Monday, July 12, 2010

Final Posting

Altogether, the previous contributions have about exhausted those topics in Stephenson's tetralogy, which had attracted my attention, while still satisfying the criteria given in my introduction. That is, they should be at an introductory level in sciences, mathematics, engineering, or technology. There are other topics which don't fit the level or the area, or for which I just wanted to add something, to go along with Stephenson's correct coverage.

As a first example, I wonder about the economics of the ship Minerva, which was inspired by the classic 'Dutch East Indiaman', as a heavily armed merchant ship. In particular, I cannot believe that Minerva's crew was just the right size at 105. That was the total number of people on board when Minerva arrived at Manila (page 700 of the Confusion), after her trip to Japan. I can think of at least one economic reason why the crew should be smaller than that. I can also think of other economic reasons, probably more cogent, why the crew should be larger than that. I doubt that I will ever know the true economics of long-distance oceanic trade, at the beginning of the eighteenth century.

A second example involves Minerva's visit to Japan, particularly the resonance between the oscillations of mercury in partially filled flasks, and the "characteristic waves" at the harbor entrance (pages 686-689 of the Confusion). Proper consideration of this phenomenon demands knowledge of hydrodynamics, which is certainly not a topic of introductory physics. We can start to see where problems may lie, by a more elementary analysis.

Enoch Root had first noted a resonance between the waves and the oil "lanthorn", hanging from the ceiling of his cabin. At a guess, the lantern may have been equivalent to a simple pendulum about 0.4 meter long, so that the natural frequency of it (and of the waves) was about 0.8 hertz. Enoch 'detuned' it by shortening its chain "a few inches".

A flask is described as "an egg of fired clay: .., stoppered at one end by a wooden bung." I have a mental image of it as resembling an American football. If the inside of each flask was indeed the same size as a football, it would hold about 60 kilograms of mercury, when full. The filling fraction was not specified, but 1/2 or 2/3 might be reasonable. Jack Shaftoe was able to hold a partially filled flask "at arm's length", "though it took the strength of both arms." It is difficult to know the physical capabilities of a former galley slave, then about a decade out of training. That is an awkward position, especially since he "was too tall to stand upright in the cabin." My guess that it would be effectively impossible to perform that feat, if the actual flask was as much as twenty percent larger in each dimension, than a football.

Thus the first hydrodynamics problem is: "Can a partially filled flask, of reasonable size and shape, have a fundamental mode at 0.8 hertz?" The second hydrodynamics problem is: "What must be the boundary conditions of the harbor shoreline and seabed, and of the wind across the water surface, so that a characteristic wave at 0.8 hertz is strongly excited?" Note that this is a higher frequency than the roll of the ship itself, which could also possibly be in resonance with a different characteristic wave.

The solution proposed by Enoch would indeed solve the problem of a dangerous resonance. When each flask was made "brim-full", it was effectively not an oscillating system at all. A much quicker and easier solution was also available. If each flask, as received, was 'stood on end' and wedged into that position with straw, it would still be an oscillatory system, but its natural frequency would be quite different from 0.8 hertz. Only if that frequency also matched a different characteristic wave at the harbor entrance, could a resonance occur.

My final comment is about the analemma, which is a plot of the 'equation of time' (difference between mean solar time and apparent solar time), versus declination of the sun, over the course of a year. Stephenson made it a sufficiently important topic in Anathem, that a multiple-exposure version of it appears on the cover or dust jacket of the book. In its own way, Anathem is as much about the history of science, albeit on a different planet, as are the books of his earlier tetralogy. I am such a technology freak, that I had always considered that the analemma depended on knowing mean time, as measured by a sufficiently good clock. This is because I was never exposed to a proper course in History of Astronomy, with the mathematics. I now know that Ptolemy understood the difference between mean solar time and apparent solar time in the second century AD, based on earlier observations. I have still not personally read his Almagest, wherein he apparently did not consider mean solar time to be very important.

Thus it is quite reasonable, that on the parallel planet Arbre of Anathem, an analemma (large enough to be visible from space) was laid out at the temple of Orithena, in about the year - 2850 (all dates from pages xiv-xvii). It, along with the temple, was buried by the volcanic eruption of -2621. The laws of dynamics, which allowed the construction of good clocks, were discovered sometime after - 500. (Note that the passage of 2350 years on Arbre, between the discovery of the analemma and the invention of clocks, is comparable to the passage of about 1550 years between the same events on Earth.) The volcanic ash which covered the analemma on Arbre, had been dug away fairly recently before the events of Anathem (+ 3689). Thus, the ancient analemma was about 6500 years old, when Fraa Erasmas saw its image (page 405), and the analemma itself (page 526).

The diagram of the analemma on page 405 is notable, because of its exact bilateral symmetry. This implies that the perihelion and aphelion exactly coincide with the solstices. (I use the terms appropriate for Earth, although we are considering the orbit of Arbre.) The analemma for Earth is only approximately symmetric, because those orbital points are about 12 degrees apart, rather than exactly coincident. The analemma on the cover/dust jacket is also bilaterally symmetric, but its creation, as multiple images, requires technology at the level of 20th-century Earth.

