Our Earliest Shadows

Projection at work: A 3D tree becomes a 2D image through the projection of its shadow. (Photo by the author)

The above image of a tree and its shadow is such a raw example of not only how projection works, but of how ancient civilizations may have come to realize projection’s potential utility. The elements present to make this effect are essentially unchanged today: a three-dimensional object, a source for the projector rays (the sun), and a screen or 2D plane to “catch” the vector information (the canvas). The resulting effect is so fundamental to understanding the history of projection, that it may be useful to consider the history of the shadow. This topic is covered elsewhere in this blog, namely through the discussion of Pliny’s tale of the Origin of Painting. That tale, written in ancient Roman times, is a fable, useful to uncover ancient attitudes about the shadow’s utility, but not to elucidate the history of discovery via shadow projection.

For that, we should look to the Greeks, who established many of the fundamental scientific and artistic standards the West has inherited*. The Greek civilization looms large for several basic reasons. Many primary documents survive, allowing us to attribute specific discoveries and theorems to individuals. Secondary sources also provide pedigree, establishing inheritance of ideas from teachers to students of the era.

Given this fairly linear record, consider the following projection milestones using light and shadow, described here in chronological order.

Thales (636-546 B.C.) measures the pyramids. Thales of Miletus is considered the father of philosophy; Bertrand Russell said “Western philosophy begins with Thales”. While today philosophy is associated with metaphysical concerns (the nature of existence and thought), ancient philosophy was closely tied to the physical (explaining the natural world). Thales and his descendants were proto-scientists, using reason and observation to begin unraveling the mysteries of the universe.

While great thinkers like Russell and Aristotle (who, living 250 years after Thales, called him the first of the philosophers) credit Thales with establishing a tradition of thought, he is clearly pivotal in the history of projection. Thales measured the height of the Great Pyramid at Giza by using its shadow. There are several accounts of how he did this, but all agree that he observed that at certain moments, his shadow would equal his height, making a right isosceles triangle (45º-45º-90º).

Shadow of the Great Pyramid at Giza. Photo by the author.

Plutarch’s account is most revealing: “…without trouble or the assistance of any instrument [Thales] merely set up a stick at the extremity of the shadow cast by the pyramid and, having thus made two triangles by the impact of the sun’s rays, … showed that the pyramid has to the stick the same ratio which the shadow [of the pyramid] has to the shadow [of the stick].”

Thales’ experiment for measuring the height of the Great Pyramid. [Image from mathforum.org]

The shadow here becomes, for the first time, an agent in generating scientific data and knowledge. The shadow here is, centuries before Euclid and Pythagoras, at the heart of geometry** and the basic formation of math and science to come in the Classical period. The shadow here prefaces the application of abstract vectors empirically translating information through multiple dimensions as we understand it today.

Anaximander (c.611-c.547 B.C.) introduces the gnomon to Greece. A gnomon is the element that casts a shadow in a sundial. Anaximander of Miletus, a younger contemporary of Thales, and teacher of Anaximenes and Pythagoras, is credited with introducing the gnomon to Greece. He did not invent the sundial or its essential components. The Babylonians documented the passage of time using sundials, but Anaximander is thought to be the first to accurately calibrate sundials (as they are sensitive to different latitudes) and measure solstices and equinoxes.

A basic gnomon

The gnomon is a potent evolution of Thales’ measurement. By inscribing shadow indicators, a simple vertical element can passively inform not only time, but direction (the shadow traces a perfect east-west path). It is also tempting, considering Anaximader’s famous pupil Pythagoras, that the gnomon, with its right angle, vertical element, horizontal shadow, and invisible hypotenuse, that this device leads directly to the geometry of triangles. It does not take an enormous leap to imagine the application of a cast shadow into the drawing of vectors, the translation of shapes and angles, and the general shift to precise graphic representation.

Eratosthenes (276-194 B.C.) measures the Earth’s circumference. Eratosthenes of Cyrene was well known in his time for a number of scientific and intellectual achievements and may be best known for being the third Librarian at Alexandria under the reign of Ptolemy III. He is credited with key measurements, methods, and models of determining large-scale data***, such as celestial distances and inventing the armillary sphere.

His most notable projection achievement is his remarkably accurate calculation of the earth’s circumference. Around 240 B.C., Eratosthenes used his knowledge of shadows and the sun to execute a surprisingly simple experiment:

Using the sun's parallel light rays, a well, and an obelisk, as well as accepted knowledge that the Earth is round (sorry, Mr. Columbus), Eratosthenes calculates the circumference of the Earth. His calculation of 39,690km (converted from his standard unit of "Egyptian stadia") has a percent error of less than 1%.

Eratosthenes established the following givens: The city of Cyrene (modern Aswan) lies on the Tropic of Cancer, meaning that on the summer solstice, the sun is directly overhead. In Alexandria, on the same day, the sun is not directly overhead, and the shadows cast are longer than the non-existing shadows in Cyrene. Eratosthenes also knows the distance overland between the two cities (estimated at around 800km). He also assumes the sun’s rays are parallel as they strike the Earth.

With this knowledge, he devises the experiment depicted above. At noon on the summer solstice, a well/pit in Cyrene confirms the perpendicularity of the sun, as there is no shadow cast by the well walls. In Alexandria, an obelisk, known to be quite plumb (truly vertical), casts a shadow. The measurement records a 7º12′ deviation from vertical, using the obelisk and the ground as the legs of a right triangle, and the sun’s ray as the hypotenuse.  Knowing the overland distance, Eratosthenes correctly assumes that the degree measurement, at around 1/50th of a full circle, corresponds to an overall circumference of around 800km x 50. His total of 36,690km is within 1% of the actual total of 40,040 (average)****

WHILE it is virtually impossible to find smoking guns when it comes to our ancestors, it is compelling to consider the shadow’s long history of utility and its possible progeny in the form of geometry and the many branches of science and mathematics still using projection methods today. The shadow, and the sun that generates it, is one of the few things we can confidently say has not changed from the earliest days of human awareness. As a result, the sciagraphic paradigm, in a scientific context, is one that does not waver and does not fall victim to cultural or historical interpretation.*****


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*The Egyptians were able to build one of the earliest, most advanced civilizations on Earth, and their use of solar and celestial projection was unparalleled in the ancient world. Construction of the pyramids, agricultural technology, calendar calibration and other projection-related technologies were well-established by 2500BC, long before the Greeks made their discoveries. I will go into Egyptian technologies in a future post, but for this post, weight is given to the Greek development of projection as a direct ancestor of our current application methods. Many Egyptian methods are lost to time, with only scholarly speculation to describe their achievements.

**”Geometry” literally translates to “earth-measuring”, but this association has been lost, unlike “geography” (“earth-drawing”)

***Eratosthenes is credited with not only making one of the first maps of the known world, but also for inventing the term “geography”. Despite his early attempt at mapmaking, it would not be until Ptolemy (Claudius Ptolemaeus) invents the first map projection published in Geographia (c.150A.D.) that cartography is codified.

****One source of error is that Eratosthenes assumed the Earth to be a sphere, where it is in fact an oblate spheroid, flatter at the poles like a tomato (as opposed to a prolate spheroid, narrower at the equator like an egg). As a result, the Earth’s actual circumference varies from 40,075km (east-west equatorial) to 40,007km (north-south meridional) for an average of 40,040km.

*****For cultural or interpretive or metaphoric studies of the shadow, consider the excellent “Shadows: The Depiction of Cast Shadows in Western Art”, by E.H. Gombrich and “A Short History of the Shadow” by Victor Stoichita.

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