Metrics and Nature's Gifts

The demand for mechanical precision, particularly in the areas of time keeping and the measurement of angular displacement, originated in Europe's astronomical observatories and first showed to good effect there . . . or so we were told.

Some 2,500 years ago the Greek historian Herodotus visited Egypt and reported that the Great Pyramid, one of seven wonders of the ancient world, was used as a celestial observatory during the course of its construction. We’ll take the man from Halicarnassus at his word. Because we find these features true today, it must have been important to the pyramid’s builders that the growing mound of limestone blocks be aligned precisely to true north, that its base be level and square, that the entry or “descending” passageway parallel the Earth’s axis, that the “ascending” passageway be reflected from the former and, finally, that the top of the pyramid be centered directly over the intersection of the diagonals of its base. A tall order it would seem, but was it really?

Let's give it a go, relying only on tools nature provides.

Steps 1 and 2—pick a construction site that is reasonably flat with clear site lines to the north; lay out a true north-south line.

For this exercise we’ll use an available field near Las Vegas’s McCarran International Airport. Fortunately, we have a reference for true north, the North or Pole Star, Polaris, readily identified in the night sky. (The two stars that form the lip of the cup of the Big Dipper constellation point to Polaris, itself the last star in the handle of the Little Dipper.) If Polaris were not almost perfectly situated along a projection of Earth’s spin axis, it would describe a small circle in the sky as the heavens wheel. A bisection of that circle—or the circle described by any other neighboring star—does lay on Earth’s spin axis. Drive a stake into the ground at what will become our pyramid’s northeast corner. Drive a second stake into the ground a little beyond what will become our southeast corner such that the two stakes line up with yonder Pole Star. Stretch a string between the two stakes and, behold, in one act, a true north-south reference line and one boundary of our structure.

Steps 3 and 4—lay out a square base and level it.

Roughly stretch a second string at right angles to the first, passing the base of the northeast corner stake. Since it has been known from ancient times that a triangle with legs in the ratio 3:4:5 forms a right triangle—three squared plus four squared equals five squared—adjust the second string so the rule prevails and fix it in place. Now measure off two equal sides of the pyramid starting at the northeast corner. Translate both strings and lengths to their opposite sides and “square up” the construction boundaries by comparing diagonal measurements. The diagonals should be—must be—equal. Trench around the boundaries of our construction site then within the boundaries on a grid of, say, three feet by three feet. Temporarily fill the connected trenches with water. Dig out the trench bottoms until the bottoms are everywhere equally flooded. Drain the trench system and plane the surface of the construction site to meet that datum. We are now aligned, squared and level, with nothing but nature’s bounty—Polaris, gravity and the properties of a taught line.

Steps 5 and 6—construct an entry or “descending” passageway parallel to the Earth’s axis; at some point along that passageway, reflect its orientation and construct an ascending passageway.

There are two points we must determine and preserve on the square surface that is to be the base of our pyramid. One is the intersection of the square’s two diagonals—done—and the other lies somewhere on a true north-south line that passes through the first point. Let’s choose a spot roughly halfway between the first point and the north edge of the square. Precision in this placement is not important to this exercise. (Doubtless, the ancient Egyptians chose purposely.) At both locations carefully center stone basins designed to hold a few inches of water with centers marked by fine bronze pins. Lay up a few courses of limestone block—or Styrofoam block if time is pressing—leaving open a shaft to the north that allows light from the Pole Star to strike the head of the pin in basin two. Roll something flat and pliable, a piece of papyrus paper perhaps, into a straw-like tube and poke it through a curtain limiting the rays from Polaris to only those that strike the head of the pin. Fill the basin with water to just cover the pin. Backing up from the basin while facing north and adjusting height-of-eye, there is one alignment and only one that reveals Polaris reflected off the basin’s surface, and that alignment guides construction of the ascending passage. More blocks, now, being careful to maintain vertical access to the first basin so that it may serve its intended purpose.

