People From Lyon

"Great iniquities still exist in the world. Slavery has been holding her ground for the last eighty years in the United States, and was a stain upon the American Republic long before becoming the instrument of her ruin, or at least of a scandalous dissolution. In Sweden, civil liberty of conscience is still proscribed by laws of another age, and is only bought by exile or loss of property. Perfidy and violence are going hand in hand to destroy the nationality of Poland, and crush in her the last struggles of Catholic faith by the united force of schism and despotism. In fine, there is in the East of Europe an anti-Christian empire crumbling away visibly under the weight of its own corruption; whose protestations do not hinder indignant Europe from requiring an account of the thousands of victims butchered by Moslem fanaticism almost under our eyes, which empire is, nevertheless, protected by the mutual jealousy and distrust of western nations. All these facts are so many outrages upon justice, and so many insolent triumphs of might over right."

"Gothic art affords the greatest number and the best representations of animal forms. The great cathedrals, especially those of the Isle of France, where sculpture reached its highest point of excellence, are a sort of encyclopedia of the knowledge of the time."

"His administration was marked by piety, zeal, and providential prudence."

"During nearly twenty years of devoted work for the church he won the esteem of thousands of its best citizens."

"At first the Mangamuka natives seemed inclined to expel "the Papists" and destroy their crucifixes and images. Ultimately, however, they were brought to kindlier dispositions, more probably as the result of Bishop Pompallier's reasonings."

"We are happy that God has given us a common mission here in Algeria. And this mission is all the more important at a time when the pressure of globalisation comes up against reaffirmation of the specific identity of different civilisations, nations and cultures. We are convinced that by means of the "sacrament of encounter" we are called to operate together with our Muslim friends in their diversity, welcoming those among them who wish to have a better understanding of who we are."

"For the poorest of the poor the Church is Mother, the Church is the bearer of a message of peace and love. We hope the new Pontiff will continue with interreligious dialogue, to have special relationship with young people and to fight against secularisation. In Algeria Catholics live in a difficult context but we offer our contribution towards building harmony among the peoples and the religions of the world."

"The Islamic-Christian dialogue may appear overcome when Islamic movements in certain Muslim countries stir up armed groups and set up their followers against whoever does not share their ideas. The Church of Algeria is undergoing a trial of this kind. Fidelity to their vocation - in a difficult context - as priests, religious brothers, religious sisters and lay missionaries shows that this development of fundamentalist movements of Islam does not justify abandoning dialogue."

"He was a brother to me. He opened and paved a way and we will continue down the same path."

"We are also in the Arab world, which has its specific problems, and it seems important to us that the universal Church pause to take into account in a truly particular way the problems of Christians who are minorities in the Arab world, beginning with those in the Middle East."

"Invisibly our Eucharistic celebrations gather a still absent people, one of those who are searching for God in the righteousness of their hearts. For a particular Church, the way of living the Eucharist cannot be separated from its concrete history with the people she was given to by the Lord."

"We may talk, but dialogue is not about one truth versus another. Fraternity is not about understanding who is wrong and who is right, but about discovering together how each one of us, through our respective faiths, can become true brothers and sisters."

"God has filled our hearts with desires so vast and so magnificent that nothing in creation is capable of satisfying them. Thus it is that in the hope of finding some pleasure, we attach ourselves to created objects and that we have no sooner possessed and sampled that which we have so ardently desired than we turn to something else, hoping to find what we wanted. We are, then, through our own experience, constrained to admit that it is but useless for us to want to derive our happiness here below from transient things. If we hope to have any consolation in this world, it will only be by despising the things which are passing and which have no lasting value and in striving towards the noble and happy end for which God has created us. Do you want to be happy, my friends? Fix your eyes on Heaven; it is there that your hearts will find that which will satisfy them completely."

"Philibert de l'Orme 1518-77. If Lescot and Bullant were at least as much decorators as builders, Philibert de l'Orme was less an architect than an engineer; construction and not decoration was the important thing to him. The works he designed as an artist he usually executed as a builder."

