Cryptographers

"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning... The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings."

"Instruction tables will have to be made up by mathematicians with computing experience and perhaps a certain puzzle-solving ability. There need be no real danger of it ever becoming a drudge, for any processes that are quite mechanical may be turned over to the machine itself."

"A man provided with paper, pencil, and rubber, and subject to strict discipline, is in effect a universal machine."

"There is a remarkably close parallel between the problems of the physicist and those of the cryptographer. The system on which a message is enciphered corresponds to the laws of the universe, the intercepted messages to the evidence available, the keys for a day or a message to important constants which have to be determined. The correspondence is very close, but the subject matter of cryptography is very easily dealt with by discrete machinery, physics not so easily."

"This is only a foretaste of what is to come, and only the shadow of what is going to be. We have to have some experience with the machine before we really know its capabilities. It may take years before we settle down to the new possibilities, but I do not see why it should not enter any one of the fields normally covered by the human intellect, and eventually compete on equal terms."

"Science is a differential equation. Religion is a boundary condition."

"The Exclusion Principle is laid down purely for the benefit of the electrons themselves, who might be corrupted (and become dragons or demons) if allowed to associate too freely."

"Let us now assume, for the sake of argument, that these machines are a genuine possibility, and look at the consequences of constructing them. To do so would of course meet with great opposition, unless we have advanced greatly in religious toleration from the days of Galileo. There would be great opposition from the intellectuals who were afraid of being put out of a job. It is probable though that the intellectuals would be mistaken about this. There would be plenty to do, trying to understand what the machines were trying to say, i.e. in trying to keep one’s intelligence up to the standard set by the machines, for it seems probable that once the machine thinking method had started, it would not take long to outstrip our feeble powers. There would be no question of the machines dying, and they would be able to converse with each other to sharpen their wits. At some stage therefore we should have to expect the machines to take control, in the way that is mentioned in Samuel Butler’s “Erewhon”."

"The "computable" numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. ...According to my definition, a number is computable if its decimal can be written down by a machine. ...I show that certain large classes of numbers are computable. They include, for instance, the real parts of all s, the real parts of the zeros of the Bessel functions, the numbers π, e, etc. The computable numbers do not, however, include all definable numbers. ...[C]onclusions are reached which are superficially similar to those of Gödel. ...[I]t is shown ...that the Hilbertian can have no solution. In a recent paper ... reaches similar conclusions..."

"We may compare a man in the process of computing a real number to a machine which is only capable of a finite number of conditions q1, q2, ..., qK which will be called " m-configurations "."

"The machine is supplied with a "tape" (the analogue of paper) running through it, and divided into sections (called "squares") each capable of bearing a "symbol"."

"The "scanned symbol" is the only one of which the machine is... "directly aware". However, by altering its m-configuration the machine can effectively remember some of the symbols which it has "seen" (scanned) previously."

"In some of the configurations in which the scanned square is blank... the machine writes down a new symbol on the scanned square: in other configurations it erases the scanned symbol."

"The machine may also change the square which is being scanned, but only by shifting it one place to right or left."

"[T]he m-configuration may be changed."

"Some of the symbols written down will form the sequence of figures which is the decimal of the real number... being computed. The others are just rough notes to "assist the memory ". ...[O]nly ...these rough notes ...will be liable to erasure."

"[T]hese operations include all those which are used in the computation..."

"For some purposes we might use machines (choice... or c-machines) whose motion is only partially determined by the configuration... When such a machine reaches... ambiguous configurations, it cannot go on until some arbitrary choice has been made by an external operator. This would be the case if we were using machines to deal with axiomatic systems."

"In this paper I deal only with automatic machines, and will therefore often omit the prefix ɑ-."

"If an ɑ-machine prints two kinds of symbols, of which the first kind (called figures) consists entirely of 0 and 1 (the others being called symbols of the second kind), then the machine will be called a computing machine."

"It will be useful to put... tables into a... standard form. ...The lines of the table are... of form m-config. | Symbol | Operations | Final m-config. In this way we obtain a complete description of the machine. ...This new description of the machine may be called the standard description (S.D). ...[W]e shall have a description of the machine in the form of an arabic numeral. The integer represented by this numeral may be called a description number (D.N) of the machine. The D.N determine the S.D and the structure of the machine uniquely. The machine whose D.N is n may be described as \mathcal{M}(n)."

"To each computable sequence there corresponds at least one description number, while to no description number does there correspond more than one computable sequence. The computable sequences and numbers are therefore enumerable."

"It is possible to invent a single machine which can be used to compute any computable sequence. If this machine \mathcal{U} is supplied with a tape on the beginning of which is written the S.D of some computing machine \mathcal{M}, then \mathcal{U} will compute the same sequence as \mathcal{M}."

"Can machines think?"... The new form of the problem can be described in terms of a game which we call the 'imitation game." It is played with three people, a man (A), a woman (B), and an interrogator (C) who may be of either sex. The interrogator stays in a room apart from the other two. The object of the game for the interrogator is to determine which of the other two is the man and which is the woman. He knows them by labels X and Y, and at the end of the game he says either "X is A and Y is B" or "X is B and Y is A." The interrogator is allowed to put questions to A and B... We now ask the question, "What will happen when a machine takes the part of A in this game?" Will the interrogator decide wrongly as often when the game is played like this as he does when the game is played between a man and a woman? These questions replace our original, "Can machines think?"

"We do not wish to penalise the machine for its inability to shine in beauty competitions, nor to penalise a man for losing in a race against an aeroplane. The conditions of our game make these disabilities irrelevant."

"May not machines carry out something which ought to be described as thinking but which is very different from what a man does?"

"We are not asking whether all digital computers would do well in the game nor whether the computers at present available would do well, but whether there are imaginable computers which would do well."

"The idea behind digital computers may be explained by saying that these machines are intended to carry out any operations which could be done by a human computer."

"A digital computer can usually be regarded as consisting of three parts: (i) Store. (ii) Executive unit. (iii) Control. ...The executive unit is the part which carries out the various individual operations involved in a calculation. ...It is the duty of the control to see that...[the table of] instructions are obeyed correctly and in the right order. ...A typical instruction might say—"Add the number stored in position 6809 to that in 4302 and put the result back into the latter storage position." Needless to say it would not occur in the machine expressed in English. It would more likely be coded in a form such as 6809430217. Here 17 says which of various possible operations [add] is to be performed on the two numbers. ...It will be noticed that the instruction takes up 10 digits and so forms one packet of information..."

"Suppose Mother wants Tommy to call at the cobbler's every morning on his way to school to see if her shoes are done, she can ask him afresh every morning. Alternatively she can stick up a notice once and for all in the hall which he will see when he leaves for school and which tells him to call for the shoes, and also to destroy the notice when he comes back if he has the shoes with him."

"If one wants to make a machine mimic the behaviour of the human computer in some complex operation one has to ask him how it is done, and then translate the answer into the form of an instruction table. Constructing instruction tables is usually described as "programming.""

"I believe that at the end of the century the use of words and general educated opinion will have altered so much that one will be able to speak of machines thinking without expecting to be contradicted."

"I am not very impressed with theological arguments whatever they may be used to support. Such arguments have often been found unsatisfactory in the past. In the time of Galileo it was argued that the texts, "And the sun stood still... and hasted not to go down about a whole day" (Joshua x. 13) and "He laid the foundations of the earth, that it should not move at any time" (Psalm cv. 5) were an adequate refutation of the Copernican theory."

"Machines take me by surprise with great frequency."

"The view that machines cannot give rise to surprises is due, I believe, to a fallacy to which philosophers and mathematicians are particularly subject. This is the assumption that as soon as a fact is presented to a mind all consequences of that fact spring into the mind simultaneously with it. It is a very useful assumption under many circumstances, but one too easily forgets that it is false. A natural consequence of doing so is that one then assumes that there is no virtue in the mere working out of consequences from data and general principles."

"Another simile would be an atomic pile of less than critical size: an injected idea is to correspond to a neutron entering the pile from without. Each such neutron will cause a certain disturbance which eventually dies away. If, however, the size of the pile is sufficiently increased, the disturbance caused by such an incoming neutron will very likely go on and on increasing until the whole pile is destroyed. Is there a corresponding phenomenon for minds, and is there one for machines? There does seem to be one for the human mind. The majority of them seem to be "sub-critical," i.e., to correspond in this analogy to piles of sub-critical size. An idea presented to such a mind will on average give rise to less than one idea in reply. A smallish proportion are super-critical. An idea presented to such a mind may give rise to a whole "theory" consisting of secondary, tertiary and more remote ideas. Animals minds seem to be very definitely sub-critical. Adhering to this analogy we ask, "Can a machine be made to be super-critical?""

"Presumably the child-brain is something like a note-book as one buys it from the stationer's. Rather little mechanism, and lots of blank sheets."

"We can only see a short distance ahead, but we can see plenty there that needs to be done."

"It is customary... to offer a grain of comfort, in the form of a statement that some peculiarly human characteristic could never be imitated by a machine. ... I cannot offer any such comfort, for I believe that no such bounds can be set."

"Alan Turing was the first to make a careful analysis of the potential capabilities of machines, inventing his famous "Turing machines" for the purpose. He argued that if any machine could perform a computation, then some Turing machine could perform it. The argument focuses on the assertion that any machine's operations could be simulated, one step at a time, by certain simple operations, and that Turing machines were capable of those simple operations. Turing's first fame resulted from applying this analysis to a problem posed earlier by Hilbert, which concerned the possibility of mechanizing mathematics. Turing showed that in a certain sense, it is impossible to mechanize mathematics: We shall never be able to build an "oracle" machine that can correctly answer all mathematical questions presented to it with a "yes" or "no" answer. In another famous paper Turing went on to consider the somewhat different question, "Can machines think?." It is a different question, because perhaps machines can think, but they might not be any better at mathematics than humans are; or perhaps they might be better at mathematics than humans are, but not by thinking, just by brute-force calculation power. These two papers of Turing lie near the roots of the subjects today known as automated deduction and artificial intelligence."

"I’d keep hearing: 'Oh, Alan Turing? Wasn’t he something to do with Bletchley Park? Didn’t he create the Apple logo? Didn’t he bite an apple?' But not many people know the whole story, that he invented the computer, broke the Enigma code, was prosecuted for being a homosexual and underwent a year of oestrogen injections before dying in 1954. The irony and sickness of it was extraordinary! He wasn’t shouting from the rooftops or trying to start a cause, he was just a man who was gay. He’s such a quiet hero."

"Beyond any doubt, the most important thing that has happened in cognitive science was Turing’s invention of the notion of mechanical rationality."

"And if the physicists who understood its potential had not told their generals and politicians, these would certainly have remained in ignorance, unless they were themselves postgraduate physicists, which was very unlikely. Again, Alan Turing's celebrated paper of 1935, which was to provide the foundation of modern computer theory, was originally written as a speculative exploration for mathematical logicians. The war gave him and others the occasion to translate theory into the beginnings of practice for the purpose of code-breaking, but when it appeared nobody except a handful of mathematicians even read, let alone took notice of Turing's paper. Even in his own college this clumsy-looking pale-faced genius, then a junior fellow with a taste for jogging, who posthumously became a sort of icon among homosexuals, was not a figure of any prominence; at least I do not remember him as such."

"Actually, with one alleged exception, the links of King’s dons with intelligence were with the British rather than the Soviet secret services. Kingsmen, headed by the small, roly-poly later professor of ancient history, F. E. Adcock, had set up the British codebreaking establishment in the First World War, and at least seventeen King’s dons were recruited by Adcock for the much more famous establishment at Bletchley during the Second World War, including probably the only genius at King’s in my undergraduate years, the mathematical logician Alan Turing, whom I recall as a clumsy-looking, pale-faced young fellow given to what would today be called jogging."

"From a very young age, I knew about the legend of Alan Turing – among awkward, nerdy teenagers, he is a patron saint. He never fit in, but accomplished these wonderful things, as part of a secret queer history of computer science. And so I always dreamt of writing something about him, and I thought that there had never been a proper narrative treatment of his life, that he deserved. I by chance met the producers of the film at a party, and one of them told me they had optioned a biography. When I asked who it was, they said, ‘it’s a mathematician that you’ve never heard of.’ When they told me it was Alan Turing, I almost tackled them, and I told them I’d do anything to write this film, I’d write it for free. It was all about luck and passion."

"Here’s the thing. Alan Turing never got to stand on a stage like this and look out at all of these disconcertingly attractive faces. I do. And that’s the most unfair thing I’ve ever heard. So in this brief time here, what I wanted to do was say this: When I was 16-years-old, I tried to kill myself because I felt weird and I felt different, and I felt like I did not belong. And now I’m standing here — and so I would like this moment to be for this kid out there who feels like she’s weird or she’s different or she doesn’t fit in anywhere: Yes, you do. I promise you do. Stay weird, stay different — and then, when it’s your turn, and you are standing on this stage, please pass the same message to the next person who comes along."

"I’m not gay, but I don’t think you have to be gay to have a gay hero. Growing up, Alan Turing was certainly mine. … I’m also not the greatest mathematician of my generation. We have lots of biographical differences, but nonetheless I always identified with him so much."

