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\section{What is Maxima?}
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Maxima (pronounced \maxp\footnote{The acronym \Max\ is the corruption of the main project name MACSYMA, which stands for Project MAC's
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SYmbolic MAnipulation System. {\it MAC} itself is an acronym, usually
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cited as meaning Man and Computer or Machine Aided Cognition. The
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Laboratory for Computer Science at the Massachusetts Institute of Technology
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was known as Project MAC during the initial development of MACSYMA. The name
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MACSYMA is now trademarked by Macsyma Inc.})
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is a large computer program designed for the manipulation
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of algebraic expressions. You can use \Max\
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for manipulation of algebraic expressions involving
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constants, variables, and functions. It can
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differentiate, integrate, take limits, solve equations,
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factor polynomials, expand functions in power series, solve differential
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equations in closed form, and perform many other
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operations. It also has a programming language that you can use to
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extend Maxima's capabilities.
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\subsection*{The Dangers of Computer Algebra}
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With all this marvelous capability, however, you must bear in mind the
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limitations inherent in any such tool. Those considering the use of
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computers to do mathematics, particularly
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students, must be warned that these systems are no substitute for
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hands on work with equations and struggling with concepts. These systems
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do not build your mathematical intuition, nor will they strengthen
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your core skills. This will matter a great deal down the road, especially
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to those of you who wish to break new ground in theoretical mathematics
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and science. Do not use a computer as a substitute for your basic
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By the same token, however, proficiency with computers and computer
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based mathematics is crucial for attacking the many problems which
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literally cannot be solved by pencil and paper methods. In many cases
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problems which would take years by hand can be reduced to seconds
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by powerful computers. Also, in the course of a long derivation, it
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is sometimes useful for those who have already mastered the fundamentals
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to do work in these systems as a guard against careless errors, or
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a faster means than a table of deriving some particular result. Also,
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in case of an error, fixing the resulting error can often be much
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quicker and simpler courtesy of a mathematical notebook, which can
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be reevaluated with the correct parameters in place.
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But just as a computer can guard against human error, the human must
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not trust the computer unquestioningly. All of these systems have
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limits, and when those limits are reached it is quite possible for
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bizarre errors to result, or in some cases answers which are actually
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wrong, to say nothing of the fact that the people who programmed these
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systems were human, and make mistakes. To illustrate the limits of
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computer algebra systems, we take the following example: when given the
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integral \verb@Integrate 1/sqrt(2-2*cos(x))@ \verb@from x=-pi/2 to pi/2@,
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Mathematica 4.1 gives, with no warnings, \verb@\!\(2\ Log[4] - 2\ Log[Cos[\[Pi]\/8]]@
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\verb@+ 2\ Log[Sin[\[Pi]\/8]]\)@ which \verb@N[%]@ evalutates numerically to
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give 1.00984. Maxima 5.6 returns the integral unevaluated, the commercial
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Macsyma says the integral is divergent, and Maple 7 says infinity. (Cite Maxima
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Had the person who wished to learn the result blindly trusted most of the
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systems in question, he might have been misled. So remember to think about
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the results you are given. The computer is not always necessarily right,
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and even if it gives a correct answer that answer is not necessarily complete.
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\section{A Brief History of Macsyma}
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The birthplace of Macsyma, where much of the original coding took place, was
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Project MAC at MIT in the late 1960s and earlier 1970s. Project MAC was an MIT
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research unit, which was folded into the current Laboratory for Computer
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Science. Research support for Macsyma included the Advanced Research Projects
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Agency(ARPA), Department of Defense, the US Department of Energy, and other
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government and private sources.
