Mathematicians From Switzerland

"Longomontanus and Byrgius, are all whom Dr Hutton can find to represent his learned calculators of the sixteenth and seventeenth centuries, who anticipated or coincided with Napier in the discovery. ...But he is contradicted by the history of science, ancient and modern, and by every philosopher of greatest name, both in Napier's time and ours. Among the finest characteristics of our philosopher's invention was the unhoped-for manner in which it removed a pressure, long and severely felt, and which might have crushed the temple of science, had that not possessed such a pillar as Kepler. To use the expressions of a distinguished writer, " What all mathematicians were now wishing for, the genius of Neper enabled him to discover; and the invention of Logarithms introduced into the calculations of a degree of simplicity and ease, which no man had been so sanguine as to expect." Kepler, Ursine, Speidell, Gunter, Briggs, Vlacq, [Petrus] Cugerus, Cavalieri, Wolff, Wallis, Halley, Keill, and a host of others, all bear witness against Dr Hutton, in the honourable and enthusiastic manner they acknowledge Napier as the only author of that revolution in science."

"Had it appeared a century before Napier, would not physical astronomy have been as far advanced in his time as it was a century after, and would not NAPIER have been NEWTON? But there were many persons having thoughts of such a table of numbers besides the few who are said to have attempted it! Dr Hutton, in support of this assertion... clings to Byrgius;"Kepler also says, that one Juste Byrge, assistant astronomer to the Landgrave of Hesse, invented or projected Logarithms long before Neper did, but that they had never come abroad on account of the great reservedness of their author with regard to his own compositions."But Hutton, though he suppresses what... qualifies the words of Kepler, and ventures not into the slightest examination of the pretension for Byrgius (who never made it for himself) is fond of the story, and does what he can to fix it upon the legislator of the stars as an unqualified assertion of his; for, speaking of the Rudolphine Tables, our author takes occasion to repeat,"and here it is that he (Kepler) mentions Justus Byrgius as having had Logarithms before Napier published them.""

"We... add the name... of another distinguished historian of science... carried by this groundless pretension, which was probably a villanous though weak attempt to wrest the laurels from the grave of a foreigner. M. Kluegel, in his philosophical dictionary, a work of great ability, records, that"Neper in Scotland, and Jobst Byrg in Germany, were the first who, without any intercommunication, calculated tables of Logarithms." ...But how happened it, we would ask M. Kluegel, that Kepler gave all the glory to Napier, and none to his own countryman? This same author expresses most graphically the enthusiastic zeal with which the legislator of the stars rushed upon the Logarithms; "Kepler ergriff Nepers Erfindung mit Eifer,"—[translation] Kepler seized Napier's discovery with enthusiasm,—now Kepler expressly regards the speculation of Byrgius with contempt."

"Our authority is the letter from Kepler to Napier, with which these Memoirs conclude, and which Montucla had never seen. So the "homo cunctator" calculated tables of Logarithms in 1609, and then cast them among the rubbish of his study; in the year 1617 a copy of Napier's Canon is laid, as the wonder of the day, before Kepler himself, the oracle of European science, in the city of Prague; from that moment Kepler's whole existence is identified with his love of Logarithms, and all that he ever says for his friend Byrgius is, that he did not make the discovery; in 1619 (two years after Napier's death,) the "homo cunctator" has his portrait engraved; in 1620 he is said to have printed at Prague some isolated and useless fragment of a table, but it is not even pretended that he put forth any claim; ten years afterwards, namely, in 1630, Bramer, brother-in-law to the "homo cunctator," has the effrontery to announce, and without so much as a detailed or explicit account in support of his allegation, that Justus Byrgius, and not John Napier, is the inventor of Logarithms."

"[T]hough Montucla was not aware of the fact... the... place where Kepler himself first saw a copy of John Napier's Canon Mirificus was THE ANCIENT CITY OF PRAGUE, and this was in the year 1617."