"His high-pitched voice already stood out above the general murmur of well-behaved junior executives grooming themselves for promotion within the Bell corporation. Then he was suddenly heard to say: "No, I'm not interested in developing a powerful brain. All I'm after is just a mediocre brain, something like the President of the American Telephone and Telegraph Company.""

"He wondered what one might ask in a structured conversation to decide if one's interlocutor was a human being or a computer. ...but the ultimate Turing test might be to pose the question "How would a guilt-stricken homosexual commit suicide?" Would a computer ever conceive of eating an apple laced with cyanide?"

"Although a mathematician, Turing took quite an interest in the engineering side of computer design. There was some discussion in 1947 as to whether a cheaper substance than mercury could not be found for use as an ultrasonic delay medium. Turing's contribution to this discussion was to advocate the use of gin, which he said contained alcohol and water in just the right proportions to give a zero temperature coefficient of propagation velocity at room temperature."

"Turing had a strong predeliction for working things out from first principles, usually in the first instance without consulting any previous work on the subject, and no doubt it was this habit which gave his work that characteristically original flavor. I was reminded of a remark which Beethoven is reputed to have made when he was asked if he had heard a certain work of Mozart which was attracting much attention. He replied that he had not, and added "neither shall I do so, lest I forfeit some of my own originality.""

"He was particularly fond of little programming tricks (some people would say that he was too fond of them to be a "good" programmer) and would chuckle with boyish good humor at any little tricks I may have used."

"Anyone, from the most clueless amateur to the best cryptographer, can create an algorithm that he himself can't break."

"There are two kinds of cryptography in this world: cryptography that will stop your kid sister from reading your files, and cryptography that will stop major governments from reading your files."

"Attacks always get better, they never get worse."

"The lesson here is that it is insufficient to protect ourselves with laws; we need to protect ourselves with mathematics. Encryption is too important to be left solely to governments."

"A few years ago I heard a quotation, and I am going to modify it here: If you think technology can solve your security problems, then you don't understand the problems and you don't understand the technology."

"It's certainly easier to implement bad security and make it illegal for anyone to notice than it is to implement good security."

"Digital files cannot be made uncopyable, any more than water can be made not wet."

"Every time I write about the impossibility of effectively protecting digital files on a general-purpose computer, I get responses from people decrying the death of copyright. "How will authors and artists get paid for their work?" they ask me. Truth be told, I don't know. I feel rather like the physicist who just explained to a group of would-be interstellar travelers, only to be asked: "How do you expect us to get to the stars, then?" I'm sorry, but I don't know that, either."

"Against the average user, anything works; there's no need for complex security software. Against the skilled attacker, on the other hand, nothing works."

"Elections serve two purposes. The first, and obvious, purpose is to accurately choose the winner. But the second is equally important: to convince the loser."

"It is poor civic hygiene to install technologies that could someday facilitate a police state."

"I mean, the computer industry promises nothing. Did you ever read a shrink-wrapped license agreement? You should read one. It basically says, if this product deliberately kills your children, and we knew it would, and we decided not to tell you because it might harm sales, we're not liable. I mean, it says stuff like that. They're absurd documents. You have no rights."

"Beware the Four Horsemen of the Information Apocalypse: terrorists, drug dealers, kidnappers, and child pornographers. Seems like you can scare any public into allowing the government to do anything with those four."

"Chaos is hard to create, even on the Internet. Here's an example. Go to Amazon.com. Buy a book without using SSL. Watch the total lack of chaos."

"When my mother gets a prompt 'Do you want to download this?' she's going to say yes. It's disingenuous for Microsoft to give you all of these tools [in Internet Explorer] with which to hang yourself, and when you do, then say it's your fault."

"More people are killed every year by pigs than by sharks, which shows you how good we are at evaluating risk."

"The very definition of news is something that hardly ever happens. If an incident is in the news, we shouldn't worry about it. It's when something is so common that its no longer news – car crashes, domestic violence – that we should worry."

"… if anyone thinks they can get an accurate picture of anyplace on the planet by reading news reports, they're sadly mistaken."

"We can't keep weapons out of prisons; we can't possibly expect to keep them out of airports."

"The point of terrorism is to cause terror, sometimes to further a political goal and sometimes out of sheer hatred. The people terrorists kill are not the targets; they are collateral damage. And blowing up planes, trains, markets or buses is not the goal; those are just tactics. The real targets of terrorism are the rest of us: the billions of us who are not killed but are terrorized because of the killing. The real point of terrorism is not the act itself, but our reaction to the act. And we're doing exactly what the terrorists want."

"Well-designed security systems fail gracefully."

"Not being angels is expensive"

"Technical problems can be remediated. A dishonest corporate culture is much harder to fix."

"Only amateurs attack machines; professionals target people."

"In China, programs have to be certified by the government in order to be used on computers there, which sounds an awful lot like the Apple store."

"If privacy is outlawed, only outlaws will have privacy."

"At no time in the past century has public distrust of the government been so broadly distributed across the political spectrum, as it is today."

"The natural flow of technology tends to move in the direction of making surveillance easier."

"There may be occasions when it is best to behave irrationally, but whether there are should be decided rationally."

"Let an ultraintelligent machine be defined as a machine that can far surpass all the intellectual activities of any man however clever. Since the design of machines is one of these intellectual activities, an ultraintelligent machine could design even better machines; there would then unquestionably be an "intelligence explosion", and the intelligence of man would be left far behind. Thus the first ultraintelligent machine is the last invention that man need ever make, provided that the machine is docile enough to tell us how to keep it under control. It is curious that this point is made so seldom outside of science fiction. It is sometimes worthwhile to take science fiction seriously."

"The minimum I. Q. necessary for one to grasp the concepts of Statistics required for the undergraduate degree is 120.(paraphrased)"

"That Homeric epic and the earliest Hebrew poetry were the results of long and rich developments should have been apparent to anyone who realizes that artistic perfection is never created ex nihilo."

"Prior to 1952, when M. Ventris first published his decipherment of the 'Linear B' tablets from Crete and Greece, showing that they were Greek, everyone assumed that Hebrew was recorded in writing before Greek. But now... we are reading Linear B Greek texts, written before the birth of Abraham (let alone before the date of any known Hebrew text)."

"While Ugarit is revolutionizing the problem of Old Testament origins, the Dead Sea scrolls are doing the same for the New Testament. How fortunate is this generation to live at a time when the sources of our culture—sacred and profane—are illuminated in a brighter light of history than our forefathers imagined possible!"

"Mesopotamian merchants spread their commercial institutions far and wide, into Western Asia, Egypt and Europe. The ancient inhabitants of Babylonia used the word qaqqadum, 'head', in the sense of 'principal'... our English word 'capital' (via Latin caput [head]) reflects ancient Mesoptamian usage. ...our financial system, that reckons with interest on principal, harks back to the land between the Tigris and Euphrates rivers."

"Traders (like the Phoenicians) carried their methods as well as their wares to Europe by ship."

"Babylonian mathematics and astronomy have left an indelible impression on our exact sciences. We still call some of the planets by their Babylonian names in translation."

"The Sabbath, perhaps the most important labour legislation next to the abolition of slavery, is a Hebrew institution."

"For every mound excavated in the Near East, a hundred remain untouched. ...most of the excavated mounds have been dug only in small part."

"I hope that the reader will not regard the contents of this book as an escape from the present world but rather as a key part of it."

"The competent archeologist can date pottery much as some of us can date cars or dresses of our own century."

"Transjordan, or Palestine east of the Jordan Rift is not sufficiently known and has therefore been in need of archeological study. ...these nations in antiquity belonged to a group of people called the Canaanites. Culturally and linguistically they were practically identical with the Judean and Israelite 'Canaanites' west of the Rift."

"It was Yawhism that distinguished the two Hebrew nations from the other Canaanites and it was the great Hebrew prophets who transformed their little 'Canaanite' people into one of the great factors of world history."

"It has been said that the Bedouin Arab is a parasite that lives on the camel, and this to a great extent is true. It is the camel that carries him about; it is the camel's hair that supplies him with both his clothes and his tent; the camel's dung is the fuel of the desert; it is the camel's meat that supplies food for his banquets; the camel's milk is his beverage; and I could go on enumerating the basic gifts of the camel to his Arab master."

"Solomon was a 'copper king', and all along that Araba, on both sides, we found many copper mines and smelting stations, all attributable to Solomon and his immediate successors."

"At Aqaba we were received in the most hospitable manner of the Arabs. We were put up in the police station there. The prisoners, oddly enough, were walking about enjoying apparent freedom. They were used as waiters and servants instead of being shut up in cells. ...I could detect no trace of bullying of even of discourtesy to the prisoners."

"By examining the pottery on any given site you can tell during which periods it has been occupied."

"The excavators cleared out one of the ancient cisterns, and a few of the winter rains sufficed to fill the cistern with enough water to supply the expedition with water for the whole season. This illustrates the possibilities of almost any country, provided the right kind of people are there. With energetic people, the few, but heavy, winter rains and be stretched a long, long way."

"The nomadic Semite... roves as a herdsman, partaking of Allah's hospitality."

"Archeological discoveries at sites like Ugarit prevent us from regarding Greece as the hermetically sealed Olympian miracle, or Israel as the vacuum-packed miracle from Sinai."

"The thesis of this book is simply that Greek and Hebrew civilisations are parallel structures built upon the same East Mediterranean foundation. ...the evidence is so abundant that our problem is one of selection and arrangement."

"Homer and Bible... towered above their predecessors and contemporaries."

"For centuries scholars have been forced to grapple with the problem of accounting for the parallels between Greek literature and the Bible. Did Greece borrow from Israel? Or did Israel borrow from Greece? Can the parallels be accidental, do they obliterate the uniqueness of both Israel and Greece?"

"V. Bérard attributed the [Greece and Israel] links mainly to the role of the Phoenicians. P. Jensen explained matters through the diffusion of the Gilgamesh Epic. ...but their one-sidedness and exaggeration brought them, and indeed the problem itself, into disrepute among critical scholars. The history of the problem has been ably documented by W. Baumgartner..."

"The prevailing attitude (which is gradually losing its grip) may be described as the tacit assumption that ancient Israel and Greece are two water-tight compartments... One is said to be sacred; the other, profane; one, Semitic; the other, Indo-European. One, Asiatic and Oriental; the other, European and Occidental. But the fact is that both flourished during the same centuries, in the same East Mediterranean corner of the globe, with both ethnic groups in contact with each other from the start."

"It seems strange that so many generations of Old Testament scholars, trained in Greek as well as Hebrew literature, have managed to keep their Greek and Hebrew studies rigidly compartmentalised."

"We absorb attitudes as well as subject matter in the learning process. ...the attitudes tend to determine what we see, and what we fail to see, in the subject matter. This is why attitude is just as important as content in the educational process."

"The Greeks viewed the Mediterranean not as a barrier but as a network of routes connecting people who dwelt along its shores. This is familiar to any student of Greece. ...the Hebrews express themselves similarly in passages like Psalm 8: 9 ("crossing the paths of the seas")."

"The priestly guilds were highly mobile, with the result that cultic practices crossed ethnic lines over wide areas."

"Manslaughter was requited through blood revenge. Accordingly the offender, to escape the avenger, would be forced to flee, cut off from his land and people, at the mercy of strangers far from home. [Examples are] 2 Samuel (14: 5-7)... Iliad a6: 571-574... Odyssey (15: 271-278)... (Genesis 4: 14)... (Genesis 4: 15)"

"The text of Homer about the Mycenaen Age with its memories of the Trojan War, and the Hebrew text covering from the Conquest through David's reign, cover ground with much in common geographically, chronologically and ethnically."

"Just as Mycenaean civilisation is a development of the Minoan, so too the Philistines who came to Palestine from Caphtor were an offshoot of the same general Aegean civilisation. ...until David's victories around 1000 B.C., the Philistines dominated the Hebrews, so that, militarily at least, the Davidic monarchy was the Hebrew response to the Philistine stimulus. ...the Achaeans, Trojans, Philistines and Hebrews during the closing centuries of the second millennium belonged to the same international complex of peoples, sharing many conventions and institutions, specifically in military matters."

"The customs of both the Greeks and Hebrews in that heroic age were often alien to their respective descendants in the classical periods. We shall have to bear in mind that the gulf separating classical Israel (of the great Prophets) from classical Greece (of the scientists and philosophers) must not be read back into the heroic age when both peoples formed part of the same international complex."

"It would be foolhardy to swell the pages of this book with an exhaustive list of Greco-Hebrew differences. Everyone knows that Homer is very different from the Bible."

"The parallels that form the core of this book fit into a historical framework in the wake of the Armana Age during the closing centuries of the second millenium. Prior to the Armana Age (i.e., before 1400 B.C.) Egyptian, Canaanite, Mesopotamian, Anatolian, Aegean and other influences met around the East Mediterranean to form an international order, by which each was in turn affected."

"Out of the Armana Age synthesis emerged the earliest traditions of Israel and Greece."

"That both the Gilgamesh Epic and the Odyssey deal with the episodic wanderings of a hero, would not be sufficiently specific to establish a genuine relation between them. But when both epics begin with the declaration that the hero gained experience from his wide wanderings, and end with his homecoming, a relationship dimly appears. ...when we note that whole episodes are in essential agreement, we are on firmer ground. For instance, both Gilgamesh and Odysseus reject a goddess's proposal for marriage; and each of the heroes interviews his dead companion in Hades."