"If the use of iron in building does not enable us to exceed these dimensions at a decidedly less cost, then indeed we are inferior to our ancestors. In fact the great builders of the Middle Ages, like those of the Renaissance, were eminently men of subtle, active, and inventive intellect. I say inventive intellect, for that is the ruling characteristic of the works bequeathed to us by those old builders. It is apparent in the structure of our mediaeval buildings, and only ceases to manifest itself when the material becomes inadequate. It is apparent in the attempts of the Renaissance; for apart from the superficial imitation of classic forms which the architects of the latter period affected, they did not adhere to this imitation in the construction of their buildings and in the methods they employed. Without reference to the buildings of that epoch, we may find the proof of this fact in the written works of several of those architects, such as Albert Dürer, Serlio, Philibert de l'Orme, etc. On every page of their writings we find some original idea, or new adaptation; and as in the case of their predecessors, their ingenuity is circumscribed only by the inadequacy of their materials."

"Have I not also done a great service in having brought into France the fashion of good building, done away with barbarous manners, and great gaping joints in masonry, shown to all how one should observe the measures of architecture, and made the best workmen of the day, as they admit themselves? Let people recollect how they built when I began Saint-Maur for my lord the Cardinal du Bellay ...Moreover, let it be recollected that all I have ever done has been found to be very good and to give great contentment to all."

"Mr. G. Rennie said, he believed that few, if any, examples of oblique bridges existed in England prior to those which had been mentioned, and the extreme obliquity of Mr. Storey's bridge rendered it very interesting; such bridges had long been constructed in Italy, and in France. Vasari mentioned an oblique bridge over the Mugnone near Florence, erected in 1530. In a curious old work intituled "L'Architecture des Voutes," [a treatise on stereotomy] par Derand, (folio, 1645) diagrams were given of the oblique, as well as of almost every other kind of arch. Philibert de L'Orme, and subsequent French architects, seemed also to have been fond of oblique arches. Nicholson, who was quoted by Mr. Buck as having first explained the method of constructing the oblique arch must, Mr. Rennie conceived, have seen Derand's work."

"French architects and engineers in the 16th, 17th, and 18th centuries occupied themselves a good deal with roofs with curved ribs, and two systems of constructing the rib were worked out. In the most modern of them, that invented by Colonel Emy, the ribs were constructed of a series of thicknesses of bent timber, one on the back of another, and held together by bolts. In the older system that of Philibert de l'Orme, the ribs were also built up, but the pieces composing them are placed side by side, and either form a polygon approaching a semicircle or are cut to bring them to a curve. thumb|Bourse de commerce (dome of the Paris Corn Market)In fact, the ribs are very much such as... used for the great dome of the Paris Corn Market. There is, however, a great difference between a dome—the strongest of all forms—and one permitting the introduction of as many rings of ties as may be desired; and a roof over an ordinary oblong space, where no such binding together is admissible, and where straight rafters may have to be used, which loads the rib at certain points only. In the latter case, a good many precautions have, generally speaking, to be taken to prevent the rib from being unequally loaded, and so either spreading or losing its shape in some other way. The rib made of unbent timber, side by side, on De l'Orme's plan, is admitted to be stronger than the one made of bent timbers laid one on the back of the other; but both have been largely used, and good examples of both may be met with..."

"Fortunate indeed is the man who has found wisdom and who is full of that discretion which is better than all the acquiring, trafficking, and possession of gold and silver. ... I dwell (so says Wisdom) in good and salutary counsel, and am present at learned and wise cogitations. Therefore must a man seek this Wisdom and, having found it, take care to hold it well, that in its time and place it may be of help to him. The ensuing representation will set before your eyes the treatise which I have propounded."

"On January 8, I570, Philibert Delorme died in his canon's house of Notre-Dame. He had played a considerable part in the life of his times, he had written an immense book, and he had designed some of the most notable buildings in France. In his own opinion he had simply re-established architecture in France. ...There are two woodcuts at the conclusion of Philibert Delorme's Livre d'architecture. One shows a figure without eyes and hands moving aimlessly across a Gothic landscape. Behind him stands a medieval castle with its moat and turrets, a cloudburst filling the sky above it. This is his concept of the Bad Architect. The other is a scene of classic architecture, fruitful vines, and playing fountains. The sky is serene, and in the ordered court stands the Good Architect, triple-eyed and double-handed, presenting a roll of plans to a willing workman. ...Could it have been a sketch for a self-portrait that Messer Philibert Delorme was setting before our eyes?"

