Academics From Switzerland

"God, the pure limit and pure beginning of all that we are, have, and do, standing over in infinite qualitative difference to man and all that is human, nowhere and never identical with that which we call God, experience, surmise, and pray to as God, the unconditioned Halt as opposed to all human rest, the Yes in our No and the No in our Yes, the first and last and as such unknown, but nowhere and never a magnitude amongst others in the medium known to us, God the Lord, the Creator and Redeemer . . . that is the living God."

"Religion is the possibility of the removal of every ground of confidence except confidence in God alone. Piety is the possibility of the removal of the last traces of a firm foundation upon which we can erect a system of thought."

"The revelation in Jesus, just because it is the revelation of the righteousness of God is at the same time the strongest conceivable veiling and unknowableness of God. In Jesus, God really becomes a mystery, makes himself known as the unknown, speaks as the eternally Silent One."

"We know that God is He whom we do not know, and that our ignorance is precisely the problem and the source of our knowledge. The Epistle to the Romans is a revelation of the unknown God; God chooses to come to man, not man to God. Even after the revelation man cannot know God, for he is ever the unknown God. In manifesting himself to man he is farther away than before."

"The power of God can be detected neither in the world of nature nor in the souls of men. It must not be confounded with any high, exalted, force, known or knowable."

"Form - as a pattern of perception - can motivate the use of architecture. Architecture provides a complete service for its users, if function (practical purpose) coincides with use motivated by form. (Mit der Gestalt als Wahrnehmungsstruktur lässt sich der Gebrauch einer Architektur motivieren. Wenn die Funktion (der praktische Zweck) mit der (gestaltbezogenen) Motivation zum Gebrauch übereinstimmt, ist die Dienstleistung der Architektur für den Menschen komplett.)"

"Function and purpose define architecture as a practical arrangement, while form defines architecture as a manifestation of apparition. The world of appariation that we cannot exist without (Max Horkheimer), is a psychological issue related to form. (Funktion und Zweck definieren Architektur als praktisches Bauwerk, die Gestalt definiert Architektur als Erscheinungsform. Die Welt der Erscheinungen, ohne die wir nicht leben können (Max Horkheimer), ist ein gestaltpsychologisches Problem.)"

"Per definition, architecture is a service for the whole human being. As such, architecture includes a material and an immaterial aspect; it has to meet rational and irrational requirements. (Architektur versteht sich als Dienstleistung für den ganzen Menschen. Als solche hat sie eine materielle und eine immaterielle Komponente; es sind rationale und irrationale Bedürfnisse zu befriedigen.)"

"The central problem of architecture is space. Space is essential for the individual and for the community. This is equally true for the small space (the capsule) as well as for the large space (the city)."

"The creation of space incorporates the debate about the dialogue between dream and reality. We must use the hidden surrealistic potential of our environment to awaken basic emotions. (Raumgebung beinhaltet die Auseinandersetzung mit dem dialogischen Verhältnis von Traum und Wirklichkeit. Wir müssen das surrealistische Potenzial ausschöpfen, welches in unserer Umwelt verborgen ist. Es lassen sich damit Basisgefühle wecken.)"

"Spaces that trigger emotions alter the behaviour of people. Architecture aims at influencing human behaviour by space creations. (Räume, die Empfindungen auslösen, verändern das Verhalten des Menschen. Die Verhaltensbeeinflussung des Menschen durch die gestaltete Umwelt ist eine qualitative Zielsetzung des architektonischen Entwurfs.)"

"The world has arisen in some way or another. How it originated is the great question, and Darwin's theory, like all other attempts to explain the origin of life, is thus far merely conjectural. I believe he has not even made the best conjecture possible in the present state of our knowledge."

"The facts will eventually test all our theories, and they form, after all, the only impartial jury to which we can appeal."

