Although Atanasoff made every effort to find out about what calculating machines were being invented, and Alan Turing was as well connected as a mathematician could be, neither one of them was, or perhaps could have been, familiar with Konrad Zuse, an inventor who was working in Berlin. Zuse, born on June 22, 1910, was two years older than Turing and, unlike the others involved in the invention of the computer, he wrote his autobiography, entitled The Computer—My Life. Konrad Zuse, born in Berlin, was the son of a Prussian civil servant. Of his ancestors, he writes, “I have traced my ancestry back to my great-grandparents, who lived in the village of Voigtshagen in Pomerania. Many a shepherd is said to be among their forefathers. Perhaps this explains my inclination toward introversion.” The elder Zuse’s work with the civil service took his family to the small city of Braunsberg in East Prussia, south of Rostock, before the First World War, but even before that departure, the little boy was fascinated with architecture, noticing the railroad bridges in Berlin and the patterns they made as they overlapped one another in his childish gaze. Zuse’s earliest strong memories were of the dangers and fears of World War I—of the influx of refugees from the eastern front, where the German armies were fighting Russia, and of fires in Braunsberg itself, especially in the medieval section of the town.
Zuse’s earliest aptitudes were not for math or engineering but for performance and visual art. As a child, he loved traveling circus troupes that performed in the post office square in Braunsberg, and he emulated them by perfecting his own routine—doing tricks while balancing on an empty oil drum. His schooling was standard for the time and place—his worst subsequent memories were of eight hours of Latin class every week. Like Atanasoff, he was something of a disruptive influence in class when he was not merely inattentive—the margins of his Latin textbook were filled with drawings, and one teacher nicknamed him “Dozy.” His drawing skills were appreciated, though—the art teacher saw some of his work and persuaded his father to give him higher-quality drawing paper. During his teens, the family moved from Braunsberg to Hoyerswerda, not far from Dresden, where the gymnasium provided a more modern education, with younger teachers. Hoyerswerda was in an industrial area of Germany, which piqued Zuse’s interest in technology, and his great hobby, in addition to drawing, was his Stabilbauskasten, or Erector Set. At one point, he built a model of a large crane in his room and then sketched a picture of himself lounging underneath it, with his feet on the desk. Like Atanasoff, he graduated early. Around the same time, he acquired a bicycle that was bent to the right side. Zuse attached a string to the left side of the handlebars so that he could ride with no hands, like his friends did. When the bicycle repairman could not fix the gearing mechanism, he fixed it himself with some pieces from his building set.
Undecided about whether to pursue art or engineering, he pursued engineering, but with a continued interest in design—for his senior school project, he had designed a city of the future (à la Fritz Lang’s Metropolis) based on a hexagonal grid. Like Alan Turing, Zuse was educated in a system that focused on a child’s emotional and philosophical life as well as his intellectual life, and at the end of school, like Turing, Zuse found himself to be something of an outsider—to the disappointment of his very conventional parents, he no longer believed in God or religion.
In 1927, at the same time that Turing was making his difficult way through the Sherborne School, Zuse entered the Technische Universität in Berlin and took up residence in the city of his birth, a sociable young man in an exciting and rapidly changing urban environment. He was immediately fascinated once again by the bridge building that was going on, a fascination that was encouraged by the requirement that students at the Technische Universität had to have practical experience in ironwork or carpentry or bricklaying. Zuse’s experience in these trades served to break down class barriers somewhat, but he remained a thinker more than a builder—interested in photography, movies, drawing, performance. When he became intrigued by a technological question, such as how to build a rocket that might head to a distant star, it was more often through some form of art, such as science fiction, than it was through science itself. He does not mention taking an interest, per se, in physics or mathematics or cosmology, as Atanasoff and Turing did. He writes, “Given my many detours and by-ways, I am still amazed that I earned a diploma at all.” (And then he goes on to recount how he was lucky that his mathematics examiner asked a particular question—as he was eavesdropping upon the questions that the other students were asked, he realized that he could not have answered any of them.)
According to Zuse, amid all the busyness, freedom, and pleasure of his university and postuniversity life, there was not much understanding about what the Nazis were up to. While Zuse himself was reading Das Kapital and the autobiography of Henry Ford, neither he nor his friends paid much attention to those who were reading Mein Kampf. Zuse, as a son of the Prussian civil servant class, felt more inspired by the writings of Oswald Spengler than of Hitler, especially Spengler’s anti-Marxist 1920 political tract Prussianism and Socialism. Even so, Zuse found the Marxists he knew friendly and interested in discussion.