"According to Bramer, his kinsman had calculated tables... more than twenty years before 1630. As he has not fixed the date, we take the assumption as referring to the year 1609. "But," says Kepler, writing in... 1624, and without the slightest notice of Byrgius, "a certain Scotchman, so early as the year 1594, wrote to Tycho a promise of that wonderful canon." According to Bramer, his kinsman, the "homo cunctator," [a hesitant man] did so far bestir himself as to have his portrait engraved, in the year 1619, for a frontispiece to his great discoveries, among which, and probably the least, were the Logarithms! In 1620 the fragment of his tables was printed at Prague, but without frontispiece or anything else."

"The miserable fragment of miscalculated tables discovered by Kästner proves nothing, for there is neither description nor claim attached to them, and their date is 1620; and any support which the claim attempted to be reared upon that fragment may seem to obtain from the notice of Kepler (also very vague) is more than neutralized by Kepler himself."

"The value of Byrgius's share of any honour in the matter may be expressed by that ghostly symbol which is the soul of Arabic notation, 0. We might say so upon the evidence adduced in his favour, which is totally inadequate to sustain his claim. His brother-in-law is, under the circumstances, not competent evidence; for the peremptory manner in which he springs from so vague a statement to the astounding conclusion, that Byrgius, and not Napier, is the Inventor of Logarithms, proves Bramer to have been either an idiot or a false witness."

"Montucla then proceeds to give a specimen of the fragment of Byrgius taken from M. Kastner, and concludes... We must remark at the same time, that it would be unjust to conclude, from the work printed in 1620, that Byrge had invented Logarithms before Neper; for the work of Neper appeared in 1614, and it is the priority of dates of works which determines at the bar of public opinion the anteriority of the invention. How then does Bramer from that date, 1620, arrive at the conclusion, that his brother-in-law had made the discovery long before Napier? It is well known, that the date of an invention requiring much calculation is necessarily anterior to that of publication, and Neper is equally entitled to the assumption, that his invention existed in his head for several years before he published it; and besides, in a court of law itself, Byrge would lose his suit, for, according to the strictest administration of justice, a date of publication anterior by six years must be held to have afforded an opportunity of becoming acquainted with the discovery, and disguising it under another form. Let us be contented, therefore, with associating at a distance, and to a certain extent only, Byrge with the honour of that ingenious invention; but the glory must always belong to Neper.""

"Montucla continues..."But the work of this geometer was nowhere to be found, and probably would never have been discovered had not the passage led M. Kästner to recognize these tables among some old mathematical works which he had purchased. They bore this title in German: Tables of Arithmetical and Geometrical Progressions, with an introduction explanatory of their meaning and use in all manner of Calculations, by J. B. printed in the ancient city of Prague, 1620. The tables contain seven leaves and a-half, printed in folio, but the introduction announced is awanting, which leads to the conjecture, that some peculiar circumstances had stopped the progress of the work; and, indeed, Bramer informs us in another of his own works, that Juste Byrge contemplated the publication of several of his inventions, and, for that purpose, had his portrait engraved in the year 1619, but the thirty years' war, which unhappily desolated Germany, opposed an obstacle to his design."

"There is a geometer," says Montucla,"to whom we must here give a place, and that is, Juste Byrge. That which chiefly renders him worthy of notice is the fact, that he invented and constructed tables of Logarithms simultaneously with Napier. Kepler represents him to us as a man of considerable genius, but thinking so modestly of his own inventions, and so indifferent about them, as to suffer them to be buried in the dust of his study; and, says Kepler, for that reason he never gave any thing to the public through the medium of the press.But Kepler was in error when he said so, and we shall proceed to unfold a tale... Notwithstanding what Kepler says of J. Byrge, bears witness to the fact, that... Byrge... did publish something relative to Logarithms. That author in a German work... Description of an Instrument very useful for perspective and drawing plans, (...1630, 4to,) says..."It was upon these principles that my dear brother-in-law and master, Juste Byrge, constructed, more than twenty years ago, a beautiful table of progressions, with their differences from 10 to 10, calculated to 9 places, and which he caused to be printed at Prague in 1620, so that the invention of Logarithms is not Neper's, but was made by Juste Byrge long before him."

"In such a way, with much trouble and labor, the whole Canon has been established. For many hundreds of years, up to now, our ancestors have been using this method because they were not able to invent a better one. However, this method is uncertain and dilapidated as well as cumbersome and laborious. Therefore we want to perform this in a different, better, more correct, easier and more cheerful way. And we want to point out now how all sines can be found without the troublesome inscription [of polygons], namely by dividing a right angle into as many parts as one desires."

