History Of Mathematics

"Speaking... from personal experience, one of the effects of prolonged study of some of the more abstract branches of Mathematics, as for example the Theory of Functions, is that one begins to take the greatest interest in, and to be most attracted by... aspects of the subject which are most remote from the interests of the Physicist. One gets into an attitude... in which the kind of well-behaved functions, without abnormal singularities... appear to have a somewhat bourgeois aspect, in their comparatively uninteresting respectability. In the mind of one who makes a minute and prolonged study of the peculiarities which Fourier's series may present, a[n]... effect of that study is that the ordinary Fourier's series, which converge everywhere... normally, begin to acquire a certain tameness... which deprives them of interest. The failure of convergence of Taylor's series becomes to some Mathematical students a matter of greater interest than that presented by the series in the ordinary cases in which... they are fitted for purposes of application."

"The Mathematical Physicist plays a part of supreme importance as an intermediary and interpreter between the Pure Mathematician and the experimental Physicist. ...he must follow... progress both in Mathematics... and in experimental Physics. ...[I]n spite of some brilliant exceptions, the Mathematical Physicist does not... take as prominent a part as was formerly the case, especially during the nineteenth century, the age of Maxwell, Kelvin, Stokes, Helmholtz. ...In earlier times, when ...molar mechanics, and especially , occupied the centre of the interests... the passage from the observation of concrete phenomena to their abstract Mathematical representation was comparatively easy. The observational work was simpler and less technical... highly equipped physical laboratories had not yet come into existence... The Mathematical Physicists and Astronomers of the eighteenth century were largely engaged in working out the detailed implications of the law of gravitation, and had commenced, largely under the influence of the idea of , to work out problems such as... the vibrations of strings and other bodies."

"In the... [early] nineteenth century the centre of physical interest passed on to such subjects as Hydrodynamics, the Conduction of Heat, and Elasticity, in which an abstract representation of a body as a continuous plenum... made the problems readily accessible to continuous Mathematical Analysis. ...[M]uch attention was given to... Electricity and Magnetism, and much of the Mathematical Analysis which had been devised for...dealing with problems of gravitational attractions, vibrations, etc., was found, with further development, to be applicable to the new problems... Much of the work, such as that of Ampère... was still carried out under the influence of the idea of , first brought into prominence in connection with the Newtonian law of gravitation, but the idea of the continuous medium gradually became the dominating notion."

"Descartes published his "Géométrie" as an application of his general method of unification, in this case the unification of algebra and geometry."

"It is true that [analytic geometry] eventually evolved under the influence of Descartes' book, but the "Géométrie"... can hardly be considered the first textbook on this subject. There are no "Cartesian" axes and no equations of the straight line and of conic sections are derived, though a particular equation of the second degree is interpreted as denoting a conic section. Moreover, a large part of the book consists of a theory of algebraic equations, containing the "rule of Descartes" to determine the number of positive and negative roots."

"The gradual evolution of calculus was considerably stimulated by the publication of Descartes' "Géométrie"... which brought the whole field of classical geometry within the scope of algebraists."

"Descartes' recognized that the points of intersection of two curves are given by solving their equations simultaneously. The last implies... a major advance over all who had previously used coordinates: Descartes saw that an infinity of distinct curves can be referred to one system of coordinates. In this... he was far ahead of Fermat..."

"Descartes separated all curves into two classes, the "geometrical" and the "mechanical" ...according as (in our terminology) dy/dx is an algebraic or a transcendental function. ...this classification was abandoned long ago... The current definition... [a curve] which intersects some straight line in an infinity of points was given by Newton in his work on cubics."

"Descartes' merits lie above all in his consistent application of the well developed algebra of the early Seventeenth century to the geometric analysis of the Ancients, and by this, in the enormous widening of its applicability. A second merit is Descartes' final rejection of the homogeneity restrictions of his predecessors which even vitiated Viète's "logica speciosa," so that x2, x3, xy were now considered line segments. An algebraic equation became a relation between numbers, a new advance in mathematical abstraction necessary for the treatment of algebraic curves."

"It is evident from Descartes' explanation of his method that he had an intuitive grasp of the elusive concepts of 'variable' and 'function,' both of which are basic in analysis. Moreover, he intuited continuous variation."

