Non Fiction Authors From Germany

"The beauty of [ Eudoxus' ] theory of proportions [ expounded in Book V of Euclid's Elements ] was its adaptability to this new climate. ...The length \sqrt2 is determined by the two sets of positive rationalsL_\sqrt2 = \{r: r^2 < 2\}, \qquad U_\sqrt2 = \{r: r^2 > 2\}Dedekind... decided to let \sqrt2 be this pair of sets! In general, let any partition of the positive rationals into sets L, U such that any member of L is less than any member of U be a positive real number. This idea, now known as the Dedekind cut, is more than just a twist of Eudoxus; it gives a complete and uniform construction of all real numbers, or points on a line, using just the discrete, finally resolving the fundamental conflict in Greek mathematics."

"In discussing the notion of the approach of a variable magnitude to a fixed limiting value, [Dedekind] had recourse, as had Cauchy before him, to the evidence of the geometry of continuous magnitude. ...Dedekind's approach was somewhat different from that of Weierstrass, Méray, Heine, and Cantor in that, instead of considering in what manner the irrationals are to be defined so as to avoid the vicious circle of Cauchy, he asked himself... what is the nature of continuity? ...The philosophy and mathematics of Leibniz had led him to agree with Galileo that continuity was a property concerning conjunctive aggregation, rather than a unity or coincidence of parts. Leibniz had regarded a set as forming a continuum if between any two elements there was always another element of the set. ...Ernst Mach likewise regarded this property of denseness of an assemblage as constituting its continuity, but... rational numbers... possess the property of denseness and yet do not constitute a continuum. Dedekind...found the essence of... continuity, not by a vague hang-togetherness, but in the nature of the division of the line by a point. ...in any division of the points into two classes such that every point of the one is to the left of every point of the other, there is one and only one point which produces this division. This is not true of the ordered system of rational numbers."

"The modern theory of functions of one real variable was first worked out by H. Hankel, Dedekind, G. Cantor, Dini, and Heine, and then carried further, principally, by Weierstrass, Schwarz, Du Bois-Reymond, Thomae, and Darboux. Hankel established the principle of the condensation of singularities; Dedekind and Cantor gave definitions for irrational numbers..."

"The above comparison of the domain R of rational numbers with a straight line has led to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the former, while we ascribe to the straight line completeness, absence of gaps, or continuity. In what then does this continuity consist? Everything must depend on the answer to this question, and only through it shall we obtain a scientific basis for the investigation of all continuous domains."

"In the preceding section attention was called to the fact that every point p of the straight line produces a separation of the same into two portions such that every point of one portion lies to the left of every point of the other. I find the essence of continuity in the converse, i.e., in the following principle: "If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions." ...every one will at once grant the truth of this statement; the majority of my readers will be very much disappointed in learning that by this commonplace remark the secret of continuity is to be revealed."

"Although the real theory might have been less useful than the complex in obtaining properties of special functions, its significance for the development of mathematics as a whole has been incomparably greater. It was in the real variable that the necessity for a rigorous theory of the number system of analysis was first recognized. ...the reconstruction of the real number system by Weierstrass in the 1860's and by Dedekind and Cantor in the 1870's led in the last three decades of the nineteenth century, to a profound reconsideration of the nature of all mathematical reasoning. This in turn initiated some of the most searching examinations of all deductive reasoning since the days or Aristotle. Thus the theory of the functions of a real variable since the 1870's has increasingly acquired more than merely a local interest: its problems, solved and unsolved, are significant in fields far distant from technical mathematics."

"The system R forms a well-arranged domain of one dimension extending to infinity on two opposite sides. What is meant by this is sufficiently indicated by my use of expressions borrowed from geometric ideas; but just for this reason it will be necessary to bring out clearly the corresponding purely arithmetic properties in order to avoid even the appearance as if arithmetic were in need of ideas foreign to it."

"Addition is the combination of any arbitrary repetitions of the above-mentioned simplest act into a single act; from it in a similar way arises multiplication. While the performance of these two operations is always possible, that of the inverse operations, subtraction and division, proves to be limited. Whatever the immediate occasion may have been, whatever comparisons or analogies with experience, or intuition, may have led thereto; it is certainly true that just this limitation in performing the indirect operations has in each case been the real motive for a new creative act; thus negative and fractional numbers have been created by the human mind; and in the system of all rational numbers there has been gained an instrument of infinitely greater perfection. This system, which I shall denote by R, possesses first of all a completeness and self-containedness which I have designated... as characteristic of a body of numbers [Zahlkőrper] and which consists in this, that the four fundamental operations are always performable with any two individuals in R, i.e., the result is always an individual of R, the single case of division by the number zero being excepted."