"Historical, sociological, literary, linguistic, archeological and other techniques must be brought to bear when they are applicable to the material at hand."

"Homeric tradition has its own way of telling us that Minoan/Mycenaean civilisation was intertwined with the culture of the Semitic Phoenicians. Iliad 14: 321-322 makes Phoenix (named after the ancestor of the Phoenicians) the maternal grandfather of Minos. ...Archaeology bears out early cultural connections between the two."

"Epic poetry is divinely inspired (Iliad I: I) and as such is just as true as oracles, and for the same reason. It is no accident that oracles (such as those at Delphi) were enunciated in the same dactylic hexameter as the epic."

"Music was an art fostered by the mightiest of heroes. Achilles is represented as entertaining himself with his lyre. (Iliad 9: 185-6). We compare David, the warrior skilled in poetry, singing and musical instruments."

"When Zeus' son Sarpedon meets his fate, Zeus expresses grief for his dead son by causing blood to rain (Iliad 16: 459-461). In Egypt, the function of rain is replaced by the Nile which fructifies the soil. Accordingly, the Biblical Plague of Blood (Exodus 7: 19-25) is the Egyptian equivalent of the bloody rainfall in the Iliad."

"Despite the polytheism of the East Mediterranean nations, monotheistic trends were always present even in such crass polytheisms such as we find in Homer and in Egyptian literature."

"When the characters of epic and heroic saga are on significant missions, they are led divinely."

"The notion of a language of the gods appears in Sanskrit, Greek, Old Norse and Hittite cultures."

"The warriors who constituted the aristocracy were awarded land grants to recompense them for their share in conquering the country. Both in Greece and in Israel, the theory of society was basically the same. The conquerors were the fighting and ruling stratum; the conquered natives were degraded to the labouring class. In Sparta the latter were called Helots. In Israel the Canaanites were the "hewers of wood and the drawers of water.""

"The fitness (physical and moral) of kings were serious matters, for they were believed to bring on a corresponding state of land and people."

"The ancient theory of heroic genealogy... reflects paternity at two levels: human and divine. A man's inheritance comes from his human father, but his qualitative superiority among mortals comes from his divine father. When Odysseus is called Zeus-born (diognēs) this does not mean that the poet has forgotten... that he is the son of human Laertes. ...Zeus is often described as impregnating noble ladies, not so much to gratify his lust for women, but because divine parentage was a necessity among the claims of the aristocracy. Odysseus is a superhuman because he is diogenēs; but he is king of Ithaca because of his human father Laertes. Jesus is divine because of his heavenly Father; but he derives his kingship of the Jews from the mortal Joseph, who was heir to the throne (Matthew I). While normative Judaism has has tried to strip the Old Testament of this phenomenon, vestiges have nevertheless remained in the text."

"If the entire aristocracy is of divine descent, Zeus (or El) cannot save the human son without upsetting the order of things. ...Hera reminds Zeus that many sons of gods are fighting around Troy, and that if Zeus spares his son, other gods will do the same for their sons, so that the earthly system will cease (Iliad 16: 445-449)"

"When a new religion supplants an old religion, the gods of the old often survive as the demons of the new."

"There is a large corpus of magical texts from Babylonia of the Sassanian Era, designed to exorcise demons. In these texts, which are mostly Jewish and Christian, the Indo-Iranian deities called daiva appear as demons. ...the demons of these texts are constantly appearing to women in the form of their husbands, and impregnating them. As a result, the names of the clients are always matronymic because no one could be sure of his paternity. ...both the Greeks and Iranians had such notions."

"The ancients were not as denominationally minded as we in matters of their clergy. They were more concerned with obtaining services of a bona fide professional member of a priestley guild who was qualified to intercede between mortals and immortals, than with finding a religious leader whose sole qualification was like-mindedness."

"The older cultures did not develop the concept of canonical writings. There is no Bible in Egypt or Mesopotamia. Neither country had a collection of sacred writings that excluded other writings from comparable status. ...there was never an official "Book of the Dead" in Egypt."

"Only two people in East Mediterranean antiquity developed [parallel tendecies towards] "canonical" Scripture: the Greeks and the Jews. The Greeks treated Homer as their Scripture par excellence, much as the Jews regarded the Bible. ...Hebrew and pagan Greek scriptures were each considered the divinely inspired guide for life."

"Minos has rightly been compared with Moses. Both are greater-than-life-size figures who received the law from the supreme god on the sacred mountain (see Dionysius of Halicarnassus, Roman Antiquities 2: 61 concerning Minos)."

"The Book [of Judges] as a whole gives a coherent picture of an era and propounds the thesis that the institutions of pre-monarchic Israel were so chaotic... that centralized, hereditary kingship was necessary."

"The incorporation of... earlier sources does not mean that the Pentateuch or Former Prophets is the work of an editor who pasted together various docuements. Once we view the work as a whole, we see that it is a fresh creation though not a creatio ex nihilo. The same holds for Homeric Epic that has been subjected to the same kinds of modern literary criticism."

"Heroic epic and saga (Indic as well as Greek and Hebrew, etc.) combine action with genealogy. This is necessary because the action is performed by aristocrats who require genealogies."

"The most important document at Ugarit for both Biblical and Homeric studies is the Epic of Kret. It anticipates the Helen-of-Troy motif in the Iliad and Genesis, thus bridging the gap between the two literatures."

"Once we recognize the factor of royal epic in Genesis, we see that the Helen-of-Troy motif permeates the Patriarchal Narratives. ...Like Helen and Hurrai, Sarah and Dinah are heroines according to the standards of royal epic."

"Like Helen, Sarah is wonderously fair and ageless. ...Like Helen, Sarah's name means "princess" in normal Hebrew, and "queen" in Akkadian. It is conceivable that (like David afterwards, whose name dāvîd means "leader, chief") her title came to be used as her name."

"The [Judaic] Patriarchs are depicted as Arameans as long as they remained in their native lands."

"Scripture makes it clear that unlike the conceptions of Abraham and of Jacob, Isaac was conceived through divine agency. Like the Mycenaean Greek heroes, Isaac could claim paternity at two levels; the human and the divine. ...Normative Judaism has divested itself of this approach to the paternity of heroes, in spite of the tell-tale text in Genesis. Midrash does not hesitate to call Moses half-god and half-man. ...The Church tradition that connects the sacrifice of Isaac with the sacrifice of Christ apparently rests on a sound exegesis, for the sacrifice of Isaac would have meant not only the sacrifice of Abraham's son but of God's."

"The Samaritans (whose beginnings were pre-Josianic) have a Pentateuch quite similar to the familiar Jewish Pentateuch. ...our Pentateuchal text was fairly well established before the rift between the Samaritans and Judeans."

"The function of reciting (actually chanting—for Scripture and national epic were sung, not read) Pentateuch and Homer at national reunions is the same in both cases. The narrative knits the segments of the nation together telling how they achieved their place in history in the course of a great event (The Exodus or the Trojan War). All of the tribes and their leaders are heroic. The text brings in each tribe by name. ...there must be an honoured place for all."

"The Conquest of Joshua could not have been a primitive assault, because a civilized land like Canaan with well-fortified cities could easily have repulsed an attack that was militarily naïve. ...Spies were sent to search out the land and lay the groundwork."

"Battles ended with sunset or dusk; so heroes, on special occasions when they needed more time, were vouchsafed victory by the stoppage of the sun in Greek as well as Hebrew saga."

"The central problem of the Greek tragedies is why we suffer so at the hands of God. The movement that evoked Greek tragedy in the fifth century B.C. was spread over the East Mediterranean evoking a parallel response in Israel. ...And as in Greek tragedy, Job deals with the problem of why man suffers so at the had of God."

"Aristocrats (among Hebrews and Greeks) often had harems that included women of common or even servile origin, as well as well-born aristocratic ladies. Normally, the successors would be chosen from the sons born by ladies; but on occasion those born by servile or common wives achieved the ascendency. In the latter case, tradition could dwell on the phenomenon as "worthy of saga.""

"The prevailing view is simply that the Judges were inspired, not hereditary leaders. But this misses the point; the Judges were normally from the ruling aristocracy, quite like the kings in Homer. ...The kings did not necessarily inherit rulership from their fathers but sometimes did, like Odysseus from Laertes, or Abimelech from Gideon. ...the kings came from the fighting and landed aristocracy..."

"If archeology had yielded only the Epic of Kret, we would have enough to bridge the gap between the Iliad and Genesis. But... our new sources are so rich that we have only begun... The years ahead bid fair to be the most fruitful in the annals of Classical and Biblical scholarship. Our debt to the Bible and Classics is so great that this type of research will deepen our understanding of our culture and of ourselves."

"Cyrus Gordon is a brilliant linguist and one of the greatest living Semitists. Despite attempts by his enemies to replace it, his pioneering Ugaritic Grammar remains the standard work on the first new Semitic language to be discovered this century. Nevertheless, for the past thirty years he has been on the fringes of academia and most scholars consider him to be a crank. This is partly because his sins or errors are not ones of omission – towards which academia is extremely lenient – but of commission, which are considered irredeemably heinous. Moreover, his attempts to demonstrate the existence of Phoenician or even early Jewish influence on America are so far from conventional wisdom as to make him appear ludicrous. This means that all his original work can be, and has been, brushed aside with contempt."

"Dr. Gordon... contended that Hebrew inscriptions many centuries old had been found at two sites in the southeastern United States. Frank Moore Cross said... that Dr. Gordon was "in many ways a great scholar" but that this belief "simply did not make sense.""

"In 1894 Cyrus Thomas, a Smithsonian Institution archeologist, identified the Bat Creek site as a Cherokee burial ground. That identification has been challenged in the twentieth century by various writers including the irrepressible Cyrus Gordon, professor of Semitic languages. They claim that the Bat Creek inscription is Hebrew and related to the Bar Kochba rebellion that took place during AD 135 in Roman Judea. Gordon attempted to bolster the theory by pointing out that the Bat Creek inscription ties in quite nicely with various finds of Roman and Bar Kochba coins in the Kentucky and Tennessee area. Unfortunately, experts consider these finds to be fakes. Gordon's willingness to consider the possibility that these inscriptions were made by refugees from the defeat of the Jewish Revolt in AD 70 does not help his case because the arguments against it are almost as strong as those against the Bar Kochba rebellion."

"During the 1960's Cyrus Gordon, a respected professor of the Semitic languages and an ardent diffusionist, revived the Paraíba Stone's claims to authenticity. Basically Gordon asserted that the Paraíba inscription contained Phoenician grammatical constructions unknown in 1872. These same constructions were originally used in the 1870's to argue against the stone's authenticity. Subsequent research during the twentieth century, Gordon said, revealed that the anomalous grammatical usages in the Paraíba Stone were genuine. Other equally qualified specialists disagree with his conclusions and continue to declare the Paraíba Stone a hoax. That opinion remains the judgement of archeologists and historians in general."

"Professor Gordon has made himself at home in both the Semitic and Indo-European compartments of philology. This makes it possible for him to do things and to see things that are beyond a single compartment scholar's horizon."

"[Mathematics were] scarce looked upon as Academical studies but rather Mechanical... And among more than two hundred students (at that time) in our college, I do not know of any two (perhaps not any) who had more of Mathematicks than I, (if so much) which was then but little; and but very few, in that whole university. For the study of Mathematicks was at that time more cultivated in London than in the universities."

"Mathematicks were not, at the time, looked upon as Accademical Learning, but the business of Traders, Merchants, Seamen, Carpenters, land-measurers, or the like; or perhaps some Almanak-makers in London. And of more than 200 at that time in our College, I do not know of any two that had more of Mathematicks than myself, which was but very little; having never made it my serious studie (otherwise than as a pleasant diversion) till some little time before I was designed for a Professor in it."

"However, it is not unlikely that the Arabs, who received from the Indians the numeral figures (which the Greeks knew not), did from them also receive the use of them, and many profound speculations concerning them, which neither Latins nor Greeks know, till that now of late we have learned them from thence. From the Indians also they might learn their algebra, rather than from Diophantus."

"‘Yet some few of such investigations we have in the five first propositions of Euclid’s thirteenth book … seems to be the work of Theo, […] rather than of Euclid himself.’"

"Perhaps it would have been more prudent, if I were only writing to seek fame, to have presented some few particular propositions—as something admirable or stupefying—with apagogic proofs, concealing the method by which they were reached... Quite often they [the ancients] seem to have thought of doing this in order that others would marvel at them rather than understand; at least, so that these others, being compelled, produce their assent to those utterances of the mathematicians rather than understand a genuine investigation of the problem."

"You may find this work (if I judge rightly) quite new. For I see no reason why I should not proclaim it; nor do I believe that others will take it wrongly. ...it teaches all by a new method, introduced by me for the first time into geometry, and with such clarity that in these more abstruse problems no-one (as far as I know) has used..."

"This method of mine takes its beginnings where Cavalieri ends his Method of indivisibles. ...for as his was the Geometry of indivisibles, so I have chosen to call my method the Arithmetic of infinitesimals."

