Some Contemporaries of Descartes, Fermat, Pascal and Huygens

Bachet

Claude Gaspard Bachet de Méziriac was born at Bourg in 1581, and died in 1638. He wrote the Problèmes plaisants, of which the first edition was issued in 1612, a second and enlarged edition was brought out in 1624; this contains an interesting collection of arithmetical tricks and questions, many of which are quoted in my Mathematical Recreations and Essays. He also wrote Les éléments arithmétiques, which exists in manuscript; and a translation of the Arithmetic of Diophantus. Bachet was the earliest writer who discussed the solution of indeterminate equations by means of continued fractions.

Mersenne

Marin Mersenne, born in 1588 and died at Paris in 1648, was a Franciscan friar, who made it his business to be acquainted and correspond with the French mathematicians of that date and many of their foreign contemporaries. In 1634 he published a translation of Galileo's mechanics; in 1644 he issued his Cogita Physico-Mathematica, by which he is best known, containing an account of some experiments in physics; he also wrote a synopsis of mathematics, which was printed in 1664.

The preface to the Cogitata contains a statement (possibly due to Fermat) that, in order that may be prime, the only values of p, not greater than 257, which are possible are 1, 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257; the number 67 is probably a misprint for 61. With this correction the statement appears to be true, and it has been verified for all except twenty-one values of p, namely 71, 89, 101, 103, 107, 109, 127, 137, 139, 149, 157, 163, 167, 173, 181, 193, 199, 227, 229, 241, and 257. Of these values, Mersenne asserted that p = 127 and p = 257 make prime, and that the other nineteen values make a composite number. It has been asserted that the statement has been verified when p = 89 and 127, but these verifications rest on long numerical calculations made by single computators and not published; until these demonstrations have been confirmed we may say that twenty-one cases still await verification or require further investigation. The factors of when p = 89 are not known, the calculation merely showing that the number could not be prime. It is most likely that these results are particular cases of some general theorem on the subject which remains to be discovered.

The theory of perfect numbers depends directly on that of Mersenne's numbers. It is probable that all perfect numbers are included in the formula , where is a prime. Euclid proved that any number of this form is perfect. Euler shewed that the formula includes all even perfect numbers; and there is reason to believe - though a rigid demonstration is wanting - that an odd number cannot be perfect. If we assume that the last of these statements is true, then every perfect number is of the above form. Thus if p = 2, 3, 5, 7, 13, 17, 19, 31, 61, then, by Mersenne's rule, the corresponding values of are prime; they are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951; and the corresponding perfect numbers are 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, and 2658455991569831744654692615953842176.

Roberval

Gilles Personier (de) Roberval, born at Roberval in 1602 and died at Paris in 1675, described himself from the place of his birth as de Roberval, a seigniorial title to which he had no right. He discussed the nature of the tangents to curves, solved some of the easier questions connected with the cycloid, generalized Archimedes's theorems on the spiral, wrote on mechanics, and on the method of indivisibles, which he rendered more precise and logical. He was a professor in the university of Paris, and in correspondence with nearly all the leading mathematicians of his time.

Van Schooten

Frans van Schooten, to whom we owe an edition of Vieta's works, succeeded his father (who had taught mathematics to Huygens, Hudde, and Sluze) as professor at Leyden in 1646. He brought out in 1659 a Latin translation of Descartes's Géométrie, and in 1657 a collection of mathematical exercises in which he recommended the use of co-ordinates in space of three dimensions. He died in 1661.

Saint-Vincent

Grégoire de Saint-Vincent, a Jesuit, born at Bruges in 1584 and died at Ghent in 1667, discovered the expansion of log(1 + x) in ascending powers of x. Although a circle-squarer he is worthy of mention for the numerous theorems of interest which he discovered in his search after the impossible, and Montucla ingeniously remarks that ``no one ever squared the circle with so much ability or (except for his principal object) with so much success.'' He wrote two books on the subject, one published in 1647 and the other in 1668, which cover some two or three thousand closely printed pages; the fallacy in the quadrature was pointed out by Huygens. In the former work he used indivisibles. An earlier work entitled Theoremata Mathematica, published in 1624, contains a clear account of the method of exhaustions, which is applied to several quadratures, notably that of the hyperbola.

Torricelli

Evangelista Torricelli, born at Faenza on Oct. 15, 1608, and died at Florence in 1647, wrote on the quadrature of the cycloid and conics; the rectifications of the logarithmic spiral; the theory of the barometer; the value of gravity found by observing the motion of two weights connected by a string passing over a fixed pulley; the theory of projectiles; and the motion of fluids.

Hudde

Johann Hudde, burgomaster of Amsterdam, was born there in 1633, and died in the same town in 1704. He wrote two tracts in 1659: one was on the reduction of equations which have equal roots; in the other he stated what is equivalent to the proposition that if f(x,y) = 0 be the algebraical equation of a curve, then the subtangent is

but being ignorant of the notation of the calculus his enunciation is involved.

Frénicle

Bernard Frénicle de Bessy, born in Paris circ. 1605 and died in 1670, wrote numerous papers on combinations and on the theory of numbers, also on magic squares. It may be interesting to add that he challenged Huygens to solve the following system of equations in integers, x² + y² = z², x² = u² + v², x - y = u - v. A solution was given by M. Pépin in 1880.

De Laloubère

Antoine de Laloubère, a Jesuit, born in Languedoc in 1600 and died at Toulouse in 1664, is chiefly celebrated for an incorrect solution of Pascal's problems on the cycloid, which he gave in 1660, but he has a better claim to distinction in having been the first mathematician to study the properties of the helix.

N. Mercator

Nicholas Mercator (sometimes known as Kauffmann) was born in Holstein about 1620, but resided most of his life in England. He went to France in 1683, where he designed and constructed the fountains at Versailles, but the payment agreed on was refused unless he would turn Catholic; he died of vexation and poverty in Paris in 1687. He wrote a treatise on logarithms entitled Logarithmo-technica, published in 1668, and discovered the series

he proved this by writing the equation of a hyperbola in the form

to which Wallis's method of quadrature could be applied. The same series had been independently discovered by Saint-Vincent.

Wren

Sir Christopher Wren was born at Knoyle, Wiltshire, on October 20, 1632, and died in London on February 25, 1723. Wren's reputation as a mathematician has been overshadowed by his fame as an architect, but he was Savilian professor of astronomy at Oxford from 1661 to 1673, and for some time president of the Royal Society. Together with Wallis and Huygens he investigated the laws of collision of bodies; he also discovered the two systems of generating lines on the hyperboloid of one sheet, though it is probable that he confined his attention to a hyperboloid of revolution. Besides these he wrote papers on the resistance of fluids, and the motion of the pendulum. He was a friend of Newton and (like Huygens, Hooke, Halley, and others) had made attempts to shew that the force under which the planets move varies inversely as the square of the distance from the sun.

Wallis, Brouncker, Wren, and Boyle (the last-named being a chemist and physicist rather than a mathematician) were the leading philosophers who founded the Royal Society of London. The society arose from the self-styled ``indivisible college'' in London in 1645; most of its members moved to Oxford during the civil war, where Hooke, who was then an assistant in Boyle's laboratory, joined in their meetings; the society was formally constituted in London in 1660, and was incorporated on July 15, 1662. The French Academy was founded in 1666, and the Berlin Academy in 1700. The Accademia dei Lincei was founded in 1603, but was dissolved in 1630.