"If a, c are two different numbers, there are infinitely many different numbers lying between a, c."

"What advantage will be gained by even a purely abstract definition of real numbers of a higher type, I am as yet unable to see, conceiving as I do of the domain of real numbers as complete in itself."

"I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an already-formed individual to the consecutive new one to be formed. The chain of these numbers forms in itself an exceedingly useful instrument for the human mind; it presents an inexhaustible wealth of remarkable laws obtained by the introduction of the four fundamental operations of arithmetic."

"If a is any definite number, then all numbers of the system R fall into two classes, A1 and A2, each of which contains infinitely many individuals; the first class A1 comprises all numbers a1 that are < a, the second class A2 comprises all numbers a2 that are > a; the number a itself may be assigned at pleasure to the first or second class, being respectively the greatest number of the first class or the least of the second. In every case the separation of the system R into the two classes A1, A2 is such that every number of the first class A1 is less than every number of the second class A2."

"As professor in the Polytechnic School [autumn of 1858] in Zurich I found myself for the first time obliged to lecture upon the elements of the differential calculus and felt, more keenly than ever before, the lack of a really scientific foundation for arithmetic. In discussing the notion of the approach of a variable magnitude to a fixed limiting value, and especially in proving the theorem that every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value, I had recourse to geometric evidences. Even now such resort to geometric intuition in a first presentation of the differential calculus, I regard as exceedingly useful, from the didactic standpoint, and indeed indispensable, if one does not wish to lose too much time. But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny. For myself this feeling of dissatisfaction was so overpowering that I made the fixed resolve to keep meditating on the question till I should find a purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis."

"The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. Among these, for example, belongs the above mentioned theorem, and a more careful investigation convinced me that this theorem, or any one equivalent to it, can be regarded in some way as a sufficient basis for infinitesimal analysis. It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858."

"The way in which the irrational numbers are usually introduced is based directly upon the conception of extensive magnitudes—which itself is nowhere carefully defined—and explains number as the result of measuring such a magnitude by another of the same kind. Instead of this I demand that arithmetic shall be developed out of itself."

"Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must endeavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this."

"That such comparisons with non-arithmetic notions have furnished the immediate occasion for the extension of the number-concept may, in a general way, be granted (though this was certainly not the case in the introduction of complex numbers); but this surely is no sufficient ground for introducing these foreign notions into arithmetic, the science of numbers."

"Was dich die Liebe nicht lehrt, das sollst du nicht wissen."

"Die Wolken gehören zur Erde, nicht zum Himmel."

"Christian identity can be understood only as an act of identification with the crucified Christ, to the extent to which one has accepted the proclamation that in him God has identified himself with the godless and those abandoned by God."

"One cannot grasp freedom in faith without hearing simultaneously the categorical imperative: One must serve through bodily, social and political obedience the liberation of the suffering creation out of real affliction. ... Consequently, the missionary proclamation of the cross of the Resurrected One is not an opium of the people which intoxicates and incapacitates, but the ferment of new freedom. It leads to the awaking of that revolt which, in the "power of the resurrection" ... follows the categorical imperative to overthrow all conditions in which man is a being who labors and is heavily laden,"

"Even the sharpest problem focus cannot help but sharpen the problem."

"Unsere übertragungen, auch die besten, gehen von einem falschen grundsatz aus, sie wollen das indische, griechische, englische verdeutschen, anstatt das deutsche zu verindischen, vergriechischen, verenglischen. ... Der grundsätzliche irrtum des übertragenden ist, daß er den zufälligen stand der eigenen sprache festhält, anstatt sie durch die fremde gewaltig bewegen zu lassen."

"In its resistance to the great attack on traditional dogma launched by the Enlightenment, the Protestant theology of the 19th century … had very largely agreed in taking up a defensive position in which the truth of Christian faith was to be proved by demonstrating the indispensability of religious feeling to all higher humanity. … Yet on this basis it was not possible to refute the contention of Ludwig Feuerbach that the secret of theology is anthropology. … The secret of God was in truth that of self-glorifying man speaking of God and extolling his own divinity."

"The attack and claim which are so unmistakable in the New Testament were muted and neutralized by the official recognition, the cultural integration and the social control of Christianity as a middle-class institution."

