Inventors

"Book 1 deals with logistic numbers, and the calculation of sines. “Logistic numbers” are the numbers... used for astronomical calculations. ...Bürgi explains the four basic operations of arithmetic and the extraction of roots. ...The topic of Chapter 3 is . ...The remaining 10 chapters ...are on sines and ...calculation of sine values. Chapters 11 and 12... explains here in detail his own method for computing... all sine values from 0 to 90 degrees... The result is a sine table for every minute with 5-7 places."

"In Chapter 11 Bürgi also deals with... how to calculate sine values for every minute. ...He divides the known value for sin 1° by 60 to receive an approximation for sin 1′. This he improves by... two corrections and obtains a sufficiently exact value... With... trigonometric relations he then computes sin 2′, etc. ...[U]sing first and higher order differences of consecutive sine values he derives a simple relation for producing the further sines... The result is Bürgi’s sine table... 5400 entries... in his '."

"All [previous] procedures for calculating chords and sines... [were] based in principle on the method which Ptolemy had presented in his '. Totally different... is a procedure which Jost Bürgi...invented... [H]e was able to compute the sine of each angle with any desired accuracy in a... short time... Bürgi explained his procedure in ...'."

"Jost Bürgi... invented logarithms independently of John Napier..."

"Book 2 deals with the calculation of triangles. The first four chapters are on plane triangles and chapters 5-11 on spherical ones."

"Bürgi’s algorithm... reverses the process of forming second differences, i.e., performs up to sign some form of two-fold discrete integration—with the right normalization at the start and end of the sequence. Of course, our Perron-Frobenius eigenvector v of M is also an of M−1, but now for the smallest eigenvalue. Bürgi’s insight must have been that the study of iterations of M is much more useful than those of M−1 in order to approximate the entries of the critical eigenvector. ...[T]his process has unexpected stability properties leading to a quickly convergent sequence of vectors that approximate this eigenvector and hence the sine-values. The reason for its convergence is more subtle than just some geometric principle such as exhaustion, monotonicity, or , it rather relies on the equidistribution of a diffusion process over time—an idea which was later formalized as the and studied... in the theory of s. ...Bürgi’s insight anticipates some aspects of ideas and developments that came to full light only at the beginning of the 20th century."

"In general the French writers... paid no attention to any of the laws except that of multiplication, while the German writers, following the lead of Stifel, took the broader view of the theory. ...[I]n general the German writers were in the lead. ...particularly This is particularly true of Simon Jacob (1565), who followed Stifel closely, recognizing all four [fundamental] laws [of logarithms], and... influencing Jobst Bürgi. These writers did not use the general exponents essential to logarithms, but the recognition of the four laws is significant."

"It is difficult to say who it was who first recognized the advantage of always equating to zero in the study of the general equation. It may... have been Napier, for he wrote... De Arte Logistica before 1594 (although... first printed... 1839), and in this there is evidence that he understood the advantage... Bürgi (c. 1619) also recognized the value of making the second member zero, Harriot (c. 1621) may have done the same, and the influence of Descartes (1637) was such that the usage became fairly general."

"Disputes like this and the other over the calculus have made more than one man of science envy his successors of ten thousand years hence, to whom Newton and Leibniz, Napier and Bürgi, and scores of lesser contestants for individual fame will be semimythical figures as indistinct as Pythagoras."

"... [i]n his unprinted manual of trigonometry... expounded the prosthaphæretic method aimed at simplifying... trigonometric computations. This method first appeared in print in 1588, in Nicolas Reimerus (Ursus)' '. It was of great value... and was going to be a direct competitor to the method of logarithms."

"[T]he autograph Fundamentum Astronomiæ... contained the author’s lost algorithm for computing sine tables..."

"In 1616 Kepler wrote a work on mensuration in which he distinctly took up the decimal fraction, using both a decimal point (comma) and the parentheses to separate the fractional part. He stated it as his opinion that these fractions were due to Bürgi, although it seems strange that he was not familiar with the work of Stevin."

"The Artificium and the whole autograph were decrypted and edited by Dieter Launert... The first proof of convergence was given by Andreas Thom using the . Another proof with matrices by Jörg Waldvogel is more elementary and determines the rate of convergence. In this paper we present a new proof of convergence that encompasses a whole family of related methods..."