Times soon changed, and “on all sides now Germans were being forced into line and marched off,” yet Zuse and his fellow students seem to still have had the feeling that they had some freedom of opinion, some future in terms of working choices. And then, on the night of June 30 (the Night of the Long Knives), Hitler used his personal bodyguard, the SS, to purge Ernst Röhm, an enemy in the von Hindenburg government, and two hundred of his allies in the armed forces. When von Hindenburg died a month later, Hitler made himself president and head of the armed forces, which were henceforth to pledge allegiance to him personally rather than simply to the state. After von Hindenburg’s funeral, Hitler assumed the title “führer,” but, perhaps as an indication of Zuse’s ongoing focus on other things, he writes, “The psychological effect was that one assumed the impetuous and hysterical period would now be followed by a period of common sense and work.” Zuse belonged to a fraternity of long standing at the university. When the three Jewish members were required by the Hitler government to leave the club, the club decided to disband but ultimately did not do so, only because the Jewish members asked them not to.
At the end of his university career, Zuse idled about Berlin for a year, undecided about what to do next, but in 1935, aged twenty-five, he took an engineering job with the new aircraft division of Henschel and Son, a locomotive corporation that was to produce several types of planes for the Luftwaffe. Zuse, apparently alerted by his new job to the sorts of calculating problems that aircraft design required, quit almost immediately to begin his own project—a computer.
According to Zuse’s account, he started from scratch. “When I began to build my own computer, I neither understood anything about computing machines nor had I ever heard of Babbage.” Zuse is not clear about why he decided to build a computer, or the theoretical basis of the machine, but it seems to have grown out of his talent for and interest in design rather than the desire to solve a particular kind of mathematical problem. His first attempt at a machine had nothing to do with mathematics—it was a skeletal vending machine “which took money and gave mandarin [oranges], and sometimes, indeed, returned the money with the mandarins.”
Zuse’s working space was the living room of his parents’ apartment and his capital (amounting to several thousand marks at the most) came from his father’s and sister’s paychecks and the contributions of his friends who had managed to find jobs or had a bit of money. His collaborators were his friends from the technical school, who received their pay in the form of meals that Zuse’s mother provided. His raw materials were bought piecemeal when he had the money, and they were simple ones. One friend describes how he made the mechanical relays—Zuse would draw the pattern on a piece of paper, then the friend “pasted the paper on a small plywood board, then fixed the necessary number of metal sheets between it and a second board that lay under it.” He then “screwed the two boards together with threaded screws, and sawed out the form of the relays with a small, electric fretsaw.” He “made these relays by the thousand.”1
Zuse seems to have built on the loyalty he developed in his college fraternity to accrue dedicated student helpers, who, like Atanasoff and his students (and Turing, too) recognized that the sorts of calculations they were required to do normally with analog desk calculators were much easier with Zuse’s machine. But the project was secret (Zuse does not say why, but possibly the authorities would have looked with suspicion upon a project that was diverting parts and supplies from war preparations). Those working on it declared when asked that they were attempting to build an aircraft tank gauge, because the German Air Ministry was at the time sponsoring a contest to build such a machine.
The basis of Zuse’s original design was electromechanical, akin to telephone relays, with which Zuse happened to be familiar, but, like other pioneers, he soon realized that the number of relays required in even a small-capacity machine was impractically enormous—the machine he was building so filled the family apartment that a friend who was working on it later wrote, “It took up almost the entire living room. It was a permanent fixture in the apartment. I think that it was only after the house was bombed during the war that the first Zuse Universal Computing Machine could be moved into the museum.” In his first effort, Zuse had some success with his electromechanical ideas and was able to build a flexible enough device so that he could use it to test his ideas about switching and build his understanding of mathematical logic.
