Non Fiction Authors From The United States

"Characterization is an accident that flows out of action and dialogue."

"Professor Florian Cajori died August 14, 1934. In May of the following year I was invited by the University of California Press to edit this work. ...this is a revision of Motte's translation of the Principia. From many conversations with Professor Cajori, I know that he had long cherished the idea of revising Newton's immortal work by rendering certain parts into modern phraseology (e.g., to change the reading of "reciprocally in the subduplicate ratio of " to "inversely as the square root of") and to append historical and critical notes which would provide instruction to some readers and interest to all. This is his last work; one of the most fitting to crown a life devoted to investigation and to the history of the sciences in his chosen field."

"As a mathematician Dean Cajori has achieved a name which very few in this world can equal, a name which is respected all over the globe. His text books and his writings have been published all over the world. We are proud of all the achievements of our "Caj", of course, but we are especially proud of what he has done for us here, and it is for this reason that we shall always hold him in our memory. As a friend and as an instructor he has been more to us than we can ever measure, and we shall always look back upon the days when we had "Caj"."

"They [the students] looked upon "Caj" as one of their best friends in all the College, and thought of him as the one responsible for their later success. He has had a personal appeal to a great many students... It was the appeal of the one with a human interest in what somebody else is doing, the appeal of the true friend and the hearty well-wisher. It was the appeal of "Caj"."

"What is perhaps the greatest blow that has ever come to the student body of Colorado College came last Friday when it was announced that Dean Florian Cajori, for about thirty years the best-known and best-liked professor in the College, had resigned and will not be back with us next year. It was not only on account of the value of his service as an instructor that the students felt such a sense of loss at the announcement, but more on account of the friendship and intimate relationship which he has shown to us. "Caj"… has been closer to this student body than any other one man. It was usually "Caj" who made the speech at the Barbecue, it was "Caj" who talked upcoming events in chapel, it was "Caj" who was always out there at the picnic or the Festival or the ball game. No form of student activity has seemed entirely complete unless our "Caj" has been there or has had something to do with it."

"The grandest achievement of the Hindus and the one which, of all mathematical inventions, has contributed most to the general progress of intelligence, is the invention of the principle of position in writing numbers. Generally we speak of our notation as the “Arabic” notation, but it should be called the “Hindu” notation, for the Arabs borrowed it from the Hindus. That the invention of this notation was not so easy as we might suppose at first thought, may be inferred from the fact that, of other nations, not even the keen-minded Greeks possessed one like it."

"J. J. Sylvester was an enthusiastic supporter of reform [in the teaching of geometry]. The difference in attitude on this question between the two foremost British mathematicians, J. J. Sylvester, the algebraist, and Arthur Cayley, the algebraist and geometer, was grotesque. Sylvester wished to bury Euclid "deeper than e'er plummet sounded" out of the schoolboy's reach; Cayley, an ardent admirer of Euclid, desired the retention of Simson's Euclid. When reminded that this treatise was a mixture of Euclid and Simson, Cayley suggested striking out Simson's additions and keeping strictly to the original treatise."

"Most of his [Euler's] memoirs are contained in the transactions of the Academy of Sciences at St. Petersburg, and in those of the Academy at Berlin. From 1728 to 1783 a large portion of the Petropolitan transactions were filled by his writings. He had engaged to furnish the Petersburg Academy with memoirs in sufficient number to enrich its acts for twenty years a promise more than fulfilled, for down to 1818 [Euler died in 1793] the volumes usually contained one or more papers of his. It has been said that an edition of Euler's complete works would fill 16,000 quarto pages."

"In 1735 the solving of an astronomical problem, proposed by the Academy, for which several eminent mathematicians had demanded several months' time, was achieved in three days by Euler with aid of improved methods of his own... With still superior methods this same problem was solved by the illustrious Gauss in one hour."

"Fermat died with the belief that he had found a long-sought-for law of prime numbers in the formula 2^{2^n} + 1 = a prime, but he admitted that he was unable to prove it rigorously. The law is not true, as was pointed out by Euler in the example 2^{2^5} + 1 = 4,294,967,297 = 6,700,417 times 641. The American lightning calculator Zerah Colburn, when a boy, readily found the factors but was unable to explain the method by which he made his marvellous mental computation."

"The miraculous powers of modern calculation are due to three inventions : the Arabic Notation, Decimal Fractions and Logarithms."

"The Elements has been considered as offering models of scrupulously rigorous demonstrations. It is certainly true that in point of rigour it compares favourably with its modern rivals; but when examined in the light of strict mathematical logic, it has been pronounced by C.S. Peirce to be "riddled with fallacies." The results are correct only because the writer's experience keeps him on his guard."

