The 23 Diagrams of Calcidius


Calcidius presented what might be described as the world's first known slideshow during his lectures on the Timaeus by Plato. The diagrams helped his audience visualize some of the difficult ideas in a book which was already 750 years old and had recently gained renewed interest. The Timaeus starts from the perspective that the world is not chaotic, but explicable by physical principles, for example that attributes are quantifiable, that matter is composed of elements and that motion follows predictable paths.

In about 400 CE, Calcidius (for whom we possess no biographical data) translated part of the Timaeus from Greek to Latin (probably in hope of a grant or legacy from one wealthy Osius) and wrote a commentary with a series of visual analogies to both elucidate and modernize these ancient ideas for contemporary students who knew some mathematics, but probably very little else about ancient Greek thought.

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01 and 02 and 03

In three visualizations, Calcidius interprets Plato as stating that certain attributes of things, at least in the scientific sense, can be greater or smaller and thus present a continuum. So where not only opposite qualities are identifiable, but also some halfway house between them exists, we have an attribute on a continuum, and this will have stations or "intervals" along it.

Calcidius’s first analogy expressing this idea uses area: the left rectangle has an area of 6, the right rectangle an area of 24. The middle term, with an area of 12, demonstrates that area is a true attribute. In addition, the three areas can be made proportionate, the insight Calcidius wants to lead us to. The grey and blue rectangles are each twice the size of that to their left.

The second and third examples are geometrical. If you take the red and the blue parallelogram, an affine and intermediate parallelogram (grey) can be drawn in the bay between. Note that in antiquity, the audience did not need to see an actual crooked parallelogram: it was sufficient to say the square was a parallelogram. In the third diagram (the tertia), the red and blue triangles have the grey triangle in their bay at left as their middle term. In both cases, this is achieved by simply growing some straight lines. This crystalline quality of shapes is another important type of neo-Platonic intermediacy.

Calcidius diagram


The next visualization is not Euclidean geometry as one might suppose at first glance, but an analogy to help make sense of Plato's claim that the cosmos consists of two opposites, fire and earth, with air and water as middle terms. According to Calcidius, two things cannot be linked in two-dimensional space without a third as their bond. For three-dimensional space, yet another bond is required. Thus, the entire physical world is made up of two extremes and two bonds, making four fundamental elements.

Calcidius solves this riddle by positing that the earth : water : air : fire relationship is analogous to a series of terms, 24:48:96:192, visualized as a sequence of four solid shapes in the order of normal reading. In the diagram, the first and last solids have fronts with areas of 2*3 = 6 and 4*6 = 24, like the rectangles of Diagram 01. On each solid, that front area has to be multiplied by two “middle-term” factors (here framed in yellow) to reach the solid’s full value. Calcidius proudly notes that the highest middle term (4) of Solid One multiplied by the lowest middle term of Solid Four (12) makes 48, the number pertaining to Solid 2. This new example extends his concept of orderly proportions in a series. [Chapter 14]

Calcidius diagram


Having introduced interwoven series, Calcidius observes that a proportionate series can be continuous, as at left, or involve discrete pairs, as at right, a situation he just introduced with the solids. Continuous proportion means the numbers make a “chain”, whereas discrete proportion brings about what he calls disrupted sections, such as the regression in the series 3 .. 6 .. 4 .. 8. This will be important later, when Calcidius tries to explain the apparent retrograde motion of the planets in the night sky.

Calcidius diagram


This figure could be regarded as a conversion of Diagram 04 to Euclidean space. This time round, Calcidius wants to show by a geometrical analogy how fire and earth, represented by the sticking-out cubes, can be mutual opposites with nothing in common, yet be connected by the two middle terms, air and water. [Chapter 17]

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07 and 08 and 09

Next, three lambda diagrams containing number series visualize Plato's obscure account of how regularities were part of the plan when the World Soul was first formed. Note the common feature among the three diagrams: a left arm with numbers increasing from 1 by factor 2 and a right arm by factor 3.