Great Pyramid Passageways

All this trouble might strike the novice astronomer as odd until the novice is informed that most telescopes are mounted “equatorially,” taking into account the latitude of the observation site. The angle between the base plane of our laboratory pyramid and the ray of light connecting Polaris and the pin in our second basin is equal to—yes, equal to—our site’s latitude, in this case the latitude of McCarran International Airport or 36°-04'-47" N.

Step 7—center the top of the pyramid directly over the intersection of the diagonals of its base.

After each course of blocks is laid, a new square, smaller and higher, may be established with the intersection of its diagonals falling precisely over the intersection of the base diagonals. This is achieved by suspending a plumb bob over the bronze pin in the first basin. (Filling the basin with water will help dampen the bob in its swings, and a shout of “Mark!” will tell fellow workers the bob’s nether-most point is directly above the centering pin.) With a final course of blocks squared and trued, the pyramidion cap piece may be sky-craned into place. Job done.

Among other interesting tidbits, Herodotus reported that the Great Pyramid’s pyramidion was made of solid gold. Solid gold or just gold clad, it must have been a magnificent sight. Suppose the priests in charge of the finished complex linked the pyramidion with an insulated gold wire dropped down through the plumb bob shaft and anchored to ground—what a lightning rod! Throw in a switch in a basement control room, no telling what miracles might be pulled off.

Herodotus (c. 484-425 BCE)

Hotel Luxor in Las Vegas

Here and There

While training on California's San Clemente Island—this was in the 60's, about the time the Beach Boys were surrendering top pop-chart honors to the British Invasion—first light found this future gunnery officer scoping a U.S. Navy destroyer on the southern horizon, the ship's silhouette gradually taking shape some seven miles distant. A call for fire was ordered, and I was about to witness a remarkable series of events. The target, a stack of 55-gallon drums a few hundred yards out in front of our pillbox observation post, had already been selected, plotted and forwarded to the slowly steaming firing platform. "Sandals, Sandals, this is Blockhouse One. You have your coordinates. One round, high explosive, point-detonating . . . fire for effect, over."

"Roger, Blockhouse One. One round, high explosive, point-detonating . . . fire for effect." And then, "Round one on the way." Through my scope I watched a single puff of smoke billow up from the ship's forward gun mount, a clear signal the round was indeed on its way.

Let's call the overall distance to target, along the parabolic flight path the projectile had to take, an even eight miles. At an average velocity of 2,300 feet-per-second (the gun's muzzle velocity might have been as high as 2,500 ft/sec) I could expect impact in approximately 16 to 18 seconds, and that's about the time it took. Lucky first round—in eerie silence the stack of targeted drums suddenly flew apart. A moment later, just as the drums were crashing back to earth, the thunderclap of 40 pounds of high explosive going off slammed the blockhouse.

But the show was not over. My instructor redirected my attention to the distant ship where the smoke from its gunfire was finally dissipating. More seconds passed—maybe half a minute from trigger's release—before the rumble of gunfire reached me.

At sea level, on this standard of days, sound traveled much slower than the Navy's 5-inch diameter projectile—half as fast, in fact. Everything is catty-wampus to the bombardment observer. Smoke on the horizon first, the target out in front taking a hit some handful of seconds later, the sound of the target taking a hit following closely, and finally—an absurd amount of time later—the sound of the projectile being fired.

Even the ancients knew to gauge the distance of a lightning flash by the delay in thunderclap's arrival. (We've all counted the seconds between flash and clap, but few of us know what to do with the result. Dividing seconds by 5 produces distance in miles.) The ancients assumed the lightning's flash arrived instantaneously or at a speed so very fast there was no practical way to measure it.

Sound does have a measurable speed. It's plenty fast in air (1,125 ft/sec in standard conditions), faster in water (4.3 times as fast) and faster yet in solid objects such as a struck table top. The point is, sound takes a fixed amount of time to travel from place to place, except in space where it simply refuses to propagate—no molecules to compress, it turns out.