To me, the most interesting thing about the analemma, is that it changes its shape with time. This change is driven mainly by the 'precession of the equinoxes', which is the manifestation of the precession of Earth's axis of rotation, in response to tidal torques exerted on Earth by Sun and Moon. There is also a slower change of Earth's elliptical orbit, relative to the fixed stars, caused by the varying gravitational forces exerted on Earth by the other planets. I don't know of any 'fossil' analemma on Earth, as much as 500 years old. That is not a long enough time for the change of shape of the analemma to be grossly obvious.

However, the situation on Arbre is quite different, because of the 6500-year age of the original analemma. In fact, that age is almost exactly 1/4 of the length of the precession cycle on Earth. Earth and Arbre are such close homologues, according to Stephenson, that it should also be 1/4 of the precession cycle on Arbre. In that case, at the time of the story, the perihelion and aphelion should exactly coincide with the equinoxes. The analemma would have a very different symmetry. The two loops of the 'figure eight' would be identical, so that the analemma is invariant under a rotation by 180 degrees around its crossing point. However, the analemma would seem to 'lean over', rather than standing erect along the meridian.

This is undoubtedly more that most readers would want to know. Even I would have been quite content with a mere half sentence: "... , different than the analemma that Fraa Erasmas knew."

Well, that's it. I will consider other topics from Neal Stephenson's work to appear here, only if someone asks me to.


Wednesday, June 16, 2010

How Much Solomonic Gold Was There?

One of the featured items in the continuity among these four novels, is the "heavy gold" or "Solomonic gold". Some of it ended up as thousands of perforated square cards, which were last seen about to be deposited in the vault at Kinakuta (page 1074 of Cryptonomicon). Its nature was not revealed there, but only gradually through the books of The Baroque Cycle. All of the mentions of large quantities of the gold appear in the Confusion.

The first explicit appearance of the gold was on the Viceroy's brig (pages 192 ff). It had just been captured by the "Cabal" of galley slaves, as the first step in their "Plan" to win their freedom. There was not enough information given to allow any reasonable estimate of the amount. We are told only that it was so massive that "Most of their previous cargo and ballast had been thrown overboard ...."

The next appearance was after the galleot arrived at Cairo (pages 227 ff). The augmented Cabal (now fifteen), and Nyazi's clansmen (perhaps about three dozen): "... all got busy pulling the gold-crates out of the galleot and putting them on the camels, which took no more than half an hour." As they travelled through Cairo: "Nyazi's caravan, three dozen horses and camels strong, armed to the teeth, laden with tons of gold, was nothing here." This does provide information for an estimate.

There are conflicting statements online about the capabilities of pack animals, and almost none of them cite an original source. An intermediate value appears at . "A fully grown camel can weigh up to 700kg/1542lbs" "A camel can carry as much as 450kg/990lbs, but a usual and more comfortable cargo weight is 150kgs/330lbs." This maximum load seems rather large at about 64% of body weight, but it may be possible for the short trip involved in the story.

I don't know where the horses came from, but they may have been carrying gold too. Perhaps the most authoritative source for their capacity was President (General) U. S. Grant, in his Personal Memoirs. He asserted that a pack horse could carry 300 pounds across rough terrain. This was undoubtedly a conservative value. Perhaps for this short trip along the streets of Cairo, a horse could manage 500 pounds. If we assume 4 horses (definitely plural) and 32 camels, their total load was about 34,000 pounds. However, that load could not have been all gold. There had to be packsaddles or frames, padding, ropes, etc., as well as the crates. This 'tare' may have been 15% or more, so that the net mass of gold was less than 13 tonnes.

All of this gold was hidden among the haystacks in the stables of the caravanserai. At one point during the battle and fire, Jack "...estimated that somewhat more than half of the gold had been recovered." However, his companions ".., knew where every last bar was hid, and were making sure that none were missing." After heavy fighting, "... they were able to drive away from the stables--a cyclone of flame now--with four of the original six gold-carts." Only those four carts reached the boat. After the boat escaped down the Nile, it entered "... the marshy expanses to the east. At the end of that day, they made rendezvous with a small caravan of Nyazi's people, and there a share of the gold was loaded onto their camels."

It is difficult to know how much that share may have been. Several of the original Cabal were dead or missing, and ten men remained on the boat, three of them recently added. Nyazi certainly deserved a full share, because without the help of his clansmen at the caravanserai, much less of the gold could have been brought out. There must have been less than 8 tonnes of gold still on the boat, as it entered the Red Sea.

The capture of the boat, by the Malabar pirates of Queen Kottakkal, was described only in conversations later (pages 454-456). There was no description of the transfer of the gold to her treasury, nor of the negotiations by which she was convinced to invest in the Cabal's Plan, by helping them to build the ship Minerva.