Plato’s story attached to this says the creator of the universe shaped the World Soul by first taking a fragment of soul [size 1], then a second part which was double the first [size 2], a third part which was half as much again as the second and three times as much as the first [size 3], a fourth part which was twice as much as the second [2 x 2 = 4], and a fifth part which was three times the third [3 x 3 = 9], and a sixth part which was eight times the first [size 8], and a seventh part which was twenty-seven times the first [size 27] and composed these to give ordered life to the universe.

The second diagram visualizes Plato's concept of intervals occurring in the same series, explained orally by analogy to the intervals of the notes on a multi-stringed musical instrument. (Plato does in fact claim that the sun, moon and planets make sounds by their motions and that these are in harmony.)

Calcidius is not offering any empirical evidence for this: he is simply elucidating for his students the Pythagorean number series which Plato had in mind.

The third lambda diagram continues in the same vein, with a mixed series, starting at the apex of the triangle with 192. Calcidius says he chose large numbers to illustrate "the density of the full sound, its full packing in as it were" of the World Soul, but was perhaps also testing if his students were alert. The two exceptional jumps in the series are a piece of Pythagorean reasoning which Calcidius had to explain. Within the numbers, S is Latin notation for half. This is the last of the mathematical diagrams. [Chapter 32 etc]

Calcidius diagram


Plato states that what holds for the whole world holds also for the parts. This visualization offers a simple example: the surface of the sea or any body of water on the earth is never flat, but visibly bulges up to block your view into the far distance, which is empirically observed in the fact that you can see further from the mast-top than from the deck of a ship. Hence the sphericity of the Earth applies both locally and globally. This should not be understood as an attack as such on flat-earthers: Calcidius is merely guiding us to the insight that scientific knowledge corrects what we see at first glance.

Calcidius diagram


The first of the astronomical diagrams provides a freshener to Plato’s very out-of-date astronomical knowledge of 750 years earlier. Plato made no use of terms such as arctic or antarctic. State of the art in 400 CE meant some form of late Hellenistic theory, though probably not yet that of the still-obscure Ptolemy of Alexandria.

This diagram is of the heavens, not of the Earth, as indicated by labels for the zodiac, the Milky Way, the Pole Star of the north and the horizons and including parts of the sky, such as the Australis never seen by dwellers on the Mediterranean. Colurus is a term for a meridian, but other designations such as tropicus denote belts, not lines. Calcidius does slip into asides about the land, as when he says the arctic and antarctic sky cannot be seen from "the central region of the Earth, that is where the sun-scorched and thus uninhabitable zone (the Saharan and Arabian deserts) extends beneath the equatorial belt." [Chapter 67]

Calcidius diagram


In the Commentary, Calcidius compares Plato’s teaching about the orbit heights of the “wandering stars” (the Sun, Moon and five planets visible to the naked eye) with the views of the geographer Eratosthenes, and visualizes here the Platonic view: the Moon is lowest, then the Sun, then Mercury and so on, up to the Aplanes, the fixed firmament. “In the Republic he observes not only the order of the planets, but their individual magnitudes, velocities and colors ... likening the celestial axis to a spindle (on a wool-spinning wheel) and the orbits revolving around it ... to its flywheels.” The diagram, however does not seem to reflect this, with Plato’s order not depicted. Nor is there any mention in the diagram of color. Possible this diagram comes from an astronomical book. [Chapter 73]

Calcidius diagram


With this diagram of the varying lengths of the seasons, Calcidius begins an argument to explain the apparent retrograde motion of the planets. The outer ring shows the perceived displacement of the sun in the space of the year through the twelve constellations of the zodiac, while the inner notation gives the elapsed time for each quarter from the vernal equinox (top) to the summer solstice (left): 94½ days. The succeeding intervals are 92½ to the autumnal equinox, 88 and one eighth to the winter solstice and 90 and one eighth back to the spring equinox. In the notation, S denotes semis (half) and ς denotes one eighth. [Chapter 80]

