A Cosmos in Bronze

The Antikythera mechanism

This file is a self-contained kinematic reconstruction: one input drives the astronomical cycles, while a native WebGL scene presents an idealized bronze-and-wood restoration. The complete mechanism engine and graphics are embedded in the HTML. The bibliographic links are optional reading and require internet access.

Quick operation

1. Open the downloaded HTML directly in desktop Chrome. No installation, account or API key is needed for the mechanism. Its calculations and graphics work offline; optional Google translation and Analytics use online services. Allow WebGL / graphics acceleration for the 3D view.
2. Drag the mechanism to orbit it. Wheel or trackpad pinch changes distance. Right-drag, Shift-drag, or a two-finger gesture pans. Rotation is unrestricted. Double-click fits the initial camera. Focus the canvas and use arrow keys to orbit; + and − zoom.
3. Use Front, Back, Gearwork and Exploded to inspect different aspects. The sliders separate layers and the checkboxes remove the enclosure or plates. Restore Front to return to an assembled view.
4. Press Play to animate. Playback is elapsed days per real second, with backward options. The engine is manually driven rather than a self-running clock. The original hand drive is represented by the draggable circular crank.
5. Choose a step size and use − or +. A month means a mean synodic month of this mechanism. One Saros advances 223 such months, so its pointer returns while Exeligmos advances one sector.
6. Enter any supported elapsed day directly, or use the timeline. “Center here” moves the timeline window to the current time; changing the window does not change the date.
7. The Dial reader provides legible front/rear views, current cells, a pin-and-slot diagram, the reconstructed eclipse register, and the declared civil-calendar completion. Click Go on an eclipse row to move to its cell in the current Saros cycle.
8. The Gear laboratory exposes the ratios and planetary velocities. Retrograde indicates negative apparent longitude velocity, not reverse rotation of every intermediate wheel.
9. Calibration controls the independent phase settings and date reference. Save a JSON session to preserve calibration, notes, elapsed time, display layers, ring option, camera, timeline window and playback-speed selection; reload it later. Loading pauses playback and validates the whole file before applying any changes. The file does not silently save changes into your browser or rewrite itself.
10. Use Verification to rerun the embedded checks. Export their results, export the eclipse register, save a PNG of the 3D view, or print this manual.

Global Space toggles playback; Left/Right step time when a form or the 3D canvas does not hold focus. Background tabs pause elapsed-time advancement. Unsaved session changes disappear when the file is closed.

Evidence, reconstruction and scope

The mechanism was recovered from a wreck near Antikythera in the early twentieth century. Its corroded remains are kept in Athens. Tomography and surface imaging revealed gearing, scales and previously hidden writing; these observations transformed the reconstruction of its astronomical functions [1, 2]. The identity of the maker, original use context and complete front assembly remain unresolved.

Different investigators offer competing explanations for missing parts. For example, Voulgaris and collaborators argue for a draconic assembly and a different interpretation of the instrument [8, 9]. These are competing hypotheses, not extra gears that can safely be added to every other reconstruction. This file does not merge incompatible assemblies into a supposedly certain original.

The front ring is disputed

Woan and Bayley’s statistical analysis favors about 354–355 holes under its assumptions, rather than 365 [7]. Hole count alone does not provide a complete alternative mechanism. The selector therefore changes the displayed registration pattern while preserving the declared annual drive. The 354/355 modes are comparisons of the physical ring hypothesis, not validated lunar-calendar restorations.

Mechanical mathematics

Let d be elapsed days and Y the selected number of days per mechanism year. Then t = d/Y is the annual input, measured in full revolutions. Angles are calculated directly from this input. Moving forward and then backward returns to the same state; no differential-equation integration is used.

For an external mesh, tooth counts determine speed. If wheel A has NA teeth and wheel B has NB, their angular speeds satisfy the following relation. Coaxial wheels fixed together share angular speed.

These equations are constraints, not empirical curve fitting. An idler changes sense while cancelling from the magnitude of the final ratio. An epicyclic system must be evaluated in the carrier frame before its motion is converted back to the fixed frame. Ignoring that frame change is a common source of plausible-looking but wrong models.

The compound lunar and calendar relationships are supported by the research literature [1, 2, 6]. Physical viewing direction can reverse the apparent sign of a dial. The engine separates these displayed positive cycle magnitudes from the alternating signs of intermediate external meshes.