The final appearance of all the gold in one place was as the sheathing on the hull of Minerva, as she departed from Qwghlm (page 796). Stephenson gave us no explicit information about the dimensions of Minerva, other than that she carried 44 guns. Perhaps we can compare her to another well known ship of literature. That was Lydia, the 36-gun frigate commanded by Horatio Hornblower, in the first-written story of that series by Cecil Scott Forester. Forester actually didn't do any better with dimensions than Stephenson, but the omission was corrected by C. Northcote Parkinson, in his novel The Life and Times of Horatio Hornblower (Little, Brown and Company, 1970, page 124). "The Lydia was a middle-sized Fifth Rate built at Woolwich in 1796 to the design of Sir William Rule. She measured 951 tons and 143 feet long, mounted twenty-six 18-pounders on the gun-deck, eight 9-pounders on the quarterdeck, and two 12-pounders on the forecastle. She was established for a crew of 274 men ...."

Minerva's original guns were described inadequately as "... forty-four large naval cannons, preferably of most modern and finest sort, ..." (page 578). I am sure that Jan Vroom specified exactly the caliber of the guns for which he was designing Minerva. The hull had to be strong enough to manage their recoil, but not stronger than necessary. The guns on Minerva would have to be spaced about 20% closer together than those on Lydia, in order to be accommodated on the same length of deck. In that case, they would be smaller than 18-pounders, and could not appropriately be called "large naval cannons".

Minerva under construction (page 577) was described as "Being so long and so rakish, she had to be narrow--quite a bit of volume was sacrificed for that." "Such a ship can pay for her upkeep only if she is hauling items of small bulk and great value." On the other hand, Lydia was strictly a warship. She had to carry only those things necessary for her to fight: crew, gunpowder, shot, food, water, firewood, and materials to make repairs. Minerva had to carry all of that, plus a cargo, however small. Thus Minerva seems to have been both longer and wider than Lydia. In addition, Minerva was notably V-shaped near the keel (page 576). Altogether, the area of Lydia's sheathing (copper), would be a lower limit for the area of Minerva's sheathing.

From Lydia's length and displacement, one can estimate the average cross-sectional area of her hull, below the waterline, as about 20 square meters. The shortest curve which can enclose that area, between vertical sides of the hull above water, is a semicircle (flat near the keel). Therefore, the smallest possible wetted area for the hull was about 500 square meters. Using 1/8 inch as the thickness of the sheathing (page 145 of The System of the World), and 19.3 as the specific gravity of the gold, there must have been at least 30 tonnes of gold on Minerva's hull.

This is much larger than the amount of Solomonic gold which the Cabal possessed, before it was captured by Queen Kottakkal's pirates. She must have supplied the extra gold from her own treasury, but that would have been ordinary gold. Thus one is faced with the question, were the two types of gold mixed together ("confused") before being hammered to the desired thickness, or were some of the sheathing plates made of one type of gold, and some of the other?

Saturday, April 17, 2010

Stealing and Decrypting The Arethusa Intercepts

The messages, which were encrypted under the Arethusa algorithm, interconnect more of the principal characters in Cryptonomicon, than almost any other factor in the story. The cards first appear (pages 813, 814) in a trunk that had belonged to Lawrence Waterhouse. His grandson Randy Waterhouse digs through it, "to reveal a stack of bricks, neatly wrapped in paper which has gone golden with age, each consisting of a short stack of ETC cards, and each labelled ARETHUSA INTERCEPTS with a date from 1944 or '45." Those cards are read out on one of Chester's antique ETC card readers, with the help of Chester's "card man", and transferred to a floppy disk, and ultimately to Randy's laptop computer.

While flying to Kinakuta a few days later (pages 876-882), Randy has a long telephone conversation with Enoch Root, who had been told by the "card man" about Randy's Arethusa cards. In the conversation, Enoch tells the history of the study of Arethusa at NSA, which had shown that the cards at NSA had been generated by a particular random-number generator, and were not message intercepts at all. After arrival at Kinakuta, Randy discovers that the Arethusa files on his hard drive are different from those at NSA (pages 906, 907).

While in prison in Manila, with Enoch Root in the next cell (pages 981 ff), Randy learns enough cryptography to break the Arethusa encryptions, without ever displaying the files on the screen of his computer. Along the way, he uses the suspected relation of Arethusa to the algorithm Azure, which used zeta functions in the generation of daily one-time pads (pages 1005, 1006). His success is described (page 1025) as: "He comes up ... with A(x) = K, such that for any given date x, he can figure out what K, the keystream for that day would be; ... it worked in each case." The most important message, by far (page 1028), includes: "THE PRIMARY IS CODE NAMED GOLGOTHA. COORDINATES OF THE MAIN DRIFT ARE AS FOLLOWS: LATITUDE NORTH (etc.)"

All of the other mentions of Arethusa occur later in Cryptonicon, but in flashbacks to World War II. On pages 1028-1034, Lawrence Waterhouse breaks the Azure/Pufferfish algorithm, which is a zeta function with the date as one of its input data. [I particularly admire Stephenson's choice of date (6 August 1945), for which the slaves had been evaluating the algorithm.] On pages 1094-1097, Lawrence and Rudy von Hacklheber discuss both Arethusa and Azure/Pufferfish, which differ mainly in that Arethusa does not use the date. On one occasion, for "the long message", it used "his military serial number" from "Shaftoe's headstone". Lawrence also had many probable words in the message, to use as a crib in selecting the proper zeta-function algorithm.

