A Short Account of the History of Mathematics · W. W. Rouse Ball
Where no other references are given, see parts xii, xiii, xiv, and the early chapters of part xv of Cantor's Vorlesungen; on the Italian mathematicians of this period see also G. Libri, Histoire des sciences mathématiques en Italie, 4 vols., Paris, 1838--1841.
The last chapter is a digression from the chronological arrangement to which, as far as possible, I have throughout adhered, but I trust by references in this chapter to keep the order of events and discoveries clear. I return now to the general history of mathematics in western Europe. Mathematicians had barely assimilated the knowledge obtained from the Arabs, including their translations of Greek writers, when the refugees who escaped from Constantinople after the fall of the eastern empire brought the original works and the traditions of Greek science into Italy. Thus by the middle of the fifteenth century the chief results of Greek and Arabian mathematics were accessible to European students.
The invention of printing about that time rendered the dissemination of discoveries comparatively easy. It is almost a truism to remark that until printing was introduced a writer appealed to a very limited class of readers, but we are perhaps apt to forget that when a medieval writer "published" a work the results were known to only a few of his contemporaries. This had not been the case in classical times, for then and until the fourth century of our era Alexandria was the recognized centre for the reception and dissemination of new works and discoveries. In medieval Europe, on the other hand, there was no common centre through which men of science could communicate with one another, and to this cause the slow and fitful development of medieval mathematics may be partly ascribed.
The introduction of printing marks the beginning of the modern world in science as in politics; for it was contemporaneous with the assimilation by the indigenous European school (which was born from scholasticism, and whose history was traced in chapter viii) of the results of the Indian and Arabian schools (whose history and influence were traced in chapters ix and x), and of the Greek schools (whose history was traced in chapters ii to v).
The last two centuries of this period of our history, which may be described as the renaissance, were distinguished by great mental activity in all branches of learning. The creation of a fresh group of universities (including those in Scotland), of a somewhat less complex type than the medieval universities above described, testify to the general desire for knowledge. The discovery of America in 1492 and the discussions that preceded the Reformation flooded Europe with new ideas which, by the invention of printing, were widely disseminated; but the advance in mathematics was at least as well marked as that in literature and that in politics.
During the first part of this time the attention of mathematicians was to a large extent concentrated on syncopated algebra and trigonometry; the treatment of these subjects is discussed in the first section of this chapter, but the relative importance of the mathematicians of this period is not very easy to determine. The middle years of the renaissance were distinguished by the development of symbolic algebra: this is treated in the second section of this chapter. The close of the sixteenth century saw the creation of the science of dynamics: this forms the subject of the first section of chapter xiii. About the same time and in the early years of the seventeenth century considerable attention was paid to pure geometry: this forms the subject of the second section of chapter xiii.
Regiomontanus.
Amongst the many distinguished writers of this time Johann Regiomontanus was the earliest and one of the most able. He was born at Königsberg on June 6, 1436, and died at Rome on July 6, 1476. His real name was Johannes Müller, but, following the custom of that time, he issued his publications under a Latin pseudonym which in his case was taken from his birthplace. To his friends, his neighbours, and his tradespeople he may have been Johannes Müller, but the literary and scientific world knew him as Regiomontanus, just as they knew Zepernik as Copernicus, and Schwarzerd as Melanchthon. It seems as pedantic as it is confusing to refer to an author by his actual name when he is universally recognized under another: I shall therefore in all cases as far as possible use that title only, whether latinized or not, by which a writer is generally known.
Regiomontanus studied mathematics at the university of Vienna, then one of the chief centres of mathematical studies in Europe, under Purbach who was professor there. His first work, done in conjunction with Purbach, consisted of an analysis of the Almagest. In this the trigonometrical functions sine and cosine were used and a table of natural sines was introduced. Purbach died before the book was finished: it was finally published at Venice, but not till 1496. As soon as this was completed Regiomontanus wrote a work on astrology, which contains some astronomical tables and a table of natural tangents: this was published in 1490.
Leaving Vienna in 1462, Regiomontanus travelled for some time in Italy and Germany; and at last in 1471 settled for a few years at Nuremberg, where he established an observatory, opened a printing-press, and probably lectured. Three tracts on astronomy by him were written here. A mechanical eagle, which flapped its wings and saluted the Emperor Maximilian I. on his entry into the city, bears witness to his mechanical ingenuity, and was reckoned among the marvels of the age. Thence Regiomontanus moved to Rome on an invitation from Sixtus IV. who wished him to reform the calendar. He was assassinated, shortly after his arrival, at the age of 40.
