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.
Saturday, April 17, 2010
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.
“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., http://www.britannica.com/bps/additionalcontent/18/21080238/CATADIOPTRICS-AND-COMMERCE-IN-EIGHTEENTHCENTURY-LONDON .)
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 (http://encyclopedia.com/doc/1O80-Porroprism.html ), 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.
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., http://www.britannica.com/bps/additionalcontent/18/21080238/CATADIOPTRICS-AND-COMMERCE-IN-EIGHTEENTHCENTURY-LONDON .)
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 (http://encyclopedia.com/doc/1O80-Porroprism.html ), 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.
Friday, December 18, 2009
Appearance of the Moon
In the Confusion, two different observations of the Moon were reported. On page 24 (October 1689): “After a long series of wrestling bouts, .... The night was nearly moonless, with only the merest crescent creeping across the sky– ....” On page 188 (5 August 1690): “There was a half-moon that night, and as they drifted into the Gulf Jack watched it chasing the lost sun towards the western ocean, all aglow on its underside, ....”
The analysis of the first observation requires estimates of two different times. One of these is the elapsed time after sunset. The first bout happened soon after sunset, in that the galley slaves could still see light in the sky as they were walking to the banyolar, but the courtyard had to be illuminated so that the crowd of spectators could see the activities. The first bout lasted perhaps 20 to 30 minutes, what with the preliminary activities, the fighting itself, and the settling of bets afterward. If the “long series” included about eight bouts of about the same length, then the lunar observation took place about three to four hours after sunset. The “merest crescent” moon would have been waxing after New Moon, visible in the west to southwest after sunset. An equivalent waning crescent moon would be visible in the east to southeast before sunrise, or well after midnight.
The second time to be estimated is the 'age' of the new moon on the evening in question, or equivalently, the angle between moon and sun as viewed from earth. Unfortunately, the term “merest crescent” is poetic but imprecise. A rough table relating age, angle from sun, fraction of the apparent radius illuminated at the western limb, and time between sunset and moonset follows.
Age ``` Elongation ``` Illumination ``` Time in Sky
days ``` degrees ```````fraction ````````hours
1 ``````` 12 ``````````` 0.023 ````````` 0.8
2 ``````` 24 ``````````` 0.09 `````````` 1.6
3 ``````` 37 ``````````` 0.19 ``````````` 2.4
4 ``````` 49 ``````````` 0.31 ``````````` 3.3
I would say that a four-day-old moon is already a fairly “fat” crescent, but a three-day-old moon may qualify as “mere” if not “merest”. Unfortunately, it would probably already have set before the galley slaves reached the roof. There would be no real problem if the moon had been described as “setting”, rather than “creeping across the sky”.
Stephenson’s description of the “half-moon” is also poetic, and rather good. It would be about 90 degrees east of the sun, and could be seen until about midnight. At that time of year, the moon’s declination would be well south of the sun’s. As viewed from the latitude of the gulf of Cadiz, the illuminated half of the moon would be the lower right, rather than exactly “its underside”. The problem here is the exact date. The site http://eclipse.gsfc.nasa.gov/phase/phasecat.html offers tables of eclipses and lunar phases over a 6000 year period. In 1690, First Quarter actually happened on 11 August Gregorian [1 August Julian]. Throughout The Baroque Cycle, Stephenson uses Julian dates, as used in Great Britain during those years. On 5 August Julian, the moon would actually have been about midway between First Quarter and Full Moon, with about 3/4 of its face illuminated.
The analysis of the first observation requires estimates of two different times. One of these is the elapsed time after sunset. The first bout happened soon after sunset, in that the galley slaves could still see light in the sky as they were walking to the banyolar, but the courtyard had to be illuminated so that the crowd of spectators could see the activities. The first bout lasted perhaps 20 to 30 minutes, what with the preliminary activities, the fighting itself, and the settling of bets afterward. If the “long series” included about eight bouts of about the same length, then the lunar observation took place about three to four hours after sunset. The “merest crescent” moon would have been waxing after New Moon, visible in the west to southwest after sunset. An equivalent waning crescent moon would be visible in the east to southeast before sunrise, or well after midnight.