"I came across the mathematical writings of Torricelli... which... I read in... 1651... where... he expounds the geometry of indivisibles of Cavalieri. ...His method, as taught by Torricelli... was indeed all the more welcome to me because I do not know that anything of that kind was observed in the thinking of almost any mathematician I had previously met; for what holds for most... concerning the circle... usually had by polygons with an infinite number of sides, and... the circumference by... an infinite number of infinitely short lines... could.., it seemed to me, with... changes, be... adjusted to other problems; and... by that means examine... Euclid, Appolonius and especially... Archimedes. ...I began to think ...whether this might bring ...light to the quadrature of the circle."

"I imagined... it was possible... to establish by what means the circle could be squared, or... that it could... not, or... something would emerge... worthwhile."

"I... began... with simple series... of quantities in arithmetic proportion, or... their squares, cubes, etc. and then... their square roots, cube roots, etc. and powers composed of these... square roots of cubes etc. or... whatever... composites, whether the power was rational or... irrational. ...Whence a general theorem emerged... Proposition 64. But also... the quadrature... of the simple parabola... of all higher parabolas, and their complements, which no-one before... achieved. I... had enlarged geometry; for... there may now be taught by a single proposition the quadrature or all higher of infinitely many kinds... by one general method. ...I felt it would be welcome ...to the mathematical world ...also I saw ...the same doctrine widened ...I have related everything, whether conoids or pyramids, either erect or inclined, to cylinders and prisms. ...I saw ...as a direct consequence an almost completed teaching of spirals; and indeed I have taught the comparison with a circle... But also that teaching... was capable of extension..."

"Passing then to augmented series... and diminished... or altered... constituted from sums or differences of two or more other series. ...[I]t was not too difficult to relate everything to series of equals... I have continued the investigation with the same success not only for these series, ...but also for those which are as the squares, cubes, or any higher power... Where at the same time we made use of the figurate numbers, thus triangular, pyramidal, etc... and their distinguishing features were unexpectedly uncovered."

"[W]hereas Nature, in propriety of Speech, doth not admit more than Three (Local) Dimensions, (Length, Breadth and Thickness, in Lines, Surfaces and Solids;) it may justly seem improper to talk of a Solid (of three Dimensions) drawn into a Fourth, Fifth, Sixth, or further Dimension. A Line drawn into a Line, shall make a Plane or Surface; this drawn into a Line, shall make a Solid. But if this Solid be drawn into a Line, or this Plane into a Plane, what shall it make? A Plano-plane? This is a Monster in Nature, and less possible than a Chimera or a Centaure. For Length, Breadth and Thickness, take up the whole of Space. Nor can our Fansie imagine how there should be a Fourth Local Dimension beyond these Three."

"Let as many Numbers, as you please, be proposed to be Combined: Suppose Five, which we will call a b c d e. Put, in so many Lines, Numbers, in duple proportion, beginning with 1. The Sum (31) is the Number of Sumptions, or Elections; wherein, one or more of them, may several ways be taken. Hence subduct (5) the Number of the Numbers proposed; because each of them may once be taken singly. And the Remainder (26) shews how many ways they may be taken in Combination; (namely, Two or more at once.) And, consequently, how many Products may be had by the Multiplication of any two or more of them so taken. But the same Sum (31) without such Subduction, shews how many Aliquot Parts there are in the greatest of those Products, (that is, in the Number made by the continual Multiplication of all the Numbers proposed,) a b c d e. For every one of those Sumptions, are Aliquot Parts of a b c d e, except the last, (which is the whole,) and instead thereof, 1 is also an Aliquot Part; which makes the number of Aliquot Parts, the same with the Number of Sumptions. Only here is to be understood, (which the Rule should have intimated;) that, all the Numbers proposed, are to be Prime Numbers, and each distinct from the other. For if any of them be Compound Numbers, or any Two of them be the same, the Rule for Aliquot Parts will not hold."

"Suppose we a certain Number of things exposed, different each from other, as a, b, c, d, e, &c. The question is, how many ways the order of these may be varied? as, for instance, how many changes may be Rung upon a certain Number of Bells; or, how many ways (by way of Anagram) a certain Number of (different) Letters may be differently ordered? frameless|left|upright=.45|Alt.1,21) If the thing exposed be but One, as a, it is certain, that the order can be but one. That is 1. 2) If Two be exposed, as a, b, it is also manifest, that they may be taken in a double order, as ab, ba, and no more. That is 1 x 2 = 2. frameless|right|upright=.45|Alt.3 3) If Three be exposed; as a, b, c: Then, beginning with a, the other two b, c, may (by art. 2,) be disposed according to Two different orders, as bc, cb; whence arise Two Changes (or varieties of order) beginning with a as abc, acb: And, in like manner it may be shewed, that there be as many beginning with b; because the other two, a, c, may be so varied, as bac, bca. And again as many beginning with c as cab, cba. And therefore, in all, Three times Two. That is 1 x 2, x 3 = 6. frameless|left|upright=.7|Alt.34) If Four be exposed as a, b, c, d; Then, beginning with a, the other Three may (by art. preceeding) be disposed six several ways. And (by the same reason) as many beginning with b, and as many beginning with c, and as many beginning with d. And therefore, in all, Four times six, or 24. That is, the Number answering to the case next foregoing, so many times taken as is the Number of things here exposed. That is 1 x 2 x 3, x 4 = 6 x 4 = 24. 5) And in like manner it may be shewed, that this Number 24 Multiplied by 5, that is 120 = 24 x 5 = 1 x 2 x 3 x 4 x 5, is the number of alternations (or changes of order) of Five things exposed. (Or, the Number of Changes on Five Bells.) For each of these five being put in the first place, the other four will (by art. preceeding) admit of 24 varieties, that is, in all, five times 24. And in like manner, this Number 120 Multiplied by 6, shews the Number of Alternations of 6 things exposed; and so onward, by continual Multiplication by the conse quent Numbers 7, 8, 9, &c. 6) That is, how many so ever of Numbers, in their natural Consecution, beginning from 1, being continually Multiplied, give us the Number of Alternations (or Change of order) of which so many things are capable as is the last of the Numbers so Multiplied. As for instance, the Number of Changes in Ringing Five Bells, is 1 x 2 x 3 x 4 x 5 = 120. In Six Bells, 1 x 2 x 3 x 4 x 5 x 6 = 120 x 6 = 720. In Seven Bells, 720 x 7 = 5040. In Eight Bells, 5040 x 8 = 40320, And so onward, as far as we please."

"Logarithms was first of all Invented (without any Example of any before him, that I know of) by John Neper... And soon after by himself (with the assistance of Henry Briggs...) reduced to a better form, and perfected. The invention was greedily embraced (and deservedly) by Learned Men. ...in a short time, it became generally known, and greedily embraced in all Parts, as of unspeakable Advantage; especially for Ease and Expedition in Trigonometrical Calculations."

"These Exponents they call Logarithms, which are Artificial Numbers, so answering to the Natural Numbers, as that the addition and Subtraction of these, answers to the Multiplication and Division of the Natural Numbers. By this means, (the Tables being once made) the Work of Multiplication and Division is performed by Addition and Subtraction; and consequently that of Squaring and Cubing, by Duplication and Triplication; and that of Extracting the Square and Cubic Root, by Bisection and Trisection; and the like in the higher Powers."

"It was always my affectation even from a child, in all pieces of Learning or Knowledge, not merely to learn by rote, which is soon forgotten, but to know the grounds or reasons of what I learn; to inform my Judgement, as well as furnish my Memory; and thereby, make a better Impression on both."

"At Christmass 1631, (a season of the year when Boys use to have a vacancy from School,) I was, for about a fortnight, at home with my Mother at Ashford. I there found that a younger Brother of mine (in Order to a Trade) had, for about 3 Months, been learning (as they call'd it) to Write and Cipher, or Cast account, (and he was a good proficient for that time,) When I had been there a few days; I was inquisitive to know what it was, they so called. And (to satisfie my curiosity) my Brother did (during the Remainder of my stay there before I return'd to School) shew me what he had been Learning in those 3 Months. Which was (besides the writing a fair hand) the Practical part of Common Arithmetick in Numeration, Addition, Substraction, Multiplication, Division, The Rule of Three (Direct and Inverse) the Rule of Fellowship (with and without, Time) the Pule of False-Position, Rules of Practise and Reduction of Coins, and some other little things. Which when he had shewed me by steps, in the same method that he had learned them; and I had wrought over all the Examples which he before had done in his book; I found no difficulty to understand it, and I was very well pleased with it: and thought it ten days or a fortnight well spent. This was my first insight into Mathematicks; and all the Teaching I had."

"This suiting my humor so well; I did thenceforth prosecute it, (at School and in the University) not as a formal Study, but as a pleasing Diversion, at spare hours; as books of Arithmetick or others Mathematical fel occasionally in my way. For I had none to direct me, what books to read, or what to seek, or in what Method to proceed. For Mathematicks, (at that time, with us) were scarce looked upon as Academical Studies, but rather Mechanical; as the business of Traders, Merchants, Seamen, Carpenters, Surveyors of Lands, or the like; and perhaps some Almanack-makers in London."

"I made no Scruple of diverting (from the common Road of Studies then in fashion) to any part of Useful Learning. Presuming, that Knowledge is no Burthen; and, if of any part thereof I should afterwards have no occasion to make use, it would at least do me no hurt; And what of it l might or might not have occasion for, I could not then foresee."

"As to Divinity, (on which I had an eye from the first,) l had the happiness of a strict and Religious Education, all along from a Child: Whereby I was not only preserved from vicious Courses, and acquainted with Religious Exercises; but was early instructed in the Principles of Religion, and Catachetical Divinity, and the frequent Reading of Scripture, and other good Books, and diligent attendance on Sermons. (And whatever other Studies I followed, I was careful not to neglect this.) And became timely acquainted with Systematick and Polemick Theology. And had the repute of a good Proficient therein."

"In Hilary Term 1636, 7. I took the Degree of Batchelor of Arts; and in 1640, the Degree of Master of Arts, and then left Emanuel College; and the same year I entered into Holy Orders, ordained by Bishop Curle, then Bishop of Winchester. I then lived a Chaplain for about a year, in the house of Sr. Richard Darley, (an antient worthy Knight,) at Buttercramb in Yorkshire, and then, for two years more, with the Lady Vere, (the Widdow of the Lord Horatio Vere,) partly in London, and partly at Castlc-Hedingham in Essex, the antient seat of the Earls of Oxford."

"The Occasion of that Assembly was this; The Parliament which then was, (or the prevailing part of them,) were ingaged in a War with the King. ...The Issue of which War, proved very different from what was said to be at first intended. As is usual in such cases; the power of the sword frequently passing from hand to hand and those who begin a War, not being able to foresee where it wil end."

"About the beginning of our Civil Wars, in the year 1642, a Chaplain of Sr. Will. Waller's (one evening as we were sitting down to Supper at the Lady Vere's in London, with whom I then dwelt,) shewed me an intercepted Letter written in Cipher. He shewed it me as a Curiosity (and it was indeed the first thing I had ever seen written in Cipher.) And asked me between jeast and earnest, whether I could make any thing of it. And he was surprised when I said (upon the first view) perhaps I might, if it proved no more but a new Alphabet. It was about ten a clock when we rose from Supper. I then withdrew to my chamber to consider of it. And by the number of different Characters therein, (not above 22 or 23:) I judged that it could not be more than a new Alphabet, and in about 2 hours time (before I went to bed) I had deciphered it; and I sent a Copy of it (so deciphered) the next morning to him from whom I had it. And this was my first attempt at Deciphering."

"Being encouraged by... success, beyond expectation; I afterwards ventured on many others and scarce missed of any, that I undertook, for many years, during our civil Wars, and afterwards. But of late years, the French Methods of Cipher are grown so intricate beyond what it was wont to be, that I have failed of many; tho' I have master'd divers of them. Of such deciphered Letters, there be copies of divers remaining in the Archives of the Bodleyan Library in Oxford; and many more in my own Custody, and with the Secretaries of State."

"On March 4. 1644, 5. I married Susanna daughter of John and Rachel Glyde of Northjam in Sussex; born there about the end of January 1621, 2. and baptised Feb. 3 following. By whom I have (beside other children who died young) a Son and two Daughters now surviving; John born Dec. 26 1650. Anne born June 4. 1656. and Elizabeth born Sept. 23 1658. ...My Wife died at Oxford Mar. 17. 1686, 7. after we had been married more than 42 years."

"About the year 1645 while, I lived in London (at a time, when, by our Civil Wars, Academical Studies were much interrupted in both our Universities:) beside the Conversation of divers eminent Divines, as to matters Theological; I had the opportunity of being acquainted with divers worthy Persons, inquisitive into Natural Philosophy, and other parts of Humane Learning; And particularly of what hath been called the New Philosophy or Experimental Philosophy. We did by agreement, divers of us, meet weekly in London on a certain day, to treat and discourse of such affairs. ...Some of which were then but New Discoveries, and others not so generally known and imbraced, as now they are, with other things appertaining to what hath been called The New Philosophy; which, from the times of Galileo at Florence, and Sr. Francis Bacon (Lord Verulam) in England, hath been much cultivated in Italy, France, Germany, and other Parts abroad, as well as with us in England. About the year 1648, 1649, some of our company being removed to Oxford (first Dr. Wilkins, then I, and soon after Dr. Goddard) our company divided. Those in London continued to meet there as before... Those meetings in London continued, and (after the King's Return in 1660) were increased with the accession of divers worthy and Honorable Persons; and were afterwards incorporated by the name of the Royal Society, &c. and so continue to this day."

