• M. Josh Roberts

The Heavens at Hand

The technology I would like to describe is one that originates in the ancient past, likely as old as astronomy itself. While a few instruments will be described, they are merely different takes on the same technique and as such, seemed to cleve closely enough together to be described as a single technology, evolving and growing organically along a complex path. Like the evolution of a species, there is no clear progression towards an end; many offshoots were successful in their own way but have faded over time. I will explore the means by which the angle between objects in the sky might be measured and better understood by the hand, by the radius astronomicus or Jacob’s staff, by the quadrant, and finally, by the kamāl.

When we look up at the sky during a clear night, we see a scattering of stars. Seemingly random (for they are, in effect, randomly positioned) but human minds excel at recognizing patterns, even where there are none (Rizzo & Cantasano 2017). We imagine constellations and groupings that reflect the stories we tell, or else make up new stories to match the patterns in the stars we see. Season by season different stars are revealed when the Sun sets, but over the course of a single night the stars move as well. It would seem nearly impossible to determine when, in the depths of history, humans began to realize the patterns of the sky, both in its daily and annual motion, but it has been a commonality across nearly all human cultures to observe these patterns.

What might we learn from these cycles? For agricultural societies, the calendar year of the seasons was critical to know when crops should be planted and when to harvest. The stars that rose moments before the Sun (heliacal rising) would always be able to tell you what season you were in and allow you to predict when weather would turn cold or when the rainy season could be expected. But for hunter-gatherers or those who sailed, another important aspect of the night sky was the fixed point in the sky, somewhere between the stars now called Polaris, Kochab, and Thuban (CornellWeb).Today we call this the North Celestial pole (NCP) and we understand its relationship to the rotation of the Earth, but back then it was an invaluable guide for travelers. Not only was it a set directional point, but it could also indicate where you were on Earth with respect to your distance from the North pole.

Observing this motion is not a simple thing to do. The stars move slowly enough that most observers get cold and go inside before much motion can be observed.. If someone wanted to determine the direction and rate of the motion, a few strategies might come to mind: you might use a foreground stationary feature and see how the objects in the background moved with respect to it, but those can be hard to spot at night. You might place something in front of the star and see how long it took for the star to become visible again. You could also watch a star rise from the horizon to its highest point, but how do you measure that distance?

Fingers seem an obvious solution (Egler 2006) since they are always “at hand”. To date, many amateur astronomy clubs (FortWorthWeb), skywatchers and first-year astronomy students (UniCinnWeb) utilize their fingers, hands, and palms to estimate a separation between objects in the sky, as well as altitude above the horizon. This turns out to be an adequate system of measurement in the sky to allow us to find the measure of a few degrees. A finger at arm’s length measures roughly a degree, three fingers subtend 5°, a fist is 10° and fingers fully-splayed spans 20° (think “shaka/hang loose”).

Given human variation in size, one might immediately assume that this, like other body-based units of measure would be biased to the proportions of the individual (why you would always send the relative with the longest forearms to buy a few cubits of cloth) but this one usually works out pretty well. The secret is that these astronomical measures are always taken at arm’s length with the hand outstretched. People with larger/thicker fingers tend towards longer arms making the larger fingers farther from the observer and therefore cover less of the sky. While this is somewhat lacking in scientific rigour, it is still very much in use today at nearly every star party I have attended.

In my research, I have not yet found anyone willing to offer a “discovery date” for when people began using measurements like this, but I can only imagine that it was early on in the course of human astronomy. Without a tool to be uncovered, or a written record to be translated, it would be difficult to prove when the method began. We do have records dating to the 1300s that detail the proportionality that allows for the individual variation (SimonsonWeb).

Yet, with any unfixed and “body-based” measure there is a degree of unreliability (and some people really do have disproportionate finger widths!), so a more uniform system was eventually developed, wherein a given length object was set at a given distance to define a specific angular separation. This is a moment where the progress of the angle measuring technology splits, one path on precision, the other on useability.

The precision-oriented version was known as the Jacob’s staff or the radius astronomicus. The staff was invented (or at least formalized and popularized) by a French rabbi, Levi ben Gershon, in the 14th century and intended for angular measurements both astronomical and terrestrial. This was simply a long and straight stick with markings of distance upon it with a crossbar of a set length that slid up and down the length of the main staff (Egler 2006). Once two points were defined, the observer would place the staff along their line of sight to the midpoint in between the the selected objects. The measured angle between them could be found by moving the crossbar up and down the staff until it spanned the distance between them (figure A).

(figure A. radius astronomicus measuring separation of two stars)

The reading of the distance from the eye of the observer to the cross bar at the correct position would define an isosceles triangle (⃤OAB) or a right triangle (⃤OAC), and while ratios of the lengths measured (line CA and line OC) might have been used at the time, Levi ben Gershon (SimonsonWeb) wrote “Sefer milhamot Adonai” in 1329 (Brittanicaweb) which offers insight into the most sophisticated mathematical technique to determine the angle: sine and cosine tables. This simplified calculations and made the Jacob’s staff a critical instrument that freed astronomers and sky watchers from the imprecision and relatively small size of the hand.