The second time to be estimated is the 'age' of the new moon on the evening in question, or equivalently, the angle between moon and sun as viewed from earth. Unfortunately, the term “merest crescent” is poetic but imprecise. A rough table relating age, angle from sun, fraction of the apparent radius illuminated at the western limb, and time between sunset and moonset follows.
Age ``` Elongation ``` Illumination ``` Time in Sky
days ``` degrees ```````fraction ````````hours
1 ``````` 12 ``````````` 0.023 ````````` 0.8
2 ``````` 24 ``````````` 0.09 `````````` 1.6
3 ``````` 37 ``````````` 0.19 ``````````` 2.4
4 ``````` 49 ``````````` 0.31 ``````````` 3.3
I would say that a four-day-old moon is already a fairly “fat” crescent, but a three-day-old moon may qualify as “mere” if not “merest”. Unfortunately, it would probably already have set before the galley slaves reached the roof. There would be no real problem if the moon had been described as “setting”, rather than “creeping across the sky”.
Stephenson’s description of the “half-moon” is also poetic, and rather good. It would be about 90 degrees east of the sun, and could be seen until about midnight. At that time of year, the moon’s declination would be well south of the sun’s. As viewed from the latitude of the gulf of Cadiz, the illuminated half of the moon would be the lower right, rather than exactly “its underside”. The problem here is the exact date. The site http://eclipse.gsfc.nasa.gov/phase/phasecat.html offers tables of eclipses and lunar phases over a 6000 year period. In 1690, First Quarter actually happened on 11 August Gregorian [1 August Julian]. Throughout The Baroque Cycle, Stephenson uses Julian dates, as used in Great Britain during those years. On 5 August Julian, the moon would actually have been about midway between First Quarter and Full Moon, with about 3/4 of its face illuminated.
Sunday, December 6, 2009
Just How Heavy Were People?
PFC Gerald Hott, the dead butcher being stripped by Bobby Shaftoe (page 185 of Cryptonomicon), was described as “a big man, easily two-twenty-five in fighting trim, easily two-fifty now.” But when Enoch Root was weighing out meat to substitute for Hott’s body in the coffin (page 192), he had established that “the weight of Gerald Hott, converted into kilograms, is one hundred and thirty.” Unfortunately, 130 kilograms equal about 287 pounds. Thus, either Enoch Root and Bobby Shaftoe made quite different estimates of Hott’s weight, or else Neal Stephenson himself did not emulate Enoch Root, “who seems to be conversant with exotic systems of measurement [and] has made a calculation and checked it twice.”
When Günter Bischoff announced the birth weight of Günter Enoch Bobby Kivistik (page 1057 of Cryptonomicon), he almost certainly did not say, “Eight pounds, three ounces – superb for a wartime baby.” He would have used the metric units that he, Rudy von Hacklheber, and Otto Kivistik used every day. He would say, "3,714 grams", or "3.7 kilograms". I doubt that there was a scale anywhere in Norrsbruck (where the baby had been born), calibrated in English units, and who would have bothered to convert the weight? (Bobby Shaftoe would have wanted English units, but he had died without meeting Günter or Enoch in the Philippines.)
When Günter Bischoff announced the birth weight of Günter Enoch Bobby Kivistik (page 1057 of Cryptonomicon), he almost certainly did not say, “Eight pounds, three ounces – superb for a wartime baby.” He would have used the metric units that he, Rudy von Hacklheber, and Otto Kivistik used every day. He would say, "3,714 grams", or "3.7 kilograms". I doubt that there was a scale anywhere in Norrsbruck (where the baby had been born), calibrated in English units, and who would have bothered to convert the weight? (Bobby Shaftoe would have wanted English units, but he had died without meeting Günter or Enoch in the Philippines.)
Uniform Trouser Cuffs
At Bobby Shaftoe’s burial (pages 1067,1068 of Cryptonomicon), Goto Dengo picks up some loose dirt, which “trickles out from between his fingers and trails down the legs of his crisp new United States Army uniform, getting caught in the trouser cuffs.” During my service with the U.S. Army in the 1950s (Korean War), most of our uniforms were identical to those used at the end of World War II. None of those uniform trousers had cuffs. This last farewell from Dengo to his old friend Bobby was moving, but what were the non-existent cuffs supposed to contribute to the scene?