"The first important example solved by Descartes in his geometry is the "problem of Pappus"; viz. "Given several straight lines in a plane, to find the locus of a point such that the perpendiculars. or more generally, straight lines at given angles, drawn from the point to the given lines, shall satisfy the condition that the product of certain of them shall be in a given ratio to the product of the rest." Of this celebrated problem, the Greeks solved only the special case when the number of given lines is four, in which case the locus of the point turns out to be a conic section. By Descartes it was solved completely, and it afforded an excellent example of the use which can be made of his analytical method in the study of loci. Another solution was given later by Newton in the Principia."

"Methods of drawing tangents [were] invented by Roberval and Fermat. Descartes gave a third method. Of all the problems which he solved by his geometry, none gave him as great pleasure as his mode of constructing tangents. It is profound but operose, and, on that account, inferior to Fermat's. His solution rests on the method of Indeterminate Coefficients, of which he bears the honour of invention. Indeterminate coefficients were employed by him also in solving bi-quadratic equations."

"Descartes' geometry was called "analytical geometry," partly because, unlike the synthetic geometry of the ancients, it is actually analytical, in the sense that the word is used in logic; and partly because the practice had then already arisen, of designating by the term analysis the calculus with general quantities."

"It is frequently stated that Descartes was the first to apply algebra to geometry. This statement is inaccurate, for Vieta and others had done this before him. Even the Arabs sometimes used algebra in connection with geometry. The new step that Descartes did take was the introduction into geometry of an analytical method based on the notion of variables and constants, which enabled him to represent curves by algebraic equations. In the Greek geometry, the idea of motion was wanting, but with Descartes it became a very fruitful conception. By him a point on a plane was determined in position by its distances from two fixed right lines or axes. These distances varied with every change of position in the point. This geometric idea of co-ordinate representation, together with the algebraic idea of two variables in one equation having an indefinite number of simultaneous values, furnished a method for the study of loci, which is admirable for the generality of its solutions. Thus the entire conic sections of Apollonius is wrapped up and contained in a single equation of the second degree."

"The Latin term for "ordinate," used by Descartes comes from the expression lineœ ordinatœ, employed by Roman surveyors for parallel lines. The term abscissa occurs for the first time in a Latin work of 1659, written by ..."

"The essays of Descartes on dioptrics and geometry were sharply criticised by Fermat, who wrote objections to the former, and sent his own treatise on "maxima" and "minima" to show that there were omissions in the geometry. Descartes thereupon made an attack on Fermat's method of tangents. Descartes was in the wrong in this attack, yet he continued the controversy with obstinacy."

"Geometry is not easy reading. An edition appeared subsequently with notes by his friend De Beaune, which were intended to remove the difficulties."

"His very first definition, that of the multiplication of two line segments, set the new geometry apart from the geometric demonstrations of algebraic results that Arabic mathematicians and Cardano had offered. In Descartes' geometry, multiplication of two line segments is a line segment. ...he stripped terms like "square" and "cube" of traditional geometric connotations but assigned them a new meaning that exploited a fundamental correspondence between arithmetic and geometry. ...he began with a problem stated geometrically; translated the problem into an equation; if necessary, reduced that equation into an irreducible one; and then geometrically constructed the root(s)... algebra was ...a means to an [geometric] end for Descartes."

"In book II, he implicitly recognized as legitimate curves of construction only those that are algebraic, that is, expressible in equations. ...discussing in book III problems for which different constructions were possible, he mandated that the simplest curves be used, and he defined those curves as those with equations of the lowest algebraic dimension. This was algebra dictating the methodology of geometry."

"The one book that turned out to be perhaps the most influential in guiding Newton's mathematical and scientific thought was none other than Descartes' La Géométrie. Newton read it in 1664 and re-read it several times until "by degrees he made himself master of the whole." ...Not only did analytic geometry pave the way for Newton's founding of calculus... but Newton's inner scientific spirit was truly set ablaze."

"Descartes' presentation differed from that now current. ...he used only an x-axis and did not refer to a y-axis. ...He considered only equations in the first quadrant, as it was thence that he translated the geometry into algebra. This... led to inexplicable anomolies in the translation back from algebra to geometry. As analytic geometry evolved and negative numbers were fearlessly used, the restriction was removed. ... The new method was not fully appreciated by Descartes' contemporaries, partly because he had deliberately adopted a rather crabbed style. When geometers did see what analytic geometry meant, it developed with great rapidity. But it was only with the development of calculus that analytic geometry came into its own."