A useful independent identity

Thus the selected lunar carrier gives precisely 239 anomaly cycles during the 223-month Saros. This is an arithmetic property of the gear model. It does not imply exact identity among the corresponding physical periods in the real sky.

Continuous versus indexed outputs

Pointers use continuous, unrounded coordinates. Text labels use a cell index. A positive Euclidean remainder is used for negative time, so a small move backward from cell 1 correctly enters the last cell of the preceding cycle. The complete unwrapped cycle count is retained separately from the visible angle.

The Moon: mean motion, anomaly and phase

The surviving lunar assembly includes an eccentric pin-and-slot device. The 2006 reconstruction reports approximately 1.1 mm between axes and 9.6 mm from pin to its rotation center [1]. This is a geometric ratio of about 0.114583, not the modern orbital eccentricity of the Moon. Confusing the two nearly halves the modeled angular variation.

To derive the implemented geometry, take the pin center as O = (0,0), the slot center as C = (e,0), pin radius r, and input anomaly A. The pin position is P = (r cos A, r sin A). A slotted arm through C must point along vector P − C:

The implementation uses the bounded correction instead of subtracting two independently wrapped angles. This avoids a false full-turn jump at an atan2 branch boundary. The numerical verification compares it with the direct vector construction at 1,201 input angles.

The phase mechanism measures the Moon relative to the mean Sun. Here its phase coordinate is D = wrap(λMoon − λMeanSun). The displayed illuminated fraction is (1 − cos D)/2, and the phase image is a lit sphere under that geometry. New, quarter and full phases follow naturally from D. The “lunar age coordinate” is a phase fraction multiplied by the mean month; it is not a root-solved time since the most recent true conjunction.

Real lunar motion contains evection, variation, inclination and further perturbations. Those corrections are deliberately not imported into the mechanical train. Doing so would create a modern ephemeris disguised as ancient gearwork. Switching off lunar anomaly is a diagnostic comparison, not the preferred historical model.

Rear calendars and spiral followers

The major rear dials reconcile time scales rather than depict spatial orbits. The Metonic spiral contains 235 lunar months in five turns; the Saros spiral contains 223 in four. The games dial follows a four-year sequence, and the Exeligmos dial supplies an eclipse-time correction [2]. A rotating arm and a radial follower allow each spiral to hold many months without requiring a huge dial.

The renderer uses an Archimedean spiral with a deliberately expanded inner opening for readability. It is not a measured tracing of the surviving uneven grooves. At the outer end, the software moves the follower to the beginning. On a physical instrument, the operator must reset the follower; a spiral is not an endless circular track.

The civil calendar is a different layer

The named months are from the Corinthian calendar family [10]. The surviving text does not uniquely settle every missing intercalation and civil-day assignment. The app supplies a labeled regularized completion: 13-month years 3, 6, 8, 11, 14, 17 and 19; an additional Apellaios in each; and a repeating 47-cell pattern containing 22 short months. This is a computational completion chosen for consistency, not a transcription of all 235 cells.

Do not silently equate that schematic 6,940-day civil cycle with 19 exact input years of 365.25 days: the latter is 6,939.75 days. Four civil Metonic cycles minus one day give 27,759 days, equal to 76 × 365.25. The 76-year dial in this model is an explicitly hypothetical subsidiary; its presence is less secure than the major spirals. The application displays cycle position, not a fabricated fully recovered ancient festival schedule.

Games interpretation

The app implements the four-year mechanical indication. Inscriptions associate this cycle with named games, but the full historical alignment and missing entries should not be inferred from a modern year number. Accordingly it reports the phase and numbered sector instead of assigning a definitive Olympiad to a modern date.

Eclipse possibilities and the Saros register

The Saros dial behaves like a read-only table selected by gearwork. Its pointer does not compute a new eclipse from a modern orbital solution at each date. Engraved cells contain predictions and references to descriptive inscriptions. Solar eclipses are highly location dependent; a recorded possibility is not a guarantee of visibility.

