Money in Eighteenth-Century England: Incomes, Prices, Buying Power – Some Problems in Cultural Economics. Huntington Library Quarterly, V77N4, pp.373-416.

Jones, A. [2017]. A Portable Cosmos: Revealing the Antikythera Mechanism, Scientific Wonder of the Ancient World. Oxford University Press.

Karl, J. [2011]. Celestial Navigation in the GPS age. Paradise Cay Publications.

Karlsen, L. [2003]. Secrets of the Viking Navigators. One Earth Press, Seattle.

Keir, B. [2010]. Captain Cook’s Longitude Determinations and the Transit of Mercury – Common Assumptions Questioned. Journal of the Royal Society of New Zealand, V40N2, pp 27-38.

Marchant, J. [2008]. Michael Wright Demonstrates His Model of the Antikythera Mechanism. http://www. youtube.com/watch?v=4eUibFQKJqI, 4 min. Accessed February 7, 2022.

Marchant, J. [2009]. Decoding the Heavens: A 2,000 Year-Old Computer and the Century-Long Search to Discover its Secrets. Da Capo Press.

NACE. [2021]. 2022 Nautical Almanac Commercial Edition. Paradise Cay Publications.

Nicastro, N. [2008]. Circumference: Eratosthenes and the Ancient Quest to Measure the Globe. St. Martin’s Press, New York.

O’Connor J. and E. Robertson. [2014]. Nevil Maskelyne. MacTutor History of Mathematics Archive. University of St. Andrews, Scotland. https://mathshistory.st-andrews. ac.uk/. Accessed February 7,

Porto Editora. [2022]. Pedro Nunes. Infopedia. https://www. infopedia.pt/$pedro-nunes. Accessed February 7, 2022.

Price, D. [1974]. Gears from the Greeks, the Antikythera mechanism: A Calendar Computer from ca. 80 BC. Transactions of the American Philosophical Society, New Ser. 64, No. 7, pp.1-70.

Reed, F. [2022] .Celestial Navigation in the Age of Sail. reednavigation.com/c19-class/

Rehm, A. [1906]. Notizbuch (unpublished notebook), research manuscripts and photographs from 1906-1906. Bayerische Staatsbibliothek, Munich, Germany, Rehmiana III/7and III/9.

Slocum, J. [1899]. Sailing Alone Around the World. Serialized in The Century Magazine.

Snowdon, K. [1984]. Captain Cook as a Hydrographer. Cook’s Log V7N4. Reprinted in https://www. captaincooksociety.com/home/detail/ captain-cook-as-a-hydrographer. Accessed February 7, 2022.

Sobel, D. [1995]. Longitude. Walker Publishing Company, New York.

Stewart, D. [2016]. Hipparchus. Famous Scientists: The Art of Genius. https:// www.famousscientists.org/hipparchus/. Accessed February 7, 2022.

Van Brummelen, G. [2009]. The Mathematics of the Heavens and the Earth: The Early History of Trigonometry. Princeton University Press.

Vanvaerenbergh, M. and P. Ifland. [2003]. Line of Position Navigation: Sumner and Saint-Hilaire, the Two Pillars of Modern Celestial Navigation. Unlimited Publishing. Bloomington, Indiana.

Werf, S. [1997]. The Lunar Distance Method in the Nineteenth Century: A Simulation of Joshua Slocum’s Observation on June 16, 1896. NAVIGATION, Journal of The Institute of Navigation, Vol. 44, No. 1: 1-14.

Wright, M. T. [2005]. Counting the Months and Years: The Upper Back Dial of the Antikythera Mechanism. Bulletin of the Scientific Instrument Society, No. 87.