This relationship between Azure/Pufferfish and Arethusa requires that we go back to the previous mentions of Azure. It first appeared as the handwritten pages, which Lawrence found when he opened the safe from U-553 (pages 375-382). They are described as: "The pages are ruled with faint horizontal and vertical lines, dividing each one into a grid, and the grids are filled in with hand-printed letters in groups of five." Sometime later, Lawrence arrives at Bletchley Park, and discusses the pages with Alan Turing (pages 418-429). The important technical material (page 424) includes Lawrence's statements: "But this message was different--it used thirty-two characters--a power of two--meaning that each character had a unique binary representation, five binary digits long." "So I converted each letter into a number between one and thirty-two, using the Baudot code." "If the first letter is R, and its Baudot code is 01011, and the second letter is F, and its code is 10111, ..." At the bottom of the page is a footnote: "Baudot code is what teletypes use. Each of the 32 characters in the teletype alphabet has a unique number assigned to it. This number can be represented as a five-digit binary number, that is, five ones or zeroes, or (more useful) five holes, or absences of holes, across a strip of paper tape. ..." Alan recognizes that the original pages were written by Rudolph von Hacklheber. Eventually, Alan convinces Lawrence that the pages make up a one-time pad, and not a message.

There is no clear statement that Arethusa uses the same message structure as Azure. The fake Arethusa messages, as quoted by Enoch, started with five-letter groups, but he did not go on to say that other characters also appeared. However, because Azure is effectively a simplification of Arethusa, this seems to be a reasonable assumption.

On pages 1114-1119, Lawrence decides to steal the Arethusa intercepts, and sets about to do so. He justifies this by the danger that he and Goto Dengo would face, if they were ever decrypted in the future. He intends to conceal the theft by replacing the cards with fakes: "It'll look like any other encrypted message, ..." Using the "Project X" encrypted telephone circuit, Alan Turing gives Lawrence the parameters of a particularly effective zeta function, for generating pseudo-random numbers. Lawrence then prepares his digital computer to generate the replacement cards. "In order to make the computer execute Alan's random number function, he even has to design a new circuit board on the fly, and solder it together."

Before he can start the computer, the last two Arethusa intercept sheets arrive, and he punches their information onto cards. "Waterhouse goes to the oven and takes out a brick of hot, blank ETC cards. He has learned that he must keep the cards hot, or else they wiil soak up the tropical humidity and jam the machinery; ..." "He takes those cards out of the puncher's output tray and places them neatly in the box with the cards containing all of the previous Arethusa messages. He then takes the entire contents of this box--a brick of messages about a foot thick--and puts them in his attache case."

The production of the replacement cards is described as: "He dumps a foot-thick stack of hot blank cards into the input hopper of the card punch." "Then he starts the program he has written, the one that generates random numbers according to Turing's function. Lights flash, and the card reader whirrs, as the program is loaded into the computer's RAM. Then it pauses, waiting for input: ..." "Waterhouse thinks about it for a moment, and then types in COMSTOCK. The card punch rumbles into action. The stack of blanks begins to get shorter. Punched cards skitter into the output tray." "He puts this stack of freshly punched cards into the box labeled ARETHUSA INTERCEPTS, and puts it back in its place on the shelf." He then burns all of the original intercept sheets.

Unfortunately, this set of events constitutes perhaps the worst breakdown of continuity in these four novels. The description of Randy's decryption of the intercepts would apply to Azure, but not to Arethusa. In particular, the long message, with the location of Golgotha, was encrypted using Bobby Shaftoe's USMC service number, not a date shortly after his funeral. Are we to believe the 1 chance in 100,000,000 that they exactly matched? The description of the cards that went into Lawrence's attache case does not match the description of the cards that came out of his trunk fifty years later. How could Lawrence have reconstructed the dates of the messages, to be written on the wrappers? He had burned the intercept sheets, which undoubtedly did include the dates.

In the quotation: "It'll look like any other encrypted message," the pronoun "it" suffers from 'indefinite antecedent'. If "it" refers to a single card, then any one individual fake card may indeed look like rather like an individual real card, in that they have the same format. However, if "it" refers to one complete message, then the description of the production of the fakes is inadequate to guarantee the truth of the quotation.

A real message consisted of a particular number of alphabetical characters. It may or may not have always ended with a group of the characteristic size (i.e., 5), depending upon how Rudy constructed his algorithm. It very probably did not consist of exactly enough groups to fill an integer number of ETC cards. Depending upon the format used, one card might hold 12 to 16 five-character groups. Rudy's algorithm was so labor-intensive, that the users would have been reluctant even to encrypt meaningless 'filler characters' (e.g., ZZZ), to fill out the last group. They would certainly have balked at encrypting 'filler groups'. Thus, the 'average' real message, transcribed to cards, would end with a last card that was about half blank. The 'average' fake message must end the same way, and the set of all fake messages should have last cards with a reasonable distribution of numbers of characters or groups. For the best apparent equivalence, each fake message should have exactly the same length as the corresponding real message.