The embedded register reproduces the 2014 EYM integer construction [4]. The 2019 follow-up [5] revises index-letter group assignments and interpretation of associated inscriptions. The register here contains only the 2014 possibility marks, not those letter groups; it must not be mistaken for a complete transcription of the back plate. The model uses 38 Eclipse Year units (EYu) per month and 446 per eclipse year. For cell n:

The cyclic nearest-node distance is computed before applying the north/south test. The generated result must contain 38 lunar possibilities, 28 solar possibilities and 51 occupied cells because some cells contain both. This count is a regression check of this implementation against the selected research model. It is not evidence that all generated marks survive on the artifact.

Why three Saros cycles?

The approximate one-third-day excess in a Saros shifts the hour of the next corresponding eclipse. Exeligmos tracks three repeats. The software returns corrections of 0, 8 or 16 hours and wraps to zero at the next Exeligmos. This reproduces the approximate mechanical correction rather than substituting a modern, variable timing result.

The hour calculator takes a user-supplied base reading. It does not invent missing glyph times, colours, magnitudes or alphabetical references. A complete certain restoration of all of those lost inscriptions is not possible from the evidence. The 2014 zigzag timing theory is discussed in the cited paper but is not substituted for an authenticated complete inscription table.

What the calendar date does not tell you

By default the Saros plate starts at the reference day only as a laboratory convention. The EYM pattern and the modern mean lunar phases are independent calibrations. To discuss an ancient proposed epoch, first select and document that proposal and then set the corresponding offsets. Without that step, a marked cell near today is not a forecast for today.

A conjectural restoration of the cosmos

The 2021 reconstruction proposes a compact planetary assembly and concentric ring display. It uses long synodic period relations, including the inscribed Venus and Saturn periods [3]. The app adopts those period relations for the mathematical outputs. Its 3D spatial arrangement is its own explanatory design, not the authors’ CAD model or a measured replica of their 69-wheel proposal.

The radii and initial mean planetary longitudes are modern calibration choices taken from JPL’s approximate-elements table [11]. They are not claimed as recovered ancient settings. They supply a reproducible circular model while leaving every phase editable. JPL’s eccentricity, inclination and secular-element evolution are not used in this ancient-style calculation.

The following independent vector construction is the kinematic equivalent used by the app. It makes retrograde motion transparent. Represent the mean geocentric Sun by unit vector S, and the planet’s circular heliocentric position by vector P of radius R. Adding S and P translates the heliocentric planet into the geocentric frame:

The time derivative is analytic, so a transition to retrograde is not a sampling artifact. For Mercury and Venus, the sum produces oscillation about the Sun while the mean longitude completes an annual circuit. For the superior planets, changing relative geometry produces opposition, stationary points and retrograde intervals.

A geometrically exact circular model can still be astronomically inaccurate. Mercury and Mars especially cannot be represented accurately over all elongations with one fixed circular epicycle. The intended fidelity is to the declared mechanical model. Conjunction or opposition in this app is a model configuration, not an authoritative modern observational prediction.

Solar anomaly and nodes

The optional Sun uses the same bounded geometric correction with a default epicycle/deferent ratio of 1/24, a declared ancient-style parameter. The optional node indication uses 1 − (49/62)(64/48) = −5/93, giving a retrograde 18.6-year period [3]. These additions are explicitly conjectural and can be hidden or disabled as appropriate. Changing a display checkbox does not erase its mathematical output.

Calibration, absolute dates and numerical conventions

The default datum is JD 2451545.0, labeled as 2000-01-01 at noon in the proleptic Gregorian calendar. Initial mean lunar and solar phases are conventional J2000-style mean arguments; they are not intended as a precise apparent-coordinate ephemeris. A Julian Date here is a uniformly counted day coordinate. No ΔT, leap-second, local civil-time or time-zone model is applied.

All UI date inputs use astronomical year numbering. The year before 1 CE is 0; 205 BCE is −204. Gregorian and Julian conversion are separate explicit functions, so the application never introduces a hidden historical cutover. A Gregorian date before 1582 is a proleptic label, not a claim that historical Greeks used that calendar.

Epoch date and gear phase are independent concepts. Changing the reference JD changes the label attached to day zero; it does not re-observe the sky and automatically solve new phase constants. Changing the days-per-year parameter changes rates per day but leaves ratios per input turn intact. Changing the rear offsets rotates a calendar relative to the common drive without altering its speed.

Research on proposed epochs must distinguish the time represented at dial start from the manufacturing date and the wreck date. These are not interchangeable [12]. This is why the app exposes offsets instead of asserting one unique original start date.

