Rangekeeper

Rangekeepers were electromechanical fire control computers used primarily during the early part of the 20th century. They were sophisticated analog computers whose development reached its zenith following World War II, specifically the Computer Mk 47 in the Mk 68 Gun Fire Control system. During World War II, rangekeepers directed gunfire on land, sea, and in the air. While rangekeepers were widely deployed, the most sophisticated rangekeepers were mounted on warships to direct the fire of long range guns. Technically, it would be more accurate to use the term "rifle" for long-range ship-board cannon. However, the term "gun" is commonly used and that nomenclature is maintained here.]

These warship-based computing devices needed to be sophisticated because the problem of calculating gun angles in a naval engagement is very complex. In a naval engagement, both the ship firing the gun and the target are moving with respect to each other. In addition, the ship firing its gun is not a stable platform because ships roll, pitch, and yaw due to wave action, ship change of direction, and effect of board firing. The rangekeeper also performed the required ballistics calculations associated with firing a gun. This article will focus on US Navy shipboard rangekeepers, but the basic principles of operation are applicable to all rangekeepers regardless of where they are deployed.

A rangekeeper is defined as an analog fire control system that performed three functions: cite book
title = Naval Ordnance and Gunnery
origdate = 1950
url = http://www.eugeneleeslover.com/FIRE-CONTROL-PAGE.html
format = HTML
accessdate = 2006-08-26
year = 1958
publisher = United States Naval Academy
location = Annapolis, MA
id = NavPers 10798-A
chapter = Chapter 19: Surface Fire Control Problem
chapterurl = http://www.eugeneleeslover.com/USNAVY/CHAPTER-19-F.html ]
* Target Tracking :The rangekeeper continuously computed the current target bearing. This is a difficult task because both the target and the ship firing (generally referred to as "own ship") are moving. This requires knowing the target's range, course, and speed accurately. It also requires accurately knowing the own ship's course and speed.
* Target Position Prediction:When a gun is fired, it takes time for the projectile to arrive at the target. The rangekeeper must predict where the target will be at the time of projectile arrival. This is the point at which the guns are aimed.
* Gunfire Correction:Directing the fire of a long range weapon to deliver a projectile to a specific location requires many calculations. The projectile point of impact is a function of many variables, including: gun azimuth, gun elevation, wind speed and direction, air resistance, gravity, latitude, gun/sight parallax, barrel wear, powder load, and projectile type.

During WWII, all the major warring powers developed rangekeepers to different levels.cite web
url = http://www.navweaps.com/index_tech/tech-086.htm
title = Overview of USN and IJN Warship Ballistic Computer Design
accessdate = 2006-08-26
author = Bradley Fischer
date = 2003-09-09
format = HTML
publisher = NavWeaps] Rangekeepers were only one member of a class of electromechanical computers used for fire control during World War II. Related analog computing hardware used by the United States included:
* Norden bombsight: US bombers used the Norden bombsight, which used similar technology to the rangekeeper for predicting bomb impact points.
* Torpedo Data Computer (TDC): US submarines used the TDC to compute torpedo launch angles. This device also had a rangekeeping function that was referred to as "position keeping." This was the only submarine-based fire control computer during World War II that performed target tracking. Because space within a submarine hull is limited, the TDC designers overcame significant packaging challenges in order to mount the TDC within the allocated volume.
* M-9/SCR-584 Anti-Aircraft System:This equipment was used to direct air defense artillery. It made a particularly good account of itself against the V-1 flying bombs.cite book
last = Mindell
first = David
title = Between Human and Machine
publisher = Johns Hopkins
date = 2002
location = Baltimore
pages = p. 254
id = ISBN 0-8018-8057-2 ] During World War II, rangekeeper capabilities were expanded to the point where the name rangekeeper was deemed to be inadequate. The name computer, which had been reserved for human calculators, then began to be applied to the rangekeeper equipment. After World War II, digital computers began to replace rangekeepers. However, components of the analog rangekeeper system continued in service with the US Navy until the 1990s. cite news
title = Older weapons hold own in high-tech war
url = http://www.dogtagsrus.com/p-38%20can%20opener%20articles.htm
format = html
publisher = Dallas Morning News
date = 1991-02-10
accessdate = 2006-09-30]