Regiomontanus was among the first to take advantage of the recovery of the original texts of the Greek mathematical works in order to make himself acquainted with the methods of reasoning and results there used; the earliest notice in modern Europe of the algebra of Diophantus is a remark of his that he had seen a copy of it at the Vatican. He was also well read in the works of the Arab mathematicians.
The fruit of his study was shewn in his De Triangulis written in 1464. This is the earliest modern systematic exposition of trigonometry, plane and spherical, though the only trigonometrical functions introduced are those of the sine and cosine. It is divided into five books. The first four are given up to plane trigonometry, and in particular to determining triangles from three given conditions. The fifth book is devoted to spherical trigonometry. The work was printed at Nuremberg in 1533, nearly a century after the death of Regiomontanus.
As an example of the mathematics of this time I quote one of his propositions at length. It is required to determine a triangle when the difference of two sides, the perpendicular on the base, and the difference between the segments into which the base is thus divided are given [book ii, prop. 23]. The following is the solution given by Regiomontanus.
Sit talis triangulus ABG , cujus duo latera AB et AG differentia habeant nota HG , ductaque perpendiculari AD duorum casuum BD et DG , differentia sit EG : hae duae differentiae sint datae, et ipsa perpendicularis AD data. Dico quod omnia latera trianguli nota concludentur. Per artem rei et census hoc problema absolvemus. Detur ergo differentia laterum ut 3, differentia casuum 12, et perpendicularis 10. Pono pro basi unam rem, et pro aggregato laterum 4 res, nae proportio basis ad congeriem laterum est ut HG ad GE , scilicet unius ad 4. Erit ergo BD 1/2 rei minus 6, sed AB erit 2 res demptis 3/2 . Duco AB in se, producuntur 4 census et 21/4 demptis 6 rebus. Item BD in se facit 1/4 census et 36 minus 6 rebus: huic addo quadratum de 10 qui est 100. Colliguntur 1/4 census et 136 minus 6 rebus aequales videlicet 4 censibus et 21/4 demptis 6 rebus. Restaurando itaque defectus et auferendo utrobique aequalia, quemadmodum ars ipsa praecipit, habemus census aliquot aequales numero, unde cognitio rei patebit, et inde tria latera trianguli more suo innotescet.
To explain the language of the proof I should add that Regiomontanus calls the unknown quantity res, and its square census or zensus; but though he uses these technical terms he writes the words in full. He commences by saying that he will solve the problem by means of a quadratic equation (per artem rei et census); and that he will suppose the difference of the sides of the triangle to be 3, the difference of the segments of the base to be 12, and the altitude of the triangle to be 10. He then takes for his unknown quantity (unam rem or x ) the base of the triangle, and therefore the sum of the sides will be 4x . Therefore BD will be equal to 1/2x - 6 ( 1/2 rei minus 6), and AB will be equal to 2x - 3/2 (2 res demptis 3/2 ); hence AB^2 ( AB in se) will be 4x^2 + 21/4 - 6x (4 census et 21/4 demptis 6 rebus), and BD^2 will be 1/4x^2 + 36 - 6x . To BD^2 he adds AD^2 (quadratum de 10) which is 100, and states that the sum of the two is equal to AB^2 . This he says will give the value of x^2 (census), whence a knowledge of x (cognitio rei) can be obtained, and the triangle determined.
To express this in the language of modern algebra we have
AG^2 - DG^2 & = AB^2 - DB^2, AG^2 - AB^2 & = DG^2 - DB^2,
but by the given numerical conditions
AG - AB & = 3 = 14(DG - DB), AG + AB & = 4(DG + DB) = 4x.
Therefore AB = 2x - 3/2 , and BD = 1/2x - 6 . Hence (2x - 3/2)^2 = (1/2x-6)^2 + 100 . From which x can be found, and all the elements of the triangle determined.
It is worth noticing that Regiomontanus merely aimed at giving a general method, and the numbers are not chosen with any special reference to the particular problem. Thus in his diagram he does not attempt to make GE anything like four times as long as GH , and, since x is ultimately found to be equal to 1/3sqrt(321) , the point D really falls outside the base. The order of the letters ABG , used to denote the triangle, is of course derived from the Greek alphabet.
Some of the solutions which he gives are unnecessarily complicated, but it must be remembered that algebra and trigonometry were still only in the rhetorical stage of development, and when every step of the argument is expressed in words at full length it is by no means easy to realize all that is contained in a formula.
It will be observed from the above example that Regiomontanus did not hesitate to apply algebra to the solution of geometrical problems. Another illustration of this is to be found in his discussion of a question which appears in Brahmagupta's Siddhanta. The problem was to construct a quadrilateral, having its sides of given lengths, which should be inscribable in a circle. The solution given by Regiomontanus was effected by means of algebra and trigonometry.