"Divide a right angle in as many parts as you want and construct herefrom the sine table. (Einen rechten Winckell in also viel theile theilenn alß man will, vnnd aus demselben den Canonen Sinuum vermachenn.)"

"[D]ue to a lack of languages, the door to... authors has not always been open to me, as... to others, I have had to follow my own thoughts a little more than the learned and well-read, and seek new paths. (Und weil mir auß mangel der sprachen die thür zu den authoribus nit alzeitt offen gestanden, wie andern, hab jch etwas mehr, als etwa die glehrte vnd belesene meinen eigenen gedanckhen nachhengen vnd newe wege suechen müessen.)"

"Justus Byrgius is the solitary mathematician for whom any thing like an independent claim to the invention has been set up betwixt the time of Archimedes and Napier. Not that it has ever been said that our philosopher borrowed any thing from the German; for the priority of Napier's publication, and the surpassing beauty of his algebraic method, has never met with contradiction. But there is a story that Kepler's friend had actually computed tables of Logarithms years before Napier published his canon, and, consequently, that the German stands nearly in the same relation to this great discovery that Newton himself does to the infinitesmal calculus, in the celebrated competition with Leibnitz. It would, indeed, be singular, if this public astronomer had computed such tables without giving them to the world, or ever himself pretending to the discovery."

"The mathematician whose claim we are considering ranked not meanly in science; he was instrument-maker and astronomer to the Landgrave of Hesse, and must have been well known to Kepler; he may have been "homo cunctator," [an indolent, or hesitant man] but he was not so foolish as to have cast aside his own immortality had he really extended the Archimedean principle in any remarkable manner; he was a public astronomer, under high patronage, in a country teeming with rivals in science, and where a great mathematical discovery was the means of obtaining rank, wealth, and adoration; it is absolutely impossible, therefore, that...[he] could have calculated tables of Logarithms... and then have cast them aside; there was the gulf of ignorance betwixt him and Logarithms, and so we must construe the expressions of Kepler, "fœtum in partu destituit, non ad usos publicos educavit [instead of rearing up his child for the public benefit, he deserted it in the birth]." Supposing him even to have observed all the curious properties of a corresponding series, under the fertile and flexible Arabic notation,—the parent of progressions,—he would not have been singular in thus obtaining a glimpse of Logarithms without knowing them; and there would still be this distinction betwixt Byrgius and Napier, that the former, neither seeking nor dreaming of such a power, stumbled upon a natural tract in the system of notation, which might have led him, but did not, to an imperfect and accidental developement of Logarithms; whereas the latter saw that the power was wanted, that calculation was impeded, and, to use his own words, "began therefore to consider in my mind by what certain and ready art I might remove those hindrances," and in doing so sought no easy path pointed out to him by the progressive power of cyphers, but, plunging at once into the algebraic depth of his own original fluxionary system, took the very path which Newton and Leibnitz would have taken, and returned leading the whole system of Numbers captive to the properties of progressions."

"Kepler meant no honour to his friend to the prejudice of Napier. On the contrary, the spirit... is, that Byrgius had substantially failed to perceive that a chapter of algebra might be composed in which that property of progressions would be reared into vast importance; an importance never felt until Napier demonstrated it by a method far more nearly allied to the profound algebraic views of Newton, than those easy progressions,—so obvious in the Arabic scale itself, and through which, perhaps, Byrgius had been unwittingly on a tract to Logarithms,—are to Napier's system."