"I made it my business to examine things to the bottom; and reduce effects to their first principles and original causes. Thereby the better to understand the true ground of what hath been delivered to us from the Antients, and to make further improvements of it. What proficiency I made therein, I leave to the Judgement of those who have thought it worth their while to peruse what I have published therein from time to time; and the favorable opinion of those skilled therein, at home and abroad."

"In the year 1660 being importuned by some friends of his, I undertook so to teach Mr. Daniel Whalley of Northampton, who had been Deaf and Dumb from a Child. I began the work in 1661, and in little more than a year's time, I had taught him to pronounce distinctly any words, so as I directed him... and in good measure to understand a Language and express his own mind in writing; And he had in that time read over to me distinctly (the whole or greatest part of) the English Bible; and did pretty well understand (at least) the Historical part of it. In the year 1662 I did the like for Mr. Alexander Popham... I have since that time (upon the same account) taught divers Persons (and some of them very considerable) to speak plain and distinctly, who did before hesitate and stutter very much; and others, to pronounce such words or letters, as before they thought impossible for them to do: by teaching them how to rectify such mistakes in the formation, as by some natural impediment, or acquired Custome, they had been subject to."

"It hath been my Lot to live in a time, wherein have been many and great Changes and Alterations. It hath been my endeavour all along, to act by moderate Principles, between the Extremities on either hand, in a moderate compliance with the Powers in being, in those places, where it hath been my Lot to live, without the fierce and violent animosities usual in such Cases, against all, that did not act just as I did, knowing that there were many worthy Persons engaged on either side. And willing whatever side was upmost, to promote (as I was able) any good design for the true Interest of Religion, of Learning, and the publick good; and ready so to do good Offices, as there was Opportunity; And, if things could not be just, as I could wish, to make the best of what is: And hereby, (thro' God's gracious Providence) have been able to live easy, and useful, though not Great."

"Thus in Compliance with your repeated desires, I have given you a short account of divers passages of my life, 'till I have now come to more than fourscore years of age. How well I have acquitted my self in each, is for others rather to say, than for Your friend and servant John Wallis. Oxford January 29. 1696, 7."

"It is not unknown to those who know any Thing of publike Affairs, of how great Concernment it is, especially in civill Commotions, for those who are to manage such Transactions, to be furnished with continuall Intelligence from their Correspondents, yet so as to conceal their Councells and Resolutions from the adverse Party. And to this Purpose, in all Ages, much Care and lndustry hath been still used, how in Matters of Consequence, to convey Intelligence safely and secretly to those with whom they hold Correspondence, so as not to bee intercepted by the Enemy, or if intercepted, at least not discovered. And as this is no where of more Concernment, so no where more difficult, than in civill Wars, where the intermingling of opposite Parties makes it difficult, if not impossible, to distinguish Friends and Foes."

"Upon this Occasion many Methods have been invented of secret Writing, or Writing in Cipher, a Thing heretofore scarce known to any but the Secretaries of Princes, or others of like Condition; but of late Years, during our Commotions and civill Wars in England, grown very common and familiar, so that now there is scarce a Person of Quality, but is more or lesse acquainted with it, and doth as there is Occasion, make use of it."

"If any ask, with what Confidence I durst adventure upon a Task so unusuall, as interpreting of Letters committed to Cipher; I shall only give this plain Account thereof."

"Partly out of my owne Curiosity, partly to satisfy the Gentleman's Importunity that did request it, I resolved to try what I could do in it: And having projected the best Methods I could think of for the effecting it, I found yet so hard a Task, that I did divers Times give it over as desperate: Yet, after some Intermissions, resuming it againe, I did at last overcome the Difficulty; but with so much Paines and Expense of Time as I am not willing to mention; though yet I did not repent of that Labour, when I had discovered thereby, that it was a Businesse, which though with much Difficulty, was yet capable to bee effected."

"I was... informed, that Baptista Porta, and one or two more, had written somewhat of that Subject, upon this Information I was willing to see whether I might from any of them find any Directions, that might help mee, if I should afterwards have the like Occasion: But I found very little in any of them for my Purpose. Their Businesse being for the most Part, onely to shew how to write in Cipher, (which was not my Work,) and that Things so written were beyond the Skill of Men to decipher. Onely in Baptista Porta (who alone if I mistake not, hath written any Thing to Purpose about deciphering, and was it seemes famous in his Time for his Abilities that Way;) I found that there were some general Directions, such as were obvious from the Nature of the Thing, and which I had before of myself taken Notice of, and made use of so far as the Nature of an intricate Cipher would permit. But the Truth of it is, there are scarce any of his Rules, which the present Way of Cipher (which is now much improved, beyond what, it seemes, it was in his Days) doth not in a Manner wholly elude..."

"I saw, there was little or no Help to bee exspected from others; but that if I should have further Occasions of that Kind, I must trust to my owne Industry, and such Observations as the present Case should afford. And indeed the Nature of the Thing is scarce capable of any other Directions; every new Cipher allmost being contrived in a new Way, which doth not admit any constant Method for the finding of it out: But hee that will do any Thing in it, must first furnish himself with Patience and Sagacity, as well as hee may, and then Consilium in arenâ capere, and make the best Conjectures hee can, till hee shall happen upon something that hee may conclude for Truth."

"Of the Oxford mathematician John Wallis... Sorbière wrote that his appearance inclined one to laughter and that he suffered from bad breath that was "noxious in conversation." Wallis' only hope, according to Sorbière, was to be purified by the "Air of the court of London." For the Society's nemesis Thomas Hobbes, however, who was also Wallis's personal enemy, Sorbière had only praise."

"Accountants eventually became comfortable with using negative numbers... but for a long time mathematicians remained wary... the negatives were known as absurd numbers—numeri absurdi...Consider this equation:\frac{-1}{\quad 1} = \frac{\quad 1}{-1}...it states that the ratio of a smaller number, -1, to a larger number, 1, is equal to the ratio of a larger number, 1, to a smaller one, -1. The paradox was much discussed... To make sense of negative numbers, many mathematicians, including Leonhard Euler, came to the bizarre conclusion that they were larger than infinity. ...One voice of clarity among the confusion belonged to... John Wallis, who devised a powerful visual interpretation for the negative numbers. In his 1685 work A Treatise of Algebra, he first described the ","... By replacing the idea of quantity with the idea of position, Wallis argued that negative numbers were neither "Unuseful [nor] Absurd,"...It took a few years for Wallis' idea to hit the mainstream, but... it is the most successful explicatory diagram of all time."

"Vieta died in 1603, Porta died in 1615, and Dr. Wallis was born in 1617. So that, in all Probability, here is a great Deduction to be made from the many hundred Years, in which we were to have understood that the Art of Decyphering had been in being before Dr. Wallis was born."

"How the Dr. first came to apply himself to this Art, we shall have from himself; what use he made of it, we have had, tho unfairly, from other Hands. But as his Skill in the Art of Decyphering has no Relation to his Political Principles, he might be very sagacious in one Respect, and very erroneous in the other. I wish, both for his own Credit and the publick Good, that he had employed his Skill in the Service of the King; but it is too Well known, that he did not so. He was publickly charged in his Life-time by Henry Stubbe, and from him by Anthony Wood, with "having decyphered (besides others, to the Ruin of many loyal Persons) the King's Cabinet taken at Naseby and as a Monument of his noble Performances, depositing the Original, with the Decyphering, in the publick Library at Oxford."

"It is not that I do not approve it, but all his propositions could be proved in the usual, regular Archimedian way in many fewer words than this book [Arithmetica Infinitorum] contains. I do not know why he has preferred this method with algebraic notation to the older way which is both more convincing and more elegant."

"[W]e advise that you would lay aside (for some time at least) the Notes, Symbols, or Analytick Species (now since Vieta's time, in frequent use,) in the construction and demonstration of Geometrick Problems, and perform them in such method as Euclide and Apollonius were wont to do; that the neatness and elegance of Construction and Demonstrations, by them so much affected, do not by any degrees grow into disuse."

"It is customary to consider Chebyshev, Gauss, Jacobi, and Legendre as the main creators of the theory of orthogonal polynomials. However, their contributions were directly influenced by Brouncker and Wallis who, in March of 1655, made discoveries which influenced the development of analysis for the next hundred years. Namely, Wallis found an infinite product of rational numbers converging to 4/π and Brouncker gave a remarkable continued fraction for this quantity. ...The only mathematician who understood the importance of these discoveries was Euler. ...he felt that the recovery of the original Brouncker's proof could open up new perspectives for analysis. As usual, Euler was right."

"By March of 1655 John Wallis had almost completed his Arithmetica Infinitorum in which he promoted an important method of interpolation. This was a great work. ...Wallis discovered that analytic formulas can be interpolated by their values at integer numbers. ...Wallis successfully applied his interpolation to find formulas for the areas under many curves. Only one curve remained uncovered. It was the unit circle. In 1593 Viète had found the formula \frac{2}{\pi} = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2+\sqrt{2}}}{2}\cdot\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdot\cdots. Since the multipliers in Viète's formula are algebraic irrationalities of increasing order, it was not the formula which could meet Wallis' requirements. Finally in March of 1655, Wallis obtained his now well-known formula \frac{2}{\pi} = \frac{1\cdot3}{2\cdot2}\cdot\frac{3\cdot5}{4\cdot4}\cdot\frac{5\cdot7}{6\cdot6}\cdot\cdots\frac{(2n-1)\cdot(2n+1)}{2n\cdot2n}\cdot\cdots."

"Wallis' mathematical work, most notably his Arithmetica Infinitorum, was the polemic target of Pierre de Fermat and Thomas Hobbes. ...the letters of the French mathematician were reproduced in Wallis' Commercium Epistolicum (1658) ...One of the criticisms leveled at Wallis concerned the validity of induction. The fact that a proposition is proven true for a few numbers... does not imply that it is true for all... as Fermat, a master of number theory, knew too well. Fermat invited Wallis to devote himself to number theory, but Wallis found it of little interest. Number theory struck him as something of little use in applications, in other words, as a useless inquiry. ...Wallis ...claimed that induction methods were not his invention but had been employed both recently by Henry Briggs and Viète and in the ancient world by Euclid."

"During the wars between Charles I and Cromwell, Wallis's sympathies were with Cromwell, and he was of great service in reading royalist dispatches written in cipher. In fact, he was one of the most famous cryptologists of his day."

"Wallis did not become interested in mathematics till the age of thirty-one, but devoted himself to the subject for the rest of his life. One of the earliest and most important books on algebra ever written in English was his treatise published in 1685. It contains a brief historical sketch of the subject which is unfortunately not entirely accurate, but his treatment of the theory and practice of arithmetic and algebra has made the book a standard work for reference ever since."

"In that part... of my book where I treat of geometry, I thought it necessary in my definitions to express those motions by which lines, superficies, solids, and figures were drawn and described, little expecting that any professor of geometry should find fault therewith, but on the contrary supposing I might thereby not only avoid the cavils of the sceptics, but also demonstrate divers propositions which on other principles are indemonstrable. And truly, if you shall find those my principles of motion made good, you shall find also that I have added something to that which was formerly extant in geometry. For first, from the seventh chapter of my book De Corpore, to the thirteenth, I have rectified and explained the principles of the science; id est, I have done that business for which Dr. Wallis receives the wages."

"You can see without admonition, what effect this false ground of yours will produce in the whole structure of your Arithmetica Infinitorum; and how it makes all that you have said unto the end of your thirty-eighth proposition, undemonstrated, and much of it false. The thirty-ninth is this other lemma: "In a series of quantities beginning with a point or cypher and proceeding according to the series of the cubic numbers as 0.1.8.27.64, &c. to find the proportion of the sum of the cubes to the sum of the greatest cube, so many times taken as there be terms." And you conclude that "they have a proportion of 1 to 4;" which is false. ... And yet there is grounded upon it all that which you have of comparing parabolas and paraboloeides with the parallelograms wherein they are accommodated. ... Besides, any man may perceive that without these two lemmas (which are mingled with all your compounded series with their excesses) there is nothing demonstrated to the end of your book: which to prosecute particularly, were but a vain expense of time. Truly, were it not that I must defend my reputation, I should not have showed the world how little there is of sound doctrine in any of your books. For when I think how dejected you will be for the future, and how the grief of so much time irrecoverably lost, together with the conscience of taking so great a stipend, for mis-teaching the young men of the University, and the consideration of how much your friends will be ashamed of you, will accompany you for the rest of your life, I have more compassion for you than you have deserved. Your treatise of the Angle of Contact, I have before confuted in a very few leaves. And for that of your Conic Sections, it is so covered over with the scab of symbols, that I had not the patience to examine whether it be well or ill demonstrated."