The Algorithmus Demonstratus of Jordanus, described above, which was first printed in 1534, was formerly attributed to Regiomontanus.
Regiomontanus was one of the most prominent mathematicians of his generation, and I have dealt with his works in some detail as typical of the most advanced mathematics of the time. Of his contemporaries I shall do little more than mention the names of a few of those who are best known; none were quite of the first rank, and I should sacrifice the proportion of the parts of the subject were I to devote much space to them.
Purbach. I may begin by mentioning George Purbach, first the tutor and then the friend of Regiomontanus, born near Linz on May 30, 1423, and died at Vienna on April 8, 1461, who wrote a work on planetary motions which was published in 1460; an arithmetic, published in 1511; a table of eclipses, published in 1514; and a table of natural sines, published in 1541.
Cusa. Next I may mention Nicolas de Cusa, who was born in 1401 and died in 1464. Although the son of a poor fisherman and without influence, he rose rapidly in the church, and in spite of being "a reformer before the reformation" became a cardinal. His mathematical writings deal with the reform of the calendar and the quadrature of the circle; in the latter problem his construction is equivalent to taking 3/4(sqrt 3 + sqrt 6) as the value of pi . He argued in favour of the diurnal rotation of the earth.
Chuquet. I may also here notice a treatise on arithmetic, known as Le Triparty, by Nicolas Chuquet, a bachelor of medicine in the university of Paris, which was written in 1484. This work indicates that the extent of mathematics then taught was somewhat greater than was generally believed a few years ago. It contains the earliest known use of the radical sign with indices to mark the root taken, 2 for a square-root, 3 for a cube-root, and so on; and also a definite statement of the rule of signs. The words plus and minus are denoted by the contractions p , m . The work is in French.
Introduction of signs + and - . In England and Germany algorists were less fettered by precedent and tradition than in Italy, and introduced some improvements in notation which were hardly likely to occur to an Italian. Of these the most prominent were the introduction, if not the invention, of the current symbols for addition, subtraction, and equality.
The earliest instances of the regular use of the signs + and - of which we have any knowledge occur in the fifteenth century. Johannes Widman of Eger, born about 1460, matriculated at Leipzig in 1480, and probably by profession a physician, wrote a Mercantile Arithmetic, published at Leipzig in 1489 (and modelled on a work by Wagner printed some six or seven years earlier): in this book these signs are used merely as marks signifying excess or deficiency; the corresponding use of the word surplus or overplus was once common and is still retained in commerce.
It is noticeable that the signs generally occur only in practical mercantile questions: hence it has been conjectured that they were originally warehouse marks. Some kinds of goods were sold in a sort of wooden chest called a lagel, which when full was apparently expected to weigh roughly either three or four centners; if one of these cases were a little lighter, say 5 lbs., than four centners, Widman describes it as weighing 4c - 5 lbs.: if it were 5 lbs. heavier than the normal weight it is described as weighing 4c nblongplus 5 lbs. The symbols are used as if they would be familiar to his readers; and there are some slight reasons for thinking that these marks were chalked on the chests as they came into the warehouses. We infer that the more usual case was for a chest to weigh a little less than its reputed weight, and, as the sign - placed between two numbers was a common symbol to signify some connection between them, that seems to have been taken as the standard case, while the vertical bar was originally a small mark super-added on the sign - to distinguish the two symbols. It will be observed that the vertical line in the symbol for excess, printed above, is somewhat shorter than the horizontal line. This is also the case with Stifel and most of the early writers who used the symbol: some presses continued to print it in this, its earliest form, till the end of the seventeenth century. Xylander, on the other hand, in 1575 has the vertical bar much longer than the horizontal line, and the symbol is something like nbshortplus .
Another conjecture is that the symbol for plus is derived from the Latin abbreviation & for et; while that for minus is obtained from the bar which is often used in ancient manuscripts to indicate an omission, or which is written over the contracted form of a word to signify that certain letters have been left out. This view has been often supported on a priori grounds, but it has recently found powerful advocates in Professors Zangmeister and Le Paige who also consider that the introduction of these symbols for plus and minus may be referred to the fourteenth century.
These explanations of the origin of our symbols for plus and minus are the most plausible that have been yet advanced, but the question is difficult and cannot be said to be solved. Another suggested derivation is that + is a contraction of -0.2ex the initial letter in Old German of plus, while - is the limiting form of m (for minus) when written rapidly. De Morgan proposed yet another derivation: the Hindoos sometimes used a dot to indicate subtraction, and this dot might, he thought, have been elongated into a bar, and thus give the sign for minus; while the origin of the sign for plus was derived from it by a super-added bar as explained above; but I take it that at a later time he abandoned this theory for what has been called the warehouse explanation.
