Mathematicians From Greece

"But how much more splendid the methods which reduce the problems which seem to be hardly capable of solution even with the help of surds in such a way that, while the surds, when bidden (so to speak) to plough the arithmetic soil, become true to their name and deaf to entreaty, they are not so much as mentioned in these most ingenious solutions."

"The solution of the higher indeterminates depends almost entirely on very favourable numerical conditions and his methods are defective. But the extraordinary ability of Diophantus appears rather in the other department of his art, namely the ingenuity with which he reduces every problem to an equation which he is competent to solve."

"Diophantus shows great Adroitness in selecting the unknown, especially with a view to avoiding an adfected quadratic. ...The most common and characteristic of Diophantus' methods is his use of tentative assumptions which is applied in nearly every problem of the later books. It consists in assigning to the unknown a preliminary value which satisfies one or two only of the necessary conditions, in order that, from its failure to satisfy the remaining conditions, the operator may perceive what exactly is required for that purpose. ...a third characteristic of Diophantus [is] ...the use of the symbol for the unknown in different senses. ...The use of tentative assumptions leads again to another device which may be called... the method of limits. This may best be illustrated by a particular example. If Diophantus wishes to find a square lying between 10 and 11, he multiplies these numbers by successive squares till a square lies between the products. Thus between 40 and 44, 90 and 99 no square lies, but between 160 and 176 there lies the square 169. Hence x^2 = \tfrac{169}{16} will lie between the proposed limits."

"It may be in some measure due to the defects of notation in his time that Diophantos will have in his solutions no numbers whatever except rational numbers, in [the non-numbers of] which, in addition to surds and imaginary quantities, he includes negative quantities. ...Such equations then as lead to surd, imaginary, or negative roots he regards as useless for his purpose: the solution is in these cases ὰδοπος, impossible. So we find him describing the equation 4 = 4x + 20 as ᾰτοπος because it would give x = -4. Diophantos makes it throughout his object to obtain solutions in rational numbers, and we find him frequently giving, as a preliminary, conditions which must be satisfied, which are the conditions of a result rational in Diophantos' sense. In the great majority of cases when Diophantos arrives in the course of a solution at an equation which would give an irrational result he retraces his steps and finds out how his equation has arisen, and how he may by altering the previous work substitute for it another which shall give a rational result. This gives rise, in general, to a subsidiary problem the solution of which ensures a rational result for the problem itself. Though, however, Diophantos has no notation for a surd, and does not admit surd results, it is scarcely true to say that he makes no use of quadratic equations which lead to such results. Thus, for example, in v. 33 he solves such an equation so far as to be able to see to what integers the solution would approximate most nearly."

"For not more than six books [of Diophantus] are found, though in the proëmium he promises thirteen. If this book, a wonderful and difficult work, could be found entire, I should like to translate it into Latin, for the knowledge of Greek I have lately acquired would suffice for this."

"With Diophantus the history of Greek arithmetic comes to an end. No original work, that we know of, was done afterwards."

"No one has yet translated from the Greek into Latin the thirteen books of Diophantus, in which the very flower of the whole of arithmetic lies hid, the ars rei et census [the art of the evaluation of wealth or tax] which to-day they call by the Arabic name of Algebra."

"Diophantos' work is so unique among the Greek treatises which we possess, that he cannot be said to recall the style or subject-matter of any other author, except, indeed, in the fragment on Polygonal Numbers; and even there the reference to Hypsikles is the only indication we can lay hold of."

"The Alexandrians used fractions as numbers in their own right, whereas the mathematicians of the classical period spoke only of a ratio of integers, not of parts of a whole and the ratios were used only in proportions. However, even in the classical period genuine fractions... as entities in their own right, were used in commerce. In the Alexandrian period, Archimedes, Heron, Diophantus, and others used fractions freely and performed operations with them. Though... they did not discuss the concept of fractions, apparently these were intuitively sufficiently clear..."

"He appears to be the first who could perform such operations as (x - 1) \times (x - 2) without reference to geometry. Such identities as (a + b)^2 = a^2 + 2ab + b^2, which with Euclid appear in the elevated rank of geometric theorems, are with Diophantus the simplest consequences of the algebraic laws of operation."