"The non-Christian can put no question which is not necessarily that of the Christian dogmatician as well. ... In solidarity with the non-Christian, he has no prior knowledge. He must begin from the very beginning with no presupposed consciousness of God. He must learn to spell out the message which comes to him as to every man."

"Over this period many issues have been clarified. Today we know very much more about these twenty-seven short writings from a little religious sub-culture in the Roman empire than ever before. Nevertheless, they risk being forgotten - partly because the link with the Christian history which they have influenced has been broken, partly because many of our educated contemporaries come from other religious and cultural traditions, and partly because the results of historical-critical research are so complex that many people are deterred from trying to grasp them."

"The lukewarm baths at 25 to 27° serve to arouse the slumbering sensibility of fibre in the apparent dead (frozen, drowned, suffocated) which benumbed the sensation of the nerves. Though only palliative, still they often prove themselves sufficiently active, especially when given in conjunction with coffee and rubbing with the hands. They may give homœopathic aid in cases where the irritability is very unevenly distributed and accumulated too unevenly in some organs as is the case in certain hysteric spasms and infantile convulsions. In the same way, cold baths 10 to 6° in persons cured medically of chronic diseases and with deficiency of vital heat, act as an homœopathic aid. By instantaneous and later with repeated immersions they act as a palliative restorative of the tone of the exhausted fibre. For this purpose, such baths are to be used for more than momentary duration, rather for minutes and of gradually lowered temperature, they are a palliative, which, since it acts only physically has no connection with the disadvantage of a reverse action to be feared afterwards, as takes place with dynamic medicinal palliatives."

"If the physician clearly perceives what is to be cured in diseases, that is to say, in every individual case of disease (knowledge of disease, indication), if he clearly perceives what is curative in medicines, that is to say, in each individual medicine (knowledge of medical powers), and if he knows how to adapt, according to clearly defined principles, what is curative in medicines to what he has discovered to be undoubtedly morbid in the patient, so that the recovery must ensue - to adapt it, as well in respect to the suitability of the medicine most appropriate according to its mode of action to the case before him (choice of the remedy, the medicine indicated), as also in respect to the exact mode of preparation and quantity of it required (proper dose), and the proper period for repeating the dose; - if, finally, he knows the obstacles to recovery in each case and is aware how to remove them, so that the restoration may be permanent, then he understands how to treat judiciously and rationally, and he is a true practitioner of the healing art ."

"A physical-mental characteristic of mine is a certain passive magnetism of the animal world... The mental-physical passive animal magnetism mentioned is passive, not active, for the reason that the person for whom it is a characteristic does not attract, but rather feels himself attracted, just as a passive magnetism dwells in a piece of soft iron, since it does not attract, but is attracted by the steel magnet, whereas active magnetism is in the attracting steel magnet (perhaps a passive magnetism as well, but at least an active is there)."

"Until now science has not sought to investigate this passive animal magnetism (by no means an isolated phenomenon), although the doctor, the anthropologist and physiologist, the jurist, the psychologist, and the moralist could cultivate an entirely new field. In fact they have made not the slightest effort to investigate its nature: rather (misled by poorly understood Bible passages and by laws based on such Bible passages—laws whose moral value stands on the same level as those against witchcraft and heresy in the Middle Ages) they have believed they should ignore or disdain it with hatred and scorn, examples of which are in scientific books."

"Sunt mihi barba maris, artus, corpusque virile, His inclusa quidem. Sed sum maneoque puella"

"The Urning is not by a hair’s breadth any more dangerous to immature boys than the genuine man is to immature girls. For the rest, I gladly leave the child molester to his deserved punishment by the law. Let the integrity of a will-less minor be sacred to every Urning."

"Karl Heinrich Ulrichs was the first to formulate a scientific theory of homosexuality. Indeed, his theory implicated, as Klaus Müller has emphasized, “the first scientific theory of sexuality altogether” (1990, 100). It was set forth and elaborated in five writings published in 1864 to 1865. The series of writings was continued—there were twelve in all, the last appearing in 1879—but with only slight revisions in the theory, Ulrichs’s intention in his writings was not merely explanatory, but also—and especially—emancipatory. This was based on his view that the condition of being homosexual is inborn. This was a major departure from previous and subsequent theories that saw the practice of homosexuality/“sodomy” as an acquired vice. In this Ulrichs was the first in a long and continuing line of researchers who believe that a proof of the “naturaless” of homosexuality, that is, the discovery of a biological basis for it, will lead to equal legal and social treatment of hetero- and homosexuals."