I should perhaps here add that till the close of the sixteenth century the sign + connecting two quantities like a and b was also used in the sense that if a were taken as the answer to some question one of the given conditions would be too little by b . This was a relation which constantly occurred in solutions of questions by the rule of false assumption.
Lastly, I would repeat again that these signs in Widman are only abbreviations and not symbols of operation; he attached little or no importance to them, and no doubt would have been amazed if he had been told that their introduction was preparing the way for a revolution of the processes used in algebra.
The Algorithmus of Jordanus was not published till 1534; Widman's work was hardly known outside Germany; and it is to Pacioli that we owe the introduction into general use of syncopated algebra; that is, the use of abbreviations for certain of the more common algebraical quantities and operations, but where in using them the rules of syntax are observed.
Pacioli. Lucas Pacioli, sometimes known as Lucas di Burgo, and sometimes, but more rarely, as Lucas Paciolus, was born at Burgo in Tuscany about the middle of the fifteenth century. We know little of his life except that he was a Franciscan friar; that he lectured on mathematics at Rome, Pisa, Venice, and Milan; and that at the last-named city he was the first occupant of a chair of mathematics founded by Sforza: he died at Florence about the year 1510.
His chief work was printed at Venice in 1494 and is termed Summa de arithmetica, geometria, proporzioni e proporzionalita. It is divided into two parts, the first dealing with arithmetic and algebra, the second with geometry. This was the earliest printed book on arithmetic and algebra. It is mainly based on the writings of Leonardo of Pisa, and its importance in the history of mathematics is largely due to its wide circulation.
In the arithmetic Pacioli gives rules for the four simple processes, and a method for extracting square roots. He deals pretty fully with all questions connected with mercantile arithmetic, in which he works out numerous examples, and in particular discusses at great length bills of exchange and the theory of book-keeping by double entry. This part was the first systematic exposition of algoristic arithmetic, and has been already alluded to in chapter xi. It and the similar work by Tartaglia are the two standard authorities on the subject.
Many of his problems are solved by "the method of false assumption," which consists in assuming any number for the unknown quantity, and if on trial the given conditions be not satisfied, altering it by a simple proportion as in rule of three. As an example of this take the problem to find the original capital of a merchant who spent a quarter of it in Pisa and a fifth of it in Venice, who received on these transactions 180 ducats, and who has in hand 224 ducats. Suppose that we assume that he had originally 100 ducats. Then if he spent 25+20 ducats at Pisa and Venice, he would have had 55 ducats left. But by the enunciation he then had 224-180 , that is, 44 ducats. Hence the ratio of his original capital to 100 ducats is as 44 to 55. Thus his original capital was 80 ducats.
The following example will serve as an illustration of the kind of arithmetical problems discussed.
I buy for 1440 ducats at Venice 2400 sugar loaves, whose nett weight is 7200 lire; I pay as a fee to the agent 2 per cent.; to the weighers and porters on the whole, 2 ducats; I afterwards spend in boxes, cords, canvas, and in fees to the ordinary packers in the whole, 8 ducats; for the tax or octroi duty on the first amount, 1 ducat per cent.; afterwards for duty and tax at the office of exports, 3 ducats per cent.; for writing directions on the boxes and booking their passage, 1 ducat; for the bark to Rimini, 13 ducats; in compliments to the captains and in drink for the crews of armed barks on several occasions, 2 ducats; in expenses for provisions for myself and servant for one month, 6 ducats; for expenses for several short journeys over land here and there, for barbers, for washing of linen, and of boots for myself and servant, 1 ducat; upon my arrival at Rimini I pay to the captain of the port for port dues in the money of that city, 3 lire; for porters, disembarkation on land, and carriage to the magazine, 5 lire; as a tax upon entrance, 4 soldi a load which are in number 32 (such being the custom); for a booth at the fair, 4 soldi per load; I further find that the measures used at the fair are different to those used at Venice, and that 140 lire of weight are there equivalent to 100 at Venice, and that 4 lire of their silver coinage are equal to a ducat of gold. I ask, therefore, at how much I must sell a hundred lire Rimini in order that I may gain 10 per cent. upon my whole adventure, and what is the sum which I must receive in Venetian money?
In the algebra he discusses in some detail simple and quadratic equations, and problems on numbers which lead to such equations. He mentions the Arabic classification of cubic equations, but adds that their solution appears to be as impossible as the quadrature of the circle. The following is the rule he gives for solving a quadratic equation of the form x^2 + x = a : it is rhetorical and not syncopated, and will serve to illustrate the inconvenience of that method.