Hooke

Robert Hooke, born at Freshwater on July 18, 1635, and died in London on March 3, 1703, was educated at Westminster, and Christ Church, Oxford, and in 1665 became professor of geometry at Gresham College, a post which he occupied till his death. He is still known by the law which he discovered, that the tension exerted by a stretched string is (within certain limits) proportional to the extension, or, in other words, that the stress is proportional to the strain. He invented and discussed the conical pendulum, and was the first to state explicitly that the motions of the heavenly bodies were merely dynamical problems. He was as jealous as he was vain and irritable, and accused both Newton and Huygens of unfairly appropriating his results. Like Huygens, Wren, and Halley, he made efforts to find the law of force under which the planets move about the sun, and he believed the law to be that of the inverse square of the distance. He, like Huygens, discovered that the small oscillations of a coiled spiral spring were practically isochronous, and was thus led to recommend (possibly in 1658) the use of the balance spring in watches. He had a watch of this kind made in London in 1675; it was finished just three months later than a similar one made in Paris under the directions of Huygens.

Collins

John Collins, born near Oxford on March 5, 1625, and died in London on November 10, 1683, was a man of great natural ability, but of slight education. Being devoted to mathematics, he spent his spare time in correspondence with the leading mathematicians of the time, for whom he was always ready to do anything in his power, and he has been described - not inaptly - as the English Mersenne. To him we are indebted for much information on the details of the discoveries of the period.

Pell

Another mathematician who devoted a considerable part of his time to making known the discoveries of others, and to correspondence with leading mathematicians, was John Pell. Pell was born in Sussex on March 1, 1610, and died in London on December 10, 1685. He was educated at Trinity College, Cambridge; he occupied in succession the mathematical chairs at Amsterdam and Breda; he then entered the English diplomatic service; but finally settled in 1661 in London, where he spent the last twenty years of his life. His chief works were an edition, with considerable new matter, of the Algebra by Branker and Rhonius, London, 1668; and a table of square numbers, London, 1672.

Sluze

René François Walther de Sluze (Slusius), canon of Liége, born on July 7, 1622, and died on March 19, 1685, found for the subtangent of a curve f(x,y) = 0 an expression which is equivalent to

he wrote numerous tracts, and in particular discussed at some length spirals and points of inflexion.

Viviani

Vincenzo Viviani, a pupil of Galileo and Torricelli, born at Florence on April 5, 1622, and died there on September 22, 1703, brought out in 1659 a restoration of the lost book of Apollonius on conic sections, and in 1701 a restoration of the work of Aristaeus. He explained in 1677 how an angle could be trisected by the aid of the equilateral hyperbola or the conchoid. In 1692 he proposed the problem to construct four windows in a hemispherical vault so that the remainder of the surface can be accurately determined; a celebrated problem, of which analytical solutions were given by Wallis, Leibnitz, David Gregory, and James Bernoulli.

Tchirnhausen

Ehrenfried Walther von Tschirnhausen was born at Kislingswalde on April 10, 1631, and died at Dresden on October 11, 1708. In 1682 he worked out the theory of caustics by reflexion, or, as they were usually called, catacaustics, and shewed that they were rectifiable. This was the second case in which the envelope of a moving line was determined. He constructed burning mirrors of great power. The transformation by which he removed certain intermediate terms from a given algebraical equation is well known; it was published in the Acta Eruditorum for 1683.

De la Hire

Philippe De la Hire (or Lahire), born in Paris on March 18, 1640, and died there on April 21, 1719, wrote on graphical methods, 1673; on the conic sections, 1685; a treatise on epicycloids, 1694; one on roulettes, 1702; and, lastly, another on conchoids, 1708. His works on conic sections and epicycloids were founded on the teaching of Desargues, whose favourite pupil he was. He also translated the essay of Moschopulus on magic squares, and collected many of the theorems on them which were previously known; this was published in 1705.

Roemer

Ole Roemer, born at Aarhuus on September 25, 1644, and died at Copenhagen on September 19, 1710, was the first to measure the velocity of light; this was done in 1675 by means of the eclipses of Jupiter's satellites. He brought the transit and mural circle into common use, the altazimuth having been previously generally employed, and it was on his recommendation that astronomical observations of stars were subsequently made in general on the meridian. He was also the first to introduce micrometers and reading microscopes into an observatory. He also deduced from the properties of epicycloids the form of the teeth in toothed-wheels best fitted to secure a uniform motion.

Rolle

Michel Rolle, born at Ambert on April 21, 1652, and died in Paris on November 8, 1719, wrote an algebra in 1689, which contains the theorem on the position of the roots of an equation which is known by his name. He published in 1696 a treatise on the solutions of equations, whether determinate or indeterminate, and he produced several other minor works. He taught that the differential calculus, which, as we shall see later, had been introduced towards the close of the seventeenth century, was nothing but a collection of ingenious fallacies.

René Descartes (1596 - 1650)

We may consider Descartes as the first of the modern school of mathematics. René Descartes was born near Tours on March 31, 1596, and died at Stockholm on February 11, 1650; thus he was a contemporary of Galileo and Desargues. His father, who, as the name implies, was of good family, was accustomed to spend half the year at Rennes when the local parliament, in which he held a commission as councillor, was in session, and the rest of the time on his family estate of Les Cartes at La Haye. René, the second of a family of two sons and one daughter, was sent at the age of eight years to the Jesuit School at La Flêche, and of the admirable discipline and education there given he speaks most highly. On account of his delicate health he was permitted to lie in bed till late in the mornings; this was a custom which he always followed, and when he visited Pascal in 1647 he told him that the only way to do good work in mathematics and to preserve his health was never to allow anyone to make him get up in the morning before he felt inclined to do so; an opinion which I chronicle for the benefit of any schoolboy into whose hands this work may fall.

On leaving school in 1612 Descartes went to Paris to be introduced to the world of fashion. Here, through the medium of the Jesuits, he made the acquaintance of Mydorge, and renewed his schoolboy friendship with Mersenne, and together with them he devoted the two years of 1615 and 1616 to the study of mathematics. At that time a man of position usually entered either the army or the church; Descartes chose the former profession, and in 1617 joined the army of Prince Maurice of Orange, then at Breda. Walking through the streets there he saw a placard in Dutch which excited his curiosity, and stopping the first passer, asked him to translate it into either French or Latin. The stranger, who happened to be Isaac Beeckman, the head of the Dutch College at Dort, offered to do so if Descartes would answer it; the placard being, in fact, a challenge to all the world to solve a certain geometrical problem. Descartes worked it out within a few hours, and a warm friendship between him and Beeckman was the result. This unexpected test of his mathematical attainments made the uncongenial life of the army distasteful to him, but under family influence and tradition he remained a soldier, and was persuaded at the commencement of the Thirty Years' War to volunteer under Count de Bucquoy in the army of Bavaria. He continued all this time to occupy his leisure with mathematical studies, and was accustomed to date the first ideas of his new philosophy and of his analytical geometry from three dreams which he experienced on the night of November 10, 1619, at Neuberg, when campaigning on the Danube. He regarded this as the critical day of his life, and one which determined his whole future.

He resigned his commission in the spring of 1621, and spent the next five years in travel, during most of which time he continued to study pure mathematics. In 1626 we find him settled at Paris, ``a little well-built figure, modestly clad in green taffety, and only wearing sword and feather in token of his quality as a gentleman.'' During the first two years there he interested himself in general society, and spent his leisure in the construction of optical instruments; but these pursuits were merely the relaxations of one who failed to find in philosophy that theory of the universe which he was convinced finally awaited him.

In 1628 Cardinal de Berulle, the founder of the Oratorians, met Descartes, and was so much impressed by his conversation that he urged on him the duty of devoting his life to the examination of truth. Descartes agreed, and the better to secure himself from interruption moved to Holland, then at the height of his power. There for twenty years he lived, giving up all his time to philosophy and mathematics. Science, he says, may be compared to a tree; metaphysics is the root, physics is the trunk, and the three chief branches are mechanics, medicine, and morals, these forming the three applications of our knowledge, namely, to the external world, to the human body, and to the conduct of life.