"Comparatively few of the propositions and proofs in the Elements are his [Euclid's] own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him."

"It is a remarkable fact in the history of geometry, that the Elements of Euclid, written two thousand years ago, are still regarded by many as the best introduction to the mathematical sciences."

"The history of mathematics is important also as a valuable contribution to the history of civilization. Human progress is closely identified with scientific thought. Mathematical and physical researches are a reliable record of intellectual progress."

"The history of mathematics may be instructive as well as agreeable ; it may not only remind us of what we have, but may also teach us to increase our store. Says De Morgan, "The early history of the mind of men with regards to mathematics leads us to point out our own errors; and in this respect it is well to pay attention to the history of mathematics." It warns us against hasty conclusions; it points out the importance of a good notation upon the progress of the science; it discourages excessive specialization on the part of the investigator, by showing how apparently distinct branches have been found to possess unexpected connecting links; it saves the student from wasting time and energy upon problems which were, perhaps, solved long since; it discourages him from attacking an unsolved problem by the same method which has led other mathematicians to failure; it teaches that fortifications can be taken by other ways than by direct attack, that when repulsed from a direct assault it is well to reconnoitre and occupy the surrounding ground and to discover the secret paths by which the apparently unconquerable position can be taken."

"As regards algebra, the early Arabs failed to adopt either the Diophantine or the Hindu notations. An examination of [the algebra of Al-Khwarizmi] shows that the exposition was altogether rhetorical, i.e., devoid of all symbolism."

"My quotations from Newton suggest the motive which induced him to take a stand against the use of hypotheses, namely, the danger of becoming involved in disagreeable controversies. ...Newton could no more dispense with hypotheses in his own cogitations than an eagle can dispense with flight. Nor did Newton succeed in avoiding controversy."

"Our so-called "Arabic" notation owes its excellence to the application of the principle of local value and the use of a symbol for zero. It is now conclusively established that the principle of local value was used by the ns much earlier than by the Hindus and that the Maya of Central America used the principle and symbols for zero in a well-developed numeral system of their own. The notation of Babylonia used the scale of 60, that of the Maya, the scale 20 (except in one step). It follows, therefore, that the present controversy on the origin of our numerals does not involve the question of the first use of local value and symbols for zero; it concerns itself only with the time and place of the first application of local value to the decimal scale and with the origin of the forms or shapes of our ten numerals. ... Hurt by the alleged arrogance of certain Greek scholars, Sebokht praises the science of the Hindus and speaks of "their valuable methods of computation. . . . I wish only to say that this computation is done by means of nine signs." Unfortunately, he leaves it to us to guess whether or not he used the zero. The passage, written about 662 A.D., is the earliest reference that has been found outside of India to our numerals. ...The form of the symbols with the zero, used in India, differed so widely from the old forms without the zero used there, that the former seem to have had an independent origin and to have been imported into India. ...The following are outstanding facts: 1. The earliest reliable record of the use of our numerals with zero is an inscription of 867 A.D. in India. 2. The validity of the testimony of early Arabic writers ascribing to India the numerals with zero is shaken, but not destroyed. 3. There is not a scintilla of evidence in the form of old manuscripts or numeral inscriptions to support the Greek origin of our numerals. 4. At present the hypothesis of the Hindu origin of our numerals stands without serious rival. But this hypothesis is by no means firmly established."

"Professor Sylvester's first high class at the new university Johns Hopkins consisted of only one student, G. B. Halsted, who had persisted in urging Sylvester to lecture on the modern algebra. The attempt to lecture on this subject led him into new investigations in quantics."

"The opinion is widely prevalent that even if the subjects are totally forgotten, a valuable mental discipline is acquired by the efforts made to master them. While the Conference admits that, considered in itself this discipline has a certain value, it feels that such a discipline is greatly inferior to that which may be gained by a different class of exercises, and bears the same relation to a really improving discipline that lifting exercises in an ill-ventilated room bear to games in the open air. The movements of a race horse afford a better model of improving exercise than those of the ox in a tread-mill."

"Bob Machol's life involved a number of strands — aviation, scientific writing, systems engineering, chemistry, research, applying OR to sports, computing and mushrooms — that intertwined over the years. Consider his involvement with aviation. It started in 1940 when, fresh out of Harvard, Bob enlisted in the Marines, intent on becoming an aviator. Although Bob didn't earn his pilot's wings, he did emerge from World War II holding the rank of lieutenant commander. Following the war, Bob became involved with research organizations (the Operations Evaluation Group and the University of Michigan's Willow Run Laboratories) that were looking for improved ways of defending the United States against air attack. This work led to Bob's groundbreaking book, "Systems Engineering," co-authored with the late Harry H. Goode."