As his work progressed, Zuse decided he needed more reliable financing. In 1937, he got in touch with a Dr. Kurt Pannke, who manufactured calculators. Pannke told the young man, “I don’t want to discourage you from continuing work as an inventor and from developing new ideas, but I must go ahead and tell you one thing: in the field of computing machines, practically everything has been researched and perfected to the last detail.” When Zuse told Pannke that his prototype could multiply, Pannke was silent for a long time and then came for a visit to the machine. Zuse demonstrated that because (like Atanasoff) he was using binary numbers (only the digits 1 and 0), adding and multiplying amounted (literally) to the same thing. In his autobiography, Zuse demonstrates why this is. When a calculator uses ten digits (0–9), the number of different keys required to represent the multiplication table is unwieldy—0 × 1 = 0, 2 × 2 = 4, 6 × 6 = 36, 8 × 8 = 64, with each digit represented by a key of its own. As we saw in third grade, when we were learning the multiplication tables in the back of our arithmetic books, between 0 × 0 and 9 × 9, there are a hundred different numbers. In a binary system, 0 × 0 = 0, 1 × 0 = 0, 0 × 1 = 0, and 1 × 1 = 1. Only two digits are needed. The problem for Pannke, as a businessman, was that calculators that multiplied by repeated adding were cheaper to build than calculators that attempted to multiply; there was a limited market for calculators, so adding was good enough. Zuse points out, “To construct large and expensive computing machines for scientists, for mathematicians and engineers, appeared absurd, and above all held no promise of commercial success. These people didn’t have any money.” But Pannke gave Zuse about 7,000 reichsmarks, and he began to work on his second prototype, the Z2.
In his home workshop and at school, as well as in plans and diagrams, Atanasoff was trying this and that. The work was taxing and frustrating mostly because there was no apparent place to begin. Every idea he came up with immediately branched into a tangle of relationships that were complex and contradictory. And he had to factor in available hardware. Like other inventors of the computer, he knew that rods and gears and motors were reliable and much more precise than they had ever been, thanks to advances in machining and production—Atanasoff was tempted by these advances to pursue the analog path. But he was strongly drawn to the speed that the novel, but as yet unreliable, technology of electronics offered.
Atanasoff’s interest in binary system was not based on quite the same reasoning as Zuse’s interest—IBM had, after all, introduced a multiplying calculator in 1931. What he suspected was that using a binary number system would make it possible to use vacuum tubes for actual calculating. The vacuum tubes would be arranged inside a processing unit and different arrangements of on and off tubes would stand for different numbers—any number could be represented by a row of on-off vacuum tubes. At the same time, although he himself was perfectly familiar with binary counting systems, he knew that not many other people were (something Zuse’s experience also demonstrated)—even most mathematicians were uncomfortable operating outside of the decimal system. The prevailing wisdom was that translating from the binary system to the decimal system would pose an enormous difficulty—a decimal number would have to be entered somehow, turning on the tubes and turning them off, then, when the calculation had been performed, the result would have be communicated to some sort of output mechanism that would translate the binary number to a decimal number.
And there would have to be “memory.” In the most advanced IBM tabulator of the day, there were two types of memory. The first comprised the set of instructions that the tabulator used to carry out operations. If a byte in today’s computer terminology consists of 8 bits of storage capacity,2 the first type of memory belonging to the IBM tabulator of Atanasoff’s day had 266 bits or 33.25 bytes of memory. That tabulator’s memory hardly bears comparison with what we are familiar with in 2010—a single page of text saved as a file in Microsoft Word that includes all preference settings, containing 514 words and forty lines of English text in 14-point type, uses 28 kilobytes, or 28,662 bytes, or 862 times the capacity of the IBM tabulator Atanasoff was familiar with. A single 3.2 megabyte digital photograph (3.2 million bytes) uses almost 96,400 times the IBM’s memory capacity. The IBM’s second type of memory was larger, but external to the machine—it was the record of calculations produced, punch cards that could then be fed back into the machine and used for future tabulations. The punch cards, of course, were kept track of by the operator, not the machine.
Modern computers still have two types of memory. The first type is called the RAM, or random-access memory, which the computer uses while it is turned on for operations, applications, and frequently accessed data. The second type is the storage memory, which the computer has access to and is stored externally to the main operating system on hard disk drives, floppy disks, magnetic tape, and so on. Although today, at least in personal computers, they are both inside the computer, the two kinds of memory follow Atanasoff’s (and IBM’s) ideas by being separate but communicating with each other.