"Si res et census numero coaequantur, a rebus "dimidio sumpto censum producere debes, "addereque numero, cujus a radice totiens "tolle semis rerum, census latusque redibit."
He confines his attention to the positive roots of equations.
Though much of the matter described above is taken from Leonardo's Liber Abaci, yet the notation in which it is expressed is superior to that of Leonardo. Pacioli follows Leonardo and the Arabs in calling the unknown quantity the thing, in Italian cosa--hence algebra was sometimes known as the cossic art--or in Latin res, and sometimes denotes it by co or R or Rj. He calls the square of it census or zensus, and sometimes denotes it by ce or Z; similarly the cube of it, or cuba, is sometimes represented by cu or C; the fourth power, or censo di censo, is written either at length or as ce di ce or as ce ce. It may be noticed that all his equations are numerical, so that he did not rise to the conception of representing known quantities by letters as Jordanus had done and as is the case in modern algebra; but Libri gives two instances in which in a proportion he represents a number by a letter. He indicates addition by p or p , the initial letter of the word plus, but he generally evades the introduction of a symbol for minus by writing his quantities on that side of the equation which makes them positive, though in a few places he denotes it by m for minus or by de for demptus. Similarly, equality is sometimes indicated by ae for aequalis. This is a commencement of syncopated algebra.
There is nothing striking in the results he arrives at in the second or geometrical part of the work; nor in two other tracts on geometry which he wrote and which were printed at Venice in 1508 and 1509. It may be noticed, however, that, like Regiomontanus, he applied algebra to aid him in investigating the geometrical properties of figures.
The following problem will illustrate the kind of geometrical questions he attacked. The radius of the inscribed circle of a triangle is 4 inches, and the segments into which one side is divided by the point of contact are 6 inches and 8 inches respectively. Determine the other sides. To solve this it is sufficient to remark that rs = Delta = sqrt(s(s-a)(s-b)(s-c)) which gives 4s = sqrt(s × (s-14) × 6 × 8) , hence s=21 ; therefore the required sides are 21 - 6 and 21 - 8 , that is, 15 and 13. But Pacioli makes no use of these formulae (with which he was acquainted), but gives an elaborate geometrical construction, and then uses algebra to find the lengths of various segments of the lines he wants. The work is too long for me to reproduce here, but the following analysis of it will afford sufficient materials for its reproduction. Let ABC be the triangle, D , E , F the points of contact of the sides, and O the centre of the given circle. Let H be the point of intersection of OB and DF , and K that of OC and DE . Let L and M be the feet of the perpendiculars drawn from E and F on BC . Draw EP parallel to AB and cutting BC in P . Then Pacioli determines in succession the magnitudes of the following lines: (i) OB , (ii) OC , (iii) FD , (iv) FH , (v) ED , (vi) EK . He then forms a quadratic equation, from the solution of which he obtains the values of MB and MD . Similarly he finds the values of LC and LD . He now finds in succession the values of EL , FM , EP , and LP ; and then by similar triangles obtains the value of AB , which is 13. This proof was, even sixty years later, quoted by Cardan as "incomparably simple and excellent, and the very crown of mathematics." I cite it as an illustration of the involved and inelegant methods then current. The problems enunciated are very similar to those in the De Triangulis of Regiomontanus.
Leonardo da Vinci. The fame of Leonardo da Vinci as an artist has overshadowed his claim to consideration as a mathematician, but he may be said to have prepared the way for a more accurate conception of mechanics and physics, while his reputation and influence drew some attention to the subject; he was an intimate friend of Pacioli. Leonardo was the illegitimate son of a lawyer of Vinci in Tuscany, was born in 1452, and died in France in 1519 while on a visit to Francis I. Several manuscripts by him were seized by the French revolutionary armies at the end of the last century, and Venturi, at the request of the Institute, reported on those concerned with physical or mathematical subjects.
Leaving out of account Leonardo's numerous and important artistic works, his mathematical writings are concerned chiefly with mechanics, hydraulics, and optics--his conclusions being usually based on experiments. His treatment of hydraulics and optics involves but little mathematics. The mechanics contain numerous and serious errors; the best portions are those dealing with the equilibrium of a lever under any forces, the laws of friction, the stability of a body as affected by the position of its centre of gravity, the strength of beams, and the orbit of a particle under a central force; he also treated a few easy problems by virtual moments. A knowledge of the triangle of forces is occasionally attributed to him, but it is probable that his views on the subject were somewhat indefinite. Broadly speaking, we may say that his mathematical work is unfinished, and consists largely of suggestions which he did not discuss in detail and could not (or at any rate did not) verify.