"The creation of the formal language of mathematics is identical with the foundation of modern algebra. ...As far as Greek sources are concerned, the special influence of the Arithmetic of Diophantus on the content, but even more so on the form, of this Arabic science is unmistakable. ...concurrently with the elaboration... of the theory of equations which the Arabs had passed on to the West, the original text of Diophantus began, as early as the fifteenth century, to become well known and influential. But it was not until the last quarter of the sixteenth century that Vieta undertook to modify Diophantus' technique in a really critical way. He thereby became the true founder of modern mathematics."

"This work of Diaphantus... was the first Greek mathematics, if indeed it was Greek, to show a genuine talent for algebra. ...He had begun to use symbols operationally. This long stride forward is all the more remarkable because his algebraic notation... was almost as awkward as Greek logistic. That he accomplished what he did with the available techniques places him beyond question among the great algebraists."

"As a square number is known to be the product of a number multiplied by itself, so every polygonal number, multiplied by one number and added to another, both of which depend upon the number of its angles, produces a square number. I shall prove this, and shall show also how from a given side to find its polygon and conversely. Some auxiliary propositions must first be proved."

"Diophantus himself, it is true, gives only the most special solutions of all the questions which he treats, and he is generally content with indicating numbers which furnish one single solution. But it must not be supposed that his method was restricted to these very special solutions. In his time the use of letters to denote undetermined numbers was not yet established, and consequently the more general solutions which we are now enabled to give by means of such notation could not be expected from him. Nevertheless, the actual methods which he uses for solving any of his problems are as general as those which are in use to-day; nay, we are obliged to admit, that there is hardly any method yet invented in this kind of analysis of which there are not sufficiently distinct traces to be discovered in Diophantus."

"If we arrive at an equation containing on each side the same term but with different coefficients, we must take equals from equals until we get one term equal to another term. But, if there are on one or on both sides negative terms, the deficiencies must be added on both bides until all the terms on both sides are positive. Then we must take equals from equals until one term is left on each side."

"Perhaps the topic [of this book] will appear fairly difficult to you because it is not yet familiar knowledge and the understanding of beginners is easily confused by mistakes; but with your inspiration and my teaching it will be easy for you to master, because clear intelligence supported by good lessons is a fast route to knowledge."

"His work as a whole and his comments shew that he was a geometrician of power; but it was his misfortune to live at a time when but little interest was taken in geometry, and when the subject, as then treated, had been practically exhausted."

"He discovered the directrix in the conic sections, but he investigated only a few isolated properties: the earliest comprehensive account was given by Newton and Boscovich. As an illustration of his power I may mention that he solved [book VII, prop. 107] the problem to inscribe in a given circle a triangle whose sides produced shall pass through three collinear points. This question was in the eighteenth century generalised by Cramer... It was sent in 1742 as a challenge to Castillon, and in 1776 he published a solution. Lagrange, Euler, Lhulier, Fuss, and Lexell also gave solutions in 1780. A few years later the problem was set to a Neapolitan lad A. Giordano, who was only 16 but... he extended it to the case of a polygon of n sides which pass through n given points and gave a solution both simple and elegant. Poncelet extended it to conics of any species and subject to other restrictions."

"The last two books contain a good deal of original work by Pappus ...it should be remarked that in two or three cases he has been detected in appropriating proofs from earlier authors, and it is possible he may have done this in other cases. Subject to this suspicion we may say that Pappus's best work is in geometry."

"In mechanics Pappus shewed that the centre of mass of a triangular lamina is the same as that of an inscribed triangular lamina whose vertices divide each of the sides of the original triangle in the same ratio. He also discovered the two theorems on the surface and volume of a solid of revolution which are still quoted in text-books under his name: these are that the volume generated by the revolution of a curve about an axis is equal to the product of the area of the curve and the length of the path described by its centre of mass; and the surface is equal to the product of the perimeter of the curve and the length of the path described by its centre of mass."

"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."

"In Pascal's wonderful work on conics, written at the age of sixteen and now lost, were given the theorem on the anharmonic ratio, first found in Pappus."