"It was the use of algebra in geometry that he undertook to exploit. He saw fully the power of algebra and its superiority over the Greek geometrical methods in providing a broad methodology. He... stressed the generality of algebra and its value in mechanizing the reasoning processes and minimizing the work in solving problems. He saw its potential as a universal science of method. The product of his application of algebra to geometry was La Géométrie."

"Much of the obscurity was deliberate. Descartes boasted that few mathematicians in Europe would understand his work. He indicated the constructions and demonstrations, leaving to others to fill in the details. ...Many explanatory commentaries were written to make Descartes's book clear. ...He says that he omits the the demonstrations of most of his general statements because if one takes the trouble to examine systematically these examples, the demonstrations of the general results will become apparent, and it is of more value to learn them that way."

"Suppose 3, 4, 5 or a greater number of lines to be given in position, required a point from which, drawing lines to the given lines, each making a given angle with them, the rectangles of two lines thus drawn from the given point may have a given ratio to the square on the third, if there are three; or to the rectangle of the two others, if there are four; or again, if there are five lines, that the of the two remaining lines, together with a third given line, or to the parallelopiped composed of the three others, if there are six; or again, if there are seven, that the algebraic product of the three others and a given line, or to the four others, if there are eight, and so on. This was a problem which very much perplexed the ancient geometricians. Pappus says that neither Euclid nor Apollonius could give a solution. He himself knew that when there are only three or four lines the locus was a , but he could not describe it, much less could he tell what the curve would be when the number of lines were more than four. When the number of lines were seven or eight, the ancients could scarcely enunciate the problem, for there are no figures beyond solids, and without the aid of algebra, it is impossible to conceive what the product of four lines can mean. It was this problem which Descartes successfully attacked, and which, most probably led him to apply algebra generally to geometry. The following solution is that given by Descartes with a few abbreviations: AB, AD, EF and GH (fig. 2) are the given lines, C the required point from which are drawn the lines CB, CD, CF and CH making given angles CBA, CDA, CFE, and CHG. AB (=x) and BC (=y) are the principal lines to which all the others will be referred. Suppose the given lines to meet CB in the points R, S, T, and AB in the points A, E and G. Let AE = c and AG = d... By the... method he found the equation to bey^2 + xy + x^2 - 2y -5x = 0;which he showed belonged to a circle."

"Book 1 of Géométrie explains how to translate a geometrical problem into an equation ...by making a revolutionary break with the tradition of the theory of proportions to which Galileo adhered. ...Descartes interpreted algebraic operations as closed operations on segments. For instance, if a and b are segments, the product ab is not conceived by Descartes as representing an area, but rather another segment. Prior to the Géométrie the multiplication of two segments... would have been taken as the representation of the area of a rectangle... Descartes' interpretation of algebraic operations was a huge innovation... From Descartes' new viewpoint, ratios are considered as quotients, and proportions as equations. Consequently, the direct multiplication of ratios is allowed... The homogeneity of geometrical dimensions is no longer a constraint for the formation of ratios and proportions... multiplication of time and speed or division of weight by surface's area are all possible. Descartes' approach won great success."

"Since there is always an infinite number of different points satisfying these requirements, it is also required to discover and trace the curve containing all such points. Pappus says that when there are only three or four lines given, this line is one of the three conic sections, but he does not undertake to determine, describe, or explain the nature of the line required when the question involves a greater number of lines. He only adds that the ancients recognized one of them which they had shown to be useful, and which seemed the simplest, and yet was not the most important. This led me to find out whether, by my own method, I could go as far as they had gone."

"I could give here several other ways of tracing and conceiving a series of curved lines, each curve more complex than any preceding one, but I think the best way to group together all such curves and them classify them in order, is by recognizing the fact that all points of those curves which we may call "geometric," that is, those which admit of precise and exact measurement, must bear a definite relation to all points of a straight line, and that this relation must be expressed by a single equation. If this equation contains no term of higher degree than the rectangle of two unknown quantities, or the square of one, the curve belongs to the first and simplest class, which contains only the circle, the parabola, the hyperbola, and the ellipse; but when the equation contains one or more terms of the third or fourth degree in one or both of the two unknown quantities (for it requires two unknown quantities to express the relation between two points) the curve belongs to the second class; and if the equation contains a term of the fifth or sixth degree in either or both of the unknown quantities the curve belongs to the third class, and so on indefinitely."