Dürer. Albrecht Dürer was another artist of the same time who was also known as a mathematician. He was born at Nuremberg on May 21, 1471, and died there on April 6, 1528. His chief mathematical work was issued in 1525, and contains a discussion of perspective, some geometry, and certain graphical solutions; Latin translations of it were issued in 1532, 1555, and 1605.
Copernicus. An account of Nicolaus Copernicus, born at Thorn on Feb. 19, 1473, and died at Frauenberg on May 7, 1543, and his conjecture that the earth and planets all revolved round the sun, belong to astronomy rather than to mathematics. I may, however, add that Copernicus wrote on trigonometry, his results being published as a text-book at Wittenberg in 1542; it is clear though it contains nothing new. It is evident from this and his astronomy that he was well read in the literature of mathematics, and was himself a mathematician of considerable power. I describe his statement as to the motion of the earth as a conjecture, because he advocated it only on the ground that it gave a simple explanation of natural phenomena. Galileo in 1632 was the first to try to supply a proof of this hypothesis.
By the beginning of the sixteenth century the printing-press began to be active, and many of the works of the earlier mathematicians became now for the first time accessible to all students. This stimulated inquiry, and before the middle of the century numerous works were issued which, though they did not include any great discoveries, introduced a variety of small improvements all tending to make algebra more analytical.
Record. The sign now used to denote equality was introduced by Robert Record. Record was born at Tenby in Pembrokeshire about 1510, and died at London in 1558. He entered at Oxford, and obtained a fellowship at All Souls College in 1531; thence he migrated to Cambridge, where he took a degree in medicine in 1545. He then returned to Oxford and lectured there, but finally settled in London and became physician to Edward VI. and to Mary. His prosperity must have been short-lived, for at the time of his death he was confined in the King's Bench prison for debt.
In 1540 he published an arithmetic, termed the Grounde of Artes, in which he employed the signs + to indicate excess and - to indicate deficiency; " + whyche betokeneth too muche, as this line - plaine without a crosse line betokeneth too little." In this book the equality of two ratios is indicated by two equal and parallel lines whose opposite ends are joined diagonally, ex. gr. by . A few years later, in 1557, he wrote an algebra under the title of the Whetstone of Witte. This is interesting as it contains the earliest introduction of the sign = for equality, and he says he selected that particular symbol because than two parallel straight lines "noe 2 thynges can be moare equalle." M. Charles Henry has, however, asserted that this sign is a recognized abbreviation for the word est in medieval manuscripts; and, if this be established, it would seem to indicate a more probable origin. In this work Record shewed how the square root of an algebraic expression could be extracted. He also wrote an astronomy. These works give a clear view of the knowledge of the time.
Rudolff. Riese. About the same time in Germany, Rudolff and Riese took up the subjects of algebra and arithmetic. Their investigations form the basis of Stifel's well-known work. Christoff Rudolff published his algebra in 1525; it is entitled Die Coss, and is founded on the writings of Pacioli, and perhaps of Jordanus. Rudolff introduced the sign of sqrt for the square root, the symbol being a corruption of the initial letter of the word radix, similarly sqrt sqrt sqrt denoted the cube root, and sqrt sqrt the fourth root. Adam Riese was born near Bamberg, Bavaria, in 1489, of humble parentage, and after working for some years as a miner set up a school; he died at Annaberg on March 30, 1559. He wrote a treatise on practical geometry, but his most important book was his well-known arithmetic (which may be described as algebraical), issued in 1536, and founded on Pacioli's work. Riese used the symbols + and - .
Stifel. The methods used by Rudolff and Riese and their results were brought into general notice through Stifel's work, which had a wide circulation. Michael Stifel, sometimes known by the Latin name of Stiffelius, was born at Esslingen in 1486, and died at Jena on April 19, 1567. He was originally an Augustine monk, but he accepted the doctrines of Luther, of whom he was a personal friend. He tells us in his algebra that his conversion was finally determined by noticing that the pope Leo X. was the beast mentioned in the Revelation. To shew this, it was only necessary to add up the numbers represented by the letters in Leo decimus (the m had to be rejected since it clearly stood for mysterium) and the result amounts to exactly ten less than 666, thus distinctly implying that it was Leo the tenth. Luther accepted his conversion, but frankly told him he had better clear his mind of any nonsense about the number of the beast.
Unluckily for himself Stifel did not act on this advice. Believing that he had discovered the true way of interpreting the biblical prophecies, he announced that the world would come to an end on October 3, 1533. The peasants of Holzdorf, of which place he was pastor, aware of his scientific reputation, accepted his assurance on this point. Some gave themselves up to religious exercises, others wasted their goods in dissipation, but all abandoned their work. When the day foretold had passed, many of the peasants found themselves ruined. Furious at having been deceived, they seized the unfortunate prophet, and he was lucky in finding a refuge in the prison at Wittenberg, from which he was after some time released by the personal intercession of Luther.