"In 1870, Ulrichs gestured toward likely biological underpinnings of homosexuality as an argument for gay rights, arguing that a male homosexual has inalienable rights. His sexual orientation is a right established by nature. Legislators have no right to veto nature; no right to persecute nature in the course of its work; no right to torture living creatures who are subject to those drives nature gave them."

"The prohibition of the expression of the sex drive, i.e., between consenting adults in private, lies outside the legal sphere."

"Until my dying day I will look back with pride that I found the courage to come face to face in battle against the spectre which for time immemorial has been injecting poison into me and into men of my nature. Many have been driven to suicide because all their happiness in life was tainted. Indeed, I am proud that I found the courage to deal the initial blow to the hydra of public contempt."

"Theological condemnation of others, which breaks off fellowship in either judgment or contempt, is impermissible."

"Leupold is also credited as an early inventor of air pumps. He designed his first pump in 1705, and in 1707 he published a book Antlia pneumatica illustrata. In 1711 following an advice of its president Wilhelm Leibniz, Prussian Academy of Sciences acquired Leupold's pump. In 1720 Leupold started to work on the manuscript of his prominent encyclopedie Theatrium machinarum, a nine-volume series on machine design and technology, published between 1724 and 1739 . It was the first systematic analysis of mechanical engineering in the world."

"Jacob Leupold (1674-1727) German engineer who collected, for the first time in print, the basic principles of mechanical engineering."

"[His work is addressed]... not to the learned and experienced mathematicians who are already, or should be, better acquainted with them... [and most of whom] have studied mechanics more as a subject of curiosity and a hobby, than with any view of service to the public. The people we had in mind were rather the mechanic, handicraftsman and the like, who, without education or knowledge of foreign languages have no access to many sources of information..."

"Most often it is the case that people know that something big can be manipulated with it [i.e., the screw], but not how and in what way it is connected to time, and that untold time, and finally such force of machines, wheels, and shafts is necessary as cannot be produced nor be had."

""Theatrum machiuamm universale," &c. by Jacob Leupold, Leipsic, seven volumes, folio, 1724, 1727,1774. This is the greatest and most complete work of this kind that ever was published. The first volume is little more than an introduction to the work; the second and third volumes contain a description of hydraulic machines; the next two volumes relate to machines for raising weights, the theory of levelling, and other subjects; and the sixth treats principally on machines connected with the construction of bridges; the seventh volume is entitled, "Theatre arithmetico geometrique," where the author treats of all instruments employed in these two sciences This work would have been much more considerable if its author had lived to complete the immense task he had undertaken."

"I had not only opportunity of seeing how different things have been made, but also manual work made me strong."

"In the Histoire de l'Academie for the year 1725, p. 78, it is stated that when M. du Fay was at Strasbourg, M. Jacob Leupold had a pump which threw water in a continuous stream, using only one piston, and that he made a great mystery of it; but that M. du Fay immediately stated the reason of it."

"Another Dominican mystic of this period is the Blessed Henry Suso, or Heinrich Seuse, to give him his original German name. The son of a noble Swabian family, he was born near Lake Constance, on the border between Switzerland and Germany. His father was very worldly, his mother deeply devout. As he tells us in his own Life, written in later years, one of his earliest memorable experiences occurred on the death of his mother when he was still young. She appeared in a vision and told him to love God, then kissed and blessed him, and disappeared. Suso’s sense of loneliness and abandonment, his excessive asceticism in later life, harshly maltreating his body in imitation of Christ’s suffering, and his expressions of tender love addressed to God may all have been linked to “starved human affections seeking an outlet,” as Evelyn Underhill has suggested. Suso entered the Dominican order at the age of thirteen, but found monastic life rather difficult until he experienced a conversion and spiritual awakening. He subsequently studied under Eckhart in Cologne and became a devoted follower and great admirer of his beloved teacher. By 1326 Suso was back in Constance, where he wrote his famous Büchlein der Wahrheit, or Little Book of Truth, which is full of mystical reflection...Suso experienced intense mystical states and visions that made him see ultimate reality as eternal, uncreated truth in which all things have their source and being. He goes even beyond Eckhart in his understanding of divine and human oneness—a state in which “something and nothing are the same.”...Suso preached widely in the Upper Rhineland and Switzerland, enjoying great popularity wherever he went...The savage asceticism and austerities that he practiced over many years are vividly described in his Life, where he speaks of himself in the third person...But after some twenty years of severe ascetic practices he abandoned them as nothing more than a beginning on the way to the highest knowledge of God, whose overwhelming beauty he praised with great tenderness: “Ah, gentle God, if Thou art so lovely in Thy creatures, how exceedingly beautiful and ravishing Thou must be in Thyself!… Praise and honor be to the unfathomable immensity that is in Thee!” Suso must have left a deep impression on his contemporaries, for the veneration of the “Blessed Henry Suso” began soon after his death, although officially the Church did not beatify him until 1831."