"I had professor Machol at Kellogg in 1985 for Operations. Professor Machol was intimidating. Rumor around the class was that he had been in charge of logistics for the U.S. Navy in the Pacific theatre during WWII. He was teaching EOQ modeling in one early class, leading me to pose a question, ‘Professor Machol, would it not make sense to do … for the sake of consistency’. His reply is oh so memorable almost 40 years ago.. “Mr. Clark, consistency is the bugaboo of small minds” (ouch!). No doubt relative to this giant, my mind was and remains small indeed!"

"I recount this as it reminds me of some of the lessons Bob preached and practiced throughout his professional career:"

"At an age when many people consider retiring, Robert E. Machol stopped teaching at Northwestern University and started a new career as chief scientist for the Federal Aviation Administration. There, while in his 70s, he predicted "catastrophe" after studying the turbulence created by the jet engines of 757 airplanes--work that predicted fatal crashes and eventually led to a change in federal aviation policy... "I was the first guy within the agency who got up and said, `We're likely to have a catastrophe, a real catastrophe … if we don't do something," Mr. Machol told the Los Angeles Times in 1994. Eventually, the agency ordered landing aircraft to maintain a greater distance behind 757s to avoid the jet's dangerous "wake vortex." But the policy change came only after crashes in Billings, Mont., and Santa Ana, Calif., that killed 13 people."

"The pressure to generate the ideas and methods attributed to Systems Engineering stems directly from the needs of 20th century society. As our frontiers have disappeared, man has turned to technology to furnish the "good life" in a rapidly shrinking, crowded world. Our interdependence upon one another has increased in direct proportion to the population increase. The race to maintain or improve the operating efficiency of society has required that the systems and mechanisms that serve the society also become increasingly complex and interdependent. Goode and Machal have provided statistics to illustrate the above. They note that the world population increased from 800 million in 1750, to 1200 million in 1850, and 2400 million in 1950. Maximum transportation speeds went from 40 mph in 1850, and 100 mph in 1900, to commercial transport speed of 350 mph in 1950 and supersonic transport planes of over 1200 mph in the 1960's. Our communication systems are a good indication of increasing complexity. U.S. telephones jumped from 350,000 in 1900, to 55 million in 1955."

"Most accidents in well-designed systems involve two or more events of low probability occurring in the worst possible combination."

"Sometimes, where a complex problem can be illuminated by many tools, one can be forgiven for applying the one he knows best."

"There comes a time when one must stop suggesting and evaluating new solutions, and get on with the job of analyzing and implementing one pretty good solution."

"If the assumptions are wrong, the conclusions aren't likely to be very good."

"The conclusions of most good operations research studies are obvious."

"The ideal system engineer is an engineer thoroughly versed in his field but conversant with and knowledgeable of other fields. You have to have the capability and desire to become a 'six-month expert'... You've got to want to become a generalist, too."

"Scientists possess healthy skepticism. They realize that you've got to know the answer before you measure it."

"Mathematicians are there to find the constraints and to eliminate those things that aren't constraints... I know this will surprise many of you, but they are useful!"

"The purpose and real value of systems engineering is... to keep going around the loop; find inadequacies and make improvements."

"Everyone knows what engineering is. All that's left is to define systems, and I'm not fool enough to do that."

"Sometimes it is more difficult to formulate the criterion for a problem than to state the question itself."

"[There is a] basic problem … of building a mathematical model of thought processes, and in particular of those aspects of thought which are concerned with information and decision processes. The perceptron is one type of model -- a set of memory devices connected in random fashion-- which has not yet achieved useful results but certainly seems to be a promising approach. The self-adaptive feedback control system which goes beyond the normal servo function of controlling its output, and in addition controls the parameters by which it controls its output is another which has already achieved pragmatic results in equipment control. It may be that the question of self-adaptation is a key to the whole question of how the human functions in a decisioning situation. For in many cases the ability of the human mind to adapt itself to a changing and complex environment is beyond our present aims in model construction."

"We assert that it is possible to describe analytically any human function which can be reasonably defined in objective terms and we specifically include in such functions "thinking" insofar as that term is definable. If by "thinking" one means being able to do arithmetic, or play a good game of chess, or learn from experience, or make optimal decisions in exceedingly complex situations, then we assert that thinking can be described analytically. And there are two important corollaries: if It can be described analytically, it can be simulated; and if it can be simulated, it can be performed mechanically."

