When the worksheets returned to the planning committee, they were checked for errors. Blanch and Lowan went to extraordinary lengths to find every possible mistake, including errors created in transcribing the values to the final table. Behind their efforts was the knowledge, rarely mentioned, that one poorly computed table could permanently damage their reputation and limit their acceptance within the scientific community. “We had very little equipment,” reported Gertrude Blanch, “and a lot of supervisors who thought that this was another boondoggle project.”68 The first defense against errors was found within the worksheets. Blanch designed them in such a way that elementary mistakes in calculation could be quickly identified. The computers filled in a grid of numbers. The last column of this grid had to match a set of predetermined values, numbers that had been computed by the planning committee or one of the special computing groups. If they did not match, the sheet contained a mistake in calculation.
Some computers attempted to avoid the discipline imposed by the worksheets. Recognizing the structure of the sheet, a worker could put the predetermined values in the last column and then fill the remaining blanks with any values that seemed appropriate. This did not happen often, given the quality of the final tables, but each member of the planning committee could recall at least one incident of cheating. Blanch remembered two women who submitted falsified worksheets. “You can’t cheat in those things,” she said flatly. “It comes out almost immediately. Well, these girls were fired immediately.”69 The other examples of cheating or carelessness also ended in dismissals. One resulted in the departure of a planning committee member.70
Blanch checked each calculation with six or eight different procedures. Some of the procedures were based upon the mathematical properties of the function they were tabulating. All tables were checked with the technique known as differencing. Differencing had been used by the computers of de Prony and Maskelyne, but it reached a new level of sophistication in the hands of Gertrude Blanch. It reversed the method on which Charles Babbage based his difference engine. Babbage had shown that many functions could be reduced, through the method of finite differences, to a series of repeated additions of constant terms. Blanch used this technique to check tables by repeatedly taking the differences of adjoining terms. In a table of cubes, the first five values are the numbers 1, 8, 27, 64, and 125. The first differences are 8 − 1, 27 − 8, 64 − 27, and 125 − 64, or 7, 19, 37, and 61. If we repeat the process and take adjoining differences of these values, we get 19 − 7, 37 − 19, and 61 − 37, or 12, 18, and 24. In one final set of differences we get 18 − 12 and 24 − 18, or the numbers 6 and 6. If one of these third differences were not 6, then one of the original values in the table would be in error. For most functions tabulated by the Mathematical Tables Project, the sixth or seventh repeated difference would produce a list of nearly constant numbers. Any significant variation would indicate an error in the calculations.
By WPA regulation, the human computers worked thirty-two hours a week. They were supposed to spend the remaining time looking for permanent employment. This limitation on the program gave Blanch the opportunity to study the worksheets when the computing floor was silent. She spent many an evening hour checking calculations and preparing new worksheets for the next day. She seemed to enjoy the quiet time after the computers left for the day, when there were no workers to encourage, no questions to answer, no formulas to explain. The office was quiet, and she could concentrate on mathematics.
The Mathematical Tables Project finished the table of powers in early April 1938, printed the manuscript with a mimeograph stencil on rough sheets of paper, and bound the pages with a cardboard cover. A few of the numbers were a little difficult to read. Prominently displayed on the face of the book was a WPA seal, a small rectangle that identified it as the product of a relief effort. Lowan and Blanch claimed that there were no errors in the table, a claim that was tested when a copy reached the desk of L. J. Comrie in England. Comrie reviewed the entire work, checked every value with a difference test, and concurred that the table was entirely free of errors.71
Finishing the first table was a relief to the entire group and created what Blanch called a sense of “heartwarming camaraderie” among the staff.72 It steeled their courage for their second project, creating tables of the exponential function, ex. The exponential function is harder to compute than a simple power of an integer. It is the inverse of the logarithm and can be expressed as the sum of an infinite number of terms, ex = x/1 + x2/2 + x3/6 + x4/24 and so on, continuing in the same pattern. In practice, a computer does not need to deal with an infinite number of operations, as the terms eventually become so small that they can be safely ignored.
The computers began working on the exponential calculations in the early spring, while they were still working on the powers of integers. It seemed important that this project be under way when the WPA reviewed the progress of the group in June. For all the anxiety among the planning staff, the review proved to be quick and perfunctory. The WPA officials reported that they were satisfied with the accomplishments of Blanch, Lowan, and the human computers and approved another six months of budget for the group.73 With the future of the project on a stronger financial foundation, Lowan was able to arrange for Columbia University Press to publish the tables from the project.
While Blanch was preparing computing plans and directing the daily operations of the human computers, Lowan was attempting to bring the Mathematical Tables Project to the attention of American scientists and engineers. His first problem was to overcome the stigma of the WPA. His second was to show that his group could equal the accomplishments of the two other scientific computing organizations in New York City. Thornton Fry, Clara Froelich, and the other Bell Telephone Laboratory computers worked two miles south of the Mathematical Tables Project office. Wallace Eckert and the Columbia University Astronomical Computing Bureau sat a slightly greater distance to the north. Both had reputations that Lowan would have liked to equal.