"Ptolemy had shewn not only that geometry could be applied to astronomy, but had indicated how new methods of analysis like trigonometry might be thence developed. He found however no successors to take up the work he had commenced so brilliantly, and we must look forward 150 years before we find another geometrician of any eminence. That geometrician was Pappus... We know that he had numerous pupils, and it is probable that he temporarily revived an interest in the study of geometry."

"Pappus wrote several books, but the only one which has come down to us is his Συναɣωɣή, a collection of mathematical papers arranged in eight books of which the first and part of the second have been lost. This collection was intended to be a synopsis of Greek mathematics together with comments and additional propositions... it is trustworthy, and we rely largely on it for our knowledge of other works now lost. ...it is most likely that it gives roughly the order in which the classical authors were read at Alexandria. Probably the first book, which is now lost. was on arithmetic. The next four books deal with geometry exclusive of conic sections; the sixth with astronomy including, as subsidiary subjects, optics and trigonometry; the seventh with analysis, conics, and porisms; and the eighth with mechanics."

"He is known in three instances to have copied theorems without giving due credit and that he may have done the same tiling in other cases in which we have no data by which to ascertain the real discoverer"

"Pappus proved... that the centre of gravity of a triangle is that of another triangle whose vertices lie upon the sides of the first and divide its three sides in the same ratio."

"We shall now cite the more important of those theorems in the Mathematical Collections which are supposed to be original with Pappus. First of all ranks the elegant theorem re-discovered by Guldin, over 1000 years later, that the volume generated by the revolution of a plane curve which lies wholly on one side of the axis, equals the area of the curve multiplied by the circumference described by its centre of gravity."

"In the fourth book are new and brilliant propositions on the quadratrix which indicate an intimate acquaintance with curved surfaces. He generates the quadratrix..."

"Pappus, probably born about 340 A.D., in Alexandria, was the last great mathematician of the Alexandrian school. His genius was inferior to that of Archimedes, Apollonius, and Euclid, who flourished over 500 years earlier. But living, as he did, at a period when interest in geometry was declining, he towered above his contemporaries "like the peak of Teneriffa above the Atlantic." He is the author of a Commentary on the Almagest, a Commentary on Euclid's Elements, a Commentary on the Analemma of Diodorus,—a writer of whom nothing is known. All these works are lost. Proclus, probably quoting from the Commentary on Euclid, says that Pappus objected to the statement that an angle equal to a right angle is always itself a right angle."

"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."

"The only work of Pappus still extant is his Mathematical Collections. This was originally in eight books, but the first and portions of the second are now missing. The Mathematical Collections seems to have been written by Pappus to supply the geometers of his time with a succinct analysis of the most difficult mathematical works and to facilitate the study of them by explanatory lemmas. But these lemmas are selected very freely, and frequently have little or no connection with the subject on hand. However, he gives very accurate summaries of the works of which he treats. The Mathematical Collections is invaluable to us on account of the rich information it gives on various treatises by the foremost Greek mathematicians, which are now lost. Mathematicians of the last century considered it possible to restore lost works from the résumé by Pappus alone."

"Pappus considers curves of double curvature still further. He produces a spherical spiral by a point moving uniformly along the circumference of a great circle of a sphere, while the great circle itself revolves uniformly around its diameter. He then finds the area of that portion of the surface of the sphere determined by the spherical spiral..."

"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..."

"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."

"Pappus proceeds as follows: ...If three straight lines are given in position, and if straight lines be drawn from one and the same point, making given angles with three given lines; and if there be given the ratio of the rectangle contained by two of the lines so drawn to the square of the other, the point lies on a solid locus given in position, namely, one of the three conic sections Again, if lines be drawn making given angles with four straight lines given in position, and if the rectangle of two of the lines so drawn bears a given ratio to the rectangle of the other two; then, in like manner, the point lies on a conic section given in position. It has been shown that to only two lines there corresponds a plane locus. But if there be given four lines, the point generates loci not known up to the present time (that is, impossible to determine by common methods), but merely called 'lines'. It is not clear what they are, or what their properties. One of them, not the first but the most manifest, has been examined, and this has proved to be helpful. These, however, are the propositions concerning them. If from any point straight lines be drawn making given angles with five straight lines given in position, and if the solid rectangular parallelepiped contained by three of the lines so drawn bears a given ratio to the sold rectangular parallelepiped contained by the other two and any given line whatever, the point lies on a 'line' given in position. Again, if there be six lines, and if the solid contained by three of the lines bears a given ratio to the solid contained by the other three lines, the point also lies on a 'line' given in position. But if there be more than six lines, we cannot say whether a ratio of something contained by four lines is given to that which is contained by the rest, since there is no figure of four dimensions."