"The true "principle of number," for Wallis as for Stevin, is the "nought". It is the sole numerical analogue of the geometric point (just as the instant is the temporary analogue... Wallis expressly rejects the accusation that he is relinquishing the unanimous opinion of the ancients and the moderns, who all saw the unit as the element of number. ...the traditional opinion can be brought into accord with his own if the following distinction is taken account of: Something can be a "principle" of something (1) which is the "first which is such" (primum quod sic) as to be of the same nature as the thing itself and (2) which is the last which is not" (ultimum quod non) such as to be of the same nature of the thing itself. In the first sense the unit may indeed be called the "principle of number," while the nought is a "principle" in the second sense. ...The ancients... overlooked the fact that the analogy which exists is not between the "point" and the "unit," but between the point and the "nought." For this reason they were able to develop their algebra only for "geometric magnitudes"..."

"Paralleling what happened in France, an English group centered about John Wallis began in 1645 to hold meetings in Gresham College, London, These men emphasized mathematics and astronomy. The group was given a formal charter by Charles II in 1662 and adopted the name of the Royal Society of London for the Promotion of Natural Knowledge."

"Before Newton and Leibniz, the man who did most to introduce analytical methods in the calculus was John Wallis. Though he did not begin to learn mathematics until he was about twenty—his university education at Cambridge was devoted to theology—he became professor of geometry at Oxford and the ablest British mathematician of the century, next to Newton. In his Arithmetica Infinitorum (1655), he applied analysis and the method of indivisibles to effect many quadratures and obtain broad and useful results."

"[E]arly analytic geometers—Descartes in particular—did not accept that geometry could be based on numbers or algebra. Perhaps the first to take the idea of arithmetizing geometry seriously was Wallis... [(1657) Mathesis universalis. Opera 1, 11-228.] Chs. XXIII and XXV, gave the first arithmetic treatment of Euclid's Books II and V, and he had earlier given purely algebraic treatment of s [(1655) De sectionibus conicus. Opera 1, 291-364.]. He initially derived equations from classical definitions by sections of the cone but then proceeded to derive their properties from the equations, "without the embranglings of the cone," as he put it."

"The greatest of modern have been so far from adding any thing of importance to the discoveries of ancient mathematicians, that some of their most splendid inventions are either wholly erroneous or remarkable instances of the possibility of deducing true conclusions from unscientific and false principles. Strange, however as this assertion may seem, the following elementary treatise demonstrates it to be true; by showing that all the leading propositions of the Arithmetic of Infinites of Dr. Wallis are false, and that the Doctrine of Fluxions is a baseless fabric, and in the language of the ingenious Bishop Berkley, "must be considered only as a presumption, as a knack, an art, or rather an artifice, but not a scientific demonstration."

"Wallis, whether by his own efforts or not, acquired sufficient mathematics at Cambridge to be ranked as the equal of mathematicians such as Descartes, Pascal, and Fermat."

"There was then no professorship in mathematics and no opening for a mathematician to a career at Cambridge; and so Wallis reluctantly left the university. In 1649 he was appointed to the Savilian chair of geometry at Oxford, where he lived until his death on Oct. 28, 1703. It was there that all his mathematical works were published. Besides those he wrote on theology, logic, and philosophy; and was the first to devise a system for teaching deaf mutes."

"The most notable of these [his mathematical works] was his Arithmetica infinitorum, which was published in 1656. It is prefaced by a short tract on conic sections which was subsequently expanded into a separate treatise. He then established the law of indices, and shewed that x^{-n} stood for the reciprocal of x^n and that x^\frac{p}{q} stood for the q^{th} root of x^p. He next proceeded to find by the method of indivisibles the area enclosed between the curve y = x^m, the axis of x, and any ordinate x = h; and he proved that this was to the parallelogram on the same base and of the same altitude in the ratio 1:m + 1. He apparently assumed that the same result would also be true for the curve y = ax^m, where a is any constant. In this result m may be any number positive or negative, and he considered in particular the case of the parabola in which m = 2, and that of the hyperbola in which m = -1: in the latter case his interpretation of the result is incorrect. He then shewed that similar results might be written down for any curve of the form y = \sum{ax^m}; so that if the ordinate y of a curve could be expanded in powers of the abscissa x, its quadrature could be determined. Thus he said that if the equation of a curve was y = x^0 + x^1 + x^2 +... its area would be y = x + \frac{1}{2}x^2 + \frac{1}{3}x^3 +... He then applied this to the quadrature of the curves y = (1 - x^2)^0, y = (1 - x^2)^1, y = (1 - x^2)^2, y = (1 - x^2)^3, &c. taken between the limits x = 0 and x = 1: and shewed that the areas are respectively1,\quad \frac{2}{3},\quad \frac{8}{15},\quad \frac{16}{35},\quad \&c."

"He next considered curves of the form y = x^\frac{1}{m} and established the theorem that the area bounded by the curve, the axis of x, and the ordinate x = 1 is to the area of the rectangle on the same base and of the same altitude as m:m + 1. This is equivalent to finding the value of \int_{0}^{1}x^\frac{1}{m}dx. He illustrated this by the parabola in which m = 2. He stated but did not prove the corresponding result for a curve of the form y = x^\frac{p}{q}."

"As he was unacquainted with the he could not effect the quadrature of the circle, whose equation is y = (1 - x^2)^\frac{1}{2}, since he was unable to expand this in powers of x. He laid down however the principle of interpolation. He argued that as the ordinate of the circle is the geometrical mean between the ordinates of the curves y = (1 - x^2)^0 and y = (1 - x^2)^1, so as an approximation its area might be taken as the geometrical mean between 1 and \frac{2}{3}. This is equivalent to taking 4\sqrt{\frac{2}{3}} or 3.26... as the value of \pi. But, he continued, we have in fact a series 1, \frac{2}{3}, \frac{8}{15}, \frac{16}{35},... and thus the term interpolated between 1 and \frac{2}{3} ought to be so chosen as to obey the law of this series. This by an elaborate method leads to a value for the interpolated term which is equivalent to making\pi = 2\frac{2\cdot2\cdot4\cdot4\cdot6\cdot6\cdot8\cdot8...}{1\cdot3\cdot3\cdot5\cdot5\cdot7\cdot7\cdot9...}The subsequent mathematicians of the seventeenth century constantly used interpolation to obtain results which we should attempt to obtain by direct algebraic analysis."

"The Arithmetica infinitorum was followed in 1656 by a tract on the angle of contact; in 1657 by the Mathesis universalis; in 1658 by a correspondence with Fermat; and by a long controversy with Hobbes on the quadrature of the circle."

"In 1659 Wallis published a tract on s in which incidentally he explained how the principles laid down in his Arithmetica infinitorum could be applied to the rectification of s: and in the following year one of his pupils, by name William Neil, applied the rule to rectify the x^3 = ay^2. This was the first case in which the length of a curved line was determined by mathematics, and as all attempts to rectify the ellipse and hyperbola had (necessarily) been ineffectual, it had previously been generally supposed that no curves could be rectified."

"In 1665 Wallis published the first systematic treatise on Analytical conic sections. Analytical geometry was invented by Descartes and the first exposition of it was given in 1637: that exposition was both difficult and obscure, and to most of his contemporaries, to whom the method was new, it must have been incomprehensible. Wallis made the method intelligible to all mathematicians. This is the first book in which these curves are considered and defined as curves of the second degree and not as sections of a cone."

"In 1668 he laid down the principles for determining the effects of the collision of imperfectly elastic bodies. This was followed in 1669 by a work on statics (centres of gravity) and in 1670 by one on dynamics: these provide a convenient synopsis of what was then known on the subject."

"In 1686 Wallis published an Algebra, preceded by a historical account of the development of the subject which contains a great deal of valuable information... This algebra is noteworthy as containing the first systematic use of formulae."

"A particle moving with a uniform velocity would be denoted by Wallis by the formula s = vt, ...while previous writers would have denoted the same relation by stating what is equivalent to the proposition s1 : s2 = v1t1 : v2t2 (see e.g. Newton's Principia, bk. I. sect. I., lemma 10 or 11)."

"Wallis rejected as absurd and inconceivable the now usual idea of a negative number as being less than nothing, but accepted the view that it is something greater than infinity."

"The writings of Wallis published between 1655 and 1665 revealed and explained to all students the principles of those new methods which distinguish modern from classical mathematics. His reputation has been somewhat overshadowed by that of Newton, but his work was absolutely first class in quality. Under his influence a brilliant mathematical school arose at Oxford. In particular I may mention Wren, Hooke, and Halley as among the most eminent of his pupils. But the movement was shortlived, and there were no successors of equal ability to take up their work."

"The foundations of the new analysis were laid in the second half of the seventeenth century when Newton... and Leibnitz... founded the Differential and Integral Calculus, the ground having been to some extent prepared by the labours of Huyghens, Fermat, Wallis, and others. By this great invention of Newton and Leibnitz, and with the help of the brothers James Bernoulli... and John Bernoulli... the ideas and methods of the Mathematicians underwent a radical transformation which naturally had a profound effect upon our problem. The first effect of the new analysis was to replace the old geometrical or semi-geometrical methods of calculating \pi by others in which analytical expressions formed according to definite laws were used, and which could be employed for the calculation of \pi to any assigned degree of approximation."

"The first result of this kind was due to John Wallis... undergraduate at Emmanuel College, Fellow of Queen's College, and afterwards Savilian Professor of Geometry at Oxford. He was the first to formulate the modern arithmetic theory of limits, the fundamental importance of which, however, has only during the last half century received its due recognition; it is now regarded as lying at the very foundation of analysis. Wallis gave in his Arithmetica Infinitorum the expression\frac{\pi}{2} = \frac {2}{1}\cdot\frac {2}{3}\cdot\frac {4}{3}\cdot\frac {4}{5}\cdot\frac {6}{5}\cdot\frac {6}{7}\cdot\frac {8}{7}\cdot\frac {8}{9}\cdotsfor \pi as an infinite product, and he shewed that the approximation obtained at stopping at any fraction in the expression on the right is in defect or in excess of the value \frac{\pi}{2} according as the fraction is proper or improper. This expression was obtained by an ingenious method depending on the expression for \frac{\pi}{8} the area of a semi-circle of diameter 1 as the definite integral \int\limits_{0}^{1}\sqrt{x-x^2}dx. The expression has the advantage over that of Vieta that the operations required are all rational ones."

"Lord Brounckner... communicated without proof to Wallis the [] expression\frac{4}{\pi} = 1 + \frac {1}{2 +} \frac {9}{2 +} \frac {25}{2 +} \frac {49}{2 +}\cdots,a proof of which was given by Wallis in his Arithmetica Infinitorum. It was afterwards shewn by Euler that Wallis' formula could be obtained from the development of the sine and cosine in infinite products, and that Brounckner's expression is a particular case of much more general theorems."

"Of the contemporaries of Newton one of the most prominent was John Wallis. ...Wallis was a voluminous writer, and not only are his writings erudite, but they show a genius in mathematics... He was one of the first to recognize the significance of the generalization of exponents to include negative and fractional as well as positive and integral numbers. He recognized also the importance of Cavalieri's method of indivisibles, and employed it in the quadrature of such curves as y=xn, y=x1/n, and y=x0 + x1 + x2 +... He failed in his attempts at the approximate quadrature of the circle by means of series because he was not in possession of the general form of the binomial theorem. He reached the result, however, by another method. He also obtained the equivalent of ds = \!dx \sqrt{1+(\frac{dy}{dx})^2} for the length of an element of a curve, thus connecting the problem of rectification with that of quadrature."

"In 1673 he wrote his great work De Algebra Tractatus; Historicus & Practicus, of which an English edition appeared in 1685. In this there is seen the first serious attempt in England to write on the history of mathematics, and the result shows a wide range of reading of classical literature of the science. This work is also noteworthy because it contains the first of an effort to represent the imaginary number graphically by the method now used. The effort stopped short of success but was an ingenious beginning."

"Wallis was in sympathy with Greek mathematics and astronomy, editing parts of the works of Archimedes, Eutocius, Ptolemy, and Aristarchus; but at the same time he recognized the fact that the analytic method was to replace the synthetic, as when he defined a conic as a curve of the second degree instead of as a section of a cone, and treated it by the aid of coordinates."

"His writings include works on mechanics, sound, astronomy, the tides, the laws of motion, the Torricellian tube, botany, physiology, music, the calendar (in opposition to the Gregorian reform), geology, and the compass,—a range too wide to allow of the greatest success in any of the lines of his activity. He was also an ingenious cryptologist and assisted the government in deciphering diplomatic messages."

"Among his interesting discoveries was the relation \frac{4}{\pi} = \frac32\cdot\frac34\cdot\frac54\cdot\frac56\cdot\frac76\cdot\frac78\cdots one of the early values of π involving infinite products."

"In his 1657 Mathesis universalis... Wallis printed geometric representations of the first 10 propositions of book II of the Elements alongside symbolic 'arithmetical' demonstrations and worked examples in numbers; he believed that the symbolic treatment was more general."

"Wallis’s 1656 Arithmetic of infinities was a crucial text for Newton’s development of calculus, a fact that Newton... related to Gottfried Leibniz in a... letter."

"Wallis... denigrat[ed] synthetic demonstration for its failure to bring clarity to mathematical investigation."

"The association between geometric synthesis and 'common' understanding was not Newton’s innovation: many of Newton’s predecessors, including Wallis, endorsed it."

"Seventeenth-century proponents of symbols frequently advanced the fecundity of their approach as evidence for its superiority over classical mathematics: Wallis’s Arithmetic of infinities displayed results 'neither discovered by nor known to others', and René Descartes’s 1637 Geometry announced its general solution to a family of problems that the Greeks had left mostly unsolved. ...Newton, inspired ...by these ...texts ...used symbols to invent an algorithmic version of the calculus."