Beginning in the fall of 1938, Lowan began to promote his project as a general-purpose computing office for the nation’s engineers and scientists. He prepared mimeographed circulars, using the same machines that reproduced his computing sheets, and mailed them to universities, government offices, manufacturers, and every scientist he could name.74 In general, these efforts at self-promotion were frustrating affairs, as he might have learned from anyone who dealt with direct mail advertisement. He often posted one hundred or two hundred letters in a month and received not a single reply. When he did receive a reply, he was usually disappointed. Lowan received requests to prepare salary tables, to tabulate data, and to do simple calculations for other WPA projects. In general, he tried to decline such work by citing the WPA requirement that all products needed to be “widely usable.”75 However, he quickly learned that he was part of a government organization that could be subject to political pressure. More than once, a rejected request returned to his office in the form of a letter from Lyman Briggs. These letters reminded Lowan that the Mathematical Tables Project received government funds and that it would be well advised to honor the request, no matter how uninteresting it might be.76
The few letters from well-reputed scientists could be tantalizingly sad, like the gentleman caller who offers a moment of hope and happiness and promise before disappearing. In response to a circular, John von Neumann wrote, “Many thanks for the announcement of your project. I am much interested in your program and should like to get your material.” He ended the letter, “I may have some remarks and suggestions in connection with these things and will write to you concerning them before long.”77 As Lowan had known von Neumann at the Institute for Advanced Study, he had some reason to expect that the mathematician would act on his promise, but the weeks passed, and the Mathematical Tables Project received no further correspondence.
The one scientist who offered more than hope was Philip Morse (1903–1985), a professor of physics at the Massachusetts Institute of Techn
ology. “I have been in charge of a number of similar projects,” he wrote, and felt “that I should get in touch with you.” With no more introduction or pleasantries, he then described a lengthy calculation, acknowledging at the end that “this calculation has no particular difficulties but requires considerable man-hours and if you are in a position to be of help, this help would be welcomed.” Such interest was well received by Lowan, Blanch, and the rest of the senior staff, but it was accompanied by an embarrassing query. Morse ended his letter by stating, “I will be very pleased to know just what sort of arrangements you make on calculating facilities and in general how you operate.”78
Lowan responded quickly, assuring Morse that his group would be happy to compute for anyone at the Massachusetts Institute of Technology. He gave a brief overview of the Mathematical Tables Project, telling Morse that “we are operating with a staff of 110 workers under the supervision of a planning section of which I am in charge.” He mentioned nothing about worksheets or relief workers. When he reached the topic of machinery, he equivocated. “Most of the work is, of course, done with the aid of calculating machines.”79 Strictly speaking, the statement was true, as the project’s three adding machines contributed indirectly to everything that the staff did. However, the bulk of the calculations were done with only paper and pencil.
We do not know when Morse learned the true nature of the Mathematical Tables Project computers. He was unusually well connected in the scientific community and may have known from the start that Lowan was stretching the truth. He never criticized Lowan, never suggested that there was anything wrong with the project. In fact, he seemed to grasp innately the nature of the group. In his second letter to Lowan, he said that he had “one or two other sets of calculations which I want to suggest to you in the next month or so” and that he would “consult with a number of the other members of the Physics and Mathematics Departments here and at Harvard in order to submit to you a program that is not a set of personal interest.”80
The last phrase in Morse’s letter was important. Lowan had to establish that every calculation was of general interest and would benefit more than a single scientist or engineer. None of the tables could be copyrighted, as they had been prepared with public money. It was one of the WPA policies that governed his actions as long as he accepted their money. The WPA also required that he use labor-intensive methods, in order to employ the greatest number of people, and restricted the number of female employees to about 20 percent of his staff, an attempt to prevent one household from receiving two sets of benefits.81 Every request for scientific calculation had to follow the WPA rules, even if that meant using methods that seemed archaic or unsuited to 1930s research.
Morse never seemed to be bothered by the WPA restrictions. He proved to be a tireless ally and a key link to the scientific community. He helped bring the group its first important scientific computation, a problem posed by the Cornell physicist Hans Bethe (1906–). Bethe was studying the processes that produce solar energy. In order to understand these reactions, he needed a table that would show the internal temperature of the sun, beginning at the outer edges and moving inward to the core. Since these measurements could not be taken directly, they had to be computed from a mathematical model.82 Bethe claimed that the calculations were not “prohibitively laborious” but were so difficult that no one had attempted the work without first substantially simplifying them.83
Lowan eagerly agreed to undertake the project, for the work would bring the Mathematical Tables Project to the attention of astrophysicists and might also show that the group was as capable as the Columbia Astronomical Computing Bureau. If he expected that the calculations would be done quickly, he was mistaken. After the planning committee reviewed the plan, they had a half dozen questions for Bethe. They asked Bethe to verify equations and check key values. The calculations involved a three-dimensional system of differential equations, the form of equations found in Richardson’s weather model. Though the equations were simpler than those posed by Richardson, they still demanded a great deal of work. If the starting values were wrong, the effort would be wasted. Not even Hans Bethe could adjust the sun and bring it into agreement with a mistaken calculation.