The performance of these analog computers was impressive. The battleship North Carolina during a 1945 test was able to maintain an accurate firing solutionThe rangekeeper in this exercise maintained a firing solution that was accurate within a few hundred yards, which is within the range needed for an effective rocking salvo. The rocking salvo was used by the US Navy to get the final corrections needed to hit the target.] on a target during a series of high-speed turns.cite journal
last = Jurens
first = W.J.
url=http://www.navweaps.com/index_inro/INRO_BB-Gunnery_p1.htm
year = 1991
title = The Evolution of Battleship Gunnery in the U.S. Navy, 1920-1945
journal = Warship International
volume = No. 3
pages = p. 255 ] It is a major advantage for a warship to be able to maneuver while engaging a target.

Night naval engagements at long range became feasible when radar data could be input to the rangekeeper. The effectiveness of this combination was demonstrated in November 1942 at the Third Battle of Savo Island when the USS Washington engaged the Japanese battlecruiser Kirishima at a range of 18,500 yards at night. The Kirishima was set aflame, suffered a number of explosions, and was scuttled by her crew. She had been hit by nine 16 inch rounds out of 75 fired (12% hit rate). cite paper
author = A. Ben Clymer
title = The Mechanical Analog Computers of Hannibal Ford and William Newell
publisher = IEEE Annals of the History of Computing
date = Vol. 15 No. 2, 1993
url = http://web.mit.edu/STS.035/www/PDFs/Newell.pdf
format = pdf
accessdate = 2006-08-26] The wreck of the Kirishima was discovered in 1992 and showed that the entire bow section of the ship was missing.cite web
url = http://www.combinedfleet.com/atully08.htm
title = Located/Surveyed Shipwrecks of the Imperial Japanese Navy
work = Mysteries/Untold Sagas Of The Imperial Japanese Navy
accessdate = 2006-09-26
author = Anthony P. Tully
date = 2003
format = html
publisher =CombinedFleet.com] The Japanese during World War II did not develop radar or automated fire control to the level of the US Navy and were at a significant disadvantage. cite book
last = Mindell
first = David
title = Between Human and Machine
publisher = Johns Hopkins
date = 2002
location = Baltimore
pages = pp. 262-263
id = ISBN 0-8018-8057-2 ] Even the British did not adopt gyroscopic stabilization of their guns until quite late in the history of rangekeepers.

Rangekeepers were very large and the ship designs needed to make provisions to accommodate them. For example, the Ford Mk 1A Computer weighed 3150 lbs. cite web
url = http://www.de220.com/Armament/Fire%20Directors/Mk%201%20Computer/Mk%201%20Computer.htm
title = Ballistic Computer
work = Destroyer Escort Central
accessdate = 2006-09-26
date = 2003
format = html
publisher =USS Francis M. Robinson (DE-220) Association, 2000] The Mk. 1/1A's mechanism support plates, some an inch thick, were made of aluminum alloy, but nevertheless, the computer is very heavy. On at least one museum ship, the destroyer U.S.S. Cassin Young (now in Boston), the computer and Stable Element more than likely still are below decks, probably because they are too difficult to remove. The rangekeepers also required a large number of electric cables for synchro data transmission links over which they received information from the various sensors (e.g. gun director, Pitometer, rangefinder, gyrocompass) and sent commands to the guns.

Background

History

The early history of naval fire control was dominated by the engagement of targets within visual range (also referred to as direct fire). In fact, most naval engagements before 1800 were conducted at ranges of 20 to 50 yards. Even during the American Civil War, the famous engagement between the USS Monitor and the CSS Virginia was often conducted at less than 100 yard range. cite web
url = http://www.monitorcenter.org/history/chronology/chronology2.php
title = Chronology of the USS Monitor: From Inception to Sinking
accessdate = 2006-08-26
format = html
work = The Mariner's Museum
publisher =USS Monitor Center] With time, naval guns became larger and had greater range. At first, the guns were aimed using the technique of artillery spotting. Artillery spotting involved firing a gun at the target, observing the projectile's point of impact, and correcting the aim based on where the shell was observed to land, which became more and more difficult as the range of the gun increased. The increasing range of the guns also forced ships to create very high observation points from which optical rangefinders and artillery spotters could see the battle. The need to spot artillery shells was one of the compelling reasons behind the development of naval aviation and early aircraft were used to spot the naval gunfire points of impact. In some cases, ships launched manned observation balloons as a way to artillery spot. Even today, artillery spotting is an important part of directing gunfire, though today the spotting is often done by unmanned aerial vehicles. For example, during Desert Storm, UAVs spotted fire for the Iowa-class battleships involved in shore bombardment.]