Depending upon programming style, each card in a real message should have included a sequence number, as well as the characters of the message itself. Again, depending upon programming style, there could be a 'start' card in front of the first card of the message, or an 'end' card after the last card, or both. Such a 'control' card could well include other information from the intercept sheet, such as date, time, origin, length of message, etc., plus 'identifiers' so that the program would not try to treat it as a character card. Each fake message must have the same structure as a real message, and whatever that takes should be mentioned.

If "it" in that quotation means the entire set of fake messages, then there is yet another possible difference from the set of real messages. The real messages were intercepted over a period of months, and presumably each was trancribed to cards immediately. Thus each message had been exposed to high humidity for a different length of time, after being heated in the oven, and could have turned a slightly different color. The fake messages were all produced at the same time, and would all have been the same color.

The overall control of Lawrence's primitive computer seemed to follow the prescription: "Leave it turned on until it runs out of cards." That may have been appropriate for the underlying fixed-program machinery, which was intended to do simple functions such as sorting, tabulating, etc. However, my exposure (in 1956) to a first-generation, main-frame, digital computer, convinced me that such a prescription is quite inappropriate for a stored-program computer. The first important thing I learned in the introductory course, was to make sure that the computer would stop in the right places, even before I learned how to make it do anything useful in between.

In that era, the programs had to be written in the 'native language' of the computer. In particular, every transfer of quantities between the arithmetic registers and the input, output, or RAM, had to be specified by the programmer. For that particular computer, the input and output were on five-row paper teletype tape. During the hours when students were scheduled to use the computer, there would typically be 3 to 20 people waiting in line with their tapes, all able to see the operator's station. If your program tape fell to the floor instead of stopping in the tape reader, it was rather likely that your program would not work. If your data tape also fell to the floor, it was almost certain that your program would not work. On the other hand, if your program was running, but got into an endless loop, you would exceed the time allowed for each student program. (That was only a few minutes, enough for a few thousand arithmetic operations.) The operator would have to press the button to stop the computer, thereby guaranteeing that your program failed. All of the people in line would be either smirking, snickering out loud, or standing with eyes closed, thinking, "Please, God, don't let that happen to me!"

This experience with teletype allowed me to see that Stephenson's descriptions on page 424 (quoted above) include a remarkable number of misstatements. The most fundamental is that five-bit numbers run from 00000 (zero) to 11111 (thirty-one), and not from one to thirty-two. Teletype can handle only 26 encoded letters, and the other six codes are necessarily non-printing. Punctuation characters and decimal digits are produced by mechanically shifting the teleprinter from LETTERS mode to FIGURES mode. [See Wikipedia for one standard assignment of numerical codes: International Telegraphy Alphabet Number 2 (ITA2).] That complete "teletype alphabet" has 50 printable characters, rather than 32, but they are all encoded with only 26 binary numbers. That Wikipedia article does not give the original Baudot coding, and anything else should be called only "Baudot-like", rather than "Baudot code".

Moreover, the encoding can be changed, in the tape perforators and the teleprinters, at the convenience of the user. I did not realize, until doing this analysis, that the encoding, used at that computer center in the 1950s and '60s, was very different from ITA2. After all the intervening years, I can remember only 19 character codes (0 - 9, +, -, F, J, K, L, N, S, and NUL), and not a single one of them matched ITA2. Stephenson actually hinted at this, when he suggested that F has code 10111. In ITA2 it is 01101, and as the sixth letter in alphabetic order it could be thought of as 00110. I remember it to be 01110. Thus, there is no such thing as a "unique binary representation", even for the alphabet. Six more characters would have to be added 'by hand', because they cannot be transmitted in a single usable teletype mode, along with the alphabet. Even after 6 particular characters have been chosen from the 14 available in ITA2 (3003 distinct combinations), they can be paired up with the six available codes in 720 different ways.

I am not particularly concerned that Lawrence P. Waterhouse is credited here with inventing the stored-program computer, which is usually ascribed to John von Neumann, after World War II. I will accept it as an earlier independent invention, which Neal Stephenson was astute enough to notice. I am more concerned that Lawrence tried to do too many different things, all at the same time. It was bad enough that he had to solder together a new circuit he had designed. It is highly unlikely that he had designed a printed-circuit board ("circuit board" for short). They did exist in 1945, typically for mass production of electronic systems that had to function under mechanically stressful conditions. However, I can't believe that a facility to make them was available in the war-time Philippines. Instead, Lawrence would have built his 'one-off' circuit in and on a sheet-metal chassis. That was the way that I built amateur-radio equipment a few years later.

Sunday, February 21, 2010

Newcomen's Steam Engine

The last technology mentioned in The Baroque Cycle is described on pages 883 ff of The System of the World. Daniel Waterhouse is visiting the steam-powered pump, at a mine in Cornwall, in the winter of 1714-15. He sees and contemplates several features of the installation, which must be compared to the true nature of an engine working on the Newcomen cycle.