"The question, then, the solution of which... was completed by no one, is this: Having three, four or more lines given in position, it is first required to find a point from which as many other lines may be drawn, each making a given angle with one of the given lines, so that the rectangle of two of the lines so drawn shall bear a given ratio to the square of the third (if there be only three); or to the rectangle of the other two (if there be four), or again, that the parallelepiped constructed upon three shall bear a given ratio to that upon the other two and any given line (if there be five); or to the parallelepiped upon the other three (if there be six);or (if there be seven) that the product obtained by multiplying four of them together shall bear a given ratio to the product of the other three, or (if there be eight) that the product of four of them shall bear a given ratio to the product of the other four. Thus the question admits of extension of any number of lines."

"This is also evident from what Pappus has done in the beginning of his seventh book, where... he refers to a question which he says that neither Euclid nor Apollonius nor any one else had been able to solve completely..."

"The considerations that forced ancient writers to use arithmetical terms in geometry, thus making it impossible for them to proceed beyond a point where they could see clearly the relation between two subjects, caused much obscurity and embarrassment, in their attempts at explanation."

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

"I have given these very simple [methods] to show that it is possible to construct all the problems of ordinary geometry by doing no more than the little covered in the four figures that I have explained. This is one thing which I believe the ancient mathematicians did not observe, for otherwise they would not have put so much labor into writing so many books in which the very sequence of the propositions shows that they did not have a sure method of finding all, but rather gathered together those propositions on which they had happened by accident."

"Often it is not necessary thus to draw the lines on paper, but it is sufficient to designate each by a single letter. ...it must be observed that by a2, b3, and similar expressions, I ordinarily mean only simple lines, which, however, I name squares, cubes, etc., so that I may make use of the terms employed in algebra."

"If... we wish to solve any problem, we first suppose the solution already affected, and give names to all the lines that seem needful for its construction,—to those that are unknown as well as to those that are known. Then, making no distinction between the known and unknown lines, we must unravel the difficulty in a way that shows most naturally the relations between these lines, until we find it possible to express a single quantity in two ways. This will constitute an equation, since the terms of one of these two expressions are together equal to the terms of the other."

"If the square root of GH is desired, I add, along the same straight line, FG equal to unity; then, bisecting FH at K, I describe the circle FIH about K as the center, and draw from G a perpendicular and extend it to I, and GI is the required root."

"La Géométrie... is... divided into three parts. The first... contains an explanation of some of the principles of algebraic geometry and shows a real advance over the Greeks. To the Greeks... the product of two variables [corresponded] to the area of some rectangle, and the product of three variables to the volume of some rectangular parallelepiped. Beyond this the Greeks could not go. To Descartes... x2 [is] the fourth term in the proportion 1 : x = x : x2... representable by an appropriate line length which can easily be constructed when x is known. ... The second part... deals with... a now obsolete classification of curves and with an interesting method of constructing tangents to curves. ... The third part...concerns... the solution of equations of degree greater than two. Use is made of what we now call "Descartes' rule of signs,"... determining limits to the number of negative and positive roots [of] a polynomial."

"Any problem in geometry can be reduced to such terms that a knowledge of the lengths of certain straight lines is sufficient for its construction."

"Let AB be taken as unity, and let it be required to multiply BD by BC. I have only to join the points A and C, and draw DE parallel to CA; then BE is the product of BD and BC. If it be required to divide BE by BD, I join E and D, and draw AC parallel to DE; then BC is the result of division."

"Thus, all unknown quantities can be expressed in terms if a single quantity, whenever the problem can be constructed by means of circles and straight lines, or by conic sections, or even by some other curve of degree not greater than the third or fourth. But I shall not stop to explain this in more detail, because I should deprive you of the pleasure of mastering it yourself, as well as of the advantage of training your mind by working over it, which is in my opinion the principle benefit to be derived from this science. Because, I find nothing here so difficult that it cannot be worked out by anyone at all familiar with ordinary geometry and with algebra, who will consider carefully all that is set forth in this treatise."

"The reader must pretty much construct the method for himself from certain isolated statements. There are... figures... but in none do we find the coordinate axes explicitly set forth. This work was written with intentional obscurity... too difficult to be widely read."

"An oral tradition makes it possible to indicate the line segments with the fingers; one can emphasize essentials and point out how the proof was found. All of this disappears in the written formulation... as soon as some external cause brought about an interruption in the oral tradition, and only books remained, it became very difficult to assimilate the work of the great predursors, and next to impossible to pass beyond it."