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The original idea, first voiced by Marvin Minsky, was to automate the kinds
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of manipulations done by mathematicians, as a step toward understanding the
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power of computers to exhibit a kind of intelligent behavior. \cite{MAC-M-124}
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The undertaking grew out of a previous effort at MITRE Corp called Mathlab,
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work of Carl Engelman and others, plus the MIT thesis work of Joel Moses on
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symbolic integration, and the MIT thesis work of William A. Martin. The new
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effort was dubbed Macsyma - Project MAC's SYmbolic MAnipulator. The original
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core design was done in July 1968, and coding began in July 1969. This was long
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before the days of personal computers and cheap memory - initial development
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was centered around a single computer shared with the Artificial Intelligence
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laboratory, a DEC PDP-6. This was replaced by newer more powerful machines
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over the years, and eventually the Mathlab group acquired its own DEC-PDP-10,
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MIT-ML running the ITS operating system. This machine became a host on the
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early ARPANET, predecessor to the internet, which helped it gain a
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wider audience. As the effort grew in scope and ability the general
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interest it created led to attempts to "port" the code - that is, to
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take the series of instructions which had been written for one machine
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and operating system and adapt them to run on another, different
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system. The earliest such effort was the running of Macsyma in a MacLisp
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environment on a GE/Honeywell Multics mainframe, another system
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at MIT. The Multics environment provided essentially unlimited address
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space, but for various reasons the system was not favored by
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programmers and the Multics implementation was never popular.
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The next effort came about when a group at MIT designed and implemented a machine
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which was based on the notion that hardware support of Lisp would make
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it possible to overcome problems that inhibited the solution of
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many interesting problems. The Lisp machine clearly had to support
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Macsyma, the largest Lisp program of the day, and the effort paid off with
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probably the best environment for Macsyma to date (although requiring something
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of an expert perspective). Lisp machines, as well as other special
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purpose hardware, tended to become slow and expensive compared to
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off-the-shelf machines built around merchant-semiconductor CPUs, and
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so the two companies that were spun off from MIT (Symbolics Inc, and
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LMI) both eventually disappeared. Texas Instruments built a machine called
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the Explorer bases on the LMI design, but also stopped production.
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Around 1980, the idea of porting Macsyma began to be more interesting,
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and the Unix based vaxima distribution, which ran on a Lisp system
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built at the University of California at Berkeley for VAX UNIX demonstrated
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that it was both possible and practical to run the software on less expensive
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systems. (This system, Franz Lisp, was implemented primarily in Lisp with
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some parts written in C.) Once the code stabilized, the new version opened up
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porting possibilities, ultimately producing at least six variations on the
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theme which included Macsyma, Maxima, Paramax/Paramacs, Punimax, Aljbar, and
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Vaxima. These have followed somewhat different paths, and most were destined
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to fade into the sunset. The two which survived obscurity, Maxima and
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Macsyma, we will discuss below. Punimax was actually an offshoot of Maxima -
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some time around 1994 Bruno Haible (author of clisp) ported maxima to clisp.
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Due to the legal concerns of Richard Petti, then the owner of the commercial
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Macsyma, the name was changed to Punimax. It has not seen much activity since
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the initial port, and although it is still available the ability of the main
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Maxima distribution to compile on Clisp makes further development of
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There is a certain surprising aspect in this multiplicity of versions
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and platforms, given how the code seemed tied to the development environment,
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which included a unique operating system. Fortunately, Berkeley's building
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a replica of the Maclisp environment on the MIT-ML PDP-10, using tools
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available in almost any UNIX/C environment, helped solve this problem.
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Complicating the matter was the eventual demise of the PDP-10 and
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Maclisp systems as Common Lisp (resembling lisp-machine lisp), influenced by
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BBN lisp and researchers at Stanford, Carnegie Mellon University, and Xerox, began to take
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hold. It seemed sensible to re-target
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the code to make it compatible with what eventually
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became the ANSI Common Lisp standard. Since almost everything needed for
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for Macsyma can be done in ANSI CL, the trend toward standardization
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made many things simpler. There are a few places
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where the language is not standardized, in particular connecting to
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modules written in other languages, but much of the power of the
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system can be expressed within ANSI CL. It is a trend the Maxima
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project is planning to carry on, to maintain and expand on this
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flexibility which has emerged.
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With all these versions, in recent history there are two which have
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been major players, due this time more to economics than to code
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quality. 1982 was a watershed year in many respects for Macsyma - it
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marks clearly the branching of Macsyma into two distinct products,
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and ultimately gave rise to the events which have made Maxima both
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possible and desirable. MIT had decided, with the gradual spread of
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computers throughout the academic world, to put Macsyma on the market
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commercially, using as a marketing partner the firm of Arthur D. Little, Inc.