"As a writer, Pappus must have been quite versatile if the following list of works attributed to him is any indication: (1) Description of the World. (2) Comments on the Four Books of the Almagest. (3) Interpretation of Dreams. (4) On the Rivers of Libya. (5) Commentary on the Analemma of Diodorus. (6) Comments on Euclid's Elements. (7) Comments on Ptolemy's Harmonica. (8) Collection. Of all these the only one extant even in part is the Collection, which is a summary in eight books of the principal works of preceding Greek mathematicians with comments and lemmas on the works in question."

"The superior lines treated of by Pappus, and other ancient writers, were the conchoid, the cissoid, the spiral, and the quadratrix; and a few others are slightly alluded to."

"One thing... ought to be emphasized in any discussion relative to the Collection and that is its remarkable suggestiveness. In order to understand this, one has only to turn the works of such men as Chasles, [Siegmund] Günther, Descartes, Newton, and Steiner, for in the writings of these men it furnished the basic ideas for analytic geometry, projective geometry, and other allied theories. And if it had so much offer these men, it ought to furnish some suggestions to careful reader of today."

"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."

"A question which was brought into prominence by Descartes and Newton is the "problem of Pappus." Given several straight lines in a plane, to find the locus of a point such that when perpendiculars (or, more generally, straight lines at given angles) are drawn from it to the given lines, the product of certain ones of them shall be in a given ratio to the product of the remaining ones."

"Many elementary textbooks of geometry contain a few remarks about analysis, synthesis, and "assuming the problem as solved." There is little doubt that this almost ineradicable tradition goes back to Pappus, although there is hardly a current textbook whose writer would show any direct acquaintance with Pappus. The circumstance alone that it is restricted to textbooks of geometry shows a current lack of understanding..."

"In the seventh book of his Collections, Pappus reports about a branch of study he calls analyomenos. We can render this name in English as "Treasury of Analysis," or as "Art of Solving Problems," or even as "Heuristic"... A good English translation of Pappus's report is easily accessible... Pappus's text is important in many ways. ...the procedures described by Pappus are by no means restricted to geometric problems; they are, in fact, not even restricted to mathematical problems. ...the paraphrase... emphasizes certain curious phrases of the original: "assume what is required to be done as already done, what is sought as found, what you have to prove is true." This is paradoxical; it is not mere self-deception..."

"One of the most celebrated geometrical problems of antiquity was the trisection of an angle. It stands side by side with those other famous problems,—the squaring of the circle and the duplication of the cube. ...modern analysis shows that the trisection of an angle is an insoluble problem if in our constructions we confine ourselves to the use of circles and straight lines. i.e. to Euclidean geometry. ...By the use of the conic sections, however, the problem is readily solved in many ways. Pappus... has given us the following beautiful reduction of the problem... "Since we can trisect a right angle," says Pappus, "it follows that the trisection of any angle can be effected if we can trisect an acute angle." ..While the geometricians quoted by Pappus could not solve the problem, Pappus himself, who lived at a time when the conic sections had been developed to some extent, fixed the position of the point E by means of an hyperbola. Pappus also claims, as his own, a solution by means of the conchoid of Nicomedes."