He spend the first four years, 1629 to 1633, of his stay in Holland in writing Le Monde, which embodies an attempt to give a physical theory of the universe; but finding that its publication was likely to bring on him the hostility of the church, and having no desire to pose as a martyr, he abandoned it: the incomplete manuscript was published in 1664. He then devoted himself to composing a treatise on universal science; this was published at Leyden in 1637 under the title Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences, and was accompanied with three appendices (which possibly were not issued till 1638) entitled La Dioptrique, Les Météores, and La Géométrie; it is from the last of these that the invention of analytical geometry dates. In 1641 he published a work called Meditationes, in which he explained at some length his views on philosophy as sketched out in the Discours. In 1644 he issued the Principia Philosophiae, the greater part of which was devoted to physical science, especially the laws of motion and the theory of vortices. In 1647 he received a pension from the French court in honour of his discoveries. He went to Sweden on the invitation of the Queen in 1649, and died a few months later of inflammation of the lungs.

In appearance, Descartes was a small man with large head, projecting brow, prominent nose, and black hair coming down to his eyebrows. His voice was feeble. In disposition he was cold and selfish. Considering the range of his studies he was by no means widely read, and he despised both learning and art unless something tangible could be extracted therefrom. He never married, and left no descendants, though he had one illegitimate daughter, who died young.

As to his philosophical theories, it will be sufficient to say that he discussed the same problems which have been debated for the last two thousand years, and probably will be debated with equal zeal two thousand years hence. It is hardly necessary to say that the problems themselves are of importance and interest, but from the nature of the case no solution ever offered is capable either of rigid proof or of disproof; all that can be effected is to make one explanation more probable than another, and whenever a philosopher like Descartes believes that he has at last finally settled a question it has been possible for his successors to point out the fallacy in his assumptions. I have read somewhere that philosophy has always been chiefly engaged with the inter-relations of God, Nature, and Man. The earliest philosophers were Greeks who occupied themselves mainly with the relations between God and Nature, and dealt with Man separately. The Christian Church was so absorbed in the relation of God to Man as entirely to neglect Nature. Finally, modern philosophers concern themselves chiefly with the relations between Man and Nature. Whether this is a correct historical generalization of the views which have been successively prevalent I do not care to discuss here, but the statement as to the scope of modern philosophy marks the limitations of Descartes's writings.

Descartes's chief contributions to mathematics were his analytical geometry and his theory of vortices, and it is on his researches in connection with the former of these subjects that his mathematical reputation rests.

Analytical geometry does not consist merely (as is sometimes loosely said) in the application of algebra to geometry; that had been done by Archimedes and many others, and had become the usual method of procedure in the works of the mathematicians of the sixteenth century. The great advance made by Descartes was that he saw that a point in a plane could be completely determined if its distances, say x and y, from two fixed lines drawn at right angles in the plane were given, with the convention familiar to us as to the interpretation of positive and negative values; and that though an equation f(x,y) = 0 was indeterminate and could be satisfied by an infinite number of values of x and y, yet these values of x and y determined the co-ordinates of a number of points which form a curve, of which the equation f(x,y) = 0 expresses some geometrical property, that is, a property true of the curve at every point on it. Descartes asserted that a point in space could be similarly determined by three co-ordinates, but he confined his attention to plane curves.

It was at once seen that in order to investigate the properties of a curve it was sufficient to select, as a definition, any characteristic geometrical property, and to express it by means of an equation between the (current) co-ordinates of any point on the curve, that is, to translate the definition into the language of analytical geometry. The equation so obtained contains implicitly every property of the curve, and any particular property can be deduced from it by ordinary algebra without troubling about the geometry of the figure. This may have been dimly recognized or foreshadowed by earlier writers, but Descartes went further and pointed out the very important facts that two or more curves can be referred to one and the same system of co-ordinates, and that the points in which two curves intersect can be determined by finding the roots common to their two equations. I need not go further into details, for nearly everyone to whom the above is intelligible will have read analytical geometry, and is able to appreciate the value of its invention.

Descartes's Géométrie is divided into three books: the first two of these treat of analytical geometry, and the third includes an analysis of the algebra then current. It is somewhat difficult to follow the reasoning, but the obscurity was intentional. ``Je n'ai rien omis.'' says he, ``qu'à dessein ... j'avois prévu que certaines gens qui se vantent de sçavoir tout n'auroient par manqué de dire que je n'avois rien écrit qu'ils n'eussent sçu auparavant, si je me fusse rendu assez intelligible pour eux.''

The first book commences with an explanation of the principles of analytical geometry, and contains a discussion of a certain problem which had been propounded by Pappus in the seventh book of his and of which some particular cases had been considered by Euclid and Apollonius. The general theorem had baffled previous geometricians, and it was in the attempt to solve it that Descartes was led to the invention of analytical geometry. The full enunciation of the problem is rather involved, but the most important case is to find the locus of a point such that the product of the perpendiculars on m given straight lines shall be in a constant ratio to the product of the perpendiculars on n other given straight lines. The ancients had solved this geometrically for the case m = 1, n = 1, and the case m = 1, n = 2. Pappus had further stated that, if m = n = 2, the locus is a conic, but he gave no proof; Descartes also failed to prove this by pure geometry, but he shewed that the curve is represented by an equation of the second degree, that is, a conic; subsequently Newton gave an elegant solution of the problem by pure geometry.

In the second book Descartes divides curves into two classes, namely, geometrical and mechanical curves. He defines geometrical curves as those which can be generated by the intersection of two lines each moving parallel to one co-ordinate axis with ``commensurable'' velocities; by which terms he means that dy/dx is an algebraical function, as, for example, is the case in the ellipse and the cissoid. He calls a curve mechanical when the ratio of the velocities of these lines is ``incommensurable''; by which term he means that dy/dx is a trancendental function, as, for example, is the case in the cycloid and the quadratrix. Descartes confined his discussion to geometrical curves, and did not treat of the theory of mechanical curves. The classification into algebraical and transcendental curves now usual is due to Newton.

Descartes also paid particular attention to the theory of the tangents to curves - as perhaps might be inferred from his system of classification just alluded to. The then current definition of a tangent at a point was a straight line through the point such that between it and the curve no other straight line could be drawn, that is, the straight line of closest contact. Descartes proposed to substitute for this a statement equivalent to the assertion that the tangent is the limiting position of the secant; Fermat, and at a later date Maclaurin and Lagrange, adopted this definition. Barrow, followed by Newton and Leibnitz, considered a curve as the limit of an inscribed polygon when the sides become indefinitely small, and stated that the side of the polygon when produced became in the limit a tangent to the curve. Roberval, on the other hand, defined a tangent at a point as the direction of motion at that instant of a point which was describing the curve. The results are the same whichever definition is selected, but the controversy as to which definition was the correct one was none the less lively. In his letters Descartes illustrated his theory by giving the general rule for drawing tangents and normals to a roulette.

The method used by Descartes to find the tangent or normal at any point of a given curve was substantially as follows. He determined the centre and radius of a circle which should cut the curve in two consecutive points there. The tangent to the circle at that point will be the required tangent to the curve. In modern text-books it is usual to express the condition that two of the points in which a straight line (such as y = mx + c) cuts the curve shall coincide with the given point: this enables us to determine m and c, and thus the equation of the tangent there is determined. Descartes, however, did not venture to do this, but selecting a circle as the simplest curve and one to which he knew how to draw a tangent, he so fixed his circle as to make it touch the given curve at the point in question, and thus reduced the problem to drawing a tangent to a circle. I should note in passing that he only applied this method to curves which are symmetrical about an axis, and he took the centre of the circle on the axis.