Saturday, December 5, 2009
Where Can Venus Be?
There are several places in The Baroque Cycle where the planet Venus is mentioned. The first occurs in late winter or early spring of 1666 (page 146 of Quicksilver). At dawn, Daniel Waterhouse “found that he was walking directly toward a blazing planet, a few degrees above the western horizon, which could only be Venus.” There was “nothing before him but the Dawn Star.”
A few days later, Daniel arrived at Woolsthorpe to assist Isaac Newton in observations of Venus (pages 150-156). The house, “made of ... soft pale stone”, had a “clear sunny exposure” at its southern end, with “almost no windows there–just a couple of them, scarcely larger than gunslits, ...” As seen inside the house, “The southern half–with just a few tiny apertures to admit the plentiful sunlight–was Isaac’s: ...” “The sun was going down, and they were preparing for Venus to wheel around into the southern sky.” “Isaac had worked out during which hours of the night Venus would be shining her perfectly unidirectional light on Woolsthorpe Manor’s south wall, and he’d done it not only for tonight but for every night in the next several weeks.” “When Daniel looked, he realized that he could see not only the spectrum from Venus, but tiny, ghostly streaks of color all over the wall: the spectra cast by the stars that surrounded Venus in the southern sky.” “The earth spun and the ribbons of color migrated across the invisible wall, an inch a minute, ...”
Finally, in the early evening of 28 July 1714 (page 539 in The System of the World), Lord Ravenscar is with Lord Bolingbroke, preparing to use Bolingbroke’s telescope. Bolingbroke says, “Presently night shall fall and Venus shall shine forth.”
Let us consider whether or not Venus could have done those things at those times.
Venus, like Mercury, is an inferior planet relative to Earth, i.e., closer to the Sun. The first thing that means is that Venus, as viewed from Earth, can never get very far from the Sun. The direct result of that, is that the simplest description of its motion is 'synodic', i.e., relative to the Sun, rather than relative to the fixed stars. The plane of the orbit of Venus is inclined at an angle of about 3.4 degrees from the plane of Earth’s orbit. Therefore, Venus is never very far from the ecliptic, the path which the Sun appears to follow throughout the year. If the point of maximum excursion of Venus from the ecliptic happens to coincide with its closest approach to Earth, then it can appear to be as much as about 9 degrees north or south of the ecliptic.
To get a reasonable approximate description of Venus’s motion along the ecliptic, let us make the simplifying assumptions that Earth and Venus move in circular orbits (rather than the actual ellipses), and at constant speeds (rather than faster when closer to the Sun). We can also ignore the inclination of Venus’s orbit, so that everything of interest happens on one plane. Earth’s sidereal period of 365.25 days (i.e., relative to the stars), and Venus’s sidereal period of 224.7 days, combine to produce a synodic period of 584 days. During that period, Earth performs 1.599 revolutions, and Venus performs 2.599 revolutions, or exactly one additional revolution. Thus, the original configuration of Sun, Venus, and Earth is exactly reproduced after any integer number of synodic periods.
In evaluating the synodic description, Earth and Sun are considered fixed at a separation of one astronomical unit (AU). Venus revolves eastward (counter-clockwise as viewed from ecliptic north) in a circle of radius 0.723 AU. Its angular speed is 0.6164 degrees per day, so that it goes once around the circle in 584 days. The 'elongation' of Venus, which is its angular displacement from the Sun as viewed from Earth, can be found at any time by solving the Sun-Earth-Venus triangle.
Let us start with zero elongation, increasing eastward. Venus is then directly on the far side of the Sun from Earth ('superior conjunction'). At that time, it has its maximum velocity in elongation, at 0.259 degrees per day. The apparent angular speed decreases continuously for about 217 days, at which point the line of sight from Venus to Earth is tangential to Venus’s orbit. This is the condition of 'greatest eastward elongation', and Venus is 46.3 degrees east of the Sun. (Note that sine 46.3 degrees = 0.723, the ratio of orbit radii.) Venus’s synodic motion is very slow at this time. It spends about 24 days within 0.5 degree of greatest elongation. (That angle is the apparent width of Sun or Moon.)