“Plenty of steam leaks out around it [the seal at the edge of the piston], but most stays where it belongs.” “This platform is dripping wet, and yet it’s warm, for the used steam exhaled by the Engine drifts round it and condenses on the planks.” “The level ground below the Engine is pocked all around, with wreckage of Newcomen’s boilers.” “He wonders if these Cornish men have the faintest idea that they are sitting around an explosive device.” “..., the seams and rivet lines joining one curved plate to the next radiate from the top center just like meridians of Longitude spreading from the North Pole.” “Below is a raging fire, and within is steam at a pressure that would blow Daniel to Kingdom Come (just like Drake) if a rivet were to give way.” “The steam is piped off to raise water, ...”

My old Encyclopaedia Britannica includes a good diagram of a Newcomen engine, but there is no adequate explanation of its operation. The following description is based mainly on what I remember, from the course where I learned about this cycle, in 1948.

A Newcomen engine does not exhale used steam. The steam enters the cylinder during the return stroke, as the piston rises and the pump rod goes down. All of that steam is supposed to be condensed inside the cylinder during the power stroke, by a spray of cold water into the cylinder. That condensation leaves a partial vacuum in the cylinder. The pressure of the ambient air on the top of the piston pushes it down, so the pump rod goes up, doing useful work by lifting a quantity of ground water. The water in the cylinder, consisting of the condensed steam and the sprayed-in water, is released during the next return stroke. The only steam which comes out of a Newcomen engine is leakage.

Because the main function of the steam is to keep air out of the cylinder, it need not be at high pressure. As I recall, a gauge pressure of 3 or 5 pounds per square inch (psi), i.e., 1/5 or 1/3 of atmospheric pressure, would be plenty. This may or may not qualify the engine as a dangerously explosive device.

The description of a succession of failed boilers suggests to me that Stephenson may have presented Newcomen as engaged in a program of increasing the boiler pressure. The Newcomen cycle is so incredibly inefficient, that the first-order effect of an increased pressure is a reduction of the efficiency. One must burn more coal to increase the temperature and pressure of the steam, but all of that added heat energy is thrown away during the condensation in the cylinder. The concept of efficiency was poorly understood at that time, and an experimenter may have felt that a possible increase in the speed of the engine were a good thing. (I can use the subjunctive mood, too.)

One cannot see the top center of the boiler on a typical Newcomen engine, because the vertical cylinder is immediately above the boiler, with a valve between. There is no pipe, to carry the steam away from the boiler.

The combination of features, use of steam at low pressure and condensation of all the steam inside the apparatus, also was employed in the engine developed by James Watt after about 1769. (See Wikipedia or Encyclopaedia Britannica for this history.) It was far more efficient than Newcomen’s engine, because the condensation took place in an external condenser, thereby allowing the cylinder to stay hot.

Altogether, Stephenson’s description comes closest to that for an 'expansion' engine, in which the full boiler pressure is admitted to the cylinder for only a fraction of the power stroke. Such an engine is more efficient if it has an external condenser, to allow expansion to below atmospheric pressure, and to recycle the feed water. However, it can be built to operate in an open cycle, in which case it does exhale used steam, after the steam has expanded to drive the piston during the remainder of the power stroke. Such an engine does operate at high pressure, perhaps hundreds of psi. Increasing the boiler pressure does directly increase the efficiency. However, that type of engine was not developed until after 1800, when Watt’s patents expired. The most familiar example (with quite different mechanical arrangements), was probably the steam locomotive, as seen in old movies.

Friday, February 19, 2010

Newtonian Reflecting Telescope

This instrument is mentioned several times by Stephenson, probably because Isaac Newton, who invented it, is a central character throughout The Baroque Cycle. The first appearance is at pages 171 ff in Quicksilver. Daniel Waterhouse has the presentation telescope out in public in 1670, before it was donated to the Royal Society. According to my Encyclopaedia Britannica, that donation actually happened in January 1672, not in August 1670. Stephenson's description matches fairly well to photographs of the real thing (as in Wikipedia), except for another, more serious, anachronism.

The concave mirror (“dish”) at the closed end of the tube is said to be made of silvered glass. The process of chemical deposition of silver onto glass was not invented until 1835. (See e.g., Wikipedia.) Newton actually used speculum metal, an alloy mostly of copper and tin. It had long been used for making hand mirrors. The major deficiencies of speculum metal are, that it reflects only a fraction of the incident light when newly polished, and that it tarnishes rapidly, with degradation of the reflectivity. Such a mirror must be repolished frequently, with the danger of damaging the desired shape ('figure') of the surface. A silver-on-glass mirror is far superior, firstly because the silver has a higher initial reflectivity, and tarnishes more slowly. Secondly, a tarnished silver coating can be dissolved off the glass, and replaced by a fresh coat, without any effect on the figure of the glass.

The use of speculum metal for telescope mirrors extended beyond the discovery of chemical silvering, such as in Lord Rosse's six-foot diameter "Leviathan", finished in 1845. (See e.g., Wikipedia.) If Newton had made a silver-on-glass mirror in 1670, the switch-over would have happened many decades earlier.

At this first appearance, and in two other places, there are descriptions of the use of a Newtonian reflector for observing events on the surface of the Earth. There are several things to be considered in evaluating the utility of a Newtonian reflector as a terrestrial telescope.