"But where were all the " learned calculators of the 16th and 17th centuries," whom Dr Hutton pictures as evolving the Logarithms by profound reasonings upon the doctrine of progressions? And who were they? Not Kepler, who, when he first heard of Napier's method, could hardly form an accurate idea of its meaning. Not Tycho, nor Longomontanus, nor Galileo, nor any one of Kepler's numerous correspondents, including... nearly all the learned calculators of the period. ...Kepler, who to his dying day never ceased to marvel at the achievement, seems a little excited by discovering that one other person had actually approached the theory without being aware of it. In his Rudolphine Tables... 1627, he remarks,"the accents in calculation led Justus Byrgius on the way to these very Logarithms many years before Napier's system appeared; but being an indolent man, and very uncommunicative, instead of rearing up his child for the public benefit, he deserted it in the birth."This was the result of Kepler's indefatigable inquiries, for nine years... and... it amounts to this, that Byrgius had made some observations upon the adaptation of an arithmetical to a geometrical progression, very naturally occurring to him in trigonometrical calculations. The Apices Logistici ["accents in calculation"], to which Kepler alludes, are those accents which the Greeks used... to change the value or mark the order of a symbol, as we use the cypher; and this is... exemplified in their sexagesimal division of the circle still in use, where the accents ′, ″, ′″, ″″, &c. of minutes, seconds, thirds, fourths, &c. are an arithmetical progression denoting the fractional orders, the values of which descend in a ratio of 60, and form the corresponding geometrical progression."

"If a hint could have urged any human mind thus rapidly upon the theory of the Logarithms, there was a hint which arose in the , which was submerged in the middle ages, and rose again with the letters of Greece; which Tycho had — which Stifellius, Byrgius [Bürgi], Longomontanus, and above all which Kepler had—and all made no more of it than Archimedes had done."

"That fine old gossip, Anthony Wood, picked up a story of Napier, Dr Craig, and the Logarithms, which he thus recorded in the Athenæ Oxonienses."It must be now known, that one Dr Craig, a Scotchman, perhaps the same mentioned in the Fasti, under the year 1605, among the incorporation, coming out of Denmark into his own country, called upon Joh. Neper, Baron of Mercheston, near Edinburgh, and told him, among other discourses, of a new invention in Denmark (by Longomontanus, as 'tis said,) to save the tedious multiplication and division in astronomical calculations. Neper being solicitous to know farther of him concerning this matter, he could give no other account of it than that it was by proportional numbers. Which hint Neper taking, he desired him at his return to call upon him again. Craig, after some weeks had passed, did so, and Neper then showed him a rude draught of what he called Canon mirabilis logarithmorum. Which draught, with some alterations, he printing in 1614, it came forthwith into the hands of our author Briggs, and into those of Will. Oughtred, from whom the relation of this matter came.""

"It is surprising that Kepler did not consider Jost Bürgi, who from around 1580 to 1592 already constructed planetary globes that were considered awesome works of art. ...In the letter 42 dated May 28th 1598, to the Duke Friederich I von Württemberg (1557–1608), Kepler writes that he will construct a globus with a planetarium. It is possible and likely that the Duke had in mind a sky globus similar to those already constructed by Eberhard Baldewein (1525-1593), Gerhard Emmoser (1556-1584) or Jost Bürgi (1552-1631), but none of these authors were cited in Kepler’s overview. These machines were designed and constructed to show the overall motion of the sky, to identify the position of the stars, to show the motion either of the Sun, the Moon or both."

"The calculation of the Canon Sinuum can be done... in the usual way, by inscribing the sides of a regular polygon into a circle... geometrically. Or... by a special way,.. dividing a right angle into as many parts as one wants... arithmetically. This has been found by Justus Bürgi... the skilful technician..."

"I do not have to explain to which level of comprehensibility this extremely deep and nebulous theory has been corrected and improved by the tireless study of my dear teacher, Justus Bürgi... by assiduous considerations and daily thought. ...Therefore neither I nor my dear teacher, the inventor and innovator of this hidden science, will ever regret the trouble and the labor which we have spent."

"Joost Bürgi... a Swiss watch and instrument maker and an assisitant to Kepler in Prague was... interested in facilitating astronomical calculations; he invented logarithms independently of Napier about 1600 but did not publish his work, Progress Tabulen, until 1620. Bürgi too was stimulated by Stifel's remarks that multiplication and division of terms in a geometric progression can be performed by adding and subtracting the exponents. His arithmetical work was similar to Napier's."