The quadrant is an ancient instrument, used by Ptolemy (Cambridgeweb) and is so named for its measure, traditionally 90° and a quarter of a full circle (figure B). Its cousin, the sextant, was a similar tool but smaller and simpler with a maximum measurement of 60°. These devices had a quarter of a circle along their measuring edge, graded very finely (perhaps a degree or two) and a plumb bob at the center of the circle defined by the outer edge. One edge had small sights along it and by aligning them with a distant star or celestial object, its altitude could be discerned from the bob’s position along the measured edge.

(figure B: the quadrant determining the angle of a star)

A less cumbersome tool to be sure, but its precision was limited by size (more measurement angle markings would be more legible on a larger model) and it could only be used to measure angular separation in one direction (down) so determining the distances between stars or planets was much more complex. For this reason it was often used alongside the Jacob’s staff by medieval astronomers. As a navigation tool there were also challenges for those at sea, since the rocking of a boat would make the plumb bob sway about, eliminating vital precision.

An close relative of the radius astronomicus was the kamāl (figure C). The kamāl was a navigation tool used by seafarers on the Indian Ocean and was in full practice well before the 15th century (McGrail 2001). The very height of simplicity to make and to use- it was a small wooden tablet with a string running through the middle. The string was adhered to the tablet itself and extended out from it. Along the string were knots, each knot corresponding to a specific distance. The tablet was inscribed with the number of the knot and the names of specific towns in the region around where the instrument was being used — in one example it gave the names of Madras, Pondicherry, Negapatam, Port Calymere, and Trincomalee (McGrail 2001) indicating its use in the Indian Ocean. Why were these names associated with knots? Because of the North Celestial Pole! Just like the staff’s crossbar, the tablet of the kamāl was brought away from the observer’s eye (where the string was being held). When the angle of sky covered by the tablet went from the NCP to the horizon, you saw on the string which knot you were closest to (thereby getting a quick measure of the string) and this would give you your latitude by way of telling you which ports were nearby.

(figure C: the kamāl measuring the angle between NCP and the horizon as a function of distance to the observer)

The kamāl is perhaps more of a navigation tool than an instrument of celestial study, but nonetheless is a triumph of utility, useability, and simple design.The biggest advantages were twofold: someone with a decent memory could memorize the order of the knots and didn’t have to read to be able to utilize the kamāl, and it could be used by touch, without a need for readings in the dark (a shortcoming of the staff; in order to extract the information from the instrument’s measurements, you needed light bright enough to read by, surely destroying the observer’s dark adaptation with each measurement). Yet in spite of these boons, the kamāl stayed a useful navigation tool widely utilized in the region but never made a jump to a celestial measuring tool for anything but the NCP.

It is interesting to note that, while the devices are very similar in principle, in form and in function they are so disparate. The kamāl was a simple device, cheap and easy to duplicate, for everyday use. It could be stowed easily in a pocket or stashed on a crowded ship and only took a few seconds to take the measure that was needed. Contrast that with the Jacob’s staff, a large and often lavishly designed piece of finished wood and brass (MuseoGalileoWeb), carefully crafted with exacting specifications. While technically “portable”, it was a tool intended for the wealthy academic. Both are a marked improvement over hand reckoning, but in very different directions. The quadrant sits in between, fairly simple, easy to construct and as accurate as a situation allows for, but not as versatile in application or function.

These instruments represent variations on a theme; clever methods of determining where objects could be located on the celestial sphere. Building a quantitative understanding of the motions of the night sky led to later breakthroughs like the astrolabe. These tools are fundamentally simple in concept and execution, which makes their capabilities all the more impressive. They also illuminate an interesting crossroads of science and commerce. Their ubiquity in the ancient world had everything to do with their useability at home and abroad, on land and at sea. By enabling trade, more resources came to building and developing these technologies and science grew. By sharing the tenets and ideas of science with more people with different backgrounds, innovation and adaptation increase as well.

Word count: 1999


Brittanicaweb: Brittanica entry for Rabbi Gershom (accessed 10/19/18)

Cambridgeweb: 10/20/18)

CornellWeb: Precession of Earth’s orbit (accessed 10/20/18)

Egler, R. 2006, JRASC, 100, 37

FortWorthWeb: Measuring the sky (accessed 10/24/18)

McGrail, S. 2001, Boats of the World (Oxford: Oxford University press)

MuseoGalileoWeb: (accessed 10/24/18)

Rizzo, C. & Cantasano, N. 2017, IJAsB, 16, 297

SimonsonWeb: The Mathematics of Levi ben Gershon (accessed 10/20/18)

UniCinnWeb: “ancient astronomy” course slides (accessed 10/20/18)

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