Cryptographers

"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."

"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."

"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."

"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 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."

"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."

"[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."

"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."

"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."