A reproducible calibration workflow

1. Specify your reconstruction model and documentary basis outside the app, or record them in a separate research note.
2. Enter its epoch as a Julian Date. Enter your observed or stipulated mean longitudes, anomaly and node phases in degrees.
3. Set the Metonic and Saros offsets in mean months and the games offset in years. Set the movable ring offset in divisions.
4. Apply, inspect two or more independently known configurations, and export the session.
5. When sharing results, share the session along with the HTML. A screenshot by itself does not contain calibration settings.

Supported elapsed time is ±3,652,500 days. Numerical availability over that range does not imply physical predictive accuracy over millennia. Ordinary double-precision arithmetic is used; tooth-count ratios are expressed directly, not rounded to a few decimal digits.

The parapegma and descriptive inscriptions

The original instrument included written astronomical information, including stellar phenomena associated with an annual sequence [13]. A parapegma is a reference list, not another live gear-computed star catalog. Surviving material mentions stars, constellations and clusters such as the Pleiades, Hyades and Arcturus. Its interpretation involves textual reconstruction as well as observational conventions.

The decorative front plate here identifies this astronomical context but does not pretend to engrave a complete recovered original manual. Its Greek headings and English labels are modern design elements. Heliacal visibility depends on observer latitude, horizon, atmosphere and what is meant by first or last visibility. An exact universal calendar day cannot responsibly be generated from a name alone.

For that reason, this app preserves the functional distinction: its pointers calculate cyclic positions, its eclipse register selects a model table, and this research manual explains the epigraphic evidence. It does not fabricate a complete parapegma or imply that local weather or religious interpretations were numerically calculated by the gears.

Precision and the limits of this restoration

Three different kinds of precision must be kept separate:

1. Arithmetic precision. The numerical engine realizes the stated ratios and geometric equations to ordinary floating-point accuracy. Reversing time, crossing cycle boundaries, and selecting a date are testable software properties.
2. Mechanical fidelity. A real device has tooth-spacing error, play, friction, flexure, nonuniform tooth profiles and imperfect axes. The app uses ideal kinematics. Triangular teeth in the visualization do not constitute a contact-dynamics simulation. Recent modeling of tooth and manufacturing errors illustrates the need to distinguish those issues [14].
3. Historical and astronomical fidelity. Missing components and inscriptions admit alternatives. Ancient circular models do not match a modern numerical ephemeris. More displayed decimal places do not remove either limitation.

The 3D scene includes procedural bronze wheels, triangular teeth, shafts, openwork, bearing tubes, plates, wooden framing, and a hand drive. Core wheels follow associated algebraic rates. Planetary wheels show the chosen compound rate structure. Some supporting parts, wheel positions, layer spacing and intermediate idlers are interpretive. Tooth engagement phase, clearances, collisions, force transmission and full mechanical constructability have not been certified. The scene is therefore not suitable for exporting dimensions to manufacture a working replica.

The high-resolution 2D readers are authoritative for cell labels and numerical interpretation; the 3D texture uses compact modernized engravings. The calendar-ring alternatives change registration marks, not a hypothetical unknown gear train. The civil-calendar completion is disclosed rather than silently passed off as a recovered transcription. Missing eclipse hours and a complete original parapegma remain unfilled.

Verification strategy

The delivery audit uses three distinct passes: algebraic identities and selected published invariants; independent geometric/numerical and calendar-boundary checks; and browser interaction/rendering checks. The runtime test ledger remains available to repeat the mathematical pass offline. This verifies implementation of a declared reconstruction, not historical uniqueness.

Browser and accessibility notes

The recommended setting is a desktop Chrome window at least 1280 pixels wide. Smaller windows reflow the panels. WebGL is required only for the 3D scene; the numerical engine and 2D views continue without it. If a graphics context is lost, save a session and reload the file; automatic graphics-context reconstruction is not implemented. The numerical readers and session controls remain usable. The app uses native controls, visible focus feedback, accessible canvas descriptions and keyboard shortcuts. Canvas engravings also have numerical/text equivalents in the readers.