The obscure style deliberately adopted by Descartes diminished the circulation and immediate appreciation of these books; but a Latin translation of them, with explanatory notes, was prepared by F. de Beaune, and an edition of this, with a commentary by F. van Schooten, issued in 1659, was widely read.

The third book of the Géométrie contains an analysis of the algebra then current, and it has affected the language of the subject by fixing the custom of employing the letters at the beginning of the alphabet to denote known quantities, and those at the end of the alphabet to denote unknown quantities. [On the origin of the custom of using x to represent an unknown example, see a note by G. Eneström in the Bibliotheca Mathematica, 1885, p. 43.] Descartes further introduced the system of indices now in use; very likely it was original on his part, but I would here remind the reader that the suggestion had been made by previous writers, though it had not been generally adopted. It is doubtful whether or not Descartes recognized that his letters might represent any quantities, positive or negative, and that it was sufficient to prove a proposition for one general case. He was the earliest writer to realise the advantage to be obtained by taking all the terms of an equation to one side of it, though Stifel and Harriot had sometimes employed that form by choice. He realised the meaning of negative quantities and used them freely. In this book he made use of the rule for finding the limit to the number of positive and of negative roots of an algebraical equation, which is still known by his name; and introduced the method of indeterminate coefficients for the solution of equations. He believed that he had given a method by which algebraical equations of any order could be solved, but in this he was mistaken. It may also be mentioned that he enunciated the theorem, commonly attributed to Euler, on the relation between the numbers of faces, edges and angles of a polyhedron: this is in one of the papers published by Careil.

Of the two other appendices to the Discours one was devoted to optics. The chief interest of this consists in the statement given of the law of refraction. This appears to have been taken from Snell's work, though, unfortunately, it is enunciated in a way which might lead a reader to suppose that it is due to the researches of Descartes. Descartes would seem to have repeated Snell's experiments when in Paris in 1626 or 1627, and it is possible that he subsequently forgot how much he owed to the earlier investigations of Snell. A large part of the optics is devoted to determining the best shape for the lenses of a telescope, but the mechanical difficulties in grinding a surface of glass to a required form are so great as to render these investigations of little practical use. Descartes seems to have been doubtful whether to regard the rays of light as proceeding from the eye and so to speak touching the object, as the Greeks had done, or as proceeding from the object, and so affecting the eye; but, since he considered the velocity of light to be infinite, he did not deem the point particularly important.

The other appendix, on meteors, contains an explanation of numerous atmospheric phenomena, including the rainbow; the explanation of the latter is necessarily incomplete, since Descartes was unacquainted with the fact that the refractive index of a substance is different for lights of different colours.

Descartes's physical theory of the universe, embodying most of the results contained in his earlier and unpublished Le Monde, is given in his Principia, 1644, and rests on a metaphysical basis. He commences with a discussion on motion; and then lays down ten laws of nature, of which the first two are almost identical with the first two laws of motion as given by Newton; the remaining eight laws are inaccurate. He next proceeds to discuss the nature of matter which he regards as uniform in kind though there are three forms of it. He assumes that the matter of the universe must be in motion, and that the motion must result in a number of vortices. He states that the sun is the centre of an immense whirlpool of this matter, in which the planets float and are swept round like straws in a whirlpool of water. Each planet is supposed to be the centre of a secondary whirlpool by which its satellites are carried: these secondary whirlpools are supposed to produce variations of density in the surrounding medium which constitute the primary whirlpool, and so cause the planets to move in ellipses and not in circles. All these assumptions are arbitrary and unsupported by any investigation. It is not difficult to prove that on his hypothesis the sun would be in the centre of these ellipses, and not at a focus (as Kepler had shewn was the case), and that the weight of a body at every place on the surface of the earth except the equator would act in a direction which was not vertical; but it will be sufficient here to say that Newton in the second book of his Principia, 1687, considered the theory in detail, and shewed that its consequences are not only inconsistent with each of Kepler's laws and with the fundamental laws of mechanics, but are also at variance with the laws of nature assumed by Descartes. Still, in spite of its crudeness and its inherent defects, the theory of vortices marks a fresh era in astronomy, for it was an attempt to explain the phenomena of the whole universe by the same mechanical laws which experiment shews to be true on the earth.

The Carter family: pioneer country music singers

Nestled in the shadow of Clinch Mountain, in the crossroads hamlet of Maces Spring, Virginia, sits one of the most rustic concert theaters in the world. It is made of rough-cut lumber with sides that swing open on hinges to admit summer breezes. In winter, the entire building is heated by wood and coal-fed iron stoves. The seating is a medley of old chairs taken from abandoned theaters, and discarded church pews. Carpeting consists of remnants and sample squares, in an infinite variety of colors and designs. It looks like the entire building was thrown together from a jumble of second thoughts. Beside the theater is a tiny white clapboard building surrounded by spreading trees. Inside are stacks of ancient 78 rpm phonograph records, thousands of faded pictures and dozens of dogeared scrapbooks. Clothing is sealed behind glass wall cases -- the expensive dress worn by June Carter Cash on a visit to Jimmy Carter’s White House and a dress suit worn by her husband, Johnny, on the same night. In stark contrast is a pair of ragged britches with broken suspenders that are attached to the waistband with a rusty nail. The little museum was once a general store and it’s proprietor was one of the true legends of country music -- A.P. Carter. He opened his business in the 1940s simply because he wanted something to do. A.P. was also the threadbare owner of the pants with the broken suspenders. The theater and museum, now called the Carter Fold, are operated by A.P.’s daughter and son, Jeanette and Joe. Although Joe designed and built the theater, Jeanette is the prime mover behind the Fold. She wanted to build a living memorial to her father, her mother Sara, and her Aunt Maybelle -- members of the first modern country singing group. The Carter Family was not the first country music group and certainly not the first to make records. But the Carter Family, according to some scholars, heralded the beginning of modern country music -- a major break from the string bands that had recorded up to that time. Ralph Peer (first working for Okeh Records, then for Victor) had been recording so-called “hillbilly groups” for five years, mostly around Atlanta. But in 1927, he decided to take his recording equipment to Bristol, Tennessee on a talent search. To flush local singers out of the hills he advertised in local newspapers that he would audition all comers. About three dozen singers and groups answered the call, including the Carters and another country legend-to-be, Jimmie Rodgers. The Carters had already been singing in local churches, schoolhouses and auditoriums for years. On a hot July day the Carter Family recorded six of their songs for Peer including “The Wandering Boy” and “Single Girl, Married Girl”. The records sold well and Victor offered the Carters a long term contract. Over the next 15 years. the Carters cut over 300 sides for nearly every major record companyin the business. Sara, A.P.’s wife, sang lead and played the autoharp. Maybelle, a tiny woman who had married A.P.’s brother Ezra, sang harmony and played a jazz-style guitar that was almost as big as she was. A.P. sang bass. A.P. constantly searched out new material for the group. He would disappear for days, roaming the Virginia mountains, seeking new songs. Then he would arrange each new song to his family’s style and copyright it -- even the folk songs. This is why his name appears on songs that he could not have possibly written -- “Wildwood Flower” or “Wreck of the Old ‘97”. But he did pen new songs of his own like “My Clinch Mountain Home” and “The Cyclone At Rye Cove”. Throughout their recording career, the Carter Family never changed their musical style. Rising stars like Roy Acuff and Gene Autry modernized Carter material but the Carter Family remained the same, thereby losing ground to new performers. The first “modern” country singers were standing stone still while the rest of the world passed them by. On top of all this, A.P. and Sara separated in 1932 (they divorced seven years later) and Maybelle and her husband had moved to Washington, D.C. The trio seldom saw each other except at recording sessions. The Carter Family was definitely on the downswing by the late 30s. The group’s popularity was temporarily revitalized when they signed a contract with powerful radio station XERF in Del Rio, Texas. The increased exposure gave the Carters a brief surge in popularity. The group continued to record until 1943 when Sara decided to retire and move to California with her new husband. Maybelle, in the meantime, joined her three daughters and began making records on her own. A.P., who could not hold his own as a solo act, retired to Maces Spring and opened his general store. In 1952, A.P. and Sara, along with their grown children Jeanette and Joe, reformed the Carter Family to record about 100 songs for Acme Records of Kentucky, but the project failed. The act broke up for a second time in 1956. When A.P. Carter died in 1960, several record companies including Columbia and RCA Victor reissued LP collections of the original Carter Family sides. New interest in the group surfaced and scholars began looking seriously at their contribution to country music. In the meantime, a new generation of warblers -- Bob Dylan, Joan Baez, and Doc Watson among them -- began issuing their own versions of Carter Family songs. Now on Saturday nights, hundreds of fans crowd the Carter Fold in Maces Spring to celebrate not only A.P., Sara and Maybelle, but traditional performances of country music as well. Both Joe and Jeanette participate in the concerts, and Carter memorabilia and records are sold in front of the stage each night. The Carter Family made comparatively little money during their career because A.P. insisted on playing smaller dates. And, after 1932, the family was hardly together except for recording sessions. Even though most of their records sold well, royalties were low and the records seldom made them much money. When A.P. returned to Virginia for the last time, there was barely enough money left to pay the bills and he was forced to live with Jeanette and her husband. Perhaps, if he had been willing to change a bit more with the times, he would have been able to afford new pair of suspenders in his old age instead of having to hold up his britches with a rusty nail.