When Atanasoff jumped in his new Ford V8 that evening in December 1937, he later testified, “I was in such a mental state that no resolution was possible. I was just unhappy to an extreme degree.” But he was pleased with his new car (Burton notes that he purchased a new car every year). He enjoyed its speed and maneuverability. He felt himself calm down, and he also felt a sort of suspension of time—“When I finally came to earth I was crossing the Mississippi River, 189 miles from my desk.” His next thought was perhaps characteristic of his practical and no-nonsense temperament: “Now you’ve got to quit this damned foolishness.”
Then he saw the tavern sign. He went in, sat down, and ordered a bourbon and soda. A radio sitting behind the bar was playing music. Almost as soon as the waitress brought him his drink, the nature of his computing system occurred to him as a logical whole, and he began envisioning both the component pieces and how the pieces could fit together. He jotted some notes down on a paper napkin, but later he didn’t need the notes because he was able to visualize and contemplate his machine so thoroughly that he had no trouble recalling what he had come up with. He sat in the bar for several hours, thinking through each of his concepts but concentrating particularly upon ideas for how the memory would work and how an electronically based on-off process would calculate.
Atanasoff’s experience is interesting on a number of levels. The way in which a state of effort followed by a state of relaxation induced an understanding of the system he wanted to build is reminiscent of what had happened to Turing and also to Henri Poincaré, the mathematician, as quoted in psychiatric researcher Nancy Andreasen’s The Creative Brain:
For fifteen days I strove to prove that there could not be any functions like those we have since called Fuchsian functions. I was very ignorant: every day, I seated myself at my work table, stayed an hour or two, tried a great number of combinations and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. By the next morning, I had established the existence of a class of Fuchsian functions … I had only to write out the results, which took but a few hours.
But what Poincaré really wants to do is to boil his results down into a principle that can be understood in relation to other well-known mathematical principles. When he then takes a trip, he manages to do this without even interrupting his conversation with another passenger: “The changes of travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go someplace or other. At the moment when I put my foot on the step, the idea came to me without anything in my former thoughts seeming to have paved the way … On my return to Caen, for conscience’s sake, I verified the result at my leisure.”
Andreasen then goes on to detail recent research (as of 2005) into how the brain is structured and how it works to create. She describes the brain as a system of sending and receiving neurons that are organized into areas that govern different functions. They connect to one another at synapses, where a tiny electric charge jumps over a tiny space. The neurons are embedded in gray matter (the cortex of the brain that contains nerve cell bodies), and the fuel of the brain is glucose. Andreasen distinguishes between ordinary creativity of the sort that is required in talking and the extraordinary creativity required for innovative or artistic thought. She points out that “most of the time that we speak, we are producing a sequence of words that we have not produced before.” But the sort of creativity that invents the computer is of a different order. The brain, she argues, is a self-organizing system “created from components that are in existence and that spontaneously reorganize themselves to create something new.” An essential part of a self-organizing system is the feedback loop—in the brain, this would consist of electrical impulses passing along neurons back and forth between one part of the brain and the others, contradicting or reinforcing earlier impulses and influencing later ones.
In order to understand how the brain creates, Andreasen distinguishes between episodic memory, used for personal reminiscence and free association of thoughts, and semantic memory, used for information storage and retrieval of thoughts and concepts not related to personal history. Using positron emission tomography (PET) to image her study subjects’ brains while they relax and free-associate, Andreasen discovered that the most active regions in her free-associating subjects’ brains were the associative regions, that is, the frontal, parietal, and temporal lobes, the most complexly structured regions, the slowest to develop, and the regions dedicated to generating connections among all the other regions of the brain. She notes that in famous recollections of creative moments by poets such as Coleridge and scientists such as Poincaré and chemist Friedrich Kekulé (who dozed off and dreamed of a snake eating its tail and came up with the structure of the benzene ring), there is often a sudden flash of insight, in which previously unconnected ideas combine into a new thing. She explains this often attested experience: “I would hypothesize that during the creative process, the brain begins by disorganizing, making links between shadowy forms of objects or symbols or words or remembered experiences that have not previously been linked. Out of this disorganization, self-organization eventually re-emerges and takes over in the brain. The result is a completely new and original thing.”