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The first of two theories which Calcidius (not Plato) offers to explain this anomaly assumes that Earth is the centre of the universe, but that the Sun has an eccentric orbit (blue). Thus, seen from Earth, the Sun would seem during the year to displace at varying speeds with respect to the dome of fixed stars (red). [Chapter 80]

Calcidius diagram


The second of the two theories is the epicycle explanation of planetary orbits, here visualized with a planar diagram of motion within motion. Suppose the Sun has quite a small orbit which the Earth is not within (the green path, direction of rotation clockwise), we would suppose from the vantage point of the Earth that the Sun was speeding up and slowing down as it moves through the zodiac in the course of the year (red path, direction anti-clockwise). Epicycles would create the impression, when seen from the centre (Earth), that an object hastens and slows. Calcidius uses this example to urge on us a broad Platonic insight: certain phenomena appear differently to us than how they occur in reality. [Chapter 81]

Calcidius diagram


In the following diagram, Calcidius proposes to explain our perception of retrograde motion of certain planets against the starry background. He marks the Earth Θ, and the three solid blue radii are lines of sight to the firmament (red). The green circle is the epicyclic orbit of a planet which takes more than a year to complete its cycle. At Ζ and Η it appears motionless during its clockwise course, while on the upper part of the circle it appears to move backwards.

Calcidius diagram


Next, in three diagrams, Calcidius digresses on the evidence provided by eclipses and shadows. In the first visualization, he shows the configuration of a solar eclipse. The diagram marks with “umbrae” the halo of light when the lunar disc is in front of the Sun, but this is not mentioned in the matching passage of Calcidius’s text, suggesting the diagram may be taken from some other book. He notes that lunar eclipse is similar, adding: “The Moon, when standing diametrically opposite the Sun and encountering Earth's shadow, is obscured, and yet not in all months, just as an eclipse of the Sun does not occur at every conjunction either.”

Calcidius diagram


The second diagram of the group sets out a theory of shadows, with the light source at bottom and a block object at middle . The result at left is a cylindrical shadow, and at right a fanning-out shadow. Calcidius reasons from this that we are unable to see certain dark objects in the firmament of the sky because of spreading shadows being cast on them from elsewhere in the universe without us on Earth being able to detect these cones of shadow. This seems tied to a theory that stars may not emit light of their own.

Calcidius diagram


By contrast, narrowing-cone shadows shrink to a vanishing point. A lunar eclipse provides an example on an extraterrestrial scale, since the Sun is the largest object (1,880 times greater than the Earth according to Hipparchus as quoted by Calcidius), the Earth intermediate in size and the Moon smallest (one twenty-seventh the size of Earth, H claimed). This is a rather strange text passage, since the relative sizes of the Solar System objects ought to be the unknown and the behavior of conic shadows is familiar, but Calcidius foregoes the chance to argue from the former to the latter. [Chapter ..., version preferred by Bakhouche and Magee]

Calcidius diagram


This is the only technical diagram of the series, illustrating without transformation the Plato text as understood by Calcidius in his translation: “(The Great Craftsman) cut (Existence) lengthwise and from one series made two, and he forced them round, middle to middle, into the shape of the Greek letter chi and bent them into circles so that their ends joined one another.” Calcidius says the proper sense is that the god bent the chi (X) “to make two interconnected circles ... and encircle these with another outer circle,” the fixed vault of the heavens.

Calcidius diagram


Here Calcidius returns to Plato’s mythic account of the making of the world and connects it to the planetary order as understood by more advanced astronomy: “The position of the planets is in evident accord also with the cutting of the parts from which he constituted the soul, since he has: first, one portion i.e., the smallest, drawn from the whole, for Earth to the Moon; second its double, i.e. the one lying between the Moon and the Sun; third, its triple, namely that of Venus; fourth, double the second, i.e. quadruple the first, that of Mercury; and the octuple portion, that of Mars, which is the fifth cut; sixth, triple the third, i.e. the region or orbit of Jupiter; and the last cut, that of Saturn, of 27 parts.”