After greatest eastward elongation, Venus moves westward (toward the Sun). Its speed in elongation increases continuously for about 75 days, until it is directly between Earth and Sun ('inferior conjunction'). The maximum westward speed is about 1.609 degrees per day. During this half-synodic-period of 292 days, Venus is east of the Sun, and is visible in the western sky after sunset (an 'evening star'). As viewed through a telescope, it exhibits phases, similar to those of the moon. Near superior conjunction, it appears 'full' (circular). It is 'waning gibbous' until greatest eastward elongation, when it appears 'half full'. Thereafter it is 'waning crescent', until it is 'dark' near inferior conjunction.
The second half-period of the synodic motion is essentially the mirror image of the first half-period. For about 75 days the apparent angular speed decreases continuously until Venus reaches greatest westward elongation at 46.3 degrees west of the Sun. It then starts moving eastward, until after another 217 days of continually increasing angular speed it reaches superior conjunction again. During this entire time it is visible in the eastern sky before sunrise (a 'morning star', or the “Dawn star”). Its phases are successively 'dark', 'waxing crescent', 'half full' (at greatest westward elongation), 'waxing gibbous', and 'full'.
The motion of Venus relative to the stars can be found by superimposing the Sun’s average motion (about 0.986 degrees per day eastward along the ecliptic) upon this synodic motion. The effects of the eccentricity of the orbits of Venus and Earth are to change the angles of greatest elongation by a degree or two, and to change the time intervals between unique configurations by a few days.
It is obvious that the first described appearance of Venus is quite impossible. Any planet which is about 180 degrees from the Sun must be superior to Earth, i.e., further from the Sun. That configuration is 'opposition', and it is the time at which such a planet is closest to Earth, and hence appears brightest. The two best candidate planets are Jupiter and Saturn, because they appear white, like Venus. Mars is less bright, and appears reddish rather than “blazing”. I tried online to find where the superior planets were at that time, but I gave up (perhaps too soon) on finding a free historical ephemeris. Apparently there is still so much money to be made in astrology, that the proprietors of ephemerides insist on being paid.
The description of Woolsthorpe Manor, given by Stephenson, is not a very good match to the photographs easily found in a search on that place name. (See e.g., Wikipedia.) The photos show several good-sized windows on the sunlit broad front of the house. However, there could have been extensive alterations in the intervening centuries.
The first problem with viewing the spectrum of Venus lies in getting the light through the wall of the house, as described by Stephenson. My guess is that the masonry might be eight inches thick. If the apertures were made like “gunslits”, they might be relieved at an angle of perhaps 45 degrees on each side. In such a case, light from only those directions between south-east and south-west could get in.
Normal window openings, perhaps two feet wide, would relieve the problem almost completely. Opaque panels over the sashes could darken the room. If such a panel were near the midpoint of the thickness of the wall, then an aperture in the center of the panel would admit light from any direction less than about 71 degrees from south.
It is difficult to reconcile Stephenson’s statement, about Venus wheeling around into the southern sky, with the above description of Venus’s synodic motion. If Venus is an evening star, then at sunset it is already as far south as it will ever be during that night. (The only possible exception is if greatest eastward elongation nearly coincides with winter solstice. At that high latitude (about 52.8 degrees), the sun then sets well south of west. Calculations of the azimuths of sunset and of Venus at that time are left to the interested reader.)
On the other hand, if Venus is a morning star, then sunset has nothing to do with the situation. The observers should go to bed early, so that they can get up before sunrise, when Venus is sufficiently south of east.
Newton’s calculations of Venus’s visibility on various nights were no big deal. Except close to inferior conjunction, Venus moves so sedately that times for visibility would change by only a few minutes over the course of a week.