Stifel wrote a small treatise on algebra, but his chief mathematical work is his Arithmetica Integra, published at Nuremberg in 1544, with a preface by Melanchthon.
The first two books of the Arithmetica Integra deal with surds and incommensurables, and are Euclidean in form. The third book is on algebra, and is noticeable for having called general attention to the German practice of using the signs + and - to denote addition and subtraction. There are traces of these signs being occasionally employed by Stifel as symbols of operation and not only as abbreviations; in this use of them he seems to have followed G. V. Hoecke. He not only employed the usual abbreviations for the Italian words which represent the unknown quantity and its powers, but in at least one case when there were several unknown quantities he represented them respectively by the letters A, B, C, &c.; thus re-introducing the general algebraic notation which had fallen into disuse since the time of Jordanus. It used to be said that Stifel was the real inventor of logarithms, but it is now certain that this opinion was due to a misapprehension of a passage in which he compares geometrical and arithmetical progressions. Stifel is said to have indicated a formula for writing down the coefficients of the various terms in the expansion of (1 + x)^n if those in the expansion of (1 + x)^n-1 were known.
In 1553 Stifel brought out an edition of Rudolff's Die Coss, in which he introduced an improvement in the algebraic notation then current. The symbols at that time ordinarily used for the unknown quantity and its powers were letters which stood for abbreviations of the words. Among those frequently adopted were R or Rj for radix or res ( x ), Z or C for zensus or census ( x^2 ), C or K for cubus ( x^3 ), &c. Thus x^2 + 5x - 4 would have been written 1 Z p. 5 R m. 4; where p stands for plus and m for minus. Other letters and symbols were also employed: thus Xylander (1575) would have denoted the above expression by 1Q + 5N - 4; a notation similar to this was sometimes used by Vieta and even by Fermat. The advance made by Stifel was that he introduced the symbols 1A , 1AA , 1AAA , for the unknown quantity, its square, and its cube, which shewed at a glance the relation between them.
Tartaglia. Niccolo Fontana, generally known as Nicholas Tartaglia, that is, Nicholas the stammerer, was born at Brescia in 1500, and died at Venice on December 14, 1557. After the capture of the town by the French in 1512, most of the inhabitants took refuge in the cathedral, and were there massacred by the soldiers. His father, who was a postal messenger at Brescia, was amongst the killed. The boy himself had his skull split through in three places, while his jaws and his palate were cut open; he was left for dead, but his mother got into the cathedral, and finding him still alive managed to carry him off. Deprived of all resources she recollected that dogs when wounded always licked the injured place, and to that remedy he attributed his ultimate recovery, but the injury to his palate produced an impediment in his speech, from which he received his nickname. His mother managed to get sufficient money to pay for his attendance at school for fifteen days, and he took advantage of it to steal a copy-book from which he subsequently taught himself how to read and write; but so poor were they that he tells us he could not afford to buy paper, and was obliged to make use of the tombstones as slates on which to work his exercises.
He commenced his public life by lecturing at Verona, but he was appointed at some time before 1535 to a chair of mathematics at Venice, where he was living, when he became famous through his acceptance of a challenge from a certain Antonio del Fiore (or Florido). Fiore had learnt from his master, one Scipione Ferro (who died at Bologna in 1526), an empirical solution of a cubic equation of the form x^3 + qx = r . This solution was previously unknown in Europe, and it is possible that Ferro had found the result in an Arab work. Tartaglia, in answer to a request from Colla in 1530, stated that he could effect the solution of a numerical equation of the form x^3 + px^2 = r . Fiore, believing that Tartaglia was an impostor, challenged him to a contest. According to this challenge each of them was to deposit a certain stake with a notary, and whoever could solve the most problems out of a collection of thirty propounded by the other was to get the stakes, thirty days being allowed for the solution of the questions proposed. Tartaglia was aware that his adversary was acquainted with the solution of a cubic equation of some particular form, and suspecting that the questions proposed to him would all depend on the solution of such cubic equations, set himself the problem to find a general solution, and certainly discovered how to obtain a solution of some if not all cubic equations. His solution is believed to have depended on a geometrical construction, but led to the formula which is often, but unjustly, described as Cardan's.
When the contest took place, all the questions proposed to Tartaglia were, as he had suspected, reducible to the solution of a cubic equation, and he succeeded within two hours in bringing them to particular cases of the equation x^3 + qx = r , of which he knew the solution. His opponent failed to solve any of the problems proposed to him, most of which were, as a matter of fact, reducible to numerical equations of the form x^3 + px^2 = r . Tartaglia was therefore the conqueror; he subsequently composed some verses commemorative of his victory.