What if light, like sound, moved from place to place with a fixed velocity, an incredibly high velocity, to be sure, but a finite one—how would one go about measuring it? As early as 1629 a questing mind (Isaac Beeckman's) wondered if the flash of a cannon firing, caught and reflected in a mirror a mile or so away, might reveal some duration to an observer standing beside the cannon—a heartbeat, perhaps, from gunfire to flash's reflected return. Within a decade Galileo Galilei (yes, the man of early telescope fame and much else besides) shared with friends the outline of an experiment he claimed to have attempted. Instead of a cannon, Galileo placed an assistant at his side and another assistant a known distance away, both equipped with signal lanterns. Galileo nodded for his first assistant to open his lantern's shutter and commenced timing. Immediately he noted a return flash from his second assistant, but sadly the interval was no greater than the interval noted when the two assistants were practicing at close quarters. Galileo was astute enough to realize this did not mean that the speed of light was infinite, only that it was so fast it was impossible to measure over short distances with the equipment at hand.

Galileo's experiment was repeated in the interval between his death in 1642 and the year 1667 by the Florentine Accademia del Cimento. Again, the base leg was about a mile, and again, the results were inconclusive. The Academy had to agree with Galileo's earlier assessment—"If not instantaneous, [the speed of light] is extraordinarily rapid."

Galileo Galilei (1564-1642)

A longer base leg might have helped, say on the order of hundreds of thousands of miles, the distance to the moon and back. And had Galileo a lantern powerful enough to flash the moon when the moon was in its dark phase, and had that flash been detectible back on earth, and had Galileo's timepiece been capable of recording the 2.5 seconds the round trip actually takes, he would have established an important fact—the speed of light is finite. The accuracy of his measurement would have been laughable by today's standards. Old Galileo had only the roughest of ideas of the distance to the moon (not only because that distance is constantly changing) and the crudest of timepieces, perhaps a cumbersome water clock better suited to recording long durations than short.

That brings us to Ole Rømer, the Dane in Paris—his defenders would say the great Dane in Paris. The year is 1675, but first, pardon a brief discussion of the developing arts of navigation and cartography—

As the sixteenth century drew to a close, increasingly a nation's wealth and prestige were linked to the success of its maritime enterprise. Knowing where one was on the planet—and how one got there—could no longer be left to "dead reckoning" and the educated guess of a captain lost at sea. Half of the position equation, latitude (degrees north or south of the equator), presented no real problem. Simply measure the angle of the Pole Star, Polaris, above the horizon—a procedure requiring nightfall, a clear sky and a protractor-like gadget called a mariner's astrolabe—or judge the elevation of the sun at the moment it reaches zenith (local noon) and refer to published tables for seasonal adjustments. The other half of the equation, longitude (degrees east or west of a favored national observatory), was the deal breaker.

Longitude is essentially a matter of time and time-keeping, and here's how it works. A reference meridian is selected, the meridian of the Paris Observatory, say, or the one the Spanish preferred at El Hierro (Ferro) in the Canary Islands or that of the Royal Observatory in Greenwich, England. This "prime" meridian is designated 0° longitude. A traveling 24-hour timepiece is matched to Observatory Standard Time and carefully maintained. The cartographer or ship's navigator determines local noon by observing the sun at its zenith and noting the hour, the minute and the second displayed on his timepiece. In the unusual circumstance that the timepiece reads exactly 12:00:00, the observer knows he is somewhere on the prime meridian. A timepiece time before Observatory Noon would indicate the observer is displaced to the east; a timepiece time later than Observatory Noon would indicate he is displaced to the west.

Meridians of Longitude

The earth turns under the sun fifteen degrees per hour, a quarter of a degree per minute and so on. The time difference between observed local noon and recorded Observatory Standard Time is thus a measure of longitude. In Ole Rømer's day (and for some years thereafter) the accuracy of traveling timepieces was abysmal, good for a day or two perhaps—upon land—but of little use in long-distance voyaging.