"[T]he work by two sixteenth century masters of stereotomic architecture – the Spanish architect (1505-1575) and the French architect Philibert De l’Orme (1515-1570) – is paradigmatic. Their cut-stone vaults and domes are an expression of the quest for the formal identification and definition of construction elements in keeping with the technical know-how and aesthetic canons of stereotomy. A comparison of two of their works is particularly interesting: the dome of the chapel of Salvador at Úbeda by Andrés de Vandelvira, built between 1536 and 1542, and the dome of the chapel at Anet by Philibert De l’Orme, built between 1548 and 1553. ...[T]he figurative solution adopted by De l’Orme... On a technical level, the juxtaposition of the decorative and the construction pattern is not casual, but geometrically controlled in order to optimise the production of the s by reducing to a minimum the number of “panneaux” needed to cut them. The system of ribbing, conceived according to the logic of this production process, is commensurate with the “metre” used in the wall assemblage and is consequently segmented in strict relation to the shape and dimension of the curved surface of the voussoirs that define the intrados of the dome."

"[Catharine] Randall, Building Codes, argues that Philibert de l'Orme was, if not a Calvanist, someone with a 'strongly evangelical stance and perhaps Calvanist sympathies'... Such Calvinist sympathies, according to Randall, are detectable in his 'stylistic idiosyncracies', which compose 'the architectural vocabulary of the late Calvinist architects',... his use (like Calvin) of the biblical text as a 'textual template for his building activity in general',... in his creation of a Protestant architectural genealogy... etc. No direct evidence exists, however, to support claims that De l'Orme was anything but Catholic—he was, after all, a priest (diocese of Lyons) and later canon (Potié, Philibert De l'Orme, 23). As Andrew Spicer notes, in his review of Randall's book, 'much of [her] evidence would seem to be circumstantial, and there are problems in equating the terms "evangelical" with "crypto-" or "proto-" Calvinist (Catholic Historical Review, 89/1 (2003), 106). I do not propose to resolve this debate here..."

"The researches of Kepler and Desargues will serve to remind us that as the geometry of the Greeks was not capable of much further extension, mathematicians were now beginning to seek for new methods of investigation, and were extending the conceptions of geometry. The invention of analytical geometry and of the infinitesimal calculus temporarily diverted attention from pure geometry, but at the beginning of the last century there was a revival of interest in it, and since then it has been a favourite subject of study with many mathematicians."

"Desrgues commences [in the Brouillon project] with a statement of the doctrine of continuity as laid down by Kepler: thus the points at the opposite ends of a straight line are regarded as coincident, parallel lines are treated as meeting at a point at infinity, and parallel planes on a line at infinity, while a straight line may be considered as a circle whose center is at infinity. The theory of involution of six points, with its special cases, is laid down, and the projective property of pencils in involution is established. The theory of polar lines is expounded and its analogue in space suggested. A tangent is defined as the limiting case of a secant, and an asymptote as a tangent at infinity. Desargues shows that the lines which join four points in a plane determine three pairs of lines in involution on any transversal, and from any conic through the four points another pair of lines can be obtained which are in involution with any two of the former. He proves that the points of intersection of the diagonals and the two pairs of opposite sides of any quadrilateral inscribed in a conic are a conjugate triad with respect to the conic, and when one of the three points is at infinity its polar is a diameter; but he fails to explain the case in which the quadrilateral is a parallelogran, although he had formed the conception of a straight line which was wholly at infinity. The book, therefore, may be fairly said to contain the fundamental theorems on involution, homology, poles and polars, and perspective."

"Hardly less interesting than the new ideas of Descartes and Cavalieri are those of their contemporary Desargues... who made important researches in geometry. But for the still more brilliant geometrical achievements of Descartes, these might have led to the immediate development of projective geometry, the elements of which are contained in Desargues's work."