Clearly, Atanasoff began his trip from Ames, Iowa, to Rock Island, Illinois, in a disorganized (and frustrated) state. Like Turing and Poincaré, though, once he was able to forget his mathematical work, ideas that had refused to come together when he was thinking about them (using his semantic memory) succeeded in coming together once he came to earth upon crossing the Mississippi and realized how far he had traveled in a dreamlike state.
What is especially intriguing, and even moving, about Atanasoff’s story is that the machine he was trying to create was intended to mimic the brain—it was to be a self-organizing system, with feedback loops. The very mechanism that he pondered most that evening in the tavern was the calculator’s “regenerative memory”—the mechanism by which the capacitors and the vacuum tubes would charge one another, in a feedback loop. And without having a concept of how the human brain works, he also understood that electricity would be the medium of memory and thought, as it is in the human brain. Turing was thinking of a machine-like human process. Atanasoff was thinking of a human-like machine process.
Another way of looking at Atanasoff is that he fits into Malcolm Gladwell’s profile of a maven, that is, a person so interested in a particular field of endeavor that he not only drives himself to become an expert in that field but also is driven to communicate what he learns and intuits about that field to others. Atanasoff was widely considered to be a dedicated and effective teacher: he was good at explaining concepts to his students; he was good at probing their depth of knowledge; and he was good at encouraging them to learn what they needed to know. When his students came up with ideas, he helped them work them out, and he learned from what the students did. Atanasoff also had productive relationships with colleagues like A. E. Brandt. First and foremost, Atanasoff wanted to invent the calculator he thought the world of physics and mathematics needed—he seems not to have given much thought at the time to who might own what piece of the equipment, unlike the IBM executives who were offended when he fiddled with their machine. It also does not seem to have occurred to him that people at IBM could be offended, just as it seems not to have occurred to him that the authorities at his daughters’ school might be offended at his oft-expressed negative views on the science curriculum there. Atanasoff was intent upon innovation.
Through 1938, Atanasoff worked out both the practical and the theoretical implications of the ideas he came up with in the tavern in Illinois. To reiterate, Atanasoff’s four linked ideas were:
1. Electronic logic circuits (which would perform a calculation simply by turning on and off)
2. Binary enumeration (using a number system with only two digits, 0 and 1, rather than ten, 0–9)
3. Capacitors for regenerative memory (a capacitor is like a battery in that it can store electrical energy while not connected to a source)
4. Computing by direct logical action and not by enumeration (that is, by counting rather than by measuring; the numbers represented by rows of 0s and 1s, or the on-off states of the vacuum tubes, would be directly added and subtracted rather than being represented by points on disks or shafts)
One important consideration was how to stabilize the electrical supply of the vacuum tubes that would be doing the calculating. Atanasoff decided to construct the operating memory (CPU, or central processing unit) and the storage memory in different ways, in this case because vacuum tubes were expensive. He decided to reserve them for the operating memory and use capacitors for the storage memory. The results (including intermediate results) would be charred onto paper cards—still another type of memory. Capacitors (also known at the time as condensers) were (and are) very simple devices that store electricity like a bottle stores water. They store electricity without converting it to anything, using two conductors separated by an insulator. If a charge is applied to one of the conductors, it stays there by electrostatic attraction but cannot jump across the insulator. The charge can be removed very quickly by completing the circuit to the other conductor. In terms of the binary operation of a computer, “charged” can represent a 1 and “not charged” can represent a 0, for example. But insulators leak slightly, so the electric charge doesn’t stay there very long; therefore, both Atanasoff’s design and modern DRAM chips have electronics to refresh the state of the capacitor periodically by detecting its charge and restoring it before it fades.3
Atanasoff was perfectly familiar with condensers—when he was considering the dielectric constant of helium in his PhD dissertation, he was calculating the reduction in electric field strength caused by the presence of helium. Alan Turing was familiar with them, too—when he could make no progress finding the dielectric constant of water—that is, in calculating how effective water is at reducing electric field intensity. In the thirties, the most common insulator in capacitors was dry paper, which has a dielectric constant of 2, meaning that it cuts electric field intensity in half. Most modern capacitors now use ceramic insulation.