"Algebra appealed to Newton’s contemporaries... for its fecundity... [and] the... presentational qualities... In... his... Treatise on conic sections [dedication], Wallis contrasted his novel treatment of the s with the diagrammatic treatment in Apollonius’s Conics... 'neglected beyond measure' by Wallis’s contemporaries 'as though it were insurmountable and full of troublesome madness'. Wallis implied that geometers read it... superficially since they feared that it would drive them mad. He contrasted his own figures with Apollonius’s... to use 'schemata as simple as possible, lest intricate leadings of lines bring in confusion'."

"Wallis saw Greek diagrams as intricate and confused. His own... clearly displayed the conic sections and the key lines characterizing... their ‘essential affections’... More importantly, he used the same diagram as often as possible and, when not... changed the diagram only minimally. ...[He] claim[ed]... his 1685 Treatise on algebra... considered the sections 'abstractly as Figures in plano, without the embranglings of the Cone'."

"By regularizing... symbols Wallis circumvented the typical role of diagrams in geometric argument. ...Wallis’s new lettering strategy meant that the reader would not have to constantly look back at the diagram to understand the argument... the reader might recognize from the letters which parts of the cone were signified."

"Newton opposed entangled and tedious algebraic calculations to simple, elegant geometric constructions; Wallis opposed difficult, embrangled geometric diagrams to simplified, rationally lettered diagrams and the symbolic expressions they enabled."

"[I]n On conic sections, Wallis... claimed that algebra would enable an 'absolute contemplation' of the conic sections by directly expressing... 'primary and essential affections' from which secondary affections could be 'deduced by calculation'. ...Both the use of symbols to express essences and their manipulation to obtain easy results characterized Wallis..."

"Based on a pattern he observed in the characters of the first few s, Wallis conceived a procedure for generating further characters. ...Wallis explained how to use figurate numbers to measure areas and volumes and, eventually, to square the circle. ...[H]e demphasized traditional forms of mathematical demonstration ...taking ...analysis as a way of finding. He believed he could promote mathematics more by exposing his own investigations than through sterile demonstrations."

"Wallis used Arithmetic of infinities to produce genuine understanding rather than brutal persuasion."

"On symbolic use of equalities and proportions. Chapter II. The analytical method accepts as proven the most famous [ as known from Euclid ] symbolic use of equalities and proportions that are found in items such as: 1. The whole is equal to the sum of its parts. 2. Quantities being equal to the same quantity have equality between themselves. [a = c & b = c => a = b] 3. If equal quantities are added to equal quantities the resulting sums are equal. 4. If equals are subtracted from equal quantities the remains are equal. 5. If equal equal amounts are multiplied by equal amounts the products are equal. 6. If equal amounts are divided by equal amounts, the quotients are equal. 7. If the quantities are in direct proportion so also are they are in inverse and alternate proportion. [a:b::c:d=>b:a::d:c & a:c::b:d] 8. If the quantities in the same proportion are added likewise to amounts in the same proportion, the sums are in proportion. [a:b::c:d => (a+c):(b+d)::c:d] 9.If the quantities in the same proportion are subtracted likewise from amounts in the same proportion, the differences are in proportion. [a:b::c:d => (a-c):(b-d)::c:d] 10. If proportional quantities are multiplied by proportional quantities the products are in proportion. [a:b::c:d & e:f::g:h => ae:bf::cg:dh] 11. If proportional quantities are divided by proportional quantities the quotients are in proportion. [a:b::c:d & e:f::g:h => a/e:b/f::c/g:d/h] 12. A common multiplier or divisor does not change an equality nor a proportion. [a:b::ka:kb & a:b::(a/k):(b/k)] 13. The product of different parts of the same number is equal to the product of the sum of these parts by the same number. [ka + kb = k(a+b)] 14. The result of successive multiplications or divisions of a magnitude by several others is the same regardless of the sequential order of quantities multiplied times or divided into that magnitude. But the masterful symbolic use of equalities and proportions which the analyst may apply any time is the following: 15. If we have three or four magnitudes and the product of the extremes is equal to the product means, they are in proportion.[ad=bc => a:b::c:d OR ac=b2 => a:b::b:c] And conversely 10. If we have three or four magnitudes and the first is to the second as the second or the third is to the last, the product of the extremes is equal to that of means. [a:b::c:d => ad=bc OR a:b::b:c => ac=b2] We can call a proportion the establishment of an equality [equation] and an equality [equation] the resolution of a proportion."

"In mathematics there is a certain way of seeking the truth, a way which Plato is said first to have discovered and which was called "analysis" by Theon and was defined by him as "taking the thing sought as granted and proceeding by means of what follows to a truth which is uncontested"; so, on the other hand, "synthesis" is "taking the thing that is granted and proceeding by means of what follows to the conclusion and comprehension of the thing sought." And although the ancients set forth a twofold analysis, the zetetic and the poristic, to which Theon's definition particularly refers, it is nevertheless fitting that there be established also a third kind, which may be called rhetic or exegetic, so that there is a zetetic art by which is found the equation or proportion between the magnitude that is being sought and those that are given, a poristic art by which from the equation or proportion the truth of the theorem set up is investigated, and an exegetic art by which from the equation set up or the proportion, there is produced the magnitude itself which is being sought. And thus, the whole threefold analytic art, claiming for itself this office, may be defined as the science of right finding in mathematics. ...the zetetic art does not employ its logic on numbers—which was the tediousness of the ancient analysts—but uses its logic through a logistic which in a new way has to do with species [of number]..."

"There is a certain way of searching for the truth in mathematics that Plato is said first to have discovered; Theon named it analysis, and defined it as the assumption of that which is sought as if it were admitted and working through its consequences to what is admitted to be true. This is opposed to synthesis, which is the assuming what is admitted and working through its consequences to arrive at and to understand that which is sought."

"Vieta presented his analytic art as "the new algebra" and took its name from the ancient mathematical method of "analysis", which he understood to have been first discovered by Plato and so named by . Ancient analysis is the 'general' half of a method of discovering the unknown in geometry; the other half, "synthesis", being particular in character. The method was defined by Theon like this: analysis is the "taking of the thing sought as granted and proceeding by means of what follows to a truth that is uncontested"'. Synthesis, in turn, is "taking the thing that is granted and proceeding by means of what follows to the conculsion and comprehension of the thing sought" (Vietae 1992: 320). The transition from analysis to synthesis was called "conversion", depending on whether the discovery of the truth of a geometrical theorem or the solution ("construction") to a geometrical problem was being demonstrated, the analysis was called respectively "theoretical" or "problematical"."

"Vieta's innovation contains three interrelated and interdependent aspects. ...methodical ...making calculation possible with both known and unknown indeterminate (and therefore 'general') numbers. ...cognitive ...resolving mathematical problems in this general mode, such that its indeterminate solution allows arbitrarily many determinate solutions based on numbers assumed at will. ...analytic ...being applicable indifferently to the numbers of traditional arithmetic and the magnitudes of traditional geometry."

"A major advance in notation with far-reaching consequences was François Viète's idea, put forward in his "Introduction to the Analytic Art"... of designating by letters all quantities, known or unknown, occurring in a problem. ...for the first time it was possible to replace various numerical examples by a single "generic" example, from which all others could be deduced by assigning values to the letters. ...by using symbols as his primary means of expression and showing how to calculate with those symbols, Viète initiated a completely formal treatment of algebraic expressions, which he called logistice speciosa (as opposed to logistice numerosa, which deals with numbers). This "symbolic logistic" gave some substance, some legitimacy to algebraic calculations, which allowed Viète to free himself from the geometric diagrams used... as justifications."

"Ars Magna, published in 1545... contains many valuable discoveries; but that which has been most celebrated is the rule for the solution of cubic equations, generally known by Cardan's name, though he had obtained it from a man of equal genius in algebraic science, Nicolas Tartaglia. ...Cossali has ingeniously attempted to trace the process by which Tartaglia arrived at this discovery; one which, when compared with the other leading rules of algebra, where the invention... has generally lain much nearer the surface, seems an astonishing effort of sagacity. Even Harriott's beautiful generalization of the composition of equations was prepared by what Cardan and Vieta had done before, or might have been suggested by observation in the less complex cases. Cardan, though not entitled to the honor of this discovery, nor even equal, perhaps, in mathematical genius to Tartaglia, made a great epoch in the science of algebra; and according to Cossali and Hutton, has a claim to much that Montucla has unfairly or carelessly attributed to his favorite, Vieta."

"Cossali has given the larger part of a quarto volume to the algebra of Cardan; his object being to establish the priority of the Italian's claim to most of the discoveries ascribed by Montucla to others, and especially to Vieta. Cardan knew how to transform a complete cubic equation into one wanting the second term; one of the flowers which Montucla has placed on the head of Vieta; and this he explains so fully, that Cossali charges the French historian of mathematics with having never read the Ars Magna."

"Rhaeticus was not a ready calculator only... Up to his time, the trigonometric functions had been considered always with relation to the arc; he was the first to construct the right triangle and to make them depend directly upon its angles. It was from the right triangle that Rhæticus got his idea of calculating the hypotenuse; i.e., he was the first to plan a table of secants. Good work in trigonometry was done also by Vieta and Romanus."

"Cardan applied the Hindoo rule of "false position" (called by him regula aurea) to the cubic, but this mode of approximating was exceedingly rough. An incomparably better method was invented by Franciscus Vieta... whose transcendent genius enriched mathematics with several important innovations... For this process, Vieta was greatly admired by his contemporaries. It was employed by Harriot, Oughtred, Pell, and others. Its principle is identical with the main principle involved in the methods of approximation of Newton and Horner. The only change lies in the arrangement of the work. This alteration was made to afford facility and security in the process of evolution of the root."

"Vieta [was] the most eminent French mathematician of the sixteenth century."

"He was employed throughout life in the service of the state, under Henry III and Henry IV. He was, therefore, not a mathematician by profession, but his love for the science was so great that he remained in his chamber studying, sometimes several days in succession, without eating and sleeping more than was necessary to sustain himself. So great devotion to abstract science is the more remarkable because he lived at a time of incessant political and religious turmoil."

"During the war against Spain, Vieta rendered service to Henry IV by deciphering intercepted letters written in a species of cipher, and addressed by the Spanish Court to their governor of Netherlands. The Spaniards attributed the discovery of the key to magic."

"An ambassador from Netherlands once told Henry IV that France did not possess a single geometer capable of solving a problem propounded to geometers by a Belgian mathematician, Adrianus Romanus. It was the solution of the equation of the forty fifth degree:—45y - 3795y^3 + 95634y^3 -\ldots+945y^{41} - 45y^{43} + y^{45} = C...Vieta, who, having already pursued similar investigations, saw at once that this awe-inspiring problem was simply the equation by which C = 2 sin φ was expressed in terms of y = 2 sin 1⁄45 φ that since 45 = 3·3·5, it was necessary only to divide an angle once into 5 equal parts, and then twice into 3,—a division which could be effected by corresponding equations of the fifth and third degrees. Brilliant was the discovery by Vieta of 23 roots to this equation, instead of only one. The reason why he did not find 45 solutions, is that the remaining ones involve negative sines, which were unintelligible to him."

"Detailed investigations on the famous old problem of the section of an angle into an odd number of equal parts, led Vieta to the discovery of a trigonometrical solution of Cardan's irreducible case in cubics. He applied the equation (2 cos 1⁄3 φ)3 - 3 (2 cos 1⁄3 cos φ) = 2 cos φ to the solution of x3 - 3 a2x = a2b, when a > ½ b, by placing x = 2 a cos 1⁄3 φ, and determining φ from b = 2a cos φ."

"The main principle employed by him in the solution of equations is that of reduction. He solves the quadratic by making a suitable substitution which will remove the term containing x to the first degree. Like Cardan, he reduces the general expression of the cubic to the form x3 + mx + n = 0; then assuming x = (1⁄3 a - z2)÷z and substituting, he gets z6 - bz3 - 1⁄27 a3 = 0. Putting z3 = y, he has a quadratic. In the solution of bi-quadratics, Vieta still remains true to his principle of reduction. This gives him the well-known cubic resolvent. He thus adheres throughout to his favourite principle, and thereby introduces into algebra a uniformity of method which claims our lively admiration."

"In Vieta's algebra we discover a partial knowledge of the relations existing between the coefficients and the roots of an equation. He shows that if the coefficient of the second term in an equation of the second degree is minus the sum of two numbers whose product is the third term, then the two numbers are roots of the equation. Vieta rejected all except positive roots; hence it was impossible for him to fully perceive the relations in question."

"The most epoch making innovation in algebra due to Vieta is the denoting of general or indefinite quantities by letters of the alphabet. To be sure, Regiomontanus and Stifel in Germany, and Cardan in Italy, used letters before him, but Vieta extended the idea and first made it an essential part of algebra. The new algebra was called by him logistica speciosa in distinction to the old logistica numerosa."

"Vieta's formalism differed considerably from that of to-day. The equation a3 + 3a2b + 3ab2 + b3 = (a + b)3 was written by him "a cubus + b in a quadr. 3 + a in b quadr. 3 + b cubo æqualia a+b cubo.""