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This version was sold to the Symbolics Inc., which, depending on your
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perspective, either turned the project into a significant marketing effort
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to help sell their high-priced lisp machines, or was a diversionary
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tactic to deny their competitors (LMI) this program. At the same
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time MIT forced UC Berkeley (Richard Fateman) to withdraw the
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copies from about 50 sites of the VAX/UNIX and VAX/VMS versions
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of Macsyma that he had distributed with MIT's consent, until some
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agreement could be reached for technology transfer. Symbolics hired some of
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the MIT staff to work at Symbolics in order to improve the code,which was
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now proprietary. The MIT-ML PDP-10 also went
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off the Arpanet in 1983. (Interestingly, the closing of the MIT Lisp
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and Macsyma efforts was a key reason Richard Stallman decided to form
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the Free Software Foundation.) Between the high prices, closed
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source code, and neglecting all platforms in favor of Lisp Machines
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pressure came to bear on MIT to release another version to accommodate
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these needs, which they did with some reluctance. The new version was distributed
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via the National Energy Software Center, and called DOE Macsyma. It
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had been re-coded in a dialect of lisp written for the VAX at MIT
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called NIL. There was never a complete implementation. At about the
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same time a VAX/UNIX version "VAXIMA" was put into the same library by
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Berkeley. This ran on any of hundreds of machines running the Berkeley version of
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VAX Unix, and through a UNIX simulator on VMS, on any VAX system.
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The DOE versions formed the basis of the subsequent non-Symbolics
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distributions. The code was made available through the National Energy
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Software Center, which in its attempt to recoup its costs, charged
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a significant fee \verb@($1-2k?).@ It provided full source, but in a
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concession to MIT, did not allow redistribution. This prohibition seems
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to have been disregarded, and especially so since NESC disappeared. Perhaps
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it didn't recoup its costs! Among all the new activity centered around DOE
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Macsyma, Prof. William Schelter began maintaining
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a version of the code at UT Austin, calling his variation Maxima. He
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refreshed the NESC version with a common-lisp compatible code version.
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There were, from the earliest days, other computer algebra systems
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including Reduce, CAMAL, Mathlab-68, PM, and ALTRAN. More
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serious competition, however, did not arrive until Maple and Mathematica were
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released, Maple in 1985 (Cite list of dates) and Mathematica in 1988 (cite
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wolfram website). These systems were inspired by Macsyma in terms of their
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capabilities, but they proved to be much better at the challenge of building
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mind-share. DOE-Macsyma, because of the nature of its users and maintainers, never
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responded to this challenge. Symbolics' successor Macsyma Inc, having
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lost market share and unable to meet its expenses, was sold in the summer of
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1999 after attempts to find endowment and academic buyers failed. (Cite Richard
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Petti usenet post.) The purchaser withdrew Macsyma from the market
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and the developers and maintainers of that system dispersed. Mathematica and Maple
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appeared to have vanquished Macsyma.
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It was at this point Maxima re-entered the game. Although it was not
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widely known in the general academic public, W. Schelter had been
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maintaining and extending his copy of the code ever since 1982. He
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had decided to see what he could do about distributing it more widely.
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He attempted to contact the NESC to request permission to distribute
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derivative works. The duties of the NESC had been assumed in 1991 by
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the Energy Science and Technology Software Center, which granted him
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virtually unlimited license to make and distribute derivative works,
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with some minor export related caveats.
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It was a significant breakthrough. While Schelter's code had been available
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for downloading for years, this
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activity became legal with the release from DOE granted in Oct. 1998, and
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Maxima began to attract more attention.
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When the Macsyma company abruptly vanished in 1999, with no warning or
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explanation, it left their customer base hanging. They began looking
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for a solution, and some drifted toward Maxima.
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Dr. Schelter maintained the Maxima system until his untimely death in
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July, 2001. It was a hard and unexpected blow, but Schelter's obtaining
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the go-ahead to release the source code saved the project and possibly
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even the Macsyma system itself. A group of users and developers who
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had been brought together by the email list for Maxima decided to try and
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form a working open source project around the Maxima system, rather
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than let it fade - which is where we are today.
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