Revised functions, precise conventions and reproducibility

Two phases and two different meanings of lunar age

The reconstructed 2021 ring mechanism explicitly supplies a mean-Sun rotation to the lunar differential [3]. This design avoids routing the nonadjacent true-Sun output back into the central Moon display. The phase sphere in this app therefore uses Moon minus mean Sun. It is deliberate that a conjunction with the separately displayed true Sun need not produce a perfectly black sphere. The Dial reader now shows the true-Sun phase and its idealized illuminated fraction separately.

The illuminated fraction assumes effectively parallel sunlight and zero lunar ecliptic latitude. It describes a projected sphere, not a photometric brightness prediction. The phase-scaled coordinate is useful as a circular reading, but variable lunar speed means it is not the time since conjunction. Version 1.1 calculates that elapsed time separately by finding the preceding model new Moon. At the lower supported time boundary a preceding event can lie outside the range; the reader then explicitly reports that it is unavailable.

Event finder

Model events is a numerical measuring aid applied to the mechanical equations. It does not add a hidden modern astronomy engine or claim that an ancient gear train executed numerical root finding. Choose Moon for new, first-quarter, full or last-quarter phases. Choose a planet for conjunction or a stationary point; Mercury and Venus also offer greatest eastern/western elongations, while the three outer planets offer opposition. The chosen solar reference affects conjunction, opposition and elongation; it does not change a planet’s stationary-longitude condition.

Search runs strictly forward or backward from the current day, with a 3,650-day horizon. It returns at most the requested number of events, bounded by the application’s supported range. It pauses playback for a reproducible start. Each row records a Julian Date and the residual of its defining equation. Go reinstates that search’s calibration before driving to its result. The export includes the search direction, limits, full calibration and all rows so that a subsequent calibration change cannot silently relabel an old result.

The cosine test rejects the antipodal sine root. Wrapped angles are not bisected directly: their 360° discontinuity would create false roots. Lunar intervals are scanned at half a day; planetary steps are no larger than one day or 1/256 of a synodic cycle. Sign-changing brackets are bisected to about 10⁻⁹ day or 60 iterations. The permitted eccentricities and rates keep the implemented event branches regular. These are numerical tolerances, not claims of millisecond astronomical prediction accuracy. Root residuals are in degrees for angular configurations and degrees/day for stations and elongation extrema. Inner-planet conjunctions are additionally classified as inferior or superior from their circular geometry relative to the mean-Sun vector; this classification does not model transits or angular separation in latitude.

Numerical and input safeguards

Calendar conversion validates actual month lengths and the distinct Gregorian and Julian leap rules, including astronomical year zero and negative years. The displayed civil date includes hours, minutes and seconds; its rounding carries across midnight correctly. Dates are uniform model-day labels: no local time zone, UT1, TT or leap seconds are inferred. Blank numeric fields are rejected instead of becoming zero. Cycle indexing uses a 32-ULP tolerance scaled by the magnitudes of the terms being added, so cancellation between a large time and an opposing calibration offset does not select the preceding cell or games year at an intended exact boundary. This numerical convention is limited to indexing; it is not a physical tolerance model. Session imports validate all required numeric ranges, Boolean switches, option values and camera settings before changing state. Format v1 remains readable; format v2 adds display and annotation state. The active event-results list itself is exported separately, not included in session files.

Camera distance stays between 180 and 1,800 scene units for wheel, touch and keyboard controls. Coincident touch points do not divide by zero. Lost pointer capture ends dragging. Both orbit angles are periodic; panning is bounded. Capture PNG refreshes the rear texture at the capture time. Lighting now uses the inverse-transpose normal transform for nonuniformly scaled objects. Small antipodal markers have been added to the planetary display; they mark the opposite longitude, not a second planet.

What remains a declared reconstruction assumption

The current verification ledger embeds the actual results of this revision’s independent tests, along with a digest of the numerical engine. Runtime checks remain available without a network connection. The tests establish specific properties of this implementation; they do not turn the assumptions above into archaeological facts.

Research bibliography

Primary research and original institutional sources consulted for this implementation, reviewed 1 October 2026, with a focused model and correction review on 3 October 2026. No external images or published reconstruction meshes are embedded. Mathematical diagrams, graphics and code are newly produced for this app.