"The idea of the logarithm probably had its source in the use of... trigonometric formulas that transformed multiplication into addition and subtraction. ...[I]f one needed to solve a triangle using the , a multiplication and division were required. ...[C]alculations were long and errors... made. Astronomers realized... multiplication and division could be replaced by additions and subtractions. To accomplish this... sixteenth century astronomers used formulas... as 2 \sin \alpha \sin \beta = cos(\alpha - \beta) - \cos (\alpha + \beta). ...A second source of the... logarithm was probably found in... algebraists as Stifel and Chuquet, who both displayed tables relating the powers of 2 to the exponents and showed that multiplication in one table corresponded to addition in the other. But because these tables had large gaps, they could not be used for necessary calculations. ...[T]wo men... independently, the Scot John Napier... and the Swiss Jobst Bürgi... came up with the idea of producing an extensive table... to multiply any... numbers... (not just powers of 2)... Napier published... first."

"Jost Burgi, a Swiss clockmaker and mathematician, invented logarithms independently of Napier and Briggs, although it is not clear when he started work on them. Some historians have suggested that Burgi may have invented logarithms earlier than Napier, but his work was not published until 1620, when the German mathematician and astronomer Johannes Kepler asked him to do so. ...six years after the publication of Napier's work."

"Napier was... the first... to publish.., but...it is possible that the idea of logarithms had occured to Bürgi as early as 1588... half a dozen years before Napier began work... However, Bürgi printed... in 1620, half a dozen years after Napier published... Descriptio. Bürgi's... book... Arithmetische und geometrische Progress-Tabulen... indicates... the influences were similar... to Napier. Both... proceeded from the properties of arithmetic and geometric sequences, spurred, probably by the method of . The differences... lie chiefly in... terminology and... numerical values..; the fundamental principles were the same. Instead of proceeding from a number a little less than one (...Napier used 1 - 10^{-7}), Bürgi... a little greater than one... 1 + 10^{-4}; and instead of multiplying powers of this number by 10^7, Bürgi multiplied... 10^8. ...[O]ne other minor difference: Bürgi multiplied all... power indices by ten... [I]f N = 10^8(1+10^{-4})^L, Bürgi called 10L the "red"... corresponding to the "black"... N. If... we were to divide all black[s]... by 10^8 and all red[s]... by 10^5, we should have... a system of s. ...Bürgi gave for the black...1,000,000,000 the red...230,270.022, which on shifting decimal points, is equivalent to... \ln 10 = 2.3027022... not a bad approximation... especially when... (1 - 10^{-4})^{10^4} is not quite the same as \lim_{n \to \infty}(1+ \frac{1}{n})^n although... values agree to four significant figures. In publishing... he had... an antilogarithmic table... The essence of the principle is there... Bürgi must be regarded as an independent discoverer who lost credit... because of Napier's priority in publication. In one respect his logarithms come closer to ours than Napier's, for as... black[s] increase, so do the red[s]..; but the two systems share the disadvantage that the logarithm of the product or quotient is not the sum or difference of the logarithms."

"[I]n 1575 Western Europe had recovered most of the major mathematical works of antiquity now extant. Arabic algebra had been... mastered and improved... through the solution of the cubic and quartic and through... partial... symbolism; and trigonometry had become an independent discipline. The time was almost ripe for rapid strides... The transition from the Renaissance to the modern world was... made through... intermediate figures, a few of the more important... Galileo Galilei... and ... from Italy; several... as .., Thomas Harriot.., and ... were English; two... Simon Stevin... and ... were Flemish; others came from varied lands—John Napier... from Scotland, Jobst Bürgi... from Switzerland, and Johann Kepler... from Germany."

"It would seem that J. Bürgi, independently of Napier, had constructed before 1611 a table of antilogarithms of a series of natural numbers: this was published in 1620. ...Bürgi also employed decimal franctions ..."

"We probably all agree that eventually reducing a difficult problem to a "nice" situation is at the heart of mathematics."

"Still another important area is Poincaré duality for groups, invented by Robert Bieri and myself. They behave like manifolds: homology, cohomology, you see, in complementary dimensions, but with another dualizing module. Many groups that are interesting in algebraic geometry, group theory or other areas are such duality groups."

"Together with many other people and after a long development I could prove that a Poincaré duality group of cohomological dimension 2 is the group of a Riemann surface. That was actually a conjecture of Jean-Pierre Serre. "You have to prove it!" he had always insisted."