When Philip III of Spain (aka Philip II of Portugal) ascended to his dual thrones in 1598 he was counseled to offer a princely sum to the man who came up with a practical means of determining longitude at sea. Galileo Galilei and a multitude of others weighed in with suggestions, none of them qualifying for the prize, though Galileo's idea was pursued to a somewhat successful conclusion by others.

Galileo is perhaps best known for his discovery and close observation of Jupiter's four largest moons, Ganymede, Callisto, Io and Europa. The fact that these heavenly bodies could be seen circling around another heavenly body gave great weight to the Copernican argument that the planets themselves circled the sun. Galileo championed this notion and suddenly found himself crosswise with the thought police of his day, the Roman Catholic Church. That surely didn't help him advance his time-keeping ideas with the Spanish government.

So regular were the orbits of Jupiter's moons, Galileo argued, that the orbiting system or any part of the system could be used by all upon Earth as a universal timepiece, a source of Absolute Time. Tables of satellite occultations could be produced by the world's observatories with the times of lunar disappearances and re-appearances predicted for a number of months or even years into the future. Back on Earth, the time difference between Absolute Time and local noon correlated with longitude. Problem solved.

Except it wasn't, not by a long shot. Taking up the case for measuring longitude where the Spanish government left off, the States General of the United Provinces of the Netherlands commissioned a new round of observations of the occultations of the moons of Jupiter. It seems the States General found Galileo's original observations wanting. Too, there was the nagging problem of timepiece accuracy. With encouragement from his government, the Dutch scientist, Christiaan Huygens, designed and built the first "natural period" pendulum clock following up a concept from that man-of-many-parts, Galileo. Huygens achieved an error rate—on land of course—of less than a minute a day, a major improvement over conventional measuring devices. Later refinements (taking into account suggestions from contemporary Ole Rømer) brought the error rate down to 10 seconds a day. Now we're talking!

Chris's Clock, Tick-Talk

Christiaan Huygens (1598-1695)

Galileo’s preliminary observations demonstrated the promise of predictable occultations of Jupiter’s moons though not the accuracy required to produce improvements in cartography and navigation. That promise stimulated further observations. Sicilian astronomer Giovanni Odierna gave it a go, but his tables, published in 1654, also failed to meet the test. By 1668, a more regular set of observations emerged at the Paris Observatory from the efforts of Gian Domenico Cassini and his staff. Cassini’s tabulations proved useful over short periods of time and over short distances—within the bounds of France itself. For the larger project, determining longitude at sea and abroad, theory was confounded. Some mysterious force or influence was inserting its hand and frustrating the most careful of observations.

Like fellow Italian Galileo, Cassini was a man of recognized talents, chief among them demonstrated prowess in the fields of mathematics, astronomy and engineering. In 1671 the French government invited Cassini to take charge of its newly founded national observatory. One year after assuming the directorship, with an eye to dimensioning the solar system itself, Cassini sent colleague Jean Richer halfway round the world to Cayenne, French Guiana. Simultaneous observations of the planet Mars, when compared using trigonometric processes, gave a distance to the planet, a first step in determining its orbit and the orbits of its fellow planets.

To understand the process involved, imagine a hole drilled through the Earth connecting observation platforms in Paris and Cayenne. Measure that distance and use it as the base leg of a triangle. Sight lines to Mars, one from Paris, the other from Cayenne, comprise the second and third legs of the triangle. Were Mars infinitely distant, as all stars appear to be, the sight lines would be parallel and the angles they make with the base leg would add up to 180°. Pull Mars close—arbitrarily close—and the sight lines toe in toward each other producing a combined angle measurably less than 180°. This is called parallax, which also rears its head in naval gunnery when the guns of a broadside, fore and aft, must be toed in to hit the same target abeam.

Giovanni Domenico Cassini (1625-1712)

Cassini deployed his observatory staff throughout France. Using occultations of Jupiter's moons as a universal time standard and the most accurate of clocks for measuring differences between that reference and local noon at each observation site, Cassini was able to chart the important features of his adopted country. The completed topographic survey of 1679 mapped a national territory sizably reduced from earlier depictions, eliciting from its bemused monarch, “This [Italian] astronomer had wrested more from my kingdom than I have won in all my wars."