In order to see a spectrum of Venus’s light with the unaided eye, Venus should be as bright as possible. Its brightness depends upon both the phase, which determines how much illuminated area is exposed, and the distance from Earth, which determines the solid angle subtended by that illuminated area. The maximum brightness occurs between greatest elongation and inferior conjunction, but is only slightly greater than the brightness at greatest elongation.
The really big question remains: where was Venus relative to the sun in the early spring of 1666? Fortunately, the free site http://www.fourmilab.ch/images/venus_daytime/ presents the article “Viewing Venus in Broad Daylight”, by John Walker. It includes a calculator for dates of greatest elongations of Venus, for arbitrary years. It does not offer dates of conjunctions. The dates which span the time of interest are as follows.
Date``````````Elongation
1665 Sep 11```46.461 deg W
1666 Nov 29```47.473 deg E
Midway between those two dates was 1666 Apr 21. That should be within perhaps 5 days of the true date of superior conjunction. Unfortunately, that website does not identify which calendar is used for those dates. I would guess that it is Gregorian (“New Style”), which was then used in most of continental Europe. In Britain, they still used the Julian calendar (“Old Style”), and called it April 11.
I don’t know exactly when apple trees bloom in Lincolnshire, but it almost has to be rather close to April 21. Unfortunately, superior conjunction is the worst possible time to make observations of Venus. The planet is as far away from Earth as it ever gets, so that it has only about one-third its maximum brightness. It is lost in the glow of the twilight sky, until it gets far enough away from the sun. At a guess, it should be at least 10 degrees away, in order to give any useful viewing time. That would take about six weeks after conjunction. As the final blow, Venus would be very close to directly west at sunset. Its light would not shine on the south wall of Woolsthorpe Manor at all, during hours of darkness.
Finally, the dates of greatest elongation which span midsummer of 1714 are as follows.
Date``````````Elongation
1713 Aug 29```45.459 deg W
1714 Nov 16```47.473 deg E
The rather close match of these dates to the previous case arises because a difference of 30 synodic periods amounts to just less than 48 Earth years. The superior conjunction for this case happened close to 1714 April 8 (Gregorian).
The evening in question [28 July (Julian) or 8 August (Gregorian)] was 122 days later, and Venus’s elongation would have been about 30.5 degrees Eastward. It was indeed an evening star on that date, and would have been clearly visible in the western sky after sunset. (Unfortunately, the seeing was probably very bad after all the bonfires were started.)
Thus Stephenson put Venus in the right place one time out of three.
A few days later, Daniel arrived at Woolsthorpe to assist Isaac Newton in observations of Venus (pages 150-156). The house, “made of ... soft pale stone”, had a “clear sunny exposure” at its southern end, with “almost no windows there–just a couple of them, scarcely larger than gunslits, ...” As seen inside the house, “The southern half–with just a few tiny apertures to admit the plentiful sunlight–was Isaac’s: ...” “The sun was going down, and they were preparing for Venus to wheel around into the southern sky.” “Isaac had worked out during which hours of the night Venus would be shining her perfectly unidirectional light on Woolsthorpe Manor’s south wall, and he’d done it not only for tonight but for every night in the next several weeks.” “When Daniel looked, he realized that he could see not only the spectrum from Venus, but tiny, ghostly streaks of color all over the wall: the spectra cast by the stars that surrounded Venus in the southern sky.” “The earth spun and the ribbons of color migrated across the invisible wall, an inch a minute, ...”
Finally, in the early evening of 28 July 1714 (page 539 in The System of the World), Lord Ravenscar is with Lord Bolingbroke, preparing to use Bolingbroke’s telescope. Bolingbroke says, “Presently night shall fall and Venus shall shine forth.”
Let us consider whether or not Venus could have done those things at those times.
Venus, like Mercury, is an inferior planet relative to Earth, i.e., closer to the Sun. The first thing that means is that Venus, as viewed from Earth, can never get very far from the Sun. The direct result of that, is that the simplest description of its motion is 'synodic', i.e., relative to the Sun, rather than relative to the fixed stars. The plane of the orbit of Venus is inclined at an angle of about 3.4 degrees from the plane of Earth’s orbit. Therefore, Venus is never very far from the ecliptic, the path which the Sun appears to follow throughout the year. If the point of maximum excursion of Venus from the ecliptic happens to coincide with its closest approach to Earth, then it can appear to be as much as about 9 degrees north or south of the ecliptic.