"... and more recently, Bernard Cache, have argued that Girard Desargues' mathematics provided a model for Leibniz's monad. ...Desargues was a founder of projective geometry, which offers a mathematical model for the intuitive notions of perspective and horizon by studying what remains invariable in projections. Outlining the concept of the "invariant," he gives his name to the "Desargues theorem," focusing on homological triangles. His disciple was the engraver, , author of a Treatise on Projections and Perspective (1665), who later taught linear perspective to stone cutters, carpenters, engravers, manufacturers of instruments and, less successfully, to painters. The perspective that Bosse teaches implicitly introduces the idea of infinity, in that he uses parallel lines with an infinitely extending vanishing point... Moreover, permeated by the knowledge of Desargues, Bosse develops a method for tracing shadows, which was inspired by his master."

"The discovery of a method for correct perspective is usually attributed to... Brunelleschi... The first published method appears in... On Painting [Della Pittura] by Alberti. ...Alberti's veil, used a piece of transparent cloth, stretched on a frame... viewing the scene with one eye... fixed... one could trace the scene directly onto the veil. ...The mathematical setting... is the family of lines ("light rays") through the point (the "eye"), together with the plane... (the "veil"). ...In this setting, the problems of perspective and were not very difficult, but the concepts were... a challenge to traditional geometric thought. Contrary to Euclid, one had... (i) Points at infinity ("vanishing points") where parallels met. (ii) Transformations that changed lengths and angles (projections). The first to construct a mathematical theory incorporating these ideas was Desargues,... although the idea of points at infinity had already been used by Kepler..."

"The famous geometer Desargues worked on the lines of Kepler and is now commonly credited with the authorship of some of the ideas of his predecessor. ...the oneness of opposite infinities followed simply and logically from a first principle of Desargues, that every two straight lines, including parallels, have or are to be regarded as having one common point and one only. A writer of his insight must have come to this conclusion, even if the paradox had not been held by Kepler, Briggs, and we know not how many others, before Desargues wrote. ...Desargues must have learned directly or indirectly from the work in which Kepler propounded his new theory of these points, first called by him the Foci (foyers), including the modern doctrine of real points at infinity."

"The book of Desargues (1639), Brouillon project d'une atteinte aux événemens des recontres du cône avec un plan (Schematic Sketch of What Happens When a Cone Meets a Plane), suffered an extreme case of delayed recognition, being completely lost for 200 years."

"The influence exerted by the lectures of Desargues on Descartes, Pascal and the French geometricians of the seventeenth century was considerable; but the subject of projective geometry soon fell into oblivion, chiefly because the analytical geometry of Descartes was so much more powerful as a method of proof or discovery."

"Nous démontrerons aussi la propriété suivante dont le premier inventeur est M. Desargues, Lyonnois, un des grands esprits de ce temps, et des plus versés aux mathématiques, et entre autres aux coniques, dont les écrits sur cette matière, quoiqu'en petit nombre, en ont donné un ample témoignage à ceux qui auront voulu en recevoir l'intelligence. Je veux bien avouer que je dois le peu que j'ai trouvé sur cette matière à ses écrits, et que j'ai tâché d'imiter, autant qu'il m'a été possible, sa méthode..."

"Girard Desargues... gave some courses of gratuitous lectures in Paris from 1626 to about 1630 which made a great impression upon his contemporaries. Both Descartes and Pascal had a high opionion of his work and abilities, and both made considerable use of the theorems he had enunciated."

"Kepler (and Desargues) regarded the two "ends" of the ["straight"] line as meeting at "infinity" so that the line has the structure of a circle. In fact, Kepler actually thought of a line as a circle with its center at infinity."

"One of the first important steps to be taken in modern times... was due to Desargues. In a work published in 1639 Desargues set forth the foundation of the theory of four harmonic points, not as done today but based on the fact that the product of the distances of two conjugate points from the center is constant. He also treated the theory of poles and polars, although not using these terms."

"In 1639, nine years after Kepler's death, there appeared in Paris a remarkably original but little-heeded treatise on the conic sections. ...The work was so generally neglected by other mathematicians that it was soon forgotten and all copies of the publication disappeared. ...in 1845 Chasles happened upon a manuscript copy... made by Desargues' pupil, ... and since that time the work has been regarded as one of the classics in the early development of synthetic projective geometry."