"In 1690... Jacob Bernoulli brought up the problem of the catenary in a memoir... in the '...Huygens' solution represents the past... a complex, though skillful, geometrical method. Leibniz, using his new [infinitesimal calculus] reaches a correct analytical formula...y/a = (b^\frac{x}{a} + b^\frac{-x}{a})/2 where a is [a] segment... and b... corresponds to... e... Johann Bernoulli ...supplied two correct constructions ...presents valid statistical arguments and... new and important... equations of equilibrium in differential form. ...In 1697-1698, Jacob Bernoulli was the first to derive the general equations that not only solved the problem, but also permitted the treatment of the more general theme of the equilibrium of a flexible rope, subject to any distribution of tangential (f_t) and normal (f_n) forces. Bernoulli's equations are...\frac{dT}{ds} + f_t = 0, \qquad \frac{T}{r} + f_n= 0where T is the tension, s the curvilinear abscissa, and r the radius of curvature."

"[P]robability as a measurable degree of certainty; necessity and chance; moral versus mathematical expectation; a priori an a posteriori probability; expectation of winning when players are divided according to dexterity; regard of all available arguments, their valuation, and their calculable evaluation; law of large numbers..."

"The art of measuring, as precisely as possible, probabilities of things, with the goal that we would be able always to choose or follow in our judgments and actions that course, which will have been determined to be better, more satisfactory, safer or more advantageous."

"Eadem mutata resurgo [Changed and yet the same, I rise again]"

"Elastic Curve is the name that James Bernoulli gave to the curve which is formed by an elastic blade, fixed horizontally by one of its extremities in a vertical plane, and loaded at the other extremity with a weight, which by its gravity bends the blade into a curve... This problem is resolved by James Bernoulli in the "Memoirs of the Acad. of Sciences for 1703;" and other solutions have been given by some of the most celebrated mathematicians of Europe..."

"The term "induction" had been used by John Wallis in 1656, in his Arithmetica infinitorum; he used the induction known to natural science. In 1686 Jacob Bernoulli criticised him for using a process which was not binding logically and then advanced in place of it the proof from n to n + 1. This is one of the several origins of the process of mathematical induction."

"The tract in which Leibnitz deals with series appeared late in the seventeenth century and was among the first on the subject. ...the question of their convergence or divergence ...was in those days more or less ignored. ...It was not until the publication of Jacques Bernoulli's work on infinite series in 1713 that a clearer insight into the problem was gained. ...Bernoulli's work directed attention towards the necessity of establishing criteria of convergence. The evanescence of the general term, i.e., of the generating sequence, is certainly a necessary condition, but this is generally insufficient. Sufficient conditions have been established by d'Alembert and Maclauren, Cauchy, Abel, and many others. ...to recognized whether a series converges or diverges is even today rather difficult in some cases."

"The great invention... Descartes gave to the world, the analytical diagram, ...gives at a glance a graphical picture of the law governing a phenomenon, or of the correlation which exists between dependent events, or of the changes which a situation undergoes in the course of time. ...the invention of Descartes not only created the important discipline of analytic geometry, but it gave Newton, Leibnitz, Euler, and the Bernoullis that weapon for the lack of which Archimedes and later Fermat had to leave inarticulate their profound and far-reaching thoughts."

"The name is due to Jacques Bernoulli. The spiral has been called also the geometrical spiral, and the proportional spiral, but more commonly, because of the property observed by Descartes, the equiangular spiral. Bernoulli (and Collins at an earlier date) noted the analogous generation of the spiral and loxodrome ("loxodromica"), the spherical curve which cuts all meridians under a constant angle. ... During 1691-93 Jacques Bernoulli gave the following theorems among others: (a) Logarithmic spirals defined [by the polar equation \rho = ke^{c\theta} of a curve cutting radial vectors (drawn from a certain fixed point 0) under a constant angle \phi , where k is constant and c = cot\phi] for different values of k are equal and have the same asymptotic point; (b) the evolute of a logarithmic spiral is another equal logarithmic spiral having the same asymptotic point; (c) the pedal of a logarithmic spiral with respect to its pole is an equal logarithmic spiral (d) the caustics by reflection and refraction of a logarithmic spiral for rays emanating from the pole as a luminous point are equal logarithmic spirals. The discovery of such "perpetual renascence" of the spiral delighted Bernoulli. "Warmed with the enthusiasm of genius he desired, in imitation of Archimedes, to have the logarithmic spiral engraved on his tomb, and directed, in allusion to the sublime tenet of the resurrection of the body, this emphatic inscription to be affixed—Eadem mutata resurgo." The engraved spiral (very inaccurately executed) and inscription in accordance with Bernoulli's desire, may be seen to-day on his tomb in the cloister of the cathedral at Basel."