While Cassini was observing and tabulating the occultations of Jupiter’s largest moons at his Paris Observatory, Ole Christensen Rømer was busy with similar observations at the Danish observatory of Uraniborg near Copenhagen. By comparing notes, the two observatories were able to determine their respective longitudes—Uraniborg conceded its position to be east of Paris at longitude 10°-23'-05" E. (Had the longitude of the Uraniborg observatory been adopted as the baseline standard, Paris would have found herself west by the same number. Today, by international agreement, both observatories find themselves east of the Royal Observatory at Greenwich, England—Paris by 2°-20'-14", Uraniborg by 12°-43'-19".)

It was Ole Rømer, by the way, who in 1675 suggested to Huygens the use of epicycloid-shaped gear teeth in his clocks, improving the accuracy of Huygens’ instruments significantly and allowing for greater precision in the determination of longitude.

Ole Rømer (1644-1710)

Romer was invited to Paris in 1672 to help resolve the still pressing issue of why the best occultation observations, when tabulated and projected into the future, were not useful for determining longitudes at distances beyond the European continent. Cassini had already noted certain “discrepancies” in measurements taken at his observatory between 1666 and 1668 and initially attributed them to the radical possibility that light had a finite speed and therefore took more or less time to cross the gap between Jupiter and Earth depending on the ever-changing distance between the two planets as they traversed their orbits. Assisting Cassini, Rømer noted that these discrepancies were regular rather than random, gradually growing larger as the Earth receded from its closest approach to Jupiter—inferior conjunction—and growing smaller as the Earth returned from its journey around the sun and again reached the position of inferior conjunction.

Cassini went so far as to suggest to the Academy of Sciences, in August of 1676, this very idea, that “discrepancies” and “irregularities” in his observations might be due to the finite speed of light. Rømer latched on to this argument and held fast despite his director’s eventual retreat. Perhaps the notion of light having a finite speed was too radical for the older man; apparently, the notion was too radical for Cassini’s contemporaries in the Academy as well.

One month later, in September of 1676, standing before the same august body and using his own Uraniborg observations, Rømer predicted that the 9 November occultation of Io would end not 5 hours, 25 minutes and 45 seconds into the day as Cassini’s tables indicated, but at 5:37:49, twelve minutes and four seconds later. Rømer’s argument and prediction was published and widely read. On the appointed day, observatories across Europe swung their telescopes in the direction of Jupiter looking for the first hint that Io had re-emerged from behind her parent planet. Cassini’s tables said re-emergence would take place at 5:27. 5:27 came and went and still no Io. Ten minutes passed, then twelve. At precisely 5 hours, 37 minutes and 49 seconds Paris time, as Danish astronomer Ole Rømer had very publicly predicted, Io re-emerged.

Rømer's Sketch of Io's Occlusion

Cassini gave many reasons for his own apparent failure and Rømer’s apparent success: Jupiter was too far distant for truly accurate observations, Jupiter’s atmosphere might have billowed up obstructing lines of sight to its moon . . . a playful God was having fun at his expense. That Rømer’s and his own earlier arguments regarding the speed of light might be valid was not to be entertained.

Using Rømer’s data, Huygens submitted an estimate of the speed of light. Other estimates followed, some reasonably close to today’s accepted value, some not so close. Much depended on the value an astronomer assigned to the diameter of Earth’s orbit and a correct interpretation of Rømer’s data. Rømer apparently did not assay his own estimate, but had he, using the then accepted value of the diameter of Earth’s orbit, Rømer might have suggested the figure 225,000 kilometers per second, not far from today’s 299,792 kps. For sticking to his guns, Ole is given credit for suspecting and determining that the speed of light was finite.

Cassini and his supporters held the day, however. Rømer gave up trying to convince them the speed of light was finite and retired to Denmark where he continued to serve the public in remarkable ways. It was 50 some years before Rømer’s arguments were finally accepted.