"We have discovered in this past decade that thinking, and decision, are not solely the province of the metaphysicist, but are appropriate subjects for scientific inquiry."

"At this time it is difficult to put one's finger on any single contribution in the decade 1950 - 1960 which is comparable to those above, and yet progress has probably been even greater. From the point of view of an educator, one cannot overlook the wide distribution which has been given to these ideas. There has been remarkable progress from analysis to synthesis, always a sign of maturity in a field of analytic endeavour. There has been consolidation, for example in the establishment of a more rigorous basis for information theory; there has been unification, for example in the demonstration of the formal similarity between game theory and ; there has been application to mathematically more difficult situations, for example nonlinear servo systems and information channels with memory; there has been implementation, as in commercially available computers which by any reasonable measure are hun- dreds of times more powerful than the primitive devices of 1950; there has been de-limitation of the boundaries of many of these fields."

"In April of 1959, ten of this country's leading scholars forgathered on the campus of Purdue University to discuss the nature of information and the nature of decision... What interests do these men have in common?... To answer these questions it is necessary to view the changing aspect of the scientific approach to epistemology, and the striking progress which has been wrought in the very recent past. The decade from 1940 to 1950 witnessed the operation of the first stored- program digital computer. The concept of information was quantified, and mathematical theories were developed for communication (Shannon) and decision (Wald). Known mathematical techniques were applied to new and important fields, as the techniques of complex- variable theory to the analysis of feedback systems and the techniques of matrix theory to the analysis of systems under multiple linear constraints. The word "cybernetics" was coined, and with it came the realization of the many analogies between control and communication in men and in automata. New terms like "operations research" and "system engineering" were introduced; despite their occasional use by charlatans, they have signified enormous progress in the solution of exceedingly complex problems, through the application of quantitative ness and objectivity."

"Management has a design and operation function, as does engineering. The design is usually done under the heading of organization. It should be noted first that the performance of a group of people is a strong function of the capabilities of the individuals and a rather weak function of the way they are organized. That is, good people do a fairly good job under almost any organization and a somewhat better one when the organization is good. Poor talent does a poor job with a bad organization, but it is still a poor job no matter what the organization. Repeated reorganizations are noted in groups of individuals poorly suited to their function, though no amount of good organization will give good performance. The best architectural design fails with poor bricks and mortar. But the payoff from good organization with good people is worthwhile."

"Every large-scale system is an . The automatic factory, the vehicular-traffic system, any military system- in the design of all these systems, primary attention should be given to the flow of information about the elements of the system."

"[This set of inputs [can be defined] as 1) input which is always the same or is of many types, 2) input which occurs periodically (or very infrequently), and 3) input which does or does not seek to destroy the system. Their rationale for developing the classification was to aid in the definition of steps to be followed in order to find the] solution of the problem of a large-scale or complex system."

"[Systems should be classified] on the basis of the types of inputs with which they must cope."

"There are four distinct bases on which a system-design book might be organized. First are the chronological phases through which the system-design effort passes, such as organization and preliminary design. Second are the logical steps such as analysis of the single thread (operation on a single input) and high traffic (methods of handling multiplex inputs). Third are the parts of the system, such as communications and displays. Fourth are the tools of system design, such as information theory and queueing theory."

"A new concept and a new method were needed. The concept from the engineering standpoint is the evolution of the engineering scientist, i.e., the scientific generalist who maintains a broad outlook. The method is that of the team approach. On large-scale-system problems, teams of scientists and engineers, generalists as well as specialists, exert their joint efforts to find a solution and physically realize it. We are led to the concept of the system-design team, a small group of engineers or scientists, to lead a large project and organize the system effort. Such men have been variously called engineering scientists, system engineers, system analysts, or large-scale-system designers. The technique has been variously called the systems approach or the team development method. It is toward this man and his teammates that these discussions are directed. With the realization that not enough can be learned in all the required fields to make him a specialist, enough is introduced to make him aware of the language and problems of the specialist. This generalist is a new quantity in the engineering world, and his education must be begun."

"The lack of formal definition does not prevent us from noting the characteristics which are frequently present in large scale systems. Each such system has a certain integrity. It may or may not be rigidly controlled from some central point, but in every case, all the parts of the system have some common purpose; in some sense, they all contribute to the production of a single set of optimum outputs from the given set of inputs, with respect to some appropriate measure of effectiveness."

"The fellow who is able to get the grants and contracts is the fellow who gets the doctoral students. Each student publishes two or three papers on his thesis, in each of which he cites his professor's work. Then he goes on and does his own subsequent work, in which, of course, he cites those thesis papers which were coauthorised by his professor."