"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."

"It was in 1691, that the penetrating genius of James Bernoulli discovered the true nature of the catenarian curve. A similar investigation was soon produced by John Bernoulli, by Huygens, and by Leibnitz."

"The first investigation of any importance is that of the elastic line or elastica by James Bernoulli in 1705, in which the resistance of a bent rod is assumed to arise from the extension and contraction of its longitudinal filaments, and the equation of the curve assumed by the axis is formed. This equation practically involves the result that the resistance to bending is a couple proportional to the of the rod when bent, a result which was assumed by Euler in his later treatment of the problems of the elastica, and of the vibrations of thin rods."

"There is... [a] need to code cheaper and accessible programs in line with using sustainable methods to better the livelihood of mankind. To address this issue a theory is formulated based on the Euler-Bernoulli beam model. This model is applicable to thin elements which include plate and membrane elements. This paper illustrates a finite element theory to calculate the master stiffness of a curved plate. The master takes into account the stiffness, the geometry and the loading of the element. The of this is established from which the load which is unknown in the matrix is evaluated by the principle of bifurcation."

"The remarkable principle of James Bernoulli consists exactly of this... namely, that the mean given by a series of trials falls near the number sought within limits so much the more narrow as the trials are more multiplied. All the properties which result from his learned researches constitute one of the most honourable monuments to his memory. But Bernoulli established his calculations on the hypothesis that the number sought was fixed and determined. ... It may happen that this quantity will experience small variations... But the principle of Bernoulli is still applicable to this case and has been demonstrated by M. Poisson by means of analysis. ...In the case before us the experiments should generally be very numerous: it is for this reason that M. Poisson has designated the extension of Bernoulli's principle as the law of great numbers."

"[T]he writings of the Bernoullis are not antiquated from loss of beauty, nor deserve neglect..."

"Notwithstanding the broad foundation for mechanics laid by Newton in his Principia, and notwithstanding the indefatigable labors of Clairaut, d'Alembert, the Bernoullis, and Euler, there was near the end of the eighteenth century no comprehensive treatise on the science. Its leading principles and methods were fairly well known, but scattered through many works, and presented from divers points of view. It remained for Lagrange to unite them into one harmonious system. Mechanics had not yet freed itself from the restrictions of geometry, though progress since Newton's time had been constantly toward analytical... methods. The emancipation came with Lagrange's Mécanique Analytique published one hundred and one years after the Principia."

"[H]e was soon seconded by two illustrious men, who adopted his method with such ardour, rendered it so completely their own, and made so many elegant applications of it that Leibnitz several times published in the journals, with a disinterestedness worthy of so great a man, that it was as much indebted to them as to himself. ...I am speaking of the two brothers James and John Bernoulli."

"We find an excellent tract by James Bernoulli concerning the elastic curve, isochronous curves, the path of mean direction in the course of a vessel, the inverse method of tangents, &c. On most of these subjects he had treated already; but here he has given them with additions, corrections, and improvements. His scientific discussions are interspersed with some historical circumstances, which will be read with pleasure. Here for the first time he repels the unjust and repeated attacks of his brother; and exhorts him to moderate his pretensions; to attach less importance to discoveries, which the instrument, with which they were both furnished, rendered easy; and to acknowledge, that, 'as quantities in geometry increase by degrees, so every man, furnished with the same instrument, would find by degrees the same results.' Very modest and remarkable expressions from the pen of one of the greatest geometricians, that ever lived. This memoir concluded with an invitation to mathematicians, to sum up a very general differential equation, of great use in analysis. The solution which James Bernoulli had found of this problem, as well as those which Leibnitz and John Bernoulli gave of it, were published in the Leipsic Transactions."