Between the American Civil War and 1905, numerous small improvements, such as telescopic sights and optical rangefinders, were made in fire control. There were also procedural improvements, like the use of plotting boards to manually predict the position of a ship during an engagement. Around 1905, mechanical fire control aids began to become available, such as the Dreyer Table, Dumaresq (which was also part of the Dreyer Table), and [http://www.dreadnoughtproject.org/tech/essays/FireControl/ArgoAimCorrector/ Argo Clock] , but these devices took a number of years to become widely deployed. cite book
last = Mindell
first = David
title = Between Human and Machine
publisher = Johns Hopkins
date = 2002
location = Baltimore
pages = pp. 25-28
id = ISBN 0-8018-8057-2 ] The reasons were for this slow deployment are complex. As in most bureaucratic environments, institutional inertia and the revolutionary nature of the change required caused the major navies to move slow in adopting the technology.] These devices were early forms of rangekeepers.

The issue of directing long-range gunfire came into sharp focus during World War I with the Battle of Jutland. While the British were thought by some to have the finest fire control system in the world at that time, during the Battle of Jutland only 3% of their shots actually struck their targets. At that time, the British primarily used a manual fire control system. The one British ship in the battle that had a mechanical fire control system turned in the best shooting results. cite book
last = Mindell
first = David
title = Between Human and Machine
publisher = Johns Hopkins
date = 2002
location = Baltimore
pages = pp. 20-21
id = ISBN 0-8018-8057-2 ] This experience contributed to rangekeepers becoming standard issue.The British fleet's performance at Jutland has been a subject of much analysis and there were many contributing factors. When compared to the long-range gunnery performance by the US Navy and Kriegsmarine, the British gunnery performance at Jutland is not that poor. In fact, long range gunnery is notorious for having a low hit percentage. For example, during exercises in 1930 and 1931, US battleships had hit percentages in the 4-6% range (Jurens).]

The US Navy's first deployment of a rangekeeper was on the USS Texas (BB-35) in 1916. Because of the limitations of the technology at that time, the initial rangekeepers were crude. For example, during World War I the rangekeepers would generate the necessary angles automatically but sailors had to manually follow the directions of the rangekeepers (a task called "pointer following" or "follow the pointer"). Pointer following could be accurate, but the crews tended to make inadvertent errors when they became fatigued during extended battles. During World War II, servomechanisms (called "power drives" in the U.S. Navy) were developed that allowed the guns to automatically steer to the rangekeeper's commands with no manual intervention. The Mk. 1 and Mk. 1A computers contained approx. 20 servomechanisms, mostly position servos, to minimize torque load on the computing mechanisms.cite web
url = http://www.navweaps.com/index_tech/tech-052.htm
title = Fire Control Systems in WWII
author = Tony DiGiulian
accessdate = 2006-09-28
format = html
work = The Mariner's Museum
date = 17 April 2001
publisher =Navweaps.com]

During their long service life, rangekeepers were updated often as technology advanced and by World War II they were a critical part of an integrated fire control system. The incorporation of radar into the fire control system early in World War II provided ships the ability to conduct effective gunfire operations at long range in poor weather and at night. The degree of updating varied by country. For example, the US Navy used servomechanisms to automatically steer their guns in both azimuth and elevation. The Germans used servomechanisms to steer their guns only in elevation, and the British did not use servomechanisms for this function at all.]

The rangekeeper's target position prediction characteristics could be used to defeat the rangekeeper. For example, many captains under long range gun attack would make violent maneuvers to "chase salvos." A ship that is chasing salvos is maneuvering to the position of the last salvo splashes. Because the rangekeepers are constantly predicting new positions for the target, it is unlikely that subsequent salvos will strike the position of the previous salvo.cite web
url = http://pages.cthome.net/boyd544/Diary03a.htm
title = Nauru Island: Enemy Action - December 8, 1943
author = Captain Robert N. Adrian
accessdate = 2006-10-06
format = html
work = U.S.S. Boyd (DD-544)
publisher =USS Boyd DD-544 Document Archive] Practical rangekeepers had to assume that targets were moving in a straight-line path at a constant speed, to keep complexity to acceptable limits. A sonar rangekeeper was built to include a target circling at a constant radius of turn, but that function had been disabled.