To get a reasonable approximate description of Venus’s motion along the ecliptic, let us make the simplifying assumptions that Earth and Venus move in circular orbits (rather than the actual ellipses), and at constant speeds (rather than faster when closer to the Sun). We can also ignore the inclination of Venus’s orbit, so that everything of interest happens on one plane. Earth’s sidereal period of 365.25 days (i.e., relative to the stars), and Venus’s sidereal period of 224.7 days, combine to produce a synodic period of 584 days. During that period, Earth performs 1.599 revolutions, and Venus performs 2.599 revolutions, or exactly one additional revolution. Thus, the original configuration of Sun, Venus, and Earth is exactly reproduced after any integer number of synodic periods.
In evaluating the synodic description, Earth and Sun are considered fixed at a separation of one astronomical unit (AU). Venus revolves eastward (counter-clockwise as viewed from ecliptic north) in a circle of radius 0.723 AU. Its angular speed is 0.6164 degrees per day, so that it goes once around the circle in 584 days. The 'elongation' of Venus, which is its angular displacement from the Sun as viewed from Earth, can be found at any time by solving the Sun-Earth-Venus triangle.
Let us start with zero elongation, increasing eastward. Venus is then directly on the far side of the Sun from Earth ('superior conjunction'). At that time, it has its maximum velocity in elongation, at 0.259 degrees per day. The apparent angular speed decreases continuously for about 217 days, at which point the line of sight from Venus to Earth is tangential to Venus’s orbit. This is the condition of 'greatest eastward elongation', and Venus is 46.3 degrees east of the Sun. (Note that sine 46.3 degrees = 0.723, the ratio of orbit radii.) Venus’s synodic motion is very slow at this time. It spends about 24 days within 0.5 degree of greatest elongation. (That angle is the apparent width of Sun or Moon.)
After greatest eastward elongation, Venus moves westward (toward the Sun). Its speed in elongation increases continuously for about 75 days, until it is directly between Earth and Sun ('inferior conjunction'). The maximum westward speed is about 1.609 degrees per day. During this half-synodic-period of 292 days, Venus is east of the Sun, and is visible in the western sky after sunset (an 'evening star'). As viewed through a telescope, it exhibits phases, similar to those of the moon. Near superior conjunction, it appears 'full' (circular). It is 'waning gibbous' until greatest eastward elongation, when it appears 'half full'. Thereafter it is 'waning crescent', until it is 'dark' near inferior conjunction.
The second half-period of the synodic motion is essentially the mirror image of the first half-period. For about 75 days the apparent angular speed decreases continuously until Venus reaches greatest westward elongation at 46.3 degrees west of the Sun. It then starts moving eastward, until after another 217 days of continually increasing angular speed it reaches superior conjunction again. During this entire time it is visible in the eastern sky before sunrise (a 'morning star', or the “Dawn star”). Its phases are successively 'dark', 'waxing crescent', 'half full' (at greatest westward elongation), 'waxing gibbous', and 'full'.
The motion of Venus relative to the stars can be found by superimposing the Sun’s average motion (about 0.986 degrees per day eastward along the ecliptic) upon this synodic motion. The effects of the eccentricity of the orbits of Venus and Earth are to change the angles of greatest elongation by a degree or two, and to change the time intervals between unique configurations by a few days.
It is obvious that the first described appearance of Venus is quite impossible. Any planet which is about 180 degrees from the Sun must be superior to Earth, i.e., further from the Sun. That configuration is 'opposition', and it is the time at which such a planet is closest to Earth, and hence appears brightest. The two best candidate planets are Jupiter and Saturn, because they appear white, like Venus. Mars is less bright, and appears reddish rather than “blazing”. I tried online to find where the superior planets were at that time, but I gave up (perhaps too soon) on finding a free historical ephemeris. Apparently there is still so much money to be made in astrology, that the proprietors of ephemerides insist on being paid.