In the end, Gertrude Blanch decided to do all the computations herself. She concluded that she could calculate the tables in the time that it would take her to prepare worksheets for the computing floor.84 The calculations took about three weeks of effort. Bethe was impressed with the results, calling them the “first modern determination of the temperature of a star.”85 The table was published in the Astrophysical Journal, which must have pleased Lowan. It was the first major publication to come from the project. Perhaps more important, it brought attention from a major physics department. Bethe reported that his colleagues had taken a look at the group’s first tables and that “the department has decided to order copies of them.”86
By this time, the project’s second volume, Tables of the Exponential Function, had appeared in print. Compared to Karl Pearson’s Tracts for Computers or H. T. Davis’s Tables of Higher Mathematical Functions, this new volume received serious attention from the scientific press. Though book reviewers inevitably mentioned that the volume had been produced as a relief project, they generally treated it with respect. The review in the widely circulated American Mathematical Monthly was positive, though perhaps not as enthusiastic as Lowan might have liked. The reviewer, John Curtiss (1909–1977) of Northwestern University, noted that the book claimed to be “entirely free from error.” If he had been L. J. Comrie, he would have tested this assertion. As he was not, he was willing to concede that “the precautions taken seem to give considerable weight to this claim.”87
John Curtiss’s review was an important milestone for Blanch and Lowan, evidence that they had done their job well, had met the requirements of the WPA, and had created an organization that could produce something of value to at least a few scientists. Nevertheless, the project was still a work relief effort. It still occupied a dirty industrial loft in an unattractive part of Manhattan. Each week, the members of the computing floor collected their wages hoping that by the next payday they would have a full-time job with some employer that was not the WPA.
CHAPTER FOURTEEN
Tools of the Trade: Machinery 1937
Some of them hated the mathematics that drove them, and some were afraid, and some worshiped the mathematics because it provided a refuge from thought and feeling.
John Steinbeck, The Grapes of Wrath (1939)
WHEN MATILDA PERSILY came to work at the Mathematical Tables Project, she followed a little ritual to prepare her tools for the day. She was part of the special computing group, one of the few who regularly used an adding machine. Her machine was a old Sunstrand, an inexpensive device that had ten keys on the top and a crank on the right side. She would first add a few meaningless numbers while sharing a moment of conversation with other members of the staff. As she talked, she would listen for the sound of grit in the gears and try to feel any catch or slip in the mechanism. She could apply some lubricating oil from a can that sat on a small shelf, but too much oil would attract the very dust she was trying to avoid. She had found that the machine was best cleaned with an orangewood stick; such sticks were sold at drugstores to groom fingernails and cuticles. Once convinced that the machine was ready, she would sit at her desk, take out her worksheets, and begin to compute.
Most of the computing machinery acquired by the Mathematical Tables Project during its first years of operation was scavenged from terminated WPA offices and other government agencies.1 Ida Rhodes described this equipment as “the most broken down, the most ancient of contraptions.”2 The majority were inexpensive, hand-cranked Sunstrand calculators. They were not designed for the repetitious work of the computing floor, nor was their crank mechanism easy for project computers to pull. Rhodes would complain, “Oh, how my arm ached by the end of the day.”3 Few of these machines were in operating condition when they arrived at the
project’s office. A couple of the senior computers had learned how to transplant gears and levers from one machine to another and were able to produce two or three working machines for every four that came to their door. Through 1938 and 1939, the group had no other way to acquire calculators. A new mechanical calculator cost $400, almost as much as the $560 annual salary paid to a WPA worker.
During the Great Depression, most computing laboratories had at least one staff member with enough mechanical expertise to repair or modify their desk calculators. Even the smallest laboratory had to follow a maintenance regimen for its machines, oiling gears and adjusting levers. Observatories relied on the same technicians that kept telescopes in working order. The larger organizations, such as the Iowa State Statistical Laboratory, kept a mechanic on their staffs. The Iowa State laboratory, formerly known as George Snedecor’s Mathematical and Statistical Service, had to maintain an unusually large number of machines, as the laboratory acted as a general computing facility for the campus. It not only did calculations for any college faculty member but also provided adding machines and calculators to campus laboratories and offices. Laboratory staff kept the machines in good repair and trained the people who would use them.4 With such a responsibility, it is perhaps not surprising that the laboratory director in 1938, Alva E. Brandt (1898–1975?), was a trained mechanical engineer and had once served as a professor of farm machinery at Oregon State University.5
The computing office at Bell Telephone Laboratories probably had the greatest access to trained engineers and inventors. Regularly, laboratory scientists turned their gaze on the computing staff of Clara Froelich and saw them as a model for some new calculating device. “In these laboratories,” observed the staff scientist George Stibitz (1904–1995), “we have 10 or more girls, including at least one Hunter graduate, who spend most of their time dealing with [complex numbers].” Stibitz, a physicist, was a new addition to the mathematical division. In spite of his patronizing language toward the women, who were often a decade or more older than himself, he was one of those rare scientists who treated the computers as individuals. He seems to have been friendly with them, pausing over their desks to share a bit of conversation, comment on the upcoming weekend, and learn how they handled their calculations. He rarely described his early research without mentioning them.
When Computers Were Human Page 28