The last combat action for the analog rangekeepers, at least for the US Navy, was in the 1991 Persian Gulf War when the rangekeepers on the Iowa-class battleships directed their last rounds in combat.

The Problem of Rangekeeping

Long range gunnery is a complex combination of art, science, and mathematics. There are numerous factors that affect the ultimate placement of a projectile and many of these factors are difficult to model accurately. As such, the accuracy of battleship guns was ~1% of range (sometimes better, sometimes worse). Shell-to-shell repeatability was ~0.4% of range.

Accurate long range gunnery requires that a number of factors be taken into account::* Target course and speed:* Own ship course and speed:*Gravity:* Coriolis effect: Because the Earth is rotating, there is an apparent force acting on the projectile. :* Internal ballistics: Guns do wear and this aging must be taken into account. There are also shot-to-shot variations due to barrel temperature and interference between guns firing simultaneously.:* External ballistics: Different projectiles have different ballistic characteristics. Also, air conditions have an effect as well (temperature, wind, air pressure).:* Parallax correction: In general, the position of the gun and target spotting equipment (radar, mounted on the gun director, pelorus, etc) are in different locations on a ship. This creates a parallax error for which corrections must be made.:* projectile characteristics (e.g. ballistic coefficient):* powder charge weight and temperature

These issues are so complicated and need to be performed so quickly that the need arose for an automated way of performing these corrections. Part of the complexity came from the amount of information that must be integrated from many different sources. For example, information from the following sensors, calculators, and visual aids must be integrated to generate a solution:

:* Gyrocompass: This device provided an accurate true north own ship course.:* Rangefinders: Optical devices for determining the range to a target.:* Pitometer Logs: These devices provided an accurate measurement of the own ship's speed.:* Range clocks: These devices provided a prediction of the target's range at the time of projectile impact if the gun was fired now. This function could be considered "range keeping".:* Angle clocks: This device provided a prediction of the target's bearing at the time of projectile impact if the gun was fired now.:* Plotting board: A map of the gunnery platform and target that allowed predictions to be made as to the future position of a target. (The compartment ("room") where the Mk.1 and Mk.1A computers was located was called "Plot" for historical reasons.):* Various slide rules: These devices performed the various calculations required to determine the required gun azimuth and elevation. :* Meteorological sensors: Temperature, wind speed, and humidity all have an effect on the ballistics of a projectile. U.S. Navy rangekeepers and analog computers did not consider different wind speeds at differing altitudes.

To illustrate the complexity, Table 1 lists the types of input for the Ford Mk 1 Rangekeeper (ca 1931).

::

An integrated solution was needed and the first rangekeepers were developed. The ultimate solution also included the automated steering of the guns to the proper azimuth and elevation through the use of servomechanisms. The first rangekeepers were being deployed during World War I. During World War II, many types of rangekeepers were in use on many types of warships.

Implementations

The implementation methods used in analog computers were many and varied. The fire control equations implemented during World War II on analog rangekeepers are the same equations implemented later on digital computers. The key difference is that the rangekeepers solved the equations mechanically. While mathematical functions are not often implemented mechanically today, mechanical methods exist to implement all the common mathematical operations. Some examples include:

:* Addition and Subtraction:Differential gears, usually referred to by technicians simply as "differentials", were often used to perform addition and subtraction operations. The Mk. 1A contained apporoximately 160 of them. The history of this gearing for computing dates to antiquity (see Antikythera mechanism).:* Multiplication by a Constant:Gear ratios were very extensively used to multiply a value by a constant.:* Multiplication of two variables: The Mk. 1 and Mk.1A computer multipliers were based on the geometry of similar triangles.:* Sine/cosine generation: These mechanisms would be called resolvers, today; they were called "component solvers" in the mechanical era. In most instances, they resolved an angle and magnitude (radius) into sine and cosine components, with a mechanism based on the Scotch yoke in steam-engine technology, but with a variable crankpin radius, so to speak.:* Integration: [http://www.scientific-computing.com/scwmayjun03computingmachines.html Disk and ball integrators (or its variants)] performed the integration operation. As well, four small Ventosa integrators in the Mk. 1 and Mk. 1A computers scaled rate-control corrections according to angles.:* Differentiation:Differentiation was performed by using an integrator in a feedback loop.:* Evaluation of Functions:Rangekeepers used a a number of cams to generate function values. For surface fire control (The Mk. 8 Range Keeper), a single flat cam was sufficient to define ballistics, but in the Mk. 1 and Mk 1A computers, four three-dimensional cams were needed. Many face cams (flat discs with wide spiral grooves) were used in both rangekeepers.