Calcidius diagram


Here the topic is Venus, a planet which can often be seen low in the morning eastern sky for an hour or so before the Sun rises and the planet becomes lost in the blue sky of day, and is thus called the Morning Star. In the opposite phase of the synodic period of 584 days, Venus is seen low in the evening western sky for an hour or so after the Sun sets, until the Evening Star sets too. In our modern heliocentric view, this is easy to understand, but before the rise of Greek astronomy it was not even settled in Europe that the Morning and Evening Star (Lucifer and Hesperus) were one. To a geocentric mindset, the phenomenon remained puzzling.

Calcidius, whose purpose here is to explain why Plato said Venus and Mercury (the other inferior planet) had a “contrary power,” ἐναντία δύναμις in the original Greek of Timaeus 38D. Calcidius, who translates this to Latin as “vis contraria,” is looking for features in contemporary astronomical theory which are congruent with Plato's terminology.

He goes back to Heraclides of Pontus, who he cites as an authority that the motions of Venus and the Sun follow epicycles sharing a common center. The top diagram exploits this to show why Venus never appears more than 50 celestial degrees either side of the Sun (ostensibly visible in front of Β), since its epicycle (blue) is of limited diameter. The outer ring (mauve) is the ecliptic, rotating once daily clockwise around the Earth, which is marked X. Venus can oscillate within bounds only 100 celestial degrees apart in the course of a 584-day synodic period.

Some have read the diagram also to mean that an earthbound observer gazing along the line ΧΓ will see the Morning Star rise into view. From the diagram, one is to understand that Venus achieves its greatest western elongation (i.e. Venus preceding sunrise by the greatest angle) when its epicycle touches the tangent ΧΓ. At the opposite phase, the Sun sets first, then Venus sets, and the viewer is left looking along XA to where they have disappeared from sight.

As drawn, K marks the common midpoint of the solar and Venusian epicycles. Whether Heraclides had actually drawn any diagram of his own is to be greatly doubted: in keeping with his own times, he probably simply described the hypothesis in words.

Calcidius adds that “Plato and others who have conducted a more careful assessment” do not require that the solar and Venusian epicycles have the same midpoint, but merely that their centers are situated along a unified line leading out from the center of the Earth (K). This would seem to be the import of the second diagram, at bottom. Here too, the epicycle of Venus is blue, while the two green lines present the major orbits of Venus and the Sun.

This double diagram has a convoluted transmission history. Scholars continue to debate the interpretation of the pair of figures: some including .. Bowen and Bruce Eastwood argue that the two diagrams are not in conflict, but show the same phenomenon in “observational” and “explanatory” fashion, to use Eastwood’s terms.

The version offered in Waszink’s edition and found in a majority of manuscripts has litle logic to it, though Béatrice Bakhouche has defended it as tendentially correct. The version shown here does not appear until the 15th century, probably as a result of deliberate editorial correction, but has the virtue of clarity. It was adopted by John Magee in his edition and English translation of Calcidius, and serves well enough to represent Calcidius’s intent as judged from the text. [Chapters 111-112]

Calcidius diagram


Calcidius ingeniously picks out phrases by Plato about planetary paths which revolve in a "spiral and, as it were, winding acanthus coil" to imply that Plato might have foreshadowed the epicycle hypothesis, and he buttresses this with diagram 23. The diagram is not entirely a success: the double-lined radii seem to have no function except to remind the reader that the context is the same as in diagram 22. Venus is shown inked in at a starting position in the constellation Aries, as stated in the text, and its retrograde motion is visualized as a planar spiral.

Calcidius uses the Greek term helix, but it is not clear if he comprehends the hypothesized motion as a screw-form spiral, since he (a) stresses its similarity to acanthus (that is to say, to curving tendrils carved in wood or stone as decoration) and (b) explicitly defines such a spiral as a two-dimensional line created by slowly shifting off-base the pin of a pair of compasses while the pen is drawing. The latter remark suggests that Calcidius was technically competent enough to draw these diagrams himself and practised this uncommon trick while writing his manuscript. [Chapter 116]

© Jean-Baptiste Piggin