Biography of Abba

ABBA was one of the most popular groups in the world throughout the 70’s and early 80’s. Many of their hits reached the top of the charts everywhere from their native Sweden to here in the U.S. All of the members had previously been involved in the show business, but only in Sweden. Agnetha was a solo singer, and Benny and Bjorn were both in groups of their own. As these groups disintegrated at the end of the 60’s, ABBA slowly came together. Surprisingly, the first time they recorded as a group, it was disliked by the members and was shelved away for awhile. In the early 70’s, they decided to try it again. The first single that the world knew about was “Waterloo” – the one that won them the Eurovision contest. A year before this, they had entered and won third – not good enough. After “Waterloo,” though, their career took off. They topped charts with the song all over the world. More singles followed this, such as “Ring Ring” and “SOS.” “SOS” was their second UK Top Ten hit, proving that they weren’t one-hit wonders after “Waterloo.” This cemented their career. The first album, hesitantly titled “Bjorn and Benny, Agnetha and Ani-Frid” was released in 1972. It was an awkward album, released prior to their success at the Eurovision contest. Also, they needed a new name. Their manager suggested ABBA, and after a “Name the Group” contest came up with the same thing, it was sealed. The second album was “Waterloo,” titled after their famous hit. It was released in 1973. This was followed by “ABBA” in the spring of 1975. By this time, of course, they were on tour. “Mamma Mia” came in the summer of 1975, reaching number one in the UK. Along with “I Do I Do I Do I Do I Do,” this song conquered Australia for the first time, spreading their popularity even further. By this time, they were famous all over the world. ABBA, as most people know, is made up of two couples – Agnetha and Bjorn, and Frida (as Ani-Frid is called) and Benny. Agnetha and Bjorn married in July of 1971. Their first child, Linda, was born in 1973. Their second child, Christian, was born in late 1977. Frida and Benny didn’t marry until much later (they’d been a couple for nine years by this time) in 1978, on the same day that the single “Summer Night City” reached number one in Sweden – the last of their songs to do so. Sadly, both couples later divorced – Agnetha and Bjorn in 1979, Frida and Benny in 1981. By the end of the 70’s and the beginning of the 80’s, they were experiencing many of their “lasts.” Some singles, such as “The Day Before You Came” were met with confusion, reaching number one in some countries, but barely making the charts in others. “The Visitors” was their last album, released in 1981. 1982 was their last year together. After that, the members felt the need to do something different for awhile. Their record of nine number one hits in the UK was a record topped only by The Beatles , Elvis Presley, and Cliff Richard . They were truly a famous group the world over – this was proved when the ABBA Gold albums were released in the early 90’s, topping charts in several countries. Since then, the group has been asked to do several revival tours (the latest in 2000, for around a billion dollars), but all have been refused.

Biography of Marian Anderson: singer

Marian Anderson was born on February 27, 1897 in South Philadelphia, Pennsylvania. Beginning at the age of six, she sang in the Union Baptist Church Choir. Her voice was classified as contralto, she could sing the high soprano notes and the low baritone notes. Marian's father died when she was a child and her mother worked as a cleaning woman and laundress to support the family. Her mother's religious faith and strength were lasting influences throughout Marian's life. The members of the Union Baptist Church gave a benefit concert to raise money for Marian to take Private singing lessons .The advertisements for the concert had a picture of Marian and the words, "Come and hear the baby contralto, ten years old." When she was nineteen, she began studying with Giuseppe Boghetti. In 1925, he helped her enter a contest in which she competed with 300 singers for the Lewisohn Stadium Concert Award. The prize was an opportunity to perform with the New York Philharmonic Orchestra. She won the contest and sang with the Orchestra on August 26, 1925. After performing with the Orchestra, she received a Rosenwald Foundation Fellowship and had the opportunity to go to England and Germany. In Germany she studied "Leider," German,songs, which became part of her repertoire. She gave concerts throughout Europe and received rave reviews and accolades for all 116 of her performances. In 1939, she planned to give a concert in the Constitution Hall in Washington, D.C. The Hall was owned by the Daughters of the American Revolution (DAR). The DAR refused to let her perform because she was African American. Franklin D. Roosevelt was president at this time and his wife, Eleanor Roosevelt, was outraged at the prejudice shown by the DAR and resigned her membership in the organization. Mrs. Roosevelt helped arrange for Anderson to give a concert outdoors at the Lincoln Memorial on Easter Sunday. Anderson performed in front of the statue of Abraham Lincoln for an audience of 75,000 people. She began the concert by singing "America." This was one of the most famous concerts given in the United States. This event helped open the doors of opportunity for other African Americans. From this time on, Anderson refused to sing at any place that was segregated. In 1943, a mural was unveiled on the wall of the Department of the Interior building depicting the concert. In her autobiography, "My Lord, What a Morning," she said: "There are many persons ready to do what is right because in their hearts they know it is right. But they hesitate, waiting for the other fellow to make the first move--and he, in turn, waits for you. The minute a person whose word means a great deal dares to take the open-hearted and courageous way, many others follow. Not everyone can be turned aside from meanness and hatred, but the great majority of Americans is heading in that direction. I have a great belief in the future of my people and my country." In 1941, Anderson received the Bok Award from the city of Philadelphia, given to the citizen of which it is the most proud. She was the first African American to receive the award. The $10,000 award was used to establish the Marian Anderson Scholarship Fund for music students of all races. In 1943, Anderson married Orpheus Fisher, an architect, who designed their home in Danbury, Connecticut named "Marrianna Farm." On January 7, 1955, Anderson performed with the Metropolitan Opera Company in New York as Ulrica, the Gypsy fortune-teller, in Verdi's opera "The Masked Ball." With this appearance, she became the first African-American to sing an important role at the Metropolitan Opera as a regular company member. In 1956, Anderson made a farewell tour throughout America and Europe. In 1957, she toured twelve Asian nations on behalf of the U.S. State Department. In 1958, she was named to the U.S. delegation to the United Nations. In 1986, Anderson received the National Medal of Arts. She received the Presidential Medal of Freedom in 1963. In 1991, she appeared at the dedication of St. Christopher's Hospital for Children's pediatric sickle-cell anemia clinic and research center, which is named in her honor. Marian Anderson died in 1993 at the age of ninety-six. During her professional singing career she was considered the world's greatest contralto.