"Direct solutions by means of conics. Pappus gives two solutions of the trisection problem in which conics are applied directly without any preliminary reduction of the problem to a νϵῡσɩς. ...The passage of Pappus from which this solution is taken is remarkable as being one of three passages in Greek mathematical works still extant (two being in Pappus and one in a fragment of Anthemius on burning mirrors) which refer to the focus-and-directrix property of conics. The second passage in Pappus comes under the heading of Lemmas to the Surface-Loci of Euclid . Pappus there gives a complete proof of the theorem that, if the distance of a point from a fixed point is in a given ratio to its distance from a fixed line, the locus of the point is a conic section which is an ellipse, a parabola, or a hyperbola according as the given ratio is less than, equal to, or greater than, unity. The importance of these passages lies in the fact that the Lemma was required for the understanding of Euclid's treatise. We can hardly avoid the conclusion that the property was used by Euclid in his Surface-Loci, but was assumed as well known. It was, therefore, probably taken from some treatise current in Euclid's time, perhaps from Aristaeus's work on Solid Loci."

"To give here an elaborate account of Pappus would be to create a false impression. His work is only the last convulsive effort of Greek geometry which was now nearly dead and was never effectually revived. It is not so with Ptolemy or Diophantus. The trigonometry of the former is the foundation of a new study which was handed on to other nations indeed but which has thenceforth a continuous history of progress. Diophantus also represents the outbreak of a movement which probably was not Greek in its origin, and which the Greek genius long resisted, but which was especially adapted to the tastes of the people who, after the extinction of Greek schools, received their heritage and kept their memory green. But no Indian or Arab ever studied Pappus or cared in the least for his style or his matter. When geometry came once more up to his level, the invention of analytical methods gave it a sudden push which sent it far beyond him and he was out of date at the very moment when he seemed to be taking a new lease of life."

"The Porisms. Our only source of information about the nature and contents of the Porisms is Pappus. ...With Pappus's account of Porisms must be compared the passages of Proclus on the same subject. ...Proclus's definition... agees well enough with the first, the 'older', definition of Pappus. A porism occupies a place between a theorem and a problem; it deals with something already existing, as a theorem does, but has to find it (e.g. the centre of a circle) and, as a certain operation is therefore necessary, it partakes to that extent of the nature of a problem, which requires us to construct or produce something not previously existing. ...all the positive information which we have about the nature of a porism and the contents of Euclid's Porisms ...is obscure and leaves great scope for speculation and controversy; naturally, therefore, the problem of restoring the Porisms has had a great fascination for distinguished mathematicians ever since the revival of learning. But it has proved beyond them all."

"Snellius, Vieta, Marinus, Ghetaldus, and Fermat, attempted, from the description of the τόωος άναλυομἐνος in the seventh book, to restore several of the lost books of Apollonius, which made a principal part of that collection. Afterwards Schooten, Dr. Wallis, and subsequently Dr. Halley, pursued the same track; and thus a number of these ancient treatises were restored, with various success indeed, and with different characters of elegance and accuracy..."

"Now analysis is of two kinds, the one directed to searching for the truth and called theoretical, the other directed to finding what we are told to find and called problematical. (1) In the theoretical kind we assume what is sought as if it were existent and true, after which we pass through its successive consequences, as if they too were true and established by virtue of our hypothesis, to something admitted: then (a), if that something admitted is true, that which is sought will also be true and the proof will correspond in the reverse order to the analysis, but (b), if we come upon something admittedly false, that which is sought will also be false. (2) In the problematical kind we assume that which is propounded as if it were known, after which we pass through its successive consequences, taking them as true, up to something admitted: if then (a) what is admitted is possible and obtainable, that is, what mathematicians call given, what was originally proposed will also be possible, and the proof will again correspond in reverse order to the analysis, but if (b) we come upon something admittedly impossible, the problem will also be impossible."

"From the defective mode of notation among the Greeks... though there be much ingenuity in some of their methods, we need not be surprised at the great inferiority of their system. Dr. Wallis remarks, that the business of the second book of Pappus appears to be nearly equivalent to what is now considered as a very simple proposition, viz. that the multiplication of any numbers... The first book he with much probability conjectures to have been employed about the simple operations of the addition and subtraction of numbers."

"waives the customary distinction between a circle, and ellipse, a parabola, and a hyperbola; these curves are simply conics, all alike. Although conics were studied by , Euclid, Archimedes and Apollonius, in the fourth and third centuries B.C., the earliest truly projective theorems were discovered by Pappus of Alexandria... and it was J. V. Poncelet... who first proved such theorems by purely projective reasoning."