The chief works of Tartaglia are as follows: (i) His Nova scienza, published in 1537: in this he investigated the fall of bodies under gravity; and he determined the range of a projectile, stating that it was a maximum when the angle of projection was 45° , but this seems to have been a lucky guess. (ii) His Inventioni, published in 1546, and containing, inter alia, his solution of cubic equations. (iii) His Trattato di numeri e misuri, consisting of an arithmetic, published in 1556, and a treatise on numbers, published in 1560; in this he shewed how the coefficients of x in the expansion of (1 + x)^n could be calculated, by the use of an arithmetical triangle, from those in the expansion of (1 + x)^n-1 for the cases when n is equal to 2, 3, 4, 5, or 6. His works were collected into a single edition and republished at Venice in 1606.
The treatise on arithmetic and numbers is one of the chief authorities for our knowledge of the early Italian algorism. It is verbose, but gives a clear account of the arithmetical methods then in use, and has numerous historical notes which, as far as we can judge, are reliable, and are ultimately the authorities for many of the statements in the last chapter. It contains an immense number of questions on every kind of problem which would be likely to occur in mercantile arithmetic, and there are several attempts to frame algebraical formulae suitable for particular problems.
These problems give incidentally a good deal of information as to the ordinary life and commercial customs of the time. Thus we find that the interest demanded on first-class security in Venice ranged from 5 to 12 per cent. a year; while the interest on commercial transactions ranged from 20 per cent. a year upwards. Tartaglia illustrates the evil effects of the law forbidding usury by the manner in which it was evaded in farming. Farmers who were in debt were forced by their creditors to sell all their crops immediately after the harvest; the market being thus glutted, the price obtained was very low, and the money-lenders purchased the corn in open market at an extremely cheap rate. The farmers then had to borrow their seed-corn on condition that they replaced an equal quantity, or paid the then price of it, in the month of May, when corn was dearest. Again, Tartaglia, who had been asked by the magistrates at Verona to frame for them a sliding scale by which the price of bread would be fixed by that of corn, enters into a discussion on the principles which it was then supposed should regulate it. In another place he gives the rules at that time current for preparing medicines.
Pacioli had given in his arithmetic some problems of an amusing character, and Tartaglia imitated him by inserting a large collection of mathematical puzzles. He half apologizes for introducing them by saying that it was not uncommon at dessert to propose arithmetical questions to the company by way of amusement, and he therefore adds some suitable problems. He gives several questions on how to guess a number thought of by one of the company, or the relationships caused by the marriage of relatives, or difficulties arising from inconsistent bequests. Other puzzles are similar to the following. "Three beautiful ladies have for husbands three men, who are young, handsome, and gallant, but also jealous. The party are travelling, and find on the bank of a river, over which they have to pass, a small boat which can hold no more than two persons. How can they pass, it being agreed that, in order to avoid scandal, no woman shall be left in the society of a man unless her husband is present?" "A ship, carrying as passengers fifteen Turks and fifteen Christians, encounters a storm; and the pilot declares that in order to save the ship and crew one-half of the passengers must be thrown into the sea. To choose the victims, the passengers are placed in a circle, and it is agreed that every ninth man shall be cast overboard, reckoning from a certain point. In what manner must they be arranged, so that the lot may fall exclusively upon the Turks?" "Three men robbed a gentleman of a vase containing 24 ounces of balsam. Whilst running away they met in a wood with a glass-seller of whom in a great hurry they purchased three vessels. On reaching a place of safety they wish to divide the booty, but they find that their vessels contain 5, 11, and 13 ounces respectively. How can they divide the balsam into equal portions?"
These problems--some of which are of oriental origin--form the basis of the collections of mathematical recreations by Bachet de Méziriac, Ozanam, and Montucla.
Cardan. The life of Tartaglia was embittered by a quarrel with his contemporary Cardan, who published Tartaglia's solution of a cubic equation which he had obtained under a pledge of secrecy. Girolamo Cardan was born at Pavia on September 24, 1501, and died at Rome on September 21, 1576. His career is an account of the most extraordinary and inconsistent acts. A gambler, if not a murderer, he was also an ardent student of science, solving problems which had long baffled all investigation; at one time of his life he was devoted to intrigues which were a scandal even in the sixteenth century, at another he did nothing but rave on astrology, and yet at another he declared that philosophy was the only subject worthy of man's attention. His was the genius that was closely allied to madness.
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End of part 15 of 28
Continue readingChapter 12: The Mathematics of the Renaissance (Part 2)
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