"In 1636 Desargues issued a work on perspective; but most of his researches were embodied in his Brouillon project on conics, published in 1639, a copy of which was discovered by Chasles in 1845."

"The beginning of the seventeenth century witnessed also a revival of . ...it remained for Girard Desargues... and for Pascal to leave the beaten track and cut out fresh paths. They introduced the important method of Perspective. All conics on a cone with circular base appear circular to an eye at the apex. Hence Desargues and Pascal conceived the treatment of the conic sections as projections of circles. Two important and beautiful theorems were given by Desargues: The one is on the "involution of the six points," in which a transversal meets a conic and an inscribed quadrangle; the other is that, if the vertices of two triangles, situated either in space or in a plane, lie on three lines meeting in a point, then their sides meet in three points lying on a line; and conversely. This last theorem has been employed in recent times by Brianchon, C. Sturm; Gergonne, and Poncelet. Poncelet made it the basis of his beautiful theory of homological figures."

"We owe to Desargues the theory of involution and of transversals; also the beautiful conception that the two extremities of a straight line may be considered as meeting at infinity, and that parallels differ from other pairs of lines only in having their points of intersection at infinity. He re-invented the and showed its application to the construction of gear teeth, a subject elaborated more fully later by La Hire."

"Desargues also gives the theory of polar lines. What is called "" in elementary works is as follows: If the vertices of two triangles, situated either in space or in a plane, lie on three lines meeting in a point, then their sides meet in three points lying on a line, and conversely. This theorem has been used since by Brianchon, Sturm, Gergonne, and others. Poncelet made it the basis of his beautiful theory of homological figures."

"Blaise Pascal... was one of the very few contemporaries who appreciated the worth of Desargues. He says in his Essais pour les coniques, "I wish to acknowledge that I owe the little that I have discovered on this subject to his writings.""

"He gives the theory of involution of six points, but his definition of "involution" is not quite the same as the modern definition, first found in Fermat, but really introduced into geometry by Chasles. On a line take the point A as origin (souche), take also the three pairs of points B and H, C and G, D and F; then, says Desargues, if AB \cdot AH = AC \cdot AG = AD \cdot AF, the six points are in "involution." If a point falls on the origin, then its partner must be at an infinite distance from the origin. If from any point P lines be drawn through the six points, these lines cut any transversal MN in six other points, which are also in involution; that is, involution is a projective relation."

"Pascal made grateful acknowlegement to Desargues for his skill in projective geometry."

"Pascal greatly admired Desargues' results... Pascal's and Desargues writings contained some of the fundamental ideas of modern synthetic geometry."

"He met Descartes while employed by Cardinal Richelieu at the , and they with others met regularly in Paris for the discussion of the new Copernican theory and other scientific problems."

"I freely confess that I never had taste for study or research either in physics or geometry except in so far as they could serve as a means of arriving at some sort of knowledge of the proximate causes... for the good and convenience of life, in maintaining health, in the practice of some art,... having observed that a good part of the arts is based on geometry, among others that cutting of stone in architecture, that of sundials, that of perspective in particular."

"Perceiving that the practitioners of these arts ["...among others, the cutting of stones in architecture, that of sun-dials, that of perspective in particular"] had to burden themselves with the laborious acquisition of many special facts in geometry, he sought to relieve them by developing more general methods and printing notes for distribution among his friends."

"Desargues the architect was doubtless influenced by what in his day was surrealism. In any event, he composed more like an artist than a geometer, inventing the most outrageous technical jargon in mathematics for the enlightenment of himself and the mystification of his disciples. Fortunately Desarguesian has long been a dead language."

"He who shall wish to disentangle this proposition will easily be able to compose a volume."

"Desargues contented himself with enunciating general principles remarking:—"He who shall wish to disentangle this proposition will easily be able to compose a volume.""

"An interesting theorem bearing his name and typical of projective geometry is as follows:—If two triangles ABC and A'B'C' are so related that lines joining corresponding vertices meet in a point O, then the intersections of corresponding sides will lie in a straight line A"B"C". It remained for Monge, the inventor of descriptive geometry... and others more than a century later to carry this development forward. Desargues's work was indeed practically lost until Poncelet in 1822 proclaimed him the Monge of his century."