Wolfgang Amadeus Mozart

Wolfgang Amadeus Mozart (baptized as Johannes Chrysostomus Wolfgangus Theophilus Mozart; January 27, 1756 – December 5, 1791) was a prolific and highly influential composer of Classical music. His enormous output of more than six hundred compositions includes works that are widely acknowledged as pinnacles of symphonic, chamber, piano, operatic, and choral music. Mozart is among the most enduringly popular of European composers, and many of his works are part of the standard concert repertoire.

Life - Family and early years

Mozart was born to Leopold and Anna Maria Pertl Mozart, in the front room of nine Getreidegasse in Salzburg, the capital of the sovereign Archbishopric of Salzburg, in what is now Austria, then part of the Holy Roman Empire. His only sibling who survived beyond infancy was an older sister: Maria Anna, nicknamed Nannerl. Mozart was baptized the day after his birth at St. Rupert's Cathedral.

The baptismal record gives his name in Latinized form as Joannes Chrysostomus Wolfgangus Theophilus Mozart. Of these names, the first two refer to John Chrysostom, one of the Church Fathers, and they were names not employed in everyday life, while the fourth, meaning "beloved of God", was variously translated in Mozart's lifetime as Amadeus (Latin), Gottlieb (German), and Amadé (French). Mozart's father Leopold announced the birth of his son in a letter to the publisher Johann Jakob Lotter with the words "...the boy is called Joannes Chrysostomus, Wolfgang, Gottlieb". Mozart himself preferred the third name, and he also took a fancy to "Amadeus" over the years.

Mozart's father Leopold (1719–1787) was one of Europe's leading musical teachers. His influential textbook Versuch einer gründlichen Violinschule, was published in 1756, the year of Mozart's birth (English, as "A Treatise on the Fundamental Principles of Violin Playing", transl. E.Knocker; Oxford-New York, 1948). He was deputy kapellmeister to the court orchestra of the Archbishop of Salzburg, and a prolific and successful composer of instrumental music. Leopold gave up composing when his son's outstanding musical talents became evident. They first came to light when Wolfgang was about three years old, and Leopold, proud of Wolfgang's achievements, gave him intensive musical training, including instruction in clavier, violin, and organ. Leopold was Wolfgang's only teacher in his earliest years. A note by Leopold in Nannerl's music book – the Nannerl Notenbuch – records that little Wolfgang had learned several of the pieces at the age of four. Mozart's first compositions, a small Andante (K. 1a) and Allegro (K. 1b), were written in 1761, when he was five years old.

The years of travel

During his formative years, Mozart made several European journeys, beginning with an exhibition in 1762 at the Court of the Elector of Bavaria in Munich, then in the same year at the Imperial Court in Vienna and Prague. A long concert tour spanning three and a half years followed, taking him with his father to the courts of Munich, Mannheim, Paris, London (where Wolfgang Amadeus played with the famous Italian cellist Giovanni Battista Cirri), The Hague, again to Paris, and back home via Zürich, Donaueschingen, and Munich. During this trip Mozart met a great number of musicians and acquainted himself with the works of other great composers. A particularly important influence was Johann Christian Bach, who befriended Mozart in London in 1764–65. Bach's work is often taken to be an inspiration for Mozart's music. They again went to Vienna in late 1767 and remained there until December 1768. On this trip Mozart contracted smallpox, and his healing was considered by Leopold as a proof of God's intentions concerning the child.

After one year in Salzburg, three trips to Italy followed: from December 1769 to March 1771, from August to December 1771, and from October 1772 to March 1773. Mozart was commissioned to compose three operas: "Mitridate Rè di Ponto" (1770), "Ascanio in Alba" (1771), and "Lucio Silla" (1772), all three of which were performed in Milan. During the first of these trips, Mozart met Andrea Luchesi in Venice and G.B. Martini in Bologna, and was accepted as a member of the famous Accademia Filarmonica. A highlight of the Italian journey, now an almost legendary tale, occurred when he heard Gregorio Allegri's Miserere once in performance in the Sistine Chapel then wrote it out in its entirety from memory, only returning to correct minor errors; thus producing the first illegal copy of this closely-guarded property of the Vatican.

On September 23, 1777, accompanied by his mother, Mozart began a tour of Europe that included Munich, Mannheim, and Paris. In Mannheim he became acquainted with members of the Mannheim orchestra, the best in Europe at the time. He fell in love with Aloysia Weber, who later broke up the relationship with him. He was to marry her sister Constanze some four years later in Vienna. During his unsuccessful visit to Paris, his mother died (1778).

Mozart in Vienna

In 1780, Idomeneo, widely regarded as Mozart's first great opera, premiered in Munich. The following year, he visited Vienna in the company of his employer, the harsh Prince-Archbishop Colloredo. When they returned to Salzburg, Mozart, who was then Konzertmeister, became increasingly rebellious, not wanting to follow the whims of the archbishop relating to musical affairs, and expressing these views, soon fell out of favor with him. According to Mozart's own testimony, he was dismissed – literally – "with a kick in the arse".[3] Mozart chose to settle and develop his own freelance career in Vienna after its aristocracy began to take an interest in him.

On August 4, 1782, against his father's wishes, he married Constanze Weber (1763–1842; her name is also spelled "Costanze"); her father Fridolin was a half-brother of Carl Maria von Weber's father Franz Anton Weber. Although they had six children, only two survived infancy. Neither of these two, Karl Thomas (1784–1858) and Franz Xaver Wolfgang (1791–1844); later a minor composer himself), married or had children who reached adulthood. Karl did father a daughter, Constanza, who died in 1833.

The year 1782 was an auspicious one for Mozart's career: his opera Die Entführung aus dem Serail ("The Abduction from the Seraglio") was a great success and he began a series of concerts at which he premiered his own piano concertos as director of the ensemble and soloist.

During 1782–83, Mozart became closely acquainted with the work of J.S. Bach and G.F. Handel as a result of the influence of Baron Gottfried van Swieten, who owned many manuscripts of works by the Baroque masters. Mozart's study of these works led first to a number of works imitating Baroque style and later had a powerful influence on his own personal musical language, for example the fugal passages in Die Zauberflöte ("The Magic Flute") and in the Symphony No. 41.

In 1783, Wolfgang and Constanze visited Leopold in Salzburg, but the visit was not a success, as his father did not open his heart to Constanze. However, the visit sparked the composition of one of Mozart's great liturgical pieces, the Mass in C Minor, which, though not completed, was premiered in Salzburg, and is now one of his best-known works. Wolfgang featured Constanze as the lead female solo voice at the premiere of the work, hoping to endear her to his father's affection.

In his early Vienna years, Mozart met Joseph Haydn and the two composers became friends. When Haydn visited Vienna, they sometimes played in an impromptu string quartet. Mozart's six quartets dedicated to Haydn date from 1782–85, and are often judged to be his response to Haydn's Opus 33 set from 1781. Haydn was soon in awe of Mozart, and when he first heard the last three of Mozart's series he told Leopold, "Before God and as an honest man I tell you that your son is the greatest composer known to me either in person or by name: He has taste, and, furthermore, the most profound knowledge of composition.