"Henry Suso is a bundle of contradictions, and a person, moreover, who has gathered legends about him like a snowball rolling downhill. He was a poet, which is not always a key to happiness in this world; a mystic of the highest order; a hard working Dominican; and a man with a positive genius for getting into embarrassing situations... It will require many years of exhaustive research to sort out the diverse elements in his personality, if, indeed, it can ever be accomplished. Poets are not easy to analyze, and Henry, before all else, was a poet...Henry was born in Switzerland, in 1290, the son of a warlike family of counts and crusaders. His father said more than once that he wished Henry had been a girl and some of his spirited daughters had been boys; for Henry was not a type to carry a sword. Henry was a gentle, dreamy lad, who liked to accompany his mother on pilgrimages and read about heroic deeds. He had taken his mother's name of Suso, perhaps out of sheer inability to live up to the warlike title of the Count von Berg...The best known work of Henry Suso is his Little Book of Eternal Wisdom, which is a classic of spiritual writing. He also composed many other short treatises on the mystical union of the soul with God, all written with the same poetic language and the same intensity of feeling. The man who had carved "the lovely name of Jesus" into the flesh over his heart was just as intense in his spiritual life."

"One thing you must know: Just as there is no comparison between actually hearing the sound of harp-strings sweetly plucked and listening to someone talking about it, so too there is no comparison between words which are received in pure grace, issuing from a living heart, spoken by living lips, and those self-same words committed to dry parchment — especially words in German. For these somehow grow chill, losing their vitality like roses cut. For the enchanting melody which, more than anything else, moves human hearts, then fades away, so that the words are received now into the dryness of dry hearts. No harp-strings were ever so sweet but, when stretched across dry timber, they fall silent. An unloving heart can no more understand a love-filled speaker than a German an Italian. Therefore, an eager enquirer should hasten to the out-flowing streams of these sweet teachings so that she may see and observe them at their source in all its living and wondrous beauty – that is, the in-flowing of present grace which is able to restore dead hearts to life."

"Bl. Henry Suso (c. 1300 - 1366) studied theology under Meister Eckhart in Cologne. But Eckhart was more than a teacher to him: there is a touching account in Suso's autobiography of how he went to Eckhart when his hypersensitive conscience was tormenting him, and how Eckhart gave him complete peace. He entered the Dominican Order in his native Constance. Some years later he had a profound religious experience which he described in great detail. It was the beginning of a great love story, told with impressive literary skill in the tender language of courtly love...The language of chivalry, parodied in a later century in Don Quixote, was still viable in Suso's century. 'Your young unruly heart,' he said to himself, 'can scarcely endure to be without a special object of love.' So he often 'meditated about her, thinking of her lovingly, and liking her full well with all his heart and soul.' The mediaeval knight delighted to suffer for the lady he worshipped. Two of his books are written as dialogue, a favourite literary form in the 14th century."

"Among all the medieval mystics, the Rhineland mystics are perhaps the ones best known today. They are widely accessible and cited more frequently than any others. Their message has a directness and freshness of expression that communicates itself across the centuries...Initially one might expect that all mystics from the Rhine region would be counted as Rhineland mystics...But the term “Rhineland mystics” is customarily used in a more restricted sense. It refers only to the mystics of the fourteenth century who lived in Germany and the Low Countries. It applies particularly to a group of several men—Meister Eckhart, Henry Suso, Johannes Tauler and Jan van Ruusbroec...The most daring and original of the Rhenish mystics was Meister Eckhart, whose disciples were Johannes Tauler and Henry Suso. These three Germans all belonged to the Dominican order. Suso is seen as the most intimate and personal mystic, Tauler was known as an inspiring preacher, whereas the Flemish mystic Jan van Ruusbroec limited the fusion of human soul and God by stating that at the summit of the ascent the soul still preserves its identity. These men, deeply involved in the theological debates of their own age, describe the deepest levels of inward experience where God is known in the inner recesses of the human soul, a most intimate presence that is also a transcendence. Their message addresses us so directly because it relates to a search for what is most essential to religion, its deepest, most inward dimension, and it emphasizes a real indwelling of ourselves in God and of God in us. The mysticism of the Rhineland is known in German as Wesensmystik, a mysticism of being or essence."