The first of these is the nature of the image formed by the mirrors, which is to be examined with the ocular lens. I first looked through a Newtonian telescope in 1946, but this is the first time I have fully analyzed the image. The concave primary mirror forms a real image, which is both inverted across the axis of the mirror, and 'flipped' by the reflection. This could be verified by examination of the primary image in an 'off axis' (Herschelian) telescope. In that configuration, the primary mirror is tipped to one side, so that the observer’s head does not block the passage of light from the object to the mirror. The flipping of the image is revealed, for example, in that the sweep-second hand of a clock would seem to be moving counterclockwise, and lettering would be reversed.

In a Newtonian telescope, a plane diagonal mirror is inserted between the primary mirror and its focal point, thereby directing the light across the tube and out its side. This second reflection flips the image again. This real image is thus direct, in that lettering would appear normal, and a sweep-second hand would move clockwise. However, the orientation of this image must be evaluated by careful ray tracing through a telescope, as it is likely to be used.

The small presentation telescope, on its ball-and-socket mount, could indeed be used on a tabletop, especially if one wished to be as discrete as possible out in public. (That is how Waterhouse set it up, in Stephenson’s description, although there are few details given.) With the tube nearly horizontal and pointed at the object of interest, the most convenient orientation of the ocular (eyepiece) would be upward. For a person seated behind the telescope, bending forward to look down into it, the real image would be inverted, or equivalently, rotated through 180 degrees. For a person seated on either side of the telescope, bending forward to look down into it without moving it, the real image would seem to be rotated through 90 degrees. Any horizontal line, such as the edge of a step, would appear to be vertical. For a person seated at the open end of the tube, bending forward to look down into it, the real image would appear to be erect. The problem, of course, is that the person would have to contort his body awkwardly, in order not to block the light from entering the tube.

Lord Bolingbroke’s telescope [pages 539 ff in The System of the World] appears to have been essentially the same as Newton’s presentation telescope, but perhaps resting on a somewhat higher surface. He was apparently standing, “hunched over the eyepiece, twiddling the tube ... this way and that.” The ball-and-socket mount would require exerting enough force to overcome the friction, in order to adjust the angle of the tube below the horizontal. Lord Ravenscar noticed “the tiny lens of the eyepiece” as he stepped up to the telescope. That description certainly applies to Newton’s telescope.

One wonders where Bolingbroke got this telescope, so that he had it in July 1714. Newtonian telescopes were apparently not commercially available in that era, probably because Newton’s original model was not a good instrument. It had a spherical surface on the concave mirror, because Newton was unable to produce a paraboloidal surface, which would focus parallel rays to a point. Especially with the small f-ratio of about 3.1, spherical aberration would produce large fuzzy images of stars, or of each point in an extended object.

In 1721, John Hadley presented a Newtonian telescope of his own construction to the Royal Society. He had developed a method of producing and testing a paraboloidal surface on the primary mirror. His telescope had image quality comparable to the best refracting telescopes of the time. The renewed interest in the reflecting telescope led a London optical firm to start making them thereafter. (See e.g., .)

The “big Newtonian reflector” on Huygens’s roof in December 1687 [pages 755 ff in Quicksilver], would have been too long for someone to bend over its closed end and see into the eyepiece. Huygens could have made it himself; he had already made refracting telescopes. It was in some form of alt-azimuth mount, on a pedestal or tripod. At any rate, it was easy to “sweep ... the instrument back and forth.” The altitude axis might have been at about shoulder height. With the tube nearly horizontal, the most convenient orientation of the ocular would be horizontal, to one side or the other. For a person standing on either side, looking horizontally into the telescope, the real image would be inverted. Thus, the Newtonian reflector typically presents an inverted, direct, real image to the person using it conveniently, just as the objective lens of a refracting telescope does.

The ocular lens, to produce a virtual image from that real image, can be chosen using the same criteria, irrespective of the nature of the device which formed the real image. A true natural philosopher (Newton, Waterhouse, or Huygens) would use a positive (Keplerian) ocular lens for looking at astronomical objects. It is placed beyond the real image, and gives a magnified virtual image of it, without changing its orientation. Such a user would simply ignore the fact that the final virtual image is inverted. This is no problem with astronomical objects, because there is no preconception as to 'which way is up', for an enlarged image of a planet, nebula, or star cluster.

That is not the case for a useful terrestrial telescope, with which one expects to see people, things, and activities 'right side up'. In the present era, there are three ways to produce an erect final virtual image. The simplest is to use a negative (Galilean) ocular lens, placed between the real image and the device which formed it. The main disadvantage is a restricted field of view, plus the impossibility of superimposing cross-hairs onto the image.

Another way is to put a prism system into the path of the light converging toward the real image, to erect it by multiple reflections. This is fine for prism binoculars, but in a reflecting telescope it would block and/or scatter some of the light enroute between the object and the primary mirror. The first such prism system was invented in 1851 ( ), and so was unavailable for Stephenson's characters.

The third way is to insert a positive lens between the primary real image and the positive ocular lens. This 'erecting lens' produces an erect real image, which in turn produces an erect virtual image. This can have about the same width of field as the simpler Keplerian telescope, but it has several disadvantages for astronomical use. It increases the light path by at least four times the focal length of the erecting lens, which must be accomodated by additional tubing. For a Newtonian telescope, the additional tubing would 'stick out' from the side of the main tube. The brightness of the virtual image would be decreased, by reflection of some of the light at the added surfaces, and by absorbtion of light in the added glass.