"Foldings of the earth's crust, low hills, extensive plains, mountain-chains and narrow valleys, broad table-lands and wide valleys, local chimneys or volcanoes, river-beds, lake-basins, inland seas,—such are some of the phenomena which, disconnected as they seem at first glance, have nevertheless been brought under certain principles, and explained according to definite physical laws."

"The eye of the trilobite tells us that the sun shone on the old beach where he lived; for there is nothing in nature without a purpose, and when so complicated an organ was made to receive light, there must have been light to enter it."

"But whatever its defects, the classification of Linnæus was the first attempt at grouping animals together according to certain common structural characters."

"The crust of our earth is a great cemetery, where the rocks are tombstones on which the buried dead have written their own epitaphs."

"Therefore, Agassiz says that when a new doctrine is presented, it must go through three stages. First, people say that it isn't true, then that it is against religion, and, in the third stage, that it has long been known."

"Agassiz... was doomed to help the cause he hated. Agassiz not only maintained the fact of the progressive advance in organisation of the inhabitants of the earth at each successive geological epoch, but he insisted upon the analogy of the steps of this progression with those by which the embryo advances to the adult condition, among the highest forms of each group. In fact, in endeavoring to support these views he went a good way beyond the limits of any cautious interpretation of the facts then known."

"There, at the table's further end I see In his old place our Poet's vis-à-vis, The great PROFESSOR, strong, broad-shouldered, square, In life's rich noontide, joyous, debonair. His social hour no leaden care alloys, His laugh rings loud and mirthful as a boy's,— That lusty laugh the Puritan forgot,— What ear has heard it and remembers not? How often, halting at some wide crevasse Amid the windings of his Alpine pass, High up the cliffs, the climbing mountaineer, Listening the far-off avalanche to hear, Silent, and leaning on his steel-shod staff, Has heard that cheery voice, that ringing laugh, From the rude cabin whose nomadic walls Creep with the moving glacier as it crawls! How does vast Nature lead her living train In ordered sequence through that spacious brain, As in the primal hour when Adam named The new-born tribes that young creation claimed!— How will her realm be darkened, losing thee, Her darling, whom we call our AGASSIZ!"

"In the subsequent meetings of the society, the geologist William Barton Rogers would skillfully dismantle most of these arguments. Rogers was the soon-to-be-president of the Massachusetts Institute of Technology. In his debates with Agassiz he employed an encyclopedic knowledge of North American geology to show that most of his opponent’s claims were either false or partially true at best. But Agassiz cared little for debating the topic; he was not interested in debating a subject about which he was already certain. As far as he was concerned, his own theory of special creation rendered Darwin’s null and void. This was one of the reasons he preferred discussing the topic with members of the Saturday Club, who were far more sympathetic to his ideas and who were also more likely to shape public opinion about Darwinian theory than a handful of scientific specialists."

"The time has come when scientific truth must cease to be the property of the few, when it must be woven into the common life of the world."

"Every great scientific truth goes through three stages. First, people say it conflicts with the Bible. Next they say it has been discovered before. Lastly they say they always believed it."

"Till now the mathematicians tried in vain to discover some order in the sequence of the prime numbers and we have every reason to believe that there is some mystery which the human mind shall never penetrate. To convince oneself, one has only to glance at the tables of primes which some people took the trouble to compute beyond a hundred thousand, and one perceives that there is no order and no rule. This is so much more surprising as the arithmetic gives us definite rules with the help of which we can continue the sequence of the primes as far as we please, without noticing, however, the least trace of order."

"Madam, I have come from a country where people are hanged if they talk."

"Now I will have less distraction."

"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate."