Historical Footnote—

Determination of longitude on the continent was one thing; its determination at sea and abroad, quite another. Europe’s seafaring powers bent to the task, often in secrecy, seeing in the eventual solution a grand strategic advantage, the equivalent, perhaps, of the invention of the telegraph more than a century in the future or the wireless “radio” some years beyond that. The greatest challenge was shipboard, where for identifying and tracking what could not be seen with the naked eye, refracting telescopes with larger fields of view were tried, and then reflecting telescopes with both wide field and greater magnification. To counter the roll and pitch of a moving platform, the British Admiralty tried placing some of her navigators in gimbaled seats. Had it not been for the vertical rise and fall that characterizes open sea—what sailors call heave—the Admiralty might have had something.

All the while great efforts were expended at improving portable timekeeping devices. Surpassing earlier prize offerings, the English, in the Longitude Act of 1714, offered—and here I use the words of Dava Sobel from her elegant and effervescent book, Longitude—“the highest bounty of all, naming a prize equal to a king’s ransom (several million dollars in today’s currency) for a ‘Practicable and Useful’ means of determining longitude [at sea]."

In 1761, clockmaker extraordinaire, John Harrison, submitted his latest design, designated simply H4, to the Board of Longitude, which immediately sent the mechanism on a trial run to Jamaica in the care of Harrison’s own son. The voyage took two months and a day. Over that span of time, H4 lost a trifling 5.1 seconds, an error rate of better than one part in a million. The marine chronometer was born and the vexing problem of determining longitude at sea was solved.

Harrison's H4 Chronometer

With a mind to putting Harrison’s design to a more severe test, Captain James Cook took with him on his second voyage to the South Seas a replica of H4 made by Larcum Kendall. That voyage was completed in 1775 at which time Cook reported that using a marine chronometer for the purposes of navigating and chart making was—in the great Captain's plain-spoken words—“entirely satisfactory."

Still, upon unmoving firmament, with unobstructed views of the ecliptic’s wide-ranging arc, the observation of one or more referenced celestial objects was the preferred method for establishing one’s longitude. Between the years of 1763 and 1767, Charles Mason and Jeremiah Dixon surveyed what was to become the notorious Mason-Dixon Line, the boundary between Pennsylvania and Maryland, between North and South. To determine exact longitudes along their route the men observed and reported occultations of Jupiter’s moons.

Charged by the British Admiralty, Captain Philip Parker King of HMS Adventure, between the years 1826 and 1830, and with HMS Beagle accompanying, did map the “coasts, harbours and channels” of the Magellan Strait, working always from baseline sites whose longitudes he determined with reference to the moons of Jupiter. A year later, Britain’s Hydrographic Department returned Captain Robert Fitzroy with the Beagle “to improve and complete the King’s charts.” On board the Beagle on this second voyage were some of the finest chronometers available and a bright young naturalist named Charles Darwin. Cartographic responsibilities took up in the city of Rio de Janerio. Darwin reports in his Beagle Diary, “In particular, the longitude of Rio de Janeiro, a starting point for these [South American] surveys, was in doubt due to discrepancies in [earlier] measurements, and an exact longitude was to be found, using calibrated chronometers, and the checking of these measurements through repeated astronomical observations."

Britain's Hydrographic Department was as much interested in the lay of North America’s northwest provinces as in the coastways of her sister continent to the south. While the captains of HMS Adventure and HMS Beagle were charting the Magellan Strait, Scottish botanical surveyor David Douglas took on the additional responsibility of surveying the heavens as seen from Fort Vancouver on the Columbia River. Celestial objects of interest included the Earth’s moon, Jupiter’s “Galilean” moons and the other visible planets. (The Hydrographic Department wanted to know the angular distances between the visible planets at a particular moment in local time, a unique set of observations and a way of cross-checking other of Douglas’s more than 600 logged observations.)

Fort Vancouver

(University of Washington Libraries)