The idea Atanasoff had that most vexed and intrigued him over the next year was that the passive capacitors could work with the vacuum tubes. He later testified, “I chose small condensers for memory because they would have the required voltage to actuate the tubes, and the plates … of the tubes would give enough power to charge the condensers.” Atanasoff called this energy reciprocation “jogging,” as in “jogging one’s memory.” He thought that jogging would make both memories more stable while also saving on expense for supplies and on electrical usage. In this context, I think it is important to remember that Atanasoff was by nature and upbringing as frugal as he was ambitious, and also that he had no access to government money or private investment funds. Frugality was part of what drove him to invent a calculator—he didn’t want to waste time calculating using the machines of the day. Frugality dictated what he could try—he and Brandt had to experiment with what they had on hand, the IBM, not a Monroe. And frugality dictated the terms of his invention—it had to be cheap to produce, easy to operate, and cheap to run.
The use of electronic components both dictated the use of a binary number system and was dictated by it. If all Atanasoff needed to indicate a number was “on” or “off,” he was free of the burden of gears, shafts, measuring, and estimating, but once he was freed of those clumsy parts, he was committed to a binary number system, which he justified in two ways at the time—that his device would prove itself by being accurate, and that his device was intended to solve various sorts of mathematical problems including but not restricted to systems of equations,4 which meant that it was most likely to be used by scientists, who were more likely to understand a binary number system. It could also be said that using a binary number system is, as Zuse was pointing out to Pannke at the same time, the frugal choice.
Atanasoff spent a good deal of 1938 thinking about a mathematical system that would enable him to understand how to compute by direct logical operation, the way a person computes
by subtracting 6 from 7 and writing 1, then moving to the left and subtracting 2 from 3, and writing 1 to the left of 2, and then moving to the left and subtracting 0 from 1 and writing 1 to the left of 1, then seeing the answer as 111. Although it looks much more complicated to those used to decimals, binary subtraction would work the same way:
What he came up with was his own form of Boolean algebra (which he, like Zuse, later said that he was unaware of at the time). Here again, Atanasoff and Turing were thinking along the same lines, but Turing, as a mathematician among mathematicians, did not have to devise his own system.
Boolean algebra is a logic system invented by George Boole (1815–1864) that posits that there are only two values in the universe. They are zero and one. On these two values, four operations can be performed: (1) “no-op” (also called identity), (2) “not” (the value is changed into its opposite), (3) “and,” and (4) “or.” The first two operate (i.e., do something to and then return a single outcome value) on a single value. The second two operate on a pair of values and then return a single outcome value.
The values do not have to be read as numbers—they can be read as “true” or “false,” or “green” or “not green,” for example. For the purposes of the computer, both Atanasoff and Zuse realized that large numbers were easier to calculate using a 1 and 0 system, but Boolean algebra also has philosophical implications about the nature of reality and how to discover if something is true or not true that Turing brought to bear on not only breaking German codes, but also on his theory of how the mind works, and how, therefore, a mind-like machine might work. Working out his own form of Boolean algebra showed Atanasoff, as it showed Zuse, that his system was manageable and would not require rooms full of hardware.
Atanasoff didn’t have the money or resources to try to build any of his components, so most of the work he did was on paper and in his head. However, in March 1939, fifteen months after the revelation in the roadhouse, Atanasoff turned in an application for a grant of $650 to hire a graduate student and attempt to build what he had conceived of. In May, his grant request was approved: $450 was salary for the student and $200 was to go for raw materials.
1. The objects he calls “relays” bear no relationship in either looks or operation to what are now known as relays. They were entirely mechanical.
2. The term “bit” is an abbreviation for “binary digit.”
3. Thanks to John Gustafson, who adds, “Whenever you scuff your shoes on a carpet in dry weather such that you get a shock when you touch something metallic, you’ve made yourself a capacitor. Rubbing shoes on the carpet scrapes electrons from one surface to the other, creating an excess electric charge. The electric charge stays there because the charge cannot jump through the air, which serves as the insulator. If you ‘close the circuit’ by touching a metal object, the charge will suddenly discharge with a painful spark. Or if you stand still long enough, the static buildup will dissipate by itself, because even air conducts a little electricity.”
4. The list he eventually came up with was: (1) multiple correlation, (2) curve fitting, (3) method of least squares, (4) vibration problems including the vibrational Raman effect, (5) electrical circuit analysis, (6) analysis of elastic structures, (7) approximate solution of many problems of elasticity, (8) approximate solutions of problems of quantum mechanics, (9) perturbation theories of mechanics, astronomy, and the quantum theory.