"All the greatest mathematicians have long since recognized that the method presented in this book is not only extremely useful in analysis, but that it also contributes greatly to the solution of physical problems. For since the fabric of the universe is most perfect, and is the work of a most wise Creator, nothing whatsoever takes place in the universe in which some relation of maximum and minimum does not appear. Wherefore there is absolutely no doubt that every effect in the universe can be explained as satisfactorily from final causes, by the aid of the method of maxima and minima, as it can from the effective causes themselves. Now there exist on every hand such notable instances of this fact, that, in order to prove its truth, we have no need at all of a number of examples; nay rather one's task should be this, namely, in any field of Natural Science whatsoever to study that quantity which takes on a maximum or a minimum value, an occupation that seems to belong to philosophy rather than to mathematics. Since, therefore, two methods of studying effects in Nature lie open to us, one by means of effective causes, which is commonly called the direct method, the other by means of final causes, the mathematician uses each with equal success. Of course, when the effective causes are too obscure, but the final causes are more readily ascertained, the problem is commonly solved by the indirect method; on the contrary, however, the direct method is employed whenever it is possible to determine the effect from the effective causes. But one ought to make a special effort to see that both ways of approach to the solution of the problem be laid open; for thus not only is one solution greatly strengthened by the other, but, more than that, from the agreement between the two solutions we secure the very highest satisfaction."

"To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be."

"La construction d'une machine propre à exprimer tous les sons de nos paroles , avec toutes les articulations , seroit sans-doute une découverte bien importante. … La chose ne me paroît pas impossible."

"It will seem a little paradoxical to ascribe a great importance to observations even in that part of the mathematical sciences which is usually called Pure Mathematics, since the current opinion is that observations are restricted to physical objects that make impression on the senses. As we must refer the numbers to the pure intellect alone, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of numbers. Yet, in fact, as I shall show here with very good reasons, the properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations; they will lead us continually to new properties which we shall endeavor to prove afterwards. The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful."

"A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities."

"Quanquam nobis in intima naturae mysteria penetrare, indeque veras caussas Phaenomenorum agnoscere neutiquam est concessum: tamen evenire potest, ut hypothesis quaedam ficta pluribus phaenomenis explicandis aeque satisfaciat, ac si vera caussa nobis esset perspecta."

"He calculated without any apparent effort, just as men breathe, as eagles sustain themselves in the air."

"The most influential mathematics textbook of ancient times is easily named, for the Elements of Euclid has set the pattern in elementary geometry ever since. The most effective textbook of the medieval age is less easily designated; but a good case can be made out for the Al-jabr of Al-Khwarizmi, from which algebra arose and took its name. Is it possible to indicate a modern textbook of comparable influence and prestige? Some would mention the Géométrie of Descartes or the Principia of Newton or the Disquisitiones of Gauss; but in pedagogical significance these classics fell short of a work by Euler titled Introductio in analysin infinitorum."

"The Introductio does not boast an impressive number of editions, yet its influence was pervasive. In originality and in the richness of its scope it ranks among the greatest of textbooks; but it is outstanding also for clarity of exposition. Published two hundred and two years ago, it nevertheless possesses a remarkable modernity of terminology and notation, as well as of viewpoint. Imitation is indeed the sincerest form of flattery."

"Of no little importance are Euler's labors in analytical mechanics. ...He worked out the theory of the rotation of a body around a fixed point, established the general equations of motion of a free body, and the general equation of hydrodynamics. He solved an immense number and variety of mechanical problems, which arose in his mind on all occasions. Thus on reading Virgil's lines. "The anchor drops, the rushing keel is staid," he could not help inquiring what would be the ship's motion in such a case. About the same time as Daniel Bernoulli he published the Principle of the Conservation of Areas and defended the principle of "least action," advanced by P. Maupertius. He wrote also on tides and on sound."

"Somebody said "Talent is doing what others find difficult. Genius is doing easily what others find impossible." ...by that definition, Euler was a genius. He could do the seemingly impossible, and he did it throughout his long and illustrious life. ...Way to Go, Uncle Leonhard!"

"Euler calculated the force of the wheels necessary to raise the water in a reservoir … My mill was carried out geometrically and could not raise a drop of water fifty yards from the reservoir. Vanity of vanities! Vanity of geometry!"