"In numerical equations the unknown quantity was denoted by N, its square by Q, and its cube by C. Thus the equation x3 - 8 x2 + 16 x = 40 was written 1 C - 8 Q - 16 N œqual. 40."

"Exponents and our symbol (=) for equality were not yet in use; but... Vieta employed the Maltese cross (+) as the short-hand symbol for addition, and the (-) for subtraction. These two characters had not been in general use before the time of Vieta."

"Vieta (c. 1590) rejected the name "algebra" as having no significance in the European languages, and proposed to use the word "analysis," and it is probably to his influence that the popularity of this term in connection with higher algebra is due."

"Vieta: 1QC - 15QQ + 85C - 225Q + 274N, aequator 120. Modern form:x^6 - 15x^4 + 85x^3 - 225x^2 + 274x = 120"

"He used capital vowels for the unknown quantities and capital consonants for the known, thus being able to express several unknowns and several knowns."

"In the work of Vieta the analytic methods replaced the geometric, and his solutions of the quadratic equation were therefore a distinct advance upon those of his predecessors. For example, to solve the equation x^2 + ax + b = 0 he placed u + z for x. He then hadu^2 + (2z + a)u +(z^2 + az + b) = 0.He now let 2z + a = 0, whence z = -\frac{1}{2}a,and this gaveu^2 - \frac{1}{4}(a^2 - 4b) = 0. u = \pm \frac{1}{2} \sqrt{a^2 - 4b}.andx = u + z = -\frac{1}{2}a \pm \sqrt{a^2 - 4b}."

"Although Cardan reduced his particular equations to forms lacking a term in x^2, it was Vieta who began with the general formx^3 + px^2 + qx + r = 0and made the substitution x = y -\frac{1}{3}p, thus reducing the equation to the formy^3 + 3by = 2c.He then made the substitutionz^3 + yz = b, or y = \frac{b - z^2}{z},which led to the formz^6 + 2cz^2 = b^2,a sextic which he solved as a quadratic."

"Vieta was the first algebraist after Ferrari to make any noteworthy advance in the solution of the biquadratic. He began with the type x^4 + 2gx^2 + bx = c, wrote it as x^4 + 2gx^2 = c - bx, added gx^2 + \frac{1}{4}y^2 + yx^2 + gy to both sides, and then made the right side a square after the manner of Ferrari. This method... requires the solution of a cubic resolvent. Descartes (1637) next took up the question and succeeded in effecting a simple solution... a method considerably improved (1649) by his commentator Van Schooten. The method was brought to its final form by Simpson (1745)."

"Letters had been used before Vieta to denote numbers, but he introduced the practice for both given and unknown numbers as a general procedure. He thus fully recognized that algebra is on a higher level of abstraction than arithmetic. This advance in generality was one of the most important steps ever taken in mathematics. The complete divorce of algebra and arithmetic was consummated only in the nineteenth century, when the postulational method freed the symbols of algebra from any necessary arithmetical connotation."

"Improving on the devices of his European predecessors, Vieta gave a uniform method for the numerical solution of algebraic equations. ...it was essentially the same as Newton's (1669)... Although Vieta's method has been displaced by others... The method applies to transcendental equations as readily as to algebraic when combined with expansions to a few terms by Taylor's or Maclaurin's series."

"An algebraic equation of degree 45 which Vieta attacked in reply to a challenge indicates the quality of his work in trigonometry. Consistently seeking the generality underlying particulars, Vieta had found how to express sin nθ (n a positive integer) as a polynomial in sin θ, cos θ. He saw at once that the formidable equation of his rival had manufactured from an equivalent of dividing the circumference of the unit circle into 45 equal parts. ...More important than this spectacular feat was Vieta's suggestion that cubics can be solved trigonometrically."

"Vieta's principle advance in trigonometry was his systematic application of algebra. ...he worked freely with all six of the usual functions, and... obtained many of the fundamental identities algebraically. With Vieta, elementary (non-analytic) trigonometry was practically completed except on the computational side. All computation was greatly simplified early in the seventeenth century by the invention of logarithms."

"It is notoriously difficult to convey the proper impression of the frontiers of mathematics to nonspecialists. Ultimately the difficulty stems from the fact that mathematics is an easier subject than the other sciences. Consequently, many of the important primary problems of the subject‍—‌that is, problems which can be understood by an intelligent outsider‍—‌have either been solved or carried to a point where an indirect approach is clearly required. The great bulk of pure mathematical research is concerned with secondary, tertiary, or higher-order problem, the very statement of which can hardly be understood until one has mastered a great deal of technical mathematics."

"Early in his college days, Minsky had had the good fortune to encounter Andrew Gleason. Gleason was only six years older than Minsky, but he was already recognized as one of the world’s premier problem-solvers in mathematics; he seemed able to solve any well-formulated mathematics problem almost instantly... “I couldn’t understand how anyone that age could know so much mathematics,” Minsky told me. “But the most remarkable thing about him was his plan. When we were talking once, I asked him what he was doing. He told me that he was working on Hilbert’s fifth problem.” Gleason said he had a plan that consisted of three steps, each of which he thought would take him three years to work out. Our conversation must have taken place in 1947, when I was a sophomore. Well, the solution took him only about five more years... I couldn’t understand how anyone that age could understand the subject well enough to have such a plan and to have an estimate of the difficulty in filling in each of the steps. Now that I’m older, I still can’t understand it. Anyway, Gleason made me realize for the first time that mathematics was a landscape with discernible canyons and mountain passes, and things like that. In high school, I had seen mathematics simply as a bunch of skills that were fun to master—but I had never thought of it as a journey and a universe to explore. No one else I knew at that time had that vision, either."

"I would have wished that I could write in some detail of the nature of our work in those wonderfully exciting days. For we were regularly reading the highest grade cipher messages passing between the German High Command and a the senior echelons of the German army, the German navy (including the U-boat fleet) and the Luftwaffe; moreover, we were reading those messages within a few hours of their original transmission. We were thus able to provide as perfect and complete picture of the enemy's plans and dispositions as any nation at war has ever had at its disposal — not lightly did Churchill described our work as his "secret weapon," far more potent than anything Werner von Braun could deploy against us. Unfortunately, the British government currently is behaving in a remarkably paranoid fashion with respect to the revelations of "secrets" by those who at some time (as, of course, I had to do) taken an oath of confidentiality."

"I also attended his eightieth birthday celebration in , in 2003. Peter gave a wonderful polished talk about his experiences at in World War II, which was informative and moving and made a political point. I noticed that he frequently paused to refer to a very small sheaf of notes in his hand. He left the papers on the rostrum after the talk, and out of curiosity I took a look. They were blank! It was a ."

"Georg Hamel was born in 1877 in Düren, Germany, and died in 1954 in Landshut, Germany. In 1897, Hamel went to the University of Berlin, where he was taught by Hermann Schwarz and Max Planck, to name two. Subsequently, he went to Göttingen University, where he studied with Felix Klein and David Hilbert. He was awarded a doctorate under the supervision of Hilbert in 1901. The subject of his dissertation was Hilbert’s fourth problem. In 1905, he went to Brno. It was during the period of his work in Brno that his 1905 paper on Hamel bases was written."

"I am one of the most active researchers in the field of information security, which supports the safety and security of the information society, and has also served as an editor of international standardization standards."

"I have been active as a pioneer of elliptic curve cryptography, and my world-first mixed coordinate, MNT (Miyaji-Takano-Nakabayashi) curve was adopted as an international standard by ISO/IEC."

"the New England natives used neither silver nor gold. Instead, they used the most appropriate money to be found in their environment – durable skeleton parts of their prey. Specifically, they used wampum, shells of the clam Venus mercenaria and its relatives, strung onto pendants."

"Only a handful of tribes, such as the Narragansetts, specialized in manufacturing wampum, while hundreds of other tribes, many of them hunter-gatherers, used it. Wampum pendants came in a variety of lengths, with the number of beads proportional to the length. Pendants could be cut or joined to form a pendant of length equal to the price paid. Once they got over their hangup about what constitutes real money, the colonists went wild trading for and with wampum. Clams entered the American vernacular as another way to say “money”. The Dutch governor of New Amsterdam (now New York) took out a large loan from an English-American bank – in wampum. After a while the British authorities were forced to go along. So between 1637 and 1661, wampum became legal tender in New England. Colonists now had a liquid medium of exchange, and trade in the colonies flourished. The beginning of the end of wampum came when the British started shipping more coin to the Americas, and Europeans started applying their mass-manufacturing techniques."

"Barter works well at small volumes but becomes increasingly costly at large volumes, until it becomes too costly to be worth the effort. If there are n goods and services to be traded, a barter market requires n^2 prices."

"On a larger scale, the Laffer curve may be the most important economic law of political history. Charles Adams uses it to explain the rise and fall of empires."

"Clock time is a fungible measure of sacrifice. Of all measurement instruments, the clock is the most valuable because so many of the things we sacrifice to create are not fungible. The massive clock towers of Europe, with their enormous loud and resonant bells, broadcasting time fairly across the town and even the countryside, rather than the last relics of the medieval, were the first building block of the wealthy modern world. The Europeans evolved their institutions and deployed two very different but complementary timekeeping devices, the sandglass and the mechanical clock, to partition the day into frequently rung and equal hours. Europe progressed in a virtuous circle where bells and clocks improved the productivity of relationships; the resulting wealthy institutions in turn funded more advances in timekeeping... The massive change on the farm, the dominant form of industry, in the 14th and successive centuries from serfdom and slavery to markets and wage labor, was caused not only by the temporary labor shortages of the Black Plague, but more fundamentally and permanently by the time-rate contract and the new ability to accurately and fairly verify its crucial measurement of sacrifice, time. Time rates also became the most common relationship for the mines, mills, factories, and other industries that rapidly grew after the advent of the clock."

"Nevertheless, this nice fiction allowed the United States to avoid creating a real locus of sovereignty by creating a fictional sovereignty to satisfy the Romanists. Sovereignty, under this doctrine, is vested in "We The People." Some of this political power is granted, charter-like, to the United States federal government. The rest is granted (per the 10th Amendment) "to the States, or to the people, respectively" -- that is to each State and to individuals. In reality, political power is distributed amongst the federal government, states, counties, and munipalities, with a variety of enumerated and unenumerated rights that these governments may not infringe being retained by private persons."

"Rug the spammers."

"All banks go bankrupt"

"In the and the of the , grandiloquent homes were built for the nation's leaders and heroes with great avenues of approach and triumphal arches. Villages which were found to stand in the way of these grandiose undertakings were removed out of sight. Sweeping changes were made at the seat of the , the victor of , which necessitated the moving of the village of in ; was destroyed in the creating of 's dramatic for the ; disappeared in the lay-out for the magnificent seat of the in . The great Whig palaces and extensive gardens at , and overran ancient villages and hamlets that stood in the way of improvements. , who had envisaged an avenue of trees between London and his , began his improvements by removing the village of which lay in the shadow of his house. The village of in was resited to give breathing space to the family of . ... By the middle of the century great gardens were being made, not only to reflect their creator's importance or political beliefs, but to demonstrate the excellence of his taste. The new vogue was not for great avenues, canals, fountains and grand parterres but for naturalized landscape. Wealthy families in every county bought up vast tracts of land to make natural gardens, which would look like landscape paintings; some took the English countryside for these picture gardens and with the help of idealized and, 'improved' it; the with memories of their s revelled in the creation of Italian classical landscapes."

"… The accepted idea of the of a building, furniture or a painting, as the rehabilitation of an object already in existence, albeit in imperfect form, cannot be applied to gardens which are by their nature organic. They have allotted life spans and have been dug up and refashioned over the centuries. ... At the has been able to restore the garden of the great from original plans, so that the design of the s and seen today is much as Evelyn described it when he visited in 1678. At in the National Trust has restored a from engravings, existing evidence and plant list which have enabled them to use contemporary plants including old cultivars of Turkish irises, apples and pears and old tulips. A current true restoration is being undertaken at , , where the poet 's famous beds, painted by in 1777, are being reinstated with authentic planting. … The ultimate in scholarly garden reconstruction is the Roman garden at executed through excavation and .."

"Mavis fell in love with her future husband, , himself one of the Bletchley “break-in” experts, after he helped her with a particularly difficult code breaking problem: “I was alone on the evening shift in the cottage and I sought the help of what called 'one of the clever Cambridge mathematicians in Hut 6’. We put our heads together and in the calmer light of logic, and much ersatz coffee, solved the problem. Dilly made no objections to my having sought such help and when I told him I was going to marry the 'clever mathematician from hut 6’ he gave us a lovely wedding present.” After the war Mavis Batey brought her indefatigability to the protection of Britain’s historical gardens. Her interest began in the late 1960s, when her husband was appointed the “Secretary of the Chest”, the chief financial officer of Oxford University. They lived in a university-owned house on the park at and she set about ensuring that the overgrown gardens were restored to their original landscaped state."

"Rule 1 of cryptanalysis: check for plaintext."

"Never underestimate the attention, risk, money, and time that an opponent will put into reading traffic."

"It is easy to run a secure computer system. You merely have to disconnect all dial-up connections and permit only direct-wired terminals, put the machine and its terminals in a shielded room, and post a guard at the door."