Even worse, in the era of these stories, the only available lenses were single pieces of glass. Each lens would contribute its own chromatic aberration to the final virtual image. Newton's main reason for developing the reflecting telescope was to eliminate chromatic aberration in the primary image. Achromatic lenses, which reduce or eliminate chromatic aberration by using two different types of glass, were not invented until 1729 (Encyclopaedia Britannica).

Thus, the only reasonable way to produce an erect virtual image in a Newtonian telescope, available to Stephenson's characters, was to replace the usual positive ocular lens with a negative lens. It is intriguing that Newton's method of adjusting the focus, by changing the length of the main tube, would make this relatively easy. A person who regularly spied on his neighbors (Bolingbroke), might keep the negative lens installed at all times.

With this background established, we are ready to analyze Stephenson’s descriptions. Initially, Waterhouse was using Newton’s presentation telescope, which almost certainly contained a Keplerian ocular lens. (I have not found any reference which states that, but every diagram I have seen for the instrument shows a biconvex ocular lens.) It obviously did not have an erecting system between the eyepiece and the diagonal mirror. Thus every thing or person Waterhouse saw would have been inverted, making it difficult to identify persons or activities. There is also the fact that Newton’s telescope had an angular magnification of about 38x (Wikipedia). That is suitable for looking at astronomical objects, but it is ridiculously large for looking at things only a hundred yards away.

When Waterhouse, Fatio and Huygens were looking at Saturn with the big Newtonian reflector, they were almost certainly using a Keplerian lens. A short time later, with no mention of changing the ocular lens, Eliza was looking at a ship on the horizon. In a brief glance, Waterhouse was able to identify its sail plan, which would be difficult for an inverted virtual image.

Thus, in his mentions of a Newtonian telescope, Stephenson has apparently ignored the problem of the orientation of the image.

The final consideration, about using a Newtonian telescope for terrestrial observations, has operational and physiological aspects, which do not apply to astronomical observations. Almost every interesting (e.g., dangerous) terrestrial object can appear unexpectedly, and thereafter move unpredictably. In contrast, every astronomical object (except meteors and near-earth artificial satellites) is both predictable and very simple in its motion across the sky. Even the 'wanderers' (Sun, Moon, planets, and comets) come very close to sharing the motion of the 'fixed' stars over an observation period of several hours. Everything moves in a circle around the celestial axis, at about 15 degrees per hour.

Even if one idealizes the reflecting telescope by providing an erecting system, a suitable angular magnification, and low-maintenance mirrors (e.g., silvered or aluminized), it has a severe disadvantage compared to a refracting terrestrial telescope. (Stephenson typically calls such a thing a perspective glass, prospective glass, or spyglass. These names do not specify the exact nature of the instrument. It was most likely to be a Galilean telescope in that era.)

It is easy to acquire the enlarged image in a spyglass, because the axis of the observer's eye is parallel to the axis of the telescope. While continuing to look directly at the object of interest with both eyes, the observer interposes the spyglass before one eye, checking its alignment with the other eye. With a few minutes of practice, it becomes almost second nature. It is similarly easy to follow a moving object: move the head and the telescope together.

It is much more difficult to acquire the enlarged image in a Newtonian reflector, because the axis of the observer's eye is necessarily 90 degrees from the axis of the telescope. If the telescope is no larger or heavier than a spyglass, it could be held freehand. If the observer decided to hold it with the ocular upward, she could face the object of interest at all times. She would hold the telescope under an armpit, guessing which way to point it. Then she would bend her head forward, to look nearly straight down into the ocular. She would have to shift the alignment, either by guesswork, or with the help of someone crouching behind her to sight along the tube, until the image of the desired object appeared in the field of view. Thereafter, it would be fairly easy to follow a moving object. If it moved to the right, she would turn to the right. If it moved upward, she would straighten up somewhat. At some point, she might develop a 'crick in the neck', by keeping her head bent down.

Alternatively, the observer could decide to hold it with the ocular horizontal. The main tube would be horizontal in front of her, with one hand under the primary mirror, and the other hand under the open end. After spotting an object of interest by naked eye, she would then turn her whole body 90 degrees to one side, while raising the ocular to her eye. There would almost have to be an assistant alongside her, to sight along the tube, in order to acquire the image. To follow horizontal motion thereafter might not be too difficult, but it would surely take lots of practice, in order to learn how to tip the tube sideways in following vertical motion.

A mount for the telescope would not solve all these problems. The user herself could sight along the tube of the telescope at the object of interest. Then she must look 90 degrees away from the object of interest, into the eyepiece. With luck, the image would be somewhere in the field of view, but if the object had moved meantime, she must guess which way to move the tube to pick it up. The other eye is of no help. In trying to follow a moving object, the telescope tube is constrained by the mount. The user’s head must move to accommodate the motion allowed by the mount.

Altogether, birdwatching with a Newtonian telescope is almost too horrible to contemplate.