"Euler lacked only one thing to make him a perfect genius: He failed to be incomprehensible."

"The study of Euler's works will remain the best school for the different fields of mathematics and nothing else can replace it."

"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."

"Following a suggestion by Daniel Bernoulli, Euler gave the first treatment of elastic lines by means of the in the Additamentum I to his Methodus inveniendi (1744...) which carries the title De curvis elasticus. Euler characterized the equilibrium position of an elastic line by the following variational principle: Among all curves of equal length, joining two points where they have prescribed tangents, to determine that which minimizes the value of the expression \int ds/\rho^2 [where \rho is the radius of curvature]. In other words, Euler interpreted an elastic line as an inextensible curve \boldsymbol\zeta with a "" of \int \kappa^2 ds, \kappa [i.e., 1/\rho] being the function of \boldsymbol\zeta, whose positions of (stable) equilibrium are characterized by the minima of the potential energy, i.e., by 's principle of . Thus the problem of the elastic line leads to the isoperimetric problem\int_{\boldsymbol \zeta} \kappa^2 ds \to min \qquad with \int_{\boldsymbol\zeta} ds = L..."

"Euler published so much and in so many different fields that an edited volume is probably the only way (at least at this time) to do him something like justice, since no one person will know enough to span all of his work."

"Galileo does not attempt any theory to account for the flexure of the beam. This theory, supplied by , was applied by Mariotte, Leibnitz, De Lahire, and Varignon, but they neglect compression of the fibres, and so place the neutral in the lower face of Galileo's beam. The true position of the neutral plane was assigned by James Bernoulli 1695, who in his investigation of the simplest case of bent beam, was led to the consideration of the curve called the "elastica." This "elastica" curve speedily attracted the attention of the great Euler (1744), and must be considered to have directed his attention to the s. Probably the extraordinary divination which led Euler to the formula connecting the sum of two elliptic integrals, thus giving the fundamental theorem of the addition equation of s, was due to mechanical considerations concerning the "elastica" curve; a good illustration of the general principle that the pure mathematician will find the best materials for his work in the problems presented to him by natural and physical questions."

"Who has studied the works of such men as Euler, Lagrange, Cauchy, Riemann, , and Weierstrass, can doubt that a great mathematician is a great artist? The faculties possessed by such men, varying greatly in kind and degree with the individual, are analogous with those requisite for constructive art. Not every mathematician possesses in a specially high degree that critical faculty which finds its employment in the perfection of form, hi conformity with the ideal of logical completeness; but every great mathematician possesses the rarer faculty of constructive imagination."

"To the reader of today much in the conception and mode of expression of that time appears strange and unusual. Between us and the mathematicians of the late seventeenth century stands Leonhard Euler... He is the real founder of our modern conception. However non-rigorous he may be in details: he ends and conquers the previous epoch of direct geometric infinitesimal considerations and introduces the period of mathematical analysis according to form and content. Whatever was written after him on the logarithmic series is necessarily based no longer on the already obscured predecessors in the receding mathematical Renaissance, but on Euler's Introductio in analysin infinitorum... in which the entire seventh chapter [De Quantitabus exponentialibus ac Logarithmis] treats of logarithms."

"Read Euler: he is our master in everything."

"He was later to write that he had made some of his best discoveries while holding a baby in his arms surrounded by playing children."

"If we compared the Bernoullis to the Bach family, then Leonhard Euler is unquestionably the Mozart of mathematics, a man whose immense output... is estimated to fill at least seventy volumes. Euler left hardly an area of mathematics untouched, putting his mark on such diverse fields as analysis, number theory, mechanics and hydrodynamics, cartography, topology, and the theory of lunar motion. ...Moreover, we owe to Euler many of the mathematical symbols in use today, among them i, π, e, and f(x). And as if that were not enough, he was a great popularizer of science..."