The description of Woolsthorpe Manor, given by Stephenson, is not a very good match to the photographs easily found in a search on that place name. (See e.g., Wikipedia.) The photos show several good-sized windows on the sunlit broad front of the house. However, there could have been extensive alterations in the intervening centuries.
The first problem with viewing the spectrum of Venus lies in getting the light through the wall of the house, as described by Stephenson. My guess is that the masonry might be eight inches thick. If the apertures were made like “gunslits”, they might be relieved at an angle of perhaps 45 degrees on each side. In such a case, light from only those directions between south-east and south-west could get in.
Normal window openings, perhaps two feet wide, would relieve the problem almost completely. Opaque panels over the sashes could darken the room. If such a panel were near the midpoint of the thickness of the wall, then an aperture in the center of the panel would admit light from any direction less than about 71 degrees from south.
It is difficult to reconcile Stephenson’s statement, about Venus wheeling around into the southern sky, with the above description of Venus’s synodic motion. If Venus is an evening star, then at sunset it is already as far south as it will ever be during that night. (The only possible exception is if greatest eastward elongation nearly coincides with winter solstice. At that high latitude (about 52.8 degrees), the sun then sets well south of west. Calculations of the azimuths of sunset and of Venus at that time are left to the interested reader.)
On the other hand, if Venus is a morning star, then sunset has nothing to do with the situation. The observers should go to bed early, so that they can get up before sunrise, when Venus is sufficiently south of east.
Newton’s calculations of Venus’s visibility on various nights were no big deal. Except close to inferior conjunction, Venus moves so sedately that times for visibility would change by only a few minutes over the course of a week.
In order to see a spectrum of Venus’s light with the unaided eye, Venus should be as bright as possible. Its brightness depends upon both the phase, which determines how much illuminated area is exposed, and the distance from Earth, which determines the solid angle subtended by that illuminated area. The maximum brightness occurs between greatest elongation and inferior conjunction, but is only slightly greater than the brightness at greatest elongation.
The really big question remains: where was Venus relative to the sun in the early spring of 1666? Fortunately, the free site http://www.fourmilab.ch/images/venus_daytime/ presents the article “Viewing Venus in Broad Daylight”, by John Walker. It includes a calculator for dates of greatest elongations of Venus, for arbitrary years. It does not offer dates of conjunctions. The dates which span the time of interest are as follows.
Date``````````Elongation
1665 Sep 11```46.461 deg W
1666 Nov 29```47.473 deg E
Midway between those two dates was 1666 Apr 21. That should be within perhaps 5 days of the true date of superior conjunction. Unfortunately, that website does not identify which calendar is used for those dates. I would guess that it is Gregorian (“New Style”), which was then used in most of continental Europe. In Britain, they still used the Julian calendar (“Old Style”), and called it April 11.
I don’t know exactly when apple trees bloom in Lincolnshire, but it almost has to be rather close to April 21. Unfortunately, superior conjunction is the worst possible time to make observations of Venus. The planet is as far away from Earth as it ever gets, so that it has only about one-third its maximum brightness. It is lost in the glow of the twilight sky, until it gets far enough away from the sun. At a guess, it should be at least 10 degrees away, in order to give any useful viewing time. That would take about six weeks after conjunction. As the final blow, Venus would be very close to directly west at sunset. Its light would not shine on the south wall of Woolsthorpe Manor at all, during hours of darkness.
Finally, the dates of greatest elongation which span midsummer of 1714 are as follows.
Date``````````Elongation
1713 Aug 29```45.459 deg W
1714 Nov 16```47.473 deg E
The rather close match of these dates to the previous case arises because a difference of 30 synodic periods amounts to just less than 48 Earth years. The superior conjunction for this case happened close to 1714 April 8 (Gregorian).
The evening in question [28 July (Julian) or 8 August (Gregorian)] was 122 days later, and Venus’s elongation would have been about 30.5 degrees Eastward. It was indeed an evening star on that date, and would have been clearly visible in the western sky after sunset. (Unfortunately, the seeing was probably very bad after all the bonfires were started.)
Thus Stephenson put Venus in the right place one time out of three.
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