During the years 1782–1785, Mozart put on a series of concerts in which he appeared as soloist in his piano concertos, widely considered among his greatest works. These concerts were financially successful. After 1785 Mozart performed far less and wrote only a few concertos. Maynard Solomon conjectures that he may have suffered from hand injuries [citation needed]; another possibility is that the fickle public ceased to attend the concerts in the same numbers.

Mozart was influenced by the ideas of the eighteenth century European Enlightenment as an adult, and became a Freemason (1784). His lodge was a specifically Catholic, rather than a deistic one, and he worked fervently and successfully to convert his father before the latter's death in 1787. Die Zauberflöte, his second last opera, includes Masonic themes and allegory. He was in the same Masonic Lodge as Haydn.

Mozart's life was occasionally fraught with financial difficulty. Though the extent of this difficulty has often been romanticized and exaggerated, he nonetheless did resort to borrowing money from close friends, some debts remaining unpaid even to his death. During the years 1784-1787 he lived in a lavish, seven-room apartment, which may be visited today at Domgasse 5, behind St Stephen's Cathedral; it was here, in 1786, that Mozart composed the opera Le nozze di Figaro.

Mozart and Prague

Mozart had a special relationship with the city of Prague and its people. The audience there celebrated the Figaro with the much-deserved reverence he was missing in his hometown Vienna. His quotation "Meine Prager verstehen mich" (My Praguers understand me) became very famous in the Bohemian lands. Many tourists follow his tracks in Prague and visit the Mozart Museum of the Villa Bertramka where they can enjoy a chamber concert. In the later years of his life, Prague provided Mozart with many financial resources from commissions [citation needed]. In Prague, Don Giovanni premiered on October 29, 1787 at the Theatre of the Estates. Mozart wrote La clemenza di Tito for the festivities accompanying Leopold II's coronation in November 1790; Mozart obtained this commission after Antonio Salieri had allegedly rejected it.

Final illness and death

Mozart's final illness and death are difficult topics for scholars, obscured by romantic legends and replete with conflicting theories. Scholars disagree about the course of decline in Mozart's health – particularly at what point (or if at all) Mozart became aware of his impending death and whether this awareness influenced his final works. The romantic view holds that Mozart declined gradually and that his outlook and compositions paralleled this decline. In opposition to this, some present-day scholars point out correspondence from Mozart's final year indicating that he was in good cheer, as well as evidence that Mozart's death was sudden and a shock to his family and friends. Mozart's attributed last words: "The taste of death is upon my lips...I feel something, that is not of this earth". The actual cause of Mozart's death is also a matter of conjecture. His death record listed "hitziges Frieselfieber" ("severe miliary fever," referring to a rash that looks like millet-seeds), a description that does not suffice to identify the cause as it would be diagnosed in modern medicine. Dozens of theories have been proposed, including trichinosis, mercury poisoning, and rheumatic fever. The practice, common at that time, of bleeding medical patients is also cited as a contributing cause.

Mozart died around 1 a.m. on December 5, 1791 in Vienna. Some days earlier, with the onset of his illness, he had largely ceased work on his final composition, the Requiem. Popular legend has it that Mozart was thinking of his own impending death while writing this piece, and even that a messenger from the afterworld commissioned it. However, documentary evidence has established that the anonymous commission came from one Count Franz Walsegg of Schloss Stuppach, and that most if not all of the music had been written while Mozart was still in good health. A younger composer, and Mozart's pupil at the time, Franz Xaver Süssmayr, was engaged by Constanze to complete the Requiem. However, he was not the first composer asked to finish the Requiem, as the widow had first approached another Mozart student, Joseph Eybler, who began work directly on the empty staves of Mozart's manuscript but then abandoned it.

Because he was buried in an unmarked grave, it has been popularly assumed that Mozart was penniless and forgotten when he died. In fact, though he was no longer as fashionable in Vienna as before, he continued to have a well-paid job at court and receive substantial commissions from more distant parts of Europe, Prague in particular [citation needed]. He earned about 10,000 florins per year[6], equivalent to at least 42,000 US dollars in 2006, which places him within the top 5% of late 18th century wage earners[6], but he could not manage his own wealth. His mother wrote, "When Wolfgang makes new acquaintances, he immediately wants to give his life and property to them." His impulsive largesse and spending often put him in the position of having to ask others for loans. Many of his begging letters survive but they are evidence not so much of poverty as of his habit of spending more than he earned. He was not buried in a "mass grave" but in a regular communal grave according to the 1784 laws in Austria.

Though the original grave in the St. Marx cemetery was lost, memorial gravestones (or cenotaphs) have been placed there and in the Zentralfriedhof. In 2005, new DNA testing was performed by Austria's University of Innsbruck and the US Armed Forces DNA Identification Laboratory in Rockville, Maryland, to determine if a skull in an Austrian Museum was actually his, using DNA samples from the marked graves of his grandmother and Mozart's niece. However, test results were inconclusive, suggesting that none of the DNA samples were related to each other.

In 1809, Constanze married Danish diplomat Georg Nikolaus von Nissen (1761–1826). Being a fanatical admirer of Mozart, he (and Constanze?) edited vulgar passages out of many of the composer's letters and wrote a Mozart biography. Nissen did not live to see his biography printed, and Constanze finished it.

Works, musical style, and innovations

Style

Mozart's music, like Haydn's, stands as an archetypal example of the Classical style. His works spanned the period during which that style transformed from one exemplified by the style galant to one that began to incorporate some of the contrapuntal complexities of the late Baroque, complexities against which the galant style had been a reaction. Mozart's own stylistic development closely paralleled the development of the classical style as a whole. In addition, he was a versatile composer and wrote in almost every major genre, including symphony, opera, the solo concerto, chamber music including string quartet and string quintet, and the piano sonata. While none of these genres were new, the piano concerto was almost single-handedly developed and popularized by Mozart. He also wrote a great deal of religious music, including masses; and he composed many dances, divertimenti, serenades, and other forms of light entertainment.

The central traits of the classical style can all be identified in Mozart's music. Clarity, balance, and transparency are hallmarks, though a simplistic notion of the delicacy of his music obscures for us the exceptional and even demonic power of some of his finest masterpieces, such as the Piano Concerto in C minor, K. 491, the Symphony in G minor, K. 550, and the opera Don Giovanni. The famed writer on music Charles Rosen has written (in The Classical Style): "It is only through recognizing the violence and sensuality at the center of Mozart's work that we can make a start towards a comprehension of his structures and an insight into his magnificence. In a paradoxical way, Schumann's superficial characterization of the G minor Symphony can help us to see Mozart's daemon more steadily. In all of Mozart's supreme expressions of suffering and terror, there is something shockingly voluptuous." Especially during his last decade, Mozart explored chromatic harmony to a degree rare at the time. The slow introduction to the "Dissonant" Quartet, K. 465, a work that Haydn greatly admired, rapidly explodes a shallow understanding of Mozart's style as light and pleasant.

From his earliest years Mozart had a gift for imitating the music he heard; since he travelled widely, he acquired a rare collection of experiences from which to create his unique compositional language. When he went to London[7] as a child, he met J.C. Bach and heard his music; when he went to Paris, Mannheim, and Vienna, he heard the work of composers active there, as well as the spectacular Mannheim orchestra; when he went to Italy, he encountered the Italian overture and the opera buffa, both of which were to be hugely influential on his development. Both in London and Italy, the galant style was all the rage: simple, light music, with a mania for cadencing, an emphasis on tonic, dominant, and subdominant to the exclusion of other chords, symmetrical phrases, and clearly articulated structures. This style, out of which the classical style evolved, was a reaction against the complexity of late Baroque music. Some of Mozart's early symphonies are Italian overtures, with three movements running into each other; many are "homotonal" (each movement in the same key, with the slow movement in the tonic minor). Others mimic the works of J.C. Bach, and others show the simple rounded binary forms commonly being written by composers in Vienna.