A note on the servomechanisms used on the Mk.1 and Mk.1A computers: These were electromechanical, using reversible two-phase capacitor-run induction motors and tungsten contacts. They were stabilized primarily by rotary magnetic drag (eddy-current) slip clutches, like high-torque versions of classical rotating-magnet speedometers. One part of the drag was geared to the motor, and the other was constrained by a fairly-stiff spring. The latter part offset the null position of the contacts by an amount proportional to motor speed, thus providing velocity feedback. Flywheels mounted on the motor shafts, but coupled by magnetic drags, prevented contact chattering when the motor was at rest. Unfortunately, they also must have slowed down the servos somewhat.A more-elaborate scheme, which placed a rather-large flywheel and differential between the motor and the magnetic drag, removed velocity error for critical data, such as gun orders.

The Mk. 1 and Mk. 1A computers used a motor with its speed regulated by a clock escapement, cam-operated contacts, and a jeweled-bearing spur-gear differential to drive the integrator discs. Although its speed cycled slightly, the total inertia made it effectively a constant-speed motor. At each tick, contacts switched on motor power, then the motor opened the contacts again. It was in effect slow pulse-width modulation of motor power according to load. When running, the computer had a unique sound as motor power was switched on and off at each tick -- dozens of gear meshes inside the cast-metal computer housing spread out the ticking into a "chunk-chunk" sound.

ome notes on the computing mechanisms

These computers had to be formidably rugged, partly to withstand the shocks created by firing the ship's own guns, and also to withstand the effects of hostile enemy hits to other parts of the ship. They not only needed to continue functioning, but also stay accurate.

The Mk. 1/1A mechanism was mounted into a pair of approximately-cubical large castings with very wide openings, the latter covered by gasketed castings. (See the Ford Instrument Company museum site, listed below under "See Also".) Individual mechanisms were mounted onto thick aluminum-alloy plates, and along with interconnecting shafts, were progressively installed into the housing. Progressive assembly meant that future access to much of the computer required progressive disassembly.

A Navy Ordnance Pamphlet (OP) (OP number [s] needed!), actually a two-volume book with several hundred pages and several hundred excellent photographs, described in great detail how to dismantle and reassemble. (That OP was a treasure!) When reassembling, shaft connections between mechanisms had to be loosened and the mechanisms mechanically moved so that an output of one mechanism was at the same numerical setting (such as zero) as the input to the other. OP 1140, cited below, gives specific procedures, but these perhaps were superseded. Most fortunately, these computers were especially well-made, and very reliable.

The mechanisms were interconnected by rotating shafts mounted in ball bearings fitted into brackets fastened to the support plates. Just about every corner was a right angle, and nearly all were done by miter gears (1:1 ratio). Contrast this with the ease of running a wire carrying data, or having a copper trace on a circuit board.

The Mk 47 computer was a radical improvement in accessibility. It was more akin to a tall, wide storage cabinet in shape, with most or all dials on the front vertical surface. Its mechanism was built in six sections, each mounted on very heavy-duty pull-out slides. Behind the panel were typically a horizontal and a vertical mounting plate, arranged in a tee. (Of course, the computer was mounted so the slides moved fore and aft! They were heavy, and a ship rolls much more off the vertical than it pitches.)

There were rotating shafts to interconnect the six sections, by way of shafts inside the back of the cabinet. However, it was not necessary to adjust the connection as described above for the Mk. 1/1A. Shrewd design meant that the data carried by these shafts had no boundaries. Only their movement was what mattered. One such sort of data could be the aided-tracking output from an integrator roller. When a section was put back into normal position, the shaft couplings apparently mated as soon as the shafts rotated.