"The notion of a "base" can no more be forced upon Bürgi's logarithms than it can be upon the logarithms in Napier's tables. In neither system is \log 1 = 0. Their logarithmic concepts were more general than those of the present day in... that by sliding one progression past the other they could select any positive number at random as the one whose logarithm is zero. We have seen that Napier originally chose \log 10^7 = 0 while Bürgi chose \log 10^8= 0. The logarithms in their tables were integral numbers. More than this, the terms of the two series could be made to increase in the same direction or in opposite directions, at pleasure. That is, if m > n, one can make \log m < \log n , or \log m > \log n , just as one may choose. Napier originally chose the first alternative, Bürgi the second."

"Simon Jacob... wrote two commercial arithmetics. Bürgi mentions Jacob’s treatment of series, and apparently the... table of antilogarithms, the Progress Tabulen, was suggested by the nature of exponents as laid down in these and similar books of the 16th century."

"Two... Swiss mathematicians of the 17th century deserve mention,—one a genius, the other a plagiarist. The genius was Jobst Bürgi, from 1579 to 1603 court watchmaker to Landgraf Wilhelm IV of Hesse, and later (until 1622) to Kaiser Rudolph II. He wrote on the proportional compasses and on astronomy, but is best known for his invention of logarithms independently of Napier. He was led to the idea by an entirely different route from that taken by the latter, approaching it through the theory of exponents. He did not publish anything upon the subject until after Napier had made known his discovery, and when he finally concluded to print his work it was in the form of a small table of antilogarithms, issued anonymously at Prag in 1620. The book never attracted any attention and remained practically unknown except to historians of mathematics."

"Bürgi’s example of his skillful method, the Artificium, explains the calculation for the multiples of 10°, the ninth parts of the right angle."

"Bürgi, Joost (Jobst). Born at Lichtensteig, St. Gall, Switzerland, 1552; died at Cassel in 1632. One of the first to suggest a system of logarithms. The first to recognize the value of making the second member of an equation zero."

"From certain passages in authors like Stifel one might be tempted to say that the logarithmic concept really existed before the time of Napier and Bürgi. Yet how much of a novelty the logarithms of Napier really were to the foremost mathematicians of his day can be realized by the enthusiasm with which Briggs and Kepler took up the new topic."

"In 1620 appeared in Prag the Progress-Tabulen, containing Bürgi's logarithmic tables, but omitting the explanations of them that were promised on the title-page. Hence his logarithms were unintelligible to the ordinary reader."

"Common to Bürgi and Napier was the use of progressions in defining logarithms. In Bürgi's tables the numbers in the were printed in red, the numbers in the were in black. The relation between Bürgi's logarithms, 10n, and their antilogarithms is expressed in modern notation by the equation 10n = \log[10^8(1 + \frac{1}{10^4})^n], \qquad n = 1, 2, 3, \cdots ."

"The other Swiss writer was of a different character. He was a professor while Bürgi was a watchmaker; his name has been known for three centuries, while Bürgi’s has been almost forgotten; but he was a plagiarist, while Bürgi was a genius. began his work as a goldsmith. He later entered the Jesuit order, lived for a long time in Rome, and became professor of mathematics at the and later at Gratz."

"Ursus... and Bürgi became good friends..."

"Bürgi not only used this method, but... improved it. He found the second formula, for Brahe and Wittich only knew the first. In addition, he improved the computation of the ...\cos c = \cos a \cos b + \sin a \sin b \cos CUsing the method of prosthaphæresis... cos a cos b and sin a sin b... could be computed but two new multiplications were... left... Bürgi realized that... prosthaphæresis could be used a second time, and... all multiplications could be replaced by additions or subtractions."

"Another treatise of Bürgi is... his algebraic work Coss... Coss is not restricted to algebra, but... treats the division of an angle and the associated.... calculation of chords and sines."

"[L]ogarithms are one of the most disorderly battlegrounds in mathematical history. ... [A]s adjudicated in 1914... Napier's priority ...is undisputed; J. Bürgi ...independently invented logarithms and constructed a table between 1603 and 1611, while "Napier worked on logarithms probably as early as 1594 ...; therefore, Napier began working on logarithms probably much earlier than Bürgi.""

"Forerunners of Bürgi. Napier approached logarithms from the standpoint of geometry, whereas... we approach the subject from the relation a^m a^n=a^{a+m}. This relation was known to Archimedes and to various other writers. More generally, if we take the two series"

"In 1620 Jobst Bürgi published his Progress Tabulen, a work conceived some years earlier. ...[H]e was influenced by Simon Jacob's work. The tables were printed at Prague and are simply lists of antilogarithms with base 1.0001. The logarithm is printed in red in the top line and the left-hand column, and the antilogarithms are in black, and hence Bürgi calls the logarithm Die Rothe Zahl [The Red Number]. The first part of his table is as follows:"

"In 1588, when Ursus published the method of prosthaphæresis, he did not give any sources. But he acknowledged his debt to Wittich and Bürgi in... De astronomicis hypothesibus [1597]... Once Ursus... published the method... spread... and was improved by other mathematicians, in particular Clavius."