"Thales and Pythagoras took their start from Babylonian mathematics but gave it a very different... specifically Greek character... in the Pythagorean school and outside, mathematics was brought to... ever higher development and began gradually to satisfy the demands of stricter logic... through the work of Plato's friends Theaetetus and Eudoxus, mathematics was brought to a state of perfection, beauty and exactness, which we admire in the elements of Euclid. ...the mathematical method of proof served as a prototype for Plato's dialectics and for Aristotle's logic."

"Plato is very fond of appealing to mathematics to show that exact reasoning is possible, not about things which are seen and heard, but about ideal objects which exist in thought only. ... Plato is apparently convinced that the mathematicians will agree with him. And indeed, when we deal with line segments which one sees and which one measures empirically, the question as to the existence of a common measure has no sense; a hair's breadth will measure integrally every line that is drawn. The question of commensurability makes sense only for line segments as objects of thought. It is therefore clear that Theodorus could not appeal to intuition to prove the incommensurability of the sides of his squares."

"was the author of a book purporting to be a manual of mathematical subjects such as a student would require to enable him to understand Plato."

"Euclid, the author of the incomparable Elements, wrote on almost all the other branches of mathematics known in his day. Archimedes's work, all original and set forth in treatises which are models of scientific exposition, perfect in form and style, was even wider in its range of subjects. The imperishable and unique monuments of the genius of these two men must be detached from their surroundings and seen as a whole if we would appreciate to the full the pre-eminent place which they occupy, and will hold for all time, in the history of science."

"It is a defect in the existing histories that, while they state generally the contents of, and the main propositions proved in, the great treatises of Archimedes and Apollonius, they make little attempt to describe the procedure by which the results are obtained. I have therefore taken pains, in the most significant cases, to show the course of the argument in sufficient detail to enable a competent mathematician to grasp the method used and to apply it, if he will, to other similar investigations."

"After Apollonius Greek mathematics came to a dead stop. It is true that there were some epigones, such as Diocles and Zenodorus... But apart from trigonometry, nothing great nothing new appeared. The geometry of the conics remained in the form Apollonius gave it, until Descartes. ...The "Method" of Archimedes was lost sight of, and the problem of integration remained where it was, until it was attacked anew in the 17th century... Germs of projective geometry were present, but it remained for Desargues and Pascal to bring these to fruition. ...Higher plane curves were studied only sporadically... Geometric algebra and the theory of proportions were carried over into modern times as inert traditions, of which the inner meaning was no longer understood. The Arabs started algebra anew, from a much more primitive point of view... Greek geometry had run into a blind alley."

"Dr. James Gow did a great service by the publication in 1884 of his Short History of Greek Mathematics, a scholarly and useful work which has held its own and has been quoted with respect and appreciation by authorities on the history of mathematics in all parts of the world. At the date when he wrote, however, Dr. Gow had necessarily to rely upon the works of the pioneers Bretschneider, Hankel, Allman, and (first edition). Since then the subject has been very greatly advanced... scholars and mathematicians... have thrown light on many obscure points. It is therefore high time for the complete story to be rewritten."

"vi. The angle in a semicircle is a right angle. It is believed that Thales proved this proposition in the following manner: Let ABCH be a circle of which the diameter is BC, and the centre E. ...Draw AE and produce BA to F. Because BE is equal to EA [both being radii of the circle], the angle EAB is equal to EBA; also, because AE is equal to EC, the angle EAC is equal to ECA [being angles at the base of an isosceles triangle]; wherefore, the whole angle BAC is equal to the two angles ABC, ACB. But FAC, the exterior angle of the triangle ABC, is also equal to the two angles ABC, ACB [since the sum of the three angles of the triangle is equal to two right angles, i.e., a straight line]; therefore the angle BAC is equal to the angle FAC, and each of them is therefore a right angle; wherefore the angle BAC in a semicircle is a right angle. Thales's demonstration, if we may call this his, is quite different from the one given in modern text-books; but it is certainly neither less rigid nor less beautiful. The demonstration is the one given in Euclid, but his work, we must remember, is to a large extent compiled from the works of previous writers. It will be seen, however, that this demonstration implies a knowledge of a seventh proposition,—"If one side of a triangle be produced, the exterior angle is equal to the sum of the two interior and opposite angles." Thales must have been familiar with this truth."

"For the mathematician the important consideration is that the foundations of mathematics and a great portion of its content are Greek. The Greeks laid down the first principles, invented the methods ab initio, and fixed the terminology. Mathematics in short is a Greek science, whatever new developments modern analysis has brought or may bring."