1. Freeth et al. (2006). Decoding the ancient Greek astronomical calculator known as the Antikythera Mechanism. Nature 444, 587–591. DOI. Lunar train, pin-and-slot geometry and early tomography reconstruction.
2. Freeth, Jones, Steele & Bitsakis (2008). Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism. Nature 454, 614–617. DOI. Rear calendars, games and eclipse indications.
3. Freeth, Higgon, Dacanalis, MacDonald, Georgakopoulou & Wojcik (2021). A Model of the Cosmos in the ancient Greek Antikythera Mechanism. Scientific Reports 11, 5821. Paper and supplements; Author correction. Proposed planetary architecture and period relations. The supplement contains detailed gear proofs and dimensions; the app is not a reproduction of its CAD.
4. Freeth (2014). Eclipse Prediction on the Ancient Greek Astronomical Calculating Machine Known as the Antikythera Mechanism. PLOS ONE 9(7), e103275. Open-access paper. Source for the selected EYM integer pattern.
5. Freeth (2019). Revising the eclipse prediction scheme in the Antikythera mechanism. Palgrave Communications 5, 7. Paper. Later revision of index-letter grouping and associated inscriptions; the app retains the explicitly identified 2014 possibility table.
6. Spinellis (2008). The Antikythera Mechanism: A Computer Science Perspective. Computer. Author’s full text. Gear-chain arithmetic and computational interpretation.
7. Woan & Bayley (2024). An improved calendar ring hole-count for the Antikythera mechanism. Research manuscript. Statistical alternative to the traditional 365-hole interpretation.
8. Voulgaris, Mouratidis & Vossinakis (2021). The Draconic gearing of the Antikythera Mechanism: Assembling the Fragment D, its role and operation. Research manuscript. Competing interpretation of Fragment D.
9. Voulgaris and collaborators (2024; 2025). Lost parts of b1 and its cover disc; Central front dial and zodiac ring. Alternative constructional arguments; not adopted as established fact.
10. Iversen (2017). The Calendar on the Antikythera Mechanism and the Corinthian Family of Calendars. Hesperia 86(1), 129–203. Publisher. Calendar context and proposed synchronization.
11. NASA/JPL Solar System Dynamics. Approximate Positions of the Planets. Modern semimajor axes and J2000 mean longitudes used only as explicit circular-model calibration constants.
12. Jones (2020). The Epoch Dates of the Antikythera Mechanism (With an Appendix on its Authenticity). ISAW Papers 17. Author’s paper. Distinction among dial epochs and historical dating.
13. Bitsakis & Jones (2016). The Front Dial and Parapegma Inscriptions. Almagest 7(1). Inscriptions corpus. Front scales and stellar reference text.
14. Szigety & Arenas (2025). The Impact of Triangular-Toothed Gears on the Functionality of the Antikythera Mechanism. Research manuscript. Conditional modeling of manufacturing errors, not proof that the ancient machine could never work.

Source availability and manuscript status vary. Competing proposals are included to make the limits of reconstruction visible, not to imply scholarly consensus on every part.

Languages and browser operation

The top language selector uses Google Website Translator, the same service used on nonlinearplatform.com. English is the original language. Choosing another language loads the translator only on demand; interface text and this manual are translated automatically. Mathematical formulas, numerical readouts, source titles and the rendered canvas engravings preserve their original representation. Canvas diagrams have accompanying text and numerical readers.

A language switch pauses playback, preserves the current session, form edits, event results and scroll position, then reloads the page. The chosen language is remembered locally and is also represented by the page’s lang query parameter. Returning to English restores the original document. If Google is unavailable or blocked, an explicit Retry control appears; the mechanism continues to operate. Translation requires internet access and is most reliable on a hosted HTTPS page. Machine translation can misinterpret specialist historical and astronomical terminology; use the English source and cited papers for exact interpretation.

This revision includes the Google Analytics tag G-PM09XGNS48, loaded asynchronously. Loading or blocking either external service does not block the mechanism. Analytics reporting depends on network access and browser settings. The historical offline audit in the Verification tab describes the previous core release; this revision intentionally adds these two optional external services.

The interface scales for desktop and mobile screens, with touch orbit/pinch and keyboard controls. High-density mobile rendering uses a bounded resolution to reduce memory and GPU work. The renderer initializes after the first page paint, large tables are populated when first needed, and animation frames stop when the display is idle or the page is hidden. WebGL availability depends on the browser and device; the 2D readers remain available when 3D acceleration is unavailable.