As Mozart matured, he began to incorporate some features of Baroque styles into his music. For example, the Symphony No. 29 in A Major K. 201 uses a contrapuntal main theme in its first movement, and experimentation with irregular phrase lengths. Some of his quartets from 1773 have fugal finales, probably influenced by Haydn, who had just published his opus 20 set. The influence of the Sturm und Drang ("Storm and Stress") period in German literature, with its brief foreshadowing of the Romantic era to come, is evident in some of the music of both composers at that time.

Over the course of his working life Mozart switched his focus from instrumental music to operas, and back again. He wrote operas in each of the styles current in Europe: opera buffa, such as The Marriage of Figaro, Don Giovanni, or Così fan tutte; opera seria, such as Idomeneo; and Singspiel, of which Die Zauberflöte is probably the most famous example by any composer. In his later operas, he developed the use of subtle changes in instrumentation, orchestration, and tone colour to express or highlight psychological or emotional states and dramatic shifts. Here his advances in opera and instrumental composing interacted. His increasingly sophisticated use of the orchestra in the symphonies and concerti served as a resource in his operatic orchestration, and his developing subtlety in using the orchestra to psychological effect in his operas was reflected in his later non-operatic compositions.

Influence

Mozart's legacy to subsequent generations of composers (in all genres) is immense.

Many important composers since Mozart's time have expressed profound appreciation of Mozart. Rossini averred, "He is the only musician who had as much knowledge as genius, and as much genius as knowledge." Ludwig van Beethoven's admiration for Mozart is also quite clear. Beethoven used Mozart as a model a number of times: for example, Beethoven's Piano Concerto No. 4 in G major demonstrates a debt to Mozart's Piano Concerto in C major, K. 503. A plausible story – not corroborated – regards one of Beethoven's students who looked through a pile of music in Beethoven's apartment. When the student pulled out Mozart's A major Quartet, K. 464, Beethoven exclaimed "Ah, that piece. That's Mozart saying 'here's what I could do, if only you had ears to hear!' "; Beethoven's own Piano Concerto No. 3 in C minor is an obvious tribute to Mozart's Piano Concerto No. 24 in C minor, and yet another plausible – if unconfirmed – story concerns Beethoven at a concert with his sometime-student Ferdinand Ries. As they listened to Mozart's Piano Concerto No. 24, the orchestra reached the quite unusual coda of the last movement, and Beethoven whispered to Ries: "We'll never think of anything like that!" Beethoven's Quintet for Piano and Winds is another obvious tribute to Mozart, similar to Mozart's own quintet for the same ensemble. Beethoven also paid homage to Mozart by writing sets of variations on several of his themes: for example, the two sets of variations for cello and piano on themes from Mozart's Magic Flute, and cadenzas to several of Mozart's piano concertos, most notably the Piano Concerto No. 20 K. 466. A famous legend asserts that, after the only meeting between the two composers, Mozart noted that Beethoven would "give the world something to talk about." However, it is not certain that the two ever met. Tchaikovsky wrote his Mozartiana in praise of Mozart; and Mahler's final word was alleged to have been simply "Mozart". The theme of the opening movement of the Piano Sonata in A major K. 331 (itself a set of variations on that theme) was used by Max Reger for his Variations and Fugue on a Theme of Mozart, written in 1914 and among Reger's best-known works.

In addition, Mozart received outstanding praise from several fellow composers including Frédéric Chopin, Franz Schubert, Peter Ilich Tchaikovsky, Robert Schumann, and many more.

Mozart has remained an influence in popular contemporary music in varying genres ranging from Jazz to modern Rock and Heavy metal. An example of this influence is the jazz pianist Chick Corea, who has performed piano concertos of Mozart and was inspired by them to write a concerto of his own.

The Köchel catalogue

In the decades after Mozart's death there were several attempts to catalogue his compositions, but it was not until 1862 that Ludwig von Köchel succeeded in this enterprise. Many of his famous works are referred to by their Köchel catalogue number; for example, the Piano Concerto in A major (Piano Concerto No. 23) is often referred to simply as "K. 488" or "KV. 488". The catalogue has undergone six revisions, labeling the works from K. 1 to K. 626.

Myths and controversies

Mozart is unusual among composers for being the subject of an abundance of legend, partly because none of his early biographers knew him personally. They often resorted to fiction in order to produce a work. Many myths began soon after Mozart died, but few have any basis in fact. An example is the story that Mozart composed his Requiem with the belief it was for himself. Sorting out fabrications from real events is a vexing and continuous task for Mozart scholars mainly because of the prevalence of legend in scholarship. Dramatists and screenwriters, free from responsibilities of scholarship, have found excellent material among these legends.

An especially popular case is the supposed rivalry between Mozart and Antonio Salieri, and, in some versions, the tale that it was poison received from the latter that caused Mozart's death; this is the subject of Aleksandr Pushkin's play Mozart and Salieri, Nicolai Rimsky-Korsakov's opera Mozart and Salieri, and Peter Shaffer's play Amadeus. The last of these has been made into a feature-length film of the same name. Shaffer's play attracted criticism for portraying Mozart as vulgar and loutish, a characterization felt by many to be unfairly exaggerated, but in fact frequently confirmed by the composer's letters and other memorabilia. For example, Mozart wrote canons on the words "Leck mich im Arsch" ("Lick my arse") and "Leck mich im Arsch recht fein schön sauber" ("Lick my arse nice and clean") as party pieces for his friends. The Köchel numbers of these canons are 231 and 233.

Another debate involves Mozart's alleged status as a kind of superhuman prodigy, from childhood right up until his death. While some have criticised his earlier works as simplistic or forgettable, others revere even Mozart's juvenilia. In any case, several of his early compositions remain very popular. The motet Exultate, jubilate (K. 165), for example, composed when Mozart was seventeen years old, is among the most frequently recorded of his vocal compositions. It is also mentioned that around the time when he was five or six years old, he could play the piano blindfolded and with his hands crossed over one another.

Benjamin Simkin, a medical doctor, argues in his book Medical and Musical Byways of Mozartiana[9] that Mozart had Tourette syndrome. However, no Tourette syndrome expert, organization, psychiatrist or neurologist has stated that there is credible evidence that Mozart had this syndrome, and several have stated now that they do not believe there is enough evidence to substantiate the claim.

Amadeus (1984)

Milos Forman’s 1984 motion picture Amadeus, based on the play by Peter Shaffer, won eight Academy Awards and was one of the year’s most popular films. While the film did a great deal to popularize Mozart’s work with the general public, it has been criticized for its historical inaccuracies, and in particular for its portrayal of Antonio Salieri’s intrigues against Mozart, for which little historical evidence can be found. On the contrary, it is likely that Mozart and Salieri regarded each other as friends and colleagues: it is well documented, for instance, that Salieri frequently lent Mozart musical scores from the court library, that he often chose compositions by Mozart for performance at state occasions, and Salieri taught Mozart's son, Franz Xaver.

The idea that he never revised his compositions, dramatized in the film, is easily exploded by even a cursory examination of the autograph manuscripts, which contain many revisions. Mozart was a studiously hard worker, and by his own admission his extensive knowledge and abilities developed out of many years' close study of the European musical tradition. In fairness, Schaffer and Forman never claimed that Amadeus was intended to be an accurate biographical portrait of Mozart. Rather, as Shaffer reveals on the DVD release of the film, the dramatic narrative was inspired by the biblical story of Cain and Abel — one brother loved by God, and the other scorned.