"In 1603, Bürgi was called to the imperial court in Prague... There he... received the praise of Kepler who wrote that Bürgi would sometime be as famous in his art as Dürer is in painting..."

"In 1597, in a letter to Kepler, Ursus wrote that Bürgi was on the "same level as Archimedes and Euclides""

"The use of the prosthaphæretic method required a table of sines. This is likely... why Bürgi constructed a Canon sinuum... sine table... Bürgi... seems to have been reluctant at publishing it and in 1592, Brahe wrote that he did not understand why he was keeping the table hidden, after... a look at it."

"She did the next best thing, she says. She drove to Disney World in Florida and asked if she could play the role of Goofy."

"“I’m a really bad test taker and I ended up bombing the LSATs twice and I was not able to get into a law school,” she says."

"We may say most aptly that the Analytical Engine weaves algebraical patterns just as the Jacquard-loom weaves flowers and leaves."

"Changing an industry requires a willingness to take risks and challenge the status quo."

"The portrait of Jacquard was, in fact, a sheet of woven silk, framed and glazed, but looking so perfectly like an engraving, that it has been mistaken for such by two members of the Royal Academy."

"Jacquard was a man who was most at home among workmen. He was always happiest in their company, and to know him as he really was one had to see him in his ordinary clothes in a weaver's studio, giving the weavers instruction on how to make best use of his loom."

"The machine that I created is simple, costs virtually nothing to maintain, and only requires being kept clean, protected from rust and dust. … This is not all: the fabric is created with speed and perfection, the designs go with the warp of the thread, and the outlines are imperceptible and blended, like in painting."

"If we’re [kids] called creative, why can’t we use that creativity to solve some of the world’s biggest problems?"

"It was not worth engaging in small-scale work, but necessary to conduct the work broadly, with Russian scope. . . . It was not necessary to seek cheaper paths."

"If in reality eka-osmium possesses the same properties as uranium-235, it will be possible to extract it from the "uranium boiler" and use it as a material for an "eka-osmium" bomb. The bomb will therefore be made from an "unearthly" material, which has vanished from our planet."

"Human life is not eternal, but science and knowledge cross the threshold of centuries."

"In work, Comrade Stalin said, it is necessary to move decisively, with the investment of a decisive quantity of resources, but in the basic directions. It is also necessary to use Germany to the utmost; there, there are people, and equipment, and experience, and factories. Comrade Stalin asked about the work of German scholars and the benefits which they brought to us."

"Physics is indebted to the founder of nuclear physics, Ernest Rutherford, for information regarding the interaction of deuterons. In one Of his last investigations Rutherford studied the nuclear reactions that occur when two deuterons collide. It was difficult to suspect at that time that the new facts discovered by him would help realize our hope of mastering the energy sources of the hot interior of the sun and distant stars that shine overhead."

"In any case, it is important to prioritize. Otherwise, the secondary, although necessary, will take all your strength and will not allow you to reach the main one."

"Of foremost significance among the more important problems of modern engineering science is utilization of the energy of thermonuclear reactions. Physicists the world over are attracted by the extraordinarily interesting and very difficult task of controlling thermonuclear reaction."

"On the one hand, there are the approaches that lead to stationary thermonuclear reactions, and on the other hand, those that are based on the idea of utilizing an instantaneous temperature rise in transient processes of very brief duration. However, irrespective of the way the investigation is carried out, there is one problem that is inevitably encountered; namely, the insulation of the plasma, which is heated to a high temperature, from the walls of the vessel in which it is confined, In other words, a means must be found to keep the fast particles within the plasma over a period sufficient for the particles to have a good chance to react with each other."

"As is known, thermonuclear reactions can arise if the temperature of matter is sufficiently high for atomic nuclei to surmount the forces of the Coulomb barrier with appreciable probability daring thermal collisions. The excitation of thermonuclear reaction in deuterium or in a mixture of deuterium and tritium is especially interesting since in this case a noticeable effect should be obtainable at relatively low temperatures."

"In a sufficiently strong magnetic field, electrons and ions can move freely only along the lines of magnetic force. In a plane normal to these lines of force the particles will move along circles of small radius. The positions of the centers of these circles can vary only as a result of collisions, each collision displacing the center by a distance of the same order of magnitude as the radius of curvature of the particle trajectory."