Typical mechanisms in the Mk. 1/1A were lots of miter-gear differentials, a group of four 3-D cams, some disk-ball-roller integrators, and servo motors with their associated mechanism; all of these had bulky shapes. However, most of the computing mechanisms were thin stacks of wide plates of various shapes and functions. A given mechanism might be an inch thick, possibly less, and more than a few were maybe 14 inches across. Thinness meant that they took up less space, while width permitted a total range of movement much greater than slight looseness in sliding parts; that width enhanced accuracy.

The Mk. 47 had gears and shafts, differentials (although fewer), totally-enclosed disk-ball-roller integrators, but no mechanical multipliers or resolvers ("component solvers"); they were electrical. (Precision potentiometers did the multiplying.) It was an hybrid, doing some computing electrically, and the rest mechanically.

In the Mk. 1/1A, however, except for the electrical (not electronic) servos, all computing was mechanical. For a truly excellent and possibly very interesting set of illustrations and explanations, see Chapter 2 of the Navy manual OP 1140, cited below under "See Also".

The integrators had rotating discs and a full-width roller mounted in a hinged casting (!), pulled down toward the disc by two strong springs. Twin balls permitted free movement of the radius input with the disk stopped, something done at least daily for static tests. Integrators were made with discs of 3, 4 and 5 inch diameters, the larger being more accurate. Ford Instrument Company integrators had a clever mechanism for minimizing wear when the ball-carrier carriage was in one position for extended periods.

Resolvers, called "component solvers" back then, did polar-to-rectangular conversion. One input was an angle, and the other, the magnitude, expressed as a radius.

Steam enthusiasts know of the Scotch yoke, and a common type of this resolver mechanism could be described as crossed Scotch yokes at 90 degrees, with a variable-radius crankpin.

Component integrators were essentially Ventosa integrators. all enclosed. Think of a traditional heavy-ball computer mouse and its pickoff rollers at right angles to each other. Underneath the ball is a roller that turns to rotate the mouse ball. However, the shaft of that roller can be set to any angle you want. In the Mk. 1/1A, a rate-control correction (keeping the sights on target) rotated the ball, and the two pickoff rollers at the sides distributed the movement appropriately according to angle. That angle depended upon the geometry of the moment, such as which way the target was heading.

Three-dimensional cams for ballistic computation rotated on their axis for one input. The other input moved a ball follower along the length of the cam.

The four cams in the Mk. 1/1A computer provided mechanical time fuse setting, time of flight (this time is from firing to bursting at or near the target), time of flight divided by predicted range, and superelevation combined with vertical parallax correction. (Superelevation is essentially the amount the gun barrel needs to be raised to compensate for gravity drop.)

Of course, inside the computer were many synchros, both to receive and to send data.

Some minor corrections and other notes:

The specifications stated (on the order of 100 A.) for current consumption of the Mk. 1 are vastly more than what's required for normal computer operation. It's likely that the servos and time motor could be powered by a 15-ampere 115 V 60 Hz circuit. Apparently, the stated figure was worst-case for planning purposes, and probably referred to current drawn by all synchro transmitters with their receiver rotors locked in worst-case misalignment, or their stator leads short-circuited. (Computer power was the same as ordinary utility power in the USA, 115 V 60 Hz.)

Another item of confusion is that the discs in the integrators are not nearly as thick as one inch, or even a half inch; they are more like 1/4 inch thick. (They are also removable and reversible should they become worn).

Some National Archives photos of the insides of the Mk. 1 are rotated a quarter turn; the top of the computer is to the left, on at least one. The flywheels for the servos, and those with mimimal velocity error are easy to see (when you know what to look for!).

ee also

:* [http://www.fischer-tropsch.org/primary_documents/gvt_reports/USNAVY/USNTMJ%20Reports/USNTMJ-200F-0086-0124%20Report%20O-32.pdf USN Report on IJN Technology] :* [http://www.ijnhonline.org/volume1_number2_Oct02/articles/article_hone1_doctrine.doc.htm Excellent article on the performance of long-range gunnery between the World Wars.] :* [http://www.hnsa.org/doc/br224/index.htm British fire control] :*British fire control expert:* [http://dcoward.best.vwh.net/analog/ford.htm Ford Instrument Company museum site. Ford built rangekeepers for the US Navy during World Wars I and II] :* [http://hnsa.org/doc/op1140a/index.htm OP1140, a superb Navy manual. Chapter 2 has many fine illustrations and clearly-written text.]

References/Endnotes